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### ELECTRICITY

Electricity

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"Electric" redirects here. For other uses, see Electric (disambiguation).

Lightning is one of the most dramatic effects of electricity

Lightning is one of the most dramatic effects of electricity

Electricity (from New Latin ēlectricus, "amber-like") is a general term that encompasses a variety of phenomena resulting from the presence and flow of electric charge. These include many easily recognizable phenomena such as lightning and static electricity, but in addition, less familiar concepts such as the electromagnetic field and electromagnetic induction.

In general usage, the word 'electricity' is adequate to refer to a number of physical effects. However, in scientific usage, the term is vague, and these related, but distinct, concepts are better identified by more precise terms:

* Electric charge – a property of some subatomic particles, which determines their electromagnetic interactions. Electrically charged matter is influenced by, and produces, electromagnetic fields.

* Electric current – a movement or flow of electrically charged particles, typically measured in amperes.

* Electric field – an influence produced by an electric charge on other charges in its vicinity.

* Electric potential – the capacity of an electric field to do work, typically measured in volts.

* Electromagnetism – a fundamental interaction between the electric field and the presence and motion of electric charge.

Electricity has been studied since antiquity, though scientific advances were not forthcoming until the seventeenth and eighteenth centuries. It would not be until the late nineteenth century, however, that engineers were able to put electricity to industrial and residential use. This period witnessed a rapid expansion in the development of electrical technology. Electricity's extraordinary versatility as a source of energy means it can be put to an almost limitless set of applications which include transport, heating, lighting, communications, and computation. The backbone of modern industrial society is, and for the foreseeable future can be expected to remain, the use of electrical power.[1]

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Contents

[hide]

* 1 History

* 2 Concepts

o 2.1 Electric charge

o 2.2 Electric current

o 2.3 Electric field

o 2.4 Electric potential

o 2.5 Electromagnetism

* 3 Electric circuits

* 4 Production and uses

o 4.1 Generation

o 4.2 Uses

* 5 Electricity and the natural world

o 5.1 Physiological effects

o 5.2 Electrical phenomena in nature

* 6 See also

* 7 References

* 8 Bibliography

* 9 External links

[edit] History

Thales, the earliest researcher into electricity

Thales, the earliest researcher into electricity

Main article: History of electricity

See also: Etymology of electricity

Knowledge of electric discharge from electric fishes was first reported in 2750 BC by the ancient Egyptians, who referred to it as the "thunderer of the Nile". They were again reported millennia later by ancient Greek, Roman and Arabic naturalists and physicians.[2] Several ancient writers, such as Pliny the Elder and Scribonius Largus, attested to the numbing effect of electric shocks delivered by catfish and torpedo rays, and knew that such shocks could travel along conducting objects.[3] Patients suffering from ailments such as gout or headache were directed to touch electric fish in the hope that the powerful jolt might cure them.[4] Similar observations were later reported by Al-Jahiz in medieval Egypt.[5]

That certain objects such as rods of amber could be rubbed with cat's fur and attract light objects like feathers was known to ancient cultures around the Mediterranean. Thales of Miletos made a series of observations on static electricity around 600 BC, from which he believed that friction rendered amber magnetic, in contrast to minerals such as magnetite, which needed no rubbing.[6][7] Thales was incorrect in believing the attraction was due to a magnetic effect, but later science would prove a link between magnetism and electricity.

According to a controversial theory, the Parthians may have had knowledge of electroplating, based on the 1936 discovery of the Baghdad Battery, which resembles a galvanic cell, though it is uncertain whether the artefact was electrical in nature.[8] In the 9th century AD, the Andalusian engineer, Abbas Ibn Firnas, invented an artificial weather simulation room, in which spectators saw stars and clouds, and were astonished by artificial thunder and lightning, which were produced by mechanisms hidden in his basement laboratory.[9][10]

Benjamin Franklin conducted extensive research on electricity in the 18th century

Benjamin Franklin conducted extensive research on electricity in the 18th century

Electricity would remain little more than an intellectual curiosity for millennia until 1600, when the English physician William Gilbert made a careful study of electricity and magnetism, distinguishing the lodestone effect from static electricity produced by rubbing amber.[6] He coined the New Latin word electricus ("of amber" or "like amber", from ήλεκτρον [elektron], the Greek word for "amber") to refer to the property of attracting small objects after being rubbed.[11] This association gave rise to the English words "electric" and "electricity", which made their first appearance in print in Thomas Browne's Pseudodoxia Epidemica of 1646.[12]

Further work was conducted by Otto von Guericke, Robert Boyle, Stephen Gray and C. F. du Fay. In the 18th century, Benjamin Franklin conducted extensive research in electricity, selling his possessions to fund his work. In June 1752 he is reputed to have attached a metal key to the bottom of a dampened kite string and flown the kite in a storm-threatened sky.[13] He observed a succession of sparks jumping from the key to the back of his hand, showing that lightning was indeed electrical in nature.[14]

In 1791 Luigi Galvani published his discovery of bioelectricity, demonstrating that electricity was the medium by which nerve cells passed signals to the muscles.[15] Alessandro Volta's battery, or voltaic pile, of 1800, made from alternating layers of zinc and copper, provided scientists with a more reliable source of electrical energy than the electrostatic machines previously used.[15] André-Marie Ampère discovered the relationship between electricity and magnetism in 1820; Michael Faraday invented the electric motor in 1821, and Georg Ohm mathematically analysed the electrical circuit in 1827.[15]

While it had been the early 19th century that had seen rapid progress in electrical science, the late 19th century would see the greatest progress in electrical engineering. Through such people as Nikola Tesla, Thomas Edison, George Westinghouse, Ernst Werner von Siemens, Alexander Graham Bell and Lord Kelvin, electricity was turned from a scientific curiosity into an essential tool for modern life, becoming a driving force for the Second Industrial Revolution.[16]

[edit] Concepts

[edit] Electric charge

Main article: Electric charge

See also: electron, proton, and ion

Electric charge is a property of certain subatomic particles, which gives rise to and interacts with, the electromagnetic force, one of the four fundamental forces of nature. Charge originates in the atom, in which its most familiar carriers are the electron and proton. It is a conserved quantity, that is, the net charge within an isolated system will always remain constant regardless of any changes taking place within that system.[17] Within the system, charge may be transferred between bodies, either by direct contact, or by passing along a conducting material, such as a wire.[18] The informal term static electricity refers to the net presence (or 'imbalance') of charge on a body, usually caused when dissimilar materials are rubbed together, transferring charge from one to the other.

Charge on a gold-leaf electroscope causes the leaves to visibly repel each other

Charge on a gold-leaf electroscope causes the leaves to visibly repel each other

The presence of charge gives rise to the electromagnetic force: charges exert a force on each other, an effect that was known, though not understood, in antiquity.[19] A lightweight ball suspended from a string can be charged by touching it with a glass rod that has itself been charged by rubbing with a cloth. If a similar ball is charged by the same glass rod, it is found to repel the first: the charge acts to force the two balls apart. Two balls that are charged with a rubbed amber rod also repel each other. However, if one ball is charged by the glass rod, and the other by an amber rod, the two balls are found to attract each other. These phenomena were investigated in the late eighteenth century by Charles-Augustin de Coulomb, who deduced that charge manifests itself in two opposing forms, leading to the well-known axiom: like-charged objects repel and opposite-charged objects attract.[19]

The force acts on the charged particles themselves, hence charge has a tendency to spread itself as evenly as possible over a conducting surface. The magnitude of the electromagnetic force, whether attractive or repulsive, is given by Coulomb's law, which relates the force to the product of the charges and has an inverse-square relation to the distance between them.[20][21] The electromagnetic force is very strong, second only in strength to the strong interaction,[22] but unlike that force it operates over all distances.[23] In comparison with the much weaker gravitational force, the electromagnetic force pushing two electrons apart is 1042 times that of the gravitational attraction pulling them together.[24]

The charge on electrons and protons is opposite in sign, hence an amount of charge may be expressed as being either negative or positive. By convention, the charge carried by electrons is deemed negative, and that by protons positive, a custom that originated with the work of Benjamin Franklin.[25] The amount of charge is usually given the symbol Q and expressed in coulombs;[26] each electron carries the same charge of approximately −1.6022×10−19 coulomb. The proton has a charge that is equal and opposite, and thus +1.6022×10−19 coulomb. Charge is possessed not just by matter, but also by antimatter, each antiparticle bearing an equal and opposite charge to its corresponding particle.[27]

Charge can be measured by a number of means, an early instrument being the gold-leaf electroscope, which although still in use for classroom demonstrations, has been superseded by the electronic electrometer.[18]

[edit] Electric current

Main article: Current (electricity)

The movement of electric charge is known as an electric current, the intensity of which is usually measured in amperes. Current can consist of any moving charged particles; most commonly these are electrons, but any charge in motion constitutes a current.

By historical convention, a positive current is defined as having the same direction of flow as any positive charge it contains, or to flow from the most positive part of a circuit to the most negative part. Current defined in this manner is called conventional current. The motion of negatively-charged electrons around an electric circuit, one of the most familiar forms of current, is thus deemed positive in the opposite direction to that of the electrons.[28] However, depending on the conditions, an electric current can consist of a flow of charged particles in either direction, or even in both directions at once. The positive-to-negative convention is widely used to simplify this situation. If another definition is used—for example, "electron current"—it needs to be explicitly stated.

An electric arc provides an energetic demonstration of electric current

An electric arc provides an energetic demonstration of electric current

The process by which electric current passes through a material is termed electrical conduction, and its nature varies with that of the charged particles and the material through which they are travelling. Examples of electric currents include metallic conduction, where electrons flow through a conductor such as metal, and electrolysis, where ions (charged atoms) flow through liquids. While the particles themselves can move quite slowly, sometimes with a average drift velocity only fractions of a millimetre per second,[18] the electric field that drives them itself propagates at close to the speed of light, enabling electrical signals to pass rapidly along wires.[29]

Current causes several observable effects, which historically were the means of recognising its presence. That water could be decomposed by the current from a voltaic pile was discovered by Nicholson and Carlisle in 1800, a process now known as electrolysis. Their work was greatly expanded upon by Michael Faraday in 1833.[30] Current through a resistance causes localised heating, an effect James Prescott Joule studied mathematically in 1840.[30] One of the most important discoveries relating to current was made accidentally by Hans Christian Ørsted in 1820, when, while preparing a lecture, he witnessed the current in a wire disturbing the needle of a magnetic compass.[31] He had discovered electromagnetism, a fundamental interaction between electricity and magnetics.

In engineering or household applications, current is often described as being either direct current (DC) or alternating current (AC). These terms refer to how the current varies in time. Direct current, as produced by example from a battery and required by most electronic devices, is a unidirectional flow from the positive part of a circuit to the negative.[32] If, as is most common, this flow is carried by electrons, they will be travelling in the opposite direction. Alternating current is any current that reverses direction repeatedly; almost always this takes the form of a sinusoidal wave.[33] Alternating current thus pulses back and forth within a conductor without the charge moving any net distance over time. The time-averaged value of an alternating current is zero, but it delivers energy in first one direction, and then the reverse. Alternating current is affected by electrical properties that are not observed under steady state direct current, such as inductance and capacitance.[34] These properties however can become important when circuitry is subjected to transients, such as when first energised.

[edit] Electric field

Main article: Electric field

See also: Electrostatics

The concept of the electric field was introduced by Michael Faraday. An electric field is created by a charged body in the space that surrounds it, and results in a force exerted on any other charges placed within the field. The electric field acts between two charges in a similar manner to the way that the gravitational field acts between two masses, and like it, extends towards infinity and shows an inverse square relationship with distance.[23] However, there is an important difference. Gravity always acts in attraction, drawing two masses together, while the electric field can result in either attraction or repulsion. Since large bodies such as planets generally carry no net charge, the electric field at a distance is usually zero. Thus gravity is the dominant force at distance in the universe, despite being much the weaker.[24]

Field lines emanating from a positive charge above a plane conductor

Field lines emanating from a positive charge above a plane conductor

An electric field generally varies in space,[35] and its strength at any one point is defined as the force (per unit charge) that would be felt by a stationary, negligible charge if placed at that point.[36] The conceptual charge, termed a 'test charge', must be vanishingly small to prevent its own electric field disturbing the main field and must also be stationary to prevent the effect of magnetic fields. As the electric field is defined in terms of force, and force is a vector, so it follows that an electric field is also a vector, having both magnitude and direction. Specifically, it is a vector field.[36]

The study of electric fields created by stationary charges is called electrostatics. The field may be visualised by a set of imaginary lines whose direction at any point is the same as that of the field. This concept was introduced by Faraday,[37] whose term 'lines of force' still sometimes sees use. The field lines are the paths that a point positive charge would seek to make as it was forced to move within the field; they are however an imaginary concept with no physical existence, and the field permeates all the intervening space between the lines.[37] Field lines emanating from stationary charges have several key properties: first, that they originate at positive charges and terminate at negative charges; second, that they must enter any good conductor at right angles, and third, that they may never cross nor close in on themselves.[38]

The principals of electrostatics are important when designing items of high-voltage equipment. There is a finite limit to the electric field strength that may withstood by any medium. Beyond this point, electrical breakdown occurs and an electric arc causes flashover between the charged parts. Air, for example, tends to arc at electric field strengths which exceed 30 kV per centimetre across small gaps. Over larger gaps, its breakdown strength is weaker, perhaps 1 kV per centimetre.[39] The most visible natural occurrence of this is lightning, caused when charge becomes separated in the clouds by rising columns of air, and raises the electric field in the air to greater than it can withstand. The voltage of a large lightning cloud may be as high as 100 MV and have discharge energies as great as 250 kWh.[40]

The field strength is greatly affected by nearby conducting objects, and it is particularly intense when it is forced to curve around sharply pointed objects. This principle is exploited in the lightning conductor, the sharp spike of which acts to encourage the lightning stroke to develop there, rather than to the building it serves to protect.[41]

An electric field is zero inside a conductor. This is because the net charge on a conductor only exists on the surface. External electrostatic fields are always perpendicular to the conductors surface. Otherwise this would produce a force on the charge carriers inside the conductor and so the field would not be static as we assume.

[edit] Electric potential

Main article: Electric potential

See also: Voltage

A pair of AA cells. The + sign indicates the polarity of the potential differences between the battery terminals.

A pair of AA cells. The + sign indicates the polarity of the potential differences between the battery terminals.

The concept of electric potential is closely linked to that of the electric field. A small charge placed within an electric field experiences a force, and to have brought that charge to that point against the force requires work. The electric potential at any point is defined as the energy required to bring a unit test charge from an infinite distance slowly to that point. It is usually measured in volts, and one volt is the potential for which one joule of work must be expended to bring a charge of one coulomb from infinity.[42] This definition of potential, while formal, has little practical application, and a more useful concept is that of electric potential difference, and is the energy required to move a unit charge between two specified points. An electric field has the special property that it is conservative, which means that the path taken by the test charge is irrelevant: all paths between two specified points expend the same energy, and thus a unique value for potential difference may be stated.[42] The volt is so strongly identified as the unit of choice for measurement and description of electric potential difference that the term voltage sees greater everyday usage.

For practical purposes, it is useful to define a common reference point to which potentials may be expressed and compared. While this could be at infinity, a much more useful reference is the Earth itself, which is assumed to be at the same potential everywhere. This reference point naturally takes the name earth or ground. Earth is assumed to be an infinite source of equal amounts of positive and negative charge, and is therefore electrically uncharged – and unchargeable.[43]

Electric potential is a scalar quantity, that is, it has only magnitude and not direction. It may be viewed as analogous to temperature: as there is a certain temperature at every point in space, and the temperature gradient indicates the direction and magnitude of the driving force behind heat flow, similarly, there is an electric potential at every point in space, and its gradient, or field strength, indicates the direction and magnitude of the driving force behind charge movement. Equally, electric potential may be seen as analogous to height: just as a released object will fall through a difference in heights caused by a gravitational field, so a charge will 'fall' across the voltage caused by an electric field.[44]

The electric field was formally defined as the force exerted per unit charge, but the concept of potential allows for a more useful and equivalent definition: the electric field is the local gradient of the electric potential. Usually expressed in volts per metre, the vector direction of the field is the line of greatest gradient of potential.[18]

[edit] Electromagnetism

Main article: Electromagnetism

Magnetic field circles around a current

Magnetic field circles around a current

Ørsted's discovery in 1821 that a magnetic field existed around all sides of a wire carrying an electric current indicated that there was a direct relationship between electricity and magnetism. Moreover, the interaction seemed different from gravitational and electrostatic forces, the two forces of nature then known. The force on the compass needle did not direct it to or away from the current-carrying wire, but acted at right angles to it.[31] Ørsted's slightly obscure words were that "the electric conflict acts in a revolving manner." The force also depended on the direction of the current, for if the flow was reversed, then the force did too.[45]

Ørsted did not fully understand his discovery, but he observed the effect was reciprocal: a current exerts a force on a magnet, and a magnetic field exerts a force on a current. The phenomenon was further investigated by Ampère, who discovered that two parallel current carrying wires exerted a force upon each other: two wires conducting currents in the same direction are attracted to each other, while wires containing current flowing in opposite directions are forced apart.[46] The interaction is mediated by the magnetic field each current produces and forms the basis for the international definition of the ampere.[46]

The electric motor exploits an important effect of electromagnetism: a current flowing through a magnetic field experiences a force at right angles to both the field and current

The electric motor exploits an important effect of electromagnetism: a current flowing through a magnetic field experiences a force at right angles to both the field and current

This relationship between magnetic fields and currents is extremely important, for it led to Michael Faraday's invention of the electric motor in 1821. Faraday's homopolar motor consisted of a permanent magnet sitting in a pool of mercury. A current was allowed to flow through a wire suspended from a pivot above the magnet and dipped into the mercury. The magnet exerted a tangential force on the wire, making it circle around the magnet for as long as the current was maintained.[47]

Experimentation by Faraday in 1831 revealed that a wire moving perpendicular to a magnetic field developed a potential difference between its ends. Further analysis of this process, known as electromagnetic induction, enabled him to state the principal, now known as Faraday's law of induction, that the potential difference induced in a closed circuit is proportional to the rate of change of magnetic flux through the loop. Exploitation of this discovery enabled him to invent the first electrical generator in 1831, in which he converted the mechanical energy of a rotating copper disc to electrical energy.[47] Faraday's disc was inefficient and of no use as a practical generator, but it showed the possibility of generating electric power using magnetism, a possibility that would be taken up by those that followed on from his work.

Faraday's and Ampère's work showed that a time-varying magnetic field acted as a source of an electric field, and a time-varying electric field was a source of a magnetic field. Thus, when either field is changing in time, then a field of the other is necessarily induced.[48] Such a phenomenon has the properties of a wave, and is naturally referred to as an electromagnetic wave. Electromagnetic waves were analysed theoretically by James Clerk Maxwell in 1864. Maxwell discovered a set of equations that could unambiguously describe the interrelationship between electric field, magnetic field, electric charge, and electric current. He could moreover prove that such a wave would necessarily travel at the speed of light, and thus light itself was a form of electromagnetic radiation. Maxwell's Laws, which unify light, fields, and charge are one of the great milestones of theoretical physics.[48]

[edit] Electric circuits

Main article: Electric circuit

A basic electric circuit. The voltage source V on the left drives a current I around the circuit, delivering electrical energy into the resistance R. From the resistor, the current returns to the source, completing the circuit.

A basic electric circuit. The voltage source V on the left drives a current I around the circuit, delivering electrical energy into the resistance R. From the resistor, the current returns to the source, completing the circuit.

An electric circuit is an interconnection of electric components, usually to perform some useful task, with a return path to enable the charge to return to its source.

The components in an electric circuit can take many forms, which can include elements such as resistors, capacitors, switches, transformers and electronics. Electronic circuits contain active components, usually semiconductors, and typically exhibit non-linear behavior, requiring complex analysis. The simplest electric components are those that are termed passive and linear: while they may temporarily store energy, they contain no sources of it, and exhibit linear responses to stimuli.[49]

The resistor is perhaps the simplest of passive circuit elements: as its name suggests, it resists the flow of current through it, dissipating its energy as heat. Ohm's law is a basic law of circuit theory, stating that the current passing through a resistance is directly proportional to the potential difference across it. The ohm, the unit of resistance, was named in honour of Georg Ohm, and is symbolised by the Greek letter Ω. 1 Ω is the resistance that will produce a potential difference of one volt in response to a current of one amp.[49]

The capacitor is a device capable of storing charge, and thereby storing electrical energy in the resulting field. Conceptually, it consists of two conducting plates separated by a thin insulating layer; in practice, thin metal foils are coiled together, increasing the surface area per unit volume and therefore the capacitance. The unit of capacitance is the farad, named after Michael Faraday, and given the symbol F: one farad is the capacitance that develops a potential difference of one volt when it stores a charge of one coulomb. A capacitor connected to a voltage supply initially causes a current to flow as it accumulates charge; this current will however decay in time as the capacitor fills, eventually falling to zero. A capacitor will therefore not permit a steady state current to flow, but instead blocks it.[49]

The inductor is a conductor, usually a coil of wire, that stores energy in a magnetic field in response to the current flowing through it. When the current changes, the magnetic field does too, inducing a voltage between the ends of the conductor. The induced voltage is proportional to the time rate of change of the current. The constant of proportionality is termed the inductance. The unit of inductance is the henry, named after Joseph Henry, a contemporary of Faraday. One henry is the inductance that will induce a potential difference of one volt if the current through it changes at a rate of one ampere per second.[49] The inductor's behaviour is in some regards converse to that of the capacitor: it will freely allow an unchanging current to flow, but opposes the flow of a rapidly changing one.

[edit] Production and uses

[edit] Generation

Main article: Electricity generation

Wind power is of increasing importance in many countries

Wind power is of increasing importance in many countries

Thales' experiments with amber rods were the first studies into the production of electrical energy. While this method, now known as the triboelectric effect, is capable of lifting light objects and even generating sparks, it is extremely inefficient.[50] It was not until the invention of the voltaic pile in the eighteenth century that a viable source of electricity became available. The voltaic pile, and its modern descendant, the electrical battery, store energy chemically and make it available on demand in the form of electrical energy.[50] The battery is a versatile and very common power source which is ideally suited to many applications, but its energy storage is finite, and once discharged it must be disposed of or recharged. For large electrical demands electrical energy must be generated and transmitted in bulk.

Electrical energy is usually generated by electro-mechanical generators driven by steam produced from fossil fuel combustion, or the heat released from nuclear reactions; or from other sources such as kinetic energy extracted from wind or flowing water. Such generators bear no resemblance to Faraday's homopolar disc generator of 1831, but they still rely on his electromagnetic principle that a conductor linking a changing magnetic field induces a potential difference across its ends.[51] The invention in the late nineteenth century of the transformer meant that electricity could be generated at centralised power stations, benefiting from economies of scale, and be transmitted across countries with increasing efficiency.[52][53] Since electrical energy cannot easily be stored in quantities large enough to meet demands on a national scale, at all times exactly as much must be produced as is required.[52] This requires electricity utilities to make careful predictions of their electrical loads, and maintain constant co-ordination with their power stations. A certain amount of generation must always be held in reserve to cushion an electrical grid against inevitable disturbances and losses.

Demand for electricity grows with great rapidity as a nation modernises and its economy develops. The United States showed a 12% increase in demand during each year of the first three decades of the twentieth century,[54] a rate of growth that is now being experienced by emerging economies such as those of India or China.[55][56] Historically, the growth rate for electricity demand has outstripped that for other forms of energy, such as coal.[57]

Environmental concerns with electricity generation have led to an increased focus on generation from renewable sources, in particular from wind- and hydropower. While debate can be expected to continue over the environmental impact of different means of electricity production, its final form is relatively clean.[58]

[edit] Uses

The light bulb, an early application of electricity, operates by Joule heating: the passage of current through resistance generating heat

The light bulb, an early application of electricity, operates by Joule heating: the passage of current through resistance generating heat

Electricity is an extremely flexible form of energy, and has been adapted to a huge, and growing, number of uses.[59] The invention of a practical incandescent light bulb in the 1870s led to lighting becoming one of the first publicly available applications of electrical power. Although electrification brought with it its own dangers, replacing the naked flames of gas lighting greatly reduced fire hazards within homes and factories.[60] Public utilities were set up in many cities targeting the burgeoning market for electrical lighting.

The Joule heating effect employed in the light bulb also sees more direct use in electric heating. While this is versatile and controllable, it can be seen as wasteful, since most electrical generation has already required the production of heat at a power station.[61] A number of countries, such as Denmark, have issued legislation restricting or banning the use of electric heating in new buildings.[62] Electricity is however a highly practical energy source for refrigeration,[63] with air conditioning representing a growing sector for electricity demand, the effects of which electricity utilities are increasingly obliged to accommodate.[64]

Electricity is used within telecommunications, and indeed the electrical telegraph, demonstrated commercially in 1837 by Cooke and Wheatstone, was one of its earliest applications. With the construction of first intercontinental, and then transatlantic, telegraph systems in the 1860s, electricity had enabled communications in minutes across the globe. Optical fibre and satellite communication technology have taken a share of the market for communications systems, but electricity can be expected to remain an essential part of the process.

The effects of electromagnetism are most visibly employed in the electric motor, which provides a clean and efficient means of motive power. A stationary motor such as a winch is easily provided with a supply of power, but a motor that moves with its application, such as an electric vehicle, is obliged to either carry along a power source such as a battery, or by collecting current from a sliding contact such as a pantograph, placing restrictions on its range or performance.

Electronic devices make use of the transistor, perhaps one of the most important inventions of the twentieth century,[65] and a fundamental building block of all modern circuitry. A modern integrated circuit may contain several billion miniaturised transistors in a region only a few centimetres square.[66]

[edit] Electricity and the natural world

[edit] Physiological effects

Main article: Electric shock

A voltage applied to a human body causes an electric current to flow through the tissues, and although the relationship is non-linear, the greater the voltage, the greater the current.[67] The threshold for perception varies with the supply frequency and with the path of the current, but is about 1 mA for mains-frequency electricity.[68] If the current is sufficiently high, it will cause muscle contraction, fibrillation of the heart, and tissue burns.[67] The lack of any visible sign that a conductor is electrified makes electricity a particular hazard. The pain caused by an electric shock can be intense, leading electricity at times to be employed as a method of torture. Death caused by an electric shock is referred to as electrocution. Electrocution is still the means of judicial execution in some jurisdictions, though its use has become rarer in recent times.[69]

[edit] Electrical phenomena in nature

The electric eel, Electrophorus electricus

The electric eel, Electrophorus electricus

Electricity is by no means a purely human invention, and may be observed in several forms in nature, a prominent manifestation of which is lightning. The Earth's magnetic field is thought to arise from a natural dynamo of circulating currents in the planet's core.[70] Certain crystals, such as quartz, or even sugarcane, generate a potential difference across their faces when subjected to external pressure.[71] This phenomenon is known as piezoelectricity, from the Greek piezein (πιέζειν), meaning to press, and was discovered in 1880 by Pierre and Jacques Curie. The effect is reciprocal, and when a piezoelectric material is subjected to an electric field, a small change in physical dimensions take place.[71]

Some organisms, such as sharks, are able to detect and respond to changes in electric fields, an ability known as electroreception,[72] while others, termed electrogenic, are able to generate voltages themselves to serve as a predatory or defensive weapon.[3] The order Gymnotiformes, of which the best known example is the electric eel, detect or stun their prey via high voltages generated from modified muscle cells called electrocytes.[4][3] All animals transmit information along their cell membranes with voltage pulses called action potentials, whose functions include communication by the nervous system between neurons and muscles.[73] (Because of this principle, an electric shock can induce temporary or permanent paralysis by "overloading" the nervous system.) They are also responsible for coordinating activities in certain plants.[73]

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"Electric" redirects here. For other uses, see Electric (disambiguation).

Lightning is one of the most dramatic effects of electricity

Lightning is one of the most dramatic effects of electricity

Electricity (from New Latin ēlectricus, "amber-like") is a general term that encompasses a variety of phenomena resulting from the presence and flow of electric charge. These include many easily recognizable phenomena such as lightning and static electricity, but in addition, less familiar concepts such as the electromagnetic field and electromagnetic induction.

In general usage, the word 'electricity' is adequate to refer to a number of physical effects. However, in scientific usage, the term is vague, and these related, but distinct, concepts are better identified by more precise terms:

* Electric charge – a property of some subatomic particles, which determines their electromagnetic interactions. Electrically charged matter is influenced by, and produces, electromagnetic fields.

* Electric current – a movement or flow of electrically charged particles, typically measured in amperes.

* Electric field – an influence produced by an electric charge on other charges in its vicinity.

* Electric potential – the capacity of an electric field to do work, typically measured in volts.

* Electromagnetism – a fundamental interaction between the electric field and the presence and motion of electric charge.

Electricity has been studied since antiquity, though scientific advances were not forthcoming until the seventeenth and eighteenth centuries. It would not be until the late nineteenth century, however, that engineers were able to put electricity to industrial and residential use. This period witnessed a rapid expansion in the development of electrical technology. Electricity's extraordinary versatility as a source of energy means it can be put to an almost limitless set of applications which include transport, heating, lighting, communications, and computation. The backbone of modern industrial society is, and for the foreseeable future can be expected to remain, the use of electrical power.[1]

Look up Electricity in

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Contents

[hide]

* 1 History

* 2 Concepts

o 2.1 Electric charge

o 2.2 Electric current

o 2.3 Electric field

o 2.4 Electric potential

o 2.5 Electromagnetism

* 3 Electric circuits

* 4 Production and uses

o 4.1 Generation

o 4.2 Uses

* 5 Electricity and the natural world

o 5.1 Physiological effects

o 5.2 Electrical phenomena in nature

* 6 See also

* 7 References

* 8 Bibliography

* 9 External links

[edit] History

Thales, the earliest researcher into electricity

Thales, the earliest researcher into electricity

Main article: History of electricity

See also: Etymology of electricity

Knowledge of electric discharge from electric fishes was first reported in 2750 BC by the ancient Egyptians, who referred to it as the "thunderer of the Nile". They were again reported millennia later by ancient Greek, Roman and Arabic naturalists and physicians.[2] Several ancient writers, such as Pliny the Elder and Scribonius Largus, attested to the numbing effect of electric shocks delivered by catfish and torpedo rays, and knew that such shocks could travel along conducting objects.[3] Patients suffering from ailments such as gout or headache were directed to touch electric fish in the hope that the powerful jolt might cure them.[4] Similar observations were later reported by Al-Jahiz in medieval Egypt.[5]

That certain objects such as rods of amber could be rubbed with cat's fur and attract light objects like feathers was known to ancient cultures around the Mediterranean. Thales of Miletos made a series of observations on static electricity around 600 BC, from which he believed that friction rendered amber magnetic, in contrast to minerals such as magnetite, which needed no rubbing.[6][7] Thales was incorrect in believing the attraction was due to a magnetic effect, but later science would prove a link between magnetism and electricity.

According to a controversial theory, the Parthians may have had knowledge of electroplating, based on the 1936 discovery of the Baghdad Battery, which resembles a galvanic cell, though it is uncertain whether the artefact was electrical in nature.[8] In the 9th century AD, the Andalusian engineer, Abbas Ibn Firnas, invented an artificial weather simulation room, in which spectators saw stars and clouds, and were astonished by artificial thunder and lightning, which were produced by mechanisms hidden in his basement laboratory.[9][10]

Benjamin Franklin conducted extensive research on electricity in the 18th century

Benjamin Franklin conducted extensive research on electricity in the 18th century

Electricity would remain little more than an intellectual curiosity for millennia until 1600, when the English physician William Gilbert made a careful study of electricity and magnetism, distinguishing the lodestone effect from static electricity produced by rubbing amber.[6] He coined the New Latin word electricus ("of amber" or "like amber", from ήλεκτρον [elektron], the Greek word for "amber") to refer to the property of attracting small objects after being rubbed.[11] This association gave rise to the English words "electric" and "electricity", which made their first appearance in print in Thomas Browne's Pseudodoxia Epidemica of 1646.[12]

Further work was conducted by Otto von Guericke, Robert Boyle, Stephen Gray and C. F. du Fay. In the 18th century, Benjamin Franklin conducted extensive research in electricity, selling his possessions to fund his work. In June 1752 he is reputed to have attached a metal key to the bottom of a dampened kite string and flown the kite in a storm-threatened sky.[13] He observed a succession of sparks jumping from the key to the back of his hand, showing that lightning was indeed electrical in nature.[14]

In 1791 Luigi Galvani published his discovery of bioelectricity, demonstrating that electricity was the medium by which nerve cells passed signals to the muscles.[15] Alessandro Volta's battery, or voltaic pile, of 1800, made from alternating layers of zinc and copper, provided scientists with a more reliable source of electrical energy than the electrostatic machines previously used.[15] André-Marie Ampère discovered the relationship between electricity and magnetism in 1820; Michael Faraday invented the electric motor in 1821, and Georg Ohm mathematically analysed the electrical circuit in 1827.[15]

While it had been the early 19th century that had seen rapid progress in electrical science, the late 19th century would see the greatest progress in electrical engineering. Through such people as Nikola Tesla, Thomas Edison, George Westinghouse, Ernst Werner von Siemens, Alexander Graham Bell and Lord Kelvin, electricity was turned from a scientific curiosity into an essential tool for modern life, becoming a driving force for the Second Industrial Revolution.[16]

[edit] Concepts

[edit] Electric charge

Main article: Electric charge

See also: electron, proton, and ion

Electric charge is a property of certain subatomic particles, which gives rise to and interacts with, the electromagnetic force, one of the four fundamental forces of nature. Charge originates in the atom, in which its most familiar carriers are the electron and proton. It is a conserved quantity, that is, the net charge within an isolated system will always remain constant regardless of any changes taking place within that system.[17] Within the system, charge may be transferred between bodies, either by direct contact, or by passing along a conducting material, such as a wire.[18] The informal term static electricity refers to the net presence (or 'imbalance') of charge on a body, usually caused when dissimilar materials are rubbed together, transferring charge from one to the other.

Charge on a gold-leaf electroscope causes the leaves to visibly repel each other

Charge on a gold-leaf electroscope causes the leaves to visibly repel each other

The presence of charge gives rise to the electromagnetic force: charges exert a force on each other, an effect that was known, though not understood, in antiquity.[19] A lightweight ball suspended from a string can be charged by touching it with a glass rod that has itself been charged by rubbing with a cloth. If a similar ball is charged by the same glass rod, it is found to repel the first: the charge acts to force the two balls apart. Two balls that are charged with a rubbed amber rod also repel each other. However, if one ball is charged by the glass rod, and the other by an amber rod, the two balls are found to attract each other. These phenomena were investigated in the late eighteenth century by Charles-Augustin de Coulomb, who deduced that charge manifests itself in two opposing forms, leading to the well-known axiom: like-charged objects repel and opposite-charged objects attract.[19]

The force acts on the charged particles themselves, hence charge has a tendency to spread itself as evenly as possible over a conducting surface. The magnitude of the electromagnetic force, whether attractive or repulsive, is given by Coulomb's law, which relates the force to the product of the charges and has an inverse-square relation to the distance between them.[20][21] The electromagnetic force is very strong, second only in strength to the strong interaction,[22] but unlike that force it operates over all distances.[23] In comparison with the much weaker gravitational force, the electromagnetic force pushing two electrons apart is 1042 times that of the gravitational attraction pulling them together.[24]

The charge on electrons and protons is opposite in sign, hence an amount of charge may be expressed as being either negative or positive. By convention, the charge carried by electrons is deemed negative, and that by protons positive, a custom that originated with the work of Benjamin Franklin.[25] The amount of charge is usually given the symbol Q and expressed in coulombs;[26] each electron carries the same charge of approximately −1.6022×10−19 coulomb. The proton has a charge that is equal and opposite, and thus +1.6022×10−19 coulomb. Charge is possessed not just by matter, but also by antimatter, each antiparticle bearing an equal and opposite charge to its corresponding particle.[27]

Charge can be measured by a number of means, an early instrument being the gold-leaf electroscope, which although still in use for classroom demonstrations, has been superseded by the electronic electrometer.[18]

[edit] Electric current

Main article: Current (electricity)

The movement of electric charge is known as an electric current, the intensity of which is usually measured in amperes. Current can consist of any moving charged particles; most commonly these are electrons, but any charge in motion constitutes a current.

By historical convention, a positive current is defined as having the same direction of flow as any positive charge it contains, or to flow from the most positive part of a circuit to the most negative part. Current defined in this manner is called conventional current. The motion of negatively-charged electrons around an electric circuit, one of the most familiar forms of current, is thus deemed positive in the opposite direction to that of the electrons.[28] However, depending on the conditions, an electric current can consist of a flow of charged particles in either direction, or even in both directions at once. The positive-to-negative convention is widely used to simplify this situation. If another definition is used—for example, "electron current"—it needs to be explicitly stated.

An electric arc provides an energetic demonstration of electric current

An electric arc provides an energetic demonstration of electric current

The process by which electric current passes through a material is termed electrical conduction, and its nature varies with that of the charged particles and the material through which they are travelling. Examples of electric currents include metallic conduction, where electrons flow through a conductor such as metal, and electrolysis, where ions (charged atoms) flow through liquids. While the particles themselves can move quite slowly, sometimes with a average drift velocity only fractions of a millimetre per second,[18] the electric field that drives them itself propagates at close to the speed of light, enabling electrical signals to pass rapidly along wires.[29]

Current causes several observable effects, which historically were the means of recognising its presence. That water could be decomposed by the current from a voltaic pile was discovered by Nicholson and Carlisle in 1800, a process now known as electrolysis. Their work was greatly expanded upon by Michael Faraday in 1833.[30] Current through a resistance causes localised heating, an effect James Prescott Joule studied mathematically in 1840.[30] One of the most important discoveries relating to current was made accidentally by Hans Christian Ørsted in 1820, when, while preparing a lecture, he witnessed the current in a wire disturbing the needle of a magnetic compass.[31] He had discovered electromagnetism, a fundamental interaction between electricity and magnetics.

In engineering or household applications, current is often described as being either direct current (DC) or alternating current (AC). These terms refer to how the current varies in time. Direct current, as produced by example from a battery and required by most electronic devices, is a unidirectional flow from the positive part of a circuit to the negative.[32] If, as is most common, this flow is carried by electrons, they will be travelling in the opposite direction. Alternating current is any current that reverses direction repeatedly; almost always this takes the form of a sinusoidal wave.[33] Alternating current thus pulses back and forth within a conductor without the charge moving any net distance over time. The time-averaged value of an alternating current is zero, but it delivers energy in first one direction, and then the reverse. Alternating current is affected by electrical properties that are not observed under steady state direct current, such as inductance and capacitance.[34] These properties however can become important when circuitry is subjected to transients, such as when first energised.

[edit] Electric field

Main article: Electric field

See also: Electrostatics

The concept of the electric field was introduced by Michael Faraday. An electric field is created by a charged body in the space that surrounds it, and results in a force exerted on any other charges placed within the field. The electric field acts between two charges in a similar manner to the way that the gravitational field acts between two masses, and like it, extends towards infinity and shows an inverse square relationship with distance.[23] However, there is an important difference. Gravity always acts in attraction, drawing two masses together, while the electric field can result in either attraction or repulsion. Since large bodies such as planets generally carry no net charge, the electric field at a distance is usually zero. Thus gravity is the dominant force at distance in the universe, despite being much the weaker.[24]

Field lines emanating from a positive charge above a plane conductor

Field lines emanating from a positive charge above a plane conductor

An electric field generally varies in space,[35] and its strength at any one point is defined as the force (per unit charge) that would be felt by a stationary, negligible charge if placed at that point.[36] The conceptual charge, termed a 'test charge', must be vanishingly small to prevent its own electric field disturbing the main field and must also be stationary to prevent the effect of magnetic fields. As the electric field is defined in terms of force, and force is a vector, so it follows that an electric field is also a vector, having both magnitude and direction. Specifically, it is a vector field.[36]

The study of electric fields created by stationary charges is called electrostatics. The field may be visualised by a set of imaginary lines whose direction at any point is the same as that of the field. This concept was introduced by Faraday,[37] whose term 'lines of force' still sometimes sees use. The field lines are the paths that a point positive charge would seek to make as it was forced to move within the field; they are however an imaginary concept with no physical existence, and the field permeates all the intervening space between the lines.[37] Field lines emanating from stationary charges have several key properties: first, that they originate at positive charges and terminate at negative charges; second, that they must enter any good conductor at right angles, and third, that they may never cross nor close in on themselves.[38]

The principals of electrostatics are important when designing items of high-voltage equipment. There is a finite limit to the electric field strength that may withstood by any medium. Beyond this point, electrical breakdown occurs and an electric arc causes flashover between the charged parts. Air, for example, tends to arc at electric field strengths which exceed 30 kV per centimetre across small gaps. Over larger gaps, its breakdown strength is weaker, perhaps 1 kV per centimetre.[39] The most visible natural occurrence of this is lightning, caused when charge becomes separated in the clouds by rising columns of air, and raises the electric field in the air to greater than it can withstand. The voltage of a large lightning cloud may be as high as 100 MV and have discharge energies as great as 250 kWh.[40]

The field strength is greatly affected by nearby conducting objects, and it is particularly intense when it is forced to curve around sharply pointed objects. This principle is exploited in the lightning conductor, the sharp spike of which acts to encourage the lightning stroke to develop there, rather than to the building it serves to protect.[41]

An electric field is zero inside a conductor. This is because the net charge on a conductor only exists on the surface. External electrostatic fields are always perpendicular to the conductors surface. Otherwise this would produce a force on the charge carriers inside the conductor and so the field would not be static as we assume.

[edit] Electric potential

Main article: Electric potential

See also: Voltage

A pair of AA cells. The + sign indicates the polarity of the potential differences between the battery terminals.

A pair of AA cells. The + sign indicates the polarity of the potential differences between the battery terminals.

The concept of electric potential is closely linked to that of the electric field. A small charge placed within an electric field experiences a force, and to have brought that charge to that point against the force requires work. The electric potential at any point is defined as the energy required to bring a unit test charge from an infinite distance slowly to that point. It is usually measured in volts, and one volt is the potential for which one joule of work must be expended to bring a charge of one coulomb from infinity.[42] This definition of potential, while formal, has little practical application, and a more useful concept is that of electric potential difference, and is the energy required to move a unit charge between two specified points. An electric field has the special property that it is conservative, which means that the path taken by the test charge is irrelevant: all paths between two specified points expend the same energy, and thus a unique value for potential difference may be stated.[42] The volt is so strongly identified as the unit of choice for measurement and description of electric potential difference that the term voltage sees greater everyday usage.

For practical purposes, it is useful to define a common reference point to which potentials may be expressed and compared. While this could be at infinity, a much more useful reference is the Earth itself, which is assumed to be at the same potential everywhere. This reference point naturally takes the name earth or ground. Earth is assumed to be an infinite source of equal amounts of positive and negative charge, and is therefore electrically uncharged – and unchargeable.[43]

Electric potential is a scalar quantity, that is, it has only magnitude and not direction. It may be viewed as analogous to temperature: as there is a certain temperature at every point in space, and the temperature gradient indicates the direction and magnitude of the driving force behind heat flow, similarly, there is an electric potential at every point in space, and its gradient, or field strength, indicates the direction and magnitude of the driving force behind charge movement. Equally, electric potential may be seen as analogous to height: just as a released object will fall through a difference in heights caused by a gravitational field, so a charge will 'fall' across the voltage caused by an electric field.[44]

The electric field was formally defined as the force exerted per unit charge, but the concept of potential allows for a more useful and equivalent definition: the electric field is the local gradient of the electric potential. Usually expressed in volts per metre, the vector direction of the field is the line of greatest gradient of potential.[18]

[edit] Electromagnetism

Main article: Electromagnetism

Magnetic field circles around a current

Magnetic field circles around a current

Ørsted's discovery in 1821 that a magnetic field existed around all sides of a wire carrying an electric current indicated that there was a direct relationship between electricity and magnetism. Moreover, the interaction seemed different from gravitational and electrostatic forces, the two forces of nature then known. The force on the compass needle did not direct it to or away from the current-carrying wire, but acted at right angles to it.[31] Ørsted's slightly obscure words were that "the electric conflict acts in a revolving manner." The force also depended on the direction of the current, for if the flow was reversed, then the force did too.[45]

Ørsted did not fully understand his discovery, but he observed the effect was reciprocal: a current exerts a force on a magnet, and a magnetic field exerts a force on a current. The phenomenon was further investigated by Ampère, who discovered that two parallel current carrying wires exerted a force upon each other: two wires conducting currents in the same direction are attracted to each other, while wires containing current flowing in opposite directions are forced apart.[46] The interaction is mediated by the magnetic field each current produces and forms the basis for the international definition of the ampere.[46]

The electric motor exploits an important effect of electromagnetism: a current flowing through a magnetic field experiences a force at right angles to both the field and current

The electric motor exploits an important effect of electromagnetism: a current flowing through a magnetic field experiences a force at right angles to both the field and current

This relationship between magnetic fields and currents is extremely important, for it led to Michael Faraday's invention of the electric motor in 1821. Faraday's homopolar motor consisted of a permanent magnet sitting in a pool of mercury. A current was allowed to flow through a wire suspended from a pivot above the magnet and dipped into the mercury. The magnet exerted a tangential force on the wire, making it circle around the magnet for as long as the current was maintained.[47]

Experimentation by Faraday in 1831 revealed that a wire moving perpendicular to a magnetic field developed a potential difference between its ends. Further analysis of this process, known as electromagnetic induction, enabled him to state the principal, now known as Faraday's law of induction, that the potential difference induced in a closed circuit is proportional to the rate of change of magnetic flux through the loop. Exploitation of this discovery enabled him to invent the first electrical generator in 1831, in which he converted the mechanical energy of a rotating copper disc to electrical energy.[47] Faraday's disc was inefficient and of no use as a practical generator, but it showed the possibility of generating electric power using magnetism, a possibility that would be taken up by those that followed on from his work.

Faraday's and Ampère's work showed that a time-varying magnetic field acted as a source of an electric field, and a time-varying electric field was a source of a magnetic field. Thus, when either field is changing in time, then a field of the other is necessarily induced.[48] Such a phenomenon has the properties of a wave, and is naturally referred to as an electromagnetic wave. Electromagnetic waves were analysed theoretically by James Clerk Maxwell in 1864. Maxwell discovered a set of equations that could unambiguously describe the interrelationship between electric field, magnetic field, electric charge, and electric current. He could moreover prove that such a wave would necessarily travel at the speed of light, and thus light itself was a form of electromagnetic radiation. Maxwell's Laws, which unify light, fields, and charge are one of the great milestones of theoretical physics.[48]

[edit] Electric circuits

Main article: Electric circuit

A basic electric circuit. The voltage source V on the left drives a current I around the circuit, delivering electrical energy into the resistance R. From the resistor, the current returns to the source, completing the circuit.

A basic electric circuit. The voltage source V on the left drives a current I around the circuit, delivering electrical energy into the resistance R. From the resistor, the current returns to the source, completing the circuit.

An electric circuit is an interconnection of electric components, usually to perform some useful task, with a return path to enable the charge to return to its source.

The components in an electric circuit can take many forms, which can include elements such as resistors, capacitors, switches, transformers and electronics. Electronic circuits contain active components, usually semiconductors, and typically exhibit non-linear behavior, requiring complex analysis. The simplest electric components are those that are termed passive and linear: while they may temporarily store energy, they contain no sources of it, and exhibit linear responses to stimuli.[49]

The resistor is perhaps the simplest of passive circuit elements: as its name suggests, it resists the flow of current through it, dissipating its energy as heat. Ohm's law is a basic law of circuit theory, stating that the current passing through a resistance is directly proportional to the potential difference across it. The ohm, the unit of resistance, was named in honour of Georg Ohm, and is symbolised by the Greek letter Ω. 1 Ω is the resistance that will produce a potential difference of one volt in response to a current of one amp.[49]

The capacitor is a device capable of storing charge, and thereby storing electrical energy in the resulting field. Conceptually, it consists of two conducting plates separated by a thin insulating layer; in practice, thin metal foils are coiled together, increasing the surface area per unit volume and therefore the capacitance. The unit of capacitance is the farad, named after Michael Faraday, and given the symbol F: one farad is the capacitance that develops a potential difference of one volt when it stores a charge of one coulomb. A capacitor connected to a voltage supply initially causes a current to flow as it accumulates charge; this current will however decay in time as the capacitor fills, eventually falling to zero. A capacitor will therefore not permit a steady state current to flow, but instead blocks it.[49]

The inductor is a conductor, usually a coil of wire, that stores energy in a magnetic field in response to the current flowing through it. When the current changes, the magnetic field does too, inducing a voltage between the ends of the conductor. The induced voltage is proportional to the time rate of change of the current. The constant of proportionality is termed the inductance. The unit of inductance is the henry, named after Joseph Henry, a contemporary of Faraday. One henry is the inductance that will induce a potential difference of one volt if the current through it changes at a rate of one ampere per second.[49] The inductor's behaviour is in some regards converse to that of the capacitor: it will freely allow an unchanging current to flow, but opposes the flow of a rapidly changing one.

[edit] Production and uses

[edit] Generation

Main article: Electricity generation

Wind power is of increasing importance in many countries

Wind power is of increasing importance in many countries

Thales' experiments with amber rods were the first studies into the production of electrical energy. While this method, now known as the triboelectric effect, is capable of lifting light objects and even generating sparks, it is extremely inefficient.[50] It was not until the invention of the voltaic pile in the eighteenth century that a viable source of electricity became available. The voltaic pile, and its modern descendant, the electrical battery, store energy chemically and make it available on demand in the form of electrical energy.[50] The battery is a versatile and very common power source which is ideally suited to many applications, but its energy storage is finite, and once discharged it must be disposed of or recharged. For large electrical demands electrical energy must be generated and transmitted in bulk.

Electrical energy is usually generated by electro-mechanical generators driven by steam produced from fossil fuel combustion, or the heat released from nuclear reactions; or from other sources such as kinetic energy extracted from wind or flowing water. Such generators bear no resemblance to Faraday's homopolar disc generator of 1831, but they still rely on his electromagnetic principle that a conductor linking a changing magnetic field induces a potential difference across its ends.[51] The invention in the late nineteenth century of the transformer meant that electricity could be generated at centralised power stations, benefiting from economies of scale, and be transmitted across countries with increasing efficiency.[52][53] Since electrical energy cannot easily be stored in quantities large enough to meet demands on a national scale, at all times exactly as much must be produced as is required.[52] This requires electricity utilities to make careful predictions of their electrical loads, and maintain constant co-ordination with their power stations. A certain amount of generation must always be held in reserve to cushion an electrical grid against inevitable disturbances and losses.

Demand for electricity grows with great rapidity as a nation modernises and its economy develops. The United States showed a 12% increase in demand during each year of the first three decades of the twentieth century,[54] a rate of growth that is now being experienced by emerging economies such as those of India or China.[55][56] Historically, the growth rate for electricity demand has outstripped that for other forms of energy, such as coal.[57]

Environmental concerns with electricity generation have led to an increased focus on generation from renewable sources, in particular from wind- and hydropower. While debate can be expected to continue over the environmental impact of different means of electricity production, its final form is relatively clean.[58]

[edit] Uses

The light bulb, an early application of electricity, operates by Joule heating: the passage of current through resistance generating heat

The light bulb, an early application of electricity, operates by Joule heating: the passage of current through resistance generating heat

Electricity is an extremely flexible form of energy, and has been adapted to a huge, and growing, number of uses.[59] The invention of a practical incandescent light bulb in the 1870s led to lighting becoming one of the first publicly available applications of electrical power. Although electrification brought with it its own dangers, replacing the naked flames of gas lighting greatly reduced fire hazards within homes and factories.[60] Public utilities were set up in many cities targeting the burgeoning market for electrical lighting.

The Joule heating effect employed in the light bulb also sees more direct use in electric heating. While this is versatile and controllable, it can be seen as wasteful, since most electrical generation has already required the production of heat at a power station.[61] A number of countries, such as Denmark, have issued legislation restricting or banning the use of electric heating in new buildings.[62] Electricity is however a highly practical energy source for refrigeration,[63] with air conditioning representing a growing sector for electricity demand, the effects of which electricity utilities are increasingly obliged to accommodate.[64]

Electricity is used within telecommunications, and indeed the electrical telegraph, demonstrated commercially in 1837 by Cooke and Wheatstone, was one of its earliest applications. With the construction of first intercontinental, and then transatlantic, telegraph systems in the 1860s, electricity had enabled communications in minutes across the globe. Optical fibre and satellite communication technology have taken a share of the market for communications systems, but electricity can be expected to remain an essential part of the process.

The effects of electromagnetism are most visibly employed in the electric motor, which provides a clean and efficient means of motive power. A stationary motor such as a winch is easily provided with a supply of power, but a motor that moves with its application, such as an electric vehicle, is obliged to either carry along a power source such as a battery, or by collecting current from a sliding contact such as a pantograph, placing restrictions on its range or performance.

Electronic devices make use of the transistor, perhaps one of the most important inventions of the twentieth century,[65] and a fundamental building block of all modern circuitry. A modern integrated circuit may contain several billion miniaturised transistors in a region only a few centimetres square.[66]

[edit] Electricity and the natural world

[edit] Physiological effects

Main article: Electric shock

A voltage applied to a human body causes an electric current to flow through the tissues, and although the relationship is non-linear, the greater the voltage, the greater the current.[67] The threshold for perception varies with the supply frequency and with the path of the current, but is about 1 mA for mains-frequency electricity.[68] If the current is sufficiently high, it will cause muscle contraction, fibrillation of the heart, and tissue burns.[67] The lack of any visible sign that a conductor is electrified makes electricity a particular hazard. The pain caused by an electric shock can be intense, leading electricity at times to be employed as a method of torture. Death caused by an electric shock is referred to as electrocution. Electrocution is still the means of judicial execution in some jurisdictions, though its use has become rarer in recent times.[69]

[edit] Electrical phenomena in nature

The electric eel, Electrophorus electricus

The electric eel, Electrophorus electricus

Electricity is by no means a purely human invention, and may be observed in several forms in nature, a prominent manifestation of which is lightning. The Earth's magnetic field is thought to arise from a natural dynamo of circulating currents in the planet's core.[70] Certain crystals, such as quartz, or even sugarcane, generate a potential difference across their faces when subjected to external pressure.[71] This phenomenon is known as piezoelectricity, from the Greek piezein (πιέζειν), meaning to press, and was discovered in 1880 by Pierre and Jacques Curie. The effect is reciprocal, and when a piezoelectric material is subjected to an electric field, a small change in physical dimensions take place.[71]

Some organisms, such as sharks, are able to detect and respond to changes in electric fields, an ability known as electroreception,[72] while others, termed electrogenic, are able to generate voltages themselves to serve as a predatory or defensive weapon.[3] The order Gymnotiformes, of which the best known example is the electric eel, detect or stun their prey via high voltages generated from modified muscle cells called electrocytes.[4][3] All animals transmit information along their cell membranes with voltage pulses called action potentials, whose functions include communication by the nervous system between neurons and muscles.[73] (Because of this principle, an electric shock can induce temporary or permanent paralysis by "overloading" the nervous system.) They are also responsible for coordinating activities in certain plants.[73]

### REFRACTION

Refraction

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For the property of metals, see refraction (metallurgy).

For the magic effect, see David Penn (magician).

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

The straw seems to be broken, due to refraction of light as it emerges into the air.

The straw seems to be broken, due to refraction of light as it emerges into the air.

Refraction is the change in direction of a wave due to a change in its speed. This is most commonly seen when a wave passes from one medium to another. Refraction of light is the most commonly seen example, but any type of wave can refract when it interacts with a medium, for example when sound waves pass from one medium into another or when water waves move into water of a different depth. Refraction is described by Snell's law, which states that the angle of incidence is related to the angle of refraction by

\frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1}

or

n_1\sin\theta_1 = n_2\sin\theta_2\

where

v1 and v2 are the wave velocities through the respective media.

θ1 and θ2 are the angles between the normal (to the interface) plane and the incident waves respectively.

n1 and n2 are the refractive indices

Contents

[hide]

* 1 Explanation

* 2 Optometry

* 3 Acoustics

* 4 See also

* 5 Reference

* 6 External links

[edit] Explanation

In optics, refraction occurs when light waves travel from a medium with a given refractive index to a medium with another. At the boundary between the media, the wave's phase velocity is altered, it changes direction, and its wavelength increases or decreases but its frequency remains constant. For example, a light ray will refract as it enters and leaves glass; understanding of this concept led to the invention of lenses and the refracting telescope

Refraction of light waves in water. The dark rectangle represents the actual position of a pencil sitting in a bowl of water. The light rectangle represents the apparent position of the pencil. Notice that the end (X) looks like it is at (Y), a position that is considerably shallower than (X).

Refraction of light waves in water. The dark rectangle represents the actual position of a pencil sitting in a bowl of water. The light rectangle represents the apparent position of the pencil. Notice that the end (X) looks like it is at (Y), a position that is considerably shallower than (X).

Photograph of refraction of waves in a ripple tank

Photograph of refraction of waves in a ripple tank

Refraction can be seen when looking into a bowl of water. Air has a refractive index of about 1.0003, and water has a refractive index of about 1.33. If a person looks at a straight object, such as a pencil or straw, which is placed at a slant, partially in the water, the object appears to bend at the water's surface. This is due to the bending of light rays as they move from the water to the air. Once the rays reach the eye, the eye traces them back as straight lines (lines of sight). The lines of sight (shown as dashed lines) intersect at a higher position than where the actual rays originated. This causes the pencil to appear higher and the water to appear shallower than it really is. The depth that the water appears to be when viewed from above is known as the apparent depth. This is an important consideration for spearfishing from the surface because it will make the target fish appear to be in a different place, and the fisher must aim lower to catch the fish.

Diagram of refraction of water waves

Diagram of refraction of water waves

The diagram on the right shows an example of refraction in water waves. Ripples travel from the left and pass over a shallower region inclined at an angle to the wavefront. The waves travel more slowly in the shallower water, so the wavelength decreases and the wave bends at the boundary. The dotted line represents the normal to the boundary. The dashed line represents the original direction of the waves. The phenomenon explains why waves on a shoreline never hit the shoreline at an angle. Whichever direction the waves travel in deep water, they always refract towards the normal as they enter the shallower water near the beach.

Solar glory at the steam from hot springs at Yellowstone National Park.A glory is an optical phenomenon produced by light backscattered (a combination of diffraction, reflection and refraction) towards its source by a cloud of uniformly-sized water droplets.

Solar glory at the steam from hot springs at Yellowstone National Park.A glory is an optical phenomenon produced by light backscattered (a combination of diffraction, reflection and refraction) towards its source by a cloud of uniformly-sized water droplets.

Refraction is also responsible for rainbows and for the splitting of white light into a rainbow-spectrum as it passes through a glass prism. Glass has a higher refractive index than air and the different frequencies of light travel at different speeds (dispersion), causing them to be refracted at different angles, so that you can see them. The different frequencies correspond to different colors observed.

While refraction allows for beautiful phenomena such as rainbows, it may also produce peculiar optical phenomena, such as mirages and Fata Morgana. These are caused by the change of the refractive index of air with temperature.

Refraction in a Perspex (acrylic) block.

Refraction in a Perspex (acrylic) block.

Snell's law is used to calculate the degree to which light is refracted when traveling from one medium to another.

Recently some metamaterials have been created which have a negative refractive index. With metamaterials, we can also obtain the total refraction phenomena when the wave impedances of the two media are matched. There is no reflected wave.

Also, since refraction can make objects appear closer than they are, it is responsible for allowing water to magnify objects. First, as light is entering a drop of water, it slows down. If the water's surface is not flat, then the light will be bent into a new path. This round shape will bend the light outwards and as it spreads out, the image you see gets larger.

A useful analogy in explaining the refraction of light would be to imagine a marching band as they march from pavement (a fast medium) into mud (a slower medium) The marchers on the side that runs into the mud first will slow down first. This causes the whole band to pivot slightly toward the normal (make a smaller angle from the normal).

[edit] Optometry

In medicine, particularly optometry and ophthalmology, refraction (also known as refractometry) is a clinical test in which a phoropter may used by an optometrist to determine the eye's refractive error and the best corrective lenses to be prescribed. A series of test lenses in graded optical powers or focal lengths are presented to determine which provide the sharpest, clearest vision.[1]

[edit] Acoustics

In underwater acoustics, refraction is the bending or curving of a sound ray that results when the ray passes through a sound speed gradient from a region of one sound speed to a region of a different speed. The amount of ray bending is dependent upon the amount of difference between sound speeds, that is, the variation in temperature, salinity, and pressure of the water.[2] Similar acoustics effects are also found in the Earth's atmosphere. The phenomenon of refraction of sound in the atmosphere has been known for centuries;[3] however, beginning in the early 1970s, widespread analysis of this effect came into vogue through the designing of urban highways and noise barriers to address the meteorological effects of bending of sound rays in the lower atmosphere.[4]Snell's law

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In optics and physics, Snell's law (also known as Descartes' law or the law of refraction), is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves, passing through a boundary between two different isotropic media, such as water and glass. The law says that the ratio of the sines of the angles of incidence and of refraction is a constant that depends on the media.

In optics, the law is used in ray tracing to compute the angles of incidence or refraction, and in experimental optics to find the refractive index of a material.

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

Named after Dutch mathematician Willebrord Snellius, one of its discoverers, Snell's law states that the ratio of the sines of the angles of incidence and refraction is equivalent to the ratio of velocities in the two media, or equivalent to the opposite ratio of the indices of refraction:

\frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1}

or

n_1\sin\theta_1 = n_2\sin\theta_2\ .

Snell's law follows from Fermat's principle of least time, which in turn follows from the propagation of light as waves.

Contents

[hide]

* 1 History

* 2 Explanation

o 2.1 Total internal reflection and critical angle

o 2.2 Derivations

* 3 Vector form

* 4 Dispersion

* 5 See also

* 6 References

* 7 External links

[edit] History

Ptolemy, of ancient Greece, had, through experiment, found a relationship regarding refraction angles, but which was inaccurate for angles that were not small. Ptolemy was confident he had found an accurate empirical law, partially as a result of fudging his data to fit theory (see: confirmation bias).[1]

An 1837 view of the history of "the Law of the Sines"

An 1837 view of the history of "the Law of the Sines"[2]

Snell's law was first described in a formal manuscript in a 984 writing by Ibn Sahl,[3][4] who used it to work out the shapes of lenses that focus light with no geometric aberrations, known as anaclastic lenses.

It was described again by Thomas Harriot in 1602,[5] who did not publish his work.

In 1621, Willebrord Snellius (Snel) derived a mathematically equivalent form, that remained unpublished during his lifetime. René Descartes independently derived the law using heuristic momentum conservation arguments in terms of sines in his 1637 treatise Discourse on Method, and used it to solve a range of optical problems. Rejecting Descartes' solution, Pierre de Fermat arrived at the same solution based solely on his principle of least time.

According to Dijksterhuis[6], "In De natura lucis et proprietate (1662) Isaac Vossius said that Descartes had seen Snell's paper and concocted his own proof. We now know this charge to be undeserved but it has been adopted many times since." Both Fermat and Huygens repeated this accusation that Descartes had copied Snell.

In French, Snell's Law is called "la loi de Descartes" or "loi de Snell-Descartes."

Huygens's construction

Huygens's construction

In his 1678 Traité de la Lumiere, Christiaan Huygens showed how Snell's law of sines could be explained by, or derived from, the wave nature of light, using what we have come to call the Huygens–Fresnel principle.

Although he spelled his name "Snel", as noted above, it has conventionally been spelled "Snell", apparently by misinterpreting the Latin form of his name, "Snellius".[7]

[edit] Explanation

Snell's law is used to determine the direction of light rays through refractive media with varying indices of refraction. The indices of refraction of the media, labeled n1,n2 and so on, are used to represent the factor by which a light ray's speed decreases when traveling through a refractive medium, such as glass or water, as opposed [8] to its velocity in a vacuum.

As light passes the border between media, depending upon the relative refractive indices of the two media, the light will either be refracted to a lesser angle, or a greater one. These angles are measured with respect to the normal line, represented perpendicular to the boundary. In the case of light traveling from air into water, light would be refracted towards the normal line, because the light is slowed down in water; light traveling from water to air would refract away from the normal line.

Refraction between two surfaces is also referred to as reversible because if all conditions were identical, the angles would be the same for light propagating in the opposite direction.

Snell's law is generally true only for isotropic or specular media (such as glass). In anisotropic media such as some crystals, birefringence may split the refracted ray into two rays, the ordinary or o-ray which follows Snell's law, and the other extraordinary or e-ray which may not be co-planar with the incident ray.

When the light or other wave involved is monochromatic, that is, of a single frequency, Snell's law can also be expressed in terms of a ratio of wavelengths in the two media, λ1 and λ2:

\frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2} = \frac{\lambda_1}{\lambda_2}

[edit] Total internal reflection and critical angle

An example of the angles involved within total internal reflection.

An example of the angles involved within total internal reflection.

Main article: Total internal reflection

When light moves from a dense to a less dense medium, such as from water to air, Snell's law cannot be used to calculate the refracted angle when the resolved sine value is higher than 1. At this point, light is reflected in the incident medium, known as internal reflection. Before the ray totally internally reflects, the light refracts at the critical angle; it travels directly along the surface between the two refractive media, without a change in phases like in other forms of optical phenomena.

As an example, a ray of light is incident at 50o towards a water–air boundary. If the angle is calculated using Snell's Law, then the resulting sine value will not invert, and thus the refracted angle cannot be calculated by Snell's law, due to the absence of a refracted outgoing ray:

\theta_2 = \sin^{-1} \left(\frac{n_1}{n_2}\sin\theta_1\right) = \sin^{-1} \left(\frac{1.333}{1.000}0.766\right) = \sin^{-1} 1.021

In order to calculate the critical angle, let θ2 = 90o and solve for θcrit:

\theta_{\mathrm{crit}} = \sin^{-1} \left( \frac{n_2}{n_1} \right)

When θ1 > θcrit, no refracted ray appears, and the incident ray undergoes total internal reflection from the interface medium.

[edit] Derivations

Wavefronts due to a point source in the context of Snell's law (the region below the gray line has a higher index of refraction than the region above it).

Wavefronts due to a point source in the context of Snell's law (the region below the gray line has a higher index of refraction than the region above it).

Snell's law may be derived from Fermat's principle, which states that the light travels the path which takes the least time. By taking the derivative of the optical path length, the stationary point is found giving the path taken by the light (though it should be noted that the result does not show light taking the least time path, but rather one that is stationary with respect to small variations as there are cases where light actually takes the greatest time path, as in a spherical mirror). In a classic analogy by Richard Feynman, the area of lower refractive index is replaced by a beach, the area of higher refractive index by the sea, and the fastest way for a rescuer on the beach to get to a drowning person in the sea is to run along a path that follows Snell's law.

Alternatively, Snell's law can be derived using interference of all possible paths of light wave from source to observer—it results in destructive interference everywhere except extrema of phase (where interference is constructive)—which become actual paths.

Another way to derive Snell’s Law involves an application of the general boundary conditions of Maxwell equations for electromagnetic radiation.

[edit] Vector form

Given a normalized light vector l (pointing from the light source toward the surface) and a normalized plane normal vector n, one can work out the normalized reflected and refracted rays:

\cos\theta_1=\mathbf{n}\cdot (-\mathbf{l})

\cos\theta_2=\sqrt{1-\left(\frac{n_1}{n_2}\right)^2\left(1-\left(\cos\theta_1\right)^2\right)}

\mathbf{v}_{\mathrm{reflect}}=\mathbf{l}+\left(2\cos\theta_1\right)\mathbf{n}

\mathbf{v}_{\mathrm{refract}}=\left(\frac{n_1}{n_2}\right)\mathbf{l} + \left( \frac{n_1}{n_2}\cos\theta_1 - \cos\theta_2\right)\mathbf{n}

Note: \mathbf{n}\cdot(-\mathbf{l}) must be positive. Otherwise, use

\mathbf{v}_{\mathrm{refract}}=\left(\frac{n_1}{n_2}\right)\mathbf{l} + \left(\frac{n_1}{n_2}\cos\theta_1 + \cos\theta_2\right)\mathbf{n}.

Example:

\mathbf{l}=\{0.707107, -0.707107\},~\mathbf{n}=\{0,1\},~\frac{n_1}{n_2}=0.9

\mathbf{~}\cos\theta_1=0.707107,~\cos\theta_2=0.771362

\mathbf{v}_{\mathrm{reflect}}=\{0.707107, 0.707107\} ,~\mathbf{v}_{\mathrm{refract}}=\{0.636396, -0.771362\}

The cosines may be recycled and used in the Fresnel equations for working out the intensity of the resulting rays. During total internal reflection an evanescent wave is produced, which rapidly decays from the surface into the second medium. Conservation of energy is maintained by the circulation of energy across the boundary, averaging to zero net energy transmission.

[edit] Dispersion

Main article: Dispersion (optics)

In many wave-propagation media, wave velocity changes with frequency or wavelength of the waves; this is true of light propagation in most transparent substances other than a vacuum. These media are called dispersive. The result is that the angles determined by Snell's law also depend on frequency or wavelength, so that a ray of mixed wavelengths, such as white light, will spread or disperse. Such dispersion of light in glass or water underlies the origin of rainbows, in which different wavelengths appear as different colors.

In optical instruments, dispersion leads to chromatic aberration, a color-dependent blurring that sometimes is the resolution-limiting effect. This was especially true in refracting telescopes, before the invention of achromatic objective lenses.Diffraction

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The intensity pattern formed on a screen by diffraction from a square aperture

The intensity pattern formed on a screen by diffraction from a square aperture

Colors seen in a spider web are partially due to diffraction, according to some analyses.

Colors seen in a spider web are partially due to diffraction, according to some analyses.[1]

Diffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle. It is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings.[2] Very similar effects are observed when there is an alteration in the properties of the medium in which the wave is travelling, for example a variation in refractive index for light waves or in acoustic impedance for sound waves and these can also be referred to as diffraction effects. Diffraction occurs with all waves, including sound waves, water waves, and electromagnetic waves such as visible light, x-rays and radio waves. As physical objects have wave-like properties, diffraction also occurs with matter and can be studied according to the principles of quantum mechanics.

While diffraction occurs whenever propagating waves encounter such changes, its effects are generally most pronounced for waves where the wavelength is on the order of the size of the diffracting objects. The complex patterns resulting from the intensity of a diffracted wave are a result of the superposition, or interference of different parts of a wave that traveled to the observer by different paths.

The formalism of diffraction can also describe the way in which waves of finite extent propagate in free space. For example, the expanding profile of a laser beam, the beam shape of a radar antenna and the field of view of an ultrasonic transducer are all explained by diffraction theory.

Contents

[hide]

* 1 Examples of diffraction in everyday life

* 2 History

* 3 The mechanism of diffraction

* 4 Diffraction systems

o 4.1 Single-slit diffraction

o 4.2 Diffraction Grating

o 4.3 Diffraction by a circular aperture

o 4.4 Propagation of a laser beam

o 4.5 Diffraction limited imaging

o 4.6 Speckle patterns

* 5 Common features of diffraction patterns

* 6 Particle diffraction

* 7 Bragg diffraction

* 8 Coherence

* 9 References

* 10 See also

* 11 External links

[edit] Examples of diffraction in everyday life

Solar glory at the steam from hot springs at Yellowstone National Park.A glory is an optical phenomenon produced by light backscattered (a combination of diffraction, reflection and refraction) towards its source by a cloud of uniformly-sized water droplets.

Solar glory at the steam from hot springs at Yellowstone National Park.A glory is an optical phenomenon produced by light backscattered (a combination of diffraction, reflection and refraction) towards its source by a cloud of uniformly-sized water droplets.

The effects of diffraction can be readily seen in everyday life. The most colorful examples of diffraction are those involving light; for example, the closely spaced tracks on a CD or DVD act as a diffraction grating to form the familiar rainbow pattern we see when looking at a disk. This principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired; the hologram on a credit card is an example. Diffraction in the atmosphere by small particles can cause a bright ring to be visible around a bright light source like the sun or the moon. A shadow of a solid object, using light from a compact source, shows small fringes near its edges. The speckle pattern which is observed when laser light falls on an optically rough service is also a diffraction phenomenon. All these effects are a consequence of the fact that light is a wave.

Diffraction can occur with any kind of wave. Ocean waves diffract around jetties and other obstacles. Sound waves can diffract around objects, this is the reason we can still hear someone calling us even if we are hiding behind a tree. Diffraction can also be a concern in some technical applications; it sets a fundamental limit to the resolution of a camera, telescope, or microscope.

[edit] History

Thomas Young's sketch of two-slit diffraction, which he presented to the Royal Society in 1803

Thomas Young's sketch of two-slit diffraction, which he presented to the Royal Society in 1803

The effects of diffraction of light were first carefully observed and characterized by Francesco Maria Grimaldi, who also coined the term diffraction, from the Latin diffringere, 'to break into pieces', referring to light breaking up into different directions. The results of Grimaldi's observations were published posthumously in 1665.[3][4] Isaac Newton studied these effects and attributed them to inflexion of light rays. James Gregory (1638–1675) observed the diffraction patterns caused by a bird feather, which was effectively the first diffraction grating. In 1803 Thomas Young did his famous experiment observing interference from two closely spaced slits. Explaining his results by interference of the waves emanating from the two different slits, he deduced that light must propagate as waves. Augustin-Jean Fresnel did more definitive studies and calculations of diffraction, published in 1815 and 1818, and thereby gave great support to the wave theory of light that had been advanced by Christiaan Huygens and reinvigorated by Young, against Newton's particle theory.

[edit] The mechanism of diffraction

Photograph of single-slit diffraction in a circular ripple tank

Photograph of single-slit diffraction in a circular ripple tank

Diffraction arises because of the way in which waves propagate; this is described by the Huygens–Fresnel principle. The propagation of a wave can be visualized by considering every point on a wavefront as a point source for a secondary radial wave. The subsequent propagation and addition of all these radial waves form the new wavefront. When waves are added together, their sum is determined by the relative phases as well as the amplitudes of the individual waves, an effect which is often known as wave interference. The summed amplitude of the waves can have any value between zero and the sum of the individual amplitudes. Hence, diffraction patterns usually have a series of maxima and minima.

To determine the form of a diffraction pattern, we must determine the phase and amplitude of each of the Huygens wavelets at each point in space and then find the sum of these waves. There are various analytical models which can be used to do this including the Fraunhoffer diffraction equation for the far field and the Fresnel Diffraction equation for the near-field. Most configurations cannot be solved analytically; solutions can be found using various numerical analytical methods including Finite element and boundary element methods

[edit] Diffraction systems

It is possible to obtain a qualitative understanding of many diffraction phenomena by considering how the relative phases of the individual secondary wave sources vary, and in particular, the conditions in which the phase difference equals half a cycle in which case waves will cancel one another out.

The simplest descriptions of diffraction are those in which the situation can be reduced to a two dimensional problem. For water waves, this is already the case, water waves propagate only on the surface of the water. For light, we can often neglect one direction if the diffracting object extends in that direction over a distance far greater than the wavelength. In the case of light shining through small circular holes we will have to take into account the full three dimensional nature of the problem.

Some of the simpler cases of diffraction are considered below.

[edit] Single-slit diffraction

Main article: Diffraction formalism

Numerical approximation of diffraction pattern from a slit of width four wavelengths with an incident plane wave. The main central beam, nulls, and phase reversals are apparent.

Numerical approximation of diffraction pattern from a slit of width four wavelengths with an incident plane wave. The main central beam, nulls, and phase reversals are apparent.

Graph and image of single-slit diffraction

Graph and image of single-slit diffraction

A long slit of infinitesimal width which is illuminated by light diffracts the light into a series of circular waves and the wavefront which emerges from the slit is a cylindrical wave of uniform intensity.

A slit which is wider than a wavelength has a large number of point sources spaced evenly across the width of the slit. The light at a given angle is made up of contributions from each of these point sources and if the relative phases of these contributions vary by more than 2π, we expect to find minima and maxima in the diffracted light.

We can find the angle at which a first minimum is obtained in the diffracted light by the following reasoning. The light from a source located at the top edge of the slit interferes destructively with a source located at the middle of the slit, when the path difference between them is equal to λ/2. Similarly, the source just below the top of the slit will interfere destructively with the source located just below the middle of the slit at the same angle. We can continue this reasoning along the entire height of the slit to conclude that the condition for destructive interference for the entire slit is the same as the condition for destructive interference between two narrow slits a distance apart that is half the width of the slit. The path difference is given by (d sinθ)/2 so that the minimum intensity occurs at an angle θmin given by

d \sin \theta_{min} = \lambda \,

where d is the width of the slit.

A similar argument can be used to show that if we imagine the slit to be divided into four, six eight parts, etc, minima are obtained at angles θn given by

d \sin \theta_{n} = n\lambda \,

where n is an integer greater than zero.

There is no such simple argument to enable us to find the maxima of the diffraction pattern. The intensity profile can be calculated using the Fraunhofer diffraction integral as

I(\theta)\, = I_0 {\left[ \mathrm{sinc} \left( \frac{\pi d}{\lambda} \sin \theta \right) \right] }^2

where the sinc function is given by sinc(x)=sin(x)/x.

It should be noted that this analysis applies only to the far field, i.e a significant distance from the diffracting slit.

2-slit and 5-slit diffraction of red laser light

2-slit and 5-slit diffraction of red laser light

[edit] Diffraction Grating

Main article: Diffraction grating

A diffraction grating is an optical component with a regular pattern. The form of the light diffracted by a grating depends on the structure of the elements and the number of elements present, but all gratings have intensity maxima at angles θm which are given by the grating equation

d \left( \sin{\theta_m} + \sin{\theta_i} \right) = m \lambda.

where θi is the angle at which the light is incident, d is the separation of grating elements and m is an integer which can be positive or negative.

The light diffracted by a grating is found by summing the light diffracted from each of the elements, and is essentially a convolution of diffraction and interference patterns.

The figure shows the light diffracted by 2-element and 5-element gratings where the grating spacings are the same; it can be seen that the maxima are in the same position, but the detailed structures of the intensities are different.

A computer-generated image of an Airy disk

A computer-generated image of an Airy disk

[edit] Diffraction by a circular aperture

Main article: Airy disk

The far-field diffraction of a plane wave incident on a circular aperture is often referred to as the Airy Disk. The variation in intensity with angle is given by

I(\theta) = I_0 \left ( \frac{2 J_1(ka \sin \theta)}{ka \sin \theta} \right )^2

where a is the radius of the circular aperture, k is equal to 2π/λ and J1 is a Bessel function. The smaller the aperture, the larger the spot size at a given distance, and the greater the divergence of the diffracted beams.

[edit] Propagation of a laser beam

The way in which the profile of a laser beam changes as it propagates is determined by diffraction. The output mirror of the laser is an aperture, and the subsequent beam shape is determined by that aperture. Hence, the smaller the output beam, the quicker it diverges. Diode lasers have much greater divergence than He-Ne lasers for this reason.

Paradoxically, it is possible to reduce the divergence of a laser beam by first expanding it with one convex lens, and then collimating it with a second convex lens whose focal point is coincident with that of the first lens. The resulting beam has a larger aperture, and hence a lower divergence.

[edit] Diffraction limited imaging

Main article: Diffraction-limited system

The Airy disc around each of the stars from the 2.56m telescope aperture can be seen in this lucky image of the binary star zeta Boötis.

The Airy disc around each of the stars from the 2.56m telescope aperture can be seen in this lucky image of the binary star zeta Boötis.

The ability of an imaging system to resolve detail is ultimately limited by diffraction. This is because a plane wave incident on a circular lens or mirror is diffracted as described above. The light is not focused to a point but forms an Airy pattern with a central spot of diameter

d = 1.22 \lambda \frac{f}{a},\,

where λ is the wavelength of the light, f is the focal length of the lens, and a is the diameter of the beam of light, or (if the beam is filling the lens) the diameter of the lens.

This is why telescopes have very large lenses or mirrors, and why optical microscopes are limited in the detail which they can see.

[edit] Speckle patterns

Main article: speckle pattern

The speckle pattern which is seen when using a laser pointer is another diffraction phenomenon. It is a result of the superpostion of many waves with different phases, which are produced when a laser beam illuminates a rough surface. They add together to give a resultant wave whose amplitude, and therefore intensity varies randomly.

[edit] Common features of diffraction patterns

Several qualitative observations can be made of diffraction in general:

* The angular spacing of the features in the diffraction pattern is inversely proportional to the dimensions of the object causing the diffraction, in other words: the smaller the diffracting object the 'wider' the resulting diffraction pattern and vice versa. (More precisely, this is true of the sines of the angles.)

* The diffraction angles are invariant under scaling; that is, they depend only on the ratio of the wavelength to the size of the diffracting object.

* When the diffracting object has a periodic structure, for example in a diffraction grating, the features generally become sharper. The third figure, for example, shows a comparison of a double-slit pattern with a pattern formed by five slits, both sets of slits having the same spacing, between the center of one slit and the next.

[edit] Particle diffraction

See also: neutron diffraction and electron diffraction

Quantum theory tells us that every particle exhibits wave properties. In particular, massive particles can interfere and therefore diffract. Diffraction of electrons and neutrons stood as one of the powerful arguments in favor of quantum mechanics. The wavelength associated with a particle is the de Broglie wavelength

\lambda=\frac{h}{p} \,

where h is Planck's constant and p is the momentum of the particle (mass × velocity for slow-moving particles) . For most macroscopic objects, this wavelength is so short that it is not meaningful to assign a wavelength to them. A Sodium atom traveling at about 3000 m/s would have a De Broglie wavelength of about 5 pico meters.

Because the wavelength for even the smallest of macroscopic objects is extremely small, diffraction of matter waves is only visible for small particles, like electrons, neutrons, atoms and small molecules. The short wavelength of these matter waves makes them ideally suited to study the atomic crystal structure of solids and large molecules like proteins.

Relatively recently, larger molecules like buckyballs,[5] have been shown to diffract. Currently, research is underway into the diffraction of viruses, which, being huge relative to electrons and other more commonly diffracted particles, have tiny wavelengths so must be made to travel very slowly through an extremely narrow slit in order to diffract.

[edit] Bragg diffraction

Following Bragg's law, each dot (or reflection), in this diffraction pattern forms from the constructive interference of X-rays passing through a crystal. The data can be used to determine the crystal's atomic structure.

Following Bragg's law, each dot (or reflection), in this diffraction pattern forms from the constructive interference of X-rays passing through a crystal. The data can be used to determine the crystal's atomic structure.

For more details on this topic, see Bragg diffraction.

Diffraction from a three dimensional periodic structure such as atoms in a crystal is called Bragg diffraction. It is similar to what occurs when waves are scattered from a diffraction grating. Bragg diffraction is a consequence of interference between waves reflecting from different crystal planes. The condition of constructive interference is given by Bragg's law:

m \lambda = 2 d \sin \theta \,

where

λ is the wavelength,

d is the distance between crystal planes,

θ is the angle of the diffracted wave.

and m is an integer known as the order of the diffracted beam.

Bragg diffraction may be carried out using either light of very short wavelength like x-rays or matter waves like neutrons (and electrons) whose wavelength is on the order of (or much smaller than) the atomic spacing[6]. The pattern produced gives information of the separations of crystallographic planes d, allowing one to deduce the crystal structure. Diffraction contrast, in electron microscopes and x-topography devices in particular, is also a powerful tool for examining individual defects and local strain fields in crystals.

[edit] Coherence

Main article: Coherence (physics)

The description of diffraction relies on the interference of waves emanating from the same source taking different paths to the same point on a screen. In this description, the difference in phase between waves that took different paths is only dependent on the effective path length. This does not take into account the fact that waves that arrive at the screen at the same time were emitted by the source at different times. The initial phase with which the source emits waves can change over time in an unpredictable way. This means that waves emitted by the source at times that are too far apart can no longer form a constant interference pattern since the relation between their phases is no longer time independent.

The length over which the phase in a beam of light is correlated, is called the coherence length. In order for interference to occur, the path length difference must be smaller than the coherence length. This is sometimes referred to as spectral coherence as it is related to the presence of different frequency components in the wave. In the case light emitted by an atomic transition, the coherence length is related to the lifetime of the excited state from which the atom made its transition.

If waves are emitted from an extended source this can lead to incoherence in the transversal direction. When looking at a cross section of a beam of light, the length over which the phase is correlated is called the transverse coherence length. In the case of Young's double slit experiment this would mean that if the transverse coherence length is smaller than the spacing between the two slits the resulting pattern on a screen would look like two single slit diffraction patterns.

In the case of particles like electrons, neutrons and atoms, the coherence length is related to the spacial extent of the wave function that describes the particle.

[edit] ReferencesAtmospheric refraction

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Photograph of the full Moon partly obscured by Earth's atmosphere. Note the deviation from circular in the Moon's lower edge, caused by refraction.

Photograph of the full Moon partly obscured by Earth's atmosphere. Note the deviation from circular in the Moon's lower edge, caused by refraction.

diagram showing displacement of the Sun's image at sunrise and sunset

diagram showing displacement of the Sun's image at sunrise and sunset

Atmospheric refraction is the deviation of light or other electromagnetic wave from a straight line as it passes through the atmosphere due to the variation in air density as a function of altitude. Atmospheric refraction near the ground produces mirages and can make distant objects appear to shimmer or ripple. The term also applies to the refraction of sound.

Atmospheric refraction causes astronomical objects to appear higher in the sky than they are in reality. It affects not only lightrays but all electromagnetic radiation, although in varying degrees (see dispersion (optics)). For example in visible light, blue is more affected than red. This may cause astronomical objects to be spread out into a spectrum in high-resolution images.

Whenever possible astronomers will always schedule their observations around the time of culmination of an object when it is highest in the sky. Likewise sailors will never shoot a star which is not at least 20° or more above the horizon. If observations close to the horizon cannot be avoided, it is possible to equip a telescope with control systems to compensate for the shift caused by the refraction. If the dispersion is a problem too, (in case of broadband high-resolution observations) atmospheric refraction correctors can be employed as well (made from pairs of rotating glass prisms). But as the amount of atmospheric refraction is function of temperature and pressure as well as humidity (the amount of water vapour especially important at mid-infrared wavelengths) the amount of effort needed for a successful compensation can be prohibitive.

It gets even worse when the atmospheric refraction is not homogenous, when there is turbulence in the air for example. This is the cause of twinkling of the stars and deformation of the shape of the sun at sunset and sunrise.

Contents

[hide]

* 1 Values

* 2 Random refraction effects

* 3 See also

* 4 External links

[edit] Values

The atmospheric refraction is zero in the zenith, is less than 1′ (one arcminute) at 45° altitude, still only 5′ at 10° altitude, but then quickly increases when the horizon is approached. On the horizon itself it is about 34′ (according to FW Bessel), just a little bit larger than the apparent size of the sun. Therefore if it appears that the setting sun is just above the horizon, in reality it has already set. Formulae to calculate the times of sunrise and sunset do not calculate the moment that the sun reaches altitude zero, but when its altitude is −50′: 16′ for the radius of the sun (solar positions are for the centre of the sun-disc, but sunrise and sunset usually refer to the appearance and disappearance of the upperlimb) plus 34′ for the refraction. In the case of the Moon one should apply additional corrections for the horizontal parallax of the moon, its apparent diameter and its phase, although the latter is seldom done.

The refraction is also a function of temperature and pressure. The values given above are for 10 °C and 100.3 kPa. Add 1 % to the refraction for every 3 °C colder, subtract if hotter (hot air is less dense, and will therefore have less refraction). Add 1 % for every 0.9 kPa higher pressure, subtract if lower. Evidently day to day variations in the weather will affect the exact times of sunrise and sunset as well as moonrise and moonset, and for that reason are never given more accurately than to the nearest whole minute in the almanacs.

As the atmospheric refraction is 34′ on the horizon itself, but only 29′ above it, the setting or rising sun seems to be flattened by about 5′ (about 1/6 of its apparent diameter).

[edit] Random refraction effects

Turbulence in the atmosphere magnifies and de-magnifies star images, making them appear brighter and fainter on a time-scale of milliseconds. The slowest components of these fluctuations are visible to the eye as twinkling (also called “scintillation”).

Turbulence also causes small random motions of the star image, and produces rapid changes in its structure. These effects are not visible to the naked eye, but are easily seen even in small telescopes. They are called “seeing” by astronomers.Refractive index

From Wikipedia, the free encyclopedia

Jump to: navigation, search

The refractive index (or index of refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves) is reduced inside the medium. For example, typical glass has a refractive index of 1.5, which means that in glass, light travels at 1 / 1.5 = 0.67 times the speed of light in a vacuum. Two common properties of glass and other transparent materials are directly related to their refractive index. First, light rays change direction when they cross the interface from air to the material, an effect that is used in lenses and glasses. Second, light reflects partially from surfaces that have a refractive index different from that of their surroundings.

Definition: The refractive index n of a medium is defined as the ratio of the phase velocity c of a wave phenomenon such as light or sound in a reference medium to the phase velocity vp in the medium itself:

n = \frac{c}{v_{\mathrm {p}}}

It is most commonly used in the context of light with vacuum as a reference medium, although historically other reference media (e.g. air at a standardized pressure and temperature) have been common. It is usually given the symbol n. In the case of light, it equals

n=\sqrt{\epsilon_r\mu_r},

where εr is the material's relative permittivity, and μr is its relative permeability. For most materials, μr is very close to 1 at optical frequencies, therefore n is approximately \sqrt{\epsilon_r}. Contrary to a widespread misconception, n may be less than 1, for example for x-rays.[1]. This has practical technical applications, such as effective mirrors for x-rays based on total external reflection.

The phase velocity is defined as the rate at which the crests of the waveform propagate; that is, the rate at which the phase of the waveform is moving. The group velocity is the rate that the envelope of the waveform is propagating; that is, the rate of variation of the amplitude of the waveform. Provided the waveform is not distorted significantly during propagation, it is the group velocity that represents the rate that information (and energy) may be transmitted by the wave, for example the velocity at which a pulse of light travels down an optical fiber.

Contents

[hide]

* 1 Speed of light

* 2 Negative Refractive Index

* 3 Dispersion and absorption

* 4 Relation to dielectric constant

* 5 Anisotropy

* 6 Nonlinearity

* 7 Inhomogeneity

* 8 Relation to density

* 9 Momentum Paradox

* 10 Applications

* 11 See also

* 12 References

* 13 External links

[edit] Speed of light

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

The speed of all electromagnetic radiation in vacuum is the same, approximately 3×108 meters per second, and is denoted by c. Therefore, if v is the phase velocity of radiation of a specific frequency in a specific material, the refractive index is given by

n =\frac{c}{v}

or inversely

v =\frac{c}{n}

This number is typically greater than one: the higher the index of the material, the more the light is slowed down (see Cherenkov radiation). However, at certain frequencies (e.g. near absorption resonances, and for X-rays), n will actually be smaller than one. This does not contradict the theory of relativity, which holds that no information-carrying signal can ever propagate faster than c, because the phase velocity is not the same as the group velocity or the signal velocity.

Sometimes, a "group velocity refractive index", usually called the group index is defined:

n_g=\frac{c}{v_g}

where vg is the group velocity. This value should not be confused with n, which is always defined with respect to the phase velocity. The group index can be written in terms of the wavelength dependence of the refractive index as

n_g = n - \lambda\frac{dn}{d\lambda},

where λ is the wavelength in vacuum. At the microscale, an electromagnetic wave's phase velocity is slowed in a material because the electric field creates a disturbance in the charges of each atom (primarily the electrons) proportional to the permittivity of the medium. The charges will, in general, oscillate slightly out of phase with respect to the driving electric field. The charges thus radiate their own electromagnetic wave that is at the same frequency but with a phase delay. The macroscopic sum of all such contributions in the material is a wave with the same frequency but shorter wavelength than the original, leading to a slowing of the wave's phase velocity. Most of the radiation from oscillating material charges will modify the incoming wave, changing its velocity. However, some net energy will be radiated in other directions (see scattering).

If the refractive indices of two materials are known for a given frequency, then one can compute the angle by which radiation of that frequency will be refracted as it moves from the first into the second material from Snell's law.

If in a given region the values of refractive indices n or ng were found to differ from unity (whether homogeneously, or isotropically, or not), then this region was distinct from vacuum in the above sense for lacking Poincaré symmetry.

[edit] Negative Refractive Index

Recent research has also demonstrated the existence of negative refractive index which can occur if the real parts of both εr and μr are simultaneously negative, although such is a necessary but not sufficient condition. Not thought to occur naturally, this can be achieved with so-called metamaterials and offers the possibility of perfect lenses and other exotic phenomena such as a reversal of Snell's law. [1] [2]

[edit] Dispersion and absorption

The variation of refractive index vs. wavelength for various glasses.

The variation of refractive index vs. wavelength for various glasses.

In real materials, the polarization does not respond instantaneously to an applied field. This causes dielectric loss, which can be expressed by a permittivity that is both complex and frequency dependent. Real materials are not perfect insulators either, i.e. they have non-zero direct current conductivity. Taking both aspects into consideration, we can define a complex index of refraction:

\tilde{n}=n+i\kappa

Here, n is the refractive index indicating the phase velocity as above, while κ is called the extinction coefficient, which indicates the amount of absorption loss when the electromagnetic wave propagates through the material. (See the article Mathematical descriptions of opacity.) Both n and κ are dependent on the frequency (wavelength). Note that the sign of the complex part is a matter of convention, which is important due to possible confusion between loss and gain. The notation above, which is usually used by physicists, corresponds to waves with time evolution given by e − iωt.

The effect that n varies with frequency (except in vacuum, where all frequencies travel at the same speed, c) is known as dispersion, and it is what causes a prism to divide white light into its constituent spectral colors, explains rainbows, and is the cause of chromatic aberration in lenses. In regions of the spectrum where the material does not absorb, the real part of the refractive index tends to increase with frequency. Near absorption peaks, the curve of the refractive index is a complex form given by the Kramers–Kronig relations, and can decrease with frequency.

Since the refractive index of a material varies with the frequency (and thus wavelength) of light, it is usual to specify the corresponding vacuum wavelength at which the refractive index is measured. Typically, this is done at various well-defined spectral emission lines; for example, nD is the refractive index at the Fraunhofer "D" line, the centre of the yellow sodium double emission at 589.29 nm wavelength.

The Sellmeier equation is an empirical formula that works well in describing dispersion, and Sellmeier coefficients are often quoted instead of the refractive index in tables. For some representative refractive indices at different wavelengths, see list of indices of refraction.

As shown above, dielectric loss and non-zero DC conductivity in materials cause absorption. Good dielectric materials such as glass have extremely low DC conductivity, and at low frequencies the dielectric loss is also negligible, resulting in almost no absorption (κ ≈ 0). However, at higher frequencies (such as visible light), dielectric loss may increase absorption significantly, reducing the material's transparency to these frequencies.

The real and imaginary parts of the complex refractive index are related through use of the Kramers–Kronig relations. For example, one can determine a material's full complex refractive index as a function of wavelength from an absorption spectrum of the material.

[edit] Relation to dielectric constant

The dielectric constant (which is often dependent on wavelength) is simply the square of the (complex) refractive index. The refractive index is used for optics in Fresnel equations and Snell's law; while the dielectric constant is used in Maxwell's equations and electronics.

Where \tilde\epsilon, ε1, ε2, n, and κ are functions of wavelength:

\tilde\epsilon=\epsilon_1+i\epsilon_2= (n+i\kappa)^2

Conversion between refractive index and dielectric constant is done by:

ε1 = n2 − κ2

ε2 = 2nκ

n = \sqrt{\frac{\sqrt{\epsilon_1^2+\epsilon_2^2}+\epsilon_1}{2}} [2]

\kappa = \sqrt{ \frac{ \sqrt{ \epsilon_1^2+ \epsilon_2^2}- \epsilon_1}{2}}[3]

[edit] Anisotropy

A calcite crystal laid upon a paper with some letters showing birefringence

A calcite crystal laid upon a paper with some letters showing birefringence

The refractive index of certain media may be different depending on the polarization and direction of propagation of the light through the medium. This is known as birefringence or anisotropy and is described by the field of crystal optics. In the most general case, the dielectric constant is a rank-2 tensor (a 3 by 3 matrix), which cannot simply be described by refractive indices except for polarizations along principal axes.

In magneto-optic (gyro-magnetic) and optically active materials, the principal axes are complex (corresponding to elliptical polarizations), and the dielectric tensor is complex-Hermitian (for lossless media); such materials break time-reversal symmetry and are used e.g. to construct Faraday isolators.

[edit] Nonlinearity

The strong electric field of high intensity light (such as output of a laser) may cause a medium's refractive index to vary as the light passes through it, giving rise to nonlinear optics. If the index varies quadratically with the field (linearly with the intensity), it is called the optical Kerr effect and causes phenomena such as self-focusing and self-phase modulation. If the index varies linearly with the field (which is only possible in materials that do not possess inversion symmetry), it is known as the Pockels effect.

[edit] Inhomogeneity

A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens.

A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens.

If the refractive index of a medium is not constant, but varies gradually with position, the material is known as a gradient-index medium and is described by gradient index optics. Light travelling through such a medium can be bent or focussed, and this effect can be exploited to produce lenses, some optical fibers and other devices. Some common mirages are caused by a spatially-varying refractive index of air.

[edit] Relation to density

Relation between the refractive index and the density of silicate and borosilicate glasses ().

Relation between the refractive index and the density of silicate and borosilicate glasses ([4]).

In general, the refractive index of a glass increases with its density. However, there does not exist an overall linear relation between the refractive index and the density for all silicate and borosilicate glasses. A relatively high refractive index and low density can be obtained with glasses containing light metal oxides such as Li2O and MgO, while the opposite trend is observed with glasses containing PbO and BaO as seen in the diagram at the right.

[edit] Momentum Paradox

The momentum of a refracted ray, p, was calculated by Hermann Minkowski[5] in 1908, where E is energy of the photon, c is the speed of light in vacuum and n is the refractive index of the medium.

p=\frac{nE}{c}

In 1909 Max Abraham[6] proposed

p=\frac{E}{nc}

Rudolf Peierls raises this in his "More Surprises in Theoretical Physics" Princeton (1991). Ulf Leonhardt, Chair in Theoretical Physics at the University of St Andrews has discussed[7] this including experiments to resolve.

[edit] Applications

The refractive index of a material is the most important property of any optical system that uses refraction. It is used to calculate the focusing power of lenses, and the dispersive power of prisms.

Since refractive index is a fundamental physical property of a substance, it is often used to identify a particular substance, confirm its purity, or measure its concentration. Refractive index is used to measure solids (glasses and gemstones), liquids, and gases. Most commonly it is used to measure the concentration of a solute in an aqueous solution. A refractometer is the instrument used to measure refractive index. For a solution of sugar, the refractive index can be used to determine the sugar content (see Brix).Refractive index

From Wikipedia, the free encyclopedia

Jump to: navigation, search

The refractive index (or index of refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves) is reduced inside the medium. For example, typical glass has a refractive index of 1.5, which means that in glass, light travels at 1 / 1.5 = 0.67 times the speed of light in a vacuum. Two common properties of glass and other transparent materials are directly related to their refractive index. First, light rays change direction when they cross the interface from air to the material, an effect that is used in lenses and glasses. Second, light reflects partially from surfaces that have a refractive index different from that of their surroundings.

Definition: The refractive index n of a medium is defined as the ratio of the phase velocity c of a wave phenomenon such as light or sound in a reference medium to the phase velocity vp in the medium itself:

n = \frac{c}{v_{\mathrm {p}}}

It is most commonly used in the context of light with vacuum as a reference medium, although historically other reference media (e.g. air at a standardized pressure and temperature) have been common. It is usually given the symbol n. In the case of light, it equals

n=\sqrt{\epsilon_r\mu_r},

where εr is the material's relative permittivity, and μr is its relative permeability. For most materials, μr is very close to 1 at optical frequencies, therefore n is approximately \sqrt{\epsilon_r}. Contrary to a widespread misconception, n may be less than 1, for example for x-rays.[1]. This has practical technical applications, such as effective mirrors for x-rays based on total external reflection.

The phase velocity is defined as the rate at which the crests of the waveform propagate; that is, the rate at which the phase of the waveform is moving. The group velocity is the rate that the envelope of the waveform is propagating; that is, the rate of variation of the amplitude of the waveform. Provided the waveform is not distorted significantly during propagation, it is the group velocity that represents the rate that information (and energy) may be transmitted by the wave, for example the velocity at which a pulse of light travels down an optical fiber.

Contents

[hide]

* 1 Speed of light

* 2 Negative Refractive Index

* 3 Dispersion and absorption

* 4 Relation to dielectric constant

* 5 Anisotropy

* 6 Nonlinearity

* 7 Inhomogeneity

* 8 Relation to density

* 9 Momentum Paradox

* 10 Applications

* 11 See also

* 12 References

* 13 External links

[edit] Speed of light

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

The speed of all electromagnetic radiation in vacuum is the same, approximately 3×108 meters per second, and is denoted by c. Therefore, if v is the phase velocity of radiation of a specific frequency in a specific material, the refractive index is given by

n =\frac{c}{v}

or inversely

v =\frac{c}{n}

This number is typically greater than one: the higher the index of the material, the more the light is slowed down (see Cherenkov radiation). However, at certain frequencies (e.g. near absorption resonances, and for X-rays), n will actually be smaller than one. This does not contradict the theory of relativity, which holds that no information-carrying signal can ever propagate faster than c, because the phase velocity is not the same as the group velocity or the signal velocity.

Sometimes, a "group velocity refractive index", usually called the group index is defined:

n_g=\frac{c}{v_g}

where vg is the group velocity. This value should not be confused with n, which is always defined with respect to the phase velocity. The group index can be written in terms of the wavelength dependence of the refractive index as

n_g = n - \lambda\frac{dn}{d\lambda},

where λ is the wavelength in vacuum. At the microscale, an electromagnetic wave's phase velocity is slowed in a material because the electric field creates a disturbance in the charges of each atom (primarily the electrons) proportional to the permittivity of the medium. The charges will, in general, oscillate slightly out of phase with respect to the driving electric field. The charges thus radiate their own electromagnetic wave that is at the same frequency but with a phase delay. The macroscopic sum of all such contributions in the material is a wave with the same frequency but shorter wavelength than the original, leading to a slowing of the wave's phase velocity. Most of the radiation from oscillating material charges will modify the incoming wave, changing its velocity. However, some net energy will be radiated in other directions (see scattering).

If the refractive indices of two materials are known for a given frequency, then one can compute the angle by which radiation of that frequency will be refracted as it moves from the first into the second material from Snell's law.

If in a given region the values of refractive indices n or ng were found to differ from unity (whether homogeneously, or isotropically, or not), then this region was distinct from vacuum in the above sense for lacking Poincaré symmetry.

[edit] Negative Refractive Index

Recent research has also demonstrated the existence of negative refractive index which can occur if the real parts of both εr and μr are simultaneously negative, although such is a necessary but not sufficient condition. Not thought to occur naturally, this can be achieved with so-called metamaterials and offers the possibility of perfect lenses and other exotic phenomena such as a reversal of Snell's law. [1] [2]

[edit] Dispersion and absorption

The variation of refractive index vs. wavelength for various glasses.

The variation of refractive index vs. wavelength for various glasses.

In real materials, the polarization does not respond instantaneously to an applied field. This causes dielectric loss, which can be expressed by a permittivity that is both complex and frequency dependent. Real materials are not perfect insulators either, i.e. they have non-zero direct current conductivity. Taking both aspects into consideration, we can define a complex index of refraction:

\tilde{n}=n+i\kappa

Here, n is the refractive index indicating the phase velocity as above, while κ is called the extinction coefficient, which indicates the amount of absorption loss when the electromagnetic wave propagates through the material. (See the article Mathematical descriptions of opacity.) Both n and κ are dependent on the frequency (wavelength). Note that the sign of the complex part is a matter of convention, which is important due to possible confusion between loss and gain. The notation above, which is usually used by physicists, corresponds to waves with time evolution given by e − iωt.

The effect that n varies with frequency (except in vacuum, where all frequencies travel at the same speed, c) is known as dispersion, and it is what causes a prism to divide white light into its constituent spectral colors, explains rainbows, and is the cause of chromatic aberration in lenses. In regions of the spectrum where the material does not absorb, the real part of the refractive index tends to increase with frequency. Near absorption peaks, the curve of the refractive index is a complex form given by the Kramers–Kronig relations, and can decrease with frequency.

Since the refractive index of a material varies with the frequency (and thus wavelength) of light, it is usual to specify the corresponding vacuum wavelength at which the refractive index is measured. Typically, this is done at various well-defined spectral emission lines; for example, nD is the refractive index at the Fraunhofer "D" line, the centre of the yellow sodium double emission at 589.29 nm wavelength.

The Sellmeier equation is an empirical formula that works well in describing dispersion, and Sellmeier coefficients are often quoted instead of the refractive index in tables. For some representative refractive indices at different wavelengths, see list of indices of refraction.

As shown above, dielectric loss and non-zero DC conductivity in materials cause absorption. Good dielectric materials such as glass have extremely low DC conductivity, and at low frequencies the dielectric loss is also negligible, resulting in almost no absorption (κ ≈ 0). However, at higher frequencies (such as visible light), dielectric loss may increase absorption significantly, reducing the material's transparency to these frequencies.

The real and imaginary parts of the complex refractive index are related through use of the Kramers–Kronig relations. For example, one can determine a material's full complex refractive index as a function of wavelength from an absorption spectrum of the material.

[edit] Relation to dielectric constant

The dielectric constant (which is often dependent on wavelength) is simply the square of the (complex) refractive index. The refractive index is used for optics in Fresnel equations and Snell's law; while the dielectric constant is used in Maxwell's equations and electronics.

Where \tilde\epsilon, ε1, ε2, n, and κ are functions of wavelength:

\tilde\epsilon=\epsilon_1+i\epsilon_2= (n+i\kappa)^2

Conversion between refractive index and dielectric constant is done by:

ε1 = n2 − κ2

ε2 = 2nκ

n = \sqrt{\frac{\sqrt{\epsilon_1^2+\epsilon_2^2}+\epsilon_1}{2}} [2]

\kappa = \sqrt{ \frac{ \sqrt{ \epsilon_1^2+ \epsilon_2^2}- \epsilon_1}{2}}[3]

[edit] Anisotropy

A calcite crystal laid upon a paper with some letters showing birefringence

A calcite crystal laid upon a paper with some letters showing birefringence

The refractive index of certain media may be different depending on the polarization and direction of propagation of the light through the medium. This is known as birefringence or anisotropy and is described by the field of crystal optics. In the most general case, the dielectric constant is a rank-2 tensor (a 3 by 3 matrix), which cannot simply be described by refractive indices except for polarizations along principal axes.

In magneto-optic (gyro-magnetic) and optically active materials, the principal axes are complex (corresponding to elliptical polarizations), and the dielectric tensor is complex-Hermitian (for lossless media); such materials break time-reversal symmetry and are used e.g. to construct Faraday isolators.

[edit] Nonlinearity

The strong electric field of high intensity light (such as output of a laser) may cause a medium's refractive index to vary as the light passes through it, giving rise to nonlinear optics. If the index varies quadratically with the field (linearly with the intensity), it is called the optical Kerr effect and causes phenomena such as self-focusing and self-phase modulation. If the index varies linearly with the field (which is only possible in materials that do not possess inversion symmetry), it is known as the Pockels effect.

[edit] Inhomogeneity

A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens.

A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens.

If the refractive index of a medium is not constant, but varies gradually with position, the material is known as a gradient-index medium and is described by gradient index optics. Light travelling through such a medium can be bent or focussed, and this effect can be exploited to produce lenses, some optical fibers and other devices. Some common mirages are caused by a spatially-varying refractive index of air.

[edit] Relation to density

Relation between the refractive index and the density of silicate and borosilicate glasses ().

Relation between the refractive index and the density of silicate and borosilicate glasses ([4]).

In general, the refractive index of a glass increases with its density. However, there does not exist an overall linear relation between the refractive index and the density for all silicate and borosilicate glasses. A relatively high refractive index and low density can be obtained with glasses containing light metal oxides such as Li2O and MgO, while the opposite trend is observed with glasses containing PbO and BaO as seen in the diagram at the right.

[edit] Momentum Paradox

The momentum of a refracted ray, p, was calculated by Hermann Minkowski[5] in 1908, where E is energy of the photon, c is the speed of light in vacuum and n is the refractive index of the medium.

p=\frac{nE}{c}

In 1909 Max Abraham[6] proposed

p=\frac{E}{nc}

Rudolf Peierls raises this in his "More Surprises in Theoretical Physics" Princeton (1991). Ulf Leonhardt, Chair in Theoretical Physics at the University of St Andrews has discussed[7] this including experiments to resolve.

[edit] Applications

The refractive index of a material is the most important property of any optical system that uses refraction. It is used to calculate the focusing power of lenses, and the dispersive power of prisms.

Since refractive index is a fundamental physical property of a substance, it is often used to identify a particular substance, confirm its purity, or measure its concentration. Refractive index is used to measure solids (glasses and gemstones), liquids, and gases. Most commonly it is used to measure the concentration of a solute in an aqueous solution. A refractometer is the instrument used to measure refractive index. For a solution of sugar, the refractive index can be used to determine the sugar content (see Brix).Refractive index

From Wikipedia, the free encyclopedia

Jump to: navigation, search

The refractive index (or index of refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves) is reduced inside the medium. For example, typical glass has a refractive index of 1.5, which means that in glass, light travels at 1 / 1.5 = 0.67 times the speed of light in a vacuum. Two common properties of glass and other transparent materials are directly related to their refractive index. First, light rays change direction when they cross the interface from air to the material, an effect that is used in lenses and glasses. Second, light reflects partially from surfaces that have a refractive index different from that of their surroundings.

Definition: The refractive index n of a medium is defined as the ratio of the phase velocity c of a wave phenomenon such as light or sound in a reference medium to the phase velocity vp in the medium itself:

n = \frac{c}{v_{\mathrm {p}}}

It is most commonly used in the context of light with vacuum as a reference medium, although historically other reference media (e.g. air at a standardized pressure and temperature) have been common. It is usually given the symbol n. In the case of light, it equals

n=\sqrt{\epsilon_r\mu_r},

where εr is the material's relative permittivity, and μr is its relative permeability. For most materials, μr is very close to 1 at optical frequencies, therefore n is approximately \sqrt{\epsilon_r}. Contrary to a widespread misconception, n may be less than 1, for example for x-rays.[1]. This has practical technical applications, such as effective mirrors for x-rays based on total external reflection.

The phase velocity is defined as the rate at which the crests of the waveform propagate; that is, the rate at which the phase of the waveform is moving. The group velocity is the rate that the envelope of the waveform is propagating; that is, the rate of variation of the amplitude of the waveform. Provided the waveform is not distorted significantly during propagation, it is the group velocity that represents the rate that information (and energy) may be transmitted by the wave, for example the velocity at which a pulse of light travels down an optical fiber.

Contents

[hide]

* 1 Speed of light

* 2 Negative Refractive Index

* 3 Dispersion and absorption

* 4 Relation to dielectric constant

* 5 Anisotropy

* 6 Nonlinearity

* 7 Inhomogeneity

* 8 Relation to density

* 9 Momentum Paradox

* 10 Applications

* 11 See also

* 12 References

* 13 External links

[edit] Speed of light

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

The speed of all electromagnetic radiation in vacuum is the same, approximately 3×108 meters per second, and is denoted by c. Therefore, if v is the phase velocity of radiation of a specific frequency in a specific material, the refractive index is given by

n =\frac{c}{v}

or inversely

v =\frac{c}{n}

This number is typically greater than one: the higher the index of the material, the more the light is slowed down (see Cherenkov radiation). However, at certain frequencies (e.g. near absorption resonances, and for X-rays), n will actually be smaller than one. This does not contradict the theory of relativity, which holds that no information-carrying signal can ever propagate faster than c, because the phase velocity is not the same as the group velocity or the signal velocity.

Sometimes, a "group velocity refractive index", usually called the group index is defined:

n_g=\frac{c}{v_g}

where vg is the group velocity. This value should not be confused with n, which is always defined with respect to the phase velocity. The group index can be written in terms of the wavelength dependence of the refractive index as

n_g = n - \lambda\frac{dn}{d\lambda},

where λ is the wavelength in vacuum. At the microscale, an electromagnetic wave's phase velocity is slowed in a material because the electric field creates a disturbance in the charges of each atom (primarily the electrons) proportional to the permittivity of the medium. The charges will, in general, oscillate slightly out of phase with respect to the driving electric field. The charges thus radiate their own electromagnetic wave that is at the same frequency but with a phase delay. The macroscopic sum of all such contributions in the material is a wave with the same frequency but shorter wavelength than the original, leading to a slowing of the wave's phase velocity. Most of the radiation from oscillating material charges will modify the incoming wave, changing its velocity. However, some net energy will be radiated in other directions (see scattering).

If the refractive indices of two materials are known for a given frequency, then one can compute the angle by which radiation of that frequency will be refracted as it moves from the first into the second material from Snell's law.

If in a given region the values of refractive indices n or ng were found to differ from unity (whether homogeneously, or isotropically, or not), then this region was distinct from vacuum in the above sense for lacking Poincaré symmetry.

[edit] Negative Refractive Index

Recent research has also demonstrated the existence of negative refractive index which can occur if the real parts of both εr and μr are simultaneously negative, although such is a necessary but not sufficient condition. Not thought to occur naturally, this can be achieved with so-called metamaterials and offers the possibility of perfect lenses and other exotic phenomena such as a reversal of Snell's law. [1] [2]

[edit] Dispersion and absorption

The variation of refractive index vs. wavelength for various glasses.

The variation of refractive index vs. wavelength for various glasses.

In real materials, the polarization does not respond instantaneously to an applied field. This causes dielectric loss, which can be expressed by a permittivity that is both complex and frequency dependent. Real materials are not perfect insulators either, i.e. they have non-zero direct current conductivity. Taking both aspects into consideration, we can define a complex index of refraction:

\tilde{n}=n+i\kappa

Here, n is the refractive index indicating the phase velocity as above, while κ is called the extinction coefficient, which indicates the amount of absorption loss when the electromagnetic wave propagates through the material. (See the article Mathematical descriptions of opacity.) Both n and κ are dependent on the frequency (wavelength). Note that the sign of the complex part is a matter of convention, which is important due to possible confusion between loss and gain. The notation above, which is usually used by physicists, corresponds to waves with time evolution given by e − iωt.

The effect that n varies with frequency (except in vacuum, where all frequencies travel at the same speed, c) is known as dispersion, and it is what causes a prism to divide white light into its constituent spectral colors, explains rainbows, and is the cause of chromatic aberration in lenses. In regions of the spectrum where the material does not absorb, the real part of the refractive index tends to increase with frequency. Near absorption peaks, the curve of the refractive index is a complex form given by the Kramers–Kronig relations, and can decrease with frequency.

Since the refractive index of a material varies with the frequency (and thus wavelength) of light, it is usual to specify the corresponding vacuum wavelength at which the refractive index is measured. Typically, this is done at various well-defined spectral emission lines; for example, nD is the refractive index at the Fraunhofer "D" line, the centre of the yellow sodium double emission at 589.29 nm wavelength.

The Sellmeier equation is an empirical formula that works well in describing dispersion, and Sellmeier coefficients are often quoted instead of the refractive index in tables. For some representative refractive indices at different wavelengths, see list of indices of refraction.

As shown above, dielectric loss and non-zero DC conductivity in materials cause absorption. Good dielectric materials such as glass have extremely low DC conductivity, and at low frequencies the dielectric loss is also negligible, resulting in almost no absorption (κ ≈ 0). However, at higher frequencies (such as visible light), dielectric loss may increase absorption significantly, reducing the material's transparency to these frequencies.

The real and imaginary parts of the complex refractive index are related through use of the Kramers–Kronig relations. For example, one can determine a material's full complex refractive index as a function of wavelength from an absorption spectrum of the material.

[edit] Relation to dielectric constant

The dielectric constant (which is often dependent on wavelength) is simply the square of the (complex) refractive index. The refractive index is used for optics in Fresnel equations and Snell's law; while the dielectric constant is used in Maxwell's equations and electronics.

Where \tilde\epsilon, ε1, ε2, n, and κ are functions of wavelength:

\tilde\epsilon=\epsilon_1+i\epsilon_2= (n+i\kappa)^2

Conversion between refractive index and dielectric constant is done by:

ε1 = n2 − κ2

ε2 = 2nκ

n = \sqrt{\frac{\sqrt{\epsilon_1^2+\epsilon_2^2}+\epsilon_1}{2}} [2]

\kappa = \sqrt{ \frac{ \sqrt{ \epsilon_1^2+ \epsilon_2^2}- \epsilon_1}{2}}[3]

[edit] Anisotropy

A calcite crystal laid upon a paper with some letters showing birefringence

A calcite crystal laid upon a paper with some letters showing birefringence

The refractive index of certain media may be different depending on the polarization and direction of propagation of the light through the medium. This is known as birefringence or anisotropy and is described by the field of crystal optics. In the most general case, the dielectric constant is a rank-2 tensor (a 3 by 3 matrix), which cannot simply be described by refractive indices except for polarizations along principal axes.

In magneto-optic (gyro-magnetic) and optically active materials, the principal axes are complex (corresponding to elliptical polarizations), and the dielectric tensor is complex-Hermitian (for lossless media); such materials break time-reversal symmetry and are used e.g. to construct Faraday isolators.

[edit] Nonlinearity

The strong electric field of high intensity light (such as output of a laser) may cause a medium's refractive index to vary as the light passes through it, giving rise to nonlinear optics. If the index varies quadratically with the field (linearly with the intensity), it is called the optical Kerr effect and causes phenomena such as self-focusing and self-phase modulation. If the index varies linearly with the field (which is only possible in materials that do not possess inversion symmetry), it is known as the Pockels effect.

[edit] Inhomogeneity

A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens.

A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens.

If the refractive index of a medium is not constant, but varies gradually with position, the material is known as a gradient-index medium and is described by gradient index optics. Light travelling through such a medium can be bent or focussed, and this effect can be exploited to produce lenses, some optical fibers and other devices. Some common mirages are caused by a spatially-varying refractive index of air.

[edit] Relation to density

Relation between the refractive index and the density of silicate and borosilicate glasses ().

Relation between the refractive index and the density of silicate and borosilicate glasses ([4]).

In general, the refractive index of a glass increases with its density. However, there does not exist an overall linear relation between the refractive index and the density for all silicate and borosilicate glasses. A relatively high refractive index and low density can be obtained with glasses containing light metal oxides such as Li2O and MgO, while the opposite trend is observed with glasses containing PbO and BaO as seen in the diagram at the right.

[edit] Momentum Paradox

The momentum of a refracted ray, p, was calculated by Hermann Minkowski[5] in 1908, where E is energy of the photon, c is the speed of light in vacuum and n is the refractive index of the medium.

p=\frac{nE}{c}

In 1909 Max Abraham[6] proposed

p=\frac{E}{nc}

Rudolf Peierls raises this in his "More Surprises in Theoretical Physics" Princeton (1991). Ulf Leonhardt, Chair in Theoretical Physics at the University of St Andrews has discussed[7] this including experiments to resolve.

[edit] Applications

The refractive index of a material is the most important property of any optical system that uses refraction. It is used to calculate the focusing power of lenses, and the dispersive power of prisms.

Since refractive index is a fundamental physical property of a substance, it is often used to identify a particular substance, confirm its purity, or measure its concentration. Refractive index is used to measure solids (glasses and gemstones), liquids, and gases. Most commonly it is used to measure the concentration of a solute in an aqueous solution. A refractometer is the instrument used to measure refractive index. For a solution of sugar, the refractive index can be used to determine the sugar content (see Brix).Refractive index

From Wikipedia, the free encyclopedia

Jump to: navigation, search

The refractive index (or index of refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves) is reduced inside the medium. For example, typical glass has a refractive index of 1.5, which means that in glass, light travels at 1 / 1.5 = 0.67 times the speed of light in a vacuum. Two common properties of glass and other transparent materials are directly related to their refractive index. First, light rays change direction when they cross the interface from air to the material, an effect that is used in lenses and glasses. Second, light reflects partially from surfaces that have a refractive index different from that of their surroundings.

Definition: The refractive index n of a medium is defined as the ratio of the phase velocity c of a wave phenomenon such as light or sound in a reference medium to the phase velocity vp in the medium itself:

n = \frac{c}{v_{\mathrm {p}}}

It is most commonly used in the context of light with vacuum as a reference medium, although historically other reference media (e.g. air at a standardized pressure and temperature) have been common. It is usually given the symbol n. In the case of light, it equals

n=\sqrt{\epsilon_r\mu_r},

where εr is the material's relative permittivity, and μr is its relative permeability. For most materials, μr is very close to 1 at optical frequencies, therefore n is approximately \sqrt{\epsilon_r}. Contrary to a widespread misconception, n may be less than 1, for example for x-rays.[1]. This has practical technical applications, such as effective mirrors for x-rays based on total external reflection.

The phase velocity is defined as the rate at which the crests of the waveform propagate; that is, the rate at which the phase of the waveform is moving. The group velocity is the rate that the envelope of the waveform is propagating; that is, the rate of variation of the amplitude of the waveform. Provided the waveform is not distorted significantly during propagation, it is the group velocity that represents the rate that information (and energy) may be transmitted by the wave, for example the velocity at which a pulse of light travels down an optical fiber.

Contents

[hide]

* 1 Speed of light

* 2 Negative Refractive Index

* 3 Dispersion and absorption

* 4 Relation to dielectric constant

* 5 Anisotropy

* 6 Nonlinearity

* 7 Inhomogeneity

* 8 Relation to density

* 9 Momentum Paradox

* 10 Applications

* 11 See also

* 12 References

* 13 External links

[edit] Speed of light

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

The speed of all electromagnetic radiation in vacuum is the same, approximately 3×108 meters per second, and is denoted by c. Therefore, if v is the phase velocity of radiation of a specific frequency in a specific material, the refractive index is given by

n =\frac{c}{v}

or inversely

v =\frac{c}{n}

This number is typically greater than one: the higher the index of the material, the more the light is slowed down (see Cherenkov radiation). However, at certain frequencies (e.g. near absorption resonances, and for X-rays), n will actually be smaller than one. This does not contradict the theory of relativity, which holds that no information-carrying signal can ever propagate faster than c, because the phase velocity is not the same as the group velocity or the signal velocity.

Sometimes, a "group velocity refractive index", usually called the group index is defined:

n_g=\frac{c}{v_g}

where vg is the group velocity. This value should not be confused with n, which is always defined with respect to the phase velocity. The group index can be written in terms of the wavelength dependence of the refractive index as

n_g = n - \lambda\frac{dn}{d\lambda},

where λ is the wavelength in vacuum. At the microscale, an electromagnetic wave's phase velocity is slowed in a material because the electric field creates a disturbance in the charges of each atom (primarily the electrons) proportional to the permittivity of the medium. The charges will, in general, oscillate slightly out of phase with respect to the driving electric field. The charges thus radiate their own electromagnetic wave that is at the same frequency but with a phase delay. The macroscopic sum of all such contributions in the material is a wave with the same frequency but shorter wavelength than the original, leading to a slowing of the wave's phase velocity. Most of the radiation from oscillating material charges will modify the incoming wave, changing its velocity. However, some net energy will be radiated in other directions (see scattering).

If the refractive indices of two materials are known for a given frequency, then one can compute the angle by which radiation of that frequency will be refracted as it moves from the first into the second material from Snell's law.

If in a given region the values of refractive indices n or ng were found to differ from unity (whether homogeneously, or isotropically, or not), then this region was distinct from vacuum in the above sense for lacking Poincaré symmetry.

[edit] Negative Refractive Index

Recent research has also demonstrated the existence of negative refractive index which can occur if the real parts of both εr and μr are simultaneously negative, although such is a necessary but not sufficient condition. Not thought to occur naturally, this can be achieved with so-called metamaterials and offers the possibility of perfect lenses and other exotic phenomena such as a reversal of Snell's law. [1] [2]

[edit] Dispersion and absorption

The variation of refractive index vs. wavelength for various glasses.

The variation of refractive index vs. wavelength for various glasses.

In real materials, the polarization does not respond instantaneously to an applied field. This causes dielectric loss, which can be expressed by a permittivity that is both complex and frequency dependent. Real materials are not perfect insulators either, i.e. they have non-zero direct current conductivity. Taking both aspects into consideration, we can define a complex index of refraction:

\tilde{n}=n+i\kappa

Here, n is the refractive index indicating the phase velocity as above, while κ is called the extinction coefficient, which indicates the amount of absorption loss when the electromagnetic wave propagates through the material. (See the article Mathematical descriptions of opacity.) Both n and κ are dependent on the frequency (wavelength). Note that the sign of the complex part is a matter of convention, which is important due to possible confusion between loss and gain. The notation above, which is usually used by physicists, corresponds to waves with time evolution given by e − iωt.

The effect that n varies with frequency (except in vacuum, where all frequencies travel at the same speed, c) is known as dispersion, and it is what causes a prism to divide white light into its constituent spectral colors, explains rainbows, and is the cause of chromatic aberration in lenses. In regions of the spectrum where the material does not absorb, the real part of the refractive index tends to increase with frequency. Near absorption peaks, the curve of the refractive index is a complex form given by the Kramers–Kronig relations, and can decrease with frequency.

Since the refractive index of a material varies with the frequency (and thus wavelength) of light, it is usual to specify the corresponding vacuum wavelength at which the refractive index is measured. Typically, this is done at various well-defined spectral emission lines; for example, nD is the refractive index at the Fraunhofer "D" line, the centre of the yellow sodium double emission at 589.29 nm wavelength.

The Sellmeier equation is an empirical formula that works well in describing dispersion, and Sellmeier coefficients are often quoted instead of the refractive index in tables. For some representative refractive indices at different wavelengths, see list of indices of refraction.

As shown above, dielectric loss and non-zero DC conductivity in materials cause absorption. Good dielectric materials such as glass have extremely low DC conductivity, and at low frequencies the dielectric loss is also negligible, resulting in almost no absorption (κ ≈ 0). However, at higher frequencies (such as visible light), dielectric loss may increase absorption significantly, reducing the material's transparency to these frequencies.

The real and imaginary parts of the complex refractive index are related through use of the Kramers–Kronig relations. For example, one can determine a material's full complex refractive index as a function of wavelength from an absorption spectrum of the material.

[edit] Relation to dielectric constant

The dielectric constant (which is often dependent on wavelength) is simply the square of the (complex) refractive index. The refractive index is used for optics in Fresnel equations and Snell's law; while the dielectric constant is used in Maxwell's equations and electronics.

Where \tilde\epsilon, ε1, ε2, n, and κ are functions of wavelength:

\tilde\epsilon=\epsilon_1+i\epsilon_2= (n+i\kappa)^2

Conversion between refractive index and dielectric constant is done by:

ε1 = n2 − κ2

ε2 = 2nκ

n = \sqrt{\frac{\sqrt{\epsilon_1^2+\epsilon_2^2}+\epsilon_1}{2}} [2]

\kappa = \sqrt{ \frac{ \sqrt{ \epsilon_1^2+ \epsilon_2^2}- \epsilon_1}{2}}[3]

[edit] Anisotropy

A calcite crystal laid upon a paper with some letters showing birefringence

A calcite crystal laid upon a paper with some letters showing birefringence

The refractive index of certain media may be different depending on the polarization and direction of propagation of the light through the medium. This is known as birefringence or anisotropy and is described by the field of crystal optics. In the most general case, the dielectric constant is a rank-2 tensor (a 3 by 3 matrix), which cannot simply be described by refractive indices except for polarizations along principal axes.

In magneto-optic (gyro-magnetic) and optically active materials, the principal axes are complex (corresponding to elliptical polarizations), and the dielectric tensor is complex-Hermitian (for lossless media); such materials break time-reversal symmetry and are used e.g. to construct Faraday isolators.

[edit] Nonlinearity

The strong electric field of high intensity light (such as output of a laser) may cause a medium's refractive index to vary as the light passes through it, giving rise to nonlinear optics. If the index varies quadratically with the field (linearly with the intensity), it is called the optical Kerr effect and causes phenomena such as self-focusing and self-phase modulation. If the index varies linearly with the field (which is only possible in materials that do not possess inversion symmetry), it is known as the Pockels effect.

[edit] Inhomogeneity

A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens.

A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens.

If the refractive index of a medium is not constant, but varies gradually with position, the material is known as a gradient-index medium and is described by gradient index optics. Light travelling through such a medium can be bent or focussed, and this effect can be exploited to produce lenses, some optical fibers and other devices. Some common mirages are caused by a spatially-varying refractive index of air.

[edit] Relation to density

Relation between the refractive index and the density of silicate and borosilicate glasses ().

Relation between the refractive index and the density of silicate and borosilicate glasses ([4]).

In general, the refractive index of a glass increases with its density. However, there does not exist an overall linear relation between the refractive index and the density for all silicate and borosilicate glasses. A relatively high refractive index and low density can be obtained with glasses containing light metal oxides such as Li2O and MgO, while the opposite trend is observed with glasses containing PbO and BaO as seen in the diagram at the right.

[edit] Momentum Paradox

The momentum of a refracted ray, p, was calculated by Hermann Minkowski[5] in 1908, where E is energy of the photon, c is the speed of light in vacuum and n is the refractive index of the medium.

p=\frac{nE}{c}

In 1909 Max Abraham[6] proposed

p=\frac{E}{nc}

Rudolf Peierls raises this in his "More Surprises in Theoretical Physics" Princeton (1991). Ulf Leonhardt, Chair in Theoretical Physics at the University of St Andrews has discussed[7] this including experiments to resolve.

[edit] Applications

The refractive index of a material is the most important property of any optical system that uses refraction. It is used to calculate the focusing power of lenses, and the dispersive power of prisms.

Since refractive index is a fundamental physical property of a substance, it is often used to identify a particular substance, confirm its purity, or measure its concentration. Refractive index is used to measure solids (glasses and gemstones), liquids, and gases. Most commonly it is used to measure the concentration of a solute in an aqueous solution. A refractometer is the instrument used to measure refractive index. For a solution of sugar, the refractive index can be used to determine the sugar content (see Brix).Spectrum

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This article deals with the general meaning of "spectrum" and the history of its use. For other meanings and specific uses, see Spectrum (disambiguation).

The spectrum in a rainbow

The spectrum in a rainbow

A spectrum (plural spectra or spectrums[1]) is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. The word saw its first scientific use within the field of optics to describe the rainbow of colors in visible light when separated using a prism; it has since been applied by analogy to many fields. Thus one might talk about the spectrum of political opinion, or the spectrum of activity of a drug, or the autism spectrum. In these uses, values within a spectrum are not necessarily precisely defined numbers as in optics; exact values within the spectrum are not precisely quantifiable. Such use implies a broad range of conditions or behaviors grouped together and studied under a single title for ease of discussion.

In most modern usages of spectrum there is a unifying theme between extremes at either end. Some older usages of the word did not have a unifying theme, but they led to modern ones through a sequence of events set out below. Modern usages in mathematics did evolve from a unifying theme, but this may be difficult to recognize.

Contents

[hide]

* 1 Origins

* 2 Modern meaning in the physical sciences

* 3 See also

* 4 References

* 5 External links

[edit] Origins

In Latin spectrum means "image" or "apparition", including the meaning "spectre". Spectral evidence is testimony about what was done by spectres of persons not present physically, or hearsay evidence about what ghosts or apparitions of Satan said. It was used to convict a number of persons of witchcraft at Salem, Massachusetts in the late 17th century.

[edit] Modern meaning in the physical sciences

The spectrum of a star of spectral type K4III

The spectrum of a star of spectral type K4III

In the 17th century the word spectrum was introduced into optics, referring to the range of colors observed when white light was dispersed through a prism. Soon the term referred to a plot of light intensity or power as a function of frequency or wavelength, also known as a spectral density.

The term spectrum was soon applied to other waves, such as sound waves, and now applies to any signal that can be decomposed into frequency components. A spectrum is a usually 2-dimensional plot, of a compound signal, depicting the components by another measure. Sometimes, the word spectrum refers to the compound signal itself, such as the "spectrum of visible light", a reference to those electromagnetic waves which are visible to the human eye. Looking at light through a prism separates visible light into its colors according to wavelength. It separates them according to its dispersion relation and a grating separates according to the grating equation and if massive particles are measured often their speed is measured. To get a spectrum, the measured function has to be transformed in their independent variable to frequencies and the dependent variable has to be reduced in regions, where the independent variable is stretched. For this imagine that the spectrum of pulse with a finite number of particles is measured on a film or a CCD. Assuming no particles are lost, any nonlinearity (compared to frequency) on the spectral separation concentrates particles at some points of the film. The same is true for taking a spectrum by scanning a monochromator with a fixed slit width. Violet at one end has the shortest wavelength and red at the other end has the longest wavelength of visible light. The colors in order are violet, blue, green, yellow, orange, red. As the wavelengths get bigger below the red visible light they become infrared, microwave, and radio. As the wavelengths get smaller above violet light, they become ultra-violet, x-ray, and gamma ray.

[edit] See alsoDispersion (optics)

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In a prism, material dispersion (a wavelength-dependent refractive index) causes different colors to refract at different angles, splitting white light into a rainbow.

In a prism, material dispersion (a wavelength-dependent refractive index) causes different colors to refract at different angles, splitting white light into a rainbow.

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency.[1] Media having such a property are termed dispersive media.

The most familiar example of dispersion is probably a rainbow, in which dispersion causes the spatial separation of a white light into components of different wavelengths (different colors). However, dispersion also has an impact in many other circumstances: for example, it causes pulses to spread in optical fibers, degrading signals over long distances; also, a cancellation between dispersion and nonlinear effects leads to soliton waves. Dispersion is most often described for light waves, but it may occur for any kind of wave that interacts with a medium or passes through an inhomogeneous geometry (e.g. a waveguide), such as sound waves. Dispersion is sometimes called chromatic dispersion to emphasize its wavelength-dependent nature.

There are generally two sources of dispersion: material dispersion and waveguide dispersion. Material dispersion comes from a frequency-dependent response of a material to waves. For example, material dispersion leads to undesired chromatic aberration in a lens or the separation of colors in a prism. Waveguide dispersion occurs when the speed of a wave in a waveguide (such as an optical fiber) depends on its frequency for geometric reasons, independent of any frequency dependence of the materials from which it is constructed. More generally, "waveguide" dispersion can occur for waves propagating through any inhomogeneous structure (e.g. a photonic crystal), whether or not the waves are confined to some region. In general, both types of dispersion may be present, although they are not strictly additive. Their combination leads to signal degradation in optical fibers for telecommunications, because the varying delay in arrival time between different components of a signal "smears out" the signal in time.

Contents

[hide]

* 1 Material dispersion in optics

* 2 Group and phase velocity

* 3 Dispersion in waveguides

* 4 Dispersion in gemology

* 5 Dispersion in imaging

* 6 Dispersion in pulsar timing

* 7 See also

* 8 References

* 9 External links

[edit] Material dispersion in optics

The variation of refractive index vs. wavelength for various glasses. The wavelengths of visible light are shaded in red.

The variation of refractive index vs. wavelength for various glasses. The wavelengths of visible light are shaded in red.

Influences of selected glass component additions on the mean dispersion of a specific base glass (nF valid for λ = 486 nm (blue), nC valid for λ = 656 nm (red))

Influences of selected glass component additions on the mean dispersion of a specific base glass (nF valid for λ = 486 nm (blue), nC valid for λ = 656 nm (red))[2]

Material dispersion can be a desirable or undesirable effect in optical applications. The dispersion of light by glass prisms is used to construct spectrometers and spectroradiometers. Holographic gratings are also used, as they allow more accurate discrimination of wavelengths. However, in lenses, dispersion causes chromatic aberration, an undesired effect that may degrade images in microscopes, telescopes and photographic objectives.

The phase velocity, v, of a wave in a given uniform medium is given by

v = \frac{c}{n}

where c is the speed of light in a vacuum and n is the refractive index of the medium.

In general, the refractive index is some function of the frequency f of the light, thus n = n(f), or alternately, with respect to the wave's wavelength n = n(λ). The wavelength dependency of a material's refractive index is usually quantified by an empirical formula, the Cauchy or Sellmeier equations.

The most commonly seen consequence of dispersion in optics is the separation of white light into a color spectrum by a prism. From Snell's law it can be seen that the angle of refraction of light in a prism depends on the refractive index of the prism material. Since that refractive index varies with wavelength, it follows that the angle that the light is refracted will also vary with wavelength, causing an angular separation of the colors known as angular dispersion.

For visible light, most transparent materials (e.g. glasses) have:

1 < n(\lambda_{\rm red}) < n(\lambda_{\rm yellow}) < n(\lambda_{\rm blue})\ ,

or alternatively:

\frac{{\rm d}n}{{\rm d}\lambda} < 0,

that is, refractive index n decreases with increasing wavelength λ. In this case, the medium is said to have normal dispersion. Whereas, if the index increases with increasing wavelength the medium has anomalous dispersion.

At the interface of such a material with air or vacuum (index of ~1), Snell's law predicts that light incident at an angle θ to the normal will be refracted at an angle arcsin( sin (θ) / n) . Thus, blue light, with a higher refractive index, will be bent more strongly than red light, resulting in the well-known rainbow pattern.

[edit] Group and phase velocity

Another consequence of dispersion manifests itself as a temporal effect. The formula above, v = c / n calculates the phase velocity of a wave; this is the velocity at which the phase of any one frequency component of the wave will propagate. This is not the same as the group velocity of the wave, which is the rate that changes in amplitude (known as the envelope of the wave) will propagate. The group velocity vg is related to the phase velocity by, for a homogeneous medium (here λ is the wavelength in vacuum, not in the medium):

v_g = c \left( n - \lambda \frac{dn}{d\lambda} \right)^{-1}.

The group velocity vg is often thought of as the velocity at which energy or information is conveyed along the wave. In most cases this is true, and the group velocity can be thought of as the signal velocity of the waveform. In some unusual circumstances, where the wavelength of the light is close to an absorption resonance of the medium, it is possible for the group velocity to exceed the speed of light (vg > c), leading to the conclusion that superluminal (faster than light) communication is possible. In practice, in such situations the distortion and absorption of the wave is such that the value of the group velocity essentially becomes meaningless, and does not represent the true signal velocity of the wave, which stays less than c.

The group velocity itself is usually a function of the wave's frequency. This results in group velocity dispersion (GVD), which causes a short pulse of light to spread in time as a result of different frequency components of the pulse travelling at different velocities. GVD is often quantified as the group delay dispersion parameter (again, this formula is for a uniform medium only):

D = - \frac{\lambda}{c} \, \frac{d^2 n}{d \lambda^2}.

If D is less than zero, the medium is said to have positive dispersion. If D is greater than zero, the medium has negative dispersion.If a light pulse is propagated through a normally dispersive medium, the result is the higher frequency components travel slower than the lower frequency components. The pulse therefore becomes positively chirped, or up-chirped, increasing in frequency with time. Conversely, if a pulse travels through an anomalously dispersive medium, high frequency components travel faster than the lower ones, and the pulse becomes negatively chirped, or down-chirped, decreasing in frequency with time.

The result of GVD, whether negative or positive, is ultimately temporal spreading of the pulse. This makes dispersion management extremely important in optical communications systems based on optical fiber, since if dispersion is too high, a group of pulses representing a bit-stream will spread in time and merge together, rendering the bit-stream unintelligible. This limits the length of fiber that a signal can be sent down without regeneration. One possible answer to this problem is to send signals down the optical fibre at a wavelength where the GVD is zero (e.g. around ~1.3-1.5 μm in silica fibres), so pulses at this wavelength suffer minimal spreading from dispersion—in practice, however, this approach causes more problems than it solves because zero GVD unacceptably amplifies other nonlinear effects (such as four wave mixing). Another possible option is to use soliton pulses in the regime of anomalous dispersion, a form of optical pulse which uses a nonlinear optical effect to self-maintain its shape—solitons have the practical problem, however, that they require a certain power level to be maintained in the pulse for the nonlinear effect to be of the correct strength. Instead, the solution that is currently used in practice is to perform dispersion compensation, typically by matching the fiber with another fiber of opposite-sign dispersion so that the dispersion effects cancel; such compensation is ultimately limited by nonlinear effects such as self-phase modulation, which interact with dispersion to make it very difficult to undo.

Dispersion control is also important in lasers that produce short pulses. The overall dispersion of the optical resonator is a major factor in determining the duration of the pulses emitted by the laser. A pair of prisms can be arranged to produce net negative dispersion, which can be used to balance the usually positive dispersion of the laser medium. Diffraction gratings can also be used to produce dispersive effects; these are often used in high-power laser amplifier systems. Recently, an alternative to prisms and gratings has been developed: chirped mirrors. These dielectric mirrors are coated so that different wavelengths have different penetration lengths, and therefore different group delays. The coating layers can be tailored to achieve a net negative dispersion.

[edit] Dispersion in waveguides

Optical fibers, which are used in telecommunications, are among the most abundant types of waveguides. Dispersion in these fibers is one of the limiting factors that determine how much data can be transported on a single fiber.

The transverse modes for waves confined laterally within a waveguide generally have different speeds (and field patterns) depending upon their frequency (that is, on the relative size of the wave, the wavelength) compared to the size of the waveguide.

In general, for a waveguide mode with an angular frequency ω(β) at a propagation constant β (so that the electromagnetic fields in the propagation direction z oscillate proportional to ei(βz − ωt)), the group-velocity dispersion parameter D is defined as:[3]

D = -\frac{2\pi c}{\lambda^2} \frac{d^2 \beta}{d\omega^2} = \frac{2\pi c}{v_g^2 \lambda^2} \frac{dv_g}{d\omega}

where λ = 2πc / ω is the vacuum wavelength and vg = dω / dβ is the group velocity. This formula generalizes the one in the previous section for homogeneous media, and includes both waveguide dispersion and material dispersion. The reason for defining the dispersion in this way is that |D| is the (asymptotic) temporal pulse spreading Δt per unit bandwidth Δλ per unit distance travelled, commonly reported in ps / nm km for optical fibers.

A similar effect due to a somewhat different phenomenon is modal dispersion, caused by a waveguide having multiple modes at a given frequency, each with a different speed. A special case of this is polarization mode dispersion (PMD), which comes from a superposition of two modes that travel at different speeds due to random imperfections that break the symmetry of the waveguide.

[edit] Dispersion in gemology

In the technical terminology of gemology, dispersion is the difference in the refractive index of a material at the B and G Fraunhofer wavelengths of 686.7 nm and 430.8 nm and is meant to express the degree to which a prism cut from the gemstone shows "fire", or color. Dispersion is a material property. Fire depends on the dispersion, the cut angles, the lighting environment, the refractive index, and the viewer.

[edit] Dispersion in imaging

In photographic and microscopic lenses, dispersion causes chromatic aberration, distorting the image, and various techniques have been developed to counteract it.

[edit] Dispersion in pulsar timing

Pulsars are spinning neutron stars that emit pulses at very regular intervals ranging from milliseconds to seconds. It is believed that the pulses are emitted simultaneously over a wide range of frequencies. However, as observed on Earth, the components of each pulse emitted at higher radio frequencies arrive before those emitted at lower frequencies. This dispersion occurs because of the ionised component of the interstellar medium, which makes the group velocity frequency dependent. The extra delay added at frequency ν is

D = 4.15 ms (\frac{\nu}{GHz})^{-2} \times (\frac{DM}{cm^{-3} pc})

where the dispersion measure DM is

DM = \int_0^d{n_e\;dl}

is the integrated free electron column density ne out to the pulsar at a distance d[4].

Of course, this delay cannot be measured directly, since the emission time is unknown. What can be measured is the difference in arrival times at two different frequencies. The delay ΔT between a high frequency νhi and a low frequency νlo component of a pulse will be

\Delta T = 4.15 ms [(\frac{\nu_{lo}}{GHz})^{-2} - (\frac{\nu_{hi}}{GHz})^{-2} ] \times (\frac{DM}{cm^{-3} pc})

and so DM is normally computed from measurements at two different frequencies. This allows computation of the absolute delay at any frequency, which is used when combining many different pulsar observations into an integrated timing solution.

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For the property of metals, see refraction (metallurgy).

For the magic effect, see David Penn (magician).

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

The straw seems to be broken, due to refraction of light as it emerges into the air.

The straw seems to be broken, due to refraction of light as it emerges into the air.

Refraction is the change in direction of a wave due to a change in its speed. This is most commonly seen when a wave passes from one medium to another. Refraction of light is the most commonly seen example, but any type of wave can refract when it interacts with a medium, for example when sound waves pass from one medium into another or when water waves move into water of a different depth. Refraction is described by Snell's law, which states that the angle of incidence is related to the angle of refraction by

\frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1}

or

n_1\sin\theta_1 = n_2\sin\theta_2\

where

v1 and v2 are the wave velocities through the respective media.

θ1 and θ2 are the angles between the normal (to the interface) plane and the incident waves respectively.

n1 and n2 are the refractive indices

Contents

[hide]

* 1 Explanation

* 2 Optometry

* 3 Acoustics

* 4 See also

* 5 Reference

* 6 External links

[edit] Explanation

In optics, refraction occurs when light waves travel from a medium with a given refractive index to a medium with another. At the boundary between the media, the wave's phase velocity is altered, it changes direction, and its wavelength increases or decreases but its frequency remains constant. For example, a light ray will refract as it enters and leaves glass; understanding of this concept led to the invention of lenses and the refracting telescope

Refraction of light waves in water. The dark rectangle represents the actual position of a pencil sitting in a bowl of water. The light rectangle represents the apparent position of the pencil. Notice that the end (X) looks like it is at (Y), a position that is considerably shallower than (X).

Refraction of light waves in water. The dark rectangle represents the actual position of a pencil sitting in a bowl of water. The light rectangle represents the apparent position of the pencil. Notice that the end (X) looks like it is at (Y), a position that is considerably shallower than (X).

Photograph of refraction of waves in a ripple tank

Photograph of refraction of waves in a ripple tank

Refraction can be seen when looking into a bowl of water. Air has a refractive index of about 1.0003, and water has a refractive index of about 1.33. If a person looks at a straight object, such as a pencil or straw, which is placed at a slant, partially in the water, the object appears to bend at the water's surface. This is due to the bending of light rays as they move from the water to the air. Once the rays reach the eye, the eye traces them back as straight lines (lines of sight). The lines of sight (shown as dashed lines) intersect at a higher position than where the actual rays originated. This causes the pencil to appear higher and the water to appear shallower than it really is. The depth that the water appears to be when viewed from above is known as the apparent depth. This is an important consideration for spearfishing from the surface because it will make the target fish appear to be in a different place, and the fisher must aim lower to catch the fish.

Diagram of refraction of water waves

Diagram of refraction of water waves

The diagram on the right shows an example of refraction in water waves. Ripples travel from the left and pass over a shallower region inclined at an angle to the wavefront. The waves travel more slowly in the shallower water, so the wavelength decreases and the wave bends at the boundary. The dotted line represents the normal to the boundary. The dashed line represents the original direction of the waves. The phenomenon explains why waves on a shoreline never hit the shoreline at an angle. Whichever direction the waves travel in deep water, they always refract towards the normal as they enter the shallower water near the beach.

Solar glory at the steam from hot springs at Yellowstone National Park.A glory is an optical phenomenon produced by light backscattered (a combination of diffraction, reflection and refraction) towards its source by a cloud of uniformly-sized water droplets.

Solar glory at the steam from hot springs at Yellowstone National Park.A glory is an optical phenomenon produced by light backscattered (a combination of diffraction, reflection and refraction) towards its source by a cloud of uniformly-sized water droplets.

Refraction is also responsible for rainbows and for the splitting of white light into a rainbow-spectrum as it passes through a glass prism. Glass has a higher refractive index than air and the different frequencies of light travel at different speeds (dispersion), causing them to be refracted at different angles, so that you can see them. The different frequencies correspond to different colors observed.

While refraction allows for beautiful phenomena such as rainbows, it may also produce peculiar optical phenomena, such as mirages and Fata Morgana. These are caused by the change of the refractive index of air with temperature.

Refraction in a Perspex (acrylic) block.

Refraction in a Perspex (acrylic) block.

Snell's law is used to calculate the degree to which light is refracted when traveling from one medium to another.

Recently some metamaterials have been created which have a negative refractive index. With metamaterials, we can also obtain the total refraction phenomena when the wave impedances of the two media are matched. There is no reflected wave.

Also, since refraction can make objects appear closer than they are, it is responsible for allowing water to magnify objects. First, as light is entering a drop of water, it slows down. If the water's surface is not flat, then the light will be bent into a new path. This round shape will bend the light outwards and as it spreads out, the image you see gets larger.

A useful analogy in explaining the refraction of light would be to imagine a marching band as they march from pavement (a fast medium) into mud (a slower medium) The marchers on the side that runs into the mud first will slow down first. This causes the whole band to pivot slightly toward the normal (make a smaller angle from the normal).

[edit] Optometry

In medicine, particularly optometry and ophthalmology, refraction (also known as refractometry) is a clinical test in which a phoropter may used by an optometrist to determine the eye's refractive error and the best corrective lenses to be prescribed. A series of test lenses in graded optical powers or focal lengths are presented to determine which provide the sharpest, clearest vision.[1]

[edit] Acoustics

In underwater acoustics, refraction is the bending or curving of a sound ray that results when the ray passes through a sound speed gradient from a region of one sound speed to a region of a different speed. The amount of ray bending is dependent upon the amount of difference between sound speeds, that is, the variation in temperature, salinity, and pressure of the water.[2] Similar acoustics effects are also found in the Earth's atmosphere. The phenomenon of refraction of sound in the atmosphere has been known for centuries;[3] however, beginning in the early 1970s, widespread analysis of this effect came into vogue through the designing of urban highways and noise barriers to address the meteorological effects of bending of sound rays in the lower atmosphere.[4]Snell's law

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In optics and physics, Snell's law (also known as Descartes' law or the law of refraction), is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves, passing through a boundary between two different isotropic media, such as water and glass. The law says that the ratio of the sines of the angles of incidence and of refraction is a constant that depends on the media.

In optics, the law is used in ray tracing to compute the angles of incidence or refraction, and in experimental optics to find the refractive index of a material.

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

Named after Dutch mathematician Willebrord Snellius, one of its discoverers, Snell's law states that the ratio of the sines of the angles of incidence and refraction is equivalent to the ratio of velocities in the two media, or equivalent to the opposite ratio of the indices of refraction:

\frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1}

or

n_1\sin\theta_1 = n_2\sin\theta_2\ .

Snell's law follows from Fermat's principle of least time, which in turn follows from the propagation of light as waves.

Contents

[hide]

* 1 History

* 2 Explanation

o 2.1 Total internal reflection and critical angle

o 2.2 Derivations

* 3 Vector form

* 4 Dispersion

* 5 See also

* 6 References

* 7 External links

[edit] History

Ptolemy, of ancient Greece, had, through experiment, found a relationship regarding refraction angles, but which was inaccurate for angles that were not small. Ptolemy was confident he had found an accurate empirical law, partially as a result of fudging his data to fit theory (see: confirmation bias).[1]

An 1837 view of the history of "the Law of the Sines"

An 1837 view of the history of "the Law of the Sines"[2]

Snell's law was first described in a formal manuscript in a 984 writing by Ibn Sahl,[3][4] who used it to work out the shapes of lenses that focus light with no geometric aberrations, known as anaclastic lenses.

It was described again by Thomas Harriot in 1602,[5] who did not publish his work.

In 1621, Willebrord Snellius (Snel) derived a mathematically equivalent form, that remained unpublished during his lifetime. René Descartes independently derived the law using heuristic momentum conservation arguments in terms of sines in his 1637 treatise Discourse on Method, and used it to solve a range of optical problems. Rejecting Descartes' solution, Pierre de Fermat arrived at the same solution based solely on his principle of least time.

According to Dijksterhuis[6], "In De natura lucis et proprietate (1662) Isaac Vossius said that Descartes had seen Snell's paper and concocted his own proof. We now know this charge to be undeserved but it has been adopted many times since." Both Fermat and Huygens repeated this accusation that Descartes had copied Snell.

In French, Snell's Law is called "la loi de Descartes" or "loi de Snell-Descartes."

Huygens's construction

Huygens's construction

In his 1678 Traité de la Lumiere, Christiaan Huygens showed how Snell's law of sines could be explained by, or derived from, the wave nature of light, using what we have come to call the Huygens–Fresnel principle.

Although he spelled his name "Snel", as noted above, it has conventionally been spelled "Snell", apparently by misinterpreting the Latin form of his name, "Snellius".[7]

[edit] Explanation

Snell's law is used to determine the direction of light rays through refractive media with varying indices of refraction. The indices of refraction of the media, labeled n1,n2 and so on, are used to represent the factor by which a light ray's speed decreases when traveling through a refractive medium, such as glass or water, as opposed [8] to its velocity in a vacuum.

As light passes the border between media, depending upon the relative refractive indices of the two media, the light will either be refracted to a lesser angle, or a greater one. These angles are measured with respect to the normal line, represented perpendicular to the boundary. In the case of light traveling from air into water, light would be refracted towards the normal line, because the light is slowed down in water; light traveling from water to air would refract away from the normal line.

Refraction between two surfaces is also referred to as reversible because if all conditions were identical, the angles would be the same for light propagating in the opposite direction.

Snell's law is generally true only for isotropic or specular media (such as glass). In anisotropic media such as some crystals, birefringence may split the refracted ray into two rays, the ordinary or o-ray which follows Snell's law, and the other extraordinary or e-ray which may not be co-planar with the incident ray.

When the light or other wave involved is monochromatic, that is, of a single frequency, Snell's law can also be expressed in terms of a ratio of wavelengths in the two media, λ1 and λ2:

\frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2} = \frac{\lambda_1}{\lambda_2}

[edit] Total internal reflection and critical angle

An example of the angles involved within total internal reflection.

An example of the angles involved within total internal reflection.

Main article: Total internal reflection

When light moves from a dense to a less dense medium, such as from water to air, Snell's law cannot be used to calculate the refracted angle when the resolved sine value is higher than 1. At this point, light is reflected in the incident medium, known as internal reflection. Before the ray totally internally reflects, the light refracts at the critical angle; it travels directly along the surface between the two refractive media, without a change in phases like in other forms of optical phenomena.

As an example, a ray of light is incident at 50o towards a water–air boundary. If the angle is calculated using Snell's Law, then the resulting sine value will not invert, and thus the refracted angle cannot be calculated by Snell's law, due to the absence of a refracted outgoing ray:

\theta_2 = \sin^{-1} \left(\frac{n_1}{n_2}\sin\theta_1\right) = \sin^{-1} \left(\frac{1.333}{1.000}0.766\right) = \sin^{-1} 1.021

In order to calculate the critical angle, let θ2 = 90o and solve for θcrit:

\theta_{\mathrm{crit}} = \sin^{-1} \left( \frac{n_2}{n_1} \right)

When θ1 > θcrit, no refracted ray appears, and the incident ray undergoes total internal reflection from the interface medium.

[edit] Derivations

Wavefronts due to a point source in the context of Snell's law (the region below the gray line has a higher index of refraction than the region above it).

Wavefronts due to a point source in the context of Snell's law (the region below the gray line has a higher index of refraction than the region above it).

Snell's law may be derived from Fermat's principle, which states that the light travels the path which takes the least time. By taking the derivative of the optical path length, the stationary point is found giving the path taken by the light (though it should be noted that the result does not show light taking the least time path, but rather one that is stationary with respect to small variations as there are cases where light actually takes the greatest time path, as in a spherical mirror). In a classic analogy by Richard Feynman, the area of lower refractive index is replaced by a beach, the area of higher refractive index by the sea, and the fastest way for a rescuer on the beach to get to a drowning person in the sea is to run along a path that follows Snell's law.

Alternatively, Snell's law can be derived using interference of all possible paths of light wave from source to observer—it results in destructive interference everywhere except extrema of phase (where interference is constructive)—which become actual paths.

Another way to derive Snell’s Law involves an application of the general boundary conditions of Maxwell equations for electromagnetic radiation.

[edit] Vector form

Given a normalized light vector l (pointing from the light source toward the surface) and a normalized plane normal vector n, one can work out the normalized reflected and refracted rays:

\cos\theta_1=\mathbf{n}\cdot (-\mathbf{l})

\cos\theta_2=\sqrt{1-\left(\frac{n_1}{n_2}\right)^2\left(1-\left(\cos\theta_1\right)^2\right)}

\mathbf{v}_{\mathrm{reflect}}=\mathbf{l}+\left(2\cos\theta_1\right)\mathbf{n}

\mathbf{v}_{\mathrm{refract}}=\left(\frac{n_1}{n_2}\right)\mathbf{l} + \left( \frac{n_1}{n_2}\cos\theta_1 - \cos\theta_2\right)\mathbf{n}

Note: \mathbf{n}\cdot(-\mathbf{l}) must be positive. Otherwise, use

\mathbf{v}_{\mathrm{refract}}=\left(\frac{n_1}{n_2}\right)\mathbf{l} + \left(\frac{n_1}{n_2}\cos\theta_1 + \cos\theta_2\right)\mathbf{n}.

Example:

\mathbf{l}=\{0.707107, -0.707107\},~\mathbf{n}=\{0,1\},~\frac{n_1}{n_2}=0.9

\mathbf{~}\cos\theta_1=0.707107,~\cos\theta_2=0.771362

\mathbf{v}_{\mathrm{reflect}}=\{0.707107, 0.707107\} ,~\mathbf{v}_{\mathrm{refract}}=\{0.636396, -0.771362\}

The cosines may be recycled and used in the Fresnel equations for working out the intensity of the resulting rays. During total internal reflection an evanescent wave is produced, which rapidly decays from the surface into the second medium. Conservation of energy is maintained by the circulation of energy across the boundary, averaging to zero net energy transmission.

[edit] Dispersion

Main article: Dispersion (optics)

In many wave-propagation media, wave velocity changes with frequency or wavelength of the waves; this is true of light propagation in most transparent substances other than a vacuum. These media are called dispersive. The result is that the angles determined by Snell's law also depend on frequency or wavelength, so that a ray of mixed wavelengths, such as white light, will spread or disperse. Such dispersion of light in glass or water underlies the origin of rainbows, in which different wavelengths appear as different colors.

In optical instruments, dispersion leads to chromatic aberration, a color-dependent blurring that sometimes is the resolution-limiting effect. This was especially true in refracting telescopes, before the invention of achromatic objective lenses.Diffraction

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The intensity pattern formed on a screen by diffraction from a square aperture

The intensity pattern formed on a screen by diffraction from a square aperture

Colors seen in a spider web are partially due to diffraction, according to some analyses.

Colors seen in a spider web are partially due to diffraction, according to some analyses.[1]

Diffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle. It is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings.[2] Very similar effects are observed when there is an alteration in the properties of the medium in which the wave is travelling, for example a variation in refractive index for light waves or in acoustic impedance for sound waves and these can also be referred to as diffraction effects. Diffraction occurs with all waves, including sound waves, water waves, and electromagnetic waves such as visible light, x-rays and radio waves. As physical objects have wave-like properties, diffraction also occurs with matter and can be studied according to the principles of quantum mechanics.

While diffraction occurs whenever propagating waves encounter such changes, its effects are generally most pronounced for waves where the wavelength is on the order of the size of the diffracting objects. The complex patterns resulting from the intensity of a diffracted wave are a result of the superposition, or interference of different parts of a wave that traveled to the observer by different paths.

The formalism of diffraction can also describe the way in which waves of finite extent propagate in free space. For example, the expanding profile of a laser beam, the beam shape of a radar antenna and the field of view of an ultrasonic transducer are all explained by diffraction theory.

Contents

[hide]

* 1 Examples of diffraction in everyday life

* 2 History

* 3 The mechanism of diffraction

* 4 Diffraction systems

o 4.1 Single-slit diffraction

o 4.2 Diffraction Grating

o 4.3 Diffraction by a circular aperture

o 4.4 Propagation of a laser beam

o 4.5 Diffraction limited imaging

o 4.6 Speckle patterns

* 5 Common features of diffraction patterns

* 6 Particle diffraction

* 7 Bragg diffraction

* 8 Coherence

* 9 References

* 10 See also

* 11 External links

[edit] Examples of diffraction in everyday life

Solar glory at the steam from hot springs at Yellowstone National Park.A glory is an optical phenomenon produced by light backscattered (a combination of diffraction, reflection and refraction) towards its source by a cloud of uniformly-sized water droplets.

Solar glory at the steam from hot springs at Yellowstone National Park.A glory is an optical phenomenon produced by light backscattered (a combination of diffraction, reflection and refraction) towards its source by a cloud of uniformly-sized water droplets.

The effects of diffraction can be readily seen in everyday life. The most colorful examples of diffraction are those involving light; for example, the closely spaced tracks on a CD or DVD act as a diffraction grating to form the familiar rainbow pattern we see when looking at a disk. This principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired; the hologram on a credit card is an example. Diffraction in the atmosphere by small particles can cause a bright ring to be visible around a bright light source like the sun or the moon. A shadow of a solid object, using light from a compact source, shows small fringes near its edges. The speckle pattern which is observed when laser light falls on an optically rough service is also a diffraction phenomenon. All these effects are a consequence of the fact that light is a wave.

Diffraction can occur with any kind of wave. Ocean waves diffract around jetties and other obstacles. Sound waves can diffract around objects, this is the reason we can still hear someone calling us even if we are hiding behind a tree. Diffraction can also be a concern in some technical applications; it sets a fundamental limit to the resolution of a camera, telescope, or microscope.

[edit] History

Thomas Young's sketch of two-slit diffraction, which he presented to the Royal Society in 1803

Thomas Young's sketch of two-slit diffraction, which he presented to the Royal Society in 1803

The effects of diffraction of light were first carefully observed and characterized by Francesco Maria Grimaldi, who also coined the term diffraction, from the Latin diffringere, 'to break into pieces', referring to light breaking up into different directions. The results of Grimaldi's observations were published posthumously in 1665.[3][4] Isaac Newton studied these effects and attributed them to inflexion of light rays. James Gregory (1638–1675) observed the diffraction patterns caused by a bird feather, which was effectively the first diffraction grating. In 1803 Thomas Young did his famous experiment observing interference from two closely spaced slits. Explaining his results by interference of the waves emanating from the two different slits, he deduced that light must propagate as waves. Augustin-Jean Fresnel did more definitive studies and calculations of diffraction, published in 1815 and 1818, and thereby gave great support to the wave theory of light that had been advanced by Christiaan Huygens and reinvigorated by Young, against Newton's particle theory.

[edit] The mechanism of diffraction

Photograph of single-slit diffraction in a circular ripple tank

Photograph of single-slit diffraction in a circular ripple tank

Diffraction arises because of the way in which waves propagate; this is described by the Huygens–Fresnel principle. The propagation of a wave can be visualized by considering every point on a wavefront as a point source for a secondary radial wave. The subsequent propagation and addition of all these radial waves form the new wavefront. When waves are added together, their sum is determined by the relative phases as well as the amplitudes of the individual waves, an effect which is often known as wave interference. The summed amplitude of the waves can have any value between zero and the sum of the individual amplitudes. Hence, diffraction patterns usually have a series of maxima and minima.

To determine the form of a diffraction pattern, we must determine the phase and amplitude of each of the Huygens wavelets at each point in space and then find the sum of these waves. There are various analytical models which can be used to do this including the Fraunhoffer diffraction equation for the far field and the Fresnel Diffraction equation for the near-field. Most configurations cannot be solved analytically; solutions can be found using various numerical analytical methods including Finite element and boundary element methods

[edit] Diffraction systems

It is possible to obtain a qualitative understanding of many diffraction phenomena by considering how the relative phases of the individual secondary wave sources vary, and in particular, the conditions in which the phase difference equals half a cycle in which case waves will cancel one another out.

The simplest descriptions of diffraction are those in which the situation can be reduced to a two dimensional problem. For water waves, this is already the case, water waves propagate only on the surface of the water. For light, we can often neglect one direction if the diffracting object extends in that direction over a distance far greater than the wavelength. In the case of light shining through small circular holes we will have to take into account the full three dimensional nature of the problem.

Some of the simpler cases of diffraction are considered below.

[edit] Single-slit diffraction

Main article: Diffraction formalism

Numerical approximation of diffraction pattern from a slit of width four wavelengths with an incident plane wave. The main central beam, nulls, and phase reversals are apparent.

Numerical approximation of diffraction pattern from a slit of width four wavelengths with an incident plane wave. The main central beam, nulls, and phase reversals are apparent.

Graph and image of single-slit diffraction

Graph and image of single-slit diffraction

A long slit of infinitesimal width which is illuminated by light diffracts the light into a series of circular waves and the wavefront which emerges from the slit is a cylindrical wave of uniform intensity.

A slit which is wider than a wavelength has a large number of point sources spaced evenly across the width of the slit. The light at a given angle is made up of contributions from each of these point sources and if the relative phases of these contributions vary by more than 2π, we expect to find minima and maxima in the diffracted light.

We can find the angle at which a first minimum is obtained in the diffracted light by the following reasoning. The light from a source located at the top edge of the slit interferes destructively with a source located at the middle of the slit, when the path difference between them is equal to λ/2. Similarly, the source just below the top of the slit will interfere destructively with the source located just below the middle of the slit at the same angle. We can continue this reasoning along the entire height of the slit to conclude that the condition for destructive interference for the entire slit is the same as the condition for destructive interference between two narrow slits a distance apart that is half the width of the slit. The path difference is given by (d sinθ)/2 so that the minimum intensity occurs at an angle θmin given by

d \sin \theta_{min} = \lambda \,

where d is the width of the slit.

A similar argument can be used to show that if we imagine the slit to be divided into four, six eight parts, etc, minima are obtained at angles θn given by

d \sin \theta_{n} = n\lambda \,

where n is an integer greater than zero.

There is no such simple argument to enable us to find the maxima of the diffraction pattern. The intensity profile can be calculated using the Fraunhofer diffraction integral as

I(\theta)\, = I_0 {\left[ \mathrm{sinc} \left( \frac{\pi d}{\lambda} \sin \theta \right) \right] }^2

where the sinc function is given by sinc(x)=sin(x)/x.

It should be noted that this analysis applies only to the far field, i.e a significant distance from the diffracting slit.

2-slit and 5-slit diffraction of red laser light

2-slit and 5-slit diffraction of red laser light

[edit] Diffraction Grating

Main article: Diffraction grating

A diffraction grating is an optical component with a regular pattern. The form of the light diffracted by a grating depends on the structure of the elements and the number of elements present, but all gratings have intensity maxima at angles θm which are given by the grating equation

d \left( \sin{\theta_m} + \sin{\theta_i} \right) = m \lambda.

where θi is the angle at which the light is incident, d is the separation of grating elements and m is an integer which can be positive or negative.

The light diffracted by a grating is found by summing the light diffracted from each of the elements, and is essentially a convolution of diffraction and interference patterns.

The figure shows the light diffracted by 2-element and 5-element gratings where the grating spacings are the same; it can be seen that the maxima are in the same position, but the detailed structures of the intensities are different.

A computer-generated image of an Airy disk

A computer-generated image of an Airy disk

[edit] Diffraction by a circular aperture

Main article: Airy disk

The far-field diffraction of a plane wave incident on a circular aperture is often referred to as the Airy Disk. The variation in intensity with angle is given by

I(\theta) = I_0 \left ( \frac{2 J_1(ka \sin \theta)}{ka \sin \theta} \right )^2

where a is the radius of the circular aperture, k is equal to 2π/λ and J1 is a Bessel function. The smaller the aperture, the larger the spot size at a given distance, and the greater the divergence of the diffracted beams.

[edit] Propagation of a laser beam

The way in which the profile of a laser beam changes as it propagates is determined by diffraction. The output mirror of the laser is an aperture, and the subsequent beam shape is determined by that aperture. Hence, the smaller the output beam, the quicker it diverges. Diode lasers have much greater divergence than He-Ne lasers for this reason.

Paradoxically, it is possible to reduce the divergence of a laser beam by first expanding it with one convex lens, and then collimating it with a second convex lens whose focal point is coincident with that of the first lens. The resulting beam has a larger aperture, and hence a lower divergence.

[edit] Diffraction limited imaging

Main article: Diffraction-limited system

The Airy disc around each of the stars from the 2.56m telescope aperture can be seen in this lucky image of the binary star zeta Boötis.

The Airy disc around each of the stars from the 2.56m telescope aperture can be seen in this lucky image of the binary star zeta Boötis.

The ability of an imaging system to resolve detail is ultimately limited by diffraction. This is because a plane wave incident on a circular lens or mirror is diffracted as described above. The light is not focused to a point but forms an Airy pattern with a central spot of diameter

d = 1.22 \lambda \frac{f}{a},\,

where λ is the wavelength of the light, f is the focal length of the lens, and a is the diameter of the beam of light, or (if the beam is filling the lens) the diameter of the lens.

This is why telescopes have very large lenses or mirrors, and why optical microscopes are limited in the detail which they can see.

[edit] Speckle patterns

Main article: speckle pattern

The speckle pattern which is seen when using a laser pointer is another diffraction phenomenon. It is a result of the superpostion of many waves with different phases, which are produced when a laser beam illuminates a rough surface. They add together to give a resultant wave whose amplitude, and therefore intensity varies randomly.

[edit] Common features of diffraction patterns

Several qualitative observations can be made of diffraction in general:

* The angular spacing of the features in the diffraction pattern is inversely proportional to the dimensions of the object causing the diffraction, in other words: the smaller the diffracting object the 'wider' the resulting diffraction pattern and vice versa. (More precisely, this is true of the sines of the angles.)

* The diffraction angles are invariant under scaling; that is, they depend only on the ratio of the wavelength to the size of the diffracting object.

* When the diffracting object has a periodic structure, for example in a diffraction grating, the features generally become sharper. The third figure, for example, shows a comparison of a double-slit pattern with a pattern formed by five slits, both sets of slits having the same spacing, between the center of one slit and the next.

[edit] Particle diffraction

See also: neutron diffraction and electron diffraction

Quantum theory tells us that every particle exhibits wave properties. In particular, massive particles can interfere and therefore diffract. Diffraction of electrons and neutrons stood as one of the powerful arguments in favor of quantum mechanics. The wavelength associated with a particle is the de Broglie wavelength

\lambda=\frac{h}{p} \,

where h is Planck's constant and p is the momentum of the particle (mass × velocity for slow-moving particles) . For most macroscopic objects, this wavelength is so short that it is not meaningful to assign a wavelength to them. A Sodium atom traveling at about 3000 m/s would have a De Broglie wavelength of about 5 pico meters.

Because the wavelength for even the smallest of macroscopic objects is extremely small, diffraction of matter waves is only visible for small particles, like electrons, neutrons, atoms and small molecules. The short wavelength of these matter waves makes them ideally suited to study the atomic crystal structure of solids and large molecules like proteins.

Relatively recently, larger molecules like buckyballs,[5] have been shown to diffract. Currently, research is underway into the diffraction of viruses, which, being huge relative to electrons and other more commonly diffracted particles, have tiny wavelengths so must be made to travel very slowly through an extremely narrow slit in order to diffract.

[edit] Bragg diffraction

Following Bragg's law, each dot (or reflection), in this diffraction pattern forms from the constructive interference of X-rays passing through a crystal. The data can be used to determine the crystal's atomic structure.

Following Bragg's law, each dot (or reflection), in this diffraction pattern forms from the constructive interference of X-rays passing through a crystal. The data can be used to determine the crystal's atomic structure.

For more details on this topic, see Bragg diffraction.

Diffraction from a three dimensional periodic structure such as atoms in a crystal is called Bragg diffraction. It is similar to what occurs when waves are scattered from a diffraction grating. Bragg diffraction is a consequence of interference between waves reflecting from different crystal planes. The condition of constructive interference is given by Bragg's law:

m \lambda = 2 d \sin \theta \,

where

λ is the wavelength,

d is the distance between crystal planes,

θ is the angle of the diffracted wave.

and m is an integer known as the order of the diffracted beam.

Bragg diffraction may be carried out using either light of very short wavelength like x-rays or matter waves like neutrons (and electrons) whose wavelength is on the order of (or much smaller than) the atomic spacing[6]. The pattern produced gives information of the separations of crystallographic planes d, allowing one to deduce the crystal structure. Diffraction contrast, in electron microscopes and x-topography devices in particular, is also a powerful tool for examining individual defects and local strain fields in crystals.

[edit] Coherence

Main article: Coherence (physics)

The description of diffraction relies on the interference of waves emanating from the same source taking different paths to the same point on a screen. In this description, the difference in phase between waves that took different paths is only dependent on the effective path length. This does not take into account the fact that waves that arrive at the screen at the same time were emitted by the source at different times. The initial phase with which the source emits waves can change over time in an unpredictable way. This means that waves emitted by the source at times that are too far apart can no longer form a constant interference pattern since the relation between their phases is no longer time independent.

The length over which the phase in a beam of light is correlated, is called the coherence length. In order for interference to occur, the path length difference must be smaller than the coherence length. This is sometimes referred to as spectral coherence as it is related to the presence of different frequency components in the wave. In the case light emitted by an atomic transition, the coherence length is related to the lifetime of the excited state from which the atom made its transition.

If waves are emitted from an extended source this can lead to incoherence in the transversal direction. When looking at a cross section of a beam of light, the length over which the phase is correlated is called the transverse coherence length. In the case of Young's double slit experiment this would mean that if the transverse coherence length is smaller than the spacing between the two slits the resulting pattern on a screen would look like two single slit diffraction patterns.

In the case of particles like electrons, neutrons and atoms, the coherence length is related to the spacial extent of the wave function that describes the particle.

[edit] ReferencesAtmospheric refraction

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Photograph of the full Moon partly obscured by Earth's atmosphere. Note the deviation from circular in the Moon's lower edge, caused by refraction.

Photograph of the full Moon partly obscured by Earth's atmosphere. Note the deviation from circular in the Moon's lower edge, caused by refraction.

diagram showing displacement of the Sun's image at sunrise and sunset

diagram showing displacement of the Sun's image at sunrise and sunset

Atmospheric refraction is the deviation of light or other electromagnetic wave from a straight line as it passes through the atmosphere due to the variation in air density as a function of altitude. Atmospheric refraction near the ground produces mirages and can make distant objects appear to shimmer or ripple. The term also applies to the refraction of sound.

Atmospheric refraction causes astronomical objects to appear higher in the sky than they are in reality. It affects not only lightrays but all electromagnetic radiation, although in varying degrees (see dispersion (optics)). For example in visible light, blue is more affected than red. This may cause astronomical objects to be spread out into a spectrum in high-resolution images.

Whenever possible astronomers will always schedule their observations around the time of culmination of an object when it is highest in the sky. Likewise sailors will never shoot a star which is not at least 20° or more above the horizon. If observations close to the horizon cannot be avoided, it is possible to equip a telescope with control systems to compensate for the shift caused by the refraction. If the dispersion is a problem too, (in case of broadband high-resolution observations) atmospheric refraction correctors can be employed as well (made from pairs of rotating glass prisms). But as the amount of atmospheric refraction is function of temperature and pressure as well as humidity (the amount of water vapour especially important at mid-infrared wavelengths) the amount of effort needed for a successful compensation can be prohibitive.

It gets even worse when the atmospheric refraction is not homogenous, when there is turbulence in the air for example. This is the cause of twinkling of the stars and deformation of the shape of the sun at sunset and sunrise.

Contents

[hide]

* 1 Values

* 2 Random refraction effects

* 3 See also

* 4 External links

[edit] Values

The atmospheric refraction is zero in the zenith, is less than 1′ (one arcminute) at 45° altitude, still only 5′ at 10° altitude, but then quickly increases when the horizon is approached. On the horizon itself it is about 34′ (according to FW Bessel), just a little bit larger than the apparent size of the sun. Therefore if it appears that the setting sun is just above the horizon, in reality it has already set. Formulae to calculate the times of sunrise and sunset do not calculate the moment that the sun reaches altitude zero, but when its altitude is −50′: 16′ for the radius of the sun (solar positions are for the centre of the sun-disc, but sunrise and sunset usually refer to the appearance and disappearance of the upperlimb) plus 34′ for the refraction. In the case of the Moon one should apply additional corrections for the horizontal parallax of the moon, its apparent diameter and its phase, although the latter is seldom done.

The refraction is also a function of temperature and pressure. The values given above are for 10 °C and 100.3 kPa. Add 1 % to the refraction for every 3 °C colder, subtract if hotter (hot air is less dense, and will therefore have less refraction). Add 1 % for every 0.9 kPa higher pressure, subtract if lower. Evidently day to day variations in the weather will affect the exact times of sunrise and sunset as well as moonrise and moonset, and for that reason are never given more accurately than to the nearest whole minute in the almanacs.

As the atmospheric refraction is 34′ on the horizon itself, but only 29′ above it, the setting or rising sun seems to be flattened by about 5′ (about 1/6 of its apparent diameter).

[edit] Random refraction effects

Turbulence in the atmosphere magnifies and de-magnifies star images, making them appear brighter and fainter on a time-scale of milliseconds. The slowest components of these fluctuations are visible to the eye as twinkling (also called “scintillation”).

Turbulence also causes small random motions of the star image, and produces rapid changes in its structure. These effects are not visible to the naked eye, but are easily seen even in small telescopes. They are called “seeing” by astronomers.Refractive index

From Wikipedia, the free encyclopedia

Jump to: navigation, search

The refractive index (or index of refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves) is reduced inside the medium. For example, typical glass has a refractive index of 1.5, which means that in glass, light travels at 1 / 1.5 = 0.67 times the speed of light in a vacuum. Two common properties of glass and other transparent materials are directly related to their refractive index. First, light rays change direction when they cross the interface from air to the material, an effect that is used in lenses and glasses. Second, light reflects partially from surfaces that have a refractive index different from that of their surroundings.

Definition: The refractive index n of a medium is defined as the ratio of the phase velocity c of a wave phenomenon such as light or sound in a reference medium to the phase velocity vp in the medium itself:

n = \frac{c}{v_{\mathrm {p}}}

It is most commonly used in the context of light with vacuum as a reference medium, although historically other reference media (e.g. air at a standardized pressure and temperature) have been common. It is usually given the symbol n. In the case of light, it equals

n=\sqrt{\epsilon_r\mu_r},

where εr is the material's relative permittivity, and μr is its relative permeability. For most materials, μr is very close to 1 at optical frequencies, therefore n is approximately \sqrt{\epsilon_r}. Contrary to a widespread misconception, n may be less than 1, for example for x-rays.[1]. This has practical technical applications, such as effective mirrors for x-rays based on total external reflection.

The phase velocity is defined as the rate at which the crests of the waveform propagate; that is, the rate at which the phase of the waveform is moving. The group velocity is the rate that the envelope of the waveform is propagating; that is, the rate of variation of the amplitude of the waveform. Provided the waveform is not distorted significantly during propagation, it is the group velocity that represents the rate that information (and energy) may be transmitted by the wave, for example the velocity at which a pulse of light travels down an optical fiber.

Contents

[hide]

* 1 Speed of light

* 2 Negative Refractive Index

* 3 Dispersion and absorption

* 4 Relation to dielectric constant

* 5 Anisotropy

* 6 Nonlinearity

* 7 Inhomogeneity

* 8 Relation to density

* 9 Momentum Paradox

* 10 Applications

* 11 See also

* 12 References

* 13 External links

[edit] Speed of light

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

The speed of all electromagnetic radiation in vacuum is the same, approximately 3×108 meters per second, and is denoted by c. Therefore, if v is the phase velocity of radiation of a specific frequency in a specific material, the refractive index is given by

n =\frac{c}{v}

or inversely

v =\frac{c}{n}

This number is typically greater than one: the higher the index of the material, the more the light is slowed down (see Cherenkov radiation). However, at certain frequencies (e.g. near absorption resonances, and for X-rays), n will actually be smaller than one. This does not contradict the theory of relativity, which holds that no information-carrying signal can ever propagate faster than c, because the phase velocity is not the same as the group velocity or the signal velocity.

Sometimes, a "group velocity refractive index", usually called the group index is defined:

n_g=\frac{c}{v_g}

where vg is the group velocity. This value should not be confused with n, which is always defined with respect to the phase velocity. The group index can be written in terms of the wavelength dependence of the refractive index as

n_g = n - \lambda\frac{dn}{d\lambda},

where λ is the wavelength in vacuum. At the microscale, an electromagnetic wave's phase velocity is slowed in a material because the electric field creates a disturbance in the charges of each atom (primarily the electrons) proportional to the permittivity of the medium. The charges will, in general, oscillate slightly out of phase with respect to the driving electric field. The charges thus radiate their own electromagnetic wave that is at the same frequency but with a phase delay. The macroscopic sum of all such contributions in the material is a wave with the same frequency but shorter wavelength than the original, leading to a slowing of the wave's phase velocity. Most of the radiation from oscillating material charges will modify the incoming wave, changing its velocity. However, some net energy will be radiated in other directions (see scattering).

If the refractive indices of two materials are known for a given frequency, then one can compute the angle by which radiation of that frequency will be refracted as it moves from the first into the second material from Snell's law.

If in a given region the values of refractive indices n or ng were found to differ from unity (whether homogeneously, or isotropically, or not), then this region was distinct from vacuum in the above sense for lacking Poincaré symmetry.

[edit] Negative Refractive Index

Recent research has also demonstrated the existence of negative refractive index which can occur if the real parts of both εr and μr are simultaneously negative, although such is a necessary but not sufficient condition. Not thought to occur naturally, this can be achieved with so-called metamaterials and offers the possibility of perfect lenses and other exotic phenomena such as a reversal of Snell's law. [1] [2]

[edit] Dispersion and absorption

The variation of refractive index vs. wavelength for various glasses.

The variation of refractive index vs. wavelength for various glasses.

In real materials, the polarization does not respond instantaneously to an applied field. This causes dielectric loss, which can be expressed by a permittivity that is both complex and frequency dependent. Real materials are not perfect insulators either, i.e. they have non-zero direct current conductivity. Taking both aspects into consideration, we can define a complex index of refraction:

\tilde{n}=n+i\kappa

Here, n is the refractive index indicating the phase velocity as above, while κ is called the extinction coefficient, which indicates the amount of absorption loss when the electromagnetic wave propagates through the material. (See the article Mathematical descriptions of opacity.) Both n and κ are dependent on the frequency (wavelength). Note that the sign of the complex part is a matter of convention, which is important due to possible confusion between loss and gain. The notation above, which is usually used by physicists, corresponds to waves with time evolution given by e − iωt.

The effect that n varies with frequency (except in vacuum, where all frequencies travel at the same speed, c) is known as dispersion, and it is what causes a prism to divide white light into its constituent spectral colors, explains rainbows, and is the cause of chromatic aberration in lenses. In regions of the spectrum where the material does not absorb, the real part of the refractive index tends to increase with frequency. Near absorption peaks, the curve of the refractive index is a complex form given by the Kramers–Kronig relations, and can decrease with frequency.

Since the refractive index of a material varies with the frequency (and thus wavelength) of light, it is usual to specify the corresponding vacuum wavelength at which the refractive index is measured. Typically, this is done at various well-defined spectral emission lines; for example, nD is the refractive index at the Fraunhofer "D" line, the centre of the yellow sodium double emission at 589.29 nm wavelength.

The Sellmeier equation is an empirical formula that works well in describing dispersion, and Sellmeier coefficients are often quoted instead of the refractive index in tables. For some representative refractive indices at different wavelengths, see list of indices of refraction.

As shown above, dielectric loss and non-zero DC conductivity in materials cause absorption. Good dielectric materials such as glass have extremely low DC conductivity, and at low frequencies the dielectric loss is also negligible, resulting in almost no absorption (κ ≈ 0). However, at higher frequencies (such as visible light), dielectric loss may increase absorption significantly, reducing the material's transparency to these frequencies.

The real and imaginary parts of the complex refractive index are related through use of the Kramers–Kronig relations. For example, one can determine a material's full complex refractive index as a function of wavelength from an absorption spectrum of the material.

[edit] Relation to dielectric constant

The dielectric constant (which is often dependent on wavelength) is simply the square of the (complex) refractive index. The refractive index is used for optics in Fresnel equations and Snell's law; while the dielectric constant is used in Maxwell's equations and electronics.

Where \tilde\epsilon, ε1, ε2, n, and κ are functions of wavelength:

\tilde\epsilon=\epsilon_1+i\epsilon_2= (n+i\kappa)^2

Conversion between refractive index and dielectric constant is done by:

ε1 = n2 − κ2

ε2 = 2nκ

n = \sqrt{\frac{\sqrt{\epsilon_1^2+\epsilon_2^2}+\epsilon_1}{2}} [2]

\kappa = \sqrt{ \frac{ \sqrt{ \epsilon_1^2+ \epsilon_2^2}- \epsilon_1}{2}}[3]

[edit] Anisotropy

A calcite crystal laid upon a paper with some letters showing birefringence

A calcite crystal laid upon a paper with some letters showing birefringence

The refractive index of certain media may be different depending on the polarization and direction of propagation of the light through the medium. This is known as birefringence or anisotropy and is described by the field of crystal optics. In the most general case, the dielectric constant is a rank-2 tensor (a 3 by 3 matrix), which cannot simply be described by refractive indices except for polarizations along principal axes.

In magneto-optic (gyro-magnetic) and optically active materials, the principal axes are complex (corresponding to elliptical polarizations), and the dielectric tensor is complex-Hermitian (for lossless media); such materials break time-reversal symmetry and are used e.g. to construct Faraday isolators.

[edit] Nonlinearity

The strong electric field of high intensity light (such as output of a laser) may cause a medium's refractive index to vary as the light passes through it, giving rise to nonlinear optics. If the index varies quadratically with the field (linearly with the intensity), it is called the optical Kerr effect and causes phenomena such as self-focusing and self-phase modulation. If the index varies linearly with the field (which is only possible in materials that do not possess inversion symmetry), it is known as the Pockels effect.

[edit] Inhomogeneity

A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens.

A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens.

If the refractive index of a medium is not constant, but varies gradually with position, the material is known as a gradient-index medium and is described by gradient index optics. Light travelling through such a medium can be bent or focussed, and this effect can be exploited to produce lenses, some optical fibers and other devices. Some common mirages are caused by a spatially-varying refractive index of air.

[edit] Relation to density

Relation between the refractive index and the density of silicate and borosilicate glasses ().

Relation between the refractive index and the density of silicate and borosilicate glasses ([4]).

In general, the refractive index of a glass increases with its density. However, there does not exist an overall linear relation between the refractive index and the density for all silicate and borosilicate glasses. A relatively high refractive index and low density can be obtained with glasses containing light metal oxides such as Li2O and MgO, while the opposite trend is observed with glasses containing PbO and BaO as seen in the diagram at the right.

[edit] Momentum Paradox

The momentum of a refracted ray, p, was calculated by Hermann Minkowski[5] in 1908, where E is energy of the photon, c is the speed of light in vacuum and n is the refractive index of the medium.

p=\frac{nE}{c}

In 1909 Max Abraham[6] proposed

p=\frac{E}{nc}

Rudolf Peierls raises this in his "More Surprises in Theoretical Physics" Princeton (1991). Ulf Leonhardt, Chair in Theoretical Physics at the University of St Andrews has discussed[7] this including experiments to resolve.

[edit] Applications

The refractive index of a material is the most important property of any optical system that uses refraction. It is used to calculate the focusing power of lenses, and the dispersive power of prisms.

Since refractive index is a fundamental physical property of a substance, it is often used to identify a particular substance, confirm its purity, or measure its concentration. Refractive index is used to measure solids (glasses and gemstones), liquids, and gases. Most commonly it is used to measure the concentration of a solute in an aqueous solution. A refractometer is the instrument used to measure refractive index. For a solution of sugar, the refractive index can be used to determine the sugar content (see Brix).Refractive index

From Wikipedia, the free encyclopedia

Jump to: navigation, search

The refractive index (or index of refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves) is reduced inside the medium. For example, typical glass has a refractive index of 1.5, which means that in glass, light travels at 1 / 1.5 = 0.67 times the speed of light in a vacuum. Two common properties of glass and other transparent materials are directly related to their refractive index. First, light rays change direction when they cross the interface from air to the material, an effect that is used in lenses and glasses. Second, light reflects partially from surfaces that have a refractive index different from that of their surroundings.

Definition: The refractive index n of a medium is defined as the ratio of the phase velocity c of a wave phenomenon such as light or sound in a reference medium to the phase velocity vp in the medium itself:

n = \frac{c}{v_{\mathrm {p}}}

It is most commonly used in the context of light with vacuum as a reference medium, although historically other reference media (e.g. air at a standardized pressure and temperature) have been common. It is usually given the symbol n. In the case of light, it equals

n=\sqrt{\epsilon_r\mu_r},

where εr is the material's relative permittivity, and μr is its relative permeability. For most materials, μr is very close to 1 at optical frequencies, therefore n is approximately \sqrt{\epsilon_r}. Contrary to a widespread misconception, n may be less than 1, for example for x-rays.[1]. This has practical technical applications, such as effective mirrors for x-rays based on total external reflection.

The phase velocity is defined as the rate at which the crests of the waveform propagate; that is, the rate at which the phase of the waveform is moving. The group velocity is the rate that the envelope of the waveform is propagating; that is, the rate of variation of the amplitude of the waveform. Provided the waveform is not distorted significantly during propagation, it is the group velocity that represents the rate that information (and energy) may be transmitted by the wave, for example the velocity at which a pulse of light travels down an optical fiber.

Contents

[hide]

* 1 Speed of light

* 2 Negative Refractive Index

* 3 Dispersion and absorption

* 4 Relation to dielectric constant

* 5 Anisotropy

* 6 Nonlinearity

* 7 Inhomogeneity

* 8 Relation to density

* 9 Momentum Paradox

* 10 Applications

* 11 See also

* 12 References

* 13 External links

[edit] Speed of light

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

The speed of all electromagnetic radiation in vacuum is the same, approximately 3×108 meters per second, and is denoted by c. Therefore, if v is the phase velocity of radiation of a specific frequency in a specific material, the refractive index is given by

n =\frac{c}{v}

or inversely

v =\frac{c}{n}

This number is typically greater than one: the higher the index of the material, the more the light is slowed down (see Cherenkov radiation). However, at certain frequencies (e.g. near absorption resonances, and for X-rays), n will actually be smaller than one. This does not contradict the theory of relativity, which holds that no information-carrying signal can ever propagate faster than c, because the phase velocity is not the same as the group velocity or the signal velocity.

Sometimes, a "group velocity refractive index", usually called the group index is defined:

n_g=\frac{c}{v_g}

where vg is the group velocity. This value should not be confused with n, which is always defined with respect to the phase velocity. The group index can be written in terms of the wavelength dependence of the refractive index as

n_g = n - \lambda\frac{dn}{d\lambda},

where λ is the wavelength in vacuum. At the microscale, an electromagnetic wave's phase velocity is slowed in a material because the electric field creates a disturbance in the charges of each atom (primarily the electrons) proportional to the permittivity of the medium. The charges will, in general, oscillate slightly out of phase with respect to the driving electric field. The charges thus radiate their own electromagnetic wave that is at the same frequency but with a phase delay. The macroscopic sum of all such contributions in the material is a wave with the same frequency but shorter wavelength than the original, leading to a slowing of the wave's phase velocity. Most of the radiation from oscillating material charges will modify the incoming wave, changing its velocity. However, some net energy will be radiated in other directions (see scattering).

If the refractive indices of two materials are known for a given frequency, then one can compute the angle by which radiation of that frequency will be refracted as it moves from the first into the second material from Snell's law.

If in a given region the values of refractive indices n or ng were found to differ from unity (whether homogeneously, or isotropically, or not), then this region was distinct from vacuum in the above sense for lacking Poincaré symmetry.

[edit] Negative Refractive Index

Recent research has also demonstrated the existence of negative refractive index which can occur if the real parts of both εr and μr are simultaneously negative, although such is a necessary but not sufficient condition. Not thought to occur naturally, this can be achieved with so-called metamaterials and offers the possibility of perfect lenses and other exotic phenomena such as a reversal of Snell's law. [1] [2]

[edit] Dispersion and absorption

The variation of refractive index vs. wavelength for various glasses.

The variation of refractive index vs. wavelength for various glasses.

In real materials, the polarization does not respond instantaneously to an applied field. This causes dielectric loss, which can be expressed by a permittivity that is both complex and frequency dependent. Real materials are not perfect insulators either, i.e. they have non-zero direct current conductivity. Taking both aspects into consideration, we can define a complex index of refraction:

\tilde{n}=n+i\kappa

Here, n is the refractive index indicating the phase velocity as above, while κ is called the extinction coefficient, which indicates the amount of absorption loss when the electromagnetic wave propagates through the material. (See the article Mathematical descriptions of opacity.) Both n and κ are dependent on the frequency (wavelength). Note that the sign of the complex part is a matter of convention, which is important due to possible confusion between loss and gain. The notation above, which is usually used by physicists, corresponds to waves with time evolution given by e − iωt.

The effect that n varies with frequency (except in vacuum, where all frequencies travel at the same speed, c) is known as dispersion, and it is what causes a prism to divide white light into its constituent spectral colors, explains rainbows, and is the cause of chromatic aberration in lenses. In regions of the spectrum where the material does not absorb, the real part of the refractive index tends to increase with frequency. Near absorption peaks, the curve of the refractive index is a complex form given by the Kramers–Kronig relations, and can decrease with frequency.

Since the refractive index of a material varies with the frequency (and thus wavelength) of light, it is usual to specify the corresponding vacuum wavelength at which the refractive index is measured. Typically, this is done at various well-defined spectral emission lines; for example, nD is the refractive index at the Fraunhofer "D" line, the centre of the yellow sodium double emission at 589.29 nm wavelength.

The Sellmeier equation is an empirical formula that works well in describing dispersion, and Sellmeier coefficients are often quoted instead of the refractive index in tables. For some representative refractive indices at different wavelengths, see list of indices of refraction.

As shown above, dielectric loss and non-zero DC conductivity in materials cause absorption. Good dielectric materials such as glass have extremely low DC conductivity, and at low frequencies the dielectric loss is also negligible, resulting in almost no absorption (κ ≈ 0). However, at higher frequencies (such as visible light), dielectric loss may increase absorption significantly, reducing the material's transparency to these frequencies.

The real and imaginary parts of the complex refractive index are related through use of the Kramers–Kronig relations. For example, one can determine a material's full complex refractive index as a function of wavelength from an absorption spectrum of the material.

[edit] Relation to dielectric constant

The dielectric constant (which is often dependent on wavelength) is simply the square of the (complex) refractive index. The refractive index is used for optics in Fresnel equations and Snell's law; while the dielectric constant is used in Maxwell's equations and electronics.

Where \tilde\epsilon, ε1, ε2, n, and κ are functions of wavelength:

\tilde\epsilon=\epsilon_1+i\epsilon_2= (n+i\kappa)^2

Conversion between refractive index and dielectric constant is done by:

ε1 = n2 − κ2

ε2 = 2nκ

n = \sqrt{\frac{\sqrt{\epsilon_1^2+\epsilon_2^2}+\epsilon_1}{2}} [2]

\kappa = \sqrt{ \frac{ \sqrt{ \epsilon_1^2+ \epsilon_2^2}- \epsilon_1}{2}}[3]

[edit] Anisotropy

A calcite crystal laid upon a paper with some letters showing birefringence

A calcite crystal laid upon a paper with some letters showing birefringence

The refractive index of certain media may be different depending on the polarization and direction of propagation of the light through the medium. This is known as birefringence or anisotropy and is described by the field of crystal optics. In the most general case, the dielectric constant is a rank-2 tensor (a 3 by 3 matrix), which cannot simply be described by refractive indices except for polarizations along principal axes.

In magneto-optic (gyro-magnetic) and optically active materials, the principal axes are complex (corresponding to elliptical polarizations), and the dielectric tensor is complex-Hermitian (for lossless media); such materials break time-reversal symmetry and are used e.g. to construct Faraday isolators.

[edit] Nonlinearity

The strong electric field of high intensity light (such as output of a laser) may cause a medium's refractive index to vary as the light passes through it, giving rise to nonlinear optics. If the index varies quadratically with the field (linearly with the intensity), it is called the optical Kerr effect and causes phenomena such as self-focusing and self-phase modulation. If the index varies linearly with the field (which is only possible in materials that do not possess inversion symmetry), it is known as the Pockels effect.

[edit] Inhomogeneity

A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens.

A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens.

If the refractive index of a medium is not constant, but varies gradually with position, the material is known as a gradient-index medium and is described by gradient index optics. Light travelling through such a medium can be bent or focussed, and this effect can be exploited to produce lenses, some optical fibers and other devices. Some common mirages are caused by a spatially-varying refractive index of air.

[edit] Relation to density

Relation between the refractive index and the density of silicate and borosilicate glasses ().

Relation between the refractive index and the density of silicate and borosilicate glasses ([4]).

In general, the refractive index of a glass increases with its density. However, there does not exist an overall linear relation between the refractive index and the density for all silicate and borosilicate glasses. A relatively high refractive index and low density can be obtained with glasses containing light metal oxides such as Li2O and MgO, while the opposite trend is observed with glasses containing PbO and BaO as seen in the diagram at the right.

[edit] Momentum Paradox

The momentum of a refracted ray, p, was calculated by Hermann Minkowski[5] in 1908, where E is energy of the photon, c is the speed of light in vacuum and n is the refractive index of the medium.

p=\frac{nE}{c}

In 1909 Max Abraham[6] proposed

p=\frac{E}{nc}

Rudolf Peierls raises this in his "More Surprises in Theoretical Physics" Princeton (1991). Ulf Leonhardt, Chair in Theoretical Physics at the University of St Andrews has discussed[7] this including experiments to resolve.

[edit] Applications

The refractive index of a material is the most important property of any optical system that uses refraction. It is used to calculate the focusing power of lenses, and the dispersive power of prisms.

Since refractive index is a fundamental physical property of a substance, it is often used to identify a particular substance, confirm its purity, or measure its concentration. Refractive index is used to measure solids (glasses and gemstones), liquids, and gases. Most commonly it is used to measure the concentration of a solute in an aqueous solution. A refractometer is the instrument used to measure refractive index. For a solution of sugar, the refractive index can be used to determine the sugar content (see Brix).Refractive index

From Wikipedia, the free encyclopedia

Jump to: navigation, search

The refractive index (or index of refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves) is reduced inside the medium. For example, typical glass has a refractive index of 1.5, which means that in glass, light travels at 1 / 1.5 = 0.67 times the speed of light in a vacuum. Two common properties of glass and other transparent materials are directly related to their refractive index. First, light rays change direction when they cross the interface from air to the material, an effect that is used in lenses and glasses. Second, light reflects partially from surfaces that have a refractive index different from that of their surroundings.

Definition: The refractive index n of a medium is defined as the ratio of the phase velocity c of a wave phenomenon such as light or sound in a reference medium to the phase velocity vp in the medium itself:

n = \frac{c}{v_{\mathrm {p}}}

It is most commonly used in the context of light with vacuum as a reference medium, although historically other reference media (e.g. air at a standardized pressure and temperature) have been common. It is usually given the symbol n. In the case of light, it equals

n=\sqrt{\epsilon_r\mu_r},

where εr is the material's relative permittivity, and μr is its relative permeability. For most materials, μr is very close to 1 at optical frequencies, therefore n is approximately \sqrt{\epsilon_r}. Contrary to a widespread misconception, n may be less than 1, for example for x-rays.[1]. This has practical technical applications, such as effective mirrors for x-rays based on total external reflection.

The phase velocity is defined as the rate at which the crests of the waveform propagate; that is, the rate at which the phase of the waveform is moving. The group velocity is the rate that the envelope of the waveform is propagating; that is, the rate of variation of the amplitude of the waveform. Provided the waveform is not distorted significantly during propagation, it is the group velocity that represents the rate that information (and energy) may be transmitted by the wave, for example the velocity at which a pulse of light travels down an optical fiber.

Contents

[hide]

* 1 Speed of light

* 2 Negative Refractive Index

* 3 Dispersion and absorption

* 4 Relation to dielectric constant

* 5 Anisotropy

* 6 Nonlinearity

* 7 Inhomogeneity

* 8 Relation to density

* 9 Momentum Paradox

* 10 Applications

* 11 See also

* 12 References

* 13 External links

[edit] Speed of light

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

The speed of all electromagnetic radiation in vacuum is the same, approximately 3×108 meters per second, and is denoted by c. Therefore, if v is the phase velocity of radiation of a specific frequency in a specific material, the refractive index is given by

n =\frac{c}{v}

or inversely

v =\frac{c}{n}

This number is typically greater than one: the higher the index of the material, the more the light is slowed down (see Cherenkov radiation). However, at certain frequencies (e.g. near absorption resonances, and for X-rays), n will actually be smaller than one. This does not contradict the theory of relativity, which holds that no information-carrying signal can ever propagate faster than c, because the phase velocity is not the same as the group velocity or the signal velocity.

Sometimes, a "group velocity refractive index", usually called the group index is defined:

n_g=\frac{c}{v_g}

where vg is the group velocity. This value should not be confused with n, which is always defined with respect to the phase velocity. The group index can be written in terms of the wavelength dependence of the refractive index as

n_g = n - \lambda\frac{dn}{d\lambda},

where λ is the wavelength in vacuum. At the microscale, an electromagnetic wave's phase velocity is slowed in a material because the electric field creates a disturbance in the charges of each atom (primarily the electrons) proportional to the permittivity of the medium. The charges will, in general, oscillate slightly out of phase with respect to the driving electric field. The charges thus radiate their own electromagnetic wave that is at the same frequency but with a phase delay. The macroscopic sum of all such contributions in the material is a wave with the same frequency but shorter wavelength than the original, leading to a slowing of the wave's phase velocity. Most of the radiation from oscillating material charges will modify the incoming wave, changing its velocity. However, some net energy will be radiated in other directions (see scattering).

If the refractive indices of two materials are known for a given frequency, then one can compute the angle by which radiation of that frequency will be refracted as it moves from the first into the second material from Snell's law.

If in a given region the values of refractive indices n or ng were found to differ from unity (whether homogeneously, or isotropically, or not), then this region was distinct from vacuum in the above sense for lacking Poincaré symmetry.

[edit] Negative Refractive Index

Recent research has also demonstrated the existence of negative refractive index which can occur if the real parts of both εr and μr are simultaneously negative, although such is a necessary but not sufficient condition. Not thought to occur naturally, this can be achieved with so-called metamaterials and offers the possibility of perfect lenses and other exotic phenomena such as a reversal of Snell's law. [1] [2]

[edit] Dispersion and absorption

The variation of refractive index vs. wavelength for various glasses.

The variation of refractive index vs. wavelength for various glasses.

In real materials, the polarization does not respond instantaneously to an applied field. This causes dielectric loss, which can be expressed by a permittivity that is both complex and frequency dependent. Real materials are not perfect insulators either, i.e. they have non-zero direct current conductivity. Taking both aspects into consideration, we can define a complex index of refraction:

\tilde{n}=n+i\kappa

Here, n is the refractive index indicating the phase velocity as above, while κ is called the extinction coefficient, which indicates the amount of absorption loss when the electromagnetic wave propagates through the material. (See the article Mathematical descriptions of opacity.) Both n and κ are dependent on the frequency (wavelength). Note that the sign of the complex part is a matter of convention, which is important due to possible confusion between loss and gain. The notation above, which is usually used by physicists, corresponds to waves with time evolution given by e − iωt.

The effect that n varies with frequency (except in vacuum, where all frequencies travel at the same speed, c) is known as dispersion, and it is what causes a prism to divide white light into its constituent spectral colors, explains rainbows, and is the cause of chromatic aberration in lenses. In regions of the spectrum where the material does not absorb, the real part of the refractive index tends to increase with frequency. Near absorption peaks, the curve of the refractive index is a complex form given by the Kramers–Kronig relations, and can decrease with frequency.

Since the refractive index of a material varies with the frequency (and thus wavelength) of light, it is usual to specify the corresponding vacuum wavelength at which the refractive index is measured. Typically, this is done at various well-defined spectral emission lines; for example, nD is the refractive index at the Fraunhofer "D" line, the centre of the yellow sodium double emission at 589.29 nm wavelength.

The Sellmeier equation is an empirical formula that works well in describing dispersion, and Sellmeier coefficients are often quoted instead of the refractive index in tables. For some representative refractive indices at different wavelengths, see list of indices of refraction.

As shown above, dielectric loss and non-zero DC conductivity in materials cause absorption. Good dielectric materials such as glass have extremely low DC conductivity, and at low frequencies the dielectric loss is also negligible, resulting in almost no absorption (κ ≈ 0). However, at higher frequencies (such as visible light), dielectric loss may increase absorption significantly, reducing the material's transparency to these frequencies.

The real and imaginary parts of the complex refractive index are related through use of the Kramers–Kronig relations. For example, one can determine a material's full complex refractive index as a function of wavelength from an absorption spectrum of the material.

[edit] Relation to dielectric constant

The dielectric constant (which is often dependent on wavelength) is simply the square of the (complex) refractive index. The refractive index is used for optics in Fresnel equations and Snell's law; while the dielectric constant is used in Maxwell's equations and electronics.

Where \tilde\epsilon, ε1, ε2, n, and κ are functions of wavelength:

\tilde\epsilon=\epsilon_1+i\epsilon_2= (n+i\kappa)^2

Conversion between refractive index and dielectric constant is done by:

ε1 = n2 − κ2

ε2 = 2nκ

n = \sqrt{\frac{\sqrt{\epsilon_1^2+\epsilon_2^2}+\epsilon_1}{2}} [2]

\kappa = \sqrt{ \frac{ \sqrt{ \epsilon_1^2+ \epsilon_2^2}- \epsilon_1}{2}}[3]

[edit] Anisotropy

A calcite crystal laid upon a paper with some letters showing birefringence

A calcite crystal laid upon a paper with some letters showing birefringence

The refractive index of certain media may be different depending on the polarization and direction of propagation of the light through the medium. This is known as birefringence or anisotropy and is described by the field of crystal optics. In the most general case, the dielectric constant is a rank-2 tensor (a 3 by 3 matrix), which cannot simply be described by refractive indices except for polarizations along principal axes.

In magneto-optic (gyro-magnetic) and optically active materials, the principal axes are complex (corresponding to elliptical polarizations), and the dielectric tensor is complex-Hermitian (for lossless media); such materials break time-reversal symmetry and are used e.g. to construct Faraday isolators.

[edit] Nonlinearity

The strong electric field of high intensity light (such as output of a laser) may cause a medium's refractive index to vary as the light passes through it, giving rise to nonlinear optics. If the index varies quadratically with the field (linearly with the intensity), it is called the optical Kerr effect and causes phenomena such as self-focusing and self-phase modulation. If the index varies linearly with the field (which is only possible in materials that do not possess inversion symmetry), it is known as the Pockels effect.

[edit] Inhomogeneity

A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens.

A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens.

If the refractive index of a medium is not constant, but varies gradually with position, the material is known as a gradient-index medium and is described by gradient index optics. Light travelling through such a medium can be bent or focussed, and this effect can be exploited to produce lenses, some optical fibers and other devices. Some common mirages are caused by a spatially-varying refractive index of air.

[edit] Relation to density

Relation between the refractive index and the density of silicate and borosilicate glasses ().

Relation between the refractive index and the density of silicate and borosilicate glasses ([4]).

In general, the refractive index of a glass increases with its density. However, there does not exist an overall linear relation between the refractive index and the density for all silicate and borosilicate glasses. A relatively high refractive index and low density can be obtained with glasses containing light metal oxides such as Li2O and MgO, while the opposite trend is observed with glasses containing PbO and BaO as seen in the diagram at the right.

[edit] Momentum Paradox

The momentum of a refracted ray, p, was calculated by Hermann Minkowski[5] in 1908, where E is energy of the photon, c is the speed of light in vacuum and n is the refractive index of the medium.

p=\frac{nE}{c}

In 1909 Max Abraham[6] proposed

p=\frac{E}{nc}

Rudolf Peierls raises this in his "More Surprises in Theoretical Physics" Princeton (1991). Ulf Leonhardt, Chair in Theoretical Physics at the University of St Andrews has discussed[7] this including experiments to resolve.

[edit] Applications

The refractive index of a material is the most important property of any optical system that uses refraction. It is used to calculate the focusing power of lenses, and the dispersive power of prisms.

Since refractive index is a fundamental physical property of a substance, it is often used to identify a particular substance, confirm its purity, or measure its concentration. Refractive index is used to measure solids (glasses and gemstones), liquids, and gases. Most commonly it is used to measure the concentration of a solute in an aqueous solution. A refractometer is the instrument used to measure refractive index. For a solution of sugar, the refractive index can be used to determine the sugar content (see Brix).Refractive index

From Wikipedia, the free encyclopedia

Jump to: navigation, search

The refractive index (or index of refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves) is reduced inside the medium. For example, typical glass has a refractive index of 1.5, which means that in glass, light travels at 1 / 1.5 = 0.67 times the speed of light in a vacuum. Two common properties of glass and other transparent materials are directly related to their refractive index. First, light rays change direction when they cross the interface from air to the material, an effect that is used in lenses and glasses. Second, light reflects partially from surfaces that have a refractive index different from that of their surroundings.

Definition: The refractive index n of a medium is defined as the ratio of the phase velocity c of a wave phenomenon such as light or sound in a reference medium to the phase velocity vp in the medium itself:

n = \frac{c}{v_{\mathrm {p}}}

It is most commonly used in the context of light with vacuum as a reference medium, although historically other reference media (e.g. air at a standardized pressure and temperature) have been common. It is usually given the symbol n. In the case of light, it equals

n=\sqrt{\epsilon_r\mu_r},

where εr is the material's relative permittivity, and μr is its relative permeability. For most materials, μr is very close to 1 at optical frequencies, therefore n is approximately \sqrt{\epsilon_r}. Contrary to a widespread misconception, n may be less than 1, for example for x-rays.[1]. This has practical technical applications, such as effective mirrors for x-rays based on total external reflection.

The phase velocity is defined as the rate at which the crests of the waveform propagate; that is, the rate at which the phase of the waveform is moving. The group velocity is the rate that the envelope of the waveform is propagating; that is, the rate of variation of the amplitude of the waveform. Provided the waveform is not distorted significantly during propagation, it is the group velocity that represents the rate that information (and energy) may be transmitted by the wave, for example the velocity at which a pulse of light travels down an optical fiber.

Contents

[hide]

* 1 Speed of light

* 2 Negative Refractive Index

* 3 Dispersion and absorption

* 4 Relation to dielectric constant

* 5 Anisotropy

* 6 Nonlinearity

* 7 Inhomogeneity

* 8 Relation to density

* 9 Momentum Paradox

* 10 Applications

* 11 See also

* 12 References

* 13 External links

[edit] Speed of light

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the phase velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

The speed of all electromagnetic radiation in vacuum is the same, approximately 3×108 meters per second, and is denoted by c. Therefore, if v is the phase velocity of radiation of a specific frequency in a specific material, the refractive index is given by

n =\frac{c}{v}

or inversely

v =\frac{c}{n}

This number is typically greater than one: the higher the index of the material, the more the light is slowed down (see Cherenkov radiation). However, at certain frequencies (e.g. near absorption resonances, and for X-rays), n will actually be smaller than one. This does not contradict the theory of relativity, which holds that no information-carrying signal can ever propagate faster than c, because the phase velocity is not the same as the group velocity or the signal velocity.

Sometimes, a "group velocity refractive index", usually called the group index is defined:

n_g=\frac{c}{v_g}

where vg is the group velocity. This value should not be confused with n, which is always defined with respect to the phase velocity. The group index can be written in terms of the wavelength dependence of the refractive index as

n_g = n - \lambda\frac{dn}{d\lambda},

where λ is the wavelength in vacuum. At the microscale, an electromagnetic wave's phase velocity is slowed in a material because the electric field creates a disturbance in the charges of each atom (primarily the electrons) proportional to the permittivity of the medium. The charges will, in general, oscillate slightly out of phase with respect to the driving electric field. The charges thus radiate their own electromagnetic wave that is at the same frequency but with a phase delay. The macroscopic sum of all such contributions in the material is a wave with the same frequency but shorter wavelength than the original, leading to a slowing of the wave's phase velocity. Most of the radiation from oscillating material charges will modify the incoming wave, changing its velocity. However, some net energy will be radiated in other directions (see scattering).

If the refractive indices of two materials are known for a given frequency, then one can compute the angle by which radiation of that frequency will be refracted as it moves from the first into the second material from Snell's law.

If in a given region the values of refractive indices n or ng were found to differ from unity (whether homogeneously, or isotropically, or not), then this region was distinct from vacuum in the above sense for lacking Poincaré symmetry.

[edit] Negative Refractive Index

Recent research has also demonstrated the existence of negative refractive index which can occur if the real parts of both εr and μr are simultaneously negative, although such is a necessary but not sufficient condition. Not thought to occur naturally, this can be achieved with so-called metamaterials and offers the possibility of perfect lenses and other exotic phenomena such as a reversal of Snell's law. [1] [2]

[edit] Dispersion and absorption

The variation of refractive index vs. wavelength for various glasses.

The variation of refractive index vs. wavelength for various glasses.

In real materials, the polarization does not respond instantaneously to an applied field. This causes dielectric loss, which can be expressed by a permittivity that is both complex and frequency dependent. Real materials are not perfect insulators either, i.e. they have non-zero direct current conductivity. Taking both aspects into consideration, we can define a complex index of refraction:

\tilde{n}=n+i\kappa

Here, n is the refractive index indicating the phase velocity as above, while κ is called the extinction coefficient, which indicates the amount of absorption loss when the electromagnetic wave propagates through the material. (See the article Mathematical descriptions of opacity.) Both n and κ are dependent on the frequency (wavelength). Note that the sign of the complex part is a matter of convention, which is important due to possible confusion between loss and gain. The notation above, which is usually used by physicists, corresponds to waves with time evolution given by e − iωt.

The effect that n varies with frequency (except in vacuum, where all frequencies travel at the same speed, c) is known as dispersion, and it is what causes a prism to divide white light into its constituent spectral colors, explains rainbows, and is the cause of chromatic aberration in lenses. In regions of the spectrum where the material does not absorb, the real part of the refractive index tends to increase with frequency. Near absorption peaks, the curve of the refractive index is a complex form given by the Kramers–Kronig relations, and can decrease with frequency.

Since the refractive index of a material varies with the frequency (and thus wavelength) of light, it is usual to specify the corresponding vacuum wavelength at which the refractive index is measured. Typically, this is done at various well-defined spectral emission lines; for example, nD is the refractive index at the Fraunhofer "D" line, the centre of the yellow sodium double emission at 589.29 nm wavelength.

The Sellmeier equation is an empirical formula that works well in describing dispersion, and Sellmeier coefficients are often quoted instead of the refractive index in tables. For some representative refractive indices at different wavelengths, see list of indices of refraction.

As shown above, dielectric loss and non-zero DC conductivity in materials cause absorption. Good dielectric materials such as glass have extremely low DC conductivity, and at low frequencies the dielectric loss is also negligible, resulting in almost no absorption (κ ≈ 0). However, at higher frequencies (such as visible light), dielectric loss may increase absorption significantly, reducing the material's transparency to these frequencies.

The real and imaginary parts of the complex refractive index are related through use of the Kramers–Kronig relations. For example, one can determine a material's full complex refractive index as a function of wavelength from an absorption spectrum of the material.

[edit] Relation to dielectric constant

The dielectric constant (which is often dependent on wavelength) is simply the square of the (complex) refractive index. The refractive index is used for optics in Fresnel equations and Snell's law; while the dielectric constant is used in Maxwell's equations and electronics.

Where \tilde\epsilon, ε1, ε2, n, and κ are functions of wavelength:

\tilde\epsilon=\epsilon_1+i\epsilon_2= (n+i\kappa)^2

Conversion between refractive index and dielectric constant is done by:

ε1 = n2 − κ2

ε2 = 2nκ

n = \sqrt{\frac{\sqrt{\epsilon_1^2+\epsilon_2^2}+\epsilon_1}{2}} [2]

\kappa = \sqrt{ \frac{ \sqrt{ \epsilon_1^2+ \epsilon_2^2}- \epsilon_1}{2}}[3]

[edit] Anisotropy

A calcite crystal laid upon a paper with some letters showing birefringence

A calcite crystal laid upon a paper with some letters showing birefringence

The refractive index of certain media may be different depending on the polarization and direction of propagation of the light through the medium. This is known as birefringence or anisotropy and is described by the field of crystal optics. In the most general case, the dielectric constant is a rank-2 tensor (a 3 by 3 matrix), which cannot simply be described by refractive indices except for polarizations along principal axes.

In magneto-optic (gyro-magnetic) and optically active materials, the principal axes are complex (corresponding to elliptical polarizations), and the dielectric tensor is complex-Hermitian (for lossless media); such materials break time-reversal symmetry and are used e.g. to construct Faraday isolators.

[edit] Nonlinearity

The strong electric field of high intensity light (such as output of a laser) may cause a medium's refractive index to vary as the light passes through it, giving rise to nonlinear optics. If the index varies quadratically with the field (linearly with the intensity), it is called the optical Kerr effect and causes phenomena such as self-focusing and self-phase modulation. If the index varies linearly with the field (which is only possible in materials that do not possess inversion symmetry), it is known as the Pockels effect.

[edit] Inhomogeneity

A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens.

A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens.

If the refractive index of a medium is not constant, but varies gradually with position, the material is known as a gradient-index medium and is described by gradient index optics. Light travelling through such a medium can be bent or focussed, and this effect can be exploited to produce lenses, some optical fibers and other devices. Some common mirages are caused by a spatially-varying refractive index of air.

[edit] Relation to density

Relation between the refractive index and the density of silicate and borosilicate glasses ().

Relation between the refractive index and the density of silicate and borosilicate glasses ([4]).

In general, the refractive index of a glass increases with its density. However, there does not exist an overall linear relation between the refractive index and the density for all silicate and borosilicate glasses. A relatively high refractive index and low density can be obtained with glasses containing light metal oxides such as Li2O and MgO, while the opposite trend is observed with glasses containing PbO and BaO as seen in the diagram at the right.

[edit] Momentum Paradox

The momentum of a refracted ray, p, was calculated by Hermann Minkowski[5] in 1908, where E is energy of the photon, c is the speed of light in vacuum and n is the refractive index of the medium.

p=\frac{nE}{c}

In 1909 Max Abraham[6] proposed

p=\frac{E}{nc}

Rudolf Peierls raises this in his "More Surprises in Theoretical Physics" Princeton (1991). Ulf Leonhardt, Chair in Theoretical Physics at the University of St Andrews has discussed[7] this including experiments to resolve.

[edit] Applications

The refractive index of a material is the most important property of any optical system that uses refraction. It is used to calculate the focusing power of lenses, and the dispersive power of prisms.

Since refractive index is a fundamental physical property of a substance, it is often used to identify a particular substance, confirm its purity, or measure its concentration. Refractive index is used to measure solids (glasses and gemstones), liquids, and gases. Most commonly it is used to measure the concentration of a solute in an aqueous solution. A refractometer is the instrument used to measure refractive index. For a solution of sugar, the refractive index can be used to determine the sugar content (see Brix).Spectrum

From Wikipedia, the free encyclopedia

Jump to: navigation, search

This article deals with the general meaning of "spectrum" and the history of its use. For other meanings and specific uses, see Spectrum (disambiguation).

The spectrum in a rainbow

The spectrum in a rainbow

A spectrum (plural spectra or spectrums[1]) is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. The word saw its first scientific use within the field of optics to describe the rainbow of colors in visible light when separated using a prism; it has since been applied by analogy to many fields. Thus one might talk about the spectrum of political opinion, or the spectrum of activity of a drug, or the autism spectrum. In these uses, values within a spectrum are not necessarily precisely defined numbers as in optics; exact values within the spectrum are not precisely quantifiable. Such use implies a broad range of conditions or behaviors grouped together and studied under a single title for ease of discussion.

In most modern usages of spectrum there is a unifying theme between extremes at either end. Some older usages of the word did not have a unifying theme, but they led to modern ones through a sequence of events set out below. Modern usages in mathematics did evolve from a unifying theme, but this may be difficult to recognize.

Contents

[hide]

* 1 Origins

* 2 Modern meaning in the physical sciences

* 3 See also

* 4 References

* 5 External links

[edit] Origins

In Latin spectrum means "image" or "apparition", including the meaning "spectre". Spectral evidence is testimony about what was done by spectres of persons not present physically, or hearsay evidence about what ghosts or apparitions of Satan said. It was used to convict a number of persons of witchcraft at Salem, Massachusetts in the late 17th century.

[edit] Modern meaning in the physical sciences

The spectrum of a star of spectral type K4III

The spectrum of a star of spectral type K4III

In the 17th century the word spectrum was introduced into optics, referring to the range of colors observed when white light was dispersed through a prism. Soon the term referred to a plot of light intensity or power as a function of frequency or wavelength, also known as a spectral density.

The term spectrum was soon applied to other waves, such as sound waves, and now applies to any signal that can be decomposed into frequency components. A spectrum is a usually 2-dimensional plot, of a compound signal, depicting the components by another measure. Sometimes, the word spectrum refers to the compound signal itself, such as the "spectrum of visible light", a reference to those electromagnetic waves which are visible to the human eye. Looking at light through a prism separates visible light into its colors according to wavelength. It separates them according to its dispersion relation and a grating separates according to the grating equation and if massive particles are measured often their speed is measured. To get a spectrum, the measured function has to be transformed in their independent variable to frequencies and the dependent variable has to be reduced in regions, where the independent variable is stretched. For this imagine that the spectrum of pulse with a finite number of particles is measured on a film or a CCD. Assuming no particles are lost, any nonlinearity (compared to frequency) on the spectral separation concentrates particles at some points of the film. The same is true for taking a spectrum by scanning a monochromator with a fixed slit width. Violet at one end has the shortest wavelength and red at the other end has the longest wavelength of visible light. The colors in order are violet, blue, green, yellow, orange, red. As the wavelengths get bigger below the red visible light they become infrared, microwave, and radio. As the wavelengths get smaller above violet light, they become ultra-violet, x-ray, and gamma ray.

[edit] See alsoDispersion (optics)

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In a prism, material dispersion (a wavelength-dependent refractive index) causes different colors to refract at different angles, splitting white light into a rainbow.

In a prism, material dispersion (a wavelength-dependent refractive index) causes different colors to refract at different angles, splitting white light into a rainbow.

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency.[1] Media having such a property are termed dispersive media.

The most familiar example of dispersion is probably a rainbow, in which dispersion causes the spatial separation of a white light into components of different wavelengths (different colors). However, dispersion also has an impact in many other circumstances: for example, it causes pulses to spread in optical fibers, degrading signals over long distances; also, a cancellation between dispersion and nonlinear effects leads to soliton waves. Dispersion is most often described for light waves, but it may occur for any kind of wave that interacts with a medium or passes through an inhomogeneous geometry (e.g. a waveguide), such as sound waves. Dispersion is sometimes called chromatic dispersion to emphasize its wavelength-dependent nature.

There are generally two sources of dispersion: material dispersion and waveguide dispersion. Material dispersion comes from a frequency-dependent response of a material to waves. For example, material dispersion leads to undesired chromatic aberration in a lens or the separation of colors in a prism. Waveguide dispersion occurs when the speed of a wave in a waveguide (such as an optical fiber) depends on its frequency for geometric reasons, independent of any frequency dependence of the materials from which it is constructed. More generally, "waveguide" dispersion can occur for waves propagating through any inhomogeneous structure (e.g. a photonic crystal), whether or not the waves are confined to some region. In general, both types of dispersion may be present, although they are not strictly additive. Their combination leads to signal degradation in optical fibers for telecommunications, because the varying delay in arrival time between different components of a signal "smears out" the signal in time.

Contents

[hide]

* 1 Material dispersion in optics

* 2 Group and phase velocity

* 3 Dispersion in waveguides

* 4 Dispersion in gemology

* 5 Dispersion in imaging

* 6 Dispersion in pulsar timing

* 7 See also

* 8 References

* 9 External links

[edit] Material dispersion in optics

The variation of refractive index vs. wavelength for various glasses. The wavelengths of visible light are shaded in red.

The variation of refractive index vs. wavelength for various glasses. The wavelengths of visible light are shaded in red.

Influences of selected glass component additions on the mean dispersion of a specific base glass (nF valid for λ = 486 nm (blue), nC valid for λ = 656 nm (red))

Influences of selected glass component additions on the mean dispersion of a specific base glass (nF valid for λ = 486 nm (blue), nC valid for λ = 656 nm (red))[2]

Material dispersion can be a desirable or undesirable effect in optical applications. The dispersion of light by glass prisms is used to construct spectrometers and spectroradiometers. Holographic gratings are also used, as they allow more accurate discrimination of wavelengths. However, in lenses, dispersion causes chromatic aberration, an undesired effect that may degrade images in microscopes, telescopes and photographic objectives.

The phase velocity, v, of a wave in a given uniform medium is given by

v = \frac{c}{n}

where c is the speed of light in a vacuum and n is the refractive index of the medium.

In general, the refractive index is some function of the frequency f of the light, thus n = n(f), or alternately, with respect to the wave's wavelength n = n(λ). The wavelength dependency of a material's refractive index is usually quantified by an empirical formula, the Cauchy or Sellmeier equations.

The most commonly seen consequence of dispersion in optics is the separation of white light into a color spectrum by a prism. From Snell's law it can be seen that the angle of refraction of light in a prism depends on the refractive index of the prism material. Since that refractive index varies with wavelength, it follows that the angle that the light is refracted will also vary with wavelength, causing an angular separation of the colors known as angular dispersion.

For visible light, most transparent materials (e.g. glasses) have:

1 < n(\lambda_{\rm red}) < n(\lambda_{\rm yellow}) < n(\lambda_{\rm blue})\ ,

or alternatively:

\frac{{\rm d}n}{{\rm d}\lambda} < 0,

that is, refractive index n decreases with increasing wavelength λ. In this case, the medium is said to have normal dispersion. Whereas, if the index increases with increasing wavelength the medium has anomalous dispersion.

At the interface of such a material with air or vacuum (index of ~1), Snell's law predicts that light incident at an angle θ to the normal will be refracted at an angle arcsin( sin (θ) / n) . Thus, blue light, with a higher refractive index, will be bent more strongly than red light, resulting in the well-known rainbow pattern.

[edit] Group and phase velocity

Another consequence of dispersion manifests itself as a temporal effect. The formula above, v = c / n calculates the phase velocity of a wave; this is the velocity at which the phase of any one frequency component of the wave will propagate. This is not the same as the group velocity of the wave, which is the rate that changes in amplitude (known as the envelope of the wave) will propagate. The group velocity vg is related to the phase velocity by, for a homogeneous medium (here λ is the wavelength in vacuum, not in the medium):

v_g = c \left( n - \lambda \frac{dn}{d\lambda} \right)^{-1}.

The group velocity vg is often thought of as the velocity at which energy or information is conveyed along the wave. In most cases this is true, and the group velocity can be thought of as the signal velocity of the waveform. In some unusual circumstances, where the wavelength of the light is close to an absorption resonance of the medium, it is possible for the group velocity to exceed the speed of light (vg > c), leading to the conclusion that superluminal (faster than light) communication is possible. In practice, in such situations the distortion and absorption of the wave is such that the value of the group velocity essentially becomes meaningless, and does not represent the true signal velocity of the wave, which stays less than c.

The group velocity itself is usually a function of the wave's frequency. This results in group velocity dispersion (GVD), which causes a short pulse of light to spread in time as a result of different frequency components of the pulse travelling at different velocities. GVD is often quantified as the group delay dispersion parameter (again, this formula is for a uniform medium only):

D = - \frac{\lambda}{c} \, \frac{d^2 n}{d \lambda^2}.

If D is less than zero, the medium is said to have positive dispersion. If D is greater than zero, the medium has negative dispersion.If a light pulse is propagated through a normally dispersive medium, the result is the higher frequency components travel slower than the lower frequency components. The pulse therefore becomes positively chirped, or up-chirped, increasing in frequency with time. Conversely, if a pulse travels through an anomalously dispersive medium, high frequency components travel faster than the lower ones, and the pulse becomes negatively chirped, or down-chirped, decreasing in frequency with time.

The result of GVD, whether negative or positive, is ultimately temporal spreading of the pulse. This makes dispersion management extremely important in optical communications systems based on optical fiber, since if dispersion is too high, a group of pulses representing a bit-stream will spread in time and merge together, rendering the bit-stream unintelligible. This limits the length of fiber that a signal can be sent down without regeneration. One possible answer to this problem is to send signals down the optical fibre at a wavelength where the GVD is zero (e.g. around ~1.3-1.5 μm in silica fibres), so pulses at this wavelength suffer minimal spreading from dispersion—in practice, however, this approach causes more problems than it solves because zero GVD unacceptably amplifies other nonlinear effects (such as four wave mixing). Another possible option is to use soliton pulses in the regime of anomalous dispersion, a form of optical pulse which uses a nonlinear optical effect to self-maintain its shape—solitons have the practical problem, however, that they require a certain power level to be maintained in the pulse for the nonlinear effect to be of the correct strength. Instead, the solution that is currently used in practice is to perform dispersion compensation, typically by matching the fiber with another fiber of opposite-sign dispersion so that the dispersion effects cancel; such compensation is ultimately limited by nonlinear effects such as self-phase modulation, which interact with dispersion to make it very difficult to undo.

Dispersion control is also important in lasers that produce short pulses. The overall dispersion of the optical resonator is a major factor in determining the duration of the pulses emitted by the laser. A pair of prisms can be arranged to produce net negative dispersion, which can be used to balance the usually positive dispersion of the laser medium. Diffraction gratings can also be used to produce dispersive effects; these are often used in high-power laser amplifier systems. Recently, an alternative to prisms and gratings has been developed: chirped mirrors. These dielectric mirrors are coated so that different wavelengths have different penetration lengths, and therefore different group delays. The coating layers can be tailored to achieve a net negative dispersion.

[edit] Dispersion in waveguides

Optical fibers, which are used in telecommunications, are among the most abundant types of waveguides. Dispersion in these fibers is one of the limiting factors that determine how much data can be transported on a single fiber.

The transverse modes for waves confined laterally within a waveguide generally have different speeds (and field patterns) depending upon their frequency (that is, on the relative size of the wave, the wavelength) compared to the size of the waveguide.

In general, for a waveguide mode with an angular frequency ω(β) at a propagation constant β (so that the electromagnetic fields in the propagation direction z oscillate proportional to ei(βz − ωt)), the group-velocity dispersion parameter D is defined as:[3]

D = -\frac{2\pi c}{\lambda^2} \frac{d^2 \beta}{d\omega^2} = \frac{2\pi c}{v_g^2 \lambda^2} \frac{dv_g}{d\omega}

where λ = 2πc / ω is the vacuum wavelength and vg = dω / dβ is the group velocity. This formula generalizes the one in the previous section for homogeneous media, and includes both waveguide dispersion and material dispersion. The reason for defining the dispersion in this way is that |D| is the (asymptotic) temporal pulse spreading Δt per unit bandwidth Δλ per unit distance travelled, commonly reported in ps / nm km for optical fibers.

A similar effect due to a somewhat different phenomenon is modal dispersion, caused by a waveguide having multiple modes at a given frequency, each with a different speed. A special case of this is polarization mode dispersion (PMD), which comes from a superposition of two modes that travel at different speeds due to random imperfections that break the symmetry of the waveguide.

[edit] Dispersion in gemology

In the technical terminology of gemology, dispersion is the difference in the refractive index of a material at the B and G Fraunhofer wavelengths of 686.7 nm and 430.8 nm and is meant to express the degree to which a prism cut from the gemstone shows "fire", or color. Dispersion is a material property. Fire depends on the dispersion, the cut angles, the lighting environment, the refractive index, and the viewer.

[edit] Dispersion in imaging

In photographic and microscopic lenses, dispersion causes chromatic aberration, distorting the image, and various techniques have been developed to counteract it.

[edit] Dispersion in pulsar timing

Pulsars are spinning neutron stars that emit pulses at very regular intervals ranging from milliseconds to seconds. It is believed that the pulses are emitted simultaneously over a wide range of frequencies. However, as observed on Earth, the components of each pulse emitted at higher radio frequencies arrive before those emitted at lower frequencies. This dispersion occurs because of the ionised component of the interstellar medium, which makes the group velocity frequency dependent. The extra delay added at frequency ν is

D = 4.15 ms (\frac{\nu}{GHz})^{-2} \times (\frac{DM}{cm^{-3} pc})

where the dispersion measure DM is

DM = \int_0^d{n_e\;dl}

is the integrated free electron column density ne out to the pulsar at a distance d[4].

Of course, this delay cannot be measured directly, since the emission time is unknown. What can be measured is the difference in arrival times at two different frequencies. The delay ΔT between a high frequency νhi and a low frequency νlo component of a pulse will be

\Delta T = 4.15 ms [(\frac{\nu_{lo}}{GHz})^{-2} - (\frac{\nu_{hi}}{GHz})^{-2} ] \times (\frac{DM}{cm^{-3} pc})

and so DM is normally computed from measurements at two different frequencies. This allows computation of the absolute delay at any frequency, which is used when combining many different pulsar observations into an integrated timing solution.

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