Photon Polarization

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Photon polarizationFrom Wikipedia, the free encyclopedia

Photon polarization is the quantum mechanical description of the classical polarized sinusoidal planeelectromagnetic wave. Individual photons are completely polarized. Their polarization state can be linear or circulaor it can be elliptical, which is anywhere in between linear and circular polarization.

The description contains many of the physical concepts and much of the mathematical machinery of more involvedquantum descriptions, such as the quantum mechanics of an electron in a potential well, and forms a fundamental

basis for an understanding of more complicated quantum phenomena.

Much of the mathematical machinery of quantum mechanics, such as state vectors, probability amplitudes, unitaryoperators, and Hermitian operators, emerge naturally from the classical Maxwell's equations in the description.

The quantum polarization state vector for the photon, for instance, is identical with the Jones vector, usually used tdescribe the polarization of a classical wave.

Unitary operators emerge from the classical requirement of the conservation of energy of a classical wavepropagating through media that alter the polarization state of the wave. Hermitian operators then follow forinfinitesimal transformations of a classical polarization state.

Many of the implications of the mathematical machinery are easily verified experimentally. In fact, many of theexperiments can be performed with two pairs (or one broken pair) of polaroid sunglasses.

The connection with quantum mechanics is made through the identification of a minimum packet size, called aphoton, for energy in the electromagnetic field. The identification is based on the theories of Planck and theinterpretation of those theories by Einstein. The correspondence principle then allows the identification ofmomentum and angular momentum (called spin), as well as energy, with the photon.

Contents

1 Polarization of classical electromagnetic waves1.1 Polarization states

1.1.1 Linear polarization

1.1.2 Circular polarization1.1.3 Elliptical polarization

1.1.4 Geometric visualization of an arbitrary polarization state2 Energy, momentum, and angular momentum of a classical electromagnetic wave

2.1 Energy density of classical electromagnetic waves2.1.1 Energy in a plane wave

2.1.2 Fraction of energy in each component2.2 Momentum density of classical electromagnetic waves2.3 Angular momentum density of classical electromagnetic waves

3 Optical filters and crystals3.1 Passage of a classical wave through a polaroid filter3.2 Example of energy conservation: Passage of a classical wave through a birefringent crystal


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Effect of a polarizer on reflection from mud flats. In the first picture,

the polarizer is rotated to minimize the effect; in the second it is

rotated 90 to maximize it: almost all reflected sunlight is eliminated.

3.2.1 Initial and final states3.2.2 Dual of the final state

3.2.3 Unitary operators and energy conservation3.2.4 Hermitian operators and energy conservation

4 Photons: The connection to quantum mechanics4.1 Energy, momentum, and angular momentum of photons

4.1.1 Energy

4.1.2 Momentum4.1.3 Angular momentum and spin

4.1.3.1 Spin operator4.1.3.2 Spin states4.1.3.3 Spin and angular momentum operators in differential form

4.2 The nature of probability in quantum mechanics

4.2.1 Probability for a single photon4.2.2 Probability amplitudes

4.3 Uncertainty principle4.3.1 Mathematical preparation

4.3.2 Application to angular momentum4.4 States, probability amplitudes, unitary and Hermitian operators, and eigenvectors

5 See also

6 References

Polarization of classical electromagnetic waves

Main article: Sinusoidal plane-wave solutions of the electromagnetic wave equation

Polarization states

Linear polarization

Main article: Linear polarization

The wave is linearly polarized (or planepolarized) when the phase anglesare equal,

This represents a wave with phasepolarized at an angle with respect to thex axis. In that case the Jones vector can bewritten


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The state vectors for linear polarization in x or y are special cases of this state vector.

If unit vectors are defined such that

and

then the linearly polarized polarization state can written in the "x-y basis" as

Circular polarization

Main article: Circular polarization

If the phase angles and differs by exactly and the x amplitude equals the y amplitude the wave is

circularly polarized. The Jones vector then becomes

where the plus sign indicates right circular polarization and the minus sign indicates left circular polarization. In thecase of circular polarization, the electric field vector of constant magnitude rotates in the x-y plane.

If unit vectors are defined such that

and

then an arbitrary polarization state can written in the "R-L basis" as


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where

and

We can see that

Elliptical polarization

Main article: Elliptical polarization

The general case in which the electric field rotates in the x-y plane and has variable magnitude is called ellipticalpolarization. The state vector is given by

Geometric visualization of an arbitrary polarization state

To get an understanding of what a polarization state looks like, one can observe the orbit that is made if thepolarization state is multiplied by a phase factor of and then having the real parts of its components interprete

as x and y coordinates respectively. That is:

If only the traced out shape and the direction of the rotation of(x(t),y(t)) is considered when interpreting thepolarization state, i.e. only

(wherex(t) andy(t) are defined as above) and whether it is overall more right circularly or left circularly polarized(i.e. whether|

R| > |

L| or vice versa), it can be seen that the physical interpretation will be the same even if the

state is multiplied by an arbitrary phase factor, since

and the direction of rotation will remain the same. In other words, there is no physical difference between two


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polarization states and , between which only a phase factor differs.

It can be seen that for a linearly polarized state,Mwill be a line in the xy plane, with length 2 and its middle in theorigin, and whose slope equals to tan(). For a circularly polarized state,Mwill be a circle with radius 1/2 andwith the middle in the origin.

Energy, momentum, and angular momentum of a classical

electromagnetic wave

Energy density of classical electromagnetic waves

Energy in a plane wave

Main article: Energy density

The energy per unit volume in classical electromagnetic fields is (cgs units)

For a plane wave, this becomes

where the energy has been averaged over a wavelength of the wave.

Fraction of energy in each component

The fraction of energy in the x component of the plane wave is

with a similar expression for the y component resulting in .

The fraction in both components is

Momentum density of classical electromagnetic waves

The momentum density is given by the Poynting vector


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Linear polarization

For a sinusoidal plane wave traveling in the z direction, the momentum is in the z direction and is related to theenergy density:

The momentum density has been averaged over a wavelength.

Angular momentum density of classical electromagnetic waves

Electromagnetic waves can have both orbital and spin angular momentum.[1] The total angular momentum density

For a sinusoidal plane wave propagating along axis the orbital angular momentum density vanishes. The spinangular momentum density is in the direction and is given by

where again the density is averaged over a wavelength.

Optical filters and crystals

Passage of a classical wave through a polaroid filter

A linear filter transmits one component of a plane wave and absorbs

the perpendicular component. In that case, if the filter is polarized inthe x direction, the fraction of energy passing through the filter is

Example of energy conservation: Passage of a

classical wave through a birefringent crystal

An ideal birefringent crystal transforms the polarization state of anelectromagnetic wave without loss of wave energy. Birefringent

crystals therefore provide an ideal test bed for examining theconservative transformation of polarization states. Even though thistreatment is still purely classical, standard quantum tools such as unitary and Hermitian operators that evolve thestate in time naturally emerge.

Initial and final states

A birefringent crystal is a material that has an optic axis with the property that the light has a different index ofrefraction for light polarized parallel to the axis than it has for light polarized perpendicular to the axis. Light

polarized parallel to the axis are called "extraordinary rays" or "extraordinary photons", while light polarized


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

double refraction

perpendicular to the axis are called "ordinary rays" or "ordinary photons". If a linearly polarized wave impingeson the crystal, the extraordinary component of the wave will emerge from the crystal with a different phase than theordinary component. In mathematical language, if the incident wave is linearly polarized at an angle with respectto the optic axis, the incident state vector can be written

and the state vector for the emerging wave can be written

While the initial state was linearly polarized, the final state is elliptically polarized. The birefringent crystal alters thecharacter of the polarization.

Dual of the final state

The initial polarization state is transformedinto the final state with the operator U. Thedual of the final state is given by

where is the adjoint of U, the complex

conjugate transpose of the matrix.

Unitary operators and energy

conservation

The fraction of energy that emerges fromthe crystal is

In this ideal case, all the energy impinging on the crystal emerges from the crystal. An operator U with the property

that

where I is the identity operator and U is called a unitary operator. The unitary property is necessary to ensureenergy conservation in state transformations.

Hermitian operators and energy conservation

If the crystal is very thin, the final state will be only slightly different from the initial state. The unitary operator will b


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Doubly refracting Calcite from

Iceberg claim, Dixon, New Mexico.

This 35 pound (16 kg) crystal, on

display at the National Museum of

Natural History, is one of the largest

single crystals in the United States.

close to the identity operator. We can define the operator H by

and the adjoint by

Energy conservation then requires

This requires that

Operators like this that are equal to their adjoints are called Hermitian or self-adjoint.

The infinitesimal transition of the polarization state is

Thus, energy conservation requires that infinitesimal transformations of a polarization state occur through the actionof a Hermitian operator.

Photons: The connection to quantum mechanicsMain article: Photon

Energy, momentum, and angular momentum of photons

Energy

The treatment to this point has been classical. It is a testament, however, to the generality of Maxwell's equationsfor electrodynamics that the treatment can be made quantum mechanical with only a reinterpretation of classical


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quantities. The reinterpretation is based on the theories of Max Planck and the interpretation by Albert Einstein ofthose theories and of other experiments.

Einsteins's conclusion from early experiments on the photoelectric effect is that electromagnetic radiation iscomposed of irreducible packets of energy, known as photons. The energy of each packet is related to the angularfrequency of the wave by the relation

where is an experimentally determined quantity known as Planck's constant. If there are photons in a box ofvolume , the energy in the electromagnetic field is

and the energy density is

The energy of a photon can be related to classical fields through the correspondence principle which states that fora large number of photons, the quantum and classical treatments must agree. Thus, for very large , the quantumenergy density must be the same as the classical energy density

The number of photons in the box is then

Momentum

The correspondence principle also determines the momentum and angular momentum of the photon. Formomentum

which implies that the momentum of a photon is

Angular momentum and spin

Similarly for the spin angular momentum


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which implies that the spin angular momentum of the photon is

the quantum interpretation of this expression is that the photon has a probability of of having a spin angula

momentum of and a probability of of having a spin angular momentum of . We can therefore think

of the spin angular momentum of the photon being quantized as well as the energy. The angular momentum of

classical light has been verified.[2] Photons have only been observed to have spin angular momenta of

.[citation needed]

Spin operator

The spin of the photon is defined as the coefficient of in the spin angular momentum calculation. A photon has

spin 1 if it is in the state and -1 if it is in the state. The spin operator is defined as the outer product

The eigenvectors of the spin operator are and with eigenvalues 1 and -1, respectively.

The expected value of a spin measurement on a photon is then

An operator S has been associated with an observable quantity, the spin angular momentum. The eigenvalues of thoperator are the allowed observable values. This has been demonstrated for spin angular momentum, but it is ingeneral true for any observable quantity.

Spin states

We can write the circularly polarized states as

where s=1 for

and s= -1 for

An arbitrary state can be written


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where

Spin and angular momentum operators in differential form

When the state is written in spin notation, the spin operator can be written

The eigenvectors of the differential spin operator are

To see this note

The spin angular momentum operator is

The nature of probability in quantum mechanics

Probability for a single photon

There are two ways in which probability can be applied to the behavior of photons; probability can be used tocalculate the probable number of photons in a particular state, or probability can be used to calculate the likelihoodof a single photon to be in a particular state. The former interpretation violates energy conservation. The latterinterpretation is the viable, if nonintuitive, option. Dirac explains this in the context of the double-slit experiment:

Some time before the discovery of quantum mechanics people realized that the connection betweenlight waves and photons must be of a statistical character. What they did not clearly realize, however,was that the wave function gives information about the probability ofone photon being in a particular

place and not the probable number of photons in that place. The importance of the distinction can bemade clear in the following way. Suppose we have a beam of light consisting of a large number of

photons split up into two components of equal intensity. On the assumption that the beam is connectedwith the probable number of photons in it, we should have half the total number going into each


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Cauchy-Schwarz inequality in Euclidean space.

component. If the two components are now made to interfere, we should require a photon in onecomponent to be able to interfere with one in the other. Sometimes these two photons would have toannihilate one another and other times they would have to produce four photons. This wouldcontradict the conservation of energy. The new theory, which connects the wave function with

probabilities for one photon gets over the difficulty by making each photon go partly into each of thetwo components. Each photon then interferes only with itself. Interference between two different

photons never occurs.

Paul Dirac, The Principles of Quantum Mechanics, Fourth Edition, Chapter 1

Probability amplitudes

The probability for a photon to be in a particular polarization state depends on the fields as calculated by theclassical Maxwell's equations. The polarization state of the photon is proportional to the field. The probability itselis quadratic in the fields and consequently is also quadratic in the quantum state of polarization. In quantummechanics, therefore, the state or probability amplitude contains the basic probability information. In general, therules for combining probability amplitudes look very much like the classical rules for composition of probabilities:[The following quote is from Baym, Chapter 1]

1. The probability amplitude for two successive probabilities is the product of amplitudes for the individualpossibilities. For example, the amplitude for the x polarized photon to be right circularly polarizedand for thright circularly polarized photon to pass through the y-polaroid is the product of the

individual amplitudes.2. The amplitude for a process that can take place in one of several indistinguishable ways is the sum of

amplitudes for each of the individual ways. For example, the total amplitude for the x polarized photon topass through the y-polaroid is the sum of the amplitudes for it to pass as a right circularly polarized photon,

plus the amplitude for it to pass as a left circularly polarized photon,

3. The total probability for the process to occur is the absolute value squared of the total amplitude calculatedby 1 and 2.

Uncertainty principle

Main article: Uncertainty principle

Mathematical preparation

For any legal operators the following

inequality, a consequence of the Cauchy-Schwarz inequality, is true.


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This implies

IfB A andA B are defined then

where

is the operator mean of observableXin the system state and

Here

is called the commutator of A and B.

This is a purely mathematical result. No reference has been made to any physical quantity or principle. It simplystates that the uncertainty of an operator acting on a state times the uncertainty of another operator acting on thestate is not necessarily zero.

Application to angular momentum

The connection to physics can be made if we identify the operators with physical operators such as the angularmomentum and the polarization angle. We have then

which simply states that angular momentum and the polarization angle cannot be measured simultaneously withinfinite accuracy.

States, probability amplitudes, unitary and Hermitian operators, and eigenvectors

Much of the mathematical apparatus of quantum mechanics appears in the classical description of a polarizedsinusoidal electromagnetic wave. The Jones vector for a classical wave, for instance, is identical with the quantum

polarization state vector for a photon. The right and left circular components of the Jones vector can be interpreted


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.

with a unitary operation. This implies that infinitesimal transformations are transformed with a Hermitian operator.These conclusions are a natural consequence of the structure of Maxwell's equations for classical waves.

Quantum mechanics enters the picture when observed quantities are measured and found to be discrete rather thancontinuous. The allowed observable values are determined by the eigenvalues of the operators associated with theobservable. In the case angular momentum, for instance, the allowed observable values are the eigenvalues of thespin operator.

These concepts have emerged naturally from Maxwell's equations and Planck's and Einstein's theories. They havebeen found to be true for many other physical systems. In fact, the typical program is to assume the concepts of thsection and then to infer the unknown dynamics of a physical system. This was done, for instance, with thedynamics of electrons. In that case, working back from the principles in this section, the quantum dynamics of

particles were inferred, leading to Schrdinger's equation, a departure from Newtonian mechanics. The solution ofthis equation for atoms led to the explanation of the Balmer series for atomic spectra and consequently formed a

basis for all of atomic physics and chemistry.

This is not the only occasion in which Maxwell's equations have forced a restructuring of Newtonian mechanics.Maxwell's equations are relativistically consistent. Special relativity resulted from attempts to make classicalmechanics consistent with Maxwell's equations (see, for example, Moving magnet and conductor problem).

See also

SternGerlach experiment

Waveparticle dualityDouble-slit experiment

Theoretical and experimental justification for the Schrdinger equation

References

1. ^ Allen, L.; Beijersbergen, M.W.; Spreeuw, R.J.C.; Woerdman, J .P. (June 1992). "Orbital angular momentum oflight and the transformation of Laguerre-Gaussian laser modes" (http://pra.aps.org/abstract/PRA/v45/i11/p8185_1

Physical Review A45 (11): 81869. Bibcode:1992PhRvA..45.8185A(http://adsabs.harvard.edu/abs/1992PhRvA..45.8185A). doi:10.1103/PhysRevA.45.8185(http://dx.doi.org/10.1103%2FPhysRevA.45.8185). PMID 9906912 (//www.ncbi.nlm.nih.gov/pubmed/9906912).

2. ^ Beth, R.A. (1935). "Direct detection of the angular momentum of light"

(http://prola.aps.org/abstract/PR/v48/i5/p471_1). Phys. Rev.48 (5): 471. Bibcode:1935PhRv...48..471B(http://adsabs.harvard.edu/abs/1935PhRv...48..471B). doi:10.1103/PhysRev.48.471

(http://dx.doi.org/10.1103%2FPhysRev.48.471).

Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X.Baym, Gordon (1969).Lectures on Quantum Mechanics. W. A. Benjamin. ISBN 0-8053-0667-6.

Dirac, P. A. M. (1958). The Principles of Quantum Mechanics, Fourth Edition. Oxford. ISBN 0-19-851208-2.

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