Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.
The description of photon polarization contains many of the physical concepts and much of the mathematical machinery of more involved quantum descriptions, such as the quantum mechanics of an electron in a potential well. Polarization is an example of a qubit degree of freedom, which 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, unitary operators, 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 to describe the polarization of a classical wave. Unitary operators emerge from the classical requirement of the conservation of energy of a classical wave propagating through lossless media that alter the polarization state of the wave. Hermitian operators then follow for infinitesimal transformations of a classical polarization state.
Many of the implications of the mathematical machinery are easily verified experimentally. In fact, many of the experiments can be performed with polaroid sunglass lenses.
The connection with quantum mechanics is made through the identification of a minimum packet size, called a photon, for energy in the electromagnetic field. The identification is based on the theories of Planck and the interpretation of those theories by Einstein. The correspondence principle then allows the identification of momentum and angular momentum (called spin), as well as energy, with the photon.
Polarization of classical electromagnetic waves
Polarization states
Linear polarization
The wave is linearly polarized (or plane polarized) when the phase angles are equal,
This represents a wave with phase polarized at an angle with respect to the x axis. In this case the Jones vector
can be written with a single phase:
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 be written in the "x–y basis" as
Circular polarization
If the phase angles and differ 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 left circular polarization and the minus sign indicates right circular polarization. In the case 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 be written in the "R–L basis" as
where
and
We can see that
Elliptical polarization
The general case in which the electric field rotates in the x–y plane and has variable magnitude is called elliptical polarization. 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 the polarization state is multiplied by a phase factor of and then having the real parts of its components interpreted as x and y coordinates respectively. That is:
If only the traced out shape and the direction of the rotation of is considered when interpreting the polarization state, i.e. only
(where and are defined as above) and whether it is overall more right circularly or left circularly polarized (i.e. whether 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 polarization states and , between which only a phase factor differs.
It can be seen that for a linearly polarized state, M will be a line in the xy plane, with length 2 and its middle in the origin, and whose slope equals to . For a circularly polarized state, M will be a circle with radius and with 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
The energy per unit volume in classical electromagnetic fields is (cgs units) and also Planck 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
For a sinusoidal plane wave traveling in the z direction, the momentum is in the z direction and is related to the energy 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. The total angular momentum density is
For a sinusoidal plane wave propagating along axis the orbital angular momentum density vanishes. The spin angular 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 in the 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 an electromagnetic wave without loss of wave energy. Birefringent crystals therefore provide an ideal test bed for examining the conservative transformation of polarization states. Even though this treatment is still purely classical, standard quantum tools such as unitary and Hermitian operators that evolve the state 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 of refraction 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 perpendicular to the axis are called "ordinary rays" or "ordinary photons". If a linearly polarized wave impinges on the crystal, the extraordinary component of the wave will emerge from the crystal with a different phase than the ordinary component. In mathematical language, if the incident wave is linearly polarized at an angle with respect to 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 the character of the polarization.
Dual of the final state
The initial polarization state is transformed into the final state with the operator U. The dual 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 from the 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 ensure energy 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 be 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 action of a Hermitian operator.
Photons: connection to quantum mechanics
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 equations for electrodynamics that the treatment can be made quantum mechanical with only a reinterpretation of classical quantities. The reinterpretation is based on the theories of Max Planck and the interpretation by Albert Einstein of those theories and of other experiments.
Einstein's conclusion from early experiments on the photoelectric effect is that electromagnetic radiation is composed of irreducible packets of energy, known as photons. The energy of each packet is related to the angular frequency of the wave by the relationwhere is an experimentally determined quantity known as the reduced Planck constant. If there are photons in a box of volume , the energy in the electromagnetic field isand the energy density is
The photon energy can be related to classical fields through the correspondence principle that states that for a large number of photons, the quantum and classical treatments must agree. Thus, for very large , the quantum energy 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. For momentumwhere is the wave number. This implies that the momentum of a photon is
Angular momentum and spin
Similarly for the spin angular momentumwhere is field strength. This implies that the spin angular momentum of the photon isthe quantum interpretation of this expression is that the photon has a probability of of having a spin angular 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. A photon that is linearly polarized (plane polarized) is in a superposition of equal amounts of the left-handed and right-handed states. Upon absorption by an electronic state, the angular momentum is "measured" and this superposition collapses into either right-hand or left-hand, corresponding to a raising or lowering of the angular momentum of the absorbing electronic state, respectively.
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. These values are based on the point of view of the source as the convention to define circular polarization handedness.
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 the operator are the allowed observable values. This has been demonstrated for spin angular momentum, but it is in general true for any observable quantity.
Spin states