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Photon polarization

Photon polarization is a fundamental quantum mechanical property of photons, the quanta of electromagnetic radiation, describing the orientation of the associated electric field vector perpendicular to the direction of propagation.^[1] In classical optics, polarization arises from the coherent oscillation of the electric field in a specific direction, but at the quantum level, it manifests as the intrinsic angular momentum or helicity of the photon, which can take values of ±ħ along the propagation axis.^[2] This property enables photons to exist in various polarization states, including linear polarizations (such as horizontal or vertical, defined relative to a chosen basis) and circular polarizations (right-handed or left-handed, corresponding to clockwise or counterclockwise rotation of the field vector).^[3] Unlike classical waves, where polarization is deterministic, photon polarization exhibits probabilistic behavior upon measurement, as described by quantum superposition and the projection postulate.^[1] For instance, a photon in a linearly polarized state at an angle θ to a polarizer's axis has a transmission probability of cos²θ through that polarizer, illustrating Malus's law in a quantum context.^[3] Unpolarized light, in quantum terms, consists of an incoherent mixture of orthogonal polarization states, resulting in a 50% average transmission probability through any linear polarizer.^[3] Polarization states form a two-dimensional Hilbert space for a given propagation direction, spanned by basis states like |H⟩ (horizontal) and |V⟩ (vertical), allowing general states to be expressed as superpositions such as |ψ⟩ = cosθ |H⟩ + sinθ |V⟩ for linear polarization at angle θ.^[3] Circular states, |R⟩ and |L⟩, are eigenstates of helicity and can be written as superpositions of linear states: |R⟩ = \frac{1}{\sqrt{2}} (|H⟩ + i |V⟩) and |L⟩ = \frac{1}{\sqrt{2}} (|H⟩ - i |V⟩).^[2] Measurement of polarization collapses the superposition into one of the basis states, highlighting the role of observation in quantum mechanics.^[1] In quantum optics, photon polarization underpins phenomena such as entanglement, where paired photons exhibit correlated polarizations that violate classical inequalities, enabling applications in quantum information processing.^[4] It is also crucial for quantum cryptography protocols like BB84, which use polarization states to encode qubits,^[5] and for experiments demonstrating superposition, such as photons passing through crossed polarizers with an intermediate 45° polarizer, where 25% transmission occurs due to quantum interference.^[3] Birefringent materials, like calcite, further exploit polarization by splitting light into ordinary and extraordinary rays based on the angle relative to the optic axis, with refractive indices n_o ≈ 1.658 and n_e ≈ 1.486.^[2]

Classical Polarization of Electromagnetic Waves

Linear, Circular, and Elliptical Polarization States

Polarization of electromagnetic waves refers to the orientation and behavior of the electric field vector as the wave propagates. In classical optics, light waves are transverse, meaning their electric and magnetic fields oscillate perpendicular to the direction of propagation.^[6] This transverse nature restricts polarization to the plane normal to the wave's path, allowing for distinct states based on the field's oscillation pattern.^[7] Linear polarization occurs when the electric field vector oscillates along a fixed straight line within the transverse plane. For instance, horizontal linear polarization has the field vibrating parallel to the ground, while vertical linear polarization aligns it perpendicular to the ground.^[6] This state arises from a single component of the field dominating or when orthogonal components are in phase.^[7] Circular polarization describes a situation where the electric field vector rotates at a constant magnitude in the transverse plane as the wave advances, tracing a circle. Right-handed (or right-circular) polarization involves clockwise rotation when viewed facing the oncoming wave, whereas left-handed (or left-circular) polarization rotates counterclockwise.^[6] The field completes one full rotation per wavelength along the propagation direction.^[7] Elliptical polarization represents the most general form, where the electric field traces an ellipse in the transverse plane, combining elements of linear and circular motion. The ellipse is characterized by its major and minor axes, which indicate the varying amplitudes of the orthogonal field components, and a tilt angle defining the ellipse's orientation relative to reference axes.^[6] This state results from the superposition of two orthogonal linear components with a phase difference between 0° and 180° that is neither 0°/180° (linear) nor exactly 90° with equal amplitudes (circular).^[7] Linear and circular polarizations are special cases of elliptical polarization, with the ellipse degenerating to a line or circle, respectively.^[6] The concept of polarization as evidence for the transverse wave nature of light was established in the 1820s by Augustin-Jean Fresnel, who through experiments on interference and reflection demonstrated that light's vibrations occur in planes perpendicular to its propagation direction.^[8] Fresnel's work, including the 1822 discovery of circular polarization via decomposition of linearly polarized light, solidified the transverse model over longitudinal alternatives.^[9]

Mathematical Representation Using Jones Vectors

The polarization state of a fully coherent, monochromatic plane electromagnetic wave propagating along the z-axis can be described using Jones vectors, a formalism introduced by R. C. Jones in 1941. The Jones vector is a two-component complex column vector representing the electric field amplitudes in the horizontal (H) and vertical (V) linear polarization basis:

\mathbf{E} = \begin{pmatrix} E_H \\ E_V \end{pmatrix},

where E_H and E_V are complex numbers encoding both the magnitudes and relative phase of the field components.^[6] For fully polarized light, the Jones vector is typically normalized such that |E_H|^2 + |E_V|^2 = 1, ensuring the vector lies on the unit circle in the complex plane; the relative phase difference \delta = \arg(E_V) - \arg(E_H) then determines whether the polarization is linear (\delta = 0 or \pi), circular (\delta = \pm \pi/2), or elliptical (general \delta).^[10] Specific polarization states correspond to standard normalized Jones vectors in the H-V basis. Horizontal linear polarization is represented by \begin{pmatrix} 1 \\ 0 \end{pmatrix}, vertical linear by \begin{pmatrix} 0 \\ 1 \end{pmatrix}, and 45° linear (diagonal) by \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix}.^[6] Right-circular polarization, defined for light propagating toward the observer with the electric field rotating clockwise, is given by \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -i \end{pmatrix}, while left-circular is \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ i \end{pmatrix}.^[10] Conversions between states, such as from linear to circular polarization, can be achieved by introducing a relative phase shift of \pi/2; for example, applying a quarter-wave retarder to horizontal linear light (\begin{pmatrix} 1 \\ 0 \end{pmatrix}) with fast axis at 45° yields the right-circular state.^[6] Under a rotation of the coordinate system by an angle \theta (counterclockwise when looking toward the source), the Jones vector transforms via the unitary rotation matrix:

R(\theta) = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix},

such that the new vector is R(\theta) \mathbf{E}; this preserves the polarization ellipse's shape and orientation relative to the new axes.^[10] For instance, rotating horizontal linear polarization by 45° yields the diagonal state: R(45^\circ) \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix}.^[6] The Jones vector formalism is limited to fully polarized, coherent plane waves and does not apply to partially polarized or incoherent light, for which the Mueller-Stokes calculus is required instead.^[10]

Geometric Visualization of Polarization

The Poincaré sphere provides a geometric representation of the polarization states of fully polarized light on the surface of a unit sphere in three-dimensional space.^[11] Developed by Henri Poincaré in the late 19th century as a tool for analyzing the evolution of light's polarization through interactions with matter, the sphere maps elliptical polarization states using spherical coordinates related to the orientation and ellipticity of the polarization ellipse.^[12] The north pole corresponds to right-circular polarization, the south pole to left-circular polarization, the equator to all linear polarization states (with horizontal and vertical polarizations at opposite points), and latitudes away from the equator represent elliptical states, where the angle from the equator quantifies the degree of ellipticity.^[11] Jones vectors, which algebraically describe polarization in the linear basis, map to points on the Poincaré sphere via stereographic projection from the complex plane of normalized Jones vectors to the sphere's surface.^[13] This projection identifies the Jones vector components with coordinates on the sphere, and the Cartesian axes (S₁, S₂, S₃) of the sphere directly correspond to the normalized Stokes parameters, satisfying S₁² + S₂² + S₃² = 1 for fully polarized light:

\begin{align*} S_1 &= \cos(2\chi) \cos(2\psi), \\ S_2 &= \cos(2\chi) \sin(2\psi), \\ S_3 &= \sin(2\chi), \end{align*}

where ψ is the orientation angle and χ is the ellipticity angle of the polarization ellipse.^[11]^[14] Polarization transformations, such as those induced by wave plates (retarders) or optical rotators, appear as rotations of points on the sphere, with great circles tracing the paths of evolving states under linear retardance or rotation.^[15] For instance, a quarter-wave retarder rotates states around the S₁ axis by 90°, converting linear to circular polarization. This geometric view simplifies the analysis of sequential optical elements, as compositions of transformations correspond to successive rotations on the sphere.^[16] The Poincaré sphere's advantages lie in its intuitive visualization of complex elliptical states and its extension to partially polarized light, where points inside the sphere (with S₁² + S₂² + S₃² < 1) represent mixtures of polarization states.^[15] It also serves as a natural introduction to the Mueller calculus, which generalizes Stokes parameters to describe transformations for both polarized and unpolarized light using 4×4 matrices that act as rotations and scalings in this space, though without delving into matrix details here.^[17]

Physical Quantities in Polarized Electromagnetic Waves

Energy Density and Distribution

The energy density of an electromagnetic wave in free space is given by the sum of the electric and magnetic field contributions, u = \frac{1}{2} \epsilon_0 | \mathbf{E} |^2 + \frac{1}{2 \mu_0} | \mathbf{B} |^2, where \epsilon_0 and \mu_0 are the permittivity and permeability of free space, respectively.^[18] For plane waves propagating in vacuum, the magnitudes of the electric and magnetic fields are related by | \mathbf{B} | = | \mathbf{E} | / c, where c = 1 / \sqrt{\epsilon_0 \mu_0} is the speed of light, leading to equal partitioning of the energy between the electric and magnetic fields: u = \epsilon_0 | \mathbf{E} |^2 = | \mathbf{B} |^2 / \mu_0.^[18] This equal partition holds regardless of the wave's polarization state, as the total energy density depends on the overall field strengths. Polarization influences the distribution of energy within the electric field components transverse to the propagation direction. For a plane wave propagating along the z-axis, the electric field can be decomposed into x- and y-components, \mathbf{E} = (E_x, E_y, 0), such that | \mathbf{E} |^2 = | E_x |^2 + | E_y |^2. The fraction of energy associated with each component is proportional to the square of its amplitude; for instance, in a linearly polarized wave at 45° to the x-axis, | E_x | = | E_y | = | \mathbf{E} | / \sqrt{2}, resulting in 50% of the electric energy in each component.^[19] In circular polarization, the amplitudes are equal but phases differ by 90°, maintaining the same total energy density while distributing it equally over time between the components. The flow of this energy is described by the Poynting vector, \mathbf{S} = \mathbf{E} \times \mathbf{H}, which represents the instantaneous power flux density.^[20] For monochromatic plane waves, the time-averaged intensity is \langle \mathbf{S} \rangle = \frac{1}{2} \Re ( \mathbf{E} \times \mathbf{H}^* ), directed along the propagation axis. In free space, \mathbf{H} = \mathbf{E} / Z_0, where Z_0 = \sqrt{\mu_0 / \epsilon_0} \approx 377 \, \Omega is the impedance of free space, yielding \langle S \rangle = \frac{1}{2} \epsilon_0 c | \mathbf{E} |^2.^[21] Notably, for a fixed total electric field magnitude | \mathbf{E} |, the time-averaged energy flux is independent of polarization, as it depends only on the overall intensity rather than the orientation or ellipticity of the field components.^[19]

Momentum Density

In classical electromagnetism, the linear momentum density \mathbf{g} of electromagnetic fields is given by \mathbf{g} = \frac{1}{c^2} \mathbf{S}, where \mathbf{S} is the Poynting vector representing the energy flux and c is the speed of light in vacuum.^[22] This expression arises from the relativistic stress-energy-momentum tensor for the electromagnetic field, linking momentum directly to the flow of energy.^[23] For time-harmonic fields, the time-averaged momentum density follows similarly from the real part of the complex Poynting vector. For a plane electromagnetic wave propagating in vacuum, the magnitude of the momentum density simplifies to g = \frac{u}{c}, where u is the energy density of the wave. This result holds because the Poynting vector magnitude S = c u for such waves, leading to g = \frac{S}{c^2} = \frac{u}{c} directed along the propagation axis.^[24] Notably, g is independent of the polarization state—whether linear, circular, or elliptical—as long as the wave intensity (and thus u) remains the same, since the transverse orientation of the electric and magnetic fields does not affect the longitudinal energy flow in isotropic media.^[6] The momentum carried by polarized waves manifests physically through radiation pressure, the force exerted upon momentum transfer to matter. For a plane wave at normal incidence on a perfectly absorbing surface, the time-averaged pressure is P = \frac{I}{c}, where I is the intensity (equal to the time-averaged S).^[25] This pressure arises from the complete absorption of the wave's momentum flux, with no dependence on polarization in isotropic absorbers. In structured or anisotropic media, however, polarization can influence momentum transfer via spin-momentum locking, where the wave's spin (related to circular polarization) couples to its linear momentum direction, leading to transverse shifts or directional selectivity in the classical limit. Conservation of total linear momentum in electromagnetic systems requires considering both field and mechanical contributions, as isolated field momentum is not conserved. For a localized electromagnetic wave packet, the total field momentum is the volume integral \mathbf{P} = \int \mathbf{g} \, dV = \frac{1}{c^2} \int \mathbf{S} \, dV, which balances any mechanical momentum changes during interactions like emission or absorption.^[26] This integral form ensures overall momentum conservation in closed systems, consistent with Noether's theorem applied to space-translation symmetry in Maxwell's equations.

Angular Momentum Density

The spin angular momentum density of electromagnetic fields originates from the rotational character of the electric and magnetic field vectors, particularly in non-linear polarization states. For circularly polarized plane waves, this density manifests along the direction of propagation due to the helical rotation of the field vectors as the wave advances, imparting a torque on absorbing matter. A general expression for the spin angular momentum density is \mathbf{l}_s = \epsilon_0 \Im (\mathbf{E}^* \times \mathbf{A}), where \epsilon_0 is the vacuum permittivity, \mathbf{E} is the complex electric field, and \mathbf{A} is the vector potential; however, for plane waves, it simplifies to l_z = \pm \frac{u}{\omega}, with u denoting the energy density and \omega the angular frequency, the sign indicating right- or left-handed circular polarization.^[27]^[28] In linearly polarized waves, the spin angular momentum density is zero, as the field vectors oscillate without net rotation. For circular polarization, it reaches its maximum value of \pm u / \omega, while elliptical polarization yields an intermediate magnitude proportional to the degree of circularity \sigma (where |\sigma| \leq 1), given by l_z = \sigma \frac{u}{\omega}. The total angular momentum density of the field includes both spin and orbital components, but for uniformly polarized plane waves, the orbital contribution vanishes, leaving the spin as the sole contributor.^[27]^[29] Experimental verification of spin angular momentum transfer has been achieved using optical tweezers, where circularly polarized light rotates trapped birefringent particles, such as calcite microspheres, at rates consistent with the absorption of spin angular momentum from the beam. In these setups, the rotation direction and speed align with the polarization handedness, confirming the classical torque exerted by the field's spin density on matter.

Interactions of Polarized Waves with Matter

Passage Through Polarizing Filters

When a beam of polarized electromagnetic radiation encounters an ideal linear polarizing filter, the filter transmits the component of the electric field vector parallel to its transmission axis while absorbing the orthogonal component. This interaction is governed by Malus's law, which states that the transmitted intensity I is related to the incident intensity I_0 by I = I_0 \cos^2 \theta, where \theta is the angle between the polarization direction of the incident light and the filter's axis.^[30] Étienne-Louis Malus formulated this law in 1810 based on experiments with reflected light and calcite crystals, establishing the foundational principle for intensity variation in polarized light transmission.^[30] In the Jones calculus framework, the action of an ideal linear polarizer can be represented by a projection matrix that selects the parallel field component. For a horizontal polarizer (transmission axis along the x-direction), the Jones matrix is

\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix},

which, when multiplied by the input Jones vector describing the incident field's horizontal (E_x) and vertical (E_y) components, yields the transmitted field as \begin{pmatrix} E_x \\ 0 \end{pmatrix}.^[31] For a polarizer at an arbitrary angle \phi, the matrix is obtained by rotating the horizontal form using the rotation matrix R(\phi) = \begin{pmatrix} \cos \phi & \sin \phi \\ -\sin \phi & \cos \phi \end{pmatrix}, resulting in J_p(\phi) = R(\phi) \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} R(-\phi).^[32] This formalism, introduced by R. Clark Jones in 1941, facilitates the modeling of sequential optical elements by matrix multiplication.^[31] The absorbed portion of the incident energy, corresponding to I_0 \sin^2 \theta, is converted into heat within the filter material, ensuring conservation of energy as thermal dissipation.^[33] For unpolarized incident light, which has equal intensities in all polarization directions, an ideal linear polarizer transmits on average 50% of the incident intensity, as the random orientations average to \langle \cos^2 \theta \rangle = 1/2.^[32] In contrast, for linearly polarized input aligned with the axis (\theta = 0), transmission is complete (I = I_0), while crossed alignment (\theta = 90^\circ) results in zero transmission.^[30] Real-world polarizing filters, such as Polaroid sheets, approximate ideal behavior through dichroic absorption. These consist of polyvinyl alcohol films stretched to align polymer chains, into which iodine molecules are embedded, preferentially absorbing light polarized perpendicular to the alignment direction due to resonance with the molecular transitions.^[34] Edwin H. Land developed this H-type polarizer in 1938, enabling practical, low-cost polarization control for applications like sunglasses and optical instruments.^[34] While ideal models assume perfect transmission and absorption, actual devices exhibit slight imperfections, such as minor leakage (extinction ratios typically around 10^2 to 10^3) and wavelength-dependent efficiency, but they faithfully reproduce Malus's law over visible and near-infrared ranges.^[33]

Propagation in Birefringent Crystals

Birefringent crystals exhibit an optical property known as birefringence, where the refractive index depends on the polarization direction of the incident light, leading to double refraction. In uniaxial crystals such as calcite, light propagating along a direction not aligned with the optic axis splits into two orthogonally polarized rays: the ordinary ray (o-ray), which experiences a constant refractive index n_o regardless of direction, and the extraordinary ray (e-ray), which has a direction-dependent refractive index n_e that differs from n_o. For example, in calcite, n_o = 1.658 and n_e = 1.486 at 589 nm, resulting in a birefringence magnitude |n_e - n_o| \approx 0.172.^[35]^[36] When linearly polarized light enters a birefringent crystal at an angle to the optic axis, it decomposes into orthogonal o- and e-ray components that propagate with different velocities due to their distinct refractive indices, a phenomenon called double refraction. The o-ray follows the standard laws of refraction, while the e-ray deviates slightly, causing spatial separation of the beams inside the crystal. This splitting occurs because the crystal's anisotropic structure imposes different phase velocities on the two polarization components, without absorption.^[37]^[38] As the o- and e-rays traverse the crystal of thickness d, they accumulate a relative phase difference \delta = \frac{2\pi d}{\lambda} (n_o - n_e), where \lambda is the wavelength in vacuum.^[39] This phase retardation alters the output polarization state; for instance, an input linear polarization can emerge as elliptically polarized light, with the ellipticity depending on \delta and the input orientation relative to the optic axis. The propagation through a birefringent retarder can be described using Jones calculus, where the Jones matrix for a linear retarder aligned with the principal axes is diagonal:

\begin{pmatrix} e^{i\delta/2} & 0 \\ 0 & e^{-i\delta/2} \end{pmatrix},

assuming equal transmission amplitudes and a common phase factor. This matrix applies a differential phase shift between the o- and e-components. A specific case is the quarter-wave plate, where \delta = \pi/2, which converts linearly polarized light incident at 45° to the fast axis into circularly polarized light by equalizing the amplitudes while introducing a 90° phase difference.^[40]^[41] Waveplates, fabricated from birefringent crystals like quartz or mica, exploit this effect for precise polarization control in optical systems, such as rotating linear polarization or generating specific elliptical states. Historically, the Wollaston prism, invented by William Hyde Wollaston around 1803, uses two cemented calcite prisms to spatially separate o- and e-rays, enabling applications in polarimetry and beam splitting.^[42]^[43]

Energy Conservation in Optical Interactions

In the Jones calculus framework, energy conservation in lossless optical interactions involving polarized light is ensured through unitary transformations represented by Jones matrices. A unitary Jones matrix U satisfies U^\dagger U = I, where U^\dagger is the conjugate transpose and I is the identity matrix, preserving the norm of the input Jones vector \mathbf{E}_{in}. For an output vector \mathbf{E}_{out} = U \mathbf{E}_{in}, the squared magnitude |\mathbf{E}_{out}|^2 = \mathbf{E}_{out}^\dagger \mathbf{E}_{out} = \mathbf{E}_{in}^\dagger U^\dagger U \mathbf{E}_{in} = |\mathbf{E}_{in}|^2, which corresponds to the total intensity or energy flux of the electromagnetic wave remaining unchanged.^[44] This norm preservation reflects the absence of absorption or dissipation in the system, applicable to devices such as waveplates or retarders that alter polarization without energy loss. Hermitian operators play a complementary role in describing observables within these interactions; for instance, the intensity measurement operator is self-adjoint (H = H^\dagger), guaranteeing real eigenvalues that quantify measurable energy distributions, such as the projected intensity along a specific polarization axis.^[45]^[46] A practical example is the interaction of linearly polarized light with a birefringent crystal acting as a unitary retarder. An input Jones vector representing horizontal linear polarization, \mathbf{E}_{in} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, transforms via the retarder's Jones matrix—typically diagonal with phase shifts e^{i\delta/2} and e^{-i\delta/2} along the fast and slow axes—into an elliptical polarization state, such as circular for a quarter-wave retardance (\delta = \pi/2). Despite the change in polarization geometry, the total energy remains conserved, as |\mathbf{E}_{out}|^2 = 1, matching the input.^[47]^[48] Unitary transformations also maintain the orthogonality of dual polarization states, where "dual" refers to pairs of orthogonal basis vectors, such as horizontal and vertical linear polarizations. If two input states \mathbf{E}_1 and \mathbf{E}_2 satisfy \mathbf{E}_1^\dagger \mathbf{E}_2 = 0, their outputs U \mathbf{E}_1 and U \mathbf{E}_2 retain this property, with swapped or rotated components but no cross-energy transfer, upholding conservation in multi-component systems.^[44] In contrast, absorbing elements like ideal polarizers introduce non-unitary Jones matrices, incorporating explicit loss terms that reduce the output norm; for a horizontal polarizer, the matrix \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} projects the field, halving the intensity for diagonal input polarization and violating unitarity due to energy dissipation as heat.^[46]^[45]

Quantum Description of Photon Polarization

Energy, Momentum, and Spin of Single Photons

In quantum electrodynamics, the energy of a single photon is given by E = h \nu = \hbar \omega, where h is Planck's constant, \nu is the frequency, \hbar = h / 2\pi is the reduced Planck's constant, and \omega = 2\pi \nu is the angular frequency; this energy is independent of the photon's polarization state. The quantization of light energy into discrete photon packets was first proposed by Einstein to explain the photoelectric effect, establishing photons as fundamental quanta of the electromagnetic field. The linear momentum of a photon is \mathbf{p} = \hbar \mathbf{k}, where \mathbf{k} is the wave vector with magnitude k = 2\pi / \lambda and \lambda the wavelength, yielding a momentum magnitude p = h \nu / c = E / c directed along the propagation direction; like energy, this momentum does not depend on polarization. This relation was experimentally confirmed through the Compton effect, where X-ray photons scatter off electrons, transferring momentum consistent with particle-like behavior. Photons, as massless spin-1 bosons, possess intrinsic spin angular momentum with helicity—the projection of spin along the momentum direction—taking values \pm \hbar, corresponding to right- and left-handed circular polarization states, respectively. These two helicity eigenstates fully describe the transverse polarization degrees of freedom for photons in quantum electrodynamics, arising from the representation theory of the Poincaré group. In the classical limit of many photons, the total energy and momentum densities recover the Poynting vector, while the spin angular momentum density aligns with the field's circular polarization components.

Probability Amplitudes and Quantum States

In the quantum mechanical description of photon polarization, the state is represented in a two-dimensional Hilbert space with orthonormal basis vectors |H⟩ and |V⟩ corresponding to horizontal and vertical linear polarizations, respectively.^[49] This basis captures the two possible helicity states of the photon's spin angular momentum along the propagation direction, as discussed in prior sections on single-photon properties. A general polarization state for a single photon is a coherent superposition given by

|\psi\rangle = \alpha |H\rangle + \beta |V\rangle,

where \alpha and \beta are complex coefficients satisfying the normalization condition |\alpha|^2 + |\beta|^2 = 1 to ensure the total probability is unity.^[49] Such superpositions enable the photon to exist simultaneously in multiple polarization configurations until measured, embodying the principle of quantum indeterminacy. Upon measurement in the {|H⟩, |V⟩} basis—typically via a polarizing beam splitter or filter—the probability of detecting horizontal polarization is P(H) = |\alpha|^2, and vertical polarization is P(V) = |\beta|^2, as dictated by the Born rule.^[49] This probabilistic outcome arises because the measurement collapses the superposition to one of the basis states, with the squared modulus of the amplitude determining the likelihood. Single-photon experiments vividly illustrate these probability amplitudes, particularly in setups where polarization serves as which-path information. In a double-slit interference experiment, mutually perpendicular polarizers placed over the slits (vertical over one and horizontal over the other) imprint orthogonal polarization states on photons traversing each path, rendering the paths distinguishable and suppressing the interference pattern in favor of classical single-slit diffraction profiles.^[50] If the incident photons are in a polarization superposition, the detection probabilities at the screen reflect the |α|^2 and |β|^2 amplitudes, modulated by the path-encoded states, demonstrating how measurement outcomes encode the quantum interference. For single photons, the focus remains on pure states, which preserve full phase coherence and allow maximal superposition effects, unlike mixed states that arise in incoherent ensembles and lack such predictable interference.^[51] Preparation techniques, such as spontaneous parametric down-conversion, routinely generate single photons in these pure polarization states for precise control in quantum optics experiments.^[51]

Unitary and Hermitian Operators in Polarization Measurements

In quantum optics, unitary operators describe the evolution of photon polarization states under lossless transformations, such as those induced by wave plates or beam splitters, ensuring the preservation of total probability and energy. These operators satisfy U^\dagger U = I, where I is the identity, guaranteeing that the norm of the state vector remains unchanged: if |\psi\rangle is the initial state, then \|\ U |\psi\rangle \ \| = 1. For example, a rotation of the polarization by an angle \theta around the propagation axis is represented by the unitary operator U(\theta) = \exp(-i \theta \sigma_y / 2), where \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} is the Pauli-y matrix in the horizontal-vertical basis. This operator can be explicitly written as

U(\theta) = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix},

and is physically realized by a half-wave plate oriented at angle \theta/2 relative to the horizontal. Similarly, a polarizing beam splitter acts as a unitary transformation that separates orthogonal polarizations into different spatial modes while preserving the overall photon number.^[52]^[53]^[54] Hermitian operators, which are self-adjoint (H = H^\dagger), represent observables in polarization measurements, yielding real eigenvalues that correspond to measurable outcomes such as detection probabilities. For a projective measurement in a specific polarization basis, the operator is a projector onto the desired state, such as H = |H\rangle\langle H| for horizontal polarization, where |H\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}. This operator has eigenvalues 1 (for the horizontal eigenstate) and 0 (for the orthogonal vertical state), reflecting the binary outcome of transmission or absorption in a polarizer. The expectation value \langle H \rangle = \langle \psi | H | \psi \rangle gives the probability of detecting a horizontally polarized photon. In the circular basis, which serves as the eigenbasis for the photon's helicity or spin angular momentum along the propagation direction, the states are |R\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -i \end{pmatrix} (right-circular, eigenvalue +1 for spin) and |L\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ i \end{pmatrix} (left-circular, eigenvalue -1), with the corresponding projectors |R\rangle\langle R| and |L\rangle\langle L|. These bases are complete and orthogonal, allowing measurements of spin components.^[55]^[52]^[56] According to the measurement postulate of quantum mechanics, performing a measurement with a Hermitian operator H collapses the state |\psi\rangle to one of its eigenvectors |e_k\rangle with probability | \langle e_k | \psi \rangle |^2, and the observable's value is the corresponding eigenvalue. Post-measurement, the state updates to |e_k\rangle (normalized), altering the polarization description for subsequent operations. For instance, measuring in the horizontal basis with projector H on a state |\psi\rangle = \cos\alpha |H\rangle + \sin\alpha |V\rangle yields horizontal polarization with probability \cos^2\alpha, collapsing to |H\rangle. This process is irreversible, unlike unitary evolutions. In classical optics, unitary operators analogize to lossless transformations like rotations that preserve wave intensity, while Hermitian projectors correspond to real-valued intensity measurements after filters, where outcomes reflect absorbed or transmitted power without quantum superposition.^[55]^[53]

Uncertainty Principle Applications

The Heisenberg uncertainty principle, in its general form for two non-commuting observables represented by Hermitian operators \hat{A} and \hat{B}, states that the product of their standard deviations satisfies \Delta A \Delta B \geq \frac{1}{2} |\langle [\hat{A}, \hat{B}] \rangle|, where [\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A} is the commutator and the expectation value is taken over the quantum state.^[57] This relation imposes fundamental limits on the simultaneous measurability of incompatible properties, arising from the non-commutativity inherent in quantum mechanics. In the context of photon polarization, this principle applies to the quantum operators describing polarization states, preventing precise joint knowledge of certain polarization components. For photon polarization, the Stokes operators \hat{S}_1, \hat{S}_2, and \hat{S}_3—which generalize the classical Stokes parameters to the quantum domain—obey the commutation relations [\hat{S}_1, \hat{S}_2] = 2i \hat{S}_3 and cyclic permutations thereof, forming an \mathfrak{su}(2) algebra.^[57] Consequently, the uncertainty relation for, say, \hat{S}_1 (measuring the difference between horizontal and vertical linear polarizations) and \hat{S}_2 (measuring the difference between +45° and -45° linear polarizations) is \Delta S_1 \Delta S_2 \geq |\langle \hat{S}_3 \rangle|, where \hat{S}_3 corresponds to the circular polarization component. This implies that knowledge of the ellipticity (quantified by \langle \hat{S}_3 \rangle) introduces unavoidable uncertainty in the linear polarization components; for a state with significant circular polarization (large |\langle \hat{S}_3 \rangle|), one cannot precisely determine both \langle \hat{S}_1 \rangle and \langle \hat{S}_2 \rangle simultaneously. For a single photon, the maximum value of |\langle \hat{S}_3 \rangle| is 1 (in units where the total photon number is normalized), saturating the bound in pure states like right- or left-circular polarization.^[58] The spin angular momentum of a photon, associated with its polarization, provides another direct application. The components \hat{L}_x, \hat{L}_y, and \hat{L}_z (transverse and longitudinal spin) satisfy [\hat{L}_x, \hat{L}_y] = i \hbar \hat{L}_z and cyclic relations, leading to the uncertainty relation \Delta L_x \Delta L_y \geq \frac{\hbar}{2} |\langle L_z \rangle|. Here, \langle L_z \rangle = \pm \hbar for a circularly polarized photon, with zero variance in L_z, but the transverse components exhibit uncertainties \Delta L_x = \Delta L_y = \hbar / \sqrt{2}, exactly meeting the bound. This underscores the impossibility of knowing the full linear and circular polarization precisely at once, as linear polarization relates to transverse spin components while circular corresponds to the longitudinal one. These uncertainty relations manifest experimentally in single-photon interferometry, where polarization measurements impose limits on phase sensitivity and interference visibility. For instance, in a Mach-Zehnder interferometer with polarized single photons, attempting to resolve which-path information via polarization projections disturbs the transverse coherence, reducing visibility in accordance with the uncertainty bounds on Stokes operators; experiments have verified such trade-offs, with noise in linear polarization components limiting ellipticity discrimination to the quantum limit of \Delta S_1 \Delta S_2 \geq 1 for single-photon states.^[59]

Advanced Topics in Photon Polarization

Stokes Parameters for Partial Polarization

The Stokes parameters provide a mathematical framework for describing the polarization state of light that extends beyond fully polarized waves to include partially polarized and unpolarized light, representing the light as an incoherent superposition of polarized components. Introduced by George Gabriel Stokes in 1852, these parameters are particularly useful for analyzing real-world optical phenomena where coherence is incomplete, such as scattered light in natural environments.^[60]^[61] The Stokes vector is defined as \mathbf{S} = (S_0, S_1, S_2, S_3), where S_0 represents the total intensity of the light beam, S_1 quantifies the difference between horizontally and vertically linearly polarized components, S_2 measures the difference between linearly polarized components at +45° and -45°, and S_3 captures the difference between right- and left-circularly polarized components. These parameters are derived from time-averaged intensities measurable in the laboratory and form a real-valued vector that fully characterizes the polarization state for any degree of coherence. For fully polarized light, the relation S_0^2 = S_1^2 + S_2^2 + S_3^2 holds, but for partial polarization, S_0^2 > S_1^2 + S_2^2 + S_3^2.^[62]^[61] The degree of polarization P is given by

P = \frac{\sqrt{S_1^2 + S_2^2 + S_3^2}}{S_0},

which ranges from 0 for completely unpolarized light (where S_1 = S_2 = S_3 = 0) to 1 for fully polarized light. This metric indicates the fraction of the total intensity that arises from the coherent, polarized portion of the beam, with $1 - P representing the unpolarized contribution.^[61]^[62] In the context of coherence theory, the Stokes parameters are directly related to the coherency matrix, a 2×2 Hermitian matrix \mathbf{J} formed from the correlation functions of the electric field components E_x and E_y:

\mathbf{J} = \begin{pmatrix} \langle E_x E_x^* \rangle & \langle E_x E_y^* \rangle \\ \langle E_y E_x^* \rangle & \langle E_y E_y^* \rangle \end{pmatrix},

where the angle brackets denote time averages. The elements of \mathbf{J} yield the Stokes parameters via S_0 = J_{11} + J_{22}, S_1 = J_{11} - J_{22}, S_2 = 2\Re(J_{12}), and S_3 = 2\Im(J_{12}), providing a bridge between field correlations and observable intensities for partially coherent light.^[63] Stokes parameters are measured experimentally by passing the light through combinations of quarter-wave plates and linear polarizers, with intensities recorded at specific orientations to isolate each component; for instance, S_3 requires a quarter-wave plate to convert circular polarization differences into linear ones detectable by the polarizer. Optical transformations, such as passage through birefringent elements or scattering media, are described using Mueller matrices, which are 4×4 real matrices that map input Stokes vectors to output ones, enabling prediction of polarization changes in complex systems.^[64]^[65] A key application of Stokes parameters arises in analyzing natural light sources like sunlight, which becomes partially polarized due to single scattering by atmospheric molecules, typically exhibiting a degree of polarization up to around 80% in the band 90° from the sun, with the location of maximum polarization shifting from near the horizon (when the sun is high) to near the zenith (when the sun is low on the horizon), as observed in clear skies.^[66]^[67] This partial polarization facilitates remote sensing of atmospheric properties and biological navigation in animals, highlighting the parameters' utility in bridging classical optics with practical observations.

Polarization Entanglement in Quantum Optics

Polarization entanglement occurs when two or more photons share a quantum state such that their polarization degrees of freedom are correlated in a way that cannot be described by classical physics, leading to non-local correlations observable in joint measurements. This phenomenon is a cornerstone of quantum optics, enabling the creation of maximally entangled states known as Bell states, such as the state |\Phi^+\rangle = \frac{1}{\sqrt{2}} (|HH\rangle + |VV\rangle), where |H\rangle and |V\rangle denote horizontal and vertical polarization, respectively. These states were first realized experimentally using spontaneous parametric down-conversion (SPDC) in the 1990s, marking a significant advancement in generating high-fidelity entangled photon pairs for quantum information tasks.^[68] SPDC involves the nonlinear optical process in a birefringent crystal where a pump photon splits into a pair of lower-energy signal and idler photons, conserving energy and momentum. In type-II phase-matching configurations, such as those using beta-barium borate (BBO) crystals, the signal and idler photons emerge with orthogonal polarizations—typically one horizontal and one vertical—naturally forming an entangled state upon proper alignment and compensation for walk-off effects. This method, pioneered in high-intensity sources, produces polarization-entangled pairs with fidelities exceeding 90% and has become the standard for laboratory-scale quantum optics experiments since the mid-1990s.^[68] The nonlocality of polarization entanglement manifests in violations of Bell inequalities, which test the incompatibility of quantum mechanics with local hidden variable theories. In landmark experiments from the early 1980s, Alain Aspect and collaborators used entangled photon pairs from atomic cascades to measure polarization correlations, achieving a clear violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality with a value of S = 2.697 \pm 0.015, surpassing the classical bound of 2. Subsequent adaptations to SPDC sources in the 1990s confirmed these violations with polarization-entangled photons, solidifying the empirical evidence for quantum nonlocality in optical systems.^[69] Polarization-entangled states are highly sensitive to decoherence, where interactions with the environment—such as birefringence in optical fibers or scattering in free space—disrupt the phase coherence between the polarization components, reducing entanglement fidelity. For instance, propagation through dispersive media can introduce polarization mode dispersion, leading to stochastic decoherence that scales with distance and wavelength mismatch, necessitating active compensation techniques like wave plates or feedback loops to preserve entanglement for practical quantum protocols. Recent studies show that atmospheric turbulence can degrade entanglement fidelity over propagation distances, often reducing Bell state visibility significantly without mitigation.^[70] As of 2025, polarization entanglement has found critical applications in satellite-based quantum key distribution (QKD), where entangled photon pairs are transmitted over global distances to enable secure communication. The 2017 Micius satellite demonstrated the distribution of polarization-entangled photons over 1,200 km, achieving CHSH violations in space-to-ground links, and ongoing missions continue to refine this for practical networks, with recent entanglement-based QKD protocols viable up to 400 km in low-Earth orbit under daylight conditions.^[71]

Applications in Quantum Information Processing

Photon polarization plays a central role in quantum key distribution (QKD), particularly through the BB84 protocol, where information is encoded in the polarization states of single photons. In this protocol, the sender (Alice) prepares photons in one of four polarization states: horizontal (H), vertical (V), or the diagonal states at 45° and 135°. Alice randomly chooses between two bases—rectilinear (H/V) or diagonal—for each photon, while the receiver (Bob) measures in a randomly selected basis, discarding mismatched measurements post-sifting. The security arises from the quantum no-cloning theorem and the disturbance caused by any eavesdropping attempt, which introduces detectable errors exceeding 25% in the quantum bit error rate (QBER), allowing Alice and Bob to verify privacy amplification.^[72] In quantum computing, photon polarization serves as a qubit encoding in linear optical systems, enabling universal quantum gates through interferometric setups. The Knill-Laflamme-Milburn (KLM) scheme demonstrates how polarization-encoded qubits, combined with beam splitters and single-photon detectors, can implement deterministic two-qubit gates like the controlled-NOT (CNOT) with high fidelity by probabilistically teleporting nonlinear interactions. This approach leverages post-selection on measurement outcomes to overcome the limitations of linear optics, which cannot directly create entanglement, achieving fault-tolerant computation with success probabilities approaching 1 using ancillary photons. Polarization encoding reduces the required number of spatial modes compared to path-based schemes, facilitating scalable integration.^[73] Quantum teleportation utilizes polarization-entangled photon pairs to transfer an unknown polarization state from one photon to another without physical transmission of the carrier. In the protocol, the sender performs a Bell-state measurement on the input photon and one half of the entangled pair, classically communicating the result (two bits) to the receiver, who applies a corrective Pauli operator to reconstruct the state on the distant photon. Experimental demonstrations began in 1998, achieving fidelities above 0.7 for polarization states over laboratory distances using spontaneous parametric down-conversion sources. These implementations have since scaled to network distances, enabling quantum repeaters and distributed computing.^[74] In quantum metrology, polarization enhances sensing precision beyond classical limits, as seen in ghost imaging protocols with entangled photons. Here, spatial and polarization correlations between signal and idler photons from parametric down-conversion allow reconstruction of an object's polarization-dependent transmission profile using only bucket detection on one arm and a reference scan on the other, achieving sub-shot-noise resolution. This entanglement-based method surpasses classical ghost imaging by factors of up to √2 in signal-to-noise ratio, with applications in remote sensing of birefringent materials. Despite these advances, challenges persist in deploying polarization-based systems, particularly high losses and state decoherence in optical fibers due to birefringence and environmental perturbations, limiting transmission to tens of kilometers without active stabilization. Free-space links mitigate these issues, offering polarization stability over atmospheric paths up to 100 km, as demonstrated in satellite-based QKD experiments. As of 2025, integrated photonic chips address scalability by enabling on-chip polarization rotation and manipulation with losses below 1 dB, using silicon or lithium niobate platforms for compact quantum processors and networks.^[75]^[76]^[77]

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