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Stokes parameters

The Stokes parameters are a set of four real-valued quantities that completely describe the polarization state of electromagnetic radiation, such as light, for both fully polarized and partially polarized (including unpolarized) cases.^[1] Introduced by the mathematician and physicist George Gabriel Stokes in his 1852 paper on the composition of polarized light streams, these parameters provide a mathematically convenient framework because they transform linearly under rotations and are additive for the incoherent superposition of light beams.^[2]^[3] The parameters, often denoted as S_0, S_1, S_2, S_3, are defined in terms of the electric field components of the light wave. Specifically, S_0 = \langle E_x^2 \rangle + \langle E_y^2 \rangle represents the total intensity, S_1 = \langle E_x^2 \rangle - \langle E_y^2 \rangle measures the excess of horizontal over vertical linear polarization, S_2 = 2 \langle E_x E_y \rangle \cos \delta captures the difference between +45° and -45° linear polarizations (where \delta is the phase difference), and S_3 = 2 \langle E_x E_y \rangle \sin \delta quantifies right- versus left-handed circular polarization, with angle brackets denoting time averages.^[1] For fully polarized light, the relation S_0^2 = S_1^2 + S_2^2 + S_3^2 holds, and the degree of polarization is given by P = \sqrt{S_1^2 + S_2^2 + S_3^2} / S_0, ranging from 0 (unpolarized) to 1 (fully polarized).^[1] These definitions stem from Stokes' original formulation using intensities measured through polarizers at different orientations, avoiding the need for complex amplitudes.^[2] In practice, Stokes parameters are represented as a vector \mathbf{S} = (S_0, S_1, S_2, S_3)^T, which can be mapped to the Poincaré sphere for geometric visualization of polarization states, where S_0 is the radius and the vector (S_1, S_2, S_3) points to a location on the sphere's surface for fully polarized light.^[3] This representation facilitates analysis in fields like optics, where they enable the design of polarimeters for measuring polarization; astronomy, for studying cosmic microwave background radiation and interstellar dust; and biomedical imaging, such as polarization-sensitive optical coherence tomography for tissue characterization.^[4] Their utility extends to modern applications in quantum optics and remote sensing, underscoring their enduring role in quantifying polarization phenomena.^[3]

Background on Light Polarization

Electromagnetic Nature of Light

Light is fundamentally an electromagnetic wave, consisting of oscillating electric and magnetic fields that propagate through space perpendicular to the direction of wave travel.^[5] The electric field vector \mathbf{E} oscillates in a plane transverse to the propagation direction, while the magnetic field \mathbf{B} oscillates in a perpendicular plane, with both fields mutually orthogonal to the wave's velocity vector.^[6] This transverse nature arises from Maxwell's equations, which dictate that electromagnetic waves in free space cannot support longitudinal components without charges or currents.^[7] For a basic description, the electric field of a monochromatic plane wave propagating in the z-direction can be expressed as \mathbf{E}(t) = \mathbf{E_0} \cos(\omega t + \phi), where \mathbf{E_0} is the amplitude vector lying in the xy-plane, \omega is the angular frequency, and \phi is the phase.^[8] Monochromatic plane waves assume a single frequency and uniform phase across the wavefront, simplifying analysis of wave interactions. The time-averaged intensity I of such a wave, which quantifies its energy flux, is given by I = \frac{1}{2} c \epsilon_0 |\mathbf{E_0}|^2, where c is the speed of light and \epsilon_0 is the vacuum permittivity; this average arises because the instantaneous Poynting vector oscillates but delivers a net constant power over each cycle.^[7] Polarization describes the orientation and coherence of the electric field oscillations in these waves. Fully polarized light occurs when the electric field vector maintains a fixed orientation or follows a definite elliptical path relative to the propagation direction.^[9] Unpolarized light, typical from natural sources like the sun, features electric field vectors with random, uncorrelated orientations and phases, resulting in no preferred direction.^[10] Partially polarized light represents an intermediate case, combining components of coherent (polarized) and incoherent (unpolarized) fields, often quantified by the degree of polarization ranging from 0 (fully unpolarized) to 1 (fully polarized).^[10]

Classical Polarization States

Classical polarization states refer to the well-defined orientations and phases of the electric field vector in coherent, fully polarized electromagnetic waves propagating in free space. These states arise from the transverse nature of light waves, where the electric field oscillates perpendicular to the direction of propagation. In the early 19th century, French physicist Augustin-Jean Fresnel made seminal discoveries on polarization phenomena, including the explanation of reflection and refraction laws for polarized light, which supported the wave theory and demonstrated that light vibrations are transverse rather than longitudinal.^[11] Fresnel's work, building on earlier observations by Étienne-Louis Malus in 1808, laid the foundation for understanding these states through interference and birefringence experiments.^[12] Linear polarization occurs when the electric field oscillates along a fixed direction in the plane perpendicular to propagation. For horizontal linear polarization, the field vibrates parallel to the x-axis (assuming propagation along z), represented by the Jones vector \begin{pmatrix} 1 \\ 0 \end{pmatrix}.^[6] Vertical linear polarization has the field along the y-axis, with Jones vector \begin{pmatrix} 0 \\ 1 \end{pmatrix}.^[6] Diagonal linear polarizations, such as at 45° to the x-axis, involve equal amplitudes in x and y with no phase difference, given by the normalized Jones vector \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix}.^[6] These states are produced, for example, by passing unpolarized light through a polarizer aligned with the desired axis. Circular polarization describes states where the electric field vector rotates with constant magnitude as the wave propagates, resulting from equal x and y amplitudes with a π/2 phase difference. Right-handed circular polarization features clockwise rotation when viewed against the direction of propagation (i.e., facing the oncoming wave), represented by the normalized Jones vector \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -i \end{pmatrix}.^[6] Left-handed circular polarization rotates counterclockwise under the same viewing convention, with Jones vector \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ i \end{pmatrix}.^[6] These can be generated using quarter-wave plates on linear polarized input, and they exhibit unique properties like immunity to rotation in isotropic media. The Jones vector formalism applies specifically to coherent, fully polarized light, where the phase relationship between field components is fixed.^[13] However, natural light sources often produce partially polarized or unpolarized beams due to incoherent superpositions of multiple polarization states, requiring generalized parameters to describe such cases beyond the limitations of Jones vectors.^[13]

Definition of Stokes Parameters

The Stokes Vector

The Stokes vector provides a complete mathematical description of the polarization state of electromagnetic waves, encompassing fully polarized, partially polarized, and unpolarized light, through four real-valued components. It is defined as \mathbf{S} = (S_0, S_1, S_2, S_3)^T, where S_0 represents the total intensity, and the remaining parameters capture differences and correlations in the orthogonal electric field components.^[14] These components are derived from time averages of the squared magnitudes and cross terms of the electric field, assuming propagation along the z-direction with orthogonal x and y components E_x(t) and E_y(t) = |E_y| \cos(\omega t + \delta). Specifically,

\begin{align*} S_0 &= \langle |E_x|^2 \rangle + \langle |E_y|^2 \rangle, \\ S_1 &= \langle |E_x|^2 \rangle - \langle |E_y|^2 \rangle, \\ S_2 &= 2 \langle |E_x| |E_y| \cos \delta \rangle, \\ S_3 &= 2 \langle |E_x| |E_y| \sin \delta \rangle, \end{align*}

where \langle \cdot \rangle denotes the time average over many optical cycles, and \delta is the phase difference between E_x and E_y./04:_Stokes_Parameters_for_Describing_Polarized_Light/4.01:_Polarized_Light_and_the_Stokes_Parameters) This formulation arises from the coherency matrix of the field, enabling the representation of incoherent superpositions unlike the Jones vector approach. The parameters were originally introduced by George Gabriel Stokes in 1852, who characterized polarization via measurable intensities of light passed through polarizing prisms and analyzers, laying the foundation for quantifying mixtures of polarized beams from independent sources.^[2] A key quantity derived from the Stokes vector is the degree of polarization P = \sqrt{S_1^2 + S_2^2 + S_3^2}/S_0 \leq 1, which indicates the proportion of S_0 attributable to fully polarized light, with P = 1 for complete polarization and P = 0 for unpolarized light.^[14]

Component Interpretations

The Stokes parameter S_0 represents the total intensity of the light beam, encompassing both polarized and unpolarized components, and is independent of the specific polarization state.^[15] This parameter is equivalent to the sum of the intensities measured through orthogonal linear polarizers, providing a measure of the overall energy flux without regard to orientation.^[16] The parameter S_1 quantifies the difference in intensities between light linearly polarized in the horizontal direction and that polarized in the vertical direction.^[15] Specifically, S_1 = I_H - I_V, where I_H and I_V are the intensities of horizontally and vertically polarized components, respectively; a positive value indicates dominance of horizontal polarization, while a negative value signifies vertical dominance.^[16] This difference arises from projections onto orthogonal linear bases aligned with the coordinate axes. Similarly, S_2 captures the intensity difference between light linearly polarized at +45° and that at -45° relative to the horizontal axis.^[15] Expressed as S_2 = I_{+45^\circ} - I_{-45^\circ}, it highlights the contribution of diagonal linear polarizations, with positive values favoring the +45° orientation and negative values the -45° orientation.^[16] These measurements are obtained by passing the light through polarizers oriented at 45° and -45°. The parameter S_3 measures the difference between the intensities of right-circularly polarized and left-circularly polarized light components.^[15] Defined as S_3 = I_R - I_L, where I_R and I_L denote right- and left-circular intensities, a positive S_3 indicates right-circular dominance, and a negative value left-circular dominance; this is assessed using quarter-wave retarders combined with linear polarizers.^[16] For unpolarized light, the Stokes vector takes the form (I, 0, 0, 0), where I is the total intensity, reflecting equal contributions from all polarization states with no net differences.^[15] In contrast, fully horizontally polarized light is represented as (I, I, 0, 0), where S_0 = S_1 = I and S_2 = S_3 = 0, indicating complete alignment along the horizontal axis with no diagonal or circular components.^[16]

Physical Representations

Relation to the Polarization Ellipse

The polarization state of fully polarized light can be geometrically represented by an ellipse, where the orientation angle \psi and ellipticity angle \chi of the ellipse are directly related to the Stokes parameters S_1, S_2, and S_3. Specifically, the orientation angle is given by \psi = \frac{1}{2} \tan^{-1}(S_2 / S_1), which describes the angle of the major axis of the ellipse relative to the reference axes, while the ellipticity angle is \chi = \frac{1}{2} \tan^{-1}\left( S_3 / \sqrt{S_1^2 + S_2^2} \right), indicating the shape from linear (\chi = 0) to circular (\chi = \pm \pi/4) polarization.^[17] This geometric mapping allows the Stokes parameters to visualize the polarization as a point on the Poincaré sphere, a unit sphere where the vector (S_1/S_0, S_2/S_0, S_3/S_0) lies on the surface for fully polarized light, with the north and south poles corresponding to right- and left-circular polarizations (S_3 = \pm S_0) and the equator representing linear polarizations (S_3 = 0). The latitude on the sphere relates to the ellipticity $2\chi, and the longitude to the orientation $2\psi.^[18]^[17] The Poincaré sphere provides an intuitive tool for understanding polarization transformations, such as those induced by retarders or rotators, which correspond to rotations on the sphere. This representation was originally introduced by Henri Poincaré in 1892 as a geometrical method to track the evolution of light polarization states interacting with matter.^[18] For partially polarized light, where the degree of polarization p = \sqrt{S_1^2 + S_2^2 + S_3^2}/S_0 < 1, the polarization ellipse describes only the fully polarized fraction p S_0, with the remaining (1 - p) S_0 representing an unpolarized component; on the Poincaré sphere, the point lies inside the sphere at a distance p from the center.^[17]

Transformations Between Bases

The Stokes parameters provide a complete description of the polarization state in a given reference frame, but measurements or analyses often require transforming the Stokes vector \mathbf{S} = (S_0, S_1, S_2, S_3)^T to a different polarization basis, such as when rotating the analyzer or coordinate system. Under a rotation of the polarization basis by an angle \theta, the transformed Stokes vector is given by \mathbf{S}' = R(\theta) \mathbf{S}, where the rotation matrix R(\theta) acts on the linear polarization components S_1 and S_2 while leaving the total intensity S_0 and circular polarization component S_3 unchanged:

R(\theta) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos 2\theta & \sin 2\theta & 0 \\ 0 & -\sin 2\theta & \cos 2\theta & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}.

The factor of $2\theta arises because polarization states, being traceless in the coherency matrix representation, exhibit double-angle dependence under rotations, analogous to spin-1 particles in quantum mechanics. This transformation preserves the degree of polarization and the overall intensity, ensuring that the Poincaré sphere representation rotates rigidly around the S_3 axis.^[19] A practical example is the transformation from the standard linear basis (horizontal-vertical for S_1, 45° linear for S_2) to a basis aligned with ±45° linear polarizations, which can be achieved by applying R(\theta) at \theta = 45^\circ. Here, \cos 2\theta = 0 and \sin 2\theta = 1, yielding S_1' = S_2, S_2' = -S_1, S_3' = S_3, and S_0' = S_0. This reorients the linear components to align with the 45°-135° axes, facilitating analysis in setups involving linear polarizations at those angles. Such basis changes are essential for matching experimental setups to specific polarization ellipse orientations without altering the underlying physical state.^[19] In isotropic media, where the refractive index is uniform and independent of polarization, the Stokes vector remains invariant during free propagation, as no birefringence or dichroism introduces differential phase shifts or absorption between orthogonal components. This property holds for plane waves in non-absorbing, homogeneous environments, simplifying polarization tracking in optical systems like free space or simple lenses. For coherent light, these basis transformations in the Stokes formalism correspond directly to those in Jones calculus, where the Jones matrix \mathbf{J} for the electric field evolves the polarization state, and the associated Mueller matrix (derived from \mathbf{J}) applies the equivalent transformation to the Stokes vector via \mathbf{S}' = \mathbf{M} \mathbf{S}, with \mathbf{M} constructed as \mathbf{M} = \mathbf{A} (\mathbf{J} \otimes \mathbf{J}^*) \mathbf{A}^{-1} using a suitable basis matrix \mathbf{A}. This linkage enables seamless analysis of coherent propagation effects, such as retarder-induced rotations, within the partially polarized framework of Stokes parameters.^[20]^[21]

Mathematical Properties

Invariance and Mueller Calculus

The Stokes parameters possess certain invariant properties that reflect fundamental physical constraints on the polarization state of light. In lossless optical systems, where no energy is absorbed or scattered away from the beam, the zeroth Stokes parameter S_0, representing the total intensity, remains conserved such that S_{0,\text{out}} = S_{0,\text{in}}.^[22] Additionally, the parameters satisfy the inequality S_1^2 + S_2^2 + S_3^2 \leq S_0^2, with equality holding for fully polarized light and strict inequality indicating partial polarization; this relation defines the degree of polarization as P = \sqrt{S_1^2 + S_2^2 + S_3^2}/S_0 \leq 1.^[23] These invariance properties underpin the Mueller calculus, a matrix formalism developed in the 1940s by Hans Mueller to extend the original work of George Gabriel Stokes on polarization description to arbitrary linear optical transformations, including those involving partially polarized or depolarizing media.^[24] In this framework, the output Stokes vector \mathbf{S}_{\text{out}} is obtained by linearly transforming the input Stokes vector \mathbf{S}_{\text{in}} via a 4×4 real Mueller matrix \mathbf{M}:

\mathbf{S}_{\text{out}} = \mathbf{M} \mathbf{S}_{\text{in}}.

The elements of \mathbf{M} encode the polarization-altering effects of an optical element or system, such as diattenuation, retardance, and depolarization.^[25] For non-absorbing (lossless) systems, the Mueller matrix exhibits specific structural properties, including unimodularity where \det \mathbf{M} = \pm 1, ensuring conservation of the degree of polarization for fully polarized input light and alignment with energy preservation.^[26] This contrasts with absorbing systems, where \det \mathbf{M} < 1, reflecting partial loss of intensity. Mueller matrices for cascaded optical elements multiply in sequence, allowing complex systems to be modeled as the product of individual matrices.^[27] A representative example is the Mueller matrix for an ideal polarizer, which selectively transmits light aligned with its transmission axis characterized by the unit vector components (p_1, p_2, p_3):

\mathbf{M}_p = \frac{1}{2} \begin{pmatrix} 1 & p_1 & p_2 & p_3 \\ p_1 & p_1^2 & p_1 p_2 & p_1 p_3 \\ p_2 & p_1 p_2 & p_2^2 & p_2 p_3 \\ p_3 & p_1 p_3 & p_2 p_3 & p_3^2 \end{pmatrix}.

For a linear horizontal polarizer, where p_1 = 1, p_2 = 0, p_3 = 0, this simplifies to a matrix that projects the input Stokes vector onto horizontal linear polarization while halving the intensity.^[28]

Connection to Coherency Matrix

The coherency matrix offers a fundamental statistical framework for describing the polarization of quasi-monochromatic light, capturing both coherent and incoherent contributions through ensemble averages of the electric field components. It is defined as \mathbf{J} = \langle \mathbf{E} \mathbf{E}^\dagger \rangle, where \mathbf{E} = (E_x, E_y)^T is the Jones vector representing the transverse electric field, and the angle brackets denote a time average over fluctuations. In component form, this yields the Hermitian matrix

\mathbf{J} = \begin{pmatrix} \langle |E_x|^2 \rangle & \langle E_x E_y^* \rangle \\ \langle E_y E_x^* \rangle & \langle |E_y|^2 \rangle \end{pmatrix},

with J_{11} = \langle |E_x|^2 \rangle, J_{22} = \langle |E_y|^2 \rangle, J_{12} = \langle E_x E_y^* \rangle, and J_{21} = J_{12}^*. The Stokes parameters emerge directly from this matrix, providing a real-valued vector representation of the same information: 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}). These relations transform the complex coherency matrix into the observable Stokes vector \mathbf{S} = (S_0, S_1, S_2, S_3)^T, where S_0 quantifies total intensity, and the differences and real/imaginary parts encode linear and circular polarization degrees, respectively. This connection is particularly advantageous for partially coherent light, where the off-diagonal elements J_{12} and J_{21} reflect the mutual coherence between orthogonal components, enabling quantification of the degree of polarization p = \sqrt{S_1^2 + S_2^2 + S_3^2}/S_0 \leq 1. In such cases, the van Cittert–Zernike theorem extends to polarized fields, linking the spatial variation of the coherency matrix elements—and thus the Stokes parameters—to the intensity distribution across an incoherent source, facilitating predictions of polarization coherence in the far field. The coherency matrix formalism, including its ties to Stokes parameters, was formalized in the mid-20th century within optical coherence theory, notably by Emil Wolf and collaborators starting in the 1950s.^[29]

Experimental Measurement

Theoretical Principles

The theoretical foundations for measuring Stokes parameters trace back to George Gabriel Stokes' seminal 1852 work, where he proposed experimental methods to quantify the polarization state of light using Nicol prisms as polarizers and analyzers. In this approach, Stokes suggested passing light through a fixed Nicol prism (acting as a polarizer) and then analyzing the transmitted intensity with a second rotatable Nicol prism to determine the relative intensities of orthogonally polarized components, enabling the resolution of mixed polarization streams into their linear and circular elements. This laid the groundwork for later refinements, emphasizing intensity differences as proxies for the parameters without direct field measurements. The standard theoretical principle for complete Stokes parameter measurement relies on acquiring six intensities from a combination of linear polarizers and a quarter-wave plate, assuming ideal, monochromatic light and perfect optical elements.^[30] Specifically, the linear components S_0, S_1, and S_2 are obtained using polarizers oriented at 0° (horizontal), 90° (vertical), 45°, and 135° (or -45°), where the total intensity is S_0 = I_{0^\circ} + I_{90^\circ}, the horizontal-vertical difference is S_1 = I_{0^\circ} - I_{90^\circ}, and the ±45° difference is S_2 = I_{45^\circ} - I_{135^\circ}.^[30] For the circular component S_3, a quarter-wave plate is inserted with its fast axis at 45° to the reference axis, converting right- and left-circular polarizations into linear ones detectable by the 0° and 90° polarizers, yielding S_3 = I_R - I_L, where I_R and I_L are the intensities after the retarder for the respective circular states.^[30] These relations assume the incident light is quasi-monochromatic and the setup is aligned such that the polarizer transmission axes are precisely orthogonal, with the quarter-wave plate providing exactly \pi/2 phase retardation. Theoretical error sources in these measurements arise primarily from the wavelength dependence of the quarter-wave plate's retardance and effects from finite spectral bandwidth.^[31] The retardance \delta of a wave plate varies with wavelength as \delta = \frac{2\pi d (n_e - n_o)}{\lambda}, where d is thickness, n_e and n_o are extraordinary and ordinary refractive indices, and \lambda is wavelength; thus, deviation from the design wavelength \lambda_0 (where \delta = \pi/2) introduces phase errors that couple linear and circular components, distorting S_3 and partially affecting S_1 and S_2.^[31] For finite bandwidth \Delta\lambda, the effective retardance averages over the spectrum, leading to reduced modulation efficiency and crosstalk between Stokes parameters, with error magnitudes scaling as \Delta\delta \approx \frac{\delta \Delta\lambda}{\lambda} for small bandwidths.^[32] These effects are negligible for narrowband sources but necessitate corrections or achromatic elements for broadband light.

Practical Techniques and Instruments

Practical techniques for measuring Stokes parameters often rely on sequential or simultaneous intensity measurements through polarizing elements, building on the theoretical principles of projecting the polarization state onto basis vectors. One common approach involves rotating polarizer and analyzer setups, where a linear polarizer or analyzer is rotated in steps (typically at 0°, 45°, 90°, and 135°) in front of a detector to capture intensities that directly relate to the Stokes components S₀, S₁, and S₂; a quarter-wave plate is added before the analyzer to measure circular polarization S₃ by converting it to linear components.^[1] This method, known as the rotating analyzer technique, is straightforward for laboratory use but requires mechanical rotation, limiting its speed to the rotation rate, often around 1-10 Hz.^[33] For improved efficiency, the rotating quarter-wave plate method rotates a quarter-wave plate at a known speed while fixing the analyzer, allowing all four Stokes parameters to be extracted from the modulated intensity signal using Fourier analysis, achieving simultaneous measurement with a single detector.^[31] To enable faster, real-time measurements without mechanical rotation, modulation techniques employ photoelastic modulators (PEMs), which are resonant devices that introduce a time-varying retardance at high frequencies (typically 20-50 kHz) using piezoelectric-driven stress on an optical material like fused silica. In a single-PEM setup, the modulator is oriented at 45° to a fixed analyzer, yielding signals at the fundamental (1f) and second harmonic (2f) frequencies that provide S₃ and a combination of S₁ and S₂, respectively; full separation requires an additional rotation or wave plate adjustment.^[34] Dual-PEM configurations, with modulators tuned to slightly different frequencies (e.g., 50 kHz and 52 kHz) and axes at 0° and 45°, demodulate the intensity into distinct harmonics for all four Stokes parameters simultaneously, offering polarization sensitivity down to 10⁻⁶ of the total intensity and enabling broadband operation from UV to mid-IR.^[35] These systems are particularly valued for their stability and low noise, as the high modulation frequency rejects low-frequency drifts. For simultaneous, non-scanning measurements, division-of-amplitude polarimeters (DOAPs) split the incident beam into four paths using polarizing beam splitters and wave plates, each path projecting a unique Stokes component onto a dedicated detector; for example, one path measures horizontal-vertical linear difference (S₁), another at 45° (S₂), a third with quarter-wave retardance (S₃), and the fourth total intensity (S₀).^[36] This approach provides real-time full-Stokes vectors at rates up to kilohertz, with open-source implementations achieving better than 1% accuracy in polarization degree across UV to IR wavelengths using standard optics.^[37] Extending this to spatial resolution, imaging polarimeters integrate DOAP principles with pixelated sensors or micro-optics arrays; division-of-focal-plane sensors, for instance, assign sub-pixel polarizers to measure local Stokes parameters per frame, while channeled systems encode modulation spatially for snapshot full-Stokes imaging.^[38] Advanced designs using switchable liquid crystal cells avoid resolution loss from fixed micropolarizers, preserving full camera spatial resolution (e.g., 1080p) while measuring S₀, S₁, and S₂ with over 80% transmission efficiency.^[39] Recent developments as of 2025 include metasurface-based polarimeters, which enable compact, single-shot full-Stokes detection even through scattering media or with wavelength insensitivity, using nanostructured elements for integrated polarization analysis.^[40] Calibration of these instruments is essential to correct for misalignments, retardance errors, and detector crosstalk, typically involving a least-squares fit of measured intensities against known input polarization states generated by a calibration polarizer and rotatable wave plates. Procedures often include self-calibration using a quarter-wave plate of known fast axis or reference light with predefined Stokes vectors, minimizing in situ time to minutes while reducing uncertainties from temperature variations or component drifts.^[41] High-end systems, such as those for solar observations, employ multi-parameter telescope models and gain corrections to achieve polarization fraction accuracies of 0.1% for circular components and 0.4% for linear, with net-linear signals recoverable to 10⁻³ of continuum intensity against backgrounds of 10⁻².^[42] These techniques have supported polarimetric remote sensing applications since the 1970s, where Stokes measurements from satellites enhance aerosol and ocean color retrievals.^[43]

Quantum Interpretation

Link to Density Operators

In quantum mechanics, the classical Stokes parameters describing light polarization acquire a precise operational meaning through the density matrix formalism, which treats photon polarization states as mixed quantum states in a two-level system. This link was established in the mid-1950s by Ugo Fano, who extended the classical Stokes method to quantum calculations of polarization effects using operator techniques based on Pauli spin matrices. The approach formalizes polarization as an observable property of quantum ensembles, bridging statistical optics with quantum state descriptions. Photon polarization is modeled as a pseudospin-1/2 system, with the two basis states corresponding to horizontal (|H⟩) and vertical (|V⟩) polarizations. The Stokes operators, which generalize the classical parameters to quantum observables, are expressed in terms of the Pauli matrices in this basis:

\begin{align} \hat{S}_1 &= \hat{\sigma}_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \\ \hat{S}_2 &= \hat{\sigma}_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \\ \hat{S}_3 &= \hat{\sigma}_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. \end{align}

These operators satisfy the su(2) algebra of angular momentum, with expectation values bounded by the total photon number, analogous to spin components. For a general polarization state represented by the density matrix \rho (a positive semidefinite operator with \Tr(\rho) = 1 for a normalized single-photon ensemble), the Stokes parameters emerge as the traces of \rho with the Pauli matrices:

S_i = \Tr(\rho \hat{S}_i), \quad i=1,2,3; \quad S_0 = \Tr(\rho).

This yields the Bloch vector representation \vec{S} = (S_1, S_2, S_3), where |\vec{S}| \leq S_0, with the inequality reflecting partial coherence or mixedness. Pure states (fully polarized light) correspond to |\vec{S}| = S_0, while unpolarized light maps to \vec{S} = 0. Fano's operator techniques further allow computation of polarization evolution under interactions, such as scattering, by propagating the density matrix. This quantum framework generalizes partially polarized classical light to ensembles of photons in mixed states, where the degree of polarization quantifies the state's deviation from complete randomness (maximal mixedness \rho = I/2). The analogy parallels the classical coherency matrix but incorporates quantum superposition and entanglement effects inherent to the density operator.

Applications in Quantum Optics

In quantum optics, the Stokes parameters provide a powerful framework for representing polarization-entangled states, such as Bell states, on the Poincaré sphere. For two-photon polarization-entangled states generated via spontaneous parametric down-conversion, the polarization states of the entangled photons are correlated such that their representations on the Poincaré sphere exhibit specific geometric relations. The singlet Bell state |ψ⁻⟩ = (1/√2)(|HV⟩ - |VH⟩), where H and V denote horizontal and vertical polarizations, corresponds to the two photons having antipodal points on the sphere (θ₁ + θ₂ = π, φ₁ - φ₂ = π mod 2π), reflecting perfect anticorrelation in their Stokes vectors. In contrast, the triplet state |ψ⁺⟩ = (1/√2)(|HV⟩ + |VH⟩) maps to points at the same latitude (θ₁ = θ₂, φ₁ + φ₂ = π mod 2π), indicating correlated but phase-shifted polarizations. These representations extend the classical Poincaré sphere to quantum superpositions, enabling visualization of entanglement correlations beyond single-photon states.^[44] The quantum generalization of Stokes parameters manifests as expectation values of Hermitian Stokes operators, which are bilinear combinations of creation and annihilation operators for orthogonal polarization modes. Specifically, the operators are defined as Ŝ₀ = â_H† â_H + â_V† â_V (total photon number), Ŝ₁ = â_H† â_H - â_V† â_V, Ŝ₂ = â_H† â_V + â_V† â_H, and Ŝ₃ = i(â_V† â_H - â_H† â_V), all of which are Hermitian (Ŝ_i† = Ŝ_i) and correspond to measurable observables in polarization experiments. For a quantum state described by the density operator ρ, the Stokes parameters are S_i = ⟨Ŝ_i⟩ = Tr(ρ Ŝ_i), linking the quantum description directly to experimentally accessible quantities while preserving the real-valued nature of classical Stokes parameters. This Hermitian structure ensures that the parameters lie within the quantum Poincaré sphere of radius equal to the mean photon number, with uncertainties governed by commutation relations [Ŝ₁, Ŝ₂] = 2i Ŝ₃ (and cyclic permutations). Quantum state tomography leverages Stokes measurements to reconstruct the full density matrix of polarization-encoded photonic qubits or qudits. By performing projections onto multiple bases corresponding to the Stokes operators—such as measuring photon counts after wave plates and polarizers—one obtains the expectation values S_i, which parameterize the state on the Poincaré sphere. For single-qubit tomography, six measurements (two per Stokes parameter) suffice to reconstruct ρ, as the parameters fully specify the Bloch vector for pure or mixed states. In multi-photon cases, such as two-qubit entangled states, joint Stokes measurements on both photons yield the 15 parameters needed for complete tomography, enabling verification of entanglement fidelity and Bell inequality violations. This approach has been experimentally realized for photonic Bell states, achieving reconstruction fidelities above 99% with standard polarimetric setups. Post-2000 advances have integrated Stokes polarimetry into quantum key distribution (QKD) protocols, enhancing security and robustness against polarization disturbances. In polarization-encoded QKD schemes, such as those using coherent states, the key is encoded in the Stokes parameters of the transmitted pulses, allowing detection via homodyne measurements of Ŝ_i and enabling secure key distribution over optical fibers, while compensating for channel-induced rotations such as birefringence.^[45] In quantum sensing, Stokes parameters facilitate high-precision polarimetry for detecting weak fields, such as in biological samples, where quantum-enhanced estimation of Stokes vector rotations achieves Heisenberg-limited sensitivity, improving resolution in magnetometry and ellipsometry by factors of √N (N photons) over classical limits. These applications underscore the transition from classical polarimetry to quantum-enhanced protocols, with ongoing developments in satellite-based QKD incorporating adaptive Stokes compensation for atmospheric turbulence.

References

[1]
[PDF] Measuring the Stokes polarization parameters
The Stokes formulation for representing polarized light is discussed along with the classical measurement method for determining the Stokes polarization ...
[2]
On the Composition and Resolution of Streams of Polarized Light ...
On the Composition and Resolution of Streams of Polarized Light from different Sources. Published online by Cambridge University Press: 07 September 2010.
[3]
Polarization and the Stokes Parameters | American Journal of Physics
The Stokes parameters have been found to offer a very convenient method for the description of polarization of both electromagnetic radiation and elementary ...Missing: review | Show results with:review
[4]
Review of polarization sensitive optical coherence tomography and ...
Jul 1, 2002 · In this review paper the authors describe theory and calculation of the Stokes vectors in more detail than that provided in our early reports.
[5]
Anatomy of an Electromagnetic Wave - NASA Science
Aug 3, 2023 · One of the physical properties of light is that it can be polarized. Polarization is a measurement of the electromagnetic field's alignment. In ...
[6]
[PDF] Chapter 12 - Polarization - MIT OpenCourseWare
Polarization is a general feature of transverse waves in three dimensions. The general elec- tromagnetic plane wave has two polarization states, ...
[7]
[PDF] Lecture 14: Polarization
This equation implies that the magnetic field in a plane wave is completely determined by the electric field. In particular, it implies that their ...
[8]
The Feynman Lectures on Physics Vol. I Ch. 33: Polarization
Light is linearly polarized (sometimes called plane polarized) when the electric field oscillates on a straight line; Fig. 33–1 illustrates linear polarization.
[9]
[PDF] Chapter 6 - Polarization of Light - PhysLab
Such sources are commonly referred to as unpolarized. It is common to have a mixture of unpolarized and polarized light, called partially polarized light.
[10]
[PDF] What is Polarization?
Whenever DOP=0, light is said to be unpolarized, and whenever. DOP=1, it is totally polarized. Intermediate cases correspond to partially polarized light.
[11]
July 1816: Fresnel's Evidence for the Wave Theory of Light
In July 1816, Fresnel published preliminary results on diffraction, aiming to develop a full theory, building on Young's work and embracing the wave theory.
[12]
200 Years of Fresnel's Legacy - Optics & Photonics News
Sep 1, 2015 · Fresnel discovered the basic equations that describe how light behaves when it moves through media of different refractive indices; his ...
[13]
[PDF] Physics of Light and Optics
... Jones Vectors for Representing Polarization. R. Clark Jones (1916–2004 ... fully polarized (such that Iun = 0), the degree of polarization is one. A ...
[14]
Polarization: Stokes Vectors - Ocean Optics Web Book
Oct 13, 2021 · Stokes vectors are a quantitative way to specify the state of polarization of light, using an array of four real numbers.
[15]
The Stokes Polarization Parameters - SPIE
The first Stokes parameter S0 describes the total intensity of the optical beam; the second parameter S1 describes the preponderance of LHP light over LVP light ...<|control11|><|separator|>
[16]
https://www.meadowlark.com/wp-content/uploads/2022/04/Polarimetry-using-liquid-crystal-variable-retarders.pdf
[17]
[PDF] Polarized light and the STOKES PARAMETERS - UVIC
Its wavelength or frequency. Its wavelength depends upon the refractive index of the material in which it is travelling, whereas its frequency does not.Missing: interpretation | Show results with:interpretation
[18]
https://opg.optica.org/ao/fulltext.cfm?uri=ao-47-22-4009
[19]
Stokes Parameters
Under a counterclockwise rotation of the axes through an angle $\psi$ the intensity ${\bf T}$ transforms as ${\bf T}'={\bf R}{\bf . $\Theta$ and V remain ...
[20]
[PDF] Chapter 7. Electromagnetic Wave Propagation
We will start from the simplest, plane waves in uniform and isotropic media, and then proceed to a discussion of nonuniform systems, bringing up such effects as ...
[21]
Stokes-Algebra Formalism - Optics Letters - Optica
The advantage of the new ordering of the Stokes parameters is that familiar relationships are preserved; the rotation matrix looks like a rotation matrix, ...Missing: bases | Show results with:bases
[22]
Theory of lossless polarization attraction in telecommunication fibers
The Stokes parameters of the signal and pump beam are normalized with respect to S 0 + ( z , t ) and S 0 − ( z , t ) , respectively. Download Full Size | PDF.
[23]
Statistics of the Stokes parameters - Optica Publishing Group
The probability-density functions and lower-order moments of the four Stokes parameters are obtained as a function of the degree of polarization and the mean ...
[24]
[PDF] OF THE PHOTO SHUTTER - Mueller Matrix Polarimetry
1). POLARIZATION OPTICS. Poincaré's representation of polarized light. A polarized light beam is usually described by giving its electrical vector. E. X. E. P₁ ...Missing: original | Show results with:original
[25]
https://opg.optica.org/josaa/abstract.cfm?uri=josaa-13-5-1106
[26]
[PDF] Arbitrary decomposition of a Mueller matrix - arXiv
In analogy to the fact that a Stokes vector can be pure (fully polarized) or not, Mueller matrices that preserve the degree of polarization of totally polarized ...
[27]
Measurement of polarized light interactions via the Mueller matrix
A new instrument for rapid and accurate measurement of the Mueller matrix is described. Distinct measurements of all sixteen elements are made ...
[28]
The Mueller Matrices for Polarizing Components - SPIE
An explanation of the Mueller Matrices for Polarizing Components from the Field Guide to Polarization, SPIE Press.Missing: example | Show results with:example
[29]
[PDF] C:\Users\Lenovo\Desktop\Emil Wolf - Introduction to the Theory of ...
EMIL WOLF. D. Page 2. Introduction to the Theory of Coherence and Polarization of Light ... 9.5 Generalized Stokes parameters. Problems ... coherency matrix," a 2 × ...
[30]
Classical Measurement of the Stokes Parameters - SPIE Digital Library
The four Stokes parameters of a polarized beam can be measured by passing a beam sequentially through two polarizing elements known as a wave plate and a ...
[31]
[PDF] Rotating-wave-plate Stokes polarimeter for differential group delay ...
An error in Aλ therefore produces a proportional er- ror in DGD. This error increases as the wavelength step size decreases.Missing: finite | Show results with:finite
[32]
Analysis of errors in polarimetry using a rotating waveplate
In terms of geometric dimensions, three modulation error sources are analyzed: the waveplate axial error, waveplate rotation axis tip–tilt error (zenithal error) ...
[33]
[PDF] Measurement of the Stokes parameters of light
Stokes parameters. The technique which can be fully automated incorporates a monochromator and single. photon counting detection and can thus be applied over a ...
[34]
[PDF] Stokes Polarimetry | Hinds Instruments
Two photoelastic modulators may be used to provide a polarimeter with “real time” measurement capability. The two modulators are mounted with their modulator ...
[35]
Stokes polarimeter using two photoelastic modulators
This Stokes polarimeter employs two low birefringence photoelastic modulators (PEMs) operating at different resonant frequencies. The two PEMs in the instrument ...
[36]
Division-of-amplitude Photopolarimeter (DOAP) for the ...
Nov 14, 2010 · To measure all four Stokes parameters of a light beam simultaneously, the beam is divided into four separate beams using a beamsplitter and two ...Missing: polarimeter | Show results with:polarimeter
[37]
Homemade open-source full-Stokes polarimeter based on division ...
Sep 16, 2024 · We present an open-source homemade Stokes polarimeter for real-time determination of the state and degree of polarization of a light beam by measuring all four ...
[38]
Signal-to-noise analysis of Stokes parameters in division of focal ...
This high-resolution polarimetric sensor extracts the first three Stokes parameters of the incident light for every acquired frame on a neighborhood of pixels.
[39]
Polarimetric imaging with high spatial resolution | Scientific Reports
Jul 5, 2025 · ... Stokes parameters. The method does not use pixels nor traditional polarizers, thus providing very high spatial resolution and high light ...
[40]
[PDF] Simple self-calibrating polarimeter for measuring the Stokes ...
Nov 15, 2021 · The relative Stokes parameters can thus be calculated from a set of Fourier components determined at a single linear polarizer orientation ˜α.
[41]
Advanced Stokes Polarimeter Calibration - IOP Science
The Advanced Stokes Polarimeter is calibrated using a seven-parameter telescope model, polarizing optics, and a least-squares solution, also correcting for ...
[42]
Principle and Implementation of Stokes Vector Polarization Imaging ...
Polarization imaging technology has been studied since the 1970s in foreign countries. ... The Stokes parameters obtained by these schemes have been increased ...Missing: history | Show results with:history
[43]
https://www.mdpi.com/2076-3417/12/13/6613
[44]
Quantum key distribution using polarized coherent states - arXiv
Jun 9, 2006 · ... Quantum key distribution using polarized coherent states. ... The key encoding is performed using the variables known as Stokes parameters, rather ...