Coherence-Stokes Vector in Optical Fields

• Coherence-Stokes vector is a two-point generalization of traditional Stokes parameters that describes the cross-correlations of electromagnetic field components.
• It unifies polarization and coherence by using a matrix framework derived from the cross-spectral density, separating intensity scaling from genuine coherence properties.
• Experimental implementations via interferometry and visibility measurements validate its practical use in imaging, fiber optics, and quantum induced-coherence protocols.

Searching arXiv for recent and foundational papers on coherence-Stokes vector and closely related formalisms. arxiv_search(query="coherence Stokes vector optical fields polarization", max_results=10) A coherence-Stokes vector is a Stokes-vector representation of second-order correlations in electromagnetic fields, obtained by extending the ordinary single-point Stokes parameters to two-point quantities in space and/or time. In this formulation, the usual polarization observables are promoted to correlation observables built from the cross-correlations of orthogonal field components, so that partial coherence and partial polarization are described within a single matrix/vector framework (Kanseri et al., 2021). Across adjacent literatures, closely related constructions also appear under different names, notably the coherency vector for nondepolarizing media, visibility Stokes parameters for induced-coherence protocols, and Stokes-like encodings of local correlation structure (José et al., 2020, Kysela et al., 2024, Li, 6 Jun 2025).

1. Terminological scope

The most direct use of the expression refers to the collection of generalized, two-point Stokes parameters associated with the cross-spectral density matrix or beam coherence-polarization matrix. In that sense, the coherence-Stokes vector is not a new replacement for the Stokes vector, but its two-point generalization (Kanseri et al., 2021). By contrast, some nearby works use Stokes-linked vector objects for other purposes.

Literature	Object	Role
(Kanseri et al., 2021, Laatikainen et al., 2023)	Coherence Stokes parameters / coherence-Stokes vector	Two-point description of coherence and polarization
(José et al., 2020)	Coherency vector	Representation of a nondepolarizing medium
(Kysela et al., 2024)	Visibility Stokes parameters	Visibility-defined analogue for undetected photons
(Li, 6 Jun 2025)	Stokes-parameter correlation map	Spatially resolved local correlation encoding

This distribution of terminology matters conceptually. In the two-point optical-coherence literature, the vector components describe correlations of field components at two arguments. In polarimetric transformation theory, the vector represents the medium rather than the field state. In induced-coherence quantum protocols, the Stokes-like quantities are inferred from visibilities and depend on source coherence and environment. A plausible implication is that “coherence-Stokes vector” is best treated as a family resemblance term rather than a universally fixed object.

2. Two-point vectorial coherence formalism

For a random electromagnetic field with transverse components $E_x(\mathbf r,\omega)$ and $E_y(\mathbf r,\omega)$, the basic object is the electric cross-spectral density matrix

$$\mathbf W(\mathbf r_1,\mathbf r_2,\omega)=\left[W_{ij}(\mathbf r_1,\mathbf r_2,\omega)\right],\qquad W_{ij}(\mathbf r_1,\mathbf r_2,\omega)=\left\langle E_i^*(\mathbf r_1,\omega)E_j(\mathbf r_2,\omega)\right\rangle,$$

with $i,j\in\{x,y\}$. In the space-time domain, the analogous object is the beam coherence-polarization matrix

$$\mathbf J(\mathbf r_1,\mathbf r_2)= \begin{pmatrix} J_{xx}(\mathbf r_1,\mathbf r_2) & J_{xy}(\mathbf r_1,\mathbf r_2)\ J_{yx}(\mathbf r_1,\mathbf r_2) & J_{yy}(\mathbf r_1,\mathbf r_2) \end{pmatrix}, \qquad J_{ij}(\mathbf r_1,\mathbf r_2)=\left\langle E_i^*(\mathbf r_1,t)E_j(\mathbf r_2,t)\right\rangle.$$

At $\mathbf r_1=\mathbf r_2$, these reduce to the familiar single-point coherency matrix of polarization optics (Kanseri et al., 2021).

The generalized Stokes parameters are linear combinations of the matrix elements, exactly paralleling ordinary Stokes calculus: $$S_0(\mathbf r_1,\mathbf r_2,\omega)= \left\langle E_x^*(\mathbf r_1,\omega)E_x(\mathbf r_2,\omega)\right\rangle+ \left\langle E_y^*(\mathbf r_1,\omega)E_y(\mathbf r_2,\omega)\right\rangle,$$

$$S_1(\mathbf r_1,\mathbf r_2,\omega)= \left\langle E_x^*(\mathbf r_1,\omega)E_x(\mathbf r_2,\omega)\right\rangle- \left\langle E_y^*(\mathbf r_1,\omega)E_y(\mathbf r_2,\omega)\right\rangle,$$

$$S_2(\mathbf r_1,\mathbf r_2,\omega)= \left\langle E_x^*(\mathbf r_1,\omega)E_y(\mathbf r_2,\omega)\right\rangle+ \left\langle E_y^*(\mathbf r_1,\omega)E_x(\mathbf r_2,\omega)\right\rangle,$$

$$S_3(\mathbf r_1,\mathbf r_2,\omega)= i\left[ \left\langle E_x^*(\mathbf r_1,\omega)E_y(\mathbf r_2,\omega)\right\rangle- \left\langle E_y^*(\mathbf r_1,\omega)E_x(\mathbf r_2,\omega)\right\rangle \right].$$

At two points, these are correlation functions rather than merely polarization descriptors. The physical interpretation given in the review literature is explicit: $E_y(\mathbf r,\omega)$0 is the total summed correlation between the $E_y(\mathbf r,\omega)$1 and $E_y(\mathbf r,\omega)$2 components across the two points; $E_y(\mathbf r,\omega)$3, $E_y(\mathbf r,\omega)$4, and $E_y(\mathbf r,\omega)$5 are basis-resolved differences of those correlations in linear, $E_y(\mathbf r,\omega)$6, and circular bases, respectively (Kanseri et al., 2021).

This is the core unification: polarization concerns relations between field components, coherence concerns relations between field values at different points or times, and for vector fields both are encoded by the same correlation matrix. The coherence-Stokes vector is the vectorized expression of that fact.

3. Normalization, contrasts, and cross-spectral purity

To separate coherence from trivial intensity scaling, the generalized Stokes parameters are normalized by the local spectral densities

$E_y(\mathbf r,\omega)$7

The normalized generalized Stokes parameters are

$E_y(\mathbf r,\omega)$8

When the two interfering beams have equal intensities, the modulus

$E_y(\mathbf r,\omega)$9

acts as a contrast parameter with $$\mathbf W(\mathbf r_1,\mathbf r_2,\omega)=\left[W_{ij}(\mathbf r_1,\mathbf r_2,\omega)\right],\qquad W_{ij}(\mathbf r_1,\mathbf r_2,\omega)=\left\langle E_i^*(\mathbf r_1,\omega)E_j(\mathbf r_2,\omega)\right\rangle,$$0, and the electromagnetic degree of coherence is

$$\mathbf W(\mathbf r_1,\mathbf r_2,\omega)=\left[W_{ij}(\mathbf r_1,\mathbf r_2,\omega)\right],\qquad W_{ij}(\mathbf r_1,\mathbf r_2,\omega)=\left\langle E_i^*(\mathbf r_1,\omega)E_j(\mathbf r_2,\omega)\right\rangle,$$1

The last identity makes the vectorial content explicit: full electromagnetic coherence is not exhausted by a single scalar fringe visibility, but is resolved across all four Stokes-like modulations (Kanseri et al., 2021).

For random nonstationary electromagnetic beams, the formalism is sharpened by introducing the intensity-normalized time-integrated coherence Stokes parameters

$$\mathbf W(\mathbf r_1,\mathbf r_2,\omega)=\left[W_{ij}(\mathbf r_1,\mathbf r_2,\omega)\right],\qquad W_{ij}(\mathbf r_1,\mathbf r_2,\omega)=\left\langle E_i^*(\mathbf r_1,\omega)E_j(\mathbf r_2,\omega)\right\rangle,$$2

Within a Young two-pinhole geometry, cross-spectral purity of a Stokes parameter $$\mathbf W(\mathbf r_1,\mathbf r_2,\omega)=\left[W_{ij}(\mathbf r_1,\mathbf r_2,\omega)\right],\qquad W_{ij}(\mathbf r_1,\mathbf r_2,\omega)=\left\langle E_i^*(\mathbf r_1,\omega)E_j(\mathbf r_2,\omega)\right\rangle,$$3 is defined by

$$\mathbf W(\mathbf r_1,\mathbf r_2,\omega)=\left[W_{ij}(\mathbf r_1,\mathbf r_2,\omega)\right],\qquad W_{ij}(\mathbf r_1,\mathbf r_2,\omega)=\left\langle E_i^*(\mathbf r_1,\omega)E_j(\mathbf r_2,\omega)\right\rangle,$$4

with frequency-independent constants $$\mathbf W(\mathbf r_1,\mathbf r_2,\omega)=\left[W_{ij}(\mathbf r_1,\mathbf r_2,\omega)\right],\qquad W_{ij}(\mathbf r_1,\mathbf r_2,\omega)=\left\langle E_i^*(\mathbf r_1,\omega)E_j(\mathbf r_2,\omega)\right\rangle,$$5. The central result is the reduction formula

$$\mathbf W(\mathbf r_1,\mathbf r_2,\omega)=\left[W_{ij}(\mathbf r_1,\mathbf r_2,\omega)\right],\qquad W_{ij}(\mathbf r_1,\mathbf r_2,\omega)=\left\langle E_i^*(\mathbf r_1,\omega)E_j(\mathbf r_2,\omega)\right\rangle,$$6

which the authors identify as equivalent to cross-spectral purity under the stated conditions (Laatikainen et al., 2023).

For $$\mathbf W(\mathbf r_1,\mathbf r_2,\omega)=\left[W_{ij}(\mathbf r_1,\mathbf r_2,\omega)\right],\qquad W_{ij}(\mathbf r_1,\mathbf r_2,\omega)=\left\langle E_i^*(\mathbf r_1,\omega)E_j(\mathbf r_2,\omega)\right\rangle,$$7, the reduction is scalar-like. For $$\mathbf W(\mathbf r_1,\mathbf r_2,\omega)=\left[W_{ij}(\mathbf r_1,\mathbf r_2,\omega)\right],\qquad W_{ij}(\mathbf r_1,\mathbf r_2,\omega)=\left\langle E_i^*(\mathbf r_1,\omega)E_j(\mathbf r_2,\omega)\right\rangle,$$8, the reduction contains an additional factor that depends exclusively on the polarization properties. This is the principal refinement over scalar coherence theory: vectorial cross-spectral purity factorizes into spatial and temporal coherence contributions, but polarization-state structure remains an independent multiplicative ingredient (Laatikainen et al., 2023).

4. Distinction from coherency-vector representations of media

A recurring source of confusion is the relation between a coherence-Stokes vector and the coherency vector formalism for polarimetric transformations. In the latter, the field state still uses the ordinary Stokes vector, but a pure, nondepolarizing medium is represented by a four-component complex vector derived from the Jones matrix expansion

$$\mathbf W(\mathbf r_1,\mathbf r_2,\omega)=\left[W_{ij}(\mathbf r_1,\mathbf r_2,\omega)\right],\qquad W_{ij}(\mathbf r_1,\mathbf r_2,\omega)=\left\langle E_i^*(\mathbf r_1,\omega)E_j(\mathbf r_2,\omega)\right\rangle,$$9

The associated Jones matrix is recovered as

$i,j\in\{x,y\}$0

The standard coherency matrix of the field is written

$i,j\in\{x,y\}$1

and the Stokes vector remains the representation of the polarization state through the unitary mapping

$i,j\in\{x,y\}$2

In this framework, the medium—not the state—is encoded by $i,j\in\{x,y\}$3 (José et al., 2020).

The distinctive algebraic ingredient is the coherency product

$i,j\in\{x,y\}$4

which makes the serial composition of nondepolarizing media vectorial: $i,j\in\{x,y\}$5 The induced Stokes transformation can then be written compactly as

$i,j\in\{x,y\}$6

The formalism is associative, noncommutative, and compatible with inversion and scalar multiplication. It also extends to depolarizing media via coherency matrices formed from partially coherent sums of pure coherency vectors (José et al., 2020).

This usage is closely related to Stokes calculus but is not identical to the two-point coherence-Stokes vector. The distinction is structural: one object parameterizes field correlations, the other parameterizes medium transformations.

5. Measurement and Stokes-space manifestations

The generalized coherence-Stokes quantities are experimentally accessible through interferometric methods. Modified Young interferometers with polarizers and wave plates can isolate the elements of the cross-spectral density matrix; generalized Stokes parameters can also be inferred directly from observed interference contrasts; and temporal coherence variants can be measured with Hanbury Brown–Twiss-type intensity interferometry. In these settings, fringe visibility, spectral modulation, and intensity correlations reveal the underlying two-point matrix elements and their phases (Kanseri et al., 2021).

A distinct but related manifestation occurs when coherence loss is viewed geometrically in Stokes space rather than through field-phase interferometry. In nonlinear depolarization of light in fiber, a fully polarized continuous-wave probe co-propagating with unpolarized ASE noise undergoes a fast random walk on the Poincaré sphere. The normalized Stokes vector

$i,j\in\{x,y\}$7

then occupies a finite region rather than a single point. The measured quantity is a Stokes-space spread expressed as a solid angle or corresponding apex angle, with the net nonlinear contribution defined by

$i,j\in\{x,y\}$8

The reported scattering angles over trans-Pacific distances are within a few degrees, and the spread increases with propagation distance while decreasing with repeater output power (Moeller, 2020).

Although that work does not define a formal “coherence-Stokes vector,” it provides a geometric operational picture of polarization coherence: coherent evolution corresponds to a sharply localized Stokes trajectory, while incoherent evolution appears as a scattered cloud on the sphere. This suggests a complementary viewpoint in which coherence is inferred from the concentration, spread, and spectral content of the Stokes trajectory rather than directly from field-correlation matrices.

6. Extensions to local-correlation mapping and quantum induced coherence

Outside classical coherence theory proper, Stokes-like coherence vectors have been adapted to other correlation problems. An adjacency-based data-visualization method maps local pairwise correlations between two fields into Stokes-like quantities

$i,j\in\{x,y\}$9

followed by local sums $$\mathbf J(\mathbf r_1,\mathbf r_2)= \begin{pmatrix} J_{xx}(\mathbf r_1,\mathbf r_2) & J_{xy}(\mathbf r_1,\mathbf r_2)\ J_{yx}(\mathbf r_1,\mathbf r_2) & J_{yy}(\mathbf r_1,\mathbf r_2) \end{pmatrix}, \qquad J_{ij}(\mathbf r_1,\mathbf r_2)=\left\langle E_i^*(\mathbf r_1,t)E_j(\mathbf r_2,t)\right\rangle.$$0. The correlation angle

$$\mathbf J(\mathbf r_1,\mathbf r_2)= \begin{pmatrix} J_{xx}(\mathbf r_1,\mathbf r_2) & J_{xy}(\mathbf r_1,\mathbf r_2)\ J_{yx}(\mathbf r_1,\mathbf r_2) & J_{yy}(\mathbf r_1,\mathbf r_2) \end{pmatrix}, \qquad J_{ij}(\mathbf r_1,\mathbf r_2)=\left\langle E_i^*(\mathbf r_1,t)E_j(\mathbf r_2,t)\right\rangle.$$1

encodes correlation type, and the scalar $$\mathbf J(\mathbf r_1,\mathbf r_2)= \begin{pmatrix} J_{xx}(\mathbf r_1,\mathbf r_2) & J_{xy}(\mathbf r_1,\mathbf r_2)\ J_{yx}(\mathbf r_1,\mathbf r_2) & J_{yy}(\mathbf r_1,\mathbf r_2) \end{pmatrix}, \qquad J_{ij}(\mathbf r_1,\mathbf r_2)=\left\langle E_i^*(\mathbf r_1,t)E_j(\mathbf r_2,t)\right\rangle.$$2 encodes correlation degree, with $$\mathbf J(\mathbf r_1,\mathbf r_2)= \begin{pmatrix} J_{xx}(\mathbf r_1,\mathbf r_2) & J_{xy}(\mathbf r_1,\mathbf r_2)\ J_{yx}(\mathbf r_1,\mathbf r_2) & J_{yy}(\mathbf r_1,\mathbf r_2) \end{pmatrix}, \qquad J_{ij}(\mathbf r_1,\mathbf r_2)=\left\langle E_i^*(\mathbf r_1,t)E_j(\mathbf r_2,t)\right\rangle.$$3 meaning no correlation and $$\mathbf J(\mathbf r_1,\mathbf r_2)= \begin{pmatrix} J_{xx}(\mathbf r_1,\mathbf r_2) & J_{xy}(\mathbf r_1,\mathbf r_2)\ J_{yx}(\mathbf r_1,\mathbf r_2) & J_{yy}(\mathbf r_1,\mathbf r_2) \end{pmatrix}, \qquad J_{ij}(\mathbf r_1,\mathbf r_2)=\left\langle E_i^*(\mathbf r_1,t)E_j(\mathbf r_2,t)\right\rangle.$$4 meaning perfect correlation. The method is explicitly designed to respect the two-fold symmetry of the correlation vector and to reveal subregions with different correlation regimes (Li, 6 Jun 2025). In this setting, the Stokes structure is applied to correlation regularity rather than optical polarization.

In induced-coherence quantum optics, polarization information of undetected photons is inferred from measured visibilities. The visibility Stokes parameters are defined as

$$\mathbf J(\mathbf r_1,\mathbf r_2)= \begin{pmatrix} J_{xx}(\mathbf r_1,\mathbf r_2) & J_{xy}(\mathbf r_1,\mathbf r_2)\ J_{yx}(\mathbf r_1,\mathbf r_2) & J_{yy}(\mathbf r_1,\mathbf r_2) \end{pmatrix}, \qquad J_{ij}(\mathbf r_1,\mathbf r_2)=\left\langle E_i^*(\mathbf r_1,t)E_j(\mathbf r_2,t)\right\rangle.$$5

and assembled into a visibility Bloch vector

$$\mathbf J(\mathbf r_1,\mathbf r_2)= \begin{pmatrix} J_{xx}(\mathbf r_1,\mathbf r_2) & J_{xy}(\mathbf r_1,\mathbf r_2)\ J_{yx}(\mathbf r_1,\mathbf r_2) & J_{yy}(\mathbf r_1,\mathbf r_2) \end{pmatrix}, \qquad J_{ij}(\mathbf r_1,\mathbf r_2)=\left\langle E_i^*(\mathbf r_1,t)E_j(\mathbf r_2,t)\right\rangle.$$6

For pure idler states, these coincide with the Bloch-vector coordinates. For mixed states, however, the quantities depend on environmental overlaps such as

$$\mathbf J(\mathbf r_1,\mathbf r_2)= \begin{pmatrix} J_{xx}(\mathbf r_1,\mathbf r_2) & J_{xy}(\mathbf r_1,\mathbf r_2)\ J_{yx}(\mathbf r_1,\mathbf r_2) & J_{yy}(\mathbf r_1,\mathbf r_2) \end{pmatrix}, \qquad J_{ij}(\mathbf r_1,\mathbf r_2)=\left\langle E_i^*(\mathbf r_1,t)E_j(\mathbf r_2,t)\right\rangle.$$7

so the measured vector is a coherence-based descriptor of the induced-coherence experiment rather than a universally unique state vector (Kysela et al., 2024).

Taken together, these extensions show that the coherence-Stokes idea is broader than a single formalism. Its common content is the representation of coherence, correlation, or indistinguishability through Stokes-like components that preserve basis structure and symmetry. In classical vector optics, this yields a unified treatment of coherence and polarization; in polarimetric media theory, it yields a vector algebra for nondepolarizing transformations; in Stokes-space diagnostics, it yields geometric measures of polarization randomness; and in quantum induced-coherence protocols, it yields visibility-defined analogues of Stokes or Bloch coordinates.