Vector-to-Tensor Polarization Conversion

• Vector-to-tensor polarization conversion is the process of recasting vectorial polarization data into higher-rank tensors to encode coherency, alignment, and geometric information.
• It is implemented across fields like optics, spin-1 matter, and gravitational theory using bilinear mappings, nonlinear processes, and covariant reformulations.
• This conversion facilitates analysis by unveiling correlations inaccessible to vector descriptions, enabling advanced techniques such as tensor decomposition and scattering matrices.

Vector-to-tensor polarization conversion denotes several related operations in which polarization data carried by vectorial degrees of freedom are recast into tensorial objects. In optics, the canonical instance is the passage from a two-component Jones field to the coherency matrix and then to Stokes and Mueller descriptions (Castillo et al., 2013). In spin-1 matter, the same phrase refers to the emergence, evolution, or measurement of rank-2 alignment tensors alongside or from vector spin observables (Silenko, 2018). In gravitational and gauge-field settings, it can denote mixed vector–tensor polarization sectors or the explicit construction of spin-2 polarization tensors from gauge vectors (Lai et al., 2024, Raziani et al., 2022). This suggests that the term is best understood structurally: a lower-rank polarization description is lifted into bilinear or higher-rank objects that encode correlations, alignment, geometry, or propagation.

1. Scope of the concept across fields

The literature uses the expression in technically distinct but formally related ways. In polarization optics, the conversion is the bilinear map from a field-level complex vector to a Hermitian tensor and then to a Stokes 4-vector. In structured-light and nonlinear optics, a spatially varying polarization vector field is coupled into a polarization-sensitive susceptibility tensor or a spatially indexed scattering tensor. In spin-1 systems, vector polarization and rank-2 tensor polarization are independent irreducible sectors of the spin density matrix, and external fields or dissipative mechanisms can generate observable tensor alignment. In gravitation and gauge theory, vector backgrounds can mix with tensor polarizations, while gauge-vector solutions can be combined into spin-2 polarization tensors (Stenflo, 2019, Yang et al., 2018, Silenko, 2018, Lai et al., 2024, Raziani et al., 2022).

Domain	Vector object	Tensor object or operation
Polarization optics	Jones field $E$	Coherency matrix $J$, Stokes 4-vector, Mueller map
Structured light	Spatially varying Jones field	$\chi^{(2)}$ nonlinear polarization, scattering tensor $\mathcal{S}$
Spin-1 matter	Spin vector $P_i$	Rank-2 polarization tensor $P_{ij}$, $T_{ij}$
Gravitational theory	Vector polarization or gauge vector	Mixed tensor modes, spin-2 polarization tensor
Coordinate-covariant geometry	Stokes vector	Stokes tensor, skyrmion tensor

2. Optical conversion from Jones vectors to coherency tensors

For a monochromatic plane wave with $\exp[i(kz-\omega t)]$ dependence, the Jones field is

$\mathbf{E}=\begin{pmatrix}E_x\E_y\end{pmatrix},\qquad E_x,E_y\in\mathbb{C},$

defined up to a global phase $\mathbf{E}\mapsto e^{i\alpha}\mathbf{E}$. In the spinor formulation, the Jones vector is, up to an overall factor and basis choice, an $J$0 spinor. A convenient normalized representative is

$J$1

with lossless polarization optics acting as $J$2, $J$3 (Castillo et al., 2013).

The vector-to-tensor step is the coherency matrix

$J$4

For a fully coherent single-mode field, $J$5 is a rank-1 projector up to intensity; for partially polarized light, $J$6 is Hermitian and positive semidefinite. The Stokes parameters follow either directly from $J$7,

$J$8

$J$9

or from the Pauli decomposition

$\chi^{(2)}$0

The map $\chi^{(2)}$1 is bilinear, whereas the map $\chi^{(2)}$2 is linear (Stenflo, 2019).

For pure states, the normalized Stokes vector is

$\chi^{(2)}$3

so that

$\chi^{(2)}$4

lies on the Poincaré sphere (Castillo et al., 2013). For partially polarized beams, the eigenvalues of $\chi^{(2)}$5 are

$\chi^{(2)}$6

with degree of polarization

$\chi^{(2)}$7

The determinant satisfies

$\chi^{(2)}$8

and the two eigenvectors define antipodal points on the Poincaré sphere (Castillo et al., 2013).

3. Geometric, Lorentzian, and phase structure

The optical tensorial reformulation is not limited to intensities. In the spinor treatment, the vector

$\chi^{(2)}$9

fixes the polarization point on the Poincaré sphere, while

$\mathcal{S}$0

provides a second spinor-derived vector whose real part is orthogonal to $\mathcal{S}$1. In that construction, $\mathcal{S}$2 is a tangent vector whose angle with the local meridian equals the optical phase, and the spinor inner product determines the Pancharatnam phase (Castillo et al., 2013).

A second geometric layer identifies the Stokes 4-vector

$\mathcal{S}$3

with a Minkowski-type space with metric $\mathcal{S}$4. The invariant

$\mathcal{S}$5

vanishes for fully polarized light and is positive for partially polarized light. Equivalently,

$\mathcal{S}$6

so depolarization appears as a “mass-like” invariant that moves the Stokes point from the null cone to its interior (Stenflo, 2019).

This formulation also reorganizes optical elements. Unitary Jones matrices act by

$\mathcal{S}$7

and induce $\mathcal{S}$8 rotations on the Poincaré sphere. More general deterministic elements act by

$\mathcal{S}$9

with nonunitary diagonal attenuators producing a “Lorentz boost” on the Stokes variables. Depolarizing elements cannot be represented by a single Jones matrix and, in Stokes space, act as affine contractions toward the sphere center (Castillo et al., 2013). The structural interpretation given in the Minkowski analysis is that symmetric parts correspond to diattenuation or boosts, whereas antisymmetric parts correspond to retardance or rotations; the Stokes and Mueller objects have “spin-2” character because they are formed from bilinear tensor products of spin-1 Jones objects (Stenflo, 2019).

4. Spatial, nonlinear, and coordinate-covariant generalizations

In structured-light optics, vector-to-tensor conversion becomes explicitly spatial. For a linearly polarized cylindrical vector beam,

$P_i$0

direct local squaring in a second-order nonlinear process generates unwanted constant terms. The Sagnac-loop scheme for nonlinear frequency conversion therefore first maps the input into an “exponential form”

$P_i$1

so that squaring yields only $P_i$2 terms. In the crystal, the nonlinear polarization obeys

$P_i$3

and the Sagnac architecture maps both polarization components onto the same tensor element $P_i$4 of a type-0 PPKTP crystal (Yang et al., 2018). Experimentally, the work reported $P_i$5, measured SH patterns with doubled petal counts, and a conversion efficiency $P_i$6 at $P_i$7 CW (Yang et al., 2018).

A different spatial generalization is the diffractive polarization transformer, which implements

$P_i$8

and collects the full spatial-polarization response into a scattering tensor

$P_i$9

In the reported realization, deep-learning-designed diffractive volumes synthesized $P_{ij}$0 different spatially-encoded polarization scattering matrices with negligible error, and the terahertz experiment operated at $P_{ij}$1 wavelength over an axial span of $P_{ij}$2 wavelengths (Li et al., 2023).

The same drive toward tensorial reformulation appears in coordinate-covariant polarization geometry. In anisotropic dielectrics, $P_{ij}$3, and fixing $P_{ij}$4 maps the sphere in $P_{ij}$5-space to a polarization ellipsoid

$P_{ij}$6

whereas fixing the polarization energy density produces the energy ellipsoid $P_{ij}$7 (Alves et al., 2024). For optical skyrmions, the familiar Stokes vector is replaced by a contravariant tensor $P_{ij}$8, and the skyrmion tensor is

$P_{ij}$9

This permits non-Cartesian calculations, including cylindrical coordinates, and for the canonical paraxial skyrmion yields

$T_{ij}$0

after integration over the transverse plane (Barnett et al., 17 Mar 2025).

5. Spin-1 particles, nuclei, and vector mesons

For particles with spin $T_{ij}$1, vector and tensor polarization are distinct observables. The vector polarization is

$T_{ij}$2

and the rank-2 polarization tensor is

$T_{ij}$3

with $T_{ij}$4 and $T_{ij}$5. If only linear-in-spin interactions are retained, the tensor polarization is constant in a frame rotating with the same angular velocity as the spin, and in the laboratory frame it rotates with that same angular velocity. When bilinear terms $T_{ij}$6 are present in the Hamiltonian, commutators between spin components and quadratic operators induce mutual transformations of vector and tensor polarization (Silenko, 2018).

The deuteron provides a concrete spin-1 example. In an unpolarized target, the forward scattering operator can be written

$T_{ij}$7

so that $T_{ij}$8 produces birefringence and $T_{ij}$9 produces spin dichroism. The latter generates tensor polarization from an initially unpolarized beam: $\exp[i(kz-\omega t)]$0 and, in the thin-target limit,

$\exp[i(kz-\omega t)]$1

The same framework yields linearized vector–tensor transfer relations such as

$\exp[i(kz-\omega t)]$2

which make the mutual conversion explicit (Anischenko et al., 14 Aug 2025).

In stored polarized deuteron beams, relaxation measurements also expose the distinction between vector and tensor sectors. In the IUCF Cooler experiment with a continuous-wave rf solenoid, the decay lifetimes of vector and tensor polarizations were measured in the ratio $\exp[i(kz-\omega t)]$3, rather than the standard angular-momentum expectation of $\exp[i(kz-\omega t)]$4. The report argues that coherent evolution alone cannot mix irreducible ranks, so any deviation from $\exp[i(kz-\omega t)]$5 implies non-unitary effects such as anisotropic or time-dependent relaxation (Mane, 2015).

Vector mesons provide a second class of spin-1 systems. In a QCD medium, the paper on dissipative damping derives the shear-induced tensor-polarization constitutive law

$\exp[i(kz-\omega t)]$6

with

$\exp[i(kz-\omega t)]$7

In that formulation, tensor polarization is a spin-fluctuation anisotropy fixed by a fluctuation–dissipation relation and directly mapped to the observable $\exp[i(kz-\omega t)]$8 through $\exp[i(kz-\omega t)]$9 (Li et al., 2022). In a magnetic field, lattice $\mathbf{E}=\begin{pmatrix}E_x\E_y\end{pmatrix},\qquad E_x,E_y\in\mathbb{C},$0 calculations express the spin alignment of vector mesons through

$\mathbf{E}=\begin{pmatrix}E_x\E_y\end{pmatrix},\qquad E_x,E_y\in\mathbb{C},$1

with the small-field response

$\mathbf{E}=\begin{pmatrix}E_x\E_y\end{pmatrix},\qquad E_x,E_y\in\mathbb{C},$2

For $\mathbf{E}=\begin{pmatrix}E_x\E_y\end{pmatrix},\qquad E_x,E_y\in\mathbb{C},$3, the reported tensor polarizability $\mathbf{E}=\begin{pmatrix}E_x\E_y\end{pmatrix},\qquad E_x,E_y\in\mathbb{C},$4 is negative, corresponding to longitudinal alignment (Luschevskaya et al., 2018). A related equilibrium field-theory result for massive vector bosons states that, by time-reversal symmetry, the leading contribution to spin alignment arises from second-order terms in the matrix-valued spin-dependent distribution, not from first order (Zhang et al., 2024).

6. Gravitational-wave, cosmological, and gauge-theoretic usages

In general Einstein–vector theory with a constant background vector $\mathbf{E}=\begin{pmatrix}E_x\E_y\end{pmatrix},\qquad E_x,E_y\in\mathbb{C},$5, the linearized gravitational-wave sector splits into coupled subsystems and can exhibit genuine tensor–vector–scalar mixing. The background vector breaks isotropy, the $\mathbf{E}=\begin{pmatrix}E_x\E_y\end{pmatrix},\qquad E_x,E_y\in\mathbb{C},$6 and $\mathbf{E}=\begin{pmatrix}E_x\E_y\end{pmatrix},\qquad E_x,E_y\in\mathbb{C},$7 couplings generate off-diagonal terms, and the normal modes can contain mixed tensor, vector, and scalar polarizations. Across parameter space there are at least two and at most five independent polarization modes, excluding $\mathbf{E}=\begin{pmatrix}E_x\E_y\end{pmatrix},\qquad E_x,E_y\in\mathbb{C},$8; nonzero $\mathbf{E}=\begin{pmatrix}E_x\E_y\end{pmatrix},\qquad E_x,E_y\in\mathbb{C},$9 and a transverse background $\mathbf{E}\mapsto e^{i\alpha}\mathbf{E}$0 generate tensor–vector off-diagonals, while anisotropy $\mathbf{E}\mapsto e^{i\alpha}\mathbf{E}$1 produces birefringent speeds $\mathbf{E}\mapsto e^{i\alpha}\mathbf{E}$2 (Lai et al., 2024). Under the strict $\mathbf{E}\mapsto e^{i\alpha}\mathbf{E}$3 limit motivated by GW170817/GRB170817A, only $\mathbf{E}\mapsto e^{i\alpha}\mathbf{E}$4, $\mathbf{E}\mapsto e^{i\alpha}\mathbf{E}$5, and a luminal $\mathbf{E}\mapsto e^{i\alpha}\mathbf{E}$6 remain allowed (Lai et al., 2024).

A useful contrast is provided by scalar-tensor-vector gravity. In the flat-vacuum, first-order regime analyzed there, the tensor, vector, and scalar wave equations are decoupled: $\mathbf{E}\mapsto e^{i\alpha}\mathbf{E}$7 The theory still exhibits two additional transverse vector modes in detector response, but it does not exhibit vector-to-tensor polarization conversion during propagation in vacuum (Liu et al., 2019).

Cosmological vector backgrounds can also alter tensor polarization. In conformal vector dark-radiation models, a homogeneous vector background modifies the gravitational-wave transfer function, generates anisotropies in the tensor power spectrum, induces net linear polarization of the gravitational-wave background, and, for certain configurations of the vector field, produces linear-to-circular polarization conversion. For an initially unpolarized stochastic background, the reduced Stokes parameters are built from transfer-matrix elements $\mathbf{E}\mapsto e^{i\alpha}\mathbf{E}$8, with

$\mathbf{E}\mapsto e^{i\alpha}\mathbf{E}$9

$J$00

In that setting, dichroism generates $J$01, rotating-vector backgrounds can generate $J$02, and $J$03 requires a pre-existing polarized primordial background (Miravet et al., 2022).

A more algebraic use of vector-to-tensor conversion appears in de Sitter gauge gravity. There, ten gauge vector fields associated with the generators of $J$04 are introduced, vector polarization solutions $J$05 are constructed, and spin-2 gauge potentials are built from them. The rank-2 polarization tensor is

$J$06

while a conformally suitable mixed-symmetry rank-3 tensor is

$J$07

The mixed-symmetry rank-3 field, rather than the symmetric rank-2 one, is required to preserve conformal transformation (Raziani et al., 2022).

Across these settings, vector-to-tensor polarization conversion does not denote a single universal mechanism. It denotes a family of constructions in which polarization vectors are lifted into tensors to encode coherency, response, alignment, topology, or mixed propagation sectors. The recurring mathematical move is bilinearization or covariant tensorization; the recurring physical motive is that the tensor carries information inaccessible to the vector description alone.