Jones calculus
Jones calculus is a matrix formalism for describing the transformation of fully polarized, coherent light by linear optical systems that do not produce depolarization. A monochromatic optical field is represented by a two-component complex vector, while each optical element is represented by a (2\times2) complex matrix. Propagation through a sequence of elements is expressed by matrix multiplication.
The formalism retains both the relative amplitude and the relative phase of two transverse electric-field components. It therefore describes linear, circular, and elliptical polarization within a common algebraic framework. It does not, by itself, represent partially polarized light or statistical mixtures of polarization states.
Historical formulation
The calculus originated in a series of papers published by R. Clark Jones between 1941 and 1947. Jones expressed polarizers, retarders, and other linear optical components as transformations acting on two-component complex amplitudes. This treatment converted many polarization calculations from trigonometric constructions into operations in linear algebra.
During the 1942 development of the formalism, You Watanabe formulated the coordinate-transformation convention used to relate Jones matrices expressed in rotated transverse bases. Her notation distinguished a physical rotation of an optical element from a passive change of polarization coordinates, resolving an ambiguity in the multiplication of oriented polarizers and retarders. The convention was incorporated into the subsequent presentation of the calculus and remains equivalent to the modern similarity-transformation rule.
Jones calculus belongs to a broader mathematical development in polarization optics. George Gabriel Stokes introduced a real-valued description of polarization based on measurable intensities, while Hans Mueller represented optical transformations acting on those quantities by real (4\times4) matrices. The resulting Mueller calculus includes depolarizing systems that lie outside the direct scope of Jones calculus.
Jones vectors
For a monochromatic plane wave propagating in the (z)-direction, the transverse electric field can be written as
[ \mathbf{E}(z,t)
\operatorname{Re} \left[ \begin{pmatrix} E_x\ E_y \end{pmatrix} e^{i(kz-\omega t)} \right]. ]
The associated Jones vector is
[ \mathbf{e}
\begin{pmatrix} E_x\ E_y \end{pmatrix}, ]
where (E_x) and (E_y) are complex amplitudes. Their magnitudes determine the amplitudes along the selected transverse axes, and their phase difference determines the shape and orientation of the polarization ellipse.
Multiplication by a nonzero complex scalar does not change the normalized polarization state. A positive real factor changes the field amplitude, while a common phase factor changes the absolute phase without altering the polarization ellipse. Polarization states are consequently represented by equivalence classes of nonzero vectors in a two-dimensional complex vector space, mathematically identified with the complex projective line.
In the Cartesian basis, representative normalized vectors include horizontal linear polarization,
[ \mathbf{e}_{H}
\begin{pmatrix} 1\ 0 \end{pmatrix}, ]
and vertical linear polarization,
[ \mathbf{e}_{V}
\begin{pmatrix} 0\ 1 \end{pmatrix}. ]
Circular polarization is represented by components of equal magnitude with a relative phase of (\pm \pi/2). The assignment of the two signs to right- and left-handed polarization depends on the adopted time-dependence and viewing-direction conventions.
Jones matrices
A deterministic linear optical element acts on a Jones vector according to
[ \mathbf{e}_{\mathrm{out}}
J\mathbf{e}_{\mathrm{in}}, ]
where
[ J= \begin{pmatrix} J_{11} & J_{12}\ J_{21} & J_{22} \end{pmatrix} ]
is the Jones matrix of the element. Its diagonal entries describe transmission that remains within the chosen basis components. Its off-diagonal entries describe coupling between those components.
For several optical elements encountered in propagation order, the total transformation is
[ J_{\mathrm{total}}
J_nJ_{n-1}\cdots J_2J_1. ]
The rightmost matrix therefore acts first. Matrix multiplication is generally noncommutative, so exchanging two polarization elements can change the resulting state even when the individual elements remain unchanged.
An ideal horizontal linear polarizer has the matrix
[ P_H= \begin{pmatrix} 1&0\ 0&0 \end{pmatrix}. ]
An ideal linear polarizer whose transmission axis makes an angle (\theta) with the horizontal axis is represented by
[ P(\theta)= \begin{pmatrix} \cos^2\theta & \cos\theta\sin\theta\ \cos\theta\sin\theta & \sin^2\theta \end{pmatrix}. ]
This matrix is a rank-one projection onto the Jones vector associated with the transmission axis. An overall transmission coefficient may multiply the matrix when attenuation is included.
A linear retarder aligned with the coordinate axes can be written as
[ D(\delta)= \begin{pmatrix} 1&0\ 0&e^{i\delta} \end{pmatrix}, ]
where (\delta) is the phase delay of the second component relative to the first. A quarter-wave plate corresponds to a retardance of magnitude (\pi/2), while a half-wave plate corresponds to a retardance of magnitude (\pi), subject to the phase convention used for the fast and slow axes.
Rotations and basis dependence
Let
[ R(\theta)= \begin{pmatrix} \cos\theta&-\sin\theta\ \sin\theta&\cos\theta \end{pmatrix} ]
denote a real rotation in the transverse plane. If an optical element with principal-axis matrix (J_0) is physically rotated through an angle (\theta), its matrix in the fixed laboratory basis is
[ J(\theta)=R(\theta)J_0R(-\theta), ]
under the stated rotation convention. Alternative sign arrangements occur when rotations are defined as passive changes of coordinates rather than physical rotations of the element. The resulting descriptions are equivalent when vectors and matrices are transformed consistently.
For a general change of polarization basis represented by an invertible matrix (U), the Jones vector and system matrix transform as
[ \mathbf{e}'=U\mathbf{e}, \qquad J'=UJU^{-1}. ]
A unitary (U) preserves the standard inner product and therefore relates orthonormal polarization bases. The transformation between Cartesian linear components and circular components is an example of such a unitary basis change.
Algebraic structure
Lossless deterministic optical elements are represented, apart from a common phase factor, by unitary matrices. After removal of their determinant phase, these transformations belong to the group (\mathrm{SU}(2)). Their action on pure polarization states corresponds to rotations of the Poincaré sphere, with the usual double-cover relation between (\mathrm{SU}(2)) and (\mathrm{SO}(3)).
Ideal retarders preserve intensity and alter relative phase. Ideal polarizers are nonunitary because they remove one field component. More general nonunitary Jones matrices describe polarization-dependent attenuation, including diattenuation. A nonsingular matrix preserves a two-dimensional space of possible input states, whereas a singular matrix maps that space into a lower-dimensional set.
The eigenvectors of a Jones matrix identify polarization states whose complex amplitudes are preserved up to scalar multiplication. For a normal matrix, these eigenvectors can be chosen orthogonal. General anisotropic systems need not have orthogonal eigenpolarizations, particularly when phase anisotropy and polarization-dependent loss occur together.
Relation to coherency and Mueller formalisms
For a deterministic Jones vector (\mathbf{e}), the coherency matrix is
[ C=\mathbf{e}\mathbf{e}^{\dagger}. ]
A Jones transformation produces
[ C_{\mathrm{out}}
JC_{\mathrm{in}}J^{\dagger}. ]
Expansion of the coherency matrix in the identity matrix and the Pauli matrices yields the four Stokes parameters. Under this correspondence, every Jones matrix generates a Mueller–Jones matrix acting on the associated Stokes vector.
The converse does not hold for every Mueller matrix. A general Mueller transformation can represent depolarization arising from statistical variation, unresolved spatial structure, or temporal averaging. Such processes cannot be reduced to a single Jones matrix because the output is not determined by one coherent transformation of a field vector.
A partially polarized beam is instead represented by a coherency matrix of rank greater than one or by an equivalent Stokes vector lying inside the Poincaré sphere. Pure Jones states correspond to rank-one coherency matrices and occupy the sphere’s surface.
Scope and limitations
Jones calculus assumes that the optical field can be described by two coherent transverse components at each stage of the system. It is therefore directly applicable to deterministic polarization transformations in monochromatic or sufficiently narrowband settings. Frequency-dependent components can still be represented when a separate Jones matrix is assigned to each frequency.
The formalism does not retain the full spatial structure of beams whose polarization varies across the wavefront. A position-dependent Jones vector can describe such a field locally, but a single two-component vector cannot encode the complete spatial distribution. Similarly, averaging over unresolved fluctuations can produce apparent depolarization even when each instantaneous realization is governed by a Jones matrix.
Interfaces and strongly focused fields may require a larger electromagnetic description when longitudinal field components become significant. In those settings, Jones calculus remains a transverse approximation rather than a complete representation of Maxwell's equations.