Jones calculus is a compact matrix method for describing fully polarized monochromatic light.
Instead of tracking the full electromagnetic field, it represents the transverse electric field by a two-component complex vector,
usually written in the \((x,y)\) basis as
\[
\mathbf{E}=
\begin{bmatrix}
E_x\\
E_y
\end{bmatrix},
\]
where \(E_x\) and \(E_y\) are generally complex.
Their magnitudes determine the component amplitudes, and their phase difference determines whether the light is linear, circular, or elliptical.
Because an overall complex phase does not change the observed polarization state, Jones vectors that differ only by a global phase represent the same physical polarization.
Optical elements are represented by \(2\times 2\) Jones matrices.
If \(\mathbf{J}\) is the matrix of the element and \(\mathbf{E}_{in}\) is the incident Jones vector, then the output field is
\[
\mathbf{E}_{out}=\mathbf{J}\mathbf{E}_{in}.
\]
This simple matrix multiplication makes Jones calculus very powerful for polarization optics.
Linear polarizers, quarter-wave plates, half-wave plates, and more general retarders can all be modeled by appropriate matrices.
For a linear retarder whose fast axis is aligned with the \(x\)-axis, the basic Jones matrix is
\[
\mathbf{J}_0=
\begin{bmatrix}
1 & 0\\
0 & e^{i\delta}
\end{bmatrix},
\]
where \(\delta\) is the retardance.
For a quarter-wave plate, \(\delta=90^\circ\) or \(\pi/2\), and for a half-wave plate, \(\delta=180^\circ\) or \(\pi\).
If the optic axis is rotated by angle \(\alpha\), the rotated matrix becomes
\[
\mathbf{J}(\alpha)=\mathbf{R}(-\alpha)\mathbf{J}_0\mathbf{R}(\alpha),
\]
with the rotation matrix
\[
\mathbf{R}(\alpha)=
\begin{bmatrix}
\cos\alpha & -\sin\alpha\\
\sin\alpha & \cos\alpha
\end{bmatrix}.
\]
A linear polarizer aligned with the \(x\)-axis has the matrix
\[
\mathbf{P}_0=
\begin{bmatrix}
1 & 0\\
0 & 0
\end{bmatrix},
\]
and when rotated by angle \(\alpha\), it becomes
\[
\mathbf{P}(\alpha)=\mathbf{R}(-\alpha)\mathbf{P}_0\mathbf{R}(\alpha).
\]
The sample case in this calculator is especially important in polarization optics:
horizontal input light sent through a quarter-wave plate at \(45^\circ\).
Horizontal input corresponds to the Jones vector
\[
\mathbf{E}_{in}=
\begin{bmatrix}
1\\
0
\end{bmatrix}.
\]
For a quarter-wave plate at \(45^\circ\), the output becomes proportional to
\[
\mathbf{E}_{out}\propto
\begin{bmatrix}
1\\
i
\end{bmatrix}.
\]
This is the standard Jones-vector form of circular polarization, up to a possible convention-dependent handedness label.
In this calculator, that state is labeled right-circular.
The important physical point is that the quarter-wave plate has converted a linear state into a circular one by introducing a \(90^\circ\) phase shift between orthogonal components of equal amplitude.
To classify the polarization state in a way that connects directly with geometry, one may compute the normalized Stokes parameters from the Jones vector:
\[
s_1=\frac{|E_x|^2-|E_y|^2}{|E_x|^2+|E_y|^2},
\qquad
s_2=\frac{2\operatorname{Re}(E_xE_y^*)}{|E_x|^2+|E_y|^2},
\qquad
s_3=\frac{-2\operatorname{Im}(E_xE_y^*)}{|E_x|^2+|E_y|^2}.
\]
From these, the polarization azimuth \(\psi\) and ellipticity angle \(\chi\) are
\[
\psi=\frac{1}{2}\tan^{-1}\!\left(\frac{s_2}{s_1}\right),
\qquad
\chi=\frac{1}{2}\sin^{-1}(s_3).
\]
If \(\chi=0^\circ\), the state is linear.
If \(|\chi|=45^\circ\), the state is circular.
Intermediate values indicate elliptical polarization.
This makes Jones calculus and Stokes analysis a very natural pair: the Jones matrix tells you how the optic transforms the field, and the Stokes parameters help interpret the resulting state geometrically.
In practical optics, Jones matrices are used for wave plates, polarizers, polarization rotators, and optical systems that preserve full coherence and full polarization.
At a more advanced university level, one extends this to chains of elements, partially polarized light, and Mueller matrices.
But for many laboratory and classroom problems, the core idea is simply matrix multiplication of a complex two-component vector.
\[
\mathbf{E}_{out}=\mathbf{J}\mathbf{E}_{in},
\qquad
\mathbf{J}(\alpha)=\mathbf{R}(-\alpha)\mathbf{J}_0\mathbf{R}(\alpha),
\qquad
\mathbf{P}(\alpha)=\mathbf{R}(-\alpha)\mathbf{P}_0\mathbf{R}(\alpha).
\]
These equations let you predict whether a given optical element leaves the state unchanged, rotates linear polarization, converts linear to circular, converts circular to linear, or partially extinguishes the beam.
That is why Jones calculus remains one of the most useful tools in classical polarization optics.
