The Stokes vector is one of the most useful ways to describe polarization because it works not only for fully polarized light,
but also for partially polarized and even unpolarized mixtures.
Instead of storing the electric field directly as a complex Jones vector, the Stokes formalism summarizes the state with four real numbers:
\(I\), \(Q\), \(U\), and \(V\).
Together these four quantities form the Stokes vector
\[
\mathbf{S}=
\begin{bmatrix}
I\\
Q\\
U\\
V
\end{bmatrix}.
\]
The first parameter \(I\) is the total intensity.
The parameter \(Q\) measures preference for horizontal versus vertical linear polarization.
The parameter \(U\) measures preference for linear polarization at \(45^\circ\) versus \(135^\circ\).
The parameter \(V\) measures the circular component and therefore captures handedness under a chosen sign convention.
In this calculator, positive \(V\) is labeled right-handed or right-circular.
If the polarization state is given as a Jones vector
\(\mathbf{E}=\begin{bmatrix}E_x\\E_y\end{bmatrix}\),
then the Stokes parameters are
\[
I=|E_x|^2+|E_y|^2,
\qquad
Q=|E_x|^2-|E_y|^2,
\]
\[
U=2\operatorname{Re}(E_xE_y^\*),
\qquad
V=-2\operatorname{Im}(E_xE_y^\*).
\]
These formulas are valid for a fully polarized monochromatic field.
They are especially convenient because they convert the complex Jones description into four real quantities with a direct physical interpretation.
If the state is instead described by the polarization ellipse, using total intensity \(I\), azimuth \(\psi\), and ellipticity angle \(\chi\),
then the Stokes parameters are
\[
Q=I\cos 2\chi \cos 2\psi,
\qquad
U=I\cos 2\chi \sin 2\psi,
\qquad
V=I\sin 2\chi.
\]
This form makes the geometry of the state immediately visible.
The angle \(\psi\) determines the orientation of the ellipse, while \(\chi\) determines its shape.
When \(\chi=0^\circ\), the state is linear.
When \(|\chi|=45^\circ\), the state is circular.
Intermediate values correspond to elliptical polarization.
One of the most important derived quantities is the degree of polarization, or DOP:
\[
\mathrm{DOP}=\frac{\sqrt{Q^2+U^2+V^2}}{I}.
\]
If \(\mathrm{DOP}=1\), the light is fully polarized.
If \(\mathrm{DOP}=0\), the light is completely unpolarized.
Intermediate values correspond to partial polarization.
This is one of the major strengths of the Stokes formalism over the Jones formalism: it remains meaningful even when the light is not fully coherent or not fully polarized.
Closely related quantities are the degree of linear polarization and the degree of circular polarization:
\[
\mathrm{DoLP}=\frac{\sqrt{Q^2+U^2}}{I},
\qquad
\mathrm{DoCP}=\frac{V}{I}.
\]
These help separate how much of the polarization is linear and how much is circular.
For example, a pure linear state has \(\mathrm{DoLP}=1\) and \(\mathrm{DoCP}=0\), while a pure circular state has \(\mathrm{DoLP}=0\) and \(|\mathrm{DoCP}|=1\).
Consider the sample input of linear polarization at \(45^\circ\).
In ellipse language, this means
\(I=1\),
\(\psi=45^\circ\),
and
\(\chi=0^\circ\).
Substituting into the ellipse formulas gives
\[
Q=1\cdot \cos 0^\circ \cos 90^\circ = 0,
\]
\[
U=1\cdot \cos 0^\circ \sin 90^\circ = 1,
\]
\[
V=1\cdot \sin 0^\circ = 0.
\]
Therefore the Stokes vector is
\[
\mathbf{S}=
\begin{bmatrix}
1\\
0\\
1\\
0
\end{bmatrix},
\]
and the degree of polarization is
\[
\mathrm{DOP}=\frac{\sqrt{0^2+1^2+0^2}}{1}=1.
\]
This confirms that the sample state is fully polarized, purely linear, and oriented at \(45^\circ\).
The normalized Stokes parameters
\(q=Q/I\),
\(u=U/I\),
and
\(v=V/I\)
are often plotted on the Poincaré sphere.
Fully polarized states lie on the sphere surface, while partially polarized states lie inside it.
Linear states lie on the equator, circular states lie at the poles, and elliptical states occupy intermediate locations.
This geometric picture is extremely useful in advanced polarization optics.
At a more advanced university level, the Stokes formalism naturally connects to Mueller matrices, depolarization, retarders, polarizers, and full polarization-state tomography.
But even at an introductory level, the Stokes vector already provides a complete real-valued description of a polarization state and a direct way to quantify how polarized the light is.
\[
\mathbf{S}=
\begin{bmatrix}
I\\Q\\U\\V
\end{bmatrix},
\qquad
\mathrm{DOP}=\frac{\sqrt{Q^2+U^2+V^2}}{I},
\qquad
\mathrm{DoLP}=\frac{\sqrt{Q^2+U^2}}{I}.
\]
These formulas make the Stokes vector one of the most versatile tools in polarization analysis, from basic linear polarization to advanced full-state descriptions.
