The Stokes Parameters and Stokes Vector

The Stokes parameters provide a way to quantify the polarization state of a light beam. The four (4) Stokes parameters are calculated from four (4) intensity measurements using unique polarization arrangements. Under natural illumination, each of these measurements would let through half of the incident light . The particular set of Stokes measurements are not unique however and other arrangements could be imagined.

We will label the Stokes parameters zero indexed as $\tilde{S}_0,\tilde{S}_1,\tilde{S}_2,\tilde{S}_3$

$\tilde{\overset{\rightharpoonup }{S}}=\left( \begin{array}{c} \tilde{S}_0 \\ \tilde{S}_1 \\ \tilde{S}_2 \\ \tilde{S}_3 \\ \end{array} \right)$

It is customary to normalized the Stokes vector composed of these parameters by $S_0$ i.e.,

$\overset{\rightharpoonup }{S}\equiv \frac{\tilde{\overset{\rightharpoonup }{S}}}{S_0}=\left( \begin{array}{c} 1 \\ \tilde{S}_1/\tilde{S}_0 \\ \tilde{S}_2/\tilde{S}_0 \\ \tilde{S}_3/\tilde{S}_0 \\ \end{array} \right)\equiv \left( \begin{array}{c} 1 \\ S_1 \\ S_2 \\ S_3 \\ \end{array} \right)$

Now that we have the general idea of how the Stokes vector looks mathematically, we'll now discuss the intensity measurements that are used to calculate the $\tilde{S}_j$.

 

The first measurement is simply an isotropic filter (not sensitive to polarization but still blocking half the light. An example is a simple neutral density filter Links to an external site.). We'll call this measurement $I_0$.

The second intensity measurement, $I_1$, is made with a linear polarized placed so that its transmission axis is horizontal.

The second intensity measurement, $I_2$, is made with a linear polarized placed so that its transmission axis is 45^o.

Both $I_1$ and $I_2$ are measured relative to some reference plane in the lab, like the optical table.

The last intensity measurement, $I_3$, is made using a circular polarizer configured to pass only right-hand circular light or $\mathcal{R}$-state light.

Together, these intensity measurements are combined to calculate the $\tilde{S}_j$:

 

$\tilde{S}_0\equiv 2I_0$

$\tilde{S}_1\equiv 2I_1-2I_0$

$\tilde{S}_2\equiv 2I_2-2I_0$

$\tilde{S}_3\equiv 2I_3-2I_0$

 

Notice that because $I_0$ blocks half of the light, $\tilde{S}_0\equiv 2I_0$ (notice the factor of 2) is the total intensity of light before the isotropic filter. The normalized version of the Stokes parameters are

 

$S_0=1$

$S_1=\frac{I_1}{I_0}-1$

$S_2=\frac{I_2}{I_0}-1$

$S_3=\frac{I_3}{I_0}-1$

 

From these definitions we can see that $S_0$ is simply the total intensity of the input light. $S_1$ is basically a measure of how horizontally polarized the light is. If the light is horizontally polarized $S_1=1$, if it is vertically polarized then $S_1=-1$, remember that $S_1=\frac{I_1}{I_0}-1$ and $I_1$ is the intensity of light that gets through a polarizer with its transmission axis horizontal. If no light gets through (and there was light to begin with i.e., $S_0\neq 0$) then the light must have been vertically polarized. If $S_1=0$ then the light could be 'un-polarized', circular, or elliptical at $\text{$\pm $45${}^{\circ}$}$.

Same story for $S_2$, which is basically a measure of how 'diagonally' polarized the light is. If the light is diagonally polarized $\text{+45${}^{\circ}$}$ then $S_2=1$. If it is diagonally polarized $\text{-45${}^{\circ}$}$ then $S_2=-1$.

Then $S_3$ is a measure of how right-hand circular polarized the light is. If the light is completely RHCP then $S_3$=1 if the light is completely LHCP then $S_3$=-1.

For completely polarized light

$S_0^2=S_1^2+S_2^2+S_3^2$

and for partially polarized light, the degree of polarization is given by

$V=\frac{1}{S_0}\sqrt{S_1^2+S_2^2+S_3^2}$

Here is a table describing some Stokes vectors for pure polarization states

It is important to note that if the input light is a superposition of incoherent light, then the intensities add, not the electric fields. In that case, the total Stokes vector is the sum of all of the individual Stokes vectors. In other words, for incoherent light

$\overset{\rightharpoonup }{S}_{\text{tot}}=\sum _j\overset{\rightharpoonup }{S}_j$