$$\begin{align}&\langle E_x(\vect{r},t)E_x(\vect{r},t)\rangle =\langle |E_x(\vect{r},t)|^2\rangle\\&\langle E_y(\vect{r},t)E_y(\vect{r},t)\rangle=\langle |E_y(\vect{r},t)|^2\rangle\\&\langle E_x(\vect{r},t)E_y^*(\vect{r},t)\rangle\end{align}$$ where $|\cdot|$ is the complex magnitude, $*$ is the complex conjugate, and $$\langle f(t)\rangle = \lim_{T \to \infty}\frac{1}{2T}\int_{-T}^T f(t)dt$$ is the time average.

When taking polarimetric measurements, we actually measure what we call the Stokes parameters, represented by $\vect{S}$:

$\begin{align*}
\vect{S} = \begin{bmatrix}s_0\\s_1\\s_2\\s_3\end{bmatrix} =
\begin{bmatrix}\langle |E_x(\vect{r},t)|^2\rangle + \langle |E_y(\vect{r},t)|^2\rangle\\
\langle |E_x(\vect{r},t)|^2\rangle – \langle |E_y(\vect{r},t)|^2\rangle\\
2\Re{\{\langle E_x(\vect{r},t)E_y^*(\vect{r},t)\rangle\}}\\
2\Im{\{\langle E_x(\vect{r},t)E_y^*(\vect{r},t)\rangle\}}\end{bmatrix} = \begin{bmatrix}\text{irradiance}\\\text{horiz. over vert.}\\+45^\circ\:\text{over}\:-45^\circ\\\text{right over left}\end{bmatrix}
\end{align*}$

where $\Re{\{\cdot\}}$ denotes the real part, $\Im{\{\cdot\}}$ denotes the imaginary part, $s_0$ represents the irradiance, $s_1$ represents the degree of horizontal polarization ($0^\circ$) over the degree of vertical polarization ($90^\circ$), $s_2$ represents the degree of $+45^\circ$ polarization over the degree of $-45^\circ$ polarization, and $s_3$ represents the degree of right circular polarization over left circular polarization.

The Stokes parameters were originally defined this way because we can manufacture (and have been able to manufacture for awhile) optical elements which can allow us to measure these quantities:

• the irradiance.
• the relative amount of horizontal polarization ($0^\circ$) over the degree of vertical polarization ($90^\circ$).
• the relative amount of $+45^\circ$ polarization over the degree of $-45^\circ$ polarization.
• the relative amount of right circular polarization over left circular polarization.

However, only in modern classical electromagnetics have we formally defined them in proper statistical terms of the electric field. Since the Stokes parameters represent measurements, they are always real valued, i.e., $\vect{S}\in\R^4$.

Aside : conventions

There are some pitfalls to the Stokes parameters, it is quite obvious that the quantities defined above are dependent on 1) the handedness of the coordinate system, 2) the phase convention, and 3) the position of the observer.

Fortunately, nearly everyone in the physics, remote sensing, radar, and optics community use a right handed coordinate system. The problems arise from differing conventions among the communities about the last two convention choices, phase convention and the position of the observer. We urge the reader to be careful using wikipedia for Jones and Stokes polarization parameter definitions as different subsections incorrectly mix up these conventions! We must choose a single convention and adhere to it, otherwise our computations will result in nonsense. We have introduced changes to the wikipedia articles to correct this, but overzealous editors (who are not professionals in optics) continue to revert the corrections and we have chosen to no longer waste our time arguing with them.

Given the two different conventions we may use, there are four possible definitions for the Stokes parameters, and we will derive them here. First we will go over what exactly phase convention and the position of the observer are.

Light is always assume to propagate in the $+z$ direction, with a right handed coordinate system attached.

Phase convention

Phase convention is a result of a duplicity of solutions which work when solving Maxwell’s differential equations. When solving the wave equation for linear media, superpositions of both $$\begin{align}\vect{E}\cdot e^{i(\vect{k}\cdot\vect{r} – \omega t)} = \begin{bmatrix}E_xe^{i(\vect{k}\cdot\vect{r} – \omega t)}\\E_ye^{i(\vect{k}\cdot\vect{r} – \omega t)}\\E_ze^{i(\vect{k}\cdot\vect{r} – \omega t)}\end{bmatrix}\end{align}$$ and $$\begin{align}\vect{E}\cdot e^{i(\omega t – \vect{k}\cdot\vect{r})} = \begin{bmatrix}E_xe^{i(\omega t – \vect{k}\cdot\vect{r})}\\E_ye^{i(\omega t – \vect{k}\cdot\vect{r})}\\E_ze^{i(\omega t – \vect{k}\cdot\vect{r})}\end{bmatrix}\end{align}$$ will work as long as we interpret the result correctly with respect to the physical phenomena. We call the $\vect{E}\cdot e^{i(\vect{k}\cdot\vect{r} – \omega t)}$ solutions the decreasing phase convention, and the $\vect{E}\cdot e^{i(\omega t – \vect{k}\cdot\vect{r})}$ solutions the increasing phase convention.

Remember that $E_x, E_y,E_z \in \C$, i.e. they are complex valued.

Because physically we usually only have one propagation direction, we must choose a phase convention and stick to it. In these articles, we will always use the decreasing phase convention. The texts of Hecht, Jackson, Goodman, Born and Wolf use the decreasing phase convention. The texts of Gaskill, Collett, Jenkins and White use the increasing phase convention.

Position of the observer

We will now go over the position of the observer. This affects our definition of direction in a clockwise or counterclockwise sense. There are two possible definitions for the observer, looking into the beam or looking down the beam.

Looking into the beam assumes that the observer is looking toward the origin of the $z$-axis from some positive point on the $z$-axis, think of a laser being pointed at you.

Looking down the beam assumes that the observer is looking away from the origin of the $z$-axis to some positive point on the $z$-axis, think of looking down a laser that you are pointing away from you.

This definition determines what we call right (clockwise) or left (counterclockwise) circular, because it depends on which position you are observing in.

Now that the problem parameters are defined, we will continue this discussion in the next post and finish deriving the Stokes parameters depending on which conventions are used.

Background

Before delving into the derivation of the Stokes parameters, we first need to make some assumptions:

• the propagation media is linear
• our coherence model is in the spatially incoherent approximation
• scattering results in fields which are ergodic at the measurement plane

The first point just means there are no nonlinear optical effects from propagation through the media.  The second means that irradiances add linearly (i.e. if we have pinholes there is no interference). The third point implies that a time average is equal to an ensemble average (described by a statistically stationary process), which is actually quite a strong assumption.

Light can be described by the following parameters from Maxwell’s equations

• $\vect{E}(\vect{r},t) $, the electric field
• $\vect{D}(\vect{r},t)$, the displacement field
• $\vect{H}(\vect{r},t)$, the magnetic field
• $\vect{B}(\vect{r},t)$, the magnetic induction field
• $\vect{P}(\vect{r},t)$, the polarization source
• $\vect{M}(\vect{r},t)$, the magnetization source
• $\vect{k}$, the vector valued Fourier space (or plane wave) variable associated with $\vect{r}$
• $\omega$, the Fourier variable associated with t, (associated with the wavelength)
• $\vect{S}(\vect{r},t) = \vect{E}(\vect{r},t) \times \vect{H}(\vect{r},t)$, the complex Poynting vector

where generically $$\begin{align*}\vect{A}(\vect{r},t) = \begin{bmatrix}A_x(x,y,z,t)\\A_y(x,y,z,t)\\A_z(x,y,z,t)\end{bmatrix}\end{align*}$$ are the complex vector valued functions with vector inputs. Note that we are ignoring current and charge sources here. These parameters are also not independent, we only need either $\vect{E},\vect{B}$ or $\vect{D},\vect{H}$ to completely describe the fields.

Given any arbitrary field in linear media, it can be represented as a superposition of plane waves by taking the Fourier transform, with each individual plane wave represented via $\vect{k}$ and $\omega$. Note that in linear, isotropic, homogeneous media $\vect{k}$ and $\vect{S}$ point in the same direction.  This is not true, however, in something like a birefringent crystal, where $\vect{k}$ and $\vect{S}$ may point in different directions.

In the optical wavelength regime, we typically measure a quantity related to $\vect{E}(\vect{r},t)$ and call it polarization. Specifically we infer the following three complex valued parameters in a special way $$\begin{align}&\langle E_x(\vect{r},t)E_x(\vect{r},t)\rangle =\langle |E_x(\vect{r},t)|^2\rangle\\&\langle E_y(\vect{r},t)E_y(\vect{r},t)\rangle=\langle |E_y(\vect{r},t)|^2\rangle\\&\langle E_x(\vect{r},t)E_y^*(\vect{r},t)\rangle\end{align}$$ where $|\cdot|$ is the complex magnitude, $*$ is the complex conjugate, and $$\langle f(t)\rangle = \lim_{T \to \infty}\frac{1}{2T}\int_{-T}^T f(t)dt$$ is the time average.

We use irradiance (a time-averaged Poynting vector) to obtain these quantities, notice that we only obtain information about the $E_x$ and $E_y$ components since the $E_z$ components are perpendicular to our detector and are not measured. We cannot obtain the complete picture, but do get some information about the polarization from these measurements.

In the next posts I will go over the Stokes parameters, how they represent the above quantities, and how irradiance modulation can give us these parameters.