$$\begin{align}&\langle E_x(\vect{r},t)E_x(\vect{r},t)\rangle =\langle |E_x(\vect{r},t)|^2\rangle\\&\langle E_y(\vect{r},t)E_y(\vect{r},t)\rangle=\langle |E_y(\vect{r},t)|^2\rangle\\&\langle E_x(\vect{r},t)E_y^*(\vect{r},t)\rangle\end{align}$$ 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.

There are ways to mitigate spectral leakage when we reconstruct our data from the channel(s), but this makes writing the reconstruction code more difficult, more bug prone, etc. These techniques are beyond the scope of this entry, but may be discussed in detail in another post.

Assumptions

Suppose that we denote our sampling rate $$f_s = n_s \frac{1}{s}, \:n_s \in \mathbb{N},$$ i.e., $n_s$ evenly spaced samples per second, where $n_s$ is an integer. Now suppose that we also have defined an ordered list of channel frequencies, $$\nu_0 \leq \nu_1 \leq \cdots \leq \nu_k \leq \frac{f_s}{2},$$ where the last frequency is necessarily less than or equal to the Nyquist frequency (caveat : when sampling at Nyquist there exist functions which cannot be reconstructed, the Nyquist condition is strictly less than, but we typically may use the equality in engineering applications). Given these assumptions, we now want to find the minimum window length which encloses an integer number of sampled periods, for each frequency. Some additional insight and graphs of what is happening in the temporal domain is discussed at GaussianWaves.

The solution

The sample increment is $$\Delta t = \frac{1}{f_s},$$ so we need to find $m$ such that $$f_0(0) = f_0(m\Delta t)\\f_1(0) = f_1(m\Delta t)\\\vdots\\f_k(0) = f_k(m\Delta t)$$ simultaneously, where $f_j(t) = \cos 2\pi(\nu_j t)$ or $f_j(t) = \sin 2\pi(\nu_j t)$. This implies that for an arbitrary $j$, $$ 1 = \cos 2\pi(\nu_j m \Delta t)\quad \text{or}\quad 0 = \sin 2\pi(\nu_j m \Delta t)$$ which then leads to the fact $$ \nu_j m \Delta t = l_j,\:\: l_j\in\mathbb{N}\\\implies m = \frac{l_j}{\nu_j\Delta t},\forall j\\\implies \boxed{m = \frac{l_j f_s}{\nu_j},\forall j} $$

This last equation ($\forall$ means for all) reveals a key piece of information; $f_s/\nu_j$ must be a rational number.

The boxed equation above can then be rearranged to solve for $l_j$, $$l_j=\frac{m\nu_j}{f_s}$$since $m$ is fixed and both $\nu_j$ and $f_s$ can be specified as in the previous post, for example $$\cos 2\pi(3.75t) + \cos2\pi(7.5t) + \cos2\pi(11.25t) + \cos2\pi(15t),$$ sampled at 30fps.

For our example, we have the following set of equations $$\begin{align}l_0 &=  \frac{3.75m}{30}=\frac{m}{8}\\\\l_1 &=  \frac{7.5m}{30}=\frac{m}{4}\\\\l_2 &=  \frac{11.25m}{30}=\frac{3m}{8}\\\\l_3 &= \frac{15m}{8} = \frac{m}{2}.\end{align}$$ By inspection we can see that $m \geq 8$, then plugging in $m=8$ does in fact seem to satisfy all of the equations, resulting in $l_0=1,l_1=2,l_2=3,l_3=4$.  For this example, the smallest window is 8 samples.

8 should not be used, however, because in this case there is not enough “distance” between the respective $\delta$-functions in the Fourier domain, so at least 16 samples should be used (any multiple of 8 will work for this specific example).

To summarize, a spectral leakage free channel construction requires:

$$\boxed{m = \frac{l_j f_s}{\nu_j},\forall j},$$ where $m$ is the minimum number of samples, $f_s$ is the sampling rate, $\nu_j$ is each of the channel frequencies, $l_j$ corresponds to the number of periods in the window for each respective $\nu_j$, and finally $f_s/\nu_j$ must be rational. Notice that, typically $f_s$ is fixed, which implies that there can never be an irrational ratio between any of the channel frequencies if we want the channels to remain free of spectral leakage.

Motivation

The application which I’m applying linear systems to is a channel design for a channeled polarimeter.  Abstractly we may think of channels in the typical communications sense, sets of $\delta$-functions in the Fourier domain which are convolved with the data (carrier channels/frequencies). The Fourier domain data around each channel can then be filtered out and deconvolved to obtain the spatio-temporal domain data for different channels.

In our polarimetric system design, we must temporally sample (via our focal plane array, i.e., a camera). We also are forced to use a window of samples, since we don’t have infinite time or computer memory. These windows must be integer periods of our carrier (or modulation) frequencies, otherwise we get spectral leakage. The seminal paper on spectral leakage and mitigating it is by Harris (1978). Basically when performing a discrete Fourier transform (DFT, including FFT), if the window is not an integer multiple of the period, then frequencies from the sharp cutoff become introduced into the DFT.

The issues

The camera on our polarimeter has a maximum framerate of 30 frames per second (fps). The bandwidth optimal system design (found so far) for this particular type of system has sets of 9 temporal channels (there are also spatial channels, and the channels are actually three dimensional spatio-temporal channels, but for simplicity here we may ignore this complexity), including the DC or 0 channel, corresponding to the temporal modulation dimension.  Due to Hermicity, this means there are really 4 unique temporal frequencies other than 0 for this design, and they are evenly spaced. The maximum frequency is limited by the physical rotation speed limitations of a retarder, and the sample rate of the camera (due to Nyquist).

The above puts constraints on the windows,  frequencies, and sampling rates that we can use without spectral leakage. It is sometimes theoretically desirable (for noise reasons in this polarimetric instrument) to sample at slightly above Nyquist, however in our case spectral leakage does not allow this.  Our window must be an integer number of periods of all 4 frequencies. Practically this means that I cannot increase the sampling rate to something like 32 fps, because of the interplay between Nyquist, spectral leakage, and channel frequency spacing.

Below is a graph of $$ f(t) = \cos 2\pi(3.75t) + \cos2\pi(7.5t) + \cos2\pi(11.25t) + \cos2\pi(15t),$$ the blue dots denote sampling at 30 fps (or once every $\frac{1}{30}s$), and the red squares denote sampling at 32 fps.

When we take the Fourier transform (I use the unitary version here) $$\mathcal{F}\{f(t)\}_{t \to \nu}$$ for the 30fps sampled case we obtain the graph in blue below. This is the correct representation, there is no imaginary part, and the discrete $\delta$-functions are located precisely at the frequencies expected from the corresponding real (non-discrete) Fourier transform results.  The red graphs show the spectral leakage issue, the imaginary part becomes non-zero, and the $\delta$-functions sort of “split” into positive and negative parts. This creates problems when trying to reconstruct data after filtering. Note that we have the leftmost $\delta$-function at twice the magnitude of the others, and a missing $\delta$-function on the right side, this is due to the channel being right at Nyquist.

In the next post I will go over precisely what the mathematical relationship is between spectral leakage, Nyquist, and channel frequency, and the constraints imposed upon us as engineers when using channeled systems.