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Since Erasmus Bartholinus first analyzed polarization in a scientific way, the polarization of light has mostly been described in terms of how we measure it. In modern classical optics we can use wave field descriptions of light, including polarization, for our optical modeling and then integrate (in a time average way) over the resulting Poynting vector to obtain irradiance. In the field of polarimetric systems and measurement, however, this is not often done, and our models still incorporate measurement parameters instead of field parameters, after all we are primarily building instruments that measure the polarization property using irradiance modulation (at least in the optical wavelength regime).

### 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.