Overview

• The Stokes vector is a four-element vector that fully defines the polarization state of a beam of light.
• The Mueller matrix is a 16-element, 4x4 transfer matrix that fully describes how an object will change the polarization state of a beam of light upon interaction.
• The Mueller matrix is great for polarization ray tracing, but the 16 individual elements are difficult to interpret by themselves. So from the Mueller matrix, we calculate a number of Reduced Parameters that have useful physical interpretations.

The Mueller Matrix

The polarization properties of any beam of light can be fully described by the four elements of the Stokes vector. But in many cases, researchers are more interested in understanding how various materials and objects will change the polarization state of an incident beam into a new polarization state for the exiting beam.

We want to know how an object will change polarization states.

Since both the input state and output polarization state are defined by 4-element Stokes vectors, it is reasonable to expect that there would be 16 parameters required two fully describe how a sample would change polarization states.

Thankfully, the mathematics for this interaction turn out to be straightforward. The interaction can be defined by a simple 4x4 transfer matrix called the Mueller matrix, M.

The Mueller matrix fully describes the polarization changing properties of a sample.  That is, M operates on any incident Stokes vector S and calculates the corresponding exiting Stokes vector S′. The matrix multiplication involved is just a dot product, like this:

How to multiply a matrix by a column vector

Once we know the Mueller matrix, we know everything about the polarization properties of a sample. But keep in mind that the Mueller matrix of a sample typically changes with wavelength (spectral variation), incident angle (field-of-view variation), and location on the sample (spatial variation).  So, depending on our application, we may need to measure a sample at multiple wavelengths, incident angles, or positions.

When multiplying matrices, it is important to remember that matrix multiplication is non-commutative, meaning that in general A·B ≠ B·A.  So be careful to get the order of multiplication correct, as shown below:

Calculating Reduced Parameters

We understand what the Mueller matrix is, but now we need to know what to do with it if we measure one. Consider the following example of a possible Mueller matrix measurement from an AxoScan system:

While M completely describes how the sample changes the polarization of light, it is really confusing and it is not particularly helpful in its raw form. We usually want to perform some additional analysis on the Mueller matrix so that understandable parameters such as polarizer axis orientation and optical retardation can be presented.

There are a few different techniques that have been used over the years for analyzing Mueller matrices when depolarization is present. Axometrics chooses to use the Lu-Chipman decomposition described in this paper.

S. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. A. 13, 1106-1113 (1996).

While the actual computation is quite complex, the basic idea is understandable. A measured Mueller matrix can be mathematically separated into three different Mueller matrices, each having only one of the three basic polarization properties: diattenuation, retardance (retardation) and depolarization. An example of result of this calculation is shown below:

Further analysis of these matrices results in the following Reduced Parameters:

Transmittance is percent of unpolarized light that transmits through sample

Diattenuation is caused by having different transmittance values for different incident polarization states.  This is the primary characteristic of polarizer. Diattenuation has two properties:

• The Polarizer Efficiency or Diattenuation Magnitude = (T_max – T_min) / (T_max + T_min)
• The Transmission axis is the incident state on Poincaré sphere that has maximum transmission T_max  

Polarizance is related to diattenuation and is the ability of a sample to create polarized light when unpolarized light is input.  Polarizance has two properties:

• The Polarizance Magnitude is the degree of polarization of the output beam when unpolarized light is incident on a sample
• The Polarizance Axis is the output polarization state on Poincaré sphere when unpolarized light is input

Retardance is caused by a phase delay between different propagating states

• The Retardance Magnitude is the total phase delay between the fastest and slowest propagating states, typically measured as phase (degrees), distance (nm) or in fractions of a wavelength
• The Fast-axis is the state on the Poincaré sphere representing the fastest-propagating polarization state through the sample

Depolarization is the ability to reduce the degree of polarization of incident light

These Reduced Parameters are automatically calculated by all Axometrics software. No assumptions or approximations are made, and the retardance, diattenuation and polarization vectors are all presented as fully arbitrary polarization states with any orientation, ellipticity and handedness.

Further Analysis

The raw Mueller matrix and reduced parameters described above are adequate for solving a large fraction of polarization measurement requirements for researchers and engineers. But sometimes, additional calculations are required.

For example, in the case of liquid crystal testing, our LCDView™ software uses the patented Mueller Matrix Method (MMM) for measuring any combination of Cell gap, Twist angle, Rubbing or alignment direction (CF and TFT) and Pre-tilt angles (CF and TFT). This works for any mode of cell, and all parameters can be measured at the same time.

Another example is ellipsometry, which is an indirect way to measure the thickness and refractive index of thin films. Ellipsometry involves measuring how a sample changes polarization of reflected light (the Mueller matrix), determining the parameters Ψ and Δ (related to diattenuation and retardance), and using our EllipsoView™ software to fit a mathematical model of film thickness and refractive index to the measurement.

And there are countless other examples. Since the Mueller matrix contains all possible polarization properties of a sample, the measurement is futureproofed. When new types of samples or new analysis requirements come up, the Mueller matrix measurement itself will not need to change.  Only additional analysis will be required.