How to use the Mueller Matrix surface

Authored By: Mumlesh Sawasiya

Abstract

The Mueller Matrix surface is a simple way to define polarizing components, especially those which may yield partially polarized light. This article provides some examples of its use.

Introduction

Polarization analysis is an extension to conventional ray tracing which considers the effects that optical coatings and reflection and absorption losses have on the propagation of light through a system.
Zemax has detailed analysis capabilities for almost any coating or birefringent medium. However, it is sometimes required to use simpler models, due to a lack of the real prescription data. For example, Zemax supports IDEAL and TABLE coatings for use when real coating data is not available. In a similar manner, the Jones Matrix & Mueller Matrix surfaces can be used to describe polarization components without doing detailed physical modelling. These surfaces can be a useful 'black box' approach to modelling some polarization effects.
This article specifically covers the Mueller Matrix surface. To learn more about the Jones Matrix surface, please refer to the article “How to use the Jones Matrix surface.”

Mueller Matrix

Mueller matrix is a 4×4 real matrix that describes how an optical element transforms the Stokes vector of incident light. Unlike the Jones Matrix (which works only for fully polarized, coherent light), the Mueller Matrix can model:

Unpolarized or partially polarized light
Depolarizing elements
Incoherent illumination

It is widely used for characterizing optical components such as polarizers, retarders, diffusers, coatings, and scattering surfaces.
This surface is used to define an arbitrary polarizing component using the Mueller matrix convention. The surface shape is always plane. The Mueller Matrix converts the input Stokes vector S_in to the output Stokes vector S_out:

Here,
    S_in = 4×1 input Stokes vector
    S_out = 4×1 output Stokes vector
    M = 4×4 Mueller Matrix of the element

where the values S₀ – S₃ describe the incident ray’s polarization state, S₀’ – S₃’ describe the outgoing ray’s polarization state, and M₀₀ - M₃₃ describe the polarizing element.
This surface supports the modeling of partially polarizing elements. When the “Depolarize” flag is enabled (i.e., set to a non-zero value), the Degree of Polarization (DoP) of the outgoing ray’s polarization state, defined by S₀’ – S₃’ is found through.

The value of the DoP determines the probability that the output ray’s polarization state is defined by the outgoing Stokes vector (S₀’ – S₃’). Rays will exit the Mueller matrix surface in a randomly-polarized state with a probability of (1 – DoP)
When the “Depolarize” flag is disabled (set to zero), then the outgoing Stokes parameter S₀’ is found as

meaning the outgoing ray’s polarization state always matches that defined by the outgoing Stokes vector. Note that for Mueller matrices which define partially polarizing elements, this means the outgoing ray intensity is reduced proportionally to the DoP defined by the outgoing ray Stokes parameters.

Here are some typical settings of the Mueller matrix coefficients, taken from the Zemax Help System

Mueller Matrix Example in Sequential Mode

The Polarization tab has been enabled in the Surface Properties for both Mueller Matrix and Jones Matrix surfaces. This tab allows you to define and configure the polarization parameters associated with a Mueller Matrix surface.
For more detailed information on the Polarization tab, please refer to the attached article.
Using the Polarization Tab for Jones and Mueller Matrix Setup – Ansys Optics

Here is an example of a Mueller Matrix surface being used as a quarter-wave plate. The sample file is included as an attachment to this article.
Seq mode.png

To aid in the definition of the quarter-wave plate, the new Polarization tab within Lens Data Editor…Surface Properties was used to define the Type of polarizing element (“Linear Retarder”) the element’s fast axis (0 degrees from the local X-axis), and the element’s retardance (90 degrees). For a full description of the new Polarization tab, please refer to the article “Using the Polarization Tab to Define Jones and Mueller Matrix Surfaces and objects.”
Note that because the Mueller Matrix surface always behaves as a plane surface, the Radius of Curvature value is disabled. The Mueller matrix coefficients are defined through the Lens Data Editor and can also be viewed or updated through various element settings in the Lens Data Editor…Surface Properties…Polarization tab. In this case, the Mueller matrix is configured to act like a quarter-wave plate in the X-direction
The easiest way to see the effect of the Jones Matrix surface is with the Polarization Pupil Map, which is located under Analyze...Polarization...Polarization Pupil Map.

The input linear polarization has been converted to circular polarization with 100% efficiency.

When Mueller Matrix is used as an HWP

If we change the Mueller matrix elements to represent a half-wave plate in x (M₀₀ = 1, M₁₁ = +1, M₂₂ = -1, M₃₃ = -1, all others zero), we get an output circular polarization with the opposite handedness. Note the direction arrow drawn on the polarization ellipses.

When Mueller Matrix is used as a QWP with a fast axis angle of 30 degrees

When the Mueller matrix is configured to model a quarter-wave plate with fast axis 30° from the local x-axis, a linearly polarized input beam is converted into elliptically polarized output.

Mueller Matrix Example in Non-Sequential Mode

Here is an example of a Mueller Matrix surface being used as a quarter-wave plate in Non-Sequential Mode. The sample file is included as an attachment to this article.
In the non-sequential file, we have a Source Ray, a Mueller Matrix surface, and a Detector Rectangle.

The Source Ray was defined to have linear +45 polarization, using Jx = Jy = 1 in the Non-Sequential Component Editor…Object Properties…Sources tab:

The quarter-wave plate was defined using the Non-Sequential Component Editor…Object Properties…Polarization tab:

To report the electric field, we use the NSRA operand, along with a set of arithmetic operands, to display the ray intensity along X, Y, and the phase difference between them:

From the Merit Function, we can observe that the electric‑field Ray intensity components Ex and Ey have the same magnitude, and there is an induced phase shift between these two components at the Detector when the ray passes through the Quarter-Wave Plate.

Summary

Muller matrix is a 4×4 matrix used in optical system to describe and analyze the polarization behavior of light. It defines how an optical element transforms the Stokes vector, allowing accurate modelling of polarization effects such as rotation, retardance, and depolarization induced by optical components.
In both Sequential and Non-Sequential Modes of Zemax, Muller Matrix surfaces can be applied to evaluate the polarization response of optical components. This article demonstrates how the Muller Matrix surface can be used to simulate common polarization elements including quarter-wave plates, half-wave plates, and partially polarized light and analyze their impact on system performance.
Polarization behavior in Zemax can be visualized and validated using tools such as the polarization pupil map and by examining the electric field components at the detector.

References

Chipman, R., Lam, W.S.T., & Young, G. (2018). Polarized Light and Optical Systems (1st ed.). CRC Press. https://doi.org/10.1201/9781351129121
Pericles S. Theocaris, Emmanuel E. Gdoutos (2013). Matrix Theory of Photoelasticity. Springer Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-35789-6
Edward Collett (2005). Field Guide to Polarization. SPIE—The International Society for Optical Engineering. https://doi.org/10.1117/3.626141