The Jones Matrix surface is a simple way to define polarizing components. This article provides some examples of its use.

**Authored By Mark Nicholson **

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## 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.

OpticStudio 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, OpticStudio supports IDEAL and TABLE coatings for use when real coating data is not available. In a similar manner, the Jones matrix can be used to describe polarization components without doing detailed physical modeling. The Jones matrix is a useful 'black box' approach to modeling some polarization effects.

## The Jones Matrix

The amplitude and polarization state of the electric field is described by a vector __E__ which has components {E

_{x}, E

_{y}, E

_{z}} which are all complex-valued. The ray propagation vector

__has components {l, m, n} where l, m, and n are the direction cosines of the ray in the x, y and z directions. The electric field vector__

*k*__must be orthogonal to the propagation vector__

*E*__so that__

*k*and therefore

Any boundary between two media can polarize a beam, and OpticStudio models this in great detail. However, OpticStudio also supports an idealized model for a general polarizing device. The model is implemented as a special "Jones Matrix" surface type for Sequential ray tracing, and a "Jones Matrix" object type for Non-Sequential ray tracing. The Jones matrix modifies a Jones vector (which describes the electric field) according to

where A, B, C and D are all complex numbers. In the lens data and in the Non-Sequential components editor, OpticStudio provides cells for defining A real, A image, etc.

It is important to note that the Jones matrix does not define what happens to the E_{z} component. This assumes therefore that rays land at normal incidence, i.e. that the idealized polarizer is being placed in a collimated beam. This is a reasonable assumption: most polarizers and waveplates are indeed used in collimated beams or in beams with only small divergence angles.

If the beam is collimated and normal to the Jones matrix, then because * k *•

__= 0 and the vector__

*E*__has components {0, 0, 1} then E__

*k*_{z}must be zero and we can specify the polarization purely in terms of E

_{x}and E

_{y}. If rays land with some arbitrary {l, m, n} then OpticStudio will adjust either E

_{z}or {E

_{x}, E

_{y}} such that

__k__*•*

__= 0 and the magnitude of__

*E*__does not increase. The adjustment may however require a reduction in the magnitude of__

*E*__, and thus an associated loss of transmitted energy.__

*E*Here are some typical settings of the Jones matrix coefficients, taken from the OpticStudio Help System

## An Example

Here is an example of a Jones Matrix surface being used as a quarter wave plate. The sample file is included as an attachment to this article.

Note that the Jones Matrix surface does not use the radius of curvature column: it is always a plane. This is consistent with its common use being in collimated light at normal incidence. The matrix elements are entered as parameter data in the Lens Data Editor. In this case, the Jones 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**:

It can be seen that the input circular polarization has been altered to a linear polarization, with 100% efficiency. If we change the Jones matrix elements to represent a half-wave plate in x (A_{real} = -1, D_{real} = +1, all others zero), we get an output circular polarization with the opposite handedness. Note the direction arrow drawn on the polarization ellipses:

If we set up the Jones matrix as an x-analyzer (A_{real} = +1, all other elements zero), then only x-polarized light is passed, and the transmission (naturally) falls to 50%

Note: all the analysis features under **Analyze...Polarization** have settings dialogs that allow the user to enter the input polarization directly. If you use other Analysis features, like say the Huygens PSF, that have a checkbox to 'Use Polarization' but do not explicitly allow you to define the polarization state of the light, the polarization is controlled via a global setting under **System Explorer...Polarization.**

KA-01441

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