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# How to Use the Jones Matrix Surface

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

Mark Nicholson
08/01/2007
Polarization and Thin Film Coatings

Introduction

Ray tracing programs generally treat rays as purely geometric entities, which have only a position, orientation, and phase. For example, a ray is completely described at a surface by the ray intercept coordinates, the direction cosines which define the angles the ray makes with respect to the local coordinate axes, and the phase, which determines the optical path length or difference along the ray.

At the boundary between two media, such as glass and air, refraction occurs according to Snell's law. Usually, the effects at the interface which do not affect beam direction are ignored. These effects include amplitude and phase variations of the electric field which depend upon the angle of incidence, the incident polarization, and the properties of the two media and any optical coatings at the interface.

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 an optical system.

Zemax has detailed analysis capabilities for almost any coating or birefringent medium. However, it is sometimes required to use simpler models, because of a lack of data 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 can be used to describe polarization components such as polarizers and retarders 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 {Ex, Ey, Ez} which are all complex-valued. The ray propagation vector k 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 E must be orthogonal to the propagation vector k so that

k.strong>E = 0

and therefore

Ex.l + Ey.m +Ez.n = 0

Any boundary between two media can polarize a beam, and Zemax models this in great detail. However, Zemax 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, Zemax provides cells for defining A real, A imag, etc.

It is important to note that the Jones matrix does not define what happens to the Ez 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.E = 0 and the vector k has components {0, 0, 1} then Ez must be zero and we can specify the polarization purely in terms of Ex and Ey. If rays land with some arbitrary {l, m, n} then Zemax will adjust either Ez or {Ex, Ey} such that  k.E = 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.

Here are some typical settings of the Jones matrix coefficients, taken from the Zemax User's Guide

An Example

Here is an example of a Jones surface being used as a quarter wave plate. The sample file is included in a zip file that can be downloaded from the link at the end of the last page of 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 Analysis...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 (Areal = -1, Dreal = +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 (Areal = +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 Analysis...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...General...Polarization.

Summary

Zemax can model polarizing components based on birefringence or polarizing thin-film coatings exactly. Sometimes, however, a simpler method of entering polarizer data is useful. The Jones Matrix surface (for sequential rays) and Jones Matrix object (for non-sequential ray-tracing) allow a simple and fast way to enter a polarizing component.

Because the Jones matrix describes the polarizer in terms of Ex and Ey only, it should be used in collimated light at normal incidence. Zemax will handle non-normal and non-collimated rays, but must compute the Ez separately to keep the E vector orthogonal to the k vector, and care should be taken in interpreting the results.