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.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: