Modelling birefringent polarizers usually requires two birefringent media, with the crystal axes of the two media rotated in some fashion. For example, the Rochon polarizer is made of two prisms of a birefringent material (KDP in this case)

In this Rochon, two prisms of KDP are cemented together with their crystal axes rotated by 90° with respect to each other. In the first, the crystal axis is oriented so its direction cosines form the vector {0,0,1}, such that the crystal axis points along the local z-axis. This is shown by the dashed red line.

In the second, the crystal axis is at {1, 0, 0} so the crystal axis lies in the x-axis. This data is entered using the parameter data of the Birefringent In surface. The crystal axis can be located at any position with respect to the surface vertex by entering the direction cosines of the axis in this manner.

Now when light propagates through birefringent media, the index of refraction of the glass is different for the S and P polarizations. {Note that the S-polarization has nothing to do with the S vector defined earlier. The S vector is the ray vector that points in the direction of energy propagation. This vector has an associated electric field which is orthogonal to S. We refer to the polarization of the S vector as a mixture of S- and P- polarization states. Note also that within the birefringent medium, the S and P polarization directions are also not in general the same as those used by the coating and Fresnel surface effects computation. Within the birefringent medium S and P refer to the perpendicular and parallel orientations relative to the crystal axis rather than the surface normal vector.}

The ordinary index is seen by the perpendicular, or S-polarized light, while the effective (angle-dependent) index is seen by the parallel or P-polarized light. The plane that contains the refracted ray and the crystal axis vector is the parallel plane; and the P- polarization lies in this plane,  normal to the ray vector S. The S- polarization is perpendicular to both the P-polarization and the ray vector S.

If the mode is 0, the ordinary ray is traced, which only has an S- component, so the P component transmission is set to zero. If the mode is 1, the extraordinary ray is traced, and the S component is therefore set to zero.

This technique yields the correct transmission results for each possible path separately. However, to get the total transmission requires analysis of each possible combination of modes for every pair of birefringent surfaces. If there are 2 pairs of birefringent surfaces in the system, 4 separate ray traces are required; and if there are 3 pairs of birefringent surfaces, 8 traces required, etc.

For the Rochon, the required combination of rays is given in the multi-configuration editor:



So for each ray we trace, we: 

• trace the ordinary component of the ray in crystal 1, and the ordinary component of the ray in crystal 2 (config 1)
• trace the ordinary component of the ray in crystal 1, and the extraordinary component in crystal 2 (config 2)
• trace the extraordinary component of the ray in crystal 1, and the ordinary component in crystal 2 (config 3)
• trace the extraordinary component of the ray in crystal 1, and the extraordinary component in crystal 2 (config 4)

We then recombine the rays by adding their field amplitudes together, not their intensities. This is a key point, and is discussed in detail on the next page.