The angular separation of two beams by a WP depends on its material and geometry as deo=2?n.tanW, with W the prism angle and ?n=(ne-no) the material birefringence. Then, the FOV is given by the separation on the sky of both rays as 206265.DpDtel-1.deo where the pupil and telescope diameter (Dp and Dtel) are in the same unit. On the other hand, the prism effect of the WP creates elongation of the image proportional to the separation of the rays. The lateral chromatism for a particular wavelength in the range between ?1 and ?3 is quantified in terms of the parameter that depends on the birefringence of the material at ?, ?1 and ?3. The third major factor to consider is the reflection at the prism interface that depends on the index of refraction. Low refraction index material accepts higher input angles. This factor and the opto-mechanical properties are major factors in the selection of the material. Once the material is chosen the optimization of the polarimetric system concerns the geometry of the WeDoWo and the size of the slots/slits. Two angles of the WeDoWo affect the separation of the beam: the angle of the edge (ß) split rays between both WP; the angle of the prism (a) defines the separation of the e-o rays in each WP. The WeDoWo can be coupled with FSi to decrease the chromatism. This layer defines a third angle (?) of the geometry. Once the material of the WeDoWo is defined the geometry of the system for a given FOV is calculated in the first order from Snell's laws for each ray. However, CIRCE is a complex system with aspheric mirrors and accurate analytical results are complex. We explain later how we obtain optimum results in the FOV from simulations and the chromatic aberrations from ZEMAX computation.
Hough et al (1994) discuss materials in the NIR range. In selecting a material, considerations must be given to its transmission, required beam separation, birefringence of the material, refractive index and wavelength dependence. The more common and cheapest material in this wavelength range are MgF2. It presents good optical transmission below ? less than 6um with typical absorption of 40e-3 cm-1 at 2.7um. MgF2 has index of refraction no=1.3836, ne=1.3957 at 0.4um with reflection loss at the 2 surfaces about 5.2% at 0.6 microns. In addition, it is moderately achromatic for NIR range with no (ne) from 1.37964 (1.38521) at 1um to 1.36000 (1.37060) at 3um. In addition, thermo-optical properties adapts well to cryogenic conditions. At 0.4um variation of the index of refraction is dno/dT=2.3e-6 and dne/dT=1.7e-6 corresponding to a variation of the deviation angle of 0.11%. On the other hand, thermal expansions are aa=13.7e-6/K; ac=8.48e-6/K that correspond to a variation of the expansion of 0.07mm for the required size of pupil. The crystal is not significantly deformed but special cement is required at liquid nitrogen temperatures.
The HWP is a 45x45mm2 octagon made of MgF2 and quartz crystal. It has a retardation of ?/2±4% in the range 700-2500nm. Because its thermal expansion coefficient is 5.5e-7 /K the contraction of the HWP is in the order of 6um, more than 10 times less than the MgF2 plate. Both plates are cemented with infrasil 302 avoiding breaking at low temperatures. This configuration results in low chromatic aberrations even though the thickness of the HWP introduces defocus in the beam. Calcium fluoride (CaF2) and infrasil are recommended for windows in the NIR range because of their good transmittance. Both material have index of refraction about 1.4 at ambient temperature in the NIR equivalent to a loss of light at the refracting surface crossing perpendicular to a plane parallel plate about 6%. However, Infrasil 301 is optically homogeneous along three axes (?n less than 5e-6). This property is required for polarization measurements. In addition, Infrasil 301 contains low OH and it has only very weak absorptions bands between 1.3 and 2.72um. Indexes of refraction of this material are from 1.45023 at 1um to 1.4372 at 2um with transmittance in this range about 99.9%
Our study consists in a multiconfiguration analysis with Zemax and data treatment with Excel. Polarimetric analysis needs different configurations for e-o rays. Multiconfiguration also includes two perpendicular positions of the HWP along the X-axis and Y-axis. Thus, the design of the WeDoWo requires 8 configurations. We change the parameters of the multiconfiguration mode using a Macro that calculates the new values of the optical design for a given geometry. We change the geometry by sampling the values of the angles of the WeDoWo within a range. Results are calculated for each configuration in J, H and K bands. We tested the case of one, two and three slots for two different materials (MgF2 and LiYF4). The resulting text file for each simulation, configuration and field contains information about the position of the ray, the scattering, vigneting and Zernikes' coefficients for each of the five wavelengths at the selected filter. An Excel macro extracts the required information from these files in order to calculate and plot the image of the slots and the chromatic aberration. Our results shows that three slots are possible but minimum degradation occurs at the central slot for a FOV of 12''.