Figure 1 (click here) is a sketch of a Wollaston prism with a schematic ray-tracing. In the first (entrance) prism the "o" light beam vibrating perpendicular to the optic axis has a refractive index while the e-ray has a refractive index . At the prisms interface the refraction index of the o-ray changes from to and the opposite occurs for the e-ray, and when exiting the second prism the angle between the two rays is increased further.
Figure 1: Schematic representation of a Wollaston prism and ray-tracing (adapted from Fig. 11 of Bennet 1995)
For most practical applications the deviation can
be considered symmetric around (i.e. )
and the separation between the o and
is given by:
where = is the birefringence index of the crystal. For astronomical instruments the separation can be most conveniently expressed in sky-projected angles
where is the telescope diameter and is the diameter of the pupil image inside the instrument. The maximum field of view which can be handled by the Wollaston is . The deviation is chromatic because varies with , this implies that broad-band images taken through a Wollaston are elongated. The amount of lateral chromatism, i.e. the image "blurring" over a given wavelength range, follows from Eq. (1) and can be conveniently quantified in terms of the parameter
The image elongation
is simply given by
and this relationship is independent on the detailed parameters of the prism such e.g. its aperture angle. In the case e.g. of a Calcite Wollaston with a sky-projected separation of (i.e. a maximum field of view of 40'') the image blurring in the H band is 0.56'' because V=36 (cf. Table 1 (click here)). Wollaston prisms from crystals with could therefore be used to image fields of view >1 arcmin with an image blurring of only <0.3''. These materials are analyzed in some details in the following subsections.
Figure 2: Predicted separations (sky projected angles) between the "`o" and "e" rays in AgGaS and LiYF (YLF) Wollaston prisms inserted at the 22 mm pupil plane of NICS, the IR instrument for the Italian 3.5 m Galileo telescope. Note the large total separation of the beams, , and the very small image elongation in the 0.9-1.1, J, H and K bands
Table 1: Refraction indices and chromatic characteristics of
IR birefringent crystals with
This crystal has a quite large birefringence () which peaks at 1.05 m and decreases very slowly at longer wavelengths. The combination of large and low chromatism makes it possible to design Wollaston prisms which can handle large fields of view with sub-seeing image quality. An example is given in Fig. 2 (click here) (left panel) where we plot the computed beam separation for an AgGaS Wollaston inserted at the 22 mm pupil plane of NICS, the IR instrument for the Italian 3.5m Galileo telescope (Oliva & Gennari 1995). The separation is and the field of view is therefore 1 arcmin with an image elongation of only in all bands.
From the practical point of view, AgGaS is considered "one of the most successful materials developed for nonlinear optical laser devices in the infrared'' (Bhar et al. 1983). The thermo-optic coefficients are quite large ( K) but the change of birefringence with temperature is small: (Bhar et al. 1983). No significant change of optical performances is therefore expected when the prism is cooled to 77 K. A potential problem is total reflection at the prisms interface which may be difficult to avoid because of the large refractive index: . Standard cements with could only be used for prisms with , and their practical use may be further limited by the fact that AgGaS crystals are significantly deformed by cooling. The thermal expansion coefficients || and to the optical axis are +29 and -19 ( ), respectively.
This material is transparent from below 3000 Å to beyond 4 m. It
displays a very low chromatism from 1 to 2 m and
also has excellent performances from 0.8 to 2.5 m.
The computed beam separation
for an YLF Wollaston are displayed in
Fig. 2 (click here) (right hand panel) where one can better visualize
the chromatic performances of this material which is comparable to AgGaS\
in all bands but K.
The thermo-optics coefficients of YLF are very small,
and respectively increase by and
between room temperature and 77 K (Barnes & Gettemy 1980).
Practical advantages of YLF are:
- The refractive index of YLF is and very close to that of standard optical cements.
- YLF crystals are only slightly deformed by cooling because the thermal expansion coefficients || and to the crystal axis differ by only 5 .
- Presently, YLF is much cheaper than AgGaS.
YVO is an interesting crystal to which attention was already drawn by Bennet & Bennet (1978) and which is now produced by several companies. It has a large birefringence () and could be therefore useful for manufacturing thin Wollastons when the physical thickness of the prism is limited e.g. by the space available in the filter wheel. Accurate measurement of the refractive indices of YVO are not available in the literature and the only information we could find is from the CASIX data sheet which gives the following relationships ( in m)
These are probably based on measurements at m
and are not necessarily accurate at longer wavelengths.
is a well known crystal for quasi-achromatic Wollastons from the ultraviolet to 1 m but has a very low birefringence ( = 0.011) and becomes quite chromatic at m ( in the K band, cf. Table 1).
CdSe has a very low chromatism in J, H, K but is not suited for applications below 1.1 m. Compared to LiYF it has a much higher refractive index () and a slightly lower birefringence ( = 0.020). Both facts make CdSe a not attractive alternative to YLF.
is a cheap compound extensively used for non linear applications and which is transparent to well beyond 2.5 m. It has a quite large birefringence () and may be useful to manufacture low cost prisms whose performances are however limited by the relatively large chromatism of the crystal ().
Rutile () is another classical material which could be a good though expensive alternative to whenever high birefringence is required.
Calcite, the most widely used birefringent crystal, has the advantage of low refractive index and high but displays a large lateral chromatism and cannot be used beyond 2.0 m because of its large opacity to the ordinary ray (cf. Sect. 3).
Figure 3: Transmission of Calcite, thickness 9 mm, at room temperature (upper panel) and 77 K (lower panel), the solid and dashed lines refer to the ordinary and extraordinary ray, respectively, while the thin lines show the transmission expected including reflection (Fresnel) losses alone
has interesting optical properties, large and in all bands, but is very hygroscopic and therefore unlikely to find practical applications in astronomical instruments.