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
e rays
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 
(silver thiogallate)
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.
(YLF)
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.
