SNAP Telescope Polarization

Patrick Jelinsky

06/09/2004

# Introduction

Reflection off mirrors at angles other than normal will have sensitivity
to the linear polarization of the light. In the SNAP telescope \[1\],
the fold flat mirror has an angle of incidence from 40° to 50°. This
will cause the telescope to have some polarization sensitivity. The
purpose of this memo is to quantify the sensitivity of the telescope to
polarization.

# Reflection

The reflection from a surface can be calculated from the Fresnel
equations \[2\] and the optical constants of the mirror. The Fresnel
equations are:

$$\begin{array}{r}
R_{} = \frac{a^{2} + b^{2} - 2a\text{cos}\theta + \text{cos}^{2}\theta}{a^{2} + b^{2} + 2a\text{cos}\theta + \text{cos}^{2}\theta} \\
R_{\text{ll}} = R_{}\frac{a^{2} + b^{2} - 2a\text{sin}\theta\text{tan}\theta + \text{sin}^{2}\theta\text{tan}^{2}\theta}{a^{2} + b^{2} + 2a\text{sin}\theta\text{tan}\theta + \text{sin}^{2}\theta\text{tan}^{2}\theta}
\end{array}$$

where *θ* is the angle of incidence, *R~⊥~* is the reflectivity for
perpendicular polarization, *R~ll~* is the reflectivity of parallel
polarization, and *a* and *b* are given by

$$\begin{array}{r}
2a^{2} = \sqrt{\left( n^{2} - k^{2} - \text{sin}^{2}\theta \right)^{2} + 4n^{2}k^{2}} + (n^{2} - k^{2} - \text{sin}^{2}\theta) \\
2b^{2} = \sqrt{\left( n^{2} - k^{2} - \text{sin}^{2}\theta \right)^{2} + 4n^{2}k^{2}} - (n^{2} - k^{2} - \text{sin}^{2}\theta)
\end{array}$$

where *n* is the real part of the index of refraction, and *k* is the
imaginary part.

The polarization sensitivity is (R~⊥~ - R~ll~ )/(R~⊥~ + R~ll~ ). This is
the modulation that a 100% linear polarized light would have.

# Results

The mirrors on SNAP will be coated with protected silver \[3\]. The
optical constants of silver \[4\] are shown in Figure 1 for the
wavelength range of SNAP. The reflectance of the flat mirror at 45° is
shown in Figure 2. The reflectance is very high above 500 nm. The
perpendicular reflectance is slightly higher than the parallel
reflectance. The polarization caused by the flat mirror is shown in
Figure 3 for incidence angles between 40° and 50°. For wavelengths
longer than 750 nm the polarization is less than 1%. The largest
polarization in the SNAP band is at 350 nm and is 9.6%. This means that
the signal from a linear polarized source will change as the telescope
is rotated. If the source is 100% linear polarized, the signal will
modulate (from telescope rotation) by 9.6 % in the blue and less than 1%
above 750 nm.

![](media/image1.wmf)

Figure 1 Optical constants of Silver \[4\].

![](media/image2.wmf)

Figure 2 Reflectance for both polarizations of a silver mirror at 45°
angle of incidence.

![](media/image3.wmf)

Figure 3 Polarization from the SNAP flat mirror for various incidence
angles.

# Conclusion

The polarization sensitivity caused by the 45° folding flat in SNAP is
fairly small. If the supernovae are polarized by less then 10% in the
blue, then the signal will change less than 1% when the telescope is
rotated. In the red, the sensitivity to polarization is 10 times less so
even a 100% linear polarized source will have it's signal change by less
than 1% as the telescope rotates.

The zodiacal light is about 20% polarized near the ecliptic poles. This
means that at the shortest wavelengths, the background will change by
about 2% as the telescope is rotated.

References

1)  SNAP Optical Telescope Assembly Definition and Requirements, M.
    Lampton, 2002,
    \<http://sprg.ssl.berkeley.edu/sprite/snap_cm/1_SNAP_Controlled_Drawings_Repository/00008-MW02-C.doc\>

2)  Handbook of Space Astronomy and Astrophysics, Martin V. Zombeck,
    Cambridge University Press, 1982, p. 243.

3)  SNAP Mirror Coating Trade Study, M. Lampton,
    \<http://sprg.ssl.berkeley.edu/sprite/snap_cm/1_SNAP_Controlled_Drawings_Repository/00009-MW02-A.doc\>

4)  Handbook of Optical Constants of Solids, ed. Edward D. Palik,
    Academic Press, 1985, p. 353

5)  Handbook of Space Astronomy and Astrophysics, Martin V. Zombeck,
    Cambridge University Press, 1982, p. 243.