FanCam Optics:
Instrumental Polarization Theory

Figure 1 below shows FanCam's four internal beam folding mirrors and their orientation.

In the general case of metallic reflection, linearly polarized light is reflected as linearly polarized light only when the position angle (direction of vibration of the electric vector) is either parallel or perpendicular to the plane of incidence. According to the classical theory of wave optics (c.f. Born & Wolf 1999), the Fresnel equations (derived from the boundary conditions for the electric and magnetic fields at a surface of discontinuity) give the relations between the incident, transmitted, and reflected waves in terms of the refractive index n′ = n + i k which includes an imaginary component k = αλ/4π defined in terms of the absorption coefficient α for the exponential decrease of wave intensity with depth x in the metallic medium I = I₀e^-αx.

The solution of the Fresnel equations gives the following different values of the reflectivity R (ratio of reflected to incident intensity) and the phase shift δ on reflection at a vacuum interface with a "thick" metallic layer (i.e. thickness at least comparable to the wavelength λ of the light) for angle of incidence θ_i, depending on the parallel (p) or perpendicular (s) orientation of linearly polarized light waves:

where κ = k / n, and

and

are the solutions of Snell's law for real numbers u and v used to represent the complex product n′cos θ_r = u + i v which describes the angle of refraction θ_r of the transmitted wave as absorbed by the metal.

Figure 2 shows the complex index of refraction of gold as a function of vacuum wavelength according to measurements provided by the Sopra S.A. Company, a French manufacturer of spectroscopic ellipsometers. The reflectivities calculated from Equations 1 and 2 for θ_i = 45° are shown on the left panel of Figure 3, with the equivalent data supplied by Edmund Scientific Co. for its "protected gold" mirror coating plotted on the right panel for comparison. The calculated phase shifts are shown in Figure 4. No equivalent phase shift data are available from Edmund. For practical purposes over the J, H, and K_S passbands, the reflectivities are indistinguishable even though the Edmund product has a protective coating. We will therefore assume that calculations based on the Sopra refractive index data should give at least a useful approximation for the polarization effects of the gold mirror reflections in FanCam.

According to Tinbergen (2007), the Mueller matrix for an inclined mirror is

where R_a = 0.5 (R_p + R_s) is the arithmetic mean of the reflectivities, R_g = √R_p R_s is their geometric mean, p = 0.5 (R_p - R_s), and Δ = δ_p - δ_s is the difference in phase shift on reflection.

For an arbitrary input Stokes vector

let us follow the polarization effects of each FanCam optical element in turn to derive the measured output Stokes vector S′. The effects of the first two fold mirrors M1 and M2 cancel since their relative orientation is to switch the s and p states.

The Stokes vector after reflection by M3 is then

The Wollaston prism polarization analyzers in FanCam are simply orthogonal beam splitters for linearly polarized light and produce no phase changes. Neglecting transmission losses, we evaluate individually the intensity of each of the four 100% linearly polarized beams emerging from the Wollaston analyzers on reflection from M4 as follows.

The Stokes vector representation of the 0° beam (with q = Q/I = 1) is

Similarly,

and

Next we apply the top row of matrix M₄ = M₃ to find the changes in each of these four beam intensities due to reflection by M4:

and

The normalized Stokes parameters as measured by FanCam are therefore

and

Thus we see that the normalized linear Stokes parameters q, u, and v of the incident light are related to the measured quantities q′ and u′ by the relations

and

which allow us to recover q from q′, but not to recover u from u′ without considering v, which in practice is usually unknown.

Values of the matrix element parameters, calculated from the Sopra data with &theta_i = 45° and averaged over the near infrared passbands, are tabulated in Table 1. The expected values of the measured polarization for an unpolarized standard star (q = u = v = 0) according to Eqns. 5 and 6 are given in Table 2.

Table 1. Mueller matrix parameters for the near infrared passbands.

Table 2. Theoretical measured polarization for unpolarized input.

Figure 5 show the correspondence between typical J-band input polarization and the expected output according to Eqns. 5 and 6, for values of q_J and u_J ranging from 0% to 10% and constant v_J = 1.0%. For comparison, a simple offset correction using the data given in Table 2 results in maximum absolute errors of dq_J = 0.016% and du_J = 0.308% (for the case in which q_J and u_J are both negative).

However, du_J increases with increasing v_J and reaches a value of 1% at v_J = 5%. Such an error is too large to safely ignore, which gives us an estimate of the degree of circular polarization that can be tolerated in using FanCam to measure linear polarization. This is somewhat discouraging given the well-established evidence for near infrared circular polarization on the order of 2% with maximum values as high as 17% in typical star formation regions (Lonsdale et al. 1980; Chrysostomou et al. 2000; Buschermöhle et al. 2005).

REFERENCES

• Born, M., & Wolf, E., 1999, in "Principles of Optics" (New York: Cambridge)

• Buschermöhle, M., Whittet, D. C. B., Chrysostomou, A., Hough, J. H., Lucas, P. W., Adamson, A. J., Whitney, B. A., & Wolff, M. J. 2005, ApJ, 624, 821

• Chrysostomou, A., Gledhill, T. M., Ménard, F., Hough, J. H., Tamura, M., & Bailey, J. 2000, MNRAS, 312, 103

• Lonsdale, C. J., Dyck, H. M., Capps, R. W., & Wolstencroft, R. D., 1980, ApJ, 238, L31

• Tinbergen, J. 2007, PASP, 119, 1371