Polarization Aberrations in Astronomical Telescopes: The Point Spread Function

Polarization measurements of astronomical sources contain substantial astrophysical information. Many stars observed in the UV, Visible, and IR are thermal emitters and their radiation at the star is unpolarized except for a minority of stars with high magnetic fields. Hiltner (1950) and Mavko et al. (1974) showed that the asymmetry of aligned dipoles in interstellar matter selectively absorbs the thermal emission from background stars. Unpolarized radiation that scatters from planetary atmospheres and circumstellar disks can become partially polarized. When one polarization is preferentially absorbed over its orthogonal state, the unpolarized starlight becomes partially polarized. Clarke (2010) and Perrin (2009a, 2009b) provide a comprehensive review of the value of precision polarization measurements to general astrophysics. Keller (2002) provides a review of spectropolarimetric instrumentation. Hines (2000) reviews the NICMOS polarimeter on the Hubble space telescope.

Analysis by Stam et al. (2004) and measurements reported by Tomasko & Doose (1984), West et al. (1983), and Gehrels et al. (1969) using data from the imaging photopolarimeters on Pioneers 10 and 11 and the Voyagers showed that Jupiter-like exoplanets will exhibit a degree of polarization (DoP) as high as 50% at a planetary phase angle near 90°. Stam et al. (2004) showed that polarization measurements of the planet's radiation in the presence of light scattered from the star reveal the presence of exoplanetary objects and provides important information on their nature. Since the first report by Berdyugina et al. (2011) of the detection of polarized scattered light from an exoplanet (HD 189733b) atmosphere, several theoretical models have been developed. de Kok, Stam & Karalidi (2012) showed that the DoP changes with wavelength across the UV, visible, and near IR band-passes to reveal the structure of the exoplanet's atmosphere. Karalidi et al. (2011) showed that polarization measurements are of value in exoplanet and climate studies. Madhusudhan & Burrows (2012) and Fluri & Berdyugina (2010) showed that orbital parameters (inclination, position angle of the ascending node, and eccentricity) could be retrieved from precision polarimetric measurements. Graham et al. (2007) have shown that a polarization signature of primordial grain growth within the AU Microscopii debris disk, provides clues to planetary formation. Perrin (2009a, 2009b) shows that imaging polarimetry provides important constraints for the analysis of circumstellar disks.

Polarimetric measurements of astronomical sources provide critical astrophysical and exoplanet information. All polarization measurements are made with telescopes and instruments that contribute their own polarization signature. Many authors discuss methods to calibrate photopolarimetric measurements for changes in polarization transmissivity. However, this article provides the tools to understand the source of this instrument polarization and to estimate the magnitude of the effect on the image quality for coronagraphy and astrometry.

The aberration of an optical system is its deviation from ideal performance. In an imaging system with ideal spherical or plane wave illumination, the desired output is spherical wavefronts with constant amplitude and constant polarization state centered on the correct image point. Deviations from spherical wavefronts arise from variations of optical path length (OPL) of rays through the optic due to the geometry of the optical surfaces and the laws of reflection and refraction. The deviations from spherical wavefronts are known as the wavefront aberration function. Deviations from constant amplitude arise from differences in reflection or refraction efficiency between rays. Amplitude variations are amplitude aberration or apodization. Polarization change also occurs at each reflecting and refracting surface due to differences between the s and p-components of the light's reflectance and transmission coefficients. Across a set of rays, the angles of incidence changes and thus the polarization varies, so that a uniformly polarized input beam has polarization variations when exiting the system (Kubota & Inoué 1959; Chipman 1989a). For many optical systems, the desired polarization output would be a constant polarization state with no polarization change transiting the system; identity Jones matrices can describe such ray paths through an optical system. Deviations from this identity matrix are referred to as polarization aberrations.

In this hierarchy, wavefront aberrations are by far the most important aberration, as variations of OPL of small fractions of a wavelength can greatly reduce the image quality. The relative priority of wavefront aberrations is so great that for the first 40 years of computer-aided optical design, and amplitude and polarization aberration were not calculated by the leading commercial optical design software packages. The variations of amplitude and polarization found in high-performance astronomical systems cause much less change to the image quality than the wavefront aberrations, but as the community prepares to image and measure the spectrum and polarization of exoplanets and similar demanding tasks, these amplitude and polarization effects can no longer be ignored. For example, Stenflo (1978) has discussed limitations on the accuracy of solar magnetic field measurements due to polarization aberration.

In a system of reflecting and refracting elements, amplitude and polarization aberration contributions arise from the Fresnel coefficients for uncoated or reflecting metal surfaces and by the related amplitude coefficients for thin film-coated interfaces. Polarization aberration, also called instrumental polarization, refers to all polarization changes of the optical system and the variations with pupil coordinate, object location, and wavelength. The term "Fresnel aberrations" refers to polarization aberrations which arise strictly from the Fresnel equations, i.e., systems of metal coated mirrors and uncoated lenses (Kubota & Inoué 1959; Chipman 1987; Chipman 1989a; McGuire & Chipman 1994a). Multilayer-coated surfaces produce polarization aberrations with similar functional forms and may have larger or smaller magnitudes, but all the polarization aberration-related image quality issues addressed here can be demonstrated with metal reflectors.

Polarization ray tracing is the technique of calculating the polarization matrices for ray paths through optical systems (Bruegge 1989; Chipman 1989a, 1989b; Waluschka 1988; Wolff & Kurlander 1990; Yun 2011a, 2011b). Diffraction image formation of polarization-aberrated beams is then handled by vector extensions to diffraction theory (Kuboda & Inouè 1959; Urbanczyk 1984, 1986; McGuire & Chipman 1990, 1991; Mansuriper 1991; Dorn 2003; Tu 2012). These polarization aberrations frequently have similar functional forms to the geometrical aberrations, since they arise from similar geometrical considerations of surface shape and angle of incidence variation (Chipman 1987; McGuire & Chipman 1987, 1990, 1991, 1994a, 1994b; Hansen 1988; Chipman & Chipman 1989; Shribak et al. 2002; Beckleyet al. 2010). Polarization aberrations can be measured by placing an optical system in the sample compartment of an imaging polarimeter and measuring images of the Jones matrices and/or Mueller matrices for a collection of ray paths through the optical system (Pezzanitti et al. 1995; McEldowney et al. 2008).

Image quality in astronomical telescopes is traditionally quantified using four metrics: wavefront aberration, the image of the point spread function (PSF), the optical transfer function (OTF), and the behavior of these metrics across the field-of-view (FOV) and with wavelength.

Conventional astronomical telescope/instrument systems today are mostly ray traced and analyzed using a scalar representation for the electromagnetic field, usually calculated by "conventional ray tracing," without regard for polarization. Very accurate simulation of high-resolution and high-contrast imaging systems, at the level of polarization artifacts comprising 10^-3 of less than the total flux, requires a vector representation of the field and a matrix representation of the optical system to account for the typically small, but increasingly important, effects of polarization aberration.

Image formation is a phenomenon of interference. Consider the image quality for the image of a star. The light must be coherent across the wavefront entering the telescope to form a diffraction-limited image. Since different wavelengths are incoherent with respect to each other, the different wavelengths essentially each form separate diffraction-limited images on top of each other; i. e., they add in intensity. For starlight, the wavefront components in two orthogonal polarizations (call them X and Y) are also incoherent with respect to each other, and also form separate diffraction-limited images on top of each other. If the star's X-polarized light is rotated to Y, it does not form fringes with the star's Y-polarized light; this is the meaning of unpolarized light. A metric of the degree to which there is good coherence from waves across the pupil is fringe contrast or the visibility of fringes.

The calculation of polarization aberration effects on image formation presented below will follow the steps shown in Figure 1. The Jones pupil is determined as an array of Jones matrix values by polarization ray tracing. The Fourier transforms of the Jones pupil elements yield an amplitude response matrix (ARM), which describes the amplitude distribution in the image of a monochromatic point source specified by a Jones vector. In what follows, bold acronyms indicate matrix functions. Conversion of the ARM's Jones matrices into Mueller matrices (Goldstein 2010; Gil 2007; Chipman 2009) yields the Point spread matrix (PSM). The image of an incoherent point source specified by a set of four Stokes parameters is obtained by matrix multiplying the Stokes parameters by the PSM, yielding the image of the point spread function in the form of Stokes parameters (McGuire & Chipman 1990).

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Volume, packaging, and mass constraints levied by space-craft structural engineers to accommodate launch vehicles now require large aperture astronomical space telescopes to have low F/# and a compact instrument optical package, which requires multiple fold mirrors in the optical path. The larger the deviation angle for each ray at a mirror reflection, the larger is the magnitude of the instrumental polarization and the impact on the PSF. Similarly, ground-based telescopes are being built very compact, with low F/# primary mirrors to minimize the cost of the telescope and instrument system structure. The current set of ground-based large telescopes under development, Giant Magellan Telescope (GMT), Thirty Meter Telescope (TMT), and the European-Extremely Large Telescope (E-ELT), use optical system architectures where radiation strikes mirror surfaces at high angles of incidence introducing polarization-induced variations to wavefronts. These telescopes function as partial polarizers, and the retardance and diattenuation at the focal plane depend where on the sky the telescope is pointing.

Drude (1900), Drude (1902), Stratton (1941), Azzam et al. (1987), Born & Wolf (1993), Ward (1988), and many others show that the polarization of light changes at each non-normal reflection; this introduces diattenuation and retardance, which apodize and change the wavefront. This causes a change in the shape of the PSF and can lead to unexpected performance for some astronomical applications. The magnitude of the degraded performance depends on the particular opto-mechanical layout selected for the optical system architecture and the mirror coatings.

Witzel et al. (2011) characterized the polarization transmissivity of the VLT; Hines et al. (2000) analyzed the HST NICMOS instrument, and Ovelar et al. (2012) modeled the ELT for instrumental polarization. These works were done for the purpose of correcting photo-polarimetric data and not, as our work is here, for the purpose of studying the PSF image structure. Breckinridge & Oppenheimer (2004) and Breckinridge (2013) established that the shape of the PSF image for the astronomical telescope depends on polarization aberrations. McGuire & Chipman (1994a, 1994b), Yun et al. (2011), and Yun et al. (2011) developed analytic tools and models to analyze polarization aberrations.

Geometrical ray tracing optimizes geometric image quality by minimizing physical optical path differences (OPD). An analysis that also takes polarization into consideration is needed to determine whether or not the wavefront is compromised by polarization such that it would not meet stringent specifications. As shown in our example, the geometric ray trace can be perfect, scalar diffraction accounted for, and the entire set of OPLs equal but the polarization aberration can still reduce image quality.

When a plane wave is incident on a metal reflector, the radiation's electric and magnetic fields drive the charges in the metal, which undergo a small oscillation at optical frequencies. These accelerating charges give rise to the reflected beam. The response of the charges, and thus the reflected beam, depends on the orientation of the electric field. The reflection process is a linear and can be completely described by the reflection of the s-component and p-component separately.

Figure 4 provides the notation to express the Fresnel equations which calculate the polarization content of the reflected beam. The complex refractive index N = n + ik, has imaginary part k and real part n. The values of n and k are given in optical materials handbooks (Palik 1961). Given the angle of incidence θ₀ and the incident medium refractive index N₀, the angle of refraction in medium 1 is found using Snell's law: N₀ sin θ₀ = N₁ sin θ₁, where N₁ and θ₁ and θ₀ are defined in Figure 4. From , the angle θ₁ is

which for metals is a complex angle. The reflectivity for p-polarized light, polarized parallel to the plane of incidence, is given by the Fresnel coefficient for p-polarized light (Azzam et al. 1987),

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Similarly the reflectivity r_s for the s-polarized light, polarized perpendicular to the plane of incidence is

The amplitude reflection components in Figure 3a describe the relative amplitude of the reflected light. The fraction of reflected flux is the amplitude squared, |r_s|² and |r_p|², which for normal incidence is 0.932² = 0.87. The remainder of the light's energy is lost to resistance from the charges moving through the metal, heating the reflector in the process. In Figure 3a, the s-reflectance is seen to be greater than the p-reflectance, leading to diattenuation. The variations of |r_s| and |r_p| with angle cause amplitude aberrations. For s-polarized light, the wavefront is brighter at larger angles of incidence while for p-polarized light the wavefront is dimmer. This apodization has a small effect on the image quality. The difference between |r_s|² and |r_p|² indicates that the reflectors act as weak polarizers. The polarization-dependent reflectance is characterized by the diattenuation D,

Metallic reflection acts as a weak polarizer, called a diattenuator after the two attenuations. Diattenuation varies from zero when all polarization states have the same reflectance or transmission (as with ideal retarders or nonpolarizing interactions) to one for ideal polarizers. When unpolarized light, such as starlight, is incident, the diattenuation is equal to the DoP of the exiting light. Due to the diattenuation, light incident in states different from s and p have some fraction of the energy coupled into the orthogonal polarization, and these orthogonally polarized components have an important role in the image formation that will be highlighted later.

Figure 3b shows the phase change on reflection, which is different for the s and p-polarizations. This phase change is a contribution the mirrors to the wavefront aberrations of the system. The phase shift δ between the s and p-reflected beams upon reflection is the retardance,

Since s and p at interfaces are linearly polarized states, this retardance is referred to linear retardance. Linear retardances with their fast axes aligned add; linear retardances with perpendicular fast axes subtract. In general the retardance of a sequence of retarders is simulated through the multiplication of Jones or Mueller matrices. Sequences of linear retardances at arbitrary orientations have elliptically polarized eigenvectors (fast and slow axes).

The polarization aberration changes the polarization state of a small fraction of the light, and as described later, that component changes the intensity and polarization distribution of the image, which can be an important factor in high contrast and resolution imaging. For small diattenuation (dimensionless) or small retardance (radians), the maximum fraction F of light which can be coupled into the orthogonal polarization state occurs for light at 45° to the diattenuation axis or retardance axis and both have the same quadratic form for F,

These equations are readily derived using the Mueller calculus by placing a diattenuator or retarder oriented at 45° between crossed polarizers and evaluating the Taylor series for the transmitted flux.

To generate scaling rules for polarization aberration, approximate forms for the diattenuation (eq. [4]) and retardance (eq. [5]) due to the Fresnel coefficients are presented in the appendix. For the two on-axis mirrors, these Fresnel coefficients have been expanded as even quadratic equations about normal incidence (eq. [A7]); for the fold mirror, these coefficients have been expanded as linear equations about the axial ray (eq. [A8]).

The phases in Figure 3b are important; they are polarization-dependent contributions to the wavefront aberration. They are perturbations to the wavefront which change depending on the metal or coating applied (McGuire & Chipman 1991). Since the fold mirror is in a converging beam, the nonzero slopes of the s and p-phases are both important and have been highlighted in Figure 3b. These slopes cause linear phase shifts, which shift the locations of the X and Y-polarized components from the geometrical image location, and since the slopes are different with opposite slopes, these components move in different directions by a small fraction of the Airy disk radius. Wavefront correction (e.g., adaptive optics) can flatten the slope of the phase through angle. Since the slopes are different for s and p-phases, wavefront correction can only correct either the s-polarized wavefront or the p-polarized wavefront, but not both simultaneously. This effect is discussed further in § 2.6 and in § 5 (design rules 8, 9, and 10).

The polarization aberration of the example telescope of Figure 2 will first be examined from maps of diattenuation and retardance and then as a Jones matrix representation at the exit pupil. The diattenuation contributions from the three mirror elements are shown in the first three panels of Figure 5. The fourth panel in Figure 5 shows the cumulative diattenuation for the entire telescope as viewed looking into the exit pupil from the image plane. Each line inside the circle shows the diattenuation magnitude and the orientation of the maximum transmission axis at a point in the pupil. The primary and secondary mirrors (the first two panels in Fig. 5) produce rotationally symmetric, tangentially oriented diattenuation with a magnitude that increases quadratically from the center of the pupil.⁵ The fold mirror introduces a horizontally oriented diattenuation with a linear and cubic variation along the vertical axis. The cumulative diattenuation map shown on the right is predominantly linear from top to bottom. Polarization aberration functions which closely fit these diattenuation maps are discussed in the appendix.

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The aluminum's retardance introduces a polarization-dependent phase contribution to the OPL differently for the s and p-components of the light. Retardance aberration thus represents a difference in the metal coating's wavefront aberration contribution as experienced by orthogonal polarization states. Figure 6 shows the individual surface contributions to the retardance aberration in the first three panels and the cumulative retardance aberration through the system in the last panel. Each line shows the retardance magnitude and fast axis orientation at a grid of locations in the beam. The primary and secondary mirrors produce a rotationally symmetric tangentially oriented fast axis, which increases quadratically from the center, while the fold mirror introduces retardance with a vertically oriented fast axis. The fold mirror has a linearly varying retardance increasing from the bottom to the top of the pupil. Since the fold mirror has the largest retardance, the resultant retardance for the entire telescope shown on the right is similar to the fold mirror retardance with contributions from the primary and secondary mirrors. The cumulative linear retardance map (the fourth panel of Fig. 6) is primarily a constant retardance, with a linear variation from bottom to top, and a variation of retardance orientation from left to right. The cumulative retardances are shown as linear retardances (lines) but because the three individual weak retardances in the first three panels are not strictly parallel or perpendicular, the fast and slow axes of the retardances in the last panel are slightly elliptical; however, the ellipticity, which has a maximum value of 0.0047, is much smaller than would be visible at this scale. Similarly the diattenuation becomes slightly elliptical when the axes are not aligned, and in the map of Figure 5 has a maximum ellipticity of 0.011.

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Polarization aberration functions which approximate these retardance maps are discussed in the appendix. Constant retardance is a constant difference in the wavefront aberration, a "piston" between polarization states; it changes polarization states but piston does not degrade image quality. The linear variation of retardance indicates a difference in the wavefront aberration tilt, X- and Y-polarizations get different linear phases, and so their images are shifted from the nominal image location by different amounts (see the end of § 2.2).

The spatial variation of the telescope's diattenuation (fourth panel of Fig. 5) and retardance (fourth panel of Fig. 6) is a low order variation which can be characterized by simple polynomials (see appendix). The retardance from the primary and secondary mirrors has a quadratic phase variation; this pattern has been named "retardance defocus" (Chipman 1989a). For X-polarized light, the relative phase is advanced quadratically moving along the x-axis from the center to the edge of the field, and is retarded quadratically moving to the edge of the field along the y-axis. This causes astigmatism arising from the different quadratic variations of ϕ_rs and ϕ_rp about the origin in Figure 3. So the X-polarized image, being astigmatic by 0.022 radians (0.012 + 0.010 or 3.4 milliwaves; see the scale of the primary mirror and secondary mirror retardances at the edge of the pupil in Fig. 6), becomes slightly elongated in opposite directions on either side of the best focus. Similarly for Y-polarized light, the relative phase is advanced moving along the y-axis from the center to the edge of the field, and is retarded moving to the edge of the field along the x-axis. So the Y-polarized image is astigmatic with the opposite sign. This concave mirror-induced astigmatism is real and has been observed with interferometers.

Unlike conventional astigmatism, which on-axis would likely be caused by a cylindrical deformation in a mirror, this coating-induced astigmatism arises from the primary and secondary mirror's retardance defocus and is tied to the polarization state of the light. Coating-induced astigmatism rotates with the polarization state, whereas for a cylindrical deformation, the astigmatism would rotate with the mirror and not with the polarization state. For unpolarized light, the coating-induced astigmatic image is the average over the PSF of all polarization components, which is also the sum of the PSF for any two orthogonal components. So the astigmatism which is seen in a single incident polarization state, such as X-polarized, when added to the astigmatism for Y-polarized light, where the astigmatism is rotated by 90°, forms a radially symmetric PSF, which is slightly larger than the unaberrated image. Inserting a polarizer will reveal the astigmatism in any particular polarization component. More information on retardance defocus and the associated astigmatism in Cassegrain telescopes is found in Reiley & Chipman (1994).

Each ray through the optical system has an associated Jones matrix, which describes the polarization change, the diattenuation and retardance, for that ray path (Jones 1941). The polarization aberration function is the set of Jones matrices expressed as a function of pupil coordinates and object coordinates (McGuire & Chipman 1990). The set of Jones matrices for a specified object point is called the Jones pupil, and has the form of a Jones matrix map over the pupil (Ruoff & Totzeck 2009). The Jones pupil is represented by the 2 × 2 Jones matrix, which contains complex components with amplitude A(x,y) and phase ϕ(x,y) at each point in the pupil (x,y),

For the X-polarized incident field at the entrance pupil, the term A_XX is the amplitude of J_XX at point (x,y) of the X-polarized field at the exit pupil. Also, for the X-polarized incident field at the entrance pupil, A_YX is the amplitude at point (x,y) of light coupled from X into the Y-polarized field at the exit pupil. The term ϕ_YX, the complex argument of J_XX, is the phase shift from the X-polarized incident field to the X-polarized exiting field due to the metal reflections. Similarly ϕ_YX is the phase shift for the X-polarized field coupled into the Y-polarized field. The field from a single point in object space maps into the 2 × 2 Jones matrix shown on the right of equation (7). Similarly, the right column of J describes the effects for the Y-polarized incident field.

The Jones pupil is calculated using the algorithms of geometrical optics (ray-tracing) augmented with polarization ray tracing. During the ray trace, the angle of incidence and orientation of the plane of incidence are evaluated at each ray intercept, and the Fresnel equations are used to calculate the s and p-reflection coefficients, as plotted in Figure 3. These coefficients are used to generate a polarization matrix for the ray intercept, such as a Jones matrix (Yun, Crabtree, & Chipman 2011). The OPL contributions are summed for each ray segment to determine the geometrical phase in the exit pupil. This is repeated for a grid of rays to calculate the geometrical wavefront aberration function. "Geometrical" here refers to the calculation of OPL, as determined by conventional ray tracing, without the influence of instrumental polarization. The polarization matrices are multiplied for each ray intercept from the entrance pupil through the exit pupil to determine the amplitudes in each polarization and the contributions of the metallic reflections to the overall phase.

A coordinate system must be chosen for the description of Jones matrices. The choice of the orthogonal basis is arbitrary. Here it is simplest to decompose the incident plane waves into a component parallel to our fold mirror's rotation axis, horizontal or X-polarized, and a vertical or Y-polarized component. The result of all flux and PSF calculations is independent of the orthogonal basis chosen.

A polarization ray trace was performed on the example telescope of Figure 2 and the values of the Jones pupil elements, given by equation (7), are displayed in Figure 7, which is color coded to show the amplitude and phase variations across the exit pupil. This Jones pupil is very close to the identity matrix times ∼0.806; the 0.806 accounts for average reflection losses from aluminum. Deviations from the identity matrix occur because the aluminum mirrors are weakly polarizing. Equation (A2) contains a closed form approximation to this Jones pupil.

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The J_XX and J_YY diagonal elements contain different amount of polarization aberrations. The overall J_XX amplitude is about 5% larger than the overall J_YY amplitude because of the s and p-reflection difference at 45° shown in Figure 3a. This will cause the x-polarized image to be about 9% brighter than the y-polarized image. From the amplitude images of the Jones pupil in Figure 7a is close to the identity matrix, only a small fraction of the light has its polarization changed. The off-diagonal J_XY and J_YX elements show the polarization coupling between orthogonal polarizations. This polarization crosstalk has relatively low amplitude compared to the diagonal elements. The amplitudes A_XX and A_YY are nearly constant (< 2%), but the A_XY and A_YX are highly apodized, showing a Maltese cross pattern (dark along x and y-axes) shifted downwards.

The phases of the four elements in the Jones pupil, shown in the Figure 7b, represent contributions to the wavefront aberration function from the aluminum mirrors. The Fresnel phase changes are different for the s and p-components leading to different wavefronts for these two components. Since the on-axis geometrical wavefront aberration of the telescope is zero, the phase variation across the pupil are wavefront contributions from the mirror coatings. ϕ_XX, the wavefront aberration function of the telescope illuminated with x-polarized light and analyzed with an x-analyzer, has an overall linear variation of about 0.008 waves with an additional deviation, which is primarily astigmatism. The diagonal elements ϕ_XX and ϕ_YY have a different wavefront tilt between the ϕ_XX and ϕ_YY wavefronts. The source of this tilt difference is shown in Figure 3b, where it is seen that about a 45° angle of incidence, the slopes of the phase change are different for s- incident polarized light than for p- polarized light at the fold mirror. If the Fresnel phases were linear about 45°, only tilt would be introduced. The deviations from linear introduce higher order aberrations including small amounts of astigmatism (from quadratic deviation), coma (from cubic deviation), and other aberrations.

In conventional scalar image formation calculations, the amplitude response function is calculated as the Fourier transform of the exit pupil function. This electric field distribution is then squared to obtain the point spread function (Goodman 2004). To evaluate the image formed by systems with polarization aberration, McGuire & Chipman (1990) introduced a Jones calculus version of the amplitude response function named the Amplitude Response Matrix (ARM),

where is a spatial Fourier transform over each of the Jones pupil elements. For a plane wave incident on the telescope with Jones vector E, the amplitude and phase of the image is given by the matrix multiplication, ARM·E. The ARM for the 3-mirror telescope of Figure 2 is shown in Figure 8. Table 1 summarizes the system and the parameters most relevant to the imaging calculations.

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The ARM's diagonal elements are close to the well-known Airy disk pattern, but are slightly larger due to the aberrations in ϕ_XX and ϕ_YY. Each is slightly astigmatic. Their centroids are slightly shifted due to the differences in their tilt. The off-diagonal elements have much lower amplitudes and contain interesting structure, mostly due to the fold mirror. We refer to these off-diagonal PSF images as the ghost PSFs.

For unpolarized illumination, the incident X- and Y-polarizations are incoherent with respect to each other. So the output components ARM_XX (X in X out) and ARM_YX (X in Y out) are coherent with each other but incoherent with ARM_XY and ARM_YY. So for unpolarized illumination, the two output X-components in the ARM are incoherent with respect to each other, as are the two output Y-components. So the point spread function for an unpolarized source has four additive components I = I_X + I_Y = (|ARM_XX|² + |ARM_XY|²) + (|ARM_YX|² + |ARM_YY|²). The polarization structure of the generic telescope PSF is explored further in the next section.

The distribution of flux and polarization in the image of an incoherent point source, such as a star, can be described with a 4 × 4 Mueller matrix Point Spread Matrix (PSM), the Mueller matrix generalization of the PSF (McGuire & Chipman 1990). This PSM is calculated by the transformation of the ARM's Jones matrices into Mueller matrix functions (Goldstein 2010; Gil 2007; Chipman 2009). The Mueller matrix representation of polarization properties is familiar to most astronomers who make or work with astrophysical measurements of the four Stokes parameters (Gehrels 1974; Hovenier et al. 2005). The example telescope's PSM, calculated from the ARM (Fig. 8), is shown in Figure 9.

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The contribution of each of the 16 elements varies across the PSM and changes depend on the incident Stokes parameters. Hence, each element of the matrix is shown with its contribution and appears as miniature PSF's with different shapes. An example of PSM measurements is found in McEldowney et al. (2008).

The PSF for unpolarized illumination is described by the Stokes parameter image in the first column (m00, m10, m20, m30) inside the red rectangle. Since m10, m20 and m30 are not zero, the PSF of an unpolarized star is not unpolarized. In this example, the Q component's 4.7 × 10^-2 contribution mostly arises from the diattenuation of the fold mirror, which is reflecting more 0° (s-polarized) light than 90° (p-polarized) polarized light. The U component (at 4.36 × 10^-3) is mostly due to the diattenuation contributions at 45° and 135° from the primary and secondary seen in the first two panels in Figure 5. The ellipticity (from the V component) arises when weakly polarized light reflected from the primary and secondary interacts with the retardance from the fold mirror. The spatial variations of Q, U, and V introduce polarization fluctuations in the region of the diffraction rings. Figure 10 maps the DoP in these zones. Such polarization fluctuations in the PSF of a star are clearly a concern when measuring the polarization of exoplanets and debris disks.

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Figure 9 contains a graphical equation describing the 4 × 4 PSM operating on an X-polarized incident beam (represented by the 4 × 1 matrix) which yields a 4 × 1 Stokes image, (I_X, Q_X, U_X, V_X), as represented by the right-most term in the equation. For an unpolarized collimated incident beam, the resulting Stokes image is contained in the first column of the PSM, shown inside the red box. The m10 element describes the ∼9% DoP for the image of the unpolarized source. The m20 element describes small variations of linear polarization orientation within the PSF while m30 characterizes even smaller ellipticity variations.

The light coupled into orthogonal components has a significant impact on the outer portions of the PSF because they arise from the highly apodized Jones pupil components A_XY and A_YX as shown in Figure 7a. To see this, compare the PSF arising from J_XX and J_YX. These PSF terms are calculated using the resultant Stokes image components on the right side of Figure 9 as:

The two terms in equation (9), I_XX and I_YX, are compared in Figure 11 where it is seen that the peak of I_YX is about 10^-5 of I_XX. This "ghost PSF" should be very important in imaging applications that require contrast ratios of 10^-8 or greater.

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Figure 12 shows the irradiance along an x-axis cross-section through the two PSFs in log ₁₀ scale at the plane drawn through the two images shown in Figure 11. This ghost PSF has its light spread away from the center. In Figure 12, the Airy disk's zeros of I_XX are not at the same location as the zeros for the cross-coupled term I_YX. Thus, the zeros of I_XX are washed out by the light leakage from the nonzero I_YX. The point spread function I_YX cannot be corrected by wavefront compensation for either the XX or YY-components alone because most of the image spread is due to I_YX's apodization (Fig. 7). A linear Polaroid placed at the image plane can pass I_XX and remove I_YX, but will still pass the other ghost I_XY, and thus will not correct for this polarization aberration.

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The shape of I_YX shows that the I_X and Q_X Airy disks are not exactly on top of each other. The image plane irradiance distribution for the I_YX term sits beneath the Airy diffraction pattern characteristic of the I_XX term. Figure 12a shows a slice normal to the axis at the RMS best focus through the PSF for I_XX and for I_YX in Figure 11. Figure 12b shows a high-dynamic range image of the irradiance across the focal plane in the vicinity of the core of the PSF for I_YX. The concentric green circles superposed on Figure 12b shows the first and second zeros of the Airy diffraction pattern of I_XX. These dark rings overlay regions with nonzero I_YX.

The right column in Figure 9, (I_X, Q_X, U_X, V_X), is the Stokes parameter PSF for the X-polarized component of an incident beam. The flux of this component is I_X = I_XX + I_YX. Similarly, the PSF I_Y for a Y-polarized incident beam is calculated by multiplying the Stokes parameters (1, -1, 0, 0) to the PSM, and I_Y = I_YY + I_XY. Finally the PSF for unpolarized incident light is (I_X + I_Y)/2 which can also be calculated by multiplying the unpolarized Stokes parameters (1, 0, 0, 0) to the PSM.

This demonstrates that for unpolarized starlight passing into a "generic" optical system like that shown in Figure 2, the PSF is the sum of two nearly Airy diffraction patterns, I_XX and I_YY, plus two secondary or "ghost" PSFs, I_YX and I_YX, which originates from system's polarization cross-talk, the off-diagonal elements in the Jones pupil.

The Jones pupil, ARM, and PSM can be calculated in other basis sets than x and y. Here x and y are aligned parallel and perpendicular to the fold mirror's s-state. The resulting Jones pupil and ARM for other bases are found by Cartesian rotation of the matrices of Figures 7 and 8. The overall flux distributions, I = I_X + I_Y, for an unpolarized source or point source of arbitrary polarization are unchanged by such a change of basis. Similarly, the PSM is rotated by the same rotation operation as Mueller matrices, and again the net flux for any source polarization is unchanged; the corresponding Stokes images are just rotated versions of the ones presented above. The advantage of the x and y basis chosen here is that the off-diagonal elements I_XY and I_YX have their smallest values in this basis. As the basis set rotates or becomes elliptical, the scale of these off-diagonal image components increases rapidly and quickly approach Airy disks due to the coupling between bright diagonal elements and weak off-diagonal elements from the rotation operation. Thus, the fold mirror's s and p-basis (our x and y) is best basis for viewing the value and functional form of the ghost PSF components for the example system of Figure 2.

Consider a Stokes imaging polarimeter located before the telescope of Figure 2's focal plane measuring the PSF of an unpolarized star as a Stokes image. The PSFs for the X-polarized, I_X = I_XX + I_YX, and Y-polarized light, I_Y = I_XY + I_YY, at the focal plane are very close in form to the classical Airy diffraction pattern because the polarization-induced wavefront aberration, ϕ_XX and ϕ_YY in Figure 7, is less than 8 milliwaves, and the amplitude apodization is less than 0.015. But these two PSF images are not exactly superposed, the peaks of I_X and I_Y are displaced from each other by 0.625 mas. The PSF cross-section through the maxima of I_X, I_Y, and I_X–I_Y (the star's Stokes Q image), are shown in Figure 13. The shift between the I_X and I_Y PSFs arises from the difference in slopes of the s- and p-phases in the Fresnel coefficients (Fig. 3b, red and blue tangent lines), which is the cause of the overall linear variations in ϕ_XX and ϕ_YY. Their difference Q = I_X - I_Y is sheared from I_X and I_Y by 5.8 mas, as shown in Figure 13, and is due to the shift between I_X and I_Y. These PSF's shifts and PSF's ellipticities are listed in Table 2 for a single 45° fold mirror before the focal plane. The ellipticity of the PSF image was calculated by fitting an ellipse to the PSF at the half power points.

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In astronomical applications involving the precise measurement of the location of the centroid of the PSF, distortions of the shape of the PSF are important. Most systems incorporate multiple folds, for example in Goullioud et al. (2014) and Witzel et al. (2011). These relay optics with multiple folds may increase the shear between PSF's polarization components. The variation of linear phase across the pupil, as seen in Figure 3, is approximately linear, thus the shear between polarization components is linear in the F/#. As Clark & Breckinridge (2011) showed, across the FOV, variations of PSF ellipticity and orientation are expected from polarization aberration.

The Fresnel polarization aberrations, unless corrected, may affect our ability to characterize exoplanets using space telescopes. At least two mitigation approaches have been suggested. These are discussed in § 4.

Many astronomical instruments require multiple fold mirrors, which often rotate with respect to each other. Consider the telescope in Figure 14 with five-fold mirrors. In the altitude axis and azimuth axis configuration shown, these folds are all coplanar, and the s-polarized light reflecting from the first mirror is s-polarized at all five mirrors; similarly p-polarized light at the first mirror is p-polarized at all five mirrors. For this configuration, the retardances for the axial ray at all five mirrors add. For general configurations where rotations have been performed on the axes, Jones matrices for a ray at all five mirrors need to be multiplied, and a data reduction step (Lu & Chipman 1994) performed to determine the retardance.

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From Figure 3 and equation (5), we see that the retardance is defined as the absolute value of the difference in the phase between s and p-polarized light. The retardance adds cumulatively. The angles of incidence of the axial ray shown in Figure 14 are all equal to 45° (deviate the light by 90°) and therefore, the retardance for collimated rays at the exit pupil will be 5 times the retardance of a single 90° deviation reflection. By equation (6), 25 times the light would be coupled into orthogonal polarization states compared to a single-fold mirror. Also, the diattenuation, given in equation (4), increases approximately five-fold in the configuration of Figure 14.

Recall the linear variation of retardance induces a small spatial shift between the XX and YY PSM components. When the five coplanar mirrors in Figure 14 are reflecting a 0.06 numerical aperture (na) converging beam (F/8, as in the system of Fig. 2) in sequence, the converging ray from the left side of the secondary has a greater angle of incidence at the first, second, and fifth fold mirrors than the axial ray, and a smaller angle of incidence at the third and fourth mirrors. The linear variation of retardance across the pupil is the sum of the linear variations of the five individual coplanar mirrors. Each fold mirror has a retardance pattern corresponding to Figure 6 (third panel for fold mirror). So the linear variation of retardance adds for mirrors one, two, and five, and subtracts for mirrors three and four. The overall linear variation would be unchanged, and be the same as the linear variation of the single-fold mirror of Figure 2. The separation between the XX and YY-polarized images (discussed in the second paragraph of § 2.2), which depends on this linear retardance variation, would be unchanged. But the ghost PSFs would be 25 times greater than those shown in Figures 11 and 12. This all changes once the telescope is steered about the azimuth axis and altitude axis. Such general cases of sequences of fold mirrors in noncollimated beams can always be analyzed by polarization ray tracing. Lam & Chipman (2014) contains additional information on the polarization aberrations for sequences of fold mirrors.

The intensity of the ghost PSF is about one part in 10^-4 of the two primary PSF images I_XX and I_YY and with spatial structure of much larger extent across the focal plane and with more structural complexity than the classical Fraunhofer scalar diffraction PSF. The exoplanet community needs to control the intensity PSF to a part in 10^-10 or 10^-6 of the magnitude below the intensity level for the ghost PSF image which has been calculated for the "typical" telescope system described here. The intensity and the structural complexity in this ghost PSF have the potential to disguise terrestrial exoplanets images and become a source of false positive discoveries and measurement errors. The "ghost" PSF overfills the image plane mask designed using scalar theory and will obscure exoplanet images. The presence of the ghost PSF and its intensity depend on the packaging configuration of mirrors and the physical optics properties of the highly reflective mirror coatings within the mechanical layout of the entire telescope/instrument optical system.

For example, Groff & Kasdin (2013), Carlotti et al. (2013), and Carlotti et al. (2011) design, build, and optimize image and pupil plane masks using scalar models of reflecting optical systems. These masks are now being developed for the new AFTA-WFIRST coronagraph. Our results here show that a modified system of stops and masks may be needed to optimize the WFIRST coronagraph mission for exoplanets.

Astrometry is the measurement of the fundamental position of objects in the sky. Applications include determining the dynamics of the galaxy and calculating the orbital elements of gravitationally interacting bodies. Astrometric precision depends on the precise measurement of the intensity centroid of each point source within an ensemble of point sources across the FOV as mapped by the telescope system onto the focal plane. Typical measurements are to better than 1% of the size of the PSF to achieve an error in the ensemble of less than 0.1%. An inherent assumption is that the intensity centroid remains constant across the FOV. However, the contents of Table 2 shows that accurate modeling of the effects of instrumental polarization indicates centroid asymmetries of about 1%. At 800 nm wavelength, the full width at half max of the PSF image is about 70 mas. McArthur et al. (2011) report absolute parallaxes of select stares in the Hyades to errors near 0.2 mas for HST data; this is 0.2% of the FWHM for stellar PSF's at 500 nm from HST.

Astrometric reference stars are chosen to be at great distances so they appear fixed over long periods of time in relation to foreground objects. The light from these distant objects becomes partially polarized by interstellar matter, particularly, in the Galactic bulge region. The PSF image for partially polarized stars becomes asymmetric due to the polarization aberration. The magnitude and orientation of this asymmetry depends on the number of fold mirrors in the telescope and the bore sight orientation of the telescope. Astrometrists need to consider polarization aberrations as sources of measurement error.

We acknowledge the prior support of the Science Foundation Arizona in the development of the polarization ray tracing program, Polaris-M, used in this analysis. We thank Karlton Crabtree for many contributions leading to this article. We also thank Dr. Keith Patterson who as a student at Caltech performed the initial numerical modeling of polarization in this optical system. We acknowledge the valuable comments provided by an anonymous reviewer.