What are the different types of polarization aberrations of telescopes?

spherical, and aspherical crystalline elements or element assemblies can provide a wide variety of compensating polarization aberrations (Chowdhury et al. 2004).

What is the resultant retardance for the telescope shown on the right?

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.

What is the phase shift between s and p-reflected beams?

The phase shift δ between the s and p-reflected beams upon reflection is the retardance,δ ¼ jϕrp ϕrsj: (5)Since s and p at interfaces are linearly polarized states, this retardance is referred to linear retardance.

What is the polarization aberration of a small fraction of light?

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.

What is the way to reduce the image quality of a telescope?

The geometric (optical path difference) ray trace of an optical system can be ideal (as in the example telescope) but the polarization content of each ray may change enough across the wavefront to reduce image quality.

What are the optical coatings for mirrors?

The optical-coating prescriptions for antireflection coatings of lenses and reflection-enhancingcoatings of mirrors provide design degrees of freedom (thicknesses and materials) to adjust the diattenuation and retardance.

What components need to be considered as potential polarization sources?

All components need to be considered as potential polarization sources, including metal and dielectric thin films, optical filters, polarization analyzers, and dispersing elements.

What is the disadvantage of a polarization wavefront division approach?

This polarization wavefront division approach has the substantial disadvantage of doubling the number of flight hardware optical components and, in theory reduces the signal-tonoise ratio (S/N) by at least a factor of 0.7, but can improve overall image quality.

What is the intensity of the ghost PSF?

The intensity of the ghost PSF is about one part in 10 4 of the two primary PSF images IXX and IY Y and with spatial structure of much larger extent across the focal plane and with more structural complexity than the classical Fraunhofer scalar diffraction PSF.

What are the slopes of the s and p phases?

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.

What are the terms that are similar to the wavefront aberrations?

These terms have functional forms similar to the wavefront aberrations piston (constant polarization), tilt (linearly varying from the origin and changing sign at the origin), and defocus (quadratically varying from the origin).

What is the average of the PSF for any two orthogonal components?

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.

(Open Access) Polarization Aberrations in Astronomical Telescopes: The Point Spread Function (2015) | James B. Breckinridge

Polarization Aberrations in Astronomical Telescopes: The Point Spread Function

Author(s): James B. Breckinridge, Wai Sze T. Lam, and Russell A. Chipman

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Vol. 127, No. 951 (May 2015),

pp. 445-468

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Polarization Aberrations in Astronomical Telescopes: The Point Spread Function

AMES

B. B

RECKINRIDGE

1,2

T. L

AND

USSELL

A. C

HIPMAN

Received 2014 October 20; accepted 2015 March 02; published 2015 March 31

ABSTRACT. Detailed knowledge of the image of the point spread function (PSF) is necessary to optimize astro-

nomical coronagraph masks and to understand potential sources of errors in astrometric measurements. The PSF for

astronomical telescopes and instruments depends not only on geometric aberrations and scalar wave diffraction but

also on those wavefront errors introduced by the physical optics and the polarization properties of reflecting and

transmitting surfaces within the optical system. These vector wave aberrations, called polarization aberrations, result

from two sources: (1) the mirror coatings necessary to make the highly reflecting mirror surfaces, and (2) the optical

prescription with its inevitable non-normal incidence of rays on reflecting surfaces. The purpose of this article is to

characterize the importance of polarization aberrations, to describe the analytical tools to calculate the PSF image, and

to provide the background to understand how astronomical image data may be affected. To show the order of magni-

tude of the effects of polarization aberrations on astronomical images, a generic astronomical telescope configuration is

analyzed here by modeling a fast Cassegrain telescope followed by a single 90° deviation fold mirror. All mirrors in

this example use bare aluminum reflective coatings and the illumination wavelength is 800 nm. Our findings for this

example telescope are: (1) The image plane irradiance distribution is the linear superposition of four PSF images: one

for each of the two orthogonal polarizations and one for each of two cross-coupled polarization terms. (2) The PSF

image is brighter by 9% for one polarization component compared to its orthogonal state. (3) The PSF images for two

orthogonal linearly polarization components are shifted with respect to each other, causing the PSF image for un-

polarized point sources to become slightly elongated (elliptical) with a centroid separation of about 0.6 mas. This

is important for both astrometry and coronagraph applications. (4) Part of the aberration is a polarization-dependent

astigmatism, with a magnitude of 22 milliwaves, which enlarges the PSF image. (5) The orthogonally polarized com-

ponents of unpolarized sources contain different wavefront aberrations, which differ by approximately 32 milliwaves.

This implies that a wavefront correction system cannot optimally correct the aberrations for all polarizations simulta-

neously. (6) The polarization aberrations couple small parts of each polarization component of the light (∼10

4

)into

the orthogonal polarization where these components cause highly distorted secondary, or “ghost” PSF images. (7) The

radius of the spatial extent of the 90% encircled energy of these two ghost PSF image is twice as large as the radius of

the Airy diffraction pattern. Coronagraphs for terrestrial exoplanet science are expected to image objects 10

10

,or6

orders of magnitude less than the intensity of the instrument-induced “ghost” PSF image, which will interfere with

exoplanet measurements. A polarization aberration expansion which approximates the Jones pupil of the example

telescope in six polarization terms is presented in the appendix. Individual terms can be associated with particular

polarization defects. The dependence of these terms on angles of incidence, numerical aperture, and the Taylor series

representation of the Fresnel equations lead to algebraic relations between these parameters and the scaling of the

polarization aberrations. These “design rules” applicable to the example telescope are collected in § 5. Currently,

exoplanet coronagraph masks are designed and optimized for scalar diffraction in optical systems. Radiation from

the “ghost” PSF image leaks around currently designed image plane masks. Here, we show a vector-wave or polari-

zation optimization is recommended. These effects follow from a natural description of the optical system in terms of

the Jones matrices associated with each ray path of interest. The importance of these effects varies by orders of mag-

nitude between different optical systems, depending on the optical design and coatings selected. Some of these effects

can be calibrated while others are more problematic. Polarization aberration mitigation methods and technologies to

minimize these effects are discussed. These effects have important implications for high-contrast imaging, coronag-

raphy, and astrometry with their stringent PSF image symmetry and scattered light requirements.

Online material: color figures

Graduate Aerospace Laboratories, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125; jbreckin@caltech.edu.

Also adjunct professor at the College of Optical Sciences, University of Arizona, Tucson, AZ 85719.

College of Optical Sciences, University of Arizona, 1630 University Blvd., Tucson, AZ 85721; waisze@optics.arizona.edu, rchipman@optics.arizona.edu.

445

UBLICATIONS OF THE

STRONOMICAL

OCIETY OF THE

ACIFIC

, 127:445–468, 2015 May

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1. INTRODUCTION

In this section, we describe briefly the value of polarization

measurements to stellar and exoplanet astronomical sciences,

summarize polarization aberrations, discuss the physical optics

of image formation in astronomical telescopes, and describe

how modern telescopes introduce polarization aberrations.

1.1. Photopolarimetry

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 selec-

tively absorbs the thermal emission from background stars. Un-

polarized radiation that scatters from planetary atmospheres and

circumstellar disks can become partially polarized. When one po-

larization 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 astro-

physics. 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 Pio-

neers 10 and 11 and the Voyagers showed that Jupiter-like exo-

planets will exhibit a degree of polarization (DoP)ashighas50%

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 de-

veloped. 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 or-

bital parameters (inclination, position angle of the ascending

node, and eccentricity) could be retrieved from precision polari-

metric 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 forma-

tion. Perrin (2009a, 2009b) shows that imaging polarimetry pro-

vides important constraints for the analysis of circumstellar disks.

Polarimetric measure ments of astronomical sources provide

critical astrophysical and exoplanet information. All polariza-

tion measuremen ts 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.

1.2. Aberration

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. De-

viations from constant amplitude arise from differences in reflec-

tion 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 in-

cidence changes and thus the polarization varies, so that a uni-

formly 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 tran-

siting 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 astro-

nomical 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 exopla-

nets and similar demanding tasks, these amplitude and polari-

zation effects can no longer be ignored. For example, Stenflo

(1978) has discussed limitations on the accuracy of solar mag-

netic 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. Po-

larization aberration, also called instrumental polarization, refers

to all polarization changes of the optical system and the varia-

tions with pupil coordinate, object location, and wavelength. The

446

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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 aberra-

tions 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 po-

larization 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 exten-

sions 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 sur-

face shape and angle of incidence variation (Chipman 1987;

McGuire & Chipman 1987, 1990, 1991, 1994a, 1994b; Hansen

1988; Chipman & Chipman 1989; Shribak et al. 2002; Beckley

et 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 op-

tical system (Pezzanitti et al. 1995; McEldowney et al. 2008).

1.3. Image Formation

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 to-

day are mostly ray traced and analyzed using a scalar represen-

tation for the electromagnetic field, usually calculated by

“conventional ray tracing,” without regard for polarization. Very

accurate simulation of high-resolution and high-contrast imag-

ing systems, at the level of polarization artifacts comprising

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, ef-

fects 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 in-

coherent 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 mean-

ing 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 trans-

forms 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 func-

tions. 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 ob-

tained 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).

1.4. Instrument-Induced Polarization

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 re-

quires 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),

.1.—A flowchart of image analysis in optical systems with polarization

aberration. ARM is the Fourier transform of the Jones pupil. The ARM converts

to the PSM as a Jones matrix converts to a Mueller matrix.

POLARIZATION ABERRATIONS IN TELESCOPES: THE PSF 447

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and the European-Extremely Large Telescope (E-ELT), use op-

tical system architectures where radiation strikes mirror surfaces

at high angles of incidence introducing polarization-induced var-

iations to wavefronts. These telescopes function as partial polar-

izers, 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 de-

graded 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 trans-

missivity 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 struc-

ture. Breckinridge & Oppenheimer (2004) and Breckinridge

(2013) established that the shape of the PSF image for the as-

tronomical 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 per-

fect, scalar diffraction accounted for, and the entire set of OPLs

equal but the polarization aberration can still reduce image

quality.

2. POLARIZATION ANALYSIS OF AN EXAMPLE

CASSEGRAIN TELESCOPE

To explain the effects of polarization aberration on the PSF

and explore the implications for astronomical imaging, a

generic telescope consisting of a primary, secondary, and fold

mirror is analyzed. It is difficult to select a single fully repre-

sentative astronomical high-resolution optical system as a polar-

ization aberration example. Further, if an example system with

many elements is chosen, it is more difficult to relate the indi-

vidual surfaces to the features in the polarization aberration and

polarized PSF, so a relatively simple system is analyzed. Quan-

titative values are calculated for this telescope’s polarization.

What is of particular interest is not these specific values but

the functional form of the image defects and their order of

magnitude. This should help the reader to assess whether these

defects are of concern for various applications.

The example Cassegrain telescope and fold mirror is shown

in Figu re 2. It is illuminated on-axis. This system has no on-

axis geometric wavefront aberrations; the OPL is equal for all

on-axis rays. Thus the on-axis image calculated by conven-

tional ray tracing is ideal, so any deviations from ideal imaging

are due to the polarization of the mirrors and is not mixed

with the effects of geometric wavefront aberration. The mirrors

are coated with bare aluminum and analyzed at 800 nm with

a complex refractive index N ¼ 2:80 þ 8:45i. The Fresnel

amplitude and phase coefficients for alum inum are plotted in

Figure 3. The remainder of this manuscript will focus on the

effect of these Fresnel coefficients on i mage formation in the

example telescope, and by extension to other image forming

systems.

2.1. The Fresnel Coefficients and Fresnel Aberrations

When a plane wave is incident on a metal reflector, the ra-

diation’s electric and magnetic fields drive the charges in the

metal, which undergo a small oscillation at optical frequencies.

These accelera ting 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 equa-

tions which calculate the polarization content of the reflected

beam. The complex refractive index N ¼ n þ ik, has imaginary

part k and real part n. The values of n and k are given in optical

.2.—An example Cassegrain telescope system with a primary mirror at

F/1.2, a Cassegrain focus of F/8, and a 90° fold mirror in the F/8 converging

beam. The 90° fold mirror is folded about the x-axis. The primary mirror

has a clear aperture of 2.4 meters. The operating wavelength is 800 nm. All

three mirrors are coated with aluminum. x and y define the coordinates for inci-

dent polarization states.

448 BRECKINRIDGE, LAM, & CHIPMAN

2015 PASP, 127:445–468

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Polarization Aberrations in Astronomical Telescopes: The Point Spread Function

Figures

Citations

Numerical modeling of the proposed WFIRST-AFTA coronagraphs and their predicted performances

About the effects of polarising optics on lidar signals and the Δ90 calibration

SPHERE/ZIMPOL high resolution polarimetric imager. I. System overview, PSF parameters, coronagraphy, and polarimetry

Polarization Effects in Reflecting Coronagraphs for White Light Applications in Astronomy

The speckle polarimeter of the 2.5-m telescope: Design and calibration

References

Handbook of Optical Constants of Solids

Principles of optics : electromagnetic theory of propagation, interference and diffraction of light

Proc of SPIE

Sharper Focus for a Radially Polarized Light Beam

A New Calculus for the Treatment of Optical SystemsI. Description and Discussion of the Calculus

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Three-dimensional polarization ray-tracing calculus I: definition and diattenuation

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First light of the Gemini Planet Imager

Frequently Asked Questions (15)

Q1. What are the contributions in "Polarization aberrations in astronomical telescopes: the point spread function" ?

Q2. What are the different types of polarization aberrations of telescopes?

Q3. What is the resultant retardance for the telescope shown on the right?

Q4. What is the phase shift between s and p-reflected beams?

Q5. What are the four metrics used to measure image quality in astronomical telescopes?

Q6. What is the polarization aberration of a small fraction of light?

Q7. What is the way to reduce the image quality of a telescope?

Q8. What are the optical coatings for mirrors?

Q9. What components need to be considered as potential polarization sources?

Q10. What is the significance of the polarization signature of primordial grain growth?

Q11. What is the disadvantage of a polarization wavefront division approach?

Q12. What is the intensity of the ghost PSF?

Q13. What are the slopes of the s and p phases?

Q14. What are the terms that are similar to the wavefront aberrations?

Q15. What is the average of the PSF for any two orthogonal components?