DIFFRACTION ANALYSIS OF TWO-DIMENSIONAL PUPIL MAPPINGFOR HIGH-CONTRAST IMAGING

Robert J. Vanderbei

Operations Research and Financial Engineering, Princeton University, Engineering Quadrangle, Princeton, NJ 08544; rvdb@princeton.edu

Received 2005 June 23; accepted 2005 September 5

ABSTRACT

Pupil mapping is a technique whereby a uniformly illuminated input pupil, such as from starlight, can be mappedinto a nonuniformly illuminated exit pupil, such that the image formed from this pupil will have suppressedsidelobes, many orders of magnitude weaker than classical Airy ring intensities. Pupil mapping is therefore a can-didate technique for coronagraphic imaging of extrasolar planets around nearby stars. Unlike most other high-contrast imaging techniques, pupil mapping is lossless and preserves the full angular resolution of the collectingtelescope. So it could possibly give the highest signal-to-noise ratio of any proposed single-telescope system fordetecting extrasolar planets. Prior analyses based on pupil-to-pupil ray-tracing indicate that a planet fainter than10�10 times its parent star, and as close as about 2k/D, should be detectable. In this paper we describe the results ofcareful diffraction analysis of pupil-mapping systems. These results reveal a serious unresolved issue. Namely, high-contrast pupil mappings distribute light from very near the edge of the first pupil to a broad area of the second pupil,and this dramatically amplifies diffraction-based edge effects, resulting in a limiting attainable contrast of about 10�5.We hope that by identifying this problem, others will provide a solution.

Subject headinggs: planetary systems — techniques: high angular resolution

1. INTRODUCTION

For roughly 10 years now astronomers have been finding ex-trasolar planets. To date, well over 100 short-period large-massplanets have been found using radial-velocity measurements(see, e.g., Cumming et al. 2003). As a result, there is widespreadinterest in finding and directly imaging Earth-like planets in thehabitable zones of nearby stars. In fact, NASA has plans to launchtwo space telescopes to aid in this search. These two space tele-scopes are called the Terrestrial Planet Finder–Coronagraph(TPF-C ) and theTerrestrial Planet Finder–Interferometer (TPF-I ).Both of these telescopes are still in the concept study phase.

Direct imaging of Earth-like extrasolar planets is an extremelychallenging problem in high-contrast imaging. If one were toview our solar system from a good distance (say, from anothernearby star system), our Sun would appear 1010 times brighterthan Earth. Hence, we need an imaging system capable of de-tecting planets that are 10 orders of magnitude fainter than thestar they orbit. Furthermore, given the distances involved, the an-gular separation formost targets is very small, requiring the largestlaunchable telescope possible.

For TPF-C, the current baseline design involves a traditionalLyot coronagraph with a modern eighth-order occulting mask at-tached to the back end of a Ritchey-Chretien telescope having an8 ; 3:5 m elliptical primary mirror (see, e.g., Kuchner et al. 2004).Alternative innovative back-end designs still being considered in-clude ‘‘shaped pupils’’ (see, e.g., Kasdin et al. 2003;Vanderbei et al.2004), a ‘‘visible nuller’’ (see, e.g., Shao et al. 2004), and ‘‘pupilmapping’’ (see, e.g., Guyon [2003], where this technique is calledphase-induced amplitude apodization, or PIAA). By pupil map-ping we mean a system of two lenses, or mirrors, that take a flatinput field at the entrance pupil and produce an output field that isamplitude-modified but still flat in phase (at least for on-axissources).

There seems to be growing concern that the baseline designmight only find a few terrestrial extrasolar planets. Hence, thealternative designs mentioned above are currently being givencareful consideration. In particular, pupil mapping is generating

the most excitement, since it uses 100% of the available light andexploits the full resolution of the optical system. While prelim-inary laboratory results presented in Galicher et al. (2004) werenot very impressive, it is widely felt that better optics will pro-duce dramatically better results. The purpose of this paper is toreport the results of a careful diffraction analysis of pupil map-ping with the aim of characterizing how well an ideal pupil map-ping system might perform.The figures of the two lenses (or mirrors) that form the pupil

mapping system are, of course, determined by ray optics. In Traub& Vanderbei (2003) and Vanderbei & Traub (2005), we deriveformulae for the shapes of the optical elements. In these papers, wealso study the off-axis performance of these systems. We show thatthe rays fail to converge for off-axis sources. That is, the ability ofthe system to form images degrades quickly as one moves off-axis.To address this problem, Guyon et al. (2005) suggested using twoidentical pupil-mapping systems as follows. The incoming beamis sent through the first pupil-mapping system. Then it is passedthrough a focusing element. At the focal plane, there is an occulterto remove the starlight. The remaining light (hopefully containingthat of a planet) is allowed to pass. After the image plane, there isanother optical element to recollimate the beam, which is thenpassed backward through the second pupil-mapping system. Thissecond pass largely undoes the distortions introduced by the firstpupil-mapping system. Hence, the beam is mostly restored, butthe starlight has been removed. A final pass through a focusingelement forms an image in which one can hope to find a planet.The analyses in these previous papers are based entirely on

ray-tracing except that the amplitude profile is chosen to be onein which the associated point-spread function (PSF; computedas the square of the magnitude of the Fourier transform of theamplitude profile) has extremelywell-suppressed sidelobes. Hence,these prior analyses are an odd mix of geometric optics and dif-fraction analysis. The concept begs for a complete diffractionanalysis. That is, given lens (or mirror) figures and an input beamthat consists of on-axis starlight and much fainter slightly off-axis planet light, one should propagate the electric field throughthe entire system to see how faint a planet can be detected. This

528

The Astrophysical Journal, 636:528–543, 2006 January 1

# 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A.

was the original plan. But, as we demonstrate below, the diffrac-tion effects from the pupil-mapping systems themselves are sodetrimental that contrast attained at the first image plane is lim-ited to 10�5. Hence, there is no point in propagating further. Un-less this problem can be resolved, pupil mapping may prove notviable for TPF-C. It is our sincere hope that someone will find aclever way to resolve this diffraction-induced problem.

2. PUPIL MAPPING VIA RAY OPTICS

We begin by summarizing the ray-optics description of pupilmapping. An on-axis ray entering the first pupil at radius r fromthe center is to be mapped to radius r ¼ R(r) at the exit pupil. Op-tical elements at the two pupils ensure that the exit ray is parallel tothe entering ray. The function R(r) is assumed to be positive andincreasing or, sometimes, negative and decreasing. In either case,the function has an inverse that allows us to recapture r as a func-tion of r: r ¼ R(r ). The purpose of pupil mapping is to createnontrivial amplitude profiles. An amplitude profile function A(r )specifies the ratio between the output amplitude at r to the inputamplitude at r (althoughwe typically assume the input amplitude isa constant). We showed in Vanderbei & Traub (2005) that for anyamplitude profile A(r ), there is a pupil-mapping function R(r ) thatachieves this profile. Specifically, the pupil mapping is given by

R(r ) ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ r

0

2A2(s)s ds

s: ð1Þ

Furthermore, if we consider the case of a pair of lenses that areplano on their outward-facing surfaces (as shown in Fig. 1),then the inward-facing surface profiles, h(r) and h(r ), that arerequired to obtain the desired pupil mapping are given by the so-lutions to the following ordinary differential equations:

@h

@r(r) ¼ r � R(r)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Q20 þ (n2 � 1) r � R(r)

� �2q ; h(0) ¼ z; ð2Þ

@h

@r(r ) ¼ R(r )� rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Q20 þ (n2 � 1) R(r )� r½ �2

q ; h(0) ¼ 0: ð3Þ

Here n is the refractive index and Q0 is a constant determinedby the distance z separating the centers (r ¼ 0, r ¼ 0) of the twolenses: Q0 ¼ �(n� 1)z.

Let S(r; r ) denote the distance between a point on the firstlens surface r units from the center and the corresponding pointon the second lens surface r units from its center. Up to an ad-ditive constant, the optical path length of a ray that exits at radiusr after entering at radius r ¼ R(r ) is given by

Q0(r ) ¼ S R(r ); r½ � þ n h(r )� h R(r )ð Þ� �

: ð4Þ

Fig. 1.—Pupil mapping via a pair of properly figured lenses.

DIFFRACTION ANALYSIS OF PUPIL MAPPING 529

In Vanderbei & Traub (2005), we showed that, for an on-axissource, Q0(r ) is constant and equal to Q0.

3. HIGH-CONTRAST AMPLITUDE PROFILES

If we assume that a collimated beam with amplitude profileA(r ), such as one obtains as the output of a pupil-mappingsystem, is passed into an ideal imaging system with focal lengthf, the electric field E(�) at the image plane is given by the Fouriertransform of A(r ):

E(�; �) ¼ E0

k f

Z Ze2�i x�þy�ð Þ=k f A

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

p� �dy dx: ð5Þ

Here E0 is the input amplitude, which, unless otherwise noted,we take to be unity. Since the optics are azimuthally symmetric,it is convenient to use polar coordinates. The amplitude profileA is a function of r ¼ x2 þ y2ð Þ1=2, and the image-plane electricfield depends only on the image-plane radius � ¼ �2 þ �2ð Þ1=2:

E(�) ¼ 1

k f

Z Ze2�i r�=k fð Þ cos (���)A(r )r d� dr ð6Þ

¼ 2�

k f

ZJ0 2�

r�

k f

� �A(r )r dr: ð7Þ

The PSF is the square of the electric field:

PSF(�) ¼ jE(�)j2: ð8Þ

For the purpose of terrestrial planet finding, it is important toconstruct an amplitude profile for which the PSF at small non-zero angles is 10 orders of magnitude reduced from its value atzero. Figure 2 shows one such profile. In Vanderbei et al. (2003),we explain how these functions are computed as solutions tocertain optimization problems.

We end this section by noting that designing customizedamplitude profiles arises in many areas usually in the context ofapodization—i.e., profiles that only attenuate the beam. Slepian(1965) was perhaps the first to study this problem carefully.For some recent applications, the reader is referred to the follow-ing papers in the area of beam shaping: Carney & Gbur (1999),Goncharov et al. (2002), and Hoffnagle & Jefferson (2003).

4. HUYGENS WAVELETS

We have designed the pupil-mapping system using simple rayoptics, but we have relied on diffraction theory to ensure that theremapped pupil provides high contrast. This begs the questionwill the desired high contrast remain after a diffraction analysisof the entire system including the two-lens pupil mapping sys-tem, or will diffraction effects in the pupil-mapping system itselfcreate ‘‘errors’’ that are great enough to destroy the high contrastthat we seek? To answer this question, we need to do a diffractionanalysis of the pupil-mapping system itself.If we assume that a flat, on-axis, electric field arrives at the

entrance pupil, then the electric field at a particular point of theexit pupil can bewell approximated by superimposing the phase-shifted waves from each point across the entrance pupil (this isthewell-knownHuygens-Fresnel principle; see, e.g., x 8.2 inBorn& Wolf 1999). That is,

Eout(x; y) ¼Z Z

1

kQ(x; y; x; y)e2�iQ(x; y; x; y)=k dy dx; ð9Þ

where

Q(x; y; x; y) ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(x� x)2 þ ( y� y)2 þ h(r)� h(r )

� �2qþ n Z � h(r)þ h(r )

� �ð10Þ

is the optical path length, Z is the distance between the planolens surfaces (i.e., a constant slightly larger than z), and where,of course, we have used r and r as shorthands for the radii in theentrance and exit pupils, respectively. As before, it is convenientto work in polar coordinates:

Eout(r ) ¼Z Z

1

kQ(r; r; �)e2�iQ(r; r; � )=kr d� dr; ð11Þ

where

Q(r; r; �) ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir 2 � 2rr cos �þ r 2 þ h(r)� h(r )

� �2qþ n Z � h(r)þ h(r )

� �: ð12Þ

For numerical tractability, it is essential to make approximationsso that the integral over � can be carried out analytically. To that

Fig. 2.—Left: Amplitude profile providing contrast of 10�10 at tight inner working angles. Right: Corresponding on-axis PSF.

VANDERBEI530 Vol. 636

end, we need tomake an appropriate approximation to the squareroot term:

S ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir 2 � 2rr cos �þ r 2 þ h(r)� h(r )

� �2q: ð13Þ

5. FRESNEL PROPAGATION

In this section we consider approximations that lead to the so-called Fresnel propagation formula. If we assume that the lensseparation is fairly large, then the (h� h)2 term dominates therest, and so we can use the first two terms of a Taylor series ap-proximation [i.e., for u small relative to a, u2 þ a2ð Þ1=2� aþu2 /2a] to get the following ‘‘large separation’’ approximation:

S � h(r)� h(r )� �

þ r 2 � 2rr cos �þ r 2

2 h(r)� h(r )� � : ð14Þ

If we assume further that the lenses are thin (i.e., that n is large),then h� h in the denominators can be approximated simply by z:

S � h(r)� h(r )þ r 2 � 2rr cos �þ r 2

2z: ð15Þ

This is called the ‘‘thin lens approximation.’’Combining the large lens separation approximation with the

thin lens approximation, we get

Eout(r ) ¼Z Z

1

kQ1(r; r; �)e2�iQ1(r; r; �)=kr d� dr ð16Þ

where

Q1(r; r; �) ¼ r 2 � 2rr cos �þ r 2

2z

þ Z þ (n� 1) Z � h(r)þ h(r )� �

: ð17Þ

Finally, we simplify the reciprocal ofQ1 by noting that the Z termdominates the other terms [i.e., for u smaller than a, 1/(uþ a) �1/a], and so we get that

1

Q1(r; r; �)� 1

Z: ð18Þ

This last approximation is called the ‘‘paraxial’’ approximation.Combining all three approximations, we now arrive at the

‘‘standard Fresnel approximation’’:

Eout(r ) ¼2�

kZe�iðr

2=zkÞþ2�i½(n�1)h(r )�=k

;

Ze�iðr

2=zkÞ�2�i (n�1)h(r)½ �=kJ0(2�rr=zk)r dr: ð19Þ

While the standard Fresnel approximation works very well inmost conventional situations, it turns out (as we show below) tobe too crude of an approximation for high-contrast pupil map-ping. It is inadequate because it does not honor the constancy ofthe optical path length Q(r; r; �) along the rays of ray-optics.That is, the fact thatQ(r; R(r ); 0) is constant has been lost in theapproximations. We should have used the ray-tracing opticalpath length as the ‘‘large quantity’’ in our large-lens-separationapproximation instead of the simpler difference h(r)� h(r ).But this seemingly simple adjustment quickly gets tedious, andsowe prefer to take a completely different (and simpler) approach,which is described in the next section.

6. AN ALTERNATIVE TO FRESNEL

Aswehave just explained andwill demonstrate below, the stan-dard Fresnel approximation does not produce good results forhigh-contrast pupil mapping computations. In this section, wepresent an alternative approximation that is slightly more com-putationally demanding but is much closer to a direct calculationof the true Huygens wavelet propagation.

As with Fresnel, we approximate the 1/Q(r; r; �) amplitude-reduction factor in equation (11) by the constant 1/Z (the paraxialapproximation). TheQ(r; r; �) appearing in the exponentialmust,on the other hand, be treated with care. Recall that Q(r; R(r ); 0)is a constant. Since constant phase shifts are immaterial, we cansubtract it from Q(r; r; �) in equation (11) to get

Eout(r ) �1

kZ

Z Ze2�i Q(r; r; � )�Q r; R(r ); 0ð Þ½ �=kr d� dr: ð20Þ

Next we write the difference in Q as follows:

Q(r; r; �)� Q(r; R(r ); 0)

¼ S(r; r; �)� S r; R(r ); 0ð Þ þ n h R(r )ð Þ � h(r)½ � ð21Þ

¼ S2(r; r; �)� S2 r; R(r ); 0ð ÞS(r; r; �)þ S r; R(r ); 0ð Þ þ n h R(r )ð Þ � h(r)½ � ð22Þ

and then we expand out the numerator and cancel large termsthat can be subtracted one from another to get

S2(r; r; �)� S2(r; R(r ); 0)

¼ r � R(r )½ � r þ R(r )½ � � 2r r cos �� R(r )½ �þ h(r)� h R(r )ð Þ½ � h(r)þ h R(r )ð Þ � 2h(r )

� �: ð23Þ

When r ¼ R(r ) and � ¼ 0, the right-hand side clearly vanishes asit should. Furthermore, for r close to R(r ) and � close to zero, theright-hand side gives an accurate formula for computing the devia-tion from zero. That is, the right-hand side is easy to program in sucha manner as to avoid subtracting one large number from another,which is always the biggest danger in numerical computation.

So far, everything is exact (except for the paraxial approxi-mation). The only further approximation we make is to replaceS(r; r; �) in the denominator of equation (22)with S(r; R(r ); 0)so that the denominator becomes just 2S(r; R(r ); 0). Putting thisall together and replacing the integral on � with the appropriateBessel function, we get a new approximation, which we refer toas the Huygens approximation:

Eout(r ) �2�

kZ

Zexp

�2�i

n½r � R(r )�½r þ R(r )� þ 2rR(r )

þ ½h(r)� h(R(r ))�½h(r)þ h(R(r ))� 2h(r )�o

; ½2S(r; R(r ); 0)��1 þ n½h(R(r ))� h(r)�=k�

; J0

�2�rr

kS(r; R(r ); 0)

r dr: ð24Þ

7. SANITY CHECKS

In this section we consider a number of examples.

7.1. Flat Glass Windows (A � 1)

We begin with the simplest example in which the amplitudeprofile function is identically equal to 1. Taking the positive root inequation (1), we get that R(r ) ¼ r. That is, the ray-optic design is

DIFFRACTION ANALYSIS OF PUPIL MAPPING 531No. 1, 2006

for the light to go straight through the system. The inverse mapR(r) is also trivial: R(r) ¼ r. Hence, the right-hand sides in the dif-ferential equations (2) and (3) vanish, and the lens figures becomeflat: h(r) � z and h(r ) � 0. In this case, Q1 r; R(r ); 0ð Þ is a con-stant (independent of r), and so the Fresnel approximation is a goodone. The Fresnel results, shown in Figure 3, should match textbookexamples for simple open circular apertures—and they do.

7.2. A Galilean Telescope (A � a > 1)

In this case, we ask for a system in which the output pupil hasbeen uniformly amplitude-intensified by a factor a > 1. If wechoose the positive root in equation (1), then we get a Galilean-style refractor telescope consisting of a convex lens at the en-trance pupil and a concave lens as an eyepiece. Specifically,we get R(r ) ¼ ar and R(r) ¼ r /a. From these it follows that ifthe aperture of the first lens is D, then the aperture of the secondis D ¼ D/a. It is easy to compute the lens figures

h(r) ¼ zþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ2

0 þ (n2 � 1)(1� 1=a)2r 2q

� jQ0j(n2 � 1)(1� 1=a)

; ð25Þ

h(r ) ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ2

0 þ (n2 � 1)(a� 1)2r2q

� jQ0j(n2 � 1)(a� 1)

: ð26Þ

If the relative index of refraction n is greater than 1 (as in air-spaced glass lenses), then these functions represent portions of ahyperbola. If, on the other hand, n < 1 (as in a glass medium be-tween the two surfaces), then the functions are ellipses. Fresnelresults for a ¼ 3 and n ¼ 1:5 are shown in Figure 4. Note thelarge error in the phase map and the fact that the computed PSFdoes not follow the usual Airy pattern. This is strong evidencethat the Fresnel approximation is too crude, since real systemsof this sort exist and exhibit the expected Airy pattern.In Figure 5 we consider the same system but use the Huygens

approximation. Note that the phase map, while not perfect, isnow much flatter. Also, the computed PSF closely matches theexpected Airy pattern.

7.3. An Ideal Lens

If we let a tend to infinity, we see that D tends to zero, and thesystem reduces to a convex lens focusing a collimated input

Fig. 3.—Fresnel analysis of a pupil-mapping system consisting of two flat pieces of n ¼ 1:5 glass in circular D ¼ 25 mm apertures separated by z ¼ 15D. Thewavelength is k ¼ 632:8 nm. Top left: Target amplitude profile (gray) and the amplitude profile computed using standard Fresnel propagation (black). Top right:Zoomed-in section of the amplitude profiles. Bottom left: Computed optical path length Q0(r) (gray) and the phase map computed using Fresnel propagation (black).Bottom right: PSF associated with the ideal amplitude profile (gray) and the PSF computed by Fresnel propagation (black). These results should match textbookexamples for simple open circular apertures—and they do.

VANDERBEI532

Fig. 4.—Same as Fig. 3, but with the apodization profile replaced with a constant value of A � 3. In this example, both lenses are hyperbolic, as shown in the topright plot. The lens profiles h and h were computed using a 5000 point discretization, and the Fresnel propagation (eq. [19]) was also carried out with a 5000 pointdiscretization. Note the large error in the phase map and the fact that the computed PSF does not follow the usual Airy pattern. This is strong evidence that the stan-dard Fresnel approximation is too crude, since real systems of this sort exist and exhibit the expected Airy pattern.

Fig. 5.—Same as Fig. 4, but computed using the Huygens approximation. Note that the phase map is now much flatter. Also, the computed PSF closely matchesthe expected Airy pattern.

beam to a point. In this case, the second lens vanishes; only thefirst lens is of interest. Its equation is

h(r) ¼ zþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ2

0 þ (n2 � 1)r 2p

� jQ0j(n2 � 1)

: ð27Þ

The plane where the second lens was is now the image plane. Tocompute the electric field here, we put h(r ) � 0 and use eitherthe Fresnel or the Huygens approximation. Since we would puta detector at this plane, we can ignore any final phase correc-tions, and both approximations reduce to the same formula:

Eout(r ) ¼2�

kZ

Ze�iðr

2=zkÞ�2�i½(n�1)h(r)=k�J0

�2�rr

zk

�r dr: ð28Þ

Substituting equation (27) into equation (28) and dropping anyunit complex numbers that factor out of the integral, we get

Eout(r ) ¼2�

kZ

Zexp

"�i

r 2

zk� 2�i

k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�n� 1

nþ 1

�2

z2 þ n� 1

nþ 1r 2

s #

; J0

�2�rr

zk

�r dr: ð29Þ

Of course, this formula is for a uniform collimated input beam.If the input beam happens to be preapodized by some upstreamoptical element, then the expression becomes

Eout(r ) ¼2�

kZ

Zexp �i

r 2

zk� 2�i

k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin� 1

nþ 1

� �2

z2 þ n� 1

nþ 1r 2

s24

35

; J0

�2�rr

zk

�A(r)r dr ð30Þ

where A(r) denotes the preapodization function. This formuladoes not agree with the Fourier transform expression given ear-lier by equation (5). However, if the square root is approximatedby the first two terms of its Taylor expansion,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n� 1

nþ 1

� �2

z2 þ n� 1

nþ 1r 2

s¼ n� 1

nþ 1zþ r 2

2z; ð31Þ

then equation (30) reduces to a Fourier transform as in equa-tion (5) (again dropping unit complex factors).

This raises an interesting question: if a high-contrast ampli-tude profile is designed based on the assumption that the focus-ing element behaves like a Fourier transform (i.e., as in eq. [5]),how well will the profile work if the true expression for the elec-tric field is closer to the one given by equation (30)? The answeris shown in Figure 6. The PSF degradation is very small.

8. HIGH-CONTRAST PUPIL MAPPING

The purpose of the examples discussed in x 7.3 was to con-vince the reader that the Huygens approximation provides areasonable estimate of the electric field at the exit pupil of apupil-mapping system. Assuming we were convincing, we nowproceed to apply the Huygens approximation to compute theelectric field and focused PSF of the pupil-mapping systemcorresponding to the amplitude profile function shown in Fig-ure 2. As before, we assume a wavelength of 0.6328 �m andlenses with aperture D ¼ 25 mm. In Figure 7, we show theresults for z ¼ 15D and a refraction index of 1.5. For thesesimulations to be meaningful, it is critical that the integrals berepresented by a sum over a sufficiently refined partition. Thebigger the disparity between wavelength k and aperture D, themore refined the partition needs to be. For the parameters wehave chosen, a partition into 5000 parts proved to be adequate.The plot in the upper left section of the figure shows the targetamplitude profile in gray and the amplitude profile computedusing the Huygens approximation in black (i.e., eq. [24]). Theplot in the upper right shows the lens profiles. The first lens isshown in black and the second in gray. The plot in the lower leftshows the computed optical path length Q0(r ) in gray. If the nu-merical computation of the lens figures had been done with in-sufficient precision, this curve would not be flat. As we see, it isflat. The lower left also shows in black the phase map as com-puted by the Huygens approximation. Note that here there arehigh-frequency oscillations everywhere and a low-frequency os-cillation that has an amplitude that increases as one moves out tothe rim of the lens. The lower-right plot shows the PSF associ-ated with the ideal amplitude profile in gray and the PSF com-puted by Huygens propagation in black.

The PSF in Figure 7 is disappointing. It is important to de-termine whether this is real or is a result of the approximationsbehind the Huygens propagation formula. As a check, we did abrute force computation of the Huygens integral equation (11).Because this integral is more difficult, we were forced to use only1000 r-values and 1000 �-values. Hence, we had to increasethe wavelength by a factor of 5. With these changes, the result isshown in Figure 8. It too shows the same amplitude and phaseoscillations. This sanity check convinces us that these effects arephysical. We need to consider changes to the physical setup thatmight mollify these oscillations. Such changes are considerednext.

There is no particular reason to make the two lenses haveequal aperture. By scaling the amplitude profile, we can easilygenerate examples with unequal aperture. One such experiment

Fig. 6.—Gray plot shows an ideal high-contrast PSF designed assuming thatthe focusing element is a parabolic mirror so that the Huygens image-planeelectric field is a Fourier transform as in eq. (5). The black line shows the PSF ifone assumes that the focusing element is actually an ideal lens (having an ellip-tical profile), as described in x 7.3. The two curves are visually identical except inthe neighborhood of the first and second minima of the Fourier (gray) curve.

DIFFRACTION ANALYSIS OF PUPIL MAPPING 535

Fig. 7.—Analysis of a pupil-mapping system using the Huygens approximation with z ¼ 15D and n ¼ 1:5. Top left: Target amplitude profile (gray) and theamplitude profile computed using the Huygens approximation (black). Top right: Lens profiles, black for the first lens and gray for the second. The lens profiles h andh were computed using a 5000 point discretization. Bottom left: Computed optical path length Q0(r) (gray) and the phase map computed using Huygens propagation(black). The Huygens propagation was carried out with a 5000 point discretization. Bottom right: PSF computed as the square of the Fourier transform of the idealamplitude profile (gray) and the PSF computed using the Huygens approximation (black).

536

Fig. 8.—Same as Fig. 7, but computed using a brute force computation of the Huygens integral eq. (11). Here we used 1000 r-values and 1000 �-values.Consequently, we needed to increase the wavelength by a factor of 5 to guarantee adequate sampling of the phase and amplitude ripples.

537

Fig. 9.—Same as Fig. 7, but with the amplitude profile normalized to a maximum value of 1. This normalization results in the two lenses having differentapertures; the second is about 4 times that of the first.

538

Fig. 10.—Same as Fig. 7, but with 2.5 m lenses (a 100-fold enlargement). The larger optical elements required a finer discretization of the integrals: 500,000points were used. The contrast improves by roughly a factor of 100.

539

Fig. 11.—Various sanity checks suggest that diffraction effects fundamentally limit the close-in contrast attainable by pupil mapping to about 10�5. So, anamplitude profile designed for 10�10 might not be the right one to use. Here we show results for an amplitude profile that only attempts to achieve 10�5. Note that thePSF via Huygens propagation (black) agrees fairly well with the ideal (Fourier-transform) PSF (gray) in the bottom right plot.

540

Fig. 12.—Given our success in Fig. 11 of achieving 10�5 contrast with an amplitude profile specifically designed for this level of contrast, we can now askwhether it is possible to put a conventional apodizer (or shaped pupil equivalent) in the exit pupil to further ‘‘convert’’ this amplitude profile into one that achieves10�10. The result is shown here. As seen in the PSF plots in the bottom right, this combination gets the contrast to 10�7:5.

541

we tried was to scale the amplitude function so that its value at thecenter is 1 (thereby creating a traditional apodization). This scal-ing results in the second lens having almost 4 times the apertureof the first lens. The results for this case are shown in Figure 9.

Another possibility is to consider larger optical elements—25 mm is rather small. In Figure 10 we show results for D ¼ 2:5m (with z ¼ 15D). The intensity of the first sidelobe relative tothe main lobe has improved by a factor of about 100 (3 ; 10�5

vs. 3 ; 10�7). Clearly, the optics would need to be orders ofmagnitude larger in order to drive the contrast to the 10�10 level.

If an amplitude profile designed for 10�10 contrast only pro-duces 10�5, one wonders how well a profile designed for 10�5

will do. The answer is shown in Figure 11. In this case, the deg-radation due to diffraction effects is very small.

9. HYBRID SYSTEMS

If a pupil-mapping system designed for 10�5 works well, per-haps it could be followed by a conventional apodizer that attemptsto bring the system from 10�5 down to 10�10. We tried this. Theresult is shown in Figure 12. As can be seen in the lower right plot,

the contrast achieved is limited to about 10�7:5. Apparently, thediffraction ripples going into the apodizer are enough to preventthe system from achieving the desired contrast.As an alternative to a downstream apodizer, one could con-

sider a preapodizer placed in front of the first lens. Since it isgenerally hard edges that create bad diffraction effects, we canimagine using the preapodizer to provide the near–outer edgeamplitude profile and allow the pupil-mapping system to providethe main body of the profile. In this way, perhaps the diffractioneffects can be minimized while at the same time maintaining asystem with high throughput. The results, shown in Figure 13,are comparable to those obtained with a downstream apodizer.

This research was partially performed for the Jet PropulsionLaboratory, California Institute of Technology, sponsored by theNationalAeronautics and SpaceAdministration as part of theTPFarchitecture studies and also under JPL subcontract 1260535. Theauthor also received support from the NSF (CCR 00-98040) andthe ONR (N00014-05-1-0206).

Fig. 13.—As an alternative to the back-end apodizer considered in Fig. 12, we consider here the possibility of ‘‘softening’’ the edge of the first lens by using apreapodizer. The dashed curve in the top left plot shows the preapodization function we used. As one can see from the bottom right plot, this preapodization techniqueallows one to get the first sidelobe almost down to the 10�7 level. This result is not quite as good as what we had with the postapodizer of Fig. 12, but it is likely to bemore manufacturable.

VANDERBEI542 Vol. 636

REFERENCES

Born, M., & Wolf, E. 1999, Principles of Optics (7th ed.; New York:Cambridge Univ. Press)

Carney, P. S., & Gbur, G. 1999, J. Opt. Soc. Am., 16, 1638Cumming, A., Marcy, G. W., Butler, R. P., & Vogt, S. S. 2003, in ASP Conf. Ser.294, Scientific Frontiers in Research on Extrasolar Planets, ed. D. Deming &S. Seager (San Francisco: ASP), 27

Galicher, R., Guyon, O., Ridgway, S., Suto, H., & Otsubo, M. 2004, in Proc.2nd TPF Darwin Conf., Dust Disks and the Formation, Evolution, and De-tection of Habital Planets (Pasadena: JPL), http://planetquest1.jpl.nasa.gov/TPFDarwinConf /index.cfm

Goncharov, A., Owner-Petersen, M., & Puryayev, D. 2002, Opt. Eng., 41, 3111Guyon, O. 2003, A&A, 404, 379

Guyon, O., Pluzhnik, E.A, Galicher, R., Martinache, R., Ridgway, S. T., &Woodruff, R. A. 2005, ApJ, 622, 744

Hoffnagle, J. A., & Jefferson, C. M. 2003, Opt. Eng., 42, 3090Kasdin, N. J., Vanderbei, R. J., Spergel, D. N., & Littman, M. G. 2003, ApJ,582, 1147

Kuchner, M. J., Crepp, J., & Ge, J. 2005, ApJ, 628, 466Shao, M., Levine, B. M., Wallace, J. K., & Liu, D. T. 2004, Proc. SPIE, 61, 5487Slepian, D. 1965, J. Opt. Soc. Am., 55, 1110Traub, W. A., & Vanderbei, R. J. 2003, ApJ, 599, 695Vanderbei, R. J., Kasdin, N. J., & Spergel, D. N. 2004, ApJ, 615, 555Vanderbei, R. J., Spergel, D. N., & Kasdin, N. J. 2003, ApJ, 599, 686Vanderbei, R. J., & Traub, W. A. 2005, ApJ, 626, 1079

DIFFRACTION ANALYSIS OF PUPIL MAPPING 543No. 1, 2006