Optimal Asymmetric Apodizations and Shaped Pupils for Planet Finding Coronagraphy
N. Jeremy Kasdin
Robert J. Vanderbei
Michael G. Littman
David N. Spergel
The realization that direct imaging of extrasolar planets could be technologically feasible within the next decade or so has inspired a great deal of recent research into high-contrast imaging. We ourselves have contributed several design ideas, all of which can be described as {\em shaped pupil coronagraphs}. In this paper, we offer a complete and unified survey of asymmetric shaped pupils designs, some of which have been published in our previous papers. We also introduce a promising new design, which we call {\em barcode masks}. These masks achieve the required contrast with a fairly large discovery zone and throughput but most importantly they are perhaps the easiest to manufacture and might therefore stand up best to a refined analysis based on vector propogation techniques.
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%\input latestmacros.tex
\newcommand{\refeqn}[1]{Eq. (\ref{#1})} \newcommand{\reffig}[1]{Fig. \ref{#1}}
\newcommand{\ld}{\lambda/D}
\newcommand{\xinaught}{\xi_{\mbox{\scriptsize iwa}}}
\newcommand{\xione}{\xi_{\mbox{\scriptsize owa}}}
\newcommand{\hoverra}{\alpha}    % h/r
\newcommand{\hoverrb}{\alpha}  % \frac{h}{r}
\newcommand{\hovertwor}{\frac{\alpha}{2}}  % \frac{h}{2r}

\bibdata{FinalReport.bib}

\begin{abstract}
The realization that direct imaging of extrasolar planets could be technologically feasible within the next decade or so has inspired a great deal of recent research into high-contrast imaging.  We ourselves have contributed several design ideas, all of which can be described as {\em shaped pupil coronagraphs}.  In this paper, we offer a complete and unified survey of asymmetric shaped pupils designs, some of which have been published in our previous papers.  We also introduce a promising new design, which we call {\em barcode masks}.  These masks achieve the required contrast with a fairly large discovery zone and throughput but most importantly they are perhaps the easiest to manufacture and might therefore stand up best to a refined analysis based
on vector propogation techniques.
\end{abstract}


\begin{document}
\title{Optimal Asymmetric Apodizations and Shaped Pupils for Planet Finding Coronagraphy}
\author{ N. Jeremy Kasdin, Robert J. Vanderbei, Michael G. Littman, David N. Spergel}
\affil{Princeton University \\ Princeton, NJ   08544}

%\renewcommand{\thepage}{\roman{page}}

\maketitle

\section{Introduction}

The discovery of more than 100 extrasolar Jupiter-sized planets in just the last decade has generated enormous interest, both among astronomers and the public, in the problem of discovering and characterizing Earthlike planets.  NASA is already planning its next large space-based observatory, the \emph{Terrestrial Planet Finder (TPF)}, with a planned launch date toward the end of the next decade.  TPF's primary objective will be to discover Earthlike planets and characterize them for indications of life.  

While the technical challenges for TPF are great,  foremost among them is the problem of high-contrast imaging.  In order to discover as many planets as possible, it is necessary to design an imaging system that achieves very high contrast between the parent star and the nearby planet.  An earlier study by \cite{ref:Brown} 
indicates that a $D = 4$m class visible-light 
(i.e. $400\text{nm} \le \lambda \le 650\text{nm}$)
instrument ought to be able to discover
about 50 extrasolar Earth-like planets if it can provide contrast
of $10^{-10}$ at an angular separation of $3\ld$ and that a $4 \times 10$m class
telescope ought to be able to discover about 150 such
planets if it can provide the same contrast at a separation of $4\ld$.

Many approaches have been examined for designing a telescope with this level of contrast.  The most promising fall into two broad categories---nulling interferometers (operating in the infrared) and coronagraphs (operating in the visible).  One important subset of coronagraphs, referred to as shaped pupils, has been gaining interest (\cite{ref:Spergel, ref:KVSL, VSK02, VSK03}).  These are apodized entrance pupils that rely solely on one/zero binary openings.   In this paper we present a unified treatment of the one-dimensional shaped pupil designs, a few of which have been previously published, and we introduce a promising new design which we call {\em barcode masks}.

%Shaped pupils coronagraphs achieve high-contrast by modifying the shape of the telescope entrance pupil in order to achieve control of the diffracted light in the image plane.  Shaped pupils are a particular class of Apodized telescopes that have shown great promise for achieving the contrast necessary for TPF.

The paper is organized as follows.  In the next section we briefly review the relationship between pupil-plane apodization 
and the corresponding image-plane electric field and point-spread function.  In the following section we discuss the various performance metrics that characterize a planet finding telescope.  In Section 4 we show how optimization problems can be formulated and used to find the ``best'' entrance pupil apodization for high contrast in one-dimensional square apertures as well as azimuthally symmetric (circular or elliptical) pupils.  In Section 5 we describe how these results can be used to find various families of assymmetric shaped pupils.  In section 6 we present a preliminary sensitivity analysis to determine the tolerances of the pupil approaches to manufacturing errors.

\section{Apodization}

The main objective of our work is to develop optimal, high-contrast, binary shaped pupils (that is, pupils that consist of only open and closed areas).  Since pupil masks can be viewed mathematically as a special case of apodized pupils (where the apodization function is simply zero-one valued), we begin by developing some notation and concepts in the general context of pupil apodization.  In this paper we use scalar far-field diffraction theory (i.e., Fraunhoffer optics).  We are currently investigating near field and vector propagation effects and will report on those more refined models in a future paper.
%That is, using the Kirchoff approximations, the scalar field through an aperture is normally given by the Rayleigh-Sommerfeld integral \cite{ref:Hecht, ref:Jackson, ref:Born}:
%%
%\begin{equation}
%E(\vec{x}) = \int_{S}{\frac{ke^{ikR}}{2\pi R^2} \left (i-\frac{1}{kR}\right) \frac{f^2}{\sqrt{f^2+|\vec{x}'|^2}}E_0e^{-ikn\delta(\vec{x}')}da'}
%\end{equation}

In the Fraunhoffer regime, the image-plane electric field $E(\xi,\zeta)$, produced by an on-axis point source and an \emph{apodization function}, $A(x,y)$, reduces to the Fourier Transform of the apodization:
%
\begin{equation}
\label{eq:squareapod}
E(\xi,\zeta)=\int{\int_{S}{e^{-2\pi i (x\xi+y\zeta)}A(x,y)dxdy}}
\end{equation}
where S denotes the overall aperture and can be taken, for example, to be
a square:
\[ S=\{(x,y):-1/2\le x \le 1/2, -1/2\le y \le 1/2\}  \]
 
 Here, $x$ and $y$ denote the coordinates in the pupil plane measured in units of the aperture $D$ and ($\xi$, $\zeta$) denote angular (radian) deviations from on-axis measured in units of wavelength over aperture ($\lambda/D$) or, equivalently, physical distance in the image plane measured in units of focal-length times wavelength over aperture ($f\lambda/D$).  

When S is a square and the apodization function, $A(x)$,  is one-dimensional and symmetric about the $y$-axis, the expression for the electric field can be simplified into a single, one-dimensional Fourier Transform:
\begin{equation}
\label{eq:onedapod}
E(\xi,\zeta)=\frac{2\sin(\pi\zeta)}{\pi\zeta}\int_0^{1/2}{A(x)\cos(2\pi \xi x) dx }
\end{equation}

In this paper we will only be considering apodizations (and shaped pupils)  along a single dimension. 
%When $S$ is instead a circle and the apodization function, $A(\sqrt{x^2+y^2})$, is circularly symmetric, it is natural to  write the electric field as:
%\begin{equation}
%E(\xi,\zeta)=\int{\int_{S}{e^{-2\pi i (x\xi+y\zeta)}A(\sqrt{x^2+y^2})dxdy}}
%\end{equation}
%where,
%\[ S=\{(x,y): 0 \le r(x,y) \le 1/2, \theta(x,y) \in [0,2\pi]\}  \]
%and $r(x,y)$ and $\theta(x,y)$ denote the polar coordinates associated with point $(x,y)$.
% For this case, the resulting image plane electric field, $E(\rho)$, is itself just a function of a the radius $\rho=\sqrt{\xi^2+\zeta^2}$ in the image plane and a straightforward change of coordinates gives the electric field as the Hankel Transform of the apodization:
% \begin{eqnarray}
%        E(\rho) 
%        &=&  
%        \int_0^{1/2} \int_0^{2\pi} 
%            e^{-2 \pi i r \rho \cos(\theta-\phi)} A(r) 
%        r d\theta dr, \nonumber \\
%        &=&  \label{eq2}
%        2 \pi \int_0^{1/2} J_0(2 \pi r \rho) A(r) r dr,
%    \end{eqnarray}
%    where $J_0$ denotes the $0$-th order Bessel function of the first kind,
%    $r$ and $\theta$ denote polar coordinates
%    in the pupil plane, 
%    and $\rho$ and $\phi$ denote polar coordinates in the image plane.   Angles $\theta$ and $\phi$ are measured in radians. 
%%    The unitless pupil-plane ``length'' $r$ is given 
%%    as a multiple of the aperture diameter $D$ and
%%    the unitless image-plane ``length'' $\rho$ is given 
%%    as a multiple of focal-length times
%%    wavelength over aperture ($f \lambda/D$) or, equivalently, as an angular
%%    measure on the sky, in which case it is a multiple of just $\lambda/D$.
    For both square and circular  apertures, the {\em point spread function} (PSF) is the square of the electric field and, for planet finding, the apodization is chosen so that the corresponding PSF has the desired contrast.

\section{Performance Metrics}

The goal of our apodizations is to create point spread functions that provide the contrast necessary to detect a planet \emph{with as short an integration time as possible}.  In order to compare different apodization approaches and to perform optimizations it is therefore necessary to introduce a few important performance metrics (\cite{VSK03}).  The first is {\em contrast}, which is a function of the linear  position in the image plane.  In rectangular coordinates we define contrast by:
        \[
    E^2(\xi,\zeta)/E^2(0,0).
    \]
    %
%    and in polar coordinates by:
%    \[
%    	E^2(\rho)/ E^2(0) .
%    \]
    We say that we have high-contrast when this ratio is very
    small.  For planet finding, the consensus is that a contrast of $10^{-10}$ outside a predefined radius is
    necessary in order to image an Earth-like planet that is $1$ AU from its
    Sun-like star.

 The second and third metrics are the {\em inner working angle}, $\xinaught$,
    and, of lesser importance, the {\em outer working angle}, $\xione$.
    We wish to design pupil masks that have high-contrast for all $\xi$'s
    in the interval $\xinaught \le \xi \le \xione$.  
    
The final performance metric is integration time, or its surrogate, throughput.  Integration time is a critical metric for any planet finding telescope such as TPF.  Smaller integration time relaxes requirements on system stability, simplifies control, allows more observations, and potentially allows return visits to interesting systems.  In ~\cite{ref:KVSL}  we described a number of different approaches to image analysis that resulted in quantifiable integration times that we used to compare different coronagraph approaches.  All are based on some type of photometry on a potential planet and a predefined signal-to-noise ratio.  While useful for comparing different coronagraphs, the resulting equations were too complicated to use in optimization work.  Therefore, we consider here slightly simpler measures of throughput. (These measures are closely related to integration time because they measure the amount of light entering the system, but they do not account for the effect of PSF shape.  While we acknowledge that PSF shape or ``sharpness'' is an important quantity in determining absolute integration times, it does not have a strong effect on the relative comparisons or optimizations.)  Even asking only for measures of throughput is not such a simple question as there are multiple ways one might formulate an answer.

Perhaps the most natural measure of throughput is the amount of energy that falls into the main lobe of the PSF relative to the total energy conveyed through a fully open aperture:
%
\begin{equation}
\mathcal{T}_{Airy} =4 \int_0^{\xi_{iwa}}{\int_0^{\zeta_{iwa}}{E^2(\xi,\zeta) d\zeta d\xi}}
\end{equation}
%
where $\xi_{\mbox{iwa}}$ and $\zeta_{\mbox{iwa}}$ denote the locations in ($\xi,\zeta$) of the first null of the PSF.  We call this the \emph{Airy-throughput}  (\cite{VSK03}).  
%For circularly symmetric PSFs $\mathcal{T}_{Airy}$ can be written more simply in polar coordinates:
%%
%\begin{equation}
%\mathcal{T}_{Airy} = \frac{\int_0^{\rho_{iwa}}{E^2(\rho))2\pi \rho d\rho}}{(\pi(1/2)^2)}=8\int_0^{\rho_{iwa}}{E^2(\rho)\rho d\rho}
%\end{equation}
%%
%where here $\rho_{iwa}$ denotes the radial location of the first null of the PSF.

There are two other relevant measures of throughput.  The {\em total throughput} of an apodization is the integral of the PSF over the entire image plane, which,  by Parseval's theorem, is the integral of the square of the apodization over the entire entrance pupil.
%For example, in polar coordinates, the total throughput is:
%\begin{equation}
%\mathcal{T}_{total} = \int_0^\infty{E^2(\rho)2\pi\rho d\rho} = \int_0^{1/2}{A^2(r) 2\pi r dr }
%\end{equation}

Finally, the electric field at ($\xi=0, \zeta=0$),
\begin{equation}
E(0,0) = 2\int_0^{1/2}{A(x)dx}
\end{equation}
for one-dimensional rectangular apodizations provides another measure of the ``central'' throughput of the apodization, since its square is the peak throughput density at the center of the Airy disk.  If $A()$ is zero-one valued, then the apodization can be realized as a shaped pupil mask.  In this case, $E(0,0)$ is precisely the open area of the mask (and is also the total-throughput).  For this reason, we call $E(0,0)$ the {\em pseudo-area} of the apodization, even when the apodization is not zero-one valued.

\section{Optimal Apodizations for High-Contrast}
\label{sec:optimal_apod}

In this section we discuss the general problem of designing apodized pupils for high contrast.  This problem is not new; an excellent survery can be found in  \citet{ref:Jacquinot} with more recent ideas in \cite{ref:Indebetouw} and \cite{ref:Watson}.   \cite{ref:Nisenson} describe their concept of using an apodized square aperture for high-contrast, where discovery is made along the diagonals.  We describe below alternative optimal apodizations that  lead naturally to various shaped pupil concepts.

\subsection{One-Dimensional Apodizations}
\label{sec:one_d}

\begin{figure}[t]
\plottwo{f3_small.eps}{f4_small.eps}
\begin{center}
%\includegraphics[width=2.5 in]{../../TPF/Planet_Finding_NRA/Phase_II/FinalReportFigs/f3.pdf}
%\includegraphics[width=2.5 in]{../../TPF/Planet_Finding_NRA/Phase_II/FinalReportFigs/f4.pdf}
\end{center}
\caption{{\em Left} The PSF for the one-dimensional prolate spheroidal apodized square
aperture~\citep{ref:Slepian} of unit area plotted
on a logarithmic scale with black areas $10^{-10}$ below brightest.
{\em Right} On-axis cross section of the PSF showing an inner working distance of
$4 \lambda/a$.  The total throughput is 25\%.}
\label{fig:OneDProlate}
\end{figure}

Based on \refeqn{eq:onedapod} we ask for the optimal one-dimensional apodization for planet finding.  Of course, there are a number of ways one can state the optimization problem, all involving shaping the point spread function (PSF) in some way.  For instance, \cite{ref:Jacquinot} describe many different approaches to designing optimal apodizations.  The most important optimal apodization is due to \cite{ref:Slepian}, who introduced the prolate spheroidal wave function as the optimal 1-D telescope apodization.  Slepian's one dimensional optimization problem  asks for the function that concentrates as much light as possible into the central lobe of the finite Fourier transform.  \cite{ref:Slepian1} derive a finite fourier transform analog of the uncertainty principal and show that the function that solves this optimization problem is the zero order prolate spheroidal wavefunction, that is, the solution to the wave equation in prolate spheroidal coordinates (there is also an elegant solution to this problem using the calculus of variations).   Their statement is equivalent to the computationally easier problem:
 \[
        \begin{array}{ll}
            \mbox{minimize } & 
	    \displaystyle \int_{\xi_{iwa}}^{\infty} E(\xi,0)^2 d\xi \\[0.2in]
	    \mbox{subject to } &
	        \setlength{\arraycolsep}{0.1em}
	        \begin{array}[t]{rcll}
	        \displaystyle
		    & A(0) & = 1 .
	        \end{array}
        \end{array}
    \]
%
Choosing $\xinaught = 4$ provides the needed $10^{-10}$ contrast for $\xi \ge \xinaught$.  \reffig{fig:OneDProlate} shows the PSF resulting from a square aperture apodized in one dimension with a prolate spheroidal wave function.  The total  throughput of this apodized system is 25\%.

We note that outside the iwa the PSF has higher than needed contrast, which leads us to a slightly different optimization problem---namely, to maximize the pseudo-area subject to contrast constraints.  This has the enormous advantage that it can be formulated as an infinite dimensional linear programming problem:

\begin{equation} \label{100}
    \begin{array}{ll}
        \mbox{maximize } & \int_0^{1/2} A(x) dx \\
	\mbox{subject to } &
	    \setlength{\arraycolsep}{0.1em}
	    \begin{array}[t]{rcll}
	    \displaystyle
	        -10^{-5} E(0,\zeta) \le & E(0,\zeta) & \le 10^{-5} E(0,\zeta), &
	            \qquad \xinaught \le \xi \le \xione , \\
		  0 \le & A(x)   & \le  1, & \qquad 0 \le x \le 1/2, \\
%%		        & A'(r)  & \le  0, & \qquad 0 \le r \le 1/2, \\
%		        & A'(0)  &  =   0, & \\
%		-50 \le & A''(r) & \le 50, & \qquad 0 \le r \le 1/2, 
	    \end{array}
    \end{array}
\end{equation}
Discretizing the sets of $x$'s and $\xi$'s and replacing the integrals with their Riemann sums, problem~(\ref{100}) is approximated by a finite dimensional linear programming problem, which can be solved to a high level of precision (see, e.g., \citet{Van01}).
The numerical solution to this problem reveals that the optimal solution is
zero-one valued.   In other words, the optimal solution is, in fact, a binary mask consisting of a series of slots in the $y$-direction of varying width alternating along the $x$-axis.  We call such a mask a {\em barcode} mask.  \reffig{fig:barcode} shows a cross section of the PSF of such a mask optimized for high contrast in the region $4 \le \xi \le 40 \mbox{ } (\lambda/D)$.  The Airy throughput of this mask is 25\%.  Note that this response is very similar to the sine-squared bandlimited coronagraph, with bands of discovery zones in the image plane (\cite{ref:kuchner}).  
%
\begin{figure}[t]
%\plottwo{../../TPF/Planet_Finding_NRA/Phase_II/FinalReportFigs/bangbangpsflarge.pdf}{../../TPF/Planet_Finding_NRA/Phase_II/FinalReportFigs/bangbangpsfsmall.pdf}
\begin{center}
\includegraphics[width=2 in]{slottedmask_small.eps}
\includegraphics[width=2 in]{slottedmaskPSF_small.eps}
\includegraphics[width=2 in, angle=90, origin=c]{bangbangpsfsmall.eps}
%\includegraphics[width=3.5 in]{FinalReportFigs/bangbangpsflarge.pdf}
%\includegraphics[width=3.5 in]{FinalReportFigs/bangbangpsfsmall.pdf}
\end{center}
\caption{{\em Left} A one-dimensional barcode mask optimized for a dark region in [0,40] (all slots are not visible because of resolution limitations). {\em Center} The 2-D point spread function of the barcode mask {\em Right} A cross section of the optimal one-dimensional PSF near the inner working angle.  The Airy throughput is 25 \%.} 
\label{fig:barcode}
\end{figure}

In order to investigate as many planets as possible (particularly at larger distances), there is a strong desire to achieve as small an inner working angle as possible.  One benefit of small inner working angle is a reduction in the size of the telescope required to image the full habitable zone of these stars.  One way to achieve smaller inner working angle is to optimize a narrower discovery region (that is, bring in the outer working angle).  \reffig{fig:3to5barcode} shows a cross section  of the PSF of such a design optimized for high contrast in the range 3 to 5 $\lambda/D$ with an Airy throughput of 33.8\%.  It also possible to design a mask for an inner working angle as small as $2\mbox{ } \lambda/D$, but the width becomes prohibitively small (2 to 2.2 $\lambda/D$).
%
\begin{figure}[t]
\begin{center}
\plottwo{3to5_2DPSF_small.eps}{3to5psf_small.eps}
%\includegraphics[width=2.75 in]{3to5_2DPSF.pdf}
%\includegraphics[width=2.75 in]{3to5psf.pdf}
\end{center}
\caption{{\em Left} The Point Spread Function of a barcode mask designed for high contrast in the range 3 to 5 $\lambda/D$.  {\em Right} The cross-section of the PSF.  This mask has an Airy throughput of 33.8 \%. }
\label{fig:3to5barcode}
\end{figure}

Perhaps a more natural, but more complicated, optimization is to maximize the total throughput rather than the pseudo-area.  This results in a nonlinear optimization that is much more difficult to solve.  Nevertheless, we examined some solutions and found the same result as the pseudo-area problem.  We also note the importance of considering the Airy throughput for these finite band masks rather than the total throughput as the light gathered from a planet with this PSF would correspond only  to the inner core.  It is thus more appropriate to ignore the light scattered outside the outer working distance.

Finally, it is possible to numerically obtain a smooth apodization similar to the one in \reffig{fig:OneDProlate} by modifying the original optimization problem.  To accomplish this, we add seemingly artificial smoothness constraints.  Motivated by the fact that optimal apodizations look qualitatively like a Gaussian function, we impose smoothness constraints that correspond to the following conditions:
\begin{eqnarray}
\log(A)'& \le & 0  \label{eq:logpcon} \\
\log(A)'' & \le & 0  \label{eq:logppcon}
\end{eqnarray}
%
The resulting max-pseudo-area optimization problem is:
\begin{equation} \label{120}
    \begin{array}{ll}
        \mbox{maximize } & \int_0^{1/2} A(x) dx \\
	\mbox{subject to } &
	    \setlength{\arraycolsep}{0.1em}
	    \begin{array}[t]{rcll}
	    \displaystyle
	        -10^{-5} E(0,\zeta) \le & E(0,\zeta) & \le 10^{-5} E(0,\zeta), &
	            \qquad \xinaught \le \xi \le \xione , \\
		  0 \le & A(x)   & \le  1, & \qquad 0 \le x \le 1/2, \\
		        & A'(x)  & \le  0, & \qquad 0 \le x \le 1/2, \\
		        & A(x) A''(x)& \le A'(x)^2, & \qquad 0 \le x \le 1/2. \\
%		        & A'(0)  &  =   0, & \\
%		-50 \le & A''(r) & \le 50, & \qquad 0 \le r \le 1/2, 
	    \end{array}
    \end{array}
\end{equation}
%
\reffig{fig:OneDSmooth} shows the resulting smooth apodization function and the corresponding PSF (i.e., the Fourier Transform).  Not surprisingly, it has a strong similarity to the prolate spheroidal wavefunction.
%
\begin{figure}[t]
\begin{center}
\plottwo{OneDSmoothApod_small.eps}{OneDSmoothPSF_small.eps}
%\includegraphics[width=2.75 in]{OneDSmoothApod.pdf}
%\includegraphics[width=2.75 in]{OneDSmoothPSF.pdf}
\end{center}
\caption{{\em Left} A smooth one-dimensional optimal apodized function.
{\em Right} The corresponding cross section of the PSF showing an inner working distance of
$4 \lambda/a$ and contrast better than $10^{-10}$.}
\label{fig:OneDSmooth}
\end{figure}

%\subsection{Two-Dimensional, ``Checkerboard'' Apertures}
%\label{sec:Checkerboard}

%Here we use the barcode mask design to dramatically reduce the inner working angle at only a slight expense in search area.  We use the fact that for square entrance pupils the electric field integral separates into the product of two one-dimensional Fourier transforms:
%\begin{equation}
%\label{eq:squaredapod}
%E(\xi,\zeta)=\int_0^{1/2}{A(x)\cos(2\pi \xi x) dx } \int_0^{1/2}{A(y)\cos(2\pi \xi y) dy }
%\end{equation}

%The previous one-dimensional barcode mask can thus be designed for a contrast of $10^{-5}$ in each of the cartesian directions; the resulting product has the required $10^{-10}$ contrast in each of the four quadrants.  This has two advantages: a $10^{-5}$ mask can be designed with a much smaller inner working angle and such a mask can much more easily be verified in the laboratory.  We call the resulting product of the two, rotated barcode masks a ``checkerboard'' mask.  \reffig{fig:checkerboard} presents the PSF of a $10^{-5}$ barcode mask optimized for 2 to 25 $\lambda/D$ and the PSF of the resulting checkerboard mask.   The inner working angle along the diagonal is $2\sqrt{2} \lambda/D$.  The Airy throughput of the checkerboard mask is 15.1 \%.
%%
%\begin{figure}[t]
%\begin{center}
%\includegraphics[width=2 in]{../../Berkeley_Project/1e5Slotted.pdf}
%\includegraphics[width=2in]{../../Berkeley_Project/1e5SlottedPSFCross.pdf}
%\includegraphics[width=2in]{../../Berkeley_Project/CheckerboardPSF.pdf}
%\end{center}
%\caption{{\em Left} A barcode mask optimized for a contrast of $10^{-5}$ and an inner working angle of $2 \lambda/D$.  {\em Center} The PSF of the barcode mask. {\em Right} The PSF of the checkerboard mask resulting from the product of two $10^{-5}$ barcode masks.  Dark areas have a contrast of $10^{-10}$.  The Airy throughput is 15.1 \%.}
%\label{fig:checkerboard}
%\end{figure}

%This mask design utilizes the same principal as the apodized square aperture of \cite{ref:Nisenson} in that it relys on the product of two one-dimensional integrals to achieve high contrast on the diagonal.  However, there are some essential improvements; it utilizes a shaped pupil with only  binary apodizations (which are far easier to manufacture) and by using an optimal design in each direction, high contrast is achieved in the entire quadrant outside the inner working angle rather than a narrow strip along the diagonal.

%\subsection{Two-Dimensional, circularly symmetric Optimal Apodizations}
%\label{sec:two_d}

%While the one-dimensional apodizations above have many desirable properties (such as higher throughput), they still need at least one rotation to view the entire planetary system.  It is interesting, then, to examine optimal apodizations that result in azimuthally symmetric dark discovery zones.

%As before, an important optimization problem was solved by  \cite{ref:Slepian}, who termed the resulting apodization function the {\em generalized prolate spheroidal wave function}.    As an analog to the one-dimensional problem, it is defined as the solution to the following optimization problem:
%    \[
%        \mbox{maximize } \frac{\int_0^{\rhonaught} E(\rho)^2 \rho d\rho}%
%                              {\int_0^{\infty}     E(\rho)^2 \rho d\rho} .
%    \]
%    An equivalent but computationally easier formulation is this radial version of the previous problem:
%    \[
%        \begin{array}{ll}
%            \mbox{minimize } & 
%	    \displaystyle \int_{\rhonaught}^{\infty} E(\rho)^2 \rho d\rho \\[0.2in]
%	    \mbox{subject to } &
%	        \setlength{\arraycolsep}{0.1em}
%	        \begin{array}[t]{rcll}
%	        \displaystyle
%		    & A(0) & = 1 .
%	        \end{array}
%        \end{array}
%    \]
%    
%    The resulting PSF of this symmetric apodization has a radial cross section almost identical to that in \reffig{fig:OneDProlate}.  
%    
%As we mentioned earlier and computed for the 1D case, the generalized prolate spheroidal wave function solution performs better than necessary.   It is computationally easier and physically more compelling to design a 
%    pupil that provides high contrast between some {\em inner working angle} and
%    a finite {\em outer working angle}.  
%   The radial optimization problem that accomplishes this is:
%    \[
%        \begin{array}{ll}
%          \mbox{maximize } & \displaystyle \int_0^{1/2} A(r) 2\pi r dr \\[0.2in]
%	  \mbox{subject to } &
%	      \setlength{\arraycolsep}{0.1em}
%	      \begin{array}[t]{rcll}
%	      \displaystyle
%	          -10^{-5} E(0) \le & E(\rho) & \le 10^{-5} E(0), &
%	              \qquad \rhonaught \le \rho \le \rhoone , \\[0.2in]
%		    0 \le & A(r)   & \le  1, & \qquad 0 \le r \le 1/2, \\
%	      \end{array}
%        \end{array}
%    \]
%    Not surprisingly, the solution to this problem also turns out to be zero-one valued.  
%    That is, it is a {\em shaped pupil mask} consisting
%    of concentric rings.
%    The number of rings tends to infinity as $\rhoone \rightarrow \infty$.  Again, the more complicated problem of maximizing the total throughput can be attempted, but the resulting apodizations and PSFs are the same yet the computational cost is severe.

%\reffig{fig:rings} shows the optimal concentric ring mask for
%    $\rhonaught = 4$ and $\rhoone = 60$.   Note that there is a bright set of
%    diffraction rings extending from $\rho = 60$ to about $\rho = 80$.  It
%    turns out that this design, and in fact designs in general that have a finite
%    outer working angle, puts about half of the transmited light into the Airy
%    disk and the other half into the outer regions.
%    The Airy throughput for this mask is $9\%$ (and the total throughput,
%    which equals the open area, is about $18\%$).
%
%    \begin{figure}[h]
%    \begin{center}
%%    \hfill
%%    \begin{minipage}{2.5in}
%%    $\rhonaught = 4$ \\ $\rhoone = 60$ \\ Airy Thruput = $9\%$.
%%    \end{minipage}
%%    \hfill
%%    \begin{minipage}{2.5in}
%    \includegraphics[width=2.0in]{../../TPF/Planet_Finding_NRA/Phase_II/Bobspie/figs.1/mask.pdf} 
%%    \end{minipage}
%%    \hfill ~
%%    \end{center}
%%    \begin{center} 
%%    \hfill
%    \includegraphics[width=2.3in]{../../TPF/Planet_Finding_NRA/Phase_II/Bobspie/figs.1/psf1_00025_0.pdf}
%%    \hfill
%    \includegraphics[width=2.0in]{../../TPF/Planet_Finding_NRA/Phase_II/Bobspie/figs.1/psf_00025_0.pdf}
%%    \hfill ~ 
%    \end{center}
%    \caption{{\em Left.} A concentric ring mask designed for $\rhonaught = 4$
%		    and $\rhoone = 60$.
%             {\em Center and Right.} The associated PSF.
%	     Note: the Airy throughput is $9\%$.
%    } \label{fig:rings}
%    \end{figure}

%Also as in the 1D case we can define an optimal apodization problem that forces a smooth constraint on the apodization function, modeled after the Gaussian function:
%%
% \[
%        (\log A)' \le 0 , \qquad (\log A)'' \le 0 .
%    \]
%    The resulting optimization problem is:
%    \[
%        \begin{array}{ll}
%            \mbox{maximize } & \int_0^{1/2} A(r) 2\pi r dr \\
%	    \mbox{subject to } &
%	        \setlength{\arraycolsep}{0.1em}
%	        \begin{array}[t]{rcll}
%	        \displaystyle
%	            -10^{-5} E(0) \le & E(\rho) & \le 10^{-5} E(0), &
%	                \qquad \rhonaught \le \rho \le \rhoone , \\
%		      0 \le & A(r)   & \le  1, & \qquad 0 \le r \le 1/2, \\
%		            & A'(r)  & \le  0, & \qquad 0 \le r \le 1/2, \\
%		            & A(r)A''(r)&\le A'(r)^2,&\qquad 0 \le r \le 1/2. \\
%	        \end{array}
%        \end{array}
%    \]
%    
%    Fig.~\ref{fig:slepian} shows this optimal apodization and its associated PSF computed using
%    $\rhonaught = 4$ and $\rhoone = 60$.  The Airy throughput is again $9\%$.
%%
%    \begin{figure}[h]
%    \begin{center} 
%%    \begin{minipage}{2.5in}
%%    $\rhonaught = 4$ \\ $\rhoone = \infty$ \\ Airy Thruput = $9\%$.
%%    \end{minipage}
%%    \begin{minipage}{2.5in}
%    \includegraphics[width=2.0in]{../../TPF/Planet_Finding_NRA/Phase_II/Bobspie/figs.1/apod.pdf} 
%%    \end{minipage}
%%    \\
%    \includegraphics[width=2.0in]{../../TPF/Planet_Finding_NRA/Phase_II/Bobspie/figs.1/apodpsf.pdf} 
%    \includegraphics[width=2.0in]{../../TPF/Planet_Finding_NRA/Phase_II/Bobspie/figs.1/generalizedProlate.pdf} 
%    \end{center}
%    \caption{{\em Left.} The optimal smooth apodization computed using
% $\rhonaught = 4$ and $\rhoone = 60$.
% {\em Center and Right.} The associated PSF.
% Note: the Airy throughput is $9\%$.}
%    \label{fig:slepian}
%    \end{figure}

\subsection{Phase Shifts in Apodized Masks}
\label{sec:phaseshifts}

The central drawback of apodized pupils, and one of the significant factors that led us to consider  binary masks, is the difficulty manufacturing them.   It is extremely hard to create a smoothly varying mask to the necessary accuracy for $10^{-10}$ contrast.  In addition, there is no known process for creating apodized masks that doesn't also introduce a phase shift as a function of the transmission.   This phase shift can dramatically reduce the performance of the apodized pupil.  To better understand the issue and in an effort to ameliorate the problem, we examined optimizing in the presence of such a phase shift.

Incorporating the effect of phase shift,
the correct expression for the electric field is
\[
    E(\xi) =  
    \int_{-1/2}^{1/2} e^{-2 \pi i x \xi + i k \log(A(x))} A(x) dx
\]
where $k$ is a constant giving the rate that lag appears as attenuation is
increased.
Note that, unlike our earlier expressions, 
this complex expression does not reduce to a real-valued one.
%
\begin{figure}[t]
\begin{center}
\includegraphics[width=3.0in, angle=90, origin=c]{A0andA_small.eps} 
\end{center}
\caption{Optimal Apodization with and without Phase Shift}
\label{PhaseShiftApodizations}
\end{figure}
%
In \reffig{PhaseShiftApodizations},
the upper apodization was computed with $k=0$ (i.e., no phase shift) and
using $\xinaught = 4$ and $\xione = 40$.
The lower apodization corresponds to a nonzero phase shift.  It was computed
using a larger inner working angle of $\xinaught = 4.5$.  
The problem seems to be infeasible with a tighter inner working angle.

If the apodization is computed assuming no phase shift but in reality there is
a phase shift, then one expects the PSF to degrade because of the shift.
\reffig{nophaseshift} shows the amount of degradation one will see with $k=0.1$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=3.0in, angle=90, origin=c]{psf0andpsf1_small.eps} 
\end{center}
\caption{PSF for Optimal Apodization without Phase Shift.
Here is shown the PSF for the apodization computed without accounting for
phase shift.  
The upper curve is the PSF one gets with the phase shift term using $k=0.1$.
}
\label{nophaseshift}
\end{figure}

%(NOTE: THIS LAST PARAGRAPH PROBABLY SHOULD BE OMITTED.)
Finally, \reffig{withphaseshift} shows the psf corresponding to the apodization computed
assuming a phase shift of $k=0.1$.  Note that the iwa is now $4.5$.
%
\begin{figure}[t]
\begin{center}
\includegraphics[width=3.0in, angle=90,origin=c]{psf_small.eps} 
\end{center}
\caption{PSF for Optimal Apodization with Phase Shift.
Here is shown the PSF for the apodization computed with accounting for
phase shift.  Inner working angle is $4.5 \lambda/D$.
}
\label{withphaseshift}
\end{figure}

\section{Asymmetric Shaped Pupils}

Because of the problems associated with manufacturing and the limitations due to phase shifts described above, apodized pupils present unique and difficult challenges as a mechanism for modifying the point spread function to achieve high contrast.  We favor, instead, shaped pupils, where manufacturing involves only cutting openings in the material at accuracies easily achieved with current technology.  We have already, in fact, presented one class of shaped pupils as the optimal solution to the high-contrast imaging problem---the barcode mask of Section~\ref{sec:one_d} (see also \cite{VSK03} and \cite{VSK02} for examples of circularly symmetric shaped pupil masks).  In the following subsections we present a collection of other asymmetric shaped pupil masks derivable from a variety of optimization problems. 

\subsection{Single Pupil}
David Spergel was the first to realize that shaped pupils could be used to achieve essentially arbitrarily deep nulls close to the central star.  His first suggestion (\cite{ref:Spergel}) was to use a gaussian-shaped mask.  Shortly thereafter, Jeremy Kasdin realized that Slepian's prolate-spheroidal wave function (\cite{ref:Slepian1}) could be used here too as the optimal shape for achieving high contrast along one axis of the image plane.  This is seen by returning to the expressions for apodizations in a square or rectangular aperture (Eqs.~\ref{eq:squareapod} and \ref{eq:onedapod}).  If in place of a smooth apodization we assume instead a single pupil opening of width $w(x)$ relative to the $y=0$ axis we have for the electric field on the $\zeta=0$ axis in the image plane:
%
\begin{equation}
E(\xi,0)=\int_{-1/2}^{1/2}{w(x)e^{-i2\pi\xi x}dx}
\end{equation}

In other words, the field on the axis of the image plane is just the one-dimensional Fourier Transform of the pupil width.  Thus, an optimization problem for that width is identical to the Slepian problem we described earlier for the one-dimensional apodization.  A single pupil with the Slepian shape would provide the needed contrast.  Such a pupil is shown in \reffig{fig1}.  It achieves a contrast of
$10^{-10}$ everywhere along the $x$-axis except within an Airy disk of radius $4 \lambda/D$.  This was our first proposed shaped pupil for TPF.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=2in]{singlepupil_small.eps} 
\includegraphics[width=2in]{Prolate_pupil_psf_small.eps}
\includegraphics[width=2.1in, angle=90, origin=c]{apodpsf_small.eps}
\end{center}
\caption{{\em Top.} The Spergel-Kasdin prolate-spheroidal mask.  
{\em Bottom.} The associated
$2$-D point spread function and its $x$-axis slice shown in decibels
($10^{-10} = -100\mbox{dB}$).}
\label{fig1}
\end{figure}

It is clear from \reffig{fig1} that the disadvantage of the Spergel-Kasdin mask is the extreme narrowness of the high-contrast region at close working angles from the central star.  Many rotations of the telescope would be necessary to image the entire discovery space, dramatically impacting integration time and mission life.  The solution to opening up this discovery space is to introduce multiple openings, what we call multi-pupil masks.  In fact, the barcode  mask introduced earlier is an example of a multi-pupil mask.    In what follows we describe several other approaches to accomplishing this end.

\subsection{Multi-Pupil Square Aperture}

The simplest possible multi-pupil mask involves just repeating the single pupil, one on top of another, in a square aperture until the desired PSF is achieved.  In this approach, rather than optimizing a design directly, we attempt to reproduce, as closely as possible, the PSF of the one-dimensional apodization described in Section \ref{sec:one_d} and shown in \reffig{fig:OneDProlate}.  To begin, we assume that each opening in the mask is identical and stacked vertically across the pupil.  The opening, $y(x)$, in the upper quadrant of a $2N$ pupil mask, can then be described by the set S given by:
\begin{eqnarray}
S & = & \{(x,y): -1/2 \le x \le 1/2, y(x) \in \mathrm{Y_u}\} \\
\mathrm{Y_u} & = & \bigcup_{n=0}^{N-1}{\left [\frac{D}{2}\frac{n}{N}+\frac{w(x)}{2}, \frac{D}{2}\frac{n+1}{N}-\frac{w(x)}{2} \right ] }
\end{eqnarray}
while for the lower pupils, $y(x) \in \mathrm{Y_l}$,
\begin{equation}
\mathrm{Y_l}  =  \bigcup_{n=0}^{N-1}{\left [-\frac{D}{2}\frac{n+1}{N}+\frac{w(x)}{2}, -\frac{D}{2}\frac{n}{N}-\frac{w(x)}{2} \right ] }
\end{equation}
%
where $w(x)$ denotes the half width of the pupil and $D$ is the horizontal width of the aperture.  For what follows, all lengths  are normalized by $D$ to simplify notation.  Note that the shape of each pupil is the same and determined by $w(x)$.  Again, the goal is to determine choices for this function that will yield an image-plane PSF matching the PSF corresponding to a given apodization (such as that in \reffig{fig:OneDProlate}).  The electric field for such a multi-opening mask can by found from \refeqn{eq:onedapod}:
\begin{equation}
E(\xi,\zeta)=\frac{2\sin(\pi\zeta)}{\pi\zeta}\int_0^{1/2}\sum_{n=0}^{N-1}{{\left [ \sin 2\pi\zeta\left(\frac{n+1}{2N}-\frac{w(x)}{2}\right) - \sin2\pi\zeta\left(\frac{n}{2N}+\frac{w(x)}{2}\right)\right]\frac{\cos(2\pi \xi x)}{\sin(\pi\zeta)} dx }}
\end{equation}
%
which can be simplified by completing the finite sum:
%
\begin{equation}
E(\xi,\zeta)=\frac{2\sin(\pi\zeta)}{\pi\zeta}\int_0^{1/2}{\left ( \cos(\pi \zeta w(x)) - \frac{[1+\cos(\pi \zeta/N)]}{\sin(\pi \zeta/N)}\sin(\pi \zeta w(x)) \right ) \cos(2\pi \xi x) dx}
\label{eq:Ew}
\end{equation}

This looks  like the field due to an arbitrary apodization $A(x)$ in \refeqn{eq:onedapod}, except with $A(x)$ replaced by the expression in parentheses.  If we expand this expression in a Taylor series about $\zeta=0$, we find that the leading term in the series is  $1-2Nw(x)$, independent of $\zeta$.  Thus, if we choose:
\begin{equation}
w(x) = \frac{1-A(x)}{2N}
\label{eq:w(x)}
\end{equation}
%
we can create an approximation to the continuous one-dimensional apodization response.  What remains is to show that the higher order terms, for large enough $N$, only produce interfering light outside some outer working angle $\zeta_{\mbox{\scriptsize owa}}$.  Replacing for $w(x)$ from \refeqn{eq:w(x)} into \refeqn{eq:Ew} results in:
%
\begin{equation}
E(\xi,\zeta)=\frac{2\sin(\pi\zeta)}{\pi\zeta}\int_0^{1/2}{\left (\frac{\sin\left (\frac{\pi \zeta (1+A(x))}{2N} \right ) - \sin \left (\frac{\pi \zeta (1-A(x))}{2N} \right )}{\sin(\pi \zeta /N)} \right ) \cos(2\pi \xi x) dx}
\label{eq:EA}
\end{equation}
%

We can now Taylor expand the term in parentheses in \refeqn{eq:EA} for small $(\pi \zeta/N)$ to find:
%
\begin{equation}
E(\xi,\zeta)=\frac{2\sin(\pi\zeta)}{\pi\zeta}\int_0^{1/2}{\left (A(x) + \mathcal{O}\left ( \frac{\pi \zeta}{N} \right )^2 \right ) \cos(2\pi \xi x) dx}
\end{equation}

Thus, for large enough N, the scattered light due to the finite number of openings can be made arbitrarily small within some range 
 $0\le\zeta\le\zeta_{\mbox{\scriptsize owa}}$.  \reffig{fig:multisame} shows the PSF resulting from such a multi-opening square aperture with 10 and 100 openings using the prolate spheroidal apodization shown in \reffig{fig:OneDProlate} for $A(x)$.
%
\begin{figure}[h]
\plottwo{10pupilPSF_small.eps}{100pupilPSF_small.eps}
%\includegraphics[width=2.75 in]{../../TPF/Planet_Finding_NRA/Phase_II/FinalReportFigs/10pupilPSF.pdf}
%\includegraphics[width=2.75 in]{../../TPF/Planet_Finding_NRA/Phase_II/FinalReportFigs/100pupilPSF.pdf}
\caption{PSFs for a multipupil square aperture associated with the apodization shown in \reffig{fig:OneDProlate}.  {\em Left} A 10 opening pupil.
{\em Right} A 100 opening pupil.}
\label{fig:multisame}
\end{figure}

While it is certainly feasible, and perhaps not overly difficult, to manufacture a mask with over 100 openings, it may be desirable to use fewer pupils.  It is thus sensible to attempt to improve the resopnse with fewer openings by relaxing the requirement that each edge of the openings be the same, hoping  that these extra degrees of freedom can be used to improve the match to the apodized pupil, albeit at the cost of a  much more difficult computational problem.

In this case, the set $\mathrm{Y_u}$ for the upper half of the pupil is replaced by:
%
\begin{equation}
\mathrm{Y_u}  =  \bigcup_{n=0}^{N-1}{\left [\frac{1}{2}\frac{n}{N}+\frac{b_n(x)}{2}, \frac{1}{2}\frac{n+1}{N}-\frac{t_n(x)}{2} \right ] }
\end{equation}
where $b_n(x)$ is the bottom edge of opening $n$ and $t_n(x)$ is the top edge of opening $n$.  A similar set holds for the lower half of the pupil.  The resulting electric field in the image plane is given by:
\begin{equation}
E(\xi,\zeta)=\frac{2\sin(\pi\zeta)}{\pi\zeta}\int_0^{1/2}\sum_{n=0}^{N-1}{{\left [ \sin \pi\zeta\left(\frac{n+1}{N}-t_n(x)\right) - \sin\pi\zeta\left(\frac{n}{N}+b_n(x)\right)\right]\frac{\cos(2\pi \xi x)}{\sin(\pi\zeta)} dx }}
\end{equation}

Comparing again to \refeqn{eq:onedapod} we find that making the correspondence:
\begin{equation}
A(x) = \sum_{n=0}^{N-1}{\frac{\sin \pi\zeta\left(\frac{n+1}{N}-t_n(x)\right) - \sin\pi\zeta\left(\frac{n}{N}+b_n(x)\right)}{\sin\pi\zeta}}
\label{eq:match}
\end{equation}
results in a multi-pupil mask reproducing the one-dimensional apodization.  We again expand the right-hand side of \refeqn{eq:match} about $\zeta=0$ and match the lowest order term (independent of $\zeta$):
\[
A(x) = 1-\sum_{n=0}^{N-1}t_n-\sum_{n=0}^{N-1}b_n
\]
Unlike the previous multi-pupil mask, we can now use the extra degrees of freedom to make the coefficients of the terms of higher order in $\zeta$ vanish.  Each additional pupil pair allows us to eliminate two higher order terms.  The computational problem to find  the shapes of the openings becomes the solution to  N simultaneous polynomial equations.   While intuitively  one would expect such extra degrees of freedom to result in a better match  to the apodized solution with fewer pupils, the surprising result is that no significant improvement is made.  \reffig{fig:6pupil} shows an example of a 6 opening pupil designed with this method, its PSF, and the PSF of a uniform 6 opening pupil. 
%
\begin{figure}[h]
\begin{center}
%\plottwo{../../TPF/Planet_Finding_NRA/Phase_II/FinalReportFigs/f13.pdf}{../../TPF/Planet_Finding_NRA/Phase_II/FinalReportFigs/f14.pdf}
\includegraphics[width=2 in]{6PupilImageCrop_small.eps}
\includegraphics[width=2 in]{6PupilPSF_small.eps}
\includegraphics[width=2 in]{6EqualPupilPSF_small.eps}
\end{center}
\caption{{\em Left} An unequal six-opening pupil inscribed in a square aperture. 
{\em Center} The Corresponding PSF plotted on a logarithmic scale with black areas $10^{-10}$ below brightest. {\em Right} The PSF of a 6 equal opening multi-pupil aperture.  Spatial Frequencies shown from -20 to 20 $\lambda/D$.}
\label{fig:6pupil}
\end{figure}
%

One of the advantages of asymmetric masks is that they  can be made to have higher throughput by reducing the size of the high contast region (smaller opening angle).  While this may be unattractive for planet detection, as many rotations become necessary to cover the discovery region, it can have large advantages for characterizing a planet at a known location.  We summarize another category of assymetric masks ideally suited to this problem in the next section.

\subsection{Multi-Pupil Circular and Elliptical Apertures}

The multi-pupil asymmetric masks above are all embedded in square or rectangular apertures.  Thus, the central core of the PSF is also square, resulting in slight degradation of the inner working angle off axis.  Alternatively, multi-pupil assymetric masks can be designed that are embedded in circular and elliptical apertures by directly optimizing the edges of each opening relative to a performance function similar to those described in Section \ref{sec:optimal_apod}.   Many such pupils are described in detail in \cite{ref:KVSL}.  \reffig{fig:mult} shows an example of a multi-openning pupil mask that was designed for an elliptical aperture.  In an elliptical aperture, the asymmetry is used to leverage the long axis for improved iwa along the corresponding axis in the image plane.  
%    to improve somewhat on the narrow high-contrast zone.
%    It can cover all angles around the central star with $10$
%    integrations.  Its Airy throughput is $12\%$.
    % (\emph{including the rotations?  Is there a huge improvement then for characterization?})
%
 \begin{figure}[h]
    \begin{center} 
    \plottwo{Ellipticalfullmask2_small.eps}{Ellipticalpsf_small.eps}
   % \includegraphics[width=2.75 in]{Ellipticalfullmask2.pdf}
    %\includegraphics[width=2.75 in]{Ellipticalpsf.pdf}
%    \hfill  
%    \includegraphics[width=2.0in]{Ellipticalfullmask2.pdf}
%    \hfill  
%    \includegraphics[width=2.0in]{Ellipticalpsf.pdf}
%    \hfill ~ 
    \end{center}
    \caption{{\em Left.} A multiopening pupil mask designed to open up the
    high-contrast region.  {\em Right.} The corresponding PSF.
    On the $x$-axis, contrast of $10^{-10}$ extends over $\rho \ge 4$.
    } 
    \label{fig:mult}
    \end{figure}
    
\section{Tolerances}
    
    A critical question  for any apodized or shaped pupil coronagraph is how well the mask needs to be made.  In this section we summarize the tolerancing requirements on an apodized pupil coronagraph and a barcode mask, insisting that any error  produce scattered light below the $10^{-10}$ contrast level.
    
    \subsection{Apodized masks}
    
    The electric field in the image plane for an apodized entrance pupil with some arbitrary error in amplitude and phase is given by a modification of \refeqn{eq:squareapod}:
    %
\begin{equation}
E(\xi,\zeta)=\int{\int_{S}{e^{\gamma(x,y)}A(x,y)e^{-2\pi i (x\xi+y\zeta)}dxdy}}
\label{eq:apoderror}
\end{equation}
%
where $\gamma(x,y)=\alpha(x,y)+i \phi(x,y)$ is an arbitrary complex function of position in the pupil representing the phase and amplitude transmission error of the mask.  For simplicity, we consider here only the amplitude error in the apodization; we briefly discussed phase errors in Section~\ref{sec:phaseshifts}.  While the small amplitude error is arbitrary, because of the finite extent of the pupil we can model the error as a periodic function with fundamental period on $[-1/2, 1/2]$ (absorbing the top hat function into the apodization $A(x,y)$).  This allows us to decompose the error into a Fourier Series:
%
\begin{equation}
e^{\alpha(x,y)} = 1+\epsilon \sum_{n=1}^{\infty}{a_n \cos(\vec{k}_n \cdot \vec{x} + \varphi_n)}
\label{eq:ampfourier}
\end{equation}
%
where $\vec{x} \equiv (x,y)$ is the position vector, $\vec{k}_n = (k_x,k_y)$ is the wave vector of the harmonic term (with $|\vec{k}_n|$ being the spatial frequency), $A_n$ and $\varphi_n$ are the amplitude and phase of each harmonic component, and $\epsilon$ is small.  This expansion allows us to write the electric field in the image plane as the nominal, designed for point spread function plus a small  perturbation, $E(\xi,\zeta) = E_0(\xi,\zeta) + \Delta E(\xi,\zeta)$.  Since $\Delta E$ is small, the intensity in the image plane is approximately given by:
%
\begin{equation}
|E(\xi,\zeta)|^2 \approx |E_0(\xi,\zeta)|^2+ 2|E_0(\xi,\zeta)||\Delta E(\xi,\zeta)| 
\label{eq:Eerror}
\end{equation}
%

Since the electric field in \refeqn{eq:Eerror} is linear in the small change in intensity, superposition holds and it suffices to consider only a single harmonic component of the apodization error in \refeqn{eq:ampfourier}.  Thus, the field error is given by:
%
\begin{equation}
\Delta E = \epsilon \int{\int_{S}{A_n \cos(\vec{k}_n \cdot \vec{x} + \varphi_n)A(x,y)e^{-2\pi i (x\xi+y\zeta)}dxdy}}
 \end{equation}
 %
 By the Fourier Shift theorem, this is just a copy of the point spread function to the location $(k_x, k_y)$ in the image plane, commonly referred to as speckle:
 %
 \[ \Delta E = \epsilon E_0(\xi-k_x, \zeta-k_y) \]
 %
 The intensity error in the high  contrast region, due to errors in manufacturing the apodization,  is thus given by:
 %
 \begin{equation}
 \Delta_A \cong 2\epsilon |E_0(\xi,\zeta)| |E_0(\xi-k_x,\zeta-k_y)| \leq 10^{-10} |E_0(0,0)|^2
 \label{eq:delta_A}
 \end{equation}
 %
 
 The requirement that $\Delta_A$ be less than $10^{-10}$ of the nominal PSF (below  the  contrast level) allows us to set a requirement on $\epsilon$, the magnitude of the apodization variation.  Since we are concerned only with the error in the high contrast region of the image plane, we can set $|E_0(\xi,\zeta)|/|E_0(0,0)|$ to $10^{-5}$ in \refeqn{eq:delta_A}, leaving:
 %
 \begin{equation}
 \epsilon \leq \frac{5 \times 10^{-6}|E_0(0,0)|}{|E_0(\xi-k_x,\zeta-k_y)|}
 \label{eq:ap_epsilon}
 \end{equation}
 %
 
 At the worst case error location of the speckle center, the ratio in \refeqn{eq:ap_epsilon} is unity, leaving a requirement on the manufacturing error of $\epsilon \le 5 \times 10^{-6}$.  Achieving a throughput variation with this accuracy is a formidable challenge.  This, in part, motivates our search for shaped pupil solutions.
 
 \subsection{Barcode Masks}
 
 While it is demonstrably easier to manufacture a shaped pupil mask than an apodized one, it is still necessary to estimate the precision required.  In this section we calculate the performance sensitivity of the barcode mask to manufacturing errors. Here we assume that the edge locations of each slot along the x-axis, $r_i$, are in error by some function $\epsilon_i f_i(y)$ on the right half and $\epsilon'_i f'_i(y)$ on the left half of the mask, where again $\epsilon$ is small.  We further assume that each $\epsilon_i$ and $\epsilon'_i$ are independent and identically distributed (i.i.d.) random variables with $\mathscr{E}\{\epsilon_i\} = 0$ and $\mathscr{E}\{\epsilon_i \epsilon_j \} = \sigma^2 \delta_{ij}$.  
 
 The error in the electric field is then given by:
 %
 \begin{equation}
 \Delta E(\xi,\zeta) = \sum_{i=1}^N{\int_{-1/2}^{1/2}{e^{-2\pi i \zeta y} \left [ (\epsilon_i f_i(y) - \epsilon'_i f'_i(y) )\sin(2\pi r_i \xi) + i(\epsilon_i f_i(y) + \epsilon'_i f'_i(y))\cos(2\pi r_i \xi) \right ] dy}}
 \label{eq:deltaE_slot}
 \end{equation}
 %
 Here, rather than use \refeqn{eq:Eerror} for  the image plane error we instead turn  to a stochastic analysis.  Since $\mathscr{E}\{\Delta E\}$ is zero, we compute the expected value of the intensity:
 %
 \begin{equation}
 \mathscr{E}\{|E(\xi,\zeta)|^2\} = |E_0(\xi,\zeta)|^2 + \mathscr{E}\{|\Delta E(\xi,\zeta)|^2 \}
 \end{equation}
 %
 From \refeqn{eq:deltaE_slot} we therefore find,
 %
 \begin{equation}
 \mathscr{E}\{|\Delta E(\xi,\zeta) |^2 \} = \sigma^2 \sum_{i=1}^N {\left \{ \left [ \int_{-1/2}^{1/2}{f_i(y)e^{-2\pi i \zeta y}dy} \right ]^2 + \left [ \int_{-1/2}^{1/2}{f'_i(y)e^{-2\pi i \zeta y}dy} \right ]^2 \right \} } \le 10^{-10} |E_0(0,0)|^2
 \end{equation}
  
  As we did with the apodization error, we can model each of the mask edge error functions, $f_i(y)$, via a harmonic  expansion.  Since the small amplitude of the error was absorbed into the $\epsilon_i$, the terms in brackets are bounded by one at the  worst locations in the image plane.  Thus, a conservative bound on the average edge error is given by:
  %
  \begin{equation}
  \sigma \le \frac{10^{-5}|E_0(0,0)|}{\sqrt{2N}}
  \end{equation}   
  
For the barcode mask  shown in Section~\ref{sec:one_d}, which has 44 edges per side, this translates into a manufacturing accuracy on each edge of the mask of approximately  $5 \times 10^{-7}$.  For a 2 inch mask, this corresponds to a 25 nm accuracy  requirement on the mask edges, something achievable with current fabrication technologies.

%\section{Circularly Symmetric Shaped Pupils}

%The main advantage of the assymetric masks is their potentially higher throughput.  In some cases, the calculations involved in deriving them are also less complex.  However, as with the one-dimensional apodization, they only produce a limited dark region in the image plane that requires rotations in order to enable planet detection in a full 360 deg azimuth about the parent star.  While these masks may be useful for characterization studies of known planets, they are thus potentially less attractive for discovery.  In this section we explore the derivation of circularly symmetric shaped pupils that allow full discovery in one-image, just as the generalized prolate spheroidal apodization improved upon the one-dimensional version (at the expense of throughput).

%\subsection{Spiderweb Masks}

%We have already presented a circularly symmetric shaped pupil mask---the concentric ring mask of Section \ref{sec:two_d}.  Unfortunately, such a mask is impossible to make without a some additional means of supporting the rings.  We feel that glass is undesirable because of the inevitable scatter that would result.  The only alternative is to use 
%    a spider vane structure. A very similar analysis to the previous one for the multi-pupil mask shows that with enough spiders, the resulting scattered light is pushed outside the desired outer working angle.
%    
%    Consider $N$ ``pie-shaped'' spider vanes each of width $\alpha$:
%    \[
%        S_n = \displaystyle \left\{ (r,\theta): \;
%    		              0 \le r \le 1/2, \;
%			       \frac{2 \pi n}{N}     + \hovertwor
%			       \le \theta \le
%			       \frac{2 \pi (n+1)}{N} - \hovertwor
%		           \right\},  \qquad n=0,1,...,N-1 .
%    \]
%    As shown in \cite{VSK02},
%    overlaying these spider vanes on an apodization $A(r)$ yields the
%    following electric field:
%    \begin{eqnarray*}
%        E(\rho,\phi)
%        &=&
%        (2\pi - N\hoverrb) \int_0^{1/2} J_0(2\pi r \rho) A(r) r dr \\
%        && - 4 \sum_{j=1}^{\infty} 
%            \int_0^{1/2} J_{jN}(2\pi r \rho)\cos(jN(\phi-\pi/2)) 
%    	                       \frac{1}{j} \sin(jN\hovertwor)
%    				A(r) rdr .
%    \end{eqnarray*}
%    It is also shown in \cite{VSK02} that,
%    for $N$ large and $\rho$ not very large, the first term in the infinite
%    sum dominates the rest.  Hence, the electric field can be approximated by
%    a combination of just two Bessel integrals:
%    \begin{eqnarray}
%        E(\rho,\phi)
%        & \approx & \nonumber
%        2 \pi \int_0^{1/2} J_0(2\pi r \rho) (1 - N\hoverrb/(2 \pi) ) A(r) rdr \\
%        && - 4 \int_0^{1/2} J_{N}(2\pi r \rho)\cos(N(\phi-\pi/2)) 
%	                           \sin(N\hovertwor) A(r) rdr .
%				   \label{eq3}
%    \end{eqnarray}
%    Note that the first term is just the original electric field given
%    in Eq.~\ref{eq2} reduced 
%    by a factor of $1 - N\hoverrb/(2\pi)$,
%    which is just the fraction of the aperture that remains uncovered by the
%    spiders.  The second term produces the well-known diffraction spikes.
%    As is well known, using
%    a small number of spider vanes makes bright diffraction spikes
%    that would destroy the required contrast and adding
%    more spiders means producing more spikes.  This sounds bad but, more spiders also
%    means that the spikes move out away from $\rho = 0$.  It turns out that
%    with enough spiders the high contrast zone can be preserved.
%    Fig.~\ref{fig:spiders} shows the PSFs one gets using $100$ and $180$ 
%    spiders.
%
%    \begin{figure}[h]
%    \begin{center}
%    \hfill
%    \includegraphics[width=2.5in]{../../TPF/Planet_Finding_NRA/Phase_II/Bobspie/figs.1/psf1_005_100.pdf} 
%    \hfill
%    \includegraphics[width=2.0in]{../../TPF/Planet_Finding_NRA/Phase_II/Bobspie/figs.1/psf_005_100.pdf} 
%    \hfill ~
%    \end{center}
%    \begin{center} 
%    \hfill
%    \includegraphics[width=2.5in]{../../TPF/Planet_Finding_NRA/Phase_II/Bobspie/figs.1/psf1_0025_180.pdf}
%    \hfill
%    \includegraphics[width=2.0in]{../../TPF/Planet_Finding_NRA/Phase_II/Bobspie/figs.1/psf_0025_180.pdf}
%    \hfill ~ 
%    \end{center}
%    \caption{{\em Top.} The PSF for a spiderweb mask with $rhonaught = 4$,
%    		$\rhoone = 60$, and $N = 100$ vanes.
%    	     {\em Bottom.} The PSF for a spiderweb mask with $rhonaught = 4$,
%		$\rhoone = 60$, and $N = 180$ vanes.
%    } \label{fig:spiders}
%    \end{figure}

%\subsection{Starshaped Masks}

%It seems wasteful to develop a high-throughput concentric-ring mask
%    only to overlay it with a throughput reducing spiderweb obstruction
%    just to offer support.  To avoid this waste, we ask whether it is possible
%    simply to adjust the width of the spiders
%    appropriately as a function of radius and dispense with the concentric 
%    rings altogether.  As shown in \cite{VSK03}, this can be done.
%    We call such masks {\em starshape} masks and recognize them as the polar analog of the 1D multi-pupil mask we designed earlier.
%    A typical mask of this type is shown in Fig.~\ref{fig:star}.
%    \begin{figure}[h]
%    \begin{center} 
%    \includegraphics[width=2.5in]{../../TPF/Planet_Finding_NRA/Phase_II/Bobspie/figs.1/star20.pdf}
%    \end{center}
%    \caption{A 20-point (or vane) starshape mask.} \label{fig:star}
%    \end{figure}
%
%    The analysis behind starshape masks 
%    is very similar to that which leads to Eq.~\ref{eq3}.
%    The only changes are that the constant $\alpha$ is now allowed to depend on
%    radius $r$ and that the apodization $A(r)$ is gone:
%    \begin{eqnarray}
%        E(\rho,\phi)
%        & \approx & \nonumber
%        2 \pi \int_0^{1/2} J_0(2\pi r \rho) (1 - N\alpha(r)/(2 \pi)) rdr \\
%        && - 4 \int_0^{1/2} J_{N}(2\pi r \rho)\cos(N(\phi-\pi/2)) 
%	                           \sin(N\alpha(r)/2) rdr .
%				   \label{eq4}
%    \end{eqnarray}
%    Our aim is for the factor $(1 - N\alpha(r)/(2 \pi))$, which depends on
%    $r$, to give the effect of any specified apodization.  
%    This is done by setting
%    \[
%        \alpha(r) = \frac{2 \pi}{N} \left( 1 - A(r) \right) .
%    \]
%    The resulting PSF turns out to be that associated with the apodization
%    overlayed with diffraction spikes caused by using a finite number of vanes.
%    Again, the diffraction spikes move out as $N$ increases and a value
%    of $N$ of about $150$ or more gives a sizeable high-contrast discovery zone.
%
%    As was the case in 1D, there is a starshape mask associated with every apodization.
%    The question arises as to which apodization we should use.  
%    We've already argued that the zero-one valued concentric-ring
%    apodization is in some sense the optimal apodization.  But, a zero-one
%    valued apodization implemented as a starshape mask is again just a set of
%    concentric rings and so the issue of how to support the rings resurfaces.
%    One possible solution to this is to put an upper bound on the apodization
%    function that is strictly less than one.  In this way, we can prevent
%    there being any rings of openness in the starshape mask.  However, such
%    an upper bound reduces throughput which was exactly what motivated us to
%    consider this family of masks.  Instead, we use the smooth
%    apodization function found via smoothness constraints in \ref{sec:two_d} and shown in \ref{fig:slepian} and that is arbitrarily close to the zero-one valued apodization
%    in terms of throughput.  
    
%    We choose the same smoothness constraints as we did in 1D, modeled after a Gaussian function:
%    \[
%        (\log A)' \le 0 , \qquad (\log A)'' \le 0 .
%    \]
%    The resulting optimization problem is:
%    \[
%        \begin{array}{ll}
%            \mbox{maximize } & \int_0^{1/2} A(r) 2\pi r dr \\
%	    \mbox{subject to } &
%	        \setlength{\arraycolsep}{0.1em}
%	        \begin{array}[t]{rcll}
%	        \displaystyle
%	            -10^{-5} E(0) \le & E(\rho) & \le 10^{-5} E(0), &
%	                \qquad \rhonaught \le \rho \le \rhoone , \\
%		      0 \le & A(r)   & \le  1, & \qquad 0 \le r \le 1/2, \\
%		            & A'(r)  & \le  0, & \qquad 0 \le r \le 1/2, \\
%		            & A(r)A''(r)&\le A'(r)^2,&\qquad 0 \le r \le 1/2. \\
%	        \end{array}
%        \end{array}
%    \]
%    Fig.~\ref{fig:smooth} shows the optimal apodization computed using
%    $\rhonaught = 4$ and $\rhoone = 60$.  The Airy throughput is again $9\%$.
%    \begin{figure}[h]
%    \begin{center} 
%    %\includegraphics[angle=-90,width=3.0in]{C:/user/rvdb/public_html/JAVA/tpf/starshape/1/apod.pdf} 
%    %\includegraphics[angle=-90,width=3.0in]{C:/user/rvdb/public_html/JAVA/tpf/starshape/1/apodpsf.pdf} 
%    \includegraphics[width=3.0in]{../../TPF/Planet_Finding_NRA/Phase_II/Bobspie/figs.1/apod_gs.pdf} 
%    \includegraphics[width=3.0in]{../../TPF/Planet_Finding_NRA/Phase_II/Bobspie/figs.1/apodpsf_gs.pdf} 
%    \end{center}
%    \caption{{\em Left.} The optimal smooth apodization computed using
%    $\rhonaught = 4$ and $\rhoone = 60$.
%     	     {\em Right.} The associated psf.
%	     Note: the Airy throughput is $9\%$.
%    } \label{fig:smooth}
%    \end{figure}
%    

%Fig.~\ref{fig:starpsf} shows the PSFs of the starshape
%    masks associated with this smooth apodization using $20$ and $150$ star points.  We see that $20$ points is too few
%    and that $150$ is adequate to provide a large dark zone in the PSF.
%
%    \begin{figure}[h]
%    \begin{center} 
%    \hfill
%    \includegraphics[width=2.0in]{../../TPF/Planet_Finding_NRA/Phase_II/Bobspie/figs.1/psf_20}
%    \hfill
%    \includegraphics[width=2.5in]{../../TPF/Planet_Finding_NRA/Phase_II/Bobspie/figs.1/psf0_20}
%    \hfill ~ \\
%    \hfill
%    \includegraphics[width=2.0in]{../../TPF/Planet_Finding_NRA/Phase_II/Bobspie/figs.1/psf_150}
%    \hfill
%    \includegraphics[width=2.5in]{../../TPF/Planet_Finding_NRA/Phase_II/Bobspie/figs.1/psf0_150}
%    \hfill ~ 
%    \end{center}
%    \caption{{\em Top.} The PSF for a 20-point starshape mask having
%	     $\rhonaught = 4$ and $\rhoone = 60$.
%    	     {\em Bottom.} The PSF for a 150-point starshape mask having
%	     $\rhonaught = 4$ and $\rhoone = 60$.
%	     In both cases, the Airy throughput is $9\%$.
%	     } \label{fig:starpsf}
%    \end{figure}

\section{Final Remarks}

In this paper we presented all the asymmetric shaped
pupil masks we have studied to date for achieving high contrast imaging.  All of these masks use apodization at a single pupil to shape the point spread function of the imaging system, achieving $10^{-10}$ contrast outside some desired inner working angle.  These masks can be used in either rectangular or circular apertures, depending upon the specific mission tradeoffs.  While the idea of using pupil apodization has been around for many years, the problems associated with manufacturing such smoothly varying masks to the accuracy needed for planet finding has yet to be solved.  We show here that any apodized pupil can be made into a shaped pupil mask at a small cost in discovery space but with enormous practical benefit.  Shaped pupils are far easier and less expensive to manufacture, are more tolerant to errors, and, in some cases, provide more throughput.  While classical Lyot coronagraphs typically have more throughput, they suffer from the same manufacturing difficulties and are far more sensitive to alignments and pointing error. 

\section*{Acknowledgements}
We gratefully acknowledge the support of the National Aeronautics and Space Administration through the Jet Propulsion Laboratory, California Institute of Technology for this work.  


%\bibliographystyle{unsrtnat}
%\bibliography{../../../TPF/Planet_Finding_NRA/Phase_II/FinalReport}

\begin{thebibliography}{13}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{{\tt #1}}\fi

\bibitem[Brown et~al.(2002)Brown, Burrows, Casertano, Clampin, Eggets, Ford,
  Jucks, Kasdin, Kilston, Kuchner, Seager, Sozzetti, Spergel, Traub, Trauger,
  and Turner]{ref:Brown}
R.~A. Brown, C.~J. Burrows, S.~Casertano, M.~Clampin, D.~Eggets, E.B. Ford,
  K.W. Jucks, N.~J. Kasdin, S.~Kilston, M.~J. Kuchner, S.~Seager, A.~Sozzetti,
  D.~N. Spergel, W.~A. Traub, J.~T. Trauger, and E.~L. Turner.
\newblock The 4-meter space telescope for investigating extrasolar earth-like
  planets in starlight: {T}{P}{F} is {H}{S}{T}2.
\newblock In {\em Proceedings of SPIE: Astronomical Telescopes and
  Instrumentation}, number~14 in 4860, 2002.

\bibitem[Spergel(2000)]{ref:Spergel}
D.~N. Spergel.
\newblock A new pupil for detecting extrasolar planets.
\newblock {\em astro-ph/0101142}, 2000.

\bibitem[Kasdin et~al.(2003)Kasdin, Vanderbei, Spergel, and Littman]{ref:KVSL}
N.~J. Kasdin, R.~J. Vanderbei, D.~N. Spergel, and M.~G. Littman.
\newblock Extrasolar planet finding via optimal apodized-pupil and shaped-pupil
  coronagraphs.
\newblock {\em The Astrophysical Journal}, 582:\penalty0 1147--1161, January
  2003.

\bibitem[Vanderbei et~al.(2003{\natexlab{a}})Vanderbei, Spergel, and
  Kasdin]{VSK02}
R.J. Vanderbei, D.N. Spergel, and N.J. Kasdin.
\newblock % \href{http://www.princeton.edu/~rvdb/tex/spiders/ms.pdf}%
  {Spiderweb Masks for High Contrast Imaging}.
\newblock {\em Astrophysical Journal}, 590:\penalty0 593--603, June 10
  2003{\natexlab{a}}.

\bibitem[Vanderbei et~al.(2003{\natexlab{b}})Vanderbei, Spergel, and
  Kasdin]{VSK03}
R.J. Vanderbei, D.N. Spergel, and N.J. Kasdin.
\newblock % \href{http://www.princeton.edu/~rvdb/tex/starshape/ms.pdf}%
  {Circularly Symmetric Apodization via Starshaped Masks}.
\newblock {\em Astrophysical Journal}, 599\penalty0 (1):\penalty0 686--694,
  December 10 2003{\natexlab{b}}.

\bibitem[Jacquinot and Roizen-Dossier(1964)]{ref:Jacquinot}
P.~Jacquinot and B.~Roizen-Dossier.
\newblock Apodisation.
\newblock {\em Progess in Optics}, 3:\penalty0 29--186, 1964.

\bibitem[Indebetouw(1990)]{ref:Indebetouw}
G.~Indebetouw.
\newblock Optimal apodizing properties of gaussian pupils.
\newblock {\em Journal of Modern Optics}, 37\penalty0 (7):\penalty0 1271--1275,
  1990.

\bibitem[Watson et~al.(1991)Watson, Mills, Gaiser, and Diner]{ref:Watson}
S.~M. Watson, J.~P. Mills, S.~L. Gaiser, and D.~J. Diner.
\newblock Direct imaging of nonsolar planets with infrared telescopes using
  apodized coronagraphs.
\newblock {\em Applied Optics}, 30\penalty0 (22):\penalty0 3253--3260, 1991.

\bibitem[Nisenson and Papaliolios(2001)]{ref:Nisenson}
P.~Nisenson and C.~Papaliolios.
\newblock Detection of earth-like planets using apodized telescopes.
\newblock {\em The Astrophysical Journal}, 548\penalty0 (2):\penalty0
  L201--L205, 2001.

\bibitem[Slepian(1965)]{ref:Slepian}
D.~Slepian.
\newblock Analytic solution of two apodization problems.
\newblock {\em Journal of the Optical Society of America}, 55\penalty0
  (9):\penalty0 1110--1115, 1965.

\bibitem[Slepian and Pollack(1961)]{ref:Slepian1}
D.~Slepian and H.~O. Pollack.
\newblock Prolate spheroidal wave functions, fourier analysis and
  uncertainty---i.
\newblock {\em The Bell System Technical Journal}, pages 43--84, January 1961.

\bibitem[Vanderbei(2001)]{Van01}
R.J. Vanderbei.
\newblock {\em Linear Programming: Foundations and Extensions}.
\newblock Kluwer Academic Publishers, 2nd edition, 2001.

\bibitem[Kuchner and Traub(2002)]{ref:kuchner}
M.~J. Kuchner and W.~A. Traub.
\newblock A coronagraph with a band-limited mask for finding terrestrial
  planets.
\newblock {\em The Astrophysical Journal}, 570:\penalty0 900--908, 2002.

\end{thebibliography}


\end{document}
astro-ph0404388.extracted_bib
\begin{thebibliography}{13}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{{\tt #1}}\fi

\bibitem[Brown et~al.(2002)Brown, Burrows, Casertano, Clampin, Eggets, Ford,
  Jucks, Kasdin, Kilston, Kuchner, Seager, Sozzetti, Spergel, Traub, Trauger,
  and Turner]{ref:Brown}
R.~A. Brown, C.~J. Burrows, S.~Casertano, M.~Clampin, D.~Eggets, E.B. Ford,
  K.W. Jucks, N.~J. Kasdin, S.~Kilston, M.~J. Kuchner, S.~Seager, A.~Sozzetti,
  D.~N. Spergel, W.~A. Traub, J.~T. Trauger, and E.~L. Turner.
\newblock The 4-meter space telescope for investigating extrasolar earth-like
  planets in starlight: {T}{P}{F} is {H}{S}{T}2.
\newblock In {\em Proceedings of SPIE: Astronomical Telescopes and
  Instrumentation}, number~14 in 4860, 2002.

\bibitem[Spergel(2000)]{ref:Spergel}
D.~N. Spergel.
\newblock A new pupil for detecting extrasolar planets.
\newblock {\em astro-ph/0101142}, 2000.

\bibitem[Kasdin et~al.(2003)Kasdin, Vanderbei, Spergel, and Littman]{ref:KVSL}
N.~J. Kasdin, R.~J. Vanderbei, D.~N. Spergel, and M.~G. Littman.
\newblock Extrasolar planet finding via optimal apodized-pupil and shaped-pupil
  coronagraphs.
\newblock {\em The Astrophysical Journal}, 582:\penalty0 1147--1161, January
  2003.

\bibitem[Vanderbei et~al.(2003{\natexlab{a}})Vanderbei, Spergel, and
  Kasdin]{VSK02}
R.J. Vanderbei, D.N. Spergel, and N.J. Kasdin.
\newblock % \href{http://www.princeton.edu/~rvdb/tex/spiders/ms.pdf}%
  {Spiderweb Masks for High Contrast Imaging}.
\newblock {\em Astrophysical Journal}, 590:\penalty0 593--603, June 10
  2003{\natexlab{a}}.

\bibitem[Vanderbei et~al.(2003{\natexlab{b}})Vanderbei, Spergel, and
  Kasdin]{VSK03}
R.J. Vanderbei, D.N. Spergel, and N.J. Kasdin.
\newblock % \href{http://www.princeton.edu/~rvdb/tex/starshape/ms.pdf}%
  {Circularly Symmetric Apodization via Starshaped Masks}.
\newblock {\em Astrophysical Journal}, 599\penalty0 (1):\penalty0 686--694,
  December 10 2003{\natexlab{b}}.

\bibitem[Jacquinot and Roizen-Dossier(1964)]{ref:Jacquinot}
P.~Jacquinot and B.~Roizen-Dossier.
\newblock Apodisation.
\newblock {\em Progess in Optics}, 3:\penalty0 29--186, 1964.

\bibitem[Indebetouw(1990)]{ref:Indebetouw}
G.~Indebetouw.
\newblock Optimal apodizing properties of gaussian pupils.
\newblock {\em Journal of Modern Optics}, 37\penalty0 (7):\penalty0 1271--1275,
  1990.

\bibitem[Watson et~al.(1991)Watson, Mills, Gaiser, and Diner]{ref:Watson}
S.~M. Watson, J.~P. Mills, S.~L. Gaiser, and D.~J. Diner.
\newblock Direct imaging of nonsolar planets with infrared telescopes using
  apodized coronagraphs.
\newblock {\em Applied Optics}, 30\penalty0 (22):\penalty0 3253--3260, 1991.

\bibitem[Nisenson and Papaliolios(2001)]{ref:Nisenson}
P.~Nisenson and C.~Papaliolios.
\newblock Detection of earth-like planets using apodized telescopes.
\newblock {\em The Astrophysical Journal}, 548\penalty0 (2):\penalty0
  L201--L205, 2001.

\bibitem[Slepian(1965)]{ref:Slepian}
D.~Slepian.
\newblock Analytic solution of two apodization problems.
\newblock {\em Journal of the Optical Society of America}, 55\penalty0
  (9):\penalty0 1110--1115, 1965.

\bibitem[Slepian and Pollack(1961)]{ref:Slepian1}
D.~Slepian and H.~O. Pollack.
\newblock Prolate spheroidal wave functions, fourier analysis and
  uncertainty---i.
\newblock {\em The Bell System Technical Journal}, pages 43--84, January 1961.

\bibitem[Vanderbei(2001)]{Van01}
R.J. Vanderbei.
\newblock {\em Linear Programming: Foundations and Extensions}.
\newblock Kluwer Academic Publishers, 2nd edition, 2001.

\bibitem[Kuchner and Traub(2002)]{ref:kuchner}
M.~J. Kuchner and W.~A. Traub.
\newblock A coronagraph with a band-limited mask for finding terrestrial
  planets.
\newblock {\em The Astrophysical Journal}, 570:\penalty0 900--908, 2002.

\end{thebibliography}