# Shaped pupil Lyot coronagraphs: high-contrast solutions for restricted focal planes

**Neil T. Zimmerman**,

**A. J. Eldorado Riggs**,

**N. Jeremy Kasdin**

Princeton University, Department of Mechanical and Aerospace Engineering, Engineering Quadrangle, Olden Street, Princeton, New Jersey 08544, United States

**Alexis Carlotti**

Centre National de la Recherche Scientifique, Institut de Planétologie et d’Astrophysique de Grenoble, BP 53, F-38041, Grenoble, Cédex 9, France

**Robert J. Vanderbei**

Princeton University, Department of Operations Research and Financial Engineering, Sherrerd Hall, Charlton Street, Princeton, New Jersey 08544, United States

*J. Astron. Telesc. Instrum. Syst*. 2(1), 011012 (Jan 14, 2016). doi:10.1117/1.JATIS.2.1.011012

#### Open Access

**Abstract.**
Coronagraphs of the apodized pupil and shaped pupil varieties use the Fraunhofer diffraction properties of amplitude masks to create regions of high contrast in the vicinity of a target star. Here we present a hybrid coronagraph architecture in which a binary, hard-edged shaped pupil mask replaces the gray, smooth apodizer of the apodized pupil Lyot coronagraph (APLC). For any contrast and bandwidth goal in this configuration, as long as the prescribed region of contrast is restricted to a finite area in the image, a shaped pupil is the apodizer with the highest transmission. We relate the starlight cancellation mechanism to that of the conventional APLC. We introduce a new class of solutions in which the amplitude profile of the Lyot stop, instead of being fixed as a padded replica of the telescope aperture, is jointly optimized with the apodizer. Finally, we describe shaped pupil Lyot coronagraph (SPLC) designs for the baseline architecture of the Wide-Field Infrared Survey Telescope–Astrophysics Focused Telescope Assets (WFIRST-AFTA) coronagraph. These SPLCs help to enable two scientific objectives of the WFIRST-AFTA mission: (1) broadband spectroscopy to characterize exoplanet atmospheres in reflected starlight and (2) debris disk imaging.

The last two decades have witnessed tremendous advances in high-contrast imaging technology in tandem with the emergence of exoplanet research. There is now a mature and growing assortment of instrument concepts devised to isolate the light of an exoplanet from its host star and acquire its spectrum. Stellar coronagraphs descended from Bernard Lyot’s invention represent a major component of this effort, complementing and intersecting the innovations in interferometry, adaptive optics, wavefront control, and data processing. With these tools in place, several exoplanet imaging programs at large ground-based observatories are underway.^{1}^{–}^{3} Their observations have led to discoveries and astrophysical measurements that are steering theories of planet formation, planetary system evolution, and planetary atmospheres.^{4}^{–}^{7} Meanwhile, laboratory testbeds are setting the stage for yet more ambitious instruments on new space telescopes.^{8}^{–}^{11}

A coronagraph alters the point spread function (PSF) of a telescope so that a region of the image normally dominated by starlight is darkened by destructive interference, enabling observations of faint surrounding structures and companions. Starlight cancellation is accomplished with a group of optical elements that operate on the complex field of the propagating beam. The classical Lyot coronagraph functions with a pair of simple masks: one an opaque occulting spot at the focus and second a Lyot stop to block the outer edge of the recollimated on-axis beam before it is reimaged.^{12} To take advantage of the diffraction-limited imaging capabilities of high-order adaptive optics (AO) systems, beginning in the 1990s, classical Lyot designs were revised for high-contrast stellar coronagraphy.^{13}^{–}^{18} Through Fourier optical analysis and modeling, researchers soon discovered the remarkable performance benefits of apodizing the entrance pupil of a coronagraph.^{19}^{–}^{21} Since then, the transmission profile of this apodizer mask has been a topic of vigorous study.^{22}^{–}^{27} One resulting family of designs, the apodized pupil Lyot coronagraph (APLC), has been successfully integrated with several AO-fed cameras to facilitate deep observations of young exoplanetary systems at near-infrared wavelengths.

As an alternative to a coronagraph with two or more mask planes, pupil apodization by itself is perhaps the simplest and oldest way to reject unwanted starlight from a telescope image.^{28}^{,}^{29} Fraunhofer diffraction theory dictates how any change in the shape or transmission profile of the entrance pupil redistributes a star’s energy in the image plane. This relationship can be used to design an apodizer whose PSF has a zone of high contrast near the star without additional coronagraph masks. This is the shaped pupil approach developed by Kasdin and collaborators, who pioneered the optimization of apodizers with binary-valued transmission.^{30}^{–}^{33} In recent years, shaped pupil solutions have evolved to work around arbitrary two-dimensional telescope apertures, in parallel with similar breakthroughs in APLC design.^{34}^{–}^{39} The relative simplicity of a single mask, however, comes with a sacrifice in how close the dark search region can be pushed toward the star. At the contrast levels relevant to exoplanet imaging, the smallest feasible shaped pupil inner working angle (IWA) is between 3 and $4\lambda /D$.^{33}

Shaped pupil coronagraphs (SPCs) and Lyot coronagraphs (both classical and APLC) both rely on masks that operate strictly on the transmitted amplitude of the propagating beam. Numerous coronagraph designs have been introduced that incorporate phase masks^{40}^{–}^{42} and pupil remapping via aspheric mirrors and/or static deformations.^{43}^{,}^{44} In general, coronagraphs that manipulate phase in addition to amplitude can achieve higher performance in terms of IWA and throughput than SPCs and APLCs. For example, the vector vortex coronagraph has a theoretical inner working limit of $\u223c1\lambda /D$ separation from a star.^{45} Recent theoretical innovations have improved the compatibility of phase-mask and pupil-remapping coronagraph concepts with segmented and obstructed telescope apertures.^{46}^{–}^{49} For broad comparisons between coronagraph design families, see Refs. ^{9}, ^{50}, ^{51}, ^{52}.

Until the past two years, all SPC testbed experiments with wavefront control used freestanding shaped pupil designs with connected obstruction patterns. In particular, experiments in Princeton’s High Contrast Imaging Laboratory^{53} and the High Contrast Imaging Testbed (HCIT)^{54} at the Jet Propulsion Laboratory (JPL) have used ripple-style SPC masks along with two deformable mirrors in series.^{55}^{–}^{57} In this issue, Cady et al.^{58} report the first experimental results with a nonfreestanding SPC design. This mask was fabricated on a silicon wafer substrate with aluminized reflective regions and highly absorptive black silicon regions; the fabrication process is described in detail in this issue by Balasubramanian et al.^{59} Cady et al. used a single deformable mirror in their experiments to create a single-sided dark hole from 4.4 to $11\lambda /D$ in a 52-deg wedge.^{58} They achieved $5.9\xd710\u22129$ contrast in a 2% bandwidth about 550 nm and $9.1\xd710\u22128$ contrast in a 10% bandwidth about 550 nm.

The Science Definition Team of NASA’s Wide-Field Infrared Survey Telescope–Astrophysics Focused Telescope Assets (WFIRST-AFTA) mission has proposed including a coronagraph instrument (CGI) to observe super-Earth and gas-giant exoplanets in reflected starlight at visible wavelengths.^{60} Coronagraph designs for WFIRST-AFTA must be compatible with its heavily obscured telescope aperture, broad filter bands, and rapid development timeline. The SPC, recognized to match these demands, was selected as one of the two baseline coronagraph technologies to undergo extensive testing at JPL in advance of the mission formulation.^{61} The method under development in parallel with the SPC is the hybrid Lyot coronagraph (HLC).^{62}^{,}^{63} The HLC departs from the classical Lyot approach by using a focal plane mask (FPM) with a complex transmission profile.^{64} The SPC and HLC can share the same optical path and wavefront control system. A third coronagraph type, the phase-induced amplitude apodization complex mask coronagraph, is being pursued in parallel as a backup option.^{48}^{,}^{65}^{–}^{67}

In the course of our efforts to improve the SPC designs already meeting the minimum performance goals, we investigate a hybrid coronagraph architecture in which a binary shaped pupil functions as the apodizer mask in an APLC-like configuration.^{68}^{,}^{69} In effect, this expands on the idea first put forward by Cady et al.,^{70} who designed a hard-edged, star-shaped apodizer for the Gemini Planet Imager’s APLC. We identify this design category as the shaped pupil Lyot coronagraph (SPLC). The SPLC offers a persuasive union of the virtues of SPC and APLC: a binary apodizer with achromatic transmission properties and promising fabrication avenues,^{59}^{,}^{71}^{,}^{72} and the relatively small IWA and robustness to aberrations of an APLC.^{73}

Although coronagraph designs for obscured apertures are the ones of highest practical interest and relevance to WFIRST-AFTA and the general community, a clear circular telescope aperture offers a natural starting point to understand how the SPLC relates functionally to the conventional APLC. For example, circular symmetry simplifies the analytical formulation, the numerical optimization problem, and the interpretation. The same qualitative relationships that occur for a simple aperture will reappear for more complicated cases [e.g., SP apodizer feature size and outer working angle (OWA)]. Furthermore, the clear circular aperture allows us to probe the limitations of pure amplitude Lyot coronagraphy, offering useful insights for exoplanet imaging mission design studies such as the recent Exo-C.^{74}

For each of our numerical SPLC experiments, we consider two forms of FPM, illustrated in Fig. 1: first, the occulting spot of the conventional APLC, with radius $\rho 0$; and second, an annular diaphragm with inner radius $\rho 0$ and outer radius $\rho 1$. In our descriptions of the on-axis field propagation, we will make use of the complement of the FPM transmission function. For the spot and diaphragm FPM cases, we label these as $Ma$ and $Mb$, respectively. In terms of the radial spatial coordinate, $\rho $, they are defined as

To find apodizer solutions for the circular SPLC, we use the same numerical optimization tools previously applied to shaped pupil mask designs. In addition to the two types of FPM above, we consider two different planes of field cancellation constraints, as diagrammed in Fig. 2. The results of all the circular SPLC trials are later summarized in Table 1. Details about our optimization method, including discrete algebraic models for the on-axis field propagation and definitions of the linear program objectives and constraints, are given in Appendix A1.

**F2 :**

A representative subset of the optimization schemes we apply to the circular aperture Lyot coronagraph. From left to right, the critical mask planes are (a) the apodizer, (b) the FPM, (c) the Lyot stop, and (d) the final image. The red (dashed) regions are where the on-axis field is constrained. The gold (solid) segments mark the masks that are optimized as free variables.

Config. | Focal plane mask (FPM) | Lyot stop | Field constraint | Bandwidth | Throughput | PSF area | Notes |
---|---|---|---|---|---|---|---|

Ia | Spot $\rho 0=1.87$ | Fixed $O.D.=1$ | $|\Psi C|\u226410\u22123$ | Mono | 0.134 | 1.84 | APLC for $\Lambda =0.999$ |

Ib | Ann $[\rho 0=1.87,\rho 1=12]$ | Fixed $O.D.=1$ | $|\Psi C|\u226410\u22123$ | Mono | 0.148 | 1.73 | 8-ring SP |

IIa | Spot $\rho 0=3$ | Fixed $O.D.=1$ | $10\u22129$ contrast, $3<\rho <12$ | 10% | 0.095 | 2.42 | 9-ring SP |

IIb | Ann $[\rho 0=3,\rho 1=12]$ | Fixed $O.D.=1$ | $10\u22129$ contrast, $3<\rho <12$ | 10% | 0.108 | 2.18 | 8-ring SP |

IIIa | Spot $\rho 0=3$ | Free mask | $10\u22129$ contrast, $3<\rho <12$ | 10% | 0.334 | 1.25 | 10-ring SP, 3-ring LS |

IIIb | Ann $[\rho 0=3,\rho 1=12]$ | Free mask | $10\u22129$ contrast, $3<\rho <12$ | 10% | 0.144 | 1.39 | 7-ring SP, 7-ring LS |

IVa | Spot $\rho 0=3$ | Fixed $[I.D.=0.1,O.D.=0.9]$ | $10\u22129$ contrast, $3<\rho <12$ | 10% | 0.317 | 1.32 | 8-ring SP |

IVb | Ann $[\rho 0=3,\rho 1=12]$ | Fixed $[I.D.=0.1,O.D.=0.9]$ | $10\u22129$ contrast, $3<\rho <12$ | 10% | 0.121 | 1.93 | 8-ring SP |

Loosely following the nomenclature that Soummer et al. formulated in Ref. ^{21}, we represent the scalar electric field in the entrance pupil, focal plane, and Lyot plane, respectively, by $\Psi A$, $\Psi B$, and $\Psi C$. In a slight departure, we define the focal plane radial coordinate $\rho $ in units of image resolution elements ($f\lambda /D$) and the radial coordinate in the two conjugate pupil planes as $r$, normalized to the aperture diameter $D$. For brevity, we implicitly apply the pupil cutoff function in all instances of $\Psi A$ and $\Psi C$, and we set $D$ to 1. These provisions allow us to succinctly express the on-axis scalar electric field in the Lyot plane after the occulting spot, in accordance with the Babinet principle:

Setting the condition for total field cancellation, $\Psi C(r)=0$, leads to an integral equation of the variable function $\Psi A(r)$. In Ref. ^{21}, Soummer et al. showed that the approximate solutions are a subset of Slepian’s circular prolate spheroidal wave functions, originally published four decades prior.^{75} The zero-order prolate spheroidal wave functions possess two exceptional apodization properties. First, they are by definition invariant to the finite Fourier transform, so the scalar field in the focal plane after the apodizer is equal to the unrestricted prolate function itself to within a scale factor. Second, the prolate apodizer maximizes the concentration of energy in the focal plane, within a radius set by the eigenvalue $0<\Lambda <1$ of the integral equation. ^{76} Therefore, once a focal plane spot radius $\rho 0$ has been chosen, the apodizer for optimum monochromatic extinction $\Psi A(r)=\Phi \Lambda (r)$ is fully determined. Invoking the finite Hankel invariance property of $\Phi \Lambda (r)$, one can start from Eq. (2) and arrive at a simple expression for the residual on-axis Lyot plane electric field:

^{19}for the opaque occulting spot, the monochromatic on-axis cancellation is never complete because no prolate solution corresponding to $\Lambda =1$ exists.

^{21}But $\Lambda $ is already 0.999 at $\rho 0=1.87\lambda /D$, for example, and it can be made arbitrarily close to unity by further widening the occulting spot at the expense of the IWA.

As a first experiment, we start with the coronagraph model portrayed at the top of Fig. 2, which we label config. Ia. For the clear circular aperture, an occulting focal plane spot of radius $\rho 0$, and a Lyot stop with the same diameter as the aperture, we ask what entrance apodizer results in a monochromatic Lyot field cancellation factor $1\u2212\Lambda $ while maximizing the overall field transmission. Our aim is to independently recover one of the canonical circular prolate apodizers presented in Ref. ^{21}, corresponding to $\Lambda =0.999$ and $\rho =1.87$. We form a linear program relating the discretized apodizer vector to the resulting Lyot field, using a Riemann sum representation of the Hankel transforms on the right-hand side of Eq. (2). For details about this procedure, see Appendix A1.

The resulting discrete apodizer solution array is $ASP(ri)$, where $ri$ is the normalized pupil radius. We check our result by evaluating the integrated energy transmission metric originally tabulated by Soummer et al.: $2\pi \u2211i=1NrriASP2(ri)$. Our integrated energy transmission is 0.193, which is in close agreement with the corresponding value of the analytical prolate solution, 0.190.^{21} A gray-scale map of the apodizer transmission is plotted on the left-hand side of Fig. 3.

**F3 :**

Circular Lyot coronagraph, config. Ia. The (a) apodizer is optimized for maximum transmission while achieving a monochromatic Lyot plane cancellation factor of $10\u22123$, with an occulting focal plane spot of radius $1.87\lambda 0/D$. The Lyot stop is fixed to the diameter of the reimaged telescope pupil. In the right-hand plot (b), the algebraic components of the Lyot plane field ($\Psi C$) are compared, namely, Eq. (4). Due to the Fourier transform invariance property of the prolate-apodized field ($\Psi A$), the filtered subtrahend component [$\Psi A*jinc(\rho 0r)$] traces the same profile within the radius of the stop, constraining the difference ($\Psi C$) near zero.

We decompose the two algebraic components of the Lyot plane field to learn how the design constraints are fulfilled. In the upper right plot of Fig. 3; the apodizer curve is drawn in blue, followed by the Hankel transform of the field inside the occulting spot in gold. Recall that the latter curve is the function subtracted in Eq. (2) to compute the resulting Lyot field. The invariance of the apodizer to the finite Hankel transform is evident by the fact that within the aperture $r<D/2$, the subtrahend curve is indistinguishable from the apodizer transmission. Outside $r=D/2$, the subtrahend remains continuous since it recovers the unrestricted prolate function. The difference of the two functions reveals a slight deviation from the analytical solution. The residual Lyot field is not shaped like the circular prolate function, as prescribed by Eq. (3). Recall, however, that we did not specify a point-wise constraint in the Lyot plane, instead imposing a less stringent requirement that $|\Psi C(r)|<1\u2212\Lambda $ for $r<D/2$.

We express the on-axis Lyot field in terms of the apodizer transmission and the FPM profile, for both FPM types. To offer a slightly more intuitive description than Eq. (2), instead of explicitly writing out the Hankel transform integrals, this time we express the Lyot plane field components in terms of pupil-plane convolutions:

^{77}In both configurations, it serves as a low-pass filter kernel on the entrance pupil field.

Equation (5) does not yield the same form of integral equation as before, since the apodizer function appears inside an integral in both terms. Therefore, the original framework for the approximate analytical solution no longer applies. In spite of this, we show now that a similar cancellation of the on-axis Lyot field is easily achievable with the annular FPM.

For config. Ib (Fig. 4), we repeat the same problem as Ia, except that now we replace the occulting spot with the annular diaphragm [as defined in the second part of Eq. (1)]. The inner and outer radii are $1.87\lambda 0/D$ and $12\lambda 0/D$, respectively. The resulting solution no longer resembles a circular prolate function, but instead a concentric ring shaped pupil mask of the kind previously described by Vanderbei et al.^{32} By blocking the outer region of the focal plane, the solver is able to take advantage of a mask whose Bessel harmonics lie outside $\rho 1$, because energy distributed there can no longer propagate on to the Lyot plane. The spatial frequencies of the strong Bessel harmonics of the concentric ring mask depend on the ring spacing and thickness. As a consequence, when we repeat the trial for larger $\rho 1$, the number of rings increases, while the overall open area decreases slightly. Conversely, when $\rho 1$ is reduced, the apodizer solution has fewer, thicker rings and higher transmission.

**F4 :**

Circular Lyot coronagraph, config. Ib. The (a) apodizer is optimized for maximum transmission while achieving a monochromatic Lyot plane cancellation factor of $10\u22123$, with an annular diaphragm FPM of inner radius $1.87\lambda 0/D$ and outer radius $12\lambda 0/D$. The Lyot stop is fixed to the diameter of the reimaged telescope pupil. In the right-hand plot (b), the algebraic components of the Lyot plane field ($\Psi C$) are compared, namelyg Eq. (5). In the image domain, $\Psi A$ concentrates most energy within $\rho 0$ (as shown in Fig. 5). Therefore, both low-pass filtered instances of the ring-apodized field [$\Psi A\xd7jinc(\rho 0r)$ and $\Psi A\xd7jinc(\rho 1r)$] are approximately equal, and the residual difference meets the design constraints inside the Lyot stop.

The plot of the Lyot field decomposition on the right-hand side of Fig. 4 reveals another interesting result. The two components of the Lyot plane field, which we expressed before in Eq. (5) in terms of convolutions between the apodizer transmission and $jinc$ functions, bear a striking resemblance to the original circular prolate function that appeared in config. Ia. Therefore, even though the apodizer is binary, the low-pass filter effect of the $jinc$ convolution recovers a rough approximation of the circular prolate function for both the inner and outer components. The residual ripple shows the two components are equal to within the $10\u22123$ field constraint specified in the design. This could only be the case for an apodizer that concentrates a great fraction of its energy within the inner edge of the annulus. We verify this characteristic in Fig. 5, where the field distributions produced in the first focal plane by the apodizers of Ia and Ib are compared. The ring apodizer (red curve) has a higher overall throughput and, therefore, a higher peak. Right outside the outer FPM edge $\rho 1$, the rejected high-frequency Bessel harmonics of the ring apodizer emerge and continue to oscillate beyond the plotted radius. By comparison, the ripple envelope of the circular prolate focal plane field (blue curve) decreases monotonically out to infinity.

**F5 :**

Comparison of the scalar focal plane fields for apodizer solutions Ia and Ib. The blue curve ($\Psi Ba$) is the on-axis field in the first focal plane after the circular prolate solution shown in Fig. 3. The red curve ($\Psi Bb$) is the focal plane field after the concentric ring mask apodizer of config. Ib (Fig. 4). Notably, the amplitude of $\Psi Bb$ rises immediately outside the outer radius of the annular FPM ($\rho 1=12$), while the ripple envelope of $\Psi Ba$ continues to fall monotonically.

While the previous trials offer a useful conceptual perspective on how binary apodizers can function in a Lyot coronagraph, we are ultimately concerned with image plane performance metrics and solutions that suppress starlight over a finite bandwidth. Therefore, the remaining trials carry the propagation to the final focal plane and constrain the contrast there to $10\u22129$ in a restricted region (details in Appendix A1). We define contrast here as the ratio of the intensity in the final image to the peak of the off-axis coronagraph PSF. Bandwidth is achieved by repeating the field constraints at three wavelength samples spanning a 10% fractional bandwidth.

For all the configurations, we compute the throughput and area of the coronagraph PSF, and assemble the results in Table 1. Following the convention of Krist et al.,^{78} throughput takes into account the overall proportion of energy from an off-axis (planet-like) point source that reaches the final image, as well as the proportion of that energy concentrated in the main lobe of the corresponding PSF. We make the assumption that only in the main lobe of the off-axis PSF is the intensity high enough to generate a useful signal. We compute the throughput by propagating an off-axis plane wave through the coronagraph model, masking off the full-width half-maximum (FWHM) region of the resulting PSF and summing the intensity there. Then we repeat the same calculation when the off-axis source is directly imaged by the telescope without a coronagraph. The ratio of these intensity sums gives a normalized metric indicating how efficiently off-axis point sources are preserved by the coronagraph. For a Lyot coronagraph (including classical, APLC, and SPLC), throughput is approximately constant over the field of view (FoV), as long as the off-axis PSF core clears line-of-sight obstruction by the FPM.

Independent of throughput, we also assess how tightly the energy is concentrated in the central lobe of the off-axis PSF based on the area of the FWHM region. A small PSF area is desirable, because for a given throughput value, a smaller area results in a higher peak signal on the detector. We again normalize this to the reference case of a PSF without a coronagraph. Since these designs have an unobstructed circular pupil, the reference telescope PSF is an Airy disk.

To enable the most meaningful comparison across the various configurations in the table, we relaxed the IWA from $1.87\lambda 0/D$ to $3\lambda 0/D$, where $\lambda 0$ is the center wavelength of the passband. This increase in the inner edge is needed because for some configurations, we failed to find any polychromatic solutions for $\rho 0$ of $2\lambda 0/D$ or below. The outer edge of the high-contrast region is arbitrarily fixed at $12\u2009\lambda 0/D$ for all designs.

The first set of trials with image plane constraints are configs. IIa and IIb, with a fixed Lyot stop again matched to the telescope aperture. Following the previous nomenclature, type “a” designs use the occulting spot FPM, and type “b” designs use the annular diaphragm FPM. For both types of FPM, the apodizer with the highest throughput is a concentric ring shaped pupil (Table 1). Even for the spot FPM, the hard outer edge of our specified dark region means that the strong Bessel harmonics of the ring apodizer are tolerated outside $\rho 1=12\u2009\lambda 0/D$. If we had constrained the contrast out to an infinite radius from the star, we would instead expect the solution to revert to a smooth apodizer with a continuous derivative. More practically, we could have included derivative constraints in the optimization program.^{33}^{,}^{79} In a recent APLC design study, for example, the contrast constraints were also imposed over a restricted area of the final image.^{27} The authors, requiring a smooth apodizer transmission profile, added constraints on the spatial derivative of the apodizer in order to avoid binary solutions.

For the conventional monochromatic APLC, the optimal Lyot stop is one exactly matched to the telescope aperture (after a 180 deg rotation, for telescope apertures lacking circular symmetry).^{25} The Lyot stop is padded only for the purpose of alignment tolerance.^{26} However, recent investigations have shown that APLC optimizations incorporating bandwidth and image constraints yield better results when the Lyot stop’s central obstruction replica is significantly oversized.^{27} For example, in the course of optimizing an APLC for an aperture with central obstruction of diameter $0.14D$, aiming for contrast $10\u22128$ over a 10% bandwidth, N’Diaye et al. found that increasing the inner diameter of the Lyot stop to $\u223c0.35D$ enabled the IWA to be reduced from $3.7\u2009\lambda 0/D$ to $2.4\u2009\lambda 0/D$ at the same throughput.^{27} Evidently, the transmission profile of the Lyot stop offers an important parameter space to survey in addition to the apodizer.

Building on this notion, for configs. IIIa and IIIb, we recast the Lyot coronagraph optimization as a nonlinear program in which both the apodizer and the Lyot stop transmission profile are free vectors, optimized simultaneously. See Appendix A1 for further description of this procedure. The program seeks to maximize the sum of the transmission of both apodizer and Lyot stop, given the same contrast and 10% bandwidth goal as before. The results are illustrated in Figs. 67 and 8 and listed in Table 1. As in the case of config. IIa (not plotted), the mismatch between the apodizer and the Babinet subtrahend profiles leads to high amplitude, sharp residual features in the Lyot plane. This time, however, a subtle rearrangement of Lyot stop obstructions is enough to enable an apodizer with far more open area. Most of the sharp residual Lyot plane features are not obstructed by the freely varying stop, contrary to what one might expect. Apparently, not even a modest level of field cancellation in the Lyot plane is required to create deep, broadband destructive interference in the image plane. The FWHM throughput of this solution is 0.33, more than triple that of the comparable clear Lyot stop configuration (IIa). The coronagraph PSF also sharpens, giving an FWHM area only 25% larger than the Airy disk. We also tested the effect of decreasing the focal occulting spot radius from $3\u2009\lambda 0/D$ to $2\u2009\lambda 0/D$ and arrived at a similar design with a throughput of 17%. The contrast curve of this design is plotted in Fig. 7, showing the intensity pattern at three wavelengths, as well as the average over five wavelength samples spanning the 10% passband.

**F6 :**

Circular aperture Lyot coronagraph, config. IIIa. The (a) apodizer and (b) Lyot stop are jointly optimized for maximum transmission while achieving a contrast of $10\u22129$ in the image plane over a 10% passband over a working angle range of 2 to $12\lambda 0/D$. The FPM is an occulting spot of radius $2\u2009\lambda 0/D$. The right-hand plot (c) shows that for this configuration, the solution depends on high-amplitude discontinuities in the Lyot plane field.

**F7 :**

On-axis intensity pattern for a circular shaped pupil Lyot coronagraph (SPLC) with a jointly optimized apodizer and Lyot stop and a working angle range of 2 to $12\lambda 0/D$. The nonlinear optimization program constrains the contrast at three wavelengths, resulting in a pseudo-achromatized dark search region fixed in sky coordinates. The black curve shows the average intensity across five wavelength samples spanning the 10% passband.

**F8 :**

Circular aperture Lyot coronagraph, config. IIIb. The (a) apodizer and (b) Lyot stop are jointly optimized for maximum transmission while achieving a contrast of $10\u22129$ in the image plane over a 10% passband over a working angle range of 2 to $12\lambda 0/D$. The FPM is an annular diaphragm with an inner radius of $2\lambda 0/D$ and outer radius of $12\lambda 0/D$. The Lyot plane field illustrated in the right-hand plot (c) differs from the case of config. IIIa in two ways: (1) it varies smoothly, due to the filtering effect of the annular FPM, and (2) field nulls coincide with gaps in the Lyot stop.

With the annular diaphragm FPM, allowing the Lyot stop transmission profile to vary also results in increased throughput, although the improvement here is less dramatic, climbing from 0.108 to 0.144 (Table 1). The apodizer and Lyot stop both have less open area than the spot FPM variant. Notably, however, the PSF is almost as sharp as the occulting spot variant, with FWHM area 39% larger than the Airy disk. The plot in Fig. 8 of the Lyot plane field alongside the Lyot stop transmission profile shows that the performance of this configuration benefits from notching out the radial peaks, which was not the case for the spot FPM. The resulting Lyot stop has five prominent opaque rings and one small dark spot at the center. Another important aspect of the Lyot plane behavior for the diaphragm FPM is the relative smoothness of the field structure as compared to the spot FPM case. This is a direct outcome of the mathematical description of the Lyot field in Eq. (5), where both instances of the apodizer transmission function are convolved with a $jinc$ function. This quality of the diaphragm FPM variant of the SPLC hints at a more generous tolerance to manufacturing and alignment. We revisit this point in Sec. 4.3, in the context of our WFIRST-AFTA designs.

From further experiments, we found that the nonlinear, nonconvex program used to derive joint shaped pupil and Lyot stop solutions only converges for one-dimensional coronagraph models. In our circular aperture case, this two-plane optimization program operates near the limit of the interior point solver’s capability, and reliable outcomes require tuning. Even for low spatial resolution versions of obstructed two-dimensional apertures, there are too many variables to extend the tactic. This obstacle is algorithmic in nature rather than one that can be surmounted by expanding the computing hardware capacity. This difficulty, combined with the practical attractions of a simpler Lyot stop, suggest one might settle for an intermediate performance level by surveying an annular Lyot stop described by only two parameters (inner and outer radius). We have not yet explored the full range of inner and outer diameter Lyot stop combinations for the circular aperture. However, we found that for an arbitrary test design with a $0.1D$ inner diameter and $0.9D$ outer diameter (configs. IVa and IVb), performance is not far from the optimized Lyot stop: for the case of the spot FPM, throughput decreases only from 0.334 to 0.317 (Table 1). For the diaphragm FPM variant, the throughput loss resulting from the switch to the annular Lyot stop is also small. However, the coronagraph PSF deteriorates significantly, jumping in area from 1.39 to 1.93 times that of the Airy core.

###### Distinction between the Shaped Pupil Lyot Coronagraph and Microdot Realizations of the Apodized Pupil Lyot Coronagraph

Microdot lithography can be used to stochastically approximate the continuous prolate apodizer solutions derived from Eqs. (2) and (3), as well as their analogs for more complicated telescope apertures.^{80}^{–}^{82} The technique stems from long-established printing processes, in which an array of black pixels with varying spatial density imitates the halftones of a gray-scale image. A microdot apodizer for a Lyot coronagraph can be manufactured with an opaque metal layer deposited on a glass substrate at the locations of black pixels.^{83}^{,}^{84} In testbed experiments, APLC designs with microdot apodizers have reached contrasts as low as $5\xd710\u22127$.^{85} Microdot APLC apodizers are core components in several on-sky, AO-fed coronagraphs.^{1}^{–}^{3}

Although the halftone microdot process results in a binary-valued transmission pattern, there is nonetheless a categorical distinction from the SPLC. A shaped pupil, rather than approximating a continuous mask solution in the apodizer plane, instead matches the desired destructive interference properties in the image plane. Consequently, on a macroscopic scale, the ring apodizer shown in Fig. 4 is qualitatively dissimilar to a halftone APLC approximation, despite solving a similar field cancellation problem. Instead, the image domain is where the strongest resemblance appears between the SPLC and APLC solutions. This is made evident by comparing their on-axis field distributions at the first focal plane within the bounded search region, shown in Fig. 5 for the most elementary design case (monochromatic cancellation in the Lyot plane).

Because the SPLC design process directly optimizes the performance, the fabrication instruction set for the apodizer realization is a one-to-one replica of the linear program solution. A microdot APLC apodizer, on the other hand, is one step removed from an underlying numerical solution. In this sense, the shaped pupil technique has a clear advantage for meeting the high precision required for the most demanding applications.

CGI proposed by the WFIRST-AFTA Science Definition Team aims to image and measure the spectra of mature, long-period gas giants in the solar neighborhood. This planet population, which at present can only be studied indirectly through radial velocity (RV) surveys, is out of reach of transit spectroscopy methods due to their strong bias toward highly irradiated planets on short orbital periods. Depending on orbital configuration and albedo characteristics, the planet-to-star contrast of an exo-Jupiter seen in reflected starlight is of the order of $10\u22128$ or below. Due to AO performance limitations, this contrast ratio may prove too extreme for ground-based imaging, regardless of telescope aperture or coronagraph design.^{86}

The reflected spectra of gas giants are sculpted by a series of methane absorption bands in the range of 600 to 970 nm. Acquiring these fingerprints for an ensemble of planets, in conjunction with mass constraints from radial velocities and astrometry, will provide a wealth of insights into the structure, composition, and evolution of gas giants.^{87}^{,}^{88} The Princeton team was tasked with providing shaped pupil designs for this characterization mode, covering the stated wavelength range in three 18% passbands, each corresponding to one filter setting of the integral field spectrograph (IFS).^{89}^{–}^{91}

The optical path of the proposed CGI is shared between SPC/SPLC and JPL’s HLC.^{62} The HLC uses an FPM with a phase- and amplitude-modulating transmission profile.^{64} In the baseline configuration of the CGI, the HLC mode operates with two imaging filters, nominally 10% bandpasses centered at 465 and 565 nm. The HLC mode is optimized for detection and color measurements of the scattered continuum, rather than spectroscopic characterization with the wider bandpass of the IFS.^{91}

In addition to exoplanets, a closely related category of scientific opportunity for WFIRST-AFTA is circumstellar debris structure. One of the goals of the CGI will be to image scattered light from low-density, solar-system-like zodiacal disks that are below the noise floor of existing instruments. In addition, thick debris disks of the kind already studied with the Hubble Space Telescope will be probed at smaller angular separations than before. This will unveil the dynamic evolution of circumstellar debris and its interaction with planets in the habitable zones of exoplanetary systems.^{92} Small angular separation observations of debris disks can be carried out with the HLC mode. However, some of the foreseen disk imaging programs require larger OWAs ($>\u223c0.5$ arc second) than those relevant to reflected starlight exoplanet detection. Therefore, we explored separate SPLC mask solutions for a dedicated, wide-field disk science coronagraph mode.

Throughout our design process, we concentrate on three essential performance metrics: contrast, IWA, and throughput. The scientific goals require all WFIRST-AFTA designs to achieve a raw contrast of $10\u22128$, defined at a given image position as the ratio of diffracted starlight intensity to the peak of the off-axis coronagraph PSF shifted to that location. We make the assumption that data postprocessing will further reduce the intensity floor, so that planets several times below this nominal contrast can be detected.^{78}^{,}^{89}^{,}^{91}^{,}^{93}

IWA is defined as the minimum angular separation from the star at which the coronagraph’s off-axis (planet) PSF core throughput reaches half-maximum.^{78} For a Lyot coronagraph, planet throughput rises steadily with increasing angular separation from the edge of the FPM, leveling off when the core of the PSF clears the line-of-sight FPM occultation. Having a small IWA is especially important for a coronagraph aiming to detect starlight reflections, because the irradiance falls off with the square of the planet–star distance. The consequence—when considered along with the distances to nearby FGK stars and their expected distributions of planet semimajor axes—is twofold: (1) the number of accessible planets rises steeply with reduced IWA and (2) those giant exoplanets at smaller angular separations tend to be the brightest targets.^{91}^{,}^{94}

As in Sec. 2, we define throughput as the ratio of energy contained within the FWHM contour of the PSF core to that of the telescope PSF with no coronagraph. Planet signal-to-noise ratios will generally be low, and the number of targets the instrument can acquire over the mission lifespan will be limited by the cumulative integration times.^{91} Detection times will depend on the total amount of planet light that survives propagation losses through the optical train and how tightly that remaining energy is concentrated on the detector. In characterization mode, the spectrograph will disperse the planet’s light over many detector pixels. Therefore, the majority of the instrument’s operational time budget will be consumed by integrations totaling one day or more per target.^{78}

The 2.4 m-diameter WFIRST-AFTA telescope aperture is illustrated in Fig. 9. Its large central obstruction ($0.31D$) and six off-center support struts, each oriented at a unique angle, pose a challenge for any coronagraph design. The SPC and SPLC use the apodization pattern of the shaped pupil to confine the diffraction effects of these obstructions outside of the optimized dark region.^{37}^{,}^{38} The penalties of this strategy are lower throughput and higher IWA than would be the case for a clear circular aperture. Alternatively, it is possible for a coronagraph to counteract these obstructions with static phase excursions applied to deformable mirrors^{44}^{,}^{62} or custom aspheric optics.^{43}^{,}^{65}^{,}^{66}^{,}^{95} However, amplitude-mask-based apodization places less demanding requirements on mirror surface manufacturing tolerances, alignment tolerances, and deformable mirror reliability, thereby mitigating the overall technological risk of our design.

The SPLC designs presented here build directly on the efforts of Carlotti et al.,^{38} who led the first shaped pupil designs for WFIRST-AFTA. Those first-generation shaped pupil coronagraphs fulfill the basic mission requirements. They were described in further detail by Riggs et al.,^{68} and in this issue, Cady et al. describe successful laboratory demonstrations of the first-generation characterization SPC design.^{58} The SPLC designs here form part of a reference design case adopted by the Science Definition Team for the purpose of technology demonstrations, mission simulations, and cost assessment.^{60} The flight design moving forward may differ significantly.

For the characterization design, the challenge is to achieve a small IWA while maintaining acceptable throughput. Although it is always desirable to create a full 360-deg dark search region around the star, we know from previous work that it is impossible for a shaped pupil alone to produce an annular FoV with IWA of $4\lambda /D$ or below with the obscurations of the WFIRST-AFTA aperture.^{38} However, knowing that the SPLC configuration should be able to reach a smaller IWA at the same contrast and throughput as the first-generation SPC, we now examine again how close we can push a 360-deg dark region in toward the star. At the same time, for effective broadband characterization, we strongly prefer a quasi-achromatic dark region,^{26} so that a target located near the IWA is detected across the full spectrograph passband. Therefore, for our parameter exploration, we always apply polychromatic image constraints, with inner and outer image radii defined in terms of central wavelength diffraction elements ($\lambda 0/D$), as we did before in Sec. 2.3, and similar to previous APLC optimizations described by N’Diaye et al.^{27} In Appendix A2, we describe the practical details of the optimization procedure used to test a given set of design parameters.

Informed by the results of our circular SPLC trials (Secs. 2.3 and 2.4), we use an occulting spot FPM, with radius either $\rho 0=2.5$ or $3.0\lambda 0/D$. We surveyed the Lyot stop parameter space by repeating optimizations with different padding levels on the inner and outer edges of the telescope aperture replica, ranging from 2 to 12% of the diameter. We also varied the outer radius of the dark region between 8, 9, and $10\lambda 0/D$. Finally, we repeated these multiparameter trials for two bandwidths: 18% (the target characterization design) and 10%. A subset of the results is summarized in Table 2. Several conclusions can be drawn. First, for the smaller inner FPM radius $\rho 0=2.5\u2009\lambda 0/D$, there are no acceptable 360-deg SPLC solutions for the full 18% characterization bandwidth. At $\rho 0=3.0\lambda /D$, however, some weak solutions begin to appear. Performance here is sensitive to the Lyot stop padding level, and for the 18% bandwidth case, the best padding levels are in the range of 8 to 10% of the pupil diameter. When the bandwidth is reduced to 10%, the improvement in throughput is dramatic. In particular, we highlight a design with a throughput of 0.14 and FPM radius of $3.0\lambda 0/D$. We did not find a strong dependence on outer dark region radius $\rho 1$ over the values we surveyed.

18% bandwidth $\rho 0$ ($\lambda 0/D$) | 10% bandwidth $\rho 0$ ($\lambda 0/D$) | |||
---|---|---|---|---|

Lyot stop padding (% diam.) | 2.5 | 3.0 | 2.5 | 3.0 |

6 | $<10\u22123$ | 0.042 | 0.040 | 0.13 |

8 | $<10\u22123$ | 0.057 | 0.060 | 0.14 |

10 | $<10\u22123$ | 0.066 | 0.062 | 0.12 |

12 | $<10\u22123$ | 0.056 | 0.043 | 0.07 |

To reach an IWA smaller than $3\lambda 0/D$, and do so over the full characterization bandwidth, we need to restrict the azimuthal span of the constrained dark region. This strategy was originally developed to design the first-generation WFIRST-AFTA SPCs, resulting in a design with $\rho 0=4\lambda /D$. In a survey aiming to discover exoplanets, bowtie-shaped dark zones have the disadvantage of requiring repeat integrations for two or more mask orientations. However, the overhead for characterizing a planet with a known position is minor. Furthermore, restricting the azimuthal and radial FoV of the image plane search area alleviates the demands placed on the wavefront correction. When the wavefront control system aims to suppress light only in a small region, there are more degrees of freedom available than when trying to suppress the full correctable region. Restricting the dark hole problem thus leaves greater tolerance for unknown aberrations in the propagation model.

For the SPLC configuration, we surveyed a range of bowtie-shaped focal plane geometries with inner radii between $2.4\lambda 0/D$ and $3.0\lambda 0/D$, and opening angles between 30 and 90 deg. The trials are repeated for 18 and 10% bandwidths. All optimization attempts at smaller inner radii, such as $2.2\lambda 0/D$, failed to give results with reasonable throughput (above 0.01). Here, instead of an occulting spot, the FPM is a bowtie-shaped aperture matched to the optimized focal plane region in the final image, as illustrated in Fig. 10. The presence of the outer edge in the first focal plane makes this design most analogous to config. IVb, among the circular SPLCs described in Sec. 2 and Table 1.

**F10 :**

Diagram of the characterization-mode SPLC mask scheme, along with the plots of the intensity of the on-axis field at each critical plane. The shaped pupil (a) forms a bowtie-shaped region of the destructive interference in the first focal plane (b), which is then occulted by a diaphragm with a matched opening. The on-axis field is further rejected by an annular stop in the subsequent Lyot plane (c) before it is reimaged at the entrance of the integral field spectrograph (d). The propagation is shown at the central wavelength of the design for the case of a perfectly flat wavefront with no planet or disk present. The flux scale bars indicate the intensity on a $log10$ scale. In the first focal plane (b), the flux scale is normalized to the point spread function (PSF) peak, whereas in the final focal plane (d), the scale is normalized to the peak of the unseen off-axis PSF, in order to map the contrast ratio in a way that accounts for the Lyot stop attenuation. The mean contrast in the dark bowtie region (averaged over azimuth angle, then averaged over wavelength, and then averaged over radial separation) is $6\xd710\u22129$.

The Lyot stop we use for the bowtie characterization design is a simple clear annulus rather than a padded replica of the telescope aperture (Fig. 10). That is because the low-pass filter effect of the bowtie FPM smears the support strut field features in the Lyot plane. We verified through separate optimization tests that there is no advantage to including matched support struts in the Lyot stop in this configuration. To survey the dependence of throughput on focal plane geometry, we fix the inner diameter of the Lyot stop annulus at $0.3D$ and the outer diameter at $0.9D$. Later, we tune the inner and outer diameters of the Lyot stop for a specific characterization design.

Some results from the focal plane geometry trials are collected in Table 3. Since we found that throughput depends relatively weakly on outer dark region radius, here we only tabulate the throughput values for $\rho 1=9\lambda 0/D$. Throughput varies steeply with opening angle. In particular, from 60 to 90 deg, the throughput decreases by a factor of $\u223c4$ to 5. At an inner radius of $2.4\lambda 0/D$, the only opening angle with throughput above 0.1 is the 30-deg bowtie.

Opening angle (deg) | |||||
---|---|---|---|---|---|

$\rho 0$ ($\lambda 0/D$) | 30 | 45 | 60 | 75 | 90 |

2.4 | 0.14 | 0.089 | 0.048 | 0.026 | 0.010 |

2.6 | 0.16 | 0.17 | 0.11 | 0.049 | 0.022 |

2.8 | 0.16 | 0.18 | 0.13 | 0.067 | 0.030 |

3.0 | 0.17 | 0.18 | 0.15 | 0.088 | 0.042 |

We find the most compelling trade-off at $\rho 0=2.6\lambda 0/D$, which has throughput a of 0.11 at an opening angle of 60 deg. Similar to the first-generation SPC, this enables the full FoV to be covered with three pairs of shaped pupils and FPMs oriented at 120-deg offsets. Due to their limited utility, the 10% bandwidth trials are not tabulated here, but we can summarize them by pointing out that throughput increases by a factor of 1.1 to 3 over the 18% bandwidth case, with the largest changes occurring for small $\rho 0$ and wide opening angle.

A full set of mask designs for a bowtie characterization SPLC have been delivered to JPL for fabrication and experiments on the HCIT. The detailed structure of the $1000\xd71000$ pixel shaped pupil apodizer array is illustrated in Fig. 11.

**F11 :**

Detail of the $1000\xd71000$ point shaped pupil mask solution for the WFIRST-AFTA characterization mode, corresponding to the design exhibited in Figs. 10 and 12. The obscurations of the WFIRST-AFTA telescope aperture are colored blue, the regions of the pupil masked by the shaped pupil apodizer in addition to the telescope pupil are colored green, and the regions of the pupil transmitted by the apodizer are colored yellow. The magnified inset shows the granular quality of the square, binary elements of the shaped pupil array. The inset also shows the gap between the edge of the telescope aperture features and the open regions of the apodizer, which is reserved in order to ease the alignment tolerance between the shaped pupil apodizer and the telescope pupil.

For this version, we raised the opening angle above 60 deg to provide a margin of FoV overlap among the three mask orientations needed to cover an annulus around the star. The overlap slightly reduces the likelihood of a scenario where the location of an exoplanet coincides with the edge of a bowtie mask, cropping the PSF core and requiring extra integration time to compensate. After the opening angle was fixed at 65 deg, we decremented $\rho 0$ from $2.6\lambda 0/D$ to the smallest radius that maintains the throughput above an arbitrary goal of 0.10, thereby revising $\rho 0$ to $2.5\lambda 0/D$. The Lyot stop is an annulus with inner diameter of $0.26D$ and outer diameter