2.1.

Focal Occulting Spot

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 ${\mathrm{\Psi }}_A$, ${\mathrm{\Psi }}_B$, and ${\mathrm{\Psi }}_C$. 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 ${\mathrm{\Psi }}_A$ and ${\mathrm{\Psi }}_C$, 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, ${\mathrm{\Psi }}_C(r)=0$, leads to an integral equation of the variable function ${\mathrm{\Psi }}_A(r)$. 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 $0<\mathrm{\Lambda }<1$ of the integral equation. ⁷⁶ Therefore, once a focal plane spot radius ${\rho }_0$ has been chosen, the apodizer for optimum monochromatic extinction ${\mathrm{\Psi }}_A(r)={\mathrm{\Phi }}_{\mathrm{\Lambda }}(r)$ is fully determined. Invoking the finite Hankel invariance property of ${\mathrm{\Phi }}_{\mathrm{\Lambda }}(r)$, one can start from Eq. (2) and arrive at a simple expression for the residual on-axis Lyot plane electric field:

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 $1-\mathrm{\Lambda }$ 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 $\mathrm{\Lambda }=0.999$ 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 ${\mathcal{A}}_{\mathrm{SP}}(r_i)$, where $r_i$ is the normalized pupil radius. We check our result by evaluating the integrated energy transmission metric originally tabulated by Soummer et al.: $2\pi \sum _{i=1}^{N_r}{r}_i{\mathcal{A}}_{\mathrm{SP}}^2(r_i)$. 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.

Fig. 3

Circular Lyot coronagraph, config. Ia. The (a) apodizer is optimized for maximum transmission while achieving a monochromatic Lyot plane cancellation factor of ${10}^{-3}$, 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 (${\mathrm{\Psi }}_C$) are compared, namely, Eq. (4). Due to the Fourier transform invariance property of the prolate-apodized field (${\mathrm{\Psi }}_A$), the filtered subtrahend component [${\mathrm{\Psi }}_A*\mathrm{jinc}({\rho }_0\mathrm{r})$] traces the same profile within the radius of the stop, constraining the difference (${\mathrm{\Psi }}_C$) 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 $|{\mathrm{\Psi }}_C(r)|<1-\mathrm{\Lambda }$ for $r<D/2$.

2.2.

Focal Diaphragm

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:

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.

Fig. 4

Circular Lyot coronagraph, config. Ib. The (a) apodizer is optimized for maximum transmission while achieving a monochromatic Lyot plane cancellation factor of ${10}^{-3}$, 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 (${\mathrm{\Psi }}_C$) are compared, namelyg Eq. (5). In the image domain, ${\mathrm{\Psi }}_A$ concentrates most energy within ${\rho }_0$ (as shown in Fig. 5). Therefore, both low-pass filtered instances of the ring-apodized field [${\mathrm{\Psi }}_A\times \mathrm{jinc}({\rho }_0\mathrm{r})$ and ${\mathrm{\Psi }}_A\times \mathrm{jinc}({\rho }_1\mathrm{r})$] 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 $\mathrm{jinc}$ 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 $\mathrm{jinc}$ 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 ${10}^{-3}$ 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.

Fig. 5

Comparison of the scalar focal plane fields for apodizer solutions Ia and Ib. The blue curve (${\mathrm{\Psi }}_{B_{\mathrm{a}}}$) is the on-axis field in the first focal plane after the circular prolate solution shown in Fig. 3. The red curve (${\mathrm{\Psi }}_{B_{\mathrm{b}}}$) is the focal plane field after the concentric ring mask apodizer of config. Ib (Fig. 4). Notably, the amplitude of ${\mathrm{\Psi }}_{B_{\mathrm{b}}}$ rises immediately outside the outer radius of the annular FPM (${\rho }_1=12$), while the ripple envelope of ${\mathrm{\Psi }}_{B_{\mathrm{a}}}$ continues to fall monotonically.

2.3.

Polychromatic Focal Plane Field Cancellation

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 ${10}^{-9}$ 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 $12\hspace{0.17em}{\lambda }_0/D$ 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 ${\rho }_1=12\hspace{0.17em}{\lambda }_0/D$. 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.

2.4.

Joint Optimization of the Apodizer and Lyot Stop

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 ${10}^{-8}$ over a 10% bandwidth, N’Diaye et al. found that increasing the inner diameter of the Lyot stop to $\sim 0.35D$ enabled the IWA to be reduced from $3.7\hspace{0.17em}{\lambda }_0/D$ to $2.4\hspace{0.17em}{\lambda }_0/D$ 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. 6Fig. 7 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 $3\hspace{0.17em}{\lambda }_0/D$ to $2\hspace{0.17em}{\lambda }_0/D$ 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.

Fig. 6

Circular aperture Lyot coronagraph, config. IIIa. The (a) apodizer and (b) Lyot stop are jointly optimized for maximum transmission while achieving a contrast of ${10}^{-9}$ 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 $2\hspace{0.17em}{\lambda }_0/D$. The right-hand plot (c) shows that for this configuration, the solution depends on high-amplitude discontinuities in the Lyot plane field.

Fig. 7

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.

Fig. 8

Circular aperture Lyot coronagraph, config. IIIb. The (a) apodizer and (b) Lyot stop are jointly optimized for maximum transmission while achieving a contrast of ${10}^{-9}$ 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 $\mathrm{jinc}$ 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.

2.5.

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 $5\times {10}^{-7}$.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.

3.1.

WFIRST-AFTA Coronagraph Instrument Concept

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 ${10}^{-8}$ 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 ($>\sim 0.5$ 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 ${10}^{-8}$, 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.

Fig. 9

The WFIRST-AFTA telescope aperture from the most recent mission design cycle, as seen on-axis.

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.

3.2.

Characterization Mode Shaped Pupil Lyot Coronagraph

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 ${\rho }_0=2.5\hspace{0.17em}{\lambda }_0/D$, 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.

Table 2

Throughput of 360-deg field-of-view WFIRST-AFTA shaped pupil Lyot coronagraph (SPLC) solutions for different combinations of inner working angle (IWA), bandwidth, and Lyot stop padding. The outer edge of the dark image constraint region is fixed at ρ1=8λ0/D.

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.

Fig. 10

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 ${\log }_{10}$ 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 $6\times {10}^{-9}$.

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 $\sim 4$ 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.

Table 3

Throughput of bowtie characterization WFIRST-AFTA SPLC solutions for different combinations of inner radius and opening angle. The optimization passband is 18%, the outer radius is fixed at ρ1=9λ/D, and the Lyot stop is a clear annulus with [I.D.,O.D.]=[0.3,0.9]. The approximate trade-off in inner radius, opening angle, and throughput we chose to pursue is highlighted in bold font.

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 $1000\times 1000$ pixel shaped pupil apodizer array is illustrated in Fig. 11.

Fig. 11

Detail of the $1000\times 1000$ 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 of $0.88D$. We stress that these design choices are provisional and that maximizing the scientific yield would require integrating the parameter survey with end-to-end observatory and data simulations.

The ideal model PSF and contrast curves with zero wavefront error are shown in Fig. 12. At center wavelength, the FWHM PSF area is 1.6 times that of the WFIRST-AFTA PSF. The contrast constraint is slightly relaxed relative to the previous parameter trials: $2\times {10}^{-8}$ for separations below $3.5{\lambda }_0/D$ and $1.5\times {10}^{-8}$ in the rest of the bowtie region. Still, the average contrast curve (here averaged over azimuth and then over wavelength) is well below the worst-case intensity values, as plotted in the right-hand side of Fig. 12, below $7\times {10}^{-9}$ at all angular separations in the range of 3 to $8{\lambda }_0/D$.

Fig. 12

(a) Ideal on-axis PSF of the characterization SPLC design, with no aberrations, shown at three wavelengths: the short-wavelength extreme of the passband (top; $\lambda /{\lambda }_0=0.91$), the center wavelength (middle; $\lambda /{\lambda }_0=1.00$), and the long-wavelength extreme (bottom; $\lambda /{\lambda }_0=1.09$). (b) Raw contrast plotted alongside the estimated contrasts and separations of the known long-period radial velocity exoplanet population. The contrast curves are shown for the SPLC design centered on ${\lambda }_0=770\mathrm{nm}$, the middle of the three spectrograph filters. For reference, we also include the contrast curve of the first-generation shaped pupil coronagraph (purple dashed curve).

The two deformable mirrors integrated with the WFIRST-AFTA CGI are expected to improve the nominal SPLC performance. Since the SPLC design optimization only makes use of amplitude operations, the extra degrees of freedom from deformable mirror (DM) phase control can yield higher contrast. To demonstrate this, we simulate the effect of DM control with an unaberrated wavefront. We simulated wavefront control on a layout similar to the actual WFIRST-AFTA CGI with two $48\times 48$ actuator DMs upstream of the SPLC. We divided the 18% passband into nine wavelength samples and weighted each equally to control the dark hole with a stroke minimization algorithm, originally described by Pueyo et al.⁹⁶ The inner region of the bowtie is most critical since more exoplanets are expected to be observed at small angular separations, so we weighted the intensity from 2.5 to $4.5{\lambda }_0/D$, three times higher for a slight improvement. The resulting contrast curve is plotted in the right-hand-side plot of Fig. 12. With DM control, the average intensity in the separation range of 2.5 to $3.5{\lambda }_0/D$ is reduced by a factor of 2, and at separations of 4 to $8{\lambda }_0/D$, by a factor of 4 or more. In addition to the azimuthally averaged contrast, in the same figure, we also plot the standard deviation of the intensity pattern as a function of separation from the star, measured in concentric annuli.

We use the distribution of known RV exoplanets to indicate where the performance of this coronagraph sits relative to plausible characterization targets. We take the same assumptions made by Traub et al.⁹¹ in their science yield calculations, resulting in a representative target population plotted on the right-hand side of Fig. 12. Planets are assumed to be on circular orbits inclined by 60 deg and observed at a favorable mean anomaly of 70 deg. The longest period RV exoplanet detections are generally on the upper end of the mass distribution, so in the absence of other constraints, they are assigned a size equal to Jupiter and a geometric albedo of 0.4. Finally, to compare the SPLC contrast curve on an angular scale, we set the central wavelength of the characterization design to 770 nm, the middle of the three nominal IFS filters.⁹⁰ It can be seen that 12 planets outside of the $2.8{\lambda }_0/D$ IWA have contrasts and separations placing them above the band-averaged contrast floor obtained from the wavefront control simulation.

The first-generation characterization SPC design for WFIRST-AFTA is overplotted as the dashed purple contrast curve in Fig. 12. With an IWA of $4\lambda /D$, that coronagraph could only access half of the exoplanets in the mock target sample. There is an additional disadvantage of the first-generation design that is not apparent in the contrast plot. The IWA of a shaped pupil PSF scales directly with wavelength. Without a Lyot stop, there is no possibility to anchor the inner radius of the dark bowtie region across the spectrograph filter bandpass, as we do to optimize the SPLC. Therefore, an exoplanet falling near the inner edge of the first-generation contrast curve would be undetected at the long-wavelength end of the filter.

We acknowledge that the raw contrast prediction for the characterization SPLC is optimistic since it does not include aberrations, tip-tilt jitter, alignment errors, and so forth. But the comparison verifies that our design functions in the regime of contrast and angular separation needed to meet the top-level mission requirement of acquiring the reflected spectra of six or more gas giants.⁹¹ In this issue, Krist et al. analyze the sensitivity of the SPLC performance to realistic aberrations and incorporate the coronagraph in an end-to-end simulation of the WFIRST-AFTA telescope for a complete observing scenario.⁷⁸

3.3.

Debris Disk Mode–Shaped Pupil Lyot Coronagraph

To design an SPLC for debris disk imaging over a much wider FoV, we carry out a parameter survey similar to that of the 360-deg characterization design trials. Assuming a deformable mirror with a $48\times 48$ actuator array, the maximum correctable aberration spatial frequency corresponds to an angular separation of $21.8{\lambda }_0/D$ at the short wavelength end of an 18% passband. To approximately match this, we fix the outer radius of the polychromatic dark region at $20{\lambda }_0/D$. Within that dark annulus, we constrain the contrast to $\le {10}^{-8}$ over an 18% bandwidth. We test FPM spot radii of 6.0, 6.5, and $7.0{\lambda }_0/D$, and Lyot stop padding levels between 2 and 8%. The throughput results are tabulated in Table 4. We highlight the solution with ${\rho }_0=6.5{\lambda }_0/D$ and padding level of 4%, since it gives a throughput of 0.23, almost as high as the best design at ${\rho }_0=7{\lambda }_0/D$. With the smaller focal occulting spot radius of ${\rho }_0=6{\lambda }_0/D$, on the other hand, there is a significant throughput drop for all the Lyot stops.

Table 4

Throughput of debris disk science WFIRST-AFTA solutions for different combinations of inner radius and Lyot stop padding. The trade-off in inner radius and throughput corresponding to the design exhibited in Figs. 13 and 14 is highlighted in bold font.

In Fig. 13, we illustrate the SPLC mask scheme for the highlighted disk science design. The apodizer maintains over 59% of the available open area around the WFIRST-AFTA pupil obscurations, and the FWHM PSF area is only 1.11 times that of the WFIRST-AFTA telescope. The on-axis PSF of the coronagraph at the center wavelength is plotted on the left-hand side of Fig. 14, along with the ideal contrast curves. Like the characterization design, the mean contrast is significantly deeper than the constraint value, due to the lumpy structure of the diffraction pattern.

Fig. 13

Mask scheme for the debris disk mode SPLC for WFIRST-AFTA, from left to right: shaped pupil apodizer, FPM, and Lyot stop. The relative sizes of the masks in conjugate planes are not shown to scale. This design produces a broadband, annular dark region with $\le {10}^{-8}$ contrast over an 18% bandwidth from $6.5{\lambda }_0/D$ to $20{\lambda }_0/D$. The FPM is an occulting spot of radius $6.5{\lambda }_0/D$. The Lyot stop is a replica of the telescope aperture, with the inner and outer edges padded at 4% of the pupil diameter.

Fig. 14

(a) The ideal, on-axis, center wavelength PSF of a broadband disk science SPLC design. On the right-hand side (b), the ideal contrast is averaged over azimuth samples and then averaged over seven wavelength samples spanning the 18% passband. Also plotted are the maximum contrasts at each separation over all azimuth and wavelength samples.

We have also optimized debris disk designs that use an annular diaphragm FPM instead of an occulting spot. The throughput in this configuration is approximately half that of the spot FPM case for the same contrast, FoV, and bandwidth parameters. However, a general advantage of SPLC designs that use the annular diaphragm FPM is their greater tolerance to Lyot stop mask misalignment, an issue we discuss in Sec. 4. Therefore, despite the lower theoretical performance, for the initial testbed implementation of the debris disk SPLC, we will use an annular FPM variant.⁹⁷

3.4.

Summary of WFIRST-AFTA Shaped Pupil Lyot Coronagraph Designs

We assemble in Table 5 the parameters and performance metrics of our candidate SPLC designs for WFIRST-AFTA. To be consistent with other coronagraph descriptions, the IWA and OWA are measured by the half-maximum crossings of the throughput curve rather than the dimensions of the FPM and optimization constraints.⁵⁰ We use the same definitions for throughput and PSF area first given in Sec. 2.3. The PSF area is the FWHM region of the PSF for an off-axis (planet-like) source, normalized to the FWHM area of the PSF of the WFIRST-AFTA telescope without a coronagraph. We note that two designs are listed for the debris disk mode. One is the occulting spot configuration described in Sec. 3.3 and Figs. 13 and 14; the other is the annular diaphragm variant that is undergoing fabrication for testbed evaluation at HCIT, described in more detail in Ref. 97.

Table 5

Summary of WFIRST-AFTA SPLC designs.

4.1.

Shaped Pupil Apodizer

For the first-generation SPC designs for WFIRST-AFTA, Riggs et al. found that the contrast performance was sufficiently robust to etching errors.⁶⁸ The current testbeds in the JPL HCIT are using shaped pupil masks with diameters between 14 and 22 mm. With 1000 optimized transmission points across the mask diameter, the binary array is therefore composed of square pixels of width 14 to 22 microns. The standard etching tolerance in JPL’s Microdevices Lab is below 1 micron, so the etching error on each SP pixel is less than $\sim 5\%$. For a 5% uniform over/underetching error, the first-generation characterization SPC had an open-loop contrast degradation of $2\times$, from 1 to $2\times {10}^{-8}$. For the mission payload, even tighter etching tolerances can be achieved, so uniform etching errors are not a major concern for the SP apodizer mask.

4.2.

Focal Plane Mask and Pointing Sensitivity

For the bowtie characterization SPLC described in Sec. 3.2, we modeled the sensitivity to systematic errors in the FPM fabrication and alignment, as well as line-of-sight pointing errors originating from spacecraft jitter. The width of the stellar PSF core in relation to the inner radius of the bowtie mask is highest at the long-wavelength end of the design passband. Any disparity from the transmission profile assumed in the on-axis optimization therefore causes unwanted starlight to leak into the inner part of the dark region. Conversely, at the short-wavelength end of the passband, the outer perimeter of the dark bowtie region is sensitive to transmission profile disparities along the outer edge of the bowtie mask.

We plot the effect of pointing errors and, in particular, their effect on the inner part of the image in Fig. 15. In our Fourier propagation model, we apply phase ramps in the apodizer plane corresponding to a set of tilt errors along the long axis of the bowtie for the characterization design with central wavelength of 660 nm. This wavelength corresponds to the bluest of the three characterization filters, chosen here because when IWA is fixed in resolution elements (${\lambda }_0/D$), the shortest-wavelength coronagraph realization—as defined by the physical scale of the FPM—is the one most sensitive to a given telescope pointing error. We apply pointing errors of 0.4, 0.8, and 1.6 milliarc second. These are on the same angular scale as the residual jitter levels that may be present on the WFIRST spacecraft.^78,89,91 At the 660 nm center wavelength of the characterization passband, these tilts translate to focal plane offsets of $7\times {10}^{-3}{\lambda }_0/D$, $1.4\times {10}^{-2}{\lambda }_0/D$, and $2.8\times {10}^{-2}{\lambda }_0/D$, respectively. To show the upper bound of the impact on the polychromatic coronagraph PSF, in Fig. 15, we plot the contrast only at the long-wavelength end of the passband, 719 nm, where the effect is worst. At each separation, we plot the azimuthal average of the contrast in the degraded half of the bowtie region.

Fig. 15

Contrast degradation of WFIRST-AFTA characterization design for three pointing errors, shown at $\lambda =719\mathrm{nm}$, the red end of the design centered on the ${\lambda }_0=660\mathrm{nm}$ spectrograph filter.

The results plotted in Fig. 15 show that contrast outside of $4{\lambda }_0/D$ is not degraded significantly for pointing errors up to 1.6 milliarc second, roughly equivalent to an inner FPM radius error of $\sim 3\times {10}^{-2}{\lambda }_0/D$. At the interior, the long-wavelength intensity increments for 0.4, 0.8, and 1.6 milliarc second are, respectively, $6\times {10}^{-9}$, $1.4\times {10}^{-8}$, and $3.6\times {10}^{-8}$, as expressed in units of contrast. We emphasize that these values only indicate the contrast degradation at the red edge of the characterization filter. The impact over the rest of the band is much smaller: with a 1.6 milliarc second pointing error, for example, the intensity increment at the interior of the bowtie averaged over seven wavelength samples is $3.6\times {10}^{-9}$, a factor of 10 below the increment at the red extreme. It remains to be seen through integrated modeling on the full scope of the observatory, including low-order wavefront sensing and control and data postprocessing, how tip-tilt error affects the science yield of the characterization SPLC. For example, there may be an advantage in redesigning the apodizer for a passband extended slightly beyond the actual spectrograph filter in order to provide a buffer against pointing or alignment errors, at some cost in throughput.

Errors in the lateral alignment between the apodizer and the FPM have a comparable impact on the on-axis intensity as line-of-sight pointing errors. We summarize the results of trial offsets in Table 6. We quantify this in two ways. First, we compute the mean change in contrast over all the spatial samples and wavelength samples in the dark bowtie region. Then we isolate the region of the image that is most severely affected: the inner part of the bowtie (within $3.8{\lambda }_0/D$), at the red edge of the bandpass. Offsets of up to $4\times {10}^{-2}{\lambda }_0/D$ cause a mean contrast degradation of only 2 to $4\times {10}^{-9}$. However, in the most sensitive part of the image, for a horizontal offset of $2\times {10}^{-2}{\lambda }_0/D$, the contrast increment approaches ${10}^{-8}$.

Table 6

Contrast degradation of the characterization SPLC for a set of horizontal and vertical translation offsets of the FPM.

We have also tested the effect of bowtie FPM clocking errors on the characterization SPLC performance. The clocking angle of the bowtie FPM in the first focal plane needs to be accurate to within 0.5 deg to keep the worst contrast degradation (over all spatial samples and wavelengths) below $5\times {10}^{-9}$.

4.3.

Lyot Stop

We test the Lyot stop alignment tolerance of the WFIRST-AFTA characterization SPLC by modeling the propagation of flat, on-axis wavefronts when the Lyot stop is translated off-center. Over 7 wavelength samples and 27 angular separations spaced at $({\lambda }_0/D)/4$, we compute the azimuthally averaged contrast. Then we compute the mean and maximum increment relative to the nominal contrast values over all those wavelengths and separations. We find that the coronagraph performance is more sensitive to horizontal than vertical Lyot stop translations (here horizontal and vertical orientations are used in the same sense as the diagram in Fig. 10). In Table 7, we summarize the effect for horizontal translations of 0.5, 1.0, and 2.0% of the pupil diameter.

Table 7

Contrast degradation of WFIRST-AFTA characterization SPLC for horizontal Lyot stop alignment errors.

The FPM of the WFIRST-AFTA characterization SPLC is opaque outside the optimized bowtie region (Fig. 10). Any such diaphragm in the first focal plane has a low-pass filter effect on the morphology of the field in the Lyot plane, as we showed in Eq. (5) for the simple circular case. The effect can also be examined visually in the evaluation plots of the type-b SPLC configurations in Sec. 2. The lack of sharp field transitions near the Lyot stop edges helps to keep the alignment tolerance reasonable, in spite of the very high contrast goal in the final image.

This smooth Lyot field characteristic is not the case for the WFIRST-AFTA debris disk design we presented in Sec. 3.3, which operates with an occulting spot FPM. Initial calculations indicate that its Lyot stop alignment tolerance is at least an order of magnitude tighter than that of the diaphragm FPM variants. It is possible that an expanded optimization procedure can counteract this sensitivity. For example, if the optimizer model propagates the field not only through perfectly aligned masks but also through a set of cases with translated Lyot stops, then the final field could be constrained simultaneously for misaligned mask scenarios. However, if that approach is not feasible, then there is a substantial practical advantage for SPLC designs optimized for a diaphragm-type FPM, despite the fact that their throughput is in most cases lower for the same image constraints (e.g., comparing the metrics of configs. IVa and IVb in Table 1, and the debris disk designs in Table 5).