3.1.

Setup

In this section, we will describe our refractive setup at UCSC’s Laboratory for AO facilities developed to test FAST, which is the same setup from which subsequent results in this paper are obtained. Although we will refer to this setup as the Santa cruz Extreme AO Laboratory (SEAL), SEAL will ultimately be a mostly reflective setup designed for multipurpose AO-based wavefront sensing and control techniques, beyond the scope of just testing FAST; the higher-level setup of the full SEAL testbed, including the optical design and both current and ongoing projects (including FAST), are presented and described in Ref. 16. Note that the main facilities used here are the same as previously described in Ref. 21, including a highly stabilized granite testbed and custom-made enclosure.

Figure 2 shows our SEAL testbed setup and outlines the main hardware components relevant to FAST testing. The visible light star and planet light sources are commercial Thorlabs lasers KLS635 and HNLS008L, respectively, centered at 633 nm. The adjustable star light source is set to 0.15 mW unless otherwise noted, and the planet light source is fixed at 0.8 mW. We place a neutral density (ND) filter with an optical density (OD) of 2 on the planet source. The beamcube combining the two light sources further attenuates the planet by a factor of 9, theoretically setting the planet at a flux ratio of $6\times {10}^{-3}$ (see Sec. 4 for a measurement of the star-to-planet flux ratio). Although for most systems this flux ratio is unlikely to yield planet -mass sensitivity (i.e., more realistically in the brown dwarf mass sensitivity range), for the purposes of this paper (i.e., demonstrating real-time correction of AO residuals, not demonstrating long-exposure high-contrast capabilities), we will subsequently refer to this off-axis source as the planet light source. Additional higher OD ND filters (and/or increasing the star light source power) can further decrease this flux ratio for future high contrast experiments, but for the purposes of our main focus in this paper (i.e., real-time FAST correction of AO residuals) the above-described planet flux ratio is sufficient. Moving downstream, a set of relay lenses then images the system pupil (note that the system pupil stop is defined further downstream; see below) onto our $34\times 34$ actuator microelectrical-mechanical system DM (MEMS; 952 actuators forming a circular pupil), illuminating 29 actuators across the system beam diameter (but see the Lyot stop sizing implications of the MEMS footprint two paragraphs below). Reimaging optics then relays a pupil plane to an $11\times 11$ actuator ALPAO DM (97 actuators forming a circular pupil), also serving as the primary system pupil stop (i.e., a circular unobscured pupil geometry; backpropagation determines the relevant illuminated pupil on the MEMS). A plate beam splitter then reflects 50% of the beam through a pair of relay lenses to a commercial Thorlabs Shack Hartmann wavefront sensor (SHWFS; model WFS-20), sampling 32 lenslets across the beam diameter. The transmitted light is sent though a 500-mm focal length lens to generate an $f/37$ beam on the FPM, which is designed and fabricated specifically for that $f$-number and the 633-nm light source wavelength.

Fig. 2

An outline of our FAST SEAL setup, showing a conceptual diagram of the full setup and an inset of testbed images in two areas. In the inset, red arrows indicate light propagation (with different line styles combined/split from/into different paths). Double-sided black arrows indicate a deployable mechanism. Starting in the upper left, a planet and star light source, both aligned to the same focal plane, are collimated and then combined, with the planet source positioned slightly off-axis a few diffraction bandwidths from the star. Re-imaging optics place the pupil first on the $29\times 29$ (across the beam) MEMS DM, and then on the $11\times 11$ ALPAO DM, the latter defining the system pupil stop. A plate beam splitter (labeled as “BS”) reflects light to an SHWFS with 32 subapertures across the beam diameter and transmits light to the FAST arm. The FAST beam is then focused on our setup-specific FPM, then recollimated to a downstream coronagraphic pupil, where a deployable Lyot stop can modulate the beam (the central and off-axis coronagraphic pupil are visible on the Lyot stop in the lower left inset). Finally, downstream imaging optics enable coronagraphic focal plane (i.e., FAST) or pupil plane imaging on an Andor camera.

Figures 1(a) and 2 already show the fully reflective setup-specific FAST tip/tilt Gaussian (TG) FPM fabricated and characterized via Zygo interferometry measurements by University of Alberta’s nanoFAB laboratory, made from a nanoscribe three-dimensional (3D) printing machine and subsequently aluminum coated. This mask design only deviates from a flat surface over the central region (a $3\lambda /D$ radius from the optical axis) that redistributes the core starlight, and with this 3D printing approach, the full tilt can be made without phase wrapping (i.e., avoiding additional related chromatic effects). However, the design is also essentially a Lyot coronagraph, with no component optimized for diffracted starlight suppression, and so it is not ideal for reaching high contrasts. However, such a mask is still sufficient for the focus of this paper (i.e., ground-based observations where coronagraphic images are dominated by unpinned speckle noise). Further characterization of this mask is presented in Ref. 18, which shows that it is fabricated at sufficient quality to enable high fringe visibility SCC images to enable real-time wavefront control, which is the main focus of this paper.

An $f=200\mathrm{mm}$, 2″Ø lens then collimates the post-FPM beam, oversizing the clear aperture relative to the noncoronagraphic pupil footprint by a factor of 6.2, sufficient for the “classical” SCC.²² A Lyot stop then transmits light through the central 7.3-mm coronagraphic pupil (90% undersized) and off-axis 0.45 mm pinhole (which is the maximum pinhole diameter required to enable the first pinhole PSF Airy minimum to lie outside the DM control region). The 90% undersized Lyot stop pupil means that the 29 actuators sampled across the noncoronagraphic pupil become only 26 actuators sampled across the coronagraphic pupil. Avoiding edge effects, we found that using 24 actuators across the MEMS pupil diameter enabled sufficient performance in the FAST coronagraphic image. Although the FAST FPM prescription is designed for a theoretical pinhole–pupil separation (center-to-center) of 1.6 pupil diameters, the separation is instead empirically determined to be 1.524 pupil diameters (see Sec. 3.2). An $f=500\mathrm{mm}$ 2″Ø lens lastly focuses the post-Lyot stop beam onto our Andor Zyla 5.5 sCMOS camera, enabling a theoretical plate scale of $6\text{pixels}/\text{resel}$ (see Sec. 9.3 for a corresponding measurement), sufficiently oversampled for the SCC fringes.²² An $f=50\mathrm{mm}$ lens is placed 50 mm from the Andor detector on a flip mount to reimage the coronagraphic pupil on the same detector when deployed to be used for calibration purposes. The Lyot stop is also on a flip mount to enable coronagraphic pupil imaging both with and without the Lyot stop.

Our SEAL software infrastructure is based on the Keck pyramid wavefront sensor real-time control (RTC) architecture, described in Ref. 23. Reference 16 describes more specifically the hardware and infrastructure for SEAL software control and development, which includes a GPU-based clone of the Keck RTC in Linux (which we will refer to as the SEAL RTC) and a Windows machine for some off-the-shelf components without available Linux drivers. Low-level drivers for the MEMS and Andor camera (i.e., to get and send DM commands and to get images/sub-arrays) are written on the SEAL RTC in C and interfaced with Python to enable high-level FAST software development. Unless otherwise mentioned, for the Andor camera we use a $320\times 320$ subarray updating at 100 Hz (although in future work we will plan to demonstrate the higher-speed FAST closed-loop control: Andor benchmarks the Zyla 5.5 with $<1\mathrm{e}$- read noise at frame rates up to 1 kHz; Andor, private communication). Running in serial mode in Python (i.e., not parallelized), a 20-ms pause between grabbing Andor frames and applying DM commands is needed to enable a given image to be used to compute the corresponding DM commands. However, benchmarking tests for other projects on the SEAL RTC have already demonstrated the ability to reach $\sim 5\mathrm{ms}$ latency in Python by multithreading and synchronizing imaging-grabbing and DM-command-sending threads.²⁴ Our windows machine is used to operate the adjustable star light source, deployable pupil imaging lens, and SHWFS. For the SHWFS, although Thorlabs does not provide Linux-based drivers for the commercial WFS-20, we use the Windows serial interface in Python and an ethernet connection to the SEAL RTC to use SHWFS frames in our SEAL RTC Python interface for high-level AO software development (as with the Andor and MEMS above); current frames are limited to updating at 50 Hz (i.e., limited by the above-mentioned 20 ms pause) to interface with Python. However, separate benchmarking with this SHWFS setup shows that we can readout slopes with 23 subapertures across the beam (i.e., slightly undersampled relative to the 29 DM actuators across the beam) at 460 Hz for future “first stage AO + FAST” tests at higher speeds (see Sec. 7).

3.2.

Alignment

Off-the-shelf lenses are aligned with standard techniques, using a shear plate to collimate the beam and a knife-edge test to calibrate focal lengths but will not be discussed in detail further in this paper. The FAST FPM is first aligned manually using a series of stage adjustments and then algorithmically using the ALPAO DM. Further details of this FPM alignment process are given in Appendix Sec. 9.1. Appendix Sec. 9.2 additionally describes the steps taken after FPM alignment to empirically define the Lyot stop prescription from recorded coronagraphic pupil images.

4.1.

High Order Reconstruction

Our high order wavefront calibration and reconstruction procedure largely follows the standard methods developed for calibrating the SCC as outlined in Appendix Sec. 8.2. In addition to the overview given in Sec. 8.2, we briefly summarize these steps here, specifically noting additional setup-specific subtleties we have implemented. Low order WFS (LOWFS) calibration and linearity will be discussed next in Sec. 4.2.

To generate the high order interaction matrix (IM), sine and cosine waves are applied to the MEMS for every spatial frequency greater than five cycles per pupil (c/p) within the DM control region at intervals of 1 c/p. Applying lower order Fourier modes (i.e., those that are close to or within the edge of the FAST FPM at $3\lambda /D$) in the same IM as these mid to high order modes would increase nonlinearities by modulating the pinhole PSF throughput for only those modes. To minimize aliasing effects, we limit the highest MEMS spatial frequency along the $x$ and $y$ directions to 12 c/p, below the theoretical 13 c/p maximum.^19,20 For each DM Fourier mode, we compute $I_{-}$ [i.e., the Fourier-filtered complex-valued fringe component of the SCC image, as described in Sec. 8.1, Eq. (2), and Fig. 11] pixel values (real and imaginary) within a user-defined precomputed (DH) region (i.e., which sets the 12 c/p maximum spatial frequency limit along the $x$ direction described above) for a differential image (i.e., as in Fig. 12). As in Ref. 20, we implement an additional binary mask before storing the computed $I_{-}$ for a given mode in the IM, which isolates the given $I_{-}$-plane sine spot within a $1.3\lambda /D$ radius of its central location and sets all other pixel values within the DM control region equal to zero, minimizing cross-talk between modes. Using the same notation as Sec. 8.2, looping through all Fourier modes, these $I_{-}$ pixel values are then stored in a reference vector, $\overrightarrow{A}$, with dimensions $n\times m=456\times \mathrm{27,936}$. The $456\times \mathrm{27,936}$ command matrix (CM), $\overrightarrow{C}$, and reconstructed high order DM commands, ${\overrightarrow{D}}_n$, are then generated via Eqs. (3) and (4), respectively.

We compute the pseudoinverse via singular value decomposition (SVD), choosing the lowest SVD cutoff value (i.e., the lowest number of IM Eigenmodes conserved in the CM) that sufficiently reconstructs individual Fourier modes applied on the DM without cross-talk from other modes. Note that although with a more conventional zonal-based pupil plane WFS where the SVD cutoff is used to minimize the propagation of waffle modes onto the DM,²⁵ this waffle mode propagation is less relevant for a focal plane WFS. A binary DH mask is spatially filtering modes that the DM controls, inherently removing waffle modes at the DM Nyquist frequency.²⁰

4.2.

LOWFS Reconstruction and Linearity

Using FAST, the SCC image is also an LOWFS, as demonstrated in simulations in Ref. 20. Here, we demonstrate laboratory validation of this concept with SEAL. FAST LOWFS capabilities are enabled by the setup-specific FAST FPM, which sends the PSF core (i.e., low order pupil plane wavefront aberrations) through the Lyot stop pinhole aperture, forming fringes in the SCC image that track the amplitude and phase of low order modes (including for even modes such as focus, whose sign ambiguity is resolved from the fringes).

We apply a similar matrix vector multiplication (MVM)-based approach to generating an IM and CM for low order modes as described in Sec. 4.1 for high order modes. During the IM generation, instead of applying binary masks around individual sine spots, a binary mask in the $I_{-}$ plane uses pixel values only within a $5\lambda /D$ radius—the limiting DH inner working angle (IWA) for this system¹⁹—of the optical axis. After testing and development, we resolved to implement closed-loop LOWFS performance for three different modal groups: (1) tip/tilt/focus (TTF), (2) the next five Zernike modes, and (3) low order Fourier modes between 2 and 4 c/p with modal groups (1) and (2) deprojected from this basis. The linearity curves, showing reconstructed output wavefront error (WFE) versus input WFE amplitude for all modes applied and measured, are in Figs. 3Fig. 4–5.

Fig. 3

LOWFS linearity for tip/tilt and focus.

Fig. 4

LOWFS linearity for Zernike modes beyond TTF, labeled by their $n$ and $m$ indices.

Fig. 5

LOWFS linearity for Fourier modes between 2 and 4 c/p, indicated by the amplitude, position angle, and relative phase of their spatial frequency (labeled “freq,pa,ph,” respectively). The relative phase is reconstructed from a pair of sine and cosine modes for a given spatial frequency, as labeled. All panels show the same $x$ and $y$ scale of $\pm 0.45\mu \mathrm{m}$ WFE.

As shown, these modal groups form an excellent basis set to enable low order control, with a clear linear regime defined for each mode and minimal cross-talk between other modes in this regime. Figure 3 shows a tip/tilt capture range of $\sim 0.2\mu \mathrm{m}$ PV corresponds to $\sim 0.3\lambda /D\approx 10\mathrm{mas}$ for a 8-m telescope at $\lambda =1.6\mu \mathrm{m}$, consistent with simulations from Ref. 20 and the need for a diffraction-limited AO residual wavefront (i.e., Strehl ratio $\gtrsim 20\%$) to close the FAST loop. For each modal group, the input “poke” amplitude used to generate the IM and SVD cutoff used to generate the CM are separately optimized to produce optimal linearity and minimal cross-talk. It is for this reason that we placed focus in the same modal group as tip/tilt rather than with the other Zernike modes: a single SVD cutoff value produced good linearity for all Zernikes other than focus, where as the same tip/tilt SVD cutoff preserved good linearity for focus. We also confirmed that if all modes were combined in the same IM and CM that cross-talk between any mode and all the other modes was still negligible within the linear range of that mode (although with a single SVD cuttoff for all modal groups, linearity appeared qualitatively worse than as shown in Figs. 3Fig. 4–5, which is why we chose to separate these groups into different IMs and CMs). The robustness of our choice to use separate SVD cutoffs and IM amplitudes for each modal group is also further supported by closed-loop results in Sec. 5 (i.e., if cross-talk between modal groups remained significant, the loops would not close and/or go unstable).

Interestingly, Fig. 3 shows some conceptual discrepancies from TTF linearity simulations in Ref. 20, including a smaller capture range for tip and for tilt and cross talk in focus only from tip but not from tilt. As we have not observed this behavior in simulation and we have already optically aligned the system for these modes, we conclude (by process of elimination) that these behaviors are due to FPM fabrication defects. Reference 20 did not analyze SCC/LOWFS modes beyond TTF, so we are not able to compare simulations to Figs. 4 and 5, but in general the strong linearity and low cross talk for each mode indicates the potential for efficient closed-loop control, which is the main focus of this paper.

5.1.

Static Errors

Figure 6 shows images and contrast curves of speckle correction by DM control of static aberrations within the SEAL system. The off-axis planet source (to the lower right of the star) is visible in the middle “DH SCC image” panel. Due to the static nature of this initial DH, two control approaches produce similar results: (1) using a low gain integral controller (typically below 0.01) to ensure loop convergence over a large number of iterations (converging typically after a few hundred frames) or (2) using a higher gain but limiting the number of iterations, beyond which the loop diverges due to detector and photon noise propagation into the reconstructed DM commands (e.g., a gain of 0.1 fixed at 100 iterations typically produces similar results as option 1). Contrast curves in Fig. 6(c), calibrated to the peak off-axis stellar flux as described in Sec. 9.3, compute five times the standard deviation within a given radial annulus of the SCC image (masking the planet signal to prevent bias in computing contrast gains) that is also within the half DH region on the right-hand side of the image, showing up to a 10× contrast gain depending on separation for the shown 20 ms exposure time. In Fig. 6(c), it is also apparent that although we defined the DH control region out to 12 $\lambda /D$ along the $x$ direction (Sec. 4.1), contrast gains are negligible beyond $\sim 10\lambda /D$ due to a trade-off we found between SVD cutoff and MEMS stroke limitations: a lower SVD cutoff (i.e., including more eigen modes in the HOWFS CM) produced a deeper DH between 10 and $12\lambda /D$ but also required $\sim 2\times$ more stroke (which cannot be compensated by the higher-stroke ALPAO device due to the spatial frequencies being considered) than a higher SVD cutoff, which produced a DH as deep out to $10\lambda /D$ but not as deep between 10 and $12\lambda /D$. Given the subsequent need to use additional MEMS stroke to apply and then correct AO residual turbulence (Sec. 5.3), we chose to proceed with the higher SVD cutoff option.

Fig. 6

Speckle correction/subtraction results of quasistatic aberrations on SEAL. (a) FAST image before DM correction. (b) The same image after DM correction, generating a DH on the right side of the star with the planet visible to the lower right of the star. (c) $5\sigma$ contrast curves of the SCC image within the DH region (masking the planet signal) for before (flat) and after (DH) static correction, showing up to a 10× improvement.

5.2.

Quasistatic Errors

Figure 7 shows the temporal effects of closing the loop on air, stabilizing any quasistatic effects such as air flow through the enclosure. The histograms in Fig. 7 clearly show a gain in WFE for closed- versus open-loop control for all modal groups. The median TTF open- and closed-loop WFE are 6.2 and 1.6 nm rms, respectively, which for the latter corresponds to $0.003\lambda /D$ for tip/tilt, illustrating the highly stabilized environment the SEAL granite testbed and enclosure enables. Although this nm level of on-air stability is perhaps unrealistic for an on-sky ground-based environment due to realistic quasistatic effects (e.g., mechanical effects from telescope tracking, changing thermal environment, instrument flexure for Cassegrain instruments), it sets the stage for similar gains to be made on AO residual turbulence correction, discussed next.

Fig. 7

SEAL temporal stability before and after closing the loop on only quasistatic turbulence. (a) The statistics for a single modal within each of the four modal groups (with each group indicated by a different color; the color scheme is consistent across all panels in this figure), (b) the cumulative statistics for all modes within a given modal group. Temporal PSD plots on the left panel side of panel (a) motivate the subsequent display of low-pass filtered wavefront statistics, where the choice of temporal frequency cutoff (2 Hz) is the 0 dB bandwidth for tip. Kernel density distributions are computed using the seaborn.kdeplot package.²⁸ All panels clearly illustrate a WFE reduction and stabilization for all modes.

5.3.

Dynamic AO Residual Errors

In this section, we present an important milestone for focal plane wavefront control technologies, demonstrating FAST correction of AO residual speckles in quasi-real-time for both low- and high-order modes. An evolving full power law AO residual wavefront (i.e., including low order modes) is applied on the MEMS in open loop and then corrected by the FAST LOWFS and HOWFS in closed-loop. AO residual WFE is normalized to 100 nm rms between 0 and 16 c/p assuming $\lambda =1.6\mu \mathrm{m}$ with a $-2$ spatial power spectral density (PSD) power law. A 10-m diameter telescope, $1\mathrm{m}/\mathrm{s}$ ground layer “frozen flow” wind speed is assumed. As discussed at the beginning of Sec. 5, our Python real-time code limits operational frame rate to 50 Hz with a 1-frame delay. This limited loop speed is the main reason why a 1-m/s wind speed is used, although a more realistic 1 ms atmospheric lag, as motivated in Ref. 19, is applied at the time of DM correction. Regardless, future more-optimized real-time code running at kHz speeds will enable closed-loop correction on more typical wind speeds of order 10 m/s. With this setup implemented, Fig. 8 shows the time series and WFE distributions and Fig. 9 shows long exposure SCC images and contrast curves (computed analogously to Fig. 6 as described in Sec. 5.1) for open- and closed-loop FAST correction of evolving AO residuals.

Fig. 8

(a) Time series and (b) corresponding kernel density distributions of open- and closed-loop operations of AO residual turbulence, using the same temporal filter based on the analysis in Fig. 7. The same colors in the right panel legend also apply to the left panel, noting that high order Fourier modes are only shown for spatial frequencies between 5 and 7 c/p. Atmospheric parameters are shown in the title; the loop speed is $\sim 50\mathrm{Hz}$ with 1 frame delay. Gains and leaks for a leaky integrator controller are separately tuned for each modal group. Loop closures occur at three different times for TTF, other LOWFS modes (including Zernike and LOWFS Fourier modes), and HOWFS modes, indicated by the three different vertical lines on the left panel. Open- and closed-loop distributions in the right panel are computed with data in the left panel to the left of the solid line (TTF loops closed) and to the right of the dotted line (HOWFS loops closed), respectively.

Fig. 9

Stacked FAST images (on the same relative contrast scale) and corresponding contrast curves for open- and closed-loop (OL and CL, respectively) control of simulated AO residual turbulence (using the same frame indices to define the OL and CL regimes as in the right panel of Fig. 8), also showing the same “static DH” contrast curve from Fig. 6 for reference. The same planet source is present in the lower right side of the DH images. Closed-loop correction of AO residual turbulence is demonstrated to improve contrast by up to a factor 5 relative to no residual atmospheric correction (OL).

Figure 9 shows up to a $5\times$ contrast gain comparing open- and closed-loop correction of AO residual turbulence, illustrating the benefit of closed-loop operations at all spatial frequencies. Figure 8 shows a $1.5\times$ HOWFS median WFE gain (46 versus 30 nm rms for open versus closed-loop HOWFS WFE, respectively, for spatial frequencies between 5.5 and 7 c/p), which is consistent with the $2.3\times$ median OL versus CL contrast gains between 5.5 and $7\lambda /D$ in Fig. 9 (i.e., contrast gains should scale as the square of the WFE gains). Figure 8 also illustrates the sequential process of closing loops: in contrast to on-air control in Fig. 7, here we found that all loops cannot be closed at once, requiring initial TTF loop closure, then LOWFS (including Zernike and Fourier modes simultaneously), and finally HOWFS Fourier modes. When we tried to close all modes simultaneously, or, e.g., HOWFS modes first, the loop would go unstable shortly thereafter. This is to be expected: LOWFS modes must be closed and stabilized first (i.e., producing a stable stellar core on the FAST FPM, which is needed for HOWFS measurements), as we predicted in Ref. 20. The number of frames needed between closing the TTF, LOWFS, and HOWFS loops for the different modal groups was determined by applying AO residuals on the MEMS with only the modes from a given group applied and then determining by eye how long it took for the loop to converge; we did not further explore flexibility in this parameter space. Contrast improvement in Fig. 9 decreases with separation, however, suggesting that loop control is not yet optimized. In Fig. 8, we only plot HOWFS telemetry for spatial frequencies below 7 c/p, as including these higher order modes “dilutes” the relative gain from these plotted HOWFS modes. By applying a single gain and leak (0.3 and 0.8, respectively) to all HOWFS modes (i.e., as described in Sec. 8.2), there is an inherent nonoptimal trade-off between temporal rejection at lower spatial frequencies versus noise amplification at higher spatial frequencies;²⁷ our chosen scalar gain and leak values gave better rejection at lower spatial frequencies and more noise amplification at higher ones, consistent with the smaller contrast gains at larger separations in Fig. 9. This clearly motivates the need to implement more advanced control, such as modal gain optimization.¹⁹ Further comparing the closed-loop performance to static DH limits (dashed orange vs. dotted green curves), it is clear that across most spatial frequencies there are additional gains to be made with real-time correction of AO residuals. We discuss such further areas for contrast improvement in Sec. 6.

Regardless of additional gains to be made, Fig. 8 and particularly Fig. 9 represent an important milestone in using real-time focal plane wavefront control to boost the achievable final contrasts with exoplanet imaging instruments. To be clear, this is not the first demonstration of real-time focal plane wavefront control; Ref. 8 demonstrated on-sky linear dark field control at 20 Hz but with negligible contrast gains due to a noisy detector and insufficient bandwidth for control of residual atmospheric speckles. In this paper, however, we have demonstrated a clear raw contrast gain of up to $5\times$ for simulated AO residuals in real-time. The implications of this result scales to an up to $5\times$ final contrast gain, supported by the work from Ref. 29 (correlating AO performance to achievable raw contrast and raw contrast to final contrast). In other words, this demonstration of $5\times$ raw contrast gain with FAST translates to boosting postprocessed contrast by $5\times$ with angular differential imaging³⁰ (even in second generation high contrast imaging instruments, spectral differential imaging³¹ yields negligible contrast gains to due significant instrument chromaticity⁴).

8.1.

Image Formation and Processing

Figure 10 shows the concept of SCC image formation, both from a geometric (a) and Fourier (b) optics perspective. Reviewing a mathematical description of the SCC image, the recorded image on the focal plane detector, $I$, produced from noiseless (i.e., without photon/detector/sky background noise) propagation of WFE (i.e., from atmospheric and/or instrumental origin) through to the detector, can be described as¹⁴

Fig. 10

An simulated illustration of SCC image formation due to (a) the geometric and (b) diffractive properties of an FPM, adapted from Ref. 15. We describe the four subpanels in (b). Upper left: in an intermediate focal plane, an FPM blocks the on-axis star while transmitting remaining diffracted starlight and off-axis exoplanet light. Lower left: in the downstream pupil plane, starlight is diffracted outside the footprint of the primary mirror if no FPM was deployed (illustrated by the dashed blue circle); such diffraction effects occur for any FPM with sub-$\lambda /D$ spatial scales, in this case encoded by the hard edge of an amplitude “dot” FPM. Lower right: a Lyot stop blocks most of this diffracted light outside the pupil footprint, except for a small off-axis pinhole (whose cyan aperture footprint is also illustrated in the lower left panel) that is transmitted downstream. Upper right: The Lyot stop pinhole and pupil apertures interfere in the coronagraphic image to form fringes on sub-$\lambda /D$ spatial scales, overlaid on the existing speckle pattern (with spatial scales larger than $\lambda /D$) from the main Lyot stop pupil aperture. The electric field intensity in the pinhole aperture used to simulate the lower and upper right panels is (unphysically) enhanced by a factor of $2.5\times {10}^5$ for the purposes of this illustration.

Fig. 11

An illustration of the SCC wavefront sensing and reconstruction algorithm, adapted from Ref. 15. Starting from the recorded coronagraphic SCC image (step 1), a series of Fourier transforms (FT and IFT, the “I” representing “inverse”) and Fourier filtering steps (steps 2 to 5) enable a complex-valued measurement of the focal plane electric field (step 6) known as $I_{-}$. Note that $m_1$ is a 2D array of ones and zeros (black and white representing zeros and ones, respectively; subsequently referred to as an “algorithmic” or “binary” mask), and the “× symbol” represents a multiplication of OTF (i.e., including both real and imaginary components) by $m_1$.

8.2.

DM Control

Recording SCC images and processing them to produce $I_{-}$ can be calibrated empirically to DM commands applying sine and cosine waves over the full spatial frequency range of the DM, illustrated in Fig. 12. The Lyot stop pinhole size sets the image-plane region over which DM actuators can correct for speckles, so a higher order correction requires a smaller pinhole. A typical SCC pinhole diameter requirement is that the diagonal along the DM Nyquist region lies with the first minimum of the $I_R$ Airy function, which for a square grid DM is given in Ref. 43 [Eq. (40)].

Fig. 12

An illustration of the procedure for generating an SCC IM. The top row shows the phase applied on the DM, whereas the bottom row shows the corresponding differential SCC image (recorded image relative to a flat wavefront image). The two columns show responses for sine and cosine waves at the same spatial frequency. Such differential SCC images are recorded at the full range of controllable DM spatial frequencies to generate an IM in the complex-valued $I_{-}$ plane (see Fig. 11).

An SCC IM can be built by recording differential SCC images as shown in Fig. 12 for each DM Fourier mode [Eq. (1) for a given DM Fourier minus Eq. (1) for a flat DM] that are then converted to $I_{-}$ [Eq. (2)] and lastly multiplied by an binary DH mask. This step, not illustrated in Fig. 12, is completed by multiplying (1) the complex-valued $I_{-}$ for a differential SCC image of a given DM Fourier mode by (2) an binary DH mask (i.e., using ones to represent the region of desired image-plane speckle attenuation). Note that with only a pupil plane conjugated DM(s), speckle correction of phase and amplitude errors (including diffraction) is limited to a half DH.³⁸ A vectorized version of these $I_{-}$ DH pixel values are then used to build an IM and then CM to enable a subsequent linear least-squares-based MVM linking SCC images and DM commands as follows:

This process to generate DM commands is then applied iteratively in closed-loop until the desired DH contrast convergence in contrast achieved. The number of iterations needed to reach such is convergence system-dependent (e.g., coronagraph type, input WFE level, desired convergent DH level, available DM stroke, temporal WFE/stability environment, and more can all influence convergence timescale and achievable contrast) but can in principle be as few as 0 (i.e., only one image). Figure 13 shows this DH generation process. Note that in the context of this section serving as a high-level summary of SCC DM control, many additional low-level details used for further improving achievable contrast are not described in this appendix section, such as optimizing/tuning the CM pseudoinverse SVD cutoff, IM Fourier modal amplitude, additional IM binary masks, the optimal basis set for low order modes, and more.

Fig. 13

A simulated half DH on a diffraction-limited (i.e., no WFE) coronagraphic image (using the apodized Lyot coronagraph from Ref. 39). (a) The raw APLC SCC image (the fringe visibility is too low to see fringes by eye). (b) The binary mask (i.e., defining the DH region with ones and elsewhere with zeros) used to generate $m{I}_{-}$ pixel values. (c) Image after DH generation, after 15 iterations (the upper left and right panels are shown on the same scale). (d) Contrast within the DH at $10\lambda /D$ versus iteration number, showing that the majority of contrast gains occur after just two iterations. (e) DM commands used to generate the upper right DH image, illustrating the relative minimal amount of DM stroke needed for this SCC configuration.

9.1.

FPM Alignment

FAST FPM alignment is carried out in multiple steps: initial coarse adjustment with a 5-degree-of-freedom stage ($x\text{–}y$-pitch-yaw-roll; focus is predetermined by a knife edge test as described above). With the Lyot stop in place, pitch and yaw (i.e., tip and tilt) of the FPM mount are adjusted until the noncoronagraphic pupil aperture is centered on the intended central aperture of the Lyot stop (which displays a 0.8-mm-wide ring around the aperture edge due to the 90% undersizing). Then, a pinhole aperture is placed on the focusing lens just upstream of the FPM to simulate a flat field. The FPM core in this mode is visible as a dark circular shadow on the otherwise illuminated flat field; $x$ and $y$ positions are then adjusted so the center of this black dot is positioned at the coordinates of the normal Airy disk’s center. After removing the pinhole/flatfield, a somewhat-aligned coronagraphic pupil is visible, by eye, on the Lyot stop (as in Figs. 1 and 2); further $x\text{–}y$ adjustment can then enable visually maximizing the off-axis pupil intensity into the pinhole. The FPM roll is then adjusted to the desired position angle (i.e., defined by the Lyot stop pinhole’s position angle), after which the above $x\text{–}y$ positioning steps are repeated (since the setup-specific shape on the FPM is not exactly in the center of the optic’s clear aperture). Lastly, fine $x\text{–}y$ adjustment is performed in software with the DM tip/tilt: a grid search is performed in a square region around the estimated best alignment position, optimizing a combination of maximal fringe visibility and minimal raw contrast. Although a fringe visibility metric naively seems like a sufficient metric to align the FAST FPM (since once fully aligned the maximal amount of light is transmitted through the Lyot stop off-axis pinhole), fringe intensities are equally weighted between the focal plane amplitudes from the Lyot stop pinhole and pupil [see Eq. (1)], which causes fringe intensities to increase when the FPM is misaligned because the amount of starlight increase is greater than the amount of pinhole light decrease. This is mitigated as described above by requiring minimal starlight (measured in the image plane) in addition to maximal fringe visibilities (measured by integrated flux on the MTF sidelobe).

9.2.

Lyot Stop Design

The pupil-to-pinhole separation for our Lyot stop prescription is determined empirically, as was done in Ref. 40 and as we will illustrate should be a standard process for future FAST FPMs for laboratory and observatory instrument projects, by taking coronagraphic pupil images with the Lyot stop out and fitting the centers of each pupil (where a separate system is used with DMs replaced by fold mirrors for this characterization to enable a lower WFE characterization). As to be discussed next in Sec. 9.3, we are only able to reach 40-nm rms closed-loop WFE with the ALPAO DM while in a separate system with only fold mirrors produces $\sim \mathrm{nm}$ rms level WFE, making the latter more optimal to characterize the quality of the fabricated FAST FPM (i.e., avoiding ambiguity issues between system WFE and FPM-specific fabrication defects). The off-axis pupil center in this setup with only fold mirrors is determined by fitting a 2D Gaussian, whereas the central coronagraphic pupil position is determined by eye (incurring $\sim 1\text{–}2\text{pixel}$ error $\approx 2\%$ pupil diameter error $\approx 16\%$ pinhole diameter error). Measuring this separation is important to enable a Lyot stop prescription that both (1) optimizes fringe visibility (i.e., limiting the pinhole position to deviate from the peak intensity) and (2) prevents a differential tilt between the pinhole PSF center and unfringed coronagraphic image center (i.e., preventing an offset between the center of the fringed and unfringed image components). We initially found that a Lyot stop prescription with a pinhole separation too discrepant from what was needed indeed caused a tilted pinhole PSF with respect to the unfringed coronagraphic image due to an amplitude gradient across the pinhole aperture causing a tilt. Although our initial pinhole separation prescription was for the intended FPM design, as discussed in Sec. 3.1, it turned out that this differed significantly enough from the fabricated mask to cause such a measurable discrepancy, but with an empirically measured Lyot stop prescription this was no longer a problem. A recorded coronagraphic pupil image should therefore be the main determinant for future SCC Lyot stop prescriptions, both for lab experiments and future FAST instruments, to avoid a tilted pinhole PSF.

9.3.

Initial Calibrations

We start bench operations by flattening the ALPAO using the SHWFS, also compensating for any corresponding common path aberrations. In reality, the ALPAO surface is not flat, but instead compensates for the unflat resting MEMS and IrisAO surfaces. Although the latter produces an 8 nm rms flat as measured by our Zygo interferometer (and so likely further compensation by the ALPAO as measured by the SHWFS is negligible), we chose to keep the MEMS in a resting nonflat position while flattening the ALPAO, enabling ALPAO compensation for MEMS aberrations below $5.5\text{cycles}/\text{pupil}$ (c/p). Although this option leaves the remaining MEMS figure errors between 5.5 and 12 c/p (the limiting spatial frequency on a side we correct for) initially uncorrected, we found this approach more favorable than using an optical flat command for the MEMS. We found that the latter approach—which we also generated with our Zygo interferometer—reaches diffraction-limited contrasts in the SCC image, requiring more MEMS stroke to dig a DH on diffracted light rather than on unpinned speckles, consistent with Ref. 41 (although note the differing contrast regimes between Ref. 41, at ${10}^{-10}$ levels, and this paper, at ${10}^{-4}$ levels). Instead, leaving the MEMS figure error initially uncorrected at these mid to high spatial frequencies significantly relaxed the subsequent MEMS DH stroke limits (and also enabling the subsequent application of AO residual turbulence without running out of stroke) while having a negligible effect on initial Strehl ratio since aberrations at these spatial frequencies are a negligible component of the overall WFE. Other ALPAO devices similar to ours have been known to drift on hours to days timescales;⁴² we found that daily generation of a new best flat delivers sufficient performance. The flattening is done in SHWFS slope space, applying a classical zonal “push minus pull” actuator pokes to generate an IM.²⁵ The CM pseudoinversion SVD cutoff is chosen to optimize the convergent closed-loop WFE as measured by the precalibrated SHWFS wavefront reconstructor (i.e., we choose the SVD cutoff value that produces the flattest wavefront), usually converging to a residual system WFE of around 40 nm rms. There are likely a few key limitations to reaching a deeper closed-loop residual WFE: (1) the presence of a pinned (bad) MEMS actuator near the pupil center, which we do not account for in SHWFS reconstruction (e.g., algorithmically masking or interpolating over this region of the pupil), and (2) segment gaps caused by the IrisAO DM, which we also do not algorithmically account for in the SHWFS IM. Prior to the IrisAO and MEMS installation into SEAL, the SHWFS driving the ALPAO typically converged to $\sim 5\mathrm{nm}\mathrm{rms}$. Note that this SHWFS flattening procedure is completed before running the DM-based FPM alignment procedure described in Sec. 3.2.

To calibrate contrast, with the best flat on the MEMS applied as described above, we tilt the PSF off the FAST FPM core to record an off-axis PSF, adjusting the exposure time to prevent the star from saturating. Coronagraphic images in units of normalized intensity and/or contrast curves shown later in this paper are derived from dividing the dark-subtracted coronagraphic image in ADUs by the aforementioned maximum flux of the (also dark-subtracted) off-axis stellar PSF, scaling to account for the different exposure times in the two images. We can also calibrate flux ratio of the off-axis planet with this approach: turning off the starlight source but leaving on the planet source and scaling for exposure time gives a normalized planet flux ratio of $2.4\times {10}^{-3}$.

To find the coronagraphic image center and empirically calibrate the plate scale, we place two sine waves simultaneously on the DM–one with a position angle offset 90 deg from the other and each at separations of $10\lambda /D$–to generate “satellite spots.” Viewing aligned coronagraphic pupil images and poking MEMS actuators confirmed the 26 actuator sampling across the Lyot stop pupil described in Sec. 3.1, which is needed to ensure a 10 c/p sine wave places spots at separations of $10\lambda /D$ in the SCC image. We also determined that the MEMS actuator grid has a 1-deg position angle offset with respect to the Andor detector pixel grid, likely due to small differences in how the two devices are mounted. We correct for this algorithmically to ensure that the MEMS Fourier modes are aligned with the Andor camera pixel grid. Averaging the four 2D Gaussian-fit positions of these spots provides the coordinates of the image center. The average separation between the sine spot pairs also provides a plate scale measurement: $6.2\text{pixels}/\text{resel}$, sufficiently oversampled for the SCC fringes.

A calibration between DM voltage units and reconstructed (reflected) wavefront units for both the ALPAO and MEMS, which we use subsequently in this paper, were obtained by measuring 4 off-axis PSFs, each at four different combinations of tip and tilt commands (repeating this for both the ALPAO and MEMS). The above-calibrated plate scale is then used in combination with the fitted positions of the four off-axis PSFs to convert differential tip and tilt commands into separations in $\lambda /D$, which relates to pupil peak-to-valley (PV) tip/tilt WFE via: $\mathrm{WFE}/\lambda =X(\lambda /D)$, where $X$ is the measured separation in the focal plane in $\lambda /D$ units. Averaging the separate tip and tilt differences produces 8 and $55\mu \mathrm{m}$ of PV WFE for 1 MEMS and ALPAO unit, respectively (i.e., the tip/tilt stroke limits of these devices). It is important to note that in reality these conversion factors are mode-dependent. E.g., the ALPAO DM waffle stroke limit is $4\mu$ PV WFE, not 55 (ALPAO, private communication). However, because the main focus of this paper is the first laboratory demonstration of FAST correction of dynamic AO residual WFEs rather than high accuracy wavefront reconstruction, for simplicity we use these scalar conversion factors for all analyses and plots in this paper.