Active Compensation of Aperture Discontinuities

• ACAD is an active wavefront-control strategy that employs two sequential deformable mirrors to synthesize lossless apodization and cancel diffraction artifacts from non-axisymmetric telescope features.
• It leverages the Monge–Ampère equation and closed-loop stroke minimization techniques (e.g., ACAD-OSM) to optimize DM surface profiles for enhanced coronagraphic performance.
• ACAD achieves raw contrast gains of several orders of magnitude while maintaining high throughput and angular resolution, making it critical for missions like WFIRST-AFTA, LUVOIR, and ground-based ELTs.

Active Compensation of Aperture Discontinuities (ACAD) refers to a class of wavefront-control strategies that utilize two sequential deformable mirrors (DMs) to actively synthesize lossless apodization in the pupil plane of high-contrast coronagraphic instruments. The technique is designed to suppress bright diffraction artifacts resulting from non-axisymmetric features such as central obscurations, spider vanes, and segment gaps in real telescope apertures, thereby unlocking raw contrast gains of up to several orders of magnitude without sacrificing scientific throughput or angular resolution. ACAD leverages nonlinear wavefront mapping, often solved via the Monge–Ampère equation, and can be further extended with closed-loop stroke-minimization (SM) approaches such as ACAD-OSM (Optimized Stroke Minimization). Its practical, algorithmic, and hardware parameters have been optimized for missions including WFIRST-AFTA, LUVOIR, and ground-based Extremely Large Telescopes (ELTs) (Pueyo et al., 2012, Mazoyer et al., 2015, Mazoyer et al., 2017, Mazoyer et al., 2017, Mazoyer et al., 2017, Mazoyer et al., 2017, Mazoyer et al., 2017).

1. Physical and Mathematical Rationale

Aperture discontinuities in telescopes—ranging from secondary mirror supports (spiders), central obscuration, and mirror segmentation gaps—introduce amplitude errors that propagate to the coronagraphic focal plane, generating diffractive sidelobes and artifacts that restrict contrast performance to levels far from those needed for direct exoplanet imaging (e.g., 10⁻⁶ contrast compared to requirements of 10⁻⁹–10⁻¹⁰). Conventional apodization via static absorbing masks or amplitude filters achieves only partial suppression and can significantly degrade throughput and widen the inner working angle (IWA).

ACAD proposes a solution by remapping the pupil's amplitude and phase distribution using two "out-of-pupil" DMs. The optical mapping is governed by energy conservation constraints, resulting in a Monge–Ampère partial differential equation for the DM surfaces:

$\det\left[I + \nabla^2 H(X, Y)\right] = A(X, Y)^2$

where $A(X, Y)$ is the desired apodization (usually near unity to preserve throughput), and $H$ is a normalized representation of the DM surface. Once solved for DM1, DM2's shape is computed to restore collimation and complete the mapping (Pueyo et al., 2012, Mazoyer et al., 2015, Mazoyer et al., 2017). These surfaces together effect a lossless apodization, effectively reshaping the beam such that diffraction from aperture discontinuities is actively canceled before entering the coronagraph.

2. Algorithmic Framework and Implementation

ACAD is typically implemented as a two-step process:

1. Open-loop nonlinear mapping: The Monge–Ampère equation is numerically solved (e.g., via Newton–Kantorovich or multi-grid Newton–Krylov methods) to obtain DM surfaces that redistribute the input amplitude toward a target, often smooth apodization (Mazoyer et al., 2015, Mazoyer et al., 2017).
2. Closed-loop stroke minimization: Residual phase and amplitude errors after the open-loop step, as well as imperfections in the DM surfaces, are further suppressed via linear optimal wavefront control (e.g., SM, Electric Field Conjugation [EFC]), exploiting the fact that the final system remains in the small-aberration regime. The control problem is formulated as:

$\Delta h = - (J^H J + \mu I)^{-1} J^H E_0$

where $J$ is the interaction matrix mapping DM actuator strokes to focal-plane electric field changes, $\mu$ is a regularization parameter, and $E_0$ is the measured field (Mazoyer et al., 2017, Mazoyer et al., 2017).

The ACAD-OSM variant extends this by iteratively recalibrating $J$ as the DM surfaces evolve into the nonlinear regime, using an adaptive gain ($\gamma$) and target contrast update scheme to ensure stability and convergence. This recentering process typically involves $\sim$8 interaction matrix rebuilds, with total iterations reaching $\sim$1000–1500 for 48-actuator DMs (Mazoyer et al., 2017, Mazoyer et al., 2017, Mazoyer et al., 2017).

3. Optical Configuration and Coronagraph Integration

The typical ACAD optical layout is: input pupil (with discontinuities) $\rightarrow$ DM1 (pupil plane) $\rightarrow$ free-space propagation $z$ $\rightarrow$ DM2 $\rightarrow$ coronagraph (apodizer, focal-plane mask, Lyot stop) $\rightarrow$ science camera. Practical choices for DM size ($D$), inter-actuator pitch (IAP), actuator count ($N_\text{act}$), and DM separation ($z$) critically impact performance.

• For WFIRST-AFTA, two Boston-Micromachines or Xinetics DMs of $D=1\,\text{cm}$, $N_\text{act}=34$–$48$, $z=0.3$–$1\,\text{m}$ have been validated (Mazoyer et al., 2015, Mazoyer et al., 2017).
• The key geometry parameter is the Fresnel number $F_0 = D^2/(\lambda_0 z)$, with optimal performance near $F_0 \sim 500$–$1000$. Too small $F_0$ leads to vignetting and throughput loss; too large induces Talbot effects and large strokes (Mazoyer et al., 2017).

ACAD and ACAD-OSM are fully compatible with various coronagraph types, including Apodized Pupil Lyot Coronagraphs (APLC), Vortex Coronagraphs (PAVC), Phase-Induced Amplitude Apodization Complex Mask Coronagraphs (PIAACMC), and shaped-pupil coronagraphs. Hybrid static/active architectures are common: static apodizers handle axisymmetric central obscuration, active DMs suppress discontinuities (Mazoyer et al., 2017).

4. Performance Characterization

Empirical and simulated studies confirm that ACAD and ACAD-OSM enable:

• Monochromatic contrast: $\sim10^{-9}$–$10^{-11}$ in the dark hole (DH) region, typically 3–15 $\lambda/D$.
• Polychromatic contrast: $\sim10^{-8}$–$10^{-10}$ for 10% bandwidths; $10^{-9}$–$10^{-10}$ up to 30% bandwidth for optimized configurations following the “Shaklan law” ($C \propto R^{-2}$, $R=\lambda/\Delta\lambda$) (Mazoyer et al., 2017, Mazoyer et al., 2017, Mazoyer et al., 2017).
• Throughput: $>50\%$ at 5 $\lambda/D$ (vortex), typically $30$–$70\%$ depending on coronagraph type and spider thickness (Mazoyer et al., 2017, Mazoyer et al., 2017, Mazoyer et al., 2017).
• DM stroke budgets: Peak-to-valley strokes are well within hardware capability ($<500$ nm, often $100$–$300$ nm with ACAD-OSM) (Mazoyer et al., 2015, Mazoyer et al., 2017).

A summary table of benchmark results:

Aperture Type	Coronagraph	Contrast 10% BW	Mean Throughput	DM Stroke (nm)
WFIRST (36% obs.)	APLC, PAVC6	$10^{-9.4}$	58–67%	200–300
SCDA (17% obs., segs.)	APLC, PAVC6	$10^{-11.2}$	68–71%	100–150
E-ELT (30% obs., segs.)	PAVC6	$10^{-10.4}$	60%	200–250
LUVOIR (14% obs., segs.)	PAVC6	$10^{-10.6}$	35%	200–400

ACAD exhibits robust correction: contrast and throughput degrade modestly for missing segments, misalignments (apodizer/Lyot stop spatial errors $<0.2\% D$), and high-order phase errors (RMS $60$ nm), with final performance near nominal after re-optimization (Mazoyer et al., 2017, Mazoyer et al., 2017).

5. Parametric and Optimization Analyses

Systematic parametric analyses reveal:

• Smaller DM aperture $D$ (for fixed $N_\text{act}$) localizes strokes and preserves throughput, especially valuable at wider working angles.
• DM–DM separation $z$ is second-order but shorter separations (e.g., $0.3$ m) help mitigate off-axis point spread function (PSF) deformation.
• Apodized coronagraphs are preferred for on-axis geometries (central obscuration, thick spiders), while segmented off-axis pupils benefit from vortex designs (Mazoyer et al., 2017, Mazoyer et al., 2017).
• For broad bandwidths, actuator count ($N_\text{act} \gtrsim 48$) sets achievable outer working angle (OWA) and contrast floor.
• Stroke-minimization alone (starting from flat DMs) attains nearly identical contrast to the geometric ACAD solution but with substantially reduced strokes ($<150$ nm), maximizing throughput (Mazoyer et al., 2017).

6. Limitations, Extensions, and Future Development

Key assumptions for ACAD include the fidelity of Fresnel propagation (valid for $|A-1| \lesssim 0.1$ and $Z/D \gg 1$), perfect focal-plane field estimation, and negligible actuator nonidealities (quantization, hysteresis, calibration errors). Bench demonstrations, notably on HiCAT, are necessary to validate performance in hardware (Mazoyer et al., 2015, Mazoyer et al., 2017).

Highly aggressive apodizations (e.g., classical PIAA, $A \rightarrow 0$ at edges) require advanced propagation models (SR-Fresnel, polar S–Huygens). For multi-band operation ($20$–$30\%$ bandwidth) and larger apertures (LUVOIR), actuator arrays may need to be expanded to $64 \times 64$ or more (Mazoyer et al., 2017). Integration with advanced coronagraphs (higher-charge vortex, PIAACMC) and improved Monge–Ampère solvers remain active research areas.

A plausible implication is that future missions will benefit from co-optimization of DM geometry, static mask parameters, and actuator density, as well as adaptive control frameworks like ACAD-OSM that are robust to evolving aperture discontinuities, segmented architectures, and temporal wavefront errors.

7. Significance and Applications

ACAD and its optimized variants represent a transformative advance in coronagraphic wavefront control for both space and ground-based telescopes. By enabling Earth-like exoplanet imaging at contrasts $10^{-10}$–$10^{-12}$ and high scientific throughput, ACAD closes the gap between theoretical coronagraph designs (which often assume clear, monolithic apertures) and the realities of highly complex, segmented, and evolving telescope pupils. The method's adaptability to arbitrary pupil features, coronagraph architectures, and field aberrations positions it as a key enabling technology for direct detection and characterization of exoplanets, biosignature searches, and fundamental studies of planetary systems in the era of WFIRST, LUVOIR, HabEx, and the ELTs (Mazoyer et al., 2015, Mazoyer et al., 2017, Mazoyer et al., 2017).

Researchers designing future high-contrast imaging instruments are encouraged to exploit ACAD/ACAD-OSM, considering DM sizing, actuator count, separation, and control algorithms as outlined, to maximize contrast, throughput, and robustness for arbitrary telescope apertures.