Roman Coronagraph: High-Order Wavefront Sensing

• Roman CGI HOWFSC is a framework for high-order wavefront sensing and control, integrating advanced algorithms, DM calibration, and GITL offload.
• It employs pairwise probing and varied DM probe designs to linearize the complex electric field while maintaining a full 360° dark hole.
• The system achieves raw contrasts ≤5×10⁻⁸ through regularized least-squares control, robust calibration, and a dual-DM architecture.

High-order wavefront sensing and control (HOWFSC) is central to the Roman Space Telescope’s Coronagraph Instrument (CGI), enabling raw contrast requirements (≤ 5×10⁻⁸) necessary for the direct imaging and spectroscopic characterization of exoplanets. HOWFSC encompasses the algorithms, sensor architectures, DM calibration and estimation strategies that iteratively sense and correct the complex electric field in the focal plane over the “dark hole” region, while simultaneously coexisting with ultra-stable low-order wavefront control. The Roman CGI implementation is a reference for space-based coronagraphy, integrating mathematical modeling, hardware design, empirical calibration, and ground-in-the-loop (GITL) computational offload.

1. System Architecture and Algorithmic Framework

The HOWFSC framework in Roman CGI consists of two tightly connected stages: electric field estimation (wavefront sensing) and DM control (wavefront correction). The algorithmic workflow can be summarized as:

• Pairwise Probing Electric Field Estimation: The focal-plane electric field E(x, y) is estimated by modulating the deformable mirrors (DMs) with designed probe patterns and recording intensity images. A set of positive and negative probes (Δψ and –Δψ) are applied. The difference between the probed images enables linearized estimation of the complex electric field, exploiting the relation:

$$\Delta I(x, y) = I_{+}(x, y) - I_{-}(x, y) \approx 4\operatorname{Re}\left\{i E(x, y) \left[\mathcal{C}\{\Psi\}\right]^*\right\}$$

where $\mathcal{C}$ is the coronagraph propagation operator and $\Psi$ the probe.

• Electric Field Conjugation (EFC): Once the field is estimated, actuator updates $\Delta a$ are computed via a regularized least-squares solution:

$$\Delta a = - \left(J^\top J + \lambda I\right)^{-1} J^\top E$$

$J$ is the Jacobian (response matrix), $\lambda$ the Tikhonov regularization parameter, and $E$ the estimated field.

• Real and Imaginary Splitting: All matrices and observations are split into real and imaginary components, doubling the number of equations, to linearize the complex-valued system.
• Ground-in-the-Loop (GITL): High-performance algorithms (field estimation, Jacobian updates, SVD-based mode selection, and regularization) are executed on ground-based servers. Only compressed focal-plane images and telemetry are downlinked for processing; optimized DM control vectors and settings are uplinked. This reduces spacecraft computational demands, mitigates risk, and facilitates algorithmic evolution during the mission (Cady et al., 31 Jul 2025).
• Modularity: The HOWFSC optical model, mask definitions, DM configuration, and control logic are defined via configuration files (e.g., in YAML), enabling rapid reconfiguration and test of coronagraph modes (Hybrid Lyot, Shaped Pupil, etc.).

2. Probe Design and Electric Field Estimation

The choice of DM probe patterns is critical in achieving linearity, high SNR, and robust convergence:

• Classic Sinc–Sinc–Sine Probes: These are designed to achieve targeted rectangular modulations in the DH but are sensitive to non-linearities at large amplitudes, resulting in estimation bias when SNR is low or strong probing is required.
• Single-Actuator Probes: By “poking” individual DM actuators, the field modulation is broad and uniform, yielding lower non-linear contributions even at high amplitudes. Hardware testing and simulations on THD2 indicate these designs expedite convergence and allow higher probe strength for the same contrast degradation, minimizing exposure time and loop iteration count (Laginja et al., 1 Apr 2025).
• Sharp Sinc Probes: Undersampled sinc functions, which lie between the above in robustness to amplitude-driven non-linearity.
• Linearity Considerations: All probe designs rely on the first-order expansion $e^{i\Psi} \approx 1 + i\Psi$, valid when probe amplitude is sufficiently small. Higher probe amplitudes introduce non-linear (second/third-order) bias; single-actuator probes are most robust.

3. Deformable Mirror Modeling, Calibration, and Response Matrices

• DM Physical Modeling: Calibrations include lateral offsets, scales, actuator influence functions, voltage–displacement relationships, measured gain variations, dead/weak actuators, and spatial coupling. DM calibration experiments typically reach 10% accuracy for small displacements—sufficient for sub-nanometer stability in closed-loop operation (0911.1307, Krist et al., 2023).
• Jacobians and Influence Functions: The response matrix $J$ is updated at each iteration, either utilizing fast localized influence functions or SVD-based recalculation. Regularization (often with adaptive “beta bumping”) is essential for stability in the presence of low-SNR, DM non-linearities, and coherent/incoherent light separation (Cady et al., 31 Jul 2025).
• Empirical vs Model-Based Calibration: Classical methods rely on a precomputed optical/diffraction model for $J$. Model-free approaches such as implicit EFC (iEFC) empirically calibrate the full instrument response, including optical misalignments, using double-difference images for each DM mode. iEFC is robust to model uncertainties (e.g., unexpected DM rotation/translation after launch), but at the cost of increased calibration time and exposure requirements, particularly in broadband and high-noise regimes (Milani et al., 6 May 2024, Milani et al., 2023).

4. Two-Tiered Correction: Dual Deformable Mirrors, Loop Synchronization, and Control Regions

• Two-DM Architecture: Roman CGI deploys two DMs: DM1 is pupil conjugate, correcting dominant phase errors, while DM2 is out-of-pupil, introducing amplitude modulations via Fresnel propagation. The concatenated Jacobian $G_{\text{EFC}} = [G_1, G_2]$ governs the system’s response. Two-DM control is necessary for simultaneously correcting phase and amplitude aberrations and for forming an annular, 360° dark hole (Krist et al., 2023, Mazoyer et al., 2017).
• Spatial Control Region: The dark hole in both HLC and SPC modes is fully annular (3–9 λ/D for HLC, 6–20 λ/D for SPC), requiring the probing, estimation, and control vectors to sample a 360° region.
• Loop Synchronization: High-order (HOWFSC, running at ~0.1 Hz) and low-order (LOWFSC, correcting tip/tilt and other bulk errors, typically at ~5–80 Hz via Zernike phase-contrast) loops operate simultaneously, coordinated to ensure that low-order drift correction does not counteract dark hole digging. Strategies such as reference offsetting and careful calibration of both control bases are necessary to decouple the targets of each loop (Milani et al., 2 Sep 2025, Soummer et al., 19 Sep 2024).

5. Performance Demonstration and Influencing Factors

• Laboratory Measured Contrasts: During thermal vacuum testing, Roman CGI achieved total static raw contrasts better than $5\times10^{-8}$ in both HLC and SPC architectures, with coherent components near $10^{-8}$ and incoherent light (from ghosts, polarization, residual static errors) near $2-3\times10^{-8}$ (Cady et al., 31 Jul 2025). No technical “floor” was observed, indicating further iteration would likely yield deeper contrast. High throughput (∼94%) and stability at the $3.5 \times 10^{-9}$ level over hours were demonstrated in precursor PIAA laboratory experiments (0911.1307).
• Convergence and Calibration: The full loop—including early Jacobian calculation, probe optimization, DM update, and exposure/band selection—can be closed in several minutes for moderate SNR, but calibration and noise mitigation in iEFC may require total exposure times of ∼7 hours (e.g., for 4096 probe modes at dark-hole depths near $1\times10^{-8}$ under realistic photon/detector noise and ζ Puppis reference star) (Milani et al., 6 May 2024).
• Influences on Performance: Final contrast is driven by the combination of model precision (for classical EFC), calibration accuracy (for iEFC), probe design non-linearities, noise properties (photon, read, and dark current), and control region geometry. Regularization parameter scheduling, aggressive β bumping for SVD stabilization, and probe modulation amplitude tuning are required to avoid stagnation and to suppress both static and dynamic speckles (Laginja et al., 1 Apr 2025).

6. Future Directions and Advanced Enhancements

• Model-Free and Hybrid Control: iEFC is a baseline alternative for scenarios where optical model inaccuracies are dominant or evolve unpredictably in flight. Its robustness is partially offset by long calibration time; hybrid strategies where initial dark hole formation is done via iEFC and subsequent depth by classical EFC may be optimal (Milani et al., 2023, Milani et al., 6 May 2024).
• Probe Optimization and Non-Linearity Management: Single-actuator DM probing is recommended for resilience to non-linearities, higher achievable probe amplitudes, and efficient operation in photon-limited regimes. Other enhancements (Hadamard/frequency-limited probe sets, sharp sinc patterns) provide secondary gains (Laginja et al., 1 Apr 2025).
• Simulation and Validation Infrastructure: End-to-end modeling using physical optics codes such as PROPER and POPPY, coupled to thermal/structural finite-element modeling (STOP), underpins both hardware validation and flight operations. Realistic noise propagation, wavefront error propagation, and DM influence calibration are built into the control pipeline (Krist et al., 2023, Milani et al., 2021).
• 360° Dark Hole and Dual Coronagraph Architecture: Having both HLC and SPC architectures, each capable of creating a full 360° dark hole, allows operational redundancy and flexibility for unexpected in-flight perturbations.
• Extensions for Maintainability: Strategies such as spatial linear dark field control (sLDFC), which uses the measured response of the bright field outside the DH to maintain contrast without costly recalibration, represent practical enhancements under active development (Currie et al., 10 Sep 2025).

7. Summary Table: Key Elements of Roman CGI HOWFSC Implementation

Aspect	Implementation Detail / Outcome	Reference
Sensing Technique	Pairwise probing; single-actuator preferred for linearity, iEFC for model-free operation	(Laginja et al., 1 Apr 2025Milani et al., 6 May 2024)
Control Law	Regularized least-squares (EFC), SVD-based mode weighting, β bumping	(Cady et al., 31 Jul 2025)
Calibration	Empirical (iEFC) or model-based Jacobian; 2 DMs (34×34 actuators)	(Milani et al., 6 May 2024Cady et al., 31 Jul 2025)
Loop Execution	Ground-in-the-loop (GITL) for expensive computations; on-board for command execution	(Cady et al., 31 Jul 2025)
Achieved Raw Contrast	$<5\times10^{-8}$ (static, coherent+incoherent), 3–20 λ/D annular DH, 360° (verified in TVAC)	(Cady et al., 31 Jul 2025)
Calibration Overhead	Several minutes per iteration (classical); several hours total (iEFC with 4096 modes, SNR-governed)	(Milani et al., 6 May 2024)
Future Enhancements	Use of sLDFC, efficient bright field control, advanced probe designs, hybrid EFC approaches	(Currie et al., 10 Sep 2025Laginja et al., 1 Apr 2025)

Conclusion

The Roman CGI HOWFSC architecture integrates regularized, empirically-calibrated DM control with advanced probe engineering, robust GITL offload of computationally intensive routines, and validated physical optics modeling. Laboratory thermal-vacuum testing demonstrates requirements-exceeding contrast, showing no hard technical floor within tested parameter space. The coming generation of space coronagraphs, and especially future missions aiming for even deeper contrasts, will build directly on these validated strategies, with particular emphasis on probe linearity, model-free control, and synergistic coordination of high- and low-order correction on shared hardware platforms.