High-Order Wavefront Sensing & Control

• High-Order Wavefront Sensing and Control is a set of methods designed to detect and correct mid-to-high spatial frequency optical aberrations in advanced coronagraph systems.
• The approach integrates focal-plane electric field estimation with both model-based and empirical control laws to maintain deep dark holes and achieve contrasts below 10⁻¹⁰.
• Implementation challenges include real-time computational demands on space-qualified hardware, requiring algorithmic innovations and co-designed processor architectures.

High-Order Wavefront Sensing and Control (HOWFSC) is a class of methodologies and system architectures designed to detect and correct mid- to high-spatial-frequency aberrations in next-generation optical systems, particularly coronagraphs intended for direct exoplanet imaging. HOWFSC enables the creation and maintenance of deep "dark holes" in coronagraphic science images, suppressing stellar leakage via high-order deformable mirror (DM) control. Modern HOWFSC systems are characterized by their integration of focal-plane-based electric field estimation, model-based or empirical control laws, and stringent requirements for stability, computational throughput, and minimal non-common-path errors. Achieving and holding contrasts below $10^{-9}$–$10^{-10}$ is central for missions such as the Habitable Worlds Observatory (HWO) (Milani et al., 26 Jun 2024).

1. Optical System Architecture and Error Sources

HOWFSC frameworks operate in complex coronagraphic systems, typically involving:

• Telescope: Off-axis, unobscured three-mirror-anastigmat (TMA) designs for minimal diffraction, with apertures up to 6–15 m. Wavefront error (WFE) sources include mid–high-spatial-frequency surface errors from polishing/coating, thermal and structural drifts, and pointing-induced beamwalk. The dominant WFE spectral power follows a power-law $PSD\propto f^{-2.7}$.
• Coronagraph: Modern architectures include charge-6 vortex coronagraphs (VVCs), shaped-pupil coronagraphs (SPC) and hybrid Lyot-type masks. DMs are located upstream of the focal-plane mask, often in dual-DM planes, to correct both phase and amplitude aberrations over a 2–10~$\lambda$/D "dark hole".
• Contrast Goals: Roman CGI targets raw contrast $\approx10^{-8}$ in its dark hole, while HWO is baseline-designed to maintain $C\leq10^{-10}$ over several hours.
• Quasi-static Error Timescales: Errors evolve on timescales from tens of minutes (thermal drift) to seconds (pointing). Dynamic range requirements and loop response set the necessary bandwidth for HOWFSC at $\sim$0.1–10 Hz (Milani et al., 26 Jun 2024).

2. Physical and Computational Modeling

HOWFSC performance prediction and optimization require sophisticated, multi-domain propagation modeling:

• Hybrid Ray-Wave Propagation: Raytracing (e.g., Zemax) yields footprints, local f/#, and prescription mapping. Surface roughness/PSD maps are optically propagated via Fresnel integrals (e.g., with POPPY), incorporating mid-high-frequency error features and manufacturing statistics.
• Talbot Effect and Beamwalk: Periodic polishing ripple is self-imaged (Talbot effect), introducing spatially coherent phase structures downstream. Pointing-induced pupil shifts (beamwalk) are modeled as sub-pixel shifts on WFE maps, e.g. for an M2 mirror a factor $S=0.084\,\mathrm{\mu m/mas}$ is applied to small line-of-sight tilts, implemented via Lanczos/FFT interpolation (Milani et al., 26 Jun 2024).
• FFT/Fraunhofer Models: For rapid control loop iteration, compact FFT-based coronagraph models are injected with pre-determined WFE. Fresnel propagation between two pupil planes separated by $z$ is expressed as

$$E_2(x,y) = e^{ikz} e^{i\frac{k}{2z}(x^2+y^2)} \otimes E_1(x,y),$$

with kernel $H(x,y;z) = \exp(i\frac{k}{2z}(x^2+y^2))$. Loop optical physics thus enters both field estimation and DM control law mapping.

3. Algorithms for High-Order Sensing and Control

Modern HOWFSC implements a suite of model-based or model-free estimation and control techniques:

• Electric Field Estimation: Complex field estimation in the dark hole is primarily realized by pair-wise probing (PWP), wherein small, orthogonal DM shapes (probes) are sequentially added, yielding differential images

$$\Delta I_j = I^+_j - I^-_j \approx 4\operatorname{Re}[E\cdot a_j^*].$$

Field $E$ is recovered by linear inversion across all probe pairs.

• EFC Control Law: Electric Field Conjugation (EFC) computes the DM voltage vector update

$\Delta u = -(J^H J + \alpha I)^{-1} J^H E$

where $J_{mn}=\partial E(f_m)/\partial u_n$ is the Jacobian, $E$ is the current field in the dark hole, and $\alpha$ is Tikhonov regularization. For iteration $k$:

$$u_{k+1} = u_k - \lambda(J^H J + \alpha I)^{-1} J^H E_k, \quad 0<\lambda\leq1.$$

"Relinearization" of the Jacobian is often only performed every $N$ iterations.

• Empirical Algorithms (iEFC): Implicit EFC (iEFC) dispenses with a physics-based model, instead empirically calibrating the response $(G_{IEFC})$ of focal-plane intensity difference images to DM modal pokes using in-situ measurements. The control step becomes:

$$m_c = - (G_{IEFC}^T G_{IEFC} + \lambda I)^{-1} G_{IEFC}^T \delta,$$

directly mapping difference images to DM commands. This approach is robust to unmodeled NCPAs and model uncertainties but requires longer on-sky or in-lab calibrations (Milani et al., 6 May 2024, Milani et al., 2 Sep 2025).

4. Implementation on Space-Qualified and Embedded Processors

The stringent computational requirements of rapid, high-dimensional field estimation and control necessitate careful hardware/firmware design:

• Processor Classes: Architectures under study include BAE RAD750 (200 MHz, 1.4 GFLOPS peak), radiation-qualified RISC-V, and Jetson/NVIDIA embedded GPUs (200–500 GFLOPS). For a representative 34$\times$34 DM and 100$\times$100-pixel dark hole, one EFC iteration demands $\sim$10–20 GFLOP and $\sim$160 MB memory for the Jacobian alone (Milani et al., 26 Jun 2024).
• Real-Time Constraints: Maintenance of deep dark holes requires that control cycles complete in $\leq$5 s, ideally $\leq$1 s to guard against thermally/pointing-induced speckle drift. On a RAD750-class CPU, naive EFC runs take $\gg$60 s per iteration; sparse/algorithmically compressed or AD/Jacobian-free variants are required.
• Hardware Acceleration and Optimization: Modal EFC, modal PWP with actuator-mode reduction, and Jacobian-free nonlinear optimization via autodifferentiation (AD/E-FC) can reduce effective computational load. Dedicated radiation-hardened GPUs or FPGAs (e.g. Xilinx Virtex) can achieve $>20$ GFLOPS sustained performance within onboard power budgets.

5. Performance Metrics and Simulation Results

Simulation and lab benchmarks establish key system-level performance characteristics:

• Dark Hole Creation and Recovery: Starting from raw WFE (RMS $\sim$100 nm PV), 18 iterations of monochromatic EFC with one Jacobian relinearization (at $k=9$) yield mean contrast $\lesssim 10^{-10}$ (3–10 $\lambda/D$) in both 34$\times$34 and 68$\times$68 DM configurations (Milani et al., 26 Jun 2024).
• Dynamic Disturbance Injection: 15 mas pointing errors ("beamwalk drift") re-inject speckle leakage, degrading contrast by $\sim$1 dex to $10^{-9}$. Three additional EFC iterations restore $10^{-10}$ performance in simulated real-time operation.
• Convergence Law: Contrast exhibits exponential decay with iteration number:

$C(k) \simeq C_0 (1-\gamma)^k,$

where typical $\gamma\sim0.2$–$0.3$; floor is set by regularization $\alpha$ and DM stroke.

• Temporal Maintenance: Real-time EFC cycles of 3 s enable contrast stabilization at 10 s response to drift events—compatible with speckle dynamics in realistic environments.

6. Strategic Recommendations for HWO-Scale Missions

From system- and mission-level analyses, clear recommendations emerge:

• Continuous Operation: Active HOWFSC must run continuously at $\leq$10 s cadence (and as fast as 1 s) to maintain $10^{-10}$ contrast against both slow quasi-static drifts and faster pointing-induced errors (Milani et al., 26 Jun 2024).
• Processor Design: Marginal performance is attainable with high-end rad-hard CPUs alone; robust, scalable operation at $\sim$5 s loop times requires either radical algorithmic simplification (e.g. modal/empirical control) or dedicated co-processor hardware.
• Algorithmic Advances: Modal dimensionality reduction, AD/Jacobian-free EFC, and adaptive regularization can offset hardware constraints. Hierarchical loop partitioning (fast LOWFSC for low-order tip/tilt/focus, HOWFSC for mid-high orders) is essential.
• Hardware/Algorithm Co-Design: System-level trades between actuator count, model fidelity, and update bandwidth must be mapped to embedded processor/accelerator design as a joint optimization (Milani et al., 26 Jun 2024).

7. Synthesis, Impact, and Future Directions

HOWFSC constitutes the enabling subsystem for deep-contrast coronagraphy (direct exo-Earth imaging, biosignature spectroscopy) on both flagship space and advanced ground-based observatories. Modeling by Milani et al. (Milani et al., 26 Jun 2024) demonstrates that current field-estimation and control-loop formalisms, when executed with sufficiently high-fidelity optics and processor capability, can simultaneously suppress quasi-static and drift-induced speckle to $C\lesssim10^{-10}$—provided continuous, multi-second cadence maintenance. Integration of Talbot/beamwalk physics, Fresnel-propagation-based models, and adaptive control algorithms positions HOWFSC as the fundamental bridge between optical engineering and mission-level science yield. Future work will focus on scaling empirical calibration methods (iEFC) to larger DM mode counts, deploying advanced hardware architectures, and further refining spatio-temporal control laws to meet the anticipated demands of HWO and successors in the direct imaging of habitable exoplanets.