Focal Error Diversity in Wavefront Sensing

• Focal Error Diversity is a technique that introduces controlled focal-plane perturbations to convert ambiguous, nonlinear measurements into well-conditioned, linear estimation problems.
• It employs methods like pairwise deformable mirror probing, defocus, vortex phase, sequential diversity, and temporal amplitude modulation to make intensity-only data informative.
• This approach enhances imaging contrast and enables precise aberration correction in applications such as high-contrast coronagraphy, optical cophasing, and non-common-path aberration control.

Focal error diversity denotes the deliberate introduction or exploitation of a known perturbation so that intensity-only focal-plane measurements become informative about otherwise ambiguous phase or complex-field variables. In focal-plane wavefront sensing and control, this perturbation may be a deformable-mirror probe, defocus, a vortex-induced focal-plane asymmetry, a sequential DM command, or temporal amplitude modulation with an optical chopper. Its role is to convert a nonlinear or sign-ambiguous inverse problem into one that is linearized, better conditioned, or temporally identifiable, thereby enabling estimation of the focal-plane electric field, pupil-plane phase, or discontinuous aberrations needed for high-contrast imaging, cophasing, and non-common-path aberration correction (Groff et al., 2012, Vievard et al., 2020, Nousiainen et al., 1 Apr 2026, Gerard et al., 2023).

1. Optical definition and observability

In focal-plane estimation for coronagraphy, let $E \in \mathbb{C}$ denote the speckle field at a science pixel. The measured intensity is $I = |E|^2$, which is quadratic and insensitive to the sign of the field’s phase; the mapping from $E$ to $I$ is many-to-one. This is the fundamental observability problem that motivates focal error diversity. Applying a known probe field $E_p$ by commanding a DM shape $p$ and forming two images with $\pm p$ yields

$$I^{\pm} = |E \pm E_p|^2 = |E|^2 + |E_p|^2 \pm 2 \operatorname{Re}(E^* E_p),$$

so that differencing cancels $|E|^2$ and $|E_p|^2$ and isolates the linear cross term

$I = |E|^2$0

Writing $I = |E|^2$1 and $I = |E|^2$2 gives

$I = |E|^2$3

which is linear in the unknown real and imaginary parts. Stacking pixels and probe pairs yields a linear system $I = |E|^2$4 whose conditioning is set directly by the chosen diversity (Groff et al., 2012).

An analogous ambiguity appears in phase retrieval. With a single in-focus PSF and a centrosymmetric pupil, even pupil-plane phase modes are sign ambiguous: intensity-only measurements cannot distinguish $I = |E|^2$5 from $I = |E|^2$6. The sequential phase-diversity formulation states this explicitly as

$I = |E|^2$7

and notes that the ambiguity is stronger for perfect-coronagraph images in the small-aberration regime. Small-aberration non-coronagraphic imaging, coronagraphic imaging, and vortex coronagraphs differ in how strongly this degeneracy appears, but all require some known diversity to recover sign and amplitude information (Nousiainen et al., 1 Apr 2026, Quesnel et al., 2022).

The same observability issue underlies focal-plane sensing with an optical chopper. For an even amplitude pupil, even modes are invisible at first order in a single in-focus PSF because the first-order perturbation

$I = |E|^2$8

vanishes for even phase modes. Alternating between an unobstructed pupil and a partially blocked, non-centrosymmetric pupil makes the reference field complex and restores first-order sensitivity to the sign of even modes (Gerard et al., 2023).

2. Physical mechanisms used to create diversity

Pairwise DM probing introduces conjugate $I = |E|^2$9 probe shapes on one or more deformable mirrors. In the Princeton HCIL formulation, the probe-induced field $E$0 enters each measurement row as

$E$1

so pairwise probing makes the measurement effectively linear in the field and enables electric-field estimation in the dark hole. This is the canonical “DM diversity” construction in focal-plane wavefront correction (Groff et al., 2012).

Defocus diversity is the classical phase-diversity mechanism. Linearized Analytic Phase Diversity uses a focused image and a defocused image produced by a known defocus phase term $E$2,

$E$3

and exploits the differing response of the two planes to recover piston, tip, and tilt of segmented or multiple-aperture telescopes. In that context, focal error diversity is explicitly identified with defocus diversity (Vievard et al., 2020). The same mechanism is used for external phase discontinuities in SPHERE and Keck, where a single diverse image or a focused/defocused pair constrains low-wind-effect aberrations and segment piston errors (Lamb et al., 2017).

Vortex phase diversity uses the azimuthal phase ramp of a scalar or vector vortex coronagraph to break the $E$4 ambiguity without defocus exposures. In the vector-vortex case, the two circular polarizations provide two complementary post-coronagraphic PSFs with opposite effective charges, while the scalar-vortex case provides a single post-coronagraphic PSF that still contains usable focal-plane diversity. The reported formulation states that both SVC and VVC can lift the sign ambiguity, with the VVC dual-polar case offering two diversity channels (Quesnel et al., 2022).

Sequential phase diversity uses the natural time sequence of DM commands. In extremely fast focal-plane sensing, the DM itself introduces the known diversity between successive frames, and the diversity is chosen to reduce the aberration while simplifying the inverse problem in the small-aberration regime. In model-based reinforcement learning for non-common-path aberrations, the same principle appears as temporal diversity encoded in the state

$E$5

where the previous image and previous command, together with the current image, supply the information needed to resolve phase ambiguity (Keller et al., 2012, Nousiainen et al., 1 Apr 2026).

Temporal amplitude diversity replaces phase diversity by amplitude modulation. The optical chopper alternates between two pupil amplitude states, one of which is intentionally non-centrosymmetric. Synchronous demodulation isolates a modulated term that is linearly sensitive to even modes while leaving the un-chopped image usable for science. This differs from classical phase diversity by generating diversity in the pupil amplitude rather than through added defocus (Gerard et al., 2023).

3. Estimation and control formulations

The Kalman-filter formulation treats the focal-plane field as a quasi-static state with DM-driven evolution,

$E$6

and measurements

$E$7

Prediction and update are

$E$8

$E$9

$I$0

$I$1

Within this framework, diversity sets $I$2, and therefore directly sets the information added by the new exposure. The paper emphasizes that poor-SNR measurements cannot worsen $I$3 because the update is a positive semi-definite information add (Groff et al., 2012).

LAPD linearizes the focused/defocused PSF pair about a current estimate $I$4:

$I$5

Under independent Gaussian noise, the criterion becomes quadratic,

$I$6

and the analytic update is

$I$7

The pseudoinverse is computed by SVD, with small singular values thresholded so that global piston, global tip, and global tilt are automatically filtered (Vievard et al., 2020).

Sequential phase-diversity wavefront sensing uses an explicit odd/even PSF decomposition. With small aberrations and a second-order correction, the PSF is written as

$I$8

The odd component is estimated directly from $I$9, while the sign of the even component is determined from the known DM-induced diversity $E_p$0 using

$E_p$1

This yields a one-image-per-update reconstruction with one complex FFT per iteration (Keller et al., 2012).

In model-based reinforcement learning, the estimator is implicit rather than explicit. The learned world model $E_p$2 predicts the next preprocessed image, and the policy $E_p$3 optimizes a reward defined on focal-plane residual intensity,

$E_p$4

with $E_p$5. The paper states that estimation is embodied in the policy’s use of $E_p$6 rather than in an explicit reconstruction of $E_p$7 (Nousiainen et al., 1 Apr 2026).

4. Information content, conditioning, and probe design

The choice of diversity determines not only identifiability but estimator quality. In the Kalman formulation, the Fisher information contributed by a measurement is

$E_p$8

and the posterior covariance is

$E_p$9

Probe fields that produce rows in $p$0 that are large in norm, well distributed across spatial frequencies in the dark hole, and mutually orthogonal in the real-imag sense increase the eigenvalues of $p$1, improve conditioning, and reduce $p$2. Poorly chosen probes produce near-collinear rows and leave combinations of $p$3 and $p$4 unobservable (Groff et al., 2012).

This conditioning argument governs practical design rules. Batch least-squares generally needs at least $p$5–$p$6 probe pairs for stable inversion, whereas a Kalman filter can operate with $p$7–$p$8 pairs per iteration because prior information supplies rank. Probe amplitudes should produce sufficient $p$9 SNR while staying within the linear regime of the DM-to-field model. Two DMs, or propagation that produces both phase and amplitude content, are needed to observe both real and imaginary parts across a symmetric dark hole; collapsing both control shapes onto one DM reduces diversity and limits contrast (Groff et al., 2012).

Control-shaped probes make this co-design explicit. Stroke minimization chooses DM commands to minimize $\pm p$0 subject to a dark-hole contrast target, using the quadratic approximation

$\pm p$1

Using the control shape itself, and its conjugate, as the estimation probe concentrates modulation where speckles are bright and reduces the number of exposures. The reported preliminary test showed asymmetries when both DM shapes were collapsed onto one DM, and the paper states that reformulating $\pm p$2 to incorporate both DMs addresses this limitation (Groff et al., 2012).

Defocus diversity has analogous design trade-offs. In LAPD, both LAPD and classic PD showed decreasing estimation error as the defocus amplitude increased, reaching a plateau slightly above $\pm p$3 RMS when defocus $\pm p$4; the paper adopts $\pm p$5 defocus. Larger defocus improves diversity and capture range symmetry, but very large defocus can reduce sensitivity and broaden PSFs, affecting SNR per pixel (Vievard et al., 2020).

Amplitude-modulation diversity has its own design window. For the optical-chopper method, simulations reported that blocking approximately $\pm p$6–$\pm p$7 of the pupil provides the best balance between sensitivity and linearity, while a straight-edge chopping geometry retains sensitivity up to approximately $\pm p$8–$\pm p$9 cycles per pupil (Gerard et al., 2023).

5. Demonstrated performance and operational trade-offs

The reported literature shows that focal error diversity changes exposure count, capture range, convergence rate, and achievable residual. The following representative outcomes were reported under the corresponding system assumptions.

Context	Diversity mechanism	Representative reported result
Princeton HCIL coronagraph	Kalman filter with pairwise DM probing	$$I^{\pm} = |E \pm E_p|^2 = |E|^2 + |E_p|^2 \pm 2 \operatorname{Re}(E^* E_p),$$0 in 30 iterations with 2 pairs; $$I^{\pm} = |E \pm E_p|^2 = |E|^2 + |E_p|^2 \pm 2 \operatorname{Re}(E^* E_p),$$1 in 30 iterations using 86 estimation images with 1 pair
Princeton HCIL coronagraph	Control-as-probe	Preliminary suppression to $$I^{\pm} = |E \pm E_p|^2 = |E|^2 + |E_p|^2 \pm 2 \operatorname{Re}(E^* E_p),$$2
LAPD cophasing	Focused + defocused images with $$I^{\pm} = |E \pm E_p|^2 = |E|^2 + |E_p|^2 \pm 2 \operatorname{Re}(E^* E_p),$$3 defocus	Final wavefront dispersion $$I^{\pm} = |E \pm E_p|^2 = |E|^2 + |E_p|^2 \pm 2 \operatorname{Re}(E^* E_p),$$4 RMS
Model-based RL for NCPAs	Sequential phase diversity in state $$I^{\pm} = |E \pm E_p|^2 = |E|^2 + |E_p|^2 \pm 2 \operatorname{Re}(E^* E_p),$$5	Convergence in $$I^{\pm} = |E \pm E_p|^2 = |E|^2 + |E_p|^2 \pm 2 \operatorname{Re}(E^* E_p),$$6 steps per episode; inference time $$I^{\pm} = |E \pm E_p|^2 = |E|^2 + |E_p|^2 \pm 2 \operatorname{Re}(E^* E_p),$$7 ms
Vortex phase diversity with CNN	SVC or VVC focal-plane diversity	Residuals as low as $$I^{\pm} = |E \pm E_p|^2 = |E|^2 + |E_p|^2 \pm 2 \operatorname{Re}(E^* E_p),$$8 nm RMS from 70 nm RMS input for a bright source
Optical chopper FP-WFS	Temporal amplitude modulation	Strehl increased from $$I^{\pm} = |E \pm E_p|^2 = |E|^2 + |E_p|^2 \pm 2 \operatorname{Re}(E^* E_p),$$9 to $|E|^2$0
SPHERE/Keck discontinuity sensing	Defocus diversity and single-image PD	LWE estimated to 30 nm RMS WFE; Keck piston error estimated to 29 nm RMS WFE

In the Princeton HCIL experiments, standard pairwise least-squares with four probe pairs used eight images per iteration and achieved contrast $|E|^2$1 in $|E|^2$2 iterations and $|E|^2$3 in $|E|^2$4 iterations. The Kalman filter retained similar or better final contrast while reducing exposures: with three pairs it reached $|E|^2$5 in $|E|^2$6 iterations, with two pairs it reached $|E|^2$7 in $|E|^2$8 iterations after tuning $|E|^2$9, $|E_p|^2$0, and $|E_p|^2$1, and with one probe pair it reached $|E_p|^2$2 in $|E_p|^2$3 iterations by cycling probe shapes over time (Groff et al., 2012).

For fine cophasing, LAPD with a unilateral defocus of $|E_p|^2$4 reached a final wavefront dispersion of $|E_p|^2$5 RMS on ONERA’s BRISE bench. Its linear capture range for piston expanded from roughly $|E_p|^2$6 without internal iterations to about $|E_p|^2$7 with three internal iterations; for tilt, three iterations yielded approximately $|E_p|^2$8, essentially matching classic PD. The reported implementation ran about three times faster than classic iterative PD (Vievard et al., 2020).

Sequential phase diversity in model-based reinforcement learning was evaluated in static and dynamic non-common-path aberration regimes. In static non-coronagraphic simulations, Strehl improved from approximately $|E_p|^2$9 to approximately $I = |E|^2$00; in static perfect-coronagraph simulations, total residual flux was $I = |E|^2$01 versus $I = |E|^2$02 for fitting error projection. In dynamic SI, long-exposure Strehl was $I = |E|^2$03, matching the delay-compensated modal least-squares baseline closely, and the method remained effective for the ELT pupil, vector vortex coronagraph, and photon and background noise (Nousiainen et al., 1 Apr 2026).

The deep-learning vortex-diversity study reported that both SVC and VVC dual-polar inputs lift the sign ambiguity and achieve nearly identical residuals to classical defocus phase diversity in the low-aberration regime across the tested SNR range. For a bright source in K band, the VVC dual-polar case yielded residuals as low as $I = |E|^2$04 nm RMS from a 70 nm RMS input. The principal operational claim is a 100% science duty cycle for instruments using a vortex coronagraph, with no additional hardware required in the SVC case (Quesnel et al., 2022).

The optical-chopper implementation provided simultaneous science imaging at up to a 50% duty cycle and demonstrated closed-loop stabilization on SEAL. For DM-injected residual AO turbulence normalized to 100 nm RMS, telemetry showed approximately $I = |E|^2$05 reduction in total WFE in closed loop, and science-image Strehl increased from approximately $I = |E|^2$06 to approximately $I = |E|^2$07. The same study also reported low-order linearity for Zernike modes $I = |E|^2$08 over less than or equal to 200 nm PV per modal group (Gerard et al., 2023).

For aberrations external to AO systems, classic phase diversity on simulated SPHERE low-wind-effect data estimated LWE to within 30 nm RMS WFE, which the paper identifies as within the allowable tolerances for a target SPHERE contrast of $I = |E|^2$09. Single-image PD using DTTS’s natural defocus yielded approximately 62 nm RMS on the LWE case, while a single diverse Keck/NIRC2 image with 1.5 waves PV of focus estimated segment piston errors to within 29 nm RMS WFE and would increase Strehl by approximately 12% under perfect correction (Lamb et al., 2017).

6. Broader uses of “focal” and “error diversity” outside focal-plane optics

Outside focal-plane wavefront sensing, related papers use the same vocabulary for error mechanisms rather than focal-plane perturbations. In selective-fading multiple-access MIMO, the “dominant (focal) error event region” is the subset of users whose error probability decays slowest with SNR; the optimal diversity order is

$I = |E|^2$10

so system diversity is set by the focal error mechanism rather than by optical diversity injection (0805.0131). A related rate-dependent phenomenon appears for MMSE MIMO receivers, where diversity varies with spectral efficiency and transitions from ML-like maximal diversity at sufficiently low rates to ZF-like minimal diversity at high rates (Mehana et al., 2011).

In ensemble learning, “error diversity” denotes complementary failures across models. For unsupervised dependency parsing, society entropy measures the dispersion of predicted heads,

$I = |E|^2$11

and ensemble selection explicitly rewards sets that avoid error accumulation (Shayegh et al., 2024). In deep-ensemble pruning, focal diversity metrics compute diversity only on the focal model’s error samples, formalized through focal versions of CK, BD, KW, and GD and aggregated into a weighted focal diversity score used by hierarchical pruning (Wu et al., 2023). FusionShot adopts an episode-based focal negative correlation,

$I = |E|^2$12

and uses the ensemble-level average $I = |E|^2$13 to rank few-shot ensembles (Tekin et al., 2024). MARL-Focal defines focal diversity from failure-conditioned co-failure probabilities and uses it inside a decider agent that selects complementary LLM-based agents (Tekin et al., 6 Feb 2025).

In generative modeling, the term is mapped to diversity error or diversity bias: a systematic shortfall in the diversity of generated samples relative to the data distribution. The cited paper studies entropy-based diversity scores such as Vendi,

$I = |E|^2$14

and RKE,

$I = |E|^2$15

and shows that expected log-Vendi increases monotonically with sample size, implying finite-sample underestimation of entropy-based diversity (Farnia et al., 16 Feb 2026).

This broader record suggests that “focal error diversity” has become a domain-dependent term. In optics it denotes a controlled diversity channel that makes an inverse problem observable; in communications it denotes the dominant error-event subset that fixes the diversity exponent; and in machine learning it denotes complementary or underrepresented error structure that can be quantified, selected, or regularized.