Focal-Plane Wavefront Sensing Methods

• FPWFS is a technique that measures intensity at the focal plane to directly infer aberrations affecting image quality.
• It encompasses methods such as probe-based phase retrieval, Fourier-based sensors, and data-driven approaches that resolve phase ambiguity.
• Applications include enhancing high-contrast imaging and adaptive optics in telescopes, integrating with coronagraph systems for precise control.

A focal-plane wavefront sensor (FPWFS) is a class of wavefront-sensing methods that infers aberrations from intensity measurements formed at, or directly derived from, the science focal plane itself. In high-contrast imaging and adaptive optics, this placement is decisive because non-common-path aberrations and quasi-static speckles are sensed where they limit the final image, rather than on an upstream pupil-plane sensor. The term encompasses several distinct but related families: direct phase retrieval from focal-plane images, probe-based complex-field estimation, Fourier-based sensors that encode phase through a focal-plane mask, coronagraph-integrated sensors, and more recent data-driven and photonic implementations (Dou et al., 2015, Janin-Potiron et al., 2019, Quesnel et al., 2022).

1. Formal basis and measurement model

The common forward model is Fourier-optical. For a pupil transmission $P(\boldsymbol{\xi})$ and pupil-plane phase $\phi(\boldsymbol{\xi})$, the focal-plane field and intensity are written as

$$E_f(\boldsymbol{\rho}) = \mathcal{F}\{P(\boldsymbol{\xi}) e^{i\phi(\boldsymbol{\xi})}\}, \qquad I(\boldsymbol{\rho}) = |E_f(\boldsymbol{\rho})|^2.$$

In the small-phase regime, $e^{i\phi}\approx 1+i\phi$, so intensity perturbations become approximately linear in the phase, which yields the standard first-order sensitivity operator used across many FPWFS formulations. The same linearization underlies modal reconstructions in Zernike bases, weak-phase feedback methods, and several phase-diversity solvers (Xivry et al., 2021).

A broader class of Fourier-based FPWFS inserts a focal-plane mask with complex transmission $M(u)$ and re-images a pupil plane after the mask. In that case,

$$e_m(x)=\mathcal{F}^{-1}\{M(u)\,E_f(u)\}=h(x) * \big(P(x)e^{i\varphi(x)}\big), \qquad I(x)=|e_m(x)|^2,$$

with $h(x)=\mathcal{F}^{-1}\{M(u)\}$. This formulation includes the Zernike wavefront sensor, the pyramid wavefront sensor, generalized $n$-faced pyramids, axicone-like masks, and flattened pyramids. The focal-plane mask determines the impulse response, hence the sensitivity, linearity range, and noise propagation (Janin-Potiron et al., 2019).

Coronagraph-integrated FPWFS introduces an explicit coronagraph operator. In vortex-based formulations, for example, a pupil field $P(x)e^{i\phi(x)}$ is propagated through a focal-plane vortex mask and Lyot stop, yielding a post-coronagraphic intensity of the form

$$I(\alpha)=\left|\mathcal{F}\!\left[C\{P(x)e^{i\phi(x)}\}\right]\right|^2.$$

This makes FPWFS particularly attractive for high-contrast systems, because the sensed quantity is already downstream of the coronagraph and science optics, where residual aberrations actually set the achievable contrast (Quesnel et al., 2022).

2. Ambiguity, diversity, and identifiability

A central obstacle in FPWFS is that intensity-only measurements do not, in general, uniquely determine phase. For centrosymmetric pupils, the even component of the pupil phase has a sign ambiguity: if $\phi(\boldsymbol{\xi})$0, then intensity-only measurements cannot distinguish $\phi(\boldsymbol{\xi})$1 from $\phi(\boldsymbol{\xi})$2. In the vortex formulation this appears explicitly as

$\phi(\boldsymbol{\xi})$3

A common misconception is therefore that a single in-focus focal-plane image always yields an absolute phase estimate. For symmetric pupils it does not; some form of diversity or symmetry breaking is required (Quesnel et al., 2022).

Classical phase diversity lifts this degeneracy by introducing a known aberration such as defocus. This is the basis of many phase-retrieval methods and of analytic or iterative variants used in adaptive optics. The drawback is operational: a defocused image consumes observing time or a dedicated sensing channel. This has motivated alternatives in which diversity is intrinsic to the optical train rather than added externally (Xivry et al., 2021).

Several such alternatives now exist. Vortex coronagraphs provide an azimuthal phase ramp that can serve as phase diversity. In a scalar vortex coronagraph, a single post-coronagraphic PSF can already lift the ambiguity in most regimes; in a vector vortex coronagraph, splitting the circular polarization states produces two complementary PSFs, $\phi(\boldsymbol{\xi})$4 and $\phi(\boldsymbol{\xi})$5, associated with $\phi(\boldsymbol{\xi})$6, and their joint use robustly resolves even-mode signs even at low signal-to-noise ratio (Quesnel et al., 2022). Amplitude asymmetry is another route: asymmetric-pupil Fourier sensing on SCExAO uses a deliberately asymmetric mask so that the transfer matrix between pupil phase and Fourier phase becomes invertible for island modes and related discontinuities (Vievard et al., 2020). Temporal amplitude diversity provides a third route: an optical chopper placed in a pupil plane breaks symmetry between chopped and un-chopped frames, and synchronized differencing yields a linear phase-sensitive signal for even as well as odd modes, while preserving up to a 50% science duty cycle (Gerard et al., 2023).

These approaches differ in implementation cost and robustness, but they address the same identifiability problem. This suggests that “phase diversity” in FPWFS is better understood as a structural requirement than as a single optical recipe.

3. Estimation and control algorithms

One major FPWFS family estimates the complex focal-plane electric field directly from multiple images obtained with known perturbations. A compact example is the three-image algorithm: one image without deformable-mirror actuation and two with distinct DM phase maps. If the probe fields are $\phi(\boldsymbol{\xi})$7 and $\phi(\boldsymbol{\xi})$8, then at each pixel the quantities $\phi(\boldsymbol{\xi})$9 satisfy a $$E_f(\boldsymbol{\rho}) = \mathcal{F}\{P(\boldsymbol{\xi}) e^{i\phi(\boldsymbol{\xi})}\}, \qquad I(\boldsymbol{\rho}) = |E_f(\boldsymbol{\rho})|^2.$$0 linear system for the real and imaginary parts of the unknown field. The determinant

$$E_f(\boldsymbol{\rho}) = \mathcal{F}\{P(\boldsymbol{\xi}) e^{i\phi(\boldsymbol{\xi})}\}, \qquad I(\boldsymbol{\rho}) = |E_f(\boldsymbol{\rho})|^2.$$1

must be nonzero, which means the two probe fields must be linearly independent in complex space. This yields a per-pixel closed-form estimate of the focal-plane field, followed by inverse propagation and phase extraction in the pupil (Dou et al., 2015).

A related but more general framework is pair-wise probing and Electric Field Conjugation. Here the DM produces conjugate probes $$E_f(\boldsymbol{\rho}) = \mathcal{F}\{P(\boldsymbol{\xi}) e^{i\phi(\boldsymbol{\xi})}\}, \qquad I(\boldsymbol{\rho}) = |E_f(\boldsymbol{\rho})|^2.$$2, and the differenced intensity

$$E_f(\boldsymbol{\rho}) = \mathcal{F}\{P(\boldsymbol{\xi}) e^{i\phi(\boldsymbol{\xi})}\}, \qquad I(\boldsymbol{\rho}) = |E_f(\boldsymbol{\rho})|^2.$$3

is linear in the unknown coherent field. Space-based work has extended this with explicit noise models for detector noise, photon noise, and DM-induced noise, and derived an optimal probing contrast $$E_f(\boldsymbol{\rho}) = \mathcal{F}\{P(\boldsymbol{\xi}) e^{i\phi(\boldsymbol{\xi})}\}, \qquad I(\boldsymbol{\rho}) = |E_f(\boldsymbol{\rho})|^2.$$4 together with an adaptive exposure-time policy. In simulation with a vortex coronagraph, optimized probe shapes and exposure scheduling reduced the number of probe pairs, shortened the total imaging time, and achieved contrast on the order of $$E_f(\boldsymbol{\rho}) = \mathcal{F}\{P(\boldsymbol{\xi}) e^{i\phi(\boldsymbol{\xi})}\}, \qquad I(\boldsymbol{\rho}) = |E_f(\boldsymbol{\rho})|^2.$$5, with continued improvement below $$E_f(\boldsymbol{\rho}) = \mathcal{F}\{P(\boldsymbol{\xi}) e^{i\phi(\boldsymbol{\xi})}\}, \qquad I(\boldsymbol{\rho}) = |E_f(\boldsymbol{\rho})|^2.$$6 when adaptive exposure was maintained (Sun et al., 2019).

A distinct line of work uses existing AO residual speckles as the probing field instead of injecting dedicated DM probes. In the on-sky demonstration by Codona and Kenworthy, synchronized AO telemetry was used to estimate the rapidly varying focal-plane speckle field, which then acted as a known-but-random interferometric reference against short-exposure science images. This enabled estimation of the static complex halo and, after back-propagation, of non-common-path aberrations without extra hardware or probe light (Codona et al., 2013).

Weak-phase iterative methods form another influential family. “Fast & Furious” reconstructs odd and even phase components from successive focal-plane images using a weak-aberration model and the previous correction as the next phase diversity. Its extension FF-GS relaxes the even-pupil-amplitude assumption by using Gerchberg–Saxton-style error reduction to estimate pupil amplitudes. In high-resolution control experiments at $$E_f(\boldsymbol{\rho}) = \mathcal{F}\{P(\boldsymbol{\xi}) e^{i\phi(\boldsymbol{\xi})}\}, \qquad I(\boldsymbol{\rho}) = |E_f(\boldsymbol{\rho})|^2.$$7 pixels, these methods increased Strehl from approximately $$E_f(\boldsymbol{\rho}) = \mathcal{F}\{P(\boldsymbol{\xi}) e^{i\phi(\boldsymbol{\xi})}\}, \qquad I(\boldsymbol{\rho}) = |E_f(\boldsymbol{\rho})|^2.$$8 to $$E_f(\boldsymbol{\rho}) = \mathcal{F}\{P(\boldsymbol{\xi}) e^{i\phi(\boldsymbol{\xi})}\}, \qquad I(\boldsymbol{\rho}) = |E_f(\boldsymbol{\rho})|^2.$$9–$e^{i\phi}\approx 1+i\phi$0, with residual wavefront RMS estimates of approximately $e^{i\phi}\approx 1+i\phi$1 rad for FF and approximately $e^{i\phi}\approx 1+i\phi$2 rad for FF-GS (Korkiakoski et al., 2014). For very high-order AO calibration, algorithms similar to Fast & Furious were found to be the most practical and robust among the tested approaches, whereas Gerchberg–Saxton became brittle at high spatial frequencies and convex-lifted phase retrieval was computationally prohibitive at realistic sizes (Korkiakoski et al., 2014). On sky at Keck, F&F improved point-spread-function quality in only a few iterations and was demonstrated in both narrowband and broadband operation without defocused images, meaning it could run concurrently with science (Bos et al., 2021).

4. Sensor architectures and system integration

FPWFS does not denote a single optical architecture. One branch comprises Fourier-based sensors in which a focal-plane mask converts phase into intensity after reimaging to a pupil plane. On the LOOPS testbed, a programmable spatial light modulator reproduced the Zernike WFS, the standard and generalized pyramid WFS, the axicone limit, and flattened-pyramid variants. The same platform showed how modulation radius trades linearity against optical gain, how zeroth-order leakage from the SLM must be shifted off the detector, and how coherent overlap in a flattened pyramid modifies the phase-to-intensity mapping (Janin-Potiron et al., 2019).

The flattened pyramid wavefront sensor exemplifies an important design shift within this family. Instead of separating four pupil images as in a classical pyramid, it reduces the apex angle so that the four pupils overlap into a single intensity map. In the reported comparison, FPWFS photon usage was $e^{i\phi}\approx 1+i\phi$3 for the flattened pyramid, versus $e^{i\phi}\approx 1+i\phi$4 for the non-modulated pyramid and $e^{i\phi}\approx 1+i\phi$5 for a $e^{i\phi}\approx 1+i\phi$6 modulated pyramid. It also required approximately $e^{i\phi}\approx 1+i\phi$7 pixels per sampling unit versus approximately $e^{i\phi}\approx 1+i\phi$8 for the classical pyramid, and its noise propagation became nearly as low as the Zernike WFS for radial orders approximately $e^{i\phi}\approx 1+i\phi$9 to $M(u)$0 (Fauvarque et al., 2015).

Another variant uses a focal-plane image to assist a pyramid WFS rather than replace it. The focal-plane assisted pyramid wavefront sensor adds a beam splitter and a gain-scheduling camera before the pyramid mask, records the modulated focal-plane intensity $M(u)$1, reconstructs the instantaneous impulse response, and derives a frame-by-frame optical-gain matrix. In simulation, scaling non-common-path-aberration references frame by frame with the estimated optical gains stabilized the loop and produced a final average Strehl ratio of approximately $M(u)$2, compared with approximately $M(u)$3 for time-averaged scaling under a $M(u)$4 nm rms NCPA injection (Chambouleyron et al., 2021).

Coronagraph-integrated FPWFS is another major branch. The vector apodizing phase plate can be designed so that its dual coronagraphic PSFs themselves serve as the sensor. On SCExAO, this enabled absolute phase retrieval without dedicated holograms or DM modulation, and on sky it improved raw contrast by a factor of approximately $M(u)$5 between $M(u)$6 and $M(u)$7 after $M(u)$8 iterations of closed-loop correction (Bos et al., 2019). Behind a vAPP on MagAO-X, two complementary focal-plane loops were developed: a modal low-order wavefront sensor using vAPP-generated modal PSFs, and spatial linear dark field control using bright-field pixels. The system-level objective was continuous maintenance of the vAPP raw contrast level of approximately $M(u)$9 across a $$e_m(x)=\mathcal{F}^{-1}\{M(u)\,E_f(u)\}=h(x) * \big(P(x)e^{i\varphi(x)}\big), \qquad I(x)=|e_m(x)|^2,$$0–$$e_m(x)=\mathcal{F}^{-1}\{M(u)\,E_f(u)\}=h(x) * \big(P(x)e^{i\varphi(x)}\big), \qquad I(x)=|e_m(x)|^2,$$1 dark hole (Miller et al., 2018).

SCExAO also serves as a comparative platform. It hosts APF-WFS/ZAP, linearized analytic phase diversity, single-image phase diversity, Fast & Furious, neural-network PSF prediction from PyWFS telemetry, and the “Dr WHO” reference-optimization scheme. The resulting picture is not that one sensor dominates, but that different FPWFS embodiments occupy different niches: island-mode sensing, quasi-static NCPA tracking, low-order stabilization near coronagraphs, or computationally cheap real-time correction (Vievard et al., 2020).

5. Data-driven and photonic FPWFS

Deep learning has become a major FPWFS strategy because the inverse mapping from focal-plane intensity to wavefront is non-linear, high-dimensional, and often instrument-specific. In idealized simulations at $$e_m(x)=\mathcal{F}^{-1}\{M(u)\,E_f(u)\}=h(x) * \big(P(x)e^{i\varphi(x)}\big), \qquad I(x)=|e_m(x)|^2,$$2m, ResNet-50 and U-Net were trained either to regress Zernike coefficients or to reconstruct the phase map directly. For $$e_m(x)=\mathcal{F}^{-1}\{M(u)\,E_f(u)\}=h(x) * \big(P(x)e^{i\varphi(x)}\big), \qquad I(x)=|e_m(x)|^2,$$3 modes and $$e_m(x)=\mathcal{F}^{-1}\{M(u)\,E_f(u)\}=h(x) * \big(P(x)e^{i\varphi(x)}\big), \qquad I(x)=|e_m(x)|^2,$$4 photons per image, single-iteration inference reduced the wavefront to $$e_m(x)=\mathcal{F}^{-1}\{M(u)\,E_f(u)\}=h(x) * \big(P(x)e^{i\varphi(x)}\big), \qquad I(x)=|e_m(x)|^2,$$5, and over a broad range of fluxes the residuals tracked the photon-noise limit until low-flux regularization or high-flux training-set limitations became dominant (Xivry et al., 2021).

Vortex phase diversity has been combined with modern convolutional networks in a particularly direct form. Using EfficientNet-B4 on simulated post-coronagraphic PSFs, a scalar vortex coronagraph could lift the sign ambiguity with a single PSF, while a vector vortex coronagraph used two polarization channels with complementary $$e_m(x)=\mathcal{F}^{-1}\{M(u)\,E_f(u)\}=h(x) * \big(P(x)e^{i\varphi(x)}\big), \qquad I(x)=|e_m(x)|^2,$$6 phase ramps. In the reported example for a bright star at $$e_m(x)=\mathcal{F}^{-1}\{M(u)\,E_f(u)\}=h(x) * \big(P(x)e^{i\varphi(x)}\big), \qquad I(x)=|e_m(x)|^2,$$7m with $$e_m(x)=\mathcal{F}^{-1}\{M(u)\,E_f(u)\}=h(x) * \big(P(x)e^{i\varphi(x)}\big), \qquad I(x)=|e_m(x)|^2,$$8 nm RMS input wavefront error, the dual-polarization VVC network achieved a residual of approximately $$e_m(x)=\mathcal{F}^{-1}\{M(u)\,E_f(u)\}=h(x) * \big(P(x)e^{i\varphi(x)}\big), \qquad I(x)=|e_m(x)|^2,$$9 nm RMS after correction. Because the vortex itself provides the phase diversity, the technique offers a $h(x)=\mathcal{F}^{-1}\{M(u)\}$0 science duty cycle, and for the scalar vortex no additional hardware is required (Quesnel et al., 2022).

Machine-learning FPWFS has also been specialized to low-order infrared sensing. A ResNet18 trained on simulated AO telemetry for Keck I estimated defocus from single focal-plane PSFs in the presence of a fixed $h(x)=\mathcal{F}^{-1}\{M(u)\}$1 nm RMS astigmatism diversity. In realistic under-sampled and noisy simulated conditions, and in K-band bench validation, the reported fits reached $h(x)=\mathcal{F}^{-1}\{M(u)\}$2 values up to $h(x)=\mathcal{F}^{-1}\{M(u)\}$3 with RMSE down to approximately $h(x)=\mathcal{F}^{-1}\{M(u)\}$4 nm. Even in the most difficult simulated case the predictions remained monotonic over a $h(x)=\mathcal{F}^{-1}\{M(u)\}$5 nm RMS focus range, which is operationally important for slow-focus tracking loops (Taheri et al., 2024).

Photonic FPWFS replaces a camera image with an optical mode-conversion measurement. A 19-core photonic lantern can map the multimode focal-plane field into 19 single-mode outputs whose intensities encode the incident wavefront. Neural-network reconstruction from those 19 intensities achieved an RMSE of $h(x)=\mathcal{F}^{-1}\{M(u)\}$6 rad for petal modes and $h(x)=\mathcal{F}^{-1}\{M(u)\}$7 rad for low-wind-effect modes in the non-linear regime with average incident RMS WFE of $h(x)=\mathcal{F}^{-1}\{M(u)\}$8 rad (Wei et al., 2023). A related short-multimode-fiber FPWFS couples the focal-plane field into a weakly guiding step-index fiber of length below $h(x)=\mathcal{F}^{-1}\{M(u)\}$9 cm so that modal interference survives over a $n$0 nm bandwidth near $n$1m. Inference with a CNN recovered the first 11 Zernike coefficients with per-coefficient RMSE between $n$2 and $n$3 rad and a latency below $n$4 ms per frame on the reported hardware (Padrón-Brito et al., 3 Oct 2025).

These developments indicate a broad shift in FPWFS from explicit inverse modeling toward learned or hybrid optical-computational encoders. A plausible implication is that future FPWFS architectures will be differentiated as much by the optical encoding of information as by the reconstruction algorithm.

6. Performance regimes, control architectures, and open problems

Published FPWFS results span very different operating regimes. Probe-optimized dark-hole control in space-based simulations reached contrast on the order of $n$5 and continued below $n$6 with adaptive exposure (Sun et al., 2019). Weak-phase high-order correction with Fast & Furious achieved Strehl ratios of $n$7–$n$8 on a $n$9-pixel corrector (Korkiakoski et al., 2014). Coronagraph-integrated vAPP sensing improved raw contrast by a factor of approximately $P(x)e^{i\phi(x)}$0 between $P(x)e^{i\phi(x)}$1 and $P(x)e^{i\phi(x)}$2 after $P(x)e^{i\phi(x)}$3 iterations on sky (Bos et al., 2019). Optical-chopper FPWFS demonstrated approximately $P(x)e^{i\phi(x)}$4 wavefront-error reduction in laboratory AO-residual conditions while preserving up to a $P(x)e^{i\phi(x)}$5 science duty cycle (Gerard et al., 2023).

Operationally, FPWFS is increasingly being integrated into hierarchical control. At Keck I, focal-plane slow-focus sensing with Gerchberg–Saxton used the TRICK tip-tilt camera and existing astigmatism diversity to track sodium-layer-induced focus drift. On-sky tests at $P(x)e^{i\phi(x)}$6 Hz successfully compensated deliberately introduced focus errors; excluding the first $P(x)e^{i\phi(x)}$7 s of the run, the GS estimates had mean $P(x)e^{i\phi(x)}$8 nm and standard deviation $P(x)e^{i\phi(x)}$9 nm RMS (Salgueiro et al., 17 Feb 2026). In dual-sensor single-conjugate AO, a slow focal-plane sensor can be fused with a fast pupil-plane WFS on one DM. The reported control analysis showed that a simple AR(1) high-pass filter on the fast arm mitigates inter-arm non-common-path transfer and can reduce science-plane residuals under strong NCPA from approximately $$I(\alpha)=\left|\mathcal{F}\!\left[C\{P(x)e^{i\phi(x)}\}\right]\right|^2.$$0 rad rms to approximately $$I(\alpha)=\left|\mathcal{F}\!\left[C\{P(x)e^{i\phi(x)}\}\right]\right|^2.$$1 rad rms (Sengupta et al., 20 Jan 2026).

The main limitations are recurrent across the literature. Performance depends strongly on model fidelity or training-distribution match; amplitude aberrations are omitted in several studies; chromatic effects are often only partially modeled; detector systematics, polarization leakage, and registration errors can dominate practical error budgets; and many algorithms still rely on a small-phase regime or on iterative closure from a sufficiently good initial state. Even when a method is probe-free in principle, stability may still depend on calibration of diversity, pupil geometry, or optical gain. These are not marginal implementation details; they define the boundary between laboratory performance and on-sky routine use (Quesnel et al., 2022, Sun et al., 2019).

FPWFS is therefore best understood not as a single sensor but as a design space. Within that space, the unifying principle is constant: aberrations are measured where they matter optically most. The distinguishing questions are then how phase diversity is created, how identifiability is secured, how the inverse map is regularized or learned, and how the resulting estimate is embedded in a stable control loop. On current 8–10 m systems and in planned ELT and space architectures, those questions increasingly determine not whether FPWFS is used, but which FPWFS is used.