Multi-Plane Phase Retrieval Sensors

• Multi-plane phase retrieval sensors are intensity-only systems that recover both amplitude and phase by exploiting propagation diversity across multiple measurement planes.
• They encompass varied architectures including curvature sensors, inline holography setups, and diffractive optical processors, each tailored for specific imaging needs.
• Reconstruction methods, notably iterative Gerchberg–Saxton algorithms combined with robust calibration techniques, enable accurate phase retrieval even under low-photon or misalignment challenges.

Searching arXiv for papers on multi-plane phase retrieval sensors and related architectures. Multi-plane phase retrieval sensors are intensity-only optical sensing systems that infer a complex field—amplitude and phase—from measurements acquired at several axially separated planes, or from formally equivalent diversity channels produced by propagation, wavelength, media, or structured diffractive hardware. In adaptive optics, they include curvature and nonlinear curvature wavefront sensors; in microscopy and inline holography, they include multi-height quantitative phase imagers; and in computationally engineered hardware, they include diffractive processors that map phase information into directly measurable intensity outputs (Abbott et al., 12 Aug 2025, Aisher et al., 2012, Wang et al., 2024, Descloux et al., 2017, Crepp et al., 12 Mar 2026, Shen et al., 2024, Li et al., 2024). Across these instantiations, the common inverse problem is to recover a pupil- or object-plane field $U_0(x,y)$ from intensities $I_n(x,y)=|U_n(x,y;z_n)|^2$ measured after propagation to multiple planes $z_n$, with the propagation typically modeled by Fresnel or angular-spectrum operators (Abbott et al., 12 Aug 2025, Wang et al., 2024).

1. Physical basis and measurement models

The defining principle is path-length or propagation diversity. A single complex field is observed after several distinct forward operators, and the resulting intensity evolution along $z$ encodes information that is absent from a single intensity image. In the standard paraxial formulation used by multi-plane phase retrieval sensors, one writes

$$U_n(x,y;z_n)=\mathcal{P}_{z_n}\{U_0(x,y)\},\qquad I_n(x,y)=|U_n(x,y;z_n)|^2,$$

and seeks a field $U_0$ whose propagated intensities match all measured planes simultaneously (Abbott et al., 12 Aug 2025, Wang et al., 2024). In curvature sensing, a linearized relation near the pupil connects defocused intensity differences to the Laplacian of phase, $\Delta I(x,y)=C\,\nabla^2\phi(x,y)$; nonlinear curvature sensing instead uses the full multi-plane nonlinear diffraction physics and an iterative phase retrieval algorithm (Crepp et al., 12 Mar 2026).

This formulation generalizes naturally. In multi-frequency phase retrieval, the diversity index is wavelength rather than axial plane, and the forward operator is written $$\mathbf{u}_{s,\lambda}=\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda}$$ with noisy intensity observations $$y_{s,\lambda,i}\sim p(y_{s,\lambda,i}\mid |(\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda})_i|^2)$$ (Katkovnik et al., 2018). In media-diversity phase retrieval, the same incident field is propagated through different linear or nonlinear media, producing intensities $I_{m,p}(\bx)=|u_m(z_p,\bx)|^2$; the diversity then arises from the medium parameters $I_n(x,y)=|U_n(x,y;z_n)|^2$0 rather than from plane spacing alone (Cheng et al., 2024). In partially coherent multi-channel formulations, multiple coherently linked measurements define block-structured operators $I_n(x,y)=|U_n(x,y;z_n)|^2$1 and permit null-space formulations such as $I_n(x,y)=|U_n(x,y;z_n)|^2$2 or $I_n(x,y)=|U_n(x,y;z_n)|^2$3, where coherent linking reduces the number of unknown phase variables (Kornprobst et al., 2020).

A recurrent theme is that “multi-plane” is not restricted to literal stacks of defocused images. The same inverse structure appears in four-plane curvature sensors, translated-sensor inline holography, prism-split simultaneous multi-plane microscopes, convergent-beam layouts mapped to free-space equivalents, and wavelength-multiplexed diffractive processors (Aisher et al., 2012, Wang et al., 2024, Descloux et al., 2017, Crepp et al., 12 Mar 2026, Shen et al., 2024).

2. Representative architectures

The literature spans several distinct hardware realizations that implement the same underlying diversity principle.

Architecture	Diversity mechanism	Representative details
Four-plane nonlinear curvature WFS	Four defocused pupil-adjacent planes	Classical layout at $I_n(x,y)=|U_n(x,y;z_n)|^2$4; laboratory nlCWFS at $I_n(x,y)=|U_n(x,y;z_n)|^2$5 cm and $I_n(x,y)=|U_n(x,y;z_n)|^2$6 cm (Aisher et al., 2012, Abbott et al., 12 Aug 2025)
Align-free inline holography	Single sensor translated to multiple heights	Three planes at $I_n(x,y)=|U_n(x,y;z_n)|^2$7 mm in simulation; experimental spacing $I_n(x,y)=|U_n(x,y;z_n)|^2$8 mm (Wang et al., 2024)
Simultaneous multi-plane QPI with prism	Beam splitting into several conjugate image planes	Eight planes acquired simultaneously, $I_n(x,y)=|U_n(x,y;z_n)|^2$9 nm in object space, up to 200 Hz (Descloux et al., 2017)
Convergent-beam nlCWFS	Four contemporaneous planes in a focused beam	Physical planes $z_n$0 mm for $z_n$1 mm, mapped to $z_n$2 mm (Crepp et al., 12 Mar 2026)
Diffractive optical processors	Multiple learned diffractive layers with output-channel encoding	Five-layer amplitude/phase imager over $z_n$3; ten-layer wavelength-multiplexed multi-plane QPI processor (Li et al., 2024, Shen et al., 2024)

The classical adaptive-optics lineage is exemplified by the four-plane nonlinear curvature wavefront sensor. In one formulation, the sensor records two planes near the pupil and two farther from it, so that the inner planes retain strong near-pupil sensitivity while the outer planes sample lower Fresnel-number structure and encode low-order variance more strongly (Aisher et al., 2012, Crepp et al., 12 Mar 2026). The 2025 jitter-sensing study uses four measurement planes at $z_n$4 cm and $z_n$5 cm from the pupil and reconstructs with a modified Gerchberg–Saxton algorithm using up to five iterations (Abbott et al., 12 Aug 2025).

A second lineage is lensless or minimal-optics inline holography. Here a single 2D sensor is translated axially to acquire a stack $z_n$6, typically $z_n$7, and computational calibration is required because lateral shifts, small rotations, scaling, and field-of-view loss invalidate naive pixelwise correspondence across planes (Wang et al., 2024). The same paper models inter-plane distortions with a general $z_n$8 projective transformation and shows that raw-hologram homography is inadequate because diffraction changes the intensity nonlinearly with $z_n$9 (Wang et al., 2024).

A third lineage is simultaneous multi-plane microscopy. In the white-light quantitative phase tomography platform that combines phase imaging with SOFI, a customized prism splits the detection beam into eight conjugate planes recorded on two synchronized sCMOS cameras, yielding an effective inter-plane spacing of $z$0 nm in object space and enabling 3D phase imaging at up to 200 Hz (Descloux et al., 2017). A different single-shot strategy replaces multiple measured planes with multiple internal diffractive layers: the output plane then contains dedicated amplitude and phase channels, or wavelength-multiplexed channels corresponding to different axial object planes (Li et al., 2024, Shen et al., 2024).

3. Reconstruction algorithms and inverse formulations

The dominant reconstruction family is multi-plane Gerchberg–Saxton. In its standard form, one propagates a current estimate between planes, replaces the amplitude in each measurement plane with the measured $z$1, back-propagates, and iterates. One explicit update is

$z$2

after which the constrained fields are propagated back to the pupil and combined (Abbott et al., 12 Aug 2025, Wang et al., 2024). The four-plane nonlinear curvature literature develops Gerchberg–Saxton variants with feedback terms—Error-Reduction, Input-Output, Output-Output, and In-Out-Out—parameterized through coefficients $z$3 and step-size $z$4, with propagation implemented by two-step Fresnel operators $z$5 (Aisher et al., 2012).

In lensless multi-height imaging, alignment becomes part of the inverse problem. The Adaptive Cascade Calibrated strategy performs autofocusing of each measured hologram by minimizing a Laplacian sharpness metric,

$z$6

then detects SIFT features in the refocused object-space images, estimates homographies between neighboring planes, cascades them to a common reference, warps the measured intensities into a shared coordinate system, and only then applies an energy-conserved multi-plane Gerchberg–Saxton algorithm (Wang et al., 2024). This changes the computational role of calibration: the registration is performed in refocused object space rather than directly on the raw holograms.

More statistical formulations replace hard amplitude replacement with explicit likelihoods. In the multi-frequency framework, Poisson and Gaussian negative log-likelihoods are combined with BM3D priors on both complex field and phase, producing an alternating-projection or ADMM-like scheme with forward propagation, measurement-domain data fitting, backward propagation, and complex/phase denoising (Katkovnik et al., 2018). The paper argues that the same ML+BM3D structure transfers directly to multi-plane sensors by substituting propagation operators $z$7 for the diversity operators $z$8 (Katkovnik et al., 2018).

At the opposite end of the spectrum are explicit, non-iterative reconstructions derived from uniqueness theorems. For phase retrieval via media diversity, the linear Schrödinger and Gross–Pitaevskii cases use multiple $z$9 media to recover the phase gradient $$U_n(x,y;z_n)=\mathcal{P}_{z_n}\{U_0(x,y)\},\qquad I_n(x,y)=|U_n(x,y;z_n)|^2,$$0 up to a constant, while the quadratic nonlinear Schrödinger case uses multiple $$U_n(x,y;z_n)=\mathcal{P}_{z_n}\{U_0(x,y)\},\qquad I_n(x,y)=|U_n(x,y;z_n)|^2,$$1 media to reconstruct the full phase without global-phase ambiguity, under full-rank conditions on matrices $$U_n(x,y;z_n)=\mathcal{P}_{z_n}\{U_0(x,y)\},\qquad I_n(x,y)=|U_n(x,y;z_n)|^2,$$2 or $$U_n(x,y;z_n)=\mathcal{P}_{z_n}\{U_0(x,y)\},\qquad I_n(x,y)=|U_n(x,y;z_n)|^2,$$3 (Cheng et al., 2024). These algorithms operate on estimates of $$U_n(x,y;z_n)=\mathcal{P}_{z_n}\{U_0(x,y)\},\qquad I_n(x,y)=|U_n(x,y;z_n)|^2,$$4 and $$U_n(x,y;z_n)=\mathcal{P}_{z_n}\{U_0(x,y)\},\qquad I_n(x,y)=|U_n(x,y;z_n)|^2,$$5 at $$U_n(x,y;z_n)=\mathcal{P}_{z_n}\{U_0(x,y)\},\qquad I_n(x,y)=|U_n(x,y;z_n)|^2,$$6, extracted from near-field multi-plane data (Cheng et al., 2024).

Diffractive optical processors shift the inversion almost entirely into the optical front-end. In one design, successive learned diffractive layers produce two output intensity channels, one approximating the input amplitude and the other the input quantitative phase, with no iterative per-measurement phase retrieval (Li et al., 2024). In another, a wavelength-multiplexed ten-layer diffractive network maps several axial input phase planes onto a single output field of view, with one wavelength channel per plane and a final normalization

$$U_n(x,y;z_n)=\mathcal{P}_{z_n}\{U_0(x,y)\},\qquad I_n(x,y)=|U_n(x,y;z_n)|^2,$$7

that directly approximates the phase map of plane $$U_n(x,y;z_n)=\mathcal{P}_{z_n}\{U_0(x,y)\},\qquad I_n(x,y)=|U_n(x,y;z_n)|^2,$$8 (Shen et al., 2024). This suggests that multi-plane phase retrieval can be embedded into hardware as a fixed learned operator rather than executed numerically after measurement.

4. Calibration, alignment, and auxiliary low-order sensing

Practical multi-plane sensors are limited as much by calibration and registration as by inversion. In translated-sensor holography, lateral shifts dominate, but small rotations, scaling, perspective terms, and field-of-view loss also matter; the estimated transformation matrices for plant-stem data show diagonal terms near 1, off-diagonal rotational terms of order $$U_n(x,y;z_n)=\mathcal{P}_{z_n}\{U_0(x,y)\},\qquad I_n(x,y)=|U_n(x,y;z_n)|^2,$$9, and translation terms of order $U_0$0 pixels (Wang et al., 2024). Because diffraction patterns at different $U_0$1 are not related by a simple homography, direct geometric registration on raw holograms is inadequate (Wang et al., 2024).

Nonlinear curvature sensors add another calibration dimension: low-order pointing. Tip and tilt appear as lateral shifts of the recorded intensity patterns, and each measurement plane encodes this shift with a lever arm proportional to $U_0$2. The weighted-average centroid at plane $U_0$3 is

$U_0$4

and pixel shifts are converted to angular shifts through

$U_0$5

Reference centroids are calibrated by a Hough–Canny procedure: Canny edge detection identifies the circular beam boundary, a Circular Hough Transform fits the circle, and the fitted center becomes the reference centroid (Abbott et al., 12 Aug 2025). In the reported laboratory geometry, the outer planes correspond to $U_0$6 per pixel while the inner planes correspond to $U_0$7 per pixel, so the outer planes are more sensitive to small angular shifts (Abbott et al., 12 Aug 2025).

This capability challenges a common architectural assumption. The 2025 jitter-sensing work demonstrates that image jitter may be sensed and compensated for using a fast steering mirror and the wavefront sensor alone, without peripheral quad-cells or access to a separate scientific imaging channel (Abbott et al., 12 Aug 2025). In effect, the same multi-plane data stream supports both high-order phase retrieval and low-order tip/tilt control. A related misconception is that low-order estimation is independent of high-order reconstruction; the reported reconstructions show instead that small tip/tilt errors can cascade into large uncertainties in higher-order phase retrieval, producing branch cuts and irregularities until the low-order shift is corrected (Abbott et al., 12 Aug 2025).

5. Performance across operating regimes

Reported performance varies strongly with optical regime, photon budget, and algorithmic assumptions. In the align-free multi-plane holography study, the Adaptive Cascade Calibrated pipeline improves reconstruction quality substantially in simulation: for crystal data, PSNR/SSIM increase from $U_0$8 dB / $U_0$9 without calibration to $\Delta I(x,y)=C\,\nabla^2\phi(x,y)$0 dB / $\Delta I(x,y)=C\,\nabla^2\phi(x,y)$1 with ACC; for onion data, from $\Delta I(x,y)=C\,\nabla^2\phi(x,y)$2 dB / $\Delta I(x,y)=C\,\nabla^2\phi(x,y)$3 to $\Delta I(x,y)=C\,\nabla^2\phi(x,y)$4 dB / $\Delta I(x,y)=C\,\nabla^2\phi(x,y)$5 (Wang et al., 2024). Under pure translation errors, naive reconstructions degrade as plane shifts grow to $\Delta I(x,y)=C\,\nabla^2\phi(x,y)$6 and $\Delta I(x,y)=C\,\nabla^2\phi(x,y)$7 pixels, whereas ACC remains at roughly $\Delta I(x,y)=C\,\nabla^2\phi(x,y)$8–$\Delta I(x,y)=C\,\nabla^2\phi(x,y)$9 dB PSNR and $$\mathbf{u}_{s,\lambda}=\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda}$$0–$$\mathbf{u}_{s,\lambda}=\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda}$$1 SSIM across all tested shifts (Wang et al., 2024).

In nonlinear curvature sensing, performance is often controlled by photon starvation and by the coupling between tip/tilt and higher-order modes. The four-plane low-light analysis shows that preprocessing by Gaussian convolution can reduce by an order of magnitude the photon flux required for accurate phase retrieval of low-order errors, and identifies differentially blurred Gaussian convolution as especially effective when its blur scales match the speckle size in the inner and outer planes (Aisher et al., 2012). The same study compares ER, IO, OO, and IOO feedback schemes and finds that no single step-size is uniformly optimal across photon levels (Aisher et al., 2012). In the 2025 jitter experiments, weighted-average centroiding on the outer planes recovers injected tip and tilt to within $$\mathbf{u}_{s,\lambda}=\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda}$$2 on average for an unaberrated beam; inner-plane estimates are worse at $$\mathbf{u}_{s,\lambda}=\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda}$$3, and accuracies better than $$\mathbf{u}_{s,\lambda}=\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda}$$4 remain achievable in the presence of aberrations (Abbott et al., 12 Aug 2025). The same paper reports that diffraction-limited reconstruction required tip/tilt accuracy of approximately $$\mathbf{u}_{s,\lambda}=\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda}$$5 for one aberration and approximately $$\mathbf{u}_{s,\lambda}=\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda}$$6 for two others (Abbott et al., 12 Aug 2025).

The convergent-beam four-plane nlCWFS study reports a simulated RMS wavefront error of approximately $$\mathbf{u}_{s,\lambda}=\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda}$$7 waves ($$\mathbf{u}_{s,\lambda}=\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda}$$8 radians) after rescaling the measured convergent-beam intensities into free-space-equivalent planes and running a standard multi-plane Gerchberg–Saxton reconstruction (Crepp et al., 12 Mar 2026). In the laboratory, the same method recovers a nearly $$\mathbf{u}_{s,\lambda}=\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda}$$9-wave peak-to-valley horizontal coma imposed by a $$y_{s,\lambda,i}\sim p(y_{s,\lambda,i}\mid |(\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda})_i|^2)$$0 deformable mirror, using a compact module that fits inside a standard 2-inch Thorlabs mount (Crepp et al., 12 Mar 2026).

In white-light multi-plane quantitative phase tomography, the eight-plane prism system reaches lateral and axial phase-imaging resolutions of $$y_{s,\lambda,i}\sim p(y_{s,\lambda,i}\mid |(\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda})_i|^2)$$1 nm and $$y_{s,\lambda,i}\sim p(y_{s,\lambda,i}\mid |(\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda})_i|^2)$$2 nm, respectively, and records live-cell 3D phase data at up to 200 Hz (Descloux et al., 2017). For a nanometric staircase sample, the retrieved phase varies linearly with AFM-measured height according to

$$y_{s,\lambda,i}\sim p(y_{s,\lambda,i}\mid |(\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda})_i|^2)$$3

supporting the quantitative character of the reconstruction (Descloux et al., 2017).

Learned diffractive processors currently occupy a different performance niche. The all-optical amplitude/phase imager reports test-set PSNRs of $$y_{s,\lambda,i}\sim p(y_{s,\lambda,i}\mid |(\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda})_i|^2)$$4 dB for amplitude and $$y_{s,\lambda,i}\sim p(y_{s,\lambda,i}\mid |(\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda})_i|^2)$$5 dB for phase in one spatially multiplexed design, rising to $$y_{s,\lambda,i}\sim p(y_{s,\lambda,i}\mid |(\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda})_i|^2)$$6 dB and $$y_{s,\lambda,i}\sim p(y_{s,\lambda,i}\mid |(\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda})_i|^2)$$7 dB in a combined spatial-and-wavelength-multiplexed design (Li et al., 2024). The wavelength-multiplexed multi-plane QPI processor reports average PCC of approximately $$y_{s,\lambda,i}\sim p(y_{s,\lambda,i}\mid |(\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda})_i|^2)$$8 for five non-overlapping planes at matched phase contrast, decreasing when planes overlap laterally or when test phase contrast departs from the training regime (Shen et al., 2024).

6. Limitations, misconceptions, and future directions

Several limitations recur across the literature. First, identifiability often depends on assumptions that are stronger than the phrase “phase retrieval” suggests. The media-diversity uniqueness results assume known, non-vanishing amplitude $$y_{s,\lambda,i}\sim p(y_{s,\lambda,i}\mid |(\mathbf{H}_{s,\lambda}\mathbf{u}_{0,\lambda})_i|^2)$$9, exactly known $I_{m,p}(\bx)=|u_m(z_p,\bx)|^2$0 and $I_{m,p}(\bx)=|u_m(z_p,\bx)|^2$1, and local-in-$I_{m,p}(\bx)=|u_m(z_p,\bx)|^2$2 access to intensity derivatives near the incidence plane (Cheng et al., 2024). The white-light tomography method assumes weak scattering and the first Born approximation; it also suppresses low spatial frequencies by applying an axial high-pass filter $I_{m,p}(\bx)=|u_m(z_p,\bx)|^2$3, and with only eight planes rather than the approximately eighteen suggested by the sampling analysis, additional high-pass behavior appears laterally (Descloux et al., 2017). The diffractive-processor paradigm replaces iterative inversion with a fixed learned optical transform, but its performance depends on the training distribution, the designed wavelengths, and fabrication/alignment tolerances (Li et al., 2024, Shen et al., 2024).

Second, compactness does not eliminate sampling constraints. The convergent-beam mapping

$I_{m,p}(\bx)=|u_m(z_p,\bx)|^2$4

shows that a short physical distance $I_{m,p}(\bx)=|u_m(z_p,\bx)|^2$5 can emulate a long free-space propagation $I_{m,p}(\bx)=|u_m(z_p,\bx)|^2$6, but the effective Fresnel number and the sampling condition still depend on $I_{m,p}(\bx)=|u_m(z_p,\bx)|^2$7, and operation too close to focus drives $I_{m,p}(\bx)=|u_m(z_p,\bx)|^2$8 and $I_{m,p}(\bx)=|u_m(z_p,\bx)|^2$9 (Crepp et al., 12 Mar 2026). The same paper notes that interpolation during intensity rescaling can distort noise statistics, particularly in photon-limited regimes (Crepp et al., 12 Mar 2026).

Third, alignment-free does not mean calibration-free. The ACC method removes the need for calibration targets or markers, but it still depends on successful autofocus, feature extraction, and robust homography estimation in refocused object space; feature-poor or strongly scattering samples can defeat this stage (Wang et al., 2024). Similarly, nonlinear curvature jitter sensing eliminates dedicated tip/tilt hardware, but it depends on robust centroid calibration, known $I_n(x,y)=|U_n(x,y;z_n)|^2$00-distances, and adequate outer-plane sampling (Abbott et al., 12 Aug 2025).

Several misconceptions are addressed directly by the published results. One is that multi-plane phase retrieval sensors measure only high-order phase and therefore require a separate low-order channel; the nonlinear curvature literature shows that tip/tilt is embedded in the multi-plane intensities themselves and can be estimated with the wavefront sensor alone (Abbott et al., 12 Aug 2025). Another is that “multi-plane” necessarily denotes literal stacks of defocused sensor images; the diffractive-processor and media-diversity results show instead that multi-plane phase retrieval is better understood as a diversity principle that can be realized through wavelength, medium, or learned optical transformations (Cheng et al., 2024, Shen et al., 2024, Li et al., 2024). A plausible implication is that future systems will increasingly decouple the source of diversity from the numerical inversion method, combining compact optical front-ends with calibration-aware or learned reconstructors rather than treating defocus stacks as the only viable architecture.

The field’s current trajectory therefore combines three directions. One is improved physical modeling and calibration for compact sensors, including convergent-beam layouts and simultaneous low-order/high-order control (Crepp et al., 12 Mar 2026, Abbott et al., 12 Aug 2025). A second is statistically grounded inversion, including likelihood-based phase retrieval with object-domain priors and coherent multi-channel null-space formulations (Katkovnik et al., 2018, Kornprobst et al., 2020). The third is optical compilation of the inverse problem into diffractive hardware, reducing or eliminating per-frame digital optimization for specific imaging tasks (Li et al., 2024, Shen et al., 2024). Taken together, these developments indicate that multi-plane phase retrieval sensors now constitute a broad technical class rather than a single instrument design: a family of intensity-only sensors that trade optical diversity, calibration structure, and computational strategy against one another in order to recover complex fields under increasingly constrained photon, size, and latency budgets.