Spatial Demultiplexing (SPADE) in Optical Imaging

• Spatial demultiplexing (SPADE) is a technique that projects the image-plane field onto an orthonormal PSF-adapted mode basis, bypassing conventional pixel-based imaging.
• It exploits Hermite–Gaussian and other mode bases to re-encode subdiffraction details into mode occupancies, thereby overcoming Rayleigh’s curse.
• SPADE is implemented with various receiver architectures—including binary-SPADE, iSPADE, and PM-SPADE—for superresolution imaging, spectroscopy, and motion sensing.

Spatial-mode demultiplexing, commonly abbreviated SPADE, is a far-field optical measurement scheme in which the image-plane field is not measured in the position basis but is instead projected onto an orthonormal basis of spatial modes matched to the point-spread function (PSF), followed by photon counting or intensity readout. In the subdiffraction regime, this re-encodes separation, size, and higher-order structural information into mode occupancies rather than into small distortions of a blurred intensity profile. Foundational analyses showed that, for incoherent imaging, SPADE can estimate second and higher moments of an object much more precisely than direct imaging under photon shot noise, and can approach the quantum-optimal precision for location and scale parameters (Tsang, 2016, Tsang, 2017).

1. Measurement principle

In the standard incoherent-imaging model, the one-photon state at the image plane is written as

$$\rho_1 = \int d^2\mathbf R\, F(\mathbf R)\, |\psi_{\mathbf R}\rangle\langle \psi_{\mathbf R}|,$$

with $F(\mathbf R)$ the normalized source intensity distribution and $|\psi_{\mathbf R}\rangle$ the shifted PSF state. Direct imaging measures the position-basis intensity

$$f(\mathbf r)=\langle \mathbf r|\rho_1|\mathbf r\rangle,$$

whereas SPADE measures mode probabilities

$$p(j)=\langle \phi_j|\rho_1|\phi_j\rangle = \int d^2\mathbf R\,F(\mathbf R)\,|\langle \phi_j|\psi_{\mathbf R}\rangle|^2$$

for a chosen orthonormal mode set $\{|\phi_j\rangle\}$ (Tsang, 2016).

For a Gaussian PSF, the natural basis is the Hermite–Gaussian (HG), or equivalently TEM, basis. In that setting, the fundamental mode is matched to the on-axis PSF, while small displacements or subdiffraction structure populate first-order and higher-order modes. This is the operational reason SPADE avoids the usual image-plane collapse of information: the parameter of interest is mapped into power in modes that are orthogonal to the dominant background (Tsang, 2017).

A particularly transparent formulation appears for two incoherent point sources with known centroid. Expanding the field in the HG basis, the first-order modes $HG_{10}$ and $HG_{01}$ encode the $x$- and $y$-components of separation. In practical terms, SPADE is therefore a mode-resolved intensity measurement rather than a pixel-resolved image measurement (Boucher et al., 2020).

This mode-based viewpoint generalizes beyond separation estimation. For a one-photon state from two incoherent sources, one formulation writes

$F(\mathbf R)$0

with $F(\mathbf R)$1 an orthonormal mode basis and $F(\mathbf R)$2 the source separation. The observable is the mode index and its count, not the image-plane coordinate (Titov, 21 Sep 2025).

2. Information-theoretic structure and Rayleigh’s curse

The central statistical contrast between direct imaging and SPADE is expressed through Fisher information. In direct imaging, when two incoherent point sources become very close, the image intensity changes only weakly with separation, and the Fisher information tends to zero. This is the standard Rayleigh curse. For example, in a direct-imaging model of two equally bright sources separated by $F(\mathbf R)$3, the Fisher information behaves as $F(\mathbf R)$4 for $F(\mathbf R)$5, so the Cramér–Rao bound diverges as $F(\mathbf R)$6 (Len et al., 2019).

By contrast, for Gaussian PSFs and ideal HG-mode demultiplexing, the SPADE Fisher information remains finite in the sub-Rayleigh regime. One formulation gives

$F(\mathbf R)$7

independent of separation and equal to the quantum Fisher information in that model (Titov, 21 Sep 2025). In a related derivation for two incoherent beams with known centroid,

$F(\mathbf R)$8

showing that ideal SPADE reaches the quantum-limited scaling in the small-displacement regime (Boucher et al., 2020).

This advantage extends from separation to moment estimation. In the subdiffraction limit, direct imaging estimates high-order spatial moments poorly, with relative-error scaling

$F(\mathbf R)$9

whereas SPADE and iSPADE achieve

$|\psi_{\mathbf R}\rangle$0

which is the quantum-optimal scaling stated for that regime (Kurdzialek, 25 Nov 2025). Foundational analyses therefore treat diffraction not as a fundamental limit on information extraction, but as a consequence of measuring in the position basis rather than in a PSF-adapted mode basis (Tsang, 2016).

3. Bases, receiver architectures, and practical implementations

The canonical SPADE receiver uses HG modes, but several experimentally relevant variants appear in the literature. Full SPADE measures many orthogonal modes; binary SPADE coarse-grains this to a null mode and its orthogonal complement, as in

$|\psi_{\mathbf R}\rangle$1

which is often sufficient for subdiffraction detection and hypothesis testing (Sakuldee et al., 2024).

Interferometric extensions are also important. Tsang’s original framework introduced interferometric TEM measurements to access odd and mixed moments through two-mode superpositions (Tsang, 2016). More recent work describes iSPADE, where neighboring HG modes are interfered so that odd moments can be recovered in addition to the even moments available from plain HG sorting (Kurdzialek, 25 Nov 2025).

For noise-robust motion estimation, a two-mode implementation called PM-SPADE uses the “plus-minus” modes

$|\psi_{\mathbf R}\rangle$2

designed so that each output has occupancy near $|\psi_{\mathbf R}\rangle$3 for small displacements, improving robustness to background and dark counts (Hu et al., 6 Apr 2025).

Experimentally, SPADE has been realized with several mode-sorting technologies. A Multi-Plane Light Conversion (MPLC) device from Cailabs was used as a 9-mode demultiplexer converting free-space HG modes into spatially separated outputs (Boucher et al., 2020). A later MPLC implementation based on a folded SLM architecture used wavefront matching to sort HG modes and reported simulated average overlap around 98%, experimental average overlap around 88%, experimental average sorting fidelity around 90.6%, and simulated fidelity around 98.3% (Titov, 21 Sep 2025). Commercial HG demultiplexers such as Cailabs PROTEUS-C were used in spectroscopy and bright-source experiments (Amato et al., 2024).

A simplified two-channel architecture is the double-clad fiber (DCF) receiver, which separates the fundamental mode into a single-mode core and higher-order content into a multimode cladding. This implements a binary-SPADE-type measurement and was used in the first on-sky binary-source hypothesis-testing demonstration (Wallis et al., 16 Jun 2026).

4. Imperfections, crosstalk, and nonstationary scenes

The ideal SPADE advantage is not unconditional. Detector noise, crosstalk, misalignment, and source dynamics each modify the asymptotic picture.

For noisy detectors, the informative antisymmetric or first-order mode acquires a background floor. In a Poissonian model with background mean $|\psi_{\mathbf R}\rangle$4 and signal mean $|\psi_{\mathbf R}\rangle$5, the smallest resolvable separation scales as

$|\psi_{\mathbf R}\rangle$6

more precisely, under the paper’s half-resolution convention,

$|\psi_{\mathbf R}\rangle$7

Thus, ideal arbitrarily-small-separation superresolution is lost once detector noise is included (Len et al., 2019).

Mode-sorting crosstalk is another dominant limitation. For asymmetric source discrimination with balanced crosstalk

$|\psi_{\mathbf R}\rangle$8

one analysis derived the threshold

$|\psi_{\mathbf R}\rangle$9

below which SPADE remains asymptotically superior to direct imaging in the relevant small-separation, low-intensity-ratio regime (Amato et al., 15 May 2026). In separation estimation with arbitrary brightness imbalance, the picture is more nuanced: ideal SPADE has constant Fisher information $$f(\mathbf r)=\langle \mathbf r|\rho_1|\mathbf r\rangle,$$0, but for crosstalk-affected SPADE and any source imbalance, the paper states that SPADE performs worse than ideal direct imaging in the asymptotic limit $$f(\mathbf r)=\langle \mathbf r|\rho_1|\mathbf r\rangle,$$1; nevertheless, for practical sub-Rayleigh separations with $$f(\mathbf r)=\langle \mathbf r|\rho_1|\mathbf r\rangle,$$2, SPADE is effectively optimal (Linowski et al., 2022).

The literature also contains explicit noise-mitigation proposals. For random-unitary mode-mixing noise generated by polynomials of creation and annihilation operators, repeated demultiplexers interlaced with phase-space rotations can decouple the noise in the limit of large repetition number and small noise strength. For displacement noise, a two-step sequence with parity,

$$f(\mathbf r)=\langle \mathbf r|\rho_1|\mathbf r\rangle,$$3

cancels identical displacement errors exactly when the noise is frozen between the two steps (Sakuldee et al., 2024).

Dynamic scenes introduce a different limitation. Under Brownian motion of the source centroid, adaptive SPADE retains a short-timescale regime with $$f(\mathbf r)=\langle \mathbf r|\rho_1|\mathbf r\rangle,$$4, but in the long-timescale regime the Fisher information scales as $$f(\mathbf r)=\langle \mathbf r|\rho_1|\mathbf r\rangle,$$5, restoring Rayleigh-like behavior and giving $$f(\mathbf r)=\langle \mathbf r|\rho_1|\mathbf r\rangle,$$6 (Schlichtholz, 2024). By contrast, for rotational and oscillatory dynamics of the source pair, HG-SPADE can remain free from Rayleigh’s curse: for isotropic random orientation one reported small-separation behavior is

$$f(\mathbf r)=\langle \mathbf r|\rho_1|\mathbf r\rangle,$$7

which stays finite as $$f(\mathbf r)=\langle \mathbf r|\rho_1|\mathbf r\rangle,$$8 (Schlichtholz et al., 2024).

5. Separation estimation, discrimination, spectroscopy, and motion sensing

A substantial experimental literature treats SPADE as a practical superresolution instrument rather than only an asymptotic construct. Simultaneous multimode demultiplexing was used to estimate two-dimensional transverse separation with sensitivity around $$f(\mathbf r)=\langle \mathbf r|\rho_1|\mathbf r\rangle,$$9 in $$p(j)=\langle \phi_j|\rho_1|\phi_j\rangle = \int d^2\mathbf R\,F(\mathbf R)\,|\langle \phi_j|\psi_{\mathbf R}\rangle|^2$$0, useful performance up to $$p(j)=\langle \phi_j|\rho_1|\phi_j\rangle = \int d^2\mathbf R\,F(\mathbf R)\,|\langle \phi_j|\psi_{\mathbf R}\rangle|^2$$1, and raw scans from $$p(j)=\langle \phi_j|\rho_1|\phi_j\rangle = \int d^2\mathbf R\,F(\mathbf R)\,|\langle \phi_j|\psi_{\mathbf R}\rangle|^2$$2 to $$p(j)=\langle \phi_j|\rho_1|\phi_j\rangle = \int d^2\mathbf R\,F(\mathbf R)\,|\langle \phi_j|\psi_{\mathbf R}\rangle|^2$$3 (Boucher et al., 2020).

In the bright-source regime, SPADE has also been used for estimating both separation and relative intensity. For two bright incoherent point sources, one experiment reported a resolving power of about

$$p(j)=\langle \phi_j|\rho_1|\phi_j\rangle = \int d^2\mathbf R\,F(\mathbf R)\,|\langle \phi_j|\psi_{\mathbf R}\rangle|^2$$4

in units of beam waist, and for relative-intensity estimation found a detection limit

$$p(j)=\langle \phi_j|\rho_1|\phi_j\rangle = \int d^2\mathbf R\,F(\mathbf R)\,|\langle \phi_j|\psi_{\mathbf R}\rangle|^2$$5

compared with

$$p(j)=\langle \phi_j|\rho_1|\phi_j\rangle = \int d^2\mathbf R\,F(\mathbf R)\,|\langle \phi_j|\psi_{\mathbf R}\rangle|^2$$6

for direct imaging, i.e. about 21 times worse (Santamaria et al., 2022).

A major recent application is asymmetric source discrimination motivated by exoplanet detection. In one universal, parameter-independent test, photons in $$p(j)=\langle \phi_j|\rho_1|\phi_j\rangle = \int d^2\mathbf R\,F(\mathbf R)\,|\langle \phi_j|\psi_{\mathbf R}\rangle|^2$$7 are counted and compared with a threshold set from the $$p(j)=\langle \phi_j|\rho_1|\phi_j\rangle = \int d^2\mathbf R\,F(\mathbf R)\,|\langle \phi_j|\psi_{\mathbf R}\rangle|^2$$8 distribution at $$p(j)=\langle \phi_j|\rho_1|\phi_j\rangle = \int d^2\mathbf R\,F(\mathbf R)\,|\langle \phi_j|\psi_{\mathbf R}\rangle|^2$$9. With crosstalk $\{|\phi_j\rangle\}$0, the paper reports that SPADE requires roughly

$\{|\phi_j\rangle\}$1

times fewer photons than direct imaging in the asymptotic regime (Amato et al., 15 May 2026). A related imperfection analysis found that noisy SPADE and direct imaging can share the same realistic $\{|\phi_j\rangle\}$2 scaling, but SPADE retains a superior prefactor and remains the most efficient method under the stated sub-Rayleigh noise conditions (Linowski et al., 30 Apr 2025). The first on-sky binary-SPADE instrument, based on a DCF receiver, demonstrated binary-star detection of Alpha Centauri below the diffraction limit in a photon-starved regime and reported type-II error always lower than perfect direct imaging, although heavily limited by unbalanced loss in the coupler (Wallis et al., 16 Jun 2026).

SPADE has also been extended from imaging to spectroscopy. In a proof-of-principle star–planet experiment, light was sorted into HG modes before spectral analysis, so that the star was concentrated in $\{|\phi_j\rangle\}$3 while the off-axis planet coupled preferentially to first-order modes. The paper reports that spectral discrimination becomes clear for $\{|\phi_j\rangle\}$4 in its setup, whereas simulated direct detection needs roughly $\{|\phi_j\rangle\}$5 for comparable separation (Amato et al., 2024).

Another application is micro-oscillation frequency estimation. In that setting, PM-SPADE outperformed direct imaging in the presence of background noise because it used only two outputs and maintained substantial occupancy near small displacement. The paper reports that, without background noise, both PM-SPADE and direct imaging approach the QCRB, but with background noise PM-SPADE remains much more stable and significantly outperforms direct imaging (Hu et al., 6 Apr 2025).

6. Extensions, finite-sample subtleties, and acronym disambiguation

Several extensions push SPADE beyond the standard single-photon, static, Gaussian scenario. In a biphoton setting based on SPDC, coincidence measurements after projection of both photons onto HG modes yielded a Fisher-information enhancement proportional to $\{|\phi_j\rangle\}$6, where $\{|\phi_j\rangle\}$7 is the Schmidt number. In the ideal full-basis theory,

$\{|\phi_j\rangle\}$8

reducing to the ordinary SPADE benchmark $\{|\phi_j\rangle\}$9 when $HG_{10}$0 (Grenapin et al., 2022).

Temporal fluctuations can also be exploited as a resource. SOFSPADE combines SPADE with emitter blinking and uses temporal cumulants of detector outputs. The paper reports that temporal fluctuations both improve precision and simplify hardware, because in the presence of blinking full even-moment information can be recovered with image inversion interferometry (III) instead of a full HG sorter. For the 8th moment, SOFSPADE reaches the same precision with roughly $HG_{10}$1 fewer frames than mean-signal SPADE in the reported simulation (Kurdzialek, 25 Nov 2025).

The finite-sample literature adds an important caution. Under ideal alignment, SPADE attains the quantum-optimal large-sample Stein exponent for one-versus-two source discrimination, but a singular-learning analysis showed that ideal aligned benchmarks do not automatically survive misalignment. In a binary-SPADE reduction with offset $HG_{10}$2, the paper identifies an exact blind separation

$HG_{10}$3

at which the coarse-grained null and alternative become identical and the test power collapses to $HG_{10}$4. On the plotted Neyman–Pearson grids, direct imaging was stronger than misaligned binary-SPADE despite the latter’s distinctive local scaling structure (Kariya, 14 May 2026). This suggests that ideal aligned asymptotics should not be conflated with finite-$HG_{10}$5, imperfectly aligned performance.

The acronym SPADE is also overloaded outside this imaging literature. The temporal-to-spatial multiphoton-routing device in “Single-active-element demultiplexed multi-photon source” is explicitly not the spatial-mode demultiplexing scheme used in optical imaging, even though both involve demultiplexing into spatial outputs (Hansen et al., 2023). Likewise, “SPADE: Spatial Transcriptomics and Pathology Alignment Using a Mixture of Data Experts for an Expressive Latent Space” uses the same acronym for a pathology foundation model rather than for optical mode demultiplexing (Redekop et al., 27 Jun 2025).

Across these variants, the stable core of SPADE remains the same: project the field onto a PSF-adapted orthonormal mode basis, measure the resulting mode occupancies, and use the fact that subdiffraction information is often concentrated in a small set of orthogonal modes rather than in the intensity profile of a blurred image.