SPADE Optical Demultiplexing

• SPADE is an optical measurement technique that projects the image-plane light onto an orthonormal spatial-mode basis, such as Hermite–Gaussian modes for a Gaussian PSF.
• It reassigns photon detection into discrete mode channels, concentrating subdiffraction information and avoiding the vanishing Fisher information of direct imaging.
• SPADE enables superresolution imaging, spectroscopy, and communications by efficiently extracting spatial moments while suppressing background noise.

Searching arXiv for recent and foundational papers on SPADE to ground the encyclopedia entry. arXiv search query: spatial-mode demultiplexing SPADE incoherent imaging superresolution Spatial-mode demultiplexing (SPADE) is a class of optical measurement schemes that replaces position-resolved image-plane detection by projection of the optical field onto an orthonormal spatial-mode basis followed by mode-resolved photon counting. In far-field incoherent imaging, SPADE is typically constructed from modes matched to the point-spread function (PSF), most often Hermite–Gaussian (HG) modes for a Gaussian PSF. Its central role is to concentrate subdiffraction information—especially separation, scale, and higher spatial moments—into a small number of informative channels while suppressing the bright, parameter-insensitive background that dominates direct imaging. In the canonical problem of discriminating or estimating properties of closely spaced incoherent sources, SPADE avoids the vanishing Fisher information of direct imaging at small separations, attains quantum-optimal scaling under ideal conditions, and has become a unifying framework across superresolution imaging, asymmetric hypothesis testing, spectroscopy, microscopy, and related mode-sorting architectures (Tsang, 2016, Tsang, 2017, Kariya, 14 May 2026).

1. Measurement principle and optical construction

SPADE operates by projecting the image-plane field onto an orthonormal set of spatial modes tailored to the imaging system’s PSF, and then detecting the discrete mode label or mode-resolved intensity. For a Gaussian PSF, the natural choice is the HG basis. In one dimension, the Gaussian amplitude PSF may be written as

$$\psi(x)=\left(\frac{1}{2\pi\sigma^2}\right)^{1/4}\exp\!\left(-\frac{x^2}{4\sigma^2}\right),$$

with $\int dx\,|\psi(x)|^2=1$. In the corresponding HG-SPADE receiver, each detected photon is assigned to a mode $q\in\{0,1,2,\dots\}$, with projectors $\Pi_q=|\phi_q\rangle\langle\phi_q|$ and detection probabilities

$P_a(q)=\operatorname{Tr}(\Pi_q \rho_a),$

where $\rho_a$ denotes the one-photon state under hypothesis $a$ (Kariya, 14 May 2026).

A general construction beyond the Gaussian case uses PSF-adapted derivative-like modes. In the semiclassical formulation, one begins from the Fourier transform $\Psi(k)$ of the PSF and constructs orthogonal polynomials $g_q(k)$ with respect to the weight $|\Psi(k)|^2$. The resulting point-spread-function-adapted (PAD) modes are

$\int dx\,|\psi(x)|^2=1$0

which remain orthonormal in real space (Tsang, 2017). For Gaussian $\int dx\,|\psi(x)|^2=1$1, the $\int dx\,|\psi(x)|^2=1$2 reduce to scaled Hermite polynomials, and the real-space PAD modes become the familiar TEM/HG modes (Tsang, 2017).

The physical significance of this basis choice is that spatial perturbations of a centered source couple preferentially into higher-order modes. Under perfect alignment and a single on-axis source, all photons ideally populate the lowest mode $\int dx\,|\psi(x)|^2=1$3. Small displacements cause “leakage” out of $\int dx\,|\psi(x)|^2=1$4, with the first nontrivial contribution appearing in $\int dx\,|\psi(x)|^2=1$5. This leakage is the core signal exploited by SPADE in sub-Rayleigh problems (Kariya, 14 May 2026). A widely used coarse graining is binary-SPADE, which records only whether a photon is in the fundamental mode $\int dx\,|\psi(x)|^2=1$6 or in the complement $\int dx\,|\psi(x)|^2=1$7. Near alignment, this retains the full leading $\int dx\,|\psi(x)|^2=1$8 leakage contrast, while redistribution into higher modes enters only at $\int dx\,|\psi(x)|^2=1$9 (Kariya, 14 May 2026).

Closely related architectures include interferometric PAD (iPAD), where superpositions of two spatial modes are interfered to access off-diagonal coherences in the modal mutual coherence matrix. If $q\in\{0,1,2,\dots\}$0, then the measured powers encode $q\in\{0,1,2,\dots\}$1, extending SPADE from diagonal moment access to full moment tomography (Tsang, 2017).

2. Statistical model and relation to direct imaging

In direct imaging, the measurement outcome is the photon position $q\in\{0,1,2,\dots\}$2, with image-plane distribution

$q\in\{0,1,2,\dots\}$3

for the asymmetric one-versus-two-source problem, where $q\in\{0,1,2,\dots\}$4 is the separation, $q\in\{0,1,2,\dots\}$5 the PSF width, and $q\in\{0,1,2,\dots\}$6 the brightness ratio of the secondary source (Kariya, 14 May 2026). The data are i.i.d. samples $q\in\{0,1,2,\dots\}$7, and the likelihood over $q\in\{0,1,2,\dots\}$8 detections is $q\in\{0,1,2,\dots\}$9 (Kariya, 14 May 2026).

SPADE changes only the measurement, not the underlying optical state. In the same problem, the quantum states per detected photon are

$\Pi_q=|\phi_q\rangle\langle\phi_q|$0

$\Pi_q=|\phi_q\rangle\langle\phi_q|$1

A SPADE receiver measures the induced classical distribution over mode labels $\Pi_q=|\phi_q\rangle\langle\phi_q|$2, producing i.i.d. outcomes $\Pi_q=|\phi_q\rangle\langle\phi_q|$3, or equivalently multinomial counts after $\Pi_q=|\phi_q\rangle\langle\phi_q|$4 detections (Kariya, 14 May 2026).

In a broader semiclassical treatment of incoherent imaging, one writes the mutual coherence function as

$\Pi_q=|\phi_q\rangle\langle\phi_q|$5

and the average image intensity as

$\Pi_q=|\phi_q\rangle\langle\phi_q|$6

with $\Pi_q=|\phi_q\rangle\langle\phi_q|$7 the object intensity function. After linear-optical projection onto modes $\Pi_q=|\phi_q\rangle\langle\phi_q|$8, the mean power in channel $\Pi_q=|\phi_q\rangle\langle\phi_q|$9 is

$P_a(q)=\operatorname{Tr}(\Pi_q \rho_a),$0

and photon counts are independent Poisson variables with means $P_a(q)=\operatorname{Tr}(\Pi_q \rho_a),$1, where $P_a(q)=\operatorname{Tr}(\Pi_q \rho_a),$2 (Tsang, 2017). This Poisson channel model underlies both moment estimation and Fisher-information calculations.

The statistical contrast between direct imaging and SPADE is therefore not a change of source model but a change of sufficient statistics. Direct imaging samples a PSF-dominated intensity profile, whereas SPADE samples a discrete mode distribution in which the background-dominated fundamental mode can be rejected or separated from the informative leakage channels (Tsang, 2017).

3. Superresolution, Fisher information, and quantum limits

SPADE is most closely associated with the removal of Rayleigh’s curse. In direct imaging of two equally bright incoherent point sources with Gaussian PSF, the Fisher information for separation vanishes as the separation tends to zero. In one formulation, for a Gaussian PSF with width $P_a(q)=\operatorname{Tr}(\Pi_q \rho_a),$3,

$P_a(q)=\operatorname{Tr}(\Pi_q \rho_a),$4

while HG-SPADE yields a separation-independent value

$P_a(q)=\operatorname{Tr}(\Pi_q \rho_a),$5

matching the quantum Fisher information over the full separation range in the ideal Gaussian case (Titov, 21 Sep 2025). An earlier moment-based treatment expresses the same phenomenon as

$P_a(q)=\operatorname{Tr}(\Pi_q \rho_a),$6

for small $P_a(q)=\operatorname{Tr}(\Pi_q \rho_a),$7, again showing direct-imaging collapse and constant SPADE information (Tsang, 2016).

The mechanism is transparent in the small-displacement expansion of the modal probabilities. For a displaced Gaussian source, the HG-mode probabilities are Poisson in the mode index: $P_a(q)=\operatorname{Tr}(\Pi_q \rho_a),$8 Thus,

$P_a(q)=\operatorname{Tr}(\Pi_q \rho_a),$9

so the first-order leakage channel alone already contains the leading $\rho_a$0 separation signal (Kariya, 14 May 2026). This explains why binary-SPADE and first-order HG detection can perform nearly optimally in the sub-Rayleigh regime.

In the language of object moments, SPADE also linearizes access to higher-order structure. For a subdiffraction object with support width $\rho_a$1, the PAD-mode mutual coherence matrix satisfies

$\rho_a$2

so $\rho_a$3 is chiefly sensitive to the moment $\rho_a$4 at leading order. The resulting estimator variances scale as

$\rho_a$5

whereas direct-imaging Cramér–Rao bounds do not improve with shrinking object size in the same way (Tsang, 2017). For Gaussian PSF and two equal incoherent point sources, SPADE yields a finite separation Fisher information $\rho_a$6 as $\rho_a$7, while direct imaging gives $\rho_a$8 (Tsang, 2017).

The same logic extends to other parameters. For location and scale parameters of subdiffraction objects, SPADE approaches the optimal precision allowed by quantum mechanics in the small-object limit (Tsang, 2016). This suggests that SPADE is not merely a special-purpose separation estimator but a measurement paradigm aligned with the quantum geometry of incoherent imaging.

4. Singular asymptotics and finite-photon hypothesis testing

Beyond Fisher information and large-sample exponents, recent work has analyzed SPADE near singular model boundaries, where regular asymptotics break down. In the asymmetric one-versus-two-source discrimination problem, the null hypothesis is realized at the singular point $\rho_a$9, so the Kullback–Leibler function vanishes on a nonregular boundary. Singular learning theory then describes the Bayes free energy through the real log canonical threshold (RLCT) $a$0 and multiplicity $a$1 (Kariya, 14 May 2026).

For the aligned Gaussian problem, direct imaging and SPADE have different local divergences,

$a$2

leading to zeta functions

$a$3

Both schemes share the same rightmost pole $a$4, hence the same RLCT $a$5, but their multiplicities differ: $a$6 This yields Bayes free-energy asymptotics

$a$7

so aligned SPADE has a universal subleading advantage in the local prior-weighted singular regime (Kariya, 14 May 2026).

In the same setting, the ideal aligned SPADE classical Stein exponent matches the quantum benchmark to leading order. Direct imaging has

$a$8

whereas the quantum Stein exponent is

$a$9

and aligned SPADE achieves

$\Psi(k)$0

matching the quantum limit to leading order (Kariya, 14 May 2026).

However, finite-photon performance is not determined solely by asymptotic exponents. In misaligned binary-SPADE, the local Kullback–Leibler divergence scales as $\Psi(k)$1, and nontrivial local power first appears on the intrinsic scale $\Psi(k)$2, whereas direct imaging appears on $\Psi(k)$3 (Kariya, 14 May 2026). This suggests a formal asymptotic advantage for misaligned binary-SPADE, but numerical Neyman–Pearson comparisons under common physical conditions in that analysis show direct imaging stronger on the plotted grids and reveal an exact blind separation $\Psi(k)$4, at which the binary-SPADE power collapses to the test size $\Psi(k)$5 (Kariya, 14 May 2026). A plausible implication is that singular asymptotics organize the local theory, but experimental utility depends sensitively on which asymptotic regime is actually entered.

5. Imperfections, crosstalk, misalignment, and robustness

SPADE’s theoretical advantage is contingent on mode fidelity. Several distinct imperfections recur across the literature: misalignment between the sorter basis and the source centroid, crosstalk between modal channels, background and dark noise, finite bandwidth, and dynamic centroid motion.

Misalignment

Misalignment replaces the ideal support mismatch of aligned SPADE by a genuine fluctuating null. In the binary-SPADE reduction with detector offset $\Psi(k)$6, the $\Psi(k)$7 success probabilities are

$\Psi(k)$8

Near alignment, the Bernoulli contrast obeys

$\Psi(k)$9

with

$g_q(k)$0

This preserves the leading $g_q(k)$1 leakage contrast but removes the ideal null singularity (Kariya, 14 May 2026). In adaptive or Brownian settings, centroid fluctuations can reintroduce Rayleigh-type scaling if the measurement interval is too long. For a centroid undergoing Brownian motion with diffusion constant $g_q(k)$2, and segment duration $g_q(k)$3, the dimensionless parameter $g_q(k)$4 controls the regime. For short times $g_q(k)$5, where $g_q(k)$6, the per-photon FI remains approximately constant, but for long times $g_q(k)$7, the FI scales as $g_q(k)$8, so Rayleigh’s curse reappears (Schlichtholz, 2024). Adaptive realignment with $g_q(k)$9 and small $|\Psi(k)|^2$0 preserves near-optimal performance (Schlichtholz, 2024).

Crosstalk

Crosstalk has been analyzed both phenomenologically and experimentally. In an asymmetric one-versus-two-source test using the combined odd channel, balanced crosstalk with probability $|\Psi(k)|^2$1 modifies the Bernoulli distributions via

$|\Psi(k)|^2$2

The resulting leading SPADE error exponent becomes

$|\Psi(k)|^2$3

for small $|\Psi(k)|^2$4, while direct imaging has $|\Psi(k)|^2$5. Balanced SPADE remains exponentially superior provided

$|\Psi(k)|^2$6

At $|\Psi(k)|^2$7, SPADE requires about $|\Psi(k)|^2$8, or roughly $|\Psi(k)|^2$9 fewer photons, to achieve the same error rate (Amato et al., 15 May 2026). This threshold result is notable because it identifies a concrete imperfection level below which SPADE’s practical advantage survives.

A different crosstalk analysis for resolving separation between unbalanced sources finds that, for any imbalance, crosstalk-affected SPADE performs worse than ideal direct imaging in the strict vanishing-separation limit, although it remains superior over several orders of magnitude of sub-Rayleigh separations above an imbalance-dependent threshold $\int dx\,|\psi(x)|^2=1$00 (Linowski et al., 2022). This is not a contradiction: one result addresses asymmetric hypothesis testing with calibrated odd-channel statistics, while the other studies asymptotic estimation FI for unbalanced two-source separation. Together they indicate that the effect of crosstalk depends materially on the inference task.

Detector and background noise

With noisy detection, SPADE’s resolution becomes signal-to-noise-ratio limited. In a binary-SPADE or first-order mode detector with background $\int dx\,|\psi(x)|^2=1$01, the small-separation FI is suppressed by the factor $\int dx\,|\psi(x)|^2=1$02, where $\int dx\,|\psi(x)|^2=1$03, so the minimum resolvable separation scales as

$\int dx\,|\psi(x)|^2=1$04

rather than vanishing arbitrarily (Len et al., 2019). Photon counting retains the largest advantage; homodyne and heterodyne variants also beat direct imaging in an SNR-dependent window but are limited to a maximum FI of $\int dx\,|\psi(x)|^2=1$05 (Len et al., 2019).

Noise suppression protocols

Noise need not only be mitigated passively. For random unitary noise generated by polynomials of creation and annihilation operators, repeated demultiplexers interleaved with rotation operators can decouple the noise in the large-repetition, weak-noise limit. For displacement noise, a simpler two-pass protocol with an interleaved parity operator can achieve perfect decoupling if the noise configuration is frozen between passes, restoring SPADE’s constant small-separation FI (Sakuldee et al., 2024). This suggests a direct connection between SPADE robustness and dynamical-decoupling ideas from quantum control.

6. Implementations, applications, and extensions

SPADE has moved well beyond the canonical two-source thought experiment. Its implementations now span mode sorters, interferometric schemes, multi-plane light conversion (MPLC), digital holography, double-clad fibers, and spectroscopy platforms.

Multi-plane light conversion and mode sorters

MPLC is a prominent implementation technology because it realizes arbitrary unitary spatial-mode transforms with cascaded phase profiles and free-space propagation. Reflective folded MPLC architectures have been used to sort low-order HG modes with experimental average fidelity $\int dx\,|\psi(x)|^2=1$06 for a three-mode sorter and simulated diagonal fidelities $\int dx\,|\psi(x)|^2=1$07–$\int dx\,|\psi(x)|^2=1$08 for a six-mode design (Titov, 21 Sep 2025). These devices support direct SPADE measurements for superresolution experiments and make the ideal HG-basis picture experimentally concrete.

A larger 45-mode MPLC multiplexer/demultiplexer based on a separable HG basis achieves average insertion loss of about $\int dx\,|\psi(x)|^2=1$09 dB and average crosstalk of about $\int dx\,|\psi(x)|^2=1$10 dB across the C band (Bade et al., 2018). Although developed for space-division multiplexing rather than imaging, it implements precisely the kind of high-dimensional orthonormal projection required by SPADE, and therefore functions as a scalable mode-sorting backend for large-mode-count SPADE architectures.

Distance and intensity metrology

In the bright-source regime, SPADE can estimate transverse separation and relative intensity of two incoherent point sources with large dynamic range. In one implementation using a commercial HG demultiplexer at $\int dx\,|\psi(x)|^2=1$11, the measured odd-mode signals $\int dx\,|\psi(x)|^2=1$12 and $\int dx\,|\psi(x)|^2=1$13 scaled quadratically with sub-Rayleigh displacement, yielding a resolving power $\int dx\,|\psi(x)|^2=1$14 of the beam waist and a relative-intensity detection limit $\int dx\,|\psi(x)|^2=1$15, about $\int dx\,|\psi(x)|^2=1$16 better than direct imaging in the same apparatus (Santamaria et al., 2022).

A simultaneous nine-mode HG demultiplexing experiment likewise demonstrated 2D distance estimation beyond the Rayleigh limit with a large dynamic range, while also showing that calibrated crosstalk acts as an additive offset in first-order channels and must be explicitly modeled (Boucher et al., 2020).

Asymmetric source discrimination and exoplanet-inspired tests

In asymmetric one-versus-two-source detection, SPADE is especially attractive because odd modes or leakage channels suppress the bright on-axis source. A 2026 tabletop experiment implemented a parameter-independent test based on odd-channel counts and showed that SPADE retains an advantage over direct imaging at measured crosstalk $\int dx\,|\psi(x)|^2=1$17, with false-negative rates well below direct imaging in the small-separation, low-brightness-ratio regime (Amato et al., 15 May 2026).

This theme extends to spectroscopy. A proof-of-principle experiment sorted light from overlapping incoherent “star” and “planet” sources into HG modes and measured spectra in each mode. The star coupled almost entirely to $\int dx\,|\psi(x)|^2=1$18, while the off-axis planet preferentially populated $\int dx\,|\psi(x)|^2=1$19. In the overlapping-PSF regime, the Fisher information for planet spectroscopy scaled as $\int dx\,|\psi(x)|^2=1$20 under HG-SPADE but as $\int dx\,|\psi(x)|^2=1$21 under direct detection, a quadratic gain in information (Amato et al., 2024). The implementation used an MPLC sorter and SNSPDs, with measured first-order cross-talk $\int dx\,|\psi(x)|^2=1$22 (Amato et al., 2024).

An on-sky binary-SPADE demonstration has now been reported using a double-clad fiber coupler that separates a PSF-matched fundamental mode from its complement. In the photon-starved regime, with fewer than $\int dx\,|\psi(x)|^2=1$23 photons per measurement, the instrument detected a binary star below the diffraction limit and achieved type-II error rates consistently lower than perfect direct imaging, though unbalanced loss in the coupler substantially limited the scaling relative to ideal SPADE (Wallis et al., 16 Jun 2026). This constitutes the first on-sky demonstration of binary-source hypothesis testing using SPADE-based detection.

Dynamic sources and motion estimation

SPADE remains useful for dynamic scenes. For rotating or oscillating incoherent source pairs with Gaussian PSF, the small-separation SPADE FI remains strictly positive under broad orientation dynamics, while direct imaging often reverts to $\int dx\,|\psi(x)|^2=1$24-scaling (Schlichtholz et al., 2024). Under random orientation on the sphere,

$\int dx\,|\psi(x)|^2=1$25

with $\int dx\,|\psi(x)|^2=1$26 (Schlichtholz et al., 2024). The same work proposes a rotation-averaging algorithm that removes the need to estimate the source orientation angle while preserving near-quantum-limited precision (Schlichtholz et al., 2024).

SPADE has also been used for frequency estimation of a micro-oscillating point source. A two-mode “plus-minus” SPADE, using the superpositions $\int dx\,|\psi(x)|^2=1$27, concentrates position information into only two channels. In the presence of background, its Fisher-information penalty scales as $\int dx\,|\psi(x)|^2=1$28, making it substantially more robust than direct imaging and standard HG-SPADE for estimating oscillation frequency (Hu et al., 6 Apr 2025).

Fluctuation-enhanced SPADE

Temporal fluctuations can simplify as well as improve SPADE. With blinking emitters, temporal cumulants of low-order SPADE outputs encode higher spatial moments. In the subdiffraction regime, for SPADE mode $\int dx\,|\psi(x)|^2=1$29,

$\int dx\,|\psi(x)|^2=1$30

so higher-order cumulants improve higher-moment estimation (Kurdzialek, 25 Nov 2025). Remarkably, in the presence of fluctuations, full even-moment recovery can be obtained with the much simpler image inversion interferometer, because

$\int dx\,|\psi(x)|^2=1$31

allowing fluctuation-enhanced parity sorting to replace full SPADE for even moments (Kurdzialek, 25 Nov 2025). This suggests that blinking can act as an informational resource rather than merely a nuisance.

Quantum and communication extensions

SPADE has also entered explicitly quantum regimes. For spatially entangled bi-photons, coincidence detection after HG projection yields a Fisher-information enhancement by $\int dx\,|\psi(x)|^2=1$32, where $\int dx\,|\psi(x)|^2=1$33 is the Schmidt number of the two-photon state. In the 2D case,

$\int dx\,|\psi(x)|^2=1$34

showing that entanglement enhances superresolution beyond ordinary SPADE (Grenapin et al., 2022).

Outside imaging, the same projective mode-sorting principle appears in communications-oriented demultiplexers. A tri-degree-of-freedom multi-vortex geometric beam basis supports a SPADE-like matched-filter receiver with high divergence degeneracy and low bit-error rate, illustrating that the abstract notion of SPADE extends naturally to large-scale mode demultiplexing beyond superresolution imaging (Wan et al., 2021).

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SPADE is therefore best understood not as a single instrument but as a measurement principle: project onto a basis adapted to the PSF and the inference task, then count in the resulting mode channels. In ideal aligned Gaussian settings, this principle yields quantum-optimal discrimination and estimation, eliminates Rayleigh’s curse, and gives direct access to low-order object moments (Tsang, 2016, Tsang, 2017, Kariya, 14 May 2026). In realistic systems, performance depends critically on alignment, crosstalk, noise, and temporal dynamics; under sufficiently good calibration, those imperfections can be tolerated, modeled, or partially corrected (Amato et al., 15 May 2026, Sakuldee et al., 2024). The present research trajectory points toward robust mode sorters, adaptive and on-sky receivers, joint spectral-spatial schemes, and task-specific reduced measurements that preserve SPADE’s core informational advantage while relaxing its experimental demands (Amato et al., 2024, Wallis et al., 16 Jun 2026, Kurdzialek, 25 Nov 2025).