Spatial Mode-Based Imaging Techniques

• Spatial mode-based imaging is a technique that decomposes light into orthogonal spatial modes, enabling extraction of both amplitude and phase information beyond direct intensity measurements.
• It employs methods like spatial light modulators, holographic filters, and multi-plane light conversion to achieve precise mode sorting and address classical resolution limitations.
• The approach preserves quantum Fisher information at sub-Rayleigh separations, facilitating applications in super-resolution imaging, quantum metrology, and high-dimensional optical communication.

Spatial mode-based imaging encompasses a class of optical and quantum optical techniques in which the spatial structure of the light field—its decomposition into orthogonal spatial modes such as Hermite-Gaussian, Laguerre-Gaussian, or custom-adapted bases—forms the fundamental basis for detection and information retrieval. Spatial mode analysis, sorting, and detection allow extraction of phase and amplitude information inaccessible to direct intensity measurements, enabling new quantum protocols, super-resolution imaging, enhanced metrology, and high-dimensional optical communication.

1. Principles of Spatial Mode-Based Imaging

Spatial mode-based imaging replaces or augments pixel-by-pixel intensity readout by projecting the collected optical field onto an orthonormal set of spatial modes. These modes may be defined by the point-spread function (PSF) of the system (e.g., Hermite-Gauss for a Gaussian PSF, Zernike for a circular aperture, or numerically optimized "point-spread-adapted" (PAD) modes for complex apertures). The key insight, derived from quantum estimation theory, is that the optimal measurement can exploit both amplitude and phase information of the field—encoded in its spatial mode structure—to reach or approach the quantum Fisher information limit for parameter estimation.

In applications such as resolving sub-Rayleigh separations of incoherent sources, spatial mode projection allows demultiplexing the field into channels carrying information about high-order moments of the object, which remain robustly accessible even as sources are brought arbitrarily close (Tsang, 2016, Tsang, 2017, Titov, 21 Sep 2025). In quantum imaging, paired photons with strong spatial correlations or entanglement can be analyzed in a spatial mode basis, yielding sub-shot-noise sensitivity and super-resolution (Brida et al., 2010, Grenapin et al., 2022).

2. Realization: Mode Decomposition, Sorting, and Detection

Practical realization of mode-based imaging requires mode decomposition and sorting mechanisms. Linear optics methods include the use of spatial light modulators (SLMs), holographic filters, multi-plane light conversion (MPLC), and custom metasurfaces.

• SLMs and Holographic Methods: SLMs encode phase (and optionally amplitude) masks to transform or filter desired spatial modes. For higher fidelity, combined phase and amplitude modulation is superior to phase-only modulation in minimizing cross-mode contamination. Tomographic calibration, using displaced Gaussian beams as probe states, is essential for reconstructing the POVM of real holographic detectors (Bobrov et al., 2014).
• MPLC and Metasurfaces: MPLC achieves universal linear optical transformations—arbitrary mode sorting—by interleaving phase masks and free-space propagation, delivering reconfigurable mode demultiplexing for up to 25 modes in experiment (Defienne et al., 2020). Metasurfaces, fabricated with subwavelength resolution, further minimize cross-coupling and scale mode decomposition to high precision, with demonstrated mode-weight sensitivity reaching sub-ppm levels (Jones et al., 2021).
• Nonlinear Methods: Sum-frequency generation in nonlinear crystals enables direct, mode-selective upconversion of spatial modes, providing both modal analysis and cross-spectral image transfer from the IR to visible (Sephton et al., 2018).
• Quantum Detector Tomography: Calibration via quantum detector tomography quantifies imperfections in projective filtering, offering robust statistical tools for optimizing and interpreting spatial mode measurements in practical devices (Bobrov et al., 2014).

3. Surpassing Classical Resolution Limits and Quantum Advantages

Spatial mode-based imaging fundamentally circumvents Rayleigh's curse by transforming the limitation imposed by PSF overlap. In direct imaging, the Fisher information for estimating parameters such as source separation drops to zero as separation decreases; SPADE (spatial-mode demultiplexing) preserves constant Fisher information at all separations, saturating the quantum Cramér–Rao bound (Tsang, 2016, Tsang, 2017, Titov, 21 Sep 2025). Summary relationships include:

• In direct imaging, $CRB_{\mu\mu}^{(direct)} \propto \mu!/N$ for a moment $\theta_\mu$ (Tsang, 2016).
• For SPADE with a Gaussian PSF, $CRB_{\mu\mu}^{(SPADE)} \propto 1/N$, independent of separation, and $FI^{(SPADE)} = N/(4\sigma^2)$ (Tsang, 2016, Tsang, 2017).
• Biphoton (SPDC) entanglement further enhances Fisher information, scaling as $FI_{coinc}^{\textrm{tot}} = (1/2)\sqrt{K}$—surpassing the standard quantum limit (SQL) by $\sqrt{K}$, where $K$ is the Schmidt number (Grenapin et al., 2022).

Shot-noise suppression in quantum imaging schemes using multi-mode spatial correlations, such as twin-beam protocols, enables sub-shot-noise quantum imaging (SSNQI), with experimentally observed noise reduction factors $\sigma$ well below unity (e.g., $\sigma\approx0.5$ for large superpixels) (Brida et al., 2010).

4. Experimental Implementations and Applications

Experimental demonstrations span a wide range:

• Quantum Imaging: Sub-shot-noise, high-fidelity imaging using PDC-generated twin beams and spatially correlated detection, outperforming both direct and differential classical imaging in terms of SNR for weak absorbers (Brida et al., 2010).
• Super-Resolution of Incoherent Sources: SPADE implemented with MPLC or holographic sorters enables estimation of sub-Rayleigh scale separations with mean-square error and variance far below those attainable with direct imaging, using two-stage adaptive protocols for unknown centroids (Grace et al., 2019, Ozer et al., 6 Sep 2024, Titov, 21 Sep 2025).
• Single-Mode Squeezing: Amplitude-squeezed light is efficiently converted to arbitrary spatial modes via phase-only SLMs, supporting quantum-enhanced imaging and metrology—though ultimate squeezing levels are limited by diffractive and reflection losses in the mode conversion stage (Semmler et al., 2016).
• Multiplexed Communications: Arbitrary mode sorting in multimode fibers and high-dimensional QKD protocols are enabled by programmable sorters, with sorting ability depending on system degree-of-control and mode number (Defienne et al., 2020).
• Optomechanical Imaging: Spatial mode sorting retrieves nanomechanical motion with quantum-limited imprecision, by reading out mode amplitudes corresponding to mechanical mode shapes; the SPADE protocol directly links spatial decomposition to the optomechanical coupling and the imprecision–backaction product at the SQL (Choi et al., 7 Nov 2024, Pluchar et al., 9 Jul 2024).
• Near-Field and Quantum Sensing: Intermittent-contact scanning NV center electrometry achieves 10 nm spatial resolution in electric field imaging of ferroelectrics, exploiting sub-nanometer oscillation amplitudes for confined spatial sensitivity (Cheng et al., 15 Sep 2025).

5. Performance Limits, Resolution, and Noise

While SPADE theoretically overcomes classical limits, practical performance is limited by noise, alignment, and cross-talk:

• Detection Noise: The minimum resolvable separation under shot noise and dark counts scales as $d_{1/2} \sim 2\sigma/\sqrt{SNR}$ (Len et al., 2019). In quadrature (homodyne or heterodyne) measurements, the maximum Fisher information is a factor of 1/4 of the quantum optimum, with $d_{1/2}$ also scaling as $SNR^{-1/2}$.
• Mode Cross-Talk: Precision in mode decomposition—critical for imaging and communications—is fundamentally dictated by the optical quality of the mode conversion and the pixel size of the modulating element (SLM or metasurface). Subwavelength metasurfaces can suppress cross-coupling down to the ppm level (Jones et al., 2021).
• Sensitivity to Alignment: Mode-sorting receivers are highly sensitive to misalignment with the object centroid; two-stage adaptive protocols that first use direct imaging for centroid estimation before deploying the mode sorter mitigate this issue and maintain quantum-limited estimation performances (Grace et al., 2019, Ozer et al., 6 Sep 2024).
• Photon Budget and Modal Efficiency: In mode conversion systems, squeezing efficiency and super-resolution performance both degrade with diffractive losses, limited reflectivity, or incomplete modal coverage. Loss parameters $\eta$ directly determine the maintenance of quantum-enhanced noise properties (Semmler et al., 2016).

6. Extensions, Future Directions, and Applications

Broad implications for future work include:

• Scalability: Challenges remain in scaling mode-sorter implementations to high channel numbers with robust alignment and tolerance. Advanced optimization in MPLC design, metasurface fabrication, and integration with photon-number resolving detectors are active areas (Titov, 21 Sep 2025).
• Adaptive and Programmable Systems: Programmable SLM-based mode sorters, and ultimately metasurfaces, are rapidly increasing the flexibility and adaptability of spatial mode-based imaging in real time (Ozer et al., 6 Sep 2024).
• Quantum Information and Sensing: The ability to generate, manipulate, and detect spatially entangled (or squeezed) light states in arbitrary transverse modes is directly relevant for quantum-enhanced sensor networks, distributed quantum imaging, and high-capacity communication.
• Cross-Domain and Multimodal Sensing: Approaches such as time-of-flight imaging with data-driven inversion expand the concept of spatial mode-based imaging to situations without spatially structured detection—reconstructing spatial information purely from temporal data and learned priors (Turpin et al., 2019).

7. Summary Table: Key Spatial Mode-Based Imaging Modalities

Modality/Platform	Core Principle	Main Advantage(s)
SPADE (Spatial-mode demultiplexing)	Modal projection (PSF-adapted)	Super-resolution, robust moment estimation
SSNQI (Twin-beam quantum imaging)	Spatial quantum correlation	Sub-shot-noise imaging, higher SNR
Adaptive Mode-Sorting with MPLC	Programmable modal demux	Surpass diffraction limit, reconfigurable
NV Center Scanning Electrometry	Quantum probe + scanning	10 nm electric field imaging, nanoscale access
Squeezing in Arbitrary Modes	SLM-based mode conversion	Quantum-enhanced noise in tailored modes
Metasurface Mode Decomposition	Subwavelength phase encoding	High-precision, minimal cross-talk
Upconversion Mode Detection	Nonlinear sum-frequency gen.	Modal analysis + IR-to-visible image translation

In conclusion, spatial mode-based imaging systematically exploits the full optical field—both amplitude and phase via its modal decomposition—to access quantum-limited measurement, surpass classical resolution bounds, and enable new forms of multiplexed sensing and communication. The interplay of quantum estimation, advanced optical engineering, and precise calibration underpins ongoing advances in this rapidly evolving field.