Planar Diffractive Neural Networks

• Planar Diffractive Neural Networks (PDNNs) are multilayer optical architectures that leverage engineered phase modulation and controlled diffraction for high-speed, passive inference.
• They enable efficient, parallel processing for tasks such as image classification, mode sorting, wavelength-multiplexed transforms, and RF spatial modulation.
• Their training uses gradient-based optimization through differentiable physical models, ensuring robustness against fabrication and misalignment errors.

Planar Diffractive Neural Networks (PDNNs) are multilayer, optically-implemented neural-network architectures in which wave propagation and engineered phase modulation across planar surfaces collectively perform inference directly in the physical (electromagnetic) domain. Distinct from digital or bulk 3D optical neural networks, PDNNs exploit cascaded, engineered phase plates (diffractive optical elements, DOEs) or equivalent structures, enabling all-optical or RF signal processing, inference, imaging, and communications. PDNNs are differentiated by their physical planarity, compatibility with large-area parallel fabrication (e.g., on printed circuits or wafers), and their support for high-efficiency, ultra-fast, passive “computing-by-propagation.” Core applications span computational imaging, classification, universal linear transformation, wavelength-multiplexed and multispectral processing, mode sorting, and RF spatial modulation.

1. Physical and Mathematical Framework

PDNNs model light or EM propagation as a sequence of alternations between on-plane phase (or amplitude-phase) modulations and free-space or guided-wave propagation. The field $E_m(x, y)$ just before plate $m$ undergoes phase modulation:

$E_m^+(x, y) = e^{i\phi_m(x, y)} E_m(x, y)$

Propagation over distance $z$ is described by the Fresnel (paraxial) or Rayleigh–Sommerfeld integrals:

$$E_{m+1}(x, y) = \frac{e^{ikz}}{i\lambda z} \iint E_m^+(x', y') e^{ \frac{ik}{2z}((x - x')^2 + (y - y')^2)} dx' dy'$$

or, equivalently, via angular-spectrum methods in the Fourier domain:

$$E_{m+1}(x, y) = \mathcal{F}^{-1} \left\{ \mathcal{F}\{E_m^+\} \cdot H(f_x, f_y; z) \right\}$$

$$H(f_x, f_y; z) = \exp \left[ i 2\pi z \sqrt{1/\lambda^2 - f_x^2 - f_y^2} \right]$$

Each layer’s phase profile $\phi_m(x, y)$ is discretized (e.g., 200×200 or 512×512 grid), with pixel sizes set to match system numerical aperture and Nyquist sampling constraints (Bearne et al., 27 Aug 2025, Lin et al., 2018, Soshnikov et al., 23 Jul 2024).

2. Architectural Paradigms: Planar Optical and RF Implementations

Optical PDNNs are assembled from $M$ parallel phase plates, spaced centimeters apart (or down to hundreds of microns for compact systems), with phase profiles realized by high-resolution lithography, SLMs, or metasurfaces (Chen et al., 2022, Rahman et al., 15 Jun 2024). In the RF domain, planar transmission-line networks implement analogous systems, where each layer consists of a diffractive coupling matrix $\mathbf{W}^{(l)}$ and programmable phase-shifters $\boldsymbol{\Phi}^{(l)}$:

$$\mathbf{x}^{(l)} = f^{(l)}\left( \boldsymbol{\Phi}^{(l)} \mathbf{W}^{(l)} \mathbf{x}^{(l-1)} \right)$$

This abstraction subsumes optical diffraction in linear circuit models for planar, PCB-compatible systems (Teng et al., 30 Nov 2025). Both approaches propagate an input field (or signal vector) through a cascade of phase modulations and mixing, analogously to layers in an artificial neural network, but performed at the speed of light.

3. Training Methodologies and Loss Functionals

PDNN parameters (phase profiles and, where relevant, detection regions or mask weights) are optimized offline by gradient-based methods, using fully differentiable models of optical or EM propagation. Canonical losses include:

• Cross-entropy over output intensities assigned to labeled detector regions for classification tasks:

$$L = -\sum_i\sum_c y^{(i)}_c\log\left(\frac{I_c^{(i)}}{\sum_{c'} I_{c'}^{(i)}}\right)$$

• Quadratic or custom losses for multi-focus, multi-wavelength, or mode-sorting efficiency:

$$L = \sum_{\lambda, n} w_{\lambda, n} (I_{\lambda}(x_{\lambda, n}, y_{\lambda, n}) - I^{\mathrm{target}}_{\lambda, n})^2 + R(\{\phi_m\})$$

• Efficiency/crosstalk losses for mode sorting:

$$\mathbf{Loss}_\mathrm{eff} = \frac{1}{n}\sum_{i=1}^n (1 - \eta_i), \quad \mathbf{Loss}_\mathrm{xtalk} = \frac{1}{n}\sum_{i=1}^n \chi_i$$

where $\eta_i$ and $\chi_i$ are per-mode efficiencies and crosstalk (Bearne et al., 27 Aug 2025).

Optimization is conducted via stochastic gradient descent, Adam, or AdamW, with chain-rule back-propagation through the cascade of phase-plate and propagation operators. Physical constraints and regularizations (phase quantization, thickness bounds, smoothness) are applied as box constraints or penalty terms as appropriate (Soshnikov et al., 23 Jul 2024, Chen et al., 2022).

4. Advanced Features: Multiplexing, Robustness, and Programmability

PDNNs support a diverse range of functional extensions. Wavelength-multiplexed PDNNs assign distinct linear transformations or inference tasks to multiple spectral channels, leveraging the dispersion dependence of phase modulation:

$N \gtrsim 2 N_w N_i N_o$

where $N$ is the total number of neurons (pixels), $N_w$ is the number of spectral channels, and $N_i, N_o$ are input/output field dimensions (Li et al., 2022). Multi-task, multi-band, and multi-detection-region architectures can be encoded by augmenting the loss to cover all desired functions and spectral settings (Chen et al., 2022, Motz et al., 23 Jul 2024).

Spatial robustness to misalignment and environmental imperfections is engineered by “vaccination”—i.e., sampling, during training, random transverse and longitudinal shifts for each DOE. The expected error functional over these shifts is minimized by averaging gradients across perturbed realizations, yielding PDNNs with substantial tolerance (17 wavelengths or more) to fabrication and assembly errors (Soshnikov et al., 23 Jul 2024). Similarly, shift-, scale-, and rotation-invariance is obtained by randomizing input image transformations during optimization (Mengu et al., 2020).

Optical PDNNs can be fabricated as static devices (phase-only polymers, metasurfaces) or as programmable/reconfigurable systems using SLMs or tunable meta-atoms for in-situ phase updates. Hybrid optical-digital PDNNs combine engineered diffraction with electronic classifiers or decoders for increased flexibility, e.g., compressive spectral encoding with subsequent digital reconstruction (Rahman et al., 15 Jun 2024, Li et al., 2020).

5. Core Applications and Experimental Performance

The flexibility and speed of PDNNs support a variety of all-optical, RF, and hybrid tasks:

• Image Classification: Five-layer THz PDNNs (40k pixels/layer) achieve 91.75% MNIST accuracy (test set) with energy collection rates of 34–53% in the correct detector (Lin et al., 2018).
• Mode Sorting: Flexible detection region architectures surpass crosstalk-limited efficiency plateaus in traditional methods, e.g., achieving ⟨η⟩ ≈ 30% efficiency at ⟨χ⟩ ≈ 1% crosstalk for 25 Hermite–Gaussian modes (Bearne et al., 27 Aug 2025).
• Wavelength-Multiplexed Linear Transform: Broadband PDNNs implement hundreds to thousands of unique optical transforms in a single device, provided $N \gtrsim 2 N_w N_i N_o$ holds (Li et al., 2022).
• Single-Pixel, Spectrally-Encoded Vision: PDNNs map class information onto spectral bands, enabling classification with a single-pixel detector (e.g., 96.07% MNIST accuracy, 10 wavelength bins) (Li et al., 2020).
• Multi-Task, Multi-Band Classification: Architectures capable of solving MNIST, Fashion-MNIST, and EMNIST concurrently at $\lambda_1, \lambda_2, \lambda_3$ reach test set accuracies up to 97.9% for MNIST in cascades of three DOEs (Motz et al., 23 Jul 2024).
• RF Spatial Modulation: Planar PDNNs (RF domain) jointly perform beamforming, modulation, and detection, achieving theoretical error rates near orthogonal, non-coherent M-FSK in SSK systems, all in the analog wave domain (Teng et al., 30 Nov 2025).

Experimental reports validate modeled performance, demonstrating high-fidelity agreement between simulation and hardware in both optical (Lin et al., 2018, Bearne et al., 27 Aug 2025) and THz settings (Li et al., 2020).

6. Practical Fabrication, Implementation, and Deployment Considerations

PDNN fabrication leverages 3D printing (THz), high-resolution lithography (visible/IR), or PCB processes (RF/microwave). Material choices are dictated by target band, with metasurface variants supporting sub-100-nm features for visible wavelengths (Chen et al., 2022). Layer-to-layer alignment and pixel fidelity critically impact performance, but robust design and vaccination strategies relax tolerances substantially (mechanical positioning errors tolerated up to 17λ) (Soshnikov et al., 23 Jul 2024, Li et al., 2022). Incoherent or partially coherent illumination can be handled by coherence-aware training: the mutual intensity propagation model and randomization of spatial and temporal coherence parameters during optimization yield “coherence-blind” PDNNs effective under environmental drift (Kleiner et al., 13 Aug 2024).

Programmability via SLMs or tunable elements unlocks multi-task, dynamic, or in-situ calibrated PDNNs, with switching times limited by modulator technology. Hybrid digital back-ends can further increase task generality, reduce detection hardware complexity, and extend to compressive and reconstruction tasks (Rahman et al., 15 Jun 2024).

7. Limitations, Extensions, and Outlook

PDNN performance is fundamentally governed by system degrees of freedom (phase pixels, layer count), physical phenomena (wavelength-specificity, coherence, losses), and task complexity. Limitations include:

• Finite pixel/height quantization and sampling limiting spatial bandwidth (Chen et al., 2022).
• Wavelength and polarization sensitivity.
• Sensitivity to deviations from scalar, paraxial, or single-frequency approximations in demanding applications.

The extensibility of PDNNs to new domains is under active investigation, including amplitude-phase modulation, wavefront shaping with arbitrarily engineered point-spread functions, universal linear operator realization, and RF systems with nonlinearity and non-coherent detection (Teng et al., 30 Nov 2025). The core methodology—differentiable end-to-end modeling and training with augmented physical constraints—suggests a continued broadening of the PDNN paradigm, blurring the conventional separation between photonic or RF hardware and machine learning architectures (Lin et al., 2018).

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Key References:

• "Diffractive neural networks for mode-sorting with flexible detection regions" (Bearne et al., 27 Aug 2025)
• "All-Optical Machine Learning Using Diffractive Deep Neural Networks" (Lin et al., 2018)
• "Designing robust diffractive neural networks with improved transverse shift tolerance" (Soshnikov et al., 23 Jul 2024)
• "Integration of Programmable Diffraction with Digital Neural Networks" (Rahman et al., 15 Jun 2024)
• "Inverse design of ultracompact multi-focal optical devices by diffractive neural networks" (Chen et al., 2022)
• "Design of diffractive neural networks solving different classification problems at different wavelengths" (Motz et al., 23 Jul 2024)
• "Spectrally-Encoded Single-Pixel Machine Vision Using Diffractive Networks" (Li et al., 2020)
• "Massively Parallel Universal Linear Transformations using a Wavelength-Multiplexed Diffractive Optical Network" (Li et al., 2022)
• "Coherence Awareness in Diffractive Neural Networks" (Kleiner et al., 13 Aug 2024)
• "Planar Diffractive Neural Networks Empowered Communications: A Spatial Modulation Scheme" (Teng et al., 30 Nov 2025)