Multi-Layer Diffractive Wavefront Decoding

• Multi-layer diffractive wavefront decoding is an optical method that uses axially cascaded phase-modulating layers, optimized via deep learning, to transform complex wavefronts without additional digital processing.
• It enables precise multi-plane quantitative phase imaging, super-resolution projection, and universal linear optical transforms by mapping high-dimensional inputs into defined intensity patterns.
• Demonstrated applications include subwavelength imaging, unidirectional imaging, and phase conjugation, highlighting its potential to enhance optical computation and imaging fidelity.

Multi-layer diffractive wavefront decoding refers to a class of optical systems in which multiple axially cascaded transmissive diffractive layers, optimized via deep learning, are used to perform deterministic, all-optical transformation or “decoding” of input optical wavefronts. This approach leverages cascaded phase modulations and free-space propagation to enable operations such as multi-plane quantitative phase imaging, super-resolved image synthesis, 3D volumetric display, unidirectional imaging, subwavelength feature reconstruction, and universal linear optical transforms. The core differentiator is the ability to map a complex high-dimensional input wavefront (often containing phase, amplitude, and/or depth-encoded information) to an output intensity pattern, or to a set of patterns across multiple channels (spatial, spectral, or depth), without the need for post-detection digital computation beyond optional normalization. Large-scale trainable diffractive “neurons” (phase features) within each layer are numerically optimized using end-to-end differentiable physics-based forward models and task-specific loss functions.

1. Physical and Computational Model of Multi-Layer Diffractive Decoding

The physical system consists of $K$ transmissive diffractive layers, each discretized into arrays of phase-modulating features. For a monochromatic field, the transmission of each layer is locally

$t_k(x,y) = \exp[i\varphi_k(x,y)]$

where $\varphi_k(x,y)$ results from local thickness $h_k(x,y)$ and material index $n(\lambda)$ through $$\varphi_k(x,y) = (2\pi/\lambda)[n(\lambda)-1]h_k(x,y)$$. The field undergoes a sequence of layer transmissions and free-space propagations (using angular spectrum or Fresnel formalism) as

$$E_{k+1}(x,y,\lambda) = \mathcal{F}^{-1}\{\mathcal{F}\{E_k(x,y,\lambda) t_k(x,y)\}\cdot H(f_x,f_y;\lambda,\Delta z_k)\}$$

where $H$ is the propagation kernel. Ultimately, the output plane intensity $I_{\mathrm{out}}(x,y,\lambda)$ is mapped via joint optimization of all $\{\varphi_k\}$. Multi-channel variants leverage wavelength multiplexing, with each $\lambda_w$ addressing a distinct input or function.

The forward operator is fully differentiable, enabling stochastic gradient descent–based training of the phase maps. Loss functions are crafted to enforce task fidelity (e.g., mean squared error to a ground truth), efficiency, spectral or spatial separation, and hardware constraints (phase quantization, fabrication tolerances).

2. Multi-Plane Quantitative Phase Imaging and Wavelength Multiplexing

One primary application of multi-layer diffractive decoding is multi-plane quantitative phase imaging (QPI). Here, $M$ individual 2D phase-only objects at distinct axial positions $\{z_1,\dots, z_M\}$ are each assigned a unique wavelength $\lambda_w$. The diffractive processor, trained numerically with deep learning, can transform the stack of phase distributions $\{\phi_w(x,y)\}$ into multiplexed intensity patterns $\{I(x,y,\lambda_w)\}$ at the output sensor plane. For QPI, decoding is performed by illuminating at $\lambda_w$ and normalizing the output intensity by a reference region, yielding an accurate reconstruction of $\phi_w(x,y)$. The approach achieves simultaneous, all-optical, multi-plane phase mapping without iterative digital retrieval. Numerical and experimental results at terahertz frequencies demonstrate Pearson correlation coefficients (PCC) up to 0.993 for non-overlapping inputs, spatial resolution down to 5.2 µm line widths, and axial discrimination for interplane separation as small as $16\lambda$ (Shen et al., 2024).

3. Super-Resolution, Depth Multiplexing, and Hybrid Architectures

Hybrid encoder-decoder systems further extend the capability of diffractive wavefront decoding. In these systems, a digital encoder (usually a CNN) produces a compact wavefront encoding of a high-dimensional input (e.g., multi-plane images, super-resolved content), which is then optically decoded by the diffractive stack:

• Super-resolution image projection employs phase-only SLMs with low SBP, transmitting phase-encoded information, which is then decoded by multilayer diffractive stacks to yield pixel super-resolved outputs, overcoming diffraction and SBP limits. Gains up to 16× in SBP are demonstrated with three diffractive layers, using end-to-end error and power-regularization losses (Chen et al., 4 Oct 2025, Isil et al., 2022).
• Snapshot volumetric imaging and 3D projection are enabled by encoder networks that jointly process multiple axial slices and compress depth information into a unified phase map on a SLM. The diffractive decoder, optimized over the full 3D stack, reconstructs each target image at its respective depth with high fidelity and minimal crosstalk down to wavelength-scale axial separations (Isil et al., 23 Dec 2025).

Layer depth, SLM SBP, diffraction efficiency, and the density of axial encoding/training are all significant parameters that affect the tradeoff between output fidelity (PSNR, SSIM, PCC) and system robustness.

4. Universal Linear Optical Transformations and Spectral Multiplexing

A key generalization is the realization of broadband multi-function diffractive processors capable of universally approximating arbitrary linear transformations between input and output fields. By assigning each target transformation to a distinct wavelength channel ($\lambda_w$), and appropriately scaling the number of diffractive neurons ($N \gtrsim 2N_w N_i N_o$, where $N_i$, $N_o$ are pixel counts), multi-layer diffractive networks can all-optically implement large groups of independent matrix-vector mappings. Empirical evaluation shows that with order $10^5$–$10^6$ phase features, up to $\sim2000$ unique transforms can be multiplexed with negligible error (MSE $\lesssim 10^{-3}$), regardless of material dispersion (Li et al., 2022). This enables high-throughput parallel optical computation, hyperspectral processing, and inversion of arbitrary scattering operations for computational imaging.

5. Specialty Application Domains: Subwavelength Imaging, OPC, and Unidirectionality

• Subwavelength imaging is achieved by integrating a high-index solid-immersion encoder, which extends the transmittable spatial frequency range, with a jointly trained multi-layer diffractive decoder. This combination transcodes high-$f$ features into propagating modes, enabling reconstructions of features as small as $\sim0.29\lambda$ in intensity, experimentally demonstrated at THz (Hu et al., 2024).
• All-optical phase conjugation (OPC) uses multi-layer diffractive networks to learn the inverse mapping of phase aberrations, yielding output fields with phase conjugate to the input. Trained over large Zernike (and arbitrary) phase distortions, these stacks achieve phase MAE $<2\%$ for $\phi_{max}\leq\pi$ and up to $92\%$ diffraction efficiency, outperforming conventional OPC setups in compactness and simplicity (Shen et al., 2023).
• Unidirectional imaging in the visible employs wafer-scale nano-fabrication of multi-layer fused silica processors optimally designed such that forward transmission yields high-fidelity images while backward transmission is attenuated and distorted, with PCC > 0.86 (forward) and < 0.58 (backward) over 450–650 nm. This is accomplished via multi-objective deep learning optimizing for direction-specific NMSE, PCC, and diffraction efficiency under fabrication-constrained phase quantization (Shen et al., 2024).

6. Fabrication, Experimental Realizations, and Robustness Considerations

Device fabrication spans 3D-printing (PolyJet), wafer-scale lithography with sub-micron pitch, and monolithic multi-layer stacking, supporting lateral feature pitches from $<$1 μm (visible) to $>0.5$ mm (THz). Large-scale systems (>500 million phase features per wafer) are enabled by multi-level photoresist etching in HPFS for visible, ensuring thermal and chemical stability. Tolerance to misalignment and fabrication nonidealities is systematically baked into training through random shift augmentation (“misalignment vaccination”), quantization-aware optimization, and spectral perturbation during batch updates. Experimental validation across THz, visible, and near-IR suggests close agreement between simulated and measured performance for all the described tasks (Shen et al., 2024, Isil et al., 23 Dec 2025, Shen et al., 2024, Shen et al., 2023, Hu et al., 2024).

7. Limitations, Trade-Offs, and Prospective Directions

Key trade-offs for multi-layer diffractive decoders include:

• Layer number: More layers afford higher task complexity and degree-of-freedom but increase physical footprint and processing loss.
• Super-resolution/DOF scaling: Higher-resolution and greater depth-multiplexing demand denser phase features and deeper architecture.
• Efficiency/fidelity: Penalties for output power (diffraction efficiency) allow tuning from maximal fidelity (task-matched loss) to maximal throughput.
• Fabrication scale and quantization: Robustness to phase quantization is achieved with ≥4–6 bits for most applications; scaling to visible or IR introduces additional constraints on minimum feature dimensions, alignment, and material dispersion.

Current frontiers include integration with on-chip photonics, broadband multi-functionality (multi-task in a single device), generalized physical inversion for arbitrary scattering or dispersive scenarios, and extensibility to partially coherent or nonlinear wavefront processing.

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References:

• Multi-plane QPI and wavelength multiplexing: (Shen et al., 2024)
• Hybrid, DOF-extended and super-resolved projection: (Chen et al., 4 Oct 2025, Isil et al., 2022)
• Snapshot 3D projection and depth encoding: (Isil et al., 23 Dec 2025)
• Universal linear transforms and spectral multiplexing: (Li et al., 2022)
• Subwavelength imaging with solid immersion encoding: (Hu et al., 2024)
• Unidirectional imaging and wafer-scale fabrication: (Shen et al., 2024)
• All-optical phase conjugation: (Shen et al., 2023)