Anti-Interference Diffractive Deep Neural Network

• The paper introduces a single-layer dual-wavelength differential D2NN that achieves high classification accuracy by mitigating inter-layer misalignment and common-mode noise.
• It employs a Rayleigh–Sommerfeld diffraction model with differential readout, leveraging complementary intensity maps from two wavelengths for enhanced anti-interference performance.
• The architecture demonstrates superior robustness and efficiency, outperforming multi-layer cascaded D2NNs while using significantly fewer parameters and reducing system complexity.

to=arxiv_search.search 天天彩票与你同行json {"query":"(Wang et al., 23 Jul 2025) Anti-Interference Diffractive Deep Neural Network single-layer dual-wavelength differential D2NN misalignment resilient diffractive optical networks orthogonality diffractive deep neural networks", "max_results": 10} to=arxiv_search.search 天天种彩票json {"query":"(Wang et al., 23 Jul 2025)", "max_results": 5} to=arxiv_search.search 大发分分彩json {"query":"(Wang et al., 23 Jul 2025, Huang et al., 9 Jul 2025, Mengu et al., 2020, Zheng et al., 2018, Shi, 2020, Mengu et al., 2020, Feng et al., 2024, Lin et al., 2018)", "max_results": 20} Anti-Interference Diffractive Deep Neural Network (AI D2NN) denotes a robustness-oriented class of diffractive optical neural networks in which the optical architecture, readout rule, and training procedure are designed to preserve inference fidelity under perturbations such as inter-layer misalignment, coherent artifacts, illumination fluctuations, detector bias, fabrication phase errors, occlusions, and input noise. In its most specific 2025 formulation, AI D2NN refers to a compact single-layer dual-wavelength differential diffractive network that shares one trainable mask between $\lambda_+=1064\,\mathrm{nm}$ and $\lambda_-=532\,\mathrm{nm}$, generates two complementary detector-plane intensity maps, and classifies by normalized differential integration over class-specific subregions; the design was introduced as a visible-light alternative to cascaded multi-layer D2NNs whose alignment sensitivity becomes severe at small feature sizes (Wang et al., 23 Jul 2025).

1. Origins within diffractive optical computing

Diffractive deep neural networks (D2NNs) were introduced as all-optical systems in which passive diffractive layers, trained by backpropagation, implement inference through coherent propagation and interference rather than electronic multiply-accumulate operations. In the original five-layer THz demonstrations, phase-only transmissive masks performed handwritten digit classification and imaging by directing optical energy toward task-specific detector regions; the framework established the basic D2NN paradigm of thin-mask modulation plus free-space diffraction (Lin et al., 2018).

A theoretical foundation for D2NN behavior was later formalized through inner-product invariance and unitarity laws. Under coherent, monochromatic, scalar, lossless, phase-only assumptions, the cascade operator $U$ satisfies

$$\langle U E_1, U E_2 \rangle = \langle E_1, E_2 \rangle,$$

so modal orthogonality is preserved through propagation and phase modulation. The same analysis showed that spatially separated output intensities imply orthogonality of the underlying input fields, which explains why diffractive processors can realize low-crosstalk mode conversion, multiplexing, and mode recognition when the optical conditions remain close to the ideal unitary model (Zheng et al., 2018).

AI D2NN arises from the gap between that idealized picture and physical deployment. Conventional cascaded D2NNs are highly sensitive to inter-layer axial and lateral shifts, phase fabrication errors, optical-path occlusions, speckle, and detector-level fluctuations. The single-layer dual-wavelength differential architecture addresses that gap structurally by removing inter-layer registration altogether, algorithmically by exploiting wavelength-division multiplexing and differential readout, and during training by incorporating resilience to perturbations (Wang et al., 23 Jul 2025).

2. Optical model and dual-wavelength differential principle

The optical forward model of the single-layer AI D2NN uses Rayleigh–Sommerfeld diffraction implemented numerically by the angular spectrum method for each wavelength $\lambda \in \{\lambda_+,\lambda_-\}$. For a mask

$t(x,y)=A(x,y)e^{j\phi(x,y)},$

the post-mask field is

$U_1(x,y;\lambda)=t(x,y)\,U_0(x,y;\lambda),$

and propagation over distance $z$ is written as

$$U(x,y;\lambda)=\mathcal{F}^{-1}\!\left\{H(f_x,f_y;\lambda,z)\,\mathcal{F}\{U_1(x,y;\lambda)\}\right\},$$

with transfer function

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

Under the Fresnel approximation, the transfer function becomes

$\lambda_-=532\,\mathrm{nm}$0

An equivalent validation form uses the spatial-domain impulse response

$\lambda_-=532\,\mathrm{nm}$1

The intensity for each channel is

$\lambda_-=532\,\mathrm{nm}$2

(Wang et al., 23 Jul 2025).

The decisive departure from a conventional shallow D2NN lies in the readout. The detector plane is partitioned into 10 class subregions. For class $\lambda_-=532\,\mathrm{nm}$3, the intensities under $\lambda_-=532\,\mathrm{nm}$4 and $\lambda_-=532\,\mathrm{nm}$5 are summed over the corresponding subregion, and the temperature-scaled normalized differential score is

$\lambda_-=532\,\mathrm{nm}$6

with $\lambda_-=532\,\mathrm{nm}$7 and $\lambda_-=532\,\mathrm{nm}$8. Classification is performed by $\lambda_-=532\,\mathrm{nm}$9 (Wang et al., 23 Jul 2025).

This differential formulation serves two distinct purposes. First, it mitigates the non-negativity constraint of intensity-only single-wavelength readout by allowing signed responses through $U$0. The reported interpretation is that a shallow architecture thereby gains antagonistic feature coding, with positive support at $U$1 and negative support at $U$2. Second, if both channels are affected by a shared perturbation $U$3, then subtraction cancels the common term:

$U$4

Under stochastic noise $U$5 with variances $U$6 and correlation $U$7, the differential variance becomes

$U$8

so the noise penalty decreases as $U$9. The corresponding differential signal-to-noise estimate is

$$\langle U E_1, U E_2 \rangle = \langle E_1, E_2 \rangle,$$0

This common-mode rejection is central to the anti-interference designation (Wang et al., 23 Jul 2025).

The use of two wavelengths also changes the network’s spectral selectivity. Because the propagation phase depends on wavelength through the Fresnel factor $$\langle U E_1, U E_2 \rangle = \langle E_1, E_2 \rangle,$$1, the two channels impose different quadratic phase chirps. The 2025 implementation emphasizes that the longer wavelength, $$\langle U E_1, U E_2 \rangle = \langle E_1, E_2 \rangle,$$2, yields a stronger phase rotation per unit spatial frequency than $$\langle U E_1, U E_2 \rangle = \langle E_1, E_2 \rangle,$$3 at fixed distance, so the detector receives complementary spatial-frequency content that a single mask can exploit jointly (Wang et al., 23 Jul 2025).

3. Physical architecture, parameterization, and training

The single-layer AI D2NN is physically compact. The input plane contains $$\langle U E_1, U E_2 \rangle = \langle E_1, E_2 \rangle,$$4 pixels with $$\langle U E_1, U E_2 \rangle = \langle E_1, E_2 \rangle,$$5 pitch and uses amplitude encoding. The distance from input plane to mask is $$\langle U E_1, U E_2 \rangle = \langle E_1, E_2 \rangle,$$6. The diffractive mask is either a $$\langle U E_1, U E_2 \rangle = \langle E_1, E_2 \rangle,$$7 array for the 40k-parameter case or a $$\langle U E_1, U E_2 \rangle = \langle E_1, E_2 \rangle,$$8 array for the 10k-parameter case, again with $$\langle U E_1, U E_2 \rangle = \langle E_1, E_2 \rangle,$$9 pitch. The detector plane is $\lambda \in \{\lambda_+,\lambda_-\}$0 downstream of the mask and is partitioned into 10 class subregions, each $\lambda \in \{\lambda_+,\lambda_-\}$1 in size. The mask can be phase-only or complex-amplitude; phase is constrained to $\lambda \in \{\lambda_+,\lambda_-\}$2 through a sigmoid mapping. The optical channels are separated with spectral filtering to mitigate cross-talk (Wang et al., 23 Jul 2025).

The training workflow uses MNIST and Fashion-MNIST, with 50,000 training samples and 10,000 test samples. Inputs are upsampled from $\lambda \in \{\lambda_+,\lambda_-\}$3 to $\lambda \in \{\lambda_+,\lambda_-\}$4 by nearest-neighbor interpolation, zero-padded to $\lambda \in \{\lambda_+,\lambda_-\}$5, and amplitude-encoded. Optimization uses Adam with learning rate $\lambda \in \{\lambda_+,\lambda_-\}$6, per-epoch decay $\lambda \in \{\lambda_+,\lambda_-\}$7, batch size 32, and 50 epochs to convergence. The loss is softmax cross-entropy computed on the 10-dimensional differential score vector, with both wavelengths jointly included in each forward pass (Wang et al., 23 Jul 2025).

Gradient propagation follows the differentiable optics of the thin-mask model. With transmission $\lambda \in \{\lambda_+,\lambda_-\}$8 and local field $\lambda \in \{\lambda_+,\lambda_-\}$9 at pixel $t(x,y)=A(x,y)e^{j\phi(x,y)},$0, the post-mask field is $t(x,y)=A(x,y)e^{j\phi(x,y)},$1, and the elementary derivatives are

$t(x,y)=A(x,y)e^{j\phi(x,y)},$2

Applying the chain rule through angular-spectrum propagation and intensity formation yields

$t(x,y)=A(x,y)e^{j\phi(x,y)},$3

where $t(x,y)=A(x,y)e^{j\phi(x,y)},$4 is the linear ASM operator defined by Fourier-domain multiplication with $t(x,y)=A(x,y)e^{j\phi(x,y)},$5 (Wang et al., 23 Jul 2025).

A notable architectural consequence is that the anti-interference mechanism is embedded directly in the optics rather than delegated to electronic post-processing. The design uses a single physical diffractive plane, dual coherent sources, spectral filtering, and normalized differential energy integration. This suggests a reallocation of complexity: mechanical stacking is minimized, while source calibration and channel separation become the primary engineering constraints.

4. Accuracy, robustness, and anti-interference behavior

The reported classification results are summarized below.

Model	MNIST	Fashion-MNIST
Single-layer dual-wavelength differential, 40k parameters	98.59%	90.4%
Single-layer dual-wavelength differential, 10k parameters	97.95%	88.7%
Single-layer single-wavelength baseline	78.39%	76.50%
Five-layer cascaded D2NN, 200k neurons	91.33%	83.67%

With 40k trainable parameters, the single-layer dual-wavelength system exceeds the five-layer cascaded baseline by $t(x,y)=A(x,y)e^{j\phi(x,y)},$6 on MNIST and $t(x,y)=A(x,y)e^{j\phi(x,y)},$7 on Fashion-MNIST while using only 20% of the parameter count. The 10k-parameter version remains strong at 97.95% and 88.7%, which is presented as evidence that the architecture retains substantial expressivity even after a fourfold parameter reduction (Wang et al., 23 Jul 2025).

A modulation ablation further specifies that complex amplitude plus phase modulation outperforms phase-only modulation at lower densities. For a $t(x,y)=A(x,y)e^{j\phi(x,y)},$8 mask, the dual-wavelength single-layer configuration achieved 98.0% on MNIST and 88.9% on Fashion-MNIST. At $t(x,y)=A(x,y)e^{j\phi(x,y)},$9, both modulation types converged to 98.59% and 90.4%, indicating that increased sampling density can partially compensate for reduced modulation freedom (Wang et al., 23 Jul 2025).

The anti-interference characterization rests on perturbation studies. For fabrication-style phase errors, mask perturbations were modeled as $U_1(x,y;\lambda)=t(x,y)\,U_0(x,y;\lambda),$0, i.i.d. per neuron. Resilience-trained models retained high accuracy up to approximately $U_1(x,y;\lambda)=t(x,y)\,U_0(x,y;\lambda),$1, whereas non-resilient models declined sharply as $U_1(x,y;\lambda)=t(x,y)\,U_0(x,y;\lambda),$2 increased. For optical occlusion, an opaque blocker of width $U_1(x,y;\lambda)=t(x,y)\,U_0(x,y;\lambda),$3 was placed between input and mask. Reported accuracy peaked at 98.1% on MNIST near $U_1(x,y;\lambda)=t(x,y)\,U_0(x,y;\lambda),$4 and about 90.2% on Fashion-MNIST near $U_1(x,y;\lambda)=t(x,y)\,U_0(x,y;\lambda),$5–0.25; the five-layer baseline was about 80% on MNIST at $U_1(x,y;\lambda)=t(x,y)\,U_0(x,y;\lambda),$6. For salt-and-pepper input noise with ratio $U_1(x,y;\lambda)=t(x,y)\,U_0(x,y;\lambda),$7, the resilience-trained single-layer differential network sustained more than 50% accuracy at $U_1(x,y;\lambda)=t(x,y)\,U_0(x,y;\lambda),$8, whereas non-resilient baselines collapsed toward zero (Wang et al., 23 Jul 2025).

These results define “anti-interference” in a specific technical sense. Shared illumination fluctuations and sensor biases are canceled by the normalized differential score; coherent artifacts common to both channels are reduced by $U_1(x,y;\lambda)=t(x,y)\,U_0(x,y;\lambda),$9 subtraction; inter-layer misalignment is eliminated by construction because only one diffractive plane is used; fabrication-related phase perturbations are addressed by resilience training; and occlusion and impulse-noise robustness are demonstrated empirically (Wang et al., 23 Jul 2025).

5. Relation to other robustness-oriented D2NN paradigms

AI D2NN is not a single method in the broader literature but a family of robustness strategies. One line of work addresses misalignment directly through stochastic training. “Vaccinated D2NN” models lateral and axial layer displacements as random variables during optimization, sampling $z$0, $z$1, and $z$2 from uniform distributions in each mini-batch. In a five-layer THz MNIST system, vaccinating at $z$3 reduced nominal accuracy from 97.77% to 96.1% but improved misaligned accuracy at $z$4 from 38.40% to 94.44%; hybrid optical-electronic versions remained stronger under severe lateral shifts, reaching about 79.6% at $z$5 where the non-vaccinated all-optical network was about 12.8% (Mengu et al., 2020).

A second line treats fabrication uncertainty as weight perturbation. Weight-noise-injection training adds Gaussian phase noise to diffractive weights during optimization, effectively minimizing the expected loss over perturbed phase masks and favoring flatter minima. In a five-layer phase-only THz D2NN, the method substantially improved robustness to injected phase noise, printer Z-axis errors, frequency shifts, and spacing deviations; at $z$6 printer precision, a conventional DNN lost 35.4% accuracy relative to $z$7 precision, whereas SRNN(0.3) lost only 8.2% (Shi, 2020).

A third line targets geometric nuisance factors at the input rather than hardware errors. Scale-, shift-, and rotation-invariant diffractive networks sample translations, rotations, and isotropic scalings inside the optical forward model during training. For a five-layer MNIST D2NN, small to moderate invariance ranges often improved peak blind accuracy, while larger ranges traded peak performance for flatter robustness; differential detection and wider layers partially compensated for that trade-off (Mengu et al., 2020).

A fourth line uses “anti-interference” in the context of multi-object scenes. A two-layer THz AI D2NN for multi-object recognition trained targets and interference with different objectives so that target digits $z$8–$z$9 were mapped into six detection windows while 40 categories of interference were diffused into low-energy background noise. Reported numerical blind accuracies were 90.1% for intra-class interference, 89.7% for inter-class interference, and 87.4% for dynamic multi-object scenes; experimental blind accuracy was 86.7% (Huang et al., 9 Jul 2025).

A fifth line introduces latent-space filtering through an all-optical autoencoder. By encoding a wavefield into a compact diffractive latent space and decoding it through the same hardware in reverse, the system suppresses perturbations that do not match the latent prior. In that framework, denoising on MNIST under salt-and-pepper noise at $$U(x,y;\lambda)=\mathcal{F}^{-1}\!\left\{H(f_x,f_y;\lambda,z)\,\mathcal{F}\{U_1(x,y;\lambda)\}\right\},$$0 improved median PSNR from 7.81 dB for the noisy inputs to 17.3 dB with SOAE and 18.6 dB with DSOAE; noise-resistant diffractive classifiers built on frozen encoders also exceeded ordinary seven-layer diffractive classifiers under severe corruption (Feng et al., 2024).

Taken together, these studies separate several anti-interference mechanisms that are often conflated: structural suppression of alignment error by reducing layer count, stochastic tolerance learning through perturbation-aware optimization, differential detection for common-mode rejection and signed coding, scene-level interference rejection by explicit target-versus-clutter objectives, and latent-space priors for denoising. The single-layer dual-wavelength AI D2NN belongs primarily to the first and third categories, while borrowing resilience training from the second (Wang et al., 23 Jul 2025).

6. Implementation constraints, limitations, and future directions

The single-layer dual-wavelength AI D2NN reduces system depth and alignment degrees of freedom, but it does not remove all practical constraints. It requires two coherent sources at $$U(x,y;\lambda)=\mathcal{F}^{-1}\!\left\{H(f_x,f_y;\lambda,z)\,\mathcal{F}\{U_1(x,y;\lambda)\}\right\},$$1 and $$U(x,y;\lambda)=\mathcal{F}^{-1}\!\left\{H(f_x,f_y;\lambda,z)\,\mathcal{F}\{U_1(x,y;\lambda)\}\right\},$$2, or a wavelength-switching mechanism. Illumination may be sequential or simultaneous, provided spectral filtering at detection is sufficient to separate channels and minimize cross-talk. Coherence length and stability must match the approximately $$U(x,y;\lambda)=\mathcal{F}^{-1}\!\left\{H(f_x,f_y;\lambda,z)\,\mathcal{F}\{U_1(x,y;\lambda)\}\right\},$$3 total optical path length so that the assumed transfer functions remain valid. Differential integration reduces sensitivity to absolute power scaling and sensor offsets, but detector dynamic range and calibration remain important, especially under low-light or high-contrast conditions (Wang et al., 23 Jul 2025).

The principal architectural limitation is bounded expressivity. The same 2025 study states that a single physical layer, while robust and compact, remains less expressive than deep stacks for extremely complex tasks. The dual-source requirement adds calibration overhead, and material dispersion becomes relevant when the same mask is used at two wavelengths. These constraints distinguish the design from idealized unitary D2NN analyses, which assume monochromatic operation and lossless phase-only modulation (Wang et al., 23 Jul 2025).

Several extensions are explicitly proposed. These include using more than two wavelengths for multi-spectral coding, polarization multiplexing, jointly coded multi-plane masks implemented on a single substrate, hybrid electronic readout such as weighted differential integration, and error-correcting coding at the optics layer. Broader adjacent work also indicates that AI D2NN concepts can be extended to dynamic scenes, additional nuisance transformations, and denoising-oriented latent-space processors, suggesting that anti-interference is becoming a systems-level design principle rather than a single architecture class (Wang et al., 23 Jul 2025).

In that sense, AI D2NN design can be understood as the convergence of three ideas: optics-aware physical modeling, perturbation-aware training, and task-specific detector engineering. The single-layer dual-wavelength differential implementation is a particularly compact realization of that convergence, because it converts anti-interference from an after-the-fact robustness adjustment into a property of the optical computation itself.