Diffractive Optical Processors

• Diffractive optical processors are systems that utilize engineered diffractive surfaces to convert input electromagnetic fields into specified output distributions via free-space propagation and interference.
• They employ phase-only or complex-amplitude modulation across cascaded layers, optimized by inverse design techniques and backpropagation for tasks like imaging and sensing.
• Recent innovations enable multiplexing, reconfigurability, and hybrid optical-digital implementations, enhancing performance metrics such as diffraction efficiency, imaging fidelity, and error robustness.

Diffractive optical processors are optical computing and imaging systems in which one or more spatially engineered diffractive surfaces transform an incident electromagnetic field into a desired output field or intensity distribution through free-space propagation, interference, and detection. In the recent literature, they appear under closely related labels such as diffractive optical processors, diffractive optical networks, diffractive deep neural networks, and diffractive neural networks. Reported implementations span passive phase-only stacks, complex-amplitude metasurfaces, reconfigurable spatial-light-modulator platforms, and hybrid optical-digital systems jointly optimized for imaging, sensing, linear transforms, encryption, and machine learning tasks (Rahman et al., 2024, Shen et al., 2024).

1. Physical principles and forward models

The standard forward model treats each diffractive layer as a thin transmissive mask and the spaces between layers as free-space propagation segments. In angular-spectrum form, if $U_l(x,y;\lambda)$ denotes the optical field after layer $l$, propagation over distance $z$ is commonly written as

$$U_{l+1}(x,y;\lambda)=\mathcal{F}^{-1}\left\{\mathcal{F}[U_l(x,y;\lambda)]\cdot H(f_x,f_y;\lambda,z)\right\},$$

with

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

or, in equivalent Rayleigh–Sommerfeld form, as a convolution with the free-space kernel (Shen et al., 2024). Closely related formulations are used in monochrome linear-transform processors, multiplexed permutation devices, hybrid optical-digital systems, and complex-field imagers (Wang et al., 7 Dec 2025, Ma et al., 2024, Rahman et al., 2024, Li et al., 2024).

Layer modulation is most often phase-only, with transmission written as $t_\ell(x,y)=\exp[j\phi_\ell(x,y)]$, although amplitude-phase parameterizations of the form $t_\ell(x,y)=a_\ell(x,y)\exp[i\psi_\ell(x,y)]$ are also reported (Bai et al., 2022, Wang et al., 7 Dec 2025). In fabricated devices, the trainable variable is usually local thickness, which sets the phase delay through the material refractive index and, when absorption is non-negligible, the amplitude response as well (Li et al., 2024, Shen et al., 2023). This makes the optical processor a cascaded linear operator in the complex field domain.

A persistent conceptual point is that linearity depends on what is treated as the signal. Many diffractive processors are linear with respect to the complex optical field, but the measured output is often an intensity, $I_{\rm out}=|U_{\rm out}|^2$, which introduces a nonlinear mapping between input representation and detector readout (Bai et al., 2022). Under spatially incoherent illumination, the processor is described instead by an intensity point-spread function $H(m,n;m',n')=|h(m,n;m',n')|^2$, yielding

$$I_{\rm out}(m,n)=\sum_{m',n'} |h(m,n;m',n')|^2\,I_{\rm in}(m',n'),$$

so the diffractive volume acts as a linear operator on time-averaged intensity rather than on coherent field amplitudes (Rahman et al., 2023).

2. Inverse design, optimization, and robustness

Recent diffractive optical processors are typically obtained by end-to-end inverse design. The optical forward model is embedded in an autodifferentiable framework, and layer parameters are optimized by backpropagation using task-dependent losses. Reported software stacks include PyTorch, TensorFlow, and direct in situ optimization on physical hardware (Shen et al., 2024, Li et al., 2022, Li et al., 8 Jul 2025).

Training objectives vary with the task. Imaging systems use MSE, NMSE, PCC, SSIM-related terms, or diffraction-efficiency penalties; classification systems use cross-entropy or detector-energy losses; linear-transform processors use output-field or transformation-matrix MSE; and hybrid systems jointly update optical and digital parameters with a common end-to-end loss (Shen et al., 2024, Wang et al., 7 Dec 2025, Behroozinia et al., 2024, Wang et al., 3 Jun 2025). A representative visible unidirectional imager combined NMSE, PCC, and energy-throughput terms over both forward and backward directions,

$l$0

with random wavelength sampling from red, green, and blue bands to enforce broadband operation (Shen et al., 2024).

Robustness to fabrication and alignment errors is a recurring design requirement. Several works inject random axial and lateral shifts during training, a procedure often termed “vaccination,” to improve tolerance to misalignment, fabrication error, and lifetime drift (Shen et al., 2024, Bai et al., 2022, Shen et al., 2023). A distinct calibration strategy appears in the reconfigurable diffractive processing unit, where in-silico pre-training is followed by layer-wise adaptive fine-tuning using experimentally measured intermediate fields to compensate aberration, misalignment, and non-ideal modulator response (Zhou et al., 2020).

When accurate physical modeling is difficult, reported in situ learning methods optimize the hardware directly. A model-free reinforcement-learning approach based on Proximal Policy Optimization treats the optical system as an environment, updates phase patterns from measured rewards, reuses each in situ batch for multiple gradient steps, and experimentally shows better convergence and performance in tasks including energy focusing through a random diffuser, holographic image generation, aberration correction, and optical image classification (Li et al., 8 Jul 2025). This suggests that diffractive optical processors increasingly occupy a continuum between fully modeled inverse design and hardware-in-the-loop optimization.

3. Expressivity, universality, multiplexing, and reconfiguration

A major theoretical theme is the representation of arbitrary linear operators. Under spatially coherent light, a phase-only diffractive network can implement arbitrary complex-valued linear transformations between input and output fields-of-view when the total number of trainable diffractive features satisfies $l$1; under spatially incoherent monochromatic light, the same scaling is reported for arbitrary linear intensity transformations (Rahman et al., 2023). For polarization-multiplexed diffractive computing, the reported feature-count scaling is $l$2, where $l$3 is the number of distinct transformations assigned to input-output polarization pairs (Li et al., 2022). For illumination phase multiplexing, a monochrome diffractive network is reported to realize $l$4 distinct complex-valued linear transformations with $l$5, with numerical demonstration of $l$6 transformations at negligible error (Wang et al., 7 Dec 2025).

These results connect to a broader family of multiplexing schemes. Reported mechanisms include polarization, wavelength, bidirectional propagation, mechanical rotation of layers, and input phase diversity. A mechanically reconfigurable $l$7-layer permutation processor performs up to $l$8 independent permutation operations because each layer can take four discrete orientations $l$9 (Ma et al., 2024). A bilayer cascaded-metasurface classifier uses wavelength and polarization degrees of freedom to perform dual-task or tri-task recognition on MNIST, FMNIST, and KMNIST, with tri-task accuracies remaining greater than $z$0 and improving under end-to-end joint optimization of physically realizable meta-atoms (Behroozinia et al., 2024). An all-optical autoencoder exploits bidirectional multiplexing so that the same stack functions as an encoder in one propagation direction and as a decoder in the opposite direction, defining a diffractive latent space with compression ratios of approximately $z$1 or $z$2, depending on latent geometry (Feng et al., 2024).

Multiplexing or reconfiguration mechanism	Reported capability	Paper
Polarization encoding	Multiple arbitrary linear transforms with $z$3	(Li et al., 2022)
Illumination phase keys	$z$4 complex-valued transforms in one monochrome network	(Wang et al., 7 Dec 2025)
Layer rotations	Up to $z$5 permutation operations	(Ma et al., 2024)
Wavelength/polarization metasurfaces	Dual-task and tri-task classification	(Behroozinia et al., 2024)
Bidirectional propagation	Optical encoding and decoding in one stack	(Feng et al., 2024)

A common misconception is that a fabricated diffractive stack can perform only one fixed function. The reported literature shows instead that a single physical processor can execute multiple transformations when distinct channels or control variables are available, such as input polarization, wavelength, illumination phase profile, mechanical rotation state, or propagation direction (Li et al., 2022, Wang et al., 7 Dec 2025, Ma et al., 2024, Feng et al., 2024).

4. Imaging, wavefront processing, and computational sensing

Diffractive optical processors have been used to implement a wide range of imaging and sensing modalities. A two-layer visible-spectrum unidirectional imager fabricated on high-purity fused silica forms high-fidelity images in the forward direction while generating weak, distorted patterns in the backward direction. Over 200 test wavelengths from 450 to 650 nm, the reported two-layer design achieved forward PCC $z$6, backward PCC $z$7, forward diffraction efficiency $z$8, and backward diffraction efficiency $z$9; a three-layer variant improved these figures to forward PCC $$U_{l+1}(x,y;\lambda)=\mathcal{F}^{-1}\left\{\mathcal{F}[U_l(x,y;\lambda)]\cdot H(f_x,f_y;\lambda,z)\right\},$$0, backward PCC $$U_{l+1}(x,y;\lambda)=\mathcal{F}^{-1}\left\{\mathcal{F}[U_l(x,y;\lambda)]\cdot H(f_x,f_y;\lambda,z)\right\},$$1, forward efficiency $$U_{l+1}(x,y;\lambda)=\mathcal{F}^{-1}\left\{\mathcal{F}[U_l(x,y;\lambda)]\cdot H(f_x,f_y;\lambda,z)\right\},$$2, and backward efficiency $$U_{l+1}(x,y;\lambda)=\mathcal{F}^{-1}\left\{\mathcal{F}[U_l(x,y;\lambda)]\cdot H(f_x,f_y;\lambda,z)\right\},$$3 (Shen et al., 2024).

Several reported systems target direct optical recovery of information that conventional intensity sensors do not natively provide. A complex-field imager uses successive diffractive surfaces to create two output channels that perform amplitude-to-amplitude and phase-to-intensity transformations, thereby enabling snapshot imaging of both amplitude and quantitative phase without digital reconstruction, within an axial span of approximately $$U_{l+1}(x,y;\lambda)=\mathcal{F}^{-1}\left\{\mathcal{F}[U_l(x,y;\lambda)]\cdot H(f_x,f_y;\lambda,z)\right\},$$4 wavelengths (Li et al., 2024). A multispectral quantitative phase imaging processor uses 10 phase-only layers to encode phase profiles at 9 or 16 visible wavelengths into spatially separated intensity patterns on a monochrome focal-plane array, yielding $$U_{l+1}(x,y;\lambda)=\mathcal{F}^{-1}\left\{\mathcal{F}[U_l(x,y;\lambda)]\cdot H(f_x,f_y;\lambda,z)\right\},$$5 dB PSNR and $$U_{l+1}(x,y;\lambda)=\mathcal{F}^{-1}\left\{\mathcal{F}[U_l(x,y;\lambda)]\cdot H(f_x,f_y;\lambda,z)\right\},$$6 SSIM for the 9-channel design, and $$U_{l+1}(x,y;\lambda)=\mathcal{F}^{-1}\left\{\mathcal{F}[U_l(x,y;\lambda)]\cdot H(f_x,f_y;\lambda,z)\right\},$$7 dB PSNR and $$U_{l+1}(x,y;\lambda)=\mathcal{F}^{-1}\left\{\mathcal{F}[U_l(x,y;\lambda)]\cdot H(f_x,f_y;\lambda,z)\right\},$$8 SSIM for the 16-channel design on unseen MNIST-based phase objects (Shen et al., 2023). A solid-immersion diffractive optical processor couples a high-index encoder to decoder layers in air to resolve subwavelength phase and amplitude features; the reported terahertz proof of concept experimentally resolved $$U_{l+1}(x,y;\lambda)=\mathcal{F}^{-1}\left\{\mathcal{F}[U_l(x,y;\lambda)]\cdot H(f_x,f_y;\lambda,z)\right\},$$9 mm, described as sub-Rayleigh by approximately $$H(f_x,f_y;\lambda,z)=\exp\!\left[i\,2\pi z\sqrt{1/\lambda^2-f_x^2-f_y^2}\right],$$0 (Hu et al., 2024).

Wavefront engineering is another major application. A diffractive optical phase-conjugation processor trained on random Zernike-aberrated inputs approximates the conjugate phase distribution and was experimentally validated in the terahertz regime. In the reported $$H(f_x,f_y;\lambda,z)=\exp\!\left[i\,2\pi z\sqrt{1/\lambda^2-f_x^2-f_y^2}\right],$$1 transmissive design, blind two-mode tests yielded phase MAE $$H(f_x,f_y;\lambda,z)=\exp\!\left[i\,2\pi z\sqrt{1/\lambda^2-f_x^2-f_y^2}\right],$$2 and amplitude MAE $$H(f_x,f_y;\lambda,z)=\exp\!\left[i\,2\pi z\sqrt{1/\lambda^2-f_x^2-f_y^2}\right],$$3; depth also improved diffraction efficiency, from approximately $$H(f_x,f_y;\lambda,z)=\exp\!\left[i\,2\pi z\sqrt{1/\lambda^2-f_x^2-f_y^2}\right],$$4 at $$H(f_x,f_y;\lambda,z)=\exp\!\left[i\,2\pi z\sqrt{1/\lambda^2-f_x^2-f_y^2}\right],$$5 to approximately $$H(f_x,f_y;\lambda,z)=\exp\!\left[i\,2\pi z\sqrt{1/\lambda^2-f_x^2-f_y^2}\right],$$6 at $$H(f_x,f_y;\lambda,z)=\exp\!\left[i\,2\pi z\sqrt{1/\lambda^2-f_x^2-f_y^2}\right],$$7 (Shen et al., 2023).

Diffractive processors also appear as optical preconditioners for difficult propagation environments. An interleaved diffractive network for information transfer through random diffusers inserts trainable layers within a volumetric scattering medium and reports PCC improvement from approximately $$H(f_x,f_y;\lambda,z)=\exp\!\left[i\,2\pi z\sqrt{1/\lambda^2-f_x^2-f_y^2}\right],$$8 at $$H(f_x,f_y;\lambda,z)=\exp\!\left[i\,2\pi z\sqrt{1/\lambda^2-f_x^2-f_y^2}\right],$$9 to approximately $t_\ell(x,y)=\exp[j\phi_\ell(x,y)]$0 at $t_\ell(x,y)=\exp[j\phi_\ell(x,y)]$1 on unseen diffusers, with a best PCC of approximately $t_\ell(x,y)=\exp[j\phi_\ell(x,y)]$2 at the smallest interplane spacing of $t_\ell(x,y)=\exp[j\phi_\ell(x,y)]$3; a jointly trained hybrid system with a U-Net-style backend of approximately $t_\ell(x,y)=\exp[j\phi_\ell(x,y)]$4k parameters further improves robustness to random rotations, shifts, and scaling (Li et al., 9 Mar 2026). In structural health monitoring, a single reflective diffractive layer jointly optimized with shallow neural networks remotely encodes 3D structural vibration spectra into four detector signals; the reported spectral MSE in the $t_\ell(x,y)=\exp[j\phi_\ell(x,y)]$5 Hz band was $t_\ell(x,y)=\exp[j\phi_\ell(x,y)]$6 for the jointly optimized diffractive layer, compared with $t_\ell(x,y)=\exp[j\phi_\ell(x,y)]$7 for a separately optimized diffractive layer, $t_\ell(x,y)=\exp[j\phi_\ell(x,y)]$8 for a Fresnel lens array, and $t_\ell(x,y)=\exp[j\phi_\ell(x,y)]$9 for a random diffuser (Wang et al., 3 Jun 2025).

The same general framework has also been used for class-specific all-optical encryption and decryption, permutation-based encryption, and diffractive latent-space processing for denoising, classification, and image generation (Bai et al., 2022, Ma et al., 2024, Feng et al., 2024). This suggests that “imaging” in the diffractive-processor literature often includes learned transforms that mix sensing, coding, inference, and optical encryption rather than conventional image formation alone.

5. Fabrication, materials, and hardware embodiments

Fabrication strategies span wafer-scale nanofabrication, metasurface manufacturing, two-photon polymerization, stereolithography, PolyJet printing, and programmable optoelectronic platforms. A visible-spectrum unidirectional imager was fabricated on a 6-inch high-purity fused silica wafer using 4× projection lithography and Cl$t_\ell(x,y)=a_\ell(x,y)\exp[i\psi_\ell(x,y)]$0 dry etching to realize 16 discrete phase levels with approximately $t_\ell(x,y)=a_\ell(x,y)\exp[i\psi_\ell(x,y)]$1 nm step size. The process achieved lateral registration below $t_\ell(x,y)=a_\ell(x,y)\exp[i\psi_\ell(x,y)]$2m between front and back surfaces, $t_\ell(x,y)=a_\ell(x,y)\exp[i\psi_\ell(x,y)]$3-$t_\ell(x,y)=a_\ell(x,y)\exp[i\psi_\ell(x,y)]$4 etch-depth error across the wafer, and throughput of approximately $t_\ell(x,y)=a_\ell(x,y)\exp[i\psi_\ell(x,y)]$5 diffractive-processor chips per wafer, totaling approximately $t_\ell(x,y)=a_\ell(x,y)\exp[i\psi_\ell(x,y)]$6 billion diffractive features (Shen et al., 2024).

In metasurface implementations, the trainable optical response is tied to a library of unit cells. A reported multi-task classifier uses TiO$t_\ell(x,y)=a_\ell(x,y)\exp[i\psi_\ell(x,y)]$7 nanofins on glass with fixed height $t_\ell(x,y)=a_\ell(x,y)\exp[i\psi_\ell(x,y)]$8 nm, lattice period $t_\ell(x,y)=a_\ell(x,y)\exp[i\psi_\ell(x,y)]$9 nm, and trainable in-plane widths $I_{\rm out}=|U_{\rm out}|^2$0 to realize wavelength- and polarization-dependent complex transmittances derived from full-wave simulations at $I_{\rm out}=|U_{\rm out}|^2$1, $I_{\rm out}=|U_{\rm out}|^2$2, and $I_{\rm out}=|U_{\rm out}|^2$3 nm (Behroozinia et al., 2024). This differs from phase-only free-space stacks, but it remains within the diffractive-processor paradigm because cascaded diffraction and trainable local transmission still define the end-to-end operator.

Terahertz proof-of-concept systems frequently use 3D printing because the larger wavelength relaxes fabrication tolerances. Reported examples include class-specific encryption networks fabricated by two-photon polymerization in IP-Dip photoresist and tested at $I_{\rm out}=|U_{\rm out}|^2$4 nm (Bai et al., 2022); monolithic solid-immersion encoder-decoder pairs printed in VeroBlackPlus for terahertz subwavelength imaging (Hu et al., 2024); stereolithography-fabricated phase-conjugation processors in isotropic polymer with $I_{\rm out}=|U_{\rm out}|^2$5 and $I_{\rm out}=|U_{\rm out}|^2$6 at $I_{\rm out}=|U_{\rm out}|^2$7 mm (Shen et al., 2023); and complex-field imagers, rotation-multiplexed permutation devices, and all-optical autoencoders validated with 3D-printed diffractive layers in the terahertz band (Li et al., 2024, Ma et al., 2024, Feng et al., 2024).

A separate hardware lineage emphasizes programmable rather than fixed optics. The reconfigurable diffractive processing unit combines a digital micromirror device for amplitude encoding, a phase-only spatial light modulator with 8-bit phase resolution and $I_{\rm out}=|U_{\rm out}|^2$8 pixels for synaptic weights, and an sCMOS detector array for optical readout and square-law nonlinearity. By time-multiplexing layers, the system supports diffractive feedforward and recurrent neural networks, experimentally achieving $I_{\rm out}=|U_{\rm out}|^2$9 MNIST accuracy for a 3-layer D2NN after adaptive training and video-accuracy up to $H(m,n;m',n')=|h(m,n;m',n')|^2$0 on Weizmann with a D-RNN++ configuration (Zhou et al., 2020).

Integration with conventional imaging and photonic hardware is an explicit design goal in several reports. The visible unidirectional processor has an axial thickness of approximately $H(m,n;m',n')=|h(m,n;m',n')|^2$1 mm and lateral aperture of approximately $H(m,n;m',n')=|h(m,n;m',n')|^2$2m per processor, described as compatible with CMOS image sensors with $H(m,n;m',n')=|h(m,n;m',n')|^2$3m pixels, and potential on-chip integration includes silicon-photonic waveguide coupling and monolithic hybrid optoelectronic modules (Shen et al., 2024).

6. Linearity, nonlinear computation, limitations, and future directions

Diffractive optical processors are often described as linear optical systems, but the literature now distinguishes several routes to nonlinear computation. One route uses square-law detection together with suitable input encoding. Under phase encoding of scalar or vector inputs, a passive phase-only diffractive processor can approximate bandlimited nonlinear functions at the output intensity. A reported framework establishes universal approximation for arbitrary sets of bandlimited nonlinear functions, including multivariate and complex-valued functions, and numerically demonstrates one million distinct nonlinear functions computed in parallel by a diffractive processor (Rahman et al., 11 Jul 2025). A closely related incoherent-light framework uses intensity-only encoding and differential detector readout to realize universal nonlinear approximation under spatially incoherent or partially coherent illumination, again with numerical results showing snapshot computation of up to one million distinct nonlinear functions in a single forward pass (Chen et al., 31 Mar 2026).

A second route introduces explicit input-dependent optical modulation. The recurrent diffractive optical neural processor, ReDON, senses a fraction of the propagating field, processes it with a lightweight parametric function, and uses the result for electro-optic self-modulation in later layers. This reconfigurable self-modulated nonlinearity is combined with recurrence through hardware reuse of a fixed passive metasurface stack. On image recognition and segmentation benchmarks, the reported gains reach up to $H(m,n;m',n')=|h(m,n;m',n')|^2$4 in test accuracy or mIoU relative to prior diffractive optical neural networks with comparable model complexity, while the self-modulation overhead remains below $H(m,n;m',n')=|h(m,n;m',n')|^2$5 mW compared with more than $H(m,n;m',n')=|h(m,n;m',n')|^2$6 mW laser power (Yin et al., 27 Feb 2026).

A third route is hybrid optical-digital inference. The survey on programmable diffraction describes such systems as establishing a “diffractive language” between analog wave processing and digital neural networks, and reports applications spanning classification, computational imaging, single-pixel sensing, and programmable metasurface sensing (Rahman et al., 2024). In individual case studies, jointly optimized optical front ends and shallow or compact digital back ends improve reconstruction fidelity, classification accuracy, or task robustness while keeping the optical front end passive (Wang et al., 3 Jun 2025, Li et al., 9 Mar 2026).

Several limitations recur across the literature. Phase-only modulation is widely preferred in practice but doubles the degrees-of-freedom requirement relative to full complex modulation for universal linear transforms (Rahman et al., 2023). Static fabricated processors can require redesign or refabrication to change target functions unless reconfiguration is introduced through SLMs, phase keys, multiplexing, or mechanical rotation (Li et al., 2022, Wang et al., 7 Dec 2025, Ma et al., 2024). Alignment sensitivity, fabrication error, hardware drift, and model mismatch remain central concerns, motivating vaccination during training, adaptive experimental fine-tuning, and model-free in situ learning (Shen et al., 2024, Zhou et al., 2020, Li et al., 8 Jul 2025).

A further misconception is that diffractive optical processors are only useful under fully coherent laser illumination. Reported systems operate under coherent, partially coherent, and spatially incoherent conditions, provided the forward model and training objective are matched to the source statistics and detection physics (Rahman et al., 2024, Rahman et al., 2023, Chen et al., 31 Mar 2026). A plausible implication is that the field is moving from narrowly defined optical neural networks toward a broader class of learned wave processors in which fabrication method, illumination coherence, multiplexing strategy, and optical-digital partitioning are all design variables rather than fixed assumptions.