Differentiable Lithography Techniques

• Differentiable lithography is a set of techniques that models lithographic processes as differentiable operations, allowing gradients to be propagated for mask, source, and process optimization.
• It employs modules for imaging, resist simulation, and inverse mask design, reducing computational complexity while co-optimizing optical parameters and fabrication variability.
• This approach enhances yield and design fidelity by integrating simulation with fabrication, driving innovations in semiconductor, photonics, EUV mask, and volumetric lithography applications.

Differentiable lithography is a family of methods that represents lithographic image formation, resist response, fabrication transfer, or related manufacturing distortions as differentiable operators inside an optimization loop, so gradients can be propagated to masks, sources, optical parameters, device geometries, or fabrication layouts. In semiconductor computational lithography, this usually takes the form of differentiable Abbe or Hopkins imaging, differentiable resist surrogates, and gradient-based OPC or ILT. In computational optics, it includes learned fabrication “digital twins” that map intended diffractive layouts to predicted as-fabricated $2.5$D or $3$D structures, allowing optical objectives to be evaluated on fabricated rather than ideal geometry. In photonics and EUV mask design, it extends to differentiable fabrication twins, physics-grounded inverse lithography, and rigorous electromagnetic solvers embedded in automatic differentiation loops.

1. Scope and taxonomy

Differentiable lithography is not a single algorithmic template. The literature spans at least four distinct but connected formulations: differentiable lithography imaging for semiconductor manufacturing, differentiable mask-geometry optimization, fabrication-aware differentiable optics for manufactured diffractive elements, and domain-specific extensions in photonics, EUV masks, and computed axial lithography. A useful distinction is whether the differentiable object is the imaging pipeline, the mask parameterization, the fabrication process, or the full electromagnetic forward model (Chen et al., 2024, Wei et al., 28 May 2025, Zhou et al., 17 Feb 2026).

Strand	Core differentiable object	Representative papers
Semiconductor imaging and resist	Illumination–projector–mask–resist graph	(Chen et al., 2024, Wang et al., 6 Feb 2025)
Mask optimization	Pixels, edges, level sets, B-splines	(Chen et al., 2024, Yi et al., 16 Apr 2025, Ma et al., 2023, Yang et al., 2024)
Fabrication-aware computational optics	Layout-to-fabricated-topography digital twins	(Zheng et al., 2023, Wei et al., 28 May 2025)
Photonics, EUV, CAL	Fabrication twins, rigorous EM solvers, blur-calibrated tomography	(Zhou et al., 17 Feb 2026, Es'kin et al., 24 Jun 2026, Ye et al., 2 Jun 2026)

A second distinction concerns scope. Some works differentiate through the full physical or learned process used in optimization; others are adjacent rather than core. The metalens study for zone-plate-array lithography is explicitly an instance of inverse electromagnetic design for lithography hardware, but it does not differentiate through resist chemistry, exposure thresholding, pattern fidelity, or process models (Chung et al., 2022). Likewise, LithoDreamer is a learned world model with differentiable latent dynamics for the “Layout-Mask-Resist Image-After Development Image” pipeline, but it does not embed explicit optical imaging equations or resist PDEs in the classical sense (Jiang et al., 25 Jun 2026).

2. Differentiable forward models and gradient propagation

A canonical semiconductor formulation decomposes lithography into parameterized differentiable modules for illumination $f_i$, projector transfer $f_h$, mask $f_m$, and resist $f_r$, and writes the overall forward model as

$$f = \eta \circ f_i \circ f_h \circ f_m \circ f_r \Rightarrow Z = f(\cdot, \mathbf{\theta}) + \gamma,$$

with $Z$ the final resist image and $\eta,\gamma$ denoting signal-dependent and signal-independent noise. Within this formulation, Abbe imaging, Hopkins imaging, and SOCS/TCC-SVD approximations can all be placed inside automatic differentiation. The same framework states source optimization as

$$(\theta_{i\_opt}, \theta_{h\_opt}) := \underset{\mathbf{\theta}_i, \mathbf{\theta}_h}{\operatorname{argmin}}|f(\cdot;\mathbf{\theta}_i, \mathbf{\theta}_h) - Z_t|^2,$$

mask optimization as

$3$0

and SMO as

$3$1

In this setting, Hopkins/SOCS reduces complexity from $3$2 to $3$3, while Abbe remains attractive when gradients with respect to source variables are required (Chen et al., 2024).

A major unresolved component in many imaging-only formulations is the resist stage. TorchResist addresses this by making a compact analytical resist simulator differentiable. Its exposure model uses

$3$4

and its development model uses the Mack-style rate

$3$5

To calibrate the model against binary wafer labels, it replaces non-differentiable thresholding and $3$6 mismatch by the surrogate

$3$7

On LithoBench MetalSet, TorchResist reports 0.22 pixel difference, 0.73 nm EPE-mean, and 2.87 nm EPE-max, versus 0.49, 1.21 nm, and 3.95 nm for the variable-threshold baseline; it also provides depth simulation and runs in 0.04 s per $3$8 patch at 7 nm/pixel on a 3090 GPU (Wang et al., 6 Feb 2025).

This family of formulations establishes the basic differentiable lithography principle: the objective is evaluated on the final printed or resist image, while gradients are propagated through FFTs, convolutions, source summations, threshold surrogates, and optionally resist kinetics, instead of stopping at the aerial image. A plausible implication is that wafer-level objectives become substantially more trustworthy when the resist stage is no longer reduced to fixed thresholding.

3. Mask parameterizations and inverse optimization

Early differentiable lithography work often optimized raster masks directly, but several later methods shift the optimization variable from pixels to geometry. DiffOPC is an explicit example of differentiable edge-based OPC. It represents a mask by boundary segments

$3$9

constrains each segment to move along a legal normal-direction velocity $f_i$0 satisfying $f_i$1, uses straight-through rounding for integer-grid coordinates, and backpropagates lithography gradients to segment motion via midpoint sampling. Its total objective combines nominal fidelity, process variation band, and a differentiable EPE surrogate,

$f_i$2

with $f_i$3, $f_i$4, and $f_i$5. On ICCAD13 metal layers, DiffOPC reports $f_i$6, $f_i$7, $f_i$8, and $f_i$9, compared with MultiILT at $f_h$0, $f_h$1, $f_h$2, and $f_h$3; it reports zero MRC violations on ICCAD13, the larger dataset, and the via set (Chen et al., 2024).

A curvilinear alternative is the B-spline/Delaunay formulation, which makes the chain

$f_h$4

explicitly differentiable. The boundary is represented by periodic B-splines,

$f_h$5

the enclosed domain is approximated by Delaunay triangles, and the coherent image amplitude is evaluated by Gaussian quadrature over those triangles. The loss is

$f_h$6

with $f_h$7. The distinguishing contribution is the derivation of explicit formulas for $f_h$8 and $f_h$9, so control-point gradients can be computed analytically rather than by finite differences (Yi et al., 16 Apr 2025).

Level-set parameterizations form a third line. ILDLS represents the mask boundary by a signed-distance level set, uses a Hopkins model with $f_m$0 coherent kernels,

$f_m$1

replaces hard thresholding by

$f_m$2

and injects the level-set ILT correction gradient into UNet training. The final training objective is

$f_m$3

with $f_m$4. In test results, ILDLS reports 5.89 nm AEDE versus 7.79 nm for pure UNet, while ILDLS+ILT reports 4.02 nm versus 5.67 nm for ILT; for process window, ILDLS reports 133.9 DOF@5%EL versus 116.0 for ILT-PV (Ma et al., 2023).

A different position in the design space is ILILT, which frames inverse lithography as a learned recurrent optimizer with lithography-conditioned recurrence,

$f_m$5

It is relevant because the lithography simulator is queried at every step, but it is not primarily a differentiable forward-lithography paper. The method is trained mainly by supervision from solver-produced masks rather than by direct wafer-loss minimization, although it reports EPE = 0.08 and PVB = 4695 for ILILT-P2PHD-8, compared with 0.21 and 4656 for GPU-ILT (Yang et al., 2024).

4. Fabrication-aware differentiable optics and neural lithography

A major expansion of differentiable lithography occurred in computational optics, where the central problem is not mask-to-wafer imaging but layout-to-fabricated-device mismatch. “Neural Lithography” introduces a learned fabrication simulator $f_m$6 so that inverse design solves

$f_m$7

rather than assuming $f_m$8. The simulator is trained from 96 layout–print pairs fabricated on a Nanoscribe Photonic Professional GT2 two-photon lithography system and measured by AFM, with a 72/24 train/validation split. Its PBL architecture decomposes photolithography into optical exposure, thresholding, diffusion, shrinkage, and mismatch correction; quantized $f_m$9-level layouts are handled with the Gumbel–Softmax reparameterization. The reported PBL forward prediction error is $f_r$0, while printed flat-surface roughness is $f_r$1 and clean substrate roughness is $f_r$2. The framework improves fabricated HOEs and MDLs relative to conventional design and to simpler lithography-aware baselines (Zheng et al., 2023).

The large-area extension makes this fabrication-aware view explicitly mass-production compatible. In that pipeline, a master DOE is fabricated by direct-write grayscale lithography using positive AZ® 4562 photoresist on a Heidelberg Instruments DWL 66+, then replicated by nanoimprint lithography in UV-curable OrmoComp resin using an Obducat Eitre 3. The optical design is reformulated from

$f_r$3

to

$f_r$4

where $f_r$5 is a fully differentiable learned map from lithography layout $f_r$6 to fabricated height map $f_r$7. The model is super-resolved: calibration maps a $f_r$8-sampled design layout to AFM-measured topography at about 200 nm sampling pitch, i.e. $f_r$9 spatial super-resolution. On held-out patterns, the neural lithography model reaches 35.20 dB PSNR and 2.45% NRMSE, compared with 27.36 dB PSNR and 7.27% NRMSE for an MTF-based physical model. The system is combined with distributed FFTs, tensor-parallel convolutions, JAX, and GSPMD, allowing simulation grids up to 128,640 \times 85,760 for a 32.16 mm $$f = \eta \circ f_i \circ f_h \circ f_m \circ f_r \Rightarrow Z = f(\cdot, \mathbf{\theta}) + \gamma,$$0 21.44 mm DOE on 16 A100 GPUs. Fabrication-aware optimization yields a nearly speckle-free coherent hologram, a beam splitter with 53% higher overall spot intensity than the conventional design, and a single-DOE broadband imaging system whose measured PSF is close enough to the simulated PSF to support one-step Wiener restoration (Wei et al., 28 May 2025).

Taken together, these papers redefine differentiable lithography as fabrication-aware co-design: the optimized variable is no longer the nominal structure alone, but the input actually sent to the fabrication system, and the objective is evaluated on the predicted as-fabricated geometry.

5. Photonics, EUV masks, and volumetric lithography

In photonic integrated circuits, PRISM adapts inverse lithography to the fact that geometry fidelity is not a sufficient proxy for optical function. Its differentiable fabrication twin factorizes printed geometry into a continuous exposure field and a soft threshold: $$f = \eta \circ f_i \circ f_h \circ f_m \circ f_r \Rightarrow Z = f(\cdot, \mathbf{\theta}) + \gamma,$$1 For DUV, the exposure model is Hopkins/SOCS-based; for EBL, it is PSF-based. The ILT objective is not plain contour matching but a weighted $$f = \eta \circ f_i \circ f_h \circ f_m \circ f_r \Rightarrow Z = f(\cdot, \mathbf{\theta}) + \gamma,$$2 wafer loss,

$$f = \eta \circ f_i \circ f_h \circ f_m \circ f_r \Rightarrow Z = f(\cdot, \mathbf{\theta}) + \gamma,$$3

where the weight map is derived from normalized photonic adjoint sensitivity,

$$f = \eta \circ f_i \circ f_h \circ f_m \circ f_r \Rightarrow Z = f(\cdot, \mathbf{\theta}) + \gamma,$$4

normalized to $$f = \eta \circ f_i \circ f_h \circ f_m \circ f_r \Rightarrow Z = f(\cdot, \mathbf{\theta}) + \gamma,$$5. Under 193 nm DUV, PRISM-SOCS improves yield$$f = \eta \circ f_i \circ f_h \circ f_m \circ f_r \Rightarrow Z = f(\cdot, \mathbf{\theta}) + \gamma,$$6/yield$$f = \eta \circ f_i \circ f_h \circ f_m \circ f_r \Rightarrow Z = f(\cdot, \mathbf{\theta}) + \gamma,$$7 for crossing from 11%/0% to 100%/100%, for optical diode from 0%/0% to 90%/85%, and for $$f = \eta \circ f_i \circ f_h \circ f_m \circ f_r \Rightarrow Z = f(\cdot, \mathbf{\theta}) + \gamma,$$8 MDM mux from 0%/0% to 98%/98%; harder devices such as WDM mux remain challenging even after correction (Zhou et al., 17 Feb 2026).

For EUV masks, a distinct line of work treats the full diffraction engine itself as the differentiable object. The waveguide-method framework solves inverse lithography by differentiating through a rigorous layered periodic Maxwell solver. The absorber permittivity is parameterized by a density field,

$$f = \eta \circ f_i \circ f_h \circ f_m \circ f_r \Rightarrow Z = f(\cdot, \mathbf{\theta}) + \gamma,$$9

with either pixel-wise or Fourier parameterization, and optimized against

$Z$0

The experiments use a realistic EUV mask at $Z$1 nm with a 30 Ru/Be/Sr period multilayer, 60 nm absorber thickness, and 6^\circ incidence angle. For absorber materials, the paper reports normalized central peak intensities $Z$2 of 0.64 for TaBN, 1.99 for La, and 0.45 for U; La gives the strongest central maximum, while U gives the closest overall field match. A 3D extension is also demonstrated (Es'kin et al., 24 Jun 2026).

Computed Axial Lithography provides a volumetric analogue. There, the standard backprojected dose $Z$3 is augmented by a blur kernel,

$Z$4

and the adjoint is modified consistently: $Z$5 The kernel is fitted experimentally from micro-CT data and uncorrected prints, with a notional diffusivity near

$Z$6

while corrected-print trends suggest a peak near

$Z$7

The paper shows that co-optimizing projections under this blurred forward model improves fidelity over Richardson–Lucy target deconvolution; for the RL baseline it reports ASSD $Z$8 and MS-SSIM $Z$9 (Ye et al., 2 Jun 2026).

These extensions show that differentiable lithography is not limited to planar semiconductor mask optimization. It also encompasses photonics-informed mask correction, rigorous EUV electromagnetic inverse design, and volumetric dose optimization under experimentally identified blur.

6. Limitations, misconceptions, and research directions

A persistent misconception is that any gradient-based design related to lithography is “differentiable lithography.” The literature is more specific. The high-NA metalens study for zone-plate-array lithography performs adjoint electromagnetic optimization of a lithography-critical optical component and reports 85.50% transmission normalized focusing efficiency at 0.60 NA and 405 nm, but it does not differentiate through resist chemistry, pattern fidelity, or lithographic process transfer; it is best classified as inverse-designed lithography hardware rather than end-to-end differentiable lithography (Chung et al., 2022). LithoDreamer likewise provides differentiable latent rollouts and inverse planning in a world-model framework, but its “physics-informed” character derives from stage decomposition, latent subspace priors, and process conditioning rather than from explicit Abbe, Hopkins, or resist equations (Jiang et al., 25 Jun 2026).

Another recurring limitation is process specificity. The learned fabrication simulators in computational optics are calibrated on small, process-specific datasets: 96 layout–print pairs in the TPL-based neural lithography work, 20 fabricated random patterns in the large-area grayscale-lithography study, and compact but process-specific calibration reticles in PRISM (Zheng et al., 2023, Wei et al., 28 May 2025, Zhou et al., 17 Feb 2026). Those papers explicitly note intrinsic stochastic variability, process dependence, and the possibility of out-of-distribution corrected masks. PRISM argues that physics-grounded twins are often more reliable than unconstrained neural predictors under limited data and OOD ILT masks, especially for DUV (Zhou et al., 17 Feb 2026).

Computational cost remains another fault line. The large-area diffractive-optics pipeline requires custom distributed operators and multi-GPU infrastructure; the rigorous EUV mask solver differentiates through eigenproblems, scattering-matrix assembly, and field reconstruction; CAL co-optimization introduces additional FFT-based object-space convolution and experimentally fitted blur. These costs do not invalidate differentiability, but they move practical performance questions from pure algorithm design to systems engineering (Wei et al., 28 May 2025, Es'kin et al., 24 Jun 2026, Ye et al., 2 Jun 2026).

A further limitation is model fidelity at the chemistry and process-transfer level. Semiconductor imaging frameworks centered on scalar Abbe or Hopkins models remain strongest on optics and mask/source optimization, while resist modeling may be simplified, abstract, or delegated to compact surrogates (Chen et al., 2024, Wang et al., 6 Feb 2025). In computational optics, the fabrication-aware models capture effective fabrication well enough for demonstrated tasks, but they remain empirical surrogates rather than mechanistic simulators with explicit latent variables for beam shape, material batch variation, or stochastic development fluctuations (Zheng et al., 2023, Wei et al., 28 May 2025). This suggests that future progress will likely depend on combining process-calibrated differentiable twins with stronger physical priors and, where available, hardware-in-the-loop correction.

The process chain itself may also change. Resistless EUV patterning on HF-treated Si(100) replaces the usual photoresist route with EUV-induced surface oxidation and selective TMAH etching, achieving SiO$\eta,\gamma$0/Si gratings with 75 nm half-pitch and 31 nm height. The paper is experimental rather than differentiable, but it shows that the relevant forward model could shift from “aerial image $\eta,\gamma$1 resist chemistry $\eta,\gamma$2 development” to “aerial dose $\eta,\gamma$3 oxidation-state field $\eta,\gamma$4 etch-resistance field $\eta,\gamma$5 topography” (Tseng et al., 2023). This suggests a broader research direction: differentiable lithography need not be tied to chemically amplified resist models if the manufacturing physics itself changes.

Across these lines of work, the unifying idea is stable: lithography or fabrication is treated as an optimization-native physical process rather than as a post hoc verification step. What differs is the differentiable state variable—pixels, edges, level sets, spline control points, absorber permittivity, projection images, or fabrication layouts—and the fidelity of the forward model through which gradients are propagated.