🔬 optics

Inverse mask design for interference lithography using automatic differentiable wave propagation

This paper presents a gradient-based optimization framework using automatic differentiation and the differentiable angular spectrum method to design binary interference lithography masks that successfully generate complex, non-periodic nanostructures with minimal defects while employing memory-efficient techniques like shifted ASM and gradient checkpointing to scale to large mask sizes.

Original authors: Chuntian Cao, Jangwoon Sung, Jack Griffiths, Yuan Gao, Xi Yu, Paul Baity, Nikhil Tiwale, Zhitian Shi, Juhong Ahn, Shinjae Yoo, Yong S. Chu, Chang-Yong Nam

Published 2026-08-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Chuntian Cao, Jangwoon Sung, Jack Griffiths, Yuan Gao, Xi Yu, Paul Baity, Nikhil Tiwale, Zhitian Shi, Juhong Ahn, Shinjae Yoo, Yong S. Chu, Chang-Yong Nam

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to print a picture using only a flashlight and a piece of cardboard. In the world of making tiny things—like the circuits inside your phone or the patterns on a microchip—scientists use a technique called Interference Lithography. Think of it like this: instead of shining a light through a stencil to get a shadow, you shine two or more laser beams so they crash into each other. Where the waves of light meet just right, they create bright and dark stripes, like ripples in a pond colliding. This is fantastic for making perfect, repeating patterns (like a grid of dots) because the math is simple and predictable.

But what if you want to print something messy and unique, like a logo or a complex drawing, instead of just a repeating grid? That's where things get tricky. The light waves don't just follow a simple rule anymore; they get tangled. To get the right picture on the "film" (a special light-sensitive layer), you have to design a very specific, weird-looking mask (the cardboard with holes in it) that twists the light exactly how you need it. The problem is, figuring out what that mask should look like is like trying to guess the shape of a rock just by looking at its shadow. It's a puzzle with no obvious solution, and for a long time, making these custom masks for complex shapes was incredibly hard.

This is where a team of researchers from Brookhaven National Laboratory steps in with a clever new trick. They didn't just guess; they built a super-smart computer program that learns how to design these masks by trial and error, but with a superpower: automatic differentiation.

Here's how their "magic" works. Imagine you are trying to sculpt a statue out of clay, but you can only see the shadow it casts on a wall. You want the shadow to look exactly like a picture of a cat. In the past, you might have guessed, looked at the shadow, and guessed again, hoping to get closer. This new method is like having a robot that can instantly tell you exactly which tiny speck of clay to move to make the shadow look better. The computer simulates the light passing through a digital mask, sees how the resulting "shadow" (the pattern) compares to the target picture, and then uses math to figure out exactly how to tweak the mask to fix the mistakes. It does this thousands of times in a row, getting closer and closer to perfection.

The researchers used a technique called the Angular Spectrum Method to simulate how light travels. Think of this as a super-accurate video game engine that predicts exactly how light waves will dance and interfere with each other. By making this engine "differentiable," they allowed the computer to work backward from the mistake to the cause. They started with a random, messy digital mask and let the computer "backpropagate" the errors, adjusting the mask's pixels until the simulated light pattern matched their target design almost perfectly.

The results were impressive. They designed a mask that could print a complex pattern (based on the Brookhaven National Laboratory logo) with incredibly sharp details. The final pattern had only 0.1% of its pixels wrong—basically, just a few tiny, scattered specks of dust in an otherwise perfect image. Even cooler, the mask they designed had pixels that were 800 nanometers wide, yet it could print features as small as 400 nanometers. That's like using a coarse paintbrush to paint a line half the width of the brush's bristles, all thanks to the way the light waves interfere with each other.

However, there was a catch. Simulating this light dance for a large mask requires a massive amount of computer memory, like trying to hold a giant ocean in a single cup. To solve this, the team invented a "shifted" version of their simulation. Imagine trying to paint a huge mural. Instead of trying to hold the whole canvas in your mind at once, you break it into small squares, paint each square separately, and then tape them together. This "patchwork" approach allowed them to simulate masks that were 10.2 times larger than what their computers could normally handle. They even tested running this on four graphics cards at once, which made the process 2.5 times faster.

One important detail they found is that this magic only works if the light is very pure. If you use a flashlight with a mix of colors (like white light), the pattern gets blurry and washed out. They found that the light needs to be nearly a single color (monochromatic), like a laser, to keep the sharp edges. If the light's color varies too much, the delicate interference pattern falls apart.

In the end, this paper doesn't just show a new way to make masks; it opens the door to printing any shape you can imagine using interference lithography. By combining physics with machine learning, they turned a nearly impossible math problem into a solvable puzzle. While they proved this works in simulations, the path is now clear to build real-world masks for creating complex, non-repeating nanostructures, potentially revolutionizing how we make tiny, intricate devices in the future.

Technical Summary: Inverse Mask Design for Interference Lithography Using Automatic Differentiable Wave Propagation

Problem Statement
Interference lithography (IL) is a standard technique for fabricating high-resolution periodic nanostructures and qualifying photoresists, particularly at extreme ultraviolet (EUV) wavelengths. However, designing photomasks to generate arbitrary, non-periodic patterns remains a significant challenge. While the forward propagation from a mask to an aerial image can be computed analytically, the inverse problem—determining the mask required to produce a specific target aerial image—is ill-posed due to the lack of phase information in the intensity-based aerial patterns. Existing methods often struggle to scale to large mask sizes or to resolve features smaller than the mask pixel pitch without prohibitive computational costs.

Methodology
The authors propose a gradient-based optimization framework that utilizes automatic differentiation to solve the inverse mask design problem. The core of the approach involves a differentiable forward model based on the Angular Spectrum Method (ASM), a standard technique for simulating near- and intermediate-field diffraction.

Forward Model: The ASM computes the propagated field at the image plane by performing a Fourier transform of the mask field, multiplying by a transfer function (which includes free-space propagation and band-limiting to suppress evanescent and aliased components), and applying an inverse Fourier transform. This process is fully differentiable, allowing gradients to be backpropagated.
Optimization Formulation: The mask design is treated as an optimization problem over continuous "logits" ( $\ell$ ). These logits are converted into a soft binary mask via a sigmoid function with a temperature parameter ( $\tau$ ), which is annealed during training to approach a hard binary mask. The loss function ( $\mathcal{L}$ ) combines a Pearson correlation term between the simulated field amplitude and the target pattern with regularization terms: Total Variation ( $L_{TV}$ ) to penalize high-frequency spatial variations and Sparsity ( $L_{sp}$ ) to promote manufacturable, sparse masks.
Scalability (Shifted ASM): To address the $O(N^2)$ memory scaling of standard ASM for large masks, the authors employ the Shifted ASM. This method partitions the mask into smaller patches (e.g., $4 \times 4$ ), propagates them independently using transfer functions that account for lateral offsets, and sums the resulting complex fields at the observation plane. This reduces peak memory requirements to scale with the patch size, $O((N/P)^2)$ .
Parallelization: The framework supports multi-GPU distribution, where patches are assigned to different GPUs. Communication is limited to two all-reduce operations: summing partial fields in the forward pass and summing logit gradients in the backward pass.

Key Results
The authors validated the framework using a target pattern derived from the Brookhaven National Laboratory (BNL) logo, featuring vertical stripes with a minimum feature size of 400 nm.

Resolution and Accuracy: Using a mask with an 800 nm pixel pitch, the optimized design successfully resolved features of 400 nm at the image plane, demonstrating a 2 $\times$ resolution enhancement through coherent diffraction. The resulting pattern achieved a Pearson correlation coefficient of 0.95 with the target. When binarized to simulate resist development, the pattern exhibited only 0.1% isolated pixel-level defects.
Memory and Runtime Efficiency: For a 3.84 mm $\times$ 3.84 mm mask (19,200 $\times$ 19,200 pixels), the Shifted ASM with 16 patches reduced peak GPU memory by 3.8 $\times$ compared to standard ASM, with only a 1.3 $\times$ increase in runtime. With gradient checkpointing enabled, peak memory was reduced by 7.4 $\times$ at a 2 $\times$ runtime cost.
Large-Scale Capability: The Shifted ASM approach enabled the optimization of masks up to 12.8 mm $\times$ 12.8 mm, an area 10.2 $\times$ larger than the maximum size achievable with standard ASM on the same hardware. Distributing the optimization across 4 GPUs further accelerated the process, achieving a 2.5 $\times$ speedup relative to standard ASM.
Spectral Sensitivity: Simulations indicated that while periodic IL is tolerant to broadband sources, arbitrary non-periodic patterns require near-monochromatic illumination. The study estimated a bandwidth tolerance of $\Delta\lambda/\lambda \lesssim 5.76 \times 10^{-4}$ to maintain pattern fidelity, a constraint satisfied by lasers and synchrotron beams but not by broadband sources.

Significance and Claims
The paper establishes a physics-informed, machine learning-driven approach for IL mask design, moving beyond periodic structures to complex, non-periodic patterns. The authors claim that this differentiable optimization framework successfully bridges the gap between forward wave propagation simulation and inverse design. By leveraging the Shifted ASM and multi-GPU parallelization, the method overcomes the memory bottlenecks that previously limited the size of optimizable masks.

The work highlights the potential for "degeneracy" in inverse problems, suggesting that the high degree of freedom in mask design can be exploited to optimize for additional properties, such as robustness to experimental imperfections (e.g., distance errors or beam non-idealities) or manufacturability (e.g., sparsity), albeit potentially at the cost of image quality or intensity. The authors conclude that this framework is applicable to any inverse problem where a differentiable forward model is available, with source code provided to facilitate further research.

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