Physics-informed neural networks (PINNs) and neural operators (NOs) for solving the problem of diffraction of Extreme Ultraviolet (EUV) electromagnetic waves from contemporary lithography masks are presented. A novel hybrid Waveguide Neural Operator (WGNO) is introduced, based on a waveguide method with its most computationally expensive components replaced by a neural network. To evaluate performance, the accuracy and inference time of PINNs and NOs are compared against modern numerical solvers for a series of problems with known exact solutions. The emphasis is placed on investigation of solution accuracy by considered artificial neural systems for 13.5 nm and 11.2 nm wavelengths. Numerical experiments on realistic 2D and 3D masks demonstrate that PINNs and neural operators achieve competitive accuracy and significantly reduced prediction times, with the proposed WGNO architecture reaching state-of-the-art performance. The presented neural operator has pronounced generalizing properties, meaning that for unseen problem parameters it delivers a solution accuracy close to that for parameters seen in the training dataset. These results provide a highly efficient solution for accelerating the design and optimization workflows of next-generation lithography masks.
Figures & tables
Figure 1: EUV lithography with (a) and without (b) optical proximity correction of mask.
Figure 2: Geometry of the problem
Figure 3: Schematic diagram of the waveguide method. Here “F” is forward Fourier transform, “Eigen” is calculating eigenvalues and eigenvectors of system ( 13 ) for every layer, “Field calculation” is calculating field with Equations ( 14 ) – ( 16 ).
Figure 4: Schematic diagram of the general PINN method.
Figure 5: Schematic diagram of the waveguide neural operator. Here “F” is forward Fourier transform, “Eigen” is calculating eigenvalues and eigenvectors of system ( 13 ) for every layer, “Field calculation” is calculating field with Equations ( 14 ) – ( 16 ).
Figure 6: Problems with exact solutions: (a) is problem of electromagnetic waves reflection from a single interface, (b) is problem of electromagnetic waves reflection from a dielectric layer.
Problem
Waveguide
FreeFem++
Rel. L2 Error
Time (s)
Rel. L2 Error
Time (s)
1. Wave propagation
2.814×10−15
1.0×10−3
2.323×10−3
3.467
2. Wave scattering on interface
2.730×10−15
2.0×10−3
3.818×10−3
3.385
3. Reflection from layer
6.110×10−15
6.0×10−3
8.085×10−3
5.033
Problem
PINN
WGNO
Rel. L2 Error
Training (s)
Rel. L2 Error
Training (s)
Table 1: Performance comparison on 2D test problems. Inference times are 1.2×10−3 s and 1.7×10−4 s for the PINN and the WGNO, respectively.
Figure 7: Dependence of reflectivity of TE EUV wave (wavelength 13.5 nm) on incident angle for Mo/Si mirror of 60 layers. Thicknesses of Mo and Si layers are given in Table 2 . Maximum of reflectivity is 0.75 at incident angel 8.86°.
Method
Properties
Media
Layer
TaBO
TaBN
Ru
Mo
Si
ε
0.857−i0.079
0.861−i0.071
0.785−i0.030
0.853−i0.012
0.998−i0.004
PINN
thickness
10 nm
10 nm
2 nm
3 nm
4 nm
WGNO
thickness
10 nm
60 nm
2 nm
3 nm
4 nm
Table 2: Parameters of the mask for wavelength λ=13.5 nm which used to calculations.
Figure 8: (a) is real part of k in the media normalized to k0 . Comparison of reference WG solution (b) and PINN solution (c) for a 2D mask at 13.5 nm, (d) is absolute error.
Figure 9: (a) is real part of k in the media normalized to k0 . Comparison of reference WG solution (b) and WGNO solution (c) for a 2D mask at 13.5 nm, (d) is absolute error.
Mirror
Properties
Media
Layer
Mo
Ru
Be
Sr
ε
0.91−i0.009
0.872−i0.012
1.025−i0.003
0.986−i0.002
Mo/Be
thickness
2.22 nm
–
3.5 nm
–
Ru/Be
thickness
–
2.01 nm
3.72 nm
–
Ru/Be/Sr
thickness
–
1.7 nm
2.7 nm
1.34 nm
Table 3: Parameters of layers of mirror for wavelength λ=11.2 nm.
Figure 10: Dependences of reflectivity of TE EUV wave (wavelength 11.2 nm) on incident angle for Mo/Be, Ru/Be and Ru/Be/Sr mirrors of 60 layers. Thicknesses of layers are given in Table 3 . Maximum of reflectivities are: 0.77 at incident angel 10.71°for Mo/Be, 0.79 at incident angel 11.06°for Ru/Be, and 0.81 at incident angel 10.96°for Ru/Be/Sr.
Figure 11: (a) is real part of k in the media normalized to k0 . Comparison of reference WG solution (b) and PINN solution (c) for a 2D mask at 11.2 nm, (d) is absolute error.
Figure 12: (a) is real part of k in the media normalized to k0 . Comparison of reference WG solution (b) and WGNO solution (c) for a 2D mask at 11.2 nm, (d) is absolute error.
λ (nm)
Method
Rel. L2 Error
Training (s)
13.5
PINN
3.1×10−2
1003
WGNO
3.8×10−6
4.3
11.2
PINN
3.3×10−2
1252
WGNO
3.5×10−7
7.7
Table 4: Performance on 2D lithography mask simulation. Inference times are 1.2×10−3 s and 1.8×10−4 s for PINN and WGNO, respectively.
Figure 13: Comparison of reference WG solution ((a),(d)) and WGNO solution ((b),(e)) for a 3D mask at 13.5 nm, (c) and (f) are absolute errors. (a)–(c) are in cross-section y=0 , (d)–(f) are in the cross-section x=0 .
Figure 14: Comparison of reference WG solution ((a),(d)) and WGNO solution ((b),(e)) for a 3D mask at 11.2 nm, (c) and (f) are absolute errors. (a)–(c) are in cross-section y=0 , (d)–(f) are in the cross-section x=0 .
Wavelength
Rel. L2 Error
Inference (s)
Training time (s)
13.5 nm
3.1×10−7
2.24×10−4
19.4
11.2 nm
2.8×10−6
2.19×10−4
18.9
Table 5: Performance of WGNO on 3D lithography mask simulation. Training time less 20 s.
Nq
Measure of generalization μ
5
0.94225
10
0.98473
20
0.99607
50
0.99922
100
0.99978
Table 6: Dependence of measure of generalization μ of WGNO on the number of partitions Nq of the area of change of the parameter a .
Appendix figures & tables4 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 15: FreeFem++ solver. The left panels show the exact solutions, the middle panels show the solutions calculated using the given method, and the right panels show the absolute difference between the exact and calculated solutions. The relative error ϵerror are given in the table 1 . Panels (a), (b) and (c) correspond the problems of wave propagation in a homogeneous medium, reflection of an electromagnetic waves from a interface between two media and reflection from a dielectric layer, respectively.
Figure 16: The same is shown in Figure 15 for the waveguide method.
Figure 17: Approach based on the PINN. The left panels show the exact solutions, the middle panels show the calculated solutions using the given method, right panels are absolute difference between exact solutions and calculates solutions. The relative error ϵerror are given in the table 1 . Panels (a), (b) and (c) correspond the problems of wave propagation in a homogeneous medium, reflection of an electromagnetic waves from a interface between two media and reflection from a dielectric layer, respectively.
Figure 18: The same is shown in Figure 17 for the waveguide neural operator.
Neural PDE solvers use trainable models to approximate solutions or solution operators for partial differential equations, often with physical constraints.
We present a physics-informed neural operator (PINO) trained with pseudo-spectral frequency-domain (PSFD) equations for electromagnetic (EM) scattering problems in EUV lithography. The Fourier neural operator is factorized into a two-dimensional lateral (xy) branch and a one-dimensional axial (z) branch and is trained self-consistently with background decomposition.Thus, the full-vector coupling between the mask and the multilayer response is retained without invoking a finite-order Born approximation. In this way, the computational domain size is significantly reduced, thereby lowering the computational cost. The PINO is trained on approximately 16,000 mask designs from the LithoBench library sampled randomly at each training iteration without using precomputed EM field solutions. The PINO surrogate model yields predictions with a mean absolute error of about 7×10−3 for the scattered intensity of held-out mask patterns relative to the reference PSFD solution. Combined with spectral damping, the PINO warm-start initialization accelerates the background-decomposed PSFD solver on finer discretizations.
Gradient-based inverse lithography technology~(ILT) for extreme ultraviolet~(EUV) masks is presented. A novel framework treats the differentiable waveguide method and the recently proposed waveguide neural operator~(WGNO) as end-to-end physics engines, recovering the permittivity of the absorber of the mask through automatic differentiation of the full forward diffraction model. Numerical experiments on realistic 2D and 3D absorbers of the mask (TaBN, La, U) at λ=11.2~nm show that the considered ILT methods make it possible to obtain a mask structure that achieves the desired field on the wafer.
Vasiliy A. Es'kin, Egor V. Ivanov
Department of Radiophysics, University of Nizhny Novgorod 23 Gagarin Ave., Nizhny Novgorod 603022, Russia
We present a global-in-time spectral Duhamel-Picard method with polynomial expansions for post-exposure bake (PEB) simulation. The coupled acid-quencher reaction-diffusion equations are reformulated as nonlinear Volterra integral equations using the Duhamel principle. Full trajectories are represented by finite polynomial expansions, with temporal coefficient fields determined over the bake interval by Anderson-accelerated Picard iteration, thereby avoiding sequential time marching. In 3-D PEB simulations, we employ Chebyshev and Legendre temporal bases while evaluating the Duhamel integrals using the corresponding Gaussian quadrature rules. We demonstrate that the proposed method simulates a large domain of 1200×1200×100 voxels on the order of one minute. The representation is further incorporated into a physics-informed neural operator (PINO) with a mixed-transform variant of a factorized Fourier neural operator, which is trained without precomputed solution trajectories. Together, the integral formulation and its PINO extension enable accelerated, time-marching-free prediction of concentration fields at arbitrary times within the bake interval.