Continuous Neural Operators for
Electromagnetic Maxwell Solvers

Classical optical proximity correction (OPC) solves the scalar Helmholtz equation under Abbe's formulation using finite-difference time-domain (FDTD) discretizations:

When the physical feature dimension \( d \ll \lambda \) (where \(\lambda = 13.5\,\text{nm}\) and \(d \approx 1.4\text{–}2\,\text{nm}\)), scalar approximations fail catastrophically due to non-negligible 3D mask topography effects, oblique incidence shadowing, and polarization-dependent transmission. Conventional solvers require \(O(N^3)\) operations on discretized grids with trillions of spatial voxels.

LithoCore AI replaces discrete spatial numerical integrators with a continuous Fourier Neural Operator (FNO) that directly parameterizes the solution operator mapping between the complex dielectric permittivity function \(\epsilon(x, y, z)\) and the electromagnetic wavefront \(E(x, y)\):

By computing the integral kernel \(\kappa\) in the Fourier domain via Fast Fourier Transforms (FFT), the operator evaluates infinite spatial resolutions with zero-shot generalization across any wafer grid step size.

Traditional EDA software enforces Manhattan (orthogonal 90-degree) geometric rules because legacy electron-beam mask writers were constrained to rasterized rectangles. Modern multi-beam mask writers (MBMW) support arbitrary curved shapes.

CurviMask™ formulates mask synthesis as a continuous variational optimization problem on a level-set function \(\phi(x, y)\):

Because our Maxwell-FNO forward solver is completely differentiable with analytical gradient backpropagation, the loss function converges in fewer than 40 gradient descent iterations—yielding masks with 38% larger exposure latitude and complete elimination of corner-rounding hotspots.

High-NA EUV scanners introduce anamorphic optics—magnifying by 4x in the horizontal (X) axis and 8x in the vertical (Y) axis to prevent extreme ray angles from clipping on the mask absorber. This breaks all symmetry assumptions used in classical OPC algorithms.

LithoCore AI natively embeds anisotropic tensor coordinate transformations into the spectral operator layers, naturally handling anamorphic field stitches and half-field reticle borders without computational penalties.