Fig. 3: Illustration of computational lithography algorithm by elucidating the 2D constructive and vacant interference in n = ∞ interference patterns.

a Schematic diagram illustrating the probability for material deposited at a ∞ given position \({{{\bf{x}}}}(x,y)\) on the substrate plane is proportional to the 2D convolution of the offset trajectory function \({{{\bf{F}}}}(x,y)\) and the nanoaperture pattern function \({{{\bf{M}}}}(x,y)\). b Geometric relationship for the constructive interference at the position directly underneath the center of a nanopore, in which the angular beams come from the 1^st and 2^nd nearest nanopores for the square lattice and the 2^nd and 3^rd nearest nanopores for the hexagonal lattice. c–f Dimensional analysis for the deposition probability at the nanopore center position, \({P}_{c}\), as a function of two dimensionless variables, \(R/r\) and \(L/r\), for square (c) and hexagonal (e) nanopore lattices, revealing that the constructive interference takes place along the lines of \(R/L={c}_{i}\), where \({c}_{i}=\{1,\sqrt{2},2,\sqrt{5},\,\ldots \}\) and \({c}_{i}=\{1,\sqrt{3},2,\sqrt{7},\,\ldots \}\) for square and hexagonal nanopore lattices, respectively. Along a given horizontal cut, \(R/r=8\), we present a number of simulated 2D interference patterns corresponding to different \(R/L\) values for square (d) and hexagonal nanopore lattices (f). The white circles denote the nanopores location. The colors in each figure indicate the intensity of normalized probability \({P}_{{{{\rm{c}}}}}\) from CL model. Scale bars: 10\(r\).