All-optical synthesis of an arbitrary linear transformation using diffractive surfaces

Spatially-engineered diffractive surfaces have emerged as a powerful framework to control light-matter interactions for statistical inference and the design of task-specific optical components.Here,we report the design of diffractive surfaces to all-optically perform arbitrary complex-valued linear transformations between an input(Ni)and output(No),where Ni and No represent the number of pixels at the input and output fields-of-view(FOVs),respectively.First,we consider a single diffractive surface and use a matrix pseudoinverse-based method to determine the complex-valued transmission coefficients of the diffractive features/neurons to all-optically perform a desired/target linear transformation.In addition to this data-free design approach,we also consider a deep learning-based design method to optimize the transmission coefficients of diffractive surfaces by using examples of input/output fields corresponding to the target transformation.We compared the all-optical transformation errors and diffraction efficiencies achieved using data-free designs as well as data-driven(deep learning-based)diffractive designs to all-optically perform(i)arbitrarily-chosen complex-valued transformations including unitary,nonunitary,and noninvertible transforms,(ii)2D discrete Fourier transformation,(iii)arbitrary 2D permutation operations,and(iv)high-pass filtered coherent imaging.Our analyses reveal that if the total number(N)of spatially-engineered diffractive features/neurons is≥Ni×No,both design methods succeed in all-optical implementation of the target transformation,achieving negligible error.However,compared to data-free designs,deep learning-based diffractive designs are found to achieve significantly larger diffraction efficiencies for a given N and their all-optical transformations are more accurate for N<Ni×No.These conclusions are generally applicable to various optical processors that employ spatially-engineered diffractive surfaces.