Universal linear intensity transformations using spatially incoherent diffractive processors

Abstract

Under spatially coherent light, a diffractive optical network composed of structured surfaces can be designed to perform any arbitrary complex-valued linear transformation between its input and output fields-of-view (FOVs) if the total number (N) of optimizable phase-only diffractive features is ≥~2NiNo, where Ni and No refer to the number of useful pixels at the input and the output FOVs, respectively. Here we report the design of a spatially incoherent diffractive optical processor that can approximate any arbitrary linear transformation in time-averaged intensity between its input and output FOVs. Under spatially incoherent monochromatic light, the spatially varying intensity point spread function (H) of a diffractive network, corresponding to a given, arbitrarily-selected linear intensity transformation, can be written as H(m, n; m′, n′) = |h(m, n; m′, n′)|2, where h is the spatially coherent point spread function of the same diffractive network, and (m, n) and (m′, n′) define the coordinates of the output and input FOVs, respectively. Using numerical simulations and deep learning, supervised through examples of input-output profiles, we demonstrate that a spatially incoherent diffractive network can be trained to all-optically perform any arbitrary linear intensity transformation between its input and output if N ≥ ~2NiNo. We also report the design of spatially incoherent diffractive networks for linear processing of intensity information at multiple illumination wavelengths, operating simultaneously. Finally, we numerically demonstrate a diffractive network design that performs all-optical classification of handwritten digits under spatially incoherent illumination, achieving a test accuracy of >95%. Spatially incoherent diffractive networks will be broadly useful for designing all-optical visual processors that can work under natural light.

Complex-valued universal linear transformations and image encryption using spatially incoherent diffractive networks

“…For a phase-only diffractive network, i.e., only the phase profile of each diffractive layer is trainable, the sufficient condition becomes N≥2NiNo due to the reduced degrees of freedom within the diffractive volume. Similar conclusions can be reached for a diffractive network that operates under spatially incoherent illumination: Rahman et al 26 . demonstrated that a diffractive network can be optimized to perform an arbitrary nonnegative linear transformation of optical intensity through phase-only diffractive processors with N≥2NiNo.…”

“…S1 in the Supplementary Material), forming complex-number-based images. To accurately model the spatially incoherent propagation 26 of light through the D2NN, we averaged the output intensities over a large number of Nφ=20,000 of randomly generated 2D phase profiles at the input (see Sec. 4 for details).…”

“…An alternative to this data-free PSF-based optimization approach is to train the diffractive network in an end-to-end manner, using a data-driven direct training approach. 26 This strategy advances by minimizing the differences between the outputs and the targets on a large number of randomly generated examples, thereby learning the spatially varying PSFs implicitly from numerous input-target intensity patterns corresponding to the desired taskinstead of learning from an explicitly predetermined A r . This direct approach, however, requires a longer training time, necessitating the simulation of incoherent propagation for each training sample on a large data set.…”

“…In this article, we employed a data-free PSF-based D2NN optimization method (see Sec. 4), 26 since we can determine the nonnegative intensity transformation bold-italicAbold-italicr from the target complex-valued transformation A based on the mosaicking and demosaicking schemes; the columns of bold-italicAbold-italicr represent the desired spatially varying PSFs of the D2NN. The advantage of this data-free learning-based D2NN optimization approach is that computationally demanding simulation of wave propagation with large Nφ is not required during the training.…”

“…On the other hand, this approach necessitates prior knowledge of bold-italicAbold-italicr, which might not always be available, e.g., for tasks such as data classification. An alternative to this data-free PSF-based optimization approach is to train the diffractive network in an end-to-end manner, using a data-driven direct training approach 26 . This strategy advances by minimizing the differences between the outputs and the targets on a large number of randomly generated examples, thereby learning the spatially varying PSFs implicitly from numerous input-target intensity patterns corresponding to the desired task – instead of learning from an explicitly predetermined bold-italicAbold-italicr.…”