We report a polarization-encoded diffractive network to perform multiple arbitrary complex-valued linear transforms within a single diffractive processor. An array of pre-selected linear polarizers is placed between the trainable isotropic diffractive layers, and distinct complex-valued linear transformations are individually assigned to different combinations of input/output polarization states. A polarization-encoded diffractive network performs the target linear transforms with negligible error when N ≥ P x I x O, where N is the number of trainable diffractive features/neurons, I and O denote the number of pixels at the input and output fields-of-view, respectively, and P represents the number of target linear transforms.
We present both data-free and data-driven methods for the all-optical synthesis of an arbitrary complex-valued linear transformation using diffractive surfaces. Our analyses reveal that if the total number (N) of spatially-engineered diffractive features/neurons is larger than a threshold, dictated by the multiplication of the number of pixels at the input (I) and output (O) fields-of-views, i.e., N>IxO, both methods succeed in all-optical implementation of the target transformation. However, compared to data-free designs, deep learning-based diffractive designs with multiple diffractive layers are found to achieve significantly larger diffraction efficiencies and their all-optical transformations are much more accurate when N< IxO.
We analyze the information processing capacity of diffractive optical networks to reveal that increasing the total number of diffractive features, i.e., neurons, within a network linearly increases the dimensionality of the complex-valued linear transformation space of the network, up to a limit dictated by the input and output fields-of-view. We further show that deeper diffractive neural networks formed by larger numbers of diffractive surfaces can cover a higher-dimensional subspace of the complex-valued linear transformations between a larger input field-of-view and a larger output field-of-view, increasing the learning capability and approximation power of the optical network.
We analyze the information processing capacity of coherent optical networks formed by trainable diffractive surfaces to prove that the dimensionality of the solution space describing the set of all-optical transformations established by a diffractive network increases linearly with the number of diffractive surfaces, up to a limit determined by the size of the input/output fields-of-view. Deeper diffractive networks formed by larger numbers of trainable diffractive surfaces span a broader subspace of the complex-valued transformations between larger input/output fields-of-view, and present major advantages in terms of their function approximation power, inference accuracy and learning/generalization capabilities compared to a single diffractive surface.
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