Unitary learning for diffractive deep neural network

•Different from the real-valued Backpropagation, we formulate a unitary learning protocol for diffractive deep neural network under compatible condition, encapsulating the fundamental sigmoid, tanh and quasi-ReLu in complex space as nonlinear activations available in complex-valued Backpropagation, in which implements the concept of conjugation substitution significance in real-valued BackPropagation. The weight banks are unitary. It is launched for the first time. The modulation mechanism is introduced into a multilayer coherent diffraction to develop a diffractive deep neural network, which could circumvent the nonlinear activations implementation in physical realization.•In fact, the compatible condition overcomes the prolonged ambiguity on the differentiability of nonlinear activations in a single channel complex space training, such as fundamental sigmoid, tanh. Unitary learning is evolved from compatible learning that feeds Euclidean gradient for Riemannian gradient. Unitary learning has a competence in single channel training, and a practicability of physical unitary prior. Qausi-phase relu is especially proposed for unitary learning and profoundly investigated in different potential occasions. The classification and recognition tasks on two benchmark training sets are performed with diffractive deep neural network. And the differences between backpropagation and phase retrieval for training are presented.