Unitary learning for diffractive deep neural network

Realization of deep learning with coherent diffraction has achieved remarkable development nowadays, which benefits on the fact that matrix multiplication can be optically executed in parallel with high band-with and low latency. Coherent optical field in the form of complex-valued entity can be manipulated into a task-oriented output. In this paper, a modulation mechanism is established by implementing the equivalence between a digital deep unitary neural network and optical coherent diffraction. We present a unitary learning avenue on diffractive deep neural network, meeting the physical unitary prior in coherent diffraction. The Unitary learning is a Backpropagation serving to unitary weights update through the gradient translation from Euclidean to Riemannian space. The temporal-space evolution characteristics in unitary learning are formulated and elucidated. And a compatible condition on how to select the nonlinear activation in complex space is unveiled, encapsulating the fundamental sigmoid, tanh and quasi-ReLu in complex space available in a single channel training. The performance of phase-ReLu is particularly emphasized. As a preliminary application, diffractive deep neural network with unitary learning is tentatively implemented on the 2D classification and verification tasks.