Fig. 1: Schematic and logic diagram of on-chip diffractive optical neural network (DONN).

a Schematic of an on-chip DONN, each diffractive unit on a given layer acts as a secondary wave source, the amplitude and phase of which are determined by the product of the input wave and the complex-valued transmission at that unit. Each diffractive unit (DU) is a slot group composed of three identical silicon slots that are filled with silicon dioxide; each DU represents a single neuron in the on-chip DONN. b Logic diagram of Fig.1a that mathematically describes the physical calculation process of the on-chip DONN. The formula shown between Fig. 1a and Fig. 1b is the mathematical expression of DONN, where “T” represents matrix transposition; \({{{{{{\rm{diag}}}}}}}\left({e}^{\, j{\phi }_{11}},\cdots,{e}^{\, j{\phi }_{1n}}\right)\), \({{{{{{\rm{diag}}}}}}}\left({e}^{\, j{\phi }_{21}},\cdots,{e}^{\, j{\phi }_{2n}}\right)\), and \({{{{{{\rm{diag}}}}}}}\left({e}^{\, j{\phi }_{31}},\cdots,{e}^{\, j{\phi }_{3n}}\right)\) refer to a diagonal matrix, that is, a matrix in which the elements outside the main diagonal are all 0, where the phase values \(\left({\phi }_{11},\cdots,{\phi }_{1n},{\phi }_{21},{\cdots,\phi }_{2n},{\phi }_{31},\cdots,{\phi }_{3n}\right)\) are generated by the corresponding DUs; \({W}^{(k)}\) represents the \({k}-{{{{{{\rm{th}}}}}}}\) diffraction matrix derived from the on-chip electromagnetic propagation model (Eq. (1)); \(\left({x}_{1},{x}_{2},{x}_{3},{x}_{4}\right)\) represents the input; and \(\left({y}_{1},{y}_{2},{y}_{3}\right)\) represents the output.