Fig. 1: Planar photonic chip for spatial differentiation.

a Schematic of a photonic chip acting as a spatial differentiator that transforms an image into its first-, second- and even higher order derivative. The chip is made from a well-designed dielectric multilayer structure. b Simulated color-coded transmission coefficient amplitude as a function of frequency and in-plane wavenumber for the \(s\)- and \(p\)-polarizations. c Simulated transmission BFP image through the dielectric multilayer at an incident wavelength of 643 nm. The double arrow-headed black line at the top left corner indicates the polarization orientation of the incident light. The colored one-way arrows indicate the polarization direction of transmitted light in the momentum space, the polarization deflection increases as the color change from white to red. d Simulated transmission BFP image when an analyzer is placed after the dielectric multilayer. The polarization orientation of the analyzer lies perpendicular to the incident polarization orientation, as indicated by the double arrow-headed line. The green trapezoidal regions are designed to perform first-order differentiation, and the yellow circular region for second-order differentiation. The green dotted lines and the yellow dotted lines indicate the directions of first-order differentiation and second-order differentiation, respectively. e Simulated optical transfer function \(\left|t\left({k}_{r}\right)\right|\) for the second-order differentiation at \(\lambda\) = 643 nm in the case where \(\varphi={45}^{\circ }\) and for the quadratic fitting using the form \(\left|t\left({k}_{r}\right)\right |=a{k}_{r}^{2}\). The abbreviation “Sim.” has a full name of “Simulation”. f Simulated optical transfer function \(\left|t\left({k}_{x}\right)\right|\) (along the direction \({k}_{y}=-0.18{k}_{0}\), which is marked using the green line in (d)) for first-order differentiation and for linear fitting using the form \(\left|t\left({k}_{x}\right)\right |=b\left|{k}_{x}\right|\).