Edge detection technology based on optical analog differentiation is widely employed in fields such as microscopic imaging, data processing, and machine vision. In recent years, with the development of nanofabrication technology, various optical analog differentiators utilizing metasurfaces and metamaterials have been invented, which highly reduces the space required for optical imaging systems. We plan to propose a topological differentiator with a topological charge of 2 based on an all-dielectric one-dimensional photonic crystal (1DPC). By properly designing the bandgap structure of the 1DPC and the polarization state of input and output light fields, this topological device can perform isotropic two-dimensional second-order differential operations on the input optical field, leading to edge enhancement imaging and highly efficient edge information extraction.

The photonic chip is fabricated via PECVD (Oxford System 100, UK) of SiO₂ and Si₃N₄ layers on a standard microscope cover slip. All experiments are performed using a modified upright optical microscope (Ti2-U, Nikon, Japan), and the illumination beam with a central wavelength of 643 nm and a bandwidth of 2 nm is emitted from a supercontinuum fiber laser (SuperK EXU-6, NKT Photonics, Denmark). Left-handed circularly polarized input light incident on the objects of interest is placed on the photonic chip, and the right-handed circularly polarized component of output light passing through the photonic chip is filtered out with imaging conducted onto the detector. We introduce a spherical reference light to interfere with the output field to measure the winding number of topological charge.

According to the intensity profile in every direction of the back focal plane of the fabricated photonic chip, the optical transfer function of the chip satisfies the form required for second-order differentiation, which means this photonic chip can implement isotropic second-order differentiation. The interference result between reference light and output light represents that there is a second-order topological charge in the expression of optical transfer functions, which leads to isotropic differentiation. The USAF resolution test chart is adopted to demonstrate the performance of this photonic chip. Two peaks at the location of the chart's edge mean second-order differentiation is implemented. Additionally, the edge detection on biological objects indicates that this photonic chip can also be applied to the biological field.

We design a two-dimensional second-order topological differential optical chip operating in the transmission mode. The optical chip is composed of all-dielectric one-dimensional photonic crystals. By adjusting the structural parameters of photonic crystals, the optical transfer function required for second-order differential operation can be achieved. When the polarization states of the incident light and the output light are left-handed circularly polarized and right-handed circularly polarized respectively, a second-order topological charge is generated in the optical transfer function to achieve isotropic two-dimensional differential operation. We demonstrate the differential operation effect of the prepared optical chip on the incident light field by employing the USAF resolution test chart and biological sample. This topological differential device characterized by high throughput, high speed, and easy fabrication will have potential applications in optical computing, imaging, and sensing.