Design of ultracompact broadband focusing spectrometers based on deep diffractive neural networks

We propose the inverse design of ultracompact, broadband focusing spectrometers based on adaptive deep diffractive neural networks (a-D${}^2$NNs). Specifically, we introduce and characterize two-layer diffractive devices with engineered angular dispersion that focus and steer broadband incident radiation along predefined focal trajectories with desired bandwidth and $5$ nm spectral resolution. Moreover, we systematically study the focusing efficiency of two-layer devices with side length $L=100\mu \mathrm{m}$ and focal length $f=300\mu \mathrm{m}$ across the visible spectrum and we demonstrate accurate reconstruction of the emission spectrum from a commercial superluminescent diode. The proposed a-D${}^2$NNs design method extends the capabilities of efficient multi-focal diffractive optical devices to include single-shot focusing spectrometers with customized focal trajectories for applications to ultracompact multispectral imaging and lensless microscopy.