Inverse design for robust inference in integrated computational spectrometry

TL;DR

The paper tackles robust spectral inference in integrated computational spectrometry by inverse-designing the scattering medium with topology optimization guided by a nuclear-norm objective $||(F\sqrt{W})^+||_*$. It decouples scatterer design from the reconstruction algorithm and introduces a regularized Chebyshev interpolation approach for smooth spectra, aided by Gauss–Legendre quadrature for frequency discretization. Numerically, the inverse-designed devices show order-of-magnitude gains in noise robustness over random scatterers and improved reconstruction accuracy for finite sensor counts, highlighting the value of deterministic FOM-based design over fully end-to-end strategies. The work provides a general, deterministic framework that can extend to broader computational-imaging and inference problems, offering a complementary tool to end-to-end co-design methods and potential theoretical bounds on attainable performance.

Abstract

We propose an inverse-design approach for computational spectrometers in which the scattering media are topology-optimized to achieve better performance in inference of unknown spectra. Unlike traditional end-to-end approaches, our inverse design of the scattering media does not need a training set of spectra, a distribution of detector noise, or an inference algorithm. Our approach allows the selection of the inference algorithm to be decoupled from that of the scatterer. For smooth spectra, we additionally devise a regularized reconstruction algorithm based on Chebyshev interpolation, which yields higher accuracy compared with conventional methods in which the spectra are sampled at equally spaced frequencies or wavelengths with equal weights. Our approaches are numerically demonstrated via inverse design of integrated computational spectrometers and reconstruction of example spectra. The inverse-designed spectrometers exhibit significantly better performance in the presence of noise than their counterparts with random scatterers. Our method provides a useful complement to end-to-end co-design methods.

Figures (11)

• Figure 1: Sketch of computational spectrometry: forward process and inverse problem. In the forward process, input waves pass through a scatterer and form frequency-dependent patterns on sensors. If this dependence is calibrated beforehand, one may reconstruct unknown spectra from the signals recorded by the sensors, which is an inverse problem.
• Figure 2: Framework of design methods. (a) Inverse design of a nanophotonic structure. The frequencies at which the spectral--spatial mapping matrix (also called the measurement matrix in this paper) is computed are determined by the frequency range and the quadrature scheme. This mapping matrix and the quadrature weights determine the objective function. (b) Selection of a reconstruction algorithm. After the optimized scatterer is obtained, given prior knowledge of input spectra, one can select a reconstruction algorithm based on reconstruction error.
• Figure 3: Inverse design of an integrated spectrometer. (a) Structure of the spectrometer. This 2d device consists of an input waveguide, a wedge structure, a design region, and twelve output waveguides, with the solid material having a relative permittivity $\approx12$. The width of all the waveguides is 0.2 nm. Adjacent output waveguides are separated by 0.64 nm. (b) Transmittance of the optimized spectrometer at each output waveguide across the frequency range of interest. (c) Frequency-averaged transmittance of the optimized spectrometer at each output waveguide. The total transmittance is 58.1%. (d) Objective function ($\sum_j\sigma_j^{-1}$) and singular values of $F\sqrt{W}$ during optimization. (e) Condition number and total transmittance during optimization, computed from the $12\times7$ spectral--spatial mapping matrix.
• Figure 4: Comparison of performances of random and optimized structures. The horizontal and vertical axes represent the total transmittance and the condition number, respectively. The bottom right area in this coordinate system is associated with lower condition numbers, higher collection efficiencies, and hence better performance. The round green and black dots correspond to the optimized and random structures with the same minimum lengthscale, while the square/triangular symbols with light colors correspond to structures in which solid regions are dilated/eroded by 10 nm. The optimized structure and two random structures are illustrated on the middle panel. Their corresponding $12\times7$ spectral--spatial mapping matrices are on the right panel.
• Figure 5: Spectral reconstruction with four examples. The ground truth is plotted as thick black curves. The red dots represent reconstructed spectra at 7 equally spaced frequencies, chosen according the rectangular rule. The green dots represent reconstructed spectra at 7 frequencies chosen according to Gauss–Legendre nodes, while their Lagrange interpolating polynomials are plotted as green curves. The cyan curves represent reconstructed spectra as a linear combination of the first 7 Chebyshev polynomials of the first kind.