Machine learning (ML) techniques, such as neural networks, have emerged as powerful tools for the inverse design of nanophotonic structures. However, this innovative approach suffers some limitations. A primary one is the nonuniqueness problem, which can prevent ML algorithms from properly converging because vastly different designs produce nearly identical spectra. Here, we introduce a mixture density network (MDN) approach, which models the design parameters as multimodal probability distributions instead of discrete values, allowing the algorithms to converge in cases of nonuniqueness without sacrificing degenerate solutions. We apply our MDN technique to inversely design two types of multilayer photonic structures consisting of thin films of oxides, which present a significant challenge for conventional ML algorithms due to a high degree of nonuniqueness in their optical properties. In the 10-layer case, the MDN can handle transmission spectra with high complexity and under varying illumination conditions. The 4-layer case tends to show a stronger multimodal character, with secondary modes indicating alternative solutions for a target spectrum. The shape of the distributions gives valuable information for postprocessing and about the uncertainty in the predictions, which is not available with deterministic networks. Our approach provides an effective solution to the inverse design of photonic structures and yields more optimal searches for the structures with high degeneracy and spectral complexity.
Keywords: artificial neural networks; deep learning; inverse design; multilayer structures; nanophotonics; nonuniqueness.