Gradient-based inverse design in photonics has already achieved remarkable results in designing small-footprint, high-performance optical devices. The adjoint variable method, which allows for the efficient computation of gradients, has played a major role in this success. However, gradient-based optimization has not yet been applied to the mode-expansion methods that are the most common approach to studying periodic optical structures like photonic crystals. This is because, in such simulations, the adjoint variable method cannot be defined as explicitly as in standard finite-difference or finite-element time- or frequency-domain methods. Here, we overcome this through the use of automatic differentiation, which is a generalization of the adjoint variable method to arbitrary computational graphs. We implement the plane-wave expansion and the guided-mode expansion methods using an automatic differentiation library, and show that the gradient of any simulation output can be computed efficiently and in parallel with respect to all input parameters. We then use this implementation to optimize the dispersion of a photonic crystal waveguide, and the quality factor of an ultra-small cavity in a lithium niobate slab. This extends photonic inverse design to a whole new class of simulations, and more broadly highlights the importance that automatic differentiation could play in the future for tracking and optimizing complicated physical models.

中文翻译：

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通过自动微分逆向设计光子晶体

光子学中基于梯度的逆向设计已经在设计小尺寸、高性能光学器件方面取得了显著成果。允许有效计算梯度的伴随变量方法在这一成功中发挥了重要作用。然而，基于梯度的优化尚未应用于模式扩展方法，这是研究光子晶体等周期性光学结构的最常用方法。这是因为，在此类模拟中，不能像标准有限差分或有限元时域或频域方法那样明确定义伴随变量方法。在这里，我们通过使用自动微分来克服这个问题，这是伴随变量方法对任意计算图的推广。我们使用自动微分库实现了平面波扩展和导模扩展方法，并表明可以有效地计算任何模拟输出的梯度，并且可以相对于所有输入参数并行计算。然后，我们使用此实现来优化光子晶体波导的色散以及铌酸锂板中超小腔的品质因数。这将光子逆向设计扩展到了一个全新的模拟类别，并更广泛地强调了自动微分在未来跟踪和优化复杂物理模型方面的重要性。然后，我们使用此实现来优化光子晶体波导的色散以及铌酸锂板中超小腔的品质因数。这将光子逆向设计扩展到了一个全新的模拟类别，并更广泛地强调了自动微分在未来跟踪和优化复杂物理模型方面的重要性。然后，我们使用此实现来优化光子晶体波导的色散以及铌酸锂板中超小腔的品质因数。这将光子逆向设计扩展到了一个全新的模拟类别，并更广泛地强调了自动微分在未来跟踪和优化复杂物理模型方面的重要性。