{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "Sensitivity Analysis via Perturbation Theory\n", "--------------------------------------------\n", "\n", "For a given mode of the ring resonator, it is often useful to know how sensitive the resonant frequency $\\omega$ is to small changes in the ring radius $r$ by computing its derivative $\\partial\\omega/\\partial r$. The gradient is also a useful quantity for shape optimization because it can be paired with fast iterative methods such as [BFGS](https://en.wikipedia.org/wiki/Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm) to find local optima. The \"brute-force\" approach for computing the gradient is via a finite-difference approximation requiring *two* simulations of the (1) unperturbed [$\\omega(r)$] and (2) perturbed [$\\omega(r+\\Delta r)$] structures. Since each simulation is potentially costly, an alternative approach based on semi analytics is to use [perturbation theory](https://en.wikipedia.org/wiki/Perturbation_theory) to obtain the gradient from the fields of the unperturbed structure. This involves a single simulation and is often more computationally efficient than the brute-force approach although some care is required to set up the calculation properly. (More generally, [adjoint methods](https://math.mit.edu/~stevenj/18.336/adjoint.pdf) can be used to compute any number of derivatives with a single additional simulation.)\n", "\n", "[Pertubation theory for Maxwell equations involving high index-contrast dielectric interfaces](http://math.mit.edu/~stevenj/papers/KottkeFa08.pdf) is reviewed in Chapter 2 of [Photonics Crystals: Molding the Flow of Light, 2nd Edition (2008)](http://ab-initio.mit.edu/book/). The formula (equation 30 on p.19) for the frequency shift $\\Delta \\omega$ resulting from the displacement of a block of $\\varepsilon_1$-material towards $\\varepsilon_2$-material by a distance $\\Delta h$ (perpendicular to the boundary) is:\n", "\n", "