# Computation of High-Order Electromagnetic Field Derivatives with FDTD and the Complex-Step Derivative Approximation

## TL;DR

This paper presents a novel method integrating the complex-step derivative approximation into FDTD simulations to accurately compute electromagnetic field derivatives of any order with respect to design parameters, enhancing sensitivity analysis and optimization.

## Contribution

The paper introduces a finite-difference free approach using complex-step derivatives within FDTD for precise, high-order electromagnetic field derivatives in a single simulation.

## Key findings

- Accurate derivatives for small parameter perturbations
- Enables sensitivity analysis and optimization
- Applicable to multi-parametric electromagnetic modeling

## Abstract

This paper introduces a new approach for the computation of electromagnetic field derivatives, up to any order, with respect to the material and geometric parameters of a given geometry, in a single Finite-Difference Time-Domain (FDTD) simulation. The proposed method is based on embedding the complex-step derivative (CSD) approximation into the standard FDTD update equations. Being finite-difference free, CSD provides accurate derivative approximations even for very small perturbations of the design parameters, unlike finite-difference approximations that are prone to subtractive cancellation errors. The availability of accurate approximations of field derivatives with respect to design parameters enables studies such as sensitivity analysis of multiple objective functions (as derivatives of those can be derived from field derivatives via the chain rule), uncertainty quantification, as well as multi-parametric modeling and optimization of electromagnetic structures. The theory, FDTD implementation and applications of this technique are presented.