Is tidy3d's adjoint method in time-domain or frequency-domain?

[Guizhixing]

Hi, all.
I recently learned the lecture of Adjoint Method in learning center. The derivation of adjoint variable method i.e. AVM is starting from the solution of Maxwell’s equations in the frequency domain. In lecture, the objective function, the ajoint current source and gradient equation are expressed in frequency domain.
Tidy3d is FDTD based program. As far as my knowledge of the FDTD based AVM，it considers the time marching scheme in FDTD as a dynamic system and uses time reversal to simulate the adjoint system. The objective function is restricted to a time-domain integral function. May I ask how tidy3d gets the gradient function by transfering frequency domain to time domain, or by any other means?

[tylerflex]

Hi @Guizhixing

You are correct, in the lectures we present the adjoint method in the frequency-domain. The reason is that it's much easier to reason about the adjoint method when the electric field e is written as the solution to a linear system Ae=b. This is a pretty common and natural way to write frequency-domain Maxwell's equations (although you can write time domain problems like this too).

In fact, you can extend the same adjoint analysis to the time domain, with a bit more complication. There is a paper we wrote a few years ago that derives everything for time and frequency domain here: https://arxiv.org/abs/1908.10507. The basic idea is the same, except technically the adjoint FDTD is run backwards in time and also the adjoint source is time-dependent, among other details. The time domain adjoint often also suffers from high memory requirements due to the need to store the entire time series info.

In tidy3d's adjoint plugin, we try to take the best of both worlds approach: We run FDTD in the time-domain, but use the td.FieldMonitor to store the electromagnetic fields in the frequency-domain at the frequencies of interest in the objective function. These fields are computed by performing a running Fourier transform of the time series, which is memory efficient. Then, we use the frequency-domain adjoint formalism to compute the gradients.

Hopefully that makes sense but let me know if you need clarification or have other questions!

[Guizhixing]

I see how the Tidy3D works, thank you very much for your replies.