Non density-based sensitivity analysis using adjoint solver

[alexthunderk]

Hi meep community,

Just recently, I started using meep and it’s a great tool. Thanks for developing and maintaining it.
I came across the adjoint optimization and wanted to use it for a sensitivity analysis of design parameters, e.g., period, fill factor etc. of a grating coupler. Is that possible in meep? I found examples only for topology optimization using a material grid and haven’t been able to adapt it so far.
I created a class with an update-method so that it can be used as the design variables. However, the optimizer doesn’t update the parameters. I guess that’s because the derivative w.r.t. the parameters is not defined. Is that correct and is there a way to adapt the adjoint optimizer for this use case?

[lucasgrjn]

Hi,

I am actually interested in the same thing as you. Indeed, I need to reproduce the results of this article https://opg.optica.org/prj/fulltext.cfm?uri=prj-8-4-528&id=429433 (Inverse design of digital nanophotonic devices using the adjoint method) but unfortunately, until now, I have not been able to... (BTW, if I succeed, it can make a beautiful example).

If you succeed, please, share your solution !

A possible way of resolution for your issue will be to use the library autograd or jax. It will allow your simulation to include a gradient.

[smartalecH]

Is that correct and is there a way to adapt the adjoint optimizer for this use case?

Sure, this is possible. But there are a few caveats concerning how you implement this within the current framework.

Note that to obtain sensitivity information w.r.t. the parameters you're describing, you need to use a levelset. Technically, the adjoint solver was meant to work with density-based methods (which are starkly different than levelsets). But we recently added some features that hybridize the two frameworks, provided the following conditions are met:

1. The parameters of interest are "mapped" to an underlying grid (essentially a density-grid). For example, your "mapping function" could simply look like

import autograd.numpy as npa
x = npa.linspace(0,L,N) # where N is the number of MaterialGrid DOF, and L is grating length
def fx(Λ):
    return npa.sin(2*npa.pi/Λ*x) # where Λ is grating period

where the output of fx corresponds to the MaterialGrid design variables.

2. This function mapping must be "smooth" (you can't "threshold" or "project" yet).
3. the MaterialGrid's beta parameter is set to beta=np.inf. In other words, the projection happens "under the hood".

The adjoint solver will provide the gradient, ∂F/∂fx (where F is your objective function). You can then backpropagate the gradient through the mapping function to obtain the final sensitivity ∂F/∂Λ (or any other additional parameters you may want).

[stevengj]

In particular, for a given shape (e.g. a polygon), if you define the material grid to be 0.5 + a signed distance function, and then set β=infinity, then the subpixel smoothing and the gradients will work.

[alexthunderk]

Hi,

Thanks a lot for your answers!
I’m still stuck on this topic and I don’t really know who to implement that properly.
I attached the code how I simulate the grating coupler currently (just looping over the parameters). If the simulation is correct, that would be a good tool to simulate regular grating couplers (In the attached file, I got a strange error while plotting the simulation setup, but the simulation seems to work).
However, I would like to change and optimize the gaps and grating lines individually, which is why I thought that the adjoint method would be beneficial. I don’t know if that would work with a function like you proposed, @smartalecH.
If the optimization of the gaps, grating lines, fill factor and BOX thickness works, it would make a nice example, as @Dj1312 said, and a great tool.

FGC_2D_fiber.txt

[smartalecH]

However, I would like to change and optimize the gaps and grating lines individually

Take a look at #2011 (idea 3). Yes, it's possible to do this too. But the constraints listed above must still be met.

[alexthunderk]

Thanks again!
Looking forward to the tutorial

[wllshr123]

Hello! I'm a novice user of meep. You said you have examples only for topology optimization using a material grid. Can you share it with me? Thank you very much! Or do you have other optimization examples of vertical coupler with meep?
Thanks a lot!
I'm looking forward to your reply!

[smartalecH]

Please keep questions about another discussion (#2393) within that discussion. Similarly, please don't spam other discussions in hopes of increasing engagement.