adjoint for ldos

[mochen4]

Implements the adjoint for ldos. Closes #2026 and closes #1401.

[mochen4]

Based on my testing so far, the ldos adjoint works for Gaussian sources, but not for EigenModeSource. The following issues are observed:

1. In 2D, if the polarization of the EigenModeSource is specified to be TM, everything is fine; however, when the polarization is TE, finite different gradients agree with adjoint gradients at some resolutions (e.g. 15 and 25 in my test case), but differ by a factor at other resolutions (factor of ~0.8 at resolution=20, factor of ~0.9 at resolution =30). It does look like increasing resolution will eventually lead to complete agreement.

2. If the EigenModeSource has mp.ALL_COMPONENT, then the adjoint gradients still differ by some factor from the finite different gradients. Such factor seems to be proportional to 1/resolution^2, but it may have different sign as the resolution changes.

3. As the case for FourierFields and Near2farFields, the adjoint simulation requires one more sign flip for adjoint electric sources than adjoint magnetic sources. This also seems to be the case for LDOS with single component, but not for mp.ALL_COMPONENT.

[stevengj]

I think the derivative of the work done by an eigenmode source (or any Equivalence Principle) source should be zero (with respect to material perturbations that don't intersect the source plane). So maybe you are just looking at discretization errors (and it's not surprising that they don't agree)?

Would be nice to check this, e.g. compute the power P for different perturbations of the material grid and verify that |δP/P| << 1

[mochen4]

1. In 2D, if the polarization of the EigenModeSource is specified to be TM, everything is fine; however, when the polarization is TE, finite different gradients agree with adjoint gradients at some resolutions (e.g. 15 and 25 in my test case), but differ by a factor at other resolutions (factor of ~0.8 at resolution=20, factor of ~0.9 at resolution =30). It does look like increasing resolution will eventually lead to complete agreement.

Correction: actually, the gradients are consistent using electric components but not magnetic components, regardless of the polarization.

I think the derivative of the work done by an eigenmode source (or any Equivalence Principle) source should be zero (with respect to material perturbations that don't intersect the source plane). So maybe you are just looking at discretization errors (and it's not surprising that they don't agree)?

Would be nice to check this, e.g. compute the power P for different perturbations of the material grid and verify that |δP/P| << 1

I found that the gradients with EigenModeSource is around the same magnitude as with GaussianSource, and in both cases, gradients get smaller with larger resolutions. Since the gradients with GaussianSource make sense, I think the gradients with EigenmodeSource also needs to make sense.

3. As the case for FourierFields and Near2farFields, the adjoint simulation requires one more sign flip for adjoint electric sources than adjoint magnetic sources. This also seems to be the case for LDOS with single component, but not for mp.ALL_COMPONENT.

I checked that using mp.ALL_COMPONENTS was indeed the same as using the corresponding individual components together; however, it was not consistent with the running simulations with individual components separately. Specifically, at eig_band = 2, LDOS with mp.Hz alone is 1.705 and with mp.Ey alone is 0.2, but for mp.ALL_COMPONENTS is 0.236.

[smartalecH]

I found that the gradients with EigenModeSource is around the same magnitude as with GaussianSource, and in both cases, gradients get smaller with larger resolutions. Since the gradients with GaussianSource make sense, I think the gradients with EigenmodeSource also needs to make sense.

By GaussianSource, do you actually mean GaussianBeamSource? Note that the EigenModeSource is a spatial definition, while the GaussianSource is a time profile. As you know, an EigenModeSource can also be defined by a GaussianSource pulse...

[mochen4]

By GaussianSource, do you actually mean GaussianBeamSource? Note that the EigenModeSource is a spatial definition, while the GaussianSource is a time profile. As you know, an EigenModeSource can also be defined by a GaussianSource pulse...

Sorry for the confusion. By GaussianSource I mean a plane wave with Gaussian as the time profile. By EigenModeSource I mean an eigenmode as the space profile and Gaussian as the time profile.

[stevengj]

Looks good! Just needs some documentation?