Proper LDOS enhancement problems

[mfschubert]

• 3D FMMAX implementation
• 2D version (shoot for sub-second cycle time)

[mfschubert]

@oskooi @stevengj

We discussed that problems where large LDOS enhancement is sought would be of interest, similar to the cavity design problem in the photonics-opt-testbed.

A 3D version could be adapted from the existing "photon extractor" challenge. In that problem, one also seeks LDOS enhancement, but the structure precludes large enhancements from being achieved. (Specifically, the material within which the dipole is embedded is unpatterned, which limits the enhancement increase). A modified version could allow patterning of the host material.

An implementation based on the extractor should take 10s of seconds per iteration. It would also be interesting to have 2D versions of the problem, which run in sub-second timescales.

[stevengj]

This is a good challenge problems for optimization problems, because LDOS becomes a very ill-conditioned function (specifically an ill-conditioned Hessian) of the geometry when Q becomes large, for reasons explained in Liang & Johnson (2013). Traditional optimization algorithms (CCSA, LBFGS) really struggle for Q ≳ 10⁴.

(Going to a high Q requires a sufficiently large design domain, which can be expensive in 3d.)

[stevengj]

Note also that the LDOS will diverge (in 3d or for the 2d Hz polarization) if the design is allowed to make an arbitrarily sharp tip at the location of the dipole source. You need to regularize the problem by either:

1. Imposing a minimum lengthscale constraint similar to https://github.com/NanoComp/photonics-opt-testbed/tree/main/cavity_design
2. Staying in 2d with the Ez polarization
3. or simply excluding the design from changing the materials within a given radius of the source