PhysicalBounds

Overview

This package computes performance bounds for physical design problems for the paper Bounds on Efficiency Metrics in Photonics. These design problems are modeled using WaveOperators.jl.

For usage and examples, check out the documentation.

Physical constraints

We assume that all fields z must satisfy the Helmhotlz equation, which we will write as

z + G diag(θ)z = b

where θ are the design parameters (e.g., permittivities), G is a discretization of Green's function, and b is a term that comes from the excitation. WaveOperators.jl builds these matrices/vectors automatically and returns an IntegralEquation, which contains all physical information about the system.

Bounds

We assume that the permittivities θ lie in one of two constraint sets Θ, which can be specified by the user:

• θᵢ ∈ {-1, 1} gives a discrete choice (e.g., silicon or air) at each point in the design region

• -1 ≦ θᵢ ≦ 1 allows us to vary the permittivity between an upper and lower bound at each point in the design region

Note that we can assume WLOG θ_max = 1 and θ_min = -1, since wave equation can always be scaled such that this is true.

Efficiency bounds

The focusing efficiency and related objective functions can be modeled as

f(z) = (zᵀPz + 2pᵀz + r) / (zᵀQz + 2qᵀz + s).

Using the package

Several examples can be found in the documentation.

Citing

If you find this package useful in your work, please cite our paper

@article{angeris2022bounds,
  title={Bounds on Efficiency Metrics in Photonics},
  author={Angeris, Guillermo and Diamandis, Theo and Vu{\v{c}}kovi{\'c}, Jelena and Boyd, Stephen},
  journal={arXiv preprint arXiv:2204.05243},
  year={2022}
}