Circulant preconditioning in the volume integral equation method for silicon photonics

Area of Science:

• Photonics
• Computational Electromagnetics
• Numerical Analysis

Background:

• The volume integral equation (VIE) approach is a powerful simulation tool for silicon photonics.
• Iterative solvers for VIE face slow convergence issues in high-frequency, strong-contrast photonic problems.
• Uniform discretization leads to Toeplitz matrices, suitable for circulant approximation preconditioning.

Purpose of the Study:

• To develop an effective circulant preconditioning strategy for VIE in silicon photonics.
• To address slow convergence of iterative solvers for complex photonic structures.
• To enable efficient numerical exploration of prototype photonic devices.

Main Methods:

• Applied the multilevel circulant preconditioner of Chan and Olkin.
• Generalized the strategy using geometrical partitioning and homogenization for complex structures.
• Introduced a novel memory reduction technique for large-scale simulations.

Main Results:

• Demonstrated ideal performance for unidirectional propagation in uniform waveguides.
• Successfully generalized preconditioning for Bragg gratings, directional couplers, and disk resonators.
• Achieved manageable memory footprint for extremely long structures.

Conclusions:

• The preconditioned VIE approach offers a fast and highly useful method for silicon photonics simulations.
• The developed strategy effectively overcomes convergence limitations for diverse photonic devices.
• This work facilitates the numerical exploration and design of advanced photonic devices.