High-accuracy eigenmode computation in slab waveguides with arbitrary material profiles: a transfer matrix approach.

Area of Science:

• Computational photonics
• Optical engineering
• Applied mathematics

Background:

• Solving complex modes in waveguides with arbitrary material profiles is a significant challenge.
• Traditional analytical methods for Maxwell's equations in optical waveguides suffer from complex formulations and numerical errors.

Purpose of the Study:

• To present an analytical unified mathematical framework for one-dimensional waveguide problems.
• To develop a hybrid root-finding framework for efficient computation of complex roots.

Main Methods:

• The local higher-order transfer matrix method is employed, compatible with Perfectly Matched Layer boundary conditions.
• A hybrid root-finding framework is introduced for computing complete complex roots of nonlinear systems.

Main Results:

• The proposed method seamlessly adapts to diverse boundary conditions for computing guided, leaky, and Berenger modes.
• The hybrid solver demonstrates exceptional computational speed for nonlinear systems.
• The framework is fundamentally generalizable to arbitrary nonlinear equations.

Conclusions:

• The presented analytical framework offers an elegant and efficient solution for complex mode analysis in waveguides.
• The hybrid root-finding solver provides a significant advancement in computational speed and generalizability for nonlinear problems in photonics.