[Paper Review] Elastic Wave Eigenmode Solver for Acoustic Waveguides

Nathan Dostart, Yangyang Liu+1 more|arXiv (Cornell University)|Jul 19, 2017

Mechanical and Optical Resonators47 references3 citations

TL;DR

This paper presents a finite-difference eigenmode solver for elastic waves in acoustic waveguides with inhomogeneous cross-sections, using a staggered-grid scheme analogous to Yee's method in electromagnetics. It enables accurate simulation of guided, leaky, and radiative modes with perfectly matched layers (PMLs), validated against analytical solutions and 3D FEM solvers, and is optimized for integration with electromagnetic solvers in optomechanical device design.

ABSTRACT

A numerical solver for the elastic wave eigenmodes in acoustic waveguides of inhomogeneous cross-section is presented. Operating under the assumptions of linear, isotropic materials, it utilizes a finite-difference method on a staggered grid to solve for the acoustic eigenmodes of the vector-field elastic wave equation. Free, fixed, symmetry, and anti-symmetry boundary conditions are implemented, enabling efficient simulation of acoustic structures with geometrical symmetries and terminations. Perfectly matched layers are also implemented, allowing for the simulation of radiative (leaky) modes. The method is analogous to eigenmode solvers ubiquitously employed in electromagnetics to find waveguide modes, and enables design of acoustic waveguides as well as seamless integration with electromagnetic solvers for optomechanical device design. The accuracy of the solver is demonstrated by calculating eigenfrequencies and mode shapes for common acoustic modes in several simple geometries and comparing the results to analytical solutions where available or to numerical solvers based on more computationally expensive methods.

Motivation & Objective

• Address the need for efficient, accurate numerical tools to design chip-scale acoustic waveguides for optomechanical and RF MEMS applications.
• Overcome limitations of full 3D solvers by developing a 2+1D eigenmode formulation that reduces computational cost while preserving accuracy.
• Enable simulation of leaky modes and radiation losses using perfectly matched layers (PMLs) for realistic waveguide termination and energy leakage modeling.
• Facilitate seamless integration with electromagnetic solvers by using a Yee-like staggered grid, supporting co-design of acousto-optic and optomechanical devices.
• Provide a robust, second-order accurate finite-difference method for the vectorial elastic wave equation in isotropic, inhomogeneous media.

Proposed method

• Formulates the linear, isotropic elastic wave equation as a generalized eigenvalue problem with propagation constant β as input and modal frequency ω as output.
• Employs a staggered-grid finite-difference method to discretize the vectorial wave equation, preserving second-order accuracy for all field components.
• Implements free, fixed, symmetry, and anti-symmetry boundary conditions to exploit geometric symmetries and reduce simulation domain size.
• Introduces perfectly matched layers (PMLs) at domain boundaries to absorb outgoing waves and simulate leaky modes without reflections.
• Uses a 2+1D formulation to reduce the 3D waveguide problem to a 2D eigenvalue problem, focusing on cross-sectional modes at a given β.
• Applies the Yee-grid-like staggered arrangement to align with electromagnetic solvers, enabling multi-physics co-simulation in optomechanical systems.

Experimental results

Research questions

• RQ1 Can a finite-difference eigenmode solver be effectively adapted to the elastic wave equation in inhomogeneous, isotropic waveguides with arbitrary cross-sections?
• RQ2 How accurately can the solver compute eigenfrequencies and mode shapes for fundamental acoustic modes compared to analytical and 3D FEM solutions?
• RQ3 To what extent can PMLs in the solver accurately simulate radiation losses and leaky modes in acoustic waveguides?
• RQ4 Can the solver efficiently handle symmetric and anti-symmetric boundary conditions to reduce computational cost in symmetric waveguide geometries?
• RQ5 How well does the solver integrate with electromagnetic solvers using the Yee grid for optomechanical device co-design?

Key findings

• The solver achieves second-order accuracy in spatial discretization, validated through convergence studies and comparison with analytical solutions for simple geometries.
• Eigenfrequencies and mode shapes for quasi-Love and Rayleigh-like modes in silica-silicon waveguides were computed with high accuracy, matching analytical predictions within 0.1% error.
• The implementation of PMLs successfully absorbed outgoing waves, enabling accurate simulation of leaky modes with radiation losses, validated against a 3D FEM solver (COMSOL).
• Propagation loss in a leaky waveguide with variable silicon barrier widths showed excellent agreement between the mode solver and 3D FEM, confirming PML accuracy.
• Fourier analysis of the leaky mode’s radiated field revealed two dominant radiation components, corresponding to quasi-Love and quasi-Rayleigh waves in the silica slabs.
• The solver successfully simulated tunneling of acoustic energy across a silicon barrier into radiation modes, with wavefronts and displacement patterns consistent with theoretical expectations.