Acoustic Waveguide Eigenmode Solver Based on a Staggered-Grid Finite-Difference Method

Nathan Dostart,Miloš A Popović,Yangyang Liu

doi:10.1038/s41598-017-17511-x

Nathan Dostart, Miloš A Popović + Show 1 more

Open Access

https://doi.org/10.1038/s41598-017-17511-x

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Abstract

A numerical method of solving for the elastic wave eigenmodes in acoustic waveguides of arbitrary 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 (field and frequency) of the vector-field elastic wave equation with a given propagation constant. 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 that in 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 across four orders of magnitude in frequency in several simple geometries and comparing the results to analytical solutions where available or to numerical solvers based on more computationally expensive methods. The solver is utilized to demonstrate a novel type of leaky-guided acoustic wave that couples simultaneously to two independent radiation channels (directions) with different polarizations – a ‘bi-leaky’ mode.

Highlights

Previous work has predominantly focused on the full-wave simulation of source-driven responses in 2D and 3D domains. 3D solvers of this kind have been developed for anisotropic, heterogeneous domains using both finite-difference[26] and finite-element[27] methods (FDM, FEM) and are the predominant method for designing acoustic devices in the GHz frequency range
FEM is more sophisticated and general than FDM, FDM on a uniform grid has a number of strengths when it comes to the design of nano-scale photonic, and we believe by extension phononic, devices and circuits
We develop and implement an acoustic waveguide mode solver which solves the linear isotropic elastic wave equation based on FDM which is comparably accurate to 3D FEM and more computationally efficient

Summary

OPEN Acoustic Waveguide Eigenmode

A numerical method of solving for the elastic wave eigenmodes in acoustic waveguides of arbitrary 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 (field and frequency) of the vector-field elastic wave equation with a given propagation constant. An efficient and computationally stable implementation is to remove the corresponding boundary-normal displacements un (since they are the only ones coincident with the boundary) from the matrix operator and solution vector, which is the method used here The same geometry is implemented in the commercial 3D solver with the beam length chosen equal to the simulated wavelength and Floquet periodic boundary conditions applied to the z-oriented boundaries Both solvers are used to find these four acoustic modes and corresponding frequencies across a wavelength range of 100 μm to 0.5 μm. For this case we halve (or quarter) the simulation domain (a) Waveguide

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