A coupled-mode theory for two-dimensional exterior Helmholtz problems based on the Neumann and Dirichlet normal mode expansion

by   Kei Matsushima, et al.

This study proposes a novel coupled-mode theory for two-dimensional exterior Helmholtz problems. The proposed approach is based on the separation of the entire space R2 into a fictitious disk and its exterior. The disk is allocated in such a way that it comprises all the inhomogeneity; therefore, the exterior supports cylindrical waves with a continuous spectrum. For the interior, we expand an unknown wave field using normal modes that satisfy some auxiliary boundary conditions on the surface of the disk. For the interior expansion, we propose combining the Neumann and Dirichlet normal modes. We show that the proposed expansion sacrifices L2 orthogonality but significantly improve the convergence. Finally, we present some numerical verifications of the proposed coupled-mode theory.



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1 Introduction

Coupled-mode theories have been used for investigating wave propagation and scattering in open quantum, electromagnetic, and acoustic systems (haus1991coupled). The separation of the scattering process into resonance and radiation is the primary concept of coupled-mode theories. This approach enables us to understand the mechanism of various anomalous wave phenomena that occur during the resonant-scattering process, such as bound states in the continuum (hsu2016bound) and exceptional points in non-Hermitian physics (miri2019exceptional).

The analysis of wave propagation in a waveguide with an inhomogeneity or one attached to a resonator is a typical application of coupled-mode theories. We call such systems waveguide-resonator systems. If an unknown wave field in the vicinity of a resonator is expanded using a complete set of basis functions, called an interior modal expansion, its coefficients are obtained from a solution of a linear algebraic system. The linear system is called a coupled-mode equation and is obtained by connecting the interior modal expansion and radiating fields, expressed in terms of plane waves. In quantum mechanics, Pichugin et al. (pichugin2001effective) formulated a waveguide-resonator system using normal modes of a closed system generated by imposing the Neumann or Dirichlet condition on the interface between the resonator and waveguide. Similar approaches have recently been proposed in the field of acoustics (maksimov2015coupled; lyapina2015bound; tong2017modal).

The same concept can be applied to exterior scattering problems, where a bounded inhomogeneity exists in the homogeneous background medium , where

is the dimension of the space. However, the suitable basis functions for exterior scattering problems remain unclear. The most natural choice would be quasinormal modes, which are the eigenmodes of the entire open system. Due to the intrinsic radiation loss, a corresponding eigenvalue (eigenfrequency) has a nonzero imaginary part. This imaginary part represents the linewidth of the corresponding resonance and induces Lamb’s exponential catastrophe

(lamb1900peculiarity). Although many studies have been devoted to quasinormal mode expansions, such as (hamam2007coupled; cador2009theory; muljarov2011brillouin; dai2012generalized; ruan2012temporal; doost2013resonant; doost2014resonant; hsu2014theoretical; alpeggiani2017quasinormal; weiss2018how), it is inconvenient for describing a coupled-mode relation because the completeness of quasinormal modes is guaranteed only inside an inhomogeneity (leung1994completeness). Furthermore, the computation of quasinormal modes is more difficult than solving standard Hermitian eigenvalue problems due to the exponential catastrophe.

In this study, we propose a novel approach based on normal mode expansion with auxiliary boundary conditions to develop a coupled-mode theory for an exterior Helmholtz scattering problem instead of quasinormal mode expansions. The underlying concept of the proposed method is to separate the entire space into a ball that encloses all inhomogeneity (scatterer) and its exterior . In the exterior, the incidence and radiation can be expressed in terms of cylindrical/spherical wave functions. Inside the ball

, we use Laplacian eigenfunctions in

to expand the unknown solution. The main difficulty here is that, unlike the waveguide-resonator systems, the physical boundary of the scatterer does not correspond to any portion of the surface of the resonator. This is problematic because, even if the completeness is guaranteed, the mismatch between the auxiliary boundary condition on and the actual behavior of the unknown solution causes extremely slow convergence of the interior modal expansion. For example, if we impose the homogeneous Neumann boundary condition for the interior normal modes on , the convergence would be prohibitively slow unless the solution satisfies the same Neumann boundary condition on , which is not the case.

One solution is to use a Robin boundary condition as the auxiliary boundary condition on instead of the Neumann or Dirichlet boundary condition (xu2010hybrid). However, to simulate the radiation loss on , the impedance coefficient for the Robin condition must be complex and dependent on the operating frequency, which is not computationally preferred. Another approach is using both the Neumann and Dirichlet normal modes for the basis functions (beckemeyer1978boundary; maurel2014improved). This approach would enable us to achieve rapid convergence even if the solution does not satisfy the Neumann or Dirichlet boundary condition.

This study adopts the latter approach and develop a novel coupled-mode theory for the 2-D exterior Helmholtz problem. We briefly introduce a coupled-mode theory for a waveguide-resonator system using the variational formulation of the Helmholtz equation before describing the proposed approach. Subsequently, we explain the proposed coupled-mode theory using the same variational formulation. Finally, we illustrate some numerical examples of the scattering analysis to confirm the validity of the proposed method.

2 waveguide-resonator system

Figure 1: Two-dimensional waveguide-resonator system. The semi-infinite planar waveguide is attached to a resonator (cavity) .

In this section, we consider the Helmholtz equation


in the planar semi-infinite waveguide and resonator (cavity) filled with acoustic medium whose mass density and bulk modulus are and , respectively, as shown in Figure 1. For simplicity, we assume that the medium is homogeneous within , i.e., and for some and . The interface is assumed to be a straight line segment , where is the thickness of the waveguide placed parallel to the axis. Furthermore, we impose the Neumann boundary condition


on the entire surface, where

is the unit normal vector.

2.1 Coupled-mode theory

Following the underlying concept of coupled-mode theories, we express a solution using two different series expansions: exterior and interior modal expansions. The exterior modal expansion represents incident and radiating fields, while the interior one represents a resonance with no energy leakage.

2.1.1 Exterior modal expansion

Inside the semi-infinite waveguide , we have the following well-known modal expansion with coefficients :


where are the guided waves propagating in the positive/negative directions and defined by


with transverse modal shape


and wavenumber , where is the Kronecker delta.

2.1.2 Interior modal expansion

For the resonator , we aim to establish an expansion with coefficients :


where are appropriate basis functions independent of the operating angular frequency . To consider the inhomogeneity in and boundary condition on , we define as the real-valued normal modes that satisfy the following eigenvalue problem:


where the eigenvalues , indexed by , are ordered such that . The Sturm–Liouville theory shows that the eigenmodes form a complete and orthogonal set in the sense. Furthermore, the series creftype 6 inherits the Neumann boundary condition from creftype 8, which contributes to a rapid convergence. However, the boundary condition creftype 9 is unassociated with the original problem creftypeplural 2 and 1. This auxiliary condition is not a unique choice and can be replaced with the homogeneous Dirichlet or Robin boundary condition (pichugin2001effective).

2.1.3 Coupled-mode equation

Now, we are assuming that a solution is written as


with unknown coefficients , , and . Considering radiation and incidence, we assume that one linear relation between the sequences and exists in advance. For example, in the case of no incidence from the waveguide, we obtain .

To develop two more linear equations, we use the following assumptions:

  1. The solution satisfies the Helmholtz equation in at the operating angular frequency in the weak sense, i.e.,


    where the symbol (resp. ) denotes the trace from the interior (resp. exterior).

  2. is continuous across the interface in a weak sense, i.e.,

  3. is continuous across the interface in a weak sense, i.e.,


First, let us consider creftype 11. Substituting the interior modal expansion creftype 6 into creftype 11, we obtain


Here, we have used the following variational equation for at :


which is deduced from the eigenvalue problem creftypeplural 9, 8 and 7. Using the orthonormality written as


we further simplify creftype 14 as follows:


Applying the second assumption creftype 12 to creftype 17, it follows that


where the matrix is defined by


Another equation is derived from the third assumption creftype 13. After a simple calculation, we obtain


Combining creftypeplural 20 and 18, we obtain the following coupled-mode equation:


where and are the diagonal matrices defined by


The coupled-mode equation creftype 21, originally derived by (maksimov2015coupled), solves the unknown coefficients , when the incident wave is given. In quantum mechanics, the matrix is called an effective non-Hermitian Hamiltonian. The non-Hermiticity is the natural consequence of the radiation loss through the waveguide.

3 Exterior Helmholtz problem

Figure 2: Scattering by an object in the two-dimensional space .

In the previous section, we established that the waveguide-resonator system can be separated into interior and exterior systems based on the variational formulation. In this section, we propose a similar approach for analyzing exterior scattering problems.

As in the previous section, we consider the following Helmholtz problem with angular frequency :


where is a given incident wave. The material parameters and are homogeneous in the exterior of a bounded domain , i.e., we assume that and have compact supports on . In the homogeneous exterior, the wavenumber is given by , where is the speed of sound in ambient space.

We aim to develop a numerical solution to the exterior Helmholtz problem creftypeplural 24 and 23. Based on the underlying concept of the coupled-mode approach, we separate the entire system into resonant and radiating parts. To accomplish this, we allocate a fictitious disk centered at the origin with a radius that encloses the scatterer as shown in Figure 2. The choice of is arbitrary as long as it satisfies .

3.1 Coupled-mode theory

3.1.1 Exterior of the fictitious disk

It is well-known that a solution of the Helmholtz equation in can be written in the following form:


with coefficients (), where the functions and are given by the Hankel functions (resp. ) of the first (resp. second) kind and order as


with .

For example, when the incident wave is a plane wave, we have


where the unit vector is the direction of propagation.

As the scattered wave satisfies the radiation condition creftype 24, there exists a unique sequence such that


For sufficiently large , we obtain the following plane-wave expansion (ammari2018mathematical):


The function is called the far-field pattern and calculated from the coefficients as


which gives the scattering cross section as


When the medium is lossless and illuminated by an incident plane wave , we have the following optical theorem:


3.1.2 Interior of the fictitious disk

We want to determine an interior expansion for a solution within the fictitious disk , i.e.,


where are unknown coefficients. The main issue here is how to choose the basis functions .

In analogy with the waveguide-resonator case, discussed in Section 2.1.2, we consider the following eigenvalue problem:


where is a linear operator, and is the eigenvalue corresponding to . The Neumann boundary condition as in Section 2.1.2 is the simplest choice because it enables us to use the completeness and orthogonality of the normal modes. Unlike the waveguide-resonator case, the auxiliary boundary condition creftype 36 does not describe the true behavior of a solution to the original scattering problem creftypeplural 24 and 23. This mismatch results in an extremely slow convergence of the interior modal expansion.

Since a solution should have both the nonzero Neumann and Dirichlet data on , we propose combining the Neumann and Dirichlet normal modes as follows:


where and denote unknown coefficients. Here, the normal modes and satisfies the interior Neumann problem


and Dirichlet problem


with eigenvalues and , respectively.

Since the orthogonality does not hold between and , this approach costs additional computational effort to expand a given function. However, in most cases, this additional cost is trivial because the computation of normal modes is far more time-consuming than solving a linear system for the non-orthogonal expansion.

3.1.3 Coupling between the exterior and interior expansions

Here, we summarize the interior and exterior modal expansion for the exterior Helmholtz problem as follows:


As we did in Section 3.1.3, we assume the following three conditions to derive linear relations for the coefficients , , and :

  1. The solution satisfies the Helmholtz equation in at the operating angular frequency in the weak sense, i.e.,


    for all and .

  2. is continuous across the interface in a weak sense, i.e.,


    for all and .

  3. is continuous across the interface in a weak sense, i.e.,


From the first assumption creftype 43, we have the following equations:


where the matrices and are defined as follows:


Here, we used the following identities:


with the following orthonormalities:


Using the second assumption creftype 44, the last term in the right-hand side of creftype 46 turns into


where is defined as


From the third assumption creftype 45, we obtain the following equation:


Combining creftypeplural 57, 55, 47 and 46, we finally obtain a coupled-mode equation as


where the matrix and are defined as


Instead of solving the coefficients and , we can develop the following equation:


where is called the scattering matrix and written as


with diagonal matrix defined by




4 Numerical examples

4.1 Interior modal expansion of a given function

First, for a given function , we check the convergence of the interior modal expansion creftype 37. In this subsection, we choose and compute the relative error of the expansion against the exact values. Since is independent of , it suffices to consider the monopolar Neumann eigenmodes and Dirichlet eigenmodes , where and are the zeros of and , respectively. The zeros are calculated by Wolfram Mathematica (Mathematica) and listed in Table 1.

Table 1: First 10 zeros of and .

Figure 4 shows the relative error for different wavenumber and fixed and . From this result, we see that the large wavenumbers degrade the accuracy of the expansion. However, the expansion provides a good approximation even for the high-frequency regime when sufficiently large and are provided.

To confirm the convergence, we fix and plot the relationship between the relative error and truncation number and in Figure 3. Since the function doen not satisfies the Neumann condition or Dirichlet condition at , the pure Neumann expansion () and Dirichlet expansion give poor convergence rates. However, by combining the Neumann and Dirichlet expansions, we have a faster convergence until the error reaches approximately .

Figure 3: Relative error for various .
Figure 4: Relative error for fixed and various and .

4.2 Minnaert resonance

Next, we check the performance of the proposed coupled-mode theory for the simplest case where the scatterer is a disk of radius . To observe a strong resonance, we assume that the disk is an air bubble immersed in water, i.e., the material parameters are given by


Using the continuity of and at the interface , we obtain that is a diagonal matrix whose entries are written as


where is the acoustic impedance, and is the wavenumber in .

For the Neumann eigenvalue problem, the eigenvalues and eigenmodes are characterized by


whose coefficients , , and are nontrivial solutions of


where is the Bessel function of the second kind and order .

For the Dirichlet eigenvalue problem, we have the same expansion creftype 68; however, the coefficients are defermined using the following linear system:


Here, we use the rotational symmetry of to limit the consideration to the monopolar mode . In this case, the scattering matrix is simply a scalar value . We compute the value of using the coupled-mode theory creftype 62 and compare it to the exact expression creftype 67.

Figure 5: Spectrum of computed by the coupled-mode theory creftype 62 and exact expression creftype 67.
Table 2: First 10 eigenvalues for the coupled-mode analysis.

For various parameters and