---
title: Modal Expansion Methods (MEMs)
url: https://www.emergentmind.com/topics/modal-expansion-methods-mems
type: topic
---

# Modal Expansion Methods (MEMs)

Searching arXiv for the specified MEM-related papers and closely related terminology.
arxiv_search(query="2204.04583 OR 1804.09246 OR 1311.3244 modal expansion methods quasimodal expansion", max_results=10, sort_by="relevance")
Retrieving bibliographic matches for the elastic, electromagnetic-power-based, and quasimodal expansion papers.
arxiv_search(query="id:2204.04583", max_results=5, sort_by="relevance")
arxiv_search(query="id:1804.09246", max_results=5, sort_by="relevance")
arxiv_search(query="id:1311.3244", max_results=5, sort_by="relevance")
Modal Expansion Methods (MEMs) are spectral representations in which an unknown field or current is written as a superposition of modes associated with an operator adapted to the governing wave problem. In the literature considered here, MEMs appear in elastodynamic scattering by metamaterial quasiparticles, in electromagnetic scattering by perfect electric conductors (PECs), and in open electromagnetic resonators treated by perfectly matched layers (PMLs). Across these settings, the common structure is an expansion ansatz, a spectral problem that defines the modal basis, and a coefficient-extraction rule; when the operator is non-self-adjoint, the expansion is built on a bi-orthogonal direct–adjoint pair rather than on a single orthogonal basis [2204.04583, 1804.09246, 1311.3244].

## 1. Conceptual framework

MEMs organize wave solutions into modal contributions that are intended to separate geometry, material response, and excitation dependence. In the PEC formulation, the total induced surface-current density is expanded as
$$
\mathbf J(\mathbf r)=\sum_n a_n\,\mathbf J_n(\mathbf r),
$$
or, in matrix form, $\mathbf J=\mathbf \Phi\,\mathbf a$, where $\mathbf \Phi$ is the modal-shape matrix. In the elastodynamic formulation, the scattered field is represented through a layer-potential density that is itself expanded in a modal basis. In the quasimodal formulation for open structures, the diffracted field is expanded over discrete quasimodes together with radiation-mode contributions [1804.09246, 2204.04583, 1311.3244].

| Setting | Expanded quantity | Spectral structure |
|---|---|---|
| Elastic scattering by a metamaterial quasiparticle | Boundary density and scattered field | Static Neumann–Poincaré basis and polariton poles |
| PEC scattering | Induced surface current | Characteristic modes of the MoM impedance matrix |
| Open 2-D electromagnetic structures | Diffracted field | Direct–adjoint quasimodes with PMLs |

This suggests that MEMs are less a single algorithm than a class of operator-based reductions. The precise meaning of “mode” depends on the problem: an eigenfunction of a static Neumann–Poincaré operator, a characteristic current of a Hermitian generalized eigenproblem, or a leaky quasimode of a non-self-adjoint PML-transformed operator. What remains invariant is the idea that the physically observed solution is reconstructed from spectrally distinguished building blocks.

## 2. Operator formulations and modal bases

For time-domain elastic scattering, the construction begins after a Fourier transform in $t$. The scattered field $u^{(s)}(\omega,x)$ satisfies, for $x\notin D$,
$$
\mathcal L_{\lambda,\mu}u+\omega^2u=0,
$$
together with transmission and radiation conditions at $\partial D$. It is represented by a single-layer potential
$$
u^{(s)}(x)=S_D^\omega[\phi](x),\qquad
S_D^\omega[\phi](x)=\int_{\partial D}\Gamma^\omega(x,y)\,\phi(y)\,d\sigma(y),
$$
where $\Gamma^\omega$ is the Kupradze matrix-fundamental solution. The unknown density solves a boundary system $A^\omega[\phi]=F^\omega$, where $A^\omega$ involves the frequency-dependent Neumann–Poincaré operator $K^\omega$. In the static limit, one introduces
$$
K^0:L^2(\partial D)^3\to L^2(\partial D)^3,
$$
and, on a sphere $\partial D$, $K^0$ is compact and self-adjoint with eigenpairs indexed by the three families $\{\phi_{T,n,m},\phi_{M,n,m},\phi_{N,n,m}\}$ [2204.04583].

For PEC objects, the operator framework is the Method of Moments (MoM) impedance matrix
$$
Z=R+jX.
$$
The associated power-based modal basis is commonly chosen through Characteristic-Mode Theory (CMT), which solves
$$
X\,I_n=\lambda_n\,R\,I_n,\qquad n=1,\dots,N.
$$
These mode currents satisfy
$$
I_m^HRI_n=0,\quad I_m^HXI_n=0,\qquad m\neq n.
$$
With the normalization $I_n^HRI_n=1$ and $I_n^HXI_n=\lambda_n$, the basis simultaneously diagonalizes radiated-power and stored-power bilinear forms in the sense used by the electromagnetic-power-based MEM [1804.09246].

For open two-dimensional resonators, the spectral problem is modified by PMLs. The direct problem is written as
$$
\mathcal L(v_n)\equiv-\nabla\!\cdot\bigl(\bm\xi(\mathbf r)\,v_n(\mathbf r)\bigr)
=\Lambda_n\,\chi(\mathbf r)\,v_n(\mathbf r),
$$
and the adjoint problem as
$$
\mathcal L^\dagger(w_m)\equiv-\nabla\!\cdot\bigl(\bm\xi^\star(\mathbf r)\,w_m(\mathbf r)\bigr)
=\overline{\Lambda_m}\,\overline{\chi(\mathbf r)}\,w_m(\mathbf r).
$$
Because the PML coordinate stretch is non-real, $\mathcal L\neq\mathcal L^\dagger$, so classical modal expansion fails. The spectrum is then complex and contains discrete quasi-normal modes $\omega_n=\omega_n'+i\omega_n''$ with $\omega_n''<0$, together with a discretized approximation of the continuous radiation spectrum, the so-called PML or Bérenger modes [1311.3244].

## 3. Modal coefficients, orthogonality, and poles

In the elastic construction, the density is expanded in the static Neumann–Poincaré basis,
$$
\phi=\sum_{i\in\{T,M,N\}}\sum_{n=1}^\infty\sum_m c_{i,n,m}(\omega)\,\phi_{i,n,m},
$$
and orthogonality of the basis yields scalar equations for the modal coefficients
$$
a_{i,n,m}(\omega):=c_{i,n,m}(\omega)
=\frac{\langle F^\omega,\phi_{i,n,m}\rangle_{\partial D}}{A_{i,n}(\omega)}.
$$
The scattered field then takes the form
$$
u^{(s)}(\omega,x)=\sum_{i,n,m}a_{i,n,m}(\omega)\,S_D^\omega[\phi_{i,n,m}](x).
$$
This construction already exhibits the characteristic MEM pattern: modal numerators encode excitation overlap, while modal denominators encode spectral response [2204.04583].

In the PEC formulation, the complex power delivered by the total electric field to the current is
$$
P=\frac12\,\mathbf J^H Z\mathbf J\equiv \mathbf a^H P\,\mathbf a,
$$
with
$$
P_{\rm rad}\coloneqq \frac{P+P^H}{2}=R\ge 0,\qquad
P_{\rm sto}\coloneqq \frac{P-P^H}{2j}=X.
$$
If the modes are normalized by $I_n^HRI_n=1$, then the projection of any excitation-solution onto mode $n$ is
$$
a_n=I_n^H R\,\mathbf a.
$$
In the algorithmic workflow, after solving $Z\,\mathbf a=V$, one expands the physical solution as
$$
\mathbf a=\sum_n c_n\,\mathbf a_n,\qquad c_n=\mathbf a_n^H R\,\mathbf a.
$$
Here, orthogonality is not merely algebraic convenience; it ties each coefficient directly to radiated-power weighting [1804.09246].

In QMEM, the direct and adjoint eigenmodes satisfy the bi-orthogonality relation
$$
\langle \chi\,v_n,w_m\rangle
=\int_\Omega \chi(\mathbf r)\,v_n(\mathbf r)\,\overline{w_m(\mathbf r)}\,d\mathbf r
=K_n\,\delta_{nm},
$$
with
$$
K_n=\int_\Omega \chi\,v_n\,\overline{w_n}\,d\mathbf r.
$$
For the scattering problem $\mathcal L(u^d)=S$, the diffracted field is expanded as
$$
u^d(\mathbf r,\omega)=\sum_{n=1}^\infty P_n(\omega)\,v_n(\mathbf r)
+\int_{\Gamma_c}P_\nu(\omega)\,v_\nu(\mathbf r)\,d\nu,
$$
and the discrete coefficients are
$$
P_n(\omega)=\frac{1}{K_n}\,\langle \chi\,u^d,w_n\rangle
=\frac{J_n(\omega)}{\omega^2-\omega_n^2}.
$$
Each resonance $\omega_n$ therefore appears as a simple pole of $P_n(\omega)$ [1311.3244].

## 4. Classification, subspaces, and common misconceptions

The electromagnetic-power-based MEM gives a detailed modal classification. For a mode $\mathbf a_n$, one defines
$$
p_{\rm rad}^{(n)}=\mathbf a_n^H P_{\rm rad}\mathbf a_n\ge 0,\qquad
p_{\rm sto}^{(n)}=\mathbf a_n^H P_{\rm sto}\mathbf a_n.
$$
A mode is radiative if $p_{\rm rad}^{(n)}>0$ and non-radiative if $p_{\rm rad}^{(n)}=0$, equivalently $P_{\rm rad}\mathbf a_n=0$. It is capacitive if $p_{\rm sto}^{(n)}<0$, resonant if $p_{\rm sto}^{(n)}=0$, and inductive if $p_{\rm sto}^{(n)}>0$. The distinction between intrinsic and non-intrinsic resonance is sharper: intrinsic resonance means $P_{\rm sto}\mathbf a_n=0$, whereas non-intrinsic resonance means $p_{\rm sto}=0$ but $P_{\rm sto}\mathbf a_n\neq 0$. The resulting subclasses include non-radiative intrinsically resonant, radiative intrinsically resonant, radiative non-intrinsically resonant, radiative intrinsically capacitive, radiative non-intrinsically capacitive, radiative intrinsically inductive, and radiative non-intrinsically inductive modes [1804.09246].

A central structural result is that only some of these classes form linear spaces. The set of all non-radiative modes is the null-space of $P_{\rm rad}$ and therefore a linear subspace. The set of all intrinsically resonant modes is the null-space of $P_{\rm sto}$ and is likewise a linear subspace. Their intersection, the set of non-radiative intrinsically resonant modes, is also a linear subspace. By contrast, neither the set of all resonant modes defined by $p_{\rm sto}=0$ nor the sets of all capacitive or all inductive modes are closed under addition. The same source also states that, by including the mode $0$ into the intrinsically capacitive mode set and the intrinsically inductive mode set, these two modal sets become linear spaces respectively [1804.09246].

This directly addresses a recurrent misconception: a physically meaningful modal label does not automatically define a vector space. Another recurrent misconception appears in open resonators, where it might be tempting to use ordinary orthogonal modes despite leakage; the PML-transformed operator is explicitly non-self-adjoint, and the quasimodal expansion therefore requires the adjoint problem and bi-orthogonality rather than a classical orthogonal decomposition [1311.3244].

## 5. Static limits, perturbation theory, and time-domain reconstruction

The elastic MEM is notable for connecting a time-harmonic expansion to a time-domain approximation with explicit control of small parameters. In the static validation step,
$$
A^\omega=A^0+O(\omega^2),\qquad
A^0=c_0 I+(c_0-1)K^0,\qquad c_0=-a,
$$
and Taylor expansions of layer potentials show that $A^\omega$ converges in norm to $A^0$. Since $A^0$ is diagonalizable in the $\phi$-basis, the truncated modal series
$$
u_N^{(s)}(\omega,x):=\sum_{i,n\le N,m} a_{i,n,m}(\omega)\,S^\omega[\phi_{i,n,m}](x)
$$
converges to the exact $u^{(s)}(\omega,x)$ uniformly for small $\omega$, and as $\omega\to 0$ one recovers the static expansion
$$
u^{(s)}(0,x)=\sum_{i,n,m}\frac{(F^0,\phi_{i,n,m})}{\lambda_{i,n}}\,S^0[\phi_{i,n,m}](x),
$$
with $L^2$-convergence of the boundary densities [2204.04583].

For finite-mode perturbative approximation, one fixes $N$ and writes
$$
\phi=\sum_{i,n\le N,m} c_{i,n,m}(\omega)\,\phi_{i,n,m}+R_N(\omega),
$$
with $R_N$ small in $L^2(\partial D)$ if $F^\omega$ is smooth on $\partial D$. The operator admits the perturbative form
$$
A^\omega=A^0+(c(\omega)-1)\delta^2A_1+O(\delta^3),
$$
hence
$$
A^\omega[\phi_{i,n,m}]
=
\bigl[\lambda_{i,n}+(c(\omega)-1)\delta^2Q_{i,n}+O(\delta^3)\bigr]\phi_{i,n,m}.
$$
Accordingly,
$$
A_{i,n}(\omega)=\lambda_{i,n}+(c(\omega)-1)\delta^2Q_{i,n}+\cdots,
$$
and
$$
a_{i,n,m}(\omega)
=
a_{i,n,m}^{(0)}+\delta^2(c(\omega)-1)\,Q_{i,n}/\lambda_{i,n}^2+O(\delta^3,\omega^2).
$$
The source describes this as yielding a fast convergent finite-mode approximation for $u^{(s)}(\omega)$ [2204.04583].

The same framework identifies polariton resonances. A static polariton resonance occurs when $\lambda_{i,n}=0$. More generally, the full denominator $A_{i,n}(\omega)$ has simple zeros $\omega=\Omega_{i,n}(\delta)$ solving
$$
\lambda_{i,n}+(c(\Omega)-1)\delta^2Q_{i,n}=0,
$$
with
$$
\Omega_{i,n}(\delta)=\Omega_{i,n}^0+i\Gamma_{i,n}+O(\delta^2).
$$
Each $\Omega_{i,n}$ is a simple pole of the scattering operator, and near $\omega\approx \Omega_{i,n}$ the corresponding modal coefficient has the resonant form $a_{i,n,m}(\omega)\simeq R_{i,n,m}/(\omega-\Omega_{i,n})$ [2204.04583].

Time-domain reconstruction proceeds by inverse Fourier transform of the low-frequency part,
$$
u^{(s)}_{\le p}(t,x)=\frac{1}{2\pi}\int_{|\omega|\le p}u^{(s)}(\omega,x)e^{-i\omega t}\,d\omega.
$$
Because each modal term factorizes, the inversion yields convolution in time:
$$
m_{i,n,m}(t):=\frac{1}{2\pi}\int a_{i,n,m}(\omega)e^{-i\omega t}\,d\omega.
$$
By closing the contour into the lower half-plane one picks up the pole contributions and, for $t>t_0$,
$$
m_{i,n,m}(t)\simeq R_{i,n,m}e^{-i\Omega_{i,n}(t-t_0)}.
$$
The low-frequency scattered field can therefore be written as a sum of damped resonant exponentials plus a remainder. The sharp error estimate of Theorem 4.10 states that, if $P_{N,p}$ retains modes $i,n\le N$ and frequencies $|\omega|\le p$, then for $t\ge t_0$,
$$
\|u^{(s)}(t,\cdot)-P_{N,p}[u^{(s)}](t,\cdot)\|_{L^2(K)}
\le C\,(\delta p)^{-M}\,t^{-1},
$$
for any large integer $M$, with $K\subset\mathbb R^3\setminus D$ a compact observation region. The constants depend on the $L^2$–Sobolev regularity of the incident pulse, the retained number of modes $N$, and the bandwidth $p$ [2204.04583].

The reduced-order viewpoint also appears in QMEM. There, the continuous sum/integral is replaced in practice by
$$
u^d(\mathbf r,\omega)\approx \sum_{n=1}^M P_n(\omega)\,v_n(\mathbf r).
$$
Once the eigenpairs $(\omega_n,v_n,w_n)$ are computed by a single FEM spectral solve with PMLs, evaluating $P_n(\omega)$ at many frequencies and angles reduces to inner products and algebraic formulas. The source further states that QMEM converges rapidly near resonances, since the nearest poles dominate [1311.3244].

## 6. Representative applications, performance, and limitations

The electromagnetic-power-based MEM is illustrated on canonical PEC scatterers. For a PEC sphere of radius $32\ \mathrm{mm}$ with a mesh of approximately $2000$ RWG elements, at $f=4.10\ \mathrm{GHz}$ the method identifies a $3$-fold degenerate set of non-radiative modes with $p_{\rm rad}\approx 0$ and $p_{\rm sto}=0$ that exactly match the analytic $\mathrm{TE}_{n1}(n=1)$ internal-resonant eigenmodes, together with a second $3$-fold set of radiative intrinsically resonant modes with the same $p_{\rm sto}=0$ but $p_{\rm rad}>0$. For a PEC circular cylinder $(20\ \mathrm{mm}\times 20\ \mathrm{mm})$, at $f=5.75\ \mathrm{GHz}$ there is one non-radiative mode, $\mathrm{TE}_{111}$, with $p_{\rm rad}\approx 0$ and $p_{\rm sto}=0$, while at $f=6.54\ \mathrm{GHz}$ there is one radiative intrinsically resonant mode. The same study reports that away from resonant frequencies typically only a handful of radiative capacitive modes carry most of the scattered power, whereas near resonances one “turns on” one or two intrinsically resonant modes to capture the large reactive-field portions [1804.09246].

The quasimodal approach is illustrated on open dielectric and periodic structures. For a triangular dielectric rod in vacuum, $300$ eigenfrequencies are computed in the complex plane; the two principal leaky modes have $\omega_1=(1.77-0.036)\times 10^{13}\,{\rm s}^{-1}$ with $\lambda_1=10.61\,\mu{\rm m}$ and $\omega_2=(1.90-0.10)\times 10^{13}\,{\rm s}^{-1}$ with $\lambda_2=9.89\,\mu{\rm m}$. The modal coupling maps $|P_n(\lambda,\theta_0)|$ show strong angular dependence, and QMEM reconstruction with as few as $M=10$–$50$ modes yields sub-percent accuracy with integrated relative error smaller than $10^{-5}$. For LDOS, evaluation on a $50\times 50$ grid requires only the single spectral solve plus rapid modal summations instead of $2\,500$ separate FEM source solves. For a mono-periodic Ge-slit grating on a ZnS substrate, $801$ eigenmodes are computed for $\alpha=0$ and $10^5\,{\rm m}^{-1}$; six leaky modes in the atmospheric-IR window $8$–$14\,\mu{\rm m}$ have quality factors ranging from a few to $10^{10}$, and QMEM-reconstructed Fresnel amplitudes agree with full-wave FEM to absolute errors smaller than $5\times 10^{-4}$ and integrated field error smaller than $10^{-5}$ when all $801$ modes are used to represent the continuum well [1311.3244].

The elastic MEM is specialized to a small metamaterial quasiparticle $D$ of radius $\delta$ in $\mathbb R^3$, and its final justification is explicitly tied to the subwavelength, low-frequency regime. The source states that the low-frequency part of the time-domain scattered field can be computed by a small number of damped resonant exponentials, the polariton modes, plus a rapidly decaying remainder. This indicates both the reach and the limitation of the construction: it provides explicit, sharp error controls in $\delta$, $N$, and $p$, but the approximation target is the low-frequency component $u^{(s)}_{\le p}$ rather than an unrestricted full-band representation [2204.04583].

Taken together, these formulations show that MEMs serve several distinct but related purposes: spectral interpretation, model reduction, resonance identification, and physically structured decomposition. The differences among them are not superficial. In closed or effectively self-adjoint settings, orthogonality with respect to a Hermitian form can be exploited directly; in open systems, bi-orthogonality replaces ordinary orthogonality; and in perturbative time-domain scattering, the modal picture is coupled to residue calculus and explicit truncation estimates. This suggests that “modal expansion” is best understood as a family of operator-specific representations rather than as a single universally transferable recipe.

Source: https://www.emergentmind.com/topics/modal-expansion-methods-mems