---
title: Multipole-Frequency Decomposition
url: https://www.emergentmind.com/topics/multipole-frequency-decomposition
type: topic
---

# Multipole-Frequency Decomposition

Multipole-frequency decomposition refers, in the literature summarized here, to a set of frequency-domain constructions in which a response function, operator family, or scattered field is represented through multipolar channels, complex poles, or analytically extracted oscillatory factors. In metallic many-body perturbation theory it appears as the multipole approximation for dielectric screening in full-frequency \(GW\); in boundary-element discretizations of the Helmholtz equation it appears as frequency extraction through a Hadamard-factorized phase; and in nano-optics and photonics it appears as exact field-, source-, and current-based multipole expansions that resolve frequency-dependent resonances, mode densities, and illumination dependence [2301.02282], [2012.14287], [2204.13402], [2411.05657], [2508.16545], [2510.20749].

## 1. Recurring mathematical structure

Across these domains, the common operation is not a single universal formula but a related pattern: the explicitly frequency-dependent part of a problem is isolated into a representation that is cheaper to sample, easier to interpret, or directly tied to modal content. In some cases the reduced representation is a pole sum, in some a phase-extracted matrix family, and in some a multipole basis expansion of radiated or induced fields.

| Setting | Core representation | Stated role |
|---|---|---|
| Metallic \(GW\) | \(\Pi_{GG'}(q,\omega)\approx \sum_{p=1}^{N_p}\dfrac{2\,R_p^{GG'}(q)\,\Omega_p(q)}{\omega^2-[\Omega_p(q)]^2}\) or \(W_{GG'}(q,\omega)=v_{GG'}(q)+\sum_{p=1}^{N_p}\dfrac{B_p^{GG'}(q)}{\omega-\omega_p(q)}\) | Accurate and efficient alternative to current full-frequency methods |
| Helmholtz BEM | \(B(\kappa)=H(\kappa)\circ \widehat B(\kappa)\) with \(H_{ij}(\kappa)=e^{\imath\kappa d_{ij}}\) in the far-field | Data-sparse representation valid for a range of frequencies |
| Generalized polarizabilities | \(M^K(\omega)=\sum_{j=1}^{N_c}\alpha_j^K(\omega)\cdot E_{0j}(\omega)\) | Exact multipole decomposition for any illumination |
| 2-D photonics | \(\mathbf E_{\mathrm{scat}}(\rho,\varphi;\omega)=\sum_m\big[-iZ_0B_m(\omega)\mathbf M_m^{(4)}+A_m(\omega)\mathbf N_m^{(4)}\big]\) | Full frequency-dependent multipole decomposition and scattering spectrum |
| Exact current multipoles | \(\mathbf M^{(l)}_{\rm exact}=\dfrac{i}{\omega}\dfrac{(2l-1)!!}{(l-1)!}\int_{\mathbb R^3}\big[\mathbf J(\mathbf r)\otimes \mathbf r^{\otimes(l-1)}\big]\dfrac{j_{l-1}(kr)}{(kr)^{l-1}}\,d^3r\) | Exact and general current-moment description beyond the point-multipole approximation |

This comparison suggests an umbrella usage of the term: a multipole-frequency decomposition is any formulation that rewrites a frequency-dependent problem into a small set of poles, harmonics, rational functions, or multipolar amplitudes whose variation with \(\omega\) or \(\kappa\) can be sampled or interpreted directly [2301.02282], [2012.14287], [2204.13402], [2411.05657], [2508.16545].

## 2. Full-frequency \(GW\) for metals: multipole approximation of screening

In the \(GW\) approximation, the self-energy is
\[
\Sigma^{GW}(r,r',\omega)=\frac{i}{2\pi}\int_{-\infty}^{+\infty}d\omega'\,G(r,r',\omega-\omega')\,W(r,r',\omega')\,e^{-i\eta\omega'},
\]
with the screened Coulomb interaction obtained from the Dyson equation for the polarizability. In reciprocal space one computes the irreducible RPA polarizability \(\Pi^0_{GG'}(q,\omega)\), solves \(\Pi(q,\omega)=\Pi^0(q,\omega)+\Pi^0(q,\omega)\,v(q)\,\Pi(q,\omega)\), and forms \(W(q,\omega)=v(q)+v(q)\Pi(q,\omega)v(q)=\epsilon^{-1}(q,\omega)\,v(q)\). The metallic multipole approximation replaces dense real-frequency sampling by a small number \(N_p\sim 10\text{--}20\) of complex poles, either in a Lehmann-like form for \(\Pi\) or in a linear-pole form for \(W\), and the two forms are stated to be fully equivalent for a given \(N_p\) [2301.02282].

For metals, the numerical procedure uses a “double-parallel” sampling of \(2N_p\) complex frequencies,
\[
z_j\in \{\omega_j+i\varpi_1\}_{j=1\ldots N_p}\cup \{\omega_j+i\varpi_2\}_{j=1\ldots N_p},
\]
with typical \(\varpi_1\approx 10^{-5}\,\mathrm{Ha}\), \(\varpi_2\approx 1\,\mathrm{Ha}\), and a nonuniform “power-\(\alpha\)” grid
\[
\omega_j=\omega_{\max}\times \frac{(j-1)^\alpha}{(N_p-1)^\alpha},\qquad \alpha=2
\]
for metals. The nonlinear fit for \(\{\Omega_p,R_p\}\) is solved by linearization, with an iterative “secant” or Newton-Raphson scheme in \(\{\Omega_p\}\) and a linear least-squares solution for \(\{R_p\}\) at each step; the stated convergence criterion is a residual \(\|A-LR\|\lesssim 10^{-8}\). The metallic case also requires a treatment of the \(\mathbf q\to 0\) intraband contribution. Two ab-initio routes are described: an \(f\)-sum-rule extrapolation that reconstructs a Drude term in the head of \(Y_{00}(q,\omega)=[\epsilon^{-1}-1]_{00}\), and a constant-dielectric approximation that simply replaces \(Y_{GG'}(q=0,\omega)\leftarrow Y_{GG'}(q_{\min},\omega)\) [2301.02282].

The reported validation is unusually explicit. For simple metals such as Al and Na, using \(N_p\approx 8\) poles and the constant-dielectric intraband correction on a \(16^3\) \(k\)-mesh, MPA quasiparticle energies differ by \(<8\,\mathrm{meV}\) from a converged full-frequency contour-deformation approach. For states far from \(E_F\), plasmon-pole errors are \(\sim 50\text{--}100\,\mathrm{meV}\), while MPA reduces them to few meV. In Cu, single-pole PPA fails, with \(\sim 0.4\,\mathrm{eV}\) errors in quasiparticle energies together with wrong \(\Sigma(\omega)\) shape and spurious satellites; with \(N_p\approx 12\) poles and quadratic grid \(\alpha=2\) up to \(\omega_{\max}\approx 80\,\mathrm{eV}\), MPA reproduces full-frequency \(\Sigma(\omega)\) and \(A(\omega)\) while using only \(\sim 24\) frequency points versus \(1000\) in full frequency, corresponding to a \(\sim 40\times\) speed-up. Converged Cu quasiparticle energies agree with experiment to within \(0.1\,\mathrm{eV}\). The physical interpretation is also made explicit: in Cu the fitted MPA poles of \(Y_{00}(q=0)\) coincide within \(0.2\,\mathrm{eV}\) with peaks in the measured electron-energy-loss spectrum, and the four strongest loss peaks at \(2.1\,\mathrm{eV}\), \(4.5\,\mathrm{eV}\), \(8.9\,\mathrm{eV}\), and \(19.5\,\mathrm{eV}\) match the real parts of the first four MPA poles at \(2.0\), \(4.6\), \(9.0\), and \(19.3\,\mathrm{eV}\) [2301.02282].

## 3. Frequency extraction for Helmholtz BEM matrices

For the 3-D scalar Helmholtz equation, the discretized Galerkin BEM matrix \(B(\kappa)\in \mathbb C^{I\times J}\) is written entrywise as
\[
B(\kappa)=H(\kappa)\circ \widehat B(\kappa),
\]
where the far-field phase factor is
\[
H_{ij}(\kappa)=e^{\imath\kappa d_{ij}},\qquad d_{ij}=\|\xi_i-\eta_j\|,
\]
and \(\widehat B(\kappa)\) is the de-oscillated remainder. The phase is exactly the phase of the Helmholtz Green’s function evaluated at the cluster-center distance. The paper states that rigorous estimates show that every far-field block of \(\widehat B(\kappa)\) has numerical rank growing only like
\[
\mathcal O\bigl(\log^6\epsilon/\log^2(\kappa h)\bigr)
\]
when \(\kappa\) increases, with mesh size \(h\) [2012.14287].

The construction proceeds in two layers. First, \(\widehat B(\kappa)\) at any fixed \(\kappa\) is approximated by a standard \(\mathcal H\)-matrix using geometric cluster trees, an admissibility criterion, and Adaptive Cross Approximation on admissible blocks,
\[
\widehat B_{t\times s}\approx X\,Y^T,\qquad X\in \mathbb C^{t\times r},\;Y\in \mathbb C^{s\times r}.
\]
Because \(\widehat B(\kappa)\) is far less oscillatory, the numerical ranks \(r\) remain essentially constant in \(\kappa\), so the same block skeletons can be reused for all \(\kappa\). Second, each admissible block is treated as a matrix-valued function of \(\kappa\) and approximated by a low-tensor-rank expansion
\[
\widehat{\mathcal B}_{t\times s}(\kappa)\approx \sum_{k=1}^{R_T}M_k\otimes f_k(\kappa),
\]
where the \(f_k\) are rational functions without poles on the interval \([a,b]\). The algorithm is an alternating ACA-AAA procedure: a pivot \((i_k,j_k)\) and frequency \(\kappa_k\) are chosen, the scalar function \(\kappa\mapsto \widehat B_{i_k,j_k}(\kappa)\) is passed to AAA, a block sample at \(\kappa_k\) is obtained by ACA, and orthogonalization plus a residual-based stopping criterion in Frobenius norm determine termination [2012.14287].

The computational consequences are reported quantitatively. Sampling a new frequency requires evaluating the scalar weights \(f_k(\kappa)\) in \(\mathcal O(R_T\cdot \#S)\), forming \(\mathbf m=C\boldsymbol f\) in \(\mathcal O(R_T^2)\), reconstructing each far block in \(\mathcal O(R_T(\#t+\#s)r)\), and then multiplying entrywise by \(H_{ij}(\kappa)\). Four 3-D meshes—sphere, Netgen crankshaft, Stanford bunny, and NASA almond—of size \(N\sim 10^5\) were tested for \(\kappa\cdot\mathrm{diam}\in[10,100]\). Over that range, the classical \(\mathcal H\)-matrix memory grows roughly \(\mathcal O(N\kappa)\), whereas the extracted \(\mathcal H\)-matrix stays essentially flat; build time is stated to be \(2\times\text{--}3\times\) faster at high \(\kappa\). A compact rational model on \([10,100]\), using \(16\) Chebyshev samples, degree-\(8\) rationals, and tolerance \(10^{-4}\), took about \(4\text{--}6\) times the cost of one \(\mathcal H\)-matrix build, but thereafter any new frequency could be recovered in a few seconds, often yielding a \(10\times\text{--}30\times\) speedup compared to rebuilding the \(\mathcal H\)-matrix. Asymptotic tests on refined spheres with constant \(\kappa h=0.8\) confirm \(\mathcal O(N\log N)\) scaling of memory and time for the extracted method, with nearly constant ACA ranks [2012.14287].

## 4. Illumination-independent multipole spectra in nanophotonics

In the generalized polarizabilities formalism, the starting point is the exact size-dependent multipole expansion together with a generalized field propagator derived from the Lippmann-Schwinger integral equation. After discretization into \(N_c\) cubical cells of volume \(V_c\), the total polarization obeys
\[
P_i(\omega)=\sum_{j=1}^{N_c}V_c\,\chi_i(\omega)\,K_{ij}(\omega)\,E_{0j}(\omega),
\]
where \(M:=I-V_c\,G\,\chi\) and \(K=M^{-1}\) is the generalized propagator. Inserting the exact multipole kernels into the discretized multipole integrals yields generalized polarizability tensors \(\alpha_j^K\) such that, for each multipole \(K\),
\[
M^K(\omega)=\sum_{j=1}^{N_c}\alpha_j^K(\omega)\cdot E_{0j}(\omega).
\]
The electric dipole is split into a “primary” part \(p_0\) and a toroidal part \(p_t\), and the magnetic dipole \(m\) is given in exact, non-long-wavelength form through spherical-Bessel-weighted volume integrals over the induced polarization \(P(r',\omega)\) [2204.13402].

A central consequence of this construction is the definition of a multipole-mode density that does not depend on a chosen illumination:
\[
\rho_K(\omega):=\sum_{j=1}^{N_c}\|\alpha_j^K(\omega)\|_F^2.
\]
The paper identifies \(\rho_K(\omega)\) with the total density of mode \(K\) at frequency \(\omega\), and states that it is proportional to the maximum energy that can be radiated by mode \(K\), irrespective of illumination. Once \(\{\alpha_j^K(\omega)\}\) are known, the multipole moment for any illumination is obtained by an \(O(N_c)\) tensor-vector contraction. Optimal illumination for a given multipole under the constraint \(\sum_j |E_{0j}|^2=1\) follows from the leading eigenvector of
\[
H_K=\sum_{j=1}^{N_c}(\alpha_j^K)^\dagger \alpha_j^K,
\]
with the largest eigenvalue giving the maximum achievable \(|M^K|^2\) [2204.13402].

The formalism is presented as a remedy for a specific limitation of the conventional multipole expansion: the latter gives access only to the modes excited by a specific illumination, and the study of various illuminations requires multiple, costly numerical simulations. After a single pre-computation of the generalized propagator, the generalized polarizabilities allow one to obtain the exact multipole decomposition for any illumination, the total density of available multipole modes, and the optimum illumination field distributions that maximally couple to specific multipole modes. The reported validation includes scattering spectra and radiation patterns of a \(240\times 240\times 330\,\mathrm{nm}^3\) cuboid, for which full-field direct simulation and the \(\alpha\)-tensor reconstruction show perfect agreement up to quadrupolar order, as well as mode-density spectra \(\rho_p(\omega)\), \(\rho_m(\omega)\), \(\rho_{Q_e}(\omega)\), and \(\rho_{Q_m}(\omega)\) for a \(240\times 120\times 120\,\mathrm{nm}\) spheroid in air [2204.13402].

## 5. Electromagnetic multipole decompositions in three and two dimensions

For arbitrarily shaped magneto-dielectric scatterers, the scattered electromagnetic field can be expanded in vector spherical harmonics with multipole amplitudes \(a_E(\ell,m;\omega)\) and \(a_H(\ell,m;\omega)\). The formulation explicitly includes both polarization currents \(J_P=-i\omega P\) and magnetization currents \(M\), yielding exact volume-integral expressions
\[
a_E(\ell,m;\omega)=a_E^P(\ell,m;\omega)+a_E^M(\ell,m;\omega),\qquad
a_H(\ell,m;\omega)=a_H^P(\ell,m;\omega)+a_H^M(\ell,m;\omega),
\]
and from these one forms multipolar scattering efficiencies \(Q_\ell^{(e)}(\omega)\) and \(Q_\ell^{(m)}(\omega)\). In Faraday geometry, magnetic circular dichroism is decomposed into multipolar contributions by writing \(CD_{\rm ext}(\omega)\), \(CD_{\rm sca}(\omega)\), and \(CD_{\rm abs}(\omega)\) as sums over \(\ell\), with separate dielectric, magnetic, and interference terms. The paper states that multipole resonances associated with magnetization currents can be even stronger than multipole contributions from conventional dielectric currents, and that the analytical results were verified through comparison with finite-element simulations in COMSOL Multiphysics [2510.20749].

For two-dimensional photonics, the full-wave theory is based on an expansion of the outgoing scattered electric field in fourth-kind cylindrical vector wave functions,
\[
\mathbf{E}_{\mathrm{scat}}(\rho,\varphi;\omega)=
\sum_{m=-\infty}^{\infty}
\Big[-i\,Z_0\,B_m(\omega)\,\mathbf M_m^{(4)}(k_0\rho,\varphi)+A_m(\omega)\,\mathbf N_m^{(4)}(k_0\rho,\varphi)\Big].
\]
The coefficients \(A_m(\omega)\) and \(B_m(\omega)\) are obtained either by contour integrals over an enclosing circle or by explicit source-current volume integrals after inserting an electric-field volume integral equation with equivalent electric and magnetic currents. Once the coefficients are known, the partial scattering cross section is
\[
\sigma_m(\omega)=\frac{4}{k_0(\omega)}|a_m(\omega)|^2,
\]
and the total two-dimensional scattering cross section is
\[
Q_{\rm sc}(\omega)=\frac{4}{k_0(\omega)}\sum_{m=-\infty}^{\infty}|a_m(\omega)|^2.
\]
The stated motivation is a “critical gap” in the literature, namely that existing multipole theories are primarily for three-dimensional particles of isotropic materials, whereas the new formulation addresses isolated, arbitrarily shaped, inhomogeneous, anisotropic cylinders or collections thereof [2411.05657].

A third line of development replaces field amplitudes by exact current multipoles. The exact Cartesian current-multipole tensor of order \(l\) is
\[
\mathbf M^{(l)}_{\rm exact}=
\frac{i}{\omega}\frac{(2l-1)!!}{(l-1)!}
\int_{\mathbb R^3}
\big[\mathbf J(\mathbf r)\otimes \mathbf r^{\otimes(l-1)}\big]
\frac{j_{l-1}(kr)}{(kr)^{l-1}}\,d^3r,
\]
which reduces to the familiar point-multipole moment in the long-wavelength limit. Closed-form linear combinations of the tensor elements reproduce the classical electric and magnetic multipole coefficients \(a_E(l,m)\) and \(a_M(l,m)\), allowing scattering and extinction cross sections to be decomposed by multipole order. The paper emphasizes that current multipoles include nonradiating current configurations absent in the classical field-based expansion; certain tensor combinations appear in \(a_E\) or \(a_M\), while others do not appear in any classical coefficient and are therefore strictly dark. This gives, in the authors’ formulation, a unique, unambiguous decomposition of \(\mathbf J(\mathbf r)\) at all frequencies [2508.16545].

## 6. Interpretation, validation, and recurrent misconceptions

Several recurrent misconceptions are explicitly addressed in this literature. In metallic \(GW\), the complex poles of the multipole approximation are not presented as arbitrary fitting parameters: in Cu, comparison with electron-energy-loss spectra is used to argue that each MPA pole corresponds to a bona-fide collective excitation rather than a mere numerical artifact, with the imaginary parts reproducing finite widths without ad-hoc damping parameters [2301.02282].

In nano-optics, the conventional multipole expansion is not an illumination-independent modal inventory. The generalized polarizability formalism states that the conventional expansion only gives access to modes excited by a specific illumination, whereas the generalized tensors \(\alpha_j^K\) allow the total density of available multipole modes to be computed regardless of illumination and directly provide the optimum illumination for specific modes [2204.13402].

In current-based electromagnetic theory, field multipoles are not a complete description of internal current structure. The exact current-multipole expansion states that dark or nonradiating modes, including the anapole as a special combination of dipole and octupole currents, are absent from the classical field-based expansion and can store energy while influencing the frequency dependence and linewidth of radiative multipole resonances [2508.16545].

In boundary-element frequency extraction, the method is not merely a restatement of low-rank compression for oscillatory matrices. The essential step is the extraction of the known phase of the Green’s function, after which the de-oscillated matrix \(\widehat B(\kappa)\) is approximated by a frequency-independent \(\mathcal H\)-matrix skeleton combined with adaptive rational approximation in the continuous frequency dimension [2012.14287].

Taken together, these works indicate a shared computational strategy. This suggests that, despite the different operators and physical observables involved, multipole-frequency decomposition is most useful when it relocates the hard part of frequency dependence into a representation with low pole count, low tensor rank, or directly interpretable multipolar amplitudes, while preserving the quantities of interest—quasiparticle energies, spectral functions, BEM matrices, scattering spectra, mode densities, or dichroic observables.

Source: https://www.emergentmind.com/topics/multipole-frequency-decomposition