Pole-Expansion of the T-Matrix Based on a Matrix-Valued AAA-Algorithm

Abstract

The transition matrix (T-matrix) is a complete description of an object's linear scattering response. As such, it has found wide adoption for the theoretical and computational description of multiple-scattering phenomena. In its original form, the T-matrix describes the interaction of a scatterer with a monochromatic source. In practice, however, information about the T-matrix is usually needed in an extended spectral domain. To access the frequency-dispersion, one might naively sample T-matrices over a finely resolved set of discrete frequencies and store one T-matrix per frequency. This approach has multiple drawbacks: it is computationally expensive, requires excessive memory, and it disregards the physical origin of the spectral features, weakening physical interpretability. To overcome these major limitations, we leverage a pole-expansion technique to represent the T-matrix with arbitrary frequency resolution within a selected frequency domain via a set of resonant contributions. A matrix-valued variant of the recently established adaptive Antoulas-Anderson (AAA) algorithm for rational approximation enables us to compute the pole-expansion at minimal computational cost using only a small number of direct evaluations. We demonstrate the benefits of such a representation with examples ranging from semi-analytically accessible scatterers to quasi-dual bound states in the continuum. To allow the wider community to capitalize on these findings, we provide open-source tools to perform the presented pole-expansion of the T-matrix.

Figures (17)

• Figure 1: Schematic representation of the introduced method to obtain a pole-expansion of the T-matrix. Samples of the T-matrix at different frequencies (the first four matrix elements $|T_{ij}|^2$ are indicated in the left panel) are jointly fed to a matrix-valued variant of the adaptive Antoulas-Anderson (AAA) algorithm (tensorAAA). The result is a rational approximation of the T-matrix, decomposing it into contributions of simple poles. The dominant poles coincide with the resonant states of the scatterer, while others approximate a slowly varying background term [see Eq. (\ref{['eq-pole-exp-T']})]. The central panel shows the dominant poles in the complex plane, schematically indicating the matrix valued residues by the size of four triangle markers corresponding to the four entries of the T-matrix shown in the other panels. The three poles are labeled a,b, and c. As shown in the right panel, this representation can be evaluated for arbitrary frequencies practically for free. The poles correspond to peaks (same labels) in the spectrum, where the width is determined by the pole's imaginary part and the intensity is jointly determined by the residue and imaginary part. To emphasize the remarkable efficiency regarding the number of required sample points for the tensorAAA, the samples from the left panel are overlayed for the matrix elements colored in red. To highlight the accuracy, reference solutions with fine frequency resolution are shown as a black dashed line for the same matrix element. This figure is meant purely as a high-level conceptual introduction of our method. The treated scatterer is a sphere with a chiral shell mixing the electric and magnetic scattering coefficients. Further details can be found in Supplement I ('Sphere with Chiral Shell').
• Figure 2: Convergence study for the T-matrix of a scatterer consisting of four spheres with different radii (100, 110, 120, and 130) arranged in the order indicated in the inset (the white rods only serve as a guide to the eye). The spheres are placed at the corners of a regular tetrahedron with 300 sidelength. As all geometrical symmetries are broken, the T-matrix is densely populated. (a) The magnitude of the individual entries of a converged pole-expansion (tol AAA = 10-8 and sufficient samples) of the T-matrix ($\ell_\mathrm{max} = 3$). Vertical green lines indicate the positions of the poles found by the AAA-algorithm, as shown in (b). The matrix elements are color-coded according to the multipolar type (electric: blues; magnetic: reds) and degree of the scattered field (see inset legend). In an attempt to reduce visual clutter, differenent incident multipoles are not discriminated, i.e., only the index $i$ in $T_ {ij}$ determines the color. To highlight the electric and magnetic z-oriented dipole coefficients, reference samples of the directly evaluated scattering coefficients are indicated as black pluses and crosses, respectively, exemplifying the excellent agreement. (b) Joint poles of all entries of the T-matrix (i.e., poles of the T-matrix) as found by the converged tensorAAA algorithm (green dots). The dot size vizualises the poles' influence on the real axis. It is computed as $\mathrm{area}_\mathrm{dot} \sim |\boldsymbol{\mathcal{R}}_n|_\mathrm{HS} / \Im\{k_n\}^2$. (c) Convergence of the pole-expansion with increasing number of samples $N$ used in the AAA-algorithm. The relative error is determined from the squared Hilbert-Schmidt norm of the difference between the pole-expansion and additional samples not seen by the AAA-algorithm according to Eq. (\ref{['eq:error']}). Different colors indicate different tolerance criteria for terminating the AAA iteration. Solid lines correspond to pole-expansions constructed from the tensorAAA algorithm, while dashed lines show results from applying the standard AAA-algorithm to each element of the T-matrix separately [this approach is further discussed in Supplement V ('Approximating Individual Matrix Elements Separately')].
• Figure 3: Convergence study for a cylinder of 220 height and 55 radius, as depicted in the inset. (a) The spectra of all elements of the $\ell_\mathrm{max} = 3$ T-matrix are shown. As in Figure \ref{['fig-tetra']} the z-oriented electric and magnetic dipole coefficients are indicated by black pluses and crosses, respectively. (b) The joint poles obtained from the tensorAAA are shown in analogy to Figure \ref{['fig-tetra']}(b). (c) The convergence of the pole-expansion of the T-matrix with the number of considered frequency samples. We use the same error measure as in Figure \ref{['fig-tetra']}.
• Figure 4: Symmetry-protected quasi-bound states in the continuum (qBICs) of an infinite square lattice of cylinders with different lattice spacing $\Lambda$ under slightly off-axis TM illumination (component of the wavevector in the x-direction $k_{x} =$0.006; which corresponds to an incidence angle $\theta$ of 0.04297° to 0.02292° within the shown frequency window). The pole-expansion of the T-matrix enables a dense frequency sampling of the transmission spectrum $T$, which is needed to faithfully resolve the qBIC lineshape (see inset). The computational advantage is particularly pronounced in the presented example, as it enables leveraging the cylindrical symmetry of the meta-atom despite the lattice breaking the symmetry.
• Figure 5: Finetuning of the lattice constant to enforce the coincidence of BICs, considering the effect of higher multipoles. When supporting a BIC lattice-resonance, the lattice T-matrix $\mathbf{T}_\mathbf{k_\parallel}(k)$ becomes singular at a real frequency. As such, we can track the BICs through the largest singular value $\sigma_{\text{max}}$ of $\mathbf{T}_\mathbf{k_\parallel}(k)$. The gray horizontal line indicates the lattice constant $\Lambda_\mathrm{dipole}$ predicted from dipolar considerations only, while $\Lambda_\text{¤} \approx$348.64 marks the lattice constant for which the two quasi BICs actually coincide.