SEMPO -- Retrieving poles, residues and zeros in the complex frequency plane from an arbitrary spectral response

TL;DR

SEMPO tackles the challenge of extracting complex poles $p^{(\ell)}$, zeros $z^{(\ell})$, and residues $r^{(\ell)}$ of a transfer function $h(\omega)$ from real-frequency spectra by two complementary routes: an enhanced Cauchy-based rational approximation (ADC) and an auto-differentiation–driven optimization. The study introduces physics-informed corrections (Hermitian symmetry, stability constraints) and a Generalized Drude-Lorentz (GDL) reinterpretation, and it demonstrates that combining ADC with AutoDiff yields fast yet robust reconstructions across wide spectral windows. Benchmarking against AAA and VF shows that AutoDiff uniquely provides Hermitian-symmetric, stable poles and reliable natural-pole identification, while ADC offers competitive accuracy at lower computational cost. The methods are demonstrated on optical metasurfaces and dielectric permittivity of gold, with results applicable to a broad class of wave-based transfer functions beyond optics.

Abstract

The Singularity Expansion Method Parameter Optimizer -- SEMPO -- is a toolbox to extract the complex poles, zeros and residues of an arbitrary response function acquired along the real frequency axis. SEMPO allows to determine this full set of complex parameters of linear physical systems from their spectral responses only, without prior information about the system. The method leverages on the Singularity Expansion Method of the physical signal. This analytical expansion of the meromorphic function in the complex frequency plane motivates the use of the Cauchy method and auto-differentiation-based optimization approach to retrieve the complex poles, zeros and residues from the knowledge of the spectrum over a finite and real spectral range. Both approaches can be sequentially associated to provide highly accurate reconstructions of physical signals in large spectral windows. The performances of SEMPO are assessed and analysed in several configurations that include the dielectric permittivity of materials and the optical response spectra of various optical metasurfaces.

Figures (10)

• Figure 1: (a) Si nanodisks of diameter $d=160$ nm and height $h=130$ nm, arranged in a 2D array of period $p=420$ nm over a substrate of glass. The signal $\mathbf{h}$ is the $0^{th}$-order optical reflection coefficient at normal incidence in air. (b) Relative singular values of the matrix $\mathbf{C}=[\mathbf{A}~ -\mathbf{B}]$ with $\mathbf{A}$ and $\mathbf{B}$ defined in Equations \ref{['eq:5_8']} and \ref{['eq:5_9']}. The singular values are sorted by decreasing value, and divided by the first one, $\sigma_{max}$. In the classical Cauchy method, the rank $r$ is set to the index of the smallest singular value after which no significant variation of the ratio $\sigma_i/\sigma_{max}$ is observed. (c) Reflection coefficient $h(\omega)=\tilde{r}_{00}(\omega)$ retrieved via the classical Cauchy method (red curve), and compared to simulated data (blue markers). The absolute error $|h_i - \hat{h}_i|$, with $h_i$ the target and $\hat{h}_i$ the reconstructed value, is plotted (black curve), and shows the high accuracy of the method, with a relative $L_2$ error $e^{(2)}_{22,11,10}=4.55\times 10^{-1}$%. (d) Distribution of poles and zeros retrieved by the Cauchy method in the complex $\omega$ plane. Some poles and zeros are located outside the spectral window of interest.
• Figure 2: Reconstruction of the relative permittivity $\varepsilon_{\mathrm{Au}}(\omega_i)=h(\omega)$ of $\mathrm{Au}$, using the classical Cauchy method (dashed, red curve) and the ADC method (blue curve). A better reconstruction is obtained with the ADC method, which yields a relative $L_2$ error $e^{(2)}_{18,9,8}=2.53\times 10^{-3}$%, lower than that obtained via the classical Cauchy method, which is $e^{(2)}_{20,10,9}=5.67\times 10^{-3}$%.
• Figure 3: (a) 1D gold grating over a substrate of gold, with height $h=60$ nm, width $w=100$ nm, and period $p=200$ nm. The studied function is the $0^{th}$ order reflection coefficient $r_0(\omega)=h(\omega)$ at normal incidence in air. (b) Approximation of $h(\omega)$, using the ADC method (blue curve, ADC) or the physics-informed version (red curve, Herm. sym. ADC). (c) Distributions of poles in the complex $\omega$ plane using the two approaches. While both versions provide similar results, i.e. an accurate fitting of the experimental data (black markers), the physics-informed method fulfils the Hermitian symmetry and requires fewer pairs of poles and zeros.
• Figure 4: (a) 2D array of square Ag pillars of width $w=85$ nm and height $h=35$ nm, and period $p=130$ nm over a substrate of Ag. (b) The signal is the $0$-th order reflection coefficient $h(\omega)=r_{00}(\omega)$ of the 2D array depicted in (a) illuminated in normal incidence reconstructed with the SEM in Equation \ref{['eq:5_24']} (red curve), and compared to simulated data (blue markers). The absolute error $|h_i - \hat{h}_i|$ (black curve) is less than $10^{-3}$ at all frequencies, and a relative $L_2$ error of $4.32\times 10^{-3}$ % is obtained. (c) Distribution of poles in the complex $\omega$ plane.
• Figure 5: (a) Lattice of gold nanodisks with height $h=50$nm, diameter $d=80$nm and period $p=500$ nm, bearing on a substrate of glass. The signal is the $0^{th}$-order transmission coefficient $h(\omega)=t_{00}(\omega)$ for a normal incidence in air. (b) Reconstruction of $h(\omega)$ with the GDL in Equation \ref{['eq:5_43']} (red curve), and compared to simulated data (blue markers). A relative $L_2$ error $e^{(2)}=6.31 \times 10^{-3}$ % is obtained. (c) Distribution of the poles in the complex $\omega$ plane.