SEMPO: Singularity Expansion Method Optimizer

• SEMPO is a toolbox that extracts the complex singularity structure of a response function using meromorphic properties and a combination of Cauchy and auto-differentiation methods.
• The method first employs a fast rational approximation (ADC) for initialization, then refines the solution with gradient-based optimization that enforces physical constraints and stability.
• SEMPO is applied to linear wave systems in photonics and optics, delivering high reconstruction accuracy even with limited data, and effectively handling spectral variations.

SEMPO, the Singularity Expansion Method Parameter Optimizer, is a toolbox for retrieving the complex poles, zeros, and residues of an arbitrary response function from spectral data acquired on the real-frequency axis only, without prior information about the underlying system. It is designed for linear physical systems whose spectral responses are meromorphic in the complex frequency plane, and it combines two complementary retrieval strategies: a Cauchy-method-based rational approximation and an auto-differentiation-based optimization of the Singularity Expansion Method (SEM). The paper emphasizes that these strategies can also be used sequentially, so that a fast rational initialization is refined into a highly accurate reconstruction over large spectral windows (Soltane et al., 14 Apr 2025).

1. Problem formulation and physical significance

SEMPO addresses the inverse problem of inferring the complex singularity structure of a response function when only a finite set of samples on a real interval $[\omega_A,\omega_B]\subset\mathbb{R}$ is available. In the formulation given in the paper, the target quantities are the poles, zeros, and residues of the meromorphic continuation of the measured spectral response. The problem is described as difficult because it is ill-conditioned, sensitive to sampling, and often underdetermined unless one imposes structure (Soltane et al., 14 Apr 2025).

The physical motivation is rooted in wave physics and electromagnetism. The paper states that poles correspond to resonant states or natural modes, zeros often govern reflection cancellation, Fano interference, and absorption conditions, and residues quantify the strength of each pole’s contribution. This makes the pole-zero-residue representation useful for a compact rational model of the response, for physical insight into resonances and antiresonances, for fast broadband reconstruction from limited data, and for analytical continuation into the complex plane. A plausible implication is that SEMPO functions both as a system-identification tool and as a reduced-order modeling framework, although the paper itself frames the method primarily in terms of retrieval and reconstruction (Soltane et al., 14 Apr 2025).

A central claim of the work is that these spectral singularities are treated as intrinsic properties of the system. The method is therefore not limited to pointwise interpolation on the measured interval; rather, it seeks a parametric description whose analytic structure can remain meaningful across a broader spectral window.

2. Meromorphic representation and the Singularity Expansion Method

The mathematical basis of SEMPO is the Singularity Expansion Method. In the paper’s formulation, the response $h(\omega)$ is represented as a non-resonant term plus a sum over poles weighted by residues. The corresponding Singularity and Zero Factorization (SZF) is written as

$$h(\omega) = \eta_0 \frac{\prod_{\ell}(\omega - z^{(\ell)})}{\prod_{\ell}(\omega - p^{(\ell)})},$$

where $z^{(\ell)}$ are the zeros, $p^{(\ell)}$ the poles, and $\eta_0$ a constant depending on the poles, zeros, and a reference value of the response (Soltane et al., 14 Apr 2025).

This representation is possible because the class of responses considered is meromorphic: analytic everywhere in the complex frequency plane except at isolated poles. The paper explicitly interprets SEM as the analytical continuation of the measured real-frequency response into the complex plane. In this viewpoint, the measured spectrum is not the final object of interest; it is evidence from which the singularity structure is inferred.

The paper also establishes a connection to a generalized Drude–Lorentz model. By transforming pole-residue parameters into oscillator-like variables, the SEM can be recast into a form resembling a generalized Drude–Lorentz model (GDL), which is presented as useful for material dispersion and oscillator-like subunits. This connection provides a bridge between an abstract meromorphic expansion and a more familiar oscillator-based physical interpretation (Soltane et al., 14 Apr 2025).

3. Cauchy-method retrieval and the accuracy-driven Cauchy variant

The first retrieval engine in SEMPO is a Cauchy-method-based rational approximation. The response is assumed to admit a rational representation

$h(\omega)=\frac{f(\omega)}{g(\omega)},$

with $f$ and $g$ polynomials whose roots are the zeros and poles, respectively. The paper writes these polynomials in vector form using monomial bases and derives a linear homogeneous system of the form

$$[\mathbf{A}, -\mathbf{B}] \begin{bmatrix} \mathbf{a}\ \mathbf{b} \end{bmatrix} = \mathbf{0},$$

from which polynomial coefficients are recovered; poles and zeros then follow as the roots of $h(\omega)$0 and $h(\omega)$1. The matrix rank is related to the number of unknowns and the kernel dimension $h(\omega)$2 through

$h(\omega)$3

and the classical Cauchy procedure targets the case $h(\omega)$4, yielding a solution unique up to scaling (Soltane et al., 14 Apr 2025).

The paper extends this into an accuracy-driven Cauchy method (ADC). Instead of fixing a single model order, ADC starts from a large initial number of poles and zeros, estimates maximal rank via SVD, sweeps candidate pairs $h(\omega)$5, enforces $h(\omega)$6, reconstructs the response for each candidate, and selects the model with minimum relative $h(\omega)$7 error. The optimal candidate is defined as

$h(\omega)$8

This makes ADC, in the paper’s wording, “by construction” at least as accurate as the classical approach for the same initial pole count (Soltane et al., 14 Apr 2025).

The method is further modified by physics-informed corrections. The paper imposes or favors Hermitian symmetry, stability through lower-half-plane poles, treatment of far-away singularities as contributions to a modified offset, and a penalty for unstable poles. These corrections are intended to preserve physical plausibility while maintaining spectral accuracy.

Several quantitative examples are reported. For the reflection coefficient of a Si nanodisk array, the Cauchy method reconstructs the response from only 30 real-frequency samples over a wide interval with relative error $h(\omega)$9. For the dielectric response of gold permittivity, the fit improves from classical Cauchy $$h(\omega) = \eta_0 \frac{\prod_{\ell}(\omega - z^{(\ell)})}{\prod_{\ell}(\omega - p^{(\ell)})},$$0 to ADC $$h(\omega) = \eta_0 \frac{\prod_{\ell}(\omega - z^{(\ell)})}{\prod_{\ell}(\omega - p^{(\ell)})},$$1. For a 1D gold grating, the physics-informed ADC version preserves Hermitian symmetry and uses fewer singularities while maintaining a very accurate fit (Soltane et al., 14 Apr 2025).

4. Auto-differentiation-based SEM optimization

The second retrieval engine fits the SEM directly using auto-differentiation and gradient-based optimization. The paper rewrites the truncated SEM in a Hermitian-symmetric form suitable for real-valued physical response functions. Because current autodiff tools are described as handling complex numbers imperfectly, the optimization is reformulated in terms of real variables, with purely imaginary poles and complex-conjugate pole pairs parameterized separately (Soltane et al., 14 Apr 2025).

The optimization seeks a parameter set $$h(\omega) = \eta_0 \frac{\prod_{\ell}(\omega - z^{(\ell)})}{\prod_{\ell}(\omega - p^{(\ell)})},$$2 minimizing a composite loss built from several error measures:

• relative $$h(\omega) = \eta_0 \frac{\prod_{\ell}(\omega - z^{(\ell)})}{\prod_{\ell}(\omega - p^{(\ell)})},$$3 error
• relative $$h(\omega) = \eta_0 \frac{\prod_{\ell}(\omega - z^{(\ell)})}{\prod_{\ell}(\omega - p^{(\ell)})},$$4 error
• real-part mismatch
• imaginary-part mismatch

The paper defines the relative norm errors as

$$h(\omega) = \eta_0 \frac{\prod_{\ell}(\omega - z^{(\ell)})}{\prod_{\ell}(\omega - p^{(\ell)})},$$5

and combines them in a weighted loss. The stated advantage of this design is flexibility: the user can prioritize global fit, peak accuracy, or separate real and imaginary behavior (Soltane et al., 14 Apr 2025).

The paper characterizes the autodiff method as more flexible, easier to constrain physically, better at capturing sharp spectral variations, and capable of directly enforcing symmetry and stability. Its limitations are equally explicit: it is slower, sensitive to initialization, and dependent on hyperparameters such as learning rate, iteration count, and the initial pole distribution.

Representative results show high reconstruction accuracy. For a 2D Ag pillar array, the reflection coefficient is reconstructed with $$h(\omega) = \eta_0 \frac{\prod_{\ell}(\omega - z^{(\ell)})}{\prod_{\ell}(\omega - p^{(\ell)})},$$6, $$h(\omega) = \eta_0 \frac{\prod_{\ell}(\omega - z^{(\ell)})}{\prod_{\ell}(\omega - p^{(\ell)})},$$7, $$h(\omega) = \eta_0 \frac{\prod_{\ell}(\omega - z^{(\ell)})}{\prod_{\ell}(\omega - p^{(\ell)})},$$8, and $$h(\omega) = \eta_0 \frac{\prod_{\ell}(\omega - z^{(\ell)})}{\prod_{\ell}(\omega - p^{(\ell)})},$$9. For a gold nanodisk lattice, the transmission coefficient is reconstructed with $z^{(\ell)}$0, $z^{(\ell)}$1, $z^{(\ell)}$2, and $z^{(\ell)}$3. The same $z^{(\ell)}$4 accuracy is also reported for the generalized Drude–Lorentz representation over the $z^{(\ell)}$5–$z^{(\ell)}$6 nm interval (Soltane et al., 14 Apr 2025).

5. Sequential use of ADC and autodiff

A major practical conclusion of the paper is that the two SEMPO engines are most effective when used sequentially. The rationale is asymmetric: ADC is fast and yields a rough but useful approximation, whereas autodiff is slower but more accurate and better suited to enforcing physical constraints. The proposed workflow is therefore to use ADC to produce an initial singularity distribution and then refine it with the gradient-based SEM optimizer (Soltane et al., 14 Apr 2025).

For spectrally difficult cases, the paper introduces chained ADC over subwindows. The spectral interval is partitioned, each subwindow is fitted separately, and the resulting poles are filtered using window-based weights $z^{(\ell)}$7 derived from the variation induced by each resonant term and from its contribution to the total response. The retained poles and residues are then concatenated into a global initialization for the full-range autodiff stage. The paper notes that subwindow fitting alone can introduce too many poles and physically questionable results; the filtering stage is therefore part of the practical workflow rather than an optional embellishment.

The reported performance of the combined strategy is summarized below.

Configuration	System/example	Reported outcome
Cauchy	Si nanodisk array reflection	$z^{(\ell)}$8
Classical Cauchy	Gold permittivity	$z^{(\ell)}$9
ADC	Gold permittivity	$p^{(\ell)}$0
Autodiff SEM	2D Ag pillar array reflection	$p^{(\ell)}$1
Autodiff SEM	Gold nanodisk lattice transmission	$p^{(\ell)}$2
SEM alone	TiO$p^{(\ell)}$3 slab reflection	$p^{(\ell)}$4
ADC + SEM	TiO$p^{(\ell)}$5 slab reflection	$p^{(\ell)}$6

On the TiO$p^{(\ell)}$7 slab reflection coefficient, the combined method reduces the error from $p^{(\ell)}$8 for SEM alone to $p^{(\ell)}$9 for ADC + SEM, while requiring about half the time of the plain SEM optimization and using fewer singularities. This suggests that initialization quality is not merely a numerical convenience but a substantive factor in the recoverability of a compact and physically meaningful singularity structure (Soltane et al., 14 Apr 2025).

6. Application domain, limitations, and terminological scope

The systems used to assess SEMPO are all in optics/photonics. The paper lists the dielectric permittivity of materials, especially gold, 2D silicon nanodisk arrays, 1D gold gratings, 2D silver pillar metasurfaces, lattices of gold nanodisks, and TiO$\eta_0$0 slab reflection spectra. These examples are chosen to span smooth dispersion, sharp resonances, large spectral windows, and rapid spectral variations, and the paper presents this diversity as evidence that SEMPO applies to any linear wave system whose response is meromorphic (Soltane et al., 14 Apr 2025).

The limitations are stated with unusual clarity. The Cauchy/ADC method is fast but can struggle with strong spectral variations, large windows, very strict physical constraints, and sensitivity to sampling. The autodiff method is more robust but slower, dependent on initial pole placement, sensitive to hyperparameter choices, and potentially prone to overfitting if the loss is not chosen carefully. Accordingly, SEMPO is not presented as a universal solver; the paper frames it as a toolbox in which method choice depends on spectral complexity and on the desired level of physical constraint (Soltane et al., 14 Apr 2025).

The name should also be distinguished from unrelated acronymic uses in other literatures. Within the materials and photonics context, SEMPO denotes the Singularity Expansion Method Parameter Optimizer (Soltane et al., 14 Apr 2025). This is distinct from SEM as search engine marketing, which concerns SEO, paid search marketing, and user knowledge gain in web search (Schultheiß, 2023); from SEM as a reinforcement-learning framework for Search-Efficient LLMs (Sha et al., 12 May 2025); and from a thesis-level use of SEMPO for Semantic, Efficient, and Secure Search over Encrypted Cloud Data in searchable encryption (Boucenna, 2020). The shared acronymic surface is therefore incidental rather than conceptual.

In its primary sense, SEMPO is a retrieval framework for the meromorphic structure of spectral responses. Its defining contribution is methodological: it links analytic continuation, rational approximation, and gradient-based optimization into a single workflow for extracting poles, zeros, and residues from real-axis data alone.