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SEMPO - Retrieving complex poles, residues and zeros from arbitrary real spectral responses
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DOI：10.1016/j.cpc.2025.109929.png)
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Abstract
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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 an accuracy-driven improved version of the Cauchy method constrained by properties of physical systems, as well as an auto-differentiation-based optimization approach. 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. SEMPO’s performances are thoroughly analyzed and benchmarked with other state-of-the-art methods to highlight its capability to retrieve the natural poles of a physical system. Program summary • Program title: Singularity Expansion Method Parameter Optimizer (SEMPO) • Library link to program files: • Developer’s repository link: https://doi.org/10.5281/zenodo.15210008 • Licensing provisions: Creative Commons Attribution 4.0 International • Programming language: Python • Nature of problem: Spectral functions of scattering coefficients of physical systems can be rigorously expanded through the Singularity Expansion Method. This expansion method permits to cast the spectral function through two equivalent forms: a sum of complex rational functions depending on complex poles and residues, and a factorized expression that depends on complex poles and zeros. Spectral functions are usually acquired along the real frequency axis, the challenge is thus to identify the set of complex poles, zeros and residues from these acquisitions at real frequencies. • Solution method: SEMPO is a numerical toolbox that aims at extracting the complex poles, zeros and residues from arbitrary spectral functions. SEMPO relies on both an improved algebraic Cauchy method associated with the factorized expression, and an open-source auto-differentiation method associated with the expansion in terms of poles and residues. Both methods are based on the Singularity expansion method and leverage the properties of physical systems and signals. The accuracy of SEMPO is thoroughly assessed and analysed in different configurations including the spectral responses of optical metasurfaces and the dielectric permittivity of optical materials.
Keywords:
Auto-differentiation
Cauchy method
complex analysis
poles
residues
singularity expansion method
zeros
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