Generalization of Kramers-Krönig relations for evaluation of causality in power-law media

Gulgowski, Jacek; Stefański, Tomasz P.

doi:10.1016/j.cnsns.2020.105664

Cited by 11 publications

(17 citation statements)

References 32 publications

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“…whereF + ∈ C is a certain complex constant. Its value does not influence the causality of the transform, i.e., it can be demonstrated that the transform e −zc −1 β (jω) β is causal iff β ∈ (0, 1) [55]. However, this transform is not relativistically causal, i.e., when t < 0 then the inverse Fourier transform of e −zc −1 β (jω) β is equal to zero, whilst it is different then zero for e −zc −1 β (jω) β e jωzc −1 .…”

Section: Fundamental Solutionsmentioning

confidence: 99%

“…Theorem 1 (see Theorem 1.6.1 in [66] and Theorem 2 in [55]). If a square-integrable function G(ω) fulfills one of the four conditions below, then it fulfills all four of them:…”

Section: Causalitymentioning

confidence: 99%

“…However, since we operate outside L 2 (R), we may not directly apply classical methods and we should treat the function G r (ω) as a tempered distribution. Fortunately, there are tools that may be applied here as well: we are going to refer to the theorem proved in [68] and rephrased as Theorem 7 and discussed in [55].…”

Section: Causalitymentioning

confidence: 99%

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Formulation of Time-Fractional Electrodynamics Based on Riemann-Silberstein Vector

Stefański

Gulgowski

2021

Entropy

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In this paper, the formulation of time-fractional (TF) electrodynamics is derived based on the Riemann-Silberstein (RS) vector. With the use of this vector and fractional-order derivatives, one can write TF Maxwell’s equations in a compact form, which allows for modelling of energy dissipation and dynamics of electromagnetic systems with memory. Therefore, we formulate TF Maxwell’s equations using the RS vector and analyse their properties from the point of view of classical electrodynamics, i.e., energy and momentum conservation, reciprocity, causality. Afterwards, we derive classical solutions for wave-propagation problems, assuming helical, spherical, and cylindrical symmetries of solutions. The results are supported by numerical simulations and their analysis. Discussion of relations between the TF Schrödinger equation and TF electrodynamics is included as well.

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Section: Fundamental Solutionsmentioning

confidence: 99%

“…Theorem 1 (see Theorem 1.6.1 in [66] and Theorem 2 in [55]). If a square-integrable function G(ω) fulfills one of the four conditions below, then it fulfills all four of them:…”

Section: Causalitymentioning

confidence: 99%

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Formulation of Time-Fractional Electrodynamics Based on Riemann-Silberstein Vector

Stefański

Gulgowski

2021

Entropy

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“…It is based on the assumption that the subtracted logarithm of the response is causal, but the considered response function does not have to be causal. Therefore, we address this issue in [25], where the K-K relations are generalized towards non square-integrable functions. That is, the K-K relations with one and two subtractions are formulated.…”

Section: Introductionmentioning

confidence: 99%

“…Their validity and satisfaction of additional assumptions imply causality of the considered function. Then, the formulated theory is applied to the analysis of electromagnetic media characterized by power-law frequency dispersion [25] and fractional-order (FO) models [26]. However, in our opinion, it might still be unclear for community members how to apply the developed mathematical tools to causality evaluation.…”

Section: Introductionmentioning

confidence: 99%

Analytical Methods for Causality Evaluation of Photonic Materials

2022

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We comprehensively review several general methods and analytical tools used for causality evaluation of photonic materials. Our objective is to call to mind and then formulate, on a mathematically rigorous basis, a set of theorems which can answer the question whether a considered material model is causal or not. For this purpose, a set of various distributional theorems presented in literature is collected as the distributional version of the Titchmarsh theorem, allowing for evaluation of causality in complicated electromagnetic systems. Furthermore, we correct the existing material models with the use of distribution theory in order to obtain their causal formulations. In addition to the well-known Kramers–Krönig (K–K) relations, we overview four further methods which can be used to assess causality of given dispersion relations, when calculations of integrals involved in the K–K relations are challenging or even impossible. Depending on the given problem, optimal approaches allowing us to prove either the causality or lack thereof are pointed out. These methodologies should be useful for scientists and engineers analyzing causality problems in electrodynamics and optics, particularly with regard to photonic materials, when the involved mathematical distributions have to be invoked.

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Diffuse reflectance and terahertz phase transition in a cubic Al2O3@ZnTiO3 lower-temperature perovskite

Abouelnaga,

Hammad

2024

J Mater Sci: Mater Electron

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Generalization of Kramers-Krönig relations for evaluation of causality in power-law media

Cited by 11 publications

References 32 publications

Formulation of Time-Fractional Electrodynamics Based on Riemann-Silberstein Vector

Formulation of Time-Fractional Electrodynamics Based on Riemann-Silberstein Vector

Analytical Methods for Causality Evaluation of Photonic Materials

Diffuse reflectance and terahertz phase transition in a cubic Al2O3@ZnTiO3 lower-temperature perovskite

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