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Quasi-Herglotz functions and convex optimization
Linnéuniversitetet, Fakulteten för teknik (FTK), Institutionen för fysik och elektroteknik (IFE).ORCID-id: 0000-0002-3928-6064
Stockholm University, Sweden.
Lund University, Sweden.
KTH Royal institute of technology, Sweden.
Visa övriga samt affilieringar
(Engelska)Manuskript (preprint) (Övrigt vetenskapligt)
Abstract [en]

We introduce the set of quasi-Herglotz functions and demonstrate that it has properties useful in the modeling of non-passive systems. The linear space of quasi-Herglotz functions constitutes a natural extension of the convex cone of Herglotz functions. It consists of differences of Herglotz functions, and we show that several of the important properties and modeling perspectives of Herglotz functions are inherited by the new set of quasi-Herglotz functions. In particular, this applies to their integral representations, the associated integral identities or sum rules (with adequate additional assumptions), their boundary values on the real axis and the associated approximation theory. Numerical examples are included to demonstrate the modeling of a non-passive gain media formulated as a convex optimization problem,where the generating measure is modeled by using a finite expansion of B-splines and point masses.

Nyckelord [en]
Quasi-Herglotz functions, non-passive systems, approximation, convex optimization, sum rules
Nationell ämneskategori
Annan elektroteknik och elektronik Annan matematik
Forskningsämne
Fysik, Vågor och signaler; Fysik, Elektroteknik alt Electrical engineering
Identifikatorer
URN: urn:nbn:se:lnu:diva-90218OAI: oai:DiVA.org:lnu-90218DiVA, id: diva2:1372014
Forskningsfinansiär
Stiftelsen för strategisk forskning (SSF), AM13-0011Tillgänglig från: 2019-11-21 Skapad: 2019-11-21 Senast uppdaterad: 2019-12-18Bibliografiskt granskad
Ingår i avhandling
1. Optimization and Physical Bounds for Passive and Non-passive Systems
Öppna denna publikation i ny flik eller fönster >>Optimization and Physical Bounds for Passive and Non-passive Systems
2019 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Abstract [en]

Physical bounds in electromagnetic field theory have been of interest for more than a decade. Considering electromagnetic structures from the system theory perspective, as systems satisfying linearity, time-invariance, causality and passivity, it is possible to characterize their transfer functions via Herglotz functions. Herglotz functions are useful in modeling of passive systems with applications in mathematical physics, engineering, and modeling of wave phenomena in materials and scattering. Physical bounds on passive systems can be derived in the form of sum rules, which are based on low- and high-frequency asymptotics of the corresponding Herglotz functions. These bounds provide an insight into factors limiting the performance of a given system, as well as the knowledge about possibilities to improve a desired system from a design point of view. However, the asymptotics of the Herglotz functions do not always exist for a given system, and thus a new method for determination of physical bounds is required. In Papers I–II of this thesis, a rigorous mathematical framework for a convex optimization approach based on general weighted Lp-norms, 1≤p≤∞, is introduced. The developed framework is used to approximate a desired system response, and to determine an optimal performance in realization of a system satisfying the target requirement. The approximation is carried out using Herglotz functions, B-splines, and convex optimization. 

Papers III–IV of this thesis concern modeling and determination of optimal performance bounds for causal, but not passive systems. To model them, a new class of functions, the quasi-Herglotz functions, is introduced. The new functions are defined as differences of two Herglotz functions and preserve the majority of the properties of Herglotz functions useful for the mathematical framework based on convex optimization. We consider modeling of gain media with desired properties as a causal system, which can be active over certain frequencies or  frequency intervals.  Here, sum rules can also be used under certain assumptions.

In Papers V–VII of this thesis, the optical theorem for scatterers immersed in lossy media is revisited. Two versions of the optical theorem are derived: one based on internal equivalent currents and the other based on external fields in terms of a T-matrix formalism, respectively. The theorems are exploited to derive fundamental bounds on absorption by using elementary optimization techniques. The theory has a potential impact in applications where the surrounding losses cannot be neglected, e.g., in medicine, plasmonic photothermal therapy, radio frequency absorption of gold nanoparticle suspensions, etc.  In addition to this, a new method for detection of electrophoretic resonances in a material with Drude-type of dispersion, which is placed in a straight waveguide, is proposed.

Ort, förlag, år, upplaga, sidor
Växjö, Sweden: Linnaeus University Press, 2019. s. 217
Serie
Linnaeus University Dissertations ; 373/2019
Nyckelord
Convex optimization, physical bounds, Herglotz functions, quasi-Herglotz functions, passive systems, non-passive systems, approximation, absorption in lossy media
Nationell ämneskategori
Annan elektroteknik och elektronik
Forskningsämne
Fysik, Vågor och signaler
Identifikatorer
urn:nbn:se:lnu:diva-90223 (URN)978-91-89081-23-9 (ISBN)978-91-89081-24-6 (ISBN)
Disputation
2019-12-13, Newton, Hus C, Växjö, 09:15 (Engelska)
Opponent
Handledare
Forskningsfinansiär
Stiftelsen för strategisk forskning (SSF), AM13-0011
Tillgänglig från: 2019-11-22 Skapad: 2019-11-21 Senast uppdaterad: 2019-11-22Bibliografiskt granskad

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Ivanenko, YevhenNordebo, Sven

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