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Optimization and Physical Bounds for Passive and Non-passive Systems
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.ORCID iD: 0000-0002-3928-6064
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
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.

Place, publisher, year, edition, pages
Växjö, Sweden: Linnaeus University Press, 2019. , p. 217
Series
Linnaeus University Dissertations ; 373/2019
Keywords [en]
Convex optimization, physical bounds, Herglotz functions, quasi-Herglotz functions, passive systems, non-passive systems, approximation, absorption in lossy media
National Category
Other Electrical Engineering, Electronic Engineering, Information Engineering
Research subject
Physics, Waves and Signals
Identifiers
URN: urn:nbn:se:lnu:diva-90223ISBN: 978-91-89081-23-9 (print)ISBN: 978-91-89081-24-6 (electronic)OAI: oai:DiVA.org:lnu-90223DiVA, id: diva2:1372058
Public defence
2019-12-13, Newton, Hus C, Växjö, 09:15 (English)
Opponent
Supervisors
Funder
Swedish Foundation for Strategic Research , AM13-0011Available from: 2019-11-22 Created: 2019-11-21 Last updated: 2019-11-22Bibliographically approved
List of papers
1. Passive Approximation and Optimization Using B-Splines
Open this publication in new window or tab >>Passive Approximation and Optimization Using B-Splines
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2019 (English)In: SIAM Journal on Applied Mathematics, ISSN 0036-1399, E-ISSN 1095-712X, Vol. 79, no 1, p. 436-458Article in journal (Refereed) Published
Abstract [en]

A passive approximation problem is formulated where the target function is an arbitrary complex-valued continuous function defined on an approximation domain consisting of a finite union of closed and bounded intervals on the real axis. The norm used is a weighted L-p-norm where 1 <= p <= infinity. The approximating functions are Herglotz functions generated by a measure with Holder continuous density in an arbitrary neighborhood of the approximation domain. Hence, the imaginary and the real parts of the approximating functions are Holder continuous functions given by the density of the measure and its Hilbert transform, respectively. In practice, it is useful to employ finite B-spline expansions to represent the generating measure. The corresponding approximation problem can then be posed as a finite-dimensional convex optimization problem which is amenable for numerical solution. A constructive proof is given here showing that the convex cone of approximating functions generated by finite uniform B-spline expansions of fixed arbitrary order (linear, quadratic, cubic, etc.) is dense in the convex cone of Herglotz functions which are locally Holder continuous in a neighborhood of the approximation domain, as mentioned above. As an illustration, typical physical application examples are included regarding the passive approximation and optimization of a linear system having metamaterial characteristics, as well as passive realization of optimal absorption of a dielectric small sphere over a finite bandwidth.

Keywords
approximation, Herglotz functions, B-splines, passive systems, convex optimization, sum rules
National Category
Mathematics
Research subject
Mathematics, Applied Mathematics
Identifiers
urn:nbn:se:lnu:diva-81228 (URN)10.1137/17M1161026 (DOI)000460127100021 ()2-s2.0-85063407473 (Scopus ID)
Available from: 2019-03-22 Created: 2019-03-22 Last updated: 2019-11-21Bibliographically approved
2. Passive Approximation with High-Order B-Splines
Open this publication in new window or tab >>Passive Approximation with High-Order B-Splines
2019 (English)In: Analysis, Probability, Applications, and Computation / [ed] Karl‐Olof Lindahl, Torsten Lindström, Luigi G. Rodino, Joachim Toft, Patrik Wahlberg, Birkhäuser Verlag, 2019, p. 83-94Conference paper, Published paper (Refereed)
Abstract [en]

Convex optimization has emerged as a well-suited tool for passive approximation. Here, it is desired to approximate some pre-defined non-trivial system response over a given finite frequency band by using a passive system. This paper summarizes some explicit results concerning the Hilbert transform of general B-splines of arbitrary order and arbitrary partitions that can be useful with the convex optimization formulation. A numerical example in power engineering is included concerning the identification of some model parameters based on measurements on high-voltage insulation materials.

Place, publisher, year, edition, pages
Birkhäuser Verlag, 2019
Series
Trends in Mathematics, ISSN 2297-0215, E-ISSN 2297-024X
National Category
Mathematical Analysis
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-82770 (URN)10.1007/978-3-030-04459-6_8 (DOI)2-s2.0-85065446429 (Scopus ID)978-3-030-04458-9 (ISBN)978-3-030-04459-6 (ISBN)
Conference
11th ISAAC Congress, Växjö (Sweden) 2017
Available from: 2019-05-27 Created: 2019-05-27 Last updated: 2019-11-21Bibliographically approved
3. Quasi-Herglotz functions and convex optimization
Open this publication in new window or tab >>Quasi-Herglotz functions and convex optimization
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(English)Manuscript (preprint) (Other academic)
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.

Keywords
Quasi-Herglotz functions, non-passive systems, approximation, convex optimization, sum rules
National Category
Other Electrical Engineering, Electronic Engineering, Information Engineering Other Mathematics
Research subject
Physics, Waves and Signals
Identifiers
urn:nbn:se:lnu:diva-90218 (URN)
Funder
Swedish Foundation for Strategic Research , AM13-0011
Available from: 2019-11-21 Created: 2019-11-21 Last updated: 2019-11-21
4. Non-passive approximation as a tool to study the realizability of amplifying media
Open this publication in new window or tab >>Non-passive approximation as a tool to study the realizability of amplifying media
2019 (English)Conference paper, Published paper (Refereed)
Abstract [en]

Non-passive approximation is presented as a tool to study the realizability of amplifying media. As an interesting physical example, we derive first a suitable approximation of the plasmonic singularity of a dielectric sphere with respect to a hypothetical amplifying background medium. A non-passive approximation based on convex optimization is then employed to investigate the necessary bandwidth requirements to achieve the approximate pole singularity.

National Category
Other Electrical Engineering, Electronic Engineering, Information Engineering Other Mathematics
Research subject
Physics, Waves and Signals
Identifiers
urn:nbn:se:lnu:diva-90222 (URN)
Conference
URSI EM Theory Symposium, EMTS 2019, San Diego, CA, 27–31 May 2019
Funder
Swedish Foundation for Strategic Research , AM13-0011
Available from: 2019-11-21 Created: 2019-11-21 Last updated: 2019-11-21
5. On the physical limitations for radio frequency absorption in gold nanoparticle suspensions
Open this publication in new window or tab >>On the physical limitations for radio frequency absorption in gold nanoparticle suspensions
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2017 (English)In: Journal of Physics D: Applied Physics, ISSN 0022-3727, E-ISSN 1361-6463, Vol. 50, no 15, article id 155401Article in journal (Refereed) Published
Abstract [en]

This paper presents a study of the physical limitations for radio frequency absorption in gold nanoparticle (GNP) suspensions. A spherical geometry is considered consisting of a spherical suspension of colloidal GNPs characterized as an arbitrary passive dielectric material which is immersed in an arbitrary lossy medium. A relative heating coefficient and a corresponding optimal near field excitation are defined, taking the skin effect of the surrounding medium into account. The classical Mie theory for lossy media is also revisited, and it is shown that the optimal permittivity function yielding a maximal absorption inside the spherical suspension is a conjugate match with respect to the surrounding lossy material. A convex optimization approach is used to investigate the broadband realizability of an arbitrary passive material to approximate the desired conjugate match over a finite bandwidth, similar to the approximation of a metamaterial. A narrowband realizability study shows that for a surrounding medium consisting of a weak electrolyte solution, the electromagnetic heating, due to the electrophoretic (plasmonic) resonance phenomena inside the spherical GNP suspension, can be significant in the microwave regime, provided that the related Drude parameters can be tuned into (or near to) resonance. As a demonstration, some realistic Drude parameters are investigated concerning the volume fraction, mass, and friction constant of the GNPs. The amount of charge that can be accommodated by the GNPs is identified as one of the most important design parameters. However, the problem of reliably modelling, measuring and controlling the charge number of coated GNPs is not yet fully understood, and is still an open research issue in this field. The presented theory and related physical limitations provide a useful framework for further research in this direction. Future research is also aimed at an expansion towards arbitrary suspension geometries and the inclusion of thermodynamical analysis.

Place, publisher, year, edition, pages
Institute of Physics Publishing (IOPP), 2017
National Category
Signal Processing
Research subject
Physics, Waves and Signals
Identifiers
urn:nbn:se:lnu:diva-65659 (URN)10.1088/1361-6463/aa5a89 (DOI)2-s2.0-85016154351 (Scopus ID)
Available from: 2017-06-20 Created: 2017-06-20 Last updated: 2019-11-21Bibliographically approved
6. On the plasmonic resonances in a layered waveguide structure
Open this publication in new window or tab >>On the plasmonic resonances in a layered waveguide structure
2018 (English)In: 2018 12th International Congress on Artificial Materials for Novel Wave Phenomena (Metamaterials), IEEE, 2018, p. 188-190Conference paper, Published paper (Refereed)
Abstract [en]

An optimal plasmonic resonance and the associated Fröhlich resonance frequency are derived for a thin layer in a straight waveguide in TM mode. The layer consists of an arbitrary composite material with a Drude type of dispersion. The reflection and transmission coefficients of the layer are analyzed in detail. To gain insight into the behavior of a thin plasmonic layer, an asymptotic expansion to the first order is derived with respect to the layer permittivity.

Place, publisher, year, edition, pages
IEEE, 2018
National Category
Other Physics Topics
Research subject
Physics, Waves and Signals
Identifiers
urn:nbn:se:lnu:diva-82120 (URN)10.1109/MetaMaterials.2018.8534151 (DOI)000495100200061 ()2-s2.0-85058549846 (Scopus ID)978-1-5386-4703-5 (ISBN)978-1-5386-4702-8 (ISBN)978-1-5386-4701-1 (ISBN)
Conference
12th International Congress on Artificial Materials for Novel Wave Phenomena-Metamaterials, 27 Aug.-1 Sept. 2018, Espoo, Finland
Available from: 2019-04-24 Created: 2019-04-24 Last updated: 2019-11-21Bibliographically approved
7. Optical theorems and physical bounds on absorption in lossy media
Open this publication in new window or tab >>Optical theorems and physical bounds on absorption in lossy media
2019 (English)In: Optics Express, ISSN 1094-4087, E-ISSN 1094-4087, Vol. 27, no 23, p. 1-20, article id 377068Article in journal (Refereed) Published
Abstract [en]

Two different versions of an optical theorem for a scattering body embedded inside a lossy background medium are derived in this paper. The corresponding fundamental upper bounds on absorption are then obtained in closed form by elementary optimization techniques. The first version is formulated in terms of polarization currents (or equivalent currents) inside the scatterer and generalizes previous results given for a lossless medium. The corresponding bound is referred to here as a variational bound and is valid for an arbitrary geometry with a given material property. The second version is formulated in terms of the T-matrix parameters of an arbitrary linear scatterer circumscribed by a spherical volume and gives a new fundamental upper bound on the total absorption of an inclusion with an arbitrary material property (including general bianisotropic materials). The two bounds are fundamentally different as they are based on different assumptions regarding the structure and the material property. Numerical examples including homogeneous and layered (core-shell) spheres are given to demonstrate that the two bounds provide complimentary information in a given scattering problem.

Place, publisher, year, edition, pages
Optical Society of America, 2019
Keywords
Material properties; Mie theory; Photon counting; Radiative transfer; Refractive index; Scattering
National Category
Other Physics Topics
Research subject
Physics, Waves and Signals
Identifiers
urn:nbn:se:lnu:diva-89962 (URN)10.1364/OE.27.034323 (DOI)000495871300120 ()
Funder
Swedish Foundation for Strategic Research , AM13-0011
Available from: 2019-11-08 Created: 2019-11-08 Last updated: 2019-11-29Bibliographically approved

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