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MAXIMIZING the ELECTROMAGNETIC CHIRALITY of THIN DIELECTRIC TUBES \Ast SIAM J. APPL.MATH. © 2021 Society for Industrial and Applied Mathematics Vol. 81, No. 5, pp. 1979{2006 MAXIMIZING THE ELECTROMAGNETIC CHIRALITY OF THIN DIELECTRIC TUBES \ast y y \" y TILO ARENS , ROLAND GRIESMAIER , AND MARVIN KNOLLER Abstract. Any time-harmonic electromagnetic wave can be uniquely decomposed into left and right circularly polarized components. The concept of electromagnetic chirality (em-chirality) de- scribes differences in the interaction of these two components with a scattering object ormedium. Such differences can be quantified by means of em-chirality measures. These measures attain their minimal value zero for em-achiral objects or media that interact essentially in the same way with left and right circularly polarized waves. Scattering objects or media with positive em-chirality measure interact qualitatively differently with left and right circularly polarized waves, and maximally em- chiral scattering objects or media would not interact with fields of either positive or negative helicity. This paper examines a shape optimization problem, where the goal is to determine thin tubular structures consisting of dielectric isotropic materials that exhibit large measures of em-chirality at a given frequency. We develop a gradient based optimization scheme that uses an asymptotic rep- resentation formula for scattered waves due to thin tubular scattering objects. Numerical examples suggest that thin helical structures are at least locally optimal among this class of scattering objects. Key words. electromagnetic scattering, chirality, shape optimization, maximally chiral objects AMS subject \bfc \bfl \bfa \bfs \bfifi\bfc\bfa\bft\bfo\bfn. 78M50, 49Q10, 78A45 DOI. 10.1137/21M1393509 1. Introduction. The traditional notion of chirality defines chiral objects as those that cannot be superimposed on their mirror images. Accordingly, nonchiral objects are called achiral. Chirality is frequently observed in nature, and chiral phe- nomena have been intensively studied in the past two centuries in various scientific disciplines, e.g., biology, chemistry, and physics. Currently, a very active research field is the interaction of chiral objects with electromagnetic fields, which is also the topic of this work. The simple geometric definition of chirality stated above hides signifi- cant difficulties concerning quantifying the degree of chirality of an object (see,e.g., [18]). This lack of an unambiguous ranking for the magnitude of chirality is a hand- icap in the practical design of chiral objects that create extreme chiral effects in the interaction with electromagnetic waves. We follow a different, fairly new, approach that was proposed in [16] to circumvent these difficulties. The key idea is to quantify chiral effects directly in terms of the object's interaction with electromagnetic fields instead of quantifying them only in terms of geometric considerations. This work is concerned with scattering of time-harmonic electromagnetic waves by a compactly supported isotropic dielectric object in three-dimensional free space. Us- ing Maxwell's equations to model electromagnetic wave propagation, we can uniquely decompose any incident field, as well as the corresponding scattered field, awayfrom the scatterer into left and right circularly polarized components. The concept of elec- tromagnetic chirality (em-chirality) compares the interaction of these two components with the scattering object. Broadly speaking, a scatterer is called electromagnetically \ast Received by the editors January 21, 2021; accepted for publication (in revised form) May 10, 2021; published electronically September 16, 2021. https://doi.org/10.1137/21M1393509 Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms Funding: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Founda- tion) - Project-ID 258734477 - SFB 1173. yInstitut f\"urAngewandte und Numerische Mathematik, Karlsruher Institut f\"urTechnologie, 76131 Karlsruhe, Germany ([email protected], [email protected], [email protected]). 1979 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 1980 TILO ARENS, ROLAND GRIESMAIER, AND MARVIN KNOLLER\" achiral (em-achiral) if it scatters incident fields of one helicity in the same wayas incident fields of the opposite helicity up to a unitary transformation that swapshe- licity. If this is not the case, then the scatterer is called electromagnetically chiral (em-chiral). A precise definition of em-chirality will be given below. In the following we associate scattering objects with far field operators that map superpositions of plane wave incident fields to the far field patterns of the correspond- ing scattered waves. Based on this identification, scalar measures of em-chirality were recently introduced in [16] (see also [5, 24]). These measures quantify the degree of em-chirality of a scattering object in terms of the singular values of suitable projec- tions of the associated far field operator onto subspaces of left and right circularly polarized fields. They allow comparison of degrees of em-chirality of different scat- tering objects. The scalar measures of em-chirality are zero for em-achiral scatterers and strictly positive for em-chiral scatterers, and they would attain their maximum for scatterers that do not interact with either left or right circularly polarized elec- tromagnetic waves, i.e., scatterers that are invisible to incident fields of one helicity. It is unknown whether such maximally em-chiral scatterers exist, but even scattering objects that possess sufficiently large measures of em-chirality at optical frequen- cies have a number of interesting applications in photonic metamaterials (see, e.g., [14, 17, 19, 28, 32, 33, 37, 39]). Throughout this work, we consider scattering objects that consist of isotropic materials, and the chiral effect merely results from the particular shapes of the scat- terers. A different approach is studied in [6, 7, 36], where electromagnetic scattering problems with scatterers that consist of chiral materials are discussed. A link between these two perspectives is provided in [2], where the Drude{Born{Fedorov constitutive relations governing the propagation of electromagnetic waves in chiral media have been derived from the linear constitutive relations for homogeneous isotropic media. This is achieved by embedding a large number of regularly spaced, randomly oriented chiral objects that are made of isotropic materials similar to the ones considered in this work. We study a shape optimization problem, where the goal is to determine compactly supported dielectric scattering objects that possess comparatively large measures of em-chirality. Since thin helical structures have been proposed as candidates for highly em-chiral objects in the literature (see, e.g., [2, 16, 19] and the references therein), we focus on shape optimization for scatterers that are supported on thin tubular neigh- borhoods of smooth curves. The objective functional in this optimization problem is based on an em-chirality measure, and the evaluation of its shape derivative requires an approximation of the shape derivative of the complete far field operator. Accord- ingly, evaluating the shape derivatives in a traditional shape optimization scheme for electromagnetic scattering problems (see, e.g., [25, 26, 27, 34, 35]) would require solv- ing a large number of Maxwell systems in each iteration step of the algorithm, which would be rather expensive. Using an asymptotic representation formula for scattered fields due to thin tubular dielectric structures that was recently established in[12] (see also [1, 8, 21]), we develop a quasi-Newton scheme that does not require solving a single Maxwell system during the optimization procedure. A similar approach was used in [12] to construct an inexpensive Gau{\ss}--Newton reconstruction method foran inverse scattering problem with thin tubular scatterers. We also refer the reader to [8, 20, 22] for related work on electrical impedance tomography with thin tubular Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms conductivity inclusions. The asymptotic representation formula from [12] gives an explicit approximation of the far field operator corresponding to thin tubular scattering objects. Theac- Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. MAXIMIZING THE EM-CHIRALITY OF THIN TUBES 1981 curacy of this approximation increases with decreasing thickness of the thin tubular scattering objects. Replacing the far field operator in the objective functional of the shape optimization problem by the leading order term in the asymptotic representa- tion formula, we derive an explicit formula for the shape derivative of this modified objective functional as well. Using vector spherical harmonics expansions of the lead- ing order term in the asymptotic expansion of the far field operator and of its shape derivative, the modified objective functional in the shape optimization scheme canbe evaluated efficiently. We stabilize the optimization procedure by adding proper regu- larization terms, and we apply the final algorithm to provide examples of optimized thin tubular em-chiral structures. This paper is organized as follows. In the next section we introduce the mathemat- ical setting, and we briefly review some facts concerning the notion of electromagnetic chirality.
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