MAXIMIZING the ELECTROMAGNETIC CHIRALITY of THIN DIELECTRIC TUBES \Ast

MAXIMIZING THE ELECTROMAGNETIC CHIRALITY OF THIN DIELECTRIC TUBES \ast

TILO ARENS\dagger , ROLAND GRIESMAIER\dagger , AND MARVIN KNOLLER\" \dagger

\bfA \bfb \bfs\bft\bfr\bfa\bfc. 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 representation 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.

\bfK \bfe \bfy\bfw\bfo\bfr\bfd\bfs. electromagnetic scattering, chirality, shape optimization, maximally chiral objects

\bfA \bfM \bfS \bfs \bfu \bfb\bfj\bfe\bfc\bft\bfl\bfa\bfs\bfifi\bfc\bfo\bfn. 78M50, 49Q10, 78A45

\bfD \bfO\bfI. 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 electromagnetic 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 \bfF \bfu \bfn\bfd\bfi\bfg: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Founda- tion) - Project-ID 258734477 - SFB 1173. \dagger Institut f\"urAngewandte und Numerische Mathematik, Karlsruher Institut f\"urTechnologie, 76131 Karlsruhe, Germany ([email protected], [email protected], [email protected]).

1979

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 corresponding 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 projections 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 scattering 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 electromagnetic 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 frequencies 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 scatterers. 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 neighborhoods 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 solving 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-

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 representation 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 regularization 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 mathematical setting, and we briefly review some facts concerning the notion of electromagnetic chirality. In section 3 we consider the asymptotic representation formula for far field operators corresponding to thin tubular scattering objects. In section 4 we establish the shape derivative of the leading order term in this asymptotic expansion, and in

section 5 we develop the shape optimization scheme. Numerical results are discussed in section 6, and in the appendix we provide explicit representations for the derivatives of spherical vector wave functions that are required for the numerical implementation of the optimization algorithm.

2. Electromagnetic chirality. We consider time-harmonic electromagnetic wave propagation in a homogeneous background medium in \BbbR 3 with constant elec-

tric permittivity \varepsilon 0 > 0 and constant magnetic permeability \mu0 > 0. Throughout we will work with electric fields only, but the associated magnetic field can be retrieved i from the corresponding first order Maxwell systems. An incident field \bfitE is an entire solution to the Maxwell equations

i 2 i 3 (2.1a) curl curl \bfitE - k \bfitE = 0 in \BbbR , \surd where k = \omega \varepsilon 0\mu0 denotes the wave number at frequency \omega > 0. We suppose that this incident field is scattered by a bounded penetrable dielectric scattering object D \subseteq \BbbR 3, and that the relative electric permittivity and the relative magnetic permeability satisfy

\Biggl\{ \Biggl\{ \varepsilon r , \bfitx \in D, \mur , \bfitx \in D, \varepsilonr,D (\bfitx ) := and \mur,D (\bfitx) := 1 , \bfitx \in \BbbR 3 \setminus D, 1 , \bfitx \in \BbbR 3 \setminus D,

for some \varepsilon r, \mur > 0. Accordingly, the total electric field \bfitE solves

\bigl( - 1 \bigr) 2 3 (2.1b) curl \mur,D curl \bfitE - k \varepsilon r,D\bfitE = 0 in \BbbR ,

and the scattered electric field

(2.1c) \bfitE s = \bfitE - \bfitE i

satisfies the Silver--M\"uller radiation condition

Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms the asymptotic behavior

\infty uniformly in \bfitx\widehat = \bfitx/ |\bfitx | (see, e.g., [13, Thm. 6.9]). The vector function \bfitE \in 2 2 3 2 2 3 Lt (S , \BbbC ) is called the electric far field pattern, where as usual Lt (S , \BbbC ) denotes the vector space of square integrable tangential vector fields on the unit sphere. A plane wave with direction of propagation \bfittheta \in S2 and a polarization vector \bfitA \in 3, which must satisfy \bfitA \cdot \bfittheta = 0, can be described by a matrix \bfitE i( \cdot ; \bfittheta ) that \BbbC satisfies

i ik\bfittheta \cdot\bfitx 3 (2.2) \bfitE (\bfitx ; \bfittheta )\bfitA := \bfitA e , \bfitx \in \BbbR .

Because of the linearity of (2.1) with respect to the incident field, the scattered electric field and the corresponding electric far field pattern can also be expressed bymatrices, s \infty and we denote them by \bfitE ( \cdot ; \bfittheta )\bfitA and \bfitE ( \cdot ; \bfittheta )\bfitA , respectively. A Herglotz wave 2 2 3 with density \bfitA \in Lt (S , \BbbC ) is a superposition of plane waves

\int i ik \bfittheta \cdot\bfitx 3 (2.3) \bfitE [\bfitA ](\bfitx) := \bfitA (\bfittheta ) e ds(\bfittheta ) , \bfitx \in \BbbR . S2

We denote the corresponding total electric field by \bfitE [\bfitA ] and the scattered electric field s \infty by \bfitE [\bfitA ]. Electric far field patterns \bfitE [\bfitA ] excited by Herglotz waves as incident 2 2 3 2 2 3 fields are fully described by the far field operator \scrF D : Lt (S , \BbbC ) \rightarrow Lt (S , \BbbC ), which is defined by

\int \infty 2 (\scrF D\bfitA )(\bfitx\widehat ) := \bfitE (\bfitx\widehat ; \bfittheta )\bfitA (\bfittheta ) ds(\bfittheta ) , \bfitx\widehat \in S . S2

For later reference we note that \scrFD is an integral operator with smooth kernel (see, \infty e.g., [13, Thm. 6.9]). By linearity we have that \bfitE [\bfitA ] = \scrF D\bfitA . In the following we discuss the physical notion of chirality for such electromagnetic scattering problems and give a short synopsis of the concepts and results that have recently been developed in [5, 16]. The following traditional notion of chirality is

concerned with the geometry of objects: An object D is called geometrically achiral 3 3\times3 if there exist \bfitx 0 \in \BbbR and an orthogonal matrix J \in \BbbR with det J = - 1 such that D = \bfitx 0 + JD, and D is called geometrically chiral if this is not the case. In the context of electromagnetic scattering, one has to take into consideration that applying a reflection by a plane also changes the incident field. A plane wave

is said to be left or right circularly polarized if its polarization vector performs a counterclockwise or clockwise circular motion (one turn per wave length) along the direction of propagation, respectively. This is equivalent to the relation

(2.4) \bfitA \pm i (\bfittheta \times \bfitA ) = 0

in (2.2). Under a reflection by a plane, such a plane wave is mapped to another plane wave, but this field is then of opposite helicity. We note also that (2.4) is equivalent to the eigenvalue relation

k 1- curl \bfitE i(\cdot; \bfittheta )\bfitA = \pm \bfitE i(\cdot; \bfittheta )\bfitA ,

Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms which, on the other hand, can immediately be extended to more general fields. A solution to

2 3 (2.5) curl curl \bfitU - k \bfitU = 0 in \Omega \subseteq \BbbR

1- is said to have helicity \pm 1 if \bfitU is an eigenfunction of the operator k curl for the eigenvalue \pm 1, respectively. Using the Beltrami fields

(2.6) \bfitU \pm k - 1 curl \bfitU ,

it is easily checked that every solution to (2.5) can be decomposed into a sum of two fields of helicity +1 and - 1, respectively. Returning to the scattering problem (2.1) with Herglotz waves as incident fields, we see that both the incident field \bfitE i[\bfitA ] and the far field pattern \bfitE \infty [\bfitA ] are uniquely 2 2 3 determined by the density \bfitA \in Lt (S , \BbbC ). Since Rellich's lemma implies a one-to-one s 3 correspondence between the scattered field \bfitE [\bfitA ] in \BbbR \setminus D and its far field pattern \infty s \bfitE [\bfitA ] (see, e.g., [13, Thm. 6.10]), the scattered field \bfitE [\bfitA ]| 3 also is uniquely \BbbR \setminus D i s determined by \bfitA . To describe the helicities of \bfitE [\bfitA ] and \bfitE [\bfitA ]| 3 in terms of \bfitA , we \BbbR \setminus D 2 2 3 2 2 3 generalize (2.4) and introduce the self-adjoint operator \scrC : Lt (S , \BbbC ) \rightarrow Lt (S , \BbbC ) with

\scrC \bfitA (\bfittheta ) := i \bfittheta \times \bfitA (\bfittheta ) , \bfittheta \in S2 .

We note that its eigenspaces

\pm 2 2 3 (2.7) V := \{\bfitA \pm \scrC\bfitA | \bfitA \in Lt (S , \BbbC )\}

2 2 3 corresponding to the eigenvalues \pm 1 are orthogonal in Lt (S , \BbbC ) and satisfy

2 2 3 + - Lt (S , \BbbC ) = V \oplus V .

It has been shown in [16, 5] that \bfitE i[\bfitA ] has helicity \pm 1 if and only if \bfitA \in V \pm ,

s \infty \pm \bfitE [\bfitA ]| 3\setminus D has helicity \pm 1 if and only if \bfitE [\bfitA] \in V . \BbbR Electromagnetic chirality is a concept describing the difference in the interaction of a scattering object D with incident fields of opposite helicities. To give an accurate \pm 2 2 3 2 2 3 definition, we denote by \scrP : Lt (S , \BbbC ) \rightarrow Lt (S , \BbbC ) the orthogonal projections \pm onto V , and, accordingly, we decompose ++ + - +- - - (2.8) \scrF D = \scrFD + \scrF D + \scrFD + \scrFD , pq with \scrF := \scrP p\scrF \scrP q for p, q \in+ \{ , -. \} It has been observed in [16, 5] that if a scat- D D terer D is geometrically achiral, then there exists a unitary operator 2 2 3 2 2 3 \scrU : Lt (S , \BbbC ) \rightarrow Lt (S , \BbbC ) such that

\scrC \scrU = - \scrU \scrC and \scrF \scrU = \scrU \scrF . D D This says that \scrFD is equivalent to itself by means of a unitary transform \scrU that swaps helicity. An immediate consequence is that \scrF ++ = \scrU \scrF - -\scrU \ast and \scrF +- = \scrU \scrF + - \scrU \ast . D D D D Based on this observation, the following more general definition of electromagnetic chirality was introduced in [16]. Definition 2.1. A scattering object D is called electromagnetically achiral (or

em-achiral) if there exist unitary operators \scrU (j) : L2(S2, 3) \rightarrow L2(S2, 3) satisfying Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms t \BbbC t \BbbC \scrU (j)\scrC = - \scrC \scrU (j), j = 1,..., 4, such that

\scrF ++ = \scrU (1)\scrF - -\scrU (2) , \scrF - + = \scrU (3)\scrF + - \scrU (4) . D D D D

If this is not the case, we call the scattering object D electromagnetically chiral (or em-chiral). An immediate consequence of this definition is that for an em-achiral scatterer ++ D, the singular values (in decreasing order and repeated with multiplicity) (\sigma j )j\in \BbbN ++ - - - - of \scrFD coincide with the singular values (\sigma j )j\in \BbbN of \scrF D , and, analogously, the singular values (\sigma + - ) of \scrF + - coincide with the singular values (\sigma +- ) of \scrF +- . j j\in\BbbN D j j\in \BbbN D This leads to the idea of quantifying the degree of em-chirality of a scattering object by means of the distance of the corresponding sequences of singular values. Following [16], we define the chirality measure \chi2 of a scatterer D associated to the far field operator \scrFD as

Since the far field operator \scrF D is an integral operator with smooth kernel, its singular values are decreasing exponentially, and (2.9) is well defined. In particular \scrFD is a Hilbert--Schmidt operator. The chirality measure \chi 2 is closely connected to the Hilbert--Schmidt norm of \scrF D. In fact,

2 2 \sum\bigl( ++ - - - + + - \bigr) 2 (2.10) \chi2 (\scrF D) = \| \scrFD\|HS - 2 \sigma j \sigma j + \sigma j \sigma j \leq \|\scrFD\|HS , j\in \BbbN ++ +- and it follows immediately that the upper bound is attained when \scrF D = \scrFD = 0 - - + - or \scrF D = \scrFD = 0, i.e., when the scatterer D does not scatter incident fields of either positive or negative helicity. If, in addition, the reciprocity principle holds,

which is the case for the setting considered in this work (see, e.g., [13, Thm. 9.6]), then this invisibility property is known to be equivalent to equality in (2.10). The Hilbert--Schmidt norm \| \scrFD\|HS of the far field operator is sometimes called the total interaction cross-section of the scattering object D.

Definition 2.2. A scattering object D is said to be maximally em-chiral if \chi2 (\scrF D) = \| \scrFD\|HS .

(2.12) \chi HS(\scrF D) \leq \chi 2(\scrF D) ,

and comparing (2.10) and (2.11), we find that

\chi (\scrF ) = \| \scrF \| if and only if \chi (\scrF ) = \| \scrF \| . HS D D HS 2 D D HS Moreover, \chiHS is differentiable on

2 2 3 pq (2.13) X = \{ \scrG \in HS(Lt (S , \BbbC )) | \chi HS(\scrG ) \not = 0 and \| \scrG \|HS > 0 , p, q \in+ \{ , - \} , 2 2 3 2 2 3 where HS(Lt (S , \BbbC )) denotes the space of Hilbert--Schmidt operators on Lt (S , \BbbC ). 2 2 3 Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms Given \scrG \in X and \scrH \in HS(Lt (S , \BbbC )), the derivative is given by pq \sum pq pq \| \scrG \| HS Re\langle \scrG, \scrHHS \rangle - p,q\in+ \{ , - \} Re\langle \scrG , \scrH \rangleHS pq (2.14) (\chi )\prime [\scrG ]\scrH = \| \scrG \| HS , HS \chi (\scrG ) HS

where p := - p and q := q- . The remainder of this article is devoted to designing scatterers D that exhibit comparatively large values of \chi 2(\scrF D) and \chiHS (\scrF D). Since numerical examples in [5] suggest that \chi2 is not differentiable, we use the chirality measure \chiHS in the gradient based shape optimization scheme that we develop in section 5 below. Furthermore,

we will focus on thin tubular scattering objects, and in the next section we discuss an asymptotic representation formula for electric far field patterns due to such thin structures from [12] and apply it to our setting.

3. Far field operators of thin tubular scatterers. Before we specify the class of thin tubular scattering objects that will be considered in the following, we introduce a set of admissible parametrizations for the center curves of these scatterers by

\bigl\{ 3 3 \bigm| \prime \bigr\} (3.1) \scrU ad := \bfitp \in C ([0, 1], \BbbR ) \bigm| \bfitp ([0, 1]) is simple and \bfitp (t) \not = 0 for all t \in [0, 1] .

3 For any center curve \Gamma \subseteq \BbbR with parametrization \bfitp \Gamma \in ad \scrU we denote by (\bfitt\Gamma , \bfitn\Gamma , \bfitb \Gamma ) \prime \prime \prime the corresponding Frenet--Serret frame. For instance, if \bfitp\Gamma (t) \times \bfitp \Gamma (t) \not = 0 for all t \in [0, 1], then

\prime \prime \prime \prime \prime \bfitp \Gamma (\bfitp \Gamma \times \bfitp \Gamma ) \times \bfitp \Gamma \bfitt\Gamma = \prime , \bfitn\Gamma = \prime \prime \prime \prime , \bfitb \Gamma = \bfitt\Gamma \times \bfitn\Gamma . |\bfitp \Gamma | | (\bfitp \Gamma \times \bfitp \Gamma ) \times \bfitp \Gamma |

3 We will consider thin tubular scattering objects D \rho \subseteq \BbbR with a circular cross- section of radius \rho > 0, which are described by

\bigl\{ \bigm| \bigr\} (3.2) D\rho := \bfitp\Gamma (s) + \bfitn\Gamma (s)\eta + \bfitb \Gamma (s)\zeta \bigm| s \in (0, 1) , | (\eta ,\zeta)| < \rho .

The relative electric permittivity and the relative magnetic permeability are given by \Biggl\{ \Biggl\{ \varepsilon r , \bfitx \in D\rho , \mur , \bfitx \in D\rho , \varepsilon r,D (\bfitx ) := and \mur,D (\bfitx ) := \rho 1 , \bfitx \in 3 \setminus D , \rho 1 , \bfitx \in 3 \setminus D , \BbbR \rho \BbbR \rho

for some \varepsilonr , \mur > 0. For each \rho > 0 we denote the electric far field pattern of the i \infty solution to (2.1) with D replaced by D\rho and a Herglotz incident field \bfitE [\bfitA ] by \bfitE\rho [\bfitA ],

and, accordingly, we write \scrFD \rho for the corresponding far field operator. In this work we focus on shape optimization for scatterers that are supported on thin tubular neighborhoods D of smooth curves as in (3.2), i.e., when the param- \rho eter \rho > 0 is small with respect to the wave length \lambda = 2\pi /k of the incident field. The computational complexity of the iterative shape optimization scheme developed in section 5 below is heavily dominated by the evaluation of far field operators and of shape derivatives of far field operators corresponding to thin tubular scattering objects. The numerical simulation of far field operators and of their shape derivatives

with traditional finite element or boundary element methods becomes increasingly difficult for decreasing values of \rho > 0 because finer and finer meshes have tobe used. To facilitate these computations we refrain from finite element and boundary element methods and apply instead the following asymptotic perturbation formula for the electric far field pattern \bfitE \infty as \rho \rightarrow 0, which has recently been established \rho in [12] (see also [1, 21] and [3, 4, 8, 9, 10, 11] for earlier contributions in this direc- Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms tion). For a particular choice of k, \rhor ,\varepsilon , \mur , and \Gamma the accuracy of this approximation can be estimated by suitably accurate reference simulations using a finite element or boundary element solver (see also section 6 below).

3 Theorem 3.1. Let \Gamma \subseteq \BbbR be an admissible center curve with \bfitp\Gamma \inad \scrU , and denote by \varepsilon r, \mur > 0 the relative electric permittivity and the relative magnetic per- 2 2 3 mittivity of a thin tubular scatterer D\rho for some \rho > 0 . Suppose that \bfitA \in Lt (S , \BbbC ) is the density of a Herglotz incident wave \bfitE i[\bfitA ]. Then the electric far field pattern of the corresponding solution to the scattering problem (2.1) satisfies, for each \bfitx \in S2, \widehat \int \biggl( \infty 2 - ik\bfitx\widehat \cdot\bfity \bigl( \bigr) \varepsilon i (3.3) \bfitE \rho [\bfitA ](\bfitx\widehat ) = (k\rho) \pi (\varepsilon r - 1)e (\bfitx\widehat \times \BbbI3 ) \times \bfitx\widehat \BbbM (\bfity )\bfitE [\bfitA ](\bfity ) \Gamma

\Bigl( \Bigr)\biggr) - ik\bfitx \cdot\bfity \bigl( \bigr) \mu i i \bigl( 2\bigr) + (\mur - 1)e \widehat \bfitx \times \BbbI 3 \BbbM (\bfity ) curl \bfitE [\bfitA ](\bfity ) ds(\bfity ) + o (k\rho) \widehat k

as \rho \rightarrow 0. The matrix valued functions \BbbM\varepsilon , \BbbM \mu \in L2 (\Gamma, \BbbR 3\times 3) are the electric and magnetic polarization tensors, respectively. These are given by

\gamma (\bfitp (s)) = V (s)M \gamma V (s)\top for a.e. s \in [0, 1] and \gamma \in\varepsilon \{ \} ,\mu , \BbbM \Gamma \bfitp \Gamma \bfitp \Gamma

where M \gamma := diag(1, 2/(\gamma + 1), 2/(\gamma + 1)) \in 3\times3 , and V := \bigl[ \bfitt \bigm| \bfitn \bigm| \bfitb \bigr] \in r r \BbbR \bfitp\Gamma \Gamma \bigm| \Gamma \bigm| \Gamma C([0, 1], \BbbR 3\times 3) is the matrix valued function containing the components of the Frenet-- Serret frame (\bfitt\Gamma , \bfitn\Gamma , \bfitb \Gamma ) for \Gamma as its columns.

Note that the cross product between a vector and a matrix in (3.3) denotes the matrix of cross products between the vector and thecolumns of the original matrix. 2 2 2 The term o((k\rho ) ) in (3.3) is such that \|o ((k\rho ) )\| L\infty ( S2)/(k\rho ) converges to zero for i i any fixed \bfitE [\bfitA ], and the dependence on \bfitE [\bfitA ] is only a dependence on \|\bfitA \|L 2(S2, 3). t \BbbC Next we introduce the operator \scrT : L2(S2, 3) \rightarrow L2(S2, 3), which is defined D\rho t \BbbC t \BbbC by

\int \biggl( (3.4) (\scrT \bfitA )(\bfitx) := (k\rho )2\pi (\varepsilon - 1)e i- k\bfitx\widehat \cdot\bfity \bigl( (\bfitx \times ) \times \bfitx \bigr) \varepsilon (\bfity )\bfitE i[\bfitA ](\bfity ) D\rho \widehat r \widehat \BbbI 3 \widehat \BbbM \Gamma \biggr) i- k\bfitx\cdot \bfity \mu \Bigl( i i \Bigr) + (\mu - 1)e \widehat \bigl( \bfitx \times \bigr) (\bfity ) curl \bfitE [\bfitA ](\bfity ) ds(\bfity ) . r \widehat \BbbI3 \BbbM k

From Theorem 3.1 (and the remark about the remainder term) it follows that

\bigl( 2\bigr) (3.5) \scrFD = \scrT D + o (k\rho ) as \rho \rightarrow 0 , \rho \rho and the term o((k\rho )2) in (3.5) is such that \bigm\|o \bigl( (k\rho) 2\bigr) \bigm\| /(k\rho )2 converges to zero. In \bigm\| \bigm\| HS the following we approximate the far field operator \scrF by \scrT . D \rho D\rho For numerical implementations it will be convenient to have an explicit repre- 2 2 3 m sentation of \scrT D\rho in terms of a complete orthonormal system of Lt (S , \BbbC ). Let Yn , m = n,- . . . , n, n \in \BbbN , denote any complete orthonormal system of spherical harmonics of order n in L2(S2). A particular choice that is used for the computations in

our numerical examples is given in (A.2). Then, the vector spherical harmonics,

m 1 m m m 2 (3.6) \bfitUn (\bfittheta ) = GradS2 Yn (\bfittheta ) , \bfitV n (\bfittheta ) = \bfittheta \times \bfitUn (\bfittheta ) , \bfittheta \in S , \sqrt{}n (n + 1)

2 2 3 for m = n,- . . . , n, n = 1, 2,... , form a complete orthonormal system in Lt (S , \BbbC ). Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms Accordingly we deduce that the circularly polarized vector spherical harmonics,

m 1 m m m 1 m m (3.7) \bfitA n := \surd (\bfitUn + i\bfitVn ) and \bfitB n := \surd (\bfitUn - i\bfitV n ) , 2 2

for m = - n, . . . , n, n = 1, 2,... , form a complete orthonormal system in the subspace V + and V - from (2.7), respectively. In particular,

m m m m 2 (3.8) i\bfittheta \times \bfitAn (\bfittheta ) = \bfitA n (\bfittheta ) , i\bfittheta \times \bfitBn (\bfittheta ) = - \bfitBn (\bfittheta ) , \bfittheta \in S .

We also consider the spherical vector wave functions,

m m 3 (3.9) \bfitMn (\bfitx) := - jn(k| \bfitx| )\bfitV n (\bfitx\widehat ) , \bfitx \in \BbbR , m = - n, . . . , n , n = 1, 2, 3,...,

where jn denotes the spherical Bessel function of degree n. Note that the normaliza- tion factors used here differ from what is used elsewhere in the literature (see, e.g., [13, p. 255]). The corresponding Beltrami fields as defined in (2.6), which we will call circularly polarized spherical vector wave functions in the following, are given by

m m - 1 m m m 1- m (3.10) \bfitP n := \bfitM n + k curl \bfitMn , \bfitQn := \bfitMn - k curl \bfitM n

for m = - n, . . . , n, n = 1, 2,... . We recall that

m m m m (3.11) curl \bfitP n = k \bfitP n , curl \bfitQ n = - k \bfitQ n .

Lemma 3.2. Let \bfitA \in L2(S2, 3) with t \BbbC \infty n \sum \sum (3.12) \bfitA = \bigl( am\bfitA m + bm\bfitB m\bigr) . n n n n n=1 m= - n

Then \infty n \sum \sum (3.13) \scrT \bfitA = \bigl( cm\bfitA m + d m\bfitB m\bigr) D\rho n n n n n=1 m= - n with

\infty n\prime m \sum \sum \Bigl( m\prime \bigl\langle m\prime m\bigr\rangle m\prime \bigl\langle m\prime m\bigr\rangle \Bigr) (3.14a) cn = an\prime \scrT D\rho \bfitA n\prime , \bfitAn 2 2 3 + bn\prime \scrT D\rho \bfitB n\prime , \bfitA n 2 2 3 , Lt (S ,\BbbC ) Lt (S ,\BbbC ) n\prime =1m\prime = n- \prime \infty n\prime \Bigl( \prime \prime \prime \prime \Bigr) m \sum \sum m \bigl\langle m m\bigr\rangle m \bigl\langle m m\bigr\rangle (3.14b) dn = an\prime \scrT D\rho \bfitA n\prime , \bfitB n 2 2 3 + bn\prime \scrT D\rho \bfitB n\prime , \bfitB n 2 2 3 . Lt (S ,\BbbC ) Lt (S ,\BbbC ) n\prime =1m\prime = n- \prime

3 Introducing, for any \bfitU , \bfitV \in C(\Gamma, \BbbC ), the expressions \int \pm 3 2 \Bigl( \varepsilon \mu \Bigr) \scrJ (\bfitU , \bfitV ) := 8\pi (k\rho ) \pm (\varepsilon r - 1)\bfitV \cdot \BbbM \bfitU + (\mur - 1)\bfitV \cdot \BbbM \bfitU ds , \Gamma

we have

\bigl\langle m\prime m\bigr\rangle n\prime n- + m\prime m (3.15a) \scrT D\rho \bfitA n\prime , \bfitA n L2(S2, 3) = i \scrJ (\bfitP n\prime , \bfitPn ) , t \BbbC \bigl\langle m\prime m\bigr\rangle n\prime n- - m\prime m (3.15b) \scrT D\rho \bfitB n\prime , \bfitA n 2 2 3 = i \scrJ (\bfitQn \prime , \bfitP n ) , Lt (S , ) Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms \BbbC \bigl\langle m\prime m\bigr\rangle n\prime n- - m\prime m (3.15c) \scrT D\rho \bfitA n\prime , \bfitBn L2(S2, 3) = i \scrJ (\bfitP n\prime , \bfitQ n ) , t \BbbC \bigl\langle m\prime m\bigr\rangle n\prime n- + m\prime m (3.15d) \scrT D\rho \bfitB n\prime , \bfitBn L2(S2, 3) = i \scrJ (\bfitQ n\prime , \bfitQ n ) . t \BbbC

Proof. The expansions (3.13) and (3.14) follow by linearity. Furthermore, it follows immediately from [13, Thm. 6.29] that for any \bfittheta \in S2 and \bfitp \in \BbbC 3 with \bfitp \cdot \bfittheta = 0,

\infty n - ik\bfittheta \cdot\bfity \sum n \sum m \bigl( m \bigr) \bfitp e = - 4\pi ( - i) \bfitMn (\bfity ) \bfitV n (\bfittheta ) \cdot \bfitp n=1 m= - n \infty n 4\pi \sum \sum + ( - i)n - 1 curl \bfitM m(\bfity )\bigl( \bfitU m(\bfittheta ) \cdot \bfitp \bigr) . k n n n=1 m= - n

Hence, we find that

4\pi (3.16a) \bigl\langle e i- k\bfittheta \cdot\bfity , \bfitU m(\bfittheta )\bigr\rangle = ( i)- n - 1 curl \bfitM m(\bfity ) , n L2(S2) k n \bigl\langle - ik\bfittheta \cdot\bfity m \bigr\rangle n m (3.16b) e , \bfitV n (\bfittheta ) L2(S2) = 4- \pi ( - i) \bfitM n (\bfity ) ,

with the scalar product between a scalar and a vector understood to be taken com- ponentwise, and recalling (3.7) and (3.10) this shows that \surd \bigl\langle - ik\bfittheta \cdot\bfity m \bigr\rangle n - 1 m (3.17a) e , \bfitAn (\bfittheta ) L2(S2) = 8\pi ( i)- \bfitPn (\bfity ) , \surd (3.17b) \bigl\langle e - ik\bfittheta \cdot\bfity , \bfitB m(\bfittheta )\bigr\rangle = - 8\pi ( - i)n - 1\bfitQ m(\bfity ) . n L2(S2) n Therefore,

\surd (3.18a) \bfitE i[\bfitA m](\bfity ) = \bigl\langle eik\bfittheta \cdot\bfity , \bfitA m(\bfittheta )\bigr\rangle = 8\pi in - 1 \bfitP m(\bfity ) , n n L2(S2) n \surd (3.18b) \bfitE i[\bfitB m](\bfity ) = \bigl\langle eik\bfittheta \cdot\bfity , \bfitB m(\bfittheta )\bigr\rangle = - 8\pi in 1- \bfitQ m(\bfity ) , n n L2(S2) n

and applying (3.11) gives \surd i m n - 1 m (3.19a) curl \bfitE [\bfitA n ](\bfity ) = 8\pi k i \bfitP n (\bfity ) , \surd i m n - 1 m (3.19b) curl \bfitE [\bfitB n ](\bfity ) = 8\pi k i \bfitQ n (\bfity ) .

Similarly, applying (3.8), we obtain that

\surd \top \bigl\langle i- k\bfittheta \cdot\bfity \bigl( \bigr) m \bigr\rangle n 1- m (3.20a) e (\bfittheta \times \BbbI 3) \times \bfittheta , \bfitA n (\bfittheta ) L2(S2, 3) = 8\pi ( - i) \bfitP n (\bfity ) , t \BbbC \surd \top \bigl\langle - ik\bfittheta \cdot\bfity \bigl( \bigr) m \bigr\rangle n - 1 m (3.20b) e (\bfittheta \times \BbbI3 ) \times \bfittheta , \bfitB n (\bfittheta ) 2 2 3 = - 8\pi ( - i) \bfitQ n (\bfity ) , Lt (S ,\BbbC ) \surd \top \bigl\langle i- k\bfittheta \cdot \bfity m \bigr\rangle n m (3.20c) e (\bfittheta \times \BbbI 3), \bfitA n (\bfittheta ) 2 2 3 = 8\pi ( - i) \bfitP n (\bfity ) , Lt (S ,\BbbC ) \surd \top \bigl\langle - ik\bfittheta \cdot\bfity m \bigr\rangle n m (3.20d) e (\bfittheta \times \BbbI3 ), \bfitBn (\bfittheta ) 2 2 3 = 8\pi ( - i) \bfitQn (\bfity ) , Lt (S ,\BbbC )

with the scalar product between a matrix and a vector understood to be taken column by column. Finally, combining the identities in (3.16)--(3.20) with the integral representation in (3.4) gives (3.15).

m m Remark 3.3. The circularly polarized vector spherical harmonics \bfitA n and \bfitB n , m = n,- . . . , n, n = 1, 2,..., in (3.7) have been constructed in such a way that they span the subspace V + and V - from (2.7), respectively. Thus the expansion in Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms Lemma 3.2 immediately gives corresponding basis representations of the projected operators \scrT pq for p, q \in \{+, - \}, which are defined analogously to (2.8) with \scrF D\rho D\rho replaced by \scrT D . \rho

In numerical implementations the series over n in all these basis representations have to be truncated at some N \in \BbbN . In [23] the authors studied the singular value decomposition of the linear operator that maps current densities supported in the ball BR(0) of radius R around the origin to their radiated far field patterns and showed that for all practically relevant source distributions the radiated far field pattern is

well approximated by a vector spherical harmonics expansion of order N \gtrsim kR. This suggests choosing the truncation index N in the series representations of \scrT D\rho and \scrT pq, p, q \in+ \{ , -, \} such that N kR, where B (0) denotes the smallest ball around D\rho \gtrsim R the origin that contains the scattering object D\rho .

4. The shape derivative of \bfscrT \bfitD \bfitrho . In the previous section we discussed the

asymptotic behavior of the far field operator \scrFD \rho corresponding to a thin tubular scattering object D with a circular cross-section of radius \rho > 0 and a fixed cen- \rho ter curve \Gamma as \rho tends to zero. In this section we fix the radius \rho > 0 and discuss the Fr\'echet differentiability of the leading order term \scrT D\rho in the asymptotic expansion (3.5) with respect to the center curve \Gamma . Recalling the set of admissible parametrizations \scrU ad from (3.1) and denoting by 2 2 3 2 2 3 HS(Lt (S , \BbbC )) the space of Hilbert--Schmidt operators on Lt (S , \BbbC ), we define a nonlinear operator \bfitT : \scrU \rightarrow HS(L2(S2, 3)) by \rho ad t \BbbC

(4.1) \bfitT (\bfitp ) := \scrT , \rho \Gamma D\rho

where D\rho is the thin tubular scattering object from (3.2) with center curve \Gamma parametrized by \bfitp\Gamma . Before we establish the Fr\'echet derivative of \bfitT\rho in Theorem 4.2 below, we discuss the Fr\'echet differentiability of the polarization tensor \BbbM \gamma , \gamma \in\mu \{ \} ,\varepsilon , with respect to \Gamma . Recalling Theorem 3.1 we can identify the parametrized form \gamma := \BbbM \bfitp\Gamma \gamma \circ \bfitp of the polarization tensor with a continuous function in C([0, 1], 3). The \BbbM \Gamma \BbbR following lemma has been established in [22, Lem. 4.1].

4.1. The mapping \bfitp \mapsto \rightarrow \gamma is Fr\'echet differentiable from \scrU Lemma \Gamma \BbbM\bfitp \Gamma ad to C([0, 1], 3\times 3), and its Fr\'echetderivative at \bfitp \in \scrU is given by \bfith \mapsto \rightarrow ( \gamma )\prime \BbbR \Gamma ad \BbbM\bfitp \Gamma ,\bfith with

\gamma \prime \prime \gamma \top \gamma \prime \top 3 3 ( ) = V M V + V\bfitp M (V ) , \bfith \in C ([0, 1], ) , \BbbM \bfitp\Gamma ,\bfith \bfitp\Gamma ,\bfith \bfitp \Gamma \Gamma \bfitp\Gamma ,\bfith \BbbR

where the matrix valued function V \prime is defined columnwise by \bfitp \Gamma ,\bfith

\prime 1 \Bigl[ \prime \prime \bigm| \prime \bigm| \prime \Bigr] V = (\bfith \cdot \bfitn\Gamma )\bfitn\Gamma + (\bfith \cdot \bfitb \Gamma )\bfitb \Gamma \bigm| - (\bfith \cdot \bfitn\Gamma )\bfitt\Gamma \bigm| - (\bfith \cdot \bfitb \Gamma )\bfitt\Gamma . \bfitp \Gamma ,\bfith \prime \bigm| \bigm| |\bfitp \Gamma |

Next we establish the Fr\'echet differentiability of \bfitT \rho .

Theorem 4.2. The operator \bfitT \rho is Fr\'echetdifferentiable, and its Fr\'echetderiva- \prime 3 3 2 2 3 tive at \bfitp \Gamma \inad \scrU is given by \bfitT \rho [\bfitp\Gamma ]: C ([0, 1], \BbbR ) \rightarrow HS( Lt (S , \BbbC )) with

Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms 4 4 \biggl( \sum \sum \biggr) (4.2) \bfitT \prime [\bfitp ]\bfith = (k\rho )2\pi (\varepsilon - 1) \bfitT \prime [\bfitp ]\bfith + (\mu - 1) \bfitT \prime [\bfitp ]\bfith , \rho \Gamma r \rho ,\varepsilon ,j \Gamma r \rho ,\mu ,j \Gamma j=1 j=1

2 2 3 where, for any \bfitA \in Lt (S , \BbbC ),

\int 1 (4.3a) \bigl( \bigl(\bfitT \prime [\bfitp ]\bfith\bigr) \bfitA \bigr) (\bfitx ) = - ik(\bfitx \cdot \bfith) e - ik\bfitx\widehat \cdot\bfitp \Gamma \varepsilon \bfitE i[\bfitA ](\bfitp ) | \bfitp \prime | dt, \rho ,\varepsilon,1 \Gamma \widehat \widehat \BbbP3 ,\bfitx\widehat \BbbM\bfitp \Gamma \Gamma \Gamma 0 \int 1 (4.3b) \bigl( \bigl(\bfitT \prime [\bfitp ]\bfith\bigr) \bfitA \bigr) (\bfitx ) = e - ik\bfitx\widehat \cdot\bfitp \Gamma ( \varepsilon )\prime \bfitE i[\bfitA ](\bfitp ) |\bfitp \prime | dt, \rho ,\varepsilon,2 \Gamma \widehat \BbbP3 ,\bfitx\widehat \BbbM\bfitp \Gamma , \bfith \Gamma \Gamma 0 1 \int \prime \bigl( \bigl( \prime \bigr) \bigr) - ik\bfitx\cdot \bfitp \Gamma \varepsilon \bigl( i \bigr) \prime (4.3c) \bfitT [\bfitp \Gamma ]\bfith \bfitA (\bfitx ) = e \widehat \bfitE [\bfitA ] [\bfitp \Gamma ]\bfith |\bfitp | dt, \rho ,\varepsilon,3 \widehat \BbbP3 ,\bfitx\widehat \BbbM\bfitp \Gamma \Gamma 0 \int 1 \bfitp\prime \cdot \bfith\prime (4.3d) \bigl( \bigl(\bfitT \prime [\bfitp ]\bfith\bigr) \bfitA \bigr) (\bfitx ) = e - ik\bfitx\widehat \cdot\bfitp \Gamma \varepsilon \bfitE i[\bfitA ](\bfitp ) \Gamma dt \rho4 ,\varepsilon, \Gamma \widehat \BbbP3 ,\bfitx\widehat \BbbM\bfitp \Gamma \Gamma \prime 0 |\bfitp \Gamma |

and

\int 1 (4.4a) \bigl( \bigl(\bfitT \prime [\bfitp ]\bfith\bigr) \bfitA \bigr) (\bfitx) = - ik(\bfitx \cdot \bfith) e - ik\bfitx\widehat \cdot\bfitp \Gamma (\bfitx \times ) \mu \bfitH i[\bfitA ](\bfitp )|\bfitp \prime | dt, \rho ,\mu,1 \Gamma \widehat \widehat \widehat \BbbI3 \BbbM\bfitp \Gamma \widetilde \Gamma \Gamma 0 \int 1 \bigl( \bigl( \prime \bigr) \bigr) - ik\bfitx \cdot\bfitp \Gamma \mu \prime i \prime (4.4b) \bfitT [\bfitp \Gamma ]\bfith \bfitA (\bfitx) = e \widehat (\bfitx \times 3)( ) \bfitH\widetilde [\bfitA ](\bfitp \Gamma )| \bfitp | dt, \rho ,\mu,2 \widehat \widehat \BbbI \BbbM \bfitp\Gamma ,\bfith \Gamma 0 1 \int \prime (4.4c) \bigl( \bigl(\bfitT \prime [\bfitp ]\bfith\bigr) \bfitA \bigr) (\bfitx) = e - ik\bfitx\widehat \cdot\bfitp \Gamma (\bfitx \times ) \mu \bigl( \bfitH i[\bfitA ]\bigr) [\bfitp ]\bfith| \bfitp \prime | dt, \rho ,\mu,3 \Gamma \widehat \widehat \BbbI 3 \BbbM\bfitp \Gamma \widetilde \Gamma \Gamma 0 \int 1 \bfitp \prime \cdot \bfith\prime \bigl( \bigl( \prime \bigr) \bigr) - ik\bfitx \cdot\bfitp \Gamma \mu i \Gamma (4.4d) \bfitT [\bfitp\Gamma ]\bfith \bfitA (\bfitx) = e \widehat (\bfitx \times 3) \bfitH\widetilde [\bfitA ](\bfitp\Gamma ) dt. \rho ,\mu,4 \widehat \widehat \BbbI \BbbM\bfitp \Gamma |\bfitp \prime | 0 \Gamma

i i i Here we used the notation \BbbP 3,\bfitx := (\bfitx \times \BbbI3 ) \times \bfitx and \bfitH\widetilde [\bfitA ] := curl \bfitE [\bfitA ]. \widehat \widehat \widehat k

Proof. Let \bfitp \Gamma \in \scrUad. Then there exists \delta > 0 such that \bfitp\Gamma + \bfith \in \scrUad for 3 3 3 3 all \bfith \in C ([0, 1], \BbbR ) satisfying \| \bfith\| C ([0,1],\BbbR ) \leq \delta . We have to show that

Using (3.4), (4.2)--(4.4), and (2.3), the Hilbert--Schmidt operators \bfitT\rho (\bfitp \Gamma +\bfith), \bfitT\rho (\bfitp \Gamma ), \prime 2 2 3 and \bfitT\rho [\bfitp \Gamma ]\bfith can be written as integral operators such that for any \bfitA \in Lt (S , \BbbC ),

\int \bigl( \bfitT (\bfitp + \bfith) \bfitA \bigr) (\bfitx ) = K (\bfitx , \bfittheta )\bfitA (\bfittheta ) ds(\bfittheta ) , \rho \Gamma \widehat \bfitp\Gamma +\bfith \widehat S2 \int \bigl( \bfitT (\bfitp )\bfitA \bigr) (\bfitx ) = K (\bfitx , \bfittheta )\bfitA (\bfittheta ) ds(\bfittheta ) , \rho \Gamma \widehat \bfitp\Gamma \widehat S2 \int \bigl( \bigl(\bfitT \prime [\bfitp ]\bfith\bigr) \bfitA \bigr) (\bfitx ) = K\prime (\bfitx , \bfittheta )\bfitA (\bfittheta ) ds(\bfittheta ) , \rho \Gamma \widehat \bfitp\Gamma ,\bfith \widehat Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms S2

\prime 2 2 2 3\times3 with smooth kernels K\bfitp \Gamma +\bfith , K\bfitp\Gamma , and K\bfitp ,\bfith in L (S \times S , \BbbC ). Using the complete \Gamma

orthonormal system of vector spherical harmonics from (3.6), we obtain that

Thus, Parseval's identity shows that

3\times 3 where \| \cdot\|F denotes the Frobenius norm on \BbbC . Proceeding as in the proof of [22, Thm. 4.2] and applying Taylor's theorem, it follows that

Together with (4.7) this implies (4.5).

As already done for \scrT in Lemma 3.2 we next derive an explicit basis represen- D\rho \prime tation of the Fr\'echet derivative \bfitT \rho [\bfitp \Gamma ]\bfith in terms of the circularly polarized vector m m spherical harmonics \bfitA n and \bfitBn , m = - n, . . . , n, n = 1, 2,..., from (3.7).

2 2 3 Remark 4.3. Let \bfitA \in Lt (S , \BbbC ) with

\infty n \sum \sum (4.8) \bfitA = \bigl( am\bfitA m + bm\bfitB m\bigr) . n n n n n=1 m= - n

Then

Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms \infty n \sum \sum (4.9) \bigl( \bfitT \prime [\bfitp ]\bfith\bigr) \bfitA = \bigl( cm\bfitA m + dm\bfitB m\bigr) , \rho \Gamma n n n n n=1 m= - n

with \infty n\prime m \sum \sum \Bigl( m\prime \bigl\langle \bigl( \prime \bigr) m \prime m\bigr\rangle (4.10a) cn = an\prime \bfitT\rho [\bfitp \Gamma ]\bfith \bfitA n\prime , \bfitA n 2 2 3 Lt (S ,\BbbC ) n\prime =1 m\prime = - n\prime

\prime \prime \Bigr) m \bigl\langle \bigl( \prime \bigr) m m\bigr\rangle + bn\prime \bfitT \rho [\bfitp \Gamma ]\bfith \bfitB n\prime , \bfitA n L2(S2, 3) , t \BbbC

\prime \infty n m \sum \sum \Bigl( m\prime \bigl\langle \bigl( \prime \bigr) m\prime m\bigr\rangle (4.10b) dn = an\prime \bfitT\rho [\bfitp \Gamma ]\bfith \bfitA n \prime , \bfitB n 2 2 3 Lt (S ,\BbbC ) n\prime =1 m\prime = - n\prime

m\prime \bigl\langle \bigl( \prime \bigr) m\prime m\bigr\rangle \Bigr) + bn\prime \bfitT \rho [\bfitp \Gamma ]\bfith \bfitB n\prime , \bfitB n 2 2 3 . Lt (S ,\BbbC )

The inner products in (4.10) can be evaluated using (4.3)--(4.4), the identities (3.18)-- (3.20), and \surd \bigl( i m \bigr) \prime n - 1 m \prime \bfitE [\bfitAn ] [\bfity ] = 8\pi i (\bfitP n ) [\bfity ] , \surd \bigl( i m \bigr) \prime n - 1 m \prime \bfitE [\bfitB n ] [\bfity ] = - 8\pi i (\bfitQn ) [\bfity ] , \surd \bigl( i m \bigr) \prime n 1- m \prime curl \bfitE [\bfitA n ] [\bfity ] = 8\pi k i (\bfitPn ) [\bfity ] , \surd \bigl( i m \bigr) \prime n 1- m \prime curl \bfitE [\bfitB n ] [\bfity ] = 8\pi k i ( \bfitQn ) [\bfity ] ,

and \surd \top \bigl\langle ik(\bfitx \cdot \bfith) e i- k\bfitx\widehat \cdot\bfity \bigl( (\bfitx \times ) \times \bfitx \bigr) , \bfitA m(\bfitx )\bigr\rangle = 8\pi ( - i)n 1- (\bfitP m)\prime [\bfity ]\bfith , \widehat \widehat \BbbI3 \widehat n \widehat L2(S2, 3) n t \BbbC \surd \top \bigl\langlei k(\bfitx \cdot \bfith) e - ik\bfitx\widehat \cdot\bfity \bigl( (\bfitx \times ) \times \bfitx\bigr) , \bfitB m(\bfitx )\bigr\rangle = - 8\pi ( - i)n - 1 (\bfitQ m)\prime [\bfity ]\bfith , \widehat \widehat \BbbI 3 \widehat n \widehat L2(S2, 3) n t \BbbC \surd \top \bigl\langle - ik\bfitx\widehat \cdot\bfity m \bigr\rangle n m \prime ik(\bfitx \cdot \bfith) e (\bfitx \times \BbbI3 ), \bfitA n (\bfitx ) 2 2 3 = 8\pi ( - i) (\bfitP n ) [\bfity ]\bfith , \widehat \widehat \widehat Lt (S ,\BbbC ) \surd \top \bigl\langle - ik\bfitx\widehat \cdot\bfity m \bigr\rangle n m \prime ik(\bfitx \cdot \bfith) e (\bfitx \times \BbbI 3), \bfitB n (\bfitx ) 2 2 3 = 8\pi ( - i) (\bfitQ n ) [\bfity ]\bfith . \widehat \widehat \widehat Lt (S ,\BbbC )

m \prime m \prime Explicit formulas for the derivatives (\bfitP n ) and (\bfitQn ) of the circularly polarized m m spherical vector wave functions \bfitP n and \bfitQn , m = - n, . . . , n, n = 1, 2,... , from (3.10) can be found in Appendix A.

5. Shape optimization for thin em-chiral structures. We develop a shape optimization scheme to determine dielectric thin tubular scattering objects D\rho as

in (3.2) that exhibit comparatively large measures of em-chirality \chiHS at a given frequency. In addition to the frequency, we also fix the material parameters \varepsilon , \mu and r r the length | \Gamma| of the center curve \Gamma of D\rho before we start the optimization process. Furthermore, we assume that the radius \rho > 0 of the circular cross-section of D\rho is sufficiently small with respect to the wave length of the incident fields, such thatthe

leading order term \scrT D\rho in the asymptotic expansion (3.5) gives a good approximation of the far field operator \scrF . D\rho Combining (2.10) and (2.12) we find that

2 2 3 Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms 0 \leq \chiHS (\scrG ) \leqHS \|\scrG for any \scrG \in HS( Lt (S , \BbbC )) .

Accordingly, recalling the definition of the nonlinear operator \bfitT \rho in (4.1), we normalize the chirality measure \chiHS and consider the bounded objective functional

JHS : \scrU ad \rightarrow [0, 1], which is given by

\chi \bigl( \bfitT (\bfitp )\bigr) (5.1) J (\bfitp ) := HS \rho \Gamma . HS \Gamma \|\bfitT (\bfitp )\| \rho \Gamma HS

We discuss the optimization problem \bigl( \bigr) (5.2) find arg min - JHS(\bfitp \Gamma ) subject to |\Gamma | = L \bfitp \Gamma \in \scrUad

for some prescribed length L > 0. Remark 5.1. Since the leading order term in the asymptotic expansion of the

electric far field pattern due to a thin tubular scattering object in (3.3) is homogeneous of degree two with respect to the radius \rho of the cross-section of the scatterer D\rho , the \bigl( \bigr) same is true for \bfitT \rho (\bfitp\Gamma ) as well as for \chi HS \bfitT\rho (\bfitp \Gamma ) and \|\bfitT \rho (\bfitp \Gamma )\|HS with \bfitp\Gamma \inad \scrU . In particular, the relative chirality measure JHS(\bfitp \Gamma ) and thus also (local) minimizers for (5.2) are independent of \rho .

Below we rewrite (5.2) as an unconstrained optimization problem and apply a quasi-Newton method to approximate a (local) minimizer. This requires the Fr\'echet derivative of the objective functional J , which for any \bfitp \in \scrU such HS \Gamma ad that \bfitT \rho (\bfitp \Gamma ) \in X, where the space X has been introduced in (2.13), and for any \bfith \in C3([0, 1], \BbbR 3) is given by

\prime \bigl[ \bigr] \prime \bigl[ \bigr] \bigl\langle \prime \bigr\rangle (\chi HS) \bfitT\rho (\bfitp \Gamma ) (\bfitT [\bfitp \Gamma ]\bfith) \chi HS \bfitT \rho (\bfitp \Gamma ) Re \bfitT \rho (\bfitp \Gamma ), \bfitT [\bfitp\Gamma ]\bfith J \prime [\bfitp ]\bfith = \rho - \rho HS . HS \Gamma \| \bfitT (\bfitp )\| \| \bfitT (\bfitp )\| 3 \rho \Gamma HS \rho \Gamma HS \prime \prime The Fr\'echet derivatives\chi ( HS ) and \bfitT \rho have already been established in (2.14) and in Theorem 4.2, respectively.

5.1. Discretization and regularization. In the numerical implementation of the optimization algorithm, we approximate admissible center curves \Gamma of thin tubular scatterers D\rho using interpolating cubic splines with the not-a-knot condition at the end points. We consider a partition

\bigtriangleup := \{0 = t1 < t2 < \cdot \cdot < tn = 1 \} \subseteq [0, 1]

and denote the not-a-knot spline that interpolates the curve \Gamma with parametrization (j) #” #” 3n \bfitp \Gamma \in \scrUad at the nodes \bfitx = \bfitp \Gamma (tj), j = 1, . . . , n, by \bfitp\bigtriangleup [\bfitx ], where \bfitx \in \BbbR is the vector that contains the coordinates of the nodes \bfitx(1) ,..., \bfitx (n). The space of all

not-a-knot splines with respect to \bigtriangleup is denoted by \scrP \bigtriangleup \notad \subseteq\scrU . To stabilize the optimization procedure, and to incorporate the constraint in (5.2), we include two penalty terms in the objective functional. The total squared curvature functional \Psi 1 : \scrP \bigtriangleup \rightarrow \BbbR is defined by

\int 1 2 \prime \Psi 1(\bfitp \bigtriangleup ) := \kappa (s) | \bfitp\bigtriangleup (s) | ds , 0

Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms where

denotes the curvature of the curve \Gamma that is parametrized by \bfitp \bigtriangleup . Adding \alpha1 \Psi 1 with a suitable regularization parameter \alpha1 > 0 as apenalty term to - JHS prevents minimizers from being too strongly entangled.

Furthermore, we define2 \Psi : \scrP \bigtriangleup \rightarrow \BbbR by

\Phi : \scrP \bigtriangleup \rightarrow \BbbR ,

(5.3) \Phi (\bfitp ) := - J (\bfitp ) + \alpha \Psi (\bfitp ) + \alpha \Psi (\bfitp ) , \bigtriangleup HS \bigtriangleup 1 1 \bigtriangleup 2 2 \bigtriangleup and we consider the unconstrained optimization problem

\ast (5.4) find \bfitp \bigtriangleup := arg min \Phi (\bfitp\bigtriangleup ) . \bfitp \in \scrP \bigtriangleup \bigtriangleup Before we describe the quasi-Newton optimization scheme, we discuss the Fr\'echet derivatives of the functionals \Psi and \Psi . A short calculation gives the following 1 2 result. Lemma 5.2. The mappings \Psi1 and \Psi 2 are Fr\'echet differentiable from 2 3 \scrP \bigtriangleup \subseteq C ([0, 1], \BbbR ) to \BbbR . Their Fr\'echetderivatives at \bfitp \bigtriangleup \in \scrP\bigtriangleup are given by \prime \Psi 1[\bfitp\bigtriangleup ]: \scrP \bigtriangleup \rightarrow \BbbR with

\int 1\biggl( \prime \prime \prime \prime \prime \prime 2 \prime \prime \bfitp \bigtriangleup \cdot \bfith\bigtriangleup | \bfitp \bigtriangleup | (\bfitp\bigtriangleup \cdot \bfith\bigtriangleup ) \Psi\prime [\bfitp ]\bfith = 2 - 3 1 \bigtriangleup \bigtriangleup |\bfitp \prime |3 |\bfitp \prime |5 0 \bigtriangleup \bigtriangleup \bigl( \prime \prime \prime \prime \prime \prime \bigr) \prime \prime \prime \prime \prime \prime \prime \prime 2 \bfitp \cdot \bfith + \bfitp \cdot \bfith (\bfitp \cdot \bfitp ) (\bfitp \cdot \bfith )(\bfitp \cdot \bfitp ) \biggr) - 2 \bigtriangleup \bigtriangleup \bigtriangleup \bigtriangleup \bigtriangleup \bigtriangleup + 5 \bigtriangleup \bigtriangleup \bigtriangleup \bigtriangleup ds , |\bfitp \prime |5 | \bfitp \prime | 7 \bigtriangleup \bigtriangleup \prime and \Psi2 [\bfitp \bigtriangleup ]: \scrP \bigtriangleup \rightarrow \BbbR with n - 1 \biggl( \int tj+1 \prime \prime \biggr) \biggl( \int tj+1 \biggr) \sum \bfitp\bigtriangleup \cdot \bfith\bigtriangleup L \Psi\prime [\bfitp ]\bfith = - 2 ds - | \bfitp\prime | ds . 2 \bigtriangleup \bigtriangleup |\bfitp \prime | n - 1 \bigtriangleup j=1 tj \bigtriangleup tj

5.2. The BFGS scheme for the regularized optimization problem. We apply a BFGS scheme with an inexact Armijo-type line search and a cautious update rule as described in [30] to approximate a solution to (5.4). #” Choosing an initial guess \bfitx for the coordinates of the nodes of the center #” 0 spline \bfitp [\bfitx ], the BFGS iteration for (5.4) is given by \bigtriangleup 0 #” #” (5.5) \bfitx \ell+1 = \bfitx \ell + \lambda \ell \bfitd\ell , \ell = 0 , 1,...,

where \bfitd\ell is obtained by solving the linear system

Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms \bigl( #” \bigr) H\ell \bfitd\ell = - \nabla\Phi \bfitp \bigtriangleup [\bfitx \ell ] ,

2 \bigl( #” \bigr) and H\ell is an approximation to the Hessian matrix \nabla \Phi \bfitp \bigtriangleup [\bfitx \ell ] .

We start with H0 = \BbbI 3n, and after each iteration we use the cautious update rule from [30], which is given by

\Biggl\{ H \bfs \bfs \top H \bfity \bfity \top \bfity \top \bfs #” \ell \ell \ell \ell \ell \ell \ell \ell \bigm| \bigl( \bigr) \bigm| H\ell - \bfs\top H \bfs + \bfity \top \bfs if | \bfs |2 > \varepsilon \bigm| \nabla \Phi \bfitp \bigtriangleup [\bfitx \ell ] \bigm| , (5.6) H\ell +1 = \ell \ell \ell \ell \ell \ell H\ell otherwise ,

where

#” #” \bigl( #” \bigr) \bigl( #” \bigr) s\ell := \bfitx \ell+1 - \bfitx \ell , \bfity \ell := \nabla \Phi \bfitp \bigtriangleup [\bfitx \ell +1 ] - \nabla\Phi \bfitp \bigtriangleup [\bfitx \ell ] ,

and \varepsilon > 0 is a parameter. This ensures positive definiteness of H\ell throughout the iteration (cf. [30]).

As suggested in [30], we use an inexact Armijo-type line search to determine the step size \lambda\ell in (5.5). Choosing parameters \sigma \in (0, 1) and \delta \in (0, 1), we identify the smallest integer j = 0, 1,..., such that \delta j satisfies

\bigl( #” j \bigr) \bigl( #” \bigr) j \bigl( #” \bigr) \top (5.7) \Phi \bfitp\bigtriangleup [\bfitx \ell ] + \delta \bfitd \ell \leq \Phi \bfitp\bigtriangleup [\bfitx \ell ] + \sigma \delta \nabla \Phi \bfitp \bigtriangleup [\bfitx \ell ] \bfitd\ell .

j Then we set \lambda\ell = \delta . In the numerical examples presented in section 6 below, we use the parameters 5- - 4 \varepsilon = 10 , \sigma = 10 , and \delta = 0.9 in (5.6) and (5.7). Denoting by N \in \BbbN the maximal degree of vector spherical harmonics that are used in the basis representation of the \prime operators \bfitT\rho (\bfitp \bigtriangleup ) (see Lemma 3.2) and \bfitT\rho [\bfitp \bigtriangleup ]\bfith (see Remark 4.3), we consider discrete approximations \bfitT (\bfitp ) \in Q\timesQ and \bfitT \prime [\bfitp ]\bfith \in Q\times Q with Q = 2N(N + 2). \rho ,N \bigtriangleup \BbbC \rho ,N \bigtriangleup \BbbC #” We approximate the integrals over the parameter range [0, 1] of the spline \bfitp [\bfitx ] in #” \bigtriangleup \ell the evaluation of \nabla \Phi (\bfitp\bigtriangleup [\bfitx \ell ]), \ell = 0, 1,..., using a composite Simpson's rule with M = 5 nodes on each subinterval of the partition \bigtriangleup . We stop the BFGS iteration #” #” #” - 4 when | \bfitx \ell +1 - \bfitx \ell | /| \bfitx \ell | < 10 . The fact that not a single partial differential equation has to be solved during the optimization process makes this algorithm particularly efficient.

6. Numerical results. In the numerical examples below we use k = 1 for the wave number; i.e., the wave length of the incident fields is \lambda \approx 6.28. To assess the numerical accuracy of the asymptotic representation formula (3.3), we have compared

numerical approximations of electric far field patterns corresponding to thin tubular scattering objects D\rho that have been computed, on the one hand, using the C++ boundary element library Bempp [38] and, on the other hand, using the leading order term in the asymptotic perturbation formula (3.3). This limited study in [12] suggests that the approximations obtained from the leading order term in (3.3) are accurate

within a relative error of less than 5\% when the radius \rho of the thin tube D\rho is less than 1.5\% of the wave length of the incident field, i.e., when \rho \lesssim 0.1 in our setting. This is also the range of radii, where we expect the results of the following examples to be applicable.

Example 6.1. In the first example, we consider the material parameters \varepsilonr = 5 and \mu = 1. We use \alpha = 0.0005 and \alpha = 0.5 for the regularization parameters r 1 2 #” in (5.3), and the length constraint is chosen to be L = 6. For the initial guess \bfitx 0 we consider n = 20 equidistant nodes on the straight line segment between (0, 0, 3)-

and (0, 0, 3), and then we slightly perturb the first two components of each node by Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms adding random numbers between - 0.02 and 0.02. We note that the nodes cannot be exactly on the straight line segment, because then the thin tubular scattering object #” with center curve \bfitp\bigtriangleup [\bfitx 0] would be em-achiral, and thus the objective functional \Phi would not be differentiable at the initial guess.

Fig. 6.1. Convergence history for Example 6.1. Top left: Initial guess. Bottom right: Final result.

Remark 3.3 recommends that the maximal degree N of vector spherical harmonics

used in the basis representations of the operator \bfitT\rho (\bfitp \bigtriangleup ) (see Lemma 3.2) and of its \prime Fr\'echet derivative \bfitT \rho [\bfitp\bigtriangleup ]\bfith (see Remark 4.3) should be greater than R, where BR(0) is the smallest ball centered at the origin that contains the scattering object D\rho . For this example we use N = 5. In Figure 6.1 we show the initial guess (top left), some intermediate results that are obtained after \ell = 10, 30, 50, 70 iterations, and the final result (bottom right) that

is obtained after \ell = 88 iterations of the BFGS scheme. In each of these plots we also included the corresponding value of the relative chirality measure JHS. During the optimization process the almost straight initial guess for the center curve winds up to a helix. The orientation of this helix in space and whether it is left or right turning #” depend on the orientation of the initial curve \bfitp [\bfitx ] and on the particular values of \bigtriangleup 0 #” the random perturbations that are used to set up the initial guess \bfitx 0. In Figure 6.2 (left) we show the evolution of the relative chirality measure JHS during the optimization process. For comparison, we also include the corresponding values of the functional J2 : \scrU ad \rightarrow [0, 1],

\bigl( \bigr) \chi 2 \bfitT \rho (\bfitp\Gamma ) J2(\bfitp \Gamma ) := , \|\bfitT \rho (\bfitp\Gamma )\|HS

which is defined analogously to (5.1) but corresponds to the chirality measure \chi2 from (2.9) instead of \chi from (2.11). We observe that both functionals are increasing by HS several orders of magnitude during the optimization process and that, in accordance Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms with (2.12), JHS \leq J2. In Figure 6.2 (center) we show plots of JHS and J2 for the optimized structure from Figure 6.1 (bottom right) as a function of \varepsilonr , and in Figure 6.2 (right) we show

0.3 0.3 0.3

0.25 0.25 0.25

0.2 0.2 0.2

0.15 0.15 0.15

0.1 0.1 0.1

0.05 0.05 0.05

0 0 0 0 20 40 60 80 10 20 30 40 50 0.5 1 1.5 2

Fig. 6.2. Normalized chirality measures JHS and J2 for Example 6.1 with L = 6. Left: Evo- lution during BFGS iteration. Center: Sensitivity with respect to relative permittivity \varepsilonr . Right: Sensitivity with respect to wave number k.

Fig. 6.3. Optimal structures for different length constraints L = 4, 6, and 8 (left to right) in Example 6.1. Top row: Initial guesses. Bottom row: Final results.

corresponding plots of JHS and J2 as a function of k. The vertical lines in these plots indicate the values of \varepsilon and k that have been used in the shape optimization r (i.e., \varepsilon r = 5 and k = 1). We observe that the relative chirality measures JHS and J2 are monotonically increasing in \varepsilonr . On the other hand, JHS reaches a local maximum at the wave number k = 1 that has been used in the shape optimization, and there is a local maximum of J2 close to this wave number. This suggests that the outcome of the optimization process is sensitive to the wave length of the incident field and that

the obtained optimality property is restricted to a rather narrow band of frequencies. Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms To study the dependence of the optimized center curve on the length constraint L in (5.2), we repeat the shape optimization with L = 4 and L = 8 instead of L = 6. The corresponding initial splines are shown in Figure 6.3 (top left and top right).

Fig. 6.4. Convergence history for Example 6.2. Top left: Initial guess. Bottom right: Final result.

In accordance with Remark 3.3, we choose N = 4 when L = 4 and N = 6 when L = 8 for the maximal degree N of vector spherical harmonics that is used in the \prime basis representations of the operator \bfitT\rho (\bfitp \bigtriangleup ) and of its Fr\'echet derivative \bfitT\rho [\bfitp \bigtriangleup ]\bfith. In Figure 6.3 (bottom left and bottom right) we show the final results that are obtained

by the optimization procedure after 68 iterations (for L = 4) and after 92 iterations (for L = 8) of the BFGS scheme. For comparison we have also included the initial guess and the final result for L = 6 from Figure 6.1 in the second column of Figure 6.3. It is interesting to note that the diameters and the pitches of the three helices that are found by the optimization procedure are basically the same for the three different

values of L, and that just the number of turns of each helix increases with increasing length constraint L. The relative chirality measures JHS and J2 attain essentially the same values for these three structures, but the total interaction cross-section increases with increasing values of L (not shown).

Example 6.2. In this second example we use \varepsilonr = 30 and \mur = 1; i.e., the permittivity contrast is much larger than in the first example. We choose \alpha1 = 0.0001 and \alpha 2 = 0.1 for the regularization parameters in (5.3), and the length constraint is L = 20. #” For the initial guess \bfitx 0 we consider n = 45 nodes on a curve given by two parallel line segments connected by a half circle as shown in Figure 6.4 (top left). The distance between the two vertical line segments is 2. Again we slightly perturb the first two components of each node by adding random numbers between - 0.02 and 0.02 to obtain a well-defined gradient of the objective functional at the initial

guess. In accordance with Remark 3.3 we use N = 6 for the maximal degree of vector Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms spherical harmonics that is used in the basis representations of the operator \bfitT \rho (\bfitp \Gamma ) \prime and of its Fr\'echet derivative \bfitT\rho [\bfitp \Gamma ]\bfith. In Figure 6.4 we show the initial guess (top left), some intermediate results that

0.4 0.4 0.4

0.35 0.35 0.35

0.3 0.3 0.3 0.25 0.25 0.25 0.2 0.2 0.2

0.15 0.15 0.15 0.1 0.1 0.1 0.05 0.05 0.05

0 0 0 0 20 40 60 80 100 120 140 160 180 10 20 30 40 50 0.5 1 1.5 2

Fig. 6.5. Normalized chirality measures JHS and J2 for Example 6.2 with L = 20. Left: Evolution during BFGS iteration. Center: Sensitivity with respect to relative permittivity \varepsilonr . Right: Sensitivity with respect to wave number k.

are obtained after \ell = 10, 30, 50, 70 iterations, and the final result (bottom right) that is obtained after \ell = 189 iterations of the BFGS scheme. During the optimization process the U-shaped initial guess winds up to a double helix. In Figure 6.5 (left) we show the evolution of the relative chirality measures J HS and J2 during the optimization process. As in Example 6.1, both functionals increase by several orders of magnitude during the optimization process. Figure 6.5 (center) shows plots of JHS and J2 for the optimized structure from Figure 6.4 (bottom right) as a function of \varepsilon r, and in Figure 6.5 (right) we show corresponding plots of JHS and J as a function of k. The vertical lines in these plots indicate the values of \varepsilon 2 r and k that have been used in the shape optimization (i.e., \varepsilon r = 30 and k = 1). The relative chirality measures JHS and J2 are monotonically increasing in \varepsilon r. On the other hand, JHS reaches a local maximum at k = 1, which is the wave number that was used in the shape optimization, and there is a local maximum of J2 close to this wave number. The sensitivity of both relative chirality measures with respect to the wave number is more pronounced than in Example 6.1.

To study the dependence of the optimized center curve on the length constraint L, we repeat the shape optimization with L = 15 and L = 25 instead of L = 20. The corresponding initial splines are shown in Figure 6.6 (top left and top right). In accordance with Remark 3.3, we choose N = 5 when L = 15 and N = 7 when L = 25 for the maximal degree N of vector spherical harmonics that is used in the \prime basis representations of the operator \bfitT\rho (\bfitp \bigtriangleup ) and of its Fr\'echet derivative \bfitT\rho [\bfitp \bigtriangleup ]\bfith. In Figure 6.6 (bottom left and bottom right) we show the final results that are obtained by the optimization procedure after 109 iterations (for L = 15) and after 158 iterations (for L = 25) of the BFGS scheme. For comparison we have also included the initial guess and the final result for L = 20 from Figure 6.4 in the second column of Figure 6.6.

As we already observed in Example 6.1 for the helix, the diameters and the pitches of the three double-helices that are found by the optimization procedure are basically the same for the three different values of L, and just the number of turns of this double- helix increases with increasing length constraint L. The relative chirality measures J and J attain essentially the same values for these three structures, but the total HS 2 interaction cross-section increases with increasing values of L.

Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms Conclusions. Electromagnetic chirality measures quantify differences in the re- sponse of scattering objects or media due to left and right circularly polarized incident waves. We have considered the shape optimization problem to design dielectric thin

Fig. 6.6. Optimal structures for different length constraints L = 15, 20, and 25 (left to right) in Example 6.2. Top row: Initial guesses. Bottom row: Final results.

tubular scattering objects with comparatively large measures of electromagnetic chi-

rality. We have applied an asymptotic representation formula for the scattered electromagnetic field due to such thin tubular structures to develop an efficient iterative shape optimization scheme. Our numerical results suggest that thin helical structures are candidates for optimal thin tubular scatterers, and that high electric permittivity contrast increases the chiral effect. We also found that the chirality measure of opti-

mized structures decays rather quickly if a different frequency than the one used for the shape optimization is considered. We have restricted the discussion to dielectricscattering objects because the asymptotic representation formula from [12] has so far only been justified in this case. An extension of the asymptotic representation formula to metallic scatterers

and the shape optimization for metallic thin tubular structures will be the subject of future work.

Appendix A. Derivatives of spherical vector wave functions. The ex- \prime plicit basis representation of the Fr\'echet derivative \bfitT\rho [\bfitp \Gamma ]\bfith in Remark 4.3 contains m m derivatives of the circularly polarized spherical vector wave functions \bfitP n and \bfitQ n , m m m = - n, . . . , n, n = 1, 2,.... Recalling the definition of \bfitPn and \bfitQ n in (3.10), we pro- m vide a detailed discussion of the derivatives of the spherical vector wave functions \bfitMn

Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms from (3.9) and

\sqrt{} m n(n + 1) m 1 \prime m 3 curl \bfitMn (\bfitx) = jn(kr)Yn (\bfitx\widehat )\bfitx\widehat + (jn(kr) + krjn(kr))\bfitUn (\bfitx\widehat ) , \bfitx \in \BbbR r r

(see, e.g., [29, Thm. 2.43]). Both functions are best expressed in spherical coordinates, \left[ \right] \left[ \right] x1 sin(\theta ) cos(\varphi ) \bfitx = x2 = r sin(\theta ) sin(\varphi) =: \bfitpsi (r, \theta ,\varphi) , r > 0 , \theta \in [0, \pi ] , \varphi \in [0, 2\pi ) , x3 cos(\theta )

and consist of terms of the form \bfitF (\bfitx ) = J(r)\bfitW (\theta ,\varphi), where \bfitW is one of the vector m m m spherical harmonics Yn \bfitx\widehat , \bfitUn , or \bfitV n . Using the chain rule

(\bfitF \circ \bfitpsi )\prime = (\bfitF \prime \circ \bfitpsi )\bfitpsi \prime

and observing that (\bfitpsi \prime ) - 1 is known explicitly, it suffices to compute the partial derivatives of \bfitM m and curl \bfitM m with respect to the spherical coordinates. More precisely, n n with the spherical unit coordinate vectors \left[ \right] \left[ \right] \left[ \right] sin(\theta ) cos(\varphi) cos(\theta ) cos(\varphi) - sin(\varphi ) \bfitx\widehat := sin(\theta ) sin(\varphi) , \bfittheta\widehat := cos(\theta ) sin(\varphi ) , \bfitvarphi\widehat := cos(\varphi) , cos(\theta ) - sin(\theta ) 0

we obtain \left[ \right] 1 0 0 \prime \bigl[ \bigm| \bigm| \bigr] \bfitpsi (r, \theta ,\varphi) = \bfitx\widehat \bigm| \bfittheta\widehat \bigm| \bfitvarphi\widehat 0 r 0 , 0 0 r sin(\theta )

Note that throughout this appendix, we will suppress the dependence on r, \theta , and \varphi of the unit coordinate vectors and most other functions. m m We start with the factors in \bfitMn and curl \bfitMn that depend only on the angular m variables \theta and \varphi and express their derivatives in terms of the spherical harmonics Yn and the partial derivative of Y m with respect to \theta . First, we note that the derivatives n of the unit coordinate vectors satisfy

\partial \theta \bfitx = \bfittheta\widehat , \partial \theta \bfittheta\widehat = \bfitx- , \partial \theta \bfitvarphi = 0 , \widehat \widehat \widehat \partial\varphi \bfitx\widehat = sin(\theta )\bfitvarphi\widehat , \partial \varphi \bfittheta\widehat = cos(\theta )\bfitvarphi\widehat , \partial \varphi \bfitvarphi\widehat = - sin(\theta )\bfitx\widehat - cos(\theta )\bfittheta\widehat . m A particular choice of spherical harmonics Yn , m = - n, . . . , n, n = 0, 1,..., is obtained from the definition \sqrt{} 2n + 1 (n - |m|)! (A.2) Y m := CmP |m | (cos(\theta ))eim\varphi with Cm := , n n n n 4\pi (n + |m |)!

m 2 m/2 m where Pn (t) := (1 - t ) (d/dt) Pn(t), m = 0, . . . , n, denote the associated Le- m gendre functions (see, e.g., [29, p. 41]). Derivatives of Yn with respect to \varphi just amount to multiplications with powers of im. The first derivative of Y m with respect n to \theta is calculated explicitly from m m m+1 dPn 2 (m - 2)/2 d Pn 2 m/2 d Pn Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms (t) = - mt(1 - t ) m (t) + (1 - t ) m+1 (t) dt dt dt 1 m+1 mt m = Pn (t) - Pn (t) , n \in \BbbN , m = 0, . . . , n , (1 - t2)1/2 1 - t2

which gives m m m Cn i- \varphi m+1 - m m (A.3) \partial\theta Yn = m cot(\theta )Yn - m+1 e Yn and \partial \theta Yn = \partial \theta Yn Cn for n \in and m = 0, . . . , n. Here, we use P n+1 := 0 and Y n+1 := 0 for convenience \BbbN n n of notation. As spherical harmonics are eigenfunctions of the Laplace--Beltrami operator on the unit sphere,

1 \bigl( m\bigr) 1 2 m m (A.4) \partial\theta sin(\theta )\partial \theta Yn + \partial\varphi Yn = - n(n + 1)Yn sin \theta sin2(\theta ) m (see, e.g., [29, p. 41]), we can compute the second derivative of Yn with respect to \theta as \biggl( 2 \biggr) 2 m m m m (A.5) \partial\theta Yn = - cot(\theta )\partial\theta Yn + 2 - n(n + 1) Yn . sin (\theta ) We apply these formulas to find expressions for the derivatives of the vector m m m spherical harmonics Yn \bfitx\widehat , \bfitUn , and \bfitVn with respect to \theta and \varphi. For the radially m oriented Yn \bfitx\widehat we obtain m m m (A.6a) \partial \theta (Yn \bfitx\widehat ) = \partial\theta Yn \bfitx\widehat + Yn \bfittheta\widehat , 1 im (A.6b) \partial (Y m\bfitx ) = Y m\bfitx + Y m\bfitvarphi . sin(\theta ) \varphi n \widehat sin(\theta ) n \widehat n \widehat

m m From the definition (3.6) we find for \bfitUn and \bfitV n that

m 1 \Bigl( m im m \Bigr) (A.7a) \bfitU = \partial\theta Y \bfittheta\widehat + Y \bfitvarphi , n \sqrt{} n(n + 1) n sin( \theta ) n \widehat

m 1 \Bigl( m im m \Bigr) (A.7b) \bfitV = \partial\theta Y \bfitvarphi - Y \bfittheta\widehat . n \sqrt{} n(n + 1) n \widehat sin( \theta ) n

We further deduce m 1 \Bigl( m 2 m (A.8a) \partial\theta \bfitUn = \sqrt{} - \partial\theta Yn \bfitx\widehat + \partial\theta Y n \bfittheta\widehat n(n + 1) im \Bigr) - \bigl( cot(\theta )Y m - \partial Y m\bigr) \bfitvarphi , sin(\theta ) n \theta n \widehat

1 m 1 \Bigl( im m im \bigl( m m\bigr) (A.8b) \partial\varphi \bfitU = - Y \bfitx + \partial\theta Y - cot(\theta )Y \bfittheta\widehat sin(\theta ) n \sqrt{}n (n + 1) sin(\theta ) n \widehat sin(\theta ) n n

2 \Bigl( m m m\Bigr) \Bigr) + cot(\theta )\partial\theta Y - Y \bfitvarphi , n sin2(\theta ) n \widehat

m 1 \Bigl( im m im \bigl( m m\bigr) (A.8c) \partial\theta \bfitV = Y \bfitx + cot(\theta )Y - \partial\theta Y \bfittheta\widehat n \sqrt{}n (n + 1) sin(\theta ) n \widehat sin( \theta ) n n

\Bigr) + \partial 2Y m\bfitvarphi , \theta n \widehat

\Bigl( \Bigl( 2 \Bigr) 1 m 1 m m m m (A.8d) \partial \varphi \bfitVn = \sqrt{} - \partial\theta Yn \bfitx\widehat + 2 Yn - cot(\theta )\partial\theta Yn \bfittheta\widehat Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms sin(\theta ) n(n + 1) sin (\theta ) im \Bigr) \bigl( m m\bigr) + \partial\theta Yn - cot(\theta )Yn \bfitvarphi\widehat . sin(\theta )

The representations (A.3)--(A.8) contain several terms that are ill-suited tonu- merical evaluation for \theta close to 0 or \pi . These are

2 1 m 1 \bigl( m m\bigr) m m m (A.9) Y , cot(\theta )Y - \partial \theta Y , Y - cot(\theta )\partial\theta Y . sin(\theta ) n sin(\theta ) n n sin2(\theta ) n n

Note that the first two expressions in (A.9) always appear in combination witha

factor m in (A.3)--(A.8) and thus are only relevant for m \not = 0. We will only consider m \geq 0 in the following paragraphs, as the corresponding formulas for negative m can immediately be obtained by complex conjugation. To rewrite the first term in (A.9), we use the recurrence relation

m Pn (t) 1 \bigl( m+1 m - 1 \bigr) \surd = Pn (t)+(n+m)(n - m+1)Pn (t) , n \geq 2 , m = 1, . . . , n - 1 , 1 - t2 2mt

for the associated Legendre functions (see, e.g., [29, p. 35]). Inserting this into (A.2) gives

Y m Cm \Bigl( e i- \varphi (n + m)(n - m + 1)ei\varphi \Bigr) (A.10) n = n Y m+1 + Y m 1- sin(\theta ) 2m cos(\theta ) m+1 n m 1- n Cn Cn

for n \geq 2 and m = 1, . . . , n - 1. Furthermore, differentiating Rodrigues' formula for the associated Legendre functions (see, e.g., [29, Thm. 2.6]) n times shows that n (2n)! n Pn (cos(\theta )) = n sin (\theta ). Therefore, 2 n! n Yn n (2n)! n - 1 in\varphi (A.11) = Cn n sin (\theta )e , n \in \BbbN . sin(\theta ) 2 n!

For the second term in (A.9), from (A.3) we have that

m m+1 m 1 \bigl( m m\bigr) Cn - i\varphi Yn Yn (A.12) cot(\theta )Y - \partial\theta Y = e - (m - 1) cos(\theta ) . sin(\theta ) n n Cm+1 sin( \theta ) sin2(\theta ) n

For m = 1, this can be evaluated using (A.10). For n \geq 2 and m = 2, . . . , n, - 2 m expressions for sin (\theta )Yn are immediately obtained from (A.10) and (A.11). Finally, the third term in (A.9) satisfies

2 m m m (A.13) Y - cot(\theta )\partial\theta Y sin2(\theta ) n n Cm Y m+1 Y m n i- \varphi n \bigl( 2 2 \bigr) n = m+1 e cos(\theta ) + m - m cos (\theta ) 2 . Cn sin(\theta ) sin (\theta )

For n, m \geq 2, no new issues arise, and for m = 1, the last term on the right-hand 1 side of (A.13) reduces to Yn . In (A.13) we also have to consider the case m = 0, where (A.2) gives

0 0 0 \prime (A.14) - cot(\theta )\partial\theta Yn = - Cn cot(\theta )\partial\theta Pn(cos(\theta )) = Cn cos(\theta )Pn(cos(\theta )) .

Downloaded 09/16/21 to 129.13.172.58 Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms For numerical implementations of (A.3), (A.5)--(A.8) we suggest using the expressions directly when \theta \in [\pi4 / , 3\pi4] / and replacingthe problematic terms with the expressions from (A.10)--(A.14) when \theta \in [0, \pi4) / or \theta \in (3\pi4 / , \pi ].

m m We continue with the factors in \bfitM n and curl \bfitM n that depend only on the radial variable r, i.e.,

j (kr) j (kr) + kr j\prime (kr) (A.15) j (kr) , n , n n . n r r We require the derivatives

\partial j (kr) = kj\prime (kr) , r n n \prime jn(kr) krj (kr) - jn( kr) \partial = n , r r r2 \prime 2 \prime \prime \prime \Bigl( jn(kr) + krjn(kr)\Bigr) (kr) jn(kr) + krjn(kr) - jn(kr) \partial r = . r r2 These may be simplified using the spherical Bessel differential equation

2 \prime \prime \prime 2 t jn(t) + 2tjn(t) + (t - n(n + 1)) jn(t) = 0

(see, e.g., [29, p. 54]) and the recurrence relation

\prime n jn(t) = jn(t) - jn+1(t ) t (see, e.g., [31, 10.51.2]) to obtain

n (A.16a) \partial j (kr) = j (kr) - kj (kr) , r n r n n+1 j (kr) (n - 1)j (kr) - krj (kr) (A.16b) \partial n = n n+1 , r r r2

j (kr) + krj\prime (kr) - krj\prime (kr) + (n(n + 1) - 1 - (kr)2)j (kr) (A.16c) \partial n n = n n r r r2 2 2 krjn+1(kr) + (n - 1 - (kr) )jn(kr) = 2 . r For small values of r > 0, the expansion of the spherical Bessel functions in powers of r (see, e.g., [29, Def. 2.26]) should be inserted into (A.15) and (A.16) and truncated

to a finite sum for numerical evaluation. In particular, we note thatfor n = 1, m negative powers of r seem to remain in (A.1) when the two summands in curl \bfitMn are inserted separately. However, some tedious calculations show that these terms cancel as expected when the sum is formed. Hence, for numerical evaluation in the case n = 1, all terms of order r - 1 should be left out of the calculation to avoid cancellation effects.

Acknowledgments. The authors would like to thank I. Fernandez-Corbaton, X. Garcia-Santiago, and C. Rockstuhl for many interesting discussions related to the

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