MAXIMIZING THE ELECTROMAGNETIC CHIRALITY OF THINDIELECTRIC TUBES\ast
TILO ARENS\dagger , ROLAND GRIESMAIER\dagger , AND MARVIN KN\"OLLER\dagger
\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . Any time-harmonic electromagnetic wave can be uniquely decomposed into left andright circularly polarized components. The concept of electromagnetic chirality (em-chirality) de-scribes differences in the interaction of these two components with a scattering object or medium.Such differences can be quantified by means of em-chirality measures. These measures attain theirminimal value zero for em-achiral objects or media that interact essentially in the same way with leftand right circularly polarized waves. Scattering objects or media with positive em-chirality measureinteract 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 tubularstructures consisting of dielectric isotropic materials that exhibit large measures of em-chirality ata 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 examplessuggest that thin helical structures are at least locally optimal among this class of scattering objects.
1. Introduction. The traditional notion of chirality defines chiral objects asthose that cannot be superimposed on their mirror images. Accordingly, nonchiralobjects are called achiral. Chirality is frequently observed in nature, and chiral phe-nomena have been intensively studied in the past two centuries in various scientificdisciplines, e.g., biology, chemistry, and physics. Currently, a very active research fieldis the interaction of chiral objects with electromagnetic fields, which is also the topicof 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 theinteraction with electromagnetic waves. We follow a different, fairly new, approachthat was proposed in [16] to circumvent these difficulties. The key idea is to quantifychiral effects directly in terms of the object's interaction with electromagnetic fieldsinstead of quantifying them only in terms of geometric considerations.
This work is concerned with scattering of time-harmonic electromagnetic waves bya compactly supported isotropic dielectric object in three-dimensional free space. Us-ing Maxwell's equations to model electromagnetic wave propagation, we can uniquelydecompose any incident field, as well as the corresponding scattered field, away fromthe scatterer into left and right circularly polarized components. The concept of elec-tromagnetic chirality (em-chirality) compares the interaction of these two componentswith 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\bfF \bfu \bfn \bfd \bfi \bfn \bfg : Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Founda-
1980 TILO ARENS, ROLAND GRIESMAIER, AND MARVIN KN\"OLLER
achiral (em-achiral) if it scatters incident fields of one helicity in the same way asincident fields of the opposite helicity up to a unitary transformation that swaps he-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 mapsuperpositions 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 wererecently introduced in [16] (see also [5, 24]). These measures quantify the degree ofem-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 circularlypolarized 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 scatterersand strictly positive for em-chiral scatterers, and they would attain their maximumfor 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 scatteringobjects 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 isotropicmaterials, 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 scatteringproblems with scatterers that consist of chiral materials are discussed. A link betweenthese two perspectives is provided in [2], where the Drude--Born--Fedorov constitutiverelations governing the propagation of electromagnetic waves in chiral media havebeen derived from the linear constitutive relations for homogeneous isotropic media.This is achieved by embedding a large number of regularly spaced, randomly orientedchiral objects that are made of isotropic materials similar to the ones considered inthis work.
We study a shape optimization problem, where the goal is to determine compactlysupported dielectric scattering objects that possess comparatively large measures ofem-chirality. Since thin helical structures have been proposed as candidates for highlyem-chiral objects in the literature (see, e.g., [2, 16, 19] and the references therein), wefocus on shape optimization for scatterers that are supported on thin tubular neigh-borhoods of smooth curves. The objective functional in this optimization problem isbased on an em-chirality measure, and the evaluation of its shape derivative requiresan approximation of the shape derivative of the complete far field operator. Accord-ingly, evaluating the shape derivatives in a traditional shape optimization scheme forelectromagnetic 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, whichwould be rather expensive. Using an asymptotic representation formula for scatteredfields 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 solvinga single Maxwell system during the optimization procedure. A similar approach wasused in [12] to construct an inexpensive Gau{\ss}--Newton reconstruction method for aninverse scattering problem with thin tubular scatterers. We also refer the reader to[8, 20, 22] for related work on electrical impedance tomography with thin tubularconductivity inclusions.
The asymptotic representation formula from [12] gives an explicit approximationof the far field operator corresponding to thin tubular scattering objects. The ac-
curacy of this approximation increases with decreasing thickness of the thin tubularscattering objects. Replacing the far field operator in the objective functional of theshape optimization problem by the leading order term in the asymptotic representa-tion formula, we derive an explicit formula for the shape derivative of this modifiedobjective 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 shapederivative, the modified objective functional in the shape optimization scheme can beevaluated efficiently. We stabilize the optimization procedure by adding proper regu-larization terms, and we apply the final algorithm to provide examples of optimizedthin 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 electromagneticchirality. In section 3 we consider the asymptotic representation formula for far fieldoperators corresponding to thin tubular scattering objects. In section 4 we establishthe shape derivative of the leading order term in this asymptotic expansion, and insection 5 we develop the shape optimization scheme. Numerical results are discussedin section 6, and in the appendix we provide explicit representations for the derivativesof spherical vector wave functions that are required for the numerical implementationof the optimization algorithm.
2. Electromagnetic chirality. We consider time-harmonic electromagneticwave propagation in a homogeneous background medium in \BbbR 3 with constant elec-tric permittivity \varepsilon 0 > 0 and constant magnetic permeability \mu 0 > 0. Throughout wewill work with electric fields only, but the associated magnetic field can be retrievedfrom the corresponding first order Maxwell systems. An incident field \bfitE i is an entiresolution to the Maxwell equations
(2.1a) curl curl\bfitE i - k2\bfitE i = 0 in \BbbR 3 ,
where k = \omega \surd \varepsilon 0\mu 0 denotes the wave number at frequency \omega > 0. We suppose that this
incident field is scattered by a bounded penetrable dielectric scattering objectD \subseteq \BbbR 3,and that the relative electric permittivity and the relative magnetic permeability satisfy
1982 TILO ARENS, ROLAND GRIESMAIER, AND MARVIN KN\"OLLER
uniformly in \widehat \bfitx = \bfitx /| \bfitx | (see, e.g., [13, Thm. 6.9]). The vector function \bfitE \infty \in L2t (S
2,\BbbC 3) is called the electric far field pattern, where as usual L2t (S
2,\BbbC 3) denotesthe 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 \BbbC 3, which must satisfy \bfitA \cdot \bfittheta = 0, can be described by a matrix \bfitE i( \cdot ;\bfittheta ) thatsatisfies
Because of the linearity of (2.1) with respect to the incident field, the scattered electricfield and the corresponding electric far field pattern can also be expressed by matrices,and we denote them by \bfitE s( \cdot ;\bfittheta )\bfitA and \bfitE \infty ( \cdot ;\bfittheta )\bfitA , respectively. A Herglotz wavewith density \bfitA \in L2
We denote the corresponding total electric field by\bfitE [\bfitA ] and the scattered electric fieldby \bfitE s[\bfitA ]. Electric far field patterns \bfitE \infty [\bfitA ] excited by Herglotz waves as incidentfields are fully described by the far field operator \scrF D : L2
For later reference we note that \scrF D is an integral operator with smooth kernel (see,e.g., [13, Thm. 6.9]). By linearity we have that \bfitE \infty [\bfitA ] = \scrF D\bfitA .
In the following we discuss the physical notion of chirality for such electromagneticscattering problems and give a short synopsis of the concepts and results that haverecently been developed in [5, 16]. The following traditional notion of chirality isconcerned with the geometry of objects: An object D is called geometrically achiralif there exist \bfitx 0 \in \BbbR 3 and an orthogonal matrix J \in \BbbR 3\times 3 with det J = - 1 such thatD = \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 considerationthat applying a reflection by a plane also changes the incident field. A plane waveis said to be left or right circularly polarized if its polarization vector performs acounterclockwise or clockwise circular motion (one turn per wave length) along thedirection 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 planewave, but this field is then of opposite helicity. We note also that (2.4) is equivalentto the eigenvalue relation
is said to have helicity \pm 1 if \bfitU is an eigenfunction of the operator k - 1 curl for theeigenvalue \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 twofields 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 uniquelydetermined by the density\bfitA \in L2
t (S2,\BbbC 3). Since Rellich's lemma implies a one-to-one
correspondence between the scattered field \bfitE s[\bfitA ] in \BbbR 3 \setminus D and its far field pattern\bfitE \infty [\bfitA ] (see, e.g., [13, Thm. 6.10]), the scattered field \bfitE s[\bfitA ]| \BbbR 3\setminus D also is uniquely
determined by\bfitA . To describe the helicities of\bfitE i[\bfitA ] and\bfitE s[\bfitA ]| \BbbR 3\setminus D in terms of\bfitA , we
generalize (2.4) and introduce the self-adjoint operator \scrC : L2t (S
(2.7) V \pm := \{ \bfitA \pm \scrC \bfitA | \bfitA \in L2t (S
2,\BbbC 3)\}
corresponding to the eigenvalues \pm 1 are orthogonal in L2t (S
2,\BbbC 3) and satisfy
L2t (S
2,\BbbC 3) = 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 ,
\bfitE s[\bfitA ]| \BbbR 3\setminus D has helicity \pm 1 if and only if \bfitE \infty [\bfitA ] \in V \pm .
Electromagnetic chirality is a concept describing the difference in the interactionof a scattering object D with incident fields of opposite helicities. To give an accuratedefinition, we denote by \scrP \pm : L2
t (S2,\BbbC 3) \rightarrow L2
t (S2,\BbbC 3) the orthogonal projections
onto V \pm , and, accordingly, we decompose
(2.8) \scrF D = \scrF ++D + \scrF + -
D + \scrF - +D + \scrF - -
D ,
with \scrF pqD := \scrP p\scrF D\scrP q for p, q \in \{ +, - \} . It has been observed in [16, 5] that if a scat-
terer D is geometrically achiral, then there exists a unitary operator\scrU : L2
t (S2,\BbbC 3) \rightarrow L2
t (S2,\BbbC 3) such that
\scrC \scrU = - \scrU \scrC and \scrF D \scrU = \scrU \scrF D .
This says that \scrF D is equivalent to itself by means of a unitary transform \scrU that swapshelicity. An immediate consequence is that \scrF ++
D = \scrU \scrF - - D \scrU \ast and \scrF - +
D = \scrU \scrF + - D \scrU \ast .
Based on this observation, the following more general definition of electromagneticchirality was introduced in [16].
Definition 2.1. A scattering object D is called electromagnetically achiral (orem-achiral) if there exist unitary operators \scrU (j) : L2
t (S2,\BbbC 3) \rightarrow L2
t (S2,\BbbC 3) satisfying
\scrU (j)\scrC = - \scrC \scrU (j), j = 1, . . . , 4, such that
1984 TILO ARENS, ROLAND GRIESMAIER, AND MARVIN KN\"OLLER
If this is not the case, we call the scattering object D electromagnetically chiral (orem-chiral).
An immediate consequence of this definition is that for an em-achiral scattererD, the singular values (in decreasing order and repeated with multiplicity) (\sigma ++
j )j\in \BbbN
of \scrF ++D coincide with the singular values (\sigma - -
j )j\in \BbbN of \scrF - - D , and, analogously, the
D coincide with the singular values (\sigma - +j )j\in \BbbN of \scrF - +
D .This leads to the idea of quantifying the degree of em-chirality of a scattering objectby means of the distance of the corresponding sequences of singular values.
Following [16], we define the chirality measure \chi 2 of a scatterer D associated tothe far field operator \scrF D as
Since the far field operator \scrF D is an integral operator with smooth kernel, its singularvalues are decreasing exponentially, and (2.9) is well defined. In particular \scrF D isa Hilbert--Schmidt operator. The chirality measure \chi 2 is closely connected to theHilbert--Schmidt norm of \scrF D. In fact,
and it follows immediately that the upper bound is attained when \scrF ++D = \scrF - +
D = 0or \scrF - -
D = \scrF + - D = 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). TheHilbert--Schmidt norm \| \scrF D\| HS of the far field operator is sometimes called the totalinteraction cross-section of the scattering object D.
Definition 2.2. A scattering object D is said to be maximally em-chiralif \chi 2(\scrF D) = \| \scrF D\| HS.
Another possible choice for a chirality measure, which has been proposed in [24],is
\chi HS(\scrF D) :=\bigl( \bigl( \| \scrF ++
D \| HS - \| \scrF - - D \| HS
\bigr) 2+\bigl( \| \scrF - +
D \| HS - \| \scrF + - D \| HS
\bigr) 2\bigr) 12
=\bigl( \| \scrF D\| 2HS - 2
\bigl( \| \scrF ++
D \| HS\| \scrF - - D \| HS + \| \scrF - +
D \| HS\| \scrF + - D \| HS
\bigr) \bigr) 12 ,
(2.11)
Von Neumann's trace inequality (see, e.g., [15, Lem. XI.9.14]) shows that
(2.12) \chi HS(\scrF D) \leq \chi 2(\scrF D) ,
and comparing (2.10) and (2.11), we find that
\chi HS(\scrF D) = \| \scrF D\| HS if and only if \chi 2(\scrF D) = \| \scrF D\| 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 \chi HS(\scrF D). Since numerical examples in [5]suggest that \chi 2 is not differentiable, we use the chirality measure \chi HS in the gradientbased 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 discussan asymptotic representation formula for electric far field patterns due to such thinstructures from [12] and apply it to our setting.
3. Far field operators of thin tubular scatterers. Before we specify the classof thin tubular scattering objects that will be considered in the following, we introducea set of admissible parametrizations for the center curves of these scatterers by
(3.1) \scrU ad :=\bigl\{ \bfitp \in C3([0, 1],\BbbR 3)
\bigm| \bigm| \bfitp ([0, 1]) is simple and \bfitp \prime (t) \not = 0 for all t \in [0, 1]\bigr\} .
For any center curve \Gamma \subseteq \BbbR 3 with parametrization \bfitp \Gamma \in \scrU ad we denote by (\bfitt \Gamma ,\bfitn \Gamma , \bfitb \Gamma )the corresponding Frenet--Serret frame. For instance, if \bfitp \prime \Gamma (t) \times \bfitp \prime \prime \Gamma (t) \not = 0 for allt \in [0, 1], then
for some \varepsilon r, \mu r > 0. For each \rho > 0 we denote the electric far field pattern of thesolution to (2.1) withD replaced byD\rho and a Herglotz incident field \bfitE i[\bfitA ] by \bfitE \infty
\rho [\bfitA ],and, accordingly, we write \scrF D\rho for the corresponding far field operator.
In this work we focus on shape optimization for scatterers that are supported onthin tubular neighborhoods D\rho of smooth curves as in (3.2), i.e., when the param-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 developedin section 5 below is heavily dominated by the evaluation of far field operators andof shape derivatives of far field operators corresponding to thin tubular scatteringobjects. The numerical simulation of far field operators and of their shape derivativeswith traditional finite element or boundary element methods becomes increasinglydifficult for decreasing values of \rho > 0 because finer and finer meshes have to beused. To facilitate these computations we refrain from finite element and boundaryelement methods and apply instead the following asymptotic perturbation formulafor the electric far field pattern \bfitE \infty
\rho as \rho \rightarrow 0, which has recently been establishedin [12] (see also [1, 21] and [3, 4, 8, 9, 10, 11] for earlier contributions in this direc-tion). For a particular choice of k, \rho , \varepsilon r, \mu r, and \Gamma the accuracy of this approximationcan be estimated by suitably accurate reference simulations using a finite element orboundary element solver (see also section 6 below).
1986 TILO ARENS, ROLAND GRIESMAIER, AND MARVIN KN\"OLLER
Theorem 3.1. Let \Gamma \subseteq \BbbR 3 be an admissible center curve with \bfitp \Gamma \in \scrU ad, anddenote by \varepsilon r, \mu r > 0 the relative electric permittivity and the relative magnetic per-mittivity of a thin tubular scatterer D\rho for some \rho > 0. Suppose that \bfitA \in L2
t (S2,\BbbC 3)
is the density of a Herglotz incident wave \bfitE i[\bfitA ]. Then the electric far field pattern ofthe corresponding solution to the scattering problem (2.1) satisfies, for each \widehat \bfitx \in S2,
\bigr) as \rho \rightarrow 0. The matrix valued functions \BbbM \varepsilon ,\BbbM \mu \in L2(\Gamma ,\BbbR 3\times 3) are the electric andmagnetic polarization tensors, respectively. These are given by
(s)\top for a.e. s \in [0, 1] and \gamma \in \{ \varepsilon , \mu \} ,
where M\gamma := diag(1, 2/(\gamma r + 1), 2/(\gamma r + 1)) \in \BbbR 3\times 3, and V\bfitp \Gamma :=
\bigl[ \bfitt \Gamma
\bigm| \bigm| \bfitn \Gamma
\bigm| \bigm| \bfitb \Gamma \bigr] \in 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 thematrix of cross products between the vector and the columns of the original matrix.The term o((k\rho )2) in (3.3) is such that \| o((k\rho )2)\| L\infty (S2)/(k\rho )
2 converges to zero forany fixed \bfitE i[\bfitA ], and the dependence on \bfitE i[\bfitA ] is only a dependence on \| \bfitA \| L2
t (S2,\BbbC 3).
Next we introduce the operator \scrT D\rho : L2t (S
From Theorem 3.1 (and the remark about the remainder term) it follows that
(3.5) \scrF D\rho = \scrT D\rho
+ o\bigl( (k\rho )2
\bigr) as \rho \rightarrow 0 ,
and the term o((k\rho )2) in (3.5) is such that\bigm\| \bigm\| o\bigl( (k\rho )2\bigr) \bigm\| \bigm\|
HS/(k\rho )2 converges to zero. In
the following we approximate the far field operator \scrF D\rho by \scrT D\rho
.For numerical implementations it will be convenient to have an explicit repre-
sentation of \scrT D\rho in terms of a complete orthonormal system of L2t (S
2,\BbbC 3). Let Y mn ,
m = - n, . . . , n, n \in \BbbN , denote any complete orthonormal system of spherical har-monics of order n in L2(S2). A particular choice that is used for the computations inour numerical examples is given in (A.2). Then, the vector spherical harmonics,
n (\widehat \bfitx ) , \bfitx \in \BbbR 3 , 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 callcircularly polarized spherical vector wave functions in the following, are given by
(3.10) \bfitP mn := \bfitM m
n + k - 1 curl\bfitM mn , \bfitQ m
n := \bfitM mn - k - 1 curl\bfitM m
n
for m = - n, . . . , n, n = 1, 2, . . . . We recall that
(3.11) curl\bfitP mn = k\bfitP m
n , curl\bfitQ mn = - k\bfitQ m
n .
Lemma 3.2. Let \bfitA \in L2t (S
2,\BbbC 3) with
(3.12) \bfitA =
\infty \sum n=1
n\sum m= - n
\bigl( amn \bfitA
mn + bmn \bfitB
mn
\bigr) .
Then
(3.13) \scrT D\rho \bfitA =
\infty \sum n=1
n\sum m= - n
\bigl( cmn \bfitA
mn + dmn \bfitB
mn
\bigr) with
cmn =
\infty \sum n\prime =1
n\prime \sum m\prime = - n\prime
\Bigl( am
\prime
n\prime
\bigl\langle \scrT D\rho \bfitA
m\prime
n\prime ,\bfitA mn
\bigr\rangle L2
t (S2,\BbbC 3)
+ bm\prime
n\prime
\bigl\langle \scrT D\rho \bfitB
m\prime
n\prime ,\bfitA mn
\bigr\rangle L2
t (S2,\BbbC 3)
\Bigr) ,(3.14a)
dmn =
\infty \sum n\prime =1
n\prime \sum m\prime = - n\prime
\Bigl( am
\prime
n\prime
\bigl\langle \scrT D\rho \bfitA
m\prime
n\prime ,\bfitB mn
\bigr\rangle L2
t (S2,\BbbC 3)
+ bm\prime
n\prime
\bigl\langle \scrT D\rho
\bfitB m\prime
n\prime ,\bfitB mn
\bigr\rangle L2
t (S2,\BbbC 3)
\Bigr) .(3.14b)
Introducing, for any \bfitU ,\bfitV \in C(\Gamma ,\BbbC 3), the expressions
1988 TILO ARENS, ROLAND GRIESMAIER, AND MARVIN KN\"OLLER
Proof. The expansions (3.13) and (3.14) follow by linearity. Furthermore, it fol-lows immediately from [13, Thm. 6.29] that for any \bfittheta \in S2 and \bfitp \in \BbbC 3 with \bfitp \cdot \bfittheta = 0,
\bfitp e - ik\bfittheta \cdot \bfity = - 4\pi
\infty \sum n=1
( - i)nn\sum
m= - n
\bfitM mn (\bfity )
\bigl( \bfitV mn (\bfittheta ) \cdot \bfitp
\bigr) +
4\pi
k
\infty \sum n=1
( - i)n - 1n\sum
m= - n
curl\bfitM mn (\bfity )
\bigl( \bfitU m
n (\bfittheta ) \cdot \bfitp \bigr) .
Hence, we find that\bigl\langle e - ik\bfittheta \cdot \bfity ,\bfitU m
n (\bfittheta )\bigr\rangle L2(S2)
=4\pi
k( - i)n - 1 curl\bfitM m
n (\bfity ) ,(3.16a) \bigl\langle e - ik\bfittheta \cdot \bfity ,\bfitV m
n (\bfittheta )\bigr\rangle L2(S2)
= - 4\pi ( - i)n\bfitM mn (\bfity ) ,(3.16b)
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\bigl\langle
e - ik\bfittheta \cdot \bfity ,\bfitA mn (\bfittheta )
\bigr\rangle L2(S2)
=\surd 8\pi ( - i)n - 1\bfitP m
n (\bfity ) ,(3.17a) \bigl\langle e - ik\bfittheta \cdot \bfity ,\bfitB m
n (\bfittheta )\bigr\rangle L2(S2)
= - \surd 8\pi ( - i)n - 1\bfitQ m
n (\bfity ) .(3.17b)
Therefore,
\bfitE i[\bfitA mn ](\bfity ) =
\bigl\langle eik\bfittheta \cdot \bfity ,\bfitA m
n (\bfittheta )\bigr\rangle L2(S2)
=\surd 8\pi in - 1\bfitP m
n (\bfity ) ,(3.18a)
\bfitE i[\bfitB mn ](\bfity ) =
\bigl\langle eik\bfittheta \cdot \bfity ,\bfitB m
n (\bfittheta )\bigr\rangle L2(S2)
= - \surd 8\pi in - 1\bfitQ m
n (\bfity ) ,(3.18b)
and applying (3.11) gives
curl\bfitE i[\bfitA mn ](\bfity ) =
\surd 8\pi k in - 1\bfitP m
n (\bfity ) ,(3.19a)
curl\bfitE i[\bfitB mn ](\bfity ) =
\surd 8\pi k in - 1\bfitQ m
n (\bfity ) .(3.19b)
Similarly, applying (3.8), we obtain that\bigl\langle e - ik\bfittheta \cdot \bfity \bigl( (\bfittheta \times \BbbI 3)\times \bfittheta
\bigr) ,\bfitA m
n (\bfittheta )\bigr\rangle L2
t (S2,\BbbC 3)
=\surd 8\pi ( - i)n - 1\bfitP m
n (\bfity )\top ,(3.20a) \bigl\langle
e - ik\bfittheta \cdot \bfity \bigl( (\bfittheta \times \BbbI 3)\times \bfittheta \bigr) ,\bfitB m
with the scalar product between a matrix and a vector understood to be taken col-umn by column. Finally, combining the identities in (3.16)--(3.20) with the integralrepresentation in (3.4) gives (3.15).
Remark 3.3. The circularly polarized vector spherical harmonics \bfitA mn and \bfitB m
n ,m = - n, . . . , n, n = 1, 2, . . . , in (3.7) have been constructed in such a way thatthey span the subspace V + and V - from (2.7), respectively. Thus the expansion inLemma 3.2 immediately gives corresponding basis representations of the projectedoperators \scrT pq
D\rho for p, q \in \{ +, - \} , which are defined analogously to (2.8) with \scrF D\rho
In numerical implementations the series over n in all these basis representationshave to be truncated at some N \in \BbbN . In [23] the authors studied the singular valuedecomposition of the linear operator that maps current densities supported in the ballBR(0) of radius R around the origin to their radiated far field patterns and showedthat for all practically relevant source distributions the radiated far field pattern iswell approximated by a vector spherical harmonics expansion of order N \gtrsim kR. Thissuggests choosing the truncation index N in the series representations of \scrT D\rho
and\scrT pqD\rho
, p, q \in \{ +, - \} , such that N \gtrsim kR, where BR(0) denotes the smallest ball aroundthe origin that contains the scattering object D\rho .
4. The shape derivative of \bfscrT \bfitD \bfitrho . In the previous section we discussed theasymptotic behavior of the far field operator \scrF D\rho
corresponding to a thin tubularscattering object D\rho with a circular cross-section of radius \rho > 0 and a fixed cen-ter curve \Gamma as \rho tends to zero. In this section we fix the radius \rho > 0 and discussthe Fr\'echet differentiability of the leading order term \scrT D\rho
in the asymptotic expan-sion (3.5) with respect to the center curve \Gamma .
Recalling the set of admissible parametrizations \scrU ad from (3.1) and denoting byHS(L2
t (S2,\BbbC 3)) the space of Hilbert--Schmidt operators on L2
t (S2,\BbbC 3), we define a
nonlinear operator \bfitT \rho : \scrU ad \rightarrow HS(L2t (S
where D\rho is the thin tubular scattering object from (3.2) with center curve \Gamma parame-trized 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 \} , withrespect to \Gamma . Recalling Theorem 3.1 we can identify the parametrized form \BbbM \gamma
\bfitp \Gamma :=
\BbbM \gamma \circ \bfitp \Gamma of the polarization tensor with a continuous function in C([0, 1],\BbbR 3). Thefollowing lemma has been established in [22, Lem. 4.1].
to C([0, 1],\BbbR 3\times 3), and its Fr\'echet derivative at \bfitp \Gamma \in \scrU ad is given by \bfith \mapsto \rightarrow (\BbbM \gamma \bfitp \Gamma ,\bfith
Next we establish the Fr\'echet differentiability of \bfitT \rho .
Theorem 4.2. The operator \bfitT \rho is Fr\'echet differentiable, and its Fr\'echet deriva-tive at \bfitp \Gamma \in \scrU ad is given by \bfitT \prime
Here we used the notation \BbbP 3,\widehat \bfitx := (\widehat \bfitx \times \BbbI 3)\times \widehat \bfitx and \widetilde \bfitH i[\bfitA ] := ik curl\bfitE i[\bfitA ].
Proof. Let \bfitp \Gamma \in \scrU ad. Then there exists \delta > 0 such that \bfitp \Gamma + \bfith \in \scrU ad forall \bfith \in C3([0, 1],\BbbR 3) satisfying \| \bfith \| C3([0,1],\BbbR 3) \leq \delta . We have to show that
where \| \cdot \| F denotes the Frobenius norm on \BbbC 3\times 3. Proceeding as in the proof of [22,Thm. 4.2] and applying Taylor's theorem, it follows that
Explicit formulas for the derivatives (\bfitP mn )\prime and (\bfitQ m
n )\prime of the circularly polarizedspherical vector wave functions \bfitP m
n and \bfitQ mn , 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 shapeoptimization scheme to determine dielectric thin tubular scattering objects D\rho asin (3.2) that exhibit comparatively large measures of em-chirality \chi HS at a givenfrequency. In addition to the frequency, we also fix the material parameters \varepsilon r, \mu r andthe 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 issufficiently small with respect to the wave length of the incident fields, such that theleading order term \scrT D\rho
in the asymptotic expansion (3.5) gives a good approximationof the far field operator \scrF D\rho
.Combining (2.10) and (2.12) we find that
0 \leq \chi HS(\scrG ) \leq \| \scrG \| HS for any \scrG \in HS(L2t (S
2,\BbbC 3)) .
Accordingly, recalling the definition of the nonlinear operator \bfitT \rho in (4.1), we normalizethe chirality measure \chi HS and consider the bounded objective functional
JHS : \scrU ad \rightarrow [0, 1], which is given by
(5.1) JHS(\bfitp \Gamma ) :=\chi HS
\bigl( \bfitT \rho (\bfitp \Gamma )
\bigr) \| \bfitT \rho (\bfitp \Gamma )\| HS
.
We discuss the optimization problem
(5.2) find argmin\bfitp \Gamma \in \scrU ad
\bigl( - JHS(\bfitp \Gamma )
\bigr) subject to | \Gamma | = L
for some prescribed length L > 0.
Remark 5.1. Since the leading order term in the asymptotic expansion of theelectric far field pattern due to a thin tubular scattering object in (3.3) is homogeneousof degree two with respect to the radius \rho of the cross-section of the scatterer D\rho , thesame is true for \bfitT \rho (\bfitp \Gamma ) as well as for \chi HS
\bigl( \bfitT \rho (\bfitp \Gamma )
\bigr) and \| \bfitT \rho (\bfitp \Gamma )\| HS with \bfitp \Gamma \in \scrU ad.
In particular, the relative chirality measure JHS(\bfitp \Gamma ) and thus also (local) minimizersfor (5.2) are independent of \rho .
Below we rewrite (5.2) as an unconstrained optimization problem and apply aquasi-Newton method to approximate a (local) minimizer. This requires the Fr\'echetderivative of the objective functional JHS, which for any \bfitp \Gamma \in \scrU ad suchthat \bfitT \rho (\bfitp \Gamma ) \in X, where the space X has been introduced in (2.13), and forany \bfith \in C3([0, 1],\BbbR 3) is given by
The Fr\'echet derivatives (\chi HS)\prime and \bfitT \prime
\rho have already been established in (2.14) and inTheorem 4.2, respectively.
5.1. Discretization and regularization. In the numerical implementation ofthe optimization algorithm, we approximate admissible center curves \Gamma of thin tubularscatterers D\rho using interpolating cubic splines with the not-a-knot condition at theend points. We consider a partition
and denote the not-a-knot spline that interpolates the curve \Gamma with parametrization\bfitp \Gamma \in \scrU ad at the nodes \bfitx (j) = \bfitp \Gamma (tj), j = 1, . . . , n, by \bfitp \bigtriangleup [ #”\bfitx ], where #”\bfitx \in \BbbR 3n isthe vector that contains the coordinates of the nodes \bfitx (1), . . . ,\bfitx (n). The space of allnot-a-knot splines with respect to \bigtriangleup is denoted by \scrP \bigtriangleup \not \subseteq \scrU ad.
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 curvaturefunctional \Psi 1 : \scrP \bigtriangleup \rightarrow \BbbR is defined by
We add \alpha 2\Psi 2 with a suitable regularization parameter \alpha 2 > 0 as a penalty term to - JHS+\alpha 1\Psi 1 to enforce the constraint | \Gamma | = L in (5.2), and to promote uniformly dis-tributed nodes along the spline approximation of \Gamma during the minimization process.
Altogether, we obtain the regularized discrete nonlinear objective functional\Phi : \scrP \bigtriangleup \rightarrow \BbbR ,
Before we describe the quasi-Newton optimization scheme, we discuss the Fr\'echetderivatives of the functionals \Psi 1 and \Psi 2. A short calculation gives the followingresult.
Lemma 5.2. The mappings \Psi 1 and \Psi 2 are Fr\'echet differentiable from\scrP \bigtriangleup \subseteq C2([0, 1],\BbbR 3) to \BbbR . Their Fr\'echet derivatives at \bfitp \bigtriangleup \in \scrP \bigtriangleup are given by\Psi \prime
1[\bfitp \bigtriangleup ] : \scrP \bigtriangleup \rightarrow \BbbR with
5.2. The BFGS scheme for the regularized optimization problem. Weapply a BFGS scheme with an inexact Armijo-type line search and a cautious updaterule as described in [30] to approximate a solution to (5.4).
Choosing an initial guess #”\bfitx 0 for the coordinates of the nodes of the centerspline \bfitp \bigtriangleup [ #”\bfitx 0], the BFGS iteration for (5.4) is given by
and \varepsilon > 0 is a parameter. This ensures positive definiteness of H\ell throughout theiteration (cf. [30]).
As suggested in [30], we use an inexact Armijo-type line search to determine thestep size \lambda \ell in (5.5). Choosing parameters \sigma \in (0, 1) and \delta \in (0, 1), we identify thesmallest integer j = 0, 1, . . . , such that \delta j satisfies
Then we set \lambda \ell = \delta j .In the numerical examples presented in section 6 below, we use the parameters
\varepsilon = 10 - 5, \sigma = 10 - 4, and \delta = 0.9 in (5.6) and (5.7). Denoting by N \in \BbbN the maximaldegree of vector spherical harmonics that are used in the basis representation of theoperators \bfitT \rho (\bfitp \bigtriangleup ) (see Lemma 3.2) and \bfitT \prime
\rho [\bfitp \bigtriangleup ]\bfith (see Remark 4.3), we consider discrete
We approximate the integrals over the parameter range [0, 1] of the spline \bfitp \bigtriangleup [ #”\bfitx \ell ] inthe evaluation of \nabla \Phi (\bfitp \bigtriangleup [ #”\bfitx \ell ]), \ell = 0, 1, . . . , using a composite Simpson's rule withM = 5 nodes on each subinterval of the partition \bigtriangleup . We stop the BFGS iterationwhen | #”\bfitx \ell +1 - #”\bfitx \ell | /| #”\bfitx \ell | < 10 - 4. The fact that not a single partial differential equationhas to be solved during the optimization process makes this algorithm particularlyefficient.
6. Numerical results. In the numerical examples below we use k = 1 for thewave number; i.e., the wave length of the incident fields is \lambda \approx 6.28. To assess thenumerical accuracy of the asymptotic representation formula (3.3), we have comparednumerical approximations of electric far field patterns corresponding to thin tubularscattering 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 orderterm in the asymptotic perturbation formula (3.3). This limited study in [12] suggeststhat the approximations obtained from the leading order term in (3.3) are accuratewithin a relative error of less than 5\% when the radius \rho of the thin tube D\rho is lessthan 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 examplesto be applicable.
Example 6.1. In the first example, we consider the material parameters \varepsilon r = 5and \mu r = 1. We use \alpha 1 = 0.0005 and \alpha 2 = 0.5 for the regularization parametersin (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 byadding random numbers between - 0.02 and 0.02. We note that the nodes cannot beexactly on the straight line segment, because then the thin tubular scattering objectwith center curve \bfitp \bigtriangleup [ #”\bfitx 0] would be em-achiral, and thus the objective functional \Phi would not be differentiable at the initial guess.
1996 TILO ARENS, ROLAND GRIESMAIER, AND MARVIN KN\"OLLER
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 harmonicsused in the basis representations of the operator \bfitT \rho (\bfitp \bigtriangleup ) (see Lemma 3.2) and of itsFr\'echet derivative \bfitT \prime
\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 . Forthis example we use N = 5.
In Figure 6.1 we show the initial guess (top left), some intermediate results thatare obtained after \ell = 10, 30, 50, 70 iterations, and the final result (bottom right) thatis obtained after \ell = 88 iterations of the BFGS scheme. In each of these plots we alsoincluded the corresponding value of the relative chirality measure JHS. During theoptimization process the almost straight initial guess for the center curve winds up toa helix. The orientation of this helix in space and whether it is left or right turningdepend on the orientation of the initial curve \bfitp \bigtriangleup [ #”\bfitx 0] and on the particular values ofthe 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 correspondingvalues of the functional J2 : \scrU ad \rightarrow [0, 1],
J2(\bfitp \Gamma ) :=\chi 2
\bigl( \bfitT \rho (\bfitp \Gamma )
\bigr) \| \bfitT \rho (\bfitp \Gamma )\| HS
,
which is defined analogously to (5.1) but corresponds to the chirality measure \chi 2 from(2.9) instead of \chi HS from (2.11). We observe that both functionals are increasing byseveral orders of magnitude during the optimization process and that, in accordancewith (2.12), JHS \leq J2.
In Figure 6.2 (center) we show plots of JHS and J2 for the optimized structurefrom Figure 6.1 (bottom right) as a function of \varepsilon r, and in Figure 6.2 (right) we show
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 \varepsilon r. 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) inExample 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 theseplots indicate the values of \varepsilon r and k that have been used in the shape optimization(i.e., \varepsilon r = 5 and k = 1). We observe that the relative chirality measures JHS and J2are monotonically increasing in \varepsilon r. On the other hand, JHS reaches a local maximumat the wave number k = 1 that has been used in the shape optimization, and there isa local maximum of J2 close to this wave number. This suggests that the outcome ofthe optimization process is sensitive to the wave length of the incident field and thatthe obtained optimality property is restricted to a rather narrow band of frequencies.
To study the dependence of the optimized center curve on the length constraint Lin (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).
1998 TILO ARENS, ROLAND GRIESMAIER, AND MARVIN KN\"OLLER
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 whenL = 8 for the maximal degree N of vector spherical harmonics that is used in thebasis representations of the operator \bfitT \rho (\bfitp \bigtriangleup ) and of its Fr\'echet derivative \bfitT \prime
\rho [\bfitp \bigtriangleup ]\bfith . InFigure 6.3 (bottom left and bottom right) we show the final results that are obtainedby 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 initialguess 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 thatare found by the optimization procedure are basically the same for the three differentvalues of L, and that just the number of turns of each helix increases with increasinglength constraint L. The relative chirality measures JHS and J2 attain essentially thesame values for these three structures, but the total interaction cross-section increaseswith increasing values of L (not shown).
Example 6.2. In this second example we use \varepsilon r = 30 and \mu r = 1; i.e., the per-mittivity contrast is much larger than in the first example. We choose \alpha 1 = 0.0001and \alpha 2 = 0.1 for the regularization parameters in (5.3), and the length constraintis L = 20.
For the initial guess #”\bfitx 0 we consider n = 45 nodes on a curve given by twoparallel 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 perturbthe first two components of each node by adding random numbers between - 0.02and 0.02 to obtain a well-defined gradient of the objective functional at the initialguess. In accordance with Remark 3.3 we use N = 6 for the maximal degree of vectorspherical harmonics that is used in the basis representations of the operator \bfitT \rho (\bfitp \Gamma )and of its Fr\'echet derivative \bfitT \prime
\rho [\bfitp \Gamma ]\bfith .In Figure 6.4 we show the initial guess (top left), some intermediate results that
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 \varepsilon r. Right:Sensitivity with respect to wave number k.
are obtained after \ell = 10, 30, 50, 70 iterations, and the final result (bottom right) thatis obtained after \ell = 189 iterations of the BFGS scheme. During the optimizationprocess 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 JHS
and J2 during the optimization process. As in Example 6.1, both functionals increaseby 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 J2 as a function of k. The vertical lines in these plots indicate the values of \varepsilon rand k that have been used in the shape optimization (i.e., \varepsilon r = 30 and k = 1). Therelative chirality measures JHS and J2 are monotonically increasing in \varepsilon r. On theother hand, JHS reaches a local maximum at k = 1, which is the wave number thatwas used in the shape optimization, and there is a local maximum of J2 close to thiswave number. The sensitivity of both relative chirality measures with respect to thewave 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. Thecorresponding initial splines are shown in Figure 6.6 (top left and top right). Inaccordance with Remark 3.3, we choose N = 5 when L = 15 and N = 7 whenL = 25 for the maximal degree N of vector spherical harmonics that is used in thebasis representations of the operator \bfitT \rho (\bfitp \bigtriangleup ) and of its Fr\'echet derivative \bfitT \prime
\rho [\bfitp \bigtriangleup ]\bfith . InFigure 6.6 (bottom left and bottom right) we show the final results that are obtainedby 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 initialguess 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 ofthe three double-helices that are found by the optimization procedure are basically thesame 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 measuresJHS and J2 attain essentially the same values for these three structures, but the totalinteraction cross-section increases with increasing values of L.
Conclusions. Electromagnetic chirality measures quantify differences in the re-sponse of scattering objects or media due to left and right circularly polarized incidentwaves. We have considered the shape optimization problem to design dielectric thin
2000 TILO ARENS, ROLAND GRIESMAIER, AND MARVIN KN\"OLLER
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 electro-magnetic field due to such thin tubular structures to develop an efficient iterativeshape optimization scheme. Our numerical results suggest that thin helical structuresare candidates for optimal thin tubular scatterers, and that high electric permittivitycontrast 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 forthe shape optimization is considered.
We have restricted the discussion to dielectric scattering objects because theasymptotic representation formula from [12] has so far only been justified in thiscase. An extension of the asymptotic representation formula to metallic scatterersand the shape optimization for metallic thin tubular structures will be the subject offuture work.
Appendix A. Derivatives of spherical vector wave functions. The ex-plicit basis representation of the Fr\'echet derivative \bfitT \prime
\rho [\bfitp \Gamma ]\bfith in Remark 4.3 containsderivatives of the circularly polarized spherical vector wave functions \bfitP m
n and \bfitQ mn ,
m = - n, . . . , n, n = 1, 2, . . . . Recalling the definition of \bfitP mn and\bfitQ m
n in (3.10), we pro-vide a detailed discussion of the derivatives of the spherical vector wave functions\bfitM m
Note that throughout this appendix, we will suppress the dependence on r, \theta , and \varphi of the unit coordinate vectors and most other functions.
We start with the factors in\bfitM mn and curl\bfitM m
n that depend only on the angularvariables \theta and \varphi and express their derivatives in terms of the spherical harmonics Y m
n
and the partial derivative of Y mn with respect to \theta . First, we note that the derivatives
The representations (A.3)--(A.8) contain several terms that are ill-suited to nu-merical evaluation for \theta close to 0 or \pi . These are
(A.9)1
sin(\theta )Y mn ,
1
sin(\theta )
\bigl( cot(\theta )Y m
n - \partial \theta Ymn
\bigr) ,
m2
sin2(\theta )Y mn - cot(\theta )\partial \theta Y
mn .
Note that the first two expressions in (A.9) always appear in combination with afactor m in (A.3)--(A.8) and thus are only relevant for m \not = 0. We will only considerm \geq 0 in the following paragraphs, as the corresponding formulas for negative m canimmediately be obtained by complex conjugation.
To rewrite the first term in (A.9), we use the recurrence relation
Pmn (t)\surd 1 - t2
=1
2mt
\bigl( Pm+1n (t)+(n+m)(n - m+1)Pm - 1
n (t)\bigr) , n \geq 2 , m = 1, . . . , n - 1 ,
for the associated Legendre functions (see, e.g., [29, p. 35]). Inserting this into (A.2)gives
(A.10)Y mn
sin(\theta )=
Cmn
2m cos(\theta )
\Bigl( e - i\varphi
Cm+1n
Y m+1n +
(n+m)(n - m+ 1)ei\varphi
Cm - 1n
Y m - 1n
\Bigr) for n \geq 2 and m = 1, . . . , n - 1. Furthermore, differentiating Rodrigues' formulafor the associated Legendre functions (see, e.g., [29, Thm. 2.6]) n times shows that
Pnn (cos(\theta )) =
(2n)!2nn! sin
n(\theta ). Therefore,
(A.11)Y nn
sin(\theta )= Cn
n
(2n)!
2nn!sinn - 1(\theta )ein\varphi , n \in \BbbN .
For the second term in (A.9), from (A.3) we have that
(A.12)1
sin(\theta )
\bigl( cot(\theta )Y m
n - \partial \theta Ymn
\bigr) =
Cmn
Cm+1n
e - i\varphi Ym+1n
sin(\theta ) - (m - 1) cos(\theta )
Y mn
sin2(\theta ).
For m = 1, this can be evaluated using (A.10). For n \geq 2 and m = 2, . . . , n,expressions for sin - 2(\theta )Y m
n are immediately obtained from (A.10) and (A.11).Finally, the third term in (A.9) satisfies
(A.13)m2
sin2(\theta )Y mn - cot(\theta )\partial \theta Y
mn
=Cm
n
Cm+1n
e - i\varphi cos(\theta )Y m+1n
sin(\theta )+
\bigl( m2 - m cos2(\theta )
\bigr) Y mn
sin2(\theta ).
For n, m \geq 2, no new issues arise, and for m = 1, the last term on the right-handside of (A.13) reduces to Y 1
n . In (A.13) we also have to consider the case m = 0,where (A.2) gives
(A.14) - cot(\theta )\partial \theta Y0n = - C0
n cot(\theta )\partial \theta Pn(cos(\theta )) = C0n cos(\theta )P
\prime n(cos(\theta )) .
For numerical implementations of (A.3), (A.5)--(A.8) we suggest using the ex-pressions directly when \theta \in [\pi /4, 3\pi /4] and replacing the problematic terms with theexpressions from (A.10)--(A.14) when \theta \in [0, \pi /4) or \theta \in (3\pi /4, \pi ].
(see, e.g., [29, p. 54]) and the recurrence relation
j\prime n(t) =n
tjn(t) - jn+1(t)
(see, e.g., [31, 10.51.2]) to obtain
\partial rjn(kr) =n
rjn(kr) - kjn+1(kr) ,(A.16a)
\partial rjn(kr)
r=
(n - 1)jn(kr) - krjn+1(kr)
r2,(A.16b)
\partial rjn(kr) + krj\prime n(kr)
r=
- krj\prime n(kr) + (n(n+ 1) - 1 - (kr)2)jn(kr)
r2(A.16c)
=krjn+1(kr) + (n2 - 1 - (kr)2)jn(kr)
r2.
For small values of r > 0, the expansion of the spherical Bessel functions in powers ofr (see, e.g., [29, Def. 2.26]) should be inserted into (A.15) and (A.16) and truncatedto a finite sum for numerical evaluation. In particular, we note that for n = 1,negative powers of r seem to remain in (A.1) when the two summands in curl\bfitM m
n
are inserted separately. However, some tedious calculations show that these termscancel as expected when the sum is formed. Hence, for numerical evaluation in thecase n = 1, all terms of order r - 1 should be left out of the calculation to avoidcancellation effects.
Acknowledgments. The authors would like to thank I. Fernandez-Corbaton,X. Garcia-Santiago, and C. Rockstuhl for many interesting discussions related to thesubject of this paper.
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