Optical theorems and physical bounds on absorption in lossy media
OOptical theorems and physical bounds on absorption in lossy media
Yevhen Ivanenko ∗ Department of Physics and Electrical Engineering,Linnæus University, 351 95 V¨axj¨o, Sweden.
Mats Gustafsson † Department of Electrical and Information Technology,Lund University, Box 118, 221 00 Lund, Sweden.
Sven Nordebo ‡ Department of Physics and Electrical Engineering,Linnæus University, 351 95 V¨axj¨o, Sweden. (Dated: August 27, 2019)Two different versions of an optical theorem for a scattering body embedded inside a lossy back-ground medium are derived in this paper. The corresponding fundamental upper bounds on absorp-tion are then obtained in closed form by elementary optimization techniques. The first version isformulated in terms of polarization currents (or equivalent currents) inside the scatterer and gener-alizes previous results given for a lossless medium. The corresponding bound is referred to here as avariational bound and is valid for an arbitrary geometry with a given material property. The secondversion is formulated in terms of the T-matrix parameters of an arbitrary linear scatterer circum-scribed by a spherical volume and gives a new fundamental upper bound on the total absorption ofan inclusion with an arbitrary material property (including general bianisotropic materials). Thetwo bounds are fundamentally different as they are based on different assumptions regarding thestructure and the material property. Numerical examples including homogeneous and layered (core-shell) spheres are given to demonstrate that the two bounds provide complimentary information ina given scattering problem.
I. INTRODUCTION
Fundamental limits on the scattering and absorp-tion in resonant electromagnetic structures have beenconsidered in various formulations and applicationssuch as with small dipole scatterers , antennas ,radar absorbers , high-impedance surfaces , passivemetamaterials and optical systems . Notably,these problems are almost always formulated for a loss-less background medium such as vacuum. In most casesthere are very good reasons for doing this, at leastwhen the background losses are sufficiently small. Infact, it turns out that the presence of the lossy back-ground medium not only implies an obstructive compli-cation of the analytical derivations, it also nullifies thevalidity of many powerful theorems and constraints, see e.g. , with references. In essence,the problem is that any optical theorem for a scatteringbody embedded in a lossy medium will depend on thegeometry of the scatterer. A vivid illustration of thisis the optical theorem and the associated upper boundson dipole scattering and absorption of a small (dipole)scatterer in a lossless medium which are based solelyon the polarizability of the scatterer , and which nolonger is valid for a lossy background . Hence, fora lossy background medium the associated optical theo-rems must be modified, and it can be expected that anyanalytical results regarding the optimal absorption willbecome dependent on the geometry of the scatterer, see e.g. , . There is a number of application areas where the sur-rounding losses clearly cannot be neglected. This in-cludes typically medical applications such as localizedelectrophoretic heating of a bio-targeted and electri-cally charged gold nanoparticle suspension as a radio-therapeutic hyperthermia-based method to treat cancer, cf. , . Corresponding applications in the opticaldomain are concerned with light in biological tissue ,and the use of gold nanoparticles for plasmonic pho-tothermal therapy . Surface-enhanced biological sens-ing with molecular monolayer spectroscopy is anotherrelated application, see e.g. , . In photonic applicationsand in plasmonics, dielectric substrates based on poly-meric media are usually considered to be lossless at op-tical frequencies . However, some of the substancesthat are used can also show significant losses such aswith z-doped PMMA materials . Another importantmedical application is concerned with implantable an-tennas that are used in telemetry applications as a partof communication link of health-monitoring and health-care systems . The presence of losses in the back-ground medium affects the reliability of such links, es-pecially the performance of in-body antennas. In thisapplication, however, the aim is to reduce the amountof power absorbed by a human body, and this can beachieved by encapsulation of the implant by a biocom-patible insulator . Finally, we mention the terrestrialgaseous atmosphere which has a diversity of rotational-vibrational absorption bands ranging from microwaveto optical frequencies . Important applications in- a r X i v : . [ phy s i c s . op ti c s ] A ug clude antennas and short range communications at 60GHz (absorption bands of oxygen) as well asthe study of radiative transfer in the presence of aerosolsand cloud particles in the atmosphere .In this paper we present two different versions of an op-tical theorem and the associated absorption bounds fora scattering body embedded inside a lossy backgroundmedium. These versions are based on the interior andthe exterior fields, i.e. , the equivalent (polarization) cur-rents inside the scatterer and the T-matrix parametersof the scatterer, respectively. The two versions of the op-tical theorem express the same power balance, and yetthey are fundamentally different (and hence complimen-tary), since they are based on different assumptions re-garding the properties of the scatterer. The first versionis an extension to lossy media regarding the absorptionbounds given in , and is valid for an arbitrary geometrywith a given material property. Even though the basicoptimization technique is the same as in , it is demon-strated how the seemingly trivial extension to lossy me-dia insidiously requires a careful analysis where it is notsufficient just to replace a real-valued wave number for acomplex-valued one. In particular, the new optical theo-rem shows that the extinct power must be expressed asan affine form in the equivalent currents implying sub-tle changes in the final form of the fundamental boundon absorption. The second version is a refinement of thefundamental bounds on multipole absorption given in ,and is valid for a spherical geometry with an arbitrarymaterial property. In particular, we formulate an opti-cal theorem and derive the associated absorption boundfor the total fields including all the electric and mag-netic multipoles. It is proved that the bound is valid notonly for a rotationally invariant sphere as in , but alsofor general heterogeneous bianisotropic materials. Weprove also that the new bound, which is given by a mul-tipole summation formula, is convergent whenever thereare non-zero losses in the exterior domain. In this way,the results also provide a new way to determine the num-ber of useful multipoles in a given scattering problem, i.e. , as a function of the electrical size of the scatterer aswell as of the losses in the exterior domain. Through thenumerical examples, we show that the new fundamentalbounds give complementary information on the absorp-tion of scattering objects in lossy media. The derivedbounds are applicable for arbitrary objects made of arbi-trary materials, which gives a possibility to find such anelectrically small structure that has an absorption peakclose to the fundamental bounds.The rest of the paper is organized as follows: In SectionII is given the optical theorem based on the interior fieldsand in Section III the corresponding bounds on absorp-tion by variational calculus. In Section IV, we considerthe optical theorem and the associated bounds based onthe exterior fields using the T-matrix formalism. In Sec-tion V is illustrated the numerical examples, and the pa-per is summarized in Section VI. Finally, in Appendix Ais shown the derivation of the maximal absorbed power based on calculus of variations, and in Appendix B isput the most important definitions and formulas that areused regarding the spherical vector wave expansion. II. OPTICAL THEOREM BASED ON THEINTERIOR FIELDSA. Notation and conventions
The electric and magnetic field intensities E and H are given in SI-units and the time convention for timeharmonic fields (phasors) is given by e − i ωt , where ω isthe angular frequency and t the time. Let µ , (cid:15) , η andc denote the permeability, the permittivity, the waveimpedance and the speed of light in vacuum, respectively,and where η = (cid:112) µ /(cid:15) and c = 1 / √ µ (cid:15) . The wavenumber of vacuum is given by k = ω √ µ (cid:15) , and hence ωµ = k η and ω(cid:15) = k η − . The real and imaginaryparts and the complex conjugate of a complex number ζ are denoted by Re { ζ } , Im { ζ } and ζ ∗ , respectively. Fordyadics, the notation ( · ) † denotes the Hermitian trans-pose. B. Extinction, scattering and absorption
Consider a scattering problem consisting of a scatter-ing body V bounded by the surface ∂V and which isembedded in an infinite homogeneous and isotropic back-ground medium having relative permeability µ b and rel-ative permittivity (cid:15) b , see Fig. 1. The scatterer V con-sists of a linear material bounded by a finite open setwith volume denoted by the same letter V . The back-ground medium also consists of a passive material andhence Im { µ b } ≥ { (cid:15) b } ≥
0. The incident (i)and scattered (s) fields satisfy the following Maxwell’sequations with respect to the background medium (cid:40) ∇ × E { i , s } = i k η µ b H { i , s } , ∇ × H { i , s } = − i k η − (cid:15) b E { i , s } , (1)in the exterior region R \ V and the total fields are de-noted E = E i + E s and H = H i + H s . It is noted thatthe incident fields satisfy (1) in the whole of R .The interior scattering medium is characterized by thefollowing constitutive relations for a general bianisotropiclinear material (cid:40) D = (cid:15) (cid:15) · E + χ em · H , B = χ me · E + µ µ · H , (2)where B is the magnetic flux density and D the electricflux density and where the relative permittivity and per-meability dyadics are (cid:15) = (cid:15) b I + χ ee and µ = µ b I + χ mm ,respectively, and where χ ee , χ mm , χ em and χ me aredimensionless susceptibility dyadics. By following the (cid:15) b , µ b (cid:15) , χ em χ me , µ V E i , H i ˆ n -(cid:27) a FIG. 1. Problem setup. Here, (cid:15) b and µ b denote the rela-tive permittivity and permeability of the passive backgroundmedium, respectively, and ˆ n the outward unit vector. standard volume equivalence principles , the Maxwell’sequations for the interior region V (cid:40) ∇ × E = i k χ me · E + i k η µ · H , ∇ × H = − i k η − (cid:15) · E − i k χ em · H , (3)can now be reformulated in terms of the backgroundmedium as (cid:40) ∇ × E = i k η µ b H − J m , ∇ × H = − i k η − (cid:15) b E + J e , (4)which is just (3) rewritten based on the equivalent electricand magnetic contrast currents (cid:40) J e = − i k η − χ ee · E − i k χ em · H , J m = − i k χ me · E − i k η χ mm · H . (5)The power balance at the surface ∂V just outside V (where E = E i + E s ) is obtained by using the corre-sponding Poynting’s vectors and can be expressed as P a = − P s + P t + P i , (6)where P a , P s , P t and P i are the absorbed, scattered, ex-tinct (total) and the incident powers, respectively, de-fined by P a = −
12 Re (cid:26)(cid:90) ∂V E × H ∗ · ˆ n d S (cid:27) , (7) P s = 12 Re (cid:26)(cid:90) ∂V E s × H ∗ s · ˆ n d S (cid:27) , (8) P t = −
12 Re (cid:26)(cid:90) ∂V ( E i × H ∗ s + E s × H ∗ i ) · ˆ n d S (cid:27) , (9) P i = −
12 Re (cid:26)(cid:90) ∂V E i × H ∗ i · ˆ n d S (cid:27) , (10)and where the surface integrals are defined with an out-ward unit normal ˆ n , see also (Eq. (3.19)). Based onthe Poyntings theorem (the divergence theorem) (cid:82) ∂V E × H ∗ · ˆ n d S = (cid:82) V ( H ∗ · ∇ × E − E · ∇ × H ∗ ) d v , and byemploying the following identities on ∂V (cid:40) ˆ n × ( E i + E s ) = ˆ n × E , ˆ n × ( H i + H s ) = ˆ n × H , (11) the vector identity ˆ n · X × Y = ˆ n × X · Y and theMaxwell’s equations (3), it is possible to show that (6)gives an optical theorem for the lossy background where P a = k η Im (cid:26)(cid:90) V F ∗ · M a · F d v (cid:27) , (12) P t = k η Im (cid:26)(cid:90) V F ∗ i · M t · F d v (cid:27) − P i , (13) P i = k η Im (cid:26)(cid:90) V F ∗ i · M b · F i d v (cid:27) , (14)and which are based solely on the interior fields. Here,the field quantities are defined as F = (cid:18) E η H (cid:19) , F i = (cid:18) E i η H i (cid:19) , (15)and the material dyadics are given by M a = (cid:18) (cid:15) χ em χ me µ (cid:19) = χ + M b (16)where χ = (cid:18) χ ee χ em χ me χ mm (cid:19) , M b = (cid:18) (cid:15) b I µ b I (cid:19) , (17)and M t = (cid:18) (cid:15) − (cid:15) ∗ b I χ em χ me µ − µ ∗ b I (cid:19) = χ + i2Im { M b } . (18)It is noted that (12) through (18) generalizes previousexpressions which have been given for a lossless exteriormedium (Eqs. (4) through (7) on p. 3338) where P i = 0,and both M a and M t given above can be replaced by thesusceptibility dyadic χ .It is observed that P a is represented by a positive defi-nite (strictly convex) quadratic form and P t by an affineform in the field quantities. Note in particular the addi-tional power balancing term − P i that is present in (13).Finally, it is noted that in the present formulation it issufficient to derive three terms as in (12) through (14)since the fourth term P s will then be given by the opticaltheorem (6). III. FUNDAMENTAL BOUNDS ONABSORPTION BY VARIATIONAL CALCULUS
The fundamental bounds on absorption derived in are generalized below for the case with a lossy back-ground medium. The derivation is based on the opticaltheorem expressed in (6) together with (12) through (18)above. A. General bianisotropic media
The optimization problem of interest is given bymaximize P a subject to P s ≥ , (19)where the optimization is with respect to the interiorfields F of the structure, and where the scattered power P s = − P a + P t + P i is used as the non-negative constraint.In this case, the constraint is organized as − P a + P t +2 P i − P i ≥ P a is the positive definite quadratic formexpressed in (12), P t +2 P i is the linear form given by (13)and P i is given by (14). This is a convex maximizationproblem having a unique solution at the boundary of thefeasible region (active constraint), cf. , .By using the method of Lagrange multipliers and vari-ational calculus, it can be shown that the optimal boundon absorbed power P opta is given by P opta = k α η (cid:90) V F ∗ i · M t · (Im { M a } ) − · M † t · F i d v, (20)see the detailed derivation of the result (A5) in Ap-pendix A 1. The parameter α is found by inserting thestationary solution (A3) into the active constraint in (19),yielding the quadratic equation α + 2 α = q, (21)where q = 4 (cid:90) V F ∗ i · Im { M b } · F i d v (cid:90) V F ∗ i · M t · (Im { M a } ) − · M † t · F i d v . (22)The denominator in (22) is convex in M a for Im { M a } >
0, and thus by its minimization, it can be shown that(22) is maximal for M a = M b implying that q ≤
1, seethe proof in Appendix A 2. The maximizing root of (21)is hence given by α = − − (cid:112) − q. (23)The expression (20) together with (16) through (18),(22) and (23) generalizes the previous result in whichhas been given for a lossless exterior medium. In par-ticular, by considering a lossless exterior medium with e.g. , µ b = (cid:15) b = 1, it is seen that q = 0 (which im-plies that 0 ≤ q ≤ α = − λ = 2, M t = χ andIm { M a } = Im { χ } , so that (20) reproduces the corre-sponding result in (Eq. (23b) on p. 3342). Optimiza-tion of the scattered power can be treated similarly. B. Piecewise homogeneous and isotropic dielectricstructures
Important special cases are with the optimal absorp-tion of piecewise homogeneous and isotropic dielectricstructures in a lossy surrounding dielectric medium. Inthis case, the problem only involves the electric lossesand we can simplify the notation by writing F = E , F i = E i and χ i = χ ee ,i = ( (cid:15) i − (cid:15) b ) I , where i = 1 , . . . , N is related to the corresponding homogeneous component of the composed scatterer. Note that for N = 1, theproblem simplifies to the homogeneous structure. Theassociated material dyadics are given by M a ,i = (cid:15) i I , M b = (cid:15) b I , M t ,i = ( (cid:15) i − (cid:15) ∗ b ) I , (24)and the expression (20) becomes P vara = k α η N (cid:88) i =1 | (cid:15) i − (cid:15) ∗ b | Im { (cid:15) i } (cid:90) V i | E i ( r ) | d v, (25)for the total volume of scattering body V = (cid:80) i V i , i =1 , . . . , N , and where α is given by (23), and q is obtainedfrom (22) as q = 4Im { (cid:15) b } (cid:90) V | E i ( r ) | d v N (cid:88) i =1 | (cid:15) i − (cid:15) ∗ b | Im { (cid:15) i } (cid:90) V i | E i ( r ) | d v. (26)Assume now that the scatterer is an N -layered sphere V a of total radius a , N ≥
1, and the incident field is a planewave E i ( r ) = E e i k b ˆ k · r with vector amplitude E , prop-agation direction ˆ k and where k b = k √ (cid:15) b is the wavenumber of the background medium. By expanding theplane wave in regular spherical vector waves as expressedin (B1), it can readily be shown that (cid:90) V a | E i ( r ) | d v = | E | π (cid:88) τ =1 ∞ (cid:88) l =1 (2 l + 1) W τl ( k b , a ) , (27)with W τl ( k b , a ) defined in (B14), and where we have em-ployed the orthogonality relationships (B13) and (B14),as well as (B21) and (B22). It is observed that for elec-trically small objects of size k a <
1, the influence of thebackground medium can be appropriately neglected, andthus the incident field can be assumed to have a constantamplitude E i ( r ) = E . Hence, the relationship (27) sim-plifies as (cid:90) V a | E i ( r ) | d v = | E | V a , (28)where V a = (cid:80) Ni V i = 4 πa / k b is real-valued ( k b = k ∗ b ).The variational upper bound on the absorption crosssection σ vara is obtained by normalizing with the intensityof the plane wave at the origin r = , i.e. , I i = | E | Re {√ (cid:15) b } / η , (29)giving σ vara = k Re {√ (cid:15) b } α N (cid:88) i =1 | (cid:15) i − (cid:15) ∗ b | Im { (cid:15) i } (cid:90) V i (cid:12)(cid:12)(cid:12) e i k b ˆ k · r (cid:12)(cid:12)(cid:12) d v. (30)To give an explicit formula for (30), it is more convenientto express the normalized absorption cross section Q vara = σ vara /πa for the N -layered sphere as Q vara = k a Re {√ (cid:15) b } α N (cid:88) i =1 | (cid:15) i − (cid:15) ∗ b | Im { (cid:15) i } πa (cid:90) V i (cid:12)(cid:12)(cid:12) e i k b ˆ k · r (cid:12)(cid:12)(cid:12) d v, (31)where1 πa (cid:90) V i (cid:12)(cid:12)(cid:12) e i k b ˆ k · r (cid:12)(cid:12)(cid:12) d v = 2 (cid:88) τ =1 ∞ (cid:88) l =1 (2 l + 1) a [ W τl ( k b , a i ) − W τl ( k b , a i − )] , (32)and where a i is the radius of each subsphere for i =1 , . . . , N , a N = a and a = 0. Here, W l ( k b , a i ) = a i Im { k b j l +1 ( k b a i )j ∗ l ( k b a i ) } Im { k } , (33)and W l ( k b , a i ) = ( l + 1) W ,l − ( k b , a i ) + lW ,l +1 ( k b , a i )(2 l + 1) , (34)are readily obtained from (B15) and (B16). Note thatfor i = 1, the last term in (32) vanishes because of W τl ( k b , a ) = 0 for a = 0, see (33) and (34), respec-tively.In the case of a homogeneous ( N = 1) sphere in alossless medium where Im { (cid:15) b } = 0, we have q = 0, α = − P vara = k η | (cid:15) − (cid:15) b | Im { (cid:15) } | E | V a , (35)and which reproduces the corresponding result in (Eq. (32b) on p. 3345). IV. OPTICAL THEOREM BASED ON THEEXTERIOR FIELDSA. Notation and conventions
The definition of the spherical vector waves and their most important properties employed in this pa-per are summarized in Appendix B. In particular, theregular spherical Bessel functions, the Neumann func-tions, the spherical Hankel functions of the first kind andthe corresponding Riccati-Bessel functions are denotedj l ( z ), y l ( z ), h (1) l ( z ) = j l ( z ) + iy l ( z ), ψ l ( z ) = z j l ( z ) and ξ l ( z ) = z h (1) l ( z ), respectively, all of order l . B. Optical theorem and physical bounds for aspherical region in a lossy medium
We consider the physical bounds on absorption thatcan be derived from the optical theorem when it is formu-lated in terms of the multipole coefficients of a scatteringproblem. In particular, the scatterer is here embeddedin a spherical region surrounded by a lossy medium, asshown in Fig. 1. Hence, the scatterer may consist of ageneral bianisotropic linear material and is bounded bya spherical surface of radius a . The surrounding mediumis an infinite homogeneous and isotropic dielectric freespace having relative permittivity (cid:15) b and wave number k b = k √ (cid:15) b . For simplicity, it is assumed that the back-ground is non-magnetic (a magnetic background with rel-ative permeability µ b (cid:54) = 1 can straightforwardly be addedto the analysis if required). The background is further-more assumed to be passive, and possibly lossy, so thatIm { (cid:15) b } ≥
0, and with permittivity (cid:15) b that does not resideat the negative part of the real axis, which correspondsto the branch cut of the square root.The optical theorem is once again given by the powerbalance (6) with the absorbed, scattered, extinct (total)and the incident powers defined by (7) through (10), re-spectively. Let a i τml and f τml denote the multipole coef-ficients of the incident (regular) and the scattered (out-going) spherical vector waves, respectively, as defined in(B1). Based on the orthogonality of the spherical vectorwaves on the spherical surface ∂V a as given by (B17) and(B18), it can be shown that P s = Re {√ (cid:15) b } | k b | η (cid:88) τ,m,l A τl | f τml | , (36) P t = Re {√ (cid:15) b } | k b | η (cid:88) τ,m,l { B τl a i ∗ τml f τml } , (37) P i = Re {√ (cid:15) b } | k b | η (cid:88) τ,m,l C τl (cid:12)(cid:12) a i τml (cid:12)(cid:12) , (38)where A τl = 1Re { k b } (cid:40) − Im { k ∗ b ξ l ξ (cid:48)∗ l } τ = 1 , Im { k ∗ b ξ (cid:48) l ξ ∗ l } τ = 2 , (39) B τl = 1i2Re { k b } (cid:40) k ∗ b ξ l ψ (cid:48)∗ l − k b ψ ∗ l ξ (cid:48) l τ = 1 , − k ∗ b ξ (cid:48) l ψ ∗ l + k b ψ (cid:48)∗ l ξ l τ = 2 , (40) C τl = 1Re { k b } (cid:40) Im { k ∗ b ψ l ψ (cid:48)∗ l } τ = 1 , − Im { k ∗ b ψ (cid:48) l ψ ∗ l } τ = 2 , (41)for τ = 1 , l = 1 , . . . , ∞ , and where the argumentsof the Riccati-Bessel functions are z = k b a , see also (Eqs. (8) through (11)) and (Eqs. (6) and (7)). By ap-plying Poynting’s theorem to the scattered and the inci-dent powers defined by (8) and (10), it follows that P s ≥ P i ≥ A τl > C τl ≥
0. Note that B τl is a complex-valued constant. For a lossless medium with Im { k b } = 0,we can employ the Wronskian of the Riccati-Bessel func-tions ψ l ξ (cid:48) l − ψ (cid:48) l ξ l = i and use ξ ∗ l = 2 ψ l − ξ l to showthat the coefficients defined in (39) through (41) become A τl = 1, B τl = − / C τl = 0 in agreement with e.g. , (Eq. (7.18)), see also (Eqs. (9) through (12)).
1. Optimal absorption of an arbitrary linear scatterercircumscribed by a sphere
Consider the contribution to the absorbed power froma single partial wave with fixed multi-index ( τ, m, l ), P a ,τml = Re {√ (cid:15) b } | k b | η (cid:104) − A τl | f τml | +2Re { B τl a i ∗ τml f τml } + C τl (cid:12)(cid:12) a i τml (cid:12)(cid:12) (cid:105) , (42)where we have employed the optical theorem (6) as wellas (36) through (38). Let the scattering coefficients f τml be given by the T-matrix (Eq. (7.34)) for an arbitrarylinear scatterer inside the spherical surface ∂V a , so that f n = (cid:88) n (cid:48) T n,n (cid:48) a i n (cid:48) , (43)and where we have introduced the multi-index notation n = ( τ, m, l ). It is observed that (42) is a concave func-tion of the complex-valued variables T n,n (cid:48) with respectto the primed index n (cid:48) . Differentiating (42) with respectto T n,n (cid:48) (for fixed n ) gives the condition for stationarity A τl a i n (cid:48) (cid:88) n (cid:48)(cid:48) a i ∗ n (cid:48)(cid:48) T ∗ n,n (cid:48)(cid:48) = B τl a i ∗ n a i n (cid:48) , (44)which is an infinite-dimensional linear system of equa-tions in the double-primed indices of T n,n (cid:48)(cid:48) . The corre-sponding system matrix a i n (cid:48) a i ∗ n (cid:48)(cid:48) is of rank one and is ingeneral unbounded. Assuming that this matrix is eitherbounded, or is truncated to some finite dimension, thecorresponding matrix norm is given by g = (cid:88) τ,m,l (cid:12)(cid:12) a i τml (cid:12)(cid:12) , (45)and the unique minimum norm (pseudo-inverse) T-matrix solution to (44) is given by T n,n (cid:48) = B ∗ τl A τl g a i n a i ∗ n (cid:48) . (46)By inserting (43) into (42) and completing the squaresusing (46), it can be shown that P a ,τml = Re {√ (cid:15) b } | k b | η (cid:40) − A τl (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) τ (cid:48) m (cid:48) l (cid:48) (cid:18) T τml,τ (cid:48) m (cid:48) l (cid:48) − B ∗ τl a i τml a i ∗ τ (cid:48) m (cid:48) l (cid:48) A τl g (cid:19) a i τ (cid:48) m (cid:48) l (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:32) | B τl | A τl + C τl (cid:33) (cid:12)(cid:12) a i τml (cid:12)(cid:12) (cid:41) . (47) Due to the concavity of this expression ( A τl > T τml,τ (cid:48) m (cid:48) l (cid:48) , gives the optimal absorption. Summing overthe τ ml -indices, the optimal absorption is hence obtainedas P opta = Re {√ (cid:15) b } | k b | η (cid:88) τ,m,l (cid:32) | B τl | A τl + C τl (cid:33) (cid:12)(cid:12) a i τml (cid:12)(cid:12) . (48)It is emphasized that the infinite dimensional matrixequation in (44) in general is related to an unbounded op-erator where the series in (45) does not converge (the cor-responding matrix norm does not exist). However, this ismerely a mathematical subtlety that does not pose anyreal problem here. Hence, considering that the T-matrixin (43) can be truncated to a finite size L with l, l (cid:48) ≤ L ,the bound in (48) can be interpreted as the optimal ab-sorption with respect to all incident and scattered fieldsup to multipole order L , as L → ∞ . Note in particularthat the individual terms appearing in (48) do not dependon the truncation order, and it is only the interpretationof the partial sums that depend on L .For an incident plane wave where E i ( r ) = E e i k b ˆ k · r ,the multipole coefficients a i τml are given by (B21), andthe optimal bound becomes P opta = π Re {√ (cid:15) b } | E | | k b | η (cid:88) τ =1 ∞ (cid:88) l =1 (2 l + 1) (cid:32) | B τl | A τl + C τl (cid:33) , (49)where we have made use of the sum identities (B22) forthe vector spherical harmonics. The corresponding opti-mal normalized absorption cross section Q opta is obtainedby normalizing with the intensity I i of the plane waveat the origin r = given by (29), as well as with thegeometrical area cross section of the sphere πa , giving Q opta = 2 | k b a | (cid:88) τ =1 ∞ (cid:88) l =1 (2 l + 1) (cid:32) | B τl | A τl + C τl (cid:33) . (50)In the next section we will show that (50) convergeswhenever there are losses in the exterior medium andIm { k b } >
0. In Fig. 2 is illustrated the convergence ofthe expression (50) by plotting the partial sums againstthe number of included multipoles L . The calculationsare for electrical sizes k a ∈ { . , , } and with back-ground losses (cid:15) (cid:48)(cid:48) b ∈ { − , − , − } where (cid:15) b = 1 + i (cid:15) (cid:48)(cid:48) b .Clearly, with increasing external losses there are fewermodes that can contribute to the absorption inside thesphere, which is due to the interaction of the reactivenear-fields of the higher order modes with the lossy ex-terior domain.In the lossless case, when Im { k b } = 0, the truncatedpartial sums of (50) can be calculated as Q opt ,L a = 2( k b a ) (cid:88) τ =1 L (cid:88) l =1 l + 14 = 1( k b a ) L ( L + 2) , (51) − Number of multipoles L Optimal normalized absorption cross section Q opta (cid:15) b k a − − − FIG. 2. Optimal normalized absorption cross section Q opta of a sphere in a lossy medium, plotted as a function of thenumber of included multipoles L . where L is the truncated maximal multipole order,which can be determined by the method proposed in (p. 1508). Interestingly, the obtained result in (51) issimilar as the expression for maximum gain derived byHarrington in (Eq. (11) on p. 221).
2. Proof of convergence
To prove that (50) converges for a lossy medium whereIm { k b } >
0, we consider the following power series ex-pansions of the regular Riccati-Bessel functions ψ l ( z ) = ∞ (cid:88) k =0 α kl z l +1+2 k , (52)and the singular Riccati-Hankel functions ξ l ( z ) = i l (cid:88) k =0 β kl z − l +2 k + α l z l +1 + O{ z l +2 } , (53)where α kl = ( − / k /k !(2 l + 2 k + 1)!! and β kl = − (1 / k (2 l − k − /k ! cf. , (Eqs. (10.53.1) and(10.53.2)) and where O{·} denotes the big ordo definedin (p. 4). By inserting (52) and (53) into (39) through(41) and retaining only the most dominating terms forfixed z = k b a and increasing l , it is found that A τl ∼ β l β l sin 2 θ | z | l − cos θ τ = 1 ,lβ l sin 2 θ | z | l +1 cos θ τ = 2 , (54) B τl ∼ −
12 e − i θ (2 l +1) cos θ τ = 1 , − l e − i θ (2 l +3) + ( l + 1)e − i θ (2 l − (2 l + 1) cos θ τ = 2 , (55) C τl ∼ − α l α l | z | l +3 sin 2 θ cos θ τ = 1 , ( l + 1) α l | z | l +1 sin 2 θ cos θ τ = 2 , (56)and where z = | z | e i θ . Note that α l = 1 / (2 l + 1)!!, α l = − (1 / / (2 l + 3)!!, β l = − (2 l − β l = − (1 / l − A τl co-efficients, the boundedness of the B τl coefficients and thefactorial decrease of the C τl coefficients for large l . In thelossless case when θ = 0, we have A τl = 1, B τl = − / C τl = 0 and (50) is divergent as demonstrated in(51). V. NUMERICAL EXAMPLES
In this section, we illustrate the theory that has beendeveloped in Sections III and IV in comparison with thenormalized absorption cross sections of spherical objectsembedded in a lossy medium. As objects of study, ho-mogeneous and layered (core-shell) spheres are selected.The dielectric background medium is characterized bypermittivity (cid:15) b = (cid:15) (cid:48) b + i (cid:15) (cid:48)(cid:48) b , where the choice of (cid:15) (cid:48)(cid:48) b isbased on the skin depth of human skin α = 2 k (cid:15) (cid:48)(cid:48) b , α − ∈ (10 − , − ) cm, see (Table 3.2 on p. 49). Notethat the real part of the background permittivity doesnot play the key role in comparisons presented below,and thus we consistently choose (cid:15) (cid:48) b = 1 despite that therefractive index of human tissue n ≈ . (Table 3.8on p. 63). In addition to this investigation, the absorp-tion of spherical objects embedded in an almost losslessmedium with relative permittivity (cid:15) b = 1 + i10 − is alsoconsidered.In Fig. 3 is shown a comparison of the optimal nor-malized absorption cross section Q opta given by (50), theabsorption of a homogeneous sphere made of gold Q Aua (full Mie solution) obtained by normalization of (42) withthe intensity I i of the plane wave at the origin (29) andthe geometrical area cross section πa , and the corre-sponding variational bound Q vara for a homogeneous ob-ject given by (31). The background relative permittivityis (cid:15) b = 1 + i (cid:15) (cid:48)(cid:48) b with various levels of background loss: (cid:15) (cid:48)(cid:48) b ∈ { − , − , − } . The calculations of Q Aua and Q vara are for two different radii of gold spheres (20 nmand 89 nm) and the same photon energy range 1–5 eV(corresponds to the wavelength range 248–1240 nm) ac-cording to the Brendel-Bormann (BB) model fitted to ex-perimental data as in (the dielectric model in Eq. (11)with parameter values from Table 1 and Table 3). Thesphere of radius a = 89 nm has been tuned to optimalelectric-dipole absorption for a lossless background asin (Fig. 4). It is noted that the variational bound Q vara depends very weakly on the background loss for these pa-rameter ranges, and the bound is therefore plotted onlyfor (cid:15) (cid:48)(cid:48) b = 10 − (the plots for (cid:15) (cid:48)(cid:48) b ∈ { − , − } almost co-incide). As can be seen in this plot, the two bounds Q opta and Q vara , which are derived under different assumptions(arbitrary structure of linear bianisotropic materials in-side the sphere vs arbitrary structure of gold inside thesphere), give complementary information about the up-per bounds on absorption. At the same time, the nor-malized absorption cross section Q Aua is not tight with re-spect to the optimal bounds as shown in Fig. 3, despite itis known that its electric-dipole contribution approachesthe bound for electric-dipole absorption when the radiusof object is 89 nm, see (Fig. 4). . − Electrical size k a Normalized absorption cross section Q a Q opta (cid:15) b = 10 − Q opta (cid:15) b = 10 − Q opta (cid:15) b = 10 − Q vara ( a , a ) (cid:15) b = 10 − Q Aua ( a , a ) (cid:15) b = 10 − Q Aua ( a , a ) (cid:15) b = 10 − -(cid:27) a FIG. 3. Comparison between the optimal normalized ab-sorption cross section Q opta , the absorption of a homogeneoussphere made of gold Q Aua , and the corresponding variationalbound Q vara ; all plotted as functions of the electrical size k a .The plots are for various levels of background loss (cid:15) (cid:48)(cid:48) b , and thecalculations of Q vara and Q Aua are for two different radii of thesphere a = 20 nm (to the left) and a = 89 nm (to the right). Now, we would like to find such spherical objects whichare resonant at small electrical size, but at the same timeare of a reasonable physical size and have a resonance ab-sorption peak close to the optimal absorption bound. Tofit these requirements, one way is to “tune” a homoge-neous sphere to the resonance at the desirable electricalsize. An alternative approach is to consider a layeredsphere constructed of a dielectric core and coated with ametallic shell.In Fig. 4a is shown a comparison of the various up-per bounds on absorption and the absorption of a spheretuned to optimal electric (plasmonic) dipole resonance.Here, Q a denotes the full Mie solution for a homoge-neous sphere with a (hypothetical) fixed value of per-mittivity (cid:15) = − (cid:15) ∗ b − (cid:15) ∗ (0 . + i2 (cid:15) ∗ (cid:112) (cid:15) ∗ b (0 . whichhas been tuned to optimal electric-dipole resonance at k a = 0 . cf. , (Eq. (55)). The optimal normalized ab-sorption cross section Q opta is given by (50), Q opta , and Q opta , denote the corresponding electric multipole con-tribution ( τ = 2 , l = 1 , , , . . . ) and the normalizedelectric-dipole absorption cross section ( τ = 2 , l = 1),respectively. The corresponding variational bound Q vara is given by (31). All calculations have been made for (cid:15) (cid:48)(cid:48) b = 10 − where (cid:15) b = 1+i (cid:15) (cid:48)(cid:48) b . As can be seen in Fig. 2, forthis combination of electrical size and background loss, itis (almost) sufficient to consider the dipole ( l = 1) contri- .
01 0 . − Electrical size k a a) Normalized absorption cross section Q a Q opta Q opta , Q opta , Q vara Q a -(cid:27) a .
01 0 . − Electrical size k a b) Normalized absorption cross section Q a Q opta Q opta , Q opta , Q vara Q a -(cid:27) a FIG. 4. Comparison of the various upper bounds Q opta , Q opta , (the electric multipole contribution), Q opta , (the electric-dipoleupper bound), Q vara and the absorption of a sphere Q a tunedto optimal electric (plasmonic) dipole resonance at k a = 0 . (cid:15) (cid:48)(cid:48) b = 10 − where (cid:15) b = 1 + i (cid:15) (cid:48)(cid:48) b , and the permittivityof the sphere is (cid:15) = − .
024 + i0 . (cid:15) (cid:48)(cid:48) b = 10 − where (cid:15) b = 1+i (cid:15) (cid:48)(cid:48) b , and the permittivity of the sphere is (cid:15) = − . . bution at resonance, which explains why the Mie solution Q a in Fig. 4a is (almost) tight with the upper bounds Q opta , and Q opta , , respectively. In Fig. 4b is shown thesame calculations, except that here the background lossis given by (cid:15) (cid:48)(cid:48) b = 10 − . Again, as can be seen in Fig. 2,with such small background losses the optimal Q opta isbased on at least multipole orders up to L = 3, whichexplains why the Mie solution Q a in Fig. 4b is not tightwith the corresponding upper bound Q opta , . Interestinglythat at the same time, the Mie solution can approach theupper electric dipole bound Q opta , when such an amountof losses in the background takes place.Fig. 5 depicts a comparison of the normalized absorp-tion cross section Q opta and its electric-dipole component Q opta , with the total absorption Q a of spherical objectsof the total radius a = 89 nm. In this plot, the follow-ing objects have been considered: a homogeneous spheremade of gold Q Aua (special case with ratio r/d = 0) andthree designs of a layered sphere, where the core is madeof silicon, and it is coated by gold. The absorption of thelayered sphere is based on the normalization of (42) simi- . . . . . − − Electrical size k a Normalized absorption cross section Q a Q opta Q opta , Ratio r/d Q Si , Aua ( r, d )
6? - r (cid:27) d -(cid:27) a FIG. 5. Comparison between the optimal total normalizedabsorption cross section Q opta , the total absorption of a spheremade of gold Q Aua (ratio r/d = 0), and the total absorptionof a multilayered sphere made of silicon (core of radius r ) andgold (shell of thickness d ) Q Si , Aua ; all are plotted as functionsof the electrical size k a . The plots are made for a fixed levelof a background loss (cid:15) (cid:48)(cid:48) b = 10 − and a fixed total radius ofspheres a = 89 nm. larly as in the previous examples, but here, the scatteringcoefficients f τml (43) are expressed in terms of the transi-tion matrices t ( i ) τl for layered spherical objects, see (B20)in Appendix B 5. The designs have been considered forthree different ratios between the radius of core r and thethickness of shell d : r/d ∈ { , , } . Note that r and thetotal radius a coincide with a and a ( N = 2), respec-tively, introduced in Appendix B 5. The dielectric prop-erties of silicon are represented by Drude-Lorentz modelthat fits the measurement data and valid in the photonrange 1 − (the dielectric model in Eq. (4) withparameter values in Table 1). The permittivity of back-ground is (cid:15) b = 1 + i10 − . As can be seen from Fig. 5,by replacing a part of the metallic sphere with silicon,it is possible to obtain a plasmonic resonance at smallerelectrical sizes k a . This is a magnetic dipole resonance,which is inherent in dielectric materials . It should benoted that by increasing the ratio between the radius ofthe silicon core and the thickness of the gold shell, thecomposed sphere becomes resonant at smaller electricalsizes: e.g. , for r/d = 5, the layered sphere is resonantat k a ≈ .
49, while the sphere with r/d = 2 and thegold sphere are resonant at k a ≈ .
64 and k a ≈ . .In Figs. 6a-c is shown a comparison of the vari-ous upper bounds on absorption, and the absorptionof a 2-layered core-shell sphere of different designs forthree different levels of losses in the background: (cid:15) (cid:48)(cid:48) b ∈{ − , − , − } , where (cid:15) b = 1 + i (cid:15) (cid:48)(cid:48) b . The considereddesigns are with cores of radii r ∈ { , , } nm and thecorresponding thicknesses of shells d ∈ { , , } nm.The core is made of germanium (Ge), which is charac- . . . . − − Electrical size k a a) Normalized absorption cross section Q a . . . . − − Electrical size k a b) Normalized absorption cross section Q a . . . . − − Electrical size k a c) Normalized absorption cross section Q a Q opta Q opta , r (nm), d (nm) ,
10 60 ,
20 70 , Q vara ( r, d ) Q Ge , Aua ( r, d )
6? - r (cid:27) d -(cid:27) a FIG. 6. Comparison between the optimal total normalizedabsorption cross section Q opta , variational bound on the nor-malized absorption cross section Q vara , and total absorption ofa multilayered sphere made of germanium (core of radius r )and gold (shell of thickness d ) Q Ge , Aua . All the results are plot-ted as functions of the electrical size k a for different levels oflosses in the background medium with relative permittivity (cid:15) b = 1 + i (cid:15) (cid:48)(cid:48) b : a) (cid:15) (cid:48)(cid:48) b = 10 − ; b) (cid:15) (cid:48)(cid:48) b = 10 − ; c) (cid:15) (cid:48)(cid:48) b = 10 − . terized by Drude-Lorentz model (the dielectric modelin Eq. (4) with parameter values in Table 1) valid in thephoton energy range 0 . − . . The corresponding variational upper bound forthe 2-layered sphere Q vara has been obtained by (31). Ascan be seen from these figures, the composed structurebased on Ge and Au is resonant at small electrical sizes0 k a , but the magnitude of these resonances is not tightto none of the multipole upper bounds, even when theamount of the background losses is high, see Fig. 6awith results for (cid:15) (cid:48)(cid:48) b = 10 − . By comparison of the re-sults on the upper bounds for absorption Q opta obtainedby (50) and Q vara for different amounts of losses in thebackground in Figs. 6a-c, it should be noted that thesebounds provide a complementary information on absorp-tion, and thus this conclusion is valid both for homoge-neous and layered spherical objects. It can be concludedthat for backgrounds with strong losses, the multipolebound Q opta brings more information on absorption limi-tations, while the variational bound Q vara is more tight forsmaller objects that are embedded in low loss surround-ing media, which complements the results obtained by Q opta , see e.g. , Q vara ( r, d ) for r = 30 nm and d = 10 nm inFig. 6c. VI. SUMMARY AND CONCLUSIONS
In this paper, two fundamental multipole bounds onabsorption of scattering objects embedded in a lossy sur-rounding medium have been derived. The derivation ofthese bounds have been made under two fundamentallydifferent assumptions: based on equivalent currents in-side the scatterer, and with respect to the external fieldsusing the T-matrix parameters, respectively. The firstbound depends on the material properties of scattereras well as on its shape, while the second bound is ap-plicable to spherical objects made of an arbitrary ma-terial. Through the numerical examples, it has been il-lustrated that the derived bounds can complement eachother, depending on the amount of losses in the surround-ing medium.
ACKNOWLEDGMENTS
This work has been supported by the Swedish Founda-tion for Strategic Research (SSF), grant no. AM13-0011under the program Applied Mathematics and the projectComplex Analysis and Convex Optimization for EM De-sign.
Appendix A: Derivations based on calculus ofvariation1. Optimal power absorption
Consider the Lagrangian functional for the optimiza-tion problem (19) which is given by L ( F , λ ) = (1 − λ )Im (cid:26)(cid:90) V F ∗ · M a · F d v (cid:27) + λ Im (cid:26)(cid:90) V F ∗ i · M t · F d v (cid:27) − λ Im (cid:26)(cid:90) V F ∗ i · M b · F i d v (cid:27) , (A1)where λ is the Lagrange multiplier. Taking the first vari-ation of (A1) yields δ L ( F , λ ) = Im (cid:26)(cid:90) V δ F ∗ · (cid:104) (1 − λ ) (cid:0) M a − M † a (cid:1) · F − λ M † t · F i (cid:105) d v (cid:27) , (A2)where ( · ) † denotes the Hermitian transpose. Hence, astationary solution with δ L ( F , λ ) = 0 is given by F = α
2i (Im { M a } ) − · M † t · F i , (A3)where α = λ/ (1 − λ ), andIm { M a } = M a − M † a . (A4)Inserting the solution (A3) into (12) gives the optimalabsorption P opta = k α η (cid:90) V F ∗ i · M t · (Im { M a } ) − · M † t · F i d v. (A5)
2. Maximization of parameter q Consider the function f ( M a ) f ( M a ) = (cid:90) V F ∗ i · M t · (Im { M a } ) − · M † t · F i d v, (A6)which represents the denominator of parameter q definedin (22), and where M t = M a − M † b is based on definitions(16) through (18). The function (A6) is convex in M a for Im { M a } >
0, and hence in order to maximize q in(22), f ( M a ) has to be minimized. The first variation of1(A6) is given by δf ( M a ) = (cid:90) V F ∗ i · δ M a · (Im { M a } ) − · (cid:16) M a − M † b (cid:17) † · F i d v − (cid:90) V F ∗ i · (cid:16) M a − M † b (cid:17) · (Im { M a } ) − · Im { δ M a }· (Im { M a } ) − · (cid:16) M a − M † b (cid:17) † · F i d v + (cid:90) V F ∗ i · (cid:16) M a − M † b (cid:17) · (Im { M a } ) − · δ M † a · F i d v, (A7)and where we have employed the relation δ (Im { M a } ) − = − (Im { M a } ) − · Im { δ M a } · (Im { M a } ) − . (A8)The stationarity condition δf ( M a ) = 0 for all δ M a andall F i gives the simplified conditionRe (cid:26) δ M a · (Im { M a } ) − · (cid:16) M a − M † b (cid:17) † − (cid:16) M a − M † b (cid:17) · (Im { M a } ) − · δ M a · (Im { M a } ) − · (cid:16) M a − M † b (cid:17) † (cid:27) = , (A9)and which can be reorganized asRe (cid:40)(cid:34)(cid:16) M a − M † b (cid:17) · (Im { M a } ) − − I (cid:35) · δ M a · (Im { M a } ) − · (cid:16) M a − M † b (cid:17) † (cid:27) = . (A10)Assuming that both Im { M a } > { M b } > M a (cid:54) = M † b , the optimal solution is obtainedby requiring that the first line within parenthesis abovevanishes, yieldingIm { M a } = M a − M † b , (A11)or equivalently that M a = M b . This solution impliesthat the minimum of f ( M a ) in (A6) is given by f ( M a ) | M a = M b = (cid:90) V F ∗ i · (cid:16) M b − M † b (cid:17) · (Im { M b } ) − · (cid:16) M b − M † b (cid:17) † · F i d v = 4 (cid:90) V F ∗ i · Im { M b } · F i d v. (A12)By substitution of the result (A12) into the denominatorof (22), it is finally seen that q reaches its maximum at1, and hence q ≤ Appendix B: Spherical vector waves1. Definition of spherical vector waves
Consider a source-free homogeneous and isotropicmedium with wave number k = k √ µ(cid:15) . The electro-magnetic field can then be expanded in spherical vectorwaves as E ( r ) = (cid:88) τ,m,l a τml v τml ( k r ) + f τml u τml ( k r ) , H ( r ) = 1i η η (cid:88) τ,m,l a τml v ¯ τml ( k r ) + f τml u ¯ τml ( k r ) , (B1)where v τml ( k r ) and u τml ( k r ) are the regular and theoutgoing spherical vector waves, respectively, and a τml and f τml the corresponding multipole coefficients, see e.g. , . Here, l = 1 , , . . . , is the multipole order, m = − l, . . . , l , the azimuthal index and τ = 1 ,
2, where τ = 1 indicates a transverse electric (TE) magnetic mul-tipole and τ = 2 a transverse magnetic (TM) electricmultipole, and ¯ τ denotes the dual index, i.e. , ¯1 = 2 and¯2 = 1.The solenoidal (source-free) regular spherical vectorwaves are defined here by v ml ( k r ) = 1 (cid:112) l ( l + 1) ∇ × ( r j l ( kr )Y ml (ˆ r ))= j l ( kr ) A ml (ˆ r ) , (B2)and v ml ( k r ) = 1 k ∇ × v ml ( k r )= ( kr j l ( kr )) (cid:48) kr A ml (ˆ r ) + (cid:112) l ( l + 1) j l ( kr ) kr A ml (ˆ r ) , (B3)where Y ml (ˆ r ) are the spherical harmonics, A τml (ˆ r ) thevector spherical harmonics and j l ( x ) the spherical Besselfunctions of order l , cf. , . Here, ( · ) (cid:48) denotes adifferentiation with respect to the argument of the spher-ical Bessel function. The outgoing (radiating) spher-ical vector waves u τml ( k r ) are obtained by replacingthe regular spherical Bessel functions j l ( x ) above withthe spherical Hankel functions of the first kind, h (1) l ( x ),see . It can be shown that any one of the vec-tor spherical waves w τml ( k r ) defined above satisfy thefollowing curl properties ∇ × w τml ( k r ) = k w ¯ τml ( k r ) , (B4)and hence the source-free Maxwell’s equations (vectorHelmholtz equation) in free space, i.e. , ∇ × ∇ × w τml ( k r ) = k w τml ( k r ) . (B5)2The vector spherical harmonics A τml (ˆ r ) are given by A ml (ˆ r ) = 1 (cid:112) l ( l + 1) ∇ × ( r Y ml (ˆ r ))= 1 (cid:112) l ( l + 1) ∇ Y ml (ˆ r ) × r , A ml (ˆ r ) = ˆ r × A ml (ˆ r )= 1 (cid:112) l ( l + 1) r ∇ Y ml (ˆ r ) , A ml (ˆ r ) = ˆ r Y ml (ˆ r ) , (B6)where τ = 1 , ,
3, and where the spherical harmonicsY ml (ˆ r ) are given byY ml (ˆ r ) = (cid:114) l + 14 π (cid:115) ( l − m )!( l + m )! P ml (cos θ )e i mφ , (B7)and where P ml ( x ) are the associated Legendrefunctions . The vector spherical harmonicsare orthonormal on the unit sphere, and hence (cid:90) Ω A ∗ τml (ˆ r ) · A τ (cid:48) m (cid:48) l (cid:48) (ˆ r ) dΩ = δ ττ (cid:48) δ mm (cid:48) δ ll (cid:48) , (B8)where Ω denotes the unit sphere and dΩ = sin θ d θ d φ .
2. Lommel integrals for spherical Bessel functions
Let s l ( kr ) denote an arbitrary linear combinationof spherical Bessel and Hankel functions. Based onthe two Lommel integrals for cylinder functions, cf. , (Eqs. (10.22.4) and (10.22.5) on p. 241) and (Eqs. (8)and (10) on p. 134), the following indefinite Lommel in-tegrals can be derived for spherical Bessel functions (cid:90) | s l ( kr ) | r d r = r Im { k s l +1 ( kr )s ∗ l ( kr ) } Im { k } , (B9)where k is complex-valued ( k (cid:54) = k ∗ ), cf. , (Eq. (A.15)on p. 11), and (cid:90) s l ( kr ) r d r = 12 r (cid:0) s l ( kr ) − s l − ( kr )s l +1 ( kr ) (cid:1) , (B10)where k is either real-valued or complex-valued. Further-more, by using the recursive relationships s l ( kr ) kr = 12 l + 1 (s l − ( kr ) + s l +1 ( kr )) , s (cid:48) l ( kr ) = 12 l + 1 ( l s l − ( kr ) − ( l + 1)s l +1 ( kr )) , (B11)where l = 0 , ± , ± , . . . , cf. , (Eqs. 10.1.19-20), it can beshown that (cid:90) (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) s l ( kr ) kr + s (cid:48) l ( kr ) (cid:12)(cid:12)(cid:12)(cid:12) + l ( l + 1) (cid:12)(cid:12)(cid:12)(cid:12) s l ( kr ) kr (cid:12)(cid:12)(cid:12)(cid:12) (cid:33) r d r = 12 l + 1 (cid:90) (cid:16) ( l + 1) | s l − ( kr ) | + l | s l +1 ( kr ) | (cid:17) r d r, (B12) for all values of l = 0 , ± , ± , . . . . It can be shown simi-larly that (B12) is valid with | · | replaced for ( · ) .
3. Orthogonality over a spherical volume
Due to the orthonormality of the vector spherical har-monics (B8), it follows that the regular spherical vectorwaves defined in (B2) and (B3) are orthogonal over aspherical volume V a of radius a , yielding (cid:90) V a v ∗ τml ( k r ) · v τ (cid:48) m (cid:48) l (cid:48) ( k r ) d v = δ ττ (cid:48) δ mm (cid:48) δ ll (cid:48) W τl ( k, a ) , (B13)where W τl ( k, a ) = (cid:90) V a | v τml ( k r ) | d v, (B14)for τ = 1 , l ≥ v = r dΩ d r . Forcomplex-valued arguments k (cid:54) = k ∗ , W l ( k, a ) is obtainedfrom (B9) as W l ( k, a ) = (cid:90) a | j l ( kr ) | r d r = a Im { k j l +1 ( ka )j ∗ l ( ka ) } Im { k } , (B15)and from (B12) follows that W l ( k, a )= (cid:90) a (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) j l ( kr ) kr + j (cid:48) l ( kr ) (cid:12)(cid:12)(cid:12)(cid:12) + l ( l + 1) (cid:12)(cid:12)(cid:12)(cid:12) j l ( kr ) kr (cid:12)(cid:12)(cid:12)(cid:12) (cid:33) r d r = 12 l + 1 [( l + 1) W ,l − ( k, a ) + lW ,l +1 ( k, a )] . (B16)For real-valued arguments ( k ∗ = k ), W l ( k, a ) and W l ( k, a ) can be calculated similarly by using (B10) and(B16), respectively.
4. Orthogonality over a spherical surface
Based on the properties of the spherical vector wavesdescribed in Appendix B 1, the following orthogonalityrelationships regarding their cross products on a sphericalsurface can be derived as (cid:90) ∂V a w τml ( k r ) × z ∗ ¯ τm (cid:48) l (cid:48) ( k r ) · ˆ r d S = a δ mm (cid:48) δ ll (cid:48) w l ( ka ) (cid:18) ( kaz l ( ka )) (cid:48) ka (cid:19) ∗ τ = 1 , − (cid:18) ( kaw l ( ka )) (cid:48) ka (cid:19) z ∗ l ( ka ) τ = 2 , (B17)and (cid:90) ∂V a w τml ( k r ) × z ∗ τm (cid:48) l (cid:48) ( k r ) · ˆ r d S = 0 , (B18)3for τ = 1 ,
2. Here, ∂V a is the spherical surface of radius a , w l ( ka ) and z l ( ka ) are either of j l ( ka ) or h (1) l ( ka ), and w τml ( k r ) and z τml ( k r ) are the corresponding sphericalvector waves, respectively.
5. Mie theory
Consider the scattering of the electromagnetic fielddue to a layered sphere consisting of N layers made ofisotropic materials. Let a i , (cid:15) i , µ i , k i = k √ µ i (cid:15) i and η i = (cid:112) µ i /(cid:15) i (for i = 1 , . . . , N ) denote the radii, the rel-ative permittivities, the relative permeabilities, the wavenumbers, and the relative wave impedances of each of N layers of the sphere, respectively. The scatterer of thetotal radius a = a N is embedded in the medium char-acterized by the wave number k b = k √ µ b (cid:15) b and therelative wave impedance η b = (cid:112) µ b /(cid:15) b , where (cid:15) b and µ b denote the permittivity and the permeability of the sur- rounding medium, respectively.The electric and magnetic fields in each layer, for a i − < r < a i , i = 1 , . . . , N , can be expanded in sphericalvector waves as E ( r ) = (cid:88) τ,m,l A ( i ) τml (cid:16) v τml ( k i r ) + t ( i − τl u τml ( k i r ) (cid:17) , H ( r ) = 1i η η i (cid:88) τ,m,l A ( i ) τml (cid:16) v τml ( k i r ) + t ( i − τl u τml ( k i r ) (cid:17) , (B19)where A ( i ) τml and t ( i ) τl are the corresponding multipole co-efficients and transition matrices, respectively, see e.g. , (Eq. (8.16) on p. 436).By matching the boundary conditions for the tangen-tial electric and magnetic fields at each boundary inter-face a i , the transition matrix for scattering in the corre-sponding layer i = 1 , . . . , N can be determined as t ( i ) τl = − m ( i ) τ (cid:16) ψ l ( x i ) + t ( i − τl ξ ( x i ) (cid:17) ψ (cid:48) l ( y i ) − (cid:16) ψ (cid:48) l ( x i ) + t ( i − τl ξ (cid:48) ( x i ) (cid:17) ψ l ( y i ) m ( i ) τ (cid:16) ψ l ( x i ) + t ( i − τl ξ ( x i ) (cid:17) ξ (cid:48) l ( y i ) − (cid:16) ψ (cid:48) l ( x i ) + t ( i − τl ξ (cid:48) ( x i ) (cid:17) ξ l ( y i ) , (B20)for τ = 1 ,
2, where m ( i )1 = η i /η i +1 and m ( i )2 = η i +1 /η i , x i = k i a i and y i = k i +1 a i denote the electrical radius interms of the material paratmers on the internal and ex-ternal sides of the interface a i , respectively, see (p. 437).Note that k N +1 = k b , η N +1 = η b , and t (0) τl = 0 due tothe non-singular behavior of the electric field at the ori-gin, i.e. , r = . Hereby, the expression for the transi-tion matrix (B20) reduces to the result for an homoge-neous isotropic sphere, see (Eqs. (4.52) on p. 100) and (Eq. (8.7) on p. 420), and the multipole coefficients A (1) τml agree with the coefficients a τml in (B1).Let E i ( r ) = E e i k ˆ k · r describe a plane wave with vec-tor amplitude E and propagation direction ˆ k . It can be shown that the corresponding multipole expansion coef-ficients are given by a i τml = 4 π i l − τ +1 E · A ∗ τml (ˆ k ) , (B21)for τ = 1 , l = 1 , , . . . , and m = − l, . . . , l , and wherethe vector spherical harmonics A τml (ˆ k ) are defined asin (B6), see also (Eq. (7.28) on p. 375). Based onthe sum identities for the vector spherical harmonics (Eqs. (A17) and (A18)), it can be shown that l (cid:88) m = − l (cid:12)(cid:12) a i τml (cid:12)(cid:12) = 2 π (2 l + 1) | E | , (B22)for τ = 1 , ∗ [email protected] † [email protected] ‡ [email protected] M. Abramowitz and I. A. Stegun, editors.
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