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On the quasistatic optimal plasmonic resonances in lossy media

This paper discusses and analyzes the quasistatic optimal plasmonic dipole resonance of a small dielectric particle embedded in a lossy surrounding medium. The optimal resonance at any given frequency is defined by the complex valued dielectric constant that maximizes the absorption of the particle under the quasistatic approximation and a passivity constraint. In particular, for an ellipsoid aligned along the exciting field, the optimal material property is given by the complex conjugate of the pole position associated with the polarizability of the particle. In this paper, we employ classical Mie theory to analyze this approximation for spherical particles in a lossy surrounding medium. It turns out that the quasistatic optimal plasmonic resonance is valid provided that the electrical size of the particle is sufficiently small at the same time as the external losses are sufficiently large. Hence, it is important to note that this approximation can not be used for a lossless medium, and which is also obvious since the quasistatic optimal dipole absorption becomes unbounded for this case. Moreover, it turns out that the optimal normalized absorption cross section area of the small dielectric sphere has a very subtle limiting behavior, and is in fact unbounded even in full dynamics when both the electrical size as well as the exterior losses tend to zero at the same time. A detailed analysis is carried out to assess the validity of the quasistatic estimation of the optimal resonance and numerical examples are included to illustrate the asymptotic results.


I. INTRODUCTION
The classical theories as well as most of the recent theoretical studies and experiments on plasmonic resonances reported in the literature are concerned with metal nanoparticles where the exterior domain is lossless, see e.g., 1-6 with references. As e.g., in 5 , a variational approach is employed in connection with a generalized optical theorem for scattering, absorption and extinction 7 , to obtain an upper bound on the absorption that can be achieved inside a scatterer with arbitrary geometry and with a given volume and material property. Furthermore, in 4 is given an upper bound on the absorption of an electrically small structure (or particle) with arbitrary geometry and structural parameters, and which is obtained by optimizing directly in terms of the complex valued polarizability of the particle. Both results in 4 and 5 are valid for lossless surrounding media only.
However, there are several application areas of plasmonics where the exterior losses must be taken into account. One potential application in the medical area is the localized electrophoretic heating of a bio-targeted and electrically charged gold nanoparticle (GNP) suspension as a radiotherapeutic hyperthermia based method to treat cancer, cf., [8][9][10][11][12] . Other potential application areas include plasmonic nanoparticle enhancement of light absorption for solar cells 13 , plasmon waveguides, aperture arrays, extraordinary transmission, etc., and where small external losses can in fact be utilized to increase the available bandwidth of resonance, scattering and absorption.
In this paper, we are concerned with the optimal absorption (or optimal plasmonic resonance) that can be achieved by an electrically small spherical object embedded in a lossy surrounding medium. For simplicity, we are considering only the important special case with nonmagnetic, dielectric or metallic materials which are common in plasmonic applications. Magnetic materials can be treated similarly. We are also considering only a single electric dipole resonance, and we are discarding any possibilities of having an unbounded absorption due to multiple mode super resolution effects, etc., see e.g., [14][15][16] . A quasistatic theory has been developed in 11,12 giving the optimal plasmonic dipole resonance of small dielectric ellipsoids (including spheroids and spheres) in terms of an optimal conjugate match with respect to the background loss. The quasistatic paradox is that this theory predicts an unbounded absorption when the exterior losses approach zero, cf., e.g., 12 (Eq. (33) on p. 5). Notably, for a lossless exterior medium and under the quasistatic approximation, the maximal absorption given in 5 (Eqs. (32a) and (32b) on p. 3345) and which is achievable by an optimized ellipsoid, is also unbounded as the internal losses approach zero. However, these observations are in contradiction to the fact that the absorption cross section of a small dipole scatterer in a lossless exterior medium is bounded by 3λ 2 /8π, cf., 4 (Eq. (16) on p. 937).
The quasistatic paradox can be resolved by realizing that the plasmonic singularity of the sphere is a phenomenon that can take place only in the sense of a limit as the size of the particle approaches zero. In fact, it turns out that the dipole resonance of a small sphere has a very subtle limiting behavior as the electrical size approaches zero. As e.g., in 6 Padé approximants are used to reveal new scattering aspects of the sphere. In this paper, an asymptotic analysis based on Mie theory is employed to study the limiting behavior of the quasistatic optimal resonance 11 , as the electrical size of the sphere as well as the external losses tend to zero. The quasistatic paradox is then finally resolved by providing explicit formulas for the validity of the quasistatic approximation. An improved asymptotic formula for the optimal plasmonic resonance of the sphere is also given, and which is valid for small spheres as well as for small losses. In essence, the quasistatic approximation is missing the scattering loss, an error which becomes significant when the external losses are small.
The rest of the paper is organized as follows. In Section II is given a comprehensive statement of the quasistatic paradox relating to previous results concerning absorption, scattering and extinction. The paradox is then analyzed in detail and resolved in Section III. Numerical examples are given in Section IV and the vector spherical waves are defined in Appendix A.

A. Notation and conventions
The following notations and conventions are used in this paper. Classical electrodynamics is considered where the electric and magnetic field intensities E and H are given in SI-units 17 . The time convention for time harmonic fields (phasors) is given by e −iωt where ω is the angular frequency and t the time. Consequently, the relative permittivity of a passive isotropic dielectric material has positive imaginary part. Let µ 0 , 0 , η 0 and c 0 denote the permeability, the permittivity, the wave impedance and the speed of light in vacuum, respectively, and where η 0 = µ 0 / 0 and c 0 = 1/ √ µ 0 0 . The wavenumber of vacuum is given by k 0 = ω √ µ 0 0 . The wave number of a homogeneous and isotropic medium with relative permeability µ and permittivity is given by k = k 0 √ µ and the wavelength λ is defined by kλ = 2π.
The wave impedance of the same medium is given by η 0 η where η = µ/ is the relative wave impedance. In the following, we will consider only non-magnetic, homogeneous and isotropic dielectric materials, and hence µ = 1 from now on. The spherical coordinates are denoted by (r, θ, φ), the corresponding unit vectors (r,θ,φ), and the radius vector r = rr. Finally, the real and the imaginary parts, and the complex conjugate of a complex number ζ are denoted Re {ζ}, Im {ζ} and ζ * , respectively.

B. Formulation of the quasistatic paradox
In the quasistatic approximation the optimal absorption of a small dielectric sphere can readily be calculated as follows. Let us consider a small, homogeneous and isotropic dielectric sphere with radius a and relative permittivity , and which is embedded in a lossy dielectric background medium with relative permittivity b . The sphere is considered to be small in the sense that k 0 √ b a 1. Let E i = E 0ê denote the applied electric field which is assumed to be uniform throughout the whole space. The resulting dipole moment is given by p = 0 b αE i where α is the polarizability of the small sphere given by the well-known expression and which is valid to the lowest order in k 0 a (in the quasistatic approximation), see e.g., 1 . The dipole moment of the sphere can also be expressed as p = Va 0 ( − b )Edv where V a denotes the spherical domain of radius a and E the resulting internal field. Since the resulting field E is a constant vector parallel to E i 1 , it follows immediately that The absorbed power of the sphere can now be readily calculated from Poynting's theorem as The absorption cross section C abs of the sphere is obtained by normalizing with the power intensity of the incoming plane wave, yielding where Let o denote a fixed complex valued constant with Im{ o } > 0 and consider the real valued function where is a complex valued variable with Im{ } > 0. It can be shown that the function f ( ) has a local maximum at o cf., 11 (Sect. 2.5, Eqs. (15) through (17)). Hence, for the sphere the maximizer of C abs is given by and the corresponding maximal absorption is cf., 11 . The solution (6) is referred to as an optimal conjugate match and can be interpreted in terms of an optimal plasmonic resonance for the sphere in a lossy surrounding medium 11 . Note that when Re{ b } > 0 and Im{ b } > 0, as with a "normal" surrounding medium, we have that Re{ o } < 0 and Im{ o } > 0 which is characteristic for a passive medium in plasmonic resonance and which can be achieved e.g., by tuning a Drude model 11 . Similar results are also readily obtained for a homogeneous ellipsoid aligned along the incident field, in which case o = − * b (1 − L)/L where 0 < L < 1 is the corresponding depolarizing factor 1 , cf., 12 (Eqs. (23) and (33) on pp. [4][5].
Finally, we put our attention to a new fundamental bound on the absorption of a general scatterer in a lossless surrounding medium, recently obtained in 5 . Under the quasistatic approximation where E i is uniform, and for a homogeneous and isotropic dielectric body, the bound in 5 is given by where V is the volume of the scattering body, see 5 (Eqs. (32a) and (32b) on p. 3345). The bound (8) is furthermore achievable by optimizing the depolarizing factor of an ellipsoid, cf., 5 (Eq. (41) on p. 3349).
The important observation to make here is that (7) (as well as the similar expression for the ellipsoid 12 (Eq. (33) on p. 5)) is unbounded when the exterior medium becomes lossless and Im{ b } → 0. Furthermore, for a lossless surrounding medium, the bound (8) which is achievable by an ellipsoid, is unbounded as Im{ } → 0. As will be demonstrated in the next section, the quasistatic paradox is that all these conclusions are in contradiction to the fact that for a fixed frequency ω and particle size (radius a or volume V ), the absorption cross section of a small dipole scatterer in a lossless medium is bounded by 3λ 2 /8π, cf., e.g., 4 (Eq. (16) on p. 937).

C. Optimal absorption of a small dipole in a lossless background medium
For a lossless background medium with relative permittivity b , it is possible to show that for a small dipole scatterer with polarizability α, the extinction and scattering cross sections are given by the following (exact) formulas where k b = k 0 √ b and b is real valued and positive, see e.g., 1 (pp. 71 and 140). The absorption cross section of the dipole scatterer is readily obtained as C abs = C ext − C s . The expressions in (9) are derived under the assumption that the boundary surface S of the scatterer can be extended to a spherical surface at an arbitrary radius R without changing the value of the absorbed power P abs = P ext − P s , and hence that the far-field can be exploited by letting R → ∞. Note, however, that this assumption is true only if the medium outside the scatterer is lossless. Hence, for a lossless medium the expressions in (9) are valid for small dipole scatterers of arbitrary shape and with arbitrary structural parameters as long as it is the dipole field that dominates the radiation field and higher order modes can be neglected (no super resolution, etc.), cf., 1,4 .
Based on the expression C abs = C ext − C s where C ext and C s are given by (9), it can be shown that C abs can be maximized directly in terms of the complex valued polarizability α expressed as where the last term is due to the scattering loss and where the optimum is given by ξ = 0 and ξ = k 3 b /6π, cf., 4 (Eq. (15) on p. 937). Hence, the following upper bound is obtained for the absorption of an arbitrary electric dipole scatterer in a lossless background medium cf., 4 (Eq. (16) on p. 937). It is interesting to note that the maximal absorption (11) can be achieved by a small dielectric sphere. Hence, by employing the inverse polarizability of the dielectric sphere given by cf., 18 (Eq. (8.10)), and equating this expression to the optimal value −ik 3 b /3π, the following solution is obtained and which is accurate up to the third order in k b a, cf., 6 (Eq. (11) on p. 3) and 4 (note that the expression (13) contains a small correction of the corresponding formula given in 4 (Eq. (22) on p. 938)). We note that including only the second order dynamic correction to the polarizability of the sphere is not enough to capture the physics of ultimate absorption: From the above expression we see that the second order correction gives only a correction to the resonant value of the real part of the permittivity. If we neglect the third order term, at this resonant frequency the absorption cross section is not optimal, and is in fact unbounded in the quasistatic approximation. It is also interesting to note that a Drude model can always be tuned to the optimal solution (13). Hence, by expressing the Drude model as where ω p is the plasma frequency and τ the relaxation time, and by equating real and imaginary parts in (13) and (14) up to the third order in k b a for a given tuning frequency ω c , the following (unique) solution can be derived similar to tuning the Drude model to the optimal conjugate match as in 11,12 , and where k b is evaluated at ω c .
Note that with small k b a the solution (15) yields ω c τ 1 and The main observation in this section is that for a fixed frequency ω and particle size a, the result (11) contradicts the unboundedness of the quasistatic results given by (7) and (8). This is the quasistatic paradox. In essence, this means that for a fixed frequency and particle size, there is no plasmonic singularity of the dielectric sphere as one may think when considering only the quasistatic perspective, cf., also 6 . There is only a maximal absorption characterized by (13) and bounded by (11). The plasmonic singularity should be understood only in the sense of the limit as k 0 a → 0 and where Q dip

D. The lossy case revisited
To conclude this section, a quasistatic paradox has been formulated indicating that the calculation of the absorption cross section of a small scatterer in a lossy background medium is a very subtle issue, and in particular when the background losses are small. Hence, a complete electrodynamic solution is warranted in order to asses the validity of the quasistatic approximation expressed in (4) and (7). However, neither the simple expressions in (9) nor the relation C abs = C ext −C s are valid for a lossy surrounding medium 1 . Great care must therefore be taken to properly calculate C abs = C ext + C i − C s , where the cross sections are calculated directly on the surface of the scattering body and where C i is a correction term relating to the power absorbed by the incident wave in the surrounding medium, cf., 1,9,11,12,19,20 . However, here it seems to be a more practical approach to solve the scattering problem directly in terms of the internal fields of the sphere by employing general Mie theory and an adequate use of the first Lommel integral 21,22 for spherical Bessel functions with complex valued arguments, see also 11 . This is the topic of the next section.

A. Electrodynamic solution
The complete electrodynamic solution for the internal absorption of a small dielectric sphere in a lossy background medium is developed below. The definition of the vector spherical waves, the spherical Bessel and Hankel functions and the related Lommel integrals are given in Appendix A, see also 11 .
Consider the scattering of the electromagnetic field due to a homogeneous dielectric sphere of radius a, permittivity , and wavenumber k = k 0 √ . The medium surrounding the sphere is characterized by the permittivity b and the wavenumber k b = k 0 √ b . The incident and the scattered fields for r > a are expressed as in (A1) with multipole coefficients a τ ml and b τ ml , respectively, and the interior field is similarly expressed using regular vector spherical waves for r < a with multipole coefficients a int τ ml . By matching the tangential electric and magnetic fields at the boundary of radius a, it can be shown that b τ ml = t τ l a τ ml , for τ = 1, 2, l = 1, 2, . . ., and m = −l, . . . , l, and where t τ l and r τ l are transition matrices for scattering and absorption, respectively, see e.g., 1 (Eqs. (4.52) and (4.53) on p. 100). In particular, for the internal fields of the sphere the corresponding electric multipole coefficients are given by .
It can be shown that the multipole expansion coefficients for a plane wave E i (r) = E 0 e ik bk ·r are given by for τ = 1, 2, l = 1, 2, . . ., and m = −l, . . . , l, and where the vector spherical harmonics A τ ml (k) are defined as in A 1, cf., e.g., 23 Based on Poynting's theorem and the orthogonality of the vector spherical waves the absorbed power can now be calculated as where V a denotes the spherical volume of radius a and where W τ l (k, a) = Va |v τ ml (kr)| 2 dv is the volume integral of the regular vector spherical waves, cf., (A13) and (A14). In (21), the relations (17) and (20) have also been employed.
The absorption cross section (C abs = P abs /I i ) based solely on the dominating Transverse Magnetic (TM) dipole fields (τ = 2 and l = 1) can now be expressed as where is given by (18), and is obtained from (A15) and (A16).

B. Asymptotic analysis
To analyze the asymptotic properties of the dynamic solution C dyn abs , the expression (22) is first rewritten in the following area normalized form An asymptotic analysis can now be carried out based on the asymptotic expansions of the spherical Bessel and Hankel functions for small arguments, see e.g., 22 . Furthermore, with = + i and b = b + i b and small loss factors and b , it is readily seen that where O{·} denotes the big ordo 24 (p. 4), and where it is assumed that = 0 and b = 0 are fixed. From this observation, it is also noted that for integer m O{(ka) m } = O{(k 0 a) m }, independent of the (small) values of and b . Based on the asymptotic tools mentioned above, the different factors of (25) can now be analyzed in detail. We start by studying the normalized factor where is given by (24). Note that the spherical Bessel functions j n (x) of order n are analytic functions with a Taylor series expansion starting at the polynomial order n. The functions j n (x) are furthermore odd for odd orders and even for even orders. Hence, it is readily seen that the function F (k 0 a, ) has a convergent power series expansion in terms of and * with real valued coefficients, and that and Im{F (k 0 a, )} = k 0 a 2 3 + O{(k 0 a) 3

}O{ }. (31)
This means that the expression in (28) is well behaved when k 0 a as well as approaches zero, and that independent of . This observation can also be derived directly from the definition (A16) and the regularity of the spherical Bessel function j 1 (x). A detailed asymptotic study of (23) shows that where It can safely be assumed that C = 0 and D = 0 since b = is not an interesting case and b + /10 = 0 for lossy and passive media. For fixed and b with + 2 b = 0, and by using (32) and (33), it is now concluded that (22) is given by which is in agreement with the quasistatic approximation given in (4). However, when and b are fixed at = −2 b , and at the same time k 0 a as well as and b can be chosen freely, the area normalized absorption cross section (25) turns out to be unbounded. To study this behavior in detail, it is noted that to the lowest order in (k 0 a, , b ), the expression (25) can be approximated by where C 0 = ( b − )( b + /10) and the symbol ∼ indicates an asymptotic approximation in the sense of 24 (p. 4). Obviously, for any small but finite value of k 0 a = 0 the real part of the expression under the absolute-value sign can be brought to zero, and we see that in this approximation the absorption cross section is unbounded at and b tending to zero. We again confirm that the second-order correction is not enough to reveal the dynamic absorption bound. Interestingly, the absorption cross section can be unbounded even without tuning to the dynamically-corrected resonant value of the permittivity. In particular, by considering the special case of interest, i.e., the quasistatic optimal conjugate match where where C 0 = 12 b 2 /5 and where it is assumed that b = 0. Note that the assumption b (k 0 a) 2 C 0 /4 will restore the expression for the maximal absorption given by (7). Hence, it is observed that the quasistatic theory (7) is valid provided that k 0 a is sufficiently small at the same time as the background loss factor b is sufficiently large.
The factor governing the convergence of (37) and (25) is given by By choosing where A is an arbitrary constant and α > 0, this factor becomes A detailed study of the expression (40) for small b > 0 reveals the condition for convergence of (25), which can be summarized as where k 0 a = A b α , b is fixed and = −2 * b (the quasistatic optimal conjugate match).
For a fixed background loss parameter b > 0, the factor (38) can furthermore be maximized with respect to the electrical size k 0 a, yielding k 0 a = 2 For fixed b , the relation (42) expresses a stationary point for (37) regarded as a function of k 0 a, and hence an indicator of the domain of validity of the quasistatic approximation (7).
In conclusion, it has been shown that the area normalized absorption cross section (25) has a subtle limiting behavior and is in fact unbounded as (k 0 a, , b ) → (0, 0, 0) for fixed and b . It is also shown that if k 0 a is sufficiently small, the optimal conjugate match = −2 * b defined in (6) yields an accurate quasistatic approximation where (42) and C 0 = 12 b 2 /5 have been used. It should be noted that a direct maximization of (25) with respect to would be possible by numerical optimization techniques, but is not necessary if the quasistatic approximation is valid. And the validity of the quasistatic approximation can be assessed by checking the criteria (43) as well as by a direct comparison of (4) and (7) with the electrodynamic solution (25).
C. Optimal absorption of the small dielectric sphere in a lossy media Finally, the asymptotic analysis above with (33) and (34) makes it possible to analyze the pole structure of (25) in full dynamics. By making the following ansatz for the pole p = a 0 + a 1 k 0 a + a 2 (k 0 a) 2 + a 3 (k 0 a) 3 , and identifying terms up to the third order in the denominator of (33), it is found that (25) can be approximated for small k 0 a as and where the pole is given by see also 4 , and 6 (Eq. (11) on p. 3). The expression (44) is of the form (5) where Im{ p } < 0 for small b , and hence it can be concluded that the maximal absorption is achieved at o = * p , yielding Note that the result (46) generalizes the previous quasistatic result (6) for lossy media, as well as the result (13) for lossless media. The expression is valid for small k 0 a as well as for small loss factors b . In particular, the optimal absorption of the sphere is given by and in the limit as the exterior losses vanish, we have that where k b = k 0 b , and which is tight with the upper bound (11) given in 4 . Finally, it should be noted that there are major discrepancies in the asymptotic analysis of the full dynamic solution performed in this section in comparison to the analysis of the quasistatic approximation as in (36) and (37) where and b are fixed and = −2 b . In essence, the quasistatic approximation (6) is missing the second order term giving a shift in the resonance frequency as well as the third order term taking the scattering loss into account. These errors are negligible only when the external losses are large enough to make the second order term redundant as expressed in (43), at the same time as the electrical size of the sphere is small enough to make the scattering loss insignificant. Hence, when the external losses are too small in relation to the electrical size of the sphere, the quasistatic approximation o = −2 * b is not the correct choice to maximize the absorption and one should instead use (46).

IV. NUMERICAL EXAMPLES
In Figures 1 through 4 are shown the area normalized absorption cross section Q abs and its behavior for (k 0 a, b ) close to (0, 0). Here, Q dyn abs is the electrodynamic solution given by (25) and Q o abs corresponds to the optimal quasistatic solution given by (7), both of which are calculated for the background permittivity b = b + i b and the quasistatic optimal conjugate match = −2 * b , yielding + 2 b = i4 b . In Figure 1 is also shown the normalized absorption cross section Q dyn,o abs corresponding to the dynamic optimal solution (46), as well as the break points for the quasistatic approximation given by (42). As a comparison with the case with a lossless background ( b = 0) the optimal dipole absorption Q dip abs corresponding to (11) is also shown in Figure 1. In Figures  1 through 3 the background is defined by b = 1, and in Figure 4 the background corresponds to a saline water with b = 80.
As can be seen in the Figures 1 through 3 the normalized absorption cross section Q dyn abs has a very subtle behavior for (k 0 a, b ) close to the plasmonic singularity at (0, 0) where = −2. Note that Q dyn abs → 0 as k 0 a → 0 ( b > 0 fixed) and Q dyn abs → 0 as b → 0 (k 0 a > 0 fixed), cf., also the approximate expression in (37). However, as illustrated in Figure 3, Q dyn abs is unbounded in any neighborhood where (k 0 a, b ) is close to (0, 0) in accordance with the analysis given in section III B. It should be noted that this behavior is perfectly consistent with the bound for a lossless background C abs ≤ 3λ 2 /8π given in 4 since C abs /a 2 ≤ (3/8π)/(a/λ) 2 is unbounded as a/λ approaches zero.
In Figures 1 through 3 are illustrated how the approximation Q dyn abs ∼ Q o abs becomes valid when k 0 a is sufficiently small at the same time as b is sufficiently large. In Figure 1 is also illustrated that the break points given by (42) (or (43)) give a very accurate estimate for the validity of the quasistatic approximation (7).
The whole feasibility investigation regarding the quasistatic approximation above can readily be executed similarly for any other background permittivity b . As e.g., in Figure 4 is shown a comparison between Q dyn abs and Q o abs with b = 80 corresponding to the permittivity of biological tissue for frequencies in the lower GHz region, cf., 25 . In this frequency region the corresponding dielectric losses will be at least b > 10. Hence, the evaluation shown in Figure 4 verifies that the previous investigations made in 11,12 are safely in the quasistatic regime if e.g., k 0 a < 10 −2 , and which is certainly the case with cellular (µm) structures in the GHz frequency range.

V. SUMMARY AND CONCLUSIONS
A quasistatic paradox has been formulated referring to the unboundedness of the maximal absorption of a small dielectric sphere in a lossy surrounding medium under the quasistatic approximation. The paradox is resolved  by using general Mie theory and an asymptotic analysis to give explicit formulas to assess the validity of the quasistatic approximation. In particular, it turns out that the quasistatic theory is valid only provided that the electrical size of the sphere is sufficiently small at the same time as the exterior losses are sufficiently large. Moreover, it has been shown that the optimal area normalized absorption cross section of the small dielectric sphere has a very subtle limiting behavior, and is in fact unbounded when both the electrical size as well as the exterior losses tend to zero. Finally, an improved asymptotic formula based on full dynamics has been given for the optimal plasmonic dipole absorption of the sphere, and which is valid for small spheres as well as for small losses. An interesting topic for future research is to de- velop a more complete theory and characterization of the optimal plasmonic multipole absorption and scattering of a small sphere embedded in lossy media.
The solenoidal (source-free) regular vector spherical waves are defined here by v 1ml (kr) = 1 l(l + 1) ∇ × (rj l (kr)Y ml (r)) = j l (kr)A 1ml (r), (A3) and where Y ml (r) are the spherical harmonics, A τ ml (r) the vector spherical harmonics and j l (x) the spherical Bessel functions of order l, cf., 17,22,23,26,27 . Here, (·) denotes a differentiation with respect to the argument of the spherical Bessel function. The outgoing (radiating) vector spherical waves u τ ml (kr) are obtained by replacing the regular spherical Bessel functions j l (x) above for the spherical Hankel functions of the first kind, h l (x), see 17,22,23 .