Extracting Metamaterial Properties of Negative-Index and Plasmonic scatterers from the Mie Coefficients

An alternative theoretical approach is presented for improving the understanding of the scattering properties of simple negative-index and plasmonic spheres. The Mie scattering coefficients are broken up into two terms, the first representing the expected fields of a corresponding positive-index sphere, the second emerging from the intrinsic metamaterial nature of the scatterer. Some limiting cases are considered, e.g. dipole approximation), together with their circuital representation in terms of equivalent lumped elements. This formalism may be adopted, for example, in the evaluation of the particular internal and scattered electromagnetic field patterns observed for these particular classes of particles.

index (PRI) and a NRI medium, one observes (in comparison with a PRI-PRI interface) the inversion of Snell's law [1].Therefore, if  r is the transmitted angle, there is a 2 r offset between the transmitted rays on PRI-NRI and PRI-PRI interfaces possessing the same (absolute) electromagnetic parameters, and we can naturally ask ourselves what are the additional properties or constraints of a PRI-NRI interface responsible for this offset given that the refracted ray along a PRI-PRI is known.
The same may be asked for scatterers in the Mie or Rayleigh optical regimes in terms of EM fields.
Since the electric (E) and magnetic (H) fields are the fundamental quantities associated with light scattering phenomena, one can answer the above questions by exploiting the additional fields (E add ,H add ) that emerge from the replacement PRI-PRI → PRI-NRI, i.e., when the relative refractive index M of the particle changes sign.This problem, in turn, should be equivalent to the expansion of MCs in such a way that these additional fields come to light in a natural fashion.Specifically, because of the linear relation between (E,H) and MCs, we should be able to rewrite such coefficients (without introducing any new poles or singularities) as a sum of two terms: the first being the expected one for a PRI particle, the second the responsible for the emergence of E add and H add .This work is therefore devoted to the task of providing an alternative theoretical formalism for the understanding of the scattering properties of NRI spheres in terms of such additional contributions which arises exclusively from the metamaterial nature of the scatterers.It may be of interest in the theoretical and numerical analysis of optical forces and torques, EM fields, scattering amplitudes and cross-sections and any other optical parameter or property of interest.It may be further exploited in future works by finding equivalent Debye series, thus helping elucidating some physical aspects of such additional MCs.

II. THEORETICAL BACKGROUND
In the framework of the Lorenz-Mie theory (LMT), there is a linear relationship between EM fields (E,H) and MCs [12].This simply means that splitting MCs into two additive terms is equivalent to doing the same with (E,H) and, therefore, we shall focus our analysis only on MCs.According to one of our previous works, they can be compactly written, for a PRI spherical scatterer, as [15]  where n is a positive integer,  = ka,  = k sp a, k and k sp being, respectively, the wave numbers of the host medium and the particle of radius a. Besides, being the ratio between the intrinsic impedances of the external medium () and of the particle ( sp ).The Ricatti-Bessel functions (RBFs) and their derivatives (with respect to their arguments) are written using current terminology [10].In order to proceed further, the following relations for the RBFs, valid for real x, are introduced [16]: The wave number of a lossless NRI particle is negative, so and the set of relations (3) may be substituted in (1) (had we assumed a PRI particle, then  = +||;  is taken as positive real number): Just as for a single ray impinging on the surface of a NRI particle, we can ask ourselves what is the "pure" NRI contribution to the EM fields arising from the changes of sign in (3) which modify (1) in accordance with (4).Because EM fields are directly proportional to MCs, the answer to this question comes from rewriting (4) as follows: For instance, consider the scattered electric field components in spherical coordinates (r,,) for TM modes, where E 0 is the field strength, m TM n g , are the beam-shape coefficients, pw n c is a constant for a given n and are associated and generalized Legendre functions.Similar expressions hold for magnetic fields, and TE modes contain the transverse electric MCs b n [12].
From ( 5)-( 8) one clearly sees that the field E s and H s scattered by an NRI sphere are, in fact, a sum of two terms: one that comes from imposing (1) in ( 6)-( 8) assuming a PRI scatterer with both the same geometry and absolute value of electromagnetic parameters, and a second one that emerges as an additional contribution due to the metamaterial nature of the particle.
In this way, let us explicitly evaluate add n a and add n b in (5).By subtracting (4) and ( 5) one finds, after some algebra, and the additional EM fields are found by replacing (a n ,b n ) by ( add n a , add n b ) in ( 6)-( 8) and all other equivalent expressions for TM modes and H field. Notice that ( 5) and ( 9) do not explicitly involve negative refractive indices, even though we implicitly made use of such a condition in deriving (4).
Due to a typo in a previous work (which nonetheless did not change the results and conclusions), the numerator in ( 9) is slightly different from the one found in equation ( 8) of [15].
The same procedure, once applied to internal fields, leads to the following expressions for the internal additional MCs: where n sp is the refractive index of the scatterer.
These are quite interesting results by themselves.In fact, one can forget about negative refraction altogether and assert that a lossless NRI sphere is simply the one which replaces an equivalent PRI particle and modifies the EM fields according to ( 9) and (10).In other words, every time we are able to produce the EM fields imposed by ( 9) and ( 10) we should, in fact, create a theoretical NRI metamaterial.This is an alternative way of thinking about a NRI material in terms of scattering or internal fields.
Another possible interpretation of ( 9) can be appreciated as follows.Consider a given PRI sphere.
If, by any means, we are capable of generating the electric and magnetic scattering fields associated with (9), such fields will superpose those scattered by the corresponding PRI particle and a far-field detector should conclude that they come from some sort of lossless NRI material with spherical symmetry.
For very small particles and far from resonances, the dipole approximation asserts that only the MCs a 1 and b 1 significantly contribute to E s and H s .Accordingly, c 1 and d 1 are the only nonnegligible internal MCs.Therefore, ( 9) and (10) (11) and we may now assume a 1 add and b 1 add in (11) to be identical to the dipole terms expected for some particular PRI scatterer:  being the relative permittivity (for a 1 ) or permeability (b 1 ).Equating (11) and (12)  (the same being valid for | rel |).If any of such conditions are matched then (13) ensures that the scattering field of a NRI nano-sphere can be theoretically represented by the sum of the scattering fields produced by its corresponding PRI equivalent plus those produced by a PRI dipole with dipole terms satisfying (12).
Notice that references to a negative index have been suppressed and are no longer necessary.
Incidentally, a NRI particle can then be viewed as a structure which simultaneously contains, in some limited region of space, two homogenous PRI structures with distinct EM parameters and without field interferences.They produce identical far-field patterns when compared to the original NRI scatterer.By the same token, repetition of the above procedure for internal fields gives, instead of ( , for 2

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, for 2 and so the pure metamaterial contribution to the internal fields may be obtained by placing, at the origin of the coordinate system, an hypothetical additional PRI sphere with electromagnetic parameters constrained between 0 <  rel < 0.828 and 0 <  rel < 0.828.The above derivations may be extended to include lossy (active) NRI scatterers with the additional EM fields generated by such metamaterial particles theoretically originating from active (lossy) PRI structures.This can be verified by constructing the Ricatti-Bessel functions from the following identities [17]: and similar for modified Bessel and Hankel functions.
In (15), a > 0, b > 0 and z = a + ib, i.e., z is associated with an active PRI structure.Finally, note that lossless plasmonic scatterers with arbitrary radius have analytical expressions involving Bessel functions of purely imaginary arguments.This poses additional mathematical difficulties [one can easily check, however, that for plasmonic dipoles ( 11)-( 13) remain valid for the electric coefficient . Nonetheless, because real plasmonic materials always present absorption, this should not be a problem in modeling and studying practical situations.
In the particular case of dipole approximation, some situations may require equivalent descriptions in terms of RLC ladder networks.The MCs are then represented as combinations of lumped circuit elements.This has been done, for instance, for both PRI and NRI spheres [14,18].Focusing on the additional MCs, it is easy to see that RLC networks associated with (11) are exactly those found for the PRI case [18], with the proviso that both permeability and permittivity satisfies ( 12) and ( 13).
Our formalism may help in deriving equivalent electrical circuits with lumped nanocapacitors, nanoinductors and nanoresistors for lossless plasmonic nanospheres [6,19].Indeed, from Mie theory and imposing  rel = 1 in (11), the equivalent internal displacement current density can be rewritten as and similarly for the average potential difference so that, from ( 16) and ( 17), one finds the following nano-impedance: as expected so as to generate the inductive behavior observed for plasmonic nanoscatterers.
According to (18), sp Z for a dipolar plasmonic particle can be represented not only by   instead, by a PRI nanocapacitor PRI sp Z plus an additional nanocapacitor add sp Z that comes into play because of the excitation of surface plasmons [6].

III. NUMERICAL EXAMPLES
The preceding section dealt with the theoretical background for the analysis of the scattering properties of metamaterial spheres giving emphasis to the additional fields which superpose those of PRI scatterers.As mentioned, MCs can be split into a sum of two terms, a very interesting trick considering the fact that some of the fundamental physical quantities are directly proportional to them.
Even quantities like force, torque, scattering cross sections, etc, after some straightforward manipulation, may be analogously decomposed in such a form.
For a +z-propagating (x-polarized) plane wave, Figs. 1 and 2 reveal the TM scattered E x -field as composed by its additional and PRI contributions in  = /2 plane (x-y plane) for a radial distance r = 2a and as a function of  rel and azimuth angle .The TE field may be directly evaluated using expressions analogous to (6)-( 8) [12].In both plots, | rel | > 6.606 in order to include only additional fields generated by "physical" PRI structures, as mentioned in section II.
    In view of ( 13), ( 19)-(22) provide very distinct numerical outcomes as compared to those found in [18], even if their expressions look quite similar (here, in fact, the original  rel is simply replaced by ).In fact, some of the capacitors in the original PRI RLC circuit may have to be replaced by inductors in the RLC circuit representation for add a 1 depending on the plus or minus signs carried by each of such passive circuit element due to the presence of additional reactances, with a direct consequence on the values of the NRI reactances as well.This is illustrated in Figs. 3 and  For particles whose dimensions are comparable or larger than the wavelength, (4), ( 5) and ( 9) can be used to evaluate NRI, PRI and additional Mie coefficients individually.Therefore, it is possible to study the individual properties of their corresponding scattered fields.For example, assuming  = 1064 nm, MC a 1 NRI for a particle with  = - 0 presents an oscillatory behavior as a function of both a and  rel , as revealed in Fig. 6.If it is broken up in its two terms as suggested by ( 5), then its PRI and additional dipole coefficients are as shown in Figs.7 and 8, respectively.It becomes clear the new rearrangement of the ripples induced by virtue of a 1 add .Even if a direct physical interpretation cannot be readily extracted from such plots, expressing each MC as a Debye series could furnish interesting conclusions and insights for the equivalent series of reflection/refractions of a wave field at the surface of a NRI particle [15].

IV. CONCLUSION
An analytical method is presented for expressing scattering properties of NRI particles as superpositions of two distinct terms: the first associated with a PRI particle carrying exactly the same absolute values of the electromagnetic parameters, the second reflecting the pure metamaterial properties of the original scatterer.Under certain conditions, the latter can also be represented as a function of (scattering) MCs representing dielectric scatterers of positive permittivity.
Within some specific range of values of permittivity and assuming Rayleigh scatterers far from any resonances, this formalism asserts that the expected NRI scattering properties may be equivalently interpreted as nothing but the result of field superpositions between two spatially coincident (or very close together) PRI particles.Additional EM fields are then the physical quantities which ultimately

Fig. 1 .
Fig. 1.Additional (pure metamaterial part of) TM E x -field in the  = /2 plane and at r = 2a for a NRI sphere with ka = 0.01,  rel = - 0 under plane wave illumination.The wave propagates along +z and is x-polarized.

Fig. 2 .
Fig. 2. PRI E x -field counterpart for the NRI sphere of Fig. 1.RLC ladder network circuits for the dipole coefficients also change radically for specific radius and particular relative permittivities.From (11)-(13), assuming an admittance model, the first four reactances for 1 a are 0.01.The corresponding electric dipole coefficients are shown in Fig. 5.
reduce to the much simpler expressions