Mode spectra of magnetic and dielectric plasmon spheres obtained from Mie scattering and free oscillation theories

This paper presents Mie scattering theory as compared to rigorous electromagnetic theory of free oscillations in magnetic and electric plasmon spheres. It is shown that the maxima of Mie scattering and absorption spectra well correspond to resonance frequencies of plasmon modes occurring in dielectric and magnetic spheres, similarly as it takes place for ordinary dielectric resonator modes. Mie theory is well applicable to determine resonance frequencies and scattering parameters of spherical plasmons. However, this theory cannot be applied to determine intrinsic properties of modes induced in the object by the incident plane wave, like quality factors. On the contrary, rigorous electromagnetic theory of free oscillations allows one to determine the complex resonance frequency of each mode that can occur in a given object, and the corresponding quality factor accounting for various kind of losses, including medium and radiation losses. The advantage of the free oscillations theory, as shown in this paper, is in the determination of the quality factors of modes occurring in magnetic plasmon spheres made of a strongly dispersive magnetic medium.


Introduction
Scattering of electromagnetic (EM) waves on dielectric objects is a well-recognized physical phenomenon. A rigorous theory of scattering on spherical-shaped objects was first described in [1] and subsequently addressed in hundreds of papers and several books [2][3][4][5][6]. The relationship between Mie scattering ripples and free oscillation resonances has been already rigorously explained by several authors, e.g., [7][8][9][10][11][12][13]. The scattering problem has an analytical solution in the form of an infinite series for ellipsoidal samples made of isotropic and uniform media. Depending on the diameter of the spherical sample, d, relative to the freespace wavelength of the incident plane wave, λ, three kinds of scattering are distinguished: Rayleigh scattering (d << λ), Mie scattering (d ∼ λ), and geometric scattering (d >> λ). Mie scattering theory (Mie) is sometimes referred to as resonance scattering, because both scattered and absorbed waves create discrete spectra due to modes excited in the sphere. In fact, spherical samples are EM resonators whose free oscillations (FO) are characterized by an infinite series of complex resonance frequencies. The boundary value problem for Maxwell's curl equations has analytical solutions for isotropic spheres, which allows determining the resonance frequencies and Q-factors (i.e., losses) of the modes occurring in the object.
Although those two methods deal with the sphere in a completely different manner, as Mie is a deterministic problem and the boundary value problem solves for FO, very good correlations between resonance frequencies of a dielectric sphere and singularities in Mie absorption spectra occur. The main goal of this paper is to compare resonance frequencies obtained with the two methods for plasmonic modes. Newly recognized magnetic plasmon resonances (MPRs) occurring in ferrimagnetic spheres will be especially investigated in this paper. According to the authors' knowledge, such a comparative study of the two theories for magnetic plasmon modes has never been undertaken before. For historical reasons, comparative computations are also performed for ordinary resonances occurring in dielectric spheres with focus on their quality factors due to radiation losses.

Ordinary dielectric spherical resonators
All computations of scattering efficiency presented in this paper have been performed employing functions developed in [3,4]. Analytical solutions of the eigenvalue problem for Maxwell's curl equations in spherical dielectric resonators have been presented in [14] for homogeneous dielectrics, in [15] for multilayered dielectrics, and in [16][17][18][19] for magneto-dielectric materials (e.g., Ordinary EM resonances appear when both permittivity and permeability are positive, whereas plasmon type resonances appear when one of the two quantities is negative. Recently, it has been shown that resonances observed and employed in YIG (yttrium iron garnet) filters and oscillators are, in fact, MPRs for circularly polarized (CP) modes [16]. The effective permeability tensor corresponding to that mode can be considered isotropic and dispersive in a plane transverse to the static magnetic field bias. Formulae for the effective permeability corresponding to the clockwise CP MPR are given in Appendix A. More theoretical details about MPRs can be found in [16][17][18][19].
Results of computations of normalized resonance frequencies and Q-factors due to radiation for TE n01 modes occurring in spherical dielectric samples located in vacuum as a function of its relative permittivity are shown in Fig. 1. As it is seen in Fig. 1b, the Q-factors due to radiation increase by several orders of magnitude with the increasing elevation mode index, n, as the mode becomes more and more confined within the sphere, which in turn suppress radiation losses. The tendency is enhanced in dielectric spheres with high permittivity. For low-permittivity samples, radiation losses are relatively large and the corresponding Q-factors due to radiation are small. Commercially available cylindrical dielectric resonators with permittivity in the range 20-40 operate on a low-order mode and have to be situated in a metal shield to avoid radiation losses. Subsequently, Mie extinction coefficients, Q ext , which account for both scattering and absorption, were computed using functions available in [4] for ε r = 4 and ε r = 10 and the results are shown in Fig. 2 and in Table 1. Maxima of Q ext correspond well to the eigen-frequencies of dielectric resonator modes computed with the relevant TDE. Agreement between spectral locations of Q ext maxima and computed eigenfrequencies are to within 4-5 significant digits (see Table 1), except for the first two modes which are highly radiative, as it is seen in Fig. 1b, so their resonance frequencies are slightly deviated due to losses.

Magnetic and electric plasmon resonators
In a similar way as for ordinary resonances in nondispersive dielectric spheres, solutions of the TDE [i.e., d/λ values being the solutions of (B.1) or (B.2)-Appendix B] can be compared with μ r or ε r corresponding to the maxima of Q ext obtained in plasmonic regime, i.e., for μ r < 0 or for ε r < 0. According to the magneto-static and the electro-static models, plasmonic resonances for a small spherical magnetic (dielectric) sample situated in vacuum appear if the relative permeability (permittivity) of the sample is equal to .. Electrodynamic theory of plasmon resonances in spheres agrees with the electro-and magneto-static theories only in the Rayleigh scattering regime when d << λ. Electrodynamic theory of electric plasmon resonances (EPRs) is well established (e.g., for noble metal spheres at optical frequencies), while theory of MPRs in gyromagnetic spheres has only been recently developed [16][17][18][19]. It has been shown that MPRs correspond to the TE n01 modes of an isotropic spherical sample having an effective permeability as in (A.3) (see Appendix A). It should be underlined that the effective permeability of such a material is highly dispersive (i.e., frequencydependent).
For a given radius of the sample, its permittivity, internal static magnetic field, saturation magnetiza-  tion, and Gilbert damping factor (or ferromagnetic linewidth, ΔH ), it is possible to obtain solutions of the TDE, as in (B.1), being the resonance frequencies of various TE n0p modes. In addition, the frequencydependent effective permeability values at these frequencies are obtained. The lowest frequency TE 101 mode corresponds to the so-called mode of uniform precession (following simplified magneto-static theory of resonances [20]) for which μ r = −2 (for the lossless case). This mode is commonly employed in YIG filters and oscillators. One can easily derive from (A.4) that for this mode w−H 0r = 1 1−μr = 1/3, which shows that the normalized resonance frequency, w, linearly depends on the normalized static magnetic field, H 0r , as long as d << λ (see definitions in the Appendix A). Numerical solutions of TDE, as in (B.1), for larger d/λ are shown in Fig. 3. It can be noted that as d/λ increases, w−H 0r values converge to 0 and the effective μ r converges to −∞ (which is the ferromagnetic resonance condition in an unbounded medium).
It should be noted that the approximate Mie scattering results [21,22] well agree with the TDE solutions for small d/λ. These days more accurate Mie scattering computer programs are available (more terms in the EM field expansion series), so the agreement between FO and Mie results takes place in a broad range of d/λ values. Comparisons of Mie results based on Mie scattering software and TDE solutions are not straightforward. For a given H 0r column vector, the TDE pro-  fixed values of ε r and μ r . Mie computations of the plasmonic resonance frequencies require two steps. In the first step, Q ext is computed for an assumed d/λ vector and fixed values of ε r and μ r (μ r is taken from the TDE computations). In the second step, the normalized resonance frequencies d/λ are obtained as values corresponding to the Q ext maxima. Figure 4a shows computation results of Q ext versus d/λ for ε r = 16 and μ r = − 2.05e. Normalized resonance frequencies, d/λ, presented in Fig. 4b were obtained as the d/λ values corresponding to the maxima of several Q ext plots (for different μ r ) and are listed in Table 2 for the TE 101 mode. As it is seen in Table 2, agreement between Mie and FO (TDE) results is excellent to a relative accuracy better than 0.1 %. In Fig. 5a, computation results of Q ext versus d/λ for an electric plasmon sample having ε r = −2.05 and μ r = 1 are presented, whereas Fig. 5b shows the dependence of the normalized resonance frequencies (d/λ) versus effective permeability. Similarly, as for the MPRs, agreement between Mie and FO results for the EPRs is excellent. However, the spectrum of Q ext (see Fig. 5a) is different than for the MPRs (compare Fig. 4a). Moreover, normalized resonance frequencies d/λ (see Fig. 5b) are different for ε r < 0 as compared to the corresponding negative values of μ r (see Fig. 4b). These differences are related to the fact that the magnetic plasmon sample exhibits large permittivity (ε r = 16 for YIG), whereas the electric plasmon sample is assumed to be non-magnetic (μ r = 1). Agreement between Mie and FO computations for the lowest order TE 101 plasmon mode is much better than for the ordinary mode (compare results in Tables 1 and 2). This is related to much smaller radiation losses for plasmon modes due to the very small size of plasmonic samples. More details on this topic will be presented in the next section.

Experiments and remarks on Q-factor computations
Computations of Q-factors in resonators containing a dispersive medium with the effective permeability as in (A.2) can performed in two alternative ways. In the first approach, the complex frequency is used to extract the Q-factor, as it has been described in Sect. 2.1. In the second approach, the frequency is real in all formulae. It should be also emphasized that due to dispersive properties of μ r , magnetic energy stored in the resonator is proportional to ∂ ω μ r (ω) ∂ω and not to μ r . The final expressions for the Q-factor related to losses in a magneto-dielectric medium with dispersive permeability and non-dispersive permittivity read The denominator in (1) denotes the average total power dissipated in the resonator, P loss , divided by ω, whereas the numerator can be interpreted as the average total EM energy stored in the resonator. For the MPRs, the magnetic energy stored in the resonator per cycle can be even a few orders of magnitude larger than the average electric energy, especially for small samples. Agreement between these two approaches of Q-factor computations is excellent [19]. More information on this topic can be found in [17,19]. Figure 6a shows measurement results of the transmission spectrum of a monocrystalline YIG sphere (M S = 140 kA/m, d = 0.5 mm, ε r = 16) under a static externally applied magnetic field bias (H ext ≈ 217 kA/m). Details of the experimental setup have been presented in an earlier paper [23]. It is seen that highest peaks on the transmission spectrum correspond to the first three plasmon modes. From the measured resonance frequencies corresponding to these peaks, the saturation magnetization of the sample has been determined with a high precision [23].
Results of measurements, after [18], and FO computations of Q-factors for the YIG sphere are shown in Fig. 6b. The resonance frequency of the TE 101 mode has been tuned by over a decade by varying the DC magnetic field bias. Computations of the total Qfactors have been performed assuming the ferromagnetic linewidth of the sample ΔH = 0.5 Oe (solid lines) and computations of the Q-factors due to radiation losses (broken lines) assuming ΔH = 0. Additionally, Q-factors due to magnetic losses, Q m , are presented for the sample having d = 0.5 mm. It should be noted that Q-factors due to radiation losses for the plasmonic TE 101 mode in a YIG sphere are few orders of magnitude larger than for the ordinary TE 101 mode in a dielectric sample having the same permittivity as YIG (Fig. 1b). This is related to the size of the sample [18] which, for a given frequency, is several times smaller for the magnetic plasmon sample than for the dielectric one. This is confirmed by Mie computations where the peak of Q ext in Fig. 4a for the first plasmon mode is much sharper than the Q ext peak for the same mode in the dielectric sample (see Fig. 2b).

Conclusions
Mie and FO theories agree in the determination of the resonance frequencies for both plasmon and ordinary modes occurring in isotropic spheres. Agreement is up to 4-5 significant digits, except for highly radiative ordinary modes. Computations of the resonance frequencies with the aid of the Mie theory are typically faster than those based on the FO theory, and provide immediate results for scattering and absorption spectra of magneto/dielectric objects. However, FO theory allows one not only to accurately determine the resonance frequencies but also the Q-factors, especially for resonators containing dispersive media.