МІКРОХВИЛЬОВА ЕЛЕКТРОДИНАМІКА On some behavioral peculiarities of magnetic-type eigenmodes of a spherical particle with arbitrarily valued material parameters

Subject and Purpose. The spectral characteristics (eigenfrequencies, eigenmodes, Q-factors) of a spherical particle with arbitrarily valued permittivity and permeability are considered to take a further look into some important features of their behavior. The real and imaginary parts of the material parameters of the particle can be both positive and negative. The emphasis is on magnetic-type modes. Methods and Methodology. The spectral problem is solved using the electromagnetic ﬁ eld expansion in vector spherical wave functions. Results. The ﬁ rst eigenfrequencies of a spherical particle have been calculated depending on its relative permittivity  1 and relative permeability  1 whose real and imaginary parts can take both positive and negative values. The eigenmodes split into two, internal and external, eigenmode families. The internal eigenmodes bear an independent, associated with eigenmode structure labeling in each quadrant of the plane (  1 ,  1 ). The external eigenmodes, on the contrary, have a uniform labeling throughout the whole (  1 ,  1 ) plane and bear a structural resemblance to surface plasmon oscillations distributed in the vicinity of the particle surface or outside it. In the ﬁ rst quadrant of the plane (  1 ,  1 ), the external eigenmodes repeatedly interact with the internal eigenmodes, leading to either mode hybridization or mode type exchange. In the third quadrant of the plane (  1 ,  1 ), the external eigenmodes can interact with one another. The anomalous behavior of the spectral characteristics of a spherical particle corresponds to the already known phenomenon of wave mode coupling described in the scienti ﬁ c literature well enough. Conclusion. The performed study has revealed some new behavioral patterns as to the spectral characteristics of a spherical particle with arbitrarily valued permittivity and permeability. Fig. 5. Ref.: 13 items.

The study of spherical particles with arbitrarily valued material parameters is facilitated by the fact that any formula obtained for an ordinary matter can be applied to the case when permittivity and permeability can take arbitrary values. The eigenfrequency equations of spherical dielectric resonators were fi rst obtained by Mie andDebye in 1908-1909 [1,2]. Over a wide permittivity range, numerical calculations of complex-valued eigenfrequencies were fi rst performed in [3]. An excellent eigenmode analysis of an isotropic spherical dielectric resonator was carried out in [4]. The electrodynamic properties of a spherical particle with arbitrary real-valued material parameters were fi rst considered in [5]. That time the particle resonant properties were studied by analysis of refl ection coeffi cient resonances at a real-valued wave number. In [4,5], it was noted, in particular, that in the particle of the kind, there are modes demonstrating abnormal (i.e. qualitatively diff erent from normal or ordinary) behavior of the spectral curves. The described in [4,5] abnormal behavior of electric-type mode spectral curves was thoroughly studied in [6] for a dielectric ball and in [7] for a spherical particle with simultaneously negative permittivity and permeability. The spectral curves reported in [4][5][6][7] belong to the spectral curve class of electrodynamic structures under the Yu.V. Svishchov wave mode coupling conditions [8]. This circumstance is a driving factor to take a further look into the resonant properties of a spherical particle by analysis of its spectral characteristics in the complex frequency range.
The present work is concerned with behavioral patterns of spectral characteristics of a spherical particle having arbitrarily valued permittivity and permeability. The emphasis is on magnetic-type modes.
No restrictions are imposed on the permittivity and permeability, enabling simultaneously diff erent signs of their real and imaginary parts. Correspondingly, a question arises as to the refractive index sign choice for a medium with complexvalued material parameters. A complete analysis of refractive index signs for loss and gain media can be found in [9] (see, also, [10]). In the present problem solution, a simple formula from [11] is adopted. The fi nal results coincide with the data obtained by other algorithms of the refractive index calculation [10] (in view of the active media comments in [10]).
Mathematical problems arise in the refractive index sign determination and refer to the question of complex square root calculation because the complex plane can be cut into two Riemannian surface sheets in diff erent ways, each suitable for a certain medium type. In the general case, however, it is hard to choose one. The mentioned ambiguity prevents us from linking a proper value of the refractive index to one of the two branches of the complex square root function. Therefore the refractive index sign determination should be performed based on some additional physical considerations. Normally, refractive index sign is sought in conjunction with complex impedance sign. A condition is proposed [11] that the sign of the refractive index imaginary part and the sign of the loss or gain energy density be coincident.
1. Calculation method. Let us consider a spherical particle of radius a (Fig. 1), relative permittivity 1 1 1 i ε ε ε = + ′ ′′ and relative permeability ′′ The relative permittivity and the relative permeability of the surrounding space are, respectively,  0  0 and  0  0. The wave number inside and outside the particle is , s s s k k ε μ = s  0,1, where k   / c is the wave number in vacuum. A spherical coordinate system (r,  ,  ) with the origin at the center of the particle is introduced, where r  is the radius-vector of the observation point. The electromagnetic fi eld inside and outside the particle (in the incident fi eld absence) is sought as    (3) accurate to the normalization factors adopted in the work are defi ned in [6,7]. From here on, the time dependence of the fi elds is assumed to be exp(-i  t).
With the boundary condition that the tangential components of the E  and H  fi elds are continuous on the particle surface (r  a), we obtain the dispersion equation of the magnetic-type eigenmodes (the TE or H modes)  ε μ by the formula [11] (see above) where the square roots are considered in the principal value sense when the cut is along the negative half-axis of the real axis ( 0, z ≥ z is a non-negative real number). This principal value selection is used to advantage in the most important mathematical Application Packages.

Brief eigenmode analysis.
A distinctive feature of magnetic-type modes is a nonzero radial component H r of the magnetic fi eld (E r  0). Accordingly, magnetic type modes are described (see (1)  Hence, while on the subject of the TE 0 nq mode structure, the spatial distribution of the E  component is meant.
The eigenmode spectrum of a spherical particle splits into internal (the fi eld inside the particle) and external (the fi eld in the vicinity of the particle surface and outside it) modes (Fig. 2). It consists of two families determined by diff erent dependences on the particle permittivity. So, the roots of the characteristic equations of the internal and external modes are suitable to label diff erently. To avoid confusion, we leave the labeling TE mnq to the inter- Yu.V. Svishchov ber of | | E  antinodes along the coordinate  in the spherical coordinate system, and index q  denotes the root number of equation (4). For the internal eigenmodes TE 0 nq , index q denotes the number of | | E  antinodes along the coordinate r in the spherical coordinate system. In the present work (unlike [3,4]), the mode labeling is defi ned by the electromagnetic fi eld structure rather than the eigenfrequency real part increase. The latter is acceptable when the dependences of the eigenfrequency real parts do not meet. The grounds for the classifi cation by electromagnetic fi eld structure are given by the eigenmode wave coupling described in [6,7].
3. Behavioral peculiarities of spectral characteristics. Let us consider the eigenfrequency behavior in the fi rst quadrant of the plane (d ), and 4.0 (e). The symbols like 3 + 1  or 3  mark hybrid-type eigenmodes [6]. The plots are similar to those obtained for electric-type modes in [6]. We claim that the discovered and investigated in [6] phenomenon of wave mode coupling in a dielectric sphere with 1 1 ′′ takes place for magnetic-type modes, too. The sphere relative permeability 1 μ′ is a control parameter of this phenomenon (with 1 ε ′ fi xed). The external mode TE 011  comes into interaction with the internal modes.
The Re( ) ka and lg( ) Q of the modes TE 011  , TE 011 , and TE 012 are plotted in Fig. 4 as functions of two, 1 μ′ and 1 , ε ′ variables. As seen, in the vicinities of some points of the plane 1 1 ( , ), μ ε ′ ′ two eigenfrequencies locally form a single double-sheet surface. The cuts of this surface by the 1 const ε = ′ plane show that the real and imaginary parts of the eigenmodes behave diff erently, as described in [6]. In particular, both mode hybridization and mode type exchange are locally seen. Fig. 4 shows multiple mode transformations in the plane It is safe to assume that via a proper 1 μ′′ and 1 ε ′′ selection, the eigenmode degeneration in the complex frequency domain can be achieved [4]. In Fig. 3, one can also observe lg(Q) resonance minima. As  nal modes and attach the labeling TE mnq  with index q primed to the external modes. For the axially symmetric modes TE 0 nq  , index n denotes the num-On some behavioral peculiarities of magnetic-type eigenmodes of a spherical particle... zero (lg( ) Q −  and the fi eld is ejected out of the resonator as vacuum is approached (the oscillation amplitude therewith tends to zero). As 1 ε ′ increases, this limiting case is drawn away. The fi eld is partially ejected from the dielectric sphere, causing lg(Q) minima. As 1 ε ′ increases, these minima gradually fade. The oscillations in the Q minimum vicinity resemble the TE 011  and the TE 01q mode structure (depending on the extent to which the antinodes of the original eigenmodes TE 011  and TE 01q are ejected out of the resonator). In this connection, they are called hybrid modes. When moving away from the lg(Q) minimum towards smaller 1 μ′ values, hybrid oscillations with the TE 01q and TE 011  mode features appear, too. These oscillations have antinodes inside the ball and an antinode with a signifi cantly lower amplitude in the ball surface vicinity. These two hybridity types fundamentally diff er from the oscillation hybridity under the wave mode coupling condition when hybrid oscillations have features of two oscillations entering wave mode coupling.
Equation (4) fi nds the eigenfrequencies ka whose real parts give electromagnetic fi eld eigenfrequency values and whose imaginary parts describe attenuation of the modes. Calculations of the eigenfrequencies in the third quadrant of the plane increases in absolute value, then, starting with a certain point, the ka imaginary part gets negative, suggesting that the solution becomes physically correct. It should also be noted that the existence of physically incorrect solutions of (4) leads to resonances of the Mie coeffi cients [5], which enables the author of [5] to draw a conclusion about internal mode existence in the third quadrant of the plane 1 1 ( , ).  and, fi nally, returns to the fi rst quadrant (but now with a zero-valued real part). That is, this eigenmode frequency behaves as if it belonged to some helical surface with its axis at the point (  0 ,  0 ). In this case, during the passage from the fourth to the third quadrant (over some 1 μ′ variation interval), as well as during the passage from the second to the fi rst quadrant, the eigenfrequency real part vanishes. The eigenfrequency behavior in the fi rst quadrant (at 1 0 ) ε ε ≥ ′ was described above (Figs. 3 and 4). During the passage from the fi rst quadrant of the plane the Im( ) ka is at its minimum. With a further 1 μ′ decrease, the fi eld is restored and has a form of a "classical" plasmon eigenmode whose Q-factor increases during the passage to the second quadrant.
We have considered some behavioral peculiarities of the n  1 eigenmodes, the simplest to analyze. As n increases, the eigenfrequency and eigenmode behavior can get more complexity (from the visualization standpoint), especially under wave mode coupling conditions. Thus, by analogy with electric-type modes [7], it has been found that the TE 031  and TE 032  eigenmodes come into interaction in the third quadrant of the plane 1 1 ( , ).
μ ε ′ ′ As a consequence, when the particle material parameters vary in the critical point vicinity [7], either mode hybridization or mode type exchange is locally observed. Unlike the TE 011  oscillation, the real part of the plasmon oscillation frequency in the fourth to the third quadrant passage is other than zero, the imaginary part of the frequency therewith has a minimum, as for the TE 011  oscillation. In the (  0 ,  0 ) point vicinity, the TE 031  oscillation frequency changes into the TE 032  oscillation frequency as if it followed a helical path with the axis at the point (  0 ,  0 ).
Conclusions. The spectral problem solution for a spherical particle has been obtained, based on the electromagnetic fi eld expansion in vector spherical wave functions.
The fi rst eigenfrequencies (magnetic-type eigenmodes) of a spherical particle have been calculated depending on its relative permittivity  1 and relative permeability 1 μ′ whose real and imaginary parts can take both positive and negative values. The eigenmodes split into two, internal and external, eigenmode families. The internal eigenmodes in each quadrant of the plane (  1 ,  1 ) have a local, independent labeling based on the oscillation structure. Unlike the internal eigenmodes, the external eigenmodes bear a single labeling throughout the plane (  1 ,  1 ).
The external eigenmodes structurally resemble surface plasmon oscillations. In the fi rst quadrant of the plane (  1 ,  1 ), they repeatedly come into the interaction with the internal eigenmodes, resulting in either eigenmode hybridization or eigenmode type exchange. In the third quadrant of the plane (  1 ,  1 ), the external eigenmodes can interact with one another. The anomalous behavior of the spectral characteristics of a spherical particle corresponds to the wave mode coupling phenomenon well described in the scientifi c literature.