Magnetoquasistatic Resonances of Small Dielectric Objects

A small dielectric object with positive permittivity may resonate when the free-space wavelength is large in comparison with the object dimensions if the permittivity is sufficiently high. We show that these resonances are all magnetoquasistatic in nature being associated to values of permittivities and frequencies for which source-free quasistatic magnetic fields exist. They are connected to the eigenvalues of the magnetostatic operator expressing the magnetic vector potential in the Coulomb gauge as a function of the current density. These eigenvalues are independent of the size, frequency, and material permittivity. We present the general physical properties of magnetostatic resonances in dielectrics. Our findings improve the understanding of resonances in high-index dielectric objects, and provide a powerful tool that greatly simplifies the analysis and design of high index resonators.

It is well established that small metal (plasmonic) objects with negative permittivity may resonate when the free-space wavelength is large in comparison with their dimensions [1,2]. These resonances have an electrostatic nature, being associated to the values of permittivity for which source-free electrostatic fields exist [1,2].
Small dielectric objects with positive permittivity may also resonate when the free-space wavelength is large in comparison with their dimensions, providing that their permittivity is sufficiently high [3,4]. Specifically, for nanoscale high-permittivity objects, such as AlGaAs, Si, Ge nanoparticles, these resonances occur in the visible and NIR spectral ranges, and have been observed experimentally, e.g. [5][6][7]. High-index resonators concentrate both electric and magnetic fields, enahancing nonlinear optical processes [8] and weak magnetic transitions of molecules [9][10][11][12][13]. They constitute a promising platform for biosensing [14], and may support either directional scattering cancellation or increased scattering directionality [15]. General physical properties of resonances in high permittivity dielectric particles have not been studied yet. Moreover, the existing techniques for the direct calculation of them (e.g. quasi-normal modes [16], material-independent-modes [17], etc.) are very compli-cated being based on the full-wave Maxwell equations, and are not able to distill the essential physical nature of these resonances.
These resonances are currently known as "Mie resonances" (see for instance [7]) and, as the name suggests, it is currently believed that they can only be described in the framework of the Mie theory, or other full-Maxwell formulations. In this letter, we show that the resonances in the electromagnetic scattering from high permittivity dielectric objects have a magnetoquasistatic origin, being associated to the existence of source-free quasistatic magnetic fields. We introduce a tecnique for the direct calculation of the resonant frequency and of the corresponding modes by computing the spectrum of a magnetostatic integral operator. Irrespectively of the shape of the object, the resonance frequencies are inversely proportional to its characteristic size, and inversely proportional to its refractive index. The magnetoquasistatic modes are orthogonal in the usual sense. We validate the proposed technique by considering a sphere, then we predict the occurrence of such resonances for a finite-size cylinder for which no analytic solution exists.
Let us consider a homogeneous dielectric object with a bounded arbitrary shape Ω and relative dielectric con-arXiv:1907.02950v1 [physics.optics] 5 Jul 2019 stant ε R . We make the following ansatz: in the subwavelenght regime the electromagnetic scattering from high permittivity dielectric objects is primarily determined by the displacement current density field induced inside the object itself. Therefore, we look for the values of the parameter β = (ω/c 0 ) √ ε R for which there exists a non-trivial solution of the source-free magnetoquasistatic problem [19] where Π Ω is the characteristic function on the set Ω Π Ω (r) = 1, for r ∈ Ω 0, for r / ∈ Ω (2) ω is the frequency, and c 0 = 1/ √ ε 0 µ 0 is the light velocity in vacuum; the magnetoquasistatic vector potential A satisfies the Coulomb gauge in Ω and R 3 \Ω; A and the quasistatic magnetic field H are regular at infinity. In other words, we are neglecting the displacement current density field in vacuum. The continuity of the tangential components of A and H imply, respectively, the continuity of the normal components of H and J across the boundary ∂Ω of Ω. Since the normal component of the current density field J at the boundary ∂Ω is equal to zero, the current density field J is solenoidal everywhere in R 3 ; instead, the normal component of the vector potential at ∂Ω may be discontinuous. It is convenient to scale the spatial coordinates by the characteristic size of the object l c , r = l cr . Thus, we denote withΩ the scaled domain Ω. Then, problem 1a-1c is solved by expressing the vector potential A in terms of the current density J as: where we have introduced the magnetostatic integral operator and g 0 (r) = 1/(4πr) is the static Green function. In 4 there is the static Green function because we are neglecting the displacement current density in vacuum. By combining Eqs. 1c and 3, we obtain the linear eigenvalue problem where the parameter y is given by x denotes the size-parameter of the object defined as x = l c (ω/c 0 ). Equation 5 holds in the weak form in the functional space constituted by the functions which are solenoidal withinΩ and having zero normal component on ∂Ω, and equipped with the inner product: w, v Ṽ =Ṽ w * · v dṼ . Thus, source-free magnetoquasistatic fields may exist only for the values of frequency ω, of relative permittivity ε R , and of characteristic lenght l c for which the integral equation 5 has non trivial solutions. In other words, the resonances and the resonant magnetoquasistatic modes are given by the eigenvalues and the eigenfunctions of operator L. The operator L is compact, positive-definite, and self-adjoint. Equation 5 in weak form admits a countable set of eigenvalues {y n } n∈N and corresponding current density modes J n (eigenvectors). The eigenvalues y n are real and positive. The current density modes constitute a complete basis.
We denote with A n the magnetic vector potential produced by J n in the whole space. The eigenmodes J n and J m corresponding to different eigenvalues y n and y m are orthogonal in the usual sense. The corresponding magnetic vector potentials A n and A m are orthogonal in both the domains Ω and R 3 \Ω. The eigenvalue y n is proportional to the magnetic energy stored in the whole space by the corresponding mode: where Once the eigenvalues y n are obtained by solving the eigenvalue problem of Eq. 5, the resonant frequencies ω n are readly calculated for a given permittivity ε R It is also worth to point out that the mathematical structure of the integral equation 4 is invariant with respect to the scaling of l c . This fact combined with Eq. 7 leads to the unique property of the magnetoquasistatic resonances: independently of the shape of the object the resonance frequencies are always inversely proportional to the characteristic size of the object l c , and inversely proportional to its refractive index. In a "material resonance picture" [17,20], once the operating frequency is assigned, the resonant permittivities can be calculated as: Since the operator L is positive-definite, source-free configurations of the magnetoquasistatic field may exist only for positive values of permittivity. Moreover, the resonant permittivities ε R,n are inversely proportional to the square of the size parameter x. It is now apparent the difference between the magnetoquasistatic resonances and the plasmon resonances, which only exist for negative permittivities, and are size-independent [2].
To verify our ansatz we now consider a dielectric sphere with a positive and large relative permittivity ε R and with radius R much smaller than the operating wavelength, such that x 1. In the literature, the resonant conditions of a sphere are typically evaluated by finding the poles of the Mie coefficients [3]. In the limit x → 0 the resonances occur at y n = r n,l for the TM multipoles and at y n = r n−1,l for the TE multipoles, for any n, l ∈ N, where r m,l denotes the l-th zero of the spherical Bessel function j m [3]. Differently, we calculate these resonances as the eigenvalues of the magnetostatic integral operator L through a finite element numerical code [21]. In Figure  1 (a) we compare the first 100 magnetostatic eigenvalues with the poles of the Mie coefficients, while in Table I we show the corresponding values. We found very good agreement between the two approaches. In Fig. 2 we show some representative examples of the current modes. The first eigenvalue y = r 01 is associated to three degenerate TE current modes, M  o11 (r 01r ) in Fig. 2 (a). The second eigenvalue has an eight fold degeneracy: it is associated to five TE modes M    p = e, o (in Fig. 2 The third eigenvalue is associated to seven TE modes M   Fig. 2 (d)). The fifth eigenvalue is associated to nine TE modes M pm4 (r 31 r ) with m = 0 − 4 and p = e, o (in Fig. 2    e13 (r 31r ) is shown). The magnetostatic modes coincides with the photonic subset of the material independent modes introduced in Ref. [22] in the limit x → 0. As x increases, the material independent modes slowly change with respect to the magnetostatic limit, but their fundamental characteristics, including the angular dependence, the number of vortices, and the number of maxima along the radial direction are preserved.
We now investigate a cylinder of radius R and height h = R. We assume a characteristic size l c = R. This shape is very recurrent among nanofabbricated structures, since it is compatible with a planar nanofabrication process. In Fig. 1 (b) we show the first 100 magnetoquasistatic eigenvalues of the investigated cylinder, while in Tab. ?? we list some of their values.
We now show that the magnetostatic eigenvalues predict the occurrence of the resonance peaks in the scattering response of high-permittivity small objects. We investigate a sphere of radius R and permittivity ε R , centered in the origin of a Cartesian reference system, and excited by an electric dipole oriented alongx, i.e. N (3) e11 , oscillating at tunable frequency and located above the sphere at position (0, 0, 1.5R). In Fig. 4 we show the power absorbed by the sphere, normalized by the geometrical cross section πR 2 , as a function of y = x √ ε R . First, in Fig. 4 (a) we consider the case of ε R = 10 4 + i10 2 . We find very good agreement between the position of the resonances predicted by the eigenvalues of the operator L, shown with vertical dashed lines, and the absorption peaks. Specifically, in Fig. 4 (a), the four peaks from the left correspond to the modes shown in Fig. 2 (a-d). Good agreement is also found when ε R = 10 2 + i ( Fig. 4 (b)), even though we note a redshift of the peaks with respect to the magnetostatic prediction. Eventually, for the case of a silicon nanoparticle, shown in Fig. 4 (c), the resonance behaviour is still present, nevertheless the peaks undergo a broadening and a red-shift, so the magnetoquasistatic prediction leads to an overestimation of the actual resonance position. It is apparent that the higher the permittivity of the object the smaller the deviation between the expected peak position and the magnetostatic eigenvalues. Next, we study the power absorbed by the cylinder of radius R and height h = R, and different values of ε R . The cylinder is located at the origin of a Cartesian reference system, its axis is oriented alongẑ, and it is excited by an electric dipole directed alongŷ, i.e. N (3) o11 , located at position (R, 0, 1.5R). The absorbed power has been calculated by a surface integral equation approach [23]. We first investigate in Fig. 5 (a) the case in which ε R = 10 4 + 10 2 i: The peaks of the absorbtion occur in exact correspondence of the magnetostatic eigenvalues. Specifically, the first four peaks from the left correspond to the four modes shown in Fig. 3. Next, in Fig. 5 (b) we descrease the permittivity to ε R = 10 2 + 10i: we note a red-shift and a broadening of the peaks. Eventually in Fig 5 (c), we consider the case of a silicon cylinder with ε R = 15.45 + 0.1456i. The curve still resemble the two previous cases, but the shift and the brodening of the peaks are now significant. In conclusion, there exist two dual fundamental mechanisms through which a small homogeneous object may resonate. The first is the electroquasistatic (plasmonic) resonance [1] where the electric charge plays a central role. Each resonance is characterized by a negative eigenpermittivity, which is size-independent. The resonant electroquasistatic fields are both curl-free and div-free within the particle, but have a non-vanishing normal component to the particle surface. The second mechanism is the magnetoquasistatic resonance, described in this paper, where the electric currents are the main player. These resonances are only possible for positive permittivity. Irrespectively of the shape of the object, the resonance frequencies are inversely proportional to the characteristic size of the object, and inversely proportional to its refractive index. The resonant currents have a non-zero curl within the particle, but are div-free and have a vanishing normal component on the particle surface.