Leaky Modes of Solid Dielectric Spheres
11 Leaky Modes of Solid Dielectric Spheres
Masud Mansuripur † , Miroslav Kolesik † , and Per Jakobsen ‡ † College of Optical Sciences, The University of Arizona, Tucson ‡ Department of Mathematics and Statistics, University of Tromsø, Norway [Published in
Physical Review A , 013846, pp 1-14 (2017). DOI: 10.1103/PhysRevA.96.013846] Abstract . In the absence of external excitation, light trapped within a dielectric medium generally decays by leaking out, and also by getting absorbed within the medium. We analyze the leaky modes of solid dielectric spheres by examining solutions of Maxwell’s equations for simple homogeneous, isotropic, linearly dispersive media that admit complex-valued oscillation frequencies. We show that, under appropriate circumstances, these leaky modes constitute a complete set into which an initial electromagnetic field distribution inside a dielectric sphere can be expanded. We provide the outline of a completeness proof, and also present results of numerical calculations that illustrate the close relationship between the leaky modes and the resonances of solid dielectric spherical cavities.
1. Introduction . A well-polished, solid, smooth, homogeneous, and transparent glass sphere is a good example of material bodies which, when continually illuminated, admit and accommodate some of the incident light, eventually reaching a steady state where the rate of the incoming light equals that of the outgoing. By properly adjusting the frequency of the incident light, one can excite resonances, thus arriving at conditions under which the optical intensity inside the dielectric host exceeds, often by a large factor, that of the incident light beam [1,2]. If now the incident beam is suddenly terminated, the light trapped within the host medium begins to leak out, and, eventually, that portion of the electromagnetic (EM) energy which is not absorbed by the host, returns to the surrounding environment. The so-called leaky modes of a dielectric body are characterized by a unique set of complex-valued frequencies 𝜔𝜔 𝑞𝑞 = 𝜔𝜔 𝑞𝑞′ + i𝜔𝜔 𝑞𝑞″ , where the index 𝑞𝑞 is used here to enumerate the modes [3-7]. The imaginary part 𝜔𝜔 𝑞𝑞″ of each such frequency signifies the decay rate of the leaky mode, and (aside from a numerical coefficient) the corresponding quality-factor is given by 𝑄𝑄 = |𝜔𝜔 𝑞𝑞′ 𝜔𝜔 𝑞𝑞″ ⁄ | . The leaky modes of dielectric waveguides and cavities have been studied for many years, and a considerable volume of results pertaining to these modes exists in the literature. In addition to their applications in computational photonics and electromagnetics [8,9], such states of the EM field also pose questions of fundamental interest. Specifically, the problem of completeness and the general mathematical properties of these so-called “quasi-normal modes” have been broadly investigated. Of particular relevance to the present paper are the results reported in [5], which show that the leaky modes of a dielectric cavity can serve as a basis to represent arbitrary functions but only inside the cavity. In [6] it was shown that the set of leaky modes remains complete in the aforementioned sense even when the host medium exhibits losses as well as some chromatic dispersion limited to finite frequencies. Considering that the inclusion of chromatic dispersion and optical loss complicates the problem considerably, most of the pertinent mathematical analysis to date has been limited to one-dimensional systems, with rigorous results usually associated with cases in which chromatic dispersion and/or optical loss have been absent [10,11]. A specific application of leaky modes is the evaluation of the Purcell spontaneous emission enhancement factor when a dipole oscillator is coupled to a nearby cavity or a plasmonic resonator. Recent publications [12,13] have shown that the Purcell factor can be estimated from one (or a few) leaky modes, thus showcasing the need for delving into its detailed derivation, which exploits the expansion of Green’s tensor in terms of the leaky modes. The completeness of the leaky modes is assumed in the aforementioned papers, and references are given to the published literature where completeness has been discussed. In the present paper we emphasize that completeness should not be assumed but must be proven, that the expansion of an initial field distribution into a superposition of leaky modes is not trivial but involves subtleties due to the unusual behavior of the leaky modes in the vicinity of the singularities of the refractive index, and that the expansion coefficients obtained without proper accounting for such singularities could be wrong, resulting in a non-convergent series expansion. We will see how the leaky modes accumulate near one of the singular points of the refractive index. Consequently, if and when the excitation frequency happens to be close to the pole(s) of the refractive index, the assumption that one (or at most a few) leaky modes are sufficient to expand a given field distribution would become questionable. One of the goals of the present paper is to generalize the previous results in several ways. In particular, we study dielectric spheres with chromatic dispersion and loss properties in the framework of a Lorentz oscillator model, which also properly accounts for the behavior at high frequencies. This is an issue of fundamental importance, because high-frequency asymptotics in fact determine the conditions under which a set of leaky modes can be considered complete. As the frequency increases, the refractive index approaches unity and, for all practical purposes, high-frequency propagating waves cease to experience the presence of the cavity. While this is an important feature that was not part of the analysis in [6], one could reasonably argue that, in the absence of confinement for high-frequency EM waves, leaky modes could only provide approximations but not true resonant-mode expansions for arbitrary functions. Yet another difficulty arising from the dispersive properties of the medium is that the Lorentz oscillator model of the refractive index introduces branch-cuts into the analytic structure of the leaky-mode expansion. While the role of this singularity has been related to the over-completeness of the resonant states [14], the existence of branch-cuts can potentially invalidate arguments supporting completeness. We address these questions rigorously, and demonstrate the convergence of the leaky-mode expansion inside the cavity when a realistic high-frequency wave behavior is properly taken into consideration. In doing so, we also show that the leaky-mode expansion can be constructed in a way that eliminates the potential problem caused by the branch-cuts associated with the refractive index. The present paper contributes to the mathematics of open systems by providing an alternative completeness proof for leaky modes of solid, homogeneous, isotropic, dispersive dielectric spheres. We put forward a new approach to the completeness analysis that might find applications elsewhere in mathematical physics as well. The main tools of the trade in the existing literature on quasi-normal and resonant modes are invariably Green’s functions. In contrast to such conventional approaches, we present a method that relies solely on the analytic properties of the scattering states, thus avoiding any reliance on Green’s functions. Ours is a straightforward approach that simplifies the analysis of the leaky-mode expansion in comparison to conventional methods. Also possible are similar proofs of completeness for the leaky modes of parallel-plate dielectric slabs and infinitely-long dielectric cylinders, which we have recently reported in a conference proceedings paper [15]. An important question with regard to the completeness issue is the space of functions that can be expressed as a superposition of leaky modes. Interestingly, this is actually rarely addressed in the context of optical cavities. Note that our analysis is concerned with inherently lossy systems; in other words, not only is the system under investigation open, but also its EM energy content can be dissipated throughout the host medium. Thus, the problem being non- Hermitian, one cannot rely on the completeness of the scattering states as a point of departure when attempting to prove that the set of leaky modes forms a basis for expansion. To address this issue, we provide a constructive specification of the function space that is spanned by the leaky modes. It is specified as the space of EM fields excited within the optical cavity under external illumination. Assuming the external excitation has reached a steady state (after a sufficiently long time) when it is terminated, we proceed to show that the subsequent evolution of the EM field left inside the cavity can be represented by a convergent superposition of the leaky modes. Our approach provides a direct link between the various ways in which resonant modes can be detected and studied. In particular, the correspondence between resonance conditions, line shapes, and 𝑄𝑄 -factors of a spherical cavity can be readily explored. We present numerical results to illustrate certain general properties of spherical cavities. Last but not least, our results provide insight into features of resonant modes that are of practical interest. By comparing the properties of idealized spherical cavities with those made of realistic (i.e., lossy and dispersive) materials, we will show the way in which the losses inherent to the host material impose limits on the achievable 𝑄𝑄 -factors of solid dielectric spheres. In the following sections, we analyze the EM structure of the leaky modes of solid dielectric spheres, and examine the conditions under which certain initial field distributions can be decomposed into a superposition of leaky modes. We also present numerical results where the resonance conditions, line shapes, and quality-factors of a spherical cavity are computed; the correspondence between these and the leaky-mode frequencies is subsequently explored. We begin by describing in Sec. 2 the dispersive properties of linear, isotropic, homogeneous dielectric media whose electric permittivity and magnetic permeability each follow a single Lorentz oscillator model. Then, in Sec. 3, after a summary presentation of vector spherical harmonics, we demonstrate the completeness of the leaky modes of solid dielectric spheres for a special class of initial distributions residing within the spherical cavity. Numerical results showing the connection between the resonances of a dielectric sphere (when illuminated by a tunable source) and the corresponding leaky mode frequencies are presented in Sec. 4. Section 5 provides a summary of the main results of the paper followed by a few concluding remarks.
2. Refractive index model for a dispersive dielectric . The simplest dispersive dielectric is a medium whose electric and magnetic dipoles behave as independent Lorentz oscillators, each having their own resonance frequency 𝜔𝜔 𝑟𝑟 , plasma frequency 𝜔𝜔 𝑝𝑝 , and damping coefficient 𝛾𝛾 [16,17]. The electric and magnetic susceptibilities of the material will then be given by 𝜒𝜒 𝑒𝑒 (𝜔𝜔) = 𝜔𝜔 𝑝𝑝𝑝𝑝2 𝜔𝜔 𝑟𝑟𝑝𝑝2 − 𝜔𝜔 − i𝛾𝛾 𝑝𝑝 𝜔𝜔 , (1a) 𝜒𝜒 𝑚𝑚 (𝜔𝜔) = 𝜔𝜔 𝑝𝑝𝑝𝑝2 𝜔𝜔 𝑟𝑟𝑝𝑝2 − 𝜔𝜔 − i𝛾𝛾 𝑝𝑝 𝜔𝜔 . (1b) The corresponding refractive index, which is also a function of the frequency 𝜔𝜔 , will then be 𝑛𝑛(𝜔𝜔) = √𝜇𝜇𝜇𝜇 = �(1 + 𝜒𝜒 𝑚𝑚 )(1 + 𝜒𝜒 𝑒𝑒 ) = �1 + 𝜔𝜔 𝑝𝑝𝑝𝑝2 𝜔𝜔 𝑟𝑟𝑝𝑝2 −𝜔𝜔 −i𝛾𝛾 𝑝𝑝 𝜔𝜔 × �1 + 𝜔𝜔 𝑝𝑝𝑝𝑝2 𝜔𝜔 𝑟𝑟𝑝𝑝2 −𝜔𝜔 −i𝛾𝛾 𝑝𝑝 𝜔𝜔 = � (𝜔𝜔−Ω )(𝜔𝜔−Ω )(𝜔𝜔−Ω )(𝜔𝜔−Ω ) × � (𝜔𝜔−Ω )(𝜔𝜔−Ω )(𝜔𝜔−Ω )(𝜔𝜔−Ω ) , (2a) where Ω = ±�𝜔𝜔 𝑟𝑟2 + 𝜔𝜔 𝑝𝑝2 − ¼𝛾𝛾 − ½i𝛾𝛾 , (2b) Ω = ±�𝜔𝜔 𝑟𝑟2 − ¼𝛾𝛾 − ½i𝛾𝛾 . (2c) Assuming that 𝛾𝛾 ≪ 𝜔𝜔 𝑟𝑟 , the poles and zeros of 𝜇𝜇(𝜔𝜔) and 𝜇𝜇(𝜔𝜔) will be located in the lower-half of the complex 𝜔𝜔 -plane, as shown in Fig.1. The dashed line-segments in the figure represent branch-cuts that are needed to uniquely specify each square-root function appearing on the right- hand side of Eq.(2a). For the sake of simplicity, one might further assume that the branch-cuts of √𝜇𝜇 and those of √𝜇𝜇 do not overlap, although, strictly speaking, this restriction is not necessary. Whenever 𝜔𝜔 crosses (i.e., moves from immediately above to immediately below) one of these four branch-cuts, the refractive index 𝑛𝑛(𝜔𝜔) is multiplied by −1 . Note also that, in the limit when |𝜔𝜔| → ∞ (along any straight line originating at 𝜔𝜔 = 0 ), the complex entities 𝜇𝜇(𝜔𝜔) , 𝜇𝜇(𝜔𝜔) , and the refractive index 𝑛𝑛(𝜔𝜔) will all approach , while (𝜔𝜔) approaches (𝜔𝜔 𝑝𝑝𝑚𝑚2 + 𝜔𝜔 𝑝𝑝𝑒𝑒2 ) 𝜔𝜔 ⁄ . Fig.1 . Locations in the 𝜔𝜔 -plane of the poles and zeros of 𝜇𝜇(𝜔𝜔) , whose square root contributes to the refractive index 𝑛𝑛(𝜔𝜔) in accordance with Eq.(2). A similar set of poles and zeros, albeit at different locations in the 𝜔𝜔 -plane, represents 𝜇𝜇(𝜔𝜔) . The dashed lines connecting pairs of adjacent poles and zeros constitute branch-cuts for the function 𝑛𝑛(𝜔𝜔) . In accordance with the Cauchy-Goursat theorem [18], the integral of a meromorphic function, such as 𝑓𝑓(𝜔𝜔) , over a circle of radius 𝑅𝑅 𝑐𝑐 is times the sum of the residues of the function at the poles of 𝑓𝑓(𝜔𝜔) that reside within the circle.
3. Leaky modes of a solid dielectric sphere . The vector spherical harmonics of the EM field within a homogeneous, isotropic, linear medium having permeability 𝜇𝜇 𝜇𝜇(𝜔𝜔) and permittivity 𝜇𝜇 𝜇𝜇(𝜔𝜔) are found by solving Maxwell’s equations in spherical coordinates [16,17]. The electric and magnetic field profiles for Transverse Electric (TE) and Transverse Magnetic (TM) modes of the EM field are found to be 𝑚𝑚 = 0 TE mode (𝐸𝐸 𝑟𝑟 = 0) : 𝑬𝑬(𝒓𝒓, 𝑡𝑡) = 𝐸𝐸 𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟)√𝑘𝑘𝑟𝑟 𝑃𝑃 ℓ1 (cos 𝜃𝜃) exp(−i𝜔𝜔𝑡𝑡)𝝋𝝋� . (3) 𝑯𝑯(𝒓𝒓, 𝑡𝑡) = 𝐸𝐸 𝜇𝜇 𝜇𝜇(𝜔𝜔)𝑟𝑟𝜔𝜔 � 𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟)i√𝑘𝑘𝑟𝑟 [cot 𝜃𝜃 𝑃𝑃 ℓ1 (cos 𝜃𝜃) − sin 𝜃𝜃 𝑃𝑃̇ ℓ1 (cos 𝜃𝜃)]𝒓𝒓� − 𝑘𝑘𝑟𝑟𝐽𝐽̇ ℓ+½ (𝑘𝑘𝑟𝑟)+½𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟)i√𝑘𝑘𝑟𝑟 𝑃𝑃 ℓ1 (cos 𝜃𝜃)𝜽𝜽�� exp(−i𝜔𝜔𝑡𝑡) . (4) 𝑚𝑚 ≠ 0 TE mode (𝐸𝐸 𝑟𝑟 = 0) : 𝑬𝑬(𝒓𝒓, 𝑡𝑡) = 𝐸𝐸 � 𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟)√𝑘𝑘𝑟𝑟 𝑃𝑃 ℓ𝑝𝑝 (cos 𝜃𝜃)sin 𝜃𝜃 𝜽𝜽� + 𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟)i𝑚𝑚√𝑘𝑘𝑟𝑟 sin 𝜃𝜃 𝑃𝑃̇ ℓ𝑚𝑚 (cos 𝜃𝜃)𝝋𝝋� � exp[i(𝑚𝑚𝑚𝑚 − 𝜔𝜔𝑡𝑡)] . (5) 𝑯𝑯(𝒓𝒓, 𝑡𝑡) = − 𝐸𝐸 𝜇𝜇 𝜇𝜇(𝜔𝜔)𝑟𝑟𝜔𝜔 � ℓ(ℓ+1)𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟)𝑚𝑚√𝑘𝑘𝑟𝑟 𝑃𝑃 ℓ𝑚𝑚 (cos 𝜃𝜃)𝒓𝒓� − 𝑘𝑘𝑟𝑟𝐽𝐽̇ ℓ+½ (𝑘𝑘𝑟𝑟)+½𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟)𝑚𝑚√𝑘𝑘𝑟𝑟 sin 𝜃𝜃 𝑃𝑃̇ ℓ𝑚𝑚 (cos 𝜃𝜃)𝜽𝜽� − 𝑘𝑘𝑟𝑟𝐽𝐽̇ ℓ+½ (𝑘𝑘𝑟𝑟)+½𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟)i√𝑘𝑘𝑟𝑟 𝑃𝑃 ℓ𝑝𝑝 (cos 𝜃𝜃)sin 𝜃𝜃 𝝋𝝋� � exp[i(𝑚𝑚𝑚𝑚 − 𝜔𝜔𝑡𝑡)] . (6) × × 𝜔𝜔 ′ 𝜔𝜔 ″ Ω Ω Ω Ω 𝑅𝑅 𝑐𝑐 𝑚𝑚 = 0 TM mode (𝐻𝐻 𝑟𝑟 = 0) : 𝑬𝑬(𝒓𝒓, 𝑡𝑡) = − 𝐻𝐻 𝜀𝜀 𝜀𝜀(𝜔𝜔)𝑟𝑟𝜔𝜔 � 𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟)i√𝑘𝑘𝑟𝑟 �cot 𝜃𝜃 𝑃𝑃 ℓ1 (cos 𝜃𝜃) − sin 𝜃𝜃 𝑃𝑃̇ ℓ1 (cos 𝜃𝜃)�𝒓𝒓� − 𝑘𝑘𝑟𝑟𝐽𝐽̇ ℓ+½ (𝑘𝑘𝑟𝑟)+½𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟)i√𝑘𝑘𝑟𝑟 𝑃𝑃 ℓ1 (cos 𝜃𝜃)𝜽𝜽�� exp(−i𝜔𝜔𝑡𝑡) . (7) 𝑯𝑯(𝒓𝒓, 𝑡𝑡) = 𝐻𝐻 𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟)√𝑘𝑘𝑟𝑟 𝑃𝑃 ℓ1 (cos 𝜃𝜃) exp(−i𝜔𝜔𝑡𝑡) 𝝋𝝋� . (8) 𝑚𝑚 ≠ 0 TM mode (𝐻𝐻 𝑟𝑟 = 0) : 𝑬𝑬(𝒓𝒓, 𝑡𝑡) = 𝐻𝐻 𝜀𝜀 𝜀𝜀(𝜔𝜔)𝑟𝑟𝜔𝜔 � ℓ(ℓ+1)𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟)𝑚𝑚√𝑘𝑘𝑟𝑟 𝑃𝑃 ℓ𝑚𝑚 (cos 𝜃𝜃)𝒓𝒓� − 𝑘𝑘𝑟𝑟𝐽𝐽̇ ℓ+½ (𝑘𝑘𝑟𝑟)+½𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟)𝑚𝑚√𝑘𝑘𝑟𝑟 sin 𝜃𝜃 𝑃𝑃̇ ℓ𝑚𝑚 (cos 𝜃𝜃)𝜽𝜽� − 𝑘𝑘𝑟𝑟𝐽𝐽̇ ℓ+½ (𝑘𝑘𝑟𝑟)+½𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟)i√𝑘𝑘𝑟𝑟 𝑃𝑃 ℓ𝑝𝑝 (cos 𝜃𝜃)sin 𝜃𝜃 𝝋𝝋� � exp[i(𝑚𝑚𝑚𝑚 − 𝜔𝜔𝑡𝑡)] . (9) 𝑯𝑯(𝒓𝒓, 𝑡𝑡) = 𝐻𝐻 � 𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟)√𝑘𝑘𝑟𝑟 𝑃𝑃 ℓ𝑝𝑝 (cos 𝜃𝜃)sin 𝜃𝜃 𝜽𝜽� + 𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟)i𝑚𝑚√𝑘𝑘𝑟𝑟 sin 𝜃𝜃 𝑃𝑃̇ ℓ𝑚𝑚 (cos 𝜃𝜃)𝝋𝝋� � exp[i(𝑚𝑚𝑚𝑚 − 𝜔𝜔𝑡𝑡)] . (10) In the above equations, the Bessel function 𝐽𝐽 𝜈𝜈 (𝑧𝑧) and its derivative with respect to 𝑧𝑧 , 𝐽𝐽̇ 𝜈𝜈 (𝑧𝑧) , could be replaced by a Bessel function of the second kind, 𝑌𝑌 𝜈𝜈 (𝑧𝑧) , and its derivative, 𝑌𝑌̇ 𝜈𝜈 (𝑧𝑧) , or by Hankel functions of type 1 or type 2, namely, ℋ 𝜈𝜈(1,2) (𝑧𝑧) , and corresponding derivatives ℋ̇ 𝜈𝜈(1,2) (𝑧𝑧) . The (complex) field amplitudes are denoted by 𝐸𝐸 and 𝐻𝐻 . In our spherical coordinate system, the point 𝒓𝒓 is at a distance 𝑟𝑟 from the origin, its polar and azimuthal angles being 𝜃𝜃 and 𝑚𝑚 . The oscillation frequency is 𝜔𝜔 , and the wave-number 𝑘𝑘 is defined as 𝑘𝑘(𝜔𝜔) = 𝑛𝑛(𝜔𝜔)𝑘𝑘 , where 𝑘𝑘 = 𝜔𝜔 𝑐𝑐⁄ , and 𝑛𝑛(𝜔𝜔) = �𝜇𝜇(𝜔𝜔)𝜇𝜇(𝜔𝜔) is the refractive index of the host medium. The integers ℓ ≥ 1 , and 𝑚𝑚 (ranging from −ℓ to +ℓ ) specify the polar and azimuthal mode numbers. 𝑃𝑃 ℓ𝑚𝑚 (𝜁𝜁) is an associated Legendre function, while 𝑃𝑃̇ ℓ𝑚𝑚 (𝜁𝜁) is its derivative with respect to 𝜁𝜁 . Note that, for a given 𝑚𝑚 , the TM mode may be obtained from the corresponding TE mode by substituting 𝑬𝑬 for 𝑯𝑯 , and −𝑯𝑯 for 𝑬𝑬 , keeping in mind that 𝑟𝑟𝜔𝜔 = 𝑘𝑘𝑟𝑟 �𝜇𝜇 𝜇𝜇 𝜇𝜇(𝜔𝜔)𝜇𝜇(𝜔𝜔)⁄ , and that the 𝐸𝐸 𝐻𝐻⁄ amplitude ratio for each mode is always given by �𝜇𝜇 𝜇𝜇(𝜔𝜔) 𝜇𝜇 𝜇𝜇(𝜔𝜔)⁄ . Finally, the various Bessel functions of half-integer order are defined by the following formulas [19]: 𝐽𝐽 ℓ+½ (𝑧𝑧) = � �sin(𝑧𝑧 − ½ℓ𝜋𝜋) � (−1) 𝑘𝑘 (ℓ+2𝑘𝑘)!(2𝑘𝑘)!(ℓ−2𝑘𝑘)! � � + cos(𝑧𝑧 − ½ℓ𝜋𝜋) � (−1) 𝑘𝑘 (ℓ+2𝑘𝑘+1)!(2𝑘𝑘+1)!(ℓ−2𝑘𝑘−1)! � � � . (11) 𝑌𝑌 ℓ+½ (𝑧𝑧) = (−1) ℓ−1 � �cos(𝑧𝑧 + ½ℓ𝜋𝜋) � (−1) 𝑘𝑘 (ℓ+2𝑘𝑘)!(2𝑘𝑘)!(ℓ−2𝑘𝑘)! � � − sin(𝑧𝑧 + ½ℓ𝜋𝜋) � (−1) 𝑘𝑘 (ℓ+2𝑘𝑘+1)!(2𝑘𝑘+1)!(ℓ−2𝑘𝑘−1)! � � � . (12) ℋ ℓ+½(1) (𝑧𝑧) = � exp{i[𝑧𝑧 − ½(ℓ + 1)𝜋𝜋]} � (ℓ+𝑘𝑘)!𝑘𝑘!(ℓ−𝑘𝑘)! � i2𝜋𝜋 � 𝑘𝑘ℓ𝑘𝑘=0 . (13) Note that √𝑧𝑧𝐽𝐽 ℓ+½ (𝑧𝑧) is an even function of 𝑧𝑧 when ℓ = 1, 3, 5, ⋯ , and an odd function when ℓ = 2, 4, 6, ⋯ . This fact will be needed later on, when we try to argue that certain branch-cuts in the complex 𝜔𝜔 -plane are inconsequential. Also, the following alternative representation of Bessel functions of the first kind, order 𝜈𝜈 , will be found useful: 𝐽𝐽 𝜈𝜈 (𝑧𝑧) = (𝑧𝑧 2⁄ ) 𝜈𝜈 � (−1) 𝑘𝑘 (𝜋𝜋 2⁄ ) 𝑘𝑘! Γ(𝜈𝜈+𝑘𝑘+1)∞𝑘𝑘=0 · (14) Given that 𝜈𝜈 = ℓ + ½ ≥ 3 2⁄ for spherical harmonics, Eq.(14) reveals that 𝐽𝐽 ℓ+½ (𝑧𝑧) 𝑧𝑧⁄ → 0 when 𝑧𝑧 → 0 . Consider now a solid dielectric sphere of radius 𝑅𝑅 , relative permeability 𝜇𝜇(𝜔𝜔) , and relative permittivity 𝜇𝜇(𝜔𝜔) . Inside the particle, the radial dependence of the TE mode is governed by a Bessel function of the first kind, 𝐸𝐸 𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟) , and its derivative. The refractive index of the spherical particle being 𝑛𝑛(𝜔𝜔) = �𝜇𝜇(𝜔𝜔)𝜇𝜇(𝜔𝜔) , the corresponding wave-number inside the particle is 𝑘𝑘(𝜔𝜔) = 𝑛𝑛(𝜔𝜔)𝑘𝑘 = 𝑛𝑛(𝜔𝜔)𝜔𝜔 𝑐𝑐⁄ . The particle is surrounded by free space, which is host to an outgoing spherical harmonic whose radial dependence is governed by a type 1 Hankel function, 𝐸𝐸 ℋ ℓ+½(1) (𝑘𝑘 𝑟𝑟) , and its derivative. Invoking the Bessel function identity 𝑧𝑧𝐽𝐽̇ 𝜈𝜈 (𝑧𝑧) = 𝜈𝜈𝐽𝐽 𝜈𝜈 (𝑧𝑧) −𝑧𝑧𝐽𝐽 𝜈𝜈+1 (𝑧𝑧) — which applies to 𝑌𝑌 𝜈𝜈 (𝑧𝑧) and ℋ 𝜈𝜈(1,2) (𝑧𝑧) as well — we find, upon matching the boundary conditions at 𝑟𝑟 = 𝑅𝑅 , that the following two equations must be simultaneously satisfied: 𝐸𝐸 𝐽𝐽 ℓ+½ (𝑛𝑛𝑘𝑘 𝑅𝑅)�𝑛𝑛𝑘𝑘 𝑅𝑅 = 𝐸𝐸 ℋ ℓ+½(1) (𝑘𝑘 𝑅𝑅)�𝑘𝑘 𝑅𝑅 , (15a) 𝐸𝐸 [(ℓ+1)𝐽𝐽 ℓ+½ (𝑛𝑛𝑘𝑘 𝑅𝑅) − 𝑛𝑛𝑘𝑘 𝑅𝑅𝐽𝐽 ℓ+3 2⁄ (𝑛𝑛𝑘𝑘 𝑅𝑅)]𝜇𝜇(𝜔𝜔)�𝑛𝑛𝑘𝑘 𝑅𝑅 = 𝐸𝐸 [(ℓ+1)ℋ ℓ+½(1) (𝑘𝑘 𝑅𝑅) − 𝑘𝑘 𝑅𝑅ℋ ℓ+3 2⁄(1) (𝑘𝑘 𝑅𝑅)]�𝑘𝑘 𝑅𝑅 · (15b) Streamlining the above equations, we arrive at � 𝐽𝐽 ℓ+½ (𝑛𝑛𝑘𝑘 𝑅𝑅) −√𝑛𝑛ℋ ℓ+½(1) (𝑘𝑘 𝑅𝑅)(ℓ + 1)𝐽𝐽 ℓ+½ (𝑛𝑛𝑘𝑘 𝑅𝑅) − 𝑛𝑛𝑘𝑘 𝑅𝑅𝐽𝐽 ℓ+3 2⁄ (𝑛𝑛𝑘𝑘 𝑅𝑅) −𝜇𝜇√𝑛𝑛�(ℓ + 1)ℋ ℓ+½(1) (𝑘𝑘 𝑅𝑅) − 𝑘𝑘 𝑅𝑅ℋ ℓ+3 2⁄(1) (𝑘𝑘 𝑅𝑅)�� �𝐸𝐸 𝐸𝐸 � = 0 . (16) A non-trivial solution for 𝐸𝐸 and 𝐸𝐸 thus exists if and only if the determinant of the coefficient matrix in Eq.(16) vanishes, that is, 𝐹𝐹(𝜔𝜔) = 𝑛𝑛𝑘𝑘 𝑅𝑅ℋ ℓ+½(1) (𝑘𝑘 𝑅𝑅)𝐽𝐽 ℓ+3 2⁄ (𝑛𝑛𝑘𝑘 𝑅𝑅) + �(𝜇𝜇 − 1)(ℓ + 1)ℋ ℓ+½(1) (𝑘𝑘 𝑅𝑅) − 𝜇𝜇𝑘𝑘 𝑅𝑅ℋ ℓ+3 2⁄(1) (𝑘𝑘 𝑅𝑅)� 𝐽𝐽 ℓ+½ (𝑛𝑛𝑘𝑘 𝑅𝑅) = 0 . (17) This is the characteristic equation for leaky TE modes, whose solutions comprise the entire set of leaky frequencies 𝜔𝜔 𝑞𝑞 . (The index 𝑞𝑞 is used here to enumerate the various leaky-mode frequencies.) For TM modes, 𝜇𝜇(𝜔𝜔) in Eq.(17) must be replaced by 𝜇𝜇(𝜔𝜔) . Equation (17) must be solved numerically for complex frequencies 𝜔𝜔 𝑞𝑞 ; these being characteristic frequencies of the spherical particle’s leaky modes, one expects (on physical grounds) to find all the roots 𝜔𝜔 𝑞𝑞 of 𝐹𝐹(𝜔𝜔) in the lower-half of the complex plane. Note that √𝑛𝑛𝐹𝐹(𝜔𝜔) is an even function of 𝑛𝑛 when ℓ = 1, 3, 5, ⋯ , and an odd function when ℓ = 2, 4, 6, ⋯ . This is because successive Bessel functions 𝐽𝐽 ℓ+½ and 𝐽𝐽 ℓ+3 2⁄ alternate between odd and even parities. Note also that 𝐹𝐹(𝜔𝜔) vanishes at the zeros of 𝑛𝑛(𝜔𝜔) , that is,
𝐹𝐹(Ω ) = 𝐹𝐹(Ω ) = 0 ; see Eq.(2b). Nevertheless, Ω and Ω do not represent leaky-mode frequencies, because setting 𝑛𝑛(Ω ) = 0 in Eqs.(3) -(10) extinguishes the EM field throughout the dielectric sphere. At the poles of 𝑛𝑛(𝜔𝜔) , namely, 𝜔𝜔 = Ω and 𝜔𝜔 = Ω given by Eq.(2c), the function 𝐹𝐹(𝜔𝜔) is undefined, but an arbitrarily small circle centered at Ω (or Ω ) can be shown to contain an infinite number of the zeros of 𝐹𝐹(𝜔𝜔) . One could argue that, throughout the dielectric sphere, the EM fields associated with the Ω frequencies should be negligible, although the mathematical reasoning behind this argument is not straightforward. Finally, when 𝜔𝜔 → 0 , 𝐹𝐹(𝜔𝜔) approaches a constant (see the Appendix), and when |𝜔𝜔| → ∞ , 𝜇𝜇(𝜔𝜔) → 1 − (𝜔𝜔 𝑝𝑝𝑚𝑚 𝜔𝜔⁄ ) and 𝜇𝜇(𝜔𝜔) → 1 − (𝜔𝜔 𝑝𝑝𝑒𝑒 𝜔𝜔⁄ ) , thus allowing the asymptotic behavior of 𝐹𝐹(𝜔𝜔) to be determined from Eqs.(11) and (13). Our goal is to express an initial field distribution inside the spherical particle (e.g., one of the spherical harmonic waveforms given by Eqs.(3) -(10), which oscillate at a real-valued frequency 𝜔𝜔 ) as a superposition of leaky modes, each having its own complex frequency 𝜔𝜔 𝑞𝑞 . To this end, we must form a meromorphic function 𝐺𝐺(𝜔𝜔) incorporating the following features: i) The function
𝐹𝐹(𝜔𝜔) of Eq.(17) appears in the denominator of
𝐺𝐺(𝜔𝜔) , thus causing the zeros of
𝐹𝐹(𝜔𝜔) to act as poles for
𝐺𝐺(𝜔𝜔) . ii) A desired initial waveform, say, 𝐽𝐽 ℓ+½ [𝜔𝜔𝑛𝑛(𝜔𝜔)𝑟𝑟 𝑐𝑐⁄ ] , appearing in the numerator of 𝐺𝐺(𝜔𝜔) . iii) The real-valued frequency 𝜔𝜔 associated with the initial waveform acting as a pole for 𝐺𝐺(𝜔𝜔) . iv) In the limit when |𝜔𝜔| → ∞ , 𝐺𝐺(𝜔𝜔) → 0 exponentially, so that ∮ 𝐺𝐺(𝜔𝜔)d𝜔𝜔 over a circle of large radius 𝑅𝑅 𝑐𝑐 would vanish. A simple (although by no means the only) such function is 𝐺𝐺(𝜔𝜔) = √𝜔𝜔 exp(i𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟)(𝜔𝜔−𝜔𝜔 )𝐹𝐹(𝜔𝜔) · (18) With reference to Eq.(11), note that the pre-factor of the Bessel function in the numerator of 𝐺𝐺(𝜔𝜔) cancels the corresponding pre-factor that accompanies the denominator. The remaining part of the Bessel function in the numerator will then have the same parity with respect to 𝑛𝑛(𝜔𝜔) as the function that appears in the denominator. Consequently, switching the sign of 𝑛𝑛(𝜔𝜔) does not alter
𝐺𝐺(𝜔𝜔) , indicating that the branch-cuts associated with 𝑛𝑛(𝜔𝜔) in the complex 𝜔𝜔 -plane do not introduce discontinuities into 𝐺𝐺(𝜔𝜔) . The presence of √𝜔𝜔 exp(i𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) in the numerator of
𝐺𝐺(𝜔𝜔) is intended to eliminate certain undesirable features of the Hankel functions appearing in the denominator. The function
𝐺𝐺(𝜔𝜔) is thus analytic everywhere except at its poles, where its denominator vanishes. The poles, of course, consist of 𝜔𝜔 = 𝜔𝜔 , which is the frequency of the initial EM field residing inside the spherical particle at 𝑡𝑡 = 0 , and 𝜔𝜔 = 𝜔𝜔 𝑞𝑞 , which are the leaky-mode frequencies found by solving Eq.(17) — or its TM mode counterpart. The zeros of the refractive index 𝑛𝑛(𝜔𝜔) , namely, 𝜔𝜔 = Ω , do not become poles of 𝐺𝐺(𝜔𝜔) , because the numerator of
𝐺𝐺(Ω ) also equals zero. At the poles 𝜔𝜔 = Ω of the refractive index, 𝐺𝐺(𝜔𝜔) is undefined, but it is well-behaved in the sense that the integral of
𝐺𝐺(𝜔𝜔) around a small circle centered at 𝜔𝜔 = Ω , whose radius passes between consecutive poles, approaches zero as the radius of the circle goes to zero. For this reason, one can invoke Cauchy’s theorem in order to construct a leaky-mode expansion for the dielectric sphere, even though the integrand,
𝐺𝐺(𝜔𝜔) , has non-isolated singularities at Ω (and also when 𝜔𝜔 → ∞ ). These mathematical details will be addressed in a forthcoming paper. In the limit |𝜔𝜔| → ∞ , where 𝜇𝜇(𝜔𝜔) → 1 − (𝜔𝜔 𝑝𝑝𝑚𝑚 𝜔𝜔⁄ ) and 𝜇𝜇(𝜔𝜔) → 1 − (𝜔𝜔 𝑝𝑝𝑒𝑒 𝜔𝜔⁄ ) , we find that 𝐺𝐺(𝜔𝜔) approaches zero exponentially. Thus, the vanishing of ∮ 𝐺𝐺(𝜔𝜔)d𝜔𝜔 around a circle of large radius 𝑅𝑅 𝑐𝑐 ensures that all the residues of 𝐺𝐺(𝜔𝜔) add up to zero, that is, �𝜔𝜔 exp(i𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )𝐽𝐽 ℓ+½ [𝜔𝜔 𝑛𝑛(𝜔𝜔 )𝑟𝑟 𝑐𝑐⁄ ]𝐹𝐹(𝜔𝜔 ) + � �𝜔𝜔 𝑞𝑞 exp(i𝑅𝑅𝜔𝜔 𝑞𝑞 𝑐𝑐⁄ )𝐽𝐽 ℓ+½ [𝜔𝜔 𝑞𝑞 𝑛𝑛(𝜔𝜔 𝑞𝑞 )𝑟𝑟 𝑐𝑐⁄ ](𝜔𝜔 𝑞𝑞 − 𝜔𝜔 )𝐹𝐹 ′ �𝜔𝜔 𝑞𝑞 �𝑞𝑞 = 0 . (19) The initial field distribution 𝐽𝐽 ℓ+½ [𝜔𝜔 𝑛𝑛(𝜔𝜔 )𝑟𝑟 𝑐𝑐⁄ ] may thus be expanded as the following superposition of all the leaky modes: 𝐽𝐽 ℓ+½ [𝜔𝜔 𝑛𝑛(𝜔𝜔 )𝑟𝑟 𝑐𝑐⁄ ] = � (𝜔𝜔 𝑞𝑞 𝜔𝜔 ⁄ ) ½ exp[i𝑅𝑅(𝜔𝜔 𝑞𝑞 − 𝜔𝜔 ) 𝑐𝑐⁄ ]𝐹𝐹(𝜔𝜔 )�𝜔𝜔 − 𝜔𝜔 𝑞𝑞 �𝐹𝐹 ′ �𝜔𝜔 𝑞𝑞 � × 𝑞𝑞 𝐽𝐽 ℓ+½ [𝜔𝜔 𝑞𝑞 𝑛𝑛(𝜔𝜔 𝑞𝑞 )𝑟𝑟 𝑐𝑐⁄ ] . (20) To incorporate into the initial distribution the denominator √𝑘𝑘𝑟𝑟 , which accompanies all the field components in Eqs.(3) -(10), we modify Eq.(20) — albeit trivially — as follows: 𝐽𝐽 ℓ+½ [𝜔𝜔 𝑛𝑛(𝜔𝜔 )𝑟𝑟 𝑐𝑐⁄ ]�𝜔𝜔 𝑛𝑛(𝜔𝜔 )𝑟𝑟 𝑐𝑐⁄ = � �𝜔𝜔 𝑞𝑞 𝜔𝜔 ⁄ ��𝑛𝑛(𝜔𝜔 𝑞𝑞 ) 𝑛𝑛(𝜔𝜔 )⁄ �½ exp�i𝑅𝑅�𝜔𝜔 𝑞𝑞 − 𝜔𝜔 � 𝑐𝑐⁄ �𝐹𝐹(𝜔𝜔 )�𝜔𝜔 − 𝜔𝜔 𝑞𝑞 �𝐹𝐹 ′ �𝜔𝜔 𝑞𝑞 �𝑞𝑞 × 𝐽𝐽 ℓ+½ [𝜔𝜔 𝑞𝑞 𝑛𝑛(𝜔𝜔 𝑞𝑞 )𝑟𝑟 𝑐𝑐⁄ ]�𝜔𝜔 𝑞𝑞 𝑛𝑛(𝜔𝜔 𝑞𝑞 )𝑟𝑟 𝑐𝑐⁄ · (21) The above formula is a central result of the present paper, indicating that a general EM field distribution excited from outside the cavity can be represented by a superposition of leaky modes. Indeed, upon termination of the external excitation, the field that remains within the cavity is, in general, a superposition of functions similar to that appearing on the left-hand-side of Eq.(21), with the spectral weight associated with each such function depending on its oscillation frequency 𝜔𝜔 . Thus, with the important caveat discussed in the following paragraph, Eq.(21) provides an explicit formula for computing the leaky-mode expansion coefficients corresponding to the post-excitation evolution of the intra-cavity field. Without going into details, it must be pointed out that the argument for the vanishing of the contour integral around a large circle in the 𝜔𝜔 -plane contains a couple of subtleties. One is that the integration contour must pass between the poles that represent the very resonances used for the expansion. While the straightforward reasoning about the exponential decay of the integrand cannot be applied to such a portion of the integral, it can be shown that its contribution does indeed vanish in the limit 𝑅𝑅 𝑐𝑐 → ∞ if our choice for 𝐺𝐺(𝜔𝜔) as given by Eq.(18) is somewhat modified in such a way as to accelerate its approach to zero when |𝜔𝜔| → ∞ . The second issue is that, besides 𝜔𝜔 → ∞ being an accumulation point for the singularities of
𝐺𝐺(𝜔𝜔) , there exist other such points, namely, the poles Ω of the Lorentzian refractive index; see Fig.1. In this case, it can be shown that the requirements for the series convergence are less restrictive than those pertaining to 𝜔𝜔 → ∞ . In fact, one can introduce additional poles into 𝐺𝐺(𝜔𝜔) by multiplying its denominator with (𝜔𝜔 − Ω )(𝜔𝜔 − Ω ) and still obtain a convergent series. These convergence issues are brought about by the dispersion properties of the refractive index together with the fact that 𝑛𝑛(𝜔𝜔) → 1 when |𝜔𝜔| → ∞ , issues that, to the best of our knowledge, have not been discussed in the existing literature concerning leaky-mode expansion of dispersive optical cavities. Unfortunately, a detailed exposition of the convergence proof is beyond the scope of the present paper and must be presented elsewhere. The bottom line is that the convergence of the series can be guaranteed if the leaky-mode expansion coefficients in Eq.(21) are multiplied by the additional factor (𝜔𝜔 − Ω )(𝜔𝜔 − Ω ) [(𝜔𝜔 𝑞𝑞 − Ω )(𝜔𝜔 𝑞𝑞 − Ω )]⁄ . Taking advantage of the flexibility of
𝐺𝐺(𝜔𝜔) , we now extend the same treatment to the remaining components of the EM field. For instance, if we choose
𝐺𝐺(𝜔𝜔) = √𝜔𝜔 exp(i𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) 𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟)𝜔𝜔(𝜔𝜔 − 𝜔𝜔 )(𝜔𝜔 − Ω )(𝜔𝜔 − Ω )𝜇𝜇(𝜔𝜔)𝐹𝐹(𝜔𝜔) , (22) then 𝐺𝐺(𝜔𝜔) → 0 exponentially in the limit when |𝜔𝜔| → ∞ , resulting in a vanishing integral around the circle of large radius 𝑅𝑅 𝑐𝑐 in the 𝜔𝜔 -plane. We thus arrive at an alternative form of Eq.(21), which is useful for expanding the field component 𝐻𝐻 𝑟𝑟 appearing in Eqs.(4) and (6), that is, 𝐽𝐽 ℓ+½ [𝜔𝜔 𝑛𝑛(𝜔𝜔 )𝑟𝑟 𝑐𝑐⁄ ]𝜇𝜇(𝜔𝜔 )𝑟𝑟𝜔𝜔 �𝜔𝜔 𝑛𝑛(𝜔𝜔 )𝑟𝑟 𝑐𝑐⁄ = � (𝜔𝜔 𝑞𝑞 𝜔𝜔 ⁄ )[𝑛𝑛(𝜔𝜔 𝑞𝑞 ) 𝑛𝑛(𝜔𝜔 )⁄ ] ½ exp[i𝑅𝑅(𝜔𝜔 𝑞𝑞 − 𝜔𝜔 ) 𝑐𝑐⁄ ]𝐹𝐹(𝜔𝜔 )(𝜔𝜔 − 𝜔𝜔 𝑞𝑞 )𝐹𝐹 ′ (𝜔𝜔 𝑞𝑞 )𝑞𝑞 × (𝜔𝜔 −Ω )(𝜔𝜔 −Ω )(𝜔𝜔 𝑞𝑞 −Ω )(𝜔𝜔 𝑞𝑞 −Ω ) × 𝐽𝐽 ℓ+½ [𝜔𝜔 𝑞𝑞 𝑛𝑛(𝜔𝜔 𝑞𝑞 )𝑟𝑟 𝑐𝑐⁄ ]𝜇𝜇(𝜔𝜔 𝑞𝑞 )𝑟𝑟𝜔𝜔 𝑞𝑞 [𝜔𝜔 𝑞𝑞 𝑛𝑛(𝜔𝜔 𝑞𝑞 )𝑟𝑟 𝑐𝑐⁄ ]½ · (23) Similarly, if we choose 𝐺𝐺(𝜔𝜔) = √𝜔𝜔 exp(i𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )[𝑘𝑘𝑟𝑟𝐽𝐽̇ ℓ+½ (𝑘𝑘𝑟𝑟)+½𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟)]𝜔𝜔(𝜔𝜔 − 𝜔𝜔 )(𝜔𝜔 − Ω )(𝜔𝜔 − Ω )𝜇𝜇(𝜔𝜔)𝐹𝐹(𝜔𝜔) , (24) it continues to be meromorphic (i.e., free of branch-cuts), and will have a vanishing integral over a large circle of radius 𝑅𝑅 𝑐𝑐 in the limit when 𝑅𝑅 𝑐𝑐 → ∞ . The relevant expansion of the field components 𝐻𝐻 𝜃𝜃 and 𝐻𝐻 𝜑𝜑 appearing in Eqs.(4) and (6) will then be obtained from 𝐺𝐺(𝜔𝜔) of Eq.(24). In this way, one can expand into a superposition of leaky modes the various 𝐸𝐸 - and 𝐻𝐻 -field components that comprise an initial distribution. It will then be possible to follow each leaky mode as its phase evolves while its amplitude decays with the passage of time. As for the beam that leaks out of the cavity and into the free-space region 𝑟𝑟 > 𝑅𝑅 , it can be shown that the fields grow exponentially along the radial direction, but of course this exponential growth terminates at 𝑟𝑟 = 𝑐𝑐𝑡𝑡 , where the leaked beam meets up with the tail end of the beam that was originally reflected from the surface of the sphere (i.e., prior to the abrupt termination of the incident beam at 𝑡𝑡 = 0 ). The EM energy in the region 𝑅𝑅 < 𝑟𝑟 < 𝑐𝑐𝑡𝑡 is just the energy that has leaked out of the spherical cavity, with the exponential decline of the field amplitude in time compensating for the expansion of the region “illuminated” by the leaked beam. Before concluding this section, a note concerning over-completeness might be in order. It is known that resonant modes are subject to sum rules which make it possible to create nontrivial linear combinations that sum-up to zero [20,21]. Our method also allows derivation of such sum rules. To this end it is sufficient to remove the factor 𝜔𝜔 (𝜔𝜔 − 𝜔𝜔 )⁄ from the function 𝐺𝐺(𝜔𝜔) . This will not modify the asymptotic behavior at infinity, but it eliminates the contribution of the pole at 𝜔𝜔 , thus giving rise to an over-completeness relation.
4. Numerical results . As pointed out earlier, the zeros 𝜔𝜔 𝑞𝑞 of the characteristic function 𝐹𝐹(𝜔𝜔) appearing in Eq.(17) must be confined to the lower-half of the complex 𝜔𝜔 -plane. This is because, when the incident beam is removed, the time-dependence factor exp(−i𝜔𝜔 𝑞𝑞 𝑡𝑡) of the corresponding leaky modes inside and outside the cavity can only decrease with time. Also, considering that 𝜇𝜇(−𝜔𝜔 𝑞𝑞∗ ) = 𝜇𝜇 ∗ (𝜔𝜔 𝑞𝑞 ) , and 𝜇𝜇(−𝜔𝜔 𝑞𝑞∗ ) = 𝜇𝜇 ∗ (𝜔𝜔 𝑞𝑞 ) , and 𝑛𝑛(−𝜔𝜔 𝑞𝑞∗ ) = 𝑛𝑛 ∗ (𝜔𝜔 𝑞𝑞 ) , the zeros of 𝐹𝐹(𝜔𝜔) always appear in pairs such as 𝜔𝜔 𝑞𝑞 and −𝜔𝜔 𝑞𝑞∗ . Consequently, leaky frequencies appear in the third and fourth quadrants of the 𝜔𝜔 -plane as mirror images of each other. Trivial leaky modes occur at 𝜔𝜔 𝑞𝑞 = Ω and Ω (with their twins occurring at −𝜔𝜔 𝑞𝑞∗ = Ω and Ω ), where 𝑛𝑛(Ω ) = 0 . Substitution into Eqs.(3)- (10) reveals that, for these trivial leaky 0 modes, which are associated with the zeros of the refractive index 𝑛𝑛(𝜔𝜔) , both 𝐸𝐸 and 𝐻𝐻 fields inside and outside the cavity vanish. Finally, referring to the complex 𝜔𝜔 -plane of Fig.1, note that when 𝜔𝜔 crosses (i.e., moves from immediately above to immediately below) one of the branch-cuts, 𝑛𝑛(𝜔𝜔) gets multiplied by −1 , which causes 𝐹𝐹(𝜔𝜔) of Eq.(17) to be multiplied by ±i (depending on the value of ℓ being even or odd). The contour plots in Fig.2 show, within two segments of the fourth quadrant of the 𝜔𝜔 -plane, the zeros of Re[𝐹𝐹(𝜔𝜔)] in red (solid) lines and the zeros of
Im[𝐹𝐹(𝜔𝜔)] in blue (dashed) lines. Both
Re(𝜔𝜔) and
Im(𝜔𝜔) are normalized by the (arbitrarily chosen) reference frequency 𝜔𝜔 ref =1.216 × 10 rad/sec, which corresponds to the free-space wavelength 𝜆𝜆 ref = 1.55 𝜇𝜇𝑚𝑚 . The chosen value of ℓ for the plots of Fig.2 is , the dielectric sphere has radius 𝑅𝑅 = 1.55 𝜇𝜇𝑚𝑚 , permeability 𝜇𝜇(𝜔𝜔) = 1.0 , and the refractive index, 𝑛𝑛(𝜔𝜔) = �𝜇𝜇(𝜔𝜔) , is governed by a single Lorentz oscillator having 𝜔𝜔 𝑟𝑟 = 2𝜔𝜔 ref , 𝜔𝜔 𝑝𝑝 = 5𝜔𝜔 ref , and 𝛾𝛾 = 0.02𝜔𝜔 ref . The 4 th quadrant pole and zero of 𝑛𝑛(𝜔𝜔) are thus located at Ω ≅ (2.0 − 0.01i)𝜔𝜔 ref and Ω ≅ (5.385 − 0.01i)𝜔𝜔 ref , respectively. The parameter values chosen here do not necessarily represent a realistic cavity such as a fused silica micro-sphere. Nevertheless, we have chosen these values with the following illustration in mind. Despite being artificial, they preserve the “topology” of the resonant pole distribution in the 𝜔𝜔 -plane, while allowing a reasonable visualization. The small size of the cavity, together with a strongly lossy and dispersive medium, effectively isolates the important features that we would like to show. Fig.2 . Contours in the complex 𝜔𝜔 -plane representing regions where Re[𝐹𝐹(𝜔𝜔)] = 0 (solid red lines) and
Im[𝐹𝐹(𝜔𝜔)] = 0 (dashed blue lines). The real and imaginary axes are normalized by the reference frequency 𝜔𝜔 ref = 1.216 × 10 rad/sec. Where a solid red and a dashed blue curve cross, 𝐹𝐹(𝜔𝜔) vanishes; these crossing points (some of them marked with small circles) correspond to TE leaky-mode frequencies 𝜔𝜔 𝑞𝑞 = 𝜔𝜔 𝑞𝑞′ + i𝜔𝜔 𝑞𝑞″ of the spherical cavity at the chosen value of ℓ = 10 . The spherical particle has radius 𝑅𝑅 = 1.55 𝜇𝜇𝑚𝑚 , permeability 𝜇𝜇(𝜔𝜔) = 1.0 , and refractive index 𝑛𝑛(𝜔𝜔) = �𝜇𝜇(𝜔𝜔) governed by a single Lorentz oscillator. The 4 th quadrant pole and zero of 𝑛𝑛(𝜔𝜔) are, respectively, at Ω ≅ (2.0 − 0.01i)𝜔𝜔 ref and Ω ≅ (5.385 − 0.01i)𝜔𝜔 ref . The points where the contours depicted in Fig.2 cross each other — several crossing points are circled in the plot — represent the zeros of
𝐹𝐹(𝜔𝜔) , which we have denoted by 𝜔𝜔 𝑞𝑞 = 𝜔𝜔 𝑞𝑞′ + i𝜔𝜔 𝑞𝑞″ and referred to as leaky-mode frequencies. The region of the 𝜔𝜔 -plane depicted in Fig.2(a) contains the 4 th quadrant leaky-mode frequencies to the left of Ω ; a large number of such frequencies are seen to accumulate in the vicinity of 𝜔𝜔 = Ω , where the coupling of the incident light to the cavity is weak, and the damping within the sphere is dominated by absorption losses. (a) (b) Re(𝜔𝜔 𝜔𝜔 ref ⁄ ) Re(𝜔𝜔 𝜔𝜔 ref ⁄ ) I m ( 𝜔𝜔𝜔𝜔 r e f ⁄ ) I m ( 𝜔𝜔𝜔𝜔 r e f ⁄ ) The region of the 𝜔𝜔 -plane depicted in Fig.2(b) contains the 4 th quadrant leaky-mode frequencies to the right of Ω . The imaginary part 𝜔𝜔 𝑞𝑞″ of these leaky frequencies is seen to acquire large negative values as the corresponding real part 𝜔𝜔 𝑞𝑞′ increases. No leaky frequencies were found in the upper-half of the 𝜔𝜔 -plane, nor were there any in the strip between Ω and Ω . As mentioned earlier, symmetry considerations ensure that the poles in the third and fourth quadrants are mirror-images of each other. As will be seen shortly, when the dielectric sphere is illuminated with a real-valued excitation frequency 𝜔𝜔 , resonances occur in the vicinity of 𝜔𝜔 = 𝜔𝜔 𝑞𝑞′ , i.e., at and around the real parts of the various leaky mode frequencies. Note that the leftmost zero-crossing shown in Fig.2(b) represents a zero of the refractive index 𝑛𝑛(𝜔𝜔) , which has multiplicity equal to the order of the Bessel function associated with the modal field. However, this zero of the function 𝐹𝐹(𝜔𝜔) is cancelled out by the numerator of
𝐺𝐺(𝜔𝜔) , as can be readily seen by expanding in the vicinity of the complex zero of 𝑛𝑛(𝜔𝜔) . As such, the leftmost zero-crossing in Fig.2(b) does not contribute to the leaky-mode expansion. To investigate the resonant behavior of the dielectric sphere described in conjunction with Fig.2, we pick a real-valued frequency 𝜔𝜔 , then select a mode consisting of incoming and outgoing Hankel functions outside the sphere, matched to a Bessel function of the first kind residing inside. The resulting equations do not depend on the azimuthal mode number 𝑚𝑚 , which indicates that, for a given integer ℓ , the modes associated with all values of 𝑚𝑚 between −ℓ and ℓ are degenerate. Figure 3 shows the computed amplitude ratio of the 𝐸𝐸 -field inside the sphere to the incident 𝐸𝐸 -field, plotted versus the normalized excitation frequency 𝜔𝜔/𝜔𝜔 ref . Here the 𝐸𝐸 -field amplitude is defined as the magnitude of 𝐸𝐸 in Eq.(5). As before, 𝑅𝑅 = 1.55 𝜇𝜇𝑚𝑚 , 𝜇𝜇(𝜔𝜔) = 1.0 , 𝜇𝜇(𝜔𝜔) follows a single Lorentz oscillator model (𝜔𝜔 𝑟𝑟 = 2𝜔𝜔 ref , 𝜔𝜔 𝑝𝑝 = 5𝜔𝜔 ref , 𝛾𝛾 = 0.02𝜔𝜔 ref ) , and the selected TE mode has ℓ = 10 . In the interval [Ω , Ω ] between the pole and zero of the refractive index (see Fig.1), the field amplitude inside the cavity is seen to be vanishingly small. Outside this “forbidden” zone, the field has resonance peaks at specific frequencies, and the ratio 𝐸𝐸 inside 𝐸𝐸 incident ⁄ can vary significantly between adjacent peaks and valleys. For the chosen set of parameters in Fig.3, the minimum resonance frequency occurs at 𝜔𝜔 ≅ 0.76253𝜔𝜔 ref . Fig.3 . Logarithmic plot of the ratio of the 𝐸𝐸 -field inside the sphere to the incident 𝐸𝐸 -field, for a dielectric sphere of radius 𝑅𝑅 = 1.55 𝜇𝜇𝑚𝑚 at ℓ = 10 . The horizontal axis represents the normalized excitation frequency 𝜔𝜔 𝜔𝜔 ref ⁄ . The refractive index 𝑛𝑛(𝜔𝜔) = �𝜇𝜇(𝜔𝜔) of the spherical particle is governed by a single Lorentz oscillator. The 4 th quadrant pole and zero of 𝑛𝑛(𝜔𝜔) are at Ω ≅ (2.0 − 0.01i)𝜔𝜔 ref and Ω ≅(5.385 − 0.01i)𝜔𝜔 ref , respectively. The EM field hardly penetrates the dielectric sphere in the frequency interval between the pole and zero of 𝑛𝑛(𝜔𝜔) . Outside this “forbidden” interval, the 𝐸𝐸 -field amplitude ratio exhibits sharp peaks at certain frequencies, which is indicative of resonant behavior. A comparison of Fig.2 with Fig.3 reveals a close relationship between the leaky mode frequencies and the resonances of the dielectric sphere. Resonances occur at or near the (real-valued) frequencies 𝜔𝜔 = 𝜔𝜔 𝑞𝑞′ , and the height and width of a resonance line are, by and large, determined by the decay rate 𝜔𝜔 𝑞𝑞″ of the corresponding leaky mode — unless the leaky mode frequency happens to be so close to the pole(s) of the refractive index 𝑛𝑛(𝜔𝜔) that the strong absorption within the medium would suppress the resonance. It must be emphasized that the presence of a gap in the frequency domain (such as that between 𝜔𝜔 = Re(Ω ) and 𝜔𝜔 = 𝜔𝜔 𝜔𝜔 ref ⁄ l o g ( 𝐸𝐸 i n s i d e 𝐸𝐸 i n c i d e n t ⁄ ) Re(Ω ) in the present example) should not prevent the leaky modes from forming a complete basis. This is because one expects, on physical grounds, that the ensemble of leaky modes would carry all the spatial frequencies needed to capture the various features of an arbitrary initial EM field distribution. For leaky TE modes, the radial dependence of the 𝐸𝐸 -field inside and outside the dielectric sphere are given by 𝐸𝐸 𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟) √𝑘𝑘𝑟𝑟⁄ and 𝐸𝐸 ℋ ℓ+½(1) (𝑘𝑘 𝑟𝑟) �𝑘𝑘 𝑟𝑟� , respectively. Here 𝑘𝑘(𝜔𝜔) =𝑛𝑛(𝜔𝜔)𝜔𝜔 𝑐𝑐⁄ and 𝑘𝑘 (𝜔𝜔) = 𝜔𝜔 𝑐𝑐⁄ . Plots of the 𝐸𝐸 -field amplitude for several leaky ℓ = 10 TE modes of a sphere of radius
𝑅𝑅 = 1.55 𝜇𝜇𝑚𝑚 are shown in Fig.4. The refractive index of the dielectric material at the reference frequency 𝜔𝜔 ref = 1.216 × 10 rad/sec is 𝑛𝑛(𝜔𝜔 ref ) = 3.055 + 0.0091i . The fields are plotted as functions of the normalized radial coordinate 𝑟𝑟/𝑅𝑅 , with frames (a) and (b) depicting the real and imaginary components of the 𝐸𝐸 -field. The solid (black) curve, the dashed (red) curve, and the dash-dotted (blue) curve correspond to 𝜔𝜔 𝜔𝜔 ref ⁄ = 0.76253 +0.00128i , , and , respectively. Figure 5 provides a comparison between the 𝐸𝐸 -field inside a spherical cavity and its expansion in terms of the leaky-modes ( 𝑅𝑅 = 1.55 𝜇𝜇𝑚𝑚 , ℓ = 10 TE mode, 𝜔𝜔 = 1.8𝜔𝜔 ref ). The solid black and solid red lines show, respectively, the real and imaginary parts of the target solution, 𝐸𝐸 inside (𝑟𝑟) , whereas the symbols superposed on these solid lines represent the leaky-mode expansion, ∑ 𝐸𝐸 leaky (𝑟𝑟) 𝑞𝑞 , of the target function composed of 100 terms. The convergence is seen to be rather poor near the surface of the sphere (0.9 < 𝑟𝑟 𝑅𝑅⁄ < 1.0) . Fig.4 . The 𝐸𝐸 -field amplitude inside and outside a solid dielectric sphere of radius 𝑅𝑅 = 1.55 𝜇𝜇𝑚𝑚 , plotted versus the normalized radial coordinate 𝑟𝑟/𝑅𝑅 for three ℓ = 10
TE modes. The real part of the field is shown in (a), while its imaginary part appears in (b). The solid black, dashed red, and dash-dotted blue curves correspond, respectively, to 𝜔𝜔 𝜔𝜔 ref ⁄ = 0.76253 + 0.00128i , , and ( 𝜔𝜔 ref = 1.216 × 10 rad/sec). The particle, whose permeability is 𝜇𝜇(𝜔𝜔) = 1.0 , has a refractive index 𝑛𝑛(𝜔𝜔) = �𝜇𝜇(𝜔𝜔) governed by a single Lorentz oscillator ( 𝜔𝜔 𝑟𝑟 = 2𝜔𝜔 ref , 𝜔𝜔 𝑝𝑝 = 5𝜔𝜔 ref , and 𝛾𝛾 = 0.02𝜔𝜔 ref ). The error of the leaky-mode expansion depicted in Fig.5 (superposed on the function being expanded) illustrates that the gap between the expansion and its target function, while indiscernible at small radii, grows in the vicinity of the boundary of the cavity. This behavior is generic, and a manifestation of the fact that our leaky-mode expansion converges rather slowly. In fact, adding hundreds or even thousands of terms to the expansion only results in a minuscule reduction in the residual error. We have traced this behavior to the fact that the terms in the expansion do not enter their asymptotic regime until their count is on the order of . The (a) (b) 3 practical consequence here is that, while a good approximation can be achieved with a fairly small number of terms, suppressing the error below a few parts in a thousand becomes utterly impractical. That being said, one should keep in mind that the expansion error is due primarily to those basis functions that decay rapidly upon termination of the excitation. In other words, due to the large imaginary parts 𝜔𝜔 𝑞𝑞″ of their eigen-frequencies, the contribution of high-order leaky modes will disappear almost instantly once the excitation is terminated. If, for some applications, accuracy beyond a few parts in a thousand turns out to be necessary, it is worth noting that, with the asymptotic information about convergence rates that can easily be determined for these series, it is highly likely that convergence accelerating re-summation methods can be deployed.
Fig.5 . Comparison of the 𝐸𝐸 -field inside a spherical cavity with its expansion as a superposition of leaky modes ( 𝑅𝑅 = 1.55 𝜇𝜇𝑚𝑚 , ℓ = 10 TE mode, 𝜔𝜔 = 1.8𝜔𝜔 ref ). The real and imaginary parts of 𝐸𝐸 inside (𝑟𝑟) are shown as solid lines — black and red (grey), respectively. The superposed symbols (i.e., small solid circles) represent the result of leaky-mode expansion, ∑ 𝐸𝐸 leaky (𝑟𝑟) 𝑞𝑞 , composed of 100 terms. For the remaining set of figures, we shall ignore the dispersive nature of the dielectric host and simply assume that 𝜇𝜇(𝜔𝜔) = 1.0 and 𝜇𝜇(𝜔𝜔) = 2.25 at and around the reference frequency 𝜔𝜔 ref = 1.216 × 10 rad sec⁄ (corresponding to the vacuum wavelength 𝜆𝜆 ref = 1.55 𝜇𝜇𝑚𝑚 ). This is tantamount to confining the frequency range of interest to Ω ≪ 𝜔𝜔 ≪ Ω . Unlike the previous example in which the parameter selection was driven by the visualization needs, the parameter values in the following examples are comparable to those found in actual experiments [1,2]. Figure 6 shows the resonances of a dielectric sphere of radius 𝑅𝑅 = 50𝜆𝜆 ref and refractive index 𝑛𝑛 = 1.5 for the ℓ = 340
TE and TM modes. The contours of real and imaginary parts of the characteristic equation
𝐹𝐹(𝜔𝜔) = 0 have been plotted in the 𝜔𝜔 -plane, as was done for a different set of parameters in Fig.2. Where the contours cross each other, the function 𝐹𝐹(𝜔𝜔) vanishes, indicating the existence of a leaky mode at the crossing frequency 𝜔𝜔 𝑞𝑞 = 𝜔𝜔 𝑞𝑞′ + i𝜔𝜔 𝑞𝑞″ . The ratio |𝜔𝜔 𝑞𝑞′ 𝜔𝜔 𝑞𝑞″ ⁄ | is a measure of the 𝑄𝑄 -factor of the spherical cavity at (or near) the excitation frequency 𝜔𝜔 = 𝜔𝜔 𝑞𝑞′ . Fig. 6 . Computed 𝑄𝑄 -factor versus the resonance frequency for a solid dielectric sphere ( 𝑅𝑅 = 77.5 𝜇𝜇𝑚𝑚 , 𝜇𝜇 = 1 , 𝑛𝑛 = 1.5 ) in the vicinity of 𝜔𝜔 ref = 1.216 × 10 rad sec⁄ . The leaky frequencies 𝜔𝜔 𝑞𝑞 = 𝜔𝜔 𝑞𝑞′ + i𝜔𝜔 𝑞𝑞″ are solutions of 𝐹𝐹(𝜔𝜔) = 0 , which have been found numerically. The ratio |𝜔𝜔 𝑞𝑞′ 𝜔𝜔 𝑞𝑞″ ⁄ | is used as a measure of the cavity 𝑄𝑄 -factor at the excitation frequency 𝜔𝜔 = 𝜔𝜔 𝑞𝑞′ . Shown are computed 𝑄𝑄 -factors for both TE and TM modes (solid blue squares for TE, open red circles for TM) at several resonance frequencies of the dielectric sphere corresponding to ℓ = 340 . Shown in Fig.6 are the computed 𝑄𝑄 -factors of the spherical cavity for both TE and TM modes at the various resonance frequencies corresponding to ℓ = 340 . (Note that the characteristic equation does not depend on 𝑚𝑚 , which indicates that, for a given integer ℓ , the modes associated with all 𝑚𝑚 between −ℓ and ℓ are degenerate.) The lowest resonance frequency r / R E i n s i d e ( r ) , E e xp a nd e d ( r ) 𝜔𝜔 𝜔𝜔 ref ⁄ l o g ( 𝑄𝑄 ) occurs at 𝜔𝜔 ≅ 0.78𝜔𝜔 ref . The large values of 𝑄𝑄 seen in Fig.6 are a consequence of the fact that the refractive index 𝑛𝑛 is assumed to be purely real; later, when absorption is incorporated into the model via the imaginary part of 𝑛𝑛 , the 𝑄𝑄 -factors will drop to more reasonable values. The direct method of determining the resonances of the spherical cavity involves the computation of the amplitude ratio 𝐸𝐸 inside /𝐸𝐸 incident for an incident Hankel function of type 2 (incoming wave) and a fixed mode number ℓ . (As pointed out earlier, the amplitude of each 𝐸𝐸 -field is defined as the magnitude of the corresponding 𝐸𝐸 in Eq.(5), with the radial dependence of the inside field being given in terms of the Bessel function 𝐽𝐽 ℓ+½ (𝑘𝑘𝑟𝑟) , while that of the incident field outside the sphere involves the Hankel function ℋ 𝜈𝜈(2) (𝑘𝑘 𝑟𝑟) . ) Once again, the results are independent of the azimuthal mode number 𝑚𝑚 , as the modes associated with 𝑚𝑚 = −ℓ to ℓ are all degenerate. Figure 7 shows plots of 𝐸𝐸 inside /𝐸𝐸 incident for the spherical cavity of radius 𝑅𝑅 = 50𝜆𝜆 ref , refractive index 𝑛𝑛 = 1.5 , and mode number ℓ = 340 , at and around 𝜔𝜔 ref = 1.216 × 10 rad sec⁄ ; the results for both TE and TM modes are presented in the figure. The resonances are seen to be strong, with narrow linewidths. Fig. . Plots of the amplitude ratio of the 𝐸𝐸 -field inside the dielectric sphere (𝑅𝑅 = 77.5 𝜇𝜇𝑚𝑚, 𝜇𝜇 = 1, 𝑛𝑛 = 1.5) to the incident 𝐸𝐸 -field for the ℓ = 340 spherical harmonic. The horizontal axis represents the excitation frequency 𝜔𝜔 normalized by 𝜔𝜔 ref = 1.216 × 10 rad sec⁄ . (a) TE mode. (b) TM mode. Note that the cutoff frequency for both modes is 𝜔𝜔 ≅ 0.78𝜔𝜔 ref , below which no resonances are excited. Above the cutoff, in between adjacent resonances, the field amplitude inside the cavity drops to exceedingly small values. The occurrence of extremely large resonance peaks in these plots is due to the assumed value of the refractive index 𝑛𝑛 being purely real. (c) Close-up view of the resonance lines of the glass ball for the ℓ = 340 spherical harmonic, showing the TM resonances (dashed red lines) being slightly shifted away from the TE resonances (solid black lines). (d) Magnified view of an individual TE resonance line centered at 𝜔𝜔 𝑅𝑅 = 1.00207𝜔𝜔 ref . (a) (b) (d) (c) l o g ( 𝐸𝐸 i n s i d e 𝐸𝐸 i n c i d e n t ⁄ ) l o g ( 𝐸𝐸 i n s i d e 𝐸𝐸 i n c i d e n t ⁄ ) | 𝐸𝐸 i n s i d e 𝐸𝐸 i n c i d e n t ⁄ | 𝜔𝜔 𝜔𝜔 ref ⁄ (𝜔𝜔 − 𝜔𝜔 𝑅𝑅 ) 𝜔𝜔 ref ⁄ 𝜔𝜔 𝜔𝜔 ref ⁄ 𝜔𝜔 𝜔𝜔 ref ⁄ −2 × 10 −8
0 2 × 10 −8 l o g ( 𝐸𝐸 i n s i d e 𝐸𝐸 i n c i d e n t ⁄ ) Outside the resonance peaks and especially at lower frequencies, it is seen that the coupling of the incident beam to the cavity is extremely weak. The TE and TM modes are quite similar in their coupling efficiencies and resonant line-shapes, their major difference being the slight shift of TM resonances toward higher frequencies, as can be seen in Fig.7(c). Figure 7(d) is a magnified view of the line-shape for a single TE resonant line centered at 𝜔𝜔 = 1.00207𝜔𝜔 ref . To gain an appreciation for the effect of the mode number ℓ on the resonant behavior of our spherical cavity, we show in Fig.8 the computed ratio 𝐸𝐸 inside /𝐸𝐸 incident for ℓ = 10, 20 and . It is observed that, with an increasing mode number ℓ , the lowest accessible resonance moves to higher frequencies, and that the 𝑄𝑄 -factor associated with individual resonance lines tends to rise. Fig. 8 . Excitation frequency dependence of the ratio of the 𝐸𝐸 -field inside a glass sphere to the incident 𝐸𝐸 -field for ℓ = 10 (solid black), ℓ = 20 (dashed blue), and ℓ = 25 (dash-dotted red) TE spherical harmonics. Finally, Fig.9 shows computed 𝑄𝑄 -factors (𝑄𝑄 = |𝜔𝜔 𝑞𝑞′ 𝜔𝜔 𝑞𝑞″ ⁄ |) for a spherical cavity having 𝑅𝑅 =77.5 µm , 𝜇𝜇 = 1.0 , 𝑛𝑛 = 𝑛𝑛 ′ + i𝑛𝑛 ″ , and ℓ = 340 . Setting 𝑛𝑛 ′ = 1.5 allows a comparison between the results depicted in Fig.6, where 𝑛𝑛 ″ = 0 , and those in Fig.9, which correspond to 𝑛𝑛 ″ = 10 −8 (blue squares), −7 (red circles), and −6 (black diamonds). These positive values of 𝑛𝑛 ″ account for the presence of small amounts of absorption within the dielectric sphere. Compared to the case of 𝑛𝑛 ″ = 0 , the resonance frequencies in Fig.9 have not changed by much, but the 𝑄𝑄 -factors of the various resonances are seen to have declined substantially. As expected, the greatest drop in the 𝑄𝑄 -factor is associated with the largest value of 𝑛𝑛 ″ . This is a practically interesting finding, especially in light of the previous result on the 𝑄𝑄 -factors of an idealized cavity, which could reach exceedingly high values. Here we see that accounting for realistic values of optical loss brings down the computed 𝑄𝑄 -factor to the levels observed in experiments [1,2]. This also indicates that the limiting factor in the best spherical resonators available today is most likely the medium properties rather than roughness and other cavity imperfections as one might reasonably presume. Considering that the measured absorption coefficients (e.g., 𝑛𝑛″ ≅10 −7 for a fused silica micro-sphere in the visible optical range) are comparable to the theoretical values of 𝑛𝑛″ needed to bring the 𝑄𝑄 -factor of a perfectly spherical dielectric resonator to within the range of the highest 𝑄𝑄 -factors that are currently accessible to experiments, it is reasonable to conclude that the 𝑄𝑄 -factor-limiting physical effect is in fact absorption within the micro-sphere. Needless to say, scattering from surface roughness and also deleterious effects of inclusions, impurities, and material inhomogeneities could result in mode-mixing, which causes further reduction of the 𝑄𝑄 -factor. Nevertheless, the purity and the polish quality of existing dielectric micro-spheres are such that their observed 𝑄𝑄 -factors indeed appear to be limited by the absorption coefficient 𝑛𝑛″ of the host material. ℓ = 10 ℓ = 20 ℓ = 25 𝜔𝜔 𝜔𝜔 ref ⁄ l o g ( 𝐸𝐸 i n s i d e 𝐸𝐸 i n c i d e n t ⁄ ) 𝜔𝜔 𝜔𝜔 ref ⁄ l o g ( 𝑄𝑄 ) Fig. 9 . Similar to Fig. 6, except that the refractiv e index 𝑛𝑛 = 𝑛𝑛 ′ + i𝑛𝑛 ″ of the dielectric sp here is now allowed to have a small nonzero imaginary part, 𝑛𝑛 ″ , representing absorption within the material.
5. Concluding remarks . Leaky modes contain a wealth of information about the resonant behavior of dielectric cavities, including the lifetimes associated with the light trapped inside the cavity immediately after the source of excitation is turned off. Listed below is a summary of the main results of the present paper, with emphasis placed not only on mathematical aspects but also on the physical attributes of our findings. 1. A dielectric or metallic sphere, when illuminated from the outside or excited internally, contains EM fields. Once the excitation is terminated, the trapped fields inside the sphere decay by leaking out and/or by being absorbed within the sphere. We have identified the complete set of leaky modes, and shown the conditions under which a trapped field can be expressed as a superposition of these leaky modes. 2. We have proven the completeness of these leaky modes under special circumstances, although completeness under more general conditions remains to be demonstrated. We have modelled the dielectric function 𝜇𝜇(𝜔𝜔) = 𝑛𝑛 (𝜔𝜔) of the spherical particle as a single Lorentz oscillator, thereby treating dispersion and absorption of the material medium in a simple yet physically realistic way. While we have assumed that the sphere is surrounded by free space, the results can be readily extended to the case of a surrounding dielectric medium. 3. Our completeness proof rigorously accounts for realistic dispersion effects, including absorption losses, the existence of branch-cuts associated with the Lorentz oscillator model of the refractive index, and the fact that infinitely many complex poles accumulate in the vicinity of the singular point(s) of the refractive index. 4. We did not invoke the Green function (or tensor) that has been traditionally used to analyze this type of problem. Instead, we relied on the exact solutions of Maxwell’s equations to identify the leaky modes, then constructed the modal expansion of an initial field distribution using a straightforward application of the Cauchy theorem of complex analysis; see, e.g., Eq.(19). The explicit formulas derived here for the expansion coefficients allow easy evaluation of the relative contributions to an arbitrary initial distribution (inside the spherical particle) of the various leaky modes; see Eq.(23). 5. With regard to the conventional Green’s function approach, we note that one can certainly rewrite Maxwell’s equations into an integral equation, and the boundary conditions at infinity are carried by the choice of the Green function. Usually the waves at infinity are either outgoing or incoming, since, for these boundary conditions, Green’s functions are easy to find. However, for the specific goal of obtaining leaky-mode expansions of the fields, one also needs to find the leaky modes, then express Green’s functions as sums of the leaky modes. This can certainly be done at a formal level, but there are two problems that have to be faced. The first is to find an effective way to calculate the coefficients of the expansion; this, in general, is not a simple problem, and numerical methods might have to be deployed. The second is that the question of convergence of the resulting expansion must be treated separately, as there is nothing in the Green function formulation that would guarantee the convergence of the leaky-mode expansion. To the best of our knowledge, the arguments given in favor of the convergence in the literature do not constitute a rigorous proof, if only because the question of accumulation points of the resonance poles has so far not been analyzed within the Green function framework. 6. As a matter of fact, most works utilizing Green’s functions seem to target primarily applications to cavity perturbations rather than address questions of convergence [8,22-27]. In contrast, the approach taken in the present paper is to i) find the leaky modes, ii) find the explicit 7 expansion coefficients of the functions of interest with respect to the modes, and iii) decide on the convergence of the series. Solving Maxwell’s equations using scattering boundary conditions in conjunction with Cauchy’s theorem addresses the three aforementioned goals in a well-designed, easy-to-use package. Ours is a highly flexible approach in which the design of the leaky modes and the corresponding expansion coefficients are guided by the question of convergence. In fact, rather than being some late-comer to the game, in our approach convergence is actually a design tool. 7. Our numerical results have intimated a close association between resonant behavior and the leaky eigen modes of dielectric spheres. The fact that spherical harmonics with large ℓ values are associated with high- 𝑄𝑄 resonances hints at the importance of electromagnetic angular momentum in relation to the long lifetimes of the modes trapped inside these cavities. In other words, there appears to be a connection between the strength of the circular motion of EM energy inside a cavity and the time it takes for this energy to leak out. We have seen a similar relation between the azimuthal mode number 𝑚𝑚 and the cavity 𝑄𝑄 -factor in the case of cylindrical cavities [15]. In fact, when the radius 𝑅𝑅 and the refractive index 𝑛𝑛 of a dielectric cylinder are the same as those of a sphere, and when the mode number 𝑚𝑚 for the cylinder is the same as the mode number ℓ for the sphere (𝑚𝑚 = ℓ ≫ 1) , the plots of 𝑄𝑄 -factor versus resonance frequency for the two cavities are found to be nearly identical. 8. Leaky modes are often characterized as “unphysical” because they seem to carry infinite energy. We have emphasized that the EM field distribution outside the sphere grows exponentially with radial distance, while decaying exponentially with time. The exponential growth with distance, however, is not unphysical, because the fields only extend to a distance 𝑟𝑟 = 𝑐𝑐𝑡𝑡 from the sphere’s surface, where 𝑡𝑡 is the time elapsed since the external/internal excitation of the spherical particle was terminated. Considering that the leaky modes exist only after the termination of the excitation, the outer tails of the leaky modes within the surrounding medium do not extend to infinity and, therefore, the well-known exponential growth of the field amplitude with distance does not constitute a violation of the law of conservation of energy. (Note that the situation discussed here is completely analogous to that in quantum mechanics; see, e.g., [28].) 9. In Fig.2, we presented a typical map of the leaky frequencies 𝜔𝜔 𝑞𝑞 in the complex 𝜔𝜔 -plane, and drew attention to the singular points of this map, which are located at the pole(s) and zero(s) of the refractive index 𝑛𝑛(𝜔𝜔) of the spherical particle. It must be emphasized that, when the excited field has a frequency close to the pole(s) of the refractive index, there will be a large number of closely-spaced leaky frequencies that must be included in any physically meaningful expansion of the initial field distribution. 10. We have provided several numerical examples, some with artificial parameter values to emphasize the mathematical aspects of the leaky mode expansion (e.g., Figs. 2-5), and some with physically realistic parameter values (e.g., Figs. 6-9) in order to draw attention to the behavior of leaky modes in problems of practical interest. 11. Finally, it is interesting to note that small amounts of absorption or loss can dramatically suppress the 𝑄𝑄 -factors of a solid dielectric sphere at large ℓ and in the vicinity of the cutoff frequency, as revealed by a comparison between Fig.6 and Fig.9. This finding indicates that the 𝑄𝑄 -factors occurring in practice might be actually limited by the material properties rather than the particle’s surface quality. 8 In conclusion, the present paper has described a general approach to analyzing and computing the leaky modes of solid dielectric spheres. In Sec.3, we presented the outlines of a completeness proof for expanding certain initial field distributions as a sum over leaky modes. Mathematical details and some of the subtleties associated with the series convergence were either skipped over or mentioned only briefly. These subtleties, which revolve around the behavior of the accumulated poles of the function
𝐺𝐺(𝜔𝜔) of Eq.(18) when 𝜔𝜔 approaches the poles Ω of the refractive index 𝑛𝑛(𝜔𝜔) , and also when 𝜔𝜔 → ∞ , will be the subject of a forthcoming mathematics-oriented paper. Acknowledgement . This work has been supported in part by the AFOSR grant No. FA9550-13-1-0228.
Appendix
We show that
𝐹𝐹(𝜔𝜔) of Eq.(17) approaches a constant when 𝜔𝜔 → 0 . In the limit 𝑧𝑧 → 0 , we have 𝐽𝐽 𝜈𝜈 (𝑧𝑧) → (𝜋𝜋 2⁄ ) 𝜈𝜈 Γ(𝜈𝜈+1) · (A1) 𝑌𝑌 𝜈𝜈 (𝑧𝑧) → (𝜋𝜋 2⁄ ) 𝜈𝜈 tan(𝜈𝜈𝜋𝜋)Γ(1+𝜈𝜈) − (𝜋𝜋 2⁄ ) −𝜈𝜈 sin(𝜈𝜈𝜋𝜋)Γ(1−𝜈𝜈) ; (𝜈𝜈 ≠ an integer) . (A2) Therefore, when 𝜔𝜔 → 0 , considering that 𝑘𝑘 = 𝜔𝜔 𝑐𝑐⁄ → 0 , we will have 𝐹𝐹(𝜔𝜔) = 𝑛𝑛𝑘𝑘 𝑅𝑅ℋ ℓ+½(1) (𝑘𝑘 𝑅𝑅)𝐽𝐽 ℓ+3 2⁄ (𝑛𝑛𝑘𝑘 𝑅𝑅) + [(𝜇𝜇 − 1)(ℓ + 1)ℋ ℓ+½(1) (𝑘𝑘 𝑅𝑅) − 𝜇𝜇𝑘𝑘 𝑅𝑅ℋ ℓ+3 2⁄(1) (𝑘𝑘 𝑅𝑅)] 𝐽𝐽 ℓ+½ (𝑛𝑛𝑘𝑘 𝑅𝑅) = 𝑛𝑛𝑘𝑘 𝑅𝑅𝐽𝐽 ℓ+½ (𝑘𝑘 𝑅𝑅)𝐽𝐽 ℓ+3 2⁄ (𝑛𝑛𝑘𝑘 𝑅𝑅) + [(𝜇𝜇 − 1)(ℓ + 1)𝐽𝐽 ℓ+½ (𝑘𝑘 𝑅𝑅) − 𝜇𝜇𝑘𝑘 𝑅𝑅𝐽𝐽 ℓ+3 2⁄ (𝑘𝑘 𝑅𝑅)] 𝐽𝐽 ℓ+½ (𝑛𝑛𝑘𝑘 𝑅𝑅) +i𝑛𝑛𝑘𝑘 𝑅𝑅𝑌𝑌 ℓ+½ (𝑘𝑘 𝑅𝑅)𝐽𝐽 ℓ+3 2⁄ (𝑛𝑛𝑘𝑘 𝑅𝑅) + i[(𝜇𝜇 − 1)(ℓ + 1)𝑌𝑌 ℓ+½ (𝑘𝑘 𝑅𝑅) − 𝜇𝜇𝑘𝑘 𝑅𝑅𝑌𝑌 ℓ+3 2⁄ (𝑘𝑘 𝑅𝑅)] 𝐽𝐽 ℓ+½ (𝑛𝑛𝑘𝑘 𝑅𝑅) → i(−1)ℓ+1𝑛𝑛𝑘𝑘 𝑅𝑅Γ(½−ℓ)Γ(ℓ+5 2⁄ ) (½𝑘𝑘 𝑅𝑅) −(ℓ+½) (½𝑛𝑛𝑘𝑘 𝑅𝑅) ℓ+3 2⁄ + iΓ(ℓ+3 2⁄ ) � (−1) ℓ+1 (𝜇𝜇−1)(ℓ+1)Γ(½−ℓ) (½𝑘𝑘 𝑅𝑅) −(ℓ+½) − (−1) ℓ 𝜇𝜇𝑘𝑘 𝑅𝑅Γ(−½−ℓ) (½𝑘𝑘 𝑅𝑅) −(ℓ+3 2⁄ ) � (½𝑛𝑛𝑘𝑘 𝑅𝑅) ℓ+½ → i(−1) ℓ+1 𝑛𝑛ℓ+½Γ(ℓ+3 2⁄ ) � (𝜇𝜇−1)(ℓ+1)Γ(½−ℓ) + � = i(−1) ℓ [1 + ℓ + ℓ𝜇𝜇(0)][𝑛𝑛(0)] ℓ+½ (ℓ+½)Γ(½ + ℓ)Γ(½ − ℓ) · (A3) The identity Γ(𝑥𝑥 + 1) = 𝑥𝑥Γ(𝑥𝑥) has been used in the above derivation. We may now invoke the identity
Γ(½ + 𝑥𝑥)Γ(½ − 𝑥𝑥) = 𝜋𝜋 cos(𝜋𝜋𝑥𝑥)⁄ to arrive at lim 𝜔𝜔→0
𝐹𝐹(𝜔𝜔) = i��𝜇𝜇(0)𝜇𝜇(0) � ℓ+½ [1 + ℓ + ℓ𝜇𝜇(0)] [(ℓ + ½)𝜋𝜋]� · (A4) It is seen that
𝐹𝐹(𝜔𝜔) has no poles at 𝜔𝜔 = 0 , which indicates that, in the vicinity of 𝜔𝜔 = 0 , the function
𝐺𝐺(𝜔𝜔) is not singular.
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