Uniform approximation to Mie resonances

Uniform asymptotic approximations for the positions and widths of Mie resonances are reported . The results improve the accuracy of previous approximations, typically, by several orders of magnitude . They are suitable for fast numerical evaluation, allowing every resonance to be located well within its width, for all resonances of practical interest . The relative errors are very small even for the lowest angular momenta (size parameters) and they decrease at least as fast as the inverse square of the size parameter . Accuracies of the order of parts per billion for typical microsphere sizes are readily attained . A second-order Wentzel-Kramers-Brillouin approximation is also developed . Though far less accurate, it yields useful analytic estimates of many resonance features . 1 . Introduction The narrow resonances that occur in the interaction of light with dielectric microspheres are finding an increasing number of applications in high-precision particle size and refractive index characterization, nonlinear optics [1], cavity QED in the visible [2], strong localization of photons [3] and optical computing [4] . The physical interpretation of the resonances is well known [5] . Light gets trapped inside a microsphere with relative complex refractive index N (Re N> 1, 0 < Im N<< 1) by multiple internal reflections at its surface, beyond the critical angle . At the geometrical-optic level, neglecting absorption, this would be a bound state of light (whispering-gallery mode) . Wave-optical effects convert it into a long-lived metastable state, with slow leakage outside arising from the frustration of total reflection by the surface curvature . This may also be described as tunnelling through the centrifugal barrier [5] . Conversely, efficient outside excitation of resonant modes takes place by tunnelling (evanescent-wave coupling) . The strong internal field that builds up within the sphere at resonance, concentrated in a layer near its surface, is responsible for the observed nonlinear optical effects, such as lasing [6] and stimulated Raman scattering [7] . For a given sphere radius a, polarization (TE or TM), and multipole order 1, there are usually several resonant modes, characterized by an integer n (mode order) that represents the number of nodes of the internal radial field amplitude in the limit of vanishing leakage; equivalently, n is related to the number of internal radial intensity peaks . It may be likened to the principal quantum number of a bound state . A pictorial representation of the effective radial potential (sum of an attractive square well, associated with the sphere, with the centrifugal barrier), and of the 0950-0340/94 $1000 © 1994 Taylor & Francis Ltd . 626 L. G. Guimaraes and H. M. Nussenzveig