## Geometrical optics model of Mie resonances

JOSA A, Vol. 17, Issue 7, pp. 1301-1311 (2000)

http://dx.doi.org/10.1364/JOSAA.17.001301

Acrobat PDF (405 KB)

### Abstract

The geometrical optics model of Mie resonances is presented. The ray path geometry is given and the resonance condition is discussed with special emphasis on the phase shift that the rays undergo at the surface of the dielectric sphere. On the basis of this model, approximate expressions for the positions of first-order resonances are given. Formulas for the cavity mode spacing are rederived in a simple manner. It is shown that the resonance linewidth can be calculated regarding the cavity losses. Formulas for the mode density of Mie resonances are given that account for the different width of resonances and thus may be adapted to specific experimental situations.

© 2000 Optical Society of America

**OCIS Codes**

(080.0080) Geometric optics : Geometric optics

(080.1510) Geometric optics : Propagation methods

(140.4780) Lasers and laser optics : Optical resonators

(260.5740) Physical optics : Resonance

(290.0290) Scattering : Scattering

(290.4020) Scattering : Mie theory

**Citation**

Günter Roll and Gustav Schweiger, "Geometrical optics model of Mie resonances," J. Opt. Soc. Am. A **17**, 1301-1311 (2000)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-17-7-1301

Sort: Year | Journal | Reset

### References

- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
- S. C. Hill and R. E. Benner, “Morphology-dependant resonances,” in Optical Effects Associated with Small Particles, P. W. Barber and R. K. Chang, eds. (World Scientific, Singapore, 1988).
- R. K. Chang and A. J. Campillo, eds., Optical Processes in Microcavities (World Scientific, Singapore, 1996).
- S. C. Hill and R. K. Chang, “Nonlinear optics in droplets,” in Studies in Classical and Quantum Nonlinear Optics, O. Keller, ed. (Nova Science, New York, 1995).
- H. B. Lin and A. J. Campillo, “Radial profiling of microdroplets using cavity-enhanced Raman spectroscopy,” Opt. Lett. 20, 1589–1591 (1995).
- T. Kaiser, G. Roll, and G. Schweiger, “Investigation of coated droplets in an optical trap: Raman scattering, elastic light scattering and evaporation characteristics,” Appl. Opt. 35, 5918–5924 (1996).
- J. L. Huckaby, A. K. Ray, and B. Das, “Determination of size, refractive index, and dispersion of single droplets from wavelength-dependent scattering spectra,” Appl. Opt. 33, 7112–7125 (1994).
- G. Chen, M. M. Mazumder, R. K. Chang, J. C. Swindal, and W. P. Acker, “Laser diagnostics for droplet characterization: application of morphology dependent resonances,” Prog. Energy Combust. Sci. 22, 163–188 (1996).
- J. B. Keller and S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (New York) 9, 24–75 (1960).
- S. J. Maurer and L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
- G. Roll, T. Kaiser, S. Lange, and G. Schweiger, “Ray interpretation of multipole fields in spherical dielectric cavities,” J. Opt. Soc. Am. A 15, 2879–2891 (1998).
- G. Roll, T. Kaiser, and G. Schweiger, “Eigenmodes of spherical dielectric cavities: coupling of internal and external rays,” J. Opt. Soc. Am. A 16, 882–895 (1999).
- J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
- This statement holds only for light with the frequency of the illuminating plane wave; secondary fields, e.g., excited Raman or fluorescence fields, are not restricted to modes with azimuthal mode number m=1.
- M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1993).
- Throughout this paper we mean the trigonometric expansions of Bessel functions [compare expressions (18) and (19)] when we use the term Debye expansion. This must not be confused with the Debye series expansion of the scattering coefficients of Mie theory. Both the Debye expansion of Bessel functions and the Debye series expansion of the Mie coefficients play an important role in the ray description of the interaction of light with dielectric spheres. The former expansions are approximations to the exact functions, which break down in regions where the argument approaches the order. These expansions are found when the field within an illuminated sphere is investigated by means of geometrical optics.11 The Debye series expansion, on the other hand, denotes a mathematically exact technique to formulate the scattering process in terms of surface interactions of multipole waves. The Debye series expansion is equivalent to Mie theory but shows many similarities to the multiple-interference description of geometrical optics.17
- J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A 5, 2032–2044 (1988).
- N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, London, 1972), p. 126.
- P. Chýlek, “Resonance structure of Mie scattering: distance between resonances,” J. Opt. Soc. Am. A 7, 1609–1613 (1990).
- J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. A 1, 822–830 (1984).
- P. M. Aker, P. A. Moortgat, and J. X. Zhang, “Morphology-dependent stimulated Raman scattering imaging. I. Theoretical aspects,” J. Chem. Phys. 105, 7268–7275 (1996).
- C. C. Lam, P. T. Leung, and K. Young, “Explicit asymptotic formulas for the positions, widths, and strength of resonances in Mie scattering,” J. Opt. Soc. Am. B 9, 1585–1592 (1992).
- B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A 10, 343–352 (1993).
- At first glance this may be surprising, since we always assumed the rays to be confined by total internal reflection, which should mean T=0. However, owing to the curved surface, the energy confinement is not perfect, but there is a so-called evanescent leakage. Consequently the energy loss—although small—is not zero.
- G. Roll, T. Kaiser, and G. Schweiger, “Controlled modification of the expansion order as a tool in Mie computations,” Appl. Opt. 37, 2483–2492 (1998).
- The fact that the transmission coefficient t is complex in the case of total reflection indicates the fact that the evanescent field is phase shifted with respect to the incident wave. The phase shift is half as large as that of the reflected wave.
- A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.