Abstract

The extremely narrow morphology-dependent resonances of a microdroplet seen in Mie scattering are important for the nonlinear optical phenomena in these microdroplets, the large storage time being responsible for optical feedback. A formalism is developed within the scalar wave analog to calculate the change in the widths caused by a small perturbation, such as a distortion of the microdroplet shape from perfect sphericity. This formalism may be viewed as the generalization of the usual Rayleigh–Schrödinger perturbation scheme to the imaginary part of the energy. The results are checked against T-matrix calculations. In particular, it is proved that any perturbation can only increase the widths of these resonances, provided that terms of O(1/Q0) can be ignored in the first- and second-order corrections, where Q0 is the original quality factor of the resonance. The result can be interpreted in terms of a generalized golden rule and is relevant to similar problems involving quasi-normal modes in quantum mechanics. The full theory beyond the scalar wave approximation can be developed, and it is expected that all qualitative features will survive.

© 1991 Optical Society of America

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    [CrossRef] [PubMed]
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  15. P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990); P. W. Barber, S. C. Hill, Department of Electrical and Computer Engineering, Clarkson University, Potsdam, New York 13676) (personal communication, August1989).
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    [CrossRef] [PubMed]
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  19. The expansion on the right-hand side also contains second-order (and higher-order) terms in the shape distortion, proportional to derivatives of the delta function. These need be treated only to first order in the perturbative formalism, and we have already shown that the imaginary parts are not affected by first-order contributions.
  20. H. M. Lai, P. T. Leung, K. Young, “Thermal spectrum in leaky cavities: a string model,” Phys. Lett. 119, 337–339 (1987); S. C. Ching, H. M. Lai, K. Young, “Dielectric microspheres as optical cavities: thermal spectrum and density of states,” J. Opt. Soc. Am. B 4, 1995–2003 (1987).
    [CrossRef]
  21. The absolute value of μ appears for the following reason: Changing the sign of μ would change a prolate ellipse into an oblate ellipse; but prolate and oblate ellipses have the same eccentricity, with only a 90° rotation of the major and minor axes. In other words, ellipses cannot be distinguished as being prolate or oblate, unlike spheroids.
  22. P. T. Leung, K. Young, “Time-independent perturbation theory for quasinormal modes in quantum mechanics,” Phys. Rev. A (to be published).

1990 (4)

H. M. Lai, P. T. Leung, K. Young, “Limitations on the photon storage lifetime in electromagnetic resonances of highly transparent microdroplets,” Phys. Rev. A 41, 5199–5204 (1990).
[CrossRef] [PubMed]

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

S. Arnold, D. E. Spock, L. M. Folan, “Electric field modulated light scattering near a morphological resonance of a trapped aerosol particle,” Opt. Lett. 15, 1111–1113 (1990).
[CrossRef] [PubMed]

A. L. Huston, H.-B. Lin, J. D. Eversole, A. J. Campillo, “Nonlinear Mie scattering: electrostrictive coupling of light to droplet acoustic modes,” Opt. Lett. 15, 1176–1178 (1990).
[CrossRef] [PubMed]

1988 (2)

1987 (2)

1984 (1)

1973 (1)

A. Ashkin, J. M. Dziedzic, “Radiation pressure on a free liquid surface,” Phys. Rev. Lett. 30, 139–142 (1973); J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. 8, 14–21 (1973); P. I. Singh, C. J. Knight, “Pulsed laser-induced shattering of water drops,” AIAA J. 18, 96–100 (1980); H. M. Lai, W. M. Suen, K. Young, “Microscopic derivation of the Helmholtz force density,” Phys. Rev. Lett. 47, 177–179 (1981); “Microscopic derivation of the force on a dielectric fluid in an electromagnetic field,” Phys. Rev. A 25, 1755–1763 (1982); H.-M. Tzeng, M. B. Long, R. K. Chang, P. W. Barber, “Laser-induced shape distortions of flowing droplets deduced from morphology-dependent resonances in fluorescence spectra,” Opt. Lett. 10, 209–211 (1985); M. Autric, P. Vigliano, D. Dufresne, J. P. Caressa, Ph. Bournot, “Pulsed CO2 laser-induced effects on water droplets,” AIAA J. 26, 65–71 (1988); C. F. Wood, D. H. Leach, J.-Z. Zhang, R. K. Chang, P. W. Barber, “Time-resolved shadowgraphs of large individual water and ethanol droplets vaporized by a pulsed CO2 laser,” Appl. Opt. 27, 2279–2286 (1988); H. M. Lai, P. T. Leung, K. L. Poon, K. Young, “Electrostrictive distortion of a micrometer-sized droplet by a laser pulse,” J. Opt. Soc. Am. B 6, 2430–2437 (1989); B.-S. Park, A. Biswas, R. L. Armstrong, R. G. Pinnick, “Delay of explosive vaporization in pulsed laser-heated droplets,” Opt. Lett. 15, 206–208 (1990).
[CrossRef] [PubMed]

1964 (1)

1960 (1)

Ya. B. Zeldovich, “On the theory of unstable states,” Zh. Eksp. Teor. Fiz. 39, 776–780 (1960) [Sov. Phys. JETP 12, 542–545 (1961)].

1908 (1)

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908); H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957); M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Arnold, S.

Ashkin, A.

A. Ashkin, J. M. Dziedzic, “Radiation pressure on a free liquid surface,” Phys. Rev. Lett. 30, 139–142 (1973); J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. 8, 14–21 (1973); P. I. Singh, C. J. Knight, “Pulsed laser-induced shattering of water drops,” AIAA J. 18, 96–100 (1980); H. M. Lai, W. M. Suen, K. Young, “Microscopic derivation of the Helmholtz force density,” Phys. Rev. Lett. 47, 177–179 (1981); “Microscopic derivation of the force on a dielectric fluid in an electromagnetic field,” Phys. Rev. A 25, 1755–1763 (1982); H.-M. Tzeng, M. B. Long, R. K. Chang, P. W. Barber, “Laser-induced shape distortions of flowing droplets deduced from morphology-dependent resonances in fluorescence spectra,” Opt. Lett. 10, 209–211 (1985); M. Autric, P. Vigliano, D. Dufresne, J. P. Caressa, Ph. Bournot, “Pulsed CO2 laser-induced effects on water droplets,” AIAA J. 26, 65–71 (1988); C. F. Wood, D. H. Leach, J.-Z. Zhang, R. K. Chang, P. W. Barber, “Time-resolved shadowgraphs of large individual water and ethanol droplets vaporized by a pulsed CO2 laser,” Appl. Opt. 27, 2279–2286 (1988); H. M. Lai, P. T. Leung, K. L. Poon, K. Young, “Electrostrictive distortion of a micrometer-sized droplet by a laser pulse,” J. Opt. Soc. Am. B 6, 2430–2437 (1989); B.-S. Park, A. Biswas, R. L. Armstrong, R. G. Pinnick, “Delay of explosive vaporization in pulsed laser-heated droplets,” Opt. Lett. 15, 206–208 (1990).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt.20, 1803–1814 (1981); J. F. Owen, P. W. Barber, B. J. Messinger, R. K. Chang, “Determination of optical fiber diameter from resonances in the elastic scattering spectrum,” Opt. Lett. 6, 272–274 (1981); C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983); P. R. Conwell, C. K. Rushforth, R. E. Benner, S. C. Hill, “Efficient automated algorithm for the sizing of dielectric microspheres using the resonance spectrum,” J. Opt. Soc. Am. A 1, 1181–1187 (1984); P. W. Barber, S. C. Hill, “Effects of particle nonsphericity on light scattering,” in Proceedings of the International Symposium on Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988), pp. 43–53; K. H. Fung, I. N. Tang, H. R. Munkelwitz, “Study of condensation growth of water droplets by Mie resonance spectroscopy,” Appl. Opt. 26, 1282–1287 (1987); H. M. Lai, P. T. Leung, K. L. Poon, K. Young, “Characterization of the internal energy density in Mie scattering,” J. Opt. Soc. Am. A (to be published).
[CrossRef] [PubMed]

Barber, P. W.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

P. R. Conwell, P. W. Barber, C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A 1, 62–67 (1984); R. Thurn, W. Kiefer, “Structural resonances observed in the Raman spectra of optically levitated liquid droplets,” Appl. Opt. 24, 1515–1519 (1985).
[CrossRef] [PubMed]

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990); P. W. Barber, S. C. Hill, Department of Electrical and Computer Engineering, Clarkson University, Potsdam, New York 13676) (personal communication, August1989).

Campillo, A. J.

Chang, R. K.

Chiao, R. Y.

Conwell, P. R.

Cooney, J.

J. Cooney, A. Gross, “Coherent anti-Stokes Raman scattering by droplets in the Mie size range,” Opt. Lett.7, 218–220 (1982); J. F. Owen, R. K. Chang, P. W. Barber, “Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles,” Aerosol Sci. Technol. 1, 293–302 (1982); S. C. Hill, R. E. Benner, “Morphology-dependent resonances associated with stimulated processes in microspheres,” J. Opt. Soc. Am. B 3, 1509–1514 (1986); J. B. Snow, S. -X. Qian, R. K. Chang, “Stimulated Raman scattering from individual water and ethanol droplets at morphology-dependent resonances,” Opt. Lett. 10, 37–39 (1985); “Nonlinear optics with a micrometer-size droplet,” Opt. News 12(5), 5–7 (1986); S. -X. Qian, J. B. Snow, R. K. Chang, “Coherent Raman mixing and coherent anti-Stokes Raman scattering from individual micrometer-sized droplets,” Opt. Lett. 10, 499–501 (1985); “Nonlinear optical processes in micron-size droplets,” in Proceedings of SEICOLS ’85, T. W. Hänsch, Y. R. Shen, eds. (Springer-Verlag, Berlin, 1985), pp. 204–209; S. -X. Qian, R. K. Chang, “Multiorder Stokes emission from micrometer-size droplets,” Phys. Rev. Lett. 56, 926–929 (1986); “Phase-modulation-broadened line shapes from micrometer size CS2 droplets,” Opt. Lett. 11, 371–373 (1986); H. -M. Tzeng, K. F. Wall, M. B. Long, R. K. Chang, “Laser emission from individual droplets at wavelengths corresponding to morphology-dependent resonances,” Opt. Lett. 9, 499–501 (1984); S. -X. Qian, J. B. Snow, H. -M. Tzeng, R. K. Chang, “Lasing droplets: Highlighting the liquid–air interface by laser emission,” Science 231, 486–488 (1986); R. K. Chang, S. -X. Qian, J. H. Eickmans, “Stimulated Raman scattering, phase modulation, and coherent anti-Stokes Raman scattering from single micrometer-size liquid droplets,” in Proceedings of the Methods of Laser Spectroscopy Symposium, Y. Prior, A. Ben-Reuven, M. Rosenbluh, eds. (Plenum, New York, 1986), pp. 249–258; H. -B. Lin, A. L. Huston, B. L. Justus, A. J. Campillo, “Some characteristics of a droplet whispering-gallery mode laser,” Opt. Lett. 11, 614–616 (1986); R. G. Pinnick, A. Biswas, P. Chylek, R. L. Armstrong, H. Latifi, E. Creegan, V. Srivastava, M. Jarzembski, G. Fernandez, “Stimulated Raman scattering in micrometer-sized droplets: Time-resolved measurements,” Opt. Lett. 13, 494–496 (1988); N. M. Belov, V. Yu. Dubrovskii, F. K. Kosyrev, V. A. Motyagin, A. E. Negin, M. A. Iordanskii, V. E. Kostromin, “Nonlinear scattering and self-focusing of laser radiation in an aerosol,” Sov. J. Quantum Electron. 15, 1150–1151 (1985); L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985); S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987); P. T. Leung, K. Young, “Theory of enhanced energy transfer in an aerosol particle,” J. Chem. Phys. 89, 2894–2899 (1988); S. Arnold, L. M. Folan, “Energy transfer and the photon lifetime within an aerosol particle,” Opt. Lett. 14, 387–389 (1989).
[CrossRef] [PubMed]

Dziedzic, J. M.

A. Ashkin, J. M. Dziedzic, “Radiation pressure on a free liquid surface,” Phys. Rev. Lett. 30, 139–142 (1973); J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. 8, 14–21 (1973); P. I. Singh, C. J. Knight, “Pulsed laser-induced shattering of water drops,” AIAA J. 18, 96–100 (1980); H. M. Lai, W. M. Suen, K. Young, “Microscopic derivation of the Helmholtz force density,” Phys. Rev. Lett. 47, 177–179 (1981); “Microscopic derivation of the force on a dielectric fluid in an electromagnetic field,” Phys. Rev. A 25, 1755–1763 (1982); H.-M. Tzeng, M. B. Long, R. K. Chang, P. W. Barber, “Laser-induced shape distortions of flowing droplets deduced from morphology-dependent resonances in fluorescence spectra,” Opt. Lett. 10, 209–211 (1985); M. Autric, P. Vigliano, D. Dufresne, J. P. Caressa, Ph. Bournot, “Pulsed CO2 laser-induced effects on water droplets,” AIAA J. 26, 65–71 (1988); C. F. Wood, D. H. Leach, J.-Z. Zhang, R. K. Chang, P. W. Barber, “Time-resolved shadowgraphs of large individual water and ethanol droplets vaporized by a pulsed CO2 laser,” Appl. Opt. 27, 2279–2286 (1988); H. M. Lai, P. T. Leung, K. L. Poon, K. Young, “Electrostrictive distortion of a micrometer-sized droplet by a laser pulse,” J. Opt. Soc. Am. B 6, 2430–2437 (1989); B.-S. Park, A. Biswas, R. L. Armstrong, R. G. Pinnick, “Delay of explosive vaporization in pulsed laser-heated droplets,” Opt. Lett. 15, 206–208 (1990).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt.20, 1803–1814 (1981); J. F. Owen, P. W. Barber, B. J. Messinger, R. K. Chang, “Determination of optical fiber diameter from resonances in the elastic scattering spectrum,” Opt. Lett. 6, 272–274 (1981); C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983); P. R. Conwell, C. K. Rushforth, R. E. Benner, S. C. Hill, “Efficient automated algorithm for the sizing of dielectric microspheres using the resonance spectrum,” J. Opt. Soc. Am. A 1, 1181–1187 (1984); P. W. Barber, S. C. Hill, “Effects of particle nonsphericity on light scattering,” in Proceedings of the International Symposium on Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988), pp. 43–53; K. H. Fung, I. N. Tang, H. R. Munkelwitz, “Study of condensation growth of water droplets by Mie resonance spectroscopy,” Appl. Opt. 26, 1282–1287 (1987); H. M. Lai, P. T. Leung, K. L. Poon, K. Young, “Characterization of the internal energy density in Mie scattering,” J. Opt. Soc. Am. A (to be published).
[CrossRef] [PubMed]

Eversole, J. D.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Folan, L. M.

Gouesbet, G.

Grehan, G.

Gross, A.

J. Cooney, A. Gross, “Coherent anti-Stokes Raman scattering by droplets in the Mie size range,” Opt. Lett.7, 218–220 (1982); J. F. Owen, R. K. Chang, P. W. Barber, “Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles,” Aerosol Sci. Technol. 1, 293–302 (1982); S. C. Hill, R. E. Benner, “Morphology-dependent resonances associated with stimulated processes in microspheres,” J. Opt. Soc. Am. B 3, 1509–1514 (1986); J. B. Snow, S. -X. Qian, R. K. Chang, “Stimulated Raman scattering from individual water and ethanol droplets at morphology-dependent resonances,” Opt. Lett. 10, 37–39 (1985); “Nonlinear optics with a micrometer-size droplet,” Opt. News 12(5), 5–7 (1986); S. -X. Qian, J. B. Snow, R. K. Chang, “Coherent Raman mixing and coherent anti-Stokes Raman scattering from individual micrometer-sized droplets,” Opt. Lett. 10, 499–501 (1985); “Nonlinear optical processes in micron-size droplets,” in Proceedings of SEICOLS ’85, T. W. Hänsch, Y. R. Shen, eds. (Springer-Verlag, Berlin, 1985), pp. 204–209; S. -X. Qian, R. K. Chang, “Multiorder Stokes emission from micrometer-size droplets,” Phys. Rev. Lett. 56, 926–929 (1986); “Phase-modulation-broadened line shapes from micrometer size CS2 droplets,” Opt. Lett. 11, 371–373 (1986); H. -M. Tzeng, K. F. Wall, M. B. Long, R. K. Chang, “Laser emission from individual droplets at wavelengths corresponding to morphology-dependent resonances,” Opt. Lett. 9, 499–501 (1984); S. -X. Qian, J. B. Snow, H. -M. Tzeng, R. K. Chang, “Lasing droplets: Highlighting the liquid–air interface by laser emission,” Science 231, 486–488 (1986); R. K. Chang, S. -X. Qian, J. H. Eickmans, “Stimulated Raman scattering, phase modulation, and coherent anti-Stokes Raman scattering from single micrometer-size liquid droplets,” in Proceedings of the Methods of Laser Spectroscopy Symposium, Y. Prior, A. Ben-Reuven, M. Rosenbluh, eds. (Plenum, New York, 1986), pp. 249–258; H. -B. Lin, A. L. Huston, B. L. Justus, A. J. Campillo, “Some characteristics of a droplet whispering-gallery mode laser,” Opt. Lett. 11, 614–616 (1986); R. G. Pinnick, A. Biswas, P. Chylek, R. L. Armstrong, H. Latifi, E. Creegan, V. Srivastava, M. Jarzembski, G. Fernandez, “Stimulated Raman scattering in micrometer-sized droplets: Time-resolved measurements,” Opt. Lett. 13, 494–496 (1988); N. M. Belov, V. Yu. Dubrovskii, F. K. Kosyrev, V. A. Motyagin, A. E. Negin, M. A. Iordanskii, V. E. Kostromin, “Nonlinear scattering and self-focusing of laser radiation in an aerosol,” Sov. J. Quantum Electron. 15, 1150–1151 (1985); L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985); S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987); P. T. Leung, K. Young, “Theory of enhanced energy transfer in an aerosol particle,” J. Chem. Phys. 89, 2894–2899 (1988); S. Arnold, L. M. Folan, “Energy transfer and the photon lifetime within an aerosol particle,” Opt. Lett. 14, 387–389 (1989).
[CrossRef] [PubMed]

Hill, S. C.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990); P. W. Barber, S. C. Hill, Department of Electrical and Computer Engineering, Clarkson University, Potsdam, New York 13676) (personal communication, August1989).

Huston, A. L.

Lai, H. M.

H. M. Lai, P. T. Leung, K. Young, “Limitations on the photon storage lifetime in electromagnetic resonances of highly transparent microdroplets,” Phys. Rev. A 41, 5199–5204 (1990).
[CrossRef] [PubMed]

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

H. M. Lai, P. T. Leung, K. Young, “Electromagnetic decay into a narrow resonance in an optical cavity,” Phys. Rev. A 37, 1597–1606 (1988).
[CrossRef] [PubMed]

H. M. Lai, P. T. Leung, K. Young, “Thermal spectrum in leaky cavities: a string model,” Phys. Lett. 119, 337–339 (1987); S. C. Ching, H. M. Lai, K. Young, “Dielectric microspheres as optical cavities: thermal spectrum and density of states,” J. Opt. Soc. Am. B 4, 1995–2003 (1987).
[CrossRef]

Lam, C. C.

C. C. Lam, “Wave scattering from slightly deformed spheres,” M.Phil. thesis (The Chinese University of Hong Kong, in preparation).

Leung, P. T.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

H. M. Lai, P. T. Leung, K. Young, “Limitations on the photon storage lifetime in electromagnetic resonances of highly transparent microdroplets,” Phys. Rev. A 41, 5199–5204 (1990).
[CrossRef] [PubMed]

H. M. Lai, P. T. Leung, K. Young, “Electromagnetic decay into a narrow resonance in an optical cavity,” Phys. Rev. A 37, 1597–1606 (1988).
[CrossRef] [PubMed]

H. M. Lai, P. T. Leung, K. Young, “Thermal spectrum in leaky cavities: a string model,” Phys. Lett. 119, 337–339 (1987); S. C. Ching, H. M. Lai, K. Young, “Dielectric microspheres as optical cavities: thermal spectrum and density of states,” J. Opt. Soc. Am. B 4, 1995–2003 (1987).
[CrossRef]

P. T. Leung, K. Young, “Time-independent perturbation theory for quasinormal modes in quantum mechanics,” Phys. Rev. A (to be published).

Lin, H.-B.

Maheu, B.

Mie, G.

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908); H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957); M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Rose, M. E.

M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957).

Rushforth, C. K.

Spock, D. E.

Stoicheff, B. P.

Young, K.

H. M. Lai, P. T. Leung, K. Young, “Limitations on the photon storage lifetime in electromagnetic resonances of highly transparent microdroplets,” Phys. Rev. A 41, 5199–5204 (1990).
[CrossRef] [PubMed]

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

H. M. Lai, P. T. Leung, K. Young, “Electromagnetic decay into a narrow resonance in an optical cavity,” Phys. Rev. A 37, 1597–1606 (1988).
[CrossRef] [PubMed]

H. M. Lai, P. T. Leung, K. Young, “Thermal spectrum in leaky cavities: a string model,” Phys. Lett. 119, 337–339 (1987); S. C. Ching, H. M. Lai, K. Young, “Dielectric microspheres as optical cavities: thermal spectrum and density of states,” J. Opt. Soc. Am. B 4, 1995–2003 (1987).
[CrossRef]

P. T. Leung, K. Young, “Time-independent perturbation theory for quasinormal modes in quantum mechanics,” Phys. Rev. A (to be published).

Zeldovich, Ya. B.

Ya. B. Zeldovich, “On the theory of unstable states,” Zh. Eksp. Teor. Fiz. 39, 776–780 (1960) [Sov. Phys. JETP 12, 542–545 (1961)].

Zhang, J.-Z.

Ann. Phys. (Leipzig) (1)

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908); H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957); M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Lett. (3)

Phys. Lett. (1)

H. M. Lai, P. T. Leung, K. Young, “Thermal spectrum in leaky cavities: a string model,” Phys. Lett. 119, 337–339 (1987); S. C. Ching, H. M. Lai, K. Young, “Dielectric microspheres as optical cavities: thermal spectrum and density of states,” J. Opt. Soc. Am. B 4, 1995–2003 (1987).
[CrossRef]

Phys. Rev. A (3)

H. M. Lai, P. T. Leung, K. Young, “Limitations on the photon storage lifetime in electromagnetic resonances of highly transparent microdroplets,” Phys. Rev. A 41, 5199–5204 (1990).
[CrossRef] [PubMed]

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

H. M. Lai, P. T. Leung, K. Young, “Electromagnetic decay into a narrow resonance in an optical cavity,” Phys. Rev. A 37, 1597–1606 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

A. Ashkin, J. M. Dziedzic, “Radiation pressure on a free liquid surface,” Phys. Rev. Lett. 30, 139–142 (1973); J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. 8, 14–21 (1973); P. I. Singh, C. J. Knight, “Pulsed laser-induced shattering of water drops,” AIAA J. 18, 96–100 (1980); H. M. Lai, W. M. Suen, K. Young, “Microscopic derivation of the Helmholtz force density,” Phys. Rev. Lett. 47, 177–179 (1981); “Microscopic derivation of the force on a dielectric fluid in an electromagnetic field,” Phys. Rev. A 25, 1755–1763 (1982); H.-M. Tzeng, M. B. Long, R. K. Chang, P. W. Barber, “Laser-induced shape distortions of flowing droplets deduced from morphology-dependent resonances in fluorescence spectra,” Opt. Lett. 10, 209–211 (1985); M. Autric, P. Vigliano, D. Dufresne, J. P. Caressa, Ph. Bournot, “Pulsed CO2 laser-induced effects on water droplets,” AIAA J. 26, 65–71 (1988); C. F. Wood, D. H. Leach, J.-Z. Zhang, R. K. Chang, P. W. Barber, “Time-resolved shadowgraphs of large individual water and ethanol droplets vaporized by a pulsed CO2 laser,” Appl. Opt. 27, 2279–2286 (1988); H. M. Lai, P. T. Leung, K. L. Poon, K. Young, “Electrostrictive distortion of a micrometer-sized droplet by a laser pulse,” J. Opt. Soc. Am. B 6, 2430–2437 (1989); B.-S. Park, A. Biswas, R. L. Armstrong, R. G. Pinnick, “Delay of explosive vaporization in pulsed laser-heated droplets,” Opt. Lett. 15, 206–208 (1990).
[CrossRef] [PubMed]

Zh. Eksp. Teor. Fiz. (1)

Ya. B. Zeldovich, “On the theory of unstable states,” Zh. Eksp. Teor. Fiz. 39, 776–780 (1960) [Sov. Phys. JETP 12, 542–545 (1961)].

Other (9)

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990); P. W. Barber, S. C. Hill, Department of Electrical and Computer Engineering, Clarkson University, Potsdam, New York 13676) (personal communication, August1989).

A. Ashkin, J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt.20, 1803–1814 (1981); J. F. Owen, P. W. Barber, B. J. Messinger, R. K. Chang, “Determination of optical fiber diameter from resonances in the elastic scattering spectrum,” Opt. Lett. 6, 272–274 (1981); C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983); P. R. Conwell, C. K. Rushforth, R. E. Benner, S. C. Hill, “Efficient automated algorithm for the sizing of dielectric microspheres using the resonance spectrum,” J. Opt. Soc. Am. A 1, 1181–1187 (1984); P. W. Barber, S. C. Hill, “Effects of particle nonsphericity on light scattering,” in Proceedings of the International Symposium on Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988), pp. 43–53; K. H. Fung, I. N. Tang, H. R. Munkelwitz, “Study of condensation growth of water droplets by Mie resonance spectroscopy,” Appl. Opt. 26, 1282–1287 (1987); H. M. Lai, P. T. Leung, K. L. Poon, K. Young, “Characterization of the internal energy density in Mie scattering,” J. Opt. Soc. Am. A (to be published).
[CrossRef] [PubMed]

J. Cooney, A. Gross, “Coherent anti-Stokes Raman scattering by droplets in the Mie size range,” Opt. Lett.7, 218–220 (1982); J. F. Owen, R. K. Chang, P. W. Barber, “Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles,” Aerosol Sci. Technol. 1, 293–302 (1982); S. C. Hill, R. E. Benner, “Morphology-dependent resonances associated with stimulated processes in microspheres,” J. Opt. Soc. Am. B 3, 1509–1514 (1986); J. B. Snow, S. -X. Qian, R. K. Chang, “Stimulated Raman scattering from individual water and ethanol droplets at morphology-dependent resonances,” Opt. Lett. 10, 37–39 (1985); “Nonlinear optics with a micrometer-size droplet,” Opt. News 12(5), 5–7 (1986); S. -X. Qian, J. B. Snow, R. K. Chang, “Coherent Raman mixing and coherent anti-Stokes Raman scattering from individual micrometer-sized droplets,” Opt. Lett. 10, 499–501 (1985); “Nonlinear optical processes in micron-size droplets,” in Proceedings of SEICOLS ’85, T. W. Hänsch, Y. R. Shen, eds. (Springer-Verlag, Berlin, 1985), pp. 204–209; S. -X. Qian, R. K. Chang, “Multiorder Stokes emission from micrometer-size droplets,” Phys. Rev. Lett. 56, 926–929 (1986); “Phase-modulation-broadened line shapes from micrometer size CS2 droplets,” Opt. Lett. 11, 371–373 (1986); H. -M. Tzeng, K. F. Wall, M. B. Long, R. K. Chang, “Laser emission from individual droplets at wavelengths corresponding to morphology-dependent resonances,” Opt. Lett. 9, 499–501 (1984); S. -X. Qian, J. B. Snow, H. -M. Tzeng, R. K. Chang, “Lasing droplets: Highlighting the liquid–air interface by laser emission,” Science 231, 486–488 (1986); R. K. Chang, S. -X. Qian, J. H. Eickmans, “Stimulated Raman scattering, phase modulation, and coherent anti-Stokes Raman scattering from single micrometer-size liquid droplets,” in Proceedings of the Methods of Laser Spectroscopy Symposium, Y. Prior, A. Ben-Reuven, M. Rosenbluh, eds. (Plenum, New York, 1986), pp. 249–258; H. -B. Lin, A. L. Huston, B. L. Justus, A. J. Campillo, “Some characteristics of a droplet whispering-gallery mode laser,” Opt. Lett. 11, 614–616 (1986); R. G. Pinnick, A. Biswas, P. Chylek, R. L. Armstrong, H. Latifi, E. Creegan, V. Srivastava, M. Jarzembski, G. Fernandez, “Stimulated Raman scattering in micrometer-sized droplets: Time-resolved measurements,” Opt. Lett. 13, 494–496 (1988); N. M. Belov, V. Yu. Dubrovskii, F. K. Kosyrev, V. A. Motyagin, A. E. Negin, M. A. Iordanskii, V. E. Kostromin, “Nonlinear scattering and self-focusing of laser radiation in an aerosol,” Sov. J. Quantum Electron. 15, 1150–1151 (1985); L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985); S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987); P. T. Leung, K. Young, “Theory of enhanced energy transfer in an aerosol particle,” J. Chem. Phys. 89, 2894–2899 (1988); S. Arnold, L. M. Folan, “Energy transfer and the photon lifetime within an aerosol particle,” Opt. Lett. 14, 387–389 (1989).
[CrossRef] [PubMed]

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957).

The expansion on the right-hand side also contains second-order (and higher-order) terms in the shape distortion, proportional to derivatives of the delta function. These need be treated only to first order in the perturbative formalism, and we have already shown that the imaginary parts are not affected by first-order contributions.

C. C. Lam, “Wave scattering from slightly deformed spheres,” M.Phil. thesis (The Chinese University of Hong Kong, in preparation).

The absolute value of μ appears for the following reason: Changing the sign of μ would change a prolate ellipse into an oblate ellipse; but prolate and oblate ellipses have the same eccentricity, with only a 90° rotation of the major and minor axes. In other words, ellipses cannot be distinguished as being prolate or oblate, unlike spheroids.

P. T. Leung, K. Young, “Time-independent perturbation theory for quasinormal modes in quantum mechanics,” Phys. Rev. A (to be published).

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Figures (4)

Fig. 1
Fig. 1

The multiplet of resonant modes splits apart by Δω, while each mode has a width γ.

Fig. 2
Fig. 2

Value of 1/Q versus μ, where μ is the amplitude of the shape deformation of the droplet with L = 2. The points are numerical results from the T-matrix method for a droplet with refractive index n = 2 and a leading resonance of l = 10 at approximately x = 6.8263. The two curves and the line are the results obtained from Eq. (1.3), with C2 given by Eq. (3.16), for which m = 0, 6, and 10, respectively.

Fig. 3
Fig. 3

The points show the m dependence of the fitted values of C2 from T-matrix calculations; the curve is a fit to relation (3.19)

Fig. 4
Fig. 4

(a) Real and (b) imaginary parts of the pole positions due to degenerate perturbation versus μΔ, according to Eq. (4.5) with parameters ξ ˜ 1 = (144.5, −8.0 × 10−9) and ξ ˜ 2 = (144.0, −1.0 × 10−8). The imaginary part shows level attraction.

Equations (92)

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r / a = 1 + μ F ( θ , φ ) ,
ɛ ( r ) = n 2 + μ F ( r , θ , φ ) ,
1 / Q = 1 / Q 0 + μ C 1 + μ 2 C 2 + ,
C 1 = 0 ,             C 2 0
ɛ ( r ) 2 ψ / t 2 = 2 ψ ,
ɛ ( r ) = { 1 r > a n 2 + μ F ( r , θ ) r < a .
ψ r ( r = a , θ , φ ) = d S D ( r , r ) ψ ( r = a , θ , φ ) ,
ψ ( r , θ , φ ) = l f l ( r ) Y l m ( θ , φ ) ,             r < a .
d f l ( r = a ) d r = k l D l l f l ( a ) ,
D ^ ( z ) = D ^ ( 0 ) ( z ) + μ D ^ ( 1 ) ( z ) + μ 2 D ^ ( 2 ) ( z ) + ,
ψ = l [ a l h l ( 2 ) ( k r ) + b l h l ( 1 ) ( k r ) ] Y l m ( θ , φ ) ,
b l = l S l l a l .
a l h l ( 2 ) + b l h l ( 1 ) = l D l l [ a l h l ( 2 ) + b l h l ( 1 ) ] ,
b l h l ( 1 ) = b ˜ l ,             a l h l ( 2 ) = a ˜ l
H l l ( z ) = δ l l h l ( 1 ) ( z ) / h l ( 1 ) ( z ) .
( D ^ - H ^ ) b ˜ = - ( D ^ - H ^ * ) a ˜ ,
det M ^ = 0 ,             M ^ = D ^ ( z ) - H ^ ( z ) .
M ^ = M ^ ( 0 ) + μ M ^ ( 1 ) + μ 2 M ^ ( 2 ) + = M ^ ( 0 ) + Q .
λ l = M l l ( 0 ) + Q l l + l l Q l l Q l l [ M l l ( 0 ) - M l l ( 0 ) ] - 1 + ,
0 = Q l l - l l Q l l Q l l M l l ( 0 ) +
0 = M l l ( 1 ) ,
0 = M l l ( 2 ) - l l M l l ( 1 ) M l l ( 1 ) M l l ( 0 ) .
z = z 0 + μ z 1 + μ 2 z 2 + ,
D ^ ( z ) = [ D ^ ( 0 ) ( z 0 ) + ( μ z 1 + μ 2 z 2 ) D ^ ( 0 ) ( z 0 ) + 1 2 ( μ z 1 ) 2 D ^ ( 0 ) ( z 0 ) ] + μ [ D ^ ( 1 ) ( z 0 ) + μ z 1 D ^ ( 1 ) ( z 0 ) ] + μ 2 [ D ^ ( 2 ) ( z 0 ) ] + O ( μ 3 ) ,
H ^ ( z ) = H ^ ( z 0 ) + ( μ z 1 + μ 2 z 2 ) H ^ ( z 0 ) + 1 2 ( μ z 1 ) 2 H ^ ( z 0 ) + O ( μ 3 ) .
M ^ ( 0 ) = D ^ ( 0 ) - H ^ ,
M ^ ( 1 ) = [ D ^ ( 0 ) - H ^ ] z 1 + D ^ ( 1 ) ,
z 1 = - D l l ( 1 ) [ D l l ( 0 ) - H l l ] - 1 .
C 1 = O ( 1 / Q 0 ) ,
Im M l l ( 2 ) = [ D l l ( 0 ) - H l l ] y 2 = - ( n 2 - 1 ) y 2 .
( n 2 - 1 ) y 2 = - l l D l l ( 1 ) 2 Im { [ D l l ( 0 ) - H l l ] - 1 } .
Im { [ D l l ( 0 ) - H l l ] - 1 } = [ j l ( n x 0 ) / x 0 ] 2 W l ( x 0 ) - 2 0 ,
W l ( x ) = j l ( n x ) h l ( 1 ) ( x ) - n j l ( n x ) h l ( 1 ) ( x ) .
C 2 = - 2 y 2 x 0 = 2 ( n 2 - 1 ) x 0 3 l l D l l ( 1 ) 2 | j l ( n x 0 ) W l ( x 0 ) | 2 0
ψ = ψ 0 + μ ψ 1 + μ 2 ψ 2 + ,
ψ i = l f l , i ( r ) Y l m ( θ , φ ) .
( 2 + n 2 k 2 ) ψ 0 = 0 ,
( 2 + n 2 k 2 ) ψ 1 = - k 2 F ( r , θ ) ψ 0 .
f l , 0 ( a ) = s l ,             f l , 1 ( a ) = 0 ,
d f l , 1 ( r = a ) d r = k l D l l ( 1 ) s l .
f l , 0 ( r ) = s l j l ( n k r ) / j l ( n k a ) .
[ 1 r 2 d d r r 2 d d r - l ( l + 1 ) r 2 + n 2 k 2 ] f l , 1 ( r ) = - k 2 l [ d Ω Y l m * F ( r , θ ) Y l m ] f l , 0 ( r ) .
F ( r , θ ) = L F L ( r ) Y L 0 ( θ , φ ) .
- k 2 l A ( l , l ) s l [ j l ( n k r ) j l ( n k a ) ] F L ( r ) ,
A ( l , l ) = [ ( 2 L + 1 ) ( 2 l + 1 ) 4 π ( 2 l + 1 ) ] 1 / 2 C ( l L l , 000 ) C ( l L l , m 0 m ) ,
[ 1 r 2 d d r r 2 d d r - l ( l + 1 ) r 2 + n 2 k 2 ] G l ( r , r ) = δ ( r - r ) ,
D l l ( 1 ) = - k A ( l , l ) [ j l ( n k a ) ] - 1 × 0 a d r d G l ( r = a , r ) d r F L ( r ) j l ( n k r ) .
d G l ( r = a , r ) d r = ( r a ) 2 j l ( n k r ) j l ( n k a ) ,
D l l ( 1 ) = - k a A ( l , l ) [ a 3 j l ( n k a ) j l ( n k a ) ] - 1 × 0 a d r r 2 j l ( n k r ) F L ( r ) j l ( n k r ) ,
r / a = 1 + μ Y L 0 ( θ , φ ) .
μ F ( r , θ , φ ) = ( n 2 - 1 ) [ θ ( a + μ a Y L 0 - r ) - θ ( a - r ) ] ( n 2 - 1 ) δ ( r - a ) μ a Y L 0 ,
C 2 = n 2 - 1 2 π x 0 2 L + 1 2 l + 1 l l ( 2 l + 1 ) C ( l L l , 000 ) 2 × C ( l L l , m 0 m ) 2 | j l ( n x 0 ) W l ( x 0 ) | 2 .
W l ( x 0 ) - 2 = ( π / 2 k 2 ) x 0 4 ρ l ,
r p - r e a = 3 4 ( 5 π ) 1 / 2 μ .
C 2 ( 1 - m 2 / l 2 ) 2 .
Re ω 1 Re ω 0 = - ( 5 4 π ) 1 / 2 μ C ( l 2 l , 000 ) C ( l 2 l , m 0 m ) = - μ 4 ( 5 4 π ) 1 / 2 [ 1 - 3 m 2 l ( l + 1 ) ] .
s = s 0 [ 1 + μ 4 ( 5 4 π ) 1 / 2 ( 1 - 3 m 2 l 2 ) ] ,
e 2 = 3 ( 5 / 4 π ) 1 / 2 μ ( 1 - m 2 / l 2 ) ,
| D 11 - H 11 D 12 D 21 D 22 - H 22 | = 0.
( D i i - H i i ) z ( D i i ( 0 ) - H i i ) z + μ D i i ( 1 ) ( z ) ( D i i ( 0 ) - H i i ) ξ i ( z - ξ i ) + μ D ^ i i ( 1 ) ( ξ i ) = - ( n 2 - 1 ) ( z - ξ i - μ Δ ξ i ) ,
Δ ξ i = - D i i ( 1 ) [ D i i ( 0 ) - H i i ] - 1
( n 2 - 1 ) 2 ( z - ξ ˜ 1 ) ( z - ξ ˜ 2 ) - [ μ D 12 ( 1 ) ] 2 = 0 ,
z = ξ ˜ 1 + ξ ˜ 2 2 ± [ ( ξ ˜ 1 - ξ ˜ 2 2 ) 2 + μ 2 Δ 2 ] 1 / 2 ,
z = { ξ ˜ 1 + ( μ Δ ) 2 / ( ξ ˜ 1 - ξ ˜ 2 ) ξ ˜ 2 - ( μ Δ ) 2 / ( ξ ˜ 1 - ξ ˜ 2 ) .
x = { Re ( ξ 1 + μ Δ ξ 1 ) + ( μ Δ ) 2 Re ξ ˜ 1 - Re ξ ˜ 2 ξ ˜ 1 - ξ ˜ 2 2 Re ( ξ 2 + μ Δ ξ 2 ) + ( μ Δ ) 2 Re ξ ˜ 2 - Re ξ ˜ 1 ξ ˜ 1 - ξ ˜ 2 2
y = { Im ( ξ 1 + μ Δ ξ 1 ) - ( μ Δ ) 2 Im ξ ˜ 1 - Im ξ ˜ 2 ξ ˜ 1 - ξ ˜ 2 2 Im ( ξ 2 + μ Δ ξ 2 ) - ( μ Δ ) 2 Im ξ ˜ 2 - Im ξ ˜ 1 ξ ˜ 1 - ξ ˜ 2 2 .
z = ( ξ ˜ 1 + ξ ˜ 2 ) / 2 ± μ Δ .
C 2 = 2 Δ 2 ξ 1 Im ξ 1 - Im ξ 2 ξ 1 - ξ 2 2 < 0.
Γ = μ 2 ω 0 C 2 ,
Γ = 2 π i ϕ U ψ i 2 δ ( ω - ω i ) ,
2 t 2 [ ϕ ( r ) g ( t ) exp ( - i ω t ) ] ( zero - order term ) - 2 i ω ( d g d t ) ϕ ( r ) exp ( - i ω t ) + ,
U = ( ω / 2 ) μ F .
i = V d 3 k ( 2 π ) 3 ,
ψ i ( r ) = β l ( k ) ψ ˜ i ( r ) ,
ψ ˜ i ( r ) = j l ( k r ) Y l m ( θ , φ ) ,             r < a ,
β l ( k ) = ( 2 Λ ) 1 / 2 k ( k a ) 2 1 W l ( k a ) .
i = l Λ π d k ,
Γ = 2 l l Λ β l ( k ) 2 | ϕ | ω 2 μ F | ψ ˜ l | 2 .
ϕ ϕ = 1.
ϕ ~ exp ( i k r ) / r ;
ϕ ϕ = d 2 r ϕ 2 ~ d r r 2 exp ( i k r ) / r 2 =
ϕ ϕ = 0 R d 3 r ϕ ( r ) ϕ ( r ) + i 2 k R d S ϕ ( r ) ϕ ( r ) ,
observed cross section = average cross section averaged over incident profile sin 2 δ × Γ 1 / Q ,
d f l , 0 ( r = a ) d r = s l n k j l ( n x ) j l ( n x ) = k D l l ( 0 ) s l ,
D l l ( 0 ) = n j l ( n x ) j l ( n x ) .
n j l ( n z 0 ) j l ( n z 0 ) = h l ( 1 ) ( z 0 ) h l ( 1 ) ( z 0 ) ,
d d z [ D l l ( 0 ) - H l l ] z 0 = n 2 j l ( n z 0 ) j l ( n z 0 ) - h l ( 1 ) ( z 0 ) h l ( 1 ) ( z 0 ) .
d d z [ D l l ( 0 ) - H l l ] z 0 = - ( n 2 - 1 ) .
M l l ( 2 ) = z 2 d d z [ D l l ( 0 ) - H l l ] z 0 + ( z 1 2 2 ) d 2 d z 2 [ D l l ( 0 ) - H l l ] z 0 + z 1 D l l ( 1 ) ( z 0 ) + D l l ( 2 ) ( z 0 ) .
Im M l l ( 2 ) = y 2 d d z [ D l l ( 0 ) - H l l ] z 0 + O ( 1 / Q 0 ) - ( n 2 - 1 ) y 2 .
Im { [ D l l ( 0 ) - H l l ] - 1 } = j l ( n x ) Im [ h l ( 1 ) ( x ) n j l ( n x ) h l ( 1 ) ( x ) - j l ( n x ) h l ( 1 ) ( x ) ] = - j l ( n x ) 2 { Im [ h l ( 1 ) ( x ) h l ( 2 ) ( x ) ] } W l ( x ) - 2 .
Im [ h l ( 1 ) ( x ) h l ( 2 ) ( x ) ] = - 1 / x 2 ,

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