Abstract

A high-order asymptotic expansion for the size parameters xn(l) of electromagnetic resonances of large spherical dielectric scatterers is reported. The expansion is calculated up to order n−8/3, where n is the mode number and is valid for small orders l. Its accuracy is studied as a function of n, l, and the refractive index. For the l = 1 mode, the relative accuracy is better than 10−4 for n > 50.

© 1993 Optical Society of America

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References

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  1. S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.
  2. R. D. Richtmyer, “Dielectric resonators,” J. Appl. Phys. 10, 391–398 (1939).
    [Crossref]
  3. See, for example, G. Schweiger, S. Lange, U. Spengler, “Observation of input and output structural resonances in the Raman spectrum of a single spheroidal dielectric microparticle,” Opt. Lett. 15, 156–158 (1990); A. Ashkin, J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20, 1803–1814 (1981); R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980); A. Ashkin, J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
    [Crossref] [PubMed]
  4. A. J. Campillo, J. D. Eversole, H.-B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in micro-droplets,” Phys. Rev. Lett. 67, 437–440 (1991).
    [Crossref] [PubMed]
  5. V. B. Braginsky, M. L. Gorodetsky, V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering gallery modes,” Phys. Lett. A 137, 393–397 (1989).
    [Crossref]
  6. S. Schiller, R. L. Byer, “High-resolution spectroscopy of whispering-gallery modes in large dielectric spheres,” Opt. Lett. 16, 1138–1140 (1991).
    [Crossref] [PubMed]
  7. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 3.3.
  8. For references on whispering-gallery modes in systems other than droplets or spheres, see J. P. Braud, P. L. Hagelstein, “Whispering-gallery laser resonators. I. Diffraction of whispering gallery modes,” IEEE J. Quantum Electron. 27, 1069–1077 (1991).
    [Crossref]
  9. L. A. Weinstein, Open Resonators and Open Waveguides (Golem, Boulder, Colo., 1969), Sec. 58.
  10. C. C. Lam, P. T. Leung, K. Young, “Explicit asymptotic formulas for the positions, widths and strengths of resonances in Mie scattering,” J. Opt. Soc. Am. B 9, 1585–1592 (1992).
    [Crossref]
  11. Rayleigh scattering represents a lower limit for loss in liquid and glass spheres, Q < Qbulk = 2πns/λ0α; in fused silica α ≃ 2 × 10−6/cm at λ0 = 1 μm.
  12. P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1991).
  13. The asymptotic formula, Eq. (1), is complementary to those in J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. A 1, 822–830 (1984), which are applicable for large order l.
    [Crossref]
  14. P. Chylek, “Resonance structure of Mie scattering: distance between resonances,” J. Opt. Soc. Am. A 7, 1609–1613 (1990), and references therein.
    [Crossref]
  15. Expansion equation (1) was calculated up to order kmax = 1, inclusively, in Ref. 9.
  16. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 9.
  17. Equations (9.3.8) and (9.3.12) of Ref. 16.
  18. Equations (9.3.23) and (9.3.27) of Ref. 16.
  19. Equations (9.5.14) and (9.5.16) of Ref. 16.
  20. Equations (9.3.35) and (9.3.43) of Ref. 16.

1992 (1)

1991 (3)

S. Schiller, R. L. Byer, “High-resolution spectroscopy of whispering-gallery modes in large dielectric spheres,” Opt. Lett. 16, 1138–1140 (1991).
[Crossref] [PubMed]

A. J. Campillo, J. D. Eversole, H.-B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in micro-droplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[Crossref] [PubMed]

For references on whispering-gallery modes in systems other than droplets or spheres, see J. P. Braud, P. L. Hagelstein, “Whispering-gallery laser resonators. I. Diffraction of whispering gallery modes,” IEEE J. Quantum Electron. 27, 1069–1077 (1991).
[Crossref]

1990 (2)

1989 (1)

V. B. Braginsky, M. L. Gorodetsky, V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[Crossref]

1984 (1)

1939 (1)

R. D. Richtmyer, “Dielectric resonators,” J. Appl. Phys. 10, 391–398 (1939).
[Crossref]

Barber, P. W.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1991).

Benner, R. E.

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.

Braginsky, V. B.

V. B. Braginsky, M. L. Gorodetsky, V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[Crossref]

Braud, J. P.

For references on whispering-gallery modes in systems other than droplets or spheres, see J. P. Braud, P. L. Hagelstein, “Whispering-gallery laser resonators. I. Diffraction of whispering gallery modes,” IEEE J. Quantum Electron. 27, 1069–1077 (1991).
[Crossref]

Byer, R. L.

Campillo, A. J.

A. J. Campillo, J. D. Eversole, H.-B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in micro-droplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[Crossref] [PubMed]

Chylek, P.

Eversole, J. D.

A. J. Campillo, J. D. Eversole, H.-B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in micro-droplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[Crossref] [PubMed]

Gorodetsky, M. L.

V. B. Braginsky, M. L. Gorodetsky, V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[Crossref]

Hagelstein, P. L.

For references on whispering-gallery modes in systems other than droplets or spheres, see J. P. Braud, P. L. Hagelstein, “Whispering-gallery laser resonators. I. Diffraction of whispering gallery modes,” IEEE J. Quantum Electron. 27, 1069–1077 (1991).
[Crossref]

Hill, S. C.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1991).

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.

Ilchenko, V. S.

V. B. Braginsky, M. L. Gorodetsky, V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[Crossref]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 3.3.

Lam, C. C.

Lange, S.

Leung, P. T.

Lin, H.-B.

A. J. Campillo, J. D. Eversole, H.-B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in micro-droplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[Crossref] [PubMed]

Probert-Jones, J. R.

Richtmyer, R. D.

R. D. Richtmyer, “Dielectric resonators,” J. Appl. Phys. 10, 391–398 (1939).
[Crossref]

Schiller, S.

Schweiger, G.

Spengler, U.

Weinstein, L. A.

L. A. Weinstein, Open Resonators and Open Waveguides (Golem, Boulder, Colo., 1969), Sec. 58.

Young, K.

IEEE J. Quantum Electron. (1)

For references on whispering-gallery modes in systems other than droplets or spheres, see J. P. Braud, P. L. Hagelstein, “Whispering-gallery laser resonators. I. Diffraction of whispering gallery modes,” IEEE J. Quantum Electron. 27, 1069–1077 (1991).
[Crossref]

J. Appl. Phys. (1)

R. D. Richtmyer, “Dielectric resonators,” J. Appl. Phys. 10, 391–398 (1939).
[Crossref]

J. Opt. Soc. Am. A (2)

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

Opt. Lett. (2)

Phys. Lett. A (1)

V. B. Braginsky, M. L. Gorodetsky, V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[Crossref]

Phys. Rev. Lett. (1)

A. J. Campillo, J. D. Eversole, H.-B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in micro-droplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[Crossref] [PubMed]

Other (11)

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 3.3.

L. A. Weinstein, Open Resonators and Open Waveguides (Golem, Boulder, Colo., 1969), Sec. 58.

Expansion equation (1) was calculated up to order kmax = 1, inclusively, in Ref. 9.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 9.

Equations (9.3.8) and (9.3.12) of Ref. 16.

Equations (9.3.23) and (9.3.27) of Ref. 16.

Equations (9.5.14) and (9.5.16) of Ref. 16.

Equations (9.3.35) and (9.3.43) of Ref. 16.

Rayleigh scattering represents a lower limit for loss in liquid and glass spheres, Q < Qbulk = 2πns/λ0α; in fused silica α ≃ 2 × 10−6/cm at λ0 = 1 μm.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1991).

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Figures (2)

Fig. 1
Fig. 1

Accuracy of the l = 1, 2, 3 WGM frequencies, which were computed by using the asymptotic expansion Eq. (1), with kmax = 8, as a function of the mode number for three different indices. The numbers labeling the curves refer to the order l.

Fig. 2
Fig. 2

Relative error Δx60(l)/x60(l) of the asymptotic approximation Eq. (1) as a function of the expansion order for low-order modes. Refractive index, m = 1.363. (a) TM resonances and, in the inset, their radial field components Er(r) ≃ jn[mxn(l)r/R]/[mxn(l)r/R]; (b) TE resonances.

Tables (2)

Tables Icon

Table 1 Comparison between Exact and Asymptotic Size Parameters an(l) = xn(l) at Which TM Resonances Occur for Various Mode and Order Numbers and Refractive Index m = 1.363a

Tables Icon

Table 2 Same as Table 1 but for TE Resonances bn(l) = xn(l)

Equations (5)

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x n ( l ) = ν m ζ l m ( ν 2 ) 1 / 3 + k = 0 k max d k ( m , ζ l ) ν k / 3 ( m 2 1 ) ( k + 1 ) / 2 .
d 0 = p , d 1 = 2 1 / 3 3 ( m 2 1 ) ζ l 2 / ( 20 m ) , d 2 = 2 2 / 3 m 2 p ( 3 + 2 p 2 ) ζ l / 6 , d 3 = 350 m 4 ( 1 p ) p ( 1 + p + p 2 ) + ( m 2 1 ) 2 ( 10 + ζ l 3 ) 700 m , d 4 = 2 1 / 3 m 2 ζ l 2 ( 4 m 2 + e 4 ) 20 , d 5 = ζ l [ 40 ( 1 + 3 m 2 3 m 4 + 351 m 6 ) 479 ( m 2 1 ) 3 ζ l 3 e 5 ] 2 4 / 3 63 , 000 m , d 6 = 5 m 2 ( 13 16 m 2 + 4 m 4 ) + 2 m 2 ( 128 4 m 2 + m 4 ) ζ l 3 e 6 1400 , d 7 = ζ l 2 [ 100 ( 551 + 2204 m 2 3306 m 4 73 , 256 m 6 + 10 , 229 m 8 ) 20 , 231 ( m 2 1 ) 4 ζ l 3 + e 7 ] 2 2 / 3 16 , 170 , 000 m , d 8 = m 2 ζ l [ 10 ( 11 , 082 + 44 , 271 m 2 288 m 4 + 7060 m 6 ) 3 ( 52 , 544 + 48 , 432 m 2 11 , 496 m 4 + 2395 m 6 ) ζ l 3 ] + e 8 2 10 / 3 141 , 750 .
e 4 = ( 8 + 12 m 4 + m 8 ) / m 8 , e 5 = 7000 m 6 ( 28 m 2 + 56 m 4 16 m 6 7 m 8 + 2 m 10 ) , e 6 = m 8 [ 5 ( 200 32 m 2 + 526 m 4 226 m 6 99 m 8 + 62 m 10 + 4 m 12 ) + 2 ( 400 + 272 m 2 + 744 m 4 424 m 6 366 m 8 2 m 10 + m 12 ) ζ l 3 ] , e 7 = 269 , 500 m 8 ( 232 + 160 m 2 + 543 m 4 447 m 6 186 m 8 + 165 m 10 15 m 12 + 4 m 14 ) , e 8 = m 10 ζ l [ 10 ( 459 , 200 + 286 , 000 m 2 + 1 , 360 , 312 m 4 1 , 305 , 476 m 6 433 , 952 m 8 + 717 , 562 m 10 209 , 039 m 12 21 , 542 m 14 + 7060 m 16 ) + 3 ( 336 , 000 441 , 600 m 2 626 , 496 m 4 + 891 , 008 m 6 + 306 , 416 m 8 505 , 696 m 10 72 , 488 m 12 7664 m 14 + 2395 m 16 ) ζ l 3 ] .
Y ˜ ν ( x ) [ p 1 2 J ν ( m x ) + p m x J v ( m x ) ] ( ν 2 x 2 ) 1 / 2 J ν ( m x ) Y ˜ v ( x ) = 0 ,
Y ˜ ν ( x ) = ( π ν / 2 ) 1 / 2 [ 1 ( x / ν ) 2 ] 1 / 4 × exp ( ν { [ 1 ( x / ν ) 2 ] 1 / 2 arccosh ( ν / x ) } ) Y ν ( x ) , Y ˜ v ( x ) = x ( π / 2 ν ) 1 / 2 [ 1 ( x / ν ) 2 ] 1 / 4 × exp ( ν { [ 1 ( x / ν ) 2 ] 1 / 2 arccosh ( ν / x ) } ) Y v ( x ) ,

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