Abstract

In the geometrical-optics framework, the internal fields near morphology-dependent resonances (MDR) of dielectric spheres are represented by rays undergoing total internal reflection at the sphere surface. The round-trip path length of rays circumnavigating the sphere is used to compute the mode spacing of MDR’s. The Goos–Hänchen shift of the total internally reflected rays at the sphere surface is included in the ray picture to explain the qualitative behavior of the MDR frequency spacing in the Lorentz-Mie formalism for the entire size-parameter (circumference/wavelength) range. The MDR’s are characterized by a radial distance rm. A connection between the ray picture and the Mie theory is established, based on rm.

© 1994 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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1993 (3)

1992 (3)

D. Q. Chowdhury, P. W. Barber, S. C. Hill, “Energy-density distribution inside large nonabsorbing spheres by using Mie theory and geometrical optics,” Appl. Opt. 31, 3518–3523 (1992).
[Crossref] [PubMed]

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. A 9, 1585–1592 (1992).
[Crossref]

V. V. Datsyuk, “Some characteristics of resonant electromagnetic modes in a dielectric sphere,” Appl. Phys. B B-54, 184–187 (1992).
[Crossref]

1991 (6)

1990 (1)

1989 (4)

1988 (1)

H. Chew, “Radiation lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
[Crossref] [PubMed]

1987 (3)

1986 (3)

1985 (3)

1984 (1)

1979 (1)

1977 (1)

V. Khare, H. M. Nussenzveig, “Theory of glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[Crossref]

1974 (1)

Acker, W. P.

Armstrong, R. L.

A. Biswas, H. Latifi, R. L. Armstrong, R. G. Pinnick, “Double-resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. A 40,7413–7416 (1989).
[Crossref]

Barber, P. W.

Bar-Ziv, E.

W. M. Greene, R. E. Spjut, E. Bar-Ziv, A. F. Sarofim, J. P. Longwell, “Photophoresis of irradiated spheres: absorption center,” J. Opt. Soc. Am. A 2, 998–1004 (1985);erratum, 4, 864–865 (1988).
[Crossref]

Benincasa, D. S.

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;J.-Z. Zhang, D. H. Leach, R. K. Chang, “Photon lifetime within a droplet: temporal determination of elastic and stimulated Raman scattering,” Opt. Lett. 13,270–272 (1988).
[Crossref] [PubMed]

Biswas, A.

A. Biswas, H. Latifi, R. L. Armstrong, R. G. Pinnick, “Double-resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. A 40,7413–7416 (1989).
[Crossref]

Burke, J. J.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 74.

Byer, R. L.

Campillo, A. J.

Chang, R. K.

Cheng, F. C.

Chew, H.

H. Chew, “Radiation lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
[Crossref] [PubMed]

Ching, S. C.

Chowdhury, D. Q.

Chylek, P.

Datsyuk, V. V.

V. V. Datsyuk, “Some characteristics of resonant electromagnetic modes in a dielectric sphere,” Appl. Phys. B B-54, 184–187 (1992).
[Crossref]

Dean, C. E.

Dobson, C. C.

Fiedler-Ferrari, N.

N. Fiedler-Ferrari, H. M. Nussenzveig, W. J. Wiscombe, “Theory of near-critical angle scattering from curved surfaces,” Phys. Rev. A 43, 1005–1038 (1991).
[Crossref] [PubMed]

Greene, W. M.

W. M. Greene, R. E. Spjut, E. Bar-Ziv, A. F. Sarofim, J. P. Longwell, “Photophoresis of irradiated spheres: absorption center,” J. Opt. Soc. Am. A 2, 998–1004 (1985);erratum, 4, 864–865 (1988).
[Crossref]

Harrick, N. J.

N. J. Harrick, Internal Reflection Spectroscopy (Wiley, New York, 1967).

Hill, S. C.

D. H. Leach, R. K. Chang, W. P. Acker, S. C. Hill, “Third-order sum-frequency generation in droplets: experimental results,” J. Opt. Soc. Am. B 10, 34–45 (1993).
[Crossref]

D. Q. Chowdhury, P. W. Barber, S. C. Hill, “Energy-density distribution inside large nonabsorbing spheres by using Mie theory and geometrical optics,” Appl. Opt. 31, 3518–3523 (1992).
[Crossref] [PubMed]

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;J.-Z. Zhang, D. H. Leach, R. K. Chang, “Photon lifetime within a droplet: temporal determination of elastic and stimulated Raman scattering,” Opt. Lett. 13,270–272 (1988).
[Crossref] [PubMed]

Houston, A. L.

Hovenac, E. A.

Hsieh, W. F.

Jarzembski, M. A.

Johnson, B. R.

Justus, B. L.

Kapany, N. S.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 74.

Khare, V.

V. Khare, H. M. Nussenzveig, “Theory of glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[Crossref]

Kogelnik, H.

Lai, H. M.

Lam, C. C.

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. A 9, 1585–1592 (1992).
[Crossref]

Latifi, H.

A. Biswas, H. Latifi, R. L. Armstrong, R. G. Pinnick, “Double-resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. A 40,7413–7416 (1989).
[Crossref]

Leach, D. H.

Leung, P. T.

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. A 9, 1585–1592 (1992).
[Crossref]

H. M. Lai, P. T. Leung, K. L. Poon, K. Young, “Characteristics of the internal energy density in Mie scattering,” J. Opt. Soc. Am. A 8, 1553–1558 (1991).
[Crossref]

Lewis, J. W. L.

Lin, H. B.

Lock, J. A.

Long, M. B.

Longwell, J. P.

W. M. Greene, R. E. Spjut, E. Bar-Ziv, A. F. Sarofim, J. P. Longwell, “Photophoresis of irradiated spheres: absorption center,” J. Opt. Soc. Am. A 2, 998–1004 (1985);erratum, 4, 864–865 (1988).
[Crossref]

Marston, P. L.

Midwinter, J. E.

J. E. Midwinter, Optical Fibers for Transmission (Wiley, New York, 1979).

Nussenzveig, H. M.

N. Fiedler-Ferrari, H. M. Nussenzveig, W. J. Wiscombe, “Theory of near-critical angle scattering from curved surfaces,” Phys. Rev. A 43, 1005–1038 (1991).
[Crossref] [PubMed]

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979).
[Crossref]

V. Khare, H. M. Nussenzveig, “Theory of glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[Crossref]

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, 1992), pp. 48 and 89.

Pinnick, R. G.

A. Biswas, H. Latifi, R. L. Armstrong, R. G. Pinnick, “Double-resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. A 40,7413–7416 (1989).
[Crossref]

Poon, K. L.

Qian, S. X.

Sarofim, A. F.

W. M. Greene, R. E. Spjut, E. Bar-Ziv, A. F. Sarofim, J. P. Longwell, “Photophoresis of irradiated spheres: absorption center,” J. Opt. Soc. Am. A 2, 998–1004 (1985);erratum, 4, 864–865 (1988).
[Crossref]

Schiller, S.

Snow, J. B.

Snow, J. R.

Spjut, R. E.

W. M. Greene, R. E. Spjut, E. Bar-Ziv, A. F. Sarofim, J. P. Longwell, “Photophoresis of irradiated spheres: absorption center,” J. Opt. Soc. Am. A 2, 998–1004 (1985);erratum, 4, 864–865 (1988).
[Crossref]

Srivastava, V.

Tang, W. K.

Tzeng, H. M.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Wall, K. F.

Weber, H. P.

Wiscombe, W. J.

N. Fiedler-Ferrari, H. M. Nussenzveig, W. J. Wiscombe, “Theory of near-critical angle scattering from curved surfaces,” Phys. Rev. A 43, 1005–1038 (1991).
[Crossref] [PubMed]

Young, K.

Zhang, J. Z.

Appl. Opt. (5)

Appl. Phys. B (1)

V. V. Datsyuk, “Some characteristics of resonant electromagnetic modes in a dielectric sphere,” Appl. Phys. B B-54, 184–187 (1992).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (8)

J. Opt. Soc. Am. B (3)

Opt. Lett. (8)

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

V. Srivastava, M. A. Jarzembski, “Laser-induced stimulated Raman scattering in the forward direction of a droplet: comparison of Mie theory with geometrical optics,” Opt. Lett. 16, 126–128 (1991).
[Crossref] [PubMed]

W. P. Acker, D. H. Leach, R. K. Chang, “Third-order optical sum-frequency generation in micrometer-sized droplets,” Opt. Lett. 14, 402–404 (1989).
[Crossref] [PubMed]

S. X. Qian, J. B. Snow, R. K. Chang, “Coherent Raman mixing and coherent anti-Stokes Raman scattering from individual micrometer-size droplets,” Opt. Lett. 10, 499–501 (1985).
[Crossref] [PubMed]

S. X. Qian, R. K. Chang, “Phase-modulation-broadened line shapes from micrometer-size CS2droplets,” Opt. Lett. 11, 371–373 (1986).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

H. B. Lin, A. L. Houston, B. L. Justus, A. J. Campillo, “Some characteristics of a droplet whispering-gallery-mode laser,” Opt. Lett. 11, 614–616 (1986).
[Crossref] [PubMed]

J. R. 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).
[Crossref] [PubMed]

Phys. Rev. A (3)

A. Biswas, H. Latifi, R. L. Armstrong, R. G. Pinnick, “Double-resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. A 40,7413–7416 (1989).
[Crossref]

H. Chew, “Radiation lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
[Crossref] [PubMed]

N. Fiedler-Ferrari, H. M. Nussenzveig, W. J. Wiscombe, “Theory of near-critical angle scattering from curved surfaces,” Phys. Rev. A 43, 1005–1038 (1991).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

V. Khare, H. M. Nussenzveig, “Theory of glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[Crossref]

Other (7)

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, 1992), pp. 48 and 89.

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

N. J. Harrick, Internal Reflection Spectroscopy (Wiley, New York, 1967).

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 74.

J. E. Midwinter, Optical Fibers for Transmission (Wiley, New York, 1979).

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;J.-Z. Zhang, D. H. Leach, R. K. Chang, “Photon lifetime within a droplet: temporal determination of elastic and stimulated Raman scattering,” Opt. Lett. 13,270–272 (1988).
[Crossref] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Mode spacing between MDR’s of consecutive mode numbers but of the same order and polarization (TE) as shown as a function of size parameter x. (a) Mode spacing xn+1,lxn.l for several orders of MDR; (b) quality factor Q for the modes in (a), as a function of x. The refractive index of the sphere is 1.35.

Fig. 2
Fig. 2

Mode spacing for the 9th-order TE and TM modes (l = 9), computed with use of Mie theory, is shown as a function of size parameter. Mode spacing computed with use of the two asymptotic methods is also shown over the same size-parameter range. The refractive index of the sphere is mr = 1.35.

Fig. 3
Fig. 3

Geometrical-optics model showing one of the rays, RS, representing a MDR. The radius of the sphere is a.

Fig. 4
Fig. 4

Angle-averaged intensity contribution by resonant modes (TE) of different orders near size parameter 250 is shown along with the locations rm and dp. rm is computed with the relation rm = n/mrx, and dp is the location at which the external intensity is 1/e of that at the surface. The radial axis is normalized by the sphere radius a. The refractive index of the sphere is 1.35.

Fig. 5
Fig. 5

Radial distance rm and penetration depth dp are plotted for the mode orders shown in Fig. 1, as a function of the size parameter x. Penetration depth dp is computed with use of the Mie theory. Results for several orders of TE MDR’s are shown. The refractive index of the sphere is mr = 1.35.

Fig. 6
Fig. 6

Mode spacing Δx and penetration depth dp, computed with use of Mie theory and the geometrical-optics model, are shown as a function of rm. Results are shown for several orders of TE MDR’s. The refractive index of the sphere is mr = 1.35.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

δ = S ( θ i ) 2 tan θ i ,
S ( θ i ) = σ ( a 1 ) 1 ( a 2 ) ,
exp ( i π / 4 ) × exp ( β 0 2 / 4 ) [ B D 1 / 2 ( β 0 ) + i 2 E D 1 / 2 ( β 0 ) / k σ ] + i 2 C / k σ 2 [ k σ A + B exp ( i π / 4 ) exp ( β 0 2 / 4 ) D 1 / 2 ( β 0 ) ] ,
B exp ( i π / 4 ) exp ( β 0 2 / 4 ) D 3 / 2 ( β 0 ) 4 A k σ ,
2 ξ cos θ i sin 2 θ i ,
2 ξ sin θ i sin 2 θ i × [ ξ 2 ( cos 2 θ i + cos 2 θ c ) cos 2 θ i ] / G ,
Δ = ( sin 2 θ i sin 2 θ c ) / sin 2 θ i .
d = S ( θ i ) k 2 x tan θ i = 2 π x ( a 1 ) / [ 1 ( a 2 ) ] 2 x tan θ i
υ ϕ = c m r sin θ i ,
r m = n m r x
r r t = ( 2 π / 2 β ) R S .
c i = ( 1 r m 2 ) 1 / 2 , c e = [ ( 1 + d 2 ) r m 2 ] 1 / 2 ( 1 r m 2 ) 1 / 2 .
π β c o p = n λ , π β 2 a ( c e + m r c i ) = 2 n π a x n , x n = n β ( c e + m r c i )
Δ x = x n + 1 , l x n , l = β c e + m r c i ,
n x n + 1 x + Δ x = n + 1 x + 1 / m r

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