Abstract

The distribution of the internal field energy in nonresonant Mie scattering is shown to exhibit certain regularities not previously noticed. The distribution, when suitably expressed, is independent of the size of the spherical scatterer and obtainable from geometric optics, with the average over any spherical surface calculable in closed form. A discontinuity at a radius r = a/n is found and described, where a is the radius of the sphere and n the refractive index.

© 1991 Optical Society of America

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References

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  1. G. Mie “Beitrage zur Optik truber Medien, speziell kolloidaler Metaalosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908);M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969);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);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).
    [CrossRef] [PubMed]
  2. 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);J.-Z. Zhang, R. K. Chang, “Shape distortion of a single water droplet by laser-induced electrostriction,” Opt. Lett. 13, 916–918 (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).
    [CrossRef] [PubMed]
  3. A. L. Huston, H.-B. Lin, J. D. Eversole, A. J. Campillo, “Nonlinear Mie scattering: electrostrictive coupling of light of droplet accoustic modes,” Opt. Lett. 15, 1176–1178 (1990).
    [CrossRef] [PubMed]
  4. R. Y. Chiao, C. H. Townes, B. P. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12, 592–595 (1964);J.-Z. Zhang, R. K. Chang, “Generation and suppression of stimulated Brillouin scattering in single liquid droplets,” J. Opt. Soc. Am. B 6, 151–153 (1989);P. T. Leung, K. Young, “Doubly resonant stimulated Brillouin scattering in a microdroplet,” Phys. Rev. A (to be published).
    [CrossRef]
  5. 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);S.-X. Qian, J. B. Snow, R. K. Chang, “Coherent Raman mixing and coherent anti-Stokes Raman scattering form individual micrometer-sized droplets,” Opt. Lett. 10, 499–501 (1985);S.-X. Qian, R. K. Chang, “Multiorder Stokes emission from micrometer-sized droplets,” Phys. Rev. Lett. 56, 926–929 (1986).
    [CrossRef] [PubMed]
  6. D. S. Benincasa, P. W. Barber, J.-Z. Zhang, W-F. Hsieh, R. K. Chang, “Spatial distribution of the internal and near-field intensity of large cylindrical and spherical scatterers,” Appl. Opt. 26, 1348–1356 (1987).
    [CrossRef] [PubMed]
  7. P. Chylek, J. D. Pendleton, R. G. Pinnick, “Internal and near-surface scattered field of a spherical particle at resonance conditions,” Appl. Opt. 24, 3940–3942 (1985).
    [CrossRef] [PubMed]
  8. W. Hartle, “Zur Theorie der Lichtstreuung durch trube Schichten, besonders Trubglaser,” Licht 10, 141–143 (1940).
  9. C. M. Chu, S. W. Churchill, “Representation of the angular distribution of radiation scattered by a spherical particle,” J. Opt. Soc. Am. 45, 958–962 (1955).
    [CrossRef]
  10. D. Q. Chowdhury, P. W. Barber, S. C. Hill, “Energy–density distribution inside large nonabsorbing spheres via Mie theory and geometric optics,” submitted to Appl. Opt.
  11. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975);P. W. Barber, R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, Singapore, 1988).
  12. S. C. Ching, H. M. Lai, K. Young, “Dielectric microsphere as optical cavities: thermal spectrum and density of states,” J. Opt. Soc. Am. B 4, 1995–2003 (1987).
    [CrossRef]
  13. M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957);A. R. Edmonds, Angular Momentum in Quantum Mechanics, 2nd ed. (Princeton U. Press, Princeton, N.J., 1960).
  14. V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
    [CrossRef]
  15. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

1990 (1)

1987 (2)

1985 (3)

1977 (1)

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

1964 (1)

R. Y. Chiao, C. H. Townes, B. P. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12, 592–595 (1964);J.-Z. Zhang, R. K. Chang, “Generation and suppression of stimulated Brillouin scattering in single liquid droplets,” J. Opt. Soc. Am. B 6, 151–153 (1989);P. T. Leung, K. Young, “Doubly resonant stimulated Brillouin scattering in a microdroplet,” Phys. Rev. A (to be published).
[CrossRef]

1955 (1)

1940 (1)

W. Hartle, “Zur Theorie der Lichtstreuung durch trube Schichten, besonders Trubglaser,” Licht 10, 141–143 (1940).

1908 (1)

G. Mie “Beitrage zur Optik truber Medien, speziell kolloidaler Metaalosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908);M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969);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);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).
[CrossRef] [PubMed]

Barber, P. W.

Benincasa, D. S.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Campillo, A. J.

Chang, R. K.

Chiao, R. Y.

R. Y. Chiao, C. H. Townes, B. P. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12, 592–595 (1964);J.-Z. Zhang, R. K. Chang, “Generation and suppression of stimulated Brillouin scattering in single liquid droplets,” J. Opt. Soc. Am. B 6, 151–153 (1989);P. T. Leung, K. Young, “Doubly resonant stimulated Brillouin scattering in a microdroplet,” Phys. Rev. A (to be published).
[CrossRef]

Ching, S. C.

Chowdhury, D. Q.

D. Q. Chowdhury, P. W. Barber, S. C. Hill, “Energy–density distribution inside large nonabsorbing spheres via Mie theory and geometric optics,” submitted to Appl. Opt.

Chu, C. M.

Churchill, S. W.

Chylek, P.

Eversole, J. D.

Hartle, W.

W. Hartle, “Zur Theorie der Lichtstreuung durch trube Schichten, besonders Trubglaser,” Licht 10, 141–143 (1940).

Hill, S. C.

D. Q. Chowdhury, P. W. Barber, S. C. Hill, “Energy–density distribution inside large nonabsorbing spheres via Mie theory and geometric optics,” submitted to Appl. Opt.

Hsieh, W-F.

Huston, A. L.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975);P. W. Barber, R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, Singapore, 1988).

Khare, V.

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

Lai, H. M.

Lin, H.-B.

Long, M. B.

Mie, G.

G. Mie “Beitrage zur Optik truber Medien, speziell kolloidaler Metaalosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908);M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969);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);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).
[CrossRef] [PubMed]

Nussenzveig, H. M.

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

Pendleton, J. D.

Pinnick, R. G.

Qian, S.-X.

Rose, M. E.

M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957);A. R. Edmonds, Angular Momentum in Quantum Mechanics, 2nd ed. (Princeton U. Press, Princeton, N.J., 1960).

Snow, J. B.

Stoicheff, B. P.

R. Y. Chiao, C. H. Townes, B. P. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12, 592–595 (1964);J.-Z. Zhang, R. K. Chang, “Generation and suppression of stimulated Brillouin scattering in single liquid droplets,” J. Opt. Soc. Am. B 6, 151–153 (1989);P. T. Leung, K. Young, “Doubly resonant stimulated Brillouin scattering in a microdroplet,” Phys. Rev. A (to be published).
[CrossRef]

Townes, C. H.

R. Y. Chiao, C. H. Townes, B. P. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12, 592–595 (1964);J.-Z. Zhang, R. K. Chang, “Generation and suppression of stimulated Brillouin scattering in single liquid droplets,” J. Opt. Soc. Am. B 6, 151–153 (1989);P. T. Leung, K. Young, “Doubly resonant stimulated Brillouin scattering in a microdroplet,” Phys. Rev. A (to be published).
[CrossRef]

Tzeng, H.-M.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Young, K.

Zhang, J.-Z.

Ann. Phys. (Leipzig) (1)

G. Mie “Beitrage zur Optik truber Medien, speziell kolloidaler Metaalosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908);M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969);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);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).
[CrossRef] [PubMed]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

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

Licht (1)

W. Hartle, “Zur Theorie der Lichtstreuung durch trube Schichten, besonders Trubglaser,” Licht 10, 141–143 (1940).

Opt. Lett. (3)

Phys. Rev. Lett. (2)

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

R. Y. Chiao, C. H. Townes, B. P. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12, 592–595 (1964);J.-Z. Zhang, R. K. Chang, “Generation and suppression of stimulated Brillouin scattering in single liquid droplets,” J. Opt. Soc. Am. B 6, 151–153 (1989);P. T. Leung, K. Young, “Doubly resonant stimulated Brillouin scattering in a microdroplet,” Phys. Rev. A (to be published).
[CrossRef]

Other (4)

D. Q. Chowdhury, P. W. Barber, S. C. Hill, “Energy–density distribution inside large nonabsorbing spheres via Mie theory and geometric optics,” submitted to Appl. Opt.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975);P. W. Barber, R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, Singapore, 1988).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957);A. R. Edmonds, Angular Momentum in Quantum Mechanics, 2nd ed. (Princeton U. Press, Princeton, N.J., 1960).

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Figures (5)

Fig. 1
Fig. 1

Two hot spots inside the droplet that are due to the focusing of light at the curved surfaces.

Fig. 2
Fig. 2

Fl(r, X) for n = 1.33 versus r/a. The points are calculated by physical optics and refer to different sizes: squares, X = 300; circles, X = 500; triangles, X = 1000. Except near the rim, the results are independent of size. The curves are calculated by geometric optics.

Fig. 3
Fig. 3

(a) Position of the kink r/a is related to the refractive index n. The points are the physical-optics results for X = 1000; the line is r/a = 1/n. (b) Value of F for r/a < 1/n versus n. The points are F0(r/a = 0.3) calculated by physical optics for X = 1000; the line is F0 = n.

Fig. 4
Fig. 4

Schematic showing a ray at impact parameter b entering the droplet of radius a (large circle). The ray is required to evaluate the energy density on a sphere of radius r (small circle). The beam enters the sphere of radius r at the points Ak.

Fig. 5
Fig. 5

Relative energy distribution in the equatorial plane from Eq. (1) with terms up to l = 40 and coefficients evaluated by geometric optics. The direction of incidence is shown by the arrow.

Equations (19)

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E 2 = = 0 F ( r , θ , X ) ,
F ( r , θ , X ) = l F l ( r , X ) P l ( cos θ ) .
F l ( r , X ) = F l ( r / α )
E = l m α l m j l ( n k r ) X l m ( θ , φ ) + ,
F l ( r , x ) = 2 l + 1 4 π d Ω P l ( cos θ ) E * E .
F l ( r , X ) = l , l 2 α l 1 * α l 2 j l 1 ( n k r ) j l 2 ( n k r ) × d Ω P l ( cos θ ) X l 1 * ( θ , φ ) X l 2 * ( θ , φ ) + ,
[ ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) ] 1 / 2 2 l + 1 C ( l 1 l 2 l , 000 ) 2 cos 2 θ 12 ,
cos θ 12 = l 1 ( l 1 + l 1 ) + l 2 ( l 2 + 1 ) l ( l + 1 ) 2 [ l 1 ( l 1 + 1 ) l 2 ( l 2 + 1 ) ] 1 / 2 .
F 0 ( r / a ) = n , r / a < 1 / n .
F 0 = 2 4 π r 2 b k E k ( b ) 2 Δ S k ( b ) ,
Φ k ( b ) = ( c / n ) n 2 E k ( b ) 2 Δ S k ( b ) cos β ,
Φ k ( b ) = ( c n ) 2 π b Δ b cos ϕ cos θ n 2 T R k .
F 0 = 1 r 2 d b b T 1 R cos ϕ cos β cos θ .
T 1 R = cos θ n cos ϕ ,
F 0 = n [ 1 Θ ( a / n r ) ( 1 a 2 / n 2 r 2 ) 1 / 2 ] ,
/ 0 = F = 3 0 1 d ( r / a ) ( r / a ) 2 F 0 ( r / a ) = n [ 1 ( 1 n 2 ) 3 / 2 ] .
F l ( r a ) = 2 l + 1 2 r 2 0 n r d b b T × cos ϕ cos β cos θ k R k P l [ cos θ k ( b ) ] + ,
θ k ( b ) = θ 0 ( b ) + k ( π 2 ϕ ) ,
F l ( r a ) = 3 r 2 0 n r d b b T cos ϕ cos β cos θ Im ( sin β e i θ e i ϕ + R e i ϕ ) ,

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