Abstract

Mie theory and geometrical-optics ray tracing are used to obtain the distribution of electric energy density inside a nonabsorbing micrometer-sized sphere illuminated by a polarized plane wave. The Mie solution shows the multiply reflected geometrical-optics rays inside a sphere having a diameter of ~ 150 free-space wavelengths (size parameter = circumference/wavelength = 500). The geometrical-optics result shows the major features of the Mie solution and provides a physical interpretation of the electromagnetic interactions that result in the observed energy-density distributions. Both solutions show internal on-axis energy-density maxima inside the shadow surface of the sphere. The region of greatest enhanced energy density is approximately one internal wavelength in diameter and approximately twenty internal wavelengths in length.

© 1992 Optical Society of America

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References

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  1. 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); J.-Z. Zhang, R. K. Chang, “Shape distortion of a single water droplet by laser-induced electrostriction,” Opt. Lett. 13, 916–918 (1988).
    [CrossRef] [PubMed]
  2. R. G. Pinnick, P. Chylek, M. Jarzembski, E. Creegan, V. Srivastava, G. Fernandez, J. D. Pendleton, A. Biswas, “Aerosol-induced laser breakdown thresholds: wavelength dependence,” Appl. Opt. 27, 987–996 (1988); J.-B. Zheng, W.-F. Hsieh, S.-C. Chen, R. K. Chang, “Temporally and spatially resolved spectroscopy of laser-induced plasma from a droplet,” Opt. Lett. 13, 559–561 (1988).
    [CrossRef] [PubMed]
  3. 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).
    [CrossRef] [PubMed]
  4. 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 from individual micrometer-size droplets,” Opt. Lett. 10, 499–501 (1985); S.-X. Qian, R. K. Chang, “Multi-order Stokes emission from micrometer-size droplets,” Phys. Rev. Lett. 56, 926–929 (1986).
    [CrossRef] [PubMed]
  5. 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); S. M. Chitvanis, C. D. Cantrell, “Simple approach to stimulated Brillouin scattering in glass aerosols,” J. Opt. Soc. Am. B 6, 1326–1331 (1989).
    [CrossRef]
  6. W. P. Acker, D. H. Leach, R. K. Chang, “Third-order optical sum-frequency generation in micrometer-sized liquid droplets,” Opt. Lett. 14, 402–404 (1989).
    [CrossRef] [PubMed]
  7. W. M. Greene, R. E. Spjut, E. Bar-Ziv, A. F. Sarofim, J. P. Longwell, “Photophoresis of irradiated spheres: absorption centers,” J. Opt. Soc. Am. A 2, 998–1004 (1985); erratum 4, 864–865 (1988).
    [CrossRef]
  8. D. S. Benincasa, P. W. Barber, J. Z. Zhang, W. F. Hsieh, R. K. Chang, “Spatial distribution of the internal and near-field intensities of large cylindrical and spherical scatterers,” Appl. Opt. 26, 1348–1356 (1987).
    [CrossRef] [PubMed]
  9. C. C. Dobson, J. W. L. Lewis, “Survey of the Mie problem source function,” J. Opt. Soc. Am. A 6, 463–466 (1989).
    [CrossRef]
  10. 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]
  11. 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 8, 1553–1558 (1991).
    [CrossRef]
  12. 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).
  13. P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1991).
  14. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  15. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).
  16. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  17. A. R. Steinhardt, L. Fukshansky, “Geometrical optics approach to the intensity distribution in finite cylindrical media,” Appl. Opt. 26, 3778–3789 (1987).
    [CrossRef] [PubMed]
  18. M. A. Jarzembski, V. Srivastava, “Electromagnetic field enhancement in small liquid droplets using geometric optics,” Appl. Opt. 28, 4962–4965 (1989).
    [CrossRef] [PubMed]

1991 (2)

1989 (5)

1988 (1)

1987 (2)

1985 (2)

1984 (1)

Acker, W. P.

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 centers,” 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).

Biswas, A.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

Chang, R. K.

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

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); S. M. Chitvanis, C. D. Cantrell, “Simple approach to stimulated Brillouin scattering in glass aerosols,” J. Opt. Soc. Am. B 6, 1326–1331 (1989).
[CrossRef]

D. S. Benincasa, P. W. Barber, J. Z. Zhang, W. F. Hsieh, R. K. Chang, “Spatial distribution of the internal and near-field intensities of large cylindrical and spherical scatterers,” Appl. Opt. 26, 1348–1356 (1987).
[CrossRef] [PubMed]

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 from individual micrometer-size droplets,” Opt. Lett. 10, 499–501 (1985); S.-X. Qian, R. K. Chang, “Multi-order Stokes emission from micrometer-size droplets,” Phys. Rev. Lett. 56, 926–929 (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); 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).
[CrossRef] [PubMed]

Chylek, P.

Creegan, E.

Dobson, C. C.

Fernandez, G.

Fukshansky, L.

Greene, W. M.

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

Hsieh, W. F.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

Jarzembski, M.

Jarzembski, M. A.

Lai, H. M.

Leach, D. H.

Leung, P. T.

Lewis, J. W. L.

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 centers,” J. Opt. Soc. Am. A 2, 998–1004 (1985); erratum 4, 864–865 (1988).
[CrossRef]

Pendleton, J. D.

Pinnick, R. G.

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 centers,” J. Opt. Soc. Am. A 2, 998–1004 (1985); erratum 4, 864–865 (1988).
[CrossRef]

Snow, J. B.

Spjut, R. E.

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

Srivastava, V.

Steinhardt, A. R.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

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.

Young, K.

Zhang, J. Z.

Zhang, J.-Z.

Appl. Opt. (4)

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

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

Opt. Lett. (4)

Other (5)

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).

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

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

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

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Figures (10)

Fig. 1
Fig. 1

Geometry of the sphere of radius a illuminated by a plane wave propagating in the +z direction. the polarization of the incident wave is parallel or perpendicular to the xz plane.

Fig. 2
Fig. 2

Wedge-shaped region inside which the rays are confined: (a) inside the sphere; (b) in the xy plane for which the symbol ⊕ denotes the entering rays.

Fig. 3
Fig. 3

Geometrical-optics rays in the xz plane for a sheet of perpendicularly polarized rays incident from the left. The index of refraction is 1.332. Only the p = 1 rays are shown.

Fig. 4
Fig. 4

Mie solution for the source function from the forward direction (θ = 0°) to the shadow boundary (θ = 90°) for perpendicularly polarized incident light. The geometrical-optics critical angle is θc.

Fig. 5
Fig. 5

Geometrical-optics rays in the xz plane for a sheet of perpendicularly polarized rays incident from the left. The incident intensity is 30 times the incident intensity in Fig. 3 and the index of refraction is 1.332: (a) the p = 2 and subsequent rays, θ1 = 20° and θ2 = 70°; (b) the p = 1 and subsequent rays showing only those rays that are incident near the shadow boundary of the sphere.

Fig. 6
Fig. 6

Surface graph of the geometrical-optics solution for the source function in the xz plane for perpendicularly polarized light incident from the upper left. The index of refraction is 1.332. The center of the sphere is located at the center of the grid. The dimensions in the horizontal plane are r/a = ±1 in both directions. The maximum is 252.2.

Fig. 7
Fig. 7

Surface graph of the Mie solution for the source function in the xz plane for perpendicularly polarized light incident from the upper left. The size parameter is 500 and the index of refraction is 1.332. The center of the sphere is located at the center of the grid. The dimensions in the horizontal plane are r/a = ±1 in both directions. The maximum is 253.7.

Fig. 8
Fig. 8

Contour graph of the Mie solution for the same case as shown in Fig. 7. Nonuniform contour intervals were selected to highlight the individual rays.

Fig. 9
Fig. 9

Mie solution for the source function along the central line of Fig. 7. The normalized radius r/a varies from −1 at the illuminated face to +1 at the shadow face.

Fig. 10
Fig. 10

Mie solution for the source function perpendicular to the central line of Fig. 7 at r/a = 0.6317 (solid curve) and r/a = 0.75 (dashed curve). The normalized transverse dimension is x/a.

Equations (4)

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E ( m k r ) = E 0 n = 1 ( c n M o 1 n 1 + d n N e 1 n 1 ) ,
E ( m k r ) = E 0 n = 1 ( - c n M e 1 n 1 + d n N o 1 n 1 ) ,
θ c = 180 ° - 2 sin - 1 [ 1 - ( sin γ c / m ) 2 ] 1 / 2 - γ c ,
γ c = sin - 1 [ ( 4 - m 2 ) / 3 ] 1 / 2 .

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