Abstract

The work of a previous paper [Appl. Opt. 31, 7132 (1992)] is extended to the case of polarized inelastic (Raman and fluorescent) scattering from molecules embedded in spherical particles of large optical size parameters. The hybrid modeling technique, which combines the Lorenz–Mie theory with a geometric optics method, accounts for the contributions of directly transmitted rays as well as reflected–transmitted rays of secondary emissions. Coherent effects of light rays emitted from a single point source are considered in the model. The angular scattering patterns predicted with the modeling technique are consistent with expected physical behavior and results from classical solutions. This work has a direct impact on studies of particle thermometry and particle diagnostics in which inelastic-scattering techniques are beginning to be applied.

© 1992 Optical Society of America

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References

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  1. J. Zhang, D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: unpolarized emissions,” Appl. Opt. 31, 7132–7139 (1992).
    [CrossRef] [PubMed]
  2. M. R. Wells, L. A. Melton, “Temperature measurements of falling droplets,” Trans. ASME 112, 1008–1013 (1990).
    [CrossRef]
  3. M. Seaver, J. R. Peele, “Noncontact fluorescence thermometry of acoustically levitated water drops,” Appl. Opt. 29, 4956–4961 (1990).
    [CrossRef] [PubMed]
  4. M. Winter, “Measurement of the temperature field inside a falling droplet,” presented at the Fourth Annual Conference of the Institute of Liquid Atomization and Sprays—North and South America, Hartford, Conn., 1990.
  5. J. Zhang, “Fluorescence methods for determination of temperature in aerosol particles,” Ph.D. dissertation (Department of Mechanical Engineering, University of Nebraska at Lincoln, Lincoln, 1991), pp. 56–79.
  6. A. S. Kwok, C. F. Wood, R. K. Chang, “Fluorescence imaging of CO2 laser-heated droplets,” Opt. Lett. 15, 664–666 (1990).
    [CrossRef] [PubMed]
  7. H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phy. Rev. A 13, 396–404 (1976).
    [CrossRef]
  8. M. Kerker, P. J. McNulty, M. Sculley, H. Chew, D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: numerical results for incoherent optical processes,” J. Opt. Soc. Am. 68, 1676–1686 (1978).
    [CrossRef]
  9. H. Chew, M. Sculley, M. Kerker, P. J. McNulty, D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: numerical results for coherent optical processes,” J. Opt. Soc. Am. 68, 1686–1689 (1978).
    [CrossRef]
  10. M. Kerker, S. D. Druger, “Raman and fluorescent scattering by molecules embedded in spheres with radii up to several multiples of the wavelength,” Appl. Opt. 18, 1172–1179 (1979).
    [CrossRef] [PubMed]
  11. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 200–207.
  12. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 120–121.
  13. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 81–83.
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 21–25.
  15. A. Ashkin, J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20, 1803–1814 (1981).
    [CrossRef] [PubMed]
  16. S.-X. Qing, 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]
  17. P. Chylek, “Partial-wave resonances and the ripple structure in the Mie normalized extinction cross section,” J. Opt. Soc. Am. 66, 285–287 (1976).
    [CrossRef]
  18. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal field of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
    [CrossRef]
  19. H.-B. Lin, A. L. Huston, B. L. Justus, A. J. Camplillo, “Some characteristics of droplet whispering-gallery-mode laser,” Opt. Lett. 11, 614–616 (1986).
    [CrossRef] [PubMed]
  20. M. Winter, T. J. Anderson, “Measurements of the effect of acoustic disturbances on droplet valorization rates,” Presented at the 28th Joint Army–Navy–NASA–Air Force Combustion Meeting, San Antonio, Tex., 1991.

1992 (1)

1990 (3)

1989 (1)

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal field of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

1986 (2)

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

S.-X. Qing, 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]

1981 (1)

1979 (1)

1978 (2)

1976 (2)

P. Chylek, “Partial-wave resonances and the ripple structure in the Mie normalized extinction cross section,” J. Opt. Soc. Am. 66, 285–287 (1976).
[CrossRef]

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phy. Rev. A 13, 396–404 (1976).
[CrossRef]

Alexander, D. R.

J. Zhang, D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: unpolarized emissions,” Appl. Opt. 31, 7132–7139 (1992).
[CrossRef] [PubMed]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal field of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

Anderson, T. J.

M. Winter, T. J. Anderson, “Measurements of the effect of acoustic disturbances on droplet valorization rates,” Presented at the 28th Joint Army–Navy–NASA–Air Force Combustion Meeting, San Antonio, Tex., 1991.

Ashkin, A.

Barton, J. P.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal field of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 81–83.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 120–121.

Camplillo, A. J.

Chang, R. K.

A. S. Kwok, C. F. Wood, R. K. Chang, “Fluorescence imaging of CO2 laser-heated droplets,” Opt. Lett. 15, 664–666 (1990).
[CrossRef] [PubMed]

S.-X. Qing, 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]

Chew, H.

Chylek, P.

Cooke, D. D.

Druger, S. D.

Dziedzic, J. M.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 21–25.

Huston, A. L.

Justus, B. L.

Kerker, M.

Kwok, A. S.

Lin, H.-B.

McNulty, P. J.

Melton, L. A.

M. R. Wells, L. A. Melton, “Temperature measurements of falling droplets,” Trans. ASME 112, 1008–1013 (1990).
[CrossRef]

Peele, J. R.

Qing, S.-X.

S.-X. Qing, 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]

Schaub, S. A.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal field of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

Sculley, M.

Seaver, M.

Snow, J. B.

S.-X. Qing, 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]

Tzeng, H.-M.

S.-X. Qing, 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]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 200–207.

Wells, M. R.

M. R. Wells, L. A. Melton, “Temperature measurements of falling droplets,” Trans. ASME 112, 1008–1013 (1990).
[CrossRef]

Winter, M.

M. Winter, T. J. Anderson, “Measurements of the effect of acoustic disturbances on droplet valorization rates,” Presented at the 28th Joint Army–Navy–NASA–Air Force Combustion Meeting, San Antonio, Tex., 1991.

M. Winter, “Measurement of the temperature field inside a falling droplet,” presented at the Fourth Annual Conference of the Institute of Liquid Atomization and Sprays—North and South America, Hartford, Conn., 1990.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 81–83.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 120–121.

Wood, C. F.

Zhang, J.

J. Zhang, D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: unpolarized emissions,” Appl. Opt. 31, 7132–7139 (1992).
[CrossRef] [PubMed]

J. Zhang, “Fluorescence methods for determination of temperature in aerosol particles,” Ph.D. dissertation (Department of Mechanical Engineering, University of Nebraska at Lincoln, Lincoln, 1991), pp. 56–79.

Appl. Opt. (4)

J. Appl. Phys. (1)

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal field of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Lett. (2)

Phy. Rev. A (1)

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phy. Rev. A 13, 396–404 (1976).
[CrossRef]

Science (1)

S.-X. Qing, 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]

Trans. ASME (1)

M. R. Wells, L. A. Melton, “Temperature measurements of falling droplets,” Trans. ASME 112, 1008–1013 (1990).
[CrossRef]

Other (7)

M. Winter, “Measurement of the temperature field inside a falling droplet,” presented at the Fourth Annual Conference of the Institute of Liquid Atomization and Sprays—North and South America, Hartford, Conn., 1990.

J. Zhang, “Fluorescence methods for determination of temperature in aerosol particles,” Ph.D. dissertation (Department of Mechanical Engineering, University of Nebraska at Lincoln, Lincoln, 1991), pp. 56–79.

M. Winter, T. J. Anderson, “Measurements of the effect of acoustic disturbances on droplet valorization rates,” Presented at the 28th Joint Army–Navy–NASA–Air Force Combustion Meeting, San Antonio, Tex., 1991.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 200–207.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 120–121.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 81–83.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 21–25.

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Figures (9)

Fig. 1
Fig. 1

Schematic showing the relationship between the observation coordinate system ABC and the A′–B′–C′ systems.

Fig. 2
Fig. 2

Schematic showing the coordinate system used for the ART method.

Fig. 3
Fig. 3

Distributions of polarized scattering functions on the BC plane. n1 = 1.333, α = 1500. Detector is located on the bottom of the page viewing upward. (a) SAV(R), M = 0; (b) SAV(R), M = 10; (c) SBH(R), M = 0; (d) SBH(R), M = 10.

Fig. 4
Fig. 4

Samples of directly transmitted rays with deviation angle β = 0; n1 = 1.333.

Fig. 5
Fig. 5

G ¯ SAV and normalized computation time versus M; α = 1500, n1 = 1.333, (left scale is for G ¯ SAV, right scale is for CPU time).

Fig. 6
Fig. 6

PXV(θ) and PYH(θ) versus the observation angle θ. Excitation parameters: n ^ 1 = 1.333 + 1 × 10−6 i, φ1 = 500. Emission parameters: n1 = 1.333, α2 = 436, M = 0.

Fig. 7
Fig. 7

UXV(θ) and UYH(θ) versus the observation angle θ. Excitation parameters: n ^ 1 = 1.333 + 1 × 10−6 i, α1 = 500. Emission parameters: n1 = 1.333, α2 = 436, M = 0.

Fig. 8
Fig. 8

PXV(θ) and PYH(θ) versus θ (input resonance). α1 = 499.82928; α2 = 435.85113. The dashed curve PXV(θ) and the lowest curve PYH(θ) depict the nonresonance case.

Fig. 9
Fig. 9

UXV(θ) and UYH(θ) versus θ (input resonance). α1 = 499.82928, α2 = 435.85113. The dashed curve PXV(θ) and the lowest curve PYH(θ) depict the nonresonance case.

Equations (22)

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P j i ( θ ) = Ω { [ S ( R ) · C ( R ) · E ( R ) ] j i 2 } θ d v ,
φ 2 m = φ 1 - ( 2 m + 1 ) sin - 1 ( b sin φ 1 ) + sin - 1 ( n 1 b sin φ 1 ) + m π ,
φ 2 , = { 2 k π + ϑ , 2 ( k + 1 ) π - ϑ ,             k = 0 , 1 , , 0 < φ 2 m < M 1 π ,             { M 1 = m + 1 M 1 2 ,
S ( R ) = m = 1 M k exp [ i ϕ m ( k ) ] [ D m ( k ) ] 1 / 2 × P 2 m ( k ) · T m ( k ) · P 1 m ( k ) .
S j i ( R ) = [ S ( R ) · E ^ j ] i 2 ,
T m = [ T m 0 0 T m ] ,
T m = t ( r ) m ,
T m = t ( r ) m .
r = sin ( θ 2 - θ 1 ) sin ( θ 2 + θ 1 ) ,
r = tan ( θ 1 - θ 2 ) tan ( θ 1 + θ 2 ) ,
t = ( 1 - r 2 ) 1 / 2 ,
t = ( 1 - r 2 ) 1 / 2 .
D m = | d φ 2 m d φ 1 · sin φ 2 m sin φ 1 | - 1 ,
d φ 2 m d φ 1 = 1 + b n 1 cos φ 1 ( 1 - b 2 n 1 2 sin 2 φ 1 ) 1 / 2 - ( 2 m + 1 ) b cos φ 1 ( 1 - b 2 sin 2 φ 1 ) 1 / 2 .
P 1 = [ cos Φ 0 - sin Φ sin Φ cos ψ - sin ψ cos Φ cos ψ ] ,
P 2 = [ cos Φ sin Φ - sin Φ cos Φ ]
ϕ m = N π 2 - α ( n 1 L m + 1 - cos θ 2 ) ,
L m = ( 2 m + 1 ) cos θ 1 - b cos φ 1 ,
N = m + q + s 2 ( 1 - s 1 2 ) ,
s 1 = sign of ( d φ 2 m d φ 1 ) ,
s 2 = sign of ( φ 1 - π 2 ) ,
q = truncated integer of ( φ 2 m π ) .

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