Abstract

A hybrid modeling technique is reported for studying inelastic (Raman and fluorescent) scattering from molecules embedded in spherical particles of large optical size parameters. The modeling technique, which combines the Lorenz–Mie theory (for determination of the incident excitation field) with a geometric optics formulation (for determination of an inelastic-scattering efficiency function), permits predictions of a weighting function inside a particle and also the angular scattering patterns. These calculations provide insight into the scattering processes and may serve as a theoretical basis for guiding experiments and interpreting results in aerosol particle thermometry by using inelastic-scattering techniques.

© 1992 Optical Society of America

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References

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  3. B.-S. Park, R. L. Armstrong, “Laser droplet heating: fast and slow heating regimes,” Appl. Opt. 28, 3671–3680 (1989).
    [Crossref] [PubMed]
  4. N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in the micrometer range,” in Proceedings of the International Conference on Optical Methods in Flow and Particle Diagnostics (Laser Institute of America, Toledo, Oh., 1988), p. 294.
  5. M. Seaver, J. R. Peele, “Noncontact fluorescence thermometry of acoustically levitated water drops,” Appl. Opt. 29, 4956–4961 (1990).
    [Crossref] [PubMed]
  6. M. R. Wells, L. A. Melton, “Temperature measurements of falling droplets,” Trans. ASME 112, 1008–1013 (1990).
    [Crossref]
  7. M. Winter, “Measurement of the temperature field inside a falling droplet,” presented at the Fourth Annual Conference, (Institute of Liquid Atomization and Spray Systems, North and South America, Hartford, Conn., 21–23 May 1990).
  8. J. Zhang, “Fluorescence methods for determination of temperature in aerosol particles,” Ph.D. dissertation (Department of Mechanical Engineering, University of Nebraska at Lincoln, Lincoln, Neb., 1991), pp. 56–79.
  9. A. S. Kwok, C. F. Wood, R. K. Chang, “Fluorescence imaging of CO2 laser-heated droplets,” Opt. Lett. 15, 664–666 (1990).
    [Crossref] [PubMed]
  10. H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
    [Crossref]
  11. 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]
  12. 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]
  13. 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]
  14. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 200–214.
  15. J. Zhang, D. R. Alexander, “Hybrid inelastic scattering models for particle thermometry: polarized emissions,” Appl. Opt. 31, 7140–7146 (1992).
    [Crossref] [PubMed]
  16. C. A. Parker, Photoluminescence of Solutions (Elsevier, New York, 1968), p. 59.
  17. G. Mie, “Bieträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
    [Crossref]
  18. P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
    [Crossref]
  19. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 81–83.
  20. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 21–25.

1992 (1)

1990 (3)

1989 (1)

1987 (1)

1985 (1)

1979 (1)

1978 (2)

1976 (1)

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

1909 (1)

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
[Crossref]

1908 (1)

G. Mie, “Bieträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[Crossref]

Alexander, D. R.

Anders, K.

N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in the micrometer range,” in Proceedings of the International Conference on Optical Methods in Flow and Particle Diagnostics (Laser Institute of America, Toledo, Oh., 1988), p. 294.

Armstrong, R. L.

Born, M.

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

Chang, R. K.

Chew, H.

Cooke, D. D.

Debye, P.

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
[Crossref]

Druger, S. D.

Frohn, A.

N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in the micrometer range,” in Proceedings of the International Conference on Optical Methods in Flow and Particle Diagnostics (Laser Institute of America, Toledo, Oh., 1988), p. 294.

Goodman, J. W.

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

Gossage, H. E.

Kerker, M.

Kwok, A. S.

McNulty, P. J.

Melton, L. A.

Mie, G.

G. Mie, “Bieträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[Crossref]

Murray, A. M.

Park, B.-S.

Parker, C. A.

C. A. Parker, Photoluminescence of Solutions (Elsevier, New York, 1968), p. 59.

Peele, J. R.

Roth, N.

N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in the micrometer range,” in Proceedings of the International Conference on Optical Methods in Flow and Particle Diagnostics (Laser Institute of America, Toledo, Oh., 1988), p. 294.

Sculley, M.

Seaver, M.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 200–214.

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, “Measurement of the temperature field inside a falling droplet,” presented at the Fourth Annual Conference, (Institute of Liquid Atomization and Spray Systems, North and South America, Hartford, Conn., 21–23 May 1990).

Wolf, E.

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

Wood, C. F.

Zhang, J.

J. Zhang, D. R. Alexander, “Hybrid inelastic scattering models for particle thermometry: polarized emissions,” Appl. Opt. 31, 7140–7146 (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, Neb., 1991), pp. 56–79.

Ann. Phys. (2)

G. Mie, “Bieträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[Crossref]

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
[Crossref]

Appl. Opt. (6)

J. Opt. Soc. Am. (2)

Opt. Lett. (1)

Phys. Rev. A (1)

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[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, (Institute of Liquid Atomization and Spray Systems, North and South America, Hartford, Conn., 21–23 May 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, Neb., 1991), pp. 56–79.

N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in the micrometer range,” in Proceedings of the International Conference on Optical Methods in Flow and Particle Diagnostics (Laser Institute of America, Toledo, Oh., 1988), p. 294.

C. A. Parker, Photoluminescence of Solutions (Elsevier, New York, 1968), p. 59.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 200–214.

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 (11)

Fig. 1
Fig. 1

Incident coordinate system XYZ, and the observation coordinate system ABC.

Fig. 2
Fig. 2

Observation coordinate system ABC for evaluating S(R). A point light source P(R) with an example ray PM through M(R0) on the spherical surface and the transmitted ray MQ are shown.

Fig. 3
Fig. 3

Schematic showing the window area, the window center α0, and the trial window.

Fig. 4
Fig. 4

Distributions of the excitation intensity I(R) on the YZ plane. n ^ 1 = 1.333 + 1.0 × 10−8 i. The laser beam is propagating upward (+Z direction). (a) α = 2.0; (b) α = 15; (c) α = 150; (d) α = 1500.

Fig. 5
Fig. 5

Distributions of the scattering function S(R) on the BC plane. Detector is located on the bottom of the page viewing upward. (a) n1 = 1.333 (water), f/28; (b) n1 = 1.333 (water), f/2.8; (c) n1 = 1.200, f/5; (d) n1 = 1.409 (decane), f/5.

Fig. 6
Fig. 6

Distributions of the weighting function in the YZ plane. Parameters used for I(R): n ^ 1 = 1.333 + 1.0 × 10−6 i (transparent particle), α = 1500; for S(R): n1 = 1.333, f/2.8. The laser beam is propagating upward (+Z direction). (a) θ = 0°, detector views in the −Z direction; (b) θ = 90°, detector views in the −Y direction; (c) θ = 180°, detector views in the +Z direction.

Fig. 7
Fig. 7

(a)–(c) Distributions of the weighting function and (d) excitation intensity I(R) in the YZ plane. Parameters used for I(R): n ^ 1 = 1.333 + 1.2 × 10−3 i (absorbing particle), α = 2098; for S(R): n1 = 1.333, f/2.8. The laser beam is propagating upward (+Z direction). (a) θ = 0°, detector views in the −Z direction; (b) θ = 90°, detector views in the −Y direction; (c) θ = 180°, detector views in the +Z direction.

Fig. 8
Fig. 8

P(θ) and U(θ) versus observation angle θ. n ^ 1 = 1.333 + 1 × 10−6 i, α = 1500, β0 = 5.71, f/5. [Left scale is for P(θ), right scale is for U(θ)].

Fig. 9
Fig. 9

P(θ) and U(θ) versus observation angle θ. n ^ 1 = 1.333 + 1.2 × 10−3 i, α = 2098, β0 = 5.71, f/5. [Left scale is for P(θ), right scale is for U(θ)].

Fig. 10
Fig. 10

Excitation intensity, scattering function, and weighting function along the OZ axis. Parameters for I(R): n ^ 1 = 1.409 + 0.5 × 10−3 i (absorbing particle), α = 8850; for S(R): n1 = 1.409, f/5, θ = 90°.

Fig. 11
Fig. 11

Excitation intensity, scattering function, and weighting function along the OZ axis. Parameters for I(R): n ^ 1 = 1.409 + 1.0 × 10−6 i (transparent particle), α = 8850; for S(R): n1 = 1.409, f/5, θ = 90°.

Equations (9)

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P ( θ ) = Ω [ S ( R ) C ( R ) I ( R ) ] θ d ν ,
i ( R 0 ) = D T i ^ exp ( - χ l ) ,
θ 1 = cos - 1 { ( a - x 0 ) x 0 + ( b - y 0 ) y 0 + ( c - z 0 ) z 0 [ ( a - x 0 ) 2 + ( b - y 0 ) 2 + ( c - z 0 ) 2 ] 1 / 2 ( x 0 2 + y 0 2 + z 0 2 ) 1 / 2 } ,
θ 2 = cos - 1 { ( p - x 0 ) x 0 + ( q - y 0 ) y 0 + ( r - z 0 ) z 0 [ ( p - x 0 ) 2 + ( q - y 0 ) 2 + ( r - z 0 ) 2 ] 1 / 2 ( x 0 2 + y 0 2 + z 0 2 ) 1 / 2 } .
( c y 0 - b z 0 ) p + ( a z 0 - c x 0 ) q + ( b x 0 - a y 0 ) r = 0.
β = cos - 1 { q - y 0 [ ( p - x 0 ) 2 + ( q - y 0 ) 2 + ( r - z 0 ) 2 ] 1 / 2 } .
S ( R ) = - Δ Φ + Δ Φ α 0 - Δ α α 0 + Δ α w ( R 0 ) 1 2 h 2 × ( T + T ) cos θ 1 sin Φ d Φ d α ,
w ( R 0 ) = { 1 if β β 0 0 otherwise ,
tan [ α 0 - sin - 1 ( n 2 sin α 0 n 1 ) ] - sin α 0 - c cos α 0 - b = 0 ,

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