Abstract

By means of geometrical optics we present an approximation method for acceleration of the computation of the scattering intensity distribution within a forward angular range (0–60°) for gradient-index spheres illuminated by a plane wave. The incident angle of reflected light is determined by the scattering angle, thus improving the approximation accuracy. The scattering angle and the optical path length are numerically integrated by a general-purpose integrator. With some special index models, the scattering angle and the optical path length can be expressed by a unique function and the calculation is faster. This method is proved effective for transparent particles with size parameters greater than 50. It fails to give good approximation results at scattering angles whose refractive rays are in the backward direction. For different index models, the geometrical-optics approximation is effective only for forward angles, typically those less than 60° or when the refractive-index difference of a particle is less than a certain value.

© 2007 Optical Society of America

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References

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  1. C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics Fundamentals and Applications (Springer, 2002).
  2. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  9. A. Belafhal, M. Ibnchaikh, and K. Nassim, "Scattering amplitude of absorbing and nonabsorbing spheroidal particles in the WKB approximation," J. Quant. Spectrosc. Radiat. Transfer 72, 385-402 (2002).
    [CrossRef]
  10. M. Min, J. W. Hovenier, and A. de Koter, "Scattering and absorption cross sections for randomly oriented spheroids of arbitrary size," J. Quant. Spectrosc. Radiat. Transfer 79-80, 939-951 (2003).
    [CrossRef]
  11. Ye. Grynko and Yu. Shkuratov, "Scattering matrix calculated in geometric optics approximation for semitransparent particles faceted with various shapes," J. Quant. Spectrosc. Radiat. Transfer 78, 319-340 (2003).
    [CrossRef]
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  13. F. Xu, K. F. Ren, and X. S. Cai, "Extension of geometrical-optics approximation to on-axis Gaussian beam scattering. I. By a spherical particle,"Appl. Opt. 45, 4990-4999 (2006).
    [CrossRef] [PubMed]
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2006 (1)

2005 (1)

2004 (1)

2003 (3)

M. Min, J. W. Hovenier, and A. de Koter, "Scattering and absorption cross sections for randomly oriented spheroids of arbitrary size," J. Quant. Spectrosc. Radiat. Transfer 79-80, 939-951 (2003).
[CrossRef]

Ye. Grynko and Yu. Shkuratov, "Scattering matrix calculated in geometric optics approximation for semitransparent particles faceted with various shapes," J. Quant. Spectrosc. Radiat. Transfer 78, 319-340 (2003).
[CrossRef]

F. Xu, X. S. Cai, and J. Shen, "Geometric approximation of light scattering in arbitrary diffraction regime for absorbing particles: application in laser particle sizing," Acta Opt. Sin. 23, 1464-1469 (2003).

2002 (2)

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics Fundamentals and Applications (Springer, 2002).

A. Belafhal, M. Ibnchaikh, and K. Nassim, "Scattering amplitude of absorbing and nonabsorbing spheroidal particles in the WKB approximation," J. Quant. Spectrosc. Radiat. Transfer 72, 385-402 (2002).
[CrossRef]

2001 (1)

R. Descartes, Discourse on Method, Optics, Geometry, and Meteorology, revised edition translated by P.J.OlscampHackett, (2001).

1997 (1)

F. Corbin, X. Han, Z. S. Wu, K. F. Ren, G. Grehan, A. Garo, and G. Gouesbet, "Rainbow refractometry: application to nonhomogeneous scatters," presented at Third International Conference on Fluid Dynamic Measurement and Its Applications, 14-17 October 1997, Beijing, China, pp. 39-44.

1996 (1)

R. Xu, "Particle size distribution analysis using light scattering," in Liquid and Surfaceborne Particle Measurement Handbook, J. Z. Knapp, T. A. Barber, and A. Lieberman, eds. (Marcel Dekker, 1996), pp. 745-777.

1992 (1)

1987 (1)

1983 (1)

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

1981 (3)

L. P. Bayvel and A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, 1981).

W. J. Glantschnig and S.-H. Chen, "Light scattering from water droplets in the geometrical optics approximation," Appl. Opt. 20, 2499-2509 (1981).
[CrossRef] [PubMed]

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

1956 (1)

L. I. Schiff, "Approximation method for short wavelength or high-energy scattering," Phys. Rev. 104, 1481-1485 (1956).
[CrossRef]

Bao, C.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics Fundamentals and Applications (Springer, 2002).

Bayvel, L. P.

L. P. Bayvel and A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, 1981).

Belafhal, A.

A. Belafhal, M. Ibnchaikh, and K. Nassim, "Scattering amplitude of absorbing and nonabsorbing spheroidal particles in the WKB approximation," J. Quant. Spectrosc. Radiat. Transfer 72, 385-402 (2002).
[CrossRef]

Bohren, C. F.

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

Cai, X.

Cai, X. S.

F. Xu, K. F. Ren, and X. S. Cai, "Extension of geometrical-optics approximation to on-axis Gaussian beam scattering. I. By a spherical particle,"Appl. Opt. 45, 4990-4999 (2006).
[CrossRef] [PubMed]

F. Xu, X. S. Cai, and J. Shen, "Geometric approximation of light scattering in arbitrary diffraction regime for absorbing particles: application in laser particle sizing," Acta Opt. Sin. 23, 1464-1469 (2003).

Chen, S.-H.

Chen, T. W.

Corbin, F.

F. Corbin, X. Han, Z. S. Wu, K. F. Ren, G. Grehan, A. Garo, and G. Gouesbet, "Rainbow refractometry: application to nonhomogeneous scatters," presented at Third International Conference on Fluid Dynamic Measurement and Its Applications, 14-17 October 1997, Beijing, China, pp. 39-44.

de Koter, A.

M. Min, J. W. Hovenier, and A. de Koter, "Scattering and absorption cross sections for randomly oriented spheroids of arbitrary size," J. Quant. Spectrosc. Radiat. Transfer 79-80, 939-951 (2003).
[CrossRef]

Descartes, R.

R. Descartes, Discourse on Method, Optics, Geometry, and Meteorology, revised edition translated by P.J.OlscampHackett, (2001).

Garo, A.

F. Corbin, X. Han, Z. S. Wu, K. F. Ren, G. Grehan, A. Garo, and G. Gouesbet, "Rainbow refractometry: application to nonhomogeneous scatters," presented at Third International Conference on Fluid Dynamic Measurement and Its Applications, 14-17 October 1997, Beijing, China, pp. 39-44.

Glantschnig, W. J.

Gomez-Reino, C.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics Fundamentals and Applications (Springer, 2002).

Gouesbet, G.

F. Corbin, X. Han, Z. S. Wu, K. F. Ren, G. Grehan, A. Garo, and G. Gouesbet, "Rainbow refractometry: application to nonhomogeneous scatters," presented at Third International Conference on Fluid Dynamic Measurement and Its Applications, 14-17 October 1997, Beijing, China, pp. 39-44.

Grehan, G.

F. Corbin, X. Han, Z. S. Wu, K. F. Ren, G. Grehan, A. Garo, and G. Gouesbet, "Rainbow refractometry: application to nonhomogeneous scatters," presented at Third International Conference on Fluid Dynamic Measurement and Its Applications, 14-17 October 1997, Beijing, China, pp. 39-44.

Grynko, Ye.

Ye. Grynko and Yu. Shkuratov, "Scattering matrix calculated in geometric optics approximation for semitransparent particles faceted with various shapes," J. Quant. Spectrosc. Radiat. Transfer 78, 319-340 (2003).
[CrossRef]

Han, X.

F. Corbin, X. Han, Z. S. Wu, K. F. Ren, G. Grehan, A. Garo, and G. Gouesbet, "Rainbow refractometry: application to nonhomogeneous scatters," presented at Third International Conference on Fluid Dynamic Measurement and Its Applications, 14-17 October 1997, Beijing, China, pp. 39-44.

Hovenac, E. A.

Hovenier, J. W.

M. Min, J. W. Hovenier, and A. de Koter, "Scattering and absorption cross sections for randomly oriented spheroids of arbitrary size," J. Quant. Spectrosc. Radiat. Transfer 79-80, 939-951 (2003).
[CrossRef]

Huffman, D. R.

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

Ibnchaikh, M.

A. Belafhal, M. Ibnchaikh, and K. Nassim, "Scattering amplitude of absorbing and nonabsorbing spheroidal particles in the WKB approximation," J. Quant. Spectrosc. Radiat. Transfer 72, 385-402 (2002).
[CrossRef]

Jones, A. R.

L. P. Bayvel and A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, 1981).

Lock, J. A.

Min, M.

M. Min, J. W. Hovenier, and A. de Koter, "Scattering and absorption cross sections for randomly oriented spheroids of arbitrary size," J. Quant. Spectrosc. Radiat. Transfer 79-80, 939-951 (2003).
[CrossRef]

Nassim, K.

A. Belafhal, M. Ibnchaikh, and K. Nassim, "Scattering amplitude of absorbing and nonabsorbing spheroidal particles in the WKB approximation," J. Quant. Spectrosc. Radiat. Transfer 72, 385-402 (2002).
[CrossRef]

Perez, M. V.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics Fundamentals and Applications (Springer, 2002).

Ren, K.

Ren, K. F.

F. Xu, K. F. Ren, and X. S. Cai, "Extension of geometrical-optics approximation to on-axis Gaussian beam scattering. I. By a spherical particle,"Appl. Opt. 45, 4990-4999 (2006).
[CrossRef] [PubMed]

F. Corbin, X. Han, Z. S. Wu, K. F. Ren, G. Grehan, A. Garo, and G. Gouesbet, "Rainbow refractometry: application to nonhomogeneous scatters," presented at Third International Conference on Fluid Dynamic Measurement and Its Applications, 14-17 October 1997, Beijing, China, pp. 39-44.

Riethmuller, M. L.

Schiff, L. I.

L. I. Schiff, "Approximation method for short wavelength or high-energy scattering," Phys. Rev. 104, 1481-1485 (1956).
[CrossRef]

Shen, J.

F. Xu, X. S. Cai, and J. Shen, "Geometric approximation of light scattering in arbitrary diffraction regime for absorbing particles: application in laser particle sizing," Acta Opt. Sin. 23, 1464-1469 (2003).

Shkuratov, Yu.

Ye. Grynko and Yu. Shkuratov, "Scattering matrix calculated in geometric optics approximation for semitransparent particles faceted with various shapes," J. Quant. Spectrosc. Radiat. Transfer 78, 319-340 (2003).
[CrossRef]

van Beeck, J. P. A. J.

van de Hulst, H. C.

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

Vetrano, M. R.

Wu, Z. S.

F. Corbin, X. Han, Z. S. Wu, K. F. Ren, G. Grehan, A. Garo, and G. Gouesbet, "Rainbow refractometry: application to nonhomogeneous scatters," presented at Third International Conference on Fluid Dynamic Measurement and Its Applications, 14-17 October 1997, Beijing, China, pp. 39-44.

Xu, F.

Xu, R.

R. Xu, "Particle size distribution analysis using light scattering," in Liquid and Surfaceborne Particle Measurement Handbook, J. Z. Knapp, T. A. Barber, and A. Lieberman, eds. (Marcel Dekker, 1996), pp. 745-777.

Acta Opt. Sin. (1)

F. Xu, X. S. Cai, and J. Shen, "Geometric approximation of light scattering in arbitrary diffraction regime for absorbing particles: application in laser particle sizing," Acta Opt. Sin. 23, 1464-1469 (2003).

Appl. Opt. (4)

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

J. Quant. Spectrosc. Radiat. Transfer (3)

A. Belafhal, M. Ibnchaikh, and K. Nassim, "Scattering amplitude of absorbing and nonabsorbing spheroidal particles in the WKB approximation," J. Quant. Spectrosc. Radiat. Transfer 72, 385-402 (2002).
[CrossRef]

M. Min, J. W. Hovenier, and A. de Koter, "Scattering and absorption cross sections for randomly oriented spheroids of arbitrary size," J. Quant. Spectrosc. Radiat. Transfer 79-80, 939-951 (2003).
[CrossRef]

Ye. Grynko and Yu. Shkuratov, "Scattering matrix calculated in geometric optics approximation for semitransparent particles faceted with various shapes," J. Quant. Spectrosc. Radiat. Transfer 78, 319-340 (2003).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

L. I. Schiff, "Approximation method for short wavelength or high-energy scattering," Phys. Rev. 104, 1481-1485 (1956).
[CrossRef]

Other (7)

R. Descartes, Discourse on Method, Optics, Geometry, and Meteorology, revised edition translated by P.J.OlscampHackett, (2001).

F. Corbin, X. Han, Z. S. Wu, K. F. Ren, G. Grehan, A. Garo, and G. Gouesbet, "Rainbow refractometry: application to nonhomogeneous scatters," presented at Third International Conference on Fluid Dynamic Measurement and Its Applications, 14-17 October 1997, Beijing, China, pp. 39-44.

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

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics Fundamentals and Applications (Springer, 2002).

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

R. Xu, "Particle size distribution analysis using light scattering," in Liquid and Surfaceborne Particle Measurement Handbook, J. Z. Knapp, T. A. Barber, and A. Lieberman, eds. (Marcel Dekker, 1996), pp. 745-777.

L. P. Bayvel and A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, 1981).

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Figures (5)

Fig. 1
Fig. 1

Comparison of the light path inside a sphere with a uniform refractive index (solid curves) and the light path inside a sphere with a spherically symmetric refractive index m ( r ˜ ) (dashed curves): (a) m 0 > m 1 and (b) m 0 < m 1 .

Fig. 2
Fig. 2

Comparison of intensity of calculation by other theories and by the GOA method: (a) R = 10 μm , m 0 = 1.33 , m 1 = 1.30 and (b) R = 10 μm , m 0 = 1.30 , m 1 = 1.33 .

Fig. 3
Fig. 3

Path of light rays through a GRIN particle with a large index difference.

Fig. 4
Fig. 4

Valid range of the GOA method for a particle with an index model expressed by Eq. (15): R = 15 μm , m 0 = 4.1 , m 1 = 1.1 .

Fig. 5
Fig. 5

Comparison of intensity of calculation by other theories and by the GOA method: (a) R = 10 μm , m 0 = 1.49 , m 1 = 1.47 and (b) R = 10 μm , m 0 = 1.47 , m 1 = 1.49 .

Tables (1)

Tables Icon

Table 1 Comparison of Speed of Calculation

Equations (19)

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

s = s diffraction + s refraction + s reflection .
θ = 2 τ 2 ( N 1 ) τ ,
θ = 2 τ 2 ( N 1 ) [ χ ( r = R ) χ ( r = r m ) ] .
χ ( r ) = χ 0 + r m r e r [ m ( r ˜ ) r ˜ ] 2 e 2 d r ,
[ m ( r ˜ ) r ˜ ] 2 e 2 = 0.
i 1 , 2 = α 2 k 1 , 2 2 D ,
k 1 , 2 = ε 1 , 2 for   N = 1 , k 1 , 2 = ( 1 ε 1 , 2 2 ) ( ε 1 , 2 ) N 2 for   N = 2 , 3 , 4  …  .
D = sin   τ   cos   τ sin   θ | d θ / d τ | ,
ε 1 = sin   τ m 1   sin   τ sin   τ + m 1   sin   τ , ε 2 = m 1   sin   τ sin   τ m 1   sin   τ + sin   τ .
σ reflection 1 , 2 = π / 2 + 2 α   sin ( τ reflection ) + ψ 1 , 2 ,
σ refraction 1 , 2 = 3 π / 2 + 2 α   sin ( τ ) 2 π L λ ,
L = r m R 2 m ( r ˜ ) 2 r ˜ [ m ( r ˜ ) r ˜ ] 2 e 2 d r .
s 1 , 2 = i 1 , 2   exp ( j σ 1 , 2 ) .
s = s reflection + s refraction + s diffraction , θ [ 0 ° , 20 ° ] , s = s reflection + s refraction , θ [ 20 ° , 60 ° ] ,
I ( θ ) = λ 2 I 0 8 π 2 f 2 ( | s 1 ( θ ) | 2 + | s 2 ( θ ) | 2 ) .
m ( r ˜ ) = m 0 + ( m 1 m 0 ) e b r ˜ 1 e b 1 .
m ( r ˜ ) = m 0 ( 1 m 0 2 m 1 2 m 0 2 r ˜ 2 ) 1 / 2 .
χ ( R ) = 1 / 2 sin 1 [ m 0 2 2 e 2 m 0 4 4 ( m 0 2 m 1 2 ) e 2 ] + π 4 .
L = R 2 { m 1 2 e 2 m 0 2 2 m 0 2 m 1 2 [ sin 1 2 m 1 2 m 0 2 m 0 4 4 ( m 0 2 m 1 2 ) e 2 π 2 ] } , m 0 > m 1 , L = R 2 m 1 2 e 2 + R m 0 2 4 m 1 2 m 0 2 { ln [ 2 m 1 2 m 0 2 2 m 1 2 m 0 2 + m 1 2 e 2 ] ln [ m 0 2 4 ( m 0 2 m 1 2 ) e 2 2 m 1 2 m 0 2 ] } , m 0 < m 1 .

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