Abstract

By means of geometrical optics we present an approximation algorithm with which to accelerate the computation of scattering intensity distribution within a forward angular range (0°–60°) for coated particles illuminated by a collimated incident beam. Phases of emerging rays are exactly calculated to improve the approximation precision. This method proves effective for transparent and tiny absorbent particles with size parameters larger than 75 but fails to give good approximation results at scattering angles at which refractive rays are absent. When the absorption coefficient of a particle is greater than 0.01, the geometrical optics approximation is effective only for forward small angles, typically less than 10° or so.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  2. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  3. R. Xu, “Particle size distribution analysis using light scattering,” in Liquid and Surfaceborne Particle Measurement Handbook, J. Z. Knapp, T. A. Barber, A. Lieberman, eds. (Marcel Dekker, New York, 1996), pp. 745–777.
  4. L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
    [CrossRef]
  5. L. I. Schiff, “Approximation method for short wavelength or high-energy scattering,” Phys. Rev. 104, 1481–1485 (1956).
    [CrossRef]
  6. T. W. Chen, “Scattering of light by a stratified sphere in high energy approximation,” Appl. Opt. 26, 4155–4158 (1987).
    [CrossRef] [PubMed]
  7. A. Belafhal, M. Ibnchaikh, K. Nassim, “Scattering amplitude of absorbing and nonabsorbing spheroidal particles in the WKB approximation,” J. Quant. Spectrosc. Radiat. Transfer 72, 385–402 (2002).
    [CrossRef]
  8. W. J. Glantschnig, S.-H. Chen, “Light scattering from water droplets in the geometrical optics approximation,” Appl. Opt. 20, 2499–2509 (1980).
    [CrossRef]
  9. M. Min, J. W. Hovenier, 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]
  10. Ye. Grynko, 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]
  11. F. Xu, X. S. Cai, 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).
  12. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  13. A. Ghatak, Optics (Tata McGraw-Hill, New Delhi, 1977).
  14. H. G. Barth, Modern Methods of Particle Size Analysis (Wiley, New York, 1984).

2003

M. Min, J. W. Hovenier, 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, 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, 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

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

1987

1980

1956

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

1951

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Aden, A. L.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Barth, H. G.

H. G. Barth, Modern Methods of Particle Size Analysis (Wiley, New York, 1984).

Bayvel, L. P.

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

Belafhal, A.

A. Belafhal, M. Ibnchaikh, 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, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Cai, X. S.

F. Xu, X. S. Cai, 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.

de Koter, A.

M. Min, J. W. Hovenier, 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]

Ghatak, A.

A. Ghatak, Optics (Tata McGraw-Hill, New Delhi, 1977).

Glantschnig, W. J.

Grynko, Ye.

Ye. Grynko, 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]

Hovenier, J. W.

M. Min, J. W. Hovenier, 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, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Ibnchaikh, M.

A. Belafhal, M. Ibnchaikh, 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, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
[CrossRef]

Kerker, M.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Min, M.

M. Min, J. W. Hovenier, 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, K. Nassim, “Scattering amplitude of absorbing and nonabsorbing spheroidal particles in the WKB approximation,” J. Quant. Spectrosc. Radiat. Transfer 72, 385–402 (2002).
[CrossRef]

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, 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, 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 de Hulst, H. C.

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

Xu, F.

F. Xu, X. S. Cai, 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).

Xu, R.

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

Acta Opt. Sin.

F. Xu, X. S. Cai, 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.

J. Appl. Phys.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

M. Min, J. W. Hovenier, 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, 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]

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

Phys. Rev.

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

Other

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

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

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

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

A. Ghatak, Optics (Tata McGraw-Hill, New Delhi, 1977).

H. G. Barth, Modern Methods of Particle Size Analysis (Wiley, New York, 1984).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Path of light rays through a coated particle for (a) m 2 > m 1 > m 0, (b) m 2 < m 1 < m 0, (c) m 1 > m 0 and m 1 > m 2, and (d) m 1 < m 0 and m 1 < m 2.

Fig. 2
Fig. 2

Comparison of A–K theory and the GOA method for calculation of the intensity of nonabsorbent particles: (a) m 1 = 1.33, m 2 = 1.6, R = 10 μm, r = 0.5 μm, F AK = 4.28 × 10-9, F GOA = 4.38 × 10-9, and F FD = 4.10 × 10-9; (b) m 1 = 0.8, m 2 = 0.6, R = 20 μm, r = 15 μm, F AK = 3.55 × 10-8, F GOA = 3.36 × 10-8, and F FD = 3.28 × 10-8; (c) m 1 = 1.33, m 2 = 1.1, R = 20 μm, r = 5 μm, F AK = 3.34 × 10-8, F GOA = 3.30 × 10-8, and F FD = 3.28 × 10-8; (d) m 1 = 0.6, m 2 = 1.2, R = 15 μm, r = 7.5 μm, F AK = 1.41 × 10-8, F GOA = 1.40 × 10-8, and F FD = 1.38 × 10-8.

Fig. 3
Fig. 3

Comparison of A–K theory and the GOA method for calculation of the intensity of coated absorbent particles: (a) m 1 = 1.33 + 0.002i, m 2 = 1.6 + 0.001i, R = 10 μm, r = 0.5 μm, F AK = 4.30 × 10-9, F GOA = 4.30 × 10-9, and F FD = 4.10 × 10-9; (b) m 1 = 0.8 + 0.008i, m 2 = 0.6 + 0.004i, R = 20 μm, r = 15 μm, F AK = 3.40 × 10-8, F GOA = 3.33 × 10-8, and F FD = 3.28 × 10-8; (c) m 1 = 1.33 + 0.005i, m 2 = 1.1 + 0.01i, R = 20 μm, r = 5 μm, F AK = 3.42 × 10-8, F GOA = 3.31 × 10-8, and F FD = 3.28 × 10-8; (d) m 1 = 0.6 + 0.002i, m 2 = 1.2 + 0.004i, R = 15 μm, r = 7.5 μm, F AK = 1.43 × 10-8, F GOA = 1.41 × 10-8, and F FD = 1.38 × 10-8.

Fig. 4
Fig. 4

Scattering pattern of an ensemble of particles characterized by a bimodal distribution with 1 = 20 μm, k 1 = 8, 2 = 0.2 μm, k 2 = 4, N 1 = N 2 = 1000, F AK = 3.25 × 10-6, F GOA = 3.01 × 10-6, and F FD = 2.92 × 10-6. The scattering intensity of the particle with size parameter less than 75 is all calculated by A–K theory.

Fig. 5
Fig. 5

Comparison of GOA methods by use of FD theory in the ranges 0°–20° and 0°–60°: (a) m 1 = 1.33 + 0.005i, m 2 = 1.1 + 0.01i, R = 40 μm, r = 10 μm; (b) m 1 = 0.6 + 0.01i, m 2 = 1.2 + 0.01i, R = 15 μm, r = 7.5 μm.

Fig. 6
Fig. 6

Comparison of Mie theory and the GOA method for calculation of the intensity of homogenous particles: (a) m 1 = 1.5, R = 10 μm, F Mie = 4.39 × 10-9, F GOA = 4.13 × 10-9, and F FD = 4.12 × 10-9; (b) m 1 = 0.5, R = 10 μm, F Mie = 4.25 × 10-9, F GOA = 4.17 × 10-9, and F FD = 4.10 × 10-9; (c) m 1 = 1.5 + 0.005i, R = 10 μm, F Mie = 4.36 × 10-9, F GOA = 4.15 × 10-9, and F FD = 4.12 × 10-9, (d) m 1 = 0.5 + 0.003i, R = 10 μm, F Mie = 4.30 × 10-9, F GOA = 4.19 × 10-9, and F FD = 4.10 × 10-9.

Fig. 7
Fig. 7

Comparison of the GOA with the proposed and the usual reflection phase-shift calculation methods. Curves marked by open circles are the same as those in Figs. 6(a) and Fig. 6(b), and their phase shifts were calculated from Eqs. (7) and (8); the phase shifts of the solid curves were calculated by simple substitution of π for m i < m j and 0 for m i > m j . (a) m 1 = 1.5 and R = 10 μm, (b) m 1 = 0.5 and R = 10 μm.

Fig. 8
Fig. 8

Comparison of the GOA curve with that of the exact theory for a highly absorbent homogeneous particle. Shown is the scattering pattern of a particle with m 1 = 0.5 + 0.1i, R = 25.0 μm, F Mie = 6.63 × 10-8, F GOA = 6.48 × 10-8, and F FD = 6.41 × 10-8.

Fig. 9
Fig. 9

Comparison of speed of calculation by the A–K theory and by the GOA method. (The time consumed by the GOA method remains 1.15 s.)

Tables (1)

Tables Icon

Table 1 Validity Range for the GOA Methoda

Equations (26)

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

s=sdiff+srefraction+sreflection.
θa=π-2θi1,θa=|θa|;
θb=2θr1-2θi1,θb=|θb|;
θc=2θr1+2θr2-2θi1-2θi2,θc=|θc|;
θd=π+2θr1-2θi1-2θi2,θd=|θd|.
iad1,2=α12kad1,2Gad.
Gad=sin θi1 cos θi1sin θad|dθad/dθi1|.
ka1,2=ε1,22,kb1,2=1-ε1,222,kc1,2=1-ε1,221-ε1,222,kd1,2=1-ε1,222ε1,22,
ε1=cos θi1-m1/m0 cos θr1cos θi1+m1/m0 cos θr1,ε2=m1/m0 cos θi1-cos θr1m1/m0 cos θi1+cos θr1;
ε1=cos θi2-m2/m1 cos θr2cos θi2+m2/m1 cos θr2,ε2=m2/m1 cos θi2-cos θr2m2/m1 cos θi2+cos θr2.
ψ1=π, ψ2=π  θi>θp,ψ1=0, ψ2=π  θi<θp,
ψ1=2 tan-1mimj2sin2 θi-sin2 θc1/2cos θi,ψ2=2 tan-1sin2 θi-sin2 θc1/2cos θiθi>θc,ψ1=0,ψ2=0θi<θc,
σa1,2=π/2+2α1 cos θi1+ψa1,2,σb1,2=3π/2+2α1cos θi1-m1 cos θr1,σc1,2=3π/2+2α1 cos θi1-2α2m1 sinθi2-θr1/sin θr1+m2 cos θr2,σd1,2=π/2+2α1 cos θi1-2α2m1 sinθi2-θr1/sin θr1+ψd1,2.
σa1,2=π/2+2α1 cos θi1+ψa1,2,σb1,2=π/2+2α1cos θi1-m1 cos θr1,σc1,2=π/2+2α1 cos θi1-2α2m1 sinθi2-θr1/sin θr1+m2 cos θr2,σd1,2=π+2α1 cos θi1-2α2m1 sinθi2-θr1/sin θr1+ψd1,2.
σa1,2=π/2+2α1 cos θi1+ψa1,2,σb1,2=3π/2+2α1cos θi1-m1 cos θr1,σc1,2=π1+sc/4-qc/4+2α1 cos θi1-2α2m1 sinθi2-θr1/sin θr1+m2 cos θr2,σd1,2=πsd/4-qd/4+2α1 cos θi1-2α2m1 sinθi2-θr1/sin θr1+ψd1,2.
σa1,2=π/2+2α1 cos θi1+ψa1,2,σb1,2=π/2+2α1cos θi1-m1 cos θr1,σc1,2=π1+sc/4-qc/4+2α1 cos θi1-2α2m1 sinθi2-θr1/sin θr1+m2 cos θr2,σd1,2=π1+sd/4-qd/4+2α1 cos θi1-2α2m1 sinθi2-θr1/sin θr1+ψd1,2.
sad1,2=iad1,2expiσad1,2.
sad1,2=iad1,2 expξad1/2 expiσad1,2,
ξa=1,ξb=-2γ1R cos θr2,ξc=-2γ1r sinθi2-θr1/sin θr1-2γ2r cos θr2,ξd=-2γ1r sinθi2-θr1/sin θr1.
γ1=4πη1/λ, γ2=4πη2/λ.
s1,2=sdiff+sa1,2+sb1,2+sc1,2+sd1,2 θ0°, 20°,s1,2=sa1,2+sb1,2+sc1,2+sd1,2θ20°, 60°,
Iθ=λ2I08π2f2|s1θ|2+|s2θ|2.
dNdD=6πD3k/D¯D/D¯k-1 exp-D/D¯k,
σa=π/2+ψa+d1×2π cos θi1/λ,σb=3π/2+d1×2πcos θi1-m1 cos θr1/λ,σc=3π/2+d1×2πcos θi1-C×m1 sinθi2-θr1/sin θr1+m2 cos θr2/λ,σd=π/2+ψd+d1×2πcos θi1-C×m1 sinθi2-θr1/sin θr1/λ.
sa=d1×π/λkaGa1/2 expπ/2+ψaipa×exp2πi cos θi1/λd1qa,sb=d1×π/λkbGb expξb1/2 exp3πi/2pb×exp2πicos θi1-m1 cos θr1/λd1qb,sc=d1×π/λkcGc expξc1/2 exp3πi/2pc×exp2πicos θi1-C×m1 sinθi2-θr1/sin θr1+m2 cos θr2/λd1qc,sd=d1×π/λkdGd expξd1/2 expπ/2+ψdipd×exp2πicos θi1-C×m1 sinθi2-θr1/sin θr1/λd1qd.
sasbscsd=d1pa0000pb0000pc0000pdqaqbqcqdd1.

Metrics