Abstract

Light scattering from cylindrical particles has been described with geometric optics. The feasibility of determining the particle diameter with a planar phase Doppler anemometer has been examined by simulations and experiments. In particular, the influence of particle orientation on measurability and measurement accuracy has been investigated. Some recommendations for realizing a practical-measurement instrument have been presented.

© 1996 Optical Society of America

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References

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  1. “Optoelekronisches Meßsystem zur berührungslosen Messung von Geschwindigkeit und Länge,” in the catalog of Laser Applikation, Goseburgstrasse 27, 2120 Lüneburg, Germany (1995).
  2. H. Wang, R. Valdivia-Hernandez, “Laser scanner and diffraction pattern detection: a novel concept for dynamic gauging of fine wire,” Meas. Sci. Technol. 6, 452–457 (1995).
    [CrossRef]
  3. “Laser-Diffraction Meßsystem, System SK 9003,” in the catalog of Schäfter & Kirchoff, Opto-sensorik und Messtechnik, Celsiusweg 15, 22761 Hamburg, Germany (1995).
  4. D. Lebrun, S. Belaid, C. Özkul, K. F. Ren, G. Gréhan, “Enhancement of wire diameter measurements: comparison between Fraunhofer diffraction and Lorenz–Mie theory,” Opt. Eng. 35, 946–950 (1996).
    [CrossRef]
  5. E. Zimmermann, R. Dändliker, N. Souli, B. Kratinger, “Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach,” J. Opt. Soc. Am. A 12, 398–403 (1995).
    [CrossRef]
  6. G. Gouesbet, G. Gréhan, “Interaction between shaped beams and an infinite cylinder, including a discussion of Gaussian beams,” Part. Part. Syst. Charact. 11, 299–308 (1994).
    [CrossRef]
  7. G. Gouesbet, G. Gréhan, “On the interaction between a Gaussian beam and an infinite cylinder, using non sigma-separable potentials,” J. Opt. Soc. Am. A 11, 3261–3273 (1994).
    [CrossRef]
  8. G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” Part. Part. Syst. Charact. 26, 225–239 (1995).
  9. G. Gouesbet, “Scattering of a first order Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation,” to be published in Part. Part. Syst. Charact. (1996).
  10. F. Onofri, H. Mignon, G. Gouesbet, G. Gréhan, “On the extension of phase Doppler anemometry to the sizing of spherical multilayered particles and cylindrical particles,” in Partec 95, Fourth International Congress on Optical Particle Sizing (Nürnberg Messe GmbH, Nürnberg, 1995), pp. 275–284.
  11. R. W. Sellens, “A derivation of phase Doppler measurement relations for an arbitrary geometry,” Exp. Fluids 8, 165–168 (1989).
    [CrossRef]
  12. C. Tropea, T. H. Xu, F. Onofri, G. Gréhan, P. Haugen, M. Stieglmeier, “Dual mode phase Doppler anemometer,” Part. Part. Syst. Charact. 13, 165–170 (1996).
    [CrossRef]

1996

D. Lebrun, S. Belaid, C. Özkul, K. F. Ren, G. Gréhan, “Enhancement of wire diameter measurements: comparison between Fraunhofer diffraction and Lorenz–Mie theory,” Opt. Eng. 35, 946–950 (1996).
[CrossRef]

C. Tropea, T. H. Xu, F. Onofri, G. Gréhan, P. Haugen, M. Stieglmeier, “Dual mode phase Doppler anemometer,” Part. Part. Syst. Charact. 13, 165–170 (1996).
[CrossRef]

1995

H. Wang, R. Valdivia-Hernandez, “Laser scanner and diffraction pattern detection: a novel concept for dynamic gauging of fine wire,” Meas. Sci. Technol. 6, 452–457 (1995).
[CrossRef]

G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” Part. Part. Syst. Charact. 26, 225–239 (1995).

E. Zimmermann, R. Dändliker, N. Souli, B. Kratinger, “Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach,” J. Opt. Soc. Am. A 12, 398–403 (1995).
[CrossRef]

1994

G. Gouesbet, G. Gréhan, “Interaction between shaped beams and an infinite cylinder, including a discussion of Gaussian beams,” Part. Part. Syst. Charact. 11, 299–308 (1994).
[CrossRef]

G. Gouesbet, G. Gréhan, “On the interaction between a Gaussian beam and an infinite cylinder, using non sigma-separable potentials,” J. Opt. Soc. Am. A 11, 3261–3273 (1994).
[CrossRef]

1989

R. W. Sellens, “A derivation of phase Doppler measurement relations for an arbitrary geometry,” Exp. Fluids 8, 165–168 (1989).
[CrossRef]

Belaid, S.

D. Lebrun, S. Belaid, C. Özkul, K. F. Ren, G. Gréhan, “Enhancement of wire diameter measurements: comparison between Fraunhofer diffraction and Lorenz–Mie theory,” Opt. Eng. 35, 946–950 (1996).
[CrossRef]

Dändliker, R.

Gouesbet, G.

G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” Part. Part. Syst. Charact. 26, 225–239 (1995).

G. Gouesbet, G. Gréhan, “Interaction between shaped beams and an infinite cylinder, including a discussion of Gaussian beams,” Part. Part. Syst. Charact. 11, 299–308 (1994).
[CrossRef]

G. Gouesbet, G. Gréhan, “On the interaction between a Gaussian beam and an infinite cylinder, using non sigma-separable potentials,” J. Opt. Soc. Am. A 11, 3261–3273 (1994).
[CrossRef]

F. Onofri, H. Mignon, G. Gouesbet, G. Gréhan, “On the extension of phase Doppler anemometry to the sizing of spherical multilayered particles and cylindrical particles,” in Partec 95, Fourth International Congress on Optical Particle Sizing (Nürnberg Messe GmbH, Nürnberg, 1995), pp. 275–284.

G. Gouesbet, “Scattering of a first order Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation,” to be published in Part. Part. Syst. Charact. (1996).

Gréhan, G.

C. Tropea, T. H. Xu, F. Onofri, G. Gréhan, P. Haugen, M. Stieglmeier, “Dual mode phase Doppler anemometer,” Part. Part. Syst. Charact. 13, 165–170 (1996).
[CrossRef]

D. Lebrun, S. Belaid, C. Özkul, K. F. Ren, G. Gréhan, “Enhancement of wire diameter measurements: comparison between Fraunhofer diffraction and Lorenz–Mie theory,” Opt. Eng. 35, 946–950 (1996).
[CrossRef]

G. Gouesbet, G. Gréhan, “Interaction between shaped beams and an infinite cylinder, including a discussion of Gaussian beams,” Part. Part. Syst. Charact. 11, 299–308 (1994).
[CrossRef]

G. Gouesbet, G. Gréhan, “On the interaction between a Gaussian beam and an infinite cylinder, using non sigma-separable potentials,” J. Opt. Soc. Am. A 11, 3261–3273 (1994).
[CrossRef]

F. Onofri, H. Mignon, G. Gouesbet, G. Gréhan, “On the extension of phase Doppler anemometry to the sizing of spherical multilayered particles and cylindrical particles,” in Partec 95, Fourth International Congress on Optical Particle Sizing (Nürnberg Messe GmbH, Nürnberg, 1995), pp. 275–284.

Haugen, P.

C. Tropea, T. H. Xu, F. Onofri, G. Gréhan, P. Haugen, M. Stieglmeier, “Dual mode phase Doppler anemometer,” Part. Part. Syst. Charact. 13, 165–170 (1996).
[CrossRef]

Kratinger, B.

Lebrun, D.

D. Lebrun, S. Belaid, C. Özkul, K. F. Ren, G. Gréhan, “Enhancement of wire diameter measurements: comparison between Fraunhofer diffraction and Lorenz–Mie theory,” Opt. Eng. 35, 946–950 (1996).
[CrossRef]

Mignon, H.

F. Onofri, H. Mignon, G. Gouesbet, G. Gréhan, “On the extension of phase Doppler anemometry to the sizing of spherical multilayered particles and cylindrical particles,” in Partec 95, Fourth International Congress on Optical Particle Sizing (Nürnberg Messe GmbH, Nürnberg, 1995), pp. 275–284.

Onofri, F.

C. Tropea, T. H. Xu, F. Onofri, G. Gréhan, P. Haugen, M. Stieglmeier, “Dual mode phase Doppler anemometer,” Part. Part. Syst. Charact. 13, 165–170 (1996).
[CrossRef]

F. Onofri, H. Mignon, G. Gouesbet, G. Gréhan, “On the extension of phase Doppler anemometry to the sizing of spherical multilayered particles and cylindrical particles,” in Partec 95, Fourth International Congress on Optical Particle Sizing (Nürnberg Messe GmbH, Nürnberg, 1995), pp. 275–284.

Özkul, C.

D. Lebrun, S. Belaid, C. Özkul, K. F. Ren, G. Gréhan, “Enhancement of wire diameter measurements: comparison between Fraunhofer diffraction and Lorenz–Mie theory,” Opt. Eng. 35, 946–950 (1996).
[CrossRef]

Ren, K. F.

D. Lebrun, S. Belaid, C. Özkul, K. F. Ren, G. Gréhan, “Enhancement of wire diameter measurements: comparison between Fraunhofer diffraction and Lorenz–Mie theory,” Opt. Eng. 35, 946–950 (1996).
[CrossRef]

Sellens, R. W.

R. W. Sellens, “A derivation of phase Doppler measurement relations for an arbitrary geometry,” Exp. Fluids 8, 165–168 (1989).
[CrossRef]

Souli, N.

Stieglmeier, M.

C. Tropea, T. H. Xu, F. Onofri, G. Gréhan, P. Haugen, M. Stieglmeier, “Dual mode phase Doppler anemometer,” Part. Part. Syst. Charact. 13, 165–170 (1996).
[CrossRef]

Tropea, C.

C. Tropea, T. H. Xu, F. Onofri, G. Gréhan, P. Haugen, M. Stieglmeier, “Dual mode phase Doppler anemometer,” Part. Part. Syst. Charact. 13, 165–170 (1996).
[CrossRef]

Valdivia-Hernandez, R.

H. Wang, R. Valdivia-Hernandez, “Laser scanner and diffraction pattern detection: a novel concept for dynamic gauging of fine wire,” Meas. Sci. Technol. 6, 452–457 (1995).
[CrossRef]

Wang, H.

H. Wang, R. Valdivia-Hernandez, “Laser scanner and diffraction pattern detection: a novel concept for dynamic gauging of fine wire,” Meas. Sci. Technol. 6, 452–457 (1995).
[CrossRef]

Xu, T. H.

C. Tropea, T. H. Xu, F. Onofri, G. Gréhan, P. Haugen, M. Stieglmeier, “Dual mode phase Doppler anemometer,” Part. Part. Syst. Charact. 13, 165–170 (1996).
[CrossRef]

Zimmermann, E.

Exp. Fluids

R. W. Sellens, “A derivation of phase Doppler measurement relations for an arbitrary geometry,” Exp. Fluids 8, 165–168 (1989).
[CrossRef]

J. Opt. Soc. Am. A

Meas. Sci. Technol.

H. Wang, R. Valdivia-Hernandez, “Laser scanner and diffraction pattern detection: a novel concept for dynamic gauging of fine wire,” Meas. Sci. Technol. 6, 452–457 (1995).
[CrossRef]

Opt. Eng.

D. Lebrun, S. Belaid, C. Özkul, K. F. Ren, G. Gréhan, “Enhancement of wire diameter measurements: comparison between Fraunhofer diffraction and Lorenz–Mie theory,” Opt. Eng. 35, 946–950 (1996).
[CrossRef]

Part. Part. Syst. Charact.

G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” Part. Part. Syst. Charact. 26, 225–239 (1995).

C. Tropea, T. H. Xu, F. Onofri, G. Gréhan, P. Haugen, M. Stieglmeier, “Dual mode phase Doppler anemometer,” Part. Part. Syst. Charact. 13, 165–170 (1996).
[CrossRef]

G. Gouesbet, G. Gréhan, “Interaction between shaped beams and an infinite cylinder, including a discussion of Gaussian beams,” Part. Part. Syst. Charact. 11, 299–308 (1994).
[CrossRef]

Other

“Laser-Diffraction Meßsystem, System SK 9003,” in the catalog of Schäfter & Kirchoff, Opto-sensorik und Messtechnik, Celsiusweg 15, 22761 Hamburg, Germany (1995).

G. Gouesbet, “Scattering of a first order Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation,” to be published in Part. Part. Syst. Charact. (1996).

F. Onofri, H. Mignon, G. Gouesbet, G. Gréhan, “On the extension of phase Doppler anemometry to the sizing of spherical multilayered particles and cylindrical particles,” in Partec 95, Fourth International Congress on Optical Particle Sizing (Nürnberg Messe GmbH, Nürnberg, 1995), pp. 275–284.

“Optoelekronisches Meßsystem zur berührungslosen Messung von Geschwindigkeit und Länge,” in the catalog of Laser Applikation, Goseburgstrasse 27, 2120 Lüneburg, Germany (1995).

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Figures (16)

Fig. 1
Fig. 1

Schematic representation of light scattering from a cylindrical particle.

Fig. 2
Fig. 2

Planar phase Doppler geometry.

Fig. 3
Fig. 3

Trajectory of an arbitrary incident beam traversing a cylindrical particle.

Fig. 4
Fig. 4

Projection in the YZ plane of the beams shown in Fig. 3.

Fig. 5
Fig. 5

Schematic location of the two scattered light cones from a tilted cylinder.

Fig. 6
Fig. 6

Location of the scattered cones.

Fig. 7
Fig. 7

X displacement in millimeters of the scattering cone from beam 1 at the detector 1 location for d = 8 μm and a 160-mm receiving lens.

Fig. 8
Fig. 8

Distance in millimeters between the scattering cones from beams 1 and 2 at the detector 1 location for d = 8 μm and a 160-mm receiving lens.

Fig. 9
Fig. 9

Phase difference as a function of the orientation of a glass fiber (d = 8 μm) with a 160-mm receiving lens, Phase factor, 5.083°/μm.

Fig. 10
Fig. 10

X displacement in millimeters of a scattering cone from beam 1 at the detector 1 location for d = 30 μm and a 400-mm receiving lens.

Fig. 11
Fig. 11

Distance in millimeters between the scattering cones from beams 1 and 2 at the detector 1 location for d = 30 μm and a 400-mm receiving lens.

Fig. 12
Fig. 12

Phase difference as a function of the orientation of a glass fiber (d = 30 μm) with a 400-mm receiving lens, Phase factor, 2.034°/μm.

Fig. 13
Fig. 13

Dimensions of the receiving apertures.

Fig. 14
Fig. 14

Simulated and measured cylinder for an 8-μm glass fiber and a 160-mm receiving focal lens.

Fig. 15
Fig. 15

Simulated and measured cylinder for a 15-μm glass fiber and a 160-mm receiving focal lens.

Fig. 16
Fig. 16

Simulated and measured cylinder for a 30-μm glass fiber and a 400-mm receiving focal lens.

Tables (1)

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Table 1 Summary of System Parameters for Simulation

Equations (19)

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ξ = π 2 - tan - 1 ( 1 m - cos β 2 sin β 2 ) ,
Γ = sin - 1 ( sin Γ m ) .
L B = ( B 0 + B 2 ) n c + B 1 n d ,
L R = ( R 0 + R 2 + d ) n c ,
Δ L B = L B - L R = ( B 0 + B 2 - R 0 - R 2 - d ) n c + B 1 n d .
B 0 = A 0 cos Γ ,             B 1 = A 1 cos Γ ,             B 2 = A 2 cos Γ .
A 0 = R 0 + Δ R 0 ,             A 2 = R 2 + Δ R 2 .
Δ R 0 = Δ R 2 = Δ R = d ( 1 - cos ξ ) 2 ;
A 1 = d cos C = d ( 1 - sin 2 ξ m 2 ) 1 / 2 .
Δ L B = [ ( R 0 + R 2 ) ( 1 cos Γ - 1 ) + d ( 1 - cos ξ cos Γ ) - d ] n c + d cos Γ n d ( 1 - sin 2 ξ m 2 ) 1 / 2 .
S ^ 1 n = M · S ^ 1 ,
S ^ 1 n = | S ^ X 1 S ^ Y 1 S ^ Z 1 = [ cos γ sin γ cos τ sin γ sin τ - sin γ cos γ cos τ cos γ sin τ 0 - sin γ cos τ ] × [ 0 - sin θ cos θ ] ,
S ^ 1 n = | S ^ X 1 S ^ Y 1 S ^ Z 1 = [ - sin γ cos τ sin θ + sin γ sin τ cos θ - cos γ cos τ sin θ + cos γ sin τ cos θ sin τ sin θ + cos τ cos θ ] .
Γ 1 = π 2 - cos - 1 ( S ^ X 1 ) .
β 1 , 1 = ϕ 1 + θ , β 1 , 2 = ϕ 2 + θ , β 2 , 1 = ϕ 1 - θ , β 2 , 2 = ϕ 2 - θ .
Δ Φ 1 , 2 = Φ 1 - Φ 2 = 2 π λ [ ( Δ L 12 - Δ L 11 ) + ( Δ L 21 - Δ L 22 ) ] ,
Δ L i , j = [ ( R 0 + R 2 ) ( 1 cos Γ i - 1 ) + d ( 1 - cos ξ i , j cos Γ i ) - d ] n c + d cos Γ i n d ( 1 - sin 2 ξ i , j m 2 ) ,
ξ i , j = π 2 - tan - 1 ( | 1 m - cos β i , j 2 | sin β i , j 2 ) .
X i = R 2 tan ( ψ i - Γ i ) ( i = 1 , 2 ) ,

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