Abstract

We present a generalized theoretical model for the response of the phase/Doppler (P/D) measurement system to light scattered by cylindrical fibers. This theoretical model is valid for arbitrary fiber diameters and refractive indices, for Gaussian incident beams, and it accounts for arbitrary fiber orientations, fiber positions, and effects that are due to the two-dimensional receivers. The generalized P/D computer model (GPDCM) is the extension of an earlier study by the authors, combining past P/D simulation methodology with recent developments in modeling light scattering by tilted cylindrical fibers. A fortran computer program that implements the GPDCM theoretical development was written and tested against known P/D results and physical expectations. To illustrate the capabilities of the GPDCM, we present computation results, comparing the effect of fiber tilt, fiber position, and receiver aperture on the performance of P/D systems configured in backscatter and sidescatter arrangements. Calculations show that the effects of fiber tilt and position are most pronounced in the backscatter P/D arrangement, resulting in broadening of the measured phase distribution. The calculated mean phase shifts, however, were found to be essentially independent of the above factors. Computational results also showed that the effect of fiber tilt and position on phase-distribution measurements can be reduced through proper choice of aperture shape and by imposition of threshold criteria on measurable signal characteristics such as the amplitude ratio and visibilities. The GPDCM provides a computational tool that will be valuable in the design, optimization, and evaluation of P/D fiber measurement systems.

© 1998 Optical Society of America

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References

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  1. M. Saffman, P. Buchhave, H. Tanger, “Simultaneous measurement of size, concentration and velocity of spherical particles by a laser Doppler method,” in Laser Anemometry in Fluid Mechanics II, R. J. Adrian, D. F. Durao, F. Durst, H. Mishina, J. Whitelaw, eds. (LADOAN, Lisbon, Portugal, 1984), pp. 85–103.
  2. S. V. Sankar, W. D. Bachalo, “Response characteristics of the phase Doppler particle analyzer for sizing particles larger than the light wavelength,” Appl. Opt. 30, 1487–1496 (1991).
    [CrossRef] [PubMed]
  3. A. Naqwi, F. Durst, “Analysis of laser light-scattering interferometric devices for in-line diagnostics of moving particles,” Appl. Opt. 32, 4003–4018 (1993).
    [PubMed]
  4. S. V. Sankar, B. J. Weber, D. Y. Damemoto, W. D. Bachalo, “Sizing fine particles with the phase Doppler interferometric technique,” Appl. Opt. 30, 4914–4920 (1991).
    [CrossRef] [PubMed]
  5. S. A. Schaub, D. R. Alexander, J. P. Barton, “Theoretical analysis of the effects of particle trajectory and particle resonances on the performance of a phase-Doppler particle analyzer,” Appl. Opt. 33, 473–483 (1994).
    [CrossRef] [PubMed]
  6. S. A. Schaub, A. A. Naqwi, F. L. Harding, “Design of a phase/Doppler light-scattering system for measurement of small-diameter glass fibers during fiberglass manufacturing,” Appl. Opt. 37, 573–585 (1998).
    [CrossRef]
  7. H. Mignon, G. Grehan, G. Gouesbet, T. H. Xu, C. Tropea, “Measurement of cylindrical particles with phase Doppler anemometry,” Appl. Opt. 35, 5180–5190 (1996).
    [CrossRef] [PubMed]
  8. A. Naqwi, L. M. Jensen, “Device for interferometric measurements with compensation for tilt and position of measured cylindrical objects,” U.S. Patent5,513,004 (30April1996).
  9. A. Naqwi, T. Mahon, D. Havir, P. Tsai, C. Hassenboehler, L. Wadsworth, “On-line sizing of meltblown and spunbond fibers using adaptive phase/Doppler velocimeter (APV) method,” in Book of Papers, INDA-TEC 95 (Association of Nonwoven Fabrics Industry, Cary, N.C., 1995), pp. 167–184.
  10. J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder,” J. Opt. Soc. Am. A 14, 640–652 (1997).
    [CrossRef]
  11. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  12. J. B. Keller, H. B. Keller, “Determination of reflected and transmitted fields by geometrical optics,” J. Opt. Soc. Am. 40, 48–52 (1950).
    [CrossRef]

1998 (1)

1997 (1)

1996 (1)

1994 (1)

1993 (1)

1991 (2)

1950 (1)

Alexander, D. R.

Bachalo, W. D.

Barton, J. P.

Bohren, C. F.

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

Buchhave, P.

M. Saffman, P. Buchhave, H. Tanger, “Simultaneous measurement of size, concentration and velocity of spherical particles by a laser Doppler method,” in Laser Anemometry in Fluid Mechanics II, R. J. Adrian, D. F. Durao, F. Durst, H. Mishina, J. Whitelaw, eds. (LADOAN, Lisbon, Portugal, 1984), pp. 85–103.

Damemoto, D. Y.

Durst, F.

Gouesbet, G.

Grehan, G.

Harding, F. L.

Hassenboehler, C.

A. Naqwi, T. Mahon, D. Havir, P. Tsai, C. Hassenboehler, L. Wadsworth, “On-line sizing of meltblown and spunbond fibers using adaptive phase/Doppler velocimeter (APV) method,” in Book of Papers, INDA-TEC 95 (Association of Nonwoven Fabrics Industry, Cary, N.C., 1995), pp. 167–184.

Havir, D.

A. Naqwi, T. Mahon, D. Havir, P. Tsai, C. Hassenboehler, L. Wadsworth, “On-line sizing of meltblown and spunbond fibers using adaptive phase/Doppler velocimeter (APV) method,” in Book of Papers, INDA-TEC 95 (Association of Nonwoven Fabrics Industry, Cary, N.C., 1995), pp. 167–184.

Huffman, D. R.

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

Jensen, L. M.

A. Naqwi, L. M. Jensen, “Device for interferometric measurements with compensation for tilt and position of measured cylindrical objects,” U.S. Patent5,513,004 (30April1996).

Keller, H. B.

Keller, J. B.

Lock, J. A.

Mahon, T.

A. Naqwi, T. Mahon, D. Havir, P. Tsai, C. Hassenboehler, L. Wadsworth, “On-line sizing of meltblown and spunbond fibers using adaptive phase/Doppler velocimeter (APV) method,” in Book of Papers, INDA-TEC 95 (Association of Nonwoven Fabrics Industry, Cary, N.C., 1995), pp. 167–184.

Mignon, H.

Naqwi, A.

A. Naqwi, F. Durst, “Analysis of laser light-scattering interferometric devices for in-line diagnostics of moving particles,” Appl. Opt. 32, 4003–4018 (1993).
[PubMed]

A. Naqwi, T. Mahon, D. Havir, P. Tsai, C. Hassenboehler, L. Wadsworth, “On-line sizing of meltblown and spunbond fibers using adaptive phase/Doppler velocimeter (APV) method,” in Book of Papers, INDA-TEC 95 (Association of Nonwoven Fabrics Industry, Cary, N.C., 1995), pp. 167–184.

A. Naqwi, L. M. Jensen, “Device for interferometric measurements with compensation for tilt and position of measured cylindrical objects,” U.S. Patent5,513,004 (30April1996).

Naqwi, A. A.

Saffman, M.

M. Saffman, P. Buchhave, H. Tanger, “Simultaneous measurement of size, concentration and velocity of spherical particles by a laser Doppler method,” in Laser Anemometry in Fluid Mechanics II, R. J. Adrian, D. F. Durao, F. Durst, H. Mishina, J. Whitelaw, eds. (LADOAN, Lisbon, Portugal, 1984), pp. 85–103.

Sankar, S. V.

Schaub, S. A.

Tanger, H.

M. Saffman, P. Buchhave, H. Tanger, “Simultaneous measurement of size, concentration and velocity of spherical particles by a laser Doppler method,” in Laser Anemometry in Fluid Mechanics II, R. J. Adrian, D. F. Durao, F. Durst, H. Mishina, J. Whitelaw, eds. (LADOAN, Lisbon, Portugal, 1984), pp. 85–103.

Tropea, C.

Tsai, P.

A. Naqwi, T. Mahon, D. Havir, P. Tsai, C. Hassenboehler, L. Wadsworth, “On-line sizing of meltblown and spunbond fibers using adaptive phase/Doppler velocimeter (APV) method,” in Book of Papers, INDA-TEC 95 (Association of Nonwoven Fabrics Industry, Cary, N.C., 1995), pp. 167–184.

Wadsworth, L.

A. Naqwi, T. Mahon, D. Havir, P. Tsai, C. Hassenboehler, L. Wadsworth, “On-line sizing of meltblown and spunbond fibers using adaptive phase/Doppler velocimeter (APV) method,” in Book of Papers, INDA-TEC 95 (Association of Nonwoven Fabrics Industry, Cary, N.C., 1995), pp. 167–184.

Weber, B. J.

Xu, T. H.

Appl. Opt. (6)

J. Opt. Soc. Am. (1)

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

Other (4)

M. Saffman, P. Buchhave, H. Tanger, “Simultaneous measurement of size, concentration and velocity of spherical particles by a laser Doppler method,” in Laser Anemometry in Fluid Mechanics II, R. J. Adrian, D. F. Durao, F. Durst, H. Mishina, J. Whitelaw, eds. (LADOAN, Lisbon, Portugal, 1984), pp. 85–103.

A. Naqwi, L. M. Jensen, “Device for interferometric measurements with compensation for tilt and position of measured cylindrical objects,” U.S. Patent5,513,004 (30April1996).

A. Naqwi, T. Mahon, D. Havir, P. Tsai, C. Hassenboehler, L. Wadsworth, “On-line sizing of meltblown and spunbond fibers using adaptive phase/Doppler velocimeter (APV) method,” in Book of Papers, INDA-TEC 95 (Association of Nonwoven Fabrics Industry, Cary, N.C., 1995), pp. 167–184.

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

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Figures (6)

Fig. 1
Fig. 1

Schematic of laboratory (L) and theory (T) coordinate systems used in theoretical modeling.

Fig. 2
Fig. 2

Typical Doppler bursts illustrating the measurable quantities of interest.

Fig. 3
Fig. 3

Computed scattered-light distribution for an 18.3-μm diameter fiber: λ = 514.5 nm, 1.52 refractive index, 500 mm focal lengths, 17-mm beam separation, w 0 = 150 μm, parallel e-field polarization.

Fig. 4
Fig. 4

Phase shift as a function of diameter computed with the plane wave and the GPDCM for both backscatter and sidescatter P/D arrangements.

Fig. 5
Fig. 5

Schematic illustrating the semicircular receiver aperture and the reduced aperture generated by application of masks M1 and M2.

Fig. 6
Fig. 6

Simulated phase distributions for fibers in the 14.5–15.5-μm-diameter range. The P/D parameters are given in Table 1; the total counts for all distributions are 500.

Tables (1)

Tables Icon

Table 1 Instrument Parameters for Backscatter and Sidescatter P/D Studies

Equations (56)

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k ˆ j , L inc = cos   α u ˆ xL ± sin   α u ˆ yL ,
x L d = ρ L   cos   χ   cos ν - x rel   sin ν - y rel   sin   χ     cos ν ,
y L d = ρ L   cos   χ   sin ν + x rel   cos ν - y rel   sin   χ   sin ν ,
z L d = ρ L   sin   χ + y rel   cos   χ .
k ˆ j , T inc = cos   ξ u ˆ xT - sin   ξ u ˆ zT
x L d = x L d - x fib ,
y L d = y L d - y fib ,
z L d = z L d .
ρ L = x L d 2 + y L d 2 + z L d 2 1 / 2 ,
tan   χ = z L d x L d 2 + y L d 2 1 / 2 ,
tan   ν = y L d x L d .
R j = R zL - ϕ R yL - γ R zL - δ
cos   ϕ = cos   β   cos   γ / cos 2   β   cos 2   γ + sin 2   β 1 / 2 ,
sin   ϕ = - sin   β / cos 2   β   cos 2   γ + sin 2   β 1 / 2 ,
β = δ α .
cos   ξ = cos 2   β   cos 2   γ + sin 2   β 1 / 2 ,
sin   ξ = - cos   β   sin   γ .
E ˆ j , L e , inc E ˆ j , L m , inc = cos   ψ sin   ψ - sin   ψ cos   ψ E ˆ j , T , inc E ˆ j , T μ , inc ,
B ˆ j , L e , inc B ˆ j , L m , inc = cos   ψ sin   ψ - sin   ψ cos   ψ B ˆ j , T , inc B ˆ j , T μ , inc ,
cos   ψ = cos   γ / cos   ξ ,
sin   ψ = sin   β   sin   γ / cos   ξ .
x 0 y 0 z 0 = R j - x fib - y fib 0 ,
x T d y T d z T d = R j x L d - x fib y L d - y fib z L d .
cos   θ = x T d x T d 2 + y T d 2 1 / 2 ,
sin   θ = y T d x T d 2 + y T d 2 1 / 2 ,
tan   η = z T d - z 0 x T d 2 + y T d 2 1 / 2 - x 0 ,
ρ T = ρ L .
A l h = cos   ξ 2 s 1 - h 2 π F 1 / 2 × exp - ik hz 0 + 1 - h 2 1 / 2 x 0 × exp - h   cos   ξ + 1 - h 2 1 / 2 sin   ξ 2 / 4 s 2 × exp - s 2 ky 0 + l 1 - h 2 - 1 / 2 2 / F ,
B l h = 0 ,
s = 1 / kw 0 ,
F = cos   ξ cos   ξ - h 1 - h 2 - 1 / 2   sin   ξ - 2 is 2 kx 0 / 1 - h 2 1 / 2 .
S θ ,   h = l = - exp il θ a l h × exp - s 2 ky 0 + l 1 - h 2 - 1 / 2 2 / F ,
S μ θ ,   h = l = - exp il θ b l h × exp - s 2 ky 0 + l 1 - h 2 - 1 / 2 2 / F ,
S q θ ,   h = l = - exp il θ q l h × exp - s 2 ky 0 + l 1 - h 2 - 1 / 2 2 / F .
cos   η = 1 - h 2 1 / 2 ,
sin   η = h ,
F = cos   ξ   cos ξ + η - 2 is 2 kx 0 / cos   η ,
0 = F - 1 / 2   exp - sin 2 ξ + η / 4 s 2 exp ik ρ T - x 0   cos   η - z 0   sin   η .
E j , T scatt = 0 E ˆ j , T , scatt S   cos   ψ + S q   sin   ψ + 0 E ˆ j , T μ , scatt - S q   cos   ψ + S μ   sin   ψ .
E ˆ j , L , scatt = R j - 1 E ˆ j , T , scatt ,
E L total = E 1 , L scatt + E 2 , L scatt ,
B L total = B 1 , L scatt + B 2 , L scatt .
S total = Re cos   χ   cos   ν E y , L total * B z , L total - E z , L total * B y , L total + cos   χ   sin   ν E z , L total * B x , L total - E x , L total * B z , L total + sin   χ E x , L total * B y , L total - E y , L total * B x , L total .
P n = aperture   S total d x rel d y rel ,
P n = A n + B n   sin ( φ B - ε n ) ,
V n = P max - P min P max + P min = A n + B n - A n - B n A n + B n + A n - B n = B n A n .
τ 12 = | ε 2 - ε 1 | .
tan   ε n = sin   φ 1 - sin   φ 3 - P ˜ n sin   φ 2 - sin   φ 3 P ˜ n cos   φ 3 - cos   φ 2 - cos   φ 3 - cos   φ 1 ,
P ˜ n = P n φ 1 - P n φ 3 P n φ 2 - P n φ 3 .
B n = P n φ 2 - P n φ 3 sin φ 2 - ε n - sin φ 3 - ε n ,
A n = P n φ 3 - B n   sin φ 3 - ε n .
Δ η = 2 s ,
γ = 2 1 / 2 s / α 1.52 °
Δ z T = z 0 + x 0   tan   ξ = - γ y fib   sin   δ ± γ α x fib   sin   δ + y fib   cos   δ .
γ = 2 1 / 2 s / α ρ T / ρ fib ,
ρ fib = x fib 2 + y fib 2 1 / 2 .

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