Abstract

The amount of light occluded by a fiber as it passes through a laser beam can be used as the basis for fiber-diameter measurement. This technique is analyzed with a two-dimensional rigorous model. The occlusion seen for dielectric fibers as a function of their diameter is highly oscillatory owing to interference between the light transmitted by the fiber and the rest of the diffracted field. Scalar diffraction theory is shown to be adequate in modeling this effect. The oscillation sets a limit to the accuracy of simple diameter measurement systems and is confirmed experimentally for glass fibers. However, wool fibers are found to be better treated as an absorbing material. The effect of beam polarization is investigated and found to be negligible for dielectric fibers but significant for metal fibers of small diameter.

© 1998 Optical Society of America

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References

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  1. D. Lebrun, S. Belaid, C. Özkul, K. F. Ren, G. Gréhand, “Enhancement of wire diameter measurements: comparison between Fraunhofer diffraction and Lorentz-Mie theory,” Opt. Eng. 35, 946–950 (1996).
    [CrossRef]
  2. H. Wang, R. Valdivia-Hernandez, “Laser scanner and diffraction pattern detection: a novel concept for dynamic gauging of fine wires,” Meas. Sci. Technol. 6, 452–457 (1995).
    [CrossRef]
  3. P. Cielo, G. Vaudreuil, “Optical inspection of industrial materials by unidimensional Fourier transform,” Appl. Opt. 27, 4645–4652 (1988).
    [CrossRef] [PubMed]
  4. L. S. Watkins, “Scattering from side-illuminated clad glass fibers for determination of fiber parameters,” J. Opt. Soc. Am. 64, 767–772 (1974).
    [CrossRef]
  5. M. Glass, T. B. Dabbs, P. W. Chudleigh, “The optics of the wool fiber diameter analyser,” Textile Res. J. 65, 85–94 (1995).
    [CrossRef]
  6. M. Glass, “Fresnel diffraction from curved fiber snippets with application to fiber diameter measurement,” Appl. Opt. 35, 1605–1616 (1996).
    [CrossRef] [PubMed]
  7. E. Zimmermann, R. Dändliker, N. Souli, “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). [Note that Eq. (11) of this reference contains a sign error—the second term in the square brackets should be subtracted from the first term.]
  8. M. Glass, “Diffraction of a Gaussian beam around a strip mask,” Appl. Opt. 37, 2550–2562 (1998).
    [CrossRef]
  9. R. G. Greenler, J. W. Hable, P. O. Slane, “Diffraction around a fine wire: how good is the single-slit approximation?” Am. J. Phys. 58, 330–331 (1990).
    [CrossRef]
  10. D. H. Smithgall, L. S. Watkins, R. E. Frazee, “High-speed noncontact fiber-diameter measurement using forward light scattering,” Appl. Opt. 16, 2395–2402 (1977).
    [CrossRef] [PubMed]
  11. The rigorous solution for a plane wave incident on a cylinder is given in H. C. van de Hulst , Light Scattering by Small Particles ( Wiley, New York, 1957), Chap. 15, p. 300.
  12. The rigorous solution for a Gaussian beam incident on a cylinder may be found in, for example, S. Kozaki , “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).
  13. W. J. Lentz, “Generating Bessel functions in Mie scattering calculations using continued fractions,” Appl. Opt. 15, 668–671 (1976).
    [CrossRef] [PubMed]
  14. B. Chen, Diffraction Tomography and its Applications for Optical Imaging in Random Media, M. S. thesis (Department of Physics, University of Bergen, Norway, 1996). Chap. 1, p. 14.
  15. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989), Chap. 1, p. 40.
  16. The refractive index of Cu was taken from E. D. Palik , ed., Tables of Optical Constants of Solids I (Academic, Orlando, Fla., 1985), p. 285. The values in the tables were interpolated from neighboring wavelengths to obtain those at 633 nm.
  17. See the section on saccharimetry in E. W. Washburn , ed., International Critical Tables of Numerical Data, Physics, Chemistry and Technology (McGraw-Hill, New York, 1933), Vol. 2, p. 334.

1998 (1)

1996 (2)

M. Glass, “Fresnel diffraction from curved fiber snippets with application to fiber diameter measurement,” Appl. Opt. 35, 1605–1616 (1996).
[CrossRef] [PubMed]

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

1995 (3)

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

E. Zimmermann, R. Dändliker, N. Souli, “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). [Note that Eq. (11) of this reference contains a sign error—the second term in the square brackets should be subtracted from the first term.]

M. Glass, T. B. Dabbs, P. W. Chudleigh, “The optics of the wool fiber diameter analyser,” Textile Res. J. 65, 85–94 (1995).
[CrossRef]

1990 (1)

R. G. Greenler, J. W. Hable, P. O. Slane, “Diffraction around a fine wire: how good is the single-slit approximation?” Am. J. Phys. 58, 330–331 (1990).
[CrossRef]

1988 (1)

1982 (1)

The rigorous solution for a Gaussian beam incident on a cylinder may be found in, for example, S. Kozaki , “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).

1977 (1)

1976 (1)

1974 (1)

Belaid, S.

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

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989), Chap. 1, p. 40.

Chen, B.

B. Chen, Diffraction Tomography and its Applications for Optical Imaging in Random Media, M. S. thesis (Department of Physics, University of Bergen, Norway, 1996). Chap. 1, p. 14.

Chudleigh, P. W.

M. Glass, T. B. Dabbs, P. W. Chudleigh, “The optics of the wool fiber diameter analyser,” Textile Res. J. 65, 85–94 (1995).
[CrossRef]

Cielo, P.

Dabbs, T. B.

M. Glass, T. B. Dabbs, P. W. Chudleigh, “The optics of the wool fiber diameter analyser,” Textile Res. J. 65, 85–94 (1995).
[CrossRef]

Dändliker, R.

Frazee, R. E.

Glass, M.

Greenler, R. G.

R. G. Greenler, J. W. Hable, P. O. Slane, “Diffraction around a fine wire: how good is the single-slit approximation?” Am. J. Phys. 58, 330–331 (1990).
[CrossRef]

Gréhand, G.

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

Hable, J. W.

R. G. Greenler, J. W. Hable, P. O. Slane, “Diffraction around a fine wire: how good is the single-slit approximation?” Am. J. Phys. 58, 330–331 (1990).
[CrossRef]

Kozaki, S.

The rigorous solution for a Gaussian beam incident on a cylinder may be found in, for example, S. Kozaki , “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).

Lebrun, D.

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

Lentz, W. J.

Özkul, C.

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

Ren, K. F.

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

Slane, P. O.

R. G. Greenler, J. W. Hable, P. O. Slane, “Diffraction around a fine wire: how good is the single-slit approximation?” Am. J. Phys. 58, 330–331 (1990).
[CrossRef]

Smithgall, D. H.

Souli, N.

Valdivia-Hernandez, R.

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

van de Hulst, H. C.

The rigorous solution for a plane wave incident on a cylinder is given in H. C. van de Hulst , Light Scattering by Small Particles ( Wiley, New York, 1957), Chap. 15, p. 300.

Vaudreuil, G.

Wang, H.

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

Watkins, L. S.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989), Chap. 1, p. 40.

Zimmermann, E.

Am. J. Phys. (1)

R. G. Greenler, J. W. Hable, P. O. Slane, “Diffraction around a fine wire: how good is the single-slit approximation?” Am. J. Phys. 58, 330–331 (1990).
[CrossRef]

Appl. Opt. (5)

J. Appl. Phys. (1)

The rigorous solution for a Gaussian beam incident on a cylinder may be found in, for example, S. Kozaki , “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).

J. Opt. Soc. Am. (1)

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

Meas. Sci. Technol. (1)

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

Opt. Eng. (1)

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

Textile Res. J. (1)

M. Glass, T. B. Dabbs, P. W. Chudleigh, “The optics of the wool fiber diameter analyser,” Textile Res. J. 65, 85–94 (1995).
[CrossRef]

Other (5)

The rigorous solution for a plane wave incident on a cylinder is given in H. C. van de Hulst , Light Scattering by Small Particles ( Wiley, New York, 1957), Chap. 15, p. 300.

B. Chen, Diffraction Tomography and its Applications for Optical Imaging in Random Media, M. S. thesis (Department of Physics, University of Bergen, Norway, 1996). Chap. 1, p. 14.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989), Chap. 1, p. 40.

The refractive index of Cu was taken from E. D. Palik , ed., Tables of Optical Constants of Solids I (Academic, Orlando, Fla., 1985), p. 285. The values in the tables were interpolated from neighboring wavelengths to obtain those at 633 nm.

See the section on saccharimetry in E. W. Washburn , ed., International Critical Tables of Numerical Data, Physics, Chemistry and Technology (McGraw-Hill, New York, 1933), Vol. 2, p. 334.

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Figures (11)

Fig. 1
Fig. 1

Diffraction geometry and coordinate system. A Gaussian beam that propagates along the z axis is focused onto the cylinder from the left, producing a diffraction pattern to the right.

Fig. 2
Fig. 2

Ray trace for a cylinder with n 2 = 1.457 in water (n 1 = 1.333).

Fig. 3
Fig. 3

Path of a ray, initially parallel to the optical axis, through a dielectric cylinder. Modeling the phase variation imparted by a dielectric cylinder, we calculate a phase distribution across z = 0, which, when propagated geometrically to the exit point of the cylinder, matches the phase predicted at that point by geometric optics.

Fig. 4
Fig. 4

Comparison of the irradiance predicted by (a) the rigorous model, (b) the lens model, and (c) the black-strip model, for the diffraction of light by a dielectric cylinder. In (b) and (c) the field has been calculated only for z > a. The parameters are n 1 = 1.333, n 2 = 1.457, λ0 = 633 nm, w 0 = 6.0 μm, and a = 1.0 μm. In each image, z (horizontal) ranges from -5 to 25 μm and x (vertical) ranges from -10 to 10 μm.

Fig. 5
Fig. 5

Comparison of the rigorous model (circles), lens model (solid curve) and black-strip model (dotted curve) of diffraction by a dielectric cylinder in the far field. Here the index of the cylinder is 1.457, the vacuum wavelength is 633 nm, W 0 = 152 μm, and the distance from the cylinder is 437 mm. The medium index and the cylinder radius are shown in the graphs.

Fig. 6
Fig. 6

Occlusion as a function of fiber radius for a purely dielectric cylinder, which is calculated with the rigorous model with either orientation of incident polarization. Here the fiber is glass and the medium is water.

Fig. 7
Fig. 7

(a) Upper and lower limits to the response for a homogeneous dielectric cylinder, plotted with the true response (containing the oscillations) and the response for a black-strip mask (running through the center). (b) Percentage spread in the measured fiber radius caused by light transmitted by the fiber.

Fig. 8
Fig. 8

(a) Occlusion as a function of fiber radius for Cu wire (n 2 = 0.249 + 3.41i) in air for two orthogonal polarizations. (b) Difference in measured fiber diameter between the two polarizations: for Cu wires in air (solid curve) and for Cu wires in water (dashed curve). Note that the detector size differs depending on whether the medium is air or water so that the detector receives 78% of the incident, unoccluded beam. The absolute difference is shown in place of the relative error because it is interesting to see that it is typically not far from λ1/2.

Fig. 9
Fig. 9

Experimental arrangement for measuring the occlusion of a fiber.

Fig. 10
Fig. 10

Detector signal as a function of lateral position of the fiber. The minimum is taken to be the measured occlusion for this fiber diameter.

Fig. 11
Fig. 11

Occlusion as a function of refractive index of the medium for a glass fiber (circles) and for a wool fiber (squares). Asterisks indicate points that were rejected owing to incomplete mixing in the sugar solution. The solid curve is the occlusion for a glass fiber calculated with the lens model, where the fiber radius has been fitted to obtain agreement with the experimental data.

Equations (39)

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u r ,   θ = u 0 m = - exp - im θ - π / 2 × J m n 1 k 0 r - H m 1 n 1 k 0 r γ m A m ,
A m = w 0 π 0 exp - w 0 2 ξ 2 / 4 cos m   arcsin ξ / k 0 n 1 d ξ ,
γ m = n 2 J m n 1 k 0 a J m n 2 k 0 a J m n 2 k 0 a - n 1 J m n 1 k 0 a n 2 H m 1 n 1 k 0 a J m n 2 k 0 a J m n 2 k 0 a - n 1 H m 1 n 1 k 0 a , E   parallel   to   fiber ,
γ m = n 1 J m n 1 k 0 a J m n 2 k 0 a J m n 2 k 0 a - n 2 J m n 1 k 0 a n 1 H m 1 n 1 k 0 a J m n 2 k 0 a J m n 2 k 0 a - n 2 H m 1 n 1 k 0 a , H   parallel   to   fiber .
E = in 2 k 0 y ˆ     E   parallel to fiber ,
E = - imu r r ˆ - u r   θ ˆ     H   parallel to fiber ,
| E | 2     | u | 2 .
J m z 2 π z 1 / 2 cos z - m π / 2 - π / 4 ,
Y m z 2 π z 1 / 2 sin z - m π / 2 - π / 4 .
F a = - d d   | u r ,   θ ,   a | 2 d x - d d   | u r ,   θ ,   0 | 2 d x ,
u x ,   z = exp i ψ λ 1 z 1 / 2 -   u x ,   0 × exp π ix 2 λ 1 z exp - 2 π ixx λ 1 z d x ,
u x ,   z = u d + u t ,
u d = 2 u 0 exp i ψ λ 1 z 1 / 2 a exp - x 2 / w 0 2 × cos 2 π xx λ 1 z d x ,
u t = u 0 exp i ψ λ 1 z 1 / 2 -   A x exp 2 π i δ x / λ 0 - 2 π ixx / λ i z d x .
P = -   | u x ,   z | 2 d x = | u 0 | 2 w 0 π / 2 1 / 2 .
F T a = 1 P - d d   | u x ,   z | 2 d x = 2 π 1 / 2 1 λ 1 zw 0 - d d   Ī x ,   z d x .
Ī x ,   z = 2 a exp - x 2 / w 0 2 cos 2 π xx 2 λ 1 z d x + -   A x exp 2 π i δ x / λ 0 - 2 π ixx / λ 1 z d x 2 .
F a = F T a / F T 0 .
I x ,   y ,   z = | u 0 | 2 π w 0 2 λ 1 z 2   Ī x ,   z exp - y 2 / w 2 ,
F a = 2 π 1 / 2 1 λ 1 zw 0 - d d erf 2 d 2 - 2 x 2 1 / 2 / w Ī x ,   z d x ,
w = λ 1 z / π w 0 .
i = arcsin x 1 / a ,     r = arcsin x 1 / na ,
x = a   sin   i cos 2 i - 2 r .
ϕ 2 = n 2 l - n 1 a   cos   i .
ϕ 2 = δ x + n 1 s ,
δ x / n 1 = nl - s - a   cos   i ,
l = 2 a   cos   r ,     s = z 2 2 + x - x 2 2 1 / 2 ,
z 2 = a   cos 2 r - i ,     x 2 = a   sin 2 r - i .
g x 1 = exp - x 1 2 / w 0 2 ,
q x = dx 1 dx 1 cos 2 i - 2 r 1 / 2 ,
fr x 1 = 4 n   cos   i   cos   r n   cos   i + cos   r 2 E   parallel to fiber 4 n   cos   i   cos   r cos   i + n   cos   r 2 H   parallel to fiber .
A x = g x 1 q x fr x 1 .
x max = an 2 / 2 - n 2 ,     n < 2 .
δ x / n 1 2 a n - 1 - n - 1 an   x 2 ,
f = a 2 1 - 1 / n ,
Ī x ,   z = 2   a exp - x 2 / w 0 2 cos 2 π xx λ 1 z d x 2 .
I upper x ,   z = | u d | 2 + | u t | 2 + 2 | u d | | u t | ,
I lower x ,   z = | u d | 2 + | u t | 2 - 2 | u d | | u t | .
Δ F T a = 4 2 π 1 / 2 1 λ 1 zw 0 - d d   | u d | | u t | d x .

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