Abstract

We present a model to determine the far-field diffraction pattern of a metallic cylinder of infinite length when it is illuminated in oblique incidence. This model is based on the Helmholtz–Kirchhoff integral using the Beckmann conditions for reflection. It considers the three-dimensional nature of the diffracting object as well as the material of which the cylinder is made. This model shows that the diffraction orders are placed in a cone of light. The amplitude at the far field can be divided into three terms: the first term accounts for Babinet’s principle, that is, the contribution of the cylinder projection; the second term accounts for the three dimensionality of the cylinder; and the third term accounts for the material of which the cylinder is made. This model is applied to the diameter estimation of the cylinder. Since the amplitude of the Babinet contribution is much larger than the light reflected by the surface, the cylinder diameter can be obtained in a simple way. With this approximation, the locations of the diffraction minima do not vary when the cylinder is inclined. On the other hand, when the reflected light is considered the location of the minima and, hence, the estimation of the diameter, varies. Also, a modification of the diffraction minima is produced by the material of which the cylinder is made. Experimental results are also obtained that corroborate the theoretical approach.

© 2008 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics 6th ed. (Pergamon, 1980).
  2. J. F. Fardeau, “New laser sensors for wire diameter measurement,” Wire J. Int. 22, 42-51 (1989).
  3. T. K. Millard and T. A. Herchenreder, “Automatic diameter measurement: state of the art,” Wire J. Int. 24, 61-69 (1991).
  4. I. Serroukh, J. C. Martinez-Anton, and E. Bernabeu, “Accuracy of the thin metallic wires diameter using Fraunhofer diffraction technique,” Proc. SPIE 4099, 255-266 (2000).
    [CrossRef]
  5. J. C. Martinez-Anton, I. Serroukh, and E. Bernabeu, “On Babinet's principle and a diffraction-interferometric technique to determine the diameter of cylindrical wires,” Metrologia 38, 125-134 (2001).
    [CrossRef]
  6. S. Schmidt, “Ein beitrag zur erklärung der lichtbeugung am metallischen krieszylinder,” PTB Mitteilungen 86, 239-247(1976).
  7. E. Bernabeu, I. Serroukh, and L. M. Sanchez-Brea, “A geometrical model for wire optical diffraction selected by experimental statistical analysis,” Opt. Eng. 38, 1319-1325 (1999).
    [CrossRef]
  8. W. Tang, Y. Zhou, and J. Zhang, “Improvement on theoretical model for thin-wire and slot measurement by optical diffraction,” Meas. Sci. Technol. 10, N119-N123 (1999).
    [CrossRef]
  9. Lord Rayleigh, “The dispersal of light by a dielectric cylinder,” Philos. Mag. 36, 365-376 (1918).
  10. H. Hönl, A. W. Maue, and K. Westpfahl, Handbuch der Physik Vol. XXV (Springer Verlag, 1961).
  11. J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189-195(1955).
    [CrossRef]
  12. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  13. L. M. Sanchez-Brea, “Diameter estimation of cylinders by rigorous diffraction model,” J. Opt. Soc. Am. A 22, 1402-1407(2005).
    [CrossRef]
  14. J. Xie, Y. Qiu, H. Ming, and C. Li “Light polarization effect in measurement of thin wire diameter by laser diffraction and its explanation with boundary diffraction wave,” J. Appl. Phys. 69, 6899-6903 (1991).
    [CrossRef]
  15. L. M. Sanchez-Brea, P. Siegmann, M. A. Rebollo, and E. Bernabeu, “Optical technique for the automatic detection and measurement of surface defects on thin metallic wires, Appl. Opt. 39, 539-545 (2000).
    [CrossRef]
  16. L. M. Sanchez-Brea , and E. Bernabeu, “Diffraction by cylinders illuminated in oblique, off-axis incidence,” Optik 112, 169-174 (2001).
    [CrossRef]
  17. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987).
  18. J. J. Stamnes, Waves in Focal Regions (Adam Hilger, 1986).

2005 (1)

2001 (2)

L. M. Sanchez-Brea , and E. Bernabeu, “Diffraction by cylinders illuminated in oblique, off-axis incidence,” Optik 112, 169-174 (2001).
[CrossRef]

J. C. Martinez-Anton, I. Serroukh, and E. Bernabeu, “On Babinet's principle and a diffraction-interferometric technique to determine the diameter of cylindrical wires,” Metrologia 38, 125-134 (2001).
[CrossRef]

2000 (2)

I. Serroukh, J. C. Martinez-Anton, and E. Bernabeu, “Accuracy of the thin metallic wires diameter using Fraunhofer diffraction technique,” Proc. SPIE 4099, 255-266 (2000).
[CrossRef]

L. M. Sanchez-Brea, P. Siegmann, M. A. Rebollo, and E. Bernabeu, “Optical technique for the automatic detection and measurement of surface defects on thin metallic wires, Appl. Opt. 39, 539-545 (2000).
[CrossRef]

1999 (2)

E. Bernabeu, I. Serroukh, and L. M. Sanchez-Brea, “A geometrical model for wire optical diffraction selected by experimental statistical analysis,” Opt. Eng. 38, 1319-1325 (1999).
[CrossRef]

W. Tang, Y. Zhou, and J. Zhang, “Improvement on theoretical model for thin-wire and slot measurement by optical diffraction,” Meas. Sci. Technol. 10, N119-N123 (1999).
[CrossRef]

1991 (2)

T. K. Millard and T. A. Herchenreder, “Automatic diameter measurement: state of the art,” Wire J. Int. 24, 61-69 (1991).

J. Xie, Y. Qiu, H. Ming, and C. Li “Light polarization effect in measurement of thin wire diameter by laser diffraction and its explanation with boundary diffraction wave,” J. Appl. Phys. 69, 6899-6903 (1991).
[CrossRef]

1989 (1)

J. F. Fardeau, “New laser sensors for wire diameter measurement,” Wire J. Int. 22, 42-51 (1989).

1976 (1)

S. Schmidt, “Ein beitrag zur erklärung der lichtbeugung am metallischen krieszylinder,” PTB Mitteilungen 86, 239-247(1976).

1955 (1)

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189-195(1955).
[CrossRef]

1918 (1)

Lord Rayleigh, “The dispersal of light by a dielectric cylinder,” Philos. Mag. 36, 365-376 (1918).

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987).

Bernabeu, E.

L. M. Sanchez-Brea , and E. Bernabeu, “Diffraction by cylinders illuminated in oblique, off-axis incidence,” Optik 112, 169-174 (2001).
[CrossRef]

J. C. Martinez-Anton, I. Serroukh, and E. Bernabeu, “On Babinet's principle and a diffraction-interferometric technique to determine the diameter of cylindrical wires,” Metrologia 38, 125-134 (2001).
[CrossRef]

I. Serroukh, J. C. Martinez-Anton, and E. Bernabeu, “Accuracy of the thin metallic wires diameter using Fraunhofer diffraction technique,” Proc. SPIE 4099, 255-266 (2000).
[CrossRef]

L. M. Sanchez-Brea, P. Siegmann, M. A. Rebollo, and E. Bernabeu, “Optical technique for the automatic detection and measurement of surface defects on thin metallic wires, Appl. Opt. 39, 539-545 (2000).
[CrossRef]

E. Bernabeu, I. Serroukh, and L. M. Sanchez-Brea, “A geometrical model for wire optical diffraction selected by experimental statistical analysis,” Opt. Eng. 38, 1319-1325 (1999).
[CrossRef]

Bohren, C. F.

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

Born, M.

M. Born and E. Wolf, Principles of Optics 6th ed. (Pergamon, 1980).

Fardeau, J. F.

J. F. Fardeau, “New laser sensors for wire diameter measurement,” Wire J. Int. 22, 42-51 (1989).

Herchenreder, T. A.

T. K. Millard and T. A. Herchenreder, “Automatic diameter measurement: state of the art,” Wire J. Int. 24, 61-69 (1991).

Hönl, H.

H. Hönl, A. W. Maue, and K. Westpfahl, Handbuch der Physik Vol. XXV (Springer Verlag, 1961).

Huffman, D. R.

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

Li, C.

J. Xie, Y. Qiu, H. Ming, and C. Li “Light polarization effect in measurement of thin wire diameter by laser diffraction and its explanation with boundary diffraction wave,” J. Appl. Phys. 69, 6899-6903 (1991).
[CrossRef]

Martinez-Anton, J. C.

J. C. Martinez-Anton, I. Serroukh, and E. Bernabeu, “On Babinet's principle and a diffraction-interferometric technique to determine the diameter of cylindrical wires,” Metrologia 38, 125-134 (2001).
[CrossRef]

I. Serroukh, J. C. Martinez-Anton, and E. Bernabeu, “Accuracy of the thin metallic wires diameter using Fraunhofer diffraction technique,” Proc. SPIE 4099, 255-266 (2000).
[CrossRef]

Maue, A. W.

H. Hönl, A. W. Maue, and K. Westpfahl, Handbuch der Physik Vol. XXV (Springer Verlag, 1961).

Millard, T. K.

T. K. Millard and T. A. Herchenreder, “Automatic diameter measurement: state of the art,” Wire J. Int. 24, 61-69 (1991).

Ming, H.

J. Xie, Y. Qiu, H. Ming, and C. Li “Light polarization effect in measurement of thin wire diameter by laser diffraction and its explanation with boundary diffraction wave,” J. Appl. Phys. 69, 6899-6903 (1991).
[CrossRef]

Qiu, Y.

J. Xie, Y. Qiu, H. Ming, and C. Li “Light polarization effect in measurement of thin wire diameter by laser diffraction and its explanation with boundary diffraction wave,” J. Appl. Phys. 69, 6899-6903 (1991).
[CrossRef]

Rayleigh, Lord

Lord Rayleigh, “The dispersal of light by a dielectric cylinder,” Philos. Mag. 36, 365-376 (1918).

Rebollo, M. A.

Sanchez-Brea, L. M.

Sanchez-Brea , L. M.

L. M. Sanchez-Brea , and E. Bernabeu, “Diffraction by cylinders illuminated in oblique, off-axis incidence,” Optik 112, 169-174 (2001).
[CrossRef]

Sanchez-Brea, L. M.

L. M. Sanchez-Brea, P. Siegmann, M. A. Rebollo, and E. Bernabeu, “Optical technique for the automatic detection and measurement of surface defects on thin metallic wires, Appl. Opt. 39, 539-545 (2000).
[CrossRef]

E. Bernabeu, I. Serroukh, and L. M. Sanchez-Brea, “A geometrical model for wire optical diffraction selected by experimental statistical analysis,” Opt. Eng. 38, 1319-1325 (1999).
[CrossRef]

Schmidt, S.

S. Schmidt, “Ein beitrag zur erklärung der lichtbeugung am metallischen krieszylinder,” PTB Mitteilungen 86, 239-247(1976).

Serroukh, I.

J. C. Martinez-Anton, I. Serroukh, and E. Bernabeu, “On Babinet's principle and a diffraction-interferometric technique to determine the diameter of cylindrical wires,” Metrologia 38, 125-134 (2001).
[CrossRef]

I. Serroukh, J. C. Martinez-Anton, and E. Bernabeu, “Accuracy of the thin metallic wires diameter using Fraunhofer diffraction technique,” Proc. SPIE 4099, 255-266 (2000).
[CrossRef]

E. Bernabeu, I. Serroukh, and L. M. Sanchez-Brea, “A geometrical model for wire optical diffraction selected by experimental statistical analysis,” Opt. Eng. 38, 1319-1325 (1999).
[CrossRef]

Siegmann, P.

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987).

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Adam Hilger, 1986).

Tang, W.

W. Tang, Y. Zhou, and J. Zhang, “Improvement on theoretical model for thin-wire and slot measurement by optical diffraction,” Meas. Sci. Technol. 10, N119-N123 (1999).
[CrossRef]

Wait, J. R.

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189-195(1955).
[CrossRef]

Westpfahl, K.

H. Hönl, A. W. Maue, and K. Westpfahl, Handbuch der Physik Vol. XXV (Springer Verlag, 1961).

Wolf, E.

M. Born and E. Wolf, Principles of Optics 6th ed. (Pergamon, 1980).

Xie, J.

J. Xie, Y. Qiu, H. Ming, and C. Li “Light polarization effect in measurement of thin wire diameter by laser diffraction and its explanation with boundary diffraction wave,” J. Appl. Phys. 69, 6899-6903 (1991).
[CrossRef]

Zhang, J.

W. Tang, Y. Zhou, and J. Zhang, “Improvement on theoretical model for thin-wire and slot measurement by optical diffraction,” Meas. Sci. Technol. 10, N119-N123 (1999).
[CrossRef]

Zhou, Y.

W. Tang, Y. Zhou, and J. Zhang, “Improvement on theoretical model for thin-wire and slot measurement by optical diffraction,” Meas. Sci. Technol. 10, N119-N123 (1999).
[CrossRef]

Appl. Opt. (1)

Can. J. Phys. (1)

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189-195(1955).
[CrossRef]

J. Appl. Phys. (1)

J. Xie, Y. Qiu, H. Ming, and C. Li “Light polarization effect in measurement of thin wire diameter by laser diffraction and its explanation with boundary diffraction wave,” J. Appl. Phys. 69, 6899-6903 (1991).
[CrossRef]

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

Meas. Sci. Technol. (1)

W. Tang, Y. Zhou, and J. Zhang, “Improvement on theoretical model for thin-wire and slot measurement by optical diffraction,” Meas. Sci. Technol. 10, N119-N123 (1999).
[CrossRef]

Metrologia (1)

J. C. Martinez-Anton, I. Serroukh, and E. Bernabeu, “On Babinet's principle and a diffraction-interferometric technique to determine the diameter of cylindrical wires,” Metrologia 38, 125-134 (2001).
[CrossRef]

Opt. Eng. (1)

E. Bernabeu, I. Serroukh, and L. M. Sanchez-Brea, “A geometrical model for wire optical diffraction selected by experimental statistical analysis,” Opt. Eng. 38, 1319-1325 (1999).
[CrossRef]

Optik (1)

L. M. Sanchez-Brea , and E. Bernabeu, “Diffraction by cylinders illuminated in oblique, off-axis incidence,” Optik 112, 169-174 (2001).
[CrossRef]

Philos. Mag. (1)

Lord Rayleigh, “The dispersal of light by a dielectric cylinder,” Philos. Mag. 36, 365-376 (1918).

Proc. SPIE (1)

I. Serroukh, J. C. Martinez-Anton, and E. Bernabeu, “Accuracy of the thin metallic wires diameter using Fraunhofer diffraction technique,” Proc. SPIE 4099, 255-266 (2000).
[CrossRef]

PTB Mitteilungen (1)

S. Schmidt, “Ein beitrag zur erklärung der lichtbeugung am metallischen krieszylinder,” PTB Mitteilungen 86, 239-247(1976).

Wire J. Int. (2)

J. F. Fardeau, “New laser sensors for wire diameter measurement,” Wire J. Int. 22, 42-51 (1989).

T. K. Millard and T. A. Herchenreder, “Automatic diameter measurement: state of the art,” Wire J. Int. 24, 61-69 (1991).

Other (5)

M. Born and E. Wolf, Principles of Optics 6th ed. (Pergamon, 1980).

H. Hönl, A. W. Maue, and K. Westpfahl, Handbuch der Physik Vol. XXV (Springer Verlag, 1961).

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987).

J. J. Stamnes, Waves in Focal Regions (Adam Hilger, 1986).

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

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Figures (6)

Fig. 1
Fig. 1

Scheme showing the setup and the different parameters involved.

Fig. 2
Fig. 2

Far-field diffraction pattern of a 20 μm diameter cylinder when it is illuminated with a Gaussian beam whose wavelength is λ = 0.5 μm . (a)  ω z = 100 μm , (b)  ω z = 10 μm . The angle between the beam and the cylinder axis is α = 45 ° .

Fig. 3
Fig. 3

Comparison between I D and I R for different values of the radius a 0 . (a)  a 0 = 10 μm , (b)  a 0 = 100 μm . The wavelength is λ = 0.5 μm . The angle between the beam and the cylinder axis is α = 45 ° .

Fig. 4
Fig. 4

Diffraction patterns for two cylinders, (a) gold and (b) aluminum. Several models have been considered, such as the Babinet model, Eq. (25), where the reflected beam is not considered, and the general model, Eq. (20), for p and s polarizations. The angle of incidence is α = 45 ° , the cylinder diameter is D = 30 μm , and the wavelength is λ = 0.632 μm . For this wavelength we have considered the following refraction index: n = 0.22 + 3.0 i for gold and n = 1.19 + 7.25 i for aluminum. In both cases we can see that when the contribution of the reflected beam is considered, the locations of the diffraction minima vary with respect to Babinet’s model.

Fig. 5
Fig. 5

Difference between Babinet’s model and the one proposed in this work in terms of the cylinder radius for p and s polarizations. The angle of incidence is α = 45 ° and the wavelength is λ = 0.632 μm . (a) Gold and order 3, (b) aluminum and order 3, (c) gold and order 7, d) aluminum and order 7.

Fig. 6
Fig. 6

(a) Experimental diffraction pattern obtained for a steel cylinder whose diameter is D = 125 μm . The angle of incidence is α = 10 ° . The curvature of the diffraction pattern is observed. (b) and (c) Diffraction patterns for different angles of incidence, from α = 10 ° to α = 50 ° .

Equations (30)

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I ( ξ ) = I 0 sinc 2 ( a 0 k sin ξ ) ,
D = 2 a 0 = λ sin ξ m sin ξ m 1 ,
U ( P ) = 1 4 π s ( U ( P ) n G U ( P ) G n ) d S ,
E | S = ( 1 + R ) E 1 | S , E / n | S = ( 1 R ) E 1 | S k 1 · n ,
k 1 = k ( 0 , sin α , cos α ) .
k 2 = k ( sin θ cos ϕ , sin θ sin ϕ , cos θ ) .
E 2 ( ϕ , θ ) = C S E 1 ( R v p ) · n exp ( i v · r ) d S ,
E 1 ( φ , z ) = E 0 e ( a 0 cos φ ω x ) 2 ( sin α z ω z ) 2 .
E 2 ( ϕ , θ ) U z ( ϕ , θ ) U φ ( ϕ , θ ) .
U z ( ϕ , θ ) = e ( sin α z / ω z ) 2 e i k v z z d z
U z ( ϕ , θ ) = π ω z sin α e ( k v z ω z 2 sin α ) 2 .
U φ ( ϕ , θ ) = g ( φ ) e i k f ( φ ) d φ ,
R = R ± + R φ ± φ ,
U φ ( ϕ , θ ) = U φ 1 ( ϕ , θ ) + U φ 2 ( ϕ , θ ) ,
U φ 1 ( ϕ , θ ) = ( 1 + R ± ) sin θ 0 π cos ( ϕ φ ) e i k a 0 [ cos ( φ ϕ ) sin θ + sin α sin ϕ ] d ϕ + ( 1 R ± ) sin α 0 π sin ϕ e i k a 0 [ cos ( ϕ φ ) sin θ + sin α sin φ ] d φ ,
U φ 2 ( ϕ , θ ) = R φ ± sin θ 0 π φ cos ( ϕ φ ) e i k a 0 [ cos ( ϕ φ ) sin θ + sin α sin φ ] d φ R φ ± sin α 0 π sin φ e i k a 0 [ cos ( ϕ φ ) sin θ + sin α sin φ ] d φ .
U φ 1 ( ϕ , θ ) = S 1 [ ( R ± 1 ) sin β sin α ( 1 + R ± ) sin θ cos ( β + ϕ ) ] S 2 [ ( R ± 1 ) cos β sin α + ( 1 + R ± ) sin θ sin ( β + ϕ ) ] ,
S 2 = 2 J 0 ( A ) cos β 4 l = 1 J 2 l ( A ) cos β cos ( 2 l β ) + l sin β sin ( 2 l β ) 4 l 2 1 .
U z ( θ ) = π ω z sin α exp [ ( k v z ω z 2 sin α ) 2 ] 2 π k | sin α | δ ( θ α ) .
U φ 1 ( ξ ) = 2 sin α [ S 1 cos ( ξ / 2 ) + R ± S 2 sin ( ξ / 2 ) ] ,
E 2 ( ξ , θ ) = [ U D ( ξ ) + R ± U R ( ξ ) ] U z ( θ ) ,
U D ( ξ ) = 2 k a 0 sin ( k a 0 sin α sin ξ ) tan ( ξ / 2 ) ,
U R ( ξ ) sin α = 4 J 0 ( A ) sin 2 ( ξ / 2 ) + l = 1 ( 1 ) l 2 ( l + 1 ) 4 l 2 1 { cos [ ( l 1 ) ξ ] 2 l + 1 cos ( l ξ ) l 1 l + 1 cos [ ( l + 1 ) ξ ] } J 2 l ( A ) .
U φ 2 ( φ , θ ) = g 2 ( φ ) e i k f ( φ ) d φ ,
U φ 2 ( φ , θ ) ( π k | f 2 | ) 1 / 2 g 0 exp [ i ( k f 0 + π 4 arg f 2 2 ) ] ,
U φ 2 ( φ , θ ) e i π 4 R φ ± ( 2 π sin α k a 0 ) 1 / 2 sin 3 / 2 ( ξ 2 ) exp [ 2 i k a 0 sin α sin ( ξ / 2 ) ] ,
E 2 ( ξ , θ ) = [ U D ( ξ ) + R ± U R ( ξ ) + R φ ± U M ( ξ ) ] U z ( θ ) ,
I ( ξ , θ ) = I 0 cos 4 ( ξ / 2 ) sinc 2 ( a 0 k sin τ ) δ ( θ α ) ,
sin τ m = m λ 2 a 0 .
D = 2 a 0 = λ sin τ m sin τ m 1

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