Abstract

We propose a model for determining the far-field diffraction pattern of wires with waviness. Analytical solutions are obtained by means of the stationary phase method, which allows us to determine dimensional parameters such as wire diameter and waviness factor. Experimental results are presented, which are in accordance with our theoretical description.

© 2004 Optical Society of America

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References

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  1. E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, “Classification of surface structures on fine and ultra fine wires,” Appl. Surf. Sci. 180, 191–199 (2001).
    [CrossRef]
  2. L. M. Sanchez-Brea, P. Siegmann, M. A. Rebollo, E. Bernabeu, “Optical technique for the automatic detection and measurement of surface defects on thin metallic wires,” Appl. Opt. 39, 539–545 (2000).
    [CrossRef]
  3. L. M. Sanchez-Brea, P. Siegmann, E. Bernabeu, M. A. Rebollo, F. Pérez-Quintián, C. A. Raffo, “Detection of surface defects on thin metallic wires by geometrical conical refraction,” Wire J. Int. 33, 124–127 (2000).
  4. J. C. Martínez-Antón, P. Siegmann, L. M. Sanchez Brea, E. Bernabeu, “In-line detection and evaluation of surface defects on thin metallic wires,” in Optical Measurement Systems for Industrial Inspection II: Applications in Production Engineering, R. Hoefling, W. P. Jueptner, eds., Proc. SPIE4399, 27–31 (2001).
    [CrossRef]
  5. R. Berlasso, F. Perez-Quintián, M. Rebollo, N. Gaggioli, L. M. Sanchez-Brea, E. Bernabeu, “Speckle size of light scattered from slightly rough cylindrical surfaces,” Appl. Opt. 41, 2020–2027 (2002).
    [CrossRef] [PubMed]
  6. E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, Surface Structures on Fine and Ultra Fine Wires (Editorial Complutense, Madrid, 2002).
  7. D. Lebrun, S. Belaid, C. Ozkul, K. F. Ren, G. Grehan, “Enhancement of wire diameter measurements: comparison between Fraunhofer diffraction and Lorenz-Mie theory,” Opt. Eng. 35, 946–950 (1996).
    [CrossRef]
  8. E. Bernabeu, I. Serroukh, L. M. Sanchez-Brea, “Geometrical model for wire optical diffraction selected by experimental statistical analysis,” Opt. Eng. 38, 1319–1325 (1999).
    [CrossRef]
  9. J. C. Martínez-Antón, I. Serroukh, E. Bernabeu, “On Babinet’s principle and a diffraction-interferometric technique to determine the diameter of cylindrical wires,” Metrologia 38, 125–134 (2001).
    [CrossRef]
  10. S. L. Prosvirnin, S. A. Tretyakov, P. L. Mladyonov, “Electromagnetic wave diffraction by planar periodic gratings of wavy metal strips,” J. Electromagn. Waves Appl. 16, 421–435 (2002).
    [CrossRef]
  11. E. Kreyszig, Differential Geometry (Dover, New York, 1991), pp. 24–25.
  12. J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, 1986).

2002 (2)

S. L. Prosvirnin, S. A. Tretyakov, P. L. Mladyonov, “Electromagnetic wave diffraction by planar periodic gratings of wavy metal strips,” J. Electromagn. Waves Appl. 16, 421–435 (2002).
[CrossRef]

R. Berlasso, F. Perez-Quintián, M. Rebollo, N. Gaggioli, L. M. Sanchez-Brea, E. Bernabeu, “Speckle size of light scattered from slightly rough cylindrical surfaces,” Appl. Opt. 41, 2020–2027 (2002).
[CrossRef] [PubMed]

2001 (2)

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, “Classification of surface structures on fine and ultra fine wires,” Appl. Surf. Sci. 180, 191–199 (2001).
[CrossRef]

J. C. Martínez-Antón, I. Serroukh, 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)

L. M. Sanchez-Brea, P. Siegmann, E. Bernabeu, M. A. Rebollo, F. Pérez-Quintián, C. A. Raffo, “Detection of surface defects on thin metallic wires by geometrical conical refraction,” Wire J. Int. 33, 124–127 (2000).

L. M. Sanchez-Brea, P. Siegmann, M. A. Rebollo, 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 (1)

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

1996 (1)

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

Belaid, S.

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

Berlasso, R.

Bernabeu, E.

R. Berlasso, F. Perez-Quintián, M. Rebollo, N. Gaggioli, L. M. Sanchez-Brea, E. Bernabeu, “Speckle size of light scattered from slightly rough cylindrical surfaces,” Appl. Opt. 41, 2020–2027 (2002).
[CrossRef] [PubMed]

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, “Classification of surface structures on fine and ultra fine wires,” Appl. Surf. Sci. 180, 191–199 (2001).
[CrossRef]

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

L. M. Sanchez-Brea, P. Siegmann, E. Bernabeu, M. A. Rebollo, F. Pérez-Quintián, C. A. Raffo, “Detection of surface defects on thin metallic wires by geometrical conical refraction,” Wire J. Int. 33, 124–127 (2000).

L. M. Sanchez-Brea, P. Siegmann, M. A. Rebollo, 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, L. M. Sanchez-Brea, “Geometrical model for wire optical diffraction selected by experimental statistical analysis,” Opt. Eng. 38, 1319–1325 (1999).
[CrossRef]

J. C. Martínez-Antón, P. Siegmann, L. M. Sanchez Brea, E. Bernabeu, “In-line detection and evaluation of surface defects on thin metallic wires,” in Optical Measurement Systems for Industrial Inspection II: Applications in Production Engineering, R. Hoefling, W. P. Jueptner, eds., Proc. SPIE4399, 27–31 (2001).
[CrossRef]

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, Surface Structures on Fine and Ultra Fine Wires (Editorial Complutense, Madrid, 2002).

Gaggioli, N.

Gomez-Pedrero, J. A.

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, “Classification of surface structures on fine and ultra fine wires,” Appl. Surf. Sci. 180, 191–199 (2001).
[CrossRef]

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, Surface Structures on Fine and Ultra Fine Wires (Editorial Complutense, Madrid, 2002).

Grehan, G.

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

Hermann, H.

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, “Classification of surface structures on fine and ultra fine wires,” Appl. Surf. Sci. 180, 191–199 (2001).
[CrossRef]

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, Surface Structures on Fine and Ultra Fine Wires (Editorial Complutense, Madrid, 2002).

Hildebrand, M.

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, “Classification of surface structures on fine and ultra fine wires,” Appl. Surf. Sci. 180, 191–199 (2001).
[CrossRef]

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, Surface Structures on Fine and Ultra Fine Wires (Editorial Complutense, Madrid, 2002).

Koenders, L.

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, “Classification of surface structures on fine and ultra fine wires,” Appl. Surf. Sci. 180, 191–199 (2001).
[CrossRef]

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, Surface Structures on Fine and Ultra Fine Wires (Editorial Complutense, Madrid, 2002).

Kreyszig, E.

E. Kreyszig, Differential Geometry (Dover, New York, 1991), pp. 24–25.

Lebrun, D.

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

Martínez-Antón, J. C.

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

J. C. Martínez-Antón, P. Siegmann, L. M. Sanchez Brea, E. Bernabeu, “In-line detection and evaluation of surface defects on thin metallic wires,” in Optical Measurement Systems for Industrial Inspection II: Applications in Production Engineering, R. Hoefling, W. P. Jueptner, eds., Proc. SPIE4399, 27–31 (2001).
[CrossRef]

Mladyonov, P. L.

S. L. Prosvirnin, S. A. Tretyakov, P. L. Mladyonov, “Electromagnetic wave diffraction by planar periodic gratings of wavy metal strips,” J. Electromagn. Waves Appl. 16, 421–435 (2002).
[CrossRef]

Müller, F.

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, “Classification of surface structures on fine and ultra fine wires,” Appl. Surf. Sci. 180, 191–199 (2001).
[CrossRef]

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, Surface Structures on Fine and Ultra Fine Wires (Editorial Complutense, Madrid, 2002).

Ozkul, C.

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

Perez-Quintián, F.

Pérez-Quintián, F.

L. M. Sanchez-Brea, P. Siegmann, E. Bernabeu, M. A. Rebollo, F. Pérez-Quintián, C. A. Raffo, “Detection of surface defects on thin metallic wires by geometrical conical refraction,” Wire J. Int. 33, 124–127 (2000).

Prosvirnin, S. L.

S. L. Prosvirnin, S. A. Tretyakov, P. L. Mladyonov, “Electromagnetic wave diffraction by planar periodic gratings of wavy metal strips,” J. Electromagn. Waves Appl. 16, 421–435 (2002).
[CrossRef]

Raffo, C. A.

L. M. Sanchez-Brea, P. Siegmann, E. Bernabeu, M. A. Rebollo, F. Pérez-Quintián, C. A. Raffo, “Detection of surface defects on thin metallic wires by geometrical conical refraction,” Wire J. Int. 33, 124–127 (2000).

Rebollo, M.

Rebollo, M. A.

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

L. M. Sanchez-Brea, P. Siegmann, E. Bernabeu, M. A. Rebollo, F. Pérez-Quintián, C. A. Raffo, “Detection of surface defects on thin metallic wires by geometrical conical refraction,” Wire J. Int. 33, 124–127 (2000).

Ren, K. F.

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

Sanchez Brea, L. M.

J. C. Martínez-Antón, P. Siegmann, L. M. Sanchez Brea, E. Bernabeu, “In-line detection and evaluation of surface defects on thin metallic wires,” in Optical Measurement Systems for Industrial Inspection II: Applications in Production Engineering, R. Hoefling, W. P. Jueptner, eds., Proc. SPIE4399, 27–31 (2001).
[CrossRef]

Sanchez-Brea, L. M.

R. Berlasso, F. Perez-Quintián, M. Rebollo, N. Gaggioli, L. M. Sanchez-Brea, E. Bernabeu, “Speckle size of light scattered from slightly rough cylindrical surfaces,” Appl. Opt. 41, 2020–2027 (2002).
[CrossRef] [PubMed]

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, “Classification of surface structures on fine and ultra fine wires,” Appl. Surf. Sci. 180, 191–199 (2001).
[CrossRef]

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

L. M. Sanchez-Brea, P. Siegmann, E. Bernabeu, M. A. Rebollo, F. Pérez-Quintián, C. A. Raffo, “Detection of surface defects on thin metallic wires by geometrical conical refraction,” Wire J. Int. 33, 124–127 (2000).

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

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, Surface Structures on Fine and Ultra Fine Wires (Editorial Complutense, Madrid, 2002).

Serroukh, I.

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

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

Siegmann, P.

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, “Classification of surface structures on fine and ultra fine wires,” Appl. Surf. Sci. 180, 191–199 (2001).
[CrossRef]

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

L. M. Sanchez-Brea, P. Siegmann, E. Bernabeu, M. A. Rebollo, F. Pérez-Quintián, C. A. Raffo, “Detection of surface defects on thin metallic wires by geometrical conical refraction,” Wire J. Int. 33, 124–127 (2000).

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, Surface Structures on Fine and Ultra Fine Wires (Editorial Complutense, Madrid, 2002).

J. C. Martínez-Antón, P. Siegmann, L. M. Sanchez Brea, E. Bernabeu, “In-line detection and evaluation of surface defects on thin metallic wires,” in Optical Measurement Systems for Industrial Inspection II: Applications in Production Engineering, R. Hoefling, W. P. Jueptner, eds., Proc. SPIE4399, 27–31 (2001).
[CrossRef]

Stamnes, J. J.

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

Tretyakov, S. A.

S. L. Prosvirnin, S. A. Tretyakov, P. L. Mladyonov, “Electromagnetic wave diffraction by planar periodic gratings of wavy metal strips,” J. Electromagn. Waves Appl. 16, 421–435 (2002).
[CrossRef]

Wilkening, G.

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, “Classification of surface structures on fine and ultra fine wires,” Appl. Surf. Sci. 180, 191–199 (2001).
[CrossRef]

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, Surface Structures on Fine and Ultra Fine Wires (Editorial Complutense, Madrid, 2002).

Appl. Opt. (2)

Appl. Surf. Sci. (1)

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, “Classification of surface structures on fine and ultra fine wires,” Appl. Surf. Sci. 180, 191–199 (2001).
[CrossRef]

J. Electromagn. Waves Appl. (1)

S. L. Prosvirnin, S. A. Tretyakov, P. L. Mladyonov, “Electromagnetic wave diffraction by planar periodic gratings of wavy metal strips,” J. Electromagn. Waves Appl. 16, 421–435 (2002).
[CrossRef]

Metrologia (1)

J. C. Martínez-Antón, I. Serroukh, 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. (2)

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

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

Wire J. Int. (1)

L. M. Sanchez-Brea, P. Siegmann, E. Bernabeu, M. A. Rebollo, F. Pérez-Quintián, C. A. Raffo, “Detection of surface defects on thin metallic wires by geometrical conical refraction,” Wire J. Int. 33, 124–127 (2000).

Other (4)

J. C. Martínez-Antón, P. Siegmann, L. M. Sanchez Brea, E. Bernabeu, “In-line detection and evaluation of surface defects on thin metallic wires,” in Optical Measurement Systems for Industrial Inspection II: Applications in Production Engineering, R. Hoefling, W. P. Jueptner, eds., Proc. SPIE4399, 27–31 (2001).
[CrossRef]

E. Bernabeu, L. M. Sanchez-Brea, P. Siegmann, J. A. Gomez-Pedrero, G. Wilkening, L. Koenders, F. Müller, M. Hildebrand, H. Hermann, Surface Structures on Fine and Ultra Fine Wires (Editorial Complutense, Madrid, 2002).

E. Kreyszig, Differential Geometry (Dover, New York, 1991), pp. 24–25.

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

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Figures (5)

Fig. 1
Fig. 1

Description of a (a) helicoidal wire and (b) undulated wire and parameters involved.

Fig. 2
Fig. 2

Numerical simulation of the far-field diffraction pattern by use of Eq. (3) for helicoidal wire with parameters R = 50 μm, r = 88 μm, and p = 2525 μm. The wavelength considered is λ = 0.6328 μm. This diffraction pattern has been calculated by means of a fast-Fourier-transform algorithm.

Fig. 3
Fig. 3

Intensity distribution obtained by means of expression (8) for two wires with waviness (a) r = 40 μm, p = 2000 μm, and R = 100 μm and (b) r = 300 μm, p = 4000 μm, and R = 30 μm. In both cases the wavelength is λ = 0.6328 μm.

Fig. 4
Fig. 4

Experimental setup for measuring the far-field diffraction pattern of wavy wires.

Fig. 5
Fig. 5

Far-field diffraction patterns obtained for two wires with waviness: (a) R = 50 μm, r = 88 μm, and p = 2525 μm and (b) R = 75 μm, r = 67 μm, and p = 2400 μm.

Equations (16)

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

x=r cos2πτ-Rpν sin2πτsin χ+cos2πτcos χ,y=pτ-2πrRν sin χ,z=r sin2πτ+R2πνcos2πτsin χ-sin2πτcos χ,
x, y  wire surface Λy-Γy<x<Λy+Γy,
Λy=r sin2πyp, Γy=R1+2πrp2 cos22πyp1/2.
Uu, v=CkE0-Λy-ΓyΛy+Γyexp-ik×ux+vydxdy,
Uu, v=2Cu E0-sinkΓyu×exp-ikvy+uΛydy.
UP=- gyexpikfydy,
Uu, vπkfys1/2gysexpikfys+π4+argfys2,
Uu, v2CE0 expikgys+π/4k1/2pu×sinkRu2+v22πru2-pv21/4.
Iu, v4C2I0kpu2sin2kRu2+v22πru2-pv21/2,-1<vp/2πur<1
=0 elsewhere.
Iρ, φ4C2I0p2R2cos2 φ2πr cos φ2-p sin φ21/2×sinc2kRρρ,
x=r sin2πτp+R sin χ1+2πr/p2 cos22πτ/p1/2,y=τ-2πr/p cos2πτ/p1+2πr/p2 cos22πτ/p1/2 R sin χ,z=R cos χ,
tan φ0=±2πrp,
Iρ, φ=04C2p2R2kρsinc2kRρ2πr.
Iρ= Iρ, φdφ4C2p2R2ρsinc2×kRρ-φ0φ01cos2 φ2πr cos φ2-p sin φ21/2dφ=8C2R2pE-2πrp2sinc2kRρρ,
R=λρm-ρm-1.

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