Abstract

This research is an extension of the optical method of quality control presented in a previous paper [Appl. Opt. 39, 5811 (2000)] to the case of slightly rough cylindrical surfaces. Applying the Kirchhoff scalar diffraction theory yields an analytical expression of the autocorrelation function of the intensity scattered from slightly rough cylindrical surfaces. This function, which is related to speckle size and shape, is shown to depend on the surface correlation length, unlike for plane surfaces for which the speckle depends on the illuminated area only. The theoretical expression is compared with that for the speckle produced by the light scattered from a cylindrical bearing and from various high-quality wires, showing that the method allows the correlation lengths of high-quality cylindrical surfaces to be determined.

© 2002 Optical Society of America

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References

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  1. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963).
  2. J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).
  3. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).
  4. J. M. Bennett, Surface Finish and Its Measurement, Vol. 2 of OSA Collected Works in Optics Series (Optical Society of America, Washington, D.C., 1992).
  5. C. Babu Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Localization of surface roughness of thin wires using laser scattering,” in Proceedings of the 14th World Conference on Nondestructive Testing (Ashgate, Hampshire, UK, 1996), Vol. 3, pp. 1517–1520.
  6. C. Babu Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Laser scattering from the surface of thin wires at oblique illumination,” in Optics and Optoelectronics: Theory, Devices and Applications, O. Nijhawan, A. Gupta, A. K. Musla, K. Sing, eds. (Narosa House, New Delhi, 1999), pp. 298–301.
  7. G. Da Costa, J. Ferrari, “Anisotropic speckle patterns in the light scattered by rough cylindrical surfaces,” Appl. Opt. 36, 5231–5237 (1997).
    [CrossRef] [PubMed]
  8. F. Perez Quintián, M. A. Rebollo, N. G. Gaggioli, “Optical methods for on line surface wire testing,” in Proceedings of the 7th European Conference on Nondestructive Testing (7th ECNDT, Broendby, Denmark, 1998), pp. 2920–2926; available on line at http://www.ndt.net .
  9. F. Perez Quintiaán, M. A. Rebollo, R. Berlasso, N. G. Gaggioli, “Roughness measurement on cylindrical surfaces by optical methods,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.48–5.57.
  10. F. Perez Quintián, M. A. Rebollo, M. T. Bernal, C. A. Raffo, “Measurements of High Quality Wire Roughness,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.58–5.64.
  11. R. Berlasso, F. Perez Quintián, M. A. Rebollo, C. A. Raffo, N. G. Gaggioli, “Study of speckle size of light scattered from cylindrical rough surfaces,” Appl. Opt. 39, 5811–5819 (2000).
    [CrossRef]
  12. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics Series (Springer-Verlag, Berlin, 1975), pp. 9–75.
    [CrossRef]
  13. A. Papoulis, Probability Random Variables and Stochastic Processes, international student ed. (McGraw-Hill, New York, 1965), p. 226, Eq. (7–114).
  14. L. M. Sanchez-Brea, J. A. Gomez-Pedrero, E. Bernabeu, “Measurement of surface defects on thin metallic wires by atomic force microscopy,” Appl. Surf. Sci. 150, 125–130 (1999).
    [CrossRef]
  15. Commission of the European Community, Surface Structures on Fine and Ultrafine Wires, edited on behalf of the European Commission (Editorial Complutense, Madrid, 2002).

2000 (1)

R. Berlasso, F. Perez Quintián, M. A. Rebollo, C. A. Raffo, N. G. Gaggioli, “Study of speckle size of light scattered from cylindrical rough surfaces,” Appl. Opt. 39, 5811–5819 (2000).
[CrossRef]

1999 (1)

L. M. Sanchez-Brea, J. A. Gomez-Pedrero, E. Bernabeu, “Measurement of surface defects on thin metallic wires by atomic force microscopy,” Appl. Surf. Sci. 150, 125–130 (1999).
[CrossRef]

1997 (1)

G. Da Costa, J. Ferrari, “Anisotropic speckle patterns in the light scattered by rough cylindrical surfaces,” Appl. Opt. 36, 5231–5237 (1997).
[CrossRef] [PubMed]

Ananthalakshmi, A. V.

C. Babu Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Localization of surface roughness of thin wires using laser scattering,” in Proceedings of the 14th World Conference on Nondestructive Testing (Ashgate, Hampshire, UK, 1996), Vol. 3, pp. 1517–1520.

C. Babu Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Laser scattering from the surface of thin wires at oblique illumination,” in Optics and Optoelectronics: Theory, Devices and Applications, O. Nijhawan, A. Gupta, A. K. Musla, K. Sing, eds. (Narosa House, New Delhi, 1999), pp. 298–301.

Babu Rao, C.

C. Babu Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Laser scattering from the surface of thin wires at oblique illumination,” in Optics and Optoelectronics: Theory, Devices and Applications, O. Nijhawan, A. Gupta, A. K. Musla, K. Sing, eds. (Narosa House, New Delhi, 1999), pp. 298–301.

C. Babu Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Localization of surface roughness of thin wires using laser scattering,” in Proceedings of the 14th World Conference on Nondestructive Testing (Ashgate, Hampshire, UK, 1996), Vol. 3, pp. 1517–1520.

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963).

Bennett, J. M.

J. M. Bennett, Surface Finish and Its Measurement, Vol. 2 of OSA Collected Works in Optics Series (Optical Society of America, Washington, D.C., 1992).

Berlasso, R.

R. Berlasso, F. Perez Quintián, M. A. Rebollo, C. A. Raffo, N. G. Gaggioli, “Study of speckle size of light scattered from cylindrical rough surfaces,” Appl. Opt. 39, 5811–5819 (2000).
[CrossRef]

F. Perez Quintiaán, M. A. Rebollo, R. Berlasso, N. G. Gaggioli, “Roughness measurement on cylindrical surfaces by optical methods,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.48–5.57.

Bernabeu, E.

L. M. Sanchez-Brea, J. A. Gomez-Pedrero, E. Bernabeu, “Measurement of surface defects on thin metallic wires by atomic force microscopy,” Appl. Surf. Sci. 150, 125–130 (1999).
[CrossRef]

Bernal, M. T.

F. Perez Quintián, M. A. Rebollo, M. T. Bernal, C. A. Raffo, “Measurements of High Quality Wire Roughness,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.58–5.64.

Da Costa, G.

G. Da Costa, J. Ferrari, “Anisotropic speckle patterns in the light scattered by rough cylindrical surfaces,” Appl. Opt. 36, 5231–5237 (1997).
[CrossRef] [PubMed]

Ferrari, J.

G. Da Costa, J. Ferrari, “Anisotropic speckle patterns in the light scattered by rough cylindrical surfaces,” Appl. Opt. 36, 5231–5237 (1997).
[CrossRef] [PubMed]

Gaggioli, N. G.

R. Berlasso, F. Perez Quintián, M. A. Rebollo, C. A. Raffo, N. G. Gaggioli, “Study of speckle size of light scattered from cylindrical rough surfaces,” Appl. Opt. 39, 5811–5819 (2000).
[CrossRef]

F. Perez Quintiaán, M. A. Rebollo, R. Berlasso, N. G. Gaggioli, “Roughness measurement on cylindrical surfaces by optical methods,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.48–5.57.

Gomez-Pedrero, J. A.

L. M. Sanchez-Brea, J. A. Gomez-Pedrero, E. Bernabeu, “Measurement of surface defects on thin metallic wires by atomic force microscopy,” Appl. Surf. Sci. 150, 125–130 (1999).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics Series (Springer-Verlag, Berlin, 1975), pp. 9–75.
[CrossRef]

Kesavamoorthy, R.

C. Babu Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Laser scattering from the surface of thin wires at oblique illumination,” in Optics and Optoelectronics: Theory, Devices and Applications, O. Nijhawan, A. Gupta, A. K. Musla, K. Sing, eds. (Narosa House, New Delhi, 1999), pp. 298–301.

C. Babu Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Localization of surface roughness of thin wires using laser scattering,” in Proceedings of the 14th World Conference on Nondestructive Testing (Ashgate, Hampshire, UK, 1996), Vol. 3, pp. 1517–1520.

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).

Papoulis, A.

A. Papoulis, Probability Random Variables and Stochastic Processes, international student ed. (McGraw-Hill, New York, 1965), p. 226, Eq. (7–114).

Perez Quintiaán, F.

F. Perez Quintiaán, M. A. Rebollo, R. Berlasso, N. G. Gaggioli, “Roughness measurement on cylindrical surfaces by optical methods,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.48–5.57.

Perez Quintián, F.

R. Berlasso, F. Perez Quintián, M. A. Rebollo, C. A. Raffo, N. G. Gaggioli, “Study of speckle size of light scattered from cylindrical rough surfaces,” Appl. Opt. 39, 5811–5819 (2000).
[CrossRef]

F. Perez Quintián, M. A. Rebollo, M. T. Bernal, C. A. Raffo, “Measurements of High Quality Wire Roughness,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.58–5.64.

Raffo, C. A.

R. Berlasso, F. Perez Quintián, M. A. Rebollo, C. A. Raffo, N. G. Gaggioli, “Study of speckle size of light scattered from cylindrical rough surfaces,” Appl. Opt. 39, 5811–5819 (2000).
[CrossRef]

F. Perez Quintián, M. A. Rebollo, M. T. Bernal, C. A. Raffo, “Measurements of High Quality Wire Roughness,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.58–5.64.

Rebollo, M. A.

R. Berlasso, F. Perez Quintián, M. A. Rebollo, C. A. Raffo, N. G. Gaggioli, “Study of speckle size of light scattered from cylindrical rough surfaces,” Appl. Opt. 39, 5811–5819 (2000).
[CrossRef]

F. Perez Quintiaán, M. A. Rebollo, R. Berlasso, N. G. Gaggioli, “Roughness measurement on cylindrical surfaces by optical methods,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.48–5.57.

F. Perez Quintián, M. A. Rebollo, M. T. Bernal, C. A. Raffo, “Measurements of High Quality Wire Roughness,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.58–5.64.

Sanchez-Brea, L. M.

L. M. Sanchez-Brea, J. A. Gomez-Pedrero, E. Bernabeu, “Measurement of surface defects on thin metallic wires by atomic force microscopy,” Appl. Surf. Sci. 150, 125–130 (1999).
[CrossRef]

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963).

Appl. Opt. (2)

G. Da Costa, J. Ferrari, “Anisotropic speckle patterns in the light scattered by rough cylindrical surfaces,” Appl. Opt. 36, 5231–5237 (1997).
[CrossRef] [PubMed]

R. Berlasso, F. Perez Quintián, M. A. Rebollo, C. A. Raffo, N. G. Gaggioli, “Study of speckle size of light scattered from cylindrical rough surfaces,” Appl. Opt. 39, 5811–5819 (2000).
[CrossRef]

Appl. Surf. Sci. (1)

L. M. Sanchez-Brea, J. A. Gomez-Pedrero, E. Bernabeu, “Measurement of surface defects on thin metallic wires by atomic force microscopy,” Appl. Surf. Sci. 150, 125–130 (1999).
[CrossRef]

Other (12)

Commission of the European Community, Surface Structures on Fine and Ultrafine Wires, edited on behalf of the European Commission (Editorial Complutense, Madrid, 2002).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics Series (Springer-Verlag, Berlin, 1975), pp. 9–75.
[CrossRef]

A. Papoulis, Probability Random Variables and Stochastic Processes, international student ed. (McGraw-Hill, New York, 1965), p. 226, Eq. (7–114).

F. Perez Quintián, M. A. Rebollo, N. G. Gaggioli, “Optical methods for on line surface wire testing,” in Proceedings of the 7th European Conference on Nondestructive Testing (7th ECNDT, Broendby, Denmark, 1998), pp. 2920–2926; available on line at http://www.ndt.net .

F. Perez Quintiaán, M. A. Rebollo, R. Berlasso, N. G. Gaggioli, “Roughness measurement on cylindrical surfaces by optical methods,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.48–5.57.

F. Perez Quintián, M. A. Rebollo, M. T. Bernal, C. A. Raffo, “Measurements of High Quality Wire Roughness,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.58–5.64.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963).

J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).

J. M. Bennett, Surface Finish and Its Measurement, Vol. 2 of OSA Collected Works in Optics Series (Optical Society of America, Washington, D.C., 1992).

C. Babu Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Localization of surface roughness of thin wires using laser scattering,” in Proceedings of the 14th World Conference on Nondestructive Testing (Ashgate, Hampshire, UK, 1996), Vol. 3, pp. 1517–1520.

C. Babu Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Laser scattering from the surface of thin wires at oblique illumination,” in Optics and Optoelectronics: Theory, Devices and Applications, O. Nijhawan, A. Gupta, A. K. Musla, K. Sing, eds. (Narosa House, New Delhi, 1999), pp. 298–301.

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Figures (10)

Fig. 1
Fig. 1

Schematic of light scattering from a cylinder. A plane wave represented by wave vector k i is incident over a rough cylinder. The scattered waves are represented by the vectors k and also by the directions (θ, ϕ), the usual spherical coordinates, each of which determines a point on the observation plane at the far field. Dashed circumference, θ = α.

Fig. 2
Fig. 2

Vector r = [h(ϕ, z), ϕ, z], which ends on the cylindrical rough surface where function h(ϕ, z) is assumed to be a random process.

Fig. 3
Fig. 3

Variables τ and ζ over the cylinder surface, which are used instead of the usual cylindrical coordinates (ϕ, z) to solve the integrals involved in the calculation.

Fig. 4
Fig. 4

Experimental setup for registering the speckle pattern. The lens behind the pupil is placed such as to make valid the Fraunhofer approximation at the plane where the camera is placed.

Fig. 5
Fig. 5

Graphics created from the data obtained from a stylus profilometer of sample R01 along the z direction: (a) profile (the mean radius is taken as zero), (b) histogram of (a) with the corresponding Gaussian fit, (c) autocorrelation function with the corresponding Gaussian fit.

Fig. 6
Fig. 6

Graphics created from the data obtained from an AFM image of sample E14: (a) three-dimensional reconstruction of the surface, (b) histogram with the corresponding Gaussian fit, (c) bidimensional autocorrelation function.

Fig. 7
Fig. 7

Photograph of the speckle produced by sample R01 at ϕ = 140°.

Fig. 8
Fig. 8

Autocorrelation functions of the speckle pattern shown in Fig. 5 along the radial (horizontal in Fig. 5) and azimuthal (vertical in Fig. 5) directions.

Fig. 9
Fig. 9

Experimental autocorrelation functions along the azimuthal direction for sample R01 at various values of ϕ, with their respective theoretical fitting curves. The graphs for the other samples are similar.

Fig. 10
Fig. 10

Widths W, in pixels, of the speckle grains along the ϕ direction for the six samples as a function of scattered angle ϕ.

Tables (1)

Tables Icon

Table 1 Comparison of Characteristics of Samples Used in the Experimenta

Equations (29)

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γ AB θ ,   ϕ ,   θ ,   ϕ = I A θ ,   ϕ I B θ ,   ϕ - I A θ ,   ϕ I B θ ,   ϕ I A 2 θ ,   ϕ - I A θ ,   ϕ 2 I B 2 θ ,   ϕ - I A θ ,   ϕ 2 1 / 2 = E A θ ,   ϕ E B * θ ,   ϕ 2 I A θ ,   ϕ I B θ ,   ϕ ,
E θ ,   ϕ = K θ ,   ϕ ,   α φ z × exp ikc α ,   θ z exp ikh φ ,   z C φ ,   α ,   θ ,   ϕ C φ ,   α ,   θ ,   ϕ × h φ ,   z d z d φ ,
K θ ,   ϕ = - iE 0 k 2 π exp ikR 0 R 0 a α ,   θ ,   ϕ sin   α - c α ,   θ cos   α ;
C φ ,   α ,   θ ,   ϕ = a α ,   θ ,   ϕ cos   φ + b θ ,   ϕ sin   φ ;
a α ,   θ ,   ϕ = - sin   θ   cos   ϕ - sin   α ,   b θ ,   ϕ = - sin   θ   sin   ϕ ,   c α ,   θ = cos   α - cos   θ .
E θ ,   ϕ E * θ ,   ϕ = K θ ,   ϕ K * θ ,   ϕ z 1 φ 1 z 2 φ 2 exp ik cz 2 - c z 1   × h 1 h 2   exp ik h 2 C φ 2 - h 1 C φ 1 C φ 2 C φ 1 d z 1 d φ 1 d z 2 d φ 2 ,
c = cos   α - cos   θ = cos   α - cos θ + δ θ cos   α - cos   θ + δ θ   sin   θ ,   C φ 1 = a θ + δ θ ,   ϕ + δ ϕ cos   φ 1 + b θ + δ θ ,   ϕ + δ ϕ sin   φ 1   C φ 1 + sin   θ   sin ϕ - φ 1 δ ϕ - cos   θ   cos ϕ - φ 1 δ θ ,   C φ 2 = a θ ,   ϕ cos   φ 2 + b θ ,   ϕ sin   φ 2 C φ 1 + τ ϕ h b   cos   φ 1 - a   sin   φ 1 ,
τ   cos   ζ = τ z = z 2 - z 1 ,   τ   sin   ζ = τ φ = h φ 2 - φ 1 .
E θ ,   ϕ E * θ ,   ϕ     h   z 1 φ 1 τ φ τ z exp ikc τ z exp - ikcz 1 δ θ   sin   θ × exp ik h 2 C φ 1 + τ φ / h d φ 1 - h 1 C φ 1 + Δ φ 1 C φ 1 + Δ φ 1 C φ 1 + τ φ / h d φ 1 d z 1 d φ 1 d τ φ d τ z .
Δ φ 1 sin   θ   sin ϕ - φ 1 δ ϕ - cos   θ   cos ϕ - φ 1 δ θ ,   d φ 1 b   cos   φ 1 - a   sin   φ 1 ,
- L / 2 L / 2 exp - ikcz 1   sin   θ δ θ d z 1 = L   sin kc δ θ L / 2   sin   θ kc δ θ L / 2   sin   θ .
p H h = 1 2 π σ exp - h 2 2 σ 2 ,
exp ik h 2 C φ 1 + τ φ h   d φ 1   - h 1 C φ 1   + Δ φ 1 = exp ik h C φ 1 + τ φ h   d φ 1 - C φ 1 + Δ φ 1 exp - 1 2   σ 2 k 2 C φ 1 + τ φ h   d φ 1 2 - 2 γ surf τ φ ,   τ z C φ 1 + τ φ h   d φ 1 × C φ 1 + Δ φ 1 + C φ 1 + Δ φ 1 2 ,
exp ik h 2 C φ 1 + τ ϕ h   d φ 1 - h 1 C φ 1 + Δ φ 1   = exp ik τ ϕ d φ 1 exp - ik h Δ φ 1 × exp - σ 2 k 2 C 2 φ 1 1 - γ surf τ φ ,   τ z .
E θ ,   ϕ E * θ ,   ϕ = h L   sin kc   sin   θ δ θ L / 2 kc   sin   θ δ θ L / 2 × φ 1 τ ϕ τ z exp ik c τ z + d φ 1 τ φ exp - ik h Δ φ 1 × exp - σ 2 k 2 C 2 φ 1 1 - γ surf τ φ ,   τ z C 2 φ 1 d φ 1 d τ φ d τ z .
γ Surf τ φ ,   τ z = exp - τ z 2 T z 2 exp - τ φ 2 T φ 2 ,
exp - σ 2 k 2 C 2 φ 1 1 - γ Surf τ φ ,   τ z = exp - σ 2 k 2 C 2 φ 1 n = 0 σ 2 n k 2 n C 2 n φ 1 n ! × exp - n τ z 2 T z 2 exp - n τ φ 2 T φ 2 .
E θ ,   ϕ E * θ ,   ϕ = 2 π h L × sin kc   sin   θ δ θ L / 2 kc   sin   θ δ θ L / 2 n = 0 φ 1 τ φ τ z × exp - n τ z 2 T z 2 exp - n τ φ 2 T φ 2 × exp ik c τ z + d φ 1 τ φ d τ φ d τ z   exp - ik h Δ φ 1 × exp - σ 2 k 2 C 2 φ 1 C 2 φ 1 σ 2 n k 2 n C 2 n φ 1 n ! d φ 1 .
E θ ,   ϕ E * θ ,   ϕ = 2 π h L σ 2 k 2 × sin kc   sin   θ δ θ L / 2 kc   sin   θ δ θ L / 2 φ 1 τ φ τ z exp ikc τ z × exp ikd φ 1 τ φ exp - τ z 2 T z 2 exp - τ φ 2 T φ 2 × exp - ik h Δ φ 1 exp - σ 2 k 2 C 2 φ 1 d τ φ d τ z d φ 1 ,
E θ ,   ϕ E * θ ,   ϕ = π h L   sin kc   sin   θ δ θ L / 2 kc   sin   θ δ θ L / 2 × T z T φ σ 2 k 2   exp - k 2 T z 2 c 2 4 φ exp - ik h Δ φ 1 × exp - σ 2 k 2 C 2 φ 1 exp - k 2 T φ 2 d 2 φ 1 4 d φ 1 .
φ 1 = ϕ / 2 + ε .
C φ 1 = C ϕ / 2 cos   ε ,   d φ 1 = C ϕ / 2 sin   ε ,   Δ φ 1 = - Δ 0   cos δ + ϕ / 2 - ε ,
Δ 0 δ ϕ 2   sin 2   θ + δ θ 2   cos 2   θ 1 / 2 ,   cos   δ δ θ   cos   θ Δ 0 ,   sin   δ δ ϕ   sin   θ Δ 0 .
E θ ,   ϕ E * θ ,   ϕ = π h L × sin kc   sin   θ δ θ L / 2 kc   sin   θ δ θ L / 2 × T z T φ σ 2 k 2   exp - k 2 T z 2 c 2 4 ε exp - k 2 C 2 ϕ / 2 × σ 2   cos 2   ε + T φ 2 4 sin 2   ε exp ik h Δ 0 × cos   ε   cos δ + ϕ / 2 + sin Δ + ϕ / 2 sin   ε d ε .
E θ ,   ϕ E * θ ,   ϕ     sin kc   sin   θ δ θ L / 2 kc   sin   θ δ θ L / 2 × ε exp - k 2 C 2 ϕ / 2 T φ 2 4   ε 2 × exp - ik h Δ 0   sin δ + ϕ / 2 ε d ε ,
γ I θ ,   ϕ ,   θ + δ θ ,   ϕ + δ ϕ = sin 2 kc   sin   θ δ θ L / 2 kc   sin   θ δ θ L / 2 2 × exp - 2 h Δ 0   sin δ + ϕ / 2 C ϕ / 2 T φ 2 .
γ I θ ,   ϕ ,   θ ,   ϕ + δ ϕ = exp - 2 h Δ 0   cos ϕ / 2 C ϕ / 2 T φ 2 = exp - 2 h δ ϕ   sin   θ sin   θ + sin   α T φ 2 ,
γ I θ ,   ϕ ,   θ ,   ϕ + δ ϕ = exp - 2 h δ ϕ   sin   θ sin   θ + sin   α T φ 2 = exp - 2 x W 2 ,
δ ϕ   sin   θ = x p / D ,   T φ sin   θ + sin   α h D p = W ,

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