Abstract

We present a model to determine the light scattered by a metallic cylinder with longitudinal structures when the cylinder is illuminated by a Gaussian light beam in oblique incidence. The model is based on an approximate solution to the Helmholtz–Kirchhoff integral by means of the stationary-phase method. We have studied the variations of the diffraction pattern in terms of the size of the defect and other geometrical parameters. The width of the beam and the misalignment between the beam and the cylinder have also been considered, as well as the optical properties of the surface.

© 2004 Optical Society of America

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  1. C. B. Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Manifestation of surface roughness of thin wires in laser scattering,” in Proceedings of the National Laser Symposium (Instrument Research and Development Establishment, Dehradun, India, 1995), pp. 330–331.
  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. Intern. 33, 124–127 (2000).
  3. L. M. Sanchez-Brea, E. Bernabeu, “Diffraction by cylinders illuminated in oblique, off-axis incidence,” Optik 112, 169–174 (2001).
    [CrossRef]
  4. J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 188–195 (1955).
  5. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  6. C. B. Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Localization of surface roughness of thin wires using laser scattering,” in Trends in NDE Science and Technology, Proceedings of the 14th World Conference on Non-Destructive Testing, C. G. K. Nair, B. Raj, C. R. L. Nurthy, T. Jayakumar, eds. (Indian Society for Non-Destructive Testing, Chennai, India, 1996), Vols. 1–5, pp. 1517–1520.
  7. F. Perez Quintián, M. A. Rebollo, N. G. Gaggioli, C. A. Raffo, “Optical methods for on-line surface wire testing,”in Proceedings of the 7th European Conference on Non-Destructive Testing (Danish Society for Nondestructive Testing, Copenhagen, 1998), Vol. 3, No. 8.
  8. S. A. Stefani, C. R. Nagarajah, R. Willgoss, “A surface inspection technique for continuously extruded cylindrical products,” Meas. Sci. Technol. 10, N21–N25 (1999).
    [CrossRef]
  9. 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]
  10. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Norwood, Mass., 1987).
  11. L. M. Sanchez-Brea, J. A. Gomez-Pedrero, E. Bernabeu, “Measurement of surface defects on thin steel wires by atomic force microscopy,” Appl. Surf. Sci. 150, 125–130 (1999).
    [CrossRef]
  12. J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986).
  13. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).
  14. L. M. Sanchez-Brea, J. A. Gómez-Pedrero, E. Bernabeu, “Analysis and characterization of surface defects on thin steel wires by atomic force microscopy,” in Proceedings of the Wire and Cable Technical Symposium (Wire Association International, Guillford, Conn., 1999), pp. 189–192.

2001 (1)

L. M. Sanchez-Brea, E. Bernabeu, “Diffraction by cylinders illuminated in oblique, off-axis incidence,” Optik 112, 169–174 (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. Intern. 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 (2)

S. A. Stefani, C. R. Nagarajah, R. Willgoss, “A surface inspection technique for continuously extruded cylindrical products,” Meas. Sci. Technol. 10, N21–N25 (1999).
[CrossRef]

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

1955 (1)

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 188–195 (1955).

Ananthalakshmi, A. V.

C. B. Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Localization of surface roughness of thin wires using laser scattering,” in Trends in NDE Science and Technology, Proceedings of the 14th World Conference on Non-Destructive Testing, C. G. K. Nair, B. Raj, C. R. L. Nurthy, T. Jayakumar, eds. (Indian Society for Non-Destructive Testing, Chennai, India, 1996), Vols. 1–5, pp. 1517–1520.

C. B. Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Manifestation of surface roughness of thin wires in laser scattering,” in Proceedings of the National Laser Symposium (Instrument Research and Development Establishment, Dehradun, India, 1995), pp. 330–331.

Beckmann, P.

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

Bernabeu, E.

L. M. Sanchez-Brea, E. Bernabeu, “Diffraction by cylinders illuminated in oblique, off-axis incidence,” Optik 112, 169–174 (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. Intern. 33, 124–127 (2000).

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

L. M. Sanchez-Brea, J. A. Gómez-Pedrero, E. Bernabeu, “Analysis and characterization of surface defects on thin steel wires by atomic force microscopy,” in Proceedings of the Wire and Cable Technical Symposium (Wire Association International, Guillford, Conn., 1999), pp. 189–192.

Bohren, C. F.

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

Born, M.

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

Gaggioli, N. G.

F. Perez Quintián, M. A. Rebollo, N. G. Gaggioli, C. A. Raffo, “Optical methods for on-line surface wire testing,”in Proceedings of the 7th European Conference on Non-Destructive Testing (Danish Society for Nondestructive Testing, Copenhagen, 1998), Vol. 3, No. 8.

Gomez-Pedrero, J. A.

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

Gómez-Pedrero, J. A.

L. M. Sanchez-Brea, J. A. Gómez-Pedrero, E. Bernabeu, “Analysis and characterization of surface defects on thin steel wires by atomic force microscopy,” in Proceedings of the Wire and Cable Technical Symposium (Wire Association International, Guillford, Conn., 1999), pp. 189–192.

Huffman, D. R.

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

Kesavamoorthy, R.

C. B. Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Manifestation of surface roughness of thin wires in laser scattering,” in Proceedings of the National Laser Symposium (Instrument Research and Development Establishment, Dehradun, India, 1995), pp. 330–331.

C. B. Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Localization of surface roughness of thin wires using laser scattering,” in Trends in NDE Science and Technology, Proceedings of the 14th World Conference on Non-Destructive Testing, C. G. K. Nair, B. Raj, C. R. L. Nurthy, T. Jayakumar, eds. (Indian Society for Non-Destructive Testing, Chennai, India, 1996), Vols. 1–5, pp. 1517–1520.

Nagarajah, C. R.

S. A. Stefani, C. R. Nagarajah, R. Willgoss, “A surface inspection technique for continuously extruded cylindrical products,” Meas. Sci. Technol. 10, N21–N25 (1999).
[CrossRef]

Perez Quintián, F.

F. Perez Quintián, M. A. Rebollo, N. G. Gaggioli, C. A. Raffo, “Optical methods for on-line surface wire testing,”in Proceedings of the 7th European Conference on Non-Destructive Testing (Danish Society for Nondestructive Testing, Copenhagen, 1998), Vol. 3, No. 8.

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. Intern. 33, 124–127 (2000).

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. Intern. 33, 124–127 (2000).

F. Perez Quintián, M. A. Rebollo, N. G. Gaggioli, C. A. Raffo, “Optical methods for on-line surface wire testing,”in Proceedings of the 7th European Conference on Non-Destructive Testing (Danish Society for Nondestructive Testing, Copenhagen, 1998), Vol. 3, No. 8.

Rao, C. B.

C. B. Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Manifestation of surface roughness of thin wires in laser scattering,” in Proceedings of the National Laser Symposium (Instrument Research and Development Establishment, Dehradun, India, 1995), pp. 330–331.

C. B. Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Localization of surface roughness of thin wires using laser scattering,” in Trends in NDE Science and Technology, Proceedings of the 14th World Conference on Non-Destructive Testing, C. G. K. Nair, B. Raj, C. R. L. Nurthy, T. Jayakumar, eds. (Indian Society for Non-Destructive Testing, Chennai, India, 1996), Vols. 1–5, pp. 1517–1520.

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. Intern. 33, 124–127 (2000).

F. Perez Quintián, M. A. Rebollo, N. G. Gaggioli, C. A. Raffo, “Optical methods for on-line surface wire testing,”in Proceedings of the 7th European Conference on Non-Destructive Testing (Danish Society for Nondestructive Testing, Copenhagen, 1998), Vol. 3, No. 8.

Sanchez-Brea, L. M.

L. M. Sanchez-Brea, E. Bernabeu, “Diffraction by cylinders illuminated in oblique, off-axis incidence,” Optik 112, 169–174 (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. Intern. 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]

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

L. M. Sanchez-Brea, J. A. Gómez-Pedrero, E. Bernabeu, “Analysis and characterization of surface defects on thin steel wires by atomic force microscopy,” in Proceedings of the Wire and Cable Technical Symposium (Wire Association International, Guillford, Conn., 1999), pp. 189–192.

Siegmann, P.

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. Intern. 33, 124–127 (2000).

Spizzichino, A.

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

Stamnes, J. J.

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

Stefani, S. A.

S. A. Stefani, C. R. Nagarajah, R. Willgoss, “A surface inspection technique for continuously extruded cylindrical products,” Meas. Sci. Technol. 10, N21–N25 (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, 188–195 (1955).

Willgoss, R.

S. A. Stefani, C. R. Nagarajah, R. Willgoss, “A surface inspection technique for continuously extruded cylindrical products,” Meas. Sci. Technol. 10, N21–N25 (1999).
[CrossRef]

Wolf, E.

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

Appl. Opt. (1)

Appl. Surf. Sci. (1)

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

Can. J. Phys. (1)

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 188–195 (1955).

Meas. Sci. Technol. (1)

S. A. Stefani, C. R. Nagarajah, R. Willgoss, “A surface inspection technique for continuously extruded cylindrical products,” Meas. Sci. Technol. 10, N21–N25 (1999).
[CrossRef]

Optik (1)

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

Wire J. Intern. (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. Intern. 33, 124–127 (2000).

Other (8)

C. B. Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Manifestation of surface roughness of thin wires in laser scattering,” in Proceedings of the National Laser Symposium (Instrument Research and Development Establishment, Dehradun, India, 1995), pp. 330–331.

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

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

C. B. Rao, A. V. Ananthalakshmi, R. Kesavamoorthy, “Localization of surface roughness of thin wires using laser scattering,” in Trends in NDE Science and Technology, Proceedings of the 14th World Conference on Non-Destructive Testing, C. G. K. Nair, B. Raj, C. R. L. Nurthy, T. Jayakumar, eds. (Indian Society for Non-Destructive Testing, Chennai, India, 1996), Vols. 1–5, pp. 1517–1520.

F. Perez Quintián, M. A. Rebollo, N. G. Gaggioli, C. A. Raffo, “Optical methods for on-line surface wire testing,”in Proceedings of the 7th European Conference on Non-Destructive Testing (Danish Society for Nondestructive Testing, Copenhagen, 1998), Vol. 3, No. 8.

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

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

L. M. Sanchez-Brea, J. A. Gómez-Pedrero, E. Bernabeu, “Analysis and characterization of surface defects on thin steel wires by atomic force microscopy,” in Proceedings of the Wire and Cable Technical Symposium (Wire Association International, Guillford, Conn., 1999), pp. 189–192.

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Figures (9)

Fig. 1
Fig. 1

Experimental far-field diffraction pattern (a) for a cylinder without surface structures (diameter 200 μm); the zeroth diffraction order has been masked to avoid camera damage (bottom), and the other diffraction orders cannot be seen owing to a poor sampling; the shadow at right is due to the cylinder holder; (b) for a metallic wire with several surface structures; (c) for a single longitudinal structure over a metallic wire (diameter 200 μm). An interference effect appears; the diffraction cone is wider than for (a) and (b) because the laser beam was broadened with a cylindrical lens.

Fig. 2
Fig. 2

Sketch for diffraction by a cylinder of a light beam at oblique incidence showing the parameters involved.

Fig. 3
Fig. 3

Far-field diffraction patterns for a cylinder with radius a0=100 μm that presents a defect width σ=2 μm, defect position t=1, and N=2 [see Eq. (23)]. The incidence angle of the light beam is α=π/8. The horizontal axis represents the depth/height of the defect (a negative value represents a scratch, a positive value represents a protuberance), and the vertical axis represents the observation angle ϕ. Around the defect is seen an interference effect. In all cases, the reflection coefficient R=-1 (for a deeper discussion on how this parameter affects the light distribution see Sanchez-Brea and Bernabeu3) and λ=0.6328μm.

Fig. 4
Fig. 4

Far-field diffraction patterns obtained for wire with the longitudinal defect depicted in Fig. 3 for (a) p=0.5 μm and (b) p=-0.5 μm.

Fig. 5
Fig. 5

Intensity distribution for a cylinder with radius a0=100 μm, defect position t=1, and N=2. The incidence angle of the light beam is α=π/8. The x axis represents the width of the defect and the y axis the observation angle ϕ. (a) Defect depth p=2 μm. (b) Defect depth p=-2 μm.

Fig. 6
Fig. 6

(a) Profile obtained from Fig. 5(a) when p=2 μm and σ=8 μm. (b) Profile obtained from Fig. 5(b) when p=-2 μm and σ=8 μm.

Fig. 7
Fig. 7

Intensity profile for a cylinder with radius a0=100 μm, defect position t=0.5, and shape N=2, depth p=-0.25 μm and width σ=2 μm. The incidence angle of the light beam is α=π/4, and the wavelength λ=0.6328 μm.

Fig. 8
Fig. 8

Confocal image for a steel wire (refractive index n=2.485-1.381i, radius a0=100 μm) that presents a longitudinal defect. The experimental parameters for such longitudinal defect are depth p=-0.2 μm, width σ=2.1 μm, shape N=2.1, and position t=1.

Fig. 9
Fig. 9

Comparison between the numerical results (dashed curve) obtained by using relation (21) for the parameters of Fig. 8 and experimental angular intensity distribution (solid curve) when the wire of Fig. 8 is illuminated with a laser beam of wavelength λ=0.6328 μm and incidence angle α=π/4.

Equations (34)

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

r=[r(φ)cos φ, r(φ)sin φ, z],
E1(φ, z)=E0exp-1w2 {[r(φ)cos φ-x0]2+(z sin α)2}.
E|S=(1+R)E1|S,
EnS=(1-R)E1|Sk1  n,
E2(ϕ, θ)SE1(Rv-p)nexp(iv  r)dS,
v=k1-k2=k(vx, vy, vz)=k(-sin θ cos ϕ, -sin θ sin ϕ-sin α, cos α-cos θ),
p=k1+k2=k(-vx, -vy-2 sin α, cos α+cos θ),
n=(r˙×r¨)×r˙(r˙×r¨)×r˙
n=11+P2(φ) [cos φ+P(φ)sin φ, sin φ-P(φ)cos φ, 0],
P(φ)=1r(φ)dr(φ)dφ.
E2(ϕ, θ)=CkE0U1U2.
U1(ϕ, θ)=-exp-sin αzw2exp(ikvzz)dz=πwsin αexp-kvzw2 sin α2.
U2(ϕ, θ)=g(φ)exp[ikf(φ)]dφ,
g(φ)=exp{-[r(φ)cos φ-x0]2/w}[1+P2(φ)]1/2 r(φ)G(φ),
f(φ)=r(φ)(vxcos φ+vysin φ),
G(φ)=[(R+1)(vxcos φ+vysin φ)+2 sin α sin φ]+P(φ)[(R+1)(vxsin φ-vycos φ)-2 sin α cos φ].
U2(ϕ, θ)πkf(φs)1/2g(φs)×expikf(φs)+π4-arg[f(φs)]2,
P(φs)=vxsin φs-vycos φsvxcos φs+vysin φs,
U1(θ)=limkw U1(ϕ, θ)=2πk sin α δ(θ-α),
v=-k sin α(cos ϕ, 1+sin ϕ, 0),
cos ϕ=[1-P2(φs)]sin 2φs-2P(φs)cos 2φs1+P2(φs),
sin ϕ=-[1-P2(φs)]cos 2φs+2P(φs)sin 2φs1+P2(φs).
cos φs=sin[1/2(π/2-ϕ)]-P(φs)cos[1/2(π/2-ϕ)][1+P2(φs)]1/2,
sin φs=cos[1/2(π/2-ϕ)]+P(φs)sin[1/2(π/2-ϕ)][1+P2(φs)]1/2,
g(φs)=-2R sin α cosπ/2-ϕ2r(φs)exp[-W(φs)2],
f(φs)=-2 sin α cosπ/2-ϕ2r(φs)[1+P2(φs)]1/2,
f(φs)=2 sin α cosπ/2-ϕ2r(φs)[1+P2(φs)]1/2 ×[1+2P2(φs)-Q(φs)],
W(φs)=1wr(φs)sinπ/2-ϕ2-P(φs)cosπ/2-ϕ2[1+P2(φs)]1/2-x0,Q(φ)=1r(φ)d2r(φ)dφ2.
E2(ϕ; θ)E0sin1/2 αcos1/2π/2-ϕ2NϕA(φsj)exp-i2k sin αr(φsj)cosπ/2-ϕ2[1+P2(φsj)]1/2-π4δ(θ-α),
A(φs)=r1/2(φs)[1+P2(φs)]1/4[1+2P2(φs)-Q(φs)]1/2 R(φs)exp[-W2(φs)].
I2(ϕ)=E2(ϕ; θ)E2*(ϕ; θ)I0sin αcosπ/2-ϕ2×NϕA2(φsj)+2i<jA(φsi)×A(φsj)cos2kΦ(ϕ)sin α cosπ/2-ϕ2,
Φ(ϕ)={r(φsj)/[1+P2(φsj)]1/2-r(φsi)/[1+P2(φsi)]1/2}.
I2(ϕ; θ=α)I0a0sin αRπ/2-ϕ22×exp-2a0sinπ/2-ϕ2-x0w2×cosπ/2-ϕ2,
r(φ)=a0±p exp-12φ-tπ/2σN,

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