Abstract

The extinction theorem has been applied to the study of wave scattering from a non-single-valued surface composed of an infinitely long cylinder on a flat substrate, both of which are assumed to be perfect conductors. Cylinder diameters ranging from 0.1 to 4λ (λ being the incident wavelength) are considered. The calculation method is discussed for this kind of geometry. The study has been performed for both S (perpendicular) and P (parallel) polarizations when the direction of an incident Gaussian beam is perpendicular to the cylinder axis and the direction of the scattered wave is in the plane of incidence. The surface-current density on the flat substrate and on the cylinder has also been analyzed.

© 1994 Optical Society of America

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References

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  1. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, London, 1991), Chap. 6.
  2. M. A. Biot, “Some new aspects of the reflection of electromagnetic waves on a rough surface,” J. Appl. Phys. 28, 1455–1463 (1957).
    [CrossRef]
  3. V. Twersky, “On scattering and reflection of electromagnetic waves by rough surfaces,” IRE Trans. Antennas Propag. AP-5, 81–90 (1957).
    [CrossRef]
  4. V. Twersky, “Multiple scattering of waves and optical phenomena,” J. Opt. Soc. Am. 52, 145–171 (1962).
    [CrossRef] [PubMed]
  5. Lord Rayleigh, “On the light dispersed from fine lines ruled upon reflecting surfaces or transmitted by very narrow slits,” in Scientific Papers (Cambridge University, Cambridge, 1912), Vol. 5.
  6. R. P. Young, “Low-scatter mirror degradation by particle contamination,” Opt. Eng. 15, 516–520 (1976).
  7. A. J. Pidduck, D. J. Robbins, I. M. Young, A. G. Cullis, A. S. R. Martin, “The formation of dislocations and their in-situ detection during silicon vapor phase epitaxy at reduced temperature,” Mater. Sci. Eng. B4, 417–422 (1989).
    [CrossRef]
  8. P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).
  9. P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres—comparison with an ellipsometric experiment,” Physica 137A, 243–257 (1986).
  10. H. Weyl, “Ausbreitung elektromagnetisher Wellen ueber einem ebenen Leiter,” Ann. Phys. 60, 481–500 (1919).
    [CrossRef]
  11. K. B. Nahm, W. L. Wolfe, “Light scattering models for spheres on a conducting plane: comparison with experiment,” Appl. Opt. 26, 2995–2999 (1987).
    [CrossRef] [PubMed]
  12. D. C. Weber, E. D. Hirleman, “Light scattering signatures of individual spheres on optical smooth conducting surfaces,” Appl. Opt. 27, 4019–4026 (1988).
    [CrossRef] [PubMed]
  13. G. Videen, W. S. Bickel, V. J. Iafelice, D. Abromson, “Experimental light-scattering Mueller matrix for a fiber on a reflecting optical surface as a function of incident angle,” J. Opt. Soc. Am. A 9, 312–315 (1992).
    [CrossRef]
  14. I. V. Lindell, A. H. Sihlova, K. O. Muinonen, P. W. Barber, “Scattering by a small object close to an interface. I. Exact-image theory formulation,” J. Opt. Soc. Am. A 8, 472–476 (1991); K. O. Muinonen, A. H. Sihlova, I. V. Lindell, K. A. Lumme, “Scattering by a small object close to an interface. II. Study of backscattering,” J. Opt. Soc. Am. A 8, 477–482 (1991).
    [CrossRef]
  15. T. C. Rao, R. Barakat, “Plane-wave scattering by a conducting cylinder partially buried in a ground plane 1: TM case,” J. Opt. Soc. Am. A 6, 1270–1280 (1989).
    [CrossRef]
  16. T. C. Rao, R. Barakat, “Plane-wave scattering by a conducting cylinder partially buried in a ground plane 2: TE case,” J. Opt. Soc. Am. A 8, 1986–1990 (1991).
    [CrossRef]
  17. B. R. Johnson, “Light scattering from a spherical particle on a conducting plane: I. Normal incidence,” J. Opt. Soc. Am. A 9, 1341–1351 (1992).
    [CrossRef]
  18. G. Videen, M. G. Turner, V. J. Iafelice, W. S. Bickel, W. L. Wolfe, “Scattering from a small sphere near a surface,” J. Opt. Soc. Am. A 10, 118–126 (1993).
    [CrossRef]
  19. M. A. Taubenblatt, “Light scattering from cylindrical structures on surfaces,” Opt. Lett. 15, 255–257 (1990).
    [CrossRef] [PubMed]
  20. M. A. Kolbehdari, H. A. Auda, A. Z. Elsherbeni, “Scattering from dielectric cylinder partially embedded in a perfectly conducting ground plane,” J. Electromag. Waves Appl. 3, 531–554(1989).
  21. E. Wolf, “A generalized extinction theorem and its role in scattering theory,” in Coherence and Quantum Optics, L. Mandel, E. Wolf, eds. (Plenum, New York, 1973), pp. 339–357.
    [CrossRef]
  22. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chaps. 1 and 7.
  23. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
    [CrossRef]
  24. Ref. 1, Chap. 3, pp. 68–70 and references therein.
  25. K. K. Mei, J. G. Van Bladel, “Scattering by perfectly conducting rectangular cylinder,” IEEE Trans. Antennas Propag. AP-11, 185–192 (1963).
    [CrossRef]
  26. K. A. O’Donnell, E. R. Méndez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
    [CrossRef]
  27. J. M. Soto-Crespo, M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep reflection gratings,” J. Opt. Soc. Am. A 6, 367–384 (1989).
    [CrossRef]
  28. F. Moreno, F. González, J. M. Saiz, P. J. Valle, D. L. Jordan, “Experimental study of copolarized light scattering by spherical metallic particles on conducting flat substrates,” J. Opt. Soc. Am. A 10, 141–149 (1993).
    [CrossRef]
  29. A. K. Fung, M. F. Chen, “Numerical simulation of scattering from simple and composite random surfaces,” J. Opt. Soc. Am. A 2, 2274–2284 (1985).
    [CrossRef]
  30. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 17.

1993 (2)

1992 (2)

1991 (2)

1990 (2)

M. A. Taubenblatt, “Light scattering from cylindrical structures on surfaces,” Opt. Lett. 15, 255–257 (1990).
[CrossRef] [PubMed]

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

1989 (4)

J. M. Soto-Crespo, M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep reflection gratings,” J. Opt. Soc. Am. A 6, 367–384 (1989).
[CrossRef]

M. A. Kolbehdari, H. A. Auda, A. Z. Elsherbeni, “Scattering from dielectric cylinder partially embedded in a perfectly conducting ground plane,” J. Electromag. Waves Appl. 3, 531–554(1989).

T. C. Rao, R. Barakat, “Plane-wave scattering by a conducting cylinder partially buried in a ground plane 1: TM case,” J. Opt. Soc. Am. A 6, 1270–1280 (1989).
[CrossRef]

A. J. Pidduck, D. J. Robbins, I. M. Young, A. G. Cullis, A. S. R. Martin, “The formation of dislocations and their in-situ detection during silicon vapor phase epitaxy at reduced temperature,” Mater. Sci. Eng. B4, 417–422 (1989).
[CrossRef]

1988 (1)

1987 (2)

1986 (2)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres—comparison with an ellipsometric experiment,” Physica 137A, 243–257 (1986).

1985 (1)

1976 (1)

R. P. Young, “Low-scatter mirror degradation by particle contamination,” Opt. Eng. 15, 516–520 (1976).

1963 (1)

K. K. Mei, J. G. Van Bladel, “Scattering by perfectly conducting rectangular cylinder,” IEEE Trans. Antennas Propag. AP-11, 185–192 (1963).
[CrossRef]

1962 (1)

1957 (2)

M. A. Biot, “Some new aspects of the reflection of electromagnetic waves on a rough surface,” J. Appl. Phys. 28, 1455–1463 (1957).
[CrossRef]

V. Twersky, “On scattering and reflection of electromagnetic waves by rough surfaces,” IRE Trans. Antennas Propag. AP-5, 81–90 (1957).
[CrossRef]

1919 (1)

H. Weyl, “Ausbreitung elektromagnetisher Wellen ueber einem ebenen Leiter,” Ann. Phys. 60, 481–500 (1919).
[CrossRef]

Abromson, D.

Auda, H. A.

M. A. Kolbehdari, H. A. Auda, A. Z. Elsherbeni, “Scattering from dielectric cylinder partially embedded in a perfectly conducting ground plane,” J. Electromag. Waves Appl. 3, 531–554(1989).

Barakat, R.

Barber, P. W.

Bickel, W. S.

Biot, M. A.

M. A. Biot, “Some new aspects of the reflection of electromagnetic waves on a rough surface,” J. Appl. Phys. 28, 1455–1463 (1957).
[CrossRef]

Bobbert, P. A.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres—comparison with an ellipsometric experiment,” Physica 137A, 243–257 (1986).

Chen, M. F.

Cullis, A. G.

A. J. Pidduck, D. J. Robbins, I. M. Young, A. G. Cullis, A. S. R. Martin, “The formation of dislocations and their in-situ detection during silicon vapor phase epitaxy at reduced temperature,” Mater. Sci. Eng. B4, 417–422 (1989).
[CrossRef]

Elsherbeni, A. Z.

M. A. Kolbehdari, H. A. Auda, A. Z. Elsherbeni, “Scattering from dielectric cylinder partially embedded in a perfectly conducting ground plane,” J. Electromag. Waves Appl. 3, 531–554(1989).

Fung, A. K.

González, F.

Greef, R.

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres—comparison with an ellipsometric experiment,” Physica 137A, 243–257 (1986).

Hirleman, E. D.

Iafelice, V. J.

Johnson, B. R.

Jordan, D. L.

Kolbehdari, M. A.

M. A. Kolbehdari, H. A. Auda, A. Z. Elsherbeni, “Scattering from dielectric cylinder partially embedded in a perfectly conducting ground plane,” J. Electromag. Waves Appl. 3, 531–554(1989).

Lindell, I. V.

Maradudin, A. A.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Martin, A. S. R.

A. J. Pidduck, D. J. Robbins, I. M. Young, A. G. Cullis, A. S. R. Martin, “The formation of dislocations and their in-situ detection during silicon vapor phase epitaxy at reduced temperature,” Mater. Sci. Eng. B4, 417–422 (1989).
[CrossRef]

McGurn, A. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Mei, K. K.

K. K. Mei, J. G. Van Bladel, “Scattering by perfectly conducting rectangular cylinder,” IEEE Trans. Antennas Propag. AP-11, 185–192 (1963).
[CrossRef]

Méndez, E. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

K. A. O’Donnell, E. R. Méndez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
[CrossRef]

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Moreno, F.

Muinonen, K. O.

Nahm, K. B.

Nieto-Vesperinas, M.

O’Donnell, K. A.

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, London, 1991), Chap. 6.

Pidduck, A. J.

A. J. Pidduck, D. J. Robbins, I. M. Young, A. G. Cullis, A. S. R. Martin, “The formation of dislocations and their in-situ detection during silicon vapor phase epitaxy at reduced temperature,” Mater. Sci. Eng. B4, 417–422 (1989).
[CrossRef]

Rao, T. C.

Rayleigh, Lord

Lord Rayleigh, “On the light dispersed from fine lines ruled upon reflecting surfaces or transmitted by very narrow slits,” in Scientific Papers (Cambridge University, Cambridge, 1912), Vol. 5.

Robbins, D. J.

A. J. Pidduck, D. J. Robbins, I. M. Young, A. G. Cullis, A. S. R. Martin, “The formation of dislocations and their in-situ detection during silicon vapor phase epitaxy at reduced temperature,” Mater. Sci. Eng. B4, 417–422 (1989).
[CrossRef]

Saiz, J. M.

Sihlova, A. H.

Soto-Crespo, J. M.

Taubenblatt, M. A.

Turner, M. G.

Twersky, V.

V. Twersky, “Multiple scattering of waves and optical phenomena,” J. Opt. Soc. Am. 52, 145–171 (1962).
[CrossRef] [PubMed]

V. Twersky, “On scattering and reflection of electromagnetic waves by rough surfaces,” IRE Trans. Antennas Propag. AP-5, 81–90 (1957).
[CrossRef]

Valle, P. J.

Van Bladel, J. G.

K. K. Mei, J. G. Van Bladel, “Scattering by perfectly conducting rectangular cylinder,” IEEE Trans. Antennas Propag. AP-11, 185–192 (1963).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 17.

Videen, G.

Vlieger, J.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres—comparison with an ellipsometric experiment,” Physica 137A, 243–257 (1986).

Weber, D. C.

Weyl, H.

H. Weyl, “Ausbreitung elektromagnetisher Wellen ueber einem ebenen Leiter,” Ann. Phys. 60, 481–500 (1919).
[CrossRef]

Wolf, E.

E. Wolf, “A generalized extinction theorem and its role in scattering theory,” in Coherence and Quantum Optics, L. Mandel, E. Wolf, eds. (Plenum, New York, 1973), pp. 339–357.
[CrossRef]

Wolfe, W. L.

Young, I. M.

A. J. Pidduck, D. J. Robbins, I. M. Young, A. G. Cullis, A. S. R. Martin, “The formation of dislocations and their in-situ detection during silicon vapor phase epitaxy at reduced temperature,” Mater. Sci. Eng. B4, 417–422 (1989).
[CrossRef]

Young, R. P.

R. P. Young, “Low-scatter mirror degradation by particle contamination,” Opt. Eng. 15, 516–520 (1976).

Ann. Phys. (2)

H. Weyl, “Ausbreitung elektromagnetisher Wellen ueber einem ebenen Leiter,” Ann. Phys. 60, 481–500 (1919).
[CrossRef]

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

K. K. Mei, J. G. Van Bladel, “Scattering by perfectly conducting rectangular cylinder,” IEEE Trans. Antennas Propag. AP-11, 185–192 (1963).
[CrossRef]

IRE Trans. Antennas Propag. (1)

V. Twersky, “On scattering and reflection of electromagnetic waves by rough surfaces,” IRE Trans. Antennas Propag. AP-5, 81–90 (1957).
[CrossRef]

J. Appl. Phys. (1)

M. A. Biot, “Some new aspects of the reflection of electromagnetic waves on a rough surface,” J. Appl. Phys. 28, 1455–1463 (1957).
[CrossRef]

J. Electromag. Waves Appl. (1)

M. A. Kolbehdari, H. A. Auda, A. Z. Elsherbeni, “Scattering from dielectric cylinder partially embedded in a perfectly conducting ground plane,” J. Electromag. Waves Appl. 3, 531–554(1989).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (10)

G. Videen, W. S. Bickel, V. J. Iafelice, D. Abromson, “Experimental light-scattering Mueller matrix for a fiber on a reflecting optical surface as a function of incident angle,” J. Opt. Soc. Am. A 9, 312–315 (1992).
[CrossRef]

I. V. Lindell, A. H. Sihlova, K. O. Muinonen, P. W. Barber, “Scattering by a small object close to an interface. I. Exact-image theory formulation,” J. Opt. Soc. Am. A 8, 472–476 (1991); K. O. Muinonen, A. H. Sihlova, I. V. Lindell, K. A. Lumme, “Scattering by a small object close to an interface. II. Study of backscattering,” J. Opt. Soc. Am. A 8, 477–482 (1991).
[CrossRef]

T. C. Rao, R. Barakat, “Plane-wave scattering by a conducting cylinder partially buried in a ground plane 1: TM case,” J. Opt. Soc. Am. A 6, 1270–1280 (1989).
[CrossRef]

T. C. Rao, R. Barakat, “Plane-wave scattering by a conducting cylinder partially buried in a ground plane 2: TE case,” J. Opt. Soc. Am. A 8, 1986–1990 (1991).
[CrossRef]

B. R. Johnson, “Light scattering from a spherical particle on a conducting plane: I. Normal incidence,” J. Opt. Soc. Am. A 9, 1341–1351 (1992).
[CrossRef]

G. Videen, M. G. Turner, V. J. Iafelice, W. S. Bickel, W. L. Wolfe, “Scattering from a small sphere near a surface,” J. Opt. Soc. Am. A 10, 118–126 (1993).
[CrossRef]

K. A. O’Donnell, E. R. Méndez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
[CrossRef]

J. M. Soto-Crespo, M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep reflection gratings,” J. Opt. Soc. Am. A 6, 367–384 (1989).
[CrossRef]

F. Moreno, F. González, J. M. Saiz, P. J. Valle, D. L. Jordan, “Experimental study of copolarized light scattering by spherical metallic particles on conducting flat substrates,” J. Opt. Soc. Am. A 10, 141–149 (1993).
[CrossRef]

A. K. Fung, M. F. Chen, “Numerical simulation of scattering from simple and composite random surfaces,” J. Opt. Soc. Am. A 2, 2274–2284 (1985).
[CrossRef]

Mater. Sci. Eng. (1)

A. J. Pidduck, D. J. Robbins, I. M. Young, A. G. Cullis, A. S. R. Martin, “The formation of dislocations and their in-situ detection during silicon vapor phase epitaxy at reduced temperature,” Mater. Sci. Eng. B4, 417–422 (1989).
[CrossRef]

Opt. Eng. (1)

R. P. Young, “Low-scatter mirror degradation by particle contamination,” Opt. Eng. 15, 516–520 (1976).

Opt. Lett. (1)

Physica (2)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres—comparison with an ellipsometric experiment,” Physica 137A, 243–257 (1986).

Other (6)

E. Wolf, “A generalized extinction theorem and its role in scattering theory,” in Coherence and Quantum Optics, L. Mandel, E. Wolf, eds. (Plenum, New York, 1973), pp. 339–357.
[CrossRef]

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chaps. 1 and 7.

Lord Rayleigh, “On the light dispersed from fine lines ruled upon reflecting surfaces or transmitted by very narrow slits,” in Scientific Papers (Cambridge University, Cambridge, 1912), Vol. 5.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, London, 1991), Chap. 6.

Ref. 1, Chap. 3, pp. 68–70 and references therein.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 17.

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Figures (12)

Fig. 1
Fig. 1

Geometry of the scattering problem.

Fig. 2
Fig. 2

Semilogarithmic plot of the scattering cross section versus the scattering angle of a cylinder (D/λ = 1) on a flat substrate for three values of the half-width, w, of an S-polarized incident Gaussian beam. From top to bottom w/λ = 3, 4, and 8. (a) θ0 = 0°, (b) θ0 = 60°.

Fig. 3
Fig. 3

Semilogarithmic plot of the scattering cross section versus the scattering angle of a cylinder on a flat substrate for five sizes of the cylinder diameter. From top to bottom D/λ = 0.2, 0.5, 1, 2, and 4. Angle of incidence θ0 = 0°. (a) S polarization, (b) P polarization.

Fig. 4
Fig. 4

Same as Fig. 3 for angle of incidence θ0 = 30°.

Fig. 5
Fig. 5

Same as Fig. 3 for angle of incidence θ0 = 60°.

Fig. 6
Fig. 6

(a) Semilogarithmic plot of the backscattering cross section versus the angle of incidence of a cylinder on a flat substrate for D/λ = 0.5. The dotted–dashed curve corresponds to S polarization and the solid curve corresponds to P polarization. (b) Backscattering cross-section S to P ratio obtained from (a).

Fig. 7
Fig. 7

Same as Fig. 6 for D/λ = 1.

Fig. 8
Fig. 8

Same as Fig. 6 for D/λ = 2.

Fig. 9
Fig. 9

Diagram of the different contributions (1–5) to the back-scattered wave.

Fig. 10
Fig. 10

Semilogarithmic plot of the scattering cross section versus the scattering angle of a cylinder on a flat substrate for D/λ = 4/π and θ0 = 0°. The dotted–dashed curve corresponds to S polarization and solid curve corresponds to P polarization.

Fig. 11
Fig. 11

Plot of the modulus of the surface-current density on the flat substrate. In each case the unrippled curve corresponds to a flat substrate with no cylinder on it, and the rippled curve corresponds to a flat substrate with one cylinder of D/λ = 1 located on it at x = 0. (a) S polarization, (b) P polarization.

Fig. 12
Fig. 12

Polar plots of the modulus of the surface-currrent density on the cylinders of diameters D/λ = 0.5, 1, and 2 (bottom right) and three angles of incidence θ0 = 0°, 30°, and 60° (indicated by arrows). (a) S polarization, (b) P polarization.

Equations (18)

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

L = L d s .
L i = 1 N Δ s i ,
E y ( 0 ) ( r ; θ 0 ) = exp [ - ( x cos θ 0 + z sin θ 0 ) 2 / w 2 ] × exp [ i k ( x sin θ 0 - z cos θ 0 ) ( 1 + W ) ] ,
W = [ 2 ( x cos θ 0 + z sin θ 0 ) 2 w 2 - 1 ] 1 k 2 w 2 , r ( x , z ) ,             k = 2 π / λ ,
E y ( 0 ) ( r ; θ 0 ) = π k c L J y ( s ) H 0 ( 1 ) ( k r - s ) d s ,
E y ( s ) ( θ ) = Q π k c L J y ( s ) exp ( - i k × s ) d s ,
Q = [ 4 2 w k π 3 / 2 ( 2 w 2 k 2 - 1 - 2 tan 2 θ 0 ) ] 1 / 2 .
- π / 2 π / 2 E y ( s ) ( θ ) 2 d θ = 1.
I s ( s ) ( θ ) = E y ( s ) ( θ ) [ E y ( s ) ( θ ) ] * .
H y ( 0 ) ( r ; θ 0 ) = exp [ - ( x cos θ 0 + z sin θ 0 ) 2 / w 2 ] × exp [ i k ( x sin θ 0 - z cos θ 0 ) ( 1 + W ) ] ,
H y ( 0 ) ( r ; θ 0 ) = - i π k c L J x ( s ) H 1 ( 1 ) ( k r - s ) n ^ ( s ) × ( r - s ) r - s d s ,
n ^ ( s ) = v [ 1 + D 2 ( s ) ] - 1 / 2 [ - D ( s ) , 1 ] ,
H y ( s ) ( θ ) = Q π k c L J x ( s ) exp ( - i k × s ) × [ cos θ - D ( s ) sin θ ] d s ,
- π / 2 π / 2 H x ( s ) ( θ ) 2 d θ = 1.
I p ( s ) ( θ ) = H y ( s ) ( θ ) [ H y ( s ) ( θ ) ] * .
g = w / cos θ 0 .
J = c 4 π ( n ^ · H ) ,
H = H ( 0 ) + H ( d ) ,

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