Abstract

We describe a way to combine the method of fictitious sources and the scattering-matrix method. The resulting method presents concurrently the advantages of these two rigorous methods. It is able to solve efficiently electromagnetic problems in which the structure is made up of a jacket containing an arbitrary set of scatterers. The method is described in a two-dimensional case, but the basic ideas could be easily extended to three-dimensional cases.

© 2004 Optical Society of America

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References

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  1. D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
    [CrossRef]
  2. D. Felbacq, E. Centeno, “Theory of diffraction for 2D photonic crystals with a boundary,” Opt. Commun. 199, 39–45 (2001).
    [CrossRef]
  3. T. P. White, B. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, L. C. Botten, “Multipole method for microstructured optical fibers. I formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002).
    [CrossRef]
  4. D. Maystre, M. Saillard, G. Tayeb, “Special methods of wave diffraction,” in Scattering, P. Sabatier, E. R. Pike, eds. (Academic, London, 2001).
  5. G. Tayeb, R. Petit, M. Cadilhac, “Synthesis method applied to the problem of diffraction by gratings: the method of fictitious sources,” in Proceedings of the International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 95–105 (1991).
    [CrossRef]
  6. G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” Special issue on Generalized Multipole Techniques (GMT) of Appl. Computat. Electromagn. Soc. J. 9, 90–100 (1994).
  7. F. Zolla, R. Petit, M. Cadilhac, “Electromagnetic theory of diffraction by a system of parallel rods: the method of fictitious sources,” J. Opt. Soc. Am. A 11, 1087–1096 (1994).
    [CrossRef]
  8. F. Zolla, R. Petit, “Method of fictitious sources as applied to the electromagnetic diffraction of a plane wave by a grating in conical diffraction mounts,” J. Opt. Soc. Am. A 13, 796–802 (1996).
    [CrossRef]
  9. Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model,” IEEE Trans. Antennas Propag. AP-35, 1119–1127 (1987).
    [CrossRef]
  10. A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
    [CrossRef]
  11. A. Boag, Y. Leviatan, A. Boag, “Analysis of diffraction from echelette gratings using a strip-current model,” J. Opt. Soc. Am. A 6, 543–549 (1989).
    [CrossRef]
  12. A. Boag, Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from doubly periodic nonplanar surfaces using a patch-current model,” IEEE Trans. Antennas Propag. AP-41, 732–738 (1993).
    [CrossRef]
  13. C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House, Boston, Mass., 1990).
  14. C. Hafner, “Multiple multipole program computation of periodic structures,” J. Opt. Soc. Am. A 12, 1057–1067 (1995).
    [CrossRef]
  15. V. D. Kupradze, “On the approximate solution of problems in mathematical physics,” original (Russian), Uspekhi Mat. Nauk 22(2), 59–107 (1967); English translation, Russian Mathematical Surveys 22, 58–108 (1967).
    [CrossRef]
  16. D. Kaklamani, H. Anastassiu, “Aspects of the method of auxiliary sources (MAS) in computational electromagnetics,” IEEE Antennas Propag Mag.June2002, pp. 48–64.
  17. W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes in FORTRAN: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).
  18. A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, N. A. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E 63, 046612 (2001).
    [CrossRef]
  19. M. Abramovitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).
  20. D. Maystre, M. Cadilhac, “Singularities of the continuation of the fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26, 2201–2204 (1985).
    [CrossRef]
  21. http://institut.fresnel.free.fr/fs_ssm/index.htm , or contact the authors.

2002 (2)

T. P. White, B. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, L. C. Botten, “Multipole method for microstructured optical fibers. I formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002).
[CrossRef]

D. Kaklamani, H. Anastassiu, “Aspects of the method of auxiliary sources (MAS) in computational electromagnetics,” IEEE Antennas Propag Mag.June2002, pp. 48–64.

2001 (2)

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, N. A. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E 63, 046612 (2001).
[CrossRef]

D. Felbacq, E. Centeno, “Theory of diffraction for 2D photonic crystals with a boundary,” Opt. Commun. 199, 39–45 (2001).
[CrossRef]

1996 (1)

1995 (1)

1994 (3)

1993 (1)

A. Boag, Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from doubly periodic nonplanar surfaces using a patch-current model,” IEEE Trans. Antennas Propag. AP-41, 732–738 (1993).
[CrossRef]

1989 (1)

1988 (1)

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

1987 (1)

Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model,” IEEE Trans. Antennas Propag. AP-35, 1119–1127 (1987).
[CrossRef]

1985 (1)

D. Maystre, M. Cadilhac, “Singularities of the continuation of the fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26, 2201–2204 (1985).
[CrossRef]

1967 (1)

V. D. Kupradze, “On the approximate solution of problems in mathematical physics,” original (Russian), Uspekhi Mat. Nauk 22(2), 59–107 (1967); English translation, Russian Mathematical Surveys 22, 58–108 (1967).
[CrossRef]

Abramovitz, M.

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Anastassiu, H.

D. Kaklamani, H. Anastassiu, “Aspects of the method of auxiliary sources (MAS) in computational electromagnetics,” IEEE Antennas Propag Mag.June2002, pp. 48–64.

Asatryan, A. A.

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, N. A. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E 63, 046612 (2001).
[CrossRef]

Boag, A.

A. Boag, Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from doubly periodic nonplanar surfaces using a patch-current model,” IEEE Trans. Antennas Propag. AP-41, 732–738 (1993).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from doubly periodic nonplanar surfaces using a patch-current model,” IEEE Trans. Antennas Propag. AP-41, 732–738 (1993).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of diffraction from echelette gratings using a strip-current model,” J. Opt. Soc. Am. A 6, 543–549 (1989).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of diffraction from echelette gratings using a strip-current model,” J. Opt. Soc. Am. A 6, 543–549 (1989).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model,” IEEE Trans. Antennas Propag. AP-35, 1119–1127 (1987).
[CrossRef]

Botten, L. C.

T. P. White, B. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, L. C. Botten, “Multipole method for microstructured optical fibers. I formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002).
[CrossRef]

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, N. A. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E 63, 046612 (2001).
[CrossRef]

Busch, K.

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, N. A. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E 63, 046612 (2001).
[CrossRef]

Cadilhac, M.

F. Zolla, R. Petit, M. Cadilhac, “Electromagnetic theory of diffraction by a system of parallel rods: the method of fictitious sources,” J. Opt. Soc. Am. A 11, 1087–1096 (1994).
[CrossRef]

D. Maystre, M. Cadilhac, “Singularities of the continuation of the fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26, 2201–2204 (1985).
[CrossRef]

G. Tayeb, R. Petit, M. Cadilhac, “Synthesis method applied to the problem of diffraction by gratings: the method of fictitious sources,” in Proceedings of the International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 95–105 (1991).
[CrossRef]

Centeno, E.

D. Felbacq, E. Centeno, “Theory of diffraction for 2D photonic crystals with a boundary,” Opt. Commun. 199, 39–45 (2001).
[CrossRef]

de Sterke, C. M.

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, N. A. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E 63, 046612 (2001).
[CrossRef]

Felbacq, D.

D. Felbacq, E. Centeno, “Theory of diffraction for 2D photonic crystals with a boundary,” Opt. Commun. 199, 39–45 (2001).
[CrossRef]

D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
[CrossRef]

Flannery, B.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes in FORTRAN: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Hafner, C.

C. Hafner, “Multiple multipole program computation of periodic structures,” J. Opt. Soc. Am. A 12, 1057–1067 (1995).
[CrossRef]

C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House, Boston, Mass., 1990).

Kaklamani, D.

D. Kaklamani, H. Anastassiu, “Aspects of the method of auxiliary sources (MAS) in computational electromagnetics,” IEEE Antennas Propag Mag.June2002, pp. 48–64.

Kuhlmey, B.

Kupradze, V. D.

V. D. Kupradze, “On the approximate solution of problems in mathematical physics,” original (Russian), Uspekhi Mat. Nauk 22(2), 59–107 (1967); English translation, Russian Mathematical Surveys 22, 58–108 (1967).
[CrossRef]

Leviatan, Y.

A. Boag, Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from doubly periodic nonplanar surfaces using a patch-current model,” IEEE Trans. Antennas Propag. AP-41, 732–738 (1993).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of diffraction from echelette gratings using a strip-current model,” J. Opt. Soc. Am. A 6, 543–549 (1989).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model,” IEEE Trans. Antennas Propag. AP-35, 1119–1127 (1987).
[CrossRef]

Martijn de Sterke, C.

Maystre, D.

T. P. White, B. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, L. C. Botten, “Multipole method for microstructured optical fibers. I formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002).
[CrossRef]

D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
[CrossRef]

D. Maystre, M. Cadilhac, “Singularities of the continuation of the fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26, 2201–2204 (1985).
[CrossRef]

D. Maystre, M. Saillard, G. Tayeb, “Special methods of wave diffraction,” in Scattering, P. Sabatier, E. R. Pike, eds. (Academic, London, 2001).

McPhedran, R. C.

T. P. White, B. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, L. C. Botten, “Multipole method for microstructured optical fibers. I formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002).
[CrossRef]

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, N. A. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E 63, 046612 (2001).
[CrossRef]

Nicorovici, N. A.

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, N. A. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E 63, 046612 (2001).
[CrossRef]

Petit, R.

F. Zolla, R. Petit, “Method of fictitious sources as applied to the electromagnetic diffraction of a plane wave by a grating in conical diffraction mounts,” J. Opt. Soc. Am. A 13, 796–802 (1996).
[CrossRef]

F. Zolla, R. Petit, M. Cadilhac, “Electromagnetic theory of diffraction by a system of parallel rods: the method of fictitious sources,” J. Opt. Soc. Am. A 11, 1087–1096 (1994).
[CrossRef]

G. Tayeb, R. Petit, M. Cadilhac, “Synthesis method applied to the problem of diffraction by gratings: the method of fictitious sources,” in Proceedings of the International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 95–105 (1991).
[CrossRef]

Press, W.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes in FORTRAN: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Renversez, G.

Saillard, M.

D. Maystre, M. Saillard, G. Tayeb, “Special methods of wave diffraction,” in Scattering, P. Sabatier, E. R. Pike, eds. (Academic, London, 2001).

Stegun, I.

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Tayeb, G.

D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
[CrossRef]

G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” Special issue on Generalized Multipole Techniques (GMT) of Appl. Computat. Electromagn. Soc. J. 9, 90–100 (1994).

D. Maystre, M. Saillard, G. Tayeb, “Special methods of wave diffraction,” in Scattering, P. Sabatier, E. R. Pike, eds. (Academic, London, 2001).

G. Tayeb, R. Petit, M. Cadilhac, “Synthesis method applied to the problem of diffraction by gratings: the method of fictitious sources,” in Proceedings of the International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 95–105 (1991).
[CrossRef]

Teukolsky, S.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes in FORTRAN: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Vetterling, W.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes in FORTRAN: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

White, T. P.

Zolla, F.

Appl. Computat. Electromagn. Soc. J. (1)

G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” Special issue on Generalized Multipole Techniques (GMT) of Appl. Computat. Electromagn. Soc. J. 9, 90–100 (1994).

IEEE Antennas Propag Mag. (1)

D. Kaklamani, H. Anastassiu, “Aspects of the method of auxiliary sources (MAS) in computational electromagnetics,” IEEE Antennas Propag Mag.June2002, pp. 48–64.

IEEE Trans. Antennas Propag. (2)

A. Boag, Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from doubly periodic nonplanar surfaces using a patch-current model,” IEEE Trans. Antennas Propag. AP-41, 732–738 (1993).
[CrossRef]

Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model,” IEEE Trans. Antennas Propag. AP-35, 1119–1127 (1987).
[CrossRef]

J. Math. Phys. (1)

D. Maystre, M. Cadilhac, “Singularities of the continuation of the fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26, 2201–2204 (1985).
[CrossRef]

J. Opt. Soc. Am. A (5)

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

Opt. Commun. (1)

D. Felbacq, E. Centeno, “Theory of diffraction for 2D photonic crystals with a boundary,” Opt. Commun. 199, 39–45 (2001).
[CrossRef]

Phys. Rev. E (1)

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, N. A. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E 63, 046612 (2001).
[CrossRef]

Radio Sci. (1)

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

Uspekhi Mat. Nauk (1)

V. D. Kupradze, “On the approximate solution of problems in mathematical physics,” original (Russian), Uspekhi Mat. Nauk 22(2), 59–107 (1967); English translation, Russian Mathematical Surveys 22, 58–108 (1967).
[CrossRef]

Other (6)

C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House, Boston, Mass., 1990).

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes in FORTRAN: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

D. Maystre, M. Saillard, G. Tayeb, “Special methods of wave diffraction,” in Scattering, P. Sabatier, E. R. Pike, eds. (Academic, London, 2001).

G. Tayeb, R. Petit, M. Cadilhac, “Synthesis method applied to the problem of diffraction by gratings: the method of fictitious sources,” in Proceedings of the International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 95–105 (1991).
[CrossRef]

http://institut.fresnel.free.fr/fs_ssm/index.htm , or contact the authors.

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Figures (11)

Fig. 1
Fig. 1

Description of the problem.

Fig. 2
Fig. 2

Sources Se,n (represented by dots) radiate the fields Fe,n(r) used to represent the scattered field uscat in Ωe; sources Si,n (represented by stars) radiate the fields Fi,n(r) used to represent the total field uint in Ωi.

Fig. 3
Fig. 3

Cross section of the cylinder and the two sets of sources. The profile is given by Eq. (10) and the values c-5=-0.1134+i0.1310, c-4=-0.0297-i0.3238, c-3=-0.4117-i0.0973, c-2=-0.1260+i1.4149, c-1=-2.3936+i2.4031, c0=0.5714+i0.5000, c1=1.5568+i0.1876, c2=-0.1212-i0.0197, c3=-0.8158+i0.2155, c4=0.2772+i0.1039, and c5=-0.1532-i0.0102.

Fig. 4
Fig. 4

Scattered intensity at infinity for both polarizations.

Fig. 5
Fig. 5

Modulus of the total field in p polarization.

Fig. 6
Fig. 6

Set of Nc=3 parallel cylinders in a medium with index ni in the case in which the incident field is created by a line source.

Fig. 7
Fig. 7

Circle Dj surrounding the cylinder Cj and local coordinate system.

Fig. 8
Fig. 8

Setting of the problem to obtain the functions Fi,n(r).

Fig. 9
Fig. 9

Field modulus in the same conditions as in Ref. 2. The gray levels are chosen to match as closely as possible those of Ref. 2.

Fig. 10
Fig. 10

Modulus of the total field for a scatterer with four inclusions.

Fig. 11
Fig. 11

Modulus of the total field. Above the slab, the slanted line shows the locus of the maximum of the Gaussian incident beam. Below the slab, it shows the locus of the maximum of the transmitted field. Above the slab, the structure of the field is due to the interference between the incident and the reflected fields.

Equations (22)

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

u=uinc+uscatinΩe(outside C0)u=uintinΩi(inside C0).
u˜scat(r)=defn=1Nce,nFe,n(r),inΩe,
u(r)uinc(r)+n=1Nce,nFe,n(r),inΩe.
u˜int(r)=defn=1Nci,nFi,n(r),
Fe,n(r)=H0(1)(ke|r-re,n|),
Fi,n(r)=H0(1)(ki|r-ri,n|).
uinc(r)+uscat(r)-uint(r)=0on C0Duinc(r)+Duscat(r)-pDuint(r)=0on C0,
p=1inthes-polarizationcasep=(ne/ni)2inthep-polarizationcase.
uinc(r)+n=1Nce,nFe,n(r)-n=1Nci,nFi,n(r)Duinc(r)+n=1Nce,nDFe,n(r)-pn=1Nci,nDFi,n(r).
z(t)=x(t)+iy(t)=n=-5,5cnexp(in2πt).
uscat(r)g(θ) exp(iker)r,
D(θ)=2π|g(θ)|2.
u(P)=m=-,+[aj,mJm(kirj(P))+bj,mHm(1)(kirj(P))]exp[imθj(P)].
aj=Qj+kjTj,kbk,
bj=Sjaj.
I-S1T1,2-S1T1,N-S2T2,1I-S2T2,N-SNTN,1-SNTN,2I b1b2bN
=S1Q1S2Q2SNQN,
S-1B=A
B=SA.
u(P)=uinc(P)+j=1Ncm=-+bj,mHm(1)(kirj(P))exp[imθj(P)].
uinc(x, y)=-+A(α)exp[iαx-iβ(α)y]dα,
A(α)=W2πexp-(α-α0)2W24.

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