Abstract

For any oblique incidence and arbitrarily high order, lattice sums for one-dimensional gratings can be expressed in terms of exponentially convergent series. The scattering Green’s function can be efficiently evaluated also in the grating plane. Numerical implementation of the method is 200 times faster than for the previous best result.

© 2001 Optical Society of America

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References

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  1. R. Lampe, P. Klock, and P. Mayes, IEEE Trans. Microwave Theory Tech. 33, 734 (1985).
    [CrossRef]
  2. R. E. Jorgenson and R. Mitra, IEEE Trans. Antennas Propag. 38, 633 (1990).
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  3. S. Singh and R. Singh, IEEE Trans. Microwave Theory Tech. 38, 1746 (1990).
    [CrossRef]
  4. S. Singh, W. F. Richards, J. R. Zinecker, and D. R. Watson, IEEE Trans. Antennas Propag. 38, 1958 (1991).
    [CrossRef]
  5. S. Singh and R. Singh, IEEE Trans. Antennas Propag. 39, 1514 (1991).
    [CrossRef]
  6. E. I. Veliev and A. Oksasoglu, J. Electromagn. Waves Appl. 10, 493 (1996).
    [CrossRef]
  7. C. M. Linton, J. Eng. Math. 33, 377 (1998).
    [CrossRef]
  8. C. M. Linton, Appl. Ocean Res. 21, 17 (1999).
    [CrossRef]
  9. N. A. Nicorovici and R. C. McPhedran, Phys. Rev. E 50, 3143 (1994).
    [CrossRef]
  10. K. Yasumoto and K. Yoshitomi, IEEE Trans. Antennas Propag. 47, 1050 (1999).
    [CrossRef]
  11. A. M. Ozorio de Almeida, Acta Cryst. A 31, 435 (1975).
    [CrossRef]
  12. M. V. Berry, Ann. Phys. (NY) 131, 163 (1981), Apps.  A and B.
    [CrossRef]
  13. F. S. Ham and B. Segall, Phys. Rev. B 124, 1786 (1961).
    [CrossRef]
  14. P. P. Ewald, Ann. Phys. 64, 253 (1921).
    [CrossRef]
  15. H. van der Lem and A. Moroz, J. Opt. A 2, 395 (2000).
    [CrossRef]
  16. A. Moroz and C. Sommers, J. Phys. Condens. Matter 11, 997 (1999).
    [CrossRef]
  17. K. Kambe, Z. Naturforsch. Teil A 22, 322 (1967).
  18. M. Abramowitch and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1973).

2000 (1)

H. van der Lem and A. Moroz, J. Opt. A 2, 395 (2000).
[CrossRef]

1999 (3)

A. Moroz and C. Sommers, J. Phys. Condens. Matter 11, 997 (1999).
[CrossRef]

K. Yasumoto and K. Yoshitomi, IEEE Trans. Antennas Propag. 47, 1050 (1999).
[CrossRef]

C. M. Linton, Appl. Ocean Res. 21, 17 (1999).
[CrossRef]

1998 (1)

C. M. Linton, J. Eng. Math. 33, 377 (1998).
[CrossRef]

1996 (1)

E. I. Veliev and A. Oksasoglu, J. Electromagn. Waves Appl. 10, 493 (1996).
[CrossRef]

1994 (1)

N. A. Nicorovici and R. C. McPhedran, Phys. Rev. E 50, 3143 (1994).
[CrossRef]

1991 (2)

S. Singh, W. F. Richards, J. R. Zinecker, and D. R. Watson, IEEE Trans. Antennas Propag. 38, 1958 (1991).
[CrossRef]

S. Singh and R. Singh, IEEE Trans. Antennas Propag. 39, 1514 (1991).
[CrossRef]

1990 (2)

R. E. Jorgenson and R. Mitra, IEEE Trans. Antennas Propag. 38, 633 (1990).
[CrossRef]

S. Singh and R. Singh, IEEE Trans. Microwave Theory Tech. 38, 1746 (1990).
[CrossRef]

1985 (1)

R. Lampe, P. Klock, and P. Mayes, IEEE Trans. Microwave Theory Tech. 33, 734 (1985).
[CrossRef]

1981 (1)

M. V. Berry, Ann. Phys. (NY) 131, 163 (1981), Apps.  A and B.
[CrossRef]

1975 (1)

A. M. Ozorio de Almeida, Acta Cryst. A 31, 435 (1975).
[CrossRef]

1967 (1)

K. Kambe, Z. Naturforsch. Teil A 22, 322 (1967).

1961 (1)

F. S. Ham and B. Segall, Phys. Rev. B 124, 1786 (1961).
[CrossRef]

1921 (1)

P. P. Ewald, Ann. Phys. 64, 253 (1921).
[CrossRef]

Berry, M. V.

M. V. Berry, Ann. Phys. (NY) 131, 163 (1981), Apps.  A and B.
[CrossRef]

Ewald, P. P.

P. P. Ewald, Ann. Phys. 64, 253 (1921).
[CrossRef]

Ham, F. S.

F. S. Ham and B. Segall, Phys. Rev. B 124, 1786 (1961).
[CrossRef]

Jorgenson, R. E.

R. E. Jorgenson and R. Mitra, IEEE Trans. Antennas Propag. 38, 633 (1990).
[CrossRef]

Kambe, K.

K. Kambe, Z. Naturforsch. Teil A 22, 322 (1967).

Klock, P.

R. Lampe, P. Klock, and P. Mayes, IEEE Trans. Microwave Theory Tech. 33, 734 (1985).
[CrossRef]

Lampe, R.

R. Lampe, P. Klock, and P. Mayes, IEEE Trans. Microwave Theory Tech. 33, 734 (1985).
[CrossRef]

Linton, C. M.

C. M. Linton, Appl. Ocean Res. 21, 17 (1999).
[CrossRef]

C. M. Linton, J. Eng. Math. 33, 377 (1998).
[CrossRef]

Mayes, P.

R. Lampe, P. Klock, and P. Mayes, IEEE Trans. Microwave Theory Tech. 33, 734 (1985).
[CrossRef]

McPhedran, R. C.

N. A. Nicorovici and R. C. McPhedran, Phys. Rev. E 50, 3143 (1994).
[CrossRef]

Mitra, R.

R. E. Jorgenson and R. Mitra, IEEE Trans. Antennas Propag. 38, 633 (1990).
[CrossRef]

Moroz, A.

H. van der Lem and A. Moroz, J. Opt. A 2, 395 (2000).
[CrossRef]

A. Moroz and C. Sommers, J. Phys. Condens. Matter 11, 997 (1999).
[CrossRef]

Nicorovici, N. A.

N. A. Nicorovici and R. C. McPhedran, Phys. Rev. E 50, 3143 (1994).
[CrossRef]

Oksasoglu, A.

E. I. Veliev and A. Oksasoglu, J. Electromagn. Waves Appl. 10, 493 (1996).
[CrossRef]

Ozorio de Almeida, A. M.

A. M. Ozorio de Almeida, Acta Cryst. A 31, 435 (1975).
[CrossRef]

Richards, W. F.

S. Singh, W. F. Richards, J. R. Zinecker, and D. R. Watson, IEEE Trans. Antennas Propag. 38, 1958 (1991).
[CrossRef]

Segall, B.

F. S. Ham and B. Segall, Phys. Rev. B 124, 1786 (1961).
[CrossRef]

Singh, R.

S. Singh and R. Singh, IEEE Trans. Antennas Propag. 39, 1514 (1991).
[CrossRef]

S. Singh and R. Singh, IEEE Trans. Microwave Theory Tech. 38, 1746 (1990).
[CrossRef]

Singh, S.

S. Singh, W. F. Richards, J. R. Zinecker, and D. R. Watson, IEEE Trans. Antennas Propag. 38, 1958 (1991).
[CrossRef]

S. Singh and R. Singh, IEEE Trans. Antennas Propag. 39, 1514 (1991).
[CrossRef]

S. Singh and R. Singh, IEEE Trans. Microwave Theory Tech. 38, 1746 (1990).
[CrossRef]

Sommers, C.

A. Moroz and C. Sommers, J. Phys. Condens. Matter 11, 997 (1999).
[CrossRef]

van der Lem, H.

H. van der Lem and A. Moroz, J. Opt. A 2, 395 (2000).
[CrossRef]

Veliev, E. I.

E. I. Veliev and A. Oksasoglu, J. Electromagn. Waves Appl. 10, 493 (1996).
[CrossRef]

Watson, D. R.

S. Singh, W. F. Richards, J. R. Zinecker, and D. R. Watson, IEEE Trans. Antennas Propag. 38, 1958 (1991).
[CrossRef]

Yasumoto, K.

K. Yasumoto and K. Yoshitomi, IEEE Trans. Antennas Propag. 47, 1050 (1999).
[CrossRef]

Yoshitomi, K.

K. Yasumoto and K. Yoshitomi, IEEE Trans. Antennas Propag. 47, 1050 (1999).
[CrossRef]

Zinecker, J. R.

S. Singh, W. F. Richards, J. R. Zinecker, and D. R. Watson, IEEE Trans. Antennas Propag. 38, 1958 (1991).
[CrossRef]

Acta Cryst. A (1)

A. M. Ozorio de Almeida, Acta Cryst. A 31, 435 (1975).
[CrossRef]

Ann. Phys. (1)

P. P. Ewald, Ann. Phys. 64, 253 (1921).
[CrossRef]

Ann. Phys. (NY) (1)

M. V. Berry, Ann. Phys. (NY) 131, 163 (1981), Apps.  A and B.
[CrossRef]

Appl. Ocean Res. (1)

C. M. Linton, Appl. Ocean Res. 21, 17 (1999).
[CrossRef]

IEEE Trans. Antennas Propag. (4)

R. E. Jorgenson and R. Mitra, IEEE Trans. Antennas Propag. 38, 633 (1990).
[CrossRef]

S. Singh, W. F. Richards, J. R. Zinecker, and D. R. Watson, IEEE Trans. Antennas Propag. 38, 1958 (1991).
[CrossRef]

S. Singh and R. Singh, IEEE Trans. Antennas Propag. 39, 1514 (1991).
[CrossRef]

K. Yasumoto and K. Yoshitomi, IEEE Trans. Antennas Propag. 47, 1050 (1999).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

R. Lampe, P. Klock, and P. Mayes, IEEE Trans. Microwave Theory Tech. 33, 734 (1985).
[CrossRef]

S. Singh and R. Singh, IEEE Trans. Microwave Theory Tech. 38, 1746 (1990).
[CrossRef]

J. Electromagn. Waves Appl. (1)

E. I. Veliev and A. Oksasoglu, J. Electromagn. Waves Appl. 10, 493 (1996).
[CrossRef]

J. Eng. Math. (1)

C. M. Linton, J. Eng. Math. 33, 377 (1998).
[CrossRef]

J. Opt. A (1)

H. van der Lem and A. Moroz, J. Opt. A 2, 395 (2000).
[CrossRef]

J. Phys. Condens. Matter (1)

A. Moroz and C. Sommers, J. Phys. Condens. Matter 11, 997 (1999).
[CrossRef]

Phys. Rev. B (1)

F. S. Ham and B. Segall, Phys. Rev. B 124, 1786 (1961).
[CrossRef]

Phys. Rev. E (1)

N. A. Nicorovici and R. C. McPhedran, Phys. Rev. E 50, 3143 (1994).
[CrossRef]

Z. Naturforsch. Teil A (1)

K. Kambe, Z. Naturforsch. Teil A 22, 322 (1967).

Other (1)

M. Abramowitch and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1973).

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Figures (1)

Fig. 1
Fig. 1

Plane wave with wave vector k incident upon 1D grating Λ oriented along the x axis, with an incidence angle θ.

Equations (11)

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

Ks=σ2-k+ks21/2σ>k+ksik+ks2-σ21/2σ<k+ks,
G0 Λσ,k,R=rsΛG0σ,R+rsexp-ik·rs,
DΛσ,k,R=G0 Λσ,k,R-G0pσ,R,
DΛσ,k,R=l=-Dlσ,kJlσRYlR,
Dlσ,k=limR01JlσRYl*RDΛσ,k,RdΩR,
Dl=Dl1+Dl2+Dl3,
Dl1σ,k=-il+1l!2σv0ksΛ*n=0l/2122nn!×Γ1/2-n,exp-πiKs2η2×k+ks/σl-2nKs/σ2n-1l-2n!×exp-il-2nϕk+ksl0expil-2nϕk+ksl<0,
Dl2σ,k=--1l2|l|+12πrsΛexp-ik·rsσrsl×exp-ilϕrs0αu-l-1expu-σ2rs24udu,
Dl3σ=-122πγ+lnσ2η/2+n=1σ2η/2nn!nδl0,
Γ1/2,x=π-20xexp-t2dt=πerfcxargx=0π+2i0-xexpt2dtargx=-π.
σrs22Ul+1=lUl-Ul-1+α-l×expα-σ2rs2/4α,

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