Abstract

A systematic approach based on group theory is established to deal with diffraction problems of crossed gratings by exploiting symmetries. With this approach, a problem in an asymmetrical incident mounting can be decomposed into a superposition of several symmetrical basis problems so that the computation efficiency is improved effectively. This methodology offers a convenient and succinct way to treat all possible symmetry cases by following only several mechanical steps instead of intricate mathematical considerations or physical intuition. It is also general, applicable to both scalar-wave and vector-wave problems and in principle can be easily adapted to any numerical method. A numerical example is presented to show its application and effectiveness.

© 2004 Optical Society of America

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  1. P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
    [CrossRef]
  2. D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
    [CrossRef]
  3. S. T. Han, Y. L. Tsao, R. M. Walser, M. F. Becker, “Electromagnetic scattering of two-dimensional surface-relief dielectric gratings,” Appl. Opt. 31, 2343–2352 (1992).
    [CrossRef] [PubMed]
  4. D. C. Dobson, J. A. Cox, “An integral equation method for biperiodic diffraction structures,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, M. R. McKinney, eds., Proc. SPIE1545, 106–113 (1991).
    [CrossRef]
  5. G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
    [CrossRef]
  6. R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
    [CrossRef]
  7. J. B. Harris, T. W. Preist, J. R. Sambles, R. N. Thorpe, R. A. Watts, “Optical response of bigratings,” J. Opt. Soc. Am. A 13, 2041–2049 (1996).
    [CrossRef]
  8. G. Granet, “Analysis of diffraction by surface-relief crossed gratings with use of the Chandezon method: application to multilayer crossed gratings,” J. Opt. Soc. Am. A 15, 1121–1131 (1998).
    [CrossRef]
  9. R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
    [CrossRef]
  10. E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
    [CrossRef]
  11. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [CrossRef]
  12. J. J. Greffet, C. Baylard, P. Versaevel, “Diffraction of electromagnetic waves by crossed gratings: a series solution,” Opt. Lett. 17, 1740–1742 (1992).
    [CrossRef] [PubMed]
  13. V. Bagnoud, S. Mainguy, “Diffraction of electromagnetic waves by dielectric crossed gratings: a three-dimensional Rayleigh–Fourier solution,” J. Opt. Soc. Am. A 16, 1277–1285 (1999).
    [CrossRef]
  14. O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
    [CrossRef]
  15. O. P. Bruno, F. Reitich, “Calculation of electromagnetic scattering via boundary variations and analytic continuation,” Appl. Comput. Electromagn. Soc. J. 11, 17–31 (1996).
  16. P. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  17. G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
  18. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  19. G. Granet, J. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A Pure Appl. Opt. 4, S145–S149 (2002).
    [CrossRef]
  20. L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A Pure Appl. Opt. 5, 345–355 (2003).
    [CrossRef]
  21. P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592–1598 (1997).
    [CrossRef]
  22. P. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
    [CrossRef]
  23. C. Zhou, L. Li, “Formulation of Fourier modal method of symmetric crossed gratings in symmetric mountings,” J. Opt. A Pure Appl. Opt. 6, 43–50 (2004).
    [CrossRef]
  24. B. Y. Kinber, A. B. Kotlyar, “Use of symmetry in solving diffraction problems,” Radio Eng. Electron. Phys. 16, 581–587 (1971).
  25. W. Ludwig, C. Falter, Symmetries in Physics: Group Theory Applied to Physical Problems (Springer, Berlin, 1988).
  26. J. V. Smith, Geometrical and Structural Crystallography (Wiley, New York, 1982).
  27. J. F. Cornwell, ed., “Appendix C: Character tables for the crystallographic point groups,” in Group Theory in Physics: an Introduction (Academic, San Diego, Calif., 1997), pp. 299–318.
  28. T. Hahn, ed., International Tables for Crystallography, Vol. A (Reidel, Dordrecht, The Netherlands, 1983), pp. 92–109.

2004 (1)

C. Zhou, L. Li, “Formulation of Fourier modal method of symmetric crossed gratings in symmetric mountings,” J. Opt. A Pure Appl. Opt. 6, 43–50 (2004).
[CrossRef]

2003 (1)

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A Pure Appl. Opt. 5, 345–355 (2003).
[CrossRef]

2002 (1)

G. Granet, J. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A Pure Appl. Opt. 4, S145–S149 (2002).
[CrossRef]

1999 (1)

1998 (1)

1997 (2)

1996 (6)

1994 (1)

1993 (2)

1992 (2)

1982 (1)

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

1979 (1)

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

1978 (2)

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

1971 (1)

B. Y. Kinber, A. B. Kotlyar, “Use of symmetry in solving diffraction problems,” Radio Eng. Electron. Phys. 16, 581–587 (1971).

Bagnoud, V.

Baylard, C.

Becker, M. F.

Bräuer, R.

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Bruno, O. P.

O. P. Bruno, F. Reitich, “Calculation of electromagnetic scattering via boundary variations and analytic continuation,” Appl. Comput. Electromagn. Soc. J. 11, 17–31 (1996).

O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
[CrossRef]

Bryngdahl, O.

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Cox, J. A.

D. C. Dobson, J. A. Cox, “An integral equation method for biperiodic diffraction structures,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, M. R. McKinney, eds., Proc. SPIE1545, 106–113 (1991).
[CrossRef]

Derrick, G. H.

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Dobson, D. C.

D. C. Dobson, J. A. Cox, “An integral equation method for biperiodic diffraction structures,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, M. R. McKinney, eds., Proc. SPIE1545, 106–113 (1991).
[CrossRef]

Falter, C.

W. Ludwig, C. Falter, Symmetries in Physics: Group Theory Applied to Physical Problems (Springer, Berlin, 1988).

Granet, G.

Greffet, J. J.

Guizal, B.

Han, S. T.

Harris, J. B.

Kinber, B. Y.

B. Y. Kinber, A. B. Kotlyar, “Use of symmetry in solving diffraction problems,” Radio Eng. Electron. Phys. 16, 581–587 (1971).

Kotlyar, A. B.

B. Y. Kinber, A. B. Kotlyar, “Use of symmetry in solving diffraction problems,” Radio Eng. Electron. Phys. 16, 581–587 (1971).

Lalanne, P.

Lemercier-Lalanne, D.

P. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

Li, L.

C. Zhou, L. Li, “Formulation of Fourier modal method of symmetric crossed gratings in symmetric mountings,” J. Opt. A Pure Appl. Opt. 6, 43–50 (2004).
[CrossRef]

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A Pure Appl. Opt. 5, 345–355 (2003).
[CrossRef]

L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
[CrossRef]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

Ludwig, W.

W. Ludwig, C. Falter, Symmetries in Physics: Group Theory Applied to Physical Problems (Springer, Berlin, 1988).

Mainguy, S.

Maystre, D.

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

McPhedran, R. C.

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Morris, G. M.

Nevière, M.

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

Noponen, E.

Plumey, J.

G. Granet, J. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A Pure Appl. Opt. 4, S145–S149 (2002).
[CrossRef]

Preist, T. W.

Reitich, F.

O. P. Bruno, F. Reitich, “Calculation of electromagnetic scattering via boundary variations and analytic continuation,” Appl. Comput. Electromagn. Soc. J. 11, 17–31 (1996).

O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
[CrossRef]

Sambles, J. R.

Smith, J. V.

J. V. Smith, Geometrical and Structural Crystallography (Wiley, New York, 1982).

Thorpe, R. N.

Tsao, Y. L.

Turunen, J.

Versaevel, P.

Vincent, P.

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Walser, R. M.

Watts, R. A.

Zhou, C.

C. Zhou, L. Li, “Formulation of Fourier modal method of symmetric crossed gratings in symmetric mountings,” J. Opt. A Pure Appl. Opt. 6, 43–50 (2004).
[CrossRef]

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

O. P. Bruno, F. Reitich, “Calculation of electromagnetic scattering via boundary variations and analytic continuation,” Appl. Comput. Electromagn. Soc. J. 11, 17–31 (1996).

Appl. Opt. (1)

Appl. Phys. (1)

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

J. Mod. Opt. (1)

P. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

J. Opt. (Paris) (2)

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

J. Opt. A Pure Appl. Opt. (3)

C. Zhou, L. Li, “Formulation of Fourier modal method of symmetric crossed gratings in symmetric mountings,” J. Opt. A Pure Appl. Opt. 6, 43–50 (2004).
[CrossRef]

G. Granet, J. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A Pure Appl. Opt. 4, S145–S149 (2002).
[CrossRef]

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A Pure Appl. Opt. 5, 345–355 (2003).
[CrossRef]

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

P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592–1598 (1997).
[CrossRef]

J. B. Harris, T. W. Preist, J. R. Sambles, R. N. Thorpe, R. A. Watts, “Optical response of bigratings,” J. Opt. Soc. Am. A 13, 2041–2049 (1996).
[CrossRef]

G. Granet, “Analysis of diffraction by surface-relief crossed gratings with use of the Chandezon method: application to multilayer crossed gratings,” J. Opt. Soc. Am. A 15, 1121–1131 (1998).
[CrossRef]

P. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[CrossRef]

G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
[CrossRef]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
[CrossRef]

L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
[CrossRef]

V. Bagnoud, S. Mainguy, “Diffraction of electromagnetic waves by dielectric crossed gratings: a three-dimensional Rayleigh–Fourier solution,” J. Opt. Soc. Am. A 16, 1277–1285 (1999).
[CrossRef]

O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
[CrossRef]

Opt. Commun. (2)

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Opt. Lett. (1)

Radio Eng. Electron. Phys. (1)

B. Y. Kinber, A. B. Kotlyar, “Use of symmetry in solving diffraction problems,” Radio Eng. Electron. Phys. 16, 581–587 (1971).

Other (5)

W. Ludwig, C. Falter, Symmetries in Physics: Group Theory Applied to Physical Problems (Springer, Berlin, 1988).

J. V. Smith, Geometrical and Structural Crystallography (Wiley, New York, 1982).

J. F. Cornwell, ed., “Appendix C: Character tables for the crystallographic point groups,” in Group Theory in Physics: an Introduction (Academic, San Diego, Calif., 1997), pp. 299–318.

T. Hahn, ed., International Tables for Crystallography, Vol. A (Reidel, Dordrecht, The Netherlands, 1983), pp. 92–109.

D. C. Dobson, J. A. Cox, “An integral equation method for biperiodic diffraction structures,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, M. R. McKinney, eds., Proc. SPIE1545, 106–113 (1991).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

One-dimensional rectangular grating under plane-wave incidence. S1 and S2 indicate the directions of two incident plane waves in the xy plane, which are mirror symmetrical across the y axis. The numbers near the arrows indicate amplitudes of the plane waves.

Fig. 2
Fig. 2

An equilateral triangle with C3v symmetry. The solid triangle indicates a threefold rotation axis, and σb, σd, and σf indicate the mirror planes containing the rotation axis. The solid and open circles represent the symmetrical points on the triangle, where solid and open forms are used to emphasize the mirror-symmetrical relation.

Fig. 3
Fig. 3

Five kinds of plane lattices for plane periodic patterns: (a) monoclinic (mp), (b) tetragonal (tp), (c) hexagonal (hp), (d) orthorhombic (op), (e) orthorhombic centered (oc).

Fig. 4
Fig. 4

Schematic illustration of a general crossed-grating problem. The grating is illuminated by a linearly polarized plane wave. Two Cartesian coordinate systems are attached to the grating plane, i.e., the rectangular system Oxyz and the oblique system Ox1x2x3. The first axis x(x1) is along one of the periodic directions, and the third axis z(x3) coincides with the normal of the grating.

Fig. 5
Fig. 5

Transformation of the position vector r and the field vector ui(r) under the symmetry operation c4.

Fig. 6
Fig. 6

Top view of the crossed grating with C2v symmetry. Two unit cells are labeled A and B.

Fig. 7
Fig. 7

Schematic illustration of the four symmetry modes in a grating problem with C2v symmetry. In each mode, the symmetrical distribution of the electric vectors projected onto the xy plane is shown by the arrows. The solid (open) arrows indicate the vectors with the same Ez components, which have the opposite sign to those indicated by the open (solid) arrows.

Fig. 8
Fig. 8

Convergence of the (0, -2)th and (-1, -1)th transmitted orders of the grating. The truncation order refers to the number of retained Fourier items N in one periodic direction. Thus the total number of retained orders in the Fourier space is N2.

Fig. 9
Fig. 9

Comparison of the computation times of the new and old algorithms.

Tables (3)

Tables Icon

Table 1 Ten Plane Point Groups and Their Elements a

Tables Icon

Table 2 Correspondence of 5 Plane Lattices, 10 Plane Point Groups and 17 Plane Groups a

Tables Icon

Table 3 Character Table of Point Group C2v a

Equations (25)

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

a*=a++a-,
u*(r)=u+(r)+u-(r).
a*=i=1Ntiai.
ai=j=1NTji(gn)aj.
[a1 a2aN]=[a1 a2aN]T(gn),
ui(r)=Mnui(r)=Mnui(Mn-1r),
ui(r)=Mnui(Mn-1r).
ui(r)=j=1NTji(gn)uj(r).
Mnui(Mn-1r)=j=1NTji(gn)uj(r),
Tn=Tn(1)Tn(2)Tn(s)j=1s  Tn(j),
anj=niNTp,q(i)¯(gn)(n=1,2,,N),
j=p+(q-1)ni+s=1i-1ns2.
Tn=i=1r  niTn(i).
tj=aniNTp,q(i)¯(e)=ani/N(whenp=q)0(whenpq),
u*(r)=i=1Ntiui(r).
a1=12 (1, 1, 1, 1),a2=12 (1, 1, -1, -1),a3=12 (1, -1, -1, 1),a4=12 (1, -1, 1, -1).
T(e)=I,T(c2)=1  1  -1  -1,
T(σx)=1  -1  -1  1,
T(σy)=1  -1  1  -1.
M(e)=I,M(c2)=-1  -1  1,
M(σx)=1  -1  1,
M(σy)=-1  1  1.
Ex1(x, y, z)=-Ex1(-x, -y, z)=Ex1(x, -y, z)=-Ex1(-x, y, z)Ey1(x, y, z)=-Ey1(-x, -y, z)=-Ey1(x, -y, z)=Ey1(-x, y, z)Ez1(x, y, z)=Ez1(-x, -y, z)=Ez1(x, -y, z)=Ez1(-x, y, z).
E*(r)=i=14tiEi(r),
ti=(a*, ai)=a/2(i=1, 2, 3, 4).

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