Abstract

We establish the most general differential equations that are satisfied by the Fourier components of the electromagnetic field diffracted by an arbitrary periodic anisotropic medium. The equations are derived by use of the recently published fast-Fourier-factorization (FFF) method, which ensures fast convergence of the Fourier series of the field. The diffraction by classic isotropic gratings arises as a particular case of the derived equations; the case of anisotropic classic gratings was published elsewhere. The equations can be resolved either through classic differential theory or through the modal method for particular groove profiles. The new equations improve both methods in the same way. Crossed gratings, among which are grids and two-dimensional arbitrarily shaped periodic surfaces, appear as particular cases of the theory, as do three-dimensional photonic crystals. The method can be extended to nonperiodic media through the use of a Fourier transform.

© 2001 Optical Society of America

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References

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  1. M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
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    [CrossRef]
  4. G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
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    [CrossRef]
  6. K. Watanabe, R. Petit, M. Nevière, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A (to be published).
  7. P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Springer Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 101–121.
  8. L. Li, “Formulation and comparison of two recursive matrix algorithms for modelling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  9. F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1988).
    [CrossRef]
  10. L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
    [CrossRef]
  11. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  12. E. Glytsis, T. Gaylord, “Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction,” J. Opt. Soc. Am. A 7, 1399–1419 (1990).
    [CrossRef]
  13. B. Chernov, M. Nevière, E. Popov, “Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings,” submitted to Opt. Commun.
  14. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
    [CrossRef]
  15. E. Popov, M. Nevière, S. Enoch, R. Reinisch, “Theory of light transmission through sub-wavelength periodic hole arrays,” Phys. Rev. B 63, 16,000–16,008 (2000).
  16. S. Enoch, E. Popov, M. Nevière, “Electromagnetic theory of 3-dimensional photonic crystals,” Proc. SPIE (to be published).
  17. D. Maystre, “Integral Method,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 3.

2000 (2)

E. Popov, M. Nevière, S. Enoch, R. Reinisch, “Theory of light transmission through sub-wavelength periodic hole arrays,” Phys. Rev. B 63, 16,000–16,008 (2000).

E. Popov, M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
[CrossRef]

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

1996 (4)

1993 (1)

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

1990 (1)

1988 (1)

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1988).
[CrossRef]

1982 (1)

1974 (1)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

Chernov, B.

B. Chernov, M. Nevière, E. Popov, “Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings,” submitted to Opt. Commun.

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Enoch, S.

E. Popov, M. Nevière, S. Enoch, R. Reinisch, “Theory of light transmission through sub-wavelength periodic hole arrays,” Phys. Rev. B 63, 16,000–16,008 (2000).

S. Enoch, E. Popov, M. Nevière, “Electromagnetic theory of 3-dimensional photonic crystals,” Proc. SPIE (to be published).

Gaylord, T.

Gaylord, T. K.

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Glytsis, E.

Granet, G.

Guizal, B.

Lalanne, P.

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Li, L.

Maystre, D.

D. Maystre, “Integral Method,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 3.

Moharam, M. G.

Montiel, F.

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1988).
[CrossRef]

Morris, G. M.

Nevière, M.

E. Popov, M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
[CrossRef]

E. Popov, M. Nevière, S. Enoch, R. Reinisch, “Theory of light transmission through sub-wavelength periodic hole arrays,” Phys. Rev. B 63, 16,000–16,008 (2000).

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1988).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

B. Chernov, M. Nevière, E. Popov, “Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings,” submitted to Opt. Commun.

K. Watanabe, R. Petit, M. Nevière, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A (to be published).

S. Enoch, E. Popov, M. Nevière, “Electromagnetic theory of 3-dimensional photonic crystals,” Proc. SPIE (to be published).

Petit, R.

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

K. Watanabe, R. Petit, M. Nevière, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A (to be published).

Peyrot, P.

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1988).
[CrossRef]

Popov, E.

E. Popov, M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
[CrossRef]

E. Popov, M. Nevière, S. Enoch, R. Reinisch, “Theory of light transmission through sub-wavelength periodic hole arrays,” Phys. Rev. B 63, 16,000–16,008 (2000).

B. Chernov, M. Nevière, E. Popov, “Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings,” submitted to Opt. Commun.

S. Enoch, E. Popov, M. Nevière, “Electromagnetic theory of 3-dimensional photonic crystals,” Proc. SPIE (to be published).

Reinisch, R.

E. Popov, M. Nevière, S. Enoch, R. Reinisch, “Theory of light transmission through sub-wavelength periodic hole arrays,” Phys. Rev. B 63, 16,000–16,008 (2000).

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Vincent, P.

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Springer Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 101–121.

Watanabe, K.

K. Watanabe, R. Petit, M. Nevière, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A (to be published).

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

J. Mod. Opt. (2)

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1988).
[CrossRef]

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Nature (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Nouv. Rev. Opt. (1)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

Phys. Rev. B (1)

E. Popov, M. Nevière, S. Enoch, R. Reinisch, “Theory of light transmission through sub-wavelength periodic hole arrays,” Phys. Rev. B 63, 16,000–16,008 (2000).

Other (5)

S. Enoch, E. Popov, M. Nevière, “Electromagnetic theory of 3-dimensional photonic crystals,” Proc. SPIE (to be published).

D. Maystre, “Integral Method,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 3.

K. Watanabe, R. Petit, M. Nevière, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A (to be published).

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Springer Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 101–121.

B. Chernov, M. Nevière, E. Popov, “Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings,” submitted to Opt. Commun.

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Figures (1)

Fig. 1
Fig. 1

Convergence rates for two rules of factorization, Eq. (32), represented by squares, and Eq. (33), represented by circles. The -1st diffracted order efficiency as a function of truncation parameter N for a gold grating (refractive index, 0.2+i6.71) consisting of lamellae slanted at 45° with filling ratio 0.5, period 1 µm, and height 0.5 µm, is shown as used in TM polarization at 0.8-µm wavelength and 30° incidence. The rigorous value was calculated with the integral formalism found in Ref. 17.

Equations (84)

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hn=m=-+ fn-mgm.
hn(N)=m=-N+N fn-mgm.
[h]=f[g].
[h]=1/f-1[g].
curl E=iωB,
curl H=-iωD
ET1=T1E,
ET2=T2E.
DN=ND=N¯¯E.
ETi=TixEx+TiyEy+TizEz,i=1, 2,
DN=Nx(xxEx+xyEy+xzEz)+Ny(yxEx+yyEy+yzEz)+Nz(zxEx+zyEy+zzEz).
F=AEAE,
A=T1xT1yT1zNxxx+Nyyx+NzzxNxxy+Nyyy+NzzyNxxz+Nyyz+NzzzT2xT2yT2z.
A=T1xT1yT1z(N¯¯)x(N¯¯)y(N¯¯)zT2xT2yT2z.
[D]=¯¯CC-1[E].
C=1ξ [(N¯¯)×T2]xNx-[(N¯¯)×T1]x[(N¯¯)×T2]yNy-[(N¯¯)×T1]y[(N¯¯)×T2]zNz-[(N¯¯)×T1]z,
[D]=Q[E],
[B]=Qμ[H],
Ezy-Eyz=iωBx,
Exz-Ezx=iωBy,
Eyx-Exy=iωBz,
Hzy-Hyz=-iωDx,
Hxz-Hzx=-iωDy,
Hyx-Hxy=-iωDz.
[Ez]y-[Ey]z=iω[Bx],
[Ex]z-[Ez]x=iω[By],
Eyx-[Ex]y=iω[Bz],
[Hz]y-Hyz=-iω[Dx],
[Hx]z-[Hz]x=-iω[Dy],
Hyx-[Hx]y=-iω[Dz],
[Dy]=Q,yx[Ex]+Q,yy[Ey]+Q,yz[Ez],
[Ey]=Q,yy-1([Dy]-Q,yx[Ex]-Q,yz[Ez]).
[Ey]=Q,yy-1iω z[Hx]-x[Hz]-Q,yx[Ex]-Q,yz[Ez].
[Hy]=Qμ,yy-1-iω z[Ex]-x[Ez]-Qμ,yx[Hx]-Qμ,yz[Hz].
ddy [Ex][Ez][Hx][Hz]=M[Ex][Ez][Hx][Hz],
M11=-iαxQ,yy-1Q,yx-iQμ,zyQμ,yy-1αz,
M12=-iαxQ,yy-1Q,yz+iQμ,zyQμ,yy-1αx,
M13=iωQμ,zyQμ,yy-1Qμ,yx-i αxωQe,yy-1αz-iωQμ,zx,
M14=iωQμ,zyQμ,yy-1Qμ,yz+i αxωQ,yy-1αx-iωQμ,zz,
M21=-iαzQ,yy-1Q,yx+iQμ,xyQμ,yy-1αz,
M22=-iαzQ,yy-1Q,yz-iQμ,xyQμ,yy-1αx,
M23=-iωQμ,xyQμ,yy-1Qμ,yx-i αzωQ,yy-1αz+iωQμ,xx,
M24=-iωQμ,xyQμ,yy-1Qμ,yz+i αzωQ,yy-1αx+iωQμ,xz,
M31=-iωQ,zyQ,yy-1Q,yx+i αxωQμ,yy-1αz+iωQ,zx,
M32=-iωQ,zyQ,yy-1Q,yz-i αxωQμ,yy-1αx+iωQ,zz,
M33=-iαxQμ,yy-1Qμ,yx-iQ,zyQ,yy-1αz,
M34=-iαxQμ,yy-1Qμ,yz+iQ,zyQ,yy-1αx,
M41=iωQ,xyQ,yy-1Q,yx+i αzωQμ,yy-1αz-iωQ,xx,
M42=iωQ,xyQ,yy-1Q,yz-i αzωQμ,yy-1αx-iωQ,xz,
M43=-iαzQμ,yy-1Qμ,yx+iQ,xyQ,yy-1αz,
M44=-iαzQμ,yy-1Qμ,yz-iQ,xyQ,yy-1αx.
C=NyNx/0-NxNy/0001,C=NyNx0-NxNy000.
C=Ny1/Nx0-Nx1/Ny0001I,
C=NyNx0-NxNy000.
C-1=Ny-Nx01/-1Nx1/-1Ny0001I,
Q=CC-1=Ny2+1/-1Nx2-(-1/-1)NxNy0-(-1/-1)NxNyNx2+1/-1Ny2000.
M=-iαxQ,yy-1Q,yx00-iωμ0+iαx2ωQ,yy-100iωμ000iωQ,zz-iαx2ωμ000-iω(Q,xx-Q,xyQ,yy-1Q,yx)00-iαxQ,xyQ,yy-1,
d2[Ez]dy2=iωμ0iω[Ez]-iαx2ωμ0[Ez]=-μ0ω2[Ez]+αx2[Ez].
d[Hz]dy=-iω{Q,xx[Ex]+Q,xy[Ey]},
[Ey]=Q,yy-1αxω[Hz]-Q,yx[Ex].
d[Ex]dy=-iωμ0[Hz]+iαx[Ey].
Q=-(-1/-1)Nx2-(-1/-1)NxNy-(-1/-1)NxNz-(-1/-1)NxNy-(-1/-1)Ny2-(-1/-1)NyNz-(-1/-1)NxNz-(-1/-1)NyNz-(-1/-1)Nz2,
y=(H/2)sin(Kxx)sin(Kzz),
f(x, y, z)(H/2)sin(Kxx)sin(Kzz)-y=0.
N=1|grad f| HKx2 cos(Kxx)sin(Kzz),-1,HKz2 sin(Kxx)cos(Kzz),
grad f|=1+HKx22 cos2(Kxx)sin2(Kzz)+HKz22 sin2(Kxx)cos2(Kzz)1/2.
[D]=¯¯CC-1[E],
[D]=¯¯[E].
(fdfc)n=m=-+ fd,n-mfc,m,
[fdfc]=fd[fc]=fc[fd].
[f1f2]=F1(f1)F2(f2),
[f1f2]=F1(f1)F2(f2).
[f1 exp(inKx)]m=[f1]m-n.
[f1 exp(inKx)]m=[F(f1)In]m=[F1(f1)]m,n.
F1(f1)=f1.
[fdfc]=fd[fc],
[f2]=1/f1[fc],
[fc]=1/f1-1[f2]or[f1f2]=1/f1-1[f2].
[D]=[ET]+1/-1[EN]=[E-N(NE)]+1/-1[N(NE)].
[D]=[E]-(-1/-1)NN[E],
-1/-1,
Q=-(-1/-1)NN.
(Q)mn,ij=mnδij-(-1/-1)mnNiNj,i,j=x, y, z,m, n=-N:N,
Q,xx=(C)xx(C-1)xx+(C)xy(C-1)yx+(C)xz(C-1)zx=NyNy+Nx(1/)-1Nx=Ny2+(1/)-1Nx2=Ny2+1/-1Nx2,

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