Abstract

We reformulate the coordinate transformation method for profiles represented by parametric equations. Numerically, the eigenvalue problem is solved by expanding both the electromagnetic field and the periodic coefficients of Maxwell’s equations into Fourier series. For trapezoidal gratings, it is shown that the convergence of the method is closely related to the representation of the profile. A proper choice of the representation permits handling profiles that previously had been out of reach owing to their sharp edges. From a practical point of view, we are now able to analyze gratings with two vertical facets by using the coordinate transformation method.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
    [CrossRef]
  2. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
    [CrossRef]
  3. Ph. 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]
  4. 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]
  5. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  6. L. Li, J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247–2255 (1996).
    [CrossRef]
  7. L. Li, “Oblique-coordinate-system based Chandezon method for modeling one-dimensionally periodic, multilayer, inhomogeneous, anisotropic gratings,” J. Opt. Soc. Am. A 16, 2521–2531 (1999).
    [CrossRef]
  8. G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16, 2510–2516 (1999).
    [CrossRef]
  9. L. Li, J. Chandezon, G. Granet, J. P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304–313 (1999).
    [CrossRef]
  10. G. Granet, J. Chandezon, “The method of curvilinear coordinates applied to the problem of scattering from surface-relief gratings defined by parametric equations: application to scattering from a cycloidal grating,” Pure Appl. Opt. 6, 727–740 (1997).
    [CrossRef]
  11. J. P. Plumey, G. Granet, “Generalization of the coordinate transformation method with application to surface-relief gratings,” J. Opt. Soc. Am. A 16, 508–516 (1999).
    [CrossRef]

1999 (4)

1997 (1)

G. Granet, J. Chandezon, “The method of curvilinear coordinates applied to the problem of scattering from surface-relief gratings defined by parametric equations: application to scattering from a cycloidal grating,” Pure Appl. Opt. 6, 727–740 (1997).
[CrossRef]

1996 (4)

1982 (1)

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
[CrossRef]

1980 (1)

J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

Chandezon, J.

L. Li, J. Chandezon, G. Granet, J. P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304–313 (1999).
[CrossRef]

G. Granet, J. Chandezon, “The method of curvilinear coordinates applied to the problem of scattering from surface-relief gratings defined by parametric equations: application to scattering from a cycloidal grating,” Pure Appl. Opt. 6, 727–740 (1997).
[CrossRef]

L. Li, J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247–2255 (1996).
[CrossRef]

J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

Gaylord, T. K.

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
[CrossRef]

Granet, G.

Guizal, B.

Lalanne, Ph.

Li, L.

Maystre, D.

J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

Moharam, M. G.

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
[CrossRef]

Morris, G. M.

Plumey, J. P.

Raoult, G.

J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

Appl. Opt. (1)

J. Opt. (Paris) (1)

J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

J. Opt. Soc. Am. A (8)

Pure Appl. Opt. (1)

G. Granet, J. Chandezon, “The method of curvilinear coordinates applied to the problem of scattering from surface-relief gratings defined by parametric equations: application to scattering from a cycloidal grating,” Pure Appl. Opt. 6, 727–740 (1997).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Grating profile function y=p(x) for the following trapezoidal grating: h=d=2, d1=0.25, d2=1, and d3=1.25 (arbitrary units).

Fig. 2
Fig. 2

Three sets of parametric representations x(u) and y(u) of the trapezoidal profile of Fig. 1: (a) η1=0.25, η2=0.5, η3=0.75, g=0; (b) η1=0.25, η2=0.5, η3=0.75, g=0.99; (c) η1=0.36, η2=0.5, η3=0.86, g=0.

Fig. 3
Fig. 3

Convergence of the -3 transmitted order in TM polarization. The grating has the following parameters: h=d=2λ, d1=0.25λ, d2=1λ, d3=1.25λ, λ=1, θ=45°, and n=1.5. g=0 for all calculations. Curve A, classical formulation, Eq. (1). Curve B, η1=0.25, η2=0.5, η3=0.75. Curve C, η1=L1/L, η2-η1=L2/L, η3-η2=L3/L, and L=L1+L2+L3+L4, where L1, L2, L3, and L4, are the lengths of the first, second, third, and fourth facets, respectively.

Fig. 4
Fig. 4

Convergence of the specularly reflected order for a conducting grating in TM polarization. All parameters, except for optical index n=0.3-i7, are the same as in Fig. 3.

Fig. 5
Fig. 5

Convergence of the specularly reflected order for a conducting grating in TM polarization. The parameters are as follows: h=d=2λ, d1=0.001λ, d3=1.25λ, d2=d3-d1, λ=1, 0=45°, and n=0.3-i7. η1, η2, and η3 are such that η1=L1/L, η2-η1=L2/L, η3-η2=L3/L, and L=L1+L2+L3+L4, where L1, L2, L3, and L4 are the lengths of the first, second, third, and fourth facets, respectively. Curve A, g=0.2. Curve B, g=0.5. Curve C, g=0.95.

Tables (2)

Tables Icon

Table 1 Efficiencies of a Dielectric Trapezoidal Grating with Sharp Edges for Various Values of d1 and Efficiencies of the Corresponding Lamellar Grating a

Tables Icon

Table 2 Efficiencies of a Conducting Trapezoidal Grating with Sharp Edges for Various Values of d1 and Efficiencies of the Corresponding Lamellar Grating a

Equations (43)

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

x=u,y=v+p(x)=v+p(u),z=w.
y=v,
x=u-dpduv=u-p˙ v.
(1+p˙p˙) 2v2-vu p˙-vp˙ u+2u2+k2n2Ψ=0,
2u2+k2n2I00I ΨΨ=u p˙+p˙ uik(I+p˙p˙)ik I0vΨΨ,
Ψ(u, v)=exp(-ikrv)ψ(u),
Ψ(u, v)=exp(-ikrv)ψ(u).
uu+k2n2I00I ψψ
=r-iku p˙+p˙ uk2(I+p˙p˙)I0 ψψ.
LAψqψq=1rqLBψqψq,
LA=-ap˙-p˙aI+p˙p˙I0,
LB=-aa+n2I00I,
p˙mn=p˙m-n=1d0ddpduexp[i2π(m-n) ud]du.
Ψ(u, v)=q sqexp(-ikrqv)m ψmqexp(-ikamu).
x=f(u),y=v+p[f(u)].
Z0Hx=ikEzy,
Z0Hy=-ikEzx,
n2Ez=ik(Z0Hx)y-(Z0Hy)x,
y=v,
x=uxu+vxv=1f˙u-1f˙dpduv,
f˙=dxdu.
f˙ u=u-dpduv=u-p˙ v.
Z0f˙Hy=-ikf˙ xEz,
n2f˙Ez=ik(Z0f˙Hx)y-f˙ x(Z0Hy).
n2f˙Ez=ikyik f˙ y+f˙ xik1f˙f˙ xEz.
f˙+p˙ 1f˙ p˙2v2-vu1f˙ p˙-vp˙ 1f˙u
+u1f˙u+k2n2f˙Ez=0.
LA=-af˙-1p˙-p˙f˙-1af˙+p˙f˙-1p˙I0,
LB=-af˙-1a+n2f˙00I,
f˙mn=f˙m-n=1d0df˙(u)expi2π(m-n) uddu.
Htan=H  t,
t=1f˙2+p˙2 (f˙ee+p˙ey).
ϕ=-f˙2+p˙2E  t,TM polarizationZ0f˙2+p˙2H  t,TE polarization.
ϕ=Z0f˙Hx+Z0p˙Hy.
ϕ=ikf˙ Ezy-p˙ 1f˙f˙ xEz.
ϕ=ikf˙ v-p˙ 1f˙u-p˙ vΨ=f˙+p˙ 1f˙ p˙Ψ-ik p˙ 1f˙u Ψ,
-af˙-1p˙-p˙f˙-1af˙+p˙f˙-1p˙I0 ψqψq
=1rq-af˙-1a+n2f˙00I ψqψq.
(-p˙f˙-1a)ψq+(f˙+p˙f˙-1p˙)ψq
=1rq (n2f˙-af˙-1a)ψq+af˙-1p˙ψq.
ϕq=1rq (n2f˙-1-af˙-1a)+af˙-1p˙ψq.
x=d1η1u-g η12πsin2πuη1if0<u<η1d1+d2-d1η2-η1(u-η1)-g η2-η12πsin2π(u-η1)η2-η1ifη1<u<η2d2+d3-d2η3-η2(u-η2)-g η3-η22πsin2π(u-η2)η3-η2ifη2<u<η3d3+d-d31-η3(u-η3)-g 1-η32πsin2π(u-η3)1-η3ifη3>u1,
y=hη1u-g η12πsin2πuη1if0<u<η1hifη1<u<η2h-hη3-η2(u-η2)-g η3-η22πsin2π(u-η2)η3-η2ifη2<u<η30ifη3<u1,

Metrics