Abstract

The differential formalism of Chandezon was extended to nonhomogeneous media. In contrast to the case of homogeneous media, we were led to a different eigenvalue problem for each polarization. As an illustration, we analyzed diffraction by a sinusoidal piecewise-homogeneous modulated layer and by an inclined lamellar grating. Results were validated by comparison with the rigorous coupled-wave analysis.

© 1997 Optical Society of America

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References

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  1. J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
    [CrossRef]
  2. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  3. E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962).
  4. S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [CrossRef]
  5. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 66, 1206–1210 (1978).
    [CrossRef]
  6. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  7. N. Chateau, J. P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321–1331 (1994).
    [CrossRef]
  8. 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]
  9. 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]
  10. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  11. N. P. K. Cotter, T. W. Preist, J. R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
    [CrossRef]
  12. G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
    [CrossRef]
  13. J. P. Plumey, B. Guizal, J. Chandezon, “The coordinate transformation method as applied to asymmetric gratings with vertical facets,” J. Opt. Soc. Am. A 14, 610–617 (1997).
    [CrossRef]
  14. R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
    [CrossRef]

1997 (1)

1996 (3)

1995 (3)

1994 (1)

1993 (1)

1982 (2)

1978 (1)

K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 66, 1206–1210 (1978).
[CrossRef]

1975 (1)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Bertoni, H. L.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Chandezon, J.

Chateau, N.

Cornet, G.

Cotter, N. P. K.

Dupuis, M. T.

Gaylord, T. K.

Granet, G.

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]

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

Guizal, B.

Hugonin, J. P.

Knop, K.

K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 66, 1206–1210 (1978).
[CrossRef]

Lalanne, P.

Li, L.

Maystre, D.

Moharam, M. G.

Morf, R. H.

Morris, G. M.

Peng, S. T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Plumey, J. P.

J. P. Plumey, B. Guizal, J. Chandezon, “The coordinate transformation method as applied to asymmetric gratings with vertical facets,” J. Opt. Soc. Am. A 14, 610–617 (1997).
[CrossRef]

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

Post, E. J.

E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962).

Preist, T. W.

Sambles, J. R.

Tamir, T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Pure Appl. Opt. (1)

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

Other (1)

E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962).

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Figures (2)

Fig. 1
Fig. 1

Modulated layer with parallel faces and piecewise-homogeneous medium.

Fig. 2
Fig. 2

Inclined lamellar grating.

Tables (3)

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Table 1 Computed Eigenvalues of the Te Matrix in the Cartesian Coordinate System and in the Translation Coordinate System with a(x)=0.1 sin(2πx) a

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Table 2 Comparison between the C Method and the RCWA for a Modulated Piecewise-Homogeneous Layer

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Table 3 Comparison between the C Method and the RCWA for an Inclined Lamellar Grating

Equations (64)

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ξijkjEk=-iωBi,
ξijkjHk=iωDi,
{i, j, k}{1, 2, 3},
ξijk=+1-10if(i, j, k)isanevenpermutationof(1, 2, 3)if(i, j, k)isanoddpermutationof(1, 2, 3)otherwise.
Di=ijEj,
Bi=μijHj.
ij=0ν2 gijg,
μij=μ0 gijg
x1=x,x2=y-a(x),x3=z.
(gij)=1-dadx0-dadx1+dadx20001.
ν2(x1, x2)=ν2(x1)=ν21ν22if0x1fdiffd<x1<d,
2(x1)=ν22(x1)=p=p=+p exp-i2πp x1d
η2(x1)=12(x1)=p=p=+ηp exp-i2πp x1d.
x1=x-y tan ϕ,x2=y,x3=z,
(gij)=1+tan2 ϕ-tan ϕ0-tan ϕ10001.
kx=k sin θ,ky=-k cos θ,kz=0.
ξijkjEk=-ikgijZHj,
ξijkjZHk=ikν2gijEj,
k=ωμ00,
Z=μ0/0.
2E3=-ikZ(g11H1+g12H2),
1ZH2-2ZH1=ikν2E3,
-1E3=-ikZ(g21H1+g22H2),
2ZH3=ikν2(g11E1+g12E2),
1E2-2E1=-ikZH3,
-1ZH3=ikν2(g21E1+g22E2),
2Ψe(x1, x2)=Le(x1)Ψe(x1, x2),
Ψe(x1, x2)=E3(x1, x2)ZH1(x1, x2),
Le(x1)=-g12g22 1-ikν2(x1)+11ik 1g22 1-ik 1g22-1 g12g22
2Ψh(x1, x2)=Lh(x1)Ψh(x1, x2),
Ψh(x1, x2)=ZH3(x1, x2)-E1(x1, x2),
Lh(x1)=-g12g22 111ikν2(x1) 1g22 1-ik-ikν2(x1) 1g22-1 g12g22
 
Ψe(x1, x2)=Φe(x1)exp(-ikrex2),
Φe(x1)=E3(x1)ZH1(x1),
Ψh(x1, x2)=Φh(x1)exp(-ikrhx2),
Φh(x1)=ZH3(x1)-E1(x1),
Φe(x1)=mΦem exp(-ikαmx1),
Φh(x1)=mΦhm exp(-ikαmx1),
αm=α0+m λd=sin θ+m λd,
Φem=E3mH1m,
Φhm=ZH3m-E1m.
f(x0+0)-f(x0-0)0g(x0+0)-g(x0-0)0,
hm=pf˜mpgp,
TeΦe=reΦe,
Te=[dm-p][αmp][m-p]-[αmp][cm-p][αmp][cm-p][αmp][dm-p],
Φe=(, E3 -m, , E3 0,  E3 +m, ;ZH1 -m, , ZH1 0,  ZH1 +m, ).
αmp=αmδmp,
δmp=10ifm=pifmp.
ThΦh=rhΦh,
Th = [αmp][dmp]δmp-[αmp][ηmp][αmp][cmp][ηmp]-1[cmp][αmp][dmp],
Φh=(, ZH3-m, , ZH3 0,  ZH3+m, ;-E1-m, ,-E1 0, -E1+m, ).
E3m=0=ZH3m=ZH1m=E1m=0if|m|>M,
rjhm2=rjem2=νj2-[sin θ+m(λ/d)]2,j=1, 3,
cos2 θj=νj2-[sin θ+mλ/d]2,j=1, 3.
req+={req,q{14M+2}/Re(req)>0
or Im(req)<0},
req-={req,q{14M+2}/Re(req)<0
or Im(req)>0}.
Ψelp(u, x)=q=12M+1Aelqp+Φelq+ exp[-ikrelq+(u-up)]+q=12M+1Aelqp-Φelq- ×exp[-ikrelq-(u-up)],
Ψhlp(u, x)=q=12M+1Ahlqp+Φh,l,q+ exp[-ikrhlq+(u-up)]+q=12M+1Ahlqp-Φhlq- ×exp[-ikrhlq-(u-up)],
(l, p){(1, 1), (2, 1), (2, 2), (2, 3)}.
eeq1=|Aeq1+|2 cos θqcos θ,eeq3=|Aeq3-|2 cos θ3 qcos θ,
ehq1=|Ahq1+|2 cos θqcos θ,ehq3=|Ahq3-|2ν3 cos θ3 qcos θ.

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