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

1996

1995

1994

1993

1982

1978

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

1975

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.

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.

J. Opt. Soc. Am. A

Pure Appl. Opt.

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

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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