Abstract

A formulation of the Chandezon method for modeling one-dimensionally periodic, multilayer, inhomogeneous, anisotropic diffraction gratings is presented. The formulation is based on an oblique Cartesian coordinate system so that blazed surface-relief profiles and slanted volume permittivity variations can be treated simultaneously. The permittivity tensors of the anisotropic materials can be arbitrary, but their spatial variations are assumed to have the same period as the surface corrugations. Some of the previous authors’ works in the above individual aspects of the subject are improved on in regard to clarity, simplicity, and generality. Most important, the theory of Fourier factorization is applied throughout the analysis to ensure fast convergence of the numerical method when the permittivity tensors or the derivatives of the grating’s surface-profile functions have discontinuities. Numerical examples, including those of two types of multilayer magneto-optic gratings, are provided.

© 1999 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. J. B. Harris, T. W. Preist, J. R. Sambles, “Differential formalism for multilayer diffraction gratings made with uniaxial materials,” J. Opt. Soc. Am. A 12, 1965–1973 (1995).
    [CrossRef]
  3. J. B. Harris, T. W. Preist, E. L. Wood, J. R. Sambles, “Conical diffraction from multicoated gratings containing uniaxial materials,” J. Opt. Soc. Am. A 13, 803–810 (1996).
    [CrossRef]
  4. M. E. Inchaussandague, R. A. Depine, “Polarization conversion from diffraction gratings made of uniaxial crystals,” Phys. Rev. E 54, 2899–2911 (1996).
    [CrossRef]
  5. M. E. Inchaussandague, R. A. Depine, “Rigorous vector theory for diffraction from gratings made of biaxial crystals,” J. Mod. Opt. 44, 1–27 (1997).
    [CrossRef]
  6. G. Granet, J. Chandezon, O. Coudert, “Extension of the C method to nonhomogeneous media: application to nonhomogeneous layers with parallel modulated faces and to inclined lamellar gratings,” J. Opt. Soc. Am. A 14, 1576–1582 (1997).
    [CrossRef]
  7. J.-P. Plumey, B. Guizal, J. Chandezon, “Coordinate transformation method as applied to asymmetric gratings with vertical facets,” J. Opt. Soc. Am. A 14, 610–617 (1997).
    [CrossRef]
  8. T. W. Preist, J. B. Harris, N. P. Wanstall, J. R. Sambles, “Optical response of blazed and overhanging gratings using oblique Chandezon transformations,” J. Mod. Opt. 44, 1073–1080 (1997).
  9. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  10. 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]
  11. L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
    [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. T. W. Preist, N. P. K. Cotter, J. R. Sambles, “Periodic multilayer gratings of arbitrary shape,” J. Opt. Soc. Am. A 12, 1740–1748 (1995).
    [CrossRef]
  14. L. Li, “Periodic multilayer gratings of arbitrary shape: comment,” J. Opt. Soc. Am. A 13, 1475–1476 (1996).
    [CrossRef]
  15. L. Li, G. Granet, J.-P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
    [CrossRef]
  16. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]

1998 (1)

L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
[CrossRef]

1997 (4)

M. E. Inchaussandague, R. A. Depine, “Rigorous vector theory for diffraction from gratings made of biaxial crystals,” J. Mod. Opt. 44, 1–27 (1997).
[CrossRef]

T. W. Preist, J. B. Harris, N. P. Wanstall, J. R. Sambles, “Optical response of blazed and overhanging gratings using oblique Chandezon transformations,” J. Mod. Opt. 44, 1073–1080 (1997).

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

G. Granet, J. Chandezon, O. Coudert, “Extension of the C method to nonhomogeneous media: application to nonhomogeneous layers with parallel modulated faces and to inclined lamellar gratings,” J. Opt. Soc. Am. A 14, 1576–1582 (1997).
[CrossRef]

1996 (7)

1995 (3)

1982 (1)

Chandezon, J.

Cornet, G.

Cotter, N. P. K.

Coudert, O.

Depine, R. A.

M. E. Inchaussandague, R. A. Depine, “Rigorous vector theory for diffraction from gratings made of biaxial crystals,” J. Mod. Opt. 44, 1–27 (1997).
[CrossRef]

M. E. Inchaussandague, R. A. Depine, “Polarization conversion from diffraction gratings made of uniaxial crystals,” Phys. Rev. E 54, 2899–2911 (1996).
[CrossRef]

Dupuis, M. T.

Granet, G.

G. Granet, J. Chandezon, O. Coudert, “Extension of the C method to nonhomogeneous media: application to nonhomogeneous layers with parallel modulated faces and to inclined lamellar gratings,” J. Opt. Soc. Am. A 14, 1576–1582 (1997).
[CrossRef]

L. Li, G. Granet, J.-P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (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.

Harris, J. B.

Inchaussandague, M. E.

M. E. Inchaussandague, R. A. Depine, “Rigorous vector theory for diffraction from gratings made of biaxial crystals,” J. Mod. Opt. 44, 1–27 (1997).
[CrossRef]

M. E. Inchaussandague, R. A. Depine, “Polarization conversion from diffraction gratings made of uniaxial crystals,” Phys. Rev. E 54, 2899–2911 (1996).
[CrossRef]

Li, L.

Maystre, D.

Plumey, J.-P.

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

L. Li, G. Granet, J.-P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (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]

Preist, T. W.

Sambles, J. R.

Wanstall, N. P.

T. W. Preist, J. B. Harris, N. P. Wanstall, J. R. Sambles, “Optical response of blazed and overhanging gratings using oblique Chandezon transformations,” J. Mod. Opt. 44, 1073–1080 (1997).

Wood, E. L.

J. Mod. Opt. (3)

M. E. Inchaussandague, R. A. Depine, “Rigorous vector theory for diffraction from gratings made of biaxial crystals,” J. Mod. Opt. 44, 1–27 (1997).
[CrossRef]

T. W. Preist, J. B. Harris, N. P. Wanstall, J. R. Sambles, “Optical response of blazed and overhanging gratings using oblique Chandezon transformations,” J. Mod. Opt. 44, 1073–1080 (1997).

L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (9)

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

G. Granet, J. Chandezon, O. Coudert, “Extension of the C method to nonhomogeneous media: application to nonhomogeneous layers with parallel modulated faces and to inclined lamellar gratings,” J. Opt. Soc. Am. A 14, 1576–1582 (1997).
[CrossRef]

J. B. Harris, T. W. Preist, E. L. Wood, J. R. Sambles, “Conical diffraction from multicoated gratings containing uniaxial materials,” J. Opt. Soc. Am. A 13, 803–810 (1996).
[CrossRef]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[CrossRef]

L. Li, “Periodic multilayer gratings of arbitrary shape: comment,” J. Opt. Soc. Am. A 13, 1475–1476 (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]

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]

T. W. Preist, N. P. K. Cotter, J. R. Sambles, “Periodic multilayer gratings of arbitrary shape,” J. Opt. Soc. Am. A 12, 1740–1748 (1995).
[CrossRef]

J. B. Harris, T. W. Preist, J. R. Sambles, “Differential formalism for multilayer diffraction gratings made with uniaxial materials,” J. Opt. Soc. Am. A 12, 1965–1973 (1995).
[CrossRef]

Phys. Rev. E (1)

M. E. Inchaussandague, R. A. Depine, “Polarization conversion from diffraction gratings made of uniaxial crystals,” Phys. Rev. E 54, 2899–2911 (1996).
[CrossRef]

Pure Appl. Opt. (2)

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]

L. Li, G. Granet, J.-P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Grating structure under study and the rectangular Cartesian coordinate system attached to it. The medium interfaces can be nonconformal. The media other than the incident one can be anisotropic. The layered medium can be inhomogeneous.

Fig. 2
Fig. 2

Relationship between the rectangular and oblique Cartesian coordinate systems.

Fig. 3
Fig. 3

Triple-layer magneto-optic grating of example 1.

Fig. 4
Fig. 4

Tilted, inhomogeneous, anisotropic grating of example 2.

Fig. 5
Fig. 5

Convergence of the transmitted -1st-order diffraction efficiencies of the two gratings in example 3. Thick solid line, ζ=10°, Eq. (34); thick dashed line, ζ=45°, Eq. (34); circles, ζ=10°, coefficient matrix (58); diamonds, ζ=45°, coefficient matrix (58).

Tables (2)

Tables Icon

Table 1 Diffraction Efficiencies and Polarization Parameters of the Magneto-Optic Gratings in Example 1

Tables Icon

Table 2 Diffraction Efficiencies and Polarization Parameters of the Tilted, Inhomogeneous, Anisotropic Grating in Example 2

Equations (87)

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x¯1=x-y tan ζ,x¯2=y sec ζ,x¯3=z.
xp1=x¯1,xp2=x¯2-ap(x¯1),x3=x¯3.
b1=x-y tan ζ,
b2=-xa˙+y(sec ζ+a˙ tan ζ),
b3=z,
b1=x(1+a˙ sin ζ)+ya˙ cos ζ,
b2=x sin ζ+y cos ζ,
b3=z.
gρσ=τ=13 xρx̲τ xσx̲τ=sec2 ζ1-(a˙+sin ζ)0-(a˙+sin ζ)1+2a˙ sin ζ+a˙2000cos2 ζ,
ε¯ρσ=x¯ρx̲τ x¯σx̲χ ε̲τχ.
{ερσ}={ε¯ρσ}-a˙0ε¯110ε¯11ε¯12+ε¯21-a˙ε¯11ε¯130ε¯310.
ξρστσHτ=-ik0gερσEσ,
ξρστσEτ=ik0gμgρσHσ.
Fσ(x1, x2, x3)=exp(iγx3)nFσn(x2)exp(iαnx1),
D1=ε11E1+ε12ε11E2+ε13ε11E3,
D2=ε21ε11D1+ε22-ε21ε12ε11E2+ε23-ε21ε13ε11E3,
D3=ε31ε11D1+ε32-ε31ε12ε11E2+ε33-ε31ε13ε11E3.
(ξρστσHτ)m=-ik0gn(Eρσ)mnEσn,
E=1ε¯11-11ε¯11-1ε12ε¯111ε¯11-1ε¯13ε¯11ε21ε¯111ε¯11-1ε21ε¯111ε¯11-1ε12ε¯11+ε¯22-ε¯21ε¯12ε¯11ε21ε¯111ε¯11-1ε¯13ε¯11+ε¯23-ε¯21ε¯13ε¯11α¯31ε¯111ε¯11-1ε¯31ε¯111ε¯11-1ε12ε¯11+ε¯32-ε¯31ε¯12ε¯11ε¯31ε¯111ε¯11-1ε¯13ε¯11+ε¯33-ε¯31ε¯13ε¯11
ε12ε¯11=ε¯12ε¯11-a˙,ε21ε¯11=ε¯21ε¯11-a˙. 
(ξρστσEτ)m=ik0gμn(Gρσ)mnHσn,
G=sec2 ζ
×1-(a˙+sin ζ)0-(a˙+sin ζ)cos2 ζ+(a˙+sin ζ)2000cos2 ζ,
H2=Y1(γE1-αE3)-Y2H1,
E2=-Z22(γH1-αH3)-Z21E1-Z23E3,
Y1=1k¯0μ (G22)-1,Y2=(G22)-1G21,
Z21=(E22)-1E21,Z22=1k¯0 (E22)-1
Z23=(E22)-1E23,
2i E3H3H1E1=-Y2α-γZ23γZ22αk02μ2Y1-γ2Z22γ(Y2-Z21)-γY1α-k¯0Z13-Z12αγ(Z12-Y2)γ2Y1-k¯0Z11-αY1α+k¯0Z33Z32α-αY2-γZ32γαY1+k¯0Z31-αZ23αZ22α-k¯0μ-γαZ22-αZ21E3H3H1E1,
Z11=E11-E12(E22)-1E21,Z12=E12(E22)-1,
Z13=E13-E12(E22)-1E23,Z31=E31-E32(E22)-1E21,
Z32=E32(E22)-1,Z33=E33-E32(E22)-1E23.
AX=λX,
-α0K-N-12n-α0K+N-12,
Syq=n Re(E3nqH1nq*-E1nqH3nq*),
F(p)=b(p-1)σFσ(p, p-1)+(xp-11, xp-12)+b(p)σFσ(p, p)-(xp1, xp2),
F(p;p-1)=b(p-1)σFσ(p, p-1)+(xp-11, xp-12)+b(p-1)σF^σ(p, p)-(xp-11, xp-12),
F(p;p)=b(p)σF^σ(p, p-1)+(xp1, xp2)+b(p)σFσ(p, p)-(xp1, xp2).
xp1=xp-11,xp2=xp-12+ap-1(xp-11)-ap(xp-11).
F^3(p, p-1)+=xp-1τxp3 Fτ(p, p-1)+=F3(p, p-1)+,
F^1(p, p-1)+=xp-1τxp1 Fτ(p, p-1)+=F1(p, p-1)++ΔpF2(p, p-1)+,
F^3(p, p)-=xpτxp-13 Fτ(p, p)-=F3(p, p)-,
F^1(p, p)-=xpτxp-11 Fτ(p, p)-=F1(p, p)--ΔpF2(p, p)-,
Δp=a˙p-a˙p-1.
Fσ(p;p-1)=m,q[Fσmq(p, p-1)+ exp(iλq(p, p-1)+xp-12)uq(p)+F^σmq(p, p)- exp(iλq(p, p)-xp-12)dq(p)]×exp(iαmxp-11),
Fσ(p;p)=m,q[F^σmq(p, p-1)+ exp(iλq(p, p-1)+xq2)uq(p)+Fσ(p, p)- exp(iλq(p, p)-xp2)dq(p)]×exp(iαmxp1),
F^3mq(p, r)±=F˜3mq(p, r)±,
F^1mq(p, p-1)+=F˜1mq(p, p-1)++nΔpmnF˜2nq(p, p-1)+,
F^1mq(p, p)-=F˜1mq(p, p)--nΔpmnF˜2nq(p, p)-.
Z0=(ε¯11a˙2-2ε¯12a˙+ε¯22)-1,
2i H3E1=gε-1BQα-k¯0P1k¯0 αQα-k¯0μαQBgε-1H3E1,
B=a˙+sin ζ,
Q=ε+Bgε-1B-1,
P=gε-1-gε-1BQBgε-1=gε+Bε-1B-1.
F3(+1),R=I(f ) exp[i(α0x¯1+β0(+1)-x¯2)]+nU+Rn(f ) exp[i(αnx¯1+βn(+1)+x¯2)],
F3(-1),R=nU-Tn(f ) exp[i(αnx¯1+βn(-1)-x¯2)],
βn(p)±=αn sin ζ±k˜p2-αn2 cos ζ,
k˜p2=k02εpμ-γ2,
[p(gρσσ)+k˜2]F3=0,
2i F3F3=01(G22)-1Γ2-(G22)-1(αG12+G21α)F3F3,
F3(±1)=F3(±1),R+m exp(iαmx1)×qV±F3mq± exp(iλq±xF2)uq(f )dq(f ),
E1H1=iτ31-τ2Ξτ1Ξiτ31E3H3,
Ξ=-ig(g211+g222),
τ1=k0εgk˜2,τ2=k0μgk˜2,τ3=γk˜2.
(β0(+1)--α0 sin ζ)(ε+1|I(e)|2+μ|I(h)|2)=-k+12,
ηn(+1)=βn(+1)+-αn sin ζk˜+12 (ε+1|Rn(e)|2+μ|Rn(h)|2),
ηn(-1)=-βn(-1)--αn sin ζk˜-12 (ε-1|Tn(e)|2+μ|Tn(h)|2).
s=k×y|k×y|,p=s×k|s×k|.
s=1k (-γb¯1+αb¯3),
p=1k0kεμg [-(γ2 sin ζ+αβ)b¯1+k2b¯2+γ(α sin ζ-β)b¯3].
F(-1)=s=34nUs-fsn exp[i(αnx¯1+βsnx¯2)]Tsn+b(-1)σmqV-Fσmq-× exp[i(αmx1+λq-x-12)]dq(-1).
fsn=fxsnx+fysny+fzsnz.
F3m(-1),R=s=34nUsLm-n(-1, -2)(βsn)fzsnTsn exp(iβsnx-12),
f1=fx+a˙-1f2,f2=fx sin ζ+fy cos ζ.
F1m(-1),R=s=34nUs-[Lm-n(-1, -2)(βsn)fxsn+la˙-1mlLl-n(-1, -2)(βsn)f2sn]×exp(iβsnx-12)Tsn.
ε˜±=ε(I-yy)±iε×(y×I)+ε yy.
α sin ζ-k0 cos ζ1ε-1cos ζk0 αε-1α-k0μ cos ζα sin ζ.
F˜σmq(p, r)±=lLm-l(p, r)(λq(p, r)±)Fσlq(p, r)±,
Lm-l(p, r)(ξ)=1d 0d exp{-i[mKx+(-1)p-r×ξ(ap-ap-1)]}dx,
E2q(p, r)±=-Z22(p, r)(γH1q(p, r)±-αH3q(p, r)±)-Z21(p, r)E1q(p, r)±-Z23(p, r)E3q(p, r)±,
H2q(p, r)±=Y1(r)(γE1q(p, r)±-αE3q(p, r)±)-Y2(r)H1q(p, r)±.
Z11=(ε¯11ε¯22-ε¯12ε¯21)Z0,
Z12=Z21=ε¯12Z0-ε¯11a˙Z0,
Z13=Z31=(ε¯13ε¯22-ε¯12ε¯23)Z0-(ε¯11ε¯23-ε¯12ε¯13)a˙Z0,
Z22=Z0/k¯0,
Z23=Z32=ε¯23Z0-ε¯13a˙Z0,
Z33=(ε¯33-ε¯31ε¯13/ε¯11)-(ε¯23ε¯32-ε¯13ε¯22ε¯31/ε¯11)Z0+2ε¯13(ε¯23-ε¯21ε¯13/ε¯11)a˙Z0.

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