Abstract

The differential formalism introduced by J. Chandezon during the seventies has been successfully applied to the study of waveguides and to diffraction problems. Until now it was believed that the method could be applied only if the interfaces between media were described by graphs of functions. We show that an eigenoperator formulation of the method allows one to solve a larger set of profiles. This theoretical result is applied to gratings having a vertical facet.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
  2. R. Dusséaux, C. Faure, J. Chandezon, F. Molinet, “New perturbation theory of diffraction gratings and its application to the study of ghosts,” J. Opt. Soc. Am. A 72, 1271–1282 (1995).
    [CrossRef]
  3. J. Chandezon, G. Cornet, “Application d’une nouvelle méthode de résolution des équations de Maxwell à l'étude de la propagation des ondes électromagnétiques dans les guides périodiques,” Ann. Telecommun. 36(5–6), 305–314 (1981).
  4. J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. A 72, 839–846 (1982).
    [CrossRef]
  5. E. Popov, L. Mashev, “Conical diffraction mounting. Generalization of a rigorous differential method,” J. Opt. (Paris), 17, 175–180 (1986).
  6. E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962).
  7. L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
    [CrossRef]
  8. 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]
  9. 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]
  10. G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
    [CrossRef]
  11. J. P. Plumey, G. Granet, J. Chandezon, “Differential covariant formalism for solving Maxwell’s equations in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 835–842 (1995).
    [CrossRef]
  12. 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]
  13. L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
    [CrossRef]

1996 (1)

1995 (5)

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]

R. Dusséaux, C. Faure, J. Chandezon, F. Molinet, “New perturbation theory of diffraction gratings and its application to the study of ghosts,” J. Opt. Soc. Am. A 72, 1271–1282 (1995).
[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]

G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

J. P. Plumey, G. Granet, J. Chandezon, “Differential covariant formalism for solving Maxwell’s equations in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 835–842 (1995).
[CrossRef]

1994 (1)

1993 (1)

1986 (1)

E. Popov, L. Mashev, “Conical diffraction mounting. Generalization of a rigorous differential method,” J. Opt. (Paris), 17, 175–180 (1986).

1982 (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. A 72, 839–846 (1982).
[CrossRef]

1981 (1)

J. Chandezon, G. Cornet, “Application d’une nouvelle méthode de résolution des équations de Maxwell à l'étude de la propagation des ondes électromagnétiques dans les guides périodiques,” Ann. Telecommun. 36(5–6), 305–314 (1981).

Chandezon, J.

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]

R. Dusséaux, C. Faure, J. Chandezon, F. Molinet, “New perturbation theory of diffraction gratings and its application to the study of ghosts,” J. Opt. Soc. Am. A 72, 1271–1282 (1995).
[CrossRef]

J. P. Plumey, G. Granet, J. Chandezon, “Differential covariant formalism for solving Maxwell’s equations in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 835–842 (1995).
[CrossRef]

J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. A 72, 839–846 (1982).
[CrossRef]

J. Chandezon, G. Cornet, “Application d’une nouvelle méthode de résolution des équations de Maxwell à l'étude de la propagation des ondes électromagnétiques dans les guides périodiques,” Ann. Telecommun. 36(5–6), 305–314 (1981).

Cornet, G.

J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. A 72, 839–846 (1982).
[CrossRef]

J. Chandezon, G. Cornet, “Application d’une nouvelle méthode de résolution des équations de Maxwell à l'étude de la propagation des ondes électromagnétiques dans les guides périodiques,” Ann. Telecommun. 36(5–6), 305–314 (1981).

Cotter, N. P. K.

Dupuis, M. T.

J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. A 72, 839–846 (1982).
[CrossRef]

Dusséaux, R.

R. Dusséaux, C. Faure, J. Chandezon, F. Molinet, “New perturbation theory of diffraction gratings and its application to the study of ghosts,” J. Opt. Soc. Am. A 72, 1271–1282 (1995).
[CrossRef]

Faure, C.

R. Dusséaux, C. Faure, J. Chandezon, F. Molinet, “New perturbation theory of diffraction gratings and its application to the study of ghosts,” J. Opt. Soc. Am. A 72, 1271–1282 (1995).
[CrossRef]

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]

J. P. Plumey, G. Granet, J. Chandezon, “Differential covariant formalism for solving Maxwell’s equations in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 835–842 (1995).
[CrossRef]

G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

Guizal, B.

Haggans, C. W.

Li, L.

Mashev, L.

E. Popov, L. Mashev, “Conical diffraction mounting. Generalization of a rigorous differential method,” J. Opt. (Paris), 17, 175–180 (1986).

Maystre, D.

J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. A 72, 839–846 (1982).
[CrossRef]

Molinet, F.

R. Dusséaux, C. Faure, J. Chandezon, F. Molinet, “New perturbation theory of diffraction gratings and its application to the study of ghosts,” J. Opt. Soc. Am. A 72, 1271–1282 (1995).
[CrossRef]

Plumey, J. P.

J. P. Plumey, G. Granet, J. Chandezon, “Differential covariant formalism for solving Maxwell’s equations in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 835–842 (1995).
[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]

Popov, E.

E. Popov, L. Mashev, “Conical diffraction mounting. Generalization of a rigorous differential method,” J. Opt. (Paris), 17, 175–180 (1986).

Post, E. J.

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

Preist, T. W.

Sambles, J. R.

Ann. Telecommun. (1)

J. Chandezon, G. Cornet, “Application d’une nouvelle méthode de résolution des équations de Maxwell à l'étude de la propagation des ondes électromagnétiques dans les guides périodiques,” Ann. Telecommun. 36(5–6), 305–314 (1981).

IEEE Trans. Antennas Propag. (1)

J. P. Plumey, G. Granet, J. Chandezon, “Differential covariant formalism for solving Maxwell’s equations in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 835–842 (1995).
[CrossRef]

J. Opt. (Paris) (1)

E. Popov, L. Mashev, “Conical diffraction mounting. Generalization of a rigorous differential method,” J. Opt. (Paris), 17, 175–180 (1986).

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

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]

G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

Other (2)

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

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).

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 (4)

Fig. 1
Fig. 1

Geometry of the problem: Two media are separated by a cylindrical infinite surface described by the equation y=a(x).

Fig. 2
Fig. 2

(a) Both the function a(x) and its derivative are periodic. (b) a(x) is not periodic but has a periodic derivative.

Fig. 3
Fig. 3

Profiles used in the numerical computation: (a) triangle with a vertical facet, (b) triangle that is not a graph of a function.

Fig. 4
Fig. 4

From the parameters c, h, and d of a given triangle in (XYZ), we determine the angle Φ and the parameters h and d of the corresponding symmetric triangle in (xyz).

Tables (3)

Tables Icon

Table 1 Computed Efficiencies for the Profile Function y=h/2 cos(2πx/d)+tan(Φ)xa

Tables Icon

Table 2 Comparison between the C Method and the CWM As Improved by Granet and Guizala

Tables Icon

Table 3 Comparison between the C Method and the CWM As Improved by Granet and Guizala

Equations (58)

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

x1=x,
x2=u=y-a(x),
x3=z.
(gij)=g11g21g31g12g22g32g13g23g33=1-a˙0-a˙1+a˙20001with a˙=da(x)dx.
ξijkjEk=-iωμ0ggijHj,ξijkjHk=iωε0ν2ggijEj,1i, j3,with g=det(gij),
ξijk
=1-10if (i, j, k) is an even permutation of (1, 2, 3)if (i, j, k) is an odd permutation of (1, 2, 3)otherwise.
Fj=Ej,
Gj=iνZHj.
ξijkjFk=-kνggijGj,ξijkjGk=-kνggijFj,1i, j3.
2F3=-kν(G1+g12G2),
-1F3=-kν(g12G1+g22G2),
1G2-2G1=-kνF3.
2F3=-g12g221F3-kν G1g22,
2G1=1kν11g221F3-1g12g22G1+kνF3.
F3(x, u)=F(x)exp(-ikνru),
G1(x, u)=G(x)exp(-ikνru).
rF(x)=ikνd(x) dF(x)dx-ic(x)G(x),
rG(x)=ik2ν2ddxc(x) dF(x)dx+iF(x)+ikνddx[d(x)G(x)],
c(x)=1g22=11+a˙2,d(x)=-g12g22=a˙1+a˙2.
LxF(x)G(x)=rF(x)G(x),
Lx=ikνd(x) ddxik2ν2ddxc(x) ddx+i-ic(x)ikνddx[d(x)]
case 1periodic:a(x)=a1(x),case 2nonperiodic:a(x)=a2(x)=a1(x)+tan(Φ)x,
Φ-π2, π2.
a1(x)=2 hdxif 0xd/2-2 hd(x-d)if d/2xd,
Φ=arctan[(c-d/2)/h],
h=h/cos(Φ),
d=d cos(Φ).
F(x+d)=exp(-iϕ)F(x),
G(x+d)=exp(-iϕ)G(x).
ϕ=kαd,α is a real number.
F(x)=n=-+Fnen(x),
G(x)=n=-+Gnen(x),
f, g=1dx0x0+dfg¯dx,
rNFmN=1νnαndm-nFnN-incm-nGnN,
rNGmN=-iν2nαnαmcm-nFnN+iFmN+1νnαmdm-nGnN,
rNFmNGmN=[TmnN]FnNGnN.
FqN(x)=-NnNFnqN(rqN)exp(-ikαnx),
GqN(x)=-NnNGnqN(rqN)exp(-ikαnx).
kx=k sin(θ+Φ),
ky=-k cos(θ+Φ).
exp(-ik·r)=exp(-ikxx)exp(-ikyu)×exp[-ikya1(x)]exp[-ikyx tan(Φ)].
exp[-ikya1(x)]=n=-+Ln(ky)exp-in 2πdx,
exp(-ik·r)=exp(-ikyu)n=-+Ln(ky)×exp-ikx+ky tan(Φ)+n 2πdx.
kx+ky tan(Φ)+n 2πd=ksin(θ)cos(Φ)+n λd.
α=sin(θ)cos(Φ).
sin(θq)=sin(θ)+n(λ/d)cos(Φ).
Vd=VdrVev.
F3(x, u)=exp[ik cos(θ+Φ)u]n=-+Fn0 exp(-ikαnx)+qVdAq exp(-ikrqu)×n=-+Fnq exp(-ikαnx),
G1(x, u)=exp[ik cos(θ+Φ)u]n=-+Gn0 exp(-ikαnx)+qVdAq exp(-ikrqu)×n=-+Gnq exp(-ikαnx),
F3ν(x, u)=qVνBq exp(-ikνrqνu)×n=-+Fnqν exp(-ikαnx),
G1ν(x, u)=qVνBq exp(-ikνrqνu)×n=-+Gnqν exp(-ikαnx),
Vν=VνtVνevif ν is real
Vν=Vνev={qZ[Im(rq)>0]}if ν is complex,
F3(x, u=0)=F3ν(x, u=0),
G1(x, u=0)=νG1ν(x, u=0).
Rp=ApA¯p Im-NnNFnpG¯npIm-NnNFn0G¯n0,pVd,
Tp=νBpB¯p Im-NnNFnpνG¯npνIm-NnNFn0G¯n0,pVνt.

Metrics