Abstract

The coordinate-transformation-based differential method, initially used by Chandezon et al. for modeling surface-relief gratings, is now known as a powerful rigorous formalism for solving diffraction problems. We explain a coordinate transformation that generalizes the original one, and we extend the formulation to a wide class of monodimensional surface shapes. The boundary-value problem turns on the same eigenvalue problem for the TE and TM polarizations.

© 1999 Optical Society of America

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References

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  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. 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]
  3. E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962).
  4. L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
    [CrossRef]
  5. E. Popov, L. Tsonev, D. Maystre, “Gratings—general properties of the Littrow mounting and energy flow distribution,” J. Mod. Opt. 37, 367–377 (1990).
    [CrossRef]
  6. E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
    [CrossRef]
  7. E. Popov, L. Tsonev, D. Maystre, “Losses of plasmon surface waves on metallic grating,” J. Mod. Opt. 37, 379–387 (1990).
    [CrossRef]
  8. E. Popov, L. Mashev, “Conical diffraction mounting. Generalization of a rigorous differential method,” J. Opt. (Paris) 17, 175–180 (1986).
    [CrossRef]
  9. 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]
  10. G. Granet, J. P. Plumey, J. Chandezon, “Scattering by periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
    [CrossRef]
  11. 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]
  12. 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]
  13. G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
    [CrossRef]
  14. J. B. Harris, T. W. Preist, J. R. Sambles, R. N. Thorpe, R. A. Watts, “Optical response of bigratings,” J. Opt. Soc. Am. A 13, 2041–2049 (1996).
    [CrossRef]
  15. 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]
  16. M. E. Inchaussange, R. A. Depine, “Polarization conversion from diffraction gratings made of uniaxial crystals,” Phys. Rev. E 54, 2899–2911 (1996).
    [CrossRef]
  17. M. E. Inchaussange, R. A. Depine, “Rigorous vector theory for diffraction from gratings made of biaxial crystals,” J. Mod. Opt. 44, 1–27 (1997).
    [CrossRef]
  18. 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]
  19. J. B. Harris, T. W. Preist, J. B. Wood, J. R. Sambles, “Conical diffraction from multicoated gratings containing uniaxial materials,” J. Opt. Soc. Am. A 13, 803–810 (1996).
    [CrossRef]
  20. 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]
  21. 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]
  22. 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).
  23. G. Granet, J. Chandezon, “The method of curvilinear coordinates applied to the problem of scattering from surface-relief gratings defined by parametric equations: applica-tion to scattering from a cycloidal grating,” Pure Appl. Opt. 6, 727–740 (1997).
    [CrossRef]
  24. G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 14, 1019–1023 (1997).
  25. R. Janaswany, “Oblique scattering from lossy periodic surfaces with application to anechoic chamber absorbers,” IEEE Trans. Antennas Propag. 40, 162–169 (1995).
    [CrossRef]

1997

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]

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

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]

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).

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

G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 14, 1019–1023 (1997).

1996

1995

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

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 periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[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]

R. Janaswany, “Oblique scattering from lossy periodic surfaces with application to anechoic chamber absorbers,” IEEE Trans. Antennas Propag. 40, 162–169 (1995).
[CrossRef]

1994

1990

E. Popov, L. Tsonev, D. Maystre, “Gratings—general properties of the Littrow mounting and energy flow distribution,” J. Mod. Opt. 37, 367–377 (1990).
[CrossRef]

E. Popov, L. Tsonev, D. Maystre, “Losses of plasmon surface waves on metallic grating,” J. Mod. Opt. 37, 379–387 (1990).
[CrossRef]

1986

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

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

1982

1980

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.

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. 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, “The method of curvilinear coordinates applied to the problem of scattering from surface-relief gratings defined by parametric equations: applica-tion 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]

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]

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 periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (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. 72, 839–846 (1982).
[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]

Cornet, G.

Cotter, N. P. K.

Coudert, O.

Depine, R. A.

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

M. E. Inchaussange, 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]

G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 14, 1019–1023 (1997).

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

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]

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

Guizal, B.

G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 14, 1019–1023 (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]

Harris, J. B.

Inchaussange, M. E.

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

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

Janaswany, R.

R. Janaswany, “Oblique scattering from lossy periodic surfaces with application to anechoic chamber absorbers,” IEEE Trans. Antennas Propag. 40, 162–169 (1995).
[CrossRef]

Li, L.

Mashev, L.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

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

Maystre, D.

E. Popov, L. Tsonev, D. Maystre, “Gratings—general properties of the Littrow mounting and energy flow distribution,” J. Mod. Opt. 37, 367–377 (1990).
[CrossRef]

E. Popov, L. Tsonev, D. Maystre, “Losses of plasmon surface waves on metallic grating,” J. Mod. Opt. 37, 379–387 (1990).
[CrossRef]

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[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. 72, 839–846 (1982).
[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]

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]

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 periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

Popov, E.

E. Popov, L. Tsonev, D. Maystre, “Gratings—general properties of the Littrow mounting and energy flow distribution,” J. Mod. Opt. 37, 367–377 (1990).
[CrossRef]

E. Popov, L. Tsonev, D. Maystre, “Losses of plasmon surface waves on metallic grating,” J. Mod. Opt. 37, 379–387 (1990).
[CrossRef]

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

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Post, E. J.

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

Preist, T. W.

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]

Sambles, J. R.

Thorpe, R. N.

Tsonev, L.

E. Popov, L. Tsonev, D. Maystre, “Gratings—general properties of the Littrow mounting and energy flow distribution,” J. Mod. Opt. 37, 367–377 (1990).
[CrossRef]

E. Popov, L. Tsonev, D. Maystre, “Losses of plasmon surface waves on metallic grating,” J. Mod. Opt. 37, 379–387 (1990).
[CrossRef]

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).

Watts, R. A.

Wood, J. B.

IEEE Trans. Antennas Propag.

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]

R. Janaswany, “Oblique scattering from lossy periodic surfaces with application to anechoic chamber absorbers,” IEEE Trans. Antennas Propag. 40, 162–169 (1995).
[CrossRef]

J. Mod. Opt.

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).

E. Popov, L. Tsonev, D. Maystre, “Losses of plasmon surface waves on metallic grating,” J. Mod. Opt. 37, 379–387 (1990).
[CrossRef]

E. Popov, L. Tsonev, D. Maystre, “Gratings—general properties of the Littrow mounting and energy flow distribution,” J. Mod. Opt. 37, 367–377 (1990).
[CrossRef]

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

J. Opt. (Paris)

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]

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

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]

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

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]

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, R. N. Thorpe, R. A. Watts, “Optical response of bigratings,” J. Opt. Soc. Am. A 13, 2041–2049 (1996).
[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]

L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
[CrossRef]

G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 14, 1019–1023 (1997).

Opt. Acta

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Phys. Rev. E

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

Pure Appl. Opt.

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

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

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

Other

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

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Figures (3)

Fig. 1
Fig. 1

Two typical profile shapes: a, x=x1+0.25 sin(2x1), y=0.5 sin x1; b, x=x1-0.9 sin(2x1), y=1+0.5 sin x1.

Fig. 2
Fig. 2

From a sinusoidal profile to a variant of a triangular profile with parametric equations: x=x1+kf sin(2x1), y=sin x1. Dashed curve: kf=0, solid curve: kf=0.25, dotted curve: kf=0.5.

Fig. 3
Fig. 3

Reflectivity versus incidence direction for different values of the kf parameter: (a) kf=0, (b) kf=0.25, (c) kf=0.45. The grating profile is given by Eqs. (49) with h=4π. The optical permittivity is ν2=1.45-i0.4. The incident wave parameters are λ=12.3685 and θ=0°.

Tables (1)

Tables Icon

Table 1 Comparison of Present Method and CWM for a Variant of a Triangular Profilea

Equations (88)

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

x=x(x1, x2),y=y(x1, x2),z=x3,
J=D(x, y, z)D(x1, x2, x3)=xx1yx2-yx1xx20.
e1=xx1ex+yx1ey,
e2=xx2ex+yx2ey,
e3=ez.
[gij(x1, x2)]=xx12+yx12xx1xx2+yx1yx20xx1xx2+yx1yx2xx22+yx220001.
x(x1, x2)=f(x1)+cxx2, y(x1, x2)=g(x1)+cyx2, z=x3,
cy df(x1)dx1-cx dg(x1)dx10,
df(x1)dx1=f,dg(x1)dx1=g.
e1=fex+gey,
e2=cxex+cyey,
e3=ez.
[gij(x1)]=f2+g2cxf+cyg0cxf+cygcx2+cy20001,
g=|cyf-cxg|.
dM=cx dx2 ex+cy dx2 ey=dx2 e2.
ξijkjEk=-ikggijZHj,
ξijkjZHk=ikν2ggijEj,1i,j3.
ξ123=ξ231=ξ312=1,ξ132=ξ213=ξ321=-1.
Z=μ0/0,k=ω0μ0,
ik(ν2-γ2)E1=γ1E3+Zg(g112H3-g121H3),
ik(ν2-γ2)E2=γ2E3+Zg(g212H3-g221H3),
ik(ν2-γ2)H1=γ1H3-ν2Zg(g112E3-g121E3),
ik(ν2-γ2)H2=γ2H3-ν2Zg(g212E3-g221E3).
11g(g221ψ-g212ψ)+21g(g112ψ-g121ψ)+k2(ν2-γ2)gψ=0,
ik(ν2-γ2)E1=γ1ψTM+Zg(g112ψTE-g121ψTE),
ik(ν2-γ2)E2=γ2ψTM+Zg(g212ψTE-g221ψTE),
E3=ψTM,
ik(ν2-γ2)H1=γ1ψTE-ν2Zg(g112ψTM-g121ψTM),
ik(ν2-γ2)H2=γ2ψTE-ν2Zg(g212ψTM-g221ψTM),
H3=ψTE.
ψ(x1, x2, x3)=ψ1(x1)ψ2(x2)ψ3(x3), ψ3(x3)=exp(-ikγx3).
1ψ11gddx1g22gdψ1dx1-1ψ2dψ2dx21ψ11gddx1g21gψ1+1ψ2ddx2dψ2dx2 g11g-1ψ2dψ2dx2g12g1ψ1dψ1dx1+k2(ν2-γ2)=0.
1ψ2ddx2dψ2dx2,1ψ2dψ2dx2
ψ2(x2)=exp(-ikrx2).
-ikrψ1=φ1, -ikrddx1g21gψ1+g12gdψ1dx1-g11gφ1=k2(ν2-γ2)gψ1+ddx1g22gddx1ψ1.
-ikrM1ψ1φ1=N1ψ1φ1,
ψ=exp(-ik·r),
k=k(αex+βey+γez)withα2+β2+γ2=ν2.
ψ=exp{-ik[αf(x1)+βg(x1)]}×exp[-ik(αcx+βcy)x2]exp(-ikγx3).
r=αcx+βcy.
k=k(αex+βey).
cos ξ=k·e2ke2=αcx+βcyν2-γ2cx2+cy2.
cos ξ=rν2-γ2cx2+cy2.
f(x1)=ax1+f1(x1),g(x1)=bx1+g1(x1),
ψ1(x1)=n=-+ψnen(x1),φ1(x1)=n=-+φnen(x1),
en(x1)=exp(-ikαnx1),αn=α0+n λd,nZ.
ψ1(x1)=exp[-ik(αa+βb)x1]×n=-n=-Ln(α, β)exp-in 2πx1d,
α0=αa+βb.
[Mmn]=ψnφn=1r[Nmn]ψnφn,
[Mmn]=αmg21gm-n+g12gm-nαn-ikg11gm-nδmn0,
[Nmn]=-(ν2-γ2)(g)m-n+αmg22gm-nαn00ikδmn.
[N]-1[M]ψnφn=1rψnφn.
ψq=exp(-ikγx3)exp(-ikrqx2)n=-n=-ψnq exp(-ikαnx1).
1g(g112ψ-g121ψ)=-ikr(ν2-γ2)gψ-ikr1g22g1ψ+1g21gψ.
E1=-iγk(ν2-γ2)1ψTM+ZL12ψTE,
E3=ψTM,
H1=-iγk(ν2-γ2)1ψTE-ν2ZL12ψTM,
H3=ψTE,
L12ψ=-1rgψ-1r1k2(ν2-γ2)1g22g1ψ-ik(ν2-γ2)1g21gψ.
x(x1, x2)=x1+f1(x1)+cxx2,
y(x1, x2)=g1(x1)+cyx2.
ϕ=(k, k),θ=(-ey, k).
kx=k cos ϕ sin θ,
ky=-k cos ϕ cos θ,
kz=k sin ϕ.
E3i=(cos ϕ sin δ)exp(-ik·r),
H3i=(cos ϕ cos δ)exp(-ik·r).
α=cos ϕ sin θ,β=-cos ϕ cos θ,γ=sin ϕ.
ψqN(x1, x2, x3)=exp(-iγx3)exp(-ikrqNx2)×n=-N+NψnqN exp(-ikαnx1),
ψN(x1, x2, x3)=qsqψqN(x1, x2, x3).
rq=αqcx+βqcywithαq2+βq2=cos2 ϕ.
kq=αqex+βqey+γez.
βqd=cos ϕ cos θqd,βqd>0,
sin θqd=sin θ+q λd cos ϕ
βqi=cos ϕ cos θqi,βqi<0,
βqt=-cos ϕ cos θqt,βqi<0,
ν sin θqt=sin θ+q λd cos ϕ.
x(x1, x2)=x1+kfa1(x1)+cxx2,
y(x1, x2)=hb1(x1)+cyx2,
b1=a1,hkf=cycx;
cx=cos α,cy=sin α,α[0, π].
u=kV=kx1,v=kR-hsin αa1(x1)=kx2.
cx=sin Φ,cy=cos Φ,Φ[-π/2,+π/2].
x=x1 cos Φ,y=x1 sin Φ+hcos Φa1(x1)+x2.
u=x2=y-hcos Φa1xcos Φ-(tan Φ)x.
x=du-ad2πsin(2πu),y=h2cos(2πu)+vd.
x1=du,x2=dv,cx=0,cy=1.
x(x1, x2)=x1+kf sin(2x1),y(x1, x2)=h2sin x1.

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