Abstract

The surface impedance boundary condition is used to include the effect of high conductivity of metals in the integral theory of perfectly conducting gratings. As an intuitive approach, the diffraction formalism proposed by Petit for the treatment of infinitely conducting gratings in P polarization is extended to highly conducting materials by introducing the concept of equivalent surface current density. Then, integral equations for both polarizations are deduced in a mathematically rigorous way. The new method is used to calculate the efficiencies of sinusoidal gratings at infrared and visible light, and the numerical results are compared with those obtained using Maxwell boundary conditions and also with the perfect conductivity model.

© 1987 Optical Society of America

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References

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  1. M. Neviere, M. Cadilhac, R. Petit, “Applications of Conformal Mappings to the Diffraction of Electromagnetic Waves by a Grating,” IEEE Trans. Antennas Propag. AP-21, 37 (1973).
    [CrossRef]
  2. R. A. Depine, J. M. Simon, “Diffraction Grating Efficiencies: Conformal Mapping Method for a Good Real Conductor,” Opt. Acta 29, 1459 (1982).
    [CrossRef]
  3. R. A. Depine, J. M. Simon, “Surface Impedance Boundary Condition for Metallic Diffraction Gratings in the Optical and Infrared Range,” Opt. Acta 30, 313 (1983).
    [CrossRef]
  4. D. Maystre, “Integral Methods,” in Electromagnetic Theory of Gratings, R. Petit, Ed. (Springer-Verlag, Berlin, 1980), p. 63.
    [CrossRef]
  5. R. Petit, “Etude numérique de la diffraction par un réseau,” C. R. Acad. Sci. 260, 4454 (1965).
  6. D. Maystre, M. Neviere, R. Petit, “Experimental Verifications and Applications of the Theory,” in Electromagnetic Theory of Gratings, R. Petit, Ed. (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  7. A. Hessel, A. A. Oliner, “A New Theory of Wood’s Anomalies on Optical Gratings,” Appl. Opt. 4, 1275 (1965).
    [CrossRef]
  8. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  9. M. A. Leontovich, “A Method for the Solution of the Problem of the Propagation of Electromagnetic Waves Along the Surface of the Earth,” Izv. Akad. Nauk SSSR Ser. Fiz. 8, 16 (1944).
  10. L. Landau, F. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), p. 281.
  11. J. Pavageau, J.Bousquet Bousquet, “Diffraction par un réseau conducteur, nouvelle méthode de résolution,” Opt. Acta 17, 469 (1970).
    [CrossRef]

1983 (1)

R. A. Depine, J. M. Simon, “Surface Impedance Boundary Condition for Metallic Diffraction Gratings in the Optical and Infrared Range,” Opt. Acta 30, 313 (1983).
[CrossRef]

1982 (1)

R. A. Depine, J. M. Simon, “Diffraction Grating Efficiencies: Conformal Mapping Method for a Good Real Conductor,” Opt. Acta 29, 1459 (1982).
[CrossRef]

1973 (1)

M. Neviere, M. Cadilhac, R. Petit, “Applications of Conformal Mappings to the Diffraction of Electromagnetic Waves by a Grating,” IEEE Trans. Antennas Propag. AP-21, 37 (1973).
[CrossRef]

1970 (1)

J. Pavageau, J.Bousquet Bousquet, “Diffraction par un réseau conducteur, nouvelle méthode de résolution,” Opt. Acta 17, 469 (1970).
[CrossRef]

1965 (2)

R. Petit, “Etude numérique de la diffraction par un réseau,” C. R. Acad. Sci. 260, 4454 (1965).

A. Hessel, A. A. Oliner, “A New Theory of Wood’s Anomalies on Optical Gratings,” Appl. Opt. 4, 1275 (1965).
[CrossRef]

1944 (1)

M. A. Leontovich, “A Method for the Solution of the Problem of the Propagation of Electromagnetic Waves Along the Surface of the Earth,” Izv. Akad. Nauk SSSR Ser. Fiz. 8, 16 (1944).

Bousquet, J.Bousquet

J. Pavageau, J.Bousquet Bousquet, “Diffraction par un réseau conducteur, nouvelle méthode de résolution,” Opt. Acta 17, 469 (1970).
[CrossRef]

Cadilhac, M.

M. Neviere, M. Cadilhac, R. Petit, “Applications of Conformal Mappings to the Diffraction of Electromagnetic Waves by a Grating,” IEEE Trans. Antennas Propag. AP-21, 37 (1973).
[CrossRef]

Depine, R. A.

R. A. Depine, J. M. Simon, “Surface Impedance Boundary Condition for Metallic Diffraction Gratings in the Optical and Infrared Range,” Opt. Acta 30, 313 (1983).
[CrossRef]

R. A. Depine, J. M. Simon, “Diffraction Grating Efficiencies: Conformal Mapping Method for a Good Real Conductor,” Opt. Acta 29, 1459 (1982).
[CrossRef]

Hessel, A.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Landau, L.

L. Landau, F. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), p. 281.

Leontovich, M. A.

M. A. Leontovich, “A Method for the Solution of the Problem of the Propagation of Electromagnetic Waves Along the Surface of the Earth,” Izv. Akad. Nauk SSSR Ser. Fiz. 8, 16 (1944).

Lifshitz, F.

L. Landau, F. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), p. 281.

Maystre, D.

D. Maystre, M. Neviere, R. Petit, “Experimental Verifications and Applications of the Theory,” in Electromagnetic Theory of Gratings, R. Petit, Ed. (Springer-Verlag, Berlin, 1980).
[CrossRef]

D. Maystre, “Integral Methods,” in Electromagnetic Theory of Gratings, R. Petit, Ed. (Springer-Verlag, Berlin, 1980), p. 63.
[CrossRef]

Neviere, M.

M. Neviere, M. Cadilhac, R. Petit, “Applications of Conformal Mappings to the Diffraction of Electromagnetic Waves by a Grating,” IEEE Trans. Antennas Propag. AP-21, 37 (1973).
[CrossRef]

D. Maystre, M. Neviere, R. Petit, “Experimental Verifications and Applications of the Theory,” in Electromagnetic Theory of Gratings, R. Petit, Ed. (Springer-Verlag, Berlin, 1980).
[CrossRef]

Oliner, A. A.

Pavageau, J.

J. Pavageau, J.Bousquet Bousquet, “Diffraction par un réseau conducteur, nouvelle méthode de résolution,” Opt. Acta 17, 469 (1970).
[CrossRef]

Petit, R.

M. Neviere, M. Cadilhac, R. Petit, “Applications of Conformal Mappings to the Diffraction of Electromagnetic Waves by a Grating,” IEEE Trans. Antennas Propag. AP-21, 37 (1973).
[CrossRef]

R. Petit, “Etude numérique de la diffraction par un réseau,” C. R. Acad. Sci. 260, 4454 (1965).

D. Maystre, M. Neviere, R. Petit, “Experimental Verifications and Applications of the Theory,” in Electromagnetic Theory of Gratings, R. Petit, Ed. (Springer-Verlag, Berlin, 1980).
[CrossRef]

Simon, J. M.

R. A. Depine, J. M. Simon, “Surface Impedance Boundary Condition for Metallic Diffraction Gratings in the Optical and Infrared Range,” Opt. Acta 30, 313 (1983).
[CrossRef]

R. A. Depine, J. M. Simon, “Diffraction Grating Efficiencies: Conformal Mapping Method for a Good Real Conductor,” Opt. Acta 29, 1459 (1982).
[CrossRef]

Appl. Opt. (1)

C. R. Acad. Sci. (1)

R. Petit, “Etude numérique de la diffraction par un réseau,” C. R. Acad. Sci. 260, 4454 (1965).

IEEE Trans. Antennas Propag. (1)

M. Neviere, M. Cadilhac, R. Petit, “Applications of Conformal Mappings to the Diffraction of Electromagnetic Waves by a Grating,” IEEE Trans. Antennas Propag. AP-21, 37 (1973).
[CrossRef]

Izv. Akad. Nauk SSSR Ser. Fiz. (1)

M. A. Leontovich, “A Method for the Solution of the Problem of the Propagation of Electromagnetic Waves Along the Surface of the Earth,” Izv. Akad. Nauk SSSR Ser. Fiz. 8, 16 (1944).

Opt. Acta (3)

J. Pavageau, J.Bousquet Bousquet, “Diffraction par un réseau conducteur, nouvelle méthode de résolution,” Opt. Acta 17, 469 (1970).
[CrossRef]

R. A. Depine, J. M. Simon, “Diffraction Grating Efficiencies: Conformal Mapping Method for a Good Real Conductor,” Opt. Acta 29, 1459 (1982).
[CrossRef]

R. A. Depine, J. M. Simon, “Surface Impedance Boundary Condition for Metallic Diffraction Gratings in the Optical and Infrared Range,” Opt. Acta 30, 313 (1983).
[CrossRef]

Other (4)

D. Maystre, “Integral Methods,” in Electromagnetic Theory of Gratings, R. Petit, Ed. (Springer-Verlag, Berlin, 1980), p. 63.
[CrossRef]

D. Maystre, M. Neviere, R. Petit, “Experimental Verifications and Applications of the Theory,” in Electromagnetic Theory of Gratings, R. Petit, Ed. (Springer-Verlag, Berlin, 1980).
[CrossRef]

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

L. Landau, F. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), p. 281.

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Figures (10)

Fig. 1
Fig. 1

Presentation of the grating problem.

Fig. 2
Fig. 2

Diffracted order efficiencies for a sinusoidal gold grating as calculated with a rigorous formalism (continuous curve), the approximated formalism (circles), and the perfect conductor formalism (crosses). The groove depth to wavelength ratio is h/λ = 0.2 and the dielectric constant is = −5.28 + i1.48 (P polarization).

Fig. 3
Fig. 3

As Fig. 2 but for = −21.60 + i1.40 which corresponds to a wavelength λ = 800 nm.

Fig. 4
Fig. 4

Diffracted order efficiencies for a sinusoidal gold grating as calculated with a rigorous formalism (continuous curve), the approximated formalism (circles), and the perfect conductor formalism (crosses). The groove depth to wavelength ratio is twice the value considered for Figs. 2 and 3. The dielectric constant is = −5.28 + i1.48 (P polarization).

Fig. 5
Fig. 5

As Fig. 4 but for = −21.60 + i1.40.

Fig. 6
Fig. 6

As Fig. 4 but for = −125.15 + i12.09.

Fig. 7
Fig. 7

+1 and −3 order efficiencies for a sinusoidal gold grating calculated with a rigorous formalism (continuous curve), the approximated formalism (circles), and the perfect conductor formalism (crosses). The groove depth to wavelength ratio is h/λ = 0.4 and the dielectric constant is = −5.28 + i1.48 (S polarization).

Fig. 8
Fig. 8

As Fig. 7, order 0.

Fig. 9
Fig. 9

As Fig. 7, order −1.

Fig. 10
Fig. 10

As Fig. 7, order −2.

Equations (29)

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E = Z n ^ × H , Z R 0 ,
Z = - 1 / 2 = n - 1 ,
K eff = 0 J ( ξ ) d ξ = c 4 π n ^ × H ,
{ 2 + ω 2 c 2 } f ( x , y ) = 0 ,             y > g ( x ) ,
f + [ x , g ( x ) ] = Z i c ω f + n ^ [ x , g ( x ) ] .
f + = f - ,
f + n ^ - f - n ^ = 4 π ω i c 2 K eff ,
f d + ( x , y ) = 4 π ω i c 2 0 d g [ x - x , y - g ( x ) ] K eff ( x ) d l ,
g ( x , y ) = 1 2 i d n = - 1 φ n exp [ i φ n y + i γ n x ] ,
γ n = ω c sin θ + 2 π n d , φ n = ( ω 2 c 2 - γ n 2 ) 1 / 2 ,             I m φ n > 0 ,
f d + n ^ [ x , g ( x ) ] = 2 π ω i c 2 K eff ( x ) + 4 π ω i c 2 0 d g n ^ [ x - x , g ( x ) - g ( x ) ] K eff ( x ) d l .
0 d G ( x , x ) ϕ ( x ) d x + exp [ - i φ 0 g ( x ) ] = Z i k ( 1 + g 2 ( x ) ) 1 / 2 { ϕ ( x ) 2 + 0 d N ( x , x ) ϕ ( x ) d x - i [ k sin θ g ( x ) + ϕ 0 ] exp [ - i ϕ 0 g ( x ) ] } ,
G ( x , x ) = 1 2 i d n = - 1 φ n exp [ i φ n g ( x ) - g ( x ) + i 2 π n d ( x - x ) ] .
N ( x , x ) = 1 2 d n = - { - γ n g ( x ) φ n + s g [ g ( x ) - g ( x ) ] × exp [ i φ n g ( x ) - g ( x ) + i 2 π n d ( x - x ) ] } ,
ϕ ( x ) = 4 π ω i c 2 ( 1 + g 2 ( x ) ) 1 / 2 exp [ - i k sin θ x ] K eff ( x ) .
f d ( x , y ) = n = - B n exp [ i φ n y + i γ n x ] ,
B n = 1 2 i d φ n 0 d exp [ - i φ n g ( x ) - i 2 π n d x ] ϕ ( x ) d x ,
e n = Re { B n 2 φ n φ n } .
0 d G ( x , x ) ϕ ( x ) d x = - exp [ - i φ 0 g ( x ) ] ,
f + [ x , g ( x ) ] = σ f + n ^ [ x , g ( x ) ] , σ = { Z i c ω P polarization , i c ω Z S polarization .
F ( x , y ) = 0 d g [ x - x , y - g ( x ) ] η ( x ) d l ,
η ( x ) = d F + d n ^ [ x , g ( x ) ] - d F - d n ^ [ x , g ( x ) ] .
F + [ x , g ( x ) ] = exp [ i k x sin θ ] 0 d G ( x , x ) n ˜ ( x ) d x ,
d F + d n ^ [ x , g ( x ) ] = exp [ i k x sin θ ] [ 1 + g 2 ( x ) ] 1 / 2 { η ˜ ( x ) 2 + 0 d N ( x , x ) η ˜ ( x ) d x } ,
η ˜ ( x ) = η ( x ) [ 1 + g 2 ( x ) ] 1 / 2 exp [ - i k x sin θ ] ,
exp [ - i φ 0 g ( x ) ] + 0 d G ( x , x ) η ˜ ( x ) d x = σ [ 1 + g 2 ( x ) ] 1 / 2 { η ˜ ( x ) 2 + 0 d N ( x , x ) η ˜ ( x ) d x - i [ k sin θ g ( x ) + φ 0 ] exp [ - i φ 0 g ( x ) ] } .
exp [ - i φ 0 g ( x ) ] + 0 d G ( x , x ) η ˜ ( x ) d x = Z i k [ 1 + g 2 ( x ) ] 1 / 2 { η ˜ ( x ) 2 + 0 d N ( x , x ) η ˜ ( x ) d x - i [ k sin θ g ( x ) + φ 0 ] exp [ - i φ 0 g ( x ) ] } ,
k Z i { exp [ - i φ 0 g ( x ) ] + 0 d G ( x , x ) η ˜ ( x ) d x } = 1 [ 1 + g 2 ( x ) ] 1 / 2 { η ˜ ( x ) 2 + 0 d N ( x , x ) η ˜ ( x ) d x - i [ k sin θ g ( x ) + φ 0 ] exp [ - i φ 0 g ( x ) ] } .
N ( x , x ) = - g ( x ) [ 1 2 d n = - γ n φ n + i 2 π k sin θ ] - 1 4 π g ( x ) 1 + g 2 ( x ) .

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