Abstract

We present a theoretical approach for calculating the fields diffracted by gratings made of highly conducting wires that have a rectangular shape. The fields between the wires are represented in terms of modal expansions that satisfy the approximated impedance boundary condition. Our results show that this procedure is particularly suited to dealing with gold gratings used in the infrared range, a spectral region where the assumption of a perfect conductor does not hold, and where the rigorous modal method assuming penetrable wires exhibits numerical instabilities linked with the high conductivity of gold. Numerical results are presented, and the theory is used to determine wire parameters by fitting theoretical and experimental data.

© 1993 Optical Society of America

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  1. M. L. Schattenburg, C. R. Canizares, D. Dewey, A. M. Levine, T. H. Markert, H. I. Smith, “Transmission grating spectroscopy and the advanced x-ray astrophysics facility (AXAF),” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 210–218 (1988).
  2. A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).
  3. H. Brauninger, H. Kraus, H. Dangschat, P. Predehl, J. Trumper, “Fabrication of transmission gratings for use in cosmic x-ray and XUV astronomy,” Appl. Opt. 18, 3502–3505 (1979).
    [CrossRef] [PubMed]
  4. H. Lochbihler, P. Predehl, “Characterization of x-ray transmission gratings,” Appl. Opt. 31, 964–971 (1992).
    [CrossRef] [PubMed]
  5. R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 26–30.
  6. J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1979).
    [CrossRef]
  7. H. Brauninger, P. Predehl, K. P. Beuermann, “Transmission grating efficiencies for wavelengths between 5.4 A and 44.8 A,” Appl. Opt. 18, 368–373 (1979).
    [PubMed]
  8. 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), pp. 159–223.
    [CrossRef]
  9. R. A. Depine, “Perfectly conducting diffraction grating formalisms extended to good conductors via the surface impedance boundary condition,” Appl. Opt. 26, 2348–2354 (1987).
    [CrossRef] [PubMed]
  10. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
    [CrossRef]
  11. L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
    [CrossRef]
  12. A. Roberts, R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt. 34, 511–538 (1987).
    [CrossRef]
  13. L. C. Botten, M. S. Craig, R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
    [CrossRef]
  14. R. A. Depine, “Surface impedance boundary conditions used to study light scattering from metallic surfaces,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 239–254.
  15. Y. Nakata, M. Koshiba, “Boundary-element analysis of plane wave diffraction from groove-type dielectric and metallic gratings,” J. Opt. Soc. Am. A 7, 1494–1502 (1990).
    [CrossRef]

1992 (1)

1990 (1)

1987 (2)

1983 (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
[CrossRef]

1981 (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

1979 (3)

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1979).
[CrossRef]

Beuermann, K. P.

Bleeker, J. A. M.

A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Brauninger, H.

Brinkman, A. C.

A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).

Canizares, C. R.

M. L. Schattenburg, C. R. Canizares, D. Dewey, A. M. Levine, T. H. Markert, H. I. Smith, “Transmission grating spectroscopy and the advanced x-ray astrophysics facility (AXAF),” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 210–218 (1988).

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Dangschat, H.

de Korte, P. A. J.

A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).

Depine, R. A.

R. A. Depine, “Perfectly conducting diffraction grating formalisms extended to good conductors via the surface impedance boundary condition,” Appl. Opt. 26, 2348–2354 (1987).
[CrossRef] [PubMed]

R. A. Depine, “Surface impedance boundary conditions used to study light scattering from metallic surfaces,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 239–254.

Dewey, D.

M. L. Schattenburg, C. R. Canizares, D. Dewey, A. M. Levine, T. H. Markert, H. I. Smith, “Transmission grating spectroscopy and the advanced x-ray astrophysics facility (AXAF),” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 210–218 (1988).

Dijkstra, J. H.

A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).

Fox, J. R.

J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1979).
[CrossRef]

Heise, J.

A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).

Koshiba, M.

Kraus, H.

Levine, A. M.

M. L. Schattenburg, C. R. Canizares, D. Dewey, A. M. Levine, T. H. Markert, H. I. Smith, “Transmission grating spectroscopy and the advanced x-ray astrophysics facility (AXAF),” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 210–218 (1988).

Lochbihler, H.

Markert, T. H.

M. L. Schattenburg, C. R. Canizares, D. Dewey, A. M. Levine, T. H. Markert, H. I. Smith, “Transmission grating spectroscopy and the advanced x-ray astrophysics facility (AXAF),” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 210–218 (1988).

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), pp. 159–223.
[CrossRef]

McPhedran, R. C.

A. Roberts, R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt. 34, 511–538 (1987).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Mewe, R.

A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).

Nakata, Y.

Neviere, M.

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), pp. 159–223.
[CrossRef]

Paerels, T.

A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).

Petit, R.

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), pp. 159–223.
[CrossRef]

R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 26–30.

Predehl, P.

Roberts, A.

A. Roberts, R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt. 34, 511–538 (1987).
[CrossRef]

Schattenburg, M. L.

M. L. Schattenburg, C. R. Canizares, D. Dewey, A. M. Levine, T. H. Markert, H. I. Smith, “Transmission grating spectroscopy and the advanced x-ray astrophysics facility (AXAF),” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 210–218 (1988).

Smith, H. I.

M. L. Schattenburg, C. R. Canizares, D. Dewey, A. M. Levine, T. H. Markert, H. I. Smith, “Transmission grating spectroscopy and the advanced x-ray astrophysics facility (AXAF),” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 210–218 (1988).

Trumper, J.

van Rooijen, J. J.

A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).

Wilson, I. J.

J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1979).
[CrossRef]

Appl. Opt. (4)

Comput. Phys. Commun. (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
[CrossRef]

J. Mod. Opt. (1)

A. Roberts, R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt. 34, 511–538 (1987).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Acta (3)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1979).
[CrossRef]

Other (5)

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), pp. 159–223.
[CrossRef]

M. L. Schattenburg, C. R. Canizares, D. Dewey, A. M. Levine, T. H. Markert, H. I. Smith, “Transmission grating spectroscopy and the advanced x-ray astrophysics facility (AXAF),” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 210–218 (1988).

A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).

R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 26–30.

R. A. Depine, “Surface impedance boundary conditions used to study light scattering from metallic surfaces,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 239–254.

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Figures (6)

Fig. 1
Fig. 1

Transmittance in normal incidence as a function of λ/d for a wire grating used in S polarization. Solid curve, experiment; dashed curve, best fit obtained by using the theory of perfect conductor (d = 0.9493 μm, b = 0.637 μm, h = 0.424 μm).

Fig. 2
Fig. 2

Presentation of the problem.

Fig. 3
Fig. 3

Trajectories of the first complex eigenvalues βnS for c/λ = 0.5 when the imaginary part of ν is decreased from ∞ to 4 while keeping the real part of ν equal to 1.5. Cases correspond to (a) n = 0 and (b) n = 1,2, 3, and 4.

Fig. 4
Fig. 4

Real and imaginary parts of the refractive index ν = νR + I as a function of wavelength for gold in the near-infrared region.

Fig. 5
Fig. 5

Transmittance in normal incidence as a function of λ/d for a wire grating used in S polarization. Solid curve, experiment; dashed curve, best fit obtained by using the improved theory (d = 0.9493 μm, b = 0.643 μm, h = 0.386 μm).

Fig. 6
Fig. 6

Transmittance in normal incidence as a function of λ/d near a Rayleigh anomaly. Comparison shown between (a) experiment (solid curve) and the theory of perfect conductor (dashed curve) with the same parameters as in Fig. 1 and (b) experiment (solid curve) and the improved theory (dashed curve) with the same parameters as in Fig. 5.

Equations (46)

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

f ( x , y ) = n = { R n exp [ i χ n ( y h / 2 ) ] + δ n , 0 exp [ i χ 0 ( y h / 2 ) ] } exp ( i α n x ) , for y h / 2 ,
f ( x , y ) = n = T n exp { i [ α n x χ n ( y + h / 2 ) ] } , for y h / 2 ,
α n = α 0 + n ( 2 π / d ) , α 0 = k 0 sin θ 0 k 0 = w / c = 2 π / λ ,
χ n = ( k 0 2 α n 2 ) 1 / 2 if | k 0 | | α n | , χ n = i ( α n 2 k 0 2 ) 1 / 2 if | k 0 | < | α n | ,
f ( x , y ) = m = 0 ϕ m ( x , y ) .
ϕ m 0 ( x , y ) = cos ( β m 0 x ) [ a m 0 sin ( μ m 0 y ) + b m 0 cos ( μ m 0 y ) ] , S modes ,
ϕ m 0 ( x , y ) = sin ( β m 0 x ) [ a m 0 sin ( μ m 0 y ) + b m 0 cos ( μ m 0 y ) ] , P modes ,
E = Z n ̂ × H ,
f n ̂ = Z k 0 i f , for S polarization ,
f = Z i k 0 f n ̂ , for P polarization .
ϕ m S ( x , y ) = u m S ( x ) [ a m S sin ( μ m S y ) + b m S cos ( μ m S y ) ] , S modes ,
ϕ m P ( x , y ) = u m P ( x ) [ a m P sin ( μ m P y ) + b m P cos ( μ m P y ) ] , P modes ,
u m S ( x ) = η S β m S sin β m S x + cos β m S x ,
u m P ( x ) = η P cos β m P x + 1 β m P sin β m P x ,
tan ( β m S c ) = 2 η S β m S β m S 2 η S 2 ,
tan ( β m P c ) = 2 η P β m P ( β m P η P ) 2 1 ,
n = ( R n + δ n , 0 ) exp ( i α n x ) = m ( a m + b m ) u m ( x )
i n = χ n ( R n δ n , 0 ) exp ( i α n x ) = m ( D 1 m a m + D 2 m b m ) u m ( x )
n = T n exp ( i α n x ) = m ( a m + b m ) u m ( x )
i n = χ n T n exp ( i α n x ) = m ( D 1 m a m D 2 m b m ) u m ( x )
a m = a m sin ( μ m h / 2 ) ,
b m = b m cos ( μ m h / 2 ) ,
D 1 m = μ m cot ( μ m h / 2 ) ,
D 2 m = μ m tan ( μ m h / 2 ) .
i n = χ n ( R n δ n , 0 ) exp ( i α n x ) = η S n = ( R n + δ n , 0 ) exp ( i α n x ) ,
i n = χ n T n exp ( i α n x ) = η S n = T n exp ( i α n x ) ,
n = ( R n + δ n , 0 ) exp ( i α n x ) = i η P n = χ n ( R n δ n , 0 ) exp ( i α n x ) ,
n = T n exp ( i α n x ) = i η P n = χ n T n exp ( i α n x ) ,
I m , j S = 0 c u m S ( x ) u j S ( x ) d x = I m S δ m , j ,
I m S = [ 1 + ( η S β m ) 2 ] c 2 + η S β m 2 .
( a j S + b j S ) I j S = n = K j , n S ( R n S + δ n , 0 ) ,
( a j S + b j S ) I j S = n = K j , n S T n S ,
K j , n S = 0 c exp ( i α n x ) u j S ( x ) d x .
i χ q ( R q S δ q , 0 ) = m J q , m S ( D 1 m a m S + D 2 m b m S ) + η S n = Q q , n S ( R n S + δ n , 0 ) ,
J q , m S = 1 d 0 c exp ( i α q x ) u m S ( x ) d x ,
Q q , n S = 1 d c d exp [ i ( α n α q ) x ] d x .
i χ q T q S = m J q , m S ( D 1 m a m S D 2 m b m S ) + η S n = Q q , n S T n S .
I m , j P = 0 c u m P ( x ) u j P ( x ) d x = I m P δ m , j ,
I m P = ( η P 2 + 1 β m 2 ) c 2 + η P β m 2 .
( D 1 m a j P + D 2 m b j P ) I j P = i n = χ n K j , n P ( R n P δ n , 0 ) ,
( D 1 m a j P D 2 m b j P ) I j P = i n = χ n K j , n P T n P ,
K j , n P = 0 c exp ( i α n x ) u j P ( x ) d x .
R q P + δ q , 0 = m J q , m P ( a m P + b m P ) + i η P n = χ n Q q , n P ( R n P δ n , 0 ) ,
J q , m P = 1 d 0 c exp ( i α q x ) u m P ( x ) d x ,
Q q , n P = 1 d c d exp [ i ( α n α q ) x ] d x ,
T q P = m J q , m P ( a m P + b m P ) + i η P n = χ n Q q , n P T n P .

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