Abstract

This paper points out a number of errors that have appeared in the literature concerning transmission-line models for metal grid reflectors (strip gratings and meshes) especially in regard to the design of laser mirrors and filter elements for use at submillimeter wavelengths. General formulas are given for the transmittance of lossy grids and for the equivalent circuit impedances to be used in these formulas for strip gratings and meshes at a plane boundary between two lossless dielectrics. The results apply for normal incidence and for wavelengths in both dielectrics greater than the grid period. Limitations of the transmission-line models for meshes at dielectric boundaries are discussed.

© 1985 Optical Society of America

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References

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  1. N. Marcuvitz, Waveguide Handbook, Vol. 10, MIT Radiation Laboratory Series (McGraw-Hill, New York, 1951), pp. 280–285.
  2. R. Ulrich, “Far-Infrared Properties of Metallic Mesh and its Complementary Structure,” Infrared Phys. 7, 37 (1967).
    [CrossRef]
  3. For the results in Table 3 of Ref. 2 to be correct Z0 must be taken as 2ω0 In csc(πa/g) for inductive meshes and the reciprocal of this expression for capacitive meshes. Equation (13) is, therefore, incorrect.
  4. R. Ulrich, “Effective Low-Pass Filters for Far-Infrared Frequencies,” Infrared Phys. 7, 65 (1967).
    [CrossRef]
  5. K. D. Möller, W. G. Rothschild, Far-Infrared Spectroscopy (Wiley-Interscience, New York, 1971).
  6. G. D. Holah, “Far-Infrared and Submillimeter Wavelength Filters,” in Infrared and Millimeter Waves, Vol. 6, K. J. Button, Ed. (Academic, New York, 1982).
  7. K. Sakai, L. Genzel, “Far-Infrared Metal Mesh Filters and Fabry-Perot Interferometry,” in Reviews of Infrared and Millimeter Waves, Vol. 1, K. J. Button, Ed. (Plenum, New York, 1983).
    [CrossRef]
  8. In Ref. 4Eq. (3) should be Z0 = [2ω0 ln csc(πa/g)]−1. In Ref. 5 Table 3.2 is correct for Ξi = ω0 ln csc(πa/g) and Ξc = [4ω0 ln csc(πa/g)]−1, contradicting Eq. (3.31). In Eq. (3.33) σ must be taken as one quarter of the dc bulk conductivity of the mesh metal. In Ref. 6 Eq. (50) should be Z0c = [2ω0 ln csc(πa/g)]−1. In Ref. 7 Table II-B is correct for Ẑi=Ξi and Ẑc=Ξc as given above, contradicting Eq. (28). In Eq. (31) σ is one quarter of the dc bulk conductivity.
  9. E. J. Danielewicz, P. D. Coleman, “Hybrid Metal Mesh-Dielectric Mirrors for Optically Pumped Far Infrared Lasers,” Appl. Opt. 15, 761 (1976).
    [CrossRef] [PubMed]
  10. S. M. Wolfe, K. J. Button, J. Waldman, D. R. Cohn, “Modulated Submillimeter Laser Interferometer System for Plasma Density Measurements,” Appl. Opt. 15, 2645 (1976).
    [CrossRef] [PubMed]
  11. D. A. Weitz, W. J. Skocpol, M. Tinkham, “Capacitive-Mesh Output Couplers for Optically Pumped Far-Infrared Lasers,” Opt. Lett. 3, 13 (1978).
    [CrossRef] [PubMed]
  12. T. Timusk, P. L. Richards, “Near Millimeter Wave Bandpass Filters,” Appl. Opt. 20, 1355 (1981).
    [CrossRef] [PubMed]
  13. S. T. Shanahan, N. R. Heckenberg, “Transmission Line Model of Substrate Effects on Capacitive Mesh Couplers,” Appl. Opt. 20, 4019 (1981).
    [CrossRef] [PubMed]
  14. In Ref. 9Eqs. (13)–(15) treat an inductive strip grating, not a mesh. The applicability of Eqs. (16) and (17) is discussed in this paper. In Ref. 10 the first of the equations on the bottom of p. 2646 is not applicable for meshes, as discussed in this paper [note the misprint in Eq. (2) the square bracket should be squared]. In Ref. 11 the curves in Fig. 1 cannot be correct because they show no shift in mesh resonance caused by the dielectric substrate [see discussion following Eqs. (6)–(8), this paper]. In Ref. 12, Fig. 1, the equations are correct for ω0 = 2π/g.(Note that the definitions of ω0 and ω = 2π/λ differ from those used here.) However, ω0 = 2π/g is not reasonable for a mesh at a dielectric boundary [see discussion following our Eqs. (6)–(8) and Ref. 12, p. 1357]. For ω0 = k × 2π/g, Y0 should be 2k(n12+n22)ln csc(πa/g). In Ref. 13Eqs. (5)–(11) are not correct, see Ref. 16.
  15. P. F. Goldsmith, “Quasi-Optical Techniques at Millimeter and Submillimeter Wavelengths,” in Infrared and Millimeter Waves, Vol. 6, K. J. Button, Ed. (Academic, New York, 1982).
  16. R. C. Compton, L. B. Whitbourn, R. C. McPhedran, “Strip Gratings at a Dielectric Interface and Application of Babinet's Principle,” Appl. Opt. 23, 3236 (1984).
    [CrossRef] [PubMed]
  17. R. C. Compton, L. B. Whitbourn, R. C. McPhedran, “Simple Formulae for the Transmittance of Strip Gratings,” Int. J. Infrared Millimeter Waves 4, 901 (1983).
    [CrossRef]
  18. R. Ulrich, T. J. Bridges, M. A. Pollack, “Variable Metal Mesh Coupler for Far IR Lasers,” Appl. Opt. 9, 2511 (1970).
    [CrossRef] [PubMed]

1984

1983

R. C. Compton, L. B. Whitbourn, R. C. McPhedran, “Simple Formulae for the Transmittance of Strip Gratings,” Int. J. Infrared Millimeter Waves 4, 901 (1983).
[CrossRef]

1981

1978

1976

1970

1967

R. Ulrich, “Far-Infrared Properties of Metallic Mesh and its Complementary Structure,” Infrared Phys. 7, 37 (1967).
[CrossRef]

R. Ulrich, “Effective Low-Pass Filters for Far-Infrared Frequencies,” Infrared Phys. 7, 65 (1967).
[CrossRef]

Bridges, T. J.

Button, K. J.

Cohn, D. R.

Coleman, P. D.

Compton, R. C.

R. C. Compton, L. B. Whitbourn, R. C. McPhedran, “Strip Gratings at a Dielectric Interface and Application of Babinet's Principle,” Appl. Opt. 23, 3236 (1984).
[CrossRef] [PubMed]

R. C. Compton, L. B. Whitbourn, R. C. McPhedran, “Simple Formulae for the Transmittance of Strip Gratings,” Int. J. Infrared Millimeter Waves 4, 901 (1983).
[CrossRef]

Danielewicz, E. J.

Genzel, L.

K. Sakai, L. Genzel, “Far-Infrared Metal Mesh Filters and Fabry-Perot Interferometry,” in Reviews of Infrared and Millimeter Waves, Vol. 1, K. J. Button, Ed. (Plenum, New York, 1983).
[CrossRef]

Goldsmith, P. F.

P. F. Goldsmith, “Quasi-Optical Techniques at Millimeter and Submillimeter Wavelengths,” in Infrared and Millimeter Waves, Vol. 6, K. J. Button, Ed. (Academic, New York, 1982).

Heckenberg, N. R.

Holah, G. D.

G. D. Holah, “Far-Infrared and Submillimeter Wavelength Filters,” in Infrared and Millimeter Waves, Vol. 6, K. J. Button, Ed. (Academic, New York, 1982).

Marcuvitz, N.

N. Marcuvitz, Waveguide Handbook, Vol. 10, MIT Radiation Laboratory Series (McGraw-Hill, New York, 1951), pp. 280–285.

McPhedran, R. C.

R. C. Compton, L. B. Whitbourn, R. C. McPhedran, “Strip Gratings at a Dielectric Interface and Application of Babinet's Principle,” Appl. Opt. 23, 3236 (1984).
[CrossRef] [PubMed]

R. C. Compton, L. B. Whitbourn, R. C. McPhedran, “Simple Formulae for the Transmittance of Strip Gratings,” Int. J. Infrared Millimeter Waves 4, 901 (1983).
[CrossRef]

Möller, K. D.

K. D. Möller, W. G. Rothschild, Far-Infrared Spectroscopy (Wiley-Interscience, New York, 1971).

Pollack, M. A.

Richards, P. L.

Rothschild, W. G.

K. D. Möller, W. G. Rothschild, Far-Infrared Spectroscopy (Wiley-Interscience, New York, 1971).

Sakai, K.

K. Sakai, L. Genzel, “Far-Infrared Metal Mesh Filters and Fabry-Perot Interferometry,” in Reviews of Infrared and Millimeter Waves, Vol. 1, K. J. Button, Ed. (Plenum, New York, 1983).
[CrossRef]

Shanahan, S. T.

Skocpol, W. J.

Timusk, T.

Tinkham, M.

Ulrich, R.

R. Ulrich, T. J. Bridges, M. A. Pollack, “Variable Metal Mesh Coupler for Far IR Lasers,” Appl. Opt. 9, 2511 (1970).
[CrossRef] [PubMed]

R. Ulrich, “Far-Infrared Properties of Metallic Mesh and its Complementary Structure,” Infrared Phys. 7, 37 (1967).
[CrossRef]

R. Ulrich, “Effective Low-Pass Filters for Far-Infrared Frequencies,” Infrared Phys. 7, 65 (1967).
[CrossRef]

Waldman, J.

Weitz, D. A.

Whitbourn, L. B.

R. C. Compton, L. B. Whitbourn, R. C. McPhedran, “Strip Gratings at a Dielectric Interface and Application of Babinet's Principle,” Appl. Opt. 23, 3236 (1984).
[CrossRef] [PubMed]

R. C. Compton, L. B. Whitbourn, R. C. McPhedran, “Simple Formulae for the Transmittance of Strip Gratings,” Int. J. Infrared Millimeter Waves 4, 901 (1983).
[CrossRef]

Wolfe, S. M.

Appl. Opt.

Infrared Phys.

R. Ulrich, “Effective Low-Pass Filters for Far-Infrared Frequencies,” Infrared Phys. 7, 65 (1967).
[CrossRef]

R. Ulrich, “Far-Infrared Properties of Metallic Mesh and its Complementary Structure,” Infrared Phys. 7, 37 (1967).
[CrossRef]

Int. J. Infrared Millimeter Waves

R. C. Compton, L. B. Whitbourn, R. C. McPhedran, “Simple Formulae for the Transmittance of Strip Gratings,” Int. J. Infrared Millimeter Waves 4, 901 (1983).
[CrossRef]

Opt. Lett.

Other

For the results in Table 3 of Ref. 2 to be correct Z0 must be taken as 2ω0 In csc(πa/g) for inductive meshes and the reciprocal of this expression for capacitive meshes. Equation (13) is, therefore, incorrect.

K. D. Möller, W. G. Rothschild, Far-Infrared Spectroscopy (Wiley-Interscience, New York, 1971).

G. D. Holah, “Far-Infrared and Submillimeter Wavelength Filters,” in Infrared and Millimeter Waves, Vol. 6, K. J. Button, Ed. (Academic, New York, 1982).

K. Sakai, L. Genzel, “Far-Infrared Metal Mesh Filters and Fabry-Perot Interferometry,” in Reviews of Infrared and Millimeter Waves, Vol. 1, K. J. Button, Ed. (Plenum, New York, 1983).
[CrossRef]

In Ref. 4Eq. (3) should be Z0 = [2ω0 ln csc(πa/g)]−1. In Ref. 5 Table 3.2 is correct for Ξi = ω0 ln csc(πa/g) and Ξc = [4ω0 ln csc(πa/g)]−1, contradicting Eq. (3.31). In Eq. (3.33) σ must be taken as one quarter of the dc bulk conductivity of the mesh metal. In Ref. 6 Eq. (50) should be Z0c = [2ω0 ln csc(πa/g)]−1. In Ref. 7 Table II-B is correct for Ẑi=Ξi and Ẑc=Ξc as given above, contradicting Eq. (28). In Eq. (31) σ is one quarter of the dc bulk conductivity.

In Ref. 9Eqs. (13)–(15) treat an inductive strip grating, not a mesh. The applicability of Eqs. (16) and (17) is discussed in this paper. In Ref. 10 the first of the equations on the bottom of p. 2646 is not applicable for meshes, as discussed in this paper [note the misprint in Eq. (2) the square bracket should be squared]. In Ref. 11 the curves in Fig. 1 cannot be correct because they show no shift in mesh resonance caused by the dielectric substrate [see discussion following Eqs. (6)–(8), this paper]. In Ref. 12, Fig. 1, the equations are correct for ω0 = 2π/g.(Note that the definitions of ω0 and ω = 2π/λ differ from those used here.) However, ω0 = 2π/g is not reasonable for a mesh at a dielectric boundary [see discussion following our Eqs. (6)–(8) and Ref. 12, p. 1357]. For ω0 = k × 2π/g, Y0 should be 2k(n12+n22)ln csc(πa/g). In Ref. 13Eqs. (5)–(11) are not correct, see Ref. 16.

P. F. Goldsmith, “Quasi-Optical Techniques at Millimeter and Submillimeter Wavelengths,” in Infrared and Millimeter Waves, Vol. 6, K. J. Button, Ed. (Academic, New York, 1982).

N. Marcuvitz, Waveguide Handbook, Vol. 10, MIT Radiation Laboratory Series (McGraw-Hill, New York, 1951), pp. 280–285.

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Figures (2)

Fig. 1
Fig. 1

Four most common grid configurations: (a) Inductive strip grating; (b) capacitive strip grating; (c) inductive mesh; (d) capacitive mesh.

Fig. 2
Fig. 2

Equivalent circuits of the four grid configurations shown in Fig. 1. The loss resistance R0 is zero for all mesh results except Eqs. (9)(12).

Equations (14)

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T ( n 1 , n 2 ) = 4 n 1 n 2 ( X / Z s ) 2 1 + ( n 2 + n 1 ) 2 ( X / Z s ) 2 .
T ( n 1 , n 2 ) = n 1 n 2 T ( 1 , 1 ) 1 + T ( 1 , 1 ) [ ( n 1 + n 2 ) 2 / 4 1 ] .
X I Z s = g λ ln csc π a g ,
X c Z s = 2 ( n 1 2 + n 2 2 ) ( 4 g λ ln csc π a g ) 1 .
X I X c = Z s 2 / 4 n 2 .
X I Z s = ( ω 0 ln csc π a g ) ( ω ω 0 ω 0 ω ) 1 ,
X c Z s = 2 n 1 2 + n 2 2 ( 4 ω 0 ln csc π a g ) 1 ( ω ω 0 ω 0 ω ) .
ω 0 = ω 0 2 n 1 2 + n 2 2 .
T = 4 n 1 n 2 [ ( R 0 / Z s ) 2 + ( X / Z s ) 2 ] [ 1 + ( n 2 + n 1 ) R 0 / Z s ] 2 + ( n 2 + n 1 ) 2 ( X / Z s ) 2 ,
R 12 = [ 1 + ( n 2 + n 1 ) R 0 / Z s ] 2 + ( n 2 n 1 ) 2 ( X / Z s ) 2 [ 1 + ( n 2 + n 1 ) R 0 / Z s ] 2 + ( n 2 + n 1 ) 2 ( X / Z s ) 2 ,
A 12 = 4 n 1 R 0 / Z s [ 1 + ( n 2 + n 1 ) R 0 / Z s ] 2 + ( n 2 + n 1 ) 2 ( X / Z s ) 2 ,
R 0 Z s = 4 π 0 c λ σ η 4 ,
a eff = a + t 2 π [ ln ( 8 π a t ) + 1 ] .
a eff = a t 2 π { ln [ 4 π ( g 2 a ) t ] 1 } .

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