Abstract

In this paper, we consider the transmission-line model used to calculate the transmittance of thin metallic strip gratings (at wavelengths longer than the grating period) to resolve a conflict of published expressions for the effect of a thick dielectric substrate on the equivalent circuit capacitance of capacitive gratings. By using rigorous diffraction theory we establish the correct expression and derive a modified form of Babinet’s principle for use with strip gratings on dielectric boundaries. It is found that the equivalent circuit capacitance of a strip grating on the boundary between media of refractive indices n1 and n2 is larger than its free-space value by a factor (n12+n22)/2. The result is applicable in general to the capacitive part of the equivalent circuit of grid reflectors, which are widely used at submillimeter wavelengths. A useful set of rigorously calculated transmission curves for strip gratings is presented, and these are used to establish the range of validity of the transmission-line model.

© 1984 Optical Society of America

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  1. R. Ulrich, K. F. Renk, L. Genzel, “Tunable Submillimeter Interferometers of the Fabry-Perot Type,” IEEE Trans. Microwave Theory Tech. MTT-11, 363 (1963).
    [CrossRef]
  2. R. Ulrich, “Far-Infrared Properties of Metallic Mesh and its Complementary Structure,” Infrared Phys. 7, 37 (1967).
    [CrossRef]
  3. R. Ulrich, “Effective Low-Pass Filters for Far-Infrared Frequencies,” Infrared Phys. 7, 65 (1967).
    [CrossRef]
  4. R. Ulrich, T. J. Bridges, M. A. Pollack, “Variable Metal Mesh Coupler for Far-IR Lasers,” Appl. Opt. 9, 2511 (1970).
    [CrossRef] [PubMed]
  5. R. J. Bell, H. V. Romero, “Study of an Array of Square Openings,” Appl. Opt. 9, 2341 (1970).
    [CrossRef] [PubMed]
  6. S. E. Whitcomb, J. Keene, “Low-Pass Interference Filters for Submillimeter Astronomy,” Appl. Opt. 19, 197 (1980).
    [CrossRef] [PubMed]
  7. E. J. Danielewicz, T. K. Plant, T. A. De Temple, “Hybrid Output Mirror for Optically Pumped Far Infrared Lasers,” Opt. Commun. 13, 366 (1975).
    [CrossRef]
  8. E. J. Danielewicz, P. D. Coleman, “Hybrid Metal Mesh-Dielectric Mirrors for Optically Pumped Far Infrared Lasers,” Appl. Opt. 15, 761 (1976).
    [CrossRef] [PubMed]
  9. 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]
  10. M. R. Schubert, M. S. Durshlag, T. A. De Temple, “Diffraction Limited CW Optically Pumped Lasers,” IEEE J. Quantum Electron. QE-13, 455 (1977).
    [CrossRef]
  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. S. T. Shanahan, N. R. Heckenberg, “Transmission Line Model of Substrate Effects on Capacitive Mesh Couplers,” Appl. Opt. 20, 4019 (1981).
    [CrossRef] [PubMed]
  13. T. Timusk, P. L. Richards, “Near Millimeter Wave Bandpass Filters,” Appl. Opt. 20, 1355 (1981).
    [CrossRef] [PubMed]
  14. N. Marcuvitz, Waveguide Handbook, Vol. 10, MIT Radiation Laboratory Series (McGraw-Hill, New York, 1951), pp. 280–285.
  15. J. A. Andrewartha, “Modal Expansion Theories for Singly Periodic Diffraction Gratings,” Ph.D. Thesis, U. Tasmania (1981).
  16. G. L. Baldwin, A. E. Heins, “On the Diffraction of a Plane Wave by an Infinite Grating,” Math. Scand. 2, 103 (1954).
  17. M. C. Hutley, Diffraction Gratings (Academic, London, 1982), pp. 175–210.
  18. E. V. Loewenstein, D. R. Smith, R. L. Morgan, “Optical Constants of Far-IR Materials. 2: Crystalline Solids,” Appl. Opt. 12, 398 (1973).
    [CrossRef] [PubMed]
  19. To derive the correct results it is necessary to go back to the results of Ulrich2 and make appropriate substitutions for the capacitive part of the equivalent circuit reactance of a mesh at a dielectric interface. The results in Table 3 of Ulrich’s paper are correct if Z0 is taken to be 2ω0 ln[csc(πa/g)] for inductive gratings and the reciprocal of this expression for capacitive gratings.
  20. For a summary of the transmission-line models for strip gratings and meshes on dielectric boundaries, see L. B. Whitbourn, R. C. Compton, submitted to Appl. Opt.
  21. V. Y. Balakhanov, “Properties of a Fabry-Perot Interferometer with Mirrors in the Form of a Backed Metal Grid,” Sov. Phys. Dokl. 10, 788 (1966).
  22. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), p. 323.
  23. G. G. MacFarlane, “Quasi-Stationary Field Theory and its Application to Diaphragms and Functions in Transmission Lines and Waveguides,” J. IEE 93, 1523 (1946).
  24. W. R. Smythe, Static and Dynamic Electricity (McGraw-Hill, New York, 1939), pp. 88–91.
  25. C. C. Chen, “Transmission through a Conducting Screen Perforated Periodically with Apertures,” IEEE Trans. Microwave Theory Tech. MTT-18, 627 (1970).
    [CrossRef]
  26. R. C. McPhedran, D. Maystre, “On the Theory and Solar Applications of Inductive Grids,” Appl. Phys. 14, 1 (1977).
    [CrossRef]

1981 (2)

1980 (1)

1978 (1)

1977 (2)

M. R. Schubert, M. S. Durshlag, T. A. De Temple, “Diffraction Limited CW Optically Pumped Lasers,” IEEE J. Quantum Electron. QE-13, 455 (1977).
[CrossRef]

R. C. McPhedran, D. Maystre, “On the Theory and Solar Applications of Inductive Grids,” Appl. Phys. 14, 1 (1977).
[CrossRef]

1976 (2)

1975 (1)

E. J. Danielewicz, T. K. Plant, T. A. De Temple, “Hybrid Output Mirror for Optically Pumped Far Infrared Lasers,” Opt. Commun. 13, 366 (1975).
[CrossRef]

1973 (1)

1970 (3)

1967 (2)

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]

1966 (1)

V. Y. Balakhanov, “Properties of a Fabry-Perot Interferometer with Mirrors in the Form of a Backed Metal Grid,” Sov. Phys. Dokl. 10, 788 (1966).

1963 (1)

R. Ulrich, K. F. Renk, L. Genzel, “Tunable Submillimeter Interferometers of the Fabry-Perot Type,” IEEE Trans. Microwave Theory Tech. MTT-11, 363 (1963).
[CrossRef]

1954 (1)

G. L. Baldwin, A. E. Heins, “On the Diffraction of a Plane Wave by an Infinite Grating,” Math. Scand. 2, 103 (1954).

1946 (1)

G. G. MacFarlane, “Quasi-Stationary Field Theory and its Application to Diaphragms and Functions in Transmission Lines and Waveguides,” J. IEE 93, 1523 (1946).

Andrewartha, J. A.

J. A. Andrewartha, “Modal Expansion Theories for Singly Periodic Diffraction Gratings,” Ph.D. Thesis, U. Tasmania (1981).

Balakhanov, V. Y.

V. Y. Balakhanov, “Properties of a Fabry-Perot Interferometer with Mirrors in the Form of a Backed Metal Grid,” Sov. Phys. Dokl. 10, 788 (1966).

Baldwin, G. L.

G. L. Baldwin, A. E. Heins, “On the Diffraction of a Plane Wave by an Infinite Grating,” Math. Scand. 2, 103 (1954).

Bell, R. J.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), p. 323.

Bridges, T. J.

Button, K. J.

Chen, C. C.

C. C. Chen, “Transmission through a Conducting Screen Perforated Periodically with Apertures,” IEEE Trans. Microwave Theory Tech. MTT-18, 627 (1970).
[CrossRef]

Cohn, D. R.

Coleman, P. D.

Compton, R. C.

For a summary of the transmission-line models for strip gratings and meshes on dielectric boundaries, see L. B. Whitbourn, R. C. Compton, submitted to Appl. Opt.

Danielewicz, E. J.

E. J. Danielewicz, P. D. Coleman, “Hybrid Metal Mesh-Dielectric Mirrors for Optically Pumped Far Infrared Lasers,” Appl. Opt. 15, 761 (1976).
[CrossRef] [PubMed]

E. J. Danielewicz, T. K. Plant, T. A. De Temple, “Hybrid Output Mirror for Optically Pumped Far Infrared Lasers,” Opt. Commun. 13, 366 (1975).
[CrossRef]

De Temple, T. A.

M. R. Schubert, M. S. Durshlag, T. A. De Temple, “Diffraction Limited CW Optically Pumped Lasers,” IEEE J. Quantum Electron. QE-13, 455 (1977).
[CrossRef]

E. J. Danielewicz, T. K. Plant, T. A. De Temple, “Hybrid Output Mirror for Optically Pumped Far Infrared Lasers,” Opt. Commun. 13, 366 (1975).
[CrossRef]

Durshlag, M. S.

M. R. Schubert, M. S. Durshlag, T. A. De Temple, “Diffraction Limited CW Optically Pumped Lasers,” IEEE J. Quantum Electron. QE-13, 455 (1977).
[CrossRef]

Genzel, L.

R. Ulrich, K. F. Renk, L. Genzel, “Tunable Submillimeter Interferometers of the Fabry-Perot Type,” IEEE Trans. Microwave Theory Tech. MTT-11, 363 (1963).
[CrossRef]

Heckenberg, N. R.

Heins, A. E.

G. L. Baldwin, A. E. Heins, “On the Diffraction of a Plane Wave by an Infinite Grating,” Math. Scand. 2, 103 (1954).

Hutley, M. C.

M. C. Hutley, Diffraction Gratings (Academic, London, 1982), pp. 175–210.

Keene, J.

Loewenstein, E. V.

MacFarlane, G. G.

G. G. MacFarlane, “Quasi-Stationary Field Theory and its Application to Diaphragms and Functions in Transmission Lines and Waveguides,” J. IEE 93, 1523 (1946).

Marcuvitz, N.

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

Maystre, D.

R. C. McPhedran, D. Maystre, “On the Theory and Solar Applications of Inductive Grids,” Appl. Phys. 14, 1 (1977).
[CrossRef]

McPhedran, R. C.

R. C. McPhedran, D. Maystre, “On the Theory and Solar Applications of Inductive Grids,” Appl. Phys. 14, 1 (1977).
[CrossRef]

Morgan, R. L.

Plant, T. K.

E. J. Danielewicz, T. K. Plant, T. A. De Temple, “Hybrid Output Mirror for Optically Pumped Far Infrared Lasers,” Opt. Commun. 13, 366 (1975).
[CrossRef]

Pollack, M. A.

Renk, K. F.

R. Ulrich, K. F. Renk, L. Genzel, “Tunable Submillimeter Interferometers of the Fabry-Perot Type,” IEEE Trans. Microwave Theory Tech. MTT-11, 363 (1963).
[CrossRef]

Richards, P. L.

Romero, H. V.

Schubert, M. R.

M. R. Schubert, M. S. Durshlag, T. A. De Temple, “Diffraction Limited CW Optically Pumped Lasers,” IEEE J. Quantum Electron. QE-13, 455 (1977).
[CrossRef]

Shanahan, S. T.

Skocpol, W. J.

Smith, D. R.

Smythe, W. R.

W. R. Smythe, Static and Dynamic Electricity (McGraw-Hill, New York, 1939), pp. 88–91.

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]

R. Ulrich, K. F. Renk, L. Genzel, “Tunable Submillimeter Interferometers of the Fabry-Perot Type,” IEEE Trans. Microwave Theory Tech. MTT-11, 363 (1963).
[CrossRef]

Waldman, J.

Weitz, D. A.

Whitbourn, L. B.

For a summary of the transmission-line models for strip gratings and meshes on dielectric boundaries, see L. B. Whitbourn, R. C. Compton, submitted to Appl. Opt.

Whitcomb, S. E.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), p. 323.

Wolfe, S. M.

Appl. Opt. (8)

Appl. Phys. (1)

R. C. McPhedran, D. Maystre, “On the Theory and Solar Applications of Inductive Grids,” Appl. Phys. 14, 1 (1977).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. R. Schubert, M. S. Durshlag, T. A. De Temple, “Diffraction Limited CW Optically Pumped Lasers,” IEEE J. Quantum Electron. QE-13, 455 (1977).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

R. Ulrich, K. F. Renk, L. Genzel, “Tunable Submillimeter Interferometers of the Fabry-Perot Type,” IEEE Trans. Microwave Theory Tech. MTT-11, 363 (1963).
[CrossRef]

C. C. Chen, “Transmission through a Conducting Screen Perforated Periodically with Apertures,” IEEE Trans. Microwave Theory Tech. MTT-18, 627 (1970).
[CrossRef]

Infrared Phys. (2)

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]

J. IEE (1)

G. G. MacFarlane, “Quasi-Stationary Field Theory and its Application to Diaphragms and Functions in Transmission Lines and Waveguides,” J. IEE 93, 1523 (1946).

Math. Scand. (1)

G. L. Baldwin, A. E. Heins, “On the Diffraction of a Plane Wave by an Infinite Grating,” Math. Scand. 2, 103 (1954).

Opt. Commun. (1)

E. J. Danielewicz, T. K. Plant, T. A. De Temple, “Hybrid Output Mirror for Optically Pumped Far Infrared Lasers,” Opt. Commun. 13, 366 (1975).
[CrossRef]

Opt. Lett. (1)

Sov. Phys. Dokl. (1)

V. Y. Balakhanov, “Properties of a Fabry-Perot Interferometer with Mirrors in the Form of a Backed Metal Grid,” Sov. Phys. Dokl. 10, 788 (1966).

Other (7)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), p. 323.

To derive the correct results it is necessary to go back to the results of Ulrich2 and make appropriate substitutions for the capacitive part of the equivalent circuit reactance of a mesh at a dielectric interface. The results in Table 3 of Ulrich’s paper are correct if Z0 is taken to be 2ω0 ln[csc(πa/g)] for inductive gratings and the reciprocal of this expression for capacitive gratings.

For a summary of the transmission-line models for strip gratings and meshes on dielectric boundaries, see L. B. Whitbourn, R. C. Compton, submitted to Appl. Opt.

W. R. Smythe, Static and Dynamic Electricity (McGraw-Hill, New York, 1939), pp. 88–91.

M. C. Hutley, Diffraction Gratings (Academic, London, 1982), pp. 175–210.

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

J. A. Andrewartha, “Modal Expansion Theories for Singly Periodic Diffraction Gratings,” Ph.D. Thesis, U. Tasmania (1981).

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Figures (10)

Fig. 1
Fig. 1

(a) Geometry of capacitive and inductive strip gratings on a dielectric boundary; (b) equivalent transmission line model for a thin grating on a dielectric boundary.

Fig. 2
Fig. 2

Zero-order transmission curve at normal incidence for inductive and capacitive strip gratings in free space. Our calculations (crosses and squares) are compared with results published by Baldwin and Heins16 (solid line).

Fig. 3
Fig. 3

Total transmittance at normal incidence for inductive and capacitive strip gratings at a dielectric boundary. The sum of these (dashed curve) does not agree with the modified form of Babinet’s principle given by Eq. (5), which is represented by the solid horizontal line.

Fig. 4
Fig. 4

Comparison of calculated transmittances of an inductive strip grating (diamonds) and a capacitive strip grating (crosses) with the two transmission-line models12,13 for normally incident radiation.

Fig. 5
Fig. 5

Comparison of normalized reactance of a capacitive strip grating, according to our results (crosses) and results predicted by the two transmission-line odels for normally incident radiation.

Fig. 6
Fig. 6

Comparison of the modified and unmodified forms of Babinet’s principle. The solid horizontal line shows the transmission through a dielectric boundary in the absence of a screen. The unmodified form of Babinet’s principle [left-hand side of Eq. (5)] is shown as crosses, and the modified application of Babinet’s principle [left-hand side of Eq. (9)] is shown as diamonds.

Fig. 7
Fig. 7

Our calculated zero-order transmittance for radiation incident normally on a range of thin capacitive strip gratings placed at a dielectric boundary. The circles show the predictions of the transmission line model (n1 = 1, n2 = 2.1).

Fig. 8
Fig. 8

Our calculated zero-order transmittance for radiation incident normally on a range of thin inductive strip gratings placed at a dielectric boundary. The circles show the predictions of the transmission line model (n1 = 1, n2 = 2.1).

Fig. 9
Fig. 9

(a) Coordinate system used for the quasi-static calculation of capacitance of a strip grating at a dielectric boundary; (b) how conducting sheets are added along equipotentials of (a). By symmetry these do not disturb the electric field distribution.

Fig. 10
Fig. 10

Three regions, I, II, and III, considered in the mode-matching method for calculating the transmission properties of a strip grating at a dielectric boundary.

Equations (27)

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T ( n 1 , n 2 ) = 4 n 1 , n 2 [ X ( n 1 , n 2 ) / Z 0 ] 2 1 + ( n 1 + n 2 ) 2 [ X ( n 1 , n 2 ) / Z 0 ] 2 ,
X L ( n , n ) Z 0 = g λ ln csc ( π a g ) ,
X C ( n , n ) Z 0 = - 1 n 2 [ 4 g λ ln csc ( π a g ) ] - 1 ,
T ( n , n ) + T ( n , n ) = 1 ,
T ( n 1 , n 2 ) + T ( n 1 , n 2 ) = 4 n 1 n 2 ( n 1 + n 2 ) 2 .
X ( n 1 , n 2 ) X ( n 1 , n 2 ) = - Z 0 2 ( n 1 + n 2 ) 2 .
X C ( n 1 , n 2 ) = 4 ( n 1 + n 2 ) 2 X C ( 1 , 1 ) .
T I ( n 1 , n 2 ; K λ ) + T C ( n 1 , n 2 ; λ ) = 4 n 1 n 2 / ( n 1 + n 2 ) 2 .
K = 2 ( n 1 2 + n 2 2 ) / ( n 1 + n 2 ) 2 .
sin ( V + j U ) = csc ( π a / g ) sin [ ( x + j y ) π / g ] .
U ( T ) - U ( S ) = T S E n d s = - 1 ɛ 0 n 1 2 Q S T , U ( R ) - U ( S ) = R S E n d s = - 1 ɛ 0 n 2 2 Q R S ,
Q tot ( top sheet ) = Q S T + Q R S = ɛ 0 ( n 1 2 + n 2 2 ) 2 [ U ( T ) - U ( R ) ] = ɛ 0 ( n 1 2 + n 2 2 ) cosh - 1 ( csc π a g cosh π L g ) , C grid = 1 ( V top - V bottom ) Q tot - C sheets = lim L 1 π [ ɛ 0 ( n 1 2 + n 2 2 ) cosh - 1 ( csc π a g cosh π L g ) - ( n 1 2 + n 2 2 ) π g ] = ( n 1 2 + n 2 2 π ) ɛ 0 ln csc π a g ,
X C Z 0 = - 2 ( n 1 2 + n 2 2 ) [ 4 g λ ln csc ( π a g ) ] - 1 .
α n = ( 2 n π ) / g , β n = k 2 - α n 2 ,             γ n = ɛ k 2 - α n 2 , μ m = 1 - m 2 π 2 4 k 2 a 2 .
H z and E x = 1 ɛ H z y
R N exp ( i β N t / 2 ) = δ 0 N exp ( - i β 0 t / 2 ) - i β N m = 0 k i k μ m [ a m cos ( k μ m t / 2 ) - b m sin ( k μ m t / 2 ) ] I m N , T N exp ( - i β N t / 2 ) = i ɛ γ N m = 0 k μ m [ a m cos ( k μ m t / 2 ) + b m sin ( k μ m t / 2 ) ] I m N ,
I m n = 1 g - a + a cos m π 2 a ( x - a ) exp ( - i α n x ) d x .
m = 0 { 2 a g η m [ a m sin ( k μ m t / 2 ) + b m cos ( k μ m t / 2 ) ] δ m M + i k μ m [ a m cos ( k μ m t / 2 ) - b m sin ( k μ m t / 2 ) ] n = - 1 β n I m n I ¯ M n } = 2 exp ( - i β 0 t ) I ¯ M 0 ,
m = 0 { 2 a g η m [ - a m sin ( k μ m t / 2 ) + b m cos ( k μ m t / 2 ) ] δ m M - i ɛ k μ m [ a m cos ( k μ m t / 2 ) + b m sin ( k μ m t / 2 ) ] n = - 1 γ n I m n I ¯ M n } = 0 ,
η m = 1 , m = 0 = ½ , m 0 } .
2 a g b 0 + i k μ 0 a 0 n = - I 0 n I ¯ 0 n β n = 2 I 00 A , 2 a g b 0 - i ɛ k μ 0 a 0 n = - I 0 n I ¯ 0 n γ n = 0 .
T 0 = 2 ɛ I 00 2 A γ 0 n = - I 0 n 2 ( 1 β n + ɛ γ n ) .
T 0 = 2 ɛ A / γ 0 ( 1 + ɛ ) / k - 2 i ( 1 + ɛ ) I 00 2 n = - n 0 I 0 n 2 g 2 π n .
T ( 1 , n ) = T 0 2 ɛ γ 0 β 0 = 4 ɛ ( 1 + ɛ ) 2 + ( 1 + ɛ ) 2 4 16 k 2 I 00 4 ( n = - n 0 I 0 n 2 g 2 π n ) 2 .
T ( 1 , n ) = 4 n [ X C ( 1 , n ) / Z 0 ] 2 1 + ( 1 + n ) 2 [ X C ( 1 , n ) / Z 0 ] 2 ,
X C ( 1 , n ) / Z 0 = ( 2 1 + n 2 ) ( 4 k I 00 2 p = - p 0 I 0 p 2 g 2 π p ) - 1 .
X C ( 1 , n ) = 2 1 + n 2 X C ( 1 , 1 ) .

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