Abstract

To use a transmission line model to calculate the optical properties of a thin metal mesh on a dielectric substrate, account must be taken not only of the different propagation conditions within the substrate and of Fabry-Perot resonances due to reflections at the second surface, but also of the effect of the dielectric on the capacitive component of the equivalent reactance of the mesh. Only when this effect is accounted for, which can be done using a simple formula based on Babinet’s principle, is good agreement obtained with experimental measurements.

© 1981 Optical Society of America

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References

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  1. D. A. Weitz, W. J. Skocpol, M. Tinkham, Opt. Lett. 3, 13 (1978).
    [CrossRef] [PubMed]
  2. R. Ulrich, K. F. Renk, L. Genzel, IEEE Trans. Microwave Theory Tech. MTT-11, 363 (1963).
    [CrossRef]
  3. S. M. Wolfe, K. J. Button, J. Waldman, D. R. Cohn, Appl. Opt. 15, 2645 (1976) [note error in Eq. (2)].
    [CrossRef] [PubMed]
  4. R. Ulrich, Infrared Phys. 7, 37 (1967).
    [CrossRef]
  5. N. Marcuvitz, Waveguide Handbook, MIT Radiation Laboratory Series, Vol. 10 (McGraw-Hill, New York, 1951).
  6. W. L. Weeks, Electromagnetic Theory for Engineering Applications (Wiley, New York, 1964), pp. 446–447.
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), p. 323.
  8. S. Ramo, J. R. Whinnery, Fields and Waves in Modern Radio (Wiley, New York, 1964), p. 477.
  9. M. S. Durschlag, T. A. DeTemple, Appl. Opt. 20, 1245 (1981).
    [CrossRef] [PubMed]
  10. Available from Buckbee-Mears Co., Micro Products Division, Saint Paul, Minn.

1981 (1)

1978 (1)

1976 (1)

1967 (1)

R. Ulrich, Infrared Phys. 7, 37 (1967).
[CrossRef]

1963 (1)

R. Ulrich, K. F. Renk, L. Genzel, IEEE Trans. Microwave Theory Tech. MTT-11, 363 (1963).
[CrossRef]

Born, M.

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

Button, K. J.

Cohn, D. R.

DeTemple, T. A.

Durschlag, M. S.

Genzel, L.

R. Ulrich, K. F. Renk, L. Genzel, IEEE Trans. Microwave Theory Tech. MTT-11, 363 (1963).
[CrossRef]

Marcuvitz, N.

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

Ramo, S.

S. Ramo, J. R. Whinnery, Fields and Waves in Modern Radio (Wiley, New York, 1964), p. 477.

Renk, K. F.

R. Ulrich, K. F. Renk, L. Genzel, IEEE Trans. Microwave Theory Tech. MTT-11, 363 (1963).
[CrossRef]

Skocpol, W. J.

Tinkham, M.

Ulrich, R.

R. Ulrich, Infrared Phys. 7, 37 (1967).
[CrossRef]

R. Ulrich, K. F. Renk, L. Genzel, IEEE Trans. Microwave Theory Tech. MTT-11, 363 (1963).
[CrossRef]

Waldman, J.

Weeks, W. L.

W. L. Weeks, Electromagnetic Theory for Engineering Applications (Wiley, New York, 1964), pp. 446–447.

Weitz, D. A.

Whinnery, J. R.

S. Ramo, J. R. Whinnery, Fields and Waves in Modern Radio (Wiley, New York, 1964), p. 477.

Wolf, E.

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

Wolfe, S. M.

Appl. Opt. (2)

IEEE Trans. Microwave Theory Tech. (1)

R. Ulrich, K. F. Renk, L. Genzel, IEEE Trans. Microwave Theory Tech. MTT-11, 363 (1963).
[CrossRef]

Infrared Phys. (1)

R. Ulrich, Infrared Phys. 7, 37 (1967).
[CrossRef]

Opt. Lett. (1)

Other (5)

Available from Buckbee-Mears Co., Micro Products Division, Saint Paul, Minn.

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

W. L. Weeks, Electromagnetic Theory for Engineering Applications (Wiley, New York, 1964), pp. 446–447.

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

S. Ramo, J. R. Whinnery, Fields and Waves in Modern Radio (Wiley, New York, 1964), p. 477.

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Figures (5)

Fig. 1
Fig. 1

Mesh geometries and equivalent circuits for (a) inductive and (b) capacitive meshes in homogeneous media.

Fig. 2
Fig. 2

Equivalent circuit for mesh of reactance X at interface between dielectrics with refractive indices n1 and n2.

Fig. 3
Fig. 3

(a) Boundary between two dielectrics. Diagram shows how transmitted field Et in second dielectric may be produced in two ways: (i) incidence of Ei on boundary through first dielectric, or (ii) by changing refractive index n1 to n2 and Ei to Et; (b) screen at dielectric boundary. Same transmitted field is produced when (i) field Ei is incident from medium 1 onto screen at boundary as when (ii) refractive index n1 is changed to n2 and incident field is changed to Et of (a). Thus, τiEi = τsEt.

Fig. 4
Fig. 4

Calculated transmissivity of 200-lpi thin capacitive mesh (g = 127 μm, 2a = 20 μm): (a) in free space; (b) on infinite quartz substrate (n = 2.106), ignoring effect on mesh reactance; (c) on infinite quartz substrate, including effect on reactance; and (d) as (c), but showing channeled spectrum due to 2-mm thick substrate.

Fig. 5
Fig. 5

Measured transmissivity and corresponding theoretical curves calculated from the data in Table I for three capacitive mesh couplers: (a) 300, (b) 200, and (c) 80 lpi.

Tables (1)

Tables Icon

Table I Mesh Parameters of Meshes Whose Transmissivity Is Shown In Fig. 5

Equations (17)

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R = Z s 2 + X 2 ( n 2 - n 1 ) 2 Z s 2 + X 2 ( n 2 + n 1 ) 2 ,
T = 4 n 1 n 2 X 2 Z s 2 + X 2 ( n 2 + n 1 ) 2 ,
R = Γ 2 ;             T = n 2 n 1 τ 2 .
inductive grid X L Z n = g λ n ln csc ( π a e g ) , capacitive grid X c Z n = [ 4 g λ n ln csc ( π a e g ) ] - 1 ,
a e = a + ( t 2 π ) ( 1 + ln 8 π a t ) ,
τ s + τ s = 1.
τ i E i + τ i E i = E t
τ i + τ i = E t E i = τ ,
τ s 2 + τ s 2 = 1 ,
τ i 2 + τ i 2 = τ 2 .
X X = - Z s 2 ( n 1 + n 2 ) 2 ,
X C B = 4 ( n 1 + n 2 ) 2 X C .
2 n 1 + n 2 ω 0 for ω 0             2 n 1 + n 2 Z 0 for Z 0 .
γ = 2 π g [ 1 - ( g λ n ) 2 ] 1 / 2 .
T = T 1 T 2 [ 1 - ( R 1 R 2 ) 1 / 2 ] 2 + 4 ( R 1 R 2 ) 1 / 2 sin 2 ( δ + ϕ Γ 2 ) , R = 1 - T ,
T 2 = 4 n ( 1 + n ) 2             R 2 = ( 1 - n 1 + n ) 2 .
δ = 2 π n l λ ,             cos ϕ Γ = 1 + R 1 - n T 1 2 R 1 1 / 2 ,

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