Abstract

The transmittance of inductive single-layer and multilayer cross-shaped metal meshes has been calculated with the Micro-Stripes software program. The effect of symmetric and asymmetric alignment of the crosses of one mesh with respect to another was studied and compared with transmission line theory, which presents the nonaligned case. Significant differences are found for small spacing at approximately 1/5 the periodicity constant, whereas the differences disappear for large spacing. A pair of coupled surface waves is used to represent the mode of a single mesh. The resulting modes corresponding to the transmittance of multilayer metal meshes are interpreted by modes composed of resonance modes of a single mesh coupled by Fabry-Perot modes depending on the separation.

© 2002 Optical Society of America

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References

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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, 363–371 (1963).
    [CrossRef]
  2. R. Ulrich, “Far-infrared properties of metallic mesh and its complementary structure,” Infrared Phys. 7, 37–55 (1967).
    [CrossRef]
  3. G. M. Ressler, K. D. Möller, “Far infrared bandpass filters and measurements on a reciprocal grid,” Appl. Opt. 6, 893–896 (1967).
    [CrossRef] [PubMed]
  4. R. Ulrich, “Interference filters for the far infrared,” Infrared Phys. 7, 1987–1996 (1967).
  5. S. T. Chase, R. D. Joseph, “Resonant array bandpass filters for the far infrared,” Appl. Opt. 22, 1775–1779 (1983).
    [CrossRef] [PubMed]
  6. M. Rebbert, P. Isaacson, J. Fischer, M. A. Greenhouse, J. Grossman, M. Peckerar, H. A. Smith, “Microstructure technology for fabrication of metal-mesh grids,” Appl. Opt. 33, 1286–1292 (1994).
    [CrossRef] [PubMed]
  7. D. W. Porterfield, J. L. Hesler, R. Densing, E. R. Mueller, T. W. Crowe, R. M. Weikle, “Resonant metal-mesh bandpass filters for the far infrared,” Appl. Opt. 33, 6046–6092 (1994).
    [CrossRef] [PubMed]
  8. K. D. Möller, J. Warren, J. B. Heaney, C. Kotecki, “Cross-shaped bandpass filters for the near-and mid-infrared wavelength region,” Appl. Opt. 35, 6210–6215 (1996).
    [CrossRef] [PubMed]
  9. “Micro-Stripes program” by Flomerics Inc., 275 Turnpike Road, Suite 100, Southborough, Massachusetts 01772.
  10. D. H. Dawes, R. C. McPhedran, L. B. Whitbourn, “Thin capacitive meshes on a dielectric boundary: theory and experiment,” Appl. Opt. 28, 3498–3510 (1989).
    [CrossRef] [PubMed]
  11. C. Compton, R. D. McPhedran, G. H. Derrick, L. C. Botten, “Diffraction properties of a bandpass grid,” Infrared Phys. 23, 239–245 (1983).
    [CrossRef]
  12. R. Ruprecht, W. Bacher, P. Bley, M. Harmening, W. K. Schomburg, KfK-Nachr. Jahrg. 23, 2–91, 18–123 (1991).
  13. K. D. Möller, K. R. Farmer, D. V. P. Ivanov, O. Sternberg, K. P. Stewart, P. Lalanne, “Thin and thick cross-shaped metal grids,” Infrared Phys. 40, 475–485 (1999).
    [CrossRef]
  14. K. D. Möller, W. G. Rothschild, Far Infrared Spectroscopy (Wiley, New York, 1971).
  15. L. B. Whitbourn, R. C. Compton, “Equivalent-circuit formulas for metal grid reflectors at dielectric boundary,” Appl. Opt. 24, 217–220 (1985).
    [CrossRef]
  16. T. Timusk, P. L. Richards, “Near millimeter wave bandpass filters,” Appl. Opt. 20, 1355–1360 (1981).
    [CrossRef] [PubMed]
  17. R. Ulrich, “Modes of propagation on an open periodic waveguide for the far infrared,” in Proceedings of the Symposium of Optical and Acoustical Micro-Electronics (Polytechnic, Brooklyn, N.Y., 1974), Vol. XXIII, pp. 359–376.

1999 (1)

K. D. Möller, K. R. Farmer, D. V. P. Ivanov, O. Sternberg, K. P. Stewart, P. Lalanne, “Thin and thick cross-shaped metal grids,” Infrared Phys. 40, 475–485 (1999).
[CrossRef]

1996 (1)

1994 (2)

1991 (1)

R. Ruprecht, W. Bacher, P. Bley, M. Harmening, W. K. Schomburg, KfK-Nachr. Jahrg. 23, 2–91, 18–123 (1991).

1989 (1)

1985 (1)

1983 (2)

S. T. Chase, R. D. Joseph, “Resonant array bandpass filters for the far infrared,” Appl. Opt. 22, 1775–1779 (1983).
[CrossRef] [PubMed]

C. Compton, R. D. McPhedran, G. H. Derrick, L. C. Botten, “Diffraction properties of a bandpass grid,” Infrared Phys. 23, 239–245 (1983).
[CrossRef]

1981 (1)

1967 (3)

R. Ulrich, “Far-infrared properties of metallic mesh and its complementary structure,” Infrared Phys. 7, 37–55 (1967).
[CrossRef]

G. M. Ressler, K. D. Möller, “Far infrared bandpass filters and measurements on a reciprocal grid,” Appl. Opt. 6, 893–896 (1967).
[CrossRef] [PubMed]

R. Ulrich, “Interference filters for the far infrared,” Infrared Phys. 7, 1987–1996 (1967).

1963 (1)

R. Ulrich, K. F. Renk, L. Genzel, “Tunable submillimeter interferometers of the Fabry-Perot type,” IEEE Trans. Microwave Theory Tech. MTT- 11, 363–371 (1963).
[CrossRef]

Bacher, W.

R. Ruprecht, W. Bacher, P. Bley, M. Harmening, W. K. Schomburg, KfK-Nachr. Jahrg. 23, 2–91, 18–123 (1991).

Bley, P.

R. Ruprecht, W. Bacher, P. Bley, M. Harmening, W. K. Schomburg, KfK-Nachr. Jahrg. 23, 2–91, 18–123 (1991).

Botten, L. C.

C. Compton, R. D. McPhedran, G. H. Derrick, L. C. Botten, “Diffraction properties of a bandpass grid,” Infrared Phys. 23, 239–245 (1983).
[CrossRef]

Chase, S. T.

Compton, C.

C. Compton, R. D. McPhedran, G. H. Derrick, L. C. Botten, “Diffraction properties of a bandpass grid,” Infrared Phys. 23, 239–245 (1983).
[CrossRef]

Compton, R. C.

Crowe, T. W.

Dawes, D. H.

Densing, R.

Derrick, G. H.

C. Compton, R. D. McPhedran, G. H. Derrick, L. C. Botten, “Diffraction properties of a bandpass grid,” Infrared Phys. 23, 239–245 (1983).
[CrossRef]

Farmer, K. R.

K. D. Möller, K. R. Farmer, D. V. P. Ivanov, O. Sternberg, K. P. Stewart, P. Lalanne, “Thin and thick cross-shaped metal grids,” Infrared Phys. 40, 475–485 (1999).
[CrossRef]

Fischer, J.

Genzel, L.

R. Ulrich, K. F. Renk, L. Genzel, “Tunable submillimeter interferometers of the Fabry-Perot type,” IEEE Trans. Microwave Theory Tech. MTT- 11, 363–371 (1963).
[CrossRef]

Greenhouse, M. A.

Grossman, J.

Harmening, M.

R. Ruprecht, W. Bacher, P. Bley, M. Harmening, W. K. Schomburg, KfK-Nachr. Jahrg. 23, 2–91, 18–123 (1991).

Heaney, J. B.

Hesler, J. L.

Isaacson, P.

Ivanov, D. V. P.

K. D. Möller, K. R. Farmer, D. V. P. Ivanov, O. Sternberg, K. P. Stewart, P. Lalanne, “Thin and thick cross-shaped metal grids,” Infrared Phys. 40, 475–485 (1999).
[CrossRef]

Joseph, R. D.

Kotecki, C.

Lalanne, P.

K. D. Möller, K. R. Farmer, D. V. P. Ivanov, O. Sternberg, K. P. Stewart, P. Lalanne, “Thin and thick cross-shaped metal grids,” Infrared Phys. 40, 475–485 (1999).
[CrossRef]

McPhedran, R. C.

McPhedran, R. D.

C. Compton, R. D. McPhedran, G. H. Derrick, L. C. Botten, “Diffraction properties of a bandpass grid,” Infrared Phys. 23, 239–245 (1983).
[CrossRef]

Möller, K. D.

K. D. Möller, K. R. Farmer, D. V. P. Ivanov, O. Sternberg, K. P. Stewart, P. Lalanne, “Thin and thick cross-shaped metal grids,” Infrared Phys. 40, 475–485 (1999).
[CrossRef]

K. D. Möller, J. Warren, J. B. Heaney, C. Kotecki, “Cross-shaped bandpass filters for the near-and mid-infrared wavelength region,” Appl. Opt. 35, 6210–6215 (1996).
[CrossRef] [PubMed]

G. M. Ressler, K. D. Möller, “Far infrared bandpass filters and measurements on a reciprocal grid,” Appl. Opt. 6, 893–896 (1967).
[CrossRef] [PubMed]

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

Mueller, E. R.

Peckerar, M.

Porterfield, D. W.

Rebbert, M.

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, 363–371 (1963).
[CrossRef]

Ressler, G. M.

Richards, P. L.

Rothschild, W. G.

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

Ruprecht, R.

R. Ruprecht, W. Bacher, P. Bley, M. Harmening, W. K. Schomburg, KfK-Nachr. Jahrg. 23, 2–91, 18–123 (1991).

Schomburg, W. K.

R. Ruprecht, W. Bacher, P. Bley, M. Harmening, W. K. Schomburg, KfK-Nachr. Jahrg. 23, 2–91, 18–123 (1991).

Smith, H. A.

Sternberg, O.

K. D. Möller, K. R. Farmer, D. V. P. Ivanov, O. Sternberg, K. P. Stewart, P. Lalanne, “Thin and thick cross-shaped metal grids,” Infrared Phys. 40, 475–485 (1999).
[CrossRef]

Stewart, K. P.

K. D. Möller, K. R. Farmer, D. V. P. Ivanov, O. Sternberg, K. P. Stewart, P. Lalanne, “Thin and thick cross-shaped metal grids,” Infrared Phys. 40, 475–485 (1999).
[CrossRef]

Timusk, T.

Ulrich, R.

R. Ulrich, “Far-infrared properties of metallic mesh and its complementary structure,” Infrared Phys. 7, 37–55 (1967).
[CrossRef]

R. Ulrich, “Interference filters for the far infrared,” Infrared Phys. 7, 1987–1996 (1967).

R. Ulrich, K. F. Renk, L. Genzel, “Tunable submillimeter interferometers of the Fabry-Perot type,” IEEE Trans. Microwave Theory Tech. MTT- 11, 363–371 (1963).
[CrossRef]

R. Ulrich, “Modes of propagation on an open periodic waveguide for the far infrared,” in Proceedings of the Symposium of Optical and Acoustical Micro-Electronics (Polytechnic, Brooklyn, N.Y., 1974), Vol. XXIII, pp. 359–376.

Warren, J.

Weikle, R. M.

Whitbourn, L. B.

Appl. Opt. (8)

IEEE Trans. Microwave Theory Tech. (1)

R. Ulrich, K. F. Renk, L. Genzel, “Tunable submillimeter interferometers of the Fabry-Perot type,” IEEE Trans. Microwave Theory Tech. MTT- 11, 363–371 (1963).
[CrossRef]

Infrared Phys. (4)

R. Ulrich, “Far-infrared properties of metallic mesh and its complementary structure,” Infrared Phys. 7, 37–55 (1967).
[CrossRef]

C. Compton, R. D. McPhedran, G. H. Derrick, L. C. Botten, “Diffraction properties of a bandpass grid,” Infrared Phys. 23, 239–245 (1983).
[CrossRef]

K. D. Möller, K. R. Farmer, D. V. P. Ivanov, O. Sternberg, K. P. Stewart, P. Lalanne, “Thin and thick cross-shaped metal grids,” Infrared Phys. 40, 475–485 (1999).
[CrossRef]

R. Ulrich, “Interference filters for the far infrared,” Infrared Phys. 7, 1987–1996 (1967).

KfK-Nachr. Jahrg. (1)

R. Ruprecht, W. Bacher, P. Bley, M. Harmening, W. K. Schomburg, KfK-Nachr. Jahrg. 23, 2–91, 18–123 (1991).

Other (3)

“Micro-Stripes program” by Flomerics Inc., 275 Turnpike Road, Suite 100, Southborough, Massachusetts 01772.

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

R. Ulrich, “Modes of propagation on an open periodic waveguide for the far infrared,” in Proceedings of the Symposium of Optical and Acoustical Micro-Electronics (Polytechnic, Brooklyn, N.Y., 1974), Vol. XXIII, pp. 359–376.

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Figures (15)

Fig. 1
Fig. 1

(a) Pattern of inductive meshes; openings are white, and metal film is black. (b) Geometrical parameters g, a, b of a cross. (c) Cross in the shifted position. (d) Cross in the lined-up position.

Fig. 2
Fig. 2

Transmittance of free-standing metal meshes of thickness 11, 20, and 29 µm. The parameters of the cross are g = 20, 2a = 1.5, and 2b = 3µm.

Fig. 3
Fig. 3

Free-standing inductive grid with parameters g = 24, 2a = 9.6, 2b = 3.6 µm, and t = 0.2 µm. Thin curve, Micro-Stripes program calculations; peak at 32.4 µm. Thick curve, transmission line theory with parameters λ0 = 32.4 µm, A1 = 0.1, a1 = 0.001.

Fig. 4
Fig. 4

Micro-Stripes program calculations of free-standing inductive grids for different sets of g, 2a, and 2b: (i) with g = 24, 2a = 10.4, and 2b = 2.0 µm; (ii) with g = 24, 2a = 9.6, and 2b = 3.6 µm; (iii) with g = 24, 2a = 8.8, and 2b = 5.2 µm.

Fig. 5
Fig. 5

Positioning of crosses in two- and four- mesh filters. Filter (A), crosses of the two meshes are lined up. Filter (B), crosses in the second mesh are shifted with respect to the first. Filter (A)(A), crosses of all meshes are lined up. Filter (B)(B), the crosses of the second and the fourth mesh are shifted.

Fig. 6
Fig. 6

Shunt impedance with incident and reflected waves on both sides.

Fig. 7
Fig. 7

Two free-standing meshes with parameters g = 24 µm, 2a = 9.6 µm, 2b = 3.6 µm, and t = 0.2 µm at distance of 4 µm. Solid thick curve, Micro-Stripes program calculation of filter (A). Solid thin curve, Micro-Stripes program calculation of filter (B). Dashed curve, transmission line theory using parameters λ0 = 32.4 µm, A1 = 0.1, a1 = 0.001.

Fig. 8
Fig. 8

Two free-standing meshes with parameters g = 24 µm, 2a = 9.6 µm, 2b = 3.6 µm, and t = 0.2 µm at distance of 8 µm. Thick solid curve, Micro-Stripes program calculation of filter (A). Thin solid curve, Micro-Stripes program calculation of filter (B). Dashed curve, transmission line theory using parameters λ0 = 32.4 µm, A1 = 0.1, a1 = 0.001.

Fig. 9
Fig. 9

Two free-standing meshes with parameters g = 24 µm, 2a = 9.6 µm, 2b = 3.6 µm, and t = 0.2 µm at distance of 12 µm. Thick solid curve, Micro-Stripes program calculation of filter (A). Thin solid curve, Micro-Stripes program calculation of filter (B). Dashed curve, transmission line theory using parameters λ0 = 32.4 µm, A1 = 0.1, a1 = 0.001.

Fig. 10
Fig. 10

Two free-standing meshes with parameters g = 24 µm, 2a = 9.6 µm, 2b = 3.6 µm, and t = 0.2 µm at distance of 16 µm. Thick solid curve, Micro-Stripes program calculation of filter (A). Thin solid curve, Micro-Stripes program calculation of filter (B). Dashed curve, transmission line theory using parameters λ0 = 32.4 µm, A1 = 0.1, a1 = 0.001.

Fig. 11
Fig. 11

Transmission line calculations for two meshes at separation d and mesh parameters g = 24 µm, 2a = 9.6 µm, 2b = 3.6 µm, and t = 0.2 µm. Peak resonance wavelength (squares) and Fabry-Perot peaks (triangles)

Fig. 12
Fig. 12

Four free-standing meshes with parameters g = 24 µm, 2a = 9.6 µm, 2b = 3.6 µm, and t = 0.2 µm at distance of 4 µm between each two. Thick solid curve, Micro-Stripes program calculation of filter (A)(A). Thin solid curve, Micro-Stripes program calculation of filter (B)(B). Dashed curve, transmission line theory using parameters λ0 = 32.4 µm, A1 = 0.1, a1 = 0.001.

Fig. 13
Fig. 13

Four free-standing meshes with parameters g = 24 µm, 2a = 9.6 µm, 2b = 3.6 µm, and t = 0.2 µm at distance of 8 µm between each two. Thick solid curve, Micro-Stripes program calculation of filter (A)(A). Thin solid curve, Micro-Stripes program calculation of filter (B)(B). Dashed curve, transmission line theory using parameters λ0 = 32.4 µm, A1 = 0.1, a1 = 0.001.

Fig. 14
Fig. 14

Four free-standing meshes with parameters g = 24 µm, 2a = 9.6 µm, 2b = 3.6 µm, and t = 0.2 µm at distance of 12 µm between each two. Thick solid curve, Micro-Stripes program calculation of filter (A)(A). Thin solid curve, Micro-Stripes program calculation of filter (B)(B). Dashed curve, transmission line theory using parameters λ0 = 32.4 µm, A1 = 0.1, a1 = 0.001.

Fig. 15
Fig. 15

Four free-standing meshes with parameters g = 24 µm, 2a = 9.6 µm, 2b = 3.6 µm, and t = 0.2 µm at distance of 16 µm between each two. Thick solid curve, Micro-Stripes program calculation of filter (A)(A). Thin solid curve, Micro-Stripes program calculation of filter (B)(B). Dashed curve, ransmission line theory using parameters λ0 = 32.4 µm, A1 = 0.1, a1 = 0.001.

Equations (8)

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λR=2g-4a+2b.
Yλ=1/a1-iω0A1/Ωλ,
Ωλ=g/λω0-λω0/g,
b1=m11 a2+m12 b2, a1=m21 a2+m22 b2.
b1/a1=m12/m22
b2/a1=1/m22.
m111=-Y/2+1,m112=-Y/2, m121=Y/2,m122=Y/2+1,
m211=exp-i2πd/λ,m212=0,m221=0,m222=expi2πd/λ.

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