Abstract

We present theoretical modeling and experimental validation of both capacitive (dot) and inductive (hole) metallic crossed gratings in the mid-infrared (2–5 μm). The gratings are fabricated by use of interferometric lithography and modeled by use of rigorous coupled-wave analysis. Our experimental and numerical investigations of the transmittance spectra of these gratings suggest that, as in inductive grids, the behavior of capacitive grids is described by the coupling of the incident light into surface plasma waves.

© 2002 Optical Society of America

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  20. R. Mittra, C. H. Chan, T. Cwik, “Techniques for analyzing frequency selective surfaces—a review,” Proc. IEEE 76, 1593–1615 (1988).
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  21. R. C. McPhedran, D. Maystre, “On the theory and solar application of inductive grids,” Appl. Phys. 14, 1–20 (1977).
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  22. G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
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    [CrossRef]
  33. P. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  34. R. Petit, G. Tayeb, “On the use of the energy balance criterion as a check of validity of computations in grating theory,” Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. SPIE815, 2–10 (1988).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2001 (1)

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

2000 (2)

E. Popov, M. Nevière, S. Enoch, R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100–16108 (2000).
[CrossRef]

A. Heinzel, V. Boerner, A. Gombert, B. Blasi, V. Wittwer, J. Luther, “Radiation filters and emitters for the NIR based on periodically structured metal surfaces,” J. Mod. Opt. 47, 2399–2419 (2000).
[CrossRef]

1999 (3)

1998 (3)

V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, “Spectral filtering with finitely conducting inductive grids,” J. Opt. Soc. Am. A 15, 2783–2785 (1998).
[CrossRef]

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. B. Movchan, “Off-axis diffraction by perfectly conducting capacitive grids: modal formulation and verification,” J. Electromagn. Waves Appl. 12, 847–882 (1998).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

1997 (1)

1996 (8)

1995 (2)

1994 (1)

1993 (4)

L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
[CrossRef]

O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
[CrossRef]

N. Chateau, J. P. Hugonin, B. Guldimann, P. Chavel, “Two-wave diffraction of quasi-monochromatic light by a volume grating deposited on a thick transparent plate,” Opt. Commun. 103, 444–452 (1993).
[CrossRef]

R. Brauer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

1988 (1)

R. Mittra, C. H. Chan, T. Cwik, “Techniques for analyzing frequency selective surfaces—a review,” Proc. IEEE 76, 1593–1615 (1988).
[CrossRef]

1979 (1)

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

1978 (1)

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

1977 (1)

R. C. McPhedran, D. Maystre, “On the theory and solar application of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

1974 (1)

C. M. Horwitz, “A new solar selective surface,” Opt. Commun. 11, 210–212 (1974).
[CrossRef]

1970 (1)

C.-C. Chen, “Transmission through a conducting screen perforated periodically with apertures,” IEEE Trans. Microwave Theory Tech. 18, 627–632 (1970).
[CrossRef]

1967 (1)

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

1965 (1)

1902 (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
[CrossRef]

Blasi, B.

A. Heinzel, V. Boerner, A. Gombert, B. Blasi, V. Wittwer, J. Luther, “Radiation filters and emitters for the NIR based on periodically structured metal surfaces,” J. Mod. Opt. 47, 2399–2419 (2000).
[CrossRef]

Boerner, V.

A. Heinzel, V. Boerner, A. Gombert, B. Blasi, V. Wittwer, J. Luther, “Radiation filters and emitters for the NIR based on periodically structured metal surfaces,” J. Mod. Opt. 47, 2399–2419 (2000).
[CrossRef]

Botten, L. C.

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. B. Movchan, “Off-axis diffraction by perfectly conducting capacitive grids: modal formulation and verification,” J. Electromagn. Waves Appl. 12, 847–882 (1998).
[CrossRef]

R. C. McPhedran, D. H. Dawes, L. C. Botten, N. A. Nicorovici, “On-axis diffraction by perfectly conducting capacitive grids,” J. Electromagn. Waves Appl. 10, 1085–1111 (1996).
[CrossRef]

Brauer, R.

R. Brauer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Brueck, S. R. J.

X. Chen, S. H. Zaidi, S. R. J. Brueck, “Interferometric lithography of sub-micrometer sparse hole arrays for field-emission display applications,” J. Vac. Sci. Technol. B 14, 3339–3349 (1996).
[CrossRef]

Bruno, O. P.

Bryngdahl, O.

R. Brauer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Chan, C. H.

R. Mittra, C. H. Chan, T. Cwik, “Techniques for analyzing frequency selective surfaces—a review,” Proc. IEEE 76, 1593–1615 (1988).
[CrossRef]

Chateau, N.

N. Chateau, J. P. Hugonin, B. Guldimann, P. Chavel, “Two-wave diffraction of quasi-monochromatic light by a volume grating deposited on a thick transparent plate,” Opt. Commun. 103, 444–452 (1993).
[CrossRef]

Chavel, P.

N. Chateau, J. P. Hugonin, B. Guldimann, P. Chavel, “Two-wave diffraction of quasi-monochromatic light by a volume grating deposited on a thick transparent plate,” Opt. Commun. 103, 444–452 (1993).
[CrossRef]

Chen, C.-C.

C.-C. Chen, “Transmission through a conducting screen perforated periodically with apertures,” IEEE Trans. Microwave Theory Tech. 18, 627–632 (1970).
[CrossRef]

Chen, X.

X. Chen, S. H. Zaidi, S. R. J. Brueck, “Interferometric lithography of sub-micrometer sparse hole arrays for field-emission display applications,” J. Vac. Sci. Technol. B 14, 3339–3349 (1996).
[CrossRef]

Cwik, T.

R. Mittra, C. H. Chan, T. Cwik, “Techniques for analyzing frequency selective surfaces—a review,” Proc. IEEE 76, 1593–1615 (1988).
[CrossRef]

Dawes, D. H.

R. C. McPhedran, D. H. Dawes, L. C. Botten, N. A. Nicorovici, “On-axis diffraction by perfectly conducting capacitive grids,” J. Electromagn. Waves Appl. 10, 1085–1111 (1996).
[CrossRef]

Derrick, G. H.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Ebbesen, T. W.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

T. J. Kim, T. Thio, T. W. Ebbesen, D. E. Grupp, H. J. Lezec, “Control of optical transmission through metals perforated with subwavelength hole arrays,” Opt. Lett. 24, 256–258 (1999).
[CrossRef]

T. Thio, H. F. Ghaemi, H. J. Lezec, P. A. Wolff, T. W. Ebbesen, “Surface-plasmon-enhanced transmission through hole arrays in Cr films,” J. Opt. Soc. Am. B 16, 1743–1748 (1999).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Enoch, S.

E. Popov, M. Nevière, S. Enoch, R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100–16108 (2000).
[CrossRef]

Garcia-Vidal, F. J.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

Ghaemi, H. F.

T. Thio, H. F. Ghaemi, H. J. Lezec, P. A. Wolff, T. W. Ebbesen, “Surface-plasmon-enhanced transmission through hole arrays in Cr films,” J. Opt. Soc. Am. B 16, 1743–1748 (1999).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Gombert, A.

A. Heinzel, V. Boerner, A. Gombert, B. Blasi, V. Wittwer, J. Luther, “Radiation filters and emitters for the NIR based on periodically structured metal surfaces,” J. Mod. Opt. 47, 2399–2419 (2000).
[CrossRef]

Grann, E. B.

Grupp, D. E.

Guldimann, B.

N. Chateau, J. P. Hugonin, B. Guldimann, P. Chavel, “Two-wave diffraction of quasi-monochromatic light by a volume grating deposited on a thick transparent plate,” Opt. Commun. 103, 444–452 (1993).
[CrossRef]

Haggans, C. W.

Harris, B.

Heinzel, A.

A. Heinzel, V. Boerner, A. Gombert, B. Blasi, V. Wittwer, J. Luther, “Radiation filters and emitters for the NIR based on periodically structured metal surfaces,” J. Mod. Opt. 47, 2399–2419 (2000).
[CrossRef]

Hessel, A.

Horwitz, C. M.

C. M. Horwitz, “A new solar selective surface,” Opt. Commun. 11, 210–212 (1974).
[CrossRef]

Hugonin, J. P.

N. Chateau, J. P. Hugonin, B. Guldimann, P. Chavel, “Two-wave diffraction of quasi-monochromatic light by a volume grating deposited on a thick transparent plate,” Opt. Commun. 103, 444–452 (1993).
[CrossRef]

Hutley, M. C.

M. C. Hutley, Diffraction Gratings (Academic, New York, 1982).

Kettunen, V.

Kim, T. J.

Kuittinen, M.

Lalanne, P.

P. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

P. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[CrossRef]

Lemercier-Lalanne, D.

P. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

Lezec, H. J.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

T. J. Kim, T. Thio, T. W. Ebbesen, D. E. Grupp, H. J. Lezec, “Control of optical transmission through metals perforated with subwavelength hole arrays,” Opt. Lett. 24, 256–258 (1999).
[CrossRef]

T. Thio, H. F. Ghaemi, H. J. Lezec, P. A. Wolff, T. W. Ebbesen, “Surface-plasmon-enhanced transmission through hole arrays in Cr films,” J. Opt. Soc. Am. B 16, 1743–1748 (1999).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Li, L.

Luther, J.

A. Heinzel, V. Boerner, A. Gombert, B. Blasi, V. Wittwer, J. Luther, “Radiation filters and emitters for the NIR based on periodically structured metal surfaces,” J. Mod. Opt. 47, 2399–2419 (2000).
[CrossRef]

Martin-Moreno, L.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

Maystre, D.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

R. C. McPhedran, D. Maystre, “On the theory and solar application of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

McPhedran, R. C.

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. B. Movchan, “Off-axis diffraction by perfectly conducting capacitive grids: modal formulation and verification,” J. Electromagn. Waves Appl. 12, 847–882 (1998).
[CrossRef]

R. C. McPhedran, D. H. Dawes, L. C. Botten, N. A. Nicorovici, “On-axis diffraction by perfectly conducting capacitive grids,” J. Electromagn. Waves Appl. 10, 1085–1111 (1996).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

R. C. McPhedran, D. Maystre, “On the theory and solar application of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

Mittra, R.

R. Mittra, C. H. Chan, T. Cwik, “Techniques for analyzing frequency selective surfaces—a review,” Proc. IEEE 76, 1593–1615 (1988).
[CrossRef]

Moharam, M. G.

M. G. Moharam, E. B. Grann, D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[CrossRef]

M. G. Moharam, “Coupled-wave analysis of two-dimensional dielectric gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. SPIE883, 8–11 (1988).
[CrossRef]

Morris, G. M.

Movchan, A. B.

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. B. Movchan, “Off-axis diffraction by perfectly conducting capacitive grids: modal formulation and verification,” J. Electromagn. Waves Appl. 12, 847–882 (1998).
[CrossRef]

Nevière, M.

E. Popov, M. Nevière, S. Enoch, R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100–16108 (2000).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Nicorovici, N. A.

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. B. Movchan, “Off-axis diffraction by perfectly conducting capacitive grids: modal formulation and verification,” J. Electromagn. Waves Appl. 12, 847–882 (1998).
[CrossRef]

R. C. McPhedran, D. H. Dawes, L. C. Botten, N. A. Nicorovici, “On-axis diffraction by perfectly conducting capacitive grids,” J. Electromagn. Waves Appl. 10, 1085–1111 (1996).
[CrossRef]

Noponen, E.

Oliner, A. A.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids, Academic Press Handbook Series (Academic, Orlando, Fla., 1985).

Pellerin, K. M.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

Pendry, J. B.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

Peng, S.

Petit, R.

R. Petit, G. Tayeb, “On the use of the energy balance criterion as a check of validity of computations in grating theory,” Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. SPIE815, 2–10 (1988).
[CrossRef]

R. Petit, Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980).

Pommet, D. A.

Popov, E.

E. Popov, M. Nevière, S. Enoch, R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100–16108 (2000).
[CrossRef]

Preist, T. W.

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, 1988).

Reinisch, R.

E. Popov, M. Nevière, S. Enoch, R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100–16108 (2000).
[CrossRef]

Reitich, F.

Sambles, J. R.

Tayeb, G.

R. Petit, G. Tayeb, “On the use of the energy balance criterion as a check of validity of computations in grating theory,” Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. SPIE815, 2–10 (1988).
[CrossRef]

Thio, T.

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Figures (7)

Fig. 1
Fig. 1

(a) Definition of the incident angles, (b) labels for the intensities carried by the propagating orders used in the finite-substrate calculations, and (c) on-axis diffraction orders included in the finite-substrate model.

Fig. 2
Fig. 2

Convergence behavior of the ISM and the FSM for circular gold scatterers placed on a silicon substrate. The wavelength λ of the normally incident illumination is 4.3 μm, and, referring to the inset, Λ=1.24 μm, d=0.6 μm, h=0.1 μm, and the index of gold at λ=4.3 μm is (2.94, -26.45). A 100×100 grid is used to approximate the periodic unit cell.

Fig. 3
Fig. 3

Transmittance and SPW coupling for an inductive grid with square air holes in gold on a silicon substrate. Referring to the inset, Λ=1.2 μm, d=0.6 μm, and h=0.1 μm. Wavelength resolution for the numerical modeling is 0.02 μm, and the index of silicon is taken as 3.44, independent of wavelength. The locations of SPW coupling for air–metal and silicon–metal are obtained from Eq. (6) and are labeled by solid vertical lines. The locations of Wood’s anomalies are obtained from Eq. (7) and are labeled by dashed vertical lines. Also shown in the figure is the absorptance of a 0.1-μm-thick gold film on a silicon substrate.

Fig. 4
Fig. 4

Transmittance and SPW coupling for a capacitive grid with square gold scatterers on a silicon substrate. Referring to the inset, Λ=1.2 μm, d=0.6 μm, and h=0.1 μm. The wavelength resolution of numerical modeling is 0.02 μm, and the index of silicon is taken as 3.44 for all wavelengths. The locations of SPW coupling for air–metal and silicon–metal are obtained from Eq. (6) and are labeled by solid vertical lines. The locations of Wood’s anomalies are obtained from Eq. (7) and are labeled by dashed vertical lines. Apart from the (0, 1) substrate–metal SPW, coupling to higher-order SPWs is quite weak for this particular capacitive grid and is only visible as discontinuity in the slope of the transmittance spectrum. Also shown is the absorptance of 0.1-μm-thick gold film on a silicon substrate.

Fig. 5
Fig. 5

Comparison between experimental and rigorous modeling results for transmission through an inductive grid of circular air holes in gold on a silicon substrate. Here Λ=1.24 μm, h=0.1 μm, and the air-hole diameter obtained from both the SEM measurement and the mean square fitting of the numerical model to the data is 0.40 μm. The wavelength step used for the numerical modeling is 0.025 μm, and modeling is limited to the wavelength region in which only the on-axis diffraction orders propagate in the silicon substrate. A 100×100 grid is used to approximate the periodic unit cell.

Fig. 6
Fig. 6

Experimental and rigorous modeling results for transmittance through a capacitive grid of circular gold scatterers on a silicon substrate. Here Λ=1.24 μm, h=0.1 μm, and the diameters of the gold scatterers obtained from the SEM measurement is 0.76 μm and from the mean-square fitting of the numerical model to the experimental data is 0.75 μm. The wavelength resolution for the numerical model is 0.025 μm, and modeling is done for the on-axis diffraction orders propagating in the silicon substrate. A 100×100 grid is used to approximate the periodic unit cell.

Fig. 7
Fig. 7

Experimental and numerically calculated dependence of transmittance through a capacitive grid with the fill factor. The transmittance is shown for the case of (0, 1) substrate–metal SPW coupling at λ=4.3 μm. Referring to the inset in Fig. 4, the fill factor for a capacitive grid is given as (π/4)(d/Λ)2. The grid consisted of circular gold scatterers on a silicon substrate and had a pitch of 1.24 μm and a thickness of 0.1 μm. A 100×100 grid is used to approximate the periodic unit cell. Note the noise in numerically calculated transmittance that is due to convergence error in the FSM. The insets in the figure also show SEM images of capacitive grids with fill factors of (a) 10.8%, (b) 19.6%, and (c) 34.3%.

Equations (17)

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E0(x, y, z)=(ex, ey, ez)f00(x, y)exp(-jkz0z).
Ei(x, y, z)=m,n(Rxmn, Rymn, Rzmn)×fmn(x, y)exp(jki,zmnz).
Et(x, y, z)=m,n(Txmn, Tymn, Tzmn)×fmn(x, y)exp(-jkt,zmnz).
(ex, ey, ez)=(cos ψ cos θ cos ϕ-sin ψ sin ϕ,cos ψ cos θ sin ϕ+sin ψ cos ϕ,-cos ψ sin θ),
k0=(kx0, ky0, kz0)=k(cos ϕ sin θ, sin ϕ sin θ, cos ϕ),fmn(x, y)=exp[-j(kxmx+kyny)],
kxm=kx0-m 2πdx,kyn=ky0-n 2πdy,m, n=0, ±1, ±2, . . . ,
kl,zmn
=(k2nl2-kxmn2-kymn2)1/2,k2nl2(kxm2+kyn2)-j(kxmn2+kymn2-k2nl2)1/2,k2nl2<(kxm2+kyn2)
forl=i, t.
Ds=ηiI00i+ηsIs.
Ds=(I-ηsρs)-1ηiI00i.
k=2πλRemedme+d1/2.
λm,nΛ=1(m2+n2)1/2Remed(me+d)1/2.
λm,nΛ=d(m2+n2)1/2.
A=1-dRd-dTd.
FFC=π4dΛ2
FFI=1-π4dΛ2

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