Abstract

An array of flared rectangular holes pierced through a conducting screen is treated herein by a rigorous full-wave modal analysis using the moment method entailing Green’s functions for rectangular cavities and planar multilayer structures in the spectral domain as well as classical Floquet theorem and the mode-matching technique. In this way, flared holes with arbitrary taper profile that may each even be composed of different dielectric sections and which perforated metal films that may be sandwiched between multiple layers of dielectric slabs on both sides is herein treated. The eclectic permutations of geometrical, structural, and material attributes thus afforded by this generic topology facilitate correspondingly diverse investigations that may prove pivotal to the success of future explorations in search for new breakthrough discoveries and innovations in the subject of extraordinary transmission through subwavelength hole arrays, to which the herein-analyzed configuration is central. Oblique angles of incidence for both principal polarizations and metal losses incurred by imperfect conducting screens are also investigated in this work, all constituting crucial aspects that may often be neglected.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]

2014 (2)

R. J. Moerland, J. E. Koskela, A. Kravchenko, M. Simberg, S. van der Vegte, M. Kaivola, A. Priimagio, and R. H. A. Ras, “Large-area arrays of three-dimensional plasmonic subwavelength-sized structures from azopolymer surface-relief gratings,” Mater. Horiz., Royal Soc. Chem 1(74), 74–80 (2014).

M. Ng Mou Kehn, “Modal analysis of substrate integrated waveguides with rectangular via-holes using cavity and multilayer Green’s functions,” IEEE Trans. Microw. Theory Tech. 62(10), 2214–2231 (2014).
[Crossref]

2013 (2)

Y. Liang, W. Peng, R. Hu, and H. Zou, “Extraordinary optical transmission based on subwavelength metallic grating with ellipse walls,” Opt. Express 21(5), 6139–6152 (2013).
[Crossref] [PubMed]

V. Delgado, R. Marques, and L. Jelinek, “Coupled-wave surface-impedance analysis of extraordinary transmission through single and stacked metallic screens,” IEEE Trans. Antenn. Propag. 61(3), 1342–1351 (2013).
[Crossref]

2012 (2)

2011 (3)

J.-C. Yang, H. Gao, J. Y. Suh, W. Zhou, M. H. Lee, and T. W. Odom, “Enhanced optical transmission mediated by localized Plasmons in anisotropic, 3D nanohole arrays,” Nano Lett. 10(8), 3173–3178 (2011).
[Crossref] [PubMed]

M. Beruete, M. Navarro-Cia, and M. S. Ayza, “Understanding anomalous extraordinary transmission from equivalent circuit and grounded slab concepts,” IEEE Trans. Microw. Theory Tech. 59(9), 2180–2188 (2011).
[Crossref]

J. Beermann, T. Sondergaard, S. M. Novikov, S. I. Bozhevolnyi, E. Devaux, and T. W. Ebbesen, “Field enhancement and extraordinary optical transmission by tapered periodic slits in gold films,” New J. Phys. 13, 063029 (2011).
[Crossref]

2010 (2)

T. Søndergaard, S. I. Bozhevolnyi, S. M. Novikov, J. Beermann, E. Devaux, and T. W. Ebbesen, “Extraordinary optical transmission enhanced by nanofocusing,” Nano Lett. 10(8), 3123–3128 (2010).
[Crossref] [PubMed]

F. Medina, F. Mesa, and D. C. Skigin, “Extraordinary transmission through arrays of slits: a circuit theory model,” IEEE Trans. Microw. Theory Tech. 58(1), 105–115 (2010).
[Crossref]

2009 (1)

2008 (1)

F. Medina, F. Mesa, and R. Marques, “Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective,” IEEE Trans. Microw. Theory Tech. 56(12), 3108–3120 (2008).
[Crossref]

2007 (1)

M. Beruete, I. Campillo, M. Navarro-Cía, F. Falcone, and M. Sorolla, “Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays,” IEEE Trans. Antenn. Propag. 55(6), 1514–1521 (2007).
[Crossref]

2006 (1)

V. Lomakin and E. Michielssen, “Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes,” IEEE Trans. Antenn. Propag. 54(3), 970–984 (2006).
[Crossref]

2005 (2)

M. Beruete, M. Sorolla, I. Campillo, and J. S. Dolado, “Increase of the transmission in cut-off metallic hole arrays,” IEEE Microw. Wirel. Compon. Lett. 15(2), 116–118 (2005).
[Crossref]

V. Lomakin and E. Michielssen, “Enhanced transmission through metallic plates perforated by arrays of subwavelength holes and sandwiched between dielectric slabs,” Phys. Rev. B Condens. Matter Mater. Phys. 71(23), 235117 (2005).
[Crossref]

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[Crossref]

1982 (1)

S. W. Lee, G. Zarrillo, and C. L. Law, “Simple formulas for transmission through periodic metal grids or plates,” IEEE Trans. Antenn. Propag. 30(5), 904–909 (1982).
[Crossref]

1971 (1)

C. C. Chen, “Diffraction of electromagnetic waves by a conducting screen perforated periodically with circular holes,” IEEE Trans. Microw. Theory Tech. 19(5), 475–481 (1971).
[Crossref]

1970 (1)

C. C. Chen, “Transmission through a conducting screen perforated periodically with apertures,” IEEE Trans. Microw. Theory Tech. 18(9), 627–632 (1970).
[Crossref]

1967 (2)

R. Ulrich, “Far-infrared properties of metallic mesh and its complementary structure,” Infrared Phys. 7(1), 37–55 (1967).
[Crossref]

G. M. Ressler and K. D. Möller, “Far infrared bandpass filters and measurements on a reciprocal grid,” Appl. Opt. 6(5), 893–896 (1967).
[Crossref] [PubMed]

1963 (1)

A. Mitsuishi, Y. Otsuka, S. Fujita, and H. Yoshinaga, “Metal mesh filters in the far infrared region,” Jpn. J. Appl. Phys. 2(9), 574–577 (1963).
[Crossref]

1962 (1)

1944 (1)

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66(7-8), 163–182 (1944).
[Crossref]

Ayza, M. S.

M. Beruete, M. Navarro-Cia, and M. S. Ayza, “Understanding anomalous extraordinary transmission from equivalent circuit and grounded slab concepts,” IEEE Trans. Microw. Theory Tech. 59(9), 2180–2188 (2011).
[Crossref]

Beermann, J.

T. Sondergaard, S. I. Bozhevolnyi, J. Beermann, S. M. Novikov, E. Devaux, and T. W. Ebbesen, “Extraordinary optical transmission with tapered slits: effect of higher diffraction and slit resonance orders,” J. Opt. Soc. Am. B 29(1), 130–137 (2012).
[Crossref]

J. Beermann, T. Sondergaard, S. M. Novikov, S. I. Bozhevolnyi, E. Devaux, and T. W. Ebbesen, “Field enhancement and extraordinary optical transmission by tapered periodic slits in gold films,” New J. Phys. 13, 063029 (2011).
[Crossref]

T. Søndergaard, S. I. Bozhevolnyi, S. M. Novikov, J. Beermann, E. Devaux, and T. W. Ebbesen, “Extraordinary optical transmission enhanced by nanofocusing,” Nano Lett. 10(8), 3123–3128 (2010).
[Crossref] [PubMed]

Beruete, M.

M. Beruete, M. Navarro-Cia, and M. S. Ayza, “Understanding anomalous extraordinary transmission from equivalent circuit and grounded slab concepts,” IEEE Trans. Microw. Theory Tech. 59(9), 2180–2188 (2011).
[Crossref]

M. Beruete, I. Campillo, M. Navarro-Cía, F. Falcone, and M. Sorolla, “Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays,” IEEE Trans. Antenn. Propag. 55(6), 1514–1521 (2007).
[Crossref]

M. Beruete, M. Sorolla, I. Campillo, and J. S. Dolado, “Increase of the transmission in cut-off metallic hole arrays,” IEEE Microw. Wirel. Compon. Lett. 15(2), 116–118 (2005).
[Crossref]

Bethe, H. A.

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66(7-8), 163–182 (1944).
[Crossref]

Bozhevolnyi, S. I.

T. Sondergaard, S. I. Bozhevolnyi, J. Beermann, S. M. Novikov, E. Devaux, and T. W. Ebbesen, “Extraordinary optical transmission with tapered slits: effect of higher diffraction and slit resonance orders,” J. Opt. Soc. Am. B 29(1), 130–137 (2012).
[Crossref]

J. Beermann, T. Sondergaard, S. M. Novikov, S. I. Bozhevolnyi, E. Devaux, and T. W. Ebbesen, “Field enhancement and extraordinary optical transmission by tapered periodic slits in gold films,” New J. Phys. 13, 063029 (2011).
[Crossref]

T. Søndergaard, S. I. Bozhevolnyi, S. M. Novikov, J. Beermann, E. Devaux, and T. W. Ebbesen, “Extraordinary optical transmission enhanced by nanofocusing,” Nano Lett. 10(8), 3123–3128 (2010).
[Crossref] [PubMed]

Campillo, I.

M. Beruete, I. Campillo, M. Navarro-Cía, F. Falcone, and M. Sorolla, “Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays,” IEEE Trans. Antenn. Propag. 55(6), 1514–1521 (2007).
[Crossref]

M. Beruete, M. Sorolla, I. Campillo, and J. S. Dolado, “Increase of the transmission in cut-off metallic hole arrays,” IEEE Microw. Wirel. Compon. Lett. 15(2), 116–118 (2005).
[Crossref]

Chen, C. C.

C. C. Chen, “Diffraction of electromagnetic waves by a conducting screen perforated periodically with circular holes,” IEEE Trans. Microw. Theory Tech. 19(5), 475–481 (1971).
[Crossref]

C. C. Chen, “Transmission through a conducting screen perforated periodically with apertures,” IEEE Trans. Microw. Theory Tech. 18(9), 627–632 (1970).
[Crossref]

Chen, H. L.

Chou, Y. F.

Chuang, S. Y.

Delgado, V.

V. Delgado, R. Marques, and L. Jelinek, “Coupled-wave surface-impedance analysis of extraordinary transmission through single and stacked metallic screens,” IEEE Trans. Antenn. Propag. 61(3), 1342–1351 (2013).
[Crossref]

Devaux, E.

T. Sondergaard, S. I. Bozhevolnyi, J. Beermann, S. M. Novikov, E. Devaux, and T. W. Ebbesen, “Extraordinary optical transmission with tapered slits: effect of higher diffraction and slit resonance orders,” J. Opt. Soc. Am. B 29(1), 130–137 (2012).
[Crossref]

J. Beermann, T. Sondergaard, S. M. Novikov, S. I. Bozhevolnyi, E. Devaux, and T. W. Ebbesen, “Field enhancement and extraordinary optical transmission by tapered periodic slits in gold films,” New J. Phys. 13, 063029 (2011).
[Crossref]

T. Søndergaard, S. I. Bozhevolnyi, S. M. Novikov, J. Beermann, E. Devaux, and T. W. Ebbesen, “Extraordinary optical transmission enhanced by nanofocusing,” Nano Lett. 10(8), 3123–3128 (2010).
[Crossref] [PubMed]

Dolado, J. S.

M. Beruete, M. Sorolla, I. Campillo, and J. S. Dolado, “Increase of the transmission in cut-off metallic hole arrays,” IEEE Microw. Wirel. Compon. Lett. 15(2), 116–118 (2005).
[Crossref]

Ebbesen, T. W.

T. Sondergaard, S. I. Bozhevolnyi, J. Beermann, S. M. Novikov, E. Devaux, and T. W. Ebbesen, “Extraordinary optical transmission with tapered slits: effect of higher diffraction and slit resonance orders,” J. Opt. Soc. Am. B 29(1), 130–137 (2012).
[Crossref]

J. Beermann, T. Sondergaard, S. M. Novikov, S. I. Bozhevolnyi, E. Devaux, and T. W. Ebbesen, “Field enhancement and extraordinary optical transmission by tapered periodic slits in gold films,” New J. Phys. 13, 063029 (2011).
[Crossref]

T. Søndergaard, S. I. Bozhevolnyi, S. M. Novikov, J. Beermann, E. Devaux, and T. W. Ebbesen, “Extraordinary optical transmission enhanced by nanofocusing,” Nano Lett. 10(8), 3123–3128 (2010).
[Crossref] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[Crossref]

Falcone, F.

M. Beruete, I. Campillo, M. Navarro-Cía, F. Falcone, and M. Sorolla, “Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays,” IEEE Trans. Antenn. Propag. 55(6), 1514–1521 (2007).
[Crossref]

Fujita, S.

A. Mitsuishi, Y. Otsuka, S. Fujita, and H. Yoshinaga, “Metal mesh filters in the far infrared region,” Jpn. J. Appl. Phys. 2(9), 574–577 (1963).
[Crossref]

Gao, H.

J.-C. Yang, H. Gao, J. Y. Suh, W. Zhou, M. H. Lee, and T. W. Odom, “Enhanced optical transmission mediated by localized Plasmons in anisotropic, 3D nanohole arrays,” Nano Lett. 10(8), 3173–3178 (2011).
[Crossref] [PubMed]

Genzel, L.

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[Crossref]

Hu, R.

Jelinek, L.

V. Delgado, R. Marques, and L. Jelinek, “Coupled-wave surface-impedance analysis of extraordinary transmission through single and stacked metallic screens,” IEEE Trans. Antenn. Propag. 61(3), 1342–1351 (2013).
[Crossref]

Kaivola, M.

R. J. Moerland, J. E. Koskela, A. Kravchenko, M. Simberg, S. van der Vegte, M. Kaivola, A. Priimagio, and R. H. A. Ras, “Large-area arrays of three-dimensional plasmonic subwavelength-sized structures from azopolymer surface-relief gratings,” Mater. Horiz., Royal Soc. Chem 1(74), 74–80 (2014).

Koskela, J. E.

R. J. Moerland, J. E. Koskela, A. Kravchenko, M. Simberg, S. van der Vegte, M. Kaivola, A. Priimagio, and R. H. A. Ras, “Large-area arrays of three-dimensional plasmonic subwavelength-sized structures from azopolymer surface-relief gratings,” Mater. Horiz., Royal Soc. Chem 1(74), 74–80 (2014).

Kravchenko, A.

R. J. Moerland, J. E. Koskela, A. Kravchenko, M. Simberg, S. van der Vegte, M. Kaivola, A. Priimagio, and R. H. A. Ras, “Large-area arrays of three-dimensional plasmonic subwavelength-sized structures from azopolymer surface-relief gratings,” Mater. Horiz., Royal Soc. Chem 1(74), 74–80 (2014).

Ku, S. L.

Kuo, S. S.

Law, C. L.

S. W. Lee, G. Zarrillo, and C. L. Law, “Simple formulas for transmission through periodic metal grids or plates,” IEEE Trans. Antenn. Propag. 30(5), 904–909 (1982).
[Crossref]

Lee, M. H.

J.-C. Yang, H. Gao, J. Y. Suh, W. Zhou, M. H. Lee, and T. W. Odom, “Enhanced optical transmission mediated by localized Plasmons in anisotropic, 3D nanohole arrays,” Nano Lett. 10(8), 3173–3178 (2011).
[Crossref] [PubMed]

Lee, S. W.

S. W. Lee, G. Zarrillo, and C. L. Law, “Simple formulas for transmission through periodic metal grids or plates,” IEEE Trans. Antenn. Propag. 30(5), 904–909 (1982).
[Crossref]

Lee, W. H.

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[Crossref]

Liang, Y.

Lomakin, V.

V. Lomakin and E. Michielssen, “Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes,” IEEE Trans. Antenn. Propag. 54(3), 970–984 (2006).
[Crossref]

V. Lomakin and E. Michielssen, “Enhanced transmission through metallic plates perforated by arrays of subwavelength holes and sandwiched between dielectric slabs,” Phys. Rev. B Condens. Matter Mater. Phys. 71(23), 235117 (2005).
[Crossref]

Maes, B.

H. Shen and B. Maes, “Enhanced optical transmission through tapered metallic gratings,” Appl. Phys. Lett. 100(24), 241104 (2012).
[Crossref]

Marques, R.

V. Delgado, R. Marques, and L. Jelinek, “Coupled-wave surface-impedance analysis of extraordinary transmission through single and stacked metallic screens,” IEEE Trans. Antenn. Propag. 61(3), 1342–1351 (2013).
[Crossref]

F. Medina, F. Mesa, and R. Marques, “Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective,” IEEE Trans. Microw. Theory Tech. 56(12), 3108–3120 (2008).
[Crossref]

Medina, F.

F. Medina, F. Mesa, and D. C. Skigin, “Extraordinary transmission through arrays of slits: a circuit theory model,” IEEE Trans. Microw. Theory Tech. 58(1), 105–115 (2010).
[Crossref]

F. Medina, F. Mesa, and R. Marques, “Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective,” IEEE Trans. Microw. Theory Tech. 56(12), 3108–3120 (2008).
[Crossref]

F. Medina, R. Rodriguez-Berral, and F. Mesa, “Circuit model for metallic gratings with tapered and stepped slits,” Proc. 42nd European Microwave Conference, Oct. 2012.
[Crossref]

Mesa, F.

F. Medina, F. Mesa, and D. C. Skigin, “Extraordinary transmission through arrays of slits: a circuit theory model,” IEEE Trans. Microw. Theory Tech. 58(1), 105–115 (2010).
[Crossref]

F. Medina, F. Mesa, and R. Marques, “Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective,” IEEE Trans. Microw. Theory Tech. 56(12), 3108–3120 (2008).
[Crossref]

F. Medina, R. Rodriguez-Berral, and F. Mesa, “Circuit model for metallic gratings with tapered and stepped slits,” Proc. 42nd European Microwave Conference, Oct. 2012.
[Crossref]

Michielssen, E.

V. Lomakin and E. Michielssen, “Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes,” IEEE Trans. Antenn. Propag. 54(3), 970–984 (2006).
[Crossref]

V. Lomakin and E. Michielssen, “Enhanced transmission through metallic plates perforated by arrays of subwavelength holes and sandwiched between dielectric slabs,” Phys. Rev. B Condens. Matter Mater. Phys. 71(23), 235117 (2005).
[Crossref]

Mitsuishi, A.

A. Mitsuishi, Y. Otsuka, S. Fujita, and H. Yoshinaga, “Metal mesh filters in the far infrared region,” Jpn. J. Appl. Phys. 2(9), 574–577 (1963).
[Crossref]

Moerland, R. J.

R. J. Moerland, J. E. Koskela, A. Kravchenko, M. Simberg, S. van der Vegte, M. Kaivola, A. Priimagio, and R. H. A. Ras, “Large-area arrays of three-dimensional plasmonic subwavelength-sized structures from azopolymer surface-relief gratings,” Mater. Horiz., Royal Soc. Chem 1(74), 74–80 (2014).

Möller, K. D.

Navarro-Cia, M.

M. Beruete, M. Navarro-Cia, and M. S. Ayza, “Understanding anomalous extraordinary transmission from equivalent circuit and grounded slab concepts,” IEEE Trans. Microw. Theory Tech. 59(9), 2180–2188 (2011).
[Crossref]

Navarro-Cía, M.

M. Beruete, I. Campillo, M. Navarro-Cía, F. Falcone, and M. Sorolla, “Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays,” IEEE Trans. Antenn. Propag. 55(6), 1514–1521 (2007).
[Crossref]

Ng Mou Kehn, M.

M. Ng Mou Kehn, “Modal analysis of substrate integrated waveguides with rectangular via-holes using cavity and multilayer Green’s functions,” IEEE Trans. Microw. Theory Tech. 62(10), 2214–2231 (2014).
[Crossref]

Novikov, S. M.

T. Sondergaard, S. I. Bozhevolnyi, J. Beermann, S. M. Novikov, E. Devaux, and T. W. Ebbesen, “Extraordinary optical transmission with tapered slits: effect of higher diffraction and slit resonance orders,” J. Opt. Soc. Am. B 29(1), 130–137 (2012).
[Crossref]

J. Beermann, T. Sondergaard, S. M. Novikov, S. I. Bozhevolnyi, E. Devaux, and T. W. Ebbesen, “Field enhancement and extraordinary optical transmission by tapered periodic slits in gold films,” New J. Phys. 13, 063029 (2011).
[Crossref]

T. Søndergaard, S. I. Bozhevolnyi, S. M. Novikov, J. Beermann, E. Devaux, and T. W. Ebbesen, “Extraordinary optical transmission enhanced by nanofocusing,” Nano Lett. 10(8), 3123–3128 (2010).
[Crossref] [PubMed]

Odom, T. W.

J.-C. Yang, H. Gao, J. Y. Suh, W. Zhou, M. H. Lee, and T. W. Odom, “Enhanced optical transmission mediated by localized Plasmons in anisotropic, 3D nanohole arrays,” Nano Lett. 10(8), 3173–3178 (2011).
[Crossref] [PubMed]

Otsuka, Y.

A. Mitsuishi, Y. Otsuka, S. Fujita, and H. Yoshinaga, “Metal mesh filters in the far infrared region,” Jpn. J. Appl. Phys. 2(9), 574–577 (1963).
[Crossref]

Peng, W.

Priimagio, A.

R. J. Moerland, J. E. Koskela, A. Kravchenko, M. Simberg, S. van der Vegte, M. Kaivola, A. Priimagio, and R. H. A. Ras, “Large-area arrays of three-dimensional plasmonic subwavelength-sized structures from azopolymer surface-relief gratings,” Mater. Horiz., Royal Soc. Chem 1(74), 74–80 (2014).

Ras, R. H. A.

R. J. Moerland, J. E. Koskela, A. Kravchenko, M. Simberg, S. van der Vegte, M. Kaivola, A. Priimagio, and R. H. A. Ras, “Large-area arrays of three-dimensional plasmonic subwavelength-sized structures from azopolymer surface-relief gratings,” Mater. Horiz., Royal Soc. Chem 1(74), 74–80 (2014).

Renk, K. F.

Ressler, G. M.

Rodriguez-Berral, R.

F. Medina, R. Rodriguez-Berral, and F. Mesa, “Circuit model for metallic gratings with tapered and stepped slits,” Proc. 42nd European Microwave Conference, Oct. 2012.
[Crossref]

Shen, H.

H. Shen and B. Maes, “Enhanced optical transmission through tapered metallic gratings,” Appl. Phys. Lett. 100(24), 241104 (2012).
[Crossref]

Simberg, M.

R. J. Moerland, J. E. Koskela, A. Kravchenko, M. Simberg, S. van der Vegte, M. Kaivola, A. Priimagio, and R. H. A. Ras, “Large-area arrays of three-dimensional plasmonic subwavelength-sized structures from azopolymer surface-relief gratings,” Mater. Horiz., Royal Soc. Chem 1(74), 74–80 (2014).

Skigin, D. C.

F. Medina, F. Mesa, and D. C. Skigin, “Extraordinary transmission through arrays of slits: a circuit theory model,” IEEE Trans. Microw. Theory Tech. 58(1), 105–115 (2010).
[Crossref]

Sondergaard, T.

T. Sondergaard, S. I. Bozhevolnyi, J. Beermann, S. M. Novikov, E. Devaux, and T. W. Ebbesen, “Extraordinary optical transmission with tapered slits: effect of higher diffraction and slit resonance orders,” J. Opt. Soc. Am. B 29(1), 130–137 (2012).
[Crossref]

J. Beermann, T. Sondergaard, S. M. Novikov, S. I. Bozhevolnyi, E. Devaux, and T. W. Ebbesen, “Field enhancement and extraordinary optical transmission by tapered periodic slits in gold films,” New J. Phys. 13, 063029 (2011).
[Crossref]

Søndergaard, T.

T. Søndergaard, S. I. Bozhevolnyi, S. M. Novikov, J. Beermann, E. Devaux, and T. W. Ebbesen, “Extraordinary optical transmission enhanced by nanofocusing,” Nano Lett. 10(8), 3123–3128 (2010).
[Crossref] [PubMed]

Sorolla, M.

M. Beruete, I. Campillo, M. Navarro-Cía, F. Falcone, and M. Sorolla, “Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays,” IEEE Trans. Antenn. Propag. 55(6), 1514–1521 (2007).
[Crossref]

M. Beruete, M. Sorolla, I. Campillo, and J. S. Dolado, “Increase of the transmission in cut-off metallic hole arrays,” IEEE Microw. Wirel. Compon. Lett. 15(2), 116–118 (2005).
[Crossref]

Su, W. F.

Suh, J. Y.

J.-C. Yang, H. Gao, J. Y. Suh, W. Zhou, M. H. Lee, and T. W. Odom, “Enhanced optical transmission mediated by localized Plasmons in anisotropic, 3D nanohole arrays,” Nano Lett. 10(8), 3173–3178 (2011).
[Crossref] [PubMed]

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[Crossref]

Ulrich, R.

R. Ulrich, “Far-infrared properties of metallic mesh and its complementary structure,” Infrared Phys. 7(1), 37–55 (1967).
[Crossref]

van der Vegte, S.

R. J. Moerland, J. E. Koskela, A. Kravchenko, M. Simberg, S. van der Vegte, M. Kaivola, A. Priimagio, and R. H. A. Ras, “Large-area arrays of three-dimensional plasmonic subwavelength-sized structures from azopolymer surface-relief gratings,” Mater. Horiz., Royal Soc. Chem 1(74), 74–80 (2014).

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[Crossref]

Yang, J.-C.

J.-C. Yang, H. Gao, J. Y. Suh, W. Zhou, M. H. Lee, and T. W. Odom, “Enhanced optical transmission mediated by localized Plasmons in anisotropic, 3D nanohole arrays,” Nano Lett. 10(8), 3173–3178 (2011).
[Crossref] [PubMed]

Yoshinaga, H.

A. Mitsuishi, Y. Otsuka, S. Fujita, and H. Yoshinaga, “Metal mesh filters in the far infrared region,” Jpn. J. Appl. Phys. 2(9), 574–577 (1963).
[Crossref]

Zarrillo, G.

S. W. Lee, G. Zarrillo, and C. L. Law, “Simple formulas for transmission through periodic metal grids or plates,” IEEE Trans. Antenn. Propag. 30(5), 904–909 (1982).
[Crossref]

Zhou, W.

J.-C. Yang, H. Gao, J. Y. Suh, W. Zhou, M. H. Lee, and T. W. Odom, “Enhanced optical transmission mediated by localized Plasmons in anisotropic, 3D nanohole arrays,” Nano Lett. 10(8), 3173–3178 (2011).
[Crossref] [PubMed]

Zou, H.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

H. Shen and B. Maes, “Enhanced optical transmission through tapered metallic gratings,” Appl. Phys. Lett. 100(24), 241104 (2012).
[Crossref]

IEEE Microw. Wirel. Compon. Lett. (1)

M. Beruete, M. Sorolla, I. Campillo, and J. S. Dolado, “Increase of the transmission in cut-off metallic hole arrays,” IEEE Microw. Wirel. Compon. Lett. 15(2), 116–118 (2005).
[Crossref]

IEEE Trans. Antenn. Propag. (4)

V. Lomakin and E. Michielssen, “Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes,” IEEE Trans. Antenn. Propag. 54(3), 970–984 (2006).
[Crossref]

M. Beruete, I. Campillo, M. Navarro-Cía, F. Falcone, and M. Sorolla, “Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays,” IEEE Trans. Antenn. Propag. 55(6), 1514–1521 (2007).
[Crossref]

V. Delgado, R. Marques, and L. Jelinek, “Coupled-wave surface-impedance analysis of extraordinary transmission through single and stacked metallic screens,” IEEE Trans. Antenn. Propag. 61(3), 1342–1351 (2013).
[Crossref]

S. W. Lee, G. Zarrillo, and C. L. Law, “Simple formulas for transmission through periodic metal grids or plates,” IEEE Trans. Antenn. Propag. 30(5), 904–909 (1982).
[Crossref]

IEEE Trans. Microw. Theory Tech. (6)

C. C. Chen, “Diffraction of electromagnetic waves by a conducting screen perforated periodically with circular holes,” IEEE Trans. Microw. Theory Tech. 19(5), 475–481 (1971).
[Crossref]

C. C. Chen, “Transmission through a conducting screen perforated periodically with apertures,” IEEE Trans. Microw. Theory Tech. 18(9), 627–632 (1970).
[Crossref]

F. Medina, F. Mesa, and R. Marques, “Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective,” IEEE Trans. Microw. Theory Tech. 56(12), 3108–3120 (2008).
[Crossref]

F. Medina, F. Mesa, and D. C. Skigin, “Extraordinary transmission through arrays of slits: a circuit theory model,” IEEE Trans. Microw. Theory Tech. 58(1), 105–115 (2010).
[Crossref]

M. Beruete, M. Navarro-Cia, and M. S. Ayza, “Understanding anomalous extraordinary transmission from equivalent circuit and grounded slab concepts,” IEEE Trans. Microw. Theory Tech. 59(9), 2180–2188 (2011).
[Crossref]

M. Ng Mou Kehn, “Modal analysis of substrate integrated waveguides with rectangular via-holes using cavity and multilayer Green’s functions,” IEEE Trans. Microw. Theory Tech. 62(10), 2214–2231 (2014).
[Crossref]

Infrared Phys. (1)

R. Ulrich, “Far-infrared properties of metallic mesh and its complementary structure,” Infrared Phys. 7(1), 37–55 (1967).
[Crossref]

J. Opt. Soc. Am. B (1)

Jpn. J. Appl. Phys. (1)

A. Mitsuishi, Y. Otsuka, S. Fujita, and H. Yoshinaga, “Metal mesh filters in the far infrared region,” Jpn. J. Appl. Phys. 2(9), 574–577 (1963).
[Crossref]

Mater. Horiz., Royal Soc. Chem (1)

R. J. Moerland, J. E. Koskela, A. Kravchenko, M. Simberg, S. van der Vegte, M. Kaivola, A. Priimagio, and R. H. A. Ras, “Large-area arrays of three-dimensional plasmonic subwavelength-sized structures from azopolymer surface-relief gratings,” Mater. Horiz., Royal Soc. Chem 1(74), 74–80 (2014).

Nano Lett. (2)

J.-C. Yang, H. Gao, J. Y. Suh, W. Zhou, M. H. Lee, and T. W. Odom, “Enhanced optical transmission mediated by localized Plasmons in anisotropic, 3D nanohole arrays,” Nano Lett. 10(8), 3173–3178 (2011).
[Crossref] [PubMed]

T. Søndergaard, S. I. Bozhevolnyi, S. M. Novikov, J. Beermann, E. Devaux, and T. W. Ebbesen, “Extraordinary optical transmission enhanced by nanofocusing,” Nano Lett. 10(8), 3123–3128 (2010).
[Crossref] [PubMed]

Nature (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[Crossref]

New J. Phys. (1)

J. Beermann, T. Sondergaard, S. M. Novikov, S. I. Bozhevolnyi, E. Devaux, and T. W. Ebbesen, “Field enhancement and extraordinary optical transmission by tapered periodic slits in gold films,” New J. Phys. 13, 063029 (2011).
[Crossref]

Opt. Express (2)

Phys. Rev. (1)

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66(7-8), 163–182 (1944).
[Crossref]

Phys. Rev. B Condens. Matter Mater. Phys. (1)

V. Lomakin and E. Michielssen, “Enhanced transmission through metallic plates perforated by arrays of subwavelength holes and sandwiched between dielectric slabs,” Phys. Rev. B Condens. Matter Mater. Phys. 71(23), 235117 (2005).
[Crossref]

Other (1)

F. Medina, R. Rodriguez-Berral, and F. Mesa, “Circuit model for metallic gratings with tapered and stepped slits,” Proc. 42nd European Microwave Conference, Oct. 2012.
[Crossref]

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Figures (11)

Fig. 1
Fig. 1 Periodic two-dimensional array of flared-open type of generally rectangular holes (although shown as square ones) pierced through a metallic sheet of thickness d, with each tunnel modeled as a connected stepped series of cavity sections. The width aι and height hι along x and y of an arbitrarily-selected section deemed as the ιth section is as indicated. Two perspectives showing front and back of the screen are displayed.
Fig. 2
Fig. 2 Schematic and geometry of rectangular hole array illuminated by incident plane wave, with plane of incidence defined by (θinc, ϕinc).
Fig. 3
Fig. 3 Interconnected series of rectangular waveguide sections modeling a flared-open hole. Grayed portions represent the conducting parts of the metallic screen. Any ιth section filled with a material of parameters: (μι, ει)
Fig. 4
Fig. 4 Five-layer configuration for (a) left semi-infinite (incident) region, and (b) right semi-infinite (transmission) region of rectangular hole-array used in core-routine of spectral multilayer Green's function.
Fig. 5
Fig. 5 Validation with CST of modal formulation for zeroth-order power transmission coefficient versus frequency for TMz polarized plane wave (with ϕinc = 90°) normally incident on ordinary square hole array with unit cell period: dx = dy = 5mm, square hole size: aι = hι = 2.5mm for all ι, and screen thickness d = 1.25mm.
Fig. 6
Fig. 6 Validation with CST of modal formulation for zeroth-order power transmission coefficient versus frequency for TMz polarized plane wave (with ϕinc = 90°) normally incident on array of linearly flared-open square holes, with unit cell period (lattice constant): dx = dy = dx&y = 5 mm, input square hole sizes: a1 = h1 = 2.5 mm, output square hole sizes: aZ = hZ = 4.5 mm, and screen thickness d = 1.25 mm: (a) without flanged terminal irises (a0 = a1, h0 = h1, aZ = aZΘ, hZ = hZΘ), (b) with flanged terminal irises (a0 = a1/2, h0 = h1/2, aZΘ = aZ/2, hZΘ = hZ/2), and (c) with flanged terminal irises (a0 = 3a1/4, h0 = 3h1/4, aZΘ = 3aZ/4, hZΘ = 3hZ/4).
Fig. 7
Fig. 7 Validation with CST of modal formulation for zeroth-order transmission coefficient versus frequency for ϕinc = 0 azimuth plane of incidence of arriving plane wave that illuminates array of square holes each divided into two sections (single-step flare), with unit cell period: dx = dy = 5 mm, input square hole sizes: a0 = a1 = h0 = h1 = 2.5 mm, output square hole sizes: aZ = aZΘ = hZ = hZΘ = 4.5 mm, thicknesses of input and output sections = 0.5 mm and 0.75 mm, respectively, total screen thickness d = 1.25 mm: (a) TEz polarized incidence, θinc = 15°, (b) TMz polarized incidence, θinc = 15°, and (c) TEz polarized incidence, θinc = 30°.
Fig. 8
Fig. 8 Graphs of transmission, conductivity, and total efficiencies versus frequency, for linearly flared hole array. Copper assumed as the lossy metal, with σ = 5.8 × 107 S/m and μcond = μ0.
Fig. 9
Fig. 9 Transmission spectra for various numbers of Floquet harmonics and cavity modes (NFloq & Ncav) considered in computation: each panel pertains to certain NFloq and displays traces for Ncav = 6, 16, 30, 48, 70 & 96 as annotated: (a) NFloq = 9, (b) NFloq = 25, (c) NFloq = 49, (d) NFloq = 81, (e) NFloq = 121, and (f) NFloq = 169.
Fig. 10
Fig. 10 Transmission spectra for various numbers of Floquet harmonics and cavity modes (NFloq & Ncav) considered in computation: each panel pertains to certain Ncav and displays traces for NFloq = 9, 25, 49, 81, 121 & 169 as annotated: (a) Ncav = 6, (b) Ncav = 16, (c) Ncav = 30, (d) Ncav = 48, (e) Ncav = 70, and (f) Ncav = 96.
Fig. 11
Fig. 11 Graph of transmission coefficient versus frequency; measured for prototype 1D array of slits (period 20 mm, groove width 14 mm, and a truncated length of 500 mm), and also computed (by code of present method) for 2D array of elongated rectangular holes (dx = 20 mm, dy = 500 mm, hole dimensions a = 14 mm, h = 0.95dx). For both theory and experiment, screen thickness is 12 mm, TMz polarized incidence with θinc = 0, ϕinc = 90°.

Equations (52)

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e t p q TE (ι) (x,y)={ x ^ 2 Z 0q TE(ι) a ι h ι S y (ι) p=0,q0 y ^ 2 Z p0 TE(ι) a ι h ι S x (ι) p0,q=0 2 k c pq (ι) Z pq TE(ι) a ι h ι ( x ^ k y q (ι) C x (ι) S y (ι) y ^ k x p (ι) S x (ι) C y (ι) ) p0,q0
h t p q TE (ι) (x,y)={ y ^ 2 Y 0q TE(ι) a ι h ι [ S y (ι) ] p=0,q0 x ^ 2 Y p0 TE(ι) a ι h ι [ S x (ι) ] p0,q=0 2 k c pq (ι) Y pq TE(ι) a ι h ι ( x ^ k x p (ι) S x (ι) C y (ι) + y ^ k y q (ι) C x (ι) S y (ι) ) p0,q0
e t p q TM (ι) (x,y)= 2 k c pq (ι) Z pq TM(ι) a ι h ι ( x ^ k x p (ι) C x (ι) S y (ι) + y ^ k y q (ι) S x (ι) C y (ι) )
h t p q TM (ι) (x,y)= 2 k c pq (ι) Y pq TM(ι) a ι h ι ( x ^ k y q (ι) S x (ι) C y (ι) y ^ k x p (ι) C x (ι) S y (ι) )
k x p (ι) = pπ/ a ι ; k y q (ι) = qπ/ h ι ; k c pq (ι) = [ k x p (ι) ] 2 + [ k y q (ι) ] 2
{ C x (ι) S x (ι) }={ cos sin }( k x p (ι) x+ pπ 2 ); { C y (ι) S y (ι) }={ cos sin }( k y q (ι) y+ qπ 2 )
Z pq TE(ι) = jω μ ι / γ z pq (ι) = ( Y pq TE(ι) ) 1 ; Z pq TM(ι) = γ z pq (ι) / jω ε ι = ( Y pq TM(ι) ) 1
γ z pq (ι) = [ k c pq (ι) ] 2 k ι 2 , k i =ω μ ι ε ι ; ω=2πf
y= h ι /2 y= h ι /2 x= a ι /2 x= a ι /2 ( e t p q r (ι) × h t u v w (ι) ) z ^ dx dy= δ p q r ,u v w ={ 1, if p=u,q=v,rw 0, otherwise
C A p B u = y= h A /2 y= h A /2 x= a A /2 x= a A /2 [ e x (A) h y (B) e y (A) h x (B) ]dxdy
e t (ι) (x,y)= pqr=1 N ι A pqr (ι) e t pqr (ι) (x,y) ; e t pqr (ι) = x ^ e x pqr (ι) + y ^ e y pqr (ι)
M { y x } z s = z ι1 | z ι =+| pqr=1 N ι1|ι A pqr ( ι1|ι) [ {±} e { x y } pqr ( ι1|ι) ]
M ˜ ˜ { y x } z s = z ι1 | z ι ( k x , k y )=+| pqr=1 N ι1|ι A pqr ( ι1|ι) [ {±} e ˜ ˜ { x y } pqr ( ι1|ι) ( k x , k y ) ]
e ˜ ˜ w pqr (ι) ( k x , k y )= y= h ι 2 y= h ι 2 x= a ι 2 x= a ι 2 e w pqr (ι) (x,y) e j( k x x+ k y y ) dxdy
Φ pqr (ι) =coth[ γ z pqr (ι) l ι ], Ω pqr (ι) =csch[ γ z pqr (ι) l ι ]
[ Φ ι ] N ι × N ι =diag( [ Φ pqr=1 N ι (ι) ] ), [ Ω ι ] N ι × N ι =diag( [ Ω pqr=1 N ι (ι) ] )
[ M ι1 (ι) ] N ι × N ι1 = [ { ( [ C ι1,ι ] N ι1 × N ι [ Ω ι ] N ι × N ι ) N ι1 × N ι } Τ ] N ι × N ι1
[ M ι (ι) ] N ι × N ι =[ C ι,ι+1 ] { ( [ C ι,ι+1 ] N ι × N ι+1 [ Φ ι+1 ] N ι+1 × N ι+1 ) N ι × N ι+1 } Τ [ Φ ι ]
[ M ι+1 (ι) ] N ι × N ι+1 = ( [ C ι,ι+1 ] N ι × N ι+1 [ Ω ι+1 ] N ι+1 × N ι+1 ) N ι × N ι+1
[ M Z (Y) ] N Y × N Z Θ = [ C YZ ] N Y × N Z ( [ C Z Θ Z ] N Z Θ × N Z [ Ω Z ] N Z × N Z ) Τ
Π ¯ ( N 1 + N 2 ++ N Y )×( N 0 + N 1 + N 2 ++ N Y + N Z Θ ) =[ M ¯ 0 (1) M ¯ 1 (1) M ¯ 2 (1) M ¯ 1 (2) M ¯ 2 (2) M ¯ 3 (2) M ¯ 2 (3) M ¯ 3 (3) M ¯ 4 (3) M ¯ W (X) M ¯ X (X) M ¯ Y (X) M ¯ X (Y) M ¯ Y (Y) M ¯ Z (Y) ]
M { y x } z s =b| b+d =|+ pqr=1 N 0| Z Θ A pqr (0| Z Θ ) [ {±} e { x y } pqr (0| Z Θ ) ]
M ˜ ˜ { y x } z s =b| b+d ( k x , k y )=|+ pqr=1 N 0| Z Θ A pqr (0| Z Θ ) [ {±} e ˜ ˜ { x y } pqr (0| Z Θ ) ( k x , k y ) ]
m { y x } pqr z= z 0 (x,y)={ } e { x y } pqr (0) (x,y); m ˜ ˜ { y x } pqr,mn z= z 0 ={ } e ˜ ˜ { x y } pqr (0) ( k x m , k y m )
k x m = k x m=0 + 2mπ/ d x and k y n = k y n=0 + 2nπ/ d y
k x m=0 = k i sin θ 0 cos ϕ 0 and k y n=0 = k i sin θ 0 sin ϕ 0
F w M x+y z=b (l)left (x,y,z)= F w M x+y z=b (l)left (z) e j( k x m x+ k y n y )
F w M x+y z=b (l)left (z)= 1 d x d y pqr=1 N 0 A pqr (0) mn=1 MN G ˜ ˜ F w (l)left ·( m ˜ ˜ x pqr,mn z i+1 ex = z 0 + m ˜ ˜ y pqr,mn z i+1 ex = z 0 )
m { y x } pqr z= z Z (x,y)={ ± } e { x y } pqr ( Z Θ ) (x,y); m ˜ ˜ { y x } pqr,mn z= z Z ={ ± } e ˜ ˜ { x y } pqr ( Z Θ ) ( k x m , k y m )
F w M x+y z=b+d (l)right (x,y,z)= F w M x+y z=b+d (l)right (z) e j( k x m x+ k y n y )
F w M x+y z=b+d (l)right (z)= 1 d x d y pqr=1 N Z Θ A pqr ( Z Θ ) mn=1 MN G ˜ ˜ F w (l)right ·( m ˜ ˜ x pqr,mn z i1 ex = z Z + m ˜ ˜ y pqr,mn z i1 ex = z Z )
x ^ [ H x M x+y z=b (l=4)left + G ˜ ˜ H x (l=4)left · Ψ σ z τ=i ex1° e j( k x 0 x+ k y 0 y ) ] z=b + y ^ [ H y M x+y z=b (l=4)left + G ˜ ˜ H y (l=4)left · Ψ σ z τ=i ex1° e j( k x 0 x+ k y 0 y ) ] z=b =+ pqr=1 N 0 A pqr (0) uvw=1 N 1 coth( γ z uvw (1) l 1 ) C 0 pqr 1 uvw h t uvw (1) ( x,y;z=b ) pqr=1 N 1 A pqr (1) csch( γ z pqr (1) l 1 ) h t pqr (1) ( x,y;z=b )
x ^ [ H x M x+y z=b+d (l=2)right ] z=b+d + y ^ [ H y M x+y z=b+d (l=2)right ] z=b+d = pqr=1 N Z csch( γ z pqr (Z) l Z ) h t pqr (Z) ( x,y;z=b+d ) mnp=1 N Y A mnp (Y) C Y mnp Z pqr pqr=1 N Z coth( γ z pqr (Z) l Z ) h t pqr (Z) ( x,y;z=b+d ) mnp=1 N Z Θ A mnp ( Z Θ ) C Z Θ mnp Z pqr
[ H y M x+y z=b (l=4)left ] z=b e ˜ ˜ x rst (0) ( k x m , k y n )+ [ G ˜ ˜ H y (l=4)left · Ψ σ z τ=i ex1° ] z=b e ˜ ˜ x rst (0) ( k x 0 , k y 0 ) [ H x M x+y z=b (l=4)left ] z=b e ˜ ˜ y rst (0) ( k x m , k y n ) [ G ˜ ˜ H x (l=4)left · Ψ σ z τ=i ex1° ] z=b e ˜ ˜ y rst (0) ( k x 0 , k y 0 ) = pqr=1 N 0 A pqr (0) uvw=1 N 1 coth( γ z uv (1) l 1 ) C 0 pqr 1 uvw C 0 rst 1 uvw pqr=1 N 1 A pqr (1) csch( γ z pq (1) l 1 ) C 0 rst 1 pqr
[ H y M x+y z=b+d (l=2)right ] z=b+d e ˜ ˜ x rst cav ( k x m , k y n ) [ H x M x+y z=b+d (l=2)right ] z=b+d e ˜ ˜ y rst cav ( k x m , k y n ) = mnp=1 N Y A mnp (Y) pqr=1 N Z C Z Θrst , Z pqr csch( γ z pqr (Z) l Z ) C Y mnp Z pqr mnp=1 N Z Θ A mnp ( Z Θ ) pqr=1 N Z coth( γ z pq (Z) l Z ) C Z Θ mnp Z pqr C Z Θrst , Z pqr
V ¯ = { [ A 1 N 0 (0) ] [ A 1 N 1 (1) ] [ A 1 N Y (Y) ] [ A 1 N Z Θ ( Z Θ ) ] } ( N 0 + N 1 ++ N Y + N Z Θ )×1
[ H ˜ ˜ _ w M x+y z=b (l=4)left (mn,pqr) ] MN× N 0 =κ [ G ˜ ˜ H w (l=4)left ·( m ˜ ˜ x pqr,mn z i+1 ex = z 0 + m ˜ ˜ y pqr,mn z i+1 ex = z 0 ) ] z=b
[ H ˜ ˜ _ w M x+y z=b+d (l=2)right (mn,pqr) ] MN× N cav =κ [ G ˜ ˜ H w (l=2)right ·( m ˜ ˜ x pqr,mn z i1 ex = z Z + m ˜ ˜ y pqr,mn z i1 ex = z Z ) ] z=b+d
[ e ˜ ˜ _ w pqr (0or Z Θ ) (pqr,mn) ] N 0or Z Θ ×MN = e ˜ ˜ w pqr (0or Z Θ ) ( k x m , k y n )
[ e ˜ ˜ _ w pqr (0or Z Θ ) (pqr,mn=00) ] N 0or Z Θ ×1 = e ˜ ˜ w pqr (0or Z Θ ) ( k x 0 , k y 0 )
[ { ϒ (1|Z) (1|Z) } ] N 1|Z × N 1|Z =diag[ { coth csch }( γ z pqr=1 N 1|Z (1|Z) l 1 | l Z ) ]
[ M 00 ] N 0 × N 0 = [ e ˜ ˜ _ x pqr (0) (pqr,mn) ] N 0 ×MN [ H ˜ ˜ _ y M x+y z=b (l=4)left (mn,pqr) ] MN× N 0 [ e ˜ ˜ _ y pqr (0) (pqr,mn) ] N 0 ×MN [ H ˜ ˜ _ x M x+y z=b (l=4)left (mn,pqr) ] MN× N 0 [ C 0,1 ] N 0 × N 1 { ( [ C 0,1 ] N 0 × N 1 [ ϒ (1) ] N 1 × N 1 ) Τ } N 1 × N 0
[ M 01 ] N 0 × N 1 = [ C 0,1 ] N 0 × N 1 [ (1) ] N 1 × N 1
[ M Z Θ Y ] N Z Θ × N Y =( [ C Z Θ ,Z ] N Z Θ × N Z [ (Z) ] N Z × N Z ){ ( [ C Y,Z ] N Y × N Z ) Τ }
[ M Z Θ Z Θ ] N Z Θ × N Z Θ = [ e ˜ ˜ _ x pqr ( Z Θ ) (pqr,mn) ] N Z Θ ×MN [ H ˜ ˜ _ y M x+y z=b+d (l=2)right (mn,pqr) ] MN× N Z Θ [ e ˜ ˜ _ y pqr ( Z Θ ) (pqr,mn) ] N Z Θ ×MN [ H ˜ ˜ _ x M x+y z=b+d (l=2)right (mn,pqr) ] MN× N Z Θ + + [ C Z Θ ,Z ] N Z Θ × N Z { ( [ C Z Θ ,Z ] N Z Θ × N Z [ ϒ (Z) ] N Z × N Z ) Τ } N Z × N Z Θ
Μ ¯ N tot × N tot ultim =[ M ¯ 00 M ¯ 01 { 0 ¯ N 0 ×( N 2 ++ N Y + N Z Θ ) } { Π ¯ ( N 1 + N 2 ++ N Y )×( N 0 + N 1 + N 2 ++ N Y + N Z Θ ) } { 0 ¯ N Z Θ ×( N 0 + N 1 ++ N X ) } M ¯ Z Θ Y M ¯ Z Θ Z Θ ]
N tot = N 0 + N 1 + N 2 ++ N Y + N Z Θ
V ¯ N tot ×1 ultim =[ A ¯ 1 N 0 (0) A ¯ 1 N 1 (1) A ¯ 1 N Y (Y) A ¯ 1 N Z Θ ( Z Θ ) ]
Λ ¯ N 0 ×1 = [ G ˜ ˜ H y (l=4)left · Ψ σ z τ=i ex1° ] z=b [ e ˜ ˜ _ x pqr (0) (pqr,00) ] N 0 ×1 [ G ˜ ˜ H x (l=4)left · Ψ σ z τ=i ex1° ] z=b [ e ˜ ˜ _ y pqr (0) (pqr,00) ] N 0 ×1
Χ ¯ N tot ×1 ultim = { ( [ Λ ¯ N 0 ×1 ] Τ ) 1× N 0 [ 0 ] 1×( N 1 + N 2 + N Y + N Z Θ ) } Τ
Μ ¯ N tot × N tot ultim V ¯ N tot ×1 ultim = Χ ¯ N 0 ×1 ultim
P cond = R s 2 S metal | H t | 2 ds ; where R s = πf μ cond /σ

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