Abstract

We show that the space mapping algorithm, originally developed for microwave circuit optimization, can enable the efficient design of nanoplasmonic waveguide devices which satisfy a set of desired specifications. Space mapping utilizes a physics-based coarse model to approximate a fine model accurately describing a device. Here the fine model is a full-wave finite-difference frequency-domain (FDFD) simulation of the device, while the coarse model is based on transmission line theory. We demonstrate that simply optimizing the transmission line model of the device is not enough to obtain a device which satisfies all the required design specifications. On the other hand, when the iterative space mapping algorithm is used, it converges fast to a design which meets all the specifications. In addition, full-wave FDFD simulations of only a few candidate structures are required before the iterative process is terminated. Use of the space mapping algorithm therefore results in large reductions in the required computation time when compared to any direct optimization method of the fine FDFD model.

© 2013 Optical Society of America

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    [CrossRef] [PubMed]

2013 (2)

W. Shin, W. Cai, P. B. Catrysse, G. Veronis, M. L. Brongersma, and S. Fan, “Broadband sharp 90-degree bends and t-splitters in plasmonic coaxial waveguides,” Nano Lett.13, 4753–4758 (2013).
[CrossRef] [PubMed]

D. Li and E.-P. Li, “Impedance calculation and equivalent circuits for metal-insulator-metal plasmonic waveguide geometries,” Opt. Lett.38, 3384–3386 (2013).
[CrossRef] [PubMed]

2012 (2)

H. Nejati and A. Beirami, “Theoretical analysis of the characteristic impedance in metal-insulator-metal plasmonic transmission lines,” Opt. Lett.37, 1050–1052 (2012).
[CrossRef] [PubMed]

Y. Huang, C. Min, L. Yang, and G. Veronis, “Nanoscale plasmonic devices based on metal-dielectric-metal stub resonators,” Int. J. Opt.2012, 372048 (2012).

2011 (3)

Y. Huang, C. Min, and G. Veronis, “Subwavelength slow-light waveguides based on a plasmonic analogue of electromagnetically induced transparency,” Appl. Phys. Lett.99, 143117 (2011).
[CrossRef]

T. Tanemura, K. C. Balram, D.-S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11, 2693–2698 (2011).
[CrossRef] [PubMed]

C. Min, L. Yang, and G. Veronis, “Microcavity enhanced optical absorption in subwavelength slits,” Opt. Express19, 26850–26858 (2011).
[CrossRef]

2010 (4)

R. M. Briggs, J. Grandidier, S. P. Burgos, E. Feigenbaum, and H. A. Atwater, “Efficient coupling between dielectric-loaded plasmonic and silicon photonic waveguides,” Nano Lett.10, 4851–4857 (2010).
[CrossRef]

C. Delacour, S. Blaize, P. Grosse, J. M. Fedeli, A. Bruyant, R. Salas-Montiel, G. Lerondel, and A. Chelnokov, “Efficient directional coupling between silicon and copper plasmonic nanoslot waveguides: toward metal-oxide-silicon nanophotonics,” Nano Lett.10, 2922–2926 (2010).
[CrossRef] [PubMed]

J. A. Schuller, E. S. Barnard, W. S. Cai, Y. C. Yun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9, 193–204 (2010).
[CrossRef] [PubMed]

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4, 83–91 (2010).
[CrossRef]

2009 (4)

2008 (5)

A. Alù, M. E. Young, and N. Engheta, “Design of nano filters for optical nanocircuits,” Phys. Rev. B77, 144107 (2008).
[CrossRef]

D. Pacifici, H. J. Lezec, L. A. Sweatlock, R. J. Walters, and H. A. Atwater, “Universal optical transmission features in periodic and quasiperiodic hole arrays,” Opt. Express16, 9222–9238 (2008).
[CrossRef] [PubMed]

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Transmission line and equivalent circuit models for plasmonic waveguide components,” IEEE J. Sel. Top. Quantum Electron.14, 1462–1472 (2008).
[CrossRef]

D. Pacifici, H. J. Lezec, H. A. Atwater, and J. Weiner, “Quantitative determination of optical transmission through subwavelength slit arrays in Ag films: Role of surface wave interference and local coupling betwen adjacent slits,” Phys. Rev. B77, 115411 (2008).
[CrossRef]

S. Koziel, Q. S. Cheng, and J. W. Bandler, “Space mapping,” IEEE Microwave Mag.9, 105–122 (2008).
[CrossRef]

2007 (2)

2006 (2)

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today9, 20–27 (2006).
[CrossRef]

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science311, 189–193 (2006).
[CrossRef] [PubMed]

2005 (2)

S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys.98, 011101 (2005).
[CrossRef]

G. Veronis and S. Fan, “Bends and splitters in subwavelength metal-dielectric-metal plasmonic waveguides,” Appl. Phys. Lett.87, 131102 (2005).
[CrossRef]

2004 (2)

J. W. Bandler, Q. S. Cheng, A. S. Dakroury, A. S. Mohamed, M. H. Bakr, K. Madsen, and J. Søndergaard, “Space mapping: The state of the art,” IEEE Trans. Microwave Theory Tech.52, 337–360 (2004).
[CrossRef]

G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Opt. Lett.29, 2288–2290 (2004).
[CrossRef] [PubMed]

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature424, 824–830 (2003).
[CrossRef] [PubMed]

2002 (1)

1995 (1)

J. W. Bandler, R. M. Biernacki, S. H. Chen, R. H. Hemmers, and K. Madsen, “Electromagnetic optimization exploiting aggressive space mapping,” IEEE Trans. Microwave Theory Tech.43, 2874–2882 (1995).
[CrossRef]

1993 (1)

J. W. Bandler, S. H. Chen, R. M. Biernacki, L. Gao, K. Madsen, and H. Yu, “Huber optimization of circuits: A robust approach,” IEEE Trans. Microwave Theory Tech.41, 2279–2287 (1993).
[CrossRef]

1989 (1)

K. Krishnakumar, “Micro-genetic algorithms for stationary and non-stationary function optimization,” Proc. SPIE1196, 289–296 (1989).
[CrossRef]

1965 (1)

C. G. Broyden, “A class of methods for solving nonlinear simultaneous equations,” Math. Comput.19, 577–593 (1965).
[CrossRef]

Alù, A.

A. Alù, M. E. Young, and N. Engheta, “Design of nano filters for optical nanocircuits,” Phys. Rev. B77, 144107 (2008).
[CrossRef]

Atwater, H. A.

R. M. Briggs, J. Grandidier, S. P. Burgos, E. Feigenbaum, and H. A. Atwater, “Efficient coupling between dielectric-loaded plasmonic and silicon photonic waveguides,” Nano Lett.10, 4851–4857 (2010).
[CrossRef]

D. Pacifici, H. J. Lezec, L. A. Sweatlock, R. J. Walters, and H. A. Atwater, “Universal optical transmission features in periodic and quasiperiodic hole arrays,” Opt. Express16, 9222–9238 (2008).
[CrossRef] [PubMed]

D. Pacifici, H. J. Lezec, H. A. Atwater, and J. Weiner, “Quantitative determination of optical transmission through subwavelength slit arrays in Ag films: Role of surface wave interference and local coupling betwen adjacent slits,” Phys. Rev. B77, 115411 (2008).
[CrossRef]

S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys.98, 011101 (2005).
[CrossRef]

Bakr, M. H.

J. W. Bandler, Q. S. Cheng, A. S. Dakroury, A. S. Mohamed, M. H. Bakr, K. Madsen, and J. Søndergaard, “Space mapping: The state of the art,” IEEE Trans. Microwave Theory Tech.52, 337–360 (2004).
[CrossRef]

Balram, K. C.

T. Tanemura, K. C. Balram, D.-S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11, 2693–2698 (2011).
[CrossRef] [PubMed]

Bandler, J. W.

S. Koziel, J. W. Bandler, and K. Madsen, “Space mapping with adaptive response correction for microwave design optimization,” IEEE Trans. Microwave Theory Tech.57, 478–486 (2009).
[CrossRef]

S. Koziel, Q. S. Cheng, and J. W. Bandler, “Space mapping,” IEEE Microwave Mag.9, 105–122 (2008).
[CrossRef]

J. W. Bandler, Q. S. Cheng, A. S. Dakroury, A. S. Mohamed, M. H. Bakr, K. Madsen, and J. Søndergaard, “Space mapping: The state of the art,” IEEE Trans. Microwave Theory Tech.52, 337–360 (2004).
[CrossRef]

J. W. Bandler, R. M. Biernacki, S. H. Chen, R. H. Hemmers, and K. Madsen, “Electromagnetic optimization exploiting aggressive space mapping,” IEEE Trans. Microwave Theory Tech.43, 2874–2882 (1995).
[CrossRef]

J. W. Bandler, S. H. Chen, R. M. Biernacki, L. Gao, K. Madsen, and H. Yu, “Huber optimization of circuits: A robust approach,” IEEE Trans. Microwave Theory Tech.41, 2279–2287 (1993).
[CrossRef]

Barnard, E. S.

J. A. Schuller, E. S. Barnard, W. S. Cai, Y. C. Yun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9, 193–204 (2010).
[CrossRef] [PubMed]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature424, 824–830 (2003).
[CrossRef] [PubMed]

Beirami, A.

Biernacki, R. M.

J. W. Bandler, R. M. Biernacki, S. H. Chen, R. H. Hemmers, and K. Madsen, “Electromagnetic optimization exploiting aggressive space mapping,” IEEE Trans. Microwave Theory Tech.43, 2874–2882 (1995).
[CrossRef]

J. W. Bandler, S. H. Chen, R. M. Biernacki, L. Gao, K. Madsen, and H. Yu, “Huber optimization of circuits: A robust approach,” IEEE Trans. Microwave Theory Tech.41, 2279–2287 (1993).
[CrossRef]

Blaize, S.

C. Delacour, S. Blaize, P. Grosse, J. M. Fedeli, A. Bruyant, R. Salas-Montiel, G. Lerondel, and A. Chelnokov, “Efficient directional coupling between silicon and copper plasmonic nanoslot waveguides: toward metal-oxide-silicon nanophotonics,” Nano Lett.10, 2922–2926 (2010).
[CrossRef] [PubMed]

Bozhevolnyi, S. I.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4, 83–91 (2010).
[CrossRef]

Briggs, R. M.

R. M. Briggs, J. Grandidier, S. P. Burgos, E. Feigenbaum, and H. A. Atwater, “Efficient coupling between dielectric-loaded plasmonic and silicon photonic waveguides,” Nano Lett.10, 4851–4857 (2010).
[CrossRef]

Brongersma, M. L.

W. Shin, W. Cai, P. B. Catrysse, G. Veronis, M. L. Brongersma, and S. Fan, “Broadband sharp 90-degree bends and t-splitters in plasmonic coaxial waveguides,” Nano Lett.13, 4753–4758 (2013).
[CrossRef] [PubMed]

T. Tanemura, K. C. Balram, D.-S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11, 2693–2698 (2011).
[CrossRef] [PubMed]

J. A. Schuller, E. S. Barnard, W. S. Cai, Y. C. Yun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9, 193–204 (2010).
[CrossRef] [PubMed]

G. Veronis, Z. Yu, S. E. Kocabas, D. A. B. Miller, M. L. Brongersma, and S. Fan, “Metal-dielectric-metal plasmonic waveguide devices for manipulating light at the nanoscale,” Chin. Opt. Lett.7, 302–308 (2009).
[CrossRef]

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today9, 20–27 (2006).
[CrossRef]

Broyden, C. G.

C. G. Broyden, “A class of methods for solving nonlinear simultaneous equations,” Math. Comput.19, 577–593 (1965).
[CrossRef]

Bruyant, A.

C. Delacour, S. Blaize, P. Grosse, J. M. Fedeli, A. Bruyant, R. Salas-Montiel, G. Lerondel, and A. Chelnokov, “Efficient directional coupling between silicon and copper plasmonic nanoslot waveguides: toward metal-oxide-silicon nanophotonics,” Nano Lett.10, 2922–2926 (2010).
[CrossRef] [PubMed]

Burgos, S. P.

R. M. Briggs, J. Grandidier, S. P. Burgos, E. Feigenbaum, and H. A. Atwater, “Efficient coupling between dielectric-loaded plasmonic and silicon photonic waveguides,” Nano Lett.10, 4851–4857 (2010).
[CrossRef]

Cai, W.

W. Shin, W. Cai, P. B. Catrysse, G. Veronis, M. L. Brongersma, and S. Fan, “Broadband sharp 90-degree bends and t-splitters in plasmonic coaxial waveguides,” Nano Lett.13, 4753–4758 (2013).
[CrossRef] [PubMed]

Cai, W. S.

J. A. Schuller, E. S. Barnard, W. S. Cai, Y. C. Yun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9, 193–204 (2010).
[CrossRef] [PubMed]

Catrysse, P. B.

W. Shin, W. Cai, P. B. Catrysse, G. Veronis, M. L. Brongersma, and S. Fan, “Broadband sharp 90-degree bends and t-splitters in plasmonic coaxial waveguides,” Nano Lett.13, 4753–4758 (2013).
[CrossRef] [PubMed]

Chandran, A.

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today9, 20–27 (2006).
[CrossRef]

Chelnokov, A.

C. Delacour, S. Blaize, P. Grosse, J. M. Fedeli, A. Bruyant, R. Salas-Montiel, G. Lerondel, and A. Chelnokov, “Efficient directional coupling between silicon and copper plasmonic nanoslot waveguides: toward metal-oxide-silicon nanophotonics,” Nano Lett.10, 2922–2926 (2010).
[CrossRef] [PubMed]

Chen, S. H.

J. W. Bandler, R. M. Biernacki, S. H. Chen, R. H. Hemmers, and K. Madsen, “Electromagnetic optimization exploiting aggressive space mapping,” IEEE Trans. Microwave Theory Tech.43, 2874–2882 (1995).
[CrossRef]

J. W. Bandler, S. H. Chen, R. M. Biernacki, L. Gao, K. Madsen, and H. Yu, “Huber optimization of circuits: A robust approach,” IEEE Trans. Microwave Theory Tech.41, 2279–2287 (1993).
[CrossRef]

Cheng, Q. S.

S. Koziel, Q. S. Cheng, and J. W. Bandler, “Space mapping,” IEEE Microwave Mag.9, 105–122 (2008).
[CrossRef]

J. W. Bandler, Q. S. Cheng, A. S. Dakroury, A. S. Mohamed, M. H. Bakr, K. Madsen, and J. Søndergaard, “Space mapping: The state of the art,” IEEE Trans. Microwave Theory Tech.52, 337–360 (2004).
[CrossRef]

Collin, R. E.

R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, 1966).

Dakroury, A. S.

J. W. Bandler, Q. S. Cheng, A. S. Dakroury, A. S. Mohamed, M. H. Bakr, K. Madsen, and J. Søndergaard, “Space mapping: The state of the art,” IEEE Trans. Microwave Theory Tech.52, 337–360 (2004).
[CrossRef]

Delacour, C.

C. Delacour, S. Blaize, P. Grosse, J. M. Fedeli, A. Bruyant, R. Salas-Montiel, G. Lerondel, and A. Chelnokov, “Efficient directional coupling between silicon and copper plasmonic nanoslot waveguides: toward metal-oxide-silicon nanophotonics,” Nano Lett.10, 2922–2926 (2010).
[CrossRef] [PubMed]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature424, 824–830 (2003).
[CrossRef] [PubMed]

Dutton, R. W.

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature424, 824–830 (2003).
[CrossRef] [PubMed]

Engheta, N.

A. Alù, M. E. Young, and N. Engheta, “Design of nano filters for optical nanocircuits,” Phys. Rev. B77, 144107 (2008).
[CrossRef]

N. Engheta, “Circuits with light at nanoscales: Optical nanocircuits inspired by metamaterials,” Science317, 1698–1702 (2007).
[CrossRef] [PubMed]

Fan, S.

W. Shin, W. Cai, P. B. Catrysse, G. Veronis, M. L. Brongersma, and S. Fan, “Broadband sharp 90-degree bends and t-splitters in plasmonic coaxial waveguides,” Nano Lett.13, 4753–4758 (2013).
[CrossRef] [PubMed]

G. Veronis, Z. Yu, S. E. Kocabas, D. A. B. Miller, M. L. Brongersma, and S. Fan, “Metal-dielectric-metal plasmonic waveguide devices for manipulating light at the nanoscale,” Chin. Opt. Lett.7, 302–308 (2009).
[CrossRef]

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Transmission line and equivalent circuit models for plasmonic waveguide components,” IEEE J. Sel. Top. Quantum Electron.14, 1462–1472 (2008).
[CrossRef]

G. Veronis and S. Fan, “Theoretical investigation of compact couplers between dielectric slab waveguides and two-dimensional metal-dielectric-metal plasmonic waveguides,” Opt. Express15, 1211–1221 (2007).
[CrossRef] [PubMed]

G. Veronis and S. Fan, “Bends and splitters in subwavelength metal-dielectric-metal plasmonic waveguides,” Appl. Phys. Lett.87, 131102 (2005).
[CrossRef]

G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Opt. Lett.29, 2288–2290 (2004).
[CrossRef] [PubMed]

Fedeli, J. M.

C. Delacour, S. Blaize, P. Grosse, J. M. Fedeli, A. Bruyant, R. Salas-Montiel, G. Lerondel, and A. Chelnokov, “Efficient directional coupling between silicon and copper plasmonic nanoslot waveguides: toward metal-oxide-silicon nanophotonics,” Nano Lett.10, 2922–2926 (2010).
[CrossRef] [PubMed]

Feigenbaum, E.

R. M. Briggs, J. Grandidier, S. P. Burgos, E. Feigenbaum, and H. A. Atwater, “Efficient coupling between dielectric-loaded plasmonic and silicon photonic waveguides,” Nano Lett.10, 4851–4857 (2010).
[CrossRef]

Gao, L.

J. W. Bandler, S. H. Chen, R. M. Biernacki, L. Gao, K. Madsen, and H. Yu, “Huber optimization of circuits: A robust approach,” IEEE Trans. Microwave Theory Tech.41, 2279–2287 (1993).
[CrossRef]

Glytsis, E. N.

Gramotnev, D. K.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4, 83–91 (2010).
[CrossRef]

Grandidier, J.

R. M. Briggs, J. Grandidier, S. P. Burgos, E. Feigenbaum, and H. A. Atwater, “Efficient coupling between dielectric-loaded plasmonic and silicon photonic waveguides,” Nano Lett.10, 4851–4857 (2010).
[CrossRef]

Grosse, P.

C. Delacour, S. Blaize, P. Grosse, J. M. Fedeli, A. Bruyant, R. Salas-Montiel, G. Lerondel, and A. Chelnokov, “Efficient directional coupling between silicon and copper plasmonic nanoslot waveguides: toward metal-oxide-silicon nanophotonics,” Nano Lett.10, 2922–2926 (2010).
[CrossRef] [PubMed]

Hemmers, R. H.

J. W. Bandler, R. M. Biernacki, S. H. Chen, R. H. Hemmers, and K. Madsen, “Electromagnetic optimization exploiting aggressive space mapping,” IEEE Trans. Microwave Theory Tech.43, 2874–2882 (1995).
[CrossRef]

Huang, X.

Huang, Y.

Y. Huang, C. Min, L. Yang, and G. Veronis, “Nanoscale plasmonic devices based on metal-dielectric-metal stub resonators,” Int. J. Opt.2012, 372048 (2012).

Y. Huang, C. Min, and G. Veronis, “Subwavelength slow-light waveguides based on a plasmonic analogue of electromagnetically induced transparency,” Appl. Phys. Lett.99, 143117 (2011).
[CrossRef]

Huber, P.

P. Huber, Robust Statistics (Wiley, 1981).
[CrossRef]

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).

Jin, X.

Kocabas, S. E.

G. Veronis, Z. Yu, S. E. Kocabas, D. A. B. Miller, M. L. Brongersma, and S. Fan, “Metal-dielectric-metal plasmonic waveguide devices for manipulating light at the nanoscale,” Chin. Opt. Lett.7, 302–308 (2009).
[CrossRef]

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Transmission line and equivalent circuit models for plasmonic waveguide components,” IEEE J. Sel. Top. Quantum Electron.14, 1462–1472 (2008).
[CrossRef]

Koziel, S.

S. Koziel, J. W. Bandler, and K. Madsen, “Space mapping with adaptive response correction for microwave design optimization,” IEEE Trans. Microwave Theory Tech.57, 478–486 (2009).
[CrossRef]

S. Koziel, Q. S. Cheng, and J. W. Bandler, “Space mapping,” IEEE Microwave Mag.9, 105–122 (2008).
[CrossRef]

Krishnakumar, K.

K. Krishnakumar, “Micro-genetic algorithms for stationary and non-stationary function optimization,” Proc. SPIE1196, 289–296 (1989).
[CrossRef]

Lerondel, G.

C. Delacour, S. Blaize, P. Grosse, J. M. Fedeli, A. Bruyant, R. Salas-Montiel, G. Lerondel, and A. Chelnokov, “Efficient directional coupling between silicon and copper plasmonic nanoslot waveguides: toward metal-oxide-silicon nanophotonics,” Nano Lett.10, 2922–2926 (2010).
[CrossRef] [PubMed]

Lezec, H. J.

D. Pacifici, H. J. Lezec, H. A. Atwater, and J. Weiner, “Quantitative determination of optical transmission through subwavelength slit arrays in Ag films: Role of surface wave interference and local coupling betwen adjacent slits,” Phys. Rev. B77, 115411 (2008).
[CrossRef]

D. Pacifici, H. J. Lezec, L. A. Sweatlock, R. J. Walters, and H. A. Atwater, “Universal optical transmission features in periodic and quasiperiodic hole arrays,” Opt. Express16, 9222–9238 (2008).
[CrossRef] [PubMed]

Li, D.

Li, E.-P.

Lin, X.

Ly-Gagnon, D.-S.

T. Tanemura, K. C. Balram, D.-S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11, 2693–2698 (2011).
[CrossRef] [PubMed]

Madsen, K.

S. Koziel, J. W. Bandler, and K. Madsen, “Space mapping with adaptive response correction for microwave design optimization,” IEEE Trans. Microwave Theory Tech.57, 478–486 (2009).
[CrossRef]

J. W. Bandler, Q. S. Cheng, A. S. Dakroury, A. S. Mohamed, M. H. Bakr, K. Madsen, and J. Søndergaard, “Space mapping: The state of the art,” IEEE Trans. Microwave Theory Tech.52, 337–360 (2004).
[CrossRef]

J. W. Bandler, R. M. Biernacki, S. H. Chen, R. H. Hemmers, and K. Madsen, “Electromagnetic optimization exploiting aggressive space mapping,” IEEE Trans. Microwave Theory Tech.43, 2874–2882 (1995).
[CrossRef]

J. W. Bandler, S. H. Chen, R. M. Biernacki, L. Gao, K. Madsen, and H. Yu, “Huber optimization of circuits: A robust approach,” IEEE Trans. Microwave Theory Tech.41, 2279–2287 (1993).
[CrossRef]

Maier, S. A.

S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys.98, 011101 (2005).
[CrossRef]

Miller, D. A. B.

T. Tanemura, K. C. Balram, D.-S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11, 2693–2698 (2011).
[CrossRef] [PubMed]

G. Veronis, Z. Yu, S. E. Kocabas, D. A. B. Miller, M. L. Brongersma, and S. Fan, “Metal-dielectric-metal plasmonic waveguide devices for manipulating light at the nanoscale,” Chin. Opt. Lett.7, 302–308 (2009).
[CrossRef]

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Transmission line and equivalent circuit models for plasmonic waveguide components,” IEEE J. Sel. Top. Quantum Electron.14, 1462–1472 (2008).
[CrossRef]

Min, C.

Y. Huang, C. Min, L. Yang, and G. Veronis, “Nanoscale plasmonic devices based on metal-dielectric-metal stub resonators,” Int. J. Opt.2012, 372048 (2012).

C. Min, L. Yang, and G. Veronis, “Microcavity enhanced optical absorption in subwavelength slits,” Opt. Express19, 26850–26858 (2011).
[CrossRef]

Y. Huang, C. Min, and G. Veronis, “Subwavelength slow-light waveguides based on a plasmonic analogue of electromagnetically induced transparency,” Appl. Phys. Lett.99, 143117 (2011).
[CrossRef]

C. Min and G. Veronis, “Absorption switches in metal-dielectric-metal plasmonic waveguides,” Opt. Express17, 10757–10766 (2009).
[CrossRef] [PubMed]

Mohamed, A. S.

J. W. Bandler, Q. S. Cheng, A. S. Dakroury, A. S. Mohamed, M. H. Bakr, K. Madsen, and J. Søndergaard, “Space mapping: The state of the art,” IEEE Trans. Microwave Theory Tech.52, 337–360 (2004).
[CrossRef]

Nejati, H.

Ozbay, E.

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science311, 189–193 (2006).
[CrossRef] [PubMed]

Pacifici, D.

D. Pacifici, H. J. Lezec, L. A. Sweatlock, R. J. Walters, and H. A. Atwater, “Universal optical transmission features in periodic and quasiperiodic hole arrays,” Opt. Express16, 9222–9238 (2008).
[CrossRef] [PubMed]

D. Pacifici, H. J. Lezec, H. A. Atwater, and J. Weiner, “Quantitative determination of optical transmission through subwavelength slit arrays in Ag films: Role of surface wave interference and local coupling betwen adjacent slits,” Phys. Rev. B77, 115411 (2008).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

Pozar, D. M.

D. M. Pozar, Microwave Engineering (Wiley, 1998).

Salas-Montiel, R.

C. Delacour, S. Blaize, P. Grosse, J. M. Fedeli, A. Bruyant, R. Salas-Montiel, G. Lerondel, and A. Chelnokov, “Efficient directional coupling between silicon and copper plasmonic nanoslot waveguides: toward metal-oxide-silicon nanophotonics,” Nano Lett.10, 2922–2926 (2010).
[CrossRef] [PubMed]

Schuller, J. A.

J. A. Schuller, E. S. Barnard, W. S. Cai, Y. C. Yun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9, 193–204 (2010).
[CrossRef] [PubMed]

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today9, 20–27 (2006).
[CrossRef]

Shin, W.

W. Shin, W. Cai, P. B. Catrysse, G. Veronis, M. L. Brongersma, and S. Fan, “Broadband sharp 90-degree bends and t-splitters in plasmonic coaxial waveguides,” Nano Lett.13, 4753–4758 (2013).
[CrossRef] [PubMed]

Søndergaard, J.

J. W. Bandler, Q. S. Cheng, A. S. Dakroury, A. S. Mohamed, M. H. Bakr, K. Madsen, and J. Søndergaard, “Space mapping: The state of the art,” IEEE Trans. Microwave Theory Tech.52, 337–360 (2004).
[CrossRef]

Sweatlock, L. A.

Taflove, A.

A. Taflove, Computational Electrodynamics (Artech House, 1995).

Tanemura, T.

T. Tanemura, K. C. Balram, D.-S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11, 2693–2698 (2011).
[CrossRef] [PubMed]

Tao, J.

Veronis, G.

W. Shin, W. Cai, P. B. Catrysse, G. Veronis, M. L. Brongersma, and S. Fan, “Broadband sharp 90-degree bends and t-splitters in plasmonic coaxial waveguides,” Nano Lett.13, 4753–4758 (2013).
[CrossRef] [PubMed]

Y. Huang, C. Min, L. Yang, and G. Veronis, “Nanoscale plasmonic devices based on metal-dielectric-metal stub resonators,” Int. J. Opt.2012, 372048 (2012).

C. Min, L. Yang, and G. Veronis, “Microcavity enhanced optical absorption in subwavelength slits,” Opt. Express19, 26850–26858 (2011).
[CrossRef]

Y. Huang, C. Min, and G. Veronis, “Subwavelength slow-light waveguides based on a plasmonic analogue of electromagnetically induced transparency,” Appl. Phys. Lett.99, 143117 (2011).
[CrossRef]

C. Min and G. Veronis, “Absorption switches in metal-dielectric-metal plasmonic waveguides,” Opt. Express17, 10757–10766 (2009).
[CrossRef] [PubMed]

G. Veronis, Z. Yu, S. E. Kocabas, D. A. B. Miller, M. L. Brongersma, and S. Fan, “Metal-dielectric-metal plasmonic waveguide devices for manipulating light at the nanoscale,” Chin. Opt. Lett.7, 302–308 (2009).
[CrossRef]

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Transmission line and equivalent circuit models for plasmonic waveguide components,” IEEE J. Sel. Top. Quantum Electron.14, 1462–1472 (2008).
[CrossRef]

G. Veronis and S. Fan, “Theoretical investigation of compact couplers between dielectric slab waveguides and two-dimensional metal-dielectric-metal plasmonic waveguides,” Opt. Express15, 1211–1221 (2007).
[CrossRef] [PubMed]

G. Veronis and S. Fan, “Bends and splitters in subwavelength metal-dielectric-metal plasmonic waveguides,” Appl. Phys. Lett.87, 131102 (2005).
[CrossRef]

G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Opt. Lett.29, 2288–2290 (2004).
[CrossRef] [PubMed]

Wahl, P.

T. Tanemura, K. C. Balram, D.-S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11, 2693–2698 (2011).
[CrossRef] [PubMed]

Walters, R. J.

Weiner, J.

D. Pacifici, H. J. Lezec, H. A. Atwater, and J. Weiner, “Quantitative determination of optical transmission through subwavelength slit arrays in Ag films: Role of surface wave interference and local coupling betwen adjacent slits,” Phys. Rev. B77, 115411 (2008).
[CrossRef]

White, J. S.

T. Tanemura, K. C. Balram, D.-S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11, 2693–2698 (2011).
[CrossRef] [PubMed]

J. A. Schuller, E. S. Barnard, W. S. Cai, Y. C. Yun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9, 193–204 (2010).
[CrossRef] [PubMed]

Wu, S. D.

Yang, L.

Y. Huang, C. Min, L. Yang, and G. Veronis, “Nanoscale plasmonic devices based on metal-dielectric-metal stub resonators,” Int. J. Opt.2012, 372048 (2012).

C. Min, L. Yang, and G. Veronis, “Microcavity enhanced optical absorption in subwavelength slits,” Opt. Express19, 26850–26858 (2011).
[CrossRef]

Young, M. E.

A. Alù, M. E. Young, and N. Engheta, “Design of nano filters for optical nanocircuits,” Phys. Rev. B77, 144107 (2008).
[CrossRef]

Yu, H.

J. W. Bandler, S. H. Chen, R. M. Biernacki, L. Gao, K. Madsen, and H. Yu, “Huber optimization of circuits: A robust approach,” IEEE Trans. Microwave Theory Tech.41, 2279–2287 (1993).
[CrossRef]

Yu, Z.

Yun, Y. C.

J. A. Schuller, E. S. Barnard, W. S. Cai, Y. C. Yun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9, 193–204 (2010).
[CrossRef] [PubMed]

Zhang, Q.

Zia, R.

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today9, 20–27 (2006).
[CrossRef]

Appl. Phys. Lett. (2)

G. Veronis and S. Fan, “Bends and splitters in subwavelength metal-dielectric-metal plasmonic waveguides,” Appl. Phys. Lett.87, 131102 (2005).
[CrossRef]

Y. Huang, C. Min, and G. Veronis, “Subwavelength slow-light waveguides based on a plasmonic analogue of electromagnetically induced transparency,” Appl. Phys. Lett.99, 143117 (2011).
[CrossRef]

Chin. Opt. Lett. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Transmission line and equivalent circuit models for plasmonic waveguide components,” IEEE J. Sel. Top. Quantum Electron.14, 1462–1472 (2008).
[CrossRef]

IEEE Microwave Mag. (1)

S. Koziel, Q. S. Cheng, and J. W. Bandler, “Space mapping,” IEEE Microwave Mag.9, 105–122 (2008).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (4)

J. W. Bandler, R. M. Biernacki, S. H. Chen, R. H. Hemmers, and K. Madsen, “Electromagnetic optimization exploiting aggressive space mapping,” IEEE Trans. Microwave Theory Tech.43, 2874–2882 (1995).
[CrossRef]

J. W. Bandler, Q. S. Cheng, A. S. Dakroury, A. S. Mohamed, M. H. Bakr, K. Madsen, and J. Søndergaard, “Space mapping: The state of the art,” IEEE Trans. Microwave Theory Tech.52, 337–360 (2004).
[CrossRef]

J. W. Bandler, S. H. Chen, R. M. Biernacki, L. Gao, K. Madsen, and H. Yu, “Huber optimization of circuits: A robust approach,” IEEE Trans. Microwave Theory Tech.41, 2279–2287 (1993).
[CrossRef]

S. Koziel, J. W. Bandler, and K. Madsen, “Space mapping with adaptive response correction for microwave design optimization,” IEEE Trans. Microwave Theory Tech.57, 478–486 (2009).
[CrossRef]

Int. J. Opt. (1)

Y. Huang, C. Min, L. Yang, and G. Veronis, “Nanoscale plasmonic devices based on metal-dielectric-metal stub resonators,” Int. J. Opt.2012, 372048 (2012).

J. Appl. Phys. (1)

S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys.98, 011101 (2005).
[CrossRef]

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

Mater. Today (1)

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today9, 20–27 (2006).
[CrossRef]

Math. Comput. (1)

C. G. Broyden, “A class of methods for solving nonlinear simultaneous equations,” Math. Comput.19, 577–593 (1965).
[CrossRef]

Nano Lett. (4)

T. Tanemura, K. C. Balram, D.-S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11, 2693–2698 (2011).
[CrossRef] [PubMed]

R. M. Briggs, J. Grandidier, S. P. Burgos, E. Feigenbaum, and H. A. Atwater, “Efficient coupling between dielectric-loaded plasmonic and silicon photonic waveguides,” Nano Lett.10, 4851–4857 (2010).
[CrossRef]

C. Delacour, S. Blaize, P. Grosse, J. M. Fedeli, A. Bruyant, R. Salas-Montiel, G. Lerondel, and A. Chelnokov, “Efficient directional coupling between silicon and copper plasmonic nanoslot waveguides: toward metal-oxide-silicon nanophotonics,” Nano Lett.10, 2922–2926 (2010).
[CrossRef] [PubMed]

W. Shin, W. Cai, P. B. Catrysse, G. Veronis, M. L. Brongersma, and S. Fan, “Broadband sharp 90-degree bends and t-splitters in plasmonic coaxial waveguides,” Nano Lett.13, 4753–4758 (2013).
[CrossRef] [PubMed]

Nat. Mater. (1)

J. A. Schuller, E. S. Barnard, W. S. Cai, Y. C. Yun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9, 193–204 (2010).
[CrossRef] [PubMed]

Nat. Photonics (1)

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4, 83–91 (2010).
[CrossRef]

Nature (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature424, 824–830 (2003).
[CrossRef] [PubMed]

Opt. Express (5)

Opt. Lett. (3)

Phys. Rev. B (2)

D. Pacifici, H. J. Lezec, H. A. Atwater, and J. Weiner, “Quantitative determination of optical transmission through subwavelength slit arrays in Ag films: Role of surface wave interference and local coupling betwen adjacent slits,” Phys. Rev. B77, 115411 (2008).
[CrossRef]

A. Alù, M. E. Young, and N. Engheta, “Design of nano filters for optical nanocircuits,” Phys. Rev. B77, 144107 (2008).
[CrossRef]

Proc. SPIE (1)

K. Krishnakumar, “Micro-genetic algorithms for stationary and non-stationary function optimization,” Proc. SPIE1196, 289–296 (1989).
[CrossRef]

Science (2)

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science311, 189–193 (2006).
[CrossRef] [PubMed]

N. Engheta, “Circuits with light at nanoscales: Optical nanocircuits inspired by metamaterials,” Science317, 1698–1702 (2007).
[CrossRef] [PubMed]

Other (6)

P. Huber, Robust Statistics (Wiley, 1981).
[CrossRef]

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).

A. Taflove, Computational Electrodynamics (Artech House, 1995).

R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, 1966).

D. M. Pozar, Microwave Engineering (Wiley, 1998).

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Figures (5)

Fig. 1
Fig. 1

(a) Schematic of a MDM plasmonic waveguide side-coupled to two MDM stub resonators. (b) Schematic of the transmission line model for the structure of Fig. 1(a). Here Z(ω) and γ(ω) are the characteristic impedance and complex propagation constant of the fundamental TM mode of a silver-air-silver plasmonic waveguide with width w. (c) Transmission line model response TTL( L TL 1 *, L TL 2 *) of the structure of Fig. 1(a) for parameters L TL 1 * = 210 nm and L TL 2 * = 384 nm obtained by optimizing the transmission line model of Fig. 1(b) (dashed blue line). We also show the transmission response calculated using FDFD, TFDFD( L 1 = L TL 1 *, L 2 = L TL 2 *) for the same parameters (solid black line). Results are shown for w = 50 nm. The red lines are the design specifications imposed on the transmission response of this structure. (d) Transmission response TFDFD(1, 2) of the structure of Fig. 1(a) calculated with FDFD for the parameters 1 = 180 nm and 2 = 351 nm obtained by the space mapping algorithm.

Fig. 2
Fig. 2

Transmission response TFDFD( L 1 ( j ), L 2 ( j )) of the structure of Fig. 1(a) calculated with FDFD for parameters obtained after the jth iteration of the space mapping algorithm. L 1 ( j ), L 2 ( j ) for j = 1, 2, 3 are given in Table 1. All other parameters are as in Fig. 1(c).

Fig. 3
Fig. 3

Transmission response TFDFD( L 1 ( 1 ), L 2 ( 1 )) of the structure of Fig. 1(a) calculated with FDFD (solid line) for parameters obtained after the first iteration of the space mapping algorithm. We also show the coarse transmission line model response TTL( L TL 1 ( 1 ), L TL 2 ( 1 )) (circles), where L TL 1 ( 1 ), L TL 2 ( 1 ) are obtained through the parameter extraction procedure described in Subsection 2.2.

Fig. 4
Fig. 4

(a) Schematic of a MDM plasmonic waveguide side-coupled to two arrays of MDM stub resonators. (b) Schematic of the transmission line model for the structure of Fig. 4(a). Here Z(ω) and γ(ω) are the characteristic impedance and complex propagation constant of the fundamental TM mode of a silver-air-silver plasmonic waveguide with width w. (c) Transmission line model response TTL( L TL 1 *, L TL 2 *, L TL 3 *) of the structure of Fig. 4(a) for parameters L TL 1 * = 183 nm, L TL 2 * = 466 nm, and L TL 3 * = 229 nm obtained by optimizing the transmission line model of Fig. 4(b) (dashed blue line). We also show the transmission response calculated using FDFD, TFDFD( L 1 = L TL 1 *, L 2 = L TL 2 *, L 3 = L TL 3 *) for the same parameters (solid black line). Results are shown for w = 50 nm. The red lines are the design specifications imposed on the transmission response of this structure. (d) Transmission response TFDFD(1, 2, 3) of the structure of Fig. 4(a) calculated with FDFD for the parameters 1 = 159 nm, 2 = 439 nm, and 3 = 196 nm obtained by the space mapping algorithm.

Fig. 5
Fig. 5

(a) Schematic of a nanoplasmonic waveguide device consisting of a silicon and a silver nanorod juxtaposed in parallel in a waveguide. The parallel-plate waveguide is bounded on top and bottom by perfect electric conductors (PEC). The nanorods are connected to a PEC protrusion attached to the bottom of the waveguide. (b) Schematic of the transmission line model for the structure of Fig. 5(a). Here Z0 is the characteristic impedance of the PEC parallel-plate waveguide. The shunt impedance Zt consists of the parallel combination of a capacitor, a resistor, and an inductor. (c) Transmission line model response TTL( L TL 1 *, L TL 2 *) of the structure of Fig. 5(a) for parameters L TL 1 * = 20 nm and L TL 2 * = 7 nm obtained by optimizing the transmission line model of Fig. 5(b) (dashed blue line). We also show the transmission response calculated using FDFD, TFDFD( L 1 = L TL 1 *, L 2 = L TL 2 *) for the same parameters (solid black line). Results are shown for w = 50 nm and t = 10 nm. The red lines are the design specifications imposed on the transmission response of this structure. (d) Transmission response TFDFD(1, 2) of the structure of Fig. 5(a) calculated with FDFD for the parameters 1 = 20 nm and 2 = 9 nm obtained by the space mapping algorithm.

Tables (3)

Tables Icon

Table 1 The design parameters L 1 ( j ), L 2 ( j ) found after jth iteration of the space mapping algorithm for the structure of Fig. 1(a).

Tables Icon

Table 2 The design parameters L 1 ( j ), L 2 ( j ), and L 3 ( j ) found after jth iteration of the space mapping algorithm for the structure of Fig. 4(a).

Tables Icon

Table 3 The design parameters L 1 ( j ), L 2 ( j ) found after jth iteration of the space mapping algorithm for the structure of Fig. 5(a).

Equations (32)

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x c = P ( x f ) ,
R c ( P ( x f ) ) R f ( x f ) .
x ¯ f P 1 ( x c * )
P ( x ¯ f ) x c * = 0 .
g ( x f ) = 0 ,
g ( x f ) P ( x f ) x c * .
J ( x f ) = [ T g ( x f ) x f ] T
x f ( 1 ) = x c * .
x f ( j + 1 ) = x f ( j ) + h ( j ) ,
B ( j ) h ( j ) = g ( j ) .
g ( j ) g ( x f ( j ) ) = P ( x f ( j ) ) x c * ,
B ( 1 ) = I ,
B ( j ) = B ( j 1 ) + g ( j ) h ( j 1 ) T h ( j 1 ) T h ( j 1 ) .
min x c H ( x c ) ,
H ( x c ) = i = 1 n ρ k ( e i ( x c ) ) .
e i ( x c ) R c ( x c , ω i ) R f ( x f , ω i ) ,
ρ k ( e i ) { e i 2 / 2 , if | e i | k k | e i | k 2 / 2 , if | e i | > k .
T > 0.75 for 180 THz < f < 200 THz ,
T < 0.2 for 130 THz < f < 160 THz and 240 THz < f < 270 THz ,
Z = γ j ω ε w ,
T TL ( L TL 1 , L TL 2 ) = | 1 + 1 2 [ tanh ( γ L TL 1 ) + tanh ( γ L TL 2 ) ] | 2 .
[ L 1 ( 1 ) L 2 ( 2 ) ] T = [ L TL 1 * L TL 2 * ] T = [ 210 nm 384 nm ] T .
T > 0.5 for 190 THz < f < 200 THz ,
T < 0.03 for 110 THz < f < 160 THz and 230 THz < f < 290 THz .
T > 0.36 for 330 THz < f < 360 THz ,
T < 0.18 for 120 THz < f < 270 THz and 420 THz < f < 630 THz .
C = ε 0 ε r L 1 t .
L = t ω 2 Re ( ε m ) L 2 ,
R = t ω Im ( ε m ) L 2 ,
Z t = ( Z C 1 + R 1 + Z L 1 ) 1 = [ j ω C + R 1 + ( j ω L ) 1 ] 1 ,
Z 0 = μ 0 ε 0 w .
T TL = | 2 Z t 2 Z t + Z 0 | 2 .

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