Abstract

Nanoantennas are key optical components for several applications including photodetection and biosensing. Here we present an array of metal nano-dipoles supporting surface plasmon polaritons (SPPs) integrated into a silicon-based Schottky-contact photodetector. Incident photons coupled to the array excite SPPs on the Au nanowires of the antennas which decay by creating ”hot” carriers in the metal. The hot carriers may then be injected over the potential barrier at the Au-Si interface resulting in a photocurrent. High responsivities of 100 mA/W and practical minimum detectable powers of −12 dBm should be achievable in the infra-red (1310 nm). The device was then investigated for use as a biosensor by computing its bulk and surface sensitivities. Sensitivities of ∼ 250 nm/RIU (bulk) and ∼ 8 nm/nm (surface) in water are predicted. We identify the mode propagating and resonating along the nanowires of the antennas, we apply a transmission line model to describe the performance of the antennas, and we extract two useful formulas to predict their bulk and surface sensitivities. We prove that the sensitivities of dipoles are much greater than those of similar monopoles and we show that this difference comes from the gap in dipole antennas where electric fields are strongly enhanced.

© 2013 OSA

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2012 (4)

2011 (8)

M. Piliarik, P. Kvasnička, N. Galler, J. R. Krenn, and J. Homola, “Local refractive index sensitivity of plasmonic nanoparticles,” Opt. Express19, 9213–9220 (2011).
[CrossRef] [PubMed]

L. Guyot, A.-P. Blanchard-Dionne, S. Patskovsky, and M. Meunier, “Integrated silicon-based nanoplasmonic sensor,” Opt. Express19, 9962–9967 (2011).
[CrossRef] [PubMed]

S. Zhu, G. Q. Lo, and D. L. Kwong, “Theoretical investigation of silicide schottky barrier detector integrated in horizontal metal-insulator-silicon-insulator-metal nanoplasmonic slot waveguide,” Opt. Express19, 15843–15854 (2011).
[CrossRef] [PubMed]

I. Goykhman, B. Desiatov, J. Khurgin, J. Shappir, and U. Levy, “Locally oxidized silicon surface-plasmon schot-tky detector for telecom regime,” Nano Letters11, 2219–2224 (2011).
[CrossRef] [PubMed]

E. S. Barnard, R. A. Pala, and M. L. Brongersma, “Photocurrent mapping of near-field optical antenna resonances,” Nat Nano6, 588–593 (2011).
[CrossRef]

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science332, 702–704 (2011).
[CrossRef] [PubMed]

F. J. Rodriguez-Fortuno, M. Martinez-Marco, B. Tomas-Navarro, R. Ortuno, J. Marti, A. Martinez, and P. J. Rodriguez-Canto, “Highly-sensitive chemical detection in the infrared regime using plasmonic gold nanocrosses,” Applied Physics Letters98, 133118 (2011).
[CrossRef]

C.-Y. Tsai, S.-P. Lu, J.-W. Lin, and P.-T. Lee, “High sensitivity plasmonic index sensor using slablike gold nanoring arrays,” Applied Physics Letters98, 153108 (2011).
[CrossRef] [PubMed]

2010 (3)

F. Mazzotta, G. Wang, C. Hgglund, F. Hk, and M. P. Jonsson, “Nanoplasmonic biosensing with on-chip electrical detection,” Biosensors and Bioelectronics26, 1131 – 1136 (2010).
[CrossRef] [PubMed]

A. Akbari, R. N. Tait, and P. Berini, “Surface plasmon waveguide schottky detector,” Opt. Express18, 8505–8514 (2010).
[CrossRef] [PubMed]

C. Scales and P. Berini, “Thin-film schottky barrier photodetector models,” Quantum Electronics, IEEE Journal of46, 633 –643 (2010).
[CrossRef]

2008 (3)

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chemical Reviews108, 462–493 (2008). PMID: .
[CrossRef] [PubMed]

L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D.-S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometre-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat Photon2, 226–229 (2008).
[CrossRef]

P. Berini, “Bulk and surface sensitivities of surface plasmon waveguides,” New Journal of Physics10, 105010 (2008).
[CrossRef]

2007 (2)

E. M. Larsson, J. Alegret, M. Kll, and D. S. Sutherland, “Sensing characteristics of nir localized surface plasmon resonances in gold nanorings for application as ultrasensitive biosensors,” Nano Letters7, 1256–1263 (2007).
[CrossRef] [PubMed]

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat Photon1, 641–648 (2007).
[CrossRef]

2006 (2)

C. L. Nehl, H. Liao, and J. H. Hafner, “Optical properties of star-shaped gold nanoparticles,” Nano Letters6, 683–688 (2006).
[CrossRef] [PubMed]

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Letters6, 827–832 (2006). PMID: .
[CrossRef] [PubMed]

2005 (1)

M. M. Miller and A. A. Lazarides, “Sensitivity of metal nanoparticle surface plasmon resonance to the dielectric environment,” The Journal of Physical Chemistry B109, 21556–21565 (2005).
[CrossRef]

2004 (1)

P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Letters4, 899–903 (2004).
[CrossRef]

2000 (2)

V. Brioude and O. Parriaux, “Normalised analysis for the design of evanescent-wave sensors and its use for tolerance evaluation,” Optical and Quantum Electronics32, 899–908 (2000).
[CrossRef]

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B61, 10484–10503 (2000).
[CrossRef]

1998 (1)

J. McSpadden, L. Fan, and K. Chang, “Design and experiments of a high-conversion-efficiency 5.8-ghz rectenna,” Microwave Theory and Techniques, IEEE Transactions on46, 2053 –2060 (1998).
[CrossRef]

1996 (2)

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett.77, 1889–1892 (1996).
[CrossRef] [PubMed]

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B54, 6227–6244 (1996).
[CrossRef]

1991 (1)

C. Daboo, M. Baird, H. H. N. Apsley, and M. Emeny, “Improved surface plasmon enhanced photodetection at an augaas schottky junction using a novel molecular beam epitaxy grown otto coupling structure,” Thin Solid Films201, 9 – 27 (1991).
[CrossRef]

1987 (1)

R. Soref and B. Bennett, “Electrooptical effects in silicon,” Quantum Electronics, IEEE Journal of23, 123 – 129 (1987).
[CrossRef]

1985 (1)

S. R. J. Brueck, V. Diadiuk, T. Jones, and W. Lenth, “Enhanced quantum efficiency internal photoemission detectors by grating coupling to surface plasma waves,” Applied Physics Letters46, 915–917 (1985).
[CrossRef]

1982 (1)

G. I. Stegeman, J. J. Burke, and D. G. Hall, “Nonlinear optics of long range surface plasmons,” Applied Physics Letters41, 906–908 (1982).
[CrossRef]

1963 (1)

R. N. Stuart, F. Wooten, and W. E. Spicer, “Mean free path of hot electrons and holes in metals,” Phys. Rev. Lett.10, 119–119 (1963).
[CrossRef]

Adato, R.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat Mater11, 69–75 (2012).
[CrossRef]

Akbari, A.

A. Akbari, R. N. Tait, and P. Berini, “Surface plasmon waveguide schottky detector,” Opt. Express18, 8505–8514 (2010).
[CrossRef] [PubMed]

A. Akbari, A. Olivieri, and P. Berini, “Sub-bandgap asymmetric surface plasmon waveguide schottky detectors on silicon,” Accepted for publication in Sel. Top. Quantum Electronics, IEEE Journal of (2013).

Alegret, J.

E. M. Larsson, J. Alegret, M. Kll, and D. S. Sutherland, “Sensing characteristics of nir localized surface plasmon resonances in gold nanorings for application as ultrasensitive biosensors,” Nano Letters7, 1256–1263 (2007).
[CrossRef] [PubMed]

Altug, H.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat Mater11, 69–75 (2012).
[CrossRef]

An, K. H.

Apsley, H. H. N.

C. Daboo, M. Baird, H. H. N. Apsley, and M. Emeny, “Improved surface plasmon enhanced photodetection at an augaas schottky junction using a novel molecular beam epitaxy grown otto coupling structure,” Thin Solid Films201, 9 – 27 (1991).
[CrossRef]

Arju, N.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat Mater11, 69–75 (2012).
[CrossRef]

Baird, M.

C. Daboo, M. Baird, H. H. N. Apsley, and M. Emeny, “Improved surface plasmon enhanced photodetection at an augaas schottky junction using a novel molecular beam epitaxy grown otto coupling structure,” Thin Solid Films201, 9 – 27 (1991).
[CrossRef]

Barnard, E. S.

E. S. Barnard, R. A. Pala, and M. L. Brongersma, “Photocurrent mapping of near-field optical antenna resonances,” Nat Nano6, 588–593 (2011).
[CrossRef]

Barnes, W. L.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B54, 6227–6244 (1996).
[CrossRef]

Bennett, B.

R. Soref and B. Bennett, “Electrooptical effects in silicon,” Quantum Electronics, IEEE Journal of23, 123 – 129 (1987).
[CrossRef]

Berini, P.

S. S. Mousavi, P. Berini, and D. McNamara, “Periodic plasmonic nanoantennas in a piecewise homogeneous background,” Opt. Express20, 18044–18065 (2012).
[CrossRef] [PubMed]

A. Akbari, R. N. Tait, and P. Berini, “Surface plasmon waveguide schottky detector,” Opt. Express18, 8505–8514 (2010).
[CrossRef] [PubMed]

C. Scales and P. Berini, “Thin-film schottky barrier photodetector models,” Quantum Electronics, IEEE Journal of46, 633 –643 (2010).
[CrossRef]

P. Berini, “Bulk and surface sensitivities of surface plasmon waveguides,” New Journal of Physics10, 105010 (2008).
[CrossRef]

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B61, 10484–10503 (2000).
[CrossRef]

A. Akbari, A. Olivieri, and P. Berini, “Sub-bandgap asymmetric surface plasmon waveguide schottky detectors on silicon,” Accepted for publication in Sel. Top. Quantum Electronics, IEEE Journal of (2013).

Bielefeldt, H.

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett.77, 1889–1892 (1996).
[CrossRef] [PubMed]

Blanchard-Dionne, A.-P.

Bogdanova, M.

Brandl, D. W.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Letters6, 827–832 (2006). PMID: .
[CrossRef] [PubMed]

Brioude, V.

V. Brioude and O. Parriaux, “Normalised analysis for the design of evanescent-wave sensors and its use for tolerance evaluation,” Optical and Quantum Electronics32, 899–908 (2000).
[CrossRef]

Brongersma, M. L.

E. S. Barnard, R. A. Pala, and M. L. Brongersma, “Photocurrent mapping of near-field optical antenna resonances,” Nat Nano6, 588–593 (2011).
[CrossRef]

Brueck, S. R. J.

S. R. J. Brueck, V. Diadiuk, T. Jones, and W. Lenth, “Enhanced quantum efficiency internal photoemission detectors by grating coupling to surface plasma waves,” Applied Physics Letters46, 915–917 (1985).
[CrossRef]

Burke, J. J.

G. I. Stegeman, J. J. Burke, and D. G. Hall, “Nonlinear optics of long range surface plasmons,” Applied Physics Letters41, 906–908 (1982).
[CrossRef]

Casalino, M.

Chang, K.

J. McSpadden, L. Fan, and K. Chang, “Design and experiments of a high-conversion-efficiency 5.8-ghz rectenna,” Microwave Theory and Techniques, IEEE Transactions on46, 2053 –2060 (1998).
[CrossRef]

Coppola, G.

Daboo, C.

C. Daboo, M. Baird, H. H. N. Apsley, and M. Emeny, “Improved surface plasmon enhanced photodetection at an augaas schottky junction using a novel molecular beam epitaxy grown otto coupling structure,” Thin Solid Films201, 9 – 27 (1991).
[CrossRef]

Deinega, A.

Desiatov, B.

I. Goykhman, B. Desiatov, J. Khurgin, J. Shappir, and U. Levy, “Locally oxidized silicon surface-plasmon schot-tky detector for telecom regime,” Nano Letters11, 2219–2224 (2011).
[CrossRef] [PubMed]

Diadiuk, V.

S. R. J. Brueck, V. Diadiuk, T. Jones, and W. Lenth, “Enhanced quantum efficiency internal photoemission detectors by grating coupling to surface plasma waves,” Applied Physics Letters46, 915–917 (1985).
[CrossRef]

Emeny, M.

C. Daboo, M. Baird, H. H. N. Apsley, and M. Emeny, “Improved surface plasmon enhanced photodetection at an augaas schottky junction using a novel molecular beam epitaxy grown otto coupling structure,” Thin Solid Films201, 9 – 27 (1991).
[CrossRef]

Fan, L.

J. McSpadden, L. Fan, and K. Chang, “Design and experiments of a high-conversion-efficiency 5.8-ghz rectenna,” Microwave Theory and Techniques, IEEE Transactions on46, 2053 –2060 (1998).
[CrossRef]

Galler, N.

Goykhman, I.

I. Goykhman, B. Desiatov, J. Khurgin, J. Shappir, and U. Levy, “Locally oxidized silicon surface-plasmon schot-tky detector for telecom regime,” Nano Letters11, 2219–2224 (2011).
[CrossRef] [PubMed]

Guyot, L.

Hafner, J. H.

C. L. Nehl, H. Liao, and J. H. Hafner, “Optical properties of star-shaped gold nanoparticles,” Nano Letters6, 683–688 (2006).
[CrossRef] [PubMed]

Halas, N. J.

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science332, 702–704 (2011).
[CrossRef] [PubMed]

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat Photon1, 641–648 (2007).
[CrossRef]

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Letters6, 827–832 (2006). PMID: .
[CrossRef] [PubMed]

Hall, D. G.

G. I. Stegeman, J. J. Burke, and D. G. Hall, “Nonlinear optics of long range surface plasmons,” Applied Physics Letters41, 906–908 (1982).
[CrossRef]

Hall, W. P.

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961), 1st ed.

Hecht, B.

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett.77, 1889–1892 (1996).
[CrossRef] [PubMed]

Hgglund, C.

F. Mazzotta, G. Wang, C. Hgglund, F. Hk, and M. P. Jonsson, “Nanoplasmonic biosensing with on-chip electrical detection,” Biosensors and Bioelectronics26, 1131 – 1136 (2010).
[CrossRef] [PubMed]

Hk, F.

F. Mazzotta, G. Wang, C. Hgglund, F. Hk, and M. P. Jonsson, “Nanoplasmonic biosensing with on-chip electrical detection,” Biosensors and Bioelectronics26, 1131 – 1136 (2010).
[CrossRef] [PubMed]

Homola, J.

M. Piliarik, P. Kvasnička, N. Galler, J. R. Krenn, and J. Homola, “Local refractive index sensitivity of plasmonic nanoparticles,” Opt. Express19, 9213–9220 (2011).
[CrossRef] [PubMed]

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chemical Reviews108, 462–493 (2008). PMID: .
[CrossRef] [PubMed]

Inouye, Y.

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett.77, 1889–1892 (1996).
[CrossRef] [PubMed]

Iodice, M.

Jones, T.

S. R. J. Brueck, V. Diadiuk, T. Jones, and W. Lenth, “Enhanced quantum efficiency internal photoemission detectors by grating coupling to surface plasma waves,” Applied Physics Letters46, 915–917 (1985).
[CrossRef]

Jonsson, M. P.

F. Mazzotta, G. Wang, C. Hgglund, F. Hk, and M. P. Jonsson, “Nanoplasmonic biosensing with on-chip electrical detection,” Biosensors and Bioelectronics26, 1131 – 1136 (2010).
[CrossRef] [PubMed]

Khanikaev, A. B.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat Mater11, 69–75 (2012).
[CrossRef]

Khurgin, J.

I. Goykhman, B. Desiatov, J. Khurgin, J. Shappir, and U. Levy, “Locally oxidized silicon surface-plasmon schot-tky detector for telecom regime,” Nano Letters11, 2219–2224 (2011).
[CrossRef] [PubMed]

Kitson, S. C.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B54, 6227–6244 (1996).
[CrossRef]

Kll, M.

E. M. Larsson, J. Alegret, M. Kll, and D. S. Sutherland, “Sensing characteristics of nir localized surface plasmon resonances in gold nanorings for application as ultrasensitive biosensors,” Nano Letters7, 1256–1263 (2007).
[CrossRef] [PubMed]

Knight, M. W.

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science332, 702–704 (2011).
[CrossRef] [PubMed]

Kocabas, S. E.

L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D.-S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometre-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat Photon2, 226–229 (2008).
[CrossRef]

Krenn, J. R.

Kvasnicka, P.

Kwong, D. L.

Lal, S.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat Photon1, 641–648 (2007).
[CrossRef]

Larsson, E. M.

E. M. Larsson, J. Alegret, M. Kll, and D. S. Sutherland, “Sensing characteristics of nir localized surface plasmon resonances in gold nanorings for application as ultrasensitive biosensors,” Nano Letters7, 1256–1263 (2007).
[CrossRef] [PubMed]

Latif, S.

L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D.-S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometre-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat Photon2, 226–229 (2008).
[CrossRef]

Lazarides, A. A.

M. M. Miller and A. A. Lazarides, “Sensitivity of metal nanoparticle surface plasmon resonance to the dielectric environment,” The Journal of Physical Chemistry B109, 21556–21565 (2005).
[CrossRef]

Le, F.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Letters6, 827–832 (2006). PMID: .
[CrossRef] [PubMed]

Lee, P.-T.

C.-Y. Tsai, S.-P. Lu, J.-W. Lin, and P.-T. Lee, “High sensitivity plasmonic index sensor using slablike gold nanoring arrays,” Applied Physics Letters98, 153108 (2011).
[CrossRef] [PubMed]

Lenth, W.

S. R. J. Brueck, V. Diadiuk, T. Jones, and W. Lenth, “Enhanced quantum efficiency internal photoemission detectors by grating coupling to surface plasma waves,” Applied Physics Letters46, 915–917 (1985).
[CrossRef]

Levy, U.

I. Goykhman, B. Desiatov, J. Khurgin, J. Shappir, and U. Levy, “Locally oxidized silicon surface-plasmon schot-tky detector for telecom regime,” Nano Letters11, 2219–2224 (2011).
[CrossRef] [PubMed]

Li, K.

P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Letters4, 899–903 (2004).
[CrossRef]

Liao, H.

C. L. Nehl, H. Liao, and J. H. Hafner, “Optical properties of star-shaped gold nanoparticles,” Nano Letters6, 683–688 (2006).
[CrossRef] [PubMed]

Lin, J.-W.

C.-Y. Tsai, S.-P. Lu, J.-W. Lin, and P.-T. Lee, “High sensitivity plasmonic index sensor using slablike gold nanoring arrays,” Applied Physics Letters98, 153108 (2011).
[CrossRef] [PubMed]

Link, S.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat Photon1, 641–648 (2007).
[CrossRef]

Lo, G. Q.

Lozovik, Y.

Lu, S.-P.

C.-Y. Tsai, S.-P. Lu, J.-W. Lin, and P.-T. Lee, “High sensitivity plasmonic index sensor using slablike gold nanoring arrays,” Applied Physics Letters98, 153108 (2011).
[CrossRef] [PubMed]

Ly-Gagnon, D.-S.

L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D.-S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometre-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat Photon2, 226–229 (2008).
[CrossRef]

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007), 1st ed.

Marti, J.

F. J. Rodriguez-Fortuno, M. Martinez-Marco, B. Tomas-Navarro, R. Ortuno, J. Marti, A. Martinez, and P. J. Rodriguez-Canto, “Highly-sensitive chemical detection in the infrared regime using plasmonic gold nanocrosses,” Applied Physics Letters98, 133118 (2011).
[CrossRef]

Martinez, A.

F. J. Rodriguez-Fortuno, M. Martinez-Marco, B. Tomas-Navarro, R. Ortuno, J. Marti, A. Martinez, and P. J. Rodriguez-Canto, “Highly-sensitive chemical detection in the infrared regime using plasmonic gold nanocrosses,” Applied Physics Letters98, 133118 (2011).
[CrossRef]

Martinez-Marco, M.

F. J. Rodriguez-Fortuno, M. Martinez-Marco, B. Tomas-Navarro, R. Ortuno, J. Marti, A. Martinez, and P. J. Rodriguez-Canto, “Highly-sensitive chemical detection in the infrared regime using plasmonic gold nanocrosses,” Applied Physics Letters98, 133118 (2011).
[CrossRef]

Mazzotta, F.

F. Mazzotta, G. Wang, C. Hgglund, F. Hk, and M. P. Jonsson, “Nanoplasmonic biosensing with on-chip electrical detection,” Biosensors and Bioelectronics26, 1131 – 1136 (2010).
[CrossRef] [PubMed]

McNamara, D.

McSpadden, J.

J. McSpadden, L. Fan, and K. Chang, “Design and experiments of a high-conversion-efficiency 5.8-ghz rectenna,” Microwave Theory and Techniques, IEEE Transactions on46, 2053 –2060 (1998).
[CrossRef]

Meunier, M.

Miller, D. A. B.

L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D.-S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometre-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat Photon2, 226–229 (2008).
[CrossRef]

Miller, M. M.

M. M. Miller and A. A. Lazarides, “Sensitivity of metal nanoparticle surface plasmon resonance to the dielectric environment,” The Journal of Physical Chemistry B109, 21556–21565 (2005).
[CrossRef]

Mousavi, S. S.

Nehl, C. L.

C. L. Nehl, H. Liao, and J. H. Hafner, “Optical properties of star-shaped gold nanoparticles,” Nano Letters6, 683–688 (2006).
[CrossRef] [PubMed]

Ng, K. K.

S. M. Sze and K. K. Ng, Physics of Semiconductor Devices (Wiley, 2006), 3rd ed.
[CrossRef]

Nordlander, P.

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science332, 702–704 (2011).
[CrossRef] [PubMed]

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Letters6, 827–832 (2006). PMID: .
[CrossRef] [PubMed]

P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Letters4, 899–903 (2004).
[CrossRef]

Novotny, L.

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett.77, 1889–1892 (1996).
[CrossRef] [PubMed]

Okyay, A. K.

L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D.-S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometre-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat Photon2, 226–229 (2008).
[CrossRef]

Olivieri, A.

A. Akbari, A. Olivieri, and P. Berini, “Sub-bandgap asymmetric surface plasmon waveguide schottky detectors on silicon,” Accepted for publication in Sel. Top. Quantum Electronics, IEEE Journal of (2013).

Ortuno, R.

F. J. Rodriguez-Fortuno, M. Martinez-Marco, B. Tomas-Navarro, R. Ortuno, J. Marti, A. Martinez, and P. J. Rodriguez-Canto, “Highly-sensitive chemical detection in the infrared regime using plasmonic gold nanocrosses,” Applied Physics Letters98, 133118 (2011).
[CrossRef]

Oubre, C.

P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Letters4, 899–903 (2004).
[CrossRef]

Pala, R. A.

E. S. Barnard, R. A. Pala, and M. L. Brongersma, “Photocurrent mapping of near-field optical antenna resonances,” Nat Nano6, 588–593 (2011).
[CrossRef]

Palik, E. D.

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

Parriaux, O.

V. Brioude and O. Parriaux, “Normalised analysis for the design of evanescent-wave sensors and its use for tolerance evaluation,” Optical and Quantum Electronics32, 899–908 (2000).
[CrossRef]

Patskovsky, S.

Piliarik, M.

Pohl, D. W.

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett.77, 1889–1892 (1996).
[CrossRef] [PubMed]

Potyrailo, R. A.

Preist, T. W.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B54, 6227–6244 (1996).
[CrossRef]

Pris, A. D.

Prodan, E.

P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Letters4, 899–903 (2004).
[CrossRef]

Rendina, I.

Rodriguez-Canto, P. J.

F. J. Rodriguez-Fortuno, M. Martinez-Marco, B. Tomas-Navarro, R. Ortuno, J. Marti, A. Martinez, and P. J. Rodriguez-Canto, “Highly-sensitive chemical detection in the infrared regime using plasmonic gold nanocrosses,” Applied Physics Letters98, 133118 (2011).
[CrossRef]

Rodriguez-Fortuno, F. J.

F. J. Rodriguez-Fortuno, M. Martinez-Marco, B. Tomas-Navarro, R. Ortuno, J. Marti, A. Martinez, and P. J. Rodriguez-Canto, “Highly-sensitive chemical detection in the infrared regime using plasmonic gold nanocrosses,” Applied Physics Letters98, 133118 (2011).
[CrossRef]

Sambles, J. R.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B54, 6227–6244 (1996).
[CrossRef]

Saraswat, K. C.

L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D.-S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometre-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat Photon2, 226–229 (2008).
[CrossRef]

Scales, C.

C. Scales and P. Berini, “Thin-film schottky barrier photodetector models,” Quantum Electronics, IEEE Journal of46, 633 –643 (2010).
[CrossRef]

Segelstein, D. J.

D. J. Segelstein, “The complex refractive index of water,” Master’s thesis, University of Missouri, Kansas City, Missouri, USA (1981).

Shappir, J.

I. Goykhman, B. Desiatov, J. Khurgin, J. Shappir, and U. Levy, “Locally oxidized silicon surface-plasmon schot-tky detector for telecom regime,” Nano Letters11, 2219–2224 (2011).
[CrossRef] [PubMed]

Shvets, G.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat Mater11, 69–75 (2012).
[CrossRef]

Sirleto, L.

Sobhani, H.

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science332, 702–704 (2011).
[CrossRef] [PubMed]

Soref, R.

R. Soref and B. Bennett, “Electrooptical effects in silicon,” Quantum Electronics, IEEE Journal of23, 123 – 129 (1987).
[CrossRef]

Spicer, W. E.

R. N. Stuart, F. Wooten, and W. E. Spicer, “Mean free path of hot electrons and holes in metals,” Phys. Rev. Lett.10, 119–119 (1963).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman, J. J. Burke, and D. G. Hall, “Nonlinear optics of long range surface plasmons,” Applied Physics Letters41, 906–908 (1982).
[CrossRef]

Stockman, M. I.

P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Letters4, 899–903 (2004).
[CrossRef]

Stuart, R. N.

R. N. Stuart, F. Wooten, and W. E. Spicer, “Mean free path of hot electrons and holes in metals,” Phys. Rev. Lett.10, 119–119 (1963).
[CrossRef]

Sutherland, D. S.

E. M. Larsson, J. Alegret, M. Kll, and D. S. Sutherland, “Sensing characteristics of nir localized surface plasmon resonances in gold nanorings for application as ultrasensitive biosensors,” Nano Letters7, 1256–1263 (2007).
[CrossRef] [PubMed]

Sze, S. M.

S. M. Sze and K. K. Ng, Physics of Semiconductor Devices (Wiley, 2006), 3rd ed.
[CrossRef]

Tait, R. N.

Tang, L.

L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D.-S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometre-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat Photon2, 226–229 (2008).
[CrossRef]

Tomas-Navarro, B.

F. J. Rodriguez-Fortuno, M. Martinez-Marco, B. Tomas-Navarro, R. Ortuno, J. Marti, A. Martinez, and P. J. Rodriguez-Canto, “Highly-sensitive chemical detection in the infrared regime using plasmonic gold nanocrosses,” Applied Physics Letters98, 133118 (2011).
[CrossRef]

Tsai, C.-Y.

C.-Y. Tsai, S.-P. Lu, J.-W. Lin, and P.-T. Lee, “High sensitivity plasmonic index sensor using slablike gold nanoring arrays,” Applied Physics Letters98, 153108 (2011).
[CrossRef] [PubMed]

Wang, G.

F. Mazzotta, G. Wang, C. Hgglund, F. Hk, and M. P. Jonsson, “Nanoplasmonic biosensing with on-chip electrical detection,” Biosensors and Bioelectronics26, 1131 – 1136 (2010).
[CrossRef] [PubMed]

Wang, H.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Letters6, 827–832 (2006). PMID: .
[CrossRef] [PubMed]

Wooten, F.

R. N. Stuart, F. Wooten, and W. E. Spicer, “Mean free path of hot electrons and holes in metals,” Phys. Rev. Lett.10, 119–119 (1963).
[CrossRef]

Wu, C.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat Mater11, 69–75 (2012).
[CrossRef]

Yanik, A. A.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat Mater11, 69–75 (2012).
[CrossRef]

Zalyubovskiy, S. J.

Zhu, S.

Applied Physics Letters (4)

F. J. Rodriguez-Fortuno, M. Martinez-Marco, B. Tomas-Navarro, R. Ortuno, J. Marti, A. Martinez, and P. J. Rodriguez-Canto, “Highly-sensitive chemical detection in the infrared regime using plasmonic gold nanocrosses,” Applied Physics Letters98, 133118 (2011).
[CrossRef]

C.-Y. Tsai, S.-P. Lu, J.-W. Lin, and P.-T. Lee, “High sensitivity plasmonic index sensor using slablike gold nanoring arrays,” Applied Physics Letters98, 153108 (2011).
[CrossRef] [PubMed]

G. I. Stegeman, J. J. Burke, and D. G. Hall, “Nonlinear optics of long range surface plasmons,” Applied Physics Letters41, 906–908 (1982).
[CrossRef]

S. R. J. Brueck, V. Diadiuk, T. Jones, and W. Lenth, “Enhanced quantum efficiency internal photoemission detectors by grating coupling to surface plasma waves,” Applied Physics Letters46, 915–917 (1985).
[CrossRef]

Biosensors and Bioelectronics (1)

F. Mazzotta, G. Wang, C. Hgglund, F. Hk, and M. P. Jonsson, “Nanoplasmonic biosensing with on-chip electrical detection,” Biosensors and Bioelectronics26, 1131 – 1136 (2010).
[CrossRef] [PubMed]

Chemical Reviews (1)

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chemical Reviews108, 462–493 (2008). PMID: .
[CrossRef] [PubMed]

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

Microwave Theory and Techniques, IEEE Transactions on (1)

J. McSpadden, L. Fan, and K. Chang, “Design and experiments of a high-conversion-efficiency 5.8-ghz rectenna,” Microwave Theory and Techniques, IEEE Transactions on46, 2053 –2060 (1998).
[CrossRef]

Nano Letters (5)

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Letters6, 827–832 (2006). PMID: .
[CrossRef] [PubMed]

C. L. Nehl, H. Liao, and J. H. Hafner, “Optical properties of star-shaped gold nanoparticles,” Nano Letters6, 683–688 (2006).
[CrossRef] [PubMed]

E. M. Larsson, J. Alegret, M. Kll, and D. S. Sutherland, “Sensing characteristics of nir localized surface plasmon resonances in gold nanorings for application as ultrasensitive biosensors,” Nano Letters7, 1256–1263 (2007).
[CrossRef] [PubMed]

I. Goykhman, B. Desiatov, J. Khurgin, J. Shappir, and U. Levy, “Locally oxidized silicon surface-plasmon schot-tky detector for telecom regime,” Nano Letters11, 2219–2224 (2011).
[CrossRef] [PubMed]

P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Letters4, 899–903 (2004).
[CrossRef]

Nat Mater (1)

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat Mater11, 69–75 (2012).
[CrossRef]

Nat Nano (1)

E. S. Barnard, R. A. Pala, and M. L. Brongersma, “Photocurrent mapping of near-field optical antenna resonances,” Nat Nano6, 588–593 (2011).
[CrossRef]

Nat Photon (2)

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat Photon1, 641–648 (2007).
[CrossRef]

L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D.-S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometre-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat Photon2, 226–229 (2008).
[CrossRef]

New Journal of Physics (1)

P. Berini, “Bulk and surface sensitivities of surface plasmon waveguides,” New Journal of Physics10, 105010 (2008).
[CrossRef]

Opt. Express (6)

Optical and Quantum Electronics (1)

V. Brioude and O. Parriaux, “Normalised analysis for the design of evanescent-wave sensors and its use for tolerance evaluation,” Optical and Quantum Electronics32, 899–908 (2000).
[CrossRef]

Phys. Rev. B (2)

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B61, 10484–10503 (2000).
[CrossRef]

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B54, 6227–6244 (1996).
[CrossRef]

Phys. Rev. Lett. (2)

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett.77, 1889–1892 (1996).
[CrossRef] [PubMed]

R. N. Stuart, F. Wooten, and W. E. Spicer, “Mean free path of hot electrons and holes in metals,” Phys. Rev. Lett.10, 119–119 (1963).
[CrossRef]

Quantum Electronics, IEEE Journal of (2)

C. Scales and P. Berini, “Thin-film schottky barrier photodetector models,” Quantum Electronics, IEEE Journal of46, 633 –643 (2010).
[CrossRef]

R. Soref and B. Bennett, “Electrooptical effects in silicon,” Quantum Electronics, IEEE Journal of23, 123 – 129 (1987).
[CrossRef]

Science (1)

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science332, 702–704 (2011).
[CrossRef] [PubMed]

The Journal of Physical Chemistry B (1)

M. M. Miller and A. A. Lazarides, “Sensitivity of metal nanoparticle surface plasmon resonance to the dielectric environment,” The Journal of Physical Chemistry B109, 21556–21565 (2005).
[CrossRef]

Thin Solid Films (1)

C. Daboo, M. Baird, H. H. N. Apsley, and M. Emeny, “Improved surface plasmon enhanced photodetection at an augaas schottky junction using a novel molecular beam epitaxy grown otto coupling structure,” Thin Solid Films201, 9 – 27 (1991).
[CrossRef]

Other (7)

S. M. Sze and K. K. Ng, Physics of Semiconductor Devices (Wiley, 2006), 3rd ed.
[CrossRef]

A. Akbari, A. Olivieri, and P. Berini, “Sub-bandgap asymmetric surface plasmon waveguide schottky detectors on silicon,” Accepted for publication in Sel. Top. Quantum Electronics, IEEE Journal of (2013).

FDTD Solutions (Lumerical Solutions Inc.).

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

D. J. Segelstein, “The complex refractive index of water,” Master’s thesis, University of Missouri, Kansas City, Missouri, USA (1981).

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007), 1st ed.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961), 1st ed.

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Figures (15)

Fig. 1
Fig. 1

(a) Array of Au dipoles on p-Si on p+-Si covered by H2O. Al Ohmic contacts are located at the bottom of the structure and an Au circular contact pad is connected to all dipole arms via optically non-invasive perpendicular Au interconnects. A plane wave source illuminates the array in the +z-direction from below. (b) Geometry of a unit cell of the array under study (interconnects are shown as well); the dipole is assumed covered by an adlayer of thickness a when the surface sensitivities are computed.

Fig. 2
Fig. 2

Distribution of | E | = | E x | 2 + | E y | 2 + | E z | 2 on xy cross-sectional planes for an Au dipole of dimensions w = 30 nm, t = 30 nm, l = 210 nm, g = 20 nm and p = q = 300 nm (a) 3 nm above the Au/Si interface, (b) 15 nm above the Au/Si interface, and (c) 3 nm above the Au surface in H2O. Computations performed at λ0 = 1353 nm (resonance wavelength).

Fig. 3
Fig. 3

Calculated transmittance (T), reflectance (R), absorptance (A) and electric field enhancement (Een) vs. free space wavelength (λ0).

Fig. 4
Fig. 4

(a) Internal quantum efficiency ( η i t) vs. λ0. (b) Responsivity (Resp) and minimum detectable power (Smin) vs. λ0.

Fig. 5
Fig. 5

(a) Absorptance (A) vs. λ0 for several cover refractive indices nc ranging from 1 to 2.75). (b) Bulk sensitivity (∂λ0r/∂nc - blue) and peak responsivity (Resp,r - red) of the rectenna as a function of nc.

Fig. 6
Fig. 6

Real part of Ez of the s a b 0 mode plotted over the cross-section of a nanowire waveguide (λ0 = 1353 nm) computed using a mode solver. (a) nc = 1 (air) and (b) nc = 2.75.

Fig. 7
Fig. 7

Effective refractive index (neff blue) and mode power attenuation (α - red) of the s a b 0 mode resonating along the dipoles as a function of nc.

Fig. 8
Fig. 8

Gap capacitance Cg (blue) and characteristic impedance Z0 of the s a b 0 mode (red) as a function of nc.

Fig. 9
Fig. 9

(a) Resonant wavelengths computed using the transmission line model and the FDTD method as a function of bulk index nc. (b) Bulk sensitivity computed using the FDTD method (dashed blue), the transmission line model (Eq. (4) - red), and the analytical solution (Eq. (20) - black).

Fig. 10
Fig. 10

(a) Absorptance A vs. λ0 for several adlayer thicknesses (a = 0 to 5 nm); the curves are offset vertically by −0.05 for clarity. (b) Surface sensitivity (∂λ0r/∂a - blue) and peak responsivity (Resp,r - red) as a function of a.

Fig. 11
Fig. 11

Real part of Ez of the s a b 0 mode plotted over the cross-section of a nanowire waveguide (λ0 = 1353 nm) computed using a mode solver. (a) a = 0 (no adlyaer) and (b) a = 5 nm.

Fig. 12
Fig. 12

Effective refractive index (neff blue) and mode power attenuation (α - red) of the s a b 0 mode resonating along the dipoles as a function of a.

Fig. 13
Fig. 13

Schematic of a dipole gap showing three plate capacitances in series.

Fig. 14
Fig. 14

Gap capacitance Cg (blue) and characteristic impedance Z0 of the s a b 0 mode (red) as a function of a.

Fig. 15
Fig. 15

(a) Resonant wavelengths computed using the transmission line model and the FDTD method as a function of adlayer thickness a. (b) Surface sensitivity computed using the FDTD method (dashed blue), the transmission line model (Eq.(4) - red), and the analytical solution (Eq.(29) - black).

Equations (32)

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T ( f ) = S Re ( P m ) . ds S Re ( P s ) . ds
A = 1 T R
R esp = κ A η i t q h ν
tan ( n eff ω 0 r ε 0 μ 0 ( d + δ m ) ) = 2 ω 0 r C g Z 0
ω 0 r = 2 π c 0 λ 0 r
C g = ε c A d g
Z 0 = f ( y , z ) Re [ Z ω ( y , z ) ] d S f ( y , z ) d S
Z ω = k ^ ( E × H * ) ( k ^ × H ) ( k ^ × H * )
f ( x , y ) = | E y ( y , z ) | 2 + | E z ( y , z ) | 2
[ ( d + δ m ) ε 0 μ 0 ω 0 r n eff n c + ( d + δ m ) ε 0 μ 0 n eff ω 0 r n c ] × [ 1 + tan 2 ( n eff ω 0 r ε 0 μ 0 ( d + δ m ) ] = 2 ω 0 r C g Z 0 n c 2 ω 0 r Z 0 C g n c 2 C g Z 0 ω 0 r n c
ζ ω ω 0 r n c = ζ n n eff n c + ζ C C g n c + ζ Z Z 0 n c
ζ ω = ( d + δ m c 0 ) n eff ( 1 + 4 ω 0 r 2 C g 2 Z 0 2 ) + 2 C g Z 0
ζ n = ( d + δ m c 0 ) n eff ( 1 + 4 ω 0 r 2 C g 2 Z 0 2 )
ζ C = 2 ω 0 r Z 0
ζ Z = 2 ω 0 r C g
C g n c = ( w t g ) ε c n c = 2 ε 0 ( w t g ) n c
ω 0 r n c = ζ ω 1 [ 4 ( w t g ) ω 0 r Z 0 ] n c + ζ ω 1 ( ζ n n eff n c + ζ Z Z 0 n c )
ω 0 r n c [ 2 w t c 0 ω 0 r g ( d + δ m ) n eff ] 2 Z 0 n c + Z 0 n c n c 2 1 + ( 2 w t g 1 ω 0 r Z 0 ) 2 n c 2 ω 0 r n eff n eff n c
λ 0 r n c = ( 2 π c 0 ω 0 r 2 ) ω 0 r n c
λ 0 r n c [ 2 w t c 0 λ 0 r g ( d + δ m ) n eff ] 2 Z 0 n c + Z 0 n c n c 2 1 + ( 4 π wt g 1 c 0 λ 0 r Z 0 ) 2 n c 2 + λ 0 r n eff n eff n c
C g = ( 2 C 1 1 + C 2 1 ) 1
C 1 = ε a ( w t a )
C 2 = ε c ( w t g 2 a )
C g = ε a ε c w t 2 ( ε a ε c ) a + ε a g
ζ ω ω 0 r a = ζ n n eff a + ζ C C g a + ζ Z Z 0 a
C g a = 2 ε 0 ε r , c ε r , a ( ε r , a ε r , c ) w t [ 2 ( ε r , a ε r , c ) a + ε r , a g ] 2
ω 0 r a = 2 ε 0 ε r , c ε r , a ( ε r , a ε r , c ) w t [ 2 ( ε r , c ε r , a ) a + ε r , a g ] 2 ζ ω 1 ζ C ω 0 r Z 0 + ζ ω 1 ( ζ n n eff a + ζ Z Z 0 a )
λ 0 r a = ( 2 π c 0 ω 0 r 2 ) ω 0 r a
λ 0 r a = 4 ε 0 ε r , c ε r , a ( ε r , c ε r , a ) w t [ 2 ( ε r , c ε r , a ) a + ε r , a g ] 2 ζ ω 1 ζ C λ 0 r Z 0 ζ ω 1 λ 0 r 2 2 π c 0 ( ζ n n eff a + ζ Z Z 0 a )
cot ( n eff ω 0 r ε 0 μ 0 ( d + δ m ) ) = 0
λ 0 r n c = λ 0 r n eff n eff n c
λ 0 r a = λ 0 r n eff n eff a

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