Abstract

Simple circuit models provide valuable insight into the properties of plasmonic resonators. Yet, it is unclear how the circuit elements can be extracted and connected in the model in an intuitive and accurate manner. Here, we present a detailed treatment for constructing such circuits based on energy and charge oscillation considerations. The accuracy and validity of this approach was demonstrated for a gold nanorod, and extended for a split-ring resonator with varying gap sizes, yielding good intuitive and quantitative agreement with full electromagnetic simulations.

© 2014 Optical Society of America

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  3. A. Alù, N. Engheta, “Input Impedance, Nanocircuit Loading, and Radiation Tuning of Optical Nanoantennas,” Phys. Rev. Lett. 101(4), 043901 (2008).
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    [CrossRef] [PubMed]
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2013 (4)

H. Caglayan, S.-H. Hong, B. Edwards, C. R. Kagan, N. Engheta, “Near-Infrared Metatronic Nanocircuits by Design,” Phys. Rev. Lett. 111(7), 073904 (2013).
[CrossRef] [PubMed]

M. Bosman, E. Ye, S. F. Tan, C. A. Nijhuis, J. K. W. Yang, R. Marty, A. Mlayah, A. Arbouet, C. Girard, M. Y. Han, “Surface Plasmon Damping Quantified with an Electron Nanoprobe,” Sci. Rep. 3, 1312 (2013).
[CrossRef] [PubMed]

M. Amin, M. Farhat, H. Baǧcı, “A dynamically reconfigurable Fano metamaterial through graphene tuning for switching and sensing applications,” Sci. Rep. 3, 2105 (2013).
[CrossRef] [PubMed]

B. Willingham, S. Link, “A Kirchhoff solution to plasmon hybridization,” Appl. Phys. B 113(4), 519–525 (2013).
[CrossRef]

2012 (3)

H. Duan, A. I. Fernández-Domínguez, M. Bosman, S. A. Maier, J. K. Yang, “Nanoplasmonics: classical down to the nanometer scale,” Nano Lett. 12(3), 1683–1689 (2012).
[CrossRef] [PubMed]

Y. Sun, B. Edwards, A. Alù, N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11(3), 208–212 (2012).
[CrossRef] [PubMed]

M. Staffaroni, J. Conway, S. Vedantam, J. Tang, E. Yablonovitch, “Circuit analysis in metal-optics,” Photon. Nanostructures 10(1), 166–176 (2012).
[CrossRef]

2010 (1)

2009 (5)

O. Sydoruk, E. Tatartschuk, E. Shamonina, L. Solymar, “Analytical formulation for the resonant frequency of split rings,” J. Appl. Phys. 105(1), 014903 (2009).
[CrossRef]

A. W. Clark, A. Glidle, D. R. S. Cumming, J. M. Cooper, “Plasmonic Split-Ring Resonators as Dichroic Nanophotonic DNA Biosensors,” J. Am. Chem. Soc. 131(48), 17615–17619 (2009).
[CrossRef] [PubMed]

E. Cubukcu, S. Zhang, Y. S. Park, G. Bartal, X. Zhang, “Split ring resonator sensors for infrared detection of single molecular monolayers,” Appl. Phys. Lett. 95(4), 043113 (2009).
[CrossRef]

C. P. Huang, X. G. Yin, H. Huang, Y. Y. Zhu, “Study of plasmon resonance in a gold nanorod with an LC circuit model,” Opt. Express 17(8), 6407–6413 (2009).
[CrossRef] [PubMed]

A. Alù, N. Engheta, “All Optical Metamaterial Circuit Board at the Nanoscale,” Phys. Rev. Lett. 103(14), 143902 (2009).
[CrossRef] [PubMed]

2008 (4)

A. Alù, N. Engheta, “Input Impedance, Nanocircuit Loading, and Radiation Tuning of Optical Nanoantennas,” Phys. Rev. Lett. 101(4), 043901 (2008).
[CrossRef] [PubMed]

A. Alù, N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics 2(5), 307–310 (2008).
[CrossRef]

A. Alù, M. Young, N. Engheta, “Design of nanofilters for optical nanocircuits,” Phys. Rev. B 77(14), 144107 (2008).
[CrossRef]

M. G. Silveirinha, A. Alù, J. Li, N. Engheta, “Nanoinsulators and nanoconnectors for optical nanocircuits,” J. Appl. Phys. 103(6), 064305 (2008).
[CrossRef]

2007 (1)

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317(5845), 1698–1702 (2007).
[CrossRef] [PubMed]

2006 (1)

F. Wang, Y. R. Shen, “General Properties of Local Plasmons in Metal Nanostructures,” Phys. Rev. Lett. 97(20), 206806 (2006).
[CrossRef] [PubMed]

2005 (1)

N. Engheta, A. Salandrino, A. Alù, “Circuit Elements at Optical Frequencies: Nanoinductors, Nanocapacitors, and Nanoresistors,” Phys. Rev. Lett. 95(9), 095504 (2005).
[CrossRef] [PubMed]

2001 (1)

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory 47(11), 2075–2084 (1999).
[CrossRef]

1972 (1)

P. B. Johnson, R. W. Christy, “Optical Constants of Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Albrektsen, O.

Alù, A.

Y. Sun, B. Edwards, A. Alù, N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11(3), 208–212 (2012).
[CrossRef] [PubMed]

A. Alù, N. Engheta, “All Optical Metamaterial Circuit Board at the Nanoscale,” Phys. Rev. Lett. 103(14), 143902 (2009).
[CrossRef] [PubMed]

M. G. Silveirinha, A. Alù, J. Li, N. Engheta, “Nanoinsulators and nanoconnectors for optical nanocircuits,” J. Appl. Phys. 103(6), 064305 (2008).
[CrossRef]

A. Alù, M. Young, N. Engheta, “Design of nanofilters for optical nanocircuits,” Phys. Rev. B 77(14), 144107 (2008).
[CrossRef]

A. Alù, N. Engheta, “Input Impedance, Nanocircuit Loading, and Radiation Tuning of Optical Nanoantennas,” Phys. Rev. Lett. 101(4), 043901 (2008).
[CrossRef] [PubMed]

A. Alù, N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics 2(5), 307–310 (2008).
[CrossRef]

N. Engheta, A. Salandrino, A. Alù, “Circuit Elements at Optical Frequencies: Nanoinductors, Nanocapacitors, and Nanoresistors,” Phys. Rev. Lett. 95(9), 095504 (2005).
[CrossRef] [PubMed]

Amin, M.

M. Amin, M. Farhat, H. Baǧcı, “A dynamically reconfigurable Fano metamaterial through graphene tuning for switching and sensing applications,” Sci. Rep. 3, 2105 (2013).
[CrossRef] [PubMed]

Arbouet, A.

M. Bosman, E. Ye, S. F. Tan, C. A. Nijhuis, J. K. W. Yang, R. Marty, A. Mlayah, A. Arbouet, C. Girard, M. Y. Han, “Surface Plasmon Damping Quantified with an Electron Nanoprobe,” Sci. Rep. 3, 1312 (2013).
[CrossRef] [PubMed]

Bagci, H.

M. Amin, M. Farhat, H. Baǧcı, “A dynamically reconfigurable Fano metamaterial through graphene tuning for switching and sensing applications,” Sci. Rep. 3, 2105 (2013).
[CrossRef] [PubMed]

Bartal, G.

E. Cubukcu, S. Zhang, Y. S. Park, G. Bartal, X. Zhang, “Split ring resonator sensors for infrared detection of single molecular monolayers,” Appl. Phys. Lett. 95(4), 043113 (2009).
[CrossRef]

Bosman, M.

M. Bosman, E. Ye, S. F. Tan, C. A. Nijhuis, J. K. W. Yang, R. Marty, A. Mlayah, A. Arbouet, C. Girard, M. Y. Han, “Surface Plasmon Damping Quantified with an Electron Nanoprobe,” Sci. Rep. 3, 1312 (2013).
[CrossRef] [PubMed]

H. Duan, A. I. Fernández-Domínguez, M. Bosman, S. A. Maier, J. K. Yang, “Nanoplasmonics: classical down to the nanometer scale,” Nano Lett. 12(3), 1683–1689 (2012).
[CrossRef] [PubMed]

Bozhevolnyi, S. I.

Caglayan, H.

H. Caglayan, S.-H. Hong, B. Edwards, C. R. Kagan, N. Engheta, “Near-Infrared Metatronic Nanocircuits by Design,” Phys. Rev. Lett. 111(7), 073904 (2013).
[CrossRef] [PubMed]

Christy, R. W.

P. B. Johnson, R. W. Christy, “Optical Constants of Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Clark, A. W.

A. W. Clark, A. Glidle, D. R. S. Cumming, J. M. Cooper, “Plasmonic Split-Ring Resonators as Dichroic Nanophotonic DNA Biosensors,” J. Am. Chem. Soc. 131(48), 17615–17619 (2009).
[CrossRef] [PubMed]

Conway, J.

M. Staffaroni, J. Conway, S. Vedantam, J. Tang, E. Yablonovitch, “Circuit analysis in metal-optics,” Photon. Nanostructures 10(1), 166–176 (2012).
[CrossRef]

Cooper, J. M.

A. W. Clark, A. Glidle, D. R. S. Cumming, J. M. Cooper, “Plasmonic Split-Ring Resonators as Dichroic Nanophotonic DNA Biosensors,” J. Am. Chem. Soc. 131(48), 17615–17619 (2009).
[CrossRef] [PubMed]

Cubukcu, E.

E. Cubukcu, S. Zhang, Y. S. Park, G. Bartal, X. Zhang, “Split ring resonator sensors for infrared detection of single molecular monolayers,” Appl. Phys. Lett. 95(4), 043113 (2009).
[CrossRef]

Cumming, D. R. S.

A. W. Clark, A. Glidle, D. R. S. Cumming, J. M. Cooper, “Plasmonic Split-Ring Resonators as Dichroic Nanophotonic DNA Biosensors,” J. Am. Chem. Soc. 131(48), 17615–17619 (2009).
[CrossRef] [PubMed]

Duan, H.

H. Duan, A. I. Fernández-Domínguez, M. Bosman, S. A. Maier, J. K. Yang, “Nanoplasmonics: classical down to the nanometer scale,” Nano Lett. 12(3), 1683–1689 (2012).
[CrossRef] [PubMed]

Edwards, B.

H. Caglayan, S.-H. Hong, B. Edwards, C. R. Kagan, N. Engheta, “Near-Infrared Metatronic Nanocircuits by Design,” Phys. Rev. Lett. 111(7), 073904 (2013).
[CrossRef] [PubMed]

Y. Sun, B. Edwards, A. Alù, N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11(3), 208–212 (2012).
[CrossRef] [PubMed]

Engheta, N.

H. Caglayan, S.-H. Hong, B. Edwards, C. R. Kagan, N. Engheta, “Near-Infrared Metatronic Nanocircuits by Design,” Phys. Rev. Lett. 111(7), 073904 (2013).
[CrossRef] [PubMed]

Y. Sun, B. Edwards, A. Alù, N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11(3), 208–212 (2012).
[CrossRef] [PubMed]

A. Alù, N. Engheta, “All Optical Metamaterial Circuit Board at the Nanoscale,” Phys. Rev. Lett. 103(14), 143902 (2009).
[CrossRef] [PubMed]

A. Alù, M. Young, N. Engheta, “Design of nanofilters for optical nanocircuits,” Phys. Rev. B 77(14), 144107 (2008).
[CrossRef]

A. Alù, N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics 2(5), 307–310 (2008).
[CrossRef]

A. Alù, N. Engheta, “Input Impedance, Nanocircuit Loading, and Radiation Tuning of Optical Nanoantennas,” Phys. Rev. Lett. 101(4), 043901 (2008).
[CrossRef] [PubMed]

M. G. Silveirinha, A. Alù, J. Li, N. Engheta, “Nanoinsulators and nanoconnectors for optical nanocircuits,” J. Appl. Phys. 103(6), 064305 (2008).
[CrossRef]

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317(5845), 1698–1702 (2007).
[CrossRef] [PubMed]

N. Engheta, A. Salandrino, A. Alù, “Circuit Elements at Optical Frequencies: Nanoinductors, Nanocapacitors, and Nanoresistors,” Phys. Rev. Lett. 95(9), 095504 (2005).
[CrossRef] [PubMed]

Farhat, M.

M. Amin, M. Farhat, H. Baǧcı, “A dynamically reconfigurable Fano metamaterial through graphene tuning for switching and sensing applications,” Sci. Rep. 3, 2105 (2013).
[CrossRef] [PubMed]

Fernández-Domínguez, A. I.

H. Duan, A. I. Fernández-Domínguez, M. Bosman, S. A. Maier, J. K. Yang, “Nanoplasmonics: classical down to the nanometer scale,” Nano Lett. 12(3), 1683–1689 (2012).
[CrossRef] [PubMed]

Girard, C.

M. Bosman, E. Ye, S. F. Tan, C. A. Nijhuis, J. K. W. Yang, R. Marty, A. Mlayah, A. Arbouet, C. Girard, M. Y. Han, “Surface Plasmon Damping Quantified with an Electron Nanoprobe,” Sci. Rep. 3, 1312 (2013).
[CrossRef] [PubMed]

Glidle, A.

A. W. Clark, A. Glidle, D. R. S. Cumming, J. M. Cooper, “Plasmonic Split-Ring Resonators as Dichroic Nanophotonic DNA Biosensors,” J. Am. Chem. Soc. 131(48), 17615–17619 (2009).
[CrossRef] [PubMed]

Han, M. Y.

M. Bosman, E. Ye, S. F. Tan, C. A. Nijhuis, J. K. W. Yang, R. Marty, A. Mlayah, A. Arbouet, C. Girard, M. Y. Han, “Surface Plasmon Damping Quantified with an Electron Nanoprobe,” Sci. Rep. 3, 1312 (2013).
[CrossRef] [PubMed]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory 47(11), 2075–2084 (1999).
[CrossRef]

Hong, S.-H.

H. Caglayan, S.-H. Hong, B. Edwards, C. R. Kagan, N. Engheta, “Near-Infrared Metatronic Nanocircuits by Design,” Phys. Rev. Lett. 111(7), 073904 (2013).
[CrossRef] [PubMed]

Huang, C. P.

Huang, H.

Johnson, P. B.

P. B. Johnson, R. W. Christy, “Optical Constants of Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Kagan, C. R.

H. Caglayan, S.-H. Hong, B. Edwards, C. R. Kagan, N. Engheta, “Near-Infrared Metatronic Nanocircuits by Design,” Phys. Rev. Lett. 111(7), 073904 (2013).
[CrossRef] [PubMed]

Li, J.

M. G. Silveirinha, A. Alù, J. Li, N. Engheta, “Nanoinsulators and nanoconnectors for optical nanocircuits,” J. Appl. Phys. 103(6), 064305 (2008).
[CrossRef]

Link, S.

B. Willingham, S. Link, “A Kirchhoff solution to plasmon hybridization,” Appl. Phys. B 113(4), 519–525 (2013).
[CrossRef]

Maier, S. A.

H. Duan, A. I. Fernández-Domínguez, M. Bosman, S. A. Maier, J. K. Yang, “Nanoplasmonics: classical down to the nanometer scale,” Nano Lett. 12(3), 1683–1689 (2012).
[CrossRef] [PubMed]

Marty, R.

M. Bosman, E. Ye, S. F. Tan, C. A. Nijhuis, J. K. W. Yang, R. Marty, A. Mlayah, A. Arbouet, C. Girard, M. Y. Han, “Surface Plasmon Damping Quantified with an Electron Nanoprobe,” Sci. Rep. 3, 1312 (2013).
[CrossRef] [PubMed]

Mlayah, A.

M. Bosman, E. Ye, S. F. Tan, C. A. Nijhuis, J. K. W. Yang, R. Marty, A. Mlayah, A. Arbouet, C. Girard, M. Y. Han, “Surface Plasmon Damping Quantified with an Electron Nanoprobe,” Sci. Rep. 3, 1312 (2013).
[CrossRef] [PubMed]

Nijhuis, C. A.

M. Bosman, E. Ye, S. F. Tan, C. A. Nijhuis, J. K. W. Yang, R. Marty, A. Mlayah, A. Arbouet, C. Girard, M. Y. Han, “Surface Plasmon Damping Quantified with an Electron Nanoprobe,” Sci. Rep. 3, 1312 (2013).
[CrossRef] [PubMed]

Park, Y. S.

E. Cubukcu, S. Zhang, Y. S. Park, G. Bartal, X. Zhang, “Split ring resonator sensors for infrared detection of single molecular monolayers,” Appl. Phys. Lett. 95(4), 043113 (2009).
[CrossRef]

Pendry, J. B.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory 47(11), 2075–2084 (1999).
[CrossRef]

Pors, A.

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory 47(11), 2075–2084 (1999).
[CrossRef]

Salandrino, A.

N. Engheta, A. Salandrino, A. Alù, “Circuit Elements at Optical Frequencies: Nanoinductors, Nanocapacitors, and Nanoresistors,” Phys. Rev. Lett. 95(9), 095504 (2005).
[CrossRef] [PubMed]

Schultz, S.

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

Shamonina, E.

O. Sydoruk, E. Tatartschuk, E. Shamonina, L. Solymar, “Analytical formulation for the resonant frequency of split rings,” J. Appl. Phys. 105(1), 014903 (2009).
[CrossRef]

Shelby, R. A.

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

Shen, Y. R.

F. Wang, Y. R. Shen, “General Properties of Local Plasmons in Metal Nanostructures,” Phys. Rev. Lett. 97(20), 206806 (2006).
[CrossRef] [PubMed]

Silveirinha, M. G.

M. G. Silveirinha, A. Alù, J. Li, N. Engheta, “Nanoinsulators and nanoconnectors for optical nanocircuits,” J. Appl. Phys. 103(6), 064305 (2008).
[CrossRef]

Smith, D. R.

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

Solymar, L.

O. Sydoruk, E. Tatartschuk, E. Shamonina, L. Solymar, “Analytical formulation for the resonant frequency of split rings,” J. Appl. Phys. 105(1), 014903 (2009).
[CrossRef]

Staffaroni, M.

M. Staffaroni, J. Conway, S. Vedantam, J. Tang, E. Yablonovitch, “Circuit analysis in metal-optics,” Photon. Nanostructures 10(1), 166–176 (2012).
[CrossRef]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory 47(11), 2075–2084 (1999).
[CrossRef]

Sun, Y.

Y. Sun, B. Edwards, A. Alù, N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11(3), 208–212 (2012).
[CrossRef] [PubMed]

Sydoruk, O.

O. Sydoruk, E. Tatartschuk, E. Shamonina, L. Solymar, “Analytical formulation for the resonant frequency of split rings,” J. Appl. Phys. 105(1), 014903 (2009).
[CrossRef]

Tan, S. F.

M. Bosman, E. Ye, S. F. Tan, C. A. Nijhuis, J. K. W. Yang, R. Marty, A. Mlayah, A. Arbouet, C. Girard, M. Y. Han, “Surface Plasmon Damping Quantified with an Electron Nanoprobe,” Sci. Rep. 3, 1312 (2013).
[CrossRef] [PubMed]

Tang, J.

M. Staffaroni, J. Conway, S. Vedantam, J. Tang, E. Yablonovitch, “Circuit analysis in metal-optics,” Photon. Nanostructures 10(1), 166–176 (2012).
[CrossRef]

Tatartschuk, E.

O. Sydoruk, E. Tatartschuk, E. Shamonina, L. Solymar, “Analytical formulation for the resonant frequency of split rings,” J. Appl. Phys. 105(1), 014903 (2009).
[CrossRef]

Vedantam, S.

M. Staffaroni, J. Conway, S. Vedantam, J. Tang, E. Yablonovitch, “Circuit analysis in metal-optics,” Photon. Nanostructures 10(1), 166–176 (2012).
[CrossRef]

Wang, F.

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Appl. Phys. B (1)

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[CrossRef]

Appl. Phys. Lett. (1)

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Figures (5)

Fig. 1
Fig. 1

The process of forming a lumped circuit model for a metal nanorod. (a) The total displacement current density, −(D), can be separated into two parts: (b) the conduction current density, (J), due to free electrons, and (c) free-space displacement current density, −(D)0, that stems from the surface charges accumulated on the nanorod [Fields in (c) were scaled to accentuate the free-space displacement current]. The electron kinetic energy and Joule heating caused by the conduction current in (b) are modeled using an inductor and a resistor in series; while the electric potential energy stored in the displacement current (c) is modeled using a capacitor. (d) Schematic of a “unit-cell” circuit. (e) Overall RLC lumped circuit formed by defining the lumped current as the net current in each branch in (d) [R′, L′, and C′ are per-unit values, while R, L and C are lumped values.]

Fig. 2
Fig. 2

Comparison between FDTD simulation and RLC circuit model. (a) Geometry of the simulated nanorod. (b) Plot of the nomarlized extinction cross-section vs. resonance energy for two different nanorods. The circuit model can reproduce the spectra by considering the power dissipation in the circuit. Solid lines show FDTD simulation results for nanorods with 100 nm (blue) and 260 nm (black). Dashed lines are the reproduced extinction spectra using the circuit model. (c) Resonance enrgy (Eres) and (d) Q-factor calculated from RLC circuit model (black solid line) well matches FDTD simulation results (red triangles).

Fig. 3
Fig. 3

Using simple formulas to estimate the circuit parameters. (a) Current inside the nanorod forms a standing wave for the fundamental mode. (b) Ohmic resistance, (c) kinetic inductance, (d) Faraday inductance, and (e) resonance energy calculated from full numeric calculation (black solid lines) and simple analytical formula (red dashed lines) are with good agreement.

Fig. 4
Fig. 4

Extending the circuit model from a nanorod to an SRR. (a) Constructing a circuit model for SRRs from nanorods with equivelent lengths. (b) The total capacitance for SRRs calculated from FDTD simulation (black triangles) is approximately equal to the sum (black dot-dash lines) of the self-capacitance of a nanorod calculated from FDTD simulation (red dashed line) and the gap capacitance estimated using Eq. (27) (blue dotted line).

Fig. 5
Fig. 5

Comparison of the circuit parameters between SRRs and nanorods with equivalent lengths. (a) Kinetic inductance and (c) ohmic resistance are comparable for the SRRs and nanorods; while (b) Faraday inductance and (d) radiative resistance are geometry dependent, and their values reduce significantly after bending. (e) Smaller gaps increase gap capacitance, and the resoannce red shifts accordingly. (f) According to circuit theory, the Q-factor is calculated as R−1(L/C)1/2. Though SRRs have larger total capacitance, their radiative resistance is significantly suppressed, giving an even higher Q-factor compared to nanorods. [Black solid lines and dots are for SRRs; red dashed lines and squares are for nanorods; lines are from circuit model, and symbols are from FDTD simulations.]

Equations (34)

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i ω D = i ω ε E = i ω ( ε ε 0 ) E i ω ε 0 E = J i ω D 0
{ D 0 =ρ μ 0 H=0 ×H=iω D 0 +J ×E=iω μ 0 H with { ρ=(ε ε 0 )E J=iω(ε ε 0 )E=σE
I 0 = US iωDndA = V (iωD) dv=0
I 1 = US JndA = V J dv=iω V ρ dv=iωQ
I 2 = US (iω D 0 )ndA = V (iω D 0 )dv =iω V ρ dv=iωQ
1 2 V J Edv = 1 2 Vmetal 1 σ | J | 2 dv = 1 2 V (×H+iω D 0 ) Edv = 1 2 ( V Sdv + 1 iω ε 0 V | iω D 0 | 2 dv+iω V μ 0 | H | 2 dv )
1 2 1 iω ε 0 V | iω D 0 | 2 dv P E + 1 2 iIm( 1 σ ) Vmetal | J | 2 dv P K 1 2 iω μ 0 V | H | 2 dv P M =0
1 2 V (S) in dv P in = 1 2 Re( 1 σ ) Vmetal | J | 2 dv P ohmic + 1 2 V (S) out dv P rad
R ohmic =2 P ohmic / | I | 2 =Re[ 1 σ ] Vmetal | J | 2 dv / | I | 2
R rad =2 P out / | I | 2 = V (S) out dv / | I | 2
C=i | I | 2 /(2ω P E )=|Q | 2 /( ε 0 V | E | 2 dv)
L K =2 P K /(iω | I | 2 )= 1 ω Im[ 1 σ ] Vmetal | J | 2 dv / | I | 2
L F =2 P M /(iω | I | 2 )= V μ 0 | H | 2 dv/ | I | 2
E int =3 ε 0 E 0 /(ε+2 ε 0 ) E ext = E 0 + E dip = E 0 +[3u(pu)p]/(4π ε 0 r 3 )
E i = E int E 0 =( ε 0 ε) E 0 /(ε+2 ε 0 ) E out = E dip
I= US JndA = US σ E i ndA= σ E i π a 2 , Q=I/(iω)
R ohmic =Re[ 1 σ ] Vmetal | J | 2 dv / | I | 2 =Re[ 1 σ ] 4 3 π a 3 | σ E i | 2 | σ E i π a 2 | 2 = 4 3πa Re[ 1 σ ]
L K = 1 ω Im[ 1 σ ] Vmetal | J | 2 dv / | I | 2 = 1 ω Im[ 1 σ ] 4 3 π a 3 | σ E i | 2 | σ E i π a 2 | 2 = 4 3ωπa Im[ 1 σ ]
C=|Q | 2 /( ε 0 V | E | 2 dv)=|Q | 2 /( ε 0 Vmetal | E i | 2 dv+ ε 0 VVmetal | E out | 2 dv)
| E out | 2 = | E dip | 2 = (3|p|cosθ|p|cosθ) 2 + (|p|sinθ) 2 (4π ε 0 r 3 ) 2 = |p | 2 +3|p | 2 cos 2 θ 16 π 2 ε 0 2 r 6
ε 0 VVmetal | E out | 2 dv= ε 0 a 0 π 0 2π |p | 2 +3|p | 2 cos 2 θ 16 π 2 ε 0 2 r 6 sinθ r 2 drdθdφ = |p | 2 6π a 3 ε 0 = |4π ε 0 a 3 E i | 2 6π a 3 ε 0 = 8 3 ε 0 π a 3 | E i | 2
C=|Q | 2 /( ε 0 Vmetal | E i | 2 dv+ ε 0 VVmetal | E out | 2 dv) = | σ E i π a 2 /ω | 2 /( 4 3 π a 3 ε 0 | E i | 2 + 8 3 ε 0 π a 3 | E i | 2 ) = πa 4 ω 2 ε 0 |σ | 2
R ohmic = 4 3πa Re[ 1 σ ], L K = 4 3ωπa Im[ 1 σ ], and C= πa 4 ω 2 ε 0 |σ | 2
R sph = (πωaIm[ε]) 1 , L sph = ( ω 2 πaRe[ε]) 1 , and C fringe =2πa ε 0
L K C= 4 3ωπa Im[ 1 σ ] πa 4 ω 2 ε 0 |σ | 2 = 1 3 ω 3 ε 0 Im[ σ * ] = 1 3 ω 3 ε 0 Im[ iω( ε +i ε ε 0 ) ]= ε ε 0 3 ω 2 ε 0
Q= R ωL = Re[ε] Im[ε] ,
Q=ω L R = Im[1/σ] Re[1/σ]
ε= ε 0 (1 ω P 2 /( ω 2 +iγω)) = ε 0 ε 0 ω P 2 /( ω 2 + γ 2 )+iγ ε 0 ω P 2 /(ω( ω 2 + γ 2 )),
1/σ=1/(iω(ε ε 0 ))=iω/( ε 0 ω P 2 )+γ/( ε 0 ω P 2 )
Q=Re[ε]/Im[ε]= ω γ (1 ω 2 ω P 2 ) γω ω P 2
Q=Im[1/σ]/Re[1/σ]=ω/γ
Q= ωd ε /dω 2 ε = ω γ+ γ 3 / ω 2
Z rod = ( 1 / σ ) l eff / A = Re [ 1 / σ ] l eff / A + i Im [ 1 / σ ] l eff / A = R i ω L
C gap = ε 0 h w g + ε 0 ( h + g + w ) + 2 ε 0 ( h + w ) π log 8 l π g

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