Abstract

The integration of optical metamaterials within silicon integrated photonic circuitry bears significantly potential in the design of low-power, nanoscale footprint, all-optical functionalities. We propose a novel concept and provide detailed analysis of an on-chip ultrafast all-optical modulator based on an hyperbolic metamaterial integrated in a silicon waveguide. The anisotropic metamaterial based on gold nanorods is placed on top of the silicon waveguide to form a modulator with a 300x440x600 nm3 footprint. For the operating wavelength of 1.5 μm, the optimized geometry of the device has insertion loss of about 5 dB and a modulation depth of 35% with a sub-ps switching rate. The switching energy estimated from nonlinear transient dynamic numerical simulations is 3.7 pJ/bit when the transmission is controlled optically at a wavelength of 532 nm, resonant with the transverse plasmonic mode of the metamaterial. The switching mechanism is based on the control of the hybridization of eigenmodes in the metamaterial slab and the Si waveguide.

© 2014 Optical Society of America

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2013 (1)

A. Poddubny, I. Iorsh, P. Belov, Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[CrossRef]

2012 (1)

A. V. Krasavin, A. V. Zayats, “Photonic signal processing on electronic scales: electro-optical field-effect nanoplasmonic modulator,” Phys. Rev. Lett. 109(5), 053901 (2012).
[CrossRef] [PubMed]

2011 (3)

A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, A. V. Zayats, “All-Plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett. 11(6), 2231–2235 (2011).
[CrossRef] [PubMed]

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6(2), 107–111 (2011).
[CrossRef] [PubMed]

L. Alloatti, D. Korn, R. Palmer, D. Hillerkuss, J. Li, A. Barklund, R. Dinu, J. Wieland, M. Fournier, J. Fedeli, H. Yu, W. Bogaerts, P. Dumon, R. Baets, C. Koos, W. Freude, J. Leuthold, “42.7 Gbit/s electro-optic modulator in silicon technology,” Opt. Express 19(12), 11841–11851 (2011).
[CrossRef] [PubMed]

2009 (1)

D. A. B. Miller, “Device requirements for optical interconnects to silicon chips,” Proc. IEEE 97(7), 1166–1185 (2009).
[CrossRef]

2007 (1)

2006 (2)

J. Elser, R. Wangberg, V. A. Podolskiy, E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89(26), 261102 (2006).
[CrossRef]

M. Haurylau, G. Chen, H. Chen, J. Zhang, N. A. Nelson, D. H. Albonesi, E. G. Friedman, P. M. Fauchet, “On-chip optical interconnect roadmap: challenges and critical directions,” IEEE J. Sel. Top. Quant. 12(6), 1699–1705 (2006).
[CrossRef]

2005 (3)

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, M. S. Sherwin, “quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[CrossRef] [PubMed]

Q. Xu, B. Schmidt, S. Pradhan, M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

L. Jiang, H.-L. Tsai, “Improved two-temperature model and its application in ultrashort laser heating of metal films,” J. Heat Transfer 127(10), 1167–1173 (2005).
[CrossRef]

2004 (1)

S. Kodama, T. Yoshimatsu, H. Ito, “500 Gbit/s optical gate monolithically integrating photodiode and electroabsorption modulator,” Electron. Lett. 40(9), 555–556 (2004).
[CrossRef]

2000 (1)

J. Hohlfeld, S. S. Wellershoff, J. Güdde, U. Conrad, V. Jähnke, E. Matthias, “Electron and lattice dynamics following optical excitation of metals,” J. Chem. Phys. 251, 237–258 (2000).

1997 (1)

D. A. B. Miller, H. M. Ozaktas, “Limit to the bit-rate capacity of electrical interconnects from the aspect ratio of the system architecture,” J. Parallel Distrib. Comput. 41(1), 42–52 (1997).
[CrossRef]

1995 (1)

J. Y. Bigot, J. Y. Merle, O. Cregut, A. Daunois, “Electron dynamics in copper metallic nanoparticles probed with femtosecond optical pulses,” Phys. Rev. Lett. 75(25), 4702–4705 (1995).
[CrossRef] [PubMed]

1983 (1)

Albonesi, D. H.

M. Haurylau, G. Chen, H. Chen, J. Zhang, N. A. Nelson, D. H. Albonesi, E. G. Friedman, P. M. Fauchet, “On-chip optical interconnect roadmap: challenges and critical directions,” IEEE J. Sel. Top. Quant. 12(6), 1699–1705 (2006).
[CrossRef]

Alloatti, L.

Baets, R.

Barklund, A.

Belov, P.

A. Poddubny, I. Iorsh, P. Belov, Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[CrossRef]

Bigot, J. Y.

J. Y. Bigot, J. Y. Merle, O. Cregut, A. Daunois, “Electron dynamics in copper metallic nanoparticles probed with femtosecond optical pulses,” Phys. Rev. Lett. 75(25), 4702–4705 (1995).
[CrossRef] [PubMed]

Birkedal, V.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, M. S. Sherwin, “quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[CrossRef] [PubMed]

Bogaerts, W.

Bolger, P. M.

A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, A. V. Zayats, “All-Plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett. 11(6), 2231–2235 (2011).
[CrossRef] [PubMed]

Carter, S. G.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, M. S. Sherwin, “quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[CrossRef] [PubMed]

Chen, G.

M. Haurylau, G. Chen, H. Chen, J. Zhang, N. A. Nelson, D. H. Albonesi, E. G. Friedman, P. M. Fauchet, “On-chip optical interconnect roadmap: challenges and critical directions,” IEEE J. Sel. Top. Quant. 12(6), 1699–1705 (2006).
[CrossRef]

Chen, H.

M. Haurylau, G. Chen, H. Chen, J. Zhang, N. A. Nelson, D. H. Albonesi, E. G. Friedman, P. M. Fauchet, “On-chip optical interconnect roadmap: challenges and critical directions,” IEEE J. Sel. Top. Quant. 12(6), 1699–1705 (2006).
[CrossRef]

Citrin, D. S.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, M. S. Sherwin, “quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[CrossRef] [PubMed]

Coldren, L. A.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, M. S. Sherwin, “quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[CrossRef] [PubMed]

Conrad, U.

J. Hohlfeld, S. S. Wellershoff, J. Güdde, U. Conrad, V. Jähnke, E. Matthias, “Electron and lattice dynamics following optical excitation of metals,” J. Chem. Phys. 251, 237–258 (2000).

Cregut, O.

J. Y. Bigot, J. Y. Merle, O. Cregut, A. Daunois, “Electron dynamics in copper metallic nanoparticles probed with femtosecond optical pulses,” Phys. Rev. Lett. 75(25), 4702–4705 (1995).
[CrossRef] [PubMed]

Daunois, A.

J. Y. Bigot, J. Y. Merle, O. Cregut, A. Daunois, “Electron dynamics in copper metallic nanoparticles probed with femtosecond optical pulses,” Phys. Rev. Lett. 75(25), 4702–4705 (1995).
[CrossRef] [PubMed]

Dickson, W.

A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, A. V. Zayats, “All-Plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett. 11(6), 2231–2235 (2011).
[CrossRef] [PubMed]

Dinu, R.

Dumon, P.

Elser, J.

J. Elser, R. Wangberg, V. A. Podolskiy, E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89(26), 261102 (2006).
[CrossRef]

Fauchet, P. M.

M. Haurylau, G. Chen, H. Chen, J. Zhang, N. A. Nelson, D. H. Albonesi, E. G. Friedman, P. M. Fauchet, “On-chip optical interconnect roadmap: challenges and critical directions,” IEEE J. Sel. Top. Quant. 12(6), 1699–1705 (2006).
[CrossRef]

Fedeli, J.

Feit, M. D.

Fleck, J. J. A.

Fournier, M.

Freude, W.

Friedman, E. G.

M. Haurylau, G. Chen, H. Chen, J. Zhang, N. A. Nelson, D. H. Albonesi, E. G. Friedman, P. M. Fauchet, “On-chip optical interconnect roadmap: challenges and critical directions,” IEEE J. Sel. Top. Quant. 12(6), 1699–1705 (2006).
[CrossRef]

Gosztola, D. J.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6(2), 107–111 (2011).
[CrossRef] [PubMed]

Güdde, J.

J. Hohlfeld, S. S. Wellershoff, J. Güdde, U. Conrad, V. Jähnke, E. Matthias, “Electron and lattice dynamics following optical excitation of metals,” J. Chem. Phys. 251, 237–258 (2000).

Haurylau, M.

M. Haurylau, G. Chen, H. Chen, J. Zhang, N. A. Nelson, D. H. Albonesi, E. G. Friedman, P. M. Fauchet, “On-chip optical interconnect roadmap: challenges and critical directions,” IEEE J. Sel. Top. Quant. 12(6), 1699–1705 (2006).
[CrossRef]

Hendren, W.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6(2), 107–111 (2011).
[CrossRef] [PubMed]

Hillerkuss, D.

Hohlfeld, J.

J. Hohlfeld, S. S. Wellershoff, J. Güdde, U. Conrad, V. Jähnke, E. Matthias, “Electron and lattice dynamics following optical excitation of metals,” J. Chem. Phys. 251, 237–258 (2000).

Iorsh, I.

A. Poddubny, I. Iorsh, P. Belov, Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[CrossRef]

Ito, H.

S. Kodama, T. Yoshimatsu, H. Ito, “500 Gbit/s optical gate monolithically integrating photodiode and electroabsorption modulator,” Electron. Lett. 40(9), 555–556 (2004).
[CrossRef]

Jähnke, V.

J. Hohlfeld, S. S. Wellershoff, J. Güdde, U. Conrad, V. Jähnke, E. Matthias, “Electron and lattice dynamics following optical excitation of metals,” J. Chem. Phys. 251, 237–258 (2000).

Jiang, L.

L. Jiang, H.-L. Tsai, “Improved two-temperature model and its application in ultrashort laser heating of metal films,” J. Heat Transfer 127(10), 1167–1173 (2005).
[CrossRef]

Kivshar, Y.

A. Poddubny, I. Iorsh, P. Belov, Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[CrossRef]

Kodama, S.

S. Kodama, T. Yoshimatsu, H. Ito, “500 Gbit/s optical gate monolithically integrating photodiode and electroabsorption modulator,” Electron. Lett. 40(9), 555–556 (2004).
[CrossRef]

Koos, C.

Korn, D.

Krasavin, A. V.

A. V. Krasavin, A. V. Zayats, “Photonic signal processing on electronic scales: electro-optical field-effect nanoplasmonic modulator,” Phys. Rev. Lett. 109(5), 053901 (2012).
[CrossRef] [PubMed]

A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, A. V. Zayats, “All-Plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett. 11(6), 2231–2235 (2011).
[CrossRef] [PubMed]

Leuthold, J.

Li, J.

Lipson, M.

Manipatruni, S.

Maslov, A. V.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, M. S. Sherwin, “quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[CrossRef] [PubMed]

Matthias, E.

J. Hohlfeld, S. S. Wellershoff, J. Güdde, U. Conrad, V. Jähnke, E. Matthias, “Electron and lattice dynamics following optical excitation of metals,” J. Chem. Phys. 251, 237–258 (2000).

Merle, J. Y.

J. Y. Bigot, J. Y. Merle, O. Cregut, A. Daunois, “Electron dynamics in copper metallic nanoparticles probed with femtosecond optical pulses,” Phys. Rev. Lett. 75(25), 4702–4705 (1995).
[CrossRef] [PubMed]

Miller, D. A. B.

D. A. B. Miller, “Device requirements for optical interconnects to silicon chips,” Proc. IEEE 97(7), 1166–1185 (2009).
[CrossRef]

D. A. B. Miller, H. M. Ozaktas, “Limit to the bit-rate capacity of electrical interconnects from the aspect ratio of the system architecture,” J. Parallel Distrib. Comput. 41(1), 42–52 (1997).
[CrossRef]

Narimanov, E. E.

J. Elser, R. Wangberg, V. A. Podolskiy, E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89(26), 261102 (2006).
[CrossRef]

Nelson, N. A.

M. Haurylau, G. Chen, H. Chen, J. Zhang, N. A. Nelson, D. H. Albonesi, E. G. Friedman, P. M. Fauchet, “On-chip optical interconnect roadmap: challenges and critical directions,” IEEE J. Sel. Top. Quant. 12(6), 1699–1705 (2006).
[CrossRef]

Ozaktas, H. M.

D. A. B. Miller, H. M. Ozaktas, “Limit to the bit-rate capacity of electrical interconnects from the aspect ratio of the system architecture,” J. Parallel Distrib. Comput. 41(1), 42–52 (1997).
[CrossRef]

Palmer, R.

Poddubny, A.

A. Poddubny, I. Iorsh, P. Belov, Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[CrossRef]

Podolskiy, V. A.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6(2), 107–111 (2011).
[CrossRef] [PubMed]

J. Elser, R. Wangberg, V. A. Podolskiy, E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89(26), 261102 (2006).
[CrossRef]

Pollard, R.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6(2), 107–111 (2011).
[CrossRef] [PubMed]

Pradhan, S.

Q. Xu, B. Schmidt, S. Pradhan, M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Schmidt, B.

Shakya, J.

Sherwin, M. S.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, M. S. Sherwin, “quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[CrossRef] [PubMed]

Tsai, H.-L.

L. Jiang, H.-L. Tsai, “Improved two-temperature model and its application in ultrashort laser heating of metal films,” J. Heat Transfer 127(10), 1167–1173 (2005).
[CrossRef]

Vo, T. P.

A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, A. V. Zayats, “All-Plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett. 11(6), 2231–2235 (2011).
[CrossRef] [PubMed]

Wang, C. S.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, M. S. Sherwin, “quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[CrossRef] [PubMed]

Wangberg, R.

J. Elser, R. Wangberg, V. A. Podolskiy, E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89(26), 261102 (2006).
[CrossRef]

Wellershoff, S. S.

J. Hohlfeld, S. S. Wellershoff, J. Güdde, U. Conrad, V. Jähnke, E. Matthias, “Electron and lattice dynamics following optical excitation of metals,” J. Chem. Phys. 251, 237–258 (2000).

Wiederrecht, G. P.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6(2), 107–111 (2011).
[CrossRef] [PubMed]

Wieland, J.

Wurtz, G. A.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6(2), 107–111 (2011).
[CrossRef] [PubMed]

Xu, Q.

Yoshimatsu, T.

S. Kodama, T. Yoshimatsu, H. Ito, “500 Gbit/s optical gate monolithically integrating photodiode and electroabsorption modulator,” Electron. Lett. 40(9), 555–556 (2004).
[CrossRef]

Yu, H.

Zayats, A. V.

A. V. Krasavin, A. V. Zayats, “Photonic signal processing on electronic scales: electro-optical field-effect nanoplasmonic modulator,” Phys. Rev. Lett. 109(5), 053901 (2012).
[CrossRef] [PubMed]

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6(2), 107–111 (2011).
[CrossRef] [PubMed]

A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, A. V. Zayats, “All-Plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett. 11(6), 2231–2235 (2011).
[CrossRef] [PubMed]

Zhang, J.

M. Haurylau, G. Chen, H. Chen, J. Zhang, N. A. Nelson, D. H. Albonesi, E. G. Friedman, P. M. Fauchet, “On-chip optical interconnect roadmap: challenges and critical directions,” IEEE J. Sel. Top. Quant. 12(6), 1699–1705 (2006).
[CrossRef]

Appl. Phys. Lett. (1)

J. Elser, R. Wangberg, V. A. Podolskiy, E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89(26), 261102 (2006).
[CrossRef]

Electron. Lett. (1)

S. Kodama, T. Yoshimatsu, H. Ito, “500 Gbit/s optical gate monolithically integrating photodiode and electroabsorption modulator,” Electron. Lett. 40(9), 555–556 (2004).
[CrossRef]

IEEE J. Sel. Top. Quant. (1)

M. Haurylau, G. Chen, H. Chen, J. Zhang, N. A. Nelson, D. H. Albonesi, E. G. Friedman, P. M. Fauchet, “On-chip optical interconnect roadmap: challenges and critical directions,” IEEE J. Sel. Top. Quant. 12(6), 1699–1705 (2006).
[CrossRef]

J. Chem. Phys. (1)

J. Hohlfeld, S. S. Wellershoff, J. Güdde, U. Conrad, V. Jähnke, E. Matthias, “Electron and lattice dynamics following optical excitation of metals,” J. Chem. Phys. 251, 237–258 (2000).

J. Heat Transfer (1)

L. Jiang, H.-L. Tsai, “Improved two-temperature model and its application in ultrashort laser heating of metal films,” J. Heat Transfer 127(10), 1167–1173 (2005).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Parallel Distrib. Comput. (1)

D. A. B. Miller, H. M. Ozaktas, “Limit to the bit-rate capacity of electrical interconnects from the aspect ratio of the system architecture,” J. Parallel Distrib. Comput. 41(1), 42–52 (1997).
[CrossRef]

Nano Lett. (1)

A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, A. V. Zayats, “All-Plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett. 11(6), 2231–2235 (2011).
[CrossRef] [PubMed]

Nat. Nanotechnol. (1)

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Figures (3)

Fig. 1
Fig. 1

(a) 3D view of the metamaterial optical modulator inserted in the Si waveguide. The silicon waveguide is 340 nm height and 300 nm width. The input mode profile shown corresponds to the simulated HE11x waveguided mode. Modulator mode profiles in (b) OFF and (c) ON states.

Fig. 2
Fig. 2

(a) Dependence of the transmission on the metamaterial parameters for the HE11,x mode through the interface between the Si waveguide and the modulator in the OFF state (electron temperature 300 K). Colour plots show modulator’s mode profiles on the first and second resonance (a). The rod diameter is 50 nm and the operating wavelength is 1500 nm. (b) Transmission variation Δ T between the ON state (electron temperature 3000 K) and the OFF state. The white cross indicates the parameters for which spectral dependence in Fig. 3 has been studied.

Fig. 3
Fig. 3

(a) Transmission spectra for the waveguide-modulator system in both ON and OFF states for the HE11,x mode simulated with both effective medium theory (EMT) and full finite-element model (FEM) of the nanorod metamaterial. The rod height and period are 100 nm, the length of the metamaterial waveguide is 600 nm, corresponding to the white cross in Fig. 2 (b). The mode profile in the waveguide-modulator-waveguide configuration in (b) ON and (c) OFF states.

Equations (5)

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k x 2 +k y 2 +k z 2 = ε x k 0 2
k x 2 +k y 2 + ε z ε x k z 2 = ε z k 0 2
T i j = | t i j t i j * | R e ( n j m o d ) n i w g t i j = 2 n i w g n j m o d + n i w g S e i w g ( x , y ) e i m o d ( x , y ) d S
ε= ε intra (ω, T L , T e )= ε ω p 2 ω(ω+i γ intra (ω, T L , T e ))
E= 1 2 C e T e 2 V mod

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