Abstract

Nanoscale slot waveguides of hyperbolic metamaterials are proposed and demonstrated for achieving large optical field enhancement. The dependence of the enhanced electric field within the air slot on waveguide mode coupling and permittivity tensors of hyperbolic metamaterials is analyzed both numerically and analytically. Optical intensity in the metamaterial slot waveguide can be more than 25 times stronger than that in a conventional silicon slot waveguide, due to tight optical mode confinement enabled by the ultrahigh refractive indices supported in hyperbolic metamaterials. The electric field enhancement effects are also verified with the realistic metal-dielectric multilayer waveguide structure.

© 2012 Optical Society of America

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M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, Nature 470, 369 (2011).
[CrossRef]

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, Proc. Natl. Acad. Sci. 108, 11327 (2011).
[CrossRef]

2010

2009

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, Nature 457, 71 (2009).
[CrossRef]

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, Nat. Photon. 3, 216 (2009).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, Nat. Photon. 3, 464 (2009).
[CrossRef]

Q. Quan, I. Bulu, and M. Loncar, Phys. Rev. A 80, 011810 (2009).
[CrossRef]

2008

Y. J. Huang, W. T. Lu, and S. Sridhar, Phys. Rev. A 77, 063836 (2008).
[CrossRef]

2007

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef]

2006

2005

J. T. Shen, P. B. Catrysse, and S. Fan, Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef]

2004

2003

D. R. Smith and D. Schurig, Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

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C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, J. Phys. Chem. 98, 2963 (1994).
[CrossRef]

Alekseyev, L. V.

Almeida, V. R.

Baets, R.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, Nat. Photon. 3, 216 (2009).
[CrossRef]

Barrios, C. A.

Bartal, G.

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, Proc. Natl. Acad. Sci. 108, 11327 (2011).
[CrossRef]

Biaggio, I.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, Nat. Photon. 3, 216 (2009).
[CrossRef]

Bogaerts, W.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, Nat. Photon. 3, 216 (2009).
[CrossRef]

Bulu, I.

Q. Quan, I. Bulu, and M. Loncar, Phys. Rev. A 80, 011810 (2009).
[CrossRef]

Catrysse, P. B.

J. T. Shen, P. B. Catrysse, and S. Fan, Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef]

Chen, W.

Choi, M.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, Nature 470, 369 (2011).
[CrossRef]

Diederich, F.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, Nat. Photon. 3, 216 (2009).
[CrossRef]

Dumon, P.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, Nat. Photon. 3, 216 (2009).
[CrossRef]

Erickson, D.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, Nature 457, 71 (2009).
[CrossRef]

Esembeson, B.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, Nat. Photon. 3, 216 (2009).
[CrossRef]

Fan, S.

J. T. Shen, P. B. Catrysse, and S. Fan, Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef]

Foss, C. A.

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, J. Phys. Chem. 98, 2963 (1994).
[CrossRef]

Freude, W.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, Nat. Photon. 3, 216 (2009).
[CrossRef]

Hornyak, G. L.

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, J. Phys. Chem. 98, 2963 (1994).
[CrossRef]

Huang, Y. J.

Y. J. Huang, W. T. Lu, and S. Sridhar, Phys. Rev. A 77, 063836 (2008).
[CrossRef]

Ishii, S.

Kang, K.-Y.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, Nature 470, 369 (2011).
[CrossRef]

Kang, S. B.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, Nature 470, 369 (2011).
[CrossRef]

Kildishev, A. V.

Kim, Y.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, Nature 470, 369 (2011).
[CrossRef]

Klug, M.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, Nature 457, 71 (2009).
[CrossRef]

Koos, C.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, Nat. Photon. 3, 216 (2009).
[CrossRef]

Kwak, M. H.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, Nature 470, 369 (2011).
[CrossRef]

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef]

Lee, S. H.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, Nature 470, 369 (2011).
[CrossRef]

Lee, Y.-H.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, Nature 470, 369 (2011).
[CrossRef]

Leuthold, J.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, Nat. Photon. 3, 216 (2009).
[CrossRef]

Li, M.

M. Li, W. H. P. Pernice, and H. X. Tang, Nat. Photon. 3, 464 (2009).
[CrossRef]

Lipson, M.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, Nature 457, 71 (2009).
[CrossRef]

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, Opt. Lett. 29, 1209 (2004).
[CrossRef]

Liu, Z.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef]

Loncar, M.

Q. Quan, I. Bulu, and M. Loncar, Phys. Rev. A 80, 011810 (2009).
[CrossRef]

Lu, W. T.

Y. J. Huang, W. T. Lu, and S. Sridhar, Phys. Rev. A 77, 063836 (2008).
[CrossRef]

Martin, C. R.

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, J. Phys. Chem. 98, 2963 (1994).
[CrossRef]

Michinobu, T.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, Nat. Photon. 3, 216 (2009).
[CrossRef]

Min, B.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, Nature 470, 369 (2011).
[CrossRef]

Moore, S. D.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, Nature 457, 71 (2009).
[CrossRef]

Narimanov, E.

Park, N.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, Nature 470, 369 (2011).
[CrossRef]

Pernice, W. H. P.

M. Li, W. H. P. Pernice, and H. X. Tang, Nat. Photon. 3, 464 (2009).
[CrossRef]

Quan, Q.

Q. Quan, I. Bulu, and M. Loncar, Phys. Rev. A 80, 011810 (2009).
[CrossRef]

Schmidt, B. S.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, Nature 457, 71 (2009).
[CrossRef]

Schurig, D.

D. R. Smith and D. Schurig, Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

Shalaev, V. M.

Shen, J. T.

J. T. Shen, P. B. Catrysse, and S. Fan, Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef]

Shin, J.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, Nature 470, 369 (2011).
[CrossRef]

Smith, D. R.

D. R. Smith and D. Schurig, Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

Sridhar, S.

Y. J. Huang, W. T. Lu, and S. Sridhar, Phys. Rev. A 77, 063836 (2008).
[CrossRef]

Stockert, J. A.

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, J. Phys. Chem. 98, 2963 (1994).
[CrossRef]

Sun, C.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef]

Tang, H. X.

M. Li, W. H. P. Pernice, and H. X. Tang, Nat. Photon. 3, 464 (2009).
[CrossRef]

Thoreson, M. D.

Vallaitis, T.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, Nat. Photon. 3, 216 (2009).
[CrossRef]

Vorreau, P.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, Nat. Photon. 3, 216 (2009).
[CrossRef]

Xiong, Y.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef]

Xu, Q.

Yang, A. H. J.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, Nature 457, 71 (2009).
[CrossRef]

Yang, X.

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, Proc. Natl. Acad. Sci. 108, 11327 (2011).
[CrossRef]

Yao, J.

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, Proc. Natl. Acad. Sci. 108, 11327 (2011).
[CrossRef]

Yin, X.

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, Proc. Natl. Acad. Sci. 108, 11327 (2011).
[CrossRef]

Zhang, X.

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, Proc. Natl. Acad. Sci. 108, 11327 (2011).
[CrossRef]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef]

J. Phys. Chem.

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, J. Phys. Chem. 98, 2963 (1994).
[CrossRef]

Nat. Photon.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, Nat. Photon. 3, 216 (2009).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, Nat. Photon. 3, 464 (2009).
[CrossRef]

Nature

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, Nature 457, 71 (2009).
[CrossRef]

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, Nature 470, 369 (2011).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

Y. J. Huang, W. T. Lu, and S. Sridhar, Phys. Rev. A 77, 063836 (2008).
[CrossRef]

Q. Quan, I. Bulu, and M. Loncar, Phys. Rev. A 80, 011810 (2009).
[CrossRef]

Phys. Rev. Lett.

J. T. Shen, P. B. Catrysse, and S. Fan, Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef]

D. R. Smith and D. Schurig, Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

Proc. Natl. Acad. Sci.

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, Proc. Natl. Acad. Sci. 108, 11327 (2011).
[CrossRef]

Science

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Schematic of the metamaterial slot waveguides. (a) Three-dimensional (3D) silver-germanium multilayer structures, and (b) approximated two-dimensional (2D) slab structures for theoretical analysis.

Fig. 2.
Fig. 2.

(a) Mode profiles of the metamaterial slot waveguide with fm=0.4 and g=10nm at λ0=1.55μm. (b) 3D surface plots of the Ey field distributions [normalized to Ey(x=0,|y|=|h+g/2|+)] for slot waveguides with g=10nm and g=5nm, respectively.

Fig. 3.
Fig. 3.

(a) The real part of the effective refractive indices real(neff,z), (b) the imaginary part of the effective refractive indices and imag(neff,z), (c) electric field enhancement factor η, (d) optical power flow in the slot region Pslot, and (e) optical intensity in the slot region Islot as a function of g for fm=0.4 and fm=0.7. Results from both FEM simulation and theoretical analysis are plotted.

Fig. 4.
Fig. 4.

Comparison between the Ey field profiles along x=0 for slot waveguides with (a) fm=0.4 and (b) fm=0.7, calculated from the EMT and realistic metal-dielectric multilayer structures (layer).

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

εx=εz=fmεm+(1fm)εd,εy=εmεdfmεd+(1fm)εm,
Ey=E0{cos(kyh2+φ)cosh(γy)cosh(γg2),0<|y|<g2,1εycos[ky(|y|h+g2)+φ],g2<|y|<g2+h,cos(kyh2+φ)exp[γ(|y|g2h)],|y|>g2+h,
β2εy+ky2εz=k02,β2γ2=k02,
tan(kyh2+φ)=γεzkytanh(γg2),tan(kyh2+φ)=γεzky.
η1+εy|εz|1+εy|εz|tanh2(γg2)
Pslotrg1+rg,Islot=Pslotwg1wgrg1+rg,rgεyhg2[1+sinh(γg)γg]1+εy|εz|sinh2(γg2),

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