## Nanoscale dielectric-graphene-dielectric tunable infrared waveguide with ultrahigh refractive indices |

Optics Express, Vol. 21, Issue 14, pp. 17089-17096 (2013)

http://dx.doi.org/10.1364/OE.21.017089

Acrobat PDF (1062 KB)

### Abstract

We propose in this paper a dielectric-graphene-dielectric tunable infrared waveguide based on multilayer metamaterials with ultrahigh refractive indices. The waveguide modes with different orders are systematically analyzed with numerical simulations based on both multilayer structures and effective medium approach. The waveguide shows hyperbolic dispersion properties from mid-infrared to far-infrared wavelength, which means the modes with ultrahigh mode indices could be supported in the waveguide. Furthermore, the optical properties of the waveguide modes could be tuned by the biased voltages on graphene layers. The waveguide may have various promising applications in the quantum cascade lasers and bio-sensing.

© 2013 OSA

## 1. Introduction

1. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science **305**(5685), 788–792 (2004). [CrossRef] [PubMed]

2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**(5781), 1780–1782 (2006). [CrossRef] [PubMed]

3. Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express **16**(20), 15439–15448 (2008). [CrossRef] [PubMed]

4. Y. He, S. He, J. Gao, and X. Yang, “Nanoscale metamaterial optical waveguides with ultrahigh refractive indices,” J. Opt. Soc. Am. B **29**(9), 2559–2566 (2012). [CrossRef]

5. F. Y. Meng, Q. Wu, and L. W. Li, “Transmission characteristics of wave modes in a rectangular waveguide filled with anisotropic metamaterial,” Appl. Phys., A Mater. Sci. Process. **94**(4), 747–753 (2009). [CrossRef]

6. X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics **6**(7), 450–454 (2012). [CrossRef]

7. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science **306**(5696), 666–669 (2004). [CrossRef] [PubMed]

8. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature **438**(7065), 197–200 (2005). [CrossRef] [PubMed]

9. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics **6**(11), 749–758 (2012). [CrossRef]

*et al*. [10

10. F. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B **87**(7), 075416 (2013). [CrossRef]

*et al*. [11

11. M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express **21**(6), 7614–7632 (2013). [CrossRef] [PubMed]

11. M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express **21**(6), 7614–7632 (2013). [CrossRef] [PubMed]

*et al*. have shown a high-performance hyperbolic metamaterial formed by Al:ZnO and ZnO as the metallic and dielectric components [12

12. G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. **109**(23), 8834–8838 (2012). [CrossRef] [PubMed]

13. G. V. Naik and A. Boltasseva, “A comparative study of semiconductor-based plasmonic metamaterials,” Metamaterials (Amst.) **5**(1), 1–7 (2011). [CrossRef]

*et al*. have proposed a hyperbolic metamaterial made of alternating semiconductor and dielectric layers [14

14. C. Rizza, A. Ciattoni, E. Spinozzi, and L. Columbo, “Terahertz active spatial filtering through optically tunable hyperbolic metamaterials,” Opt. Lett. **37**(16), 3345–3347 (2012). [CrossRef] [PubMed]

*et al*. have investigated a terahertz metamaterial with unnaturally high refractive index [15

15. 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, “A terahertz metamaterial with unnaturally high refractive index,” Nature **470**(7334), 369–373 (2011). [CrossRef] [PubMed]

*et al.*have designed three-dimensional isotropic metamaterials which possess an enhanced refractive index between 5.5~7 in the wavelength range from 3 um~6 um [16

16. J. Shin, J. T. Shen, and S. H. Fan, “Three-Dimensional Metamaterials with an Ultrahigh Effective Refractive Index over a Broad Bandwidth,” Phys. Rev. Lett. **102**(9), 093903 (2009). [CrossRef] [PubMed]

4. Y. He, S. He, J. Gao, and X. Yang, “Nanoscale metamaterial optical waveguides with ultrahigh refractive indices,” J. Opt. Soc. Am. B **29**(9), 2559–2566 (2012). [CrossRef]

5. F. Y. Meng, Q. Wu, and L. W. Li, “Transmission characteristics of wave modes in a rectangular waveguide filled with anisotropic metamaterial,” Appl. Phys., A Mater. Sci. Process. **94**(4), 747–753 (2009). [CrossRef]

*et al.*have demonstrated that a hyperbolic metamaterial waveguide could achieve ultrahigh effective refractive index up to 62.0 at near-infrared region [4

4. Y. He, S. He, J. Gao, and X. Yang, “Nanoscale metamaterial optical waveguides with ultrahigh refractive indices,” J. Opt. Soc. Am. B **29**(9), 2559–2566 (2012). [CrossRef]

17. R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature **417**(6885), 156–159 (2002). [CrossRef] [PubMed]

19. M. Beck, D. Hofstetter, T. Aellen, J. Faist, U. Oesterle, M. Ilegems, E. Gini, and H. Melchior, “Continuous Wave Operation of a Mid-Infrared Semiconductor Laser at Room Temperature,” Science **295**(5553), 301–305 (2002). [CrossRef] [PubMed]

20. R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics **4**(8), 495–497 (2010). [CrossRef]

## 2. The hyperbolic dielectric-graphene-dielectric multilayer metamaterial

21. C. Xu, Y. Jin, L. Yang, J. Yang, and X. Jiang, “Characteristics of electro-refractive modulating based on Graphene-Oxide-Silicon waveguide,” Opt. Express **20**(20), 22398–22405 (2012). [CrossRef] [PubMed]

22. G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. **103**(6), 064302 (2008). [CrossRef]

*ω*is the angular frequency,

*τ*is the relaxation time which represents the loss mechanism,

*e*is the charge of the electron,

*k*is the reduced Planck's constant,

_{B}*T*is the temperature and

*μ*

_{c}is the chemical potential, which can be defined as [21

21. C. Xu, Y. Jin, L. Yang, J. Yang, and X. Jiang, “Characteristics of electro-refractive modulating based on Graphene-Oxide-Silicon waveguide,” Opt. Express **20**(20), 22398–22405 (2012). [CrossRef] [PubMed]

*a*≈9 × 10

_{0}^{16}m

^{−1}V

^{−1},

*V*and

_{Dirac}*ν*represent the voltage offset and the Fermi velocity of Dirac fermions in Graphene, respectively. Expression

_{F}*V*, which could modify the chemical potential. Moreover, the chemical potential could also be tuned by electric field, magnetic field and chemical doping [23

_{biased}23. A. Vakil and N. Engheta, “Transformation Optics Using Graphene,” Science **332**(6035), 1291–1294 (2011). [CrossRef] [PubMed]

*ρ*(

*E*) which represents the density of states per spin per unit cell, has been introduced in the deriving process of Eq. (1) and Eq. (2) [24

24. T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B **78**(8), 085432 (2008). [CrossRef]

25. S. H. Lee, M. Choi, T. T. Kim, S. Lee, M. Liu, X. Yin, H. K. Choi, S. S. Lee, C. G. Choi, S. Y. Choi, X. Zhang, and B. Min, “Switching terahertz waves with gate-controlled active graphene metamaterials,” Nat. Mater. **11**(11), 936–941 (2012). [CrossRef] [PubMed]

21. C. Xu, Y. Jin, L. Yang, J. Yang, and X. Jiang, “Characteristics of electro-refractive modulating based on Graphene-Oxide-Silicon waveguide,” Opt. Express **20**(20), 22398–22405 (2012). [CrossRef] [PubMed]

*λ*and the biased voltage

_{0}*V*

_{biased}_{}are plotted in Figs. 1(a) and 1(c), respectively. In Fig. 1(a), the voltages are

*V*= 1.6 volt and 2.5 volt with the corresponding chemical potentials

_{biased}*μ*

_{c}= 0.4 eV and 0.5 eV. And the free-space wavelength is fixed at

*λ*= 15 um in Fig. 1(c). As shown in Fig. 1(a), with a fixed

_{0}*V*, the real part of graphene conductivity stays near the universal value,

_{biased}*σ*

_{0}=

*πe*

^{2}/2

*ћ≈*6.085x10

^{−5}S/m, and the absolute value of the imaginary part increases with the wavelength. The real part of the conductivity is insensitive to the biased voltage, and the imaginary part decreases as the voltage increases. In Fig. 1(c), the real part stays relative high when the biased voltage is below a well-defined threshold, whereas over this threshold the real part recovers the universal value. For the imaginary part, a positive peak value appears at the threshold voltage.

_{2}as the dielectric material in our subsequent simulation. The permittivity of the dielectric is

*ε*= 2.2 and the refractive index is

_{d}*n*= 1.48. The period of the structure is

_{d}*a*= 20 nm. Each period is constructed with a layer of graphene with thickness

*a*1 nm and the layer of dielectric with thickness

_{g}=*a*= 19 nm. The schematic of the waveguide is sketched in Figs. 2(a) and 2(b). The

_{d}*Lx*and

*Ly*represents horizontal and vertical dimensions of the cross section of the waveguide, while the

*Lz*is the length of the waveguide.

*a*is much less than the incident wavelength, the multilayer metamaterial could be treated as a homogeneous effective medium and the anisotropic permittivity sensor could be defined as [4

**29**(9), 2559–2566 (2012). [CrossRef]

26. C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template-synthesized nanoscopic gold Particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem. **98**(11), 2963–2971 (1994). [CrossRef]

*f*0.05, which is the fraction of the multilayer structure period occupied by graphene. The

_{g}=*ε*and

_{x}, ε_{y}*ε*represents the permittivity along

_{z}*x*,

*y*and

*z*directions. The

*ε*is the permittivity of the dielectric. And the

_{d}*ε*and

_{g,t}*ε*represent the tangential and normal components of the permittivity of the graphene layer, respectively. The

_{g,n}*ε*could be derived from the relationship between the relative permittivity and the conductivity [27],where

_{g,t}*d*is the thickness of graphene layer. Since the wavelength

*λ*in the metamaterial is significantly longer than all characteristic dimensions, the general formula thus could be simplified as

*ε*(K = 0,

*ω*) =

*ε*(

*ω*). Moreover, it should be considered that the normal electric field cannot excite any current in the graphene sheet, so the normal component of the permittivity of graphene should be

*ε*= 1. So in our multilayer metamaterial, the normal component of the effective permittivity

_{g,n}*ε*, which could be derived from Eq. (4), is only depending on the filling factor

_{y}*f*. The calculated real parts of permittivity

_{g}*ε*and

_{x}, ε_{y}*ε*as functions of free-space wavelength

_{z}*λ*and the biased voltage

_{0}*λ*= 15 um in Fig. 1(d). As shown in Fig. 1(b), when the biased voltage is fixed, the real part of

_{0}*ε*stays positive, while the real parts of

_{y}*ε*and

_{x}*ε*decrease as the wavelength increases. In Fig. 1(d), the curve represents the tangential component of the permittivity shows a similar trend with the imaginary part of conductivity under fixed wavelength, which is shown in Fig. 1(c). It has been shown that the multilayer metamaterial with anisotropic permittivity sensor is equivalent to a uniaxial anisotropic material with dispersion [28

_{z}28. J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U.S.A. **108**(28), 11327–11331 (2011). [CrossRef] [PubMed]

*k*is the free-space wave vector,

_{0}*k*,

_{x}*k*and

_{y}*k*are the wave factors along

_{z}*x*,

*y*and

*z*direction, respectively. The corresponding effective refractive indices along different directions thus could be achieved through

*k*/

_{i}*k*, where

_{0}*k*is one of the three wave factors. As is shown in Figs. 1(b) and 1(d), with capable biased voltage, the permittivity component

_{i}*ε*could be negative while the normal component

_{x}*ε*is positive, which implies that the dispersion characteristic is hyperbolic type. Furthermore, since the effective permittivity appears as functions of the biased voltages, the dispersion properties of the waveguide could be tuned by adjusting the biased voltages on graphene layers.

_{y}## 3. Dielectric-graphene-dielectric waveguides

*μ*

_{r}= 1) and the environment is free of external magnetic fields. The biased voltage is

*V*= 1.6 volt, equivalent to the chemical potential

_{biased}*μ*

_{c}= 0.4 eV. We calculate the waveguide mode profiles of different mode orders under the dimensions

*Lx*= 200 nm and

*Ly =*200 nm with free-space wavelength

*λ*= 30 μm. The distribution of field components

_{0}*E*and

_{y}*H*for the mode orders (1,

_{x}*my*), which are derived from the effective medium approach and the multilayer structures, are illustrated in Figs. 3(a) and 3(b).

*m*,

_{x}*m*), the

_{y}*m*and

_{x}*m*represent the distributions of the resonant peaks along

_{y}*x*direction and

*y*direction on the cross section. Based on the two methods, we also calculate the effective refractive indices along the propagation direction

*n*and the propagation lengths

_{eff,z}*L*as functions of free-space wavelength

_{m}*λ*, which are plotted in Figs. 3(c) and 3(d). The solid and the dot-dashed lines represent the results calculated from the effective medium approach and the multilayer structure, respectively. The propagation length

_{0}*L*is defined as

_{m}*L*= 1/2Im(

_{m}*k*) =

_{z}*λ*/4πIm(

_{0}*n*). As is shown in Fig. 3(c), the modes with a higher

_{eff,z}*m*may have larger mode indices. And each mode tends to have a smaller indice at longer wavelength. In Fig. 3(d), we could find that the modes with higher indices tend to have less propagation lengths, and the propagation lengths would increase with the wavelength.

_{y}*m*) in the subsequent investigations. Then we calculate the dependences of mode indices and propagation lengths on

_{y}*Lx*and

*Ly*. The free-space wavelength is

*λ*= 30 um. The

_{0}*Lx*or

*Ly*spans from 150 nm to 250 nm, while the other is fixed at 200 nm. The result calculated from the multilayer structure and effective medium approach is shown in Fig. 4.

*Lx*or

*Ly*increases. The propagation lengths of all the modes are insensitive to the change of horizontal dimension

*Lx*, while they all increase with the vertical dimension

*Ly*, which are presented in Figs. 4(b) and 4(d). Finally, we demonstrate the optical properties of the waveguide modes could be tuned by adjusting the biased voltages on graphene layers. The dependences of the mode indices of (1,

*m*) modes on the biased voltage are shown in Fig. 5(a). The propagation lengths as functions of the biased voltage are shown in Fig. 5(b). Here we assume the waveguide cross section is square with

_{y}*L = L*= 200 nm and the wavelength is

_{x}= L_{y}*λ*= 30 um. The biased voltage spans from 0.1 volt to 1.6 volt.

_{0}29. Y. Zou, P. Tassin, T. Koschny, and C. M. Soukoulis, “Interaction between graphene and metamaterials: split rings vs. wire pairs,” Opt. Express **20**(11), 12198–12204 (2012). [CrossRef] [PubMed]

_{2}) could be fabricated by plasma-enhanced chemical vapor deposition (PECVD) [30

30. A. Satta, E. Simoen, T. Clarysse, T. Janssens, A. Benedetti, B. De Jaeger, M. Meuris, and W. Vandervorst, “Diffusion, activation, and recrystallization of boron implanted in preamorphized and crystalline germanium,” Appl. Phys. Lett. **87**(17), 172109 (2005). [CrossRef]

31. M. Björk, J. Knoch, H. Schmid, H. Riel, and W. Riess, “Silicon nanowire tunneling field-effect transistors,” Appl. Phys. Lett. **92**(19), 193504 (2008). [CrossRef]

## 4. Conclusion

## Acknowledgment

## References and links

1. | D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science |

2. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

3. | Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express |

4. | Y. He, S. He, J. Gao, and X. Yang, “Nanoscale metamaterial optical waveguides with ultrahigh refractive indices,” J. Opt. Soc. Am. B |

5. | F. Y. Meng, Q. Wu, and L. W. Li, “Transmission characteristics of wave modes in a rectangular waveguide filled with anisotropic metamaterial,” Appl. Phys., A Mater. Sci. Process. |

6. | X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics |

7. | K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science |

8. | K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature |

9. | A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics |

10. | F. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B |

11. | M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express |

12. | G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. |

13. | G. V. Naik and A. Boltasseva, “A comparative study of semiconductor-based plasmonic metamaterials,” Metamaterials (Amst.) |

14. | C. Rizza, A. Ciattoni, E. Spinozzi, and L. Columbo, “Terahertz active spatial filtering through optically tunable hyperbolic metamaterials,” Opt. Lett. |

15. | 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, “A terahertz metamaterial with unnaturally high refractive index,” Nature |

16. | J. Shin, J. T. Shen, and S. H. Fan, “Three-Dimensional Metamaterials with an Ultrahigh Effective Refractive Index over a Broad Bandwidth,” Phys. Rev. Lett. |

17. | R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature |

18. | R. Colombelli, F. Capasso, C. Gmachl, A. L. Hutchinson, D. L. Sivco, A. Tredicucci, M. C. Wanke, A. M. Sergent, and A. Y. Cho, “Far-infrared surface-plasmon quantum-cascade lasers at 21.5 μm and 24 μm wavelengths,” Appl. Phys. Lett. |

19. | M. Beck, D. Hofstetter, T. Aellen, J. Faist, U. Oesterle, M. Ilegems, E. Gini, and H. Melchior, “Continuous Wave Operation of a Mid-Infrared Semiconductor Laser at Room Temperature,” Science |

20. | R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics |

21. | C. Xu, Y. Jin, L. Yang, J. Yang, and X. Jiang, “Characteristics of electro-refractive modulating based on Graphene-Oxide-Silicon waveguide,” Opt. Express |

22. | G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. |

23. | A. Vakil and N. Engheta, “Transformation Optics Using Graphene,” Science |

24. | T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B |

25. | S. H. Lee, M. Choi, T. T. Kim, S. Lee, M. Liu, X. Yin, H. K. Choi, S. S. Lee, C. G. Choi, S. Y. Choi, X. Zhang, and B. Min, “Switching terahertz waves with gate-controlled active graphene metamaterials,” Nat. Mater. |

26. | C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template-synthesized nanoscopic gold Particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem. |

27. | S. A. Maier, |

28. | J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U.S.A. |

29. | Y. Zou, P. Tassin, T. Koschny, and C. M. Soukoulis, “Interaction between graphene and metamaterials: split rings vs. wire pairs,” Opt. Express |

30. | A. Satta, E. Simoen, T. Clarysse, T. Janssens, A. Benedetti, B. De Jaeger, M. Meuris, and W. Vandervorst, “Diffusion, activation, and recrystallization of boron implanted in preamorphized and crystalline germanium,” Appl. Phys. Lett. |

31. | M. Björk, J. Knoch, H. Schmid, H. Riel, and W. Riess, “Silicon nanowire tunneling field-effect transistors,” Appl. Phys. Lett. |

**OCIS Codes**

(230.7370) Optical devices : Waveguides

(160.3918) Materials : Metamaterials

(310.4165) Thin films : Multilayer design

(310.6628) Thin films : Subwavelength structures, nanostructures

**ToC Category:**

Metamaterials

**History**

Original Manuscript: May 14, 2013

Revised Manuscript: June 15, 2013

Manuscript Accepted: June 28, 2013

Published: July 10, 2013

**Virtual Issues**

Vol. 8, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

Bofeng Zhu, Guobin Ren, Siwen Zheng, Zhen Lin, and Shuisheng Jian, "Nanoscale dielectric-graphene-dielectric tunable infrared waveguide with ultrahigh refractive indices," Opt. Express **21**, 17089-17096 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-14-17089

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### References

- D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685), 788–792 (2004). [CrossRef] [PubMed]
- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
- Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express16(20), 15439–15448 (2008). [CrossRef] [PubMed]
- Y. He, S. He, J. Gao, and X. Yang, “Nanoscale metamaterial optical waveguides with ultrahigh refractive indices,” J. Opt. Soc. Am. B29(9), 2559–2566 (2012). [CrossRef]
- F. Y. Meng, Q. Wu, and L. W. Li, “Transmission characteristics of wave modes in a rectangular waveguide filled with anisotropic metamaterial,” Appl. Phys., A Mater. Sci. Process.94(4), 747–753 (2009). [CrossRef]
- X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics6(7), 450–454 (2012). [CrossRef]
- K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science306(5696), 666–669 (2004). [CrossRef] [PubMed]
- K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438(7065), 197–200 (2005). [CrossRef] [PubMed]
- A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics6(11), 749–758 (2012). [CrossRef]
- F. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B87(7), 075416 (2013). [CrossRef]
- M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express21(6), 7614–7632 (2013). [CrossRef] [PubMed]
- G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A.109(23), 8834–8838 (2012). [CrossRef] [PubMed]
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