Abstract

We propose a new type of a mid-infrared ultra-compact optical modulator composed of a graphene metasurface. Unlike the previously proposed schemes based on loss variation of materials or interference, the proposed one utilizes the unique topological characteristic of the isofrequency contour in the hyperbolic metasurface to modulate the transmission. The designed modulator provides a modulation depth of 10.7 dB, the length of which is 750 nm, corresponding to ${\sim}{1/30}$ of an operating wavelength.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

The mid-infrared (IR) band has attracted great attention owing to its various applications, such as imaging, broadband optical communication, spectroscopy, biochemical sensing, and defense technology [13]. Mid-IR optical modulators are essential components for enabling the realization of these useful applications. In particular, the realization of high efficiency mid-IR modulators is important for extending operating wavelength (frequency) bandwidth for data communication from near-IR to mid-IR. In spite of this significance, a mid-IR modulator is relatively underdeveloped. Although mid-IR modulators based on the Pockels effect, Kerr effect, and carrier plasma dispersion in various waveguide platforms have been actively studied, its modulation efficiency is low due to its weak electro-optic effect at the mid-IR, which limits its applications [46].

Graphene has been widely studied due to its outstanding and unique properties. Since the conductivity of graphene can be easily tuned by applying gate voltage, many kinds of graphene-based applications has been reported, such as graphene-based tunable optical delay lines [7,8], antennas [9], and broadband modulators [10]. In particular, graphene plasmonic modes have been introduced to overcome the low efficiency issue [1113]. Since the graphene plasmonic modes support very high electric field confinement with relatively low loss, and the mode property, especially loss of the mode, can be easily tuned by electrical doping, it is suitable to be applied as an active layer in modulators. Graphene also supports high mobility. It has been reported that suspended exfoliated graphene provides mobility of ${23}\;{{\rm m}^2}/{\rm Vs}$ [14], and the mobility of ${14}\;{{\rm m}^2}/{\rm Vs}$ was achieved for graphene films grown on boron nitride substrates [15]. Thus, its tunable permittivity with high mobility is expected to achieve high speed modulation, which will be limited mainly by the capacitance of the device [16]. Since the capacitance is proportional to the footprint of the device, smaller device footprint is desirable for high speed operation as well as higher integration density. In recent years, several graphene plasmonic mode-based mid-IR modulators have been studied [1719]. Those are free-space-type optical modulators based on variations of the loss and resonant frequency, whereas waveguide-based mid-IR optical modulators have not been studied much.

In this work, we suggest a new type of mid-IR modulator composed of a graphene metasurface, in which the topological characteristic variation of the isofrequency contours in the hyperbolic metasurface results in dramatic transmission modulation. As a result, the proposed modulator provides a high modulation depth with an ultra-compact footprint.

The schematic of the proposed optical modulator is shown in the Fig. 1. It consists of the graphene ribbon array, which forms the hyperbolic metasurface, on the ${{\rm SiO}_2}$ substrate, and ion-gel is deposited on the metasurface for doping of the graphene ribbons by applying a gate voltage [20], where $P$ is the period of the graphene metasurface, and $W$ is the width of the graphene ribbon.

Both refractive indices of the ion-gel and ${{\rm SiO}_2}$ are set to be two, and conductivity of the graphene layer is obtained by the Kubo formula and random phase approximation [17]. Since the period of the graphene ribbon array is very short ($P = {150}\;{\rm nm}$), and the thickness of graphene is ultra-thin (${d_G} = {0.34}\;{\rm nm}$) compared with an operating wavelength ($\lambda = {25}\;\unicode{x00B5}{\rm m}$), the graphene metasurface can be considered as a two-dimensional (2D) homogeneous anisotropic material with effective in-plane ($x$ and $y$ direction) conductivity [2123]:

$$\begin{split}{\sigma _x}& = \frac{{W\sigma {\sigma _c}}}{{P{\sigma _c} + W\sigma}},\quad {\sigma _y} = \sigma \frac{W}{P}, \\ {\sigma _c} & = - i2\omega {\varepsilon _0}\frac{P}{\pi}\ln \left[{\csc \left({\frac{{\pi\! G}}{{2P}}} \right)} \right],\end{split}$$
where $W$ is the width of the graphene ribbon, $G$ is the separation distance between two consecutive graphene layers, $\sigma$ is the conductivity of graphene, and ${\sigma _c}$ is an equivalent conductivity. We can also obtain the effective permittivity from the effective conductivity by the following equation:
$${\varepsilon _G}\!\left(\omega \right) = 1 + \frac{{i\sigma}}{{\omega {\varepsilon _0}{d_G}}},$$
where ${d_G}$ is the thickness of graphene (${d_G} = {0.34}\;{\rm nm}$).
 

Fig. 1. Schematic of the proposed graphene metasurface.

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Figure 2(a) shows the effective in-plane permittivity for the variation of the graphene ribbon width. We assumed the Fermi level of graphene as ${E_F} = {0.5}\;{\rm eV}$. The sign of the effective permittivity for the $y$ direction remains negative (i.e., a metallic characteristic) for all of the graphene ribbon widths, whereas, for the $x$ direction, it changes from positive to negative as ${W}$ increases. Therefore, the graphene metasurface shows the hyperbolic metasurface characteristic ($\varepsilon_{x}\gt 0$, $\varepsilon_{y}\lt 0$) for $W\; \lt \;{140}\;{\rm nm}$ and the effective metal characteristic ($\varepsilon_{x}\lt 0$, $\varepsilon_{y}\lt 0$) for $W\; \gt \;{140}\;{\rm nm}$.

 

Fig. 2. (a) Effective permittivity of the proposed metasurface for ${{E}_F} = {0.5}\;{\rm eV}$ ($P = {150}\;{\rm nm}$). (b) Isofrequency contour of the hyperbolic metasurface.

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Figure 2(b) illustrates the isofrequency contour of the proposed metasurface for ${W}\; \lt \;{140}\;{\rm nm}$, corresponding to the hyperbolic metasurface, where there is no propagation mode for the incident light along the $x$ direction (${k_y} = {0}$). In contrast, when the metasurface shows the effective metallic characteristic ($W\; \gt \;{140}\;{\rm nm}$) with an elliptical isofrequency contour (not shown here), the same incident light propagates with a loss via the graphene plasmonic mode. Note that the topology of the isofrequency contour of the graphene metasurface is changed by the sign of ${\varepsilon _x}$, which can also be controlled by electrical doping of the graphene ribbon array for a fixed ${W}$. Therefore, the transmission of the incident light along the $x$ direction can be modulated by the electrical doping with a proper choice of $W$.

To prove this concept, first, we investigated the transmission of the graphene metasurface composed of five graphene ribbons ($P = {150}\;{\rm nm}$) with ${E_F} = {0.5}\;{\rm eV}$ as a function of $W$, which is plotted in Fig. 3, along with the effective in-plane permittivities (dashed lines). The calculation was carried out using the 2D finite-element method (COMSOL) at the wavelength of $\lambda = {25}\;\unicode{x00B5}{\rm m}$. We launched the graphene plasmon mode source and monitored the transmitted power to calculate the transmission. Graphene was modeled as a three-dimensional (3D) material with the thickness of 0.34 nm. As expected, the transmission is suppressed when the metasurface shows the hyperbolic characteristic. In contrast, the transmission rapidly increases when the characteristic of the metasurface changes from the hyperbolic metasurface to an effective metal ($W\; \gt \;{140}\;{\rm nm}$).

 

Fig. 3. Real part of the in-plane effective permittivity (dash line) of the proposed metasurface and the transmission (solid line) in the five periods of graphene ribbons. The Fermi level of graphene is assumed as ${{E}_F} = {0.5}\;{\rm eV}$.

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Fig. 4. Transmission of the proposed metasurface for various mobility values ($\mu$). (a) $\mu = {1}\;{{\rm m}^2}/{\rm Vs}$, (b) $\mu = {3}\;{{\rm m}^2}/{\rm Vs}$, (c) $\mu = {5}\;{{\rm m}^2}/{\rm Vs}$, and (d) $\mu = {10}\;{{\rm m}^2}/{\rm Vs}$.

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Figure 4 shows the transmission variation as a function of the graphene ribbon width for various Fermi levels ($0.{2}\;{\rm eV}\; \le \;{E_F}\; \le \;{0.6}\;{\rm eV}$) and mobility ($\mu$) of graphene: (a) $\mu = {1}\;{{\rm m}^2}/{\rm Vs}$, (b) $\mu = {3}\;{{\rm m}^2}/{\rm Vs}$, (c) $\mu = {5}\;{{\rm m}^2}/{\rm Vs}$, and (d) $\mu = {10}\;{{\rm m}^2}/{\rm Vs}$. One can see that as the Fermi level decreases, the characteristic transition from the hyperbolic metasurface to the effective metal occurs at the smaller graphene ribbon width. Similar to the previous calculation, the transmission is almost zero when the metasurface is showing the hyperbolic characteristic, and it begins to increase as the characteristic of the metasurface turns to the effective metal. One can also see that for a small $W$ around 100 nm, the transmission is slightly higher for the higher Fermi level, which is because the material loss of graphene decreases as the Fermi level of graphene increases. The transmission fluctuations are observed due to the Fabry–Perot interference by the reflection at the end of the graphene ribbon array. As the mobility of graphene increases, the loss of the graphene plasmonic mode decreases [12], resulting in higher transmission. In large, the effect of mobility on the transmission is stronger for the relatively low Fermi level.

Figure 5 shows the calculated transmission of the proposed metasurface composed of five graphene ribbons ($P = {150}\;{\rm nm}$, $W = {141}\;{\rm nm}$) as a function of the Fermi level for the different mobility values. The graphene ribbon width of $W = {141}\;{\rm nm}$ was chosen to maximize the sensitivity of the transmission to the Fermi level change. In this case, the characteristic transition from the hyperbolic metasurface to the effective metal occurs at ${E_F} =\, \sim{0.5}\;{\rm eV}$: the hyperbolic metasurface characteristic for ${E_F}\; \gt \;\sim{0.5}\;{\rm eV}$. As a result, for ${0.3}\;{\rm eV}\; \le \;{E_F}\; \le \;{0.5}\;{\rm eV}$, the rapid and monotonic transmission change is observed. For ${E_F}\; \lt \;{0.3}\;{\rm eV}$, the material loss of graphene plays an important role: the transmission tends to decrease as the Fermi level decreases due to the increased loss, which is clearer in the case of a low mobility of ${1}\;{{\rm m}^2}/{\rm Vs}$. As for the mobility effect on transmission, the higher mobility results in the higher transmission in the effective metal region and, thus, the larger modulation depth: the modulation depths are 7.63, 9.87, 10.4, and 10.7 dB for $\mu = {1}$, 3, 5, and ${10}\;{{\rm m}^2}/{\rm Vs}$, respectively. The insertion losses at ${E_F} = {0.3}\;{\rm eV}$ are 4.99, 2.25, 1.66, and 1.21 dB for $\mu = {1}$, 3, 5, and ${10}\;{{\rm m}^2}/{\rm Vs}$, respectively. When the mobility is higher than ${3}\;{{\rm m}^2}/{\rm Vs}$, the transmission and the modulation depth are not much affected by the mobility. Compared with the recently reported results (modulation depth = 7 dB in Ref. [5] and 8 dB in Ref. [6]), the proposed device provides the higher modulation depth. Above all, this performance is achieved with approximately 1/1000 device length.

 

Fig. 5. Transmission of the proposed metasurface composed of five graphene ribbons (${P} = {150}\;{\rm nm}$, ${W} = {141}\;{\rm nm}$) as a function of the Fermi level for various mobility values: $\mu = {1}\;{{\rm m}^2}/{\rm Vs}$ (black line), $\mu = {3}\;{{\rm m}^2}/{\rm Vs}$ (red line), $\mu = {5}\;{{\rm m}^2}/{\rm Vs}$ (blue line), and (d) $\mu = {10}\;{{\rm m}^2}/{\rm Vs}$ (green line). The black line represents the transmission of the uniform graphene sheet.

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In Fig. 5, the transmission in the uniform single graphene sheet ($\mu = {1}\;{{\rm m}^2}/{\rm Vs}$) is also plotted as a reference (dashed line), which shows transmission modulation of ${\sim}{5.13}\;{\rm dB}$ for the Fermi level change from 0.2 to 0.6 eV. One can see that the proposed graphene metasurface provides a much more effective means to modulate the transmission in comparison with the loss variation in the uniform graphene sheet.

We also investigated the performance for the number of pairs and period variations. Figure 6(a) shows the transmission as a function of the Fermi level for the different number of pairs. The graphene ribbon width of ${W} = {141}\;{\rm nm}$ was chosen to maximize the modulation depth for all cases. The mobility was assumed to be ${10}\;{{\rm m}^2}/{\rm Vs}$. For the two-pair case, the device length is extremely short. Therefore, the high transmission is maintained regardless of the Fermi level variation because the incident light is transmitted evanescently even if the metasurface shows the hyperbolic metasurface characteristic. In contrast, for the 10-pair case, the transmission in the hyperbolic metasurface is strongly suppressed. The modulation depth increases with the number of pairs: the modulation depths are 15.5, 10.7, and 2 dB for the 10-, 5-, and 2-pair metasurface, respectively. However, an increase of the number of pairs reduces the modulation speed and increases the device footprint, which is trade-off.

 

Fig. 6. Transmission of the proposed device as a function of the Fermi level for (a) the number of pairs for the 10-pair (red line), five-pair (black line), and two-pair (blue line), and (b) for the periods of 130 nm (red line), 150 nm (black line), and 170 nm (blue line) in the five-pair metasurface and ${P} = {130}\;{\rm nm}$ in the six-pair metasurface (red dashed line).

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Figure 6(b) shows the transmission as a function of the Fermi level for the different periods: ${P} = {130}$, 150, and 170 nm, where $W$ is assumed to be 124, 141, and 152 nm, respectively, to maximize the modulation depth. As shown in Eq. (1), the effective conductivity is a function for the ratio of $W$ and $P$. Therefore, the topological transition occurred at similar ${W}$ and ${ P}$ ratios for all the cases. Compared with the $P = {150}$ and 170 nm cases, in the $P = {130}$ case, the transmission is still high at the off state because the device length is not enough to suppress the incident light by the hyperbolic metasurface characteristic. To prove it, we calculated the transmission in the six-pair metasurface of $P = {130}\;{\rm nm}$ (red dashed line), whose device length (780 nm) is almost same as the five-pair metasurface of $P = {150}\;{\rm nm}$ (750 nm). In this case, the transmission variation of both cases with respect to the Fermi level is very similar. The modulation depths of the five-pair metasurface are 5.74, 10.7, and 11.78 dB for $P = {130}$, 150, and 170 nm, respectively.

In summary, we proposed the ultra-compact mid-IR optical modulator composed of the graphene metasurface, which utilizes the topological characteristic variation of its isofrequency contour. The proposed modulator provides an insertion loss of 1.21 dB and a modulation depth of 10.7 dB for $\mu = {10}\;{{\rm m}^2}/{\rm Vs}$. We also investigated the mobility effect in the proposed modulator, and, if the mobility is higher than $\mu = {3}\;{{\rm m}^2}/{\rm Vs}$, the performance does not show much dependence on the mobility. Whereas, for the low mobility case, $\mu = {1}\;{{\rm m}^2}/{\rm Vs}$, the loss of graphene degrades the performance of the modulator: the modulation depth and insertion loss are 7.63 dB and 4.99 dB, respectively. The effects of the period and device length are investigated. Increasing the number of pairs provides the higher modulation depth. We believe that the proposed modulation scheme can lead to a new class of compact modulators in the mid-IR wavelength region.

Funding

National Research Foundation of Korea (2019R1I1A1A01061983, 2020R1A2B5B01002681, 2018R1D1A1B07049349); Korea Institute for Advancement of Technology (P0010880); GIST Research Institute (2021).

Disclosures

The authors declare no conflicts of interest.

REFERENCES

1. T. Hu, B. Dong, X. Luo, T. Y. Liow, J. Song, C. Lee, and G. Q. Lo, Photon. Res. 5, 417 (2017). [CrossRef]  

2. S. Wartewig and R. H. H. Neubert, Adv. Drug Deliv. Rev. 57, 1144 (2005). [CrossRef]  

3. R. Soref, Nat. Photonics 4, 495 (2010). [CrossRef]  

4. T. Jin, L. Li, B. Zhang, H. Y. G. Lin, H. Wang, and P. T. Lin, ACS Appl. Mater. Interfaces 9, 21848 (2017). [CrossRef]  

5. S. Liu, K. Xu, Q. Song, Z. Cheng, and H. K. Tsang, J. Lightwave Technol. 34, 3837 (2016). [CrossRef]  

6. J. Chiles and S. Fathpour, Optica 1, 350 (2014). [CrossRef]  

7. G. Brunetti, D. Conteduca, F. Dell’Olio, C. Ciminelli, and M. N. Armenise, Opt. Express 26, 4593 (2018). [CrossRef]  

8. D. Conteduca, F. Dell’Olio, C. Ciminelli, and M. N. Armenise, IEEE Photon. J. 7, 7802409 (2015). [CrossRef]  

9. F. Zangeneh-Nejad and R. Safian, J. Nanophoton. 10, 036005 (2016). [CrossRef]  

10. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011). [CrossRef]  

11. A. N. Grigorenko, M. Polini, and K. S. Novoselov, Nat. Photonics 6, 749 (2012). [CrossRef]  

12. T. Low and P. Avouris, ACS Nano 8, 1086 (2014). [CrossRef]  

13. E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418 (2007). [CrossRef]  

14. K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, Solid State Commun. 146, 351 (2008). [CrossRef]  

15. L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013). [CrossRef]  

16. T. Q. Tran and S. Kim, Curr. Opt. Photon. 1, 65 (2017). [CrossRef]  

17. W. Zhu, I. D. Rukhlenko, and M. Premaratne, Appl. Phys. Lett. 102, 241914 (2013). [CrossRef]  

18. V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, Nano Lett. 13, 2541 (2013). [CrossRef]  

19. M. C. Sherrott, P. W. C. Hon, K. T. Fountaine, J. C. Garcia, S. M. Ponti, V. W. Brar, L. A. Sweatlock, and H. A. Atwater, Nano Lett. 17, 3027 (2017). [CrossRef]  

20. T. Fujimoto and K. Awaga, Phys. Chem. Chem. Phys. 15, 8983 (2013). [CrossRef]  

21. F. H. L. Koppens, D. E. Chang, and F. J. García De Abajo, Nano Lett. 11, 3370 (2011). [CrossRef]  

22. J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, Phys. Rev. Lett. 114, 233901 (2015). [CrossRef]  

23. J. S. Gomez-Diaz and A. Alù, ACS Photon. 3, 2211 (2016). [CrossRef]  

References

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  1. T. Hu, B. Dong, X. Luo, T. Y. Liow, J. Song, C. Lee, and G. Q. Lo, Photon. Res. 5, 417 (2017).
    [Crossref]
  2. S. Wartewig and R. H. H. Neubert, Adv. Drug Deliv. Rev. 57, 1144 (2005).
    [Crossref]
  3. R. Soref, Nat. Photonics 4, 495 (2010).
    [Crossref]
  4. T. Jin, L. Li, B. Zhang, H. Y. G. Lin, H. Wang, and P. T. Lin, ACS Appl. Mater. Interfaces 9, 21848 (2017).
    [Crossref]
  5. S. Liu, K. Xu, Q. Song, Z. Cheng, and H. K. Tsang, J. Lightwave Technol. 34, 3837 (2016).
    [Crossref]
  6. J. Chiles and S. Fathpour, Optica 1, 350 (2014).
    [Crossref]
  7. G. Brunetti, D. Conteduca, F. Dell’Olio, C. Ciminelli, and M. N. Armenise, Opt. Express 26, 4593 (2018).
    [Crossref]
  8. D. Conteduca, F. Dell’Olio, C. Ciminelli, and M. N. Armenise, IEEE Photon. J. 7, 7802409 (2015).
    [Crossref]
  9. F. Zangeneh-Nejad and R. Safian, J. Nanophoton. 10, 036005 (2016).
    [Crossref]
  10. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
    [Crossref]
  11. A. N. Grigorenko, M. Polini, and K. S. Novoselov, Nat. Photonics 6, 749 (2012).
    [Crossref]
  12. T. Low and P. Avouris, ACS Nano 8, 1086 (2014).
    [Crossref]
  13. E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418 (2007).
    [Crossref]
  14. K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, Solid State Commun. 146, 351 (2008).
    [Crossref]
  15. L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
    [Crossref]
  16. T. Q. Tran and S. Kim, Curr. Opt. Photon. 1, 65 (2017).
    [Crossref]
  17. W. Zhu, I. D. Rukhlenko, and M. Premaratne, Appl. Phys. Lett. 102, 241914 (2013).
    [Crossref]
  18. V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, Nano Lett. 13, 2541 (2013).
    [Crossref]
  19. M. C. Sherrott, P. W. C. Hon, K. T. Fountaine, J. C. Garcia, S. M. Ponti, V. W. Brar, L. A. Sweatlock, and H. A. Atwater, Nano Lett. 17, 3027 (2017).
    [Crossref]
  20. T. Fujimoto and K. Awaga, Phys. Chem. Chem. Phys. 15, 8983 (2013).
    [Crossref]
  21. F. H. L. Koppens, D. E. Chang, and F. J. García De Abajo, Nano Lett. 11, 3370 (2011).
    [Crossref]
  22. J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, Phys. Rev. Lett. 114, 233901 (2015).
    [Crossref]
  23. J. S. Gomez-Diaz and A. Alù, ACS Photon. 3, 2211 (2016).
    [Crossref]

2018 (1)

2017 (4)

T. Hu, B. Dong, X. Luo, T. Y. Liow, J. Song, C. Lee, and G. Q. Lo, Photon. Res. 5, 417 (2017).
[Crossref]

T. Jin, L. Li, B. Zhang, H. Y. G. Lin, H. Wang, and P. T. Lin, ACS Appl. Mater. Interfaces 9, 21848 (2017).
[Crossref]

T. Q. Tran and S. Kim, Curr. Opt. Photon. 1, 65 (2017).
[Crossref]

M. C. Sherrott, P. W. C. Hon, K. T. Fountaine, J. C. Garcia, S. M. Ponti, V. W. Brar, L. A. Sweatlock, and H. A. Atwater, Nano Lett. 17, 3027 (2017).
[Crossref]

2016 (3)

J. S. Gomez-Diaz and A. Alù, ACS Photon. 3, 2211 (2016).
[Crossref]

S. Liu, K. Xu, Q. Song, Z. Cheng, and H. K. Tsang, J. Lightwave Technol. 34, 3837 (2016).
[Crossref]

F. Zangeneh-Nejad and R. Safian, J. Nanophoton. 10, 036005 (2016).
[Crossref]

2015 (2)

D. Conteduca, F. Dell’Olio, C. Ciminelli, and M. N. Armenise, IEEE Photon. J. 7, 7802409 (2015).
[Crossref]

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, Phys. Rev. Lett. 114, 233901 (2015).
[Crossref]

2014 (2)

J. Chiles and S. Fathpour, Optica 1, 350 (2014).
[Crossref]

T. Low and P. Avouris, ACS Nano 8, 1086 (2014).
[Crossref]

2013 (4)

W. Zhu, I. D. Rukhlenko, and M. Premaratne, Appl. Phys. Lett. 102, 241914 (2013).
[Crossref]

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, Nano Lett. 13, 2541 (2013).
[Crossref]

L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
[Crossref]

T. Fujimoto and K. Awaga, Phys. Chem. Chem. Phys. 15, 8983 (2013).
[Crossref]

2012 (1)

A. N. Grigorenko, M. Polini, and K. S. Novoselov, Nat. Photonics 6, 749 (2012).
[Crossref]

2011 (2)

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
[Crossref]

F. H. L. Koppens, D. E. Chang, and F. J. García De Abajo, Nano Lett. 11, 3370 (2011).
[Crossref]

2010 (1)

R. Soref, Nat. Photonics 4, 495 (2010).
[Crossref]

2008 (1)

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, Solid State Commun. 146, 351 (2008).
[Crossref]

2007 (1)

E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418 (2007).
[Crossref]

2005 (1)

S. Wartewig and R. H. H. Neubert, Adv. Drug Deliv. Rev. 57, 1144 (2005).
[Crossref]

Alù, A.

J. S. Gomez-Diaz and A. Alù, ACS Photon. 3, 2211 (2016).
[Crossref]

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, Phys. Rev. Lett. 114, 233901 (2015).
[Crossref]

Armenise, M. N.

G. Brunetti, D. Conteduca, F. Dell’Olio, C. Ciminelli, and M. N. Armenise, Opt. Express 26, 4593 (2018).
[Crossref]

D. Conteduca, F. Dell’Olio, C. Ciminelli, and M. N. Armenise, IEEE Photon. J. 7, 7802409 (2015).
[Crossref]

Atwater, H. A.

M. C. Sherrott, P. W. C. Hon, K. T. Fountaine, J. C. Garcia, S. M. Ponti, V. W. Brar, L. A. Sweatlock, and H. A. Atwater, Nano Lett. 17, 3027 (2017).
[Crossref]

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, Nano Lett. 13, 2541 (2013).
[Crossref]

Avouris, P.

T. Low and P. Avouris, ACS Nano 8, 1086 (2014).
[Crossref]

Awaga, K.

T. Fujimoto and K. Awaga, Phys. Chem. Chem. Phys. 15, 8983 (2013).
[Crossref]

Bolotin, K. I.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, Solid State Commun. 146, 351 (2008).
[Crossref]

Brar, V. W.

M. C. Sherrott, P. W. C. Hon, K. T. Fountaine, J. C. Garcia, S. M. Ponti, V. W. Brar, L. A. Sweatlock, and H. A. Atwater, Nano Lett. 17, 3027 (2017).
[Crossref]

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, Nano Lett. 13, 2541 (2013).
[Crossref]

Brunetti, G.

Campos, L. M.

L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
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Ciminelli, C.

G. Brunetti, D. Conteduca, F. Dell’Olio, C. Ciminelli, and M. N. Armenise, Opt. Express 26, 4593 (2018).
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D. Conteduca, F. Dell’Olio, C. Ciminelli, and M. N. Armenise, IEEE Photon. J. 7, 7802409 (2015).
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Conteduca, D.

G. Brunetti, D. Conteduca, F. Dell’Olio, C. Ciminelli, and M. N. Armenise, Opt. Express 26, 4593 (2018).
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D. Conteduca, F. Dell’Olio, C. Ciminelli, and M. N. Armenise, IEEE Photon. J. 7, 7802409 (2015).
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E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418 (2007).
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L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
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Dell’Olio, F.

G. Brunetti, D. Conteduca, F. Dell’Olio, C. Ciminelli, and M. N. Armenise, Opt. Express 26, 4593 (2018).
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D. Conteduca, F. Dell’Olio, C. Ciminelli, and M. N. Armenise, IEEE Photon. J. 7, 7802409 (2015).
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Dong, B.

Fathpour, S.

Fountaine, K. T.

M. C. Sherrott, P. W. C. Hon, K. T. Fountaine, J. C. Garcia, S. M. Ponti, V. W. Brar, L. A. Sweatlock, and H. A. Atwater, Nano Lett. 17, 3027 (2017).
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K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, Solid State Commun. 146, 351 (2008).
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T. Fujimoto and K. Awaga, Phys. Chem. Chem. Phys. 15, 8983 (2013).
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L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
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Gao, Y.

L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
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Garcia, J. C.

M. C. Sherrott, P. W. C. Hon, K. T. Fountaine, J. C. Garcia, S. M. Ponti, V. W. Brar, L. A. Sweatlock, and H. A. Atwater, Nano Lett. 17, 3027 (2017).
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F. H. L. Koppens, D. E. Chang, and F. J. García De Abajo, Nano Lett. 11, 3370 (2011).
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M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
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Gomez-Diaz, J. S.

J. S. Gomez-Diaz and A. Alù, ACS Photon. 3, 2211 (2016).
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J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, Phys. Rev. Lett. 114, 233901 (2015).
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Grigorenko, A. N.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, Nat. Photonics 6, 749 (2012).
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Guo, J.

L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
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Hon, P. W. C.

M. C. Sherrott, P. W. C. Hon, K. T. Fountaine, J. C. Garcia, S. M. Ponti, V. W. Brar, L. A. Sweatlock, and H. A. Atwater, Nano Lett. 17, 3027 (2017).
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Hone, J.

L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
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K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, Solid State Commun. 146, 351 (2008).
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Hu, T.

Huang, P. Y.

L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
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Hwang, E. H.

E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418 (2007).
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Jang, M. S.

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, Nano Lett. 13, 2541 (2013).
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Jiang, Z.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, Solid State Commun. 146, 351 (2008).
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Jin, T.

T. Jin, L. Li, B. Zhang, H. Y. G. Lin, H. Wang, and P. T. Lin, ACS Appl. Mater. Interfaces 9, 21848 (2017).
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Ju, L.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
[Crossref]

Kim, P.

L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
[Crossref]

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, Solid State Commun. 146, 351 (2008).
[Crossref]

Kim, S.

Klima, M.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, Solid State Commun. 146, 351 (2008).
[Crossref]

Koppens, F. H. L.

F. H. L. Koppens, D. E. Chang, and F. J. García De Abajo, Nano Lett. 11, 3370 (2011).
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Lee, C.

Li, L.

T. Jin, L. Li, B. Zhang, H. Y. G. Lin, H. Wang, and P. T. Lin, ACS Appl. Mater. Interfaces 9, 21848 (2017).
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Lin, H. Y. G.

T. Jin, L. Li, B. Zhang, H. Y. G. Lin, H. Wang, and P. T. Lin, ACS Appl. Mater. Interfaces 9, 21848 (2017).
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Lin, P. T.

T. Jin, L. Li, B. Zhang, H. Y. G. Lin, H. Wang, and P. T. Lin, ACS Appl. Mater. Interfaces 9, 21848 (2017).
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Liow, T. Y.

Liu, M.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
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Liu, S.

Lo, G. Q.

Lopez, J. J.

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, Nano Lett. 13, 2541 (2013).
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Low, T.

T. Low and P. Avouris, ACS Nano 8, 1086 (2014).
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Luo, X.

Meric, I.

L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
[Crossref]

Muller, D. A.

L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
[Crossref]

Neubert, R. H. H.

S. Wartewig and R. H. H. Neubert, Adv. Drug Deliv. Rev. 57, 1144 (2005).
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Novoselov, K. S.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, Nat. Photonics 6, 749 (2012).
[Crossref]

Polini, M.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, Nat. Photonics 6, 749 (2012).
[Crossref]

Ponti, S. M.

M. C. Sherrott, P. W. C. Hon, K. T. Fountaine, J. C. Garcia, S. M. Ponti, V. W. Brar, L. A. Sweatlock, and H. A. Atwater, Nano Lett. 17, 3027 (2017).
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W. Zhu, I. D. Rukhlenko, and M. Premaratne, Appl. Phys. Lett. 102, 241914 (2013).
[Crossref]

Rukhlenko, I. D.

W. Zhu, I. D. Rukhlenko, and M. Premaratne, Appl. Phys. Lett. 102, 241914 (2013).
[Crossref]

Safian, R.

F. Zangeneh-Nejad and R. Safian, J. Nanophoton. 10, 036005 (2016).
[Crossref]

Shepard, K. L.

L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
[Crossref]

Sherrott, M.

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, Nano Lett. 13, 2541 (2013).
[Crossref]

Sherrott, M. C.

M. C. Sherrott, P. W. C. Hon, K. T. Fountaine, J. C. Garcia, S. M. Ponti, V. W. Brar, L. A. Sweatlock, and H. A. Atwater, Nano Lett. 17, 3027 (2017).
[Crossref]

Sikes, K. J.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, Solid State Commun. 146, 351 (2008).
[Crossref]

Song, J.

Song, Q.

Soref, R.

R. Soref, Nat. Photonics 4, 495 (2010).
[Crossref]

Stormer, H. L.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, Solid State Commun. 146, 351 (2008).
[Crossref]

Sweatlock, L. A.

M. C. Sherrott, P. W. C. Hon, K. T. Fountaine, J. C. Garcia, S. M. Ponti, V. W. Brar, L. A. Sweatlock, and H. A. Atwater, Nano Lett. 17, 3027 (2017).
[Crossref]

Taniguchi, T.

L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
[Crossref]

Tran, H.

L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
[Crossref]

Tran, T. Q.

Tsang, H. K.

Tymchenko, M.

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, Phys. Rev. Lett. 114, 233901 (2015).
[Crossref]

Ulin-Avila, E.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
[Crossref]

Wang, F.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
[Crossref]

Wang, H.

T. Jin, L. Li, B. Zhang, H. Y. G. Lin, H. Wang, and P. T. Lin, ACS Appl. Mater. Interfaces 9, 21848 (2017).
[Crossref]

Wang, L.

L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
[Crossref]

Wartewig, S.

S. Wartewig and R. H. H. Neubert, Adv. Drug Deliv. Rev. 57, 1144 (2005).
[Crossref]

Watanabe, K.

L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
[Crossref]

Xu, K.

Yin, X.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
[Crossref]

Zangeneh-Nejad, F.

F. Zangeneh-Nejad and R. Safian, J. Nanophoton. 10, 036005 (2016).
[Crossref]

Zentgraf, T.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
[Crossref]

Zhang, B.

T. Jin, L. Li, B. Zhang, H. Y. G. Lin, H. Wang, and P. T. Lin, ACS Appl. Mater. Interfaces 9, 21848 (2017).
[Crossref]

Zhang, X.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
[Crossref]

Zhu, W.

W. Zhu, I. D. Rukhlenko, and M. Premaratne, Appl. Phys. Lett. 102, 241914 (2013).
[Crossref]

ACS Appl. Mater. Interfaces (1)

T. Jin, L. Li, B. Zhang, H. Y. G. Lin, H. Wang, and P. T. Lin, ACS Appl. Mater. Interfaces 9, 21848 (2017).
[Crossref]

ACS Nano (1)

T. Low and P. Avouris, ACS Nano 8, 1086 (2014).
[Crossref]

ACS Photon. (1)

J. S. Gomez-Diaz and A. Alù, ACS Photon. 3, 2211 (2016).
[Crossref]

Adv. Drug Deliv. Rev. (1)

S. Wartewig and R. H. H. Neubert, Adv. Drug Deliv. Rev. 57, 1144 (2005).
[Crossref]

Appl. Phys. Lett. (1)

W. Zhu, I. D. Rukhlenko, and M. Premaratne, Appl. Phys. Lett. 102, 241914 (2013).
[Crossref]

Curr. Opt. Photon. (1)

IEEE Photon. J. (1)

D. Conteduca, F. Dell’Olio, C. Ciminelli, and M. N. Armenise, IEEE Photon. J. 7, 7802409 (2015).
[Crossref]

J. Lightwave Technol. (1)

J. Nanophoton. (1)

F. Zangeneh-Nejad and R. Safian, J. Nanophoton. 10, 036005 (2016).
[Crossref]

Nano Lett. (3)

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, Nano Lett. 13, 2541 (2013).
[Crossref]

M. C. Sherrott, P. W. C. Hon, K. T. Fountaine, J. C. Garcia, S. M. Ponti, V. W. Brar, L. A. Sweatlock, and H. A. Atwater, Nano Lett. 17, 3027 (2017).
[Crossref]

F. H. L. Koppens, D. E. Chang, and F. J. García De Abajo, Nano Lett. 11, 3370 (2011).
[Crossref]

Nat. Photonics (2)

A. N. Grigorenko, M. Polini, and K. S. Novoselov, Nat. Photonics 6, 749 (2012).
[Crossref]

R. Soref, Nat. Photonics 4, 495 (2010).
[Crossref]

Nature (1)

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
[Crossref]

Opt. Express (1)

Optica (1)

Photon. Res. (1)

Phys. Chem. Chem. Phys. (1)

T. Fujimoto and K. Awaga, Phys. Chem. Chem. Phys. 15, 8983 (2013).
[Crossref]

Phys. Rev. B (1)

E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418 (2007).
[Crossref]

Phys. Rev. Lett. (1)

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, Phys. Rev. Lett. 114, 233901 (2015).
[Crossref]

Science (1)

L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
[Crossref]

Solid State Commun. (1)

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, Solid State Commun. 146, 351 (2008).
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. Schematic of the proposed graphene metasurface.
Fig. 2.
Fig. 2. (a) Effective permittivity of the proposed metasurface for ${{E}_F} = {0.5}\;{\rm eV}$ ($P = {150}\;{\rm nm}$). (b) Isofrequency contour of the hyperbolic metasurface.
Fig. 3.
Fig. 3. Real part of the in-plane effective permittivity (dash line) of the proposed metasurface and the transmission (solid line) in the five periods of graphene ribbons. The Fermi level of graphene is assumed as ${{E}_F} = {0.5}\;{\rm eV}$.
Fig. 4.
Fig. 4. Transmission of the proposed metasurface for various mobility values ($\mu$). (a) $\mu = {1}\;{{\rm m}^2}/{\rm Vs}$, (b) $\mu = {3}\;{{\rm m}^2}/{\rm Vs}$, (c) $\mu = {5}\;{{\rm m}^2}/{\rm Vs}$, and (d) $\mu = {10}\;{{\rm m}^2}/{\rm Vs}$.
Fig. 5.
Fig. 5. Transmission of the proposed metasurface composed of five graphene ribbons (${P} = {150}\;{\rm nm}$, ${W} = {141}\;{\rm nm}$) as a function of the Fermi level for various mobility values: $\mu = {1}\;{{\rm m}^2}/{\rm Vs}$ (black line), $\mu = {3}\;{{\rm m}^2}/{\rm Vs}$ (red line), $\mu = {5}\;{{\rm m}^2}/{\rm Vs}$ (blue line), and (d) $\mu = {10}\;{{\rm m}^2}/{\rm Vs}$ (green line). The black line represents the transmission of the uniform graphene sheet.
Fig. 6.
Fig. 6. Transmission of the proposed device as a function of the Fermi level for (a) the number of pairs for the 10-pair (red line), five-pair (black line), and two-pair (blue line), and (b) for the periods of 130 nm (red line), 150 nm (black line), and 170 nm (blue line) in the five-pair metasurface and ${P} = {130}\;{\rm nm}$ in the six-pair metasurface (red dashed line).

Equations (2)

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

σ x = W σ σ c P σ c + W σ , σ y = σ W P , σ c = i 2 ω ε 0 P π ln [ csc ( π G 2 P ) ] ,
ε G ( ω ) = 1 + i σ ω ε 0 d G ,

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