Abstract

We theoretically study the topological transition of dispersion types and propose a tunable planar lens based on graphene hyperbolic metamaterials (HMMs). By tuning the chemical potential (μc) of graphene, the dispersion relation of the HMM is topologically switchable between ellipse (μc<0.6 eV) and hyperbola (μc>0.6 eV) where positive and negative refractions occur respectively. Especially, for μc>0.6 eV, a Gaussian light beam is negatively refracted twice and focuses at a far-field point finally, acting well as a planar lens. Furthermore, its focal length l can be sensitively tuned by controlling μc, and Δl reaches 260 μm (from 528 to 268 μm) while μc varies with only 0.05 eV (from 0.65 to 0.7 eV). The physical reason is attributed to the different anisotropy degrees of EFCs for different μc. Such a compact, high-speed, and sensitively tunable planar lens holds great promise in photonic integration, photonic imaging, and directional coupling applications.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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2018 (8)

A. W. Zeng, M. X. Gao, and B. Guo, “Slow light in a hyperbolic metamaterial waveguide cladded with arbitrary nonlinear dielectric materials,” Appl. Phys. B 124(7), 146 (2018).
[Crossref]

M. Kim, S. Kim, and S. Kim, “Optical bistability based on hyperbolic metamaterials,” Opt. Express 26(9), 11620–11632 (2018).
[Crossref] [PubMed]

T. Guo, L. Zhu, P. Y. Chen, and C. Argyropoulos, “Tunable terahertz amplification based on photoexcited active graphene hyperbolic metamaterials,” Opt. Mater. Express 8(12), 3941–3952 (2018).
[Crossref]

M. Sakhdari, M. Farhat, and P. Y. Chen, “PT-symmetric metasurfaces: wave manipulation and sensing using singular points,” New J. Phys. 19(6), 065002 (2018).
[Crossref]

C. Wang, W. Liu, Z. Li, H. Cheng, Z. Li, S. Chen, and J. Tian, “Dynamically tunable deep subwavelength high-order anomalous reflection using graphene metasurfaces,” Adv. Opt. Mater. 6(3), 1701047 (2018).
[Crossref]

T. Roy, S. Zhang, I. L. W. Jung, M. Troccoli, F. Capasso, and D. Lopez, “Dynamic metasurface lens based on MEMS Technology,” Apl Photonics 3(2), 021302 (2018).
[Crossref]

E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, M. Faraji-Dana, and A. Faraon, “MEMS-tunable dielectric metasurface lens,” Nat. Commun. 9(1), 812 (2018).
[Crossref] [PubMed]

D. Su, X. Y. Zhang, Y. L. Ma, F. Shan, J. Y. Wu, X. C. Fu, L. J. Zhang, K. Q. Yuan, and T. Zhang, “Real-time electro-optical tunable hyperlens under subwavelength scale,” IEEE Photonics J. 10(1), 4600109 (2018).
[Crossref]

2017 (8)

H. Zhao, Z. Chen, F. Su, G. Ren, F. Liu, and J. Yao, “Terahertz wavefront manipulating by double-layer graphene ribbons metasurface,” Opt. Commun. 402, 523–526 (2017).
[Crossref]

A. Tyszka-Zawadzka, B. Janaszek, and P. Szczepański, “Tunable slow light in graphene-based hyperbolic metamaterial waveguide operating in SCLU telecom bands,” Opt. Express 25(7), 7263–7272 (2017).
[Crossref] [PubMed]

H. Lu, X. Gan, D. Mao, and J. Zhao, “Graphene-supported manipulation of surface plasmon polaritons in metallic nanowaveguides,” Photon. Res. 5(3), 162–167 (2017).
[Crossref]

T. Gric and O. Hess, “Tunable surface waves at the interface separating different graphene-dielectric composite hyperbolic metamaterials,” Opt. Express 25(10), 11466–11476 (2017).
[Crossref] [PubMed]

S. S. Islam, M. R. I. Faruque, M. T. Islam, and M. T. Ali, “A new wideband negative refractive index metamaterial for dual-band operation,” Appl. Phys., A Mater. Sci. Process. 123(4), 252 (2017).
[Crossref]

S. Pahlavan and V. Ahmadi, “Novel optical demultiplexer design using oblique boundary in hetero photonic crystals,” IEEE Photonic. Tech. L. 29(6), 511–514 (2017).
[Crossref]

T. Gric and O. Hess, “Controlling hybrid-polarization surface plasmon polaritons in dielectric-transparent conducting oxides metamaterials via their effective properties,” J. Appl. Phys. 122(19), 193105 (2017).
[Crossref]

Z. Li, W. Liang, and W. Chen, “Switchable hyperbolic metamaterials based on the graphene-dielectric stacking structure and optical switches design,” A Letters Journal Exploring the Frontiers of Physics 120(3), 37001 (2017).

2016 (6)

H. Deng, X. Chen, B. A. Malomed, N. C. Panoiu, and F. Ye, “Tunability and robustness of Dirac points of photonic nanostructures,” IEEE J. Sel. Top. Quantum Electron. 22(5), 5000509 (2016).
[Crossref]

Y. C. Chang, C. H. Liu, C. H. Liu, S. Zhang, S. R. Marder, E. E. Narimanov, Z. Zhong, and T. B. Norris, “Realization of mid-infrared graphene hyperbolic metamaterials,” Nat. Commun. 7(1), 10568 (2016).
[Crossref] [PubMed]

P. Y. Chen and J. Jung, “PT Symmetry and Singularity-Enhanced Sensing Based on Photoexcited Graphene Metasurfaces,” Phys. Rev. Appl. 5(6), 064018 (2016).
[Crossref]

X. Shang, A. M. Trinidad, P. Joshi, J. D. Smet, D. Cuypers, and H. D. Smet, “Tunable optical beam deflection via liquid crystal gradient refractive index generated by highly resistive polymer film,” IEEE Photonics J. 8(3), 6500411 (2016).
[Crossref]

P. Wang, C. Tang, Z. Yan, Q. Wang, F. Liu, J. Chen, Z. Xu, and C. Sui, “Graphene-based superlens for subwavelength optical imaging by graphene plasmon resonances,” Plasmonics 11(2), 515–522 (2016).
[Crossref]

H. Xu, L. Wu, X. Dai, Y. Gao, and Y. Xiang, “Tunable infrared plasmonic waveguides using graphene based hyperbolic metamaterials,” Optik (Stuttg.) 127(20), 9640–9646 (2016).
[Crossref]

2015 (2)

H. Deng, F. Ye, B. A. Malomed, X. Chen, and N. C. Panoiu, “Optically and electrically tunable Dirac points and Zitterbewegung in graphene-based photonic superlattices,” Phys. Rev. B Condens. Matter Mater. Phys. 91(20), 201402 (2015).
[Crossref]

B. Gao, Z. Shi, and R. W. Boyd, “Design of flat-band superprism structures for on-chip spectroscopy,” Opt. Express 23(5), 6491–6496 (2015).
[Crossref] [PubMed]

2014 (3)

W. Li, X. Zhang, X. Lin, and X. Jiang, “Enhanced wavelength sensitivity of the self-collimation superprism effect in photonic crystals via slow light,” Opt. Lett. 39(15), 4486–4489 (2014).
[Crossref] [PubMed]

V. Purlys, L. Maigyte, D. Gailevicius, M. Peckus, R. Gadonas, and K. Staliunas, “Super-collimation by axisymmetric photonic crystals,” Appl. Phys. Lett. 104(22), 221108 (2014).
[Crossref]

Y. Xiang, X. Dai, J. Guo, H. Zhang, S. Wen, and D. Tang, “Critical coupling with graphene-based hyperbolic metamaterials,” Sci. Rep. 4(1), 5483 (2014).
[Crossref] [PubMed]

2013 (3)

2012 (3)

I. 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 Condens. Matter Mater. Phys. 87(7), 075416 (2012).
[Crossref]

M. N. Cia, M. Beruete, M. Sorolla, and N. Engheta, “Lensing system and Fourier transformation using epsilon-near-zero metamaterials,” Phys. Rev. B Condens. Matter Mater. Phys. 86(16), 165130 (2012).
[Crossref]

B. Sensale-Rodriguez, R. Yan, M. M. Kelly, T. Fang, K. Tahy, W. S. Hwang, D. Jena, L. Liu, and H. G. Xing, “Broadband graphene terahertz modulators enabled by intraband transitions,” Nat. Commun. 3(1), 780 (2012).
[Crossref] [PubMed]

2011 (1)

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[Crossref] [PubMed]

2008 (2)

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]

L. Menon, W. T. Lu, A. L. Friedman, S. P. Bennett, D. Heiman, and S. Sridhar, “Negative index metamaterials based on metal-dielectric nanocomposites for imaging applications,” Appl. Phys. Lett. 93(12), 123117 (2008).
[Crossref]

2005 (1)

Ahmadi, V.

S. Pahlavan and V. Ahmadi, “Novel optical demultiplexer design using oblique boundary in hetero photonic crystals,” IEEE Photonic. Tech. L. 29(6), 511–514 (2017).
[Crossref]

Ali, M. T.

S. S. Islam, M. R. I. Faruque, M. T. Islam, and M. T. Ali, “A new wideband negative refractive index metamaterial for dual-band operation,” Appl. Phys., A Mater. Sci. Process. 123(4), 252 (2017).
[Crossref]

Arbabi, A.

E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, M. Faraji-Dana, and A. Faraon, “MEMS-tunable dielectric metasurface lens,” Nat. Commun. 9(1), 812 (2018).
[Crossref] [PubMed]

Arbabi, E.

E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, M. Faraji-Dana, and A. Faraon, “MEMS-tunable dielectric metasurface lens,” Nat. Commun. 9(1), 812 (2018).
[Crossref] [PubMed]

Argyropoulos, C.

Belov, P. A.

I. 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 Condens. Matter Mater. Phys. 87(7), 075416 (2012).
[Crossref]

Bennett, S. P.

L. Menon, W. T. Lu, A. L. Friedman, S. P. Bennett, D. Heiman, and S. Sridhar, “Negative index metamaterials based on metal-dielectric nanocomposites for imaging applications,” Appl. Phys. Lett. 93(12), 123117 (2008).
[Crossref]

Beruete, M.

M. N. Cia, M. Beruete, M. Sorolla, and N. Engheta, “Lensing system and Fourier transformation using epsilon-near-zero metamaterials,” Phys. Rev. B Condens. Matter Mater. Phys. 86(16), 165130 (2012).
[Crossref]

Boyd, R. W.

Capasso, F.

T. Roy, S. Zhang, I. L. W. Jung, M. Troccoli, F. Capasso, and D. Lopez, “Dynamic metasurface lens based on MEMS Technology,” Apl Photonics 3(2), 021302 (2018).
[Crossref]

Chang, Y. C.

Y. C. Chang, C. H. Liu, C. H. Liu, S. Zhang, S. R. Marder, E. E. Narimanov, Z. Zhong, and T. B. Norris, “Realization of mid-infrared graphene hyperbolic metamaterials,” Nat. Commun. 7(1), 10568 (2016).
[Crossref] [PubMed]

Chen, J.

P. Wang, C. Tang, Z. Yan, Q. Wang, F. Liu, J. Chen, Z. Xu, and C. Sui, “Graphene-based superlens for subwavelength optical imaging by graphene plasmon resonances,” Plasmonics 11(2), 515–522 (2016).
[Crossref]

Chen, L.

Chen, P. Y.

T. Guo, L. Zhu, P. Y. Chen, and C. Argyropoulos, “Tunable terahertz amplification based on photoexcited active graphene hyperbolic metamaterials,” Opt. Mater. Express 8(12), 3941–3952 (2018).
[Crossref]

M. Sakhdari, M. Farhat, and P. Y. Chen, “PT-symmetric metasurfaces: wave manipulation and sensing using singular points,” New J. Phys. 19(6), 065002 (2018).
[Crossref]

P. Y. Chen and J. Jung, “PT Symmetry and Singularity-Enhanced Sensing Based on Photoexcited Graphene Metasurfaces,” Phys. Rev. Appl. 5(6), 064018 (2016).
[Crossref]

Chen, S.

C. Wang, W. Liu, Z. Li, H. Cheng, Z. Li, S. Chen, and J. Tian, “Dynamically tunable deep subwavelength high-order anomalous reflection using graphene metasurfaces,” Adv. Opt. Mater. 6(3), 1701047 (2018).
[Crossref]

Chen, W.

Z. Li, W. Liang, and W. Chen, “Switchable hyperbolic metamaterials based on the graphene-dielectric stacking structure and optical switches design,” A Letters Journal Exploring the Frontiers of Physics 120(3), 37001 (2017).

Chen, X.

H. Deng, X. Chen, B. A. Malomed, N. C. Panoiu, and F. Ye, “Tunability and robustness of Dirac points of photonic nanostructures,” IEEE J. Sel. Top. Quantum Electron. 22(5), 5000509 (2016).
[Crossref]

H. Deng, F. Ye, B. A. Malomed, X. Chen, and N. C. Panoiu, “Optically and electrically tunable Dirac points and Zitterbewegung in graphene-based photonic superlattices,” Phys. Rev. B Condens. Matter Mater. Phys. 91(20), 201402 (2015).
[Crossref]

Chen, Z.

H. Zhao, Z. Chen, F. Su, G. Ren, F. Liu, and J. Yao, “Terahertz wavefront manipulating by double-layer graphene ribbons metasurface,” Opt. Commun. 402, 523–526 (2017).
[Crossref]

Cheng, H.

C. Wang, W. Liu, Z. Li, H. Cheng, Z. Li, S. Chen, and J. Tian, “Dynamically tunable deep subwavelength high-order anomalous reflection using graphene metasurfaces,” Adv. Opt. Mater. 6(3), 1701047 (2018).
[Crossref]

Cia, M. N.

M. N. Cia, M. Beruete, M. Sorolla, and N. Engheta, “Lensing system and Fourier transformation using epsilon-near-zero metamaterials,” Phys. Rev. B Condens. Matter Mater. Phys. 86(16), 165130 (2012).
[Crossref]

Cuypers, D.

X. Shang, A. M. Trinidad, P. Joshi, J. D. Smet, D. Cuypers, and H. D. Smet, “Tunable optical beam deflection via liquid crystal gradient refractive index generated by highly resistive polymer film,” IEEE Photonics J. 8(3), 6500411 (2016).
[Crossref]

Dai, X.

H. Xu, L. Wu, X. Dai, Y. Gao, and Y. Xiang, “Tunable infrared plasmonic waveguides using graphene based hyperbolic metamaterials,” Optik (Stuttg.) 127(20), 9640–9646 (2016).
[Crossref]

Y. Xiang, X. Dai, J. Guo, H. Zhang, S. Wen, and D. Tang, “Critical coupling with graphene-based hyperbolic metamaterials,” Sci. Rep. 4(1), 5483 (2014).
[Crossref] [PubMed]

Deng, H.

H. Deng, X. Chen, B. A. Malomed, N. C. Panoiu, and F. Ye, “Tunability and robustness of Dirac points of photonic nanostructures,” IEEE J. Sel. Top. Quantum Electron. 22(5), 5000509 (2016).
[Crossref]

H. Deng, F. Ye, B. A. Malomed, X. Chen, and N. C. Panoiu, “Optically and electrically tunable Dirac points and Zitterbewegung in graphene-based photonic superlattices,” Phys. Rev. B Condens. Matter Mater. Phys. 91(20), 201402 (2015).
[Crossref]

Engheta, N.

M. N. Cia, M. Beruete, M. Sorolla, and N. Engheta, “Lensing system and Fourier transformation using epsilon-near-zero metamaterials,” Phys. Rev. B Condens. Matter Mater. Phys. 86(16), 165130 (2012).
[Crossref]

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[Crossref] [PubMed]

Fang, T.

B. Sensale-Rodriguez, R. Yan, M. M. Kelly, T. Fang, K. Tahy, W. S. Hwang, D. Jena, L. Liu, and H. G. Xing, “Broadband graphene terahertz modulators enabled by intraband transitions,” Nat. Commun. 3(1), 780 (2012).
[Crossref] [PubMed]

Faraji-Dana, M.

E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, M. Faraji-Dana, and A. Faraon, “MEMS-tunable dielectric metasurface lens,” Nat. Commun. 9(1), 812 (2018).
[Crossref] [PubMed]

Faraon, A.

E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, M. Faraji-Dana, and A. Faraon, “MEMS-tunable dielectric metasurface lens,” Nat. Commun. 9(1), 812 (2018).
[Crossref] [PubMed]

Farhat, M.

M. Sakhdari, M. Farhat, and P. Y. Chen, “PT-symmetric metasurfaces: wave manipulation and sensing using singular points,” New J. Phys. 19(6), 065002 (2018).
[Crossref]

Faruque, M. R. I.

S. S. Islam, M. R. I. Faruque, M. T. Islam, and M. T. Ali, “A new wideband negative refractive index metamaterial for dual-band operation,” Appl. Phys., A Mater. Sci. Process. 123(4), 252 (2017).
[Crossref]

Friedman, A. L.

L. Menon, W. T. Lu, A. L. Friedman, S. P. Bennett, D. Heiman, and S. Sridhar, “Negative index metamaterials based on metal-dielectric nanocomposites for imaging applications,” Appl. Phys. Lett. 93(12), 123117 (2008).
[Crossref]

Fu, X. C.

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D. Su, X. Y. Zhang, Y. L. Ma, F. Shan, J. Y. Wu, X. C. Fu, L. J. Zhang, K. Q. Yuan, and T. Zhang, “Real-time electro-optical tunable hyperlens under subwavelength scale,” IEEE Photonics J. 10(1), 4600109 (2018).
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Zhao, H.

H. Zhao, Z. Chen, F. Su, G. Ren, F. Liu, and J. Yao, “Terahertz wavefront manipulating by double-layer graphene ribbons metasurface,” Opt. Commun. 402, 523–526 (2017).
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Zhao, J.

Zhong, Z.

Y. C. Chang, C. H. Liu, C. H. Liu, S. Zhang, S. R. Marder, E. E. Narimanov, Z. Zhong, and T. B. Norris, “Realization of mid-infrared graphene hyperbolic metamaterials,” Nat. Commun. 7(1), 10568 (2016).
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Zhu, L.

A Letters Journal Exploring the Frontiers of Physics (1)

Z. Li, W. Liang, and W. Chen, “Switchable hyperbolic metamaterials based on the graphene-dielectric stacking structure and optical switches design,” A Letters Journal Exploring the Frontiers of Physics 120(3), 37001 (2017).

Adv. Opt. Mater. (1)

C. Wang, W. Liu, Z. Li, H. Cheng, Z. Li, S. Chen, and J. Tian, “Dynamically tunable deep subwavelength high-order anomalous reflection using graphene metasurfaces,” Adv. Opt. Mater. 6(3), 1701047 (2018).
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Apl Photonics (1)

T. Roy, S. Zhang, I. L. W. Jung, M. Troccoli, F. Capasso, and D. Lopez, “Dynamic metasurface lens based on MEMS Technology,” Apl Photonics 3(2), 021302 (2018).
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Appl. Phys. B (1)

A. W. Zeng, M. X. Gao, and B. Guo, “Slow light in a hyperbolic metamaterial waveguide cladded with arbitrary nonlinear dielectric materials,” Appl. Phys. B 124(7), 146 (2018).
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Appl. Phys., A Mater. Sci. Process. (1)

S. S. Islam, M. R. I. Faruque, M. T. Islam, and M. T. Ali, “A new wideband negative refractive index metamaterial for dual-band operation,” Appl. Phys., A Mater. Sci. Process. 123(4), 252 (2017).
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IEEE J. Sel. Top. Quantum Electron. (1)

H. Deng, X. Chen, B. A. Malomed, N. C. Panoiu, and F. Ye, “Tunability and robustness of Dirac points of photonic nanostructures,” IEEE J. Sel. Top. Quantum Electron. 22(5), 5000509 (2016).
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IEEE Photonic. Tech. L. (1)

S. Pahlavan and V. Ahmadi, “Novel optical demultiplexer design using oblique boundary in hetero photonic crystals,” IEEE Photonic. Tech. L. 29(6), 511–514 (2017).
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IEEE Photonics J. (2)

X. Shang, A. M. Trinidad, P. Joshi, J. D. Smet, D. Cuypers, and H. D. Smet, “Tunable optical beam deflection via liquid crystal gradient refractive index generated by highly resistive polymer film,” IEEE Photonics J. 8(3), 6500411 (2016).
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D. Su, X. Y. Zhang, Y. L. Ma, F. Shan, J. Y. Wu, X. C. Fu, L. J. Zhang, K. Q. Yuan, and T. Zhang, “Real-time electro-optical tunable hyperlens under subwavelength scale,” IEEE Photonics J. 10(1), 4600109 (2018).
[Crossref]

J. Appl. Phys. (2)

T. Gric and O. Hess, “Controlling hybrid-polarization surface plasmon polaritons in dielectric-transparent conducting oxides metamaterials via their effective properties,” J. Appl. Phys. 122(19), 193105 (2017).
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Nat. Commun. (3)

Y. C. Chang, C. H. Liu, C. H. Liu, S. Zhang, S. R. Marder, E. E. Narimanov, Z. Zhong, and T. B. Norris, “Realization of mid-infrared graphene hyperbolic metamaterials,” Nat. Commun. 7(1), 10568 (2016).
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E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, M. Faraji-Dana, and A. Faraon, “MEMS-tunable dielectric metasurface lens,” Nat. Commun. 9(1), 812 (2018).
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New J. Phys. (1)

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Opt. Commun. (1)

H. Zhao, Z. Chen, F. Su, G. Ren, F. Liu, and J. Yao, “Terahertz wavefront manipulating by double-layer graphene ribbons metasurface,” Opt. Commun. 402, 523–526 (2017).
[Crossref]

Opt. Express (8)

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M. Kim, S. Kim, and S. Kim, “Optical bistability based on hyperbolic metamaterials,” Opt. Express 26(9), 11620–11632 (2018).
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Opt. Lett. (1)

Opt. Mater. Express (1)

Optik (Stuttg.) (1)

H. Xu, L. Wu, X. Dai, Y. Gao, and Y. Xiang, “Tunable infrared plasmonic waveguides using graphene based hyperbolic metamaterials,” Optik (Stuttg.) 127(20), 9640–9646 (2016).
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Photon. Res. (1)

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Phys. Rev. B Condens. Matter Mater. Phys. (3)

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P. Wang, C. Tang, Z. Yan, Q. Wang, F. Liu, J. Chen, Z. Xu, and C. Sui, “Graphene-based superlens for subwavelength optical imaging by graphene plasmon resonances,” Plasmonics 11(2), 515–522 (2016).
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Y. Xiang, X. Dai, J. Guo, H. Zhang, S. Wen, and D. Tang, “Critical coupling with graphene-based hyperbolic metamaterials,” Sci. Rep. 4(1), 5483 (2014).
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Science (1)

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Figures (5)

Fig. 1
Fig. 1 (a) Schematic diagram of the HMM consisting of periodic graphene/dielectric layers. (b) The x-z view of the designed structure.
Fig. 2
Fig. 2 (a, b) Re(ε||) and Im(ε||) as functions of f and μc. (c) Re(ε||)-μc and Im(ε||)-μc curves at f = 30 THz.
Fig. 3
Fig. 3 EFC diagrams and Finite-Difference Time-Domain (FDTD) simulation results at f = 30 THz. (a1), (a2), (a3) and (b1), (b2), (b3) are for μc = 0.2 and 0.65 eV, respectively. k i and S i represent the directions of wave vector and energy velocity in region i (i = 1, 2, 3). The HMM is located between x = 50 and 150 μm [See the enlargements of region 2 in Figs. 3(a3) and 3(b3)]. A Gaussian beam with a waist width of 10 μm is located at x = 10 μm.
Fig. 4
Fig. 4 EFC diagrams (a1-d1) and the corresponding |E| distributions (a2-d2) for μc = 0.65, 0.75, 0.85, and 0.95 eV, respectively. The focal points are denoted by the red dashed lines. The HMM region 2 (denoted by two white dashed lines) locates between x = 50 and 150 μm.
Fig. 5
Fig. 5 (a) The relationship between the focal length l of the HMM lens and the chemical potential μc. (b) Dependence of transmittance on μc. The inset shows the enlargement of the red rectangle. (c) The |E| profile along the y direction at focal points in region 3 for μc = 0.65, 0.75, 0.85, and 0.95 eV respectively.

Equations (5)

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σ ( f , μ c , Γ , Τ ) = k B Τ e 2 j 2 π 2 ( π f + Γ j ) ( μ c k B Τ + 2 ln ( e μ c k B Τ + 1 ) ) + 2 e 2 j ( π f + Γ j ) π 2 0 f d ( E ) f d ( E ) 4 ( π f + Γ j ) 2 4 ( E ) 2 d E
E b i a s = 2 e π 2 v F 2 ε 0 ε d [ ( k B T ) 2 μ c / k B T μ c / k B T x e x + 1 d x + k B T μ c ln ( e μ c / k B T + 1 ) + k B T μ c ln ( e μ c / k B T + 1 ) ]
ε e f f = [ ε 0 0 0 ε 0 0 0 ε ]
{ ε = ε d j σ ( f , μ c , Γ , Τ ) 2 π f ε 0 d ε = ε d
k x 2 ε + k z 2 ε = k 0 2

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