Abstract

We investigate the optical bistability of graphene in parity-time-symmetric (PT-symmetric) photonic lattices incorporated with a defect at terahertz frequencies. The field localization of the defect mode can strengthen the nonlinearity of graphene to achieve low-threshold bistability. The nonlinearity is further enhanced, and the bistability threshold decreases, by increasing the gain-loss factor in the PT-symmetric structure. The interval of upper and lower bistability thresholds is broadened as the exceptional points split. Moreover, we show the phase transition between bistability and nonbistability by modulating the incident wavelength and chemical potential of graphene. The study may find great applications in all-optical switches and optical storage.

© 2019 Optical Society of America

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  3. W. Liu, X. Li, Y. Song, C. Zhang, X. Han, H. Long, B. Wang, K. Wang, and P. Lu, “Cooperative enhancement of two-photon-absorption-induced photoluminescence from a 2D perovskite-microsphere hybrid dielectric structure,” Adv. Funct. Mater. 28, 1707550 (2018).
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  4. H. Nihei and A. Okamoto, “Photonic crystal systems for high-speed optical memory device on an atomic scale,” Proc. SPIE 4416, 470–473 (2001).
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  5. M. Soljacić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
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    [Crossref]
  8. N. M. R. Peres, Y. V. Bludov, J. E. Santos, A. P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Phys. Rev. B 90, 125425 (2014).
    [Crossref]
  9. M. Sanderson, Y. S. Ang, S. Gong, T. Zhao, M. Hu, R. Zhong, X. Chen, P. Zhang, C. Zhang, and S. Liu, “Optical bistability induced by nonlinear surface plasmon polaritons in graphene in terahertz regime,” Appl. Phys. Lett. 107, 203113 (2015).
    [Crossref]
  10. L. Jiang, J. Guo, L. Wu, X. Dai, and Y. Xiang, “Manipulating the optical bistability at terahertz frequency in the Fabry-Perot cavity with graphene,” Opt. Express 23, 31181–31191 (2015).
    [Crossref]
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  13. D. Zhao, Z. Q. Wang, H. Long, K. Wang, B. Wang, and P. X. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49, 163 (2017).
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  14. J. Liu, S. Park, D. Nowak, M. Tian, Y. Wu, H. Long, K. Wang, B. Wang, and P. Lu, “Near-field characterization of graphene plasmons by photo-induced force microscopy,” Laser Photon. Rev. 12, 1800040 (2018).
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  15. D. Zhao, S. Ke, Q. Liu, B. Wang, and P. Lu, “Giant Goos-Hänchen shifts in non-Hermitian dielectric multilayers incorporated with graphene,” Opt. Express 26, 2817–2828 (2018).
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  16. P. Meng, D. Zhao, D. Zhong, and W. Liu, “Topological plasmonic modes in graphene coated nanowire arrays,” Opt. Quantum Electron. 51, 156 (2019).
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  17. E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A. Mikhailov, “Coherent nonlinear optical response of graphene,” Phys. Rev. Lett. 105, 097401 (2010).
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  19. M. Canonico, G. B. Adams, C. Poweleit, J. Menendez, J. B. Page, G. Harris, H. P. van der Meulen, J. M. Calleja, and J. Rubio, “Characterization of carbon nanotubes using Raman excitation profiles,” Phys. Rev. B 65, 201402 (2002).
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  21. S. Shareef, Y. S. Ang, and C. Zhang, “Room-temperature strong terahertz photon mixing in graphene,” J. Opt. Soc. Am. B 29, 274–279 (2012).
    [Crossref]
  22. T. Jiang, D. Huang, J. Cheng, X. Fan, Z. Zhang, Y. Shan, Y. Yi, Y. Dai, L. Shi, K. Liu, C. Zeng, J. Zi, J. E. Sipe, Y.-R. Shen, W.-T. Liu, and S. Wu, “Gate-tunable third-order nonlinear optical response of massless Dirac fermions in graphene,” Nat. Photonics 12, 430–436 (2018).
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    [Crossref]
  24. J. B. Khurgin, “Response to ‘Comment on “Graphene—a rather ordinary nonlinear optical material”’ [Appl. Phys. Lett. 111, 106101 (2017)],” Appl. Phys. Lett. 111, 106102 (2017).
    [Crossref]
  25. L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
    [Crossref]
  26. Z. Wen and Z. Yan, “Dynamical behaviors of optical solitons in parity–time (PT) symmetric sextic anharmonic double-well potentials,” Phys. Lett. A 379, 2025–2029 (2015).
    [Crossref]
  27. R. Wang, Q. Zhang, D. Li, S. Xu, P. Cao, Y. Zhou, W. Cao, and P. Lu, “Identification of tunneling and multiphoton ionization in intermediate Keldysh parameter regime,” Opt. Express 27, 6471–6482 (2019).
    [Crossref]
  28. D. Zhao, W. Liu, S. Ke, and Q. Liu, “Large lateral shift in complex dielectric multilayers with nearly parity–time symmetry,” Opt. Quantum Electron. 50, 323 (2018).
    [Crossref]
  29. Q. Liu, S. Ke, and W. Liu, “Mode conversion and absorption in an optical waveguide under cascaded complex modulations,” Opt. Quantum Electron. 50, 356 (2018).
    [Crossref]
  30. C. Zhai, Y. Zhang, and Q. Zhang, “Characterizing the ellipticity of an isolated attosecond pulse,” Opt. Commun. 437, 104–109 (2019).
    [Crossref]
  31. S. Ke, D. Zhao, J. Liu, Q. Liu, Q. Liao, B. Wang, and P. Lu, “Topological bound modes in anti-PT-symmetric optical waveguide arrays,” Opt. Express 27, 13858–13870 (2019).
    [Crossref]
  32. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
    [Crossref]
  33. W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6, 31095 (2016).
    [Crossref]
  34. Y. L. Xu, W. S. Fegadolli, L. Gan, M. H. Lu, X. P. Liu, Z. Y. Li, A. Scherer, and Y. F. Chen, “Experimental realization of Bloch oscillations in a parity-time synthetic silicon photonic lattice,” Nat. Commun. 7, 11319 (2016).
    [Crossref]
  35. X. F. Zhu, “Defect states and exceptional point splitting in the band gaps of one-dimensional parity-time lattices,” Opt. Express 23, 22274–22284 (2015).
    [Crossref]
  36. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
    [Crossref]
  37. Z. Wang, B. Wang, H. Long, K. Wang, and P. Lu, “Surface plasmonic lattice solitons in semi-infinite graphene sheet arrays,” J. Lightwave Technol. 35, 2960–2965 (2017).
    [Crossref]
  38. P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5, 5855–5863 (2011).
    [Crossref]
  39. L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76, 153410 (2007).
    [Crossref]
  40. D. Zhao, D. Zhong, Y. Hu, S. Ke, and W. Liu, “Imaginary modulation inducing giant spatial Goos–Hänchen shifts in one-dimensional defective photonic lattices,” Opt. Quantum Electron. 51, 113 (2019).
    [Crossref]
  41. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University, 2007).
  42. P. Y. Chen and J. Jung, “PT symmetry and singularity-enhanced sensing based on photoexcited graphene metasurfaces,” Phys. Rev. Appl. 5, 064018 (2016).
    [Crossref]
  43. M. Sakhdari, M. Farhat, and P. Y. Chen, “PT-symmetric metasurfaces: wave manipulation and sensing using singular points,” New J. Phys. 19, 065002 (2017).
    [Crossref]
  44. M. Sakhari, N. Estakhri, H. Bagci, and P. Y. Chen, “Low-threshold lasing and coherent perfect absorption in generalized PT-symmetric optical structures,” Phys. Rev. Appl. 10, 024030 (2018).
    [Crossref]
  45. D. A. Smirnova, I. V. Shadrivov, A. I. Smirnov, and Y. S. Kivshar, “Dissipative plasmon-solitons in multilayer graphene,” Laser Photon. Rev. 8, 291–296 (2014).
    [Crossref]
  46. S. A. Mikhailow, “Non-linear electromagnetic response of graphene,” Europhys. Lett. 79, 27002 (2007).
    [Crossref]
  47. X. K. Kong, S. B. Liu, H. F. Zhang, S. Y. Wang, B. R. Bian, and Y. Dai, “Tunable bistability in photonic multilayers doped by unmagnetized plasma and coupled nonlinear defects,” IEEE J. Sel. Top. Quantum Electron. 19, 8401407 (2013).
    [Crossref]
  48. P. Witoński, A. Mossakowska-Wyszyńska, and P. Szczepański, “Effect of nonlinear loss and gain in multilayer PT-symmetric Bragg grating,” IEEE J. Quantum Electron. 53, 2100111 (2017).
    [Crossref]
  49. L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity–time symmetry,” Nat. Photonics 11, 752–762 (2017).
    [Crossref]
  50. S. Savoia, G. Castaldi, V. Galdi, A. Alú, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89, 085105 (2014).
    [Crossref]
  51. O. V. Shramkova and G. P. Tsironis, “Scattering properties of PT-symmetric layered periodic structures,” J. Opt. 18, 105101 (2016).
    [Crossref]
  52. P. Bowlan, E. Martinez-Moreno, K. Reimann, T. Elsaesser, and M. Woerner, “Ultrafast terahertz response of multilayer graphene in the nonperturbative regime,” Phys. Rev. B 89, 041408 (2014).
    [Crossref]
  53. I. Al-Naib, J. E. Sipe, and M. M. Dignam, “High harmonic generation in undoped graphene: interplay of inter-and intraband dynamics,” Phys. Rev. B 90, 245423 (2014).
    [Crossref]
  54. I. Al-Naib, M. Poschmann, and M. M. Dignam, “Optimizing third-harmonic generation at terahertz frequencies in graphene,” Phys. Rev. B 91, 205407 (2015).
    [Crossref]
  55. N. Yoshikawa, T. Tamaya, and K. Tanaka, “High-harmonic generation in graphene enhanced by elliptically polarized light excitation,” Science 356, 736–738 (2017).
    [Crossref]

2019 (5)

P. Meng, D. Zhao, D. Zhong, and W. Liu, “Topological plasmonic modes in graphene coated nanowire arrays,” Opt. Quantum Electron. 51, 156 (2019).
[Crossref]

R. Wang, Q. Zhang, D. Li, S. Xu, P. Cao, Y. Zhou, W. Cao, and P. Lu, “Identification of tunneling and multiphoton ionization in intermediate Keldysh parameter regime,” Opt. Express 27, 6471–6482 (2019).
[Crossref]

C. Zhai, Y. Zhang, and Q. Zhang, “Characterizing the ellipticity of an isolated attosecond pulse,” Opt. Commun. 437, 104–109 (2019).
[Crossref]

S. Ke, D. Zhao, J. Liu, Q. Liu, Q. Liao, B. Wang, and P. Lu, “Topological bound modes in anti-PT-symmetric optical waveguide arrays,” Opt. Express 27, 13858–13870 (2019).
[Crossref]

D. Zhao, D. Zhong, Y. Hu, S. Ke, and W. Liu, “Imaginary modulation inducing giant spatial Goos–Hänchen shifts in one-dimensional defective photonic lattices,” Opt. Quantum Electron. 51, 113 (2019).
[Crossref]

2018 (7)

M. Sakhari, N. Estakhri, H. Bagci, and P. Y. Chen, “Low-threshold lasing and coherent perfect absorption in generalized PT-symmetric optical structures,” Phys. Rev. Appl. 10, 024030 (2018).
[Crossref]

D. Zhao, W. Liu, S. Ke, and Q. Liu, “Large lateral shift in complex dielectric multilayers with nearly parity–time symmetry,” Opt. Quantum Electron. 50, 323 (2018).
[Crossref]

Q. Liu, S. Ke, and W. Liu, “Mode conversion and absorption in an optical waveguide under cascaded complex modulations,” Opt. Quantum Electron. 50, 356 (2018).
[Crossref]

T. Jiang, D. Huang, J. Cheng, X. Fan, Z. Zhang, Y. Shan, Y. Yi, Y. Dai, L. Shi, K. Liu, C. Zeng, J. Zi, J. E. Sipe, Y.-R. Shen, W.-T. Liu, and S. Wu, “Gate-tunable third-order nonlinear optical response of massless Dirac fermions in graphene,” Nat. Photonics 12, 430–436 (2018).
[Crossref]

J. Liu, S. Park, D. Nowak, M. Tian, Y. Wu, H. Long, K. Wang, B. Wang, and P. Lu, “Near-field characterization of graphene plasmons by photo-induced force microscopy,” Laser Photon. Rev. 12, 1800040 (2018).
[Crossref]

D. Zhao, S. Ke, Q. Liu, B. Wang, and P. Lu, “Giant Goos-Hänchen shifts in non-Hermitian dielectric multilayers incorporated with graphene,” Opt. Express 26, 2817–2828 (2018).
[Crossref]

W. Liu, X. Li, Y. Song, C. Zhang, X. Han, H. Long, B. Wang, K. Wang, and P. Lu, “Cooperative enhancement of two-photon-absorption-induced photoluminescence from a 2D perovskite-microsphere hybrid dielectric structure,” Adv. Funct. Mater. 28, 1707550 (2018).
[Crossref]

2017 (8)

S. A. Mikhailov, “Comment on ‘Graphene—a rather ordinary nonlinear optical material’ [Appl. Phys. Lett. 104, 161116 (2014)],” Appl. Phys. Lett. 111, 106101 (2017).
[Crossref]

J. B. Khurgin, “Response to ‘Comment on “Graphene—a rather ordinary nonlinear optical material”’ [Appl. Phys. Lett. 111, 106101 (2017)],” Appl. Phys. Lett. 111, 106102 (2017).
[Crossref]

D. Zhao, Z. Q. Wang, H. Long, K. Wang, B. Wang, and P. X. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49, 163 (2017).
[Crossref]

M. Sakhdari, M. Farhat, and P. Y. Chen, “PT-symmetric metasurfaces: wave manipulation and sensing using singular points,” New J. Phys. 19, 065002 (2017).
[Crossref]

Z. Wang, B. Wang, H. Long, K. Wang, and P. Lu, “Surface plasmonic lattice solitons in semi-infinite graphene sheet arrays,” J. Lightwave Technol. 35, 2960–2965 (2017).
[Crossref]

P. Witoński, A. Mossakowska-Wyszyńska, and P. Szczepański, “Effect of nonlinear loss and gain in multilayer PT-symmetric Bragg grating,” IEEE J. Quantum Electron. 53, 2100111 (2017).
[Crossref]

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity–time symmetry,” Nat. Photonics 11, 752–762 (2017).
[Crossref]

N. Yoshikawa, T. Tamaya, and K. Tanaka, “High-harmonic generation in graphene enhanced by elliptically polarized light excitation,” Science 356, 736–738 (2017).
[Crossref]

2016 (4)

O. V. Shramkova and G. P. Tsironis, “Scattering properties of PT-symmetric layered periodic structures,” J. Opt. 18, 105101 (2016).
[Crossref]

P. Y. Chen and J. Jung, “PT symmetry and singularity-enhanced sensing based on photoexcited graphene metasurfaces,” Phys. Rev. Appl. 5, 064018 (2016).
[Crossref]

W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6, 31095 (2016).
[Crossref]

Y. L. Xu, W. S. Fegadolli, L. Gan, M. H. Lu, X. P. Liu, Z. Y. Li, A. Scherer, and Y. F. Chen, “Experimental realization of Bloch oscillations in a parity-time synthetic silicon photonic lattice,” Nat. Commun. 7, 11319 (2016).
[Crossref]

2015 (5)

X. F. Zhu, “Defect states and exceptional point splitting in the band gaps of one-dimensional parity-time lattices,” Opt. Express 23, 22274–22284 (2015).
[Crossref]

Z. Wen and Z. Yan, “Dynamical behaviors of optical solitons in parity–time (PT) symmetric sextic anharmonic double-well potentials,” Phys. Lett. A 379, 2025–2029 (2015).
[Crossref]

M. Sanderson, Y. S. Ang, S. Gong, T. Zhao, M. Hu, R. Zhong, X. Chen, P. Zhang, C. Zhang, and S. Liu, “Optical bistability induced by nonlinear surface plasmon polaritons in graphene in terahertz regime,” Appl. Phys. Lett. 107, 203113 (2015).
[Crossref]

L. Jiang, J. Guo, L. Wu, X. Dai, and Y. Xiang, “Manipulating the optical bistability at terahertz frequency in the Fabry-Perot cavity with graphene,” Opt. Express 23, 31181–31191 (2015).
[Crossref]

I. Al-Naib, M. Poschmann, and M. M. Dignam, “Optimizing third-harmonic generation at terahertz frequencies in graphene,” Phys. Rev. B 91, 205407 (2015).
[Crossref]

2014 (7)

P. Bowlan, E. Martinez-Moreno, K. Reimann, T. Elsaesser, and M. Woerner, “Ultrafast terahertz response of multilayer graphene in the nonperturbative regime,” Phys. Rev. B 89, 041408 (2014).
[Crossref]

I. Al-Naib, J. E. Sipe, and M. M. Dignam, “High harmonic generation in undoped graphene: interplay of inter-and intraband dynamics,” Phys. Rev. B 90, 245423 (2014).
[Crossref]

S. Savoia, G. Castaldi, V. Galdi, A. Alú, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89, 085105 (2014).
[Crossref]

D. A. Smirnova, I. V. Shadrivov, A. I. Smirnov, and Y. S. Kivshar, “Dissipative plasmon-solitons in multilayer graphene,” Laser Photon. Rev. 8, 291–296 (2014).
[Crossref]

N. M. R. Peres, Y. V. Bludov, J. E. Santos, A. P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Phys. Rev. B 90, 125425 (2014).
[Crossref]

Y. Xiang, X. Dai, J. Guo, S. Wen, and D. Tang, “Tunable optical bistability at the graphene-covered nonlinear interface,” Appl. Phys. Lett. 104, 051108 (2014).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order optical nonlinearity of graphene,” New J. Phys. 16, 053014 (2014).
[Crossref]

2013 (2)

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

X. K. Kong, S. B. Liu, H. F. Zhang, S. Y. Wang, B. R. Bian, and Y. Dai, “Tunable bistability in photonic multilayers doped by unmagnetized plasma and coupled nonlinear defects,” IEEE J. Sel. Top. Quantum Electron. 19, 8401407 (2013).
[Crossref]

2012 (3)

S. Shareef, Y. S. Ang, and C. Zhang, “Room-temperature strong terahertz photon mixing in graphene,” J. Opt. Soc. Am. B 29, 274–279 (2012).
[Crossref]

N. Brandonisio, P. Heinricht, S. Osborne, A. Amann, and S. O’Brien, “Bistability and all-optical memory in dual-mode diode lasers with time-delayed optical feedback,” IEEE Photon. J. 4, 95–103 (2012).
[Crossref]

C. Argyropoulos, P. Y. Chen, F. Monticone, G. D’Aguanno, and A. Alù, “Nonlinear plasmonic cloaks to realize giant all-optical scattering switching,” Phys. Rev. Lett. 108, 263905 (2012).
[Crossref]

2011 (2)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5, 5855–5863 (2011).
[Crossref]

2010 (2)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A. Mikhailov, “Coherent nonlinear optical response of graphene,” Phys. Rev. Lett. 105, 097401 (2010).
[Crossref]

2009 (1)

A. R. Wright, X. G. Xu, J. C. Cao, and C. Zhang, “Strong nonlinear optical response of graphene in the terahertz regime,” Appl. Phys. Lett. 95, 072101 (2009).
[Crossref]

2007 (2)

L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76, 153410 (2007).
[Crossref]

S. A. Mikhailow, “Non-linear electromagnetic response of graphene,” Europhys. Lett. 79, 27002 (2007).
[Crossref]

2003 (1)

2002 (2)

M. Soljacić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[Crossref]

M. Canonico, G. B. Adams, C. Poweleit, J. Menendez, J. B. Page, G. Harris, H. P. van der Meulen, J. M. Calleja, and J. Rubio, “Characterization of carbon nanotubes using Raman excitation profiles,” Phys. Rev. B 65, 201402 (2002).
[Crossref]

2001 (1)

H. Nihei and A. Okamoto, “Photonic crystal systems for high-speed optical memory device on an atomic scale,” Proc. SPIE 4416, 470–473 (2001).
[Crossref]

1992 (1)

J. He and M. Cada, “Combined distributed feedback and Fabry–Perot structures with a phase-matching layer for optical bistable devices,” Appl. Phys. Lett. 61, 2150–2152 (1992).
[Crossref]

Adams, G. B.

M. Canonico, G. B. Adams, C. Poweleit, J. Menendez, J. B. Page, G. Harris, H. P. van der Meulen, J. M. Calleja, and J. Rubio, “Characterization of carbon nanotubes using Raman excitation profiles,” Phys. Rev. B 65, 201402 (2002).
[Crossref]

Almeida, V. R.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Al-Naib, I.

I. Al-Naib, M. Poschmann, and M. M. Dignam, “Optimizing third-harmonic generation at terahertz frequencies in graphene,” Phys. Rev. B 91, 205407 (2015).
[Crossref]

I. Al-Naib, J. E. Sipe, and M. M. Dignam, “High harmonic generation in undoped graphene: interplay of inter-and intraband dynamics,” Phys. Rev. B 90, 245423 (2014).
[Crossref]

Alú, A.

S. Savoia, G. Castaldi, V. Galdi, A. Alú, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89, 085105 (2014).
[Crossref]

Alù, A.

C. Argyropoulos, P. Y. Chen, F. Monticone, G. D’Aguanno, and A. Alù, “Nonlinear plasmonic cloaks to realize giant all-optical scattering switching,” Phys. Rev. Lett. 108, 263905 (2012).
[Crossref]

P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5, 5855–5863 (2011).
[Crossref]

Amann, A.

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Figures (8)

Fig. 1.
Fig. 1. Schematic of PT-symmetric photonic lattices incorporated with graphene. For dielectrics A, B, A′, and B′, the thicknesses are a quarter of optical wavelength. The graphene is embedded in the center of defect layer C. The refractive indices of dielectrics A, B, A′, B′, and C are na=1.38+iq, nb=2.35−iq, na′=1.38−iq, nb′=2.35+iq, and nc=1.5, respectively.

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Fig. 2.
Fig. 2. (a), (b) Transmittance and reflectance spectrum for three different Bragg periodic numbers N=3, 4, and 5, respectively. (c) Electric field intensity (|Ez|2) distribution of the DM for N=4. The incident wavelength is λ=30.68  μm. The chemical potential of graphene is μ=0.15  eV in (a)–(c).

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Fig. 3.
Fig. 3. (a), (b) Dependence of transmittance and transmitted intensity on incident light intensity for three different Bragg periodic numbers, respectively. The wavelength of incidence light λ=30.9  μm.

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Fig. 4.
Fig. 4. (a) Reflectance of light incident from the left. (b) Reflectance of light incident from the right. (c) Transmittance of light. (d) Q factor of PT-symmetric photonic lattices and EPs splitting. The Bragg periodic number N=4.

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Fig. 5.
Fig. 5. (a), (c) Dependence of transmittance and transmitted intensity on the gain-loss factor and the incident intensity of light, respectively. (b) The maximum of transmittance and corresponding incident intensity varying with the gain-loss factor. (d) Bistability threshold varying with the gain-loss factor. (a)–(d) are with the incident wavelength λ=31  μm. The Bragg periodic number N=4, and the chemical potential μ=0.15  eV.

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Fig. 6.
Fig. 6. (a) Upper and lower thresholds of bistability varying with the wavelength of incident light. The Bragg periodic number N=4, and the gain-loss factor q=0.1. (b) Phase transition of bistability to nonbistability in the parameter space spanned by the wavelength detuning and gain-loss factor.

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Fig. 7.
Fig. 7. (a) Phase transition of bistability to nonbistability in the parameter space of the chemical potential and gain-loss factor. (b), (c), (d) Bistability threshold varying with the chemical potential for three different gain-loss factors q=0, 0.05, and 0.1, respectively. The wavelength of incident light λ=31  μm.

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Fig. 8.
Fig. 8. (a), (b), (d), (e) Transmittance and reflectance for light impinging upon the PT-symmetric photonic lattices, respectively. (c) Electric field intensity (|Ez|2) distribution of CPA laser state. Gain is only considered in dielectrics for (a)–(c), while loss is only included in dielectrics for (d) and (e).

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Equations (6)

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

σ(1)(ω,μ,τ,T)=−ie2(ω+iτ−1)πℏ2[∫−∞+∞|ε|(ω+iτ−1)2∂fd(ε)∂εdε−∫0+∞∂fd(−ε)−∂fd(ε)(ω+iτ−1)2−4(ε/ℏ)2dε],
σintra=ie2kBTπℏ2(ω+iτ−1)[μkBT+2ln(exp(−μckBT)+1)].
σinter=ie24πℏln[2|μ|−ℏ(ω+iτ−1)2|μ|+ℏ(ω+iτ−1)].
(ElHl)=(cos ϕl−iηl sin ϕl−iηl sin ϕlcos ϕl)(El+1Hl+1),
t=2η1m11η1+m12η1ηN+1+m21+m22η1,
r=m11η1+m12η1ηN+1−m21−m22ηN+1m11η1+m12η1ηN+1+m21+m22ηN+1,

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