Abstract

We propose a hybrid structure where graphene is inserted to the interface of two one-dimensional photonic crystals (1D PCs). The two PCs are designed to have opposite topological properties, and at the interface, topological edge modes exist. The edge modes exist at both the fundamental frequency (FF) and the third harmonic (TH) frequency. This double resonant structure will enhance the nonlinear responses of graphene greatly, including Kerr nonlinearity and TH generation. We discuss these two kinds of nonlinearities both at terahertz (THz) and near-infrared (NIR) frequencies. The influence of Kerr nonlinearity on the resonant frequencies is considered, when we calculate the TH generation. At THz frequency, low-threshold bistability (about 8MW/cm2) is obtained and the TH generation efficiency of 2.5% is achieved with incident intensity of 10MW/cm2. At NIR frequency, the nonlinear conductivities of graphene are about 7 orders lower. Bistability is unlikely to happen with incident intensity below 1GW/cm2. The TH generation efficiency is only about 5×10−6 with incident intensity of 25MW/cm2. The proposed structure is more suitable to work as a low-threshold saturable absorber at NIR frequency. These results may be helpful both for a better understanding of graphene’s nonlinear responses in a double resonant structure and for potential applications in THz nonlinear devices and NIR nanophotonics.

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

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References

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2019 (1)

S. A. Mikhailov, “Theory of the strongly nonlinear electrodynamic response of graphene: A hot electron model,” Phys. Rev. B 100(11), 115416 (2019).
[Crossref]

2018 (10)

K.-J. Tielrooij, N. C. H. Hesp, A. Principi, M. B. Lundeberg, E. A. A. Pogna, L. Banszerus, Z. Mics, M. Massicotte, P. Schmidt, D. Davydovskaya, D. G. Purdie, I. Goykhman, G. Soavi, A. Lombardo, K. Watanabe, T. Taniguchi, M. Bonn, D. Turchinovich, C. Stampfer, A. C. Ferrari, G. Cerullo, M. Polini, and F. H. L. Koppens, “Out-of-plane heat transfer in van der Waals stacks through electron–hyperbolic phonon coupling,” Nat. Nanotechnol. 13(1), 41–46 (2018).
[Crossref]

H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-J. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, “Extremely efficient terahertz high-harmonic generation in graphene by hot Dirac fermions,” Nature 561(7724), 507–511 (2018).
[Crossref]

G. Soavi, G. Wang, H. Rostami, D. G. Purdie, D. D. Fazio, T. Ma, B. Luo, J. Wang, A. K. Ott, D. Yoon, S. A. Bourelle, J. E. Muench, I. Goykhman, S. D. Conte, M. Celebrano, A. Tomadin, M. Polini, G. Cerullo, and A. C. Ferrari, “Broadband, electrically tunable third-harmonic generation in graphene,” Nat. Nanotechnol. 13(7), 583–588 (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(7), 430–436 (2018).
[Crossref]

N. Vermeulen, D. Castelló-Lurbe, M. Khoder, I. Pasternak, A. Krajewska, T. Ciuk, W. Strupinski, J. Cheng, H. Thienpont, and J. V. Erps, “Graphene’s nonlinear-optical physics revealed through exponentially growing self-phase modulation,” Nat. Commun. 9(1), 2675 (2018).
[Crossref]

K. Alexander, N. A. Savostianova, S. A. Mikhailov, D. Van Thourhout, and B. Kuyken, “Gate-Tunable Nonlinear Refraction and Absorption in Graphene-Covered Silicon Nitride Waveguides,” ACS Photonics 5(12), 4944–4950 (2018).
[Crossref]

D. Kundys, B. Van Duppen, O. P. Marshall, F. Rodriguez, I. Torre, A. Tomadin, M. Polini, and A. N. Grigorenko, “Nonlinear Light Mixing by Graphene Plasmons,” Nano Lett. 18(1), 282–287 (2018).
[Crossref]

C. Beckerleg, T. J. Constant, I. Zeimpekis, S. M. Hornett, C. Craig, D. W. Hewak, and E. Hendry, “Cavity enhanced third harmonic generation in graphene,” Appl. Phys. Lett. 112(1), 011102 (2018).
[Crossref]

W. Gao, X. Hu, C. Li, J. Yang, Z. Chai, J. Xie, and Q. Gong, “Fano-resonance in one-dimensional topological photonic crystal heterostructure,” Opt. Express 26(7), 8634 (2018).
[Crossref]

G. X. Ni, A. S. McLeod, Z. Sun, L. Wang, L. Xiong, K. W. Post, S. S. Sunku, B.-Y. Jiang, J. Hone, C. R. Dean, M. M. Fogler, and D. N. Basov, “Fundamental limits to graphene plasmonics,” Nature 557(7706), 530–533 (2018).
[Crossref]

2017 (7)

L. Wang, T. Wang, S. Zhang, P. Xie, and X. Zhang, “Larger enhancement in four-wave mixing from graphene embedded in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 34(9), 2000 (2017).
[Crossref]

T. Wang and X. Zhang, “Improved third-order nonlinear effect in graphene based on bound states in the continuum,” Photonics Res. 5(6), 629 (2017).
[Crossref]

J. W. You, J. You, M. Weismann, and N. C. Panoiu, “Double-resonant enhancement of third-harmonic generation in graphene nanostructures,” Philos. Trans. R. Soc., A 375(2090), 20160313 (2017).
[Crossref]

K. Alexander, N. A. Savostianova, S. A. Mikhailov, B. Kuyken, and D. Van Thourhout, “Electrically Tunable Optical Nonlinearities in Graphene-Covered SiN Waveguides Characterized by Four-Wave Mixing,” ACS Photonics 4(12), 3039–3044 (2017).
[Crossref]

G. Li, S. Zhang, and T. Zentgraf, “Nonlinear photonic metasurfaces,” Nat. Rev. Mater. 2(5), 17010 (2017).
[Crossref]

A. Marini, J. D. Cox, and F. J. G. de Abajo, “Theory of graphene saturable absorption,” Phys. Rev. B 95(12), 125408 (2017).
[Crossref]

J. Guo, L. Jiang, Y. Jia, X. Dai, Y. Xiang, and D. Fan, “Low threshold optical bistability in one-dimensional gratings based on graphene plasmonics,” Opt. Express 25(6), 5972–5981 (2017).
[Crossref]

2016 (5)

M. S. Nezami, D. Yoo, G. Hajisalem, S.-H. Oh, and R. Gordon, “Gap Plasmon Enhanced Metasurface Third-Harmonic Generation in Transmission Geometry,” ACS Photonics 3(8), 1461–1467 (2016).
[Crossref]

D. Smirnova and Y. S. Kivshar, “Multipolar nonlinear nanophotonics,” Optica 3(11), 1241–1255 (2016).
[Crossref]

M. Weismann and N. C. Panoiu, “Theoretical and computational analysis of second- and third-harmonic generation in periodically patterned graphene and transition-metal dichalcogenide monolayers,” Phys. Rev. B 94(3), 035435 (2016).
[Crossref]

S. A. Mikhailov, “Quantum theory of the third-order nonlinear electrodynamic effects of graphene,” Phys. Rev. B 93(8), 085403 (2016).
[Crossref]

K. H. Choi, C. W. Ling, K. F. Lee, Y. H. Tsang, and K. H. Fung, “Simultaneous multi-frequency topological edge modes between one-dimensional photonic crystals,” Opt. Lett. 41(7), 1644 (2016).
[Crossref]

2015 (6)

R. Yu, V. Pruneri, and F. J. G. de Abajo, “Resonant Visible Light Modulation with Graphene,” ACS Photonics 2(4), 550–558 (2015).
[Crossref]

W. Zouaghi, D. Voß, M. Gorath, N. Nicoloso, and H. G. Roskos, “How good would the conductivity of graphene have to be to make single-layer-graphene metamaterials for terahertz frequencies feasible?” Carbon 94, 301–308 (2015).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third-order nonlinearity of graphene: Effects of phenomenological relaxation and finite temperature,” Phys. Rev. B 91(23), 235320 (2015).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Numerical study of the optical nonlinearity of doped and gapped graphene: From weak to strong field excitation,” Phys. Rev. B 92(23), 235307 (2015).
[Crossref]

K. O’Brien, H. Suchowski, J. Rho, A. Salandrino, B. Kante, X. Yin, and X. Zhang, “Predicting nonlinear properties of metamaterials from the linear response,” Nat. Mater. 14(4), 379–383 (2015).
[Crossref]

Z. Mics, K.-J. Tielrooij, K. Parvez, S. A. Jensen, I. Ivanov, X. Feng, K. Müllen, M. Bonn, and D. Turchinovich, “Thermodynamic picture of ultrafast charge transport in graphene,” Nat. Commun. 6(1), 7655 (2015).
[Crossref]

2014 (2)

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface Impedance and Bulk Band Geometric Phases in One-Dimensional Systems,” Phys. Rev. X 4(2), 021017 (2014).
[Crossref]

M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89(16), 165139 (2014).
[Crossref]

2013 (3)

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys.: Condens. Matter 25(21), 215301 (2013).
[Crossref]

Y. Q. An, F. Nelson, J. U. Lee, and A. C. Diebold, “Enhanced Optical Second-Harmonic Generation from the Current-Biased Graphene/SiO2/Si(001) Structure,” Nano Lett. 13(5), 2104–2109 (2013).
[Crossref]

N. Kumar, J. Kumar, C. Gerstenkorn, R. Wang, H.-Y. Chiu, A. L. Smirl, and H. Zhao, “Third harmonic generation in graphene and few-layer graphite films,” Phys. Rev. B 87(12), 121406 (2013).
[Crossref]

2012 (5)

H. Zhang, S. Virally, Q. Bao, L. K. Ping, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856–1858 (2012).
[Crossref]

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

Q. Bao and K. P. Loh, “Graphene Photonics, Plasmonics, and Broadband Optoelectronic Devices,” ACS Nano 6(5), 3677–3694 (2012).
[Crossref]

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

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nat. Photonics 6(8), 554–559 (2012).
[Crossref]

2010 (2)

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (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(9), 097401 (2010).
[Crossref]

2009 (2)

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
[Crossref]

2008 (1)

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine Structure Constant Defines Visual Transparency of Graphene,” Science 320(5881), 1308 (2008).
[Crossref]

2007 (2)

L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76(15), 153410 (2007).
[Crossref]

S. A. Mikhailov, “Non-linear electromagnetic response of graphene,” Europhys. Lett. 79(2), 27002 (2007).
[Crossref]

2004 (1)

K. S. Novoselov, “Electric Field Effect in Atomically Thin Carbon Films,” Science 306(5696), 666–669 (2004).
[Crossref]

a, E.

G. Soavi, G. Wang, H. Rostami, A. Tomadin, O. Balci, I. Paradeisanos, and E. a. A. Pogna, G. Cerullo, E. Lidorikis, M. Polini, and A. C. Ferrari, “Hot electrons modulation of third harmonic generation in graphene,” arXiv:1903.00989 [cond-mat.mes-hall] (2019).

Alexander, K.

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M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89(16), 165139 (2014).
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Figures (9)

Fig. 1.
Fig. 1. (a) Schematic diagram of the proposed structure in THz range. The band structure of PC1 (b) and PC2 (c) with nA = 1.9, nB = 1.46, α = 1.665, the red strip represents band gap with $\zeta > 0$, the blue strip represents gap with $\zeta < 0$. The Zak phase of each band is labeled in red. (d) The reflectance spectrum of the PC1-PC2 structure, N1 = N2 = 7 and ns = 3.415.
Fig. 2.
Fig. 2. (a) The reflectance spectra around fFF with varied EF. (b) The reflectance spectra around fTH with varied EF. (c) The normalized electric field distribution with varied τ at fFF = 3.05THz and EF = 0.5eV. (d) The normalized electric field distribution with varied τ at fTH= 9.07THz and EF = 0.5eV.
Fig. 3.
Fig. 3. (a) The reflectance spectra around 3THz with varied I0, EF= 0.5eV and τ = 0.5ps. (b) The bistable reflectance with varying I0 and different τ at 3THz and EF = 0.5eV.
Fig. 4.
Fig. 4. (a) The reflectance spectra around 3THz with varied I0, EF = 0.5eV and τ = 0.5ps, the vertical red dash-dotted line corresponds to fTH/3. (b) The TH generation efficiency η around 3THz with the same parameters in (a). (c) η with increasing I0 and EF = 0.5eV at fFF= 3.023THz.
Fig. 5.
Fig. 5. (a) The electron temperature Te (blue solid line, left axis) and the ratio of $\mu /{E_F}$ (red dashed line, right axis) with varied I0. (b) The $|{\chi_K^{(3 )}} |$ (blue solid line, left axis) and $|{\chi_{TH}^{(3 )}} |$ (red dashed line, right axis) with varied Te. (c) η with increasing I0 and different Te, EF = 0.5 eV, τ = 0.5ps, and fFF= 3.023THz.
Fig. 6.
Fig. 6. (a) Schematic diagram of the proposed structure in NIR range. (b) The linear reflectance spectra with varied EF, the vertical red dash-dotted line corresponds to 3λTH = 1.55µm. (c) The normalized electric field distribution with EF = 0.5 eV and τ = 0.1ps at λFF = 1.55µm. (d) The normalized electric field distribution with EF = 0.5 eV and τ = 0.1ps at λTH = 0.517µm.
Fig. 7.
Fig. 7. (a) Graphene’s Kerr nonlinear susceptibility $ \chi _K^{(3 )}$ at 1.55µm with varying EF and τ = 0.1ps. (b) The nonlinear reflectance spectra with different I0, EF = 0.8eV and τ = 0.1ps.
Fig. 8.
Fig. 8. (a) Graphene’s TH nonlinear susceptibility $ \chi _{TH}^{(3 )} $ at 1.55µm with varying EF and τ = 0.1ps. (b) The TH generation efficiency η with different I0, EF = 0.8eV and τ = 0.1ps. (c) η with increasing I0 at 1.55µm.
Fig. 9.
Fig. 9. (a) The electron temperature Te (blue solid line, left axis) and the ratio of $\mu /{E_F}$ (red dashed line, right axis) with varied I0. (b) The $\textrm{Im}({\chi_K^{(3 )}} )$ (blue solid line, left axis) and $|{\chi_{TH}^{(3 )}} |$ (red dashed line, right axis) with varied Te. The inset is the enlarged curve near 500 K. (c) η with increasing I0 and different Te, EF = 0.8 eV, τ = 0.1ps, and λFF= 1.55µm.

Equations (14)

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cos ( q Λ ) = cos k A d A cos k B d B 1 2 ( z A z B + z B z A ) sin k A d A sin k B d B ,
sgn [ ζ ( n ) ] = ( 1 ) n ( 1 ) l exp ( i m = 0 n 1 θ m Zak ) ,
[ H y E x ] | z = 0 = M d [ H y E x ] | z = d = [ cos k z d i k 0 ε / i k 0 ε ( k z Z 0 ) ( k z Z 0 ) sin k z d i k z Z 0 / i k z Z 0 ( k 0 ε ) ( k 0 ε ) sin k z d cos k z d ] [ H y E x ] | z = d ,
[ H y 0 E x 0 ] = M g [ H y 0 + E x 0 + ] = [ 1 σ l 0 1 ] [ H y 0 + E x 0 + ] ,
[ H y 0 E x 0 ] = ( M d B / d B 2 2 M d A M d B / d B 2 2 ) N 1 M g ( M d A / d A 2 2 M d B M d A / d A 2 2 ) N 2 [ H y s E x s ] ,
[ H y 0 + E x 0 + ] = ( M d A / d A 2 2 M d B M d A / d A 2 2 ) N 2 [ ( ω ε 0 ε s / ω ε 0 ε s k z s k z s ) T T ] ,
M g = [ 1 σ l + 3 σ K ( 3 ) | E x 0 + | 2 0 1 ] ,
[ H y 0 E x 0 ] = [ ( ω ε 0 / ω ε 0 k z 0 k z 0 ) E x i E x i ] + [ ( ω ε 0 / ω ε 0 k z 0 k z 0 ) E x r E x r ] ,
{ [ H ~ y 0 E ~ x 0 ] = N ~ [ H ~ y 0 E ~ x 0 ] = [ N ~ 11 R ~ k ~ z 0 / k ~ z 0 ( ω ~ ε 0 ) N ~ 12 R ~ ( ω ~ ε 0 ) N ~ 12 R ~ N ~ 21 R ~ k ~ z 0 / N ~ 21 R ~ k ~ z 0 ( ω ~ ε 0 ) ( ω ~ ε 0 ) N ~ 22 R ~ ] = O ~ R ~ [ H ~ y 0 + E ~ x 0 + ] = M ~ [ H ~ y s E ~ x s ] = [ M ~ 11 T ~ + k ~ z s / k ~ z s ( ω ~ ε 0 ε ~ s ) M ~ 12 ( ω ~ ε 0 ε ~ s ) M ~ 12 T ~ M ~ 21 T ~ + k ~ z s / M ~ 21 T ~ + k ~ z s ( ω ~ ε 0 ε ~ s ) M ~ 22 ( ω ~ ε 0 ε ~ s ) M ~ 22 T ~ ] = P ~ T ~ ,
[ O ~ 1 O ~ 2 ] R ~ [ P ~ 1 P ~ 2 ] T ~ = [ σ ~ l O ~ 2 R ~ + σ T H ( 3 ) ( E x 0 + ) 3 0 ] ,
σ l ( ω ) = e 2 π 2 i ( ω + i τ 1 ) [ E F T 0 d E f E f E 1 4 E 2 / [ 2 ( ω + i τ 1 ) 2 ] ] ,
σ ( 3 ) ( E F , T ) = 1 k B T d x f x ( 1 f x ) σ ( 3 ) ( x , 0 ) ,
E n l ( 3 ω ) graphene σ T H ( 3 ) E x 3 ( ω ) E ~ x ( 3 ω ) d S ,
T e = T 0 + τ r α c v F Δ t

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