## Abstract

We have established the theoretical relation of nonlinear optical response with respect to the dielectric/nonlinear graphene/dielectric heterostructures and further demonstrated the tunable optical bistability at terahertz frequencies. It is shown that the hysteretic behavior is strongly dependent on the Fermi energy of graphene, and the threshold electric fields could be correspondingly adjusted with the continuous tuning of Fermi Energy level. It is clear that the bistable thresholds can be lowered dramatically by decreasing the Fermi energy of graphene, at the same time the optical hysteresis width is narrowed. Moreover, we have confirmed that the optical bistability can be tuned by adjusting the incident illumination angle, or by varying the thickness and permittivity of the dielectric slabs. Our contribution might provide a new avenue of fabricating graphene based optical switching device that could even operate at terahertz regime.

© 2015 Optical Society of America

## 1. Introduction

Optical bistability (OB) is a phenomenon characterized by certain resonant optical structures whereby it is feasible to exhibit two well-discriminated stable transmission states, depending upon the history of the input [1,2]. Such a bistable device may be useful for all-optical switching [3], optical transistor [4], and optical memory [5], and hence can be used for high-speed processing of optical signals and provide a promising alternative to their electronic counterparts [6]. Fabry-Perot cavity filled by nonlinear medium is considered as a classic nonlinear feedback configuration for achieving OB. In recent decades, researchers have focused on optimizing optical bistable devices by down-scaling the structure size, increasing the switching speed, lowering the operation power, and broadening the range of operation temperature. Both improved nonlinear materials and efficient device configurations are being sought after [1]. Recently, there has been a great deal of interest in exploring OB in nanostructures, such as, photonic crystal cavities [7], sub-wavelength metallic gratings [8], metallic gap waveguide nano-cavities [9], negative-index material [10], metamaterial [11], and nanoantenna arrays loaded with nonlinear materials [12, 13]. However, the conventional nonlinear Kerr materials display very weak nonlinear response. Hence in order to achieve large optical nonlinearity, thick nonlinear media with large Kerr nonlinear index are generally required. The bulk optical bistable device is disadvantage to application in the integrated optical element. More importantly, OB with tunable optical response is also intrinsically important for the applications in all optical devices. Based on the open and close z-scan measurement, Zhang et al. had ambiguously confirmed the large nonlinear Kerr index in graphene, which is several orders of magnitude larger than that of conventional bulk materials [14, 15]. In association with the ultrafast optical response and gate-variable optical conductivity [16, 17], graphene might be expected to open a new possibility of tunable, compact, and low threshold OB as a unique nonlinear optical material.

Graphene has attracted intensive scientific interest owing to its incredible physical properties showing great potential applications in nano-electronic devices and optoelectronic devices, and it has outstanding optical properties [18, 19], such as, strong light-graphene interaction, broadband and high-speed operation, etc. Some graphene/dielectric heterostructures are also proposed to manipulate terahertz radiations [25–27]. Especially, the electrical tunability of conductivity of graphene could potentially open a new possibility of tunable optical sensor [20], metamaterials [21,22], terahertz absorber [23], and Goos-Hänchen effect [24], etc. It provides a scheme for controlling OB via suitably varying the applied voltage on graphene. Recent investigations have demonstrated that graphene and graphene related nano-materials have superior third-order nonlinear optical properties due to the linear band structure which allows for the interband optical transitions at all photon energies [28, 29]. As a result, graphene photonics have been extended to multifunction nonlinear devices including mode-locked laser [30] and optical limiter [31]. Recently, by depositing graphene with silicon photonic crystal cavity, people are able to achieve an effective nonlinear optical device that can enable ultra-low-power resonant OB, self-induced regenerative oscillations and coherent four-wave mixing [32]. We have theoretically investigated OB of reflection at the interface between graphene and Kerr-type nonlinear substrates [33] and experimentally demonstrated the OB and all-optical switching in graphene nanobubbles [34]. Peres et al., find that a single layer of graphene shows an OB in the lower THz frequency range for nonlinear graphene suspending in air [35]. Here, we investigate theoretically OB of dielectric/nonlinear graphene/dielectric heterostructures in the THz frequencies range and demonstrate the engineering of graphene nanostructures as new optical nonlinear media for generating OB. We have confirmed an approach on how to manipulate OB through engineering the Fermi energy of graphene, adjusting the incident illumination angle, and varying the thickness and permittivity of the dielectric slabs. Graphene-based optical devices with intrinsic OB allow us to explore the promise of using such nonlinear optical elements as the building block of future digital all-optical circuit.

## 2. Theoretical models and methods

The sandwiched structure is shown in Fig. 1, which consisted of four different layers of dielectric. Dielectric 1 with permittivity *ε*_{1} and dielectric 4 with permittivity *ε*_{4} are superstrate and substrate, respectively. The dielectric slabs 2 (*ε*_{2}) and 3 (*ε*_{3}) with high dielectric constants are inserted between dielectrics 1 and 4. The thicknesses of dielectric slabs 2 and 3 are *d*_{2} and *d*_{3}, respectively. The graphene sheet is sandwiched between dielectric slabs 2 and 3. Without considering the external magnetic field and under the random-phase approximation, the isotropic surface conductivity of graphene *σ*_{0} is written as the sum of the intraband *σ _{intra}* and the interband term

*σ*, where

_{inter}*ω*is the frequency of the incident light,

*E*is the Fermi energy,

_{F}*τ*is the electron-phonon relaxation time, and

*T*is a temperature in K.

*e*,

*h̄*and

*k*are the universal constants related to the electron charge, reduced Plancks constant, and Boltzmann constant, respectively. The Fermi energy

_{B}*E*=

_{F}*h̄ν*(

_{F}*πn*

_{2D})

^{1/2}can be electrically controlled by an applied gate voltage due to the strong dependence of the carrier density

*n*

_{2D}on the gate voltage, where

*ν*= 10

_{F}^{6}m/s is the Fermi velocity of electrons.

The nonlinear conductivity coefficient *σ*_{3} for graphene is given by [14, 36],

*α*is the two photon coefficient in graphene and it will not be taken into account, hence

_{T}*α*= 0. Here, the negative imaginary part of

_{T}*σ*

_{3}shows the self-focusing type nonlinear response of graphene. Obviously, both

*σ*

_{0}and

*σ*

_{3}are highly dependent on the work frequency and Fermi energy, which could provide an effective route to achieve an electrically controlled optical bistable phenomenon.

We choose the *x* axis to be parallel to the interface, and *z* axis goes along with the direction of propagation. For TE polarization, the electric and magnetic fields from medium 1 are expresses as

*E*and

_{i}*E*are the amplitudes of incident electric and reflected electric fields, respectively. ${k}_{x}={k}_{0}\sqrt{{\epsilon}_{1}}\text{sin}\theta $, and ${k}_{jz}=\sqrt{{k}_{0}^{2}{\epsilon}_{j}-{k}_{x}^{2}}$, j=1,2,3,4.

_{R}*θ*is the incident angle, and

*μ*

_{0}is the magnetic permeability of free space.

In medium 2, the electric and magnetic fields are expresses as

*A*and

*B*are the forward and backward electric fields in the medium 2.

In medium 3, the electric and magnetic fields are expresses as

*C*and

*D*are the forward and backward electric fields in the medium 3.

In medium 4, the electric and magnetic fields are expresses as

By using the boundary conditions at *z* = −*d*_{2}, 0, and *z* = *d*_{3}, we can determine the coefficients of *A*, *B*, *C*, *D*, *E _{T}*, and

*E*. Here, at

_{R}*z*= −

*d*

_{2},

*E*

_{1y}(

*z*= −

*d*

_{2}) =

*E*

_{2y}(

*z*= −

*d*

_{2}),

*H*

_{1x}(

*z*= −

*d*

_{2}) =

*H*

_{2x}(

*z*= −

*d*

_{2}); and at

*z*=

*d*

_{3},

*E*

_{3y}(

*z*=

*d*

_{3}) =

*E*

_{4y}(

*z*=

*d*

_{3}),

*H*

_{3x}(

*z*=

*d*

_{3}) =

*H*

_{4x}(

*z*=

*d*

_{3}); however at

*z*= 0,

*E*

_{2y}(

*z*= 0) =

*E*

_{3y}(

*z*= 0)

*H*

_{2x}(

*z*= 0) −

*H*

_{3x}(

*z*= 0) =

*σE*

_{2y}(

*z*= 0) hence the magnetic field component

*H*

_{2x}is discontinuous due to the appearing of the graphene sheet. According to the above boundary conditions and after some complicated processes, the relation of incident electric and transmitted electric fields can be written as,

*E*is purely real and so that the transmitted electric field

_{i}*E*is complex. However, for simplify the discussion, here we suppose that the transmitted electric field

_{T}*E*is purely real and hence incident electric field

_{T}*E*is complex. We define that

_{i}*Y*= |

*E*|

_{i}^{2}and

*X*= |

*E*|

_{T}^{2}, then we obtain where

For TM polarization, use a similar procedure we have

where*Y*= |

*H*|

_{i}^{2},

*X*= |

*H*|

_{T}^{2}, and we assume that transmitted magnetic field

*H*is purely real, and

_{T}## 3. Results and discussions

Next, we discuss the role of the nonlinear graphene sheets in OB according to Eqs. (10) and (12). In the absence of the nonlinear graphene or appearing of the linear graphene, the incident electric and transmitted electric field show a linear relationship, and hence the optical bistable phenomenon does not happened. The nonlinear dependence of conductivity of graphene (Eq. (3)) can provide us the origin of OB. But, the nonlinear conductivity coefficient *σ*_{3} is inversely proportion to *ω*^{3} so that the conductivity coefficient *σ*_{3} will become very small in the high frequencies range (both infrared and visible light) and hence need more input power to excite OB. Therefore, in the present paper, we focus on the discussion of OB in the THz frequencies range for TE polarization, and OB for TM polarization can be discussed similarly.

For frequencies 2*E _{F}* >

*h̄ω*, the interband transitions in graphene are forbidden by the Pauli exclusion principle, hence

*σ*

_{0}≈

*σ*

_{intra}. It is seen that the imaginary part of the linear conductivity

*σ*

_{0}is positive, so that the surface plasmon polarization (SPP) cannot be excited for TE-polarization. Hence, the origin of OB in the present structure for TE polarization is not from the SPP but from the interface effect due to the introduction of nonlinear graphene. Based on this principle, Peres et al., had investigated OB in the low THz frequency (0.5THz) for nonlinear graphene suspending on air for TE polarization at the normal incidence [35]. However, electrical reconfiguration is not fully achievable in their structure. This is because graphene needs to be biased from a perpendicular electrostatic field to control its conductivity. To solve the problem, we have proposed the dielectric-nonlinear graphene-dielectric sandwiched structure, as shown in Fig. 1. In this structure, we can get more means to control OB, such as, the incident angle, the thickness and the permittivity of dielectric slabs, etc. Furthermore, in Peres et al. work, it is hard to get OB at the higher frequencies (about 3THz) at the normal incidence, however OB at the higher THz frequencies can be excited in our structure.

Before analysing optical bistable behavior, we will discuss the influence of the introduction of linear graphene sheet (*σ*_{3} = 0) on the transmission of the heterostructure. To simplify the discussion, we have neglected the losses in the graphene, Re(*σ*_{0}) = 0. Due to the appearing of graphene sheet, the interface effect is greatly modified, as shown in Fig. 2. The transmission coefficient is lowered markedly and the reflection coefficient is enhanced significantly because graphene sheet has an impact on the interface effect, as indicated in Eqs. (10) and (12). At the larger incident angle, the reflectance near reaches to 100%, and the total internal reflectance (TIR) can occur for *θ* ≥ 75°. Hence, the mode of the system can switch from transmission mode to TIR mode due to the central role of graphene in interface effect.

Further, the mode of the system can switch from TIR mode back to transmission mode if the nonlinear effect of graphene sheet is included, as shown in Fig. 3. The parameters are the following: *λ* = 100 *μ*m, *ε*_{1} = *ε*_{4} = 1, *ε*_{2} = *ε*_{3} = 2.25, d_{2}=d_{3} = 4 *μ*m, *θ* = 75°, *τ*^{−1} = 0, and T = 300 K. For *E _{F}* = 0.8

*eV*, the optical bistable hysteresis is observed from monolayer nonlinear graphene in the sandwiched structure, and a threshold electric field |

*E*|

_{i}*of about 9.85 × 10*

_{up}^{7}V/m is required for producing the optical hysteresis loop for the monolayer graphene. We can see that the transmitted electric field gradually increases at low input electric field and then suddenly increases when the incident electric field is above 9.85 × 10

^{7}V/m. This is because the transmittance resonance could not be satisfied at low intensities but it is well achieved as the input electric field is increased up to threshold electric field |

*E*|

_{i}*.*

_{up}When the incident electric field reaches the switch-up threshold intensity, the mode of the system will switch from TIR mode to the transmission mode (Fig. 3(b)), and the reflected intensity becomes very small suddenly. In contrast, upon lowering the input electric field, the system will maintain the transmission mode, and finally switches to the TIR mode. We can seen that if switch-down electric field |*E _{i}*|

*≈ 5.58 × 10*

_{down}^{7}V/m is reached, then the transmitted electric field decreases suddenly when the incident electric field is below |

*E*|

_{i}*. The hysteresis width Δ|*

_{down}*E*| = |

_{i}*E*|

_{i}*− |*

_{up}*E*|

_{i}*≈ 4.3 × 10*

_{down}^{7}V/m.

Next, we consider the dependencies of the transmitted electric field and transmittance on the input light intensity at different Fermi energy *E _{F}* of the graphene, as shown in Fig. 4. From Fig. 4 we can see that the hysteretic behavior is strongly dependent on the Fermi energy of graphene. With the continuous tuning of the Fermi Energy level, the threshold electric fields |

*E*|

_{in}*and |*

_{up}*E*|

_{i}*could be correspondingly adjusted. It is clear that if we increase the Fermi energy, both the switch-up threshold |*

_{down}*E*|

_{i}*and switch-down threshold |*

_{up}*E*|

_{i}*of the bistability move to the higher light intensity, and the optical hysteresis loop becomes large and the hysteresis width is broadened; however, if we decrease the Fermi energy, both the switch-up threshold |*

_{down}*E*|

_{i}*and switch-down threshold |*

_{up}*E*|

_{i}*of OB move to the lower light intensity, and the optical hysteresis loop becomes small and the hysteresis width is narrowed. These results can be confirmed from Fig. 4, where we have shown the dependencies of the threshold electric field |*

_{down}*E*|

_{i}*and |*

_{up}*E*|

_{i}*on the Fermi energy (0.15eV 1.2eV). OB is missing while Fermi energy*

_{down}*E*is less that 0.15 eV, however as Fermi energy

_{F}*E*increases both the switch-up and switch-down threshold electric fields shift to higher electric fields quickly and the width of hysteresis loop is also enhanced markedly.

_{F}Next, we discuss the influence of the physical properties of dielectric slabs 2 and 3 on the behavior of OB of the nonlinear graphene in the sandwiched structure. We focus on the influence of the thicknesses and the permittivities of dielectric slabs 2 and 3 on the optical bistable phenomena, as shown in Fig. 4. To simplify the discussion, we first assume that the dielectric slabs 2 and 3 have the same thicknesses and the optical bistable hysteresis has been observed as shown in Fig. 5(a). It is clear that the both switch-up threshold electric field and switch-down threshold electric field shift to higher incident electric field with the increases of the thickness.

In the meantime, the width of hysteresis loop is also enhanced. Moreover, Fig. 5(b) gives the influence of the permittivities of dielectric slabs 2 and 3 on the optical bistable phenomena. Here we also assume that the dielectric slabs 2 and 3 have the same permittivity. We can find that larger dielectric constant materials will lead to the increases of the threshold electric fields and the enhancement of the width of hysteresis loop. These results remind us that we can control OB by changing the surroundings of the sandwiched structure.

Besides the material properties in the sandwiched structure, OB is also strongly dependent on the behavior of the incident angles, the results are shown in Fig. 6. The mode of system switches from transmission mode to TIR mode at the larger incident angle due to the occurring of graphene and the mode of system switches back from TIR mode to transmission mode due to the influence of the nonlinear conductivity of graphene under the larger incident electric field. This process leads to the generating of OB at the large incident angle, as shown in Fig. 6(a) where we find that OB is missing for small incident angle (such as 30°) due to the lacking of TIR mode. Even if the incident angle increases to 60°, the hysteresis is not evident and the width of hysteresis loop is narrowed compared to the result for *θ* = 75°. Moreover, from Eq. (3) we know that the nonlinear conductivity of graphene *σ*_{3} is a inverse proportion to the third power of frequency, and hence *σ*_{3} is proportional to the work wavelength. Therefore, longer wavelength will obviously enhance the nonlinearity of graphene, and hence lower the threshold electric fields, as shown in Fig. 6(b).

In the previous study, the damping rate *τ*^{−1} has been neglected, in fact, it should be considered in discussing OB. More importantly, we find that damping rate *τ*^{−1} will play a great role in OB, as shown in Fig. 7. Figs. 7(a) and (b) give the dependencies of the transmitted electric field and transmittance on the input electric field at different electron-phonon relaxation time *τ* of the graphene for wavelength *λ* = 100*um*. Where *θ* = 75°, *E _{F}* = 0.8

*eV*, and the other parameters have the same values as those in Fig. 3. Figs. 7(c) and (d) give the dependencies of the transmitted electric field and transmittance on the input electric field at different electron-phonon relaxation time

*τ*of the graphene but for wavelength

*λ*= 60

*um*. It is clear that the damping rate

*τ*

^{−1}can modified markedly the hysteresis behavior. Here the transmittances are lowered with the increasing of the damping rate

*τ*

^{−1}. Hence, in order to excite OB it is need more input light intensity due to the energy loss in the graphene sheet. Moreover, it is demonstrated that the large damping rate

*τ*

^{−1}can cause the missing of the hysteresis loop. For

*λ*= 100

*um*, hysteresis loop is vanished for

*τ*= 100

*fs*, however for

*λ*= 60

*um*, hysteresis loop is vanished for

*τ*= 50

*fs*. These differences are directly linked to the dependence of the nonlinear conductivity

*σ*

_{3}on the wavelength of the incident light, as indicated in Eq. (3). Through the analysis of the above, we can find that optical bistable phenomena are still happened if the damping rate

*τ*

^{−1}is not too seriously in spite of the transmittance is lowered.

## 4. Conclusion

In summary, we have theoretically investigated the exotic optical bistable phenomena of the light transmitted through a sandwiched structure with the introduction of nonlinear graphene. It is found that the hysteretic response occurs at some conditions. It is very important that the hysteretic effects can be electrically controlled through electrical or chemical modification of the charge carrier density of the graphene. Moreover, we demonstrated that the hysteretic responses strongly depend on the surrounding of the sandwiched structure and the properties of incident light. Graphene optical bistable devices appear to be particularly promising because of giant optical nonlinearities, tunable optical property, ultra-fast response times and infinite small thickness permitting, and the construction of miniaturized devices in integrated optics. They could potentially open a new possibility of all-optical switching, optical transistor, optical logic, and optical memories.

## Acknowledgments

This work is partially supported by the Natural Science Foundation of SZU (Grant Nos. 201452), the Science and Technology Project of Shenzhen (grant Nos. JCYJ20140828163633996), and the Scientific Research Foundation for the returned Overseas Chinese Scholar, State Education Ministry.

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