# Optical bistability in defective photonic multilayers doped by graphene

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## Abstract

We study the optical bistability (OB) in photonic multilayers doped by graphene sheets, stacking two Bragg reflectors with a defect layer between the reflectors. OB stems from the nonlinear effect of graphene, so the local field of defect mode (DM) could enhance the nonlinearity and reduce the thresholds of bistability. The structure achieves the tunability of bistability due to that the DM frequency and transmittance could be modulated by the chemical potential. Bistability thresholds and interval of the two stable states could be remarkably reduced by decreasing the chemical potential. A lager Bragg periodic number could increase the localizing of field, but the graphene loss may decrease the intensity of transmission light. We have concluded an appropriate periodic number to achieve OB. The study suggests that the tunable bistability of the structure could be used for all-optical switches in optical communication systems.

### Keywords

Optical bistability Nonlinear optics Graphene## 1 Introduction

Optical bistability (OB) is a nonlinear phenomenon in which the transmission properties of light vary with the incident light power. The behavior can be used for all-optical switches, limiters, transistors and memory elements (Soljacić et al. 2002; Svensson et al. 1990; Yanik et al. 2003; Brandonisio et al. 2012). A typical bistable device usually consists of distributed feedback (DFB) structure and nonlinear material (He and Liu 1999). The refractive index of the nonlinear material is a nonlinear function of the incident intensity, which is the nonlinear effect OB originating from. The key point to achieve OB is that the light-induced refractive-index variation should be large enough (Noskov and Zharov 2006). Combining distributed feedback and Fabry–Perot (DFB/FP) could make the intensity thresholds of bistability lower compared to a single DFB configuration (He and Cada 1992). Photonic crystal (PC) has also been implemented for bistability both theoretically and experimentally (Wang et al. 2008; Lepeshkin et al. 2004; Husakou and Herrmann 2007). PC has a photonic band gap structure, and the bistability in PC stems from the shifting of the band edge. Due to the excitation of highly localized bulk plasmon polariton, subwavelength metal-nonlinear dielectric multilayers proceeded to reduce the thresholds of OB (Chen et al. 2009). Low threshold OB for the resonant modes was experimentally demonstrated in one-dimensional (1D) PC at visible frequencies (Sreekanth et al. 2015).

A Kerr-type nonlinear defect layer was introduced in the center of 1D PC to obtain OB curve of the transmitted light intensity (Wang et al. 1997). The physical mechanism of this doped structure for bistability is dynamic shifting of the defect mode frequency. Moreover, threshold-tunable OB was achieved in doped photonic multilayers (Kong et al. 2013; Mehdian et al. 2013; Centeno and Felbacq 2000). Optical bistability at terahertz frequencies has been realized in sandwich structures embedded with graphene (Dai et al. 2015a, b; Wang et al. 2016a, b; Solookinejad 2016), which utilized the surface plasmon polaritions (SPPs) of graphene. SPPs coupled quantum cascade detector could enhance optical absorption for mid-long-infrared pixels (Li et al. 2016). Fabry–Preot cavity with graphene (Jiang et al. 2015) and grapheme-covered nonlinear interface (Xiang et al. 2014) have also been used to achieve tunable optical bistability. Graphene possesses various unique optical, electronic, and mechanical properties (Huang et al. 2017; Qin et al. 2016; Wang et al. 2016c; Wang et al. 2017a; Ke et al. 2016) and its large third-order nonlinear coefficient has been theoretically predicted (Mikhailov 2007) and experimentally verified (Hendry et al. 2010; Hong et al. 2013). SPPs on graphene have stronger confinement and relatively less propagation loss in comparison with that on metals (Ke et al. 2015; Liu et al. 2016), so graphene provides new ways to envision light propagation at the nanoscale and ultra-fast switching (Fan et al. 2016; Wang et al. 2017b; Tong et al. 2017; Qin and Zhu 2017; Li et al. 2017; Zhang et al. 2017). The surface conductivity of graphene depends on chemical potential, which can be tuned by chemical doping or gate voltage (Huang et al. 2014).

In this work, we design a distributed-feedback Bragg structure doped by graphene sheets. The graphene sheets locate at the interfaces of each dielectric layer and the middle of the defect layer. The structure combines DFB and FP structure, which has a local enhanced effect of defect mode (DM) (also called resonant mode) in near-infrared light range. The energy of the DM is mainly located at the defect layer, which could increase the nonlinear effect of graphene. OB results from the intensity-induced change of the effective dieletric constant of graphene, so the localizing of field could decrease the threshold of OB. The frequency and the transmittance of DM can be fine-tuned by the chemical potential of graphene, which could be used for modulating the threshold of OB. We numerically study light propagation through such special photonic multilayer structure and achieve tunable OB. The tunability of OB results from the variation in surface conductivity of graphene. The difference in output intensity levels of the two stable states is greatly affected by the chemical potential. Bragg periodic number and the detuning of incidence wave to DM can also change the bistable curve profile.

## 2 Theory

*A*and

*B*are lower and higher refractive index materials respectively,

*C*is defect layer, and

*G*are graphene sheets.

*A*and

*B*are alternatively placed in the multilayer stack of dielectric materials, Graphene sheets are located at the interfaces of dielectric layer and the middle of the defect layer . The optical thickness of layers

*A*,

*B*and

*C*are quarter-wavelength with

*d*

_{ a }=

*n*

_{ a }

*λ*

_{0}/4,

*d*

_{ b }=

*n*

_{ b }

*λ*

_{0}/4 and

*d*

_{ c }=

*n*

_{ c }

*λ*

_{0}/2 respectively, where

*λ*

_{0}is the midgap wavelength of PC in normal incidence.

*ε*

_{ g }= 1 + i

*σ*

_{ g }

*η*

_{0}/(

*k*

_{0}∆) (Wang et al. 2016a, b), where

*k*

_{0}= 2

*π*/

*λ*is the incidence wave number in vacuum,

*η*

_{0}is the vacuum resistivity,

*σ*

_{ g }is the total surface conductivity, and

*λ*is the free space wavelength of the incidence wave. When considering nonlinear effect,

*σ*

_{ g }can be written as

*σ*

_{ g }=

*σ*

_{1}+

*σ*

_{3}\(\left| {E_{//} } \right|^{2}\), where

*σ*

_{1}is the linear surface conductivity,

*σ*

_{3}is the nonlinear surface conductivity coefficient, and

*E*

_{//}is the in-pane component of the electric field. The linear surface conductivity of graphene may be modeled using Kubo formula (Chen and Alù 2011; Thongrattanasiri et al. 2012), which depends on incidence wavelength

*λ*, chemical potential

*μ*

_{ c }, thickness ∆, and relax time

*τ*. We choose the excitation wavelength in the range of near-infrared light, and set the parameters as ∆ ≈ 1 nm (Wang et al. 2012), and

*τ*= 0.5 ps in the following study. The nonlinear conductivity coefficient is (Smirnova et al. 2014)

*V*

_{ F }≈ c/300 is the Fermi velocity. The dielectric constant can be rewritten as

*ε*

_{ g }=

*ε*

_{1}+

*χ*

_{ g }

^{(3)}\(\left| {E_{//} } \right|^{2}\), where linear dielectric constant is

*ε*

_{1}= 1 + i

*σ*

_{1}

*η*

_{0}/(

*k*

_{0}∆). Consequently, the cubic volume susceptibility can be expressed as

*χ*

_{ g }

^{(3)}= i

*σ*

_{3}

*η*

_{0}/(

*k*

_{0}∆).

*φ*

_{ l }= 2

*πd*

_{ l }(

*ε*

_{ l }− sin

^{2}

*θ*)

^{1/2}/

*λ*,

*d*

_{ l }is the thickness of transmission medium, and

*θ*is the incident angle. For TE wave,

*η*

_{ l }= (ε

_{0}

*/*μ

_{0})

^{1/2}(

*ε*

_{ l }− sin

^{2}

*θ*)

^{1/2}, and for TM wave,

*η*

_{ l }=

*ε*

_{ l }(ε

_{0}

*/*μ

_{0})

^{1/2}/(

*ε*

_{ l }− sin

^{2}

*θ*)

^{1/2}. It should be noted that for Kerr-type nonlinear material, since its refractive index is dependent on the localized field intensity, the transfer matrix should be solved by nonlinear method. So the nonlinear layer are usually discretized into many sublayers (Kong et al. 2013), then, the field intensity and refractive index can be considered constant in each sublayer if the number of sublayers is large enough. Here, graphene is taken as nonlinear Kerr-like homogeneous ultra-thin film material, and it is sufficiently thin corresponding to the incidence wavelength. Hence, there is no need to divide it into sublayers. The electric field intensity and refractive index of the

*l*th layer graphene can be derived by

*E*

_{ l },

*H*

_{ l },

*E*

_{ l+1}and

*H*

_{ l+1}are the electromagnetic field intensity at two interfaces of graphene respectively, and

*M*

_{ l }is the transfer matrix of the

*l*th graphene. The matrix element in the

*M*

_{ l }contains parameter

*ε*

_{ l }, which can be approximately given by

*ε*

_{ l }=

*ε*

_{1}+

*χ*

_{ g }

^{(3)}\(\left| {E_{l + 1} } \right|^{2}\). The process in essence is a nonlinear correction of linear permittivity. So, taking into account the nonlinear surface conductivity, the equivalent permittivity of graphene is intensity-dependent.

*m*

_{11},

*m*

_{12},

*m*

_{21}, and

*m*

_{22}are the matrix elements of the total transfer matrix

*M*connecting EM fields at the incident and exit ends. Therefore the transmission coefficient can be obtained by transfer matrix

*η*

_{0}=

*η*

_{ n+1}= (ε

_{0}/μ

_{0})

^{1/2}(1 − sin

^{2}θ)

^{1/2}. The transmittance parameter

*T*is identified as

*T*=

*tt** =

*I*

_{ t }/

*I*

_{ i }, where

*I*

_{ t }and

*I*

_{ i }are incident and transmitted intensities, respectively.

## 3 Results and discussions

*ω*

_{ gap }= 4

*ω*

_{0}arcsin\({{\left| {(n_{b} - n_{a} )/(n_{b} + n_{a} )} \right|^{2} } \mathord{\left/ {\vphantom {{\left| {(n_{b} - n_{a} )/(n_{b} + n_{a} )} \right|^{2} } \pi }} \right. \kern-0pt} \pi }\) (Yariv et al. 2007), where

*ω*

_{0}= 2πc/

*λ*

_{0}is the center of band gap and

*λ*

_{0}= 1.55 μm is the free space wavelength,

*A*is MgF

_{2}and

*B*is ZnS. The linear refractive index of the layer materials are

*n*

_{ a }= 1.38,

*n*

_{ b }= 2.35, and

*n*

_{ c }= 1.5. Dielectrics

*A*and

*B*alternate in Z-axis to form a symmetrical structure about the defect layers. The Bragg periodic number is

*N*= 4 in each side. A series of graphene sheets are embedded in the structure. The DM appears in the middle of the forbidden band as shown in Fig. 2a. When an incident wave has the same frequency as the DM frequency, it could pass through the structure without reflection. The intensity distribution of the DM presents that the main energy localizes at the defect layer. Light-induced refractive-index variance is proportional to the local light intensity, so the graphene embedded in the defect layer is the main source for dispersive optical bistability. The function of the graphene sheets at the interfaces of dielectrics

*A*and

*B*is tuning the system characters. That is controlling on the frequency shift of defect mode and switching thresholds of bistability.

Figure 2a presents the transmittance for *μ* _{ c } = 0.40, 0.45 and 0.50 eV. For a special incidence wavelength *λ* = 1.55 μm, the transmittance and detuning could be modulated by chemical potential as shown in Fig. 2b. The graphene thickness has little impact on the wavelength shift of defect mode. The transmittance reaches the peak value 0.96 and the detuning is zero at *μ* _{ c } = 0.58 eV. So this characteristic can be used for manipulating the thresholds of OB. It could be found that, when chemical potential is less than 0.39 eV, the transmittance of DM equals to zero, and while it exceeds 0.39 eV, the value rapidly steps up to 1 as shown (the blue line) in Fig. 2c. To explain the phenomena, we plot the curves of surface conductivity of graphene versus the chemical potential as shown in Fig. 1d. It could be seen that the blue curves in Fig. 1c and d are symmetry about the horizontal axis. As *μ* _{ c } < 0.39 eV, the real part of the surface conductivity of graphene is close to 0.6 S, which corresponds to the loss of material in physics, so the transmittance of DM is almost zero. While *μ* _{c} > 0.39 eV, the value sharply jumps down to zero, and the transmittance of DM almost steps up to 1. The frequency shift curve of DM in Fig. 2c corresponds with the imaginary part of the surface conductivity of graphene in Fig. 2d. The red curve shows that, as *μ* _{ c } < 0.41 eV, the wavelength of DM is red-shift companied with *μ* _{ c } increasing, while the wavelength of DM is blue-shift with an increasing in chemical potential as *μ* _{ c } > 0.41 eV.

*μ*

_{ c }= 0.50, 0.60 and 0.70 eV. The three resonant wavelengths are

*λ*

_{ r }= 1.5549, 1.5524 and 1.5506 μm, respectively. If the incident wavelength is exactly equal to the resonant wavelength, the OB behavior is not obtained (Wang et al. 2008). The incidence wavelength is set as

*λ*= 1.57 μm. The incidence frequency is close to the three special resonant modes and could guarantee a certain amount of detuning. As introduce the nonlinearity, the strong nonlinear effect makes the defect modes to move toward the incident wave frequency (Soljacić et al. 2002). For the nonlinear graphene embedded in our geometry, the transmittance varies with the local intensity, i.e. the incident intensity, as shown in Fig. 3a. Therefore bistable phenomena occur in the 1D PC when DM frequencies almost equal to incident light frequencies as shown in Fig. 3b. It also demonstrates that the bistability curve can be modulated by chemical potential. In the previous achievements, as the incidence wavelength is invariant, the detuning of the resonant wavelength to the incident wavelength is proportional to the chemical potential. This rightly explains the interval between the up-threshold and the down-threshold gradually expands as chemical potential increases in Fig. 3c.

For a given incidence wave, the bistability occurs only when the incidence wavelength equals to the resonant mode under considering nonlinearity influence. The resonant mode is a function of chemical potential, so there is a critical chemical potential where the detuning of the two waves is zero. It means that there is no OB below the critical chemical potential. In Fig. 3c, as the given incidence wavelength is 1.57 μm, the critical chemical potential for OB is about 0.5 eV. Figure 3d depicts the curve of the critical chemical potential versus incidence wavelength. There is no OB in the area under the critical curve. It shows that the critical chemical potential approaches the saturation value 3.9 eV as the incidence wave increases, because the real part of the surface conductivity of graphene sharply drops to zero near potential 3.9 eV as shown in Fig. 2d. In other words, a little variation in the chemical potential results in a large conductivity difference near 3.9 eV.

*N*are given in Fig. 4. For different

*N*, the results show that the transmittances of defect mode are different in Fig. 4a. It demonstrates that

*N*is bigger, the resonant peak is sharper, and the transmittance of the defect mode peak drops as

*N*increases. Distinctly, the loss is more serious as

*N*is bigger, so the transmittance of resonant mode reduces step by step with

*N*increasing as shown in Fig. 4b. Almost no light could go through the structure for

*N*= 10. The transmittance of defect mode is close to 1 and the loss is very small when

*N*is less than 2. As

*N*> 2, the resonant frequency is invariant. It shows that

*N*= 4 or 5 is the appropriate number, because the defect mode simultaneously has a narrow line width and a large transmittance.

*λ*= 1.57 μm than 1.60 μm with a fixed

*N*. It is remarkable that, for

*N*< 4, there is no OB existing in the structure with an incidence wavelength 1.57 μm. Increasing the incidence wavelength to 1.60 μm, the critical layer pair number is

*N*= 3.

## 4 Conclusion

In summary, we have investigated the bistable phenomenon in a defective PC doped by graphene in near-infrared frequencies. The frequency of DM and threshold of bistability could be separately modulated by changing the chemical potential of graphene, without reconfigurating the multilayer structure. The transmittance curve of defect mode and the frequency curve are similar to the figure of the real part and the imaginary part of the surface conductivity of graphene respectively. The threshold values can also be modulated by the Bragg periodic number. The detuning and interval of threshold are proportional to chemical potential and Bragg periodic number. The threshold can be remarkably reduced by decreasing chemical potential or Bragg periodic number. The results show that *N* = 4 or 5 is the appropriate number for achieving OB. As *N* and incidence wavelength are fixed, there exists critical chemical potential for OB. The calculations further demonstrate that, for a certain chemical potential and incidence wavelength, the threshold interval increases as *N* increases. The bistability behavior of the proposed object can be applied in communication systems, such as optical switching.

## Notes

### Acknowledgements

This work is supported by the 973 Program (Grant No. 2014CB921301), the National Natural Science Foundation of China (Grant Nos. 11674117, 11304108), Natural Science Foundation of Hubei Province (Grant No. 2015CFA040).

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