Optical bistability in defective photonic multilayers doped by graphene
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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.
KeywordsOptical bistability Nonlinear optics Graphene
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.
3 Results and discussions
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.
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.
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.
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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