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Abstract
In this paper, we have investigated optical bistability modulation of transmitted beam that can be achieved by graphene sandwich structure with topological interface modes at terahertz frequency. Graphene with strong nonlinear optical effect was combined with sandwich photonic crystal to form a new sandwich structure with topological interface modes. The light-limiting properties of the topological interface modes, as well as its high unidirectionality and high transmission efficiency, all contribute positively to the reduction of the optical bistability threshold. In addition, the topological interface modes can effectively ensure the stability of the two steady state switching in the case of external interference. Moreover, optical bistability is closely related to the incident angle, the Fermi energy, the relaxation time, and the number of layers of graphene. Through parameter optimization, optical bistability with threshold of 105 V/m can be obtained, which has reached or is close to the range of the weak field.
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1. Introduction
Optical bistability (OB) is a nonlinear optical phenomenon similar to hysteresis curves in which a system can achieve two different output states at the same input [1,2], and it is regarded as the basis of the next-generation optical communication technology. Therefore, OB shows many applications in optical memory [3], all-optical logic gates [4], all-optical switches [5], and various all-optical systems including neural networks [6]. Enhancement of nonlinear optical effects is an important means of lowering the OB threshold and is expected to enhance the transmission efficiency of the entire optical networks and optical communication system even further. In recent years, researchers have been working on exploring and discovering nonlinear optical properties based on different types of materials, such as metal-doped 15-crown-5 [7], novel salicylaldehyde-based thiosemicarbazones [8], carbon encapsulated gold nanostructures [9], deep eutectic solvents [10], titanium nitride [11] and so on. However, due to the inherent drawbacks such as the low third-order nonlinear coefficient of traditional nonlinear dielectric materials, the enhancement of nonlinear optical effects usually requires a longer interaction size or higher input optical energy. It is necessary to continuously search for new modes and pathways that can realize nonlinear optical effects enhancement, to reduce the energy demand and break the limitation of the size of optical components. On the one hand, the two-dimensional material graphene offers an excellent solution to the challenge as a single atomic layer material with excellent nonlinear optical properties and dynamically tunable characteristics [12]. Graphene is regarded as a suitable nonlinear optical material for realizing OB, and various structures involving graphene to achieve OB have been explored, including graphene-covered nonlinear interface [13], PT-symmetric Thue-Morse photonic crystals [14], photonic crystal Fabry-Perot cavity [15], 1D polymethy1-methacrylate grating [16], etc. Based on this, optical bistable devices with low-threshold and tunable characteristics are a worthwhile direction to be explored in the field of graphene OB research with valuable applications.
On the other hand, the enhancement of nonlinear optical effects can be realized by the mode of local field enhancement excited by the specific structure and combined with excellent nonlinear materials. Realizing manipulation of photons in a two-dimensional plane and developing optical bistable devices with low-intervention threshold, low loss, high transmission efficiency, and tunable characteristics have been strongly pursued by the nonlinear nanophotonics community for a long time. Integrated silicon platforms with new phenomena and mechanisms have ushered in new opportunities for development in the field of micro and nano-scale optical bistable device research with continuous breakthroughs and innovations in traditional structures. Conventional metal-insulator-metal (MIM) three-layer structures (also commonly referred to as sandwich structures), can be used to enhance nonlinear optical properties [17], but are often accompanied by significant energy losses. Recently, topological photonics has attracted a great deal of attention in integrated silicon platforms [18]. It’s unique topological edge-protected state, which confines and transmits electromagnetic wave, exhibits properties such as extremely high unidirectionality, robustness to defects, and high transmission efficiency, and has been intensively investigated for stable unidirectional transmission devices, topological lasers, as well as integrated and quantum photonic devices [19,20]. Moreover, since conventional optical bistable devices operate based on the normal mode (without topological edge-protected state), the structure may lose its stable switching characteristics when subjected to some perturbations [21]. To solve the above problems, we combined topological photonics with conventional sandwich structures, which is expected to realize the enhancement of nonlinear optical effects with low loss, high transmission efficiency and tunable photonic manipulation properties based on integrated silicon platforms. Therefore, the proposed graphene new sandwich structure with topological interface modes (TIM) properties provides a new solution for modulating and lowering the OB threshold. In addition, the new sandwich structure with TIM characteristics optimized the light transmission efficiency within the system, simplifies the structure, ensures its stable switching characteristics, and provides a feasible idea for the construction of optical bistable devices that can be further adapted to integrated optical networks and optical communication systems.
In this paper, new sandwich structures with TIM properties in terahertz bands are proposed to realize tunable low-threshold OB. Two different types of sandwich photonic crystal structures were designed and combined with graphene to build new sandwich structure with TIM properties. The graphene photonic crystal sandwich structure with TIM properties can not only realize the manipulation of light in a two-dimensional plane, but also enhance the nonlinear optical effects and finally achieve the tunable OB with low threshold and low power consumption, which will support the further improvement of the transmission efficiency of the whole optical networks and optical communication system in the future. The findings can be summarized as follows: (a) Graphene can provides strong nonlinear optical properties, and the local field enhancement at the interface of two sandwich structures can well further enhance the nonlinear optical effect and reduce the loss of light intensity within the structure. (b) The OB threshold and hysteresis width can be tuned by parameters such as graphene Fermi energy level, number of graphene layers, and incident angle. (c) The graphene photonic crystal sandwich structure with TIM properties provides the strong nonlinear optical effects required for practical optical bistable devices and is a central research direction for realizing low-threshold, tunable OB and all-optical switching.
2. Theoretical model and method
We consider a graphene sandwich structure with TIM that build by two sandwich photonic crystals (PhCs), where graphene is at the highest energy of the entire structure, as shown in Fig. 1. The topological sandwich structure consists of two kinds of PhCs (PhCp and PhCq), and defective units D1 and D2. The unit cells of PhCp and PhCq are denoted as
Fig. 1. Schematic diagram of the topological sandwich structure, graphene is inserted between two PhCs and the light illuminates from the left to the right at an incidence angle of $\theta$. Here, we employ HfO2 as A and SiO2 as B. The electric field is calculated by COMSOL Multiphysics (blue solid line).
$Ph{C_p}\textrm{ } = \textrm{ }{A_{{{d1} / 2}}}{B_{d2}}{A_{{{d1} / 2}}}$, $Ph{C_q}\textrm{ } = \textrm{ }{B_{{{d2} / 2}}}{A_{d1}}{B_{{{d2} / 2}}}$. The defect unit D is denoted as D1 = Ad, D2 = Bd. The refractive index is set to be n = 2 for medium A (HfO2), n = 1.46 for medium B (SiO2). The thicknesses are ${d_1}\textrm{ } = \textrm{ }38.35\mathrm{\ \mu m}$, ${d_2}\textrm{ } = \textrm{ }45\mathrm{\ \mu m}$ and defective units is $d\textrm{ } = \textrm{ }5\mathrm{\ \mu m}$.Here, the period of each PhC is defined as $T\textrm{ } = \textrm{ }6$, and the stacking structure built up of repeated p(q)-type unit cells can thus be described as p6 (q6). Meanwhile, graphene is embedded into the interface between D1 and D2. Thus, the whole structure can be labeled as p6Dq6. At present, the fabrication and transfer technology for multi-layer dielectric structure and graphene is relatively mature, the topological sandwich structure can be fabricated. Besides, graphene could be seen as an ultra-thin two-dimensional dielectric possessing unique electrical conductivity, we choose the terahertz band to obtain the largest possible three-order nonlinear conductivity. Under the random-phase approximation and without considering the external magnetic field, the isotropic surface conductivity of graphene can be expressed as ${\sigma _0} = {\sigma _{intra}} + {\sigma _{inter}}$, where ${\sigma _{intra}}$ is the intraband conductivity. According to the Pauli Exclusion Principle, graphene cannot generate inter-band transition in the terahertz band. Therefore, the conductivity of graphene can be summarized as [22]:
where e, $\omega$ and $\hbar$ are refer to electric charge, angular frequency, and reduced Planck constant. Here, Fermi energy is ${E_F} = \hbar {v_F}\sqrt {\pi {n_{2D}}}$; ${v_F}$ is the Fermi velocity and ${v_F} \approx {10^6}\,\textrm{m/s},{n_{2D}}$ is the carrier density. The third-order nonlinear conductivity of graphene can be expressed as [13,23]:Both linear and nonlinear conductivity coefficients are highly dependent on Fermi energy ${E_F}$ and relaxation time $\tau$, which provides us an effective way to flexibly modulate OB.
The topological sandwich structure is considered as a layered structure with each layer of the medium numbered 1 to 38 from left to right, and the background material is air, numbered 0 and 39. Here, the transmitted and reflected electromagnetic waves in each layer of the medium are marked as F and B. In this paper, we choose Z-axis as the direction of electromagnetic field propagation and the position of graphene is set at $z = 0$. Meanwhile, the conductivity of graphene is $\sigma = {\sigma _0} + {\sigma _3}{|{{E_{\textrm{19}y}}({Z = 0} )} |^2}$, ${E_{\textrm{19}y}} = {F_{\textrm{20} }} + {B_{\textrm{20}}}$. In TE polarization, the incident electromagnetic field (${E_i},\textrm{ }{H_i}$) can be expressed as:
The electromagnetic field of transmitted air layer can be expressed as:
The electromagnetic field on both sides of the graphene should be continuous, the following formula can be obtained by using the boundary conditions at position $Z = 0$.
Finally, the relationship between ${E_i}$, ${E_R}$, and ${E_t}$ can be obtained, thus OB can be generated under appropriate condition.
3. Results and discussions
By calculating the energy band of the whole structure (p6 + q6), we find that there exists a topological energy band with a frequency of 0.987 THz, in the common energy gap 0.86 THz∼1.18 THz of the p6 or q6 structure, as shown in Fig. 2(a). In addition, the distribution of electromagnetic modes corresponding to the energy bands is shown in Fig. 2(b), and it is obvious that the topological states are locally distributed at the interfaces of p6 and q6, and such localized boundary states offer the possibility of enhancing the nonlinear optical effects and lowering the OB thresholds within the system. Further, the changes of transmittance and reflectance with respect to the incident frequency for the three structures of p6, q6, and p6 + D + q6 are shown in Fig. 2(c). It can be seen that individual photonic crystals (p6 or q6) maintain a low transmittance around 1THz. But a sharp transmission peak in 1THz appears in topological sandwich structure, which is due to the excitation of TIM. To verify this, Fig. 1 also shows the electric field distribution of the state, which is a unique feature of TIM. In addition, the period of the unit cell also determines the appearance of OB, and OB of the designed sandwich structure is shown in Fig. 2(d). It can be seen that the transmitted electric field is linearly related to the incident electric field in 1, 2, and 4 periods, and exhibits a nonlinear relationship similar to a hysteresis curve only after the TIM excitation (T = 6) do they show a nonlinear relationship similar to hysteresis curve, that is, the construction of a complex topological sandwich structure is a guarantee for obtaining OB. Besides, the topological edge-protected state can effectively ensure the stability of the two steady states switching in the case of external interference.
Fig. 2. (a) The energy band of topological sandwich structure; (b) The electric fields of topological sandwich structure; (c) The transmittance spectra of p-type (blue short-dashed line), q-type (black short-dotted line) and the “p + q” heterostructure (red solid line); (d) Dependence of transmitted electric field on the incident electric field for different structure.
From Eq. (2), it can be seen that the third-order nonlinear conductivity of graphene is strong in terahertz band, and the local electric field enhancement effect generated by TIM makes the nonlinear part further prominent, providing suitable conditions for the appearance of low-threshold OB.
We exploited the strong field confinement and nonlinearity of graphene to realize OB. Due to the Fermi energy of graphene is related to carrier density ${n_{2D}}$ and ${n_{2D}}\textrm{ = }{\textrm{C}_g}V/e$, where ${\textrm{C}_g} = 115\textrm{ }aF\mathrm{\mu} {m^{ - 2}}$, V is the external voltage, the tunable OB can be implemented by varying the external voltage. From Fig. 3(a), when the Fermi energy level is 0.1 eV, ${|{{E_i}} |_{up}} = 2.805 \times \textrm{1}{\textrm{0}^\textrm{5}}\textrm{ V/m}$, ${|{{E_i}} |_{down}} = 2.645 \times \textrm{1}{\textrm{0}^\textrm{5}}\textrm{ V/m}$, the hysteresis width of the OB is $\Delta = 0.16 \times \textrm{1}{\textrm{0}^\textrm{5}}\textrm{ V/m}$. Since the conductivity of graphene is closely related to the relaxation time, increasing the relaxation time will increase the imaginary part of the linear surface conductivity and decrease its real part, so increasing the relaxation time can significantly increase the width of the hysteresis loop. On the contrary, reducing the relaxation time will directly lead to the rapid narrowing or even disappearance of the hysteresis width.
Fig. 3. (a) Dependence of the transmittance on light intensity for different Fermi Energy and relaxation time of graphene; (b) Dependence of ${|{{E_i}} |_{up}}$ and ${|{{E_i}} |_{down}}$ on different Fermi energy (black solid) and relaxation time (blue solid).
Next, the effects of incident light angle as well as the number of graphene layers on the OB behavior are discussed. The relationship between the transmitted and incident electric fields for different incidence angles is shown in Fig. 4(a). Similar to the modulation of the hysteresis curve by the Fermi energy level of graphene, the effect of the incidence angle change on the hysteresis behavior is reflected in the threshold and hysteresis width. The pink dashed line in Fig. 4(a) indicates the results of the COMSOL Multiphysics calculations. The relaxation time of 0.9 ps is chosen as a reference, when the Fermi energy level is 0.10 eV and the incidence angle degree $\theta \textrm{ = }{\textrm{0}^\circ }$.Specifically, the threshold of the transmitted electric field increases with increasing the angle of incidence. The effect of graphene layer number on the OB transmissive is shown in Fig. 4(b). Although the increase in the number of graphene layers increases the nonlinearity of the structure and makes the hysteresis width increase, considering the loss of graphene itself, as the number of layers increases, the transmittance of the whole structure decreases and the OB threshold increases, which is inconsistent with our goal of achieving a low threshold. However, in general, the OB based on the topological sandwich structure can be tuned by changing the angle of incident light as well as the number of graphene layers. Finally, the relationship between reflectivity, reflected electric field and incident electric field at different Fermi energy levels of graphene is discussed, as shown in Fig. 4(c) and Fig. 4(d). It can be observed that as the Fermi energy level of graphene increases, the OB threshold of reflectivity and reflected electric field increases and the hysteresis line width becomes wider, indicating that the Fermi energy level of graphene also has a significant modulating effect on the reflective OB phenomenon. Overall, the dynamically tunable property of graphene nonlinear conductivity also plays a very important role in low-threshold reflective OB phenomena.
Fig. 4. Dependence of ${E_t}$ on ${E_i}$ for different (a) incident angle; (b) the period of graphene sheet; (c) Dependence of reflectance on ${E_i}$; (d) Dependence of ${E_r}$ on ${E_i}$ for different Fermi energy.
4. Conclusions
In summary, we have investigated tunable low-threshold OB with TIM based on graphene photonic crystal sandwich structure. The TIM of the sandwich structure and the strong third-order nonlinear effect of graphene play a key role in facilitating the realization and tuning of the low-threshold OB. In addition, the threshold and hysteresis width are also closely related to parameters such as the incident angle, the number of graphene layers, and the Fermi energy and the relaxation time. In this paper, a hysteresis curve with a threshold of 105 V/m was obtained, which is reached or near to the weak-field range, and finally tunable OB with low threshold and low power consumption is realized. The simulation results agree well with the theoretical calculation, which further verify the accuracy. Considering the simple structure and relatively low fabrication process requirements of the proposed scheme, it is expected to be potentially applied in OB devices and other nonlinear optics fields, which provides support for further enhancing the transmission efficiency of the whole optical networks and optical communication system in the future.
Funding
Central South University of Forestry and Technology (2021YJ0056); National Natural Science Foundation of China (11404410); Natural Science Foundation of Hunan Province (2020JJ4935); Research Foundation of Education Bureau of Hunan Province (20B602, 21B0253).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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