Tunable Low-Threshold Optical Bistability in Optical Tamm Plasmon Superlattices

Abstract

We propose a scheme to obtain tunable low-threshold optical bistability of reflected beams in optical Tamm plasmon superlattices (TPS). The low-threshold optical bistability is triggered due to the strong third-order non-linearity of graphene and the local field enhancement in the TPS. Our results show that the optical Tamm plasmon superlattices have the ability to lower the bistable threshold even further than the single optical Tamm state. The results show that the hysteresis behavior and optical bistability threshold can be continuously adjusted by changing the applied voltage and the number of graphene layers (N ≤ 4). In particular, the optical bistability in the TPS is affected by the incident angle. Our results introduce a new possible route for low threshold optical bistability in the THz range and provide a new method in the field of all-optical switching applications.

1. Introduction

Optical bistability (OB) describes a hysteresis phenomenon, which refers to a typical non-linear optical phenomenon in that the output light intensity has two different stable values when the system has one stable input light intensity [1,2]. The OB phenomenon has been demonstrated to be an effective method for realizing high-speed all-optical information processing, which provides potential for its application in all-optical switches [3], phototransistors [4,5], and optical memory [6]. In the past few years, researchers have carried out numerous studies on structures that can produce the OB phenomenon, such as a Fabry–Perot cavity [7], a non-linear prism coupler [8], non-linear optical crystals [9], metamaterials [10], and a photonic crystal cavity [11]. It is worth noting that as a two-dimensional material with single atomic layer thickness, graphene has attracted extensive attention from researchers due to its excellent photoelectric properties such as broadband and high electron mobility [12,13,14,15]. Graphene has been widely used in micro-nano electronic devices and optical devices, including optical modulation [16], strong light–matter interactions [17], and other fields. Moreover, it has received close attention in the field of OB research, such as OB of one-dimensional gratings based on graphene [18], OB of graphene surface plasmon [19], graphene-controlled fiber Bragg grating [20], and so on.

Although the basic principle of OB is clear, the excitation schemes for low-threshold and simple-structured OB are still challenging work. Recently, optical Tamm states (OTSs), an electromagnetic surface mode that exists uniquely at the interface between metal thin films and one-dimensional photonic crystals, have been investigated [21]. OTSs are a kind of surface mode [22], in analogy with surface plasmon polaritons (SPPs), which have attracted researchers’ attention because of their intriguing characteristics such as local enhancement of the electric field [23]. Unlike surface plasmon excitation, which requires more stringent conditions, OTSs can be excited by transverse-electric (TE) and transverse-magnetic (TM) polarized waves, and local field enhancement effects can be generated without a specific incident angle. At present, these easy-to-implement OTSs have been successfully used in the field of enhancing the interaction between light and matter, such as optical switching [24], one-way propagation channels [25], tunable and multichannel terahertz perfect absorbers [26], and bistable logic control [27]. The excitation of traditional OTSs is mainly based on metal-distributed Bragg reflector (DBR) construction [28]. Since graphene also has metal-like properties under certain conditions, we can also excite OTSs by using graphene-based DBR structures [29]. Optical Tamm superlattices have a longer length scale and can excite optical Tamm state effects at the boundary of the superlattice structure [30]. In recent years, researchers have carried out a series of studies on the optical phenomena in the optical Tamm superlattice structure such as the generation and tunability of Tamm plasmon topological superlattices [31], topological photonic Tamm states [32], and Zak phase and topological plasmonic Tamm states [33]. The strong local field enhancement effect caused by optical Tamm plasmon superlattices creates a favorable advantage for the achievement of the OB phenomenon [34]. Thus, we can optimistically estimate that the low-threshold graphene-based OB based on the optical Tamm plasmon superlattices will be one of the feasible schemes for realizing practical optical bistable devices.

In this paper, the OB of reflected beams in graphene-based optical Tamm plasmon superlattices (TPS) are investigated. We established a graphene-based optical TPS composed of optical Tamm photonic crystals, in which metal Ag was used as the substrate. Through theoretical design, we achieve OTSs in the terahertz band, which are particularly confined modes mainly located at the adjacent boundaries and are indispensable elements in building TPS support. The strong field enhancement interaction generated by the graphene-based TPS heterostructure has a positive effect on reducing the OB threshold. At the same time, the tunable conductivity of graphene provides a feasible approach to realize the dynamic tunable OB. We believe that the adjustable low-threshold OB will contribute to advances in compact and ultrafast all-optical signal processing.

2. Theoretical Model and Method

We consider a sandwich-like structure of graphene-based TPS with Ag as the substrate, as shown in Figure 1. The DBR is considered as a layered structure, and the transmitted and reflected electromagnetic waves in each layer are marked as F and B, respectively. Here, layer A is poly 4-methyl-pentene (TPX, $n_a=1.46$) and layer B is Si ($n_b=1.9$). In the terahertz (THz) frequency range, the TPX material is a transparent material, and the nanoscale Si material also possesses excellent optical properties. The thickness of A and B are $d_a=42 \mathsf{\mu }\mathrm{m}$ and $d_b=45 \mathsf{\mu }\mathrm{m}$, and the period of n = 11. The top layer has the thickness $d_s=35 \mathsf{\mu }\mathrm{m}$ and the refractive index $n_a=1.46$. The frequency-dependent permittivity of silver is calculated by the Drude model ${\epsilon }_m\left(\omega \right)={\epsilon }_{\infty }-\frac{{\omega }_p^2}{\left({\omega }^2+i\omega \gamma \right)}$, including high-frequency constant ${\epsilon }_{\infty }=3.4$, plasmon frequency ${\omega }_p=1.39\times {10}^{16} \mathrm{rad}/\mathrm{s}$, and scattering rate $\gamma =2.7\times {10}^{13} \mathrm{rad}/\mathrm{s}$. The linear conductivity of graphene can be approximately expressed as [35]:

$${\sigma }_0\approx \frac{i{e}^2{E}_F}{\pi {\hslash }^2\left(\omega +\frac{i}{\tau }\right)}.$$

(1)

The third-order non-linear conductivity of graphene without considering the two-photon coefficient [36] can be written as follows [37]:

$${\sigma }_3=-\frac{9e^2{(e{v}_F)}^{2}i}{8\pi {\hslash }^2{E}_F{\omega }^3},$$

(2)

where $e$ and $\omega$ are the electric charge and angular frequency of the incident light, respectively. $E_F$ and $\tau$ represent the Fermi energy and the relaxation time of graphene, respectively. $\hslash$ refers to the reduced Planck constant. Here $E_F=\hslash v_F\sqrt{\pi c_{2D}}$, wherein $v_F$ is the Fermi velocity of electrons ($v_F\approx {10}^6 m/s$) and $c_{2D}$ is the carrier density. It can be clearly seen that the linear and non-linear conductivity coefficients are both highly dependent on the Fermi energy, which provides an effective way to flexibly modulate OB devices with graphene.

In this paper, we calculate the propagation of electromagnetic waves through each dielectric layer. We assume that the propagation direction of the electromagnetic field is the Z-axis and that graphene is parallel to X-axis. The incident electromagnetic field in TE polarization reads as:

$$\{\begin{array}{c}\begin{array}{c}{E}_{\left(air,i\right)y}=E_i{e}^{i{k}_{0z}\left[Z-\left(d_s+11d_a+11d_b\right)\right]}{e}^{i{k}_{x}X}\\ +E_r{e}^{-i{k}_{0z}\left[Z-\left(d_s+11d_a+11d_b\right)\right]}{e}^{i{k}_{x}X}\end{array}\\ \begin{array}{c}{H}_{\left(air,i\right)x}=-\frac{k_{0z}}{{\mu }_0\omega }{E}_i{e}^{i{k}_{0z}\left[Z-\left(d_s+11d_a+11d_b\right)\right]}{e}^{i{k}_{x}X}\\ +\frac{k_{0z}}{{\mu }_0\omega }{E}_r{e}^{-i{k}_{0z}\left[Z-\left(d_s+11d_a+11d_b\right)\right]}{e}^{i{k}_{x}X}\end{array}\\ \begin{array}{c}{H}_{\left(air,i\right)z}=\frac{k_x}{{\mu }_0\omega }{E}_i{e}^{i{k}_{0z}\left[Z-\left(d_s+11d_a+11d_b\right)\right]}{e}^{i{k}_{x}X}\\ +\frac{k_x}{{\mu }_0\omega }{E}_r{e}^{-i{k}_{0z}\left[Z-\left(d_s+11d_a+11d_b\right)\right]}{e}^{i{k}_{x}X}\end{array}\end{array}$$

(3)

The transmitted electromagnetic field satisfies:

$$\{\begin{array}{c}{E}_{\left(air,t\right)y}=E_t{e}^{i{k}_{0z}Z}{e}^{i{k}_{x}X}\\ H_{\left(air,t\right)x}=-\frac{k_{0z}}{{\mu }_0\omega }{E}_t{e}^{i{k}_{0z}Z}{e}^{i{k}_{x}X}\\ H_{\left(air,t\right)z}=\frac{k_x}{{\mu }_0\omega }{E}_t{e}^{i{k}_{0z}Z}{e}^{i{k}_{x}X}\end{array}.$$

(4)

where $E_i$, $E_r$, and $E_t$ are the amplitude of the incident, reflected, and transmitted electric field, respectively. k₀ is the wave vector in vacuum and ${\epsilon }_0$ and ${\mu }_0$ are the dielectric constant and permeability in vacuum, respectively. According to the boundary conditions, the relationship between the electromagnetic fields on both sides of graphene can be represented as:

$$\{\begin{array}{c}{E}_{\left(graphene,rigth\right)y}\left(Z\right)=E_{\left(graphene,left\right)y}\left(Z\right)\\ \begin{array}{c}{H}_{\left(graphene,left\right)x}\left(Z\right)-H_{\left(graphene,rigth\right)x}\left(Z\right)\\ =\sigma E_{\left(graphene,left\right)y}\left(Z\right)\end{array}\end{array}$$

(5)

The electromagnetic field relation at each interface can be expressed as:

$$\{\begin{array}{c}{E}_{rigthy}\left(Z\right)=E_{lefty}\left(Z\right)\\ H_{leftx}\left(Z\right)-H_{rigthx}\left(Z\right)=\sigma E_{lefty}\left(Z\right)\end{array}$$

(6)

Finally, the relationship between $E_i$ and $E_r$ can be obtained; therefore, the OB phenomenon can occur.

3. Results and Discussions

Figure 2a shows the reflectance diagram of the DBR structure combined with Ag (black dotted line). The electric field intensity distribution of the DBR-Ag structure is shown in Figure 2b. The Tamm plasmon resonance mode is excited through the DBR-Ag structure, which leads to the reduced reflectivity. When the electromagnetic waves pass through the DBR structure, it is partially localized at the interface between DBR and Ag producing a strong local field enhancement phenomenon. Figure 2a also shows the reflectance diagram of the DBR structure combined with graphene (red dashed line). The electric field intensity distribution of the graphene-DBR structure is shown in Figure 2c. Similarly, most electromagnetic waves are localized at the interface between graphene and DBR, and the reflectance drops to 0% at 1 THz. The reflectance of the resonant reflection peak of the DBR-Ag structure is about 50%, and the local field enhancement phenomenon occurs at the interface of the DBR structure and the metal Ag, which is caused by the substrate of metal Ag. In order to improve the strength of the local electric field of the graphene, we consider the graphene-DBR structure with the DBR-Ag structure to form a new heterostructure: graphene-based TPS with Ag as the substrate.

Figure 3 shows the relationship between reflectance and the local field distribution of the TPS. Compared with the DBR-Ag structure and the graphene-DBR structure Tamm plasmon resonance mode, the reflectance of the graphene-based TPS is significantly higher. The electromagnetic wave can not only achieve total reflection phenomenon, but also form strong local field enhancement effect on the surface of heterogeneous structures. This is mainly due to the characteristics of the heterostructure of the graphene-based TPS with Ag as the substrate such as the robustness and unidirectional and field enhancement that create good conditions for investigating the OB phenomena of the reflected light beam. It is conducive to the excitation of the third-order non-linear effect of graphene and the generation of OB.

Next, we consider the variation of the OB phenomenon generated by the graphene-based TPS under different conditions. Due to the highly localized field enhancement in the graphene-based TPS, when the field strength E increases, the non-linear part of the graphene conductivity $\sigma ={\sigma }_0+{\sigma }_3{\left|E\right|}^2$ cannot be ignored, which excites and generates the OB phenomenon and causes a reduction in the OB threshold. The OB phenomenon of the reflectance and incident electric field at different Fermi energy of graphene are shown in Figure 4a. The dependence of the reflected electric field on the incident electric field for different Fermi energy of graphene are shown in Figure 4b. When the graphene Fermi energy is increased from 0.98 to 1.04 eV, the OB threshold is rapidly increased, and the hysteresis loop width is significantly enhanced. By adjusting the Fermi energy of graphene, the incident electric field needed to maintain the OB can be adjusted. It can be seen from Figure 4a that the OB threshold excited in the TPS (black dash line) is lower than that excited in the graphene-DBR structure (red dash–dot line). According to Figure 4c,d, we find that OB is also sensitive to the relaxation time of graphene. It is clear that when the relaxation time $\tau =1.03 ps$, the hysteresis width $∆E=0.025\times {10}^6 V/m$. As the relaxation time increases to $\tau =1.11 ps$, the hysteresis width increases to $∆E=0.145\times {10}^6 V/m$. Overall, the OB threshold was significantly reduced, while the hysteresis curve width increased by 5.8 times. If the relaxation time is small enough, ${\left|E_i\right|}_{\mathrm{up}}$ and ${\left|E_i\right|}_{\mathrm{down}}$ have a tendency to overlap, the hysteresis curve disappears, and the OB is smeared out. The threshold value and hysteresis width of the OB in the graphene-based TPS structure are both flexible and controllable in comparison to the OB excited by conventional micro/nano metallic devices. The dynamic modulate properties of graphene provide another viable idea for the fabrication and design of OB devices. Therefore, the concept of graphene-based TPS has a positive role in reducing the threshold of the OB. The pink dotted lines in Figure 4a,b show the results of COMSOL Multiphysics. The curve plotted at the Fermi energy of grapheme E_F = 1.00 eV is taken as a reference; the finite element (FEM) method is applied to check the linear and non-linear response effects of the proposed structure, and the rigorous coupled-wave theory is adopted to verify the accuracy of the numerical results. In the FEM method, the setting of the structural parameters is consistent with the mathematical calculation. According to the figures, the calculation results of COMSOL Multiphysics are in accordance with the mathematical calculation method, which proves the accuracy of the mathematical calculation. Therefore, the concept of the TPS has a positive role and practical value in reducing the threshold of OB.

In addition to the influence of the graphene parameters, the OB is also affected by the angle of incident light, the number of graphene layers, and the thickness of top layer, as shown in Figure 5. Figure 5a shows the dependence of the OB on the number of graphene layers (N). Due to the thin thickness of the monolayer graphene (only 0.34 nm), its electrical conductivity can be approximated as $\sigma \approx N{\sigma }_0$. When the number of graphene layers N = 1, the hysteresis width is about $∆E=0.995\times {10}^6 V/m$. When the number of graphene layers is N = 2, the hysteresis width is about $∆E=3.825\times {10}^6 V/m$. Therefore, as the number of graphene layers increases, both the optical bistable threshold and the hysteresis width increase simultaneously. The above results remind us that the optical bistability can be controlled by changing the number of graphene layers. In addition, under the condition that the other parameters are not changed, it can be seen from Figure 5b that the reflected electric field and incident electric field are affected by the change in the incident angle. As the angle of incidence increases from $\theta =5^o$ to $\theta ={15}^o$, the OB threshold increases and the hysteresis width increases, as shown in Figure 5b. When the incident angle is reduced to $\theta =0^o$, the optical bistable phenomenon of the reflected electric field still exists. This is mainly because the heterstructure of the TPS is under the normal incidence of light, which leads to the local enhancement of the electric field in the graphene, thereby providing the necessary conponent for the excitation of the low threshold OB phenomenon. Finally, we discuss the influence of the top layer with different thickness (d_s) on the reflection OB phenomenon, as shown in Figure 5c. The top layer is the dielectric layer of the cavity, and its change will affect the OB phenomenon of the TPS mode excitation. Therefore, it is necessary to further analyze the influence of the top layer thickness on the variation of the hysteresis curve. As show in Figure 5c, it can be seen from observation that as the top layer thickness increases, the thresholds of the reflectance OB and the reflected electric field OB both increase, and the width of hysteresis loop widens at the same time, which shows that the top layer thickness also has an obvious regulating effect on the OB phenomenon. In general, it is very important to choose the top layer thickness reasonably, and the variation in the OB hysteresis curve caused by the change in the top layer thickness will provide an important reference for the design of appropriate OB devices.

4. Conclusions

In summary, we have theoretically studied the tunable and low-threshold OB phenomenon based on graphene-based TPS in the terahertz range. The results show that the graphene-based TPS heterostructure has a very positive impact on reducing the OB threshold, and it can greatly improve the local electric field strength through constructive interference of the coupled electric fields. The threshold value of 10⁶ V/m is finally achieved by enhancing the third-order non-linear effect of graphene. Meanwhile, through further calculations, the hysteresis width and the upper and lower thresholds of the OB phenomenon can be flexibly adjusted by the graphene conductivity and are also closely related to the number of graphene layers (N ≤ 4), the incident light angle (θ), and the thickness of the top layer (d_s). Finally, we further verified the correctness of the mathematical calculation results through numerical simulation software. The study of the tunable low-threshold OB by graphene-based optical Tamm superlattices is expected to find potential applications in related non-linear optical devices.