1. Introduction

Optical bistability (OB) is a multivalue phenomenon similar to the hysteresis loop that occurs between the output light intensity and the input light intensity in a nonlinear optical system [1]. It has two well-discriminated high and low stable states which can both be controlled by optical signals. So it is widely used in all-optical switching [2,3], optical memory [4,5], optical diode [6] and all-optical logic gates [7] etc. The generation of OB and its control methods, especially the related OB devices, play a key role in optical communication system and optical network. In particular, the micro-nano optical bistable device integrated into chip size is the key to information transmission and processing. Therefore, the generation and manipulation means of OB based on micro-nano structures have received extensive attention among researchers in recent years. The OB in hyperbolic metamaterials [8], nonlinear magnetoplasmonic nanoparticles [9], spinor polariton condensates [10] and Fabry–Pérot cavity structure [11] has been reported. Additionally, in order to reduce the threshold of OB, some approaches with nonlinear enhancement effect or local field enhancement effect (e.g. metamaterial [12], surface plasmons [13], and electromagnetically induced transparency [14], etc.) were also reported. Although the basic theory of OB seems perfect at present, more practical, low-threshold, and dynamically controllable OB device implementations still poses a big challenge. OB and its induced mechanism in new materials or novel structures have become the main research direction in the realm of OB.

Graphene has become a “star” material in the field of optoelectronics owing to its excellent photoelectric properties [15–17]. As in the field of OB, graphene is featured with ultra-high giant nonlinear refractive index [18], ultra-fast response time and modulation rate [19], ultra-wide bandwidth [20], and dynamically controllable optical conductivity [21], making it extremely advantageous in the OB field. So graphene-based OB has become the focus of researchers. OB in graphene-wrapped dielectric nanowires [22], one-dimensional gratings metasurface [23], modified Kretschmann-Raether configuration [24], grapheneon-nonlinear dielectric surface [25], and graphene dielectric heterostructures [26] have also been reported. In addition, Gu et al. reported ultralow-power resonant OB in graphene-silicon hybrid optoelectronic devices [27]. Bao et al. observed graphene-based OB in Fabry-perot cavity structures [28]. It can be predicted optimistically that graphene-based OB will be one of the feasible ways to realize the application of practical OB devices.

In recent years, Tamm plasmon, a surface wave that is confined on the interface of two different media, has grabbed the attention of researchers due to its strong locality to light and being easily excited [29,30]. Unlike surface plasmon polaritons, the excitation of the optical Tamm states (OTSs) does not require a specific incidence angle, which can be excited by TE-polarized waves and has local field enhancement effect [31]. Attracted by OTSs’ excellent properties, researchers are not only enthusiastic about realizing OTSs through various structures and methods but also inclined to put OTSs into application [32–34]. Meanwhile, these characteristics are very important to realize low-threshold OB. However, traditional OTSs are excited mainly based on metal-distributed Bragg reflector structure [35], which do not have the nonlinearity to achieve OB. Fortunately, graphene is intrinsically a semimetal with special metallic properties under certain conditions [36], OTSs can be also excited based on graphene-based Bragg reflector structure [37]. Thus, an interesting question is induced considering the advantages of graphene in achieving OB: is it possible to achieve tunable, low-threshold OB by exciting OTSs with graphene-based photonic crystal structure? To answer this question, we theoretically investigated the OB phenomenon in the terahertz band based on graphene-distributed Bragg reflector (DBR) construction. It is found that graphene-based OTSs can be used to achieve low-threshold OB. The low-threshold of OB originates from local field enhancement caused by OTSs excitation. Besides, the tunability of graphene’s conductivity has created a condition for tunable OB based on this structure. Electrically tunable optical bistable devices based on graphene OTSs allow us to find potential applications in nonlinear optical elements.

2. Theoretical model and method

Graphene-DBR structure is shown in Fig. 1. This structure is composed of a graphene sheet, a top layer and a one-dimensional photonic crystal (1D PC). Among them, the PC is formed by alternately superposed dielectric layers $A$ and $B$ with a period of $N$. The top layer is coated by monolayer graphene. Nowadays, the fabrication of 1D photonic crystal and the transfer of graphene are mature technologies. It is not hard to fabricate the proposed structure as shown in Fig. 1. Hence, this program is feasible [38,39]. Without considering the external magnetic field and under the random-phase approximation, the graphene surface conductivity in the terahertz band can be approximately expressed as [40]:

 

Fig. 1 Schematic diagram of the investigated Tamm plasmon structure, where the top layer is inserted between nonlinear graphene and 1D PC. A plane wave of amplitude $E_i$ is incident on the structure with incident angle $\theta$, giving rise to a reflected and a transmitted wave with amplitude $E_r$ and$E_t$, respectively. $F_i$ and $B_i(i=1~n)$ are the amplitudes of the transmitted and reflected waves inside the dielectric slabs.

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Meanwhile, $z$ axis is chosen as the propagation direction and the location of graphene is set to $z=0$; the $x$ axis is parallel to the plane where the graphene is located, and the conductivity of graphene can be expressed as: $\sigma ={\sigma }_0+{\sigma }_3{\left|E_{0y}\left(z=0\right)\right|}^2$. For convenience, only TE-polarized is considered in this paper, thus the electric and magnetic fields from air to the left of graphene at the incident angle $\theta$ are expressed as:

Similarly, the electric and magnetic fields of the top layer are expressed as:

For the medium m ($m=\left\{2,3,\mathrm{...}n\right\}$), the electric and magnetic fields are expressed as:

At the far right of the whole structure, the electric and magnetic fields are expressed as:

Particularly, the boundary condition at the position$z=0$ satisfies $E_{0y}\left(z=0\right)=E_{1y}\left(z=0\right)$and $H_{0x}\left(z=0\right)-H_{1x}\left(z=0\right)=\sigma E_{0y}\left(z=0\right)$. Based on the boundary conditions of different contact surfaces, the relation between $E_i$, $E_r$, $E_t$, $F_m$, $B_m,m=\left\{1,2,\mathrm{...}n\right\}$is achieved, then the relation between $E_r$and $E_i$ is obtained, thus the OB can be observed under appropriate parameter conditions.

3. Results and discussions

In this section, the variation of reflectance with incident wavelength and $E_i$ is first presented. To simplify the discussion, only TE polarization is discussed and the initial incident angle is set as $\theta ={10}^{\circ }$ in the work below. As OTSs cannot be excited in the whole structure, photonic bandgap will be exhibited in the wavelength range near the incident wavelength when there is no graphene in the structure. With the dielectric loss of medium A and B neglected, the reflectance of the structure in the photonic bandgap is almost 100%, as shown in Fig. 2(a). However, the introduction of graphene led to a significant change in the reflectance. The reflection peak distinctly appeares in the original photonic bandgap when the top layer is coated by a monolayer graphene. When $E_F=0.95\text{eV}$, the reflection peak appears at $\lambda =293.5\text{μm}$ and the position of reflection peak is also regulated by Fermi energy. It is known that the excitation of OTSs needs to satisfy $r_{Gra}{r}_{DBR}\exp \left(2i\varphi \right)\approx 1$ [30,37], where $r_{Gra}$ and $r_{DBR}$ are the reflectance of electromagnetic waves in the top layer on the graphene interface and PC surface respectively; $\varphi$ is the phase change of electromagnetic wave propagation in the top layer. After simplifying the equation, the excitation conditions can be written as $\left|r_{Gra}\right|\left|r_{DBR}\right|\approx 1$ and $Arg\left(r_{Gra}{r}_{DBR}\exp \left(2i\varphi \right)\right)\approx 0$. Here, $r_{Gra}=-0.79-0.49\text{i}$, $r_{DBR}=-0.999+0.032\text{i}$, and $Arg\left(r_{Gra}{r}_{DBR}\exp \left(2i\varphi \right)\right)\approx 0.04$ are calculated according to the initial parameters in Fig. 1. So the reflection anomaly in the Fig. 2(a) is an obvious OTSs excitation behavior. At $\lambda =293.5\text{μm}$, the normalized distribution of electric fields is plotted when $E_F=0.95\text{eV}$, as shown in Fig. 2(b) and (c). Obviously, an apparent field enhancement occurred near the graphene when it covers the top layer, which validated the reflection anomaly in Fig. 2(a). The local field enhancement of OTSs on graphene surface created acondition for low-threshold bistability. In order to obtain appropriate reflectance, the wavelength is selected as $315\text{μm}$, other parameters are the same as Fig. 2(a). It can be seen from Fig. 2(d) that a significant low-threshold bistable phenomenon occurred in the reflectance of the structure as a function of the incident electric field. According to the curve in the figure, the absence of graphene makes the reflectance insensitive to $E_i$, and the reflectance is always approximately 1 as $E_i$ changes. But in the OTSs excitation state, the $E_i$ exhibits a significant low-threshold bistable phenomenon at the order of ${10}^6\text{V/m}$, and the threshold of the bistable state is significantly regulated by the graphene Fermi energy. For the convenience of comparison, the relation diagram of reflectance with incident electric field without 1D PC is drawn. It can be seen that although the nonlinear conductivity of graphene itself can enable bistability, the threshold will be nearly five orders of magnitude higher than that after the excitation of OTSs. Therefore, OTSs excitation plays a very positive role in promoting low-threshold bistability. To verify the theoretical calculations, we simulated the hysteresis curve (Fig. 2(d)) to describe the relationship between the reflectance and the incident electric field $E_i$ at $E_F=0.95\text{eV}$ based on the FEM method (Comsol Multiphysics).

 

Fig. 2 (a) Reflectance as functions of wavelength at different Fermi energies for TE-polarized wave. The normalized electric field profile distributions in the graphene-1D PC configuration without (b) and with (c) the covering of nonlinear graphene. (d) The dependencies of the reflectance on the input light intensity at different Fermi energies of the graphene.

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By comparison, the hysteresis loop obtained in the numerical simulation is consistent with our theoretical calculations, indicating that the calculations are correct.

Furthermore, based on the calculation above, the relation between $E_r$and $E_i$ can be easily presented, as shown in Fig. 3(a). It can be seen from Fig. 3(a) that there is a simple monotonic increasing relation between $E_r$ and $E_i$ owing to the absence of graphene in the structure, namely, the hysteresis curve is not presented. The hysteresis curve of $E_r$ and $E_i$ can be clearly generated after graphene is loaded, because the high third-order nonlinear conductivity of graphene can satisfy the multi-value phenomenon between $E_r$ and $E_i$. Taking $E_F=1.0\text{eV}$as an example, a backwards S Curve can be drawn by connecting all the working points, namely, a-1-b-2-d-3-c. In this curve, $E_r$ is increasing slowly with $E_i$ (a-1-b) when $E_i$ is small, which is considered as one of the stable states. But when $E_i$ increases to ${\left|E_i\right|}_{down}=5.79\times {10}^6\text{V/m}$, $E_r$ would jump to another stable state. Even if $E_i$ is lowered at this moment, the reflected electric filed $E_r$ would not return to the first stable state immediately.

 

Fig. 3 (a) The dependencies of the reflected electric field $E_r$ on the input light intensity $E_i$at different Fermi energies of the graphene. (b) The dependencies of the switch-up and switch-down threshold electric fields on the Fermi energy of the graphene.

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Instead, it would slowly decrease with $E_i$, and eventually jump from the second stable state to the first stable state when $E_i$ reaches ${\left|E_i\right|}_{up}=5.12\times {10}^6\text{V/m}$ (c-3-d). The “b-2-d” segment is not stable. The increase of the reflected electric field $E_r$ is not stable when the input electric field $E_i$ is increasing, which cannot be observed in the experiment. Hence, there would be two stable jumps at b and d, thus forming a bistable loop. This bistable loop is also reflected in our numerical simulation. But we cannot observe the “b-2-d” segment in the simulation. The bistable working range of the structure lies between Curve ab and Curve cd, where there are one incident electric field and two stable reflected electric fields. At this time, the width of the hysteresis satisfies $\Delta \left|E_i\right|={\left|E_i\right|}_{down}-{\left|E_i\right|}_{up}=0.67\times {10}^6\text{V/m}$. This is a typical OB phenomenon. The third-order nonlinear characteristic of graphene plays a key role in the generation of OB. At the same time, the graphene-DBR structure plays an active role in reducing bistable threshold by exciting OTSs. Essentially, the graphene-DBR structure can be considered as a resonant cavity composed of graphene and 1DPC, which can achieve nonlinear enhancement by satisfying specific wave vector matching, thus creating a condition for achieving low-threshold OB phenomenon. As shown in Fig. 3(a), the absence of PC can significantly increase the threshold of OB, and ${\left|E_i\right|}_{up}=1.8\times {10}^{11}\text{V/m}$, which is very impressive. Besides, the excellent electrical controllability of the conductivity of graphene also makes it possible to realize OTSs-based tunable OB phenomenon. In Fig. 3(a), the decrease in graphene Fermi energy causes the hysteresis curve to move toward a lower incident electric field as a whole, and ${\left|E_i\right|}_{up}$ decreases to $4.23\times {10}^6\text{V/m}$ when $E_F=0.9\text{eV}$.

At the same time, the width of the hysteresis starts to shrink and narrow. The effect of Fermi energy change on the threshold value and hysteresis width is detailed, as shown in Fig. 3(b). It is found that although the reduction of Fermi energy has positive effect on further reducing the threshold of OB, it may also lead to the narrowing or even disappearance of the width of hysteresis. In general, compared with traditional metal, graphene can excite OTSs to achieve flexible control of threshold and hysteresis width, which provides a feasible idea for dynamic controllable optical bistable devices.

Next, the influence of incident angle on the behavior of OB is discussed. During calculation, $E_F$ is set as $\text{0}\text{.95}\text{eV}$, and other parameters are the same as Fig. 2. The relation between reflection coefficient and $E_r$with $E_i$ at different incident angles is shown in Fig. 4. Similar to the regulation of graphene Fermi energy on hysteresis curve, the effect of incident angle on hysteresis behavior is reflected in both threshold and hysteresis width. For reflectance, the threshold value of reflectance curve begins to increase with the incident angle. Corresponding to the reflectance curve, there is a similar law for the bistable curve of $E_r$ with $E_i$, and ${\left|E_i\right|}_{down}$ has a greater growth rate with respect to ${\left|E_i\right|}_{up}$, resulting in a significant broadening effect of the hysteresis width as the incident angle increases. For example, when the incident angle is set as ${10}^{\circ }$, ${\left|E_i\right|}_{up}$and ${\left|E_i\right|}_{down}$ will be $4.67\times {10}^6\text{V/m}$and $5.15\times {10}^6\text{V/m}$ respectively, and the hysteresis width will be: $\Delta \left|E_i\right|=0.48\times {10}^6\text{V/m}$. When $\theta {\text{=15}}^{\circ }$, ${\left|E_i\right|}_{up}$and ${\left|E_i\right|}_{down}$ will be $4.82\times {10}^6\text{V/m}$and $5.47\times {10}^6\text{V/m}$ respectively, and the hysteresis widthwill be: $\Delta \left|E_i\right|=0.65\times {10}^6\text{V/m}$. The calculation shows that the above rule will become increasingly obvious as the incident angle continues to increase. However, when the incident angle is lowered, the hysteresis width will be narrowed as the incident angle decreases but will not disappear completely. It can be seen from Fig. 4(b) that when the incident angle$\theta =0^{\circ }$, the OB phenomenon of $E_r$ with $E_i$ still exists. This is mainly because OTSs can be excited under the condition of vertical incidence which provides a condition to realize OB. Therefore, compared with the optical bistable approach to achieve low-threshold through the coupling prism excitation surface plasmon, the optical bistable approach based on OTSs excitation to achieve low-threshold is more advantageous in the incident angle, because the former needs to meet the minimum incident angle.In previous discussion, the resonant angle of OTSs is sensitive to changes in the wavelength of incident light. Meanwhile, the OB of the resonant angle under the same incident wavelength but different incident electric field fundamentally reflects the cause of the OB based on OTSs in graphene-DBR structure. Next, how to produce optimum OB with changing incident wavelengths would be the focus of our discussion. The variation curve of the reflected electric field with four different incident wavelengths changing with the incident electric field is compared as Fig. 5. According to the comparison, both ${\left|E_i\right|}_{up}$and ${\left|E_i\right|}_{down}$ decrease with the incident wavelength. But ${\left|E_i\right|}_{down}$decreases faster than${\left|E_i\right|}_{up}$, which directly leads to a rapid narrowing or even disappearance of the hysteresis width. The phenomenon of OB disappears at $\lambda =305\text{μm}$. Besides, the increase of incident wavelength will lead to the rise of ${\left|E_i\right|}_{up}$and ${\left|E_i\right|}_{down}$, and the hysteresis width also increased. This is mainly because the third-order nonlinear conductivity of graphene would increase with the working wavelength, and the enhanced third-order nonlinear conductivity of graphene could stimulate the nonlinear increase of the whole structure. However, it does not mean that the incident wavelength can be increased indefinitely for wider hysteresis. As shown in Fig. 2(a), when $\lambda >330\text{μm}$, the wavelength will exceed the range of photonic bandgap, and OB can’t be realized by exciting OTSs. In general, optimal working wavelength should be considered when using bistable devices (such as optical switches produced based on the above structure) in the all-optical network with the most reasonable threshold values and hysteresis widths of the devices.

 

Fig. 4 (a) Reflectance, and (b) reflected electric field $E_r$as functions of the input light intensity$E_i$at different incident angles.

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Fig. 5 The influence of the properties of incident light on the reflected optical bistable phenomena for different work wavelengths.

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The top layer acts as an intracavity dielectric layer to excite OTSs, and the structural parameters of the top layer will sensitively affect the angle and frequency position of OTSs. Correspondingly, these parameters are equally sensitive to the effects of OB hysteresis curves. Therefore, it is necessary to further analyze the variation law of top layer structure parameters on the hysteresis curve. This variation law will provide important reference for the design of appropriate OB devices. On the premise that other parameters are the same as that in Fig. 2, the relation between $E_r$ and $E_i$ under different $n_t$ and $d_t$ (top layer refractive indices and thicknesses) is plotted, as shown in Fig. 6. Since the lower $n_t$ will cause the reflection peak of the OTSs to move towards the short wavelength, the incident wavelength will start to move away from the resonance wavelength of the OTSs, thus both ${\left|E_i\right|}_{up}$ and ${\left|E_i\right|}_{down}$ will increase and are accompanied by a growing hysteresis width. This is consistent with the variation law of hysteresis curve caused by the increase of incident wavelength in Fig. 5. Meanwhile, $n_t$ will narrow the hysteresis width gradually and finally cause the OB to disappear. A similar phenomenon applies to the effect of changes in $d_t$ on the hysteresis curve. Due to the change in $d_t$, the wavelength of the excitation of OTSs should be controlled within the bandgap. If the thickness is too small, the wavelength of the excitation of OTSs will leave the bandgap range and the OB will disappear; similarly, too large thickness will narrow the hysteresis width and cause the structure to lose the characteristic of OB. Therefore, it is very important to reasonably select the thickness of the top layer.

 

Fig. 6 The influence of the properties of incident light on the reflected optical bistable phenomena for (a) different refractive index of top layer, and (b) different thicknesses of top layer.

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4. Conclusions

In summary, the optical bistable phenomena of the light reflected at the graphene-DBR structure were investigated to realize tunable OB devices with low-threshold value at the terahertz frequencies due to the excitation of the OTSs near the surface of nonlinear graphene. The results have shown that Tamm plasmon are strongly dependent on the structural parameters of the geometry and the optical conductivity properties of nonlinear graphene, where the resonant angle is great influenced by the thickness and refractive index of the top layer and the Fermi energy of the graphene. By setting the appropriate initial angular and wavelength offset, hysterical curves are observed, and the switching-up and switching-down threshold values are controlled by changing the initial angular and wavelength offset. Combining the mature graphene fabrication and transfer technology with the advanced 1D photonic crystal fabrication technology, it is believed that the pathway of graphene OB investigated in this paper appears to be particularly promising and is expected to have wide applications in optical logic, all-optical switching and other related nonlinear optical devices.