Nonlinear Optical Bistability in a Bragg Reflector Multilayered Structure with MoS₂

Abstract

The special band structure of bilayer MoS₂ makes it show strong nonlinear optical characteristics in the visible band, which provides a new way to develop visible nonlinear devices. In this paper, we present a theoretical analysis of the optical bistability (OB) in a silver–Bragg reflector structure by embedding bilayer MoS₂ at the visible band. The nonlinear OB phenomenon is achieved due to the nonlinear conductivity of the bilayer MoS₂ and the excitation of the optical Tamm state at the interface between the silver and the Bragg reflector. It is found that the hysteresis behavior and the threshold width of the OB can be effectively tuned by varying the incident light wavelength. In addition, the optical bistable behavior of the structure can be adjusted by varying the position of the MoS₂ inset in the defect layer, the incident angle, and the structural parameters of the spacer layer. We believe the above results can provide a new paradigm for the construction of controllable bistable devices.

1. Introduction

Optical bistability (OB) is a nonlinear optical phenomenon in which two different stable output states are produced for a given input state in an optical system [1]. The characteristic curve of OB shows a distinct hysteresis loop relationship with a delay and abrupt change, and therefore, it is widely used in all-optical switches [2,3], all-optical logic gates [4,5], biosensors [6], optical storage [7], computing [8], etc. Although OB has been around for decades since its theoretical proposal, there is still no fully commercialized all-optical bistability device because there are still gaps between two key indicators and commercialization: first, the driving optical field power required to achieve OB is in the milliwatt range or below, and to achieve such a low threshold optical bistability, the nonlinear material used needs to have a large nonlinear coefficient; The second requirement is that the state transition response time of OB should be in the picosecond range or below, which also puts higher demands on the response time of nonlinear materials. On this basis, the preparation should meet the requirements of simplicity, ease of integration, low insertion loss, and stable performance as much as possible. Due to the relatively low requirements for the steady-state conversion speed (switching speed) of OB in current optical communication networks, reducing the optical power driving the OB conversion has become the core goal of research on OB and its applications. In recent years, micro and nano technologies have become increasingly mature and developed rapidly, people start to seek OB phenomena in micro and nano structures, based on photonic crystals [9,10], the Fabry–Perot cavity [11,12], hyper-bolic metamaterials [13], and nonlinear ring resonator structures [14], and a series of OB phenomena in micro and nano structures have been proposed one after another. We know that there is a positive correlation between the threshold of OB and the nonlinear coefficient of the material, and the higher the nonlinear coefficient of the material the lower the threshold in achieving OB. In recent years, The two-dimensional materials graphene and three-dimensional Dirac semimetals stand out among the many materials due to their large nonlinear coefficients in the terahertz band [15,16], and OB studies based on graphene and Dirac semimetals have been successively proposed. For example, OB in the graphene-on-Kerr nonliner surface [17], OB in the graphene–Bragg reflector structure [18], and OB in one-dimensional photonic crystal heterostructures based on Dirac semimetals [19] have been reported. Although OB phenomena based on graphene and Dirac semimetals have been studied extensively, the research has been mostly limited to the terahertz band. In addition, it is not difficult to see that the core starting point of the above work is to combine two-dimensional materials with excellent third-order nonlinear conductivity characteristics with micro nano structures with local field enhancement, in order to seek OB schemes with the lowest possible threshold [20,21]. At present, OB research in the visible band is still in its infancy. Therefore, the search for new ways to tune OB in the visible band is being tried by researchers.

Recently, it has been found that the bilayer MoS₂ has a high third-order nonlinear coefficient in the visible band, which has attracted attention. It was found that the monolayer MoS₂ has a linear electro-optical effect (Pockels effect) [22,23] and the bilayer MoS₂ has a secondary electro-optical effect (Kerr effect) [22]. The bilayer MoS₂ gives a large nonlinear coefficient in the visible band based on the Kerr effect [24], which provides nonlinear conditions to achieve OB. Also, we know that the wave vector in a vacuum can be tuned by changing the wavelength of the incident light, thereby changing the third-order nonlinear conductivity of MoS₂, which provides the possibility for the implementation of tunable OB. Compared to terahertz-band-based graphene and 3D Dirac semimetal OB studies, this study provides a way to tune OB in the visible band based on MoS₂, which makes this work meaningful. Optical Tamm states (OTSs) are lossless interface modes localized at two interfaces with different media [25]. They have strong localization, are easily excited, and surface-constrained properties [26]. Current studies on OTS are mainly based on the metal–Bragg reflector structure [27,28] and one-dimensional photonic crystal heterostructure [29,30]. We know that local field enhancement contributes positively to the threshold reduction of OB, which is beneficial for achieving OB [31].

In this paper, we theoretically analyze the OB phenomenon in a silver–Bragg reflector structure based on bilayer nonlinear MoS₂. The results show that OTS is generated at the interface between the silver and the spacer layer, which leads to an enhancement of the local field. Furthermore, inserting the bilayer of MoS₂ into the multilayer structure provides a nonlinear condition for the generation of OB. At the same time, we can dynamically tune the hysteresis behavior and threshold width of the OB by varying the wavelength of the incident light. This OB scheme has the properties of a simple structure, easy preparation, and OB realization in the visible band, for which we believe this scheme can provide a new idea for the manipulation of OB in optical fields, such as all-optical switches and other nonlinear optical bistable devices, in the future.

2. Theoretical Model and Method

We consider a one-dimensional multilayer structure with a silver–Bragg reflector and bilayer MoS₂. As shown in Figure 1, Figure 1a shows the visual view and Figure 1b shows the side view. In this structure, the Bragg reflector consists of alternating periodic arrangements of dielectric A (SiO₂) and dielectric B (Si), and the period is set to $T=10$. The refractive index of medium A is $n_A=1.46$, and that of medium B is $n_B=2.82$. In this Bragg reflector, the central wavelength is set to ${\lambda }_c=660 \mathrm{nm}$, the thickness of medium A is $d_{\mathrm{A}}={\lambda }_{\mathrm{c}}/4n_A$, and the thickness of medium B is $d_{\mathrm{B}}={\lambda }_{\mathrm{c}}/4n_B$. The bilayer of MoS₂ is embedded between the spacer layer and the Bragg reflector. The silver placed at the top of the structure and the relative permittivity and refractive index of sliver can be expressed as follows [32]:

$${\epsilon }_{Ag}(\omega )={\epsilon }_{\infty 1}-{{\omega }_{pA\mathrm{g}}^2/(\omega }^2+i\tau \omega ),n_{Ag}(\omega )=\sqrt{{\epsilon }_{A\mathrm{g}}\left(\omega \right)},$$

(1)

where $\omega$ is the angular frequency, ${\epsilon }_{\infty 1}$ is the limit value when its angular frequency tends towards infinity, ${\omega }_{pA\mathrm{g}}$ is the plasma frequency of Ag, and $\tau$ is the damping coefficient. Here we set ${\epsilon }_{\infty 1}=5$, $\hslash {\omega }_{pA\mathrm{g}}=9 \mathrm{eV}$, $\hslash \tau =18 \mathrm{eV}$, where $\hslash$ is the reduced Planck constant. In addition, we set the thickness of silver to $d_{\mathrm{Ag}}=30 \mathrm{nm}$. The refractive index of the spacer layer HfO₂ is set to $n_{\mathrm{s}}=1.95$, and the thickness is designed to be $d_{\mathrm{s}}=70 \mathrm{nm}.$ Each monolayer MoS₂ has a thickness of $\Lambda =0.65 \mathrm{nm}$.

Based on the current mature micro and nano fabrication technology, structures with the above structural parameters can be easily manufactured. Meanwhile, the bilayer MoS₂ molecule has a strong Kerr effect in the visible band. Here, we describe MoS₂ using a dielectric constant. Neglecting the effect of external magnetic fields under random phase conditions, the linear dielectric constant of the MoS₂ molecule can be expressed as follows [33]:

$${\epsilon }_{(\omega )}^{(1)}={\epsilon }_{\infty 2}+{\displaystyle \sum _{i=0}^5\frac{a_i{\omega }_p^2}{{\omega }_i^2-{\omega }^2-i{b}_i\omega }}-{\displaystyle \frac{2\alpha }{\sqrt{\pi }}}DawsonF({\displaystyle \frac{\mu -\hslash \omega }{\sqrt{2}\sigma }})+\alpha \exp ({\displaystyle \frac{-{(\hslash \omega -\mu )}^2}{2{\sigma }^2}})$$

(2)

where ${\epsilon }_{\infty 2}=4.44$ is the direct-current dielectric constant, ${\mathrm{a}}_j$ represents the oscillator strength (specific value is ${\mathrm{a}}_0=2.00\times {10}^5,$ ${\mathrm{a}}_1=5.75\times {10}^4,$ ${\mathrm{a}}_2=8.14\times {10}^4,$ ${\mathrm{a}}_3=8.22\times {10}^4,$ ${\mathrm{a}}_4=3.31\times {10}^5,$ ${\mathrm{a}}_5=4.39\times {10}^4$). ${\mathrm{b}}_j$ represents the damping coefficient (specific value is ${\mathrm{b}}_0=1.63\times {10}^{13}\hspace{0.33em}\mathrm{rad}/\mathrm{s},$ ${\mathrm{b}}_1=8.92\times {10}^{13}\hspace{0.33em}\mathrm{rad}/\mathrm{s},$ ${\mathrm{b}}_2=1.70\times {10}^{14}\hspace{0.33em}\mathrm{rad}/\mathrm{s},$ ${\mathrm{b}}_3=1.80\times {10}^{14}\hspace{0.33em}\mathrm{rad}/\mathrm{s},$ ${\mathrm{b}}_4=4.27\times {10}^{14}\hspace{0.33em}\mathrm{rad}/\mathrm{s},$ ${\mathrm{b}}_5=1.18\times {10}^{15}\hspace{0.33em}\mathrm{rad}/\mathrm{s}$). ${\omega }_j$ represents the response frequency (specific value is ${\omega }_0=0\hspace{0.33em}\mathrm{rad}/\mathrm{s},$ ${\omega }_1=2.85\times {10}^{15}\hspace{0.33em}\mathrm{rad}/\mathrm{s},$ ${\omega }_2=3.06\times {10}^{15}\hspace{0.33em}\mathrm{rad}/\mathrm{s},$ ${\omega }_3=4.19\times {10}^{15}\hspace{0.33em}\mathrm{rad}/\mathrm{s},$ ${\omega }_4=4.39\times {10}^{15}\hspace{0.33em}\mathrm{rad}/\mathrm{s},$ ${\omega }_5=6.50\times {10}^{15}\hspace{0.33em}\mathrm{rad}/\mathrm{s}$). ${\omega }_p=7\times {10}^{12}\hspace{0.33em}\mathrm{rad}/\mathrm{s},$ is the plasma frequency [34]. DawsonF represents Dawson’s Function. $\alpha$, $\mu$, and $\sigma$ represent the maximum value, mean, and variance of a typical Gaussian distribution function, respectively. Their values are $\alpha =23.224,$ $\mu =2.7723\hspace{0.33em}\mathrm{e}\mathrm{V},$ and $\sigma =0.3089\hspace{0.33em}\mathrm{e}\mathrm{V}$, respectively. The first-order linear conductivity (${\epsilon }_{(\omega )}^{(1)}$) of MoS₂ is as follows:

$${\sigma }_{(\omega )}^{(1)}={\displaystyle \frac{\Lambda k_0}{\mathrm{i}{\eta }_0}}({\epsilon }_{(\omega )}^{(1)}-1),$$

(3)

where ${\eta }_0=377 \mathsf{\Omega }$ is the vacuum resistivity, and $k_0=\omega /c$ is the wave vector in a vacuum. The third-order nonlinear conductivity of double-layer MoS₂ can be expressed:

$${\sigma }_{(\omega )}^{(3)}={\displaystyle \frac{2\Lambda k_0}{\mathrm{i}{\eta }_0}}{\displaystyle \frac{3m_e{\omega }_0^2{\epsilon }^3}{d^2{N}^3{e}^4}}\left[{({\epsilon }_{(\omega )}^{(1)}-1)}^3\right]\left({\epsilon }_{(-\omega )}^{(1)}-1\right),$$

(4)

where $m_e$ is the mass of an electron, ${\omega }_0$ is the centrality frequency with ${\omega }_0=4.21\times {10}^{15} \mathrm{rad}/\mathrm{s}$, $d$ is of the order of the dimensions of an atom ($d=3\AA$), $N$ is the atomic number density with $N={10}^{28}\hspace{0.33em}{\mathrm{m}}^{-3}$, ${\epsilon }_0$ is the permittivity of vacuum, and $e$ is the electron charge [24]. Based on the above linear and nonlinear conductivity, the total conductivity of MoS₂ can be expressed as follows:

$${\sigma }_{(\omega )}={\sigma }_{(\omega )}^{(1)}+{\sigma }_{(\omega )}^{(3)}{\left|E_{2y}(z=d_{Ag}+d_s)\right|}^2$$

(5)

where $E_{2y}$ is the local electric field at the double-layer MoS₂. The equations of ${\sigma }_{(\omega )}^{(1)}$, ${\sigma }_{(\omega )}^{(3)}$, and ${\sigma }_{(\omega )}$ show that the electrical conductivity of MoS₂ has an important relationship with the wave vector in a vacuum. We assume that the electromagnetic wave propagates along the z-axis, the surface on which the silver is located is $z=0$, and the MoS₂ is parallel to the plane of the x-axis and y-axis. Considering only TE polarization, the electromagnetic field incident on the surface of the silver layer from the air according to the Maxwell equation can be expressed as follows:

$$\left\{\begin{array}{l}{E}_{iy}=E_i{e}^{i{k}_{iz}z}{e}^{i{k}_{x}x}+\hspace{0.33em}{E}_r{e}^{-i{k}_{iz}z}{e}^{i{k}_{x}x}\hspace{0.33em},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hfill \\ H_{ix}=-{\displaystyle \frac{k_{iz}}{{\mu }_0\omega }}{E}_i{e}^{i{k}_{iz}z}{e}^{i{k}_{x}x}+\hspace{0.33em}{\displaystyle \frac{k_{iz}}{{\mu }_0\omega }}{E}_r{e}^{-i{k}_{iz}z}{e}^{i{k}_{x}x},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hfill \\ H_{iz}={\displaystyle \frac{k_x}{{\mu }_0\omega }}{E}_i{e}^{i{k}_{iz}z}{e}^{i{k}_{x}x}+{\displaystyle \frac{k_x}{{\mu }_0\omega }}{E}_r{e}^{-i{k}_{iz}z}{e}^{i{k}_{x}x}\hspace{0.33em},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hfill \end{array}\right.$$

(6)

where $k_{0z}=k_{0}cos(\theta ),$ $k_{0x}=k_{0}sin(\theta ),$ ${\mu }_0$ denotes the vacuum permeability, E_i denotes the incident electric field, E_r denotes the reflected electric field, and $E_t$ denotes the transmitted electric field. F_m denotes the forward electric field, and B_m denotes the reverse electric field in the m-layer of the dielectric.

Similarly, the electric and magnetic fields in the silver layer can be expressed as follows:

$$\left\{\begin{array}{l}{E}_{1y}=F_1{e}^{i{k}_{1\mathrm{z}}z}{e}^{i{k}_{x}x}+B_1{e}^{-i{k}_{1\mathrm{z}}z}{e}^{i{k}_{x}x}\hspace{0.33em},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hfill \\ H_{1x}=-{\displaystyle \frac{k_{1z}}{u_0\omega }}{F}_1{e}^{i{k}_{1\mathrm{z}}z}{e}^{i{k}_{x}x}+{\displaystyle \frac{k_{1z}}{u_0\omega }}{B}_1{e}^{-i{k}_{1\mathrm{z}}z}{e}^{i{k}_{x}x},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hfill \\ H_{1\mathrm{z}}={\displaystyle \frac{k_x}{u_0\omega }}{F}_1{e}^{i{k}_{1\mathrm{z}}z}{e}^{i{k}_{x}x}+{\displaystyle \frac{k_x}{u_0\omega }}{B}_1{e}^{-i{k}_{1\mathrm{z}}z}{e}^{i{k}_{x}x},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hfill \end{array}\right.$$

(7)

The electric and magnetic fields in the HfO₂ layer can be represented as follows:

$$\left\{\begin{array}{l}{E}_{2\mathrm{y}}=F_2{e}^{i{k}_{2z}(\mathrm{z}-d_{A\mathrm{g}})}{e}^{i{k}_{x}x}+B_2{e}^{-i{k}_{2z}(\mathrm{z}-d_{A\mathrm{g}})}{e}^{i{k}_{x}x}\hspace{0.33em},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hfill \\ H_{2x}=-{\displaystyle \frac{k_{2z}}{u_0\omega }}{F}_2{e}^{i{k}_{2z}(\mathrm{z}-d_{A\mathrm{g}})}{e}^{i{k}_{x}x}+{\displaystyle \frac{k_{{}_{2z}}}{u_0\omega }}{B}_2{e}^{-i{k}_{2z}(\mathrm{z}-d_{A\mathrm{g}})}{e}^{i{k}_{x}x},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hfill \\ H_{2\mathrm{z}}={\displaystyle \frac{k_x}{u_0\omega }}{F}_2{e}^{i{k}_{2z}(\mathrm{z}-d_{A\mathrm{g}})}{e}^{i{k}_{x}x}+{\displaystyle \frac{k_x}{u_0\omega }}{B}_2{e}^{-i{k}_{2z}(\mathrm{z}-d_{A\mathrm{g}})}{e}^{i{k}_{x}x},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hfill \end{array}\right.$$

(8)

For a medium m (m = 3,4,5,6, …, 22), the electric and magnetic fields can be expressed as follows:

$$\left\{\begin{array}{l}{E}_{my}=F_m{e}^{i{k}_{jz}[\mathrm{z}-(d_{A\mathrm{g}}+d_s+\alpha d_{\mathrm{A}}+\beta d_{\mathrm{B}})]}{e}^{i{k}_{x}x}+B_m{e}^{-i{k}_{jz}[\mathrm{z}-(d_{A\mathrm{g}}+d_s+\alpha d_{\mathrm{A}}+\beta d_{\mathrm{B}})]}{e}^{i{k}_{x}x}\hspace{0.33em},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hfill \\ H_{mx}=-{\displaystyle \frac{k_{jz}}{u_0\omega }}{F}_m{e}^{i{k}_{jz}[\mathrm{z}-(d_{A\mathrm{g}}+d_s+\alpha d_{\mathrm{A}}+\beta d_{\mathrm{B}})]}{e}^{i{k}_{x}x}+{\displaystyle \frac{k_{jz}}{u_0\omega }}{B}_m{e}^{-i{k}_{jz}[\mathrm{z}-(d_{A\mathrm{g}}+d_s+\alpha d_{\mathrm{A}}+\beta d_{\mathrm{B}})]}{e}^{i{k}_{x}x},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hfill \\ H_{m\mathrm{z}}={\displaystyle \frac{k_x}{u_0\omega }}{F}_m{e}^{i{k}_{jz}[\mathrm{z}-(d_{A\mathrm{g}}+d_s+\alpha d_{\mathrm{A}}+\beta d_{\mathrm{B}})]}{e}^{i{k}_{x}x}+{\displaystyle \frac{k_x}{u_0\omega }}{B}_m{e}^{-i{k}_{jz}[\mathrm{z}-(d_{A\mathrm{g}}+d_s+\alpha d_{\mathrm{A}}+\beta d_{\mathrm{B}})]}{e}^{i{k}_x\mathrm{x}},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hfill \end{array}\right.$$

(9)

in the above equation. If m is an odd number, $j=a$, $\alpha =(m-1)/2$, and $\beta =(m-3)/2.$ However, if m is an even number, $j=b,$ $\alpha =m/2-1,$ $\beta =m/2-1,$ $k_{jz}=\sqrt{k_0^2{n}_j^2-k_x^2},$ and $j=\left\{s,a,b\right\}.$ Finally, the electric and magnetic fields in the substrate media can be expressed as follows:

$$\left\{\begin{array}{l}{E}_{(n+1)y}=E_t{e}^{i{k}_{\mathrm{o}z}[\mathrm{z}-(d_{A\mathrm{g}}+d_s+10d_{\mathrm{A}}+10d_{\mathrm{B}})]}{e}^{i{k}_{x}x}\hspace{0.33em},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hfill \\ H_{(n+1)x}=-{\displaystyle \frac{k_{\mathrm{o}z}}{u_0\omega }}{E}_t{e}^{i{k}_{oz}[\mathrm{z}-(d_{A\mathrm{g}}+d_s+10d_{\mathrm{A}}+10d_{\mathrm{B}})]}{e}^{i{k}_{x}x},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hfill \\ H_{(n+1)z}={\displaystyle \frac{k_x}{u_0\omega }}{E}_t{e}^{i{k}_{\mathrm{o}z}[\mathrm{z}-(d_{A\mathrm{g}}+d_s+10d_{\mathrm{A}}+10d_{\mathrm{B}})]}{e}^{i{k}_{x}x},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hfill \end{array}\right.$$

(10)

According to the boundary conditions at $z=0$, the electric field and the magnetic field are continuous, and there are $E_{iy}(z=0)=E_{1y}(z=0)$ and $H_{ix}(z=0)=H_{1x}(z=0).$ At $z=d_{Ag}$, there are $E_{1y}(z=0)=E_{2y}(z=0)$ and $H_{1x}(z=0)=H_{2x}(z=0).$ At $z=d_{Ag}+d_{\mathrm{s}}$, the electric field is continuous and the magnetic field is discontinuous, and there are $H_{2x}(z=d_A+d_s)-H_{3x}(z=d_A+d_s)=\sigma E_{2\mathrm{y}}(z=d_A+d_s)$ and $E_{2\mathrm{y}}(z=d_A+d_s)=E_{3y}(z=d_A+d_s).$ At $z=d_{Ag}+d_{\mathrm{s}}+\alpha d_{\mathrm{A}}+\beta d_{\mathrm{B}},$ the electric field is continuous and the condition is satisfied $E_{my}(z=d_{Ag}+d_{\mathrm{s}}+\alpha d_A+\beta d_B)=E_{(m+1)y}(z=d_{Ag}+d_{\mathrm{s}}+\alpha d_A+\beta d_B),$ and the magnetic field is continuous $H_{mx}(z=d_{Ag}+d_{\mathrm{s}}+\alpha d_A+\beta d_B)=H_{(m+1)x}(z=d_{Ag}+d_{\mathrm{s}}+\alpha d_A+\beta d_B).$ Where m = 1, 2, 3, 4, …, 22. After all, the electric field versus incident electric field and the reflectance versus the incident electric field curves for the whole structure can be obtained indirectly by substituting Equations (6)–(10) into the above boundary conditions.

It needs to be clarified that although the classical transfer matrix method can intuitively display the relationship between the incident light field and the reflected light field, the transfer matrix itself appears exceptionally complex for the multilayered structure presented in Figure 1. This will make the analytical model in matrix form more complex. Generally speaking, for layered structures, the bistable relationship between the incident light field and the reflected light field (or transmitted light field) is closely related to the number of layers in the structure. If the number of layers in the entire structure is less than three, the functional relationship between the incident light field and the reflected light field can be achieved through precise analytical expressions. However, if the number of layers in this layered structure is more than three, the functional relationship between the incident light field and the reflected light field will become exceptionally complex and therefore not suitable for expression through analytical formulas or matrix forms. At this point, starting from Maxwell’s equations, the relationship between adjacent electromagnetic fields is recursively deduced layer by layer, indirectly obtaining the corresponding concern between the incident electric field and the reflected electric field. This is also a common practice adopted for structures with multiple layers.

3. Results and Discussions

In the section, we first discuss the variation in reflectance with an incident light wavelength using the transfer matrix method. Figure 2a shows the plot when the incident light is incident vertically ($\theta =0^0$) on the multilayer structure. Both the black solid line and the red dashed line in the figure show the reflectance versus incident light wavelength curve. The black solid line shows the reflectance versus incident wavelength without the addition of MoS₂ in the multilayer structure. As can be seen from the graph, the black curve at $\mathsf{\lambda }=696.7 \mathrm{nm}$ shows a significant decrease in the reflectance peak. It is known that the OTS needs to satisfy $r_{Ag}{r}_{DBR}exp(2i\varphi )\approx 1$ when excited [35], where $r_{Ag}$ and $r_{DBR}$ represent the reflected coefficient of the electromagnetic wave at the silver interface and the surface of the Bragg reflector, respectively. $\varphi$ is the phase change of the electromagnetic wave as it propagates through the top layer. After simplifying the equations, the OTS excitation conditions can be expressed as $\left|r_{Ag}\right|\left|r_{DBR}\right|\approx 1$, $Arg(r_{Ag}{r}_{DBR}exp(2i\varphi ))\approx 0$. Keeping the original structural parameters in Figure 1 unchanged, the reflected coefficient of the silver surface and the Bragg reflector surface were calculated, and there are $r_{Ag}=-0.5604-0.7506i$ and $r_{DBR}=0.8677-0.4970i$, which can be seen that the black solid line reflection anomaly in Figure 2a is due to the excitation of the OTS. The red dashed line in Figure 2b shows the reflectance versus incident light wavelength with the addition of MoS₂ in the structure. It is easy to see that there is a significant deepening of the decrease and the minimum value of reflectance near 0.1. To better show the Tamm plasmon in the silver–Bragg reflector structure. We are set to the incident light wavelength $\mathsf{\lambda }=696.7 \mathrm{nm}$. In Figure 2b,c, the normalized field distribution for the whole structure were plotted based on the original structure parameters in Figure 1. In Figure 2b, we have divided the entire structure zones according to the proportion of the thickness of the material. In combination with the multi-color Figure 2c, it is found that the electric field shows a clear field enhancement effect at the interface between the defect layer and the Bragg reflector. By using the calculation method in the second part, we obtain the relationship between the incident electric field and reflected electric field in Figure 2d. In order to obtain a suitable reflectance, the incident light wavelength was set to $664.5 \mathrm{nm},$ and the other structural parameters were kept the same as the original parameters in Figure 1 during the calculation. The resulting curve is the solid black line in Figure 2d and shows a hysteresis line relationship. With the MoS₂ removed, the reflectance of the structure is approximately 1. As the incident electric field changes, the reflectance is almost unchanged, and there is no hysteresis curve relationship between the reflectance and the incident electric field Ei. This is because the bilayer MoS₂ has a high third-order nonlinear conductivity in the visible band. However, under the nonlinear conditions provided by the MoS₂, the reflectance and the incident electric field $E_{\mathrm{i}}$ exhibit a hysteresis relationship. With the Bragg reflector removed, the OB precursor between the reflectance and the incident electric field is starting to emerge, but it corresponds to the incident electric field of about ${10}^9 \mathrm{V}/\mathrm{m}.$ Comparing the incident electric field ${10}^8 \mathrm{V}/\mathrm{m}$ with the Bragg reflector, it is clear that the excitation of OTS has a very positive effect on the reduction of the OB threshold. Objectively speaking, the model proposed here has significant advantages in realizing optical bistable devices in the visible light band, such as a simple structure, easy preparation, and insensitivity to the polarization of incident light. However, the limitations of this model also exist. The biggest limitation of this scheme is that although it can achieve a typical OB phenomenon in the visible light band, the threshold is always too high to be lowered to a practical low threshold level. Although the excitation of the optical Tamm state has a positive effect on reducing the threshold, there is still a certain gap from the practical threshold.

From Equations (3) and (4), we know that changing the wavelength of the incident light can change the wave vector in the vacuum $k_0$, which can indirectly change the first-order linear conductivity (${\sigma }_{(\omega )}^{(1)}$) and the third-order nonlinear conductivity (${\sigma }_{(\omega )}^{(3)}$) of MoS₂. It can be clearly found that changing the wavelength of the incident light can obviously change the total conductivity of the MoS₂ from Equation (5). Based on the above theoretical analysis, we can use the incident light wavelength as a tool to regulate this OB. Next, we explore the effect of varying the wavelength of the incident light on the OB. Keeping the structural parameters of Figure 1 constant and varying only the incident light wavelength, the reflected electric field was simulated numerically with the variation of the incident electric field. By calculation, we obtain the relationship between the reflected electric field $E_{\mathrm{r}}$ and the incident electric field $E_{\mathrm{i}}$, as shown in Figure 3a. For example, taking the OB curve at the incident light wavelength of $\mathsf{\lambda }=664.5 \mathrm{nm}$, connecting the markers gives the S-curve a-1-b-2-d-3-c. When $E_{\mathrm{i}}$ is small, $E_r$ increases with $E_{\mathrm{i}}$ increases, which corresponds to the a-1-b process, which is a steady state. As the incident electric field $E_{\mathrm{i}}$ continues to increases until $|E_{\mathrm{i}}{|}_{\mathrm{down}}=7.43\times {10}^8 \mathrm{V}/\mathrm{m}$, $E_r$ jumps to another stable state, corresponding to the c-3-d process. At this point, even if $E_{\mathrm{i}}$ continues to decrease, $E_r$ does not immediately return to the stable state of a-1-b. Conversely, when $E_{\mathrm{i}}$ is large, the system is in the second steady state, $E_r$ decreases as $E_{\mathrm{i}}$ decreases, and when $E_{\mathrm{i}}$ decreases to $|E_{\mathrm{i}}{|}_{\mathrm{up}}=5.30\times {10}^8 \mathrm{V}/\mathrm{m}$, $E_r$ jumps from the second steady state c-3-d to the first steady state a-1-b. The b-2-d process is unstable, and $E_r$ increases as $E_{\mathrm{i}}$ increases, but the increase is unstable and the curve is not observed in the experiment. Thus, this creates two jump points b and d and a bistable loop. The OB of this multilayer structure operates in the a-b interval and the c-d interval, where one $E_{\mathrm{i}}$ corresponds to two $E_r$ values. This leads to a hysteresis width $\Delta \left|E_{\mathrm{i}}\right|=|E_{\mathrm{i}}{|}_{\mathrm{down}}-|E_{\mathrm{i}}{|}_{\mathrm{up}}=2.13\times {10}^8 \mathrm{V}/\mathrm{m}$. Clearly, this is the OB phenomenon that we are looking for, in which the nonlinear properties of MoS₂ play a decisive role in obtaining this OB. Furthermore, the controlled conductivity of MoS₂ in the visible band offers the nonlinear condition to achieve a tunable OB. It is generally believed that when the threshold light intensity of OB reaches or falls below 10⁶ V/m, it can be considered as a low threshold. Although similar research by us or our peers can achieve relatively low thresholds (usually around 10⁶ V/m), the implementation of these thresholds is mainly in the terahertz band (the selection of the terahertz band is mainly to make nonlinear materials have the largest possible nonlinear coefficients in this band). There is still a significant gap in practicality between this and optical bistable devices that can achieve low threshold OB in the infrared and even visible light bands. Here, the core innovation of this work is to combine nonlinear materials with high nonlinear conductivity in the visible light band to achieve optical bistable phenomena in the visible light band. On this basis, the excitation of the optical Tamm state also played a positive role in reducing the threshold. To be honest, the threshold of this work is not low compared to the previous approach, but it is a real OB in the visible light band. Of course, our next step will focus on how to further reduce the threshold while achieving OB in the visible light band.

In order to describe the relationship between $E_r$ and $E_{\mathrm{i}}$ more intuitively, the curve of the hysteresis width with the incident light wavelength is plotted, as shown in Figure 3b. As can be seen from the figure, different values of the incident light wavelength not only affect the upper and lower thresholds of OB but also have an effect on the hysteresis width. As the incident light wavelength continues to increase, the upper threshold becomes larger and the lower threshold also becomes larger. However, the upper threshold of the reflected electric field increases faster than the lower threshold. Therefore, the hysteresis width becomes progressively narrower as the wavelength of the incident light continues to increase. For example, when the incident wavelength is $\mathsf{\lambda }=664.8 \mathrm{nm}$, the upper threshold is $|E_{\mathrm{i}}{|}_{\mathrm{up}}=6.07\times {10}^8 \mathrm{V}/\mathrm{m}$, the lower threshold is $|E_{\mathrm{i}}{|}_{\mathrm{down}}=7.61\times {10}^8 \mathrm{V}/\mathrm{m}$, and the hysteresis width is $\Delta \left|E_{\mathrm{i}}\right|=1.54\times {10}^8 \mathrm{V}/\mathrm{m}$. When the incident wavelength is $\mathsf{\lambda }=665.1 \mathrm{nm},$ the upper threshold is $|E_{\mathrm{i}}{|}_{\mathrm{up}}=6.80\times {10}^8 \mathrm{V}/\mathrm{m}$, the lower threshold is $|E_{\mathrm{i}}{|}_{\mathrm{down}}=7.79\times {10}^8 \mathrm{V}/\mathrm{m}$, and the hysteresis width is $\Delta \left|E_{\mathrm{i}}\right|=0.99\times {10}^8 \mathrm{V}/\mathrm{m}$. It is easy to see that increasing the incident wavelength contributes positively to the reduction in the OB threshold. In summary, the incident light wavelength can tune the OB threshold and the hysteresis width, providing a reference method for obtaining dynamically tunable OB devices.

Next, we discuss the effect of the MoS₂ in the spacer layer HfO₂ at different positions based on the OB phenomenon. Keeping the total thickness of the defect layer HfO₂ $d_{\mathrm{s}}$ constant, the position of MoS₂ embedded in the defect layer HfO₂ is changed so that the defect layer HfO₂ is divided into an upper part $d_{\mathrm{s}1}$ and a lower part $d_{\mathrm{s}2}$. The incident light wavelength is fixed at $\mathsf{\lambda }=664.5 \mathrm{nm}$, and the other parameters are the same as those in Figure 2a. From Figure 4, it can be seen that the MoS₂ is embedded in the spacer layer HfO_2. The effect on the OB at different positions is similar to the effect of changing the wavelength of the incident light on the OB, which is clearly reflected in the changes in the upper and lower threshold and the hysteresis width of the OB. The threshold magnitude and hysteresis width of the reflected electric field curve decrease as the thickness of the defect layer in the lower half $d_{\mathrm{s}2}$ increases, as shown in Figure 4a. The relationship curve between the reflectance and incident electric field $E_{\mathrm{i}}$ follows a similar trend, as shown in Figure 4b. For example, when the thickness of the upper half of the spacer layer HfO₂ is ${\mathrm{d}}_{\mathrm{s}1}=68 \mathrm{nm}$ and the thickness of the lower half is ${\mathrm{d}}_{\mathrm{s}2}=2 \mathrm{nm}$, the upper threshold is ${\left|E_{\mathrm{i}}\right|}_{\mathrm{up}}=4.96\times {10}^8\hspace{0.33em}\mathrm{V}/\mathrm{m}$, the lower threshold is ${\left|E_{\mathrm{i}}\right|}_{\mathrm{down}}=5.89\times {10}^8\hspace{0.33em}\mathrm{V}/\mathrm{m}$, and the hysteresis width is $\Delta \left|E_i\right|=0.93\times {10}^8\hspace{0.33em}\mathrm{V}/\mathrm{m}$. When the spacer layer HfO₂ has an upper half thickness of ${\mathrm{d}}_{\mathrm{s}1}=66 \mathrm{nm}$ and a lower half thickness of ${\mathrm{d}}_{\mathrm{s}2}=4 \mathrm{nm}$, the upper threshold is ${\left|E_{\mathrm{i}}\right|}_{\mathrm{up}}=4.44\times {10}^8\hspace{0.33em}\mathrm{V}/\mathrm{m},$ the lower threshold is ${\left|E_{\mathrm{i}}\right|}_{\mathrm{down}}=4.57\times {10}^6\hspace{0.33em}\mathrm{V}/\mathrm{m},$ and the hysteresis width is $\Delta \left|E_i\right|=0.13\times {10}^6\hspace{0.33em}\mathrm{V}/\mathrm{m}.$ We found that the OB threshold and hysteresis width slowly decrease as the thickness of the lower half of the defect layer increases after inserting MoS₂ into the defect layer. However, the OB threshold and hysteresis width do not decrease with an increase in the thickness of the lower half of the spacer layer, and will disappear when the thickness of the spacer layer increases to a certain value. The above results show that the OB threshold can be effectively reduced and the hysteresis width can be controllably adjusted by adjusting the position of the MoS₂ inserts in the spacer layer, which provides another feasible method for the fabrication and design of optical bistable devices.

Immediately afterwards, we observe the effect of different angles of incidence on the OB phenomenon. The wavelength of the incident light was fixed at $\mathsf{\lambda }=664.5 \mathrm{nm}$, and the other parameters were consistent with Figure 1. From Figure 5, we can see that the bistable behavior is sensitive to changes in the angle of incidence $\theta$. In Figure 5a, the upper threshold of the reflected electric field curve becomes smaller and smaller with an increasing incident angle, and the lower threshold also becomes smaller and smaller. However, the lower limit threshold decreases faster than the upper limit threshold, which can lead to a decrease in the hysteresis width. For example, when $\theta =6°,$ there are ${\left|E_{\mathrm{i}}\right|}_{\mathrm{up}}=5.26\times {10}^8\hspace{0.33em}\mathrm{V}/\mathrm{m}$ and ${\left|E_{\mathrm{i}}\right|}_{\mathrm{down}}=7.17\times {10}^6\hspace{0.33em}\mathrm{V}/\mathrm{m},$ thus the hysteresis width is $\Delta \left|E_i\right|=1.91\times {10}^8\hspace{0.33em}\mathrm{V}/\mathrm{m}$. When $\theta =12°,$ the upper threshold is ${\left|E_{\mathrm{i}}\right|}_{\mathrm{up}}=5.10\times {10}^8\hspace{0.33em}\mathrm{V}/\mathrm{m}$ and the lower threshold is ${\left|E_{\mathrm{i}}\right|}_{\mathrm{down}}=6.34\times {10}^8\hspace{0.33em}\mathrm{V}/\mathrm{m}$, and $\Delta \left|E_i\right|=1.24\times {10}^8\hspace{0.33em}\mathrm{V}/\mathrm{m}$. As the angle of incidence increases further to a certain angle, the hysteresis width slowly disappears. The relationship curve between the reflectance and the incident electric field follows the same trend in Figure 5b.The above results show that adjusting the angle of incidence can effectively reduce the threshold of OB while achieving a controlled adjustment of the hysteresis width. Therefore, a reasonable choice of the incident angle is of great practical importance for reducing the OB threshold and achieving the actual required OB hysteresis width.

Finally, we discuss the effect of the structural parameters of the spacer layer on the OB phenomenon. The incident light wavelength is fixed at $\mathsf{\lambda }=664.5 \mathrm{nm}$, and the other parameters remain the same as in Figure 2a. We plot the reflectance versus incident electric field for different spacer layer thicknesses $d_s$ and different spacer layer refractive indexes $n_s$. As shown in Figure 6a, the upper threshold and lower threshold of OB become larger as the spacer layer thickness $d_s$ becomes larger, but the lower threshold increases faster than the upper threshold, leading to an increase in the hysteresis width. For example, when ${\mathrm{d}}_{\mathrm{s}}=73 \mathrm{nm}$, the upper threshold is ${\left|E_{\mathrm{i}}\right|}_{\mathrm{up}}=5.68\times {10}^8\hspace{0.33em}\mathrm{V}/\mathrm{m}$ and the lower threshold is ${\left|E_{\mathrm{i}}\right|}_{\mathrm{down}}=10.18\times {10}^8\hspace{0.33em}\mathrm{V}/\mathrm{m},$ and the hysteresis width is $\Delta \left|E_i\right|=4.50\times {10}^8\hspace{0.33em}\mathrm{V}/\mathrm{m}.$ When ${\mathrm{d}}_{\mathrm{s}}=76 \mathrm{nm},$ the upper threshold is ${\left|E_{\mathrm{i}}\right|}_{\mathrm{up}}=5.90\times {10}^8\hspace{0.33em}\mathrm{V}/\mathrm{m},$ the lower threshold is ${\left|E_{\mathrm{i}}\right|}_{\mathrm{down}}=13.25\times {10}^8\hspace{0.33em}\mathrm{V}/\mathrm{m}$, and the hysteresis width is $\Delta \left|E_i\right|=7.35\times {10}^8\hspace{0.33em}\mathrm{V}/\mathrm{m}.$ A similar pattern occurs when changing the refractive index of the defect layer, as shown in Figure 6b. As the refractive index of the defect layer $n_s$ becomes progressively larger, the upper threshold value of OB ${\left|E_{\mathrm{i}}\right|}_{\mathrm{up}}$ and the lower threshold value ${\left|E_{\mathrm{i}}\right|}_{\mathrm{down}}$ also become larger, but the lower threshold value increases faster than the upper threshold value, resulting in an increase in the hysteresis width. Therefore, a reasonable choice of the structural parameters of the defect layer can lead to a more reasonable OB phenomenon.

4. Conclusions

In summary, we have investigated the OB phenomenon in a silver–Bragg reflector structure based on a bilayer of MoS₂. Based on this structure, a tunable OB phenomenon in the visible band range was achieved. The results show that the OTS based on the silver and Bragg reflector structure leads to a local electric field enhancement effect, which facilitates the realization of OB. At the same time, the large nonlinear conductivity of the bilayer of MoS₂ provides the nonlinear conditions for the appearance of the OB phenomenon. Thereafter, after setting the initial parameters, we calculated the reflected electric field and the reflectance as a function of the incident electric field. The threshold and hysteresis width of the OB were found to be modulated by the wavelength of the incident light. Afterwards, the threshold and hysteresis width of OB were adjusted by adjusting the position of MoS₂ embedded in the spacer layer, the incident angle, and the structural parameters of the spacer layer. This scheme achieves the controlled tuning of OB in the visible band, and the structure of the scheme is easy to prepare. We believe that this scheme can provide a new reference for the implementation of OB in optical fields, such as all-optical switches and other optical bistable devices.