1. Introduction

Optical bistability (OB) refers to an optical phenomenon that produces two different stable state outputs for a given input. The two output states can interconvert with each other, showing an obvious hysteresis loop relationship [1]. As a typical optical nonlinear phenomenon, OB has potential applications in optical devices such as all-optical switches [2,3], optical transistors [4], optical storage [5], optical logical gates [6,7], etc., which has attracted extensive attention. However, the traditional optical bistable devices are limited by the small nonlinear coefficient of nonlinear materials, which makes the size larger, and the obvious OB phenomenon can be realized under the condition of strong incident light, which hinders the practicability of optical bistable devices. In recent years, with the development of micro-nano technology, researchers began to pay attention to OB phenomenon in micro-nano structures, such as Fabry-Perot (F-P) cavities [8], photonic crystals [9], metamaterials [10], optical ring cavities [11], etc. We know that the nonlinear refractive index of materials has an important influence on the threshold of OB. The application of materials with higher nonlinear refractive index can effectively reduce the threshold of OB. However, because the nonlinear coefficients of most materials are relatively low, it is difficult to achieve a lower threshold. In recent years, due to the excellent nonlinear characteristics of two-dimensional material graphene, the low threshold tunable OB based on graphene has been widely studied. For example, OB in photonic crystal/graphene/photonic crystal hybrid structure [12] and OB in graphene/waveguide hybrid structure [13] have been reported. The research of nonlinear graphene in the fields of OB and all-optical switch has made rapid development, but its two-dimensional properties have caused a bottleneck in large-area and high-quality preparation, which slows down the research progress of all-optical devices based on graphene [14]. Recently, the three-dimensional Dirac semimetal (3D DSM), a new strange topological quantum material, has attracted increasing attention. Its body electrons form a Dirac cone structure in three-dimensional space, so it is also called “3D graphene” [15,16]. Specifically, 3D DSM has high nonlinear refractive index, which is very similar to that of graphene. However, compared with graphene, 3D DSM has higher carrier mobility [17], higher Fermi velocity [18], easier preparation and stable performance [19]. Meanwhile, 3D DSM can also adjust the dielectric constant by changing the Fermi energy [20]. Therefore, the research of micro-nano optoelectronic devices based on 3D DSM has a wide application prospect. At present, many complexes have been proved to have 3D DSM properties, such as $\textrm{C}{\textrm{d}_\textrm{3}}\textrm{A}{\textrm{s}_\textrm{2}}$ [21], $\textrm{N}{\textrm{a}_\textrm{3}}\textrm{Bi}$ [22], $\textrm{ZrT}{\textrm{e}_\textrm{5}}$ [23], $\textrm{AlCuFe}$ [24], etc. The high nonlinear refractive index of 3D DSM is expected to play a crucial part in providing strong optical nonlinearity to realize low threshold OB.

In recent years, the topological edge state mode (TES) in photonic crystals has been widely concerned in micro-nano optoelectronic devices because of its unique topological protection and local field enhancement characteristics [25,26]. At present, researchers have proposed various structures and schemes theoretically or experimentally to construct photonic TES, such as plasmonic nanoparticles [27], optical waveguides [28], chiral hyperbolic metamaterials [29], optical ring resonators [30], two-dimensional photonic crystals (2D PhC) and three-dimensional photonic crystals (3D PhC). Although the TES of 2D and 3D PhC can enhance the local electric field of the structure, the application of TES of multi-dimensional topological PhC in integrated photonic devices is limited due to its complex design and manufacturing process. Compared with 2D and 3D PhC, one-dimensional photonic crystals (1D PhC) are easier to manufacture, and the expected local field enhancement can also be found at the interface of 1D topological PhC heterostructure [31]. This means that the emergence of TES can effectively reduce the threshold of OB. Therefore, we can consider such a question: Under the condition of TES excitation, whether OB with low threshold can be generated in 1D topological PhC based on 3D DSM, and whether the hysteresis loop of OB can be dynamically regulated by adjusting the parameters of 3D DSM.

In this paper, we study low threshold OB of transmitted light in a multilayer structure with 3D DSM embedded in two 1D PhCs. The results show that TES will be generated at the interface connecting two 1D PhCs, which significantly enhances the local electric field. Furthermore, inserting 3D DSM into the overall structure provides nonlinear conditions for the generation of OB. At the same time, the threshold of OB can be dynamically adjusted by changing the Fermi energy and relaxation time of 3D DSM. A low threshold OB scheme based on 3D DSM is proposed for the first time, which has the characteristics of simple structure and easy preparation. We believe that the low threshold OB scheme is expected to play an important role in the preparation of all-optical nonlinear photonic devices.

2. Theoretical model and method

We design a multilayered hybrid structure consisting of two 1D PhCs and a layer of 3D DSM. In this composite structure, 3D DSM is embedded between PhC 1 and PhC 2, as shown in Fig. 1. The two PhCs are composed of alternating medium A (TPX) and medium B (TiO₂), and the period of PhC is defined as $\textrm{N} = \textrm{4}$. The refractive index of medium A is set to 1.46, that is ${n_{A1}} = {n_{A2}} = {n_A} = 1.46$; the refractive index of medium B is set to 2.82, that is ${n_{B1}} = {n_{B2}} = {n_B} = 2.82$ [12,32]. For PhC 1, the thicknesses of medium A and medium B meet ${d_{A1}} = 270\;{\mathrm{\mu} \mathrm{m}}$ and ${d_{B\textrm{1}}} = 121\;{\mathrm{\mu} \mathrm{m}}$ respectively, and for PhC 2, the thicknesses of medium A and medium B meet ${d_{A2}} = 275\;{\mathrm{\mu} \mathrm{m}}$ and ${d_{B2}} = 170\;{\mathrm{\mu} \mathrm{m}}$ respectively. The thickness of 3D DSM is set to ${d_{Dir}} = 20\;\textrm{nm}$. Based on current mature preparation technology of micro-nano multilayer dielectric structure, the layered structure defined by the above structural parameters is not difficult to construct. In addition, 3D DSM as a typical kerr nonlinear medium, here we use the third-order nonlinear susceptibility and linear refractive index to represent it. In order to obtain a larger nonlinear coefficient, we only consider the case of the terahertz region. According to Ref. [33], the linear intraband optical conductivity of 3D DSM can be resolved as:

 

Fig. 1. Schematic diagram of TES based on 3D DSM/PhC heterostructure: (a) visual view and (b) side view.

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Thus, we can obtain the third-order nonlinear susceptibility ${\chi ^{(3)}}$ and linear refractive index ${n_D}$ of 3D DSM:

The transfer matrix method is used to evaluate the incident light intensity and its relationships to the transmitted light intensity and reflected light intensity. In this paper, we calculate the transmittance and reflectivity of the structure under TE polarization (TM can be obtained similarly). We assume that the 3D DSM is parallel to the plane of the x-axis and y-axis, and the electromagnetic field propagates along the z-axis. For linear medium A and medium B, we only need to know the thickness and refractive index of each layer to obtain the transfer matrix representing each layer of material [34]. As for the transfer matrix of the Dirac layer, considering that it is a Kerr type nonlinear material, its transfer matrix can be expressed as [35,36]:

Therefore the transmission coefficient and reflection coefficient can be expressed by

3. Results and discussions

We know that when the bandgaps of two PhCs share a common frequency range and the topological properties of the bandgaps are opposite, TES can be excited at the interface of PhC heterostructure [38]. In this part, we first discuss the conditions for the excitation of TES based on the proposed structure. Figure 2(a) and Fig. 2(b) show the relationship between PhC transmittance and frequency and that between PhC reflectivity and frequency, respectively, in which the red solid line and black dotted line represent the transmittance/reflectivity of PhC 1 and PhC 2 respectively. The results show that in the 0.97 ∼ 1.07 THz frequency range, both PhC 1 and PhC 2 show an obvious broad bandgap with nearly zero transmittance, and the bandgap of PhC 2 is slightly wider than that of PhC 1. Next, we continue to discuss the band structure of PhC. The dispersion relation of 1D PhC can be expressed by the following formula [39]:

 

Fig. 2. (a) Transmission spectrum and (b) reflection spectrum of PhC 1 (red solid line) and PhC 2 (black dotted line); (c) transmission spectrum and (d) reflection spectrum of PhC 1 + 3D DSM + PhC 2 heterostructure (red solid line) and PhC 1 + PhC 2 heterostructure (black dotted line); the band structures of (e) PhC 1 and (f) PhC 2.

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Further, we draw the electric field diagram of the PhC heterostructure, as shown in Fig. 3. It can be seen that there is an obvious electric field localization near the interface between PhC 1 and PhC 2, which confirms that our proposed structure can indeed excite TES. In addition, the characteristics of TES make abnormal transmission phenomenon appear on the interface of the PhC heterostructure near 1 THz. As shown in Fig. 2(c) and Fig. 2(d), there is a sharp transmission peak and reflection valley in the transmission spectrum and reflection spectrum respectively, which are represented by black dotted lines in the figure, and the central frequencies of the transmission peak and reflection valley are just located at the overlapping part of the fourth bandgap in PhC 1 and PhC 2. Since the frequency of TES in topological PhC heterostructure will shift with the incident angle, we consider the case where incident angle $\theta = {0^\circ }$. And we will discuss the influence of angle on OB in detail later. Considering the local field enhancement characteristics of TES, we introduce 3D DSM at the interface of heterostructure to facilitate the realization of low threshold OB. The red solid lines in Fig. 2(c) and Fig. 2(d) show the transmission and reflection spectra of the overall structure. It can be seen that the addition of 3D DSM would not affect the excitation of TES, and its nonlinearity and tunability can provide a basis for the emergence and regulation of OB. It is worth noting that although the defect modes of PhC can also produce similar the transmission peak, they are different in essence. When there are defects in the PhC, a new defect mode conduction band will appear in the bandgaps. The propagation of frequency occurs in the frequency range where transmission was completely prohibited before, and it shows a sharp transmission peak in the macro view.

 

Fig. 3. Electric field distribution of PhC heterostructure.

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Subsequently, we study low threshold OB based on the proposed structure above. Since the third-order nonlinear optical conductivities of Dirac material decrease with the increase of incident light frequency, we study the influence of 3D DSM on OB based on 1 THz frequency range. At this time, the third-order nonlinear coefficient of 3D DSM is strong. Generally speaking, as a typical Kerr medium, 3D DSM's complex refractive index includes linear and nonlinear parts. The relationship between complex refractive index and electric field can be expressed as $n = {n_D} + \Delta n = {n_D} + {n_2}{|E |^2}$, where ${n_2}\textrm{ = }{{{\chi ^{( \textrm{3}) }}} / {\textrm{2}{n_D}}}$ is the nonlinear refractive index. Therefore, the local electric field enhancement caused by the excitation of TES plays a positive role in the nonlinear refractive index part of 3D DSM. It is not difficult to find that we can excite low threshold tunable OB phenomenon by embedding 3D DSM in 1D PhC heterostructure under the condition of exciting the TES.

Then we discuss the influence of 3D DSM on OB phenomenon. We first explore the regulatory effect of Fermi energy of 3D DSM on OB. In Fig. 4(a), we show the relationship between the transmitted electric field and incident electric field under different Fermi energy of 3D DSM. We can see that when the Fermi energy gradually increases, the upper and lower thresholds of OB will increase, and the width of the hysteresis loop will also increase. In order to conveniently and qualitatively analyze the above OB phenomenon, we approximately equivalent 3D DSM to 2D film, similar to graphene. Then the bulk conductivity of 3D DSM would be equivalent to the surface conductivity, i.e., $\sigma \; = \;{d_{Dir}}({{\sigma^{(1 )}} + {\sigma^{(3 )}}{{|E |}^2}} )\;\textrm{ = }\;{\sigma _\textrm{1}}\textrm{ + }{\sigma _\textrm{3}}{|E |^2}$. According to Ref. [40], the expression of the transfer matrix of the equivalent 3D DSM is:

Meanwhile, we can regard the two PhCs as two Bragg mirrors, so their transfer matrix can be expressed as [41]:

 

Fig. 4. Dependence of (a) the transmitted electric field and (b) transmittance on the incident electric field for different ${E_F}$ of 3D DSM. Here, $\lambda \mathrm{\ =\ 300\ \mu m}$, $\tau \textrm{ = 0}\textrm{.8 ps}$ and $\theta \textrm{ = 0 }^\circ$.

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At this time, let $k = [{1 - (|{{r_1}} |+ |{{r_2}} |)\textrm{ + }|{{r_1}} ||{{r_2}} |} ]$ and $Z = {\mu _0}c/2$. After calculation and simplification, the above formula can be written as:

It can be seen from Eq. (14) that Y is the cubic function of X, that is, an incident electric field may have three corresponding transmission electric fields. In order to satisfy the above situation, the discriminant $\Delta $ of the derivative of Eq. (14) must be greater than 0. The derivative function $Y^{\prime}$ of Eq. (14) can be expressed as:

The discriminant $\Delta $ of derivative function $Y^{\prime}$ can be simplified as:

From the above formula, we can clearly see that $\Delta $ is always greater than 0, which meets the condition for generating OB. Further, we can obtain the maximum and minimum points of function Y, that is, the two zeros of derivative function $y$:

We study OB in the band near 1 THz where the individual PhC 1 and PhC 2 almost have total reflection, i.e., ${r_1} \approx \textrm{1}$ and ${r_2} \approx \textrm{1}$. In order to simplify the calculation, we set $1 - |{{r_1}} ||{{r_2}} |\approx \textrm{0}$. Equation (17), then, can be simplified to: ${x_1} \approx {{ - {\sigma _1}} / {3{\sigma _3}}}$ and ${x_2} \approx {{ - {\sigma _1}} / {{\sigma _3}}}$. Next, we bring the simplified maximum point and minimum point into Eq. (14) respectively to obtain the upper and lower thresholds of OB. Further, the width of hysteresis loop of OB can be expressed as:

In combination with the expressions of the linear optical conductivity and the third-order nonlinear optical conductivities of 3D DSM given by Eqs. (1) and (2), we can obtain the variation of OB hysteresis loop width with Fermi energy ${E_F}$ and relaxation time $\tau$:

Combining Eq. (17) with the above derivation and analysis, we find that the OB phenomenon is not only affected by Fermi energy, but also closely related to the relaxation time $\tau$ of 3D DSM. Figure 5 describes the functional relationship between the transmitted electric field and the incident electric field when ${E_F} = 0.07\;\textrm{eV}$.The width of hysteresis loop and the upper and lower thresholds of OB change with the relaxation time, and this changing trend is consistent with our derivation results. It is obvious from Fig. 5 that as relaxation time $\tau$ increases from $\textrm{0}\textrm{.5}\;\textrm{ps}$ to $\textrm{0}\textrm{.8}\;\textrm{ps}$, the width of hysteresis loop of OB is also increasing, while the threshold of OB is gradually decreasing. Specifically, when $\tau \;\textrm{ = }\;\textrm{0}\textrm{.5 ps}$, the width of hysteresis loop $\Delta Y\; \approx \;\textrm{1}\textrm{.20} \times \textrm{1}{\textrm{0}^\textrm{6}}\;\textrm{V/m}$ and the lower threshold ${|{{E_i}} |_{down}}\; \approx \;\textrm{9}\textrm{.72} \times \textrm{1}{\textrm{0}^\textrm{5}}\;\textrm{V/m}$; when $\tau \;\textrm{ = }\;\textrm{0}\textrm{.8 ps}$, the width of hysteresis loop $\Delta Y\; \approx \;\textrm{1}\textrm{.31} \times \textrm{1}{\textrm{0}^\textrm{6}}\;\textrm{V/m}$ and the lower threshold ${|{{E_i}} |_{down}}\; \approx \;\textrm{7}\textrm{.40} \times \textrm{1}{\textrm{0}^\textrm{5}}\;\textrm{V/m}$. Overall, the width of hysteresis loop increases by nearly $\textrm{1}\textrm{.1} \times \textrm{1}{\textrm{0}^\textrm{5}}\;\textrm{V/m}$, and the threshold of OB decreases by nearly $\textrm{2}\textrm{.32} \times \textrm{1}{\textrm{0}^\textrm{5}}\;\textrm{V/m}$. The above results show that the threshold of OB can be effectively reduced by reasonably adjusting the relaxation time of 3D DSM, and the controllability of the hysteresis loop can be achieved simultaneously, which provides a new scheme for designing tunable bistable devices. However, it should be pointed out that even though relaxation time plays an obvious role in regulating OB threshold, it still has limitations in flexibility. After all, once the structure and 3D DSM are prepared, relaxation time has no regulatory characteristics. Nevertheless, the influence of relaxation time on OB can still provide reference for designing bistable devices.

 

Fig. 5. Dependence of (a) the transmitted electric field and (b) transmittance on the incident electric field for different $\tau$ of 3D DSM. Here, $\lambda \mathrm{\ =\ 300\ \mu m}$, ${E_F}\textrm{ = 0}\textrm{.07}\;\textrm{eV}$ and $\theta \textrm{ = 0 }^\circ$.

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We have previously discussed the influence of the parameters of 3D DSM on OB. It has been determined that the upper and lower thresholds of OB can be controlled by adjusting the Fermi energy or relaxation time of 3D DSM separately. Further, we found that OB is also affected by the thickness of 3D DSM and the angle of incident light. As can be seen from Fig. 6(a), the change of thickness of 3D DSM has little effect on the overall threshold, which perfectly meets our requirements for the design of bistable devices. As we all know, it is very difficult to accurately control the thickness of the device during the manufacturing process. In other words, it is beneficial to control the overall performance of the device if the slight variation in thickness does not have a decisive effect on the threshold and width of threshold. In addition, the change of incident light angle also has a certain impact on the width of threshold. As shown in Fig. 6(b), we draw the image of OB changing with incident light angle when ${E_F}\;\textrm{ = }\;\textrm{0}\textrm{.07 eV}$ and $\tau \;\textrm{ = }\;\textrm{0}\textrm{.8 ps}$. It is not difficult to find that as the incident angle $\theta$ increases from ${\textrm{0}^\circ }$ to ${\textrm{3}^\circ }$, the lower threshold of OB changes little, while the upper threshold increases and so does the width of threshold. Therefore, when designing a bistable device, the influence of the incident light angle on the device performance should also be considered.

 

Fig. 6. Influence of (a) the thickness of 3D DSM ${d_{Dir}}$ and (b) the angle of incident light $\theta$ on OB. Here, $\lambda \mathrm{\ =\ 300\ \mu m}$, ${E_F}\textrm{ = 0}\textrm{.07}\;\textrm{eV}$ and $\tau \textrm{ = 0}\textrm{.8 ps}$.

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In addition, we also calculate the hysteresis loop relationship of reflective OB as shown in Fig. 7, where the relevant parameters are consistent with those in the calculation of transmissive OB. Figure 7(a) and Fig. 7(b) show the hysteresis loop relationships of the incident electric field to the reflected electric field and the reflectivity at $\tau \;\textrm{ = }\;\textrm{0}\textrm{.8 ps}$. When the Fermi energy increases from $\textrm{0}\textrm{.07}\;\textrm{eV}$ to $0.10\;\textrm{eV}$, the width of hysteresis loop of OB increases, and so do the upper and lower thresholds. This is the same as the variation law of transmissive OB shown in Fig. 4. Similarly, Fig. 7(c) and Fig. 7(d) are the hysteresis loop relationships of the incident electric field to the reflected electric field and the reflectivity when ${E_F}\;\textrm{ = }\;\textrm{0}\textrm{.07 eV}$. With the increase of relaxation time, the width of hysteresis loop of OB will increase, while the upper and lower thresholds will decrease, which is basically consistent with the law in Fig. 5.

 

Fig. 7. Dependence of (a) the reflected electric field and (b) reflectance on the incident electric field for different ${E_F}$ of 3D DSM; here, $\lambda \mathrm{\ =\ 300\ \mu m}$, $\tau \textrm{ = 0}\textrm{.8 ps}$ and $\theta \textrm{ = 0 }^\circ$. Dependence of (c) the reflected electric field and (d) reflectance on the incident electric field for different $\tau$ of 3D DSM; here, $\lambda \mathrm{\ =\ 300\ \mu m}$, ${E_F}\textrm{ = 0}\textrm{.07}\;\textrm{eV}$ and $\theta \textrm{ = 0 }^\circ$.

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4. Conclusion

In this paper, we are the first to achieve the dynamically tunable low threshold OB by exciting TES based on a structure composed of 3D DSM and two PhCs in the terahertz region. The results show that TES caused by PhC heterostructure can effectively enhance the local electric field, which has a positive effect on realizing low threshold OB. Meanwhile, the large nonlinear refractive index of 3D DSM provides nonlinear conditions for the emergence of OB phenomenon; the threshold and hysteresis loop width of OB can be adjusted by changing its relevant parameters. In addition, we further found that the threshold and width of threshold are regulated by the Fermi energy and relaxation time of 3D DSM, and are closely related to the angle of incident light, while the change of 3D DSM thickness has little effect on the overall threshold. The low threshold OB scheme proposed in this paper has the characteristics of simple structure and easy preparation. It is believed that this low threshold OB scheme has excellent potential for applications in related nonlinear optical devices.