Controllable Goos-Hänchen Shift in Photonic Crystal Heterostructure Containing Anisotropic Graphene

Abstract

In this study, we investigate the electrically and magnetically tunable Goos–Hänchen (GH) shift of a reflected light beam at terahertz frequencies. Our study focuses on a photonic crystal heterostructure incorporating a monolayer anisotropic graphene. We observe a tunable and enhanced GH shift facilitated by a drastic change in the reflected phase at the resonance angle owing to the excitation of the topological edge state. Considering the quantum response of graphene, we demonstrate the ability to switch positive and negative GH shifts through the manipulation of graphene’s conductivity properties. Moreover, we show that the GH shift can be actively tuned by the external electric field and magnetic field, as well as by controlling the structural parameters of the system. We believe that this tunable and enhanced GH shift scheme offers excellent potential for preparing terahertz shift devices.

1. Introduction

Goos–Hänchen (GH) shift generally refers to an optical phenomenon in which a reflected beam shifts laterally on an interface concerning the incident beam when the latter is incident upon the interface [1]. Although GH shift is a classical, simple, and intuitive optical phenomenon, it has been receiving continuous attention from researchers since its discovery due to its potential applications in a wide range of fields such as optical filtering [2], optical sensing [3], rainbow trapping [4], precision measurement [5], and all-optical switching [6]. From focusing on the formation and intrinsic mechanism of GH shift [7], to GH shift phenomena in diverse structures in combination with various advanced materials, continuous and extensive research has been conducted. For example: the regulation and enhancement of the GH shift phenomenon in structures such as photonic crystal structures [8], PT-symmetric medium [9], quantum wells structure [10], weakly absorbing structures [11], Otto configuration [12], and metasurfaces [13] have been reported. In recent years, GH shifts in structures such as metal insulator multilayer structures [14], magneto-optical metamaterials [15], and nanophotonic cavity [16] have also been proposed. It is worth mentioning that two-dimensional materials, which have received much attention in recent years, also play a very important role in the research of GH shift. Among them, graphene has received special attention. Graphene’s excellent optical characteristics such as dynamically adjustable conductivity, excitable surface plasmon, and broadband makes it well-suited for enhanced and adjustable GH shift. Consequently, there have been numerous reports of GH shift phenomena in various graphene-based micro-nano structures such as graphene-on-dielectric surface structures [17], attenuated total reflection structure [18], graphene-substrate structure [19], graphene-MoS₂ heterostructure [20], and other graphene-based micro-nano structures. The combination of enhanced and tunable GH shifts in micro-nano with graphene’s optical properties presents a promising research direction. Experimental verification of GH shift has also been a focus, with researchers like Liu et al. using the beam splitter scanning method to observe the enhanced GH shift [21]. In addition, the experimental investigation of GH phenomena has been carried out in various structures, including the Kretschmand–Raether structure [22], waveguide structures [23], and phase conjugate mirrors [24]. To obtain enhanced GH shift, researchers have also explored different combinations of resonance mechanisms such as surface plasmon [25], optical Tamm state [26], and Fano resonance [27]. This is mainly because the occurrence of various resonances is often accompanied by drastic changes in phase, which creates conditions for the enhancement of GH shift. To sum up, GH shift, being a highly classical optical phenomenon, has consistently been the focus of attention in integrating novel materials, new structures, and new mechanisms to achieve a simple-structure, tunable, and enhanced GH shift.

In recent years, the topological edge state (TES) has attracted extensive attention from researchers due to its local field enhancement and topological protection properties [28,29]. TES mainly exists in photonic crystal structures, and reports regarding TES have emerged in photonic crystals. For example, Wang et al. proposed a graphene-based 1D PC heterostructure to excite TES for achieving a multi-channel perfect absorber [30]. Li et al. combined the optical Kerr effect and TES in photonic crystal heterostructures to attain highly stable unidirectional transmission [29]. It is not difficult to perceive that TES based on 1D PC crystals is widely used in the field of nano-photonics on account of its excellent optical properties. Considering the similarities between TES and resonance characteristics, which are often accompanied by drastic phase changes, we explore the possibility of enhancing and regulating GH shift through TES. In this study, we investigate GH shift in a multi-layer structure with graphene embedded in two photonic crystal structures. In this multi-layer structure, the TES produced by the overlapping of two 1D PC bandgaps causes the phase of transmission and reflection to change drastically, thus creating conditions for the realization of enhanced GH shift. The graphene layer, embedded between two photonic crystals, provides dynamic and controllable properties for GH shift. Our findings indicate that GH shift can be dynamically regulated by changing the Fermi energy and magnetic field of graphene. This scheme has the advantages of a simple structure and good GH characteristics, making it a promising reference scheme for GH shift devices in the field of nanophotonics.

2. Theoretical Model and Method

We consider a multi-layer composite structure consisting of two nonidentical 1D PCs and a single layer of graphene. In this multi-layer structure, graphene is placed between two photonic crystals, as shown in Figure 1. For the convenience of calculation, we assume that both 1D PCs are composed of alternating media A and B. The medium A is SiO₂, and its refractive index is set to 1.46, that is, $n_{A1}=n_{A2}=n_A=1.46$. Si is adopted as medium B, and its refractive index is fixed at 2.82, namely, $n_{B1}=n_{B2}=n_B=2.82$ [31]. The periods of both 1D PCs are set to be 4. The difference between the two 1D PCs is mainly caused by the inconsistency in thickness. Here, in one of the 1D PCs, the thicknesses of medium A and medium B satisfy $d_{A1}=270\mathsf{\mu }\mathrm{m}$ and $d_{B1}=121\mathsf{\mu }\mathrm{m}$, respectively; while in the other photonic crystal, the thickness of medium A and medium B are $d_{A2}=275\mathsf{\mu }\mathrm{m}$, and $d_{B2}=170\mathsf{\mu }\mathrm{m}$, respectively. The above micro-nano structure is a very simple multi-layer structure. Given the current state of micro-nano processing technology, the preparation process for this particular structure presents no significant challenges, thus exhibiting characteristics of relatively simple preparation. Therefore, this scheme can be easily applied to experimental verification and even specific application scenarios [21].

To facilitate the calculation, we take the coordinate system in the structure shown in Figure 1 and specify and apply a uniform magnetic field B in the direction perpendicular to the interface along the z-axis. In the present paper, we focus on the discussion of GH shift in the THz frequencies range for TM polarization. When the quantum response of graphene is taken into account, we can describe the conductivity of the graphene in the form of magneto-optical conductivity [32]. When an external magnetic field perpendicular to graphene is applied, the surface conductivity of graphene $\overline{\sigma }$ has an asymmetric optical conductivity tensor. It can therefore be defined as:

$$\overline{\sigma }=\left(\begin{array}{cc}{\sigma }_{\mathrm{xx}}& {\sigma }_{\mathrm{xy}}\\ {\sigma }_{\mathrm{yx}}& {\sigma }_{\mathrm{yy}}\end{array}\right).$$

(1)

Matrix elements of $\overline{\sigma }$ are given as follows [32]:

$$\{\begin{array}{l}\begin{array}{c}{\sigma }_{\mathrm{xx}}={\displaystyle \frac{e^2{v}_f^2\left|eB\right|ħ\left(\omega +2\mathrm{i}\mathsf{\Gamma }\right)}{i\pi }}\hfill \\ \times \underset{\mathrm{n}=0}{\overset{\infty }{{\displaystyle \mathsf{\Sigma }}}}\left\{\begin{array}{l}{\displaystyle \frac{1}{M_{n+1}-M_n}}\times {\displaystyle \frac{n_F\left(M_n\right)-n_F\left(M_{n+1}\right)+n_F\left(-M_{n+1}\right)-n_F\left(-M_n\right)}{{\left(M_{n+1}-M_n\right)}^2-{ħ}^2{{\displaystyle \left(\omega +2\mathrm{i}\mathsf{\Gamma }\right)}}^2}}\\ +(M_n\to -M_n)\end{array}\right\},\hfill \end{array}\hfill \\ \begin{array}{c}{\sigma }_{\mathrm{xy}}=-{\displaystyle \frac{e^2{v}_f^2\left|eB\right|}{\pi }}\times \underset{\mathrm{n}=0}{\overset{\infty }{{\displaystyle \mathsf{\Sigma }}}}\left[n_F\left(M_n\right)-n_F\left(M_{n+1}\right)+n_F\left(-M_{n+1}\right)-n_F\left(-M_n\right)\right]\hfill \\ \times \left[{\displaystyle \frac{1}{{\left(M_{n+1}-M_n\right)}^2-{ħ}^2{{\displaystyle \left(\omega +2\mathrm{i}\mathsf{\Gamma }\right)}}^2}}+(M_n\to -M_n)\right],\hfill \end{array}\hfill \\ {\sigma }_{\mathrm{yx}}=-{\sigma }_{\mathrm{xy}},\hfill \\ {\sigma }_{\mathrm{yy}}={\sigma }_{\mathrm{xx}}.\hfill \end{array}$$

(2)

In the above equation, $n_F\left(\omega \right)=1/\left(1+\exp \left[\left(ħ\omega -\mu \right)/kT\right]\right)$ is the Fermi–Dirac distribution, and $M_n=\sqrt{{\delta }^2+2\mathrm{n}\left|\mathrm{e}B\right|ħv_f^2}$ is the Landau energy with the Landau level index n.

In expression (2), $v_f$, $e$, and $ħ$ are Fermi velocity, electron charge, and reduced Planck constant, respectively. $\mu$, $\delta$, $\mathsf{\Gamma }$, $B$, and $T$ represent the chemical potential of graphene, excitonic gap, scattering rate, magnitude of the external magnetic field, and temperature, respectively. The values of these parameters are consistent with the literature [32]. In the following calculations, we set the original values of these parameters as follows: $\omega =2\pi \times 1.05\mathrm{THz}$, $\mu =50\mathrm{meV}$, $\delta =0$, $\mathsf{\Gamma }=0.011\mathrm{meV}/ħ$, $B=10\mathrm{T}$, $T=300\mathrm{K}$, $v_f=1.02\times {10}^6\mathrm{m}/\mathrm{s}$.

To calculate the GH shift of the whole structure, we need to know its transmission and reflection coefficients. Considering the conductivity characteristics of graphene, the classic 4 × 4 transfer matrix method is adopted here [33]. Like the previous 2 × 2 transfer matrix method, the entire transfer matrix is also composed of a transmission matrix D and a propagation matrix P. Specifically, in the case of embedded graphene and considering only TM polarization, the transmission matrix between medium 1 and medium 2 satisfies the following equation [33]:

$${\mathrm{D}}_{1,2}={\displaystyle \frac{1}{2}}\left(\begin{array}{cccc}\left(1+{\displaystyle \frac{k_{2z}{\epsilon }_1}{k_{1z}{\epsilon }_2}}+{\displaystyle \frac{k_{2z}{\sigma }_{\mathrm{xx}}}{\omega {\epsilon }_0{\epsilon }_2}}\right)& \left(1-{\displaystyle \frac{k_{2z}{\epsilon }_1}{k_{1z}{\epsilon }_2}}-{\displaystyle \frac{k_{2z}{\sigma }_{\mathrm{xx}}}{\omega {\epsilon }_0{\epsilon }_2}}\right)& {\sigma }_{\mathrm{xy}}& {\sigma }_{\mathrm{xy}}\\ \left(1-{\displaystyle \frac{k_{2z}{\epsilon }_1}{k_{1z}{\epsilon }_2}}+{\displaystyle \frac{k_{2z}{\sigma }_{\mathrm{xx}}}{\omega {\epsilon }_0{\epsilon }_2}}\right)& \left(1+{\displaystyle \frac{k_{2z}{\epsilon }_1}{k_{1z}{\epsilon }_2}}-{\displaystyle \frac{k_{2z}{\sigma }_{\mathrm{xx}}}{\omega {\epsilon }_0{\epsilon }_2}}\right)& {\sigma }_{\mathrm{xy}}& {\sigma }_{\mathrm{xy}}\\ {\displaystyle \frac{{\mu }_0{k}_{2z}{\sigma }_{\mathrm{yx}}}{k_{1z}{\epsilon }_0{\epsilon }_2}}& -{\displaystyle \frac{{\mu }_0{k}_{2z}{\sigma }_{\mathrm{yx}}}{k_{1z}{\epsilon }_0{\epsilon }_2}}& {\displaystyle \frac{{\mu }_0\omega }{k_{1z}}}\left({\displaystyle \frac{k_{1z}+k_{2z}}{\omega {\mu }_0}}+{\sigma }_{\mathrm{yy}}\right)& {\displaystyle \frac{{\mu }_0\omega }{k_{1z}}}\left({\displaystyle \frac{k_{1z}+k_{2z}}{\omega {\mu }_0}}+{\sigma }_{\mathrm{yy}}\right)\\ {\displaystyle \frac{-{\mu }_0{k}_{2z}{\sigma }_{\mathrm{yx}}}{k_{1z}{\epsilon }_0{\epsilon }_2}}& {\displaystyle \frac{{\mu }_0{k}_{2z}{\sigma }_{\mathrm{yx}}}{k_{1z}{\epsilon }_0{\epsilon }_2}}& {\displaystyle \frac{{\mu }_0\omega }{k_{1z}}}\left({\displaystyle \frac{k_{1z}-k_{2z}}{\omega {\mu }_0}}-{\sigma }_{\mathrm{yy}}\right)& {\displaystyle \frac{{\mu }_0\omega }{k_{1z}}}\left({\displaystyle \frac{k_{1z}+k_{2z}}{\omega {\mu }_0}}-{\sigma }_{\mathrm{yy}}\right)\end{array}\right).$$

(3)

Based on expression (3), we can easily obtain the transmission matrix between adjacent dielectric layers. If there is no graphene embedded between adjacent layers, simply remove the graphene conductivity from the above transmission matrix to obtain the corresponding transmission matrix. When the incident light propagates in a uniform medium, the propagation matrix at z to z + Δz can be obtained by [33]:

$$P(\Delta z)=\left(\begin{array}{cccc}\exp \left(-i{k}_z\Delta z\right)& 0& 0& 0\\ 0& \exp \left(i{k}_z\Delta z\right)& 0& 0\\ 0& 0& \exp \left(-i{k}_z\Delta z\right)& 0\\ 0& 0& 0& \exp \left(i{k}_z\Delta z\right)\end{array}\right).$$

(4)

Based on transmission matrix D and propagation matrix P, we can easily get the total transmission matrix M in Figure 1 as follows [33]:

$$M=D_{air-B_1}{P}_{B_1}{M_1}^3{D}_{B_1-A_1}{P}_{A_1}{D}_{A_1-A_2}{P}_{A_2}{M_2}^3{D}_{A_2-B_2}{P}_{B_2}{D}_{B_2-air},$$

(5)

where $M_1=D_{B_1-A_1}{P}_{A_1}{D}_{A_1-B_1}{P}_{B_1}$ and $M_2=D_{A_2-B_2}{P}_{B_2}{D}_{B_2-A_2}{P}_{A_2}$. On this basis, we assume that the leftmost field coefficients (input field coefficients) are represented by $a_1$, $b_1$, $c_1$, and $d_1$, while the rightmost output field coefficients are represented by $a_{N+1}$, $b_{N+1}$, $c_{N+1}$, and $d_{N+1}$. Then, the input and output field coefficients are related to each other by the total matrix M [33]:

$$\left(\begin{array}{c}{a}_1\\ b_1\\ c_1\\ d_1\end{array}\right)=M\left(\begin{array}{c}{a}_{N+1}\\ b_{N+1}\\ c_{N+1}\\ d_{N+1}\end{array}\right).$$

(6)

When the incident wave is TM polarization, the reflected wave and transmitted wave also have two kinds of polarized waves. Based on the literature [33], we can get the field coefficients $b_1$, $d_1$, $a_{N+1}$, and $c_{N+1}$ from the M matrix, which are:

$$\begin{array}{l}{b}_1={\displaystyle \frac{M_{31}{M}_{23}-M_{21}{M}_{33}}{M_{13}{M}_{31}-M_{11}{M}_{33}}}\hfill \\ d_1={\displaystyle \frac{M_{43}{M}_{31}-M_{41}{M}_{33}}{M_{13}{M}_{31}-M_{11}{M}_{33}}}\hfill \\ a_{N+1}={\displaystyle \frac{-M_{33}}{M_{13}{M}_{31}-M_{11}{M}_{33}}}\hfill \\ c_{N+1}={\displaystyle \frac{M_{31}}{M_{13}{M}_{31}-M_{11}{M}_{33}}}\hfill \end{array},$$

(7)

On this basis, we can further obtain the reflection coefficient and transmission coefficient under TM polarization [33]:

$$\begin{array}{l}{r}_{\mathrm{pp}}=b_1,\\ r_{\mathrm{sp}}=\sqrt{\left[{\left(k_{1z}/\omega {\mu }_0\right)}^2+{\left(q/\omega {\mu }_0\right)}^2\right]}{d}_1,\\ t_{\mathrm{pp}}=a_{N+1},\\ t_{\mathrm{sp}}=\sqrt{\left[{\left(k_{N+1z}/\omega {\mu }_0\right)}^2\right]}{c}_{N+1}.\end{array}$$

(8)

In the subsequent discussion, we only need to discuss the reflection coefficient $r_{\mathrm{pp}}$ and transmission coefficient $t_{\mathrm{pp}}$. Accordingly, the transmission coefficients and reflection coefficients under TE polarization are also obtained.

For the calculation of conventional GH shift, we use the stationary phase method. For the incident beam with a sufficiently large beam waist, the GH shift of the reflected beam can be expressed as [34]:

$$D_{\mathrm{GH}}(\theta ,\omega )=-\frac{\lambda }{2\pi }\frac{\mathrm{d}{\Phi }_r}{\mathrm{d}\theta },$$

(9)

where $\theta$ represents the angle of incidence. $\lambda$ and ${\Phi }_r$ are wavelength and reflected phase, respectively. The GH shift calculated for transmitted beams can also be obtained by a similar method.

3. Results and Discussions

In this section, we will thoroughly consider the excitation of TES and the impact of the introduction of graphene on GH shift. First, we calculated the reflectance and transmittance curves of the whole structure depicted in Figure 1, considering the incident angle based on the classical transfer matrix method (see Figure 2). The excitation of TES requires the bandgap of two photonic crystals to be in the same band while possessing different topological properties [35]. In Figure 2, it is evident that photonic crystal 1 and photonic crystal 2 exhibit distinct band gaps within the wavelength range from 280 μm to 308 μm. Notably, their transmittance approaches zero while their reflectance approaches 1. The non-ideal transmittance and reflectance values shown in the figure are mainly due to the limited period values of the two photonic crystals. Further analysis of the topological properties of the two photonic crystals involves calculating the Zak phase of the isolated band within the dispersion relationship of each photonic crystal. In general, the dispersion relationship of photonic crystals can be expressed as: $\cos (q\mathsf{\Lambda })=\cos k_a{d}_a\cos k_b{d}_b-{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{z_a}{z_b}}+{\displaystyle \frac{z_b}{z_a}}\right)\sin k_a{d}_a\sin k_b{d}_b$, where $k_i=\omega n_i/c$, $z_i=\sqrt{{\mu }_i/{\epsilon }_i}$, ${\epsilon }_i=n_i^2/{\mu }_i$, and $i=a\mathrm{or}b$; $q$ and $\mathsf{\Lambda }=d_a+d_b$ are the Bloch wave vector and unit cell, respectively. By calculating the Zak phase ($0\mathrm{or}\pi$) of each isolated band, the topological phase symbol of the nth band gap can be determined by the sum of the Zak phases of all isolated bands in the photon band gap, which is: $\mathrm{sgn}\left[{\varsigma }^{\left(n\right)}\right]={(-1)}^n{(-1)}^l\exp \left(i{\displaystyle \sum _{m=0}^{n-1}{\theta }_m^{Zak}}\right)$. On this basis, it is confirmed by calculation that there are five band gaps in the frequency range of 0.4–1.3 THz. Not only do photonic crystal 1 and photonic crystal 2 have band gap overlap near 1 THz, but the topological phase symbols of the photonic crystal band gap on both sides are also opposite, so it can be determined that the structure can meet the conditions of TES excitation [31].

Upon confirming the capacity of the two-photonic-crystal composite structure (Figure 1) to excite TES, we set the incident wave frequency to 1.05 THz. Simultaneously, we plotted the relation curve of transmittance and reflectance with the incident angle, as shown in Figure 3. The black curve in the figure corresponds to the absence of embedded graphene. We can clearly see that the reflectance exhibits a distinct reflected dip near the incident angle of 32°. Correspondingly, there is also a transmission peak at the same angle. In conjunction with the previous analyses, this corresponds to the excitation of TES. Generally speaking, the excitation of the topological edge state is similar to resonance. The excitation of the topological edge state can cause a significant “dip” (or peak) in the reflectance (or transmittance) curve at a specific angle. At the same time, the reflected phase will be accompanied by a drastic change in phase around this angle. According to the calculation formula of the GH shift, the drastic change in the reflected phase corresponds exactly to the enhanced GH shift. This is why the excitation of the topological edge state can achieve enhanced GH shift. Here, the excitation of TES is presumed to align with a drastic phase change, thus creating the conditions for an enhanced GH shift.

On this basis, we embedded a single layer of graphene between two photonic crystals to enable further dynamic adjustability of the GH shift. Figure 3 also illustrates the transmittance and reflectance curves of the composite structure in this case. It is not difficult to see that the addition of graphene has minimal impact on the excitation of TES. The positions of transmission peaks and reflected peaks exhibit negligible shifts. This is mainly because the excitation of TES is predominantly affected by the structural and material parameters of photonic crystals. Graphene is a very thin two-dimensional material, and its optical properties are often reflected through optical conductivity. Therefore, in our computational model, the optical properties of graphene are reflected in the boundary conditions in the form of conductivity. Essentially, the introduction of graphene is equivalent to adding a loss between two photonic crystals. However, due to the large conductivity of graphene, its addition leads to increased overall loss within the structure, manifesting in the amplitudes of the reflected and the transmission peaks, as shown in the red line in Figure 3. Notably, the introduction of graphene minimally affects the reflected peak and simultaneously gives the GH shift of the composite structure dynamically adjustable characteristics, thereby significantly altering the regulatory characteristics of the whole structure. In fact, we do not want the addition of graphene to cause significant changes in the transmittance and reflectance of the entire structure. This is mainly because our core vision is that the introduction of graphene has almost no effect on the excitation of the topological edge state of the entire structure (the excitation of the topological edge state is a sufficient condition for achieving GH shift enhancement). At the same time, the introduction of graphene will provide a very flexible means for the dynamic controllability of GH shift in the entire structure. Based on the above considerations, the transmittance and reflectance curves in Figure 3 are in line with expectations.

In the subsequent discussion, our focus will be on regulating the whole multi-layer structure GH shift within the field. Combined with the basic theory in the second part, when considering the quantum response of graphene, the conductivity of graphene can be dynamically controlled not only by the applied electric field but also by the applied magnetic field. This phenomenon provides a unique way to realize dynamically controllable GH shift devices. We first examine the regulatory properties of the GH shift of graphene structures in the presence of an applied electric field. Given that the effect of the applied electric field is directly reflected in the change of graphene Fermi energy, we initially explore the effect of the change in graphene Fermi energy on the phase and shift of transmission and reflection coefficients, as shown in Figure 4. It is apparent from Figure 4 that the excitation of TES leads to substantial changes in the amplitude of the transmission and reflection coefficients (see Figure 3), and also triggers drastic alterations in the transmission phase and reflected phase around the incident angle of 32°. Near 32°, where TES occurs, the transmission phase exhibits an obvious monotonically decreasing trend. This, in combination with the calculation formula of the GH shift, creates conditions that are beneficial for realizing an enhanced positive GH shift peak. On this basis, we noted that the reduction of the Fermi energy of graphene makes the transmission phase steeper near the resonance angle, leading to a significantly enhanced transmission GH. For example, when the Fermi energy is E_F = 100 meV, the corresponding normalized GH shift is about 55. By lowering the Fermi energy, it is reduced to E_F = 0 meV, and the corresponding normalized GH shift increases to about 95, which is almost double. A similar enhancement effect is more obvious in reflecting GH shift. Likewise, in the vicinity of 32° where TES occurs, the reflected phase demonstrates an obvious monotonically increasing trend, which, in combination with the calculation formula of the GH shift, sets the stage for the realization of an enhanced negative GH shift peak. Like the transmission case, the reduction of the Fermi energy of graphene causes the reflected phase to become steeper near the resonance angle, resulting in a more pronounced enhancement of the reflected GH shift. For example, when the Fermi energy is E_F = 100 meV, the corresponding reflected normalized GH shift is about −50; when the Fermi energy is reduced to E_F = 0 meV, the corresponding normalized GH shift is enhanced to about −330, representing a sixfold increase. The external electric field facilitates the dynamic manipulation of the GH shift under TES excitation, a phenomenon that is not only reflected in the amplitude but also the symbol. This provides a significant prospect for realizing flexible and dynamically controllable GH shift devices.

Then, we further discuss the effect of the applied magnetic field on the GH shift of the entire structure. As mentioned earlier, when considering the quantum response of graphene, the anisotropic conductivity of graphene is also reflected as a function of the applied magnetic field. This phenomenon also provides a way to realize the dynamic micro-nano devices of magnetic field control. Compared with electric field control, the direct application of the magnetic field to the exterior of the structure has advantages, especially when it comes to avoiding direct contact with the device during GH shift regulation. Without loss of generality, we also draw the change curve of transmission and reflected phase concerning the incident angle and the corresponding GH shift, as shown in Figure 5. It should be noted that to simplify the drawing, we did not draw the curve of the resonance angle changing with the angle. Generally speaking, the enhanced GH shift peak or GH shift dip correspond to the “steepest” position of the reflected phase with angle variation, and the steepest position of the reflected phase often corresponds to the position of the resonance angle. This means that changes in magnetic field intensity not only affect GH shift, but also synchronously affect resonance angle. The former is the result of the latter, while the latter is the cause that leads to the former. The effect of applied magnetic fields on GH shift differs from changes in Fermi energy, reflecting two extremes in magnetic field regulation. The figure shows that at relatively high applied magnetic fields, the regulation of the change in the magnetic field on the GH shift is significant but relatively moderate. Analysis of the transmission near the 32° where TES occurs reveals a monotonically decreasing relationship between the transmission phase and the incident angle, corresponding to a positive GH shift peak. Here, the increase in the applied magnetic field perpendicular to the graphene surface markedly steepens the transmission phase of the whole structure near the resonance angle, leading to enhanced transmission GH shift. For example, when the applied magnetic field is B = 3 T, the corresponding normalized GH shift of transmission is about 50. Increasing the applied magnetic field to B = 10 T enhances the normalized GH shift of transmission to about 85, achieving an increase of more than 60%. Similarly, the reflected phase near the 32° at which TES occurs produces a very high slope, a monotonically increasing phenomenon near the resonance angle, indicative of a large negative GH shift. The increase in the applied magnetic field intensifies the steepness of the reflection phase near the resonance angle. For example, when the magnetic field is B = 3 T, the corresponding reflected normalized GH shift is about −30; at B = 10 T, the normalized GH shift of reflection is enhanced to about −200, more than a sixfold increase. Interestingly, when it comes to a relatively low value of applied magnetic fields, the regulation of GH shift presents the opposite extreme. As can be seen from Figure 5, when the applied magnetic field decreases to B = 0.01 T or even lower, both the GH transmission peak and reflected peak demonstrate a noticeable trend of shifting to a lower angle. Although subtle changes in the external magnetic field have no significant impacts on the transmission and reflection of GH shift, the obvious “jump” changes in the GH peak remain intriguing. From a reverse perspective, this means that the structure can indirectly measure the effect of the applied magnetic field by verifying the angle at which the GH peak is located, offering an interesting approach to the environmental magnetic field in micro and nanostructures. Different from the traditional control of GH properties in the entire structure through structural and material parameters, the introduction of graphene provides a good way to dynamically manipulate GH properties using an external field.

To further highlight the novelty and importance of our current work, we also compared it with previous representative research work, as shown in

Table 1. Comparison between different GH shift methods.

Reference	Mechanism	Structure	Maximum Value of GH Shift (/λ)	Tunability	Frequency Range
[12]	Surface plasmon polaritons	Otto configuration	−350~380	Electrically tunable	THz
[17]	None	Graphene-on-dielectric surface structure	−400~100	Electrically tunable	Near-infrared
[26]	Optical Tamm state	Graphene-Bragg reflector structure	−3000~1000	Electrically tunable	THz
[10]	Tunneling-induced	Quantum wells structure	−150~50	Intercavity medium tunable	Mid-infrared
[19]	None	graphene-substrate system	−150~150	Electrically tunable	Near-infrared
[20]	Surface plasmon resonance	graphene-MoS2 heterostructure	−80~100	Structure tunable	Visible light
This work	Topological edge state	Photonic crystal heterostructure	−330~90	Electrically tunable, Magnetically tunable	THz

. It is not difficult to see from Table 1 that the current work and the work of peers have significant differences in mechanism, but have the advantage of more diverse and flexible regulation methods. However, current work also has limitations and challenges: firstly, the flexibility and richness of dynamically regulating the GH shift can only be limited to the terahertz band. Only in the terahertz band can the conductivity characteristics of graphene be affected by both the applied voltage and the external magnetic field. If the frequency of the applied light source rises to the infrared or even visible light band, the conductivity characteristics of graphene are only affected by the applied voltage, resulting in a relative weakening of the control characteristics of the entire structure. Secondly, in practical applications, the externally applied voltage and magnetic field will make the “auxiliary equipment” of the entire structure relatively complex, thereby affecting the integration of the entire device.

4. Conclusions

In this paper, we propose a multi-layer hybrid structure consisting of two 1D PCs and a single layer of graphene. The enhanced GH shift is realized through the phase upheaval induced by the photonic crystal excitation TES. At the same time, the conductivity characteristics of monolayers in the terahertz band create conditions for the realization of dynamic electrically tunable and magnetically tunable GH shifts. The calculation results show that GH shift can be dynamically regulated by the change of graphene Fermi energy, and the applied magnetic field can also dynamically regulate the GH shift, which is highly significant for constructing dynamically adjustable GH shift devices in micro-nano structures. There are limitations in the experimental verification of this scheme, such as the controllable band being limited to the terahertz band and the complexity of auxiliary equipment. However, the above limitations will not have a decisive impact on the application of this heterostructure. Due to the simple structure of the scheme and its ability to achieve GH shift enhancement and dynamic regulation, we believe that it can find potential applications in relevant GH shift applications (such as high-resolution optical sensing, weak measurement, biosensors, etc.).