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Abstract
The resonant optical tunneling effect (ROTE) originates from the frustrated total reflection effect because unique transmission characteristics are used to study high-sensitivity sensors. In this study, we theoretically demonstrated that choosing a suitable transmission gap made it possible for the ROTE structure based on hexagonal boron nitride and graphene to obtain a large Goos–Hänchen shift as high as tens of thousands of times the incident wavelength at a specific incident angle. The amplitude of the Goos–Hänchen shift was found to be sensitive to the central layer thickness but was also modulated by the tunneling gap on both sides. In addition, adjusting the chemical potential and relaxation time of the graphene sheets could alter the Goos–Hänchen shift. Our work provides a new way to explore the Goos–Hänchen effect and opens the possibility for the application of high-precision measurement technology based on the ROTE.
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1. Introduction
The Goos–Hänchen (GH) shift refers to the lateral deviation of an actual reflected beam compared with a theoretical reflected beam in geometric optics when the beam is completely reflected at the interface of two different media. This is caused by the angular gradient of the phase of complex reflection or refraction coefficients at complex dispersive interfaces, and was verified by Goos and Hänchen in an experiment in 1947 [1]. Since then, the GH effect has attracted extensive attention in many research fields such as optical switches [2], biosensors [3], high-sensitivity temperature sensors [4,5], and image edge detection [6]. However, the magnitude of the GH shift is very small and difficult to measure, which significantly impedes its application and development. To improve the amplitude of the GH shift in practical applications, many researchers have sought new research methods. Currently, several research methods can be used to enhance GH shifts. The first method is based on novel waveguide structures. Many theoretical and experimental studies have successfully enhanced the GH shift on different structural interfaces. For example, abnormally large positive and negative lateral optical beam shifts based on surface plasmon resonance resonators were reported by Yin et al. [7], and Khan et al. [8] found that the GH shift could be greatly enhanced on an optomechanical cavity. Zheng [9] and Wu et al. [10] achieved a significant enhancement of the GH shift based on the bound states in the continuum on all-dielectric metasurfaces. In addition, large GH shifts in subwavelength-structured gratings [11], parity time-symmetric structures [12,13], and photonic crystals [14,15] have also been confirmed. The second method is based on the use of new materials. Numerous studies have proven that metamaterials can also enhance the magnitude of the GH shift, including chiral materials [16], epsilon-near-zero metamaterials [17,18] and cross-anisotropic metasurfaces [19].
Simultaneously, two-dimensional materials have been explored extensively in academic research because of their potential applications in electronics and optics. Graphene, as the most representative of these, has proven to be excellent candidate for tuning the surface plasmon resonance in the mid-infrared to terahertz bands owing to its high carrier mobility and strong interaction with photons over a wide energy range [20]. A large number of studies have demonstrated the tunability of graphene to the GH shift [21–24]. In addition, the structure of hexagonal boron nitride (hBN) is similar to that of graphene, and it is the natural partner of graphene in optically active heterostructures. Similar to traditional substrates such as SiO2, hBN supports phonon-polaritons. The mobility and chemical stability of graphene are significantly improved when placed on an hBN substrate. The application of hBN reduces the scattering loss of graphene surface plasmon resonance [25].
On the other hand, in modern optics, if an incident light beam falls at the angle of the total internal reflection when transmitted from one medium (ε1) to another (ε2), placing a smaller medium (ε3) between them makes it possible for the partial beam overcome the separating gap with the third medium and transfer to the second medium. This phenomenon is called the frustrated total internal reflection (FTIR), and is analogous to quantum mechanical tunneling [23,26–28]. An enhanced GH shift has been found in the FTIR configuration [23]. The resonant optical tunneling effect (ROTE) originates from the FTIR structure, and the simplest five-layer case is similar to placing a dielectric layer with a high refractive index in the FTIR tunneling gap. As a result, if the incident beam can meet the tunneling resonance conditions, the optical properties of the ROTE are analogous to those of electrons passing through a double potential barrier in quantum mechanics, forming an optical potential well [29]. Since it was first proved in 1988, it has been demonstrated that the ROTE has a preeminent ability to enhance the photonic spin Hall effect [30], and can be applied to high-sensitivity sensors [31,32]. Recently, some studies have shown that it is feasible to use the ROTE to enhance the GH effect [33].
In this study, we investigated the GH shift in a symmetrical ROTE structure composed of two-dimensional materials, graphene, hBN, and dielectric prisms. We conducted theoretical numerical simulation analyses, and found that when an incident light beam had an angle greater than and close to the critical angle of the prism and air, this structure could produce a large GH shift if the resonant tunneling conditions were satisfied. Furthermore, because of the tunability of the graphene plasmonic resonance, the GH shift could be tuned by adjusting the chemical potential and relaxation time of the graphene, as well as the number of graphene layers.
2. Model and theory
The configuration considered in this study is shown in Fig. 1; it consists of two dielectric prisms (n1) with an air gap (n0) between them separated by two hBN dielectric layers, with the bottom of each prism is coated with graphene. This was a symmetrical structure, where the air gap width was a, and the hexagonal boron nitride thickness on both sides was d.
In general, the optical properties of graphene are characterized by its optical conductivity [34]. The surface conductivity, σ, of graphene consists of two parts: the intra-band conductivity, σintra, and the inter-band conductivity, σinter. However, in the terahertz and far-infrared bands, the inter-band conductivity of graphene can be ignored. The surface conductivity of graphene is expressed as follows:
hBN is also known as “white graphene.” It shows natural hyperbolicity and can support two kinds of hyperbolic phonon polaritons. Its dielectric constant can be described using the Lorentz model, as shown below [35,36]:
The ROTE derived from FTIR, refer to a special phenomenon in which light waves can tunnel through classical impenetrable optical structures [31], which is analogous to the electron tunneling effect in quantum mechanics. In the simplest case, the ROTE optical structure consists of five dielectric layers with a refractive index (RI) distribution of high-low-high-low-high, which represent the input layer, first tunneling gap, central layer, second tunneling gap, and output layer, respectively. When light is incident at an angle greater than the critical angle, the light passing through the tunnel resonates in the central layer.
The light beam was incident from the dielectric prism to the surface of the multilayer structure and was then transmitted through the ROTE structure. The transfer matrix method (TMM) [30,40,41] is commonly used to determine the transmission spectra of a multilayer structure. In the following, [M] is the characteristic matrix between the connected dielectric layers:
Consider TM-polarized light incidence, where ${k_{l,z}} = {k_0}{({\varepsilon _l} - {\sin ^2}{\theta _\textrm{i}})^{1/2}}$, and ${q_l} =$ $ {({{\varepsilon_l} - {{\sin }^2}{\theta_\textrm{i}}} )^{1/2}}/{\varepsilon _l}$, with l = 1,2,…, N-1. Here, ${\varepsilon _1}_{(0)}$ is the permittivity of the dielectric prism (air gap), and ${\varepsilon _2}$ is the permittivity of hBN. The wave-vector component of the incident wave along the propagation direction is expressed as ${k_{1z}} = {k_1}\cos ({{\theta_\textrm{i}}} )$. The propagation vector of hBN along the z-direction is expressed as ${k_{2z}} = \sqrt {{\varepsilon _ \bot }{k_0}^2 - {\varepsilon _ \bot }{k_x}^2/{\varepsilon _\parallel }}$ owing to its unique spatial dispersion, where ${k_x} = {k_1}\sin ({{\theta_\textrm{i}}} )$, ${k_1} = \sqrt {{\varepsilon _1}} \omega /c$, and wave-vector ${k_{2z}}$ has hyperbolic characteristics. Using the TMM, we can obtain and $r = {M_{21}}/{M_{11}}$, $t = 1/{M_{11}}$, where r and t are the reflection and transmission coefficients of the structure, respectively, and the GH shift of the reflected beam calculated by the stationary phase approach [42,43] is expressed as follows:
3. Analysis and discussion of results
3.1 Graphene modulation
In composite ROTE structures, the GH shift is affected by several factors. First, we consider the GH shift of the reflected light beam in the low terahertz frequency region under TM polarization. In terms of the optical conductivity, graphene sheets have an advantage in controlling the GH shift based on the chemical formula and relaxation time, as seen in Figs. 2(c) and (d). The reflectance values at different chemical potentials and relaxation times are shown in Figs. 2(a) and (b). It can be found that there is a sharp change in reflectivity when the incident angle is close to and greater than the critical angle (θc = 38.5°). At the same time, when the incident angle of the light beam is greater than and close to the critical angle, a large reflected GH shift can be obtained, and the GH shifts are controlled by the chemical potential of the graphene sheet; when µ = 0.6 eV, the maximum GH shift is -350λ, where λ is the incident wavelength, as shown in Fig. 2(c). In addition, we found that when relaxation time τ decreases, the GH shift and reflectivity decrease, which is due to the energy loss being high when the electronic relaxation time is small. However, when $\omega \gg {\tau ^{ - 1}}$ is used, according to Eq. (1), the optical conductivity of graphene is a purely imaginary number, and a real but negative dielectric constant can be obtained. Under these circumstances, the GH shift is the largest. In contrast, we compared the reflected GH shifts at different frequencies. As seen in Fig. 3, a large reflected GH shift can be obtained when the incident light frequency is 1 THz and the incident angle is 41.8°. The absolute value of the reflected GH shift gradually decreases and approaches zero with a gradual decrease in the incident frequency. These results provide potential approaches to research on optical switches and optical spatial modulators.
Fig. 2. (a)(b) Reflection R, (c)(d) reflected light beam’s GH shift with respect to incident angle θi for different chemical potentials, µ, and relaxation times, τ, with graphene layer number N = 1, prism refractive index n1 = 1.605, hBN thickness d = 10 nm, air gap thickness a = 125 µm, and incident wavelength λ = 3 × 10−4 m.
Fig. 3. Dependences of the reflected light beam’s GH shift on the frequencies for different angle of incidence, where chemical potential µ = 0.6 eV, relaxation time τ = 8 ps, graphene layer number N = 1, prism refractive index n1 = 1.605, hBN thickness d = 10 nm, and air gap a = 125 µm.
3.2 Thickness dependence of the GH shift
Next, we systematically analyzed the GH shift effect in the ROTE structure. It is worth noting that the GH shifts obtained for each central layer thickness shown in Fig. 4 were conditional, which means that each GH shift was obtained at the most optimized incident angle where the GH shift was the maximum. For example, in Fig. 5, when a = 250 µm, the maximum GH shift was obtained at θi = 40°. Figure 4(a) shows the functional relationship between the reflected GH shift transmitted by the ROTE structure and the thickness of the central layer (i.e., air gap width) under a fixed hBN thickness. By comparing different central layer thicknesses, we found that the reflected GH shift increased with the air gap width under the ROTE structure and decreased after reaching an extreme value. When the air gap was close to 230 µm, the GH shift reached a value as high as -11000λ, but when a = 300 µm, the GH shift decreased to -600λ. In theory, the GH shift could approximate infinity as the central layer thickness infinitely approaches the tunneling resonance pole. Previous studies could also obtain a relatively large GH shift, but it was generally only several hundred times the incident wavelength. In Fig. 4(b), we magnified the GH shifts relative to the air gap from 200 µm to 250 µm, and found that the GH shift was regulated by the thickness of the central layer in real time. According to the physical mechanism of the ROTE, the ROTE optical system could be converted into a finite potential well of quantum mechanics [29]. When photons tunnel through the first barrier, they resonate between the two barriers (i.e., the location of the central layer), and then tunnel out of the second barrier. This strong resonance enhances the GH shifts. The resonance condition is highly dependent on the optical path length, L = n0a, where n0 is the RI of the central layer and a is the physical length of the central channel [29]. Therefore, once the thickness of the central layer is changed, the resonance condition is destroyed, and the GH effect is limited. These results are consistent with the GH shifts of electron-tunneling graphene double-barrier structures [24].
Fig. 4. Central layer thickness (air gap) dependence of the GH shifts: (a) a = 100–400 µm, and (b) enlarged drawing for a = 200–250 µm, where graphene chemical potential µ = 0.6 eV, relaxation time τ = 8 ps, and hBN dielectric layer thickness on both sides of the central layer is 10 nm.
Fig. 5. (a) GH shifts of reflected wave, (b) related reflection and transmission, (c) eigenvalues of S-matrix, and (d) differential function of the transmittance with respect to the incident angle for a = 250 µm and d = 10 nm, with the other parameters the same as those in Fig. 3.
Furthermore, in Fig. 4, we notice that a large GH shift does not occur at the Brewster angle, but varies with the thickness of the air gap. This phenomenon is due to the large GH displacement near the exceptional point (EP) [44]. Figure 5(a) shows the reflected light beam’s GH shift at incident angles ranging from 32° to 44°. There are two resonance peaks, an extremely large GH shift at approximately 40°, where θi > θc, and a small one at approximately 34.2°, where θi < θc. The transmitted GH shift was small under the same circumstances. In Fig. 5(b), we can see that the reflectivity and transmittance on both sides of the structure are completely coincident because of the symmetric structure, rR = rL and tR = tL, and the resonance peak is the sharpest at an angle of approximately 40°, corresponding to the maximum transmission angle. Based on Eq. (3), it is easy to obtain scattering matrix $S = \left[ {\begin{array}{ccc} {{t_R}}&{{r_L}}\\ {{r_R}}&{{t_L}} \end{array}} \right] = \left[ {\begin{array}{cc} {({M_{11}}{M_{22}} - {M_{12}}{M_{21}})/{M_{11}}}&{{M_{21}}/{M_{11}}}\\ { - {M_{12}}/{M_{11}}}&{1/{M_{11}}} \end{array}} \right]$, and eigenvalues given by ${S_ \pm } = t \pm \sqrt {{r_R}{r_L}}$. Figure 5(c) shows the eigenvalues of matrix S. There were two EPs at approximately 33° and 58°, and a large GH shift occurred between them. To compare the relationship between the large GH shift and the transmission angle more systematically, we differentiated the transmittance, as shown in Fig. 5(d). We found that the angle for the maximum splitting of the eigenvalues coincided with the maximum transmission. This phenomenon was the same as that mentioned in Ref. [44], and the transmission phase changed most dramatically at the angle for the maximum transmission, which led to the maximum GH shift. Therefore, there were two important features of the ROTE structure: (i) a large GH shift obtained when the incident angle was greater than the critical angle, and (ii) an extremely large GH shift at the angle for the maximum transmission and angle for the maximum splitting of the eigenvalues in the ROTE structure.
Now, we will investigate the influence of different hBN thicknesses, d, on the reflected GH shift, as shown in Fig. 6, and always consider the symmetrical ROTE structure. According to previous research results, it is practical to resonantly enhance the GH shift effect by placing a metallic quantum well at the interface between the dielectric and vacuum. However, the metallic quantum well of the transport structure is an ultrathin film, and the influence of the barrier width on the GH shift cannot be considered. Here, we compared the results of the GH shifts for different central layer gap thicknesses: a = 125 µm, a = 200 µm, a = 250 µm, and a = 300 µm. A comparison of these four cases showed that the GH shift was the largest when a = 250 µm, as shown in Fig. 4, Dr = -2500λ. As can be seen in Figs. 6(a) and (b), the reflected GH shift increased with the hBN thickness when the central layer gap was less than 250 µm. Even when a = 200 µm, a large reflected GH shift could be obtained by increasing the hBN thickness. Here, the reflected GH shift was close to -2000λ, as shown in Fig. 6(b). However, as seen in Figs. 6(c) and (d), as the hBN thickness increased, the GH shift decreased. Therefore, it could be concluded that although the GH shifts of the ROTE structure were much more sensitive to the central layer, a, they were also modulated by the tunneling gap, d.
Fig. 6. Variation of reflected light beam’s GH shift with respect to the incident angle when changing the thicknesses of hBN, under different air gap thicknesses: (a) a = 125 µm, (b) a = 200 µm, (c) a = 250 µm, and (d) a = 300 µm. The other parameters are the same as those in Fig. 3.
Based on the previous discussion, the reflected GH shift showed a significant change with the central layer width. To investigate the influence of the number of graphene layers on the GH shift, we conducted a further simulation. We fixed the hBN thickness at 10 nm and adjusted the number of graphene layer from 1 to 5, as shown in Fig. 7. We found that when there was only a single layer of graphene on both sides of the structure, a large reflected GH shift could often be obtained. And it is worth noting that the structure is always symmetrical, so the number of graphene layers on both sides is the same. This was because a monolayer of graphene not only had the advantage of adjusting the GH shift but also better met the refractive index gradient requirements of the ROTE structure compared with a multilayer graphene structure. This result is best reflected in Figs. 7(a) and (b), which show that the largest GH shift was obtained with one layer of graphene and a central layer thickness of 250 µm. The reflection GH shifts obtained by other layers were much smaller.
Fig. 7. Variation of reflected light beam’s GH shift with respect to the incident angle when changing the graphene layers and air gap thickness. The hBN thickness was fixed at 10 nm, and the other parameters were the same as those in Fig. 3.
3.3 ROTE physical mechanism
In Fig. 5(b), we can see that the calculated reflectivity drops sharply (θi = 40°) even in the total reflective region, which was caused by the resonant tunneling effect of photons. The physical mechanism behind the ROTE is the quantum potential well effect. When a light wave propagates from a high-RI medium to a low-RI medium at an angle higher than the critical angle, part of the evanescent wave can pass through the second interface to form a transmission near the interface [29]. According to the relationship between the RI and optical potential, the high RI parts had low optical potentials for the photons, whereas the low RI parts produced high optical potentials and thus optical barriers. In the ROTE structure for the light wave, the RI distribution was high-low-high-low-high, and the structure was equivalent to double potential barriers. When photons tunneled through the first barrier, they resonated between the two barriers (i.e., the location of the central layer), and then tunneled out of the second barrier. This strong resonance enhanced the GH shifts.
The electric field distribution of the ROTE structure under TM polarization was obtained using the finite-difference time-domain (FDTD) method, as shown in Fig. 8.
Fig. 8. Electric-field distributions of the ROTE structure with different incident angles: (a) θi = 40°; (b) θi = 38°; (c) θi = 42°, where a = 250 µm, d = 10 nm, and the other parameters are the same as those in Fig. 3.
We compared the electric field distributions of the ROTE structure with three different incident angles (38°, 40°, 42°) at a = 250 µm. These incident angles corresponded to three cases. The 38° angle corresponded to the case where the angle is less than the critical angle, the 40° angle corresponded to the incident angle for obtaining the maximum GH offset in total reflection domain and the 42° angle corresponded to other cases (except 40°) in the total optical reflection domain. Because the tunneling gap (i.e., d) on both sides of the structure was sufficiently small compared to the thickness (i.e., a) of the central layer, we mainly considered the electric field distribution near the central layer in the simulation process. In contrast to the traditional total reflection phenomenon, the electric-field intensity of an evanescent wave attenuates exponentially in the transverse propagation direction. In the ROTE structure, the electric field intensity of the evanescent wave gradually increased after the incident light entered the tunneling layer and showed a stable standing wave when it was transmitted to the resonant cavity. Then, the incident light passed through the last tunneling layer in the form of an evanescent wave. As seen in Fig. 8(a), the electric field was very small in the prisms on both sides; however, a local enhancement mode was formed in the central layer, and interference occurred in the resonant cavity to form a stable standing wave. This strong local electric-field distribution enhanced the GH effect. However, when θi = 38° or 42°, the reflectivity was small, as can be seen in Figs. 8(b) and (c). The electric field did not form a strong local mode in the central layer, and the evanescent wave distribution in the tunneling layer gradually became uneven. The GH shifts obtained were also relatively small.
3.4 ROTE universality
Figure 9 further demonstrates that the GH shift could be resonantly enhanced for a ROTE structure with other materials. Without loss of generality, we replaced the hBN medium on both sides of the structure with an epsilon-near-zero (ENZ) medium, where the refractive index was less than one. The large GH shift for the ENZ medium could also be obtained by choosing an appropriate central layer thickness, a. Compared with the loss layer, the gain layer could also obtain large GH shifts. And consistent with the hBN example, when the thickness of the central layer was infinitely close to the resonant pole of the tunnel, the GH shift was close to the maximum value. In addition, as shown by the blue line in Fig. 9, although no metamaterials such as hBN were used, the ROTE structure composed of ordinary materials could also enhance the GH shift. We also studied other cases, and the analyzed results confirmed this outcome. These results proved the universality of using a ROTE structure for GH shift research.
Fig. 9. GH shift versus the incidence angle in a ROTE configuration based on ENZ with loss (red) and with gain (black) in the terahertz frequency region. The loss ENZ dielectric constant is ε = 0.001 + 0.01i, a = 180 µm; the gain ENZ dielectric constant is ε = 0.001-0.01i, a = 150 µm. The ROTE configuration (blue) of the ordinary materials was also compared, where n1 = 1.605, n0 = 1.457, n2 = 3.2, and a = 400 µm, with the other parameters having the following values: µ = 0.5 eV, τ = 7 ps, and d = 10 nm.
4. Conclusions
In conclusion, we proposed a symmetric ROTE structure based on hBN materials to enhance GH shifts. When the corresponding parameter conditions were satisfied, large GH shifts could be obtained at angles greater than the critical angle. It is worth noting that the maximum GH shift could reach tens of thousands of times the incident wavelength in the ROTE structure, which was far higher than those obtained in most current enhancement schemes. Moreover, the GH shifts could be regulated by the thickness of the tunneling gap, and they were more sensitive to the thickness of the central layer than to the gap width of the first and second tunneling. We also considered the influence of parameters such as the chemical potential and relaxation time of graphene, which produced strong coupling between the incident light beam and graphene surface plasma, further enhancing the GH effect. In addition, the GH shifts could be enhanced in the ROTE structure even when no special materials such as hBN were used. These results provide a new theoretical basis for the GH effect and new possibilities for the application of optical switches and optical high-precision measurement sensing technology.
Funding
National Natural Science Foundation of China (62175070, 61774062, 11674109); the Key Project of DEGP (2022KTSCX166).
Disclosures
The authors declare no conflicts of interest.
Data availability
The authors confirm that data supporting the findings of this study are available within the article. The data underlying the results presented in this paper are available from the corresponding author upon request.
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