1. Introduction

Goos-Hänchen (GH) effect is a well-known phenomenon in optics, and it states that the reflected or refracted light of linear polarization experiences a spatially lateral displacement from the position predicted by geometry optics, where reflection or refraction happens at an interface of two media. The GH shift was first observed by F. Goos and H. Hänchen in 1947 [1] in the experiments of totally internal reflection. Since then, it has been extensively explored in a large amount of media and various configurations [2–19], such as dielectrics [4,5], metals [6,7], negative index metamaterials [9–12] and zero index metamaterials [19]. Generally, the GH effect is not obvious owing to its amplitude typically on a wavelength scale, which is a big challenge for experimental observations even with the use of the indirect weak measurement method [20]. To make a direct measurement, in the past decades, various schemes have been proposed to achieve enhanced or giant GH shifts, including loss media [2,3], photonic crystals [15], hyperbolic metamaterials [16] and metasurfaces [18].

In recent years, PT symmetric photonic systems have attracted considerable interest, owing to a large amount of novel phenomena [21,22]. To realize a PT symmetry, the refractive index profile of a photonic system is required to meet $n(\stackrel{\rightharpoonup }{r})=n^{*}(-\stackrel{\rightharpoonup }{r})$, which implies that such system is constituted by spatially modulating balanced gain and loss. A simple configuration of PT symmetric photonic systems is one dimensional (1D) bilayer structure which has enabled a number of exotic photon transport phenomena [23–34], such as laser and coherent perfect absorption (CPA) (i.e., anti-laser) [24–26], unidirectional reflectionless [28,29] and unidirectional invisibility [30,31]. In particular, at a semi-infinite PT-symmetric interface, one can find lossless and stable surface modes that propagate along the interface but exponentially decay in the vertical direction [34,35]. Such surface modes may result in sharp resonances in transmission or reflection spectra [36], which potentially provide a new way to obtain giant GH shifts. Previously, it was reported that giant GH shifts can be seen at the CPA-laser points and the exceptional points (EPs) [17] in a 1D PT symmetric multilayer-structure. Actually, such giant GH shifts at the EP points are caused by the excitations of surface modes; but they didn’t explore the surface modes and associated GH effect in details because in multilayer structures, the surface modes at the interfaces of two different layers are quite complicated. In addition, Ref [17]. only focused on a special case of epsilon-near-zero (ENZ) medium, so that some interesting results are missed. So far, the relationship between the surface modes at PT interfaces and the GH effect is still unclear.

The bilayer structure is a relatively simple configuration, thus the surface modes and their dynamic properties are easily explored. In this work, we will enrich the studies of GH effect in PT symmetric bilayer structures by considering a general case with all possible material parameters. We will first explore the surface modes therein and show that the properties of surface modes at PT bilayer interfaces are largely depending on the layer thickness, in particular when the thickness is comparable to the working wavelength. Then we will demonstrate that these surface modes can result in giant GH shifts for TM polarization (with only magnetic field along y direction), but have no influence on TE polarization (with only electric field along y direction). Interestingly, we find that for fixed material parameters, the GH shift can be significantly enhanced by only altering the geometry size, i.e. the thickness of loss/gain layer. The amplitude of the GH shift can monotonously grow with the increase of the thickness, and in theory its maximum value can go to infinity. Behind it, the physical mechanism is that the surface modes in PT interfaces are quasi-bound states in continuum, which lead to very sharp resonances in scattering spectra.

2. Results and discussions

2.1 Bound states at PT bilayer interfaces

Figure 1(a) shows the considered PT bilayer structure that consists of a loss layer and a gain layer with an identical thickness of d. Both loss and gain media are nonmagnetic and their parameters are ${\epsilon }_l={\epsilon }_r+i\gamma$ and ${\epsilon }_g={\epsilon }_r-i\gamma$, respectively. For the bilayer with infinite thickness (i.e., $d\to \infty$), it has shown that the surface modes only survive for TM wave, with their dispersion relationships [35] given as,

 

Fig. 1 (a) The schematics of a PT-symmetric bilayer structure. A linearly polarized light is incident on the bilayer structure from loss side indicated by the black arrows, or from the gain side marked by the red arrows. Both loss and gain layers with an identical thickness of d meet PT symmetry about z = 0. (b) The parameter space for surface states at PT-symmetric interfaces with infinite thickness. The red dashed line indicates the position where ${\beta }_{sw}=k_0$. The tiny circle indicate the parameters of ${\epsilon }_l=1.5+0.5i$ and ${\epsilon }_g=1.5-0.5i$. (c) The dispersion relationship. For a given working frequency ${\omega }_0$, a surface mode with ${\beta }_{sw}<k_0$ is in the continuum.

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Returning back to finite thickness cases, the surface modes at PT bilayer systems are largely thickness-dependent, which can be obtained in theory by analyzing the eigen modes at PT interfaces. To achieve this aim, the magnetic field in each region can be expressed as: when $z>d,$$\stackrel{\rightharpoonup }{H}=\widehat{y}a\exp (i{k}_{s}z+i\beta x)$ and when $z\le -d,$ $\stackrel{\rightharpoonup }{H}=\widehat{y}D\exp (-i{k}_{s}z+i\beta x)$. In the loss ($-d<z\le 0$) and gain ($0<z\le d$) media, the magnetic fields are $\stackrel{\rightharpoonup }{H}=\widehat{y}\left[b_1\exp (k_{L}z)+b_2\exp (-k_{L}z)\right]\exp (i\beta x)$ and $\stackrel{\rightharpoonup }{H}=\widehat{y}\left[c_1\exp (k_{G}z)+c_2\exp (-k_{G}z)\right]\exp (i\beta x)$, respectively. Here $k_{L(G)}={\left({\beta }^2-{\epsilon }_{l(g)}{k}_0^2\right)}^{1/2}$, $k_s={\left(k_0^2-{\beta }^2\right)}^{1/2}$ with $k_0=2\pi /\lambda$, and $\beta$is the wave vector along x direction. The corresponding electric field in each region is obtained by applying $\stackrel{\rightharpoonup }{E}=\nabla \times \stackrel{\rightharpoonup }{H}/(i\omega \epsilon )$. By matching the continuous boundary conditions of both magnetic and electric fields at $z=0$, d and –d, respectively, a set of linear equations about these unknown coefficients, i.e., a, $b_1$, $b_2$, $c_1$, $c_2$ and D, are built up. The dispersion relationships of the surface modes are achieved by calculating the determinant of this set of equations, with zero value corresponding to the nontrivial solutions. Figure 2(a) analytically shows the determinant value vs both $\beta$ and d. In calculations, ${\epsilon }_r=0.001$ and $\gamma =0.02$. The black dashed line in Fig. 2(a) indicates the positions of minimum values. It is clearly seen that when $d>\sim 0.4\lambda$, the determinant values are almost zero, which means the surface modes at this time are well confined (that is lossless and stable). In particular, when $d\ge 2\lambda$, all zero values fall on a horizontally straight line corresponding to a constant $\beta$, which indicates these surface modes are insensitive to the loss/gain layer thickness. For $d<0.4\lambda$, the minimum deviates from zero and gradually becomes larger, implying that the surface modes in this case are loss and less confined. Figure 2(b) illustrates the determinant value vs both $\beta$ and $\gamma$ for a fixed thickness $d=0.6\lambda$ and ${\epsilon }_r=0.001$, with the black dashed line pointing out the minimum positions. To make comparisons, Fig. 2(c) shows the corresponding surface modes in the infinite case with ${\epsilon }_r=0.001$. We can know from both results that when $\gamma \ge 0.01$, the black dashed line and the blue solid line are almost identical, hence the finite PT bilayer can support well defined surface modes. While for $\gamma <0.01$, the surface states in the finite case become radiative or less confined because the minimum value gradually deviates from zero. The inset of Fig. 1(c) displays the simulated field pattern of the surface mode at the infinite PT interface when ${\epsilon }_r=0.001$ and $\gamma =0.02$, which is obtained from COMSOL Multiphysics.

 

Fig. 2 The bound states at finite PT interfaces. (a) The determinant value for different $\beta$ and d when $\gamma =0.02$. (b) The determinant value for different $\beta$ and $\gamma$ when $d=0.6\lambda$. (c) The surface states based on Eq. (1) for infinite d. The inset shows the simulated field pattern for a surface state when $\gamma =0.02$. Here ${\epsilon }_r=0.001$ in (a)-(c). The black dashed curves in (a) and (b) denote the solutions of surface states.

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2.2 Analytic results for GH shifts

Based on above discussed surface modes, now let us study the GH shift in the PT symmetric bilayer system immersed in air. Consider a light beam with an incident angle ${\theta }_{in}$ illuminating on this bilayer from loss side or gain side (see the black and red arrows in Fig. 1(a)). The transmission and reflection coefficients can be calculated based on transfer matrix method [17]. The total matrix is

Figure 3 shows the analytically calculated GH shift L of both TM and TE polarizations for different loss/gain levels, where $d=0.6\lambda$ and ${\epsilon }_r=0.001$. Note that in a lossless ZIM (e.g., ${\epsilon }_r=0.001$ and $\gamma =0$), it was demonstrated that the GH shift for TM polarization is zero for all incident angles, except at the Brewster angle where the GH shift is usually enhanced [19] (see the black curve in Fig. 3(a)). This Brewster angle is small because of near zero index. For TE polarization, the GH shift is a constant (i.e., $L=\lambda /\pi$) at large incident angle (see the black curve in Fig. 3(b)). When it is turned into PT symmetric bilayer system, the results are dramatically changed owing to the added loss and gain portions. For example, for TM polarization as shown by Fig. 3(a), when the loss/gain is $\gamma =0.005$, there is a giant GH shift at ${\theta }_{GH}\approx {9.2}^{\circ }$ (the blue curve in Fig. 3(a)). Furthermore, the ${\theta }_{GH}$ for giant GH shifts will become larger with the increase of $\gamma$. When $\gamma =0.05$, the GH shifts vanish for all incident angles due to $\beta >k_0$ (see orange line), because the surface modes could not be excited by any incident wave from air. Figure 3(b) shows the corresponding GH shifts for TE polarization; all cases overlap together, which means that the added loss and gain make no difference in GH effect due to the non-existence of surface states for TE polarization.

 

Fig. 3 GH-shifts of transmitted wave in PT symmetric bilayers with different loss/gain for TM polarization (a) and TE polarization (b).

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It is known that in a loss medium, a large GH shift at pseudo-Brewster angle is usually observed [5,6]. Here we note that the giant GH shifts in Fig. 3(a) don’t happen at pseudo-Brewster angle. To confirm this, Fig. 4(a) replots the happening angle ${\theta }_{GH}$ (the blue solid circles) for the giant GH shifts in Fig. 3(a), and plot the pseudo-Brewster angle ${\theta }_{pB}$ (red hollow circles) in a loss medium (${\epsilon }_r=0.001+i\gamma$). In analysis, the thickness of the loss slab is set as $1.2\lambda$, in order to keep identical geometric size with that of PT bilayers ; ${\theta }_{pB}$ is achieved by finding the incident angle for minimum reflectance. We can see that in all considered cases, the pseudo-Brewster angles are quite small (${\theta }_{pB}<{10}^{\circ }$), largely deviating from ${\theta }_{GH}$. Therefore, the giant GH shifts for TM polarization in PT symmetric bilayers are indeed induced by the excitations of surface states.

 

Fig. 4 (a) The happening angles ${\theta }_{GH}$ of the giant GH shift for different $\gamma$ in Fig. 3(a) and the pseudo-Brewster angles ${\theta }_{pB}$ (red hollow circles) in a loss medium (${\epsilon }_r=0.001+i\gamma$) with a thickness of $1.2\lambda$. (b) The eigenvalues of S-matrix for $\gamma =0.02$ and $d=0.6\lambda$. (c) The reflection, transmission and transmission phase for $\gamma =0.02$ and $d=0.6\lambda$. (d) The enlarged drawing for $\gamma =0.02$ in Fig. 3(a).

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On the other hand, the excitations of surface modes at PT interfaces can give rise to anisotropic transmission resonances, and generate a pair of EPs [36]. We expect to state that the giant GH shifts in Fig. 3(a) don’t happen at the so-called EPs exactly, but at the angles in between. Here we take $\gamma =0.02$ case in Fig. 3(a) for illustrations. Based on Eq. (3), Fig. 4(b) shows the eigenvalues of the S-matrix, given by $s_{\pm }=t\pm \sqrt{r_R{r}_L}$, and the S-matrix is defined as $S=\left(\begin{array}{cc}t& r_L\\ r_G& t\end{array}\right)$. There are two EPs at about ${29.03}^{\circ }$ and ${29.33}^{\circ }$, respectively. Such two EPs will lead to two modes in scattering angular spectra: one mode is $\left|r_L\right|=0$ and $T=1$, and the other is $\left|r_G\right|=0$ and $T=1$, as shown by Fig. 4(c). Figure 4(d) shows the enlarged drawing of the corresponding results of GH shifts at incident angles ranging from ${28}^{\circ }$ to ${30}^{\circ }$. Clearly three curves of GH shifts in reflected and transmitted light coincide exactly, with only one resonance peak at the angle about ${29.20}^{\circ }$. Such angle is not consistent with any EPs, but falls somewhere in between. Further analysis exhibits that this angle actually coincides with that for the maximum transmission. As shown by the red dashed line in Fig. 4(c), the transmission phase changes most dramatically at the angle for the maximum transmission, which leads to the maximum GH shift. In fact, this angle for the maximum transmission also corresponds to that for the maximum splitting of the eigenvalues. Therefore, the giant GH shifts do not exactly happen at the EPs, but at the angle for the maximum transmission and also at the angle for the maximum splitting of the eigenvalues. We also study another case of $\gamma =0.03$ and $d=0.6\lambda$ (we didn’t show here), and the analyzed results also confirm this outcome.

2.3 Thickness-dependent GH effect and BICs

As mentioned above, the thickness of loss/gain layer has tremendous impact on the properties of surface modes. It is necessary to study the influence of thickness d on the GH shifts for fixed material parameters. Figure 5(a) shows the GH shifts vs the incident angle for the thickness d increasing from $0.4\lambda$ to $\lambda$. In calculations, ${\epsilon }_l=0.001+0.02i$ and ${\epsilon }_g=0.001-0.02i$. From it, we can see that as d grows, ${\theta }_{in}$ is decreasing from ${32.15}^{\circ }$ to ${27.11}^{\circ }$, close to ${\theta }_{sw}\approx {26.60}^{\circ }$ given by Eq. (1). This is not surprising because as the layer grows thicker, the surface mode is closer to that in the infinite case (see Fig. 2). But it is quite interesting that the GH shift is greatly enhanced from $L\approx 3.5\lambda$ to $L\approx 68\lambda$ as d increases from $0.4\lambda$ to $\lambda$, and at the same time, the resonance peak becomes sharper. More studies reveal that the thicker the thickness d, the larger the GH shift and the sharper the resonance peak. Similar thickness-dependent GH effect was found in previous works. For instance, in a dielectric slab, the amplitude of GH shift varies periodically as its thickness grows, which is caused by the Fabry–Pérot resonances [4]. But our findings suggest that increasing thickness generates monotonously increasing GH shifts, and their amplitude can go to infinity in theory. This is a new feature in subject of GH effect.

 

Fig. 5 (a) The giant GH shift vs the thickness d for fixed material parameters. From right to left, the peaks of the giant GH shifts are located at ${32.15}^{\circ }$, ${29.27}^{\circ }$, ${27.83}^{\circ }$ and ${27.11}^{\circ }$, with corresponding values of $L\approx 3.5\lambda$, $10\lambda$, $25\lambda$ and $68\lambda$. (b) The reflectance ${\left|r_L\right|}^2$ vs thickness d and the incident angle. Here ${\epsilon }_l=0.001+0.02i$ and ${\epsilon }_g=0.001-0.02i$. (c) The reflectance ${\left|r_L\right|}^2$ vs the incident angle for a fixed thickness of $d=2.0\lambda$. (d) Transmission phase vs the incident angle for different thickness of $d=0.4\lambda$, $0.6\lambda$, $1.0\lambda$, $1.2\lambda$and $2.0\lambda$, which are shown by black, red, green, blue, cyan and pink curves, respectively.

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The physical mechanism behind our findings is that the surface modes in PT interfaces are not regular bound modes, but bound states in the continuum (BICs) which are localized waves even if they coexist with a continuous spectrum of radiating waves [37]. As shown by the dispersion relationship in Fig. 1(c), at a given working frequency, all surface modes with $\beta <k_0$, which lead to giant GH effect, are located in an area above the light line (i.e., in the continuum). Therefore, the interaction between these surface modes and the continuous modes in air leads to sharp resonances in scattering waves. It should be noted that a true BIC actually is mathematical abstraction and is not practically implementable because it has infinite lifetime and most structures supporting BICs extend to infinity. So does it in PT bilayer systems. Let us consider the PT bilaryer with an extremely large thickness. On one hand, the surface modes at PT interfaces are perfect and propagate without any amplification and attenuation. On the other hand, when the thickness goes to infinity, the thickness-dependent radiative loss of these surface modes goes to zero. This means that these surface modes will be completely decoupled from the continuum modes in the surrounding medium (air), leading to a resonance with zero leakage and zero linewidth. This is the signature of BICs. Without doubt, this true BIC is trivial because of infinite size. However, by relaxing the infinite case to a finite geometry, we still can achieve a quasi-BIC with very high quality factor (high-Q).

To reveal the quasi-BIC features in PT bilayer systems, Fig. 5(b) shows the reflectance ${\left|r_L\right|}^2$ vs both incident angle and the thickness. For a fixed d, due to the excitations of surface modes, we can see that the reflectivity is a function of the incident angle, featured by a Fano resonance. As d increases, such Fano resonance becomes sharper and sharper, and eventually disappears with its Q factor going to infinity, e.g., see the case of $d=2.0\lambda$ in Fig. 5(c). This feature is a direct indication for a resonance with infinite lifetime. Such disappearing Fano resonances means the surface modes are more difficult to be excited by outside wave, bringing about more rapid variation in the phase of the scattered waves for a giant GH shift. The results of transmission phases for different thickness d, as shown by Fig. 5(d), also confirm this point. As d increases, the phase variation from $0.5\pi$ to $-0.5\pi$, becomes more and more intense. When $d=2.0\lambda$, it becomes almost abrupt, just like a step function.

2.4 Theoretical analysis for a general PT bilayer case

We have already discussed a special case of ENZ medium in details. In fact, for the parameters locating in the semicircle region in 2D parameter space (see Fig. 1(b), similar giant GH effect can be realized via the surface modes bounded at the constructed PT interfaces. Owing to $\beta \le k_0$, the physical mechanism of the giant GH effect also originates from the quasi-BICs. Consequently, the thickness-dependent GH shift in a bilayer configuration, just like that in Fig. 5(a), also can be predicted. To illustrate this point, we will study a more generalized example in the following. On the other hand, in foregoing ENZ case, the effective refractive index of the surface modes at PT interfaces is larger than that of ENZ media, so that the surface mode itself would be a bound state in spite of its refractive index smaller than unity. This fact might lead to some confusion whether the giant GH effect really results from the quasi-BICs. To differentiate it, in this section, the parameters are set as ${\epsilon }_l=1.5+0.5i$ and ${\epsilon }_g=1.5-0.5i$ (as indicated by a tiny circle in Fig. 1(b)), which has a refractive index greater than unity and also greater than the effective index of surface modes. It is noted that in this case ${\beta }_{sw}=0.91k_0$, which corresponds to ${\theta }_{sw}={65.9}^{\circ }$.

By using COMSOL Multiphysics, Fig. 6(a) shows the numerically simulated field pattern of the surface mode that propagates along such PT interface. In simulations, the thickness of each layer is $d=3\lambda$, and in order to excite the surface mode, a TM point source mimicked by using a tiny electric current loop with 1A is placed at the interface. The surface mode at the PT interface has well spatial mode profile, and its loss including radiative loss is extremely low although the geometric size of bilayer structure is not very large. Figure 6(b) displays the analytical results of GH shifts for different thicknesses, i.e., $d=1.2\lambda$, $1.8\lambda$, $2.4\lambda$ and $3.0\lambda$. From it, we can clearly see the thickness-dependent GH effect, and as d increases, the change tendency of angular spectrum is similar to that in ENZ case. On one hand, the resonance peak becomes narrower and narrower, accompanied by larger and larger GH shift. For instance, when $d=3\lambda$, the resonance in the angular spectrum becomes extremely sharp, with a peak value of $\sim 1300\lambda$. On the other hand, as d increases, the happening angle for maximum shift is shifted to a smaller angle, which is close to ${\theta }_{sw}={65.9}^{\circ }$owing to thickness-dependent surface modes in bilayer systems (see Fig. 2). These results also can be obtained from the transmission phase of each PT bilayer case, as shown by Fig. 6(d). Similarly, the ultimate increase of d leads to a step-like change of transmission phase at the position for the maximum GH shift. To check the quasi-BICs from the scattering wave, Fig. 6(c) plots the analytical results of the reflection ${\left|r_L\right|}^2$, from which the Fano-like resonance featured by asymmetric line shape still can be seen in each case. More importantly, as d increases, the Q factor gradually and monotonously grows, and goes to infinity in the end. Therefore, these analytical results further confirm our finding that the giant GH effect induced by surface modes in PT system originates from quasi-BICs.

 

Fig. 6 GH effect and BICs in the case of ${\epsilon }_l=1.5+0.5i$ and ${\epsilon }_g=1.5-0.5i$. (a) The numerically simulated field pattern of the surface modes at the PT-bilayer interface. In simulations, the thickness of each layer is $d=3.0\lambda$ and the source (see the five star) is mimicked by using a tiny circle with a current of 1A. (b)-(d) are the analytically calculated GH shifts, the reflection ${\left|r_l\right|}^2$and the transmission phase for different thicknesses. In (b), as d increases, the center point for maximum value goes to ${\theta }_{in}={66.00}^{\circ }$. For $d=2.4\lambda$ and $3.0\lambda$, the displayed data are real data divided by 5 and 50, respectively.

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3. Conclusion

We have demonstrated that the surface states bounded at the interface of PT symmetric bilayer can induce giant GH shifts for a specific incidence. The PT symmetry ensures that whatever the incident wave is from loss or gain side, the GH shifts of the reflected and transmitted beams have identical amplitudes. We also show that the surface states at PT symmetric interfaces are quasi-BICs that are largely dependent on the thickness in bilayer structures. Due to quasi-BICs, for fixed material parameters, the GH shifts can be further enhanced by increasing the thickness of loss/gain layers, which is a new feature in the subject of GH effect. Our results directly connect the GH effect and the surface states bounded at PT interfaces together, thus providing a way to study the surface modes or BICs in non-Hermitian optical systems and their dynamic properties.