1. Introduction

It is well known that Hermitian exhibits real energy eigenvalues, which is just one of the fundamental principles in the field of quantum mechanics. Moreover, theoretical investigations [1–3] indicated that non-Hermitian Hamiltonians can also exhibit real eigenvalues as long as they meet the weaker condition of PT symmetry. PT symmetry is the invariance of a system under the parity operations r→-r (i.e. space inversion), and time-reversal operations t→-t (or complex conjugation $\ast$ in the time-harmonic regime). In the past few years, several novel phenomena were predicted and observed in PT symmetrical structures. Although the idea of PT symmetry is derived from quantum mechanics [4–7], it was immediately developed in the field of optics—called the optical PT symmetry, which was reported in the systems with balanced gain and loss (i.e. $n\left(r\right)=n^{\ast }\left(r\right)$ [8–20]. Along this line, a series of intriguing optical effects were explored, such as negative refraction [10], double refraction [11, 12], unidirectional invisibility [13–19], and coherent perfect laser absorbers [20, 21].

On the other hand, since the lateral shifts of the reflected wave (i.e. Goos-Hänchen shift) in the interface of two different dielectric media were discovered by Goos and Hänchen in 1947 [22], much work concentrated on the GH shifts both theoretically [23–26] and experimentally [27–29]. In addition, GH shifts from some complex configurations including absorption [30, 31], nonlinear responses [32], temperature effect [33], epsilon-near-zero material [34, 35] and photonic crystals [36–40] were widely reported. In general, in conventional photonic crystals, it was found that there is an obvious GH shift near the edge of the photonic band gap structures [37], and giant lateral shift at the defect mode [38]. More recently, in comparison with the phenomena found in conventional photonic crystals, the GH shift for wave scattering from complex crystals was predicted to be extremely large inside the reflection band as the PT symmetry-breaking threshold was approached [40]. Incidentally, the influence of PT symmetry on the GH shift of atomic medium in a cavity was investigated, and a giant enhancement for the GH shift was revealed when the PT-symmetry condition was satisfied [41].

In this paper, we would like to investigate the lateral shifts of the wave reflected and transmitted through a periodic multilayer-structure with PT symmetry. Especially, we aim at studying the lateral shifts of reflected and transmitted beams near the CPA laser point and the exceptional points, exhibited by the PT-symmetry structure. To one’s interest, we shall show that although the propagation properties of PT-symmetry structure are highly direction-dependent (i.e., the reflections from two sides are quite different), their lateral shifts from two sides always coincide with each other, manifesting highly direction-independent. In addition, we shall show that at the CPA laser point, where both reflections and transmission achieve large magnitudes, the corresponding lateral shifts are very large, and even reach their negative (or positive) maxima. Particularly, they are very sensitive to the incident angle and the number of periods. We believe that these phenomena may lead to some potential applications for designing and producing optical devices in the future.

2. Model and theory

Let us consider a TM-polarized wave propagating in a periodic multilayer-structure, consisting of 2N slabs with the same thickness L. Adjacent slabs have the complex-conjugate relative permittivities such as ${\epsilon }_1=\epsilon \text{'}-i\epsilon \text{'}\text{'}$ and${\epsilon }_2=\epsilon \text{'}+i\epsilon \text{'}\text{'}$, which satisfy the PT-symmetry requirement in optics. For simplicity, the structure is surrounded by vacuum, and the incident beam is illuminated from the left (or right) side with angle θ₀, as shown in Fig. 1(a).

 

Fig. 1 (a) Geometry of the light beam propagating through a periodic multilayer-structure with complex-conjugate permittivities in linear homogeneous medium, and (b) schematic diagram of the model for S-matrix.

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For a TM wave, the magnetic field obeys the Helmholtz equation,

When a wave propagates along the z-axis illuminates the structure, the scattering properties can be calculated through transfer matrix method (TMM) [42, 43]. These amplitudes can be found by interrelating the fields at the structure outer interfaces,

In the case of injection only from the left side, the reflection coefficients coefficients with transmission are written as $r_L\text{=}{H}_b^{-}/H_f^{-},t_L=H_f^{+}/H_f^{-}$; and in the case of injection only from the right side, the reflection coefficients with transmission coefficients are $r_R\text{=}{H}_f^{+}/H_b^{+}$, $t_R\text{=}{H}_b^{-}/H_b^{+}$, where the subscript L(R) indicates the wave incident from the left (right) side. As a consequence, one yields the expressions of the complex reflection/transmission coefficients in terms of the elements of transfer matrix Q, $r_L=-Q_{21}/Q_{22},$ $r_R=-Q_{12}/Q_{22}$, $t_L=t_R=1/Q_{22}$. The reflectivity and transmissivity are $R_{L,R}={\left|r_{L,R}\right|}^2$, $T_{L,R}={\left|t_{L,R}\right|}^2$, respectively. Especially, $r_L$and$r_R$ are different while $t_L$and $t_R$ are equal in a PT symmetric structure, and these observables are related by a generalized unitarity relation $\left|{\left|t\right|}^2-1\right|=\left|r_L{r}_R\right|$ [18].

Based on the stationary phase method, the lateral shifts of the reflected and transmitted beams can be derived from [23],

In the PT symmetric system, as the simultaneous gain and loss are balanced, there is an unique feature that the growth and decay modes are tightly linked with each other due to the existence of the breaking phase. When the system permits self-lasing at some frequencies, one may observe the coherent perfect absorption [44]. For a laser oscillator without injected signal, the boundary conditions $H_f^{-}=H_b^{+}=0$ are adopted, which yields$Q_{22}=0$. In contrast, for a perfect absorber, the boundary conditions $H_b^{-}=H_f^{+}=0$ result in$Q_{11}=0$. In this connection, we take Fig. 1(b) for a better understanding. In general, there is always a real frequency${\omega }_0$at which $Q_{11}\left(\omega \right)=0,$ and it follows that at the same frequency one has $Q_{22}\left(\omega \right)=0$. In other words, the lasing medium also behaves as a CPA under the condition when $Q_{11}\left(\omega \right)=Q_{22}\left(\omega \right)=0.$

On the other hand, it is also necessary to illustrate the S-matrix, which is defined as,

3. Numerical results

We are now in a position to present numerical results on the lateral shifts of the reflected/transmitted waves from the PT symmetric multilayer-structure. At first, we aim at the lateral shift near the CPA-Laser point. For simplicity, the relevant parameters are chosen as $L=125um$,$\epsilon \text{'}=0.1,\epsilon \text{'}\text{'}=0.1$, and ${\theta }_0=5deg$, to realize the CPA-Laser condition.

In Fig. 2, we show the reflectivity/transmissivity and their lateral shifts for a PT bilayer-structure (N = 1) as a function of the frequency ranging from $\omega =1.25\times {10}^{13}Hz$ to $\omega =1.75\times {10}^{13}Hz$. We find that the lateral shifts of the reflection from two directions and the transmission are indeed the same [see Fig. 2(b)], although the reflections from both sides and the transmission are quite different [see Fig. 2(a)]. Actually, the same shifts can be understood from the same variation tendencies of the ${\phi }_{r,L},{\phi }_{r,R}$ and ${\phi }_t$ with the incident angle, as shown in the insert in Fig. 2(b). In addition, large negative shifts arise at the CPA-Laser point $\omega =1.49\times {10}^{13}Hz$. These results are quite meaningful since one may obtain large lateral shifts, accompanied with large reflections and transmission, which makes the shifts observable and detectable practically near the CPA-Laser point.

 

Fig. 2 (a) Reflectivity/transmissivity and (b) the related shifts for the PT bilayer-structure (N = 1) as a function of the frequency, for the thickness of each slab $L=125um$, the incident angle ${\theta }_0=5deg$, the permittivities ${\epsilon }_1=0.1-0.1i$and ${\epsilon }_{\text{2}}=0.1\text{+}0.1i$. The insert in Fig. 2(b) shows the phases of reflectivity/transmissivity as a function of incident angle ${\theta }_0$at the CPA-Laser point.

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For a given PT bilayer-structure, the frequency for the CPA-laser point varies slightly with different incident angle, as shown in Fig. 3(a). Correspondingly, the peak of the lateral shifts moves at the same time, as shown in Fig. 3(b). Note that for relatively large incident angle such as ${\theta }_0=5deg$, one achieves negatively large lateral shifts. On the contrary, large lateral shifts with positive value is found for relatively small incident angle. Therefore, through the fine adjustment of incident angle, one may expect novel phenomena for the lateral shifts at the CPA-laser points. For instance, there is a crossover incident angle, above and below which, the direction of the lateral shift changes at the CPA-laser point. Furthermore, near the crossover incident angle such as ${\theta }_0=2.8deg/2.9deg$, the lateral shift as a function of the frequency exhibits a Fano-like shape, whose physical mechanism deserves further careful investigations.

 

Fig. 3 (a) Reflectivity from left side and (b) the lateral shifts as a function of the frequency for different incident angles. Other parameters are $L=125um$,${\epsilon }_1=0.1-0.1i,$ ${\epsilon }_2=0.1+0.1i$.

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To discuss the effect of the stack periodicity on the lateral shifts near the CPA-laser points, we further increase the parameter N to 5 and 10, as shown in Fig. 4. It can be observed that, with increasing the number of unit cells, the CPA-laser points appear more than one, so as the poles of reflections and related shifts. Furthermore, some of the related shifts become much larger in comparison with the case for N = 1. Note that there are some minima of reflections due to the Bragg resonances, at which small maximal and positive shifts are found [see in the insert in Figs. 4(b) and 4(d)].

 

Fig. 4 Reflectivity from left side and the related lateral shifts for (a, b) N = 5 and (c, d) N = 10 as a function of the frequency. Other parameters are the same as those in Fig. 2.

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Next, we shall discuss the lateral shifts of the reflected wave from the PT symmetric structure near the exceptional points. In this case, we choose the permittivities of the structure to be $\epsilon \text{'}=0.0001,\epsilon \text{'}\text{'}=0.001$, which is in the epsilon-near-zero limit. General speaking, the exceptional points can be observed for the incident angle slightly larger than the critical incident angle ${\theta }_c=\arcsin \sqrt{\left(\epsilon {\text{'}}^2+\epsilon \text{'}{\text{'}}^2\right)/2\epsilon \text{'}}=4.08\mathrm{deg}$, at which the zero-reflection is fulfilled [19]. Hence, the incident angle is set as ${\theta }_0=5deg$.

In Fig. 5(a), we show the eigenvalues of the S-matrix as a function of the incident frequency for PT-symmetric bilayer-structures, i.e. N = 1. It is evident that $\omega =2.95\times {10}^{13}{s}^{-1}$ and $\omega =3.05\times {10}^{13}{s}^{-1}$correspond to the exceptional points, at which unidirectional invisibility appears. This can be clearly observed from the reflectivity/transmissivity spectra, as shown in Fig. 5(b). In addition, we show the phase of $r_L$and$r_R$ in Fig. 5(c). Note that the phases of $r_L$and$r_R$ at the exact exceptional points are non-continuous. This is understandable because the related reflectivity $r_L$or $r_R$is equal to zero, and the related lateral shift is meaningless at these points accordingly.

 

Fig. 5 (a) Eigenvalues of the S matrix, (b) Reflectivity/transmissivity and (c) phases of reflection coefficients as a function of the frequency, for $L=125um$, ${\theta }_0=5deg$, ${\epsilon }_1=0.0001-0.001i$and ${\epsilon }_{\text{2}}=0.0001\text{+}0.001i$.

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Figure 6 shows the reflectivity and the related lateral shifts as a function of the frequency for different stack periodicity. Again, for PT symmetric-structure, although the behavior of reflections from two sides is quite different in PT symmetric structure, the lateral shifts of reflection from the opposite sides are absolutely equal. In addition, the lateral shifts are non-monotonic with increasing the incident frequency, and achieve their positive/negative maxima near the exceptional points (marked with A, B and C). The reason is that near the exceptional points, the incident wave penetrates more light into the medium, resulting in large lateral shift.

 

Fig. 6 Reflectivity and the related shifts from left (blue solid line) and right (red dashed line) sides for (a, d)N = 1 (b, e)N = 45 and (c, f) N = 90 as a function of the frequency. Other parameters are the same as those in Fig. 5.

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From Fig. 6, we further find that the peak positions around the exceptional points do not move with the increase of the periodicity. This is due to the fact that the exceptional points are really independent of the parameter N. However, the maximal value of lateral shifts near the exceptional points is positive for N = 1, becomes negative for N = 45, and is positive again for N = 90. In other words, the directions of the lateral shifts depend on the parameter N. Besides the maximal shifts near the exceptional points, we get more maxima (both positive and negative) of the lateral shifts thanks to the Bragg resonances. As one knows, at these Bragg-resonance points, reflections are weak for both sides of incidence, when the thickness for the whole stack equals an integer number of Block half-waves, i.e., $2NL\overline{k}=\pi q\left(q=0,\pm 1,\pm 2,\dots \dots \right)$, where $\overline{k}$ is the Bloch wavenumber. Actually, this phenomenon is quite common in the case of normal 1D-photonic crystal, and the shifts reach their maxima near those resonances when the related reflections almost vanish. In general, the number of the Bragg resonant peaks depends not only on the number of periods but also on the permittivity of the periodic structure. Since the permittivity here is near-zero, one observes only a few reflectionless transmission/Bragg resonant peaks even for large number periods. On the contrary, a large number of Bragg resonances take place for the ten-layered PT structures with ${\epsilon }_1=0.1-0.1i$ and ${\epsilon }_2=0.1+0.1i$ [see Fig. 4(c)]. In addition, the maximal values of the lateral shifts in Bragg resonances are positive below the exceptional points, and negative above the exceptional points. We believe that the existence of the exceptional points changes the original tendency of both reflection phases [see Fig. 5(c)]. Such a tendency implies a sharp slope of phase from positive value to negative one, and thereby changing the usual direction of the lateral shifts. Hence, we conclude that one may change the direction of the lateral shift by adjusting the incident frequency and the periodic number.

To illustrate the effect of the stack periodicity, we pay attention to the lateral shifts near the exceptional points with frequency $\omega =3\times {10}^{13}{s}^{-1}$. From Fig. 7, we find that, with increasing the number of unit cells, the lateral shift increases at first, reaching the positive maximum, and then, the lateral shift becomes negative at layer number $Nc=14$, reaching the negative maximum. After that, the periodic variation takes place. Accordingly, the strongest shifts are periodically observed for $Nc=14,28,42$, and so on. For instance, we observe that the maximal value of the lateral shift can be as large as 10000 times of the incident wavelength for $Nc=14$. As the period of the structure increases, the Bragg resonances appear more and more, which leads to the transforms of phases of the reflections and transmission at the exceptional points, and hence the lateral shifts at the exceptional points become positive and negative.

 

Fig. 7 Dependence of the lateral shift of reflection from left on the stack periodicity N for $\omega =3\times {10}^{13}{s}^{-1}$. Other parameters are the same as those in Fig. 6.

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In the end, the reflectivity/transmissivity and the lateral shifts in both CPA-laser point and exceptional points are shown in Fig. 8. Note that, CPA-laser phenomenon occurs at ${\theta }_0=5deg$, and unidirectional invisibility phenomena occurs around ${\theta }_0=20deg$ [see Figs. 8(a) and 8(b)]. The related shifts (marked with A and B) appear to be very large, as expected.

 

Fig. 8 (a,b) Reflectivity/transmissivity and (c) the related shifts for the PT bilayer-structure (N = 1) as a function of the incident angle, for $L=125um$, $\omega =1.49\times {10}^{13}{s}^{-1}$, ${\epsilon }_1=0.1-0.1i$ and ${\epsilon }_{\text{2}}=0.1+0.1i$.

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4. Simulations

To verify the validity of our theory, we take one step forward to carry out numerical simulations by using COMSOL Multiphysics (RF module), based on finite element method. For simplicity, we consider a Gaussian beam incident from air to the PT symmetric bilayer.

Figure 9 shows the magnetic field amplitude distribution (a,c,e) and (b,d,f) at the CPA-Laser point. From Figs. 9(a), 9(c), and 9(e), one can observed that the Gaussian profile of the reflected beam and transmitted beam are both very strong, and hence the magnetic field of incident beam is much weaker to be observed. Since the incident beam is much weak in comparison with the reflected and transmitted ones, for the sake of clearness in Figs. 9(b), 9(d), and 9(f), we have rescaled the incident field by a amplification factor 10. In Figs. 9(a) and 9(b), we chose θ₀ = 2deg, and $\omega =1.452\times {10}^{13}{s}^{-1}$. It is evident that both reflected and transmitted waves exhibit large positive shifts. For instance, the simulated results show the lateral shifts of reflection and transmission are $\Delta r\text{=}22.18\lambda$ and $\Delta t\text{=}22.21\lambda$. On the other hand, the theoretically results are $\Delta r,t=25.02\lambda$. Taking into account the incident beam being a Gaussian beam, such a deviation is quite acceptable. For the other case with θ₀ = 5deg, and $\omega =1.493\times {10}^{13}{s}^{-1}$, large negative shifts are observed [see Figs. 9(c) and 9(d)]. The simulated results show the lateral shifts are $\Delta r\text{=}-14.03\lambda$ and $\Delta t\text{=}-14.12\lambda$, in comparison with the theoretical results $\Delta r,t\text{=}-15.50\lambda$. In addition, we have exchanged the permittivities of bilayer slabs with ${\epsilon }_1=0.1+0.1i$and ${\epsilon }_{\text{2}}=0.1-0.1i$, which corresponds to the right incidence on the bilayer slabs with ${\epsilon }_1=0.1-0.1i$ and ${\epsilon }_{\text{2}}=0.1+0.1i$. The results are shown in Figs. 9(e) and 9(f), and the relevant lateral shifts are almost the same with those in Figs. 9(c) and 9(d). In detail, the simulated results show the lateral shifts are$\Delta r\text{=}-14.15\lambda$ and $\Delta t\text{=}-14.20\lambda$. Therefore, the prediction that lateral shifts are direction-independent is numerically verified.

 

Fig. 9 Magnetic field amplitude distribution of all the calculated apace and in the surface of reflection and transmission for (a, b) θ₀ = 2deg, $\omega =1.452\times {10}^{13}{s}^{-1}$, (c, d) θ₀ = 5deg, $\omega =1.493\times {10}^{13}{s}^{-1}$, with${\epsilon }_1=0.1-0.1i$and ${\epsilon }_{\text{2}}=0.1\text{+}0.1i$, and (e, f) θ₀ = 5deg, $\omega =1.493\times {10}^{13}{s}^{-1}$, with${\epsilon }_1=0.1+0.1i$and ${\epsilon }_{\text{2}}=0.1-0.1i$.

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5. Conclusions

To summarize, we have outlined the stationary-phase method to the analysis of GH shift of the reflected/transmitted beams from the PT symmetric structures near the CPA-laser point and the exceptional points. It is found that the shifts of reflections from both sides have the same behaviors although the reflections behave diversely. Large shifts are evidently observed near the CPA-laser point and the exceptional points. To one’s interest, at the CPA-Laser points, the lateral shifts are quite large and even reach their negative or positive maxima, accompanied with large reflection and transmission. Moreover, they are very sensitive to the incident frequency and the number of periods. As a consequence, one may realize the switch between large positive and large negative shift by adjusting the incident frequency and the number of period slightly. Our investigation may be helpful in many potential applications for designing and producing optical devices, for instance, increasing the sensitivity of optical devices.