1. Introduction

In a Hermitian system invariant along the propagation direction, the propagation of a wave packet prepared around a single eigenstate is determined by the group velocity, because its eigenstates are orthogonal and stable. The lateral shift of the center of energy (COE) of wave packet, perpendicular to the in-plane shift generated by the group velocity, occurs only when the beam is in a superposed state of different eigenstates. The representative phenomena are Zitterbewegung (ZB) and ZB-like motions of wave packets, which are characterized by a trembling motion of the COE of wave packet, and widely emerge in the quantum and classic systems [1–9]. It is a consequence of the interference among the eigenstates occupied by the wave packet, and can be quantitatively described by non-Abelian Berry connections [5,8]. When the wave packet is prepared around a superposed state near the two-dimensional (2D) Dirac point, the relative larger non-Abelian Berry connections can lead to a ZB-like motion with a relative larger amplitude [3,8]. It was demonstrated that the photonic beams propagating in binary waveguide arrays could display the ZB-like motions [6,7]. In honeycomb-lattice waveguide arrays with the broken inversion symmetry, the helical ZB-like motions of photonic beams were observed [8,9].

The eigenstates and band structures of photons propagating in waveguide arrays could be efficiently and flexibly designed by tailoring the distributions of the refractive indexes and the positions of waveguides. It provides a feasible platform not only for investigating the optical steering by the photonic band engineering, but also for optical analogs of the dynamics of quantum systems. Non-diffractive and localized propagations of beams were realized in the Lieb-lattice waveguide arrays supporting a flat band [10,11]. Considering the inevitable dissipation of waveguides and real systems, great attentions have been attracted to the non-Hermitian waveguide arrays, particularly the arrays with passive parity-time (PT) symmetry [12]. The double refraction and power oscillations could occur during the beams propagating in waveguide arrays with passive PT symmetry [13,14]. Distinct from Hermitian systems, the non-orthogonal eigenstates of the non-Hermitian system and their associated eigenvalues can simultaneously coalesce at the exceptional points (EPs) [15]. Propagating in photonic honeycomb lattices with passive PT symmetry, the beams constructed by a Gaussian superposition of a Bloch mode around an EP presented the conical diffractions [16]. The studies of the PT symmetry and EPs in non-Hermitian systems have created new opportunities in ultrasensitive sensing, powerful manipulations of beam dynamics and the modes of lasers and waveguides [17–26].

Here, we address the dynamics of beams initially prepared around an eigenstate of non-Hermitian waveguide arrays unchanging along the propagation direction, because the beams undergo relatively weaker deformation than the beams prepared around a superposed state during their propagations. Remarkably, the beams could also show the interference effects among different eigenmodes [27]. For the coupled waveguides arranged in honeycomb lattice, the rings of 2nd-order EPs (EP rings) can be spawn from the Dirac cones at the corner points of the first Brillouin zone by introducing the dissipation of waveguides [28,29]. During the propagations in the waveguides array, the incident beam prepared in a stable eigenstate outside the EP rings displays a ZB-like COE shift in the lateral direction. Interestingly, a non-oscillating lateral shift appears when the beam is prepared around an eigenstate inside the exceptional ring. Independent from the non-Abelian Berry connections, the amplitude of the COE shift is determined by a U(1)-gauge-invariant non-Hermitian Berry connection, which is no counterpart in the Hermitian systems. With multiple eigenmodes introduced, in coupled waveguides arranged in Lieb lattices supporting 3rd-order EPs, the beams around different initial states exhibit not only multi-frequency ZB-like motions, but also disappeared lateral shifts caused by the destructive interferences. The diverse lateral shifts contradict the physical intuition that the ZB motions only appear when the beams are in a superposed state. It is demonstrated that the non-orthogonality of eigenstate of the non-Hermitian system underlies the lateral shifts.

2. Theoretical modeling

Without loss of generality, the evolutions of photonic beams in the periodic waveguide arrays along the z direction can be described by a Schrödinger-type equation under the tight-binding approximation [8],

In which, Z is the normalized coordinate z by the effective coupling coefficient $\kappa$ between the nearest neighbor waveguides as $Z = \kappa z$. k is the Bloch wave vector. The Hamiltonian ${\boldsymbol{ H}_{\boldsymbol{ k}}}$ and the state vector $|{{\boldsymbol{ u}_{\boldsymbol{ k}}}} \rangle$ are defined by the sub-lattice degrees of freedom. For the honeycomb-lattice waveguide array, the state vector is expressed as by $|{{\boldsymbol{ u}_{\boldsymbol{ k}}}} \rangle = {[\begin{array}{cc} {\chi _{\boldsymbol{ k}}^A}&{\chi _{\boldsymbol{ k}}^B} \end{array}]^T}$. $\chi _{\boldsymbol{ k}}^A$ and $\chi _{\boldsymbol{ k}}^B$ are the mode coefficients of waveguides located at the sublattice A and B, respectively. The COE position of the beam X_C can be obtained by the state average in the first Brillouin zone (FBZ) as $\boldsymbol{X}_C=\int_{\mathrm{FBZ}} d \boldsymbol{k}\left\langle\boldsymbol{u}_{\boldsymbol{k}}\left|i \partial_k\right| \boldsymbol{u}_k\right\rangle / \int_{\mathrm{FBZ}} d \boldsymbol{k}\left\langle\boldsymbol{u}_k \mid \boldsymbol{u}_k\right\rangle$.

In this paper, we only consider Gaussian beams initially prepared around an eigenstate $|{\phi_{{\boldsymbol{ k}_C},n}^R} \rangle$ (it is not the eigenstate of EP) of the non-Hermitian Hamiltonian ${\boldsymbol{ H}_{{\boldsymbol{ k}_C}}}$, namely, $|{{\boldsymbol{ u}_{\boldsymbol{ k}}}(Z = 0)} \rangle = {e^{ - {W^2}{{|{\boldsymbol{ k} - {\boldsymbol{ k}_C}} |}^2}/2}}|{\phi_{{\boldsymbol{ k}_C},n}^R} \rangle$ (W and k_C are the waist radius and central wave vector of the beam, respectively). In the limit of the beams with an infinitely large W, which is equivalent to considering only the contribution of the k_C component to the COE position, X_C can be simply expressed as [27],

3. Honeycomb-lattice waveguide arrays

Figure 1(a) shows our studied honeycomb-lattice waveguide array with two primitive vectors ${\boldsymbol{ a}_1} = {{a\left( { - {\boldsymbol{ e}_x} + \sqrt 3 {\boldsymbol{ e}_y}} \right)} / 2}$ and ${\boldsymbol{ a}_2} = {{a\left( {{\boldsymbol{ e}_x} + \sqrt 3 {\boldsymbol{ e}_y}} \right)} / 2}$ (a is the lattice constant). The waveguides with different losses located at the two sub-lattices are denoted by letters A and B, respectively. In one unit cell, their relative positions are $\boldsymbol{ X}_0^A = \left( {{\boldsymbol{ e}_x} + {{{\boldsymbol{ e}_y}} / {\sqrt 3 }}} \right){a / 2}$ and $\boldsymbol{ X}_0^B = \left( {{\boldsymbol{ e}_x} - {{{\boldsymbol{ e}_y}} / {\sqrt 3 }}} \right){a / 2}$. Under the tight-binding approximation, the Hamiltonian ${\boldsymbol{ H}_{\boldsymbol{ k}}}$ of the honeycomb-lattice array is given by [8]

 

Fig. 1. Schematic illustration of the coupled waveguides array arranged in a honeycomb lattice (a). The propagation loss of the waveguides located at the sub-lattice A is different from that of waveguides at the sub-lattice B. (b) The first Brillouin zone in the reciprocal space of the honeycomb lattice with a corner point K₁=$- $Ke_x, where, K = 4π/(3a). (c), (d) Momentum-space distributions of real (c) and imaginary (d) parts of eigenvalues of the Hamiltonian H_k (defined by Eq. (3) with the parameters χ=0, η=0.1i) show a ring of exceptional points (EP ring) centered at the point K₁.

Download Full Size | PDF

Considering an incident Gaussian beam with k_C= -0.92Ke_x outside the EP ring, both eigenmodes $|{\phi_{{\boldsymbol{ k}_C},1}^R} \rangle$ and $|{\phi_{{\boldsymbol{ k}_C},2}^R} \rangle$ are stable modes with opposite and real propagation constants (Fig.2a-2b). As expected by Eq. (2), the COE of the Gaussian beam with the initial state prepared around $|{\phi_{{\boldsymbol{ k}_C},2}^R} \rangle$ exhibits a ZB-like lateral (y-directional) shifts during their propagations, whose line shape is waist-radius-dependent. A stable ZB emerges only in the lateral shift of the extreme beam with an infinite W (Fig.2c). The identical lateral shift appears when the initial state of the Gaussian beam is prepared around $|{\phi_{{\boldsymbol{ k}_C},1}^R} \rangle$ (not shown in Fig. 2). On the other hand, letting k_C of the incident Gaussian beams inside the exceptional ring with k_C= -0.98Ke_x, the eigenvalues of the Hamiltonian H_k are imaginary (Fig.2a-2b). Two corresponding eigenmodes $|{\phi_{{\boldsymbol{ k}_C},1}^R} \rangle$ and $|{\phi_{{\boldsymbol{ k}_C},2}^R} \rangle$ are exponentially attenuated and amplified in the waveguide array, respectively. Based on Eq. (2), we know that the lateral COE shifts of beams present non-oscillating line shapes, when the initial states of incident Gaussian beams prepared around $|{\phi_{{\boldsymbol{ k}_C},2}^R} \rangle$ (Fig. 2(d)). The relatively larger shift is determined by the non-Hermitian Berry connection. Meanwhile, the incident beams around the attenuation eigenstate $|{\phi_{{\boldsymbol{ k}_C},1}^R} \rangle$ suffer from a relative obvious deforms during the propagation, where the COE and k_C cannot well capture the motions of beams.

 

Fig. 2. Lateral COE shifts of incident Gaussian beams with the initial state prepared around $|{\phi_{{\boldsymbol{ k}_C},2}^R} \rangle$ propagating in the honeycomb-lattice waveguide array with the parameter $\eta = 0.1i$. (a), (b) Details of the real (a) and imaginary (b) parts of two bands near K₁ along the x direction. (c), (d) Lateral COE shifts of beams with the central wave vector k_C equal to -0.92Ke_x (c) and -0.98Ke_x (d), and the incident waist radii W equals to 120/K (denoted by the circulars) and infinity (denoted by the squares), respectively.

Download Full Size | PDF

4. Lieb-lattice waveguide arrays

To further investigate lateral shifts induced by the interferences among the multiple nonorthogonal eigenmodes, we study the propagations of beams in the Lieb-lattice waveguide arrays (Fig.3a-3b). The waveguides with different losses located at three sub-lattices are denoted by letters A, B and C, respectively. Their relative positions in one unit cell are $\boldsymbol{ X}_0^A = {a / 2}{\boldsymbol{ e}_y}$, $\boldsymbol{ X}_0^B = 0$ and $\boldsymbol{ X}_0^C = {a / 2}{\boldsymbol{ e}_x}$ (a is the lattice constant). Under the tight-binding approximation, the Hamiltonian H_k of the Lieb-lattice array is written as

Similar to Eq. (3), ${\chi _B} ={-} {{{\beta ^B}} / \kappa }$, ${\eta _{A(C)}} = {{({\beta ^B} - {\beta ^{A(C)}})} / \kappa }$, ${S_{AB}}\textrm{ = }2\cos ({{k_y}{a / 2}} )$ and ${S_{BC}}=$ $2\cos ({{k_x}{a / 2}} )$. $\kappa$ is the effective coupling coefficient between the nearest-neighbor waveguides in the sub-lattices (A and B) and (B and C). When ${\eta _A} ={-} {\eta _C} = \eta $ and $\boldsymbol{ k} = k({\boldsymbol{ e}_x} + {\boldsymbol{ e}_y})$, three eigenstates of the Hamiltonian ${\boldsymbol{ H}_{\boldsymbol{ k}}}$ in Eq. (4) with the eigenvalues equal to ${\beta _{\boldsymbol{ k}, 1}}\textrm{ = }{\chi _B} - {\beta _{\boldsymbol{ k}}}$, ${\beta _{\boldsymbol{ k},2}} = 0$ and ${\beta _{\boldsymbol{ k},3}} = {\chi _\beta } + {\beta _{\boldsymbol{ k}}}$ (${\beta _{\boldsymbol{ k}}} = \sqrt {{\eta ^2} + 2S_{\boldsymbol{ k}}^2}$, ${S_{\boldsymbol{ k}}} = 2\cos ({{ka} / 2})$), respectively, are given by

 

Fig. 3. Schematic illustration of the coupled waveguides array arranged in a Lieb lattice (a). The waveguides with different propagation losses are located at three sub-lattices A, B, and C, respectively. (b) The first Brillouin zone in the reciprocal space of the Lieb lattice. (c), (d) Momentum-space distributions of real (c) and imaginary (d) parts of eigenvalues of the non-Hermitian Hamiltonian H_k (defined by Eq. (4) with the parameter ${\chi _B} = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\eta _A} ={-} {\eta _C} = 0.3i$) show a 3rd-order EPs located at k^EP ≈ 0.93π/a(e_x+ e_y).

Download Full Size | PDF

We consider the propagations of a Gaussian beam with an initial waist radius W = 80a and the central wave vector k_C= 0.9π/a(e_x+ e_y) near the 3rd-order EP (Fig. 4(a)–4(b)) in the Lieb lattice. The Hamiltonian H_k defined in Eq. (4) has three eigenmodes $|{\phi_{{\boldsymbol{ k}_C},1}^R} \rangle$, $|{\phi_{{\boldsymbol{ k}_C},2}^R} \rangle$ and $|{\phi_{{\boldsymbol{ k}_C},3}^R} \rangle$, whose eigenvalues ${\beta _{{\boldsymbol{ k}_C},1}}$, ${\beta _{{\boldsymbol{ k}_C},2}}$ and ${\beta _{{\boldsymbol{ k}_C},3}}$ are real and equal to -0.325, 0, and 0.325, respectively. Figure 4(c) shows the lateral (-e_x+ e_y direction) COE shifts of the beams with the initial state prepared around the three eigenstates, respectively. They are consistent with those of the extreme beam with an infinite W (denoted by the solid lines in Fig. 4(c)). It means that the contribution of the k_C component given by Eq. (2) plays the leading role in the motion of the beams with a relatively large waist radius.

 

Fig. 4. Lateral COE shifts of incident Gaussian beams prepared around the different initial states propagating in the Lieb-lattice waveguide array with the parameter ${\eta _A} ={-} {\eta _C} = 0.3i$. (a), (b) Details of the real (a) and imaginary (b) parts of three bands near the 3rd-order EP along the e_x+ e_y direction. (c) Lateral shifts of an incident Gaussian beam with the central wave vector k_C= 0.9π/a(e_x+ e_y). The dashed (solid) represent the lateral shifts of beams with the incident waist radii W equals to 80a (infinity) and the initial states prepared around $|{\phi_{{\boldsymbol{ k}_C},1}^R} \rangle$ (denoted by the lower triangles), $|{\phi_{{\boldsymbol{ k}_C},2}^R} \rangle$ (denoted by the circles), and $|{\phi_{{\boldsymbol{ k}_C},3}^R} \rangle$ (denoted by the upper triangles), respectively. (d) The Fourier spectra of the lateral shifts of beams with the initial states prepared around $|{\phi_{{\boldsymbol{ k}_C},1}^R} \rangle$ and the incident waist radii W equals to 80a (dashed line) and infinity (solid line) in the interval of Z from 0 to 250.

Download Full Size | PDF

For the beam with the initial state $|{{\boldsymbol{ u}_0}} \rangle = |{\phi_{{\boldsymbol{ k}_C},1}^R} \rangle$, Eq. (2) is rewritten as

Noting that ${\beta _{{\boldsymbol{ k}_C},1}} ={-} {\beta _{{\boldsymbol{ k}_C},3}}$ and ${\beta _{{\boldsymbol{ k}_C},2}} = 0$, Eq. (5) shows that the lateral ZB-like motion of the beam (the line denoted by the lower triangles in Fig. 4(c)) contains two spatial frequencies (${\beta _{{\boldsymbol{ k}_C},3}}$ and $2{\beta _{{\boldsymbol{ k}_C},3}}$ shown in Fig. 4(d)) generated by interferences between the modes $|{\phi_{{\boldsymbol{ k}_C},1}^R} \rangle $&$|{\phi_{{\boldsymbol{ k}_C},2}^R} \rangle $ and $|{\phi_{{\boldsymbol{ k}_C},1}^R} \rangle $&$|{\phi_{{\boldsymbol{ k}_C},3}^R} \rangle $, respectively. The former plays the leading role in determining the ZB-like motion. It is similar to those of the beam initially around the eigenstate $|{\phi_{{\boldsymbol{ k}_C},3}^R} \rangle$ (the line denoted by the upper triangles in Fig. 4(c)). But, the lateral shifts of the two beams are opposite. Interestingly, due to the destructive interferences among the modes $|{\phi_{{\boldsymbol{ k}_C},2}^R} \rangle $&$|{\phi_{{\boldsymbol{ k}_C},1}^R} \rangle $ and $|{\phi_{{\boldsymbol{ k}_C},2}^R} \rangle $&$|{\phi_{{\boldsymbol{ k}_C},3}^R} \rangle $, the lateral shift of the beam initially around the eigenstate $|{\phi_{{\boldsymbol{ k}_C},2}^R} \rangle$ disappears (the line denoted by the circles in Fig.4a).

5. Robustness of the lateral COE shifts

To study the robustness of the lateral COE shifts, under the tight-binding approximation, we simulate the propagations of Gaussian beams in a finite hexagonal array with 240000 waveguides in total (200 waveguides in each side) arranged in the honeycomb lattice (identical to that of Fig. 1 and 2). The random perturbations are introduced in the effective coupling coefficient between the nearest neighbor waveguides $\kappa$. That is, the coupling coefficient between the mth and nth waveguides (two nearest neighbor waveguides) is expressed as ${\kappa _{mn}} = \kappa ({1 + \delta \cdot rand} )$. The parameter δ indicates the strength of perturbation. For different m and n, rand is a single uniformly distributed random number in the interval (-1,1).

Similar to those of Fig. 2, the incident Gaussian beam with the waist radii W equals to 120/K and central wave-vector k_C, is prepared around the eigenstate $|{\phi_{{\boldsymbol{ k}_C},2}^R} \rangle$ of the undisturbed Hamiltonian ${\boldsymbol{ H}_{{\boldsymbol{ k}_C}}}$ [Eq. (3)]. With the perturbation of coupling coefficient up to 10% (namely, δ=0.1), the beams exhibit the similar ZB-like [ Fig. 5(a)] and non-oscillating [Fig. 5(b)] lateral COE shifts. It means the lateral shifts have a good tolerance to the perturbations of coupling coefficients.

 

Fig. 5. Robustness of the lateral COE shifts under the random perturbations of coupling coefficient between the nearest neighbor waveguides. The incident Gaussian beams with the initial state prepared around $|{\phi_{{\boldsymbol{ k}_C},2}^R} \rangle$ propagate in the finite hexagonal array consisting of 240000 waveguides arranged in the honeycomb lattice ($\eta = 0.1i$) with the randomly perturbed coupling coefficient denoted by the parameter δ. (a), (b) Lateral COE shifts of beams with the central wave vector k_C equal to -0.92Ke_x (a) and -0.98Ke_x (b). The incident waist radii W is equals to 120/K.

Download Full Size | PDF

6. Conclusion

In conclusion, by studying the propagations of the beams in coupled waveguides arranged in passive PT-symmetrical honeycomb lattices with 2nd-order EP rings and Lieb lattices with 3rd-order EPs, we theoretically demonstrate that the beams can exhibit the diverse COE shifts in the lateral direction. Single- and multi-frequencies ZB-like shifts, non-oscillating shifts, and disappeared shifts induced by destructive interferences can appear, when the beams are prepared around different eigenstates near EPs. The relatively obvious shifts arise from the relatively significant overlap of eigenstates in the vicinity of the EPs. It breaks the conventional wisdom in Hermitian realm that only the beam prepared around a superposed state can display non-zero lateral shifts. Hence, the designed band structures by tailoring the loss may provide a new opportunity to manipulate beams in both real space and momentum space. The proposed waveguide arrays can be practically fabricated by the laser-direct writing technology [30,31].