1. INTRODUCTION

The Talbot effect is referred to as the self-imaging effect, in which the wave reproduces the initial transverse periodic distribution periodically along the propagation direction at integer multiples of so-called Talbot distance ${z_T}$. Moreover, when the distances are rational multiples of ${z_T}$, the fractional Talbot effect with modified periods occurs. This near-field diffraction phenomenon was originally discovered for light waves [1,2]; in recent years, it has been extended to matter waves [3], electron beams [4], and x ray [5] and plasmonic systems [6], and it has been widely applied in a variety of fields, including structured illumination [7], optical computing [8], imaging processing [9], and optical testing [10].

Recent research on the optical Talbot effect in discrete structures has been a highlight since its introduction by ${ R}$. Iwanow et al. via considering one-dimensional waveguide arrays [11], in which the guides are usually evanescently coupled by a nearest-neighbor interaction. Light is transferred from waveguide to waveguide through optical tunneling, and the evolution of diffraction dynamics can be described by a set of coupled differential equations with Floquet–Bloch-like solutions. It is shown that the periods of the input fields must belong to 1, 2, 3, 4, 6 for the Talbot effect, which is different from the continuous systems where the Talbot revivals are possible regardless of the pattern period. To date, immense research works have been devoted to the discrete Talbot effect, from one-dimensional to two-dimensional [12], from straight waveguide arrays to helical waveguide arrays [13], from Hermitian systems to non-Hermitian systems [14]. The usual implementation of non-Hermitian Hamiltonians is realized through the use of space-dependent gain and loss, where the system commonly consists of two waveguides: the first with gain and the other with equivalent loss. It is assumed to be a parity–time (PT) symmetric dimer model [15,16]. It is demonstrated that Talbot effects below and at the PT transition threshold are possible for a finite set of periodicities $N = {1}$, 2, 3 and 1, 3, respectively.

Here, we consider a nonreciprocal Su–Schrieffer–Heeger (SSH)-like lattice model, as shown schematically in Fig. 1(a), where the source of non-Hermiticity derives not from gain and loss but from anisotropic coupling [17–19]. This dimer model has attracted huge attention because it displays a so-called non-Hermitian skin effect. Related models are relevant to some experimental platforms, such as pulse trains [20], electronic circuits [21,22], and ring resonators [23]. Both theoretical analysis and numerical simulation show that this lattice model can also support the Talbot self-imaging effect. The permitted input periods allowing the appearance of Talbot revivals are presented. In one-dimensional reciprocal (Hermitian) dimer lattices, the Talbot effect is possible when the periods of the input fields belong to 1, 2, 3, 4. For the nonreciprocal (non-Hermitian) case, the propagation constants at the edges and center of the Brillouin zone are real, and the permitted input periods are 1 and 2. Moreover, total energy of the field exhibits oscillatory behavior in the Talbot self-imaging process. When the periodic $N$ of the input pattern is low, the coupling coefficient can take continuous values, and the Talbot distance can also be manipulated in a wide range of coupling coefficients. But it only takes some discrete points when the periodic of the input pattern is large. Our findings may be directly extended to one-dimensional discrete trimer and tetramer optical systems.

 

Fig. 1. Geometry of one-dimensional discrete dimer lattice. Red (blue) circles are sites on sublattice $A\;(B)$. The dotted box indicates the unit cell. (a) and (b) represent the nonreciprocal dimer models with both intercell and intracell anisotropic couplings and only intercell anisotropic coupling, respectively. (c) is the SSH model. (d) is a simplified model of (a).

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2. THEORETICAL MODEL

We first construct a dimer lattice that consists of two types of sublattices, denoted as $A$ and $B$, respectively, as shown in Fig. 1(a). ${t_1} \pm {\gamma _2}$, ${t_2} \pm {\gamma _1}$ are the nonreciprocal intercell and intracell coupling coefficients, respectively, where the coupling strengths are anisotropic as the hopping from one-site to its neighbor is different from the hopping from its neighbor back to itself. This anisotropic, characterized by parameters ${\gamma _1}$ and ${\gamma _2}$, leads to a non-Hermiticity, even though there is neither gain nor loss in the model.

According to the coupled-mode theory, the evolution of field amplitudes ${A_n}$, ${B_n}$ at the sublattices $A$ and $B$ sites of $n$th unit cell can be described by the following coupled equations:

For an infinite discrete lattice, Eq. (1) admits the periodic solution in the form of Floquet–Bloch-like wave ${A_n} = a\exp (ink)\exp (i\lambda z)$ and ${B_n} = b\exp (ink)\exp (i\lambda z)$, where $a$ and $b$ are the eigenvectors, and $k$ and $\lambda$ are the transverse Bloch momentum and propagation constant, respectively. Substituting these expressions into Eq. (1), the following dispersion relation is obtained:

When ${\gamma _2} = 0$, the model is simplified as the nonreciprocal dimer lattice with intercell anisotropic coupling [Fig. 1(b)], and the corresponding propagation constants can be expressed as

If one further takes ${\gamma _1} = 0$, our model reduces to the widely studied SSH model [24], as shown in Fig. 1(c), and the dispersion relation is simplified as

Obviously, the gap closes at exceptional points ${t_1} + {t_2} = \pm ({{\gamma _1} - {\gamma _2}})/2$ and ${t_1} - {t_2} = \pm ({{\gamma _1} + {\gamma _2}})/2$ at $k = {0}$ and $\pi$, respectively. For the reciprocal lattice, the band structures are completely real. A bandgap with gap width equaling $|{t_1} - {t_2}|$ opens and closes at ${t_1} = {t_2}$. In the nonreciprocal case, the band structure is partially complex, as shown in Figs. 2(a)–2(f), and the imaginary part of the propagation constants is nonzero except the edges and center of the Brillouin zone. In addition, the band structures are symmetric around $\lambda = 0$.

 

Fig. 2. (a) Real and (b) imaginary parts of the spectrum as a function of ${d_1} = {\gamma _1}/{t_2}$ in nonreciprocal lattices with only intercell anisotropic coupling for ${t_2} = \pi /2$, $c = {t_1}/{t_2} = 0.5$, ${d_2} = {\gamma _2}/{t_2} = 0$. (c) and (d) show the calculated real and imaginary parts of the spectrum in nonreciprocal lattice with both intercell and intracell anisotropic couplings, respectively, as ${t_2} = \pi /2$, $c = {d_1} = 0.5$.

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We consider the necessary condition that guarantees the existence of the Talbot effect in these discrete dimer lattices. In order to realize the Talbot self-imaging effect, the transverse field distribution of the input wave must be periodic. To meet the periodic boundary condition, the transverse Bloch momentum should satisfy [11–14,25]

 

Fig. 3. Talbot effect in a reciprocal dimer lattice for (a) $N = 1$ and (b) $N = 2$. (a1) Talbot carpet with the input $({a_0},{b_0}) = (1,1)$ at $c = 0.5$. (a2) and (a3) show Talbot carpets with the input $({a_0},{b_0}) = (1,0)$ at $c = 0.5$ and 1.5, respectively. (b1) Talbot carpet with the input $({a_0},{b_0}) =(1,1)$ at $c = 1.5$. (b2) and (b3) show Talbot carpet with the input $({a_0},{b_0}) = (1,0)$ at $c = 1.5$ and 2, respectively. White dashed lines denote the recurrent locations predicted by theoretical analysis. (a4), (b4) Talbot distance ${z_T}$ versus $c$.

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3. RESULTS AND DISCUSSION

Let us first consider the Talbot self-imaging effect in one-dimensional reciprocal dimer lattices [Fig. 1(c)]. Due to the completely real band structure, the Talbot effect exists for different period input patterns. Since we are interested in the influence of relative coupling strength between intracell and intercell hopping on light evolution, we vary ${t_1}$ and fix ${t_2} = \pi /2$ in the following discussion. As is well known, evanescent field coupling between waveguides can be suitably controlled by adjusting the separation between waveguides. For ${t_1} = {t_2}$, Eq. (6) takes the simple condition cos $(l\pi /N) = \alpha /\beta$ [11], which can be described in terms of Chebyshev polynomials. It turns out that $\cos (l\pi /N)$ is rational if $\cos (\pi /N)$ is also rational. Based on the rational root theorem, it is shown that the roots of this polynomial in cos $(\pi /N)$ are rational if $N = 1$, 2, 3, 4, and 6. As a result, the possible input periodicities to realize the Talbot effect should belong to $\{{1},{2},{3},{4},{6}\}$.

For $N = 1$, only the mode with $k = 0$ is included, the corresponding propagation constants are $\lambda = \pm ({t_1} + {t_2})$, which lead with Eqs. (4)–(7) to a rational value $\alpha /\beta = 1$, and the Talbot distance is ${z_T} = \pi /({t_1} + {t_2})$. The case of $({a_0},{b_0}) = (1,1)$ represents the trivial case of a discrete plane wave solution [11,12] shown in Fig. 3(a1). Figures 3(a2) and 3(a3) illustrate the periodic intensity revivals at $c = {t_1}/{t_2} = 0.5$ and 1.5 for the initial input $({a_{n0}},{b_{n0}}) = (1,0)$, and the input patterns reappear every ${z_T} = 4/3$ and 0.8, respectively, where the numerical simulations completely agree with the analytical results. Light propagation in this lattice strongly depends on the ratio between ${t_2}$ and ${t_1}$. The dependence of ${z_T}$ on $c$ is depicted in Fig. 3(a4). It is shown that the Talbot distance decreases monotonically with $c$ increasing. In addition, the fractional Talbot effect can be easily observed at $z = (1/2 + m^\prime){z_T}$ with $m^\prime $ being the positive integer, where the self-imaging distance is the same as that in the integer case with half a period lateral shift with respect to the input. Moreover, the image at $z = {z_T}/4$, where the period of transverse field distribution is halved, is also repeated along the propagation direction.

 

Fig. 4. Talbot carpet supported by the reciprocal dimer lattice for (a)  $N = 3$  and (b) $N = 4$. (a1) Talbot carpet with $({a_0},{b_0}) = (1,1)$ and $c = 1.6$. (a2) and (a3) are Talbot carpets with $({a_0},{b_0}) = (1,0)$ at $c = 1.6$ and 4.2, respectively. (b1) Talbot carpet with $({a_0},{b_0}) = (1,1)$ and $c = 4/3$. (b2) and (b3) are Talbot carpets with $({a_0},{b_0}) = (1,0)$ at $c = 4/3$ and 2.4, respectively. (a4), (b4) Talbot distance ${z_T}$ as a function of $c$.

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For $N = 2$, one gets $l = {0}$ and 1, and the Bloch momentum takes values of $k = 0$ and $\pi$. Talbot revivals are possible provided that $c = (1 - \alpha /\beta)/(1 + \alpha /\beta)$ and the corresponding propagation constants are ${\lambda _0} = \pm ({t_1} + {t_2})$ and ${\lambda _1} = \pm ({t_1} - {t_2})$. The Talbot distance is, thus, LCM $(\pi /{t_1},\pi /{t_2})$. To verify this theoretical prediction, we first give the analytical solutions in this case of $N = 2$, where two adjacent unit cells including four waveguides (denoted by ${A_n}$, ${B_n}$, ${A_{n + 1}}$, and ${B_{n + 1}}$) are considered because of the transverse periodicity. The analytical solutions of the output intensity of these four waveguides read $|{A_n}{|^2} = \mathop {\cos}\nolimits^2 ({t_1}z) {\cos} ^2 ({t_2}z)$, $|{B_n}{|^2} = {\sin}^2 ({t_1}z) {\cos}^2 ({t_2}z)$, $|{A_{n + 1}}{|^2} = {\sin}^2 ({t_1}z) {\sin}^2 ({t_2}z)$, and $|{B_{n + 1}}{|^2} = {\cos}^2 ({t_1}z) {\sin}^2 ({t_2}z)$ for the initial pattern (${a_{n0}}$, ${b_{n0}}$, ${a_{(n + 1)0}}$, ${b_{(n + 1)0}}) = (1,0,0,0)$. Two typical Talbot carpets with $c = 1.5$ ($\alpha /\beta = 1/4$) and 2 ($\alpha /\beta = 1/3$) are displayed in Figs. 3(b2) and 3(b3), and the corresponding Talbot distance as a function of $c$ is plotted in Fig. 3(b4), which indicates that ${z_T}$ varies periodically. When $c$ is the integer, Talbot distance reaches a minimum ${z_T} = \pi /{t_2}$. The closer to the integer, the longer the Talbot distance is. For the case, (${a_{n0}},{b_{n0}},{a_{(n + 1)0}},{b_{(n + 1)0}}) = (1,1,0,0)$, it is deduced that the output intensities are $|{A_n}{|^2} = |{B_n}{|^2} = {\cos}^2 ({t_2}z)$ and $|{A_{n + 1}}{|^2} = |{B_{n + 1}}{|^2} = {\sin}^2 ({t_2}z)$, as shown in Fig. 3(b1). Talbot distance remains constant ${z_T} = 2$ for different values of $c$ and only depends on ${t_2}$, where the discrete dimer system can be reduced to the widely studied homogeneous lattice shown in Fig. 1(d) with the coupling coefficient ${t_2}$, and the Talbot distance is easily written as ${z_T} = \pi /{t_2}$. In this case, the fractional Talbot effect can be also recoded as shown in Fig. 3(b1).

We change the input period to be $N = 3$, and Bloch momentum can take three values $k = {0}$, $2\pi /3$, and $4\pi /3$. Eq. (6) is satisfied provided that $c = [(2{\alpha ^2} + {\beta ^2}) + \beta (12{\alpha ^2} - 3{\beta ^2})^{1/2}]/[2({\beta ^2} - {\alpha ^2})]$, where $\alpha /\beta \ge {1/2}$. The corresponding propagation constants ${\lambda _0} = \pm ({t_1} + {t_2})$ and ${\lambda _2} = {\lambda _1} =\def\LDeqbreak{} \pm (t_1^2 + t_2^2 - {t_1}{t_2})^{1/2}$ are calculated. Thus, we can easily get the expression of the Talbot distance ${z_T} = {\rm LCM}\{2\pi /[{t_1} + {t_2} + {(t_1^2 + t_2^2 - {t_1}{t_2})^{1/2}}],2\pi /[{t_1} + {t_2} - (t_1^2 + t_2^2 - {t_1}{t_2})^{1/2}],\pi /(t_1^2 \def\LDeqbreak{}+ t_2^2 - {t_1}{t_2})^{1/2}\}$ plotted in Fig. 4(a4), which shows that, to achieve the Talbot self-imaging effect during propagation, the relative coupling coefficient $c$ cannot take continuous values but some discrete points. For example, the Talbot distances are ${z_T} = 4$, 10, 20, 50, and 100 at $c = 1(\alpha /\beta = 1/2)$, 1.6 ($\alpha /\beta = {7/13}$), 4.2 ($\alpha /\beta = 19/26$), 168/25 ($\alpha /\beta = 157/193$), and 481/25 ($\alpha /\beta = 469/506$), respectively. Three examples of Talbot carpets with different input conditions are displayed in Figs. 4(a1)–4(a3), which confirm our theoretical predictions.

In detail, we choose the value of $N = 4$, and the modes with $k = 0$, $\pi /2$, $\pi$, and $3\pi /2$ are included in the field evolution. According to Eq. (4), one obtains ${\lambda _0} = \pm ({t_1} + {t_2})$, ${\lambda _1} = {\lambda _3} = \pm {(t_1^2 + t_2^2)^{1/2}}$, and ${\lambda _2} = \pm ({t_1} - {t_2})$. The Talbot distance is given by ${z_T } = {\rm LCM} \{2\pi /[{t_1} + {t_2} + {(t_1^2 + t_2^2)^{1/2}}]$, $2\pi /[{t_1} + {t_2} - {(t_1^2 + t_2^2)^{1/2}}],\pi /{t_1},\pi /{t_2}\}$. The Talbot effect exits only when $c$ equals some discrete values, as displayed in Fig. 4(b4), which shows that the Talbot effect can be observed at six different positions for the propagation distance less than 100. The Talbot carpets at $c = 4/3$ ($\alpha /\beta = {\lambda _2}/{\lambda _0} = 1/7$) and 2.4 ($\alpha /\beta = {\lambda _2}/{\lambda _0} = 7/17$) are illustrated in Figs. 4(b1)–4(b3), and the corresponding Talbot distances are ${z_T} = 6$, 6 and 10, respectively. For $N = 6$, Talbot revivals predicated by Eq. (7) are not supported when the propagation distance is smaller than 1000. From a practical point of view, the Talbot effect does not exist if the period of the input pattern is chosen as $N = 6$.

We are now concerned about the Talbot effect in one-dimensional dimer lattices with intercell anisotropic coupling (${\gamma _1} \ne 0$ and ${\gamma _2} =0$). To realize the strict Talbot effect, the dispersion spectrum given by Eq. (3) should be real, and the relation ${t_1}{\gamma _1}/2 \ge 0$ is also satisfied. Although the band structure is partially complex at ${\gamma _1} \ne 0$ and ${\gamma _2} = 0$, the propagation constants at the edges and center of the Brillouin zone are real [see Figs. 2(a) and 2(b)]. If the modes have propagation constants with imaginary parts, the input field distribution would experience amplification or attenuation and is impossible to recover during propagation. As a result, the Talbot effect only occurs at $k = 0$ and $\pi$. Based on Eqs. (3) and (5), it is shown that the permitted input periods allowing Talbot revivals in dimer lattices with intercell anisotropic coupling are $N = 1$ and 2. For other values of $N$, there would be some modes with $k \ne 0,\pi$, the imaginary parts of their propagation constants are nonzero, and the Talbot effect is not realized.

The case $N = 1$ corresponds to $\lambda = \pm [({t_1} + {t_2})^2 \,-\def\LDeqbreak{} \gamma _1^2/4]^{1/2}$, and one obtains the Talbot distance ${z_T} = \pi /[({t_1} + {t_2}{)^2} - \gamma _1^2{/4]^{1/2}}$. Three examples of quantum recurrence are shown in Figs. 5(a1)–5(a3) with same lattice parameters ${t_1} = \pi /4$ and ${t_2} = \pi /2$ but for different initial conditions. The onset of self-imaging is clearly observed with the Talbot distance ${z_T} = 1.345$ and 1.414 at ${d_1} = 0.4$ and 1, where the parameter ${d_1}$ represents the ratio ${\gamma _1}/{t_2}$. Unlike the Talbot effect in reciprocal lattices, where the self-imaging with conserved total energy of the central part of the model occurs during propagation, here the total energy of the field is not constant and exhibits oscillatory behavior. However, the periodic optical field pattern indeed self-reproduces itself at constant intervals. Furthermore, the optical field experiences various degrees of amplification at different propagation distances, which can be adjusted by changing ${\gamma _2}$. Figure 5(a4) depicts the dependence of ${z_T}$ versus ${d_1}$. As ${d_1}$ is increased, ${z_T}$ is longer, and the Talbot distance reaches a maximum at ${\gamma _1} = 2{t_1}$. Similar to that in the reciprocal dimer lattice, fractional Talbot revivals can occur except for the reduced intensity of the bright waveguides due to anisotropic coupling.

 

Fig. 5. Talbot effect in a nonreciprocal dimer lattice with intercell anisotropic coupling for (a) $N = 1$  and (b) $N = 2$. (a1) shows the Talbot intensity carpet with $({a_0},{b_0}) = (1,1)$ and ${d_1} = 0.4$. (a2) and (a3) show Talbot revivals for the input $({a_0},{b_0}) = (1,0)$ at ${d_1} = 0.4$ and 1, respectively. (a4) Talbot distance ${z_T}$ versus ${d_1}$. Other system parameters are ${t_1} = \pi /4$ and ${t_2} = \pi /2$ in (a). (b1) Talbot carpet with the input $({a_0},{b_0}) = (1,1)$ at $c = 3$. Talbot carpet with the input $({a_0},{b_0}) = (1,0)$ at $c =0.3$ and 0.48 are shown in (b2) and (b3), respectively. (b4) Talbot distance ${z_T}$ versus $c$.

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Fig. 6. Talbot effect in a nonreciprocal dimer lattice with both intercell and intracell anisotropic couplings for (a) $N = 1$  and (b) $N = 2$. (a1) Talbot intensity carpet with $({a_0},{b_0}) = (1,1)$ and ${d_2} = - 1.5$. (a2) and (a3) show Talbot revivals for the input $({a_0},{b_0}) = (1,0)$ at ${d_2} = -1.5$ and 0.5, respectively. (b1) Talbot carpet with the input $({a_0},{b_0}) = (1,1)$ at ${d_2} = -1.3$. Talbot carpet with the input $({a_0},{b_0}) = (1,0)$ at ${d_2} = -1.3$ and ${-}{0.25}$ are shown in (b2) and (b3), respectively. (a4), (b4) Talbot distance ${z_T}$ versus $c$. Other system parameters are ${t_1} = \pi /4$, ${t_2} = \pi /2$, and ${\gamma _1} = \pi /4$.

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For case of $N =2$, the corresponding propagation constants are ${\lambda _0} = \pm {[({t_1} + {t_2}{)^2} - \gamma _1^2/4]^{1/2}}$ and ${\lambda _1} = \pm [({t_1} - {t_2})^2 \,-\def\LDeqbreak{} \gamma _1^2/4]^{1/2}$. The available Talbot distance is LCM $\{2\pi \{[({t_1} + {t_2})^2 - \gamma _1^2/4]^{1/2} + [({t_1} - {t_2})^2 - \gamma _1^2/4]^{1/2}\},\;2\pi /[({t_1} + {t_2})^2 \,- \def\LDeqbreak{}\gamma _1^2/4]^{1/2} - [({t_1} - {t_2})^2 - \gamma _1^2/4]^{1/2}\}$. The Talbot effect may exist under the condition ${t_1} = {\gamma _1}/2$, which corresponds to the completely inequal nonreciprocal dimer with the weak intercell coupling coefficient being 0. The above expression reduces to ${z_T} = {\rm LCM}\{2\pi / [(t_2^2 + 2{t_1}{t_2})^{1/2} + (t_2^2 - 2{t_1}{t_2})^{1/2}], 2\pi / [(t_2^2 + 2{t_1}{t_2}{)^{1/2}} - (t_2^2 - 2{t_1}{t_2})^{1/2}] \}$, as shown in Fig. 5(b4), which is consistent with numerical results [Figs. 5(b1)–5(b3)]. Equation (6) is satisfied provided that $c = (1 - {\alpha ^2}/{\beta ^2})/[2(1 + {\alpha ^2}/{\beta ^2})]$. The corresponding Talbot distances are ${z_T} = 6.325$ and 10 at $c = 0.3$ ($\alpha /\beta = 1/2$) and 0.48 ($\alpha /\beta = 1/7$), respectively. The total intensity oscillations are also presented in the Talbot self-imaging process.

The Talbot effect can appear in the dimer lattice with both intercell and intracell anisotropic couplings. The permitted input periods allowing the Talbot recurrences and the corresponding Talbot distances can be obtained, which are dependent on the lattice parameters ${t_1}$, ${t_2}$, ${\gamma _1}$, and ${\gamma _2}$. Specifically, we fixed ${t_1} = \pi /4$, ${t_2} = \pi /2$, and ${\gamma _1} = \pi /4$ and investigated the evolution of the periodic pattern for different intracell anisotropic couplings ${\gamma _2}$. In this situation, the imaginary part of the propagation constant is always zero at the edges and center of the Brillouin zone, as shown in Figs. 2(c) and 2(d). As a result, the permitted input periods allowing the Talbot effect are also shown to be $N = 1$ and 2. The Talbot distance is ${z_T} = \pi {[({t_1} + {t_2}{)^2} - ({\gamma _1} - {\gamma _2})/4]^{- 1/2}}$ for $N = 4$, as shown in Fig. 6(a4). For $N = 2$, the corresponding propagation constants are ${\gamma _0} = \pm {[({t_1} + {t_2}{)^2} - ({\gamma _1} - {\gamma _2})/4]^{1/2}}$ and ${\gamma _1} = \pm {[({t_1} - {t_2}{)^2} - ({\gamma _1} + {\gamma _2})/4]^{1/2}}$, and the corresponding Talbot distance is obtained from Eq. (6), as shown in Fig. 6(b4), where ${d_2} = {\gamma _2}/{t_2}$. Some representative intensity revivals for $N = 1$ and 2 are shown in Figs. 6(a1)–6(a3), 6(b1)–6(b3), which confirm the above theoretical analysis. The total intensity oscillations are also observed in the Talbot process for the nonreciprocal dimer lattices with both intercell and intracell anisotropic couplings.

Finally, several issues merit discussion. First, the robustness of the Talbot effect against model imperfection is demonstrated by the numerical simulations. One finds that, after introducing small positional deformation (up to 5% in coupling coefficients), the slight distortion of the Talbot carpet is presented, and the Talbot distance mainly remains unchanged in moderate propagation distance, as shown in Fig. 7. Obviously, Talbot revivals with a shorter recurrence distance show stronger robustness against perturbation. Second, we suggest possible experimental implementations of the one-dimensional discrete dimer lattice for the Talbot effect observation. For the reciprocal dimer model, since the evanescent field coupling between waveguides can be suitably controlled by adjusting the separation between waveguides, the relative coupling strength can be precisely controlled by using the lithographic technique. The nonreciprocal model may be realized by employing a ring resonator with simultaneous amplitude and phase modulation [21]. Lastly, the above method can be extended to other photonic superlattices with an arbitrary number of sites in each unit cell. Through enumerating all available input periods permitted, the Talbot self-imaging effect can be obtained, and the self-imaging process can be flexibly controlled by changing the relative coupling strengths between different sublattices.

 

Fig. 7. (a1)–(a3) Talbot effect in the presence of structural imperfection under the action of 5% small positional deformation to the intercell coupling ${t_1}$, intercell anisotropic coupling ${\gamma _1}$, and intracell anisotropic coupling ${\gamma _2}$, and other parameters are the same as in Figs. 4(b1), 5(b1), and 6(b3), respectively. Corresponding intensity evolutions for a single waveguide indicated by red arrows are shown in (b1)–(b3), respectively.

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4. CONCLUSION

In summary, we have demonstrated that the Talbot self-imaging of light patterns can be effectively realized in both one-dimensional reciprocal (Hermitian) and nonreciprocal (non-Hermitian) discrete dimer lattices. Our theoretical analysis illustrates that the periods of the input pattern must belong to $\{{1},\;{2},\;{3},\;{4}\}$ in the reciprocal model. For nonreciprocal discrete dimer lattices, the band structure is partially complex, and the propagation constants at the edges and center of the Brillouin zone are real. As a result, the permitted input periods allowing Talbot revivals in dimer lattices with anisotropic coupling are $N = 1$ and 2. Furthermore, the total energy of the field is not constant and exhibits oscillatory behavior in the Talbot process for the nonreciprocal case. The Talbot distance is governed by the characteristics of the lattice, thus allowing one to manipulate the self-imaging process. When the periodic of the input pattern is low ($N = 1$, 2 for reciprocal case and $N = 1$ for nonreciprocal case), coupling strength can take continuous values, and the Talbot distance can be manipulated in a wide range of coupling coefficients. But it only takes some discrete points when the periodic of the input pattern is large. The analytical solutions are verified by the simulated results. Our results can be extended to other photonic superlattices with an arbitrary number of sites in each unit cell and may provide some potential applications in optical imaging, optical array illumination, and optical measurement.