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Abstract
We investigate the discrete Talbot self-imaging effect in Floquet superlattices based on a mesh of directional couplers with periodically varying separation between waveguides, both theoretically and numerically. The modulated discreteness of the lattices sets strong constraints to ensure the Talbot effect generation. We show that discrete Talbot effect occurs only if the incident periods are N = 1, 2, and 4 in dispersive regimes of the Hermitian superlattices. In both dynamic localized and rectification regimes, self-imaging effect can occur for arbitrary input period N. For the rectification case, Talbot distance equals the input period. In the regime of dynamical localization, the Talbot distance remains unchanged irrespective of the pattern period. For non-Hermitian Floquet superlattices, due to the non-zero imaginary part of quasi-energy spectrum arising at the center of the Brillouin zone, where the mode degeneracy occurs, Talbot revival is not preserved when the input period is an even number, and exists only as N = 1 in the dispersive regime. The theoretical calculations and numerical simulations verify each other completely.
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1. Introduction
Discretized light propagation in photonic lattices has attracted a lot of interest from both fundamental and applied viewpoints [1]. A wide variety of noteworthy effects have been identified in both linear and nonlinear light propagation [2]. Engineered this periodic structure offers the possibility of tailoring diffraction and molding the flow of the discrete light in spatial [3–5], frequency [6] and temporal domains [7]. Several types of lattice modulation including periodic axis bending, width modulation [3], refractive index modulation [4] and phase modulation [5] have been used to obtain longitudinally and transversely modulated photonic lattices. The importance of periodic driving as a powerful method to engineer band structures has been recognized in condensed matter physics [8–10] and photonics [11–18]. In particular, periodically longitudinal modulation can supply novel light dynamic behaviors, such as diffraction management in zigzag arrays [11], dynamic localization in periodically curved arrays [12], inhibition of tunneling [13], Bloch oscillation [14,15], resonant mode oscillation [16], Talbot effect [17] and topological effect in helical waveguide arrays [18]. Although, there is no static eigenmodes in these modulational photonic lattices, Floquet band structure and quasi-energy can be computed via Floquet theory to illustrate its spectrum. The above-mentioned modulated lattices have identical coupling strength between all waveguides or follow longitudinally sinusoidal variation. Recently, a new kind of Floquet sublattices based on a mesh of directional couplers with periodically varying separation between waveguides was proposed [19,20]. For appropriate system parameters, one can achieve either beam rectification without diffractive broadening or dynamic localization [19–21]. In addition, the edge state stemming from its truncation and longitudinal modulation was also demonstrated [22]. But, the dynamically discrete behaviors of periodic optical wave in modulated Floquet superlattices have not reported yet to the best of our knowledge. It has been shown that the periodic optical field pattern reappears in the straight waveguide arrays when the incident periods are 1, 2, 3, 4 and 6 [23,24]. This phenomenon is called discrete Talbot effect, which is different from that in continuous systems where the Talbot revivals can occur irrespective of the input pattern period [25,26]. Discrete Talbot effect have also been studied in helical waveguide arrays [17] and the PT-symmetric straight arrays composed of coupled dimers [27]. Moreover, investigation on discrete Talbot effect has been extended to synthetic temporal mesh lattices based on two coupled fiber loops with slightly different length [7]. Utilizing the time-multiplexing technique, a pulse spreads on this lattice with discrete arrival times being equivalent to positions in the spatial space. It is shown that the Talbot revivals occur as the period of input pulse train is chosen as one-, two- or fourfold compared to the time interval corresponding to the length difference of the two loops [28]. When the PT symmetry is imposed by temporally alternating gain and loss of the loops, PT -symmetric Talbot effect in a temporal mesh lattice can also exists when the input period are 2 and 4 folds compared to the time interval [29].
In this paper, we theoretically study the dynamically discrete behaviors of the self-imaging periodic optical wave in modulated Floquet superlattices based on a mesh of directional couplers with periodically varying separation between waveguides. A number of possibilities for efficient diffraction control can be achieved in such Floquet superlattices by adjusting the longitudinal period and coupling strength. We show that Talbot effect can occur in both Hermitian and non-Hermitian superlattices. Some unique properties of Talbot revivals are shown in the dynamic localized, rectification and dispersive regimes. Because of the nonorthogonality of the associated Floquet Bloch modes supported by the physical nonreciprocity and discreteness of the lattice, discrete Talbot effects are achieved only if the incident periods are N = 1, 2, 4 and N = 1 in dispersive regimes of the Hermitian and non-Hermitian superlattices, respectively.
2. Theoretical model
Our modulated system, based on a non-Hermitian extension of Su-Schrieffer-Heeger (SSH) model [30], consists of a mesh of directional couplers with periodically varying separation between waveguides, see Fig. 1(a). The superlattice with nearest-neighbor coupling comprises two sublattices A and B, which have a gain and loss nature, respectively. The coupling strength between nearest neighbors belonging to different sublattices is dynamically modulated in a step-like fashion. In the first longitudinal half-period of the structure, the waveguides with equal (n, n) indices are coupled with coupling strength c1, while the coupling between different (n, n+1) indices from two sublattices is suppressed, as shown in Fig. 1(b). The latter half-period is a reverse process of the first half-period, where the strength of the evanescent coupling between waveguides with equal (n, n) indices is negligible, whereas, the channels with different (n, n+1) indices from two sublattices interact with coupling strengths c2 [see Fig. 1(c)]. The photonic implementation is conducted using an array of coupled waveguides on a silicon, and can be fabricated using the method of electron beam lithography and the dry etching process, which is followed by the lift-off process to deposit the Cr stripes. The coupling between waveguides can be controlled by the separation between waveguide and the onsite damping can be implemented by Cr depositions on top of waveguides with a uniform thickness [31,32].
Fig. 1. (a) Schematic of a Floquet superlattice composed from sublattice A (gain waveguide) and B (lossy waveguide). Each dimer is distinguished by the index n. The couplings between different sublattices in first and second longitudinal half-periods are shown in (b) and (c), respectively.
Based on the tight-binding method with the nearest-neighbor approximation adopted, it is possible to simplify the description of the propagation of light in terms of coupled mode equations [22]:
Owing to the longitudinal periodicity of the Hamiltonian H(k, z + T) = H(k, z),where k is the transverse Bloch momentum, one can apply the concepts of Floquet band structure and quasi-energy ɛ(k) to illustrate its spectrum, which can be determined from an effective tight-binding Hamiltonian Heff, given by U(k, T) = exp(-iHeff T), U(k, z) is the evolution operator of the modulated superlattices and is defined as U(k, T)φ(k, 0) = φ(k, T), where φ(k, z) is the Floquet eigenstate of the structure. We use the Floquet-Bloch theory to write the field amplitudes on sites of sublattices A and B in the forms: An = aexp(i2nd0k) and Bn = bexp(i(2n+1)d0k), where 2d0 is the transverse distance between two adjacent sublattices of the same type. Substituting these expressions into Eq. (1) yields the following coupled equations:
The corresponding evolution operator of the dynamical modulation superlattices over one longitudinal period can be described as U(T) = exp(-iH1T/2)exp(-iH2T/2), where the Hamiltonians on the first and second half-periods read
3. Quasi-energy spectrum and discrete Talbot self-imaging effect in modulated Hermitian superlattices
Let us start from considering the Hermitian limit, i.e., assuming zero losses ρ=0 in all waveguides, the quasi-energy ɛ(k) (dispersion relation) spectrum of the band structure can be obtained in the form of (see Appendix)
From Eq. (4), it is clear that the energy band is a periodic function of k with a period π/d0, and of coupling strength c0 with a period Lcm [4(1+σ)-1π/T, 4(1-σ)-1π/T], where Lcm is the least common multiple. It is indicated that the band structure can be tuned through controlling the parameters of wave number k, the coupling strength c0 and parameter σ, as shown in Figs. 2(a)-(f). Quasi-energy is periodic in the coupling strength c0, the corresponding periods are 2π/T, 8π/T and 6π/T for σ=1, 1/2 and 1/3, respectively [Figs. 2(a)–2(c)]. Equation (4) also indicates that flat band appears at c0 = 2mπ/T and σc0 = 2mπ/T with m being an integer, which would support compact localized modes. If both the relations c0 = (2m+1)π/T and σc0 = (2m+1)π/T are simultaneously satisfied, the dependence ɛ(k) becomes linear [see black lines in Figs. 2(d) and 2(f)], the group velocity vg depending on the first derivative of the quasi-energy ɛ(k) is vg = ±2d0/T. This indicates that incident wave propagates at a fixed angle across the structure without diffractive broadening. This is the rectification regime. For other values of c0, the quasi-energy band is dispersive. When σ = 1, there are rectification and dynamical localization regimes at c0 = π/T and 2π/T, respectively [Fig. 2(d)], and the quasi-energy band is dispersive at other parts of the spectrum. Whereas for σ = 1/2, flat bands are shown at c0 = 2mπ/T with m being an integer and no rectification regime is observed, as illustrated in Fig. 2(e). The quasi-energy band is also dispersive at other coupling strengths. Let us fix σ=1/3, the spectrum exhibits a flat band pinned at zero energy at c0 = 6π/T, another two symmetric flat bands ɛ = ±2π/(3 T) are displayed at c0 = 2π/T and 4π/T, as seen in Fig. 2(f), which correspond to the dynamical localization regimes. To verify the above analysis, the wave transmission of a single waveguide excitation is investigated by directly solving Eq. (1), using MATLAB. At c0 = 2π/T and 4π/T, the excitation completely restores the input field at multiples of the period 1/σ = 3 [ Fig. 3(b)]. It is necessary to note that higher order localization modes appear for stronger coupling strength, light field distribution is restored after one longitudinal period at c0 = 6π/T with zero energy [Fig. 3(d)]. For c0 = 3π/T corresponding to the rectification regime, one finds the linear dispersion relation leading to directional transport across the lattice without diffraction [Fig. 3(c)], the transport direction depends on which sublattice is excited. For other values of coupling strength, the quasi-energy band is dispersive and would give rise to various diffraction properties in the structure [Fig. 3(a)]. A further increase in the coupling strength, band structure and optical propagation process will be repeated again except for rich structure of higher-order rectification and localization modes, which show an odd or even number of switching event during each semi-period of the structure [21].
Fig. 2. (a)-(c) Quasi-energy spectrum of a modulated Hermitian superlattice as a function of c0 and k at σ = 1, 1/2, and 1/3, respectively. (d)-(f) Corresponding band structures versus k for different coupling coefficient c0. Other parameters are d0 = 1 and T = 1.
Fig. 3. Light dynamics for single-site excitation in a Floquet superlattice with σ = 1/3 for (a) c0 = π, (b) c0 = 2π, (c) c0 = 3π, and (d) c0 = 6π. Other parameters are same as Fig. 2.
We now focus on the emergence of the Talbot effect, in which the self-imaging from the initial periodically distributed field pattern can periodically replicate at certain imaging planes. In discrete waveguide arrays, the effective Bloch momentum k° should be k°j = 2πj/N (j = 0, 1, … N-1) to meet the periodic boundary condition [23,24,27], where N is a positive integer and represents the period of the input optical field. In our modulated superlattices, one has k0 = 2k. To achieve the Talbot self-imaging, all modes ɛj and ɛj,l = |ɛj−ɛl | (j, l∈ {0, 1, … N-1}) contained in the field evolution should be able to exhibit recurrence during propagation, where ɛj is determined by dispersion relation Eq. (4). For a component with ɛj,l, the Talbot revival occurs when the relation ɛj,l zj,l = 2mj,l π holds, where zj,l is the revival distance and mj,l is an integer. The Talbot distance can be described as
It is shown that some unique properties of Talbot revivals are shown in the dynamic localized, rectification and dispersive regimes. In the following discussion, we fix the structure parameters T = 1, d0 = 1 and directly solve Eq. (1) for different periodic incident fields using MATLAB to check the theoretical predictions. Because of the period boundary condition, ideal Talbot effect occurs in infinite lattices. One should build large enough arrays to avoid the boundary effect. Here, more than 500 waveguide elements are considered and only the central part of the model is selected for analysis.
In the rectification regime, the lattice independently rectifies the propagation of optical wave injected into each waveguide. The excitations will propagate without diffraction at a fixed angle (transverse velocity) vg = ±2d0/T and the transport direction depends on which waveguide is initially excited. The location, where two waves with vg = ±2d0/T superimpose and interfere, determines the field recurrence distance. As a result, the self-imaging would be observed with zT = TN and occurs for arbitrary input period N. The distributions of optical field intensity as the Talbot effect takes place are displayed in Figs. 4(a) and 4(b) when the patterns of incident field are given by {1, i, 0, 0, 1, i, 0, 0, …} (N = 2) and {1, i, 0, 0, 0, 0, 1, i, 0, 0, 0, 0, …} (N = 3) with T = 1, σ = 1/3 and c0 = 3π. The corresponding Talbot distances are zT = 2 and 3, respectively.
Fig. 4. Discrete Talbot intensity carpets in the rectification regime as the patterns of incident field are given by (a) {1, i, 0, 0, 1, i, 0, 0, …} (N = 2) and (b) {1, i, 0, 0, 0, 0, 1, i, 0, 0, 0, 0, …} (N = 3). (c)-(e) Self-imaging effects in dynamical localization regime with input pattern {1, i, 0, 0, 1, i, 0, 0, …} (N = 2) for c0 = 2π, 4π and 6π, respectively. (f) Same as (e), but for the input pattern {1, i, 0, 0, 0, 0, 1, i, 0, 0, 0, 0, …} (N = 3). (g), (h) Self-imaging effects supported by 26 waveguide elements with the parameters used in (d) and (e). The white dashed lines denote the locations of self-imaging predicted by theoretical analysis.
In the dynamic localized regime, the compact localized mode is supported by the flat band of the system and the quasi-energy is related σ by the relation ɛ=0 and ±(1-σ)π /T. The Talbot distance, where the repetition of the planar field distribution appears, is given by zT = T and zT = 2 T/(1-σ) when σc0 = 2mπ/T and c0 = 2mπ/T with m being an integer, as shown in Figs. 4(c) -(f) with σ = 1/3, where self-imaging images revive at a distance of zT = 3, 3, 1 and 1for T = 1, c0 = 2π, 4π, 6π and 6π, respectively, for different input periods. That is to say, the Talbot distance in the dynamic localized regime is independent of the incident period. Talbot self-imaging recurrence always occurs, the Talbot distance remains unchanged for irrespective of the pattern period and equals to the restoration period for a single waveguide excitation, as shown in Figs. 3(b) and 3(d). It is necessary to note that when σ = 1 the Talbot distance shrinks to zT = T/(2 m) with the increasing the coupling strength c0, due to the same number of higher-order modes generated on longitudinal half-period of the system. If σ ≠ 1, different number of switching event on the first and second longitudinal half-periods leads to a constant Talbot distance. Furthermore, due to the complete band collapse, the dynamical localization is immune to the boundary of the system. As a result, the discrete Talbot effects in a finite dynamically modulated superlattice are also achieved at c0 = 4π and 6π, as shown in Figs. 4(g) and 4(f), respectively, where only 26 adjacent waveguide elements are contained in the finite discrete system. In the rectification regime, despite the absence of diffraction, the moving period waves change their propagation direction when they reach the lattice edges, a complicated field pattern would be observed and the input field distribution is impossible to recover during propagation in the finite discrete system.
In the dispersive regime, the dispersion relation in modulated superlattices follows a cosinelike function, similar to the one-dimensional straight waveguide arrays, Talbot self-imaging is only possible for a set of periodicities. According to the theory proposed by the R. Iwanow et al. [23], discrete Talbot effect in one-dimensional straight waveguide arrays occurs when the periods of input fields belong to {1, 2, 3, 4, 6} [23,24,27]. According to Eq. (4), the quasi-energy is symmetric with respective to the plane ɛ = 0, the recurrent distance zT must satisfies the relation zTɛ = 2mπ for possible mode with ɛ. As a result, the quasi-energy is also a rational multiple of π and the permitted input periods N are 1, 2 and 4. To summarize, the dynamically modulated character leads to the periods of the input fields N must belong to {1, 2, 4} for Talbot effect, which is much stricter comparing to the case in the one-dimensional straight waveguide arrays.
For N = 1, only the mode with k = 0 is included. According to Eq. (4), we have ɛ = ±c0(1+σ)/2, which lead with Eq. (5) to the Talbot distance zT = Lcm [1, 2(1+σ)-1π] shown in Fig. 5(f). To make it directly visualized, we show the Talbot distance varying with σ as c0 = 0.75π in Fig. 5(h), the Talbot distances are zT = 4/3 and 2 for σ = 1and 1/3, respectively. The corresponding optical field evolutions with the input pattern {1, 0, 1, 0, …} are displayed in Figs. 5(a) and 5(b), where the incident optical waves exhibit revival after z = mzT. In addition to the integer revivals, the fractional Talbot effects are also observed. For planes (z = zT/2) corresponding to the half-integer Talbot self-imaging, intensity pattern exhibits a half-period transverse shift compared to the input. The fractional self-images are also realized at zT /4 for σ = 1 [Fig. 5(a)] and zT /6 for σ = 1/3 [Fig. 5(b)], where both the spatial periods of intensity become half of that of original input. We also display the intensity carpet of Talbot effect with the input pattern {1, i, 1, i, …} in Fig. 5(c). Due to the constructive interference, the intensities at zT and zT/2 are same, but the amplitudes are conjugate. This phenomenon is called dual Talbot effect [33]. Two examples with the input pattern {1, 0, 1, 0, …} at c0 = 1.75π and 2.5π as σ = 1/3 are shown in Figs. 5(d) and 5(e). The corresponding Talbot distances calculated from Eq. (5) are zT = 6 and 3, respectively. The numerical simulations completely agree with the analytical results [Fig. 5(g)]. For the special case of σ = 1, same number of higher-order modes are shown on each longitudinal half-period, one has zT = π/c0, which implies the Talbot distance decreases monotonously with coupling strength c0.
Fig. 5. Talbot effects in the dispersive regime of a modulated superlattice with T = 1 and N = 1 as (a) c0 = 0.75π and σ = 1; (b), (c) c0 = 0.75π and σ = 1/3; (d) c0 = 1.75π and σ = 1/3; and (e) c0 = 2.5π and σ = 1/3. (f) Talbot distance zT versus σ and c0. (g) zT as a function of c0 for σ = 1/3. (h) zT as a function of σ for c0 = 0.75π. The white dashed lines in (a)-(d) denote the recurrent planes of self-imaging given by Eq. (5).
When the period of input field is N = 2, the corresponding quasi-energies are ɛ0 = ±c0(1+σ)/2 and ɛ1 = ±c0(1-σ)/2. From Eq. (5), we can obtain the Talbot distance versus c0 and σ, as presented in Fig. 6(d). In order to highlight the details of this dependence, we display zT varying with c0 for σ = 1/3 and with σ for c0 = 0.75π in Figs. 6(e) and 6(f), respectively. Two representative intensity revivals at σ = 1 and 1/3 for c0 = 0.75π as the input pattern {1, 0, 0, 0, 1, 0, 0, 0, …} are depicted in Figs. 6(a) and 6(b), the corresponding Talbot distances are same to be 8, which confirm the above theoretical analysis. The fractional Talbot effects also exist for N = 2 at z = zT/2, where the self-images are shifted half a period lateral with respective to the integer Talbot planes. Moreover, dual Talbot effect is also observed in Fig. 6(c) when the input pattern is given by {1, i, 0, 0, 1, i, 0, 0, …}. For N = 4, the quasi-energies are ɛ0 = ±c0(1+σ)/2, ɛ1 = ɛ3 = ±arccos [cos(c0/2)cos(σc0/2)] and ɛ2 = ±c0(1-σ)/2. The Talbot revivals predicted by Eq. (5) are supported when c0 and σ belong to a finite set [see Figs. 7(d)–7(f)]. The Talbot distances are zT = 12, 12 and 8 for σ = 1, 1/3 and 2/3 as c0 = 1.5π, and the corresponding Talbot carpets are displayed in Figs. 7(a)–7(c), respectively.
Fig. 6. Talbot effects for N = 2 in the dispersive regime of a modulated superlattice. (a) Talbot carpet for c0 = 0.75π, σ = 1 and input pattern {1, 0, 0, 0, 1, 0, 0, 0, …}. (b) Same as (a) but for σ = 1/3. (c) Same as (b) but for input pattern {1, i, 0, 0, 1, i, 0, 0, …}. (d) Talbot distance versus σ and c0. zT varying with c0 for σ = 1/3 and with σ for c0 = 0.75π are shown in (e) and (f), respectively. White dashed lines in (a)-(c) denote the locations of self-imaging given by Eq. (5).
Fig. 7. Talbot carpets at σ = 1 (a), 1/3 (b), and 2/3 (c) for c0 = 0.75π with input pattern {1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, …}. (d) Logarithm log10(zT) as a function of c0 and σ. (e) zT versus c0 for σ = 1/3. (f) zT versus σ for c0 = 1.5π. The white dashed lines in (a)-(c) denote the recurrent locations predicted by theoretical analysis.
4. Quasi-energy spectrum and discrete Talbot self-imaging effect in modulated non-Hermitian superlattices
We now focus on the Talbot effect in the non-Hermitian superlattices, which is realized through the use of space-dependent gain and loss (ρ≠0). For simplicity, we consider the special case c1 = c2 = c0 (σ = 1). In this case, the band structure of this non-Hermitian superlattices is obtained and given by (see Appendix)
Fig. 8. (a),(b) Real and imaginary of the quasi-energy spectrum in non-Hermitian superlattices as a function α as c0 = 0.75π. (c),(d) Real and imaginary of the quasi-energy spectrum versus c0 as α = 0.6.
Fig. 9. Self-imaging effects in the regime of dynamical localization with input pattern {1, 0, 1, 0, …} for α = 0.6, m = 1 (a) and 2 (b). The white dashed lines denote the recurrent locations predicted by theoretical analysis.
Here, we investigate the Talbot effect in the dispersive regime. The restriction on the periodicity in the Hermitian superlattices also holds for the non-Hermitian case. Both the nonorthogonality of the associated Floquet Bloch modes supported by the physical nonreciprocity and discreteness of the lattice set strong constraints to ensure the Talbot effect generation. A carefully examination of Eq. (6) reveals that, although quasi-energy spectrum is the complex for α<1, the energy values at the edge of the Brillouin zone are always real. If the modes have the propagation constants with imaginary parts, the input field distribution would experience amplification or attenuation and is impossible to recover during propagation. In order to realize Talbot self-imaging, the transverse Bloch momentum is only taken to be k = 0, which results in the input period N = 1. The case N = 1 corresponds to ɛ = ±c0(1-α2)1/2 calculated by Eq. (6). By means of Eq. (5), we get the expression of Talbot distance zT = π(1-α2)-1/2/c0, which is inversely proportional of the coupling strength. Due to the non-zero imaginary part of quasi-energy spectrum arise at the center of the Brillouin zone, where the mode degeneracy occurs. There would be some modes with k≠0 for the even numbered periods of the input field, the imaginary parts of their propagation constants are nonzero. Thus, Talbot revival is not preserved for the even numbered periods of the input field. That is to say, Talbot effect in the dispersive regime exists only as the period of incident filed is N = 1. For α = 0.6 and c0 = 0.5π, the calculated Talbot distance is zT = 2.5, and the intensity carpet shown in Fig. 10(a) coincides well with the theoretical analysis. Compared to the Talbot effect in the Hermitian case, one unique feature of non-Hermitian Talbot effect is that the power oscillations exit in the Talbot process, owing to gain and loss introduced in the waveguides. Based on Eqs. (5) and (6), it is shown that Talbot distance increases monotonically with the increase of α or the decrease of c0, as shown in Figs. 10(d)–10(f). The Talbot distances are zT = 5/6 and zT = 10/3 for α = 0.6, c0 = 1.5π and α = 0.8, c0 = 0.5π, respectively. The Talbot intensity carpets displayed in Figs. 10(b) and 10(c) are found to agree quite well with the above theoretical predictions.
Fig. 10. Talbot effects in non-Hermitian superlattices for N = 1. (a) α = 0.6 as c0 = 0.5π, (b) α = 0.6 as c0 = 1.5π, (c) α = 0.8 as c0 = 0.5π. (d) zT as a function of c0 and α. (e) zT versus c0 for α = 0.6. (f) zT versus α for c0 = 0.5π. The white dashed lines in (a)-(c) denote the recurrent locations predicted by theoretical analysis.
5. Conclusion
In conclusion, we have investigated the dynamical behaviors of the self-imaging periodic optical wave in modulated Floquet superlattices based on a mesh of directional couplers with periodically varying separation between waveguides. It is demonstrated that both Hermitian and non-Hermitian superlattices can support Talbot self-imaging revivals for input patterns with a specific set of periodicities. The analytical and numerical results agree with each other very well. Three sub-regimes, i.e., dynamic localized, rectification and dispersive regimes, are divided in Hermitian superlattices. In both dynamic localized and rectification regimes, self-imaging effects can occur for arbitrary input period N. For the rectification case, Talbot distance equals the input period. In the regime of dynamical localization, the Talbot distance remains unchanged for irrespective of the pattern period. In the dispersive regime, the discrete Talbot effect occurs only if the incident periods are N = 1, 2, 4 and the Talbot distance can be effectively controlled by adjusting the coupling strength. Both the fractional and dual Talbot revivals are also observed for the input periods N = 1 and 2. In the non-Hermitian case, the quasi-energy spectrum becomes complex, and the touch point at the center of the Brillouin zone splits into two separated exceptional points, where imaginary part of the quasi-energy is nonzero. As a result, Talbot revivals is not preserved for the even numbered input periods and exists only if N = 1, where the power oscillations are presented in the Talbot imaging process. Moreover, Talbot distance is found to increases monotonically with the increase of loss parameter or the decrease of coupling strength. Note that we only studied the Talbot effect in non-Hermitian Floquet superlattices with σ = 1, both theoretically and numerically. For the case σ ≠ 1, analytical insights into Talbot effect no longer hold since the close form solution for dispersion relation is not easy to find. Our results not only provide significant extensions of the famous discrete Talbot effect in straight waveguide, but also may find some potential applications in high precision interferometry and tunable intensity amplifiers.
Appendix: Derivation of dispersion relation of the Floquet superlattices
In the absence of losses (ρ=0) in all waveguides, from Eq. (3), the Hamiltonians on the first and second half-periods are written as
The corresponding evolution operator of the modulation superlattices over one longitudinal period is obtained according to U(T) = exp(-iH1T/2) exp(-iH2T/2),
We can immediately obtain the result Eq. (4).
For the non-Hermitian superlattices with σ = 1, the Hamiltonians on the first and second half-periods have the form
Based on the above equations, one gets
The band structure for the non-Hermitian superlattices given by Eq. (6) can be directly obtained by letting α=ρ/c0 and f = (1-α2)1/2 in Eq. (12).
Funding
Natural Science Foundation of Shandong Province (ZR2021MA030); Fundamental Research Funds for the Central Universities (22CX03028A); National Natural Science Foundation of China (61605251).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003). [CrossRef]
2. I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518(1-2), 1–79 (2012). [CrossRef]
3. S. Longhi and K. Staliunas, “Self-collimation and self-imaging effects in modulated waveguide arrays,” Opt. Commun. 281(17), 4343–4347 (2008). [CrossRef]
4. Y. V. Kartashov and V. A. Vysloukh, “Light tunneling inhibition in array of couplers with longitudinal refractive index modulation,” Opt. Lett. 35(2), 205–207 (2010). [CrossRef]
5. V. E. Lobanov, V. A. Vysloukh, and Y. V. Kartashov, “Inhibition of light tunneling for multichannel excitations in longitudinally modulated waveguide arrays,” Phys. Rev. A 81(2), 023803 (2010). [CrossRef]
6. C. Z. Qin, B. Wang, and P. X. Lu, “Frequency diffraction management through arbitrary engineering of photonic band structures,” Opt. Express 26(20), 25721–25735 (2018). [CrossRef]
7. A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012). [CrossRef]
8. N. H. Lindner, G. Refael, and V. Galitski, “Floquet topological insulator in semiconductor quantum wells,” Nat. Phys. 7(6), 490–495 (2011). [CrossRef]
9. N. Goldman and J. Dalibard, “Periodically driven quantum systems: effective Hamiltonians and engineered gauge fields,” Phys. Rev. X 4(3), 031027 (2014). [CrossRef]
10. M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, “Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems,” Phys. Rev. X 3(3), 031005 (2013). [CrossRef]
11. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000). [CrossRef]
12. S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96(24), 243901 (2006). [CrossRef]
13. A. Szameit, Y. V. Kartashov, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, V. A. Vysloukh, F. Lederer, and L. Torner, “Inhibition of light tunneling in waveguide arrays,” Phys. Rev. Lett. 102(15), 153901 (2009). [CrossRef]
14. G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83(5), 963–966 (1999). [CrossRef]
15. W. F. Zhang, X. Zhang, Y. V. Kartashov, X. F. Chen, and F. W. Ye, “Bloch oscillations in arrays of helical waveguides,” Phys. Rev. A 97(6), 063845 (2018). [CrossRef]
16. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Resonant mode oscillations in modulated waveguiding structures,” Phys. Rev. Lett. 99(23), 233903 (2007). [CrossRef]
17. K. Y. Zhan, L. C. Dou, R. Y. Jiao, W. Q. Zhang, and B. Liu, “Talbot effect in arrays of helical waveguides,” Opt. Lett. 46(2), 322–325 (2021). [CrossRef]
18. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013). [CrossRef]
19. S. Longhi, “Rectification of light refraction in curved waveguide arrays,” Opt. Lett. 34(4), 458–460 (2009). [CrossRef]
20. F. Dreisow, Y. V. Kartashov, M. Heinrich, V. A. Vysloukh, A. Tnnermann, S. Nolte, L. Torner, S. Longhi, and A. Szameit, “Spatial light rectification in an optical waveguide lattice,” Europhys. Lett. 101(4), 44002 (2013). [CrossRef]
21. Y. V. Kartashov, V. A. Vysloukh, V. V. Konotop, and L. Torner, “Diffraction control in PT-symmetric photonic lattices: From beam rectification to dynamic localization,” Phys. Rev. A 93(1), 013841 (2016). [CrossRef]
22. Y. Q. Zhang, Y. V. Kartashov, F. Li, Z. Y. Zhang, Y. P. Zhang, M. R. Belić, and M. Xiao, “Edge state in dynamical superlattices,” ACS Photonics 4(9), 2250–2256 (2017). [CrossRef]
23. R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Discrete Talbot effect in waveguide arrays,” Phys. Rev. Lett. 95(5), 053902 (2005). [CrossRef]
24. Z. H. Chen, Y. Zhang, and M. Xiao, “Discrete Talbot effect in two-dimensional waveguide arrays,” Opt. Express 23(11), 14724–14733 (2015). [CrossRef]
25. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9(56), 401–407 (1836).
26. L. Rayleigh, “On copying diffraction gratings and on some phenomenon connected therewith,” Philos. Mag. 11(67), 196–205 (1881). [CrossRef]
27. H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-symmetric Talbot effects,” Phys. Rev. Lett. 109(3), 033902 (2012). [CrossRef]
28. S. L. Wang, C. Z. Qin, B. Wang, and P. X. Lu, “Discrete temporal Talbot effect in synthetic mesh lattices,” Opt. Express 26(15), 19235–19246 (2018). [CrossRef]
29. S. L. Wang, B. Wang, and P. X. Lu, “PT -symmetric Talbot effect in a temporal mesh lattice,” Phys. Rev. A 98(4), 043832 (2018). [CrossRef]
30. W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979). [CrossRef]
31. S. J. Wu, W. G. Song, S. L. Gao, Y. X. Chen, S. N. Zhu, and T. Li, “Floquet π mode engineering in non-Hermitian waveguide lattices,” Phys. Rev. Res. 3(2), 023211 (2021). [CrossRef]
32. M. S. Pan, H. Zhao, P. Miao, S. Longhi, and L. Feng, “Photonic zero mode in a non-Hermitian photonic lattice,” Nat. Commun. 9(1), 1308 (2018). [CrossRef]
33. Y. Zhang, H. Zhong, M. R. Belić, X. Liu, W. Zhong, Y. Zhang, and M. Xiao, “Dual accelerating Airy-Talbot recurrence effect,” Opt. Lett. 40(24), 5742–5745 (2015). [CrossRef]
