Talbot self-imaging in $\mathcal{PT}$-symmetric complex crystals

The Talbot effect, i.e., the self-imaging property of a periodic wave in near-field diffraction, is a remarkable interference phenomenon in paraxial systems with continuous translational invariance. In crystals, i.e., systems with discrete translational invariance, self-imaging has been regarded so far as a rare effect, restricted to special sets of initial field distributions. Here it is shown that in a class of gapless $\mathcal{PT}$-symmetric complex crystals at the symmetry-breaking threshold Talbot revivals can arise for almost any initial periodic wave distribution which is commensurate with the lattice period. A possible experimental realization of commensurate Talbot self-imaging for light pulses in complex “temporal” crystals, realized in an optical dispersive fiber loop with amplitude and phase modulators, is briefly discussed.

Figure 1

(a) Schematic of Bragg scattering and self-imaging in a complex crystal. The input wave at $z=0$ is periodic with a spatial period $\mathcal{L}$ which is commensurate with the lattice period $a$. (b) Band structure of a gapless crystal with a dispersion relation that reproduces the parabolic dispersion of free-space propagation. Spectral singularities at energies $E_n={(n\pi /a)}^2$ may arise at either the band center $q=0$ (open circles, $n$ even) or at the band edge $q=-\pi /a$ (solid circles, $n$ odd). For the complex sinusoidal potential $V\left(x\right)=V_{0}exp(2\pi ix/a)$ all the energies $E_n$ $\left(n=1,2,3,...\right)$ are spectral singularities [24]; for the potential (4) there is only one spectral singularity at $E=E_1$, whereas for the potential (5) there are two spectral singularities at energies $E_1$ and $E_3$. In all cases the band dispersion diagram is the same and obtained by folding, inside the first Brillouin zone, the parabolic dispersion curve of the free-particle Hamiltonian.

Figure 2

Quantum recurrence in the Hermitian Mathieu potential $V\left(x\right)=V_{0}sin(2\pi x/a)$ for $a=2\pi$. (a) Solid curve, input field distribution with spatial period $\mathcal{L}=a$; dotted curve, the Mathieu potential $V\left(x\right)$. (b) Maps of field propagation [real and imaginary parts of $\psi (x,z)]$ for $V_0=2$. (c) Behavior of the the deviation function $\Delta \left(z\right)$, defined by Eq. (3), for $V_0=1$ (curve 1) and $V_0=2$ (curve 2). The dotted curve in (c) corresponds to $V_0=0$ (free-space propagation), with a revival period $z_T=2\pi$.

Figure 3

Talbot self-imaging in the $\mathcal{PT}$ crystal defined by Eq. (4). (a) Real and imaginary parts of the potential $\left(a=2\pi ,\rho =1\right)$. (b) Initial field distribution (solid curve) with commensurate period $\mathcal{L}=(3/2)a$. The Talbot revival period is $z_T={\left(Na\right)}^2/\left(2\pi \right)\simeq 56.55$. The dotted curve shows the field distribution (in modulus) after a propagation distance $z=z_T/2$. (c) Maps of field propagation [real and imaginary parts of $\psi (x,z)]$. (d) Behavior of the deviation function $\Delta \left(z\right)$ (solid curve). The dashed curve in the figure shows the behavior of the deviation function corresponding to propagation in the associated Hermitian lattice, obtained by considering only the real part of the potential (4). (e) As (d), but for an initial field distribution with period $\mathcal{L}=2a$. (f) As (e), but for a tilted field with the additional phase term $exp(2\pi ipx/a)$ $\left(p=1/3\right)$. Self imaging is restored at the period $z_T=N^2{a}^2/\left(2\pi p^2\right)\simeq 226.2$ according to Theorem III.

Figure 4

Talbot self-imaging in the $\mathcal{PT}$ crystal defined by Eq. (5). (a) Real and imaginary parts of the potential $\left(a=2\pi ,\rho =1\right)$. (b) Initial field distribution (solid curve) with commensurate period $\mathcal{L}=(3/2)a$. The Talbot revival period is $z_T={\left(Na\right)}^2/\left(2\pi \right)\simeq 56.55$. (c) Maps of field propagation [real and imaginary parts of $\psi (x,z)]$. (d) Behavior of the deviation function $\Delta \left(z\right)$.

Figure 5

Schematic of a recirculating fiber loop that realizes Bragg scattering of an optical pulse train in a synthetic complex temporal crystal. AM, amplitude modulator; PM, phase modulator; OA, broadband optical amplifier. The input pulse train comprises $N_p$ pulses spaced in time by $T_p$, which is commensurate with the modulation period $T_m$. $L_f$ is the total length of the ring. The evolution of the initial pulse train at successive round trips in the ring can be monitored at the output port of the fiber coupler using a fast photodetector. The amplitude and phase modulators are driven by two different but synchronized radio-frequency (rf) signals.