$\mathcal{P}\mathcal{T}$-symmetric Talbot effect in a temporal mesh lattice

We investigate the $\mathcal{P}\mathcal{T}$-symmetric Talbot effect in a temporal mesh lattice constructed by two coupled fiber loops, in which the $\mathcal{P}\mathcal{T}$ symmetry is introduced through temporally controlling the gain and loss of the loops. The Talbot self-imaging exists only if the period of input pulse train is chosen as two- or fourfold compared to the time interval caused by the length difference between the two loops. Through varying the gain and loss, we can tailor the lattice band structure and thus flexibly manipulate the Talbot distance, which can further be tuned by imposing a linear phase distribution on the input pulse train. In addition, the power oscillations are found in the Talbot imaging process, and the oscillation amplitude is associated with the gain and loss and the gradient of the linear phase modulation. Especially, the power oscillations possess significant amplitude as the modes near the exceptional points are excited. The study may find potential applications in pulse repetition rate multiplication, temporal cloaking, and tunable intensity amplifiers.

Figure 1

(a) Schematic of two coupled fiber loops. The red (purple) loop has a gain factor of $e^{\gamma /2}\left(e^{-\gamma /2}\right)$, and both loops change between gain and loss alternately after every round trip. OS1 and OS2 are optical switches, through which we inject the periodic pulse train to the long loop and couple out the pulse sequences from the loops. PD denotes photodiode. (b) Schematic of temporal mesh lattice with $\mathcal{P}\mathcal{T}$ symmetry. The pulse train is amplified in the red path and attenuated in the purple path, which mimics the antisymmetric distribution of the imaginary part of the complex potential. (c) Distribution of the phase modulation ${\phi }_n$.

Figure 2

(a)–(f) Band structures for $\gamma =0$, 0.63, 0.88, 1.06, 1.15, and 1.2, respectively. The blue and red lines represent the real and imaginary parts of the propagation constant, respectively.

Figure 3

(a) Talbot distance $z_T$ versus γ below the $\mathcal{P}\mathcal{T}$ transition threshold. (b), (c) Pulse intensity evolutions in the long loop for $\gamma =0$ and $\gamma =0.63$, respectively. The input period is chosen as $N=4$. (d), (e) Pulse intensity evolutions in the long loop for $N=8$ and $N=12$. γ is chosen as 0. The green dotted lines denote the locations of the Talbot images.

Figure 4

(a) Schematic of the linear phase modulation imposed on the input field. (b), (c) Talbot distance $z_T$ versus $|{\varphi }_0|$ for $\gamma =0$ and 0.63, respectively. The incident period is $N=4$. (d), (e) Talbot carpets for ${\varphi }_0=0.36\pi$ and ${\varphi }_0=0.46\pi$, respectively. γ is set as 0. (f), (g) Talbot carpets for ${\varphi }_0=0.27\pi$ and ${\varphi }_0=0.54\pi$, respectively. γ is chosen as 0.63.

Figure 5

(a), (b) Pulse intensity evolutions (after log transformation) for $N=4$ and 8 at the $\mathcal{P}\mathcal{T}$ transition threshold.

Figure 6

(a) Talbot distance $z_T$ and oscillation amplitude $E_{PO}$ (after log transformation) varying with $|{\varphi }_0|$. (b), (c) Pulse intensity evolutions (after log transformation) with ${\varphi }_0$ being 0.45π and 0.87π, respectively. The incident period is $N=4$.

Figure 7

(a) Talbot distance $z_T$ and oscillation amplitude $E_{PO}$ (after log transformation) varying with γ above the $\mathcal{P}\mathcal{T}$ transition threshold. The incident period is $N=4$. (b), (c) Talbot carpets (after log transformation) for $\gamma =1.09$ and 1.12, respectively.

Figure 8

(a) Band structures for $\phi =0$ and $\gamma =0.5$. (b) Band structure for $\phi =\pi /4$ and $\gamma =0$. The modes contained in the field evolution for $N=4$ and 8 are represented as the cyan triangles and black diamonds, respectively. (c), (d) Talbot carpets for $N=4$ and 8, respectively. φ and γ are chosen as π/4 and 0, respectively.