Gap solitons in $\mathcal{PT}$-symmetric lattices with a lower refractive-index core

We address the existence and stability properties of families of gap solitons in a lower refractive-index core, sandwiched between two optical lattices with a parity-time ($\mathcal{PT}$) symmetry imprinted in a defocusing medium. The scale of flat-topped complex solitons can be controlled arbitrarily by varying the embedded index core. Multipeaked solitons are found to exhibit equal peak values in the region of the index core, and they have no analog in other lattice-modulated or bulk media. We demonstrate that, in sharp contrast to solitons in regular $\mathcal{PT}$ lattices, flat-topped and multipeaked solitons are either stable or suffer a negligibly weak instability, even when the gain-loss coefficient approaches the $\mathcal{PT}$-symmetry-breaking point. Our results, thus, build a bridge between the $\mathcal{PT}$ defect solitons in a narrow index core and the $\mathcal{PT}$ kink pairs in a broad index core. We also suggest an effective way for the observation of “surface solitons” in $\mathcal{PT}$-symmetric lattices.

Figure 1

(a) Profile of defected $\mathcal{PT}$ lattice with $p=4, p_i=0.1$, and $n=7$. Solid blue line, real part; dashed red line, imaginary part. (b) Band-gap lattice spectrum versus lattice depth $p$ at $p_i=0.1$. (c) Band-gap structure of the lattice at $p=4$. Dashed line indicates the real eigenvalues of $b$ at $p_i=0.7$. (d) Imaginary part of the double-valued band at $p_i=0.7$. $\mathrm{\Omega }=2$ hereinafter. All quantities are plotted in dimensionless units.

Figure 2

(a) Power $P$ versus propagation constant $b$ for solitons in lattices with different $n$. (b, c) Profiles of solitons marked in (a) in lattices with $n=7$ and 3. (d) Transverse power flow of solitons shown in (b) across the lattice. (e) Existence regions of solitons at the $p,b$ plane for $n=7$ (solid line) and 3 (dashed line). (f) Propagation simulation of a soliton at $b=-0.6$ shown in (b). The modulus of the soliton is displayed. $p=4$ except for (e), and $p_i=0.1$ in all panels. All quantities are plotted in dimensionless units.

Figure 3

(a) Power $P$ versus propagation constant $b$ for solitons at different $p_i$. (b) Profile of soliton at $b=-0.2, p_i=0.45$. (c) Amplitude ratio of the imaginary and real parts of solitons versus $b$. (d) Instability growth rate versus $b$. (e) Unstable and (f) stable propagation of solitons marked by circles in (d). The modulus of solitons is displayed; $p=4, n=7$ in all panels. All quantities are plotted in dimensionless units.

Figure 4

(a) Moduli of flat-topped solitons at $b=-0.58$ in lattices $p_i=0.1$. (b) Profile of three-peaked solitons at $b=-0.3$ ($p_i=0.4, n=11$), (c) profile of five-peaked solitons at $b=-0.3$ ($p_i=0.4, n=15$), and (d) instability growth rate versus propagation constant $b$ ($p_i=0.4, n=15$). $p=4$ in all panels. All quantities are plotted in dimensionless units.

Figure 5

(a) Unstable and (b) stable propagations of five-peaked solitons marked by circles in Fig. 4. (c) Propagation of a five-peaked soliton at $b=-0.28$. The gain-loss component of the lattice is removed at $z=800$. All quantities are plotted in dimensionless units.