Frequency-dependent fundamental and dipole gap solitons in $\mathcal{PT}$-symmetric nonlinear metamaterials

We derive a theoretical model with frequency-dependent parameters for describing the transmission of electromagnetic waves in $\mathcal{PT}$-symmetric metamaterials (MMs). Based on the derived theoretical model, we investigate the tunable band structure and eigenvalue problem in the negative or positive index region of MMs, and predict the existence and stability of fundamental, out-of-phase, and in-phase dipole gap solitons in defocusing or focusing nonlinear $\mathcal{PT}$-symmetric MMs. Furthermore, we discuss the enhanced localization of solitons under different formation conditions. The results reveal that there exist abundant gap solitons with frequency controllability in $\mathcal{PT}$-symmetric MMs, and the established theoretical model paves the way to explore more solitons in $\mathcal{PT}$-symmetric MMs.

Figure 1

(a) Background refractive index $n_0$, loss: gain ${\alpha }_0$, (b) depth $v_1$, and (c) imaginary part $w_1$ of the $\mathcal{PT}$ potential for ${\varpi }_{\mathrm{p}}=0.9$, in which the gray areas denote the stop bands.

Figure 2

Profiles of $\mathcal{PT}$-symmetric potential with different parameters. (a) $\varpi =0.29$, ${\gamma }_0=0.025$, corresponding to $v_1=4$, $w_1=0.1$; (b) $\varpi =0.47$, ${\gamma }_0=0.04$, corresponding to $v_1=1$, $w_1=0.1$, and (c) $\varpi =1.42$, ${\gamma }_0=0.025$, corresponding to $v_1=0.3$, $w_1=0.02$, respectively.

Figure 3

Band structures in the (a) NIR and (b) PIR of $\mathcal{PT}$-symmetric MMs, (c) $\varpi =0.2$ ($v_1=7.31$, $w_1=0.52$); (d) $\varpi =0.21$ ($v_1=6.69$, $w_1=0.5$); (e) $\varpi =0.31$ ($v_1=3.01$, $w_1=0.36$); (f) $\varpi =1.77$ ($v_1=0.79$, $w_1=0.02$); the other parameters are ${\gamma }_0=0.1$, ${\varpi }_p=0.9$.

Figure 4

Power P versus the propagation constant $\mu$ in the NIR and PIR of $\mathcal{PT}$-symmetric MMs. (a) $\varpi =0.29$ ($v_1=4,w_1=0.1$); (b) $\varpi =0.47$ ($v_1=1,w_1=0.1$); (c) $\varpi =1.42$ ($v_1=0.3,w_1=0.02$); (d) $\varpi =1.89$ ($v_1=1,w_1=0.01$).The blue areas correspond to the Bloch band. The soliton profiles at the points marked by N1–N7 (in the NIR) and P1–P7 (in the PIR) will be displayed in Figs. 5 and 6, respectively.

Figure 5

Profiles, linear spectra, and nonlinear evolutions of the stable solitons at the points (a) N1: $\mu =-3.5$; (b) N2: $\mu =-1.6$; (c) N3: $\mu =-0.18$; (d) P1: $\mu =-0.33$; (e) P2: $\mu =1.7$; and (f) P3: $\mu =1$, which are marked in Fig. 4.

Figure 6

Profiles, linear spectra, and nonlinear evolutions of the unstable solitons at the points (a) N4: $\mu =-2.9$; (b) N5: $\mu =-0.96$; (c) N6: $\mu =-0.68$; (d) N7: $\mu =-2$; (e) P4: $\mu =-0.37$; (f) P5: $\mu =0.6$; (g) P6: $\mu =-0.2$; and (h) P7: $\mu =0.05$, which are marked in Fig. 4.

Figure 7

Profiles and linear spectra of fundamental, out-dipole and in-dipole solitons. (a)–(c) $\mu =-0.34$ at $\varpi =1.42$ ($v_1=0.3$, $w_1=0.02$) and (d)–(f) $\mu =0.1$ at $\varpi =1.89$ ($v_1=1$, $w_1=0.01$) for defocusing nonlinearity; (g)–(i) $\mu =0.4$ ($P=0.95$, 2.20, 3.19) and (j)–(l) $\mu =0.7$ ($P=1.86$, 3.73, 3.75) for focusing nonlinearity at $\varpi =1.42$, respectively.