Stable Multicolor Periodic-Wave Arrays

We study the existence and stability of periodic-wave arrays propagating in uniform quadratic nonlinear media and discover that they become completely stable above a threshold light intensity. To the best of our knowledge, this is the first example in physics of completely stable periodic-wave patterns propagating in conservative uniform media supporting bright solitons.

Figure 1

(a) Energy flow of dn waves versus propagation constant for various phase mismatches. (b),(c) dn-wave profiles at $\beta =0$ and $b_1=0.6$ and $b_1=1.5$, respectively. (d) Areas of existence of finite perturbations with imaginary growth rates at $\beta =0$ (shaded). (e) Areas of existence of finite perturbations with real growth rates at $\beta =0$ (shaded). Cutoff on propagation constant in (d) and (e) is given by $b_1\approx 0.341$. (f) Propagation of the dn wave with $b_1=0.6$ at $\beta =0$ in the presence of the perturbation with $\delta =0.12444$. Only the FF is shown.

Figure 2

(a) Energy flow of cn waves versus propagation constant for various phase mismatches. (b),(c) cn-wave profiles at $\beta =0$ and $b_1=0.15$ and $b_1=1.5$, respectively. (d) Areas of existence of finite perturbations with imaginary growth rates at $\beta =0$ (shaded). (e),(f) Areas of existence of finite perturbations with real growth rates (shaded) at $\beta =-3$ and $\beta =0$, respectively. (g) Maximum real part of complex growth rate versus propagation constant at $\beta =3$. (h) Curves at the complex plane showing possible increment values for various propagation constants at $\beta =3$. (i) Threshold propagation constant for stabilization versus phase mismatch.

Figure 3

(a) Propagation of the unstable cn wave with $b_1=0.15$ at $\beta =0$ in the presence of the perturbation with $\delta =0.0954$. Long-distance evolution of stable cn waves with $b_1=1.8$ at $\beta =-2$ (b), $b_1=0.5$ at $\beta =0$ (c), and $b_1=0.5$ at $\beta =1$ (d) in the presence of white noise with variance ${\sigma }_{1,2}^2=0.01$. Only the FF is shown.