Interaction of Airy-Gaussian beams in photonic lattices with defects

We investigate numerically the interaction between two finite Airy-Gaussian (AiG) beams in different media with the defected photonic lattices in one transverse dimension. We discuss that the beams with different intensities and phases launch into the different lattice structures but accelerate in opposite directions. During interactions, the interference fringe, breathers, and soliton pairs are observed. In the linear media, the initial deflection direction of the accelerated beams is changed by adjusting the phase shift and the beam interval. For a certain lattice period, the periodic interference fringe can form. A constructive or destructive interference can vary with the defect depth and phase shift. While the nonlinearity is introduced, the breathers is generated. Especially in the self-defocusing media, the appropriate AiG beam amplitude and lattice depth may lead to the formation of soliton pairs, On the contrary, the interaction of two Gaussian beams is diffraction.

Figure 1

Defect generation in an optical photonic lattice. (a) The regular lattice distribution $(\delta n=0)$. (b) The Gaussian beam intensity distribution $G\left(X\right)$. Numerical realization of (c) positive ($\delta n=0.5$) and (d) negative ($\delta n=-0.5$) defect lattices. The other physical parameters $A_1=A_2=2$, $A_n=3$, $T=0.1\mathrm{mm}$, $\delta \varphi =0$, $D=3$, $Q=0.05$, and $\gamma =0$.

Figure 2

The interaction of two AiG beams with changing the values of $\delta \varphi$ and $D$ for $A_1=A_2=2$, $A_n=3$, $T=0.2\mathrm{mm}$, $\delta n=0$, $Q=0.01$, and $\gamma =0$.

Figure 3

Intensity profiles of AiG beams with $D=3$ (a), $D=1$ (b), $D=-1$ (c), and $D=-3$ (d) for $\delta \varphi =\pi /2$. The other physical parameters are the same as Fig. 2.

Figure 4

The interaction of two AiG beams with changing the values of $\delta \varphi$, $\delta n$, and $T$ for $A_1=A_2=2$, $A_n=3$, $D=3$, $Q=0.01$, and $\gamma =0$.

Figure 5

(a) Intensity profiles of the initial input AiG beams with $\delta \varphi =0$ (red dashed line) and $\delta \varphi =\pi$ (green dash-dotted line) for $\delta n=-2$ and $T=0.1\mathrm{mm}$. The photonic lattices with defects with $\delta n=-2$ (b), $\delta n=-0.5$ (c), $\delta n=0$ (d), $\delta n=0.5$ (e), and $\delta n=2$ (f) for $T=0.1\mathrm{mm}$. The other physical parameters are the same as Fig. 4.

Figure 6

The interaction of two AiG beams with changing the values of $Q$, and $\gamma$ for $A_1=A_2=4$, $A_n=3$, $D=3$, $Q=0.01$, $\delta \varphi =0$, $T=0.1\mathrm{mm}$, and $\delta n=0.5$.

Figure 7

Intensity profile of AiG beams ${\left|q(X,0)\right|}^2$ and the lattice with defects $V\left(X\right)$. The red dashed and green dash-dotted lines denote ${\left|q(X,0)\right|}^2$ at $Q=0.1$ and $Q=1$, respectively. The blue solid line shows $V\left(X\right)$. The other physical parameters are the same as Fig. 6.

Figure 8

The interaction of two AiG beams with changing the values of $A_1$, $A_2$, $Q$, and $A_n$ for $T=0.1\mathrm{mm}$, $\gamma =-1$, $D=3$, $\delta \varphi =\pi$, and $\delta n=0.5$.

Figure 9

The interaction of two AiG beams ((a) and (b)) or two Gaussian beams [(c) and (d)] for $D=5$ (left column) and $D=-5$ (right column) at $A_1=A_2=1$, $A_n=5$, $T=0.1\mathrm{mm}$, $\gamma =1$, $\delta \varphi =\pi$, and $\delta n=0.5$.

Figure 10

The interaction of two AiG beams with changing the values of $A_1$, $A_2$, and $A_n$ for $\gamma =1$ and $Q=0.01$. The other physical parameters are the same as in Fig. 8.

Figure 11

The interaction (a)–(c) and intensity profiles of two AiG beams with changing the values of $A_2$ for $A_1=4$, $A_n=3$, $T=0.1\mathrm{mm}$, $\gamma =1$, $D=3$, $Q=0.01$, $\delta \varphi =0$, and $\delta n=0.5$.