Asymmetric Wave Propagation in Nonlinear Systems

A mechanism for asymmetric (nonreciprocal) wave transmission is presented. As a reference system, we consider a layered nonlinear, nonmirror-symmetric model described by the one-dimensional discrete nonlinear Schrödinger equation with spatially varying coefficients embedded in an otherwise linear lattice. We construct a class of exact extended solutions such that waves with the same frequency and incident amplitude impinging from left and right directions have very different transmission coefficients. This effect arises already for the simplest case of two nonlinear layers and is associated with the shift of nonlinear resonances. Increasing the number of layers considerably increases the complexity of the family of solutions. Finally, numerical simulations of asymmetric wave packet transmission are presented which beautifully display the rectifying effect.

Figure 1

Sketch of a layered photonic or phononic system. The central $N$ layers are nonlinear, nonmirror symmetric with respect to the structure center. In the limit of a vanishing width of the thinner layers the dynamics within high-frequency Bloch bands is described by a DNLS equation for the envelope ${\psi }_n$ (see Refs. 16, 18 for details).

Figure 2

DNLS dimer, $N=2$, $V_0=-2.5$, $\alpha =1$, $|k|=0.1$. Comparison between the symmetric case ($\epsilon =0$, dashed lines) and the asymmetric one ($\epsilon =0.05$, solid lines). (a) Transmission curves; the inset is an enlargement of the low-amplitude region. The vertical lines mark the turning points of the curves. Accordingly, the heavy lines on the horizontal axis are the multistability windows $W_{1,2}$ where the diode effect maximally occurs. (b) Transmission coefficients as a function of the transmitted intensity.

Figure 3

Square modulus of the solutions corresponding to the same incident amplitude $|R_0{|}^2=0.4$ (marked by full dots in the inset of Fig. 2), other parameters as in previous figure. Upper panel: The three left-propagating solutions corresponding to $|T{|}^2=0.327$, 0.377, 0.01 (top to bottom). Lower panel: The right-propagating solution corresponding to $|T{|}^2=0.01$. The oscillations are caused by the interference between the incident and reflected waves (wave number $2k$).

Figure 4

Contour plot of the rectifying factor (5) for increasing asymmetry level $\epsilon$. Other parameters as in the previous figure.

Figure 5

Upper panel: Transmission curves for DNLS with $N=4$, linearly modulated parameter $V_n=V_0[1+\epsilon \{1-2(n-1)\}/(N-1)]$, other parameters as in the previous figures. Comparison between the symmetric case ($\epsilon =0$, dashed lines) and the asymmetric one ($\epsilon =0.05$, solid lines). Lower panels: Contour plots of the rectifying factor (5) for increasing asymmetry. Similar results are obtained for different types of modulation.

Figure 6

Numerical simulations of the propagation of Gaussian wave packets, Eq. (7) impinging on a DNLS dimer. Here $V_0=-2.5$, $|k_0|=1.57$, $\epsilon =0.05$, $M=500$, $|I{|}^2=3$, $w={10}^4$ and $n_0=\mp 250$, respectively. Lower panels: Power spectra of the real part of ${\varphi }_n$ at times $t=0$ (green dashed line) and $t=250$ (red solid line).