Pattern Generation by Dissipative Parametric Instability

Nonlinear instabilities are responsible for spontaneous pattern formation in a vast number of natural and engineered systems, ranging from biology to galaxy buildup. We propose a new instability mechanism leading to pattern formation in spatially extended nonlinear systems, which is based on a periodic antiphase modulation of spectrally dependent losses arranged in a zigzag way: an effective filtering is imposed at symmetrically located wave numbers $k$ and $-k$ in alternating order. The properties of the dissipative parametric instability differ from the features of both key classical concepts of modulation instabilities, i.e., the Benjamin-Feir instability and the Faraday instabiltyity. We demonstrate how the dissipative parametric instability can lead to the formation of stable patterns in one- and two-dimensional systems. The proposed instability mechanism is generic and can naturally occur or can be implemented in various physical systems.

Figure 1

(a),(c) Floquet spectrum calculated using the Floquet stability analysis of the homogeneous solution of the CGLE; instability occurs above the horizontal continuous line. (b),(d) The dynamics of complex amplitude $a(k)$ of the most unstable mode (indicated by dashed vertical lines on the instability spectrum) is calculated by direct integration of the CGLE. Arrows indicate the direction of temporal evolution. The parameters used are $\mu =1$, $s=0.3$, $b=0.1\times {10}^{-6}$, with full integration time $T=1$. In the case of the BF instability [(a),(b)] $c=1$, $d=-3\times {10}^{-6}$. In the case of the Faraday instability [(c),(d)] $c=4.85$, $d_1=5\times {10}^{-6}$, $d_2=1\times {10}^{-6}$; a piecewise modulated diffraction coefficient is also considered, i.e., $d=d_1$ for $0<t<0.2$ [orange line on (d)], $d=d_2$ for $0.2<t<0.4$ (red line), and so on.

Figure 2

(a) Dissipative parametric instability arises if periodic-in-time losses are introduced asymmetrically in the $k$ domain, so that the modes with only positive or negative wave vectors are damped every half of the period $T_f$. (b) Dissipative parametric instability can be realized by alternating losses in the frequency domain, i.e., in a laser with detuned (in frequency) cavity mirrors. (c) A self-imaging resonator or self-imaging array of lenses with spatial filter displaced relative to the system’s axis is another possibility.

Figure 3

(a) Spectrum of the dissipative parametric instability. The dashed line indicates the most unstable mode. (b) Evolution of the absolute values of the amplitudes of the most unstable modes $a(k)$ and $a(-k)$ (red and blue lines, respectively). The losses are introduced at points $f_1$, $f_2$, etc., in time. (c) The complex amplitude of the mode $a(k)$ evolves in loops in phase space synchronized with external forcing. (d) Spectrum of the dissipative parametric instability as a function of the modulation period $T_f$; the dashed line is the analytically estimated scaling law of the instability. (e) The generalized phase $\mathrm{\Phi }$ locks to the optimum value (dashed line), the point at which the mode’s amplitudes are growing at the fastest rate, through periodic reset of the phase at instances of time at which the losses are applied. (f) Asymptotically stable pattern in a one-dimensional system.

Figure 4

(a) Zigzagging losses in wave number space ($k_x$, $k_y$), (b) the instability area in ($k_x$, $k_y$) space as obtained by the Floquet analysis, and (c) 2D intensity patterns. Parameters are $\mu =0.2$, $d=0.05$, $b=0.08$, $c=0.35$, $s=0.3$, $T_f=5\pi$, and $\sigma =1.0905$. Losses are centered at $k_{0x}=-1$, $k_{0y}=+1$. (d) By setting $b=0$, the pattern becomes irregular in space and nonstationary in time.