Pattern Stabilization through Parameter Alternation in a Nonlinear Optical System

We report the first experimental realization of pattern formation in a spatially extended nonlinear system when the system is alternated between two states, neither of which exhibits patterning. Dynamical equations modeling the system are used for both numerical simulations and a weakly nonlinear analysis of the patterned states. The simulations show excellent agreement with the experiment. The nonlinear analysis provides an explanation of the patterning under alternation and accurately predicts both the observed dependence of the patterning on the frequency of alternation and the measured spatial frequencies of the patterns.

Figure 1

Schematic of the experiment. The focal length of the lenses L1 and L2 is $f=25\mathrm{cm}$. The distances $d_1$ and $d_2$ are 20 and 15 cm, respectively, leading to an effective propagation length of $-15\mathrm{cm}$ and, thus, a self-focusing nonlinearity [6]. BS1, BS2: beam splitters; P: polarizer; P1: pinhole; HWP: half-wave plate; AOM: acousto-optic modulator. Inset: Phase response of the LCLV as a function of write light intensity and no feedback. The solid points show the experimental data, while the line is $10.5\tanh (2.5I)$, where $I$ is the intensity.

Figure 2

Output from the experiment. (a) and (b) show the homogeneous states obtained for an input intensity to the back of the LCLV of 3.9 and $0.1\mathrm{mW}/{\mathrm{cm}}^2$, respectively. (c) shows the output obtained with a constant intensity input of $2\mathrm{mW}/{\mathrm{cm}}^2$. (d)–(h) show the output obtained with the input irradiance switching between 3.9 and $0.1\mathrm{mW}/{\mathrm{cm}}^2$. The modulation frequencies are (d) 2 (image obtained when the intensity goes high), (e) 6, (f) 14, (g) 20, and (h) 200 Hz.

Figure 3

Homogeneous steady state phase as a function of input intensity for the LCLV feedback system with both the input polarization and the output polarizer oriented at 45°. Also indicated are the types of patterns seen in the experiment and in computer simulations under constant intensity illumination.

Figure 4

State map showing the regions of existence of stable patterns of rolls and hexagons as a function of spatial frequency and phase. The range of phase from about 5.5 to 9.4 is along the middle stable branch of Fig. 3. The corresponding intensities are also indicated on the diagram. We did not find any stable mixed states.

Figure 5

Amplitudes of hexagons vs time (a) after the intensity is switched from 0.1 to $3.9\mathrm{mW}/{\mathrm{cm}}^2$ and (b) after the intensity is switched from 3.9 to $0.1\mathrm{mW}/{\mathrm{cm}}^2$. The spatial frequencies correspond to ${\widehat{q}}^2=1$ (●), 2 (■), 2.7 (◆), 3 (*), 4 (×), and 5 (▼). Symbols are to guide the eye and do not represent time steps. The amplitudes of the rolls show very similar behavior.