Complex structures in media displaying electromagnetically induced transparency: Pattern multistability and competition

Spatially periodic and localized structures in the transverse plane of a medium displaying electromagnetically induced transparency in an optical cavity and under the action of two pumps are investigated. The system supports a multitude of different complex spatial structures depending on the chosen initial condition. We explore regimes of multistable patterns, filaments, stable defects, scrolling structures, nested patterns, fronts, and the spontaneous occurrence of multiple cavity solitons. To simulate realistic conditions of operation, we replace periodic boundary conditions with pumps of finite size. Many of the multistable features are recovered apart from the scrolling of patterns with defects.

Figure 1

(a) The cavity configuration with the three-level atomic medium and the holding $E_I$ and coupling $E_2$ beams. (b) The $\Lambda$ atomic scheme with two ground states $|1\rangle ,|2\rangle$, and a single excited level $|3\rangle$.

Figure 2

(a) The imaginary (solid line) and real (dashed line) parts of the complex susceptibility $\chi$ for ${\left|E\right|}^2={\left|E_2\right|}^2=1$. (b) Bistability in the input and output intensities for $\Delta =0.2,\theta =-1,|E_2{|}^2=1$, and $2C=20$.

Figure 3

Turing instability domains for pattern formation of wave vector $K$ vs (a) increasing $2C$ and (b) increasing stationary intensities. Parameter values are $\Delta =0.2,\theta =-1,|E_2{|}^2=1$. (a) $|E_s{|}^2=1$ (solid line), $|E_s{|}^2=1.1$ (dashed line), and $|E_s{|}^2=1.2$ (dotted line). (b) $2C=19$ (solid line) and $2C=20$ (dashed line). The line traced by the small circles determines the wave vector of largest growth for $2C=20$.

Figure 4

(a) Stable honeycomb structures for $2C=19.6$, (c) stable roll patterns for $2C=21.0$, and (e) cavity solitons for $2C=21.62$, with (b), (d), and (f) their far-field images, respectively. Note that in (b), (d), and (f), the central point has been removed to increase the clarity of the images.

Figure 5

The instability domain of the $2C$ parameter vs the input intensity (lower part and left axis) and the corresponding maximum intensities for different patterns (upper part and right axis).

Figure 6

The instability domain of the $2C$ parameter vs the input intensity (lower part and left axis) and the corresponding maximum intensities for different patterns (upper part and right axis) when starting at $2C=18.90$. DOS: distorted, oscillating, and scrolling.

Figure 7

(a) The distorted honeycomb pattern (left) and its far-field image (right) for $2C=18.84$. The yellow circles identify the position of the defects. Note that these patterns with defects maintain their spatial shape basically unchanged when turning into scrolling structures.

Figure 8

(a) Oscillation of the maximum intensity of a scrolling pattern with defects (solid line) and in the central point of the structure (dotted line) for $2C=19.0$. (b) Speed of the scrolling patterns vs $2C$.

Figure 9

(a) Stable fronts separating roll patterns nested in a honeycomb structure and (b) their far-field image for $2C=19.6$.

Figure 10

The sequence of patterns in input intensity vs $2C$ space and the corresponding intensities for different patterns (right axis). These were obtained for an initial value of $2C=19.60$.

Figure 11

(a) Wavy roll pattern and (b) the corresponding far-field image for $2C=19.9$.

Figure 12

Deviations of the wave vectors from that of maximum growth as a result of starting the scan at different distances from the threshold. Note that these values are for pure patterns, i.e., stable honeycombs forming before $2C=20$ and rolls thereafter. The missing parts in the curves are related to distorted honeycombs, bistable patterns, and wavy rolls.

Figure 13

Spontaneous formation of a gas of cavity solitons. We start from (a),(b) a roll solution, and then (c),(d) cross an unstable distorted hexagonal pattern to finally reach (e),(f) the stable cavity soliton gas. Parameter $2C=21.62$.

Figure 14

Formation of filaments from stable cavity soliton gas by decreasing the control parameter. (a) $2C=20.88$, (b) $2C=20.69$, and (c) $2C=20.65$.

Figure 15

Different progression of solutions as a consequence of selecting different initial conditions. The structures are stable honeycomb, stable roll, cavity solitons, distorted honeycomb, scrolling honeycombs, coexisting roll-honeycomb, and wavy rolls, respectively. Dashed arrows show the threshold for the patterns of the type which solid and longer arrows are pointed at. The sequence of scan in $2C$ for each set is shown by the arrows connecting the initial value, either from noise (points on the horizontal axis) or by using the memory of the last solution of a branch, to the set of solutions in branches. Note that cavity solitons are in a regime where the homogeneous background is stable, as seen from the right lowermost sequence of solutions, which starts from noise and shows the stability of the homogeneous branch up to $2C=20.89$, where a saddle-node bifurcation occurs.

Figure 16

Examples of transverse structures with a finite pump size with $\sigma =1.67/{\lambda }_c$ and $r_0=9{\lambda }_c$. (a) Honeycomb with defects $\left(2C=18.95\right)$, (b) coexistent honeycomb and roll structures $\left(2C=19.6\right)$, (c) filaments $\left(2C=20.50\right)$, and (d) gas of cavity solitons $\left(2C=21.55\right)$.