Pattern formation in a passive Kerr cavity

Abstract

Analytic and numerical investigations of a cavity containing a Kerr medium are reported. The mean field equation with plane-wave excitation and diffraction is assumed. Stable hexagons are dominant close to threshold for a self-focusing medium. Bistable switching frustrates pattern formation for a self-defocusing medium. Under appropriate parametric conditions that we identify, there is coexistence of a homogeneous stationary solution, of a hexagonal pattern solution and of a large (in principle infinite) number of localized structure solutions which connect the homogeneous and hexagonal state. Further above threshold, the hexagons show defects, and then break up with apparent turbulence. For Gaussian beam excitation, the different symmetry leads to polygon formation for narrow beams, but quasihexagonal structures appear for broader beams.

Pattern selection in optical bistability

“…A weakly nonlinear analysis and a relative stability analysis have been published [3,4]. It turns out that the analytic work published on patterns and pattern selection in the LL model is less developed [5,6]. It is the purpose of this paper to bridge this gap.…”

Spatial correlations in hexagons generated via a Kerr nonlinearity

“…The input fluctuation operators at RHS are in the vacuum state and have commutation relations as those in Eq. (15). This represents a 14 × 14 linear problem, which is trivial from a numeric point of view; however, when searching for some explicit combination of mode operators that has sub-shot-noise fluctuations, the problem seems too complex to find analytical solutions.…”

“…As it is well know [4,15] the instability is subcritical, and the hexagonal mode amplitude shows the typical hysteresis cycle. Figure 3b is the same for the amplitude |β 0 | of the homogeneous mode.…”

“…Among the transverse modes on the critical circle | k| = k c , the Kerr nonlinearity selects a discrete set of six wave vectors [4,15], so that immediately above the critical point the near field intensity distribution consists of a hexagonal lattice of bright spots , while the far field shows a central bright spot surrounded by six spots in hexagonal arrangement. The origin of this particular pattern can be understood by considering the microscopic processes leading to off-axis emission of photons allowed by energy-momentum conservation [16].…”

Spontaneous motion of localized structures and localized patterns induced by delayed feedback

“…They may either be isolated, randomly distributed or self-organized in clusters forming a well-defined spatial pattern. In two-dimensional (2D) settings, theoretical prediction of high degree of multistability between structures having different number of peaks was established for driven nonlinear planar cavities [6][7][8][9][10][11]. This prediction has been confirmed by experimental evidence of DLS in various nonlinear optical systems [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27].…”