A population-competition model for analyzing transverse optical patterns including optical control and structural anisotropy

Abstract

We present a detailed study of a low-dimensional population-competition (PC) model suitable for analysis of the dynamics of certain modulational instability patterns in extended systems. The model is applied to analyze the transverse optical exciton-polariton patterns in semiconductor quantum well microcavities. It is shown that, despite its simplicity, the PC model describes quite well the competitions among various two-spot and hexagonal patterns when four physical parameters, representing density saturation, hexagon stabilization, anisotropy, and switching beam intensity, are varied. The combined effects of the last three parameters are given detailed considerations here. Although the model is developed in the context of semiconductor polariton patterns, its equations have more general applicability, and the results obtained here may benefit the investigation of other pattern-forming systems. The simplicity of the PC model allows us to organize all steady state solutions in a parameter space 'phase diagram'. Each region in the phase diagram is characterized by the number and type of solutions. The main numerical task is to compute inter-region boundary surfaces, where some steady states either appear, disappear, or change their stability status. The singularity types of the boundary points, given by Catastrophe theory, are shown to provide a simple geometric overview of the boundary surfaces. With all stable and unstable steady states and the phase boundaries delimited and characterized, we have attained a comprehensive understanding of the structure of the four-parameter phase diagram. We analyze this rich structure in detail and show that it provides a transparent and organized interpretation of competitions among various patterns built on the hexagonal state space.Beyond the primary instability of the spatially uniform state, the interplay among these wave-mixing processes gives rise to a variety of competing modulational patterns which may be regarded as optical analogs of perhaps more familiar patterns in macroscopic chemical, biological, and fluid dynamics systems [26][27][28][29]. Besides being fascinating nonlinear optical phenomena, the optical patterns can be conveniently controlled by weak optical probes and hence hold promise for applications in low-intensity optical switching [20,23,[30][31][32][33][34].We consider in this paper the pattern dynamics supported by the exciton-polariton field in a semiconductor quantum-well microcavity. Numerical simulations of these optical patterns, by directly solving the appropriate dynamical field equations, serve as the validating link between basic physical models and experiments. Nevertheless, the simulation models are usually too complex to allow easy gleaning of intuitive insight into the mechanisms of pattern competition and control. For better understanding, it would be helpful to construct models that are sufficiently flexible to capture, albeit qualitatively, the essential physics of the full simulation models and, at the same time, sufficiently simple ...

Externally controlled Lotka-Volterra dynamics in a linearly polarized polariton fluid

“…Rather, we will derive a simplified population competition model for selected modes in k-space (details of the derivation are given in Appendix B) to provide further insight into the underlying phase space singularities that dictate the global behaviour and solution space of the nonlinear dynamical system studied. In a similar fashion this approach was previously applied to the switching between subsets of a hexagonal pattern [22]. We will systematically analyze the existence and stability properties of possible steady states in dependence of the strength of the different involved physical processes for a typical orthogonal switching setup comparable to the one introduced above.…”

“…We follow qualitatively the derivation of the hexagon PC model [22], but here we are including polarization effects for linearly polarized excitation, and therefore considering a different reduced k-space. We start with the coupled equations of motion (1) for the exciton polarization and the cavity field and transform them into k-space and in the linear polarization basis.…”

Formation and all-optical control of optical patterns in semiconductor microcavities

“…For the case of a circularly polarized pump, the effects of these processes are analyzed in detail in. 18,22 When sufficient numbers of polaritons in both polarization channels are excited, as by a linearly polarized pump, TE-TM splitting and the spin dependence of the two-polariton interaction are also important factors in determining instability conditions and pattern selection. 21,23,28 Our theoretical simulations are based on a model that treats the cavity photon fields in each cavity as depending only on the 2D spatial coordinates in the cavity's plane.…”

“…Detailed analysis of these processes can be found in Refs. 18,22 simulation of a switching between a hexagon pattern and a two-spot pattern by control beams is shown in Fig. 4.…”

“…[12][13][14][15][16] ) which is optically pumped and can be subjected to further optical control, and the interactions among the polaritons drive the instability and govern the selection of the patterns. Patterns such as rolls and hexagons in a quantum well microcavity have been analyzed theoretically [17][18][19][20][21][22] and recently demonstrated experimentally. 23 The feasibility of optical control holds implications for possible opto-electronic applications.…”