Bogdanov Map for Modelling a Phase-Conjugated Ring Resonator

Area of Science:

• Optics and Photonics
• Nonlinear Dynamics
• Chaos Theory

Background:

• Resonator systems are crucial in laser physics and optics.
• Understanding chaotic dynamics in optical systems is key for advanced applications.
• The Bogdanov Map is a fundamental model for studying chaos.

Purpose of the Study:

• To apply paraxial matrix optics to a ring-phase conjugated resonator with a chaos-generating element.
• To establish a connection between the resonator's behavior and the Bogdanov Map in phase space.
• To analyze the influence of intracavity element parameters on system dynamics.

Main Methods:

• Development of a theoretical model using paraxial matrix optics.
• Derivation of explicit expressions for intracavity chaos-generating matrix elements.
• Implementation of computer simulations to explore parameter configurations and bifurcation diagrams.

Main Results:

• The proposed model successfully describes the ring-phase conjugated resonator.
• Explicit matrix elements for the chaos-generating component were derived.
• Bifurcation diagrams showed transitions from periodic orbits to chaos, demonstrating rich dynamic behavior.
• A direct dependence of system dynamics on the intracavity element's parameters was confirmed.

Conclusions:

• Paraxial matrix optics provides an effective framework for analyzing chaotic resonators.
• The system's phase space dynamics are analogous to the Bogdanov Map.
• System parameters critically influence the emergence of complex dynamics, including chaos.