Farey sequences of spatiotemporal patterns in video feedback

Area of Science:

• Optoelectronics
• Nonlinear Dynamics
• Pattern Formation

Background:

• Optoelectronic systems can generate complex spatiotemporal patterns.
• Spatial transverse modes are observed in various optical devices.
• Previous observations of Farey sequences were limited to temporal dynamics.

Purpose of the Study:

• To experimentally and theoretically describe the dynamics of spatial patterns in a video feedback loop.
• To investigate the generation of n-fold symmetric stationary patterns.
• To explore the observation of Farey sequences in spatial patterns.

Main Methods:

• Utilizing a video feedback loop with a camera rotating around its optical axis.
• Controlling brightness and magnification to generate patterns.
• Analyzing pattern symmetries and rotational frequencies.
• Conducting experiments to observe patterns up to the k=6 level of the Farey hierarchy.

Main Results:

• Spatiotemporal patterns in the form of spots on a circle were produced.
• Stationary patterns with n-fold symmetries were generated at specific angles (alpha=2pi/n).
• Pattern rotation frequency was found to be proportional to the deviation from 2pi/n.
• General patterns with p-fold symmetry were observed at angles 2pi/(p/k), following the Farey algorithm.
• The Farey tree hierarchy was observed as a sequence of spatial patterns for the first time.

Conclusions:

• The video feedback loop system effectively generates complex spatial patterns.
• The observed patterns exhibit symmetries and dynamics analogous to transverse modes in other optical systems.
• This work establishes the first experimental observation of a Farey tree in spatial patterns, extending previous temporal observations.