Farey sequences of spatiotemporal patterns in video feedback.

In this paper we present an experimental and theoretical description of the dynamic of spatial patterns obtained in a video feedback loop. A video camera monitors the screen to which it is connected and can turn around its optical axis at an angle alpha. Under certain conditions of brightness and magnification, this optoelectronic system produces spatiotemporal patterns in the form of spots located on a circle on the screen. These patterns are very similar to the spatial transverse modes obtained in other optical devices such as lasers or photorefractive media. It is possible to generate stationary patterns of n-fold symmetries for angles alpha=2pi/n. When the angle alpha varies around 2pi/n, the pattern rotates with a certain frequency proportional to the difference between 2pi/n and alpha. We discover more general patterns at angles 2pi/(p/k) with p-fold symmetry, following the hierarchy of the Farey algorithm which theoretically can produce stationary patterns at any angle alpha. Very accurate experiments were performed to observe these patterns up to the level k=6. This is the first time a Farey tree has been observed as a sequence of spatial patterns to our knowledge. Previous observations of this hierarchy were made only in the temporal domain.