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IMA Journal of Applied Mathematics Advance Access originally published online on October 23, 2008
IMA Journal of Applied Mathematics 2008 73(6):902-935; doi:10.1093/imamat/hxn026
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© The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Non-linear stability analyses of optical pattern formation in an atomic sodium vapour ring cavity

D. J. Wollkind{dagger}

Department of Mathematics, Washington State University, Pullman, WA 99164-3113, USA

F. J. Alvarado

Departamento de Ciencias Basicas, Tecnologico de Monterrey, Campus Guadalajara, Zapopan, Jalisco C.P. 45201, Mexico

D. E. Edmeade

Department of Mathematics, Washington State University, Pullman, WA 99164-3113, USA

{dagger} Email: dwollkind{at}wsu.edu

Received on February 7, 2007; Revision received April 3, 2008. Accepted on July 23, 2008

The development of spontaneous stationary equilibrium patterns induced by the injection of a laser pump field into a purely absorptive two-level atomic sodium vapour ring cavity is investigated by means of various weakly non-linear stability analyses applied to the appropriate governing evolution equation for this optical phenomenon. In the quasi-equilibrium limit for its atomic variables, the mathematical system modelling that phenomenon can be reduced to a single modified Swift–Hohenberg non-linear partial differential time-evolution equation describing the intracavity field on an unbounded 2D spatial domain. Diffraction of radiation can induce transverse patterns consisting of stripes, squares and hexagonal arrays of bright spots or honeycombs in an initially uniform plane-wave configuration. Then, these theoretical predictions are compared with both relevant experimental evidence and existing numerical simulations from some recent non-linear optical pattern formation studies.

Keywords: rhombic and hexagonal optical pattern formation; purely absorptive two-level atomic sodium vapour ring cavity; Swift–Hohenberg equation; non-linear stability analyses.


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