Modelling Spatial Modes of Squeezed Vacuum -- When it Comes to Squeezing, Plane Waves Are Just Too Plain

Recent research relying on the \textit{polarization self rotation} (PSR) effect in Rb$^{\mathrm{87\thinspace }}$has revealed a squeezed vacuum field consisting of several spatial modes with various squeezing parameters [Mi Zhang, Spatial Modes of a Squeezed Vacuum Field, 2015 APS March Meeting]. In order to explain these results, we re-derive the beam propagation model describing the creation of squeezed vacuum via PSR and incorporate more realistic multimode input-output relations in the paraxial approximation. We solve the propagation equation and use it to predict the spatial distribution of squeezed vacuum via the proper Laguerre-Gauss modal structure. This modal structure is instrumental in the development of a complete second quantized beam propagation formalism also being reported at this meeting [Zhihao Xiao, Second quantization of squeezed light through non-linear medium, 2015 APS March Meeting].