Durnin-Whitney beams

The aim of the present work is to define a Durnin-Whitney beam as a nondiffracting beam such that its associated caustic locally only has singularities of the fold and cusp types. Since the caustic is structurally stable then the intensity pattern of this beam is also stable and this property is what makes its definition and its theoretical and experimental study worthwhile. These properties are important in applications such as uniform optical drilling in waveguides and communications through weak turbulent atmosphere. We find that in accordance with Whitney's theorem on the stability of maps from a two-dimensional manifold to a two-dimensional manifold the phase g ( Φ ) , of the complex function A ( Φ ) characterizing the beam, locally is given by g ( Φ ) = a Φ for a fold and g ( Φ ) = b Φ 2 for a cusp. This result implies that the Bessel beam of order zero is not stable and that any other Bessel beam is stable because locally it has a caustic of fold type. Finally, we present an example of a Durnin-Whitney beam given by g ( Φ ) = m Φ + b Φ 2 , which is a natural generalization of the Bessel beam of order m with a singularity of cusp ridge type.