The possibility of an optical vortex soliton in a bulk self-focusing Kerr medium follows trivially from the linear perspective [7]. It is well known that [18,30] the intensity of the second mode (one with one node) of a linear optical waveguide of circular symmetry appears as a local dark minimum in an otherwise bright background. Furthermore, this background is flat and of infinite extent when the mode is at its so-called cutoff frequency.
If an analogous self guided beam is to exist, then self consistency
demands its intensity be circularly symmetric and, allowing for
nonlinear Kerr material with induced birefringence [28], its
field must be linearly
or circularly polarized. From linear theory [18,30], then the
vector field
amplitude 
has the form 
for linearly polarized light, and 
for circularly polarized light. Here, 
are the radial and azimuthal position respectively and 
is a
solution of the radially symmetric wave equation, found numerically for
the Kerr nonlinearity, when 
is proportional to 
.
Numerical studies also show that no gray solitons of circular symmetry exist, only the black (vortex) soliton. A standard linear stability analysis shows that the vortex soliton is stable to arbitrary vector perturbations [7]. Vortex solitons are the only known stable soliton in a homogeneous Kerr medium. They are ideal for guiding a weak signal beam.