To be self-guided, a beam must exactly equal the mode of the linear-optical fiber that it induces. From this elementary consistency condition we can borrow solutions and their associated physics directly from the familiar literature of linear-optical waveguides. By considering a nonlinear medium characterized by ideal saturation, we present what is to our knowledge the first exact analytical solution of a two-dimensional self-guided beam. This beam is the familiar fundamental mode of a step-profile fiber. The stability of the beam is also determined.
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