  We discuss a time-harmonic inverse scattering problem for a nonlinear
Helmholtz equation with compactly supported inhomogeneous scattering objects
that are described by a nonlinear refractive index in unbounded free space.
Assuming the knowledge of a nonlinear far field operator, which maps Herglotz
incident waves to the far field patterns of corresponding solutions of the
nonlinear scattering problem, we show that the nonlinear index of refraction is
uniquely determined. We also generalize two reconstruction methods, a
factorization method and a monotonicity method, to recover the support of such
nonlinear scattering objects. Numerical results illustrate our theoretical
findings.
