  We experimentally realize higher-order catastrophic structures in light
fields to access the rich class of caustic swallowtail and butterfly beams.
These beams present solutions of paraxial diffraction catastrophe integrals
that are determined by potential functions, whose singular mapping manifests as
caustic hypersurfaces in control parameter space. We systematically analyze the
swallowtail and butterfly beams' caustics analytically and observe their field
distributions experimentally in real and Fourier space. Their spectra can be
described by polynomials or expressions with rational exponents capable to form
a cusp.
