Polarization critical points in polychromatic optical fields

The polarization singularities and stationary points (collectively critical points) of paraxial ellipse fields are well understood. But only monochromatic light can generate an ellipse field, whereas all other forms of light generate polarization figures that are more complex than ellipses. We call such figures Lissajous figures. We show that the critical points of an arbitrary, polychromatic, paraxial Lissajous field can be represented in two different, but complementary, ways: as the critical points of the phase of a complex Stokes field, and as the critical points of the coherency ellipses that characterize the parent optical field.