The electric and magnetic polarization singularities of paraxial waves

For superpositions of electromagnetic plane waves in space whose directions span a small angular range θ, the circular polarization line singularities (of the full and transverse electric and magnetic fields) form clusters, each of four closely-spaced lines. The separations of lines in each cluster are calculated analytically in terms of the transverse electric field and are of the order of λθ—much smaller than the wavelength λ and smaller still than the transverse scale λ/θ of the intensity variations of the field. The electric and magnetic surfaces of transverse linear polarization also lie close together and the separation is calculated analytically, as are the positions on these surfaces of the lines of linear polarization of the full fields. To the two lowest orders in θ, the local wavevector (geometric phase 1-form) is the same for the electric and magnetic fields. The sub-wavelength singular structures are illustrated by numerical calculations.