We present a detailed analysis of the structure of strongly focused, radially polarized electromagnetic fields. The existence of phase singularities of the two components of the electric field is demonstrated. Two different mechanisms to obtain creation or annihilation of these phase singularities are discussed. These are changing the aperture angle of the lens and the width of the beam. Also, it is shown that in the focal plane the handedness of the electric polarization ellipse is an alternating function of the radial distance. Finally, the different contributions to the electric energy density are examined. © 2006 Optical Society of America.