Focusing a TM(01) beam with a slightly tilted parabolic mirror

A parabolic mirror illuminated with an incident collimated beam whose axis of propagation does not exactly coincide with the axis of revolution of the mirror shows distortion and strong coma. To understand the behavior of such a focused beam, a detailed description of the electric field in the focal region of a parabolic mirror illuminated with a beam having a nonzero angle of incidence is required. We use the Richards-Wolf vector field equation to investigate the electric energy density distribution of a beam focused with a parabolic mirror. The explicit aberration function of this focused field is provided along with numerically calculated electric energy densities in the focal region for different angles of incidence. The location of the peak intensity, the Strehl ratio and the full-width at half-maximum as a function of the angle of incidence are given and discussed. The results confirm that the focal spot of a strongly focused beam is affected by severe coma, even for very small tilting of the mirror. This analysis provides a clearer understanding of the effect of the angle of incidence on the focusing properties of a parabolic mirror as such a focusing device is of growing interest in microscopy.