4π Focusing of TM(01) beams under nonparaxial conditions

The Richards-Wolf theory and the complex point-source method are both used to express the phasor of the electric field of tightly focused beams, but the connection between these two approaches is not straightforward. In this paper, the Richards-Wolf vector field equations are used to find the electromagnetic field of a TM(01) beam in the neighborhood of the focus of a 4π focusing system, such as a parabolic mirror with infinite transverse dimensions. Closed-form solutions are found for the distribution of the fields at any point in the vicinity of the focus; these solutions are identical to the electromagnetic field obtained with the complex source-point method in which sources are accompanied by sinks. This work thus establishes a connection between the Richards-Wolf theory and the complex sink/source model. The vector magnetic potential is introduced to simplify the computation of the six electromagnetic field components. The method is then used to find analytical expressions for the electromagnetic field of strongly focused TM(01) beams affected by primary aberrations such as curvature of field, coma, astigmatism and spherical aberration.