Expressions for the fields of transverse-magnetic laser beams in free space that are rigorous solutions to
Maxwell's equations are given in a closed form. The electric and the magnetic fields are both expressed in terms of
nonparaxial elegant Laguerre-Gaussian beams that are exact solutions of the Helmholtz equation. These solutions
involve spherical Bessel functions and associated Legendre functions. Radially polarized beams of arbitrary order are
considered and the lowest-order radially polarized beam (TM01 beam) is investigated in detail.