Expressions for the fields of TM and TE 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 well-known functions, such as spherical Bessel and associated Legendre functions. Radially and azimuthally polarized beams of arbitrary order are considered, and the lowest-order radially polarized beam (TM(01) beam) is investigated in detail.