  In [Phys. Rev. Lett. \textbf{117}, 213201 (2016)] we have determined the
angular resolved and the total energy spectrum of a positron produced via
nonlinear Breit-Wheeler pair production by a high-energy photon
counterpropagating with respect to a tightly focused laser beam. Here, we first
generalize the results in [Phys. Rev. Lett. \textbf{117}, 213201 (2016)] by
including the possibility that the incoming photon is not exactly
counterpropagating with respect to the laser field. As main focus of the
present paper, we determine the photon angular resolved and total energy
spectrum for the related process of nonlinear Compton scattering by an electron
impinging into a tightly-focused laser beam. Analytical integral expressions
are obtained under the realistic assumption that the energy of the incoming
electron is the largest dynamical energy of the problem and that the electron
is initially almost counterpropagating with respect to the laser field. The
crossing symmetry relation between the two processes in a tightly focused laser
beam is also elucidated.
