  This work contains a systematic treatment of single particle Synchrotron
Radiation and some application to realistic beams with given cross section
area, divergence and energy spread. Standard theory relies on several
approximations whose applicability limits and accuracy are often forgotten. We
begin remarking that on the one hand, a paraxial approximation can always be
applied without loss of generality and with ultra relativistic accuracy. On the
other hand, dominance of the acceleration field over the velocity part in the
Lienard-Wiechert expressions is not always granted and constitutes a separate
assumption, whose applicability is discussed. Treating Synchrotron Radiation in
paraxial approximation we derive the equation for the slow varying envelope
function of the Fourier components of the electric field vector. Calculations
of Synchrotron Radiation properties performed by others showed that the phase
of the Fourier components of the electric field vector differs from the phase
of a virtual point source. In this paper we present a systematic, analytical
description of this phase shift, calculating amplitude and phase of electric
field from bending magnets, short magnets, two bending magnet system separated
by a straight section (edge radiation) and undulator devices. We pay particular
attention to region of applicability and accuracy of approximations used.
Finally, taking advantage of results of analytical calculation presented in
reduced form we analyze various features of radiation from a complex insertion
device (set of two undulators with a focusing triplet in between) accounting
for the influence of energy spread and electron beam emittance.
