The paper presents a paraxial theory of the relativistic cross-focusing
of two
coaxial Gaussian laser beams of different frequencies in a homogeneous
plasma.
We discuss the self-focusing of a weaker laser beam in the plasma due to
the
optical inhomogeneities introduced by another stronger copropagating laser
beam. In the presence of the second stronger beam
(Pcr21<P2<Pcr22),
the
plasma behaves as an oscillatory waveguide for the first, weaker, beam
(P1<Pcr11) as it propagates
in the plasma. When both the beams are strong
(Pcr11,21<P1,2<Pcr12,22),
the nonlinearities introduced by the relativistic effect are
additive in nature, such that one beam can undergo oscillatory self-focusing
and
the other simultaneously defocusing, and vice versa. A comparison reveals
that
cross-focusing due to relativistic nonlinearity is possible for a wider
range of
powers of the laser pulses than is cross-focusing due to ponderomotive
nonlinearity. Relativistic cross-focusing is important in plasma beat-wave
excitation and collective laser particle accelerators.