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A scheme for beat wave excitation of electron plasma wave (EPW) is proposed by relativistic cross-focusing of two coaxial Cosh-Gaussian (ChG) laser beams in an under dense plasma. The plasma wave is generated on account of beating of two coaxial laser beams of frequencies ω1 and ω2. The mechanism for laser produced nonlinearity is assumed to be relativistic nonlinearity in electron mass. Following moment theory approach in Wentzel Kramers Brillouin (W.K.B) approximation, the coupled differential equations governing the evolution of spot size of laser beams with distance of propagation have been derived. The relativistic nonlinearity depends not only on the intensity of first laser beam but also on the intensity of second laser beam. Therefore, propagation dynamics of one laser beam affect that of second beam and hence cross-focusing of the two laser beams takes place. Due to non uniform intensity distribution of pump laser beams, the background electron concentration gets modified. The amplitude of EPW, which depends on the background electron concentration, thus gets nonlinearly coupled with the laser beams. The effects of relativistic electron mass nonlinearity and the cross-focusing of pump beams on excitation of EPW have been incorporated. Numerical simulations have been carried out to investigate the effect of laser as well as plasma parameters on cross-focusing of laser beams and further its effect on power of excited EPW.

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