In traarent media, nonlinear propagation of ultrashort, high-power laser pulses can result in filaments. Filamentation describes the capability of an intense ultrashort laser pulse to remain focused over a long distance. Filamentation arises from the dynamic balance between Kerr self-focusing and defocussing by negative higher- order Kerr effect or self-induced plasma. In this study, we use the vector wave equation and consider the vectorial (polarization) effects; the polarization corresponds to an ensemble average of the atomic or molecular dipole moments induced by the laser field. The growth rate of filamentation of laser pulses and their interaction with plasma have been theoretically investigated. Results show that after adding the vectorial effect, the growth rate of filamentation increased considerably. For over years, the standard explanation for multiple filamentation, has been that it is initiated by random noise in the input-beam profile. On the other hand, another explanation states that multiple filamentation is created due to vectorial effects. Numerical simulations of the scalar equation show that when the input beam is sufficiently powerful, vectorial effects lead to multiple filamentation. Regarding works performed on the phenomenon of multiple filamentation, nonlinear polarization appears to be a strong determinant of multiple filamentation. The simulations performed in this regard show that the nonlinear polarization is the main cause of multiple filamentation. In this study, we prove that vectorial effect can also lead to an increase in the growth rate of filamentation. Using the vector wave equation, the growth rate was calculated considering the vectorial effects in this paper. In fact, this study was conducted to examine the effect of nonlinear polarization on the growth rate of filamentation. To this end, the growth rate of filamentation was illustrated and evaluated in different states. Therefore, the polarization was considered as linear and nonlinear terms. First, only the linear term was considered and the growth rate was calculated. Then, the non-linear polarization was added to the calculations to see the effects of the nonlinear part. Our research question was whether the application of nonlinear polarization increases the growth rate of filamentation
