In this paper, we have studied plane symmetric cosmological model with the isotropic matter distribution. This is done under the framework of \(f\left( {R,L_{m} } \right)\) gravity theory. This modified gravity theory, represented by \(f\left( {R,L_{m} } \right) = \frac{R}{2} + L_{m}^{\alpha } + \beta\) , which is a combination of both the Ricci scalar (R) and matter Lagrangian density \((L_{m} )\) , we consider \(L_{m} = \rho\) . Our goal was to analyse the behaviour of the expansion rate of the universe over time. We solved the field equation by assuming (i) the constant deceleration parameter (DP) proposed by Berman, i.e. q = constant. (ii) a time varying deceleration parameter \(q\left( t \right) = - 1 + \frac{n\zeta }{{\left( {\zeta + t} \right)^{2} }}\) , which yields \(a\left( t \right) = \left[ {t^{\zeta } e^{t} } \right]^{\frac{1}{n}}\) .We discussed the physical properties of the models. Additionally, we use the equation of state (EoS) parameter, the jerk parameter and the statefinder parameters as analytical tools to understand the evolution of the Universe in this modified gravity framework. This research offers significant insights into the anisotropic behaviour of the Universe within the framework of \(f\left( {R,L_{m} } \right)\) gravity, enhancing our comprehension of the fundamental dynamics of cosmic evolution.
