Free electron lasers have been the subject of intensive interest during the recent decades. In this paper, free electron laser having sheet electron beam with arbitrary inhomogeneous profile of transverse distribution of the beam current density is studied in the linear regime, whereas a novel approach based on the Legendre polynomial expansion of eigenfunctions, already used in analyzing optical structures including stratified structures and diffraction gratings, is adapted to find the eigenfunctions and eigenvalues of the structure. As for this method is unconditionally stable, it works pretty well even in those cases in which the conventional transfer matrix method suffers from numerical instability, i.e. the detuning parameter is negative with a large absolute value. Though the used formulation, obtained by solving Maxwell's and Vlasov's equations simultaneously, is limited to the linear regime, it includes the effects of energy spread and space charge fields.