The influence of the order parameter's phase fluctuations on the Meissner effect is studied for strong anisotropic layered superconductors in the presence of a weak magnetic field, H, less than the lower critical field H-c1 < H-c1). The Josephson coupling between the layers can be realized in quasi-two dimensional (quasi-2d) superconductors when the interlayer tunnelling integral (J perpendicular to) satisfies the condition J(perpendicular to) < kappa T-c((2)) < epsilon(F), where epsilon(F) and T-c((2)) art the Fermi energy and the mean-field transition temperature for a single superconducting layer, respectively;. The system of equations of motion is obtained for the order parameter phases phi(j)((r) over right arrow) at the r-th point of the j-rh layer. This system of coupled equations is investigated by applying the self consistent phonon approximation (SCPA) method. There exists the plasmon mode in the Josephson coupled layered superconductors in the frame of the SCPA approximation. The square of the transverse effective velocity, v(ph,perpendicular to), of collective excitations become proportional to the interlayer phase-phase correlator [cos(phi(1)-phi(1-1))]. When this correlator approaches zero, correlations of the superconducting phases on the different layers disappear, i.e. the phase transition from quasi-2d superconducting state to a pure 2d state occurs at the critical temperature T-c1. The transverse rigidity of the system and the plasmon's effective velocity v(ph.perpendicular to)vanish at this temperature. T-c1 is less than T-c((2)) anti (T-c((2)) - T-c1)similar to T-c((2))(kappa T-c((2))(kappa T-c((2))/epsilon(p). In the temperature interval of (T-c1-T)similar to T-c1(kappa T-c1/epsilon(F)) ln (kappa-T-c1/J(perpendicular to)) below T-c1, the fluctuations of the order parameter's phases become essential in the quasi-2d superconductor. In this interval the ratio of the longitudinal lambda(parallel to) and transverse lambda(perpendicular to) components of the London penetration depths, lambda(parallel to)/lambda(perpendicular to), should exhibit strong temperature dependence, unlike the prediction of the usual Ginzburg-Landau phenomenological theory. lambda(parallel to) and lambda(perpendicular to) critical temperatures. namely at T-c2=T-c((2)) and T-c1, respectively. At T-c1 < T < T-c2 the phases phi(j) on the different layers become non-correlative and the Kosterlitz-Thouless vortices appear in each superconducting layers. (C) 1998 Elsevier Science B.V.