Intrinsic mode formalism, its asymptotical evaluations and uniform adiabatic mode concept in non-homogeneous environments

In this study, intrinsic mode representations are utilized to describe wave propagation in a three dimensional non-homogeneous medium wherein the refractive index varies parabolicaly with both transverse coordinates x, y and linearly with the longitudinal coordinate z. Assuming weak dependence on z, local separability is invoked to define adiabatic modes which are subsequently generalized into intrinsic mode spectral integrals. This uniform integral is evaluated asymptotically with stationary phase method, steepest descent path integration and uniform asymptotics. The stationary phase or the steepest descent path evaluation of the intrinsic mode spectral integral yields the adiabatic modes which can describe the wave phenomena in weakly non-separable environments up to the critical transition regions where drastic changes occur in the modal fields. The uniform asymptotic evaluation of the intrinsic mode integral yields a solution which is valid everywhere including the critical transitions. This asymptotical evaluation is used to introduce an uniform adiabatic mode concept which may be constructed in a straightforward manner directly from the adiabatic mode fields. Uniform adiabatic modes are valid as well as accurate representations which overcome the deficiency of the adiabatic modes through critical transition regions and ease the requirement of the evaluation of double, sometimes triple spectral contour integrals in intrinsic mode representation.