Theory of randomly inhomogeneous waveguides with slight material dispersion: II. Parabolic graded-index fibres

Abstract

This paper, following directly from Part I, continues to demonstrate the use of the paraxial approximation in solving the time-dependent wave equation for the propagation of signals in slightly dispersive optical materials with some Gaussian randomness of the permittivity. With the same emphasis as in Part I, a path-integral formulation is employed. Attention is focused on a parabolic graded-index fibre with ⟨1/nk 0( r )⟩=(1+αr 2 ⊥/2)/n¯k 0, α>0. Arising from the possibility of replacing the autocorrelation function of the permittivity by a quadratic function, some final concise expressions are derived. It is shown that if a signal begins propagating with a Gaussian power density ⟨|u( r , 0)|2⟩, the Gaussianity of the function ⟨|u( r , t)|2⟩ is maintained at all later times t. The longitudinal shape of the function ⟨|u( r , t)|2⟩ exhibits attenuation and broadening dependent on the dispersiveness of the optical medium under consideration. The perpendicular shape of the function ⟨|u( r , t)|2⟩ oscillates with the angular frequency Ω⊥=[cvx (k 0α/n¯ k 0 ]1/2 where c and vx (k 0) are, respectively, the velocity of light in vacuum and the axial group velocity of light corresponding to kx =k 0. Special attention is devoted to the limiting case when α→0, i.e. to propagation in a bulk dispersive optical medium.