Scattering of the dominant guided mode due to end discontinuity in step index optical fibers

The electromagnetic scattering of the dominant guided mode of a step-index optical fiber due to a smoothed end discontinuity with free space is treated rigorously by using the mode-matching technique. The total field inside the optical fiber as well as in the free space region are expressed in terms of the complete spectrum modes of the optical fiber and the radiation modes of free space, respectively. The mode matching technique is applied to match the transverse fields on the sides of the discontinuity plane resulting in a system of singular integral equations (SSIE) for the unknown reflection and transmission coefficients of the different modes. These are expressed in terms of sub-domain pulse basis functions over the domain of the radial propagation constant with unknown amplitudes. With a special treatment of the Kernel singularities the SSIE is converted into a finite system of linear equations for the unknown amplitudes of the basis functions. The far field patterns in the free space region are obtained using the saddle-point integration method. Also, the power reflected back to the optical fiber section (guided and radiated) is evaluated and found equal to that calculated using Fresnel reflection approximation. The power balance is verified, where the summation of the transmitted and reflected powers are found equal to the incident power