Analytic solutions for linearly tapered directional couplers

Analytic solutions are presented for the mode amplitudes in two-mode couplers which have a constant coupling coefficient and propagation coefficients that vary linearly. The solutions are obtained in terms of parabolic cylinder functions which reduce, asymptotically toward the ends of the coupler, to simpler forms in elementary functions. These lead to a simple analytic expression for the coupler efficiency, and also to quantitative criteria for the coupler length.