A modified WKB method for the improved phase shift at a turning point

With the presence of an index or/and index slope discontinuity in a planar waveguide, the phase shift at the turning point which is near or at the discontinuity is no longer a constant (π/2) as in the conventional WKB method. In this paper, we present analytical expressions of this phase shift. Not only the index discontinuity, but also the index slope discontinuity have been taken into account in deriving these expressions. Numerical results show a significant improvement over the conventional WKB method after a modified phase shift has been used. According to our results, the phase shift is π/6 instead of π/2 or zero at the cutoff for a waveguide with a truncated index profile. As a by-product, we found a more accurate phase shift expression for the total internal reflection at an index discontinuity where the index slopes on both sides of the discontinuity have been considered