General eigenvalue equations for optical planar waveguides with arbitrarily graded-index profiles

Accurate eigenvalue equations for planar waveguides with arbitrarily graded-index profile are derived and expressed in closed forms. A combination of the modified Airy functions and the Wenzel-Kramers-Brillouin (WKB) solutions are employed as field solutions, which turn out to represent almost exact field profiles. The use of new trial solutions enables us to calculate phase shifts at turning points very precisely, allowing us almost exact eigenvalues. It is demonstrated that the results obtained by the proposed method are in excellent agreement with those by the finite element method, achieving significant improvement over the conventional WKB method.