Perturbation theory for the wave equation and the "Effective refractive index" approach

We describe an Au and Aharonov type perturbation theory for the two-dimensional scalar wave equation. As an application, we show that the "effective refractive index" method is related to the first-order correction to the eigenstate. In addition, we compare results obtained for the gain in a typical laser structure using either the present theory or averaging the imaginary part of the refractive index over the fields ("average modal gain" ). The results agree to about 8 percent in a test case where the perturbative result is, in fact, exact. It is concluded that the perturbative approach will, in general, be more reliable.