An important step in estimating the index of refraction of electromagnetic
scattering problems is to compute the associated transmission eigenvalue problem.
We develop in this paper efficient and accurate spectral methods for computing the
transmission eigenvalues associated to the electromagnetic scattering problems. We
present ample numerical results to show that our methods are very effective for com-
puting transmission eigenvalues (particularly for computing the smallest eigenvalue),
and together with the linear sampling method, provide an efficient way to estimate the
index of refraction of a non-absorbing inhomogeneous medium.