  An efficient contour integral technique to approximate a cluster of nonlinear
eigenvalues of a polynomial eigenproblem, circumventing certain large
inversions from a linearization, is presented. It is applied to the nonlinear
eigenproblem that arises from a frequency-dependent perfectly matched layer.
This approach is shown to result in an accurate method for computing leaky
modes of optical fibers. Extensive computations on an antiresonant fiber with a
complex transverse microstructure are reported. This structure is found to
present substantial computational difficulties: Even when employing over one
million degrees of freedom, the fiber model appears to remain in a
preasymptotic regime where computed confinement loss values are likely to be
off by orders of magnitude. Other difficulties in computing mode losses,
together with practical techniques to overcome them, are detailed.
