  This article presents an $O(N\log N)$ method for numerical solution of
Maxwell's equations for dielectric scatterers using a 3D boundary integral
equation (BIE) method. The underlying BIE method used is based on a hybrid
Nystr\"{o}m-collocation method using Chebyshev polynomials. It is well known
that such an approach produces a dense linear system, which requires $O(N^2)$
operations in each step of an iterative solver. In this work, we propose an
approach using the recently introduced Interpolated Factored Green's Function
(IFGF) acceleration strategy to reduce the cost of each iteration to $O(N\log
N)$. To the best of our knowledge, this paper presents the first ever
application of the IFGF method to fully-vectorial 3D Maxwell problems. The
Chebyshev-based integral solver and IFGF method are first introduced, followed
by the extension of the scalar IFGF to the vectorial Maxwell case. Several
examples are presented verifying the $O(N\log N)$ computational complexity of
the approach, including scattering from spheres, complex CAD models, and
nanophotonic waveguiding devices. In one particular example with more than 6
million unknowns, the accelerated IFGF solver runs 42x faster than the
unaccelerated method.
