  We propose high-order FDTD schemes based on the Correction Function Method
(CFM) for Maxwell's interface problems with discontinuous coefficients and
complex interfaces. The key idea of the CFM is to model the correction function
near an interface to retain the order of a finite difference approximation. For
this, we solve a system of PDEs based on the original problem by minimizing an
energy functional. The CFM is applied to the standard Yee scheme and a
fourth-order FDTD scheme. The proposed CFM-FDTD schemes are verified in 2-D
using the transverse magnetic mode (TM$_z$). Numerical examples include
scattering of magnetic and non-magnetic dielectric cylinders, and problems with
manufactured solutions using various complex interfaces and discontinuous
piecewise varying coefficients. Long-time simulations are also performed to
provide numerical evidences of the stability of the proposed numerical
approach. The proposed CFM-FDTD schemes achieve up to fourth-order convergence
in $L^2$-norm and provide approximations devoid of spurious oscillations.
