In this paper, we give a rigorous analysis of the finite difference time domain (FDTD) method with high accuracy in time (HAIT) (named HAIT-FDTD(M)) for the three dimensional Maxwell equations, where the time discretization is based on the Taylor expansion of the form: $U^n=C^0_n + C^1_n ∆t + · · · + \frac{1}{ M!}C^M_n (∆t)^M$ to approximate the fields in time. It is proven that the solutions of the schemes and the vectors representing the coefficients are divergence free. By using the energy method, the numerical energy identities of HAIT-FDTD(M) with $3 ≤ M ≤ 8$ are derived. It is then proved that these schemes are numerically and monotonically energy conserved as the polynomial degree $M$ becomes large. With the help of the energy identities, stability conditions for the six schemes are derived, and how to select $M$ and $∆t$ in practice is given. By deriving error equations, we prove that the six schemes have convergence of the $M{\rm th}$ order in time and the second order in space. Numerical experiments are provided and confirm the analysis on free divergence, approximate energy conservation, stability, and convergence.