  We investigate a high-order homogenization (HOH) algorithm for periodic
multilayered stacks. The mathematical tool of choice is a transfer matrix
method. Expressions for effective permeability, permittivity and
magnetoelectric coupling are explored by frequency power expansions. On the
physical side, this high-order homogenization uncovers a magnetoelectric
coupling effect (odd order approximation) and artificial magnetism (even order
approximation) in moderate contrast photonic crystals. Comparing the effective
parameters' expressions of a stack with three layers against that of a stack
with two layers, we note that the magnetoelectric coupling effect vanishes
while the artificial magnetism can still be achieved in a center symmetric
periodic structure. Furthermore, we numerically check the effective parameters
through the dispersion law and transmission curves of a stack with two
dielectric layers against that of an effective bianisotropic medium: they
present a good agreement in the low frequency (acoustic) band until the first
stop band, where the analyticity of the logarithm function of transfer matrix
($\log\{T\}$) breaks down.
