  We study here the nonlinear Schrodinger Equation (NLS) as the first term in a
sequence of approximations for an electromagnetic (EM) wave propagating
according to the nonlinear Maxwell equations (NLM). The dielectric medium is
assumed to be periodic, with a cubic nonlinearity, and with its linear
background possessing inversion symmetric dispersion relations. The medium is
excited by a current $\mathbf{J}$ producing an EM wave. The wave nonlinear
evolution is analyzed based on the modal decomposition and an expansion of the
exact solution to the NLM into an asymptotic series with respect to some three
small parameters $\alpha $, $\beta $ and $\varrho $. These parameters are
introduced through the excitation current $\mathbf{J}$ to scale respectively
(i) its amplitude and consequently the magnitude of the nonlinearity; (ii) the
range of wavevectors involved in its modal composition, with $\beta ^{-1}$
scaling its spatial extension; (iii) its frequency bandwidth, with $\varrho
^{-1}$ scaling its time extension. We develop a consistent theory of
approximations of increasing accuracy for the NLM with its first term governed
by the NLS. We show that such NLS regime is the medium response to an almost
monochromatic excitation current $\mathbf{J} $. The developed approach not only
provides rigorous estimates of the approximation accuracy of the NLM with the
NLS in terms of powers of $\alpha $, $\beta $ and $\varrho $, but it also
produces new extended NLS (ENLS) equations providing better approximations.
Remarkably, quantitative estimates show that properly tailored ENLS can
significantly improve the approximation accuracy of the NLM compare with the
classical NLS.
