  Results from higher order mean field calculations of light interacting with
atom arrays are presented for calculations of one- and two-time expectation
values. The atoms are approximated as two-levels and are fixed in space.
Calculations were performed for mean field approximations that include the
expectation value of one operator (mean field), two operators (mean field-2),
and three operators (mean field-3). For the one-time expectation values, we
examined three different situations to understand the convergence with
increasing order of mean field and some limitations of higher order mean field
approximations. As a representation of a two-time expectation value, we
calculated the $g^{(2)}(\tau )$ for a line of atoms illuminated by a
perpendicular plane wave at several emission angles and two different
intensities. For many cases, the mean field-2 will be sufficiently accurate to
quantitatively predict the response of the atoms as measured by one-time
expectation values. However, the mean field-3 approximation will often be
needed for two-time expectation values.
