  Diffractive optical elements with a large diffraction angle require feature
sizes down to sub-wavelength dimensions, which require a rigorous
electromagnetic computational model for calculation. However, the computational
optimization of these diffractive elements is often limited by the large number
of design parameters, making parametric optimization practically impossible due
to large computation times. The adjoint method allows calculating the gradient
of the target function with respect to all design variables with only two
electromagnetic simulations, thus enabling gradient optimization. Here, we
present the adjoint method for modeling wide-angle diffractive optical elements
like 7x7 beam splitters with a maximum 53{\deg} diffraction angle and a
non-square 5x7 array generating beam splitter. After optimization we obtained
beam splitter designs with a uniformity error of 16.35% (7x7) and 6.98% (5x7),
respectively. After reviewing the experimental results obtained from fabricated
elements based on our designs, we found that the adjoint optimization method is
an excellent and fast method to design wide-angle diffractive fan-out
beam-splitters.
