We address the problem of achromatization of an optical system for the realization of planar-integrated, free-space optics. In particular we demonstrate an integrated optical Fourier transformation module that was achromatized for the visible spectrum by means of a diffractive lens doublet. The optical system design is studied by using the parabolic approximation of the scalar diffraction theory, including terms related to astigmatism. Based on the method of ABCD ray matrices, the optical specifications of the lens doublet are derived and the chromatic correction effect is quantified. For experimental confirmation the diffraction patterns of various grating structures are evaluated.