  Dispersion engineering of microring resonators is crucial for optical
frequency comb applications, to achieve targeted bandwidths and powers of
individual comb teeth. However, conventional microrings only present two
geometric degrees of freedom -- width and thickness -- which limits the degree
to which dispersion can be controlled. We present a technique where we tune
individual resonance frequencies for arbitrary dispersion tailoring. Using a
photonic crystal microring resonator that induces coupling to both directions
of propagation within the ring, we investigate an intuitive design based on
Fourier synthesis. Here, the desired photonic crystal spatial profile is
obtained through a Fourier relationship with the targeted modal frequency
shifts, where each modal shift is determined based on the corresponding
effective index modulation of the ring. Experimentally, we demonstrate several
distinct dispersion profiles over dozens of modes in transverse magnetic
polarization. In contrast, we find that the transverse electric polarization
requires a more advanced model that accounts for the discontinuity of the field
at the modulated interface. Finally, we present simulations showing arbitrary
frequency comb spectral envelope tailoring using our Frequency synthesis
approach.
