We describe experiments where Fourier ptychographic microscopy (FPM) and dual-space microscopy (DSM) are implemented for imaging photonic crystals using a hemispherical digital condenser (HDC). Phase-recovery imaging simulations show that both techniques should be able to image photonic crystals with a period below the Rayleigh resolution limit. However, after processing the experimental images using both phase-recovery algorithms, we found that DSM can, but FPM cannot, image periodic structures with a period below the diffraction limit. We studied the origin of this apparent contradiction between simulations and experiments, and we concluded that the occurrence of unwanted reflections in the HDC is the source of the apparent failure of FPM. We thereafter solved the problem of reflections by using a single-directional illumination source and showed that FPM can image photonic crystals with a period below the Rayleigh resolution limit.