With the increasing interest in establishing directional etching methods capable of atomic scale resolution for fabricating highly scaled electronic devices, the need for development and characterization of atomic layer etching processes, or generally etch processes with atomic layer precision, is growing. In this work, a flux-controlled cyclic plasma process is used for etching of SiO2 and Si at the Angstrom-level. This is based on steady-state Ar plasma, with periodic, precise injection of a fluorocarbon (FC) precursor (C4F8 and CHF3) and synchronized, plasma-based Ar+ ion bombardment [D. Metzler et al., J. Vac. Sci. Technol., A 32, 020603 (2014) and D. Metzler et al., J. Vac. Sci. Technol., A 34, 01B101 (2016)]. For low energy Ar+ ion bombardment conditions, physical sputter rates are minimized, whereas material can be etched when FC reactants are present at the surface. This cyclic approach offers a large parameter space for process optimization. Etch depth per cycle, removal rates, and self-limitation of removal, along with material dependence of these aspects, were examined as a function of FC surface coverage, ion energy, and etch step length using in situ real time ellipsometry. The deposited FC thickness per cycle is found to have a strong impact on etch depth per cycle of SiO2 and Si but is limited with regard to control over material etching selectivity. Ion energy over the 20-30 eV range strongly impacts material selectivity. The choice of precursor can have a significant impact on the surface chemistry and chemically enhanced etching. CHF3 has a lower FC deposition yield for both SiO2 and Si and also exhibits a strong substrate dependence of FC deposition yield, in contrast to C4F8. The thickness of deposited FC layers using CHF3 is found to be greater for Si than for SiO2. X-ray photoelectron spectroscopy was used to study surface chemistry. When thicker FC films of 11 Å are employed, strong changes of FC film chemistry during a cycle are seen whereas the chemical state of the substrate varies much less. On the other hand, for FC film deposition of 5 Å for each cycle, strong substrate surface chemical changes are seen during an etching cycle. The nature of this cyclic etching with periodic deposition of thin FC films differs significantly from conventional etching with steady-state FC layers since surface conditions change strongly throughout each cycle.