In this study, we investigate the impact of Doppler broadening on a four-level atomic system configured in a combined closed-loop \(\Delta \nabla \) scheme. The primary advantage of this scheme is the presence of microwave and radio-frequency fields, which facilitate quantum interference between various transition pathways. Using density matrix formalism and the optical Bloch equations, we analyze the absorption and dispersion characteristics under Doppler-free and Doppler-broadened conditions. Our simulations reveal the effects of thermal averaging at different temperatures and the role of control field strength, microwave, and radio-frequency fields in modifying the probe absorption and dispersion profiles. The findings highlight significant phase-dependent variations in optical coherence and demonstrate how Doppler broadening influences transparency windows, absorption peaks, and dispersion properties. These results provide valuable insights into phase and amplitude control in atomic systems, with potential applications in quantum optics and precision metrology.
