We present a detailed analysis of an application of frequency modulation (FM) spectroscopy in the high modulation depth limit. We have recently completed and reported a measurement of the Stark shift in the indium 5p 1/2 → 6s 1/2 410 nm transition using this spectroscopy method [Ranjit, et al. Phys. Rev. A 87, 032506 (2013)]. FM spectroscopy proved essential to resolve spectroscopic features in an atomic beam where the optical depth was ∼ 10 −3 . A dual-modulation scheme is described which ensures truly background-free FM signals even in the low-density limit. Lock-in detection of the FM signal was accomplished using both the fundamental (1f) and second-harmonic (2f) modulation frequencies. A derivation of both the 1f and 2f signal line shapes in the high-modulation-depth limit is presented. These line shapes form the basis for quantitative fits to a wide variety of experimental FM spectra.