  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 $5p_{1/2}\rightarrow
6s_{1/2}$ 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 $\sim
10^{-3}$. 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.
