  This paper presents an analysis of near-resonant, ro-vibrational two-photon
spectroscopy and the use of cavity ring-down spectroscopy for its detection.
Expressions are derived for the photon absorption rate of a three-level system,
correct to all orders and the simpler expressions that result from various
approximations. The analysis includes the angular momentum projection
degeneracies and linear or circular polarization of the exciting field.
Expressions are derived for the rate of two-photon power loss for light inside
a resonant cavity. Explicit calculations are made for excitation of the $\nu_3$
mode of $^{12}\textrm{C}^{16}\textrm{O}_2$ for which the two-photon excitation
spectrum is dominated by a single $v_3 = 0 \rightarrow 2, Q(16)$ line at
$\tilde{\nu} = 2335.826$\,cm$^{-1}$. This transition has an intermediate $v_3 =
0 \rightarrow 1,P(16)$ one-photon transition that is off resonance by 0.093
cm$^{-1}$ (2.8 GHz). At 1\,torr total pressure, the Q(16) two-photon transition
has a calculated cross-section of $2.99 \cdot 10^{-38}$\,cm$^4$s per CO$_2$
molecule in the $J = 16$ state or $2.24 \cdot 10^{-39}$\,cm$^4$s per CO$_2$
molecule at 300\,K is calculated. Analysis of the sensitivity limits for
2-photon cavity ring-down spectroscopy predicts a theoretical detection limit
of 32\,ppq ($10^{-15}$) Hz$^{-1/2}$ for $^{12}\textrm{C}^{16}\textrm{O}_2$,
higher sensitivity than has been realized using one-photon absorption. The
analysis predicts that most polyatomic molecules will have sparse, Doppler-Free
two-photon absorption spectra, which will dramatically increase the selectivity
of trace gas detection of samples with multiple components with overlapping
absorption bands. This is demonstrated by the predicted mid-IR two-photon
absorption spectrum of butadiene using theoretical spectroscopic constants.
