  We report Doppler-free two-photon absorption of N$_2$O at $\lambda$ = 4.53
$\mu$m, measured by cavity ring-down spectroscopy. High power was achieved by
optical self-locking of a quantum cascade laser to a linear resonator of
finesse $F$ = 22730, and accurate laser detuning over a 400 MHz range was
measured relative to an optical frequency comb. At a sample pressure of p =
0.13 kPa, we report a large two-photon cross-section of $\sigma_{13}^{(2)}$ =
8.0 $\times$ 10$^{-41}$ cm$^4$ s molecule$^{-1}$ for the $Q$(18) rovibrational
transition at a resonant frequency of $\nu_0$ = 66179400.8 MHz.
