Pulse area theorem in a single mode waveguide and its application to photon echo and optical memory in Tm3+:Y3Al5O12

We derive the area theorem for light pulses interacting with inhomogeneously broadened ensemble of two-level atoms in a single-mode optical waveguide and present its analytical solution for Gaussian-type modes, which demonstrates the significant difference from the formation of $2\pi$ pulses by plane waves. We generalize this theorem to the description of photon echo and apply it to the two-pulse (primary) echo and the revival of silenced echo (ROSE) protocol of photon echo quantum memory. For the first time, we implemented ROSE protocol in a single-mode laser-written waveguide made of an optically thin crystal $Tm^{3+}:Y_3Al_5O_{12}$. The experimental data obtained are satisfactorily explained by the developed theory. Finally, we discuss the obtained experimental results and possible applications of the derived pulse area approach.