Optical properties of photon echoes stimulated by three frequencies

A gaseous atomic or molecular system with four energy levels $E_a<E_b<E_c<E_d$ which is stimulated by resonant laser pulses at frequencies ${\omega }_1\simeq {\omega }_{\mathrm{ba}}$, ${\omega }_2\simeq {\omega }_{\mathrm{cb}}$, ${\omega }_3\simeq {\omega }_{\mathrm{dc}}$ is considered. The six photon echoes which are stimulated by first and second pulses at frequencies ${\omega }_1$, ${\omega }_2$, ${\omega }_3$ are discussed. For wave fronts which are described by $e^{i(\varphi -\omega t)}$ the echo phase is described by ${\varphi }_E=\Sigma (2\mathrm{}{\varphi }_m^{\mathrm{II}}-{\varphi }_m^{\mathrm{I}})$ at the sum frequency ${\omega }_1+{\omega }_2+{\omega }_3$ with $m=1,2,3\mathrm{}$, at ${\omega }_1+{\omega }_2$ with $m=1,2\mathrm{}$, at ${\omega }_1$ with $m=1\mathrm{}$, etc. For waves in the $z$ direction where $\varphi =\kappa \mathrm{}(x,y,z)+kz$ the gradient of the phase yields the wave-vector direction ${\overrightarrow{\mathrm{k}}}_E=\Sigma (2\mathrm{}{\overrightarrow{\mathrm{k}}}_m^{\mathrm{II}}-{\overrightarrow{\mathrm{k}}}_m^{\mathrm{I}})$ in a physically thin sample. A probe pulse at ${\omega }_2$ and ${\omega }_3$ after the second pulse can generate a probe-pulse echo at ${\omega }_1$ with phase ${\varphi }_1=\Sigma (2\mathrm{}{\varphi }_m^{\mathrm{II}}-{\varphi }_m^{\mathrm{I}})-{\varphi }_2^S-{\varphi }_3^S$, etc., for other frequency arrangements. Transverse phases ${\kappa }_m\mathrm{}\left(\mathrm{xy}\right)\mathrm{}$ can be added and subtracted, but no application is suggested for this novel feature. When all ${\kappa }_m$ except one are zero, phase conjugation or phase-adaptive optics can occur.