Nonlinear absorption and dispersion in a two-level system with permanent dipole moments

Abstract

We calculate the third-order nonlinear absorption and dispersion (self-phase-modulation and cross-phase modulation) of a homogeneously broadened two-level molecular system with permanent dipole moments interacting with (1) an optical field of frequency \ensuremath{\omega}, and (2) two optical fields of frequency ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$, i.e., the susceptibilities ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-\ensuremath{\omega};\ensuremath{\omega},-\ensuremath{\omega},\ensuremath{\omega}) and ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-${\mathrm{\ensuremath{\omega}}}_{1}$;${\mathrm{\ensuremath{\omega}}}_{1}$,-${\mathrm{\ensuremath{\omega}}}_{2}$,${\mathrm{\ensuremath{\omega}}}_{2}$). Sharp features in the nonlinear absorption and dispersion appear whenever one- or two-photon resonance conditions are met. Only features corresponding to one-photon resonance arise if the difference between the permanent dipole moments of the levels, \ensuremath{\Delta}\ensuremath{\mu}, vanishes. The ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-\ensuremath{\omega};\ensuremath{\omega},-\ensuremath{\omega},\ensuremath{\omega}) spectrum has two features, one at frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$ dominated by contributions proportional to ${\mathrm{\ensuremath{\mu}}}_{\mathit{a}\mathit{b}}^{4}$, where ${\mathrm{\ensuremath{\mu}}}_{\mathit{a}\mathit{b}}$ is the transition dipole moment of the two levels with transition frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$, and the other at frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$/2 dominated by contributions proportional to (\ensuremath{\Delta}\ensuremath{\mu}${)}^{2}$${\mathrm{\ensuremath{\mu}}}_{\mathit{a}\mathit{b}}^{2}$. Sharp features in ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-${\mathrm{\ensuremath{\omega}}}_{1}$;${\mathrm{\ensuremath{\omega}}}_{2}$,-${\mathrm{\ensuremath{\omega}}}_{2}$,${\mathrm{\ensuremath{\omega}}}_{1}$) appear when ${\mathrm{\ensuremath{\omega}}}_{2}$ is near ${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$,${\mathrm{\ensuremath{\omega}}}_{1}$,${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$, and \ensuremath{\Vert}${\mathrm{\ensuremath{\omega}}}_{1}$-${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$\ensuremath{\Vert}.