Past exam of the atomic and laser physics course of the University of Oxford / 2021 / 3 / c

For constant nonzero absorption coefficient, multiply the radiative transfer equation by the integrating factor $e^{{\kappa }_{\nu }z}$:Integrating from the entrance to $z$ givesThe boundary condition is immediate. Differentiation gives $I_{\nu }^{\prime }=-{\kappa }_{\nu }[I_{\nu }(0)-S_{\nu }]e^{-{\kappa }_{\nu }z}=-{\kappa }_{\nu }{I}_{\nu }+j_{\nu }$, checking the equation. For positive optical depth ${\kappa }_{\nu }z\gg 1$, the incident specific intensity is forgotten and $I_{\nu }\to S_{\nu }$. Negative absorption coefficient instead describes gain and does not have this relaxation limit. At ${\kappa }_{\nu }=0$ use the nonsingular form directly: $I_{\nu }(z)=I_{\nu }(0)+j_{\nu }z$.