The McCumber relation (or McCumber theory) is a relationship between the effective cross-sections of absorption and emission of light in the physics of solid-state lasers. It is named after Dean McCumber, who proposed the relationship in 1964.
Definition Let σ a ( ω ) {\displaystyle \sigma _{\rm {a}}(\omega )} be the effective absorption cross-section σ e ( ω ) {\displaystyle \sigma _{\rm {e}}(\omega )} be effective emission cross-sections at frequency ω {\displaystyle \omega } , and let T {\displaystyle ~T~} be the effective temperature of the medium. The McCumber relation is
(1) σ e ( ω ) σ a ( ω ) exp ( ℏ ω k B T ) = ( N 1 N 2 ) T = exp ( ℏ ω z k B T ) {\displaystyle {\frac {\sigma _{\rm {e}}(\omega )}{\sigma _{\rm {a}}(\omega )}}\exp \!\left({\frac {\hbar \omega }{k_{\rm {B}}T}}\right)=\left({\frac {N_{1}}{N_{2}}}\right)_{T}=\exp \!\left({\frac {\hbar \omega _{\rm {z}}}{k_{\rm {B}}T}}\right)}
where ( N 1 N 2 ) T {\displaystyle \left({\frac {N_{1}}{N_{2}}}\right)_{T}} is thermal steady-state ratio of populations; frequency ω z {\displaystyle \omega _{\rm {z}}} is called "zero-line" frequency;
ℏ {\displaystyle \hbar } is the Planck constant and
k B {\displaystyle k_{\rm {B}}} is the Boltzmann constant. Note that the right-hand side of Equation (1) does not depend on ω {\displaystyle ~\omega ~} .
Gain It is typical that the lasing properties of a medium are determined by the temperature and the population at the excited laser level, and are not sensitive to the method of excitation used to achieve it. In this case, the absorption cross-section
σ a ( ω ) {\displaystyle \sigma _{\rm {a}}(\omega )} and the emission cross-section
σ e ( ω ) {\displaystyle \sigma _{\rm {e}}(\omega )} at frequency ω {\displaystyle ~\omega ~} can be related to the lasers gain in such a way, that the gain at this frequency can be determined as follows:
(2) G ( ω ) = N 2 σ e ( ω ) − N 1 σ a ( ω ) {\displaystyle ~~~~~~~~~~~~~~~G(\omega )=N_{2}\sigma _{\rm {e}}(\omega )-N_{1}\sigma _{\rm {a}}(\omega )}
D.E.McCumber had postulated these properties and found that the emission and absorption cross-sections are not independent; they are related with Equation (1).
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