McCumber relation

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. 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. Let σ a ( ω ) {displaystyle sigma _{ m {a}}(omega )} be the effective absorption cross-section σ e ( ω ) {displaystyle sigma _{ m {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 where ( N 1 N 2 ) T {displaystyle left({frac {N_{1}}{N_{2}}} ight)_{T}} is thermal steady-state ratio of populations; frequency ω z {displaystyle omega _{ m {z}}} is called 'zero-line' frequency; ℏ {displaystyle hbar } is the Planck constant and k B {displaystyle k_{ m {B}}} is the Boltzmann constant. Note that the right-hand side of Equation (1) does not depend on   ω   {displaystyle ~omega ~} . 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 _{ m {a}}(omega )} and the emission cross-section σ e ( ω ) {displaystyle sigma _{ m {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: 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). In the case of an idealized two-level atom the detailed balance for the emission and absorption which preserves the Planck formula for the black-body radiation leads to equality of cross-section of absorption and emission. In the solid-state lasers the splitting of each of laser levels leads to the broadening which greatly exceeds the natural spectral linewidth. In the case of an ideal two-level atom, the product of the linewidth and the lifetime is of order of unity, which obeys the Heisenberg uncertainty principle. In solid-state laser materials, the linewidth is several orders of magnitude larger so the spectra of emission and absorption are determined by distribution of excitation among sublevels rather than by the shape of the spectral line of each individual transition between sublevels. This distribution is determined by the effective temperature within each of laser levels. The McCumber hypothesis is that the distribution of excitation among sublevels is thermal. The effective temperature determines the spectra of emission and absorption ( The effective temperature is called a temperature by scientists even if the excited medium as whole is pretty far from the thermal state ) Consider the set of active centers (fig.1.). Assume fast transition between sublevels within each level, and slow transition between levels.According to the McCumber hypothesis, the cross-sections σ a {displaystyle sigma _{ m {a}}} and σ e {displaystyle sigma _{ m {e}}} do not depend on the populations N 1 {displaystyle N_{1}} and N 2 {displaystyle N_{2}} .Therefore, we can deduce the relation, assuming the thermal state. Let   v ( ω )   {displaystyle ~v(omega )~} be group velocity of light in the medium, the product   n 2 σ e ( ω ) v ( ω ) D ( ω )   {displaystyle ~n_{2}sigma _{ m {e}}(omega )v(omega )D(omega )~} is spectral rate of stimulated emission, and   n 1 σ a ( ω ) v ( ω ) D ( ω )   {displaystyle ~n_{1}sigma _{ m {a}}(omega )v(omega )D(omega )~} is that of absorption; a ( ω ) n 2 {displaystyle a(omega )n_{2}} is spectral rate of spontaneous emission. (Note that in this approximation, there is no such thing as a spontaneous absorption)The balance of photons gives: