Absorption of Pump Radiation by the Gain Medium and Transfer of Energy to the Upper Laser Level

This energy transfer can be divided into two processes. The first is the absorption of pump radiation into the pump bands of the gain medium expressed by η_a, and the second is the transfer of energy from the pump band to the upper laser level expressed by η_Qη_s.
The absorption efficiency is the ratio of power P_a absorbed to power Pe entering the laser medium
η_a = P_a/P_e. ........(1)
The quantity ηa is a function of the path length and the spectral absorption coefficient of the laser medium integrated over the emission spectrum of the pump source. If the pump radiation is totally diffuse inside the laser rod, as is the case when the lateral surface is rough ground, a good approximation can be derived by expressing (1) in a different form

.......(2)

where dP_a/dV is the power absorbed per volume, I_w is the power density at the cylindrical surface of the rod, and F, V are the cylindrical rod surface and volume of the rod, respectively.
The power absorbed per volume can also be expressed by dP_a/dV = α₀ I_av, where α₀ is the absorption coefficient of the laser material averaged over the spectral emission range of the lamp and I_av is the average power density inside the rod. Radiation with an energy density W_av propagating at a velocity c has a power density of I_av = cW_av, where c = c₀/n₀ is the speed of light in the medium and n₀ is the refractive index. The energy density in the laser rod has a radial dependence due to absorption. The average value Wav can be closely approximated by taking the energy density W_s at the surface weighted by the factor exp(−α₀R), where R is the rod radius and α₀ is the absorption coefficient. It follows that this is a valid approximation over a large range of α0. From these considerations follows

...(3)

Owing to grinding the lateral surface of the rod, the pump radiation upon entering the rod will be diffused. For an enclosure with diffusely reflecting walls, we obtain from blackbody radiation theory a relationship between energy W_s inside the enclosure and the intensity I_w emitted by the walls

.......(4)

Introducing (3, 4) into (2) yields our final result

......(5)

For diode pumped lasers, the absorption efficiency can be approximated by

.....(5)

where α₀ is the absorption coefficient of the laser crystal at the wavelength emitted by the laser diode and l is the path length in the crystal. Detailed data on η_a for both flashlamp.
The upper state efficiency may be defined as the ratio of the power emitted at the laser transition to the power absorbed into the pump bands. This efficiency is the product of two contributing factors, the quantum efficiency η_Q, which is defined as the number of photons contributing to laser emission divided by the number of pump photons, and η_S, the quantum defect efficiency. The latter is sometimes referred to as the Stokes factor, which represents the ratio of the photon energy emitted at the laser transition h_νL to the energy of a pump photon h_νp, that is,

..........(6)

where λ_P and λ_L is the wavelength of the pump transition and the laser wavelength, respectively. For Nd :YAG emitting at 1064 nm that is pumped by a laser diode array at 808 nm, we obtain η_S = 0.76, and η_Q = 0.90.
In a flashlamp-pumped system, the value of η_S is an average value derived from considering the whole absorption spectrum of the laser.