Page 279 - Tunable Lasers Handbook

6 Transition Metal Solid-state Lasers

decomposed into components representing a radiative lifetime and a nonradia-
tive lifetime.
Given the population densities of the upper and lower manifolds, the
absorption and emission cross sections can be related to the absorption and
emission coefficients, up(k,v) and ep(k.v). respectively. In these expressions. k is
the wave vector indicating the direction of propagation and v is the frequency. A
subscript y is utilized since the absorption and emission may depend on the
polarization p. Given the absorption and emission coefficients, absorption and
emission cross sections can be defined by the relations

Using the principle of detailed balance. the absorption and emission cross sec-
tions are related by

In this expression, hp is the energy required to excite one active atom from the
lower level to the upper level while maintaining the lattice temperature T, In the
lowtemperature limit for any system and for any temperature in a mirror image
type of system, the parameter p is the frequency of the no phonon transition.
Using these relations, the gain coefficient g,,(k,v) as a function of awe-
length is given by

While either of these expressions could be utilized to determine the gain coeffi-
cient, the relation using the emission cross section is usually of the greater prac-
tical importance. In general, the absorption cross section is too small to be mea-
sured in a practical situation. On the other hand, the stimulated emission cross