Page 281 - Tunable Lasers Handbook

6 Transition Metal Solid-state Lasers 241

where h is the wavelength associated with the frequency v. In a practical labora-
tory system. only a fraction of the emitted radiation is collected by the fluores-
cence measurement device. If this fraction collected, R, is independent of the
wavelength, then

where G,(k.v) is the measured quantity. Using the preceding relations, the quan-
tity R can be determined using the relation between the radiative lifetime and
the fluorescence spectrum. With the measured spectrum. the emission cross sec-
tion becomes

where I,, is defined by the relation

Z,,,= -== rhGp(k,h) d31 . (17)

In Eq. (271, it has been tacitly assumed that the material is isotropic. If the mate-
rial is not isotropic, the extension to take into account the effects of anisotropy is
straightforward.
While McCumber related the gain of a transition metal to the absorption or
emission spectra, Struck and Fonger [l 11 presented a unified theory of both the
radiative emission and nonradiative decay processes. Previously, two disparate
theories had described nonradiative decay processes. One of these theories,
referred to as the activation energy relation, described the nonradiative decay
process by the relation

1 exp (-%) . (18)
L
A,,
=
In this expression, rn, is the nonradiative lifetime. An2 is a rate constant. El is an
activation energy, k is Boltzmann's constant, and T is the temperature. It can be
loosely interpreted as the number of times per second that the excited active
atom tries to escape from a potential well times the probability that it will have
energy to effect its escape.
Another theory is referred to as the niultiphonon emission fornzula. In this
formulation, the nonradiative decay rate is given by