Page 285 - Tunable Lasers Handbook

6 Transition Metal Solid-state lasers 245
upper manifold and end on any of the phonon levels in the lower manifold. It is
the variety of initial and final phonons levels that allows a wide spectrum of
phonon energies to be produced. Because the total energy associated with :he
transition is distributed between the photon and the phonons, the photon energy,
and thus the frequency, can vary over a wide range.
Shifts of the frequency from the zero phonon frequency are related to the
offset associated with the configurational coordinate. Transitions between the
upper and lower manifolds are represented by vertical lines in Fig. 8. Consider
the transition from the lowest energy level in the upper manifold to the lower
manifold. In the lowest level, the most likely position of the configurational
coordinate is in the center of the parabola. Consequently, a transition from the
lowest energy level in the upper manifold to the lowest energy in the lower mani-
fold is not probable since the overlap of their respective wave functions is small.
Far more likely is a transition to one of the higher energy levels in the lower
manifold. These levels are associated with the creation of more phonons, and the
photon energy will be lower. Thus. the emission spectra will be on the long-
wavelength side of the zero phonon line. Conversely, the absorption spectra will
be on the short-wavelength side of the zero phonon line.
Radiative and nonradiative transition rates for these processes, characteiized by
the radiative and nonradiative lifetimes T, and T,,,, respectively. can be expressed as

In these expressions, R1,,, and NL,, are constants from the electronic portion of
the transition integral and <un 1 Y~,>? is the squared overlap of the quantum-
mechanical wave functions. As the offset becomes larger, the overlap of the
quantum-mechanical wave functions decreases since the wave functions are
physically displaced. This expression is valid for a single set of levels in the
manifolds, but the total transition rates are the summation of the rates corre-
sponding to transitions between all of the levels in the manifold.
To determine the total radiative and nonradiative transition rat:s. a summa-
tion over all of the possible energy levels in both the upper and lower manifolds
must be taken into account. For arbitrary curvatures of the parabolas, the sum-
mation becomes more complicated and is beyond the scope of this chapter.
However, in the case where the parabolas have the same curvature. the radiative
and nonradiative transition rates reduce to