Page 284 - Tunable Lasers Handbook

244 Norman P. Barnes
the energy. Thus, the equilibrium position of the configurational coordinate may
be different for different manifolds. Struck and Fonger refer to the offset
between the equilibrium position of the configurational coordinate of different
manifolds as the Franck-Condon [ll] offset. Offsets are the difference in the
configurational coordinate for the two parabolas normalized by the amplitude of
the zero point motion of the quantum mechanical simple harmonic oscillator.
This normalized distance is denoted by all,.
Parabolas describing the different energy manifolds are also described by an
energy offset corresponding approximately to the energy required to raise the
active atom to the excited manifold. Energy offsets are represented as a vertical
difference in Fig. 8 in contrast to the horizontal difference corresponding to an
offset in the configurational coordinate. An exact definition of the energy offset
is the energy difference between the zero point energy of the upper manifold and
the zero point energy of the lower manifold. This energy difference is character-
ized by a zero point energy, hvzp. If the simple harmonic oscillator were not
quantized, the zero point energy would be zero and the equilibrium position
would be at the minimum of the parabola.
Energy absorption and emission between manifolds with an offset can nom7 be
associated with a change in the motion of the simple harmonic oscillator. For
example, consider transitions shown in Fig. 8. A transition from the zero point
level of the lower manifold, designated with the letter u, does not go to the zero
point level of the upper manifold, designated with the letter v. Rather, the transition
is to a higher level of the simple harmonic oscillator. Consequently, the several
quanta of simple harmonic motion become available. Quanta of simple harmonic
motion can be readily identified as phonons, establishing the correspondence
between the Struck and Fonger model and the McCumber model. Phonons, as
referred to here, are localized to the vicinity of the active atom. However, phonons
may also refer to simple harmonic motion of the entire crystal. Although localized
and distributed phonons are obviously not the same, the concept of quantized sim-
ple harmonic oscillation will be referred to as phonons.
Using the single configurational coordinate model, energy balances for
radiative and nonradiative transitions can be expressed as
hvzp = mhy - nhy, + hy,,,l .

hv-, = nzhy - nhy, = 0 , (23)

respectively. In this expression, v,,~ is the frequency of the emitted photon.
Energy differences between the zero point or zero phonon energy and the emitted
photon energy are taken up by the creation or annihilation of phonons, designated
as hvll and lzv, for the zi and v manifolds, respectively. Using this concept, the
cause of the wide absorption and emission spectra can be attributed to the multi-
tude of phonon levels associated with the configurational coordinate parabolas. In
emission, for example, the electron can start from any of the phonon levels in the