Page 278 - Tunable Lasers Handbook

238 Norman P. Barnes

extends to the long-wavelength side of the electronic transitions. On the other
hand, the vibronic absorption spectra extends to the short-wavelength side of the
electronic transition. In some cases, the absorption spectra and emission spectra
are mirror images of each other. Although in general this is not true. at any
wavelength the absorption, emission, and gain are related by the principle of
detailed balance.
Several assumptions must be met in order for the McCumber analysis to be
valid. Consider a system consisting of an upper manifold and a lower manifold.
As before, the term manifold will be used to describe a set of closely spaced
levels. To first order approximation, levels within the manifold can be associ-
ated with a simple harmonic motion of the active atom and its surrounding
atoms. While the simple harmonic oscillator energy level spacings of the upper
and lower manifolds may be the same. in general they do not have to be. Fur-
thermore, the position of the minimum of the simple harmonic potential wells
may be spatially offset from each other due to the difference in size of the
active atom in the ground level and the excited level. Population densities of
these manifolds are denoted by N, and N,. One of the assumptions used by the
theory is that a single lattice temperature-can describe the population densities
of these manifolds. For example, suppose the upper manifold consists of a
series of levels commencing with the lowest energy le\7el which is an energy
hvZp above the ground level. Levels within the manifold are separated by an
energy hvv where this energy represents a quantum of vibrational energy asso-
ciated with the simple harmonic motion of the upper level. According to this
assumption, the active atoms in the upper manifold will be distributed among
the various vibrational levels associated with the upper manifold according to a
simple Boltzmann distribution. In turn. the Boltzmann distribution can be char-
acterized by a single temperature T. Thus, with all of the vibrational levels
equally degenerate, the population of any particular vibrational level will be
given by N,exp (-JhvJkT) (1 - exp (-kv, /W)) where J is the integer denoting
the energy ievel, k is Boltzmann's constant, and T is the lattice temperature. The
last factor simply normalizes the distribution since it represents the summation
over all levels within the manifold. Furthermore, the same temperature can
describe the relative population of the levels comprising the lower manifold.
Another assumption is that the time interval required for thermal equilibrium
for the various population densities is very short compared with the lifetime of
the upper level. For example. suppose all of the population of the upper mani-
fold may be put initially in a single level by utilizing laser pumping. The sec-
ond assumption says, in essence, that the closely spaced levels achieve thermal
equilibrium in a time interval short with respect to the lifetime of the upper
manifold. A third assumption is that nonradiative transitions are negligible
compared with the transitions that produce the absorption or emission of a pho-
ton. Although this is not always true. the lifetime of the upper level may be