Mathematical model for two-photon fluorescence (upconversion or luminescence)
During the process of fluorescence, photons are absorbed by a
material, which raise the population level of a higher-than-ground
energy state. At some time after such excitation, these energy states
relax to lower energy states, releasing a characteristic photon.
During two-photon luminescence (also known as upconversion) a material
absorbs one photon to raise the population level to a semi-stable
higher-than-ground energy state. Shortly thereafter, this excited
state is further excited by a second photon (or phonon!) raising it
to an even higher energy state. After some time, this much higher
energy state relaxes, releasing a photon of more energy than either of
the excitation photons.
What is a good differential mathematical model for describing the
population levels in each of these different states during
luminescence? If I have a population of ground states N1, a
population of second energy level states N2 and third energy level N3
? is there a mathematical model for ?charging up? the third energy
state?
Another way of posing this problem is given a rate of excitation
photons, what will be the emission photon rate of a two-photon
material?
Differential equations based around a probabilistic approach would be appreciated. |
