The picture of an incident and an emitted light field that was introduced by Yablonovitch in [72] has been subsequently developed into a thermodynamic model of fluorescent concentrators e.g. [41-43, 73, 74]. This model was successfully used to describe fluorescent concentrators based on luminescent quantum dots. To include the more complex spectral characteristics of organic materials, a stack of different materials and features like diffuse reflectors and photonic structures proved to be
presented in chapter 4.4. Nevertheless, I will attempt to present a phenomenological thermodynamic model in this section, which brings together the main ideas of different theoretical discussions, that offers valuable insight into the working principles of fluorescent concentrators, and will be helpful later on in this work.
The incident light field with the intensity Binc excites the ensemble of fluorescent
molecules in the collector out of equilibrium with the ambient temperature T. Because of the fast thermal equilibration among the vibrational substates of the electronically excited state, the electrons cool down very fast to the ambient temperature. But as the molecule remains nonetheless in an electronically excited state, the electrons have a chemical potential µ > 0, just as in an illuminated semiconductor. The chemical potential is a measure of how many fluorescent molecules are excited. Similar to the discussion in section 3.1, the emission of the ensemble of the fluorescent molecules is described by the generalized Planck’s law. The number of emitted photons per time, per area, per unit solid angle, and per frequency interval is
1
exp
1
2
,
,
2 2 2 , ,¸¸
¹
·
¨¨
©
§
T
k
h
c
n
T
B
B emit emit emit emit emit pP
Q
Q
D
Q
P
Q
Q , (4.13)whereD Qemit isthe absorption coefficient.
Part of the emitted light is lost due to the escape cone of total internal reflection, but most of the light is trapped and guided in the collector to its edges. In consequence, the molecules are illuminated not only by the incident field but also by the emitted and trapped light. The higher the combined intensity Bint is at a point of the collector, the
higher is the chemical potential, and in turn also the emission of light. The chemical potential is not constant throughout the collector. For instance, close to the front surface, the chemical potential is higher because the fluorescent molecules are excited from the full incident field. Further away from the surface, part of the incident light has been absorbed and therefore intensity is lower.
4.2 Theoretical description of fluorescent concentrators
Fig. 4.10: Illustration of the main ideas of the thermodynamic model. Incident radiation with the intensity Binc excites the ensemble of fluorescent
molecules in the fluorescent collector. The fraction of excited molecules is described by the chemical potential µ of the molecule ensemble. The fluorescent molecules emit radiation with the intensity Bemit which depends
on the chemical potential. The trapped fraction of the emitted light and the incident light combine to the internal intensity Bint. This internal intensity
again determines the chemical potential. As the internal intensity is not constant throughout the collector, the chemical potential varies as well.
This picture can explain why there is a maximum possible concentration. The larger the collector is, the more photons from the incident field are collected. Thus, the intensity of the trapped light field that travels towards the edges also increases. This increases the chemical potential, and consequently, the emission of light as well. The maximum concentration is reached when the chemical potential has become so high that the emitted light lost in the escape cone equals the incident field. The limit obtained from this consideration is stricter than the limit presented in equation (4.12) [72].
The link of the maximum concentration with the Stokes shift and the problematic of reabsorption can be understood considering a simple model system that features an absorption region and an emission region (see Fig. 4.11) [30]. The absorption coefficient Dabs in the absorption region is much higher than in the emission region
Fig. 4.11: Idealized model of the absorption and emission characteristics of a fluorescent concentrator [30]. In the absorption region the absorption coefficient Dabs is high, while in the emission region the coefficient Demit is
much smaller. Following Kirchoff’s law, the emission coefficient equals the absorption coefficient. Nevertheless, emission in the emission region is much higher, because the generalized Planck’s law favors emissions at lower energies. A bandstop reflection filter that reflects in the emission region can increase efficiency and the maximum possible concentration considerably.
As described by Kirchoff’s law, the absorption and emissions coefficients are equal. In spite of Dabs>Demit, the emission in the emission region is much larger than in the
absorption region, because of the energy dependency of the generalized Planck’s law, which states that in this regime the emission at lower energies is considerably more likely than at higher energies. Hence, the larger the Stokes shift, that is the bigger the energy difference E2 - E1, the less frequent is the emission in the absorption range
relative to the emission in the emission range. With less light being emitted in the
absorption range, reabsorption becomes less likely. Because each reabsorption and re- emission again causes escape cone losses, with less reabsorption the escape cone losses are reduced as well. Less escape cone losses mean that a higher internally guided field and a higher chemical potential is possible until the emitted light lost in the escape cone equals the incident field. In consequence, a higher maximum concentration is possible.
4.2 Theoretical description of fluorescent concentrators
Because absorption and emission are linked by Kirchhoff’s law, it is not possible to eliminate reabsorption entirely. Additionally, without reabsorption the excitation of the molecules would be completely independent from the emitted light. This would allow for an infinite concentration, which is a clear contradiction of the second law of thermodynamics.
However, it is possible to reduce the escape cone losses and therefore to increase the maximum possible concentration with the addition of a band stop filter. The band stop filter should reflect in the emission range, but should transmit in the absorption range. The desirable reflection band is sketched in Fig. 4.11. Like this, only the small amount of light emitted in the absorption region can be subjected to escape cone losses. Again, this means that a higher internally guided field and a higher chemical potential is possible until the emitted light lost in the escape cone equals the incident field. In [30] it was shown that the maximum efficiency of a fluorescent concentrator system with such a band stop filter equals the Shockley-Queisser limit of a solar cell with a band- gap similar to that of the cut-off wavelength E2 of the band stop filter.