Issue 
EPJ Photovolt.
Volume 3, 2012



Article Number  30101  
Number of page(s)  11  
Section  Modelling  
DOI  https://doi.org/10.1051/epjpv/2012001  
Published online  26 June 2012 
https://doi.org/10.1051/epjpv/2012001
Simulations of geometry effects and loss mechanisms affecting the photon collection in photovoltaic fluorescent collectors
^{1} Institut für Photovoltaik, Universität Stuttgart, Pfaffenwaldring 47, 70569 Stuttgart, Germany
^{2} IEK5Photovoltaik, Forschungszentrum Jülich, 52425 Jülich, Germany
^{a}
email: liv.proenneke@ipv.unistuttgart.de
Received: 29 August 2011
Accepted: 29 February 2012
Published online: 26 June 2012
MonteCarlo simulations analyze the photon collection in photovoltaic systems with fluorescent collectors. We compare two collector geometries: the classical setup with solar cells mounted at each collector side and solar cells covering the collector back surface. For small ratios of collector length and thickness, the collection probability of photons is equally high in systems with solar cells mounted on the sides or at the bottom of the collector. We apply a photonic band stop filter acting as an energy selective filter which prevents photons emitted by the dye from leaving the collector. We find that the application of such a filter allows covering only 1% of the collector side or bottom area with solar cells. Furthermore, we compare ideal systems in their radiative limits to systems with included loss mechanisms in the dye, at the mirror, or the photonic filter. Examining loss mechanisms in photovoltaic systems with fluorescent collectors enables us to estimate quality limitations of the used materials and components.
© EDP Sciences 2012
1 Introduction
Fluorescent collectors (FCs) use organic dye molecules or inorganic fluorescent quantum dots surrounded by a dielectric material to trap and concentrate solar photons. The dye absorbs incoming photons with energy E_{1} and emits photons due to Stokes shift with E_{2} < E_{1}. The emission occurs with a randomized direction. Total internal reflection traps part of the radiation in the system and guides the photons to the collector sides. In a photovoltaic system, solar cells applied to the collector sides or the back side collect these photons and convert them into electrical energy. Already in the late 1970s and early 1980s Goetzberger, Wittwer and Greubel described the technological potential of FCs in photovoltaic systems [1, 2]. Recently, the basic idea has regained some interest in the context of building photovoltaic structures which exceed the classical efficiency limitations by using up and downconverting dyes [3, 4, 5]. Theoretical tools to describe FCs thermodynamically have been developed [6, 7, 8]. Numerical approaches analyzing the FC behavior gain more interest [9, 10]. In order to estimate theoretical limitations, the photovoltaic systems with FCs have been highly idealized. However, realistic setups show loss mechanisms which need to be considered. The classic idea of assembling an FC in a photovoltaic system is based on its behavior of guiding emitted photons to the sides. Therefore, the classical setup mounts solar cells to the collector sides [2, 11, 12, 13, 14, 15, 16, 17]. Technically, it seems less expensive to apply and connect solar cells at the bottom side of the collector. Experimentally, fluorescent collectors and photonic structures on top of solar cells prove to raise the output current by 95% compared to a nonfluorescent glass on top of the cell [18].
The present paper uses MonteCarlo raytracing simulations for a comparison of the classical sidemounted system to a system where solar cells cover the FC back side. We see that the sidemounted system performs better in most cases, especially at larger collector sizes. However, for both systems the maximum collection probability for photons p_{c} = 97% is only achieved in the presence of a back side mirror and a photonic band stop (PBS) filter at the collector top surface acting as an energy selective filter. This maximal photon collection occurs in the statistical limit. Here, numerous small solar cells cover the FC in close proximity. Such a smallscale system is more favorable than a system with few largescaled solar cells taking the same coverage fraction. The maximal number of collected photons is equivalent in a photovoltaic system with neither FC nor PBS, but our assembly saves us 99% of solar cell area.
In order to describe FCs in photovoltaic systems, we use numerical and analytical approaches based on the principle of detailed balance. Starting from ideal systems in their radiative recombination limitation, we also examine the influences of nonradiative recombination in the fluorescent dye, of nonperfect reflection at the mirrors, and of nonperfect reflection conditions at the PBS filter. The results point out that reflection losses at the back surface cause a higher decrease than losses due to nonradiative recombination in the dye. Compared to a system without applied PBS nonradiative losses induce higher decreases of photon collection in the PBS covered system.
Fig. 1 Sketch of the fluorescent collector geometries compared in the present paper (seen from the bottom). (a) Classical design with the solar cells mounted at each side of the collector with length l and thickness d. (b) Modified classical system where only a fraction of the respective collector sides is covered with a solar cell area A_{cell} = ds. (c) Collector with solar cells with an area A_{cell} = s^{2} mounted at the bottom. In all cases the (remaining) back side is covered with a mirror. (d) Systems (b) and (c) are assumed to be periodic in space. Exemplary, a detail of the bottommounted system is shown. 
2 Collector geometries and dye properties
This section describes the three photovoltaic systems discussed in this paper. A characterization of the FC dye properties as well as an explanation of the functionality of a PBS filter follows.
Figure 1a shows an FC in the classical configuration with an acrylic plate of length l and thickness d doped with fluorescent dye. The collecting solar cells are mounted at the sides of the plate. Let us define the coverage fraction f = A_{cell}/A_{coll} as the ratio between the area A_{cell} = 4dl of the solar cells in the system and the illuminated collector area A_{coll} = l^{2}. For the configuration in Figure 1a, we have f = 4dl/l^{2} = 4d/l, hence the coverage fraction depends only on the ratio between the collector thickness d and the side length l. A perfect mirror covers the FC back side. Figure 1b features a variant of the sidemounted FC where only a part of each side is covered with a solar cell. The system is repeated periodically in x and ydirection. Therefore, photons hitting a collector side experience periodic boundary conditions and enter the opposite side. In an alternative but equivalent perception perfect mirrors cover the remainders. The coverage fraction for the system in Figure 1b is f = 4ds/l^{2} with the side length s ≤ l of the solar cells. Thus, coverage fraction f and collector length l are decoupled and this geometry offers an additional degree of freedom for the collector design. Again, the FC back side is covered by a mirror. The collector design of Figure 1c uses a square solar cell with a side length s at the back side of the FC. Thus, the solar cells in this bottommounted system cover a fraction f = s^{2}/l^{2} of the surface. Figure 1d shows a detail of the bottommounted system which is also assumed to be periodically repeated in x and ydirection. As shown, square solar cells occupy the back surface of the collector with a period length l. The remaining parts of the back side are covered with a mirror.
We model an FC consisting of an acrylic plate with the refractive index n_{r} = 1.5 and embedded fluorescent dye molecules. Figure 2 depicts the absorption/emission behavior of the fluorescent dye used in the following. We assume a stepwise increase of the absorption constant α from zero at energies E < E_{2} to a value α_{2} for E > E_{2} and a further increase to α_{1} for energies E > E_{1}. The emission coefficient e is linked to the absorption coefficient α via Kirchhoffs law (1)with the black body spectrum (2)where n_{r} is the refractive index of the collector material, h is Planck’s constant, c the speed of light, and kT the thermal energy corresponding to the temperature T of the collector and its surroundings (T = 300K, throughout this paper).
Fig. 2 Sketch of the absorption and emission behavior as assumed in this paper. The dye absorption is given by a step function. Incoming photons have the energy E_{1} and a high absorption coefficient α_{1}. The lower absorption coefficient α_{2} holds for the lower energy E_{2} and leads with Kirchhoffs law (Eq. (1)) to a high emission coefficient e_{2}. The model also features the possibility of an energy selective photonic band stop (PBS) that keeps the emitted photons in the FC system. 
The absorption/emission dynamics used in the following is given by a twolevel scheme as used earlier to describe the detailed balance limit of FCs [11, 12, 19]. The choice of this simple approach ensures a certain generality of our results such that the trends caused by the collector geometries or by the introduction of loss mechanisms should be equally found in real systems with more complex spectral absorption/emission properties. For the present twolevel system we consider the emission probabilities (3)and (4)for photon emission by the fluorescent dye in the range of photon energies E > E_{1} and E_{1} > E > E_{2}, respectively. In equations (3) and (4), we use the definition (5)and the normalization factor p such that p_{1} + p_{2} = 1.
The choice of the energies E_{1}, E_{2}, and the absorption coefficients α_{1}, α_{2} leads to the emission probabilities p_{2} ≫ p_{1}, in contrast to the absorption coefficients α_{1} ≫ α_{2}. Due to the dominance of the exponential factor in equation (5) we approximate (6)Thus, a choice of an energy difference ΔE = E_{1} − E_{2} = 200meV and of absorption coefficients α_{1} = 100α_{2} still ensures p_{2} ≈ 20p_{1}. In the following, we assume E_{1} = 2.0eV, E_{2} = 1.8eV and absorption coefficients α_{1} = 3/d, α_{2} = 0.03/d. Therefore, the system provides a high emission coefficient e_{2} for photons with energy E_{2} and a significantly lower emission coefficient e_{1} for photons with high energies.
Figures 3a–3e sketch the functionality of the PBS filter. Incoming photons have the energy E_{1}. The dye absorbs these photons and emits spatially randomized photons with angles θ, φ defined in Figure 3a at a lower energy E_{2}. Figure 3b shows that emitted photons impinging at the top surface with an incident angle θ higher than the critical angle θ_{c} for total internal reflection are guided to the collector sides. Whereas photons with θ < θ_{c} leave the collector as shown in Figure 3c. The application of a PBS filter avoids this loss mechanism. Figure 3d shows the ideal PBS filter which is energy selective only (θ_{pbs} = θ_{c}). The filter has a reflection R = 1 and a transmission T = 0 for photon energies E < E_{th}. For the other part of the spectrum R = 0 and T = 1 is assumed. Therefore, E_{th} denotes the upper cutoff energy of the filter. We choose E_{th} = E_{1} throughout this paper. Two and threedimensional photonic crystals [20, 21, 22] are promising materials which might be used as omnidirectional PBS in FC systems. However, technological developments have led to dielectric mirrors used as band pass filters with almost rectangular cutoff characteristics for normal incident photons [23, 24]. These rugatefilters show a high angular dependency by blocking only photons with almost perpendicular incidence. In order to examine the influence of this angular selectivity, we vary the reflection cone of the filter. Figure 3e depicts that a PBS with θ_{pbs} < θ_{c} reflects photons with E ≤ E_{1} and θ < θ_{pbs}. Thus, rays with θ_{c} > θ > θ_{pbs} hitting the collector surface within the striped angle cone are neither reflected by the PBS nor subject to total internal reflection and leave the system.
Fig. 3 Light guiding behavior of a fluorescent collector covered with solar cells at the sides and a mirror at its back side. (a) Definition of photon ray angle θ. (b) Absorbed photons are reemitted spatially randomized. The system leads rays with θ > θ_{c} to the sides of the collector. (c) Rays with angle θ < θ_{c} for total internal reflection leave the top surface. (d) Applying a photonic band structure (PBS) keeps rays with θ < θ_{pbs} in the system as well. This PBS is energy selective with θ_{pbs} = θ_{c}. Therefore, rays with energies E ≤ E_{1} are kept in the system only. (e) For an energy and angular selective PBS a reflection cone is assumed such that only rays with E ≤ E_{1} and θ < θ_{pbs} are kept in the system. 
3 Simulation method
The MonteCarlo simulation calculates the collection probability for photons p_{c} for the different collector geometries shown in Figures 1a–1c with varied collector dimensions and component quality. In order to allow also the comparison between systems with and without PBS, we provide only incoming photons with energy E = E_{1}. All photons enter into the collector perpendicular with random coordinates (x,y) with 0 < x < l and 0 < y < l. Their statistical absorption occurs following Beer’s absorption law after a path length (7)where p_{w} is a random number 0 ≤ p_{w} ≤ 1. After its absorption a photon is reemitted with a probability p_{e} = 1 − p_{nr} with the nonradiative recombination probability p_{nr} of the fluorescent dye. According to equations (3) and (4) the energy of the reemitted photons lies with the probability p_{1} in the energy range E ≥ E_{1} and with p_{2} in the range E_{1} > E ≥ E_{2}. After reemission the photon also obtains a pair of spherical angles (θ,φ) with 0 < θ < π and 0 < φ < 2π using the probabilities p_{θ} = sin(θ)/2 and p_{φ} = 1/2π (for the definition of θ and φ , see Fig. 3a).
Subsequently, either the dye molecules reabsorb the reemitted radiation or the photons hit one of the six collector surfaces at a coordinate (x_{s},y_{s},z_{s}). At the top surface (x_{s},y_{s},0), the photon is reflected if θ > θ_{C} with sin(θ_{C}) = 1/n_{r}. In the presence of an omnidirectional PBS, the photon is reflected for all photon energies E < E_{1}. An assumed angular selectivity sets as a reflection condition E < E_{1} and θ < θ_{pbs} as defined in Figure 3e. Nonreflected photons are lost and the number N_{lost} of lost photons is increased accordingly. At the bottom surface (x_{s},y_{s},d) a mirror perfectly reflects the photons with a probability p_{r} = 1. We use p_{r} < 1 for the analysis of loss mechanisms. For the system shown in Figure 1c, the bottommounted solar cells collect photons with x_{s} ≤ s and y_{s} ≤ s. In this case, the photons add to the number N_{coll} of collected photons. A special case discussed below is the statistical limit where we simply assume that a photon hitting the bottom of the collector enters a solar cell with the probability f, the solar cell coverage fraction. Throughout this paper, we assume a collection probability of 100% for photons hitting the solar cell area with energy E higher than the solar cell band gap E_{gap}. In order to analyze the principle limitations of applying FCs to photovoltaic systems, we choose E_{gap} = 1.8eV which corresponds to the emission peak of the FC.
If photons hit the collector sides, for instance at (l,y_{s},z_{s}) for the right collector side, sidemounted solar cells collect the photons for the geometry shown in Figure 1a. The geometry depicted in Figure 1b collects photons hitting the collector right surface with y_{s} ≤ s. Otherwise, the photon is subject to periodic boundary conditions. For the bottommounted system, we apply periodic boundary conditions on each collector side, i.e. the photon is reinjected at the respective facing side with unchanged spherical angles (θ,φ).
Our ray tracing program, that typically handles a number N_{in} = 5 × 10^{4}photons in parallel, runs until all photons are collected by the solar cells, lost by reemission from the collector surface, by nonradiative recombination in the dye, by reflection losses at the mirrors or at the PBS. With N_{in} = N_{lost} + N_{coll}, we obtain the collection probability for photons p_{c} = N_{coll}/N_{in} as the final result.
4 Simulation results
In this section, we describe the simulation results. First, the classical FC system with sidemounted solar cells as shown in Figure 1a is modeled. Here, the variation of the collector dimensions modifies the coverage fraction of solar cells. Second, photon collection probabilities for the bottommounted system as depicted in Figure 1c are calculated. In these simulations, we already include the loss mechanism of nonradiative recombination in the dye. The influences of collector dimension and coverage fraction are decoupled in this system. In the third part, the approach of decoupling these two aspects also in the sidemounted system as sketched in Figure 1b allows the comparison between sidemounted and bottommounted system. We compare both systems, first at constant collector lengths for varied coverage fractions. Secondly, at constant coverage fractions under the inclusion of three loss mechanisms : a nonradiative recombination in the dye, a nonperfect mirror at the collector back side, and an angular selectivity at the photonic structure lying on top of the collector.
4.1 Classical collector geometry
This subsection studies the influence of the collector and cell geometry on the photon collection properties of a classical collector system as shown in Figure 1a. Figure 4 shows the dependence of the collection probability p_{c} on the collector length l/d normalized to the collector thickness d calculated for systems with and without PBS at the top surface. We have also considered nonradiative recombination in the dye by assuming p_{nr} = 0.02 and 0.08.
Fig. 4 The collection probability p_{c} of the classical fluorescent collector geometry (Fig. 1a) fully covered with solar cells at its sides depends on the collector length l normalized to the thickness d. A perfect mirror covers the back side. The systems with a photonic band stop filter (PBS, open symbols) have higher collection probabilities than those without PBS (filled symbols). For the system with PBS a value of p_{c} close to unity remains up to a normalized collector length l/d ≈ 500 and nonradiative recombination probability p_{nr} = 0 in the dye. In contrast, the system without PBS has a maximum p_{c} only slightly above 80% even in the radiative case with p_{nr} = 0. With increasing collector length l/d the collection probability p_{c} decreases immediately. The consideration of nonradiative recombination in the dye, i.e. p_{nr} > 0, leads to a deterioration of p_{c} in all cases. The cell coverage fraction f (top axis) is directly linked to the collector size for the chosen geometry. 
The most important feature in Figure 4 is that the systems with PBS have a considerably higher collection probability p_{c} than those without PBS. This is because the PBS decreases the emission of photons through the surface of the collector as shown in Figure 3c. For systems without PBS this nonradiative loss occurs whenever a photon falls into the critical angle θ_{c} of total reflectance. For the system with PBS the photon additionally must be emitted at an energy E ≥ E_{1}. This emission probability is low, but nonzero for reasons of detailed balance. For the same reason systems with PBS obtain the high values of p_{c} also for larger collector lengths l/d, whereas for the systems without PBS p_{c} drops considerably already upon slight increases of l/d. Furthermore, with increased l/d also the number of photons absorbed by the dye a second or third time increases. Each absorption event leads to θrandomization of the reemitted photon and, in consequence, to a certain probability that the photon is lost by emission from the collector surface.
Also shown in Figure 4 are curves that reflect nonradiative recombination in the dye (p_{nr} = 0.02 and 0.08). We see that the system with PBS is especially sensitive to nonradiative recombination in the range of large l/d. At low values of l/d a photon is most likely absorbed only once before collected by the sidemounted solar cells. The maximum of p_{c} decreases therefore only proportionally to (1 − p_{nr}). For higher values of l/d, repeated reabsorption of photons not only increases the risk for radiative but now also for nonradiative losses. As radiative losses are low in the systems with PBS the relative importance of nonradiative losses is higher. Whereas in the radiative case a value of p_{c} > 90% remains up to a normalized collector length l/d ≈ 500, for p_{nr} = 0.02 and 0.08 we have p_{c} > 90% only for l/d ≤ 100 and l/d ≤ 6, respectively. The changes that occur by nonradiative recombination in the case of systems without PBS over the whole range of l/d are less significant due to the high emission losses that are present in the system anyway.
4.2 Scaling effect
The classical sidemounted collector geometry has a strict relation between the collector length l and the cell coverage fraction f = A_{cell}/A_{coll} = 4d/l. In this subsection we examine the bottommounted system shown in Figure 1c where both quantities can be treated independently. For Figure 5 the coverage fraction f = 0.01 is fixed and we vary the collector length l. Because of f = s^{2}/l^{2}, we adjust the cell side length s to .
Figure 5 demonstrates that the collection probability p_{c} at constant coverage fraction f drastically depends on the collector length l/d. The data in Figure 5 display asymptotic behavior in both limits, for small and large ratios l/d. We observe a wide transition regime between the two limiting cases where the collection probability p_{c} changes from high values at small l/d to significantly smaller values at large l/d. Such a behavior is typical for spatially extended inhomogeneous systems. If the characteristic feature length (here the collector length l) is large with respect to the length scale that is characteristic for interactions within the system (here the mean free path of photons), the system can be looked at as a parallel connection of spatially separated subsystems without interaction. The collection probability p_{c,ls} in this large scale limit [25] is then the weighted average of a portion f that has a local p_{c} of a collector with full back coverage, i.e. p_{c} ≈ 1, for the system with p_{nr} = 0 and a portion (1 − f) with p_{c} = 0. This is why we observe in this limit that p_{c} of the system with PBS approaches the coverage fraction f.
Fig. 5 Collection probability p_{c} of the fluorescent collector geometry with solar cells mounted at the bottom of the collector (Fig. 1c). The coverage fraction f is kept constant at f = 0.01 and the normalized collector length l/d is varied. All data feature a transition between two asymptotic situations at low and high ratios l/d. Only the systems with a photonic band pass (PBS, full symbols) achieve p_{c} > 0.5 at low ratios of l/d. For these systems, the maximum p_{c} as well as the transition from the smallscale to the largescale behaviour is strongly dependent on the nonradiative recombination probability p_{nr}. The systems without PBS (open symbols) have already a relatively low p_{c} < 20% even in the more favorable case of l/d < 10. Accordingly, the sensitivity to the introduction of nonradiative recombination in the dye (p_{nr} > 0) is less pronounced. 
In contrast, approaching the small scale limit (l < 1/α_{2}, marked in Figure 5 with dashed line [26]), any ray, reflected forth and back within the collector, has often the possibility to hit a cell at the collector back side. In this situation, the system might be looked at as spatially homogeneous with statistical cell coverage at its back. In fact, the value of p_{c,ss} in the small scale limit is consistent with quasionedimensional computations that simulate the bottommounted solar cells by a probability p_{c} = f for a photon to be collected by cells at the back side. Therefore, we denote this case also as the statistical limit.
The introduction of nonradiative recombination (p_{nr} = 0.02,0.08) in Figure 5 leads to a deterioration of the collection probability in all cases. As we have already seen in Figure 4, the system with PBS is much more sensitive to losses in the dye because of its overall high collection probability in the radiative limit. Especially important is the influence of a finite p_{nr} on the transition from the smallscale to the largescale limit. Whereas with p_{nr} = 0 all systems with l/d ≤ 100 yield the same high collection probability, the limit for p_{nr} = 0.08 reduces to l/d ≤ 10.
All data in Figure 5 represent the same coverage f = 0.01, i.e. the same amount of solar cell area per unit collector area. Nevertheless, the collection probability strongly depends on the chosen size of collector and solar cell. A proper scaling of these quantities is therefore necessary to tune the collection probability and, finally, the collector performance to its optimum.
4.3 Statistical limit
The following section derives an analytical description for the statistical limit marked in Figure 5 with the dashed line [27]. This analytical description only holds for the statistical limit in the radiative case and for systems with applied photonic structure. As discussed by Markvart [28] and Rau et al. [29], using photon fluxes only describes systems with equal chemical potential μ for the incoming photons. Applying a PBS equalizes μ because the absorption coefficient for all incoming photons is now α_{1}. In the MonteCarlo simulation, we excite the system with a monochromatic beam with E = 2.22eV. The results fit the calculation of the analytical expression because both cases fulfill the condition of equalized μ. In contrast, the system without PBS does not provide a spectrally equal absorption for all incoming photons leading to an inhomogeneous μ. Additionally, photons experience a spatial inhomogeneity for systems beyond the statistical limit as depicted in the former section. This also leads to an inhomogeneous μ. Considering these limitations, we compare in the following a side and a bottommounted system in the statistical limit (l/d = 1) with applied PBS.
The MonteCarlo simulation gives us the expressions for the photon flux incident on the collector surface (8)the photon flux which is absorbed in the collector by the solar cells (9)and the photon flux leaving the collector without hitting a solar cell (10)We propose that the ratio between the incident photon flux Φ_{sun} and the photon flux Φ_{FC} kept in the collector by total internal reflection gives the expression (11)which is at the same time the maximum concentration of a concentrator based on total internal reflection only. A collector with a refractive index of n = 1.5yields a concentration c_{TIR} = 2.25. Inside a FC with a dielectric material doped with a spectral shifting dye, the overall flux (12)is composed of two fluxes Φ_{FC1} and Φ_{FC2} as depicted in Figure 2. Here, (13)denotes the maximum concentration inside the FC. By integrating over the corresponding sections on the energy axes, the analytical expressions for the two fluxes inside the collector (14)and (15)are derived. The PBS reflecting all photons with energy E_{2} < E < E_{1} limits the incident photon flux (16)With equations (14) to (16) it is possible to calculate the maximum concentration analytically (17)with kT = 0.0258. In order to reach c_{p,max}, the system has to be in open circuit condition, thus, no solar cells are present at the bottom of the collector.
In order to derive values for the fraction dependent concentration analytically, we follow the interpretation of the FC system of Glaeser and Rau [12], Meyer and Markvart [30]. Here, the collection probability p_{c} follows the expression (18)with the photon flux absorbed by the solar cells (19)and the emitted photon flux leaving the collector without hitting a solar cell (20)Note, that Markvart proposes an analytical approximation by introducing uniform chemical potentials μ for all photons in the system. This assumption is exact as long as the system is in open circuit condition. Then the system is in thermal equilibrium and the chemical potential for all photons is equal.
Fig. 6 (a) Collection probabilities p_{c} for photons derived with a MonteCarlo simulation for a side and a bottommounted system in the statistical limit and with applied PBS. (b) Symbollines are the values for the concentration c_{p} derived from p_{c} in (a). Solid lines show the analytically calculated c_{p}(f) from equation (22). Sidemounted system shows less agreement between the analytically and the numerically derived concentration because photons entering the system close to a solar cell are absorbed with a higher probability. This leads to an inhomogeneous chemical potential μ for the incoming photons which violates the assumptions for the calculation. But, the approach μ = const. is a good approximation in the statistical limit for the bottommounted solar cells [27]. 
This is the same condition for the system Rau et al. [29] describe. As derived in equation (11), the concentration is the ratio between the flux inside the collector Φ_{FC} and the incident flux Φ_{sun}. Following the approximately uniform μ proposed by Markvart et al. and comparing equation (13) to the result of dividing equation (19) by equation (20), we also understand the maximum concentration as (21)Linking the descriptions of the system via equalizing the photon fluxes in both descriptions with the values derived from the MonteCarlo simulation yields the analytical expression for the concentration (22)with the coverage fraction f = A_{cell}/A_{coll}. In Figure 6b we derive the numerical concentration from the numerical simulated collection probability p_{c} of Figure 6a and compare this with the analytically calculated concentration from equation (22). The analytical solutions excellently fit the statistically derived values for the bottommounted system. As described above, the system is described in thermal equilibrium and with same chemical potential μ for all incoming photons. In particular it holds qV_{oc} = μ for the cell at the side of the collector. Yet, under short circuit conditions, the voltage V of the cell equals zero and at the solar cell the chemical potential of the photons is μ = 0. Therefore, μ cannot be constant throughout the system. (This finding is also derived by the consideration that in short circuit condition photons enter the system and there they are transported to the solar cells.) A net flux of photons requires local differences in the chemical potential. In a more coarse resolution of the solar cells, at which the period length l exceeds the mean free path of the photons as it is the case for the sidemounted system, the results are not valid any more. Photons entering the system close to a solar cell are absorbed with a higher probability, whereas photons in areas with no solar cell are most likely reabsorbed by the dye and, with a higher probability, reemitted from the collector. Systems in Figures 1a and 1b become identical. Thus, in Figure 6b the sidemounted system shows less agreement between the analytically and the numerically derived concentration. However, the approach μ = constant is a good approximation in the statistical limit for the bottommounted solar cells[27]. We achieve the opencircuit condition by reducing the coverage fraction f of solar cell area to photovoltaic unreasonable low values. The collection probability p_{c} decreases with decreasing f, and reduces to nearly zero at f < 10^{5}. Concurrently, the concentration approaches the theoretical maximum c_{p,max} = 4251. Note, that c_{p} is reaching the maximum c_{p,max} for coverage fractions f at which p_{c} is almost zero. The thermodynamic limit of the concentration lies therefore beyond photovoltaic useful collector dimensions.
4.4 Comparison of sidemounted and bottommounted FC
This and the following section compare FCs with bottommounted to FCs with sidemounted solar cells. In order to compare the bottommounted FC, where the coverage fraction and the collector length are decoupled, with a sidemounted system, we use the modified sidemounted system displayed in Figure 1b. Keeping a constant coverage fraction f upon variation of the collector length l requires the adjustment of the solar cell side length sto (23)likewise s/d = f(l/d)^{2}/4 for the normalization of all quantities to the collector thickness d. The maximum coverage fraction f_{max} for the sidemounted system is given by (24)because in this case the side length s of the solar cell in equation (23) equals the collector length, and the systems in Figures 1a and 1b become identical.
Figures 7a, 7b compare the collection probabilities of side and bottommounted systems for fixed collector length l = 10d (Fig. 7a) and 100d (Fig. 7b). Therefore, for the sidemounted system coverage fractions up to f_{max} = 0.4 and 0.04 respectively are modeled.
Fig. 7 Comparison of FC systems with solar cells at the sides or at the back as sketched in Figures 1b and 1c. (a) Systems at collector length l = 10d. Without PBS sidemounted solar cells provide slightly higher collection probabilities for 10^{3} < f < 4 × 10^{1}. Applying PBS eliminates the difference in collection probability p_{c} for high coverage fractions. Both systems achieve approximately p_{c} = 1 for coverage fractions f ≈ 1. For coverage fractions f < 10^{2} mounting solar cells at the FC back side is of slight advantage. (b) Systems at collector length l = 100d. Mounting solar cells at collector sides leads to higher collection probabilities for all cases. Compared to Figure 7a, the sidemounted system obtains the same values. Therefore, this system works still in the smallscale limit, whereas for this collector length the bottommounted system is already in the transition regime to the largescale limit which is also shown in Figure 5. 
Figure 7a presents a comparison between bottommounted and sidemounted system, both with a collector length l = 10d. The results outline that without applied PBS the sidemounted system performs better for low coverage fractions. The application of PBS on top of the collector yields higher photon collection for the bottommounted system in this region. In the region of f > 10^{2} both systems reach collection probabilities close to 100%. As shown in Figure 5 a collector with bottommounted solar cells works in the statistical limit. Without the application of a PBS sidemounted solar cells provide slightly higher collection probabilities for 10^{3} < f < 4 × 10^{1}.
Figure 7b shows the comparison of the two systems with collector length l = 100d. Compared to Figure 7a, the sidemounted system obtains the same values for p_{c}. This implies that this system still works in the statistical limit whereas the bottommounted system passes into the largescale region. This is accordingly seen in Figure 5. Therefore, in Figure 7b the FC system with solar cells covering the sides performs better at all coverage fractions with or without PBS.
Fig. 8 Comparison of FC systems depicted in Figures 1b and 1c with a constant coverage fraction f = 0.01 analyzing influence of nonradiative losses in the dye. (a) Sidemounted system is only modeled up to a collector length l ≤ l_{max} = 4d/f corresponding to a full coverage of all collector sides. Inclusion of nonradiative losses deteriorates the photon collection p_{c} in all cases. The effect is more significant in the system with applied PBS. (b) Bottommounted system. The simulation covers a wider range of collector lengths. With PBS the bottommounted system in its radiative limit shows a significant drop in p_{c} for l/d ≥ 100. The inclusion of nonradiative losses reduces this limit to l/d ≥ 1. In comparison : for p_{nr} = 0 (only radiative losses) the sidemounted system leads to a better photon collection p_{c}, especially at large values of l. Inclusion of nonradiative losses leads to the same behavior in the largescale region, but for collector lengths in the statistical limit, the bottommounted system performs better. Without PBS the sidemounted system yields higher p_{c} at all collector lengths even with inclusion of nonradiative losses. 
4.5 Loss mechanisms
This section analyzes the systems in the radiative limit as well as the influence of nonradiative losses in the dye, of a nonperfect mirror at the back side, and of a nonideal reflection cone at the PBS filter. Unlike in Figures 7a, 7b, we now vary the collector length l/d while assuming a constant coverage fraction f = 0.01 for all simulations and an additional f = 0.9 for Figures 9a, 9b. Thus, the sidemounted system is modeled up to a maximum collector length l = l_{max} = 4d/f = 400d and 4.44d, respectively, where the side length s of the solar cells equals the collector length l.
4.5.1 Nonradiative losses in the dye
Figures 8a, 8b compare the side and the bottommounted systems in their radiative limit with a probability p_{nr} = 0 for nonradiative losses and under the influence of nonradiative recombination in the dye (p_{nr} = 0.02, 0.08 and 0.1). Figure 8a shows the results for the sidemounted system. In the radiative limit the application of a PBS increases the photon collection from p_{c} ≈ 19% to p_{c} ≈ 95%. These values remain constant up to the maximal simulated collector length l_{max} = 400d. As discussed above, this means that the reabsorption length in this system lies considerably under the collector length.
Fig. 9 Influence of nonperfect mirror at the back side of the FC in systems with a coverage fraction f = 0.01. (a) Sidemounted solar cells. Inclusion of nonradiative losses in the dye decreases the photon collection significantly. The relative deterioration in a system with applied PBS is higher than in the system without PBS. (b) Bottommounted solar cells. At small l/d the system shows a similar behavior as the sidemounted system. 
Figure 8b presents the results for the bottommounted system. Note, that the simulation covers a larger range of collector lengths l/d. As explained above, enlarging the systems degrades p_{c} even more than the inclusion of nonradiative recombination in the dye. However, the lowering in the photon collection occurs with p_{nr} = 0 at l/d = 100, but this limit reduces to l/d ≤ 10 for p_{nr} = 0.1.
Under the application of PBS in the radiative limit both systems display a similar p_{c} ≈ 96.5% in the limit of small collector lengths l/d < 10. Whereas the data for the bottommounted system considerably decay at l/d > 100, a high collection probability of p_{c} > 95% is maintained by the sidemounted system up to the maximum collector length l_{max}/d = 400. The inclusion of nonradiative losses in the calculation deteriorates the collection probability in both cases significantly. Also, the transition to largescale limit occurs at lower collection lengths. Interestingly, in the smallscale limit l/d < 10 the bottommounted system achieves a better photon collection than the sidemounted system. Here, the collection probability decreases by Δp_{c} ≈ 52% from p_{c} ≈ 98% to p_{c} ≈ 46% whereas in the sidemounted system it decreases by Δp_{c} ≈ 55%.
Comparing these two systems without the application of PBS shows that the sidemounted option displays a collection probability that is consistently higher than that of the bottom mounted systems. In addition, the transition towards the largescale limit that occurs at l/d ≈ 20 for the cells at the bottom is absent for the side mounted system. A collection probability p_{c} > 18% is maintained up to l_{max}. With included nonradiative loss p_{nr} = 0.08 the overall collection probability drops by about 3% for both options. The difference between side and bottommounted systems remains the same.
4.5.2 Reflection losses at the back side mirror
Mirrors cover the collector back sides either fully in the sidemounted system or partly in the bottommounted system. Optical losses due to nonideal reflection are a major loss mechanism especially for large collector systems. Note that other losses, especially parasitic absorption in the polymer matrix (not considered in the present paper), are expected to have comparable effects. In the following, we study the influence of reflection coefficients R < 1 in comparison to the perfect case (R = 1). We do not consider explicitly the case of an airgap between mirror and collector which would restrict the losses to angles larger than the critical angle of total internal reflection. The present approach uses Rvalues which are independent of the direction of the photons and could be looked at as directional averages.
Figures 9a, 9b depict the influences of reflection losses on the photon collection p_{c} of both systems. Figure 9a presents the results for the sidemounted system. Since less radiative losses occur in the system with PBS the contribution of nonradiative reflection losses at the back side mirror is more significant. Therefore, the decrease in the photon collection is higher.
Figure 9b shows the photon collections for the system with bottommounted solar cells. For small l/d the system behaves similar to the sidemounted system. The larger the system the more often a reflection takes place, thus, increasing the number of lost photons. Therefore, included reflection losses decrease the photon collection already at l/d = 10, whereas for a mirror with R = 1 this drop occurs at l/d = 100.
In systems with coverage fraction f = 0.01 at any of the calculated collector length the reabsorption is a rare event compared to the number of reflections at the FC back side as the following example clarifies. Most photons emitted by the dye experience the absorption coefficient α_{2} = 0.03/d as mentioned above. With d = 3cm, a typical thickness for industrialized fluorescent collectors, the photons are reabsorbed after 100 cm. A photon emitted with an angle θ = 45° hits the collector sides every 4.2 cm. Since the coverage fraction f = 0.01 is very low, most likely the photon is reabsorbed before it hits a solar cell. Such, a photon with θ = 45° is reflected 23 times before it is reabsorbed. Therefore, a nonperfect mirror has a stronger influence than an equally high nonradiative recombination loss in the dye. A reflection R = 98% at the back side mirror causes a drop Δp_{c} ≈ 70% for small l/d which is higher than the drop Δp_{c} ≈ 25% caused by the nonradiative recombination loss as shown in Figures 8a, 8b. Therefore, the mirror applied in the system has to feature a superior quality compared to the fluorescent dye.
4.5.3 Nonperfect photonic structure
In the simulations the application of a photonic band stop as an energy selective filter increases the photon collection in all cases. However, in realistic filters blocking the photons depends not only on the energy but also on the angle of incidence of the photon as schematically shown in Figure 3e. Figures 10a, 10b outline the influence of smaller reflection cones (θ_{pbs} = 10° and 20°) on side and bottommounted systems with coverage fractions f = 0.01 and 0.9.
Figure 10a depicts the results for the sidemounted system. The application of a nonperfect PBS filter decreases the photon collection at all collector lengths. However, a relatively higher drop occurs for the system with f = 0.01 than for the system with f = 0.9. Here, a higher reabsorption rate occurs due to longer distances between the cells. Therefore, photons are more often reemitted with their direction spatially randomized. The frequent randomization carried out also in unfavorable angles contributes to the number of lost photons.
Fig. 10 Influence of a photonic band stop filter with angular selectivity. (a) Sidemounted system. Coverage fraction f = 0.9 achieves higher photon collections p_{c} than f = 0.01 for all θ_{pbs}. (b) Bottommounted system. For small scales the system shows a better performance than the sidemounted system because this system benefits from the randomization of photon direction during reemission. 
Figure 10b presents the calculation results for the bottommounted system. At small l/d this system collects more photons under the application of a nonperfect PBS filter than the sidemounted system. This is due to the effect that the disadvantageous angles for the PBS are very favorable angles for solar cells at the FC back side. Therefore, the collection of a photon is more likely than in the sidemounted system. As in the sidemounted system, the collection in the system with f = 0.01 decreases more at large l/d than in the system with f = 0.9.
4.5.4 Realistic values
In order to get a more realistic picture, we perform a system simulation which includes several loss mechanisms simultaneously. While the absorption/emission in the fluorescent collector is as ideal as presented in Figure 2, the nonradiative recombination rate lies at 5%, which is a value typically achieved in industrial FCs. For the back side mirror, we assume a reflectivity R = 92%, which corresponds to the reflectivity of Aluminium. Furthermore, we assume dielectric layers as photonic structures which show transmittances T > 95% if manufactured carefully. However, they always show a blue shift in their transmittance spectrum for oblique incident angles. Thus, adjusting the transmittance spectrum properly should lead to a full band stop for all ray directions. Nevertheless, we assume θ_{pbs} = 30°. The coverage fraction is f = 0.1. Note, that for a clear acrylic glass on top of the solar cells, the coverage fraction f = 0.1 leads to collection probability p_{c} = 0.1.
Fig. 11 Sidemounted and bottommounted solar cells on fluorescent collectors including realistic values for the back side mirror, the photonic structure and the fluorescent dye. The bottommounted solar cells are in the smallscale limit for l/d < 1 and in the largescale limit from l/d > 10^{3}. For a clear acrylic glass on top of the solar cells, the coverage fraction f = 0.1 leads to a collection probability p_{c} = 0.1. Thus, a fluorescent collector system with the assumed losses increases the collection probability by 23%, with a PBS on top by 27%. The sidemounted solar cells show an interesting behavior for l/d > 11. Instead of descending into the largescale limit, p_{c} increases up to 0.47 at l/d > 40 (with PBS on top). 
Figure 11 shows that bottommounted solar cells perform as expected from the previous results : the system is in the smallscale limit for l/d < 1 and in the largescale limit from l/d > 10^{3}. Thus, a FC system with the assumed losses increases the collection probability by 23%, with a PBS on top even by 27% compared to clear acrylic glass. The sidemounted solar cells show an interesting behavior for l/d > 11. Instead of descending into the largescale limit, p_{c} increases up to 0.47 at l/d > 40 (with PBS on top). We explain this effect, which also is already slightly indicated in the data of Figure 10a, as follows : the larger l/d, the more collector edge area is covered with solar cells until they are fully covered for l/d = 40. Thus, the photon path length is larger for small l/d than for large l/d, because the probability to hit solar cell area is smaller. The longer the path length, the higher is the probability for the photon to be subject to loss mechanisms.
5 Conclusion
MonteCarlo simulations compare photovoltaic fluorescent collectors in sidemounted and bottommounted systems as well as the systems with or without the application of a photonic band stop filter on top. The filter greatly enhances the overall collection probability by suppressing emission of converted photons from the surface of the collector. Also, the sensitivity of the collector to repeated reabsorption and rerandomization of the fluorescent photons is reduced. We find that the collection probability of systems with identical coverage fraction, i.e., the same amount of solar cell area per unit collector area is heavily influenced by the scaling of the solar cell size. Many small solar cells generally perform better than few large solar cells with the same overall area (scalingeffect). The systems in their radiative limits are compared to systems which include loss mechanisms. The comparison of collectors with solar cells mounted at the collector side to a system with the cells at the bottom shows that in most cases the sidemounted system displays a higher collection efficiency. We find that a nonperfect reflection at the back side mirror causes a higher deterioration in the photon collection than a comparable nonradiative recombination in the dye. This is because in the analyzed systems the reabsorption is a rare event compared to a reflection at the back side. Therefore, the quality of the back side mirror merits especial care. Assuming a restricted reflection cone of the photonic band stop filter causes higher losses in the sidemounted system than in the bottommounted system which is caused by the randomization of the photon angles during the emission by the dye. All loss effects are especially significant for systems without a filter. We see that a good quality of the filter is especially necessary in systems with low coverage fraction of solar cells. In comparison to the classical sidemounted system, the system with solar cells at the back side performs equally well for small scales. Therefore, applying fluorescent collectors technically less expensive on top of solar cells is a promising approach, if the solar cells are properly scaled in size and distance.
Acknowledgments
This work was supported by a grant of the Deutsche Forschungsgemeinschaft (DFG, contract PAK88, ‘nanosun’). The authors wish to thank G. Bilger, C. Ulbrich, and T. Kirchartz for numerous discussions as well as J.H. Werner for continuous support.
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All Figures
Fig. 1 Sketch of the fluorescent collector geometries compared in the present paper (seen from the bottom). (a) Classical design with the solar cells mounted at each side of the collector with length l and thickness d. (b) Modified classical system where only a fraction of the respective collector sides is covered with a solar cell area A_{cell} = ds. (c) Collector with solar cells with an area A_{cell} = s^{2} mounted at the bottom. In all cases the (remaining) back side is covered with a mirror. (d) Systems (b) and (c) are assumed to be periodic in space. Exemplary, a detail of the bottommounted system is shown. 

In the text 
Fig. 2 Sketch of the absorption and emission behavior as assumed in this paper. The dye absorption is given by a step function. Incoming photons have the energy E_{1} and a high absorption coefficient α_{1}. The lower absorption coefficient α_{2} holds for the lower energy E_{2} and leads with Kirchhoffs law (Eq. (1)) to a high emission coefficient e_{2}. The model also features the possibility of an energy selective photonic band stop (PBS) that keeps the emitted photons in the FC system. 

In the text 
Fig. 3 Light guiding behavior of a fluorescent collector covered with solar cells at the sides and a mirror at its back side. (a) Definition of photon ray angle θ. (b) Absorbed photons are reemitted spatially randomized. The system leads rays with θ > θ_{c} to the sides of the collector. (c) Rays with angle θ < θ_{c} for total internal reflection leave the top surface. (d) Applying a photonic band structure (PBS) keeps rays with θ < θ_{pbs} in the system as well. This PBS is energy selective with θ_{pbs} = θ_{c}. Therefore, rays with energies E ≤ E_{1} are kept in the system only. (e) For an energy and angular selective PBS a reflection cone is assumed such that only rays with E ≤ E_{1} and θ < θ_{pbs} are kept in the system. 

In the text 
Fig. 4 The collection probability p_{c} of the classical fluorescent collector geometry (Fig. 1a) fully covered with solar cells at its sides depends on the collector length l normalized to the thickness d. A perfect mirror covers the back side. The systems with a photonic band stop filter (PBS, open symbols) have higher collection probabilities than those without PBS (filled symbols). For the system with PBS a value of p_{c} close to unity remains up to a normalized collector length l/d ≈ 500 and nonradiative recombination probability p_{nr} = 0 in the dye. In contrast, the system without PBS has a maximum p_{c} only slightly above 80% even in the radiative case with p_{nr} = 0. With increasing collector length l/d the collection probability p_{c} decreases immediately. The consideration of nonradiative recombination in the dye, i.e. p_{nr} > 0, leads to a deterioration of p_{c} in all cases. The cell coverage fraction f (top axis) is directly linked to the collector size for the chosen geometry. 

In the text 
Fig. 5 Collection probability p_{c} of the fluorescent collector geometry with solar cells mounted at the bottom of the collector (Fig. 1c). The coverage fraction f is kept constant at f = 0.01 and the normalized collector length l/d is varied. All data feature a transition between two asymptotic situations at low and high ratios l/d. Only the systems with a photonic band pass (PBS, full symbols) achieve p_{c} > 0.5 at low ratios of l/d. For these systems, the maximum p_{c} as well as the transition from the smallscale to the largescale behaviour is strongly dependent on the nonradiative recombination probability p_{nr}. The systems without PBS (open symbols) have already a relatively low p_{c} < 20% even in the more favorable case of l/d < 10. Accordingly, the sensitivity to the introduction of nonradiative recombination in the dye (p_{nr} > 0) is less pronounced. 

In the text 
Fig. 6 (a) Collection probabilities p_{c} for photons derived with a MonteCarlo simulation for a side and a bottommounted system in the statistical limit and with applied PBS. (b) Symbollines are the values for the concentration c_{p} derived from p_{c} in (a). Solid lines show the analytically calculated c_{p}(f) from equation (22). Sidemounted system shows less agreement between the analytically and the numerically derived concentration because photons entering the system close to a solar cell are absorbed with a higher probability. This leads to an inhomogeneous chemical potential μ for the incoming photons which violates the assumptions for the calculation. But, the approach μ = const. is a good approximation in the statistical limit for the bottommounted solar cells [27]. 

In the text 
Fig. 7 Comparison of FC systems with solar cells at the sides or at the back as sketched in Figures 1b and 1c. (a) Systems at collector length l = 10d. Without PBS sidemounted solar cells provide slightly higher collection probabilities for 10^{3} < f < 4 × 10^{1}. Applying PBS eliminates the difference in collection probability p_{c} for high coverage fractions. Both systems achieve approximately p_{c} = 1 for coverage fractions f ≈ 1. For coverage fractions f < 10^{2} mounting solar cells at the FC back side is of slight advantage. (b) Systems at collector length l = 100d. Mounting solar cells at collector sides leads to higher collection probabilities for all cases. Compared to Figure 7a, the sidemounted system obtains the same values. Therefore, this system works still in the smallscale limit, whereas for this collector length the bottommounted system is already in the transition regime to the largescale limit which is also shown in Figure 5. 

In the text 
Fig. 8 Comparison of FC systems depicted in Figures 1b and 1c with a constant coverage fraction f = 0.01 analyzing influence of nonradiative losses in the dye. (a) Sidemounted system is only modeled up to a collector length l ≤ l_{max} = 4d/f corresponding to a full coverage of all collector sides. Inclusion of nonradiative losses deteriorates the photon collection p_{c} in all cases. The effect is more significant in the system with applied PBS. (b) Bottommounted system. The simulation covers a wider range of collector lengths. With PBS the bottommounted system in its radiative limit shows a significant drop in p_{c} for l/d ≥ 100. The inclusion of nonradiative losses reduces this limit to l/d ≥ 1. In comparison : for p_{nr} = 0 (only radiative losses) the sidemounted system leads to a better photon collection p_{c}, especially at large values of l. Inclusion of nonradiative losses leads to the same behavior in the largescale region, but for collector lengths in the statistical limit, the bottommounted system performs better. Without PBS the sidemounted system yields higher p_{c} at all collector lengths even with inclusion of nonradiative losses. 

In the text 
Fig. 9 Influence of nonperfect mirror at the back side of the FC in systems with a coverage fraction f = 0.01. (a) Sidemounted solar cells. Inclusion of nonradiative losses in the dye decreases the photon collection significantly. The relative deterioration in a system with applied PBS is higher than in the system without PBS. (b) Bottommounted solar cells. At small l/d the system shows a similar behavior as the sidemounted system. 

In the text 
Fig. 10 Influence of a photonic band stop filter with angular selectivity. (a) Sidemounted system. Coverage fraction f = 0.9 achieves higher photon collections p_{c} than f = 0.01 for all θ_{pbs}. (b) Bottommounted system. For small scales the system shows a better performance than the sidemounted system because this system benefits from the randomization of photon direction during reemission. 

In the text 
Fig. 11 Sidemounted and bottommounted solar cells on fluorescent collectors including realistic values for the back side mirror, the photonic structure and the fluorescent dye. The bottommounted solar cells are in the smallscale limit for l/d < 1 and in the largescale limit from l/d > 10^{3}. For a clear acrylic glass on top of the solar cells, the coverage fraction f = 0.1 leads to a collection probability p_{c} = 0.1. Thus, a fluorescent collector system with the assumed losses increases the collection probability by 23%, with a PBS on top by 27%. The sidemounted solar cells show an interesting behavior for l/d > 11. Instead of descending into the largescale limit, p_{c} increases up to 0.47 at l/d > 40 (with PBS on top). 

In the text 
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