Modeling photon transport in fluorescent solar concentrators

Fluorescent solar concentrators (FSC) can concentrate light onto solar cells by trapping fluorescence through total internal reflection. In an ideal FSC, the major obstacle to efficient photon transport is the re‐absorption of the fluorescence emitted. In order to decompose the contribution of different photon flux streams within a FSC, the angular dependent re‐absorption probability is introduced and modeled in this paper. This is used to analyze the performance of different FSC configurations and is also compared with experimental results. To illustrate the application of the modeling, the collection efficiency of ideal devices has also been calculated from the re‐absorption probability and is shown to be useful for estimating non‐ideal losses such as those due to scattering or reflection from mirrors. The results also indicate that among the FSCs studied, the performance of those surrounded by four edge solar cells is close to ideal. The rapid optimization of FSCs has also been presented as another practical application of the models presented in this paper. © 2014 The Authors. Progress in Photovoltaics: Research and Applications published by John Wiley & Sons, Ltd.


INTRODUCTION
The main barrier to the widespread utilization of solar energy for electricity generation is higher costs as compared with conventional energy sources. There are a number of approaches to reduce the cost of solar energy [1]. One is to concentrate light in order to reduce the amount of solar cell material required. Concentration of light can be achieved using mirrors and lenses that require direct sunlight or through the use of a fluorescent solar concentrator (FSC) that can also concentrate diffuse light.
A FSC consists of fluorescent molecules embedded within a host material. Solar radiation is absorbed by the fluorescent material and is re-emitted isotropically [2,3]. Emitted photons incident on the top or bottom surface of the FSC at a zenith angle greater than the critical angle, θ c , will be trapped and concentrated onto the edge [2][3][4][5]. The thermodynamic efficiency limits for these devices calculated using the detailed balance limit [6,7] indicate the potential to achieve performances close to the Shockley-Queisser limit [8]. In particular, it has been shown that on incorporating a photonic band stop filter to FSCs, a theoretical efficiency limit of 26.8% when coupled to silicon solar can be achieved [7].
One technique for modeling the photon transport within FSCs is by calculating the re-absorption probability of certain photon flux streams found within FSCs [4] in order to track the effect of multiple absorption and emission events [9]. The modeling of the re-absorption probability of fluorescence within FSCs began with Weber and Lambe [4] who described a model for the re-absorption probability of the trapped fluorescence within a FSC attached to a single solar cell. The setup they considered is equivalent to a rectangular FSC with a solar cell mounted on one edge and mirrors on the other three edges. This configuration will be referred to by the term 1Cell setup.
In order to maximize edge concentration, re-absorption of trapped fluorescence needs to be avoided. This is because re-absorption can result in either re-emission of photons that escape the FSC through the top or bottom surfaces or loss of the absorbed energy through non-radiative processes. In the 1Cell setup, rays emitted parallel or close to parallel to the solar cell are required to travel a long path length in order to reach the solar cell, and therefore, their reabsorption probability is large. Furthermore, the reflection from mirrors on edges without a solar cell attached adds additional losses. The use of FSCs with four edge-mounted solar cells (4Cell setup) reduces or removes these losses and has therefore been explored extensively [10]. A comparison of the 1Cell and 4Cell setup has been shown in Figure 1.
The highest power conversion efficiencies of FSCs reported in the literature all use the 4Cell setup [11][12][13]. To our knowledge, re-absorption models have not been adapted to the 4Cell setup. This paper develops on concepts from [4] and [9] and applies them to the 4Cell setup because this is the most promising configuration for achieving high efficiency FSCs.
Angular measurements of the photon flux/fluorescence escaping the edge of FSCs have previously been measured in [14][15][16][17] to determine which angles of emission correspond to the greatest loss of photons. A cylindrical lens coupled to the FSC edge in [14,15] and [16] was used to "see" inside the FSC. Monte Carlo simulations of the photon flux exiting the FSC edge were also presented in these publications. In [17], the FSC edge was not coupled to a lens, and a simple numerical model that qualitatively explains fringe-like patterns detected for spot illumination was presented. However, to our knowledge, angular resolved measurements and modeling of FSCs with edges coupled to solar cells/mirrors have not been previously reported. Studying photon flux streams of FSCs with edges coupled to solar cells/mirrors is critical because this is how FSCs for photovoltaic applications are used in operation. This paper presents experimental measurements of the angularly resolved photon flux and fluorescence exiting the edge in two of the most common FSC configurations, that is, the 1Cell and 4Cell setups. Models have also been developed to analyze the experimental measurements.
Estimation of non-ideal losses such as due to scattering and reflection losses is shown to be a practical application of the models developed and has been demonstrated for two different methods of fabricating FSCs: that is, spin coating (Spin-FSC) and molding (Mold-FSC). The utility of these models has further been highlighted by the optimization of the dye concentration of a FSC containing Lumogen F Red 305 dye (BASF, Ludwigshafen, Germany).

THEORY
A FSC of volume V is considered. Assuming uniform illumination over the top surface area of the FSC, the re-absorption probability, r, of the trapped photon flux of wavelength λ in an ideal FSC is given by Equation (1): where P is the fraction of photons emitted within the escape cones, dV is a small volume element, dΩ is a small solid angle element, l is a measure of the path length from point of emission to a solar cell, and α em is the absorption coefficient at the emission wavelength. Ω t indicates that the integration is over the solid angle of total internal reflection (i.e., the trapped solid angle). Assuming absorption can be considered to be uniform across the thickness of the FSC and expanding dΩ, we obtain where A top is the top surface area of the FSC and dA is a small area element, ϕ and θ are the azimuthal and zenith angles of emission, respectively (with θ c denoting the critical angle for total internal reflection).
The average re-absorption probability of the trapped photon flux, R, is obtained by a weighted average of r over the normalized first fluorescence, f 1 (i.e., re-absorption free fluorescence): The re-absorption probability for different azimuthal angles of emission, Γ, follows from Equation (2): The average angular re-absorption probability Γ′ is calculated using an expression similar to Equation (3): Assuming that the wavelength distribution of fluorescence emitted is independent of the excitation wavelength, building upon expressions presented in [9], Γ′ can be connected to the average angular collection efficiency in an ideal FSC, ρ, that is, the probability that an absorbed photon results in emission at an angle ϕ that reaches a solar cell: where R is the average re-absorption probability of the escape cone photon flux and ϕ f is the quantum yield of the fluorescent molecule. The denominator in Equation (6) corresponds to the effect of photon recycling due to re-absorption. From Equation (6), it is clear that ρ(ϕ) is directly proportional to 1 À Γ′ (ϕ) in an ideal FSC. The photon flux emitted at an angle ϕ reaching the edge, N edge , can be written as where Q a is the absorption efficiency, that is, fraction of incident photons absorbed by the FSC and N inc is the photon flux incident on the FSC. In Equation (7), the term in the integral is a constant independent of ϕ; therefore, The previous equation shows that in an ideal FSC, we expect the angular distribution of the photon flux exiting the FSC edge to be proportional to 1 À Γ′ (ϕ).
In [18], the collection efficiency, Q c , that is, the probability that an absorbed excitation photon results in a photon reaching a solar cell, of a FSC is decomposed into an integration over the spectral collection efficiency, χ.
In this paper, we can similarly decompose Q c into an integration over ρ: Integrating Equation (9), expressions for Q c from [9] and [18] are reproduced: The optical quantum efficiency (OQE) is the ratio between photons concentrated onto edge-mounted solar cells and excitation photons incident on the FSC top surface. In a FSC where the wavelength distribution of fluorescence emitted can be assumed to be independent of the excitation wavelength, the OQE can be written as the product of two terms, a wavelength-dependent term related to the absorbance of the FSC and a wavelength-independent term, that is, Q c : The expressions presented do not include non-ideal losses such as due to scattering or inefficient reflection from mirrors. These non-ideal losses only affect the wavelengthindependent term, Q c .
Comparing the difference between ideal models and experimentally measured results would allow the estimation of these non-ideal transport losses in FSCs.
When we consider the 4Cell setup, it is necessary to model the system in two dimensions as can be seen in Figure 2. ϕ 1 and ϕ 2 in this figure represent the extent of the azimuthal angle of emission from a fluorescent molecule toward the solar cell placed at the edge shown at the top of this figure. The re-absorption probability therefore depends on the solar cell an emitted ray is emitted towards. In a square FSC of length L surrounded by four solar cells ( Figure 2), distance l for a ray emitted toward the top solar cell shown in Figure 2, is given by Modeling photon transport in FSCs T. S. Parel et al. Figure 2 allows ϕ 1 and ϕ 2 to be determined for emission from a given point in the xy plane as well as the two additional azimuthal angles required for a complete description. Dividing a FSC in the 4Cell setup into such distinct position-dependent angular regions for calculating the path length has also been used in [19]. That study, however, focused on the effect of photonic band stop filters coupled to FSCs. Reflection due to the application of such devices to FSCs also depend on the path length of light, that is, there are more reflection events with increasing path length (on average) for emission at a given direction.

Simple trigonometric arguments that follow from
Γ and Γ′ (and therefore R) are obtained from Equations (4) and (5), in addition to Equation (12) for the 4Cell setup and expressions detailed in [4] for the 1Cell setup. From Equation (12), we know that l depends on x and y; from Equation (4), this indicates that Γ (as well as related quantities like r, Γ′ and R) for the 4Cell setup requires integration over different points in the xy plane. The 1Cell setup, however, as explained in [4] can be simplified by an infinite strip (Figure 3) assuming the edge mirrors orthogonal to the solar cell edge have close to unity reflection. In this case, l only depends on one axis, and therefore, the different re-absorption probabilities are calculated by integration over only different points along one axis.
R is given by expressions presented in [20] weighted with an absorption distribution with depth (from the Beer Lambert's law). Q c can then be calculated and used in order to obtain the OQE of an ideal FSC.
The OQE can be then connected to the power conversion efficiency of the system η p , using where FF is the fill factor of the attached solar cells, V oc is the open circuit voltage of the solar cell, EQE is the average EQE of the solar cell to incident fluorescence, and P inc is the power of the incident light.

EXPERIMENTAL
The FSCs used in this study consists of 20 For the Spin-FSC, the top fluorescence of a low dye concentration sample was used to obtain the first fluorescence.
A solar simulator (T.S. Space Systems, Marlow, United Kingdom) equipped with a 300 W Xenon lamp approximating the AM1.5 spectrum and a filter selection wheel (350-1100 nm, step 50 nm) was used to measure the current output of the solar cell(s) coupled to the FSCs. The Xenon lamp was calibrated using a silicon standard solar cell calibrated at the National Renewable Energy Laboratory for standard test conditions (AM 1.5, 1000 W/m 2 at 25°C).
Optical gel (Thorlabs, Ely, United Kingdom) was used for index matching (refractive index of 1.46 at 589.3 nm) between the solar cell and the FSC. Index matching gel applied at the interface between the FSC and the solar cell ensures that light from a full hemisphere reaches the solar cell.
The emission was analyzed as a function of angle from the edge of the FSC by attaching a square silicon photo diode (Thorlabs, 1 mm × 1 mm) to a motorized system. The detector was set up to move in a semi-circle at a distance of 8 cm about the central point of the FSC edge, within in a plane parallel to the FSC top surface (Figures 4 and 5 show the plane of rotation). The emission spectrum of the edge emission was also analyzed as a function of the emission angle, and an iris diaphragm was attached at the detector to limit the angular range detected. A flexible silicon solar cell (Sol Expert Group Baindt, Germany) was used to measure the photon flux reaching a 1 mm thick semi-circle 8 cm from the FSC edge and used to normalize the area under the angular distribution. The FSC was illuminated uniformly at the top surface by a LED Luxeon (Philips Lumileds, San Jose, California, USA) light source. A single 15 mm focal length lens was attached at the edge of the FSC in order to directly lead the detected light toward the detector without refraction or reflection affecting its path. Optical gel was also used to couple the FSC edge to the lens. Finally, a thin black surface was used as a blind in order to block any light coming from directions other than the FSC edge from reaching the detector.

RESULTS AND DISCUSSION
(a) Photon flux and fluorescence with angle A study of the angular characteristics of the fluorescence exiting the FSC edge has been conducted for different azimuthal angles of emission (the setup is as shown in Figures 4 and 5).
The models described have been used to study the photon transport for the 1Cell and 4Cell setups. For simplicity, photon transport at only the peak emission wavelength, that is, 600 nm (absorption coefficient equal to 0.8 mm À1 ), will be described. Γ calculated using the analytical models of the two setups is shown in Figure 6 for fixed dimensions of the FSC, while in Figure 7, the gain is fixed (the gain is defined as the ratio between the FSC top surface area and edge solar cell area).
The results indicate that the variation of Γ with ϕ for the 4Cell setup is fairly constant. For the 1Cell setup, Γ is seen to go to unity for azimuthal angles of emission close to parallel to the edge coupled to a solar cell due to long path lengths.
Furthermore, there is a significant difference in Γ for fluorescence emitted toward and away from the solar cell, although the calculated Γ is large for the sample and setups considered, due to the high quantum yield of the fluorescent species re-absorption results in photon recycling. Most of the photons are, therefore, eventually emitted from the FSC, and the re-absorption probability affects only the ratio of how many leave through the entrance aperture or hit a solar cell.    Modeling photon transport in FSCs T. S. Parel et al.
For the 4Cell setup, it is noticed that photons emitted at an azimuthal angle normal to a particular solar cell surface (i.e., ϕ close to 0°, 90°, 180°, 270°) have a higher reabsorption probability. This is likely because rays emitted at these angles, irrespective of the distance between the point of emission and the relevant solar cell, can only reach one particular solar cell and therefore travel longer path lengths on average. It is noticed that when the dimensions of the FSC are kept constant, the fluorescence within the 4Cell setup experiences significantly lower re-absorption, while when the gain is fixed, the total re-absorption (as represented by the area under the curves) is similar for both setups. This indicates that in ideal FSCs, the performance should be similar for both configurations if the gain is kept constant. These results also indicate that although reabsorption increases with increasing A top , adjusting the gain accordingly, that is, by also increasing the FSC thickness, the performance of FSCs can be maintained (Q a must also be maintained by adjusting the dye concentration). Figure 8 shows the experimentally measured variation in the photon flux of emission reaching the edge(s) for the 4Cell and 1Cell setups (fixed FSC dimensions) with the azimuthal angle of incidence on the solar cell (β). The plotted curves have been normalized such that the integration over all angles gives the photon flux measured by the flexible solar (note that the total photon flux for the 4Cell setup is four times the photon flux measured by the flexible solar cell at one edge).
The angle β is as shown in Figures 4 and 5. For both configurations, it is clear that the photon flux decreases for angles close to parallel to a given edge-mounted solar cell. Furthermore, as expected (due to lower reabsorption), more photons reach the edge(s) in the 4Cell than in the 1Cell setup.
Spectroscopic measurements strongly indicate that the reduction in photon flux at certain angles for the two different configurations is for very different reasons. For the 4Cell setup, it is in fact seen that fluorescence reaching the FSC edge at azimuthal angles close to parallel to the edge (i.e., ϕ close to 0°) is blue shifted compared with fluorescence incident perpendicular to the edge (Figure 9). This indicates that re-absorption is lower at angles close to parallel to the solar cell. This could be because for emission at steep angles to reach a given solar cell, the point of emission must be very close to the FSC edge; otherwise, this emission will reach another solar cell. This also explains why the distribution of the photon flux with angle β is only large for a smaller range of angles for the 4Cell setup. The situation is seen to be reversed for the 1Cell setup where the spectra indicate that the re-absorption probability increases from emission perpendicular to parallel to the edge-mounted solar cell as seen by the corresponding red shift (Figure 10).
In [18], it is described how r can be obtained from experimental measurements of the fluorescence distribution of a photon flux stream, that is, by fitting the measured spectrum with the f 1 spectrum at long wavelengths where low re-absorption is expected and comparing the relative decrease at short wavelengths. In order to estimate the fluorescence distribution likely to be "seen" by the solar cell in the 1Cell and 4Cell setups, the reverse of this technique has been used, that is, from the theoretical calculations of r from Equation (2), f 1 has been scaled so as to obtain the   fluorescence spectrum expected to reach the solar cell. The results are shown in Figure 11. As expected, for the 1Cell setup, the spectrum reaching the solar cell should be red shifted with respect to that of the 4Cell setup (fixed FSC dimensions). Assuming that in the 4Cell setup, each solar cell sees the same distribution of the photon flux with angle β, then a summation of these photon fluxes will give the photon flux distribution according to emission angle ϕ reaching a solar cell. To compare this with the theoretical model, Γ′ has been calculated from Γ using Equation (5) for the 300 mg/l Mold-FSC. The results have been plotted in Figure 12. It is seen that the angular distribution of the photon flux shows the same pattern as the calculated angular distribution of 1 À Γ′. This is consistent with Equation (8), that is, the photon flux exiting the edge of an ideal FSC is directly proportional to 1 À Γ′(φ). The angular results are, therefore, consistent with the performance of the 4Cell Mold-FSC being well described by ideal models.

(b) Optical quantum efficiency and non-ideal losses
Comparisons between calculated and experimental OQE for 4Cell Mold-FSCs are shown in Figures 13 and  14 and show good agreement. This again indicates that the losses of these devices are described well by the ideal model.
On comparing the calculated OQE with experimental results of a Mold-FSC and Spin-FSC (both doped at 300 mg/l) in the 1Cell setup (Figures 15 and 16), the presence of non-ideal losses is apparent. For the Mold-FSC in the 1Cell setup, we notice a 22% drop in the OQE across all absorbing wavelengths. This wavelength-independent difference indicates that this is due to non-ideal transport losses most likely due to reflection at edges coupled to mirrors. For the Spin-FSC, this loss increases to 75%. Because the Spin-FSC lacks uniformity especially at the edges, one possible cause for the large increase in loss compared with the Mold-FSC could be due to scattering in the thin film layers. Indeed, a previous study of a similarly fabricated Spin-FSC (un-doped) indicated that scattering due to Figure 11. Spectra of total emission reaching the solar cell in the 1Cell and 4Cell Mold-fluorescent solar concentrators (FSCs) calculated from r and the measured first generation fluorescence, f 1 . f 1 has been normalized so as to represent a probability distribution.   Modeling photon transport in FSCs T. S. Parel et al.
surface roughness accurately described measurements of the light propagation at different angles in the device [21]. The ability to quantify these non-ideal losses would be useful in the optimization and design of non-ideal FSCs.

(c) Optimization of FSCs
The dye concentrations of FSCs doped with Lumogen F Red 305 dye has been optimized using the models developed for the 4Cell setup to illustrate their predictive ability for a practical FSC. The OQE has been calculated for different dye concentrations for Mold-FSCs as shown in Figure 17. The FSCs modeled contain dye concentrations of integer multiples of the 300 mg/l FSC fabricated and characterized. It is seen that as the dye concentrations increase, the OQE become flatter across absorbing wavelengths. A peak OQE close to 50% for excitation at 570 nm is seen on doubling the dye concentration of the 300 mg/l doped Mold-FSC.
On increasing the dye concentration further, a decrease in the peak OQE is seen because of the competing requirements for high Q a and Q c . After a certain dye concentration, the benefit of greater absorption does not compensate for the higher re-absorption resulting in a decrease in Q c (as indicated by Equations (6), (9), and (11)). Figure 18 shows the power conversion efficiency calculated from Equation (13) when coupled to four gallium arsenide solar cells. The solar cells are assumed to have an open circuit voltage equal to 1.11 and a fill factor or 0.87 [22] (it is assumed that the increase in open circuit voltage with concentration is negligible). The fluorescence incident on the solar cell is also assumed to be converted with an EQE of 0.95.
The results indicate that a power conversion efficiency of close to 3% can be obtained for the 300 mg/l doped Mold-FSC. The highest power conversion efficiency that can be obtained for a FSC based on Lumogen Red F 305 dye (for the given FSC dimensions) is close to 4.5% and requires a dye concentration four times that of the Mold-FSC doped at 300 mg/l. For even larger dye concentrations, a slight decrease in power conversion efficiency is expected because of the effect of re-absorption. The optimization results shown in Figure 18 are consistent with efficiency measurements previously reported, that is, 3.3% in [23] and 2.4% in [24], for FSCs consisting of only Lumogen Red dye. Both of these studies were carried out, however, for the 1Cell setup and used crystalline silicon solar cells. The FSCs used also had a much larger gain. These factors are all expected and seen to result in lower power conversion efficiencies compared with the maximum of the optimization results presented in Figure 18.
Two approaches are required for FSCs to achieve higher power conversion efficiencies. Firstly, the FSC needs to absorb light over a wider range of wavelengths. Scaling the theoretical results down by 22% fits the experimental measurements. This scaling factor is likely due to non-ideal transport losses.  This involves the use of multiple fluorescent species. Secondly the negative effects of re-absorption must be reduced. Indeed, the effect of re-absorption is clear in the OQE results shown in Figure 14. The OQE of the 4Cell Mold-FSC doped at 800 mg/l is flat from 400 to 600 nm unlike the OQE for the 300 mg/l doped FSC in the same configuration (i.e., Figure 13). This indicates that the wavelength-dependent term Q a from Equation (11) is a constant. Indeed, the absorbance of the 800 mg/l doped FSC is so large that nearly all the light is absorbed in this wavelength range. If there was no re-absorption, the only loss would be due to the escape cones (close to 25% for a PMMA based FSC), and because the quantum yield of the fluorescent species is high, an OQE of close to 0.75 between 400 to 600 nm would be expected. Because of the effects of re-absorption, the OQE is seen to decrease to 0.40. Clearly, the effect of re-absorption is large. The use of multiple fluorescent species with strong nonradiative energy transfer could be one strategy for decreasing the re-absorption loss [12]. Novel methods such as the use of photonic crystals doped with fluorescent molecules also represent an alternative method for achieving lower re-absorption in FSCs [25].

CONCLUSION
A model describing the re-absorption probability of the trapped photon flux in FSCs coupled to four edge-mounted solar cells has been developed. It has been used to show that the 4Cell setup is fundamentally different to the 1Cell setup in terms of its re-absorption characteristics as a function of the azimuthal angle of emission. However, it is seen that if the gain of the FSC is kept constant, both setups should result in similar levels of total re-absorption in ideal FSCs. The performance of the Mold-FSCs in the 4Cell setup is seen to be close to ideal as indicated by the angular resolved results, which show that the experimentally measured photon flux is directly proportional to calculations of 1 À Γ′ (ϕ), which is consistent with what is expected for an ideal FSC.
Calculations of the OQE from re-absorption models compared with experimental measurements also show that the 4Cell Mold-FSCs studied are well described by ideal models. Practical applications of the models developed are shown to include estimation of non-ideal losses and optimization of power conversion efficiencies. Figure 18. Power conversion efficiency for increasing dye concentrations. The numbers on the x-axis indicate the dye concentrations of Mold-fluorescent solar concentrators (FSCs) relative to the 300 mg/l sample. In this manner, the dye concentration for a FSC containing Lumogen Red F 305 dye has been optimized for maximum power conversion efficiency.
Modeling photon transport in FSCs T. S. Parel et al.