1. Introduction

Luminescent solar concentrators (LSCs) based on dye-doped poly(methyl methacrylate) (PMMA) plastic optical fibers (POFs), in combination with one or multiple silicon solar cells, constitute a promising alternative for the generation of photovoltaic energy in urban environments that is currently being investigated [1–4]. POFs doped with a suitable combination of organic dyes can be employed to absorb sunlight over a relatively broad range of wavelengths and, thanks to the subsequent emissions generated by the excited dye molecules and to the waveguide properties, light is guided to the solar cell in the form of photons of longer wavelengths. These are closer to the peak of responsivity of the solar cell than the absorbed photons.

There are several reasons for considering POFs connected to solar cells instead of conventional solar panels. POFs can be easily doped with organic dyes whose spectral characteristics are especially suitable for solar concentrators [5,6]. They can also be readily integrated into buildings, where they can be placed even on their facades and windows, thanks to the high flexibility and low weight of POFs [7]. Besides, POF-based solar concentrators do not need any sun-tracking system and they can work even under diffuse light [8]. Furthermore, undoped POFs having low attenuation in the red region of the spectrum are widely available, which could be employed to connect the doped ones to a remote solar cell [9]. Another advantage of utilizing POF-based solar concentrators is that the photovoltaic cell is not heated as much as when it receives the entire spectrum of the sun radiation, so its power-conversion efficiency is lowered to a lesser degree for temperature reasons. Additionally, cylindrical LSCs have shown a better performance than the flat rectangular LSCs in some experiments and simulations. Although a planar waveguide may serve to reduce the amount of rays that escape from the inside out of the concentrator, as well as to reduce the fraction of light that does not enter the concentrator due to external reflection, if multiple circular fibers are used together, both advantages mentioned of planar waveguides become negligible compared with the greater geometric concentration of circular optical fibers.

Two of the most promising organic dyes for POF-based solar concentrators are the so-called lumogen red and lumogen yellow, produced by the company BASF SE [10,11,5]. In addition to having a high quantum yield [12], such dyes are capable of absorbing sunlight over a relatively broad range of wavelengths in the visible region, especially when they are combined together. Besides, both types of dyes, principally lumogen red, allow emitting light at wavelengths where the responsivity of a silicon photovoltaic solar cell is high.

By using an appropriate arrangement of multiple POFs, preferably stacked in multiple layers, the desired amount of total output power can be achieved. However, a careful design of the dye concentrations and fiber lengths is needed, because overlaps between the emission and absorption cross sections of the dye or dyes employed lead to reabsorptions of the emitted photons, thus reducing the power that can reach the solar cell. Therefore, one of the main concerns is to achieve a better compromise between a low reabsorption of the light generated inside the optical fibers and a high trapping efficiency of the sunlight [13]. Since constructing prototypes for the sake of optimizing or testing them is usually very time-consuming, a better understanding of the design parameters can be achieved by developing theoretical models and obtaining and comparing multiple results through the use of computational or analytical simulations [14–16,3,17]. That approach is the one followed in this paper, which serves to complement and extend some related research works reported in the literature. For example, Monte-Carlo simulations were carried out in [3] to study the influence of doping a fixed fraction of the total fiber radius in peripherally doped optical fibers whose radius was varied while adjusting the dopant concentration in order to keep the total number of dopant molecules constant in each cross section. However, some of the concentrations considered in that work (up to 9 000 ppm in weight) would have been extremely high in the case of organic dyes. A more practical solution in such a case could be to maintain the dye concentration constant within a range of moderate values that do not compromise the quantum yield [12] and to try to optimize the fraction of the total fiber radius that corresponds to the doped layer. This paper does not only follow the latter approach, but it also presents a model and clarifying results for the case of employing two dopants simultaneously instead of only one, as was the case of the work reported in [3]. The study is performed for both single- and multi-layered solar concentrators. In addition, the fact that the results presented in this work are focused on POFs facilitates the construction of prototypes, apart from providing valuable information for a better understanding of POF-based solar concentrators incorporating promising dyes. Since the POF samples available in our laboratory were doped with lumogen molecules at moderate concentrations (e.g. 300 ppm) and no aggregation effects of the dye molecules were observed in them, the same or similar concentrations will be considered for our designs. Such optical fibers were drawn by ourselves [18].

This paper is organized as follows. Firstly, a model for the simulation of a POF-based solar concentrator with a single dye distributed either uniformly or peripherally in its cross section is presented. Next, the assumptions made for the simulation of combinations of two dyes are explained, and the validity of the model obtained with such assumptions is justified by comparing some results with experimental measurements. In the following section, which is dedicated to the results and discussion, the influence of the portion of either dye in the mixture is investigated for the case of employing combinations of lumogen red and lumogen yellow, and some clarifying results are presented. Besides, the advantages of employing multiple POFs stacked in layers, as shown in Fig. 1(a), are discussed. In all of such set-ups, the fiber centers are distributed in a hexagonal pattern, which is optimally compact. For multi-layered set-ups, computational analyses are carried out for multiple thicknesses of the part of the POF’s cross section that is doped. Specifically, the possible advantages of not doping the innermost part of each optical fiber are analyzed, as well as the influence of the thickness of the doped part.

 

Fig. 1. (a) Solar concentrator consisting of peripherally doped POFs stacked in layers, illustrated for the case of having three POFs in the top layer. The part of each cross section that is doped is represented as a grey shaded area. (b) Side view of the intersections between a line that represents a light ray and the two cylindrical interfaces.

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2. Model for POFs peripherally doped with a single dye

The fiber diameter of typical PMMA POFs (1 mm) is much larger than the wavelengths of the absorbed or emitted photons, and so is the thickness of the doped part (the difference a-r₁ in Fig. 1(b)), which will never be smaller than 1% of the total diameter (2·a) in practical situations. Therefore, it is feasible to employ a ray-tracing technique for our simulations. A Monte-Carlo ray-tracing method is the one chosen for this work, which allows the simulation of more complex 3-D geometries than an approach based on solving deterministic rate equations. The computational implementation of the method involves keeping track of a sufficiently large number of small pieces of energy into which light is discretized by the computer program, which will be called “photons”. Each of them is assigned a direction, a wavelength and an initial power, according to the spectral solar irradiance under standardized conditions (AM1.5 G). The total irradiance at all wavelengths is made to amount to the standard value of 1 mW/mm² on a clear day. To facilitate comparisons, we assume that, in all cases, the sunlight impinges on the top of the solar concentrator at right angles to the fiber axes (Fig. 1(a)), which, as we have found computationally, is the most favorable orientation overall. It is assumed, without loss of generality, that the light power is collected at one of the two ends of each fiber, since the other end would yield an identical power, owing to symmetry reasons. In addition to the paths of the photons emitted by the excited dye molecules, it is also essential to calculate the path of each sun photon until it is absorbed or it is lost, because only a fraction of the impinging irradiance can be absorbed. This fact is a consequence, in part, of the limited extension of the absorption band of the dopant, but the model must be comprehensive enough for us to be able to calculate the influence of other factors as well. For example, size constraints might impose a relatively low upper limit to the maximum number of stacked layers of optical fibers that can be utilized, especially in the case of employing a single cell to which the tips of all the optical fibers are connected instead of using an individual solar cell per fiber. Therefore, our goal has been to achieve an all-inclusive model that can be used to reach a compromise between a satisfactory trapping efficiency of the sun photons and a low loss of power produced by the absorptions and reabsorptions of the emitted photons. As for the number of optical fibers in the top layer (Fig. 1(a)), this number can be kept constant and equal to three without loss of generality, because, for example, with twice as many fibers per layer the output power that would be achieved has been checked to be twice as large.

To simulate the absorption, emission and propagation phenomena that happen in the solar concentrator, the evolution and interactions of each photon are analyzed by making it advance in steps. In each step, the photon considered is made to propagate in a straight line until it reaches any interface (e.g., between a doped zone and an undoped one, or between the air and the fiber), or until it is absorbed by the host matrix or by a dye molecule, or until it reaches the end of an optical fiber. In order to keep all photons under control, a photon never leads to two photons when a partial reflection at an interface is expected for a certain incidence direction. Instead, the photon is either refracted or reflected, i.e., one of the two phenomena is considered and the other one is neglected. This choice is made randomly by assuming probabilities that are proportional to the corresponding fractions of power that are predicted by Fresnel’s laws. Once decided, the power and direction of the refracted or reflected photon are updated.

A statistical approach is also adopted to include the effect of the attenuation by the host material. Instead of applying a certain attenuation coefficient per unit length due to the gradual loss of light power predicted by Beer-Lambert’s law, a probability function is utilized. It models the distribution of the distances travelled by the individual photons before they are absorbed, in such a way that a photon is assumed to lose all its energy abruptly after a random distance that is determined according to that probability function. The global effect when the number of photons is very large is the predicted one: the same quotient of input to output powers as a function of distance is obtained. The distribution of distances can be written as follows:

By means of a similar reasoning, we can also employ another probability function of the same type to model the distribution of the distances d_abs-dye travelled by the individual photons before they are absorbed by any of the dye molecules. If the event corresponding to the nearest of the randomly predicted event positions is the absorption of the photon by a dye molecule, then either a non-radiative decay or a radiative one will occur. The respective probabilities are given by the quantum yield of the dopant. In the non-radiative case, the photon is lost, whereas, in the radiative case, another photon is immediately created in an isotropically random direction and at a longer wavelength, with a probability proportional to the height of the spectral shape of the emission cross section normalized to 1 in the interval of allowed emission wavelengths. This assumption is validated in the next section by the agreement between the computational and available experimental results. The new power after the emission, as compared to the previous one just before the absorption, is made to decrease according to the quotient of wavelengths, because it is well known that the energies of the absorbed and emitted photons are in inverse proportion to their corresponding wavelengths. For an optical fiber whose cross section is doped only peripherally, as is the case of any of the fibers depicted in Fig. 1, the distance d_abs-dye is assigned a value of ∞ while the photon considered is travelling through the innermost part, because there cannot be absorptions by the dye molecules in that part.

The calculation of the distance d_interface to the nearest interface in the propagation direction is illustrated in Fig. 1(b). The type of interface depends on what part of the fiber the photon is crossing. We have called I₁ the fiber-air interface and I₂ the interface between the doped and undoped parts. For any photon travelling through the doped part, we must take into account that the photon could reach either I₁ or I₂ in the direction of the movement (Fig. 1(b)). To determine which of the two cases would occur, we calculate the four intersections (real or complex) between a straight line (the path of the photon) and the two cylinders (the two interfaces), and we discard the ones with an imaginary part and those that are behind the position of the photon in its propagation direction. Next, we choose the smallest value of the remaining solutions. In the case of an intersection with I₂, we must consider whether the refractive indices between the doped and undoped parts are different or not. In the former case, Fresnel’s laws should be applied, whereas, in the latter, we just need to update the parameters associated to the new medium. In our samples, α_host(λ) and the refractive indices will be assumed to be the same in both media, since the host material is the same and the dopant concentration is moderate. Consequently, the ray path in Fig. 1(b) has been drawn as a straight segment inside the fiber.

While the photon is inside one of the optical fibers, we also need to calculate the intersection of its path with the plane containing the end face of the fiber. If the distance from that point to the center of the fiber end is not greater than the fiber radius, we need to calculate the distance d_end from the photon to the point of intersection. Once the distances for each of the possible events to occur are known, the photon is made to advance the shortest one of them before updating its characteristics (direction, wavelength and power).

A photon may also exit the optical fiber through the side, which occurs if the event is an intersection with the outer interface (I₁) and the photon undergoes a refraction towards the air. In that case, we must determine if another optical fiber will be reached by calculating the nearest intersection point of the propagation direction with that fiber, which should be real. If this is the case, the calculations for the new fiber are more easily made if the origin of the Cartesian coordinate system is moved to the new fiber symmetry axis. In other words, if the absolute positions of the origins are (X_center, Y_center, 0), then any absolute point (X, Y, Z) becomes (X − X_center, Y − Y_center, Z) in the new reference system (Fig. 1(a)). In the case of a multilayered structure such as that depicted in the figure, we assign each optical fiber an integer to automate the calculation process of X_center and Y_center.

3. Model for POFs doped with two dyes

Let us now consider the case of employing two dyes simultaneously, which will be termed dye1 and dye2. The probability for a photon of wavelength λ to be absorbed by one of them is modeled by means of the following random distances for an absorption to be produced by dye1 (d_abs-dye1) or by dye2 (d_abs-dye2):

For the application of these equations, N_LY and N_LR are calculated from the respective absorption coefficients. In our model, the same absorption coefficients as in samples doped with only one dye are assumed to be applicable to samples with two dyes, since direct transitions between different dye molecules are not expected to occur with moderate dye concentrations. These absorption coefficients were measured directly, so the knowledge of the individual values of the two factors on the right hand sides of Eqs. (4) and (5) is not required. The absolute spectral curves of α_LY and α_LR in Fig. 2 were measured by using the side-illumination fluorescence (SIF) measurement [19] and a Cary 50 UV-Vis spectrophotometer (Agilent Technologies, Santa Clara, CA, USA) equipped with a fiber-optic coupler accessory [18]. The two absorption coefficients plotted in the figure correspond to samples doped with dye concentrations of 300 ppm. For the sample that was doped with lumogen yellow, the global attenuation coefficient was also measured at the wavelength of 580 nm, where it practically coincides with the value of α_host at 580 nm, since the absorption by the dye molecules is negligible at that wavelength. In this way, the host material’s absorption coefficient was estimated to be 2.52 Np/m at that wavelength, rising to approximately 3.2 Np/m at 725 nm according to the relative spectral shape of the attenuation curve of PMMA (Fig. 3) [9]. However, some of our samples presented inhomogeneities due to air bubbles and irregularities in the geometry, so the actual value of α_host in different parts of the same sample might differ from the estimated one.

 

Fig. 2. Spectral attenuation coefficients of the dyes lumogen yellow and lumogen red in POFs doped at 300 ppm, plotted together with the emission cross sections of both dyes (in arbitrary units).

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Fig. 3. Attenuation coefficient of the host material, comprising the absorption by PMMA and the scattering losses.

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Figure 2 shows α_LY(λ) and α_LR(λ) together with the relative shapes of the emission cross sections of both dyes, namely σ^e_LY and σ^e_LR, with the purpose of illustrating the overlaps between absorption and emission bands in samples doped with the two dyes. The corresponding absolute peak values of the emission cross sections are 10⁻²⁰ m² and 2 × 10⁻²⁰ m², respectively, which have been determined by means of the Füchtbauer-Ladenburg equation [20]. In our model, it is assumed that any absorption and the subsequent emission stem from transitions within the same dye molecule that caused the absorption. When a photon is absorbed and another one is created, the new photon is assigned an isotropically random direction. Consistently with the Stokes shift, the new photon is also given a longer wavelength [3], which is determined according to a probability function proportional in height to the dye’s emission cross section. Besides, our model does not need to include any quenching in the luminescent quantum yield due to concentration, because no significant quenching is to be expected for the moderate dye concentrations considered in this work [12].

Before passing to Section 4 for the design of the prototype, let us check that these assumptions about the emissions yield results that are consistent with the experimental ones available. For that purpose, in Fig. 4 we have plotted the experimental and theoretical spectra corresponding to individual POF samples of various lengths. In all cases, the results have been measured or calculated for POFs excited by the sun along their entire length, except for the fiber tip covered by the optical meter’s connector, whose length is 3.3 cm. The excitation or illumination length is referred to as z_e in the legend of Fig. 4. The POFs are uncladded ones of 1 mm in diameter. They are doped with a combination of 150 ppm of lumogen yellow (LY) and 75 ppm of lumogen red (LR), except for the two curves with circular markers, which correspond to POFs doped with one dye, either with LR (75 ppm) or with LY (150 ppm). By using the latter curve as a reference, it can be noticed that the small secondary bands of the spectra obtained for two dyes correspond to photons emitted by molecules of lumogen yellow, because they are located in the same spectral region as the reference curve. Similarly, the main spectral bands correspond to emissions from molecules of lumogen red, which may have been excited by photons emitted by molecules of lumogen yellow as well. One of the aspects shared between theoretical and experimental results is that the spectrum of the output emission of POFs doped with two dyes tends to be mostly or entirely located within the emission band of lumogen red. To explain this behavior, Fig. 2 illustrates that the emission band of lumogen yellow (see σ^e_LY in Fig. 2) completely overlaps with the absorption band of lumogen red (see σ^e_LR in Fig. 2). This fact favors the absorption by lumogen red of the majority of the photons emitted by lumogen yellow. It is also noticeable that the main peak of the output emission tends to become shifted towards longer wavelengths (red shift) as the fiber length is increased, owing to the overlap of the output spectrum with the absorption bands. Furthermore, the amount of red shift undergone by the main peak of the theoretical results as z_e is increased is approximately the same as that observed in the experimental results (the two lines with square markers in Fig. 4), which is another sign of the validity of our model.

 

Fig. 4. Theoretical and experimental spectra corresponding to individual POF samples of various lengths doped with a combination of 150 ppm of lumogen yellow (LY) and 75 ppm of lumogen red (LR), except for the two curves with circular markers, which correspond to POFs doped with one dye, either with LR (75 ppm) or with LY (150 ppm).

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Now, let us check to what extent the experimental and theoretical powers (heights of the curves) agree with each other (whether it is in terms of the relative shapes and order of magnitude or they agree more closely). For the measurements, our POFs were illuminated under the midday sun and nearly at sea level, and the results were normalized to standardized ones (AM1.5 G) using a calibrated reference cell. All the output powers in Fig. 4 correspond to AM1.5 G conditions. The experimental curves are depicted by the black and green curves with square markers, and they correspond to excitation lengths z_e of 5 cm (green) and 30 cm (black). As can be seen, for the shortest distance, z_e = 5 cm, the measured band is similar in height to the computationally obtained one (thin green curve in Fig. 4). For the longest distance, z_e = 30 cm, the differences in height between the experimental and computational bands increase. The fact that the theoretically predicted power densities for 30 cm are greater than the experimental ones might be due to the aforementioned inhomogeneities along the samples, or to having used distances around 8 cm in SIF measurements to calculate the absolute values of the attenuation coefficients, which could have led to underestimating them.

Similarly, Fig. 5 shows that the experimental and computational output powers obtained at the fiber end agree remarkably well when z_e is small, and to a lesser degree when the distances are longer and the influence of α_host becomes more relevant. Another problem of using a larger z_e for the experimental measurements is that, although longer samples could be drawn by us from our preforms, the quality of the samples was necessarily more uncertain, because of our own technical issues.

 

Fig. 5. Theoretical and experimental output powers corresponding to individual POF samples of various lengths doped either with LR (at 300 ppm) or with a combination of 150 ppm of LY and 75 ppm of LR.

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In this paper, we will use the global external quantum efficiency for some of the calculations, since it is a performance parameter of the purely photonic LSC lightguide. It is defined as the quotient between the total number of emitted photons reaching the solar cell and the total number of solar photons incident onto the LSC in the entire solar spectrum from 280 nm to 4000 nm, as is illustrated in Fig. 6. Notice that, in this figure, the number of real photons of energy hc/λ that are present in each interval of 5 nm is employed for the sums of photons of both the numerator and the denominator. Conversely, for the calculations of the output power, any discretization of the energy different than hc/λ would be equivalent to using the blue line in Fig. 6, provided that the sum of the powers of all the “photons”, although not the number of “photons”, be the same, and the curve be parallel to the blue one.

 

Fig. 6. Blue line: number of photons in each interval of 5 nm corresponding to the AM1.5 G solar spectrum, plotted in the range of visible wavelengths. Magenta line: number of photons in each interval of 5 nm corresponding to the theoretical results obtained for the sample of z_e = 5 cm in Fig. 4. Red line: Global external quantum efficiency corresponding to the same results. Green line: ratio of the total number of emitted photons reaching the solar cell to the total number of solar photons incident onto the LSC, but in the visible range instead of in the entire solar spectrum from 280 nm to 4000 nm.

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In light of the above, and taking into account that our purpose is to provide guidance in designing efficient solar concentrators, the output powers for the types of set-ups mentioned in the introduction will be compared between them in the next section for multiple combinations of the design parameters. For the simulations, we have programmed our Monte-Carlo code using Matlab 2019b. As for the simulation times, they depend on the number of photons launched, which is set heuristically a posteriori so that the fluctuations derived from the stochastic nature of the Monte-Carlo simulations are deemed acceptable. For example, for this paper we typically launch about 38 000 photons per centimeter of simulated fiber length, discretizing their wavelengths in 100 slots. The simulation time also depends on the number of optical fibers stacked in layers. For a typical figure with multiple layers in our paper, the calculation of each curve requires about 15 hours in a personal computer with a processor Intel i7-9700 @3 GHz.

4. Results and discussion

The results presented in Fig. 5 leave no doubt that a POF doped with lumogen red at a concentration of 300 ppm yields a higher output power than another POF of the same length doped with a combination of the dyes LY at 150 ppm and LR at 75 ppm. It could be argued that, in the latter case, the amount of lumogen red is much smaller, which apparently is not compensated by the addition of LY at 150 ppm. Conversely, it might also be the case that combining LY with LR could worsen the performance of LR. However, what actually happens, in the case of equal concentrations of LR, is that the combination of both dyes often yields a better performance than LR alone, irrespective of the number of layers of stacked fibers. This effect becomes clear in the case of a number of layers beyond which the output power hardly increases any more (9 layers in the case of 300 ppm of LR, as shown in Fig. 7(a)). The advisability or not of using LR alone is illustrated by the two lowermost curves in Fig. 7(b) for a low concentration of LR (75 ppm). It can be observed that, for the same number of stacked layers, combining the two dyes together leads to a higher output power than using only LR at 75 ppm, no matter how long the doped fiber is. The total POF length in each case is z_e plus a very short distance of non-illuminated fiber that would correspond to the optical meter’s connector. Alternatively, LY alone could be employed instead of LR, in which case a higher output power would be obtained with LY than with the combination of dyes, for the same POF lengths. These opposite results show that adding a low concentration of a second dye would, in some cases, worsen the performance of the solar concentrator. Such worsening can be explained by considering the overlaps between their emission and absorption bands. In the case of the two dyes analyzed in this section, the emission cross section of LY overlaps entirely with the absorption band of LR, so adding LR can become detrimental in the performance of LY, providing that its concentration is high enough for the sunlight to be absorbed adequately to obtain the desired output power. For instance, 150 ppm of LY seems to work satisfactorily, but not as well as 300 ppm, according to Fig. 7(c). If the concentration of the single dye is much lower (e.g., 75 ppm of LR alone), the trapping efficiency of the sunlight becomes too low, in which case adding LY turns out to be beneficial, because LR alone has a narrower absorption band than the combination of the two dyes. As already commented, another way to improve the output power is by adding more layers, as shown in Fig. 7(d) for a broad range of fiber lengths z_e. In all cases, the results show a tendency for the output power to saturate when z_e is increased, owing to the greater attenuation of the light that is generated farther away from the output end. The increase in the output power also tends to saturate as the number of stacked layers is increased from 1 to 9, as can be observed in Fig. 7(d). Stacking more than 9 layers would only produce a negligible improvement in the output power with concentrations of 300 ppm. This saturation effect can be due to the fact that the majority of the trapped photons of the sunlight are absorbed in the first few layers. In any case, irrespective of the number of layers, there are always photons that are lost without being absorbed by the dye molecules, because the impinging light progressively loses its power in partial reflections and absorptions by the host material. As a result, decreasing the dye concentration might be detrimental even with more than 9 layers.

 

Fig. 7. Theoretical output powers corresponding to POFs doped with LR, LY or both dyes, for the case of having several POF layers stacked as shown in Fig. 1(a) (with 3 POFs in the top layer). In (a): influence of the number of layers when using LR at 300 ppm. In (b): influence of adding a second dye. In (c), dyes employed individually are compared. In (d), the curves of the output powers are shown for 1, 4 and 9 stacked layers.

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The two lowermost curves in Fig. 7(c) illustrate that LR alone and LY alone tend to yield similar output powers when used independently at low concentrations. When the concentration is increased sufficiently (e.g., from 75 ppm to 300 ppm in the same figure), LR alone tends to become better, especially for the longer fiber lengths, owing to a smaller overlap between the absorption band and the longest wavelengths of the emission band in the case of LR. The penetration of the right tails of the absorption bands into the emission ones is depicted in Fig. 2. LR also has the advantage of having a broader absorption band. The fact that the differences in the output powers tend to become smaller at low concentrations can be explained by taking into account that the trapping efficiency would become too low for both types of dyes.

In light of the above, the question that arises is whether it would be beneficial or not to avoid doping the cross section entirely at the expense of a lower trapping efficiency in any of the set-ups analyzed in Fig. 7, especially in those in which the trapping efficiency has not prevented the achievement of a high output power. In this respect, the best of the aforementioned set-ups has been the one consisting of 9 layers of POFs doped with LR at 300 ppm. If the dye concentration could be increased sufficiently, which is not always possible in practice due to the aggregation and quenching effects, a peripheral distribution of the dye would become unquestionably beneficial, because the trapping efficiency would remain high enough not to compromise the advantage of the lower reabsorption effects [3]. However, a moderate concentration of 300 ppm might not be high enough, depending on the thickness of the doped layer. To clarify this issue, Fig. 8(a) shows the corresponding results calculated in the following conditions: when the entire cross section of each POF is doped, and when an inner part of radius r₁ is left undoped (either 50% or 70% of the fiber radius a). The output powers have been plotted against the fiber length z_e. The saturation distance is slightly larger when an inner part is left undoped: 120 cm for r₁ = 0.7a as compared to 100 cm for r₁ = 0. However, it can be observed that the lowest output powers are achieved for r₁ = 0.7a, showing that the trapping efficiency becomes too low to be compensated by the lower attenuation coefficient in the undoped part. In fact, we have checked that with much higher concentrations on the order of 10⁴ppm, which are not practical with our samples because of the formation of aggregations, the undoped inner part would be beneficial with regard to the output power, but not with the more practical concentrations for the dyes considered in this work. However, an advantage can be found even in this case. When plotting the same points in Fig. 8(a), but this time against the necessary total amount of dopant to obtain each power (Fig. 8(b)), it can be observed that a greater output power can be achieved if an inner part is left undoped, especially when the total amount of dopant available is limited.

 

Fig. 8. Theoretical output powers for the set-up depicted in Fig. 1(a) with 9 layers of POFs doped with LR (300 ppm) with the following dye distributions: when the entire cross section of each POF is doped (lines with circular markers), and when an inner part of radius r₁ is left undoped (either 50% or 70% of the core radius). In (a), the powers have been plotted against the fiber length z_e. In (b), the same powers have been plotted against the total amount of dopant over the entire fiber length necessary to obtain each power using the dye distribution considered. Each point represents the simulation of a different solar concentrator.

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As was shown in Fig. 7(c) for the case of doping the entire cross section (r₁ = 0), the employment of LR at 300 ppm tends to yield a higher output power than the use of LY with the same concentration and the same number of layers (9 in both cases). For that reason, a set-up with fewer layers of LR-doped POFs can be employed to obtain almost the same power as with 9 layers of LY-doped POFs. This is the case of the two black curves for r₁ = 0 in Fig. 9, which are the theoretical output powers corresponding to 4 layers in the case of LR and 9 layers in the case of LY. Indeed, if LR had been utilized in both set-ups, the output power with 9 layers would have been much larger than that with 4 layers: up to 25% larger, according to Fig. 7(c). However, the curve obtained for 9 layers with LY is no more than 7% higher than that calculated for 4 layers with LR, providing that r₁ = 0. For any other value of r₁, i.e. for worse sun-trapping characteristics, the absorption in only 4 layers would not be enough and the differences with respect to 9 layers would be higher. In other words, the fewer the layers, the more important it becomes to choose a dye distribution with a higher trapping efficiency or, equivalently, a dye with a broader absorption band. In this respect, LR is better than LY. Furthermore, it can be noted that the disadvantage of utilizing fewer layers gains special relevance as the radius of the undoped inner part is gradually increased from 0% to 70%. In other words, the two curves for r₁ = 0.7a in Fig. 9 are clearly further apart than those for r₁ = 0, and the differences in height increase as r₁ is increased. It can also be observed that, for a concentration of 300 ppm, the trapping efficiency rapidly decreases as r₁ is increased above r₁ = 0.3a.

 

Fig. 9. Theoretical results obtained to analyze various set-ups with 4 layers of LR-doped POFs that could be employed to achieve almost the same power as with 9 layers of LY-doped POFs.

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On the basis that, in general, a compromise should be reached between a sufficiently high trapping efficiency and a sufficiently large fraction of emitted photons that can propagate to the solar cell, it is also interesting to calculate the global quantum efficiency corresponding to the fibers of each layer. Such results are shown in Fig. 10(a) for an LSC of 9 layers when using a short excitation length (z_e) of 5 cm and a relatively high concentration of 1000 ppm of LR in the entire cross section. The reason for choosing such a short length z_e is that it facilitates the preparation of an experimental set-up, in which each of the optical fibers needs to be positioned in contact with the surrounding ones. As can be observed in the figure, which has been calculated computationally, the average global external quantum efficiency per fiber in each layer tends to decrease rapidly in the first four layers. Preliminary experimental results just carried out by us seem to corroborate such a tendency. One important reason for having a lower average quantum efficiency in the deeper layers might be that the light that reaches them tends to consist, in a great part, of the wavelengths that cannot be absorbed by the dye. Furthermore, the evolution of the quotient between the number of transmitted photons from the bottom to the number incident sun photons on the top, which is shown in Fig. 10(b), can also be explained in a similar way. Since the transmission is not negligible even with 9 layers doped at 1000 ppm, whereas the average external quantum efficiency per fiber in each layer becomes neglibible in the bottom layers, it can also be concluded that a significant amount of photons that are outside the absorption band reaches the bottom layer. As a rule of thumb for the design of an LSC, it can be concluded that two or three layers are often enough, especially when it is desired to achieve a high transmission, e.g., for the employment of LSCs on windows. Another option could be to use 2 layers, but employing a different dye in each layer in order to try to increase the trapping efficiency. For example, LR could be utilized in the first layer and LY in the second one, or vice versa. The fact that the average quantum efficiency tends to be higher in the first layer because of its position (Fig. 10(a)) suggests that the dye LR might well be the best choice for the top layer, because we have seen that it tends to perform better than LY in fibers doped with a single dye.

 

Fig. 10. (a)Global external quantum efficiency per fiber in each layer (%), calculated for the case in which the LSC has 9 layers. (b) Transmission of photons from top to bottom layers of the LSC in the visible spectrum, as a function of the number of layers employed. In both (a) and (b) the dopant is LR at 1000 ppm and z_e = 5 cm.

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The reason why we have employed 1000 ppm instead of 300 ppm for the example of Fig. 10, which corresponds to z_e = 5 cm, is illustrated in Fig. 11. It shows the external quantum efficiency for all fibers calculated for both concentrations in multiple set-ups, each one with a different number of layers. As can be observed, a shorter excitation length makes it advisable to employ a higher dye concentration, since the absorption losses for the emitted photons become less important in shorter distances. Conversely, we have checked that a length z_e = 30 cm would yield a lower external quantum efficiency than a length z_e = 5 cm when using a concentration of 1000 ppm.

 

Fig. 11. Global external quantum efficiency for all the fibers together (%), calculated for the case in which the LSC has 9 layers and z_e = 5 cm. The dopant is LR at 1000 ppm or at 300 ppm.

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5. Conclusions

Luminescent solar concentrators based on polymer optical fibers (POFs) doped with combinations of the promising dyes lumogen red and lumogen yellow have been investigated by means of a Monte-Carlo computational model developed for this purpose. Firstly, the model employed for POFs peripherally doped with a single dye and stacked in layers has been described. Secondly, the model has been generalized for the simulation of POFs doped with two dyes simultaneously, and it has been validated by the agreement between the computational and available experimental results, both in the spectral domain and with regard to the output powers. Then, several potential designs for such solar concentrators have been compared with each other by using this model. The results obtained have been reasoned qualitatively in order to achieve a greater level of understanding and generality in the conclusions. Given two combinations of dyes with the same concentration of one of the dyes, the presence of the other one can either worsen or improve the performance of the solar concentrator, which has been explained by considering the overlaps between the emission and absorption bands and the trapping efficiency achieved with each dye independently.

Further improvements in the output powers obtained with one or two dyes can be achieved by adding more layers up to a certain limit. It has been shown that, for moderate concentrations of 300 ppm of either dye, stacking more than 9 layers of uniformly-doped POFs would only produce a negligible improvement in the output power. The power achieved is slightly larger for lumogen red than for lumogen yellow, owing to the broader absorption band of the former, and, to a lesser degree, to the smaller overlap between its absorption and emission bands. Besides, the fewer the layers of stacked optical fibers, the more detrimental it becomes to have a narrower absorption band. When an inner part is left undoped, this effect is aggravated. Nevertheless, the saturation distance for the output power tends to become slightly larger, especially at higher dye concentrations. Furthermore, if the corresponding powers are plotted against the necessary total amount of dopant to obtain each power, greater values can be achieved if an inner part is left undoped, especially when a small amount of dye is available.

We have also investigated the variations in the global external quantum efficiency obtained on average in all the fibers of each layer and also the total one corresponding to the fibers in all layers as the number of layers is increased. Additionally, the transmission of photons from to the top to the bottom of the LSC has been discussed.

Finally, it has been shown that the lower the dye concentration, the more advantageous it is to use multiple layers of fibers. For the specific case of POFs doped with lumogen yellow or lumogen red at moderate concentrations, the disadvantage of utilizing fewer layers gains special relevance as the radius of the undoped inner part is gradually increased from 0% to 70%. These results serve as a guide to understand and improve the design of solar concentrators based on dye-doped POFs.