Analytical Description of a Luminescent Solar Concentrator Device

An analytical solution for the optical efficiency of a luminescent solar concentrator is presented. Due to a large number of input parameters and their complex effect on the device efficiency numerical simulations have been previously used for this purpose. The formulas, provided here, derived using a probabilistic approach, significantly reduce the complexity of the problem. The equations were validated by simulations and the obtained explicit expressions provide a clear common ground for the theoretical description of such devices. Implementation by a computer algebra program yields instant results for any set of input parameters. It allows a higher level of analysis, where an inverse task of finding parameters for a given efficiency can be readily solved.

An analytical solution for the optical efficiency of a luminescent solar concentrator is presented. Due to a large number of input parameters and their complex effect on the device efficiency numerical simulations have been previously used for this purpose. The formulas, provided here, derived using a probabilistic approach, significantly reduce the complexity of the problem. The equations were validated by simulations and the obtained explicit expressions provide a clear common ground for the theoretical description of such devices. Implementation by a computer algebra program yields instant results for any set of input parameters. It allows a higher level of analysis, where an inverse task of finding parameters for a given efficiency can be readily solved.
Radiation conversion by luminescence concentration has been considered decades ago, mainly motivated by a detector size reduction [1][2][3]. Acrylic glasses filled with organic dyes were proposed to convert incoming light to fluorescence for subsequent detection by small semiconductor photodetectors [4,5]. With the development of detector technology and due to inherent limitations of the available fluorophores this method did not gain much traction. Recently, however, it has drawn renewed attention, stimulated by advances in colloidal quantum dot synthesis [6,7].
The operation principle of a luminescent solar concentrator (LSC) is based on a total internal reflection of the re-emitted light for large angles, which is waveguided to the edges for conversion to electricity [8][9][10][11][12][13]. For this purpose a glass (plastic) slab is enriched with fluorophores, and photovoltaic cells are attached to the slab perimeter ( Fig. 1). In evaluating the performance of such a device for, e.g. building-integrated photovoltaics, the power conversion efficiency is a key parameter. Multiple loss mechanisms, however, exist in the system. The re-emitted light can be absorbed by the matrix material, by the fluorophores, or it can be scattered out of the slab. The apparent complexity of the system stimulates the use of numerical simulations for the efficiency estimations [14][15][16].
Analytical treatment, if possible, can substantially reduce the complexity of the problem in the sense of a classical definition of complexity as the minimal length of a "code" needed to attain a result [17]. More importantly, explicit expressions can facilitate a unified benchmark tool, as opposite to undisclosed user-dependent codes. Finally, expedient results from the implementation by a computer algebra program would imply possibility of a higher level of analysis, such as solving an inverse problem of determining device parameters for a desired output efficiency.
In this paper an analytical solution has been derived by probabilistic treatment, including all the main loss mechanisms. It was based on the obtained distribution of the optical path lengths for an isotropic emitter randomly placed in a rectangular slab with absorbing edges. In this framework a result can be obtained instantly by a computer algebra program on a desktop computer as a continuous function of input parameters. Obtained values compare well with numerical simulations and represent a convenient approach for the complete analysis of LSC performance.
We start by considering an isotropic emitter randomly placed inside a rectangular slab with height ℎ and width (diagonal ) (Fig. 1). Isotropic emission corresponds to the light output pattern of quantum dots (QDs), which are typically spherical without specific dipole orientation [18]. For organic dyes this condition reflects a random orientation of the molecules uniformly distributed in the slab. It can be shown that in the plane of the slab (Fig. 1, left) the properly normalized probability density function (PDF) for a photon to travel an optical path is a piece-wise function (see section S1A in the Supplementary for the derivation): This distribution is shown in Fig. 2 for two different geometries (blue and red lines). Points represent results of numerical simulations, where isotropic emitters were placed all over the slab and frequency counts for about a million of optical paths calculated. Numerical solutions indeed converge to the analytical formula ( ). Fig. 1. Total optical path distribution from isotropic emitters in a rectangular slab (height ℎ, width , and thickness ∆) stems from inplane ( ) and out-of-plane ( ) distributions.
Next we note that together with the in-plane distribution there is a spread of optical paths in the plane perpendicular to the slab, as shown in Fig. 1, right (slab thickness Δ). Here photons emitted along two directions are shown, both experiencing total internal reflection reaching the edge. Photons emitted to the escape cone (within the critical angle α ) directly contribute to the losses. For the typical refraction index of glass (polymers) = 1.5 the critical angle is α ≈ 42°. From the diagram it is clear that the thickness of the slab plays no role in the out-of-plane optical path distribution (Δ ≪ ). Regardless of the first point of the total internal reflection (slab thickness) the optical path length (red dashed lines) depends only on the angle α. For the out-of-plane distribution it can be shown that the distribution of optical path lengths for an isotropic emitter is (see section S1B in Supplementary for derivations): for a given in-plane optical path length to the edge . Convoluting both distributions one can obtain an expression for the full distribution of optical path lengths for the 3D case: Integration limits come from the escape cone, where it can be shown that < < = 3 /2 (see section S1B in the Supplementary). This integral can be solved analytically using special functions (complete and incomplete elliptic integrals) and exact solutions are provided in the Supplementary section S1C. Those are presented graphically in Fig. 2, inset, as bright blue and red lines. It is seen that the distributions are smoothened and stretched to slightly longer values as compared to the 2D case. The solutions were again validated by simulations and numerical results do converge to the analytical expressions (Fig. S1). While the exact formulas are possible to obtain, the extensive presence of special functions make them not very practical to work with. To simplify the result to elementary functions one can note that the out-ofplane distribution ( ) does not deviate far from the distance to the edge due to the limits set by the escape cone. So the average value < > can be taken instead of the distribution ( ). It can be written as ≈< >= • (see supplementary S1B), where Then the approximate solution for the 3D case can be represented simply through the 2D solution: A properly normalized PDF then can be written explicitly as: This function is shown in Fig. 2, inset, as dark blue and red lines. It reveals that the approximate solutions nearly coincide with the exact ones, indicating only a minor influence from the introduced assumption. The practical meaning of ( ) is that the probability for a photon from an isotropic emitter randomly placed in the slab to experience an optical path would be ( ) .
From this expression one can already get some insight into the effect of the slab geometry. The probability of having an optical path below the slab width (aspect ratio = /ℎ) can be calculated as ≈ 0.31 + 0.56 (see Supplementary, section S2B). It becomes clear that the rectangle width in fact limits most of the photon paths ( Fig. S1, inset). For example, for the slab with a "golden ratio" ≈ 0.62, often used in architectural design of windows, ~ 75% of photons will travel distances shorter than the slab width.
With the optical path length distribution established the effect of matrix absorption on device efficiency can be readily evaluated. Let the linear absorption coefficient of the matrix be [ ]. The probability of reaching the edge for a photon travelling distance ′ is exp (− ′). It signifies the random nature of the absorption process with a given average rate (experimental observation of this stochastic process is sometimes referred to as the Beer-Lambert law). Then the total probability of reaching the edge: Calculating this integral yields (see Supplementary section S3 for derivations): where two integrals are: Formula (3) gives instant results for the effect of matrix absorption on the efficiency for a rectangular slab of any geometry and absorption coefficient, using a computer algebra program on a desktop computer (Fig. 3, left). One can compare with existing Monte-Carlo simulations using, e.g. results from [12] for PMMA ( = 0.03 cm , blue dots) and for soda-lime glass ( = 0.5 cm , red dots) in a square slab. Thus, analytical results of Eq. (3) coincide reasonably well with numerical simulations. The limitations of numerical Monte-Carlo method also become obvious: only one data point can be obtained per run and the proper convergence needs to be verified. The analytical result (3), on the other hand, produces a full functional dependence at once and changing geometry is just a matter of entering new values.
Separate from the matrix-induced losses the fluorophores themselves may attenuate re-emitted light. Consider first the effect of scattering. Let the linear scattering coefficient be [ ]. It can be expressed through nanocrystal scattering crosssection and their concentration as = .
We invoke Rayleigh scattering on particles smaller than the wavelength, which is nearly isotropic. Probability of not being scattered after travelling distance ′ is exp(− ′). These photons will contribute to the total optical efficiency similarly to the absorption case above: In addition, there will be photons, which underwent a scattering event into the waveguiding mode. The probability of being scattered within a distance ′ is 1 − exp(− ′). Let be the probability of scattering to the waveguided mode and not to the escape cone ( ≈ 75% for = 1.5, see Supplementary S4). Then the probability to reach the edge after one scattering event is (see Supplementary section S5 for derivations): Summing up all contributions from multiple scattering events and using geometrical series one obtains: One can quickly evaluate that for = 1 (scattering without losses) the total probability is unity, as would be expected. For = 0 (scattering with a complete loss) the expression becomes the same as for the absorption case. Also if the scattering coefficient is zero the optical efficiency is unity. An assumption of the Markov process was made here, so that the same optical path length distribution ( ) could be used after every scattering event.
Similarly to the scattering loss the fluorophores can re-absorb propagating light with subsequent re-emission. To minimize this effect particles with a large Stoke shift are typically used in practice. Let the linear reabsorption coefficient be [ ]. Introducing a nanocrystal re-absorption cross-section it can be represented as = .
If this process dominates the losses (no scattering or matrix absorption) it can be evaluated similar to the scattering as where is the quantum yield ( ≤ 1). Here losses after every reabsorption event come not only from the escape cone, but also from the imperfect light conversion of emitters. An additional complication is a spectral dependence of the re-absorption coefficient = ( ), due to a wavelength-dependent overlap of the luminophore emission and absorption bands.  5)) for different loss mechanisms (lines) with numerical simulations (points) from [14,15].
A spectral convolution can be used in this case with a properly normalized luminophore emission spectrum ( ): As a first approximation, two re-absorption coefficients for the regions with strong and weak overlaps can be introduced. For example for Lumogen [14] a third of the emission band can be set experiencing = 1.5 cm , while the other two thirds are nearly reabsorption free: = 0 . The total optical efficiency in this case becomes a weighted sum ( )/3 + 2/3. Similarily for CuInSeS nanocrystals [14] a quarter of the band has = 0.3 cm and for the rest = 0, corresponding to the total efficiency ( )/4 + 3/4. In Fig. 3, right, bottom, analytical results for these luminophores are shown (red curve for CuInSeS QDs, and blue for Lumogen). It is seen that even without a proper spectral convolution the analytical results clearly reveal main features from the numerical Monte-Carlo simulations (dots).
So far we considered scenarios where a single loss mechanism dominates. In practice, they all can co-exist and their simultaneous contribution should be taken into account. Using similar combinatorics arguments as above (see Supplementary section S6 for derivations) it can be shown that for the case of scattering and matrix absorption co-existence the optical efficiency becomes: This formula turns into the expression for scattering only scenario for a non-absorbing matrix ( = 0). When re-absorption and scattering co-exist in the system: To validate this derivation and result a comparison with numerical Monte-Carlo simulations from [15] is shown in Fig. 3, right, top. Here the optical efficiency as a function of the scattering length ( = 1/ ) is presented for Si QDs with a quantum yield of 50% (red) and 100% (blue) for a square slab 1x1 m 2 using = 0.08 cm . For this type of fluorophores the wavelength dependence of the re-absorption within the emission band is small due to a very large Stoke shift [19,20]. As in the numerical Monte-Carlo simulations [15], here the optical efficiency including the first absorption event is shown, i.e. • • ( , ). Again, the agreement with numerical Monte-Carlo simulations is reasonable.
Combining all loss mechanisms a general solution becomes ( , So equations (3) and (6) fully describe the effect of propagation losses in an LSC device. Input parameters are scattering ( ), reabsorption ( ), and matrix absorption ( ) coefficients (first two can be expressed through luminophore concentration and crosssections , ), geometry of the rectangular slab: height and width (ℎ, ), fraction of the emission to the waveguiding mode ( = 75% for n = 1.5), quantum yield of the fluorophore ( ), and the correction factor for 3D geometry ≈ 1.14. These formulas can replace Monte-Carlo calculations and provide a quick and transparent tool for a thorough device analysis. Below we apply them to solve an inverse problem of finding acceptable device parameters for achieving a pre-set power output.
The optical power output of the device [ ], as collected at the edges, can be written as: where is the incoming energy flux [ / ], is a transmitted fraction of the incoming sunlight (1 − is the absorbed fraction), and = / is the energy conversion coefficient of the luminescence. Product • signifies losses after the first absorption event, i.e. before the waveguiding mode is initiated. Transmission of the visible light though the device is = exp (− ∆) = exp (− ∆), where and are the fluorophore linear absorption coefficient and the absorption crosssection in the visible range, respectively (reflection neglected). This effectively gives 3 more input parameters: , ∆, and . If necessary, spectral dependence of the re-absorption coefficient can be included by convolution with the normalized emission spectrum ( ). So considering , , and Φ as constants, in total there are 11 independent input parameters, where at least 3 are, in general, have wavelength dependence: ( ), ( ), and ( ). We plotted Eq. (7) as a function of the aspect ratio = /ℎ for a slab with ℎ = 100 and Si QDs as fluorophores (QY = 1), which are often considered for this application [15,20,21]. For these nanocrystals the photon energy conversion factor can be set ≈ 0.6 for the average solar photon energy of 2.5 eV and the fluorophore luminescence peak at 1.5 eV. Other parameters include the re-absorption coefficient = 0.03 [20] and the scattering coefficient = 0.001 [15] for 0.1 wt. % concentration (5 nm diameter QDs). The solar energy flux is taken as Φ = 0.1 W/cm and the device transmission is set to 50%. In the absence of all propagation-related losses, i.e. ( , , ) = 1, the optical power conversion efficiency equals (1 − ) = 22.5%, shown as a dashed straight line. We can also set a minimum acceptable threshold for the power conversion to 7% (lower dashed line). This roughly corresponds to 5% (50 W/m 2 ) electrical power output, taking into account conversion losses at the last stage. The grey area in between these lines then shows an acceptable working range.
Several optical power curves for different matrix absorption coefficients are shown in Fig. 4, left. An example file generating these curves for common computer algebra programs is provided in the Supplementary for reader's convenience. As expected, the deviation from the loss-free case is growing with increasing aspect ratio and stronger matrix absorption. A larger device area does not improve the output power much after a certain point, where losses from propagation start dominating. From such a plot one can graphically solve an inverse problem of finding input parameters for a given Fig. 4 (left) Device power output from the analytical solution of Eq. (7) for different parameters. (right) Critical aspect ratio , resulting in 7% efficiency, for different quantum yields and absorption coefficients. threshold efficiency (7% in this case). In Fig. 4, right, the critical aspect ratio corresponding to this efficiency is shown as a function of the matrix absorption coefficient for different quantum yield values. It is seen that for the "golden ratio" slab ≈ 0.62 and 60% quantum yield a very low matrix absorption coefficient is needed = 0.001 (as in N-BK7 glass). Increasing quantum yield only by 15% relaxes this condition by an order of magnitude.
In conclusion, analytical formulas were derived to account for different losses in a luminescent solar concentrator device. The results were validated by numerical simulations of optical path distribution and propagation losses. The obtained solutions can be used to quickly and transparently evaluate LSC device performance for different material compositions and design [22][23][24][25] as well as for the description of light propagation in solar-pumped lasers [26].

S1. Derivation of the optical path length distribution in a slab
We are interested in the probability density for a photon to travel distance to the edge of a rectangular slab for an isotropic point-like source randomly placed inside it.

A) 2D case in-plane (XY plane)
Consider edge element of a length . The fraction of the isotropic emission from a point source at distance into the edge element : 2 From the yellow triangle: For small one can approximate sin /2 /2.
For small one can simplify • cos . Then • sin • Then the fraction for a single source becomes:

• sin 2
Elementary length of the arc with radius contributing to the signal for the angle (blue segment):

• 2
If the arc is limited by angles , the total fraction of photons from the arc of a length , arriving to the element (summing up signal from all the sources at a distance ):

sin cos cos
Finally, for the total fraction of photons reaching the edge (length after travelling distance one should integrate over the whole edge length: cos cos which, after normalization, represents the probability density function. Since limiting angles , vary depending on the geometry and the exact position of several cases should be considered. For certainty a rectangular with a width smaller than the height is taken into account.

A1) For :
Three different cases can be considered: 1. For 0 the arc is from to , where cos 2. For the arc is from 0 to 3. For the arc is from 0 to , where cos , cos And the total number of photons can be calculated by integrating respective parts:

A2) For
Again three different cases can be considered: 1. For 0 the arc is from to 2. For the arc is from to 3. For the arc is from 0 to The same result as for the case above.

A3) For
Two cases are:

S4. Fraction of light emitted to the waveguiding mode
The emitted light from a fluorophore (quantum dot, organic dye, etc.) in a polymer/glass slab will experience total internal reflection for angles at the air interface larger than a critical angle αc. In the most common case for a glass or a polymer: Considering only emitted light through top and bottom facets as losses the total useful fraction of the emission is then ( 1.5):

S5. Effect of scattering by fluorophores
If the total loss is governed by the scattering instead (absorption-free matrix and re-absorption free fluorophore) then the optical efficiency can be also evaluated from the optical path length distribution. Let the linear scattering coefficient be 1/ . Probability for the photon to travel optical path ′ before reaching the edge is ′ ′. Probability of not being scattered within distance ′ is exp ′ .
These photons will contribute to the total optical efficiency similarly to the absorption case above:

′ • exp
In addition, there will be photons, which underwent scattering into the waveguiding mode. Probability of being scattered within distance ′ is 1 exp ′ . If is a fraction of waveguided light after a scattering event ( =75% for n=1.5) the probability to reach the edge after one scattering event is: A Markov process is considered, where there is no memory in the system. The total probability for a photon to reach the edge becomes then a geometrical series (sum of probabilities for no scattering, one scattering, two scattering events, etc.):

S6. Effect of several loss mechanisms present simultaneously
Now consider two processes taking place simultaneously: scattering and matrix absorption. First, photons experiencing no scattering and no absorption will contribute to the total signal:

, ′ • exp • exp
Then photons after one scattering event and without subsequent scattering and absorption. While every scattering event sets back to zero the travelled distance for scattering, the optical path for absorption S13 continues. So the exact history of scattering becomes important. To take into account this fact one can introduce a probability density to scatter at a point (in the absence of other processes): • exp which is a properly normalized probability density function. Then in the system where scattering and absorption coexist the probability density to scatter at a point without being absorbed before is: where a similar notation of the probability density is introduced for the pure absorption process.
Additional conditions of no subsequent scattering and absorption can be added as: where is a photon path taken to reach the device edge after the scattering event. If ′ varies in between (0; ) the integrated probability becomes: Finally taking into account probability to have photon path as and as (again Markov process without memory in the system considered) one obtains after integration from zero to for both path stretches , the input from the photons experienced one scattering event: , • 1 Continuing in the same manner for two scattering events without subsequent scattering and absorption:

, • 1
So the resulting probability can be again represented through geometrical series: This formula turns into the expression for scattering only scenario for a non-absorbing matrix ( 0).
A similar result can be derived for the case of re-absorption instead of scattering:

S14
When re-absorption and scattering both exist in the system one can show in a similar manner as above: A general solution for the optical efficiency, following derivations above, is: