1. INTRODUCTION

The ability of optical systems to concentrate sunlight onto small areas or within thin dielectric layers plays a paramount role in solar cell technology as it is directly related to cost savings induced by the reduced need for expensive semiconductor materials. Consequently, researchers have been devoting significant efforts to calculating the maximum concentration that can be achieved by any particular design in an attempt to assess the potential of their system to convert solar to electrical power. A solar cell made of a semiconductor material with a refractive index $n$, for example, can withhold up to $n^2$ times more intensity compared with the intensity of light in vacuum, as has been shown by both thermodynamic and statistical ray-optic methods [1,2].

Likewise, thermodynamic arguments have been applied to derive the theoretical limits of concentration in luminescent solar concentrators (LSCs) that operate by red-shifting the frequency of the incident radiation. The reduction to the photon energy during the process of light trapping reaps significant concentration gains, attributed to LSCs resembling the operation of heat pumps [3]. By converting part of the incoming power into heat, LSCs are able to increase the effective temperature of the trapped radiation to values above that of the incident solar radiation, resulting, in principle, in concentration efficiencies substantially superior to those achieved by other mechanisms. Nonetheless, an important distinction needs to be made between the concentration achieved in a semiconductor solar cell and that in an LSC. Whereas in solar cells every trapped photon can in principle be converted to a useful electrical carrier, this is certainly not the case in LSCs. Just because a photon has been confined within the LSC lightguide, it does not necessarily mean that it will actually make it to the sides of the device where the photon-to-electron conversion takes place. Luminophores intercepting the optical path of photons induce reabsorption resulting in the gradual removal of energy from the trapped photon gas either due to nonunity quantum yield or because of reemission at angles falling inside the escape cone. In effect, the concentration efficiency of the LSC and the number of photons collected are two different parameters and should not be used interchangeably. To make this point clearer, the concentration efficiency is a constant parameter independent of the geometry of the device, while the number of collected photons depends strongly on the length and the optical thickness of the LSC. While the absolute values of concentration may not reveal much about the overall performance of the LSC, it is rational to expect that the higher the concentration ability of the device, the larger the number of photons to be delivered to its edges. This is one of the key assumptions that we have challenged with our theory, and we have found it to be correct.

The predicted thermodynamic limit (in photons), $C{L}_T$, for concentrating devices with frequency shift is given by [4–6]

Recent leaps in nanotechnology have led to the demonstration of various subwavelength structures that have resulted in semiconductor solar cells reaching nearly maximal concentration [8–11]. However, this is certainly not the case for LSCs. As an example, the predicted limit for an LSC device operating in the visible region with a 0.25 eV energy shift can easily exceed ${10}^4$ according to Eq. (1), a value that has never been attained experimentally. The discrepancy between theoretical calculations and experimental measurements is typically so large that it renders the theoretical limits practically unusable. Recent work based on semi-analytical models, Monte Carlo ray-tracing methods, and more extensive thermodynamic models [12–19] has indicated that the main mechanism inhibiting the concentration in LSCs is the consecutive absorption and reemission events to which trapped photons are subjected inside the lightguide. Since the fluorescence quantum yield of the luminophores is always below 1, only a fraction of the concentrated optical energy survives and reaches the edges of the LSCs where it is being harvested. While these mechanisms are now well understood, no successful analytical theory has been developed to adapt the concentration limits to these nonideal conditions. Motivated by these observations, we have developed an alternative method based on statistical optics equations to analytically calculate the concentration ratio in LSCs. This method sheds light on the various mechanisms limiting the concentration efficiency.

In summary, the objectives of this paper are the following: (1) to investigate the physical processes that hinder the concentration ability of LSCs and particularly the interplay between reabsorption and nonunity quantum yield; (2) to understand whether the thermodynamic limits actually correspond to the absolute maximum levels of concentration that can be achieved by an LSC design or if any conditions persist under which they can be surpassed; (3) to devise a theory that extends the current concentration limits in a way that takes into account the broadband nature of photon–material interactions occurring in an LSC; and (4) to devise a set of simple rules that can assist in the design of efficient LSCs.

2. GEOMETRY OF THE PROBLEM

The geometry of the LSC investigated in this work is shown in Fig. 1. It consists of an infinite dielectric slab with refractive index $n$ and thickness $d$ in which luminophores are embedded. Typical host dielectrics include polymethyl methacrylate (PMMA) or glass, while organic dyes, rare earth ions, and quantum dots [20,21] are commonly used luminescent materials. The LSC is held at the ambient temperature $T_o=300\mathrm{K}$, which is considered constant, while it is illuminated by an isotropic field in equilibrium with a blackbody source at a temperature $T_s=5800\mathrm{K}$. The spectral radiance emitted by the source in photons per unit time, unit projected area, unit solid angle, and unit angular frequency is given by Planck’s law,

 

Fig. 1. Geometry of the investigated LSC: an LSC slab of thickness $d$ and refractive index $n$ is immersed into diffuse electromagnetic radiation characterized by isotropic radiance $S({\omega }_1;T_s)$. Also shown: energy levels of the fluorescent system under consideration indicating the possible electronic transitions.

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To calculate the photon flux (photons per unit time and unit projected area) per unit frequency of the field reaching the LSC, we need to multiply $S(\omega ;T_s)$ with the solid angle subtended by the sun at the position of the LSC. An alternative point of view that is particularly helpful when dealing with diffuse radiation, as in this paper, is to interpret $T_s$ as an effective temperature corresponding to the temperature of an equivalent blackbody at the surface of the earth that would generate the same photon flux at a given frequency as the one coming from the sun [4].

The fluorescent system under consideration consists of a ground state and an excited state (Fig. 1). Electrons are continuously excited to the upper state by absorbing photons of energy $h{\omega }_1$. Subsequently, quick thermal relaxation to the bottom of the excited state occurs, from which electrons return to the ground state by spontaneously emitting photons at energy $h{\omega }_2$. The population among the substates near the top of the excited state is assumed to undergo rapid thermal equilibration. Critically, only narrowband interaction between the radiation field and the fluorescent system is considered in order to maintain consistency with the previous studies that have dealt with concentration limits in LSCs. We therefore consider luminophores that absorb radiation within a narrow frequency range in the vicinity of ${\omega }_1$ and emit photons within a narrow range around ${\omega }_2$. Furthermore, we assume that a narrowband filter is placed between the blackbody source and the LSC. If the bandpass of such a filter is precisely matched to the absorption spectrum of the luminophores, then the removal of the residual frequencies from the incident Planck spectrum will not have any effect on the chemical potential of the system, a point first made by Ross [22]. For such essentially monochromatic absorbers and emitters the spectral radiance can be considered constant across the frequencies of interest. In Section 4 we will show how to extend the ideas presented in this work to incorporate broadband interactions as well.

A further point that needs clarification concerns the directional nature of interaction between the radiation field and the luminophores since not all luminescent particles absorb and emit photons isotropically. Some of the most popular dyes employed in LSC applications are dichroic, and consequently the full polarizability tensor is required to describe their interaction with an electromagnetic field. A number of papers have appeared recently specifically taking advantage of the dichroism of such dyes to improve concentration. In these studies, homeotropic alignment was shown to promote light trapping within the LSC lightguide [23,24]. However, in our model we assume that luminophores are oriented randomly within the host matrix. This assumption is compatible with most fabrication methods used to deposit the luminescent layer, such as spin coating, dip coating, or casting. Rather than individually examining the interaction between each dye molecule and the electromagnetic radiation, we instead segment the LSC into a large number of slices of thickness $\mathrm{\Delta }z$ and examine the collective behavior of the molecule ensembles enclosed within each infinitesimally small volume. The thickness $\mathrm{\Delta }z$ of each segment is taken on the one hand to be very thin compared with the wavelength of light, but sufficiently thick, on the other hand, for a large number of molecules to be incorporated. In this case, the directionality of absorption and emission is averaged out and these processes can be described by effective cross sections that exhibit uniform angular dependence [25]. It has to be noted that these assumptions are made again to maintain consistency with previous work and can be lifted if necessary.

3. CALCULATION OF CONCENTRATION RATIO IN LSC

Diffuse radiation characterized by spectral radiance $S_o=S({\omega }_1;T_s)$ is incident onto both sides of the LSC as shown in Fig. 1. Radiance is a conserved quantity in nonlossy media, but this is not the case here, as photons are continuously absorbed and reradiated by the luminophores with nonunity quantum yield. Moreover, photons are reflected at the dielectric–air interfaces, and, hence, to calculate the total fields inside the LSC, all absorption, frequency-shift, and heat dissipation processes need to be taken into account as well as to apply appropriate boundary conditions. Since Fresnel reflection and transmission coefficients depend upon the angle of incidence, each contribution coming from a different solid angle of the incident radiation needs to be considered independently. We further separate fields traveling in the positive from the negative $z$ direction by using $(+)$ and $(-)$ superscripts correspondingly. A significant simplification occurs if the reflection symmetry of the problem with respect to the plane $z=d/2$ is exploited, as all backward traveling parameters ${\mathcal{A}}^{-}$ are related to the forward ones ${\mathcal{A}}^{+}$ via ${\mathcal{A}}^{-}(z)={\mathcal{A}}^{+}(d-z)$.

The incident photon flux at the surface of the LSC contained within the solid angle element $d{\mathrm{\Omega }}_o$ is given by $S_{o}d{\mathrm{\Omega }}_o\cos {\theta }_o$ (Fig. 2), where ${\theta }_o$ is the polar angle measured from the LSC normal. Some of this flux is reflected back to the environment, while the rest is transmitted into the LSC. To calculate the radiance $S_1^{\pm }$ propagating inside the LSC, we need to consider the reflections at the two boundaries as well as to account for the absorption by the luminophores. This means that in contrast to $S_o$, which is a constant quantity, $S_1^{\pm }(\theta ,z)$ exhibits both angular and position dependence. If the effective molar absorption coefficient at ${\omega }_1$ is $ϵ({\omega }_1)$ and the molar concentration of the luminophores is $c_m$, then the absorption coefficient per unit length of the LSC is ${\alpha }_1=c_mϵ({\omega }_1)$, and from Beer–Lambert’s law the radiance of the transmitted fields scales according to

We now proceed to determine the total fields concentrated in the LSC. The strategy is to first calculate the photon flux $d{\mathrm{\Phi }}_{\text{tot}}/d\mathrm{\Omega }\cos \theta$ generated in the LSC by the portion of the incident radiation that is contained within the solid angle element $d\mathrm{\Omega }$ and then integrate over $d\mathrm{\Omega }\cos \theta$ to calculate the total flux. Photons from the incident field $S_1(\theta ;z)$ are absorbed by the luminophores contained within each elementary slice $\mathrm{\Delta }z$, and new ones are emerging at the shifted frequency ${\omega }_2$ with an isotropic radiance distribution due to the random orientation of the dipole moments (Fig. 3). The new photons can be reabsorbed with a finite probability, a process that is characterized by the absorption coefficient ${\alpha }_2=c_mϵ({\omega }_2)$. If with ${\mathcal{L}}^{\pm }(z)$ we denote the locally generated radiance as shown in Fig. 3, then in general we have three types of photon sources that traverse an elementary slice extending between $[z,z+\mathrm{\Delta }z]$ and that contribute to the buildup of the total fields in the LSC: (A) the source $S_1(\theta ;z)$ due to the portion of the external radiation transmitted into the LSC that contributes a flux per incident solid angle given by Eqs. (3) and (5); (B) the net photon flux distributions $\int {\mathcal{L}}^{-}(z+\mathrm{\Delta }z)\mathrm{d}{\mathrm{\Omega }}^{\prime }\cos {\theta }^{\prime }$ and $\int {\mathcal{L}}^{+}(z)\mathrm{d}{\mathrm{\Omega }}^{\prime }\cos {\theta }^{\prime }$ contributed by all LSC areas outside the elementary slice $[z,z+\mathrm{\Delta }z]$; (C) a feedback source field $S_2(z)=S_2^{+}(z)+S_2^{-}(z)$ due to reflections of the ${\mathcal{L}}^{\pm }(z)$ photons at the boundaries of the LSC. As shown in Supplement 1, the flux contribution from $S_2(z)$ equals

The parameter ${\mathcal{L}}_o$ that enters Eq. (11) via Eq. (6) is still unknown. By setting $z=d$ in Eq. (11), we get a recursive relationship that allows ${\mathcal{L}}_o$ to be calculated.

 

Fig. 2. Fields created inside the LSC due to incident radiation: the fields $S_1^{\pm }(\theta ;z)$ generated inside the LSC by the external radiation $S_o$ can be deduced by application of flux conservation (incoming $\text{flux}=\text{outgoing}$ flux) at the air–dielectric interface after considering the reflections at the boundaries and the attenuation imposed onto the traveling beams by the absorptivity of the fluorophores. ${\theta }_o$ is the polar angle measured from the LSC normal toward the air, while $\theta$ is the polar angle measured toward the dielectric. No azimuthal dependence is exhibited due to the symmetry of the problem.

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Fig. 3. Incident radiation generates three types of sources that traverse each infinitesimal slice extending between $[z,z+\mathrm{\Delta }z]$: A, radiation transmitted directly into the LSC with a photon flux per unit incident solid angle $S_1^{\pm }(\theta ;z)$; B, collective radiation contributed by all the LSC regions below $z$ and above $z+\mathrm{\Delta }z$; C, feedback source $S_2^{\pm }(z)$ due to reflections on LSC–air interfaces. The three sources (A–C) are responsible for generating a Lambertian emission profile emerging from each elementary slice with an angularly independent radiance ${\mathcal{L}}^{\pm }(z)$.

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We are now in the position to calculate the photon flux inside the LSC. If we isolate the fields at frequency ${\omega }_2$, then the concentrated flux consists of the superposition of the fluxes $\pi {\mathcal{L}}^{\pm }(z)$ and $S_2^{\pm }(z)$. As mentioned earlier, these two photon flux distributions are generated by this increment of incident radiation that is contained within the solid angle element $d\mathrm{\Omega }$. In order to calculate the total flux ${\mathrm{\Phi }}_{\text{tot}}^{\pm }$ propagating in each direction in the LSC, we need to integrate over $d\mathrm{\Omega }\cos \theta$ to get

Equation (11) is the main output of this paper. It allows for the calculation of the concentration limits in LSCs characterized by arbitrary quantum yields and values of absorption coefficients through Eqs. (5) and (6) and Eqs. (12)–(14). In the following we focus on some cases of special interest.

4. SOME SPECIAL CASES

Case A: $q=1$. The case in which the quantum yield is equal to unity is a particularly important one as it allows direct comparison with the previously derived thermodynamic limits of concentration. This is particularly true for optically thick LSCs, as will be shown. By combining Eq. (11) with Eq. (3) and Eqs. (5) and (6) and setting $q=1$, we get, for ${\mathcal{L}}_o$,

A1: Optically thick LSC ($e^{-{\alpha }_{2}d}$, $e^{-{\alpha }_{1}d}\to 0$). By substituting Eqs. (11), (12), and (15) into Eq. (13) and taking the limits $e^{-{\alpha }_{2}d}$, $e^{-{\alpha }_{1}d}\to 0$, we get the following expression for the concentration ratio $CR$:

A2: Zero frequency shift (${\omega }_1={\omega }_2$ and $e^{-{\alpha }_{1}d}\to 0$). In this case, photons solely undergo elastic scattering upon interacting with the luminophores within the dielectric medium. To calculate the total flux we now need to add to the flux given by Eqs. (12) and (13) the extra contribution from $S_1^{\pm }(z)$ as all fields are now of the same frequency ${\omega }_1$. In this case the concentration ratio becomes

A3: Negligible reabsorption losses $({\alpha }_2\to 0)$. When $({\alpha }_2\to 0)$, the parameter $\zeta$ given by Eq. (S3) goes to infinity. Consequently, the flux inside the LSC tends to infinity too by means of Eqs. (6) and (12), as waves build up from successive reflections at the boundaries without any losses. This result implies that there is no bound to the concentration limit for LSCs whose luminophores combine unity quantum yield and zero reabsorption losses. Such a result is not in contradiction with earlier work as thermodynamics also predict unbounded concentration limits when the Stokes shift is large enough for reabsorption losses to vanish, as emphasized by Ries in [4]. It is of course rather unrealistic to expect for such a situation to arise in a real system, but the main point here is that concentration limits above the values predicted by Eq. (1) are feasible. This point will be explored further in the following paragraph, where luminophores with quantum yield $q\le 1$ are investigated.

Case B: $q\le 1$. When $q\le 1$, the concentration ratio of the LSC, $C{R}_{q\le 1}$, reduces as now a portion of the photon energy from the incident radiation field is lost due to nonunity quantum yield. To demonstrate the impact of the nonideal quantum yield on the concentration ratio we applied our theory to a series of LSCs where Coumarin 6 was used as the prototype lumoniphore material. The particular choice of fluorescent material was made due to its popularity in LSC applications [13,14,26–28], although recently dyes from the Lumogen family have shown excellent stability and quantum yields close to 1 [29]. The following two subcases were studied:

B1: Narrowband interactions. Concentration ratio at a single absorption frequency and a single emission frequency.

The molar extinction coefficient of Coumarin 6 is $ϵ({\omega }_1)=12.43\times {10}^4{(\mathrm{M}·\mathrm{cm})}^{-1}$ at the peak absorption frequency ${\omega }_1=3.89\times {10}^{15}\mathrm{rad}·{\mathrm{s}}^{-1}$ (${\lambda }_1=460\mathrm{nm}$) and $ϵ({\omega }_2)=5.42\times {10}^3{(\mathrm{M}·\mathrm{cm})}^{-1}$ at the peak emission frequency ${\omega }_2=3.73\times {10}^{15}\mathrm{rad}·{\mathrm{s}}^{-1}$ (${\lambda }_2=500\mathrm{nm}$), correspondingly [30]. The concentration ratio $C{R}_{q\le 1}$ was calculated by varying $q$ in the range $0<q\le 1$. In these calculations, we also examined the effect of varying the dye concentration $\times$ LSC thickness product ($c_m\times d$). By varying $c_m\times d$ in the range ${10}^{-9}(\mathrm{M}·\mathrm{cm})\le c_m\times d\le {10}^{-1}(\mathrm{M}·\mathrm{cm})$, both optically thin and thick LSCs were emulated. Practically this can be realized by altering the concentration of the dyes or the thickness of the LSC or a combination of both. The host material was assumed to have a refractive index of $n=1.5$, typical for most transparent plastics or glass, while the index of the surrounding medium was set to $n_o=1$ corresponding to air. To avoid position-dependent terms, all results were averaged over the LSC thickness. Results are presented in the form of concentration ratio normalized over the thermodynamically predicted limit, $(C{R}_{q\le 1}/C{L}_T)$, which for the parameters of the specific system was calculated by Eq. (1) to be $C{L}_T=8.03\times {10}^3$.

Figure 4 shows that for high quantum yields and low levels of absorption $C{R}_q$ can easily exceed $C{L}_T$. This is because lowering the reabsorption results in a less random photon gas, which concentrates outside the escape cone of the LSC and so is trapped more efficiently. However, when the LSC becomes more optically thick or the quantum yield reduces, the concentration ratio drops off to values below $C{L}_T$. Within the range of validity of $C{L}_T$ for which $c_m\times d$ acquires large values, it is always the case that $C{R}_q<C{L}_T$ as expected. For perfect emitters with $q=1$, the crossover between $C{R}_q$ and $C{L}_T$ occurs at $c_m\times d={10}^{-3.4}(\mathrm{M}·\mathrm{cm})$ and as absorption increases the concentration ratio saturates to its minimum value $C{R}_{(q=1)}=0.81\times C{L}_T$. Saturation occurs when full absorption and randomization of the incident field has occurred. Beyond this point, increasing the thickness of the LSC or the luminophore concentration is not changing the internal dynamics of the photon gas inside the LSC. For all other luminophores with a quantum yield $q<1$ the concentration ratio asymptotically approaches zero as the LSC becomes more and more optically thick. This is because large values of thickness or luminophore concentration lead to all photons eventually being converted to heat. Likewise, the crossover between $C{R}_q$ and $C{L}_T$ occurs for lower values of $c_m\times d$.

 

Fig. 4. Normalized concentration ratio as a function of luminophore concentration $\times$ LSC thickness ($c_m\times d$) for various quantum yield values. The thick black line defines the values for which thermodynamic and geometrical optic limits coincide. For optically thick LSCs where the thermodynamic limits are valid, it is always $C{R}_q<C{L}_T$.

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B2: Broadband interactions. Concentration ratio by using weighted average absorption coefficients.

The narrowband interaction picture discussed in case B1 skews the effect of reabsorption. In most luminophores there is a substantial overlap between the emission and absorption spectra; see, for example, Fig. 5 for Coumarin 6, which is not fully accounted for. Moreover, absorption tends to be considerably weak and close to the experimental error at the emission peak and to vary steeply at nearby frequencies. In the case of Coumarin 6, the absorption coefficient increases by almost tenfold when calculated at 490 nm, only 10 nm away from the emission peak, and so calculations are prone to significant numerical errors. The above discussion indicates that rather than considering $ϵ({\omega }_2)$ at a single frequency, it would be more accurate to take its weighted average $\overline{ϵ({\omega }_2)}$ over an appropriately chosen spectral range. The same arguments apply to the absorption coefficient $ϵ({\omega }_1)$ calculated at the incident frequency. This is particularly true for the new generation of luminophores such as CdSe/Cds, ZnSe, and CdSe/ZnS quantum dots [33–35], whose absorption spectra differ considerably from typical Lorentzian distributions; see inset of Fig. 5. In this case, clear peaks where calculations of the concentration limits could be performed are not obvious. In addition, the aim of the LSC designer is usually to maximize absorption over as large a part of the solar spectrum as possible in order to boost the electrical power generated by the device. Calculating $ϵ({\omega }_1)$ at a sole frequency might again not render reliable results. In this case, it is sensible to use the average value $\overline{ϵ({\omega }_1)}$ over the spectrum covered by the incident photons.

 

Fig. 5. Absorption and emission spectra for Coumarin 6 dye (spectra obtained from [31]). Also shown, typical histogram of photons received at the edges of the LSC as calculated by Monte Carlo, ray-tracing method. Inset: absorption and emission spectra for PbS quantum dots (spectra obtained from [32]).

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A question arises about which photon spectrum should be considered in the calculation of the absorption coefficient $\overline{ϵ({\omega }_2)}$. The obvious choice would be to average $ϵ({\omega }_2)$ over the entire emission spectrum (Fig. 5). In this scenario, the emission spectrum should be interpreted as a probability density function (pdf) indicating how likely it is for a photon to be emitted at a certain frequency. This pdf can then be used for the calculation of the expected value of the absorption coefficient. As we will show in the following, this choice leads to erroneous results. This is because the higher energy reemitted photons suffer from disproportionately larger absorption compared to the lower energy ones and do not survive in the LSC. Consequently, the concentrated spectrum in the LSC is red-shifted compared to the emission spectrum of the luminophores, as has been theoretically calculated and experimentally verified previously [36]. A spectrum restricted over a narrower range of frequencies should then be used.

To identify the optimum spectrum for the calculation of $\overline{ϵ({\omega }_2)}$ and to validate our theory we developed a Monte Carlo statistical ray-tracing algorithm with the aid of which we modeled the LSCs examined in this work. The details of this algorithm can be found in [37–39], where we have demonstrated excellent agreement between theoretical and experimental results. The modeled LSCs had an area larger than $1\mathrm{m}\times 1\mathrm{m}$ and a thickness of $d=0.3\mathrm{cm}$ to make the results compatible with real systems. The concentration $c_m$ of the dyes was varied so that the parameter $c_m\times d$ ranged as ${10}^{-9}(\mathrm{M}·\mathrm{cm})<c_m\times d<{10}^{-2}(\mathrm{M}·\mathrm{cm})$. Photons incident onto the LSC were covering the spectral range between 350 and 510 nm, which coincided with the absorption spectrum of Coumarin 6. In all cases, $q=0.78$ was considered, which is the quantum yield usually quoted for Coumarin 6 [30]. The number of photons arriving at the edges of the LSC as well as their wavelength were recorded for each set of simulation parameters, and a histogram was built like the one shown in Fig. 5. Our Monte Carlo simulations reproduce the red-shifting of the peak wavelength for the emission spectrum, in consistence with the observations of other researchers [36]. This is because the high-energy photons undergo a large number of reabsorption and reemission events, and most of them are gradually dissipated as heat. Only photons exhibiting low values of absorption survive and are collected at the edges of the LSC. It is reasonable therefore to expect that the theory developed here be more accurate when the average absorption coefficient is calculated over the more restricted spectral range of the collected photos. In this case, the histogram of the collected photons from the Monte Carlo method can serve as the pdf against which the expectation of absorption is calculated. For LSC areas larger than $1{\mathrm{m}}^2$, the spectral histograms were all very similar although the absolute number of photons reaching the edges was lower. However, since all histograms are normalized to unit area for them to be converted to pdfs, the results presented in the following are applicable to all LSCs larger than the ones used in our Monte Carlo simulations, independently of their area.

The metric that is usually measured experimentally is the concentration gain of the LSC. This is defined as the fraction of the incident photons that reach the edges multiplied by the geometrical gain, G, of the device ($\mathrm{G}=\mathrm{Top}$ area/Side area) [15,17,37,38]. In Fig. 6 the concentration gain calculated by the Monte Carlo method is compared against the concentration ratio calculated by the weighted average (both full and restricted spectra) and single frequency approaches. Concentration metrics are this time normalized to unity peak for easier comparison. Results from the single frequency and full-spectrum weighted average models deviate significantly from the Monte Carlo simulations and are inadequate predictors of the photons collected from the LSC. However, when the restricted spectrum model is used, the concentrated ratios predicted by our theory and the concentration gains calculated by the Monte Carlo simulations become almost indistinguishable. The peaks of the two distributions coincide precisely as well as the body of the curves for all values of the $c_m\times d$ parameter smaller than the one corresponding to the peak of the distribution. A small discrepancy is observed for larger values of $c_m\times d$. This is because in our model we used for simplicity the pdf derived by the histogram of the collected photons at the peak $c_m\times d$ value only. Our analysis of the Monte Carlo simulations has shown that all photon histograms are very similar for smaller values of $c_m\times d$, but some deviations start appearing for larger values. When a different pdf is used for each $c_m\times d$, the discrepancy is alleviated. However, the agreement between the two results is already so good that we opted for the simpler method.

 

Fig. 6. Comparison between Monte Carlo, weighted average, and single frequency models. Average values for $\overline{ϵ({\omega }_2)}$ are calculated over the entire region where the absorption and emission spectra of Coumarin 6 overlap in the full-spectrum weighted average model. In contrast, only the spectra of those photons that are delivered to the sides of the LSC are used in the restricted spectrum model. Values used to obtain the results: full-spectrum weighted average frequency model, $\overline{ϵ({\omega }_1)}=52731{(\mathrm{M}·\mathrm{cm})}^{-1}$, $\overline{ϵ({\omega }_2)}=5420{(\mathrm{M}·\mathrm{cm})}^{-1}$. Restricted spectrum weighted average frequency model, $\overline{ϵ({\omega }_1)}=52731{(\mathrm{M}·\mathrm{cm})}^{-1}$, $\overline{ϵ({\omega }_2)}=271{(\mathrm{M}·\mathrm{cm})}^{-1}$.

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In terms of absolute values, according to the Monte Carlo simulations, only 3.6% of the photons that initially hit the LSC arrive at its edges. Multiplied with the geometrical gain, $\mathrm{G}=1333$, this gives a concentration gain of 48. For an equivalent $4\mathrm{m}\times 4\mathrm{m}$ LSC, the Monte Carlo method predicts a concentration gain of 87, while for a $2\mathrm{m}\times 2\mathrm{m}$ LSC, this becomes 128. On the other hand, our model predicts a concentration ratio of $C_R=4.71\times {10}^3$, while as mentioned previously, the thermodynamic limit is $C{L}_T=8.03\times {10}^3$. The foregoing discussion suggests that the absolute values of the concentration ratios that are predicted by theoretical methods such as ours or the thermodynamic model are perhaps not as important in the context of the LSCs. Instead, the maxima of such distributions as in Figs. 6 and 7 are by far more important indicators of the performance of the device.

 

Fig. 7. Heatmap of maximum concentration ratio as a function of absorption and quantum yield calculated by the weighted average model. Black dotted line corresponds to the $C{R}_q=C{L}_T$ boundary. White bar shows the range of concentrations that could be achieved by five experimental systems selected from the literature. Star sign signifies the position of optimum concentration for $q=0.78$.

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Figure 7 presents in the form of a heatmap the (weighted average) normalized concentration ratio when both quantum yield and absorption vary. The white bar indicates the range of concentration ratios for five systems reported in the literature [13,14,26–28] as predicted by our method. The star sign signifies the position of maximum concentration that could be achieved for $q=0.78$. Our analysis shows that only one system is operating at the maximum concentration ratio point, which occurs when $c_m\times d={10}^{-5.1}(\mathrm{M}·\mathrm{cm})$. Optimum values of $c_m\times d$ can be used by LSC researchers as design rules to maximize the performance of their systems.

5. CONCLUSIONS AND DISCUSSION

In conclusion, we have shown that for the thermodynamic limits of concentration to be valid, conditions of perfect absorption must be satisfied at both incident and emitted frequencies. If this is not the case, then concentration may exceed or be substantially lower than $C{L}_T$. While the standard thermodynamic model fails to capture the essence of the interactions occurring in an LSC, our theory naturally embeds the information about the geometry of the device and the properties of the materials used. The concentration ratio calculated by the thermodynamic limits or by our approach is in fact a different parameter from the concentration gain usually quoted in experimental measurements or Monte Carlo simulations. The latter metric relates to the fraction of those trapped photons that actually make it to the LSC edges, and, hence, in addition to photon concentration, it involves the effect of photon transport. Nonetheless, by using appropriate average values for the absorption coefficients of the incident and reemitted photons, almost perfect agreement between the concentration ratios obtained with our method and Monte Carlo simulations is achieved. This demonstrates that maximizing light concentration automatically results in the maximization of the number of photons being collected from the LSC. The absolute number of collected photons, though, always depends upon the size of the LSC. For the maximum concentration to be achieved, it is advisable to design LSCs that are optically thick at the incident frequencies but nearly transparent at the emitted frequencies.

There is obviously an overwhelming number of reasons to constrain the performance of LSCs and explain further discrepancies between theory and experiments. At first, the host matrix introduces absorption and scattering itself, which needs to be taken into account. Scattering is also introduced by imperfectly flat sidewalls. Second, at large concentrations, luminophores form aggregates that tend to introduce new nonradiative decay channels further limiting the quantum yield of the system. Third, employing aligned dichroic dyes will result in emission profiles that are not isotropic. Finally, at high light concentration it cannot be expected that the temperature of the LSC remains constant and equal to the ambient temperature. In theory, the equilibrium temperature of the LSC can raise to any value up to the (effective) temperature of the incident blackbody radiation. A rise to the ambient temperature will inflict severe consequences on the concentration ratio and thus on the number of collected photons.