## Abstract

Luminescent solar concentrators (LSCs) are an emerging technology that aims primarily to reduce the cost of solar energy, with great potential for building integrated photovoltaic (PV) structures. However, realizing LSCs with commercially viable efficiency is currently hindered by reabsorption losses. Here, we introduce an approach to reducing reabsorption as well as improving directional emission in LSCs by using stimulated emission. Light from a seed laser (potentially an inexpensive laser diode) passes through the entire area of the LSC panel, modifying the emission spectrum of excited dye molecules such that it is spectrally narrower, at wavelengths that minimize reabsorption to allow net gain in the system, and directed towards a small PV cell. A mathematical model, taking into account thermodynamic considerations, of such a system is presented which identifies key parameters and allows evaluation in terms of net effective output power.

© 2016 Optical Society of America

## 1. Introduction

Photovoltaic (PV) conversion of solar energy is one of the most promising ways of meeting increasing global energy demands. Luminescent solar concentrators (LSCs) are non-geometric light concentrators developed primarily to solve one of the main problems often associated with PV technologies: the cost of the PV cell [1,2]. They are now attractive candidates for building-integrated PV systems due to their versatility and ability to be incorporated into atriums, roofs or windows [2,3]. Physically, LSCs are typically large sheets of transparent polymer (or glass) doped with carefully chosen luminophores (e.g. dyes or quantum dots). These can absorb incident sunlight, becoming photo-excited, and then relax through photoluminescence processes, emitting photons some of which become trapped within wave-guided modes of the transparent substrate and guided to the narrow edges of the substrate, where narrow PV cells are optically coupled. Thus, the area of PV cell required to collect a given area of sunlight is reduced commensurate with the relative areas of the LSC.

There are multiple loss mechanisms that result in low efficiency for LSCs. The main mechanisms [2] are: (a) incident sunlight that is not absorbed by the luminophores; (b) non-radiative decay within photoexcited luminophores; (c) photoluminescence not coupled to waveguided modes and lost from the concentrator; and (d) waveguided light that is reabsorbed by the luminophores. In particular, reabsorption is the key limiting factor that significantly reduces the power conversion efficiency of LSCs [4].

To overcome reabsorption in LSCs, several strategies have been proposed. Most approaches rely on luminophores engineered to have large Stokes shifts to minimize the overlap between absorption and emission spectra; such luminophores include organic dyes [5–7], quantum dots [8,9] including colloidal quantum dots [10], and rare earth complexes [11,12]. This approach is fundamentally restricted by thermodynamics as the maximum energy concentration is reduced by large Stokes shifts [13,14]. Another approach is to optimize the structure, for example through cascade structures [15], resonance shifting structures [16], structures with inhomogeneous distribution of dyes [17–19], by means of plasmonic nanoparticles [20,21] and scatterers [22,23], and different geometries [1,24–27].

The approach we detail here is to consider stimulated emission, by using a seed laser, rather than only spontaneous emission as in standard LSCs. This reduces reabsorption losses by ensuring emission is narrowband and at an optimal wavelength, rather than broadband, and directs emission into guided modes. Our initial proposal of this approach explored the feasibility using commercially available and known photostable Perylene Red (PR) dyes [28], however the model presented here is a general one not connected to any specific material choice. Whilst the concept is clearly timely, as other researchers are exploring it [29], to date there is no detailed description of the operating principle and model for the power conversion efficiency of such a device. This work aims primarily to illustrate the principle of stimulated LSCs (s-LSCs) in detail and to develop an overall mathematical model considering the effect of different physical parameters on the power conversion efficiency. The model we develop allows s-LSCs to be evaluated, noting that they offer the prospect of realizing significant efficiency improvement at low cost by using an inexpensive seed laser and a miniature PV cell comparable to the beam size of the seed laser.

## 2. Principle

In conventional LSCs, dyes are photoexcited by sunlight and then spontaneously emit at an essentially random longer wavelength, on average, in a random direction. These photons can be reabsorbed due to the overlap of the absorption and emission bands. In s-LSCs, a seed laser with a wavelength close to the emission peak of the absorbing dyes, but far enough from theabsorption peak that reabsorption is negligible, is used to extract power from photoexcited dyes through stimulated emission, duly re-shaping the emission spectrum through spectral narrowing. In addition to narrowing the emission spectrum, this technique will also produce more spatially controlled emission such that all emitted photons are trapped and thus improves photon concentration both spectrally and geometrically.

Figure 1 shows the components of the s-LSC system concept. Sunlight incident on the solar concentrator enters through the top surface. Photons of wavelengths within the absorption spectrum excite the dye molecules, pumping the system. The light from the seed laser passes through the solar concentrator panel and when it encounters an excited dye molecule, may cause it to emit its energy into an identical photon: of the same wavelength, propagating in the same direction. The more power in this seed laser, the greater the rate of stimulated, rather than spontaneous, emission. As the seed light passes through the material and causes stimulated emission its power will increase, which in turn will increase this rate of stimulated emission. If a large fraction of the emission is stimulated rather than spontaneous, then the efficiency of the s-LSC can be substantially increased. A practical system (see Fig. 1) would have the edges largely mirrored so that the laser beam would zig-zag through the whole panel, sweeping out the entire area of the concentrator, and in turn being amplified until it is carrying much more power than was originally launched. As such, it will ideally extract the energy absorbed by the dye into a desired narrow wavelength range rather than the dye’s spontaneous emission spectrum. This stimulated emission will be directed towards a comparably very small PV cell, as opposed to spontaneous emission which occurs in random directions. The target PV cell can have an area comparable to the cross sectional area of the seed source, anticipated to be of orders 1 mm^{2}. A fraction of the PV cell’s output electrical power is fed back to drive the seed laser, so that additional power into this system is unnecessary. The remainder of the PV cell’s output is the final electrical power output of this system.

## 3. Mathematical model

The energy extracted by the seed laser, corresponding to the signal in an amplifier, and directed towards the PV cell can be quantified as a gain for the seed laser. The total gain (*G _{T}*) can be determined by considering a pumped/unpumped change (i.e. with the concentrator illuminated and not illuminated by the solar radiation, corresponding to the pump in an amplifier) in the output power of the PV cell measured in the electrical domain:

*P*is the electrical output power from the PV cell when both pump and signal are on;

_{(P + S)E}*P*and

_{PE}*P*are the electrical output power from the PV cell when only the pump and only the signal are on, respectively. The total gain can be represented as the product of two separate gain termswhere,

_{SE}*G*is the optical gain or amplification of the seed laser in the concentrator due to stimulated emission, and

_{opt}*G*is an effective gain in the PV cell, due to its nonlinear response to the optical energy incident on it (i.e. the signal would experience higher conversion efficiency in the presence of pump, and can be considered to be amplified in the electrical domain), that must be accounted for, for a complete description.

_{eff}To more fully understand the proposed system we consider the various processes that occur as illustrated in Fig. 2 in the pumped/unpumped cases, first in the optical domain and then in the electrical domain. Considering the seed laser alone, its output optical power *P _{SO}* will be attenuated due to absorption in the concentrator, such that the power reaching the PV cell will be reduced to a value

*η*, where

_{t}P_{SO}*η*describes transmission efficiency. Considering the effect of the pump incident on the concentrator alone, some fraction of it will result in spontaneous emission that reaches the small PV cell. For a pump power

_{t}*P*, an amount

_{PO}*η*will reach the PV cell, with

_{sp}P_{PO}*η*describing the efficiency of this process (corresponding to the optical efficiency (

_{sp}*η*) of the conventional LSC which will be discussed in detail at the end of this paper). Considering both pump and signal together, the light from the seed laser will experience both loss and optical gain, and both this amplified signal and the spontaneous emission background will be incident on the PV cell, giving a total incident power of

_{opt}*η*. Moving from the optical to the electrical domain,

_{t}G_{opt}P_{SO}+ η_{sp}P_{PO}*η*describes the coupling efficiency of the emitted light from the concentrator to the PV cell. The nonlinearity of the PV cell will result in a different efficiency depending on the incident optical power. This is quantified by three optical-to-electrical conversion efficiency terms

_{cp}*η*,

_{1}*η*and

_{2}*η*for optical powers of the pump, signal, and pump & signal together, respectively, which produce the corresponding electrical powers

_{3}*P*,

_{PE}*P*and

_{SE}*P*. Hence, Eq. (1) can be re-expressed in terms of the optical powers and conversion efficiencies as

_{(P + S)E}*η*=

_{1}*η*=

_{2}*η*, we obtain

_{3}*G*= 1 as expected and the total gain comes from the concentrator,

_{eff}*G*=

_{T}*G*.

_{opt}This model also facilitates a description of the net extracted power from the system. The total power extraction can also be divided into two parts: the optical power extracted due to optical amplification in the concentrator, and the effective power extracted due to nonlinearity of the PV cell leading to higher efficiency. The total extracted power *P _{TEE}* is described in the electrical domain as

*P*is described in the optical domain asand the additional power arising from the effective gain

_{EO}*P*is described in the electrical domain asIf the efficiency of the PV cell as a function of power is known, both the optical and effective gain can be determined directly from the electrical power output of the PV cell.

_{EE}Figure 3 shows the mathematical block model of the complete s-LSC system, including powering the seed laser. A small fraction of the output power (given by *P _{LE}*) is fed back to the seed laser and the remainder is available to drive the output load. The effective power output of the system available to drive the load is given by

*P*is expressed in terms of the output optical power

_{LE}*P*and the laser efficiency

_{SO}*η*, defined as

_{L}*P*=

_{SO}*η*. The other parameters in the system are as described earlier. When stimulated emission dominates, the effect of the background spontaneous emission on the

_{L}P_{LE}*P*is negligible since the PV cell area is small and thus the first term of Eq. (9) may be neglected. For the s-LSC system to operate as described there must be sufficient electrical power produced by the PV cell to drive the laser. This defines a minimum threshold condition requiringand

_{SL}*η*&

_{t}*G*are the most critical parameters to meet this condition.

_{opt}The transmission efficiency (*η _{t}*) depends on signal losses in the concentrator, specifically material absorption at the signal wavelength, scattering loss, confinement loss and reflection loss from the reflective coatings. The scattering component arises from surface roughness and inhomogeneity in the concentrator itself. The choice for the host material used for the s-LSC system should have low material absorption at the signal wavelength. The scattering component can be minimized using suitable fabrication techniques. Confinement loss and the loss due to surface roughness are minimized by controlling emission direction. The reflection loss from reflective coating can also be minimized by using highly reflective material for coating the edges of the concentrator. The effect of reflection loss on the transmission efficiency can be diminished by optimizing the device geometry so that the seed laser would experience a longer path length between successive reflections for a given surface area.

The optical gain (*G _{opt}*) of the system is a function of the absorbed pump power, the signal power, and the material properties of the luminescent material. The absorbed pump power is the integral of the solar spectrum that overlaps with the absorption of the luminescent material. Thus to achieve higher absorbed pump power, a luminescent material is required with an absorption band that matches the peak solar spectral power range. As well as pump absorption,

*G*incorporates the coupling of energy to the seed laser signal beam through stimulated emission. Hence, it contains the information about the absorption and stimulated emission cross sections, and luminescent lifetimes of the luminescent materials. As such, all the fundamental parameters describing the stimulated emission of this system as well as any intrinsic thermodynamic considerations are incorporated into the

_{opt}*G*parameter. It is a function of the pump wavelength, signal wavelength, and concentration of the molecules, and can be determined from pumped/unpumped gain measurement experiments [28]. The luminescent materials should, of course, be chosen to have high optical gain.

_{opt}It should be noted that for the realization of such a device using a two level system, applicable to most dyes, was considered in detail in the context of solar powered lasers by Roxlo and Yablonovitch [30]. They found two constraints imposed by thermodynamics: the Stokes shift must be greater than 13.3*kT* and the geometrical aspect ratio (i.e. path length of the signal/thickness of the concentrator) must be greater than exp(13.3). These requirements must be satisfied in order to achieve net gain, i.e. for *η _{t} G_{opt}* > 1, considering reabsorption loss only (a best-case scenario). The first condition restricts the seed laser wavelength, in that it must have sufficiently low reabsorption so as to allow for net gain under solar pumping conditions, in the absence of additional losses. The second condition is straightforward to fulfil by increasing the propagation length of the seed laser beam i.e. increasing the size of the device. In addition, to meet the minimum Stokes shift requirement, the long wavelength edge of the absorption band must be at least 13.3

*kT*shorter than the chosen signal wavelength. Thus, only that fraction of the solar spectrum with wavelength shorter than this will be absorbed. This is reflected in the value of

*G*, as it incorporates absorbed pump power.

_{opt}Whilst the model we present is a general one, to illustrate its utility and the potential of the s-LSCs concept we present the following example analysis based on a specific material combination, and on an assumption of constant optical gain, independent of signal power. We first consider a small area s-LSC system with a PMMA slab 76 × 20 × 1 mm in size doped with a dye, whose absorption and emission band are adjustable, that is illuminated by the standard AM 1.5G (i.e 1 KW/m^{2}) solar spectrum. This specific dimension, which represents a small area device, is mainly chosen to be able to compare the extracted power of a s-LSC system with a conventional LSC [31] with such dimensions. A 1 mW, 650 nm laser diode with 10% efficiency (i.e. *η _{L}* = 0.1) and 1 mm

^{2}spot size is considered as the seed source due to the availability of such lasers and the low loss of PMMA at this wavelength [32]. To satisfy the required minimum Stokes shift, the absorption band must extend only up to 550 nm, meaning only 25% of the standard solar spectrum will be absorbed. The seed laser is launched to sweep out the whole area of the concentrator and thus travels a ~1.52 m path before reaching the PV cell. We take the loss of the PMMA to be 0.2 dB/m around 650 nm [32] and reflection loss to be 0.6 dB/m, considering 99% reflection from the reflective coating and 20 reflections over the total propagation distance; this gives

*η*~0.76. We consider reabsorption losses to be smallenough to not contribute to

_{t}*η*. A 1 mm

_{t}^{2}PV cell with linear response (

*G*= 1) and 50% efficiency [33] (i.e.

_{eff}*η*= 0.5) at the signal wavelength is attached to the concentrator with appropriate index matching so that

_{3}*η*= 1. Thus, the optical gain required to reach the break-even threshold calculated from Eq. (9) is 0.0934 dB/cm, extracting 9.6 mW of electrical power from the system to drive the seed laser (with the remaining 0.4 mW from the seed laser itself). Note that this gain value should be realized with 25% of standard solar pump intensity.

_{cp}This model can also calculate the extracted power from a conventional LSC by setting *P _{SO}* = 0 in Eq. (9). The efficiency of the 76 × 20 × 1 mm conventional LSC is taken as

*η*= 13%, the maximum efficiency reported in [31]. The spontaneous emission efficiency (

_{sp}*η*) presented here is generally defined as the optical efficiency (

_{sp}*η*) in the context of LSC literature and can be represented as

_{opt}*η*=

_{opt}*η*(1-

_{LHE}η_{SA}η_{yield}η_{Stokes}η_{trap}η_{mat}*R*) [2,31], which includes the light harvesting efficiency (

*η*), the self-absorption (

_{LHE}*η*), the luminescence quantum yield of the dye (

_{SA}*η*), the Stokes efficiency (

_{yield}*η*), the trapping efficiency of the photons by the concentrator (

_{Stokes}*η*), the photon transport efficiency in the LSC material (

_{trap}*η*), and

_{mat}*R*is the Fresnel reflection coefficient of the LSC surface. Thus all the information is incorporated into the optical efficiency in standard LSC. Figure 4(a) shows the comparison of the total electrical power extracted from a s-LSC system with a 1 mm

^{2}PV cell and the conventional LSC having all edges equipped with identical PV cells, amounting to a total PV cell area of 192 mm

^{2}. The same amount of power is extracted by the s-LSC as the conventional LSC for a gain of 0.1668 dB/cm. Thus, at this gain, the area of the PV cell in the s-LSC is reduced by 192 times compared to the conventional LSC for the same power output.

As a second comparison, we consider a large area LSC 400 × 400 × 1 mm in size, with *η _{sp}* = 14.3% and

*η*= 28% considering GaAs cell [2], giving an overall power conversion efficiency of 4%, based on literature values [2,34]. We consider the same seed laser as previously and a linear response 1 mm

_{3}^{2}PV cell with 28% efficiency. For this device dimension, the reflection loss is 0.08 dB/m considering 99% reflection from the reflective coating and 400 reflections over the total propagation distance of 160 m. The power extracted by the s-LSC is the same as the conventional LSC when the gain is 0.0055 dB/cm, as shown in Fig. 4(b), reducing the PV cell area by 1600 times. As noted, the above analysis is based on the assumption of constant gain but in practice the gain saturates at high signal power, which will affect the total power extracted. To place the gain values from the above comparisons in context, we refer to reported net gains of 0.0425 dB/cm in perylimide dye-doped PMMA, achieved with pump intensities equivalent to 250 W/m

^{2}[35], comparable to absorbing 25% of the standard solar spectrum as considered above. In comparison, the gain required here is modest and, as is clear from Fig. 4, the performance of the s-LSC improves rapidly with gain. Therefore the effect of gain saturation on the total extracted power can be managed with higher gain material systems for the same output power. The s-LSC design is beneficial particularly for large area LSC as it may accommodate lower gain material and comparatively less efficient PV cells.

The above analysis is an indicative example of the promising aspects of the s-LSC concept. However, to realize it requires an appropriate material system which satisfies the thermodynamics constraints. It should be noted that in addition to the above fundamental constraints, in practice most dyes are unable to support continuous wave (CW) lasing due to triplet state accumulation losses, however CW gain has been observed in dyes [35]. Other material choices (e.g. not two-level systems) may offer more flexibility in this respect.

The outline presented here is general and not restricted to any specific gain medium or other material system. The purpose of this work is to illustrate the concept of the s-LSC and build a mathematical model to identify different physical parameters required for the realization of such a system. This can inform new material research for realizing s-LSCs. With the appropriate gain medium, this s-LSC concept is a very promising alternative to the existing conventional LSC technology, thus reducing the cost of PV power.

## 4. Conclusion

We have described the operating principle of a s-LSC and presented a comprehensive mathematical model that identifies different physical parameters and their effect on the power conversion efficiency of the device. This model allows comparison with conventional LSCs and facilitates the design and optimization of the different parameters for the development of these devices, which offer great promise for improved solar power.

## Acknowledgment

This work was supported by the Australian Research Council, an International Postgraduate Research Scholarship, and an Australian Postgraduate Award.

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