1. Introduction

Concentrating sunlight onto a small solar cell can reduce costs in solar energy generation. Luminescent solar concentrators (LSC) concentrate both diffuse and direct solar radiation with no need for tracking. They consist of plates of a transparent matrix doped with luminescent materials such as organic dyes [1], rare earth ions [2] or semiconductor quantum dots [3]. The luminescent material absorbs incident light and emits it at longer wavelengths. Most of the emitted light is trapped inside the plate by total internal reflection (TIR). It is guided to the edges where solar cells convert the concentrated light into electricity, as illustrated in Fig. 1(a) .

 

Fig. 1 (a) Cross-section of a conventional luminescent solar concentrator (LSC) that consists of a macroscopic plate doped with luminescent material that absorbs incoming light and emits at longer wavelengths. Most of the emitted light is trapped inside the plate and guided to solar cells at the edges. (b) The concept of the photonic luminescent solar concentrator (PLSC), where the luminescent material is embedded in a photonic structure to improve light guiding to the edges by mitigating escape cone and reabsorption losses.

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LSCs were first investigated in the early 1980s [4, 5]. It was found that their performance suffers from several losses:

New materials and advances in technology led to recent interest in LSCs. Slooff et al. reported a power conversion efficiency of 7.1% for a system of two organic dyes in one Poly(methyl methacrylate) (PMMA) plate with GaAs cells on all four edges [6]. To reduce losses in the escape cone, LSCs with frequency-selective reflectors on top were proposed by Smestad et al. [7]. Such a reflector reflects emitted light while transmitting light that can be absorbed by the luminescent material. This concept has been investigated theoretically and experimentally by different groups [8–12]. Goldschmidt et al. obtained a relative gain in system efficiency of 20% with such a design [13]. Longer path lengths due to steeper angles, however, increase path length dependent losses such as reabsorption or scattering.

In this paper we present the photonic LSC (PLSC) concept [14] that mitigates the major shortcomings by embedding the luminescent material in a photonic structure, as shown in Fig. 1(b). We demonstrate this with Bragg stacks, which feature structure sizes in the range of the wavelength of light, i.e. much smaller than in conventional LSCs. Thus, the emission is subject to photonic effects. Already the very first papers on photonic crystals by Bykov [15] and Yablonovitch [16] reported on the influence of such structures on the spontaneous emission process.

The concept aims at restricting the emission to in-plane directions, so that light guiding towards the edges is strongly improved through shorter paths and by circumventing the TIR escape cone. Further, redistribution of emission due to inhibited emission in a photonic band gap (PBG) could be utilized to reduce reabsorption: suppressing the emission in the overlap region of the absorption and emission spectra increases the effective Stokes shift and reduces reabsorption losses. Third, the effect of stimulated emission might increase the PLQY of the emitting material.

These effects promise the design of large devices with superior efficiency. In this paper, we investigate the emission spectrum and light guiding characteristics by means of electromagnetic simulations.

2. Simulation method and setup

Conventional LSCs are typically simulated using ray-tracing methods [17, 18], as their macroscopic thickness is much larger than the wavelength of light. Structures such as the PLSC with typical dimensions in the range of the wavelength, however, lie in the regime of interference and thus require wave-optical methods. Therefore, the electromagnetic wave propagation in the PLSC was simulated by employing the finite-difference time-domain (FDTD) method, which numerically solves Maxwell’s equations [19, 20]. More precisely, the Meep implementation from MIT [21] was used. As the one-dimensional photonic structure used in this work features an in-plane symmetry, the simulation was reduced to a 2D problem. This simplification has to be considered, however, when discussing the results.

Figure 2 shows three different simulation setups that were investigated. First, a reference setup with a light source in a homogeneous medium (n = 1.5) is used to obtain the emission characteristics in such a reference case. The second setup comprises a slab of thickness t_s with refractive index n_s = 1.5 in air (n_o = 1) and corresponds to a thin LSC. The PLSC setup consists of a thin active layer (thickness t_s) sandwiched between photonic structures, surrounded by medium with n_o = 1.

 

Fig. 2 Sketch of the three simulation setups (not to scale): (a) the reference setup with homogeneous medium (n = 1.5), (b) the slab setup corresponding to a microscopic LSC and (c) the PLSC setup, comprising an emitting layer sandwiched between Bragg stacks. Detector planes at the edges and top and bottom surfaces keep track of the energy fluxes to obtain the total emitted flux and the relative amount of flux guided to the edges. The position of the point-dipole source was varied in the y-direction to study the position dependent emission.

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In this paper, we used common Bragg stacks with 20 bi-layers of alternating films with refractive indices n₁ = 1.5 and n₂ = 2 and thicknesses t_i = λ₀/4n_i (i = 1, 2), where λ₀ is the design wavelength. All media were modeled to be ideal without absorption and dispersion.

In all simulation setups, perfectly matched layer (PML) boundary conditions avoid reflections from the walls of the simulation cell. The emission of light was modeled using a point-dipole source with current density J(r, t) = J₀ e_z δ(r) f(t) to obtain isotropic emission in the 2D simulation plane. Here, J₀, e_z and δ(r) denote the current amplitude, the unit vector in z-direction (perpendicular to the simulation plane) and the Dirac delta function, respectively.

Two different time functions f(t) were used: a sinusoidal function to analyze the propagation of monochromatic light, and a Gaussian enveloped sinusoidal function that exhibits a Gaussian-shaped spectrum in the frequency domain. This broad band excitation pulse enables frequency-resolved analysis in a single simulation run.

Further, the light source was placed at different positions (s_x, s_y) = (0, –½ t_s… + ½ t_s) along a line in the y-direction inside the emitting layer of slab and PLSC, as shown in Fig. 2(c), to investigate the influence of the emitter position in both simulation setups.

Detector planes placed at the left and right sides keep track of the flux of electromagnetic energy through these edge faces. Detector planes near the top and bottom surfaces measure the flux escaping from the structure. The sum of both is the total flux emitted from the source. Spectral information was obtained through Fourier transformation of the fields as described in Ref. 21.

The spacing d between the surface detector planes and the actual interface was introduced to account for evanescent fields that are guided parallel to the surface. As this light could easily be utilized by an attached solar cell (that is slightly larger than the edge face), it is not counted as lost. Detailed investigation of the near-surface field distributions (not shown here) yield d = λ₀.

The simulation cell width was set to 128 λ₀ based on a convergence analysis. Errors in light guiding to the edges due to the microscopic box width are minimized for this width, while keeping the computational effort reasonable.

3. Results and discussion

Two major quantities were obtained in the simulations: first, the relative emission E_rel, defined as the ratio of the total flux emitted in the investigated setup to that in the reference simulation:

It reveals how the surrounding geometry influences the emission spectrum due to an altered local photon density of states. Second, the flux through the edge detector planes divided by the total flux defines the light guiding efficiency LGE of the investigated simulation setup:

Both quantities are functions of the wavelength and can be obtained in frequency-resolved simulations. They depend not only strongly on the simulation setup geometry but also significantly on the source position inside the emitting layer. This phenomenon is not covered by ray-optical considerations. The coherent nature of light in FDTD simulations, however, allows coupling to optical modes that exist in such waveguide structures.

3.1. Relative emission (E_rel)

The relative emission for the slab and the PLSC setup, both with emitting layer thickness t_s = 2λ₀/n_s (arbitrarily chosen), is shown in Fig. 3(a) . Simulations with different source positions (36 random s_y values between –½t_s and +½ t_s) were averaged for this graph. This averaging corresponds to a homogeneous distribution of luminescent emitters that might be realized in experiments.

 

Fig. 3 Relative emission E_rel of slab and PLSC (a) averaged over s_y and (c) as a function of source position s_y. The small variation in the slab case is caused by waveguide modes due to the wavelength-sized thickness. Similar effects are seen for the PLSC, however, the relative emission in this case is dominated by suppression inside the photonic band gap (PBG). The band structure of an ideal (i.e. infinite) Bragg stack along its density of states, that is zero inside the PBG, is shown in (b) (calculated with the MPB Package [22]).

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In the case of the slab, E_rel varies weakly with frequency around the value of one, i.e. the emission spectrum of an emitter in the slab resembles the emission spectrum in a homogenous medium. The small variation, however, is caused by modes existing in this slab with wavelength-sized thickness.

This becomes clear when studying E_rel as a function of the source position. Figure 3(c) shows that, in specific positions, more energy is emitted than in others. This position dependent emission is due to the coupling of emission to modes in the slab. Each mode has a different intensity profile inside the layer which results in position dependent coupling strength to the modes. As the frequency increases, more positions with enhanced and suppressed emission are seen because the number of modes increases. The same trend is expected and observed for larger slab thicknesses t_s as well.

In the PLSC case, the photonic structure strongly alters the emission characteristics, especially near the design frequency f₀ = c/λ₀ of the Bragg stack (c denotes the speed of light). Inside the PBG of an ideal (i.e. infinite) Bragg stack, where the density of states is zero (see Fig. 3(b)), the emission is suppressed while outside the PBG E_rel is around one.

Cavity modes cause a peak in E_rel for frequencies slightly larger than f₀. The cavity modes depend on the thickness of the emitting layer (here, t_s = 2λ₀/n_s). Their existence inside the PBG is revealed in Fig. 3(c), where the influence of the source position s_y on E_rel for the PLSC is shown. The Bragg characteristic with suppressed emission inside the PBG is overlaid by a mode pattern similar to the one in the slab.

These results show how the surrounding geometry influences the emission behavior, as the local photon density of states is different than in homogeneous media. This can be utilized to suppress emission in unwanted spectral regions, e.g. in the overlap of absorption and emission to mitigate reabsorption losses. The specific characteristics can be tuned by the degrees of freedom in design, namely t_s and the number of Bragg layers, which is discussed in Section 3.3.

3.2. Light guiding efficiency (LGE)

The two-dimensional light guiding efficiency for both simulation setups (t_s = 2λ₀/n_s) is shown in Fig. 4(a) . It was averaged over different emitter positions s_y, as was done with E_rel above. The LGE of the slab lies around the expected value for 2D TIR guiding of 1 - θ_c/90° ≈53.5% (cf. 3D TIR: 74.5%) with critical angle θ_c = sin⁻¹(n_o/n_s) ≈41.8°. The weak variation with frequency is due to modes that exist in the slab. For thicker slabs and larger frequencies (smaller wavelength), the LGE becomes more and more independent of frequency and approaches the expected value of 53.5% for the macroscopic LSC.

 

Fig. 4 (a) Light guiding efficiency LGE of slab and PLSC simulation setup averaged over different source positions s_y. The LGE of the slab varies little around the expected value for 2D TIR due to coherence effects. For the PLSC, strongly enhanced light guiding is obtained for frequencies slightly larger than the design frequency f₀ due to the angular reflection characteristic of the Bragg stack shown in (b).

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In the PLSC case, the LGE is strongly enhanced near the design frequency f₀. For other frequencies, the LGE roughly approaches the 2D TIR limit. Maximum light guiding of up to 99.7% is obtained for frequencies slightly larger than f₀. This value corresponds to a loss of 0.3%, which is 155 times smaller than the loss from only TIR (46.5%). The shift of the LGE peak can be explained with the angular reflection characteristic of the Bragg stack shown in Fig. 4(b). Using a transfer matrix algorithm [23], the reflection was calculated for a single Bragg stack with 20 bi-layers surrounded with medium of n = 1 on one side and n = 1.5 on the other side (see graph inset). Light is incident from the medium with n = 1.5, in which the incident angle θ_i is defined. Thus total reflectivity due to TIR is obtained for θ_i > θ_c. While the Bragg reflection peak is centered for normal incidence (θ_i = 0), it shifts towards larger frequencies with increasing incident angles θ_i. The reflection for f = f₀ is therefore reduced at larger incident angles close to the critical angle. For slightly larger frequencies, high reflection is obtained even at large incident angles. Thus, optimum light guiding is observed for these frequencies.

Figure 5 shows the distribution of the energy density in the simulation cell for the three example frequencies that are indicated in Fig. 4. Monochromatic simulations with sinusoidal excitation were used to obtain the distributions once a steady state was reached. For all three frequencies, light guiding not only occurs in the emitting layer, but also in the photonic structure itself. Figure 5(a) shows the light distribution for f = f₀, where no light propagates in directions normal to the PLSC surface. Part of the energy, however, is lost at larger angles with reduced reflectivity.

 

Fig. 5 Energy density pattern obtained by monochromatic emission in the PLSC (t_s = 2λ₀) with (a) f = f₀, (b) f = 1.075 f₀, and (c) f = 0.75 f₀. While for f = f₀ perfect suppression is obtained in directions normal to the surface, the angle of guided light is larger for f = 1.075 f₀, satisfying the TIR condition and thus resulting in optimum LGE. For f = 1.075 f₀, light can also propagate in the escape cone, which results in reduced LGE in the range of 2D TIR.

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Maximum LGE is obtained for f = 1.075 f₀ as shown in Fig. 5(b): light is guided with a larger angle compared to f = f₀, which allows TIR at the surfaces. Propagation in the direction normal to the surfaces, however, is still suppressed sufficiently. Figure 5(c) shows the energy density distribution for an arbitrarily chosen frequency f = 0.75 f₀ that lies outside the LGE enhancement. In this case, light can propagate into the whole angular range of the escape cone, leading to a significant loss. TIR light guiding occurs for light at shallow angles, which results in a LGE of roughly the value obtained for 2D TIR (53.5%).

The influence of the emitter position on the LGE of both slab and PLSC is shown in Fig. 6 . As for the relative emission, modes in the structures create a pattern with enhanced and reduced LGE. For the PLSC setup, however, the Bragg characteristic clearly dominates the LGE pattern. Depending on the actual source position, the LGE enhancement decays differently towards smaller and larger frequencies. This behaviour can be tuned by geometry parameters such as t_s to allow adaption to a given emission spectrum of a luminescent material.

 

Fig. 6 Light guiding efficiency LGE of slab and PLSC setup as a function of source position s_y. Modes inside the slab cause deviations from the 2D TIR limit. For the PLSC, the LGE is dominated by the strong enhancement of the Bragg stack that overlays the mode pattern.

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3.3. Variation of design parameters

To optimize the performance of the PLSC, two design parameters were varied: the thickness t_s of the active layer in the PLSC and the number of bi-layers of the Bragg stack. In both cases, the relative emission E_rel and the light guiding efficiency LGE were evaluated by averaging over multiple simulations with different source positions s_y. As a first figure of merit, the mean value of LGE within the PBG was calculated. This value quantifies the LGE enhancement of the PLSC for a given design.

3.3.1 Number of Bragg bi-layers

The number of Bragg bi-layers was varied from one to 30, while the active layer thickness t_s was set to 2λ₀/n_s (as before). The results are shown in Fig. 7 . The characteristics of both E_rel and LGE emerge with increasing number of layers. Approaching 20 bi-layers, these quantities converge. The mean LGE inside the PBG also shows saturation, as depicted in Fig. 7(c). This investigation suggests that 20 bi-layers are sufficient to obtain the effects on emission and light guiding studied previously.

 

Fig. 7 The effect of the number of Bragg bi-layers on (a) the relative emission E_rel, (b) the light guiding efficiency LGE and (c) the mean LGE inside the PBG. Saturation is observed for more than 20 bi-layers.

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3.3.2 Active layer thickness (t_s)

The second design parameter, the active layer thickness t_s, was varied from 1/8 λ₀/n_s to 10 λ₀/n_s in 8 steps (using 20 Bragg bi-layers). The layer thickness has a significant impact on the relative emission, as shown in Fig. 8(a) : the distinctive E_rel characteristics with the suppression of emission inside the PBG become less pronounced for thicker layers. However, the light guiding efficiency depicted in Fig. 8(b) does not vary strongly with t_s. Evaluating the mean LGE in the PBG (Fig. 8(c)) does not reveal a significant trend either.

 

Fig. 8 Investigation of (a) relative emission, (b) light guiding efficiency and (c) mean LGE inside the PBG as a function of the active layer thickness t_s. This design parameter significantly influences the relative emission, whereas no strong impact on the light guiding is observed.

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For a more detailed analysis, a second figure of merit that combines the relative emission and the light guiding efficiency was obtained from these simulations. The idea is to qualitatively evaluate the effect of the active layer thickness with respect to the PLSC’s application: first, emission should be suppressed where the emission and absorption spectra overlap, assuming redistribution of emission to larger wavelengths, to reduce reabsorption. Given that absorption occurs at larger frequencies than emission, and that maximum suppression of emission is obtained at the upper band edge, we quantify this effect by the integral value of E_rel between the design frequency f₀ and the upper band edge (see integral region B indicated in Fig. 9(a) ).

 

Fig. 9 Combined qualitative evaluation of E_rel and LGE by calculating the ratio of the integral LGE in range A to E_rel integrated over range B. (a) shows example spectra of E_rel and LGE for t_s = ¼ λ₀/n_s with the integral ranges A and B. (b) plots the ratio of the integrals vs. the active layer thickness t_s. Thus, thin t_s are beneficial for the PLSC application.

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Second, the emitted light should be guided to the edges with high efficiency. Organic dyes, for instance, feature a rather broad emission spectrum. Additionally, emission suppressed at the upper band edge might be redistributed to lower frequencies. Thus, high efficiency in light guiding, even for lower frequencies outside the PBG, is important for the PLSC. The LGE was therefore integrated in a range as wide as the PBG from 0.823 f₀ to f₀ (indicated as “A” in Fig. 9(a)).

As large values of the LGE integral on the one hand, and small values for the E_rel integral on the other are beneficial for the PLSC application, the ratio of these two quantities was evaluated. Figure 9(b) shows this ratio as a function of the active layer thickness t_s. Here, this figure of merit is maximized for small thicknesses such as t_s = ¼ λ₀/n_s. With increasing thickness t_s, the ratio strongly decreases, converging for larger thickness values. This result suggests the design of rather thin active layers that are comparable to the layer thicknesses of the Bragg stack.

3.4. Discussion

As we propose a model for a practical device, the specific application needs to be discussed in more detail. While the light guiding efficiency is greatly enhanced at the upper band edge, the emission is suppressed in the same spectral region (see Fig. 9(a)). These two effects are linked so that highly efficient light guiding at certain frequencies and reduced emission at other frequencies cannot be achieved at the same time. This implies that the design of PLSC structures requires a trade-off of light guiding enhancement and reabsorption reduction.

The suppression of emission can be beneficial only if the emission is spectrally redistributed. Otherwise the energy is lost due to non-radiative processes. Redistribution to larger wavelengths is possible, when the emitter is able to emit at these lower frequencies. This can be obtained with luminescent materials that have broad emission spectra, like organic dyes. The Lumogen® Red dye, for instance, that was developed for the use in LSCs, features an emission spectrum that is ~66 nm wide (full width at half maximum) and has a slowly decaying tail towards longer wavelengths. Its absorption and emission spectra [24] are shown in Fig. 10(a) . In the same graph, the simulated relative emission of a PLSC (t_s=¼ λ₀/n_s) is plotted for λ₀ = 650nm. It can be seen, that the emission spectrum has a similar width as the PBG region. To achieve redistribution within the emission spectrum, it may be desirable to realize photonic structures with a narrower PBG region. In any case, the relative emission is only a first indicator for spectral redistribution. A complete description needs to include a detailed model of the emitting luminescent material and its interaction with the photonic structure. This will be investigated in future work.

 

Fig. 10 (a) The relative emission is plotted together with the absorption (Abs) and emission (Em) spectra of the organic dye Lumogen® Red (using λ₀ = 650nm). (b) shows the transmission T of the investigated Bragg stack from the outside to the luminescent layer along with the Lumogen® Red spectra. The reflection sidelobes in the absorption range cause severe losses which shows the need for photonic structures optimized for high transmission in the absorption range.

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Another important property of the photonic structure is its ability to transmit light in the absorption range of the luminescent material. In Fig. 10(b), the normal incidence transmission of the 20-bi-layer Bragg stack from the outside to the active layer, where light is absorbed, is plotted along with the Lumogen® Red spectra. Again, the design wavelength is chosen to be λ₀ = 650nm, so that the transmission edge is at the center of the absorption and emission overlap. It is obvious that such a Bragg stack would cause severe losses due to parasitic reflection of incident light in the absorption range. In this study, however, these Bragg stacks were used as a well-known model system to study the fundamental effects of light guiding enhancement and suppression of emission. With regard to the LSC application, the photonic structures need to be optimized for transmission in the absorption range of the luminescent material while achieving high light guiding efficiency for emitted light. This can be achieved with edge filters such as the one used in Ref. 13.

Additionally, more than one luminescent layer might be necessary to absorb the incoming light, depending on the absorption coefficient and layer thickness. Further, a more complex structure with different luminescent materials has to be considered to cover the whole relevant solar spectrum. In such a stack of PLSCs, each photonic structure has to be optimized for the specific emitter, while allowing transmission of incident light to the underlying PLSCs.

4. Conclusion

We investigated the concept of the photonic luminescent concentrator (PLSC) using two-dimensional FDTD simulations. While the emission in a slab LSC resembles the emission in homogeneous media, the photonic structure of the PLSC modifies the emission spectra by suppressing emission inside the PGB, which can be utilized to suppress reabsorption. Further, coupling to cavity modes is observed, which can be tuned by the active layer thickness.

The light guiding efficiency is of special interest in LCS applications: for the PLSC, it was found to be strongly enhanced in a broad spectral range. A maximum value of 99.7% (2D) was obtained for frequencies slightly larger than the design frequency of the Bragg stack, which can be explained with the angular reflection characteristic.

Investigating the number of bi-layers of the Bragg stack yields saturation for more than 20 bi-layers. The active layer thickness has a large influence on the emission spectrum but rather low impact on the light guiding efficiency. When both effects are taken into account, thin active layers as thick as the Bragg layers seem beneficial for the PLSC application.

Future work includes theoretical analysis of the interaction of luminescent emitters and the surrounding structure to investigate directional and spectral redistribution of emitted light. Additionally, optimized photonic structures need to be investigated, that avoid the shortcomings of the Bragg stack configuration such as parasitic reflection in the absorption range of the luminophore. Further, the presented PLSC structures will be investigated in experiments to validate the presented numerical calculations.