The incident photons to the waveguide sides are captured by the attached PV cell(s) for the generation of electricity. The ratio between the incident power (P _in ) and the electrical power generated by the side-mounted PV cell (P _out ) is the power conversion efficiency of the LSC device, as shown in Eq. (1) P C E = P out P i n = J mpp V mpp P AM1.5G (1) in which J _mpp is the current density at maximum power point (MPP) of the cell, V _mpp the voltage at MPP, and P _AM1.5G the power of the incident AM1.5G spectrum (1000 W/m ² under standard test conditions (STC)). MPP of the cell equals the product J _mpp V _mpp . In order to determine MPP, a procedure is followed based on [27]. First, the short circuit density J _sc is calculated using the spectrally resolved photon flux S(λ) [in # m ⁻² s ⁻¹) incident on the PV cell and the external quantum efficiency of the PV cell (EQE _PV (λ)), as follows: J s c = q ∫ E Q E P V λ S λ d λ (2)

Second, an iterative procedure for V _mpp is performed using the diode equation at MPP (Eq. 3) and the derivative of the equation for power with respect to the applied voltage (Eq. (4)). Solving this transcendental equation yields V _mpp , J _mpp , and P. J mpp = J 0 ⁡ exp q V mpp A k T − 1 − J gen (3) d P d V mpp = d J mpp V mpp V mpp = 0 (4) with q as the elemental charge (C), k _b the Boltzmann constant (m ² kg s ⁻² K ⁻¹), T the ambient temperature (K), J ₀ the diode saturation current of the PV cell (A m ⁻²), and A the diode ideality factor. J ₀, EQE _PV (λ) and A are obtained from the manufacturer, in this paper [26].

In order to function as a window, the LSC needs to transmit sufficient visible light. Assessing transmitted visible light is generally done with the Average Visible Transmission (AVT) which is dependent on the photopic response of the human eye. Eq. (5) details the calculation of the AVT which uses the ratio of the solar photon flux (AM1.5G) (S(λ)), the transmission (T(λ)) and the photopic response (P(λ)), adapted from Lunt [16]. For window applications an AVT between 555% and 90% is generally considered acceptable [16]. A V T = ∫ T λ P λ S λ d λ ∫ P λ S λ d λ (5)

Besides the transmission in the visible spectrum, a more complete figure of merit for light is based on the potential to accurately render the color of objects. To quantify this, the 1976 Color Rendering Index (CRI) is used as defined by the International Commission on Illumination (CIE) [28], the general authority on colorimetry. This method is suggested by [3] and extensively explained in [23], and besides LSC research [29], also used in assessing color quality of glazed windows [17–20].

Generally, a CRI above 70 is considered good quality and above 95 of excellent quality. Note that [3, 21] both use the AM1.5G as a reference. The CIE however suggests to use the D65 spectrum and to change the reference spectrum depending on correlated color temperature (CCT) [28]. This method is also adopted by glazed window color assessments [17], but without dependence on the CCT. In this article the AM1.5G spectrum has been taken as the reference spectrum without dependence on the CCT or chromaticity correction.

When the two different optimisation settings have been run the PCE, CRI and other colorimetric values can be calculated using PVtrace by Farrell [24]. The ray trace algorithm uses 10⁶ emitted photons distributed like the AM1.5G spectrum up until the found optimal emission (280–1,050 nm). From a list of LSC devices provided by Roncali [12] an average size of 0.3 × 0.3 m² is used in the ray trace algorithm. The thickness is set at 0.5 cm resulting in a geometric gain of 15. The wavelength dependent absorption spectrum for the PMMA waveguide is taken from [30] with a wavelength independent index of refraction of 1.49. The coupled PV cell is modelled after the record efficiency cell produced by [26]. This amorphous silicon hetero junction cell uses interdigitated back contacts to avoid reflection from the front, ideal for coupling with LSC devices. The ideality factor is set at 1.0, corresponding to V _OC ≈ 0.7 V as found in their paper.

Figures 4A,B show the influence of λ _c and σ on CRI and photon absorption, respectively. Visible is the increase in CRI with increasing σ and the opposite trend for the photon absorption. This is to be expected since σ determines the width of the absorption gap (see Figure 3).

In Figure 4A showing the CRI, a triangle is visible with high CRI values starting from λ _c = 540 nm, which is the top of the photopic response curve and the wavelength most sensitive to the human eye. Photon absorption increases with decreasing σ and shows a similar effect across all shown values of λ _c . The AM1.5G spectrum does not have enough variation in the range 360–830 nm to influence the effect of λ _c on photon absorption significantly.

Non-convex optimisation algorithms are prone to finding local optima. Analysing Figures 4A,B shows that global optima should occur around λ _c = 540 nm and 100 ≤ σ ≤ 200. The decision variable A λ c is not taken into account in this graph since it would only increase the effect of σ. The supplementary Excel file shows the optimal λ _c , σ and A λ c for the two types of constraints together with all the calculated values. The root mean square error (RMSE) for the values as predicted by the optimisation algorithm and the values found by the ray trace algorithm are shown in Table 2 and show good coherence.

The found PCE by the ray trace algorithm decreases with increasing AVT and CRI as shown in Figure 5. Without constraining the CRI, the highest PCE (14.3%) is obtained for the lowest AVT-constraint of 55% as shown by the grey dots in Figure 5. As expected, the lowest PCE is found for an AVT of 90% and a CRI of 98, resulting in a PCE of 9.53%.

When constraining the CRI, the relationship between AVT and PCE is mostly linear, as visible in Figure 5. The linear relationship is due to the necessity of an equal color distribution for high CRI values. This results in λ _c being mostly equal per CRI constraint with A λ c increasing for lower AVT values.

With only a constraint on the AVT, the relationship between AVT and PCE becomes logarithmic as shown by the grey dots in Figure 5. The logarithmic relationship occurs as a result of the normal distribution of the photopic response curve. To allow for a linear (i.e. 5%) increase in AVT, an increasingly larger part of the visible spectrum needs to be transmitted.

When relating the CRI to the PCE as shown in Figure 6, a logarithmic relationship appears again. Higher CRI values need more light of all wavelengths since the CRI is based on an average of test colors. The resulting relationships thus show an increasing drop of PCE with an increase in CRI. Note that the values with CRI < 90 have D _uv values above the confidence threshold of 0.000 ± 0.006 to 0.003 ± 0.006 for solid state lighting [17], see the supplementary Excel for exact values.

Also shown in Figure 5 are the values found in Figure 3B in the paper by Lunt [16]. Lunt’s approach defines a visible transmission between 435–670 nm which assures a CRI of > 95 . Actual calculations of his practical limit PCE are however hard to find. Discrepancies may arise from a different definition of AVT and assumptions about PV devices and associated losses.

The resulting transmitted colors for the AVT without a CRI constraint and with a CRI constraint are shown in Figure 7. A zoom-in of both types of constraints is shown in Figures 8A,B. Visible for the AVT-only constraint is a green color between 560–570 nm. This is to expected since the photopic response of the human eye has its peak in the green spectrum.

For the AVT and CRI constraint, the values are close together and close to the AM1.5G spectrum color indicated by the blue cross in Figures 8A,B. A high ( > 70 ) CRI value will ensure a good balance between colors and thus be close to daylight. The CRI values are shown by the numbers in Figure 8B for the dots closest, but follow the same pattern for all AVT values.

For optimising photon absorption and calculating the resulting PCE, numerous simplifications have been made. Reflection losses have been ignored. Adding an anti-reflection coating to the front of the LSC will improve the efficiency by avoiding the 4% reflection from the front. The losses from the LSC-PV interface have been ignored as well, beside the losses from the EQE _PV as provided by the article by [26], thus improving the efficiency. Further studies can improve the simulated efficiencies by adding an anti-reflection coating and implementing LSC-PV interface reflections.

In this article, most of the color quality metrics have been adopted from [16, 23], and [3]. Color quality metrics are however highly subjective to environmental factors and using an individual factor is often not enough to provide accurate predictions [32].

Generally, the CRI is used for assessing color quality, but a host of other metrics exist such as the more developed Color Quality Scale (CQS) (based on 15 test colors), or metrics based on long term color memory (Memory CRI), preference index of skin (PS) and others. See [33] for a more elaborate explanation of each metric. These metrics are most often used for LED applications however, while LSCs are better to be compared to windows.

For window applications, color comfort research has mostly been focused on glazing applications [17, 19]. Some studies have been conducted based on questionnaires and simulated lighting in buildings [34–36] for which results are mostly related to transmission and not related to color quality metrics. Based on [17] and their suggestion to not make CRI a design criterium, a need for a new quantification approach for transmitted light in windows is clear.

The found absorption spectra for an optimal luminophore are theoretical only, and don’t adhere to fundamental thermodynamic principles. The found results will thus only serve as a benchmark for (tandem) LSC devices with specific transmission properties. Combinations of luminophores, in a tandem structure, may approach the found spectra however.

[37] in 1981 proposed the tandem LSC as a means of capturing larger fractions of the incoming solar spectrum. Especially with recent advancement in quantum dots as a luminophore, the size dependence of the absorption spectra provide possibilities of combining UV and (N)IR absorbing LSCs in a tandem structure and optimal band-gap coupling with existing PV cells. This principle has already been successfully demonstrated by [38] obtaining a PCE of 3.1%.