Ray-Trace Modeling to Characterize Efficiency of Unconventional Luminescent Solar Concentrator Geometries

Luminescent solar concentrators (LSCs) are a promising technology to help integrate solar cells into the built environment, as they are colorful, semitransparent, and can collect diffuse light. While LSCs have traditionally been cuboidal, in recent years, a variety of unconventional geometries have arisen, for example, circular, curved, polygonal, wedged, and leaf-shaped designs. These new designs can help reduce optical losses, facilitate incorporation into the built environment, or unlock new applications. However, as fabrication of complex geometries can be time- and resource-intensive, the ability to simulate the expected LSC performance prior to production would be highly advantageous. While a variety of software exists to model LSCs, it either cannot be applied to unconventional geometries, is not open-source, or is not tractable for most users. Therefore, here we introduce a significant upgrade of the widely used Monte Carlo ray-trace software pvtrace to include: (i) the capability to characterize unconventional geometries and improved relevance to standard measurement configurations; (ii) increased computational efficiency; and (iii) a graphical user interface (GUI) for ease-of-use. We first test these new features against data from the literature as well as experimental results from in-house fabricated LSCs, with agreement within 1% obtained for the simulated versus measured external photon efficiency. We then demonstrate the broad applicability of pvtrace by simulating 20 different unconventional geometries, including a variety of different shapes and manufacturing techniques. We show that pvtrace can be used to predict the optical efficiency of 3D-printed devices. The more versatile and accessible computational workflow afforded by our new features, coupled with 3D-printed prototypes, will enable rapid screening of more intricate LSC architectures, while reducing experimental waste. Our goal is that this accelerates sustainability-driven design in the LSC field, leading to higher optical efficiency or increased utility.


S3 Experimental characterization of LSCs
An Abet Technologies Class ABB solar simulator was used as the light source for optical efficiency measurements. It was fitted with an AM1.5G filter to mimic the spectrum of actual sunlight. ( Figure S2). The height of the solar simulator above the sample was first calibrated using a reference solar cell of known efficiency coupled to a Keithley 2401 sourcemeter such that the intensity of light on the sample was 1000±10 W/m 2 . The LSC was placed on a custom-designed sample holder ( Figure S3) and a single edge of the LSC was placed against to the input port to an INS125 integrating sphere (225-1400 nm, International Light Technologies), that was connected that was connected to a spectroradiometer (SpectriLight ILT 950) via a fiber optic cable. All measurements were performed on a black absorptive background. The parameters used to characterize the optical performance of LSCs are the internal photon efficiency, int and external photon efficiency, ext, defined by the following equations: 3 (Eq. S1) where $%&'(# is the total number of edge-emitted photons summed over all edges of the LSC, $%&)*+ is the total number of photons absorbed by the LSC, and $%&!" is the total number of photons incident on the top surface of the LSC. $%&'(# is obtained from the sum of the output power spectra, Pi(out)(λ), measured for each edge of the LSC (in W nm -1 ), where λ is the wavelength of light (in nm). Pin(λ) is the input power spectrum from the solar simulator incident on the top surface of the LSC (in W nm -1 ), h is Planck's constant (in J s), c is the speed of light (in m s -1 ), and A(λ) is the absorption spectrum of the LSC. The integrations are performed over the full AM1.5G solar spectrum (250-1050 nm).

S-6
S4 Parallelization to reduce computing time Figure S4: Effect of parallelization on simulation times for a circular LSC of 6 cm diameter and 0.32 cm thickness, using a rectangular light mask. (a) Completion time of pvtrace script for 1,000 and 10,000 rays run on a laptop, using either serial (non-parallel, 1 core) configuration or the 2 cores available. (b) Runtime for parallelized pvtrace on a single node in the CSD3 cluster, 4 for various numbers of cores. (c) Runtime for parallelized pvtrace on multiple nodes in the CSD3 cluster. Implements the Ray package for distributed computing across various nodes. Increasing cores decreases runtime to a certain extent, but overhead from various python packages increases minimum runtime.
S-7 Figure S4: (a) Normalized absorbance (red) and emission (blue) spectra of Lumogen Red 305 (LR305). These spectra are in-built to pvtrace. (b) Attenuation coefficient of LR305, which has been backcalculated from absorbance for various optical path lengths and LR305 concentrations using the Beer-Lambert Law: = where A is the absorbance, is the attenuation coefficient, is the concentration, and is the optical path length. The attenuation coefficient allows calculation of absorption coefficients based on the input luminophore concentration (e.g. = where is the attenuation coefficient). The attenuation coefficient used in pvtrace is shown in purple, and demonstrates agreement with experimental values. Note the lower absorption at high concentrations (500 ppm) is likely due to dye aggregation effects.

S5 Optical properties of Lumogen Red
S-8 Table S2: Input parameters used in pvtrace v2.1sv simulations (based on experimental data from the literature) for bulk square LSCs based on LR305 doped in poly(methyl methacrylate) (PMMA). We note that many of the studies did not specific the background absorption coefficients, and most of those that did, reported a constant value rather than a more realistic wavelength-dependent response. Instead of importing the spectrum used in the studies (which was not available), a Planck distribution (Equation S3) at 5800K was used for the shape of the light source spectrum. Because only 10,000 -100,000 rays are generated, a Planck distribution is sufficient to mimic solar radiation or a solar simulator. The Planck distribution is defined as:

Label
where is the spectral radiance, is the wavelength, is the temperature, ℎ is Planck's constant, and is the speed of light.    Table S6: Simulated (pvtrace v2.1sv) external photon efficiency for a variety of hypothetical bulk LSC designs using enclosing box or surface normal methods of ray counting. The sample numbers correspond to the bulk LSCs illustrated in Figure 8. For all parts the following conditions were applied: (i) top surface area of 6.25 cm 2 ; (ii) thickness of 0.21 cm; (iii) LR305 concentration = 500 ppm; (iv) refractive index = 1.5; (v) waveguide parasitic absorption of 0.525 cm -1 ; (vi) incident spectrum of 300-900 nm. The x-and y-dimensions of the enclosing box were set 101% of the actual x-and ydimensions of the LSC, while the z-dimension was 110% of the actual z-dimension of the LSC. S-12 Table S7: Input parameters used in pvtrace v2.1sv simulations (based on experimental data from this work) of 3D printed LSCs. The sample numbers correspond to the printed LSCs illustrated in Figure  11; sample preparation as described in Section 2. To mimic the experimental set-up, a 1.5 mm gap between the edge of the LSC and the edge of the enclosing box was used. The surface normal method of ray counting was also evaluated. The top surface area and geometric gain are the same as Table  S4.  We can also apply pvtrace v2.1.sv to extrapolate the optical performance of higher-quality parts. The 3D printed parts had much higher parasitic absorption from the waveguide, compared to the laser cut bulk parts. This was primarily due to the in-house filament preparation process, which resulted in bubbles being formed along some parts of the filament. With industrial filament preparation processes, higher-quality filament and therefore higher-quality parts could be fabricated. Reducing the waveguide parasitic absorption term in pvtrace shows the potential performance of 3D printed parts. Further, switching to the surface normal method provides an indication of the potential performance with an optimized measurement technique. The results of this extrapolation are shown in Table S9.