## 2 Materials and Methods
### Calculation of Photosynthetic Rate from Simulation Results  
Based on the ray-tracing simulation results, \(P_{n}\) was calculated by the absorbed PPFD and photosynthesis model. For the photosynthesis model, the modified Faquhar, von Caemmerer, and Berry (FvCB) model by [PERSON] [24] was used. To obtain FvCB model parameters, the photosynthetic rate was measured for the upper and lower canopies by a portable photosynthesis system (LI-6400, LI-COR, Lincoln, NE, USA) with 4 different CO\({}_{2}\) concentrations (100, 400, 800, and 1200 \(\upmu\)mol mol\({}^{-1}\)) and 8 different light intensities (0, 50, 100, 200, 400, 600, 900, and 1200 \(\upmu\)mol m\({}^{-2}\) s\({}^{-1}\)). The leaf temperature was set to 22 \({}^{\circ}\)C, and the relative humidity ranged from 60% to 70%.  
The dark respiration rate (\(R_{d}\)) was fixed at the measured photosynthetic rate under a PPFD of 0 \(\upmu\)mol m\({}^{-2}\) s\({}^{-1}\), and these were 0.75 and 0.41 \(\upmu\)mol CO\({}_{2}\) m\({}^{-2}\) s\({}^{-1}\) for the upper and lower canopies, respectively. The \(V_{cmax}\) and \(J_{max}\) values were obtained by nonlinear regression using measured leaf photosynthesis data and equations of Qian [24], and were 68.324 and 139.851 for the upper canopy and 46.423 and 52.898 for the lower canopy, respectively. The CO\({}_{2}\) compensation points (\(\Gamma^{*}\)) were obtained by nonlinear regression using measured leaf photosynthesis data and the following Equation (1):  
\[P_{leaf}=\frac{aC_{i}}{b+C_{i}}+c \tag{1}\]  
where \(P_{leaf}\) is measured leaf photosynthetic rate and \(C_{i}\) is intercellular CO\({}_{2}\) concentration. \(a\), \(b\), and \(c\) are equation coefficients. Resultingly, \(\Gamma^{*}\) values were 42.897 and 16.923 for the upper and lower canopies, respectively.  
The efficiency of light energy conversion (\(\alpha\)) and curvature value (\(\theta\)) were fixed at empirical values of 0.18 mol (electron) mol\({}^{-1}\) (photon) and 0.7, respectively [25,26].  
The simulation results included the point cloud of the 3D-SPM (x, y, and z coordinates) and the absorbed light energy (\(W\)) that was converted to absorbed PPFD by a conversion coefficient of 5.013, considering the spectral distribution of the LEDs used. The photosynthetic rate on the \(i\)-th point cloud (\(P_{i}\), \(\upmu\)mol m\({}^{-2}\) s\({}^{-1}\)) was calculated by Equation (2):  
\[P_{i}=\min\Bigl{(}A_{c,i},\ A_{j,i}\Bigr{)} \tag{2}\]  
where \(A_{c,i}\) and \(A_{j,i}\) are the net photosynthetic rate (\(\upmu\)mol m\({}^{-2}\) s\({}^{-1}\)) limited by rubisco activity and the electron transfer rate of the \(i\)-th point cloud, respectively.  
P\({}_{n}\) was calculated by Equation (3):  
\[P_{n}=\frac{\sum_{i=1}^{n}(P_{i}\times OA_{i})}{LA} \tag{3}\]  
where \(OA_{i}\) (m\({}^{-2}\)) is the occupied area of a point cloud and was normalized to \(1\times 10^{-6}\) m\({}^{-2}\) for calculation. \(n\) and \(LA\) indicate the total point number and total leaf area (m\({}^{-2}\)), respectively, and these values varied depending on the size of the 3D-SPM. \(P_{n}\) was calculated in various environments with three different light intensities (100, 200, and 300 \(\upmu\)mol m\({}^{-2}\) s\({}^{-1}\)), three different CO\({}_{2}\) concentrations (500, 600, and 700 \(\upmu\)mol mol\({}^{-1}\)) and two different planting densities (HD and LD) for comparison with the measured values.