## 2 Materials and Methods
### Light Source Mathematical Model
#### 2.1.2 LED Mathematical Model under Irradiance System  
The photometric unit used to establish the above model was light intensity. Although the conversion of light measurement and radiation quantity requires the use of the human eye, the human eye's viewing function and plant sensitivity curve are completely different, and thus the luminous flux and illuminance are not suitable for calculating and evaluating the lighting effect of plants. At this stage, there were two basic measurement methods used to evaluate the effects of the plant illumination-radiant energy method and photon number method. The radiant energy method directly indicates how much light energy is effectively used for photosynthesis, so the above light source illuminance model was modified to become an irradiance model in order to be used in plant lighting and uniformity design. Luminous flux is integral for the radiant flux in the visible light range in the human eye, which can be expressed by the following formula [27]:  
\[\phi=a\Delta K_{m}\sum_{380}^{780}N_{i}(\lambda)V_{i}(\lambda) \tag{6}\]  
\(K_{m}\) is the light power equivalent and its value under photopic vision is \(683 lm/W,a\) is the conversion coefficient between the relative value of the spectral power distribution and the actual value, \(N(\lambda)\) is the relative spectral distribution of the light source, \(V_{i}(\lambda)\) is the visual function of the human eye, and \(\phi\) is the luminous flux of the light source. \[P=\ a\Delta\lambda\sum_{400}^{700}N_{i}(\lambda) \tag{7}\]  
\[\phi=\ PK_{m}\frac{\sum_{380}^{780}N_{i}(\lambda)V_{i}(\lambda)}{\sum_{400}^{7 00}N_{i}(\lambda)} \tag{8}\]  
\(P\) is radiant power. For the LED lights of the same batch number, \(K_{m}\frac{\sum_{380}^{780}N_{i}(\lambda)V_{i}(\lambda)}{\sum_{400}^{700}N_{i} (\lambda)}\) is constant. For the convenience of expression, replace it with \(K_{\lambda}\). It can be concluded that the luminous flux and radiant power are in direct proportion with Formula (8). \(I_{0}\) on the optical axis is expressed by the luminous flux per unit solid angle on the optical axis, and the irradiance \(D_{0}\) on the optical axis can be expressed by the radiant power per unit area  
Figure 3: Laser and LED lighting model. on the optical axis. According to Formula (8) combined with the standard uniformity Lambertian distribution, the optical axis can be obtained from the irradiance \(D_{0}\) at the vertical distance \(z_{0}\) from the light source. The light intensity \(I_{0}\) on the optical axis should satisfy the following formula:  
\[\frac{I_{0}}{S_{0}}=D_{0}K_{\lambda} \tag{9}\]  
\(S_{0}\) is the area that corresponds to the unit solid angle when the vertical distance from the light source on the optical axis is \(S_{0}=\pi/z_{0}{}^{2}\). The LED in this article is in accordance with the standard Lambertian distribution, m = 1. Combined with Formula (1), the light intensity distribution based on the Lambertian illuminant is converted into irradiance distribution. The standard Lambertian distribution can be obtained from the light source vertical distance \(z\) to the irradiation plane. Irradiance can be expressed by the following formula:  
\[D(\theta)=D\pi/z^{2}K_{\lambda}cos\theta \tag{10}\]  
\(\theta\) is the angle between the light and the optical axis, \(D\) is the irradiance at z on the optical axis, and \(D(\theta)\) is the illuminance distribution on the plane when the vertical distance from the light source is \(z\). The irradiance generated by the LED light source chip placed at the spatial point \(\text{A}(X,Y,Z)\) at the receiving plane point \(\text{B}(x,y,z)\) can be expressed as:  
\[E_{1}(x,y,z)=\frac{(z-Z)D\pi K_{\lambda}}{\left[\left(\text{x}-\text{X} \right)^{2}+\left(y-Y\right)^{2}+\left(z-Z\right)^{2}\right]^{3/2}} \tag{11}\]  
Suppose the number of each LED light source is \(C\). The light source designed in this paper is installed on the same horizontal plane.