## 2 Design Method  
This design method, built on the theoretical model from [PERSON] et al. [23], aims to create uniform ceiling illumination that emulates a natural sky-like light source, aligning with the principles of HCL. The process for developing the AL-TIR optics LED luminaire is illustrated in Figure 1, outlining the objective of engineering a luminaire that replicates the near-ideal luminous intensity curve presented in [PERSON]'s study. The flowchart details the steps from the initial concept based on ideal luminous intensity curves through simulations and the comparisons of these curves with the ideal. Following successful simulation, the design considers room and installation parameters, leading to ceiling illumination simulation and uniformity comparisons. Positive outcomes in these areas advance to a significant assessment, ensuring the final design meets environmental integration and luminance effectiveness. In the lighting model, the intensity of light at various points along the \(x\)-axis on the ceiling is represented by two logistic functions. As depicted in Figure 2a, two extensive linear light sources are mounted on opposite sides of a lengthy ceiling at a height \(H\) below the ceiling surface. Due to the significant length of the ceiling relative to its width \(W\), our 3D model has been simplified into a 2D representation by adopting the concept of linear illuminance (measured in \(\mathrm{lm/m}\)) rather than the conventional area-based illuminance (measured in \(\mathrm{lx}=\mathrm{lm/m^{2}}\)). This approach allows for a more focused analysis of light distribution along the ceiling's length without the complexity of a full three-dimensional model. In the 2D model, as illustrated in Figure 2a, the linear illuminance along the ceiling line \(x\) is generated by light sources positioned on both the right (R) and left (L) sides. The expressions for the linear illuminance from these sources are defined as follows:  
Linear illuminance from the right light source:  
\[E_{R}(x)=\frac{E_{m}}{1+e^{-k(x-x_{0})}} \tag{1}\]  
Linear illuminance from the left light source:  
\[E_{L}(x)=\frac{E_{m}}{1+e^{k(x-x_{0})}} \tag{2}\]  
where \(E_{m}\) represents the maximum linear illuminance, \(k\) is the logistic growth rate, \(x\) denotes the position on the ceiling (in meters), and \(x_{0}\) is the midpoint of the ceiling's \(x\)-value (in meters). Importantly, at any position \(x\) along the ceiling, the sum of \(E_{R}(x)\) and \(E_{L}(x)\) remains equal to \(E_{m}\) as shown in Figure 2b. Figure 1: The design flow chart of the proposed AL–TIR LED luminaires. Figure 2: (**a**) Schematic diagram of the lighting system with two LED arrays mounted on the walls; (**b**) the linear illuminance \(E_{R}(x)\), \(E_{L}(x)\) along the ceiling line \(x\) calculated for various logistic growth rates \(k=1\), \(2\), \(3\), \(6\), with \(E_{m}=10\), \(H=0.3\) m, and \(W=4\) m.  
As the value of \(k\) increases, the distribution pattern sharpens, indicating that the illuminance decreases more rapidly beyond \(x_{0}\). A preliminary value of \(k=3\) has been chosen, with further discussion on this selection to follow. In this model, the angular luminous intensity distribution is defined as the ratio of the logistic intensity to the change in angle \(\Delta\theta\), where \(\theta\) is the angle between the light ray and the vertical axis, calculated as follows:  
\[\theta(x)\approx arctan\left(\frac{x}{H}\right) \tag{3}\]  
\[D(\theta)=\frac{E(x)}{\Delta\theta(x)}=\frac{E_{m}}{\left(1+e^{-k(x-x_{0})} \right)\Delta\theta(x)} \tag{4}\]  
\[\Delta\theta(x)\approx\frac{d\theta}{dx}\Delta x=\frac{d}{dx}arctan\left( \frac{x}{H}\right)\approx\frac{1}{H\left(1+\frac{x^{2}}{H^{2}}\right)}\Delta x \tag{5}\]  
where \(\Delta\theta\) is the change in angle; \(H\) is the height from the light source to the ceiling; and \(\Delta x\) is the change in horizontal distance along the ceiling. Equation (5) assumes \(\Delta x\) is small, allowing the approximation to hold.