## 2 Theoretical Framework
### Light Transmission through a Straight Light Guide  
Tubular light guides harvest the diffuse light and sunlight and transport the photons into building interiors through the tube's inner high-reflecting coating. Following the analytical model [25], the luminous flux crossing the circular interface at the light-tube base can be written as  
\[\Phi_{D}=\frac{1}{2}\int\limits_{r_{0}=0}^{R}r_{0}dr_{0}\int\limits_{\phi_{0}= 0}^{2\pi}d\phi_{0}\int\limits_{\theta=0}^{\pi/2}\sin(2\theta)d\theta\int \limits_{\varphi=0}^{2\pi}j(\theta,\varphi,\phi_{0},r_{0})d\varphi, \tag{7}\]  
where \(R\) is the radius of the circular diffuser embedded at the light-tube base, \(r_{0}\) is the radial distance measured from the center of the diffuser to an elementary area on its surface, \(\phi_{0}\) is the polar angle of the elementary area, and, \(\theta\) and \(\varphi\) are the apparent zenith and azimuth angles of a light beam that enters the elementary area from upwards, respectively. The luminance at the light-tube base \(j(\theta,\varphi,\phi_{0},r_{0})\) is proportional to the sky luminance reduced primarily due to the multiple reflections in a tube and attenuation in the hemispherical top dome. The illuminance of the elementary surface of the diffuser in local coordinates \(\phi_{0}\), \(r_{0}\) is  
\[E_{i}(\phi_{0},r_{0})=\int\limits_{\theta=0}^{\pi/2}\int\limits_{\varphi=0}^{2 \pi}j(\theta,\varphi,\phi_{0},r_{0})\cos\theta\ \sin\theta d\theta\ d\varphi. \tag{8}\]  
For a vertical straight pipe, the polar angle of a light beam \(\theta\) is conserved in the process of multiple reflections, i.e., the exit angle equals to the entrance angle. The illuminance below the diffuser-free light-guide can be determined easily for any point \(X^{\prime},Y^{\prime},Z^{\prime}\) in the work plane:  
\[\begin{split} E_{W}(X^{\prime},\ Y^{\prime},Z^{\prime})& =\frac{t_{D}}{2\pi}\int\limits_{r_{0}=0}^{R}r_{0}dr_{0}\int\limits_{\phi_{0}= 0}^{2\pi}\frac{\cos^{2}\Theta}{D^{2}(\Theta,\Phi,r_{0},\phi_{0})}d\phi_{0}\\ &\int\limits_{\theta=0}^{\pi/2}\sin(2\theta)d\theta\int\limits_{ \varphi=0}^{2\pi}j(\theta,\varphi,\phi_{0},r_{0})d\varphi,\end{split} \tag{9}\]  
where \(D\) is the distance between a point on the light tube base and arbitrary point at the work plane. The angles \(\Theta,\Phi\) are to characterize the direction of the beam propagation below the light-pipe.  
In previous equations, the luminance \(j(\theta,\varphi,\phi_{0},r_{0})\) is obtained as a function of the zenith and azimuth angles and also depends on the point in which the beam intersects the upper interface of a light-pipe. When the sunlight is absent, we can write  
\[j(\theta,\varphi,\phi_{0},r_{0})=L_{\varphi}\rho^{N}t_{C}, \tag{10}\]  
where the luminance of sky element \(L_{\alpha}\) can be computed from the UniSky model as a sum of the first-order and the second-order luminance components (see Equation (1)). The reflectance of a light-tube interior surface is characterized by \(\rho\) that is considered to be constant. The same applies to the transmission coefficient of a hemispherical cupola (\(t_{C}\)). However, the number of reflections \(N\) depends on the geometry of the light guide and incidence angle (for more details see [25]. The contribution from the direct sunlight to the luminance is  
\[j_{S}(\theta,\varphi,\phi_{0},r_{0})=\frac{L_{V,S}}{\cos z_{S}}\rho^{N_{sum}( \varphi)}t_{C}, \tag{11}\]  
while \(L_{V,S}\) is determined from Equation (6), and \(N_{sum}\) can be computed analogously to \(N\) assuming the zenith and azimuth angles coincide with the position of sun.