## 2 Theory of ADETAM
\begin{split}\left[\begin{array}{c}E_{0R}^{+s(p)}\\ E_{0R}^{-s(p)}\end{array}\right]&=\mathbf{S}_{inc}^{s(p)}(\theta_{1}^{q1}, \theta_{2}^{q2},\cdots,\theta_{m}^{qm})\mathbf{S}_{mj}^{s(p)}\left[\begin{array} []{c}E_{jL}^{+s(p)}\\ E_{jL}^{-s(p)}\end{array}\right],\\ &=\mathbf{S}_{0j}^{s(p)}(\theta_{1}^{q1},\theta_{2}^{q2},\cdots,\theta_{m}^{qm })\left[\begin{array}{c}E_{jL}^{+s(p)}\\ E_{jL}^{-s(p)}\end{array}\right],\\ &=\left[\begin{array}{c}S_{01}^{01,s(p)}(\theta_{1}^{q1},\theta_{2}^{q2}, \cdots,\theta_{m}^{qm})\;\;S_{0j}^{12,s(p)}(\theta_{1}^{q1},\theta_{2}^{q2}, \cdots,\theta_{m}^{qm})\\ S_{0j}^{21,s(p)}(\theta_{1}^{q1},\theta_{2}^{q2},\cdots,\theta_{m}^{qm})\;\;S_{0j }^{22,s(p)}(\theta_{1}^{q1},\theta_{2}^{q2},\cdots,\theta_{m}^{qm})\end{array} \right]\left[\begin{array}{c}E_{jL}^{+s(p)}\\ E_{jL}^{-s(p)}\end{array}\right].\end{split}  \tag{7}\]  
Based on the matrix elements in Equation (7), the front transmission coefficient propagating from the layer \(0\) to the layer \(j\) is determined by  
\[ \begin{split} t_{0j}^{s(p)}(\theta_{1}^{q1},\theta_{2}^{q2}, \cdots,\theta_{m}^{qm})=\frac{E_{jL}^{+s(p)}}{E_{0R}^{+s(p)}}=\frac{1}{S_{0j}^ {11,s(p)}(\theta_{1}^{q1},\theta_{2}^{q2},\cdots,\theta_{m}^{qm})}.\end{split}  \tag{8}\]  
Similarly, the back reflection coefficient propagating from the layer j to the layer \(0\) is given by  
\[ \begin{split} r_{j0}^{s(p)}(\theta_{1}^{q1},\theta_{2}^{q2}, \cdots,\theta_{m}^{qm})=\frac{E_{jL}^{+s(p)}}{E_{jL}^{-s(p)}}=-\frac{S_{0j}^{12,s(p)}(\theta_{1}^{q1},\theta_{2}^{q2},\cdots,\theta_{m}^{qm})}{S_{0j}^{11,s(p )}(\theta_{1}^{q1},\theta_{2}^{q2},\cdots,\theta_{m}^{qm})}.\end{split}  \tag{9}\]  
When we focus on the absorption characteristics in the layer \(j\) within the coherent multilayer, the electric field intensity at the position z (\(0\leq z\leq d_{j}\)) can expressed as [4]  
\[ \begin{split}\left|\mathbf{E}_{j}^{s(p)}(z;\theta_{1}^{q1}, \quad\theta_{2}^{q2},\cdots,\theta_{m}^{qm})\right|^{2}\\ &=\left|\frac{\int_{0}^{s(p)}(\theta_{1}^{q1},\theta_{2}^{q2}, \cdots,\theta_{m}^{qm})}{1+r_{j0}^{s(p)}(\theta_{1}^{q1},\theta_{2}^{q2}, \cdots,\theta_{m}^{qm})e^{2\beta\beta^{d}_{j}j}}\right|^{2}\left|e^{i\beta jz}+ r_{j(n+1)}^{s(p)}e^{-i\beta jz}\right|^{2}\left|\mathbf{E}_{0R}^{+s(p)} \right|^{2}.\end{split}  \tag{10}\]  
Here, \(r_{j(n+1)}^{s(p)}\) is the reflection coefficient propagating from the coherent layer \(j\) to the right-hand ambient layer \(n+1\), which can be determined similarly to Equation (9) except the absence of the equispaced phase shifts. Correspondingly, the spatial distribution of light absorption within the layer \(j\) is calculated as [4; 15]  
\[ \begin{split} Q_{j}^{s(p)}(z;\theta_{1}^{q1},\theta_{2}^{q2}, \cdots,\theta_{m}^{qm})=\frac{1}{2}c\varepsilon_{0}\eta_{j}\kappa_{j}\Big{|} \mathbf{E}_{j}^{s(p)}(z;\theta_{1}^{q1},\theta_{2}^{q2},\cdots,\theta_{m}^{qm })\Big{|}^{2},\end{split}  \tag{11}\]  
where \(c\) is the speed of light in free space, \(\varepsilon_{0}\) is the electric permittivity in free space, and \(\alpha_{j}=4\pi\kappa_{j}/\lambda\) is the absorption coefficient of the layer \(j\). In the ADETAM, the incoherent characteristics of light propagation within the incoherent multilayer (\(v=1\), \(\cdots\), \(m\)) can be considered by averaging coherent calculation results over a set of equispaced phase thicknesses. The ADETAM-applied electric field intensity is given by  
\[\Big{|}\overline{E}_{j}^{(p)}(z)\Big{|}^{2}=\frac{1}{X^{m}}\sum_{v=1}^{m}\sum_{ q_{v}=1}^{X}\Big{|}\mathbb{E}_{j}(z;\theta_{1}^{q_{1}},\cdots,\theta_{v}^{q_{v}}, \cdots,\theta_{m}^{q_{m}})\Big{|}^{2}, \tag{12}\]  
where \(X^{m}\) is the number of total coherent calculations required to include the incoherent effect of m multiple incoherent layers.