## 2 Theory of ADETAM
In the layer \(j\), the electric field amplitudes at the left and right boundaries are related by a layer matrix, which is written as  
\[\left[ \begin{array}{c}E_{jL}^{+s(p)}\\ E_{jL}^{-s(p)}\end{array} \right]=\Gamma_{j}^{s(p)}\left[ \begin{array}{c}E_{ jR}^{+s(p)}\\ E_{jR}^{-s(p)}\end{array} \right]=\left[ \begin{array}{cc}e^{-i\beta_{j}d_{j}} &0\\ 0&e^{i\beta_{j}d_{j}}\end{array} \right]\left[ \begin{array}{c}E_{jR}^{+s(p)} \\ E_{jR}^{-s(p)}\end{array} \right] \tag{2}\]  
where \(\beta_{j}=k_{0}\widetilde{n}_{j}\cos\theta_{j}\) is the propagation constant at the layer \(j\). For the multiple incoherent layers (\(v=1,\,\cdots,\,m\)), light propagation in each additional phase-shift layer can be expressed as  
\[ \begin{array}{ll}\text{P}_{v}=\left[\begin{array}{cc}e^{-i\theta_{v}}&0\\ 0&e^{i\theta_{v}}\end{array}\right]=\left[\begin{array}{cc}e^{-i\theta_{v} \cos\theta_{v}b_{v}}&0\\ 0&e^{i\theta_{v}\cos\theta_{v}b_{v}}\end{array}\right],\text{a}=1,\end{array}  \tag{3}\]  
where \(\theta_{v}=k_{0}n_{v}\cos\theta_{v}b_{v}(0\leq\theta_{v}\leq\pi)\) represents a random phase shift that light experience in the incoherent layer \(v\). If the number of equispaced phase shifts for each incoherent layer is set to be \(X\), the equispaced phase thickness \(b_{v}\) can be determined by  
\[ \begin{array}{ll}b_{v}^{q}=\frac{\lambda}{2n_{v}\cos\theta_{v}}\frac{q-1}{X},q=1,\,2,\,\cdots,\,X.\,\text{a}=1,\end{array}  \tag{4}\]  
Because of the assumption that the incoherent layers are transparent, the values of the refractive index \(n_{v}\) and refraction angle \(\theta_{v}\) are real, which results in a real-valued equispaced phase thickness \(b_{v}\). In addition, the equispaced phase thickness \(b_{v}\) are the same for both polarizations. In Figure 1, the light propagation through the incoherent layers are written as  
\[\left[ \begin{array}{c}E_{0R}^{+s(p)}\\ E_{0R}^{-s(p)}\end{array} \right] =\mathbf{I}_{01}^{s(p)}\mathbf{I}_{1}^{s(p)}\text{P}_{1}(\theta_{ 1})\mathbf{I}_{12}^{s(p)}\cdots\mathbf{I}_{(m-1)n}^{s(p)}\mathbf{L}_{m}^{s(p)} \text{P}_{m}(\theta_{m})\left[ \begin{array}{c}E_{mR}^{+s(p)}\\ E_{mR}^{-s(p)}\end{array} \right] \tag{5}\] \[=\mathbf{S}_{inc}^{s(p)}(\theta_{1}^{q_{1}},\theta_{2}^{q_{2}}, \cdots,\theta_{m}^{q_{m}})\left[ \begin{array}{c}E_{mR}^{+s(p)}\\ E_{mR}^{-s(p)}\end{array} \right]\]where \(S_{inc}^{s(p)}(\theta_{1}^{q1},\theta_{2}^{q2},\cdots,\theta_{m}^{qm})\) indicates the system transfer matrix for the whole incoherent multilayer, of which the matrix elements depend on a combination of the equispaced phase shifts of \(\theta_{1}^{q1}(q_{1}=1,\cdots,X)\),\(\cdots\), and \(\theta_{m}^{qm}(q_{m}=1,\cdots,X)\). In the remaining coherent multiple layers (\(j=m+1,\cdots,n\)), the electric field amplitude propagating from the right-hand boundary of the rightmost incoherent layer (\(j=m\)) to the left-hand boundary of the layer \(j\) is described by  
\[ \begin{split}\left[\begin{array}{c}E_{mR}^{+s(p)}\\ E_{mR}^{-s(p)}\end{array}\right]&=\mathbf{I}_{m(m+1)}^{s(p)}\mathbf{I}_{m+1}^ {s(p)}\mathbf{I}_{(m+1)(m+2)}^{s(p)}\cdots\mathbf{L}_{j-1}^{s(p)}\mathbf{I}_{( j-1)j}^{s(p)}\left[\begin{array}{c}E_{jL}^{+s(p)}\\ E_{jL}^{-s(p)}\end{array}\right],\\ &=\mathbf{S}_{mj}^{s(p)}\left[\begin{array}{c}E_{jL}^{+s(p)}\\ E_{jL}^{-s(p)}\end{array}\right].\end{split}  \tag{6}\]  
Then, the total system transfer matrix from the left ambient (\(j=0\)) to the layer \(j\) is written as  
\[