graphene layer (layer absorptance $A^N_j\equiv \sigma^N_j |E_j|^2$), $$\begin{aligned}
 A^N = \sum_{j=1}^N \sigma^N_j |E_j|^2.
 \label{eq:AN}\end{aligned}$$ The field configuration $E_j$ is locally determined ($j=1,\cdots,N$) by a given set of $\sigma^N_{j=1,\cdots,N}$ because once $c_0^r$ is known, $c^t_i$ and $c^r_i$ can be calculated by using Eq. (\[eq:tmatr\]) repeatedly. Therefore, $A^N$ is actually a nonlinear equation of $\sigma^N_j$.

The transfer matrix method is applicable to the bunched conductivity. [@Heavens1960] Because light propagates “freely” through the medium with the relative permittivity $$\begin{aligned}
 \varepsilon_N(\omega) \equiv 1 + i \frac{\sigma_N(\omega)}{\epsilon_0 \omega},\end{aligned}$$ we have $$\begin{aligned}
 \begin{pmatrix}
  E'_{N}(z_N) \cr B'_{N}(z_N)
 \end{pmatrix}
 =
\begin{pmatrix}
  \cos(\sqrt{\varepsilon_N} \frac{\omega (z_N-z_1)}{c}) & i \frac{c}{\sqrt{\varepsilon_N}} \sin(\sqrt{\varepsilon_N}\frac{\omega (z_N-z_1)}{c}) \cr
  \frac{i\sqrt{\varepsilon_N}}{c} \sin(\sqrt{\varepsilon_N}\frac{\omega (z_N-z_1)}{c})  & \cos(\sqrt{\varepsilon_N}\frac{\omega (z_N-z_1)}{c})
 \end{pmatrix}
 \begin{pmatrix}
  E_{1}(z_1) \cr B_{1}(z_1)
 \end{pmatrix},\end{aligned}$$ instead of Eq. (\[eq:ebmatrix\]). The $2\times 2$ matrix is given in Eq. (\[eq:To\]) by replacing $d$ with $z_N-z_1$ and $c$ with $c/\sqrt{\varepsilon_N}$. We assume that the electromagnetic fields are continuous at the interface between the air and the material. By eliminating $c^t_N$ from the above equation, we obtain $$\begin{aligned}
 c^r_0 = \frac{i\left(\sqrt{\varepsilon_N}- \frac{1}{\sqrt{\varepsilon_N}} \right)\sin\left(\sqrt{\varepsilon_N} \frac{\omega (z_N-z_1)}{c}\right)}{2\cos\left(\sqrt{\varepsilon_N} \frac{\omega (z_N-z_1)}{c}\right) - i \left(\sqrt{\varepsilon_N}+ \frac{1}{\sqrt{\varepsilon_N}} \right)\sin\left(\sqrt{\varepsilon_N} \frac{\omega (z_N-z_1)}{c}\right)}.
 \label{eq:cr0}\end{aligned}$$ From which we can calculate reflectance and transmittance. When $N$ is sufficiently large, this result reproduces the standard formula, $R=|(\sqrt{\varepsilon_N}-1)/(\sqrt{\varepsilon_N}+1)|^2$ 