## 2 Theory
\[r_{e,-}^{\rm TE(TM)}=r_{j,0}^{\rm TE(TM)}=\frac{r_{j,-1}^{TE(TM)}+r_{j-1,0}^{TE(TM) }\exp\bigl{(}2 ik_{z,j-1}d_{j-1}\bigr{)}}{1+r_{j,j-1}^{TE(TM)}r_{j-1,0}^{TE(TM)} \exp\bigl{(}2 ik_{z,j-1}d_{j-1}\bigr{)}}, \tag{4}\]  
\[t_{e,+}^{\rm TE(TM)}=t_{j,m+1}^{\rm TE(TM)}=\frac{t_{j,j+1}^{TE(TM)}t_{j+1,m+1}^ {TE(TM)}\exp\bigl{(}ik_{z,j+1}d_{j+1}\bigr{)}}{1+r_{j,j+1}^{TE(TM)}r_{j+1,m+1}^ {TE(TM)}\exp\bigl{(}2 ik_{z,j+1}d_{j+1}\bigr{)}}, \tag{5}\]  
\[t_{e,-}^{\rm TE(TM)}=t_{j,0}^{\rm TE(TM)}=\frac{t_{j,j-1}^{TE(TM)}t_{j-1,0}^{TE (TM)}\exp\bigl{(}ik_{z,j-1}d_{j-1}\bigr{)}}{1+r_{j,j-1}^{TE(TM)}r_{j-1,0}^{TE( TM)}\exp\bigl{(}2 ik_{z,j-1}d_{j-1}\bigr{)}}. \tag{6}\]  
In order to derive the formula for the NF absorption in the MMC structure, we refer to a mathematical procedure that derives the electric field amplitude in a dielectric/metal interface system [18] and add new terms that consider two-beam and multi-beam interferences. According to the classical electromagnetism, the electric field of an electric oscillating dipole \(\mathbf{p}\exp(-i\omega t)\) can be expressed as a superposition of plane and evanescent waves [23; 24], where \(\omega\) is an angular frequency. Through the plane wave expansion, the electric fields in an infinite homogeneous medium and an MMC system can be written as  
\[\mathbf{E}_{\infty}^{\perp}(\mathbf{r})=\frac{iu_{0}\omega^{2}p}{8\pi^{2}} \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\frac{\sqrt{k_{x}^{2}+k_{y}^{ 2}}}{k_{e}k_{z,e}}\mathbf{e}_{e}^{\rm TM}\times\exp\bigl{[}i\bigl{(}k_{x}x+k_{ y}y+k_{z,e}(z-z_{ex})\bigr{)}\bigr{]}dk_{x}dk_{y}, \tag{7}\]  
\[\mathbf{E}_{\rm MMC}^{\perp}(\mathbf{r}) =\frac{iu_{0}\omega^{2}p}{8\pi^{2}}\int_{-\infty}^{+\infty}\int_{- \infty}^{+\infty}\frac{\sqrt{k_{x}^{2}+k_{y}^{2}}}{k_{e}k_{z,e}}\mathbf{e}_{e }^{\rm TM}\times\frac{\bigl{(}1+a_{+}^{\rm TM}\bigr{)}\bigl{(}1+a_{-}^{\rm TM} \bigr{)}}{1-a^{\rm TM}} \tag{8}\] \[\times\exp\bigl{[}i\bigl{(}k_{x}x+k_{y}y+k_{z,e}(z-z_{ex})\bigr{)} \bigr{]}dk_{x}dk_{y},\]  
\[\mathbf{E}_{\rm MMC}^{\parallel}(\mathbf{r}) =\frac{iu_{0}\omega^{2}p}{8\pi^{2}}\int_{-\infty}^{+\infty}\int_{- \infty}^{+\infty}\left( \begin{array}{c}-\frac{k_{x}}{k_{e}\sqrt{k_{x}^{2}+k _{y}^{2}}}\mathbf{e}_{e}^{\rm TM}+\frac{k_{y}}{k_{e}\sqrt{k_{x}^{2}+k_{y}^{2}} }\mathbf{e}_{e}^{\rm TE}\\ \phantom{-}\times\exp\bigl{[}i\bigl{(}k_{x}x+k_{y}y+k_{z,e}(z-z_{ex})\bigr{)} \bigr{]}dk_{x}dk_{y},\end{array} \right) \tag{10}\] \[\times\exp\bigl{[}i\bigl{(}k_{x}x+k_{y}y+k_{z,e}(z-z_{ex})\bigr{)} \bigr{]}dk_{x}dk_{y}.\]  
Here, \(\perp(\parallel)\) represents the vertical (horizontal) dipole orientation, and \(\mu_{0}\) is the magnetic permeability in free space. The term \(\mathbf{e}_{e}^{\rm TE(TM)}\) refers to basis vectors for TE and TM polarizations, which are  
\[\mathbf{e}_{e}^{\rm TE}=\frac{\mathbf{z}\times\mathbf{k}_{e}}{|\mathbf{z} \times\mathbf{k}_{e}|}=\left( \begin{array}{c}k_{y}\\ -k_{x}\\ 0\end{array} \right)\frac{1}{\sqrt{k_{x}^{2}+k_{y}^{2}}},\mathbf{e}_{e}^{\rm TM} =\frac{\mathbf{k}_{e}\times\mathbf{e}_{e}^{\rm TM}}{|\mathbf{k}_{e}\times \mathbf{e}_{e}^{\rm TM}|}=\left( \begin{array}{c}-k_{x}k_{z,e}\\ -k_{y}k_{z,e}\\ k_{x}^{2}+k_{y}^{2}\end{array} \right)\frac{1}{k_{e}\sqrt{k_{x}^{2}+k_{y}^{2}}}. \tag{11}\]  
In Equations (8) and (10), \(a_{+}^{\rm TE(TM)}=r_{e,+}^{\rm TE(TM)}\exp(2 ik_{z,e}z_{ex}),a_{-}^{\rm TE(TM)}=r_{e,-}^{\rm TE(TM)}\)\(\exp(2 ik_{z,e}(d_{e}-z_{ex}))\) are the reflection coefficients in reference to the location of the dipole emitter, and we have \(a^{\rm TE(TM)}=a_{+}^{\rm TE(TM)}a_{-}^{\rm TE(TM)}=r_{e,+}^{\rm TE(TM)}r_{e,-}^{ \rm TE(TM)}\exp(2 ik_{z,e}d_{e})\). We can derive the total radiated power from Equations (7)-(10). If we assume that the dipole emitter is immersed in a boundless EML, the total emitted power is expressed as [23]  
\[L_{e}=\frac{\omega}{2}\mathbf{p}\cdot\mathrm{Im}[\mathbf{E}_{\rm inf}(0,0,z_{ ex})]=\frac{ck_{e}^{4}}{12\pi\tilde{m}_{e}^{3}c_{0}}p^{2}, \tag{12}\]where \(\varepsilon_{0}\) and \(c\) are the electric permittivity and the speed of light in free space.