## 2 Theory
In the case of the MMC system, the total radiation power is derived as  
\[L_{M\mathrm{MMC}}^{\perp}=L_{e}\frac{3}{4\pi}\mathrm{Re}\left[\int_{-\infty}^{+ \infty}\int_{-\infty}^{+\infty}\frac{k_{x}^{2}+k_{y}^{2}}{k_{e}^{3}k_{x,\varepsilon }}\frac{\left(1+a_{+}^{\mathrm{TM}}\right)\left(1+a_{-}^{\mathrm{TM}}\right)}{ 1-a^{\mathrm{TM}}}dk_{x}dk_{y}\right], \tag{13}\]  
\[L_{M\mathrm{MMC}}^{\parallel}=L_{e}\frac{3}{4\pi}\mathrm{Re}\left[\int_{- \infty}^{+\infty}\int_{-\infty}^{+\infty}\left\{\frac{k_{x,\varepsilon}}{k_{x }^{2}}\frac{\left(1-a_{+}^{\mathrm{TM}}\right)\left(1-a_{-}^{\mathrm{TM}} \right)}{1-a^{\mathrm{TM}}}+\frac{1}{k_{e}k_{x,\varepsilon}}\frac{\left(1+a_{ +}^{\mathrm{TE}}\right)\left(1+a_{-}^{\mathrm{TE}}\right)}{1-a^{\mathrm{TE}} }\right\}dk_{x}dk_{y}\right]. \tag{14}\]  
Dividing Equations (13) and (14) by \(L_{e}\) in Equation (12), we obtain the dimensionless relative radiated power \(F\), also known as the Purcell factor [12], which is given by  
\[F=\frac{\omega}{2L_{e}}\mathbf{p}\cdot\mathrm{Im}[\mathbf{E}(0,0,z_{ex})]= \int_{0}^{\infty}K(u)du^{2}. \tag{15}\]  
Here, we use the normalized in-plane wave vector \(u=\sqrt{k_{x}^{2}+k_{y}^{2}}/|\mathbf{k}_{e}|\) instead of \(k_{x}\) and \(k_{y}\), which is convenient to consider the projection of the wave vector on the x-y plane and is better for simple calculations [12; 14; 18]. The differential \(dk_{x}dk_{y}\) can be replaced by \(k_{x}^{2}udud\alpha\) in the spherical coordinate, where \(\alpha\) is the azimuth angle. If we assume that the dipole is randomly oriented in the horizontal direction, the radiation loses its azimuthal dependence. In Equation (15), the so-called spectral power density per normalized in-plane wave vector \(K(u)\), which is also dimensionless, can be expressed with respect to the polarization and dipole orientation as follows [12; 14; 22]. \[K_{\parallel}^{\mathrm{TE}}(u)=\frac{3}{8}\mathrm{Re}\left[\frac{1}{\sqrt{1-u ^{2}}}\frac{\left(1+a_{+}^{\mathrm{TE}}\right)\left(1+a_{-}^{\mathrm{TE}} \right)}{1-a^{\mathrm{TE}}}\right],\ K_{\perp}^{\mathrm{TE}}(u)=0, \tag{16}\]  
\[K_{\parallel}^{\mathrm{TM}}(u)=\frac{3}{8}\mathrm{Re}\left[\sqrt{1-u^{2}} \frac{\left(1-a_{+}^{\mathrm{TM}}\right)\left(1-a_{-}^{\mathrm{TM}}\right)}{ 1-a^{\mathrm{TM}}}\right],\ K_{\perp}^{\mathrm{TM}}(u)=\frac{3}{4}\mathrm{Re} \left[\frac{u^{2}}{\sqrt{1-u^{2}}}\frac{\left(1+a_{+}^{\mathrm{TM}}\right) \left(1+a_{-}^{\mathrm{TM}}\right)}{1-a^{\mathrm{TM}}}\right]. \tag{17}\]  
If the dipole orientation is isotropic, the spectral power density can be expressed as [21; 22]  
\[K_{isotropic}(u)=\frac{2}{3}K_{\parallel}^{\mathrm{TE}}(u)+\frac{2}{3}K_{ \parallel}^{\mathrm{TM}}(u)+\frac{1}{3}K_{\perp}^{\mathrm{TM}}(u). \tag{18}\]  
Let us derive an analytical formula of the spectral response caused by the NF absorption, which becomes noticeable when the normalized in-plane wave vector \(u\) is very large. If we assume \(u\) is large enough, the \(z\) component of the total wave numbers in the \((j-1)\)-th, \(j\)-th, and \((j+1)\)-th layers can be approximately expressed as  
\[k_{z,j-1}=k_{0}\widetilde{n}_{j-1}\sqrt{1-\left(n_{e}/\widetilde{n}_{j-1} \right)^{2}u^{2}}\simeq k_{0}\widetilde{n}_{j-1}\sqrt{-\left(n_{e}/\widetilde{ n}_{j-1}\right)^{2}u^{2}}=ik_{0}n_{e}u, \tag{19}\]  
\[k_{z,j}=k_{0}\widetilde{n}_{j}\sqrt{1-\left(n_{e}/\widetilde{n}_{j}\right)^{2 }u^{2}}\simeq k_{0}\widetilde{n}_{j}\sqrt{-\left(n_{e}/\widetilde{n}_{j} \right)^{2}u^{2}}=ik_{0}n_{e}u, \tag{20}\]  
\[k_{z,j+1}=k_{0}\widetilde{n}_{j+1}\sqrt{1-\left(n_{e}/\widetilde{n}_{j+1} \right)^{2}u^{2}}\simeq k_{0}\widetilde{n}_{j+1}\sqrt{-\left(n_{e}/\widetilde{ n}_{j+1}\right)^{2}u^{2}}=ik_{0}n_{e}u. \tag{21}\]  
Here, we use the relation of \(\left(n_{e}/\widetilde{n}_{j-1}\right)^{2}u^{2}>>1\), \(\left(n_{e}/\widetilde{n}_{j}\right)^{2}u^{2}>>1\), and \(\left(n_{e}/\widetilde{n}_{j+1}\right)^{2}u^{2}>>1\), which holds when it is assumed that \(u>>1\).