# Thermal Effect of the Back Radiation from Disk to Head after Laser Heating in HAMR
## 2 Numerical Analysis
### Spectral Heat Flux
Since only positive frequencies are considered in the Fourier decomposition of the time-dependent fields into frequency-dependent quantities, the spectral heat flux is given by the following:  
\[q(z,\omega)=\langle S_{z}(\mathbf{r},\omega)\rangle=4\times\frac{1}{2}\text{ Re}\Big{[}\Big{\langle}E_{x}H_{y}^{*}-E_{y}H_{x}^{*}\Big{\rangle}\Big{]} \tag{7}\]  
The ensemble average of the spatial correlation functions of the stochastic electric and magnetic current densities are given by the following [18; 25]:  
\[\mathbf{E}^{e}(\mathbf{r},\omega)=i\mu_{v}\mu_{s}\omega\int_{V}\overline{ \mathbf{G}}^{Ec}\left(\mathbf{r},\mathbf{r}^{{}^{\prime}},\omega\right) \cdot\mathbf{j}^{e}\left(\dot{\mathbf{r}},\omega\right)dV^{\prime} \tag{1}\]0  
where \(\hbar\omega/2\) represents for the term of vacuum fluctuation, i.e., the zero-point energy, the zero-point energy of two objects can be cancelled out; therefore, it can be ignored during the calculation. Substitution of the electric and magnetic fields Equations (1)-(4) and the ensemble average of the spatial correlation functions Equations (8)-(11) into heat flux Equation (7), Equation (7) can be reduced into the following:  
\[q(z,\omega)=\frac{2k_{b}^{2}\Theta(\omega,T)}{\pi}\text{Re}\left[ \begin{array} []{c}i\text{Im}(\varepsilon_{s})\mu_{s}\int_{V}dV^{\prime}\left(\overline{ \overline{\mathbf{G}}}_{xx}^{Er}\overline{\overline{\mathbf{G}}}_{ya}^{Hes}- \overline{\overline{\mathbf{G}}}_{ya}^{Er}\overline{\overline{\mathbf{G}}}_{ xx}^{Hes}\right)\\ -i\varepsilon_{s}\text{Im}(\mu_{s})\int_{V}dV^{\prime}\left(\overline{ \overline{\mathbf{G}}}_{xx}^{Em}\overline{\overline{\mathbf{G}}}_{ya}^{Hms}- \overline{\overline{\mathbf{G}}}_{ya}^{Em}\overline{\overline{\mathbf{G}}}_{ xx}^{Hms*}\right)\end{array} \right] \tag{13}\]  
where the subscript \(\alpha\) refers to all the orthogonal components, i.e., \(\alpha=x,y,z\) in Cartesian coordinates. \(k_{v}\) represents for the wavevector in vacuum, and there is a relationship \(k_{v}^{2}=\omega^{2}\mu_{v}\varepsilon_{v}\). To solve Equation (13), the dyadic correlation term \(\int_{V}dV^{\prime}\overline{\overline{\mathbf{G}}}_{ua}^{Ec}\overline{ \mathbf{G}}_{va}^{Hes}\) and \(\int_{V}dV^{\prime}\overline{\overline{\mathbf{G}}}_{ua}^{Em}\overline{ \mathbf{G}}_{va}^{Hes}\) need to be determined first. Here, the subscripts \(u,v=x,y\), \(u\
eq v\) and \(\alpha=x,y,z\) have been discussed before. Because of the transition of the Cartesian coordinate to the cylindrical coordinate, the subscripts hereafter stand for \(u,v=\rho,\theta\), \(u\
eq v\) and \(\alpha=\rho,\theta,z\). Therefore, the dyadic correlation term in DGFs of fields generated by the electric sources leads to the following:  
\[\int_{V}dV^{\prime}\overline{\overline{\mathbf{G}}}_{ua}^{Ec}\overline{ \mathbf{G}}_{va}^{Hes}=\int_{-\infty}^{\infty}\frac{d\mathbf{k}_{p}}{(2\pi)^{ 2}}\!\!\int_{z}\!dz^{\prime}g_{ua}^{Ec}g_{ua}^{He*}=\frac{1}{2\pi}\int_{0}^{ \infty}k_{\rho}dk_{\rho}\!\int_{z}\!dz^{\prime}g_{ua}^{Ec}g_{ua}^{He*} \tag{14}\]  
Similarly, the dyadic correlation term in DGFs of fields generated by the magnetic sources leads to the following:  
\[\int_{V}dV^{\prime}\overline{\overline{\mathbf{G}}}_{ua}^{Em}\overline{ \mathbf{G}}_{va}^{Hms*}=\frac{1}{2\pi}\!\int_{0}^{\infty}k_{\rho}dk_{\rho}\! \int_{z}\!dz^{\prime}g_{ua}^{Em}g_{ua}^{Hms*} \tag{15}\]  
The source object in Figure 2 is the film 1, so that take \(s=1\) into Equation (13). And then substitution of Equations (14) and (15) into Equation (13), the monochromatic radiative heat flux under cylindrical coordinate is given by the following:  
\[q(z,\omega)=\frac{k_{v}^{2}\Theta(\omega,T)}{\pi^{2}}\text{Re}\left[