# Thermal Effect of the Back Radiation from Disk to Head after Laser Heating in HAMR
## 2 Numerical Analysis
### Dyadic Green's Function  
The sources in the film generate two kinds of electromagnetic waves, emitting to the forward and backward directions. Meanwhile, resulting from multiple reflections between the upper and lower surfaces of the film, the electromagnetic waves travel both forward and backward in the end. So that the electromagnetic waves consist of four parts, i.e., the forward travelling waves arising from a source emitting in the forward direction, the backward travelling waves arising from a source emitting in the forward direction, the forward travelling waves arising from a source emitting in the backward direction, and the backward travelling waves arising from a source emitting in the backward direction, as shown in Figure 3.  
DGF between source \(\mathbf{r}^{\prime}\) located in the magnetic recording layer film 1 and the target position \(\mathbf{r}\) located in magnetic head half space 3 is given by the following:  
\[\overline{\mathbf{G}}^{E\epsilon}\Big{(}\mathbf{r},\mathbf{r}^{\prime},\omega \Big{)}=\frac{i}{8\pi^{2}}\int\frac{d\mathbf{k}_{\mathbf{p}}}{k_{z1}}e^{i \mathbf{k}_{\mathbf{p}^{\prime}}(\cdot\mathbf{R}-\mathbf{R}^{\prime})}\left[ \begin{array}{l}\left(\mathbf{i}_{13}^{s++}\mathbf{\hat{s}}\mathbf{\hat{s}}+ \mathbf{i}_{13}^{p+}\mathbf{\hat{p}}_{3}^{\ast}\mathbf{\hat{p}}_{1}^{\ast} \right)e^{i\mathbf{k}_{z3}(z-z_{3})-ik_{z1}z^{\prime}}\\ +\left(\mathbf{i}_{13}^{s+}\mathbf{\hat{s}}\mathbf{\hat{s}}+\mathbf{i}_{13}^{p -}\mathbf{\hat{p}}_{3}^{\ast}\mathbf{\hat{p}}_{1}^{\ast}\right)e^{i\mathbf{k} _{z3}(z-z_{3})+ik_{z1}z^{\prime}}\\ +\left(\mathbf{i}_{13}^{s-}\mathbf{\hat{s}}\mathbf{\hat{s}}+\mathbf{i}_{13}^{p -}\mathbf{\hat{p}}_{3}^{\ast}\mathbf{\hat{p}}_{1}^{\ast}\right)e^{-i\mathbf{k} _{z3}(z-z_{3})+ik_{z1}z^{\prime}}\end{array}\right] \tag{17}\]  
where the term \(t_{sl}^{p+}\mathbf{\hat{p}}_{1}^{\ast}\mathbf{\hat{p}}_{s}^{\ast}\) stands for the direction changes of the polarized waves, for example, the term \(t_{13}^{p++}\mathbf{\hat{p}}_{3}^{\ast}\mathbf{\hat{p}}_{1}^{\ast}\) means the direction of p-polarized waves in film 1 is \(\mathbf{\hat{p}}_{1}^{\ast}\), and it changes to \(\mathbf{\hat{p}}_{3}^{\ast}\) in bulk 3, meanwhile the amplitude has a coefficient \(t_{13}^{p++}\), the superscript refers to the type of the electromagnetic, i.e., \(p++\) stands for the forward travelling waves arising from a source emitting in the forward direction. To make it clear, the amplitude coefficient \(t_{12}^{s,p++},t_{12}^{s,p+-},t_{12}^{s,p-+},t_{12}^{s,p--}\) can be represented as \(A^{s,p},B^{s,p},C^{s,p},D^{s,p}\). And substitution of the unit vectors of s-polarized and p-polarized into Equation (17), DGFs of the electric field due to electric sources are given by the following:  
\[\overline{\mathbf{G}}^{E\epsilon}\big{(}\mathbf{r},\mathbf{r}^{ \prime},\omega\big{)} =\frac{i}{8\pi^{2}}\int\frac{d\mathbf{k}_{\mathbf{p}}}{k_{z1}}e^{i \mathbf{k}_{\mathbf{p}^{\prime}}(\cdot\mathbf{R}-\mathbf{R}^{\prime})}\] (18) \[\times\left(\begin{array}{l}\frac{k_{z1}k  
flux Equation (16) and noting that the integral interval of the magnetic recording layer is \(z=[z_{1},z_{2}]=[0,d]\). Finally, the spectral heat flux is given by the following:  
\[q(z,\omega)=\frac{k_{\rm{p}}^{2}\Theta(\omega,T)}{\pi^{2}}\mathrm{Re}\left[ \begin{array}{c}i\mathrm{Im}(\epsilon_{1})\mu_{1}\int_{0}^{\infty}k_{\rho}dk_ {\rho}\int_{0}^{d}dz^{\prime}\left(\begin{array}{c}g_{\rho\rho}^{Ec}g_{\theta \rho}^{HRes}\\ -g_{\rho\rho}^{Ec}g_{\theta\rho}^{HRes}\\ +g_{\rho\rho}^{Ec}g_{\theta\rho}^{HRes}\end{array}\right)\\ -i\epsilon_{1}\mathrm{Im}(\mu_{1})\int_{0}^{\infty}k_{\rho}dk_{\rho}\int_{0}^{d }dz^{\prime}\left(\begin{array}{c}-g_{\theta\rho}^{Em}g_{\theta\rho}^{HRes} \\ +g_{\rho\rho}^{Em}g_{\theta\rho}^{HRes}\\ -g_{\theta\rho}^{Em}g_{\theta\rho}^{HRes}\end{array}\right)\end{array}\right] \tag{19}\]