# Thermal Effect of the Back Radiation from Disk to Head after Laser Heating in HAMR
## 2 Numerical Analysis
### Spectral Heat Flux  
The numerical problem is to discuss the radiative heat flux flow into half space 3 from film 1, as shown in Figure 2. In fluctuational electrodynamics, the source of thermal radiation is modeled as stochastic electric and magnetic current densities \(\mathbf{j}^{e}\) and \(\mathbf{j}^{m}\), respectively. The electric and magnetic fields at point \(\mathbf{r}\) due to electric source \(\mathbf{j}^{e}\) located at \(\mathbf{r}\) inside a body of volume \(V\) are given by the following:  
\[\mathbf{E}^{e}(\mathbf{r},\omega)=i\mu_{v}\mu_{s}\omega\int_{V}\overline{ \mathbf{G}}^{Ec}\left(\mathbf{r},\mathbf{r}^{{}^{\prime}},\omega\right) \cdot\mathbf{j}^{e}\left(\dot{\mathbf{r}},\omega\right)dV^{\prime} \tag{1}\]  
Figure 1: Schematic diagram of an HAMR recording system. Figure 2: Simplified model of head/disk structure. \[\mathbf{H}^{e}(\mathbf{r},\omega)=\int_{V}\overline{\mathbf{G}}^{He}\Big{(} \mathbf{r},\mathbf{r}^{{}^{\prime}},\omega\Big{)}\,\mathbf{j}^{e}\Big{(} \mathbf{r}^{{}^{\prime}},\omega\Big{)}dV^{\prime} \tag{2}\]  
Similarly, the electric and magnetic fields at point \(\mathbf{r}\) due to magnetic source \(\mathbf{j}^{m}\) located at \(\mathbf{r}^{{}^{\prime}}\) inside a body of volume \(V\) are given by the following:  
\[\mathbf{E}^{m}(\mathbf{r},\omega)=-\int_{V}\overline{\mathbf{G}}^{Em}\Big{(} \mathbf{r},\mathbf{r}^{{}^{\prime}},\omega\Big{)}\,\mathbf{j}^{m}\Big{(} \mathbf{r}^{{}^{\prime}},\omega\Big{)}dV^{\prime} \tag{3}\]  
\[\mathbf{H}^{m}(\mathbf{r},\omega)=i\varepsilon_{v}\varepsilon_{v}\omega\int_ {V}\overline{\mathbf{G}}^{Hem}\Big{(}\mathbf{r},\mathbf{r}^{{}^{\prime}}, \omega\Big{)}\,\mathbf{j}^{m}\Big{(}\mathbf{r}^{{}^{\prime}},\omega\Big{)}dV ^{\prime} \tag{4}\]  
where \(\mu_{v}\) and \(\varepsilon_{v}\) represent for the permittivity and permeability in vacuum, generally \(\mu_{v}=4\pi\times 10^{-7}\) NA\({}^{-2}\) and \(\varepsilon_{v}=8.854187817\times 10^{-12}\) F/m, respectively. Similarly, \(\mu_{s}\) and \(\varepsilon_{s}\) represent for the permittivity and permeability of media \(s\), respectively. Terms \(\overline{\mathbf{G}}^{Ec}\big{(}\mathbf{r},\mathbf{r}^{{}^{\prime}},\omega \big{)}\), \(\overline{\mathbf{G}}^{Em}\big{(}\mathbf{r},\mathbf{r}^{{}^{\prime}},\omega \big{)}\), \(\overline{\mathbf{G}}^{He}\big{(}\mathbf{r},\mathbf{r}^{{}^{\prime}},\omega \big{)}\), and \(\overline{\mathbf{G}}^{Hem}\big{(}\mathbf{r},\mathbf{r}^{{}^{\prime}},\omega \big{)}\) are the electric and magnetic DGFs relating the fields with frequency \(\omega\) at point \(\mathbf{r}\) due to a source located at \(\mathbf{r}^{{}^{\prime}}\), respectively. The first \(E/H\) in the superscript of DGFs represents for the type of the field at point \(\mathbf{r}\), and the second \(e/m\) in the superscript of DGF represents for the type of the source at point \(\mathbf{r}^{{}^{\prime}}\). The total electric field at point \(\mathbf{r}\) due to sources located at \(\mathbf{r}^{{}^{\prime}}\) inside a body of volume \(V\) is the sum of two types of sources, i.e., the sum of Equations (1) and (3), namely, the total magnetic field. Generally, the heat flux of the electromagnetic fields can be calculated by the Poynting vector, which is the rate of energy flow per unit area in a plane electromagnetic wave. The electromagnetic fields radiated from a thermal object changes over time and can be represented by the Poynting vector. So that the instantaneous energy flow density, i.e., the instantaneous heat flux, is given by the following:  
\[\mathbf{S}=\mathbf{E}\times\mathbf{H} \tag{5}\]  
The monochromatic radiative heat flux is given by the time-averaged Poynting vector as follows:  
\[\mathbf{S}(\mathbf{r},\omega)=\frac{1}{2}\text{Re}\big{[}\mathbf{E}(\mathbf{ r},\omega)\times\mathbf{H}^{*}(\mathbf{r},\omega)\big{]} \tag{6}\]  
After ignoring the horizontal boundary, there is no energy transfer along the \(x\) and \(y\) directions in Cartesian coordinates, so that only the z-direction heat flux is considered.