Generalized coupled dipole method for thermal far-field radiation

We introduce a many-body theory for thermal far-field emission of dipolar dielectric and metallic nanoparticles in the vicinity of a substrate within the framework of fluctuational electrodynamics. Our theoretical model includes the possibility to define the temperatures of each nanoparticle, the substrate temperature, and the temperature of the background thermal radiation, separately. To demonstrate the versatility of our method, we apply it in an exemplary way by discussing the thermal radiation of four particle assemblies of SiC and Ag nanoparticles above a planar SiC and Ag substrate. Furthermore, we use discrete dipole approximation to determine the thermal emission of a spherical nanoparticle in free space and close to a substrate. Finally, the calculation of the thermal far-field radiation of a sharp Si tip close to a SiC substrate using the discrete dipole approximation including the near-field scattering by the tip as well as the thermal emission of the tip and the contribution of the substrate which is partially blocked by the tip serves as another example.

Figure 1

Sketch of a DDA of a macroscopic object by $N$ voxels (a) and of two spherical nanoparticles differing in size and material (b) each emitting heat radiation into the far field. The substrate, the different nanoparticles, and the background radiation have, in general, different temperatures. The radiation is gathered at the detection plane lying parallel to the substrate. The assemblies are at edge-to-edge-distance $d$ to the substrate.

Figure 2

$P_{\mathrm{dir}}$ normalized to the exact result $P_{\mathrm{dir}}^{\mathrm{Mie}}$ in Ref. [86] for a (a) SiC nanoparticle at the localized surface mode frequency $\omega =1.787\times {10}^{14}\mathrm{rad}/s$ and (b) Ag nanoparticle at $\omega =1.0\times {10}^{14}\mathrm{rad}/s$.

Figure 3

$P_{\mathrm{dir}}$ and its compound for four nanoparticles made of SiC or Ag of radius $R=10\mathrm{nm}$ positioned at the corners of a square (as depicted in the inset) which is in a plane parallel to a SiC substrate at the distance $d=52\mathrm{nm}$ (substrate, nanoparticle's center). The temperature of the two nanoparticles on the left is $350\mathrm{K}$ and those of the two particles on the right is $310\mathrm{K}$, the background and substrate have the same temperature $T_{\mathrm{s}}=T_{\mathrm{b}}=300\mathrm{K}$.

Figure 4

$P_{\mathrm{dir}}$ like in Fig. 3 but for a silver substrate.

Figure 5

Sketch of the sphere used in our calculations approximated by 2533 voxels. We scale side length $L$ in such a way that our fixed number of voxels fits the radius of the sphere.

Figure 6

$P_{\mathrm{E},\mathrm{dir}}$ for a ${\mathrm{SiO}}_2$ sphere at $T_{\mathrm{p}}=700\mathrm{K}$ in a vacuum background with $T_{\mathrm{b}}=300\mathrm{K}$ for S-CDM. (a) Depicts the ${\mathrm{SiO}}_2$ spectrum for different radii using a DDA model for the sphere with $N=2553$ voxels (solid lines) and the full Mie result (open symbols). (b) DDA spectra when adding a SiC substrate at $T_{\mathrm{s}}=300$ K for different distances $d$ between substrate and sphere (solid lines) compared to the single-particle solution (open symbols; dashed lines in the inset). The inset magnifies the spectra within the reststrahlen band of SiC.

Figure 7

$P_{\mathrm{H},\mathrm{dir}}$ for a ${\mathrm{SiO}}_2$ sphere for identical parameters like in Fig. 6 with radius $R=30$ nm. The exact Mie result (solid red line) is compared with the single dipole result (open red symbols), the DDA result for 653 spherical nanoparticles (solid blue line), and the exact Mie result multiplied by $653/{13}^5$ (open blue symbols). Inset: Sketch of the sphere used for the DDA.

Figure 8

Sketch of the sharp tip used in our calculation. The foremost part of length 126.35 nm is approximated by 3904 voxels with side length 3.61 nm and the remaining part of length 2485.4 nm by 1484 voxels of side length 57.8 nm.

Figure 9

$P_{\mathrm{E},\mathrm{abs}}+P_{\mathrm{s}}$ (a), $P_{\mathrm{E}}$ (b), and $P_{\mathrm{E},\mathrm{scat}}$ (c) for a Si tip, small sphere with $R=63.2$ nm, and large sphere with $R=1.306\mu \mathrm{m}$ at $T_{\mathrm{p}}=0\mathrm{K}$ in a vacuum background with $T_{\mathrm{b}}=0\mathrm{K}$ close to a SiC substrate with a gap distance of $10\mathrm{nm}$ with $T_{\mathrm{s}}=573\mathrm{K}$ using a DDA model (S-CDM) with $N=2553$ voxels. Additionally, in (c) we highlighted the three frequencies corresponding to the wave numbers $\nu$ mentioned in Ref. [64].

Figure 10

Comparing scattered power according to expressions from Eq. (77) of Ref. [64] (solid line) and to Eq. (76) (dashed line) for distance $d=10$ nm between substrate and nanoparticle's surface, radius $R=3.12\mu \mathrm{m}$, $T_p=T_b=0$ K, $T_s=573$ K, and $\vartheta ={45}^{\circ }$. Both graphs are normalized to the maximum value of the solid line.