Figure 1

A schematic of the geometry considered: a dielectric sphere of radius $a$, and a dielectric plate separated by a distance $d$ through a vacuum gap. The color corresponds to electric-field intensity near the surface of the sphere and the plate, using the parameters $a=1000\mathrm{nm}\mathrm{and}d=800\mathrm{nm}$ as a representative example. The field near the sphere is much greater than that near the plate, so in order to resolve the variation in intensity, the scales (see the color bars on the right) are not the same. In all the examples in this paper, the sphere and plate are both amorphous ${\mathrm{SiO}}_2$, the sphere is at temperature $T_1=321\mathrm{K}$, and the plate is at $T_2=300\mathrm{K}$.Reuse & Permissions

Figure 2

Typical contribution to the spectral energy flux as a function of $m$ and $l$ [$a=20\mu \mathrm{m}$, $d=100\mathrm{nm}$, $\hslash \omega =0.0605\mathrm{eV}$(corresponding to a surface phonon polariton resonance), ${\varepsilon }_1={\varepsilon }_2$ corresponds to amorphous ${\mathrm{SiO}}_2$].Reuse & Permissions

Figure 3

A comparison of our numerical results to the experiments in Ref. 14. (a) Our results do not agree with the proximity approximation. (b) The proximity approximation (and the experimental data from Ref. 14) can be scaled to match our results.Reuse & Permissions

Figure 4

Heat flux as a function of $a$ for fixed $d=100\mathrm{nm},1\mu \mathrm{m},\mathrm{and}10\mu \mathrm{m}$. Insets are calculated scattered electric-field intensities (located in the figure at the appropriate value of $a,d$). Each curve is normalized by the far-field heat flux $Q_{ff}$, which is by definition independent of $d$.Reuse & Permissions

Figure 5

Numerically exact heat flux compared with dipole and proximity approximations, as a function of $d$, with fixed $a=1\mu \mathrm{m}$.Reuse & Permissions

Figure 6

A phase-space diagram of the relative accuracy of four approximations to the sphere-plate heat flux. The grayscale corresponds to the error as where corresponds to the far-field, dipole, proximity, or ‘far-field adjusted’ proximity approximations, respectively. The minimum error (corresponding to white) is 3%. The maximum error (corresponding to black) is 450%. The images are rendered from a smoothed interpolation of 147 data points.Reuse & Permissions