# Energy streamlines

The direction in near-field transfer cannot be determined by the wavevector as in the case of a propagating wave. From the wave point of view, phonon tunneling is through the coupling of evanescent waves since there exist a forward decaying and backward decaying waves in the vacuum gap, both with purely imaginary kz. In such a case, the Poynting vector represents the direction of energy flow and the trace if Poynting vectors gives the energy streamlines (ESLs), which can be used to elucidate the energy propagation like fluid flow [1]. Due to the random fluctuation of charges, the Poynting vectors are decoupled for different values of β [2,3]. The ESLs are laterally displaced as they leave the surface of the emitter and reach the surface of the receiver. This lateral displacement is called a lateral shift [2], which is different from the well-known Goos-Hänchen shift [1], may be important to determining the lateral dimension of the real system which can be modeled as infinite plates in near-field radiation.

Figure 9 shows the ESL projected to the x-z plane at λ = 10.55 μm and d = 100 nm for β = 40ω/c in all three media for p-polarized waves [3]. The magnitude of magnetic field is overlaid as depicted by the colored contours (i.e., the brighter color indicates the greater value). To calculate the magnetic field, thin-film optics is employed with an assumption that a plane wave is incident from medium 1. The emission originated deeper from the surface than the radiation penetration depth could not reach the SiC-vacuum interface. Hence, the field distribution is plotted in the vicinity of the vacuum gap. It can be seen from Fig. 1 that negative refraction of energy path occurs at the interfaces between SiC and vacuum due to the opposite sign of their dielectric functions. The energy streamlines are curved except for medium 3 where no backward waves exist. The magnetic field oscillates in the lateral direction as a result of the excitation of SPhPs.

Figure 1. Energy streamlines for TM waves in SiC-vacuum-SiC for . The magnitude of the magnetic field is denoted by colored contours and plotted along with the ESLs [3].

Basu et al. [4] applied fluctuational electrodynamics in multilayered structures to directly trace the energy streamlines not only in the gap and receiver but also in the emitter. It was found that when surface waves are excited, there is a larger lateral shift inside the emitter. Figure 10 shows the ESLs for combined TE and TM waves at d = 10 nm with different β values. Note that for propagating waves, β* = βc/ω < 1 and the shape of ESLs is independent of d in the proximity limit. For evanescent waves, TM waves dominate over TE waves. At the SPhP frequency, ESLs for the same βd value are essentially the same. The resonance conditions may be denoted by ωm and βm. When d is very small, ωm depends little on d, whereas βm is inversely proportional to d as mentioned previously. The value corresponds to $\beta _{m}^{*} =\beta _{m} c{/ \omega _{m} } =450$ when d = 10 nm. For propagating waves, all ESLs are located inside the conical surfaces bounded by the ESL at β = ω/c [2].

The ESLs inside the emitter and the vacuum gap are curved much more for evanescent waves than for propagating waves. Because it is assumed to be semi-infinite and no backward waves exist. Since the receiver is treated as non-emitting (i.e., at zero absolute temperature), the streamlines in the receiver are straight lines. Figure 2(b) suggests that the largest lateral shift occurs inside the emitter and the lateral shift increases with β. Hence, it is important to take the lateral shift inside the emitter into consideration when determining the minimum area needed for the emitter and receiver to be approximated as infinitely extended plates. In the receiver, it can be shown that for large β*,

$\theta \left(z,\omega ,\beta \right)=\tan ^{-1} \left(\frac{\varepsilon ''}{\varepsilon '} \right), when \quad {\beta} >> \textit{w}/\textit{c}$

(1)

Hence inside the receiver, ESLs for evanescent waves are parallel as seen in Fig. 2(b). But this is not so for propagating waves when ESLs can intercept each other. The results obtained from this study will facilitate the design of experiments for measuring nanoscale thermal radiation. The method discussed above can be extended to the study of energy flux and streamlines between layered structures and materials with coatings.

Figure 2. ESLs for combined TE and TM waves at the SPhP frequency for SiC with d = 10 nm at different β* = βc/ω values: (a) propagating waves; (b) evanescent waves. [4].

## References

[1] Zhang, Z. M., and Lee, B. J., 2006, “Lateral Shift in Photon Tunneling Studied by the Energy Streamline Method,” Optics Express, 14, pp. 9963-9970.

[2] Lee, B. J., and Zhang, Z. M., 2008, “Lateral Shift in Near-Field Thermal Radiation with Surface Phonon Polaritons,” Nanoscale and Microscale Thermophysical Engineering, 12, pp. 238-250.

[3] Lee, B. J., Park, K., and Zhang, Z. M., 2007, “Energy Pathways in Nanoscale Thermal Radiation,” Applied Physics Letters, 91, p. 153101.

[4] Basu, S., Wang, L. P., and Zhang, Z. M., 2011, “Direct Calculation of Energy Streamlines in Near-Field Thermal Radiation,” Journal of Quantitative Spectroscopy and Radiative Transfer, 112, pp. 1149-1155.