Fundamentals

Most modern works on NFRHT express results in the form of a Landauer formula.^[21] Specifically, the net heat power transferred from body 1 to body 2 is given by

${\displaystyle P_{\mathrm {1\rightarrow 2,net} }=\int _{0}^{\infty }\left\{{\frac {\hbar \omega }{2\pi }}\left[n(\omega ,T_{1})-n(\omega ,T_{2})\right]{\mathcal {T}}(\omega )\right\}d\omega }$,

where ${\displaystyle \hbar }$ is the reduced Planck constant, ${\displaystyle \omega }$ is the angular frequency, ${\displaystyle T}$ is the thermodynamic temperature, ${\displaystyle n(\omega ,T)=\left(1/2\right)\left[\coth {\left(\hbar \omega /2k_{b}T\right)}-1\right]}$ is the Bose function, ${\displaystyle k_{b}}$ is the Boltzmann constant, and

${\displaystyle {\mathcal {T}}(\omega )=\sum _{\alpha }\tau _{\alpha }(\omega )}$.

The Landauer approach writes the transmission of heat in terms discrete of thermal radiation channels, ${\displaystyle \alpha }$. The individual channel probabilities, ${\displaystyle \tau _{\alpha }}$, take values between 0 and 1.

NFRHT is sometimes alternatively reported as a linearized conductance, given by^[11]

${\displaystyle G_{\mathrm {1\rightarrow 2,net} }(T)=\lim _{T_{1},T_{2}\rightarrow T}{\frac {P_{\mathrm {1\rightarrow 2,net} }}{T_{1}-T_{2}}}=\int _{0}^{\infty }\left[{\frac {\hbar \omega }{2\pi }}{\frac {\partial n}{\partial T}}{\mathcal {T}}(\omega )\right]d\omega }$.

Two half-spaces

For two half-spaces, the radiation channels, ${\displaystyle \alpha }$, are the s- and p- linearly polarized waves. The transmission probabilities are given by^[6][11][21]

${\displaystyle \tau _{\alpha }(\omega )=\int _{0}^{\infty }\left[{\frac {k_{\rho }}{2\pi }}{\widehat {\tau }}_{\alpha }(\omega )\right]dk_{\rho },}$

where ${\displaystyle k_{\rho }}$ is the component of the wavevector parallel to the surface of the half-space. Further,

${\displaystyle {\widehat {\tau }}_{\alpha }(\omega )={\begin{cases}{\frac {\left(1-\left|r_{0,1}^{\alpha }\right|^{2}\right)\left(1-\left|r_{0,2}^{\alpha }\right|^{2}\right)}{\left|1-r_{0,1}^{\alpha }r_{0,2}^{\alpha }\exp {\left(2ik_{z,0}l\right)}\right|^{2}}},&{\text{if }}k_{\rho }\leq \omega /c\\{\frac {4\Im {\left(r_{0,1}^{\alpha }\right)}\Im {\left(r_{0,2}^{\alpha }\right)}\exp {\left(-2\left|k_{z,0}\right|l\right)}}{\left|1-r_{0,1}^{\alpha }r_{0,2}^{\alpha }\exp {\left(-2\left|k_{z,0}\right|l\right)}\right|^{2}}},&{\text{if }}k_{\rho }>\omega /c,\end{cases}}}$

where:

• ${\displaystyle r_{0,j}^{\alpha }}$ are the Fresnel reflection coefficients for ${\displaystyle \alpha =s,p}$ polarized waves between media 0 and ${\displaystyle j=1,2}$,
• ${\displaystyle k_{z,0}={\sqrt {(\omega /c)^{2}-k_{\rho }^{2}}}}$ is the component of the wavevector in the region 0 perpendicular to the surface of the half-space,
• ${\displaystyle l}$ is the separation distance between the two half-spaces, and
• ${\displaystyle c}$ is the speed of light in vacuum.

Contributions to heat transfer for which ${\displaystyle k_{\rho }\leq \omega /c}$ arise from propagating waves whereas contributions from ${\displaystyle k_{\rho }>\omega /c}$ arise from evanescent waves.