Dyadic Green’s function for near-field thermal radiation

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For prescribed geometric conditions and temperature distribution, Maxwell’s equations need to be solved in order to obtain the electric and magnetic fields. This can be done with the help of the dyadic Green’s function, which makes the formulations simple and compact. With the assistance of the dyadic Green’s function \overline{\overline{G}} _{e} (x,x',\omega ), the induced electric and magnetic fields due to the fluctuating current density can be expressed in the frequency domain as volume integration:

E(x,\omega )={\rm i}\omega \mu _{0} \int _{V}\bar{\bar{G}}_{e} (x,x',\omega )\cdot j(x',\omega ){\rm d}x'


H(x,\omega )=\int _{V}\bar{\bar{G}}_{h} (x,x',\omega )\cdot j(x',\omega ){\rm d}x'


where \bar{\bar{G}}_{h} (x,x',\omega )=\nabla \times \bar{\bar{G}}_{e} (x,x',\omega ) is the magnetic dyadic Green’s function, μ0 is the magnetic permeability of vacuum, and the integral is over the region V that contains the fluctuating sources. The dyadic Green’s function, \bar{\bar{G}} _{e} (x,x',\omega ) is essentially a spatial transfer function between a point source at location x' and the resultant electric field E at x [1]. Based on the ergodic hypothesis, the spectral energy flux is given by: [2]

\left\langle S(x,\omega )\right\rangle =\int _{0}^{\infty }\frac{1}{2} \left\langle Re[E(x,\omega )\times H^{*} (x,\omega ')]\right\rangle  {\rm \; d}\omega '


where S is the spectral Poynting vector, and ω (and ω') is the angular frequency. In order to compute the spectral Poynting vector at x , we should compute the cross spectral density of electric and magnetic field vectors Ei(x,ω) and Hj(x,ω). The cross spectral density can be written as

\left\langle E_{i} (x,\omega )H_{j}^{*} (x,\omega ')\right\rangle =i\omega \mu _{0} \int _{V}dx' \int _{V}dx''\left\{G_{e,im} (x,x',\omega )G_{h,jn} (x,x'',\omega ')\left\langle j_{m} (x',\omega )j_{n}^{*} (x'',\omega ')\right\rangle \right\}


With the relationship between the fluctuating current densities and the temperature of the emitting medium being established through Fluctuation-Dissipation Theorem, the spectral radiative heat flux can be calculated using Eq. (4) once the dyadic Green’s function, \bar{\bar{G}} _{e} (x,x',\omega ) is obtained. Since the dyadic Green’s function depends on the geometry of the physical system, the following sections will briefly describe the dyadic Green’s functions for two representative structures, i.e., for two semi-infinite media and multilayered media.

Two semi-infinite media

Consider near-field radiative heat transfer between two semi-infinite solids separated by a vacuum (or a dielectric medium). As shown in Fig. 1, a vacuum gap of width d separates the two parallel and smooth surfaces of the semi-infinite media at temperatures T1 and T2 , respectively, where T1 > T2 . Both media are nonmagnetic, isotropic, and homogeneous. Furthermore, each medium is assumed to be at thermal equilibrium. The random motion of the dipoles represented as ellipses in the figure results in a space-time dependent fluctuating electric field, E . Cylindrical coordinate system is used so that the space variable x = r + z, with r-direction being parallel to the interface and z-direction perpendicular to the interface. β and γj refer to the r-component and z-component of the wavevector kj, respectively. Thus,    k_{j} =\beta \hat{ r} +\gamma _{j} \hat{ z}    and   k_{j}^{2} =\beta ^{2} +\gamma _{j}^{2} ,  for   j = 0, 1 and 2 . The magnitude of kj is related to the dielectric function εj   by   k0 = ω / c k_{1} =\sqrt\varepsilon _{1} \omega /c ,   and k_{2} =\sqrt{\varepsilon _{2} }\omega /c , with c being the speed of light in vacuum and ε1 and ε2 being the dielectric functions (or relative permittivity) of medium 1 and 2, respectively. For the described two semi-infinite media, the dyadic Green’s function takes the following form [3,4]:

\overline{\overline{G}} _{e} (x,x',\omega )=\int _{{\rm \; }0}^{{\rm \; }\infty }{\rm \; }\frac{{\rm i}}{4\pi \gamma _{1} }  \left(\hat{s}t_{12}^{{\rm s}} \hat{s}+\hat{p}_{2} t_{12}^{{\rm p}} \hat{p}_{1} \right){\rm e}^{{\rm i}(\gamma _{2} z-\gamma _{1} z')} {\rm e}^{{\rm i}\beta (r-r')} \beta {\rm d}\beta


where  x= r\hat{\mathrm r}+z\hat{\mathrm z}   and   x' =r' {\hat{\mathrm r}}+z' \hat{\mathrm z} . The unit vectors are \hat{s}=\hat{r}\times \hat{z} \, ,\hat{p}_{1} =\left(\beta \hat{z}-\gamma _{1} \hat{r}\right)/k_{1}  ,   and   \hat{p}_{2} =\left(\beta \hat{z}-\gamma _{2} \hat{r}\right)/k_{2}. Note that t_{12}^{{\rm s}} and t_{12}^{{\rm p}} are the transmission coefficients from medium 1 to medium 2 for s and p polarizations, which are given by Airy’s formula [1]. Provided that t_{12}^{{\rm s}} and t_{12}^{{\rm p}} takes into account multiple reflections in the vacuum layer, the dyadic Green’s function describes the transfer of the electromagnetic fields across the vacuum layer, through propagating waves (i.e., β < kj ) and evanescent waves (i.e., β > kj ), from a point source at x' to a receving point at x.

Figure 1. Schematic of near-field radiative heat transfer between two closely spaced semi-infinite plates, at temperatures T1 and T2, separated by a vacuum gap d.

Multilayered media

Dyadic Green’s function for multilayered media is discussed in this section. Dyadic Green’s function for multilayered structures has been extensively used for calculating microwave thermal emission from layered media [5], thermal emission from 1-D photonic crystals [6], current generation in near-field TPV systems [7], and near-field energy transfer between bodies with thin film coatings [8,9]. Figure 2 shows a schematic of the multilayered structure containing N thin films sandwiched between two semi-infinite half spaces employed for the dyadic Green’s function analysis. Properties of the layers are different and are assumed to vary only in the z direction. The layers can be metallic, dielectric or even be a vacuum gap and can have a temperature gradient across them. The dyadic Green’s function between any two layers s and l in Fig. 2 is given by [7]

\bar{\bar{G}}_{e} (x,x',\omega )=\frac{{\rm i}}{4\pi } \int \frac{\beta {\rm d}\beta }{\gamma _{s} }  F(\beta ){\rm e}^{{\rm i}\beta (r-r')}


with F(\beta )=A{\rm e}^{{\rm i}(\gamma _{l} z-\gamma _{s} z')} \hat{e}_{l}^{+} \hat{e}_{s}^{+} +B{\rm e}^{{\rm i}(-\gamma _{l} z-\gamma _{s} z')} \hat{e}_{l}^{-} \hat{e}_{s}^{+} +C{\rm e}^{{\rm i}(\gamma _{l} z+\gamma _{s} z')} \hat{e}_{l}^{+} \hat{e}_{s}^{-} +D{\rm e}^{{\rm i}(-\gamma _{l} z+\gamma _{s} z')} \hat{e}_{l}^{-} \hat{e}_{s}^{-}. Here, the subscript s denotes a source layer (which can be any arbitrary layer) and l is the receiving layer. Note that \hat{e}^{+} and \hat{e}^{-} are two unit vectors, which are given by \hat{e}_{l}^{+} =\hat{e}_{l}^{-} =\hat{r}\times \hat{z}   for TE wave and   \hat{e}_{l}^{\pm } =(\beta \hat{z}\mp \gamma _{l} \hat{r})/k_{l}   for TM wave, respectively.

There are four terms in the expression of F(β) because for multilayered media, waves in each layer can be decomposed into an upward and a downward component due to multiple reflections at each layer. The first two terms account for the waves in the lth layer, which are induced by the upward waves in the source medium. Likewise, the last two terms denote the two waves in lth layer which are due to the downward waves in the source medium [10]. It should be pointed out that the coefficients for terms with \hat e_{s}^{-} are zero if the source is semi-infinite and the coefficients for terms with \hat{e}_{l}^{-} are zero if lth layer is semi-infinite or if there is free emission from multilayered structures. When both the source and receiver layers are semi-infinite, Eq. (6) will reduce to Eq. (5). The coefficients A, B, C and D can be determined using the transfer matrix formulation [1,7,11].

Figure 2. Schematic of a multilayered thin-film structure for Green’s function analysis. Each of the films may be at a different temperature.


[1] Zhang, Z. M., 2007, Nano/Microscale Heat Transfer, McGraw-Hill, New York.

[2] Basu, S., Zhang, Z. M., and Fu, C. J., 2009, "Review of Near-Field Thermal Radiation and Its Application to Energy Conversion," International Journal of Energy Research, 33, pp. 1203-1232.

[3] Fu, C. J., and Zhang, Z. M., 2006, "Nanoscale Radiation Heat Transfer for Silicon at Different Doping Levels," International Journal of Heat and Mass Transfer, 49, pp. 1703-1718.

[4] Joulain, K., Mulet, J.-P., Marquier, F., Carminati, R., and Greffet, J.-J., 2005, “Surface Electromagnetic Waves Thermally Excited: Radiative Heat Transfer, Coherence Properties and Casimir Forces Revisited in the Near-Field,” Surface Science Report, 57, pp. 59-112.

[5] Tsang, L., Njoku, E., and Kong, J. A., 1975, “Microwave Thermal Emission from a Stratified Medium with Nonuniform Temperature Distribution,” Journal of Applied Physics 46, pp. 5127-5133.

[6] Narayanaswamy, A., and Chen, G. 2005, “Thermal Radiation in 1D Photonic Crystals,” Journal of Quantitative Spectroscopy and Radiative Transfer, 93, pp. 175-183.

[7] Park, K., Basu, S., King, W. P., and Zhang, Z. M., 2008, "Performance Analysis of Near-Field Thermophotovoltaic Devices Considering Absorption Distribution," Journal of Quantitative Spectroscopy and Radiative Transfer, 109, pp. 305-316.

[8] Francoeur, M., Menguc, M. P., and Vaillon, R., 2008, “Near-Field Radiative Heat Transfer Enhancement via Surface Phonon Polaritons Coupling in Thin Films,” Applied Physics Letters, 93, p. 043109.

[9] Fu, C. J., and Tan, W. C., 2009, “Near-Field Radiative Heat Transfer Between Two Plane Surfaces with One Having a Dielectric Coating,” Journal of Quantitative Spectroscopy and Radiative Transfer, 110, pp. 1027-1036.

[10] Tsang L., Kong, J. A., and Ding, K. H., 2000, Scattering of Electromagnetic Wavs, Theories and Applications, Wiley, New York.

[11] Francoeur, M., Menguc M. P., and Vaillon, R., 2009, “Solution of Near-Field Thermal Radiation in One-Dimensional Layered Media Using Dyadic Green's Functions and the Scattering Matrix Method,” Journal of Quantitative Spectroscopy and Radiative Transfer, 110, pp. 2002-2018.