302
P. Ben-Abdallah et al.
mean Poynting vector in the z direction:
∪S
1→2
z
∼ = ∪E
f
1 × H
f
1 ∼ · e z
(8.6)
by means of the fluctuation dissipation theorem in Eq. (8.1). The resulting expression
contains the dyadic Green’s functions G E (r, r ≈≈ , ω) and G H (r, r ≈≈ , ω) for that layered
geometry with source points r ≈≈ inside medium 1 and observation points r inside the
vacuum gap. For the given layered geometry, the Green functions are well known
and can for example be found in [64]. For determining the net heat flux one has also
to consider the opposite case with T 1 = 0 so that only fluctuating currents inside
medium 2 are taken into account. The expression for the transfered heat is given by
H F (T 1 , T 2 , a) =
A
dA · ∪S
1→2
− S
2→1
∼ =
z=0
d
2 r ≥ ∪S
1→2
z
− S
2→1
z
∼, (8.7)
where r ≥ = (x, y) and S 1→2
z
is given by (8.6) and the integration can be performed
over any surface A that completely separates the bodies. For convenience, (and with
no loss of generality) we took it as the plane z = 0. By using the Green dyadic
introduced in (8.3, 8.4), we can recast the integrand of the previous expression into
∪S z ∼ =
∞
0
dω
2π
Θ(ω, T 1 ) − Θ(ω, T 2 )
∪S ω ∼,
(8.8)
where [123]:
∪S ω ∼ = 2 Re Tr
dr
≈
≥
G(r, r
≈
)∂ z ∂
≈
z G
†
(r, r
≈
) − ∂ z G
†
(r, r
≈
)∂
≈
z G(r, r
≈
)
z ≈ =z=0
,
(8.9)
and Θ(ω, T i ) was defined in (8.2).
The conclusion that we draw from Eqs. (8.7)–(8.9) is that, in order to evaluate the
heat transfer for a given geometry we have to determine the Green dyadic inside the
gap region. In most cases this is surely a formidable task, but for planar homogeneous
media, even if anisotropic, it is possible to simplify things enough so semi-analytic
expressions are obtainable. This is not to say that everything was made easy - in fact
even in this simplified case the calculations are fairly long [27, 120] or requires some
indirect arguments as in Ref. [97], so we shall just quote the final result for the Green
tensor
G(r, r
≈
) =
i
2
d
2
κ
e iκ·(r ≥ −r ≈ ≥ )
k z0
D 12
1e
ik z0 (z−z ≈ )
+ R 1 e
ik z0 (z+z ≈ )
+ D 21
R 2 R 1 e
ik z0 (z ≈ −z) e
2ik z0 d
+ R 2 e
2ik z0 d e
−ik z0 (z+z ≈ )
,
(8.10)
P. Ben-Abdallah et al.
mean Poynting vector in the z direction:
∪S
1→2
z
∼ = ∪E
f
1 × H
f
1 ∼ · e z
(8.6)
by means of the fluctuation dissipation theorem in Eq. (8.1). The resulting expression
contains the dyadic Green’s functions G E (r, r ≈≈ , ω) and G H (r, r ≈≈ , ω) for that layered
geometry with source points r ≈≈ inside medium 1 and observation points r inside the
vacuum gap. For the given layered geometry, the Green functions are well known
and can for example be found in [64]. For determining the net heat flux one has also
to consider the opposite case with T 1 = 0 so that only fluctuating currents inside
medium 2 are taken into account. The expression for the transfered heat is given by
H F (T 1 , T 2 , a) =
A
dA · ∪S
1→2
− S
2→1
∼ =
z=0
d
2 r ≥ ∪S
1→2
z
− S
2→1
z
∼, (8.7)
where r ≥ = (x, y) and S 1→2
z
is given by (8.6) and the integration can be performed
over any surface A that completely separates the bodies. For convenience, (and with
no loss of generality) we took it as the plane z = 0. By using the Green dyadic
introduced in (8.3, 8.4), we can recast the integrand of the previous expression into
∪S z ∼ =
∞
0
dω
2π
Θ(ω, T 1 ) − Θ(ω, T 2 )
∪S ω ∼,
(8.8)
where [123]:
∪S ω ∼ = 2 Re Tr
dr
≈
≥
G(r, r
≈
)∂ z ∂
≈
z G
†
(r, r
≈
) − ∂ z G
†
(r, r
≈
)∂
≈
z G(r, r
≈
)
z ≈ =z=0
,
(8.9)
and Θ(ω, T i ) was defined in (8.2).
The conclusion that we draw from Eqs. (8.7)–(8.9) is that, in order to evaluate the
heat transfer for a given geometry we have to determine the Green dyadic inside the
gap region. In most cases this is surely a formidable task, but for planar homogeneous
media, even if anisotropic, it is possible to simplify things enough so semi-analytic
expressions are obtainable. This is not to say that everything was made easy - in fact
even in this simplified case the calculations are fairly long [27, 120] or requires some
indirect arguments as in Ref. [97], so we shall just quote the final result for the Green
tensor
G(r, r
≈
) =
i
2
d
2
κ
e iκ·(r ≥ −r ≈ ≥ )
k z0
D 12
1e
ik z0 (z−z ≈ )
+ R 1 e
ik z0 (z+z ≈ )
+ D 21
R 2 R 1 e
ik z0 (z ≈ −z) e
2ik z0 d
+ R 2 e
2ik z0 d e
−ik z0 (z+z ≈ )
,
(8.10)