8 Controlling Thermal Radiation with Surface Waves
303
where R i (i = 1, 2) are the 2 × 2 reflection matrices characterizing interfaces (to be
extensively discussed in the next section) and D i j are defined by
D i j = (1 − R i R j e
2ik z0 d
)
−1
.
(8.11)
We now derive an explicit form of the energy transfer between two surfaces using
a scattering formalism that introduces reflection matrices between the bodies. Using
this formalism, we will recover the result derived by Polder and van Hove for the
particular case of two interfaces. We will then analyse in detail the maximum value
of the heat transfer [11]. Finally, we will establish a link with the Landauer theory
of charge transport commonly used in mesoscopic physics of electrons [12, 34].
When inserting Eq. (8.10) into the Poynting vector we find that [15]:
∪S ω ∼ =
d 2 κ
(2π) 2 T (ω, κ, d),
(8.12)
where
T (ω, k ≥ , d) =
Tr
(1 − R
†
2 R 2 )D 12 (1 − R
†
1 R 1 )D 12
†
,
κ<ω/c
Tr
(R
†
2 − R 2 )D 12 (R 1 − R
†
1 )D 12
†
e −2|k z0 |d , κ > ω/c
(8.13)
where Tr stands for the two-dimensional trace. From the previous equation we see
that the whole problem is now reduced essentially to the calculation of the reflection
matrices defined as
R j =
r
s,s
j (ω, κ) r
s,p
j (ω, κ)
r
p,s
j (ω, κ) r
p,p
j (ω, κ)
.
(8.14)
where the r
i,k
j denote the cross reflection coefficients at the interface of medium j
between the polarization i and k, respectively. In the particular case where both media
are isotropic r
s,p
j = r
p,s
j = 0 and the energy transmission coefficient T j (ω, κ, d)
reduces to the usual expression [99]
T j (ω, κ; d) =
(1 − |r 1
j | 2 )(1 − |r 2
j | 2 )/|D 12
j | 2 , κ < ω/c
4Im(r 1
j )Im(r 2
j )e −2|k z0 |d /|D 12
j | 2 , κ > ω/c
(8.15)
for j = {s,p} where r 1
j and r 2
j are the usual Fresnel coefficients
r
i
s (ω, κ) =
k z0 − k zi
k z0 + k zi
and
r
i
p (ω, κ) =
ε i (ω)k z0 − k zi
ε i (ω)k z0 + k zi
(8.16)
for s- and p-polarized light, where k zi =
ε i (ω)ω 2 /c 2 − κ 2 . We have also introduced here the Fabry-Perot-like denominator D 12 , defined by ( j = {s,p})
303
where R i (i = 1, 2) are the 2 × 2 reflection matrices characterizing interfaces (to be
extensively discussed in the next section) and D i j are defined by
D i j = (1 − R i R j e
2ik z0 d
)
−1
.
(8.11)
We now derive an explicit form of the energy transfer between two surfaces using
a scattering formalism that introduces reflection matrices between the bodies. Using
this formalism, we will recover the result derived by Polder and van Hove for the
particular case of two interfaces. We will then analyse in detail the maximum value
of the heat transfer [11]. Finally, we will establish a link with the Landauer theory
of charge transport commonly used in mesoscopic physics of electrons [12, 34].
When inserting Eq. (8.10) into the Poynting vector we find that [15]:
∪S ω ∼ =
d 2 κ
(2π) 2 T (ω, κ, d),
(8.12)
where
T (ω, k ≥ , d) =
Tr
(1 − R
†
2 R 2 )D 12 (1 − R
†
1 R 1 )D 12
†
,
κ<ω/c
Tr
(R
†
2 − R 2 )D 12 (R 1 − R
†
1 )D 12
†
e −2|k z0 |d , κ > ω/c
(8.13)
where Tr stands for the two-dimensional trace. From the previous equation we see
that the whole problem is now reduced essentially to the calculation of the reflection
matrices defined as
R j =
r
s,s
j (ω, κ) r
s,p
j (ω, κ)
r
p,s
j (ω, κ) r
p,p
j (ω, κ)
.
(8.14)
where the r
i,k
j denote the cross reflection coefficients at the interface of medium j
between the polarization i and k, respectively. In the particular case where both media
are isotropic r
s,p
j = r
p,s
j = 0 and the energy transmission coefficient T j (ω, κ, d)
reduces to the usual expression [99]
T j (ω, κ; d) =
(1 − |r 1
j | 2 )(1 − |r 2
j | 2 )/|D 12
j | 2 , κ < ω/c
4Im(r 1
j )Im(r 2
j )e −2|k z0 |d /|D 12
j | 2 , κ > ω/c
(8.15)
for j = {s,p} where r 1
j and r 2
j are the usual Fresnel coefficients
r
i
s (ω, κ) =
k z0 − k zi
k z0 + k zi
and
r
i
p (ω, κ) =
ε i (ω)k z0 − k zi
ε i (ω)k z0 + k zi
(8.16)
for s- and p-polarized light, where k zi =
ε i (ω)ω 2 /c 2 − κ 2 . We have also introduced here the Fabry-Perot-like denominator D 12 , defined by ( j = {s,p})