8 Controlling Thermal Radiation with Surface Waves
285
where i 2 = −1, j f
n (r, ω) is a spatial component of the fluctuating current density
at the frequency ω. The subscripts n or m stand for the x, y or z component of the
vector. ε nm (ω) is the dielectric tensor of the emitter and the function
Θ(ω, T ) =
ω
2
+
ω
e ω/(k B T ) − 1
(8.2)
is the mean energy of a harmonic oscillator in thermal equilibrium. In Eq. (8.1), the
delta-function δ(r − r ≈ ) shows up because we have neglected spatial dispersion. The
second delta function δ(ω − ω ≈ ) reflects the fact that we assume a stationary system.
Indeed, the fluctuation dissipation theorem is only valid in thermal equilibrium so
that by applying this theorem we assume that the medium containing the fluctuating
currents is in local thermal equilibrium at temperature T .
Computing the thermal emission by a body amounts to sum the contributions of
all the volume elements in the material. The electric and magnetic field are then given
by:
E
f
(r, ω) = iωμ 0
V
dr
≈≈
G
E
(r, r
≈≈
, ω) · j
f
(r
≈≈
, ω),
(8.3)
H
f
(r, ω) = iωμ 0
V
dr
≈≈
G
H
(r, r
≈≈
, ω) · j
f
(r
≈≈
, ω),
(8.4)
where the integrals are taken over the volume V which contains the fluctuating
source currents; G E and G H are the electric and magnetic Green tensors [117]. Note
in particular that each volume element can be characterized by a fluctuating dipole
p f such that −iωp f = j f (r ≈≈ , ω)dr ≈≈ . It follows that thermal radiation can be reduced
to the emission of a time-dependent dipole below an interface.
Let us consider the thermal emission in the upper half-space (z > 0) by a homogeneous material in the lower half-space (z < 0). With the electrodynamic point of
view, it appears that the field thermally emitted is due to random dipoles below the
interface. These dipoles radiate fields which decay upon propagation as the medium
is absorbing. Nonetheless, dipoles located close enough to the interface radiate fields
that can reach the interface. Then, these fields are either reflected or transmitted by
the interface. This suggests that the emissivity introduced in the phenomenological
description should be connected to the concept of transmission at an interface.
8.1.1.3 Kirchhoff’s Law: Electrodynamic and Radiometric Point of View
Let us consider radiation impinging from vacuum on an absorbing half-space. Energy
conservation of the incident beam imposes that the incident radiation is either
reflected or transmitted by the interface. The part transmitted by the interface enters
a thick absorbing medium so that it will be absorbed sooner or later. It follows that
absorptivity is equal to the interface transmissivity. In the previous paragraph, we
have pointed out the link between emissivity and interface transmissivity. In sum-
285
where i 2 = −1, j f
n (r, ω) is a spatial component of the fluctuating current density
at the frequency ω. The subscripts n or m stand for the x, y or z component of the
vector. ε nm (ω) is the dielectric tensor of the emitter and the function
Θ(ω, T ) =
ω
2
+
ω
e ω/(k B T ) − 1
(8.2)
is the mean energy of a harmonic oscillator in thermal equilibrium. In Eq. (8.1), the
delta-function δ(r − r ≈ ) shows up because we have neglected spatial dispersion. The
second delta function δ(ω − ω ≈ ) reflects the fact that we assume a stationary system.
Indeed, the fluctuation dissipation theorem is only valid in thermal equilibrium so
that by applying this theorem we assume that the medium containing the fluctuating
currents is in local thermal equilibrium at temperature T .
Computing the thermal emission by a body amounts to sum the contributions of
all the volume elements in the material. The electric and magnetic field are then given
by:
E
f
(r, ω) = iωμ 0
V
dr
≈≈
G
E
(r, r
≈≈
, ω) · j
f
(r
≈≈
, ω),
(8.3)
H
f
(r, ω) = iωμ 0
V
dr
≈≈
G
H
(r, r
≈≈
, ω) · j
f
(r
≈≈
, ω),
(8.4)
where the integrals are taken over the volume V which contains the fluctuating
source currents; G E and G H are the electric and magnetic Green tensors [117]. Note
in particular that each volume element can be characterized by a fluctuating dipole
p f such that −iωp f = j f (r ≈≈ , ω)dr ≈≈ . It follows that thermal radiation can be reduced
to the emission of a time-dependent dipole below an interface.
Let us consider the thermal emission in the upper half-space (z > 0) by a homogeneous material in the lower half-space (z < 0). With the electrodynamic point of
view, it appears that the field thermally emitted is due to random dipoles below the
interface. These dipoles radiate fields which decay upon propagation as the medium
is absorbing. Nonetheless, dipoles located close enough to the interface radiate fields
that can reach the interface. Then, these fields are either reflected or transmitted by
the interface. This suggests that the emissivity introduced in the phenomenological
description should be connected to the concept of transmission at an interface.
8.1.1.3 Kirchhoff’s Law: Electrodynamic and Radiometric Point of View
Let us consider radiation impinging from vacuum on an absorbing half-space. Energy
conservation of the incident beam imposes that the incident radiation is either
reflected or transmitted by the interface. The part transmitted by the interface enters
a thick absorbing medium so that it will be absorbed sooner or later. It follows that
absorptivity is equal to the interface transmissivity. In the previous paragraph, we
have pointed out the link between emissivity and interface transmissivity. In sum-