284
P. Ben-Abdallah et al.
on stochastic electrodynamics where thermal emission is considered to be a radiation
problem. This allows to introduce a simple picture of the thermal emission.
In radiometry, the power dQ emitted by an elementary opaque surface dS at
temperature T in a elementary frequency range dω around the circular frequency ω
in a solid angle dΩ around a direction u making an angle θ with the normal to the
surface is given by:
dQ(ω, θ ) = I
e
ω (T )dS cos θ dωdΩ
where I e
ω (T ) is the specific intensity of the emitted radiation. As a body can only
radiate less than a blackbody, it is useful to express the specific intensity emitted by
a material as
I
e
ω (T ) = E(ω, θ )I BB,ω (T )
where E(ω, θ ) is the emissivity and I BB,ω (T ) is the specific intensity at thermodynamic equilibrium (also known as blackbody specific intensity) given by:
I BB,ω (T ) =
c
4π
×
ω 2
π 2 c 3 ×
ω
exp(ω/k B T ) − 1
,
in which k B is Boltzmann’s and 2π is Planck’s constant. The first term in the
previous equation relates the specific intensity to the energy per unit volume, the
second term is the density of states in vacuum and the third is the mean energy
per mode. In what follows, we will be interested in the physical meaning of the
emissivity and the different ways to engineer this quantity. Here, we note that the
emissivity is a real number in the interval [0, 1] that depends on frequency and angle.
It characterizes the ability of a material to produce thermal radiation. In radiometry, it
is introduced as a phenomenological quantity. Using energy conservation arguments,
it can be shown that it is related to the interface reflectivity [44].
8.1.1.2 Electrodynamic Point of View
The above formalism is a purely phenomenological description of the radiative fluxes.
It is interesting to try to develop an alternative point of view where thermal radiation
is viewed as an antenna problem. In classical electrodynamics, fields are radiated by
time-dependent currents [110]. With this point of view, thermal emission appears to
be the radiation of random dipoles induced by the thermal motion of charges in the
emitter. These charges can be electrons in metals or ions in iono-covalent materials.
Note that the mean value of these random currents is zero so that the mean value of
the radiated fields is also zero. However, the power carried by the field fluctuations is
not zero. The key quantity is therefore the current density fluctuation which is given
at thermodynamic equilibrium by the fluctuation-dissipation theorem [72].
∪ j
f
n (r, ω) j
f
m (r
≈
, ω
≈
)∼ = 2πωΘ(ω, T )i
ε
∝
mn (ω)−ε nm (ω)
δ(ω−ω
≈
)δ(r − r
≈
), (8.1)
P. Ben-Abdallah et al.
on stochastic electrodynamics where thermal emission is considered to be a radiation
problem. This allows to introduce a simple picture of the thermal emission.
In radiometry, the power dQ emitted by an elementary opaque surface dS at
temperature T in a elementary frequency range dω around the circular frequency ω
in a solid angle dΩ around a direction u making an angle θ with the normal to the
surface is given by:
dQ(ω, θ ) = I
e
ω (T )dS cos θ dωdΩ
where I e
ω (T ) is the specific intensity of the emitted radiation. As a body can only
radiate less than a blackbody, it is useful to express the specific intensity emitted by
a material as
I
e
ω (T ) = E(ω, θ )I BB,ω (T )
where E(ω, θ ) is the emissivity and I BB,ω (T ) is the specific intensity at thermodynamic equilibrium (also known as blackbody specific intensity) given by:
I BB,ω (T ) =
c
4π
×
ω 2
π 2 c 3 ×
ω
exp(ω/k B T ) − 1
,
in which k B is Boltzmann’s and 2π is Planck’s constant. The first term in the
previous equation relates the specific intensity to the energy per unit volume, the
second term is the density of states in vacuum and the third is the mean energy
per mode. In what follows, we will be interested in the physical meaning of the
emissivity and the different ways to engineer this quantity. Here, we note that the
emissivity is a real number in the interval [0, 1] that depends on frequency and angle.
It characterizes the ability of a material to produce thermal radiation. In radiometry, it
is introduced as a phenomenological quantity. Using energy conservation arguments,
it can be shown that it is related to the interface reflectivity [44].
8.1.1.2 Electrodynamic Point of View
The above formalism is a purely phenomenological description of the radiative fluxes.
It is interesting to try to develop an alternative point of view where thermal radiation
is viewed as an antenna problem. In classical electrodynamics, fields are radiated by
time-dependent currents [110]. With this point of view, thermal emission appears to
be the radiation of random dipoles induced by the thermal motion of charges in the
emitter. These charges can be electrons in metals or ions in iono-covalent materials.
Note that the mean value of these random currents is zero so that the mean value of
the radiated fields is also zero. However, the power carried by the field fluctuations is
not zero. The key quantity is therefore the current density fluctuation which is given
at thermodynamic equilibrium by the fluctuation-dissipation theorem [72].
∪ j
f
n (r, ω) j
f
m (r
≈
, ω
≈
)∼ = 2πωΘ(ω, T )i
ε
∝
mn (ω)−ε nm (ω)
δ(ω−ω
≈
)δ(r − r
≈
), (8.1)