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P. Ben-Abdallah et al.
is characterized by the correlation function at the same point ∪E n (r, t)E m (r, t ≈ )∼ and
two different times.
The time-correlation functions contain information on the spectral content of the
fields. The Wiener-Khinchin theorem [43] shows that the Fourier transform of the
time correlation function is the spectral density of the field. Hence, at thermodynamic
equilibrium, we find the field correlation function:
∪E n (r, t + τ )E m (r, t)∼ = δ nm Re
∞
0
4μ 0
ω 2
6π c
ω
exp(ω/k B T ) − 1
exp(iωτ )
dω
2π
.
Here, we see that the broad spectrum entails a short correlation time. However,
by filtering the spectrum in order to reduce the bandwidth, it is possible to increase
the coherence time of a given source. Hence, designing temporally coherent sources
amounts to design spectrally narrow sources.
A similar property exists for spatial coherence in a plane. We introduce a plane
perpendicular to the propagation direction to analyse transverse spatial coherence.
When dealing with homogeneous random processes, the Fourier transform cannot be
defined for a function which is not square integrable. As usual, the spectral analysis
is done using the concept of power spectral density. For the sake of convergence of
the field Fourier transform, a new electric field E A (r / / , ω) is defined to be equal to
the random field in a square of area A and null outside. The spatial power spectral
density is given by:
lim
A→∞
1
A
∪E n,A (κ, ω)E m,A (−κ, ω)∼.
Using again Wiener-Khinchin theorem, the spatial correlation function is given
by [43]:
∪E n (r / / , ω)E m (r ≈
/ / , ω)∼ =
d 2 κ
4π 2 lim
A→∞
1
A
∪E n,A (κ, ω)E m,A (−κ, ω)∼ exp[iκ · (r ≈
/ / − r / / )].
It can be shown that the far-field intensity in a direction specified by κ is proportional to ∪E n,A (κ, ω)E m,A (−κ, ω)∼, so that it is seen that the directivity of the
source is directly related to the transverse correlation function by a Fourier transform.
Designing a transverse correlation function amounts to control the spatial spectrum
of the field at the source plane. Again, it is seen that spatial coherence can be seen
as a filtering issue in κ-space.
In summary, the coherence of thermal sources can be controlled by filtering the
directional and spectral emission properties. In what follows, we will show that by
taking advantage of surface-waves resonances, it is indeed possible to engineer these
properties.
P. Ben-Abdallah et al.
is characterized by the correlation function at the same point ∪E n (r, t)E m (r, t ≈ )∼ and
two different times.
The time-correlation functions contain information on the spectral content of the
fields. The Wiener-Khinchin theorem [43] shows that the Fourier transform of the
time correlation function is the spectral density of the field. Hence, at thermodynamic
equilibrium, we find the field correlation function:
∪E n (r, t + τ )E m (r, t)∼ = δ nm Re
∞
0
4μ 0
ω 2
6π c
ω
exp(ω/k B T ) − 1
exp(iωτ )
dω
2π
.
Here, we see that the broad spectrum entails a short correlation time. However,
by filtering the spectrum in order to reduce the bandwidth, it is possible to increase
the coherence time of a given source. Hence, designing temporally coherent sources
amounts to design spectrally narrow sources.
A similar property exists for spatial coherence in a plane. We introduce a plane
perpendicular to the propagation direction to analyse transverse spatial coherence.
When dealing with homogeneous random processes, the Fourier transform cannot be
defined for a function which is not square integrable. As usual, the spectral analysis
is done using the concept of power spectral density. For the sake of convergence of
the field Fourier transform, a new electric field E A (r / / , ω) is defined to be equal to
the random field in a square of area A and null outside. The spatial power spectral
density is given by:
lim
A→∞
1
A
∪E n,A (κ, ω)E m,A (−κ, ω)∼.
Using again Wiener-Khinchin theorem, the spatial correlation function is given
by [43]:
∪E n (r / / , ω)E m (r ≈
/ / , ω)∼ =
d 2 κ
4π 2 lim
A→∞
1
A
∪E n,A (κ, ω)E m,A (−κ, ω)∼ exp[iκ · (r ≈
/ / − r / / )].
It can be shown that the far-field intensity in a direction specified by κ is proportional to ∪E n,A (κ, ω)E m,A (−κ, ω)∼, so that it is seen that the directivity of the
source is directly related to the transverse correlation function by a Fourier transform.
Designing a transverse correlation function amounts to control the spatial spectrum
of the field at the source plane. Again, it is seen that spatial coherence can be seen
as a filtering issue in κ-space.
In summary, the coherence of thermal sources can be controlled by filtering the
directional and spectral emission properties. In what follows, we will show that by
taking advantage of surface-waves resonances, it is indeed possible to engineer these
properties.