The light electric field (E)
oscillation frequencies are extremely high, that is hundreds of terahertz for
frequencies corresponding to a period of a few femtoseconds, and conventional
detectors cannot measure directly E. For stationary fields, the time average of
E is null and this explains why detectors are set to measure a time average of
the square field amplitude called the intensity I as defined in equation (4-3)
|
(4-3) |
Standard detection systems have
averaging range of a few picoseconds to several nanoseconds. This averaging
process of the signal intensity leads to three types of coherence functions [4-5]
:
-
the temporal coherence function that
represents the ability of a light source to interfere with a delayed version of
itself
-
the spatial coherence function that
represents the ability of a light source to interfere with a spatially shifted
version of itself
-
the polarization coherence function
that represents the ability of a light source to interfere with another state
of polarization version of itself
Here, the spatial coherence is
neglected by assuming localized light source and one-dimensional propagation.
The polarization coherence function is also not considered by assuming
non-polarized light and propagation preserving the polarization.
In order to theoretically describe the
temporal coherence, the electric field E is represented in the orthogonal basis
of frequency with a single parameter of time t
|
(4-4) |
This is a continuous and infinite
decomposition, and frequency components of E cannot interfere with each other
since they are orthogonal. The intensity I(t) is then found to be
|
(4-5) |
where S(n) is defined as the
spectral power density. S(n) constant with time due to the stationary properties of the light
considered in this case.
The temporal coherence effects occur
when two delayed version of the same light are superposed. For this reason, we now
consider two light waves with electrical field E1 and E2
defined as
|
(4-6) |
where E1 and E2
represent two delayed versions of the same light, with source electric field
signal Es and intensity Is emitted at time t and
t + t. The frequency dependent phase factor f(n) is related to the
emission process. The intensity I(t), given by the superposition of E1
and E2, is independent of the time due to the stationary properties
of the light and is expressed as
where G(t)=<Es(t)Es*(t+t)> is the
autocorrelation function of the light source also called the temporal coherence
function. From equation (4-6) we can see that
|
(4-8) |
which is the well known Wiener-Khintchine
theorem, with G(t) and S(n) forming a
Fourier pair. It is often more convenient to use a normalized coherence
function g(t) called the degree of coherence and defined as
|
(4-9) |
The intensity I(t) becomes
|
(4-10) |
|
