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11 HISTOGRAMS
11.3 Correlation Measurements: HBT
To analyze some light source, one often uses the second order correlation function,
g
(2)
τ =
h
I ( t )
·
I ( t + τ )
i
/
h
I ( t )
i
2
. (4)
It basically tells you when to expect the next photon when there has just been one here.
A value of one corresponds to no dependence on the previous detection time. A value
larger than 1 means that there is some positive correlation (two photons are more likely
to be detected within the given time difference), while a value smaller than 1 tells you
that you are less likely to see these time differences (antibunching).
To measure the second order correlation function, cross-correlation histograms are em-
ployed. Neither start-stop, nor start-multistop histograms are the methods of choice,
although especially start-stop measurements are often used. The difference will be briefly
discussed here.
11.3.1 Hanbury-Brown and Twiss setup
The name of these kind of measurements stems from the setup devised by Hanbury-Brown
and Twiss shown in Figure
25 . It consists of two single photon detectors located at each
output of a 50/50 beam splitter. Time difference measurements of photon detections can
then be used to analyze the light sent into the input of the beam splitter.
Figure 25: Hanbury-Brown and Twiss setup.
11.3.2 g
(2)
function and start-stop histograms of coherent light
Photons in coherent light (e.g. that of a laser) are independent of each other, such that
there is a constant probability
p to detect a photon in each time interval. This leads to
Poissonian statistics when counting the number of photons and also to a constant second
order correlation function,
g
(2)
( τ ) = 1 , (5)
where p is normalized to 1.
In contrast, when a start-stop histogram is recorded, you will see a different picture.
Because all but one STOP-event between to START-pulses are ignored, you have the
probability
p (1
−
p )
n − 1
to capture a STOP-event in the n th bin after a START event in
bin 0. Thus, the correlation function will exhibit an exponential decrease instead of a
constant. Furthermore, if the used detectors have a dead time
τ
D
, i.e. they can’t register
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