LSM 880
Left Tool Area and Hardware Control Tools
ZEISS
10/2014 V_01
000000-2071-464
197
Definition of the cross-correlation function
The formalism for the cross-correlation function is identical to the auto-correlation function, with the
exception that the signal in one channel is not compared to itself, but to a signal in a second channel.
Lets assign the indices "r" and "b" for the red and blue channel, respectively, than the cross-correlation
function would read as follows:
( )
(
)
( )
( )
( )
(
)
( )
( )
t
I
t
I
t
I
t
I
t
I
t
I
t
I
t
I
G
r
b
b
r
r
b
r
b
I
X
⋅
+
⋅
=
⋅
+
⋅
=
t
d
d
t
d
d
t
d
)
(
(1d)
( ) (
)
( )
( )
( ) (
)
( )
( )
t
I
t
I
t
I
t
I
t
I
t
I
t
I
t
I
G
r
b
b
r
r
b
r
b
I
X
⋅
+
⋅
=
⋅
+
⋅
=
t
t
t
)
(
(1e)
Note that the software for FCS calculated G
I
functions, which do therefore converge to 1 and not
0. The acquired correlation functions are than compared to model equations.
(2)
Available model equations
In the following available equations used for the fits are given that define the accessible parameters. For
some equations useful conversions to other parameters are listed as well. The total correlation is given by
equation 2:
( )
( )
∏ ∑
⋅
+
+
+
=
k
l
l
k
I
tot
G
A
B
d
G
t
t
,
1
(2)
where
d
is the offset,
B
the background correction,
A
the amplitude and
G
k,l
(
t
)
the correlation for a
single process. The suffixes k and l signify correlation terms for dependent and independent processes,
respectively, that are multiplied with or added to each other.
The total correlation is therefore the amplitude multiplied to the product of the single correlation terms
that are dependent and hence convolute each other. This amplitude has to be corrected for background
and any offset. In cases, when the processes are independent from each other, the single correlations
terms add up, for example in cases where there is more than one component all bearing the same label
or of bunching terms that are independent from each other. If independent and dependent processes are
present, all independent terms will add up and are multiplied with the dependent terms.
One can distinguish between different classes of fluctuation processes: anti-bunching, bunching and
diffusion.
Summary of Contents for LSM 880
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