Chemistry Reference
In-Depth Information
(the field vector is within the plane of incidence or perpendicular to this plane,
respectively). At grazing incidence, however, the polarization does not have to
be considered. Only the length or amplitude of the electric field vector is
relevant for fluorescence. Consequently, only this component has to be taken
into account, but an averaging over time has to be carried out.
There are different waves with different amplitudes. The incident wave
amplitude
E
i
0
may be normalized to 1. The reflected wave amplitude may be
denoted by
E
0
. The refracted wave amplitude, which is the incident wave
amplitude in the substrate, is called
E
s
. Within the substrate, there is no
reflected wave, so that a hypothetical amplitude
E
s
can be set to zero. These
four amplitudes are connected by the so-called Born or transfer matrix
according to
m
1
m
2
m
3
m
4
E
s
0
1
E
0
(2.29)
?
The individual components of this 2
×
2 matrix are given by
α
0
α
s
m
1
m
4
(2.29a)
2
α
0
α
0
α
s
m
2
m
3
(2.29b)
2
α
0
The primary intensity
I
int
is defined as the square of the modulus of the relevant
amplitude. Because incoming and reflected beams interfere above the surface,
the intensity
I
int
at height
z
for a glancing angle
α
0
can be written as a sum of the
downgoing and upgoing plane waves:
2
E
0
exp
ik
0
α
0
z
I
int
α
0
;
z
I
0
exp
ik
0
α
0
z
i
φ
(2.30)
where
k
0
is 2
π
/
λ
,
φ
is a phase shift already defined by Equation 2.15, and
I
0
is a
measure for the incoming beam intensity. Squaring leads to
2
E
0
2
E
0
I
int
α
0
;
z
cos 2
k
0
α
0
z
φ
I
0
1
(2.31)
2
is defined as reflectivity
R
, this equation is equivalent to
expression (2.14).
The amplitude
E
0
is obtained from Equation 2.29 as the ratio of
m
3
and
m
1
.
The squared value of the modulus yields the reflectivity itself:
Because
E
0
2
α
0
α
s
R
α
0
(2.32)
α
0
α
s
which corresponds to Equation 1.75.
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