Chemistry Reference
In-Depth Information
1
exp
Q
r
ρ
1
d
1
2
E
1
X
r
α
1
;
d
1
(2.47b)
?
Q
r
1
exp
Q
rt
ρ
1
d
1
Re
E
t
*
1
E
1
?
X
rt
α
1
;
d
1
cos
φ
1
(2.47c)
Q
rt
Re means the real part of the complex number in brackets, the star means the
complex conjugate. As mentioned in Section 2.4.1, the angular divergence can
be taken into account by a convolution.
Two special cases may be of interest. For a very thin layer,
d
1
approximates
zero and the three fractions of Equations 2.47a-2.47c take on the limiting value
ρ
1
d
1
. The fluorescence intensity simply becomes
2
≅
I
n
c
A
S
x
;
E
0
ε
det
T
air
?
E
t
1
E
1
I
fluo
α
0
(2.48)
where
c
A
is the area-related mass given by the product
c
x
ρ
1
d
1
. The second
special case is a single thin layer for which vacuum or air may be regarded as a
substrate. In this case, the amplitude
E
1
nearly vanishes because the reflection
at the bottom of the layer can be neglected. Equation 2.47 is then reduced to the
first term of the sum.
2.4.3AStratifiedMediumofSeveralLayers
To be consistent, we now consider a stratified structure of several layers. It may
be composed of
N
layers with a thickness
d
v
(
v
=
1,...,
N
) that are each
homogeneous and plane parallel. The refractive indices may be
n
v
and the
densities
ρ
v
. The first layer adjoins to the vacuum or air (
v
=
0); the last layer, to
the substrate (
v
=
N
+
1). The origin of each layer is shifted right in the middle
of the respective layer.
In any layer, there is one definite glancing angle of incidence
α
v
.Itis
determined by the outer glancing angle
α
0
,andby
δ
v
and
β
v
of the layer,
according to
q
α
0
α
ν
2
δ
ν
2
i
β
ν
(2.49)
One incoming and one reflected beam interfere with each other at any point
of the
v
th layer, as shown by Figure 2.26. They overlap under the angle 2
α
ν
.
The amplitudes of the electric fields can be denoted by
E
t
v
and
E
r
v
for the
transmitted and reflected waves, respectively, in the middle of layer
v
.Asin
Section 2.4.2, the amplitudes of two adjacent layers can be connected by a
transfer matrix
E
t
v
E
r
v
M
ν
;
ν
1
E
t
ν
1
E
r
ν
1
(2.50)
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