Chemistry Reference
In-Depth Information
along the z-axis and the interaction of the sample with the traversing photons to be
very small. We consider no further propagation inside the object to occur, and thus
the electron density can be projected into a plane behind the object.
The phase profile
φ(
x
)
is then written as
2
π
λ
φ(
x
)
=−
δ
dz
=−
k
δ
L
(
x
),
(4.2)
L
0
with
0 the refractive index difference with respect to the surrounding water.
The formation of an image
E
δ <
at a position
z
d
in the detector plane
D
is given by propagating the impinging planar wave field (illumination function)
E
(
x
d
,
y
d
,
z
d
)
E
0
interacting with the complex-valued optical transmission function
of the sample [
22
,
23
]
T
(
x
,
y
,
0
)
=
e
i
φ(
x
,
y
)
along the direction
z
of the beam
(
x
,
y
)
=
B
(
x
,
y
)
·
(
,
)
φ(
,
)
=−
δ
(
)
(see Fig.
4.3
). Here
B
x
y
represents the absorption and
x
y
k
L
x
δ
the phase, which is defined by the refractive index contrast
(between sample and
surrounding water) and the phase accumulation length L(x) inside the sample, i.e.
the diameter of the lipid bilayer patch. For a membrane inside a microfluidic channel
this length can range from a few microns in the moment of monolayer fusion up to
several hundred microns depending on the microfluidic channel height.
In the following discussion we will assume
B
=1 since lipid bilayers can be
considered as weak phase objects. Furthermore, we are only interested in the inten-
sity I(x,z) along a single axis in dependence of the electron density distribution
.
Thus we will restrict our discussions to the one-dimensional case in the following
calculations and so for the wave field
E
in
in the plane behind the object:
ρ(
x
)
e
i
φ(
x
)
E
in
(
x
)
=
E
0
·
T
(
x
)
=
E
0
·
(4.3)
For a parallel beam or a beam of small divergence (illumination function considered
as a plane wave) and scattering signals at small momentum transfers, the imaging
process is described by the Kirchhoff diffraction integral using the paraxial Fresnel
approximation [
24
,
25
]:
i
λ
∞
E
in
e
i
2
z
(
x
−
x
d
)
2
E
(
x
d
,
z
)
=
dx
.
(4.4)
z
−∞
By inserting
√
i
e
i
4
,
k
2
π
λ
=
=
and Eq.
4.3
into Eq.
4.4
it transforms to
E
0
∞
k
z
e
−
i
4
e
i
φ(
x
)
e
i
2
z
(
x
−
x
d
)
2
E
(
x
d
,
z
)
=
dx
.
(4.5)
2
π
−∞
2
for each propagation (defo-
cus) distance
z
. For numerical computation of the Fresnel intensity profile
I
The detected intensity is given as
I
(
x
,
z
)
=|
E
(
x
d
,
z
)
|
(
x
,
z
)
,
