Biomedical Engineering Reference
In-Depth Information
(also supposed to be monochromatic) is collected through the second lens L
2
by a point detector. If
h
ex
and
h
em
are, respectively, the impulse response of
the lens L
1
and L
2
, the radiation distribution delivered on the sample will be
U
ex
=(
h
ex
⊗ δ
s
)(
x
)=
h
ex
, where the point-like excitation source has been
modelled with a Dirac delta.
The fluorescence emitted by each point
x
of the sample,
U
em
(
x
)
,
will be
proportional to the product of the field intensity delivered on the sample and
on the distribution of the fluorescent dye
D
(
x
),
U
em
(
x
)=
U
ex
(
x
)
D
(
x
). This
radiation will be then brought by the second lens to the point-like detector,
leading to the signal recorded by the detector,
U
det
(
x
)=[(
h
em
⊗ U
em
)
δ
d
](
x
),
where the detector has been modelled by a delta function. The overall collected
signal will be therefore
I
tot
=
U
det
(
x
)d
x
=
δ
d
(
x
)d
x
h
em
(
x − y
)
U
em
(
y
)d
y
=
δ
d
(
x
)d
x
h
em
(
x − y
)
h
ex
(
y
)
D
(
y
)d
y
=
h
ex
(
y
)
D
(
y
)d
y
δ
d
(
x
)
h
em
(
x − y
)d
x
=
h
ex
(
y
)
h
em
(
−y
)
D
(
y
)d
y.
(4.2)
If we consider the particular case of a point-like object, (4.2) provides the
impulsive response of the system, i.e. the total PSF of the confocal microscope.
By modelling the point-like object as a Dirac impulse
δ
0
(4.2) becomes
I
tot
=
h
ex
(
y
)
h
em
(
−y
)
δ
0
(
y
)d
y
=
h
ex
(0)
h
em
(0)
.
(4.3)
If we consider the confocal epi-fluorescence scheme L
1
=L
2
,andifwe
assume that
λ
ex
=
λ
em
,
1
we end up with
h
ex
=
h
em
=
h
. We can generally
extend the previous formulas for an
x
-
y
-
z
scanning coupled to the imaging
process. We therefore obtain for a general point
P
(
x, y, z
),
I
tot
=
h
2
(
x, y, z
),
which is the general expression for the PSF. The mathematical expression for
h
(
x, y, z
) can be formulated through the electromagnetic waves scalar theory
[52] and through Fraunhofer diffraction, leading to
J
0
(
vρ
)e
−
i
uρ
2
ρ
d
ρ
1
2
h
(
u, v
)
∝
,
(4.4)
0
1
The equivalence of the excitation and the emission wavelength is an approxima-
tion for fluorescence where generally
λ
ex
≤ λ
em
due to Stokes' shift. A more
precise expression for the fluorescence case
i
s to consi
der a weig
hted mean of the
excitation and emission wavelength
¯
λ
=
√
2
λ
em
λ
ex
/
√
λ
em
+
λ
ex
.
