Biomedical Engineering Reference
In-Depth Information
Therefore, the fluorescence intensity emitted by one molecule can be con-
sidered proportional to the two-photon molecular cross section
δ
2
(
λ
)andto
the square of the intensity delivered on the sample:
∝ δ
2
P
(
t
)
2
π
(NA)
2
hcλ
2
∝ δ
2
I
(
t
)
2
I
f
(
t
)
,
(4.11)
where
P
(
t
) is the laser power, and the intensity of the incoming radiation is
calculated by using the paraxial approximation in an ideal optical system.
Two-photon excitation is generally induced with ultra-fast pulsed lasers. The
time averaged intensity emitted by one molecule over a time
T
will be therefore
I
f
(
t
)d
t ∝ δ
2
π
(NA)
2
hcλ
2
T
T
1
T
1
T
P
(
t
)
2
d
t.
I
f
(
t
)
=
(4.12)
0
0
We can now consider
f
p
=1
/T
as the repetition rate of the pulsed laser
and
τ
p
as the pulse width. If we consider that the emission of the laser is
described by the profile
P
(
t
)=
P
max
for 0
<t<τ
p
P
(
t
)=0 for
τ
p
<t<T,
(4.13)
then the mean square power delivered on the sample
P
ave
=1
/T
0
P
(
t
)
2
d
t
becomes
P
ave
=
P
max
τ
p
f
p
and (4.12) becomes
π
2
∝δ
2
P
ave
τ
p
f
p
(NA)
2
hcλ
I
f
(
t
)
.
(4.14)
Writing the same relationship for a continuous wave (CW) laser, we obtain
∝δ
2
P
ave
π
2
(NA)
2
hcλ
I
f
(
t
)
.
(4.15)
Comparing (4.13) and (4.15) it is evident that the excitation of molecules
with a pulsed laser is more e
cient: in order to obtain the same fluores
cence
emission with a CW laser, it is necessary to use an average power 1
/
τ
p
f
p
higher relatively to the pulsed laser case.
From (4.13) it is possible to easily evaluate the probability
n
a
for a fluo-
rophore to absorb two photons simultaneously during a single pulse:
π
2
n
a
∝ δ
2
P
ave
τ
p
f
p
(NA)
2
hcλ
.
(4.16)
The selection of the repetition rate and of the width of the beam pulse
is related to the necessity of avoiding saturation of the fluorescence when
n
a
approaches unity. In other words, during the beam pulse the molecule must
