Measurement of two-photon excitation
cross sections of molecular fluorophores
with data from 690 to 1050 nm
발표자 : 조 기 호
◎ Analysis of two-photon excitation of fluorphores
One-photon and two-photon transitions follow different selection rule. ( ω and 2 ω)
ω
ω
The atomic transition rate R due to
two-photon absorption
2
I
R

The number of photons absorbed per molecule per unit time by means of twophoton excitation is proportional to the two-photon absorption cross section δ
and to the square of incident intensity I.
Nabs : total number of photons absorber per unit time
N abs (t)   dV δ C(r , t) I (r, t)  C0 δ I 0 (t)  dVS 2 (r )
2
V
2
(1)
V
C : function of dye concentration
V : illuminated sample volume
S(r) and Io(t) : spatial and temporal distribution of incident light
F(t) : The number of fluorescence photons collected per unit time
1
F(t)   η 2 N abs
2
(2)
η2 : fluorescence quantum efficiency of the dye
φ : fluorescence collection efficiency of measurement system
Time–averaged fluorescence photon flux
1
 F(t)   η 2 C  I 02 (t)   dVS 2 (r )
2
V
1
  η 2 gC   I 0 (t)  2  dVS 2 (r )
2
V
where
(3)
 I 02 (t) 
g
 I 0 (t)  2
g : measure of the second-order temporal coherence of the excitation source
Spatial dependence
Let z be the distance along the optical axis, ρ be the distance away from the
optical axis.
The dimensionless distance from the optic axis v and the distance from the infocus plane u are given by
v
2 ( N . A.) 

2 ( N . A.) 2 z
,u 
n
(4)
Where N.A. = n sinθ and θ is the half-angle of collection for the lens
The paraxial form of the normalized intensity point-spread function ( h2[u,v] )
for a diffraction-limited lens
S ( r )  h 2 (u , v )
1
 2  J 0 (v ) exp[ ( )iu  2 ]  d
0
2
1
2
(5)
Intensity distribution near the focal point
I(r, t)  I(u, v, t)  I0S(u, v)
(6)
Where I0 is intensity at the geometric focal point ( u = v = 0 ).
 ( N . A.) 2
I 0 (t ) 
P(t )
2

(7)
In thick sample for which the sample thickness is much greater the focal depth,
numerical calculations show that
8n
VS(r )dV   3 ( N .A.)4
3
2
(8)
Substituting Eq.(7) and (8) into Eq.(3),
1
8 n  P(t )  2
 F (t )   g  2 C 
2

(9)
Total fluorescence generation is independent of the N.A. of the
focusing lens in thick samples.
Temporal dependence
Pulsed excitation
Mode-locked laser is the periodic function in time :
m
I 0 (t )  I 0 (t  ), m  1, 2, 3, .....
f
(10)
f : pulse repetition rate
g
gp
f
1 /( 2 f )
gp


[
1 /( 2 f )
1 /( 2 f )
1 /( 2 f )
I 02 (t ) dt
I 0 (t ) dt ]2
For pulse width a Gaussian temporal
profile one finds that gp=0.664 and
for a hyperbolic-secant square pulse
one finds that gp=0.588
(11)
Combining Eq.(9) and eq.(11)
8 n  P(t ) 
 F (t )  
 2 C 
2 f

1 gp
2
(12)
The numerical value of g = gp / (fτ) for mode-locked Ti:sapphire laser is
approximately 105 ( f ~ 100 MHz and τ ~ 100 fs ).
Single-mode CW excitation
g=1
for Ideal single-mode
Single-mode cw excitation requires 102~103 times more average power
than pulsed excitation
◎ Experimental methods
A. Pulsed excitation
Average excitation power of 1 mW at the sample
( pulse intensity ~ 1028 photons/(cm2 s) at the focal point )
B. Single-mode cw two-photon excitation
Average excitation power of ~100 mW at the sample
C. System collection efficiency ( Φ ) and fluorescence quantum
efficiency ( η2 )
System collection efficiency ( Φ ) : collection efficiency of the objective lens
transmission of the optics
photocathode quantum efficiency
Fluorescence quantum efficiency ( η2 ) : η2 = η1 ( Same excited state )
◎ Results
A. Two-photon excitation spectra
Indo-1
Cascade Blue
( Solvent : Water )
Fluorescein
Rhodamine B
(Solvent : Water )
( Solvent : MeOH )
Absolute cross-section value by Eq.(9) and Eq.(12). (Assuming that η2 = η1 )
One general property conserved in all measured TPE spectra is that the TPE
peak wavelengths appear blue shifted and never red shifted relative to twice
the OPA peak wavelengths.
For TPE spectra that are blue shifted, such as for Rhodamine B, Fluorescein,
and DiI, one explanation is that some higher excited singlet states are reached
with greater probability by TPE than by OPE fluorescence.
Parity restrictions imply much larger TPE cross sections at the blue-shifted
wavelengths than at twice the OPA wavelengths.
Values of TPE cross sections depend on the polarization of the excitation light.
 ( cir )
 0.6  0.1at 768 nm (Rhodamine B, Fluorescein and DiI)
 (lin)
B. Power-squared dependence of two-photon-excited
fluorescence
( ● ) : Rhodamine B in Methanol
( + ) : Fluorescein in water
In all cases, it has power squared dependence of two-photon-excited
fluorescence. ( at this experiment )
Theoretically, two-photon-excited fluorescene should obey the square-law
dependence at low excitation. However, significant deviations from the squarelaw dependence of two-photon-excited fluorescence have been observed.
Stimulated emission
Excited-state absorption
Excited-state saturation
Lack of corrections for one-photon excitation
Intensity-dependent TPE cross section
C. Dependence of two-photon-excited fluorescence on pulse width
Two-photon-excited fluorescence is inversely proportional to τ only if τ is much
longer than the intermediate state lifetime. ( ~ 10-16 s )
Because real states with lifetimes of approximately 10-9~10-12 s serve as the
intermediate states in sequential TPE, two-photon-excited fluorescence should
be independent of τ when τ is approximately 100 fs.
Preliminary results show that intermediate state lifetimes for Rhodamine B,
Fluorescein, and Coumarin 307 are less than 100 fs.
D. One-photon- and two-photon-excited fluorescence emission
spectra compared
Emission spectra are independent of excitation wavelengths.
=>These results support the assumption that fluorescence quantum
efficiency is a constant regardless of the excitation wavelength.
E. TPE spectra excited by cw and femtosecond pulsed
laser compared