So-S^
Two-photon
absorption
dynamics
of
rhodamine
dyes
P.
SPERBER,
A.
PENZKOFER
Naturwissenschaftliche
Fakultat
II -
Physik,
Universitat
Regensburg,
8400
Regensburg,
FRG
Received
23
May;
accepted
11
July
1986
The
intensity-dependent
transmission
of
picosecond
ruby
laser
pulses
of
different
duration
through
methanolic
and
ethanolic
solutions
of
rhodamine
B
and
rhodamine
6G is
analysed.
The
transmission
is
affected
by
S
0
-S„
two-photon
absorption,
by
stimulated
emission
at
the
pump-
laser
frequency,
by
amplified
spontaneous
emission
and
by
excited-state
absorption.
Various
parameters
involving
the
two-photon
absorption
dynamics
are
determined
by
comparing
exper-
iments
with
numerical
simulations.
1. Introduction
The two-photon absorption of dye solutions
at
elevated laser intensities
has
been studied previously
by
fluorescence observation [1-14]. Two-photon absorption cross-sections
G
{2)
were determined
for
various dyes [1-7]. The fluorescence induced
by
two-photon absorption
has
found wide application
in
picosecond-pulse duration
measurements
[8]. Dye-laser pumping
by
two-photon absorption was
achieved
[9, 10].
Deviations
of the
two-photon excited-fluorescence signal from
the
expected
quadratic
dependence
on pump-laser intensity has been observed indicating the simultaneous action
of
additional spectroscopic effects
(excited-state
absorption, stimulated emission, etc.) [2,
3,
11-15].
In
this
paper
the
two-photon absorption dynamics
of
dyes
is
investigated
by
transmission
measurements.
The dyes rhodamine 6G and rhodamine
B
in methanolic
and
ethanolic solution
are
studied.
The
pump laser
is a
mode-locked ruby laser.
The
relaxation
of the
excited molecules
involves
radiationless decay, fluorescence emission, excited-state absorption, stimulated emission
at
pump-laser frequency, amplified
spontaneous
emission
and
fluorescence reabsorption. The dynamics
are simulated
with
a
realistic
level
scheme.
2. Theory
A
realistic
level
diagram
for the
S
0
-S„ two-photon-absorption dynamics
is
shown
in
Fig.
1. The
two-photon absorption excites molecules from
the S
0
ground
state
(region
1) to a
higher excited
singlet
state,
S
wl
(nl ^ 2,
level
2). The molecules in higher excited singlet
states
relax rapidly
to the
first
excited
singlet
state,
Sj
(level
3).
Direct relaxation from higher excited singlet
states
to the
ground
state,
S
0
,
is
neglected. Before relaxation
to the
S!
state,
excited-state absorption may elevate
some molecules from
S„,
to
S„
4
(level 4).
From
the first excited singlet
state,
S,
(level 3),
the
molecules
return
to the
ground
state
by
spontaneous
emission (indicated by transition
to
level
7), by radiation-
less decay,
by
stimulated emission
at the
pump-laser frequency,
v
L
(transition
to
level
8), and by
amplified
spontaneous
emission (transition
to
level
9,
frequency
v
ASE
).
Within
the S
0
state
the
population
of levels
7, 8, 9
thermalize
with
vibrational relaxation time
T
v
[16-18]. The fluorescence
emission
within
the S
0
-S
{
absorption region
is
partially reabsorbed (transition
7
3). The pump
laser
at
frequency
v
L
and the
generated
amplified
spontaneous
emission signal
at
frequency
v
ASE
suffer excited-state absorption from
S, to S
w2
and S
n3
,
respectively.
The
intersystem crossing from
0306-8919/86 $03.00 + .12 © 1986
Chapman
and
Hall Ltd.
381
u
ex,24
So
1
H3
I
c
I c
1
i
b
|
T-2
J
:
!
vl
|x
23
|
Oex.L
I I
|
V
ASE
|
°ex.ASE
1^53
I
x
63
I I
*.
t
*
Figure
1
Level
system.
singlet
states
to
triplet
states
is
neglected
because
the transmission behaviour of picosecond pulses
has been studied and on
a
picosecond time-scale the transfer
to
triplet
states
is negligibly small (for
rhodamine 6G,
rate
&
SiX
= 4.2 x
10V
1
[19],
for
rhodamine B,
£
SiT
= 1.7 x
10
6
s
_1
[20]).
The
extremely small fraction of molecules in thermally populated vibrational
states
or inhomogeneously
shifted
states
within
the S
0
band is able
to
populate directly the
S
x
state
(transition
8
3) [21, 22]
with
absorption cross-section cr
em
. Laser light scattering (cross-section
<x
SCA
)
and impurity absorp-
tion
((T,
M
) may contribute
to the
single-photon absorption
at the
laser frequency. The impurity
absorption may
be
bleached
at
high intensities. The S
0
-T singlet-triplet absorption
is
negligibly
small
for the rhodamine dyes investigated and is not included in the
level
scheme
(TJ-SQ
radiative
lifetimes:
i
ph
=
1.5 s for rhodamine 6G and
t
ph
=
1.6 s for rhodamine B, solvent ethanol [23]).
The rhodamine 6G and rhodamine
B
molecules in methanol and ethanol
are
treated
as
single
species. For the dye concentration used of 0.04moldm~
3
,
a
fraction of
about
8% of the molecules
are statistically
so
close
together
that
they interact mutually (closely spaced pairs) [24]. They have
a
double peaked S
0
-S, absorption spectrum [24] and act
as
quenching
centres
in the concentration
quenching of the fluorescence lifetime [25]. The S
0
-S„ (n
^ 2)
absorption of
these
closely spaced
pairs is found
to
be the
same
as
the monomer absorption. The influence of the fraction of mutually
interacting dye molecules
is
only taken into account
by a
reduction
of
the S
{
-state
fluorescence
lifetime.
The transitions
in the
level
system
of
Fig.
1 are
described
by the
following
system
of
rate
equations. The transformation
t' = t
— nz/c is used,
with
n the refractive index and
c
the vacuum
light
velocity.
Only
isotropic single-photon
and
two-photon absorption cross-sections
are
con-
sidered, i.e. absorption anisotropy of two-photon [3, 26] and of single photon
processes
[27, 28]
is
neglected. The system of equations
reads
dt'
°
{L)
(N
X
- N
2
)
2(hv
L
y
Z
+
°M
- N
S
)
<J
ASE
(N
3
- N
9
)
+
—N
2
- a
REA
(N
7
#3)
^REA
/*V
F
(1)
BN
2
_ - N
2
)
T2
o
a
,
24
(N
2
- N
4
)
T
1
(2)
dN
}
1
tT
ex
,
L
(JV
3
-
JV
5
)
r
<x
ex
,
ASE
(JV
3
-
AT
6
)
, o
m
(N
3
- N
t
)
w - kH
k
/ase
K
4
-
gASE(
f
3
"
^
/
ASE
+
W*
7
-
+
f
+
f
AT
6
-
I
tf
3
+
±
tf
4
/*V
ASE
AZV
F
T
53
T
63
T
F
T
43
(3)
SN4
=
V
Q
*A
N
2 - N
4
)
J
_
\_
N
^
dt' Av
T
.
L
T
43
4
8N
5
<x
ex
,
L
(JV
3
- N
5
)
R
1
N
^
(5)
hv
L
—^/L
-
N
6
)
dt' Av,.
L
T
53
dW
6
ffcAseW
~ Nt)
T
1
*
r
^
=
^ '"""^
(6)
dNl
~
eA
'
REA
*
3
- W*
7
(7)
/
L
-
^Ml
(8)
3/'
hv
h
SN
9
_
g
T
,ASE
», , 0ASE(N
3
- N
9
) N
9
- Q
9
N
{
.„
dt
T
RAD
Av
ASE
dN, <r
m
N
v
. . 1
4
+ —
JVy
(10)
I
L
-—N
y
(11)
Av
L
T,
M
8Ny
=
QmNy
j __1
3r'
Av
L
L
T
IM
- ffscA^o^L
- (r
m
N
y
I
L
(12)
r
2
N
3
hv
ASE
^5 + a
ASE
(N
3
- N
9
)I
ASE
-
<J^
ASE
(N
3
- N
6
)
I
ASE
-
<7
SCA
JV
ASE
(13)
e
T,ASE
dz
T
rad
_ ^A,REA
dz
T
rad
4/
(14)
The
initial
conditions
for the
number density (dimension,
cm
3
) of the
level
populations
are
#,(/'= -
oo,
r, z) =
JV
0
,
#
2
(-<»)
=
#
3
(-oo)
=
^(-oo)
= N
5
(-oo) =
#
6
(-oo)
= 0,
N
7
(—oo) =
Q-JNQ,
N
s
(—CO)
=
Q
S
N
0
,
and N
9
(—co) =
Q
9
N
0
.
g
7
, ^
8
and ^
9
are the
occupation
probabilities
of the
levels
7, 8 and 9
within
the S
0
band.
The
initial
light
intensities
are
I
L
{t',
r, z = 0) = I
0L
exp
(-f
/2
/tf
-
r
2
/r
0
2
), 7
ASE
(/',
r, z = 0) = 0, and
I
REA
(t\
r, z = 0) = 0.
f
0
=
Ar/[2(ln
2)
1/2
]
is
half
the
1/e
pulse
width
(At
FWHM)
and
r
0
is the
1/e
beam radius of the pump
pulse (frequency v
L
).
N
v
and N
y
are the
ground-state
and excited-state
level
populations,
respect-
ively,
of
the impurity molecules
in the
dye solution.
N
X
comprises
the
total population of the
S
0
band. The first term in Equation 1
is
responsible
for
two-photon
absorption; cr
(2)
is the
orientation-averaged two-photon-absorption cross-section. The
second term describes
the
stimulated emission. The third term
takes
amplified spontaneous emission
into
account.
The
fourth term gives
the
S,-S
0
relaxation
rate;
r
F
=
rj
F
T
rdd
is the
fluorescence
lifetime,
where
t]
F
is the
fluorescence quantum efficiency and
T
RAD
is
the
radiative lifetime. The last
term approximates
the
reabsorption
of
fluorescence light; a
REA
is the
reabsorption cross-section.
Equation
2
contains two-photon absorption,
S„,—S„
4
excited-state absorption,
and
S„,-S,
relax-
ation.
Equation
3 is
responsible
for the
St
-state
dynamics. The first term gives
the
level
population
by
S„,-S,
relaxation. The second
and
third
terms
describe excited-state absorption
at v
L
and
v
ASE
,
respectively.
The next two
terms
depopulate
level
3 by stimulated emission
at v
L
and
v
ASE
.
The sixth
term determines
the
reabsorption
of
fluorescence light.
The
last four
terms
describe relaxations.
Equations
4
to
6
handle
level
populations
by
excited-state absorption. Equation
7
considers
fluorescence
emission
into
the
reabsorption region. The first term gives
the
filling
of
level
7.
e
AREA
is
the
fraction
of
fluorescence
falling
into
the
spectral reabsorption region.
The
second term
describes
the
reabsorption
of
fluorescence, and
the
third term
takes
care of thermalization in
the S
0
band.
In
Equation
8 the
first term gives
level
population
by
stimulated emission while
the
second
term
takes
care
of
thermalization.
In
Equation
9 the
first term describes
the
fluorescence emission
in
the
amplified spontaneous emission spectral region.
e
TASE
presents
the
fraction
of
fluorescence
falling
into
the
transparent
spectral region
(e
TASE
+ e
AREA
«
1). The
second term handles
the
amplified
spontaneous emission,
and the
last term
causes
thermalization.
Equations 10 and 11 are included for the discussion of
impurity
effects. They describe the impurity
bleaching
in
a
three-level system
with
fast intermediate
state
(Fig.
lb)
[28].
The change
of
pump-laser intensity
is
described
by
Equation 12. Two-photon absorption (first
term), excited-state absorption (second
and
third terms), stimulated emission (fourth term), light
scattering
(fifth
term) and impurity absorption (last term)
are
included. The generation of
amplified
spontaneous emission
is
described
by
Equation
13: the
first term gives
the
seeding spontaneous
emission
in the
transparent
fluorescence region
(nrl/l
2
is the
solid
angle
of
efficient amplified
spontaneous emission;
/,
sample length),
the
second term describes
the
stimulated amplification
of
the fluorescence,
the
third term
takes
care
of
excited-state absorption,
and the
last term considers
scattering. Equation 14 describes
the
reabsorption
of
fluorescence along the light path:
the
first term
gives
the
spontaneous emission,
the
second term is due
to
reabsorption and
the
last term
takes
care
of
scattering.
Because
the S
n
-S
x
relaxation times
are
short compared
to the
pulse durations,
the
steady-state
solutions
of
Equations
2, 4, 5 and 6 are
used leading
to
Ni
=
W
(2)
IJN,
2(hv
L
)
2
+
r
n
a™ll
+
/*v
L
T
23
<7
eX)24
/
L
-
//v
L
T
23(
7
ex
,
24
/
L
l(l +
AVl
, ) <
15
>
N
<
= V"
(16)
1
+
T
43
°"ex,24
^3
N
>
= —-^r <
17
>
i
+
T
53
CT
ex,LA.
N
6
=
(18)
1
+
T
63°'ex,ASE-7ASE
The system
of
Equations
1, 3, 7 to 18 is
solved numerically
to
analyse
the
S
0
-S„ two-photon-
384
absorption dynamics.
In the
experiments
the
energy transmission
T
*
= JT [ L W, r, IW] Inrdrfc [ £ W, r,
0)d/']
2nrdr
(19)
is
measured
as a
function
of
input-pulse peak intensity
/
0L
.
Temporal
and
spatial gaussian input-
pulse
shapes
are
assumed.
A
comparison
of the
experimental energy transmission curves
with
calculations
allows
one to
determine
the
two-photon-absorption cross-section,
a
{2
\
if
the
other
dye
parameters
are
known.
3. Dye parameters
The
dyes rhodamine
R
dissolved
in
acidic methanol
[29, 30]
(0.003
mol
dm
-3
HC1 added)
and
ethanol
and
rhodamine 6G dissolved
in
neutral methanol
and
ethanol
are
investigated (dyes
from
Kodak).
A
picosecond ruby laser
is
used
for the
two-photon absorption measurements.
The dye
parameters
entering Equations 1
to 18 are
summarized
in
Table
I.
The data
are
independent
of the
solvent
methanol
or
ethanol
within
our
experimental accuracy
and,
therefore,
only
a
single
set of
data is
listed.
If dye
parameters
are
varied in some
calculations,
the
changes
are
explicitly
stated.
The
origins
of the
data
are
specified
in
Table
I.
Most
of the
data
of
Table
I are
obtained
from
the
absorption
and
emission spectra of
Figs
2 to
4. The two-photon absorption cross-sections and some
excited-state-absorption cross-sections
are
determined
by the
present
analysis.
The
absorption
and
emission spectra
of
rhodamine
B and
rhodamine 6G dissolved
in
methanol
are presented
in
Figs
2 and 3,
respectively. The corresponding absorption
and
emission spectra
for
the solvent ethanol
are
within
the
experimental accuracy
identical
to the
methanolic solutions.
Only
300 400 500 600 700
WAVELENGTH,
X [nm]
Figure
2
Absorption
and
emission
spectra
of
rhoda-
mine
B in
acidic
methanol.
d
A
,
apparent absorption
cross-section.
<r
th
,
shape
of
absorption
cross-section
due
to thermal level
population.
d
em
,
apparent
emission
cross-section.
Absorption
and
emission
spectrum
of
rhodamine
B in
<ethanol isjhe
same.
The
deviation
in
the
lohg-wavefength absorption spectrum
is
indi-
cated
by the
dashed
curve.