Quantitative description of two-photon absorption in dipolar molecules with two-level modelNikolay Makarov, Mikhail Drobizhev, Zhiyong Suo, Aleks Rebane
E. Scott Tarter, Benjamin D. Reeves, Brenda SpanglerFanqing Meng, Charles W. SpanglerCraig J. Wilson, Harry L. Anderson
Department of Physics, Montana State University, Bozeman, MT 59717Sensopath Technologies, Inc., Bozeman, MT 59715
MPA Technologies, Inc., Bozeman, MT 59715Department of Chemistry, University of Oxford, Mansfield, Oxford, UK
ABSTRACTHigh demand for efficient two-photon absorbing (2PA) chromophores requires better understanding of what molecular parameters are responsible for the enhancement of 2PA cross section.Here we present a systematic approach for quantitative description of 2PA cross section by using two-level approximation in low-lying transitions of dipolar molecules. In these molecules, the lowest energy transition is simultaneously allowed for 1PA and 2PA. The 2PA cross section is proportional to the square of the transition dipole moment (|01|), square of the difference in permanent dipole moments (|01|), and is inverse proportional to the absorption linewidth FWHM. The 2PA cross section also depends on the angle between 01 and 01 and is maximum if they are parallel.Three different types of molecules were studied: substituted linear diphenylaminostilbenes, linear carbazolyl-stylbenes, and push-pull porphyrins. In these types of molecules we measured 2PA cross sections, 01, 01, and linewidth. The measured 2PA cross sections do not exceed 150 GM and quantitatively agree with the quantum-mechanical expression for two-level system within experimental errors.This work shows for the first time the quantitative structure-to-property relationship for 2PA in dipolar molecules.Ideally, if the molecule has particularly large dipole moments 01=01=15 D and linewidth FWHM=1000 1/cm, the value of 2PA cross section could reach 900 GM. Higher cross sections are also possible if the higher energy levels of the molecule contribute to 2PA.
Time, ns
Nor
mal
ized
fluo
resc
ence
1 2 3 4 5 6 70
0.1
0.01
10-3
1
12, =1.14 ns11, =0.86 ns
4, =1.28 ns
6, =1.46 ns
1, =1.71 ns
9, =3.11 ns
0.0 0.1 0.2 0.3 0.4 0.50
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
pent
anal
2-ch
loro
buta
ne
isob
utyl
ace
tate
isob
utyl
isob
ytyr
ate
buty
l eth
er
n-oc
tane
Stok
es S
hift,
cm
-1
f(D)
1 4 5 10
1 10
0
2
4
6
8
10
12
14
16
7
61
9
11
52
3
4
12
8
10
|00|
| 0
1|
Spearman correlation coefficient for the data is =–0.69, which suggests existence of anti-correlation with ~98% confidence.This can be explained as follows: the absolute value of permanent dipole moment is limited. If the ground state dipole is high, the excited state dipole can be only increased up to the limiting value, so the dipole difference is low. If the ground state dipole is low, the difference can be much higher.
Compound , M-1cm-1 r , ns F FM, ns S1, cm-1 S2, cm-1 2, GM* , ns** a, Å*** |01|, D S, cm-1 |01|, D***
1 24400 0.022 1.71 0.72 4.6 3243 4520 47 3.3 5.3 (5.2) 6.4 1277 11.2 (10.8)
2 30900 0.031 1.67 0.34 3.8 3767 6030 136 1.3 6.0 (5.3) 5.4 2263 14.9 (12.6)
3 43100 0.030 1.57 0.67 18 3430 4210 69 12 5.8 (8.1) 6.5 780 8.8 (14.5)
4 42000 0.051 1.28 0.72 2.7 3230 3976 49 1.9 6.6 (5.7) 7.3 746 12.9 (10.6)
5 33300 0.053 1.46 0.64 3.5 2967 4194 70 2.2 7.0 (6.0) 6.6 1227 14.7 (11.7)
6 32600 0.031 1.46 0.51 4.9 3040 4238 40 2.5 5.7 (5.2) 6.0 1198 10.7 (9.3)
7 22000 0.055 2.73 0.11 38 34 114 23 2.9 8.3 (7.3) 4.0 80 4.5 (4.8)
8 8250 0.071 1.72 0.25 116 510 719 20 20 8.2 (7.3) 8.0 267 7.1 (6.2)
9 24900 0.029 3.11 0.19 44 149 150 12 5.8 7.1 (6.9) 5.1 1 0.4 (0.4)
10 16683 0.031 2.99 0.18 109 134 176 5.5 13 7.2 (7.0) 5.1 42 2.6 (2.5)
11 56633 0.150 0.86 0.27 12 489 617 145 2.1 9.2 (8.2) 12.2 128 6.8 (5.8)
12 65696 0.128 1.14 0.34 7.2 538 594 100 2.5 9.3 (10.0) 13.7 56 4.6 (5.1)
S1 is measured in toluene (D=2.4) for compounds 1, 3-6 and in n-octane (D=2.0) for compounds 2, 7-11; is measured in tetrahydrofuran (D=7.58) for compounds 1, 3-6 and in 2-chlorobutane (D=8.06) for compounds 2, 7-11.*Cross section at the doubled wavelength of the lowest dipole-allowed 1PA transition.**Calculated from FM and F.***Calculated from the density of the molecules.
0 100 200 300
0
100
200
300
7
211
6
9
1
8
5
4
3
12
10
2, G
M
2, GM
21cos215
22 2201
2012
44
2 gnchfa
0 100 200 300
0
100
200
300
2, GM
112
12
10
5
416
97
8
3
2, G
M
14
2
12201
14
23
01202 1096.0
PAPA
PAPASb
fnDfnfa
R2
For molecule density For anisotropy
2(a) 2
(b) 2(a) 2
(b)
fL 0.8 3.3 1.4 1.4
fO 0.6 1.8 0.8 0.9
2mod2
11
i i
iel
iR
yyN
123
32
2
22
nnfnf OL
83% of the experimental and theoretical values coincide within the error margins. To our best knowledge, this is the first demonstration of quantitative correspondence between experimental and theoretical two-level model based 2PA cross section for a broad range of different dipolar compounds.The combination of the expression for 2
(b) with the molecular radius data obtained from fluorescence anisotropy and the Onsager local field factor gives the best correlation between the experimental and theoretical cross sections.
78.1;15.15.14
O
L
O
Lf
ff
fn
2201
4
201
124
3fn
h
flFM
FFM
OD
OD
RRFRF
R
dFn
dFn
101101
2
2
Dfahc S
3
201
1212
DDDf
31
43
ANMa
14.04
33
r
kTa
IIII
r2||
||
3
0001
Ahc af D
250 300 350 400 450 500 5500
100
200
300
400
2, G
M
Wavelength, nm
Experimental 2PA spectrum Experimental 1PA spectrum Theoretical 2PA spectrum Theoretical 1PA spectrum
Theoretical fit of 2PA (thick solid line) and 1PA (dashed) of 13 using three level density matrix model. Thin solid line and squares are the normalized 1PA spectrum and 2PA spectrum. The molecular parameters from linear measurements are:
fsTfsTfsTfsTfsT
DDDDD
1000;1000;100;8.1;6.2
;9;2;7;8;10
2211120201
1202010201
Transition dipole moment between the excited states |12|=13.5 D is obtained from the best fit.
Conclusions• We show that perturbation theory applied for two-level system quantitatively predicts the 2PA cross sections, provided that the necessary molecular parameters such as transition- and permanent dipole moments are independently measured.•In most cases, the discrepancy between theory and experiment was less than 20%, and always less than 50%. This is the first time that such direct quantitative correspondence is demonstrated for a wide range of dipolar molecules.• The overall significance of this work demonstrates a practical way how a set of relatively straightforward linear spectroscopic measurements can be used to study and predict nonlinear 2PA properties.
8Qx
(2)
Qx(1)
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Wavelength, nm
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M
, 1
04 M-1cm
-1
9
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, 1
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, 1
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5
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60032000 30000 28000 26000 24000 22000 20000
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M
, 1
04 M-1cm
-1
Wavelength, nm
5
1.25
0
2.5
Frequency, cm-1
300 320 340 360 380 400 420 440
0
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10032000 30000 28000 26000 24000
2, G
M , 1
04 M-1cm
-13
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, 1
04 M-1cm
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Frequency, cm-1
4
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6
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14032000 30000 28000 26000 24000
, 1
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-1
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M
Wavelength, nm
8
6
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Wavelength, nm300 320 340 360 380 400 420 440
0
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M
, 1
04 M-1cm
-14
Substituted diphenylaminostilbenes
7
Qx(1)
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1
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Frequency, cm-1
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M
, 1
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-1
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Qx(1)
Qx(2)
1
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6020000 19000 18000 17000 16000 15000
Wavelength, nm
2, G
M
, 1
04 M-1cm
-1
4
Meso-DPAS and BDPAS-substituted porphyrins
Push-pull porphyrin monomer and push-pull porphyrin dimer
12
Qx
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Wavelength, nm650 700 750 800 850
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M
, 1
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-1cm
-1
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5
Frequency, cm-1
560 600 640 680 7200
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M
, 1
04 M
-1cm
-1
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Wavelength, nm
2, G
M
280 300 320 340 360 380 4000
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50034000 32000 30000 28000 26000
, 1
04 M
-1cm
-1
15 1.25
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M
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Frequency, cm-1
, 1
04 M
-1cm
-1
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M
, 1
04 M
-1cm
-1
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Frequency, cm-1
16
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Wavelength, nm280 320 360 400 440
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35034000 32000 30000 28000 26000 24000 22000
2, G
M
, 1
04 M
-1cm
-1
Substituted carbazolyl-stylbenes and diphenylaminostilbenes
Ground state dipole moment vs. permanent dipole moment differenceSolvatochromic shifts of some compoundsFluorescence decay of some compounds
Dipole moments were measured, and the line shape function was assumed both Gaussian and Lorentzian. More uncertainty on horizontal axes.
Extinction, central frequency, solvatochromic shifts and the molecular radii were measured, and the line shape function was assumed to be the same as in 1PA. Less uncertainty on horizontal axes.