Quantitative description of two-photon absorption in dipolar molecules with two-level model

Nikolay Makarov, Mikhail Drobizhev, Zhiyong Suo, Aleks RebaneE. Scott Tarter, Benjamin D. Reeves, Brenda Spangler

Fanqing 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

No

rmal

ized

flu

ore

scen

ce

1 2 3 4 5 6 70

0.1

0.01

10-3

1

12, =1.14 ns

11, =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 0.5

0

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

Sto

kes

Sh

ift,

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,

GM

2, GM

21cos215

22 22

01

2

012

44

2 gnch

fa

0 100 200 300

0

100

200

300

2, GM

112

12

10

5

416

9

7

8

3

2,

GM

14

2

12201

14

23

01202 1096.0

PAPA

PAPASb

fnDf

nfa

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

1

1

i i

iel

iR

yy

N

12

3

3

22

22

n

nf

nf 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

2

01

1

24

3

fn

h

flFM

FFM

OD

OD

RR

FRF

R

dFn

dFn

101

1012

2

Dfahc S

3

2

01

12

12

D

DDf

31

4

3

AN

Ma

1

4.04

33

r

kTa

II

IIr

2||

||

3

0001

Ahc a

f D

250 300 350 400 450 500 5500

100

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300

400

2,

GM

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)

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Substituted diphenylaminostilbenes

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Meso-DPAS and BDPAS-substituted porphyrins

Push-pull porphyrin monomer and push-pull porphyrin dimer

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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.