An Analytic Model for the Determination of Excited State 

Parameters from Pump-probe Measurements 

by Timothy M. Pritchett, Andrew G. Mott, 

Michael J. Ferry, and Robert C. Hoffman 

ARL-TR-3327 September 2004 

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Army Research Laboratory 

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ARL-TR-3327 September 2004 

An Analytic Model for the Determination of Excited State 

Parameters from Pump-probe Measurements 

Timothy M. Pritchett, Andrew G. Mott 

Michael J. Ferry and Robert C. Hoffman 

Sensors and Electron Devices Directorate, ARL 

Approved for public release; distribution unlimited. 
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An Analytic Model for the Determination of Excited State Parameters from 

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Timothy M. Pritchett, Andrew G. Mott, Michael J. Ferry and Robert C. Hoffman 

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14. ABSTRACT 

In order to completely and accurately characterize the materials whose nonlinear optical response makes them good candidates for use in eye and sensor 

protection applications, one must measure not only the absorption and refraction cross-sections of the various quantum energy levels involved in the optical 

absorption process, but also the decay rates of these states. This can be accomplished in a single experiment using time resolved pump-probe techniques. In a 

pump-probe experiment, a strong pump pulse excites a sample of the material under investigation, and one measures the transmittance of a weak probe as a 

function of time. In a double pump-probe experiment, two pump pulses excite the sample sequentially. The first pulse populates various molecular excited 

states, while the second induces transitions whose rates depend on the various populations and cross-sections. As in a single pump-probe experiment, one 

measures the sample transmittance of a weak probe beam or pulse as a function of time, from which one can infer the relevant excitation and decay rates. 

Parameter values are obtained by fitting a theoretical model to the experimental data, a procedure that may be carried out with particular ease if the model has an 

analytic solution. We describe the most complete model for the normalized probe transmittance that may still be solved analytically. 

15. SUBJECT TERMS

Pump-probe measurements, optical nonlinearities in organic materials 

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Timothy M. Pritchett 

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iii

Contents 

Table of Figures iii

1. Introduction 1

2. The Model 2

3. Experimental Validation of the Model 4

4. Analysis and Conclusions 5

5. References 6

Distribution List 7 

Table of Figures 

Figure 1. The five-band model.........................................................................................................2 

Figure 2. Three-band effective model.............................................................................................3 

Figure 3. Normalized transmittance of a 7-mW CW probe by a 1-mm sample of 1.39-mM 

C60 in the presence of a 9-ns FWHM, 2.85-μJ pump pulse.v....................................................5 
iv

INTENTIONALLY LEFT BLANK. 
1

1. Introduction 

The absorption of energy from an intense laser pulse propagating through a medium redistributes 

the molecules of the medium among the ground state and various excited states. In general, 

these states differ significantly in their intrinsic ability to absorb and refract optical radiation. 

Thus, population redistribution typically results in a transient alteration in the bulk optical 

properties of the medium. By monitoring the manner in which the medium’s optical properties 

evolve in time, one can infer not only the intrinsic optical properties of the various molecular 

states, but also the rates at which these states decay as the material relaxes back to its initial, 

unperturbed state. The monitoring may be accomplished using a second laser pulse (the probe), 

generally much weaker than the first (the pump). 

Pump-probe techniques using a single fast (sub-nanosecond) pump pulse have long been used to 

measure the lifetimes and absorption cross-sections of the excited singlet states (1). Swatton (2) 

pioneered use of a second picosecond pump pulse in an effort to obtain additional information 

about the triplet excited states. These states are inaccessible directly from the ground state and 

for this reason are not significantly populated until some time after the arrival of the initial laser 

pulse. By choosing the time delay between the two pump pulses to be on the order of the 

crossing time between the singlet and triplet manifolds, Swatton succeeded in measuring various 

photophysical parameters of both singlet and triplet excited states simultaneously. This is 

extremely important, since the triplet states play a dominant role in determining the nonlinear 

optical absorptive properties of many of the most promising materials for eye and sensor 

protection applications. More recently, McEwen and coworkers (3) employed a similar double 

pump – probe technique using a pair of nanosecond pump pulses and a continuous wave probe 

beam to measure the photophysical parameters of a series of porphyrin compounds. 

Nonlinear absorption in a wide variety of organic dyes is well described by a model consisting of 

five vibrational-rotational bands: a singlet ground state, two singlet excited states, and two 

triplet excited states (4). The arrangement of the various bands is shown in figure 1. Parameter 

values are obtained by fitting a theoretical model to the experimental data, a procedure that may 

be carried out with particular ease if the model has an analytic solution. Unfortunately, the rate 

equations comprising the five-band model admit no such analytic solution and must be solved 

numerically. The simplified model presented here yields an analytical expression for the 

normalized transmittance of a CW probe in the presence of a strong pump pulse. 
2

kisc

kS2 

S0

S1

S2

T1

T2

σT

σG

σS

kS

kT

kT2 

Figure 1. The five-band model. The σ’s are absorption cross sections; the k’s are 

rate constants for the indicated transitions among the various bands. 

2. The Model 

The pump and probe propagate along the z-axis through a sample of thickness L. Both are 

spatially Gaussian TEM0,0-mode beams; their spot radii are, respectively, wpump and wprobe . The 

probe is a continuous wave source while the pump is Gaussian in time. We approximate the 

pump by the temporally clipped Gaussian with irradiance 







 − < < =

−

 0, otherwise

( , ) , ( , , )

2 ( / )

ε ε

τ I r z e t t t I t r z

t

Peak . (1) 

The on-axis pump fluence at the input face of the sample is then 

1 1

2 2 2

0,

2 ( ,0,0) (1 ) [ / ],

(1 ), pump

t t

E F t erf t t t t

w

t t

ε

ε ε

ε

ε τ

π

ε

 ≤ −  = − + − << 

 − ≥ 

, (2) 

where E is the total energy delivered by an unclipped pump pulse, ε is fraction of that energy 

contained in the omitted tails, erf[x] is the Gaussian error function, τ is the HW(1/e)M temporal 

pulse width, and 1 t erf [1 ] ε τ ε − = − . 
3

In order to obtain a set of rate equations that may be solved analytically, we replace the five-band 

model with a three-band effective model consisting of a ground state and two excited states of 

equal spin multiplicity. The model is illustrated in figure 2. We assume that the decay from the 

ultrafast

σG

σex

k

0

1

2

Figure 2. Three-band effective model. 

second excited state is so rapid that the population of this state is always negligible. We further 

assume that the lifetime −1 k of the first excited state is sufficiently long ( kτ <<1) that we may 

neglect the effects of the decay from this state during the interval ε ε − t < t < t in which the pump 

irradiance is non-zero. The fraction of X of the absorbing molecules in the first excited band 

then evolves according to 

X I

t h

X G = (1− ) ∂

∂

ν

σ (3a) 

in the presence of the pump ( ε ε − t < t < t ) and according to 

kX

t

X = − ∂

∂ (3b) 

during the subsequent refractory period ( ε t ≥ t ). Here, σ G is the ground state absorption crosssection, which is related to the linear absorption coefficient α 0 and the number density N0 of 

absorbing molecules by α 0 = N0 σ G ; ν is the frequency of the pump radiation; h is Planck’s 

constant; and I the pump irradiance. Both pump and probe irradiances attenuate according to 
4

N ( ) X X I

dz

dI = − σ G − +σ ex (1 ) 0 , (4) 

where σ ex is the absorption cross-section of the excited band. 

Introducing the saturation fluence FS hν σ G = / , we assume that F(t = ∞,r = 0, z = 0), the total 

pump input fluence on-axis, is sufficiently small that the more stringent of the following 

conditions is satisfied: 

1 ( , 0, 0) 1

2

1

<<

∞















 −

ex S

G

F

F

σ

σ

 or 1 ( , 0, 0)

<<

∞

FS

F (5) 

With these assumptions, the normalized transmittance of the probe beam is 







≥

<

×

















+

−















 = − − − −

−

ε

ε

α

ε σ

σ

t t

t t

F e

F t

w

w

e

P t

P t

k t t

S

pump

probe

L

ex

G

Linear ,

( , 0, 0) 1,

1

1 1 1

( )

( )

2 ( )

0

. (6) 

It is straightforward to generalize (6) to the case of multiple pump pulses. 

3. Experimental Validation of the Model 

In order to validate the model, we conducted a series of pump-probe measurements on a 1-mm 

sample of 1.39-mM C60 in toluene. C60 was chosen for this study because its triplet yield is close 

to unity; in such a material, the actual decay rate of the lowest-lying triplet band should show 

close agreement with the decay rate of the first excited band in the model. The sample was 

pumped with 9-ns (FWHM) pulses from a Continuum Powerlite 7000 frequency-doubled, 

injection-seeded, single-longitudinal-mode Nd:YAG laser operating at a repetition rate of 10 Hz. 

The experiment employs a 514-nm, continuous-wave probe beam from a Lexel 95 Ar+

 laser. 

Telescopes set the spot sizes of the pump and probe beams at the sample at 150 μm and 42.5 μm, 

respectively. The two beams are aligned collinearly and have orthogonal polarizations. After 

the two beams pass through the sample, the probe is separated from the pump using a polarizer. 

The probe beam is then focused onto a fast, sensitive Nu-Focus photodiode by a 75-mm lens. 

An Edmund Scientific notch filter for 514 nm blocks any residual scattered pump light from 

reaching the probe photodiode. Figure 3 shows the data (gray dots) from two such 

measurements, one extending over temporal range of 70 ns and the other over 1.75 μs. In each 

case, the pump pulse delivered 2.85 μJ. The solid black line indicates the predictions of the 

model, computed from equation (6). 
5

- 20 - 10 0 10 20 30 40 50

0.88

0.9

0.92

0.94

0.96

0.98

1

 0 250 500 750 1000 1250 1500 1750

0.88

0.9

0.92

0.94

0.96

0.98

1

 Time Delay from Pump Pulse Peak (ns) 

Figure 3. Normalized transmittance of a 7-mW CW probe by a 1-mm sample of 1.39-mM C60 in the presence 

of a 9-ns FWHM, 2.85-μJ pump pulse, focused to a spot of radius 150 μm. The probe spot radius is 

42.5 μm. 

4. Analysis and Conclusions 

An independent measurement using a spectrophotometer yielded a value of 2.57 cm–1 for α 0 , the 

linear absorption coefficient of the sample, corresponding to a ground state absorption crosssection σ G of 3.1 × 10–18 cm2

. For the purposes of the calculation, we set ε = 0.01, i.e., we 

assume that the clipped pulse posited by the model contains 99% of the energy delivered by the 

pump pulse used in the experiment. The theoretical curves shown in figure 3 were generated by 

choosing the parameters σ ex and k to give the best fit to the experimental data. The resulting 

“best fit” value of 1.7 × 10–17 cm2

 for the excited state absorption cross-section σ ex in the model 

agrees extremely well with value of 1.6 × 10–17 cm2

 quoted in [4,5] for the absorption crosssection of the first singlet excited state of C60. Likewise, the “best fit” value of (340 ns)–1 for the 

excited state decay rate k shows excellent agreement with the rate of decay from the lowest-lying 

triplet state in C60 to the ground state, reported in [6] as (330 ± 25 ns)–1 in air-saturated benzene 

solution. The experimental pump pulse energy corresponds to a fluence ratio F(∞ , 0, 0) / FS of 

0.13, so the conditions of (5) are satisfied. 

In summary, we have presented a simple, analytic model for the normalized transmittance of a 

weak CW probe in the presence of a (relatively) strong pump pulse in the low fluence regime. 

Because it neglects excited state decay for the duration of the pump pulse, the model would be 

expected to provide best results in the limit of short (picosecond) pump pulse widths. 

Nonetheless, the predictions of the model show excellent agreement with the results of an 

experimental pump-probe study of C60 employing 9-ns pump pulses. Normalized Probe Transmittance 
6

5. References 

[1] Mansour, K.; Fuqua, P.; Marder, S. R.; Dunn, B.; Perry, J. W. Solid-state optical limiting 

materials based on phthalocyanine containing polymers and organically-modified sol-gels. 

in Proc. SPIE 1994, 2143, 239. 

[2] Swatton, Steward N. R.; Welford, Kevin R.; Hollins, Richard C;. Sambles, J. Roy. A time 

resolved double pump-probe experimental technique to characterize excited-state parameters 

of organic dyes. Appl. Phys. Lett. 1997, 71(1), 10–12. 

[3] McEwan, K. J.; Bourhill, G.; Robertson, J. M.; Anderson, H. L. The Nonlinear optical 

characterization of meso-substituted porphyrin dyes. J. Nonlinear Optical Physics and 

Materials 2000, 9(4), 451–486. 

[4] Perry, J. W. “Organic and metal-containing reverse saturable absorbers for optical limiters. 

in Nonlinear Optics of Organic Molecules and Polymers, H. S. Nalwa and S. Miyata, eds. 

(CRC Press, Boca Raton, Fla. 1997), pp 813–840. 

[5] McLean, D. G.; Sutherland, R. L.; Brant, M. C.; Brandelik, D. M.; Fleitz, P. A.; Pottenger, T. 

Nonlinear absorption study of a C60-toluene solution. Opt. Lett. 1993, 18(11), 858–860. 

[6] Arbogast, J. W.; Darmanyan, A. P.; Foote, C. S.; Rubin, Y.; Diederich, F. N.; Alvarez, M. 

M.; Anz, S. J.; Whetten, R. L. Photophysical Properties of C60. J. Phys. Chem. 1991, 95, 

11–12. 
7

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