Coherent effects in femtosecond infrared spectroscopy

Chemical Physics ELSEVIER

Chemical Physics 200 (1995) 415-429

Coherent effects in femtosecond infrared spectroscopy P. H a m m

+

lnstitut fdr Medizinische Optik, Ludwig Maximilians-Universitiit Miinchen, Barbarastrasse 16, 80797 Munich, Germany

Received 19 June 1995

Abstract

The accessible time resolution in femtosecond infrared experiments is shorter than the typical phase relaxation time of a vibronic transition. Therefore, coherent interaction of the light pulses with the sample may disturb the observed absorbance signals. Coherence results in an artifact known as perturbed free induction decay, which may be misinterpreted as an intrinsic incoherent temporal evolution of the sample. In the present paper, a model is presented describing this effect for the general situation, where a complex molecule containing many overlapping vibrational modes is investigated. The model leads to an efficient linear least square fit algorithm allowing the a,dlysis of huge data sets. The model and the fit algorithm are applied to transient absorbance changes observed in a large dye molecule. It is demonstrated that it is possible to separate an ultrafast energy relaxation process from the perturbed free induction decay signal. In addition, the analysis of the perturbed free induction decay effect itself allows one to obtain information on the instantaneous absorbance change of the sample.

O. Introduction

With the technical progress in the generation of stable tunable femtosecond pulses in a wide infrared spectral range covering the region between 10000 and < 1000 cm -1, femtosecond spectroscopy can now be performed routinely [1-8]. For time resolved measurements of the vibrational spectrum of a typical organic molecules, the most important spectral region is found between 1000 and 1800 cm -1 and around 3000 cm -1. A number of infrared ( I R ) probe-femtosecond experiments have been published giving new insights into structural changes of molecules during photochemical reactions. Applications are in molecular physics (see for example

* Corresponding author.

[9--14]) and biophysics (see for example [ 5 - 7 , 1 5 18]). However, together with the development of laser systems with a pulse duration considerably shorter than 1 ps, a new problem appears: The typical spectral bandwidth of vibrational modes in solution is 1 0 - 2 0 cm - t corresponding to a phase relaxation time of 5 0 0 - 1 0 0 0 fs; longer than the pulse duration of the IR probing pulse. Under these experimental conditions, the coherence of the interaction between the probing pulse and vibrational transitions has to be taken into account. The coherent interaction results in an 'artifact' known as perturbed free induction decay which may be misinterpreted as a kinetic component due to a photoreaction. Qualitatively, time resolved pump and probe spectroscopy and the appearance of the perturbed free induction decay effect can be explained as follows (see Fig. 1): In a pump and probe experiment, an intense pump pulse is used to trigger a photoreaction. A second,

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P. Hamm / Chemical Physics 200 (1995) 415-429

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(a)

Probing Pulse

f

Time

Probing Pulse

Excitation Pulse

x<0

~.

(b)

Time

Fig. 1. (a) The envelopes of the light fields of the incoming probing pulse Ei(t) and of the radiated polarization light field Er(t) (free induction decay). (b) The envelopes of the light fields of the probing pulse Ei(t) (dashed line), of the excitation pulse (solid line) and of the perturbed free induction decay Er,~(t) (solid line) where ~- is the delay time between probing pulse and the excitation pulse. The perturbed free induction decay light field Er.,(t) disappears upon electronic excitation. The difference Er,~(t)-Er(t) is recorded as the perturbed free induction decay signal. Due to the temporal asymmetry of the free induction decay light field Er(t) (a consequence of causality), the perturbed free induction decay signal is observed only at negative delay times; independent on special model assumptions.

weaker probing pulse measures the response of the sample by recording the absorbance change of the sample as a function of the delay time 7 between the exciting (pump) pulse and the probing pulse. Due to the causality principle, an absorbance change is expected only at positive delay times (probing pulse reaches the sample after the pump pulse). However, due to the long dephasing time of a vibrational transition, the IR-probing pulse may excite a coherent polarization which decays with its intrinsic dephasing time T2. This coherent polarization radiates

light - the so-called free induction decay- and the IR-detector will record both the probing pulse and the radiated polarization. If an excitation pulse reaches the sample after the probing pulse (this will be called negative delay times), it can not influence the intensity of the probing pulse itself. However, it may influence the temporal or spectral properties of the free induction decay signal of the sample, whenever the electronic excitation process modifies the strength or position of the absorption line. As a consequence, the IR-detector will record a difference signal which decays toward negative delay times with an intrinsic time constant T2. It is important to realize that the perturbed free induction decay signal is a consequence of dephasing and essentially reflects the spectral properties of the sample in the ground state and in the excited state. It does not reflect any temporal evolution of the sample related to a chemical reaction or an energy relaxation process. Therefore, a realistic description of the molecular dynamics requires one to distinguish between contributions of the difference signal caused by the perturbed free induction decay effect ('coherent artifact') and 'real' kinetic components. Perturbed free induction decay was described previously for VIS-pump-VIS-probe spectroscopy, where the bleaching of a narrow exciton transitions in a GaAs-semiconductor was investigated with a time resolution of = 100 fs [19-21]. Another description in the case of femtosecond VIS-pump-IRprobe spectroscopy was given in [22] only recently. For simplicity, the interaction of the probing pulse with only one single absorption line was considered in these publications. In practice however, the vibrational spectrum of a larger molecule with an extended "rr-electron system contains many overlapping vibrational bands changing center frequency, oscillator strength and / or bandwidth (or phase relaxation time) upon electronic excitation. As a consequence, a complicated perturbed free induction decay signal due to the superposition of many absorption lines is expected. In the first part of the paper, a simplified model will be presented describing this multi-absorption line case which leads to an efficient linear least square fit algorithm. In the second part of the paper, the ability of the model will be demonstrated by fitting experimental data obtained by pump and probe experiments investigating a large dye molecule.

P. Harem/ ChemicalPhysics 200 (1995) 415-429 1. Theoretical description 1.1. Free induction decay The following description is based on a formalism developed in Ref. [20]. In a first step, the interaction between a single homogeneously broadened IR-absorption line and a short IR-probing pulse is considered. The probing IR pulse generates a coherent polarization in the sample which subsequently will decay with its intrinsic dephasing time T 2. This coherent polarization will radiate light in the same direction as the probing pulse and therefore will be measured by the detector. The complex light field of the coherent polarization Er(t) (the so-called free induction decay, see also Fig. la) is

E~(t)= f + ° ° M ( t - t ' ) E i ( t ' ) d t

',

(1.1a)

where El(t) is the light field of the incoming probing pulse and M(t) is the response of the sample to a &shaped pulse. For a Lorentzian shaped absorption line (homogeneously broadened, phase relaxation time T2) M(t) is

M( t) = 19( t) e -`/Tz e -i'~,t,

(1.1b)

where ~9(t) is the Heaviside function and toa is the center frequency of the absorption line. In the frequency domain Eq. (1.1a) can be expressed as Er(to) = M(to) E(to).

(1.2)

Here, M(to) is a Lorentz function with center frequency %. Eq. (1.1) and (1.2) are solutions of the Bloch equations of the density matrix solved to first order in Ei(t) (low excitation limit) [23]. The intensity recorded by a frequency resolved detector is I ( w ) = IEi(w) + E~( to)l 2 = IEi( to)l 2 + let( to)l 2 + 2Re El* (to) E,(to). (1.3) For a weakly absorbing line one finds [E~(to)l << IEi(to)l and the second term of Eq. (1.3) can be neglected. Eq. (1.1) describes the interaction between a short IR probing pulse and a homogeneously broadened IR-transition. However, free induction decay is a

417

linear effect and therefore, the light field Er(t) does not depend whether a transition is homogeneously or inhomogeneously broadened. Eq. (1.2) and (1.3) are also valid in the case of an inhomogeneously broadened line or even in the case of overlapping absorption lines, provided that a suitable function M(to) related to the shape of the absorption spectrum is used. In the following, no explicit distinction between homogeneously and inhomogeneously broadened lines will be made and T2 will be identified qualitatively as the inverse of the bandwidth A v (FWHM) of a transition ( T 2 = 1/'rrA v). In VIS-pump-IR-probe experiments, the coherence of the interaction between the VIS-pump pulse and the electronic transition can be neglected at room temperature since in general, electronic dephasing (or the inverse inhomogeneous bandwidth) of a dye molecule is considerably faster (in the 10 fs regime, see for example [24]) than vibrational dephasing.

1.2. Perturbed free induction decay of a single absorption line In a next step, the interaction between the free induction decay signal and a visible excitation pulse is introduced. Due to causality an interaction between an excitation pulse and an incoming IR-probing pulse El(t) is impossible when the VIS-excitation pulse reaches the sample after the IR-probing pulse (i.e. at negative delay times). However, as indicated above, the excitation pulse may interact with the free induction decay light field E,(t). First, consider the situation where the VIS excitation pulse leads to a bleaching of the IR absorption line, i.e. where the radiated field Er(t) disappears when the excitation pulse reaches the sample (see Fig. lb). In this case, a difference signal (called the perturbed free induction decay) will be observed: A/v(to) = I~(to) - I(to)

= 2Re Ei* ( to)[ Er,~( to) - Er( to)],

(1.4)

where Er(to) and I(to) is the light field and intensity of the free induction decay of the unperturbed sample (without visible excitation pulse), Er, r ( t o ) a n d Iv(to) the light field and the intensity of the perturbed absorption line. r is the delay time between the VIS excitation pulse and the IR probing pulse.

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As an abbreviation the (in general complex) function G~(t)=1-Er, r(t)/Er(t) is introduced, which describes the change of oscillator strength of the IR-absorption line upon VIS excitation. One obtains a general expression for the difference signal:

the absorption line. Off-resonance, an additional oscillating term with a frequency equal to the detuning between line center and detection frequency is observed.

AI,(to) =

-

-

l i ( tO )

2

E,*( OA)E i ( t O )

XReE,*( ,o) f ) f O,( o¢) XM(

to -- tot) E i ( (.o -- tot)

dto'. (1.5)

For the simplified case of &shaped excitation and probing pulses, one finds G , ( t ) = O ( r + t) and Ei(oJ) = 1 and can solve Eq. (1.5) analytically:

Wavenumber[cra-~]

~

30 "~'-~ -3

ATc(to,T ; toa) ct - C / r 2 c°s[(to - t o a ) z ] / T 2

-(to

- toa) sin[(to - toa)T]

(to - toa)2 + ( l / T 2 ) 2 r
ATc(to,r ;toa) c(

(to _ toa)2+ (1/T2)2 , , > 0 .

(1.6)

A 3D-plot of Eq. (1.6) is shown in Fig. 2a for a typical dephasing time T2 = 600 fs corresponding to a band width of 18 cm -1. For positive delay times (r > 0), the difference signal is static and shows the expected Lorentzian line shape. At negative delay times (¢ < 0), an exponentially rising signal with a time constant equal to T2 is found at the center of

Fig. 2. (a) A perturbed free induction decay model calculation considering infinite time resolution (&pulses, Eq. (1.6)). (b) A model calculation assuming finite time resolution (8-excitation pulse, 300 fs Gaussian shaped probing pulse, Eq. (1.8)) and identical center frequencies for the absorption line and the probing pulses. The spectrum of the probing pulse is shown in the background. (c) The same conditions as in (b) except for a frequency shift of 25 cm-~ between the peak of the absorption line and that 0f the probing pulses. In all plots, a vibronic dephasing time of 600 fs corresponding to a typical bandwidth of 18 cm -1 was assumed. The frequency coordinates are labeled with the difference between the detection frequency and the peak position of the absorption line t o - toa in (a) and with the difference between the detection frequency and the peak position of the probing pulses to - too in (b) and (c), respectively.

i ° er/crn j

Wav~amaber[-emqj

~

30 "---'~ -3

P. Harem/ Chemical Physics 200 (1995) 415-429

Integrating function (1.6) over frequency to yields L + Q°ATC ( to,~" ; toa) d to a - O ( T ) .

(1.7)

At negative delay times, this integrated signal is zero whereas at delay zero, the signal decreases instantaneously (&shaped excitation). This behavior can be understood in the frame of the uncertainty principle: When using frequency resolved detection, one obtains information on spectral properties of the sample and therefore, the temporal information (i.e. the temporal evolution of the signal) is limited by the response time of the sample (inverse of the bandwidth of the absorption line or the dephasing time T2) and not by the time resolution of the pump-probe set-up itself. When measuring without spectral resolution, time resolution is limited only by the instrumental response function since no further spectral information is obtained. For a realistic model calculation, the finite pulse duration of the laser pulses has to be considered. In the experiments described below, the probing pulses have a pulse duration of tp---300 fs (FWHM) whereas the excitation pulses are shorter (tp-~ 150 fs). For simplicity, the shorter excitation pulses are still modeled by &shaped pulses while the probing pulses are assumed to be Gaussian shaped (without chirp). In this case one obtains from Eq. (1.5): Arc(to,r;to~) = - 2 R e

111"2 +

i ( t o - toa)

_

1 e - (~- ~°°)z/a~2

(1.8)

]

with A ( ~o, O~o, Coa ,A oj ,T: ) i~/-~ A ~o

f +~e-t~dxa~2/4 el(ca- ~o)t -oo 2/9(t- r)(e ('-')/r2 e i('°- ~°aXr-t) -- 1) + 1 2

X

lIT 2 +

i(to - toa)

dt.

Here too and A to are the center frequency and bandwidth of the probing pulse. Expression (1.8) can be evaluated efficiently by numerical integration since the first term of the integral limits the required integration range. Two 3D-plots of function (1.8) are shown in Figs. 2b and 2c for different values of toa and to0. In Fig.

419

2b, the center frequency of the absorption line and that of the probing pulse are identical to each other whereas in Fig. 2c, the center frequency of the absorption line is shifted by - 25 cm- 1 with respect to that of the probing pulse. The spectrum of the probing pulse is shown in the background. Again, a vibronic dephasing time of 600 fs was assumed. Qualitatively, the plots shown in Fig. 2a (Eq. (1.6)) and that shown in Figs. 2b and 2c (Eq. (1.8)) are similar to each other except for two details: (i) The limited time resolution smoothes signal changes at delay zero (~"= 0) (ii) The oscillations at the spectral wings of the probing pulses are more pronounced. This is a consequence of the denominator in Eq. (1.5), i.e. a consequence of the normalization of the signal to the intensity Ii(to) of the incoming probing pulse. Nevertheless, the experiments shown in the second part of the paper are not disturbed by the large amplitude of the oscillations in the spectral wings of the probing pulses since only a limited bandwidth ( < 50 cm -1 ) is used in each experiment. So far, the complete bleaching of one single absorption line has been described. Other, more general situations can be deduced from this formalism: (i) An absorption band, where the oscillator strength decreases upon electronic excitation. In this case, the perturbed free induction decay signal is identical to the signal generated by a complete bleaching of the absorption line as treated in Eq. (1.8) except of a linear scaling factor. (ii) An absorption band, where the oscillator strength is increased upon electronic excitation. This situation yields the same difference signal as (i) with an inverted sign. In case (i) as well as in case (ii), the signal depends linearly on the change of oscillator strength G~(t). (iii) An absorption band, where the frequency is shifted upon electronic excitation. This situation is more complicated since coherence may be transferred during the frequency shift [22] resulting in a z-dependent phase factor in G~(t). However, as shown in Ref. [22], this case can be treated as a superposition of case (i) and (ii) to a good approximation (oscillator strength decreases at ground state position and oscillator strength increases at excited state position), provided that the spectral shift is small compared to the spectral bandwidth of the absorption line.

P. Harem / Chemical Physics 200 (1995) 415-429

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There is a fourth type of interaction which has not been considered so far: Whenever an IR-transition does not interact with the probing pulse in the initial state (i.e. as long as the sample is in the ground state) and interaction is switched on during the excitation process, there will be no free induction decay from the ground state absorption line. This case occurs when an absorption line is generated by the optical excitation process; for example when the transition is symmetry forbidden in the ground state or when it is shifted from a frequency position outside of the spectral range of the probing pulse. In this case, one will observe an instantaneous rise of the band limited only by the instrumental response function of the setup and no perturbed free induction decay signal will be observed. This analysis shows that there is a distinct asymmetry in the perturbed free induction decay response (bleaching versus rise of an absorption line) which is a consequence of the temporal asymmetry of the pump-probe process.

Section 1.1) provided that they have the same frequency profile. Consequently, one can treat a homogeneously broadened absorption line by a linear superposition of narrow absorption lines; i.e. one can treat the free induction decay of a broad homogeneously broadened absorption line in the same way as an inhomogeneously broadened absorption line. The distribution function a(oJa) defines the instantaneous absorbance change of the sample upon electronic excitation at the frequency position ~oa, i.e. the spectrum a(oJa) can be identified with the difference spectrum at delay zero ( r = 0) provided that the transmission change is small. In other words, the perturbed free induction decay signal AT'c(~O,r) is directly coupled to the difference spectrum at delay zero a(~oa) and therefore can be used to obtain independent information on the difference spectrum at delay zero by analyzing the perturbed free induction decay signal (see Section 2.2). In order to evaluate Eq. (1.9) numerically, the integral is written as a discrete sum:

1.3. Multi-absorption line case AT~(00,r) = Ea(~oa)ATc(~o,~';~oa), As mentioned above, in larger molecules the probing pulses do not interact with only one single absorption line but with many overlapping absorption lines each changing oscillator strength, center frequency a n d / o r bandwidth. Since the response of one single absorption line (see Eq. (1.5)) is linear in G~(t) (where G~(t) describes the change of oscillator strength of a single absorption line) one can treat the multi-absorption line case by a linear superposition of several distinct absorption lines, provided that these absorption lines are not coherently coupled. Since a priori the frequency positions o.)a and phase relaxation times T2 of these absorption line are not known, the following approach is used: =

lim f+~a(w~)ATc(~O,~-;O~a)d~oa, T 2 -.--~ ~

_

(1.9) where ATc(~O,~-;oJ~) is evaluated according to Eq. (1.8) with a long dephasing time T 2. The approach of Eq. (1.9) is justified since the free induction decay Er(t) does not depend whether an absorption line is homogeneously or inhomogeneously broadened (see

(1.10)

with regular frequency points ~oa and a finite, but long, dephasing time T2. In practice, the distance between two subsequent points A ~oa should be in the order of half the bandwidth of the most narrow spectral feature in the difference spectrum a(o~a) in order to obtain an adequate description of the difference spectrum, even if no single frequency point oJa matches the peak position of a certain absorption line exactly. The bandwidth of each homogeneously broadened contribution to Eq. (1.10) A v = 1/~rT 2 should be at least two times broader than A ¢oa in order to obtain a flat difference spectrum AT'c(~O,7) when a(~oa) is fiat. In the examples shown in the second part of the paper, a value of 5 cm-1 and 0.6 ps was chosen for A ¢0a and T2, respectively. The total integration range should be at least _ 50 cm( ~ 1/,rrCtp) larger than the frequency range of the measured experimental data since absorption lines outside the investigated frequency range may cause perturbed free induction decay signals inside this frequency range. In order to simulate the further temporal evolution

P. Harem/ ChemicalPhysics 200 (1995) 415-429

of the sample, a sequence of subsequent reactions is introduced. The kinetics of these reactions are modeled by exponential functions with time constants % convoluted with the instrumental response function. For simplicity it is assumed that coherence is completely destroyed during these reactions and can be ignored. The transmission change caused by these reactions is

AT~t (tO,'/') ---- E b i ( t o ) ATR,i('r), i

(1.11a)

with ATR,i(~') =)_ C ( t ) ( 1 - e -(~-O/~')dt.

(1.11b)

Here, bi(to) is the transmission change due to the reaction from state ( i - 1) to state (i) with time constant Ti. C(t) is the cross correlation function between excitation pulse and probing pulse. Provided that the transmission changes AT'c(tO,Z) and AT'r(toO') are small, the total time and frequency dependent transmission change becomes: AT(o)O') = ATe(to,r) + ATi(toO'),

(1.12)

and absorbance changes can be expressed as A A( to O') = - AT( to O')/In(10).

(1.13)

The fit procedure described in the following paragraph is based on these expressions.

1.4. Fit procedure Inspection of Eqs. (1.10) and (1.11a) shows that the total transmission change (Eq. (1.12)) is a linear superposition of two types of different base functions: the first is given by Eq. (1.8) and describes the perturbed free induction decay effect while the second is given by Eq. (1.11b) and describes a sequence of subsequent reactions. These base functions are multiplied with linear scaling factors a(to a) and bi(to); the first describes the difference spectrum immediately after electronic excitation (~"= 0 difference spectrum) while the second describes the absorbance changes due to subsequent reactions. Therefore, in order to fit an experimental data set to

421

these base functions, i.e. in order to evaluate the difference spectra a ( % ) and bi(to), in principle, a standard linear least square algorithm can be used [25]. For a realistic data set like that shown below, the matrix necessary for the least square fit algorithm becomes extremely large leading to numerical problems with accessible computer memory, processing time and round-off errors. However, since this matrix contains only a small number of non-zero matrix elements and has a particular block structure, a special algorithm was developed in order to solve the fitting problem (see Appendix A for details). It is a robust feature of the fit procedure that the main part of the fit parameters, i.e. the absorbance changes a(toj) and hi(to j) at frequency position toj, are linear parameters leading to an algorithm insensitive to initial estimates of the parameters. Additional nonlinear model parameters, i.e. the time constants ~'i, are evaluated by a subsequent iterating least square algorithm based on the Levenberg-Marquardt method [25].

1.5. Special remarks (i) One goal in developing the fit procedure was to separate the perturbed free induction decay effect from 'real' kinetic components caused by an ultrafast initial chemical reaction or energy relaxation. In the case of infinite time resolution (&shaped pulses), this distinction would be straight forward: For the present measurement technique, the perturbed free induction decay signal is strictly observed at negative delay times, independent of special model assumptions. This is a consequence of the temporal asymmetry of the free induction decay light field Er(t), i.e. a consequence of causality (see Fig. 1). Therefore, all three processes - the perturbed free induction decay (~"< 0), the excitation process (~-= 0) and subsequent reactions (~-> 0) - are separated in time. In a realistic experiment with a finite time resolution, a mixture of all three processes occurs within the cross correlation time. However, they are still ordered in time to a certain extent. This ordering yields a specific stability of the data handling procedure. (ii) In the literature, an alternative technique for transient IR-spectroscopy [5-7] is described. Here a cw-IR-laser beam is used for probing. After

422

P. Hamm / Chemical Physics 200 (1995) 415-429

passing the sample, the transmitted probing light now modulated by the transient absorbance changes of the sample is upconverted in a non-linear crystal by a short gating pulse in order to obtain time resolution. Under these experimental conditions, coherent signals caused by dephasing effects are also observed [22]. However, there are two striking differences in the detected signal, when comparing the cw-probe technique with that one described in the present paper: - In the case of the cw-probe technique, the bleaching of a band is observed instantaneously within the instrumental response function [22] whereas the rise of a band is delayed by the dephasing effects. - In the case of the cw-probe technique, the coherent artifact is observed at positive delay times and temporally overlaps with 'real' dynamic components [22]. Consequently, it should be more difficult to distinguish between both contributions. In summary, both methods show up a distinct but inverse asymmetry. Thus they can be understood as complementary techniques. (iii) The treatment of a band shift as an absorbance decrease at the ground state position and an independent increase of the excited state position is a rough approximation since it neglects the possibility of coherence transfer [22]. For a more sophisticated description, it would be necessary to introduce a 7-dependent phase factor into G~(t). However, the following arguments show that it is experimentally impossible to distinguish between a pure frequency shift (with complete coherence transfer) and the approach used here (no coherent coupling between ground state mode and excited state mode). The complex light field Er,~(t) (or the complex function G~(t)) contains the entire information about the system since it describes the amplitude as well as the phase response of the sample. However, Er,~(t) cannot be determined in a pump and probe experiment since only the real part of the integral in Eq. (1.5) is measured and the imaginary part is unknown. The information necessary for the evaluation of both the amplitude and the phase of Er,~(t) is lost during the measurement procedure. Therefore, both approaches (complete coherence transfer and that one used in the present paper) are approximations. For fitting

the data, the second approach was chosen since it results in a more simple and robust (since linear) fit algorithm.

2. Experiments The suitability of the model will now be demonstrated by two experiments illustrating two different aspects of the theory: While the first data-set can be explained well by the single-absorption line situation (see Section 1.2), the second data-set represents the multi-absorption line situation (see Section 1.3). Both data-sets were obtained by pump and probe experiments investigating a large dye molecule in two different frequency regions. The dye molecule contains an extended "rr-electron system which is changed considerably upon electronic excitation. As a consequence, one expects changes of intensity and frequency for all vibronic modes which are coupled to the rr-electron system. The experimental setup was described in detail in Refs. [2,3]. Here only the basic properties are repeated: A standard pump and probe scheme was used. The sample (laser dye IR26, Lambda Physics) was dissolved in 1-1-dichloroethane (concentration --~ 20 mmol/l) and held in a rotating CaF2-cuvette (pathlength ~-50 txm). The sample was excited by 150 fs pulses at 870 nm. The absorbance change of the sample was recorded by tunable probing pulses generated by difference frequency mixing between the output of a regenerative Ti:sapphire amplifier (repetition rate 1 kHz) and that of a synchronized traveling wave dye laser in a AgGaS2-crystal. The IR-pulses had a pulse duration of .~ 300 fs and a spectral bandwidth of = 60 cm-1 corresponding to a bandwidth product which is = 1.5 times larger than the bandwidth limit for Gaussian shaped pulses. For spectrally resolved detection, the probing pulses were dispersed in a grating spectrometer after passing the sample. The spectral resolution of the spectrometer was ~ 5 cm-1. A 10-element IR-detector array was used to cover the complete bandwidth of the probing pulses simultaneously. As a consequence, each single 10-channel experiments covers a spectral range of .~ 50 cm -1 where the center frequency of the probing pulses is tuned to the center of the detection range. The spectra shown below are composed of a

P. Hamm / Chemical Physics 200 (1995) 415-429

423

)

series of 10-channel experiment which overlap in a range of = 10 cm -1. The cross correlation width and the delay zero point of the experimental set-up was deduced from an independent experiment observing free carrier absorption in a thin silicon sample.

2.1. Single-absorption line case The experimental data measured in the frequency region between 1460-1580 cm -1 are shown in Fig. 3a in connection with the fit function calculated according to the model described in the first part of the paper. As seen at late delay times ( r > 1 ps), there is essentially one single absorption line, which bleaches upon electronic excitation. Consequently, in this spectral range one expects a result similar to that described by Eq. (1.8) and shown in Fig. 2b. Indeed, by comparing the model function (Fig. 2b) with the experimental data (Fig. 3a) one finds a good qualitative agreement especially at negative delay times. However, there are pronounced differences between this simple model function and the experimental data around delay zero, indicating that the experimental situation is more complex: One recognizes that electronic excitation is followed by a subsequent ultrafast relaxation process which will be assigned to energy relaxation of the hot dye molecule (see Section 2.3). The kinetics related to this energy relaxation process is best seen close to the peak of the absorption band at 1530 cm-1 where a fast recovering signal is observed (see arrow in Fig. 3a). In the tails of the absorption line, this ultrafast kinetic component is partially hidden by the perturbed free induction decay effect. As a consequence, the fit function shown in Fig. 3a can not be calculated according to the most simple model function Eq. (1.8) but has to be calculated according to the model described by Eq. (1.12) taking into account (i) the possibility of more than one absorption line in the difference spectrum a(coa) (see Eq. (1.10)) and (ii) taking into account subsequent reactions leading to additional absorption changes bi(co) (see Eq. (1.11)). As seen above, it is necessary to introduce an additional ultrafast kinetic component with a time constant of = 250 + 100 fs in order to model the experimental data. In addition, electronic relaxation from the excited state $I into the ground state S O leads to

"~ 1' 158(

-I

~

llZ~Venumber[ern.~

(b)

1480

'~ =

N --

7 N-8 g

o

tJ

,

-

-2

Fig. 3. Experimental data obtained by a pump-probe experiment investigating the laser dye IR26 (Lambda Physics). (a) shows the spectral range between 1460 and 1580 cm-1, where essentially one isolated absorption line is observed (single-absorption line case). (b) shows the spectral range between 1090 and 1200 cm-1 where at least 5 absorption lines contribute (multi-absorption line case). The arrows mark those frequency positions, where the 250 fs kinetic component (energy relaxation process) is directly evident.

a second slow kinetic component with a time constant of 20 + 5 ps (data not shown). The results of the linear fitting problem, i.e. the

P. Hamm / Chemical Physics 200 (1995) 415-429

424

difference spectra a(to a) and bl(to) are shown in Fig. 4b and 4c in comparison with the steady-state absorption spectrum of the dye molecule (Fig. 4a): - Fig. 4b shows the ~"= 0 difference spectrum a(w a) connected to the perturbed free induction decay effect. This difference spectrum essentially shows the bleaching of the ground state absorption line at 1530 cm -1 (see Fig. 4a). In addition, at the low frequency side of the measurement range ( < 1460 cm-1), a wing from a second absorption band (bleaching due to electronic excitation) appears. The peak of this absorption line lies outside of the investigated spectral range ( - - 1 4 3 0 cm -1, data not shown). Nevertheless, some modification of the perturbed free induction decay signal may be caused by this absorption line even inside the investigated frequency region. Fig. 4c shows the absorbance changes related to the 250 fs kinetic component (difference spectrum bl(tO)). This spectrum is negative at the low frequency side and positive at the high frequency side (interpretation of this spectrum is given below). In order to demonstrate that the 250 fs kinetic

(a)

0.5 0.0 -0.5 -l.0 e.

o)=1480 cm-I A(o~40cm-I

" " ° ° --

(b)

:

o.o -0.5 ,,o < -1.0 -1.5

f Ao~.40cm-i w° om(o=1520

I , -1.0

,

,

,

"

.

i X~.~'¢'i 0.0

,

,

h

,

I 1.0

,

""" ,

,

DelayTime [psi Fig. 5. Broadband measurements at (a) 1480 c m - 1 , A to = 40 cm -1 and at (b) 1520 cm -1, A t o = 4 0 cm -1 proving the existence of the ultrafast energy relaxation process. Due to frequency integration, the perturbed free induction decay effect is suppressed. The amplitude of the 250 fs kinetic component is negative at 1480 c m - 1 and positive at 1520 c m - 1 .

-

x

2O

2 xv -2-- 0

(So,v=O_._).v=1) -

~

"

~

.(Sa,v=l->v=2)

<

-2 •-4

I 1460

I 1480

-

-

-

-

(b} -

~

(el

.

(Spv=0-',v= 1) I I I 1500 1520 1540

I 1560

I 1580

Wavenumber[cm"1] Fig. 4. (a) Steady-state absorption spectrum of the dye IR26 between 1460 and 1580 cm -1. (b) The (~"= 0) difference spectrum a(wa), which is the difference spectrum immediately after electronic excitation and which is connected to the perturbed free induction decay effect. (c) The absorbance changes related to the 250 fs kinetic component bl(tO).

component and the difference spectrum shown in Fig. 4c are due to a real reaction and not caused by an improper treatment of the coherent artifact, two broadband measurements centered at 1480 and 1520 cm-1 are shown in Fig. 5a and Fig. 5b, respectively. These plots are calculated by integrating the experimental data of Fig. 3a over a frequency range of .= + 2 0 cm -1, i.e. they correspond to a lowfrequency resolved experiment with a resolution of = 40 cm -1. As shown in Section 1.2 (Eq. (1.7)), the perturbed free induction decay effect is suppressed under these conditions. Indeed, as seen in Fig. 5, no difference signal at negative delay times due to a perturbed free induction decay effect can disturb the observation of the photo induced reaction. Around delay zero ( - 3 0 0 fs < ~-< + 300 fs), the instantaneous response of the molecule upon electronic excitation (convoluted with the cross correlation function off pump and probe pulse) is observed which is positive around 1480 cm -1 and negative around 1520 cm -~ (see also fit spectrum a(to a) in Fig. 4b). At late delay times (~'> 1 ps), the approximately constant signals corresponds to a bleach at 1530 and at 1430 cm-1 which recovers with electronic relaxation within 20 ps. At intermediate delay time positions (300 fs < ~- < 800 fs), a fast relaxing signal is found which is related to an absorbance decrease (negative amplitude) at 1480 cm -1 and an ab-

P. Hamm / Chemical Physics 200 (1995) 415-429

sorbance increase (positive amplitude) at 1520 cm- 1; exactly as predicted by the fit spectrum bl(tO) in Fig. 4c. In Section 2.3, the ultrafast transient will be assigned to vibrational relaxation of an anharmonic mode.

425

80

8 60 40

"~ 2o ~ o 2

2.2. Multi absorption line case Fig. 3b presents experimental data measured between 1090-1200 cm -1. As seen in the difference signal recorded 1 ps after electronic excitation in Fig. 3b, at least five partially overlapping absorption lines contribute in this spectral region: Three strong bleaching bands are found at 1085, 1130 and 1145 cm-1; one weaker band is observed as a shoulder at 1180 cm -1 and one band lying outside the investigated region at 1215 cm -1 is seen as a wing (see also Fig. 6a). Consequently, due to the interference of all these absorption lines the perturbed free induction decay signal is expected to be complicated. This is manifested in relatively small amplitude signals at negative delay times. As in the frequency range between 1460 and 1580 cm -1, an ultrafast 250 fs kinetic component has to be introduced in order to model the experimental data. Again, the 250 fs kinetic component is directly evident at certain spectral positions (see arrow in Fig. 3b). However, even under these complex conditions, the procedure described in the first part of the paper yields an excellent fit. Fig. 6b shows the instantaneous (~'= 0) difference spectrum a(to a) which essentially reflects the steady-state absorption spectrum (Fig. 6a) indicating that all absorption lines of this spectral range bleach upon electronic excitation. In addition, a small broad positive background appears. Another check of the suitability of the fit procedure is demonstrated in Fig. 6c: As explained in Section (1.3), the model function AT'c(tO,~') couples the perturbed free induction decay effect (data points at negative delay times) to the (r = 0) difference spectrum (data points around delay zero). However, as mentioned before, the amplitude of the perturbed free induction decay effect is small in Fig. 3b due to destructive interference of the contributions of several absorption lines. In order to prove that even these small signals contain sufficient information to partially reconstruct the spectral properties of the sample at delay zero, i.e. to partially recon-

0

x g -4 g

4 < 0 -2 1080

1100

1120

1140

1160

1180 1200

1220

Wavenumber [cmZ] Fig. 6. (a) Steady-state absorption spectrum of the dye IR26 between 1090 and 1200 cm -1 . (b) The 7- = 0 difference spectrum a(to a) obtained by the standard fit procedure as described in Section 1. (c) The same spectrum obtained by a modified fit procedure, where only the data points at negative delay times are significant. Both spectra are essentially the same except for a constant offset (for a detailed explanation see text).

struct the (r = 0) difference spectrum a(O)a), a modified model function is used to calculate a(to a) in a different way. a r ( , o , ~ ) = ar~(o~,~) + ar~(o~,z) + Ar~(,o,~). (2.1) Compared to Eq. (1.12), an additional degree of freedom AT'~(~o,~-) is introduced allowing instantaneous absorption changes without any perturbed free induction decay effect. AT[(oJ;~') = c(o~) f ~ c ( t )

dt.

(2.2)

This additional term completely decouples the experimental data at negative delay times from those around delay zero. In this case, the function AT'c(o~,~') reflects the response of the sample at negative delay, the function AT'l(oJ,~-) the absorbance change around delay zero and the function AT'R(co,T) the progress of the sample at later delay times. Therefore, only the data points at negative

426

P. Hamm/ Chemical Physics 200 (1995) 415-429

delay times are significant for the determination of the function AT'c(tO,r), i.e. for the determination of the modified fit spectrum a'(tOa). This spectrum (see Fig. 6c) and the spectrum a(to a) calculated according to the standard procedure (see Fig. 6b) are essentially the same with the exception of a constant offset. The offset can be understood when one recognizes that a constant offset does not cause any perturbed free induction decay signal since it corresponds to an infinitely short dephasing time. In summary, this treatment shows that it is possible to gain almost the complete information on the (~-= 0) difference spectrum (with the exception of a constant offset) only by analyzing the perturbed fee induction decay effect.

/

s' .......

•

It °Pt(~a'. I ~[

I

25.0fs /

*

exeltatlonl Ik

/ ...... 1

V=2

v=l

v----0-~v=-i

! ......... I..... v--0 /

J

1%..,'¢

So

..................

V=|

71v=0 °

2.3. Molecular interpretation

In the previous section it was shown that a fast kinetic component with a time constant of 250 fs is found in the transient absorption data. In this section, an interpretation of this process will be given. In the experiment, the sample is excited at a wavelength of 870 nm, whereas the absorption maximum of the dye molecule is found at = 1100 nm. Consequently, the molecule is excited with an excess energy of --~ 2400 cm -1. Immediately after electronic excitation this excess energy is localized in vibrational modes which are coupled to the electronic transition (Frank Condon region). Subsequently, the excess energy will be dissipated over the whole molecule, predominantly into low frequency modes which are outside the investigated spectral region. As a consequence, in the electronically excited molecules, the investigated high frequency modes reach the lowest vibronic state v = 0 after a short time. There are experimental indications that the 250 fs kinetic component can be related to this process: Immediately after electronic excitation from the ground state (e.g. S0,v = 0) into the electronically excited state, the molecules populate a vibronically excited state (e.g. Sl,V = 1). As a consequence, absorption to the next vibronic state (transition v = 1 ~ v = 2) can be observed (see Fig. 7). During the relaxation into the vibronic ground state v = 0, this absorption disappears and a new band due to the v = 0 ~ v = 1 transition appears. Because of the anharmonicity of the potential surface of the electronically excited state $1, the frequency

Fig. 7. Molecular interpretation of the 250 fs kinetic componentas energy relaxation from an vibronically excited level v = 1 into the vibronic ground state v = 0. Evidence for this interpretation is deduced from the spectral properties of the 250 fs kinetic (see Fig. 4b), which can be well explained by the anharnlonicity of the potential surface of the electronically excited state S 1 (Morse potential).

of the v = 0 ~ v = 1 transition is higher than that of the v = 1 ~ v = 2 transition. The experimental observations exactly show this frequency dependence (see Fig. 4b or Fig. 5): The dominating feature in the spectrum shown in Fig. 4b is the bleaching of the v = 0 ~ v = 1 transition of the electronic ground state S 0. Later on, as observed in Fig. 4c, an absorption band assigned to the v = 1 ~ v - - 2 transition (of the electronically excited state S 1) at - - 1 4 6 0 cm-1 disappears within the 250 fs transient whereas at a slightly higher frequency ( = 1520 c m -1 ) an new absorption band related to the v = 0 ~ v = 1 transition appears. To our knowledge, this is the first direct observation of vibrational energy relaxation of a distinct mode in the S 1 state of a dye molecule.

3. C o n c l u s i o n

It is a special property of ultrafast IR spectroscopy that due to the long dephasing times of vibrational modes, the observation of fast molecular

P. Harem/Chemical Physics 200 (1995) 415-429

reactions is strongly hindered by perturbed free induction decay. In the present paper, it was demonstrated that perturbed free induction decay of a complex molecule may be described by a simplified physical model. Since the model function is composed of linear superposition of elementary base functions, an efficient least square algorithm can be used to fit experimental data. It is shown that this algorithm provides an excellent description of the experimental data. It simultaneously allows to separate the perturbed free induction decay effect (i.e. the coherent artifact) from an ultrafast reaction (energy relaxation as a 'real' kinetic component). Furthermore, it was demonstrated that the perturbed free induction decay signal can be used to obtain independent information on the difference spectrum immediately after electronic excitation.

ATc(to,r;to ~) (see

Eq. (1.10)) and 144 base functions of the type AT~.i(T) (see Eq. (1.11)). The matrix used in the least square algorithm (the socalled design matrix) has a size of 3600 × (40 + 144) elements which requires -- 3 MB of computer memory space. Another model calculation was performed for the time resolved IR difference spectra of bacterial reaction center of Rb. sphaeroides [18]. In this case, a memory space of = 40 MB would be necessary leading to enormous problems with available computer memory, processing time and roundoff errors. Therefore, a modified linear least square algorithm was developed taking into account the particular block structure of the design matrix. This algorithm will be described in this paragraph. A least square algorithm minimizes the square of the differences between the experimental data y and the linear model function f : X 2 = lY - f l 2 = ly - Xxl 2 = min.

Acknowledgements The author gratefully thanks W. Zinth for valuable discussions and S. Wiemann and M. Zurek for considerable help with the experiments.

The time and frequency dependent transmission change AT(to,z) (see Eq. (1.12)) is a linear superposition of elementary base functions. Therefore in principle, a standard linear least square algorithm can be used to evaluate the fitting parameters a(toa) and bi(to) [25]. However, for a realistic data set as the one presented in Section 2, the matrix necessary for the fit procedure is extremely large: For the fit of the data shown in Fig. 3b 50 × 72 = 3600 data points were related to 40 base functions of the type

'''

ATc(tO1;toa,I)

=

.

arC(tom;toa.,)

(A.2)

= 0.

For the fit problem described in this paper, X contains two different types of base functions: (i) the base functions ATc(to,T;to a) describing the perturbed free induction decay effect (see Eq. (1.8)) and (ii) the base functions ATR,i(T) describing exponential reactions (see Eq. (1.11)). While the base functions ATR.i('r) are independent on frequency to (see Eq. (1.11)), the base functions ATc(to,r;to a) depend on to (see Eq. (1.10)) and mix all data points with each other. As a consequence, the design matrix X has the following specific block structure:

ATR,1 "'" TR.k

0

0

arc(to.;toa.1)

(A.1)

X is the design matrix containing the base functions, x a vector containing the linear fit parameters and y a vector containing the experimental data. Eq. (A.1) is solved by the well known so-called normal equation [25]:

XTy-- X T X x

Appendix A

A T c ( tol ; toa,1)

427

0

0

0

/

0

/"

ATR, 1 " ' " TR, k

(A.3a)

P. Harem/ Chemical Physics 200 (1995) 415-429

428

Here, each element of the matrix X symbolizes a column vector containing all delay time points ~-. For example:

Solving of Eq. (A.7b) for b and introducing it into Eq. (A.6) yields Xe =

:=

(A.3b)

The vector y containing the experimental data (see Eq. (A.1)) is a function of two variables y = y(to,r) and is organized in the following way:

y(tOm,,'rl)

• "" y( o~,.O',))

(A.4)

The parameter vector x (see Eq. (A.1)) has the form

x T = ( a ( wa.1) " " " a( tOa,t); bl(O)l)

"'" bk(O)l) "'"

bl(C.Om)..,

bk(O)m) ) (A.5)

m is the number of frequency points oJ, n the number of time points r, l is the number of base functions ATc(wO';oJa) with center frequency wa and k is the number of base functions ATR,i(~-)with time constant ~'i- The size of the matrix X is (m × n,(l + k) × m), that of the vector x is (l + k) × m and that of y is (m × n). The left side of the matrix X is related to the perturbed free induction decay effect while the right side is related to subsequent reactions. In practice, the size of the right side of the matrix X ( = (m × n,k X m)) is large compared to that of the left side ( = (m × n,l)). However, the right side has diagonal block structure with most elements equal to zero. In order to take into account this particular structure, the design matrix X in Eq. (A.1) is split into two matrices: X 2 =

lY - A a - a b [ 2 = min.

(A.6)

Here, A contains the base functions ATc(w,w;w a) (left side of (A.3a)), B contains the base functions ATR.i('r) (right side of (A.3a)), the parameter vector a contains the fit spectrum a(to a) (Eq. (A.5)) and the parameter vector b contains the spectra bi(to) (Eq. (A.5)). Solving of Eq. (A.6) yields two coupled normal equations: ATy -- ATAa - ATBb = 0,

(A.7a)

BTy --

(A.Yo)

BTBb

-

B~Aa

= 0.

CAal 2,

(A.8a)

with C = B ( B T B ) - I B T - 1.

A T c ( t.Ol ,'/'n ; tOa,l )

y T = ( y ( t 0 1 , 7 1 ) "'" y(to1,7"n) ' ' '

ICy -

(A.8b)

Eq. (A.8) is an exact solution of the problem (A.1) which now can be treated numerically: Eq. (A.8a) is of the same form as Eq. (A.1) with a modified data vector C y and a modified design matrix CA. It is solved by a standard linear least square algorithm based on single value decomposition as described in Ref. [25]. However, the size of the matrix CA ( = (m × n,l)) is small compared to that of the original design matrix X ( = (m X n, (l + k) × m)). On the other hand, since B is a diagonal block matrix, the matrix C is a diagonal block matrix as well and can be evaluated for each block separately. The matrix inversion in Eq. (A.8b) can be performed by Cholesky decomposition since the size of each block in BTB is small ( = (k × k)). In conclusion, the least square algorithm is efficient and stable against roundoff errors. The linear fit of the experimental data shown in Fig. 3b takes less than = 1 min of processing time on a DECAXP3000L workstation; fast enough to allow an additional fit of nonlinear parameters like the time constants z i by an additional iterating least square algorithm based on the Levenberg-Marquardt method [25].

References

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P. Harem / Chemical Physics 200 (1995) 415-429 [8] T.P. Dougherty and E.J. Heilweil, Opt. Letters 19 (1994) 129. [9] T. Els~isser and W. Kaiser, Chem. Phys. Letters 128 (1986) 231. [10] H. Graener, G. Seifert and A. Laubereau, Phys. Rev. Letters 66 (1991) 2092. [11] D. Raftery, E. Gooding and R.M. Hochstrasser in: Ultrafast Phenomena IX, eds. P. F. Barbara, W. H. Knox, G.A. Mourou and A.H. Zewail, Springer Series in Chemical Physics 60 (Springer, Berlin, 1994) p. 111. [12] A.G. Yodh, J.P. Culver, M. Li, L.G. Jahn and R.M. Hochstrasser in: Ultrafast Phenomena IX, eds. P.F. Barbara, W.H. Knox, G.A. Mourou and A.H. Zewail, Springer Series in Chemical Physics 60 (Springer, Berlin, 1994) p. 291. [13] T. Lian, B. Locke, Y. Khoiodenko and R.M. Hochstrasser, J. Phys. Chem. 98 (1994) 11648. [14] T.P. Dougherty and E.J. Heilweil, J. Chem. Phys. 100 (1994) 4006. [15] R. DiUer, M. lannone, B.R. Cowen, S. Maiti, R.A. Bogomolni and R.M. Hochstrasser, Biochemistry 31 (1992) 5567. [16] T.A. Jackson, M. Lim and P.A. Anfinrud, Chem. Phys. 180 (1994) 131.

429

[17] P. Harem, M. Zurek, W. M~intele, M. Meyer, H. Scheer and W. Zinth, Proc. Natl. Acad Sci. USA 92 (1995) 1826. [18] P. Harem and W. Zinth, J. Phys. Chem. 99 (1995) 13537. [19] B. Fluegel, N. Peyghambarian, G. Olbright, M. Lindberg, S.W. Koch, M. Joffre, D. Hulin, A. Migus and A. Antonetti, Phys. Rev. Letters 59 (1987) 2588. [20] M. Joffre, D. Hulin, A. Migus, A. Antonetti, C.B. Guillaume, N. Pegyhambarian, M. Lindberg and S.W. Koch, Opt. Letters 13 (1988) 276. [21] C.H. Brito Cruz, J.P. Gordon, P.C. Becker, R.L. Fork and C.V. Shank, IEEE J. Quantum Electron. QE-24 (1988) 261. [22] K. Wynne and R.M. Hochstrasser, Chem. Phys. 193 (1995) 211. [23] A. Lauberau and W. Kaiser, Rev. Mod. Phys. 50 (1978) 607. [24] P.C. Becker, H.L. Fragnito, J.Y. Bigot, C.H. Brito Cruz, R.L. Fork and C. V. Shank, Phys. Rev. Letters 31 (1989) 505. [25] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical recipes in C (Cambridge Univ. Press, Cambridge, 1992)