## Abstract

We examine the effect that pulse chirp has on the shape of two- dimensional electronic spectra through calculations and experiments. For the calculations we use a model two electronic level system with a solvent interaction represented by a simple Gaussian correlation function and compare the resulting spectra to experiments carried out on an organic dye molecule (Rhodamine 800). Both calculations and experiments show that distortions due to chirp are most significant when the pulses used in the experiment have different amounts of chirp, introducing peak shape asymmetry that could be interpreted as spectrally dependent relaxation. When all pulses have similar chirp the distortions are reduced but still affect the anti-diagonal symmetry of the peak shapes and introduce negative features that could be interpreted as excited state absorption.

© 2010 OSA

## 1. Introduction

Two-dimensional optical spectroscopy is a powerful tool for the study of ultrafast processes in the condensed phase, including energy transfer [1,2], solvation [3,4], and hydrogen-bonding dynamics [5–7]. A 2D spectrum can be regarded as a frequency resolved pump-probe experiment, where both excitation and detection frequencies are spectrally resolved, revealing correlations between the detection frequency and the initial excitation frequency [3]. In condensed-phase systems where transitions are congested, the time-dependent evolution of peak shapes in 2D spectra reveals the relaxation of the system as it interacts with its local environment. In particular, the ratio of diagonal to anti-diagonal widths of 2D spectra has been taken as a measure of the degree of inhomogeneous broadening [6,8,9]. Since the ultrafast pulses used to perform 2D experiments often have some residual chirp, understanding the effect of this chirp on the 2D peak shapes is essential for proper interpretation of 2D experiments. The potential for exploiting coherent control of 2D spectra to highlight weak features and obtain desired spectroscopic targets also motivates this study [10,11]. In this paper we present calculations on a simple two-level system to explore the effects of chirp on 2D spectra. We compare these calculations with experiments on the dye Rhodamine 800. We find that chirp effects are most severe when the different pulses used in the 2D experiment have different amounts of chirp. In this case the distortions that arise are asymmetric in nature and could be interpreted as spectrally-dependent relaxation, with the nature and degree of the distortion determined by the sign and magnitude of the pulse chirp. When all pulses are chirped equally the distortions are reduced, but an asymmetry above and below the diagonal is introduced, which sometimes produces negative signals that could be mistaken for excited-state absorption features.

## 2. Theory and calculations

Two-dimensional electronic spectra were calculated by first computing the third order polarization emitted from the sample upon excitation by a sequence of three laser pulses as depicted in Fig. 1 . The electric field of the jth pulse is given by:

*t*),

_{j}*σ*is the intensity FWHM of the pulse and ${\varphi}_{j}\left(t\right)$ is an additional temporal phase due to pulse chirp. It is often easier to express the chirp in terms of the spectral phase of the pulse, which can be expanded in a Taylor series about the pulse center frequency ω

_{0}:

**k**

_{j}, and the polarization is detected in the phase-matched direction

**k**

_{s}= -

**k**

_{1}+

**k**

_{2}+

**k**

_{3}. The emitted third order polarization is given by a triple convolution of the incident fields with the third order response of the system. Under the rotating wave approximation the polarization is given by:

_{1}, τ

_{2}and τ

_{3}are the positive times at which each pulse interacts with the system relative to the time t at which the polarization is detected. The variables T and τ are the waiting and coherence times, and are given by T ≡ t

_{3}– max(t

_{1}, t

_{2}) and τ ≡ t

_{2}– t

_{1}. While the pulses are temporally overlapped, the ordering of their interactions with the sample is arbitrary and all orderings must be taken into account [13].

By Fourier transforming with respect to t and τ, a 2D frequency domain polarization, *P*
^{(3)}(*ω _{t}, T, ω_{τ}*), is obtained for a given waiting time. In our experiment we measure the polarization heterodyned with a probe field given by the third laser pulse. We take this into account by multiplying the polarization with the frequency domain representation of the third pulse:

For a given waiting time T, the time domain polarization was calculated on a grid of points with τ ranging from −300 fs to + 300 fs (which includes both rephasing and nonrephasing pathways), and t ranging from –2x(full width at half maximum (FWHM) of longest pulse) to 600 fs. Care was taken to ensure that the signal decayed to zero at the extremes of the grid. In all cases pulse center frequencies were resonant with the bare electronic transition frequency, which removes fast optical frequency oscillations in the polarization and allows a time increment of 2 fs in both t and τ, which is more than adequate to fully sample the signal. The triple integrals were calculated using Fortran code generously provided by David Jonas and modified to include the effect of pulse chirp [12,13]. This code takes all possible pulse interaction orders into account, which is especially important when there is pulse overlap. The Fourier transforms and were carried out in separate Matlab code.

The Brownian oscillator model [14] was used to describe the model two level system. The key parameter in this model is the energy gap correlation function *M(t)*, which describes the interaction of the system with a bath of harmonic oscillators. For fast solvation dynamics, a Gaussian *M(t)* is often used [13,15]

_{g}is the correlation time and represents the time scale of the bath interaction. From this correlation function, the lineshape function in the high temperature limit can be calculated from

^{2}is the coupling strength to the bath. In the high temperature limit they are related through Δ

^{2}= 2k

_{B}Tλ/ℏ. For the calculations presented here, we chose a correlation time of τ

_{g}= 150 fs, a reorganization energy of λ = 45 cm

^{−1}, and a temperature of T = 298 K.

A common distortion to ultrafast pulses is uncompensated positive second order dispersion, leading to a positive linear chirp. In one-color experiments, residual chirp is likely to be similar for all three pulses, while in two-color experiments, chirp may be different between pump and probe pulses. We examined both of these cases through simulation. The first calculations were performed with transform limited pulses with an intensity FWHM of 30 fs for both pump and probe pulses. Absorptive 2D spectra for different waiting times were then calculated. To determine the effect of linear pump pulse chirp on the 2D line shape, a chirp of *ϕ*′′ = 1000 fs^{2} was applied to the first two pulses. This broadened the pulses from 30 fs to 97 fs, while still maintaining the same bandwidth as the initial pulses. By applying a positive chirp, the blue frequencies now arrive later in time than the red frequencies. The final (probe) pulse duration was unchanged at 30 fs. We also calculated spectra with an opposite pump pulse chirp of *ϕ*′′ = −1000 fs^{2}. In Fig. 2
we plot the calculated 2D spectra at several waiting times for negatively chirped, transform-limited, and positively chirped pump pulses. The ω_{1} and ω_{3} axes can be regarded as the initial excitation and final detection frequency of the chromophore. With all transform-limited pulses (Fig. 2 center row), the spectrum is elongated along the diagonal at T = 0 fs, indicating that there is an initial inhomogeneous distribution of frequencies. As the waiting time is increased, the spectrum begins to round, indicating a loss of memory of the initial excitation frequency (T = 60 fs.) As the waiting time is further increased to 120 fs, the spectrum looks nearly circular, showing that the system has lost all memory and can now be regarded as a homogeneous distribution. At T = 0, there is a significant distortion in the chirped-pump spectra (Fig. 2 top and bottom rows). For the negatively-chirped case (Fig. 2 top row), the anti-diagonal width at blue frequencies is now much broader than at red frequencies, giving a “teardrop” shape to the spectrum. As the waiting time is increased to 60 fs this distortion can still be seen. Even at 120 fs, a time at which the unchirped pulses showed a near circular spectrum, the chirped pulse case still has a small distortion present. In the positively-chirped pump pulse case (Fig. 2 bottom row) we see similar dynamics to the negatively-chirped case, however the initial “teardrop” shape is oriented in the opposite direction with the broadening appearing at lower frequencies.

For a 2D spectrum, the anti-diagonal width can be used as a measure of the homogeneous broadening. By taking anti-diagonal slices through the spectrum and measuring the FWHM, the relative degree of homogeneity can be determined across 2D spectrum. Plots of the anti-diagonal widths at three different frequencies are shown in Fig. 3
as a function of T for transform-limited, positively-chirped pump pulses and negatively-chirped pump pulses. The results show that with transform-limited pulses, the width increases at the same rate for all frequencies, indicating frequency independent spectral relaxation. For positively-chirped pump pulses, the width of the spectrum is broad at the lowest frequency and decreases as the frequency increases. As the waiting time is increased, the widths at each of the frequencies increase at different rates until they reach a common value, suggesting that the system exhibits spectrally heterogeneous relaxation. The negatively-chirped case shows the same trend but with the frequency dependence reversed. In both positively and negatively chirped cases the distortions due to pulse chirp have vanished by ~T = 300 fs. In Fig. 4
we show the effect of the magnitude of the pump pulse chirp on the resulting 2D spectra, indicating that even relatively small chirps of *ϕ*′′ = 500 fs^{2} can lead to distortions at early waiting times.

We also examined the effect of using transform-limited pump pulses and chirped probe pulses, a distortion that is often present in 2D experiments with a continuum probe pulse [16]. Figure 5 shows the calculated spectra for this case, where both positive and negative chirps are considered. In this case the resulting distortions are similar to those obtained with chirped pump pulses, but with an opposite frequency dependence: the distortions for positive/negative probe pulse chirp resemble those observed for negative/positive pump pulse chirp respectively.

In Fig. 6 we show calculations of the effect of using equivalently-chirped pump and probe pulses. In this case the frequency-dependence of the distortions is reduced, but an asymmetry remains with respect to the diagonal that depends on the sign of the chirp. In particular, at early times, negative features are enhanced that could be interpreted as excited-state absorption.

## 3. Experiments

For comparison with some of the calculated spectra 2D Fourier transform electronic spectroscopy experiments were performed in a pump-probe geometry, as described in detail previously [17]. Briefly, the laser source consists of two non-collinear parametric amplifiers (NOPA) [18]. The first is directed into an acousto-optic pulse-shaper (DAZZLER), which is programmed to create a pair of collinear pump pulses with a variable delay between them. This delay is the τ delay, or the coherence time in the experiment. The output of the second NOPA (the probe pulse) propagates down a variable delay line, which creates a delay T (waiting time) between the second pulse in the pump pair and the probe pulse. Both beams are crossed in the sample at an angle of ~3°. Pump and probe spot sizes were 150 μm and 90 μm, respectively. Pulse energies in both beams were 12 nJ/pulse. After the sample, the probe beam was directed into a spectrometer, which acts to spectrally resolve the signal. The τ delay was scanned from 0 to 300 fs in 1 fs steps. For each delay value, an average of ~250 spectra was recorded, using the four phase-cycling scheme implemented by the Zanni group [19]. Since the probe beam also acts as a local oscillator for heterodyne detection of the signal, properly phased absorptive 2D spectra are obtained when the data is Fourier transformed along the τ direction [13,19,20]. Both pump and probe pulses were measured with ZAP-SPIDER [21]. The pump beam was optimized by adjusting the phase applied by the DAZZLER to give transform limited 30 fs pulses. The probe beam was pre-compensated with a prism pair compressor to give near transform limited pulses of 30 fs.

We chose the organic dye Rhodamine 800 dissolved in ethanol (Exciton Inc.) as our model system. The optical density of the sample was 0.15 in a 300 μm thick sample cell. The first set of data was taken with transform-limited pump and probe pulses. 2D spectra were recorded for waiting times of 0 fs, 30 fs, 60 fs, 90 fs and 120 fs and are displayed in Fig. 7. Although the spectra exhibit some structure that we did not attempt to simulate, the general feature of interest is the shape of the main peak, which is due to the interaction of the chromophore with the solvent environment. At T = 0 fs, this peak is elongated along the diagonal, showing a large degree of inhomogeneity. As T is increased, the peak broadens along the anti-diagonal direction, becoming more circular. When T = 90 fs the spectrum is now homogeneously broadened. This initial elongation followed by the subsequent broadening is a common feature in 2D spectra with fast solvation dynamics [3].

To verify the effects of pump pulse chirp predicted by our calculations, we controlled the pump pulse chirp via adjustment of the spectral phase applied by the DAZZLER. To test the positive chirp case, a chirp of *ϕ*′′ = + 1000 fs^{2} was applied to each pump pulse. The resulting 2D spectra, shown in Fig. 7
(bottom row), have a pronounced broadening along the anti-diagonal at low frequencies as compared with high frequencies. This is most noticeable at very early times (T = 0 fs and 30 fs); by 90 fs the spectrum looks more symmetric. Applying the opposite negative chirp of −1000 fs^{2} (Fig. 7, top row), the expected pronounced broadening along the anti-diagonal at high frequencies, consistent with our calculations (see Fig. 2).

We also explored the effect of having transform-limited pump pulses but positively chirped probe pulses by adding one inch of fused silica to the otherwise transform-limited probe pulse path. This introduced an approximately linear chirp with *ϕ′′* = 2400 fs^{2}. Figure 8
shows a T scan under these conditions. Consistent with our simulations in Fig. 5, a positive probe chirp introduces a broader anti-diagonal width at higher frequencies at early times.

Finally, we also examined the effect of having equivalent chirp on the pump and probe pulses by introducing one inch of fused silica (*ϕ′′* = 2400 fs^{2}) into both paths of the otherwise transform-limited pulses. The resulting data is shown in Fig. 9
, showing that in this case the peak shape distortions are reduced compared to the asymmetric chirp case, but are still present to some degree compared to the unchirped spectra (Fig. 7, middle row). In rough agreement with the simulated spectra in Fig. 6 (top row), we see that the spectrum appears rounder above the diagonal and flattened below the diagonal at early waiting times as a result of the positive chirp. The negative feature below the diagonal in the simulations is not reproduced in the experimental data. In addition the maximum in the experimental data appears further below the diagonal than in the simulations, suggesting that the simulations underestimate the reorganization energy.

## 4. Discussion

Our simulations and experiments show significant distortions to 2D spectra introduced when the pump and probe pulses have different chirp. This is most likely to occur in two-color experiments, or where dispersion is not carefully balanced in a one-color experiment. The distortions are most apparent at early times during pulse overlap and concomitant spectral diffusion. The distortions arising from pump chirp can be intuitively understood as depicted in Fig. 10(a) . For identical linearly-chirped pump pulses and a transform-limited probe, the τ delay remains constant between individual frequencies, while the waiting time T is frequency dependent. For positive pump chirp, T will be longer for the lower frequencies, providing a longer relaxation period and leading to a broader anti-diagonal width since the degree of relaxation will be greater. The opposite effect occurs with negative pump chirp, where the relaxation period is longer for higher frequencies. When the pump pulses are transform-limited and the probe pulse is linearly chirped the opposite trend is seen as depicted in Fig. 10(b). For positive linear probe chirp, the higher frequencies experience a longer waiting time, resulting in a broader anti-diagonal width compared to lower frequencies.

When all three pulses have identical chirp, the spectral distortions are significantly reduced. This can be understood as arising from a constant coherence and relaxation period between each pulse pair: oscillators originally resonant with any frequency within the pulse spectrum have the same time to relax, reducing distortions to the spectra that occur for unequal relaxation times arising from differences in pump and probe chirp.

In both the calculations and experiments reported here, the spectral relaxation occurred on a similar time scale to the chirped pulse duration, leading to obvious distortions in the 2D spectra. Calculations with slower spectral relaxation, where the correlation time is considerably longer than the chirped pulse duration, show less distortion to the 2D peak shapes. Conversely, when the relaxation time is shorter than the chirped pulse duration, our calculations show that the distortions are considerably more noticeable at early waiting times (T<τ_{g}), even in the case of equal pump and probe chirp. Figure 11
illustrates this point, showing the resulting 2D spectra for τ_{g} = 50 fs.

## 5. Conclusions

We have presented simulations and experiments to explore the effects of uncorrected chirp on 2D electronic spectroscopy peak shapes. Using a simple model system we find that linear chirp on the pump pulses in a 2D experiment can lead to a frequency dependent asymmetry in the ratio of diagonal to anti-diagonal width, a measure that is often used to quantify inhomogeneous broadening in a system. The sign of the chirp determines whether high or low frequencies appear more or less homogeneously broadened, with positive pump pulse chirp (or negative probe pulse chirp) producing erroneously large anti-diagonal widths at low frequencies, and negative pump pulse chirp (or positive probe pulse chirp) producing the opposite effect. When the chirp of all three pulses is equivalent the spectral distortions are considerably reduced, although some asymmetry is introduced with respect to the diagonal, leading to features that could be interpreted as excited-state absorption. Experiments show similar behavior, verifying the need for careful pulse characterization for correct interpretation of frequency-dependent relaxation processes via 2D optical spectroscopy.

## Acknowledgments

The authors gratefully acknowledge the support of the Office of Basic Energy Sciences, U.S. Department of Energy (grant #DE-FG02-07ER15904). We thank David Jonas for kindly sharing his FORTRAN code.

## References and links

**1. **T. Brixner, J. Stenger, H. M. Vaswani, M. Cho, R. E. Blankenship, and G. R. Fleming, “Two-dimensional spectroscopy of electronic couplings in photosynthesis,” Nature **434**(7033), 625–628 (2005). [CrossRef]

**2. **D. Zigmantas, E. L. Read, T. Mancal, T. Brixner, A. T. Gardiner, R. J. Cogdell, and G. R. Fleming, “Two-dimensional electronic spectroscopy of the B800-B820 light-harvesting complex,” Proc. Natl. Acad. Sci. U.S.A. **103**(34), 12672–12677 (2006). [CrossRef]

**3. **D. M. Jonas, “Two-dimensional femtosecond spectroscopy,” Annu. Rev. Phys. Chem. **54**(1), 425–463 (2003). [CrossRef]

**4. **J. D. Hybl, A. Yu, D. A. Farrow, and D. M. Jonas, “Polar solvation dynamics in the femtosecond evolution of two-dimensional Fourier transform spectra,” J. Phys. Chem. A **106**(34), 7651–7654 (2002). [CrossRef]

**5. **M. L. Cowan, B. D. Bruner, N. Huse, J. R. Dwyer, B. Chugh, E. T. J. Nibbering, T. Elsaesser, and R. J. D. Miller, “Ultrafast memory loss and energy redistribution in the hydrogen bond network of liquid H2O,” Nature **434**(7030), 199–202 (2005). [CrossRef]

**6. **J. J. Loparo, S. T. Roberts, and A. Tokmakoff, “Multidimensional infrared spectroscopy of water. I. Vibrational dynamics in two-dimensional IR line shapes,” J. Chem. Phys. **125**, 194522 (2006). [CrossRef]

**7. **J. B. Asbury, T. Steinel, K. Kwak, S. A. Corcelli, C. P. Lawrence, J. L. Skinner, and M. D. Fayer, “Dynamics of water probed with vibrational echo correlation spectroscopy,” J. Chem. Phys. **121**(24), 12431–12446 (2004). [CrossRef]

**8. **J. D. Hybl, A. W. Albrecht, S. M. G. Faeder, and D. M. Jonas, “Two-dimensional electronic spectroscopy,” Chem. Phys. Lett. **297**(3-4), 307–313 (1998). [CrossRef]

**9. **K. Lazonder, M. S. Pshenichnikov, and D. A. Wiersma, “Easy interpretation of optical two-dimensional correlation spectra,” Opt. Lett. **31**(22), 3354–3356 (2006). [CrossRef]

**10. **D Abramavicius and S Mukamel, “Disentangling multidimensional femtosecond spectra of excitons by pulse shaping with coherent control,” J. Chem. Phys . **120**, 8373-8378 (2004). [CrossRef]

**11. **V. I Prokhorenko, A Halpin, and R. J. D Miller, “Coherently-controlled two-dimensional photon echo electronic spectroscopy,” Opt. Express **17**, 9764-9779 (2009). [CrossRef]

**12. **D. A. Farrow, A. Yu, and D. M. Jonas, “Spectral relaxation in pump-probe transients,” J. Chem. Phys. **118**(20), 9348–9356 (2003). [CrossRef]

**13. **S. M. G. Faeder and D. M. Jonas, “Two-dimensional electronic correlation and relaxation spectra: Theory and model calculations,” J. Phys. Chem. A **103**(49), 10489–10505 (1999). [CrossRef]

**14. **S. Mukamel, *Principles of Nonlinear Optics and Spectroscopy* (Oxford University Press, Oxford, 1995).

**15. **K. Ohta, D. S. Larsen, M. Yang, and G. R. Fleming, “Influence of intramolecular vibrations in third-order, time-domain resonant spectroscopies. II. Numerical calculations,” J. Chem. Phys. **114**(18), 8020–8039 (2001). [CrossRef]

**16. **P. F. Tekavec, J. A. Myers, K. L. M. Lewis, and J. P. Ogilvie, “Two-dimensional electronic spectroscopy with a continuum probe,” Opt. Lett. **34**(9), 1390–1392 (2009). [CrossRef]

**17. **J. A. Myers, K. L. M. Lewis, P. F. Tekavec, and J. P. Ogilvie, “Two-color two-dimensional Fourier transform electronic spectroscopy with a pulse-shaper,” Opt. Express **16**(22), 17420–17428 (2008). [CrossRef]

**18. **T. Wilhelm, J. Piel, and E. Riedle, “Sub-20-fs pulses tunable across the visible from a blue-pumped single-pass noncollinear parametric converter,” Opt. Lett. **22**(19), 1494–1496 (1997). [CrossRef]

**19. **S. H. Shim and M. T. Zanni, “How to turn your pump-probe instrument into a multidimensional spectrometer: 2D IR and Vis spectroscopies via pulse shaping,” Phys. Chem. Chem. Phys. **11**(5), 748–761 (2009). [CrossRef]

**20. **S. H. Shim, D. B. Strasfeld, Y. L. Ling, and M. T. Zanni, “Automated 2D IR spectroscopy using a mid-IR pulse shaper and application of this technology to the human islet amyloid polypeptide,” Proc. Natl. Acad. Sci. U.S.A. **104**(36), 14197–14202 (2007). [CrossRef]

**21. **P. Baum and E. Riedle, “Design and calibration of zero-additional-phase SPIDER,” J. Opt. Soc. Am. B **22**(9), 1875–1883 (2005). [CrossRef]