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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 Optical Society of America
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:
where is a Gaussian pulse with a maximum at the pulse arrival time (tj), σ is the intensity FWHM of the pulse and 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 ω0:The pulses are incident on the sample with wavevectors 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:The variables τ1, τ2 and τ3 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 ≡ t3 – max(t1, t2) and τ ≡ t2 – t1. 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)(ωt, T, ωτ), 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:
This treatment is necessary not only to account for the finite bandwidth of the heterodyne pulse, but also to include any chirp present in the 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]
Here τg 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 fromwhere λ is the reorganization energy and Δ2 is the coupling strength to the bath. In the high temperature limit they are related through Δ2 = 2kBTλ/ℏ. For the calculations presented here, we chose a correlation time of τg = 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 fs2 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 fs2. In Fig. 2 we plot the calculated 2D spectra at several waiting times for negatively chirped, transform-limited, and positively chirped pump pulses. The ω1 and ω3 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.
Fig. 2 Simulated spectra with transform-limited probe pulses and chirped pump pulses (top row: ϕ′′ = −1000 fs2, middle row: transform-limited, bottom row: ϕ′′ = + 1000 fs2) at different T delays (T = 0, 30, 60, 90, 120 fs from left to right). Contours are evenly spaced at 5% of the maximum. The color scale is chosen such that the background color is zero, while colors that are deeper blue or red are negative and positive respectively. The T = 0 transform-limited spectrum shows the locations at which the anti-diagonal widths are reported in Fig. 3.
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 fs2 can lead to distortions at early waiting times.
Fig. 3 Anti-diagonal peak width as a function of T for transform-limited pulses (left), positively-chirped pump pulses (ϕ′′ = + 1000 fs2) (middle) and negatively-chirped pump pulses (ϕ′′ = −1000 fs2) (right). The black line corresponds to the width at the center of the spectrum, while the red and blue lines correspond to widths on the red and blue sides of the spectrum at positions indicated in Fig. 2.
Fig. 4 Simulated spectra with transform-limited probe pulses and pump pulses with different amounts of chirp (ϕ′′ = −1000 fs2, −500 fs2, 0 fs2, + 500 fs2, + 1000 fs2, left to right) at T = 0 fs. Contours are evenly spaced at 5% of the maximum. The color scale is chosen such that the background color is zero, while colors that are deeper blue or red are negative and positive respectively.
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.
Fig. 5 Simulated spectra with transform-limited pump pulses and chirped probe pulses (top row: ϕ′′ = + 1000 fs2, bottom row: ϕ′′ = −1000 fs2) at different T delays (T = 0, 30, 60, 90, 120 fs from left to right). Contours are evenly spaced at 5% of the maximum. The color scale is chosen such that the background color is zero, while colors that are deeper blue or red are negative and positive 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.
Fig. 6 Simulated spectra with both pump and probe pulses chirped (top row: ϕ′′ = + 1000 fs2, bottom row: ϕ′′ = −1000 fs2) at different T delays (T = 0, 30, 60, 90, 120 fs from left to right). Contours are evenly spaced at 5% of the maximum. The color scale is chosen such that the background color is zero, while colors that are deeper blue or red are negative and positive respectively.
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 fs2 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 fs2 (Fig. 7, top row), the expected pronounced broadening along the anti-diagonal at high frequencies, consistent with our calculations (see Fig. 2).
Fig. 7 Experimental spectra with transform-limited probe pulses and chirped pump pulses (top row: ϕ′′ = −1000 fs2, middle row: transform limited, bottom row: ϕ′′ = + 1000 fs2) at different T delays (T = 0, 30, 60, 90, 120 fs from left to right). Contours are plotted at intervals of 5% of the maximum for levels above 15%. The color scale is chosen such that the background color is zero, while colors that are deeper blue or red are negative and positive respectively.
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 fs2. 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.
Fig. 8 Experimental data with transform-limited pump and probe pulses chirped by 1 inch of fused silica (ϕ′′ = 2400 fs2) for different waiting times (T=0, 30, 60, 90, 120 fs from left to right). Contours are plotted at intervals of 5% of the maximum for levels above 15%.
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 fs2) 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.
Fig. 9 Experimental data with both pump and probe pulses chirped equally by 1 inch of fused silica (ϕ′′ = 2400 fs2) for different waiting times (T=0, 30, 60, 90, 120 fs from left to right). Contours are plotted at intervals of 5% of the maximum for levels above 15%.
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.
Fig. 10 a): Schematic describing the origin of the distortions that occur in 2D spectra when the pump pulses are positively chirped and the probe pulse is transform-limited: the relaxation time for the red (Tred) and blue (Tblue) components are unequal, leading to broader anti-diagonal widths at low frequency (since Tblue < Tred). b): When the pump pulses are transform-limited and the probe is positively chirped the opposite case arises (Tblue > Tred), leading to broader anti-diagonal widths at high frequency. When negative chirps are used the opposite trends to those depicted here are seen.
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<τg), even in the case of equal pump and probe chirp. Figure 11 illustrates this point, showing the resulting 2D spectra for τg = 50 fs.
Fig. 11 Simulated spectra with for fast spectral relaxation (τg = 50 fs) at different T delays (T = 0, 15, 30, 45, 100 fs from left to right). Both pump and probe pulses are chirped (ϕ′′ = + 1000 fs2). Contours are evenly spaced at 5% of the maximum. The color scale is chosen such that the background color is zero, while colors that are deeper blue or red are negative and positive respectively.
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.
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