## Abstract

Two-dimensional Fourier transform spectra of a three level model system are simulated using a non-perturbative density matrix formalism. The electric field distortions resultant from using pixelated pulse shaping devices to produce phase-locked pulse pairs are modeled and the effects on the recovered spectra are examined. To minimize spectral distortions, a temporal filtering scheme is employed which eliminates contributions from spurious sample polarizations.

©2012 Optical Society of America

## 1. Introduction

Two-dimensional Fourier transform (2DFT) spectroscopy is a powerful technique for studying a wide range of ultrafast photophysics by correlating transition dipoles at two frequencies with sensitivity to phase and coherence information. 2DFT spectroscopy can provide insight into chromophores and their environments such as bath-induced relaxation [1, 2] and energy transfer phenomena [3, 4] that are inaccessible to conventional methods. After Warren’s early work in a fully collinear geometry [5, 6], several groups have demonstrated the implementation of 2DFTS in the IR [7, 8], visible [9–14] and UV [15, 16] spectral regions using pulse shaping devices in a partially collinear geometry. This method has many advantages over conventional techniques including intrinsically high phase stability, the ability to easily collect highly nondegenerate 2DFT spectra, and perhaps most importantly, a dramatic simplification of the experimental apparatus. By employing the methods described in the above papers, 2DFT spectroscopy can be readily performed in a pump-probe apparatus, allowing the ready adoption of this powerful technique by non-specialized groups.

The key component to performing 2DFT spectroscopy in such a straightforward manner is the use of a computer-controlled pulse-shaping device parameterized by an analytical transfer function (described in more detail below). However, it has long been understood that pixelated pulse shaping devices based on liquid-crystal spatial light modulators (LC-SLM) do not ideally replicate the transfer function and as a consequence, the shaped time-domain field is distorted [17]. These distortions often manifest in the form of spurious pulses, which cause unwanted contributions to the collected 2DFT spectrum.

In this paper, we employ non-perturbative density matrix theory, which allows an explicit inclusion of the non-ideal electric field arising from pixilation artifacts, to study these distortions on a model three level system. With knowledge of what specific interactions lead to distorted spectra in hand, we present a temporal filtering scheme, which allows the collection of high quality 2DFT spectra even with relatively poor pulse shaper resolution. The remainder of this paper is organized as follows: In *Methods,* we present the details of the calculation and model system. In *Results*, we present simulated spectra showing the nature of the distortions before and after application of the temporal filter. In the *Discussion* section, we examine in detail the spurious contributions that cause spectral distortion before finally concluding.

## 2. Methods

Collection of 2DFT spectra in the partially non-collinear geometry requires a phase-locked, collinear pulse pair (“pump” beam) which serves as the first two excitation fields and a non-collinear “probe” pulse which serves as both the third excitation field and means to self-heterodyne the detected signal. A computer-controlled pulse-shaping device, parameterized by Eqs. (1) and (2) to respectively modulate the amplitude and phase of the pump beam, trivializes the creation of a phase-locked pulse pair [9, 18].

*sgn*refers to the sign operator and ω is the frequency of the electric field. Inverse Fourier transform of the spectral electric field, modulated by Eqs. (1) and (2), results in the real electric field comprised of a phase-locked pulse pair separated by τ, ${E}_{1}\left(t-\tau \right)+{E}_{2}\left(t\right)=\frac{1}{4}\left({\tilde{E}}_{1}\left[t-\tau \right]+{\tilde{E}}_{2}\left[t\right]\right)+c.c.$, where ${\tilde{E}}_{1}\left[t-\tau \right]$ and ${\tilde{E}}_{2}\left[t\right]$ are the complex amplitudes of the first and second time domain pulses.

In the partially non-collinear implementation of 2DFT spectroscopy, the pulse pair, together with the third non-collinear pulse, induce the third-order polarization of the sample that reflects the system’s response to the three interactions with the electric field [19, 20]. In the case of ideal pulse shaping, the undistorted 2DFT spectrum can be recovered [21]. However, because pulse shaping devices utilizing pixilated LC-SLMs only approximately reproduce a desired transfer function, the shaped time-domain electric field is distorted, often manifest in the form of spurious pulses shown in red (and dotted) in Fig. 1 . These spurious pulses are located at times roughly corresponding to the inverse difference frequency between pixels [9, 17, 22]. As a consequence, additional contributions (not shown) to the third-order polarization contribute to the acquired signal, causing distortions in the 2DFT spectrum.

To simulate the effects of pixelated pulse shaping on 2DFT spectra, we chose to use the non-perturbative density matrix formalism. In such calculations, numerical propagation of the density matrix explicitly includes the time-dependent electric field, and consequently, distortions arising from pixilation effects are straightforward to include. In this section, we first outline the form of the electric field used to model the distortions that arise from pulse shaping with a pixilated LC-SLM. Following that, the model system is described and details regarding the calculation of the 2DFT spectra are presented.

Pixelation effects in LC-SLMs arise because each pixel of finite physical width imparts a constant modulation to the amplitudes and phases of all field components passing through it. Numerically, the total spectral electric field can be modeled by Eq. (3).

The spectrum of the electric field used in the manuscript was Gaussian shaped and centered at 2354.56 THz (~800 nm). For the time domain simulations, the field was defined in the spectral domain and numerically inverse Fourier transformed. To determine the set of${\omega}_{n}$, the pixel resolution in nm, ${\Delta}_{\lambda}$, of the LC-SLM is specified and a numerical algorithm was used to determine the width in frequency space (${\Delta}_{n}$) that corresponded to a given ${\Delta}_{\lambda}$ at each portion of the spectrum. These points were then binned in a “pixel.” The central wavelength of each pixel defines the parameter${\omega}_{n}$, which was used for all spectral components of the field passing through pixel n regardless of optical frequency. As an illustration, consider Fig. 2 which shows a portion of the spectral amplitude $\left|{\tilde{\u03f5}}_{out}\left(\omega ;\tau =500\text{\hspace{0.17em}}\text{fs}\right)\right|$ for ideal pulse shaping with no pixels (red) and for a pixel resolution of 0.5 nm (black). For both cases, the amplitude of the probe pulse, $\left|{\tilde{\u03f5}}_{3}\left(\omega \right)\right|$ (see Eqs. (3) and (4), has been set to 0 for clarity.

We chose to simulate a three level system, as it is the simplest model that can exhibit cross peaks in the 2DFT spectrum. The Hamiltonian of the system is given by Eq. (5) which is composed of the field free Hamiltonian, ${\widehat{H}}_{0}$, and the field interaction operator$\widehat{V}$. Here, $E\left(t\right)$is the total (pump and probe) real time-dependent electric field and $\mu $ is the transition dipole moment (set to 1 in this work). The ground state frequency, ${E}_{00}/\hslash $, was set to 0, and ${E}_{11}/\hslash =2366.40$ THz and ${E}_{22}/\hslash =2414.94$ THz give the frequencies of the two excited states.

^{−1}, were chosen to be at the low end of typical electronic dephasing rates in condensed phase systems [25], although there are a few exceptions [26] which are not represented here. Neither inhomogeneous broadening nor population relaxation effects were included in the simulation.

2DFT spectra in the partially collinear geometry are obtained by double Fourier transform (with respect to t and$\tau $) of the modulus of the third-order polarization, ${P}_{2D}^{(3)}\left(t,T,\tau \right)$. In the time domain, this polarization (Eq. (7) is given by the triple convolution of the system response function with the electric field.

The response function, $R\left({t}_{1},{t}_{2},{t}_{3}\right)$, is comprised of Liouville pathways describing specific light matter interactions such as stimulated emission, ground state bleach, and excited state absorption [19].

When the density matrix is perturbatively expanded, the separation of various orders of the polarization naturally arises so that calculation of 2DFT spectra is straightforward. However, when the density matrix is treated non-perturbatively as it is in this work, the expectation value of the dipole operator, Eq. (8), gives the *overall* time-dependent polarization, $P\left(t\right)$, which contains all orders of the light matter interaction.

To do this, we first write the total induced polarization as a sum of components,

*l*= −1,

*m*= + 1,

*n*= + 1. In a non-perturbative calculation, there is no equivalent directional separation. However, for calculation of the 2DFT spectrum the individual contributions can be separated in a similar way by first defining a phase${\varphi}_{n}={k}_{n}r$. The polarization can then be written as,

Since the induced polarization is dependent on the phase of the electric field (as shown in Eq. (7), we are able to recover ${P}_{2D}^{(3)}\left(t,T,\tau \right)$by calculating the overall polarization with different realizations of the parameters, ${\varphi}_{1}$, ${\varphi}_{2}$, and ${\varphi}_{3}$ in the excitation field. In this work, 16 different realizations of the electric field (${\varphi}_{1}$and ${\varphi}_{2}$; see Eqs. (3) and (4)) were used in the propagation of the density matrix resulting in 16 total polarizations$\tilde{P}\left(t;{\varphi}_{1},{\varphi}_{2},{\varphi}_{3}\right)$. The rephasing (l = −1, m = + 1, n = + 1 and non-rephasing (l = + 1, m = −1, n = + 1) components were then isolated by performing the discrete inverse Fourier transform (Eq. (11) on the set of overall polarizations, .

## 3. Results

In the partially collinear geometry, the fields corresponding to rephasing and nonrephasing pathways are emitted along the same wave vector and as such, are simultaneously detected. Two-dimensional spectra were obtained by Fourier transform along t of the sum of rephasing and nonrephasing pathways and a cosine transfer along positive-only $\tau $ which was scanned from 0 to 400 fs [31]. Figure 3
shows the real and imaginary parts of the 2D spectrum for T = 0 fs (top) and T = 100 fs (bottom). These spectra are generated assuming ideal shaping with no pixelation effects and recover peak shapes that are well known in the literature [32]. The T = 0 fs 2D spectrum exhibits “phase twist” due to the temporal overlap of E_{3} with E_{1} or E_{2} [21]. On the other hand, for all spectra generated with T greater than the pulse width, the expected peak shapes (resembling 4-point stars in the real spectra) are observed.

We next simulated 2DFT spectra using a distorted excitation field modeled by Eq. (3). Figure 4 (left) shows a series of spectra generated with pixel resolution, $\Delta \lambda $, of 0.5, 1.0, and 1.5 nm, which correspond to frequency differences between pixels () at the center of the spectrum of 1.47, 2.94, and 4.41 THz. These spectra exhibit peak shapes that are distorted by a high frequency modulation, primarily concentrated along the diagonal. These effects are more clearly visible in the difference spectra, plotted in Fig. 4 (right), which show the difference between distorted spectra and the ideal spectrum (Fig. 3 (bottom left)). In the case of the best pixel resolution simulated here ($\Delta \lambda $= 0.5 nm), there are deviations from ideal of up to ~5%, while in the worst case where $\Delta \lambda $ = 1.5 nm, the spectrum is distorted by almost 20% in places.

As is discussed in detail below, the distortions arise from a series of interactions of the three-level system with the field that depend specifically on the spurious pulses. Shown in black in Fig. 5 is the modulus of the calculated 3rd order polarization, $\left|{\tilde{P}}_{2D}^{\left(3\right)}(t)\right|$, for the case of $\Delta \lambda $ = 1 nm, T = 100 fs, and $\tau $ = 350 fs. The total 3rd order polarization is made up of the desired portion, which begins at t = 0, and the unwanted polarization resultant from the spurious pulses at t ~1.7 ps and t ~3.8 ps. It is notable that the unwanted polarization magnitude is highly significant even when the spurious pulses appear weak (this is shown later, for example, in Fig. 7). This is primarily a consequence of two effects. First, the desired polarization is itself weak, having decayed because the electronic dephasing time (100 fs) is significantly shorter than the value of τ (350 fs). Second, the contributions from the spurious polarization arise from pathways with a comparatively short interval (T = 100 fs) which does not change throughout the course of a measurement (i.e., as τ is varied).

Unfortunately, because the spurious pulses are “encoded” with the same phase information as the pulse pair, phase cycling schemes will be ineffective at removing the unwanted polarization from the collected signal. However, because the inverse electronic dephasing rate is typically much shorter than$\Delta {\omega}_{c}$, it is often the case that the unwanted induced polarization occurs significantly after the desired polarization has radiated its signal (such as that seen in Fig. 5). This suggests it should be possible to minimize distortion to the spectrum by performing a single point convolution of a temporal filter with the detected signal.

We have chosen to use a hyperbolic tangent filter, given by Eq. (12) and shown in green in Fig. 5, whose parameters can be adjusted to eliminate the spurious signal (which occurs later in time) while maintaining the desired signal which begins at t = 0 fs [33].

Here t^{+} and t^{-} give the cutoff times in positive and negative t, respectively, and determines how quickly the filter maps from 0 to 1 (in this case shown in Fig. 5, = 12 fs, so that the filter transitions from 0 to 1 in ~100 fs). Fourier transform of the product of the filter and the time domain third order polarization recovers nearly undistorted 2DFT spectra. This can be seen in Fig. 6
. Panels a–c show the differences between the pixelated spectra that have been temporally filtered and the ideal 2DFT spectrum with contours plotted at 0.2% intervals. Panel d shows the real 2DFT spectrum that is achievable using the temporal filter even for the worst of the three pixel resolutions ($\Delta \lambda $= 1.5 nm). Note the significant improvement relative to the real spectrum seen in Fig. 4 (bottom left panel). At the 10% contour interval, the difference between this filtered case (Fig. 6d) and the ideal case (see Fig. 3) is almost unobservable. In fact, the maximum deviation from the ideal spectrum after the filtering procedure has been applied is only 1.1%.

Experimentally, the signal is detected spectrally by frequency resolution of the probe pulse. Thus, in order to implement this technique, one would first perform an inverse Fourier transform to retrieve the time domain signal. After this the filter can be applied to remove any contributions to the signal that are from spurious pulses. Our results suggest that even with relatively poor pixel resolution, dramatically improved 2DFT spectra can be collected.

## 4. Discussion

The cause of spectral distortion lies in contributions to the polarization created by Liouville pathways that are insidiously dependent on the spurious pulses created by the pixelation of the LC-SLM. To illustrate this, consider the electric field generated by inverse Fourier transform of Eq. (3) with $\Delta \lambda $ = 1.0 nm, T = 100 fs, and $\tau $ = 350 fs shown in Fig. 7. The majority of the electric field amplitude is taken up by the desired three pulse sequence composed of the pulse pair (E_{1}, E_{2}) separated by , and the third excitation pulse E_{3}, centered at t = 0 fs. This pulse sequence is detailed in the inset of Fig. 7 where it is emphasized that each pulse in the pair created by the pulse shaper has the same wavevector k_{1}, and the third pulse propagates along a different wavevector, k_{3}. In addition to the desired field, pixelation effects from pulse shaping cause formation of spurious pulses (E_{sp1}, E_{sp2}) at ~1.7 ps and ~3.8 ps (at this particular ). The spurious pulses are located in time at integer multiples of $2\pi /\Delta {\omega}_{c}$ relative to E_{1} and are strongly chirped because of nonlinear dispersion of the spectral field in the pulse shaping apparatus. Because the spurious pulses are produced collinearly with the desired pulse pair E_{1}, E_{2}, they also have a wavevector given by k_{1}.

The three pulse sequence, E_{1}, E_{2}, E_{3} induces the desired polarization, ${P}_{2D}^{(3)}\left(t,T,\tau \right)$ (see Eq. (7), which, under the rotating frame approximation and assuming T is greater than the pulse width, arises from 16 different Liouville pathways within the context of the three level Hamiltonian. Under a relatively broad range of conditions, however, eight additional contributions that are dependent on the spurious pulses, contribute to the third order polarization. These pathways, which are represented by the double-sided Feynman diagrams in Fig. 8
, have the pulse interaction order of E_{2}, E_{3}, E_{sp} (see the right side of Fig. 8 for the pulse order from bottom to top where E_{sp} can be any of the spurious pulses, E_{sp1}, E_{sp2}, etc.) and create a polarization with exactly the same phase matching conditions as the desired signal.

The unwanted polarization arises from a different time ordering of field interactions than the desired polarization. In the first three pathways shown in Fig. 8, E_{2} (at t = -T) followed by E_{3} (at t = 0 fs) produces a population. In the first diagram, an excited-state population (${\rho}_{11}$ (shown) or (not shown)) is created, whereas in the second and third diagrams, a ground-state population,, is produced. In the fourth pathway of Fig. 6, these two interactions produce a coherence between the two excited states (${\rho}_{21}$ (shown) or ${\rho}_{12}$ (not shown)). For all of the pathways, the third field interaction due to E_{sp} occurs at $t~\left(n2\pi /\Delta {\omega}_{c}\right)-\tau $ (n = 1,2,3,...). Because excited state coherences decay rapidly for most condensed phase systems, the fourth pathway in Fig. 8 is not expected to substantially contribute to the distortion of the 2DFT spectrum. On the other hand, populations decay slowly and the first three pathways in Fig. 8 will contribute strongly to distortions.

The extent to which these components of the polarization contribute to the detected signal is dependent on the parameters of the experiment, however in nearly all cases, temporal filtering will be necessary. For example, since T always separates the first two interactions (E_{2} from E_{3}), spectra collected for T much greater than the inverse electronic dephasing rate between ground and excited states will contain minimal distortions. Unfortunately, it is impractical to restrict T to such values in many experiments, and without a temporal filtering approach, the distorting pathways described will then contribute to the detected signal. A further factor which necessitates a temporal filter involves the spectral resolution of the LC-SLM, where collection of distortion-free 2DFT spectra using visible excitation becomes increasingly difficult because of the higher energy density per unit wavelength. Nevertheless, assuming coherence times are not so long that the signals from desired and spurious polarizations overlap, a LC-SLM of reasonable pixel resolution combined with temporal filtering makes a robust and facile apparatus for collection of 2DFT spectra.

## 5. Conclusion

Simulations using non-perturbative density matrix theory were performed on a three-level model system to determine how pulse shaper pixelation effects lead to distortions of 2DFT spectra. It was determined that additional light matter interaction pathways, which depend on the spurious pulses generated by pixelated devices, distort the detected spectrum. A temporal filtering scheme is presented which is highly successful at eliminating these spurious contributions. Care must still be taken when long coherence times or poor pixel resolution result in cases where the radiated signal from the unwanted polarization temporally overlaps with that from the desired polarization. However, in nearly all condensed phase samples, applying a simple time-domain filter will easily eliminate problems associated with pixelated pulse shaping devices, making collection of high quality 2DFT spectra by non-specialized groups readily accessible.

## Acknowledgments

We gratefully acknowledge support for this work from the Chemical Sciences, Geosciences, and Biosciences Division, Office of Basic Energy Science, U.S. Department of Energy Grant DE-FG02-07ER15890. The authors would like to thank Prof. David Jonas for useful and informative conversations.

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