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We demonstrate a “drop-in” modification of the pulse-shaped pump-probe geometry two-dimensional Fourier transform spectrometer that significantly improves its performance by making the measurement background-free. The modification uses a hybrid diffractive optic/pulse-shaping approach that combines the advantages of background-free detection with the precise timing and phase-cycling capabilities enabled by pulse-shaping. In addition, we present a simple new method for accurate phasing of optically heterodyned two-dimensional spectra. We demonstrate the high quality of data obtainable with this approach by reporting two-dimensional Fourier transform electronic spectra of chlorophyll a in glycerol/water at 77 K.
© 2014 Optical Society of America
1. Introduction
Two-dimensional electronic spectroscopy (2DES) has become a popular extension of visible transient absorption (TA) spectroscopy. By resolving the excitation frequency in addition to the detection frequency, 2DES aids in understanding condensed matter processes such as solvation dynamics [1] and excitonic energy transfer in photosynthetic complexes [2, 3], J-aggregates [4] and semiconductor materials [5, 6]. In concept, 2DES expands TA by using two pump pulses with variable time delay t1 between them, and then probing the system after a waiting time t2. The result of splitting the pump pulse into two is that the measured transient signal oscillates as a function of t1, and the Fourier transform with respect to this delay produces the ω1 axis (or excitation axis). As in frequency-resolved TA, the signal is typically detected in the frequency domain, yielding the ω3 axis (detection axis). Since this concept was first experimentally demonstrated [7, 8] there have been a number of groups demonstrating different methods for collecting 2DES spectra [9–22].
Each of the various 2DES methods have advantages and disadvantages, as they attempt to solve different problems associated with the realization of 2DES. Some of the common practical problems addressed in the literature are: achieving phase stability, reducing instrument and measurement complexity, lowering acquisition time and minimizing component costs. Most setups use a fully noncollinear box-CARS arrangement in some form or another, since it gives a background free signal, and use motorized delay lines to scan t1. Compared to fully noncollinear methods, 2DES in the pulse-shaped pump-probe geometry [14, 15, 23] provides a reduction in measurement and instrument complexity. A significant benefit of the pump-probe geometry itself is that the collected signal is automatically real and absorptive, as are transient absorption measurements made in the same geometry [24]. In typical noncollinear geometries, rephasing and non-rephasing signals are collected separately and must be combined together and “phased” to unmix dispersive and absorptive components of the spectra. As will be discussed later, the unmixing process can be challenging and collecting the pathways at separate times leaves one vulnerable to noise from long-term laser fluctuations. In addition to the simplicity of the geometry itself, the use of a pulse-shaper to create the two excitation pulses eliminates uncertainties in coherence time zero that can be present in motorized setups [25] and enables phase-cycling which can aid in scatter removal and the isolation of signals of interest [13, 15, 23]. The use of a pulse-shaper also offers the flexibility to explore the effect of pulse-shape (amplitude and phase) on the measurement [20, 26]. Despite these advantages, pulse-shaped 2DES in the pump-probe geometry suffers from high component costs and offers less freedom to optimize the signal-to-noise ratio (S/N) compared to background-free geometries. In the pump-probe geometry, the signal emerges collinear with the probe and both are sent directly into the camera. As a result, the probe must be made to be weak in order to avoid saturating the detector. As the signal strength is proportional to the probe field, its reduction lowers the achievable S/N. Background free geometries are popular because they do not send the probe directly into the detector, so the S/N may be improved by increasing the probe strength and optimizing the relative strengths of signal and the local oscillator used for heterodyne detection [27]. To exploit this fact, polarization schemes have been employed in the pump probe geometry to control the local oscillator power [15, 28]. Polarization controlled pump-probe geometry 2DES, however, is limited to measuring certain tensor elements of the third order response, and cannot measure the all parallel response. Traditional box-CARS background free 2DES setups have no restriction on what tensor element is measured. We note that the achievable S/N of background-free detection is reduced if phase instability between the signal and local oscillator is significant [29], a problem that can be overcome with the use of diffractive optics [9]. Here we present a hybrid diffractive optic and pulse-shaping based approach to 2DES. It combines the advantages of background-free box-CARS detection with the precise time-delays and phase-cycling capabilities of pulse-shaping. The setup can be readily interconverted between the background free and pump-probe geometries to fit the demands of the system being studied.
2. Experimental implementation
The laser source consists of a Ti:Sapph oscillator (Spectra Physics MaiTai) seeding a regenerative amplifier (Spectra Physics Spitfire Pro). The 4 mJ, 500 kHz, 800 nm 40 fs output is split and sent into two dual-stage non-collinear optical parametric amplifiers (NOPAs) [30]. The pump beam is sent through a pre-compensating grism [31] and then into an acousto-optic pulse-shaper (Dazzler, Fastlite) where it is further compressed and split into two pulses with a programmable inter-pulse delay t1 and phase. Note that an amplitude and phase shaping pulse-shaper is required to produce a delayed pulse pair. Aside from the Fastlite product, there are other options, such as Spatial Light Modulators (SLM) and 4F acousto-optic (AO) pulse shapers [32, 33]. The critical feature of the Dazzler and other AO shapers over SLM-based shapers is the ability to change waveform rapidly, shot to shot, as this reduces the influence of long-term laser noise on the measurement. The second NOPA is compressed using a separate grism pair and is delayed by “waiting time” t2 with respect to the pump pulses using a conventional delay stage. The pump and probe beams are focused onto a diffractive optic (21 g/mm) with a spherical mirror (f = 500 mm, spotsize microns). The first order diffracted beams are imaged as shown in Fig. 1 to produce a crossing angle at the sample of approximately 1.5 degrees. For the data presented here a collinear FROG measurement of the pump pulses [34] yielded a pulse duration of 12 fs. Transient-grating-XFROG of the probe pulses gave a duration of 15 fs.
Fig. 1 Experimental Setup. (A) The diagram (not to scale) in lens notation of our interferometer. The round cornered dotted box indicates the “drop-in” addition to a traditional pulse-shaped pump-probe geometry 2DES setup. In our realization of the design, spherical mirrors are used with focal lengths f = 500 mm for the focusing mirror and f = 250 mm for the imaging mirror. (B) Four pulse timing diagrams are shown illustrating the origin of the signals emitted in the local oscillator direction. These diagrams communicate several things. The arrival time of the pulses is shown by their vertical displacement from the center of the diagram. The phase of the pump pulse is indicated by a color coordinated box and letter (x or y). The k-vector of the beam is indicated by the corner of the box on which it lies. Below the diagram is an equation giving the phase-cycling dependence of the signal.
Since the pulse-shaper introduces two pulses into both pump arms of the interferometer, four pump pulses strike the sample instead of two as in usual box-CARS arrangements. Third order signals which phase match in the direction of the local oscillator arise from pairwise interactions with two of the pump pulses and the probe pulse. Figure 1(b) depicts the four possible time ordered permutations that phase match with the local oscillator and the names of the signals each time ordering generates. Since all four signals enter the spectrometer simultaneously, a means to separate them is required. Phase-cycling, wherein a different constant phase is independently applied to the pump pulses, provides a means of separating the four signals [13]. As illustrated in Fig. 1(b), the first pulse out of the pulse-shaper is phase rotated by x, while the second pulse is rotated by y. Fortunately, the two signals we are interested in (rephasing and non-rephasing) have a unique phase dependence on x and y, while the two transient-grating signals share a common phase dependence. Isolating the rephasing signal (SR) and non-rephasing signal (SNR) therefore requires the solution of a linear system of three measurements S1, S2, and S3.
We note that although three measurements are required to separate the signals, this does not represent a loss of duty cycle as all signals of interest are contained in each laser shot, though in different complex linear combinations. Different phase-cycling schemes for 2DES in the pump-probe geometry have recently been explored [35]. In our implementation, we use the following pulse-shape sequence:
We also suppress interference scatter terms between pump, probe, and signals by collecting a second set of measurements where both pulses are shifted by π [15, 36]. The result is a total of six phase-cycles per t1 time point. Pump scatter is suppressed by chopping the probe with a mechanical shutter (Thorlabs SH05) and subtracting out the observed scatter. By using a mechanical shutter we are able to achieve good suppression of pump scatter while maintaining a duty cycle of 80%. A useful benefit of pulse-shaping based 2DES setups is the ability to observe the system of study in a rotating frame, also known as “phase-locking” [36, 37]. In phase-locking, the coherent oscillations along t1 are observed at the difference between their innate frequency and a lock frequency ωlock. This permits one to sample the coherence time much more coarsely, while still satisfying Nyquist sampling, and thereby increasing acquisition speed. With the pulse-shaper, we rapidly cycle through each of the six phase-cycles and t1 times, changing the pump waveform for every laser shot. The signal for a given t1 value is therefore isolated in 12 ms (minimally 6 ms if only 3 phase-cycles are used) when running at 500 Hz. When running at 1 kHz the signal isolation times are halved. In our measurements vide infra, we average ~150 2D spectra together for a total acquisition time of 1.5 minutes per spectrum.We note that this setup can easily convert to a pump-probe 2DES geometry by blocking one pump beam and a probe beam, using the local oscillator beam as the probe. It also facilitates polarization-dependent measurements [38].
To demonstrate the hybrid diffractive optic/pulse-shaping method we recorded 2DES data of chlorophyll a in a 50/50 (v/v) ethanol/glycerol mixture at 77° K. All chemicals were purchased from Sigma Aldrich and used as received. The sample was loaded into a custom-designed sample cell and cooled to 77 K in an Oxford Microstat N cryostat.
3. Experimental results
As this setup does not require wedge-pairs or other optics inside the imaging system, apart from the local oscillator attenuator, we are able to bring the beams very close together and make the interferometer compact. The compactness and use of a diffractive optic [9] makes the setup very passively stable, even without an enclosure to block air currents. To demonstrate the high phase stability of the setup we recorded the interference of the signal with the local oscillator. Figure 2 shows the recorded phase difference over an hour long period, where we estimate the standard deviation of the phase fluctuations to be ~λ/200.
Fig. 2 Relative phase of the signal measured at t1 = 0 delay, calculated by spectral interferometry [39]. (A) We demonstrate that the standard deviation of the signal phase is nearly independent of frequency. (B) The phase stability measurement as a function of time for the central frequency. In this plot measurements were averaged for a minute to produce the points shown in green.
In Fig. 3 we show 2DES, transient-grating, and pump-probe data of chlorophyll a in a 50/50 (v/v) mixture of ethanol/glycerol at 77 K using the hybrid diffractive optic/pulse shaping method. The sample O.D. was 0.3 at the maximum absorption. The fraction of molecules excited by a given laser pulse (bleach rate) was 0.15% for the pump and 1.4% for the probe.
Fig. 3 Demonstration of various 3rd order signals on Chlorophyll a in 50/50 (v/v) ethanol/glycerol at 77° K. The contour plots are drawn on a linear scale, with 1 contour above and below 0 omitted to suppress the noise floor. (A) The real rephasing spectrum with contours 100 contours over the range. (B) The real non-rephasing spectrum with 50 contours over the range. (C) A comparison of measurement to measurement signal variation between pump-probe and the phased transient-grating at the peak of the signal. There are three consecutive laser shots per measurement shown for the transient-grating signal (1 for each phase-cycle) and two consecutive laser shots (pump on and pump off) per measurement shown for the pump probe signal. The black line around which the signals vary represents the mean signal value in shared, but arbitrary units. The observed per measurement signal to noise ratio (S/N) (mean value divided by the standard deviation) is 32.2 for transient-grating and 1.3 for pump probe. Thus, factoring in the number of laser shots per measurement, we see a 19.5 fold S/N improvement at the peak of the signal.
Analytic phasing of 2DES data
One of the distinct advantages of the pump-probe geometry for 2DES is that it avoids the phasing problem. The probe pulse also acts as the local oscillator for heterodyne detection of the signal, yielding absorptive spectra [24]. Should it be desirable to separate rephasing and nonrephasing signals, this can be achieved with phase-cycling [15]. In the noncollinear geometry used here, the phasing problem returns due to the fact that signal and local oscillator follow different paths between the DO and the sample, imparting an unknown relative phase. To obtain absorptive spectra we phase the transient-grating signal against the pump-probe signal, an approach that has been used in time-domain heterodyne-detected transient-grating measurements [40, 41]. The transient-grating signal is generated during the course of a 2DES measurement in a noncollinear geometry when t1 = 0 and has the same unknown relative phase as the 2DES data. The frequency resolved pump-probe must be measured separately, though this is easily realized in our setup by removing the local oscillator attenuator and blocking a pump and probe beam. As noted by Singh et al. [42], the use of the transient-grating signal for phasing follows immediately from the projection slice theorem [7].
In the typical formulation of the phasing problem, finding the unknown phase is done by minimizing an objective function:
Where TG(ω) is the measured frequency resolved transient-grating signal, PP(ω) is the measured frequency resolved pump-probe signal, α is an unknown positive scaling constant and θ(ω) is the unknown phase model, typically a polynomial of first or second order in frequency. This minimization formulation is liable to give poor agreement between pump-probe and the transient-grating signal since the objective function is rugged (see Fig. 4(f)). The task of searching for a global minimum can be avoided by finding an analytic solution. We find that in certain common circumstances an analytic solution to the phasing problem is available if the ratio R = PP/|TG| reaches a global extremum at ωe. The existence of such a point implies that the absolute value transient-grating signal is purely absorptive there, ie . At this point, α = ||TG(ωe)|/PP(ωe)| allowing one to calculate the phase of the transient-grating signal . Since the inverse cosine is not unique, it should be constructed so that passing through a global extremum of R causes the domain of the arccos to pass to the next branch and the range to increment (or decrement) by π appropriately. A global extremum of R is guaranteed to exist within the detection window provided the frequency response of the transient-grating signal does not contain significant resonances outside the detection window. This requirement may be seen by using a Kramers-Kronig relation (KKR) on the pump-probe signal to recover the dispersive pump-probe spectrum, as is commonly done in the pump-probe geometry 2DES [15]. When the resonances of a signal are completely contained within the detection window, the dispersive component estimated in this way will have a zero crossing at the extremum of R. As resonant amplitude of the signal increases outside the detection window, the zero crossing of the dispersive spectrum estimated by KKR will shift away from the global extremum of R. In the worst case, where large resonances of the pump-probe signal exist outside the detection window, then the zero crossing of the dispersive spectrum estimated by the KKR will no longer be related to a global extremum of R, and the global extremum will likely exist outside the detection window. In this case, the analytic solution for the phasing problem will not apply. In these circumstances it is necessary to employ numeric methods which search for a global minimum of the objective function, the simplest of which is brute force grid-searching of the physical parameter space followed by local minimization.Fig. 4 Comparison of different phasing protocols. (A) Phased absorptive 2DES spectrum obtained from an exhaustive grid search with the quadratic phase model, and (B) the projection of the phased absorptive 2D spectrum to the spectrally resolved pump-probe. (C) Analytic phasing approach described in the text, showing the retrieved absorptive spectrum 2D spectrum, and (D) the projection of the phased absorptive 2D spectrum to the spectrally-resolved pump-probe signal. (E) The corrective phase applied under both models (solid line is the analytic model, dots for quadratic). (F) A slice of the objective function at the optimal quadratic phase (21 rad/fs2), which demonstrates the rugged nature of the objective function. The global optimum, corresponding to the phasing presented in sub-figures (A)-(E), can be seen at (0.9, 21).
The phase correction θ(ω) retrieved by the analytic solution (which has no intrinsic functional form), is dominantly described by a low order polynomial, as would be expected, since the phase correction should arise from the phase difference between the probe and the local oscillator generated by the piece of glass that acts as both an attenuator to control the local oscillator amplitude and a time delay to optimize fringe contrast for detection of the signal by spectral interferometry. The analytic phase correction is shown in Fig. 4(e). The higher order polynomial terms of the analytic phase may be due to the fact that the initial assumption that α is frequency independent may be only approximately true. For example, differences in signal phase-matching (most significant for large bandwidth pulses) and non-uniform heterodyne amplification could yield a frequency dependent α.
4. Discussion
From Fig. 3(c) it is clear that the background free geometry affords a significant advantage in achievable S/N (factor of ~19.5 at the signal peak). We attribute the S/N improvement to the larger signal field strength, which is enabled by the larger probe intensity afforded in the background free geometry. By increasing the ratio of the signal strength relative to the local oscillator, we move away from the laser noise dominated regime where the pump-probe measurements are made, towards a shot-noise limited regime [27]. The strength of the probe that can be used is limited by the maximum number of photons the detector can digitize per shot and the excitation requirements of the sample. In the data presented here the bleach rate was fairly mild, though if gentler excitation conditions are desired, the spotsize of the beams at the sample may be increased while maintaining the same photon count on the detector. Thus, for a given excitation condition and detector photon digitization rate, it will always be possible to construct a situation where a background free geometry provides more signal amplitude, given no restrictions on the power of the exciting laser. This fact is the dominant reason for pursuing a background free geometry. In a similar S/N comparison between pump-probe geometry and background free 2D IR spectroscopy, the Cheatum group concluded that phase instability was the dominant noise source in their background free geometry and therefore advocated the pump probe geometry [29]. Because we are able to collect both rephasing and non-rephasing signals simultaneously, and due to the high degree of passive phase stability afforded by the diffractive optic approach [9] (see Fig. 2), our background free setup offers significant S/N improvement over the pump-probe geometry.
The additional optics required to achieve a background free measurement from a pulse-shaped pump-probe geometry instrument are minimal, as shown in Fig. 1(a). The most significant additional optic is the diffractive optic used to split the beams. This optic can impose considerable power loss if the grating is not optimized for the frequency of light being used. In this study, we employed a grating optimized for 800 nm, and observed a ~25% efficiency of diffraction into each of the utilized beams. In the conventional implementation of diffractive-optics-based 2DES [9, 10], the bandwidth is limited by the desire to maintain near transform-limited pulses while using a refractive delay. In the setup presented here, this limitation does not apply, and we are currently limited to ~120 nm bandwidth (at 680 nm center wavelength) by the pulse-shaper. Recent upgrades to the pulse-shaper extend this significantly. While it is still easier to obtain absorptive spectra in the pump-probe geometry, we have shown that it is possible to obtain absorptive spectra from data collected in the hybrid geometry without much added difficulty. In fact, the analytic phasing method we presented grants excellent unmixing of the dispersive/absorptive components, limited only by the S/N of the pump-probe signal one can obtain. If a more traditional minimization approach to the phasing problem is taken, we see similar quality of unmixing; on par or better with results presented elsewhere [25]. To obtain a reliably higher S/N of the absorptive signal, one might employ this hybrid design in a “balanced” setup [19, 41] which permits the absorptive part of the signal to be collected background free. It is clear that a quadratic term is necessary to achieve reasonable phasing. This quadratic term originates from the dispersion (relative to the probe) of the local oscillator delay plate. In our case, the delay plate is a 0.5 mm thick piece of fused silica, which gives about an 850 fs delay and ~20 fs2 of group delay dispersion (GDD). Both phasing methods report a corrective phase with quadratic phase amplitude of ~20 fs2. A polynomial fit to the correction phase from the analytic phasing method gives higher order polynomial terms with amplitudes that are less than 3% of the quadratic amplitude (fit not shown).
5. Conclusion
We have demonstrated a hybrid diffractive-optic pulse shaping approach to 2DES. The method allows the high S/N measurements enabled by background free detection while retaining the advantages of precise timing, phase-cycling, and phase-locking afforded by a pulse-shaper. The hybrid setup can readily switch between background free detection and the pump-probe geometry to meet the needs of the experiment. While the automatic acquisition of absorptive spectra is lost in the background free method, we demonstrate an approach that makes recovering the absorptive spectrum significantly easier – in many cases without the need to use a minimizer. Furthermore, due to the fact that non-rephasing and rephasing signals are acquired simultaneously in the same detector, the phasing process is significantly more immune to laser noise and phase fluctuations than in setups where the signals must be acquired separately.
Acknowledgments
F. D. Fuller, and J. P. Ogilvie gratefully acknowledge the support of the Office of Basic Energy Sciences, U.S. Department of Energy (grant #DE-FG02-11ER15904). D. E. Wilcox, who contributed to design and debugging of the method acknowledges the support of the Center for Solar and Thermal Energy Conversion (CSTEC), an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under award DE-SC0000957.
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