Correction of spectral distortion in two- dimensional electronic spectroscopy arising from the wedge-based delay line

Unlike the probe wavelength, which is spectrally resolved by monochromator, the excitation wavelength in two-dimensional electronic spectroscopy is retrieved by means of Fourier transform of the interference signal introduced by the coherence delay time between the first and second excitation laser pulses. Hence, the calibration of delay lines would determine its accuracy. In this work, we showed that an inaccurate calibration factor of wedge-based delay line would result in a global peak shift and asymmetric spectral twists along the excitation axis. Both theoretical analysis and experiments have shown that such spectral distortions can be corrected by an accurately predetermined calibration factor. The relative accuracy of calibration factor reaches 3 × 10 in our setup. The dispersion effect of wedges also has been considered for the broadband excitation. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement


Introduction
Two-dimensional electronic spectroscopy (2DES) is an advanced spectroscopic tool for studying ultrafast energy and charge transfer possesses in condensed phase systems including biological light-harvesting systems [1][2][3][4][5][6], polymers and aggregates [7][8][9][10], as well as organic and inorganic semiconductors [11][12][13][14]. A 2DES map shows the correlation between the emission frequency (ω 3 ) and the initial excitation frequency (ω 1 ), directly revealing the coupling of different electronic states and energy relaxation pathways [15]. The excitation axis on 2D spectra is obtained by the Fourier transform of third-order nonlinear signals along the coherence time defined as the time delay between the first two excitation pulses. In order to control coherence delay time with sub-wavelength accuracy, there are three methods commonly used in 2DES setups: pulse shapers [16][17][18], a pair of movable transmitted wedges [19][20][21][22][23] and reflected mirrors [24][25][26][27][28]. These methods have their own advantages and disadvantages [29], among which the transmitted wedge-based delay line is a simple, compact and cost-effective way to provide attosecond time resolution, maintaining high optical phase stability between the pulses.
Although 2D spectroscopy obtains much more dynamic information than traditional pump-probe techniques, the reliable physical interpretation of each peak would depend exclusively on the quality of the spectra, i.e. free of unknown spectral distortions as much as possible. In general, the origins of spectral distortion, in most cases, are various for different geometrical configurations. Ogilvie et al analyzed the effects of laser pulse chirp on 2D spectra in a pump-probe geometry [30]. Jonas and Cundiff et al investigated the pulse propagation effects in a fully noncollinear geometry [31][32][33]. It is noted that the precise time zeros of the coherence time and the emission time (known as the phasing problem) are also essential to produce the correct 2D spectra in non-collinear 2DES. Recently, Jiang et al proposed new methods to improve the phasing procedure both in experiment and during data post processing [34,35]. In addition, a finite laser bandwidth, as a spectral filter, would distort the corresponding coherence peak position as well as its line shape in 2DES [36][37][38][39]. In this paper we found another spectral twist phenomenon resulting from the calibration factor (CF) of the wedge-based coherence time (τ) delay line. The calibration factor (time delay versus translation distance) as a function of wedge position is determined by the wedge angle and refractive index, which also can be measured by spectral interference [19]. To the best of our knowledge, the spectral distortion in 2DES maps due to the inaccuracy of wedge angle and the dispersion have not been reported quantitatively before. Here, we present a typical-case study of the spectral distortion arising from the inaccurate CF of glass wedges in 2D maps of light-harvesting complex II (LHCII) in photosystem II of higher plants. Consequently, we obtained the CF values experimentally with a relative accuracy of 3 × 10 −5 and provide a quantitative relation between the spectral shift and the deviation of CFs. Besides, a solution to dispersion correction is proposed to reduce the spectral distortion on 2D spectra.

Experimental setup
The schematic diagram of the 2DES setup is shown in Figs. 1(a) and 1(b). 35fs laser pulses were generated by Ti:sapphire laser system(Spitfire Ace, Spectra Physics) at a central wavelength of 800 nm (60 nm bandwidth), operating at an 1kHz repetition rate. The visible broadband light source was converted from 800 nm laser pulses by a home-built non-collinear optical parametric amplifier (NOPA), spanning the visible region (560-720 nm) with a fullwidth at half maximum (FWHM) bandwidth of 90 nm centered at 650 nm (See Fig. 1(c)). After pulse compression through a combination of chirped mirrors (Layertec) and a fused silica prism compressor, the resulting laser pulse could achieve a FWHM of 17fs. In our 2DES scheme, two broadband beam splitters (Layertec) were used to create four beams in a fully non-collinear geometry. After the first beam splitter (BS1), the input beam was separated into beam (1, 2) and beam (3,4) respectively. Beam (1,2) were delayed by a long-range translation stage ST1 (Newport) to introduce the population time (T) and the height of beam (3,4) was lowered to be different from beam (1, 2) after four reflections. Then the second splitter (BS2) separated both beams into four beams resulting in a BOXCARS geometry. The inherent passive phase stability was achieved with a value of λ/120 in 30 minutes since pairs of pulses always passed through each reflective component [25], shown in Fig. 1(d). Two pairs glass wedges having a wedge angle of 1° ( ± 3′) (MT-Optics Inc), mounted on the other two short-range translation stages (25mm, 0.1μm minimum increment, Newport), were inserted to paths of beam 1 and 2 to control the coherence time between the excitation pulses 1 and 2, respectively. The fourth beam, as a local oscillator (LO), was attenuated by 3-4 orders in magnitude using a variable neutral density filter. After a spherical mirror (f = 300 mm), four beams in a 15mm × 15mm BOXCARS geometry were focused to a spot size of 200 μm at the sample position and photon echo signals along the direction of LO beam are collected by an EMCCD coupled spectrometer (SP2358, Princeton Instrument). In order to remove the scattering contribution and speed up the data acquisition, a shot-to-shot scheme was applied which has been reported in detailed previously [40]. All the 2DES measurements were performed with a pulse energy of less than 40 nJ. During data processing, 2D maps were phased using the projection slice theorem [41], in which two phase factors (φ: phase constant; t c : time delay between beam 3 and the LO) were added to adjust the phase of 2D spectra to match the frequency-resolved pump-probe results.
Trimeric LHCII were isolated from spinach [42], dissolved in a buffer solution containing 0.03% β-dodecylmaltoside (β-DM) to an optical density (OD) at the Q y band maximum (675 nm) of 0.3 in a 0.1-mm-thick closed cell. Each monomeric unit of native LHCII binds eight chlorophylls a (Chl a) and six chlorophylls b (Chl b) pigments, whose Q y bands appear at 675 and 652 nm, respectively. The linear absorption spectrum of LHCII sample, shown in Fig.  1(c), was recorded at room temperature. The CF value we used is 26.6 fs/mm according to the nominal wedge angle (1°). Thus, the 2DES spectra of LHCII were collected by scanning the coherence time (τ) in the range of −106 -+ 106 fs with a step size of 1.33 fs at each fixed population time. A typical as-acquired 2D map of LHCII at T = 200 fs is shown in Fig. 2(a). Viewed from the emission axis (ω 3 ), two peaks along to the diagonal predominantly correspond to the Q y absorption band of Chl b and Chl a in LHCII, with the Chl b peak at 15360 cm −1 and two Chl a peaks (the main peak and the shoulder peak) at 14700 cm −1 and 14900 cm −1 , respectively. The cross peaks below the diagonal indicate the energy transfer from Chl b to Chl a, while the cross peak with negative amplitude located above the diagonal is due to the excited-state absorption from the Chl a state. These assignments have been discussed in the previous 2DES works on LHCII [43][44][45][46][47]. However, it is obvious that two peaks along the diagonal deviate from the diagonal collectively. Solvation effect or Stokes shift [41,48,49] is probably not so dominant as to generate such a large peak shift in this case since all the chlorophyll molecules are bound to the hydrophobic protein scaffold and the population time T is too short regarding to the solvation time. The laser spectrum covered the whole absorption band of LHCII so that the 2D signals are less affected by the effect of finite pulse width [36]. We further noted that the excitation energy of peaks mismatch with those in the absorption spectrum in Fig. 1(c), i.e. the spectral distortions with a global peak shift in the 2D map occur along the excitation axis. Since the excitation axis, obtained by the Fourier transform along the coherence delay time, is related to the CF of wedges, we analyze the factors of CF value in theory to explore whether if it is the possible origin of this spectral distortion.

Possible spectral distortions due to calibration factor
Two-dimensional electronic spectra are measured by exciting the sample in a certain phasematched direction with a varied temporal sequence of three ultrashort pulses [41,50]. The third-order nonlinear response in a 2D spectrum can be described as: where the coherence time τ is defined as the time delay of the first two pulses. After the population time T, the third pulse induces an electric field signal with the emission time t in the propagation direction of the fourth (LO) pulse for optical heterodyne detection (See Fig.  2 The coherence time delay can be created by a pair of glass wedges in an antiparallel orientation mounted on linear translation stages, shown in Fig. 2(c) [51]. When moving one of wedges with a certain distance (D) related to an initial position, the stage changes the optical path of a beam and creates a time delay Δτ = D (n-1) sinα/c for small wedge angle (α). n is the refractive index of wedges (typically fused silica) and c is the velocity of light in vacuum. Thus, the CF of wedges is defined as: Assuming that the refractive index is wavelength-independent in the range of laser spectrum and the nominal wedge angle α is 1°, the value of k is approximately 26.6 fs/mm as used in Fig. 2(a). However, because of the manufacturing technical limit of wedges, an inaccurate wedge angle (e.g. error with ± 3′) may introduce a distinct deviation ( ± 1.3 fs/mm) for the CF.
In order to obtain the quantitative influence of inaccurate k′ value on 2D maps, we assume a two-level system with a transition energy of ω 0 and its third-order response can be described simply as exp(-iω 0 τ). Since CF is only related to the coherence time, here we neglect the integral term of emission axis ω 3 . Then, the Fourier transform along the excitation axis with inaccurate coherence time (τ′) can be written as: From the expression of (3), it is clear that the resulting excitation energy in 2D map differs from the initial position (ω 0 ). The relative spectral shift can be described as: It is obvious that the relative spectral shift is proportional to the relative error of CF. When the inaccurate CF (k′) is larger than the actual value (k), the peak in 2DES maps has a global red shift along the excitation axis. When k′ is less than k, there is a blue shift along the excitation axis. Given ω 0 = 16000 cm −1 , k = 26.6 fs/mm and Δk = 0.2 fs/mm, the peak shift can be estimated to be 120 cm −1 , which is misinterpreted easily as the solvation effect or Stokes shift on 2D spectra. Thus, it can be seen that a minor deviation of wedge angle would lead to an apparent spectral distortion as discussed above. Therefore, an accurate calibration and correction must be taken to reduce this distortion. It is noted that these peak shifts cannot be corrected by phasing through the projection slice theorem because CF value only dominates the step size of coherence time.

Calibration and correction
Spectral interferometry is an efficient measurement for the CF of glass wedge, which was first proposed by Brixner et al [19]. First, we place a 100 μm diameter pinhole at the sample position and block the third beam and LO beam. Then both beams 1 and 2 pass through the pinhole and create a spectral interference pattern (See Fig. 3(a)) when a glass wedge is moved in one of the beams. Fourier transform and a Hamming window are used to extract the oscillation component at each detected wavelength, shown in Fig. 3(b). Finally, the phases of the oscillation are retrieved by inverse Fourier transform and the CF can be determined by fitting the slope of the unwrapped phase vs translation distance at each wavelength, shown in Fig. 3(c). In addition, the residue of phase fitting as shown in Fig. 3(c) could be used to estimate the surface roughness of wedges [52]. In our measurement, each wedge is moved with a step of 0.01 mm over 10 mm to reduce the influence from the error of the wedge surface flatness. Figure 3(a) shows the partial spectral interference pattern recorded from the wedge 1. Due to the excellent phase stability in our 2DES setup, the relative standard deviation Δk/k in our calibration is maintained below 3 × 10 −5 , corresponding to a possible spectral shift Δω of 0.5 cm −1 at 650 nm wavelength, which indicates the calibration of CFs is accurate and reliable. The final CF results are obtained in Fig. 4, which are dependent on the wavelength. Noted that the curves of wedge 1 and 2 correspond to the CFs of rephasing and nonrephasing spectra respectively. The difference of CF between wedge 1 and 2 is less than 0.05 fs/mm, indicating that both wedges are nearly identical. However, the variation of each CF is approximately 0.3 fs/mm from 560 nm to 760 nm. The wavelength dependence of CF originates from the refractive index of glass wedges n(ω) according to Eq. (2). For comparison, we utilize the three-term Sellmeier equation [53] to describe the material dispersion curve n (ω) and the theoretical CF curves are calculated with different wedge angles, shown as dashed lines in Fig. 4. The measured wavelength dependence of CFs is consistent with the calculated ones. Therefore, the actual wedge angles we calibrated are estimated to be 1° + 0.6′. Although the deviation from the nominal angle is much less than the given standard error ( ± 3′), it could lead to spectral distortions as discussed below. Because the main spectral signals of LHCII are only located in the 650 -700 nm, the wavelength-dependent CF curves, for the sake of discussion, are substituted for an averaged CF value (k A = 26.84 fs/mm) to study the spectral distortion effect. The wavelength dependence will be taken account of in the last part.  To demonstrate the CF effect on the spectra, 2D maps of LHCII having different CF values are constructed under the same experimental condition at the fixed population time (T = 200 fs), as shown in Fig. 5. In the Fig. 5(a), the same measured CF value (k = 26.84 fs/mm) is used for both rephasing and nonrephasing spectra. Two Chl a/b diagonal peaks, as expected, are relocated along the diagonal and the excitation energy also matches those in the absorption spectrum of LHCII, from which we can infer that the spectrum is appropriately corrected. Figures 5(c) and 5(d) show distorted spectra with an introduced error in calibration factor of ± 0.2 fs/mm respectively. All the peaks are either red-shift in Fig. 5(b) or blue-shift in Fig. 5(c) with a deviation about 110 cm −1 , which are in accordance with the relation in Eq. (4).
However, it should be noted that the spectral shape may undergo more complicated twists when variations in k values for rephasing and nonrephasing spectrum are different since a pure absorptive spectrum is the sum of both real-part rephasing and nonrephasing spectra. In Figs. 5(d)−5(f), three additional 2D spectra are provided in which CF values of the rephasing and nonrephasing spectra are treated individually with a slightly different modification. Clearly, the spectrum with the actual wavelength-averaged values (k 1 = 26.825 fs/mm and k 2 = 26.855 fs/mm) in Fig. 5(d) has no apparent distinction from that in Fig. 5(a), indicating that the spectrum in Fig. 5(d) is free of distortion and that the error of ± 0.02 fs/mm is small enough to neglect the resulted spectral distortion. However, by introducing a positive error of 0.1 fs/mm to the rephasing spectrum and a negative error to the nonrephasing, significant spectral twist can be produced in Fig. 5(e) and vice versa in Fig. 5(f). Because rephasing and nonrephasing spectra undergo peak shifts which are equal in magnitude but opposite in direction along the excitation axis, the resulting 2D absorptive spectra show slightly larger spectral broadening along ω 1 axis. If the error of CF became larger, the spectral broadening would be distinct enough to affect the measurement of homogeneous broadening on the diagonal peaks. In particular, an obvious peak shift in Fig. 5(f) with respect to that of Fig.  5(e), can be easily mixed up with Fig. 5(c). This precludes the attempt to refine an accurate CF without calibration from the global peak shift along the excitation axis with respect to the absorption spectrum.
In Section 3.2, we have obtained a quantitative relation between the spectral shift and the error in the calibration factor as shown in Eq. (4). Since the pure absorptive spectrum is the sum of the rephasing and nonrephasing spectrum, the spectral shift is governed by the maximum of distortion from either the rephasing or non-rephasing spectrum. Based on Eq. (4), the spectral shift in the pure absorptive spectrum is defined as: which is verified in Figs. 5(b), 5(c), 5(e) and 5(f). Finally, we also verify the CF effect with wavelength dependence by introducing different coherence time series for each excitation wavelength (λ m , m is the number of pixels), given by where x(i) is the fixed wedge displacement series for every scanning. Here, the actual CF curves k 1 (λ) and k 2 (λ) for both rephasing and nonrephasing data are considered in the resulting 2D spectrum, shown in Fig. 6(a). In order to distinguish the difference before and after modification, a comparison of slices from 2D spectra in Fig. 6(a) (solid curves) and Fig. 5(a) (dashed curves) taken along the emission axis at the 14700 cm −1 (blue line) and 15360 cm −1 (red line) is shown in Fig. 6(b). The averaged CF values result in a slight broadening along excitation axis but no evident shape distortion compared with Fig. 6(a) so that the wavelength dependence of CF is negligible for our 2DES measurements. One should note that in the given excitation range as shown in Fig. 6, k(λ) only gives rise to a change in CF about 0.15 fs/mm. When a broader spectral range is involved, the corresponding k(λ) would lead to a change of CF larger than 0.2 fs/mm, and then the wavelength-dependent CF should be used instead of the wavelength-averaged one. Other minor factors may exist which lead to distortion effects along the excitation axis. For example, Augulis et al have proposed a calibration method [52] to eliminate the spectral phase error from surface thickness variations of wedges. It is no doubt that poor surface roughness of wedges will result in shape twists, while the measured CF value is insensitive to the wedge surface because linear fitting is taken to neglect the minor phase error during the procedure in Fig. 3(c). Thus, the deviation of CF value would cause a global peak shift as above discussion, but surface roughness would only lead to local peak twists in 2D maps.

Conclusion
We have presented the theoretical analysis and 2DES implementation to explore effects of the inaccurate calibration factor for wedge-based delay lines leading to the distortion of 2D maps. Using a simple two-level system, we find the quantitative relation between the accuracy of calibration factors and the spectral shift along the excitation axis in 2D spectra. The asacquired and corrected 2DES maps of LHCII provide a typical example. A relative accuracy in calibration factor of 3 × 10 −5 is obtained to correct our spectra. Besides the global peak shift, more complex spectral twists may arise, especially when the deviations between wedge 1 and 2 are different or of opposite sign. Furthermore, after dispersion correction it is found that dispersion effects on the calibration factor make little contribution for spectral twist in 2DES measurements, but probably should be considered during data processing using Eq. (6) when a larger excitation bandwidth is involved in 2D spectra.