1. Introduction

Frequency spectrum of a time domain signal can be obtained generally by the Fourier transform (FT). When the phases of frequency components in a time trace are the same, FT can be applied to retrieve the frequency and damping function (line shape) for each component. Apodization and phase correction are usually required to retrieve the spectrum properly in FT spectroscopies. FT can also be adapted to display the spectral power distribution. For the Fourier power spectrum, however, amplitudes of frequency components are not linearly scaled, although line shapes can be retrieved correctly. In addition, FT shows low spectral resolution for closely-spaced frequency components, because the width of a spectral line is determined by the time scan range and the damping time.

For typical time domain spectroscopies such as pump/probe transient absorption (TA) and transient grating, a time-resolved signal usually contains oscillations arising from the coherent nuclear (or electronic) wave packets generated by the pump pulses [1, 2]. The phase of each frequency component varies by material parameters such as frequency, electron-phonon coupling strength, and electronic dephasing dynamics as well as experimental parameters such as pulse duration, residual chirp, and detection wavelength [3, 4]. In this situation, FT gives irregular line shapes, and therefore power spectrum is usually employed to circumvent the problem. When two (or more) frequency components overlap, that is, the frequency separation is smaller than their widths, the two components interfere to give a distorted power spectrum, which is not straightforward to interpret.

Alternative spectral analysis methods include maximum entropy method [5], maximum likelihood method [6], and linear prediction. Nonlinear least square fitting of a time trace to a model function could also be a viable option, if the model function is known a priori. For example, a time domain signal can be fitted to a sum of damped sinusoids and exponentials, provided the number of frequency components and their line shapes are known. In time-resolved spectroscopies, a signal is generally well represented by a sum of damped sinusoids and exponential functions. For the case of exponential damping function, it was recognized that a signal can be linearized. Subsequent application of the singular value decomposition method gives the amplitudes, frequencies, phases, and damping times [7]. This method, known as linear prediction singular value decomposition (LPSVD), has been used occasionally for the analyses of time-resolved spectroscopic data [8, 9] and more often in the field of nuclear magnetic resonance (NMR) [10].

In this work, we establish the condition that an FT power spectrum fails to faithfully reproduce the frequency spectrum of a time-domain signal. Furthermore, we present a robust way to perform the LPSVD analysis, and demonstrate by numerical simulation that LPSVD can be a good spectral analysis method, even when the FT is not suitable. As a specific example, we present the LPSVD analysis for the coherent phonon (CP) signal of the radial breathing mode (RBM) acquired from an ensemble of single-walled carbon nanotubes (SWCNT).

2. Numerical simulation

Here we present a simple simulation to demonstrate the difficulties in extracting the oscillation frequencies from the FT power spectrum, when the frequencies are closely spaced with variable phases. We start with a synthetic time-domain signal that consists of an exponential decay and three damped sinusoids as described by Eq. (1). The parameter values are listed in Table 1, and the three frequencies are similar to those of the RBMs of the SWCNT [11].

Table 1. Parameters of the Model Function that Consists of an Exponential Decay and Three Damped Sinusoids*

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Figure 1 shows the fast Fourier transform (FFT) power spectra of I(t) while varying the phases of the two minor (195 and 210 cm⁻¹) components. The exponential component was not included for the calculation of the FFT. When the phases of all three components are zero, the three peaks are well reproduced at the correct input frequencies, although their amplitudes are roughly squared. For different relative phases, however, the peaks are severely distorted, and the peak positions are seemingly different from the input values. The problem is augmented when a noise is added to the signal as shown below (Fig. 3(c)). The simulation clearly demonstrates the limitation of FT for the analysis of a time-resolved spectroscopic signal arising from a congested spectrum. The complication, of course, arises from the interference terms between different frequency components that fall within the bandwidth. In fact, such a distortion is significant even when the overlap is small.

 

Fig. 1 FFT power spectra of the time-domain signal that consists of three frequency components at 195, 200, and 210 cm⁻¹ (three dashed lines) having different phases. The parameters that generate these spectra are given in Table 1. Phases of the 195 and 210 cm⁻¹ components are varied from 0 to 1.5π, while that of the 200 cm⁻¹ is fixed to zero.

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LPSVD is an alternative method to analyze time-domain signals [7, 12]. In LPSVD, it is assumed that the signal is a sum of exponentials and exponentially-damped sinusoids. LPSVD has been shown to give better results than FFT in many cases [8, 9, 13]. For a signal without a noise, the overall number of exponentials and damped sinusoids is determined by the number of nonzero singular values of the properly formed data matrix [12], which is equal to the reduced order of the data matrix having Hankel structure, and the parameter values can be determined exactly in accordance with the reduced order. One sinusoid involves a rank of two (reduced order of two) because a sinusoid can be written by a sum of positive and negative frequency exponentials. For an experimental signal having noises, correct choice of the reduced order removes the noise, and the parameter values can be determined accurately. This procedure, however, introduces some uncertainty in the LPSVD analysis. When the number of damped sinusoids and exponentials in a signal is not known as is almost always the case, the outcome may vary depending on the reduced order; incorrect result and undesirable artifacts may appear by the incorrect choice of the reduced order.

The correct value of the reduced order can be determined systematically by reduced chi square (${\chi }_R^2$) test; when the signal is fitted properly, ${\chi }_R^2$ should be 1. That is, we can impose the same criteria for a good fit to LPSVD method as well. Figure 2(a) shows the time trace calculated by the same parameters given in Table 1 with the relative phase of π, but with added Gaussian random noise. Figure 2(b) shows the corresponding ${\chi }_R^2$ calculated for the LPSVD results by different reduced orders. When the reduced order is increased beyond the correct value of 7, ${\chi }_R^2$ becomes less than 1, and it hardly decreases at higher reduced orders. Therefore, the correct value of the reduced order can be determined reliably by the inspection of the plot of ${\chi }_R^2$ vs. reduced order. The parameter values retrieved by LPSVD are also listed in Table 1, which match nearly exactly with the input values including phases.

 

Fig. 2 (a) A synthesized time domain signal that consists of one exponential and three damped sinusoids with the parameters given in Table 1. Gaussian random noise is added as well. (b) ${\chi }_R^2$ from the LPSVD analysis are plotted vs. reduced order.

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In Fig. 3, the LPSVD spectrum thus obtained is compared with the FFT as well as the actual input spectrum. The LPSVD spectrum (Fig. 3(b)) reproduces the input spectrum nearly perfectly, whereas the two minor peaks are hardly discernable in FFT, even though their amplitudes comprise 40% of the main peak. Figure 3 demonstrates undoubtedly that LPSVD method is much better suited than FFT for the analysis of this simulated time trace, where the bands overlap with varying phases.

 

Fig. 3 (a) The input spectrum showing the three damped sinusoids given in Table 1. The spectrum of the time trace shown in Fig. 2(a) calculated by (b) LPSVD with reduced order 7 and (c) FFT. The exponential component was subtracted prior to the FFT.

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Additional advantages of the LPSVD analysis include; (1) exponential components can also be retrieved directly by LPSVD, whereas they should be subtracted beforehand for FFT, (2) the phase information can be obtained from the LPSVD, and (3) unlike the FFT power spectrum, peak heights in LPSVD analysis are linearly proportional to the oscillation amplitudes.

3. FFT and LPSVD analyses for the CP in SWCNT

An SWCNT sample is generally a mixture of many different nanotubes, each nanotube being characterized by its diameter and chiral angle, or equivalently by its (n,m) indices. SWCNT samples have been studied extensively mostly by frequency domain spectroscopies such as photoluminescence excitation and resonant Raman scattering [14]. The RBM frequency of an SWCNT is inversely proportional to the diameter (d) of the nanotube

Phonon spectrum can also be obtained by time-domain optical spectroscopies based on impulsive excitation of vibrations. CP wave packet can be launched by a short pulse of light resonant with an electronic transition, and the wave packet motion can be recorded by the probe pulse in a pump/probe transient absorption (TA) geometry [2, 15]. The wave packet motion leads to oscillations of the TA signal, observed ubiquitously in TA of materials. Chirality analysis of SWCNT samples has been reported through the CP spectrum of the RBMs [16, 17]. The CP method shows several advantages over the Raman spectroscopy in terms of frequency resolution, absence of Rayleigh scattering background at low frequencies, and absence of the photoluminescence noise. To compare FFT and LPSVD and to demonstrate the feasibility of the LPSVD method, TA of a micelle-suspended HipCo SWCNT solution was measured. Details of the TA experiment have been reported previously [17]. The light source was a home-built near-infrared cavity-dumped optical parametric oscillator employing a periodically poled lithium niobate crystal as a gain medium. Time resolution of the TA measurement was around 65 fs, which is high enough to resolve the wave packet motions of the RBMs.

Figure 4(a) shows the TA signal of SWCNT excited to E₁₁ band by a pump pulse centered at 1200 nm and detected at 1180 nm. For the fair comparison of the FFT and LPSVD methods, the oscillation part shown in Fig. 4(b) was obtained first by subtracting the 30 point simple moving-average smoothed data from the raw data recorded at 4 fs step size. This procedure removes low frequency component effectively including the exponential background; ~50 cm⁻¹ oscillation is attenuated by a factor of 10. Data of time delay below 1 ps was not used for the FFT and LPSVD to avoid interferences from the coherent spike and the free induction decay [17].

 

Fig. 4 (a) TA of HipCo SWCNT obtained by the pump and probes pulses centered at 1200 and 1180 nm, respectively. (b) Oscillation part of the TA signal obtained by subtracting the smoothed data.

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To determine the correct reduced order in LPSVD analysis, ${\chi }_R^2$ vs. reduced order is plotted in Fig. 5(a). Standard deviations at all data points were assumed to be constant, which is a very good approximation because of the differential nature of the TA measurement. Because E₁₁ resonance energy, electron-phonon coupling strength, and exciton dephasing dynamics are all distinct for each SWCNT, phases of the phonon wave packets for the RBMs should all be different. Figure 5(a) shows that ${\chi }_R^2$ decreases up to reduced order of 15 and stays mostly constant. Moreover, the ${\chi }_R^2$ crosses 1 at reduced order around 15, indicating that the correct reduced order should be 15. In fact, the spectra from LPSVD around the RBM frequency are the same for the reduced orders of 15~20. Figure 5(b) shows the spectrum with the reduced order of 15. There are seven damped sinusoids and one exponential in the signal above the noise level. The LPSVD spectrum thus obtained is fully reproducible including the phases. The LPSVD results are listed in Table 2, which shows that the phases are indeed different for each frequency component. Theses peaks are also consistent with the first-principles calculation [18], and mostly, but not exactly, consistent with the resonance Raman spectra [19]. Note that the CP spectrum cannot be directly compared with the Raman spectrum even when the excitation wavelength is the same, because they measure the Huang-Rhys factors in different contexts [20].

 

Fig. 5 LPSVD and FFT analyses for the oscillation part of the TA signal of HipCo SWCNT. (a) ${\chi }_R^2$ vs. reduced order obtained from the LPSVD. (b) and (c) are the spectra obtained by LPSVD and FFT, respectively. (d) FFT power spectrum of the time trace constructed from the parameters shown in Table 2. (e) Same as (d) except that the phases in Table 2 are all set to zero.

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Table 2. LPSVD Analysis for the CP Signal of a Micelle-suspended HipCo SWCNT Solution*

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The phase seems to vary randomly for each peak in Table 2. The phase of each frequency component depends on the material parameters as well as experimental parameters such as residual chirp in the pulse. As the pulse duration is nearly transform limited, the effect of residual chirp should be minor. Since the TA (and CP) experiment is based on the third order nonlinear phenomenon, the formation of the wave packets in the electronic ground and excited states involves two field-matter interactions [2]. Thus, the phase of each vibration is associated in a non-transparent manner with the Huang-Rhys factor, which is a unique parameter for each vibrational mode of each compound, electronic dephasing parameters, and phonon frequency [3, 4, 15]. In addition, since the frequency of the pulse is resonant with the lowest E₁₁ transition of the SWCNT, the CP oscillations may originate from either the electronic ground or excited states, and the phase of a CP oscillation depends critically on the electronic state in which the wave packet is created.

FFT power spectrum calculated for the same data is displayed in Fig. 5(c), and the peak frequencies are also listed in Table 2. The FFT power spectrum also displays seven peaks as in the LPSVD spectrum. However, the FFT spectrum looks rather different from the LPSVD spectrum; frequencies, amplitudes, and line shapes are very much different, although the FFT power spectrum itself is also very much reproducible indicating that this discrepancy is not due to the quality of the data. The deviation arises from the interference terms between the overlapping peaks in the calculation of the power spectrum. To demonstrate that the FFT power spectrum does not reproduce the actual input spectrum faithfully, we have calculated the FFT power spectrum of the time trace constructed from the same parameters shown in Table 2. The FFT power spectrum calculated in this way is shown in Fig. 5(d), which is virtually the same as the FFT power spectrum from the experiment (Fig. 5(c)). This proves that the actual spectrum and accurate frequencies cannot be retrieved by the FFT power spectrum for a situation where the peaks overlap with different phases. LPSVD method, of course, give the exact results for this synthetic time-domain signal. We have also calculated the FFT power spectrum by the same way used in Fig. 5(d) except that the phases in Table 2 were all forced to zero, that is, in the regime where FFT is stable. The spectrum looks much closer to the actual input spectrum than Fig. 5(d) and reproduces the frequencies within 1.3 cm⁻¹, although there are some non-negligible differences.

4. Conclusion

For a time-domain signal arising from frequency components that overlap spectrally with varying phases, we have demonstrated through simulation that Fourier power transform does not provide the correct spectrum; the FT power spectrum may be significantly distorted to give incorrect frequencies, amplitudes, and line shapes. LPSVD method, when used properly, can be a viable option to retrieve the actual spectrum. For an LPSVD analysis, correct rank of the data matrix can be determined systematically by imposing the same criteria used for the appraisal of a good fit, that is, ${\chi }_R^2$ should be close to 1. We have verified the results experimentally by the CP spectroscopy of RBMs of a SWCNT sample.