Frequency-resolved optical gating technique for retrieving the amplitude of a vibrational wavepacket

We propose a novel method to determine the complex amplitude of each eigenfunction composing a vibrational wavepacket of / molecular ions evolving with a ~10 fs time scale. We find that the two-dimensional spectrogram of the kinetic energy release (KER) of H+/D+ fragments plotted against the time delay of the probe pulse is equivalent to the spectrogram used in the frequency-resolved optical gating (FROG) technique to retrieve the complex amplitude of an ultrashort optical pulse. By adapting the FROG algorithm to the delay-KER spectrogram of the vibrational wavepacket, we have successfully reconstructed the complex amplitude. The deterioration in retrieval accuracy caused by the bandpass filter required to process actual experimental data is also discussed.


I. FOURIER TRANSFORM IN MW-FROG ALGORITHM
As we state in the main text, the discrete Fourier transform (FT) of the MW-FROG amplitude, T ex (ω u ; τ), with respect to τ in process (iv) in the MW-FROG algorithm intrinsically deviates from the actual T (ω u ; ω g ν ) because the vibrational frequencies, ω g ν , are configured with unequal separations. We show that the reduction of this deviation reduces the error in the phase retrieval of the wavepacket amplitude.
In our MW-FROG algorithm, the delay τ is discretized by using a delay step of ∆τ and a delay offset of τ offset as τ m =τ offset +m∆τ, where m is an integer ranging from 0 to N 0 − 1. The number of points used for the delay in the target spectrogram in FIG. 5(a) in the main text, N 0 , is set to 2 10 =1024. The delay step and delay offset are 0.69093984374954 fs and 0 fs, respectively. The frequency step after the discrete FT, ∆ω g /2π =1/(N 0 ∆τ), should be equal to 1.41338 THz, which is not sufficiently small to correctly express the beat frequencies listed in Table I in the main text. Therefore, we performed zero padding to increase the number of points N to 2 13 , leading to ∆ω g /2π =0.176673 THz, as a first trial implementation of the MW-FROG algorithm. We notate the discretized frequency as ω g n =ω g offset + n∆ω g , (n = 0, 1, . . . , N − 1). The resultant image of the magnitude square of T ex (ω u ; ω g n ) and its magnified view in the vicinity of a vibrational number of 3 are shown in the top panels of FIGS. S-1(a) and (b), respectively. The line profile obtained by integrating the image with respect to ω u is also depicted as dots in the same panels. In this calculation, we use the retrieved T ex (ω u ; τ m ) obtained from the target a ν and G(Ω) shown in FIGS. 5(c) and (d) in the main text. We can see from the coarse grained view that the image in the top panel of FIG. S-1(a) properly exhibits the distinct peaks of the vibrational frequencies. Nevertheless, the nearest-neighbor points from the vibrational frequencies, , is almost the same as that of the target a ν , depicted as crosses with bars, with an r.m.s. error of 3.5×10 −3 on the arbitrary unit scale of |a ν |. In contrast, the phase of a ν , depicted as solid triangles with connecting lines in the top panel of FIG. S-2(c), is considerably scattered from the target phase, depicted as crosses with connecting lines, resulting in an r.m.s. error of 200 mrad, which interferes with the observation of the phase modulation for the target a ν with a modulation depth of ±200 mrad. The functional error ∆rms, defined in the main text, is 10.7%. The retrieved magnitude and phase of the gate field, respectively depicted as solid curves in the bottom and top panels of FIG. S-2(d), are in reasonable agreement with those of the target gate field. Thus, we only needed to improve the retrieval of the phase of a ν in the MW-FROG algorithm.
This was realized by increasing the accuracy of the approximated vibrational frequencies defined in the discrete FT in process (iv). We increased the number of points, N, to 2 16 by zero padding, and thus the frequency step after the discrete FT was reduced to 1/8 (0.022084 THz) of that adopted in the previous algorithm. The magnitude square of T ex (ω u ; ω g n ) calculated under the condition of N = 2 16 and its magnified view in the vicinity of a vibrational number of 3 are shown in the bottom panels of FIGS. S-1(a) and (b), respectively. The line profile obtained by integrating the image with respect to ω u is also depicted as dots in the same panels. We can observe from these panels that the frequency step is considerably reduced. The retrieved spectrogram, a ν , and G(Ω) have already been shown in FIGS. 5(b), (c), and (d), respectively, in the main text. The retrieved magnitude and phase of a ν are also shown as hollow squares with bars (magnitude) and connecting dashed lines (phase) in the bottom and top panels in FIG. S-2(c), respectively. The scatter of the retrieved phase of a ν is sufficiently reduced to resolve the phase modulation depth of ±200 mrad. for the target a ν , as we have already stated in the main text.
We note that N = 2 16 is the maximum number of points achievable using our calculation software platform (IGOR Pro 6.35A5, WaveMetrics Inc.) based on a 32-bit architecture, and the accuracy of phase retrieval may be further improved by further increasing N by using software based on a 64-bit architecture with an increased physical memory. Nevertheless, we did not attempt this because we have already obtained sufficient accuracy for the phase retrieval of a ν and we can reduce the calculation cost without using such high-performance computing. Instead, optimization of the coefficients in the polynomial expansion of the phase of a ν in the MW-FROG algorithm helped to improve the accuracy of phase retrieval as demonstrated in FIGS. 5(c) and 6(c) in the main text and also demonstrated in the next section of this supplementary information.

II. PERFORMANCE TESTS OF MW-FROG ALGORITHM
In FIGS. S-3 and S-4 in this supplementary information, we show two examples demonstrating the feasibility of our matterwave FROG (MW-FROG) algorithm for retrieving the phases of vibrational wavepacket amplitudes using degraded delay-KER spectrograms, which simulate experimental data. We use the convergence parameter R≡ε[|T rtrvd | 2 , |T exct | 2 ]/ε[|T exprmnt | 2 , |T exct | 2 ], which is defined in refs. [36] and [37] in the main text, in order to confirm the convergence criterion of R < 2 to be satisfied, where ε[|T B | 2 , |T A | 2 ] is the functional distance between |T B (ω u ; τ)| 2 and |T A (ω u ; τ)| 2 normalized by the square root of the average of (|T , where ω u and τ are discretized with the numbers of points N and M, respectively. We notate the exact spectrogram (calculated by Eq. (16) in the main text), the simulated experimental spectrogram (obtained by applying the KER convolution and noise), and the retrieved spectrogram as |T exct (ω u ; τ)| 2 , |T exprmn (ω u ; τ)| 2 , and |T rtrvd (ω u ; τ)| 2 , respectively. In FIG. S-5, we demonstrate that the MW-FROG algorithm retrieves the wavepacket amplitude and gate field with similar accuracy to that obtained when we applied the MW-FROG algorithm to the target spectrogram shown in FIG. 5(a) in the main text, even when the gate field contains a chirp and a spectral phase offset. We degraded the target spectrogram in FIG S-5(a)   The solid curves in the bottom and top panels are the retrieved magnitude and phase, respectively. The magnitude of the gate field in the photon energy regions of both the third and fifth harmonic components is in good agreement with that of the target gate field. The phase of the gate field also exhibits a similar shape to that of the target field except for the constant offset of −π/4 in the photon energy region of the fifth harmonic component. The MW-FROG algorithm is insensitive to the offset time of the pulse train in the train envelope. In other words, we do not need to consider the change in the offset time of the pulse train in the train envelope to retrieve the phase of the vibrational wavepacket amplitude.