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Abstract
We demonstrate a method to simultaneously measure the rotational temperature and pump intensity in laser-induced molecular alignment by the time-resolved high harmonic spectroscopy (HHS). It relies on the sensitive dependence of the arising times of the local minima and maxima of the harmonic yields at the rotational revivals on the pump intensity and rotational temperature. By measuring the arising times of these local extrema from the time-resolved harmonic signals, the rotational temperature and pump intensity can be accurately measured. We have demonstrated our method using N2 molecules. The validity and robustness of our method are tested with different harmonic orders and by changing the gas pressures as well as the distance between the gas exit and the optical axis. Moreover, we have also demonstrated the versatility of our method by applying it to CO2 molecules.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Molecular alignment and orientation [1–6] by ultrashort pulse laser play an important role in molecular collision and reaction dynamics, and have been of great interests of both physicists and chemists for their widespread applications in many areas, varying from multiphoton ioization [7,8], ultrashort pulse compression [9,10], molecular orbital reconstruction [11–13], control and imaging of chemical reactions [14–16], charge migration [17] and high harmonic generation (HHG) [18–25]. Generally, an intense nonresonant laser pulse has proved to be the most versatile method to obtain molecular alignment. Impulsively excited by a pump laser, the induced molecular rotational dynamics can be well understood in two steps: First, the pump laser interacts with the molecules and create molecular rotational wave packet (RWP) through stimulated Raman transitions [1], leading to temporary alignment of the molecules in a narrow cone around the polarization direction; Second, the created RWP disperses and evolves under the field-free condition. Owing to the time-dependent phase beating of the coherently populated rotational states, recurrence of alignment and antialignment appears at the rotational revivals.
In molecular alignment, the induced rotational dynamics are mainly determined by two factors. One is the rotational temperature, which determines the initial thermal distribution of the rotational states. The other one is the pump laser, which determines the redistribution of the rotational states after the laser-molecule interaction. Previously, several methods have reported to measure the rotational temperature. For example, the coherent anti-Stokes Raman scattering (CARS) [26–28] and degenerate four-wave mixing [29] are two powerful methods. Nevertheless, they have limitations when, for instance, the signal frequencies are close to those of the laser pulses. In recent years, several studies demonstrated that the degree of molecular alignment depends sensitively on the pump intensity and the rotational temperature [30,31]. Moreover, Loriot et al. found that the time interval between the delocalization and alignment is correlated to the pump intensity and rotational temperature [32], which therefore provides a promising way for estimating the intensity and temperature in molecular alignment experiment. Recently, Yoshii et al. demonstrated the measurement of rotational temperature in the experiment of HHG from aligned molecules [33]. In their work, the rotational temperature is derived from the Fourier spectrum of measured time-resolved HHG signals by a fitting process in theory. In this regard, the parameters of the pump laser, especially the pump intensity, are assumed to be already known. Although the pulse duration can be obtained by an optical autocorrelator or a frequency-resolved optical gating (FROG) equipment [34] and the laser frequency can be measured by a spectrometer, an accurate measurement of the laser intensity has been a longstanding and arduous task [35–37].
In this paper, we propose an improved method to simultaneously measure the rotational temperature and pump intensity in molecular HHG experiment. In our experiment, we adopt a pump-probe system, where the pump pulse is applied to align the molecules and the intense probe pulse is used to interact with the molecules to generate high harmonic radiations [38,39]. The HHG signal is measured as a function of the time delay between the pump and probe pulses. From the measured HHG signals, we find that the arising times of the local maxima and minima of the HHG yields at the rotational revivals show strong dependence on the rotational temperature and pump intensity. Following this feature, one can take an in-situ measurement of both the rotational temperature and pump intensity synchronously. With this method, we have successfully measured the rotational temperature and pump intensity in a ${\textrm {N}_2}$ flow. The measured results are consistent for different harmonic orders and show good agreement with the nature of adiabatic expansion of a gas flow when the gas pressure and the distance between the gas exit and the optical axis are changed. These results prove our method to be valid and robust. Moreover, due to the universality of nonadiabatic molecular alignment by nonresonant femtosecond laser, our method can also be applied to other molecules except spherical-top molecules.
2. Results and discussion
2.1 Experimental details
Figure 1(a) displays the schematic of our experimental setup. The experiment is carried out by using a commercial Ti:sapphire laser system (Astrella-USP-1K, Coherent, Inc.). It delivers laser pulses at a repetition rate of 1 kHz. The pulse duration and the central wavelength of the laser are measured to be 35-fs and 800 nm, respectively. The output laser is split by a beam splitter to produce the pump and probe pulses. The pump pulse is used firstly to create the molecular RWP. Subsequently, the intense probe pulse with adjustable time delay interacts with the molecular ensemble to generate high harmonic radiations. These two pulses are focused into a gas nozzle by a convex lens with a 300-mm focal length. The nozzle with a diameter of 250 $\mu$m is mounted on a motorized stage, which is used to adjust the distance (defined as $x$) between the gas exit and the optical axis. The generated HHG signal is detected by a homemade flat-field soft x-ray spectrometer [24].
Fig. 1. (a) Schematic diagram of the experimental setup. (b) Measured HHG signal of H21 (blue dots) from ${\textrm {N}_2}$ molecule as a function of the pump-probe time delay. The solid line is the fitting result. (c) Same as (b), but for theoretical result simulated with the rotational temperature of 100 K and the pump intensity of 3.5 $\times 10^{13}$ W/cm$^2$ . $t_a, t_b, t_c, t_d$ pointed out by arrows are four characteristic times, which are independent of the rotational temperature and the pump intensity. The dashed lines together with arrows on top indicate the arising times of local extrema of HHG yields at rotational revivals.
To demonstrate our scheme, we choose ${\textrm {N}_2}$ molecule as a prototype. The measured time-dependent HHG signals (blue dots) of harmonic 21 (H21) is shown in Fig. 1(b). Here, the gas pressure is $8.58\times 10^4$ Pa and the two pulses are focused at the position of $x = 0.2$ mm. From Fig. 1(b), one can see obvious local maxima and minima at the full and fractional rotational revivals, e.g., around 8.4 ps (1 $T_{rev}$, where $T_{rev} = \tfrac {1}{2B_0c}$ is the revival period with $B_0$ the rotational constant and $c$ the velocity of light.) and 4.2 ps ($\tfrac {1}{2}$ $T_{rev}$). For ${\textrm {N}_2}$, the HHG signal is the most prominent when the molecular axis is parallel to the laser polarization due to the $\sigma _g$ symmetry of its highest occupied molecular orbital (HOMO) [40,41]. The local maximum implies that most molecules are aligned parallel to the pump pulse polarization, indicating an alignment revival. While the local minimum corresponds to an alignment perpendicular to the laser polarization, i.e., an antialignment revival.
2.2 Extracting rotational temperature and pump intensity from time-resolved HHG signals
To extract the pump intensity and rotational temperature from the measurements, we have calculated the time-dependent HHG yields with series of rotational temperatures ($T_{rot}$) and pump intensities ($I_{pump}$). In our calculations, we first numerically solve the time-dependent Schrödinger equation of the molecular RWP [2], which is
From the calculation results, we notice that the arising times of the local maxima ($t_{max}$) and minima ($t_{min}$) of the HHG yields [i.e., the local extrema pointed out by dashed lines and arrows on top at rotational revivals in Fig. 1(c)] show significant dependence on both the rotational temperature and pump intensity. For instance, in Figs. 2(a) and 2(b), we plot the arising times of the local maximum ($t_{max}^{1/2}$) and local minimum ($t_{min}^{1/2}$) around $\tfrac {1}{2}$ $T_{rev}$ as a function of the rotational temperature and pump intensity. It is shown that, as the rotational temperature or pump intensity increases, $t_{max}^{1/2}$ is delayed [see Fig. 2(a)]. While $t_{min}^{1/2}$ shows an opposite dependence, it is advanced as the rotational temperature or pump intensity increases [see Fig. 2(b)]. On the other hand, we find four characteristic times, i.e., $t_a$, $t_b$, $t_c$ and $t_d$ indicated by arrows in Fig. 1(c), in one revival, where $t_a$ ($t_c$ ) refers to the local minimum (maximum) at $\tfrac {1}{4}$ $T_{rev}$ ($\tfrac {3}{4}$ $T_{rev}$), $t_b$ ($t_d$) the midpoint between $t_{max}$ and $t_{min}$ at $\tfrac {1}{2}$ $T_{rev}$ (1 $T_{rev}$). These four characteristic times are found to be independent of the pump intensity and rotational temperature. As a result, they can be used to calibrate the time delays measured in the experiment. In our work, we use $t_a$ and $t_d$ to do the calibration, because they are easy to be identified (local minimum and maximum) in experiment. In Fig. 1(b), the time delay (the horizontal axis) has been already calibrated. To extract the arising times of the local maxima and minima precisely, we fit the time-dependent HHG signal by the least squares method. The fitting result is shown as the red line in Fig. 1(b). From the fitting curve, the arising times of local maximum and minimum around $\tfrac {1}{2}$ $T_{rev}$ are acquired to be 4.024 ps and 4.388 ps, respectively. Further, we pick out all combinations of the rotational temperatures and pump intensities that lead to $t_{max}^{1/2}=4.024$ ps. The result is shown as the solid line in Fig. 2(a). Similarly, we have also picked out the combinations of rotational temperatures and pump intensities for $t_{min}^{1/2}=4.388$ ps. The corresponding result is shown as the dotted line in Fig. 2(b). By putting these two contour lines together [see Fig. 2(c)], the resultant intersection point just corresponds to the pending rotational temperature and pump intensity. In our experiment, these two parameters extracted from the signal of H21 in Fig. 1(b) are $32.01\pm 1.45$ K and $(4.45\pm 0.13)\times 10^{13}$ W/cm$^2$, respectively.
Fig. 2. (a) The arising time of the local maximum $t_{max}^{1/2}$ at $\tfrac {1}{2}$ $T_{rev}$ for ${\textrm {N}_2}$ molecule as a function of the rotational temperature and pump intensity. The black line is a contour line for $t_{max}^{1/2}=4.024$ ps. (b) Same as (a), but for the arising time of local minimum $t_{min}^{1/2}$ at $\tfrac {1}{2}$ $T_{rev}$. The red dotted line is a contour line for $t_{min}^{1/2}=4.388$ ps. (c) Contour lines for $t=4.024$ ps (solid line) and $t=4.388$ ps (dotted line). The intersection indicates the pending rotational temperature and pump intensity.
2.3 Validity and universality of the method
To check the reliability of the extracted result, we apply our method repeatedly to other harmonic orders ranging from H15 to H19. The results are displayed in Fig. 3. It can be seen that the measured rotational temperatures and pump intensities are nearly the same for different harmonic orders. This result is consistent with the nature that HHG is a much faster process than molecular rotation. The rotational state of the molecules can be considered unchanged during the HHG process.
Fig. 3. The measured rotational temperature and pump intensity as a function of harmonic order. Here, the error bars of both results represent the standard deviation of five independent measurements.
Besides, we have performed more experiments to confirm the validity of our method. In Fig. 4, the rotational temperature and pump intensity have been measured for different gas pressures using the time-dependent signal of H21. One can see that the pump intensity remains constant with the gas pressure increasing, which is consistent with the experimental condition. While the measured temperature decreases with the increase of the gas pressure. In addition, we have also performed the experiment by changing the distance $x$. The measured rotational temperature and pump intensity as a function of $x$ are displayed in Fig. 5. It shows that the pump intensity is still unchanged, while the rotational temperature drops with the increase of $x$. The results are consistent with previous works [33,49–52].
Fig. 4. The measured rotational temperature and pump intensity as a function of pressure. Here, the error bars of both results represent the standard deviation of five independent measurements.
Fig. 5. The measured rotational temperature and pump intensity as a function of distance $x$. Here, the error bars of both results represent the standard deviation of five independent measurements.
Finally, our method is versatile and can be applied to other molecules. To demonstrate this point, we have performed experiments with ${\textrm {CO}_2}$ molecules. Figure 6(a) shows the measured results of H25. Here, the laser intensity and gas pressure are maintained the same as in Fig. 1(b). From Fig. 6(a), one can also see a strong delay dependence of the HHG signals at the rotational revivals (e.g. around $t=21.2$ and $42.4$ ps). Due to the more complex scattering cross section and the different symmetry of HOMO ($\pi _{\mu }$), the modulations of the time-dependent HHG signals of ${\textrm {CO}_2}$ are opposite to that of ${\textrm {N}_2}$. Similar to ${\textrm {N}_2}$, we choose the arising times of the local minimum and maximum [pointed out by black and red arrows together with dashed lines in Fig. 6(a), respectively] around $\tfrac {1}{2}$ $T_{rev}$ to do the extraction. For ${\textrm {CO}_2}$, $t_{min}^{1/2}$ and $t_{max}^{1/2}$ obtained from the fitting curve [yellow line in Fig. 6(a)] of the measured HHG signals are 21.013 ps and 21.791 ps, respectively. In Figs. 6(b) and 6(c), we show the calculated $t_{min}^{1/2}$ and $t_{max}^{1/2}$ as a function of the rotational temperature and pump intensity. One can see that $t_{min}^{1/2}$ and $t_{max}^{1/2}$ have opposite dependence as the rotational temperature or the pump intensity varies. We further pick out the combinations of rotational temperatures and pump intensities that lead to $t_{min}^{1/2}=21.013$ ps and $t_{max}^{1/2}=21.791$ ps. Corresponding results are shown as the solid and dotted lines in Figs. 6(b) and 6(c), respectively. Based on these two contour lines, the resultant rotational temperature and pump intensity are $40.28\pm 1.39$ K and $(4.58\pm 0.14)\times 10^{13}$ W/cm$^2$, respectively [see Fig. 6(d)]. It’s worth mentioning that, the pump intensity and rotational temperature can be determined by any two local extrema at the rotational revivals if they obey opposite dependence on $T_{rot}$ and $I_{pump}$. As an example, we select an alternative local extremum, i.e., the local minimum around $\tfrac {1}{4}$ $T_{rev}$ [$t_{min}^{1/4}$, indicated by green arrow together with dashed line in Fig. 6(a)], instead of $t_{max}^{1/2}$ to do the extraction. In our experiment, $t_{min}^{1/4}$ is measured to be 11.502 ps. Figure 7(a) displays the calculated $t_{min}^{1/4}$ as a function of $T_{rot}$ and $I_{pump}$. The dependence of $t_{min}^{1/4}$ on $T_{rot}$ and $I_{pump}$ is similar to that of $t_{max}^{1/2}$ in Fig. 6(c), but is opposite to that of $t_{min}^{1/2}$ in Fig. 6(b). With the contour lines of $t_{min}^{1/4}$ and $t_{min}^{1/2}$, the rotational temperature and pump intensity can also be determined [see Fig. 7(b)], which are $41.54\pm 1.46$ K and $(4.48\pm 0.18)\times 10^{13}$ W/cm$^2$, respectively. This result is quite close to that obtained by $t_{min}^{1/2}$ and $t_{max}^{1/2}$.
Fig. 6. (a) Measured HHG signal of H25 (blue dots) from ${\textrm {CO}_2}$ molecule as a function of the pump-probe time delay. The solid line is the fitting result. The dashed lines together with arrows on top indicate the arising times of local extrema of HHG yields at rotational revivals. (b) The arising time of local minimum $t_{min}^{1/2}$ at $\tfrac {1}{2}$ $T_{rev}$ as a function of the rotational temperature and pump intensity. The black line is a contour line for $t_{min}^{1/2}=21.013$ ps. (c) Same as (b), but for the arising time of local maximum $t_{max}^{1/2}$ at $\tfrac {1}{2}$ $T_{rev}$. The red dotted line is a contour line for $t_{max}^{1/2}=21.791$ ps. (d) Contour lines for $t=21.013$ ps (solid line) and $t=21.791$ ps (dotted line). The intersection indicates the pending rotational temperature and pump intensity.
Fig. 7. (a) The arising time of the local minimum $t_{min}^{1/4}$ at $\tfrac {1}{4}$ $T_{rev}$ for ${\textrm {CO}_2}$ molecule as a function of the rotational temperature and pump intensity. The red dotted line is a contour line for $t_{min}^{1/4}=11.502$ ps. (b) Contour lines for $t=21.013$ ps (solid line) and $t=11.502$ ps (dotted line). The intersection indicates the pending rotational temperature and pump intensity.
3. Conclusion
In summary, we demonstrate a technique to achieve an in-situ measurement of the rotational temperature and pump intensity synchronously in molecular HHG experiment. From the time-dependent HHG signals of ${\textrm {N}_2}$ molecule, we find that the arising times of the local minima and maxima of the harmonic yield have sensitive dependence on the rotational temperature and pump intensity. This feature enables us to directly determine the rotational temperature and pump intensity from the time-dependent HHG signals simultaneously. The accuracy of our method is confirmed by using different harmonic orders. Moreover, the measured results with different gas pressure and distances between the gas exit and the optical axis agree well with the nature of adiabatic expanding of a gas flow, which proves the validity of our method. Since the laser-induced alignment can generally be applied to different kinds of molecules, our method can be applied to other molecules (except spherical-top molecules) as well. Such versatility is confirmed by the experiment with ${\textrm {CO}_2}$ molecule. Besides, we have also proved that the rotational temperature and pump intensity can be determined by any two local extrema which follow contrary dependence on the rotational temperature and pump intensity. Such flexibility makes our method more convenient in practical application.
Funding
National Key Research and Development Program of China (2017YFE0116600, 2019YFA0308300); National Natural Science Foundation of China (11627809, 11704137, 11774109, 11874165); Fundamental Research Funds for the Central Universities (2017KFXKJC002); Program for HUST Academic Frontier Youth Team; Science and Technology Planning Project of Guangdong Province (2018B090944001).
Disclosures
The authors declare no conflicts of interest.
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