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Abstract
Efficient enhancement of harmonic brightness near the cutoff region is achieved by employing laser pulses with a small positive chirp in theory, where the laser intensity and frequency near the peak of the laser pulse are almost unchanged relative to the chirp-free field. The improvement of harmonic brightness is achieved under the condition that the ionization probability is almost unchanged. Through the analysis of the harmonics contributed by the rising and falling parts of the laser pulse, we have uncovered a "frequency compensation" mechanism that leads to an enhanced harmonic brightness near the cutoff region. Under appropriate chirp parameters, the harmonics contributed by the rising and falling parts can be constructively interfered in a smaller frequency range with greater intensity, thereby obtaining harmonics with good monochromaticity and high brightness. This study explains the mechanism of harmonic brightness enhancement from a new perspective, and provides a new idea for harmonic regulation without changing the ionization.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The investigation of the interaction between light and matter plays a crucial role in exploring the properties of matter and understanding the laws of physics [1–3]. High-order nonlinear optical effects—high-order harmonic radiation—can be produced when intense laser pulses irradiate atoms, molecules, and solids [4–11]. High-order harmonics serve as key sources for the desktop extreme ultraviolet, soft x-ray, and attosecond pulse [12–17], offering a powerful tool for detecting the structural and dynamic processes of matter [18–23]. The reason for limiting the further application of gas harmonics is that their conversion efficiency is generally low [24]. Therefore, how to improve the efficiency of gas harmonics has become a significant concern of people.
Aiming at the improvement of harmonic emission efficiency, people have developed many schemes in theory and experiment [25,26]. When the initial state of the system is prepared as the coherent superposition of the ground state and the excited state, the conversion efficiency of harmonic generation can be greatly improved [27]. By changing the internuclear distance of molecules, the increase of the harmonic intensities can be found [28,29]. Due to the high electron density of solids, higher efficiency harmonics can be obtained by irradiating solids with intense laser pulses [30,31]. In addition to using different media for efficient HHG, it is also proposed to enhance harmonic efficiency through the utilization of composite fields. By optimizing the waveform of the laser pulse using a multicolor laser field scheme, Jin et al. enhanced the harmonic efficiency by 1–2 orders of magnitude [32]. By changing the laser parameters of the two-color linearly polarized driving field, Lan et al. found that the efficiency of specific order harmonics can be regulated [33].
In addition, chirped pulses also have a significant effect on harmonic emission [34]. Chang et al. experimentally demonstrated that obvious harmonic peaks can be observed in positively chirped pulses, whereas negatively chirped pulses resulted in irregular harmonic peaks [35]. The research of Kim et al. showed that the harmonic chirp is determined by the competition between the dynamically induced negative chirp and the self-phase modulation induced positive chirp. Under certain experimental conditions, the appropriate chirped laser pulses can generate clear and bright harmonics [36]. This phenomenon has also been found in solids [37]. Considering the propagation effect, Petrakis et al. regulated the quantum trajectory of harmonic emission by chirped laser pulse [38]. References [37,38] explained the corresponding mechanism in time domain. Using chirped laser pulses, Han et al. effectively improved the high-order harmonic efficiency in inhomogeneous fields [39].
The above studies show that the chirped pulse can effectively control emission. However, it should be noted that the chirp parameters of laser pulses used in most studies are relatively large, which will lead to significant changes in the laser field waveform, thereby affecting the ionization and emission behavior of electrons, destroying the phase-matching process of electrons, and may affect the further improvement of harmonic emission. Therefore, it is worth exploring whether harmonic emission can be optimized while maintaining the light field waveform and ionization probability almost unchanged.
2. Method
By solving the 2D time-dependent Schrödinger equation [40,41], we studied the influence of linearly polarized laser pulses with small chirp parameters on H atomic harmonics in detail. As follows: (Atomic units are always used in this paper unless otherwise stated)
Here, the soft-core Coulomb potential
In our work, $a$=0.2629 and $\beta$=−0.2. The ground state energy is −0.5 (corresponding to the ground state energy of H atom). The laser electric field is as follows:
Here, the $f(t)$ is the Gaussian envelope of the laser pulse with a full width at half maximum of 10 cycles. $E$ and $\omega$ are the peak amplitude and the central frequency of the laser pulse, respectively. $\delta$ is the coefficient to adjust the laser chirp. $\delta$ > 0 represents positive chirp, and $\delta < 0$ represents negative chirp.
A time-dependent wave function can be obtained through the numerical solution of Eq. (1) with a splitting-operator fast-Fourier-transform scheme. By using the time-dependent wave function, the dipole accelerations in the x and y directions can be written as:
The corresponding harmonic spectra are calculated by
3. Results and discussion
The laser fields with chirp parameters of $\delta =0$ and $\delta = 2 {\times } 10^{-6}$ are illustrated in Fig. 1(a). The central wavelength of the laser field is 800 nm, and the field intensity amplitude is 0.09. When $\delta = 2 {\times } 10^{-6}$, the instantaneous frequency of the laser field gradually increases with time. In comparison to the case of $\delta =0$, there is minimal change near the peak of the laser field for $\delta = 2 {\times } 10^{-6}$, as depicted in Fig. 1(b). The corresponding harmonic spectra are presented in Fig. 1(c). Since the high-order harmonic is a highly nonlinear process, the harmonic change should be small when the field change is very small, as shown by the harmonics in the plateau region of Fig. 1(c). Due to the fact that only a small number of electron trajectories are generated near the cut-off position under the action of a Gaussian pulse, a limited number of contributions reduce the interference effect between different trajectories, so the harmonic peaks near the cut-off region are relatively isolated. It is worth noting that when the laser chirp parameter is $2 {\times } 10^{-6}$, although the peak of the laser field is almost unchanged compared to the case of $\delta =0$, the harmonic spectrum near the cutoff region mainly contributed by the peak of the laser pulse becomes sharper, and the harmonic brightness near the cut-off region is significantly enhanced relative to the case of $\delta =0$, as shown in Fig. 1(d). Moreover, we can see from the inset of Fig. 1(c) that the ionization probability is almost unchanged under the two laser conditions.
Fig. 1. (a) Comparison of laser fields for $\delta =0$ and $\delta = 2 {\times } 10^{-6}$; (b) magnified view of (a). (c) Comparison of harmonic spectra for $\delta =0$ and $\delta = 2 {\times } 10^{-6}$; (d) magnified view of (c). The inset of (c) shows the ionization probability for $\delta =0$ and $\delta = 2 {\times } 10^{-6}$
In order to elucidate the mechanism behind harmonic enhancement, we conducted a systematic investigation into the harmonic variation near the cut-off region by manipulating the positive and negative chirp parameters of the laser field, as depicted in Fig. 2. It can be seen from Fig. 2(a) that the harmonic spectrum peak near the cutoff region is wider when the chirp parameter is 0. As the positive chirp parameter increases, the harmonic brightness near the cutoff region gradually becomes stronger and the harmonic width gradually becomes narrower. However, after further increasing the positive chirp parameter, the harmonic brightness becomes weaker and the spectral peak becomes wider. Notably, within the range of approximately $1.7{\times }10^{-6}-3.4 {\times } 10^{-6}$ for the positive chirp parameter, harmonics with good monochromaticity and high brightness can be obtained, as highlighted in the white dotted box in Fig. 2(a). Conversely, with the increase of the negative chirp parameter, the harmonics near the cutoff region exhibit a multi-peak structure, and the harmonic brightness is not significantly enhanced, as demonstrated in Fig. 2(b).
Fig. 2. The harmonic spectra of atoms as a function of chirp parameter; (a) positive chirp; (b) negative chirp.
Considering the utilization of a Gaussian laser pulse envelope in our study, we divided the harmonic into two parts of the laser pulse rising and falling contribution by dividing the time-dependent dipole moment from the temporal center into two parts. This division allows us to illuminate the underlying mechanisms responsible for the modulation of harmonic brightness and width by the laser chirp. As shown in Figs. 3(a-c), we presented the total harmonic spectra (black solid line) for chirp parameter values of 0 and $\pm \; 2 {\times } 10^{-6}$, respectively, as well as the harmonics contributed by the rising part (red dashed line) and the harmonics contributed by the falling part (blue dotted line) of the laser pulse. Observing the spectra, it becomes evident that under both the chirp parameter values of 0 and $- 2 {\times } 10^{-6}$, the harmonics stemming from the rising part of the laser pulse exhibit a blueshift, while those originating from the falling part of the laser pulse display a redshift. This phenomenon can be attributed to the fact that in the rising part of the pulse, the effective amplitude of each cycle surpasses that of the preceding cycle, which leads to the driving force of each cycle being stronger than that of the previous one. Electrons undergo an additional acceleration before returning, and electrons can return to the nuclear region earlier than they would be in the case of a constant field amplitude, which causes a frequency blueshift in the harmonic spectrum. Similarly, the frequency red shift in the harmonic is caused by the falling part of the laser pulse [42–44]. The difference is that when the chirp parameter is $2 {\times } 10^{-6}$, the offset of the harmonics contributed by the rising part and the falling part of the laser is smaller and closer to the odd-order peak. When the chirp parameter is 0, the offset of the harmonics contributed by the rising part and the falling part of the laser pulse is greater and farther away from the odd-order peak. When the chirp parameter is $-2\; {\times }\; 10^{-6}$, the offset of the harmonics contributed by the rising part and the falling part of the laser pulse is further expanded.
Fig. 3. The harmonic spectra contributed by the rising and falling parts of the laser pulse. (a) $\delta =0$; (b) $\delta =$ 2 $\times$ 10$^{-6}$; (c) $\delta =-$2 $\times$ 10$^{-6}$. (d) The phase difference between the harmonics contributed by the rising part and falling part under different chirp parameters.
We further calculated the phase differences between the harmonics contributed by the rising and falling parts of the laser pulse when the chirp parameter are 0 and $\pm$ 2 $\times$ 10$^{-6}$, as depicted in Fig. 3(d). It is evident that near the integer orders, their phase differences are approximately 0 $\pi$, satisfying the condition of constructive interference. However, the change behavior of the phase difference of the frequency near the odd harmonics is slightly different under the different chirp parameters. When the chirp parameter is 2 $\times$ 10$^{-6}$, the phase difference slope near the odd harmonics is large, and it quickly passes through the $\pi$ = 0 line, resulting in the narrowing of the harmonic spectral peak. When the chirp parameter is $-$ 2 $\times$ 10$^{-6}$, the slope of the phase difference near the odd harmonics is small, and it is close to the $\pi$ = 0 line for a long frequency interval, resulting in the broadening of the harmonic spectral peak. Moreover, the phase difference curve near even harmonics contains several lines with a large slope when crossing the $\pi$ = 0 axis, leading to several narrow spectral peaks in these regions. Based on the above results, the introduction of a positive chirp reduces the offset of the harmonics contributed by the rising and falling parts of the laser pulse, resulting in enhanced constructive interference and yielding harmonics characterized by sharp peaks and high brightness. Conversely, the addition of a negative chirp enlarges the offset of the harmonics contributed by the rising and falling parts, thereby leading to a multi-peak structure.
In order to further clarify the mechanism behind the frequency modulation of harmonics by the laser pulse with a small chirp, we displayed the variations of the harmonics contributed by the rising and falling parts of the laser pulse with respect to the laser chirp parameter, as shown in Figs. 4(a) and (b). As the positive chirp parameter increases, the harmonics contributed by the rising part progressively shift toward the low-energy region, while those from the falling part gradually shift toward the high-energy region. Since the employed laser pulse in Figs. 4(a) and (b) possess positive chirp, implying an increasing frequency with time, the frequency of the rising part of the laser pulse is lower than that of the falling part. With an increase in the positive chirp parameter, the laser frequency of the rising part becomes smaller, whereas the laser frequency of the falling part becomes larger. Therefore, the harmonics of the rising part will move to the low-energy region, and the harmonics of the falling part will move to the high-energy region. After applying the positive chirp parameter, the rising part harmonic of the original blueshift is red-shifted, and the falling part harmonic of the original redshift is blue-shifted. When the chirp parameter is appropriate, the harmonics of the rising part and the falling part will be more concentrated near the odd-order harmonic peak, thereby increasing the harmonic brightness and narrowing the harmonic width. This phenomenon resembles a "frequency compensation" mechanism. For the case of negative chirp (Figs. 4(c) and (d)), as the negative chirp parameter increases, the harmonics contributed by the rising part progressively shift towards the high-energy region, while those from the falling part gradually shift towards the low-energy region. This causes the originally blue-shifted harmonics contributed by the rising part to continue blue-shifting, and the originally red-shifted harmonics contributed by the falling part to experience further redshift. Consequently, the harmonics from both the rising and falling parts deviate further from the odd-order harmonic peaks, resulting in a multi-peak structure.
Fig. 4. Variation of harmonics contributed by the laser pulse (a) rising part and (b) falling part with the positive chirp parameter; Variation of harmonics contributed by the laser pulse (c) rising part and (d) falling part with the negative chirp parameter. The black dashed lines trace the change in harmonic frequency of the harmonic peak around 43rd.
We observed the harmonics in the cutoff region when the chirp parameters are 0 and 2 $\times$ 10$^{-6}$ under different laser field amplitudes. It can be seen from Fig. 5 that after the introduction of the positive chirped pulse, the phenomenon that the harmonic width becomes narrower and the brightness becomes stronger can be observed in a larger laser intensity range.
Fig. 5. The harmonic spectra of atoms as a function of field intensity amplitude; (a) $\delta =0$; (b) $\delta =$ 2 $\times$ 10$^{-6}$.
Finally, we calculate the influence of the propagation effect [16] on the harmonics near the cut-off region under laser pulses with different chirp parameters. The detailed calculation formula can be found in the Supplement 1. The 800 nm infrared laser pulse with a pulse duration of 26 fs and a laser beam waist of 20 $\mu$m is used as the initial driving laser field. The peak intensity at the focus (in the vacuum) is assumed to be 3.1 $\times$ 10$^{14}$ W/cm$^{2}$. Note that these parameters might be modified when the fundamental laser propagates in the gas medium. The center of the gas target with uniform distribution is placed at the laser focus, and the length of the gas target medium is 0.5 mm. Ne atom is selected as the target atom. The simulation results are shown in Fig. 6. The black solid line, red dashed line, and blue dotted line are the harmonic spectra generated by the laser field with a chirp parameter of 0 and $\pm$ 2 $\times$ 10$^{-6}$, respectively. It can be seen that at a lower gas pressure of 20 Torr, after the positive chirp is added, the harmonic brightness near the cut-off region will increase and the harmonic peak will narrow. After adding a negative chirp, the harmonic peak will be broadened and a multi-peak structure will be generated. This phenomenon is consistent with the phenomenon we observed under the single-atom response, which indicates that our proposed scheme is not only applicable to the single-atom case, but also to the case where the propagation effect is considered at lower pressure.
Fig. 6. (a) Macroscopic harmonic spectra of Ne under different chirp parameters; (b) magnified view of (a).
4. Conclusion
In summary, we investigated the effect of laser chirp on atomic harmonics near the cutoff region through numerical simulations based on the time-dependent Schrödinger equation. Our findings reveal that by using the laser pulse with a small positive chirp, the harmonic brightness near the cut-off region can be significantly enhanced while the intensity and frequency near the laser peak are almost unchanged relative to the chirp-free field. We analyzed the harmonic changes contributed by the rising and falling parts of the laser pulse under different chirp parameters. Our analysis unveils a "frequency compensation" mechanism responsible for the observed harmonic enhancement. Specifically, the harmonic redshift caused by the positively chirped laser pulse compensates the blueshift of the harmonics contributed by the laser rising part, while the harmonic blueshift caused by the positively chirped laser pulse compensates the redshift of the harmonics contributed by the laser falling part. Moreover, through an in-depth analysis of harmonic phases, we demonstrated that the harmonics contributed by the rising and falling parts can interfere constructively at relatively large intensities under optimal chirp parameters. By proposing the utilization of the laser pulse with a small positive chirp, we presented an effective approach to enhance the harmonic brightness without changing the ionization. Our results contribute to a deeper understanding of the mechanisms underlying the enhancement of harmonic brightness by chirped fields.
Funding
National Key Research and Development Program of China (2019YFA0307700); National Natural Science Foundation of China (12074145, 11774219, 11975012, 12374029); Outstanding Youth Project of Taizhou University (2019JQ002); Natural Science Foundation of Zhejiang Province (Y23A040001); Jilin Provincial Research Foundation for Basic Research (20220101003JC).
Acknowledgments
The authors acknowledge the High Performance Computing Center of Jilin University for supercomputer time and the high performance computing cluster Tiger@IAMP. Yue Qiao thanks Shicheng Jiang and Guanglu Yuan for valuable discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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