1.College of Sciences, Jilin Institute of Chemical Technology, Jilin 132022, China 2.Jilin Provincial Key Laboratory of Applied Atomic and Molecular Spectroscopy, Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, China
Fund Project:Project supported by the National Key R&D Program of China (Grant No. 2017YFA0403300), the National Natural Science Foundation of China (Grant Nos. 11774129, 11627807, 11534004), the Natural Science Foundation of Jilin Province, China (Grant No. 20170101153JC), and the Science and Technology Project of the Jilin Provincial Education Department, China (Grant No. JJKH20190183KJ)
Received Date:22 May 2019
Accepted Date:14 June 2019
Available Online:01 September 2019
Published Online:05 September 2019
Abstract:The higher-order harmonics generated from an atom irradiated by ultarashort laser pulses is one of the important ways to obtain ultrashort attosecond pulses and coherent XUV sources. In order to produce a high-inntensity XUV source, one needs to study the mechanism of harmonic generation. The mechanism of the atomic high harmonic generation can be well understood by the semi-classical three-step model. First, the electron tunnels the barrier formed by the atomic potential and laser electric field, and then it propagates freely in the laser field and can be driven back to the mother ion where it recombines with the ground state of ion. Although the cutoff energy of the high harmonic can be predicted by this model, it cannot provide more information about the harmonic efficiency and the spectral structure. Recently, the generation mechanism of high harmonic has been studied by using the Bohmian trajectory scheme based on the time dependent wave packet. It is found that the harmonic structure can be reconstructed qualitatively by using a single Bohmian trajectory. The more accurate structure of harmonic spectrum needs more Bohmian trajectories. The calculation of these trajectories requires a lot of computation resources because the trajectory calculation is from the numerical solution for the time-dependent Schr?dinger equation. In this work, we numerically solve the time-dependent Schr?dinger equation of a model atom irradiated by ultrashort laser pulses. The time-dependent dipole moments at different spatial locations are calculated from the time-dependent wave function. The harmonic spectra are calculated from the Fourier transform of the time dipole moments. The harmonic spectra of different spatial locations show that the main emission positions of harmonic emission are near the nuclear region. One can observe the odd- and even-order harmonics at the different spatial positions. There has a larger radiation intensity for the integer-order harmonic. For the odd-order harmonics, their harmonic phases are the same on both sides of x = 0. For the even-order harmonics, their harmonic phases each have a pi difference on both side of x = 0. By using the filtering scheme, we analyze the phases of an harmonic at different spatial locations. It is found that the phase difference leads the odd-order harmonics to increase and the even-order harmonics to disappear. These findings contribute to the understanding of the physical mechanism of higher harmonic generated from an atom irradiated by strong laser pulses. Keywords:high-order harmonic emission/ spatial distribution/ wave packet
为了分析相位对谐波发射的影响, 在图3中给出了利用空间不同位置偶极矩计算得到的谐波相位. 从图中可以看出, 在不同空间位置, 偶极矩的相位分布变化较大. 但整体上还是看出具有较好的对称性, 和光谱强度分布的对称性一致. 这一对称性反映了原子波函数具有的宇称守恒特征. 在图中白色方框分别标出了11和12次谐波的主要发射区域. 对于11次谐波, 其相位在$x = 0$正负两侧变化不大, 因而将这部分谐波相干叠加, 其谐波强度将会相干增强. 对于12次谐波, 其相位在$x = 0$正负两侧具有较大改变, 相位相反, 因而将这部分谐波相干叠加, 其谐波强度将会相干相消. 对于图2中的偶次谐波发射, 在$x = \pm 1.2\;{\rm{ a}}{\rm{.u}}{\rm{.}}$附近存在极小值, 该极小值的产生可以通过谐波相位的空间分布理解. 从对应的空间位置相位变化可以看出, 在该空间位置的偶次谐波相位发生较快改变, 导致叠加后该位置谐波的强度相干相消, 出现节点. 图 3 利用${a_x}(t)$计算的高次谐波发射相位随x的改变 Figure3. Spatial distribution of the phase of harmonic emission calculated from ${a_x}(t)$.
为了从时域直观地观察到这一特征, 利用谐波的振幅和相位信息进行滤波, 选择出所关注的谐波次数, 进行傅里叶逆变换, 得到该次谐波的时间变化信息. 在图4(a)中给出了11次谐波在空间位置分别为x = 2 a.u. (黑色实线)和x = –2 a.u. (红色点线)的含时偶极矩随时间的改变. 从图中可以看出, 这两个偶极矩幅值相差不大, 相位相同, 因而这两个空间点产生的11次谐波可以相干增强. 图4(b)给出了12次谐波相空间范围内的偶极矩随时间的变化. 从图中可以看到, 该次谐波空间对称的两个点的含时偶极矩的幅值也接近, 但相位相反. 图 4 11次谐波(a)和12次谐波(b)分别在空间x = –2 a.u.和x = 2 a.u.位置的偶极矩随着时间的改变 Figure4. Time evolution of the dipole moment at x = –2 a.u. (black solid curve) and x = 2 a.u. (red dotted curve) : (a) The eleven-order harmonic; (b) the twelve-order harmonic.
根据上面的分析可以知道, 空间不同位置的谐波发射对整体的谐波贡献不同. 在原子核附近, 由于波包的布居较多, 电离电子返回核区后产生的谐波强度较大, 谐波发射也在这一区域. $x = 0$处由于其势函数导数为0, 谐波强度较弱. 对于原子核左右两侧, 不同阶次谐波的相位不同, 对于奇次谐波, 其谐波相位相同, 对于偶次谐波其相位相反. 整体的谐波发射过程可由示意图5给出. 图 5 不同空间区域发射谐波的相关过程产生了原子的谐波发射 Figure5. The harmonic emission of atoms is produced by the process of harmonic emission in different space regions.
图 1 (a) 激光脉冲辐照下原子的高次谐波发射; (b) 
图 2 利用
图 3 利用
图 4 11次谐波(a)和12次谐波(b)分别在空间x = –2 a.u.和x = 2 a.u.位置的偶极矩随着时间的改变
图 5 不同空间区域发射谐波的相关过程产生了原子的谐波发射