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Subatomic scale study of atom-generated higher-order harmonic

Liu Yan Guo Fu-Ming Yang Yu-Jun

Subatomic scale study of atom-generated higher-order harmonic

Liu Yan, Guo Fu-Ming, Yang Yu-Jun
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  • 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 Schrödinger equation. In this work, we numerically solve the time-dependent Schrödinger 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.
      Corresponding author: Yang Yu-Jun, yangyj@jlu.edu.cn
    [1]

    Ozaki T, Ganeev R A, Ishizawa A, Kanai T, Kuroda H 2002 Phys. Rev. Lett. 89 253902

    [2]

    Dromey B, Zepf M, Gopal A, Wei M S, Tatarakis M 2006 Nat. Phys. 2 456

    [3]

    Krausz F, Brabec T 1998 Opt. Photonics News 9 46

    [4]

    Emma P, Akre R, Arthur J, Bionta R, Bostedt C, BozekJ, Brachmann A, Bucksbaum P, Coffee R, Decker F G, Ding Y, Dowell D, Edstrom S 2010 Nat. Photonics 4 641

    [5]

    Gibsone A, Paul A, Wagner N 2003 Science 302 95

    [6]

    Cingoz A, Yost D C, Allison T K, Ruehl A, Fermann M E, Hart I, Ye J 2012 Nature 482 68

    [7]

    Goulielmakis E, Schultze M, Hofstetter M, Yakovlev V S, Gagnon J, Uiberacker M, Aquila A L 2008 Science 320 1614

    [8]

    Qin M, Zhu X, Zhang Q, Lu P 2012 Opt. Lett. 37 5208

    [9]

    Curkum P B 1993 Phys. Rev. Lett. 71 1994

    [10]

    Pan Y, Guo F M, Jin C, Yang Y J, Ding D J 2019 Phys. Rev. A 99 033411

    [11]

    Botheron P, Pons B 2010 Phys. Rev. A 82 021404

    [12]

    Wei S S, Li S Y, Guo F M, Yang Y J, Wang B B 2013 Phys. Rev. A 87 063418

    [13]

    Song Y, Li S Y, Liu X S, Chen J G, Zeng S L, Yang Y J 2012 Phys. Rev. A 86 033424

    [14]

    Song Y, Li S Y, Liu X S, Guo F M, Yang Y J 2013 Phys. Rev. A 88 053419

    [15]

    田原野, 郭福明, 曾思良, 杨玉军 2013 物理学报 62 113201

    Tian Y Y, Guo F M, Zeng S L, Yang Y J 2013 Acta Phys. Sin. 62 113201

    [16]

    Han J X, Wang J, Qiao Y, Liu A H, Guo F M, Yang Y J 2019 Opt. Express 27 8768

    [17]

    Yang Y J, Chen G, Chen J G, Zhu Q R 2004 Chin. Phys. Lett. 21 652

    [18]

    刘艳, 贾成, 郭福明, 杨玉军 2016 物理学报 65 033201

    Liu Y, Jia C, Guo F M, Yang Y Y 2016 Acta Phys. Sin. 65 033201

    [19]

    Tian Y Y, Li S Y, Wei S S, Guo F M, Zeng S L, Chen J G, Yang Y J 2014 Chin. Phys. B 23 053202

    [20]

    宋文娟, 郭福明, 陈基根, 杨玉军 2018 物理学报 67 033201

    Song W J, Guo F M, Chen J G, Yang Y J 2018 Acta Phys. Sin. 67 033201

    [21]

    He X, Miranda M, Schwenke J, Giulbaud O, Ruchon T, Heyl C, Georgadiou E 2009 Phys. Rev. A 79 063829

    [22]

    Chen Y J, Zhang B 2011 Phys. Rev. A 84 053402

  • 图 1  (a) 激光脉冲辐照下原子的高次谐波发射; (b) ${a_{x < 0}}(t)$计算得到的谐波谱; (c) ${a_{x > 0}}(t)$计算得到的谐波谱; (d) 三个谐波谱的对比

    Figure 1.  (a) High-order harmonic emission of atoms irradiated by laser pulses; (b) harmonic spectra calculated from ${a_{x < 0}}(t)$; (c) harmonic spectra calculated from ${a_{x > 0}}(t)$; (d) the comparison of three harmonic spectra.

    图 2  利用${a_x}(t)$计算的高次谐波发射谱随着x的改变

    Figure 2.  Spatial distribution in HHG spectra as a function of the electronic coordinate calculated from ${a_x}(t)$.

    图 3  利用${a_x}(t)$计算的高次谐波发射相位随x的改变

    Figure 3.  Spatial distribution of the phase of harmonic emission calculated from ${a_x}(t)$.

    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

    Figure 4.  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.

    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

    Figure 5.  The harmonic emission of atoms is produced by the process of harmonic emission in different space regions.

  • [1]

    Ozaki T, Ganeev R A, Ishizawa A, Kanai T, Kuroda H 2002 Phys. Rev. Lett. 89 253902

    [2]

    Dromey B, Zepf M, Gopal A, Wei M S, Tatarakis M 2006 Nat. Phys. 2 456

    [3]

    Krausz F, Brabec T 1998 Opt. Photonics News 9 46

    [4]

    Emma P, Akre R, Arthur J, Bionta R, Bostedt C, BozekJ, Brachmann A, Bucksbaum P, Coffee R, Decker F G, Ding Y, Dowell D, Edstrom S 2010 Nat. Photonics 4 641

    [5]

    Gibsone A, Paul A, Wagner N 2003 Science 302 95

    [6]

    Cingoz A, Yost D C, Allison T K, Ruehl A, Fermann M E, Hart I, Ye J 2012 Nature 482 68

    [7]

    Goulielmakis E, Schultze M, Hofstetter M, Yakovlev V S, Gagnon J, Uiberacker M, Aquila A L 2008 Science 320 1614

    [8]

    Qin M, Zhu X, Zhang Q, Lu P 2012 Opt. Lett. 37 5208

    [9]

    Curkum P B 1993 Phys. Rev. Lett. 71 1994

    [10]

    Pan Y, Guo F M, Jin C, Yang Y J, Ding D J 2019 Phys. Rev. A 99 033411

    [11]

    Botheron P, Pons B 2010 Phys. Rev. A 82 021404

    [12]

    Wei S S, Li S Y, Guo F M, Yang Y J, Wang B B 2013 Phys. Rev. A 87 063418

    [13]

    Song Y, Li S Y, Liu X S, Chen J G, Zeng S L, Yang Y J 2012 Phys. Rev. A 86 033424

    [14]

    Song Y, Li S Y, Liu X S, Guo F M, Yang Y J 2013 Phys. Rev. A 88 053419

    [15]

    田原野, 郭福明, 曾思良, 杨玉军 2013 物理学报 62 113201

    Tian Y Y, Guo F M, Zeng S L, Yang Y J 2013 Acta Phys. Sin. 62 113201

    [16]

    Han J X, Wang J, Qiao Y, Liu A H, Guo F M, Yang Y J 2019 Opt. Express 27 8768

    [17]

    Yang Y J, Chen G, Chen J G, Zhu Q R 2004 Chin. Phys. Lett. 21 652

    [18]

    刘艳, 贾成, 郭福明, 杨玉军 2016 物理学报 65 033201

    Liu Y, Jia C, Guo F M, Yang Y Y 2016 Acta Phys. Sin. 65 033201

    [19]

    Tian Y Y, Li S Y, Wei S S, Guo F M, Zeng S L, Chen J G, Yang Y J 2014 Chin. Phys. B 23 053202

    [20]

    宋文娟, 郭福明, 陈基根, 杨玉军 2018 物理学报 67 033201

    Song W J, Guo F M, Chen J G, Yang Y J 2018 Acta Phys. Sin. 67 033201

    [21]

    He X, Miranda M, Schwenke J, Giulbaud O, Ruchon T, Heyl C, Georgadiou E 2009 Phys. Rev. A 79 063829

    [22]

    Chen Y J, Zhang B 2011 Phys. Rev. A 84 053402

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  • Received Date:  22 May 2019
  • Accepted Date:  14 June 2019
  • Available Online:  26 November 2019
  • Published Online:  01 September 2019

Subatomic scale study of atom-generated higher-order harmonic

    Corresponding author: Yang Yu-Jun, yangyj@jlu.edu.cn
  • 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

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 Schrödinger equation. In this work, we numerically solve the time-dependent Schrödinger 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.

1.   引 言
  • 超短强激光与原子分子相互作用可以产生高次谐波辐射[1-3]. 由于谐波光谱强度随着谐波能量的增加存在独特的平台结构, 被用于产生相干软XUV和软X射线光源. 高次谐波具有较宽的频谱, 因而成为产生阿秒量级超短脉冲的重要方案之一[4,5]. 更重要的是高次谐波产生机制是由电离电子与母体离子的复合, 谐波信息中携带了原子、分子靶的信息, 人们可以利用谐波对原子、分子的电子轨道进行“成像”[6-8].

    目前, 对高次谐波的理解可以利用Corkum[9]提出的半经典三步模型解释: 原子在激光电场作用下, 形成一个由原子势和激光电场形成的势垒, 电子首先隧穿过这个势垒; 然后在激光电场作用下运动, 此时核的影响很小, 被忽略, 可以看作是一个经典粒子受到激光电场的驱动; 当激光电场方向改变后, 电离的电子有机会与母体离子发生重散射回到原子的基态, 发射出高能光子. 根据这一理论预言出高次谐波截止位置能量为${E_{{\rm{cutoff}}}} = {I_{\rm{P}}} + 3.17{U_{\rm{P}}}$, ${I_{\rm{p}}}$是电离能, $\displaystyle{U_{\rm{p}}} = \frac{{{E_0}}}{{4{\omega ^2}}}$是激光脉冲的有质动力能. ${E_0}$$\omega $分别是峰值场强和激光脉冲角频率. 这一公式给出的截止能量在很多情况下都可以解释实验观察到的结果[10], 因而被大家广泛接受.

    这一机制虽然可以对谐波的截止位置给出很好的描述, 对于谐波效率以及谐波的谱结构信息却不能提供更多的解释. 其原因是此机制是基于电子的经典描述, 而没有将电子看作为波包. 近期人们利用基于波包方案计算的玻姆轨迹信息对谐波的产生机制进行了研究, 发现利用单一轨迹就可以定性地重现谐波的结构[11,12]. 原因是玻姆轨迹的计算是基于波函数, 其粒子受力中包含了量子力的贡献. 然而, 进一步的研究表明, 为了与数值求解的准确的谐波相比, 达到半定量的一致, 需要考虑空间分布上更多的轨迹, 这些轨迹携带不同的相位信息. 为了和数值模拟结果达到定量的一致, 需要的玻姆轨迹数目达到上万条[13,14], 这些轨迹的计算需要非常多的计算资源. 不计算这些玻姆轨迹, 通过直接考虑波包不同空间位置的贡献来理解谐波发射过程是本文关注的内容. 本文通过数值求解含时薛定谔方程, 得到体系任意时刻的波函数. 分别计算出不同空间区域的谐波发射, 分析谐波发射在亚原子尺度的变化行为, 理解谐波发射过程. 如无特殊说明, 本文均使用原子单位.

2.   理论方法
  • 为了研究谐波发射过程, 需要数值求解原子在强激光作用下的含时薛定谔方程. 在长度规范和偶极近似下的方程表示为[15]

    其中$\displaystyle{V_{\rm a}}(x) = \frac{{ - 1}}{{\sqrt {{x^2} + a} }}$为原子势函数, 这里选择的软核势, 软化参数$a = 2.0$, 无外场时原子的基态能量为${E_0} = - 0.5$. $E(t) = {F_0}f(t)\sin (\omega t)$是线偏振入射激光电场, ${F_0} = {\rm{0}}{\rm{.0475}}$为激光电场峰值振幅, $f(t)$为梯形脉冲包络, 共有12个光学周期, 其中上升沿和下降沿各一个光学周期. 方程(1)的求解可以通过分拆算符方案实现[16-19]. 得到体系的波函数后, 可以计算加速度形式下的跃迁偶极矩:

    为了分析不同空间位置的高次谐波发射, 如x′附近的谐波发射, 将偶极矩的计算选择为x′点附近一个空间步长的贡献:

    对含时偶极矩做傅里叶变换, 可以得到相应的谐波发射谱的强度信息和相位信息[20]:

3.   结果与讨论
  • 利用激光电场与原子相互作用, 产生的谐波发射谱如图1(a)所示. 从图中可以看出, 在激光场作用下, 可以观察到清晰的平台结构, 截止能量是20次谐波附近, 与三步模型预言的结果一致. 由于原子的势函数的对称性质, 在图中可以观察到清晰的奇次谐波发射. 谐波发射过程主要是电子与母体离子重碰产生, 因而在核区附近的波包可以对谐波发射发挥重要的作用, 首先选择空间范围为$x < 0$$x > 0$的偶极矩进行分析. ${a_{x < 0}}(t)$${a_{x > 0}}(t)$[21]分别表示为:

    Figure 1.  (a) High-order harmonic emission of atoms irradiated by laser pulses; (b) harmonic spectra calculated from ${a_{x < 0}}(t)$; (c) harmonic spectra calculated from ${a_{x > 0}}(t)$; (d) the comparison of three harmonic spectra.

    图1(b)图1(c)分别给出了利用这两个部分的偶极矩计算出来的谐波谱. 从图中可以看出, 这两个谐波谱中仍然可以看到奇次谐波的发射, 但与全谱比较已经变得不明显, 同时还可以观察到强度较小的偶次谐波发射, 且谐波光谱的背景强度更大(如图1(d)所示). 将整体的谐波发射记为$P(\omega )$, ${P_1}(\omega )$${P_{\rm{2}}}(\omega )$分别是利用${a_{x < 0}}(t)$${a_{x > 0}}(t)$计算的谐波谱, 则整体的谐波发射强度可以表示为${\left| {P(\omega )} \right|^2} \,\approx \,{\left| {{P_1}(\omega )} \right|^2} \,+\, {\left| {{P_2}(\omega )} \right|^2} \,+\, 2{\rm{Re}} [{P_1}(\omega ){P_2}^*(\omega )]$, 目前的计算结果表明, 公式最后的干涉项对谐波的影响较大, 从单一部分的波包计算出的结果无法重现整体的结果.

    在研究了空间范围为$x > 0$$x < 0$的谐波基础上, 将产生谐波的空间进一步局域化, 在x = –15 a.u. 到x = 15 a.u. 空间范围内, 均匀选择300个空间点计算相应的含时偶极矩${a_x}(t)$以及谐波发射谱, 结果如图2所示. 从图中可以看出, 随着空间位置的变化, 谐波谱的结构相差很多. 谐波强度较大的区域仍然主要位于核区附近, 在x = –3 a.u.到x = 3 a.u. 之间, 超出这个区间的谐波主要是分布在整数倍谐波. 从图中还可以看出, 对于空间不同位置的谐波发射存在明显的偶次谐波, 产生这一现象的原因与非对称原子的偶次谐波产生机制一致, 即由于波包分布的不对称, 导致在一个周期才有一次主要的发射[22]. 相比于奇次谐波, 偶次谐波的强度较小, 且强度较大谐波发射的空间位置也有差别. 如果将这些谐波非相干地相加, 无法重现整体的谐波结构. 如图中黑框圈出的11次和12次谐波, 对于12次谐波非相干强度求和不为零, 这表明不同空间产生的谐波的相位对整体谐波的影响较大.

    Figure 2.  Spatial distribution in HHG spectra as a function of the electronic coordinate calculated from ${a_x}(t)$.

    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.

    Figure 3.  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.

    Figure 4.  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.

    Figure 5.  The harmonic emission of atoms is produced by the process of harmonic emission in different space regions.

4.   结 论
  • 高次谐波发射过程通常由三步模型给出, 但是其不能对谐波的光谱细节信息给出预言. 此外还有谐波的诸多解释, 如不同周期产生的辐射在时间上相长干涉和相消干涉或由角动量守恒带来的跃迁选择定则等. 本文利用数值求解含时薛定谔方程方案, 通过计算不同空间位置的含时偶极矩进而分析其谐波发射谱, 发现其产生机制可以利用空间不同位置的发光的相干性给出解释.

    感谢吉林大学超算中心的技术支持.

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