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本文中双色激光场E(t)形式为:
$ \begin{array}{l} E(t) = {E_1}\exp \left[ { - 4(\ln 2){{\left( {\frac{t}{{{\tau _1}}}} \right)}^2}} \right]\cos \left[ {{\omega _1}t + } \right.\\ \left. {{c_1}{{\left( {t - {t_{{\rm{d}}, {c_1}}}} \right)}^2}} \right] + {E_2}\exp \left[ { - 4(\ln 2){{\left( {\frac{t}{{{\tau _2}}}} \right)}^2}} \right] \times \\ \;\;\;\;\;\;\;\;\;\;\;\;\cos \left[ {2{\omega _1}t + {c_2}{{\left( {t - {t_{{\rm{d}}, {c_2}}}} \right)}^2}} \right] \end{array} $
(1) 式中,t表示时间,E1(2)为激光振幅,ω1为基频场频率,2ω1为其倍频场频率,τ1(2)为双色场半峰全宽,c1和c2为啁啾参量,td, c1和td, c2为啁啾延迟。具体来说,本文中双色场激光场选为20fs, 1600nm和10fs, 800nm,激光强度都为0.5×1014W/cm2。半周期单极场和紫外场会在后续讨论中做介绍。
原子发射高次谐波可由求解外场下含时薛定谔方程来研究,本文中选取He原子,在单电子近似和长度表象下,1维(设电子运动方向在x方向)薛定谔方程可描述为[18]:
$ {\rm{i}}\frac{{\partial \psi (x, t)}}{{\partial t}} = \left[ { - \frac{1}{2}\frac{{{{\rm{d}}^2}}}{{{\rm{d}}{x^2}}} + V(x) + xE(t)} \right]\psi (x, t) $
(2) 式中,$V(x) = - \frac{1}{{\sqrt {{x^2} + 0.484} }}$为He原子库仑势能。ψ(x, t)为含时波函数。这里初始波函数可由对角化非含时薛定谔方程获得,随后通过2阶分裂算符方法进行传播,具体详见参考文献[18]。
通过傅里叶变化可得高次谐波谱图S(ω)为:
$ S(\omega ) = {\left| {\frac{1}{{\sqrt {2{\rm{ \mathsf{ π} }}} }}\int a (t)\exp ( - {\rm{i}}\omega t){\rm{d}}t} \right|^2} $
(3) 式中, $a(t) = - \left\langle {\psi (x, t)\left| {\frac{{\partial V(x)}}{{\partial x}} + E(t)} \right|\psi (x, t)} \right\rangle $。
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众所周知[9-11],利用啁啾调频技术改变激光波形是常见的波形优化方案。因此,首先通过双色场啁啾调频方案调控半周期激光波形。本节中首先调控负向半周期激光波形。通过计算当啁啾参量为c1=-6,c2=-7,td,c1=td,c2= 0时,谐波光谱在相应负向半周期波形下会呈现截止能量延伸的效果,如图 1所示。首先分析图 1a激光波形可知,对于无啁啾双色组合场,其波形比较复杂,有许多个半周期波形组成。但是在激光上升和下降区域激光强度不高,因此,谐波能量不太。这里只考虑激光振幅附近谐波辐射过程,具体来说就是从-1T~+1T区间(T是1600nm激光场光学周期)。由分析可知,在该时间段内大致有4个半周期波形,标记为A1~A4。其在谐波辐射过程中对应谐波能量峰的P1~P4,如图 1b所示。随着啁啾参量的引入,负向半周期波形A4被明显展宽,如图 1a所示。因此,当自由电子在此半周期内加速会获得更多的动能,进而在其与原子核发生碰撞时可以发射更大光子能量的谐波。故导致其对应的谐波辐射能量峰P4得到有效延伸,如图 1c所示。并且,谐波复辐射能量峰P4是来自于负向半周期激光波形。分析高次谐波光谱可知,在波形驱动下,谐波截止能量得到延伸,进而可以获得从120次~380次的谐波光谱平台区,如图 1d所示。这里,为了后续说明方便,定义c1=-6,c2=-7,td, c1=td, c1=0时的激光场为本节中的基础场(fundamental pulse, FP)。图中, a.u.表示原子单位(atom unit), 本文中未做专门说明的物理量单位为任意单位。
Figure 1. a—laser profiles of chirp-free and chirped pulses b—time-frequency analyses of harmonics for the cases of c1=c2=0 c—time-frequency analyses of harmonics for the cases of c1=-6, c2=-7 d—high order harmonic spectra of chirp-free and chirped pulses
图 2中的分析显示,谐波光谱平台区全部来自P4,并且其来源于负向半周期波形。接下来引入半周期单极激光场(half-cycle pulse, HCP)对谐波截止能量进一步延伸。由于P4来自于负向半周期波形,因此,引入负向半周期单极场(down half-cycle pulse, DHCP)更为合适。目前,随着激光技术的发展,人们已经可以利用少周期激光场在反对称结构中传播来获得半周期单极场[19]。这里半周期激光形式选为较常用的形式:
$ \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{E_{{\rm{HCP}}}}(t) = \\ {E_{{\rm{HCP}}}}\left\{ {\frac{{400{{\left( {t - {t_{{\rm{d}}, {\rm{DHCP}}}}} \right)}^3}\exp \left[ { - 8\left( {t - {t_{{\rm{d}}, {\rm{DHCP}}}}} \right)/{\tau _{{\rm{DHCP}}}}} \right]}}{{{\tau _{{\rm{DHCP}}}}}} - } \right.\\ \;\;\;\;\;\left. {\frac{{0.004{{\left( {t - {t_{{\rm{d}}, {\rm{DHCP}}}}} \right)}^5}\exp \left[ { - \left( {t - {t_{{\rm{d}}, {\rm{DHCP}}}}} \right)/{\tau _{{\rm{DHCP}}}}} \right]}}{{{\tau _{{\rm{DHCP}}}}^5}}} \right\} \end{array} $
(4) Figure 2. a—laser profiles of combined field b—time-frequency analyses of harmonics for the cases of FP+DHCP with τDHCP=2.67fs c—time-frequency analyses of harmonics for the cases of FP+DHCP with τDHCP=5.34fs d—high order harmonic spectra of FP and combined fields
式中,EHCP为激光振幅,其强度与双色场一致; td, DHCP为负向半周期激光场延迟时间; τDHCP为半周期场脉宽。这里选用2种脉宽进行比较,分别为τDHCP=2.67fs和τDHCP=5.34fs。首先,分析图 2a中激光波形可知,当τDHCP=2.67fs,td,DHCP=0.8T或者τDHCP=5.34fs,td,DHCP= 0.6T时,组合场在0.5T~1.5T之间的负向半周期波形强度可以得到有效增强。这导致电子在此加速过程中可获得更多的能量,进而使谐波辐射能量峰P4得到明显延伸,如图 2b和图 2c所示。并且,随着半周期激光场脉宽增大,谐波截止能量会得到更大的延伸。分析图 2d可知,在适当引入负向半周期单极场后,谐波截止能量得到进一步延伸。并且,对于τDHCP=2.67fs和τDHCP=5.34fs的情况可以分别获得100次~600次以及100次~750次的谐波连续平台区。
通过对图 1和图 2的研究,谐波截止能量得到了有效延伸,即电子加速过程得到了调控。接下来需对谐波强度,即电离过程进行调控。虽然增大电离几率的方式有很多种,但是对于原子体系比较有效的方式是利用紫外光共振电离的方式增强电离几率[20]。本文中选用He原子,其基态与第一激发态之间能量差为19.8eV。考虑到基频场为1600nm场,因此,选用波长分别为λUV=61.5nm, λUV=123nm, λUV=184.5nm的紫外光(ultraviolet, UV)。选择此3种紫外光的原因在于其光子能量可以近似满足He原子基态与激发态之间的单、双、三光子共振跃迁能,这样可以满足紫外共振电离的条件,进而增大电离几率。紫外光脉宽选为1.5fs,强度为0.5×1014W/cm2。经过计算,紫外光延迟时间选为td, UV=0时比较合适(td, UV表示紫外光延迟时间)。如图 3a所示,当加入上述3种紫外光时,谐波强度有2个~3个数量级的增强。尤其是加入61.5nm和123nm紫外场时,即单、双光子共振电离时,谐波强度增强最为明显。随着紫外光波长增大,谐波强度的增强变弱。分析图 3b中的激光波形和图 3c中的电离几率可知,紫外光的引入位置大致在t=0时,因此,在此附近的电离几率会得到明显增强。同时,t=0时刻附近正是形成P4峰的电离时刻,因此导致P4峰强度得到明显增强,如图 3d所示。这是谐波强度增强的原因。再次分析电离几率可知,单、双光子共振电离时,电离几率明显增强,而对于多光子(大于3个光子)共振电离的情况,电离几率随也有增强,但与单、双光子共振电离相比有较明显差距。因此导致谐波强度随紫外光波长增大而减弱。
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本节中调控正向半周期激光波形。这里固定了啁啾参量依然为c1=-6,c2=-7,而只通过调节啁啾延迟来完成这一任务。通过计算当啁啾延迟为td,c1=-0.6T,td, c2=-0.3T时,谐波光谱在相应正向半周期波形下会呈现截止能量延伸的效果,如图 4所示。分析图 4a中的激光波形可知,当引入上述啁啾延迟时,激光波形在t=0到t=1T区间会呈现一个正向半周期波形,记为A5。与无啁啾场比较,其波形得到展宽。因此,电子在此区域加速时会获得更大的动能,进而使其对应的谐波辐射能量峰P5得到延伸,如图 4b所示。分析高次谐波光谱可知,在该波形驱动下,谐波截止能量得到延伸,进而可以获得从150次~400次的谐波光谱平台区,如图 4c所示。这里,为了后续说明方便,定义c1=-6,c2=-7,td, c1=-0.6T,td, c2=-0.3T时的激光场为本节中的基础场。
Figure 4. a—laser profiles of chirped pulses b—time-frequency analyses of harmonics for the cases of c1=-6, c2=-7, td, c1=-0.6T, td, c2=-0.3T c—high order harmonic spectrum for the case of c1=-6, c2=-7, td, c1=-0.6T, td, c2=-0.3T
图 4中的分析显示, 本节中谐波光谱平台区来自P5,并且其来源于正向半周期波形。因此,引入正向半周期单极场(up half-cycle pulse, UHCP)较为合适。这里,半周期场脉宽直选为τDHCP=5.34fs,强度与负向半周期激光场一样。经过计算,正向半周期激光场延迟时间选为td, UHCP=0.4T时较为合适(td, UHCP为正向半周期激光场延迟时间)。分析图 5a中的激光波形可知,当td, UHCP=0.4T时,组合场在0.4T~1.2T之间的正向半周期波形强度可以得到增强。这导致电子在此加速过程中可获得更多的能量,进而使谐波辐射能量峰P5得到明显延伸,如图 5b所示。分析图 5c中的谐波光谱可知,在适当引入正向半周期单极场后,谐波截止能量得到进一步延伸,进而获得并200次~800次的谐波光谱连续平台区。
Figure 5. a—laser profiles of combined field b—time-frequency analyses of harmonics for the cases of FP+UHCP with τUHCP=5.34fs c—high order harmonic spectra of FP and FP+UHCP
通过对图 4和图 5的研究,谐波截止能量在优化的正向半周期波形下得到有效延伸。与第2.1节中类似,接下来需对谐波强度进行调控。选用紫外场依然为λUV=61.5nm, λUV=123nm, λUV=184.5nm。紫外光脉宽和强度不变。经过计算,紫外光延迟时间选为td, UV=-0.6T时比较合适。如图 6a所示,当加入上述紫外光时,谐波强度有2个~4个数量级的增强。同样,在单、双光子共振电离时,谐波强度增强最为明显。随着紫外光波长增大,谐波强度的增强变弱。图 6b和图 6c中给出了激光波形和电离几率。由图可知,紫外光的引入位置大致在t=-0.6T时,因此,在此附近的电离几率会得到明显增强,进而导致能量峰P5的强度得到增强,如图 6d所示。这是谐波强度增强的原因。
Figure 6. a—high order harmonic spectra of different combined fields b—laser profiles of combined fields c—ionization probability of He atom driven by different combined fields d—time-frequency analyses of harmonics for the case of FP+UHCP+UV with λUV=61.5nm
最后,对谐波光谱平台区的谐波进行傅里叶变换并叠加可以获得阿秒量级的脉冲。具体来说:当选择负向半周期波形产生的谐波光谱连续区时(如图 3a或图 3d所示)。通过叠加谐波光谱的200次~400次、400次~600次以及600次~800次谐波,可获得3个脉宽分别在40as, 45as和40as的单个阿秒脉冲,如图 7a所示。当选择正向半周期波形产生的谐波光谱连续区时(如图 6a或图 6d所示)。通过叠加谐波光谱的200次~400次、400次~600次以及600次到800次谐波,可获得3个脉宽分别在38as, 40as和48as的单个阿秒脉冲,如图 7b所示。
半周期波形调控产生超宽谐波光谱平台区
Half-cycle waveform control for generating ultra-wide harmonic spectral plateau
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摘要: 为了获得高强度超宽谐波光谱平台区,采用数值求解含时薛定谔方程的方法,提出利用多色组合场调控半周期波形来获得最佳半周期谐波辐射条件。结果表明,通过调节双色场啁啾参量可以获得最佳的负向半周期波形;通过调节啁啾延迟可以获得最佳的正向半周期波形;在上述波形下,谐波截止能量得到延伸;引入紫外激光场后,在紫外共振增强电离的影响下,可以使谐波强度得到增强;当紫外光能量满足单、双光子共振增强电离时,谐波强度增强明显;随着紫外光子能量减小,谐波强度增强效果减弱;通过叠加谐波平台区谐波还可获得脉宽在50as以下的单个脉冲。这一结果对高次谐波调控以及阿秒脉冲产生是有帮助的。Abstract: In order to obtain intense and broad harmonic spectral plateau, by numerical solution of time-dependent Schrödinger equation, the half-cycle waveform control for producing the optimal half-cycle harmonic emission conditions was proposed by using multi-color combined field. The results show that, by controlling the chirps of two-color field, the optimal negative half-cycle waveform can be obtained; while, by controlling the chirp delay, the optimal positive half-cycle waveform can be produced. Driven by the above waveforms, the harmonic cutoffs can be extended. Further, with the introduction of ultraviolet pulse, due to the ultraviolet resonance enhanced ionization, the harmonic intensity can be enhanced. Furthermore, when the ultraviolet energy meets the single and two photon resonance enhanced ionization, the harmonic intensity is remarkably enhanced. With the decrease of ultraviolet photon energy, the enhancement of harmonic intensity decreases. Finally, the single attosecond pulses of sub-50as can be obtained by superposing harmonics on the harmonic spectral plateau. The results are helpful for the control of high-order harmonic generation and the production of attosecond pulses.
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