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H2+与激光场的相互作用可以通过数值求解核与电子耦合的非玻恩-奥本海默近似含时薛定谔方程来描述[20]:
$ \begin{array}{*{20}{c}} {{\rm{i}}\frac{{\partial \psi \left( {z,R,t} \right)}}{{\partial t}} = \left[ { - \frac{1}{{2{u_{\rm{e}}}}}\frac{{{\partial ^2}}}{{\partial {z^2}}} - \frac{1}{{2{u_{\rm{n}}}}}\frac{{{\partial ^2}}}{{\partial {R^2}}} + V\left( {z,R} \right) + } \right.}\\ {\left. {\left( {1 + \frac{1}{{{m_1} + {m_2} + 1}}} \right)zE\left( {z,t} \right)} \right]\psi \left( {z,R,t} \right)} \end{array} $
(1) 式中,m1,m2是2个核的质量; ue=(m1+m2)/(m1+m2+1)和un=m1m2/(m1+m2)表示电子和核的约化质量; R,z分别为核间距与电子坐标; t为时间,ψ(z, R, t)为波函数。H2+的软核库伦势V(z, R)=(R2+0.03)$ ^{ - \frac{1}{2}} $-$ {\left[ {{{\left( {z - R/2} \right)}^2} + 1.0} \right]^{ - \frac{1}{2}}} $-$ {\left[ {{{\left( {z + R/2} \right)}^2} + 1.0} \right]^{ - \frac{1}{2}}} $。激光场形式为:
$ \begin{array}{*{20}{c}} {E\left( {z,t} \right) = }\\ {E\left[ {1 + sg\left( z \right)} \right]\exp \left[ { - 4\left( {\ln 2} \right){t^2}/{\tau ^2}} \right]\cos \left( {{\omega _1}t} \right)} \end{array} $
(2) 式中, ω1, E和τ分别为激光场的频率、振幅和脉宽; $ g\left( x \right) = \sum\limits_{i = 1}^N {{\beta _i}{{(x + {z_0})}^i}} $表示空间非均匀形式, βi为空间非均匀参量; s为常数,s=0表示空间均匀激光场, s=1表示空间非均匀激光场; z0表示空间非均匀激光场中心与坐标原点的距离。本文中采用蝴蝶型金属纳米结构,如图 1所示,具体参量可见参考文献[21]。
Figure 1. Bowtie-shaped gold nanostructure and geometric parameters[21]
偶极加速度可以表示为[22]:
$ \begin{array}{*{20}{c}} {d\left( t \right) = \left\langle {\psi \left( {z,R,t} \right) \times } \right.}\\ {\left. {\left| { - \frac{{\partial V\left( {z,R} \right)}}{{\partial z}} + \frac{{\partial \left[ {zE\left( {z,t} \right)} \right]}}{{\partial z}}} \right|\psi \left( {z,R,t} \right)} \right\rangle } \end{array} $
(3) $ \begin{array}{*{20}{c}} {d\left( {z < 0.\;0{\rm{a}}{\rm{.}}\;{\rm{u}}{\rm{.}}\;,t} \right) = \int_{ - \infty }^0 {{\psi ^ * }\left( {z,R,t} \right)} \times }\\ {\left\{ { - \frac{{\partial V\left( {z,R} \right)}}{{\partial z}} + \frac{{\partial \left[ {zE\left( {z,t} \right)} \right]}}{{\partial z}}} \right\}\psi \left( {z,R,t} \right){\rm{d}}z} \end{array} $
(4) $ \begin{array}{*{20}{c}} {d\left( {z > 0.\;0{\rm{a}}{\rm{.}}\;{\rm{u}}{\rm{.}}\;,t} \right) = \int_0^\infty {{\psi ^ * }\left( {z,R,t} \right)} \times }\\ {\left\{ { - \frac{{\partial V\left( {z,R} \right)}}{{\partial z}} + \frac{{\partial \left[ {zE\left( {z,t} \right)} \right]}}{{\partial z}}} \right\}\psi \left( {z,R,t} \right){\rm{d}}z} \end{array} $
(5) 式中,上标*表共轭。高次谐波谱图可表示为:
$ S\left( \omega \right) = {\left| {\frac{1}{{\sqrt {2{\rm{ \mathsf{ π} }}} }}\int_0^{{T_{{\rm{total}}}}} {d\left( t \right){{\rm{e}}^{ - {\rm{i}}\omega t}}{\rm{d}}t} } \right|^2} $
(6) $ \begin{array}{*{20}{c}} {S\left( {z < 0.\;0{\rm{a}}{\rm{.}}\;{\rm{u}}{\rm{.}}\;,\omega } \right) = }\\ {{{\left| {\frac{1}{{\sqrt {2{\rm{ \mathsf{ π} }}} }}\int_0^{{T_{{\rm{total}}}}} {d\left( {z < 0.\;0{\rm{a}}{\rm{.}}\;{\rm{u}}{\rm{.}}\;,t} \right){{\rm{e}}^{ - {\rm{i}}\omega t}}{\rm{d}}t} } \right|}^2}} \end{array} $
(7) $ \begin{array}{*{20}{c}} {S\left( {z > 0.\;0{\rm{a}}{\rm{.}}\;{\rm{u}}{\rm{.}}\;,\omega } \right) = }\\ {{{\left| {\frac{1}{{\sqrt {2{\rm{ \mathsf{ π} }}} }}\int_0^{{T_{{\rm{total}}}}} {d\left( {z > 0.\;0{\rm{a}}{\rm{.}}\;{\rm{u}}{\rm{.}}\;,t} \right){{\rm{e}}^{ - {\rm{i}}\omega t}}{\rm{d}}t} } \right|}^2}} \end{array} $
(8) 式中,ω是频率,Ttotal是总传播时间。电离几率和解离几率(解离通道)可以表示为:
$ {P_{\rm{i}}}\left( t \right) = \int_0^t {{\rm{d}}t} \int_0^{R'} {j\left( {R,z',t} \right){\rm{d}}R} $
(9) $ {P_{\rm{d}}}\left( t \right) = \int_0^t {{\rm{d}}t} \int_{ - z'}^{z'} {j\left( {R',z,t} \right){\rm{d}}z} $
(10) $ {P_{ - ,{\rm{d}}}}\left( t \right) = \int_0^t {{\rm{d}}t} \int_{ - z'}^0 {j\left( {R',z,t} \right){\rm{d}}z} $
(11) $ {P_{ + ,{\rm{d}}}}\left( t \right) = \int_0^t {{\rm{d}}t} \int_0^{z'} {j\left( {R',z,t} \right){\rm{d}}z} $
(12) 式中,$j = \frac{1}{m}{\rm{Im}}\left[ {{\psi ^*}{\rm{\delta }} ({s_1} - {s_0})\frac{\partial }{{\partial s}}\psi } \right] $为几率流; R′, z′表示R, z几率流的位置; m表示核质量。当求电离几率时,m=ue, s1=z, s0=z′=25a.u.(本文中a.u.表示原子单位);当求解离几率时,m=un, s1=R, s0=R′=25a.u.。P±, d(t)为电子布局在正(P+, d(t))、负(P-, d(t))向H核的解离通道。这里定义反对称系数A(t)=P-, d(t)-P+, d(t)。当A(t) > 0,表示较多电子布局在负向H核;当A(t) < 0,表示较多电子布局在正向H核;A(t)=0,表示没有解离发生或者电子平均分布在正负向H核。
通过傅里叶变换可获得阿秒脉冲为:
$ I\left( t \right) = {\left| {\sum\limits_q {\left( {\int {d\left( t \right){{\rm{e}}^{ - {\rm{i}}q\omega t}}{\rm{d}}t} } \right){{\rm{e}}^{{\rm{i}}q\omega t}}} } \right|^2} $
(13) 式中,q为谐波次数。
空间均匀和非均匀场下H2+辐射谐波的空间分布
Spatial distribution of H2+ radiation harmonics in spatial homogeneous and inhomogeneous fields
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摘要: 为了了解H2+分子辐射谐波的空间分布,采用数值求解非玻恩-奥本海默近似的薛定谔方程,进行了空间均匀和非均匀激光场下H2+分子谐波辐射的空间分布研究。结果表明,在空间均匀场下,正向H核辐射谐波强度高于负向H核;在空间非均匀场下,由于金属结构表面出现的等离子共振增强现象,谐波截止能量得到延伸;负向H核辐射谐波强度明显高于正向H核,随后通过电离几率、电子布局、电子波函数以及谐波辐射的时频分析可以给出H2+分子谐波空间分布的合理解释。通过叠加谐波谱上的谐波,可获得一个脉宽为36as的超短孤立阿秒脉冲。该研究对分子谐波的空间分布及阿秒脉冲的输出是有帮助的。Abstract: In order to understand spatial distribution of H2+ molecular harmonics, the spatial distribution of H2+ molecular harmonic spectra in spatial homogeneous and inhomogeneous fields was studied through solving non-Bohn-Oppenheimer time-dependent Schr dinger equation. The results show that in spatial homogeneous field, harmonic intensity from positive-H nucleus is higher than that from negative-H nucleus. In spatial inhomogeneous field, due to plasma resonance on metallic nanostructure surface, harmonic cutoff is extended, and harmonic intensity from negative-H nucleus is higher than that from positive-H nucleus. Furthermore, spatial distribution of the harmonics can be explained by ionization probability, electron localization in two nuclei, time-dependent wave function and time-frequency analyses of harmonic spectra. Finally, by superposing a selected harmonics properly, an isolated ultrashort 36as pulse can be obtained. The investigation is helpful for understanding spatial distribution of molecular harmonics and producing attosecond pulses.
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Figure 1. Bowtie-shaped gold nanostructure and geometric parameters[21]
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