超短脉冲复宗量拉盖尔—高斯光束在群速度色散体介质中的传输

Propagation of Ultra-short Pulsed Elegant Laguerre-Gaussian Beams in Group Velocity Dispersive Media

基于脉冲光束在群速度色散体介质中的传输方程,通过傅里叶变换的方法,将超短脉冲复宗量拉盖尔—高斯光束的时空传输问题转换为谱域的空间传输问题.在谱域中单色连续光的复宗量拉盖尔—高斯光束解的基础上,利用拉盖尔多项式的递推关系式得到了时间域中超短脉冲复宗量拉盖尔—高斯光束的解析解.此类超短脉冲光束各频谱分量的空间分布为复宗量拉盖尔—高斯光束,并且各频谱分量具有同样的衍射程度.由于各高阶超短脉冲复宗量拉盖尔—高斯光束解仅为超短脉冲基模高斯光束解不同阶次的偏微分,可以很容易得到其时空光场分布.基于负色散介质中的(0,2)模超短脉冲复宗量拉盖尔—高斯光束,我们研究了此类超短脉冲光束的传输性质.结果表明:在传输过程中,其时间和空间分布均逐渐增大.除此之外,光束边沿还发生了相对光束中心的脉冲延迟.该延迟来源于两方面:其一为光束的非均匀群速度分布;其二为衍射导致的频率红移.随着脉冲宽度的增加,光束边沿的延迟效应迅速减弱,当脉冲宽度达到5倍振荡周期时,该效应可以忽略不计.

Based on the propagation equation which governs the propagation of ultra-short pulsed beams in group velocity dispersive media, and by utilizing the Fourier transform technique, the spatiotemporal propagation problem of the ultra-short pulsed elegant Laguerre Gaussian beams is reduced to the spatial propagation problem in spectrum domain. On the basis of the solution of the monochromatic elegant Laguerre-Gaussian beams in spectrum domain, the analytical solutions of ultra-short pulsed elegant Laguerre-Gaussian beams in dispersive media are obtained by using of the property of Laguerre polynomial. Each frequency component of an ultra-short pulsed elegant Laguerre-Gaussian beam has the same Laguerre-Gaussian profile and the same diffraction degree. The spatiotemporal distribution of higher-order ultra-short pulsed elegant Laguerre-Gaussian beams can be readily obtained from the partial differential on the ultra-short pulsed fundamental Gaussian beam. The propagation property of this type of pulsed beams is studied based on the （0, 2） mode ultra short pulsed elegant Laguerre-Gaussian beam. It shows that the spatial and temporal distribution expand in propagation. And the pulse at the beam periphery is delayed compared to that around the beam center. The delay originates from i） the non-uniform distribution of the group velocity and ii） the red-shift caused by the diffraction. The delay effect decreases with the pulse duration and is negligible when the pulse duration is up to 5 times of the oscillation periods.