为了研究超短啁啾脉冲复宗量拉盖尔-高斯光束合理的物理模型和光束参量对物理模型的影响,采用复振幅包络表示式和复解析信号表示式推导了这类脉冲光束的复振幅包络解和复解析信号解。对复振幅包络解奇异性的产生进行了详细分析,讨论了脉冲啁啾、脉冲宽度和模式对复振幅包络解奇异性的影响。计算结果表明,当脉冲光束存在啁啾时,啁啾量越大,复振幅包络解的奇异点位置离光束中心越近,而复解析信号解在任意大小啁啾的情况下都不存在奇异性;对于啁啾量相同的脉冲,脉冲宽度越小或模指数越大其奇异性位置离光束中心越近。因而要建立激光光束合理的物理模型必须根据光束参量采用合适的研究方法。

使用复解析信号法，研究了衍射长度与频率无关的超短脉冲贝塞耳－高斯光束在自由空间中的传输，推导出了解析表达式并进行了理论和计算分析．结果表明：在无衍射长度内，等衍射长度超短脉冲贝塞耳－高斯光束的横向光强分布扩展很小．在无衍射长度之外，横向光强分布有显著扩展．但是，在无衍射长度内轴上脉冲波形仍有展宽．随传输距离增加轴上蓝移单调增加逐渐达到一渐近值．随横向距离增加轴外光谱移动由蓝移变为红移．

基于瑞利-索末菲衍射积分，使用复解析信号法推导出了啁啾脉冲高斯光束在自由空间中的传输方程及其傅里叶谱，给出了远场的光场和空间光强的解析式，研究了啁啾参数C对脉冲光束传输的影响。结果表明，当啁啾参数C较小时，随啁啾参数增加，其轴上光谱蓝移增加C2倍，其轴上谱线宽度增加(1+C2)1/2倍。随衍射角增大，轴外光谱红移比无啁啾参数时快。脉冲宽度较小时，啁啾参数增大，轴上光强增大，横向光强分布越集中于传输轴附近;脉冲宽度较大时，啁啾参数增大对横向光强的影响减小。啁啾参数的正负号不影响横向光强分布和光谱分布。

Starting from paraxial wave equation, analytical propagation expressions of hyperbolic secant pulsed beams with narrowband and broadband are derived. By the use of numerical calculations, pulsed beams obtained by means of complex amplitude envelope representation and complex analytical signal representation are compared. The condition to determine what representation should be chosen is also given. It is found that pulsed beam obtained by using complex amplitude envelope representation has singularity, which is anti-beam behavior and loses physical meaning. In the case of pulsed beams with broadband, singularity of complex amplitude envelope solution locates near axis, which causes that the solution failed to describe pulsed beam. But for the case of pulsed beam with narrowband, singularity locates far from axis, which can be neglected in the study of pulsed beam. It is concluded that both of the representations are applicable to study narrowband pulse, but for broadband pulse only rigorous complex analytical signal representation should be adopted.

By using complex amplitude envelope representation and complex analytic signal representation, propagation expressions of Lorentz pulsed beams are derived. Propagation properties of them with different bandwidths are illustrated by numerical calculations. It is found that they are the same in the neighbourhood of z-axis for narrowband pulses, but in the case of broadband pulses, singularities emerge near z-axis for pulsed beams obtained by using complex amplitude envelope representation and only complex analytic signal representation is applicable. The conditions to decide what representation should be chosen to study pulsed beams are given. And the origin that results in the emergence of singularity is given.

从近轴波动方程出发推导了窄带和宽带双曲正割脉冲光束的解析传输公式。通过数值计算比较了分别采用复振幅包络(CAE)表示式和复解析信号(CAS)表示式得到的脉冲光束,得出了选择脉冲光束研究方法的条件。结果表明复振幅包络表示式得到的解会存在空间奇异性,使光束出现不符合物理意义的非光束行为。对于宽带光束,复振幅包络解的奇异点的位置距离轴中心较近,使复包络解不能正确表示脉冲光束,而对于奇异点位置远离轴中心的窄带光束,对脉冲光束产生的影响可以忽略。因此,窄带脉冲光束可以采用复振幅包络和复解析信号两种表示式来研究,而对于宽带脉冲光束,必须采用严格的复解析信号表示式。

使用复解析信号法研究了束宽与频率无关的超短脉冲贝塞尔-高斯光束在自由空间的传输特性.研究表明:在无衍射长度内,超短脉冲贝塞尔-高斯光束的横向光强分布几乎不变并集中在传输轴附近,轴上脉冲随传输距离增加而展宽.在无衍射长度外,横向光强分布显著扩展.随传输距离增加,轴上光谱蓝移先增加,达到极大值后逐渐减小,最后在远场趋于一渐近值.对单周期的超短脉冲贝塞尔-高斯光束的轴上光谱蓝移先增加,达到极大值0.50 fs-1.然后逐渐减小,在远场蓝移趋于渐近值0.16fs-1.