The Wigner distribution function of a super Lorentz–Gauss SLG_(11) beam through a paraxial ABCD optical system

An orthonormal beam family of super Lorentz-Gauss （SLG） beam model is proposed to describe the higher-order mode beams with high divergence, which are generated by a high power diode laser. Here we consider the simplest case of the SLG beams, where there are four mutually orthogonal SLG beams, namely SLG00, SLG01, SLG10, and SLGll beams. The SLG00 beam is just the Lorentz-Gauss beam. Based on the Collins integral formula and the Hermite-Gaussian expansion of a Lorentz function, an analytical expression for the Wigner distribution function （WDF） of an SLG11 beam through a paraxial ABCD optical system is derived. The properties of the WDF of an SLG11 beam propagating in free space are demonstrated. The normalized WDFs of an SLG11 beam at the different spatial points are depicted in several observation planes. The influence of the beam parameter on the WDF of an SLGI 1 beam in free space is analyzed at different propagation distances. The second-order moments of the WDF of an SLG11 beam in free space are also examined. This research reveals the propagation properties of an SLGll beam from another perspective. The WDFs of SLG01 and SLG10 beams can be easily obtained by using the WDFs of Lorentz-Gauss beam and the SLG11 beam.     

Nonparaxial propagation of a super-Lorentz－Gauss SLG01 mode beam

Based on the vectorial Rayleigh--Sommerfeld integral formulae, this paper derives the analytical nonparaxial propagation equation of a super-Lorentz--Gauss (SLG) SLG₀₁ mode beam in free space. The far field expression and the scalar paraxial result are treated with special cases of the general formulae. According to the obtained analytical representation, the nonparaxial propagation properties of the SLG₀₁ mode beam are illustrated and analysed with numerical examples. This research provides an approach to investigate the propagation of the SLG₀₁ mode beam within the framework of the nonparaxial regime.     

Characteristics of paraxial propagation of a super Lorentz-Gauss SLG₀₁ mode in uniaxial crystal orthogonal to the optical axis

Analytical propagation expression of a super Lorentz-Gauss(SLG) 01 mode in uniaxial crystal orthogonal to the optical axis is derived.The SLG 01 mode propagating in uniaxial crystal orthogonal to the optical axis mainly depends on the ratio of the extraordinary refractive index to the ordinary refractive index.The SLG 01 mode propagating in uniaxial crystals becomes an astigmatic beam.The beam spot of the SLG 01 mode in the uniaxial crystal is elongated in the x-or y-direction,which is determined by the ratio of the extraordinary refractive index to the ordinary refractive index.With the increase of the deviation of the ratio of the extraordinary refractive index to the ordinary refractive index from unity,the elongation of the beam spot also augments.In different observation planes,the phase distribution of an SLG 01 mode in the uniaxial crystal takes on different shapes.With the variation of the ratio of the extraordinary refractive index to the ordinary refractive index,the phase distribution is elongated in one transversal direction and is contracted in the other perpendicular direction.This research is beneficial to the practical applications of an SLG mode.     

Nonparaxial propagation of Hermite-Laguerre-Gaussian beams in uniaxial crystal orthogonal to the optical axis

Analytical expressions for the three components of the nonparaxial propagation of a Hermite-Laguerre-Gaussian （HLG） beam in uniaxial crystal orthogonal to the optical axis are derived. The intensity distribution of an HLG beam and its three components propagating in a uniaxial crystal orthogonal to the optical axis are demonstrated by numerical examples. Although the y and z components of an HLG beam in the incident plane are both equal to zero, they emerge upon propagation inside the uniaxial crystal. Moreover, the beam profile of the x component is relatively stable and the beam profiles of the y and z components have the same evolution law. If the ratio of the extraordinary refractive index to the ordinary refractive index is larger than unity, the beam profile of the HLG beam is elongated in the x direction and generally rotates clockwise. Otherwise, the beam profile of the HLG beam is elongated in the y direction and generally rotates anticlockwise. This research is beneficial to the optical trapping and nonlinear optics involved in the rotation of a beam profile.     

傍轴光束在单轴左手介质中传输的矢量性质

研究了沿光轴入射的傍轴光束在单轴左手介质中传输的矢量性质.通过平面角谱的方法得到了横电(TE)波和横磁(TM)波在单轴左手介质中传输的严格解，并得到了傍轴近似下的矢量传输模型.利用这一模型，分析得到垂直于光轴方向不同偏振态的入射光束在单轴左手介质中同时激发横向场和低阶纵向场.如果单轴左手介质的介电常数张量在沿垂直于光轴方向上的元素不相等，同时磁导率张量在沿垂直于光轴方向上的两元素也不相等(即z方向为光轴方向，_x≠_y,μ_x≠μ_y),那么入射光束激发的横向场的分布与入射光束的初始场分布和偏振态有关，而纵向场的分布与入射光束的偏振态无关，只决定于入射光束的初始场分布. 关键词： 傍轴传输 单轴左手介质 矢量效应     

傍轴光束在单轴晶体中传输的矢量性质

研究了垂直光轴入射的傍轴光束在单轴晶体中传输的矢量性质.通过角谱的方法得到了寻常光（o光）和非常光（e光）在晶体中传输的严格解，并得到了傍轴近似下的矢量传输模型.利用这一模型，分析得到了垂直光轴偏振的入射光束在晶体中只激发o光，而平行光轴偏振的入射光束在晶体中同时激发e光和二阶小量的o光. 关键词： 傍轴传输 单轴晶体 矢量效应     

Propagation of a Lorentz-Gauss beam through a misaligned optical system

Based on the generalized integral formula and the convolution theorem of the Fourier transform, an analytical propagation formula of a Lorentz-Gauss beam passing through a misaligned paraxial optical system is derived. As numerical examples, the propagation properties of a Lorentz-Gauss beam through a misaligned thin lens with the lateral displacement and the angle displacement are graphically illustrated, respectively. The influences of the lateral displacement and the angle displacement of the misaligned thin lens on the normalized light intensity and the phase distributions are also examined, respectively.     

椭圆高斯光束在单轴晶体中垂直于光轴的传输特性

基于光束在各向异性介质中的傍轴矢量传输理论,导出了椭圆高斯光束在各向异性单轴晶体中垂直于光轴的传输公式,并利用此解析式进行数值计算和分析,研究了各向异性介质对x方向偏振的椭圆高斯光束偏振特性的影响. 研究表明,晶体的各向异性对光束的偏振和对称性有较大影响. 关键词： 椭圆高斯光束 单轴晶体 各向异性 传输     

Nonparaxial propagation of radially polarized chirped Airy beams in uniaxial crystal orthogonal to the optical axis

The nonparaxial propagation of radially polarized chirped Airy beams(RPCAiBs) in uniaxial crystal orthogonal to the optical axis is analytically studied. The effects of the first and the second order chirp factors, and the ratio of the extraordinary refractive index to the ordinary refractive index on the nonparaxial evolution of RPCAi Bs in different observation planes are investigated in detail. The results show that when one parameter changes, different components behave differently, and even the same component has various behavior in different directions. The initial slope of RPCAi Bs in the x-direction varies more with the first order chirp factor than that in the y-direction. Meanwhile, with the second order chirp factor becoming larger, the depth of the focus of the y-component in the x-direction decreases while that in the y-direction has no difference. In addition, the different ratios of the extraordinary refractive index to the ordinary refractive index change the difference of the depth of the focus between the x-and the y-components.     

超洛伦兹-高斯光束SLG11模的分数傅里叶变换特性

 引入了一簇互相正交的超洛伦兹-高斯光束以描述半导体激光器所产生的大角度高阶模远场分布。将分数傅里叶变换应用于超洛伦兹-高斯光束SLG­₁₁模的传输特性的研究中。利用傅里叶变换的卷积原理,导出了SLG₁₁模经分数傅里叶变换系统后场分布的解析表达式。根据所得到的公式进行了数值计算,系统分析了分数傅里叶变换阶数和光束各参数对SLG₁₁模在分数傅里叶变换面上光强分布的影响。结果显示：SLG₁₁模在分数傅里叶变换面上的归一化强度分布随分数傅里叶变换的阶数呈周期性变化,周期为2;随着光束参数的增大,SLG₁₁模在分数傅里叶变换面上的光斑尺寸增大。     

超洛伦兹-高斯光束SLG11模的分数傅里叶变换特性

引入了一簇互相正交的超洛伦兹-高斯光束以描述半导体激光器所产生的大角度高阶模远场分布。将分数傅里叶变换应用于超洛伦兹-高斯光束SLG11模的传输特性的研究中。利用傅里叶变换的卷积原理,导出了SLG11模经分数傅里叶变换系统后场分布的解析表达式。根据所得到的公式进行了数值计算,系统分析了分数傅里叶变换阶数和光束各参数对SLG11模在分数傅里叶变换面上光强分布的影响。结果显示：SLG11模在分数傅里叶变换面上的归一化强度分布随分数傅里叶变换的阶数呈周期性变化,周期为2;随着光束参数的增大,SLG11模在分数傅里叶变换面上的光斑尺寸增大。