Analytical vectorial structure of non-paraxial nonsymmetrical vector Gaussian beam in the far field

Based on the vectorial structure of non-paraxial electromagnetic beam and the method of stationary phase, the analytical TE and TM terms of non-paraxial nonsymmetrical vector Gaussian beam in the far field are presented. According to the analytical electromagnetic fields of the TE and TM terms, the energy flux distributions of the whole beam, its TE and TM terms are investigated. Moreover, the influences of non-symmetries on the energy flux distributions of the whole beam, its TE and TM terms are also analyzed, respectively. This research reveals the internal vectorial structure of non-paraxial laser beam in the far field and is useful to the propagation of non-paraxial laser beam.     

Airy-Gaussian光束的解析矢量结构

采用矢量角谱的方法分析了Airy-Gaussian光束的矢量结构, 运用静态相位法得出了光束远场矢量结构的解析表达式. 根据得到的TE项和TM项的解析表达式, 分别研究了TE项和TM项及整个光束的能流分布. 更好地阐明了Airy-Gaussian光束的物理图像, 从而为操控光束提供潜在的新方法. 关键词： Airy-Gaussian光束 解析矢量结构 矢量角谱法 静态相位法     

Propagation of a vectorial Laguerre–Gaussian beam beyond the paraxial approximation

Based on the vectorial Rayleigh–Sommerfeld integrals, the analytical propagation expression of a vectorial Laguerre–Gaussian beam beyond paraxial approximation is presented. The far field expression and the scalar paraxial result are obtained as special cases of the general formulae. According to the analytical representation, the light intensity distribution of the vectorial Laguerre–Gaussian beam is depicted in the reference plane. The light intensity distribution of a vectorial Laguerre–Gaussian beam with cos m is also compared with that of a vectorial Laguerre–Gaussian beam with sin m.     

Analytical vectorial structure of Laguerre-Gaussian beam in the far field

Based on the vector angular spectrum of the electromagnetic beam and the method of stationary phase, the analytical vectorial structure of the Laguerre-Gaussian beam has been presented in the far field. According to the analytical electromagnetic representations of the TE and TM terms, the energy flux distributions of the TE term, the TM term, and the whole beam are investigated in the far field, respectively. The formulas obtained are applicable not only to the paraxial case, but also to the nonparaxial case. The physical pictures of Laguerre-Gaussian beams are well illustrated from the vectorial structure, which may provide a new approach to manipulate laser beams.     

Analytical structure of Hermite Gaussian beam in far field

Based on the vectorial structure of electromagnetic beam and the method of stationary phase, the analytical structure of Hermite Gaussian beam in far field is presented. The structural energy flux distributions are also investigated in the far field. The structural pictures of some Hermite Gaussian beams are depicted in the far field. As the structure of Hermite Gaussian beam is dominated by the transverse mode numbers and the initial transverse Gaussian half width, it is more complex than that of Gaussian beam. The ratios of the structural energy fluxes to the whole energy flux are independent of the transverse mode numbers and the initial transverse Gaussian half width. The present research reveals the internal vectorial structure of Hermite Gaussian beam from other viewpoint.\     

拉盖尔-高斯光束的近场矢量结构特征

直接以麦克斯韦方程组的解表征拉盖尔-高斯光束。基于麦克斯韦方程组解的矢量角谱表述和电磁光束的矢量结构理论,利用一些数学技巧导出了拉盖尔-高斯光束的TE项和TM项在近场的解析表达式。利用所导出的公式,在近场描绘了拉盖尔-高斯光束的TE项、TM项及整个光束的光强分布。并对角度依赖分别为余弦和正弦关系的拉盖尔-高斯光束的矢量结构进行了比较,结果显示两者整个光束的光斑完全相似,唯一的区别是子瓣的空间方位不同且两者的内部矢量结构完全不相同。     

Vectorial structure of the far field of an elegant Hermite–Gaussian beam

Based on the vector angular spectrum method and the method of stationary phase, an analytical expression for the vectorial structure of the far field of an elegant Hermite–Gaussian beam is derived. The analytical formulae of the energy flux distributions of the TE term, the TM term, and the whole beam are presented in the far field. Analytical expressions for the ratios of the powers of the TE and TM terms to those of the elegant Hermite–Gaussian beam are obtained without any approximation. The physical pictures of the far field of an elegant Hermite–Gaussian beam are demonstrated and compared with those of the far field of the corresponding standard Hermite–Gaussian beam. This research reveals the internal vectorial structure of the far field of an elegant Hermite–Gaussian beam from an alternative viewpoint.     

Analytical vectorial structure of the anomalous hollow beam in the far field

On the basis of the vector angular spectrum representation of the Maxwell equations and the method of stationary phase, the analytical vectorial structure of an anomalous hollow beam in free space is presented. The energy flux distributions of the anomalous hollow beam, its TE and TM terms in the far field are investigated. The analytical results are analyzed with numerical examples.     

Analytical structure of an apertured vector Gaussian beam in the far field

Based on the angular spectrum representation of the Maxwell’s equations and the complex Gaussian expansion of the aperture function, the structure of an apertured vector Gaussian beam in the far field is presented in the integral form. By means of the method of stationary phase, the analytical vectorial structures are obtained. According to the analytical expressions, the characteristics of vectorial structure of an apertured Gaussian beam are investigated in the far field. The influence of a linearly polarized angle on the vectorial structure is also studied in the far field. This research provides a novel approach to further comprehend the vectorial property of an apertured Gaussian beam.     

Analytical vectorial structure of Hermite-cosine-Gaussian beam in the far field

Based on the vector angular spectrum representation of electromagnetic beams and the method of stationary phase, the vectorial structure of a Hermite-cosine-Gaussian (HcosG) beam in the far field is derived in analytical form. Energy flux distributions for the TE term, TM term and the whole beam are represented. The vectorial structures of HcosG beams with different Hermite models are depicted. The effects of cosine displacement parameters on the vectorial structure are also demonstrated. The results may provide a flexible approach to manipulate HcosG beams in free space.     

Vectorial structure of Hermite–Laguerre–Gaussian beam in the far field

Starting from Maxwell's equations, a Hermite–Laguerre–Gaussian (HLG) beam is decomposed into the TE and TM terms by using the vector angular spectrum representation. By means of the method of stationary phase, the analytical TE and TM terms are presented in the far field. The energy flux distributions of the TE and TM terms are also investigated and depicted in the far field. The influences of the additional angle parameter and Gaussian waist width on the vectorial structure and energy flux pattern of HLG beam are also investigated. This research reveals the internal vectorial structure of HLG beam and may provide a new approach to the manipulation of laser beams.     

Vectorial structural properties of truncated beam generated by Gaussian mirror resonator in the near field

Internal vectorial structure of the beam generated by the Gaussian mirror resonator (GMR) diffracted through a hard-edged aperture are evaluated in the near-field. Based on the vectorial angular spectrum representation of electromagnetic fields and the paraxial approximation, TE and TM terms of intensity distributions of the beam in the near field are derived in analytical forms, respectively. Numerical results reveal that the behaviors of TE and TM terms not only depend on the parameters of mirror resonators but also relate to the truncated parameter of the aperture. It is also shown that the distributions of TE and TM terms would not keep detached from each other because of the unorthogonality of their vectorial structures in the near field.     

Analytical vectorial structure of an apertured hollow Gaussian beam in the far field

Based on the vector angular-spectrum and the complex Gaussian expansion of the aperture function, the structure of an apertured hollow Gaussian beam in the far field is given in a integral form. In virtue of the method of stationary phase approximation, the analytical vectorial structures are derived. Starting from the analytical expressions, the propagation properties of apertured hollow Gaussian beams with different order in the far field are illustrated graphically. In addition, the influence of the truncation parameter on far field distribution is studied detailedly. This research can shed light on the further understanding of the vectorial property of an apertured hollow Gaussian beam.     

Vectorial structure of helical hollow Gaussian beams in the far field

The analytical vectorial structure of helical hollow Gaussian beam (HHGB) is investigated in the far field based on the vector plane wave spectrum and the method of stationary phase. The energy flux distributions of HHGB in the far-field, which is composed of TE term with the electric field transverse to the propagation axis, and TM term with the magnetic field transverse to the propagation axis, are demonstrated. The physics pictures of HHGB is illustrated from the vectorial structure, which is important to understand the theoretical aspects of both scalar and vector HHGB propagation.     

湍流对M×N列阵双曲余弦高斯光束传输和远场光束质量的影响

基于广义惠更斯－菲涅耳原理,并采用光束的非相干合成方法,推导出了M×N双曲余弦高斯列阵光束在湍流中的三维光强传输方程.采用桶中功率、β参量和Strehl比作为光束质量的评价参量,研究了湍流大气对双曲余弦高斯列阵光束远场光束质量的影响.研究表明:在湍流大气中,双曲余弦高斯列阵光束的传输将经历三个阶段的变化,并且湍流使得光束传输经历三阶段的进程加快;湍流导致双曲余弦高斯列阵光束扩展、最大峰值光强下降,但是,β参量随光束数目M（N）、相邻子光束间距xd（yd）和光束参量δ的增加而减少,即光束扩展受湍流的影响减小;并且,存在最佳xd（yd）和δ值使得Strehl比取得极大值.因此,适当选取M（N）、xd（yd）和δ可以降低湍流对双曲余弦高斯列阵光束远场光束质量的影响.     

Vectorial structures of non-paraxial linearly polarized Gaussian beam and their beam propagation factors

Based on the vectorial structure of non-paraxial electromagnetic beam and non-paraxial vectorial moment theory, the relationship of the beam waists, the divergence angles and the beam propagation factors among non-paraxial linearly polarized Gaussian beam, its TE and TM terms have been presented, respectively. The analytical beam propagation factors are given and further discussed at the highly non-paraxial case. The maximum divergence angles in the x-direction of non-paraxial linearly polarized Gaussian beam, its TE and TM terms are all 54.7°, and those in the y-direction are limited to be 63.4°, 67.7° and 39.2°, respectively. As TE and TM terms are orthogonal and can be detached at the far field, the potential applications of the isolated TE and TM terms are deserved further investigation.     

Vectorial structure of far field of cylindrically polarized beams diffracted at a circular aperture

Based on the vector angular spectrum representation and the method of stationary phase, the analytical vectorial structure of the cylindrically polarized Laguerre-Gaussian beam diffracted at a circular aperture is derived in the far field, which provides an approach to further comprehend the vectorial properties of the apertured cylindrically polarized beams. The radially polarized, azimuthally polarized, and unapertured cases can be viewed as the special cases of our general result. The analyses show that the far-field energy flux distributions of the entire beam, the TE term, and the TM term depend on the beam order, the ratio of the waist width to the wavelength, the truncation parameter, and the angle between the electric field vector and the radial direction.