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 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 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.     

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.     

Analytical vectorial structure of truncated modified Laguerre–Gaussian beam in the far field

Based on the vectorial angular spectrum representation of Maxwell equations and the method of stationary phase, analytical expressions are derived for the TE term, TM term and whole energy flux of a hard-edged diffracted modified Laguerre–Gaussian beam (MLGBs) in the far field, respectively. It is shown that the evolutive energy flux not only depends on the beam order, initial waist width but also relates to the truncated parameter. Furthermore, influences of the truncated parameter on the discrepancies of energy flux distributions between paraxial and non-paraxial cases are also analyzed for different selections of the initial beam width.     

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。     

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。     

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.\     

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     

Investigation in the far field characteristics of TM polarized Gaussian beam from the vectorial structure

Based on the vectorial structure of electromagnetic beam and the method of stationary phase, the analytical TE and TM terms of TM polarized Gaussian beam, the rigorous solution of Maxwell's equations for a confocal resonator, have been presented in the far field. Then in terms of the vectorial structure, TM polarized Gaussian beam is compared with Gaussian TEM₀₀ mode. The TE term is located at the y-axis, and the TM term the x-axis. At the non-paraxial case, the whole beam spot is elliptical, and the long axis is located at the y-axis. Moreover, the whole beam spot of TM polarized Gaussian beam is smaller than that of Gaussian TEM₀₀ mode. At the paraxial case, the whole beam spot is circular, and TM polarized Gaussian beam reduces to be Gaussian TEM₀₀ mode.     

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.     

Exact solution of the vectorial field structure of a light beam

An alternative way of finding electromagnetic solutions based on the closest solution defined by Martinez-Herrero et al. [J. Opt. Soc. Am. A 18, 1678 (2001)] has been proposed. It is found that by identifying the components of closest solution as the components of the planewave spectrum of realistic fields at z = 0, an exact solution of the electromagnetic fields satisfying the Maxwell equations can also be obtained. This solution is applied to calculate the fields of a radially polarized Laguerre-Gaussian laser beam. It shows that although both formalisms are correct ways of finding electromagnetic solutions, as the beam waist size is less than a laser wavelength, the longitudinal electric field near the focus obtained from our solution can be significantly larger than that by Martinez-Herrero et al.     

Polarization properties of vectorial nonparaxial Gaussian beams in the far field

The polarization properties of vectorial coherent nonparaxial Gaussian beams are studied. It is shown that, when the source of a nonparaxial Gaussian beam is completely polarized, the degree of polarization of the propagation field maintains a constant value of 1. However, when the source is completely unpolarized, the degree of polarization does not maintain a constant value of 0. In the far field, vectorial nonparaxial Gaussian beams can become partially polarized, except on the propagation axis, and are spatially nonuniform across a typical cross section of the beam even though the source field is completely unpolarized.     

Vectorial structure of a hard-edged-diffracted four-petal Gaussian beam in the far field

Based on the vector angular spectrum method and the stationary phase method and the fact that a circular aperture function can be expanded into a finite sum of complex Gaussian functions, the analytical vectorial structure of a four-petal Gaussian beam (FPGB) diffracted by a circular aperture is derived in the far field. The energy flux distributions and the diffraction effect introduced by the aperture are studied and illustrated graphically. Moreover, the influence of the f-parameter and the truncation parameter on the non-paraxiality is demonstrated in detail. In addition, the approximate formulas obtained in this paper can degenerate into un-apertured case when the truncation parameter tends to infinity. This work is beneficial to strengthen the understanding of vectorial properties of the FPGB diffracted by a circular aperture.     

Analytical vectorial structure of radially polarized light beams

Starting from the vector angular spectrum of the electromagnetic beam, the analytical vectorial structure of the radially polarized beams (RPBs) is presented. The energy flux distributions of the RPBs are demonstrated. The physical pictures of the RPBs are well illustrated from the vectorial structure. This particular electromagnetic field is entirely transverse magnetic, and on axis it only has a longitudinal (z) electric-field component (i.e., no transverse electric field and no magnetic field at all on axis).     

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.     

Analytically vectorial structure of an apertured Laguerre-Gaussian beam in the far-field

Based on the angular spectrum representation of an arbitrary electromagnetic beam and the method of stationary phase, an analytically vectorial structure of an apertured Laguerre-Gaussian beam in the far-field has been derived without any approximation. The analytical expressions of the energy flux of the TE term, the TM term, and the apertured Laguerre-Gaussian beam are also presented in the far-field, respectively. The energy flux distributions of the TE term, the TM term, and the apertured Laguerre-Gaussian beam are numerically demonstrated in the far-field reference plane. The influences of the f-parameter, the truncation parameter, the radial and angular mode numbers, and the dependent relation of angle on the energy flux distributions in the far-field of the TE term, the TM term, and the apertured Laguerre-Gaussian beam are also discussed in detail.     

Structural polarization properties of vector Gaussian beam in the far field

Based on the vector angular spectrum representation of optical beam and the method of stationary phase, the analytical TE and TM terms of vector Gaussian beam have been presented in the far field. By using the local polarization matrix, the polarization properties of the TE and TM terms in the far field are investigated, and it is found that the degree of their polarization is only determined by the spatial location. When the source is completely polarized, the TE and TM terms are both completely polarized in the far field. When the source is completely unpolarized, the TE and TM terms in the far field are partially polarized. The whole beam is also partially polarized except on the propagating axis. Moreover, the degrees of polarization of TE and TM terms are both larger than that of the whole beam.[第一段]