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.     

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 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.     

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.     

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.     

Approximate analytical expressions of apertured broadband beams in the far field

The approximate analytical expressions of the apertured broadband beams in the far field with Gaussian and Laguerre-Gaussian spatial modes are presented.For the radially polarized Laguerre-Gaussian beam,the result reveals that the electromagnetic field in the far field is transverse magnetic.The influences of bandwidth(Γ) and truncation parameter(C 0) on the transverse intensity distribution of the Gaussian beam and on the energy flux distribution of radially polarized Laguerre-Gaussian beam are analysed.     

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 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.     

Energy flux of a Gaussian beam in microwave band beyond the paraxial approximation

Through the analytic method beyond the paraxial approximation, the energy flux for the general fields of a Gaussian laser beam in microwave band (∼10⁸–10¹⁰ Hz) with different diffraction angles and longitudinal positions are derived. The results indicate a Gaussian beam (GB) has a certain number of transverse energy flux, and it would be enhanced with larger diffraction angle. In the range of beyond paraxial approximation, the transverse flux plays an important role in energy distribution, so that it cannot be ignored when the GB is highly focused. Meanwhile both transverse and longitudinal energy decay along the propagation direction, and the proportion of transverse energy far from beam axis is much more than the counterpart in paraxial area. The transverse energy as mainly characteristic of GB will be much useful for the interactions between electromagnetic field and other field.     

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.     

Propagation properties of off-axis Hermite--cosh--Gaussian beam combinations through a first-order optical system

Based on the Collins integral formula, the analytic expressions of propagation of the coherent and the incoherent off-axis Hermite--cosh--Gaussian (HChG) beam combinations with rectangular symmetry passing through a paraxial first-order optical system are derived, and corresponding numerical examples are given and analysed. The resulting beam quality is discussed in terms of power in the bucket (PIB). The study suggests that the resulting beam cannot keep the initial intensity shape during the propagation and the beam quality for coherent mode is not always better than that for incoherent mode. Reviewing the numerical simulations of Gaussian, Hermite--Gaussian (HG) and cosh--Gaussian (ChG) beam combinations indicates that the Hermite polynomial exerts a chief influence on the irradiance profile of composite beam and far field power concentration.     

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     

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 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.     

横向激发线偏振高斯光束的非傍轴传输

文章提出一种横向激发的任意线偏振高斯光束,该光束场量的模与传统的任意线偏振高斯光束具有相同的模,在源场区,其满足自由空间无源场的场方程。同时利用瑞利—索末菲衍射积分对该高斯光束的非傍轴传输特性进行了解析研究,给出了空间光强分布、束腰和远场发散角的解析通式,发现该类光束的远场光强呈一平顶空心光束。     

Far-field structural property of a Gaussian beam diffracted by a phase aperture

Based on the angular spectrum representation of an arbitrary electromagnetic beam and the method of stationary phase, an analytically vectorial structure of a Gaussian beam diffracted by a phase aperture has been derived in the far-field. Moreover, the derivation is performed without any approximation. The analytical expressions of the energy flux of the TE term, the TM term, and the apertured Gaussian beam are also presented in the far-field, respectively. The influence of the phase delay on the energy flux distributions of the TE term, the TM term, and the apertured Gaussian beam is discussed in the far-field.     

Intensity Moments of Hermite-Cosh-Gaussian Laser Beams

1　Ｉｎｔｒｏｄｕｃｔｉｏｎ　　Ｉｎｒｅｃｅｎｔｙｅａｒｓ ,ａｎｉｎｔｅｒｅｓｔｉｎｇｓｏｌｕｔｉｏｎｏｆｔｈｅ ｐａｒａｘｉａｌｗａｖｅｅｑｕａｔｉｏｎ ,ｋｎｏｗｎａｓｔｈｅＨｅｒｍｉｔｅ ｓｉｎｕｓｏｉｄａｌ Ｇａｕｓｓｉａｎ (ＨＳＧ )ｂｅａｍｓ,ｈａｓｂｅｅｎｉｎｔｒｏｄｕｃｅｄｂｙＣａｓｐｅｒｓｏｎａｎｄＴｏｖａｒ[1,2 ] .ＡｓｏｎｅｏｆｔｈｅｓｐｅｃｉａｌｅｘａｍｐｌｅｉｎＨＳＧｆａｍｉｌｙ ,ｔｈｅＨｅｒｍｉｔｅ ｃｏｓｈ Ｇａｕｓｓｉａｎ(ＨＣｈＧ)ｂｅａｍｓ′ｓｆｉｅｌｄｄｉｓｔｒｉｂｕｔｉ…     

非傍轴高斯光束传输特性的研究

基于非傍轴矢量光束横截面上光强的严格定义〈S_z〉,采用桶中功率定义光束的束宽、远场发散角以及M²因子. 以非傍轴矢量高斯光束为例,对光束的传输特性进行了详细的理论分析和数值计算,并与非傍轴标量理论的结果进行了比较分析. 研究表明,光源半宽ω₀/Λ<1时,传输特性的矢量描述与非傍轴标量描述有显著差别,随着ω₀/Λ的增大,两者趋于一致,并逐步过渡到 关键词： 非傍轴高斯光束 桶中功率 束宽 发散角     

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.