A closed form of a kurtosis parameter of a hypergeometric-Gaussian type-Ⅱ beam

Based on the irradiance moment definition and the analytical expression of waveform propagation for hypergeometricGaussian type-Ⅱ beams passing through an ABCD system, the kurtosis parameter is derived analytically and illustrated numerically. The kurtosis parameters of the Gaussian beam, modified Bessel modulated Gaussian beam with quadrature radial and elegant Laguerre–Gaussian beams are obtained by treating them as special cases of the present treatment. The obtained results show that the kurtosis parameter depends on the change of the beam order m and the hollowness parameter p, such as its decrease with increasing m and increase with increasing p.     

Elegant Hermite-Laguerre-Gaussian beams

A novel family of paraxial laser beams called elegant Hermite-Laguerre-Gaussian beams (EHLGBs) are presented. The EHLGBs, a unity of elegant Hermite-Gaussian beams (EHGBs) and elegant Laguerre-Gaussian beams (ELGBs), constitute the exact and continuous transition modes between EHGBs and ELGBs when an additional parameter continuously changes. The virtual source for generation of the EHLGBs is identified. From the spectral representation of the elegant Hermite-Lauguerre-Gaussian waves, we derive the first three orders of nonparaxial corrections for the corresponding paraxial EHLGBs.     

Kurtosis parameter of truncated standard and elegant Laguerre–Gaussian beams propagating through ABCD optical systems

On the basis of the propagation equation of truncated standard and elegant Laguerre–Gaussian beams, the closed-form propagation expressions for the kurtosis parameter K of standard and elegant Laguerre–Gaussian beams passing through apertured optical systems are derived, it is shown that the kutosis parameter K of standard and elegant Laguerre–Gaussian beams depend on order p and index m of Laguerre polynomial, the beam-truncated parameter δ and the generalized Fresnel number F. Detailed numerical examples are given to illustrate the analytical results.     

Generation of dark hollow beams by using a fractional radial Hilbert transform system

The radial Hilbert transform has been extend to the fractional field, which could be called the fractional radial Hilbert transform (FRHT). Using edge-enhancement characteristics of this transform, we convert a Gaussian light beam into a variety of dark hollow beams (DHBs). Based on the fact that a hard-edged aperture can be expanded approximately as a finite sum of complex Gaussian functions, the analytical expression of a Gaussian beam passing through a FRHT system has been derived. As a numerical example, the properties of the DHBs with different fractional orders are illustrated graphically. The calculation results obtained by use of the analytical method and the integral method are also compared.     

Fractional Fourier transform for off-axis dark-hollow beams

Based on the definition of fractional Fourier transform (FRT) in the rectangular coordinate system, the FRT for off-axis circular and elliptical dark-hollow beams (DHB) is studied in detail, and analytical formulae are derived for the FRT for off-axis circular and elliptical DHB. The properties of off-axis DHB in the FRT plane are illustrated numerically by use of the derived formulae. The results show that the properties of the off-axis DHB in the FRT plane are closely related with the parameters of the beam and fractional order. The derived formula provides a convenient way for analyzing and calculating the FRT of off-axis DHBs.     

Fractional Fourier transform of Lorentz beams

This paper introduces Lorentz beams to describe certain laser sources that produce highly divergent fields. The fractional Fourier transform (FRFT) is applied to treat the propagation of Lorentz beams. Based on the definition of convolution and the convolution theorem of the Fourier transform, an analytical expression for a Lorentz beam passing through a FRFT system has been derived. By using the derived formula, the properties of a Lorentz beam in the FRFT plane are illustrated numerically.     

Propagation of the kurtosis parameter of Hermite-cosh-Gaussian beams

Starting from the propagation equation of Hermite-cosh-Gaussian (HChG) beams and the intensity moments definition, an analytical expression for the propagation of the kurtosis parameter of unapertured HChG beams passing through paraxial optical ABCD systems is derived and illustrated numerically. Special interesting cases are discussed, in particular, the kurtosis parameter of HChG beams at the waist plane is obtained readily from our general propagation expression.     

Deriving the integral representation of a fractional Hankel transform from a fractional Fourier transform

We derive the integral representation of a fractional Hankel transform (FRHT) from a fractional Fourier transform. Some basic properties of the FRHT such as Parseval's theorem and its optical implementation are discussed qualitatively.     

The fractional Fourier transform of Airy beams using Lohmann and quadratic optical systems

Based on the generalized Huygens–Fresnel integral, the analytical expressions for Airy beams through three types of fractional Fourier transform (FRFT) optical systems (Lohmann I, Lohmann II and quadratic graded-index systems) are derived. Using the two analytical expressions, the propagation properties of Airy beams for the three types of FRFT optical systems are discussed in detail with numerical examples. Results indicate that the intensity of Airy beams periodically changes and remains form-invariant with the changing of fractional order p for the three types of FRFT optical systems.     

Fractional Fourier transform for partially coherent beam in spatial-frequency domain

By using Fourier transform and the tensor analysis method, the fractional Fourier transform (FRT) in the spatial-frequency domain for partially coherent beams is derived. Based on the FRT in the spatial-frequency domain, an analytical transform formula is derived for a partially coherent twisted anisotropic Gaussian－Schell model (GSM) beam passing through the FRT system. The connections between the FRT formula and the generalized diffraction integral formulae for partially coherent beams through an aligned optical system and a misaligned optical system in the spatial-frequency domain are discussed, separately. By using the derived formula, the intensity distribution of partially coherent twisted anisotropic GSM beams in the FRT plane are studied in detail. The formula derived provide a convenient tool for analysing and calculating the FRTs of the partially coherent beams in spatial-frequency domain.     

Fractional Fourier transform of an elliptical dark-hollow beam

The fractional Fourier transform (FRT) of an elliptical dark-hollow beam (EDHB) is studied in detail. Based on the definition of FRT in the rectangular coordinate system, analytical formulae are derived for the FRT of an EDHB with the help of a tensor method. The properties of an EDHB in the FRT plane are illustrated numerically by use of the derived formulae. The results show that the properties of the EDHB in the FRT plane are closely related to the parameters of the beam and to the fractional order. The derived formula provides a convenient way for analyzing and calculating the FRT of an EDHB.     

Propagation properties of Bessel and Bessel-Gaussian beams in a fractional Fourier transform optical system

The properties of Bessel-Gaussian beams (BGBs) and Bessel beams (BBs) propagating through a fractional Fourier transform (FRT) optical system have been investigated. The analytical transformation formulae for BBs and BGBs propagation through a FRT optical system are derived based on definition of the FRT in the cylindrical coordinate system. By using the derived formula, numerical examples are illustrated.     

1维线阵离轴高斯光束的分数傅里叶变换

 为研究非相干的1维线阵离轴高斯光束通过分数傅里叶变换(FRFT)系统的传输特性,利用Collins积分公式,导出了其在FRFT面上的光强分布解析式,并利用此解析式作数值计算和分析。研究表明：非相干的1维线阵离轴高斯光束在FRFT面上的光强分布由FRFT的阶数和子光束数目共同决定,其归一化的光强分布随FRFT的阶数周期性变化,周期为2;子光束数目的大小及其奇偶性对归一化光强分布的影响取决于FRFT的阶数;轴上归一化光强分布也随FRFT的阶数周期性变化,变化周期也为2。     

分数傅里叶变换面上余弦-高斯光束的变换特性

 利用Wigner分布函数的方法，研究了余弦-高斯光束的分数傅里叶变换特性。导出了余弦-高斯光束在分数傅里叶变换面上光强分布和束宽的解析计算公式，并对此进行了数值模拟计算。研究表明：分数傅里叶变换阶数对余弦-高斯光束的光强分布有明显影响，余弦-高斯光束的轴上光强随分数傅里叶变换阶数呈周期性变化，束宽随分数傅里叶变换阶数也呈周期性变化，周期为2；对给定调制参数的余弦-高斯光束，通过适当选取分数傅里叶变化阶数可以获得平顶的光强分布。     

Studies of propagation characteristics of elliptical Gaussian beams through circular apertured optical systems

An approximate analytical propagation formula of the elliptical Gaussian beams through a paraxial ABCD optical system with a circular aperture is obtained on the basis of the expansion of a hard-edged aperture into a finite sum of complex Gaussian functions. And the parameter characteristics of the truncated beams including the beam propagation factor and the kurtosis parameter are also studied in detail by using the second-order-moments method.     

平顶光束K参数的解析传输方程

 从Li提出的平顶光束模型和Collins公式出发,推导出平顶光束通过近轴ABCD光学系统K参数的解析传输公式。平顶光束的K参数与光束阶数M、瑞利长度z_R和传输矩阵元A/B有关。另一方面,基于强度矩的传输方程,通过繁冗的积分,也推导出K参数的解析传输方程。大量数值计算结果证明二者所得结果相同。对平顶光束K参数的传输特性用数值计算例作了说明。     

The propagation properties and kurtosis parametric characteristics of Hermite-cosh-Gaussian beams passing through fractional Fourier transformation systems

Based on the Collins diffraction integral formula and irradiance moments, the propagation properties and kurtosis parametric characteristics of Hermite-cosh-Gaussian (HChG) beams passing through fractional Fourier transformation systems with spherically aberrated lenses are analyzed in detail. The closed-form expressions for the propagation equation and kurtosis parameter of HChG beams passing ideal fractional Fourier transformation systems are also obtained. By using an efficient algorithm some numerical calculations are also performed in this paper. The results show that the spherically aberrated lenses greatly influence on the intensity distribution and kurtosis parameter of HChG beams passage through fractional Fourier transformation systems. The intensity distribution and kurtosis parameter of HChG beams passage through two types of Lohmann's systems with spherically aberrated lenses, respectively, are no longer equal even in the case of the same fractional order and the same spherical aberration coefficients. The fundamental periods of the two types of Lohmann's systems with spherically aberrated lenses are different, 4 for the type I and 2 for the type II. Therefore, the two optical setups for implementing the fractional Fourier transformation are no longer equivalent in the presence of spherically aberrated lenses.     

用Wigner分布函数方法研究平顶多高斯光束的分数傅里叶变换

本文用Wigner分布函数方法分析了平顶多高斯光束的分数傅里叶变换。导出了分数傅里叶变换面上光的强度、束宽、远场发散角、M2因子和K参数的解析表达式。通过数值模拟研究了分数傅里叶变换阶数对平顶多高斯光束传输性质的影响。     

Propagation and the kurtosis parameter of Gaussian flat-topped beams in uniaxial crystals orthogonal to the optical axis

The analytical formulae for the Gaussian flat-topped beams propagating in uniaxial crystals orthogonal to the optical axis are derived. The numerical results show that the Gaussian flat-topped beams spread at different rates in the directions parallel and orthogonal to the optical axis due to anisotropic crystals. An analytical expression for the kurtosis parameter of the Gaussian flat-topped beams propagating in uniaxial crystals is derived and illustrated with numerical examples. It is shown that the evolution of the kurtosis parameters K_x and K_y depend on the ratio of extraordinary to ordinary refractive indices.     

Beam propagation of a higher-order cosh-Gaussian beam

On the basis of the second-order moment, the beam propagation of a higher-order cosh-Gaussian beam has been investigated. Two analytical expressions of the M² factor have been derived, which corresponds to the parity of the beam order n. The M² factor is determined by the beam order n and the parameter δ. The influences of the beam order n and the parameter δ on the M² factor are illustrated with numerical examples. The analytical kurtosis parameter has also been presented. The analytical formula is further simplified in the source plane and the far field, respectively. With given appropriate values of n and δ, the kurtosis parameter in the far field is smaller than 3. This research is helpful to the practical applications of a higher-order cosh-Gaussian beam.