Approximate analytical expressions of Laguerre-Gaussian beams passing through a paraxial optical system with an annular aperture

On the basis of the expansion of the hard aperture function into a finite sum of complex Gaussian functions, the approximate analytical expression of Laguerre-Gaussian beams passing through an annular apertured paraxial ABCD optical system is derived. Meanwhile, the corresponding closed-forms for the unapertured or circular apertured or circular black screen cases have also been given. Numerical examples are given to illustrate the propagation characteristics of Laguerre-Gaussian beams.     

Propagation of nonparaxial vector hollow Gaussian beams through a circular aperture

Based on the vectorial Rayleith-Sommerfeld formulae, the nonparaxial propagation properties of the vector hollow Gaussian beams (HGBs) through a circular aperture are studied in detail. We describe the derivation of the integral expressions of the propagation of nonparaxial vector HGBs through a circular aperture. The derived expression is independent the approximation of paraxial and far field, which are valid for either far and near field and for the systems in which aperture radius is comparable to or even smaller than wavelength. And it is also strict integral formula for the light field on the axis. Numerical calculation results indicate that there is no difference between derived formulae and the Collins formulae in the situation of paraxial approximation. Using the formula deduced, we calculate the propagation properties of HGBs. The calculated results indicate that the propagation field of vector hollow Gaussian beams is asymmetric in near field, while the propagation field is symmetric in far field. These research results could well shed light on the further understanding of the vectorial property of HGBs through a circular aperture, and would play a guiding role in the practical application of HGBs.     

一种新空心光束的理论及实验研究

提出了一种用来描述空心光束的新理论模型——空心高斯光束模型，通过控制光束参数可以方便地调制空心光束黑斑的大小.导出了空心高斯光束经傍轴光学系统的解析传输变换公式.利用导出的公式计算分析了空心高斯光束在自由空间的传输特性.在实验上利用矩阵本征值方法设计了能输出空心高斯光束的光学谐振腔，对实验上得到的空心高斯光束进行了传输特性的研究.结果表明，理论和实验相吻合.该空心高斯光束产生的光势阱为控制冷原子提供了一条新途径. 关键词： 空心高斯光束 光学谐振腔 传输特性     

Propagation of elegant Laguerre–Gaussian beams through an annular apertured paraxial ABCD optical system

Based on the generalized Huygens–Fresnel diffraction integral and the expansion of the hard aperture function into a finite sum of complex Gaussian functions, the approximate analytical expression of elegant Laguerre–Gaussian beams passing through a paraxial ABCD optical system with an annular aperture is derived. Meanwhile, the corresponding closed-forms for the unapertured, circular apertured or circular black screen cases are also given. The obtained results provide more convenience for studying their propagation and transformation than the usual way by using diffraction integral formula directly. Some numerical examples are given to illustrate the propagation properties of elegant Laguerre–Gaussian beams.     

Propagation of a general-type beam through apertured aligned and misaligned ABCD optical systems

By expanding the hard-aperture function into a finite sum of complex Gaussian functions, analytical formulae for the electric field of a general-type beam propagating through apertured aligned and misaligned ABCD optical systems are derived using the generalized Collins formulae, which provide a convenient way of studying the propagation of a variety of laser beams, such as Gaussian, cos-Gaussian, cosh-Gaussian, sine-Gaussian, sinh-Gaussian, flat-topped, Hermite-cosh-Gaussian, Hermite-sine-Gaussian, higher-order annular Gaussian, Hermite-sinh-Gaussian and Hermite-cos-Gaussian beams, through such optical systems. As numerical examples, the propagation properties of a cos-Gaussian beam through an apertured aligned or misaligned thin lens are studied.     

Propagation characteristics of the rectangular flattened Gaussian beams through circular apertured and misaligned optical systems

Based on the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions, the approximate analytical expression for the output field distribution of a rectangular flattened Gaussian beam passing through a circular apertured and misaligned paraxial ABCD system is derived. The result brings more convenient for studying its propagation than the usual way by using diffraction integral directly. Some numerical simulations are also given for illustrating the propagation properties of a rectangular flattened Gaussian beam through a circular apertured and misaligned optical system.     

Hollow Gaussian beams and their propagation properties

A new mathematical model, described as hollow Gaussian beams (HGBs), is proposed to describe a dark hollow laser beam (DHB). The area of the dark region across the HGBs can easily be controlled by proper choice of the beam parameters. Based on the Collins integral, an analytical propagation formula for the HGBs through a paraxial optical system is derived. The HGBs also can be expressed as a superposition of a series of Lagurerre-Gaussian modes by use of a polynomial expansion. As a numerical example, the propagation properties of a DHB in free space are illustrated graphically. The HGBs provide a convenient and powerful way to describe and treat the propagation of DHBs and can be used conveniently to analyze atoms manipulated with a DHB.     

Analytical formulas for a circular or non-circular flat-topped beam propagating through an apertured paraxial optical system

Propagation of a flat-topped beam of circular or non-circular (rectangular or elliptical) symmetry through an apertured optical system is investigated. By expanding the hard aperture function as a finite sum of complex Gaussian functions, some approximate analytical propagation formulas are derived for a flat-topped beam of circular or non-circular (rectangular or elliptical) symmetry propagating through an apertured paraxial general astigmatic (GA) optical system or an apertured paraxial misaligned stigmatic (ST) optical system. The derived formulas are very fast to compute. The results obtained by using the approximate analytical expressions are in a good agreement with those obtained by direct numerical integration. The present analytical formulas provide a convenient and effective way for studying the propagation and transformation of a circular or non-circular flat-topped beam through an apertured general optical system.     

Propagation of generalized Mathieu–Gauss beams through paraxial misaligned optical systems

Based on the generalized diffraction integral, we derive an analytical formula for generalized Mathieu–Gauss beams (gMGBs) passing through an apertured misaligned optical system. Furthermore, we use the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions to establish an approximate propagation equation of gMGBs through paraxial circularly apertured optical system. As an example, the propagation of ordinary and modified zeroth order MGBs through a misaligned thin lens is studied numerically.     

Propagation characteristics of the two-dimensional off-axial Hermite-cosh-Gaussian beams through rectangular apertured and misaligned optical systems

By introducing a hard aperture function into a finite sum of complex Gaussian functions, an approximate analytical expression for the two-dimensional off-axial Hermite-cosh-Gaussian beams passing through a rectangular apertured and misaligned paraxially ABCD optical system has been derived. The results provide more convenience for studying their propagation and transformation than the usual way by using diffraction integral directly. Some numerical simulations are also illustrated for the propagation characteristics of a two-dimensional off-axial Hermite-cosh-Gaussian beam through a rectangular apertured ABCD optical system.     

Analytical formula for a circular flattened Gaussian beam propagating through a misaligned paraxial ABCD optical system

Based on the generalized diffraction integral formula for treating the propagation of a laser beam through a misaligned paraxial ABCD optical system in the cylindrical coordinate system, analytical formula for a circular flattened Gaussian beam propagating through such optical system is derived. Furthermore, an approximate analytical formula is derived for a circular flattened Gaussian beam propagating through an apertured misaligned ABCD optical system by expanding the hard aperture function as a finite sum of complex Gaussian functions. Numerical examples are given.     

Propagation characteristics of flattened Gaussian beams through a misaligned optical system with a misaligned annular aperture

The approximate analytical formula for flattened Gaussian beams through a misaligned optical system with a misaligned annular aperture was derived by the extended Huygens–Fresnel principle. Some numerical simulations are illustrated to the effects on the propagation of flattened Gaussian beams by the misaligned annular aperture. To compare the difference between annular apertured system and circular apertured system, the circular apertured system is also studied. The results show that angle misalignments and lateral displacements of aperture create asymmetrical average intensity distribution at receiving plane z = 500. The effects on intensity distribution by angle misalignments of annular aperture were small. In annular aperture case, the smooth of intensity distribution was worse with escalating obscure ratio ? in near-field; the side-lobes increased and the central lobe decreased with escalating obscure ratio ? in far-field. At receiving plane z = 500: for circular aperture, the side-lobes decreased, even to be neglected, with the increasing of truncation parameter δ; for annular aperture, the side-lobes increased with the increasing of truncation parameter δ. In addition, it is found that the aligned thin lens can fix asymmetry of intensity distribution which was caused by the misaligned annular aperture.     

Propagation of a Laguerre–Gaussian beam through a slightly misaligned paraxial optical system

Based on the generalized diffraction integral formula for treating the propagation of a laser beam through a slightly misaligned optical system in a cylindrical coordinate system, an analytical formula for a Laguerre–Gaussian beam passing through such an optical system is derived. Furthermore, an approximate analytical formula is derived for a Laguerre–Gaussian beam passing through an apertured slightly misaligned optical system by expanding the hard aperture function as a finite sum of complex Gaussian functions. Some analytical formulas are also given for a flattened Gaussian beam by expanding its field as a superposition of a finite series of Laguerre–Gaussian beams. PACS 42.25.Bs; 41.85.Ew; 41.85.Ct     

Propagation of elliptical Gaussian beam passing through apertured paraxial optical systems

The propagation of elliptical Gaussian beam passing through paraxial optical systems with aperture is investigated analytically by using tensor method. The approximate formula for propagation of elliptical Gaussian beam through hard apertured optical systems is derived based on the fact that the circ function can be expanded into a finite sum of complex Gaussian functions. The derived formula provides a convenient tool for treating the propagation and transformation of elliptical Gaussian beam through apertured optical systems. As an application example, the propagation properties of elliptical Gaussian beam through apertured fractional Fourier systems are discussed.     

Propagation properties of apertured laser beams with amplitude modulations and phase fluctuations through atmospheric turbulence

The propagation properties of apertured laser beams with amplitude modulations (AMs) and phase fluctuations (PFs) through atmospheric turbulence are studied in detail both analytically and numerically. The analytical expressions for the average intensity, power in the bucket (PIB) and Strehl ratio (S _R ) of apertured laser beams with AMs and PFs propagating through atmospheric turbulence are derived. It is found that the worse the phase fluctuation and the higher the amplitude modulation are, the less laser beams are affected by turbulence. Furthermore, apertured Gaussian beams are more sensitive to turbulence than apertured laser beams with AMs and PFs. The average intensity of apertured laser beams with AMs and PFs may be even larger than that of apertured Gaussian beams due to turbulence. In particular, the influence of turbulence on the average maximum intensity of apertured laser beams with PFs and AMs may become serious if an unsuitable truncated parameter is chosen, which should be avoided in practice.     

Matrix formulation of axis-symmetric laser beams through a paraxial ABCD system containing hard-edged apertures: A comparative study

A matrix formulation is presented, which enables us to study the propagation of axis-symmetric beams through a paraxial optical ABCD system containing hard-edged aperture. Numerical calculation results of super-Gaussian beams passing through a multi-aperture-lens system are given to illustrate the advantage of the method. A comparison of the matrix formulation, complex Gaussian expansion and direct numerical integration of the Collins formula is made, where the propagation of apertured Laguerre-Gaussian beams is chosen as an illustrative example. It is shown that the matrix formulation provides satisfactory results in both Fraunhofer and Fresnel regions, and reduces the computational time greatly in comparison with the direct integration. However, this method is suited only to axis-symmetric optical beams and systems. By using the complex Gaussian expansion discrepancies exist in the near zone closer to the aperture, but usually its computational efficiency is higher than the matrix formulation.     

Propagation characteristics of laser beams with amplitude modulations and phase fluctuations in apertured fractional Fourier transforming system

The apertured fractional Fourier transforming system is introduced and applied to treat the propagation of Gaussian beams with amplitude modulations and phase fluctuations. Based on the treatment that a rectangular function can be expanded into an approximate sum of complex Gaussian functions with finite numbers, the analytical expressions for the mutual intensity distribution of laser beams with amplitude modulations and phase fluctuations passing through the apertured fractional Fourier transforming system are obtained. Some numerical simulations are illustrated for their propagation properties.     

贝塞耳函数调制的高斯光束通过有光阑ABCD光学系统的传输

基于惠更斯菲涅尔广义衍射积分,对变量为径向平方的贝塞耳函数调制的高斯光束(QBG光束)通过有圆孔光阑限制的一阶ABCD光学系统的传输进行了研究.采用将圆域函数可表示为复高斯函数叠加的方法,推导出了普适的场分布解析公式,以QBG光束通过有圆孔光阑限制的薄透镜系统为例进行了分析与讨论,数值计算结果证明了本文所用方法的优点,所得结果与直接用广义衍射积分公式进行复杂的数值计算结果完全一致 关键词： QBG光束 光阑 ABCD光学系统 传输特性     

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.     

Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture

Based on the vectorial Rayleigh diffraction integral, the nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture is studied. The far-field and paraxial cases are treated as special cases of our general result. It is shown that for the apertured case the f parameter still plays an important role in determining the nonparaxiality of vectorial diffracted Gaussian beams, but both the f parameter and truncation affect the beam evolution behavior.