平顶高斯光束经含失调圆孔光阑的失调光学系统的传输特性

为分析平顶高斯光束通过光学系统传输时圆孔光阑失调和光学元件失调对平顶高斯光束传输特性的影响,利用失调圆孔光阑的近似展开式和适用于失调光学系统的广义衍射公式,得出了平顶高斯光束经含失调圆孔光阑的失调光学系统传输的近似解析式,给出了输出光束场分布与光束参量、光阑孔径尺寸、光阑和光学元件失调量等的定量关系.针对特定光学系统定量分析了各失调量对输出光束场分布的影响,结果表明各元件失调都对输出光束强度分布产生较大影响.但在各失调量较小的情况下,透镜失调对输出光束传输特性的影响比光阑失调对输出光束传输特性的影响更明显.     

Paraxial propagation of a partially coherent flattened Gaussian beam through apertured ABCD optical systems

By expanding the hard aperture function into a finite sum of complex Gaussian functions, some approximate analytical formulae for the cross-spectral density of a partially coherent flattened Gaussian beam (FGB) propagating through apertured aligned and misaligned ABCD optical systems are derived based on the generalized Collins formula. The results obtained by using the approximate analytical formula are in good agreement with those obtained by using the numerical integral calculation. As a numerical example, the focusing properties (including average irradiance distribution and focal shift) of a partially coherent FGB focused by an apertured thin lens are studied, and it is found that the focusing properties of a partially coherent FGB are closely related to its initial coherence and the aperture width. Our results provide an effective and fast way for studying the paraxial propagation of a partially coherent FGB through apertured ABCD optical systems.     

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     

Generalized tensor ABCD law for an elliptical Gaussian beam passing through an astigmatic optical system in turbulent atmosphere

The propagation of an elliptical Gaussian beam (EGB) through an astigmatic ABCD optical system in a turbulent atmosphere is investigated. An analytical formula for the average intensity of an EGB and a generalized tensor ABCD law for the generalized complex curvature tensor are derived. As an application example, we derived an analytical formula for the average intensity of an elliptical flat-topped beam propagating through an astigmatic ABCD optical system in a turbulent atmosphere. As a numerical example, the focusing properties of an EGB focused by a thin lens in a turbulent atmosphere are studied. It is found that the focused beam at the focal plane becomes a circular Gaussian beam when the atmospheric turbulence is strong enough, and the beam width of the circular Gaussian beam is determined by atmospheric turbulence strength, focal length of the thin lens, and wavelength of the initial beam but is independent of the initial beam widths (i.e., initial intensity distribution).     

Circular partially coherent flattened Gaussian beam

An alternative theoretical model called circular partially coherent flattened Gaussian beam (FGB) is developed to describe a circular partially coherent beam with a flat-topped spatial profile. Explicit expression for the propagation factor of a circular partially coherent FGB is derived. We drive the analytical formulae for the cross-spectral density and mean-squared beam width of a circular partially coherent FGB propagating through a paraxial ABCD optical system based on the generalized Collins formula. The intensity, spreading and directionality properties of a circular partially coherent FGB propagating in free space are studied as numerical examples. The propagation properties of a circular partially coherent FGB agree well with those of a partially coherent flat-topped beam reported in the literature. Thus, our model provides an alternative but reliable model for describing a circular partially coherent beam with flat-topped profile.     

Propagation of high-order Bessel–Gaussian beam through a misaligned first-order optical system

The generalized diffraction integralis used to derive a generalized formula for high-order Bessel–Gaussian beams (HBGBs) through a misaligned first-order ABCD optical system. It is found that, when a HBGB propagates through a misaligned optical system, the beam shape of the output beam is unchanged. However, the center of the output beam is deviated from the optical axis, forming a decentered HBGB. The position of the output beam may be controlled by adjusting the misaligned parameters. Based on the derived formula, the diffraction patterns of HBGBs propagating through a simple misaligned lens system have been calculated numerically. These results may be useful in the application of laser beams for trapping and manipulating a wide variety of particles.     

高斯光束通过含失调窄缝光阑的失调光学系统的传输特性

 利用硬边窄缝光阑的近似展开式和适用于失调光学系统的广义衍射公式,得出了高斯光束经含失调窄缝光阑的失调光学系统传输的近似解析式。模拟结果表明输出光束场分布与光束参量、光阑尺寸、ABCD矩阵元、光阑失调量和光学系统失调量有关。针对给定的光学系统和高斯光束定量分析了各失调量对输出光束场分布的影响,结果表明:光阑横向位移、光学系统横向位移和角位移均引起垂直于z轴截面内明显的光强非轴对称分布。当光阑半宽度为1 mm时,光阑的衍射作用使腰斑半径为0.5 mm的高斯光束产生-1.586π～1.465π范围的相对相移,且光阑横向位移、光学系统横向位移和角位移均引起焦平面前后相对相移的迅速变化。随光阑宽度变小,各失调量对输出光束特性的影响越明显。     

Shaping the beam profile of a partially coherent beam by a phase aperture

By use of a tensor method, an analytical formula for a partially coherent Gaussian Schell-model (GSM) beam truncated by a circular phase aperture propagating through a paraxial ABCD optical system is derived. The propagation properties of a GSM beam truncated by a circular phase aperture in free space are studied numerically. It is found that the circular phase aperture can be used to shape the beam profile of a GSM beam and generate partially coherent dark hollow or flat-topped beam, which is useful in many applications, e.g., optical trapping, free-space optical communication, and material thermal processing. The propagation factor of a GSM beam truncated by a circular phase aperture is also analyzed.     

Propagation Effect of Hollow Gaussian Beams Passing through a Misaligned Optical System

A generalized formula of hollow Gaussian beams through the first-order misaligned ABCD systems is derived by using the generalized diffraction integral formula. It is shown that the hollow Gaussian beam passing through the misaligned system becomes a decentred hollow Gaussian beam. The propagation properties of the output beam are investigated when it propagates through a simple misaligned lens system. These results provide a powerful theoretical tool for applications of optical traps.     

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.     

Propagation of anomalous hollow beam through a misaligned first-order optical system

In this paper, we consider the propagation of an anomalous hollow beam passing through the first-order ABCD optical system. Based on the generalized diffraction integrals, a general propagating formula of anomalous hollow beam is derived. Based on the derived formula, the propagation properties of anomalous hollow beam through the misaligned optical system are numerically illustrated. From the numerical results we can get that when an anomalous hollow beam propagates through a misaligned optical system, the shape of the output beam changes as the propagation distance increases, and the center of the output beam is deviated from the optical axis. The position of the output beam is effected by the misaligned parameters, and can be controlled by adjusting them. This method provides a convenient tool for studying the propagation and transformation of an anomalous hollow beam through the misaligned optical system.     

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.     

洛伦兹光束经光阑失调傍轴光学系统的传输

基于广义衍射积分公式和光阑函数的复高斯展开,导出了一洛伦兹光束经一个带圆形光阑失调傍轴光学系统的近似解析传输公式.作为一般公式的特例,还给出了洛伦兹光束经一无光阑失调傍轴光学系统的解析传输式.作为数值计算的例子,运用所得到的公式分析了洛伦兹光束经带光阑失调薄透镜的传输特性.结果表明：不同强度的衍射即圆形光阑半径的大小明显影响衍射光束的归一化强度分布及其传输变化规律. 关键词： 洛伦兹光束 失调傍轴光学系统 光束传输     

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.     

环状平顶光束的传输特性研究

利用偏心高斯光束相干叠加方法，建立了柱坐标系下环状平顶光束模型.采用广义强度二阶矩定义，推导出环状光束广义M2因子的解析表达式.利用广义衍射积分理论，给出环状光束经过近轴ABCD光学系统的传输公式，并进一步给出了环状平顶光束修正的菲涅耳数定义.分析了环状平顶光束在自由空间的传输规律.研究结果表明，该模型可统一描述基模高斯光束、环状高斯光束以及环状平顶光束.环状平顶光束的广义M2因子和环状平顶光束的传输特性与其阶数及遮拦比有关.     

Propagation of a hollow Gaussian beam through a paraxial misaligned optical system

Based on the generalized diffraction integral formula for treating the propagation of a laser beam through a paraxial misaligned optical system in the cylindrical coordinate system, we obtain an analytical formula for a hollow Gaussian beam passing through a paraxial misaligned optical system. Furthermore, we also obtain the approximate analytical formula for a hollow Gaussian beam passing through a paraxial circularly apertured misaligned optical system by expanding the hard aperture function into a finite sum of complex Gaussian functions. As a numerical example, the propagation properties a hollow Gaussian beam through a misaligned thin lens are studied numerically.     

椭圆平顶高斯光束及其传输特性

 用张量光学方法定义了三维不可分离变量的椭圆平顶高斯光束(EFGB),导出了这种光束经过非轴对称光学系统的传输公式及相应的张量ABCD定律, 并对EFGB经过自由空间的传输作了数值计算。结果表明, EFGB的传输特性依赖于阶次, 而且耦合EFGB的光强分布在传输过程中会发生旋转。     

Propagation of a flattened Gaussian beam through multi-apertured optical ABCD systems

By expanding the hard-aperture function into a finite sum of complex Gaussian functions, the recurrence propagation formula of a flattened Gaussian beam through multi-apertured optical ABCD systems is derived, and the propagation of a Gaussian beam is regarded as a special case in our theoretical model. The extension to the three-dimensional case is described. Numerical examples of the beam propagation through a multiple aperture-lens system and an aperture-spatial filter are given and compared with those by the straightforward numerical integration of the Collins formula. It is shown that our analytical results give a good trade-off between the computational time and accuracy, and provide a useful tool for simulating the beam propagation through complicated optical systems including multiple apertures and spatial filters. Finally, a comparison with the previous work is made and the main results obtained in this paper are summarized.     

An alternative model for a partially coherent elliptical dark hollow beam

An alternative theoretical model named partially coherent hollow elliptical Gaussian beam (HEGB) is proposed to describe a partially coherent beam with an elliptical dark hollow profile. Explicit expression for the propagation factors of a partially coherent HEGB is derived. Based on the generalized Collins formula, analytical formulae for the cross-spectral density and mean-squared beam width of a partially coherent HEGB, propagating through a paraxial ABCD optical system, are derived. Propagation properties of a partially coherent HEGB in free space are studied as a numerical example.     

Focusing properties of a general-type beam in turbulent atmosphere

Based on the generalized Huygens–Fresnel integral, analytical propagation formulas for a general-type beam propagating through aligned or misaligned ABCD optical systems in turbulent atmosphere are derived. The derived formulas provide a convenient way for studying the focusing properties 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 in turbulent atmosphere. As an application example, the focused intensities of cos-Gaussian, Hermite-sine-Gaussian and flat-topped beams in turbulent atmosphere are studied numerically. Focal shift of a flat-topped beam in turbulent atmosphere is investigated. Effect of the misalignment of the thin lens on the focusing properties of a cos-Gaussian beam is also explored. Our results will be useful for the applications of the general-type beam in LIDAR systems and remote sensing operating in turbulent atmosphere, where optical elements such as aligned or misaligned thin lens are commonly encountered.