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.     

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 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.     

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.     

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     

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 of Lorentz and Lorentz-Gauss beams through an apertured fractional Fourier transform optical system

By expanding the hard-aperture function into a finite sum of complex Gaussian functions, we derive approximate analytical formulae for Lorentz and Lorentz-Gauss beams propagating through an apertured fractional Fourier transform (FRT) optical system. As an application example, properties of a Lorentz-Gauss beam in the FRT plane after propagating through a squarely or annularly apertured FRT optical system are studied numerically. The results obtained using the approximate analytical formula are in good agreement with those obtained using numerical integral calculation. The FRT optical system provides a convenient way for laser beam shaping.     

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.     

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.     

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.     

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 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     

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 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.     

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.     

Partially coherent elegant Hermite–Gaussian beams

Elegant Hermite–Gaussian beams (EHGBs) are extended to the partially coherent case. An explicit and analytical formula is derived for the cross-spectral density of a partially coherent EHGB propagating through an aligned or misaligned paraxial ABCD optical system. The propagation properties of a partially coherent EHGB in free space and its focusing properties through a thin lens are studied numerically, and are compared to those of a partially coherent standard Hermite–Gaussian beam (SHGB). It is found that the propagation and focusing properties of a partially coherent EHGB are closely related to its initial coherence. A partially coherent EHGB spreads slower than a partially coherent SHGB in free-space propagation. A partially coherent EHGB can be focused more tightly than a partially coherent SHGB.     

Propagation of the Hermite–Gaussian beams through misaligned optical system with a circular aperture

By means of expanding a hard-edged aperture into a finite sum of complex Gaussian functions, the approximate analytical formula of one kind of higher-order Gaussian beams called the Hermite–Gaussian beams (HGBs) passing through circular apertured and misaligned optical system is obtained in this paper. The result provides more convenience 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 the HGBs through the circular apertured optical systems.     

Fractional Fourier transform of apertured paraboloid refracting system

The limitation of paraxial condition of paraboloid refracting system in performing fractional Fourier transform acts like an aperture, which makes the system different from ideal systems. With aperture expanded as the sum of finite complex Gaussian terms, a more practical approximate analytical solution of fractional Fourier transform of Gaussian beam in an apertured paraboloid refracting system is obtained and also numerical investigation is presented. Complicated and practical fractional Fourier transform systems can be constructed by cascading several apertured paraboloid refracting systems which are the simplest and the most basic units for performing more precise transform.     

Propagation characteristics of linearly polarized Bessel-Gaussian beams through an annular apertured paraxial ABCD optical system

By means of Collins diffraction integral formula in the paraxial approximation and based on the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions, an approximate analytical expression for linearly polarized Bessel-Gaussian beams passing through a paraxial ABCD optical system with an annular aperture has been derived. The results provide more convenient for studying their propagation and transformation than the usual way by using diffraction integral directly. By using the analytical expression and the diffraction integral formula some numerical simulations are done to illustrate for the propagation characteristics of a linearly polarized Bessel-Gaussian beam through an optical system with an annular aperture.     

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.