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.     

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 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.     

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 Lorentz-Gauss beam through a misaligned optical system

Based on the generalized integral formula and the convolution theorem of the Fourier transform, an analytical propagation formula of a Lorentz-Gauss beam passing through a misaligned paraxial optical system is derived. As numerical examples, the propagation properties of a Lorentz-Gauss beam through a misaligned thin lens with the lateral displacement and the angle displacement are graphically illustrated, respectively. The influences of the lateral displacement and the angle displacement of the misaligned thin lens on the normalized light intensity and the phase distributions are also examined, respectively.     

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 cosh-Gaussian beams in a misaligned optical system

On the basis of the generalized diffraction integral formula for misaligned optical systems in spatial domain, an analytical propagation expression for the elliptical cosh-Gaussian beam (EChGB) passing through misaligned optical systems is obtained by using vector integration. Some numerical simulations are illustrated for the propagation properties of an EChGB through a misaligned optical transforming system with a misaligned thin lens. As an application example, the propagation of off-axial EChGBs passing through misaligned optical systems is studied, and further extensions are also pointed out.     

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.     

Propagation of flat-topped beams

The propagation of flat-topped beams passing through paraxial ABCD optical system is investigated based on the propagation formulas of Gaussian beam. The focal shift of focused coherent flat-topped beam is also studied in detail. Analytical expressions of the M² factor and the far-field intensity distribution for flat-topped beams are derived on the basis of second-order moments.     

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.     

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.     

部分偏振高斯－谢尔模光束在准直和失调光学系统中的传输

利用张量方法导出了部分偏振高斯－谢尔模光束在准直和失调光学系统中的相干－偏振矩阵的传输公式，并以经过失调薄透镜为例，详细讨论了部分偏振GSM光束的传输特性。导出的公式提供了一个研究部分偏振GSM光束在准直和失调光学系统中传输的简便方法。     

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 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 properties of partially polarized Gaussian Schell-model beams through an axis-unsymmetric paraxial ABCD system

By using the beam coherence-polarization (BCP) matrix approach, analytical propagation equations of partially polarized Gaussian Schell-model (PGSM) beams through an axis-unsymmetric paraxial optical ABCD system are derived, which enable us to study the propagation-induced polarization changes and irradiance distributions at any propagation distance of PGSM beams through axis-unsymmetric systems within the framework of the paraxial approximation. Detailed numerical results for a PGSM beam passing through a bifocal lens are presented to illustrate the propagation properties of PGSM beams. A comparison with the previous work is also made.     

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 the beam generated by Gaussian mirror resonator passing through a paraxial ABCD optical system

By expanding a hard aperture function into a finite sum of complex Gaussian functions, approximate propagation formula is derived in the situation that the beam generated by Gaussian mirror resonator passes through a paraxial ABCD optical system with an annular aperture. The corresponding forms for a circular aperture and a circular black screen are also given. Some numerical simulations are shown to illustrate propagation properties and focusing properties of the beam passing through a paraxial ABCD optical system with the three different kinds of aperture.     

复宗量双曲余弦高斯光束的束腰及其位置

 由复宗量双曲余弦高斯光束通过近轴-ABCD-光学系统传输公式，推导出了基于二阶矩定义的光斑尺寸的解析表达式，给出了复宗量双曲余弦高斯光束在自由空间传输和通过薄透镜时的光斑尺寸、束腰宽度及其位置的解析表达式，并进行了数值计算，对计算结果进行了分析和讨论，该方法可用于其它光束通过近轴ABCD光学系统的类似计算。     

Propagation of elliptical Gaussian beam through misaligned optical systems in spatial domain and spatial-frequency domain

The propagation formulae of elliptical Gaussian beam through misaligned optical systems in spatial domain and spatial-frequency domain are investigated analytically by using tensor method. The derived formulae provide a powerful tool for treating the propagation and transformation of elliptical Gaussian beam through misaligned optical systems. As an application example, the propagation properties of elliptical Gaussian beam through a misaligned thin lens are illustrated.     

Four-beamlets laser array and its propagation

A new type of laser array named four-beamlets laser array is proposed. Based on the diffraction integrals, analytical propagation formulae for the four-beamlets laser array passing through paraxial aligned and misaligned optical systems are derived, respectively. The four-beamlets laser array can also be expressed as a superposition of a series of Hermite Gaussian modes by using polynomial expansion. As a numerical example, the propagation properties of a four-beamlets laser array in free space are illustrated graphically. The fractional Fourier transform of the four-beamlets laser array is also studied. Our model provides a convenient way to describe laser array with four-beamlets.