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.     

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.     

Fractional Fourier transform for partially coherent Gaussian-Schell model beams

The fractional Fourier transform (FRT) is applied to partially coherent twisted anisotropic Gaussian-Schell model (GSM) beams based directly on the cross-spectral density. An analytical and concise formula is derived for the cross-spectral density of partially coherent twisted anisotropic GSM beams passing through a FRT system in terms of the tensor method. The corresponding tensor ABCD law for performing a FRT is obtained. The connection between the FRT formula and the generalized Collins formula for partially coherent beams is discussed. The formulas derived provide a powerful tool for analyzing and calculating the FRTs of partially coherent beams.     

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.     

Wigner Transforms and Fractional Fourier Transforms of the First Order Optical Systems

Based on the Collins formula, the relationship between the coordinate transform matrix (WCTM) of the Wigner distribution function (WDF) and the ray transfer matrix (RTM) of an arbitrary first-order optical system has been derived. By using this relation and the definition of fractional Fourier transform (FRT) in terms of WDF rotation, it is concluded that an arbitrary first-order optical system can be generally decomposed into a thin lens and a FRT sub-system whose order is not unique and depends on two concrete decomposing operations on the system. And when the system is reciprocally symmetric, a FRT can be implemented by it. In addition, the composition, that is also the decomposition condition of the complicated FRT optical system by cascading a series of FRT subsystems has also been derived by using the operations of RTM.     

Wigner Transforms and Fractional Fourier Transforms of the First Order Optical Systems

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Radiation forces of highly focused Bessel–Gaussian beams on a dielectric sphere

The radiation force of highly focused Bessel–Gaussian beams (BGBs) on a dielectric sphere in the Rayleigh scattering regime is theoretically investigated. Numerical results demonstrate that the focused BGBs can be used to trap and manipulate the particles with the refractive index lower than that of the ambient. The radiation force caused by the low-order focused BGBs has been studied under different input parameters and different focus lengths of thin lens. The stability conditions of trapping the particles are also analyzed.     

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。     

Off-axial elliptical Hermite-cosh-Gaussian beams

A new kind of light beams named the off-axial elliptical Hermite-cosh-Gaussian beams (EHChGBs) is introduced in this paper by use of tensor method. An analytical propagation expression for an off-axial EHChGB passing through an axially nonsymmetrical optical ABCD system is derived by use of vector integration. The derived formula can be easily reduced to the propagation formulas of off-axial elliptical cosh-Gaussian beams and elliptical Hermite-cosh-Gaussian beams. Some numerical simulations are illustrated for the propagation properties of off-axial EHChGBs passing through a free space and a focusing optical system, and further extensions are suggested.     

Analytical solution for an anomalous hollow beam in a fractional Fourier transforming optical system with a hard aperture

Based on the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions, the approximate analytical expressions for the output field distribution of an anomalous hollow beam (AHB) passing through an apertured fractional Fourier transform (FRT) system are derived. By using the approximate analytical formulae and diffraction integral formulae, the propagation properties of an AHB in circular and rectangular apertured FRT system are studied numerically. The results show that this method provides a convenient tool for studying the propagation properties of an AHB passing through apertured FRT system, and the apertured FRT system can be applied to laser beam shaping conveniently.     

相位翻转厄米-高斯光束的传输特性

将相位翻转高斯光束的概念推广到相位翻转高阶模光束.以相位翻转厄米-高斯光束为例,推导出相位翻转高斯光束通过近轴理想(无光阑)ABCD光学系统,以及带刀口和光阑的ABCD光学系统传输的递推方程,并用以研究相位翻转高斯光束的传输特性.特别是,得出对TEM1、TEM2和TEM3模相位翻转高斯光束,光阑效应可以忽略的条件分别是截断参量δ≥2.1,2.5和 3.0.     

Decentered twisted Gaussian Schell-model beams and their propagation through a misaligned first-order optical system

The propagation properties of decentered twisted Gaussian Schell-model (DTGSM) beams passing through a misaligned first-order optical system are studied. The explicit expressions for the cross-spectral density function and Wigner distribution function of the output beam are derived, which retain their form unchanged. It is shown that the DTGSM beams preserve their closed property. The second-order moments matrix and the Wigner distribution function evolve with the usual laws, whereas the first-order moments matrix varies, as if a ray passes through such system. The propagation of DTGSM beams through an aligned first-order optical system is treated as the limiting case that corresponds to the vanishing misalignment parameters.     

Fractional Fourier transform for a hollow Gaussian beam

The fractional Fourier transform (FRT) for a hollow Gaussian beam (HGB) is investigated. Based on the definition of FRT in the cylindrical coordinate system, analytical formulae are derived for the FRT for a HGB. By using the derived formulae, the properties of a HGB in the FRT plane are illustrated numerically. The derived formulae provide a convenient way for analyzing and calculating the FRT of a HGB.     

厄米-余弦-高斯光束的传输特性

 从广义惠更斯-菲涅耳衍射积分出发对厄米-余弦-高斯光束的传输特性作了研究，给出了厄米-余弦-高斯光束通过近轴ABCD光学系统的解析传输公式，以厄米-余弦-高斯光束通过自由空间和薄透镜系统为例做了数值计算和分析，结果表明厄米-余弦-高斯光束在自由空间传输时不能保持其形状不变。厄米-双曲余弦-高斯光束、余弦-高斯光束、厄米-高斯光束的传输公式可作为厄米-余弦-高斯光束的特例得出。     

Propagation of elegant Hermite cosine Gaussian laser beams

In this paper, a more generalized beam containing an elegant Hermite Gaussian beam and a cosine Gaussian beam, which is called elegant Hermite cosine Gaussian (EHcoG) beam, is introduced and studied. The propagation properties of the EHcoG laser beams through a first-order paraxial optical system are demonstrated analytically and numerically. The recurrence propagation equations of the EHcoG beams through a first-order paraxial optical system with and without aperture are derived, from which the analytical propagation expression for EHcoG beams of any order can be obtained by means of the recursive procedures. The validity of the analytical results is confirmed by the numerical results.     

Experimental observation of lensless coincidence fractional Fourier transform with a Gaussian Schell-model beam

A modified lensless optical system for implementing coincidence fractional Fourier transform (FRT) is proposed, and the conditions for the optical system to implement the coincidence FRT with incoherent or partially coherent light are discussed. Furthermore, we report the experimental observation of lensless coincidence FRT of an object (double slits) with a typical partially coherent beam - Gaussian Schell-model (GSM) beam. The experimental results are analyzed and agree reasonably well with the theoretical results.     

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 of a stochastic electromagnetic Gaussian Schell-model beam through an optical system in turbulent atmosphere

A generalized diffraction integral formula for stochastic electromagnetic beams propagating through an optical system in turbulent atmosphere is derived with the help of tensor method. Some analyses are illustrated by a numerical example relating to changes in the average intensity and the degree of polarization of an electromagnetic Gaussian Schell-model beam propagating through a double-lenses system. It is shown that the optical system has strong influence on the propagation properties of the beam. The method used in this paper can be widely applied to the propagation of astigmatic beams through an optical system in turbulent atmosphere.     

高斯光束经复合光学系统的聚焦特性

 讨论了高斯光束经由凹透镜，子系统和凸透镜构成的复合光学系统的聚焦特性，利用矩阵光学方法，推导出了高斯光束经复合光学系统聚焦后束腰宽度和像距的计算公式。计算例表明，选择适当参数的复合光学系统，既可以获得较小的束腰宽度，又可以获得较长的像距。     

Lensless optical implementation of the coincidence fractional Fourier transform

A lensless optical system for implementing the coincidence fractional Fourier transform (FRT) is proposed. The conditions for the lensless optical system to implement the coincidence FRT with incoherent light and entangled photon pairs are discussed. The results offer a novel scheme for FRTs and thus suggest useful applications.