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.     

Propagation of Gaussian beams family through a Kerr-type left-handed metamaterial

In this paper the propagation of elegant Hermite-cosh-Gaussian, elegant Laguerre Gaussian, and Bessel Gaussian beams through a Kerr left-handed metamaterial(LHM) slab have been studied. A split-step Fourier method is used to investigate the propagation of laser beams through this media. Numerical simulation shows that Gaussian beams have different focusing behaviors in a Kerr LHM slab with positive or negative nonlinearity. Beam focusing happens in slabs with positive nonlinearity and not in slabs with negative nonlinearity; however, negative nonlinearity is required for a Kerr LHM slab to act like a lens. Additionally, the focusing properties of beams can be controlled by controlling the thickness of the slab or the input power of the incident beam. A multilayer structure is also proposed to have beam focusing by thinner slabs and passing longer distances.     

Beam propagation factor of partially coherent elegant Hermite-cosh-Gaussian beam through non-Kolmogorov turbulence

Based on the extended Huygens–Fresnel principle and the second moments of the Wigner distribution function, an analytical expression for the propagation factors of partially coherent elegant Hermite-cosh-Gaussian beams (EHChGB) in non-Kolmogorov is derived. The M² factor of partially coherent EHChGB through non-Kolmogorov turbulence is studied analytically and numerically. The results show that partially coherent EHCHGB with higher beam order and longer wavelengths is less affected by non-Kolmogorov turbulence.     

Focal shifts in focused Hermite-cosh-Gaussian laser beams

A closed-form propagation equation of Hermite-cosh-Gaussian beams passing through an unapertured thin lens is derived. Focal shifts are analyzed by means of two different methods according to the facts that the axial intensity of some focused Hermite-cosh-Gaussian beams are null and that of some others are not null but the principal maximum intensity may be located on the axis or off the axis. Optimal focusing for the beams is studied, and the condition of optimal focusing ensuring the smallest beam width is also given.     

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.     

Fractional Fourier transforms of elliptical Hermite-cosh-Gaussian beams

A new kind of laser beams named the elliptical Hermite-sinusoidal-Gaussian beams (EHSGBs) is introduced and defined by use of tensor method, and the elliptical Hermite-cosh-Gaussian beam (EHChGB) can be regarded as special case of EHSGBs. The fractional Fourier transform (FRFT) is applied to treat the propagation of EHChGBs. An analytical expression for an EHChGB passing through an FRFT system is derived by using vector integration. Some numerical simulations are illustrated for the propagation properties of EHChGBs through FRFT systems.     

会聚厄米-双曲余弦高斯光束的光强分布和焦移

 基于厄米-双曲余弦高斯光束通过无光阑限制薄透镜聚焦的解析传输公式，研究了厄米-双曲余弦高斯光束聚焦区域的光强分布，并对光束的焦移进行了分析，讨论了偏心参数对光强主极大位置的影响。结果表明：TEM₁₁模厄米-双曲余弦高斯光束的相对焦移（绝对值）随偏心参数和菲涅尔数的减小而增大，菲涅尔数较大时相对焦移趋于零。TEM₂₂模光束在偏心参数小于0.54时，轴外与轴上光强极大值的比值大于1，此时光强主极大在轴外，偏心参数大于0.54时则相反；在偏心参数等于0.54时比值为1，此时光束有两个主极大，偏心参数愈大光强愈集中于轴上。使用LW法和GH法得到的TEM₂₂模光束的相对焦移（绝对值）随偏心参数和菲涅尔数的变化规律与TEM₁₁模光束一致，但相同参数下使用这两种方法得出的具体结果不同。     

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.     

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.     

Acceleration control of Airy beams with optically induced refractive-index gradient

We demonstrate both experimentally and theoretically controlled acceleration of one- and two-dimensional Airy beams in optically induced refractive-index potentials. Enhancement as well as reduction of beam acceleration are realized by changing the index gradient, while the beam shape is maintained during propagation through the linear optical potential. Our results of active acceleration manipulation in graded media are pertinent to Airy-type beam propagation in various environments.     

The elliptical elegant Laguerre–Gaussian beam and its propagation through aligned and misaligned paraxial optical systems

A new kind of laser beam called the elliptical elegant Laguerre–Gaussian beam (EELGB) is defined by using tensor method. By using the generalized diffraction integral formulas for light beam passing through paraxial optical system, the analytical propagation formulas for EELGB passing through paraxial aligned and misaligned optical systems are obtained through vector integration. As examples of applications, the propagation properties of EELGBs in free space propagation and through a misaligned thin lens are studied.     

Average intensity and spectral shifts of a partially coherent standard or elegant Laguerre-Gaussian beam beyond paraxial approximation

With the help of the Rayleigh-Sommerfeld diffraction integral, an analytical formula for the spectral density of a nonparaxial polychromatic partially coherent (NPPC) standard or elegant Laguerre-Gaussian (LG) beam propagating through free space is derived. The evolution properties of the intensity distribution of a NPPC standard or elegant LG beam are studied numerically. It is found that the behavior of the intensity distribution of the NPPC standard or elegant LG beam is determined by the initial beam parameters (i.e., beam waist size, spatial coherence width and beam orders). Furthermore, we also study the influence of the initial beam parameters on the spectral change of a NPPC standard or elegant LG beam upon propagation in free space. It is shown that the on-axis spectral shift of a NPPC standard LG beam is almost the same with that of a NPPC elegant LG beam when the initial beam parameters of two type beams are identical. However, the difference between the off-axis spectral shift between a NPPC standard LG beam and a NPPC elegant LG beam is enhanced with the decrease of beam waist size or with the increase of spatial coherence width and beam mode orders.     

Propagation of the kurtosis parameter of Hermite-cosh-Gaussian beams

Starting from the propagation equation of Hermite-cosh-Gaussian (HChG) beams and the intensity moments definition, an analytical expression for the propagation of the kurtosis parameter of unapertured HChG beams passing through paraxial optical ABCD systems is derived and illustrated numerically. Special interesting cases are discussed, in particular, the kurtosis parameter of HChG beams at the waist plane is obtained readily from our general propagation expression.     

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.     

Focal shift in focused off-axial Hermite-cosh-Gaussian beams

Based on the encircled-power criterion, the focal shift of off-axial Hermite-cosh-Gaussian beams passing through an apertured lens is studied and illustrated with detailed numerical examples. The dependence of focal shift on the beam and optical system parameters is discussed.     

部分相干双曲余弦厄米高斯光束在非Kolmogorov大气湍流中的传输特性

基于广义Huygens-Fresnel原理和非Kolmogorov大气湍流折射率起伏谱密度函数, 采用Wigner分布函数的二阶矩方法, 推导出了在大气湍流中传输的部分相干双曲余弦厄米高斯光束束宽和M²因子的解析表达式. 研究表明: 相对束宽和归一化M²因子随传输距离的增大而增大; 光束阶数越大、相干长度越小、双曲余弦参数越小, 相对束宽和归一化M²因子受大气湍流影响越小; 相对束宽随束腰宽度的增大存在极大值, 在一定的相干长度范围内, 归一化M²因子随束腰宽度的增大存在极小值; 相对束宽和归一化M²因子随广义指数的变化均存在极大值, 随内尺度的增大而逐渐减小, 随外尺度的增大几乎没有变化.     

Free wave propagation in two-dimensional periodic plates

The propagation of flexural waves in a two-dimensional periodic plate which rests on an orthogonal array of equi-spaced simple line supports has been investigated. A type of plane wave motion has been considered. An energy method has been developed to predict the frequency of wave propagation in terms of the propagation constants. A Galerkin type of analysis has been used, incorporating assumed complex modes of wave motion for the identical rectangular elements of the periodic plate. Expressions for the frequency have been obtained firstly by using simple polynomial modes for the plate displacements, and then (alternatively) by using characteristics beam function modes. The use of these different modes has first been demonstrated by applying them to the analysis of wave propagation in periodic beams. A single polynomial mode which satisfies the geometric and wave-boundary conditions of the periodic plate element leads to an elegant expression relating the frequency and the wave propagation constants in the first propagation band. The frequencies so obtained compare well with those found from a multi-mode, characteristic beam function analysis. The latter involves much more algebra, is solved as an eigenvalue problem, and yields the frequencies in as many propagation bands as are desired. The bounding frequencies and corresponding wave motions in the first and higher propagation bands have been identified, and it has been shown that the propagation bands can overlap. Consideration has been given to one-dimensional “strip” structures which are equivalent to the two-dimensional plate when a plane wave in a general direction is propagating. Furthermore, it is shown that the natural frequencies of finite rectangular periodic plates can be obtained very simply from the results of the wave propagation analysis.     

Circular beams

A very general beam solution of the paraxial wave equation in circular cylindrical coordinates is presented. We call such a field a circular beam (CiB). The complex amplitude of the CiB is described by either the Whittaker functions or the confluent hypergeometric functions and is characterized by three parameters that are complex in the most general situation. The propagation through complex ABCD optical systems and the conditions for square integrability are studied in detail. Special cases of the CiB are the standard, elegant, and generalized Laguerre-Gauss beams; Bessel-Gauss beams; hypergeometric beams; hypergeometric-Gaussian beams; fractional-order elegant Laguerre-Gauss beams; quadratic Bessel-Gauss beams; and optical vortex beams.     

Observation of multivortex solitons in photonic lattices

We report on the first observation of topologically stable spatially localized multivortex solitons generated in optically induced hexagonal photonic lattices. We demonstrate that topological stabilization of such nonlinear localized states can be achieved through self-trapping of truncated two-dimensional Bloch waves and confirm our experimental results by numerical simulations of the beam propagation in weakly deformed lattice potentials in anisotropic photorefractive media.