Propagation of apertured Bessel beams

The propagation features of several apertured Bessel beams are numerically calculated. The calculations show that the relations of axial intensity versus propagation distance are similar to the radial distribution of the aperture functions, which may be helpful in choosing the proper aperture functions in experiments.     

Propagation properties of cosh-Airy beams

We theoretically investigate the propagation properties of cosh-Airy beams, which can be considered as a superposition of two Airy beams with different decay factors. We find that the field distribution of cosh-Airy beams is the same as that of Airy beams. Moreover, we find that the evolution of cosh-Airy beams is determined by the parameters of the cosh modulation function, in addition to the transverse scale factor and decay factor of the Airy beams. Our results demonstrate a possible method of manipulating Airy beams in free space. They can also be extended to the study of the propagation properties of cosine-Airy beams (or sine-Airy beams).     

Propagation of broken stable beams

Beams that are stable during propagation in a homogeneous and isotropic medium are widely used in optical particle manipulation, communication, metrology, etc. In real life scattering, absorption and diffraction by obstacles produce changes of the beam structure. The propagation of these broken beams is analysed. It is shown that, similarly to the Bessel beams, they are able to reconstruct their original shape after being disturbed by obstacles of relatively small size.     

复宗量双曲正弦高斯光束的传输特性

对复宗量双曲正弦高斯(EshG)光束的一些基本特性进行了研究。由Collins公式出发推导出了光束通过近轴ABCD光学系统传输的场分布函数,根据二阶矩的定义和光束在近轴ABCD光学系统中的传输公式推导出了光束通过近轴ABCD光学系统光斑尺寸的解析表达式。得到了复宗量双曲正弦高斯光束在自由空间传输时束腰宽度和位置的解析表达式以及M2因子的表达式。对光强分布、光斑尺寸、束腰宽度及其位置和M2因子进行了数值计算,并对计算结果作了分析。本文所得结果具有广泛的意义,因为正弦高斯光束、双曲正弦高斯光束和复宗量正弦高斯光束均可视为其特例。     

Propagation of flat-topped multi-Gaussian laser beams

The multi-Gaussian beam shape is proposed as a model for aperture functions and laser beam profiles that have a nearly flat top but whose sides decrease continuously. Beams and apertures of this type represent a simple, elegant, and intuitive alternative to super-Gaussian beams, which are important in a number of applications such as laser resonator design. Analytical formulas are developed for the propagation of these beams through free space and optical systems representable by ABCD matrices.     

Propagation of polychromatic Gaussian beams through thin lenses

The transformation by a lens of a polychromatic laser beam composed of on-axis superposed monochromatic TEM00 Gaussian modes in the paraxial approximation is studied. The chromatic aberrations are described by allowing the waist position on the z axis and the Rayleigh range to depend on wavelength. The beam radius, the far-field divergence, the Rayleigh range, the beam product, the beam propagation factor, and the kurtosis parameter are calculated. The relationship between the fourth-order and the second-order moments of Hermite-Gaussian and Laguerre-Gaussian modes is obtained and is used for calculating kurtosis parameter. The results are generalized to polychromatic modes of higher orders. It is shown that the on-axis superposition of monochromatic TEM00 modes with no chromatic aberration is leptokurtic.     

Propagation of partially coherent beams through particulate media

When propagating through particulate media, optical beams are degraded owing to scattering. We found that the ratio between the width of the distorted beam and the width of the initial beam decreases when the spatial coherence of the incident beam is reduced. These experimental observations are well described within the paraxial approximation of the transport theory.     

Propagation factors of laser array beams in turbulent atmosphere

The propagation factors of phased locked laser array beams of radial and rectangular symmetries in a turbulent atmosphere are investigated based on the extended Huygens–Fresnel integral and the Wigner distribution function. Analytical propagation formulae for the propagation factors are derived and numerical examples are illustrated. We find that unlike their propagation invariant properties in free space, the propagation factors of laser array beams increase when propagating in turbulent atmosphere, and are closely related to the parameters of initial beams and the atmosphere.     

Propagation of Helmholtz-Gauss beams in absorbing and gain media

The propagation of Helmholtz-Gauss beams in media exhibiting loss or gain is studied. The general expressions for the field propagation, the time-averaged power on propagation, the trajectory of the beam centroid, the beam spreading, the nondiffracting distance, and the far field are derived and discussed. Explicit expressions of these parameters for Bessel-Gauss and cosine-Gauss beams are included. The general expressions can be applied straightforwardly to describe the propagation of Mathieu-Gauss and parabolic-Gauss beams in complex media as well.     

Propagation properties of flat-topped beams in a turbulent atmosphere

Based on the extended Huygens–Fresnel principle and the quadratic approximation for phase structure function, the analytical formulas for the average irradiance of the phase-locked and the non-phase-locked radial modified flat-topped array beams in turbulence have been derived. The propagation properties of radial modified flat-topped array beams have been investigated. It is found that the irradiance distributions of the radial flat-topped array beams propagating in turbulence are different from those in free space. The propagation properties depend on the refractive index structure constant, the beam order and the array radius. The power in the bucket (PIB) for the array beams increases with the decrease of the array radius for a given propagation distance. For a certain bucket radius, there exhibits a PIB maximum upon propagation. The optimal propagation distance becomes shorter along with the increasing refractive index structure constant, and the PIB maximum increases with bucket radius.     

Propagation of Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation

On the basis of the vectorial Rayleigh-Sommerfeld formulas and by means of the relation between Hermite and Laguerre polynomials, the analytical expressions for the propagation of the Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) beams beyond the paraxial approximation are derived, with the corresponding far-field propagation expressions and that for the Gaussian beams being given as special cases of the results. Some detailed comparisons of our results with the expansion series and paraxial expressions are made, which show the advantages of our results over the expansion series. With the results obtained, some typical intensity patterns of nonparaxial HG and LG beams are shown.     

Propagation of partially coherent flat-topped beams through a turbulent atmosphere

Based on the modified beam model for flat-topped beams and the Schell model for partially coherent light, an expression for partially coherent flat-topped (PCFT) beams has been proposed. The propagation characteristics of PCFT beams with circular symmetry through a turbulent atmosphere have been studied. By using the generalized Huygens-Fresnel integral and Fourier transform method, the expressions for the cross-spectral density function and the average intensity have been given and the analytical expression for the root-mean-square width has been derived. The effects of the beam order, the spatial coherence, and the turbulent parameter on the intensity distributions and beam spreading have been discussed in detail. Our results show that the on-axis intensity of the beams decreases with increasing turbulence and decreasing coherence of the source, whereas the on-axis intensity of the beams in the far field decreases slightly with increasing beam order. The relative spreading of PCFT beams is smaller for beams with a higher order, a lower degree of global coherence of the source, a larger inner scale, and a smaller outer scale of the turbulence.     

Propagation of anomalous hollow beams through a circular-apertured optical system

In this paper, we consider the propagation of anomalous hollow beams through a circular-apertured paraxial ABCD optical system. Based on the Huygens–Fresnel integral and the expansion of the circular aperture function into a finite sum of Gaussian functions, the approximation analytical expression of an anomalous hollow beam passing through a circular-apertured paraxial ABCD optical system is derived. Based on the approximation analytical expression, the propagation properties in free space are numerically illustrated. This method provides a convenient tool for studying the propagation and transformation properties of anomalous hollow beams passing through a circular-apertured paraxial ABCD optical system.     

Propagation of hollow Gaussian beams through apertured paraxial optical systems

On the basis of the generalized Collins formula and the expansion of the hard-aperture function into a finite sum of complex Gaussian functions, an approximate analytical formula for a hollow Gaussian beam propagating through an apertured paraxial stigmatic (ST) ABCD optical system is derived. Some numerical examples are given. Furthermore, by using a tensor method, we derive approximate analytical formulas for a hollow elliptical Gaussian beam propagating through an apertured paraxial general astigmatic ABCD optical system and an apertured paraxial misaligned ST ABCD optical system. Our results provide a convenient way for studying the propagation and transformation of a hollow Gaussian beam and a hollow elliptical Gaussian beam through an apertured general optical system.     

Propagation properties of ultrashort pulsed bessel beams in dispersive media

A detailed study of ultrashort pulsed Bessel beams in linear dispersive media is performed. The spatial and temporal parts of pulsed Jn beams are separable in dispersive media, provided that the parameter a is independent of frequency omega. The spatial part keeps the Jn shape unchanged during propagation. The temporal evolution behavior of pulsed Jn beams depends on the material's dispersion and diffraction. The pulses can be broadening and become negatively chirped while propagating in anomalous dispersive media. In normal dispersive media, the pulses can be broadening and positively or negatively chirped; even dispersion-free propagation can be achieved if the beam and material parameters are suitably chosen. The condition under which higher-order dispersive effects can be neglected is also discussed.     

Propagation of elliptical Gaussian beams modulated by an elliptical annular aperture

An analytical propagation expression for a generalized astigmatic elliptical Gaussian beam (EGB) modulated by an elliptical annular aperture and passing through an axially nonsymmetrical optical system is obtained by the use of vector integration. The derived analytical results provide more convenience for studying the propagation and transformation of EGBs than the usual method of using the diffraction integral directly, and the efficiency of the numerical calculation is significantly improved. Some numerical simulations are illustrated for the propagation properties of EGBs passing through a free space with an elliptical annular aperture, an elliptical screen, or an elliptical aperture. Further extensions are also pointed out.     

Propagation of doughnut-shaped super-Gaussian beams,convolution theorem and Hankel transform

The most direct and simple method to calculate analytically the propagation in the far field of a coherent beam with a rectangular symmetry and a super-Gaussian-like irradiance profile involves the use of the convolution theorem. In this paper we extend this method to a circularly symmetric beam, and hence from the Fourier transform to the zero-order Hankel transform. Then, we examine the more complex case of super-Gaussian-like beam irradiance shapes with an axial shadow, as those emitted by unstable resonators with diffraction output coupling or with a variable-reflectivity mirror. The analytical results obtained by approximating the super-Gaussian-like beam with a convolution function and applying the Hankel transform are in excellent agreement with those obtained by numerical simulations.     

Propagation properties of partially coherent anomalous hollow beams in uniaxial crystals

Analytical propagation formulae for partially coherent anomalous hollow beams in uniaxial crystals are derived. Paraxial propagation of partially coherent anomalous hollow beams in uniaxial crystals orthogonal to the optical axis is investigated based on the beam propagation equations. The propagation properties of partially coherent anomalous hollow beams in uniaxial crystals and in free space are studied numerically and comparatively. It is found that the propagation properties of partially coherent anomalous hollow beams in uniaxial crystals behave very differently from those in free space and are closely determined by the parameters of the uniaxial crystals and the initial coherence width. The uniaxial crystals provide an effective way for generating astigmatic beams.     

Propagation of elliptical Gaussian beams in apertured and misaligned optical systems

On the basis of the fact that a hard-edged elliptical aperture can be expanded approximately as a finite sum of complex Gaussian functions in tensor form, an analytical propagation expression for an elliptical Gaussian beam (EGB) through a misaligned optical system with an elliptical aperture is derived by use of vector integration. The approximate analytical results provide more convenience for studying the propagation and transformation of EGBs than the usual way by using a diffraction integral directly, and the efficiency of numerical calculation is improved. Some numerical simulations are illustrated for the propagation properties of EGBs through apertured optical transforming systems with misaligned thin lenses.