Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis

The properties of Lorentz beams propagating in uniaxial crystals orthogonal to the optical axis are studied. The diffraction field components and effective beam sizes of the Lorentz beams are derived in analytical forms. Numerical results show that, upon propagating in uniaxial crystals, a Lorentz beam loses its initial Lorentz type distribution. Also, after propagating for sufficient distances, the transverse components would finally convert into a specific four-petal profile with an axial shadow, which may find applications in the optical trapping of particles. It also shows that the Lorentz beam parameters w_0x, w_0y and the ratios of refractive indices have strong influences on the diffraction field components and on the effective beam sizes when propagating in uniaxial crystals.     

Propagation of confluent hypergeometric beam through uniaxial crystals orthogonal to the optical axis

Based on the theory of paraxial electromagnetic beams in anisotropic medium, propagation properties of a linearly polarized confluent hypergeometric (HyG) beam through uniaxial crystals orthogonal to the optical axis are investigated. Analytical expressions are derived for the resultant optical field components. Effects of separate beam parameters together with the ratio of refractive indices of crystals on transversal intensity distributions are revealed by numerical calculations, respectively. It is shown that the beam profile finally converts into an elliptical shape due to influences of anisotropic crystals. Moreover, it is also indicated that through suitable selection of parameters, types of the optical vortices of resultant field would change from a central dark spot to a slight dark ring or an axial shadow during the propagation. These numerical results may provide alternative advantages to the trapping of tiny particles by utilizing HyG beams in practical experiments.     

Propagation of Ince-Gaussian beams in uniaxial crystals orthogonal to the optical axis

An analytical propagation expression of an Ince-Gaussian beam in uniaxial crystals orthogonal to the optical axis is derived. The uniaxial crystal considered here has the property of the extraordinary refractive index being larger than the ordinary refractive index. The Ince-Gaussian beam in the transversal direction along the optical axis spreads more rapidly than that in the other transversal direction. With increasing the ratio of the extraordinary refractive index to the ordinary refractive index, the spreading of the Ince-Gaussian beam in the transversal direction along the optical axis increases and the spreading of the Ince-Gaussian beam in the other transversal direction decreases. The effective beam size in the transversal direction along the optical axis is always larger than that in the other transversal direction. When the even and odd modes of Ince-Gaussian beams exist simultaneously, the effective beam size in the direction along the optical axis of the odd Ince-Gaussian beam is smaller than that of the even Ince-Gaussian beam in the corresponding direction, and the effective beam size in the transversal direction orthogonal to the optical axis of the odd Ince-Gaussian beam is larger than that of the even Ince-Gaussian beam in the corresponding direction.     

Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis

Propagation properties of higher-order cosh-Gaussian (HOCG) beams in uniaxial crystals orthogonal to the optical axis are studied. Analytical formulae for electric fields and kurtosis parameters are derived. Results show that a HOCG beam with larger acentric parameters or beam orders would better preserve its intensity profile when it propagates in uniaxial crystals. At arbitrary propagation distance flat-topped beams can be shaped by selecting appropriate acentric parameters. It is also indicated that kurtosis parameters τ_x and τ_y both increase when acentric parameters take larger values, however, they show different behaviors due to effects of anisotropic crystals.     

Propagation of Gaussian array beams in uniaxial crystals orthogonal to the optical axis

The propagation of phase-locked and non-phase-locked Gaussian array beams in uniaxial crystals orthogonal to the optical axis is investigated. Analytical formulas are derived and the propagation properties of phase-locked and non-phase-locked Gaussian array beams are illustrated by numerical examples. Our results show that for non-phase-locked combination, the Gaussian array beams involve into an elliptical Gaussian-like distribution when the propagation distance is far enough, while for phase-locked combination, the Gaussian array beams involve into a central peak distribution and some small peaks are around the center peak when the propagation distance is far enough.     

Propagation of partially polarized, partially coherent beams in uniaxial crystals orthogonal to the optical axis

The propagation of partially polarized and partially coherent beams in uniaxial crystals orthogonal to the optical axis is investigated. The analytical formulae for the elements of the cross-spectral density matrix of partially polarized and partially coherent beams propagating through uniaxial crystals orthogonal to the optical are derived. The numerical results show that the degree of polarization decreases with the increase of the ratio of extraordinary to ordinary refractive indices at a certain propagation distance, and the influence of uniaxial crystals on the degree of coherent is not so evident. And the beams spread more rapidly in the direction parallel to the optical axis than orthogonal to the optical axis in positive crystal with the propagation distance increasing.     

Propagation and the kurtosis parameter of Gaussian flat-topped beams in uniaxial crystals orthogonal to the optical axis

The analytical formulae for the Gaussian flat-topped beams propagating in uniaxial crystals orthogonal to the optical axis are derived. The numerical results show that the Gaussian flat-topped beams spread at different rates in the directions parallel and orthogonal to the optical axis due to anisotropic crystals. An analytical expression for the kurtosis parameter of the Gaussian flat-topped beams propagating in uniaxial crystals is derived and illustrated with numerical examples. It is shown that the evolution of the kurtosis parameters K_x and K_y depend on the ratio of extraordinary to ordinary refractive indices.     

椭圆高斯光束在单轴晶体中垂直于光轴的传输特性

基于光束在各向异性介质中的傍轴矢量传输理论,导出了椭圆高斯光束在各向异性单轴晶体中垂直于光轴的传输公式,并利用此解析式进行数值计算和分析,研究了各向异性介质对x方向偏振的椭圆高斯光束偏振特性的影响. 研究表明,晶体的各向异性对光束的偏振和对称性有较大影响. 关键词： 椭圆高斯光束 单轴晶体 各向异性 传输     

Nonparaxial propagation properties of the chirped Airy Gaussian vortex beams in uniaxial crystals orthogonal to the optical axis

In this article, we investigate the nonparaxial propagation properties of the chirped Airy Gaussian vortex(CAiGV)beams in uniaxial crystals orthogonal to the optical axis analytically and numerically. We discuss how the linear chirp parameters, the quadratic chirp parameters, and the Gaussian factors influence the nonparaxial propagation dynamics of the CAiGV beams. The intensity, the energy flow, the beam center, and the angular momentum of the CAiGV beams are deeply investigated. It is shown that the Gaussian factors have a great effect on the intensity and the centroid positions of the CAiGV beams. With the Gaussian factors increasing, the intensity of CAiGV beams decreases rapidly. The main lobes of the transverse intensity distribution of the CAiGV beams are similar to triangles.     

Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals

Based on the paraxial vectorial theory of beams propagating in uniaxially anisotropic media, we have derived the analytical propagation equations of hollow Gaussian beams (HGBs) in uniaxial crystals, and given the typical numerical example to illustrate our analytical results. Due to the anisotropy crystals, the ordinary and extraordinary beams originated by incident HGBs propagate with different diffraction lengths, thus the linear polarization state and axial symmetry of incident HGBs do not remain during propagating in crystals.     

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.     

Wigner distribution function matrix of electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis

The Wigner distribution function matrix of electromagnetic beams propagating in uniaxial crystals is introduced, and its analytical propagation expressions through uniaxial crystals orthogonal to the optical axis are derived. The application of the Wigner distribution function matrix of the electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis is illustrated with the stochastic electromagnetic beams.     

Propagation of a Pearcey beam in uniaxial crystals

An analytical propagation expression of a Pearcey beam in uniaxial crystals orthogonal to the optical axis is derived.The propagations of the Pearcey beam in the tourmaline and the quartz are investigated. The phase distribution and the angular momentum of the Pearcey beam in the tourmaline are also performed. The result shows that the positions of the auto-focusing and the inversion simply relate to the extraordinary refractive index of the crystals. In other words, we can choose the suitable crystals to adjust the positions of auto-focusing and inversion of the Pearcey beam to meet the actual needs.     

Propagation of an Airy vortex beam in uniaxial crystals

The propagation dynamics of an Airy beam superimposed with a unit topological charge optical vortex is investigated analytically and numerically in uniaxial crystals orthogonal to the optical axis. Upon propagation, the Airy vortex beam is mainly dependent on the ratio of the extraordinary refractive index to the ordinary refractive index. Due to the anisotropic effect of the crystals, the acceleration of an Airy vortex beam in the transversal direction along the optical axis is more rapidly than that in the other transversal direction, which is alike to a conventional Airy beam but with twice the vortex acceleration velocity.     

Nonparaxial propagation of Hermite-Laguerre-Gaussian beams in uniaxial crystal orthogonal to the optical axis

Analytical expressions for the three components of the nonparaxial propagation of a Hermite-Laguerre-Gaussian （HLG） beam in uniaxial crystal orthogonal to the optical axis are derived. The intensity distribution of an HLG beam and its three components propagating in a uniaxial crystal orthogonal to the optical axis are demonstrated by numerical examples. Although the y and z components of an HLG beam in the incident plane are both equal to zero, they emerge upon propagation inside the uniaxial crystal. Moreover, the beam profile of the x component is relatively stable and the beam profiles of the y and z components have the same evolution law. If the ratio of the extraordinary refractive index to the ordinary refractive index is larger than unity, the beam profile of the HLG beam is elongated in the x direction and generally rotates clockwise. Otherwise, the beam profile of the HLG beam is elongated in the y direction and generally rotates anticlockwise. This research is beneficial to the optical trapping and nonlinear optics involved in the rotation of a beam profile.     

Propagation properties of anomalous hollow beams in a turbulent atmosphere

Propagation of circular and elliptical anomalous hollow beams in a turbulent atmosphere is investigated in detail. Based on the extended Huygens–Fresnel integral, analytical formulae for the average irradiance of circular and elliptical anomalous hollow beams propagating in a turbulent atmosphere are derived. The irradiance and spreading properties of circular and elliptical anomalous hollow beams in a turbulent atmosphere and in free space are studied numerically. It is found that circular and elliptical anomalous hollow beams at short propagation distance in turbulent atmosphere have similar propagation properties to those of free space, while at long propagation distance, circular and elliptical anomalous hollow beams eventually become circular Gaussian beams in a turbulent atmosphere, which is much different from their propagation properties in free space. The conversion from an anomalous hollow beam to a circular Gaussian beam becomes quicker and the beam spot spreads more rapidly for a larger structure constant, a shorter wavelength and a smaller waist size of the initial beam.     

Nonparaxial propagation of radially polarized chirped Airy beams in uniaxial crystal orthogonal to the optical axis

The nonparaxial propagation of radially polarized chirped Airy beams(RPCAiBs) in uniaxial crystal orthogonal to the optical axis is analytically studied. The effects of the first and the second order chirp factors, and the ratio of the extraordinary refractive index to the ordinary refractive index on the nonparaxial evolution of RPCAi Bs in different observation planes are investigated in detail. The results show that when one parameter changes, different components behave differently, and even the same component has various behavior in different directions. The initial slope of RPCAi Bs in the x-direction varies more with the first order chirp factor than that in the y-direction. Meanwhile, with the second order chirp factor becoming larger, the depth of the focus of the y-component in the x-direction decreases while that in the y-direction has no difference. In addition, the different ratios of the extraordinary refractive index to the ordinary refractive index change the difference of the depth of the focus between the x-and the y-components.     

Propagation properties of a non-paraxial hollow Gaussian beam

Based on the non-paraxial vectorial moment theory of beam propagation, the analytical expressions of the M² factors for a non-paraxial hollow Gaussian beam (HGB) have been derived. The analytical formulae are further simplified for paraxial and highly non-paraxial cases. The beam waists, the divergence angles and the beam propagation factors are also depicted as functions of the parameter w₀. The divergence angles will not exceed the maximum value of 90°. When w₀ is within the scale of one time of light wavelength, the TE polarization results in the different beam propagation factors in the two transverse directions and the beam propagation factors first increase and then decrease. When w₀ is large enough, the beam propagation factors are determined only by the beam order.     

Propagation of hollow Gaussian beam through a misaligned first-order optical system and its propagation properties

Propagation properties of hollow Gaussian beam through a misaligned first-order ABCD system is studied using the generalized Huygens-Fresnel diffraction integral, augmented matrix. It is shown that, as a hollow Gaussian beam passes through the misaligned first-order ABCD system, the beam shape is not preserved, the out-put beams have differences when passing different misaligned optical systems. We can adjust the size of dark region through adjusting the misaligned transverse vector E.