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.     

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 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 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 properties of anomalous hollow beam in uniaxial crystals orthogonal to the optical axis

The analytical formulae for anomalous hollow beam propagating in uniaxial crystals orthogonal to the optical axis are derived. The numerical results show that the anomalous hollow beam spreads at different rates in the directions along and orthogonal to the optical axis. The beam spreads more rapidly in the direction along the optical axis than orthogonal to the optical axis in positive crystal (n_e/n_o>1), and the beam spreads more rapidly in the direction orthogonal to the optical axis than along the optical axis in negative crystal (n_e/n_o<1).     

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     

Kurtosis parameter of truncated standard and elegant Laguerre–Gaussian beams propagating through ABCD optical systems

On the basis of the propagation equation of truncated standard and elegant Laguerre–Gaussian beams, the closed-form propagation expressions for the kurtosis parameter K of standard and elegant Laguerre–Gaussian beams passing through apertured optical systems are derived, it is shown that the kutosis parameter K of standard and elegant Laguerre–Gaussian beams depend on order p and index m of Laguerre polynomial, the beam-truncated parameter δ and the generalized Fresnel number F. Detailed numerical examples are given to illustrate the analytical results.     

Kurtosis parameters of super Lorentzben Gauss beams through a paraxial and real ABCD optical system

Based on the propagation equation of higher-order intensity moments, analytical propagation expressions for the kurtosis parameters of a super Lorentz-Gauss (SLG) SLG₀₁ beam through a paraxial and real ABCD optical system are derived. By replacing the parameters in the expressions of the kurtosis parameters of the SLG₀₁ beam, the kurtosis parameters of the SLG₁₀ and SLG₁₁ beams through a paraxial and real ABCD optical system can be easily obtained. The kurtosis parameters of an SLG₀₁ beam through a paraxial and real ABCD optical system depend on two ratios. One is the ratio of the transfer matrix element B to the product of the transfer matrix element A and the diffraction-free range of the super-Lorentzian part. The other is the ratio of the width parameter of the super-Lorentzian part to the waist of the Gaussian part. As a numerical example, the properties of the kurtosis parameters of an SLG₀₁ beam propagating in free space are illustrated. The influences of different parameters on the kurtosis parameters of an SLG₀₁ beam are analysed in detail.     

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 properties of beams generated by Gaussian mirror resonator in uniaxial crystals

Based on the paraxial vectorial theory of beams propagating in uniaxially anisotropic media, we have derived the analytical propagation equations of beams generated by Gaussian mirror resonator (GMR) 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 beams generated by GMR propagate with different diffraction lengths, thus the linear polarization state and axial symmetry of the incident beams generated by GMR do not remain during propagating in crystals.     

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.     

Characteristics of paraxial propagation of a super Lorentz—Gauss SLG01 mode in uniaxial crystal orthogonal to the optical axis

Analytical propagation expression of a super Lorentz-Gauss（SLG） 01 mode in uniaxial crystal orthogonal to the optical axis is derived.The SLG 01 mode propagating in uniaxial crystal orthogonal to the optical axis mainly depends on the ratio of the extraordinary refractive index to the ordinary refractive index.The SLG 01 mode propagating in uniaxial crystals becomes an astigmatic beam.The beam spot of the SLG 01 mode in the uniaxial crystal is elongated in the x-or y-direction,which is determined by the ratio of the extraordinary refractive index to the ordinary refractive index.With the increase of the deviation of the ratio of the extraordinary refractive index to the ordinary refractive index from unity,the elongation of the beam spot also augments.In different observation planes,the phase distribution of an SLG 01 mode in the uniaxial crystal takes on different shapes.With the variation of the ratio of the extraordinary refractive index to the ordinary refractive index,the phase distribution is elongated in one transversal direction and is contracted in the other perpendicular direction.This research is beneficial to the practical applications of an SLG mode.     

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.     

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.     

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.     

Propagation properties of three-dimensional flattened Gaussian beams in uniaxially anisotropic crystals

Starting from the paraxial vectorial theory of beams propagating in uniaxially anisotropic media, the analytical propagation equations of three-dimensional flattened Gaussian beams (FGBs) in uniaxial crystals are derived and illustrated numerically. It is shown that, due to the anisotropy of crystals, the ordinary and extraordinary beams originated by incident FGBs propagate with different Rayleigh lengths, thus the linear polarization state and axial symmetry of incident FGBs do not remain during propagation in crystals.     

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.     

M factor and kurtosis parameter of super-Gaussian beams passing through an axicon

The effects of axicons on the M² factor and kurtosis parameter of super Gaussian beams(SGBs) are studied in detail. The closed-form expression for the M² factor of SGBs after passing through an axicon is derived, and the reason why the convergent and divergent axicons give rise to the same increase of the M² factor is explained physically from the similarity of the resulting irradiance distributions. The analytical propagation equation of the K parameter of SGBs passing through an axicon followed by a paraxial optical ABCD system is given, some interesting special cases are discussed. In particular, it is found that even for the Gaussian special case theK parameter is no longer a constant due to the effect of the axicon. Numerical examples are also given to illustrate our analytical results.     

Propagation of Airy Gaussian vortex beams in uniaxial crystals

The propagation dynamics of the Airy Gaussian vortex beams in uniaxial crystals orthogonal to the optical axis has been investigated analytically and numerically. The propagation expression of the beams has been obtained. The propagation features of the Airy Gaussian vortex beams are shown with changes of the distribution factor and the ratio of the extraordinary refractive index to the ordinary refractive index. The correlations between the ratio and the maximum intensity value during the propagation, and its appearing distance have been investigated.     

Irradiance-moments characterization of off-axis Gaussian beam combinations by means of the Wigner distribution function

The irradiance-moments characterization of off-axis Gaussian beam combinations is presented in use of the Wigner distribution function (WDF). Analytical expressions for the propagation through an optical paraxial ABCD system, beam propagation factor (M² factor), beam width, far field divergence angle and kurtosis parameter of the resultant beam are derived and illustrated with specific numerical examples. It is reexamined that a flat-topped irradiance profile of the resultant beam can be achieved at a certain plane by appropriately choosing beamlet parameters. As compared with previous work, our results show the advantage of using the WDF and irradiance-moments methods.