Propagation properties of Airy–Gaussian beams in centrosymmetric photorefractive media

Based on the biased centrosymmetric photorefractive media, we investigate numerically the propagation and interaction properties of Airy–Gaussian beams. The single Airy–Gaussian beam forms the one-component breather with the help of the photorefractive nonlinearity. The interaction properties of two Airy–Gaussian beams can be controlled by adjusting the relative parameters, such as photorefractive nonlinearity, transverse distance and relative phase of two incident beams. The two-component breather with ladder structure can be observed for both the in-phase and out-of-phase cases when the self-acceleration property is balanced by the photorefractive nonlinearity. The one- or three-component breathers can be observed for the in-phase case only when the transverse distance takes a certain range.     

Propagation and far-field beam quality of M × N Hermite–Gaussian beams propagating through atmospheric turbulence

The analytical propagation equation of M×N Hermite–Gaussian (H–G) beams, which are combined incoherently and propagate through atmospheric turbulence, is derived, which enables us to study their propagation properties and far-field beam quality. The propagation of M×N Gaussian beams through atmospheric turbulence and M×N H–G beams in free space are treated as special cases of our general result. The power in the bucket (PIB), β-parameter and Strehl ratio are chosen as the parameters characterizing the beam quality in the far field. The dependence of PIB, β-parameter and Strehl ratio of M×N H–G beams through atmospheric turbulence on the refraction index structure constant C _n ², beam numbers M, N, mode indices m, n and separate distances x_d , y_d is illustrated numerically and interpreted physically. It is found that M×N H–G beams are less sensitive to the effects of turbulence than M×N Gaussian beams.     

Propagation of high-order Bessel–Gaussian beams through a hard-aperture misaligned optical system

In this paper, we consider the propagation of high-order Bessel–Gaussian beams (HBGBs) passing through a hard-aperture misaligned optical system. By expanding the hard-aperture function into a finite sum of complex Gaussian functions, a general propagating formula of HBGBs is derived in terms of the generalized diffraction integrals. Based on the derived formula, the diffraction properties of HBGBs propagating through a simple misaligned lens system are numerically illustrated. This method provides a convenient tool for studying the propagation and transformation properties of a high-order Bessel–Gaussian beam through an apertured misaligned optical system.     

Propagation properties of rectangular Laguerre–Gaussian-correlated Schell-model beams in atmospheric turbulence

Based on the extended Huygens–Fresnel principle and the second-order moments of the Wigner distribution function (WDF), the analytical expressions for the cross-spectral density (CSD) and the propagation factor of a rectangular Laguerre–Gaussian-correlated Schell-model (LGCSM) beam propagating in atmospheric turbulence are derived. The statistical properties, such as the average intensity, the spectral degree of coherence (SDOC) and the propagation factor, of a rectangular LGCSM beam in free space and atmospheric turbulence are comparatively analysed. It is illustrated that a rectangular LGCSM beam exhibits self-splitting and combing properties on propagation in atmospheric turbulence, and the self-splitting properties of such beam are closely related to its beam orders m and n, which is quite different from other self-splitting beams. In addition, the rectangular LGCSM beam has an advantage for reducing the turbulence-induced degradation compared with the conventional partially coherent beams.     

Representation and focusing properties of higher-order radially polarized Laguerre–Gaussian beams

A method based on higher-order Poincaré sphere was proposed to represent the states of polarization of higher-order radially polarized Laguerre–Gaussian (LG) vectorial beams (VBs). And the focusing properties of such LG beams of different radial orders focused by a high-NA lens were discussed. By tuning the ratio of the pupil radius to the waist of the incident beams, some cage-like or needlelike electric intensity field is generated in the focal region for several specific LG VBs with high order. Modulated by diffractive optical elements, the shape of the focal field shows novel behaviors such as splitting of cage-like modes, which provides potentially a more flexible control over micro-particles.     

Effective radius of curvature of partially coherent Hermite–Gaussian beams propagating through non-Kolmogorov turbulence

Based on the extended Huygens–Fresnel principle and non-Kolmogorov spectrum, the analytical expression for the effective radius of curvature of partially coherent Hermite–Gaussian (PCHG) beams propagating through non-Kolmogorov turbulence is derived, and the relative effective radius of curvature is used to describe the effect of turbulence on the effective radius of curvature. It is shown that the effective radius of curvature of PCHG beams depends on the beam and non-Kolmogorov turbulence parameters and on the propagation distance. The variation of relative effective radius of curvature with increasing generalized exponent parameter α of non-Kolmogorov turbulence is non-monotonic. The longer the propagation distance is, the larger the effect of turbulence on the effective radius of curvature of PCHG beams is. The effective radius of curvature of PCHG beams with shorter wavelength, smaller beam order, larger beam waist width or better spatial coherence is more affected by the non-Kolmogorov turbulence. The results are interpreted physically.     

A comparative study of standard and elegant Hermite–Gaussian beams propagating through turbulent atmosphere

Based on the extended Huygens–Fresnel principle and the definition of the second-order moments of the Wigner distribution function (WDF), the analytical expressions for the root-mean-square (rms) beam width and far-field divergence angle, curvature radius and M ²-factor of standard Hermite–Gaussian (SHG) and elegant Hermite–Gaussian (EHG) beams passing through turbulent atmosphere are derived and compared. It is shown that in turbulent atmosphere the far-field divergence angle of SHG and EHG beams is equal under the same conditions, but the rms beam width, curvature radius and the M ²-factor of SHG and EHG beams are different except for beam orders m?=?0 and m?=?1. The relative rms beam width, relative curvature radius and relative M ²-factor of SHG beams are less than those of EHG beams. Therefore, the conclusion that SHG beams are less influenced by turbulence than EHG beams can be drawn if we examine one of the above three relative beam parameters.     

Propagation of partially coherent Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system in a turbulent atmosphere

Based on the generalized Huygens–Fresnel integral, propagation of partially coherent Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system in a turbulent atmosphere was investigated. Analytical propagation formulae were derived for the cross-spectral densities of partially coherent Lorentz and Lorentz–Gauss beams. As an application example, the focusing properties of partially coherent Gaussian, Lorentz and Lorentz–Gauss beams in a turbulent atmosphere and in free space were studied numerically and comparatively. It is found that the focusing properties of such beams are closely related to the initial coherence length and the structure constant of turbulence. By choosing a suitable initial coherence length, a partially coherent Lorentz beam can be focused more tightly than a Gaussian or Lorentz–Gauss beam in free space or in a turbulent atmosphere with small structure constant at the geometrical focal plane.     

Propagation based on second-order moments for partially coherent Laguerre–Gaussian beams through atmospheric turbulence

Abstract

The Wigner distribution function (WDF) has been used to study the propagation properties of partially coherent Laguerre Gaussian (PCLG) beams through atmospheric turbulence. Based on the extended Huygens–Fresnel principle, an analytical formula of the propagation matrixes in terms of the second-order moments of the WDF for PCLG Beams in the receiving plane is derived. And then the analytical formulae for the curvature radii of PCLG Beams propagating in turbulence are given by the second-order moments of the WDF. The numerical results indicate that the curvature radius of PCLG Beams changes more rapidly in turbulence than that in the free space. The influence of the transverse coherence width and the beam waist width on the curvature radius of PCLG Beams is obvious, while the laser wavelength and the inner scale of turbulence have a slight effect. The study results may be useful for remote sensing and free space optical communications.     

The M 2 factor matrix of a paraxial Laguerre–Gaussian beam

The M ² factor matrix of a paraxial Laguerre–Gaussian beam has been derived based on the combination of the M ² factor definition and introduced quantities such as the coupled beam’s half width square, the coupled M ² factor and 2×2 rotational matrix, when the beam is rotated around the propagation z-axis by an arbitrary azimuth angle α. The beam’s half widths at different distances from the beam waist along the z-axis, the M ² factors of the beam with different orders, and the track of M ² factor matrix elements versus α have been obtained by numerical simulation and analytical derivation. Results illustrate that by using the invariability of the M ² factor matrix, the beam quality of the mode with an arbitrary rotational angle can be easily evaluated. Like the M ² factor matrix of the Hermite–Gaussian mode, this M ² factor matrix is meaningful in the evaluation of the beam quality of a two-dimensional astigmatic or asymmetric laser beam.     

The kurtosis parametric characterization of the passage of a standard Hermite–Gaussian beam and an elegant Hermite–Gaussian beam through a fractional Fourier transformation system with a spherically aberrated lens

The propagation characteristics of the kurtosis parameters of a standard Hermite–Gaussian (SHG) beam and of an elegant Hermite–Gaussian (EHG) beam, each passing through a fractional Fourier transformation (FRFT) system with a spherically aberrated lens, are studied in detail. Some numerical calculations are made by introducing an efficient algorithm, based on the Collins diffraction integral formula. The resulting graphs illustrate the striking difference between ideal FRFT systems and those with a spherically aberrated lens. The kurtosis parameters of both SHG and EHG beams passing through a type I Lohmann system with a spherically aberrated lens are seen to change with the fractional order periodically and the fundamental period is 4, but for type II the fundamental period is 2. Different values of spherical aberration coefficients affect the kurtosis parameters in greatly different ways. The values of the kurtosis parameters of a SHG beam passing through either type of Lohmann system with a spherically aberrated lens are no longer equal to those of an EHG beam, even when they have the same fractional orders and the same spherical aberration coefficients.     

A Hermite DRK interpolation-based collocation method for the analyses of Bernoulli–Euler beams and Kirchhoff–Love plates

A Hermite differential reproducing kernel (DRK) interpolation-based collocation method is developed for solving fourth-order differential equations where the field variable and its first-order derivatives are regarded as the primary variables. The novelty of this method is that we construct a set of differential reproducing conditions to determine the shape functions of derivatives of the Hermite DRK interpolation, without directly differentiating it. In addition, the shape function of this interpolation at each sampling node is separated into a primitive function possessing Kronecker delta properties and an enrichment function constituting reproducing conditions, so that the nodal interpolation properties are satisfied for the field variable and its first-order derivatives. A weighted least-squares collocation method based on this interpolation is developed for the static analyses of classical beams and plates with fully simple and clamped supports, in which its accuracy and convergence rate are examined, and some guidance for using this method is suggested.     

Evolution properties of the complex degree of coherence of a partially coherent Laguerre–Gaussian beam in turbulent atmosphere

Evolution properties of the complex degree of coherence of a partially coherent Laguerre–Gaussian beam (LGB) on propagation in free space and turbulent atmosphere are studied comparatively with the help of the general propagation formula for such beam. It is found that the behavior of the complex degree of coherence of a partially coherent LGB on propagation in turbulent atmosphere is much different from that in free space and is closely related to the initial beam parameters and the structure constant of the turbulent atmosphere. The distribution of the modulus of the complex degree of coherence of the partially coherent LGB finally becomes of Gaussian distribution at long propagation distance in turbulent atmosphere, and it becomes of Gaussian distribution more slowly with the increase of the mode orders, beam width and wavelength. Our results will be useful in long-distance free-space optical communications.     

Vectorial structure of Ince–Gaussian beam in the far field

By means of the vector angular spectrum representation of the electromagnetic beam and the method of stationary phase, the analytical vectorial structure of the Ince–Gaussian beam has been presented in the far field. The amplitude distributions of the Ince–Gaussian beam and its TE and TM terms are investigated in the far field. The extreme cases of the ellipticity parameter tending to infinity or zero are also considered. Although the vectorial structures of different Ince–Gaussian beams are apparently distinct, the ratios of the amplitude distributions of the TE and TM terms to the whole beam amplitude are independent of the parity and the values of the radial and angular elliptic mode numbers. This research reveals the abundant and interesting internal details of the Ince–Gaussian beam in the far field.     

Focusing of radially polarized Lorentz–Gauss beams with the power–exponent–phase vortex

Using the vector diffraction theory, the optical field of the focusing radially polarized Lorentz–Gauss beam with the power–exponent–phase vortex is derived. The normalized intensity distributions of the focusing radially polarized Lorentz–Gauss beam with the power–exponent–phase vortex are numerically demonstrated. The influences of the power order n and the topological charge m on the normalized intensity distribution are examined. The beam centre and the effective beam size, which are defined by the first- and the second-order moments of the intensity distribution, are the important parameters for focus. Therefore, the quantitative effects of the power order n and the topological charge m on the beam centre and the effective beam size are further investigated. This research is beneficial to the optical manipulation which is involved in the radially polarized Lorentz–Gauss beam with the power–exponent–phase vortex.     

Laguerre–Gaussian modes and the Wigner transform

Recent developments in laser physics have called renewed attention to Laguerre–Gaussian (LG) beams of paraxial light. In this paper we consider the corresponding LG modes for the two-dimensional harmonic oscillator, which appear in the transversal plane at the laser beam's waist. We see how they arise as Wigner transforms of Hermite–Gaussian modes, and we proceed to find a closed form for their own Wigner transforms, providing an alternative to the methods of Simon and Agarwal. Our main observation is that the Wigner transform intertwines the creation and annihilation operators for the two classes of modes.     

The vibrations of pre-twisted rotating Timoshenko beams by the Rayleigh–Ritz method

A modeling method for flapwise and chordwise bending vibration analysis of rotating pre-twisted Timoshenko beams is introduced. In the present modeling method, the shear and the rotary inertia effects on the modal characteristics are correctly included based on the Timoshenko beam theory. The kinetic and potential energy expressions of this model are derived from the Rayleigh–Ritz method, using a set of hybrid deformation variables. The equations of motion of the rotating beam are derived from the kinetic and potential energy expressions introduced in the present study. The equations thus derived are transmitted into dimensionless forms in which main dimensionless parameters are identified. The effects of dimensionless parameters such as the hub radius ratio, slenderness ration, etc. on the natural frequencies and modal characteristics of rotating pre-twisted beams are successfully examined through numerical studies. Finally the resonance frequency of the rotating beam is evaluated.     

Improvement in dynamic properties of laminated MWCNT-polystyrene composite beams via an integrated numerical–experimental approach

A new concept for the optimization of dynamic behavior of laminated nanocomposites is introduced where fiber orientation factor in continuous fiber-reinforced composites is replaced by different wt.% of carbon nanotubes (CNTs) in each layer. First, at a design concept level, an optimum distribution of multi-walled CNTs (MWCNTs) through the thickness of a typical cantilever beam is sought to achieve its highest fundamental natural frequency for a given weight percent of MWCNTs. This is done using a finite element (FE) model in ABAQUS along with a user-defined Python code. Next, based on the obtained optimum distribution, actual laminated MWCNT/polystyrene (PS) composite beams were fabricated and their effective stiffness, fundamental natural frequencies and damping ratios were measured through static deflection and free vibration tests. It was found that the optimum distribution of MWCNTs resulted in an increase of 21.9% and 10.4% in the effective Young’s modulus and the fundamental damped natural frequency values, respectively, which were almost two-fold higher than those of a beam with a uniform MWCNT distribution. In addition, compared to a pure polymer beam, 38.9% and 27.8% improvements in the damping ratio of the uniformly and optimally distributed MWCNT polymer composite beams were achieved.