Propagation properties of vector Mathieu-Gauss beams

The vector Helmholtz-Gauss (vHzG) beam is known as a general family of localized vector beam solutions of the Maxwell equations in the paraxial limit and the vector Mathieu-Gauss beam constitutes its version in elliptic cylindrical coordinates system. In this work, starting from the expansion of the scalar Mathieu-Gauss beam in term of Bessel-Gauss beams, we give a general expression of vector Mathieu-Gauss beams in cylindrical coordinates. Within the frame work of the Collins diffraction integral formula we derive the analytical expressions of transverse vector Mathieu-Gauss beams through an axisymmetric ABCD optical system. Some numerical calculations are performed to illustrate the propagation of the vector Mathieu-Gauss beam in free space and through a simple lens system. The results are analyzed and discussed.     

Flat-topped Mathieu-Gauss beam and its transformation by paraxial optical systems

In order to avoid the severe axial intensity oscillations which appear when an ideal Mathieu beam is truncated, we propose to modulate it by a flattened multi-Gaussian envelope. The obtained beam, which is referred as flattened Mathieu-Gauss (FTMG) beam, can be expanded into a finite series of Mathieu-Gauss beams with various waists. The propagation study of this beam reveals that the axial intensity is unchanged within a certain propagation distance as for super-Gaussian-Bessel beams or Gori’s flattened Bessel beams. By using Collins formula, we derive closed-form expressions of FTMG beam propagating through a paraxial axisymmetric ABCD optical system. Some numerical calculations and discussions are also given.     

Focal shift and focal switch of flat-topped Mathieu-Gaussian beams passing through an apertured lens system

By introducing a hard aperture function into a finite sum of complex Gaussian functions, an approximate analytical expression predominating the distribution of axial intensity for the flat-topped Mathieu-Gauss (FTMG) beams passing through a system with the aperture and lens separated has been derived. The focal shift and the focal switch effect of FTMG beams passing through the system is studied in detail. Numerical calculations have shown that the position of real focal plane is not coincident with the geometrical focus but is somewhat shifted toward the lens. The focal shift and focal switch of FTMG beams take place when the relative separation s/f = 1 by a suitable choice of beam parameter and truncation parameter, for example, the beam parameter is smaller than its corresponding critical value or the truncation parameter is between its two corresponding critical values.     

Four-petal Gaussian beams and their propagation

A new form of laser beams called four-petal Gaussian beams is introduced. Based on the Collins integral, two kinds of analytical propagation expressions for this new kind of beams through a paraxial ABCD optical system are derived. The propagation properties of the four-petal Gaussian beams are studied and illustrated with numerical examples. At the source plane the beam has four-petals; the space among the petals is determined by the beam order. In the far field the beam evolves into a number of mirror symmetric petals and the petals of higher order beams can be equally spaced.     

Changes in the spectrum of twist anisotropic Gaussian Schell-model beams propagating through turbulent atmosphere

Based on the extended Huygens-Fresnel principle, the spectrum of twist anisotropic Gaussian Schell-model (TAGSM) beams propagating through turbulent atmosphere is derived analytically by using the partially coherent complex curvature tensor. The relative spectral shift of TAGSM beams propagating through turbulent atmosphere is closely related with the strength of atmospheric turbulence, the beam’s parameter and the radial coordinate. The on-axis spectral shift of TAGSM beams propagating through turbulent atmosphere changes from blue shift to red shift with the increasing of the propagation distance z, and at a certain propagation distance z, there exists a rapid transition of the spectrum at the critical position r_c.     

Beam propagation factor of partially coherent beams in gain or absorbing media

The definition of second-order intensity moments in the spatial domain and spatial frequency domain has been generalized for the case that the linear gain or absorbing media are included, where the wave number is generally complex. The formula for beam propagation M²-factor of partially coherent beams in linear gain or absorbing media has been given. The partially coherent flat-topped Schell-model beam is taken as an example. The closed-form expression for the beam propagation M²-factor of partially coherent flat-topped beam in gain or absorbing media has been derived. The changes of the M²-factor in media have been discussed with numerical examples. It can be shown that the M²-factor of flat-topped Schell-model beams in gain or absorbing media depends on the coherent parameter β, the coherent length σ₀, the beam order M, the propagation distance B, the imaginary part of the wave number K_i, as well as the real part of the wave number K_r.     

Propagation properties of partially coherent Laguerre-Gaussian beams in turbulent atmosphere

To study the propagation properties of partially coherent Laguerre-Gaussian (PLG) beams through turbulent atmosphere, the analytical formulas are derived for the angular width and the beam-propagation factor (M²-factor) of PLG beams by using the extended Huygens-Fresnel principle and the second-order moments of the Wigner distribution function (WDF). The corresponding numerical results are also calculated. When propagation distance increases, the angular width is found to spread faster for PLG beams with higher beam order, smaller correlation length and bigger structure constant The angular width of PLG beams decreases with increase in waist width (w₀).The M²-factor of PLG beams with higher beam order and smaller correlation length is less affected by turbulence with increase in propagation distance. The propagation properties of the M²-factor for PLG beams with the smaller structure constant are better than that with bigger structure constant . The M²-factor of PLG beams decreases with increase in the wavelength λ, and it is also less affected by turbulence for beams with higher order and smaller correlation length. Furthermore, for the PLG beams with the same beam order, the angular width and the M²-factor keep invariable in free space.     

Influence of atmospheric turbulence on the propagation of superimposed partially coherent Hermite-Gaussian beams

The influence of atmospheric turbulence on the propagation of superimposed partially coherent Hermite-Gaussian (H-G) beams is studied in detail. The closed-form propagation equation of superimposed partially coherent H-G beams through atmospheric turbulence is derived. It is shown that the turbulence accelerates the evolution of three stages which superimposed partially coherent H-G beams undergo. The turbulence results in a beam spreading and a decrease of the maximum intensity. However, the larger the beam number M, the beam order m, the separate distance x_d, and the smaller the beam correlation length σ₀ are, the less the power focusability of superimposed partially coherent H-G beams is affected by the turbulence. Specially, superimposed partially coherent H-G beams are less sensitive to turbulence than superimposed fully ones, and than partially coherent H-G beams if the beam power focusability and the maximum intensity are taken as beam criterions. However, the maximum intensity of superimposed partially coherent H-G beams is less sensitive or more sensitive to turbulence than that of superimposed Gaussian Schell-model (GSM) beams depending on σ₀.     

Slant path intensity distribution of focusing J0-correlated Schell-model vortex beams in atmospheric turbulence

The intensity distribution of the J₀-correlated Schell-model (JSM) vortex beams focused by a lens and propagation in weak-to-strong turbulent atmosphere are investigated. It is shown that the beam spreading increases with the increase in topological charge n, the source coherent length α−1, turbulent outer scale L₀ and propagation distance z. The center hollow depth of intensity distribution of the J₀-correlated Schell-model (JSM) vortex beams decrease with the increase of topological charge n, turbulent outer scale L₀ and propagation distance z or the decrease of the source coherent length α−1.     

Propagation properties of partially polarized Gaussian Schell-model beams through an axis-unsymmetric paraxial ABCD system

By using the beam coherence-polarization (BCP) matrix approach, analytical propagation equations of partially polarized Gaussian Schell-model (PGSM) beams through an axis-unsymmetric paraxial optical ABCD system are derived, which enable us to study the propagation-induced polarization changes and irradiance distributions at any propagation distance of PGSM beams through axis-unsymmetric systems within the framework of the paraxial approximation. Detailed numerical results for a PGSM beam passing through a bifocal lens are presented to illustrate the propagation properties of PGSM beams. A comparison with the previous work is also made.     

Propagation characteristics of flattened Gaussian beams through a misaligned optical system with a misaligned annular aperture

The approximate analytical formula for flattened Gaussian beams through a misaligned optical system with a misaligned annular aperture was derived by the extended Huygens–Fresnel principle. Some numerical simulations are illustrated to the effects on the propagation of flattened Gaussian beams by the misaligned annular aperture. To compare the difference between annular apertured system and circular apertured system, the circular apertured system is also studied. The results show that angle misalignments and lateral displacements of aperture create asymmetrical average intensity distribution at receiving plane z = 500. The effects on intensity distribution by angle misalignments of annular aperture were small. In annular aperture case, the smooth of intensity distribution was worse with escalating obscure ratio ? in near-field; the side-lobes increased and the central lobe decreased with escalating obscure ratio ? in far-field. At receiving plane z = 500: for circular aperture, the side-lobes decreased, even to be neglected, with the increasing of truncation parameter δ; for annular aperture, the side-lobes increased with the increasing of truncation parameter δ. In addition, it is found that the aligned thin lens can fix asymmetry of intensity distribution which was caused by the misaligned annular aperture.     

Solutions of the full- and quasi-vector wave equations based on H-field for optical waveguides by using mapped Galerkin method

The mapped Galerkin method in solving the full-vector and quasi-vector wave equations in terms of transverse magnetic fields (H-formulation) for optical waveguides with step-index profiles is described. By transforming the whole x-y space onto a unit square and using two-dimensional Fourier series expansion, the modal distributions and propagation constants for optical waveguides are obtained in the absence of boundary truncation. Results for step-index circular fiber, buried rectangular waveguide, and optical rib waveguide are presented. Solutions are good agreed with exact solutions and numerical results by using vector nonlinear iterative method, Fourier operator transform method, and vector beam propagation method.     

Effective radius of curvature of Hermite–Gaussian array beams

Analytical expressions for the effective radius of curvature, R, of Hermite–Gaussian (H–G) array beams propagating in free space for both coherent and incoherent combinations are derived. It is shown that for the two types of beam combination a minimum of the effective radius of curvature, R_min, appears as the propagation distance z increases. For the coherent combination, R is larger than that for the incoherent combination. The position z_min where the effective radius of curvature reaches its minimum is further away from the source plane for the coherent combination than that for the incoherent combination. For the two types of beam combination, R and z_min increase with increasing beam number, increasing beam separation distance, increasing waist width, and decreasing beam order and wavelength. In particular, the R of single H–G beams is always smaller than that of H–G array beams; the R of Gaussian array beams is always larger than that of H–G array beams.     

Gyrator transform of Gaussian beams with phase difference and generation of hollow beam

The optical expression of Gaussian beams with phase difference, which is caused by gyrator transform (GT), has been obtained. The intensity and phase distribution of transform Gaussian beams are analyzed. It is found that the circular hollow vortex beam can be obtained by overlapping two GT Gaussian beams with π phase difference. The effect of parameters on the intensity and phase distributions of the hollow vortex beam are discussed. The results show that the shape of intensity distribution is significantly influenced by GT angle α and propagation distance z. The size of the hollow vortex beam can be adjusted by waist width ω₀. Compared with previously reported results, the work shows that the hollow vortex beam can be obtained without any model conversion of the light source.     

Spreading of apertured partially coherent beams in turbulent media

The closed-form expression for the mean-squared width of apertured partially coherent beams propagating through turbulent media is derived by using the integral transform technique. The influence of turbulence on the spreading of apertured partially coherent beams is studied quantitatively by examining the relative mean-squared width, which is defined as the ratio of the mean-squared width of an apertured partially coherent beam in turbulence to the mean-squared width of the same beam in free space. On the other hand, the range of turbulence-independent propagation, also a reasonable measure of the resistance of a beam to turbulence, is obtained by examining the mean-squared width. It is shown that the spreading of apertured partially coherent beams is less affected by turbulence with smaller truncation parameter δ and coherence parameter α than with larger δ and α. In addition, the influence of turbulence on the spreading of apertured partially coherent beams increases first and then decreases due to increasing waist width w₀. The results obtained are explained physically.     

The refined theory of transversely isotropic piezoelectric rectangular beams

The problem of deducing one-dimensional theory from two-dimensional theory for a transversely isotropic piezoelectric rectangular beam is investigated. Based on the piezoelasticity theory, the refined theory of piezoelectric beams is derived by using the general solution of transversely isotropic piezoelasticity and Lur’e method without ad hoc assumptions. Based on the refined theory of piezoelectric beams, the exact equations for the beams without transverse surface loadings are derived, which consist of two governing differential equations: the fourth-order equation and the transcendental equation. The approximate equations for the beams under transverse loadings are derived directly from the refined beam theory. As a special case, the governing differential equations for transversely isotropic elastic beams are obtained from the corresponding equations of piezoelectric beams. To illustrate the application of the beam theory developed, a uniformly loaded and simply supported piezoelectric beam is examined.     

Propagation and focal shift of J0-correlated Schell-model beams

Based on the propagation law of partially coherent beams, the propagation equations of J₀-correlated Schell-model (JSM) beams through a paraxial optical ABCD system are derived and expressed in closed-form. The free-space propagation and focusing of JSM beams are regarded two special cases of our result. The focusing properties, in particular, the focal shift of JSM beams are studied, and illustrative numerical results are given. It is found that the focal shift of JSM beams, which is determined by an algebraic equation of 3rd order, increases with decreasing Fresnel number N and increasing parameter σ.     

Mathieu-Gauss beams: A thermal consideration

The generation of Helmholtz-Gauss beam families such as Mathieu-Gauss (MG) and Bessel-Gauss (BG) beams can be dramatically suffered from heat-induced thermal effects. This work is devoted to the generation of MG beams under severe induced thermal loads. The output MG beams of a solid-state laser were assumed to pass through an ABCD system matrix. The simulated results which were achieved by characterization of the transfer matrix of the thermally affected inhomogeneous medium report systematic variations of the intensity profile of MG beams from pump power of 1-5 W. It is shown that for high values of pump power, the thermal effects are so influential and the beam intensity profile is so changed that the identification of MG beams is not easily possible at first hand. The results can facilitate this identification and can be utilized for deeper understanding of thermal effects phenomena.     

Beam propagation factor of partially coherent Hermite-Gaussian beams through non-Kolmogorov turbulence

Based on extended Huygens-Fresnel principle, an analytical expression of M² factor for partially coherent Hermite-Gaussian (PCHG) beams through non-Kolmogorov turbulence is obtained. Our results show that M² factor of PCHG beams depends on coherence length, beam order m, wavelength, exponent value α, outer scale L₀ and inner scale l₀. The impact of the exponent's variation on the M² factor is analyzed in detail. When the propagation distance fixed, the M² factor of PCHG beams has a maximum for alpha values lower than 11/3.     

Polarization properties of nonparaxial partially polarized Gaussian Schell-model beams

The polarization properties of nonparaxial partially polarized Gaussian Schell-model (PGSM) beams are studied. Based on the beam coherence-polarization matrix, the propagation formula of the degree of polarization for the nonparaxial PGSM beams is derived. The paraxial approximation is dealt with as special cases of our general results. Some conditions that limit the choice of the parameters are established. The sufficiency condition of polarization invariance in propagation for the nonparaxial PGSM beams is derived. By using the derived formulae, the propagation properties of the degree of polarization for the nonparaxial PGSM beam in free space are illustrated and analyzed with numerical examples. Some detailed comparisons of the obtained results with the paraxial results are made.