Controlling the Trajectory of Swallowtail Gaussian Beams in Dynamic Parabolic Potentials

The propagation dynamics of finite-energy Swallowtail beams in a dynamic parabolic potential, including uniformly moving, accelerating, and oscillating potentials, are investigated. The strong influence of dynamic potentials on the propagation trajectory of Swallowtail beams is demonstrated and various effective manipulations of the beams, including trajectory control, are validated. The intensity and the focal position can also be affected. In addition, the extension to 2D scenarios is also presented. The results theoretically provide more diverse manipulation possibilities for Swallowtail beams, and thus may broaden their potential applications in trajectory and particle manipulation.     

Application of the Wigner distribution function to complex-argument Hermite– and Laguerre–Gaussian beams beyond the paraxial approximation

The Wigner distribution function (WDF) is applied to study the propagation of complex-argument Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) beams beyond the paraxial approximation. The analytical expressions for their intensity distributions in free-space propagation are derived, which are expressed in terms of Hermite polynomials for nonparaxial complex-argument HG beams and in terms of the sum of finite Hermite polynomials for nonparaxial complex-argument LG beams. A detailed comparison of the WDF approach, series expansion method and paraxial expressions is made, which shows that in the paraxial regime the WDF approach and series expansion method deliver consistent results with that of paraxial expressions. Beyond the paraxial approximation, the WDF approach offers convergent results, whereas the series expansion method has a limited applicable range, within which it gives consistent results with that of WDF approach but beyond which it gives unrealistic and divergent results.     

A detailed study of Mathieu-Gauss beams propagation through an apertured ABCD optical system

Using Collins formula and the expansion of Mathieu beams in terms of Bessel beams we derive the exact propagation equations of Mathieu-Gauss beams through an apertured paraxial ABCD optical system. A comparison between the exact propagation equations and the approximated ones, which are derived by expanding the circ function into a finite sum of Gaussian functions, is presented. The propagation characteristics of zeroth-order Mathieu-Gauss beams in (y-z) and (x-z) planes are analyzed with detailed numerical calculations. It is found that the profile of the propagated Mathieu-Gauss beam is similar to that of Bessel-Gauss beam. Furthermore, the focalization of the Mathieu-Gauss beams through a thin lens is illustrated and analyzed with numerical results.     

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.     

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.     

Airy beams beyond the paraxial approximation

The behavior of the vectorial Airy beams beyond the paraxial approximation is investigated. Indeed, closed-form (even though non exact) expressions for the electric components of the fields generated by the same boundary conditions, which should pertain to the scalar Airy beams, are obtained on the basis of the vectorial Rayleigh-Sommerfeld diffraction integrals under suitable approximations. Such expressions may accompany more complete approaches, like that in Opt. Expr. 17, 22432 (2009), where a fully numerical analysis of the propagation of exponentially smoothed Airy beams has been presented, faithfully reproducing the conditions of their experimental demonstration as reported in Phys. Rev. Lett. 99, 213907 (2007). Comments on other well known approaches to the investigation of the nonparaxial propagation of definite paraxial beams are also given.     

Effect of chromatic aberration on Gaussian beams: non-dispersive laser resonators

The propagation of a non-monochromatic (polychromatic) TEM₀₀ Gaussian beam in vacuum, its passage through a thin plate and its transformation by a thin lens are studied in the case of a non-dispersive laser resonator. The basic assumptions of the model are as follows: optical fields are stationary and plane-polarized, the paraxial wave equation is valid, an equivalent non-dispersive hemiconfocal resonator represents the lasing medium and its stable resonator, the laser emits in a single mode. It is also assumed that the plate and the lens have large transverse dimensions. Mathematical expressions, for beam radius, divergence, radius of curvature and beam parameter product, are obtained. A beam quality factor for polychromatic Gaussian beams is defined and its value calculated in each case of interest. It is proposed to simulate a dispersive laser resonator by a non-dispersive resonator complemented with a plate and/or a thin lens.     

Bizarre alterations of the morphology of pineal synaptic bodies under constant light and an evaluation of suitable 3D-reconstruction software

Three dimensional (3D) reconstruction and modelling software was evaluated to find a procedure suitable for visualization of small subcellular structures in transmission electron microscope images. The method applied in this study demonstrates bizarre alterations of the structure of synaptic bodies (SBs) in pinealocytes of the guinea-pig pineal gland caused by constant illumination. It can, in general, be used for any 3D reconstruction from serial sections.

Pineal glands of five guinea-pigs (two kept under a LD cycle of 12:12 h; three kept in constant light, for 4 months) were investigated. SBs consist of an electron-dense centre with attached vesicles. Under normal lighting conditions most SBs are flat plates (about 35 nm in thickness), which eventually may be bent. The proteins comprising the molecular basis of SBs, mainly RIBEYE A and B are polymerised in a regular manner in these plates. This is not the case in other SBs, which appear as spheres or irregular lumps. SBs lie in groups in which usually some of the plates are arranged in parallel arrays

Constant illumination caused different changes in morphology: many of the SBs lie in ‘paired fields’, i.e. appear in groups attached to the cell membranes of two pinealocytes directly opposite to each other. Some of the SBs in such groups are strongly bent, showing blebs and irregular thickened areas, others seem to aggregate and show inclusions of cytoplasm. Further goblet-like, shield-like and other bizarre forms of SBs occurred and the relative number of spheroid and lump-like SBs increased. Protrusions on larger SBs suggest detachment or fusion of SB material to a greater extent than in the control animals. There is a reduction of areas in which the polymerisation of the SB proteins remains well ordered, i.e. where the typical thickness of 35 nm is maintained. It remains unclear why this polymerisation pattern is only partly affected by constant light.     

Generalized (two-point) stokes-parameters representation in terms of the radial and azimuthal field components

Within the paraxial framework, a generalization of the Stokes representation is introduced in terms of the radial and azimuthal field structure of the beam cross-section. These generalized parameters depend on two spatial variables and are closely related with the cross-spectral density (CSD) matrix of the field. The propagation law of such parameters through general first-order optical systems is also given. It allows to describe, in terms of the radial and azimuthal field components, the coherence and polarization behavior of partially-polarized, partially-coherent beams.     

Propagation characteristics of linearly polarized Bessel-Gaussian beams through an annular apertured paraxial ABCD optical system

By means of Collins diffraction integral formula in the paraxial approximation and based on the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions, an approximate analytical expression for linearly polarized Bessel-Gaussian beams passing through a paraxial ABCD optical system with an annular aperture has been derived. The results provide more convenient for studying their propagation and transformation than the usual way by using diffraction integral directly. By using the analytical expression and the diffraction integral formula some numerical simulations are done to illustrate for the propagation characteristics of a linearly polarized Bessel-Gaussian beam through an optical system with an annular aperture.     

Polychromatic Gaussian beams emitted by dispersive laser resonators

In-vacuum propagation of a polychromatic Gaussian beam emitted by a dispersive laser resonator and its passage through thin plates are studied under following assumptions: optical fields are stationary and plane-polarized, the paraxial wave equation is valid for each monochromatic component of the beam, the waist position and the Rayleigh range depend on wavelength and diffraction edge effects are negligible. Mathematical expressions for beam radius, divergence, beam parameter product and beam propagation factor are obtained. The use of thin plates for waist position achromatization is analyzed.     

Propagation of Bessel-modulated Gaussian beams through a paraxial ABCD optical system with an annular aperture

Based on the generalized Collins diffraction integral and the expansion of the hard aperture function into a finite sum of complex Gaussian functions, the approximate analytical expressions of Bessel-Gaussian beams and QBG beams passing through a paraxial ABCD optical system with an annual aperture are derived. As special cases, the corresponding closed-forms for the unaperture or circular aperture or circular black screen cases are also given. The results provide more convenience for studying their propagation and transformation than the usual way by using diffraction integral formula directly. Numerical examples are given to illustrate the propagation properties of Bessel-Gaussian and QBG beams.     

Polarization and intensity properties of converging J0-correlated Schell-model beams

Based on the cross-spectral density matrix, closed form result for propagation equation of electromagnetic J₀-correlated Schell-model beams (EJSMBs) through a paraxial optical system is obtained and the focusing properties are studied. Both numerical calculation and physical interpretation are obtained. It is found that a tunable dark hollow area, which has potential applications in optical trapping, can be obtained by altering the coherence parameter and the focal length. It is also shown that even though the original field is unpolarized, the beams can become fully polarized in the focal region with its width being tunable by changing the coherence parameter. The relevance of this work to applications such as coherent detection in optical communication is also discussed.     

Free-space evolution of focused Gaussian beams transformed by conical diffraction in a biaxial crystal

The intensity and polarization pattern of circularly and linearly polarized Gaussian beams transformed by the effect of conical diffraction in a biaxial crystal has been studied experimentally, and compared with the predictions of paraxial theory. Some general regularities in the relative intensity and geometry of beam structures were analyzed. The formation of Mathieu beam profiles from linearly polarized input beams was demonstrated.     

The influence of nonparaxiality on the spectral behavior in Young's experiment illuminated by partially coherent light

Starting from the Rayleigh--Sommerfeld diffraction integral, this paper studies the spectral behavior in Young's experiment illuminated by nonparaxial partially coherent light and compares with the paraxial case, where the influence of nonparaxiality of partially coherent light on the spectral shifts and spectral switches is stressed. It is shown that there is a spectral shift in the nonparaxial case relative to the paraxial one and the critical position changes, at which the spectral switch occurs. The ratio of the waist width to the central wavelength w₀/λ₀ and relative spatial correlation length \De affect the spectral difference. The smaller w₀/λ₀ is, the larger the difference between the nonparaxial and paraxial results appears. The effect of relative spatial correlation length Δ is relatively small.     

非傍轴高斯光束传输方程解的探讨

由洛伦兹规范下光场满足的矢势和标势方程出发, 探讨了初始光场具有任意对称性的一般非傍轴光束在空间的传输情况。利用角频谱并结合傅里叶变换处理得到了非傍轴光束的传输方程。利用扰动方法并结合新近似取法得到了非傍轴横向光场的扰动形式解, 该解对于近场弱非傍轴情况下的效果比强非傍轴条件下要好。给出了光束非傍轴传输高阶(八阶)修正解。对于零阶情况, 非傍轴光束退化为傍轴光束, 传输方程也相应退化为傍轴传输的模型方程。高斯解为扰动参量的零阶修正解。     

Propagation of nonparaxial vector hollow Gaussian beams through a circular aperture

Based on the vectorial Rayleith-Sommerfeld formulae, the nonparaxial propagation properties of the vector hollow Gaussian beams (HGBs) through a circular aperture are studied in detail. We describe the derivation of the integral expressions of the propagation of nonparaxial vector HGBs through a circular aperture. The derived expression is independent the approximation of paraxial and far field, which are valid for either far and near field and for the systems in which aperture radius is comparable to or even smaller than wavelength. And it is also strict integral formula for the light field on the axis. Numerical calculation results indicate that there is no difference between derived formulae and the Collins formulae in the situation of paraxial approximation. Using the formula deduced, we calculate the propagation properties of HGBs. The calculated results indicate that the propagation field of vector hollow Gaussian beams is asymmetric in near field, while the propagation field is symmetric in far field. These research results could well shed light on the further understanding of the vectorial property of HGBs through a circular aperture, and would play a guiding role in the practical application of HGBs.     

Hollow Gaussian Schell-model beam and its propagation

In this paper, we present a new model, hollow Gaussian Schell-model beams (HGSMBs), to describe the practical dark hollow beams. An analytical propagation formula for HGSMBs passing through a paraxial first-order optical system is derived based on the theory of coherence. Based on the derived formula, an application example showing the influence of spatial coherence on the propagation of beams is illustrated. It is found that the beam propagating properties of HGSMBs will be greatly affected by their spatial coherence. Our model provides a very convenient way for analyzing the propagation properties of partially coherent dark hollow beams.     

Effect of polarization on the evolution of electromagnetic hollow Gaussian Schell-model beam

Based on the theory of coherence, an analytical propagation formula for partially polarized and partially coherent hollow Gaussian Schell-model beams (HGSMBs) passing through a paraxial optical system is derived. Furthermore, we show that the degree of polarization of source may affect the evolution of HGSMBs and a tunable dark region may exist. For two special cases of fully coherent and partially coherent δ_xx = δ_yy, normalized intensity distributions are independent of the polarization of source.