Polarization properties of two Gaussian Schell-model beams in the turbulent atmosphere

General expressions are derived for the spectral degree of polarization of a beam generated by the superposition of two Gaussian Schell-model (GSM) beams propagating in a turbulent atmosphere by adopting the beam coherence-polarization matrix and Young's interference theory. We analyzed the distance of two beams, atmospheric turbulence intensity, and the location of the imaging plane affect the degree of polarization by numerical calculation. In particular, when the correlation coefficients of x and y components of the electric field are the same (i.e. δ_xx = δ_yy), the degree of polarization will obtain the same value.     

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.     

Fractal Coding Metamaterials

The concept of fractal coding metamaterials is proposed here, which can be used to design reflective metasurfaces with self‐similar pseudo‐random phase responses based on the coding strategy utilizing fractal interpolation functions (FIF). The class of fractals whose FIF is formed by a contraction mapping on the y‐coordinate and an arbitrary linear function of the x‐coordinate is considered. This formulation is used in deriving the theoretical model and in the experimental realization. An analytical relation between the reflection phase distribution in the form of the above‐mentioned FIF and the far‐field radiation pattern of the structure is derived. In the numerical example, a reflecting metasurface formed by slotted metallic patches of varying sizes is considered. A number of full wave scattering simulations for such a metamaterial reflectarray are performed and the optimal designs which make the response of the structure more accurate as compared with the analytical predictions are identified. Following the simulation results, the array is built and a number of measurements are performed. The results of this work may find applications in phased antenna array design and beam forming.     

Polarization singularities in near-field of Gaussian vortex beam diffracted by a circular aperture

Polarization singularities in the near-field of Gaussian vortex beams diffracted by a circular aperture are studied by a rigorous electromagnetic theory.It is shown that there exist C-points and L-lines,which depend on off-axis displacement parameters along the x and y directions,waist width,wavelength,and topological charge of the diffracted Gaussian vortex beam,as well as on propagation distance.The results are illustrated by numerical calculations.     

An MPI parallel level-set algorithm for propagating front curvature dependent detonation shock fronts in complex geometries

We present a parallel, two-dimensional, grid-based algorithm for solving a level-set function PDE that arises in Detonation Shock Dynamics (DSD). In the DSD limit, the detonation shock propagates at a speed that is a function of the curvature of the shock surface, subject to a set of boundary conditions applied along the boundaries of the detonating explosive. Our method solves for the full level-set function field, φ(x, y, t), that locates the detonation shock with a modified level-set function PDE that continuously renormalises the level-set function to a distance function based off of the locus of the shock surface, φ(x, y, t)=0. The boundary conditions are applied with ghost nodes that are sorted according to their connectivity to the interior explosive nodes. This allows the boundary conditions to be applied via a local, direct evaluation procedure. We give an extension of this boundary condition application method to three dimensions. Our parallel algorithm is based on a domain-decomposition model which uses the Message-Passing Interface (MPI) paradigm. The computational order of the full level-set algorithm, which is O(N ⁴), where N is the number of grid points along a coordinate line, makes an MPI-based algorithm an attractive alternative. This parallel model partitions the overall explosive domain into smaller sub-domains which in turn get mapped onto processors that are topologically arranged into a two-dimensional rectangular grid. A comparison of our numerical solution with an exact solution to the problem of a detonation rate stick shows that our numerical solution converges at better than first-order accuracy as measured by an L1-norm. This represents an improvement over the convergence properties of narrow-band level-set function solvers, whose convergence is limited to a floor set by the width of the narrow band. The efficiency of the narrow-band method is recovered by using our parallel model.     

Two-dimensional thermoelasticity solution for functionally graded thick beams

Two-dimensional thermoelasticity analysis of functionally graded thick beams is presented using the state space method coupled with the technique of differential quadrature. Material properties vary continuously and smoothly through the beam thickness, leading to variable coefficients in the state equation derived from the elasticity equations. Approximate laminate model is employed to translate the state equation into the one with constant coefficients in each layer. To avoid numerical instability, joint coupling matrices are introduced according to the continuity conditions at interfaces in the approximate model. The differential quadrature procedure is applied to discretizing the beam in the axial direction to make easy the treatment of arbitrary end conditions. A simply-supported beam with exponentially varying material properties is considered to validate the present method. Numerical examples are performed to investigate the influences of relative parameters.     

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.     

Propagation of astigmatic stochastic electromagnetic beams in oceanic turbulence

In this paper, we investigate the characteristics of astigmatic stochastic electromagnetic beams through oceanic turbulence. Taking the electromagnetic Gaussian Schell-model (GSM) beam as an example, the analytic expressions for the spectral density and the spectral degree of polarization of the beam propagating the oceanic turbulence are derived. It is indicated that the spectral density along the z-axis of the GSM beam in the oceanic turbulence is severely influenced by the source correlation properties, as well as by the sea-related parameters. We show that the characteristics of the spectral density along the x-axis, y-axis and z-axis of astigmatic electromagnetic GSM beams passing through the oceanic turbulence are qualitatively different. Furthermore, we find that as the astigmatic coefficient becomes larger, the maximum value of the spectral density along the z-axis increases rapidly and the width of the spectral density becomes shorter rapidly. Finally, the results have shown that different strengths of astigmatism have different effects on the spectral degree of polarization.     

The electromagnetic field of a guided Gaussian beam

An electromagnetic wave traveling along a dielectric slab between two parallel, plane conductors is analyzed for a special case that is particularly relevant to guided laser beams employed in integrated optics. The beam is assumed to have an approximate gaussian field dependence along the y-direction, defined to be transverse to the direction of propagation and to the normal to the conductor planes. Approximate expressions for the field in a well collimated beam are derived. Asymptotic approximations, valid far away from the focus of a beam that is not necessarily well collimated, are also derived. It is found that the TE and TM guided gaussian beams are not linearly polarized. This is unlike the case for the usual waveguide modes whose fields are constant along the y direction. It is also found that the guided gaussian beam must have a focus and that the divergence from focus is more rapid than for a similar gaussian beam in free space.     

Coherence vortex properties of partially coherent flat-topped vortex beams

The analytical propagation expression of partially coherent flat-topped vortex beams through a paraxial optical ABCD system is derived, and it then is used to investigate the coherence vortex properties of partially coherent flat-topped vortex beams in the Fourier transform and fractional Fourier transform systems. It is shown that in the Fourier transform system the coherence vortex depends on the flat-topped beam order N, spatial coherence parameter α and position (x ₁,y ₁) of the reference point, whereas in the fractional Fourier transform system the flat-topped beam order N does not affect the spectral degree of coherence. Furthermore, in both transform systems, depending on the choice of the reference point, the zero value point of the spectral degree of coherence may be present or absent. In particular, if x ₁=y ₁=0 is selected, the phase at the zero value point of the spectral degree of coherence may be determinate; thus the coherence vortex does not exist.     

Observation of polarization-controlled spatial splitting of four-wave mixing in a three-level atomic system

We report experimental observations of intensity modulation and spatial splitting of four-wave mixing (FWM) signal beams which can be effectively controlled by the polarization states of the pumping laser beams. Due to the periodic change of the pumping beam’s polarization states, the intensity of the FWM beam also evolves periodically. The periodic spatial splitting phenomenon has been observed in both x- and y-directions. The cases with/without the dressing beams are compared. Such studies can be very useful in better understanding the formation of spatial solitons and for signal processing applications, such as spatial beam splitter, routing, and switching.     

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.     

The Effect of the Spin Angular Momentum on the Tight-Focusing Vortex Hollow Gaussian Beams

The propagation properties of a hollow Gaussian beam (HGB) carrying on-axis and off-axis vortices through a high numerical aperture lens are investigated. The intensity of the focused beam in the focal plane can be controlled by choosing the different topological charges, the beam order, and the semi-aperture angle. As intrinsic properties, vortex beams possess both spin and orbital angular momenta. The spin angular momenta (SAM) density can be treated as a vector in 3D since it exists in arbitrary orientation during the beam propagation. The vectors of SAM density orientation of the focused beam in 3D rotate around the central axis whose locations mainly rely on the vortices. The magnitude of the SAM density near the focus plane abruptly varies by altering the focal length of the lens. Under tightly focusing condition, two new pairs of vortices generate alternately on x and y axes in the vectorial electric fields, while the topological charges increase by one.     

Angular momentum conversion of elliptically polarized beams focused by high numerical-aperture phase Fresnel zone plates

Based on the vectorial Debye theory, we investigate the tight focusing of elliptically polarized light beams by high numerical-aperture (NA) phase Fresnel zone plates (FZP). The conversion of the spin angular momentum (SAM) and the orbital angular momentum (OAM) in the tight focusing of the elliptically polarized beams is investigated in great detail. It is shown that the direction and magnitude of the OAM can be directly modulated by the input polarization, providing effective evidence that the SAM carried by an elliptically polarized light beam is converted into OAM in the tight focusing process. The properties of phase retardation between x and y components of the focused field are also investigated. It is found that focused field of x and y components still keeps its original elliptical polarization state, indicating that the focused field composed of x and y components still carries SAM, whereas the focused field of z component is changed into carrying OAM. The influences of the total zone number and the phase difference of binary phase FZPs with high NA on the intensity distribution in the focal plane are also investigated.     

ON THE FREE VIBRATION OF STIFFENED SHALLOW SHELLS

A finite element analysis for free vibration behaviour of doubly curved stiffened shallow shells is presented. The stiffened shell element is obtained by the appropriate combinations of the eight-/nine-node doubly curved isoparametric thin shallow shell element with the three-node curved isoparametric beam element. The shell types examined are the elliptic and hyperbolic paraboloids, the hypar and the conoidal shells. The accuracy of the formulation is established by comparing some of the authors' results of specific problems with those available in the literature. Numerical results of additional stiffened shells are also presented to study the effects of various parameters of shells and stiffeners such as orientation (i.e., along x -/y -/both x and y directions), type (concentric, eccentric at top and eccentric at bottom) and number of stiffeners, stiffener depth to shell thickness ratio, and aspect ratio, shallowness and boundary conditions of shells on free vibration characteristics.     

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.     

Bending vibrations of wedge beams with any number of point masses

The natural frequencies and mode shapes of beams with constant width and linearly tapered depth (or thickness) carrying any number of point masses at arbitrary positions along the length of the beams were investigated using the Euler-Bernoulli equation. Use of the closed-form (exact) solutions for the natural frequencies and mode shapes of the unconstrained single-tapered beam (without carrying any point masses) and incorporation of the expansion theorem, the equation of motion for the associated constrained beam (carrying any point masses) were derived. Solution of the last equation will yield the desired natural frequencies and mode shapes of the constrained single-tapered beam. The bending vibrations of a single-tapered beam with six kinds of boundary conditions were investigated. Comparison with the existing literature or the traditional finite element method results reveals that the adopted approach has excellent accuracy and simple algorithm.     

The influence of optical activity on optimal conditions of focusing of one- and two-dimensional light beams of different profiles in a cubic photorefractive crystal

We compare regular features of propagation of one- and two-dimensional light beams in an optically active photorefractive crystal placed in an external electric field. We show that there exist ranges of the crystal thickness in which the optical activity can lead to an increase in the relative intensity of either beam. We have found that, taking into account the optical activity, an optimal orientation angle at which a maximal self-focusing of light beams is observed differs from commonly used values of 35.3° and 90° for the x and y polarized beams, respectively.     

Generalized beams in ABCDGH systems

For a generalized beam at the source plane passing through co-located aperture and a propagation path consisting of an off-axis x-y asymmetric ABCDGH system, the receiver plane irradiance expression is derived using the Collins integral. A collection of source beam profiles that are obtainable from the generalized beam formulation are illustrated. Plots are given for viewing the progress of selected generalized beam types along the propagation axis, containing a single thin lens, co-centric and misaligned in the x-direction. The received power falling onto a finite aperture surface is calculated for various misalignment situations.     

Local momentum space and two-loop renormalizability of λφ 4 field theory in curved space-time

A generalization is given of some previous work in which a momentum space representation for the Feynman propagator,G(x, y), of a scalar field in an arbitrary curved space-time was obtained. The pointsx andy are allowed to vary in a normal neighborhood of an arbitrary fixed pointz which is taken as an origin of normal coordinates and the representation is obtained by Fourier transformation in the coordinate differencex -y . The generality of this representation enables it to be applied to the evaluation of the divergences in any Feynman graph. As an example, the third-order (two-loop) corrections to the four-point function of ø⁴ field theory are shown to be renormalizable in curved space-time.