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.     

Fraunhofer diffraction of a Laguerre–Gaussian laser beam by fork-shaped grating

In this article we present a theoretical study for Fraunhofer diffraction of a Laguerre–Gaussian laser beam with zeroth radial mode number and azimuthal mode number l by a diffractive grating with embedded fork-shaped dislocations of integer order p. Analytical expressions describing the diffracted wave field amplitude and intensity distributions in the Fourier plane are deduced and analyzed. They are also followed by the vortex radii expressions.     

Fractional Fourier transform of a higher-order cosh–Gaussian beam

A higher-order cosh–Gaussian beam is an appropriate model to describe the flattened laser beam. The fractional Fourier transform (FRFT) is applied to treat the propagation of higher-order cosh–Gaussian beams. An analytical expression for a higher-order cosh–Gaussian beam passing through a FRFT system has been derived. By using the derived expression, the properties of a higher-order cosh–Gaussian beam in the FRFT plane are graphically illustrated with numerical examples.     

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.     

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.     

Interaction between a vortex and an edge dislocation nested in a cos‐Gaussian beam passing through a tilted lens

The interaction between a vortex and an edge dislocation nested in a cos‐Gaussian beam passing through a tilted lens was studied analytically and numerically. It is shown that in the presence of the tilted lens, by varying the cos‐part parameter of the beam, the slope of the edge dislocation, or the tilt coefficient of the lens, the topological charge is conserved for the case of the off‐axis edge dislocation, while charge conservation does not hold true for the case of an on‐axis edge dislocation, irrespective of the on‐axis or off‐axis vortex. The relation between the transverse positions (x, y) of vortices and the tilt coefficient is linear, and is interpreted.     

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.     

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.     

Spectral shift of a partially coherent standard or elegant Laguerre–Gaussian beam in turbulent atmosphere

Spectral changes of a partially coherent standard or elegant Laguerre–Gaussian (LG) beam propagating in turbulent atmosphere were studied numerically. Our results show that the spectral changes of a partially coherent standard or elegant LG beam in turbulent atmosphere are determined by both the structure constant of the turbulent atmosphere and the initial beam parameters. Furthermore, it is found that a partially coherent elegant LG beam is less affected by the turbulent atmosphere than a partially coherent standard LG beam from the aspect of the on-axis spectral shift, and this advantage is enhanced for small structure constant, small beam waist size, large mode orders, and large transverse coherence length. Our results will be useful in long-distance free-space optical communications.     

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.     

Optimization design and analysis of reflecting polarizing beam splitter based on metal–multilayer dielectric grating for 800 nm

A new reflecting polarizing beam splitter (RPBS), based on a metal–multilayer dielectric grating (MMDG), was designed using rigorous coupled wave analysis method and a genetic algorithm. The RPBS reflects the TE wave in the ?1st order and TM wave in the 0th order for 800?nm. The optimized RPBS has an extinction ratio of over 20 dB, from 765?nm to 823?nm, and a large angle, from 43.8° to 60°. At 800?nm, the extinction ratios of the 0th and ?1st orders reach a maximum of 41.0 dB and 53.2 dB, respectively. Furthermore, analysis suggests that the designed RPBS allows sufficient manufacture tolerance. These results show the potential of MMDGs in fabrication of a RPBS, which can be widely used in optical systems.     

Microstructure and mechanical properties of a new Al–Zn–Mg–Cu alloy joints welded by laser beam

A new type of Al–Zn–Mg–Cu alloy sheets with T6 temper were welded by laser beam welding (LBW). Microstructure characteristics and mechanical properties of the joints were evaluated. Results show that grains in the heat affected zone (HAZ) exhibit an elongated shape which is almost same as the base metal (BM). A non-dendritic equiaxed grain zone (EQZ) appears along the fusion line in the fusion zone (FZ), and grains here do not appear to nucleate epitaxially from the HAZ substrate. The FZ is mainly made up of dendritic equiaxed grains whose boundaries are decorated with continuous particles, identified as the T (AlZnMgCu) phase. Obvious softening occurs in FZ and HAZ, which mainly due to the changes of nanometric precipitates. The precipitates in BM are mainly η′, while plenty of GPI zones exist in FZ and HAZ adjacent to FZ, in the HAZ farther away from FZ, η phase appears. The minimum microhardness of the joint is always obtained in FZ at different times after welding. The ultimate tensile strength of the joint is 471.1 MPa which is 69.7% of that of the BM. Samples of the tensile tests always fracture at the FZ.     

Bifurcation and stability analysis of a hybrid energy harvester with fractional-order proportional–integral–derivative controller and Gaussian white noise excitations

Hybrid energy harvester (HEH) has received much attention in the field of energy harvesting. The HEH inevitably suffers from external stochastic disturbances in the working environment such as winds, waves, and ocean currents. Only few works deal with the stochastic dynamics of a hybrid energy harvester with fractional-order proportional–integral–derivative (PID) controller. This paper aims to investigate bifurcation control and stability analysis of a HEH with fractional-order PID controller subjected to Gaussian white noise excitations. An approximately equivalent dimensionally reduced system is formulated via the variables transformation method, and its approximate stationary solutions are derived through the stochastic averaging method. One example is worked out in detail to verify the effectiveness of the proposed procedure. It is shown that the analytical results provide a good approximation to the numerical simulation results. The influences of system parameters on the stochastic bifurcation and the asymptotic Lyapunov stability are also discussed.     

Establishment of a novel surface-imprinting system for melamine recognition and mechanism of template–matrix interactions

A surface molecular-imprinting system was developed on polypropylene (PP) fiber for melamine (Mel) as an N-containing template. In this article, acrylic acid was introduced onto the surface of PP for template binding. Subsequently, binding sites on PP were stabilized by crosslinking with ethylene glycol diglycidyl ether in the presence of Mel. The imprinted fiber (MIF-Mel) prepared with the optimal 15 % crosslinking density showed best-imprinting effect, with an imprinting factor of 2.18 respect to nonimprinted fiber, and a relative selectivity coefficient k′ of 10.40 for Mel with respect to its structural analog 2,4-dinitroaniline. MIF-Mel showed higher affinity to Mel with the maximum adsorption capacity of 15.5 mg g^?1, while that on nonimprinted fiber was only 6.9 mg g^?1. Its adsorption isotherm was well described using Langmuir model. Kinetic studies showed a rapid-binding interaction and high affinity of the MIF-Mel for its template, with a 2.5 times higher in binding amount and 4.7 times faster in binding speed than those of granular molecular-imprinting polymer with the same chemical structure. High degree of fitness with pseudo-second-order model revealed chemisorption was the rate-controlling step in the template-binding process. Basic theory of matrix–template interaction in this imprinting system was clarified to be dominated by electrostatic force synergized by hydrogen bonding between deprotonated carboxyl groups and protonated N atom in the template. It suggests that extension of this novel approach or theory to other imprinting system involving nitrogenous templates is very likely.     

Applications of dual MRM for determining the natural frequencies and natural modes of an Euler–Bernoulli beam using the singular value decomposition method

In this paper, a dual multiple reciprocity method (MRM) is employed to solve the natural frequencies and natural modes for an Euler–Bernoulli beam. It is found that the conventional MRM using an essential integral equation results in spurious eigenvalues and modes. By using the natural integral equation of dual MRM, the spurious eigendata can be filtered out. Four numerical examples are given to verify the validity of the present formulation. In one of these four examples, fixed–fixed supported beam, it is found that the boundary eigenvector cannot be determined by either the essential or natural integral equation alone since the rank of the corresponding leading coefficient matrix is insufficient. The singular value decomposition method is then used to solve the eigenproblem after combining the essential and natural integral equations. This method can avoid the spurious eigenvalue problem and possible indeterminancy of boundary eigenvectors at the same time.     

Three-dimensional performance surfaces–a tool for analysing and estimation of production system performances

This paper presents a new method to describe, analyse and estimate production system performances. Work-in-process (units), lead time (number of time units spent in the production system for each unit) and throughput (number of produced units per time unit) are basic performance measures, also used in this article. It is essential for industry to know about relations between system parameters and system performances in existing systems, and in not yet implemented system alternatives. Different performances are achieved by adjusting system parameters. Trade-offs between system parameters and its different performances are necessary to stay efficient and competitive in today's market. Queuing theory and simulation can help the decision makers to estimate system performances of existing and not yet implemented systems. When the complexity increases queuing theory becomes cumbersome, very difficult and eventually impossible to use. A single simulation presents limited information. Multiple simulations are necessary to ensure that the best alternative is chosen. A high number of simulations demand a lot of computer time and resources. Reduction of runs is desirable even with cheaper computer equipment. Currently, traditional two-dimensional charts are the only tools to present and analyse system performances. This article presents a new surrogate model for easier estimation and presentation of system performances, their internal relations, and relations to the system parameters. With the new surrogate model, system performances based on simulations are presented as positions in a three-dimensional environment. Parametric curves and surfaces of Bezier type are generated and adapted to these positions. System performances of other system alternatives can then be estimated without explicit simulation. The number of simulation calculations can thereby be moderated. The method is illustrated with a small production line system.