Coupling characteristics and kurtosis parameter of partially coherent beams in turbulent atmosphere

Coupling properties and kurtosis parameter (K parameter) of arbitrary beams propagating through atmospheric turbulence are investigated. A correlation factor (C⁴-factor) is introduced to describe the influence of turbulence on coupling characteristics. The general analytical expression for C⁴-factor of arbitrary beams in atmospheric turbulence is derived. It is shown that C⁴-factor of arbitrary beams in the turbulent atmosphere depends on the initial second-order moments and fourth-order moments and turbulence quantities. Taking the partially coherent anomalous elliptical hollow Gaussian (PCAEHG) beam as an example, we can obtain that C⁴-factor decreases as structure constant of the refractive index fluctuations and inner scale increase, and waist width and transverse coherence length decrease when z?>?5?km. Moreover, K parameter of PCAEHG beam in turbulent atmosphere converges to 2 when propagation distance is large enough. It indicates that the profile of PCAEHG beams in turbulent atmosphere finally evolves into fundamental Gaussian distribution.     

Kurtosis parameter K of arbitrary electromagnetic beams propagating through non-Kolmogorov turbulence

General analytical formulae for the kurtosis parameters K (K parameters) of the arbitrary electromagnetic (AE) beams propagating through non-Kolmogorov turbulence are derived, and according to the unified theory of polarization and coherence, the effect of degree of polarization (DOP) of an electromagnetic beam on the K parameter is studied. The analytical formulae can be given by the second-order moments and fourth-order moments of the Wigner distribution function for AE beams at source plane, the two turbulence quantities relating to the spatial power spectrum, and the propagation distance. Our results can also be extended to the arbitrary beams and the arbitrary spatial power spectra of Kolmogorov turbulence or non-Kolmogorov turbulence. Taking the stochastic electromagnetic Gaussian Schell-model (SEGSM) beam as an example, the numerical examples indicate that the K parameters of a SEGSM beam in non-Kolmogorov turbulence depend on propagation distance, the beam parameters and turbulence parameters. The K parameter of a SEGM beam is more sensitive to effect of turbulence with smaller inner scale and generalized exponent parameter. A non-polarized light has the strongest ability of resisting turbulence (ART), however, a fully polarized SEGSM beam has the poorest ART.     

Arbitrary moments of elliptical Gaussian-Schell beam in turbulent atmosphere

Arbitrary moments of elliptical Gaussian-Schell beams in turbulent atmosphere is studied. An analytical formula for arbitrary moments of elliptical Gaussian-Schell beams in turbulent atmosphere is derived. As an example, the kurtosis parameter for elliptical Gaussian-Schell beams in turbulent atmosphere is investigated in detail. The comparison between the variations of the kurtosis parameter for elliptical Gaussian-Schell beams with different parameters is made. Meanwhile, the difference between the kurtosis parameter in vacuum and in turbulent atmosphere is analyzed and discussed.     

Propagation properties of cosh-Airy beams

We theoretically investigate the propagation properties of cosh-Airy beams, which can be considered as a superposition of two Airy beams with different decay factors. We find that the field distribution of cosh-Airy beams is the same as that of Airy beams. Moreover, we find that the evolution of cosh-Airy beams is determined by the parameters of the cosh modulation function, in addition to the transverse scale factor and decay factor of the Airy beams. Our results demonstrate a possible method of manipulating Airy beams in free space. They can also be extended to the study of the propagation properties of cosine-Airy beams (or sine-Airy beams).     

Beam wander and M2-factor of partially coherent electromagnetic hollow Gaussian beam propagating through non-Kolmogorov turbulence

Propagation formulae for M²-factor and beam wander of partially coherent electromagnetic hollow Gaussian (PCEHG) beam in non-Kolmogorov turbulence are derived based on the extended Huygens–Fresnel principle and the second-order moments of the Wigner distribution function. Our results indicate that the normalized M²-factors of PCEHG beam with larger beam order, waist width, inner scale of turbulence, the generalized exponent parameter, and smaller transverse coherent widths, outer scale of turbulence, the generalized structure parameter are less affected by the turbulence. The root mean square beam wander and relative beam wander are more obvious for PCEHG beam with smaller beam order, larger inner and outer scales of turbulence, exponent parameter, transverse coherent widths, and the generalized structure parameter. What is more, the beam wander properties of PCEHG beam in non-Kolmogorov turbulence are very different from M²-factor and spreading properties of beam in turbulence.     

Effect of turbulence on the beam quality of apertured partially coherent beams

The effects of turbulence on the beam quality of apertured partially coherent beams have been studied both analytically and numerically. Taking the Gaussian Schell-model (GSM) beam as a typical example of partially coherent beams, closed-form expressions for the average intensity, mean-squared beam width, power in the bucket, beta parameter, and Strehl ratio of apertured partially coherent beams propagating through atmospheric turbulence are derived. It is shown that the smaller the beam truncation parameter is, the less affected by turbulence the apertured partially coherent beams are. Furthermore, the apertured partially coherent beams are less sensitive to the effects of turbulence than unapertured ones. The main results are interpreted physically.     

Analytical expression for the kurtosis parameter of flattened Gaussian beams propagating through ABCD optical systems

The closed-form propagation expression for the kurtosis parameter of flattened Gaussian beams (FGBs) in passage through paraxial ABCD optical systems is derived on the basis of the propagation equation of FGBs and analysed with numerical examples. A comparison of our results with those obtained by using the propagation law of irradiance moments is made, showing the advantage of the method used in this paper.     

On the existence of maximum entropy distributions with four and more assigned moments

It is known that the probability distribution satisfy the Maximum Entropy Principle (MEP) if the available data consist in four moments of probability density function. Two problems are typically associated with use of MEP: the definition of the range of acceptable values for the moments M_i; the evaluation of the coefficients a_j. Both problems have already been accurately resolved by analytical procedures when the first two moments of the distribution are known.

In this work, the analytical solution in the case of four known moments is provided and a criterion for confronting the general case (whatever the number of known moments) is expounded. The first four moments are expressed in nondimensional form through the expectation and the coefficients of variation, skewness and kurtosis. The range of their acceptable values is obtained from the analytical properties of the differential equations which govern the problem and from the Schwarz inequality.     

M2-factor of truncated partially coherent beams propagating through atmospheric turbulence

A method of studying the M2-factor of truncated partially coherent beams both in free space and in turbulence is proposed, i.e., the method of the window function being expanded into a finite sum of complex-valued Gaussian functions. Taking the Gaussian Schell-model (GSM) beam as a typical example of partially coherent beams, the analytical formula of the M2-factor of truncated GSM beams propagating through atmospheric turbulence is derived. It is shown that the M2-factor decreases as the truncation parameter δ and the coherence parameter α increase. However, the M2-factor in turbulence is more sensitive to δ than that in free space. On the other hand, the M2-factor of truncated partially coherent beams with smaller δ is more affected by turbulence. In addition, the effect of turbulence on the M2-factor of truncated GSM beams is less sensitive to the coherence parameter α than that of nontruncated GSM beams.     

Mode coupling approach to beam propagation in atmospheric turbulence

A mode coupling approach based on the modal theory of coherence is suggested for the study of partially coherent beams in atmospheric turbulence. An approximate expression is derived for the mode power coupling coefficients, and some specific cases are studied using numerical methods. Several general results derived from the properties of the coupling coefficients are also presented.     

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.     

Generation of high quality Airy beams with blazed micro-optical cubic phase plates

We design and fabricate a hybrid refractive-diffractive cubic phase plate (CPP) with a combined conventional blazed grating for generating high quality Airy beams. The grating enables elimination of direct incident illumination in the reconstructed beam. The CPP is fabricated in a negative photoresist on a substrate by laser direct writing lithography with precise exposure control of gray scales. Experimentally measured intensity distribution of the Airy beam is found in good agreement with the theoretical predictions. Furthermore, self-healing and nondiffraction properties of the Airy beam are verified experimentally. The proposed method gives rise to a simple, reliable, and low-cost micro-optical element solution for the generation of high quality Airy beams.     

An analytical computation method for statistical tolerance analysis of assemblies with truncated normal mean shift

This paper proposes an analytical solution for fast tolerance analysis of the assembly of components with a mean shift or drift in the form of a doubly-truncated normal distribution. The assembly of components with a mean shift or drift in the form of a uniform distribution (the Gladman model) can be calculated by this method as well since the uniform distribution is a special form of the doubly-truncated normal distribution. Integration formulae of the first four moments of the truncated normal distribution are first derived. The first four moments of the resultant tolerance distribution are then calculated. As a result, the resultant tolerance specification is represented as a function of the standard deviation and the coefficient of kurtosis of the resultant distribution. Based on this method, the calculated resultant tolerance specification is more accurate than that predicted by the Gladman's model or the simplified truncated normal model. The difference between this model and the Monte Carlo method with 1,000,000 simulation samples is less than 0.5%. The merit of the proposed method is that it is fast and accurate which is crucial for engineering applications in tolerance analysis.     

Non-parametric estimation of conditional moments for sensitivity analysis

In this paper, we consider the non-parametric estimation of conditional moments, which is useful for applications in global sensitivity analysis (GSA) and in the more general emulation framework. The estimation is based on the state-dependent parameter (SDP) estimation approach and allows for the estimation of conditional moments of order larger than unity. This allows one to identify a wider spectrum of parameter sensitivities with respect to the variance-based main effects, like shifts in the variance, skewness or kurtosis of the model output, so adding valuable information for the analyst, at a small computational cost.     

Comparative study of the beam-width spreading of partially coherent Hermite-sinh-Gaussian beams in atmospheric turbulence

Taking the partially coherent Hermite-sinh-Gaussian (H-ShG) beam as a more general type of partially coherent beams, a comparative study of the beam-width spreading of partially coherent H-ShG beams in atmospheric turbulence is performed by using the relative width, normalized beam width, and turbulence length. It is shown that the relative width versus the beam parameters, such as the spatial correlation length sigma(0), beam orders m, n, Sh-part parameter Omega(0), and waist width w(0), provides a simple and intuitive insight into the beam-width spreading of partially coherent H-ShG beams in turbulence, and the results are consistent with those using the turbulence length. The validity of our results is interpreted physically.     

Propagation of partially coherent flat-topped beams through a turbulent atmosphere

Based on the modified beam model for flat-topped beams and the Schell model for partially coherent light, an expression for partially coherent flat-topped (PCFT) beams has been proposed. The propagation characteristics of PCFT beams with circular symmetry through a turbulent atmosphere have been studied. By using the generalized Huygens-Fresnel integral and Fourier transform method, the expressions for the cross-spectral density function and the average intensity have been given and the analytical expression for the root-mean-square width has been derived. The effects of the beam order, the spatial coherence, and the turbulent parameter on the intensity distributions and beam spreading have been discussed in detail. Our results show that the on-axis intensity of the beams decreases with increasing turbulence and decreasing coherence of the source, whereas the on-axis intensity of the beams in the far field decreases slightly with increasing beam order. The relative spreading of PCFT beams is smaller for beams with a higher order, a lower degree of global coherence of the source, a larger inner scale, and a smaller outer scale of the turbulence.     

A Method of Estimating the Process Capability Index from the First Four Moments of Non‐normal Data

A method is presented to estimate the process capability index (PCI) for a set of non‐normal data from its first four moments. It is assumed that these four moments, i.e. mean, standard deviation, skewness, and kurtosis, are suitable to approximately characterize the data distribution properties. The probability density function of non‐normal data is expressed in Chebyshev–Hermite polynomials up to tenth order from the first four moments. An effective range, defined as the value for which a pre‐determined percentage of data falls within the range, is solved numerically from the derived cumulative distribution function. The PCI with a specified limit is hence obtained from the effective range. Compared with some other existing methods, the present method gives a more accurate PCI estimation and shows less sensitivity to sample size. A simple algebraic equation for the effective range, derived from the least‐square fitting to the numerically solved results, is also proposed for PCI estimation. Copyright © 2004 John Wiley & Sons, Ltd.     

Propagation of multi-Gaussian beams in incoherent combination through turbulent atmosphere and their beam quality

Based on the extended Huygens–Fresnel principle, the propagation of M?×?N multi-Gaussian beams in incoherent combination through turbulent atmosphere, and their beam quality are studied. The power in the bucket (PIB), β parameter and strehl ratio (S_R ) are taken as the characteristic parameters of beam quality, and analytical expressions for PIB and S_R are derived. It is shown that multi-Gaussian beams in turbulent atmosphere undergo three stages of evolution with increasing propagation distance z, and turbulence accelerates the evolution of the three stages which multi-Gaussian beams undergo. The turbulence results in a degradation of the beam quality, and multi-Gaussian beams with higher numbers M and N are less sensitive to the effects of turbulence than those with lower M and N.     

Optical trapping with focused Airy beams

Airy beams are attractive owing to their two intriguing properties--self-bending and nondiffraction--that are particularly helpful for optical manipulation of particles. We perform theoretical and experimental investigations into the focusing property of Airy beams and provide insight into the trapping ability of tightly focused Airy beams. Experiment on optical tweezers demonstrates that the focused Airy beams can create multiple traps for two-dimensional confining particles, and the stable traps exist in the vicinity of the main intensity lobes in the focused beams. The trapping pattern can be varied with changes in the cross section of the focused beam. The focused Airy beam offers a novel way of optically manipulating microparticles.