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.     

Propagation properties of rectangular Laguerre–Gaussian-correlated Schell-model beams in atmospheric turbulence

Based on the extended Huygens–Fresnel principle and the second-order moments of the Wigner distribution function (WDF), the analytical expressions for the cross-spectral density (CSD) and the propagation factor of a rectangular Laguerre–Gaussian-correlated Schell-model (LGCSM) beam propagating in atmospheric turbulence are derived. The statistical properties, such as the average intensity, the spectral degree of coherence (SDOC) and the propagation factor, of a rectangular LGCSM beam in free space and atmospheric turbulence are comparatively analysed. It is illustrated that a rectangular LGCSM beam exhibits self-splitting and combing properties on propagation in atmospheric turbulence, and the self-splitting properties of such beam are closely related to its beam orders m and n, which is quite different from other self-splitting beams. In addition, the rectangular LGCSM beam has an advantage for reducing the turbulence-induced degradation compared with the conventional partially coherent beams.     

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.     

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.     

Propagation of partially coherent Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system in a turbulent atmosphere

Based on the generalized Huygens–Fresnel integral, propagation of partially coherent Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system in a turbulent atmosphere was investigated. Analytical propagation formulae were derived for the cross-spectral densities of partially coherent Lorentz and Lorentz–Gauss beams. As an application example, the focusing properties of partially coherent Gaussian, Lorentz and Lorentz–Gauss beams in a turbulent atmosphere and in free space were studied numerically and comparatively. It is found that the focusing properties of such beams are closely related to the initial coherence length and the structure constant of turbulence. By choosing a suitable initial coherence length, a partially coherent Lorentz beam can be focused more tightly than a Gaussian or Lorentz–Gauss beam in free space or in a turbulent atmosphere with small structure constant at the geometrical focal plane.     

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.     

Propagation properties of Airy–Gaussian beams in centrosymmetric photorefractive media

Based on the biased centrosymmetric photorefractive media, we investigate numerically the propagation and interaction properties of Airy–Gaussian beams. The single Airy–Gaussian beam forms the one-component breather with the help of the photorefractive nonlinearity. The interaction properties of two Airy–Gaussian beams can be controlled by adjusting the relative parameters, such as photorefractive nonlinearity, transverse distance and relative phase of two incident beams. The two-component breather with ladder structure can be observed for both the in-phase and out-of-phase cases when the self-acceleration property is balanced by the photorefractive nonlinearity. The one- or three-component breathers can be observed for the in-phase case only when the transverse distance takes a certain range.     

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.     

Representation and focusing properties of higher-order radially polarized Laguerre–Gaussian beams

A method based on higher-order Poincaré sphere was proposed to represent the states of polarization of higher-order radially polarized Laguerre–Gaussian (LG) vectorial beams (VBs). And the focusing properties of such LG beams of different radial orders focused by a high-NA lens were discussed. By tuning the ratio of the pupil radius to the waist of the incident beams, some cage-like or needlelike electric intensity field is generated in the focal region for several specific LG VBs with high order. Modulated by diffractive optical elements, the shape of the focal field shows novel behaviors such as splitting of cage-like modes, which provides potentially a more flexible control over micro-particles.     

Propagation properties of vectorial Hermite–cosine–Gaussian beams beyond the paraxial approximation

Starting from the vectorial Rayleigh–Sommerfeld diffraction integrals, the analytical propagation expressions of vectorial nonparaxial Hermite–cosine–Gaussian (HCosG) beams in free space are derived. The on-axis intensity, far-field equation and, in particular, paraxial expressions are given and treated as special cases of our result. It is shown that for vectorial nonparaxial HCosG beams, the parameter f = 1/kw ₀, with k being the wave number and w ₀ being the waist width, determines their nonparaxiality. What is more, the decentered parameter also affects their intensity distribution and nonparaxial behavior. The calculation results indicate that the position of maximum on-axis intensity changes with the mode indices.     

Atom guiding along high order Laguerre–Gaussian light beams formed by spatial light modulation

A spatial light modulator (SLM) has been used to create high quality Laguerre–Gaussian (LG) light beams, which have been used to study the guiding of cold rubidium atoms. The SLM allows real-time variation of the hollow guiding beam and permits direct comparison of the guided atom fluxes for different LG modes with minimal adjustment of the other optical components. It is demonstrated that, by increasing the azimuthal index l of the Laguerre–Gaussian beam, the radiation pressure pushing the trapped atoms may be reduced while maintaining the same guided flux. This is the first comparative study of hollow beam atom guiding, and further demonstrates the versatility of the SLM for studies in atom optics.     

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.     

Response of dynamic systems to renewal impulse processes: Generating equation for moments based on the integro-differential Chapman–Kolmogorov equations

In the present paper the method is developed for the derivation of differential equations for statistical moments of the state vector (response) of a non-linear dynamic system subjected to a random train of impulses. The arrival times of the impulses are assumed to be driven by a non-Poisson counting process. The state vector of the dynamic system is then a non-Markov process and no method is directly available for the derivation of the equations for response moments. The original non-Markov problem is converted into a Markov one by recasting the excitation process with the aid of an auxiliary, pure-jump stochastic process characterized by a Markov chain. Hence the conversion is carried out at the expense of augmentation of the state space of the dynamic system by auxiliary Markov states. For the augmented problem the sets of forward and backward integro-differential Chapman–Kolmogorov equations are formulated. The general, generating equation for moments is obtained with the aid of the forward and backward integro-differential Chapman–Kolmogorov operators. The developed method is illustrated by the examples of several renewal impulse processes.     

Error performance analysis for FSO systems with diversity reception and optical preamplification over gamma–gamma atmospheric turbulence channels

The error performance was investigated of free-space optical (FSO) systems that use optical preamplifiers and diversity reception in turbulent atmosphere with gamma–gamma distribution. More specifically, assuming that the dominant sources of the receiver noises are the background light and amplifier spontaneous emission noises, we develop an exact bit-error-rate (BER) expression for binary pulse position modulation (BPPM) by a moment-generating function (MGF) derivative-based method. To gain more insight, we also derive a closed-form asymptotic BER expression for BPPM at high received optical power, by which we obtain the diversity and coding gains of the considered systems. Our analytical results lead to the observations that the attained diversity gain only changes with the gamma-gamma distribution parameters and the number of spatial modes collected by the receiver, while independent on the number of temporal modes passed by the optical preamplifiers. We verify the analytical results by Monte Carlo simulations.     

Stress–strain curve for concrete in circular columns based on elastoplastic analysis

This paper presents a stress–strain curve that describes the axial behavior of concrete in circular columns with lateral reinforcement. As opposed to most studies on this subject, which used semiempirical methods, the proposed stress–strain relations are based on a theoretical derivation. They have been derived from analysis of the problem’s full range according to the theories of elasticity and plasticity. Based on these theories, the current study analytically examines the influence of the main variables, such as the volumetric lateral reinforcement ratio and the material properties, on the behavior of circular confined concrete columns and proposes theoretical expressions that describe their stress–strain relations. Application of the proposed curve shows good agreement with published test results. Since these expressions were derived from a theoretical analysis, they can be considered as an analytical verification of existing empirical curves, yet they are also simple enough for practical applications.     

A solution-processable D–A–D small molecule based on isoindigo for organic solar cells

A new linear D–A–D organic small molecule (M1), with triphenylamine as electron donor (D) unit and isoindigo (ID) as electron acceptor (A) unit, was synthesized by Stille coupling reactions. It exhibits broad and strong absorption (300–700 nm), a relatively low HOMO energy level (?5.30 eV), low band gap (1.69 eV), and moderate hole mobility (2.49×10^?4 cm²/Vs). Solution-processed small molecule bulk-heterojunction solar cells based on M1: PC₆₁BM (1:3, w/w) blend film exhibits a power conversion efficiency of 0.84 % with an open-circuit voltage (V _oc) of 0.78 V, under the illumination of AM1.5, 100 mW/cm².