Hypergeometric Gaussian Research Articles

Propagation of auto-focusing hypergeometric Gaussian beams along a slant path in oceanic turbulence

Compared to horizontal transmission, the oceanic dissipation rate and temperature-salinity distribution ratio are no longer constant but vary with depth, imposing greater complexity on oceanic turbulence when beams propagate through a slant path and resulting in more limitations on the performance of underwater wireless optical communication (UWOC) links. This study focuses on investigating the performance, especially the auto-focusing characteristic, of auto-focusing hypergeometric Gaussian (AHGG) beams propagating along slant paths in oceanic turbulence. We theoretically derive the spatial coherence radius and the relative probability of the orbital angular momentum (OAM) mode for AHGG beams passing through such links. Numerical simulations reveal that AHGG beams exhibit superior propagation performance compared to hypergeometric Gaussian beams. Lower beam orders and OAM numbers contribute to improved performance, while careful selection of auto-focusing length can tangibly enhance detection performance as well. Additionally, tidal velocities and wind speeds have nonnegligible effects on OAM signal probability. Our results further demonstrate that surface buoyancy flux, temperature gradients, and waterside friction velocity significantly affect beam transmission under varying wind conditions. These findings, particularly controlling the auto-focusing length of AHGG beams to match the transmission distance, provide valuable insights for enhancing the quality of UWOC links.

EXPANSIONS OF KAMP ́E DE F ́ERIET HYPERGEOMETRIC FUNCTIONS

When studying the properties of the hypergeometric functions in two variables, expansion formulas are very important, allowing one to represent a function of two variables in the form of an infinite sum of products of several hypergeometric Gaussian functions, and this in turn facilitates the process of studying the properties of functions in two variables. Burchnall and Chaundy, in 1940--41, using the symbolic method, obtained more than 15 pairs of expansions for the second-order double hypergeometric Appell and Humbert functions. In order to find expansion formulas for functions depending on three or more variables, Hasanov and Srivastava introduced symbolic operators, with the help of which they were able to expand a whole class of hypergeometric functions of several variables. Hasanov, Turaev and Choi defined so-called H-operators that make it possible to find expansions for generalized hypergeometric functions of one variable. In addition, applications of these H-operators to the expansion of the hypergeometric functions of two and three variables of second order are known. On the other hand, thanks to the Kampe de Feriet functions, solutions of the boundary value problems for some degenerate and singular partial differential equations can be written in explicit forms. In this paper, expansion formulae are obtained for the hypergeometric Kampe de Feriet functions of the superior order. Some Kampe de Feriet functions are expanded in terms of the Appell and Humbert functions as illustrative examples

Average capacity of an underwater wireless communication link with the quasi-Airy hypergeometric-Gaussian vortex beam based on a modified channel model.

Prompted by alleviating the random perturbation of underwater channel and enhancing the performance for the orbital angular momentum (OAM) -based underwater wireless optical communication (UWOC), the quasi-Airy Hypergeometric-Gaussian (QAHyGG) vortex beam is first proposed and demonstrated. Moreover, an underwater channel model is first modified for more accurate simulated results of the propagation property of various beams. Based on the modified model, the transmission and communication performance of three different OAM-carrying beams (the Gauss vortex (GV) beam, the Hypergeometric-Gaussian (HyGG) vortex beam, and the QAHyGG vortex beam) are comparatively studied. In addition, the parameters optimization of the QAHyGG vortex beam is made for further enhancing the average capacity. The results show that the QAHyGG vortex beam exhibits higher received power and lower crosstalk probability under different channel conditions. The average capacity of the QAHyGG vortex beam has enhanced by ∼8% and ∼27% compared with the HyGG vortex beam and the GV beam at 100m, respectively. The QAHyGG vortex beam is more suitable in an OAM-based UWOC system with a limited-size receiving aperture or lower transmit power. Besides, the average capacity will improve effectively at longer distances with the optimized beam parameters. These research results can provide advances in designing the practical OAM-based UWOC system.

Channel capacity and quantum entanglement of autofocusing hypergeometric-Gaussian beams through non-Kolmogorov turbulence

We establish propagation models for distributing autofocusing hypergeometric-Gaussian (HyGG) beams via non-Kolmogorov atmospheric turbulence in the single-photon and biphoton cases. Analytical expressions of the channel capacity and entanglement are employed to explore the communication performance in turbulence. It can be demonstrated that in the single-photon case, a lower hollowness parameter and larger topological charge could contribute to the anti-interference ability of HyGG beams. Such an anti-interference ability could be affected by the autofocusing distance more severely for beams with larger topological charges. In addition, the channel capacity of beams at a certain distance decays rapidly at first before it grows steadily with the increase in the power-law exponent of the non-Kolmogorov spectrum. For the biphoton case, the higher hollowness parameter, larger power-law exponent of the non-Kolmogorov spectrum and larger topological charge make HyGG beams maintain a high concurrence at longer distances. We believe that our results could serve as a reference for improving the quality of orbital-angular-momentum communication of HyGG beams via atmospheric turbulence.

Orbital angular momentum state variation of vortex beams propagating in a plasma sheath turbulence

The presence of turbulent media is one of the major problems in the free-space communications. This arises from the sever distortion of spatial modes which leads to data degradation. In recent years, utilizing pseudo-nondiffracting beams has been proposed as an efficient way for mitigating the disturbance induced by turbulence. In this regard, this paper is intended to theoretically investigate anisotropic plasma turbulence (PT) effects on the orbital angular momentum (OAM) spectra and phase evolutions of different kinds of pseudo-nondiffracting vortex beams (VB), including Bessel-Gaussian (BG) and Hypergeometric-Gaussian (HGG) beams. For this purpose, using the modified von Karman spectrum in the frame of Rytov theory, the orbital OAM-spectra of VBs propagating in anisotropic PT are derived, and the effects of anisotropy and source beams parameters on the mode probability of transmitted VBs are studied. Our numerical analysis reveals that when propagating VBs carrying OAM come across PT, under influences of such random medium, their OAM-spectra spread, and several side-band modes which induce phase perturbations are generated. As well as, the propagation performance of pseudo-nondiffracting VBs in an anisotropic PT is better than isotropic one. Furthermore, it is found out that under the equivalent PT conditions BG beams are less influenced by PT in comparison with HGG beams. We expect our results to provide advances on turbulence-resilient communications, free-space optical data transmissions, and imaging.

Enhanced detection techniques of orbital angular momentum states in the classical and quantum regimes

The orbital angular momentum (OAM) of light has been at the center of several classical and quantum applications for imaging, information processing and communication. However, the complex structure inherent in OAM states makes their detection and classification nontrivial in many circumstances. Most of the current detection schemes are based on models of the OAM states built upon the use of Laguerre–Gauss (LG) modes. However, this may not in general be sufficient to capture full information on the generated states. In this paper, we go beyond the LG assumption, and employ hypergeometric-Gaussian (HyGG) modes as the basis states of a refined model that can be used—in certain scenarios—to better tailor OAM detection techniques. We show that enhanced performances in OAM detection are obtained for holographic projection via spatial light modulators in combination with single-mode fibers (SMFs), and for classification techniques based on a machine learning approach. Furthermore, a three-fold enhancement in the SMF coupling efficiency is obtained for the holographic technique, when using the HyGG model with respect to the LG one. This improvement provides a significant boost in the overall efficiency of OAM-encoded single-photon detection systems. Given that most of the experimental works using OAM states are effectively based on the generation of HyGG modes, our findings thus represent a relevant addition to experimental toolboxes for OAM-based protocols in quantum communication, cryptography and simulation.

Research on Hypergeometric-Gaussian Vortex Beam Propagating under Oceanic Turbulence by Theoretical Derivation and Numerical Simulation

In this paper, we use two methods to research the propagation characteristics of a Hypergeometric-Gaussian (HyGG) vortex beam under oceanic turbulence. One is numerical calculation based on the Rytov approximation theory, where the theoretical detection probability equation of the HyGG vortex beam propagating through oceanic turbulence is derived. The other is numerical simulation based on random phase screens model of oceanic turbulence, where the influences generated by oceanic turbulence on the phase and intensity of the propagation beam as well as the propagation of the beam through several independent phase screens, kept at the same distance, have the same effect. The effects of oceanic turbulence parameters and initial beam parameters on the detection probability of the HyGG vortex beam at the receiver are discussed. The results of theoretical derivation are well in agreement with those of numerical simulation, which demonstrated that the numerical simulation method could effectively simulate the complex theoretical derivation. Both results show that with higher dissipation rate of kinetic energy per unit mass of fluid, smaller dissipation rate of mean-squared temperature and lower temperature-salinity contribution ratio comes the better detection probability. Meanwhile, a HyGG vortex beam with smaller topological charge and longer wavelength has a superior turbulent resistance property. It provides a promising way to estimate the propagation characteristics of the optical beams in an underwater environment.

Интегро-дифференциальные уравнения на замкнутом контуре с функцией Гаусса в ядре

Integro-differential equations with kernels including hypergeometric Gaussian function that depends on the arguments ratio are studied over a closed curve in the complex plane. Special cases of the equations considered are the special integro-differential equation with Cauchy kernel, equations with power and logarithmic kernels. By means of the curvilinear convolution operator with the kernel of special kind, the equations with derivatives are reduced to the equations without derivatives. We find out the connection between special cases of the above-mentioned convolution operator and the known integral representations of piecewise analytical functions applied in the study of boundary value problems of the Riemann type. The correct statement of Noetherian property for the investigated class of equations is given. In this case, the operators corresponding to the equations are considered acting from the space of summable functions into the space of fractional integrals of the curvilinear convolution type. Examples of integro-differential equations solvable in a closed form are given.

Propagation of hypergeometric Gaussian beams in strongly nonlocal nonlinear media

Optical vortex beams have attracted lots of interest due to its potential application in image processing, optical trapping and optical communications, etc. In this work, we theoretically and numerically investigated the propagation properties of hypergeometric Gaussian (HyGG) beams in strongly nonlocal nonlinear media. Based on the Snyder–Mitchell model, analytical expressions for propagation of the HyGG beams in strongly nonlocal nonlinear media were obtained. The influence of input power and optical parameters on the evolutions of the beam width and radius of curvature is illustrated, respectively. The results show that the beam width and radius of curvature of the HyGG beams remain invariant, like a soliton when the input power is equal to the critical power. Otherwise, it varies periodically like a breather, which is the result of competition between the beam diffraction and nonlinearity of the medium.

Mode coupling in vortex beams

We examine the mode coupling in vortex beams. Mode coupling also known as the crosstalk takes place due to turbulent characteristics of the atmospheric communication medium. This way, the transmitted intrinsic mode of the vortex beam leaks power to other extrinsic modes, thus preventing the correct detection of the transmitted symbol which is usually encoded into the mode index or the orbital angular momentum state of the vortex beam. Here we investigate the normalized power mode coupling ratios of several types of vortex beams, namely, Gaussian vortex beam, Bessel Gaussian beam, hypergeometric Gaussian beam and Laguerre Gaussian beam. It is found that smaller mode numbers lead to less mode coupling. The same is partially observed for increasing source sizes. Comparing the vortex beams amongst themselves, it is seen that hypergeometric Gaussian beam is the one retaining the most power in intrinsic mode during propagation, but only at lowest mode index of unity. At higher mode indices this advantage passes over to the Gaussian vortex beam.

Evolution of orbital angular momentum mode of the autofocusing Hypergeometric-Gaussian beams through moderate-to-strong anisotropic non-Kolmogorov turbulence

In order to investigate the evolution of mode probability density (MPD) and received power of orbital angular momentum (OAM) mode carried by autofocusing Hypergeometric-Gaussian (HyGG) beams in moderate-to-strong anisotropic non-Kolmogorov turbulence, the corresponding expressions of the spatial coherence radius and spiral spectrum of HyGG beams are achieved. Results show that the autofocusing property of HyGG beams has a significant impact on the propagation of the MPD. Smaller topological charge, Shorter wavelength, higher values of hollowness parameters and wider beam waist shrink to a narrower width of the MPD and maintain the characteristic of diffraction-free with a longer transmission distance. Moreover, smaller outer-scale, larger inner-scale, larger non-Kolmogorov parameter and larger anisotropic coefficient make the received power of OAM suffer less turbulence.

The algorithms of constructing the continued fractions for any rations of the hypergeometric Gaussian functions

Описано алгоритм побудови рекурентних спiввiдношень гiпергеометричних функцiй Гаусса, в яких змiщення параметрiв a, b, c дорiвнює 0, 1 або −1. На основi таких рекурентних спiввiдношень побудовано розвинення для вiдношення функцiй Гаусса у неперервнi дроби. Отриманi неперервнi дроби є розвиненням вiдповiдних гiпергео- метричних функцiй Гаусса, якщо параметри функцiї є цiлими числами.

Nonlinear interaction of intense hypergeometric Gaussian subfamily laser beams in plasma

Propagation of Hypergeometric-Gaussian laser beam in a nonlinear plasma medium is investigated by considering the Source Dependent Expansion method. A subfamily of Hypergeometric-Gaussian beams with a non-negative, even and integer radial index, can be expressed as the linear superposition of finite number of Laguerre–Gaussian functions. Propagation of Hypergeometric-Gaussian beams in a nonlinear plasma medium depends on the value of radial index. The bright rings’ number of these beams is changed during the propagation in plasma medium. The effect of beam vortex charge number l and initial (input) beam intensity on the self-focusing of Hypergeometric-Gaussian beams is explored. Also, by choosing the suitable initial conditions, Hypergeometric-Gaussian subfamily beams can be converted to one or more mode components that a typical of mode conversion may be occurred. The self-focusing of these winding beams can be used to control the focusing force and improve the electron bunch quality in laser plasma accelerators.

Effects of non-Kolmogorov turbulence on the spiral spectrum of Hypergeometric-Gaussian laser beams.

We study the effects of non-Kolmogorov turbulence on the orbital angular momentum (OAM) of Hypergeometric-Gaussian (HyGG) beams in a paraxial atmospheric link. The received power and crosstalk power of OAM states of the HyGG beams are established. It is found that the hollowness parameter of the HyGG beams plays an important role in the received power and crosstalk power. The larger values of hollowness parameter give rise to the higher received power and lower crosstalk power. The results also show that the smaller OAM quantum number and longer wavelength of the launch beam may lead to the higher received power and lower crosstalk power.

FOCUSING PROPERTIES OF HYPERGEOMETRIC GAUSSIAN BEAM THROUGH A HIGH NUMERICAL-APERTURE OBJECTIVE

The focusing properties of radially polarized hypergeometric Gaussian beam are studied using the Richards-Wolf vectorial diffraction model. Such a polarized beam is decomposed into radial and longitudinal polarization. With a proper combination of the beam order, beam size and imaginary parameter variables, the adjustably confined flat-topped focus and focal hole can be obtained in the focal region. Moreover, we got originality characteristic for the axial intensity distribution of two shaped symmetric light spots. The tight focusing of a hypergeometric Gaussian beam may find applications in data storage, laser drilling, optical trapping, etc.

Random wandering of laser beams with orbital angular momentum during propagation through atmospheric turbulence.

The propagation of laser beams having orbital angular momenta (OAM) in the turbulent atmosphere is studied numerically. The variance of random wandering of these beams is investigated with the use of the Monte Carlo technique. It is found that, among various types of vortex laser beams, such as the Laguerre-Gaussian (LG) beam, modified Bessel-Gaussian beam, and hypergeometric Gaussian beam, having identical initial effective radii and OAM, the LG beam occupying the largest effective volume in space is the most stable one.

Scintillation analysis of hypergeometric Gaussian beam via phase screen method

We give a scintillation treatment of hypergeometric Gaussian beams via the use of random phase screens. In particular, we analyse the on-axis, point-like and aperture averaged power scintillation characteristics of this beam that cannot be undertaken easily by analytic means. Within the range of examined source and propagation parameters, our evaluations show that there will be less scintillation, with increasing hollowness at small source sizes and zero topological charge. At larger source sizes or topological charges, this is reversed and decreasing hollowness will reduce scintillation. More or less the same trend is observed for aperture averaging such that at small source sizes and zero topological charge, increased hollowness will result in lower scintillation. At larger source size and topological charges, there will be a transition from the case of smaller values of hollowness giving rise to less scintillation at smaller aperture openings to the case of larger values of hollowness giving rise to less scintillation at larger aperture openings. In general nonzero topological charges will produces more scintillations, both in on-axis and aperture averaged cases.

Hypergeometric Gaussian beam and its propagation in turbulence

We study propagation characteristics of hypergeometric Gaussian beam in turbulence. In this context, we formulate the receiver plane intensity using extended Huygens–Fresnel integral. From the graphical results, it is seen that, after propagation, hypergeometric Gaussian will in general assume the shape of a dark hollow beam at topological charges other than zero. Increasing values of topological charge will make the beam broader with steeper walls. On the other hand, higher values of hollowness parameter will contract into a narrower shape. Raising the topological charge or the hollowness parameter individually will cause outer rings to appear. Both increased levels of turbulence and longer propagation distances will accelerate the beam evolution and help reach the final Gaussian shape sooner. At lower wavelengths, there will be less beam spreading.

Hollow sinh-Gaussian beams and their paraxial properties

A new mathematical model of dark-hollow beams, described as hollow sinh-Gaussian (HsG) beams, has been introduced. The intensity distributions of HsG beams are characterized by a single bright ring along the propagation whose size is determined by the order of beams; the shape of the ring can be controlled by beam width and this leads to the elliptical HsG beams. Propagation characteristics of HsG beams through an ABCD optical system have been researched, they can be regarded as superposition of a series of Hypergeometric-Gaussian (HyGG) beams. As a numerical example, the propagation characteristics of HsG beams in free space have been demonstrated graphically.

A nonlocal problem for the Bitsadze-Lykov equation

We study a nonlocal boundary-value problem for a degenerate hyperbolic equation. We prove that this problem is uniquely solvable if Volterra integral equations of the second kind are solvable with various values of parameters and a generalized fractional integro-differential operator with a hypergeometric Gaussian function in the kernel.