hypergeometric-Gaussian Beams 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.

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.

Certain integral transforms involving Appell and Bessel functions and their applications

Abstract This paper deals with the evaluation of certain integral transforms involving the product of certain Appell and Bessel functions with a weight e - γ ⁢ t 2 {e^{-\gamma t^{2}}} . The transformations of these integrals are evaluated in terms of the Appell, Kampé de Fériet and the triple hypergeometric functions. As an application, we studied propagation of generalized Humbert–Gaussian beams (GHGBs) and hypergeometric-Gaussian beams (HyGGBs) in turbulent atmosphere and through an ABCD paraxial optical system. The evaluation of these integral transforms has initiated a great interest in mathematical physics and its applications to laser physics and linear or non-linear optics.

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.

Knotted trajectories of neutral and charged particles in Gaussian light beams

Making use of the equivalence between paraxial wave equation and two-dimensional Schr\"odinger equation, Gaussian beams of monochromatic light, possessing knotted nodal structures are obtained in an analytical way. These beams belong to the wide class of paraxial beams called the Hypergeometric-Gaussian beams [E. Karimi, G. Zito, B. Piccirillo, L. Marrucci and E. Santamato, Opt. Lett. {\bf 32}, 3053(2007)]. Four topologies are dealt with: the unknot, the Hopf link, the Borromean rings and the trefoil. It is shown in the numerical way that neutral polarizable particles placed in such light fields, upon precise tuning of the initial conditions, can be forced to follow the identical knotted trajectories. A similar outcome is also valid for charged particles that are subject to a ponderomotive potential. This effect can serve to precisely steer particles along chosen complicated pathways exhibiting non-trivial topological character, guide them around obstacles and seems to be helpful in engineering more complex nanoparticles.

Super pulses of orbital angular momentum in fractional-order spiroid vortex beams.

We consider optical properties of hypergeometric-Gaussian beam compositions with spiral-like intensity and phase distributions called spiroid beams. Their orbital angular momentum as a function of a fractional-order topological charge has a chain of super-pulses (bursts and dips). The form of the super-pulses can be controlled by the spiral parameters. Such a phenomenon can be used in optical switches and triggers for optical devices and communication systems.

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.

Theoretical conversion of the hypergeometric-Gaussian beams family into a high-order spiraling Bessel beams by a curved fork-shaped hologram

Based on the process of the Fresnel diffraction, the possibility of generating a new type of laser beams family by illuminating a curved fork-shaped hologram, with an input hypergeometric-Gaussian beams family of orders n and m is studied in this paper. The theoretical and the numerical results showed that, at the output plane, a high order spiraling Bessel vortex beam is produced. This vortex beam is divergence or non-divergence depending upon the waist position of the input hypergeometric-Gaussian beams, regarding the plane where the curved fork-shaped hologram is situated. Analytical expressions of the amplitude and the intensity distribution of the diffracted wave field are calculated and deduced using the stationary phase method. The actual work generalizes also the Fresnel diffraction study of some subfamilies of the hypergeometric-Gaussian beams family, such as: fundamental Gaussian, hollow Gaussian, modified quadratic Bessel–Gaussian and elegant Laguerre–Gaussian beams.

Conversion of circular beams by a spiral phase plate: Generation of Generalized Humbert beams

Generalized Humbert beams (GHBs) are a novel donut beams family which created by converting of circular beams (CiBs) passing through a paraxial ABCD optical system with a spiral phase plate (SPP). The amplitude field of CiBs can be described by either the Whittaker or the confluent hypergeometric functions. The transformation and propagation of CiBs by the use of a SPP after propagating the considered optical system is presented and derived in detail. Some special cases of GHBs as standard, elegant and generalized Laguerre-Gaussian beams; Hypergeometric-Gaussian beams; Fractional order elegant Laguerre-Gaussian beams; Bessel-Gauss beams; quadratic Bessel–Gauss beams; and Whittaker-Gaussian beams are deduced in this paper. Numerical simulations are performed to study the transformation of the Whittaker-Gaussian beam by a SPP in free space as a particular example in this work.

Birth and evolution of an optical vortex.

When a phase singularity is suddenly imprinted on the axis of an ordinary Gaussian beam, an optical vortex appears and starts to grow radially, by effect of diffraction. This radial growth and the subsequent evolution of the optical vortex under focusing or imaging can be well described in general within the recently introduced theory of circular beams, which generalize the hypergeometric-Gaussian beams and which obey novel kinds of ABCD rules. Here, we investigate experimentally these vortex propagation phenomena and test the validity of circular-beam theory. Moreover, we analyze the difference in radial structure between the newly generated optical vortex and the vortex obtained in the image plane, where perfect imaging would lead to complete closure of the vortex core.

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.

Manipulating light at distance by a metasurface using momentum transformation.

A momentum conservation approach is introduced to manipulate light at distance using metasurfaces. Given a specified field existing on one side of the metasurface and specified desired field transmitted from the opposite side, a general momentum boundary condition is established, which determines the amplitude, phase and polarization transformation to be induced by the metasurface. This approach, named momentum transformation, enables a systematic way to synthesize metasurfaces with complete control over the reflected and transmitted fields. Several synthesis illustrative examples are provided: a vortex hypergeometric-Gaussian beam and a "delayed-start" accelerated beam for Fresnel region manipulation, and a pencil beam radiator and a holographic repeater for Frauenhofer region manipulation.

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.

Numerical Simulations of Fractional Fourier Transform of Hypergeometric-Gaussian Beam

The fractional Fourier transform (FRFT) of a new type of laser beams called the hypergeometric-Gaussian beam (HyGGB) is investigated in detail. The analytical expression for the FRFT of a HyGGB is derived. The properties of a HyGGB in the FRFT plane with different parameters are illustrated. The results show that the intensity distribution of a HyGGB in the FRFT plane strongly depends on the fractional order, the lens focal length and the initial beam width.

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.