Elegant Laguerre-Gaussian Beams Research Articles

Propagation properties of elegant modified Bessel Gaussian beams.

A kind of optical beam with a radially parabolic propagating manner and intensity decay inversely proportional to propagating distance in the far field is investigated. The initial complex amplitudes of this kind of beam have the form of a Gaussian function multiplied by a m/2-order modified Bessel function and a helical phase factor with topological charge m. The arguments for Bessel and Gauss parts in the propagating solutions of these beams are complex and symmetric as elegant Laguerre and Hermite Gaussian beams. As a result, the beams can be referred to as elegant modified Bessel Gauss (EMBG) beams. Similar to non-diffractive beams such as Bessel and Airy beams, the EMBG beams also carry infinite power due to a transversely slowly decaying tail of complex amplitude. The EMBG beams demonstrate intermediate propagating properties between non-diffractive and finite-power beams. Unlike non-diffractive beams that never spread their power and finite-power beams that always diverge in a linear manner and spread their power by inversely square law in the far field, the EMBG beams demonstrate a far-field parabolic propagating manner and decay their power by inversely linear law. In addition, the EMBG beams have total Gouy phase, which is only half of that of elegant Laguerre Gauss beams with the same topological charge, and have far-field intensity distributions regardless of the beam waist radius in the initial plane. The propagating and focusing properties of EMBG beams represent an intermediate status between the non-diffractive and finite-power beams.

Paraxial Dirac equation

In this work, the paraxial approximation of the free Dirac equation is examined. The results are first obtained by constructing superpositions of exact solutions with suitable profiles, which are borrowed from paraxial optics. In this manner, the paraxial Dirac beams are obtained in four cases: as Gaussian, Bessel-Gaussian, modified Bessel-Gaussian and elegant Laguerre-Gaussian beams. In the second part of the work, the paraxial Dirac equation is derived, and then its solutions in the aforementioned cases are directly obtained. All the resulting wave functions conform to those derived formerly by virtue of superpositions, except for terms that are negligible upon the assumption that the paraxial functions along the propagation axis vary only slightly over a distance equal to the de Broglie wavelength, which is the standard paraxial requirement.

Analytical analysis of the beam propagation factor of elegant Hermite-Gaussian and elegant Laguerre-Gaussian beams with astigmatism

The impact of astigmatism on the beam propagation factor (M2) of elegant Hermite–Gaussian and elegant Laguerre–Gaussian beams is examined. We derive closed-form expressions for M2 when the optical beams are aberrated with astigmatism. The analysis shows that the beam radius is crucial to the degree of impact astigmatism has on M2. To this extent, we derive the beam radius that separates the region where the M2 is negligibly affected and the region where it becomes severely affected. For the elegant Laguerre-Gaussian beams, we establish a parameter that determines a set of beams that are impacted equally by astigmatism. The analytical results are validated with numerical simulations.

Generalized Asymmetric Hermite–Gaussian and Laguerre–Gaussian Beams

We derive analytical formulae for the complex amplitudes of variants of generalized Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) beams. We reveal that, at particular values of parameters of the exact solution of the paraxial propagation equation, these generalized beams are converted into conventional elegant HG and LG beams. We also deduce variants of asymmetric HG and LG beams that are described by complex amplitudes in the form of Hermite and Laguerre polynomials whose argument is shifted into the complex plane. The asymmetric HG and LG beams are, respectively, shown to present the finite superposition of the generalized HG and LG beams. We also derive an explicit relationship for the complex amplitude of a generalized vortex HG beam, which is built as the finite superposition of generalized HG beams with phase shifts. Newly introduced asymmetric HG and LG beams show promise for the study of the propagation of beams carrying an orbital angular momentum through the turbulent atmosphere. One may reasonably believe that the asymmetric laser beams are more stable against turbulence when compared with the radially symmetric ones.

Close relationship between Bessel-Gaussian and conical refraction beams.

We demonstrate that the conical refraction of the input elegant Laguerre-Gaussian beams can be effectively described through generalized Bessel-Gaussian light beams. We performed numerical simulations and show good agreement between the exact solution and our proposed Bessel-Gaussian approximation model. Physical clarity of the proposed model has allowed us to explain the transition of the classical double-ring pattern of conical refraction in the Lloyd plane into a multi-ring one and predict new phenomenon such as the Raman spot shift and dependence of the conical refraction ring radius on the value of the orbital angular momentum.

Comparative Study of Spiral Spectrum of Elegant and Standard Laguerre–Gaussian Beams in Atmospheric Turbulence

We derive the analytical expression of the probability density of orbital angular momentum modes for elegant and standard Laguerre–Gaussian beams in turbulence. The results show that we can increase the initial topological charge of the beams to reduce the detrimental mode crosstalk induced by the turbulence. We also discuss the influence of the beam parameters on the probability of signal and the variances of crosstalk probability. The results obtained show that the beams with smaller beam width and longer wavelength have higher detection probabilities of signal and are less affected by turbulence. Comparing with the results of elegant and standard Laguerre–Gaussian beams, we find that elegant Laguerre–Gaussian beams display stronger turbulence resistance. Our results will be useful in free-space optical communications.

Coherent Vortices Properties of Partially Coherent Elegant Laguerre-Gaussian Beams in the Free Space

The analytical expression of the cross-spectral density of partially coherent elegant Laguerre-Gaussian beams propagating in free space has been derived. The coherence vortex properties of such beams have been investigated. The effect of the beam parameters, including the topological charge, radial mode index and coherence length on the coherent vortex, is analyzed. The results show that the higher order (lth) of coherent vortices split to lth first order of coherent vortex. New coherent vortices of opposite sign appear, and then pairs of coherent vortices form. The propagation distance and coherence length affect the generation of coherent vortices, while the radial mode index doesn’t affect the coherent vortices. These results might be helpful for potential application of such beams in optical communication.

Properties of partially coherent elegant Laguerre-Gaussian beam in free space and oceanic turbulence

The analytical expression of partially coherent elegant Laguerre-Gaussian (PCELG) beam generated by anisotropic Gaussian Schell-model sources is given. The analytical formulas of CSD for a PCELG beam propagating in free space and oceanic turbulence are derived. The characteristics of average intensity and spectral degree of coherence of a PCELG beam propagating in free space and oceanic turbulence are investigated by using numerical simulations. It is shown that the stronger oceanic turbulence will accelerate the evolution of a PCELG beam propagating in oceanic turbulence, and the influences of coherence length on the spectral degree of coherence will disappear as the propagation distance increases. The findings may be useful in underwater optical communications.

Channel capacity of orbital-angular-momentum-based wireless communication systems with partially coherent elegant Laguerre-Gaussian beams in oceanic turbulence.

For practical wireless communication links, one of the critical challenges is the random fluctuation of turbulence that will impair link performance. Here a transmission model of partially coherent elegant Laguerre-Gaussian (ELG) beams in oceanic turbulence is established. An analytical formula for channel capacity of a partially coherent ELG beam propagating through a turbulent ocean is derived. The effects of oceanic turbulence on the evolution of channel capacity performance are studied quantitatively in a series of numerical simulations. Research results show that decreasing the rate of dissipation of mean-square temperature and ratio of temperature to salinity, as well as increasing the dissipation rate of turbulent kinetic energy per unit mass of fluid of a turbulent ocean can significantly improve communication channel capacity. Furthermore, choosing optimum beam source parameters is favorable to mitigate the influence of oceanic turbulence. Results also show that in the underwater turbulence, the partially coherent ELG beams are more affected by turbulence as compared to the fully coherent ELG beams. These study results may provide potential help in designing the free-space optical vortex communication systems.

Propagation of a radial phase-locked partially coherent elegant Laguerre–Gaussian beam array in non-Kolmogorov medium

A model of a radial phase-locked partially coherent elegant Laguerre–Gaussian (PCELG) beam array has first been introduced in theory. The analytical propagation equation for the cross-spectral density function of a radial phase-locked PCELG beam array in non-Kolmogorov medium has been derived using the extended Huygens–Fresnel principle. The average intensity and spectral degree of coherence properties of a radial phase-locked PCELG beam array propagating in non-Kolmogorov medium have been studied in details using the numerical examples. One can find that the evolution properties of a radial phase-locked PCELG beam array propagating in non-Kolmogorov medium are affected by the initial beam parameters and the non-Kolmogorov medium, and the beam array propagating in non-Kolmogorov medium will evolve into a solid beam with Gaussian-like distribution in the far field.

Performance analysis of the OAM based optical wireless communication systems with partially coherent elegant Laguerre–Gaussian beams in oceanic turbulence

The transmission model of partially coherent elegant Laguerre–Gaussian (ELG) beams in oceanic turbulence is established by employing the spatial power spectrum of the refractive index of sea-water and the optics approximation. The mathematical expressions of mode probability, energy content, and orbital angular momentum (OAM) state detection probability for partially coherent ELG beams are obtained. The influences of beam and turbulence parameters on the propagation properties of light beam are illustrated in detail with numerical simulations. Results indicate that partially coherent ELG beams with smaller azimuthal mode, smaller OAM state difference, longer blue–green incident wavelength, and appropriate beam waist are less affected by turbulent ocean. Moreover, the evolution characteristics of laser beam may be able to keep a better quality under the condition of lower rate of dissipation of mean-square temperature, higher rate of dissipation of turbulent kinetic energy per unit mass of fluid, and smaller ratio of temperature to salinity for maritime environment. In addition, the partially coherent ELG beams exhibit more robust resistance to turbulence than the partially coherent standard Laguerre–Gaussian beams. These results are expected to be potential helpful in understanding the optical transmission process in turbulent media and designing the optical vortex communication systems.

Measuring topological charge of partially coherent elegant Laguerre-Gaussian beam.

The partially coherent elegant Laguerre-Gaussian (ELG) beam is of importance and exhibits extraordinary characteristics in many fields, such as optical communications and optical trapping. Here, we show a method to measure the topological charge of a partially coherent ELG beam. We find that the number of ring dislocations in the far-field complex degree of coherence is equal to the topological charge |l| of a partially coherent ELG beam, and which is confirmed experimentally. Our results will be useful for applications using partially coherent ELG beams.

Perturbative representation of ultrashort nonparaxial elegant Laguerre-Gaussian fields

An analytical method for calculating the electromagnetic fields of a nonparaxial elegant Laguerre-Gaussian (LG) vortex beam is presented for arbitrary pulse duration, spot size, and LG mode. This perturbative approach provides a numerically tractable model for the calculation of arbitrarily high radial and azimuthal LG modes in the nonparaxial regime, without requiring integral representations of the fields. A key feature of this perturbative model is its use of a Poisson-like frequency spectrum, which allows for the proper description of pulses of arbitrarily short duration. This model is thus appropriate for simulating laser-matter interactions, including those involving short laser pulses.

Elegant Laguerre–Gaussian beams—formulation of exact vector solution

In photonic applications of optical beams, their transverse cross-section should be often narrow, with a diameter in their waist of the order of one wavelength or even less. Within this range, the paraxial approximation of beam fields is not valid and standard corrections by field expansions with respect to a small parameter are not efficient as well. Thus, still there is a need for more accurate beam field description. In this report, an exact vector solution for free-space propagation is given in terms of elegant Laguerre–Gaussian beams. The analysis starts from the known paraxial field approximation and next, through bidirectional field transformation and application of a Hertz potential leads to an exact vector solution. The role of the paraxial solution in construction of the exact solution is elucidated. The method works well not only in cases of free-space propagation but also in description of beam interactions with planar interfaces and multilayers.

Propagation of anomalous vortex beams beyond the paraxial approximation

The propagation properties of an anomalous vortex beam(AVB) beyond the paraxial approximation are studied. Based on the vectorial Rayleigh-Sommerfeld formulas, the electric field of the AVB beyond the paraxial approximation are derived analytically. Simulations are performed to investigate the intensity distribution characteristics and the far-field behaviours of the non-paraxial AVB. The influences of the topological charge, the beam order, and the natural expansion parameter f on the propagation of the non-paraxial AVB are also explored in detail. The results show that the natural expansion parameter f plays an important role in determining whether an AVB can evolve into an elegant Laguerre-Gaussian beam in the far field. There is a critical value of the natural expansion parameter f, and the AVB can become the elegant Laguerre-Gaussian beam in the far field only when the natural expansion parameter f is less than this critical value. Furthermore, the critical value increases with smaller topological charge or larger beam order. These results provide guidelines for generation of elegant Laguerre-Gaussian beams, which has potential application in optical vortex communication systems.

Controllable accelerating and decelerating Airy-elegant-Laguerre–Gaussian wave packets in free space

The propagation of three-dimensional controllably accelerating and decelerating Airy-elegant-Laguerre–Gaussian (CAiELG) wave packets in free space is investigated theoretically and numerically by solving the ( 3 + 1 ) D Schrodinger equation in cylindric coordinates. The CAiELG wave packets are constructed with the Airy pulses with the initial velocity in temporal domain and the elegant-Laguerre–Gaussian beams in space domain. Decelerating and accelerating AiELG wave packets are obtained by selecting different initial velocities. The initial velocities can be determined by incident angle and directions. According to the intensity distribution of CAiELG wave packets at the propagating section, two special types of wave packets are accessed: one type is ring shaped with the modulation depth q = 1 and another type is necklace shaped with q = 0 . The direction of the energy flow of CAiELG wave packets is kept away from the center during propagation, and their Poynting vector snapshots at different propagating distances are shown.

Propagation of partially coherent Laguerre Gaussian beams through inhomogeneous turbulent atmosphere.

Analytical formulas for the root-mean-square (rms) spatial width, the rms angular width, and the M2-factor of partially coherent standard Laguerre Gaussian beams (PC-SLGBs) and partially coherent elegant Laguerre Gaussian beams (PC-ELGBs) in inhomogeneous turbulent atmosphere have been derived. The propagation properties of PC-SLGBs and PC-ELGBs in inhomogeneous atmospheric turbulence are studied numerically and comparatively. It can be found that the propagation of laser beams in inhomogeneous turbulence is different from that in homogeneous turbulence. It is also shown that relative rms spatial widths and M2-factors of PC-ELGBs are more affected by inhomogeneous turbulence than those of PC-SLGBs. Moreover, the relative rms spatial widths and M2-factors of PC-SLGBs and PC-ELGBs in inhomogeneous turbulent atmosphere are closely related with waist widths, coherence widths, zenith angles, inner scales, and beam orders. Furthermore, the saturation propagation distance of the relative M2-factor and rms angular width with zenith angles of π/6 are 20 km and 0.5 km, respectively.

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.

Propagation factors of standard and elegant Laguerre Gaussian beams in non-Kolmogorov turbulence

Analytical formulae are derived for the propagation factors (M2-factors) of standard Laguerre Gaussian (SLG) beam and elegant Laguerre Gaussian (ELG) beam in non-Kolmogorov turbulence based on the extended Huygens-Fresnel principle and second order moments of the Wigner distribution function. The M2-factors of SLG beam and ELG beam in non-Kolmogorov turbulence have been discussed numerically and comparatively. It is shown that the relative M2-factors of ELG beam are more affected by the turbulence and keep the better beam quality in comparison with that of SLG beam. Furthermore, the normalized M2-factors of SLG beam and ELG beam with higher orders, longer wavelength, larger waist width, larger inner scale of turbulence and the generalized exponent parameter, smaller outer scale of turbulence and smaller the generalized structure parameter are less affected by the non-Kolmogorov turbulence.