Extended Huygens-Fresnel Principle Research Articles

Polarization and coherence properties of a partially coherent elegant Laguerre–Gaussian vortex beam in turbulent plasma

The analytical expressions for the cross-spectral density matrix elements of the partially coherent elegant Laguerre–Gaussian (eLG) vortex beam propagating through anisotropic turbulent plasma were derived based on the extended Huygens-Fresnel principle and used to study the changes in the polarization degree and the coherence degree of the partially coherent eLG vortex beam in the turbulent plasma. The numerical results show that the polarization degree of a partially coherent eLG vortex beam reaches a specific value (which is equal to the degree of polarization in the source plane) after a long propagation distance in the turbulent plasma. Moreover, this value is independent of the beam order, topological charge, correlation coefficient, wavelength of the source plane, anisotropy parameter, refractive index fluctuation variance, and outer and inner scales of the turbulent plasma. The results also show that with increasing distance, the coherence degree first decreases from unity and then oscillates around zero. However, this oscillation gradually disappears after traveling a long distance. Our results intuitively present the beam polarization and coherence properties through anisotropic hypersonic turbulence, which can be useful for optical communication in hypersonic turbulent environments.

Depth of focus and intensity distribution of a lensacon illuminated by a partially coherent Gaussian Schell vortex beam

Using the extended Huygens–Fresnel principle, a cross-spectral density formula was developed for a Gaussian Schell model vortex (PCGSMV) beam diffracted through a lensacon (lens with an axicon). The intensity and depth of focus (DOF) shaped by the lensacon were calculated. Our numerical results show the relationship between the intensity distribution and depth of focus with the beam waist width as well as the spatial correlation of the coherence length. Furthermore, the relationship between the beam spot size and propagation distance was investigated. In the case of the lensacon tandem, the maximum intensity was greater than that attained by the axicon alone for the same beam parameters, and the DOF was smaller than that of the axicon alone. The vortex structure canceled out the low value of the spatial degree of coherence length. Our numerical model exhibited high-intensity values and high-quality Bessel rings along the DOF, which are critical for various applications.

Evolution properties of partially coherent electromagnetic elliptical vortex beam propagating through atmospheric turbulence

We obtained the analytical formulas of partially coherent electromagnetic elliptical vortex beam (PCEEVB) propagating through atmospheric turbulence with the help of the extended Huygens–Fresnel principle and explored the normalized average intensity distribution of PCEEVB in turbulence. It is worth noting that the PCEEVB appears new characteristics such as self-rotating, self-splitting, and self-healing in turbulence. It can be also found that the number of peaks caused by splitting of beams is relevant to the topological charge n. What’s more, reducing the ellipticity, magnifying the topological charge, and increasing the initial coherent length δ 0 can improve the anti-turbulence ability of PCEEVB in turbulence. The multi-phase screen simulation results are consistent with the theoretical results. Our results will have significant implications for laser communications, laser radar, and laser remote sensing.

Propagation properties of a partially coherent electromagnetic hyperbolic-sine-Gaussian vortex beam through anisotropic atmospheric turbulence.

Using the extended Huygens-Fresnel principle and the Rytov approximation, the analytical formula for the propagation of a partially coherent electromagnetic hyperbolic-sine-Gaussian vortex beam (PCEShVB) in anisotropic atmospheric turbulence has been theoretically derived. Detailed studies have been conducted on the evolution characteristics of the average intensity, the degree of coherence (DOC), and the degree of polarization (DOP) of the beam in turbulence. The results show that during propagation, the intensity distribution of the beam will exhibit a spiral structure, and the overall distribution of the light spots will rotate in a direction related to the sign of the topological charge. The DOC distribution of PCEShVB will display a pattern reminiscent of beam interference fringes with an increase in propagation distance, with the number of "interference fringes" greatly impacted by the hyperbolic sine parameter. Furthermore, PCEShVB with a large initial coherent length and hyperbolic sine parameter will increase the degree of separation of the spots and yield a large DOP. Finally, for the validation of the theoretical findings, the random phase screen method was employed to simulate the propagation of PCEShVB through anisotropic atmospheric turbulence. The studies revealed a consistent alignment between the simulation results and the theoretical predictions.

Underwater entanglement propagation of auto-focusing Airy beams.

In underwater wireless optical communication, orbital angular momentum (OAM) states suffer from turbulence distortions. This study aims to investigate the effectiveness of auto-focusing and OAM entanglement of the beams in reducing the turbulence effects. We implement the single-phase approximation and the extended Huygens-Fresnel principle to derive the detection probability of the entangled Airy beams under unstable oceanic turbulence. The results show that auto-focusing can protect the signal OAM mode and suppress modal crosstalks, while entangled OAM states can further enhance the resistance against oceanic turbulence around the focus position. The numerical analysis demonstrates that after the auto-focusing position, the beams evolve in completely opposite directions, indicating that the focal length should be modulated according to the length of a practical link to enhance received signals. These findings suggest that entangled auto-focusing vortex beams may be a desirable light source in underwater communication systems.

Propagation properties of partially coherent radially polarized rotationally-symmetric power-exponent-phase vortex beams in turbulent atmosphere

In this paper, the propagation properties of partially coherent radially polarized rotationally-symmetric power-exponent-phase vortex beams (RP-RSPEPVBs) in turbulent atmosphere were investigated. Based on the extended Huygens-Fresnel principle and unified theory of coherence and polarization, the theoretical propagation model was established. Then, the numerical simulations were carried out to reveal the evolution of intensity distribution, spectral degree of coherence (DOC) and spectral degree of polarization (DOP) in turbulent atmosphere. The results showed that although the partially coherent RP-RSPEPVBs diverged during the propagation, the spatial structure remains analogous, and the relatively larger waist widths gain smaller divergence. Moreover, the optical pattern can be evidently modulated by optical parameters, as the number of sidelobes can be changed with topological charge l, and the continuity of the light spot as well as the rotating speed around the optical axis can be influenced by power exponent n. Besides, the spectral DOC and DOP would decrease with larger propagation distance and refractive-index structure constant, but decrease less with a shorter coherence width. This work may be helpful in fields such as remote sensing and optical communication.

Average Intensity of a Laguerre—Gaussian Vector Vortex Beam through Inhomogeneous Atmospheric Turbulence

We investigate the evolution properties of a partially coherent Laguerre–Gaussian vector vortex (LGVV) beam through inhomogeneous atmospheric turbulence. Analytical formulae for the elements of a cross-spectral density matrix of a partially coherent LGVV beam propagating in turbulence are derived with the help of the extended Huygens–Fresnel principle. Our outcomes demonstrate that the normalized initial profile of a partially coherent beam with concentric dark rings gradually evolves into a Gaussian-like beam profile in turbulence. We also find that the beam is emitted at a large zenith angle and quickly converts to a Gaussian-like beam. Furthermore, it is also shown that a propagation beam with a large topological charge has a stronger ability to resist atmospheric turbulence. In order to confirm our numerical results, we combine the complex screen method and multi-phase screen method to simulate the propagation of a partially coherent LGVV beam in atmospheric turbulence. It is indicated that the simulation results are in good agreement with theoretical predictions. Our results will pave the way for the development of free-space optical communications and remote sensing.

Effect of oceanic turbulence on the propagation behavior of a radially polarized Laguerre-Gaussian Schell-model vortex beam.

Optical wireless communications applications are restricted by oceanic media-induced beam quality degradation. However, modulating the coherence and polarization structures of the laser beams can effectively diminish the negative influence of oceanic turbulence on the beams. The average intensity of a radially polarized Laguerre-Gaussian Schell-model vortex (RPLGSMV) beam propagating through oceanic turbulence is explored by employing the extended Huygens-Fresnel principle. We found that the average intensity of an RPLGSMV beam is greatly affected by oceanic turbulence with a large rate of dissipation of the mean-square temperature and a large relative strength of the temperature and salinity fluctuations as well as the small rate of dissipation of the turbulent kinetic energy per unit mass of fluid and small Kolmogorov microscale. It was also found that a RPLGSMV beam with a larger radial index, topological charge, initial coherent length, and beam waist has a stronger anti-turbulence ability. Our numerical findings may be of great significance for the detection and imaging of oceanic optical telecommunications links.

Quantitative Analysis of the Effect of Atmospheric Turbulence on a Bessel–Gaussian Beam

Suppressing the impact of atmospheric turbulence on laser beam propagation is a bottleneck problem in the application of free space optical communications, with the primary difficulty being the lack of a quantitative description of the effect of turbulence on a laser beam. In this paper, we propose a quantitative description of the effect of turbulence and express a quantitative analysis of the effect of atmospheric turbulence on Bessel–Gaussian beam based on the extended Huygens–Fresnel principle. The results of numerical analysis confirm the validity of the theoretical description. This study also shows that the influence of turbulence is stronger when the annular shape of the Bessel–Gaussian beam is more obvious. The method also provides theoretical guidance for reverse engineering the amplitude and phase distribution of the initial laser beam, thereby effectively reducing the impact of atmospheric turbulence on laser beam propagation.

Beam intensity and spectral coherence of Hermite-cosine-Gaussian rectangular multi-Gaussian correlated Schell-model beam in oceanic turbulence

The beam spreading and evolution behavior of a partially coherent, Hermite-cosine-Gaussian (HcosG) rectangular multi-Gaussian correlated Schell-model beam propagating in oceanic turbulence is studied. Analytical expressions for the cross-spectral density function, as well as the root mean square (rms) beam width and the spectral degree of coherence, are derived based on the extended Huygens-Fresnel principle. The HcosG rectangular multi-Gaussian correlated Schell-model beam exhibits a multi-lobe pattern at short propagation distances. The dependencies of the number, size, shape, and centroid of the lobes on displacement parameters, source size, order of field distribution, and displacement are investigated. As the propagation distance increases, the spectral coherence decreases, and the differences between the spectral coherence curves gradually diminish. Additionally, for HcosG rectangular multi-Gaussian correlated Schell-model beams, better propagation performance was found in oceanic turbulence with larger mean square temperature dissipation rate, smaller turbulent kinetic energy dissipation rate per unit mass of fluid, and larger relative strength of temperature and salinity fluctuation.

Spreading properties of partially coherent multi-Gaussian Schell-model and modified Bessel-correlated vortex beams in anisotropic turbulent plasma

We investigate the propagation characteristics of partially coherent multi-Gaussian Schell-model (MGSM) and modified Bessel-correlated (MBc) vortex beams traveling in a turbulent plasma. Based on the extended Huygens-Fresnel principle, the cross-spectral density expressions for partially coherent MGSM and MBc vortex beams propagating through turbulent plasma were derived. The results show that the dark spot at the center of the partially coherent MGSM beams disappears in the low-coherence states and remains in the high-coherence states only. In contrast, the intensity of partially coherent MBc vortex beams exists in low- and high-coherence states and does not change during propagation in a weak turbulent plasma.

Propagation of coherence-OAM matrix of an optical beam in vacuum and turbulence.

Propagation of the coherence-orbital angular momentum (COAM) matrix of partially coherent beams in homogeneous and isotropic turbulence, e.g., atmosphere, is formulated using the extended Huygens-Fresnel principle. It is found that under the effect of turbulence the elements in the COAM matrix will generally be affected by other elements, resulting in certain OAM mode dispersion. We show that if turbulence is homogeneous and isotropic, there exists an analytic "selection rule" for governing such a dispersion mechanism, which states that only the elements having the same index difference, say l - m, may interact with each other, where l and m denote OAM mode indices. Further, we develop a wave-optics simulation method incorporating modal representation of random beams, multi-phase screen method and the coordinate transformation to simulate propagation of the COAM matrix of any partially coherent beam propagating in free space or in turbulent medium. The simulation method is thoroughly discussed. As examples, the propagation characteristics of the most representative COAM matrix elements of circular and elliptical Gaussian Schell-model beams in free space and in turbulent atmosphere are studied, and the selection rule is numerically demonstrated.

Spectral properties of pulsed Laguerre higher-order cosh-Gaussian beam propagating through the turbulent atmosphere

The formula for the propagation of pulsed Laguerre higher-order cosh-Gaussian (LHChG) beam in turbulent atmosphere is derived using the extended Huygens–Fresnel principle and the Fourier Transform method. The impact of atmospheric turbulence, transverse positions, and initial beam parameters on the spectral intensity of the propagated beam has been examined using graphical representations. The relative spectral shift of pulsed LHChG beam in turbulent atmosphere with different radial coordinates has also been studied and illustrated with numerical calculations. Analysis show that the spectral intensity depends on the structure constant of the refractive index, pulse duration, and beam order. On-axis spectral intensity is blue-shifted and off-axis spectral intensity is red-shifted with increasing radial coordinate. From our main results, one derive some published particular cases of the considered beam. The research results have potential applications in information coding and transmission.

Propagation of radially polarized beams with a Hermite non-uniformly correlated array in free space and turbulent atmosphere.

We introduce what we believe to be a novel class of radially polarized partially coherent beams in which the correlation function possesses a Hermite non-uniformly correlated array. The source parameter conditions required to generate a physical beam are derived. The statistical properties of such beam propagating in free space and turbulent atmosphere are thoroughly examined using the extended Huygens-Fresnel principle. It is shown that the intensity profile of such beams presents a controllable periodic grid distribution due to its multi-self-focusing propagation property and can keep the shape in free space while propagating in turbulent atmosphere, it exhibits self-combining properties over a long-ranges. Owing to the interaction between the non-uniform correlation structure and the non-uniform polarization, this beam can locally self-recover the polarization state after propagating a long distance in a turbulent atmosphere. Furthermore, the source parameters play essential roles in determining the distribution of spectral intensity, the state of polarization, and the degree of polarization of the RPHNUCA beam. Our results may benefit multi-particle manipulation and free-space optical communication applications.

Propagation of Lorentz-Gaussian elliptical multi-Gaussian correlated Schell-Model beam in anisotropic turbulence

The beam spreading, spectral degree of coherence and evolution behavior of the intensity profile of partially coherent Lorentz-Gaussian elliptical multi-Gaussian correlation Schell-Model beam propagating in anisotropic atmospheric turbulence are studied. The analytical expressions of cross-spectral density function, as well as root mean square (rms) beam width, are derived based on the extended Huygens-Fresnel principle and the relationship of Lorentz distribution and Hermitian Gaussian function. With the increases of propagation distance, the elliptical beam will first evolve into Gaussian beam and then change back to elliptical beam. In anisotropic atmospheric turbulence, the influence of the inner scale of turbulence on the spectral degree of coherence and rms beam width are obviously greater than that of the outer scale. For Lorentz-Gaussian elliptical multi-Gaussian correlated Schell-Model beams, better propagation performance was found in anisotropic atmospheric turbulence with larger anisotropic factor and smaller inner scale.

The behavior of partially coherent twisted space-time beams in atmospheric turbulence

We study how atmospheric turbulence affects twisted space-time beams, which are non-stationary random optical fields whose space and time dimensions are coupled with a stochastic twist. Applying the extended Huygens–Fresnel principle, we derive the mutual coherence function of a twisted space-time beam after propagating a distance z through atmospheric turbulence of arbitrary strength. We specialize the result to derive the ensemble-averaged irradiance and discuss how turbulence affects the beam’s spatial size, pulse width, and space-time twist. Lastly, we generate, in simulation, twisted space-time beam field realizations and propagate them through atmospheric phase screens to validate our analysis.

Propagation of a Lorentz Non-Uniformly Correlated Beam in a Turbulent Ocean

We study the propagation characteristics (spectral intensity and degree of coherence) of a new type of Lorentz non-uniformly correlated (LNUC) beam based on the extended Huygens–Fresnel principle and the spatial power spectrum of oceanic turbulence. The effects of the oceanic turbulence parameters and initial beam parameters on the evolution propagation characteristics of LNUC beams are studied in detail by numerical simulation. The results indicate that such beams exhibit self-focusing propagation features in both free space and oceanic turbulence. Decreasing the dissipation rate of kinetic energy per unit mass of fluid and the Kolmogorov inner scale, or increasing the relative strength of temperature to salinity undulations and the dissipation rate of mean-square temperature of the turbulent ocean tends to increase the negative effects on the beams. Furthermore, we propose a strategy of increasing the beam width and decreasing the coherence length, to reduce the negative effects of the turbulence.

Theoretical and computational study of a partially coherent laser beam in a marine environment

The propagation formula of a partially coherent Generalized Flattened Hermite-Cosh-Gaussian (GFHChG) beam in maritime atmospheric turbulence is derived with the help of the extended Huygens–Fresnel principle. In addition, the analytical expression for the beam width of a partially coherent GFHChG beam in the considered environment is investigated. From the numerical results based on the analytical formulae, we find that the analyzed beam can take different shapes of the profile, depending on the turbulence and beam parameters. And also, it can significantly resist turbulence with small wavelength and waist width values. On the other hand, when the medium becomes turbulent, the beam loses its characteristics and its resistance to fluctuations. Furthermore, the results reveal that the beam spreads more rapidly with the increase of the strength of turbulence, the outer scale size, and the decrease of the inner scale size. We should mention that the results gained represent a general form of numerous partially coherent laser beams such as Generalized Flattened Hermite Gaussian, Generalized Flattened Cosh-Gaussian, Hermite-Cosh-Gaussian, Cosh-Gaussian, Hermite-Gaussian and Gaussian Schell model beams.

Propagation characteristics of radial phase-locked discrete vector beam generated from a Gaussian Schell-Model laser array in atmospheric turbulence

In this paper, combining the advantages of partial coherence, vector beam, and laser array in better resistance to atmospheric turbulence, high capacity, and high output power, we proposed a radial phase-locked discrete vector beam generated from a Gaussian Schell-Model (GSM) laser array (i.e., phase-locked GSM discrete vector beam array) and theoretically studied its propagation characteristics in atmospheric turbulence. The analytical expressions for spectral intensity, spectral degree of coherence (DOC), and spectral degree of polarization (DOP) of a phase-locked GSM discrete vector beam array were derived based on the extended Huygens-Fresnel principle. And also, based on numerical simulation, we investigated the influence of atmospheric and laser parameters on the propagation properties in detail. The results show that the beam profile will gradually evolve from hollow structure distribution, then form a flat-topped beam and finally present the Gaussian-like distribution; the spectral DOC exhibits decaying oscillation analogous to under-damping vibration; a dip appears in the distribution of the spectral DOP. The evolution rate of spectral intensity, oscillation amplitude, and period of spectral DOC and dip width of spectral DOP are determined by propagation distance, refraction structure constant, beam waist, spatial correlation length, and the number of beamlets. More importantly, by analyzing Stokes parameters, different initial polarization distributions correspond to different crosslines of S1/S0, and even with the increase of the propagation distance, the crosslines of S1/S0 still show similar performance. This work may provide novel thoughts and methods to improve propagation performance and benefit potential applications in free-space optical communication.

Propagation factor of partially coherent vector vortex beam in inhomogeneous turbulent atmosphere

We undertake the propagation properties of a partially coherent vector vortex beam (PCVVB) through the inhomogeneous atmospheric turbulence. Analytical expressions for M2-factor of partially coherent radially polarized vortex beam (PCRPVB) in turbulence with the extended Huygens–Fresnel principle and the second-order moments of Wigner distribution function (WDF) are derived. Numerical results indicate that the M2-factor and root-mean-square (rms) angular width mainly depend on the light source parameters and zenith angle. We find that M2-factor and rms angular width of PCRPVB in turbulence will become saturated with increasing propagation distance. It can be also found that the relative M2-factor and relative rms angular width will decrease with the increase of topological charge and wavelength as well as the decrease of initial coherent length, which indicates that the PCRPVB with larger topological charge and wavelength, smaller initial coherent length has a stronger anti-turbulence ability. Our studies may be useful for the lidar distance measurement and free-space optical communications.