Huygens-Fresnel Diffraction Integral Research Articles

Laser Beam Quality of Airy Beam in the Jet Engine Exhaust Induced Turbulence

Theoretical predictions of laser beam interactions with jet-engine-exhaust-induced turbulence are of importance for optimization of various optical systems, including LIDARs and airborne laser measurement systems. This paper is based on the extended Huygens–Fresnel diffraction integral and the Wigner distribution function, combined with the recently proposed power spectral density function of atmospheric turbulence induced by jet engine exhaust, to derive the formula for the quality factor of a partially coherent Airy beam in turbulent media induced by jet engine exhaust. Numerical calculations based on the analytical formula show that the smaller the structure constant and outer scale of the atmospheric turbulence induced by jet engine exhaust, the smaller the quality factor of a partially coherent Airy beam, indicating better laser beam quality. At the same time, the larger the generalized parameter and inner scale of the jet-engine-exhaust-induced turbulence and the longer the wavelength of the laser beam, the smaller the corresponding quality factor. The numerical results also show that reducing the coherence length of a partially coherent Airy beam and increasing its characteristic width are beneficial for reducing the quality factor in the atmospheric turbulence induced by jet engines. However, the numerical results show that increasing the coherence length of a partially coherent Airy beam and reducing its characteristic width are beneficial for reducing the quality factor in vacuum. This situation demonstrates significant differences between the interaction mechanisms of an Airy beam in vacuum and in the jet-engine-exhaust-induced turbulence. The results of our discussion contribute to clarifying the mechanism of interaction between an Airy laser beam and random media, as well as the application of a curved Airy laser beam in practical situations.

Propagation properties of rotationally-symmetric power-exponent-phase vortex beam through oceanic turbulence

In this paper we explored the propagation characteristics of rotationally-symmetric power-exponent-phase vortex beam (RSPEPVB) through the oceanic turbulence. Based on the extended Huygens-Fresnel diffraction integral and the oceanic turbulence theory the theoretical model of RSPEPVBs propagating in oceanic turbulence was established. Then the propagation properties of RSPEPVBs were explored by numerical simulation and the influences of the propagation distance z the rate of dissipation of turbulence kinetic energy per unit mass of fluid ε the temperature-salinity contribution ratio ω and the dissipation rate of the mean-squared temperature χT were discussed. The results illustrated that the intensity of RSPEPVBs propagating through oceanic turbulence weakened and the size of RSPEPVBs diffused with the increasing distance and strong oceanic turbulence of larger values of parameters χT and ω and the smaller value of the parameter ε. Meanwhile the coherence of RSPEPVBs would decrease in stronger oceanic turbulence. Further an experimental setup was demonstrated to confirm that the RSPEPVBs will diffuse in strong oceanic turbulence which was strengthened with the rise of salinity and the increase of propagation distance. Contrarily with the increase of temperature the loss of RSPEPVBs mitigated and the intensity increased gradually. The obtained results may provide a potential application for optical underwater communication and imaging.

Paraxial propagation of cosine-hyperbolic-Gaussian beams in a chiral medium

The properties of a linearly polarized cosine-hyperbolic-Gaussian beam (ChGB) propagating in a chiral medium are investigated theoretically by using the Huygens-Fresnel diffraction integral. Beam intensity distribution characteristics in the chiral medium are discussed with numerical examples. The linearly polarized ChGB is found to split into a superposition of left and right circularly polarized Gaussian components, and its intensity pattern exhibits a variable shape depending on the chirality factor and decentered parameter b. The output beam can exhibit a richer intensity pattern compared to the Gaussian beam. The results could be beneficial for the application of ChGB in beam shaping and optical micromanipulation.

Propagation dynamics of symmetric Pearcey-Gaussian beam with optical vortices

The propagation dynamics of symmetric Pearcey-Gaussian beam embedded with on-axis and off-axis optical vortices (OVs) is investigated. Based on the extended Huygens-Fresnel diffraction integral, influences of embedded location, the off-axis distance and topological charge (TC) of OVs on the autofocusing dynamics of the symmetric Pearcey-Gaussian vortex beam (SPGVB) are explored analytically and numerically. The results show that, the embedded location, the off-axis distance and TC of OVs can significantly affect the initial intensity distribution, the propagation trajectory, the autofocusing length and the peak intensity at the focal plane. Interestingly, the fitted expression describing the relationship between the peak intensity of SPGVB and the off-axis distance of OVs with different TC is obtained, which would provide a method to determine the off-axis distance or the TC of OV based on the intensity distribution. The results indicate that SPGVB holds potential applications in biomedical treatment, optical manipulation and optical communication.

Influences of salinity and temperature on propagation of radially polarized rotationally-symmetric power-exponent-phase vortex beams in oceanic turbulence.

In this paper, the propagation properties of radially polarized rotationally-symmetric power-exponent-phase vortex beams (RP-RSPEPVBs) in oceanic turbulence were theoretically and experimentally studied. Based on the extended Huygens-Fresnel diffraction integral and vector beams theories, the theoretical propagation model of RP-RSPEPVBs in the oceanic turbulence was established. Then, the numerical simulations were carried out to study the influences of the propagation distance z, the rate of dissipation of turbulence kinetic energy per unit mass of fluid ε, the temperature-salinity contribution ratio ω, and the dissipation rate of the mean-squared temperature χT on the optical intensity, spectral degree of polarization (DOP) and spectral degree of coherence (DOC) of RP-RSPEPVBs. Further, an experiment setup was demonstrated to confirm the influences of salinity and temperature on propagation of RP-RSPEPVBs in oceanic turbulence. The results showed that increasing salinity, propagation distance, and turbulence intensity, will result in beam diffusion and intensity reduction of the RP-RSPEPVBs, as well as depolarization and decoherence. Contrarily, high temperature mitigated the intensity loss of the RP-RSPEPVBs and the spectral DOP and spectral DOC increased when the turbulence tends to be dominated by temperature. As a vector beam, the RP-RSPEPVB shows well anti-turbulence interference characteristics, which provides a new choice for optical underwater communication and imaging.

Introduction and propagation properties of circular lorentz-bessel-gaussian beams

In this paper, circular Lorentz–Bessel–Gaussian beams (CLBGBs) are introduced as a novel member of the Lorentz-Gaussian beams family. The analytical expression of the propagation of these beams passing through a paraxial optical ABCD system is derived. The generalized Huygens-Fresnel diffraction integral of the form Collins’s formula and the expansion of the Lorentz distribution in terms of the complete orthonormal basis set of the Hermite-Gauss modes are used. The influences of the beam-order and Bessel part β, Gaussian and Lorentzian waists, and propagation distance z on the propagation of CLBGBs are investigated. Some numerical simulation results are done. The beams family in this work may be useful to the practical applications in free-space optical communications because they have vortex properties with their being experimentally realizable.

Generation and Detection of Optical Vortices with Multiple Cascaded Spiral Phase Plates

Spiral phase plate (SPP) is the widely used method in the generation of vortex beam (VB) with fixed topological charges (TCs) for specific wavelength. Although VB with large TCs can be directly generated by using the SPP with high vortex order. The fabrication of high-quality SPPs with high vortex orders usually requires complex manufacturing process and high machining accuracy. An alternative method to generate VBs with large TCs is cascaded multiple SPPs with low order. In this study, we numerically calculate the transmitted light field of cascaded multiple SPPs according to the Huygens–Fresnel diffraction integral, and perform the experimental verifications. Based on cascading 6 SPPs (3 SPPs with TCs of 2, and 3 SPPs with TCs 4, respectively), an VB with TCs as high as 18 is generated. Furthermore, The TCs of the generated VB are detected by coaxial and off-axis interfering with fundamental Gaussian beam or its conjugate beam, respectively. The generated fork and spiral patterns allow us to distinguish the value and sign of TCs carried by the VB. The experimental results coincide well with the theoretical simulations. The fork pattern shows better resolution than the spiral one, and the petal pattern with small spiral allows us to distinguish large TCs with a higher resolution.

Comparative analysis of some Schell-model beams propagating through turbulent atmosphere

In this paper, we investigate a comparative analysis of some Generalized Laguerre–Gaussian Schell-model beams through a paraxial ABCD optical system in a turbulent atmosphere. Based on the extended Huygens–Fresnel diffraction integral, analytical expressions for the spectral density in the receiver plane of the studied beams are derived in detail. By studying the effects of the source coherence parameters and the atmospheric turbulence strength, the numerical results indicate that the profile of these Schell-model beams takes different shapes during their propagation.

Propagation properties of vortex cosine-hyperbolic-Gaussian beams through oceanic turbulence

Based on the extended Huygens–Fresnel diffraction integral, the analytical expression of the average intensity for a vortex cosine hyperbolic-Gaussian beam (vChGB) propagating in oceanic turbulence is derived in detail. From the derived formula, the propagation properties of a vChGB in oceanic turbulence, including the average intensity distribution and the beam spreading, are discussed with numerical examples. It is shown that oceanic turbulence influences strongly the propagation properties of the beam. The vChGB may propagate within shorter distance in weak oceanic turbulence by increasing the dissipation rate of mean-square temperature and the ratio of temperature to salinity fluctuation or by increasing the dissipation rate of turbulent kinetic energy per unit mass of sea water. Meanwhile, the evolution properties of the vChGB in the oceanic turbulence are affected by the initial beam parameters, namely the decentered parameter b, the topological charge M, the beam waist width ω0 and the wavelength λ. The obtained results can be beneficial for applications in optical underwater communication and remote sensing domain, imaging, and so on.

Behavior of the central intensity of generalized humbert-gaussian beams against the atmospheric turbulence

In this paper, we present a theoretical study of the propagation properties for Generalized Humbert-Gaussian beams (GHGBs) passing through a turbulent atmosphere. The axial intensity distribution of beams propagating through atmospheric turbulence is evaluated analytically based on the Huygens-Fresnel diffraction integral and the Rytov theory. The impact of the incident beam parameters and the turbulent strength on the output axial intensity is investigated through numerical illustrations. The results show that the propagation of GHGBs is faster when the atmosphere is very turbulent for small wavelength and small beam waist.

Fractional Fourier transforms of vortex Hermite-cosh-Gaussian beams

In this paper, we investigate the propagation properties of a vortex Hermite-cosh-Gaussian beam (vHChGB) passing through a fractional Fourier transform (FRFT) optical system. The analytical expression of a vHChGB in FRFT plane is derived based on the Huygens–Fresnel diffraction integral. From the obtained expression, the evolution of the intensity and phase distributions of output beam are illustrated numerically as a function of the order of the FRFT and under different beam parameters conditions. The results show that a vHChGB in FRFT plane evolves gradually and periodically versus the order of the FRFT optical system. The beam shape depends strongly on the fractional order, and it is affected by the beam order, the decentered parameter and the topological vortex charge. The results obtained may be beneficial to applications in optical trapping, micro-particle manipulation, and beam shaping.

Effects of turbulent atmosphere on the propagation properties of vortex Hermite-cosine-hyperbolic-Gaussian beams

The propagation properties of a vortex Hermite-cosh-Gaussian beam (vHChGB) in atmospheric turbulence are investigated based on the extended Huygens–Fresnel diffraction integral and Rytov method. The analytical formula for the average intensity of a vHChGB propagating in turbulent atmosphere is derived in detail. The influence of the turbulence strength on the intensity distribution under the change of beam parameters conditions is illustrated numerically and discussed. Results show that the profile of the initial vHChGB remains unchanged within small propagation distance range, and at certain propagation distance a central peak intensity appears, and finally the beam evolves into Gaussian profile–like in far-field. The rising speed of the central peak intensity is faster when the turbulence strength is larger or the beam parameters such as the beam order, the vortex charge and the Gaussian waist width are smaller. With a small decentered parameter b, the beam profile changes faster as the wavelength is larger, whereas the reverse behavior occurs when b is large. The obtained results may be useful for the practical applications of vHChGB in optical communications and remote sensing.

Focusing properties and focal shift of a vortex cosine-hyperbolic Gaussian beam

In this paper, we investigate the focusing properties of a vortex-cosh-Gaussian (vChG) beam passing through a converging thin lens. Based on the Huygens–Fresnel diffraction integral, we derived the analytical propagation equation as well as the beam width expression of a focused vChGB. It is shown that the focusing properties including the focal shift of the focused vChGB are crucially dependent on the incident beam parameters namely the decentered parameter and the topological charge in addition to the Gaussian Fresnel number \(N_{F}\). From typical numerical examples, it is found that the focused vChGB is transformed into a multi-lobes structure shape at the real focus plane, and the principal maximum intensity of the beam is located away from the axis. The amount of focal shift, which is determined from the minimum spot size criterion, is strongly dependent on the Fresnel number, the decentered parameter and the vortex charge m. The obtained results may be useful for the applications of the vChGBs in beam shaping and beam focusing.

Propagation of vortex cosine-hyperbolic-Gaussian beams in atmospheric turbulence

In this paper, the propagation properties of a vortex cosh-Gaussian beam (vChGB) in turbulent atmosphere are investigated. Based on the extended Huygens–Fresnel diffraction integral and the Rytov method, the analytical expression for the average intensity of the vChGB propagating in the atmospheric turbulence is derived. The effects of the turbulent strength and the beam parameters on the intensity distribution and the beam spreading are illustrated numerically and analyzed in detail. It is shown that upon propagating, the incident vChGB keeps its initial hollow dark profile within a certain propagation distance, then the field loses gradually its central hole-intensity and transformed into a Gaussian-like beam for large propagation distance. The rising speed of the central peak is demonstrated to be faster when the constant strength turbulence or the wavelength are larger and the Gaussian width is smaller. The obtained results can be beneficial for applications in optical communications and remote sensing.

Paraxial propagation and focusing of higher-order cosh-Gaussian beams

Based on the Huygens-Fresnel diffraction integral, a concise formula for the higher-order cosh-Gaussian beams (HOChGB) propagating through a paraxial ABCD optical system is derived. The propagation formula of the HOChGB is expressed as a superposition of cosh-Gaussian modes and in terms of the Gaussian beam-parameter q. The profile and the inter-lobe space of the beam at the initial plane are analyzed with the help of the maximum intensity criterion as a function of the decentered parameter and the beam order. It is shown that during propagation the HOChGB with appropriate parameters may progressively change profile, and in far-field the amplitude of the beam is expressed as a superposition of cosine functions modulated by a Gaussian envelope. Furthermore, it is found that the focal shift of the focused HOChGB increases with the decrease of the Fresnel number, the decentered parameter and the beam order.

Introduction of the vortex Hermite-Cosh-Gaussian beam and the analysis of its intensity pattern upon propagation

In this paper, a vortex-Hermite-cosh-Gaussian beam (vHChGB) is introduced as a general form of the vortex cosh-Gaussian and vortex Hermite-Gaussian beams. Based on the Huygens-Fresnel diffraction integral, a propagation formula of the vHChGB passing through a paraxial optical system is derived. The evolution of the intensity pattern of the propagated vHChGB as a function of the beam order, decentered parameter (b parameter) and topological vortex charge is discussed with numerical results. It is shown that the vHChGB shape in the source plane depends crucially on the decentered parameter. In far-field, the multi-lobe profile obtained for small b evolves into a four-petal-like beam. Whereas the initial four-lobe profile for large b, morphs into a multi-lobes pattern. The vHChGB shows specific intensity patterns depending on the beam order and the vortex charge. This study may be beneficial for applications involving beam shaping of hollow vortex beam, beam splitting techniques and optical communications.

Evolution of the partially coherent Generalized Flattened Hermite-Cosh-Gaussian beam through a turbulent atmosphere

The current paper investigates the impact of a turbulent atmosphere on the propagation of a partially coherent Generalized Flattened Hermite Cosh-Gaussian (GFHChG) beam. The evaluation of a theoretical expression of the average intensity distribution for the partially coherent GFHChG beam is based on the extended Huygens-Fresnel diffraction integral and the Rytov’s quadratic approximation. The results of some laser beams are deduced as special cases from the present work. The behavior of the studied beam which is propagating through a turbulent atmosphere is analyzed through numerical illustrations. Results indicate that the partially coherent GFHChG beam successively changes shape from a doughnut to a flattened then to a Gaussian profile during its propagation in a turbulent atmosphere.

Intensity characteristics of double-half inverse Gaussian hollow beams through turbulent atmosphere

The propagation characteristics of a double-half inverse Gaussian hollow beam (DHIGHB) through atmospheric turbulence are investigated theoretically. The on-axis average intensity of the beam passing through a turbulent atmosphere is derived in closed-form under the framework of the Huygens–Fresnel diffraction integral and the Rytov method. Numerical examples have been performed to illustrate the dependence of the on-axis intensity on the index structure constant of the atmospheric turbulence and the parameters of the incident DHIGHB. It is shown that upon propagating the DHIGHB will have a bright central spot, and the on-axis intensity will reach a threshold value at certain propagation distance zm, and after that the beam will retrieve its original dark hollow characteristics. This conversion is quicker for large constant strength turbulence, large wavelength and small initial beam waist size.

Production of Airy-related beams by an Airy transform optical system

Based on the process of Huygens–Fresnel diffraction integral, we investigate theoretically the characteristics of fields generated by diffracted controllable dark-hollow beams (CDHBs) through an Airy transform optical system. It is shown that the CDHB is transformed into Airy-related beams and the dark central spot is converted into peak intensity. The effects of the incident CDHB parameters such as the mode number N, central spot width $$w_{0}$$ and $$\varepsilon$$ as well as the phase pattern parameters of Airy transform optical system on the generated beam are analyzed in detail with numerical examples.

Transformation of a generalized Bessel–Laguerre–Gaussian beam by a paraxial ABCD optical system

An analytical expression of a Generalized Bessel–Laguerre–Gaussian (GBLG) beam passing through a paraxial ABCD optical system is developed by using the generalized Huygens–Fresnel diffraction integral. This result is regarded as the main finding of this paper. Several studies such as the propagation of fundamental Gaussian, flat-topped Gaussian, Bessel–Gaussian, Generalized Bessel–Gaussian and Laguerre–Gaussian beams, through an ABCD optical system, are derived as particular cases of our principle investigation. Numerical simulations are performed, in this work, to illustrate the propagation properties of GBLG beam through a free space, a thin lens, a Fractional Fourier transform system and a quadratic graded-index system.