Lorentz-Gauss Vortex Beam Research Articles

Propagation intensity and phase distribution of a Partially Coherent Lorentz–Gauss Vortex Beam in a Gradient-Index Medium

The study focused on examining the characteristics of partially coherent Lorentz-Gauss vortex beams in a medium featuring a gradient refractive index. The study employed the ABCD matrix, deriving analytical expressions for the cross-spectral density function based on the extended Huygens-Fresnel principle and the relationship between the Hermitian Gaussian function and Lorentz distribution. The findings revealed that the topological charge, coherence length, and gradient refractive index coefficient are pivotal factors influencing the beam’s behaviour in the medium. An increase in topological charge led to an expansion of the dark spot in intensity, while the gradient refractive index plays a crucial role in altering the size and shape of the intensity periodically in the medium and does not affect the dark spot in the middle of the beam. The findings from this study hold potential for applications in remote sensing and optical communications, making them valuable for further research endeavors.

The Propagation Properties of a Lorentz–Gauss Vortex Beam in a Gradient-Index Medium

Based on the Huygens–Fresnel integral and ABCD matrix, the propagation equation for the Lorentz–Gauss vortex beam (LGVB) in a gradient-index medium (GRIN) is rederived. The evolution of the intensity and phase distributions of an LGVB through a GRIN medium are numerically calculated as a function of the gradient-index parameter with changes in the incident beam parameters. The results showed that the propagation path and intensity distributions changed periodically with increasing propagation distance. In contrast, phase distributions change at multiples of π/β or 2π/β, depending on whether the M values are odd or even, respectively. At the same time, the parameters of the gradient index determine the periodic values of the Lorentz–Gauss vortex beams during propagation, and as β increased, the period of evolution decreased. The Lorentz–Gauss vortex beam propagating through the gradient index will develop from a square beam to a Gaussian vortex beam more quickly with an increase of w0x=w0y. In addition, the topological charge affects the size of the dark spot at the center of the beam and the size of the beam, causing the phase distributions to change periodically in the medium. This study is beneficial for laser optics and optical communications.

Intensity and beam widths of a four-petal Lorentz-Gauss vortex beam propagating in turbulent atmosphere

The equations of average intensity and beam widths of a four-petal Lorentz-Gauss vortex (FPLGV) beam propagating in turbulent atmosphere have been derived, and the evolutions of average intensity and beam widths of a FPLGV beam in turbulent atmosphere are discussed using numerical examples. It is found that intensity distribution of a FPLGV beam at the short distance can be controlled by modulating topological chargeM and beam order N, and the FPLGV beam will translate into Gaussian beam due to the actions of atmospheric turbulence at longer distance. It is also found that the beam widths of a FPLGV beam will increase as the distance increases, and the beam propagating in stronger atmospheric turbulence will have a larger beam width at the same propagation distance.

Four-petal Lorentz-Gauss vortex beam and its propagation in free space

Considering the formations of vortex beam and four-petal Lorentz-Gauss beam produced by Liu (Optik, 179, 492, 2019), a new vortex beam called four-petal Lorentz-Gauss vortex beam is firstly introduced. The propagation equation of four-petal Lorentz-Gauss vortex beam in free space is derived, and the evolutions of intensity and phase are analyzed by using derived equations. With increasing the propagation distance, the intensity distribution with four-petal beams will be destroyed and evolve from array beam into a beam with dark hollow center, and the phase will become irregular. With increasing the topological M, the four-petal Lorentz-Gauss vortex beam will translate into a beam with larger dark hollow center, while the influences of N on the distribution can be neglected at the longer propagation distance.

Non-paraxial propagation of a circular Lorentz-Gauss vortex beam.

The optical field of a circular Lorentz-Gauss vortex beam propagating in free space is derived by using the vectorial Rayleigh-Sommerfeld integrals. The expression of the longitudinal component of the angular momentum density of a circular Lorentz-Gauss vortex beam is also presented. The normalized intensity, the three components of intensity, and the longitudinal angular momentum density of a circular Lorentz-Gauss vortex beam are demonstrated in the reference plane, respectively. The effects of linearly polarized angle, topological charge, and Lorentzian width parameter on the intensity, the three components of intensity, and the longitudinal angular momentum density are analyzed, respectively. The normalized intensity and the longitudinal angular momentum density of a paraxial circular Lorentz-Gauss vortex beam are also shown in the same reference plane. The missing longitudinal component of the optical field will result in deficient patterns of the intensity and the longitudinal angular momentum density. In terms of the longitudinal angular momentum density, the longitudinal component of a circular Lorentz-Gauss vortex beam cannot be neglected under the condition that the Lorentzian width parameter and the Gaussian waist are of the order of the optical wavelength.

Evolution properties of a partially coherent Lorentz-Gauss vortex beam in a uniaxial crystal

ABSTRACTThe propagation properties of a partially coherent Lorentz-Gauss vortex beam propagating in a uniaxial crystal orthogonal to the optical axis were studied. The analytical expressions of the partially coherent Lorentz-Gauss vortex beam propagating in a uniaxial crystal were derived. Based on the derived formulae, the average intensity properties and evolution properties of the coherent vortex of the partially coherent Lorentz-Gauss vortex beam propagating in a uniaxial crystal were illustrated using numerical examples. The influences of the uniaxial crystal and the parameters of the beam at the source plane of z = 0 on the average intensity and evolution properties of the coherent vortex for the partially coherent Lorentz-Gauss vortex beam are analyzed in detail.

Nonparaxial properties of partially coherent Lorentz-Gauss vortex beam propagating in free space

The analytical formulae of nonparaxial propagation, paraxial propagation and far field propagation for the partially coherent Lorentz-Gauss vortex beam have been derived, the propagation properties of nonparaxial, paraxial and far field propagation are analyzed using the derived formulae. One finds that the paraxial propagation can be seen as a special case of the nonparaxial propagation at the propagation distance of the Rayleigh length, and the far field propagation can also be regarded as special case of the nonparaxial propagation at the longer propagation distance.

Propagation of partially coherent Lorentz-Gauss vortex beam through oceanic turbulence.

The partially coherent Lorentz-Gauss vortex beam generated by a Schell-model source has been introduced. Based on the extended Huygens-Fresnel principle, the cross-spectral density function of a partially coherent Lorentz-Gauss vortex beam propagating in oceanic turbulence is derived. The influences of coherence length, topological charge M, and oceanic turbulence on the spreading properties and position of the coherence vortex for a partially coherent Lorentz-Gauss vortex beam are analyzed in detail. The results show that a partially coherent Lorentz-Gauss vortex beam propagating in stronger oceanic turbulence will evolve into a Gaussian-like beam more rapidly as the propagation distance increases, and the number of coherent vortices will change.

Orbital angular momentum density and spiral spectra of Lorentz–Gauss vortex beams passing through a single slit**Project supported by the National Natural Science Foundation of China (Grant No. 11574272) and Zhejiang Provincial Natural Science Foundation of China (Grant No. LY16A040014).

Based on the Hermite–Gaussian expansion of the Lorentz distribution and the complex Gaussian expansion of the aperture function, an analytical expression of the Lorentz–Gauss vortex beam with one topological charge passing through a single slit is derived. By using the obtained analytical expressions, the properties of the Lorentz–Gauss vortex beam passing through a single slit are numerically demonstrated. According to the intensity distribution or the phase distribution of the Lorentz–Gauss vortex beam, one can judge whether the topological charge is positive or negative. The effects of the topological charge and three beam parameters on the orbital angular momentum density as well as the spiral spectra are systematically investigated respectively. The optimal choice for measuring the topological charge of the diffracted Lorentz–Gauss vortex beam is to make the single slit width wider than the waist of the Gaussian part.

Orbital angular momentum density and spiral spectrum of Lorentz–Gauss vortex beams diffracted by a rectangular aperture

Based on the expansions of the Lorentz distribution and the aperture function, an analytical expression of Lorentz–Gauss vortex beams with a topological charge of ±1 diffracted by a rectangular aperture is derived. One can judge the sign of the topological charge from the normalized intensity and the phase distributions. The effects of the rectangular aperture on the orbital angular momentum density and the spiral spectrum are investigated, respectively. When the length and the width of the rectangular aperture are not equal, the orbital angular momentum density distribution becomes twisted and tilted. When the size of the rectangular aperture increases, the magnitude of the orbital angular momentum density and the weight coefficient of the dominant spectrum both increase, while the weight coefficients of other minor spectra decrease. In addition, the expansion of the spiral spectrum in the case of rectangular aperture is smaller than that in the case of the single slit. The difference between the adjacent spectra in the case of the rectangular aperture is four, which is twice the difference in the case of the single slit. Moreover, the weight coefficient of the dominant spectrum in the case of the rectangular aperture is relatively larger. To measure the topological charge of the diffracted Lorentz–Gauss vortex beam, this research denotes that the rectangular aperture is superior to the single slit.

Tunable optical gradient force of radially polarized Lorentz-Gauss vortex beam by sine-azimuthal variation wavefront

Tunable optical gradient force distributions of radially polarized Lorentz-Gauss vortex beam by sine-azimuthal variation wavefront were investigated by vector diffraction theory, which shows that many interesting force patterns may appear, including “x” shape optical trap, twines “C” shape optical trap, crescent shape optical trap, and overlapping optical gradient force traps. And whole force distributions are symmetric about two coordinate axes for case of even number value of phase parameter, while only symmetric about one coordinate axis for odd number value of phase parameter. Effect of charge number of the optical vortex on force pattern evolution is more remarkable for lower phase parameter than that for higher. Force pattern is very sensitive to phase parameter, therefore, force pattern may be used to act as one kind of measurement method of phase parameter.

Radially polarized Lorentz-Gauss vortex beam with sine-azimuthal variation wavefront

Radially polarized Lorentz-Gauss vortex beam with sine-azimuthal variation wavefront was proposed and its focusing properties were investigated by vector diffraction theory. The effect of phase parameter on focal pattern is very considerable, and some novel intensity pattern comes into being under certain focusing condition, including multiple-peaks, crescent shape, multiple intensity lines, intensity cross. Especially, the symmetry characteristics are systematical on increasing phase parameter. For case of even number of phase parameter, there are two symmetric axes for whole focal pattern, while there is only one symmetric axis for odd number of phase parameter. When topological charge is small, the intensity maximum shifts within small distance range, while intensity maximum may switch for higher topological charge in focal pattern evolution process.

Effect of Astigmatism on the Tight Focusing of Azimuthally Polarized Lorentz-Gauss Vortex Beam

Effect of Astigmatism on the Tight Focusing of Azimuthally Polarized Lorentz-Gauss Vortex Beam

Orbital angular momentum density of a general Lorentz–Gauss vortex beam

Based on the vectorial Rayleigh–Sommerfeld integral formulae, the analytical expression of a general Lorentz–Gauss vortex beam with an arbitrary topological charge is derived in free space. By using the analytical expressions of the electromagnetic field beyond the paraxial approximation, the orbital angular momentum density of a general Lorentz–Gauss vortex beam can be calculated. The effects of the linearly polarized angle and the topological charge on the three components of the orbital angular momentum density are investigated in the reference plane. The two transversal components of the orbital angular momentum are composed of two lobes with the same areas and opposite signs. The longitudinal component of the orbital angular momentum density is composed of four lobes with the same areas. The sign of the orbital angular momentum density in a pair of lobes is positive, and that of the orbital angular momentum density in the other pair of lobes is negative. Moreover, the negative magnitude of the orbital angular momentum density is larger than the positive magnitude of the orbital angular momentum density. The linearly polarized angle affects not only the shape and the location of the lobes, but also the magnitude of the three components of the orbital angular momentum density. With increasing the topological charge, the distribution of the orbital angular momentum density expands, the magnitude of the orbital angular momentum density increases, and the shape of the lobe also slightly changes.

Wigner distribution function of a Lorentz–Gauss vortex beam: alternative approach

The Wigner distribution function of a Lorentz–Gauss vortex beam with one topological charge is considered. The topics have been the object of a recent publication. However, we present an alternative compact expression that could complement the analysis developed in the aforementioned publication. The deduced expression can be usefully exploited to fully account for the Wigner dynamics of the Lorentz–Gauss vortex beam as far as the paraxial propagation through first-order optical systems is concerned.

The Wigner distribution function of a Lorentz–Gauss vortex beam passing through a paraxial ABCD optical system

Based on the Collins integral formula and the Hermite–Gaussian expansion of a Lorentz function, an analytical expression for the Wigner distribution function (WDF) of a Lorentz–Gauss vortex beam passing through a paraxial ABCD optical system is derived. The WDF properties of a Lorentz–Gauss vortex beam propagating in free space are also demonstrated. During propagation, the WDF pattern shrinks in the direction of the spatial frequency variables and elongates in the direction of the transverse spatial coordinates. With given the transverse spatial coordinates or the spatial frequency variables, the evolvement of the WDF of a Lorentz–Gauss vortex beam during propagation is examined in detail. To show the WDF application, the second-order moments of the WDF for a Lorentz–Gauss vortex beam in free space are also presented. This research exhibits the propagation properties of a Lorentz–Gauss vortex beam from the phase space.

Propagation property of a Lorentz–Gauss vortex beam in a strongly nonlocal nonlinear media

An analytical expression of a Lorentz–Gauss vortex beam with one topological charge propagating in a strongly nonlocal nonlinear media is derived. The analytical expressions of the beam width, the curvature radius, and the orbital angular momentum density for the Lorentz–Gauss vortex beam with one topological charge have been also presented. The normalized intensity distribution, the relative beam width, the curvature radius, and the orbital angular momentum density distribution of the Lorentz–Gauss vortex beam with one topological charge are demonstrated in the strongly nonlocal nonlinear media, respectively. The normalized intensity, the beam width, the curvature radius, and the orbital angular momentum density versus the axial propagation distance are all periodic and the period is T=πz0/η. The evolution of the propagation property of the Lorentz–Gauss vortex beam with one topological charge has been exhibited in the strongly nonlocal nonlinear media. When the parameter η reaches the critical value, the beam width keeps invariant upon propagation, and the corresponding curvature radius is infinite. The propagation of Lorentz–Gauss vortex beams with larger topological charge propagating in the strongly nonlocal nonlinear media can be analyzed by the same procedure as here.

Beam propagation factors and kurtosis parameters of a Lorentz-Gauss vortex beam.

Based on the second-order and the higher-order moments, analytical expressions for the beam propagation factors of a Lorentz-Gauss vortex beam with l=1 have been derived, and analytical propagation expressions for the kurtosis parameters of a Lorentz-Gauss vortex beam with l=1 through a paraxial and real ABCD optical system have also been presented. The M² factor is determined by the parameters a and b and decreases with increasing the parameter a or b. The M² factor is validated to be larger than 2. The kurtosis parameters depend on the diffraction-free ranges of the Lorentz part, the parameters a and b, and the ratio A/B. The kurtosis parameters of a Lorentz-Gauss vortex beam propagating in free space are demonstrated in different reference planes. In the far field, the kurtosis parameter K decreases with increasing one of the parameters a and b. Upon propagation, the kurtosis parameter K first decreases, then increases, and finally tends to a saturated value. In any case, the kurtosis parameter K is larger than 2. This research is beneficial to optical trapping, guiding, and manipulation of microscopic particles and atoms using Lorentz-Gauss vortex beams.

LINEAR MOMENTUM DENSITY OF A GENERAL LORENTZ-GAUSS VORTEX BEAM IN FREE SPACE

Based on the Collins integral, an analytical expression of a general Lorentz-Gauss vortex beam propagating in free space is derived, which allows one to calculate the linear momentum density of a general Lorentz-Gauss vortex beam in free space. The linear momentum density distribution of a general Lorentz-Gauss vortex beam propagating in free space is graphically demonstrated. The x- and y-components of the linear momentum density are composed of two lobes with the equivalent area and the opposite sign. Therefore, the overall x- and y-components of the linear momentum in an arbitrary reference plane are equal to zero. The longitudinal component of the linear momentum density is proportional to the intensity distribution. The in∞uences of the Gaussian waist, the width parameters of the Lorentzian part, the axial propagation distance, and the topological charge on the linear momentum density distribution of a general Lorentz-Gauss vortex beam in free space are examined in detail.

Angular momentum density of a linearly polarized Lorentz–Gauss vortex beam

Based on the vectorial Rayleigh–Sommerfeld integral formulae, the analytical propagation equation of a linearly polarized Lorentz–Gauss vortex beam is derived in free space. By taking curl of the electric field, the propagating magnetic field of the linearly polarized Lorentz–Gauss vortex beam is also presented. By using the analytical propagation equation of the Lorentz–Gauss vortex beam beyond the paraxial approximation, the analytical expression of the angular momentum density of a Lorentz–Gauss vortex beam is obtained. The effects of the linearly polarized angle and the beam parameters on the three components of the angular momentum density of a Lorentz–Gauss vortex beam are analyzed in detail. The two transversal components of the angular momentum of a Lorentz–Gauss vortex beam beyond the paraxial approximation must be zero in an arbitrary reference plane. The longitudinal component of the angular momentum is determined by the linearly polarized angle and the beam parameters. The distributions of the angular momentum densities of the Lorentz–Gauss vortex beam are also compared with those of the Gaussian vortex beam. This research is very useful to the optical trapping, the optical guiding, and the optical manipulation of microscopic particles using the single mode diode laser beams.