Paraxial ABCD Optical System Research Articles

Introduction of a modified anomalous vortex beam with self-focusing properties.

This study introduces and experimentally demonstrates the concept of a modified anomalous vortex beam (MAVB), which carries orbital angular momentum (OAM) and exhibits unique self-focusing properties. By utilizing holographic techniques and customizing phase masks, we precisely control the beam's phase and intensity distribution, enhancing self-focusing behavior while preserving traditional anomalous vortex beam features. We derive an analytical formula to describe MAVB propagation within a paraxial ABCD optical system. The self-focusing characteristics are influenced by initial parameters such as beam order, quantum number, beam waist, wavelength, and the modification parameter. Additionally, we simulate MAVB propagation and their OAM spectrum in maritime atmospheric turbulence. Through comprehensive theoretical analysis and experimental validation, we show how MAVBs achieve controlled self-focusing, leading to enhanced beam control and stability. Our study explores the mechanisms, design principles, and practical implications of MAVBs, emphasizing their potential to revolutionize optical applications.

Paraxial propagation characteristics of a controllable anomalous hollow vortex beam in free space

Propagation of a recently proposed controllable anomalous hollow vortex (CAHV) beam is investigated. Based on the integral formula of generalized Huygens–Fresnel diffraction, analytical expression for the CAHV beam through a paraxial ABCD optical system is derived. The factors that affect the intensity pattern are determined by the beam’s controllable parameters a, c x, c y, and the topological charge m. Results show that the Gaussian distribution features are controlled by parameter a, and the horizontal and vertical stretching deformations of the beam are adjusted by parameters c x and c y, respectively. For a controllable anomalous hollow (CAH) beam, when propagating in free space, it could initially maintain anomalous hollow property and the size of the spot increases with the increase of the propagation distance. Due to the CAHV beam carries the optical vortex, a dark hollow channel appears in the center of the beam during propagation, and the channel structure changes with the increase of topological charge. Additionally, the Poynting vector of CAHV beam proves the direction of energy flow corresponding to the intensity distribution. Results obtained in this paper could have potential applications in particle trapping and optical control.

Study on the construction of twisted cosine partially coherent beams and their propagation characteristics

We propose a novel Schell model source for generating twisted partially coherent beams with an initial radius of curvature, which is called a twisted flat-topped cosine Gaussian Schell-model (TFCGSM) source. The TFCGSM beam comprises a wavefront phase and a flat-top structure, with the source degree of coherence determined by two cosine functions. Based on the Huygens–Fresnel principle, the general analytical expression of the cross-spectral density function of the TFCGSM beam propagating through the paraxial ABCD optical system is derived, and then its propagation properties are studied. The results show that the conversion of the array of the beam and the non-uniform structure can be realized by adjusting the parameters in the source plane. As the propagation distance of the TFCGSM beam increases, it rotates around the axis and increases the intensity of the array distribution. Surprisingly, the initial radius of curvature can cause the beam to rotate. The unique shape and properties of the TFCGSM beam create new possibilities for optical communication and enhanced optical functions.

Introduction of Bessel higher-order cosh-Gaussian beam and its propagation through a paraxial ABCD optical system

Introduction of Bessel higher-order cosh-Gaussian beam and its propagation through a paraxial ABCD optical system

Analyzing the spectral characteristics of a pulsed Laguerre higher-order cosh-Gaussian beam propagating through a paraxial ABCD optical system

Analyzing the spectral characteristics of a pulsed Laguerre higher-order cosh-Gaussian beam propagating through a paraxial ABCD optical system

Properties of circular cosh-gaussian beams focused by a thin lens system under a turbulent atmosphere

The propagation of a Circular cosh-Gaussian beam (CiChGB) through a paraxial ABCD optical system in a turbulent atmosphere is investigated based on the extended Collins formula and Rytov theory. In particular, the analytical expression for the on-axis average intensity of a CiChGB focused by a thin lens in a turbulent atmosphere is derived in detail. From the obtained formula, the on-axis intensity distribution of the output beam is discussed numerically with a comparison to direct propagation in free space. The influences of the turbulence strength and initial beam parameters on the on-axis intensity are examined. It is found that the on-axis intensity near the focus region is less affected by atmospheric turbulence compared to the free space.

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.

Fast calculation of orbital angular momentum flux density of partially coherent Schell-model beams on propagation.

Optical coherence has recently become a degree of freedom to modulate the orbital angular momentum (OAM) flux density of a partially coherent beam during propagation. However, the calculation of the OAM flux density for the partially coherent beam involves partial differential and four-dimensional integral operations, which poses drawbacks for its fast numerical calculations. In this paper, we present an efficient numerical protocol for calculating the OAM flux density of any partially coherent Schell-model beam propagating through a paraxial ABCD optical system by only adopting two-dimensional (2D) Fourier transforms. The general formalism is established in detail for the fast numerical calculation of the OAM flux density. It is found that the operation number in the developed algorithm is independent on the spatial coherence states of the beam. To demonstrate the validity of our algorithm, we calculate the OAM flux density of the partially coherent Laguerre-Gaussian beams during propagation with both the analytical and numerical methods. The obtained results are consistent well with each other. Moreover, the OAM flux density properties of two other classes of Schell-model beams, having no analytical solutions, are investigated as the specific examples. Our method provides a convenient way for studying the correlation-induced OAM density changes for any Schell-model beam propagation through a paraxial optical system.

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.

Radially polarized twisted partially coherent vortex beams

We introduce a new type of partially coherent vector beam, named the radially polarized twisted partially coherent vortex (RPTPCV) beam. Such a beam carries the twist phase and the vortex phase simultaneously, and the initial state of polarization (SOP) is radially polarized. On the basis of the pseudo-modal expansion and fast Fourier transform algorithm, the second-order statistics such as the spectral density, the degree of polarization (DOP) and the SOP, propagation through a paraxial ABCD optical system are investigated in detail through numerical examples. The results reveal that the propagation properties of the RPTPCV beam closely depends on the handedness of the twist phase and the vortex phase. When the handedness of the two phases is same, the beam profile is easier to remain a dark hollow shape and the beam spot rotates faster during propagation, compared to the partially coherent vortex beam or the RPTPCV beam with the opposite handedness of the two phases. In addition, the same handedness of two phases resists the coherence induced de-polarization of the beam upon propagation, and the SOP is also closely related to the handedness, topological charge of the vortex phase and the twist factor of the twist phase, providing an efficient way to modulate the beam's DOP and SOP in the output plane. Moreover, we establish an experiment setup to generate the RPTPCV beam. The average spectral density and the polarization properties are examined in the experiment. The experimental results agree reasonable well with the theoretical predictions. Our results will be useful for particle manipulating, free-space optical communications, and polarization lidar systems.

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.

Statistical properties of partially coherent polarization singular vector beams

We investigate the statistical properties of partially coherent polarization singular beams embedded with a V-point polarization singularity. An analytical formula for the cross-spectral density matrix is derived for the family of partially coherent polarization singular vector beams (PSVBs) propagating through a paraxial ABCD optical system. It is observed that the far-field intensity profiles and the coherence-induced depolarization effect in partially coherent PSVBs depend on both the input spatial coherence length and the Poincar\'e-Hopf index (PHI) of the beam. Interestingly, it is found that in this process of coherence degradation, the polarization (Stokes, ${S}_{12}$) vortices are preserved. The depolarization is due to an enhanced unpolarized light field that in turn modulates the beam profile, the transverse distribution of the degree of polarization (DOP) and the degree of coherence (DOC). Furthermore, the Gaussian distribution of the DOC evolves into a non-Gaussian profile in the far-field with the number of ring dislocations equal to the magnitude of PHI of the beam. The degeneracy associated with the intensity profile, the Stokes intensity distribution, the DOP, and DOC profiles of these partially coherent PSVBs carrying opposite polarity of PHI are also discussed to complete this study. Subsequently, all of these findings are experimentally verified by generating a family of partially coherent PSVBs with controllable spatial coherence. The modulation of the spatial coherence length in the source plane leads to efficient control of its intensity, the DOC and DOP profiles on propagation, which are of importance in particle trapping, material thermal processing, free-space optical communications, and detection of a phase object.

Propagation invariance and dark hollow structures of sinh-Gaussian beams with small complex parameters

Through investigating the sinh-Gaussian beam with complex beam parameters, it is the first time to find that such beams can carry vortices and exhibit dark hollow intensity distributions when the complex beam parameters are sufficiently small. A closed-form propagation equation for sinh-Gaussian beams through paraxial ABCD optical systems is derived based on the Collins formula and illustrated with numerical methods. It is shown that the perfect hollow configuration can retain a quite long propagating distance under small complex beam parameters. The analytical discussions affirm the numerical conclusions.

Propagation analysis of some doughnut lasers beams through a paraxial ABCD optical system

We present in this work the propagation properties of Generalized Humbert-Gaussian beams (GHGBs) passing through a paraxial ABCD optical system. Based on the extended Huygens-Fresnel integral, the analytical expression of the intensity distribution for GHGBs is derived and some numerical simulations are performed for particular cases in free space, thin lens in a free space and graded refractive index system. Our results show that the comportment of several particular cases of these beams during the propagation is influenced by the beam orders and the characteristic parameters of the system.

Introduction of a new vortex cosine-hyperbolic-Gaussian beam and the study of its propagation properties in fractional Fourier transform optical system

Based on the framework of the Huygens–Fresnel diffraction, we investigate theoretically the propagation properties of a new vortex beam, which is referred to as vortex-cosh-Gaussian beam (vChGB), through a paraxial ABCD optical system. A closed-form formula of vChGB passing through the above system is derived. We show by analytical and numerical calculations that the decentered parameter and the topological vortex charge affect strongly the characteristics of the considered beam upon propagating in free space. In a fractional Fourier transform system (FrFT), it is found that the intensity and the phase distributions of the propagating vChGB evolves gradually and periodically versus the order of the FrFT. The shape of the vChGB depends on the parameters of the beam and the fractional order of the FrFT system. The results obtained may be beneficial to applications in optical trapping, optical micromanipulation and beam shaping.

Propagation propertiesof a novel generalized flattened Hermite-cosh-Gaussian light beam

In this paper, we have introduced a general formula of a new laser beam with a nearly Flattened-top profile. The propagation properties of this beam, called Generalized Flattened Hermite-cosh-Gaussian (GFHChG) beam are studied in detail. Based on the Huygens–Fresnel integral, an analytical expression of the intensity distribution for GFHChG passing through a paraxial ABCD optical system is developed. Several particular cases such as the propagation of the flattened Gaussian beam, the cosh-Gaussian beam, the higher and squared cosh-Gaussian beam and the Hermite-cosh-Gaussian beam are deduced from our result. Numerical simulations are also given in free space and through a circular thin lens in a free space to study the comportment of this beam during propagation.

Random source for generating Airy-like spectral density in the far field.

A stationary beam forming an Airy-like spectral density in the far field is analyzed theoretically and experimentally. The Schell-model source that radiates such a beam is an extended version of a recently introduced source [O. Korotkova, et al., Opt. Lett.43, 4727 (2018)10.1364/OL.43.004727; X. Chen, et al., Opt. Lett.44, 2470 (2019)10.1364/OL.44.002470, in 1D and 2D, respectively]. We show, in particular, that the source degree of coherence, being the fourth-order root of a Lorentz-Gaussian function and having linear and cubic phase terms, may be either obtained from the Fourier transform of the far-field Airy-like pattern or at the source using the sliding function method. The spectral density of the beam is analyzed on propagation through paraxial ABCD optical systems, on the basis of the generalized Collins integral, by means of the derived closed-form expression. We show that the distribution of the side lobes in the Airy beam spectral density can be controlled by the parameters of the source degree of coherence. Further, an experiment involving a spatial light modulator (SLM) is carried out for generation of such a beam. We experimentally measure the complex degree of coherence of the source and observe the gradual formation of a high-quality Airy-like spectral density towards the far field. In addition, the trajectory of the intensity maxima of the beam after a thin lens is studied both theoretically and experimentally. The random counterpart of the classic, deterministic Airy beam may find applications in directed energy, imaging, beam shaping, and optical trapping.

Radiation forces of hypergeometric-Gaussian type-II beams acting on a Rayleigh dielectric sphere

We drive the analytical expressions for the propagation of hypergeometric-Gaussian typed-II (HyGG-II) beam passing through a paraxial ABCD optical system. The radiation forces of a strongly focused HyGG-II beam exerting on a Rayleigh dielectric sphere are theoretically and numerically investigated under the dipole approximation. The influences of optical parameters on the radiation forces are also discussed in detail. The results demonstrate that the HyGG-II beams can hold the micro particles with both high and low refractive index. Finally, we analyze the necessary conditions for stably trapping the particle.

Integral transforms involving the product of Humbert and Bessel functions and its application

In this paper, we develop some integral transforms involving a product of Humbert and Bessel functions with a weight e-γx2. These integral transforms will be evaluated in terms of hypergeometric functions. Various transformation formulae are also evaluated in terms of Appell functions to complete this study. Some special cases of the evaluated integrals yield some infinite series of generalized hypergeometric and Appell functions. As application, one of our main results is investigated to give an expression of the Generalized Humbert-Gaussian beams (GHGBs) propagating through a paraxial ABCD optical system.

Twisted elliptical multi-Gaussian Schell-model beams and their propagation properties.

We introduce a new kind of partially coherent source whose cross-spectral density (CSD) function is described as the incoherent superposition of elliptical twisted Gaussian Schell-model sources with different beam widths and transverse coherence widths, named twisted elliptical multi-Gaussian Schell-model (TEMGSM) beams. Analytical expression for the CSD function propagating through a paraxial ABCD optical system is derived with the help of the generalized Collins formula. Our results show that the TEMGSM beam is capable of generating a flat-topped elliptical beam profile in the far field, and the beam spot during propagation exhibits clockwise/anti-clockwise rotation with respect to its propagation axis. In addition, the analytical expressions for the orbital angular momentum (OAM) and the propagation factor are also derived by means of the Wigner distribution function. The influences of the twisted factor and the beam index on the OAM and the propagation factor are studied and discussed in detail.