Laguerre-Gaussian Functions Research Articles

Low frequency twisted waves in a self-gravitating nonextensive complex plasma

Abstract The effects of dust–dust self-gravitational force and nonextensive characteristics of plasma species on the low frequency twisted waves owing to the helical wave structure in complex (dusty) plasmas are analyzed. The electrons and ions of the plasma are modelled by nonextensive q-distribution function while massive dust particles are Maxwellian distributed. The self-gravitational effects are incorporated in the Vlasov equation of kinetic theory where perturbed distribution function, electrostatic and gravitational potentials are expressed with Laguerre–Gauss functions. The governing equations of kinetic theory are solved together under paraxial approximations. The dispersion relations and damping rates of twisted dust-acoustic waves (TDAWs) are obtained for two situations; (a) super-extensivity (q < 1) and (b) sub-extensivity (q > 1). The effects of self-gravity, nonextensivity and twist parameter significantly modified the basic features of dust-acoustic waves. This study contributes to our understanding of the complex dynamics of TDAWs in interstellar dust clouds, considering the interplay of self-gravity, nonextensivity, and helical phase structures. The obtained theoretical and numerical results provide valuable insights into the behavior of these waves and offer a foundation for further investigations in this field. However, understanding of the topic can be enhanced through a combination of theoretical models, numerical simulations and observational data.

Twisted dust-acoustic waves in a self-gravitating dusty plasma with nonextensive q-distributed electrons and ions

We have investigated the twisted dust-acoustic waves (TDAWs) in an electrostatic self-gravitating dusty plasma whose electrons and ions are modelled by nonextensive q-distribution function while massive dust particles are Maxwellian distributed. A well-known kinetic theory is employed for this purpose where perturbed distribution function, electrostatic and gravitational potentials are expressed with Laguerre–Gauss functions. The governing equations of kinetic theory are solved together under paraxial approximations. The dispersion relations and instability growth rates are obtained for two situations; a) super-extensivity (q < 1) and b) sub-extensivity (q > 1). Significant modifications concerning the wave frequencies and growth rates are presented with respect to self-gravitation parameter, twist parameter, nonextensive parameter and streaming speed. It is observed that wave frequency and growth rate of TDAWs reduces in the presence of self-gravitating effects. Furthermore, the growth rates exhibit a significant enhancement in amplitude with the increase in twist parameter, q-parameter and streaming speed. Our present results may have applications in interstellar dust clouds and in the dusty plasma environments of Halley’s Comet.

Quasi-electrostatic instability in non-gyrotropic tri-ion electron plasma

The plasma modes are significantly influenced by the simple or/and molecular anions. The reciprocity of beam and tri-ion electron (TIE) plasmas is modeled by the kinetic theory. The set of Vlasov–Poisson equation is solved and decomposed by the Laguerre–Gaussian function under paraxial approximation. The beam plasma expedites unstable twisted modes (UTMs) in the TIE plasma. The UTM or instability is calibrated via axial and azimuthal wave numbers, wind speed, and its growth essence. The qualitative and quantitative anatomy of the instability threshold is also predicted. Our model is eminently applicable to the space plasmas and has been applied to the Ar+–F−–SF6− and Xe+–F−–SF6− electron plasmas, specifically.

Twisted waves in symmetric and asymmetric bi-ion kappa-distributed plasmas

Waves in bi-ion plasmas are affected by asymmetry. The kinetic theory of the Maxwellian and Lorentzian/kappa-distributed bi-ion plasma is ameliorated to incorporate the transfer of orbital angular momentum from the helical electric field to the plasma modes. By operating the Laguerre–Gaussian function, the perturbed distribution function and helical electric field are decomposed into characteristic axial and azimuthal components. In symmetric bi-ion plasmas, the conventional ion modes/waves are only present if both ions have similar masses and the concentration of the electrons is negligible. An imbalance of the symmetry is considered by the contamination of a small fraction of the heavy immobile ions, which urges the negative ions to become heavier than the positive ions in the bi-ion plasma system. The distinct masses of the positive and negative ions provoke mass-asymmetry in the kappa-distributed bi-ion plasmas. The signature of the unique acoustic-laden twisted modes in non-Maxwellian asymmetric bi-ion plasma is perceived by the temperature of the lighter positive ions and the dynamics of the heavier negative ion. The deliberated results of Landau damping are displayed for distinct values of the azimuthal wave-number and spectral index, temperature-variation, and mass-asymmetry.

Statistical properties of an electromagnetic Gaussian Schell-model beam propagating in uniaxial crystals along the optical axis.

Within the angular-spectrum representation, we study the partially coherent beam propagating in uniaxial crystals along the optical axis. By a method of vortex expansion, we derive the analytical solution for the cross-spectral density (CSD) function of an electromagnetic Gaussian Schell-model (EGSM) beam. We demonstrate that the analytical expression of CSD function can be written into a quasi-coherent-mode representation, whose basis vectors are constructed by the elegant Laguerre-Gaussian (eLG) functions. Several limits of the analytical solution are examined and good agreements with previous theories are obtained. Moreover, we calculate the energy density and degree of polarization (DOP) of the EGSM beam, from which the effects of coherent degree on the propagating properties are revealed. It is found that the energy conversion between circularly polarized components becomes rapid when the degree of coherence is decreasing. For all degree of coherence, the energy density and DOP exhibit a similar saturated behavior in the far field.

Laguerre-Gaussian mode expansion for arbitrary optical fields using a subspace projection method.

Laguerre-Gaussian (LG) mode decomposition has found applications in various optics fields. However, numerical LG mode expansion for arbitrary field is still a problem, since the physical dimension of LG modes would vary with three parameters-the beam waist width w, the radial index p, and the azimuthal index m, which make it difficult to determine the optimal value of w and the truncation order on p. Here a general method of LG mode expansion for an arbitrary field is developed. It is found that the local frequency distribution of the LG function consists of two parts, the quasi-periodic part and the chirped part. The effective space-bandwidth product of the LG function is defined as the product of the spatial width and frequency-domain width of the quasi-periodic part. Then, based on this space-bandwidth product definition, the criteria for determining the beam waist parameter and the truncation order of LG series expansion are given. The scheme is tested for the representation of off-axis Hermite-Gaussian mode, which shows that our method is of high accuracy.

Compton Scattering of \u03b3-Ray Vortex with Laguerre Gaussian Wave Function

In this work, we report calculation for Compton scattering of a γ-ray vortex with a wave function of Laguerre Gaussian on an electron in the framework of the relativistic quantum mechanics. We consider the coincidence measurement of the scattered photon and the scattered electron from each Compton scattering. The momentum of the scattered photon distributes outside of the reaction plane determined by the incident photon and the scattered electron, and the energy of the scattered photon also distributes, when the scattered angle of the electron is simultaneously measured. These distributions depend on the angular momentum and the node number of the Laguerre Gaussian function of the incident photon. Thus, the coincident measurement for Compton scattering is useful to identify the nature of the vortex photon wave function.

Discrete Laguerre Gaussian Transforms and Their Applications

Laguerre Gaussian functions serve as a complete and orthonormal basis for a variety of physical problems, such as 2D isotropic quantum harmonic oscillators and circularly symmetric laser modes. In this paper, we propose “discrete Laguerre Gaussian functions,” which are defined such that some elegant physical properties are preserved and a fast computation algorithm of complexity $O(N \log N)$ is available. Discrete Laguerre Gaussian transforms, as introduced in this paper, inherit nice properties from discrete Laguerre Gaussian functions and admit signal analysis over circularly symmetric patterns. It is demonstrated through examples that discrete Laguerre Gaussian transforms find applications in circular pattern keypoints selection, object detection, image compression, rotational invariance feature for pattern recognition, and rotational angle estimation.

Nonlinear interaction of intense hypergeometric Gaussian subfamily laser beams in plasma

Propagation of Hypergeometric-Gaussian laser beam in a nonlinear plasma medium is investigated by considering the Source Dependent Expansion method. A subfamily of Hypergeometric-Gaussian beams with a non-negative, even and integer radial index, can be expressed as the linear superposition of finite number of Laguerre–Gaussian functions. Propagation of Hypergeometric-Gaussian beams in a nonlinear plasma medium depends on the value of radial index. The bright rings’ number of these beams is changed during the propagation in plasma medium. The effect of beam vortex charge number l and initial (input) beam intensity on the self-focusing of Hypergeometric-Gaussian beams is explored. Also, by choosing the suitable initial conditions, Hypergeometric-Gaussian subfamily beams can be converted to one or more mode components that a typical of mode conversion may be occurred. The self-focusing of these winding beams can be used to control the focusing force and improve the electron bunch quality in laser plasma accelerators.

Electrostatic twisted modes in multi-component dusty plasmas

Various electrostatic twisted modes are re-investigated with finite orbital angular momentum in an unmagnetized collisionless multi-component dusty plasma, consisting of positive/negative charged dust particles, ions, and electrons. For this purpose, hydrodynamical equations are employed to obtain paraxial equations in terms of density perturbations, while assuming the Gaussian and Laguerre-Gaussian (LG) beam solutions. Specifically, approximated solutions for potential problem are studied by using the paraxial approximation and expressed the electric field components in terms of LG functions. The energy fluxes associated with these modes are computed and corresponding expressions for orbital angular momenta are derived. Numerical analyses reveal that radial/angular mode numbers as well as dust number density and dust charging states strongly modify the LG potential profiles attributed to different electrostatic modes. Our results are important for understanding particle transport and energy transfer due to wave excitations in multi-component dusty plasmas.

Laguerre-Gaussian functions and generalized formulation of electromagnetic Gaussian Schell-model sources.

Fourier transform of the combination of Gaussian function and power functions of even exponents 2n(n=0,1,2,…) gives rise to a class of physically realizable electromagnetic sources with correlation properties expressible in terms of Laguerre-Gaussian functions. Relatively simple mathematical expressions of the electromagnetic beams produced by the Laguerre-Gaussian Schell-model sources are given for an investigation of the coherence and polarization properties and the higher-order coherence effect of the correlations between intensity fluctuations in the Laguerre-Gaussian Schell-model beams (Laguerre-GSMB) of any order n. Results obtained are compared numerically with the predictions of the classic electromagnetic Gaussian Schell-model beam that is the lowest-order Laguerre-GSM beams.

Two-dimensional accessible solitons in PT-symmetric potentials

Two-dimensional parity-time (PT) symmetric potentials are introduced, which allow the existence of spatial solitons in the model of the strongly nonlocal nonlinear Schrodinger equation. Two-dimensional accessible solitons are found in the form of solutions separating the radial amplitude, given in terms of Laguerre polynomials, a phase function involving quadratic, linear, and constant phase shifts, and a specific azimuthal modulation function. Shape-preserving solitons are constructed from Laguerre–Gaussian functions containing the self-similar variable and an exponential form of the azimuthal modulation, containing sine and cosine functions, when a suitable PT-symmetric potential is chosen. Interesting soliton profiles and the corresponding PT-symmetric potentials are displayed for different values of the parameters.

Electron Beams with a Twist

The properties and behaviour of electron beams possessing orbital angular momentum are considered. Methods of their creation are discussed, but we concentrate mainly on the method involving binarized holographic masks. It is shown how suitable masks can be modelled using Bessel functions as well as Laguerre-Gaussian functions. The theoretical far-field diffraction patterns of these holograms are simulated using Fourier transform techniques and compared with the recently reported experimental results of electron vortex beams. The limitations of the masks and the potential of the technique for future research are pointed out and discussed.

Quantum mechanical photoncount formula derived by entangled state representation

By introducing the thermo entangled state representation, we derive four new photocount distribution formulas for a given light field density operator. It is shown that these new formulas, which are convenient to calculate the photocount, can be expressed as integrations over a Laguerre—Gaussian function with a characteristic function, Wigner function, Q-function and P-function, respectively.

Time evolution of distribution functions in dissipative environments

By introducing the thermal entangled state representation, we investigate the time evolution of distribution functions in the dissipative channels by bridging the relation between the initial distribution function and the any time distribution function. We find that most of them are expressed as such integrations over the Laguerre—Gaussian function. Furthermore, as applications, we derive the time evolution of photon-counting distribution by bridging the relation between the initial distribution function and the any time photon-counting distribution, and the time evolution of R-function characteristic of nonclassicality depth.

Modeling mode characteristics of transverse anisotropic waveguides using a vector pseudospectral approach

I extend the full vector pseudospectral-based eigenvalue scheme, based on the transverse magnetic field components, to analyze the mode behaviors of dielectric optical waveguides with transverse, nondiagonal anisotropy. One of the principal axes of the anisotropic materials is thus constrained to point in the longitudinal direction of the waveguide. I expand the guided mode fields in the interior subdomains with finite extent by using Chebyshev polynomials and those in the exterior subdomains with semi-infinite extent by using Laguerre-Gaussian functions with an accurately determined scaling factor. This study analyzes two examples: (1) the circularly-polarized modes of a magneto-optical raised strip waveguide and (2) the guided mode patterns of a nematic liquid-crystal channel waveguide under different orientations of the liquid-crystal molecule. The comparison of the numerical results with those from the vector finite difference approach demonstrates that my numerical approach has a higher computational efficiency and requires less computer memory.

Adaptive Laguerre-Gaussian variant of the Gaussian beam expansion method

A variant of the Gaussian beam expansion method consists in expanding the Bessel function J0 appearing in the Fresnel-Kirchhoff integral into a finite sum of complex Gaussian functions to derive an analytical expression for a Laguerre-Gaussian beam diffracted through a hard-edge aperture. However, the validity range of the approximation depends on the number of expansion coefficients that are obtained by optimization-computation directly. We propose another solution consisting in expanding J0 onto a set of collimated Laguerre-Gaussian functions whose waist depends on their number and then, depending on its argument, predicting the suitable number of expansion functions to calculate the integral recursively.

Statistical properties of photon-added coherent state in a dissipative channel

We adopt a new approach (using the entangled state representation) to study time evolution of the mixed state in a dissipative channel (DC) with decay rate κ. We then investigate the statistical properties of the m-photon-added coherent state (PACS) in the DC, by deriving the analytical expressions of the Wigner function (WF), which turns out to be the Laguerre–Gaussian function. The criterion time for the definite positivity of WF: , valid for arbitrary m, is revealed. Time evolution of the photocount distribution is also derived analytically, which is related to two-variable Hermite polynomials. The tomogram of PACS is also calculated through the intermediate coordinate–momentum representation.

Complex Derivative Filters

Steerable filters are a valuable tool for various low-level vision tasks. In this paper, we argue for the use of complex analysis in the context of 2-D steerable filters. In particular, we recommend the use of complex partial derivatives as a computational basis. Complex derivatives have a major advantage in comparison to real derivatives: they show a canonical rotation behavior, namely a rotation affects the derivative just by a multiplication with a complex unit number. So, the complex derivatives can be steered in a more elegant way and above that they are less expensive to compute. We present several analytical formulas for common and new filter kernels in terms of complex derivatives. Further we relate the complex derivatives of a Gaussian with the Gauss-Laguerre transform and show that the Gauss-Laguerre functions provide an optimal signal representation for local and smooth images. We discuss various finite difference schemes for the realization of the derivatives and use them in practice. In a first experiment, we use a newly introduced filter kernel for anisotropic blurring. The complex formalism offers an elegant way to locally adapt the shape and orientation of the kernel. Second, we use the proposed filters as matched filters to detect vessels in retinal images.

Semiconductor nanodevice simulation by multidomain spectral method with Chebyshev, prolate spheroidal and Laguerre basis functions

A new approach based on spectral method with efficient basis functions is proposed in the paper to simulate semiconductor nanodevice by solving the Schrödinger equation. The computational domain is partitioned at heterojunctions into a number of subdomains. The envelope functions in subdomains are expanded by various efficient basis functions and then patched by the BenDaniel–Duke boundary conditions to preserve exponential order of accuracy. Importantly, the consideration to choose the basis functions depends on the oscillatory characteristics of envelope functions. Three kinds of basis functions including prolate spheroidal wave functions, Chebyshev polynomials, and Laguerre–Gaussian functions are used according to the mathematical features in this work. In addition, the determinations of optimum values of scaling factor in Laguerre–Gaussian functions and bandwidth parameter in prolate spheroidal wave functions are also discussed in detail. Several quantum well examples are simulated to validate the effectiveness of the present scheme. The relative errors of energy levels achieve the order of 10 −12 requiring merely a few grid points.