Higher-order Gaussian Beams Research Articles

Laser Gaussian beam analysis of structure constant depends on Kolmogorov in turbulent atmosphere for a variable angle of wave propagation

Laser Gaussian beam propagation is important in optical communications; the challenges of the measured constant parameters are varied angle wave propagation, the receiver plan in the slant path of laser Gaussian beam (G) in Kolmogorov gauges, and the effect of the laser in a random medium. At this point, atmospheric turbulence is reputable based on laser Gaussian beam propagation. Moreover, the space propagation of Gaussian beam waves of higher-order Gaussian beam modes has been expanded from the effects of the scintillation index and with different zenith angles between the laser propagator and the detector (receiver) in the ramp path. Furthermore, in the Cartesian coordinate system, the scintillation index in the Kolmogorov standard is evaluated, and it employs two methods; first, Huygens–Fresnel fundamental and, second, random phase screen for work synchronization. Additionally, to analyze the contour parameters of constant construction, the intensity and receiver field is found to compute the high turbulent modes in the atmospheric deposition, which depends on propagation distances. These result in significant reserves in the matlab computation period and serve the imitation for enhancement of the link transmission; therefore, the zenith angle and scintillation index are pretentious by slant beam, and they enhance the performance that makes the comparisons found with fixed structure constant parameters C2n and the propagation distance against the scintillation index.

Hyperparameter tuning of optical neural network classifiers for high-order Gaussian beams.

High-order Gaussian beams with multiple propagation modes have been studied for free-space optical communications. Fast classification of beams using a diffractive deep neural network (D2NN) has been proposed. D2NN optimization is important because it has numerous hyperparameters, such as interlayer distances and mode combinations. In this study, we classify Hermite-Gaussian beams, which are high-order Gaussian beams, using a D2NN, and automatically tune one of its hyperparameters known as the interlayer distance. We used the tree-structured Parzen estimator, a hyperparameter auto-tuning algorithm, to search for the best model. As a result, the proposed method improved the classification accuracy in a 16 mode classification from 98.3% in the case of equal spacing of layers to 98.8%. In a 36 mode classification, the proposed method significantly improved the classification accuracy from 84.9% to 94.9%. In addition, we confirmed that accuracy by auto-tuning improves as the number of classification modes increases.

Beam propagation factor of Hollow higher-order Cosh-Gaussian beams

Based on the second-order moments definition, the beam propagation factor of a new mathematical model of Hollow higher-order Cosh-Gaussian (HhCG) beams is investigated. Analytical formula for the M2-factor of HhCG beams is derived, which depends on the hollowness parameter l, the beam order n, the parameter of the cosh part δ and the beam waist ω0. The influences of these parameters are illustrated with numerical examples. The result provides more general characteristics because the higher-order Cosh-Gaussian, Cosh-Gaussian and Gaussian beams are obtained as specials cases of HhCG beams.

All-optical tunable high-order Gaussian beam splitter based on a periodic dielectric atomic structure.

Beam splitting of high-order Gaussian (HOG) beams increases the channel capacity and improves the processing speed of the incoming information. Here a novel all-optical tunable multi-port HOG beam splitter under a periodic dielectric atomic structure is proposed and demonstrated. The original HOG beam is replicated in the output beams. A distinguishable five-port output beam is observed in the experiment, which is beneficial for high-speed optical communications. By tuning the optical properties of this periodic dielectric structure, the spatial position and intensity distribution of each output port are precisely controllable. The splitting ratio δ can be finely adjusted in the range 0 - 4.8. This work provides a new approach for multi-port HOG beam splitters and the basis for all-optical communication.

Property of high-order Hermite-Gauss beam generated by self-excitation He-Ne laser with intracavity modulator

In this paper, the change of the high-order Gaussian beam pattern generated by the external cavity laser under the condition of modulating the intracavity medium is studied. We used a specially customized multi-mode He-Ne laser tube and other optical components to build a laser experiment platform,the laser tube can generate a variety of stable, high-quality, high-brightness high-order Gaussian beams under normal working conditions. When the device normally outputs a stable beam, firstly we select several modes of high-order Gaussian beams by adjusting the cavity mirror, and then interfere the laser cavity medium with an element composed of two metal wires with 0.15-mm diameter to modulate the output beam pattern, in this process, we studied the modulation effects of the metal wires’ distance, position,angle and other factors on the beam pattern, and used a CMOS camera to record the experimental results. Due to the interference of the metal wire, the order, direction, intensity and other characteristics of the output laser beam will change to different degrees, which may have great application prospects in many fields where the laser beam needs to be encoded.

Impact of Nonlinear Kerr Effect on the Focusing Performance of Optical Lens with High-Intensity Laser Incidence

The rapid development of high-energy and high-power laser technology provides an important experimental means for the research of extreme physical state in the laboratory and for the design of large laser facilities for realizing inertial confinement fusion. However, when the incident laser field is very strong, the Kerr effect of materials affects the nominal performance of optical elements. In this work, the impact of Kerr effect on the focusing performance of an optical lens is studied by calculating and comparing the filed patterns of focal spots for three different incident laser beams together with three different levels of light intensities. The traditional transfer function of an optical lens is firstly modified according to the theory of nonlinear Kerr effect. We use the two-dimensional fast Fourier transform algorithm and angular spectrum algorithm to numerically calculate the field distributions of focal spots in the nominal focal plane of lens and its adjacent planes based on the Fresnel diffraction integral formula. The obtained results show that the Kerr effect affects the focusing characteristics of lens, especially for the incidence of high-order Gaussian beams, such as Hermite-Gaussian beams and Laguerre-Gaussian beams. At the same time, the focal length and refractive index of lens also change the field patterns of focal spots. The presented methodology is of great value in engineering applications where the practical problem with beam size up to 100 mm can be calculated using a common laptop computer. The work provides an efficient numerical technique for high-intensity incident laser beams focused by lens that takes Kerr effect into consideration, which has potential applications in high energy density physics and large laser facilities for inertial confinement fusion.

Generation of high-order Gaussian beams by resonator with deformed steel wire

In this paper, the different spot phenomena of high-order laser cavity under wire interference diffraction are verified experimentally. Firstly, we customize an external cavity He-Ne laser tube which can directly generate high-order Gaussian beams with various modes, good beam stability and outstanding quality. This device is easy to operate and easy to implement. We have done some research on this laser cavity. On this basis, we use a wire with a diameter of 0.15 mm to deform and then interfere with the beam, and use a CMOS camera to collect the diffraction pattern of the beam. Finally, the induction and summary of the experimental phenomena under different steel wires are carried out, and the change process of high-order Gaussian beams is recorded in detail, which provides an experimental precedent for future research.

Generation of singular optical beams from fundamental Gaussian beam using Sagnac interferometer

We propose a simple free-space optics recipe for the controlled generation of optical vortex beams with a vortex dipole or a single charge vortex, using an inherently stable Sagnac interferometer. We investigate the role played by the amplitude and phase differences in generating higher-order Gaussian beams from the fundamental Gaussian mode. Our simulation results reveal how important the control of both the amplitude and the phase difference between superposing beams is to achieving optical vortex beams. The creation of a vortex dipole from null interference is unveiled through the introduction of a lateral shear and a radial phase difference between two out-of-phase Gaussian beams. A stable and high quality optical vortex beam, equivalent to the first-order Laguerre–Gaussian beam, is synthesized by coupling lateral shear with linear phase difference, introduced orthogonal to the shear between two out-of-phase Gaussian beams.

Investigations on Shaped Mirror Systems in Quasi-Optical Mode Converters Based on Irradiance Moments Method

A method of transforming high-order Gaussian beams (GBs) mode into circular symmetry fundamental Gaussian beam (FGB) mode with arbitrary waist size is presented using irradiance moments method in quasi-optical (QO) mode converters. The double shaped mirrors correcting amplitude and phase simultaneously are generated by a single incidence irradiance sampling data and known ideal output FGB taking advantage of linear moment matching technique and Fresnel diffraction theory, which can be applied to a wide frequency range especially significant for terahertz band. The numerical coding procedure of creating double correcting mirrors and its fast convergence speed are discussed at 325 GHz. Numerical and experimental comparisons reveal the conclusion that enhancing surface precision and increasing moments order can improve main lobe levels.

Energy, momentum and propagation of non-paraxial high-order Gaussian beams in the presence of an aperture

Non-paraxial theories of wave propagation are essential to model the interaction of highly focused light with matter. Here we investigate the energy, momentum and propagation of the Laguerre–, Hermite– and Ince–Gaussian solutions (LG, HG, and IG) of the paraxial wave equation in an apertured non-paraxial regime. We investigate the far-field relationships between the LG, HG, and IG solutions and the vector spherical wave function (VSWF) solutions of the vector Helmholtz wave equation. We investigate the convergence of the VSWF and the various Gaussian solutions in the presence of an aperture. Finally, we investigate the differences in linear and angular momentum evaluated in the paraxial and non-paraxial regimes. The non-paraxial model we develop can be applied to calculations of the focusing of high-order Gaussian modes in high-resolution microscopes. We find that the addition of an aperture in high numerical aperture optical systems does not greatly affect far-field properties except when the beam is significantly clipped by an aperture. Diffraction from apertures causes large distortions in the near-field and will influence light–matter interactions. The method is not limited to a particular solution of the paraxial wave equation. Our model is constructed in a formalism that is commonly used in scattering calculations. It is thus applicable to optical trapping and other optical investigations of matter.

Post-selected von Neumann measurement with Hermite–Gaussian and Laguerre–Gaussian pointer states

Through the von Neumann interaction followed by post-selection, we can extract not only the eigenvalue of an observable of the measured system but also the weak value. In this post-selected von Neumann measurement, the initial pointer state of the measuring device is assumed to be a fundamental Gaussian wave function. By considering the optical implementation of the post-selected von Neumann measurement, higher-order Gaussian modes can be used. In this paper, we consider the Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) modes as pointer states and calculate the average shift of the pointer states of the post-selected von Neumann measurement by assuming the system observable with and for an arbitrary interaction strength, where represents the identity operator. Our results show that the HG and LG pointer states for a given coupling direction have advantages and disadvantages over the fundamental Gaussian mode in improving the signal-to-noise ratio. We expect that our general treatment of the weak values will be helpful for understanding the connection between weak- and strong-measurement regimes and may be used to propose new experimental setups with higher-order Gaussian beams to investigate further the applications of weak measurement in optical systems such as the optical vortex.

Numerical approximation of the Schrödinger equation with the electromagnetic field by the Hagedorn wave packets

In this paper, we approximate the semi-classical Schrödinger equation in the presence of electromagnetic field by the Hagedorn wave packets approach. By operator splitting, the Hamiltonian is divided into the modified part and the residual part. The modified Hamiltonian, which is the main new idea of this paper, is chosen by the fact that Hagedorn wave packets are localized both in space and momentum so that a crucial correction term is added to the truncated Hamiltonian, and is treated by evolving the parameters associated with the Hagedorn wave packets. The residual part is treated by a Galerkin approximation. We prove that, with the modified Hamiltonian only, the Hagedorn wave packets dynamics give the asymptotic solution with error O(ε1/2), where ε is the scaled Planck constant. We also prove that, the Galerkin approximation for the residual Hamiltonian can reduce the approximation error to O(εk/2), where k depends on the number of Hagedorn wave packets added to the dynamics. This approach is easy to implement, and can be naturally extended to the multidimensional cases. Unlike the high order Gaussian beam method, in which the non-constant cut-off function is necessary and some extra error is introduced, the Hagedorn wave packets approach gives a practical way to improve accuracy even when ε is not very small.

Quadrupolar second-harmonic generation by helical beams and vectorial vortices with radial or azimuthal polarization

We study the optical second-harmonic radiation (SHG) generated by scattering from a homogeneous centrosymmetric thin composite material illuminated by higher-order Gaussian laser beams. The induced second-order source polarization is taken as of quadrupolar type $(\mathbf{E}\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\nabla})\mathbf{E}$, which depends on the inhomogeneity of the incident electric field $\mathbf{E}$. This nonlinear source has the same form as that responsible of the SH signal observed in a composite made of Si nanocrystals embedded uniformly in a SiO${}_{2}$ matrix and that calculated for a thin disordered array of nanospheres. We calculate the SH radiation angular patterns generated by several incident combinations of spatial modes and states of polarizations. In particular, excitation with radially and azimuthally polarized doughnut modes and helical beams carrying orbital angular momentum with linear or circular polarization are considered. We found that this quadrupolar SHG depends sensitively on the transverse structure and polarization of the driving field. The response to $\ensuremath{\nabla}\mathbf{E}$ introduces a factor $\mathbf{E}(\mathbf{E}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{K})$ in the Fourier component of the SH scattering amplitude, absent in electric-dipole-allowed SHG, that can give additional nodal lines or rings in the SH angular patterns, changes of the state of polarization, or additional azimuthal phases in the harmonic radiation. For circularly polarized beams with helical phase wave front, we found a selection rule according to which the nonlinear scattering of an optical vortex with charge ${l}_{\ensuremath{\omega}}$ and spin $\ensuremath{\sigma}=\ifmmode\pm\else\textpm\fi{}1$ induces a SH vortex field with a spin-dependent charge doubling ${l}_{2\ensuremath{\omega}}=2{l}_{\ensuremath{\omega}}+\ensuremath{\sigma}$. These features may be useful to identify SHG processes of quadrupolar nature and suggest a way to produce scattered SH radiation with a desired angular pattern and state of polarization.

Modal Characterization using Principal Component Analysis: application to Bessel, higher-order Gaussian beams and their superposition

The modal characterization of various families of beams is a topic of current interest. We recently reported a new method for the simultaneous determination of both the azimuthal and radial mode indices for light fields possessing orbital angular momentum. The method is based upon probing the far-field diffraction pattern from a random aperture and using the recorded data as a ‘training set'. We then transform the observed data into uncorrelated variables using the principal component analysis (PCA) algorithm. Here, we show the generic nature of this approach for the simultaneous determination of the modal parameters of Hermite-Gaussian and Bessel beams. This reinforces the widespread applicability of this method for applications including information processing, spectroscopy and manipulation. Additionally, preliminary results demonstrate reliable decomposition of superpositions of Laguerre-Gaussians, yielding the intensities and relative phases of each constituent mode. Thus, this approach represents a powerful method for characterizing the optical multi-dimensional Hilbert space.

Semi-Eulerian and High Order Gaussian Beam Methods for the Schrödinger Equation in the Semiclassical Regime

AbstractA novel Eulerian Gaussian beam method was developed in [8] to compute the Schrödinger equation efficiently in the semiclassical regime. In this paper, we introduce an efficient semi-Eulerian implementation of this method. The new algorithm inherits the essence of the Eulerian Gaussian beam method where the Hessian is computed through the derivatives of the complexified level set functions instead of solving the dynamic ray tracing equation. The difference lies in that, we solve the ray tracing equations to determine the centers of the beams and then compute quantities of interests only around these centers. This yields effectively a local level set implementation, and the beam summation can be carried out on the initial physical space instead of the phase plane. As a consequence, it reduces the computational cost and also avoids the delicate issue of beam summation around the caustics in the Eulerian Gaussian beam method. Moreover, the semi-Eulerian Gaussian beam method can be easily generalized to higher order Gaussian beam methods, which is the topic of the second part of this paper. Several numerical examples are provided to verify the accuracy and efficiency of both the first order and higher order semi-Eulerian methods.

Matrix formulation of the beam quality of the Hermite-Gaussian beam

Based on the M 2-parameter of Hermite-Gaussian beam, we deduced the matrix formulation of the beam quality for nonsymmetric two-dimensional Hermite-Gaussian beam, which at an arbitrary azimyth angle α. And coupled beam radius square, coupled M 2-parameter also have been introduced. The beam radius versus azimuth angle α at different positions has been presented by numerical simulation; and the track of M 2-parameter versus azimuth angle α has been given. The M 2-parameter matrix is universal and available both in the theoretical analysis and the practical measurement of the beam quality for a arbitrary two-dimensional nonsymmetric higher-order Gaussian beam.

Superpositions and higher order Gaussian beams

High frequency solutions to partial differential equations (PDEs) are notoriously difficult to simulate numerically due to the large number of grid points required to resolve the wave oscillations. In applications, one often must rely on approximate solution methods to describe the wave field in this regime. Gaussian beams are asymptotically valid high frequency solutions concen- trated on a single curve through the domain. We show that one can form integral superpositions of such Gaussian beams to generate more general high frequency solutions to PDEs. As a particular example, we look at high frequency solutions to the constant coefficient wave equation and construct Gaussian beam solutions with Taylor expansions of several orders. Since this PDE can be solved via a Fourier transform, we use the Fourier transform solution to gauge the error of the constructed Gaussian beam superposition solutions. Furthermore, we look at an example for which the solution exhibits a cusp caustic and investigate the order of magnitude of the wave amplitude as a function of frequency at the tip of the cusp. We show that the observed behavior is in agreement with the predictions of Maslov theory.

Complex degree of coherence for partially coherent general beams in atmospheric turbulence

With the use of the general beam formulation, the modulus of the complex degree of coherence for partially coherent cosh-Gaussian, cos-Gaussian, Gaussian, annular and higher-order Gaussian optical beams is evaluated in atmospheric turbulence. For different propagation lengths in horizontal atmospheric links, the moduli of the complex degree of coherence at the source and receiver planes are examined when reference points are taken on the receiver axis and off-axis. In the on-axis case, it is observed that in propagation, the moduli of the complex degree of coherence are symmetrical and look like the intensity profile of the related coherent beam propagating in a turbulent atmosphere. For all the beams considered, the moduli of the complex degree of coherence profiles turn into Gaussian shapes beyond certain propagation lengths. In the off-axis case, the moduli of complex degree of coherence patterns become drifted at the earlier propagation lengths. Among the beams investigated, the cos-Gaussian beam is found to be almost independent of the changes in the source partial coherence parameter, and the annular beam seems to be affected the most against the variations of the source partial coherence parameter.

Gaussian beams in the optics course

We introduce the physics of high-order Gaussian beams to the treatment of Gaussian beams in the undergraduate optics course. Of particular interest are Laguerre-Gauss beams, which provide the basis for discussing the new and increasingly important concept of the orbital angular momentum of light. We also describe laboratory exercises that complement the class material.

Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence

The propagation characteristics of higher-order annular Gaussian (HOAG) beams in turbulence are investigated. From a HOAG source plane excitation, the average intensity of the receiver plane is developed analytically. This formulation is verified against the previously derived HOAG beam solution in free space. The graphical outputs indicate that, upon traveling in turbulent atmosphere, the HOAG beam will undergo different stages of evolution. At intermediate propagation distances, it will attempt to concentrate the energy near the origin. In this process, the appearance of the single higher-order primary beam will be encountered. Eventually HOAG originated beam will become a pure Gaussian beam after propagating an excessive distance in the turbulent medium.