Hermite-Gaussian Functions Research Articles

A robust eigenbasis generation system for the discrete Fourier transform

We have developed a systematic approach to construct an intelligible real eigenbasis for discrete Fourier transforms (DFT) by directly utilizing the eigenbases of some specific types of discrete sine and cosine transforms (DST and DCT). This methodological advancement not only enhances the comprehension of DFT spectra but also leads to a significant outcome: the identification of an explicit discrete analogue of Hermite-Gaussian functions within the context of DFT. By capitalizing on the inherent structure present in DST and DCT eigenbases, our approach facilitates a seamless transition to the domain of discrete Hermite-Gaussian functions, thereby opening up new avenues for related applications.

Orthogonal waveform-based backscattering interference suppression technique for underwater optical wireless communication

The backscattering effects degrade the detection performance in a full-duplex underwater optical wireless communication (UOWC) system when the power of received backscattering interference is comparable to that of a target signal. In this paper, the orthogonal waveforms with the Hermite–Gaussian function and prolate spheroidal wave function are first introduced for a pulse position modulation based dynamic UOWC system to alleviate the backscattering effects. Then, a joint maximum likelihood detection scheme is proposed accordingly. Experimental results verify the superiority of the proposed method with orthogonal waveforms over the scheme without orthogonal pulse shaping in a dynamic underwater bubbly channel.

On the properties of the anisotropic multivariate Hermite-Gauss functions

The Hermite-Gauss basis functions have been extensively employed in classical and quantum optics due to their convenient analytic properties. A class of multivariate Hermite-Gauss functions, the anisotropic Hermite-Gauss functions, arise by endowing the standard univariate Hermite-Gauss functions with a positive definite quadratic form. These multivariate functions admit useful applications in optics, signal analysis and probability theory, however they have received little attention in literature. In this paper, we examine the properties of these functions, with an emphasis on applications in computational optics.

Very short time Fourier transform for turbulent analysis

Nominally stationary acoustic sensors, e.g., buoy suspended or bottom anchored, are subject to various environmental flows. Direct numerical simulation of hydrodynamic flow around the acoustic sensor housing show turbulence along the housing surface. Time-space analysis of pressure variation along the surface of the housing shows significant acoustic-like surface interaction at low frequencies, even at very low mean flow velocity. In this case turbulence manifests as intermittently generated, irregular, coherent vortex structures that separate from the surface of the housing and advect downstream. These structures drive rapid surface pressure transients that propagate within the housing to the acoustic sensors as noise. Fractional Fourier Analysis provides joint time-frequency methods to extract transient features of turbulence. Use of Hermite-Gauss functions and phase relations are of particular interest. [This research is supported by 6.2 NRL base program sponsored by the Office of Naval Research. Distribution Statement A: Approved for public release. Distribution unlimited.]

Refined beam expansion method for unstable resonator modes

A refined and simplified version of the beam expansion method for computing the modes of unstable resonators is presented here.The principle of the beam expansion method is to express the field pattern on a basis of Hermite-Gaussian functions. Propagating each individual component around the resonator enables a transfer matrix to be constructed from which the eigenvalues and eigenvectors of the modes can be determined.Whereas the earlier proof-of-principle approach performed well for low Fresnel numbers, the refined technique gives good results for the challenging test case of magnification M=1.9 and equivalent Fresnel number Neq=49.4. The performance enhancement has been achieved by incorporating a number of adjustments that improve the accuracy of the calculation.

Soliton solutions of the space–time fractional nonlocal nonlinear Schrödinger equation

In this paper, the space–time fractional nonlocal nonlinear Schrödinger equations under Jumarie modified Riemann–Liouville derivative is studied. The soliton solutions are formally derived by using the extended tanh method of fractional Riccati subequation and the ansatz method combined with the Hermite–Gaussian function and Mittag-Leffler function. Different types of solutions are obtained and the necessary conditions for the existence of solutions are given. With the help of Maple, we plot figures exhibit that solitons are forming as fractional order λ increasing, and also, we noticed that at different order n of the Hermite–Gaussian function Hn(y), mainly affects the amplitude of soliton. Furthermore, we have discussed the new soliton solutions in the fractional case. It is of great significance to understand complex physical phenomena described by fractional nonlocal nonlinear equations.

Orbital angular momentum of structurally stable laser beams

For structurally stable laser beams whose amplitude can be represented as a finite sum of the Hermite-Gaussian functions with undefined weight coefficients, we obtain an analytical expression for the normalized orbital angular momentum (OAM) that is also expressed through finite sums of weight coefficients. It is shown that a certain choice of weight coefficients allows obtaining the maximal OAM, which is equal to the maximal index of the Hermite polynomial in the sum. In this case, the sum describes a single-ringed Laguerre-Gaussian beam with a topological charge equal to the maximal OAM and to the maximal order of the Hermite polynomial.

Exponential bound of the integral of Hermite functions product with Gaussian weight

In this paper we derive a bound on the integral of a product of two Hermite-Gaussian functions with a Gaussian weight. We prove that such integrals decay exponentially in the difference of the indices of the Hermite-Gaussian functions. Such integrals arise naturally in mathematical physics and applied mathematics. The estimate is applied to a variational problem related to a Strichartz functional.

Quantum harmonic oscillators with nonlinear effective masses in the weak density approximation

We study the eigen-energy and eigen-function of a quantum particle acquiring the probability density-dependent effective mass (DDEM) in harmonic oscillators. Instead of discrete eigen-energies, continuous energy spectra are revealed due to the introduction of a nonlinear effective mass. Analytically, we map this problem into an infinite discrete dynamical system and obtain the stationary solutions in the weak density approximation, along with the proof on the monotonicity in the perturbed eigen-energies. Numerical results not only give agreement to the asymptotic solutions stemmed from the expansion of Hermite-Gaussian functions, but also unveil a family of peakon-like solutions without linear counterparts. As nonlinear Schrödinger wave equation has served as an important model equation in various sub-fields in physics, our proposed generalized quantum harmonic oscillator opens an unexplored area for quantum particles with nonlinear effective masses.

A Hermite–Gaussian based exoplanet radial velocity estimation method

As the first successful technique used to detect exoplanets orbiting distant stars, the radial velocity method aims to detect a periodic Doppler shift in a stellar spectrum due to the star’s motion along the line sight. We introduce a new, mathematically rigorous approach to detect such a signal that accounts for the smooth functional relationship of neighboring wavelengths in the spectrum, minimizes the role of wavelength interpolation, accounts for heteroskedastic noise and easily allows for accurate calculation of the estimated radial velocity standard error. Using Hermite–Gaussian functions, we show that the problem of detecting a Doppler shift in the spectrum can be reduced to linear regression in many settings. A simulation study demonstrates that the proposed method is able to accurately estimate an individual spectrum’s radial velocity with precision below 0.3 m s−1, corresponding to a Doppler shift much smaller than the size of a spectral pixel. Furthermore, the new method outperforms the traditional cross-correlation function approach for estimating the radial velocity by reducing the root mean squared error up to 15 cm s−1. The proposed method is also demonstrated on a new set of observations from the EXtreme PREcision Spectrometer (EXPRES) for the host star 51 Pegasi, and successfully recovers estimates of the planetary companion’s parameters that agree well with previous studies. The method is implemented in the R package rvmethod, and supplemental Python code is also available.

Eigenvectors of Ordinary, Generalized, Centered and Offset Discrete Fourier Transforms Based on Lookup Table Methods: Efficiency and Approximation Uses

In this paper, we first propose a methodology to construct an eigenvector of the N × N discrete Fourier transform (DFT) matrix from any eigenvector of the ( k 2 N ) × ( k 2 N ) DFT matrix, where k is a positive integer. Specifically, using the procedure to down-sample by k any ( k 2 N )-point DFT eigenvector with zero as the initial index and then fold modulus N and add the resultant down-sampled ( kN )-point vector, we obtain an N -point DFT eigenvector. Computing an N -point DFT eigenvector using the proposed method is very efficient, which requires only N ( k -1) additions and no multiplications with a pre-stored lookup table of ( k 2 N )-point DFT eigenvectors. Similar efficient methods are also developed to compute N -point eigenvectors of other DFT-related transforms, including the generalized DFT (GDFT), centered DFT (CDFT) and the offset DFT (ODFT), from ( k 2 N )-point eigenvectors of their corresponding transforms or the DFT. As application examples, we apply the proposed methods to compute N -point DFT, GDFT, CDFT and ODFT eigenvectors, which are closer to samples of the continuous Hermite-Gaussian function (HGF) than existing N -point eigenvectors, from their corresponding ( k 2 N )-point eigenvectors. These improved N -point Hermite-Gaussian-like eigenvectors can be used to define fractional versions of the various N × N DFT transforms whose outputs are closer to samples of the continuous fractional Fourier transform than outputs of the corresponding existing fractional transforms. Finally, we perform computer experiments to verify the effectiveness and demonstrate applications of our proposed methods.

Spatiotemporal self-accelerating Airy–Hermite–Gaussian and Airy–helical–Hermite–Gaussian wave packets in strongly nonlocal nonlinear media

The self-accelerating Airy–Hermite–Gaussian (AiHG) and Airy–helical–Hermite–Gaussian (AiHHG) wave packets in strongly nonlocal nonlinear media (SNNM) are obtained analytically and numerically. The complex amplitudes of the AiHG and AiHHG wave packets are obtained by multiplying the self-accelerating Airy functions, the Hermite Polynomials and Gaussian functions. Furthermore, the different 3D spatiotemporal profiles and the photoelasticity can be obtained by selecting different values of the mode number, the chirp factor, the initial velocity, and the degree of the nonlocality.

Quantum temporal imaging with squeezed light

We consider a simple quantum temporal imaging system on the basis of a time lens implemented by a parametric nonlinear optical process with a chirped pump. We review the main results of the modal theory of temporal imaging, developed recently by us. We illustrate this theory by a concrete example of type-I non-collinear sum-frequency generation process, where the phase matching is limited by the temporal walk-off of the signal and the idler waves. We have shown that the temporal modal functions of such an imaging system are chirped Hermite-Gauss functions.

Stable Hermite-Gaussian solitons in optical lattices

We report a family of solitons generated by Hermite-Gaussian beams that are supported in optical lattices, also described by Hermite-Gaussian functions in combination with a harmonic potential that is modelled by a (1+1)D nonlinear Schrödinger equation. We find that this kind of solitons is stable during propagation, provided they remain below a level of the power threshold. The pure local nonlinear system studied here can mimic, up to a certain extent, a strongly nonlocal medium, thus allowing generation of accessible solitons. These Hermite-Gaussian profiles constitute a kind of uncommon analytical solitons that allow the study of nonlinear wave behavior phenomena in a more tractable and closed form.

Dilated and Shifted Discrete Hermite Functions With Minimum Time-Bandwidth Product

In this letter, a novel method is proposed to generate discrete approximations of continuous Hermite–Gaussian functions with noninteger shift and dilation. Moreover, the approximations preserve orthogonality. In addition, the discrete Hermite–Gaussian functions reach the minimum time-bandwidth product under discrete version of Gabor uncertainty principle. Two applications are demonstrated. In fractional delay application, the proposed method is compared with the windowing method and proved to have smaller delay error. Another one is the signal expansion based on the proposed complete orthogonal basis.

Обнаружение сигналов с использованием ортогонального базиса функций Гаусса - Эрмита

The analysis of the shape and parameters of signals is a key part of the signal processing. As a rule in practicable task, the shape of the input signal is known and identification of the unknown parameters (amplitude, duration, location or time of arrival, etc.) is required, then the dynamics of the signal sequence in the process under study is important as well. Thus, the problem is the most reliable detection of the desired signal on the background of random noise and the most accurate measurement of its parameters. It is known that a solution of such problems concerns optimal filtering. The main abstract of the method of the signal detection, based on orthogonal basis functions of the Gauss-Hermite functions, is described. The method is founded on the matched filtering principles, it allows tracking not only the dynamics of the signal under investigation, but also its local features, which provide additional diagnostic information. The verification method is carried out on a real signal of a phase measuring complex. The signal represents a sequence of pulses as a mixture component of the respiratory and cardiac activity of a medical patient.

Experimental realization of structured super-oscillatory pulses.

We demonstrate experimentally a generic method for the synthesis of optical femtosecond pulses based on Gaussian, Airy and Hermite-Gauss functions, which are transformed to exhibit fringes with tunable width. The width of the fringes is set in some cases to be much narrower than the inverse of the spectral bandwidth. Such pulses might be useful for ultrafast spectroscopy, coherent control and nonlinear optics.

Perturbation-iteration method for multi-peak solitons in nonlocal nonlinear media

An perturbation-iteration method is developed for the computation of the Hermite-Gaussian-like solitons with arbitrary peak numbers in nonlocal nonlinear media. This method is based on the perturbed model of the Schr\"{o}dinger equation for the harmonic oscillator, in which the minimum perturbation is obtained by the iteration. This method takes a few tens of iteration loops to achieve enough high accuracy, and the initial condition is fixed to the Hermite-Gaussian function. The method we developed might also be extended to the numerical integration of the Schr\"{o}dinger equations in any type of potentials.

Hyperbolic accelerating beams and their relation with Hermite-Gaussian beams.

We derive the initial distributions of phase and complex amplitude of accelerating beams with arbitrary predesigned hyperbolic trajectories using the caustic-design method and explore the relation between these beams and Hermite-Gaussian beams. The results show the hyperbolic accelerating beams are a larger class of beams than Hermite-Gaussian beams. When the bending parameter is an integer, the hyperbolic accelerating beams have a similar initial complex amplitude distribution and almost the same propagating characteristics as Hermite-Gaussian beams. Through the analysis of the ray-based method, we also derive an approximate expression for the initial complex amplitude of Hermite-Gaussian beams after introducing an amplitude distribution function. Although the proposed approximate expressions of complex amplitude are more complex than the usually used Hermite-Gaussian function, they explicitly indicate the information on local amplitude, wave vector, and internal ray structure (including caustics) of these beams and thus provide us clearer geometrical insights into these beams.

Discrete Fractional Fourier Transforms Based on Closed-Form Hermite–Gaussian-Like DFT Eigenvectors

In this paper, we construct discrete fractional Fourier transforms (DFrFT) using recently introduced closed-form Hermite–Gaussian-like (HGL) eigenvectors. With respect to such eigenvectors, we discuss the convergence of their components to samples of the corresponding continuous Hermite–Gaussian functions and propose solutions to deal with some restrictions related to their construction. This allows us to give new procedures for obtaining orthonormal bases of HGL eigenvectors, which are used to fractionalize the discrete Fourier transform. We illustrate the application of the resulting DFrFT in the scenario of filtering in the fractional domain and compare the results with existing DFrFT approaches.