Hermite-Gaussian Functions Research Articles

Direct sequential evaluation of optimal orthonormal eigenvectors of the discrete Fourier transform matrix by constrained optimization

The recent emergence of the discrete fractional Fourier transform has spurred research activity aiming at generating Hermite-Gaussian-like (HGL) orthonormal eigenvectors of the discrete Fourier transform (DFT) matrix F. By exploiting the unitarity of matrix F - resulting in the orthogonality of its eigenspaces pertaining to the distinct eigenvalues - the problem decouples into finding orthonormal eigenvectors for each eigenspace separately. A Direct Sequential Evaluation by constrained Optimization Algorithm (DSEOA) is contributed for the generation of optimal orthonormal eigenvectors for each eigenspace separately. This technique is direct in the sense that it does not require the generation of initial orthonormal eigenvectors as a prerequisite for obtaining the final optimal ones. The resulting eigenvectors are optimal in the sense of being as close as possible to samples of the Hermite-Gaussian functions. The technique is found to be numerically robust because total pivoting is allowed in performing the QR matrix decomposition step. The DSEOA is proved to be theoretically equivalent to each of the Gram-Schmidt algorithm (GSA) and the sequential orthogonal Procrustes algorithm (SOPA). However the three techniques are algorithmically quite distinct. An extensive comparative simulation study shows that the DSEOA is by far the most numerically robust technique among all sequential algorithms thus paying off for its relatively long computation time.

Geometric phase decomposition in the basis of Hermite-Gaussian functions

Abstract The work presents geometric phase decomposition for analytical signals using Hermite-Gaussian functions. The decomposition is based on the time-frequency distribution with reassigned and multi-tapered spectrogram resulting in increased phase estimation resolution. Numerical analysis is applied to a number of SU(2) evolutions, such as spin-1/2 particle in a static and rotating magnetic field, as well as polarization rotation of a plane wave in optically active medium. Geometric phase decomposition results are provided also for quantum harmonic oscillator and a radiation field of an electric dipole exited by a short pulse.

Solitary waves in the nonlinear Schrödinger equation with Hermite-Gaussian modulation of the local nonlinearity

We demonstrate "hidden solvability" of the nonlinear Schrödinger (NLS) equation whose nonlinearity coefficient is spatially modulated by Hermite-Gaussian functions of different orders and the external potential is appropriately chosen. By means of an explicit transformation, this equation is reduced to the stationary version of the classical NLS equation, which makes it possible to use the bright and dark solitons of the latter equation to generate solitary-wave solutions in our model. Special kinds of explicit solutions, such as oscillating solitary waves, are analyzed in detail. The stability of these solutions is verified by means of direct integration of the underlying NLS equation. In particular, our analytical results suggest a way of controlling the dynamics of solitary waves by an appropriate spatial modulation of the nonlinearity strength in Bose-Einstein condensates, through the Feshbach resonance.

Theory of quantum frequency translation of light in optical fiber: application to interference of two photons of different color

We study quantum frequency translation and two-color photon interference enabled by the Bragg scattering four-wave mixing process in optical fiber. Using realistic model parameters, we computationally and analytically determine the Green function and Schmidt modes for cases with various pump-pulse lengths. These cases can be categorized as either "non-discriminatory" or "discriminatory" in regards to their propensity to exhibit high-efficiency translation or high-visibility two-photon interference for many different shapes of input wave packets or for only a few input wave packets, respectively. Also, for a particular case, the Schmidt mode set was found to be nearly equal to a Hermite-Gaussian function set. The methods and results also apply with little modification to frequency conversion by sum-frequency conversion in optical crystals.

Rogon-like solutions excited in the two-dimensional nonlocal nonlinear Schrödinger equation

We report the analytical one- and two-rogon-like solutions for the two-dimensional nonlocal nonlinear Schrödinger equation by means of the similarity transformation. These obtained solutions can be used to describe the possible physical mechanisms for rogue-like wave phenomenon. Moreover, the free function of space y involved in the obtained solutions excites the abundant structures of rogue-like wave propagations. The Hermite–Gaussian function of space y (normalized function) is, in particular, chosen to depict the dynamical behaviors for rogue-like wave phenomenon.

Recent advances and perspectives in four-component Dirac–Kohn–Sham calculations

We review recent theoretical and computational advances in the full relativistic four-component Dirac-Kohn-Sham (DKS) approach and its application to the calculation of the electronic structure of chemical systems containing many heavy atoms. We describe our implementation of an all-electron DKS approach based on the use of G-spinor basis sets, Hermite Gaussian functions, state-of-the-art density-fitting techniques and memory distributed parallelism. This approach has enormously extended the applicability of the DKS method, including for example large clusters of heavy atoms, and opens the way for future key developments. We examine the current limitations and future possible applications of the DKS approach, including the implementation of four-current density functionals and real-time propagation schemes. This would make possible to describe molecules in strong fields, accurately accounting for relativistic kinematic effects and spin-orbit coupling.

Fractionalisation of an odd time odd frequency DFT matrix based on the eigenvectors of a novel nearly tridiagonal commuting matrix

The discrete equivalent of Hermite-Gaussian functions (HGFs) plays a critical role in the definition of a discrete fractional Fourier transform (DFRFT). The discrete equivalents are typically calculated through the eigendecomposition of a commutator matrix. In this study, the authors mainly focus on the fractionalisation of an odd time odd frequency discrete Fourier transform (O-ODFT) matrix. First, the authors propose a novel nearly tridiagonal matrix, which commutes with the O-ODFT matrix. It does not have multiple eigenvalues. The authors can determine a unique orthonormal eigenvector set based on block diagonalisation of a new commuting matrix. The eigenvectors of the new nearly tridiagonal matrix are shown to be O-ODFT eigenvectors, which are similar to the continuous HGFs. Then, the result of the eigendecomposition of the transform matrix is used in order to define the fractionalisation of O-ODFT (O-ODFRFT). The definition is exactly unitary, index additive and reduces to the O-ODFT for unit order. Finally, numerical examples are illustrated to demonstrate that the proposed O-ODFRFT is approximated to the continuous fractional Fourier transform.

Analytical solutions for laser modes in misaligned resonators

Laser modes of misaligned resonators transversally limited in gain are obtained analytically within the paraxial approximation. The eigenfunctions are shown to be complex Hermite–Gaussian functions, not centered either in intensity or in curvature center. Eigenvalues are also obtained, and the effect of resonator parameters and misalignment is analyzed, revealing the lack of selectivity of misalignment on order mode. The changes in the appearance of the modes are also shown.

GEN1INT: A unified procedure for the evaluation of one‐electron integrals over Gaussian basis functions and their geometric derivatives

AbstractWe propose a unified procedure for evaluating a variety of one‐electron integrals and their (arbitrary‐order) geometric derivatives by using a generalized one‐electron operator, which is formed as the product of four operators: (1) a scalar depending on the displacement of the two basis function centers A and B: (Ax − Bx)(Ay − By)(Az − Bz), (2) a multipole moment operator (x − Mx)(y − My)(z − Mz) around origin M, (3) an arbitrary central potential operator f(|r − C|) around center C, and (4) an electronic differential operator (∂/∂x)(∂/∂y)(∂/∂z). The use of Hermite Gaussian functions enables us to evaluate both the integrals and their geometric derivatives on a common footing. This unified computational scheme has been implemented in an open‐ended integral package GEN1INT, and interfaced to the DALTON program, using the Q5Cost library to ensure the portability of the code. Operators of the form f(|r − C|) = |r − C|−1, |r − C|−2, and Dirac delta function δ(r − C) have been implemented, and improvements in the evaluation of integrals involving the operator |r − C|−2 are proposed. The integral package GEN1INT can compute complicated one‐electron property integrals and their arbitrary‐order geometric derivatives, and is therefore expected to be a valuable tool when calculating higher order molecular properties, in particular, in combination with a recently proposed open‐ended quasi‐energy derivative approach (Thorvaldsen et al., J Chem Phys 2008, 129, 214108). © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Spectrally Efficient OFDMA Lattice Structure via Toroidal Waveforms on the Time-Frequency Plane

We investigate the performance of frequency division multiplexed (FDM) signals, where multiple orthogonal Hermite-Gaussian carriers are used to increase the bandwidth efficiency. Multiple Hermite-Gaussian functions are modulated by a data set as a multicarrier modulation scheme in a single time-frequency region constituting toroidal waveform in a rectangular OFDMA system. The proposed work outperforms in the sense of bandwidth efficiency compared to the transmission scheme where only single Gaussian pulses are used as the transmission base. We investigate theoretical and simulation results of the proposed methods.

Eigenvectors of the Discrete Fourier Transform Based on the Bilinear Transform

Determining orthonormal eigenvectors of the DFT matrix, which is closer to the samples of Hermite-Gaussian functions, is crucial in the definition of the discrete fractional Fourier transform. In this work, we disclose eigenvectors of the DFT matrix inspired by the ideas behind bilinear transform. The bilinear transform maps the analog space to the discrete sample space. As jω in the analog s-domain is mapped to the unit circle one-to-one without aliasing in the discrete z-domain, it is appropriate to use it in the discretization of the eigenfunctions of the Fourier transform. We obtain Hermite-Gaussian-like eigenvectors of the DFT matrix. For this purpose we propose three different methods and analyze their stability conditions. These methods include better conditioned commuting matrices and higher order methods.We confirm the results with extensive simulations.

Self-Similar Hermite–Gaussian Spatial Solitons in Two-Dimensional Nonlocal Nonlinear Media

We study analytically and numerically the propagation of spatial solitons in a two-dimensional strongly nonlocal nonlinear medium. Exact analytical solutions in the form of self-similar spatial solitons are obtained involving higher-order Hermite–Gaussian functions. Our theoretical predictions provide new insights into the low-energy spatial soliton transmission with high fidelity.

Hybrid FEM/BEM modeling of finite‐sized photonic crystals for semiconductor laser beams

AbstractWe propose a 2‐D finite element/boundary element hybrid method for calculating the spatial distribution and frequency response of electromagnetic waves coming from a semiconductor laser when interacting with a finite‐sized photonic crystal. We thus provide a flexible tool for the design of novel optical and microwave devices, among other applications. In opposition to current methodologies, we simultaneously take into account the laser modes, the finiteness of the crystal, and the unboundedness of the isotropic medium in which the crystal is embedded. At the laser output, instead of approximating reflected and transmitted beams by plane waves, we use the more realistic Hermite–Gauss functions. In the isotropic medium, we set an artificial boundary encircling the crystal and define exterior and interior domains. Radiating solutions for the scattered far field over the exterior are derived analytically through a series of Hankel polynomials. The interior domain is described by a finite element formulation coupled with Dirichlet‐to‐Neumann maps enforcing laser and far‐field behaviors. Results and error analyses are provided in view of future improvements. Copyright © 2010 John Wiley & Sons, Ltd.

Maximum Likelihood Orientation Estimation of 1-D Patterns in Laguerre-Gauss Subspaces

A method for measuring the orientation of linear (1-D) patterns, based on a local expansion with Laguerre-Gauss circular harmonic (LG-CH) functions, is presented. It lies on the property that the polar separable LG-CH functions span the same space as the 2-D Cartesian separable Hermite-Gauss (2-D HG) functions. Exploiting the simple steerability of the LG-CH functions and the peculiar block-linear relationship among the two expansion coefficients sets, maximum likelihood (ML) estimates of orientation and cross section parameters of 1-D patterns are obtained projecting them in a proper subspace of the 2-D HG family. It is shown in this paper that the conditional ML solution, derived by elimination of the cross section parameters, surprisingly yields the same asymptotic accuracy as the ML solution for known cross section parameters. The accuracy of the conditional ML estimator is compared to the one of state of art solutions on a theoretical basis and via simulation trials. A thorough proof of the key relationship between the LG-CH and the 2-D HG expansions is also provided.

Analysis of Nonlinear Phase Noise in Coherent Fiber-Optic Systems Based on Phase Shift Keying

Analytical expressions for the phase variance in a nonlinear fiber optic system based on phase-shift keying are developed. The Gauss-Hermite functions are used as the orthogonal basis to represent the noise field. Number of degrees of freedom (DOF) to accurately model the phase variance is estimated. The amplifier noise excites higher order Gauss-Hermite noise modes and the nonlinear mixing of a signal pulse and higher order Gauss-Hermite noise mode leads to new noise fields which enhance the nonlinear phase noise. The higher order noise modes propagate linearly and enhance the linear phase noise if the matched filter is not used at the receiver. Analytical expression for the optimum launch power is developed taking into account the linear and nonlinear phase noise.

Mode analysis and signal restoration with Kravchuk functions

When a continuous-signal field is sampled at a finite number N of equidistant sensor points, the N resulting data values can yield information on at most N oscillator mode components, whose coefficients should in turn restore the sampled signal. We compare the fidelity of the mode analysis and synthesis in the orthonormal basis of N-point Kravchuk functions with those in the basis of sampled Hermite-Gauss functions. The scale between the two bases is calibrated on the ground state of the field. We conclude that mode analysis is better approximated in the nonorthogonal sampled Hermite-Gauss basis, while signal restoration in the Kravchuk basis is exact.

Propagation and spatiotemporal coupling of iso-diffraction ultra-short pulsed Hermite Gaussian beams in free space

A family of solutions called iso-diffraction ultra-short pulsed Hermite Gaussian beams （UPHGB） are obtained based on the paraxial propagation equation. The propagation of UPHGB is investigated in detail. The examples shows that: the nonuniformlly distributed group velocity causes the time delay at the beam periphery, which in turn induces the deviation of the transverse distribution from the Hermite Gaussian function. In propagation, the number of pulse polarity inverions varies with the order of the UPHGB. And this phenomenon originates in the difference between the group velocity and the phase velocity in UPHGB.

DFT-Commuting Matrix With Arbitrary or Infinite Order Second Derivative Approximation

Recently, Candan introduced higher order DFT-commuting matrices whose eigenvectors are better approximations to the continuous Hermite-Gaussian functions (HGFs). However, the highest order 2k of the O(h2k) NtimesN DFT-commuting matrices proposed by Candan is restricted by 2k+1 les N. In this paper, we remove this order upper bound restriction by developing two methods to construct arbitrary-order DFT-commuting matrices. Computer experimental results show that the Hermite-Gaussian-like (HGL) eigenvectors of the new proposed DFT-commuting matrices outperform those of Candan. In addition, the HGL eigenvectors of the infinite-order DFT-commuting matrix are shown to be the same as those of the n2 DFT-commuting matrix recently discovered in the literature.

Fast convergent and unconditionally stable Galerkin's method with adaptive Hermite-Gauss expansion for guided-mode extraction in two-dimensional photonic crystal based waveguides

It has been recently shown that guided modes in two-dimensional photonic crystal based structures can be fast and efficiently extracted by using the Galerkin's method with Hermite-Gauss basis functions. Although quite efficient and reliable for photonic crystal line defect waveguides, difficulties are likely to arise for more complicated geometries, e.g., for coupled resonator optical waveguides. First, unwanted numerical instability may well occur if a large number of basis functions are retained in the calculation. Second, the method could have a slow convergence rate with respect to the truncation order of the electromagnetic field expansion. Third, spurious solutions are not unlikely to appear. All these three important issues are here resolved by applying the unconditionally stable S-matrix propagation method, by proposing an adaptive algorithm to expedite the convergence rate of the expansion through duly scaled Hermite-Gauss basis functions, and by providing an effective algorithm for the elimination of spurious modes.

Solving the phase-retrieval problem of a complex signal in the fractional Fourier domain by nonlinear least-squares

From the viewpoint of unconstrained optimization the phase-retrieval problem of a 1D complex signal in the fractional Fourier domain is formulated as a nonlinear least-squares problem. A definition of the discrete fractional Fourier transform (DFRFT) constructed by a discrete Hermite-Gaussian function is adopted here. The ill-posedness of the problem is stressed, and the Levenberg-Marquardt algorithm of Moré's form is used to solve it. In contrast to many published references, this method can reconstruct the phase accurately from the amplitude of the original signal and the one of its DFRFT at any order in the interval (0, 2). For amplitudes with low-level noise this method also works well.