Hermite-Gaussian Functions Research Articles

Optimized Hermite-gaussian ansatz functions for dispersion-managed solitons

By theoretical analysis, we show that the usual procedure of simply projecting the dispersion-managed (DM) soliton profile onto the basis of an arbitrary number of Hermite-gaussian (HG) polynomials leads to relatively accurate ansatz functions, but does not correspond to the best representation of DM solitons. Based on the minimization of the soliton dressing, we present a simple procedure, which provides highly accurate representation of DM solitons on the basis of a few HG polynomials only.

The line-of-sight velocity distributions of simulated merger remnants

We use Gauss-Hermite functions to study the line of sight velocity distributions in simulated merger remnants. Our sample contains sixteen remnants; eight produced by mergers between disk galaxies of equal mass and eight produced by mergers between disk galaxies with mass ratios of 3:1. The equal-mass mergers display a wide range of kinematic features, including counterrotation at large radii, orthogonally rotating cores, and misaligned rotational axes. Most of the unequal-mass remnants exhibit fairly regular disk-like kinematics, although two have kinematics more typical of the equalmass remnants. Our results may be compared to observations of early type objects, including ellipticals with misaligned kinematic axes, counterrotating systems, and S0 galaxies.

Intensity-based modal decomposition of optical beams in terms of Hermite-Gaussian functions

We show that when an arbitrary optical beam is decomposed into a superposition of Hermite-Gaussian functions, it is sufficient to record a number of intensity profiles sampled at various transverse planes to uniquely determine the relative modal weights. This result follows from the parity relation and the nature of the Gouy phase, in addition to the orthogonality of the Fourier-transformed intensity profiles associated with the Hermite-Gaussian modes.

The discrete fractional Fourier transform

We propose and consolidate a definition of the discrete fractional Fourier transform that generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform generalizes the continuous ordinary Fourier transform. This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The definition is exactly unitary, index additive, and reduces to the DFT for unit order. The fact that this definition satisfies all the desirable properties expected of the discrete fractional Fourier transform supports our confidence that it will be accepted as the definitive definition of this transform.

The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform

Certain solutions to Harper's equation are discrete analogues of (and approximations to) the Hermite-Gaussian functions. They are the energy eigenfunctions of a discrete algebraic analogue of the harmonic oscillator, and they lead to a definition of a discrete fractional Fourier transform (FT). The discrete fractional FT is essentially the time-evolution operator of the discrete harmonic oscillator.

Novel approaches to numerical modeling of periodic dispersion-managed fiber communication systems

We present two approaches to numerical modeling of periodic dispersion-managed (DM) fiberoptic communication systems. The first approach is a path-average mapping method giving analytical expression for the transfer function (over a system period) for arbitrary cascaded DM system with different periods of the amplification and dispersion compensation. The second method is an expansion of the signal in periodic DM transmission lines in the complete basis of Gauss-Hermite functions with the leading Gaussian zero mode. Theoretical results are verified by direct numerical simulations.

Time-resolved spatial structure of TEA CO2 laser pulses

The evolution of the intensity profile of TEA CO2 laser pulses along the pulse length is investigated both analytically and experimentally. A simple scalar model of the pulse amplitude is introduced, which gives the loaded-cavity modes as a linear combination of bidimensional Hermite-Gauss functions. According to this model a number of equations are derived which link the time-varying spatial structure of the pulse to its time-resolved second-order intensity moments, namely, the beam width and the M2 parameter.

Dispersion-managed soliton in fiber links with in-line filtering presented in the basis of chirped Gauss–Hermite functions

Applying the complete basis of chirped Gauss–Hermite functions, we present a theory of dispersion-managed (DM) solitons in communication systems with in-line filtering. In specific practical example we show that a DM soliton can be well described by a few modes in such an expansion (with the zero Gaussian mode), justifying the use of a Gaussian trial function in the previously developed variational approach. The method presented here is a regular way to estimate the accuracy of the approximate variational approach. An analytical formula to describe the dependence of excess gain on filter bandwidth and map parameters is derived. It is shown that use of the filters additionally increases the peak power of a DM soliton compared with that of a DM system without in-line filtering. The suggested expansion also presents a systematic method to account for the effect of practical perturbations. Theoretical results are verified by numerical simulations.

A new method for computing nonlinear carrier diffusion in semiconductor optical devices

The solution of carrier diffusion equations typically associated with semiconductor optical devices has been achieved by combining a function expansion scheme, using the Hermite-Gauss functions as the basis set, with the collocation numerical procedure. Results for a wide range of cases obtained by this new scheme compare very favorably with those calculated with other methods. Not only is the present process computationally fast and efficient, but it has the added attraction of providing the basis for conveniently solving also the nonlinear electromagnetic wave equation for the self-consistent modeling of such devices.

Path-averaged chirped optical soliton in dispersion-managed fiber communication lines

We present a comprehensive path-average theory of dispersion-managed (DM) optical pulse. Applying complete basis of the chirped Gauss-Hermite orthogonal functions, we derive a path-average propagation equation in the time domain and present an analytical description of the breathing dynamics of the chirped DM soliton. This theory describes both self-similar evolution of the central, energy-containing core and accompanying nonstationary oscillations of the far-field tails of an optical pulse propagating in a fiber line with an arbitrary dispersion map. In the case of a strong dispersion management the DM soliton is well described by a few modes in this expansion, justifying the use of a Gaussian trial function in the previously developed variational approach. Suggested expansion in the basis of chirped Gauss-Hermite functions presents a regular way to describe soliton properties for arbitrary dispersion map and to account for the effect of practical perturbations (filters, gratings, noise an so on) on the dynamics of the ideal DM soliton. We also present path-averaged propagation model in the spectral domain that could be useful for multichannel transmission applications. Theoretical results are verified by numerical simulations.

Hamiltonian averaging in soliton-bearing systems with a periodically varying dispersion

Optical pulse dynamics in dispersion-managed fiber lines is studied using a combination of a Lagrangian approach and Hamiltonian averaging. By making self-similar transform in the Lagrangian and assuming in the leading order a bell-shaped pulse dynamics, we reduce the original system to a nonautonomous Hamiltonian system with two variables. Subsequent Hamiltonian averaging gives a function of two variables whose extrema correspond to periodic pulses. To describe a fine structure of the pulse tails, we further develop Hamiltonian averaging using the complete set of Gauss-Hermite functions and also applying averaging in the spectral domain. @S1063-651X~99!50504-7#

A new discrete fractional Fourier transform based on constrained eigendecomposition of DFT matrix by Lagrange multiplier method

This paper is concerned with the definition of the discrete fractional Fourier transform (DFRFT). First, an eigendecomposition of the discrete Fourier transform (DFT) matrix is derived by sampling the Hermite Gauss functions, which are eigenfunctions of the continuous Fourier transform and by performing a novel error-removal procedure. Then, the result of the eigendecomposition of the DFT matrix is used to define a new DFRFT. Finally, several numerical examples are illustrated to demonstrate that the proposed DFRFT is a better approximation to the continuous fractional Fourier transform than the conventional defined DFRFT.

Dequantization Techniques for Weyl Quantization

By quantization we mean the linear bijection from \mathscr S'(\Pi) to \mathscr L(\mathscr S(ℝ)), \mathscr S'(ℝ)) due to Weyl, and by dequantization we mean its inverse. We propose two new, but related, dequantization schemes. The first is adapted to knowledge of the matrix elements (with respect to the Hermite–Gaussian functions) of the operator to be dequantized, while the second is adapted to its integral kernel. Our dequantization schemes are completely general. We apply these methods to the case where the operators in question are Toeplitz operators related to functions of angle on phase space. This enables us to compare the symbols of these Toeplitz operators with the functions of angle themselves.

Breathing self-similar dynamics and oscillatory tails of the chirped dispersion-managed soliton

An effective approach to describe a breathing soliton in systems with periodically varying dispersion is developed. A generalized solution of the propagation equation is presented in terms of chirped Gauss-Hermite orthogonal functions. As a particular example, developed theory describes both averaged slow evolution and rapid oscillations of the dispersion-managed soliton in fiber links. Self-similar structure of the main peak is described by a system of ordinary differential equations for root-mean-square width and integral chirp of the pulse.

Shape of stationary pulse in strong dispersion management regime

Using an expansion over a set of Hermite-Gaussian functions, the shape of the stationary pulse in a strongly dispersion-managed optical fibre is calculated.

Dynamics of self-similar dispersion-managed soliton presented in the basis of chirped Gauss-Hermite functions

Applying chirped Gauss-Hermite orthogonal functions we present an analytical description of the breathing dynamics of the chirped dispersion-managed soliton. The theory describes both self-similar evolution of the central, energy-containing core and accompanying nonstationary oscillations of the far-field tails of an optical pulse propagating in a fiber line with arbitrary dispersion map.

Hermite–sinusoidal-Gaussian beams in complex optical systems

Sinusoidal-Gaussian beams have recently been obtained as exact solutions of the paraxial wave equation for propagation in complex optical systems. Another useful set of beam solutions for Cartesian coordinate systems is based on Hermite–Gaussian functions. A generalization of these solution sets is developed here. The new solutions are referred to as Hermite–sinusoidal-Gaussian beams, because they are in the form of a product of Hermite-polynomial functions of either complex or real argument, sinusoidal functions of complex argument, and Gaussian functions of complex argument. These beams are valid for propagation through systems that can be represented in terms of complex beam matrices, and the previous beam solution sets are special cases of these more general results. Propagation characteristics and applications of these beams are discussed, including their use as a basis set for propagation of arbitrary electromagnetic beams.

Molecular integrals over Gaussian-type geminal basis functions

We present formulas for the evaluation of molecular integrals over basis functions with an explicit Gaussian dependence on interelectronic coordinates. These formulas use expansions in Hermite Gaussian functions and represent an extension to the work of McMurchie and Davidson to two-electron basis functions. Integrals that depend on the coordinates of up to four electrons are discussed explicitly. A key feature of this approach is that it allows full exploitation of the shell structure of the orbital part of the basis.

Exact solutions for linear propagation of chirped pulses using a chirped Gauss–Hermite orthogonal basis

A generalized solution of the linear propagation equation is proposed in terms of chirped Gauss-Hermite orthogonal functions. Some well-known special cases are pointed out, and the usefulness of this approach in analyzing arbitrarily shaped chirped pulses in rapidly converging series is discussed.

Properties of the fractionalization of a Fourier transform

The new fractional Fourier transform, proposed by Shih [Optics Comm. 118 (1995) 495], is shown to differ from the conventional fractional Fourier transform by its four periodic eigenvalues with respect to the order of Hermite-Gaussian functions. Simulations demonstrate that the Hermite-Gaussian superposition algorithm is still valid for this new transform. We also propose a general fractionalization procedure which includes these two definitions as two limits.