https://people.ucalgary.ca/~dfeder/535/split_operator.pdf

Solution of the Schrödinger equation by a spectral method

An accurate numerical method is described for solving the Helmholtz equation for a general class of optical fibers. The method yields detailed information about the spatial and angular properties of the propagating beam as well as the modal propagation constants for the fiber. The method is applied to a practical graded-index fiber under the assumptions of both coherent and incoherent illumination. A spectral analysis of the calculated field shows that leaky modes are lost and steady-state propagating conditions are established over a propagation distance of a fraction of a meter.

Light propagation in graded-index optical fibers

The authors derive a single-wave equation which describes transient stimulated Raman scattering in optical fibers. The equation is valid for forward traveling waves whose bandwidths are less than approximately=1/3 the carrier frequency. It correctly conserves a classical photon number and not the total optical energy as is appropriate for Raman scattering problems. From this equation, the authors derive a formula for the small-signal gain spectrum which includes the effects of Raman scattering, four-wave mixing, and modulational instabilities. Examples of numerical integrations are given which show the generation of short soliton pulses from the nonlinear evolution of a noise band around a continuous wave pump. >

Theoretical description of transient stimulated Raman scattering in optical fibers

The spectral method utilizes numerical solutions to the time‐dependent Schrodinger equation to generate the energy eigenvalues and eigenfunctions of the time‐independent Schrodinger equation. Accurate time‐dependent wave functions ψ(r, t) are generated by the split operator FFT method, and the correlation function 〈ψ(r, 0) ‖ ψ(r, t)〉 is computed by numerical integration. Fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system. Analysis of the location of these peaks reveals the eigenvalues with high accuracy. Additional Fourier transforms of ψ(r, t) with respect to time generate the eigenfunctions. Previous applications of the method were to two‐dimensional potentials. In this paper energy eigenvalues and wave functions obtained with the spectral method are presented for vibrational states of three‐dimensional Born–Oppenheimer potentials applicable to SO2, O3, and H2O. The energy eigenvalues are compared with results obtained with the variational method. It is concluded that the spectral method is an accurate tool for treating a variety of practical three‐dimensional potentials.

Solution of the Schrödinger equation by a spectral method II: Vibrational energy levels of triatomic molecules

We studied the efficiency of different implementations of the split-step Fourier method for solving the nonlinear Schro/spl uml/dinger equation that employ different step-size selection criteria. We compared the performance of the different implementations for a variety of pulse formats and systems, including higher order solitons, collisions of soliton pulses, a single-channel periodically stationary dispersion-managed soliton system, and chirped return to zero systems with single and multiple channels. We introduce a globally third-order accurate split-step scheme, in which a bound on the local error is used to select the step size. In many cases, this method is the most efficient when compared with commonly used step-size selection criteria, and it is robust for a wide range of systems providing a system-independent rule for choosing the step sizes. We find that a step-size selection method based on limiting the nonlinear phase rotation of each step is not efficient for many optical-fiber transmission systems, although it works well for solitons. We also tested a method that uses a logarithmic step-size distribution to bound the amount of spurious four-wave mixing. This method is as efficient as other second-order schemes in the single-channel dispersion-managed soliton system, while it is not efficient in other cases including multichannel simulations. We find that in most cases, the simple approach in which the step size is held constant is the least efficient of all the methods. Finally, we implemented a method in which the step size is inversely proportional to the largest group velocity difference between channels. This scheme performs best in multichannel optical communications systems for the values of accuracy typically required in most transmission simulations.

/pdf/optimization-of-the-split-step-fourier-method-in-modeling-43jowds4fr.pdf

Optimization of the split-step Fourier method in modeling optical-fiber communications systems

The computation of time-dependent three-space-dimensional laser beam propagation is described. The methods are applicable to the propagation of high energy laser beams through the atmosphere in the presence of a horizontal wind and turbulence for most situations of interest. Possible cases are propagation of cw beams through stagnation zones, multi-pulse propagation, including the self-consistent treatment of pulse self-blooming, and propagation involving transonic slewing. The solution of the Maxwell wave equation in Fresnel approximation is obtained by means of a discrete Fourier transform method, which, surprisingly, gives excellent results for diffraction problems. The latter provide a stringent test for the accuracy of any solution method. Considerable use is also made of discrete Fourier transform methods in solving the hydrodynamic equations. The treatment of turbulence is based on the generation of random phase screens at each calculation step along the propagation path. In a time-dependent calculation the random phase screens can be either made to move with the wind at a given propagation position or generated anew for each successive time.

Time-dependent propagation of high-energy laser beams through the atmosphere: II

Time-dependent propagation of high-energy laser beams through the atmosphere, 2

The propagation of a laser beam in a plasma column several meters long with a realistic electron density distribution is examined. The electron density distribution is based on laser‐beam heating at z=0, but is otherwise uncoupled to the laser beam. The aberrated nature of the resulting lenslike medium leads to essentially aperiodic beam properties, which contrast with the completely periodic properties of Gaussian beams propagating in quadratic lenslike media. The beam is nonetheless stably trapped. These aberrated‐beam properties also help to stabilize the beam against axial variations in refractive index.

M. D. Feit

Papers

Time-dependent propagation of high-energy laser beams through the atmosphere: II

Time-dependent propagation of high-energy laser beams through the atmosphere, 2

Propagation of highly aberrated laser beams in nonquadratic plasma waveguides

K6 Laser beam trapping and propagation in cylindrical plasma columns