_{1}

We apply a Fourier pseudospectral algorithm to solve a 2D nonlinear paraxial envelope-equation of laser interactions in plasmas. In this algorithm, we first use the second order Strang time-splitting method to split the envelope-equation into a number of equations, next we spatially discrete the filed quantity and its spatial derivatives in these equations in term of Fourier interpolation polynomials (FFT), finally we sequentially integrate the resultant equations by means of a discrete integration method in order to obtain the solution of the envelope-equation. We carry out several numerical tests to illustrate the efficiency and to determine accuracy of the algorithm. In addition, we conduct a number of numerical experiments to examine its performance. The numerical results have shown that the algorithm is highly efficient and sufficiently accurate to solve the 2D envelope-equation, furthermore, it yields an optimal performance in simulating fundamental phenomena in laser interactions in plasmas.

The envelope-equation is the equation that describes the evolution of the slowly varying laser-envelope under the static transformation of the laser coordinate (Quasi Statics Approximation QSA) [

Since the early studies of laser interactions in plasmas, the envelope-equation has been subject to several analytical attempts. Preliminary, the Source-Dependent Expansion (SDE) was proposed to analyze the equation, also the ray tracing was applied to follow the frequency evolution and wave number in the first order envelope-equation, as well the variation approach was used to construct an exact Hamiltonian formulation for the high order envelope-equation. It has been noted that these attempts present an insufficient analysis for the multi dimensional phenomena of the high frequency and high intensity filed, besides it comes with an ambiguous interpretation for strongly nonlinear phenomena as plasmas turbulence and plasma complexity. Because of this incomplete analysis and in order to improve the interpretation, the numerical modeling has been considered.

Over the last few years, various numerical models have been applied to solve the envelope-equation, such as the finite difference time domain (FD) [

The Fourier pseudospectral method has been verified as an accurate and effective technique for solving the envelope-equation in nonlinear optics [

The following is the envelope-equation that is employed to study the laser interactions in plasmas:

As seen above, the equation is a 2D Nonlinear Schrödinger Equation-type (2D NLSE-type), where a is vector potential, t is the time,

linear source term, n is the electron density, and

The envelope-equation is derived using three distinct approximations: first, the Quasi Static Approximation (QSA) [_{0}; third, the paraxial approximation that considers no variation along the direction of propagation

As a matter of fact, the Fourier pseudospectral algorithm (FPSA) is a modified approach for the standard Fourier Pseudospectral method [

Within the Strang time-splitting method, the envelope-equation can be optionally splitted into two/three equations, the two equations splitting is called the first order Strang time-splitting, while the three equations splitting is called the second order Strang time-splitting. It has been found that the second order splitting provides convergence accuracy when it is applied to the Shrödinger equation (the envelope-equation) [

Using the second order Strang time-splitting, our envelope-equation is splitted into the following equations:

As noted, the above equations are classified into two sets of equations: a) two ordinary differential equations (ODE); Equations (2) and (4); which include the source term, b) one 2D partial differential equation (PDE); Equation (3), that contains the spatial operators only. Each set defines different physics, and hence each requires different numerical solution procedures as it will be explained in the coming part.

In the first place, we carry out the solution procedures of Equations (2)-(4) in the time domain_{1}, m_{2} are the maximum number of grids along

for

and

Equations (5)-(7) are the final splitted form of the envelope-equation, and its solution procedures are entirely explained below.

The solution procedures of Equations (5)-(7) start at the time step n in order to obtain

Equation (5) is solved to obtain

As it is ordinary differential equation, Equation (5) has the following straight forward solution:

Equation (6) is solved to obtain

To obtain the solution of Equation (6), in the beginning the field quantities

for

and

where

and

Then, Equations (8) and (9) are substituted in Equation (6) to give

Equation (10) has the following solution:

In the above equation, the solution is presented in the wave number domain. Therefore, we have to transfer it back to the physical domain using the Inverse DFT as

Equation (7) is solved to obtain

Similarly to Equation(5), Equation (7) is ODE, so it has the solution given below

The time step n is constantly increment by one and the above three steps are sequentially repeated unit we reach the maximum computational time and allocate the field quantity

In the FPSA; as in the other spectral methods, the boundary conditions are restricted to be periodic, in addition, by these conditions the physical property of the laser-envelope should be preserved for a very long evolution time. To comply with these requirements, we introduce the following boundary conditions:

and

As it is clear above, the boundary conditions are periodic. Furthermore, within these conditions, the soliton solution is valid. It is necessary to know that, in the soliton solution the field quantity a and its spatial derivatives are vanished at

The numerical test is fundamentally required in order to illustrate the efficiency and to determine the accuracy of the FPSA, therefore in this section we conduct this test. In fact, we perform two tests, in the first test we benchmark the algorithm against the analytical solution of a 1D Cubic Nonlinear Shrödinger Equation [

The equation given below is the Cubic Nonlinear Shrödinger Equation (CNLS).

The CNLS is a general envelope-equation that describes critical phenomena in plasma, such as the optical propagation in dispersive medium, the waves generation, and the self-focusing of laser beam. As shown above, the equation is 1D time-dependent, where u is the complex amplitude,

To run the test, we numerically solved the CNLS using the FPSA in the space domain

and the following boundary conditions:

To illustrate the efficiency, we plotted the analytical solution together with the obtained numerical solution at different grid sizes in

As shown in

In

As we mentioned before, the CNLS is time-dependent equation, because of this dependence, illustrating the efficiency of the FPSA at advanced time has to be undertaken. For this purpose, we re-conducted the comparison between the analytical and the numerical solution at different advanced times, the optimum grid and time-step size previously obtained in

It is clear in

The equation listed below is the Nonlinear Shrödinger Equation (NLSE) [

The NLSE is a Gross-Pitaevskii equation that solves the Bose-Einstein condensate at a very low temperature. As noted above, the equation is 2D time-dependent, where ψ is the wave function and

To conduct the present test, we applied the FPSA to numerically solve the NLSE in the spatial domain

To determine the accuracy, we evaluated the absolute error

and the numerical solution

t | SSFD | SSFS | FPSA |
---|---|---|---|

4 | 4.057E−4 | 3.244E−12 | 3.167E−12 |

8 | 8.115E−4 | 1.250E−11 | 2.821E−12 |

12 | 1.217E−3 | 2.792E−11 | 2.246E−12 |

16 | 1.623E−3 | 4.951E−11 | 1.853E−12 |

20 | 2.029E−3 | 7.728E−11 | 1.569E−12 |

24 | 2.434E−3 | 1.113E−10 | 8.334E−11 |

28 | 2.840E−3 | 1.520E−10 | 5.821E−11 |

32 | 3.246E−3 | 2.008E−10 | 3.774E−11 |

It is noted in

After we illustrated the efficiency and determined the accuracy of the FPSA in the previous section, herein we examine its performance. To examine this performance, we carry out a number of numerical experiments [

In these experiments, we apply the FPSA to numerically solve the envelope-equation in the transverse plane (x − y plane) for an incident linearly polarized laser beam with

an initial Gaussian profile_{0} is the initial complex amplitude

and r_{0} = 1 μm is the spot size. Also in these experiments, to consider the QSA, the speed of light frame is used through the following transformation:

where_{0} is the wave number of the applied beam, c is the speed of light, and _{0}, and the power P is normalized by the critical power for self-focusing P_{cr}.

The self-focusing or channeling is a basic nonlinear phenomenon in laser interactions in plasmas [

To study the PSF, we presented in

In laser interactions in plasmas, when the power of a focused beam exceeds the critical value for beam collapse (P_{c}) in the presence of a spatial-temporal perturbation on this beam profile, the focused beam is suddenly collapsed and turned into narrow patterns of small filaments. This process is called the Multiple Filamentation (MF). Understanding the MF phenomena is essential in the supercontium radiation production and lighting control, from that point on, the MF is an extremely interested phenomenon to be studied.

To study the MF phenomena, we presented in

figure is selected to be high enough not only to demonstrate the self-focusing, but to reach a focused power above the critical power for collapse. As shown in _{c}, thus at τ = 65, the narrow filaments are more collapsed and its number is more increased. Later at τ = 90, the filaments are stagnate, as no more collapse is observed beside the filaments number remains approximately constant. Owing to the fact that, these stagnated filaments are a subject to various dynamics; depending on the distortion on the initial beam profile and the period between the beam shots, in examples including the mutual attraction where the separation-distance among the filaments deceases until the filaments are fused, or the mutual repulsion where this separation- distance increases and consequently the filaments are more widely spread. In our result, it is clear that the mutual repulsion is the most dominated dynamics, as seen at τ = 120 the narrow filaments run away from one another and spatially spread. Although the mutual repulsion would reduce the effectiveness of generation of strong filaments, this dynamics gives a better understanding for the physics of the soliton vortices and spiraling.

The filamentation of a femtosecond (fs) laser in air plasma [

To simulate the periodic self-focusing and defocussing, we applied a 1D FPSA to solve the envelope-equation. This is simply because this phenomena is a 1D dynamics which demonstrates along the direction of propagation (z−) only. In addition, in this simulation we considered the fs parameters [^{−}^{15}, the wavelength λ = 800 nm, the spot size r_{0} = 17 mm, and the critical power P_{cr} = 3.32 GW, the simulation results are presented in

_{0} = 200P_{cr}, as seen in this figure, a periodically unbalanced and instable fs filament is clearly formed, as noted the defocsuing period of this filament is longer than the self-focusing term, and at an advanced propagation-distance the formed filament is

gradually decayed and finally disappeared. The balance between the self-focusing and defocsuing period is subject to the balance between the filament intensity-converging rate during the self-focusing period and the intensity-diverging rate in the defocussing one, as long as these two rates are comparable, a periodically balanced self-focusing and defocussing periods can be achieved, otherwise, unsatisfactory results as seen in

_{0} = 500P_{cr}, as shown in this figure, a fs filament with periodically balanced self-focused and defocsued periods that stably propagates over a long propagation-distance is clearly formed. In the fs filamentation in air plasma, the stable propagation-distance is changeable from application to another; depending on the input parameters, in our experiment a stable propagation-distance of few meters is satisfactory.

We successfully applied a Fourier pseudospectral algorithm to solve a 2D nonlinear paraxial envelope-equation of laser interactions in plasmas. In this algorithm, we used the second order Strang time-splitting method to split the envelope-equation into three equations; two ordinary differential equations where the source term is included and one 2D partial differential equation (PDE) where the spatial operators only are existing. To obtain the solution, we spatially discreted the field quantity and its spatial derivatives in term of the Fourier polynomials, and then we integrated the resultant equations sequentially using the discrete time integration.

We illustrated the efficiency and determined the accuracy of the proposed algorithm in two numerical tests. In the first test, the algorithm was shown to be valid for solving the 1D Coupled Nonlinear Schrödinger Equation (CNLS) and sufficiently efficient to present stable and accurate results over a sufficiently long computational time. In the second test, the evaluated absolute error confirmed that the algorithm provides sufficiently accurate solution for the 2D Nonlinear Schrödinger Equation (2D NLSE) and compares more accurately with other available schemes.

We examined the performance of the algorithm in a series of numerical experiments. In these experiments, the algorithm showed considerable potential for studying fundamental and advanced phenomena in laser interactions in plasmas. At low input laser intensity, the algorithm efficiently depicts the ponderomotive self-focusing formation and smoothly follows the self-focusing developing and stabilization. Furthermore, at a focused power over the critical power for self-focusing, the algorithm clearly images the sudden self-focused beam collapse and the multiple filaments formation. Moreover, by tunning the input power of an input femtosecond beam, the FPSA successfully simulates the formation of a fs filament that stably propagates over a long propagation- distance in air plasma.

Mahdy, A.I. (2016) Fourier Pseudospectral Solution for a 2D Nonlinear Paraxial Envelope Equation of Laser Interactions in Plasmas. Journal of Applied Mathematics and Physics, 4, 2186- 2202. http://dx.doi.org/10.4236/jamp.2016.412213