Computer Simulation of a Cylindrical Laser Beam Self-focusing in a Plasma

D. Subbarao, S. Bhaskar , H.Singh and R. Uma
Indian Institute of Technology (Delhi)

Abstract:

Cylindrical laser beam propagation is best modelled for fast computation using a spectral method based on Hankel transform technique.A split step Hankel transform based beam propagation algorithm for a nonlinear plasma and its computational implimentation are discussed in the paper.

Many analytical techniques of solving self-focusing in a plasma have been derived by approximation of the partial differential equation of the beam[1-4]. These approximations try to reduce the partial differential equations to a suitable set of ordinary differential equations, each in a single dimension, because exact solutions can not be worked out in terms of the inverse transform technique [3,5] always. The beam in its direction of propagation evolves slowly while in the transverse direction, the same cannot be assumed.

A convenient way of taking into account the faster transverse variations in general is to use methods based on representation of the beam in convenient orthogonal spaces [4,6-9].The choice of the orthogonal transformation space depends on the symmetries and hence the inherent group-theoretical structure and nature of the partial differential equation involved. A popular choice which works reasonably almost always is angular spectrum representation in terms of Fourier transformation of the beam [2,4,7,10] and is the basis of many a computational scheme known as the beam propagation or FFT-based methods. Such simple angular spectrum representations, however, are not always the best choice and prompt one to choose other transformation spaces such as the Hermite-Gauss transformation or the Laguere-Gauss transformation for some beams [1,8,11-13]. A complete scheme of analysis is possible using any of these non-Fourier transformations as outlined in Ref.[14] but are not very convenient computationally because there are no fast techniques like the FFT for such transformations. These methods are often superior to finite difference methods to solve the Nonlinear Shcrodinger equation (see Taha and Ablowitz [15], and Weidman and Herbst [16] for details). The so-called beam propagation methods based on FFT angular spectrum expansions of a soliton [14-23] are typcal of such methods.Analytical methods mentioned above particularly Refs.[4,8-13] have shown that aberrational effects are better accounted for using such transformed spaces and could seem dramatic in terms of the improvements of the results .

In this paper an attempt has been made to solve the beam equation using beam propagation computational methods for the cylindrical geometry . Our paper is based on the cylindrical laser beam propagation formulation for a nonlinear plasma in terms of Hankel transformations by Subbarao and colleagues [4,8,10] adapting fast Hankel transformation techniques due to Siegman [24] and Lax et. al. [19] and making them more suitable for high performance computing environments. Group theoretical representations and and their use in improvements of the results are discussed as one goes along.

The representaion of beam electric field in cylindrical coordinates is made up of two parts. Firstly the Fourier mode representation in the periodic coordinate is truncated by M modes relavant to the azimuthal details expected and secondly the radial analysis is carried out using by Hankel transformation . As a result of moving over to the momentum space defined by the Hankel transformation, the Bessel operator in the linear part of the radial electric field wave equation would introduce simply a constant multiplication factor , its eigenvalue being independent of the mode of the Bessel's function taken. The operator that arises from the nonlinear portion of the dielectric function of the plasma, , is and appears as a sum with the above noncommuting linear operator in an exponent that desctribes the steady state beam envelope evolution of the scalar wave.The split-step method used by us for beam propagation is based on this exponential operator being approximated using the Baker-Hausdroff formula of Lie groups ( Tappert et.al.[17]) to split it into two separate consecutive operators.The Fast Hankel Transformation (FHT) based split step operator can, then be defined through the equation on the azimuthal mode which can explicitly be written as:

Finally, advances the electric field along propagation direction by one step. Group theoretical representation of the paraxial beam is of particular interest in this paper. The FHT procedure is equivalent to the two dimensional Fourier transformation which could be used directly. Depending on the symmetry of the beam, the present method is, however, expected to save computational time.

Address of the Principal Author: Fusion Studies Program, Plasma Sc. and Tech. Group, Centre for Energy Studies, Indian Institute of Technology (Delhi) New Delhi 110 016. INDIA. Fax: 91-11-6862037,6855227 ; Telephone: 91-11-(6861977 to 6861986) Ext.5035,8801. e-mail: dsr@ces.iitd.ernet.in
**The author H.Singh is at present with the Computer Applications R&D group of NIIT,Delhi and the author S.Bhaskar is at present with Motorola, Bangalore.
Acknowledgements: This work has been supported by the CSIR Research Scheme No.03(0815)/ 97/ EMR-II, India.

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