Wakefield generation and GeV acceleration in tapered plasma channels P. Sprangle, 1 B. Hafizi, 2 J. R. Pen ˜ ano, 3 R. F. Hubbard, 1 A. Ting, 1 C. I. Moore, 1 D. F. Gordon, 4 A. Zigler, 5 D. Kaganovich, 5 and T. M. Antonsen, Jr. 6 1 Plasma Physics Division, Naval Research Laboratory, Washington, DC 20375 2 Icarus Research, Inc., P.O. Box 30780, Bethesda, Maryland 20824-0780 3 LET Corporation, 4431 MacArthur Boulevard, Washington, DC 20007 4 Naval Research Laboratory, National Research Council, Washington, DC 20375 5 Hebrew University, Jerusalem, Israel 6 University of Maryland, College Park, Maryland 20742 Received 20 September 2000; revised manuscript received 22 November 2000; published 18 April 2001 To achieve multi-GeV electron energies in the laser wakefield accelerator LWFA, it is necessary to propagate an intense laser pulse long distances in a plasma without disruption. One of the purposes of this paper is to evaluate the stability properties of intense laser pulses propagating extended distances many tens of Rayleigh ranges in plasma channels. A three-dimensional envelope equation for the laser field is derived that includes nonparaxial effects such as group velocity dispersion, as well as wakefield and relativistic nonlinearities. It is shown that in the broad beam, short pulse limit the nonlinear terms in the wave equation that lead to Raman and modulation instabilities cancel. This cancellation can result in pulse propagation over extended distances, limited only by dispersion. Since relativistic focusing is not effective for short pulses, the plasma channel provides the guiding necessary for long distance propagation. Long pulses greater than several plasma wavelengths, on the other hand, experience substantial modification due to Raman and modulation instabilities. For both short and long pulses the seed for instability growth is inherently determined by the pulse shape and not by background noise. These results would indicate that the self-modulated LWFA is not the optimal configuration for achieving high energies. The standard LWFA, although having smaller accelerating fields, can provide acceleration for longer distances. It is shown that by increasing the plasma density as a function of distance, the phase velocity of the accelerating field behind the laser pulse can be made equal to the speed of light. Thus electron dephasing in the accelerating wakefield can be avoided and energy gain increased by spatially tapering the plasma channel. Depending on the tapering gradient, this luminous wakefield phase velocity is obtained several plasma wavelengths behind the laser pulse. Simulations of laser pulses propagating in a tapered plasma channel are presented. Experimental techniques for generating a tapered density in a capillary discharge are described and an example of a GeV channel guided standard LWFA is presented. DOI: 10.1103/PhysRevE.63.056405 PACS numbers: 52.35.Mw, 41.75.Jv, 52.50.-b I. INTRODUCTION The extremely large acceleration gradients generated by laser pulses propagating in plasmas can be used to accelerate electrons 1–7. In the standard laser wakefield accelerator LWFA a short laser pulse, on the order of a plasma wave- length long, excites a trailing plasma wave that can trap and accelerate electrons to high energy. There are a number of issues that must be resolved before a viable, practical high- energy accelerator can be developed. These include Raman, modulation, and hose instabilities that can disrupt the accel- eration process 8–16. In addition, extended propagation of the laser pulse is necessary to a achieve high electron energy. In the absence of optical guiding the acceleration distance is limited to a few Rayleigh ranges, which is far below that necessary to reach GeV electron energies 1,17. The physics of laser beams propagating in plasmas has been studied in great detail 8,18–23, and there is ample experimental con- firmation of extended guided propagation in plasmas and plasma channels 24–30. In addition to these issues, dephas- ing of electrons in the wakefield can limit the energy gain. Spatially tapering the plasma density may be useful in over- coming electron dephasing in the wakefield. This paper addresses the guiding and stability of an in- tense laser pulse in a uniform plasma channel and analyzes the wakefield acceleration process in an inhomogeneous channel. The coupled electromagnetic and plasma wave equations are derived for laser pulses propagating in a plasma channel with a parabolic radial density profile and arbitrary axial density variation. For a uniform channel, Ra- man and modulation instabilities are analyzed and numerical solutions of the three-dimensional 3D wave equation are discussed. In particular, propagation of short laser pulses over many Rayleigh ranges is demonstrated for a uniform channel. For a nonuniform channel the axial and radial elec- tric fields associated with the plasma wave are obtained in- side and behind the laser pulse. It is shown analytically and through numerical simulations that by tapering the plasma density the wakefield phase velocity several plasma wave- lengths behind the laser pulse can equal the speed of light in vacuo. Tapered density channels have been produced experi- mentally in capillary discharges, and optical guiding in these channels has been demonstrated. A variable channel density may be generated by tapering the wall radius of the capillary 27, or by applying different voltages to a segmented capil- lary. The equations for the laser envelope and wakefield are derived in Sec. II. The formulation includes the effects of plasma density inhomogeneity, diffraction, nonparaxial PHYSICAL REVIEW E, VOLUME 63, 056405 1063-651X/2001/635/05640511/$20.00 ©2001 The American Physical Society 63 056405-1