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.

A high-quality electron beam can be extracted from a channel guided laser wakefield accelerator without confining the injected particles to a small region of phase. By careful choice of the injection energy, a regime can be found where uniformly phased particles are quickly bunched by the accelerator itself and subsequently accelerated to high energy. The process is particularly effective in a plasma channel because of a favorable phase shift that occurs in the focusing fields. Furthermore, particle-in-cell simulations show that the self-fields of the injected bunches actually tend to reduce the energy spread on the final beam. The final beam characteristics can be calculated using a computationally inexpensive Hamiltonian formulation when beam-loading effects are minimal.

The ponderomotive force associated with an intense laser beam expels electrons radially and can lead to cavitation in plasma. Relativistic effects as well as ponderomotive expulsion of electrons modify the refractive index. An envelope equation for the laser spot size is derived, using the source-dependent expansion method with Laguerre-Gaussian eigenfunctions, and reduced to quadrature. The envelope equation is valid for arbitrary laser intensity within the long pulse, quasistatic approximation and neglects instabilities. Solutions of the envelope equation are discussed in terms of an effective potential for the laser spot size. An analytical expression for the effective potential is given. For laser powers exceeding the critical power for relativistic self-focusing the analysis indicates that a significant contraction of the spot size and a corresponding increase in intensity is possible.

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To achieve multi-GeV electron energies in the laser wakefield accelerator (LWFA) it is necessary to propagate an intense laser pulse long distances in plasma without disruption. A 3D envelope equation for a laser pulse in a tapered plasma channel is derived, which includes wakefields and relativistic and nonparaxial effects, such as finite pulse length and group velocity dispersion. It is shown that electron energies of approximately GeV in a plasma-channel LWFA can be achieved by using short pulses where the forward Raman and modulation nonlinearities tend to cancel. Further energy gain can be achieved by tapering the plasma density to reduce electron dephasing.