1. Introductio n The guiding of relativistic laser pulse in performed hollow plasma channels Xin Wang and Wei Yu Shanghai Institute of Optics and Fine Mechanics, CAS, Shanghai 201800, China E-mail: wxeverest@gmail.com Analytical Model PIC simulation We can see that the spot size of laser oscillates during it is propagating in the channel, which is in agreement with the analytical results. We can see that the spot size of laser oscillates during it is propagating in the channel, which is in agreement with the analytical results. But for the high density boundary condition, it is hard to form soliton structure. One can see that the laser spot size is oscillating when the laser propagating in a capillary, this process repeats in the long capillary. When we enlarge the capillary radius, we will find that the laser will propagate as soliton without spot oscillation. We averaging the laser spot size in the capillary as the final spot size in the capillary roughly. We assume the vector potential of an axis- symmetric laser pulse as: (1) The paraxial wave equation can be written as: (2) The solution of equation (2) can be written in the form, so we get: (3) Integrating the both sides of (3) with respect to from 0 to yields: (4) where, and is given by (5) Conclusio n We use self-focusing analytical model for describing the propagation of a relativistic laser pulse in a performed plasma channel of many Rayleigh lengths long. It is shown that the spot size oscillates in the channels when the channel radius is relatively small, due to the combined effects of laser diffraction in vacuum and laser focusing by the plasma boundary. The PIC simulations confirm the mechanism. It is also demonstrated that the pulse width changes slightly together with the oscillation in spot size. Curved capillary Ultra-Short Laser Pulses Recent experiments which demonstrating GeV-electron acceleration showed that the key issue for laser acceleration is how to maintain to required laser intensity beyond the Rayleigh length, which motivates the investigation on the guiding of a short, relativistic laser pulse in plasmas. Laser self-guiding in uniform plasmas and laser guiding in a preformed plasma channel are considered as the main approaches which enable the relativistic laser pulses to propagate over many Rayleigh lengths. Here we only address the propagation of the laser pulse in preformed hollow plasma channels, leaving the electron acceleration dynamics for possible future investigation. To extend the formulation to a preformed plasma channel (or a capillary), we set for and for, where the channel radius. The channel is in length, located at Evolution of the laser spot size (perpendicular to the propagation direction, blue solid line) and the laser vector potential (at plasma boundary, red dashed line) along the laser propagation axis. The solid black lines mark the plasma boundary. The average spot size of the propagating lasers in capillaries as the function of the capillary radiuses. Here we assume the capillaries are long enough, and the capillary radiuses r t are normalize by the initial spot size b 0. The 2D distribution of the electromagnetic energy density E 2 +B 2 at t=30T 0,102T 0,165T 0 and t=225T 0, respectively. Here the laser’s initial beam waist is b 0 =3λ 0, and its duration is τ=50T 0, the capillary’s radius is r=3λ 0, density is n=10n c. The dot-dashed white lines mark the capillary position initially. The distribution of electron kinetic energy Δγ=γ-1 at the capillary boundary at (a) t=102T 0 and (b) t=165T 0, respectively. Here the most outstanding feature is the two-peak structure (or ring structure in cylindrical geometry) appeared in the enlarged laser spot, which obviously disagrees with the Gaussian radial profile. When the laser pulse enters the channel with a tightly focused spot, it diffracts as in free space. The laser spot size increases while the peak intensity decreases. As laser periphery “touches” the channel wall, the difference in dielectrics leads to conversion of laser energy toward the axis. The combined effects of focusing in the periphery and diffraction in the central vacuum region can result in the two-peak structure or ring structure. E. Esarey, et al, IEEE J. Quantum Elec. 33, 11 (1997). Y. Ehrlich, et al, Phys. Rev. Lett. 77, 4186 - 4189 (1996). In addition to straight capillary, laser guiding in a curved capillary also drawn considerable interest, because of its potential application in the design of a multistage laser wake-field accelerator. We also see that the spot size of laser oscillates during its propagation in the curved capillary, just like our former results of straight channels.