By Gwenael J Fubiani; Lawrence Berkeley National Laboratory.; United States. Dept. of Energy. Office of Scientific and Technical Information.; USDOE Director. Office of Science. Office of High EnergyPhysics

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23 24 Chapter 3. 1 Introduction Plasma-based accelerators [9] are capable of producing compact and high energy electron sources in much shorter distances than conventional accelerators due to the large longitudinal electric fields that can be excited without the limitation of breakdown as in RF structures. In a plasma, the accelerating fields of a plasma wave are on the order of the cold, nonrelativistic wavebreaking field E0 = me cωp /e, or E0 [V/m] 1/2 96(n0 [cm−3 ])1/2 , where ωp = (4πn0 e2 /me ) is the plasma frequency, n0 is the plasma density, c is the speed of light, me the electron mass, and e the electron charge.

38) Inserting Eq. 37) into Eq. 39) = kp /∆k, τ = ω ˆ T t, ω ˆT = is the bounce frequency of an electron in a deeply trapped beat wave T orbit (orbits in Hamiltonian sytems will be studied in Sec. 40) , = φ0 /(1 + a20 + a21 )−1/2 and ψ˙ b stands for dψb /dτ . In this model two circularly polarized squared laser pulse profiles have been assumed along with a wakefield of the form φ = φ0 sin kp ζ. A detailed derivation of Eq. 39) is done in Appendix D. It is clear from Eq. 39) that the exact axial equation of motion has the form of a non linear equation for coupled pendulum with amplitude and velocity dependent frequencies ωT (ψb , ψ˙ b ) and ωL (ψb , ψ˙ b ) [57].

7 Beam loading considerations . . . . . . . . . 9 Conclusion 68 . . . . . . . . . . . . . . 23 24 Chapter 3. 1 Introduction Plasma-based accelerators [9] are capable of producing compact and high energy electron sources in much shorter distances than conventional accelerators due to the large longitudinal electric fields that can be excited without the limitation of breakdown as in RF structures. In a plasma, the accelerating fields of a plasma wave are on the order of the cold, nonrelativistic wavebreaking field E0 = me cωp /e, or E0 [V/m] 1/2 96(n0 [cm−3 ])1/2 , where ωp = (4πn0 e2 /me ) is the plasma frequency, n0 is the plasma density, c is the speed of light, me the electron mass, and e the electron charge.