Tev-acceleration of electrons with high-intense laser pulses xxxxin vacuum
In collaboration with Prof. Heinrich Hora, University of New South Wales, Sydney (Australia) Prof. Ho, Prof. P. X. Wang, Fudan University, IMP, Shanghai (China) Th. Häuser, N. Kowarsch, St. Lourenco, Phd- and diploma-theses, Justus-Liebig-University, Giessen (Germany)
Contents Introduction Acceleration by linearly polarized plane electromagnetic waves Three-dimensional laser pulses Approximate calculation of the propagation of three-dimensional laser pulses Summary
Introduction Acceleration of electrons by lasers in vacuum: Future development of electron accelerators with extremely high energies and short extensions in vacuum Plane electromagnetic waves can not accelerate charges in vacuum. No absorption of a photon by a free electron Laser accelerators usually work in plasmas or near surfaces.
Inverse Free Electron Laser Accelerator: Wiggling of an electron in an undulating magnetic field: Up to 170 MeV/m (Pellegrini et al.) Here we consider the acceleration of electrons by laser pulses which travel in vacuum.
2. Acceleration by linearly polarized plane electromagnetic waves Laser fields with field strengths up to 1015 V/m Classical description of motion of electrons; we neglect the reaction of electrons on the electromagnetic fields; bremsstrahlung is disregared. z-direction: direction of propagating of the xxxxxxxxxxtransversal wave x-direction: electric field of the plane wave, xxxxxxxxxxpolarization in x-direction
Plane polarized wave: The Maxwell equations are fulfilled. The equation of motion for the electron is: Here, is the Lorentz factor: . The pre-factor is a characteristic constant for the acceleration (k0=w0/c):
The constant a0 can be very large; for example, for k0=1 (mm)-1 (l0=2p mm) and an intensity of I=(e0/m0)1/2E02=1022 W/cm2 we find a0=0.38x103; g is proportional to a02. Equations of motion in components: These equations are solved with arbitrary functions Ex(k0z-w0t ). The solutions depend on the initial conditions.
We choose the following initial conditions . Definition of the integral with u=k0z-w0t: This yields: .
x E z B y Trajectory of the electron Examples for plane, linearly polarized waves: Harmonic wave packets b) Pulses with an extension of a half wave length
Harmonic wave packets: Ansatz with u= k0z-w0t : Ex=-E0sin(u)Q(u+k0l)
Length of the wave train: l Lorentz factor g = 1+ 2 a02 sin4(k0l/2) Maximal acceleration of electron only for l= (2n+1)l0/2 with n=0,1,2... Special length: half wave length (n=0) l=l0/2=p/k0; After crossing of a wave train with a length l of half wave length over an electron, this electron has received the values g=1+2a02, k0z=(3/4)pa02, k0x=-pa0 .
Example: k0=1 (mm)-1, l0=2p mm, I=6.93x1022 W/cm2 g=2x106 (TeV-Elektron) z=(1/k0)x0.75xpx106=0.75xp m x=-(1/k0)xpx103=-p mm Result: relatively short distance of acceleration, very small deflection into the transversal direction, small bremsstrahlung Energy gain per length z: T(kinetic energy)/z=(16/3)xm0c2/l0 =2.725 TeV/m for l0=1 mm
b) Pulses with an extension of a half wave length Ansatz: a=0.5
1.2 (g-1)/a02
Phase j is j >>1. The Lorentz factor results as . Maximum of (g-1)/a02 for a=0.5 with the value pxexp(-1)=1.156. Pulse with a half positive wave length has the value (g-1)/a02 =2. For a=0.5 we obtain (j=3xp/2,u=ufin=-3xp) k0z=(3xp/4)xa02x2.20 k0x=-pxa0x2.28 .
3. Three-dimensional laser pulses Starting point is the vector potential in transversal gauge. with . The vector potential can be obtained from a vector potential with , which fulfills the differential equation: . We set: with .
The electromagnetic fields can be calculated by differentiation of Uy: We use a Fourier transformation for Uy at time t=0:
Pulses which move essentially in positive z-direction have the dependence on time t: with . This integral is difficult to solve since it requires a three-dimensional integration over kx, ky und kz in general.
To study suche pulses, we choose the half-wave packet which is restricted by Gaussian functions in the x- and y- directions: . If , one gets the plane wave: with u=k0z-w0t .
The Fourier transformed is . Examples for Uy/U0 with U0=E0/(k02c) for k0w0= 1 and 1000, k0ct= 0 and 0.5
k0ct=0 k0ct=0.5 k0r k0r k0z k0z k0r k0r k0z k0z Uy/U0 Uy/U0 k0w0=1000
4. Approximate calculation of the propagation of three-dimensional laser pulses With the assumption that the taken pulse spreads out essentially in the z-direction with its wave number k0 , we expand the root of the frequency about k0. For kzk0 it yields: .
For -kzk0 it yields: These expansions can be inserted in the integral. Then we obtain:
Thereby we use the abreviation . The constants a, b and c are real. The integral over k has to be calculated numerically. Examples (for a0=1): as function of k0w0 k0z as function of k0ct for various k0w0 k0x as function of k0ct for k0w0=0.5, 1000
g
5. Summary Short and intense laser pulses of an extension of about a half wave length can be applied to the acceleration of electrons. The laser pulse overruns the electron and let it return in an accelerated state. Plane laser pulses of a half wave length with a strength parameter a0=|eE0|/(m0c2k0) lead to an acceleration of g=1+2a02 inside a short distance (in the range of meter for TEV electrons, intensities I1022-1024 W/cm2).
In contrast to various investigations in the literature, our wave packets contain many frequencies. This leads to difficult calculations (precise numerical solution of a three-dimensional integral). It is necessary that experimentally the electron acceleration with laser pulses has to be studied if one plans to construct new electron accelerators which accelerate electrons on short distances to very high energies (TeV regime). D.G.