CHAPTER 3. DESIGN TARGET PARAMETERS AND SIMULATION TOOLS/METHOD
4.7. CONCLUSION 95 comparison of theory and bandwidth leads to the determination of a laser parameter
configuration of large quality, i.e. at τ = 4 ps, w0 = 5 µm and a0 = 0.37 which is the basis for the next chapter on lens focusing. For this configuration, emittance into higher harmonics was shown to be negligible. Furthermore, at the chosen bunch-waist and -divergence values, space-charge effects can be neglected within the chosen parameter range of the bunch charge Q= 5−200pC.
Still, throughout this chapter, the laser pulse energy was fixed to Ep = 0.5 J, thus determining the relation betweena0, w0andτ. From the findings, it can be stated that the optimum laser would be of higher pulse energy, so that a broad waist and a long duration still yield a large enough a0. Then, the broad waist would lead to a nearly constant transversal a0 for moderately low bunch waists, and the weaker electron focusing would reduce the divergence, and hence photon yield and bandwidth impairment. For large a0, spectral broadening due to the longitudinal a0 variation can be reduced by choosing a small deviation from the counter-propagating interaction angle. The yield reduction would still be large (up to 50 %), but due to the proposed higher pulse energy, the initial yield would be larger. In the proposed pump-probe setup the laser pulse energy is limited, future lasers with high repetition rates and higher power, however, might allow for this.
The simulations and calculations in this chapter with electron emittance did not in-clude an electron-bunch energy spread, as this requires a specification on bunch propa-gation and focusing. This will be investigated in the next chapter, where the chromatic focusing effect of a discharge-capillary active plasma lens is applied to achieve the opti-mum electron parameters for a given laser configuration.
Chapter 5
Discharge-Capillary Active Plasma Lens
Transport and focusing of laser-wakefield-accelerated electron beams is an important as-pect of their application. As presented in the previous chapter, optimum photon yield of the X-ray Thomson source is obtained for rather small electron focal waists (∼µm). As the electron beam is divergent when exiting the plasma, an appropriate beam optic has to be found, in order to focus the electrons onto the laser focus to obtain the optimum focal waist and according divergence.
Electron focusing is typically achieved via magnetic fields, hence on the basis of the Lorentz force. Due to the energy-dependence of the Lorentz force, electrons of different kinetic energy experience different focusing strengths. This leads to energy-dependent focal lengths, as well as focal spot sizes and divergences. According to the spatial overlap and electron focal parameters, electrons of different energy contribute more or less to the spectrum. An interesting prospect is therefore the use of this chromatic focusing effect, in order to decrease the effective electron energy spread, as experimentally confirmed by Fuchs et al. [34]. Electron beam focusing or collimation is conventionally achieved via solenoids or quadrupole triplets. However, solenoids are highly chromatic with a focusing strength proportional to 1/γ2. While, with a proportionality factor of 1/γ, the chromaticity effect of quadrupole triplets is weaker, radially symmetric focusing requires an assembly of three lenses with different and opposite strengths. Consequently, despite their rather strong field gradients (≈ 500 T/m), the total focusing length is tens of centimetres. Hence, the longitudinal extent of focusing lengths of an electron bunch with energy spread σγ/γ >0is also of the order of centimetres.
A discharge-capillary active plasma lens (APL) as proposed by van Tilborg et al.
[35] holds the prospect of a chromatic focusing effect at short focusing lengths and thus smaller setups. The concept of such a lens is presented in detail in the following section 5.1. In section 5.2, the theoretical approximation of the focusing of a divergent electron beam (after plasma acceleration) and GPT simulations are presented. In order to assess
98 CHAPTER 5. DISCHARGE-CAPILLARY ACTIVE PLASMA LENS
Active Plasma Lensing for Relativistic Laser-Plasma-Accelerated Electron Beams
J. van Tilborg,
1S. Steinke,
1C. G. R. Geddes,
1N. H. Matlis,
1B. H. Shaw,
1,2A. J. Gonsalves,
1J. V. Huijts,
1K. Nakamura,
1J. Daniels,
1C. B. Schroeder,
1C. Benedetti,
1E. Esarey,
1S. S. Bulanov,
1N. A. Bobrova,
3P. V. Sasorov,
4and W. P. Leemans
1,21
Lawrence Berkeley National Laboratory, University of California, Berkeley, California 94720, USA
2
Department of Physics, University of California, Berkeley, California 94720, USA
3
Institute of Theoretical and Experimental Physics, Moscow 117218, Russia
4
Keldysh Institute of Applied Mathematics, Moscow 125047, Russia (Received 8 June 2015; published 28 October 2015)
Compact, tunable, radially symmetric focusing of electrons is critical to laser-plasma accelerator (LPA) applications. Experiments are presented demonstrating the use of a discharge-capillary active plasma lens to focus 100-MeV-level LPA beams. The lens can provide tunable field gradients in excess of 3000 T=m, enabling cm-scale focal lengths for GeV-level beam energies and allowing LPA-based electron beams and light sources to maintain their compact footprint. For a range of lens strengths, excellent agreement with simulation was obtained.
DOI: 10.1103/PhysRevLett.115.184802 PACS numbers: 41.75.Jv, 41.85.Ja, 41.85.Lc, 52.38.Kd
Laser-plasma accelerators (LPAs) [1] have produced MeV-to-multi-GeV electron beams in mm-to-cm-scale plasma structures [2 – 9]. This maturing technology is being developed for use in applications such as ultrafast electron-beam pump-probe studies [10], compact light sources including coherent x rays [11 – 13] and incoherent MeV photons [14 – 17], and high-energy particle colliders driven by multiple LPA stages [18,19]. For all of these, transport and focusing of electron beams over short, cm-scale distances is important. Traditional magnetic elements are challenging to apply: (i) Because of the 1=γ
2scaling of the focusing strength, with γ the electron relativistic Lorentz factor, solenoids have weak focusing for relativistic electrons and have, hence, only been applied to energies of a few MeV or less [20]; (ii) the strong field gradients of miniature quadru-poles (of order 500 T=m [21]) are promising, as is the more favorable 1=γ scaling of the focusing strength, but the effective field gradient is strongly reduced when one con-siders that three lenses of varying and opposite strengths need to be combined to achieve radially symmetric focusing [22].
This leads to a longer effective focal length (of order > tens of cm) with increased chromaticity.
This Letter describes recent multistage LPA experiments where we have realized strong, single-element, radially symmetric focusing of electron beams by applying a dis-charge current in a gas-filled capillary. Figure 1(a) illustrates the radial focusing force on an electron propagating collin-early to an externally driven discharge current. Such a lens is also referred to as an active plasma lens. Active plasma lenses were first discussed by Panofsky and Baker in 1950 [23], and have been extensively demonstrated on ion beams using z-pinch plasma discharges [24 – 26]. Until now, applications for electron beams have received little experimental attention. Figure 1(b) highlights the advantage of the active plasma lens, which can provide field gradients > 3000 T=m
for typical parameters considered here. The focal length F
0for 300-MeV electrons is compared for a state-of-the-art solenoid, quadrupole triplet, and active plasma lens, with values of, respectively, 500, 20, and 1.7 cm. The chromatic dependence can be expressed as the energy-dependent change in focal length j Δ F j relative to F
0, as shown in Fig. 1(b), and is much weaker for the shorter focal length of the active plasma lens (red curve). Note that plasma-wake-field lenses, where focusing wakeplasma-wake-fields are driven by either the electron beam itself [27 – 30] or a laser pulse [31,32], have been considered for their ultrastrong focusing fields, approaching even 1 T=μm [28]. However, their applicability is challenging since the focusing force has an intrinsic longitudinal variation (electrons in the head of the beam experience a different lens strength than the electrons in the tail), and tunability is limited since electron-beam parameters (charge, current profile, and size) strongly affect the focusing forces and lens aberrations.
Current Βφ(r)
z r
R Electron
beam
270 280 290 300 310 320 330
0 1 2 3 4
Electron energy [MeV]
Solenoid B=2 T, L=20 cm F0=500 cm
Chromatic dependence |∆F| / F0 [%]
Quadrupole triplet
∂B/∂r=500 T/m L=3 cm F0=20 cm Active plasma lens
∂B/∂r=2000 T/m L=3 cm F0=1.7 cm
Electron
(a) (b)
Force
Electrode Electrode
FIG. 1 (color online). (a) Schematic concept of the focusing force in an active plasma lens. (b) The focal length F
0for 300-MeV electrons and chromatic dependency jΔ F=F
0j is displayed for a state-of-the-art solenoid (black curve), quadrupole triplet (blue curve), and active plasma lens (red curve), illustrating the advantage of the active plasma lens (cm-scale focal length with reduced chromatic dependence).
PRL 115, 184802 (2015) P H Y S I C A L R E V I E W L E T T E R S
week ending 30 OCTOBER 20150031-9007=15=115(18)=184802(5) 184802-1 © 2015 American Physical Society
Figure 5.1.1: (a) Schematic display of a discharge capillary. (b) Chromatic dependence of the focal length for a solenoid (black), a quadrupole triplet (blue) and a discharge capillary active plasma lens. This figure is adapted from [35].
the effects of electron focusing on the Thomson spectrum, the interaction of a laser with a divergent electron beam behind the plasma is investigated, at first. The Thomson scattering simulation results of an divergent beam at the plasma exit, and of an APL-focused electron bunches are presented in chapters 5.3 and 5.4, respectively.