1. Matching free space propagation to plasma focusing S. Barber UCLA Dept. of Physics and Astronomy FACET II Workshop October 15, 2015

2. FrFr Inside of blowout radius there is linear transverse focusing force for electrons that depends only on ion density  ~MT/m! For plasma densities of interest (~10^16) and ultrarelativistic beams (γ>100), this implies matched beta functions on the mm scale Focusing in blown out plasma bubble

3. Density transition region Plasma density ramp aided matching  Plasma density ramp provides additional focusing, aids in matching Focusing strength inside blown out plasma region: Envelope equation: Plasma density profile

4. Example with Gaussian rise in density Gaussian density profile function given by: Plasma density profile Beam beta function evolution Virtual waist

5. Example with Gaussian rise in density Plasma density profile Beam beta function evolution Plasma density profile τ l =15 mm τ l =1.5 mm

6. Initial matched beta vs total density ramp length Peak density is chosen such that The total density transition focusing increases with increasing density ramp length, up to a point In this example, the additional focusing saturates at a ramp length of L~3τ l, or roughly 2-3 orders of magnitude in density

7. Initial matched beta vs total density ramp length For longer RMS lengths the virtual waist increases, indicating stronger focusing from the density transition region Also, for longer RMS lengths, the saturation length of increases

8. Focusing for fixed plasma density boundaries Precise control of density profile over many orders of magnitude may be difficult There is a lower limit on plasma density for which the majority of the drive beam feels the full ion focusing* Upper limit on plasma density depends on application. For optimal configuration for driving wake for witness acceleration  For SPARC beam this implies density bounds ~10 15 ->2.5x10 16, or equivalently L=2.5τ l *Barov, N., & Rosenzweig, J. B. (1994). Phys. Rev. E, 49(5)

9. Focusing for fixed plasma density boundaries If interested purely in focusing of the beam (not driving wake for acceleration), can allow Upper limit on plasma density set by. Limit is given roughly by  For SPARC beam this implies density bounds ~10 15 ->5x10 18, or equivalently L=3.5τ l

10. 3D PIC simulations: Matched for accelerator case Beam beta function evolution Emittance evolution Density profile Peak plasma density is chosen such that Using previous analysis, density ramp is chosen to produce maximum beam focusing for appropriate density range  Beam matches very well to transition focusing region with minimal emittance growth

11. Beam beta function evolution Emittance evolution  Very strong agreement with analytic envelope  Beam is focused from 2 mm waist to matched waist of 700 microns 3D PIC simulations: Matched for accelerator case

12. 3D PIC simulation: Strong lensing case Beam beta function evolution Emittance evolution  Results are considerably worse with large emittance growth and overall poor matching Peak plasma density is chosen such that Using previous analysis, density ramp is chosen to produce factor of 10 beam beta function compression

13. 3D PIC simulation: Strong lensing case Beam beta function evolution  Poor matching and emittance growth is due almost entirely to onset of hosing of beam tail

14. 3D PIC simulation: Strong lensing case Beam beta function evolution Emittance evolution  If you focus only on the first 85% of the beam, the results are more encouraging

15. 3D PIC simulation: Strong lensing case Beam beta function evolution Emittance evolution  Again, if looking only at the core of the beam, there is good agreement between envelope equation and simulation.  Core of the beam is indeed focused by the density ramp by factor of 10, with marginal emittance growth

16. Discussion Prospects for adiabatic plasma lensing experiments at SPARC are encouraging and doable in the near term Onset of hosing may be the limiting factor All of the scaling indicates that conditions improve with increasing charge, current, energy and decreasing emittance  FACET II pushes the envelope even more!

17. Thanks