1. NON-WIGGLER-AVERAGED START-TO-END SIMULATIONS OF HIGH-GAIN FREE- ELECTRON LASERS H.P. Freund Science Applications International Corp. McLean, Virginia 22102, USA Presented at the ICFA Future Light Sources Sub-Panel Mini Workshop on: Start-to-End Simulations of X-Ray FELs 18-22 August 2002 at DESY-Zeuthen

2. COLLABORATORS R. BartoliniENEA-Frascati S.G. BiedronMAX Lab G. DattoliENEA-Frascati L. GianessiENEA-Frascati S.V. MiltonAPS/ANL H.-D. NuhnSLAC P.L. OttavianiENEA-Bologna A.RenieriENEA-Frascati etc.

3. SIMULATION CODE: MEDUSA MEDUSA is the only 3-D FEL simulation code that does not employ wiggler-averaging. It has the following properties: The E & M field is a superposition of Gaussian modes Polychromatic  harmonics & sidebands. Arbitrary wiggler geometry planar, helical, field map 3-D Lorentz force equations without wiggler average step size must resolve wiggle-motion includes entry & exit tapers Arbitrary focusing fields quadrupoles, dipoles, field map Field propagation code without particles Particle tracking code without fields

4. PREVIOUS BENCHMARKS – HGHG EXPERIMENT Simulations of the experiment using MEDUSA showed good agreement for the fundamental power (28 MW). Harmonic HGHG MEDUSA 2 2.0   4   4     2   2 P harmonic /P fundamental Among the benchmarks made has been the HGHG experiment at BNL: S.G. Biedron, H.P. Freund, S.V. Milton, L.H. Yu, and X.J. Wang, NIMA 475, 118 (2001). HARMONICS FUNDAMENTAL

5. SIMULATION OF THE SPARC UNDULATOR LINE Wiggler: Amplitude: 6.12 kG Period: 3.3 cm Length: 76 w Transitions: 3 w Quadrupoles (not FODO): Gradient: 0.25 kG/cm Length: 0.132 m Electron Beam: Energy: 150 MeV Current: 150 A Emittance: 2 mm-mrad Energy Spread: 0.1% The SPARC undulator line has been studied previously, and we begin with a summary of the results. The entry & exit tapers in the wiggler segments were chosen to be 3 periods in length. This is because  B w  0 in the analytic field representation. A realistic model (field map or off- axis expansion) would cure this issue.

6. MAGNETIC TRANSPORT LINE The Twiss parameters for the initial beam have not yet been fully implemented in MEDUSA, so we start the undulator/focusing line with a half-quad. LQLQ LdLd LdLd L gap wiggler L gap = 2L d + L Q The wigglers (green) include 3 period entry & exit tapers. The quadrupoles (blue) are centered in the gaps, and alternate between focusing in the x- (wiggle-plane) and y-directions to compose a FODO lattice. L Q 2 LdLd

7. SPARC – VARIATION WITH GAP LENGTH In order to obtain a phase match between the light and electrons propagating through the gaps, the phase slippage between the light and the electrons must be an integer number of wavelengths. This means that L gap = N w (1 + K 2 /2) For the case of the SPARC design, this gives a periodicity of about 0.092 m. This is seen in simulation. The transition region can be abrupt, and care must be taken in the design.

8. At the SPARC design point (L d = 0.099 m), we see saturation after about 15 m at about 62 MW. Mode expansion is rapid in the last segment. The resonant wavelength is 529.6 nm SPARC – POWER AND MODE EVOLUTION The beam propagation is regular. The quads are chosen to give focusing in the wiggle-plane.

9. EVOLUTION OF TRANSVERSE MODE IN SPARC

10. SIMULATION OF HARMONICS in SPARC Harmonic growth lengths scale inversely with the harmonic number. Transverse coherence of harmonics is also included in simulation.

11. START-TO-END SIMULATION OF THE LCLS These are preliminary simulations of the LCLS transport line. Need to fully implement the Twiss parameters for beam initialization for more complete simulation. Wiggler: Amplitude: 13.25 kG Period: 3.0 cm Length: 118 w Transitions: 3 w Quadrupoles (FODO): Gradient: 10.8 kG/cm Length: 0.05 m Electron Beam: Energy: 14.36 GeV Current: 3272 A x-Emittance: 0.8869 mm-mrad y-Emittance: 0.8198 mm-mrad Energy Spread: 6.142  10  5 The Twiss parameters are needed to get better match to the beta functions of the transport line. Right now the beam dimensions are chosen using the beta function of the first wiggler.

12. SIMULATED BEAM PROPERTIES FOR THE LCLS LCLS Simulations of the LCLS beam yield the following slice parameters. Note that the Twiss parameters have not yet been incorporated into MEDUSA.

13. LCLS – VARIATION WITH GAP LENGTH For the LCLS w (1 + K 2 /2)  0.24 m and this periodicity is reproduced in simulation. Actual gap length for good performance needs to take wiggler entries/exits into account. “Dead-Man’s Gulch”

14. ELECTRON BEAM AND OPTICAL EVOLUTION Optical guiding depends on beam scalloping which is large for the assumed initial beam conditions. Need to implement Twiss parameters to get initial beam dimensions correct. Actual beam dimensions from the LCLS website give initial x- and y-dimensions of about 0.002 cm. MEDUSA chooses beam dimensions for a matched beam using the beta function of the 1 st wiggler.

15. SINGLE-WIGGLER-PERIOD ENTRY/EXIT TAPERS We can estimate the effect of gap spacing using entry/exit transitions of one wiggler period. zero energy spread. LCLS design point is close to the edge of Dead-Man’s Gulch. An example of the beam motion upon exit from one wiggler, propagation through the gap, and entry into the next wiggler by means of the beam centroid in the wiggler-plane is shown at left.

16. MATCHED BEAM IN LCLS WIGGLER/FODO LINE Limited use of Twiss parameters: x rms & y rms  functions  x =  y = 0 Much more uniform beam envelope Much more regular exponentiation Dead-Man’s Gulch shifts position

17. PHASE ADVANCE WAVELENGTH DEPENDENCE SASE produces a spectral width comparable to the FEL gain bandwidth. The phase advance in the gaps vary with wavelength as well as with the length of the wiggler transitions. There is a large sensitivity of the phase match for a given gap spacing for wavelengths in the LCLS gain band. This can be a source for spectral narrowing. Must be careful in designing the undulator line which has varying gap lengths not to have destructive interference between gaps.

18. SUMMARY AND CONCLUSIONS There seems to be more than adequate gain to saturate the LCLS undulator line if this initial beam can be achieved. Caution needed because the optical mode expands rapidly after saturation. The actual gap length between the wigglers needed to achieve peak performance and avoid “Dead-Man’s Gulch” depends on the length of the wiggler transitions. MEDUSA can accept a field map including all magnetic elements so that the transitions can be accurately modeled. Extreme caution required in choice of L gap. Need to run with multi-wavelengths. Harmonics have not yet been studied for the LCLS.