The X-ray FEL Oscillator: Parameters, Physics, and Performance

Transcription

1 The X-ray FEL Oscillator: Parameters, Physics, and Performance Ryan Lindberg Future Light Source Workshop Monday March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

2 Outline Introduc?on and schema?c lay out Typical parameters and performance Gain and Bragg mirrors in the x ray FEL oscillator Longitudinal dynamics and linear supermodes Some transverse cavity physics High current opera?on Harmonic genera?on R.R. Lindberg: The x ray FEL Oscillator, WG4: FEL Based Sources, FLS 1

3 Introduction and schematic lay-out ~ 3 6 m undulator for single pass FEL gain ~.3 High brightness electron beam with ~ MHz repe??on rate e - D G M H undulator M 1 H A x-rays G Grazing incidence mirrors to provide x ray focusing B H C Bragg crystals for nearnormal reflec?on of x rays in a narrow range in wavelength (Darwin width) λ λ H cos Θ R.R. Lindberg: The x ray FEL Oscillator, WG4: FEL Based Sources, FLS 1 3

4 A few canonical parameters Cavity Length ~ 1 to 3 m Repe??on rate ~ 1 to 3 MHz Low charge ~ to 5 pc Bunch lengths ~ to.1 ps peak currents ~ 5 to A High quality electron beam: ε x.1.3 mm mrad, Low single pass gain, losses: γ γ.1.% G lin.3 1., R tot.85.5 > (1 + G lin ) 1 σ e = 1 ps, ε x =.mm mrad, E beam = 7 GeV, ( γ/γ )=.% Parameter kev 1.4 kev kev λ u (cm) N u FEL K I peak (A) G linear R total L cavity (m) Bragg crystal C( ) C(4 4 4) C(5 5 9) New technology (e.g., SC undulators, etc.) may relax these parameters R.R. Lindberg: The x ray FEL Oscillator, WG4: FEL Based Sources, FLS 1 4

5 Canonical performance Saturated cavity power ~ 1 to MW Output power ~.5 to 1 MW Output ~ 1 8 to 5x1 9 photons Spectral Brightness ~ 1 31 to 1 34 Nearly Fourier limited pulses E 1 1 mev, σ e ω ω σ e = 1 ps, ε x =.mm mrad, E beam = 7 GeV, ( γ/γ )=.% Parameter kev 1.4 kev kev G linear R total P sat (MW) spectral width (mev) temporal width (ps) t ω photons/pulse peak power (MW) R.R. Lindberg: The x ray FEL Oscillator, WG4: FEL Based Sources, FLS 1 5

6 Single pass (linear) gain For no external focusing, FEL gain is maximized when beam divergence matches that of the radia?on.5 G/G max 1. Z β Z R and when the beam nearly matches the spontaneous radia?on mode size: Z β Z R L u π ZR/Lu E(t) ( 1+Ge t /σ e ) E(t) Z β /L u For a single pass, the gain (approximately) follows the current profile: Approximate using E(t) (1 + G/)E(t) Gt σe E(t) R.R. Lindberg: The x ray FEL Oscillator, WG4: FEL Based Sources, FLS 1 6

7 R = r Effect of the Bragg crystal mirror The Bragg crystals operate via the coherent reflec?on from many crystal planes Np. We approximate the reflec?vity r by r(ω) = R(ω)e iψ R To avoid gain significant reduc?on we require σ ω σ e R ψ ψ z E E (mev) π p π p π -p π -p phase ψ Losses Power (a.u) [ 1 1 σ ω Decrease cavity length by l to flaaen phase ψ ψ z Spectral filtering with σ ω N 1 p (ω ω ) ] e i(l/c)(ω ω ) Spontaneous radia?on Time delay with l N p Power (a.u) Radia?on for next pass R.R. Lindberg: The x ray FEL Oscillator, WG4: FEL Based Sources, FLS 1. 7

8 Longitudinal linear supermodes Ader n passes, the radia?on field E(t, n) is approximately described by 1 E(t, n) = n σω Bragg crystal frequency filtering t +1 R E(t, n)+g E(t, n) Gt (linear gain) (losses) with solu?on [ E m (t, n) =e Λmn e t σ ω l/ exp Gσω σ e t ] [ H m Single [ pass gain Λ m 1 G +1 R Gain reduc?on due to finite crystal bandwidth Decrease in gain due to current profile 4σ e E(t, n)+l E(t, n) t Hermite func?on of order m Timing errors between electron bunch and radia?on G 1/4 σ ω ] G (m + 1) 1 σ e σ σ ωl ω Gaussian is the mode with largest gain G. Daaoli, A. Marino, A. Renieri, F. Romanelli, IEEE J. Quant. Elect. 17, 1371 (1981); P. Elleaume, IEEE J. Quant. Elect. 1, 11 (1985) σe t ] R.R. Lindberg: The x ray FEL Oscillator, WG4: FEL Based Sources, FLS 1 8

9 Supermode decomposition We decompose the growing radia?on using the Hermite basis func?on, each of which grow according to their supermode growth rate: Λ m = 1 [ ] g (g α) (m + ) σ e σ ω Ini?al start up from noisy spontaneous radia?on ln P (arb. units) m = m =1 m = N pass Nonlinear satura?on Linear growth of the three lowest order supermodes R.R Lindberg and K. J. Kim, Phys. Rev. ST AB, 1 77 (9) R.R. Lindberg: The x ray FEL Oscillator, WG4: FEL Based Sources, FLS 1 9

10 Longitudinal evolution using GINGER Energy (µj) e N pass = 1 6 N pass = 3 3 N pass = 1 3 N pass = N pass N pass = 6,1 3 Power (W) 6 3 Power (kw) 4 Power (MW) 1 Power (MW) Power (MW) t t (ps) t t (ps) t t (ps) t t (ps) t t (ps) P(!) (arb. units) P( ) (a.u.) ω P(!) (arb. units) P( ) (a.u.) ω P(!) (arb. units) P( ) (a.u.) ω P(!) (arb. units) P( ) (a.u.) ω P(!) (arb. units) P( ) (a.u.) ω E E H (mev) E - E H (mev) E - E E - E E E H (mev) E E H (mev) R.R. Lindberg: The x ray FEL Oscillator, WG4: FEL Based Sources, FLS E - E E E H (mev) E - E E E H (mev) 1

11 Transverse physics in a simple cavity Simplest descrip?on of the transverse mode shape near exact backscaaer is the symmetric two mirror resonator cavity: C(444).1 undulator e - C(444) 1 kev x-rays Focusing mirror f Lc R 1 =.99 mirror R =.97 R =.95 9 cm T =.4 (b) Trapped radiation The stable, transverse vacuum modes are Hermite Gauss, with Set to maximize FEL gain ZR ( = L c f 1 4 L ) c Determined by electron rep rate ~ MHz Although this describes the vacuum cavity mode, GINGER simula?ons have shown that the transverse profile (i.e., the waist, Rayleigh range, shape), is closely approximated by the vacuum mode even for gains 1. R.R. Lindberg: The x ray FEL Oscillator, WG4: FEL Based Sources, FLS 1 11

12 4-crystal cavity for tunable x-rays D G M H B Parameter kev kev.514 kev FEL K ɛ x,n (mm mrad)...1 linear G Bragg crystal C(3 3 3) C(3 3 7) C(3 3 11) Tuning range 6.% 6.% 3.5% P peak (MW) spectral width (mev) t ω photons/pulse H undulator C e - M 1 H R tot =.81 R tot = A x-rays Bragg s Law: λ = λ H cos Θ Changing the incident angles of all four G crystals while adjus?ng the leg distances to keep the round trip?me invariant allows the radia?on energy to be varied. R tot =.83 R tot = R tot = R.R. Lindberg: The x ray FEL Oscillator, WG4: FEL Based Sources, FLS 1 1

13 Transverse mode structure in a more general, 4-crystal tunable cavity We have begun a study the transverse mode structure and stability in a more general 4 mirror cavity using matrix methods familiar to accelerator physics. Calculate the cavity beta and dispersion func?ons σx beam size σy Radia?on waist in center of undulator Focusing elements z (m) Asymmetric crystal loca?ons calcula?on by G. Park Asymmetric crystals make telescope transforma?on [ ] [ ][ x b x x = 1/b x ] D. M. Smilgies, Applied Physics. 47, 16 (8); Applied Physics. 47, 116 (8 R.R. Lindberg: The x ray FEL Oscillator, WG4: FEL Based Sources, FLS 1 13

14 Increasing beam current for increased gain/reduced tolerances We can compress the 5 pc beam from 1 ps to 1 fs 1 A peak current Increase in the gain relaxes tolerances on, e.g., total reflec?vity, undulator errors, etc. Alterna?vely, one could reduce injector requirements Possible 1 A opera?ng parameters for FEL 1Å ɛ x (mm mrad) σ E /E (%)...3. N u L u (m) net gain G Spectral FWHM (mev) Temporal FWHM (fs) P peak (MW) photons/pulse A R.R. Lindberg: The x ray FEL Oscillator, WG4: FEL Based Sources, FLS 1 14

15 Harmonic generation Power (MW) P (MW) The FEL process will generate radia?on at odd harmonics for free Since the penetra?on length ~ frequency, nearly all the higher harmonic radia?on is transmiaed through the thin crystal 1 36 kev Cavity P(t) output ebeam I(t) P (MW) Power (MW) Output P(t) output ebeam I(t) P (kw) t (fs) t 1 kev radia?on from a 1 fs, 1 A electron beam at 7 GeV May require switching the mirror and crystal placement (which may reduce the roundtrip angular acceptance ) R, T, P (ω) R, T, P(ω) 1. 1 Rtotal R total Tthin T output P(ω) R.R. Lindberg: The x ray FEL Oscillator, WG4: FEL Based Sources, FLS t (fs) P (ω) t (fs) E - E H E EH (mev) P (W) t (fs) kev t (fs) 15