SASE FEL PULSE DURATION ANALYSIS FROM SPECTRAL CORRELATION FUNCTION

Transcription

1 SASE FEL PULSE DURATION ANALYSIS FROM SPECTRAL CORRELATION FUNCTION Shanghai, 4. August. Alberto Lutman Jacek Krzywinski, Yuantao Ding, Yiping Feng, Juhao Wu, Zhirong Huang, Marc Messerschmidt

2 X-ray pulse duration measurement Collection Shot-by-shot spectra.. Average time profile T δω / ω x 3 4 We recover the average x-ray time profile length T from a collection of single shot spectra t [ fs] 6 8 Shanghai, 4.August.

3 Model description Electron beam current: I( t) = ( e) δ ( t tk ) N k = t k are independent random variables with probability density f( t) SASE FEL Amplifier in linear regime: + + E( t) h( t, τ ) I( τ ) dτ h ( t τ ) h ( τ ) I( τ ) dτ = = ti ( t τ z/ vg ) i + i( kz ω ( t τ )) 4σ t 3 τ = time independent h ( t ) A ( z) e e ti h ( τ ) time dependent td td 3 Shanghai, 4.August.

4 Model description Electric field spectrum: N i tk E% ( ω) = ( e) H% ( ω) h ( t ) e ω X ( t) = h ( t) f( t) td ti td k k= Average x-ray profile: First and second order correlations E% ω E% ω e NH% ω H% ω X% ω ω * ( ') ( '') = ti ( ') ti ( '') ( ' '') ( ) E% ( ω ') E% ( ω '') e N H% ( ω ') H% ( ω '') X% () X% ( ω ' ω '') 4 = ti ti + 4 Shanghai, 4.August.

5 Model description First and second order correlations functions g ( ω ', ω '') = E% % * ( ω ') E ( ω '') E% ( ω ') E% ( ω '') g ( ω ', ω '') = E% ( ω ') E% ( ω '') E% ( ω ') E% ( ω '') g g ( ω ', ω '') ( ω ' ω '') X = % X% ( ω ', ω '') = + g ( ω ', ω '') ( ) Valid in the linear regime For non-linear regime, we run numerical simulations 5 Shanghai, 4.August.

6 Model description Correlation of the intensity at the exit of the spectrometer G ( ω ) ( ) + δω S ω δω ( ω δω ) ( ) + S ω δω S ( δω) = S S( ω) ω Intensity spectrum at frequency Central frequency of amplification ω 6 Shanghai, 4.August.

7 Procedure to calculate G 8 x Single-shot Spectrum S( ω) For each collected spectrum.5 x 9.5 S ( ω ) ( ) + δω S ω δω x 4 3 Average Spectrum δω S( ω) ω G ( δω) Average on many shots And normalization δω ω ω δω Shanghai, 4.August.

8 G function for different X(t) profiles ( ω ω ) % σ ω σ a H ti a ( ω) e relative FEL bandwidth We assume that the spectrometer has Gaussian resolution function σ relative rms spectrometer resolution G m ( δω) = ( ζ δωξ ) + ο e X% ζ T πσ, ( ) dζ ξ = σ = σ a ω σ m + σ a σ σ a m σ m + σ a To find an analytical expression for G we need just to plug in the X(t) average profile ω 8 Shanghai, 4.August.

9 G with Gaussian profile X ( t, σ ) T = e t σ T πσ T G ( δω) = e t σ σt δω ξ σ + + σ σ t X-ray beam profile.8 G * * # #.. 4 µ m 3 δω Shanghai, 4.August.

10 G with flat top profile X ( t, T) T = t t T > T T ζ σ = G ( δω) e ( ζ )cos( δωξ Tζ ) dζ * X-ray beam profile.5 G. * #.5. #.5 3 µm δω 4 Shanghai, 4.August.

11 Non stationary ω and FEL gain jitter 5 x Central frequency jitters with Gaussian law with rms ω σ ω.5 ω ( ζ δωξ ) + σ e X% T ( ζ, ) G ( δω) = K( δω) dζ πσ K( δω) = ( σ ) a + σ m + σω e δω σω a + m a + m + ω 4( σ σ )( σ σ σ ) ( )( σ ) a + σ m σ a + σ m + σω Shanghai, 4.August.

12 Non stationary ω and FEL gain jitter Shot to shot gain as random variable with average G correlate spectral intensities I ' = ( G + G)( S ' + S ') I '' = ( G + G)( S '' + S '') I ' I '' S ' S '' G = + + I ' I '' S ' S '' G I ' I '' I ' I '' G ( ω ', ω '') = G + G at at ω ' ω '' Shanghai, 4.August.

13 Numerical simulations ) Verify that relations hold well enough in saturation ) Recover bunch length and spectrometer resolution - electron bunch - electron bunch Wavelength.8 nm Undulator period 3 cm - - 3µ m 3µ m σ m σ m = = P [ W] Short bunch Long bunch 4 z[ m] 9 Shanghai, 4.August.

14 Verify relation m Exponential growth At saturation g ( ω ', ω '') = g ( ω ', ω '') µ 3µ m at.5m at 37.5m g (δω) g (δω) g (δω) g (δω).5.5 Exponential growth at 7m at 45m g (δ ω) g (δω) g (δ ω) g (δω).5 Deep saturation at 75m g (δω) g (δω).5 At saturation at 9m g (δ ω) g (δω) δω x 4 4 Deep saturation δω x 4 4 Shanghai, 4.August.

15 Simulated measurements results Retrieved spectrometer resolutions Long bunch (. ±.) 4 (.5 ±.5) 4 Short bunch (.98 ±.3) 4 (.98 ±.9) 4 5 Shanghai, 4.August.

16 Experimental results Experimental demonstration performed at LCLS Photon energy.5 kev Electron charge 5 pc Different undulator length Different peak current Controlling lasing part of electron bunch with slotted foil 6 Shanghai, 4.August.

17 Different undulators distance Electron bunch length Measured spectrometer relative resolution σ m 4 (. ±.4) Measured x-rays are shorter than the electron bunch 7 Shanghai, 4.August.

18 Different peak currents Measured x-rays pulse durations are consistent with electron bunch length change. 8 Shanghai, 4.August.

19 Slotted foil measurements Electron bunch FWHM fs 8 fs 7 fs 56 fs x-ray FWHM 3 fs 4 fs 39 fs 5 fs 4 9 Shanghai, 4.August.

20 Thank you for your attention Shanghai, 4.August.