1. SASE FEL PULSE DURATION ANALYSIS FROM SPECTRAL CORRELATION FUNCTION
FEL Conference 2011
Shanghai, 24. August. 2011
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 -5 5 6 12 18
We recover the average x-ray
time profile length T
from a collection of single shot spectra
FEL Conference 2011
Shanghai, 24.August.2011

3. Model description Electron beam current:
are independent random variables with probability density
SASE FEL Amplifier in linear regime:
time independent
time dependent
FEL Conference 2011
Shanghai, 24.August.2011

4. Model description Electric field spectrum: Average x-ray profile:
First and second order correlations
FEL Conference 2011
Shanghai, 24.August.2011

5. Model description First and second order correlations functions
Valid in the linear regime
For non-linear regime, we
run numerical simulations
FEL Conference 2011
Shanghai, 24.August.2011

6. Model description
Correlation of the intensity at the exit of the spectrometer
Intensity spectrum at frequency
Central frequency of amplification
FEL Conference 2011
Shanghai, 24.August.2011

7. Procedure to calculate G2
Single-shot
Spectrum
For each
collected
spectrum
Average on many shots
And
normalization
Average
Spectrum
FEL Conference 2011
Shanghai, 24.August.2011

8. G2 function for different X(t) profiles
relative FEL bandwidth
We assume that the spectrometer has Gaussian resolution function
relative rms spectrometer resolution
To find an analytical expression for G2 we need just to plug in the X(t) average profile
FEL Conference 2011
Shanghai, 24.August.2011

9. G2 with Gaussian profile* *
X-ray
beam
profile # #
FEL Conference 2011
Shanghai, 24.August.2011

10. G2 with flat top profile * * ° ° X-ray beam profile G2 # #
FEL Conference 2011
Shanghai, 24.August.2011

11. Non stationary w0 and FEL gain jitter
Central frequency jitters with Gaussian law with rms
FEL Conference 2011
Shanghai, 24.August.2011

12. Non stationary w0 and FEL gain jitter
Shot to shot gain as random variable with average at
correlate spectral
intensities at
FEL Conference 2011
Shanghai, 24.August.2011

13. Numerical simulations
1) Verify that relations hold well enough in saturation
2) Recover bunch length and spectrometer resolution
electron bunch
Short bunch
Wavelength 0.8 nm
Undulator period 3 cm
Long bunch
FEL Conference 2011
Shanghai, 24.August.2011

14. Verify relation
FEL Conference 2011
Shanghai, 24.August.2011

15. Simulated measurements results
Retrieved spectrometer resolutions
Long bunch
Short bunch
FEL Conference 2011
Shanghai, 24.August.2011

16. Experimental results Experimental demonstration performed at LCLS
Photon energy 1.5 keV
Electron charge 250 pC
Different undulator length
Different peak current
Controlling lasing part of electron bunch with slotted foil
FEL Conference 2011
Shanghai, 24.August.2011

17. Different undulators distance
Measured spectrometer relative resolution sm
Electron bunch length
Measured
x-rays are shorter than the electron bunch
FEL Conference 2011
Shanghai, 24.August.2011

18. Different peak currents
Measured
x-rays pulse durations are consistent with electron bunch length change.
FEL Conference 2011
Shanghai, 24.August.2011

19. Slotted foil measurements
Electron bunch FWHM
x-ray
FWHM
10 fs
13 fs
18 fs
24 fs
27 fs
39 fs
56 fs
52 fs
FEL Conference 2011
Shanghai, 24.August.2011

20. Thank you for your attention
FEL Conference 2011
Shanghai, 24.August.2011

21. Flat top vs Gaussian G2 We cannot distinguish between:
- Gaussian profile with rms length
- Flat Top profile with full length
With these x-ray profiles we cannot distinguish distrubutions with same rms
FEL Conference 2011
Shanghai, 24.August.2011

22. Double horn profile
FEL Conference 2011
Shanghai, 24.August.2011

23. Double horn profile Closed form is Straightforward, but long to write
Oscillation period
Depends on distance between the horns
Interaction diminishes for longer separation
Closed form is
Straightforward,
but long to write
FEL Conference 2011
Shanghai, 24.August.2011

24. Simulations with peak current jitter
The multiplicative effect of the statistical gain is in accord with the derived formula
After saturation, the effect is much less relevant
FEL Conference 2011
Shanghai, 24.August.2011

25. Another numerical simulation
Full Length = 10 mm
Radiation Wavelength = 1.5 A
r = 4.5 x 10-4
Gain Length = 2.98 m
Undulator Length = 100 m
Number of Shots = 2000
Total Slippage Length = 0.5 mm
Simulation
Parameters
FEL Conference 2011
Shanghai, 24.August.2011

26. Another numerical simulation
At Saturation
Deep Saturation
Exponential Growth
FEL Conference 2011
Shanghai, 24.August.2011

27. Another numerical simulation
z=30m
Linear regime
Agreement:
|X(w-w’)| analytical
|g1(w-w’)| simulated
z=100m
Deep saturation
z=60m
saturation
FEL Conference 2011
Shanghai, 24.August.2011

28. Another numerical simulation
Agreement:
Z=30m
Linear regime
g2(w-w’) simulated
1+|g1(w-w’)|2 simulated
Z=60m
saturation
Z=100m
Deep saturation
FEL Conference 2011
Shanghai, 24.August.2011

29. Fourier analysis of G2 function
Only
Only X(t)
FEL Conference 2011
Shanghai, 24.August.2011

30. Fourier analysis for double horn
Gaussian in 0
Gaussian in 0
Gaussian in
Triangle in 0
FEL Conference 2011
Shanghai, 24.August.2011

31. Camera background noise
Impact on the number of modes (w1 = w2)
small
Background must be removed accurately
Impact on the offset (|w1 - w2| large)
FEL Conference 2011
Shanghai, 24.August.2011
Can eventually go to 0