As already mentioned, the characteristics of FEL radiation are high power, short pulse length, narrow bandwidth, spatial coherence and wavelength tunability. In this section, some examples will be given that shed light on the differences between the TESLA XFELs and stateoftheart synchrotron radiation sources to illustrate the exceptional properties of an FEL. Table 1 gives an example of some typical photon beam parameters for one of the planned XFELs at TESLA (a detailed discussion of the properties of all XFEL undulators at TESLA can be found in the TDR Part V, Chap. 4).
Units 
SASE1 


Wavelength* 
Å 
15 
Peak power 
GW 
37 
Average power 
W 
210 
Photon beam size (FWHM)** 
µm 
100 
Photon beam divergence (FWHM)*** 
µrad 
0.8 
Bandwidth (FWHM) 
% 
0.08 
Coherence time 
fs 
0.3 
Pulse duration (FWHM) 
fs 
100 
Min. pulse separation**** 
ns 
93 
Max. number of pulses per train**** 
# 
11500 
Repetition rate**** 
Hz 
5 
Number of photons per pulse 
# 
1.8 x 10^{12} 
Average flux of photons 
#/sec 
1.0 x 10^{17} 
Peak brilliance 
B***** 
8.7 x 10^{33} 
Average brilliance 
B***** 
4.9 x 10^{25} 
*Parameters are given for the shortest wavelength.
** Value at the exit of the undulator.
*** Far field divergence.
**** Values determined by the time structure of the electron beam in the accelerator. The average parameters for the SASE1 FEL are given for the ultimate case when only this beamline is in operation.
***** In units of photons/( sec · mrad^{2} · mm^{2} · 0.1%bandwidth).
One of the key parameters to compare different radiation sources is their brilliance^{3}. For partially coherent light sources (wigglers and undulators) the brilliance
[photons/( sec · mrad^{2} · mm^{2} · 0.1%bandwidth)] can be calculated from the spectral flux
[photons/( sec · 0.1%bandwidth)] divided by the photon beam's rms radius ∑ and divergence ∑' obtained from convolution of the electron beam and photon diffraction parameters as
= 
(5) 

= 
(6) 

= 
(7) 

In the case of full transverse coherence (e.g. FELs in saturation) ∑ and ∑ are related through 


(8) 

and Equation (5) can be transformed into 


(9) 

This means that the brilliance is simply given by the spectral flux divided by the transverse photon phase space. 


(10) 

Eight orders of magnitude more photons per mode reflect the improved coherence of the FELs compared to the spontaneous emission of third generation SR sources. In summary, extremely short pulses, high intensity and full transverse coherence are the attractive features of the FEL radiation.
^{3} Note: In the US the European "brilliance" is mostly called "brightness".