Quantum effects in high-gain FELs are ruled by the quantum FEL parameter, \u3c1 = pmcy/hkr, which is the ratio between the momentum spread at saturation and the one photon momentum recoil. It has been shown that when  3c\u3c1  64 1 the spectrum of the emitted radiation changes from the broad continuous and chaotic spectrum of the classical regime to a series of discrete and equally spaced very narrow lines, due to transitions between discrete momentum states. In this paper we show that the quantum regime can be achieved using Kilometers long magnetic wigglers or a laser wiggler. In this paper we state the scaling laws necessary to operate a Quantum SASE FEL in the Angstrom region with a laser wiggler. Specific example is given having in mind a high power Ti: Sa laser wiggler at \u3bb= 0.8 \u3bcm, in construction at LNF-INFN Laboratories, for the SPARC/PLASMON-X project

A. Schiavi

G.R.M. Robb

L. Serafini

M. Ferrario

N. Piovella

R. Bonifacio

English

AIR Universita degli studi di Milano

QUANTUM SASE FEL WITH A LASER WIGGLER* 
 
R. Bonifacio, INFN-MI, Milano, Italy and Universidade Federal Alagoas, Maceiò, Brazil  
M. Ferrario, INFN-LNF, Frascati, Italy  
G. R. M. Robb, Department of Physics, University of Strathclyde, Glasgow, Scotland, UK  
N. Piovella#, INFN-MI and Università degli Studi di Milano, Milano, Italy   
A.Schiavi, INFN-LNF, Frascati, Italy and Università di Roma “La Sapienza”, Rome, Italy 
L. Serafini, INFN-MI, Milano, Italy  
 
 
Abstract 
Quantum effects in high-gain FELs are ruled by the 
quantum FEL parameter, / rmc kρ ρ γ= h , which is the 
ratio between the momentum spread at saturation and the 
one photon momentum recoil. It has been shown that 
when 1ρ ≤   the spectrum of the emitted radiation 
changes from the broad continuous and chaotic spectrum 
of the classical regime to a series of discrete and equally 
spaced very narrow lines, due to transitions between 
discrete momentum states. In this paper we show that the 
quantum regime can be achieved using Kilometers long 
magnetic wigglers or a laser wiggler. In this paper we 
state the scaling laws necessary to operate a Quantum 
SASE FEL in the Angstrom region with a laser wiggler.  
Specific example is given having in mind a high power 
Ti:Sa laser wiggler at λ= 0.8 μm, in construction at LNF-
INFN Laboratories, for the SPARC/PLASMON_X 
project.  
 
INTRODUCTION  
 
    It has been previously recognized that quantum effects 
in SASE FEL are determined by the quantum FEL 
parameter ρ   [1], [2] defined as the usual FEL parameter 
times the ratio between the electron energy and the 
photon energy. Quantum effects become relevant when 
1ρ ≤ . However the calculations of [1],[2] are confined 
to the linear regime. In [3] we have extended the theory of 
Quantum SASE FEL to the non linear regime and we 
have shown the phenomenon of quantum purification of 
SASE spectrum: the broad superposition of chaotic series 
of random spikes predicted by the classical theory 
changes dramatically, when 1ρ ≤ , to a series of discrete 
equally spaced very narrow lines. The question is: which 
are the experimental set up and the experimental 
parameters necessary to observe Quantum SASE in the 
short wavelength region, where quantum effects are 
expected to be relevant?  
 
MAGNETIC VERSUS LASER WIGGLER 
 
   The Quantum FEL (QFEL) parameter is given by  
                                                          
 
 
* Work supported by INFN, Italy. 
# nicola.piovella@mi.infn.it 
 
r
r C
mc
k
λγρ ρ γρ λ= =h   (1) 
 
where ρ is the classical FEL parameter [4], λr and 
≈= mc/hcλ 0.024 Å are the radiation wavelength and 
the Compton wavelength, respectively, and γ is the 
resonance energy in units mc2, given by  
 
21 (1 )w
r
a
b
λγ λ= +     (2) 
 
where λ is the wiggler wavelength and 2b =  for the 
magnetic wiggler and 2b =  for an electromagnetic (e.m.) 
laser wiggler. Clearly the physical meaning of ρ  is the 
ratio between the classical momentum spread and the 
photon recoil kh , so that quantum effects become 
important when 1ρ ≤ , because in this case the 
discreteness of momentum exchange is  relevant. 
    Using Eqs. (1) and (2) the QFEL condition becomes 
 
2(1 )
c
r w
b
a
λρ
λ λ
≤
+
  (3) 
 
so that, to reach the high-gain region, one needs a number 
of wiggler periods of the order of 1/ρ, i.e., a wiggler 
length Lw given by 
 
                
c
wr
ww b
)a(
NL λ
λλ
ρ
λλ
23 1+
≥≈=   (4) 
 
In order to have ≈rλ 1 Å, using Eqs.(2), (3) and (4), one 
has the following numbers: 
Magnetic wiggler, ≈λ 1 cm, 2b = , E= 3.5 GeV, 
63.4 10ρ −≤ ⋅ , 3wL ≥  Km. 
Laser wiggler, ≈λ 1 μm, 2b = , E=50 MeV, 45.10ρ −≤ , 
2wL ≥ mm! 
    For simplicity we have assumed aw << 1. The opposite 
case would require a longer wiggler. The previous 
considerations clearly show that a QFEL with a magnetic 
Proceedings of the 27th International Free Electron Laser Conference
21-26 August 2005, Stanford, California, USA 71 JACoW / eConf C0508213
wiggler is unpractical, whereas, using a laser wiggler it 
can be a table top apparatus. 
    Therefore, we propose a typical Compton back-
scattered configuration: a low energy electron beam 
counter-propagating with respect to an electromagnetic 
wiggler (wave) provided by a high power laser.  
 
THE LASER WIGGLER 
 
    Let us consider a laser wiggler with radiation 
propagating in the z direction opposite to an electron 
beam with the following specifications: W0 is the 
minimum diameter of the laser beam in the focus, σ0 is 
the minimum radius of the electron beam, 
0
2
0 /Z Wπ λ=   is the distance in which the radiation 
beam diverges (Rayleigh range),  * 202 / nβ σ γ ε=   is the 
analogous length for the electron beam [5]. The meaning 
of Z0 and β* appears from the well known equations 
2
0 0( ) 1 ( / )W z W z z= +  and * 20( ) 1 ( / )z zσ σ β= + [5].  
In this section we follow [6], within factors 2. Let us 
define the adimensional parameters:  
 
0
1
02
W
ε
σ
=
 ;  
*
2
0Z
β
ε = .  (5) 
 
In order to ensure a good overlapping and matching 
between radiation and electron beam we must impose that 
ε1 and ε2  1≥ . Hence, because 21 2 / 2 1nε ε γλ πε= ≥ , our 
conditions guarantee that the emittance criterium is 
satisfied for the laser wiggler. 
    In [6], it has been shown that the previous relations 
leads to: 
 
( )
2 3
3 2
2 2
(1 ) ( )( ) (1 )
32
w
r w
a m
A a
λ μλ λ
ηπη
+
= = +
 (6) 
 
where 
  
2
1 2 ( . )
8
n mm mradε ε εη = .  (7) 
 
The factor 8, which does not appear in [6], is due to a 
factor 2 in the definition of ε1 and β*. We remark that Eq. 
(6) (formally independent on the electron energy) gives a 
direct relation between the radiation wavelength and the 
wiggler wavelength in terms of two geometrical 
parameters and  εn  (via the η factor), and the wiggler 
parameter. Equation (6) can be derived using the 
following chain of equations 
 
 
2 2 2 2 *
0 1 0 1
0
4 2
n
W
Z
π πε σ πε ε β
λ λ λγ
= = = =
  
1 / 22
1 2
03 / 2 2
4
1
n r
w
Z
a
πε ε ε λ
λ +
⎛ ⎞⎜ ⎟⎝ ⎠  
 
    Eliminating Z0 from the first and the last equation, we 
obtain Eq. (6). Furthermore, using Eqs.(2) and (6) one 
obtain the identity 
  
16 50 ( )m
πη ηγ λ λ μ= =   (8) 
 
Equation (8) fix the resonant energy only in terms of the 
parameter η and of the laser wiggler wavelength.  
    As an example, if we want to produce 1 Angstrom 
radiation, with aw <<1 and η =1, Eq.(6) and (8) gives a 
laser wiggler with λ=1 μm and γ = 50. 
In [6] it has been shown that the wiggler parameter is 
self consistently determined by the equation 
2
2 0
21w w
a
a
a
=
+
, 
where 
 
⎟⎟⎠
⎞
⎜⎜⎝
⎛
+
≈
ρ
ρ
λ
η
ε
ε
1
23
2
3
22
0
/
Pa   (9) 
 
where P is the peak laser power in TW and we have 
assumed a laser pulse with a gaussian transverse profile. 
Therefore, solving the previous equation, we obtain 
0 0/ ( )wa a F a=  where  
 
2
0 2
0
1 1 4( ) 1
2 w
a
F a a
+ +
= = +   (10)  
 
Note that in the limit 204 1a << , 0aaw ≈  , whereas in the 
opposite limit 0aaw ≈ . 
    Inverting Eq. (1) and using Eq.(6), one has  
 
4
2
0
5.10 ( )[ ] F am
η ρρ λ μ
−
=    (11) 
 
    Let us define the quantum gain length [6]  
 
( ) ( )3 03/ 2[ ] [ ][ ] 1 80 1 ( )8g
m m
L m F a
λ μ λ μμ ρ ρ
ηρπρ ρ
= + = +
 (12) 
 
where Eq. (11) has been used. 
    Equation (12) is not an exact equation, but is an 
interpolation formula which gives the correct behaviour in 
the quantum regime ρ <<1 [3] and the classical 
expression [4] in the opposite limit. This equation can be 
rigorously justified in the asymptotic cases ρ  very large 
or very small.  
Proceedings of the 27th International Free Electron Laser Conference
JACoW / eConf C0508213 72 21-26 August 2005, Stanford, California, USA
    We must also impose that the electron beam 
characteristic length β* is larger than the gain length, i.e., 
 
3
* 1/ gLε β= ≥ .  (13) 
 
It can be easily shown that 
  
[ ] [ ] ( ) )a(FmLm /ngn*n 023243320 122 ρρη μλεεγεεγβεμσ +≈== .  (14) 
 
Furthermore, the relative energy spread, in the quantum 
regime, is subjected to the limitation [3] 
 
3
3 / 2
2
0
2 10
4
( ) [ ]F a m
γ ηρ ρ ρ
γ λ μ
−Δ ⋅
≤ =
 (15) 
 
where Eqs. (11) has been used. 
   In [6] it has been shown that the current, I, necessary to 
achieve a given value of the QFEL parameter, ρ , is 
given by 
 
( )
2 3
2 3
5
2 0
1( ) 3 10 1( )
nI A
F aP
ε ε η ρ
ελ
⎛ ⎞
= ⋅ +⎜ ⎟⎜ ⎟⎝ ⎠
 (16) 
 
where the wiggler laser power, P,  is given in TW.  
     
 
εn (mm mrad) 0.5 1 0.5 1 
ε1 1 1 1 1 
ε2 12 6 10 5 
ε3 20 30 50 40 
Laser power (TW) 20 100 50 100
     
Current (A) 230 353 440 531
λr (Å) 1.5 1.8 2.0 2.1 
Eb (MeV) 23.4 23.4 19.5 19.5
Δγ/γ (10-4) 2.5 2.2 2.4 2.2 
τ (psec) 84 148 229 195
d(μm) = 2 σ0 7.3 13.7 13.2 17.2
Lgain (103 μm) 1.1 1.3 1.2 1.3 
N of photons (1011) 1.2 3.3 6.3 6.5 
P(MW) 1.8 2.4 2.7 3.0 
 
Table 1. Example of the various parameters for ρ =0.2 
and  λ = 0.8 μm (laser wiggler). 
 
    We remark that in the quantum limit ρ <<1 the current 
is independent on ρ , whereas in the classical limit, 
ρ >>1, the current increases linearly with ρ , making 
more problematic the use of an e.m. wiggler to see 
classical SASE. 
The minimum laser time duration required is given by: 
 
* 3
3( sec) / 3.3 10 [ ]gp c L mτ β ε μ−= = ⋅     (17) 
 
where Lg is given by Eq. (12). 
Furthermore, using Eq. (10), Eq. (6) can be rewritten as:  
 
3
02 ( )r F a
λλ
η
= .   (18)   
 
The units are: λ in μm, λr in Å and εn in mm mrad and P 
in TW. Explicit examples for a laser wiggler at 0.8 μm are 
given in Table 1. Furthermore, keeping in mind that in 
QFEL one can have maximum of the order of one photon 
per electron, the order of magnitude of the number of 
photons, N, and the radiated power, Pr, are given by 
/N I eτ≈  and /
r
P I eω≈ h .  
 
EMITTANCE LIMITATIONS 
 
   In this paragraph we will assume to be in the quantum 
regime, i.e., ρ <<1.   
    The emittance criterium is satisfied for the laser 
wavelength imposing that ε1, ε2 of Eq.(5) are larger than 
one. Roughly speaking, this is equivalent to require that 
the electron beam is contained in the laser wiggler beam, 
otherwise the electron would not be interacting with the 
wiggler. If one defines the quantity ε1, ε2, Z0 and W0 
referred to the radiation beam at the wavelength λr, then, 
in order to guarantee that the radiated field is contained in 
the radiating electron beam, we should impose that ε1 and 
ε2 are less than unity, because the field outside the 
electron beam cannot be amplified. Therefore, the 
emittance criterium, i.e. 1/2 ≤rn γλπε , should be 
reversed regarding the radiation field. However, the 
emittance cannot be arbitrarily large because the 
transverse emittance implies a spread on the resonant 
wavelength, which must be small enough not to degrade 
the FEL action. In fact, the resonance condition with off-
axis propagation reads: 
 
2 2 2
2
(1 )
4
w
r
aλ γ θλ
γ
+ +
=    (19) 
 
Because 0 / 2
r
θ ε σ≤ ≤  the maximum spread is given by 
γ
γΔ
σ
ε
λ
λΔ 2)1(4 22
2
≈
+
=
w
n
a
. Imposing condition (15), 
one has  
 
2 3
3/ 2
2 2 2
0
2 104 ( )8 (1 ) [ ]
n
w F aa m
εγ ηρ ρ ρ
γ σ λ μ
−Δ ⋅
= < =
+
. (20) 
 
Note that the emittance criterium can be written as 
 
Proceedings of the 27th International Free Electron Laser Conference
21-26 August 2005, Stanford, California, USA 73 JACoW / eConf C0508213
2 / 1
n r
X πε γλ= <   (21) 
 
whereas condition (20), using Eq.(6,8,14) with ρ <<1, 
can be written as 
  
320X ε≤    (22) 
 
where ε3 is given by Eq.(13). Hence, if ε3 >> 1 the 
emittance criterium can be relaxed by orders of 
magnitude. Note that, for ε1 = 1 one has */rX Z β= , 
where 
r
Z  is the Rayleigh length of the radiation field. In 
this case, criterium (22) can be written as  
 
4 /r gX Z L≤    (23) 
 
Finally, using Eq.(14), with ρ <<1, Eq. (20) can be 
written as  
 
 
2 2
3
0
1.6 10 ( )
n
F a
ε λ
ε
η
−
⋅
≤ .       (24) 
 
Eq.(24), using Eq.(7), is equivalent to  
 
3 0
2
1 2
( )
0.4
n
F aε
ε λ
ε ε
≤  .  (25) 
 
Eqs.(22), (24) and (25) give the emittance limitation for a 
QFEL with a laser wiggler. 
 
QFEL EXPERIMENTAL STUDIES  
 
The high brightness SPARC photo-injector [7] and the 
High Intensity Laser Laboratory (HILL) [8] under 
development at LNF, will provide an excellent facility to 
test the Quantum SASE FEL (QFEL) theoretical 
predictions [3]. The main component of this test facility is 
a 800 nm, 100 TW-class Ti:Sa laser system synchronized 
to the SPARC photo-injector. Eventually an additional 
beam line will be built in the SPARC bunker in order to 
transport the SPARC electron beam at an interaction point 
with the incident laser beam. A final focus system, 
already foreseen for the Thomson backscattering source 
[8], will allow conducting experiments of generation of 
X-ray radiation via QFEL interaction of electron bunches 
in a laser wiggler, as schematically shown in Figure 1. 
Parametric studies based on the QFEL scaling laws 
discussed in this paper, 3D simulations of the QFEL 
interaction and electron beam dynamics studies are under 
way in order to match the SPARC photo-injector 
performances to the requirement of a realistic QFEL 
experiment. 
 
 
Figure 1: Schematic experimental setup for a QFEL 
interaction of electron bunches in a laser wiggler. 
  
CONCLUSIONS 
 
    In conclusion, we have shown that the construction of a 
Quantum SASE FEL with a magnetic wiggler is very 
problematic because it would require a wiggler length 
larger than 1 Km, with electron energy in the GeV region, 
whereas with a laser wiggler of 1 μm wavelength, the 
apparatus can be table top with the wiggler length of the 
order of mm. Furthermore, we have shown that the 
criterium for emittance can be relaxed by orders of 
magnitude. 
    We have given the expression of the relevant quantities 
for the design of a Quantum SASE FEL with a laser 
wiggler in terms of the quantum FEL parameter ρ , 
showing that the required current increases linearly with 
ρ  in the classical regime, when ρ >>1, whereas is 
independent of ρ  in the quantum regime, when 1ρ ≤ . 
This fact makes more difficult the use of the laser wiggler 
in the classical regime than in the quantum regime. If a 
quantum SASE FEL can be built one would have X-ray 
coherent FEL that can be a table-top object and the 
technological problem would go from high-energy 
accelerator plus long magnetic wigglers to the widely 
used high-power laser technology. We have given an 
example assuming a Ti:Sa laser wiggler at 0.8 μm.  
 
REFERENCES 
 
[1] R. Bonifacio, F. Casagrande, Opt. Comm. 50, 251 
(1984), NIM Phys. Res. A 237, 168 (1985); R. 
Bonifacio, ibid, 400, 165 (1997) and Opt. Comm. 150, 
126 (1998). 
[2] C.B.  Schroeder, C. Pellegrini and P. Chen, Phys. Rev. 
E 64 (2001) 056502-1.  
[3] R. Bonifacio, N. Piovella, G.R.M. Robb, NIM A 543 
(2005) 645 and R. Bonifacio, N. Piovella, G.R.M. 
Robb in this Proceedings. 
[4] R. Bonifacio, C. Pellegrini, L. Narducci, Opt. Comm., 
50, 373 (1984) 
[5] E. T. Scharlemann, pg.115 and U. Amaldi, pg 172 in 
High Gain, High Power FEL: Physics and Application 
to TeV Particle Accleration, Varenna, Italy (1988).   
[6] R. Bonifacio, NIM A 546 (2005), 634.  
[7] D. Alesini et al., “Status of the SPARC project”, Proc. 
of the PAC 2005, Knoxville, Tennessee, USA 
[8] D. Alesini et al., “The project PLASMONX”, Proc. of   
the PAC 2005, Knoxville, Tennessee, USA.  
 
Proceedings of the 27th International Free Electron Laser Conference
JACoW / eConf C0508213 74 21-26 August 2005, Stanford, California, USA


Quantum SASE FEL with a laser wiggler

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Quantum SASE FEL with a laser wiggler

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