Transcript talk multiscala

Multiple scale analysis of a
single-pass free-electron lasers
Andrea Antoniazzi
(Dipartimento di Energetica, Università di Firenze)
High Intensity Beam Dynamics
September 12 - 16, 2005 Senigallia (AN), Italy
plan
1.
Single-pass FEL
• introduction to the model
• short overview of the results obtained by our group
2.
Multiple scale analysis
• introduction to this method
• application to the FEL
3.
Conclusions
1.
The single-pass FEL
Hamiltonian model
 d j
 dz  p j

 dp j
 2 I cos( j   )

 dz
 dI
 dz  2 I j cos( j   )

n
p2j
H    2 I sin( j  )
2
j1
j1
n
Bonifacio et al., Riv. del Nuovo Cimento 13, 1-69 (1990)

numerics
Conjugated to the Hamiltonian that
describes the beam-plasma instability
I
n
p2j
H 
 2 I cos( j  )
2
j1
j1
n
Results
•
Statistical mechanics prediction of the laser intensity (Large
deviation techniques and Vlasov statistics)
J. Barre’ et al., Phys. Rev. E, 69 045501
•
Derivation of a Reduced Hamiltonian (four degrees of
freedom) to study the dynamics of the saturated regime
Antoniazzi et al., Journal of Physics: Conference series 7 143-153
•
Multiple-scale approach to characterize the non linear
dynamics of the FEL
Collaborations: Florence (S.Ruffo, D. Fanelli), Lyon (T. Dauxois),
Nice (J. Barre’), Marseille (Y. Elskens)
2.
Multiple scale analysis
Multiple-scale analysis is a powerful perturbative technique
that permits to construct uniformly valid approximation to
solutions of nonlinear problems.
When studying perturbed systems with usual perturbation
expansion, we can have secular terms in the approximated
solution, which diverges in time.
The idea is to eliminate the secular contributions at all
orders by introducing an additional variable =t, defining a
longer time scale. Multiple scale analysis seeks solution
which are function of t and  treated as independent
variables
Example: approach to limit cycle
Consider the Rayleigh oscillator, whose solution approaches a limit cycle
in phase-space.
1 3

y  y    y   y  
3


Using regular perturbation expansion y(t )  y0 (t )  y1 (t )  ..
inserting in (1) and solving order by order
y0  y0  0
1
3
y1  y1  y 0   y 0 
3
The first order solution contains a term that diverges like εt
y1 (t )  At sin(t )  ..
This expression is a good approximation of the exact solution only for
short time. When t~O(1/є) the discrepancy becomes relevant
Multiple-scale analysis permits to avoid the presence of
secularities.
Assume a perturbation expansion in the form:
where =t.
Inserting the ansatz and equating coefficients of 0 and 1 we obtain:
(1)
(2)
The solution of eq (1) reads:
Observe that secular terms will arise unless the coefficients of eit
in the right hand side of eq. (2) vanish.
Setting the contribution to zero, after some algebra,
one gets:
where
Coming back to the original time variable, the solution reads :
Є=0.2
• The
approximate
solution
accounts both for the initial
growth and for the later saturated
regime
• We have obtain a zero-th order
solution that remains valid for
time at least of order 1/є, while
usual perturbation method is
valid only for t~O(1)
Multiple scale analysis of single-pass FEL
Vlasov-wave system:
where
Janssen P.et al., Phys. of Fluids, 24 268-273
plays the role of the small parameter.
Linear analysis
In the linear regime one gets:
The dispersion relation reads:
Thus motivating the introduction of the slower time scales
2=2t,
4=4t, ….
Non-linear regime
Following the prescription of the multiple-scale analysis we replace the time
derivative by:
and develop:
where:
Avoiding the secularities...
X1  ( 2 , 4 )ei t  c.c.
0
...at the third order
where:
And  is the solution of the adjoint problem
...at the fifth order
obtaining..
C
CRe
CIm
Coming back to t..
Non linear Landau
equation
where
Analytical solution
This solution account for both the exponential growth and the
limit cycle asymptotic behavior
Comparison with numerical results
• Qualitative agreement with numerical results, both for exponential
growth and saturated regime
• The saturation intensity level increases with δ as observed in
numerical simulation
• the
level of the plateau is
sensibly
higher
than
the
corresponding numerical value:
probably some approximations
need to be relaxed (quasi-linear
approximation)
3.
Conclusions and perspectives
• Developed an analytical approach to study the dynamics and
saturated intensity of a single-pass FEL in the steady-state
regime.
• Next step: improvement of the calculations to have a
quantitative matching with numerical results.
• Future direction of investigations: HMF?