Multiple scale analysis of a single-pass free-electron lasers
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Multiple scale analysis of a single-pass free-electron lasers
Andrea Antoniazzi(Dipartimento di Energetica, Università di Firenze)
High Intensity Beam DynamicsSeptember 12 - 16, 2005 Senigallia (AN), Italy
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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
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1. The single-pass FEL
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)cos(2
)cos(2
jj
j
j
j
j
Izd
dI
Izd
dp
pzd
d
Hamiltonian model
Bonifacio et al., Riv. del Nuovo Cimento 13, 1-69 (1990)
H p j
2
2 2 I sin( j )
j1
n
j1
n
numerics
H p j
2
2 2 I cos( j )
j1
n
j1
n
Conjugated to the Hamiltonian that describes the beam-plasma instability
I
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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)
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Multiple-scale analysis is a powerful perturbative technique that permits to construct uniformly valid approximation to solutions of nonlinear problems.
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
When studying perturbed systems with usual perturbation expansion, we can have secular terms in the approximated solution, which diverges in time.
2. Multiple scale analysis
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Example: approach to limit cycle
Consider the Rayleigh oscillator, whose solution approaches a limit cycle in phase-space.
Using regular perturbation expansion
inserting in (1) and solving order by order
This expression is a good approximation of the exact solution only for short time. When t~O(1/є) the discrepancy becomes relevant
..)()()( 10 tytyty
3
0011
00
3
1
0
yyyy
yy
3
3
1yyyy
..)sin()(1 ttAty
The first order solution contains a term that diverges like εt
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Multiple-scale analysis permits to avoid the presence of secularities.
where =t.
Assume a perturbation expansion in the form:
(1)
(2)
Inserting the ansatz and equating coefficients of 0 and 1 we obtain:
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The solution of eq (1) reads:
Observe that secular terms will arise unless the coefficients of eit in the right hand side of eq. (2) vanish.
Setting the contribution to zero, after some algebra, one gets:
where
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Coming back to the original time variable, the solution reads:
• 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)
Є=0.2
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Multiple scale analysis of single-pass FEL
Vlasov-wave system:
where
plays the role of the small parameter.
Janssen P.et al., Phys. of Fluids, 24 268-273
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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, ….
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Non-linear regime
Following the prescription of the multiple-scale analysis we replace the time derivative by:
and develop:
where:
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Avoiding the secularities...
...at the third order
where:
And is the solution of the adjoint problem
..),( 0
421 cceX ti
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...at the fifth order
obtaining..
C CRe CIm
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Coming back to t..
where
Non linear Landau equation
Analytical solution
This solution account for both the exponential growth and the limit cycle asymptotic behavior
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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)
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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.
• Future direction of investigations: HMF?
• Next step: improvement of the calculations to have a quantitative matching with numerical results.