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

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

)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

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)

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

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

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:

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

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

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

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...

...at the third order

where:

And is the solution of the adjoint problem

..),( 0

421 cceX ti

...at the fifth order

obtaining..

C CRe CIm

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

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.

• Future direction of investigations: HMF?

• Next step: improvement of the calculations to have a quantitative matching with numerical results.