Déstabilisation par un processus d advection dans un laser: défauts spectro temporels et structures induites par bruit

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1 Déstabilisation par un processus d advection dans un laser: défauts spectro temporels et structures induites par bruit S. Bielawski and C. Szwaj Lab. PhLAM/CERLA, université de Lille 1 (France) C. Bruni, D. Garzella, G.L. Orlandi and M.E. Couprie CEA, Gif sur Yvette (France) Laboratoire pour l utilisation du rayonnement électromagnétique (LURE) Université Paris Sud, Orsay (France) M. Hosaka, A. Mochihashi, and M Katoh UVSOR, Institute for Molecular Science, Okazaki (Japon) Rencontres du non linéaire 26

2 Impusion lumineuse amplificateur optique Miroir Miroir 1. Le laser à électrons libres: aspect physique aspect "dynamique" expérience et modèle système du type "advection diffusion" à 1d + saturation globale 2. Résultats (quoi de neuf?) dynamique spectro temporelle saturation globale vs saturation locale 3. Quels phénomènes retrouve t on dans des équations de Ginzburg Landau élémentaires?

3 e

4 e Synchrotron radiation B e B periodic (Undulator)

5 electromagnetic wave B periodic amplified wave e (Undulator)

6 electromagnetic wave B periodic amplified wave e (Undulator) M 1 Undulator system M 2 Free electron laser (FEL): tunable far infrared to UV (and X...) Super ACO (LURE, Orsay, France) UVSOR (IMS, Okazaki, Japan) UV (35 nm) UV (25 nm), visible (52 nm), etc

7 UVSOR storage ring (IMS, Okazaki)

8 M 2 e M 1 e Energy: 8 MeV Lifetime: several hours Revolution frequency: 5.6 MHz Current: tens of ma

9 Stability issues vs frequency mismatch 1) Envelope of the pulse train Desired envelope laser pulses: duration= O(1ps) period=o(1ns) t Other envelopes observed experimentaly: T (5 ms/div) 2) Internal structure of the picosecond pulses? (this talk)

10 e n ( θ ) at each round trip: e n ( θ ) loss, gain e n+1 ( θ ) "space" θ "!$# round trip #1 #2 #n discrete time (round trip n)

11 Effet du gain? θ Gain amplification + diffusion... θ

12 Effet du gain? TF θ Gain amplification + diffusion...! "$# %! " θ gain spectrum Re(G) pulse spectrum & '( ) * & ',+-)/. & '1+-)2(. & 34'1+-)2( * & 9 ';:. <>=. <>== ) ω

13 O V U Final step: continuous limit: Map > PDE! " #"$ Pulse shape T= round trips frequency mismatch losses Gain variable bunch shape (gaussian) Gain depends only on slow time T % & '"(*)+,(.-/ ):9 <;=1 ;>4?9@(*A B (C+@-D9E58+,( FHG I F GKJML N I L & O P LRQ S P F G T WYX JZ [ L W G I Z \ Gain saturation depends on overall power

14 competition between advection diffusion #%$ & '(#*) +, # & -/ # & #:);)2<& =?>A@ θ!" v large: transient growth e= is globally stable (convective instability) v not too large: (absolute instability) θ θ "space" "space" η= η= Time T (number of round trips) Time T (number of round trips) loss (=1) Basic concepts: see eg Huerre and Monkewitz, Ann. Fluid Mech. 22, 473 (199),

15 competition between advection diffusion "!#%$&'( )*+, -/.1 θ v large: transient growth e= is globally stable (convective instability) v not too large: (absolute instability) θ θ "space" "space" η= η= Time T (number of round trips) Time T (number of round trips) loss (=1) Transient growth known in mode locked lasers: Kartner et al. PRL 82, 4428 (1999) Morgner & Mitschke, PRE58, 187 (1999) Basic concepts on cv instabilities: see eg Sturrock, Phys. Rev. 5, 488 (1958) Huerre and Monkewitz, Ann. Fluid Mech. 22, 473 (199), Fluid ex.(hele Shaw cell) PRL 82, 1442 (1999) Cossu & Chomaz PRL 78, 4387 (1997)

16 Numerical results 1 (a) (d) (g) (j) θ 1 T 1 (b) T 1 (e) T 1 (h) T 1 (k) Frequency mismatch: v= v=.5 v=.7 v=3.4

17 Numerical results 1 (a) (d) (g) (j) θ 1.15 (b) (e) (h) (k) k.15 T 1 T 1 T 1 T 1

18 Numerical results 1 (a) (d) (g) (j) θ 1.15 (b) (e) (h) (k) k k (c) (f) (i) (l).15 T 1 T 1 T 1 T 1

19 amplitude k amplitude hole: e(k,t)= ~ e(k,t) phase k 2 π phase singularity time (T) "spatio temporal defect"

20 k Questions: Experimentally realistic? k 2π insights on the "origin" of these holes? time (T)

21 Real time spectrum analyzer cylindrical lens Fabry Perot Etalon Linear CCD camera (EG&G) 256 pixels 67k lines/s x FEL output e PC acquisition Flat mirrors R=97% streak camera spacing e=1 5 µ m x

22 Fast time θ (25 ps/div.) Experimental results (super ACO)

23 Experimental results (super ACO) frequency mismatch Fast time θ (25 ps/div.)

24 A /. 7 Experimental results (super ACO) frequency mismatch Fast time θ (25 ps/div.) >?@ ;=< : /1 +-,.! " #$ %& ' ()*

25 Optical spectrum versus time: Recent results at UVSOR (IMS, Japan) time (4 ms/div) time (4 ms/div)

26 Nonlinear dynamics point of view: Minimum dynamical ingredients? Part of this specific model is necessary for the instability, part is NOT e T + ve θ = e + g(t )f(θ)(e + e θθ ) + ηξ, (1) g(t ) = A σ 2 (T ) exp [ (σ 2 (T ) 1)/2 ] (2) with dσ2 dt = γ 1 σ 2 + laser pulse length bunch length: Taylor expansion of f(θ) L e(θ, T ) 2 dθ. (3) identification of slowest timescales? Usually γ (e.g., macropulse instabilities). Here?

27 ^ n ^ "Minimal" equations?! " #$ % '& ' &)(*' + &, %.-/& "& * 8:9<;>=48?A@B8:?6?C;ED " #$ L -/7c& & JdGe ]gf $ PTh FHG"IKJMLONQP/RTSU8VIW8YX[Z)\ 8 ] \ ^ i kj ', l-/+'(*'m n 9o@BL6R nlpqp ;ED n IKJsr3=utv;wtxR:P n (Global I n X Z)\ ] \ ^ convection or 2 e e or local coupling) R _Ǹ ;>a4b R _Qty;>a4b

28 ^ n ^ "Minimal" equations? ex. with global coupling! " #$ % '& ' &)(*' + &, %.-/& "& * 8:9<;>=48?A@B8:?6?C;ED " #$ L -/7c& & JdGe ]gf $ PTh FHG"IKJMLONQP/RTSU8VIW8YX[Z)\ 8 ] \ ^ i kj ', l-/+'(*'m n 9o@BL6R nlpqp ;ED n IKJsr3=utv;wtxR:P n (Global I n X Z)\ ] \ ^ convection or 2 e e or local coupling) R _Ǹ ;>a4b R _Qty;>a4b space (z) wavenumber (k) wavenumber (k) e(z,t) E(k,t) Arg(E(k,t)) T

29 ^ ^ Mechanism? (GL+global coupling)! " #$ % '& ' &)(*' + &, %.-/& "& * space (z) e(z,t) 8:9<;>=48?A@B8:?6?C;ED " #$ L -/7c& & JdGe ]gf $ PTh FHG"IKJMLONQP/RTSU8VIW8YX[Z)\ 8 ] \ ^ i kj ', l-/+'(*'m n 9o@BL6R nlpqp ;ED n IKJsr3=utv;wtxR:P n I n X Z)\ n ] \ ^ Diffusion Non uniformities of control parameters R _Ǹ ;>a4b R _Qty;>a4b Global coupling wavenumber (k) wavenumber (k) T E(k,t) Arg(E(k,t)) Open question: links with the "Riecke and Paap instability? Riecke and Paap, PRL 59, 257 (1987)

30 Differences between global and local couplings? > (1) local e 2 stable pattern infinite medium (ε=) e= absolutely unstable Vc convectively unstable v st. pattern finite medium (ε=) e= unstable Vc +Ο(ε) e= stable v finite medium +noise "clean" pattern "noisy" pattern

31 Differences between global and local couplings? > (2) global! e 2 stable pattern infinite medium (ε=) e= absolutely unstable Vc convectively unstable v finite medium finite medium +noise (ε=) st. pattern e= unstable e= stable Vc +Ο(ε) (.17) (8) "clean" pattern "noisy" pattern v

32 .5 couplage local 1.2 couplage global e e ! $ #" " % % &'!() *,+ -./* 1 * :<;=?>A@B29 CD*ECF@HGI*

33 Conclusion * Laser à électrons libres = système avec advection + saturation globale * Transition lorsque v augmente > trous spectro temporels > idem pour Ginzburg Landau avec couplage local ou global Bielawski, Szwaj, Bruni, Garzella, Orlandi, Couprie, PRL 95, 3481 (25) For other issues (FEL control), see: Bielawski, Bruni, Garzella, Orlandi, Couprie, PRE, 69, R4552 (24)

34 Conclusion * Laser à électrons libres = système avec advection + saturation globale * Transition lorsque v augmente > trous spectro temporels > idem pour Ginzburg Landau avec couplage local ou global couplage global Distinction entre: Domaines convectif/absolu Structures entretenues par le bruit Structures bruyantes poster RNL Bielawski, Szwaj, Bruni, Garzella, Orlandi, Couprie, PRL 95, 3481 (25) For other issues (FEL control), see: Bielawski, Bruni, Garzella, Orlandi, Couprie, PRE, 69, R4552 (24)

35 &!#"%$ )( * + ) $' ' 1 (a) (d) z (b) (e) k (c) (f) k -.15 T 1 T 1