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Introduction to Classical and Quantum High-Gain FEL Theory

Introduction to Classical and Quantum High-Gain FEL Theory. Rodolfo Bonifacio & Gordon Robb University of Strathclyde, Glasgow, Scotland. Outline Introductory concepts Classical FEL Model Classical SASE Quantum FEL Model Quantum SASE regime : Harmonics

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Introduction to Classical and Quantum High-Gain FEL Theory

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  1. Introduction to Classical and Quantum High-Gain FEL Theory Rodolfo Bonifacio & Gordon Robb University of Strathclyde, Glasgow, Scotland.

  2. Outline • Introductory concepts • Classical FEL Model • Classical SASE • Quantum FEL Model • Quantum SASE regime : Harmonics • Coherent sub-Angstrom (g-ray) source • Experimental evidence of QFEL in a BEC

  3. S S S N N N N N S S 1. Introduction The Free Electron Laser (FEL) consists of a relativistic beam of electrons (v≈c) moving through a spatially periodic magnetic field (wiggler). Relativistic electron beam EM radiation llw /g2 << lw Magnetostatic “wiggler” field (wavelength lw) • Principal attraction of the FEL is tunability : • - FELs currently produce coherent light from microwaves • through visible to UV • X-ray production via Self- Amplified Spontaneous Emission (SASE) (LCLS – 1.5Å)

  4. Exponential growth of the emitted radiation and bunching:

  5. Ingredients of a SASE-FEL : • High-gain (single pass) (no mirrors) • Propagation/slippage of radiation with respect to electrons • Startup from electron shot noise (no seed field) • Consequently, structure of talk is : • Recap of high-gain FEL theory (classical & quantum) • Propagation effects (slippage & superradiance) • SASE (classical & quantum)

  6. Some references relevant to this talk HIGH-GAIN AND SASE FEL with “UNIVERSAL SCALING” Classical Theory (1) R.B, C. Pellegrini and L. Narducci, Opt. Commun. 50, 373 (1984). (2) R.B, B.W. McNeil, and P. Pierini PRA 40, 4467 (1989) (3) R.B, L. De Salvo, P.Pierini, N.Piovella, C. Pellegrini, PRL 73, 70 (1994). (4, 5) R.B. et al,Physics of High Gain FEL and Superradiance, La Rivista del Nuovo Cimento vol. 13. n. 9 (1990) e vol. 15 n.11 (1992) QUANTUM THEORY • (6) R. B., N. Piovella, G.R.M.Robb, and M.M.Cola, Europhysics Letters, 69, (2005) 55 and quant-ph/0407112. • R.B., N. Piovella, G.R.M. Robb & A. Schiavi, PRST-AB 9, 090701 (2006) • R. B., N. Piovella, G.R.M.Robb, and M.M.Cola, Optics Commun. 252, 381 (2005)

  7. 2. The High-Gain FEL We consider a relativistic electron beam moving in both a magnetostatic wiggler field and an electromagnetic wave. EM wave electron beam wiggler Wiggler field (helical) : Radiation field : (circularly polarised plane wave) where

  8. Problem : What is ? (details in refs. 4,5) 2.1 Classical Electron Dynamics We want to know the beam-radiation energy exchange : Energy of the electrons is Rate of electron energy change is This must be equal to work done by EM wave on electrons i.e. The canonical momentum is a conserved quantity. i.e. Consequently :

  9. where (wiggler + EM field) Now EM field << wiggler no time dependence so the only term of interest is so (1)

  10. Whether electron gains or loses energy depends on the value of the phase variable The EM wave (w,k) and the wiggler “wave” (0,kw) interfere to produce a “ponderomotive wave” with a phase velocity From the definition of q, it can be shown that : (2) where is the resonant energy

  11. FEL resonance condition (magnetostatic wiggler ) Let: Example : for l=1A, lw=1cm, E~5GeV (electromagnetic wiggler ) Example : for l=1A, lpump=1mm, E=35MeV

  12. (details in refs. 4,5) 2.2 Field Dynamics Radiation field : (circularly polarised plane wave) The radiation field evolution is determined by Maxwell’s wave equation The (transverse) current density is due to the motion of the (point-like) electrons in the wiggler magnet. where Apply the SVEA : and average on scale of lr to give where (3)

  13. ‘Classical’ universally scaled equations A is the normalised S.V.E. A. of FEL rad. – self consistent Ref 1. 13

  14. We will now use these equations to investigate the high-gain regime. We solve the equations with initial conditions (uniform distribution of phases) (cold, resonant beam) (small input field) and observe how the EM field and electrons evolve.

  15. Strong amplification of field is closely linked to phase bunching of electrons. Bunched electrons mean that the emitted radiation is coherent. For randomly spaced electrons : intensity  N For perfectly bunched electrons : intensity ~ N2 z=0 |b|<<1 Ponderomotive potential z>0 |b|~1 It can be shown that at saturation in classical case, intensity  N4/3 As radiated intensity scales > N, this indicates collective behaviour Exponential amplification in high-gain FEL is an example of a collective instability.

  16. In FEL and CARL particles self-organize to form compact bunches ~l which radiate coherently. Collective Recoil Lasing = Optical gain + bunching bunching factor b (0<|b|<1):

  17. FEL instability animation Steady State Animation shows evolution of electron/atom positions in the dynamic pendulum potential together with the probe field intensity.

  18. Classical high-gain FEL

  19. Bonifacio, Casagrande & Casati, Optics Comm. 40 (1982) A fully Hamiltonian model of the classical FEL Steady State Defining then Defining then the FEL equations can be rewritten as where Equilibrium occurs when so BUT so i.e. GAIN

  20. |A|2 |b| z z z σp The scaled radiation power|A|2, electron bunching|b| and the energy spread σp for the classical high-gain FEL amplifier.

  21. Classical chaos in the FEL If we calculate the distance, d (z), between different trajectories in the 2-dimensional phase-space so where In the exponential regime :

  22. Linear Theory (classical) Ref(1) Linear theory runaway solution See figure (a) Maximum gain at d=0 Quantum theory: different results (see later)

  23. For long beams (L >> Lc) Seeded Superradiant Instability Ref(2): Including propagation CLASSICAL REGIME, LONG PULSE L = 30LC , resonant (d=0)

  24. CLASSICAL SASE • Ingredients of Self Amplified Spontaneous Emission (SASE) • Start up from noise • Propagation effects (slippage) • SR instability •  • The electron bunch behaves as if each cooperation • length would radiate independently a SRspike • which is amplified propagating on the other electrons • without saturating. Spiky time structure and spectrum. SASE is the basic method for producing coherent X-ray radiation in a FEL

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  26. DRAWBACKS OF ‘CLASSICAL’ SASE Time profile has many random spikes Broad and noisy spectrum at short wavelengths (x-ray FELs) simulations from DESY for the SASE experiment (λ ~ 1 A) 26 26

  27. what is QFEL?QFEL is a novel macroscopic quantum coherent effect:collective Compton backscattering of a high-power laser wiggler by a low-energy electron beam.The QFEL linewidth can be four orders of magnitude smaller than that of the classical SASE FEL Phys. Rev. ST Accel. Beams 9 (2006) 090701 Nucl. Instr. And Meth. A 593 (2008) 69 27 27

  28. Why QUANTUM FEL theory? In classical theory e-momentum recoil DP continuous variable QUANTUM THEORY WRONG: if one electron emits n photons QUANTUM FEL parameter: If CLASSICAL LIMIT If STRONG QUANTUM EFFECTS 28

  29. why QFEL requires a LASER WIGGLER? and for a laser wiggler to lase atlr=0.1 A: MAGNETIC WIGGLER: lW ~ 1cm, E ~10 GeV r~ 10-6 ,LW ~ 1Km LASER WIGGLER lL ~ 1 mm, E ~100 MeV r~ 10-4 ,LW ~ 1 mm 29 29

  30. Conceptual design of a QFEL Compton back-scattering (COLLECTIVE) lr lL If g  200 ( E  100 MeV)  lr  0.3 Å ! 30

  31. QUANTUM FEL MODEL Procedure : Describe N particle system as a Quantum Mechanical ensemble Write a Schrödinger-like equation for macroscopic wavefunction: 31 31

  32. R.Bonifacio, N.Piovella, G.Robb, A. Schiavi, PRST-AB (2006) 1D QUANTUM FEL MODEL : normalized FEL amplitude 32

  33. Madelung Quantum Fluid Description of QFEL* where *R. Bonifacio, N. Piovella, G. R. M. Robb, and A. Serbeto,  Phys. Rev. A 79, 015801 (2009) Let and See E. Madelung, Z. Phys 40, 322 (1927) Classical limit : no free parameters

  34. Wigner approach for 1D QUANTUM MODEL Introducing the Wigner function : Using the equation for we obtain a finite-difference equation for

  35. for r>>1: The Wigner equation becomes a Vlasov equation describing the evolution of a classical particle ensemble The classical model is valid when Quantum regime for

  36. Quantum Dynamics is momentum eigenstate corresponding to eigenvalue Only discrete changes of momentum are possible : pz= n (k) , n=0,±1,.. n=1 pz n=0 n=-1 probability to find a particle with p=n(ħk) 36

  37. steady-state evolution: classical limit is recovered for many momentum states occupied, both with n>0 and n<0 37

  38. Quantum bunching where : relative phase Momentum wave interference Maximum interference: Maximum bunching when 2-momentum eigenstates are equally populated with fixed relative phase 38

  39. Bunching and density grating QUANTUM REGIME r<1 CLASSICAL REGIME r>>1 39

  40. The physics of the Quantum FEL Momentum-energy levels: (pz=nħk, Enpz2 n2) (harmonics) Frequencies equally spaced by with width Increasing the lines overlap for CLASSICAL REGIME: many momentum level transitions →many spikes QUANTUM REGIME: a single momentum level transition →single spike 40

  41. Quantum Linear Theory Quantum regime for r<1 Classical limit max at width

  42. discrete frequencies as in a cavity max for = Continuous limit 42 42

  43. QUANTUM REGIME: CLASSICAL REGIME: momentum distribution for SASE Classical regime: both n<0 and n>0 occupied Quantum regime: sequential SR decay, only n<0 43 43

  44. SASE Quantum purification R.Bonifacio, N.Piovella, G.Robb, NIMA(2005) quantum regime classical regime 44 44

  45. 45 45

  46. LINEWIDTH OF THE SPIKE IN THE QUANTUM REGIME QUANTUM SINGLE SPIKE CLASSICAL ENVELOPE 46 46

  47. Energy spread : Condition to neglect diffraction : QFEL requirements Not necessary with plasma guiding (D. Jaroszynski collaboration) (thermal) Emittance: Rosenzweig et al, NIM A 593, 39 (2008) 47

  48. Harmonics Production Possible frequencies One photon recoil Larger momentum level separation quantum effects easier Extend Q.F. Model to harmonics [G Robb NIMA A 593, 87 (2008)] Results (a0 >1) Distance between gain lines: Gain bandwidth of each line: . Separated quantum lines if i.e. Possible classical behaviour for fundamental BUT quantum for harmonics 48

  49. 3rd harmonic 5th harmonic Fundamental 49 0.1A 0.06A e.g. 0.3A

  50. Main limitations in classical regime : 1. 2. 3. 4. Quantum FEL : as above with Quantum regime easier in the sub-A region and

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