Free-Electron Lasers Sven Reiche, PSI - PowerPoint Presentation

1. Free-Electron LasersSven Reiche, PSI

2. I. Motivation

3. 1st Generation: Synchrotron radiation from bending magnets in high energy physics storage rings2nd Generation: Dedicated storage rings for synchrotron radiation3rd Generation: Dedicated storage rings with insertion devices (wigglers/undulators)4th Generation: Free-Electron LasersLight Sources

4. FEL as a Brilliant Light SourceHigh photon fluxSmall freq. bandwidthLow divergenceSmall source sizeSwissFEL

5. Tunable wavelength, down to 1 ÅngstroemPulse Length less then 100 fsHigh Peak Power above 1 GWFully Transverse CoherenceTransform limited Pulses (longitudinal coherence)4th Generation Light SourcesXFELs fulfill all criteria except for the longitudinal coherence(but we are working on it  )

6. X-ray/VUV FEL Projects Around the WorldLCLSWiFelNGLSSACLAMaRIEShanghai LSFLASHEuro XFELSwissFELNLSArc en CielFERMISPARXPolFELPAL FEL

7. II. Very Basic Principles

8. Controlling the Wavelength – The IdeaDipole Radiation (Antenna)Dipole Radiation + Doppler ShiftWavefrontOscillating ElectronTrajectory(forward direction)For relativistic electrons the longitudinal velocity vz is close to c, resulting in very short wavelength (blue shift of photon energy)

9. … by injecting them into a period field of an wiggler magnet (also often called undulator).Forcing the Electrons to Wiggle…Wiggler module from the LCLS XFEL

10. Defined by a transverse magnetic field which switch polarity multiple times, defining the undulator period luOn Axis-Field (we want simple fields!!!):Wiggler FieldPlanar UndulatorHelical UndulatorNote that field is only valid on-axis. Maxwell Equation requires other field component off-axis.Valid solution for planar undulator. Impact negligible for following discussion

11. Planar UndulatorTop PartPolesMagnetsFlux

12. Helical Undulator (APPLE Type)

13. Undulator parameters put into one constant:Transverse MotionLorenz Force:Motion in Planar UndulatorDominant motion in z (~bzct)Dominant field in yLongitudinal Motion

14. Longitudinal wiggle has twice the period.Causes a figure “8” motion in the co-moving frame.The longitudinal position is effectively smeared outHas some effects on the following discussionIn the Co-moving frametrajectoryEffective positionIn a helical wiggle the longitudinal motion is constantzx

15. III. Undulator Radiation

16. Emission Direction: qqAccelerated particles are emitting radiationResonance Condition:Wavefront from Emission Point AR=cTFor small angles:

17. Lienard-Wichert Field for relativistic particles:Emitted Field within Undulator PeriodObserved time:Spiked when n and b coincide Compressed when n and b coincide TrajectoryFieldxtEtObservation in forward directionRich harmonic content but only fundamental is important for FEL

18. The wavelength can be controlled by Changing the electron beam energy,Varying the magnetic field (requires K significantly larger than 1)To reach 1 Å radiation, it requires an undulator period of 15 mm, a K-value of 1.2 and an energy of 5.8 GeV (g=11000)The FEL Resonant WavelengthThe Free-Electron Lasers are based on undulator radiation in the forward direction with the resonant wavelength:

19. IV. Interaction with Radiation Field

20. The wiggling electrons emit radiation. Some of it co-propagates along the undulator.The transverse oscillation allows the coupling of the Electrons, absorbing more photons than emitting, become faster and tend to group with electrons, which are emitting more photons than absorbing.Interaction with Co-Propagating FieldThe FEL exploits a collective process, which ends with an almost fully coherent emission at the resonant wavelength.

21. The transverse oscillation allows to couple with a co-propagating field The electron moves either with or against the field lineAfter half undulator period the radiation field has slipped half wavelength. Both, velocity and field, have changed sign and the direction of energy transfer remains.Co-Propagation of Electrons and FieldThe net energy change can be accumulated over many period.

22. Energy change:Step I : Energy Modulation=0Resonance ConditionPonderomotive Phase:transverse motionplane waveAveraging over an undulator period:Second exp-function drops out (<exp(iq-2ikuz)>=0)Longitudinal oscillation smears out position  reduced coupling fc

23. For a given wavelength l there is one energy gr, where the electron stays in phase with radiation field.Electrons with energies above the resonant energy, move faster (dq/dz > 0), while energies below will make the electrons fall back (dq/dz <0 ). For small energy deviation from the resonant energy, the change in phase is linear with Dg=(g-gr).Step II: Longitudinal Motionz > 0 < 0 = 0

24. Stable Fix Point: Dg=0, q+f=0Instable Fix Point: Dg=0, q+f=pOscillation gets faster with growing E-Field (Pendulum: shorter length L)Analogy to PendulumFEL EquationsPendulum EquationsFrequencyFrequency

25. Wavelength typically much smaller than bunch length.Electrons are spread out initially over all phases.Motion in Phasespaceq+fDgq+fDgq+fDgq+fDgElectrons are bunched on same phase after quarter rotation

26. MicrobunchingMicrobunching has periodicity of FEL wavelength. All electrons emit coherently.3D Simulation forFLASH FEL over 4 wavelengthsFrame moving with electron beam through 15 m undulatorWiggle motion is too small to see. The ‘breathing’ comes from focusing to keep beam small.Slice of electron bunch (4 wavelengths)Transverse position

27. V. Field Emission

28. The electrons are spread out over the bunch length with its longitudinal position dzj. The position adds a phase fj=kdzj to the emission of the photon.The total signal is: Coherent EmissionElectrons spread over wavelength:Phasor sum = random walk in 2DReImReImElectrons bunched within wavelength:Phasor sum = Add up in same directionPower ~ |E|2 -> Possible Enhancement: N

29. Because the degree of coherence grows the field does not stay constant (Pendulum is only valid for adiabatic changes)The change in the radiation field is given by Maxwell equation:Complete Picture: Evolving Radiation FieldSlow Varying Envelope Approximation:Period-averaged motion:

30. The Levels of the FEL ModelDiffractionEvolution along Bunch (Slippage)Evolution along UndulatorSource Term1D Model: Ignore Diffraction EffectsSteady-State Model: Ignore Time DependenceTo start-easy: 1D, steady-state FEL Model

31. Scaling of 1D FEL EquationsD=Detuningh=DeviationChoose r to eliminate constants

32. VI. Easy Solution(but limited)

33. 1D FEL EquationsFieldPositionEnergyDerivative of field eq.Derivative again.Negligible with no energy spreadAnsatz:

34. Electron beam in resonance (no detuning D=0)Only initial field present (A=A0, A’=0, A’’=0)SolutionsReImSolutions:oscillatinggrowingdecayingGeneral Solution:Initial Conditions:

35. For a long distance the growing mode dominates:Characteristic power gain length (1D):In start-up, other modes are not negligible. Growth is inhibited. This “lethargy” is about two gain lengthLinear Regime and Start-up Regime

36. The Generic Amplification ProcessExponential AmplificationSaturation (max. bunching)Start-up LethargyLethargy at start-up is the response time till some bunching has formed.Beyond saturation there is a continuous exchange of energy between electron beam and radiation beam.FEL Gain|A| ~1

37. Energy conservation and efficiency:Efficiency and BandwidthFor a given wavelength the beam can be detuned by Dg=rg and still amplify the signal.For a given energy the wavelength can be detuned by Dw=2rw and still be amplified. Bandwidth:

38. VII. SASE FELs

39. Previous solution had an initial input field.However the FEL can be started also with different initial condition:How to start the FEL differently…Initial bunching in the electron position, resulting in some emission of radiation. Due to the discrete nature of electrons this term is always present (shot noise)An energy modulation on the period of the wavelength will yield a non-vanishing term. This energy modulation will turn into a bunching over some drift length.This term is normally negligible

40. SASE FEL (Self-Amplified Spontaneous Emission)FEL Amplifier (starts with an input signal) FEL ModesSeedElectronsUndulatorsOutputElectronsUndulatorsOutputSeed must be above shot noise powerSpontaneous emission (shot nosie) is a broadband seed of “white noise”

41. Typical Growth of SASE PulseSimulation for FLASH FEL

42. FEL starts with the broadband signal of spontaneous radiation (almost a white noise signal)Within the FEL bandwidth Dw=2rw the noise is amplifiedSpikes in spectrum and time profile.SASE FELsPulse at saturationtctbSpectrum at saturation1/tb1/tc=2rwSwissFEL: Simulation for 1 Angstrom radiationCoorperation Length:

43. VIII. The Importance of r

44. Typical values r = 10-2 – 10-4ScalingBeam Requirements:The Importance of the FEL Parameter rGain lengthEfficiencyBandwidthSASE Spike LengthEnergy SpreadBeam SizeEmittance

45. Only electrons within the FEL bandwidth can contribute to FEL gain.FEL process is a quarter rotation in the separatrix of the FEL. If separatrix is filled homogeneously, no bunching and thus coherent emission can be achieved.Electron Beam Requirements: Energy SpreadSmall spreadLarge spreadStrong BunchingWeak BunchingEnergy Spread Constraint:

46. Optimizing the FocusingDecreasing the b-function (increase focusing), increases the FEL parameter r.Stronger focusing:Larger kinetic energy of betatron oscillationLess kinetic energy for longitudinal motionSmearing out of growing bunching3D Optimization for SwissFELFrom 1D Theory:

47. “Soft” constraint for emittance to be smaller than photon emittance for all electrons to contribute on the emission processQuick Overview of 3D TheoryNew free Parameter: Diffraction ParameterAssuming electron size as radiation source size:Rayleigh LengthScaling length of the FEL (2kur)xpxPhoton EmittanceDiffraction LimitedPhoton EmittanceElectron

48. FEL utilized the strong coherent emission in the collective instability with the tuning ability of the wavelength.Instability can only occur with a beam with low energy spread and emittance.SummaryInduced energy modulationIncreasing density modulationEnhanced emissionRun-away process(collective instability)FEL Process

49. IX. Some Mathematical Hardcore Stuff(Exact Solution-1D)

50. 2N Ordinary and 1 partial differential equation.Electrons are represented by the density function f(q,h,z), fulfilling Liouville’s theorem:Fourier series expansion:Equation with terms proportional to eiq:How to solve the Problem?Initial distributionCurrent modulation(Vlasov Equation)

51. Field evolution is given by density function:Including the field equation…The formal solution for f1 is:Problem reduce to integro-differential equation:

52. Laplace transformation: (Fourier transformation does not work) :FEL Equation:Solving the FEL EquationsIntegration over z by parts on both sides:Formal Solution:

53. Laplace Transformation:Dispersion EquationLaplace Tran.Completed PathDispersion EquationxxxSingularitiesRe(z)Im(z)Inv. Laplace Tran.Inverse Laplace Transformation:

54. No detuning (D=0) and no energy spread (f0=d(h))3 Solutions to:FEL ModeThe 1D SolutionOscillatingGrowingDecayingGain Length (e-folding length):

55. X. Even More Hardcore(3D Theory)

56. Scaling of 3D Field Equation1D Model3D ModelScaling:

57. Assuming a round, rigid beam with no change in transverse momenta and position.Add a normalized distribution g(r) to describe the transverse beam profileVlasov EquationFormal Solution for Field Equation:3D Vlasov Equation same as in 1D

58. Left Hand Side:Right Hand Side:Laplace TransformationDispersion Function of 1D Model

59. Effective PotentialRemarks:The constant term is the particular solution and omitted in the following discussionThe solution of the solution needs to be well defined (L2-norm is proportional to radiation power):Solution needs to drop faster than r-1Solution has no singularities at the originSimple Model:Uniform transverse distribution with sharp edge at r=1Expansion into radial and azimuthal modes.Analogy to 2D Quantum Mechanicswith

60. Differential Equation:The Solution for FEL ModeOutside SolutionInside SolutionContinuation Condition at r=1Allows for different solution (eigenvalues) for m. Solutions are ordered by the index n

61. The FEL Eigenmodes RmnIn the limit B infinity (negligible diffraction) the eigenvalues for m are degenerated with pB=D1D Solution:

62. Growth rates for FEL eigenmodes (r,f-decomposition):FEL Eigenmodes + Mode CompetionIncreased gain length due to strong diffractionMode competition and reduced coherenceOptimum growth rate of 1D modelOptimum for SASE:Only one dominating mode at saturationTransverse coherence1D Limit