Electron beam proper.es and FEL Lecture I - PowerPoint PPT Presentation

¡Electron ¡beam ¡proper.es ¡and ¡FEL ¡– ¡Lecture ¡I ¡ Massimo.Ferrario@LNF.INFN.IT ¡ Fresnel diffraction pattern at the European XFEL (30 June 2017)

LCLS at SLAC 1.5-15 Å X-FEL based on last 1-km of existing SLAC linac

Short Wavelength SASE FEL FLASH XFEL PAL LCLS FERMI SACLA SwissFel SDUV

Genera.ons ¡of ¡Synchrotron ¡Light ¡Sources ¡ I. Bending magnets in HEP rings II. Dedicated Undulators III. Optimized Rings IV. Short Wavelength FEL V. Compact Sources

A Free Electron Laser is a device that converts a fraction of the electron kinetic energy into coherent radiation via a collective instability in a long undulator 12.4 1.24 0.124 λ (nm) SPARX & ) 2 γ 2 1 + K 2 λ rad ≈ λ u 2 + γ 2 ϑ 2 ( + ' * (Tunability - Harmonics)

Electron source and acceleration

Magnetic bunch compressor (< 1 ps)

Long undulators chain

Beam separation

Experimental hall (Single Protein Imaging) http://lcls.slac.stanford.edu/AnimationViewLCLS.aspx

Transverse electron motion in an Undulator: v x c = β ⊥ = K ( ) γ cos k u z ' * 1 − 1 2 ≈ 1 − 1 1 β 2 − β ⊥ 2 = 2 β // = γ 2 − β ⊥ γ 2 + β ⊥ ) , 2 ( + % ( 2 γ 2 1 + K 2 1 // = 1 − β ' * 2 & )

Undulator Radiation 2 γ 2 1 + K 2 % ( 1 // = 1 − β ' * 2 & ) θ = 1 γ x = K ( ) ! γ cos k u z The ¡electron ¡trajectory ¡is ¡determined ¡ The ¡electron ¡trajectory ¡ by ¡ the ¡ undulator ¡ field ¡ and ¡ the ¡ K ≤ 1 is ¡inside ¡ ¡the ¡radia.on ¡ electron ¡energy ¡ cone ¡if: ¡

Relativistic Mirrors ' = λ u λ u ' ' λ rad = λ u γ // Counter propagating pseudo-radiation Thompson back-scattered radiation in the mirror moving frame ( ) Doppler effect in the laboratory frame ( ) ≈ λ u 1 − β rad 1 − β cos ϑ // cos ϑ λ rad = γ $ λ cos ϑ ≈ 1 − ϑ 2 % ( 2 γ 2 1 + K 2 1 // = 1 − β ' * 2 & ) 2 & ) 2 γ 2 1 + K 2 λ rad ≈ λ u 2 + γ 2 ϑ 2 ( + ' * Tunability & Red Shift

Resonant Wavelength Sensitivity to beam parameters " % 2 γ 2 1 + K 2 λ rad ≈ λ u 2 + γ 2 ϑ 2 $ ' # & Undulator tolerances γ + 2 K 2 K + γ 2 ϑ 2 Δ K Δ λ rad = − 2 Δ γ 1 + K 2 1 + K 2 λ rad Energy spread Beam Emittance 2 = γ 2 ε 2 2 2 = ε n γ 2 ϑ 2 = γ 2 σ ! x 2 σ x σ x

e 2 6 πε o c 3 γ 4 ˙ 2 P v Peak power of one accelerated charge: 1 = ⊥ Different electrons radiate indepedently hence the total power depends linearly on the number N e of electrons per bunch: e 2 6 πε o c 3 γ 4 ˙ 2 P T = N e v Incoherent Spontaneous Radiation Power: ⊥ 2 e 2 T = N e 6 πε o c 3 γ 4 ˙ 2 Coherent Stimulated Radiation Power: P v ⊥ Bunching on the scale of the wavelength:

Spontaneous Emission ==> Random phases N

Coherent Light ==> Stimulated Emission N 2

Radiation Simulator – T. Shintake, @ http://www-xfel.spring8.or.jp/Index.htm

N u = 5 { { { { { L pulse = N u λ rad λ rad ∝ λ u 2 γ 2

Letargy Exponential Growth Saturation Spontaneous Emission Stimulated emission Absorption Low Gain High Gain No Gain Enhanced Bunching Debunching Slow Bunching

Free Electron Laser 1D Self Consistent Model Consider“seeding”by an external light source with wavelength λ r The light wave is co-propagating with the relativistic electron beam   d γ dt = − e β = − e E ⋅ mc E ⊥ β ⊥ mc Energy exchange occurs only if there is transverse motion

Newton Lorentz Equations Problem: electrons are slower than light Question: can there be a continuous energy transfer from electron beam to E,B J ⊥ light wave? Answer: We need a Self Consistent Model Maxwell Equations (R. Bonifacio, C.Pellegrini, L.Narducci, Opt. Comm., 50, 373 (1984))

After one wiggler period the electron sees the radiation with the same phase if the flight time delay is exactly one radiation period: Δ t = t e − t ph = T rad * - 2 γ 2 1 + K 2 Δ t = λ u − λ u c = λ rad & λ rad = 1 − β & λ rad ≈ λ u β // ≈ 1 // & → λ u & & → , / c β // c 2 β + . // % ( 1 + K 2 λ u γ res ≈ ' * 2 λ rad 2 & ) The relative slippage of the radiation envelope through the electron beam can be neglected, provided that l b >>N u λ r (Steady State Regime)

( ) = E o cos k l z − ω l t + ψ o ( ) E x z,t Plane wave with constant amplitude , co-propagating with the electron c = 2 π k l = ω l beam: λ l d γ m e c E x β x = − eE o K e β x = K ( ) [ ] γ cos k u z cos ψ − cos ψ dt = − 2 γ m e c Ponderomotive potential Fast oscillating phase (we can neglect it) ( ) = k l + k u ( ) z − ω l t + ψ o Ponderomotive phase: ψ z,t In a resonant and randomly phased electron beam, nearly one half of the electrons absorbs energy and one half loses energy, with no net energy exchange.

Off-resonance electrons motion 2 γ 2 1 + K 2 " % β z = 1 − 1 $ ' 2 # & k u << k l k l c # & # & ( 1 + K 2 d ψ 1 2 − 1 ( ) v z − k l c ≈ dt = k l + k u % % ( γ 2 2 2 γ r $ ' $ ' " % 1 + K 2 λ u γ r ≈ $ ' 2 λ rad 2 # & Electrons with energies above the resonant energy d ψ dt ≈ 2 k u c γ − γ r move faster, while energies below will make the γ r electrons fall back

If the undulator is sufficiently long the energy modulation becomes a phase modulation: the electrons self-bunch on the scale of a radiation wavelength. Ponderomotive Potential t=0 t=0 ………………………… Optical t>0 potential λ The particles bunch around a phase where there is weak coupling ψ r with the radiation: # d ψ dt > 0 for γ > γ r % % d ψ dt ≈ 2 k u c γ − γ r $ d ψ γ r % dt < 0 for γ < γ r % &

The Bunching parameter: Ponderomotive Potential t=0 t=0 ………………………… Optical t>0 potential λ b ≈ 0 Spontaneous emission N ) = 1 Bunching e − i ψ j = e − i ψ j ∑ ( b z,t Parameter: N Stimulated emission b → 1 j = 1

Motion in the potential well: the electron pendulum equations For particles with off resonance energy, the ponderomotive phase is no longer constant # d ψ dt ≈ 2 k u c γ − γ r γ − γ r = 2 k u c η % << 1 η = % γ r γ r $ d η dt = 1 d γ dt = − eE o K % 2 m e c cos ψ % 2 γ r γ r & Two coupled first order differential equations

Combining the two coupled first order differential equations: & d ψ d 2 ψ d 2 ψ dt 2 = − eE o Kk u dt 2 + Ω 2 cos ψ = 0 dt = 2k u c η cos ψ ( ( 2 m e γ r ' d η eE o K ( cos ψ dt = − Ω 2 = eE o Kk u 2 m e c ( 2 γ r ) 2 m e γ r η η ψ ψ & ) eEK 2 cos ψ − ψ r Separatrix η sep = ± ( + k u m e c 2 γ r 2 ' *

Letargy Exponential Growth Saturation & Stimulated emission Absorption d ψ dt = 2k u c η ( ( High Gain No Gain ' d η eE o K ( dt = − cos ψ Enhanced Bunching Debunching 2 m e c ( 2 γ r )

High gain FEL regime (1D model ) $ ' 2 + ∂ 2 ∂ 2 ∂ z 2 − 1 ∂ j x )  ( ) = µ o E x z , t ∇ ⊥ Test solution & c 2 ∂ t 2 ∂ t % ( ( ) e i ϕ ) = E o z ( ( ) ˜ ) = ˜ ( ) e i k l z − ω l t e i k l z − ω l t ( E x z,t E x z 2 ∂ j x ) = µ o ( 2ik l ˜ " ( ) + ˜ " ] e i k l z − ω l t [ ( ) E x z E x z " ∂ t Slowly Varying Envelope Approximation (SVEA): the amplitude variation within one undulator period is very small ˜ ˜ " d˜ ( ) ( ) ( ) E x z E x z E x z = − i µ o ∂ j x ( ) ˜ " ⇒ ˜ " ∂ t e − i k l z − ω l t ( ) << ( ) << E x z E x z " λ u λ u dz 2k l To be consistent with SVEA we should average also the source term over a time in which could be considered constant ˜ T ≈ n λ l c ( ) E x z t + T ˜ 1 j ∂ ( ) dt 2ik l ˜ " e − i k l z − ω l t ∫ x E x = µ o T ∂ t t

∂  t + T t + T 1 j x ) dt = − i ω l ) dt  ( ( Integration by parts − i k l z − ω l t − i k l z − ω l t ∂ 2  ∫ ∫ e j x e j x T ∂ t T with: ∂ t 2 ≈ 0 t t Beam model N N x = e e ∑ ∑ ˜ ( ) = ( ) ( ) ( ) j v xj δ z − z j t v xj δ t − t j z S Sv z S: transverse beam area j = 1 j = 1 Exercise: verify there are not misprints (~mistakes): N t + T t + T 1 e ( ) dt ( ) dt ∫ x e − i k l z − ω l t ∫ ∑ ) e − i k l z − ω l t ˜ ( ( ) j v xj δ t − t j z = T Sv z T t t j = 1 N = e ( ) where : V = Sv z T − i k l z − ω l t j ∑ v xj e V j = 1 N = e Kc ( ) − i k l z − ω l t j ∑ ( ) e cos k u z using v xj = ..... V γ j j = 1 N N = eKc = eKc ( ) ( ) z − ω l t j − i k l + k u e − i ψ j ∑ ∑ e using γ j ≈ γ r V γ r V γ r j = 1 j = 1 = eKc = eKc n e e − i ψ j where n e = N N e − i ψ j V γ r γ r V