Physics of undulators J. Chavanne Insertion Device Group ASD -Introductory remarks -Basis of undulator radiation -Spectral properties -Source size -Present technology -Summary J.Chavanne 1 Books Undulators, wigglers The physics of The science and Technology And their applications Synchrotron radiation of Undulators and Wigglers H. Onuki, P.Elleaume A. Hofman J. A Clarke Modern theory of undulator Very accessible Very clear approach radiations Several ESRF authors J.Chavanne 2 Few preliminary remarks Many software simulations are used for undulator radiations: All simulations done using SRW Synchrotron Radiation Workshop ( O. Chubar, P.Elleaume) - wavefront propagation - near & far field - will evolve in near future B2E (B to E) also ESRF tool - field measurement analysis - undulator spectrum with field errors Unfortunately very few topics in undulator physics will be presented J.Chavanne 3 Electrons in the framework of relativity Any particle with non zero mass cannot exceed speed of light 2 Electron energy: E = ! mc = ! E0 E0 is the electron energy at rest =0.511 MeV 1 ! is the relativistic Lorentz factor also defined as ! = Speed of electron v2 1" c2 2 1 ESRF: E= 6.04 GeV so ! = E / E = 11820 v / c = !e = 1"1/# $ 1" 0 2# 2 Electron Energy E v/c Mass: m=9.10938 e-31 Kg 1 MeV 0.869 Charge: e=-1.60218 e-19 C 100 MeV 0.9999869 Speed of light in vacuum: c = 299792.45 m/s 1 GeV 0.999999869 6 GeV 0.9999999964 J.Chavanne 4 Simple periodic emitter Period = !0 e 1 2 3 ! v Points with wavefront emission Somewhere between point 1 and 2 And between point 2 and 3 J.Chavanne 5 Simple approach The question: what is the relation between λ0 and λ ? !0 e ! ? J.Chavanne 6 On axis observation Point i the electron has reached point i+1 ! 2 1 v / c = ! = 1"1/# $ 1" !0 e 2# 2 " Time taken by the electron to move from point i to point i+1: !t = 0 #ec ! ! During this time the wavefront created at point i has expanded by r = c 0 = 0 "ec "e Therefore we have: !0 !0 ! = # !0 $ 2 "e 2% Example: !0 = 28mm we get ! = 1Å with the ESRF energy (γ =11820) Remark: in the backward direction ! " 2!0 J.Chavanne 7 Off axis observation !0 1 θ !(") = $ !0 cos" % !0 (1$ cos" + 2 ) #e 2& ! 2 For small angles : cos! " 1# 2 !0 ! !(") # 0 (1+$ 2" 2 ) 2$ 2 1.0 !(0) 1 γ=11820 @ ESRF Interesting to look at = 2 2 0.8 !(") 1+# " ≈1/γ 0.6 c !(0) E (") Photon energy: E = h p 0.4 p = ! !(") Ep (0) 0.2 0.0 The radiated energy is concentrated in a narrow -2 -1 0 1 2 cone of typical angle 1/γ ! [mrad] J.Chavanne 8 Important remarks From our simple “periodic emitter ” we have seen : - Radiations at wavelength of ~ 1 Å can be produced with a spatial wavelength of few centimeters and few GeV electron beam - Emitted radiations are highly collimated (~ 1/γ ) The angular dependence of emitted wavelength has a direct consequence on associated spectrum Off axis On axis radiation ! !(") # 0 (1+$ 2" 2 ) 2$ 2 Ph/s/0.1%bw Photon Energy Presence of “tails” at low energy side on harmonics with non zero angular acceptance J.Chavanne 9 Other approaches Lorentz transform + Doppler shift: Electron frame Observer frame Lorentz transform spatial period λ1=λ0/γ spatial period λ0 angle dependent Doppler shift emitted wavelength: λ1 !(") = !1# (1$ %e cos" ) ! 2 2 example: !(") # 0 (1+$ " ) 2$ 2 undulator with λ0=28 mm has λ1= 2.36 µm , 1.6 m long undulator has a length of 0.135 mm in electron frame J.Chavanne 10 Electron and observer times ! ! v(t ) c (t ) Electron moving with speed ! = "e ! Wave emitted at time t’ by electron received at time t by observer D(t!) t = t! + c ! D(t!) n(t!) dt ! ! = 1" n(t!)#e (t!) ! dt! ! ! v(t!) R(t!) ! For ultra-relativistic electron and small angles: r dt 2 1 2 2 o = 1" # cos$ = 1" 1"1/% cos$ & (1+% $ ) Relativistic compression of time: dt! e 2% 2 Observer time evolves several orders of magnitude slower than electron time Basis for “retarded potentials” or Lienard-Wiechert potentials J.Chavanne 11 Improving oscillator model How to get periodic emission from an electron ? ! ! ! ! Apply periodic force on electron: F = !e(E + v " B) B= 1 T 300 MV/m ~ 10 MV/m with V ! c conventional technology Best option z Planar field undulator λ0 x ! ! Vertical field B = B cos 2! s / " z 0 ( 0 ) s ! Electron trajectory ! B J.Chavanne 12 Undulator equation (1) ! dP ! ! ! ! ! ! = !e(v " B) P = ! mv B = B0 cos(2! s / "0 )z dt Assumptions: γ constant , βx= vx/c,<< 1 , βz= vz/c<<1 Angular motion e 2" s K 2" s e !x (s) = B0$0 sin( ) = sin( ) K = B0"0 = 0.9336B0[T ]"0[cm] 2"# mc $0 # $0 2! mc !z (s) = cst = 0 Deflection parameter Electron trajectory K"0 2# s 2# s x(s) = ! cos( ) = !x0 cos( ) 2#$ "0 "0 γ λ0[cm] B[T] K xo[µm] z(s) = cst = 0 11820 2 1 1.87 0.5 J.Chavanne 13 Undulator equation (2) We need to know the longitudinal motion βs of the electron in the undulator to bring more consistence to our initial “naïve” device: 2 2 2 1 K 2# s Since γ is constant so is ! e = !x + !s = 1" 2 (!x (s) = sin( )) # " $0 1 K 2 K 2 4% s !s (s) " 1# 2 # 2 + 2 cos( ) 2$ 4$ 4$ &0 2 ˆ 1 K Average longitudinal relative velocity: !s " 1# 2 # 2 2$ 4$ ! K 2 Angle dependent emitted wavelength: !(") # 0 (1+ +$ 2" 2 ) 2$ 2 2 γ λ0[cm] B[T] K λ(0) 11820 2 1 1.87 1.96 Å We have now a field dependent wavelength 11820 2 0.1 0.187 0.72 Å J.Chavanne 14 The electric field ! ! The electric field E ( r , t ) seen by an observer is the relevant quantity to determine ! ! D(t!) ! ! n(t!) ! ! n(t!) Has always B field “companion”: B(r,t) = " E(r,t) ! ! c E(r,t) ! ! ! v(t!) R(t!) ! r Moving charge along arbitrary motion: Electric field includes two terms o ! ! ! ! ! ! ! ! E(r,t) = E (n(t ),v(t ), D(t )) + E (n(t ),v(t ), D(t )) 1 ! ! ! 2 ! ! ! Needs to find t’(t) to ! ! evaluate E(r,t) Velocity field or Acceleration field Coulomb field Decays as 1/D Decays as 1/D2 ! Far field approximation: drop velocity field and n (t!) constant J.Chavanne 15 Frequency domain ! ! 1 $ ! ! E(r,! ) = E(r,t)ei!t dt %#$ 2" Electric field in time domain Electric field in frequency domain Complex quantity Wavefront propagation Phase ? Coherence Etc.. ! ! 2 ! " E(r,! ) N(r,! ) = "! Number of photons at ω J.Chavanne 16 Importance of deflection parameter K Peak angular deflection angle K !1 ! electric field ±1/! Frequency time Electric field angle K > 1 Only odd harmonics ±1/! time Frequency Electric field Off axis observation with angle θ odd + even harmonics angle ±1/! +" Frequency J.Chavanne 17 On axis angular spectral flux Spectral photon flux units: Watts/eV can be translated into photons/sec/relative bandwidth Ex: 1 phot/s/0.1%bw= 1.602e-16 W/eV Angular spectral flux: photon flux/unit solid angle Usual unit is phot/sec/0.1%/mrad2: Ideal on axis angular spectral flux with filament electron beam (zero emittance) 4.0x1018 ] Only odd harmonics on axis Undulator: 2 3.0 Period λ0= 22 mm Number of period N=90 2.0 K=1.79 1.0 [Phot/s/0.1%/mrad 0.0 0 20 40 60 80 keV Harmonic n =7 Relative bandwidth at harmonic n: ] 2 18 2.0x10 ~ 1/N ∆E/E=1/nN ~ N2 1.0 Radiated power: ~ N2/N=N proportional to N 0.0 [Phot/s/0.1%/mrad 41.75 42.00 keV J.Chavanne 18 Off axis radiation 15 2 4x10 3 2 Undulator: Period λ = 22 mm 1 0 Ph/s/0.1%bw/mm Number of period N=90 0 K=1.79 Filament electron beam 5 10 15 20 25 30 35keV (zero emittance) Photon Energy z γθ Vert. angle angle Vert. Hor. angle !" x n=2 n=3 n=5 n=6 2 !0 K 2 2 2 2 2 Wavelength of harmonic n ! (") = (1+ +# " ) ! = ! x +!z n 2n# 2 2 J.Chavanne 19 Angle integrated flux Undulator: Ideal filament electron beam Period λ0= 22 mm Number of period N=90 En! En K=1.79 Harmonic n=3 12 15 800x10 4x10 On axis flux density [Phot/s/0.1%/mm 10 µrad x 10 µrad "!n 20 µrad x 20 µrad 30 µrad x 30 µrad 600 50 µrad x 50 µrad 3 All angles on axis 400 !n 2 Ph/s/0.1%bw 200 1 2 0 0 ] 17.4 17.6 17.8 18.0keV Photon Energy En energy of on axis resonance 1 2hc" 2 0.95E 2[GeV ] En (!) = 2 = 2 "!n # 2"n En! = En (1" ) K K nN # (1+ +" 2! 2 ) # [cm](1+ +" 2! 2 ) 0 2 0 2 J.Chavanne 20 Angle integrated flux (remark) A unique specificity of ESRF: ∆s Segmented independent undulators with passive phasing capability ~ all in-air segments Undulator A For a fixed energy and collecting aperture Undulator gaps are optimized for maximum flux Undulator B One undulator # E! = E (1" ) 0 ! " ! 1 n n nN Two undulators # E! = E (1" ) n n 2nN ∆s depends on period 2.5 mm for λ0=18 mm The optimum gap depends on the length of undulator 5 mm for λ0=35 mm J.Chavanne 21 Radiated power Radiated power & power density can be an issue for ESRF beamlines Total power emitted by an Insertion device: (only a fraction is generally taken by a beamline) " P[kW ] = 1.266E 2[GeV ]I[A] (B2[T ]+B2[T ])ds ID with arbitrary field # x z !" 2 2 Planar sinusoidal field undulator P[kW ] = 0.633E [Gev] B0 [T ]I[A]L[m] B0: peak field On axis power density: Undulator length 2 4 dP / d![W / mrad ] = 10.84 B0[T ]E [Gev]I[A]N N: number of periods, K>1 Ex: ESRF 6 .04 Gev with I=0.2 A 2 Period[mm] L[m] N B0[T] P[kW] Dp/dΩ[kW/mrad ] 22 2 90 0.87 7 260 27 5 185 0.52 6.7 277 With ~ all ESRF IDs at minimum gap: the total radiated power is ~ 300 kW (0.2 A, 6.04 Gev) (to be compared to ~ 1 MW for all dipoles) J.Chavanne 22 Electron beam size ! = 4nm Geometrical size of ESRF electron beam (assuming Gaussian beam) x ! z = 3pm 0.10 High beta straight 0.00 middle -0.10 Vert.