Fundamentals of Accelerators - 2012 Lecture - Day 8
William A. Barletta Director, US Particle Accelerator School Dept. of Physics, MIT Economics Faculty, University of Ljubljana
US Particle Accelerator School What do we mean by radiation?
Energy is transmitted by the electromagnetic field to infinity Applies in all inertial frames Carried by an electromagnetic wave
Source of the energy Motion of charges
US Particle Accelerator School Schematic of electric field
From: T. Shintake, New Real-time Simulation Technique for Synchrotron and Undulator Radiations, Proc. LINAC 2002, Gyeongju, Korea US Particle Accelerator School Static charge
US Particle Accelerator School Particle moving in a straight line with constant velocity
E field
Q
B field
US Particle Accelerator School Consider the fields from an electron with abrupt accelerations At r = ct, ∃ a transition region from one field to the other. At large r, the field in this layer becomes the radiation field.
US Particle Accelerator School Particle moving in a circle at constant speed
dQ = q dl
Field energy flows to infinity
US Particle Accelerator School Remember that fields add, we can compute radiation from a charge twice as long
dQ = 2q dl
The wavelength of the radiation doubles
US Particle Accelerator School All these radiate
Not quantitatively correct because E is a vector; But we can see that the peak field hits the observer twice as often
US Particle Accelerator School Current loop: No radiation
Field is static
B field
US Particle Accelerator School QED approach: Why do particles radiate when accelerated?
Charged particles in free space are “surrounded” by virtual photons Appear & disappear & travel with the particles. ! e! ! Acceleration separates the charge from the photons & “kicks” photons onto the “mass shell”
Lighter particles have less inertia & radiate photons more efficiently
In the field of the dipoles in a synchrotron, charged particles move on a curved trajectory. Transverse acceleration generates the synchrotron radiation Electrons radiate ~αγ photons per radian of turning
US Particle Accelerator School Longitudinal vs. transverse acceleration
2 2 q $ dp|| ' P|| = 2 3 & ) 2 2 6"# m c % dt ( q 2& dp" ) 0 0 P" = 2 3 % ( + 6#$0m0c ' dt * negligible!
4 c 2 (%&) ! P" = q 2 ' = curvature radius ! 6#$0 '
Radiated power for transverse acceleration increases dramatically with energy
! Limits the maximum energy obtainable with a storage ring US Particle Accelerator School Energy lost per turn by electrons
dU 2c r E 4 = "P = " e $ U = P dt energy lost per turn dt SR 2 3 2 0 % SR 3(m0c ) # finite # For relativistic electrons: 1 2r E 4 ds ds U P ds e 0 0 = # SR = 2 3 # 2 ! s = #ct " ct ! dt = c c finite " 3(m0c ) finite " "
For dipole magnets with constant radius r (iso-magnetic case):
! 4" r E 4 e2 % 4 U = e 0 = 0 2 3 3 3(m0c ) # $o # The average radiated power is given by: U 4" c r E 4 P = 0 = e 0 where L $ ring circumference SR! T 2 3 L 0 3(m0c ) # US Particle Accelerator School
! Energy loss to synchrotron radiation (practical units)
e2" 4 E(GeV)4 Uo,electron (keV ) = = 88.46 Energy Loss per turn (per particle) 3#0$ $(m)
e2" 4 E(TeV )4 Uo,proton (keV ) = = 6.03 3#0$ $(m)
! Power radiated by a beam of average ! e" 4 E(GeV)4 I(A) P (kW ) = I = 88.46 current Ib: to be restored by RF system electron b 3#0$ $(m) I T b ! rev 4 4 Ntot = e" E(TeV ) I(A) e Pproton (kW ) = Ib = 6.03 3#0$ $(m) !
Power radiated by a beam of 4 4 average current I in a dipole of ! e" L(m)I(A)E(GeV) b Pe (kW ) = 2 LIb =14.08 2 length L (energy loss per second) 6#$0% %(m)
US Particle Accelerator School ! Frequency spectrum
Radiation is emitted in a cone of angle 1/γ Therefore the radiation that sweeps the observer is emitted by the particle during the retarded time period
"t # $ ret %c Assume that γ and ρ do not change appreciably during Δt.
At the observer !
dtobs 1 "tobs = "tret = 2 "tret dtret #
Therefore the observer sees Δω ~ 1/ Δtobs
! c "# ~ % 3 $
US Particle Accelerator School
! Critical frequency and critical angle
3 2 2 2 2 d I e ( 2#& + 20 ' . 3 1 2 2 K 2 ( ) K 2 ( ) = 3 * 2 - ( + ' . ) 2 2 / 3 / + 2 2 1/ 3 / 5 d"d# 16$ %0c ) 3c' , 1 1+ ' . 4 Properties of the modified Bessel function ==> radiation intensity is negligible for x >> 1
! #$ 2 2 3 / 2 " = 1+ % & >>1 Higher frequencies 3c% 3 ( ) have smaller critical 3 c angle Critical frequency " = $ 3 c 2 # ! 3 % "rev$
1/ 3 1 & 'c # Critical angle ) c = $ ! ! ( % ' " For frequencies much larger than the critical frequency and angles much larger than the critical angle the synchrotron radiation emission is negligible
US Particle Accelerator School Integrate over all angles ==> Frequency distribution of radiation
The integrated spectral density up to the critical frequency contains half of the
total energy radiated, the peak occurs approximately at 0.3ωc where the critical photon energy is
3 hc 3 $ c = h#c = ! 2 "
50% 50% For electrons, the critical energy in practical units is
E[GeV ]3 # [keV ] = 2.218 = 0.665! E[GeV ]2 ! B[T ] c "[m]
US Particle Accelerator School Number of photons emitted
Since the energy lost per turn is
e2" 4 U ~ 0 # And average energy per photon is the
1 % 1 c ! " $ " = h c = h # 3 # 3 c 3 2 &
The average number of photons emitted per revolution is
! n" # 2$% fine"
US Particle Accelerator School ! Comparison of S.R. Characteristics
US Particle Accelerator School From: O. Grobner CERN-LHC/VAC VLHC Workshop Sept. 2008 Synchrotron radiation plays a major role in electron storage ring dynamics
• Charged particles radiate when accelerated • Transverse acceleration induces significant radiation (synchrotron radiation) while longitudinal acceleration generates negligible radiation (1/γ2).
U0 = PSR dt energy lost per turn 4 # dU 2c re E finite " = "PSR = " 3 2 dt 3 m c 2 # ( 0 ) 1 dU 1 d P E dt " D = # = [ $ SR ( 0) ] 2T0 dE E 2T0 dE re ! classical electron radius ! 0 trajectory curvature " , " damping in all planes ! " ! DX DY
" p p equilibrium momentum spread and emittances 0 ! ! X ,! Y
US Particle Accelerator School RF system restores energy loss
Particles change energy according to the phase of the field in the RF cavity
"E = eV (t) = eVo sin(!RF t) For the synchronous particle
"E = U 0 = eV0 sin(!s )
US Particle Accelerator School Energy loss + dispersion ==> Longitudinal (synchrotron) oscillations
Longitudinal dynamics are described by 1) ε, energy deviation, w.r.t the synchronous particle 2) τ, time delay w.r.t. the synchronous particle
qV0 $ c "'= [sin(#s + $%) & sin#s] and "'= # % L E s Linearized equations describe elliptical phase space trajectories ε ! e dV " ! " '= ! # '= $ c ! T0 dt Es
τ 2 !c eV& "s = angular synchrotron frequency T0 E0
US Particle Accelerator School Radiation damping of energy fluctuations
dU The derivative 0 (> 0) dE is responsible for the damping of the longitudinal oscillations
Combine the two equations for (ε, τ) in a single 2nd order differential equation 2 d ! 2 d! 2 & 4 # + +" ! = 0 + = Ae't /) s sin$ * 2 ' t +( ! 2 s $ s 2 ! dt # s dt % ) s "
2 ! eV& angular synchrotron frequency "t /! s "s = e T0 E0 1 1 dU = 0 longitudinal damping time ! s 2T0 dE
US Particle Accelerator School Damping times
The energy damping time ~ the time for beam to radiate its original energy
Typically 4" R$ Ti = 3 C# Ji E o
$5 $3 Where Je ≈ 2, Jx ≈ 1, Jy ≈1 and C" = 8.9 #10 meter $ GeV ! Note Σ Ji = 4 (partition theorem) !
US Particle Accelerator School Quantum Nature of Synchrotron Radiation
Synchrotron radiation induces damping in all planes. Collapse of beam to a single point is prevented by the quantum nature of synchrotron radiation Photons are randomly emitted in quanta of discrete energy Every time a photon is emitted the parent electron “jumps” in energy and angle Radiation perturbs excites oscillations in all the planes. Oscillations grow until reaching equilibrium balanced by radiation damping.
US Particle Accelerator School Energy fluctuations
Expected ΔEquantum comes from the deviation of
emitted in one damping time, τE
The mean energy of each quantum ~ εcrit
1/2 ==> σε = εcrit(nγ τE )
Note that nγ = Pγ/ εcrit and τE = Eo/ Pγ
US Particle Accelerator School Therefore, …
The quantum nature of synchrotron radiation emission generates energy fluctuations
1/ 2 2 "E E E Cq$ o $ # crit o # ~ E E o J% &curv E o & where Cq is the Compton wavelength of the electron
-13 Cq = 3.8 x 10 m ! Bunch length is set by the momentum compaction & Vrf % $E ( + RE " 2 = 2#' * c o z & E ) eV˙ Using a harmonic rf-cavity can produce shorter bunches
! US Particle Accelerator School Schematic of radiation cooling
Transverse cooling:
P⊥ less P remains less P less ⊥ || P|| restored
Passage Acceleration through dipoles in RF cavity Limited by quantum excitation
US Particle Accelerator School Emittance and Momentum Spread
• At equilibrium the momentum spread is given by: 2 2 2 2 3 '* p $ Cq) 0 # " p & Cq) 0 1 * ds 13 % " = = + where C = 3.84 ,10- m % " % ( 2 q & p0 # J S ( $ p0 ' JS + 1 * ds iso ! magnetic case • For the horizontal emittance at equilibrium:
! 2 H $ 3ds # 0 % 2 2 " = Cq 2 where: H(s) = "T D# + $T D + 2%T DD# JX % 1 $ ds • In the vertical plane, when no vertical bend is present, the synchrotron radiation contribution to the equilibrium emittance is very small ! ! • Vertical emittance is defined by machine imperfections & nonlinearities that couple the horizontal & vertical planes: # 1 " = " and " = " with " # coupling factor Y # +1 X # +1 US Particle Accelerator School
! ! Equilibrium emittance & ΔE
Set Growth rate due to fluctuations (linear) = exponential damping rate due to radiation
==> equilibrium value of emittance or ΔE
#2t /$ d #2t /$ d "natural = "1e + "eq (1# e )
US Particle Accelerator School
! Quantum lifetime
At a fixed observation point, transverse particle motion looks sinusoidal x = a " sin # t + $ T = x,y T n ( " n ) Tunes are chosen in order to avoid resonances. At a fixed azimuth, turn-after-turn a particle sweeps all possible positions within the envelope ! Photon emission randomly changes the “invariant” a Consequently changes the trajectory envelope as well. Cumulative photon emission can bring the envelope beyond acceptance at some azimuth The particle is lost This mechanism is called the transverse quantum lifetime
US Particle Accelerator School Several time scales govern particle dynamics in storage rings
Damping: several ms for electrons, ~ infinity for heavier particles
Synchrotron oscillations: ~ tens of ms
Revolution period: ~ hundreds of ns to ms
Betatron oscillations: ~ tens of ns
US Particle Accelerator School Interaction of Photons with Matter
Radiography
Diffraction
Photoelectric Effect Compton Scattering
US Particle Accelerator School Brightness of a Light Source
Brightness is a principal characteristic of a particle source Density of particle in the 6-D phase space Same definition applies to photon beams Photons are bosons & the Pauli exclusion principle does not apply Quantum mechanics does not limit achievable photon brightness
Brightness = # of photons in given Δλ/λ cdt sec, mrad θ, mrad ϕ, mm2 dx dθ dφ dy Flux = # of photons in given Δλ/λ dN& Flux = = Brightness dS d! sec d# "
US Particle Accelerator School Spectral brightness
Spectral brightness is that portion of the brightness lying within a relative spectral bandwidth Δω/ω:
US Particle Accelerator School How bright is a synchrotron light source?
Remark: the sources are compared at different wavelengths!
US Particle Accelerator School Angular distribution of SR
US Particle Accelerator School Energy dependence of SR spectrum
Bending Magnet
US Particle Accelerator School Spectrum available using SR
US Particle Accelerator School Two ways to produce radiation fromrelativistic electrons
US Particle Accelerator School Relativistic electrons radiate in a narrow cone
US Particle Accelerator School Third generation light sources have long straight sections and bright e-beams
US Particle Accelerator School Light sources provide three types of SR
US Particle Accelerator School Bend magnet radiation
Advantages: Broad spectral range Least expensive Most accessible • Many beamlines
Disadvantages: Limit coverage of hard X-rays Not as bright at undulator radiation
US Particle Accelerator School For brighter X-rays add the radiation from many small bends
US Particle Accelerator School Undulator radiation: What is λrad?
An electron in the lab oscillating at frequency, f, emits dipole radiation of frequency f
f
What about the relativistic electron?
US Particle Accelerator School Power in the central cone of undulator radiation
US Particle Accelerator School Spatial coherence of undulator radiation
US Particle Accelerator School US Particle Accelerator School Characteristics of wiggler radiation
For K >> 1, the radiation appears in high harmonics, and at rather large horizontal angles θ = ±K/γ One tends to use larger collection angles, which tends to spectrally merge nearby harmonics. Continuum at high photon energies, similar bend magnet radiation, • Increased by 2N (the number of magnet pole pieces).
US Particle Accelerator School X-ray beamlines transport the photons to the sample
US Particle Accelerator School A Typical Beamline: Monochromator Plus Focusing Optics
US Particle Accelerator School To get brighter beams we need another great invention The Free Electron Laser (John Madey, Stanford, 1976)
Physics basis: Bunched electrons radiate coherently
START MIDDLE END Madey’s discovery: the bunching can be self-induced!
US Particle Accelerator School Coherent emission ==> Free Electron Laser
100 MeV linac Injector High energy linac buncher
long wiggler
to experiment
Dump
See movie
US Particle Accelerator School Manipulate electrons in longitudinal phase space via interaction with laser in wiggler
V E B S - N S e N Light k k N S N S B λ E w V
Electron trajectory through wiggler with two periods
US Particle Accelerator School Fundamental FEL physics
Electrons see a potential V(x) ~ A (1" cos(x + #)) where A " Bw#w E laser and ϕ is! the phase between the electrons and the laser field
Imagine an! electron part way up the potential well but falling toward the potential minimum at θ = 0 Energy radiated by the electron increases the laser field and consequently lowers the minimum further. Electrons moving up the potential well decrease the laser field
US Particle Accelerator School The equations of motion
The electrons move according to the pendulum equation d 2 x = A sin(x + ") dt 2 The field varies as dA = "J e"ix ! dt
where x = (kw-k) z - ωt ! The simulation will show us the bunching and signal growth
DownLoad: FELOde.zip US Particle Accelerator School Basic Free Electron Laser Physics
Resonance condition: Slip one optical period per wiggler period FEL bunches beam on an optical wavelength at ALL harmonics Bonifacio et al. NIM A293, Aug. 1990 Gain-bandwidth & efficiency ~ ρ Gain induces ΔE ~ ρ 1) Emittance constraint Match beam phase area to diffraction limited optical beam 2) Energy spread condition Keep electrons from debunching 3) Gain must be faster than diffraction
US Particle Accelerator School FOM 1 from condensed matter studies: Light source brilliance v. photon energy
Duty factor correction for ERLs pulsed linacs
US Particle Accelerator School Near term: x-rays from betatron motion and Thomson scattering Betatron oscillations:
Esarey et al., Phys. Rev E (2002) Rousse et al., Phys. Rev. Lett. (2004) - laser pulse
+ channel radiation
- λ =λ /2γ2 λu x u plasma
Strength parameter
1/2 Betatron: aβ=π(2γ) rβ/λp
2 Thomson scattering: a0 = e/mc A
Radiation pulse duration = bunch duration
US Particle Accelerator School Potential Thompson source from all optical accelerator
Gas jet femtosecond laser fs - laser - pulse undulator e pulse 800 nm, 50 fs, 100 mJ
femtosecond femtosecond x-ray pulses electron beam Δθ < 20 fs 10 - 200 keV 85 MeV (γ=17 0) 90 µm ~107 photons/pulse/01.% BW 0 .3 nC/pulse Δθ ~ 3 mrad 15 f s λL λ = X 2γ 2
Experimental Schoenlein et al., Science 1996 sample Leemans et al., PRL 1996 US Particle Accelerator School