Accelerator Physics Synchrotron Radiation. G. A. Krafft Old Dominion University Jefferson Lab Lecture 17

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1 Accelerator Physics Synchrotron Radiation G. A. Krafft Old Dominion University Jefferson Lab Lecture 17

2 Relativistic Kinematics In average rest frame the insertion device is Lorentz contracted, and so its wavelength is * * * ID / The sinusoidal wiggling motion emits with angular frequency * * c / Lorentz transformation formulas for the wave vector of the emitted radiation * * * k k1 cos k k k * x * y * z k k x y * k k sin cos k sin sin * cos

3 ID (or FEL) Resonance Condition Angle transforms as Wave vector in lab frame has * * cos * k cos z k * 1 * cos k * k * * c * * 1 cos 1 cos ID In the forward direction cos θ = 1 e ID ID 1 K / *

4 Power Emitted Lab Frame Larmor/Lienard calculation in the lab frame yields P e 4 K zc ID 1 6 Total energy radiated after one passage of the insertion device e E z NK * 6 ID

5 Power Emitted Beam Frame Larmor/Lienard calculation in the beam frame yields P * e 1 ck * 6 Total energy of each photon is ħπc/λ *, therefore the total number of photons radiated after one passage of the insertion device N NK 3

6 Spectral Distribution in Beam Frame Begin with average power distribution in beam frame: dipole radiation pattern (single harmonic only when K<<1; replace γ* by γ, β* by β) dp d e c 3 * * Kk sin * * Number distribution in terms of wave number Solid angle transformation dn k k d 4 k d * * y z NK * * * d 1 cos

7 Number distribution in beam frame dn d Energy is simply E sin sin cos NK cos Number distribution as a function of normalized lab-frame energy c ˆ 1 E 1 cos 1 cos ID dn ˆ E NK 1 ˆ 3 de 4

8 Limits of integration Average Energy ˆ 1 ˆ 1 cos 1 E cos 1 E 1 1 Average energy is also analytically calculable E dn ˆ E de deˆ c/ dn de ˆ deˆ max ID E

9 Number Spectrum θ = π θ = sin θ = 1/ γ dn/de γ E und c/ ID βγ E und (1+β)βγ E und E γ

10 Nonlinear Thomson Scattering Many of the the newer Thomson Sources are based on a PULSED laser (e.g. all of the high-energy lasers are pulsed by their very nature) Have developed a general theory to cover radiation distribution calculations in the general case of a pulsed, high field strength laser interacting with electrons in a Thomson scattering arrangement. The new theory shows that in many situations the estimates people do to calculate flux and brilliance, based on a constant amplitude models, need to be modified. The new theory is general enough to cover all 1-D undulator calculations and all 1- D pulsed laser Thomson scattering calculations. The main new physics that the new calculations include properly is the fact that the electron motion changes based on the local value of the field strength squared. Such ponderomotive forces (i.e., forces proportional to the field strength squared), lead to a red-shift detuning of the emission, angle dependent Doppler shifts of the emitted scattered radiation, and additional transverse dipole emission that this theory can calculate.

11 Ancient History Early 196s: Laser Invented Brown and Kibble (1964): Earliest definition of the field strength parameters K and/or a in the literature that I m aware of ee a mc Thomson Sources Interpreted frequency shifts that occur at high fields as a relativistic mass shift. Sarachik and Schappert (197): Power into harmonics at high K and/or a. Full calculation for CW (monochromatic) laser. Later referenced, corrected, and extended by workers in fusion plasma diagnostics. Alferov, Bashmakov, and Bessonov (1974): Undulator/Insertion Device theories developed under the assumption of constant field strength. Numerical codes developed to calculate real fields in undulators. Coisson (1979): Simplified undulator theory, which works at low K and/or a, developed to understand the frequency distribution of edge emission, or emission from short magnets, i.e., including pulse effects K eb Undulators mc

12 Coisson s Spectrum of a Short Magnet Coisson low-field strength undulator spectrum* de dd rc e 1 f B 1 / f f f f f 1 1 sin cos *R. Coisson, Phys. Rev. A, 54 (1979)

13 Dipole Radiation Assume a single charge moves in the x direction ( x, y, z, t) e x d J( x, y, z, t) ed xˆ x d t t y z t y z Introduce scalar and vector potential for fields. Retarded solution to wave equation (Lorenz gauge), R r r' t' 1 R e ( t ' t R / c) r, t r ', t dx ' dy ' dz ' dt ' R c 4 R 1 R e d t ' ( t ' t R / c) Ax r t J r t dx dy dz dt R c 4 R, x ', ' ' ' ' e A B A B d t R c E t, lim lim / r r 4 R 1

14 lim B r 1 lim E r 1 I ed t r / c 4 4 cr ed t r / c cr Dipole Radiation sin ˆ xˆ sin ˆ E B 1 e d t r / c sin 3 16 cr di d 1 16 Polarized in the plane containing e d t r / c c 3 sin rˆ rˆ n ŷ and xˆ r ˆ ˆ ẑ

15 Define the Fourier Transform ~ d d( t) e Dipole Radiation it dt d( t) 1 With these conventions Parseval s Theorem is d t dt ~ d 1 ~ d 4 / de dd d 1 3 e 3 3 d( ) c sin i e t d de e d t r c dt e d d d 16 c 3 c Blue Sky! This equation does not follow the typical (see Jackson) convention that combines both positive and negative frequencies together in a single positive frequency integral. The reason is that we would like to apply Parseval s Theorem easily. By symmetry, the difference is a factor of two.

16 Dipole Radiation For a motion in three dimensions de dd e d n c 3 3 Vector inside absolute value along the magnetic field e 4 d n n e 4 n d n d de 1 1 dd 3 c 3 c Vector inside absolute value along the electric field. To get energy into specific polarization, take scalar product with the polarization vector

17 Co-moving Coordinates Assume radiating charge is moving with a velocity close to light in a direction taken to be the z axis, and the charge is on average at rest in this coordinate system For the remainder of the presentation, quantities referred to the moving coordinates will have primes; unprimed quantities refer to the lab system xˆ ˆx' z c ẑ, ẑ' ŷ ŷ' In the co-moving system the dipole radiation pattern applies

18 New Coordinates ŷ' ˆx' ' ' n ' ˆ ' ˆ ' ẑ' ŷ' ˆx ' ' n ' ' ˆ' ˆ ' ẑ' Resolve the polarization of scattered energy into that perpendicular (σ) and that parallel (π) to the scattering plane n' sin 'cos ' xˆ' sin 'sin ' yˆ' cos ' zˆ' eˆ' eˆ' n' zˆ' / n' zˆ' sin ' xˆ' cos ' yˆ' ˆ' n' eˆ' cos 'cos ' xˆ' cos 'sin ' yˆ' sin ' zˆ' ˆ'

19 Polarization It follows that ~ ~ d ' ' eˆ' d ' x ~ ~ d ' ' eˆ' d ' ' sin ' d ' ' ~ ~ cos ' ' cos 'cos ' d ' ' cos 'sin ' sin ' d ' ' x y So the energy into the two polarizations in the beam frame is de 1 e ' d ' d ' 3 c de ' e ' ' y d d' ' sin ' ' ' cos ' d y d ' d ' 3 c sin ' d ' ' x x y d' ' cos 'cos ' ' ' cos 'sin ' z ~ z

20 Comments/Sum Rule There is no radiation parallel or anti-parallel to the x-axis for x-dipole motion In the forward direction θ', the radiation polarization is parallel to the x-axis for an x-dipole motion One may integrate over all angles to obtain a result for the total energy radiated de d 1 e ' ' 4 3 ' 3 3 c x d d' ' ' ' d ' x ' d ' y ' d ' z ' de d ' ' 4 1 e ' c y ' e ' d' ' detot 1 8 d ' 3 c 3 Generalized Larmor

21 Sum Rule Total energy sum rule E ' tot e ' d' ' c d ' Parseval s Theorem again gives standard Larmor formula P' 3 3 ' de tot 1 e d ' t ' 1 e a ' t ' dt ' 6 c 6 c

22 Energy Distribution in Lab Frame 1 cos d x 4 de e ' 1 cos sin dd c d ' 1 cos y cos cos d ' x 1 cos cos 1 cos 4 de e 1 cos cos d ' 1 cos sin 3 3 y dd 3 c 1 cos By placing the expression for the Doppler shifted frequency and angles inside the transformed beam frame distribution. Total energy radiated from d' z is the same as d' x and d' y for same dipole strength. z d ' 1 cos sin 1 cos

23 Flux [ph/s/.1%bw] Brightness [ph/s/mm /mr /.1%bw] Flux [ph/s/.1%bw] Bend Undulator Wiggler e e e white source partially coherent source powerful white source

24 de dd Weak Field Undulator Spectrum ec B '/ cz ~ ikz ˆ ˆ Bk Bze dz d ' ' d ' ' x x mc ' r B e c 1 cos / c z 1 z cos r 1 cos / e c B z cz cos 1 cos 1 co z z sin de z dd s z r e 4 e 16mc cos z cos 1 z Generalizes Coisson to arbitrary observation angles

25 Strong Field Case Why is the FEL resonance condition? 1 K d dt d m c ec B dt x e mc z ' z B z dz '

26 High K z z 1 1 x z z z 1 1 e 1 K K mc 4 z 1 Bz ' dz ' 1 1 cos k z Inside the insertion device the average (z) velocity is * z 1 K 1 1 and the radiation emission frequency redshifts by the 1+K / factor

27 Thomson Scattering Purely classical scattering of photons by electrons Thomson regime defined by the photon energy in the electron rest frame being small compared to the rest energy of the electron, allowing one to neglect the quantum mechanical Dirac recoil on the electron In this case electron radiates at the same frequency as incident photon for low enough field strengths Classical dipole radiation pattern is generated in beam frame Therefore radiation patterns, at low field strength, can be largely copied from textbooks Note on terminology: Some authors call any scattering of photons by free electrons Compton Scattering. Compton observed (the so-called Compton effect) frequency shifts in X-ray scattering off (resting!) electrons that depended on scattering angle. Such frequency shifts arise only when the energy of the photon in the rest frame becomes comparable with.511 MeV. Krafft, G. A. and G. Priebe, Rev. Accel. Sci. and Tech., 3, 147 (1)

28 Simple Kinematics e - ẑ Beam Frame p ' e mc, p' E ', E ' p L L Lab Frame pe mc z, ˆ p E 1,sin yˆ cos zˆ p L p e p p mc E' L mc EL 1 cos

29 cos 1 ' E L E L In beam frame scattered photon radiated with wave vector ' ',cos 'sin ',sin 'cos 1,sin ' ' c E k L Back in the lab frame, the scattered photon energy E s is cos 1 ' ' cos 1 ' L L s E E E 1 cos 1 cos s L E E

30 Electron in a Plane Wave Assume linearly-polarized pulsed laser beam moving in the direction (electron charge is e) n inc sin yˆ A inc cos zˆ x, t A ct sin y cos zxˆ A xˆ Polarization 4-vector (linearly polarized) x,1,, Light-like incident propagation 4-vector n inc n inc 1,,sin, cos n n inc inc Krafft, G. A., Physical Review Letters, 9, 48 (4), Krafft, Doyuran, and Rosenzweig, Physical Review E, 7, 565 (5)

31 Electromagnetic Field F A A da d n n inc inc A x A x Our goal is to find x μ (τ)=(ct(τ),x(τ),y(τ),z(τ)) when the 4-velocity u μ (τ)=(cdt/dτ,dx/dτ,dy/dτ,dz/dτ)(τ) satisfies du μ /dτ= ef μν u ν /mc where τ is proper time. For any solution to the equations of motion. d n incu n F u n u n u d inc inc inc Proportional to amount frequencies up-shifted in going to beam frame

32 ξ is proportional to the proper time On the orbit ct n x d / d n u inc Integrate with respect to ξ instead of τ. Now inc d u d c df n d inc u c d f d where the unitless vector potential is f(ξ) = ea(ξ )/mc. u cf u

33 Electron Orbit u u cf u ninc u n inc c n inc f u n inc Direct Force from Electric Field Ponderomotive Force x u n inc u cu c n inc n u n inc u inc f ' d ' n inc c u n inc f ' d '

34 Energy Distribution de dd e 3 c 3 3 D t sin 1cos ;, sin D p ;, cos D t cos ;, cos 1 cos de e sin cos D ;, sin 3 3 p dd 3 c 1 cos 1 cos cos sin 1 cos 1 cos D ;, p

35 Effective Dipole Motions: Lab Frame 1 ea i, ;, Dt ;, e d 1cos mc 1 ea i, ;, Dp ;, e d 1cos mc And the (Lorentz invariant!) phase is 1 cos sincos ea ' d' 1 cos 1 cos mc, ;, c 1 sin sin sin cos cos ea' d ' 1 cos mc

36 Summary Overall structure of the distributions is very like that from the general dipole motion, only the effective dipole motion, including physical effects such as the relativistic motion of the electrons and retardation, must be generalized beyond the straight Fourier transform of the field At low field strengths (f <<1), the distributions reduce directly to the classical Fourier transform dipole distributions The effective dipole motion from the ponderomotive force involves a simple projection of the incident wave vector in the beam frame onto the axis of interest, multiplied by the general ponderomotive dipole motion integral The radiation from the two transverse dipole motions are compressed by the same angular factors going from beam to lab frame as appears in the simple dipole case. The longitudinal dipole radiation is also transformed between beam and lab frame by the same fraction as in the simple longitudinal dipole motion. Thus the usual compression into a 1/γ cone applies

37 Weak Field Thomson Backscatter With Φ = π and f <<1 the result is identical to the weak field undulator result with the replacement of the magnetic field Fourier transform by the electric field Fourier transform Undulator Thomson Backscatter Driving Field Forward Frequency cb 1 cos / c y z z ~ E x 1 cos / c1 z 4 z Lorentz contract + Doppler Double Doppler

38 Photon Number Spectrum θ = π θ = sin θ = 1/ γ dn/de γ Compton Edge (1+β)γ E laser (1+β) γ E laser E γ

39 High Field Thomson Backscatter For a flat incident laser pulse the main results are very similar to those from undulaters with the following correspondences Undulator Thomson Backscatter Field Strength K a Forward Frequency 1 K a 1 4 Transverse Pattern * cos ' z 1 cos ' NB, be careful with the radiation pattern, it is the same at small angles, but quite a bit different at large angles

40 Forward Direction: Flat Laser Pulse -period equivalent undulator: A cos / A x 1 z c / 4 c /, a ea / mc

41 1/ 1 a /

42 Realistic Pulse Distribution at High a In general, it s easiest to just numerically integrate the labframe expression for the spectrum in terms of D t and D p. A 1 5 to 1 6 point Simpson integration is adequate for most purposes. Flat pulses reproduce previously known results and to evaluate numerical error, and Gaussian amplitude modulated pulses. One may utilize a two-timing approximation (i.e., the laser pulse is a slowly varying sinusoid with amplitude a(ξ)), and the fundamental expressions, to write the energy distribution at any angle in terms of Bessel function expansions and a ξ integral over the modulation amplitude. This approach actually has a limited domain of applicability (K,a<.1)

Forward Direction: Gaussian Pulse A x cos / A exp z / 8.

43 Forward Direction: Gaussian Pulse A x cos / A exp z / peak a ea / mc peak peak A peak and λ chosen for same intensity and same rms pulse length as previously

44 Radiation Distributions: Backscatter Gaussian Pulse σ at first harmonic peak de 5 d d de d d x y Courtesy: Adnan Doyuran (UCLA)

45 Radiation Distributions: Backscatter Gaussian π at first harmonic peak de 5 d d de d d x y Courtesy: Adnan Doyuran (UCLA)

46 Radiation Distributions: Backscatter Gaussian σ at second harmonic peak de 1-43 d d - x y - Courtesy: Adnan Doyuran (UCLA)

47 de 9 Degree Scattering e D 3 3 t 3 c d d 1 ;, sin D ;, cos D t cos ;, cos 1 cos de e 1 cos D ;, sin 3 3 p dd 3 c 1 cos D p ;, 1 cos p sin

48 And the phase is, ;, 9 Degree Scattering 1 ea i, ;, Dt ;, e d mc 1 ea i, ;, Dp ;, e d mc sincos ea ' 1 cos d ' mc c 1 sinsin ea' d ' mc

49 Radiation Distribution: 9 Degree Gaussian Pulse σ at first harmonic peak de4 1-4 d d de 1-4 d d.5 x y Courtesy: Adnan Doyuran (UCLA)

50 Radiation Distributions: 9 Degree Gaussian Pulse π at first harmonic peak de4 1-4 d d de4 1-4 d d x. -. y Courtesy: Adnan Doyuran (UCLA)

51 Polarization Sum: Gaussian 9 Degree de 1-4 d d.5 x y Courtesy: Adnan Doyuran (UCLA)

52 Radiation Distributions: 9 Degree Gaussian Pulse second harmonic peak de d 5d de d 5 d x y x y Second harmonic emission on axis from ponderomotive dipole! Courtesy: Adnan Doyuran (UCLA)

53 Compensating the Red Shift Add possibility of frequency modulation (Ghebregziabher, et al.) Wave phase A a cos f / x f / Wave (angular) frequency and wave number t k z

54 Double Doppler to Constant Frequency In beam frame 1 k 1 * * * * k Doppler shift to lab frame (θ=) * 1 a * a 1 1 / 1 1 * d d * C 1 d d Compensation occurs when f a d 1 / 1 / a

55 Compensation By Chirping Terzic, Deitrick, Hofler, and Krafft, Phys. Rev. Lett., 11, 7481 (14)

56 Hajima, et al. Uranium Detection Hajima, et al., NIM A, 68, S57 (9) TRIUMF Moly Source?

57 THz Source

58 Radiation Distributions for Short High-Field Magnets a z / D i z dz t ;, = exp, ;, z a z / 1 1 D ;, exp, ;, p i z dz z z z And the phase is 1 cos z 1 1 c c z ' z z, z;, dz ' z z z z sin cos a z'/ c a z'/ z dz '. Krafft, G. A., Phys. Rev. ST-AB, 9, 171 (6)

59 Wideband THz Undulater Primary requirements: wide bandwidth and no motion and deflection. Implies generate A and B by simple motion. One half an oscillation is highest bandwidth! a z x z exp z / ea z z exp z / mc B z da z B dz z eb peak K mc 1 exp / peak

60 THz Undulator Radiation Spectrum

61 Total Energy Radiated Lienard s Generalization of Larmor Formula (1898!) de 1 e 6 1 e 4 dt 6 c 6 c Barut s Version E dx 3 de 1 e dt d x d 6 c d d d e 1 cos df f df d 6 d d Usual Larmor term From ponderomotive dipole

62 Some Cases Total radiation from electron initially at rest E e df f df 6 d d d For a flat pulse exactly (Sarachik and Schappert) de dt 1 1 e c a 1 a / 8

63 For Circular Polarization Only other specific case I can find in literature completely calculated has usual circular polarization and flat pulses. The orbits are then pure circles Sokolov and Ternov, in Radiation from Relativistic Electrons, give de ' 1 e ' dt ' 6 c (which goes back to Schott and the turn of the th century!) and the general formula checks out a 1a

64 Conclusions I ve shown how dipole solutions to the Maxwell equations can be used to obtain and understand very general expressions for the spectral angular energy distributions for weak field undulators and general weak field Thomson Scattering photon sources A new calculation scheme for high intensity pulsed laser Thomson Scattering has been developed. This same scheme can be applied to calculate spectral properties of short, high-k wigglers. Due to ponderomotive broadening, it is simply wrong to use single-frequency estimates of flux and brilliance in situations where the square of the field strength parameter becomes comparable to or exceeds the (1/N) spectral width of the induced electron wiggle The new theory is especially useful when considering Thomson scattering of Table Top TeraWatt lasers, which have exceedingly high field and short pulses. Any calculation that does not include ponderomotive broadening is incorrect.

65 Conclusions Because the laser beam in a Thomson scatter source can interact with the electron beam non-colinearly with the beam motion (a piece of physics that cannot happen in an undulator), ponderomotively driven transverse dipole motion is now possible This motion can generate radiation at the second harmonic of the up-shifted incident frequency on axis. The dipole direction is in the direction of laser incidence. Because of Doppler shifts generated by the ponderomotive displacement velocity induced in the electron by the intense laser, the frequency of the emitted radiation has an angular asymmetry. Sum rules for the total energy radiated, which generalize the usual Larmor/Lenard sum rule, have been obtained.

66 Conventions on Fourier Transforms For the time dimensions it f f t e dt 1 it f t f e d Results on Dirac delta functions 1 it t e d t e it dt 1

67 For the three spatial dimensions ik x 3 f k f x e d x 1 ik x 3 f x f k e d k ik x 3 3 x x e d k

68 Green Function for Wave Equation Solution to inhomogeneous wave equation 1 x y z c t G x, t; x, t x x t t 4 Will pick out the solution with causal boundary conditions, i. e. G x, t; x, t t t This choice leads automatically to the so-called Retarded Green Function

69 In general G x, t; x, t t t G x, t; x, t ik xt ik xt 3 A k e B k e d k t t because there are two possible signs of the frequency for each value of the wave vector. To solve the homogeneous wave equation it is necessary that k k c i.e., there is no dispersion in free space.

70 Continuity of G implies Integrate the inhomogeneous equation between t t 1 G x, t; x, t 4 x x c t A k e it ik xt ik xt 3 ia k e i B k e d k c ik x A k e e i B k e t it 4 c x x it t t and

71 G x, t; x, t c i ik xx c e i t t c e dk x x x x / c t t x x 1 ik xxtt ik xxtt 3 e e d k ik xx e x x t t i tt e dk t t Called retarded because the influence at time t is due to the source evaluated at the retarded time t t x x / c

Accelerator Physics NMI and Synchrotron Radiation. G. A. Krafft Old Dominion University Jefferson Lab Lecture 16

Accelerator Physics NMI and Synchrotron Radiation G. A. Krafft Old Dominion University Jefferson Lab Lecture 16 Graduate Accelerator Physics Fall 17 Oscillation Frequency nq I n i Z c E Re Z 1 mode has