Energy Recovered Linacs and Ponderomotive Spectral Broadening G
Energy Recovered Linacs and Ponderomotive Spectral Broadening • G. A. Krafft • Jefferson Lab Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy
Outline APS Talk January 13, 2006 • • Recirculating linacs defined and described Review of recirculating SRF linacs Energy recovered linacs as light sources Ancient history: lasers and electrons Dipole emission from a free electron Thomson scattering Motion of an electron in a plane wave 1. Equations of motion 2. Exact solution for classical electron in a plane wave • Applications to scattered spectrum 1. General solution for small a 2. Finite a effects 3. Ponderomotive broadening 4. Sum Rules • Conclusions Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 2
Schematic Representation of Accelerator Types APS Talk January 13, 2006 RF Installation Beam injector and dump Beamline Ring Linac Recirculating Linac Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 3
Why Recirculate? APS Talk January 13, 2006 • Performance upgrade of a previously installed linac - Stanford Superconducting Accelerator and MIT Bates doubled their energy this way • Cheaper design to get a given performance - Microtrons, by many passes, reuse expensive RF many times to get energy up. Penalty is that the average current has to be reduced proportional to 1/number passes, for the same installed RF. - Jefferson Lab CEBAF type machines: add passes until the “decremental” gain in RF system and operating costs no longer pays for an additional recirculating loop - Jefferson Lab FEL and other Energy Recovered Linacs (ERLs) save the cost of higher average power RF equipment (and much higher operating costs) at higher CW operating currents by “reusing” beam energy through beam recirculation. Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 4
Beam Energy Recovery APS Talk January 13, 2006 Recirculation path length in standard configuration recirculated linac. For energy recovery choose it to be (n + 1/2)λRF. Then Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 5
Beam Energy Recovery APS Talk January 13, 2006 Recirculation Path Length Linac Cavity Center Recirculation path length in herring-bone configuration recirculated linac. For energy recovery choose it to be nλRF. Note additional complication: path length has to be an integer at each and every different accelerating cavity location in the linac. Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 6
Comparison between Linacs and Storage Rings • APS Talk January 13, 2006 Advantage Linacs - Emittance dominated by source emittance and emittance growth down linac - Beam polarization “easily” produced at the source, switched, and preserved - Total transit time is quite short - Beam is easily extracted. Utilizing source laser control, flexible bunch patterns possible - Long undulaters are a natural addition - Bunch durations can be SMALL (10 -100 fsec) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 7
Comparison Linacs and Storage Rings • APS Talk January 13, 2006 Advantage Storage Rings - Up to now, the stored average current is much larger - Very efficient use of accelerating voltage - Technology well developed and mature • Disadvantage Storage Rings - Technology well developed and mature (maybe!) - The synchrotron radiation damping equilibrium, and the emittance and bunch length it generates, must be accepted Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 8
Power Multiplication Factor APS Talk January 13, 2006 • Energy recovered beam recirculation is nicely quantified by the notion of a power multiplication factor: where Prf is the RF power needed to accelerate the beam • By the first law of thermodynamics (energy conservation!) k < 1 in any linac not recirculated. Beam recirculation with beam deceleration somewhere is necessary to achieve k > 1 • If energy IS very efficiently recycled from the accelerating to the decelerating beam Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 9
High Multiplication Factor Linacs Recirculated Linacs Normal Conducting Recirculators k<<1 LBNL Short Pulse X-ray Facility (proposed) k=0. 1 CEBAF (matched beam load) k=0. 99; (typical) k=0. 8 JLAB IR DEMO k=16 JLAB 10 k. W Upgrade k=33 Cornell/JLAB ERL k=200 (proposed) BNL PERL k=500 (proposed) High Multiplication Factor Superconducting Linacs Will use the words “High Multiplication Factor Linac” for those designs that feature high k. Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy APS Talk January 13, 2006 Page 10
Comparison Accelerator Types APS Talk January 13, 2006 High Energy Electron Linac High k Superconducting Linac Ring Accelerating Gradient[MV/m] >50 10 -20 NA Duty Factor <1% 1 1 Average Current[m. A] <1 10 going to 1000 Average Beam Power[MW] 0. 5 1. 0 going to 700 3000 Multiplication Factor <1 33 going to 200 1000 Normalized Emittance[mm mrad] 1 1 4 100 fsec 20 psec Parameter Bunch Length Typical results by accelerator type Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 11
More Modern Reason to Recirculate! APS Talk January 13, 2006 • A renewed general interest in beam recirculation has arisen due to the success of Jefferson Lab’s high average current energy recovered Free Electron Lasers (FELs), and the broader realization that it may be possible to achieve beam parameters “Unachievable” in storage rings or linacs without recirculation. ERL synchrotron source: Beam power in a typical synchrotron source is (100 m. A)(5 Ge. V)=500 MW. Realistically, even the federal govt. will be unable to provide a third of a nuclear plant to run a synchrotron source. Idea is to use the high multiplication factor possible in energy recovered designs to reduce the power load. Pulse lengths of order 100 fsec or smaller may in be possible in an ERL source; “impossible” at a storage ring. Better emittance may be possible too. The limits, in particular the average current carrying capacity of possible recirculated linac designs, are not yet determined and may be far in excess of what the FELs can do! Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 12
Challenges for Beam Recirculation APS Talk January 13, 2006 • Additional Linac Instability - Multipass Beam Breakup (BBU) - Observed first at Illinois Superconducting Racetrack Microtron - Limits the average current at a given installation - Made better by damping non-fundamental electromagnetic High Order Modes (HOMs) in the cavities - Best we can tell at CEBAF, threshold current is around 20 m. A, measured to be several m. A in the FEL - Changes based on beam recirculation optics • Turn around optics tends to be a bit different than in storage rings or more conventional linacs. Longitudinal beam dynamics gets coupled more strongly to the transverse dynamics and nonlinear corrections different • HOM cooling will perhaps limit the average current in such devices. Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 13
Challenges for Beam Recirculation APS Talk January 13, 2006 • High average current sources needed to provide beam - Right now, looks like a good way to get there is with DC photocathode sources as we have in the Jefferson Lab FEL. - Need higher fields in the acceleration gap in the gun. - Need better vacuum performance in the beam creation region to increase the photocathode lifetimes. - Goal is to get the photocathode decay times above the present storage ring Toushek lifetimes • Beam dumping of the recirculated beam can be a challenge. Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 14
Recirculating SRF Linacs Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy APS Talk January 13, 2006 Page 15
The CEBAF at Jefferson Lab § § APS Talk January 13, 2006 Most radical innovations (had not been done before on the scale of CEBAF): • choice of Superconducting Radio Frequency (SRF) technology • use of multipass beam recirculation Until LEP II came into operation, CEBAF was the world’s largest implementation of SRF technology. Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 16
CEBAF Accelerator Layout* APS Talk January 13, 2006 *C. W. Leemann, D. R. Douglas, G. A. Krafft, “The Continuous Electron Beam Accelerator Facility: CEBAF at the Jefferson Laboratory”, Annual Reviews of Nuclear and Particle Science, 51, 413 -50 (2001) has a long reference list on the CEBAF accelerator. Many references on Energy Recovered Linacs may be found in a recent ICFA Beam Dynamics Newsletter, #26, Dec. 2001: http: //icfausa/archive/newsletter/icfa_bd_nl_26. pdf Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 17
CEBAF Beam Parameters APS Talk January 13, 2006 Beam energy 6 Ge. V Beam current A 100 μA, B 10 -200 n. A, C 100 μA Normalized rms emittance Repetition rate 1 mm mrad 500 MHz/Hall Charge per bunch < 0. 2 p. C Extracted energy spread < 10 -4 Beam sizes (transverse) < 100 microns Beam size (longitudinal) <100 microns (330 fsec) Beam angle spread < 0. 1/γ Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 18
Short Bunches in CEBAF APS Talk January 13, 2006 Wang, Krafft, and Sinclair, Phys. Rev. E, 2283 (1998) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 19
Short Bunch Configuration APS Talk January 13, 2006 Kazimi, Sinclair, and Krafft, Proc. 2000 LINAC Conf. , 125 (2000) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 20
dp/p data: 2 -Week Sample Record APS Talk January 13, 2006 Energy Spread less than 50 ppm in Hall C, 100 ppm in Hall A Primary Hall (Hall C) X Position => relative energy Drift rms X width => Energy Spread 1. 2 0. 8 0. 4 Energy spread X and sigma X in mm 1. 2 Energy drift Secondary Hall (Hall A) 0. 8 Energy drift 0. 4 1 E-4 23 -Mar 27 -Mar 31 -Mar 0 4 -Apr Energy spread 0 23 -Mar 27 -Mar Date 31 -Mar 4 -Apr Time Courtesy: Jean-Claude Denard Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 21
Energy Recovered Linacs APS Talk January 13, 2006 § The concept of energy recovery first appears in literature by Maury Tigner, as a suggestion for alternate HEP colliders* § There have been several energy recovery experiments to date, the first one in a superconducting linac at the Stanford SCA/FEL** § Same-cell energy recovery with cw beam current up to 10 m. A and energy up to 150 Me. V has been demonstrated at the Jefferson Lab 10 k. W FEL. Energy recovery is used routinely for the operation of the FEL as a user facility * Maury Tigner, Nuovo Cimento 37 (1965) ** T. I. Smith, et al. , “Development of the SCA/FEL for use in Biomedical and Materials Science Experiments, ” NIMA 259 (1987) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 22
The SCA/FEL Energy Recovery Experiment § § The former Recyclotron beam recirculation system could not be used to obtain the peak current required for FEL lasing and was replaced by a doubly achromatic single-turn recirculation line. Same-cell energy recovery was first demonstrated in an SRF linac at the SCA/FEL in July 1986 Beam was injected at 5 Me. V into a ~50 Me. V linac (up to 95 Me. V in 2 passes) Nearly all the imparted energy was recovered. No FEL inside the recirculation loop. T. I. Smith, et al. , NIM A 259, 1 (1987) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 23 APS Talk January 13, 2006
CEBAF Injector Energy Recovery Experiment N. R. Sereno, “Experimental Studies of Multipass Beam Breakup and Energy Recovery using the CEBAF Injector Linac, ” Ph. D. Thesis, University of Illinois (1994) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 24 APS Talk January 13, 2006
Instability Mechanism APS Talk January 13, 2006 Courtesy: N. Sereno, Ph. D. Thesis (1994) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 25
Threshold Current APS Talk January 13, 2006 Growth Rate where If the average current exceeds the threshold current have instability (exponentially growing cavity amplitude!) Krafft, Bisognano, and Laubach, unpublished (1988) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 26
Jefferson Lab IR DEMO FEL APS Talk January 13, 2006 Wiggler assembly Neil, G. R. , et. al, Physical Review Letters, 84, 622 (2000) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 27
FEL Accelerator Parameters Parameter Designed Measured Kinetic Energy 48 Me. V 48. 0 Me. V Average current 5 m. A 4. 8 m. A Bunch charge 60 p. C Up to 135 p. C Bunch length (rms) <1 ps 0. 4 0. 1 ps Peak current 22 A Up to 60 A Trans. Emittance (rms) <8. 7 mmmr 7. 5 1. 5 mmmr Long. Emittance (rms) 33 ke. Vdeg 26 7 ke. Vdeg Pulse repetition frequency (PRF) 18. 7 MHz, x 2 18. 7 MHz, x 0. 25, x 0. 5, x 2, and x 4 Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy APS Talk January 13, 2006 Page 28
ENERGY RECOVERY WORKS APS Talk January 13, 2006 Gradient modulator drive signal in a linac cavity measured without energy recovery (signal level around 2 V) and with energy recovery (signal level around 0). Courtesy: Lia Merminga Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 29
Longitudinal Phase Space Manipulations APS Talk January 13, 2006 Simulation calculations of longitudinal dynamics of JLAB FEL Piot, Douglas, and Krafft, Phys. Rev. ST-AB, 6, 0030702 (2003) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 30
Phase Transfer Function Measurements APS Talk January 13, 2006 Krafft, G. A. , et. al, ERL 2005 Workshop Proc. in NIMA Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 31
Longitudinal Nonlinearities Corrected by Sextupoles Off Nominal Settings Basic Idea is to use sextupoles to get T 566 in the bending arc to compensate any curvature in the phase space. Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 32 APS Talk January 13, 2006
IR FEL Upgrade Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy APS Talk January 13, 2006 Page 33
IR FEL 10 k. W Upgrade Parameters Parameter Kinetic Energy Design Value 160 Me. V Average Current 10 m. A Bunch Charge 135 p. C Bunch Length <300 fsec 10 mm mrad Transverse Emittance Longitudinal Emittance Repetition Rate Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy 30 ke. V deg 75 MHz Page 34 APS Talk January 13, 2006
ERL X-ray Source Conceptual Layout Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Talk CHESS / LEPP January. APS 13, 2006 Page 35
Why ERLs for X-rays? Talk CHESS / LEPP January. APS 13, 2006 ESRF 6 Ge. V @ 200 m. A ERL 5 Ge. V @ 10 -100 m. A εx = 4 nm mrad εy = 0. 02 nm mrad B ~ 1020 ph/s/mm 2/mrad 2/0. 1%BW LID = 5 m εx = εy 0. 01 nm mrad B ~ 1023 ph/s/mm 2/mrad 2/0. 1%BW LID = 25 m ERL (no compression) ERL (w/ compression) ESRF t Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 36
Brilliance Scaling and Optimization Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 37 APS Talk January 13, 2006
ERL Phase II Sample Parameters Parameter Beam Energy Average Current Fundamental frequency Charge per bunch Injection Energy Normalized emittance Energy spread Bunch length in IDs Total radiated power CHESS / LEPP Value 5 -7 100 / 10 1. 3 77 / 8 10 2 / 0. 2* 0. 02 -0. 3* 0. 1 -2* 400 Unit Ge. V m. A GHz p. C Me. V μm % ps k. W * rms values Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 38 APS Talk January 13, 2006
ERL X-ray Source Average Brilliance and Flux CHESS / LEPP APS Talk January 13, 2006 Courtesy: Qun Shen, CHESS Technical Memo 01 -002, Cornell University Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 39
ERL Peak Brilliance and Ultra-Short Pulses CHESS / LEPP APS Talk January 13, 2006 Courtesy: Q. Shen, I. Bazarov Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 40
Cornell ERL Phase I: Injector CHESS / LEPP APS Talk January 13, 2006 Injector Parameters: Beam Energy Range Max Average Beam Current Max Bunch Rep. Rate @ 77 p. C Transverse Emittance, rms (norm. ) Bunch Length, rms Energy Spread, rms 5 – 15 a Me. V 100 m. A 1. 3 GHz < 1 b mm 2. 1 ps 0. 2 % a at reduced average current b Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy corresponds to 77 p. C/bunch Page 41
Beyond the space charge limit Cornell ERL Prototype Injector Layout Solenoids CHESS / LEPP APS Talk January 13, 2006 2 -cell SRF cavities 500 -750 k. V DC Photoemission Gun Injector optimization Merger dipoles into ERL linac Buncher 0. 1 mm-mrad, 80 p. C, 3 ps Courtesy of I. Bazarov Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 42
Nonlinear Thomson Scattering • • • APS Talk January 13, 2006 Many of 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 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 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. Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 43
Ancient History • • • APS Talk January 13, 2006 Early 1960 s: 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 Interpreted frequency shifts that occur at high fields as a “relativistic mass shift”. Sarachik and Schappert (1970): 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 Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 44
Coisson’s Spectrum from a Short Magnet APS Talk January 13, 2006 Coisson low-field strength undulator spectrum* *R. Coisson, Phys. Rev. A 20, 524 (1979) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 45
Dipole Radiation APS Talk January 13, 2006 Assume a single charge moves in the x direction Introduce scalar and vector potential for fields. Retarded solution to wave equation (Lorenz gauge), Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 46
Dipole Radiation Polarized in the plane containing Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy APS Talk January 13, 2006 and Page 47
Dipole Radiation APS Talk January 13, 2006 Define the Fourier Transform With these conventions Parseval’s Theorem is 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. Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 48
Dipole Radiation APS Talk January 13, 2006 For a motion in three dimensions Vector inside absolute value along the magnetic field Vector inside absolute value along the electric field. To get energy into specific polarization, take scaler product with the polarization vector Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 49
Co-moving Coordinates APS Talk January 13, 2006 • 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 • In the co-moving system the dipole radiation pattern applies Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 50
New Coordinates APS Talk January 13, 2006 Resolve the polarization of scattered energy into that perpendicular (σ) and that parallel (π) to the scattering plane Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 51
Polarization APS Talk January 13, 2006 It follows that So the energy into the two polarizations in the beam frame is Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 52
Comments/Sum Rule APS Talk January 13, 2006 • There is no radiation parallel or anti-parallel to the x-axis for xdipole motion • In the forward direction θ'→ 0, 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 Generalized Larmor Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 53
Sum Rule APS Talk January 13, 2006 Total energy sum rule Parseval’s Theorem again gives “standard” Larmor formula Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 54
Energy Distribution in Lab Frame APS Talk January 13, 2006 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. Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 55
Bend Undulator Wiggler APS Talk January 13, 2006 e– e– white source partially coherent source Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Flux [ph/s/0. 1%bw] Brightness [ph/s/mm 2/mr 2/0. 1%bw] e– powerful white source Page 56
Weak Field Undulator Spectrum Generalizes Coisson to arbitrary observation angles Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 57 APS Talk January 13, 2006
Strong Field Case APS Talk January 13, 2006 Why is the FEL resonance condition? Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 58
High K APS Talk January 13, 2006 Inside the insertion device the average (z) velocity is and the radiation emission frequency redshifts by the 1+K 2/2 factor Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 59
Thomson Scattering • • • APS Talk January 13, 2006 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 socalled 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 0. 511 Me. V. Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 60
Simple Kinematics APS Talk January 13, 2006 e- Beam Frame Lab Frame Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 61
APS Talk January 13, 2006 In beam frame scattered photon radiated with wave vector Back in the lab frame, the scattered photon energy Es is Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 62
Electron in a Plane Wave APS Talk January 13, 2006 Assume linearly-polarized pulsed laser beam moving in the direction (electron charge is –e) Polarization 4 -vector Light-like incident propagation 4 -vector Krafft, G. A. , Physical Review Letters, 92, 204802 (2004), Krafft, Doyuran, and Rosenzweig , Physical Review E, 72, 056502 (2005) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 63
APS Talk January 13, 2006 Electromagnetic Field Our goal is to find xμ(τ)=(ct(τ), x(τ), y(τ), z(τ)) when the 4 -velocity uμ(τ)=(cdt/dτ, dx/dτ, dy/dτ, dz/dτ)(τ) satisfies duμ/dτ= –e. Fμνuν/mc where τ is proper time. For any solution to the equations of motion. Proportional to amount frequencies up-shifted going to beam frame Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 64
ξ is exactly proportional to the proper time On the orbit Integrate with respect to ξ instead of τ. Now where the unitless vector potential is f(ξ)=-e. A(ξ )/mc 2. Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 65 APS Talk January 13, 2006
Electron Orbit Direct Force from Electric Field Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy APS Talk January 13, 2006 Ponderomotive Force Page 66
Energy Distribution Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy APS Talk January 13, 2006 Page 67
Effective Dipole Motions: Lab Frame And the (Lorentz invariant!) phase is Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 68 APS Talk January 13, 2006
Summary APS Talk January 13, 2006 • 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 Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 69
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 Lorentz contract + Doppler Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Double Doppler Page 70 APS Talk January 13, 2006
High Field Strength Thomson Backscatter APS Talk January 13, 2006 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 Forward Frequency Transverse Pattern NB, be careful with the radiation pattern, it is the same at small angles, but quite a bit different at large angles Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 71
Forward Direction: Flat Laser Pulse 20 -period equivalent undulator: Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 72 APS Talk January 13, 2006
APS Talk January 13, 2006 Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 73
Realistic Pulse Distribution at High a APS Talk January 13, 2006 In general, it’s easiest to just numerically integrate the lab-frame expression for the spectrum in terms of Dt and Dp. A 105 to 106 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<0. 1) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 74
Forward Direction: Gaussian Pulse APS Talk January 13, 2006 Apeak and λ 0 chosen for same intensity and same rms pulse length as previously Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 75
Radiation Distributions: Backscatter APS Talk January 13, 2006 Gaussian Pulse σ at first harmonic peak Courtesy: Adnan Doyuran (UCLA) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 76
Radiation Distributions: Backscatter APS Talk January 13, 2006 Gaussian π at first harmonic peak Courtesy: Adnan Doyuran (UCLA) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 77
Radiation Distributions: Backscatter APS Talk January 13, 2006 Gaussian σ at second harmonic peak Courtesy: Adnan Doyuran (UCLA) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 78
90 Degree Scattering Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy APS Talk January 13, 2006 Page 79
90 Degree Scattering APS Talk January 13, 2006 And the phase is Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 80
Radiation Distribution: 90 Degree APS Talk January 13, 2006 Gaussian Pulse σ at first harmonic peak Courtesy: Adnan Doyuran (UCLA) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 81
Radiation Distributions: 90 Degree APS Talk January 13, 2006 Gaussian Pulse π at first harmonic peak Courtesy: Adnan Doyuran (UCLA) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 82
Polarization Sum: Gaussian 90 Degree APS Talk January 13, 2006 Courtesy: Adnan Doyuran (UCLA) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 83
Radiation Distributions: 90 Degree APS Talk January 13, 2006 Gaussian Pulse second harmonic peak Second harmonic emission on axis from ponderomotive dipole! Courtesy: Adnan Doyuran (UCLA) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 84
THz Source Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy APS Talk January 13, 2006 Page 85
Radiation Distributions for Short High-Field Magnets APS Talk January 13, 2006 And the phase is Krafft, G. A. , to be published Phys. Rev. ST-AB (2006) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 86
Wideband THz Undulater APS Talk January 13, 2006 Primary requirements: wide bandwidth and no motion and deflection. Implies generate A and B by simple motion. “One half” an oscillation is highest bandwidth! Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 87
THz Undulator Radiation Spectrum Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 88 APS Talk January 13, 2006
Total Energy Radiated APS Talk January 13, 2006 Lienard’s Generalization of Larmor Formula (1898!) Barut’s Version Usual Larmor term From ponderomotive dipole Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 89
Some Cases APS Talk January 13, 2006 Total radiation from electron initially at rest For a flat pulse exactly (Sarachik and Schappert) Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 90
For Circular Polarization APS Talk January 13, 2006 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 (which goes back to Schott and the turn of the 20 th century!) and the general formula checks out Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 91
Conclusions APS Talk January 13, 2006 • Recent development of superconducting cavities has enabled CW operation at energy gains in excess of 20 MV/m, and acceleration of average beam currents of 10 s of m. A. • The ideas of Beam Recirculation and Energy Recovery have been introduced. How these concepts may be combined to yield a new class of accelerators that can be used in many interesting applications has been discussed. I’ve given you some indication about the historical development of recirculating SRF linacs. • The present knowledge on beam recirculation and its limitations in a superconducting environment, leads us to think that recirculating accelerators of several Ge. V energy, and with beam currents approaching those in storage ring light sources, are possible. Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 92
Conclusions APS Talk January 13, 2006 • 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 singlefrequency 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 Tera. Watt lasers, which have exceedingly high field and short pulses. Any calculation that does not include ponderomotive broadening is incorrect. Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 93
Conclusions APS Talk January 13, 2006 • 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. Thomas Jefferson National Accelerator Facility Operated by the Southeastern Universities Research Association for the U. S. Department of Energy Page 94
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