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Undulator Radiation & FEL U.S. Particle Accelerator School January 25 – February 19, 2021 VUV and X-ray Free-Electron Lasers Introduction, Electron Motions in an Undulator, Undulator Radiation & FEL Dinh C. Nguyen,1 Petr Anisimov,2 Nicole Neveu1 1 SLAC National Accelerator Laboratory 2 Los Alamos National Laboratory LA-UR-21-20610 Monday (Jan 25) Lecture Outline Time • VUV and X-ray FELs in the World 10:00 – 10:30 • Properties of electromagnetic radiation 10:30 – 10:50 • Break 10:50 – 11:00 • Electron motions in an undulator 11:00 – 11:20 • Undulator radiation 11:20 – 11:40 • Introduction to FELs 11:40 – Noon 2 VUV and X-ray FELs in the World 3 World Map of VUV and X-ray FELs FLASH FERMI European XFEL Italy Germany SPARC LCLS Italy SwissFEL POLFEL LCLS-II & LCLS-II-HE Switzerland Poland USA PAL XFEL S. Korea SACLA Japan SDUV-SXFEL SHINE China Blue=VUV to Soft X-ray 4 Purple=Soft to Hard X-ray Sub-systems of an RF Linac Driven X-ray FEL An RF-linac driven XFEL has the following sub-systems in order to produce Low-emittance • PHOTOINJECTOR to generate low-emittance electrons in ps bunches electron beams • RF LINAC to accelerate the electron beams to GeV energy • BUNCH COMPRESSORS to shorten the bunches and produce kA current High peak current • LASER HEATER to reduce the microbunching instabilities • BEAM OPTICS to transport the electron beams to the undulators • UNDULATORS to generate and amplify the radiation in a single pass Single-pass, high- gain X-ray FEL • DIAGNOSTICS to characterize the electron & FEL beams Laser Photoinjector heater L2 BC2 L1 L3 Optics Undulators L0 X-rays electrons Layout of the sub-systems of the LCLS first X-ray FEL 5 Linac Coherent Light Source LCLS-II LCLS-II-HE Photoinjector The last 1 km of the SLAC linac accelerates electrons up to 15 GeV Starting at Sector 20, electron bunches are Linac-to-Undulator Transport (340m) injected at 135 MeV 140-m long undulator hall with 2 side-by-side undulators, Hard X-ray (HXR) and Soft X-ray (SXR) Near Experimental Hall Far Experimental Hall 6 RF-linac Driven FEL Pulse Format Low-repetition-rate Mode (e.g., LCLS CuRF) ~8 ms (1/120 Hz) ~10 fs Burst Mode (e.g., Eu-XFEL) ~1 ms macropulse ~1 ms macropulse ~100 ms (1/10 Hz) ~200ns micropulses Continuous-Wave Mode (e.g., LCLS-II/HE, SHINE) ~1 ms (1/MHz) 7 LCLS-II-HE Soft X-ray Undulator Planar Undulators strongback magnetic field electron beam magnets 푢 X-ray polarization (same direction as Undulator magnetic field varies sinusoidally electron oscillations) with z and points in the y direction B = 퐵0푠푖푛 푘푢푧 풚ෝ Planar undulators produce linearly (plane) polarized radiation at the fundamental frequency and also at harmonic frequencies. 8 Helical Superconducting Undulators Superconducting coils are wound around the electron beam pipe in a helical pattern. Undulator magnetic field varies sinusoidally with z and points in both x and y directions in a helical fashion. B = 퐵0 푐표푠 푘푢푧 풙ෝ + 푠푖푛 푘푢푧 풚ෝ 2휋 Undulator wavenumber 푘푢 = 푢 Helical undulators produce circularly Snapshots of the helical polarized radiation at the fundamental undulator magnetic field at frequency. Helical undulators do not different locations in one produce undulator radiation at the undulator period. harmonic frequencies. 9 Delta Undulators (Variable Polarization) Varying the linear positions of the magnet jaws Delta undulator cross-section changes radiation polarization from linear to circular 10 X-ray FEL Wavelength for a Planar Undulator Undulator period FEL resonant wavelength 2 푢 퐾 Undulator parameter = 2 1 + 2훾 2 푒퐵0 푒푢퐵0 K = = Lorentz relativistic factor 푘 푚 푐 2휋푚 푐 (Dimensionless beam energy) 푢 푒 푒 Dimensionless K parameter is a measure 퐸푡표푡푎푙 of how much the electron beam is K = 0.934 퐵 훾 = 2 0 푢 푚푒푐 deflected transversely in the undulator 퐾 휃푚푎푥 = 푥 훾 B0 in tesla 푢 in cm radiation wave u + 휃푚푎푥 푧 e- trajectory u 11 Electron Beam Kinematics Dimensionless electron beam energy Electron velocity relative to the speed of light 푣 퐸푡표푡푎푙 훽 = 훾 = 2 푐 푚푒푐 Using the g-b relation, we can calculate b 2 퐸푡표푡푎푙 = 퐸푘 + 푚푒푐 1 1 훾 = 훽 = 1 − 1 − 훽2 2 Rest mass energy 훾 Kinetic energy 0.511 MeV Approximate b for 1 Approximate g for GeV electrons GeV electrons 훽 ≈ 1 − 2훾2 퐸푡표푡푎푙 ≈ 퐸푘 = 퐸푏 We shall see later the longitudinal component of 훾 ≈ 1957 퐸푏 퐺푒푉 the velocity, ʋz is reduced when the electrons undergo transverse oscillations in the undulator. 12 X-ray FEL Wavelength Tuning The FEL x-ray wavelength can be tuned by one of the following methods 1. varying the electron beam energy, Eb and thus the beam g 2. varying the gap by moving the magnet jaws symmetrically in and out, thus changing the K value 2 The on-axis magnetic field amplitude depends 푔 푔 on the gap-to-period ratio. The remanence 퐵0 푔, 푢 = 3.13퐵푟 푒푥푝 −5.08 + 1.54 푢 푢 magnetic field, 퐵 푟 of NdFeB is about 1.2 tesla. Small gap – Large K – Long Larger gap – Smaller K – Shorter gap gap e- beam e- beam poles magnets period 13 LCLS-II-HE Variable-Gap Soft X-ray Undulator SXR K parameterK vs gap (mm) This example illustrates how we change the LCLS X-ray FEL 10 9 wavelength by varying the magnet gap of the Soft X-ray 8 (SXR) undulator which has an undulator period of 56 mm. 7 6 The undulator K parameter decreases and photon energy K 5 4 increases as we increase the gap (right figures), using the 3 LCLS-II-HE 8-GeV electron beams as an example. 2 1 However, opening the gap to increase the photon energy 0 also reduces the FEL output pulse energy (below). 7.2 9.2 11.2 13.2 15.2Photon17.2 19.2 energy21.2 23.2 25.2 27.2 29.2 31.2 5000 3.5 4500 Photon energy (eV) vs gap (mm) 3 4000 2.5 3500 3000 Output 2 Photon 2500 pulse 1.5 energy energy (eV) 2000 1 (mJ) 1500 0.5 1000 0 500 500 750 250 0 1250 1500 1750 2000 2250 2500 2750 3000 3250 3500 3750 4000 4250 4500 4750 5000 5000 1000 7.2 9.2 11.2 13.2 15.2 17.2 19.2 21.2 23.2 25.2 27.2 29.2 31.2 Photon energy (eV) 14 gap (mm) Properties of Electromagnetic Radiation 15 EM Radiation in Free Space ℎ = 4.1357 ∙ 10−15 푒V ∙ 푠 푐 = 2.9979 ∙ 108 푚/푠 Low frequency Long wavelength ∙ ℎ휈 = ℎ푐 = 1.23984 ∙ 10−6푒V−m 1239.84 푒푉 Photon energy ℎ휈 = λ 푛푚 푥 푐 푦 푧 High frequency Short wavelength 16 Accelerated charged particles emit EM radiation Radiation from circular motions Bremsstrahlung Synchrotron radiation Radiation from oscillatory motions 17 Generations of Beam-based Radiation Sources • X-ray Tubes – X-ray tubes emit Bremsstrahlung and characteristic peaks – Characteristic peaks are narrow-line atomic transitions • Synchrotron Radiation (1st and 2nd Generation Light Sources) – Electrons going around bends produce synchrotron radiation – First generation SR operated as parasitic radiation devices – Second generation SR are dedicated to radiation production • Synchrotron Radiation (3rd Generation Light Sources) – 3rd Gen SR have low-emittance lattice with straight sections – Bending magnet and wiggler radiation is broadband – Undulator radiation has narrow spectral lines • X-ray Free-Electron Lasers (4th Generation Light Sources) – XFEL produce coherent, tunable, narrow-band x-rays – X-ray pulses are typically a few femtoseconds long – FEL peak brilliance is typically 10 orders of magnitude brighter than third-generation SR. 18 Electric Field of a Gaussian Wave-Packet Complex electric 푡−푡 2 − 0 field of a Gaussian 2 퐸(푧, 푡) = 퐸 푒 푘푧−휔푡+휓 푒 2휎푡 wave-packet 0 + c.c. Amplitude Phase Gaussian envelope rms temporal width Wavenumber Angular frequency 2휋 2휋푐 Radiation intensity 푘 = 휔 = 1 퐼 = 퐸 푧, 푡 2 2푍0 Normalized Fast carrier waves at w intensity envelope 1 2 퐼0 = 퐸0 2푍0 Gaussian envelope Impedance of free space = 120p 19 Fourier Transform a Gaussian Pulse Consider only the time-dependent part of a 푡2 Gaussian wave-packet. Its Fourier Transform 퐸0 − 2 ℰ(휔) = න 푒− 휔−휔푟 푡푒 2휎푡 푑푡 is a Gaussian spectrum centered at ± 휔푟 2휋 2 2 푡 휔−휔푟 − 2 퐸0 − 2 −휔푟푡 2휎 ℰ(휔) = 2휎휔 퐸(푡) = 퐸0푒 푒 푡 푒 2휎휔 Fourier Transform limit 2휎푡 of time-bandwidth product (rms widths) ℰ 휔 2 휎휔휎푡 = ½ 2휎휔 In general, TBW is larger than 1/2 휔푟 Time domain Frequency domain 20 Radiation Pulse & Time-Bandwidth Product Linear frequency Full-width-at-half-maximum (FWHM) in FWHM time 훿 푡 and linear frequency domain 훿휈 휔 휈 = 2휋 • Time-bandwidth product for a Gaussian pulse 2 4푙푛2 퐸 푡 d푡 훿휈 ∙ 훿푡 = 휎 휎 = 0.44 훿푡 = 2 2푙푛2 휎 휋 휔 푡 푡 • Multiply both sides by the Planck’s constant in eV-s -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 푡 − 푡0 ℎ = 4.136 ∙ 10−15 푒V-s 휎휔 ℎ훿휈 ∙ 훿푡 = 1.82 푒푉 ∙ 푓푠 훿휈 = 2 2푙푛2 2휋 훿휈 Energy (eV) – time FWHM product ℰ 휈 2 훿휀 ∙ 훿푡 ≥ 1.82 푒푉 ∙ 푓푠 휈 − 휈푟 21 Wave Equation & Helmholtz Equation • Wave equation 1 휕2 2 − 푢 풓, 푡 = 0 푐2 휕푡2 −휔 푡 • Solution to the wave equation 푢 풓, 푡 = Re 풓 푒 푟 Time-independent Time-dependent wave amplitude oscillatory term • Helmholtz equation for the time-independent wave amplitude 2 + 푘2 풓 = 0 풓 = 퐴 풓 푒풌·풓 22 Paraxial Approximation Paraxial wave equation for a wave 2 휕 propagating in the z direction 푇 − 2푖푘 퐴 푥, 푦, 푧 = 0 휕푧 Gaussian beam transverse amplitude T denotes transverse spreading due (beam propagating in the z direction) to optical diffraction 휕2 휕2 2 = + 2 2 2 푇 2 2 and 푘 = 푘푥 + 푘푦 + 푘푧 휕푥 휕푦 2 2 2 Paraxial approximation 푘푥 + 푘푦 ≪ 푘푧 푦 For axisymmetric Gaussian beams, 푘푥 = 푘푦 푟2 푟2 푤 − 푘 퐸(푟, 푧) = 퐸 0 푒 푤2 푧 푒 푘푧−휔푡+휓 푧 푒 2푅 푧 0 푤 푧 푥 Gouy phase shift 푤0 is the radius where the field Radius of curvature of the decays to 1/e of E0 at the beam waist 푤 푧 z Gaussian beam wavefront is the 1/e radius at location 23 Gaussian Beam Intensity & Diffraction 퐼 • 1 0 Optical intensity 퐼 푟, 푧 = 퐸 푟, 푧 2 2푍0 2푟2 퐼 푟 • Gaussian beam 푤 2 − 0 푒 푤2 퐼(푟, 푧) = 퐼0 푤0 푤 퐼0 푧2 푒2 푤 = 푤0 1 + 2 -4.0 -3.0 -2.0 -1.0 0.0
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