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 • Properties of electromagnetic radiation • Break 10:00 – 10:30 10:30 – 10:50 10:50 – 11:00 11:00 – 11:20 11:20 – 11:40 11:40 – Noon • Electron motions in an undulator • Undulator radiation • Introduction to FELs 2VUV and X-ray FELs in the World 3World Map of VUV and X-ray FELs FLASH FERMI European XFEL Italy Germany SPARC Italy LCLS SwissFEL POLFEL Poland LCLS-II & LCLS-II-HE USA Switzerland PAL XFEL S. Korea SACLA Japan SDUV-SXFEL SHINE China Blue=VUV to Soft X-ray Purple=Soft to Hard X-ray 4Sub-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 Single-pass, high- • UNDULATORS to generate and amplify the radiation in a single pass gain X-ray FEL • DIAGNOSTICS to characterize the electron & FEL beams Laser <ul style="display: flex;"><li style="flex:1">Photoinjector heater </li><li style="flex:1">BC2 </li></ul>L2 L3 Undulators Optics L1 L0 X-rays electrons 5Layout of the sub-systems of the LCLS first X-ray FEL 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 injected at 135 MeV Linac-to-Undulator Transport (340m) 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 6RF-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) <ul style="display: flex;"><li style="flex:1">~1 ms macropulse </li><li style="flex:1">~1 ms macropulse </li></ul>~100 ms (1/10 Hz) ~200ns micropulses Continuous-Wave Mode (e.g., LCLS-II/HE, SHINE) ~1 ms (1/MHz) 7LCLS-II-HE Soft X-ray Undulator Planar Undulators strongback magnetic field electron beam magnets 푢 X-ray polarization (same direction as electron oscillations) Undulator magnetic field varies sinusoidally with z and points in the y direction ෝB = 퐵0푠푖푛 푘푢푧 풚 Planar undulators produce linearly (plane) polarized radiation at the fundamental frequency and also at harmonic frequencies. 8Helical 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. <ul style="display: flex;"><li style="flex:1">ෝ</li><li style="flex:1">ෝ</li></ul>B = 퐵0 푐표푠 푘푢푧 풙 + 푠푖푛 푘푢푧 풚 2휋 Undulator wavenumber 푘푢 = 푢 Helical undulators produce circularly polarized radiation at the fundamental frequency. Helical undulators do not produce undulator radiation at the harmonic frequencies. Snapshots of the helical undulator magnetic field at different locations in one undulator period. 9Delta Undulators (Variable Polarization) Varying the linear positions of the magnet jaws changes radiation polarization from linear to circular Delta undulator cross-section 10 X-ray FEL Wavelength for a Planar Undulator Undulator period FEL resonant wavelength 푢 2훾2 퐾2 2Undulator parameter  = 1 + ꢃ퐵0 ꢃ푢퐵0 K = =Lorentz relativistic factor (Dimensionless beam energy) <ul style="display: flex;"><li style="flex:1">푘푢ꢂ푒푐 </li><li style="flex:1">2휋ꢂ푒푐 </li></ul>Dimensionless K parameter is a measure of how much the electron beam is deflected transversely in the undulator 퐸푡ꢁ푡푎푙 훾 = K = 0.934 퐵0푢 ꢂ푒푐2 퐾휃푚푎ꢀ =푥훾B0 in tesla 푢in cm u +  radiation wave 휃푚푎ꢀ 푧e- trajectory u 11 Electron Beam Kinematics Dimensionless electron beam energy Electron velocity relative to the speed of light 푣 훽 = 푐퐸푡ꢁ푡푎푙 훾 = ꢂ푒푐2 Using the g-b relation, we can calculate b 퐸푡ꢁ푡푎푙 = 퐸ꢄ + ꢂ푒푐2 11훾2 훾 = 훽 = 1 − 1 − 훽2 Rest mass energy Kinetic energy 0.511 MeV Approximate b for GeV electrons 1Approximate g for GeV electrons 훽 ≈ 1 − 2훾2 퐸푡ꢁ푡푎푙 ≈ 퐸ꢄ = 퐸푏 We shall see later the longitudinal component of the velocity, ʋz is reduced when the electrons undergo transverse oscillations in the undulator. 훾 ≈ 1957 퐸푏 퐺ꢃ푉 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 2The on-axis magnetic field amplitude depends on the gap-to-period ratio. The remanence <ul style="display: flex;"><li style="flex:1">푔</li><li style="flex:1">푔</li></ul>퐵0 푔, 푢 = 3.13퐵푟 ꢃ푥푝 −5.08 + 1.54 <ul style="display: flex;"><li style="flex:1">푢 </li><li style="flex:1">푢 </li></ul>퐵magnetic field, of NdFeB is about 1.2 tesla. 푟Larger gap – Smaller K – Shorter  Small gap – Large K – Long  gap gap e- beam e- beam magnets poles 13 period LCLS-II-HE Variable-Gap Soft X-ray Undulator KSXR K parameter vs gap (mm) 10 This example illustrates how we change the LCLS X-ray FEL 9wavelength by varying the magnet gap of the Soft X-ray (SXR) undulator which has an undulator period of 56 mm. 8765KThe undulator K parameter decreases and photon energy 4increases as we increase the gap (right figures), using the LCLS-II-HE 8-GeV electron beams as an example. 321However, opening the gap to increase the photon energy also reduces the FEL output pulse energy (below). 07.2 9.2 11.2 13.2 15P.2h1o7t.2o1n9.2en21e.2rg23y.2 25.2 27.2 29.2 31.2 5000 3.5 4500 Photon energy (eV) vs gap (mm) 3 2.5 24000 3500 3000 2500 2000 1500 1000 500 Photon energy (eV) Output pulse energy (mJ) 1.5 10.5 0 0 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 14 Photon energy (eV) gap (mm) Properties of Electromagnetic Radiation 15 EM Radiation in Free Space ℎ = 4.1357 ∙ 10−15 ꢃV ∙ 푠 푐 = 2.9979 ∙ 108 ꢂ/푠 <ul style="display: flex;"><li style="flex:1">Low frequency </li><li style="flex:1">Long wavelength </li></ul>= 1.23984 ∙ 10−6ꢃV−m  ∙ ℎ휈 = ℎ푐 1239.84 ꢃ푉 ℎ휈 = Photon energy λ 푛ꢂ 푥푐푦 푧<ul style="display: flex;"><li style="flex:1">High frequency </li><li style="flex:1">Short wavelength </li></ul>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 2Complex electric field of a Gaussian wave-packet ꢈ−ꢈ 0−2<ul style="display: flex;"><li style="flex:1">퐸(푧, ꢅ) = 퐸0ꢃꢆ ꢄꢇ−휔푡+휓 </li><li style="flex:1">ꢃ</li></ul>+ c.c. 2휎 ꢈPhase Amplitude Gaussian envelope rms temporal width Wavenumber Angular frequency 2휋 푘 = 2휋푐 ꢉ = Radiation intensity 12<ul style="display: flex;"><li style="flex:1">퐼 = </li><li style="flex:1">퐸 푧, ꢅ </li></ul>2푍0 Normalized intensity envelope Fast carrier waves at w 12퐼0 = 퐸0 2푍0 Gaussian envelope Impedance of free space = 120p  19 Fourier Transform a Gaussian Pulse 2Consider only the time-dependent part of a 푡−퐸0 න ꢃ−ꢆ 휔−휔 푡 <ul style="display: flex;"><li style="flex:1">ꢃ</li><li style="flex:1">2 푑ꢅ </li></ul>Gaussian wave-packet. Its Fourier Transform 2ꢋ ꢊℰ(ꢉ) = ꢈ2휋 ꢉis a Gaussian spectrum centered at ± 푟2 2휔−휔 푡ꢊꢃ− −퐸0 2 22ꢋ <ul style="display: flex;"><li style="flex:1">퐸(ꢅ) = 퐸0ꢃ−ꢆ휔 푡 </li><li style="flex:1">ꢃ</li></ul>2ꢋ <ul style="display: flex;"><li style="flex:1">ꢍ</li><li style="flex:1">ꢊ</li></ul>ℰ(ꢉ) = ꢈ2ꢌ휔 Fourier Transform limit of time-bandwidth 2ꢌ푡 product (rms widths) 2ℰ ꢉ ꢌ휔ꢌ푡 = ½ 2ꢌ휔 In general, TBW is larger than 1/2 ꢉ푟 Time domain Frequency domain 20 Radiation Pulse & Time-Bandwidth Product Linear frequency Full-width-at-half-maximum (FWHM) in FWHM ꢉ<ul style="display: flex;"><li style="flex:1">훿ꢅ </li><li style="flex:1">훿휈 </li></ul>time and linear frequency domain 휈 = 2휋 • Time-bandwidth product for a Gaussian pulse 2퐸 ꢅ dꢅ 4ꢎ푛2 <ul style="display: flex;"><li style="flex:1">훿휈 ∙ 훿ꢅ = </li><li style="flex:1">ꢌ휔ꢌ푡 = 0.44 </li></ul>훿ꢅ = 2 2ꢎ푛2 ꢌ푡 휋• Multiply both sides by the Planck’s constant in eV-s <ul style="display: flex;"><li style="flex:1">-4.0 </li><li style="flex:1">-3.0 </li><li style="flex:1">-2.0 </li><li style="flex:1">-1.0 </li><li style="flex:1">0.0 </li><li style="flex:1">1.0 </li><li style="flex:1">2.0 </li><li style="flex:1">3.0 </li><li style="flex:1">4.0 </li></ul>ꢅ − ꢅ0 ℎ = 4.136 ∙ 10−15 ꢃV-s ꢌ휔 2휋 훿휈 = 2 2ꢎ푛2 ℎ훿휈 ∙ 훿ꢅ = 1.82 ꢃ푉 ∙ 푓푠 훿휈 Energy (eV) – time FWHM product 2ℰ 휈 훿휀 ∙ 훿ꢅ ≥ 1.82 ꢃ푉 ∙ 푓푠 <ul style="display: flex;"><li style="flex:1">-4.0 </li><li style="flex:1">-3.0 </li><li style="flex:1">-2.0 </li><li style="flex:1">-1.0 </li><li style="flex:1">0.0 </li><li style="flex:1">1.0 </li><li style="flex:1">2.0 </li><li style="flex:1">3.0 </li><li style="flex:1">4.0 </li></ul>휈 − 휈푟 21 Wave Equation & Helmholtz Equation 1 휕2 푐2 휕ꢅ2 • Wave equation 2 − ꢏ 풓, ꢅ = 0 ꢏ 풓, ꢅ = Re  풓 ꢃ−ꢆ휔 푡 ꢊ• Solution to the wave equation Time-independent wave amplitude Time-dependent oscillatory term • Helmholtz equation for the time-independent wave amplitude 2 + 푘2  풓 = 0  풓 = 퐴 풓 ꢃꢆ풌·풓 22 Paraxial Approximation Paraxial wave equation for a wave 휕ꢒ푇 − 2푖푘 퐴 푥, 푦, 푧 = 0 propagating in the z direction 휕푧 Gaussian beam transverse amplitude (beam propagating in the z direction) T denotes transverse spreading due to optical diffraction <ul style="display: flex;"><li style="flex:1">휕ꢒ </li><li style="flex:1">휕ꢒ </li></ul>ꢒ푇 = +휕푥ꢒ 휕푦ꢒ and 푘 = 푘ꢀꢒ + 푘ꢐꢒ + 푘ꢇꢒ 푘ꢀꢒ + 푘ꢐꢒ ≪ 푘ꢇꢒ Paraxial approximation 푦푘 = 푘 For axisymmetric Gaussian beams, <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢐ</li></ul>2 2<ul style="display: flex;"><li style="flex:1">푟</li><li style="flex:1">푟</li></ul>ꢃ− ꢃꢆ ꢄꢇ−휔푡+휓 ꢇ ꢃꢆ ꢄ ꢑ02<ul style="display: flex;"><li style="flex:1">ꢑ ꢇ </li><li style="flex:1">ꢒ푅 ꢇ </li></ul>퐸(ꢓ, 푧) = 퐸0 ꢑ ꢇ 푥Gouy phase shift 푤0 is the radius where the field Radius of curvature of the Gaussian beam wavefront decays to 1/e of E0 at the beam waist 푤 푧 is the 1/e radius at location z 23 Gaussian Beam Intensity & Diffraction 퐼0 1• Optical intensity ꢒ<ul style="display: flex;"><li style="flex:1">퐼 ꢓ, 푧 = </li><li style="flex:1">퐸 ꢓ, 푧 </li></ul>2푍0 퐼 ꢓ 2ꢒ푟 ꢑ ꢒ• Gaussian beam ꢃ− ꢑ0 2퐼(ꢓ, 푧) = 퐼0 푤0 퐼0 ꢑꢃꢒ 푧ꢒ 푧Rꢒ <ul style="display: flex;"><li style="flex:1">-4.0 </li><li style="flex:1">-3.0 </li><li style="flex:1">-2.0 </li><li style="flex:1">-1.0 </li><li style="flex:1">0.0 </li><li style="flex:1">1.0 </li><li style="flex:1">2.0 </li><li style="flex:1">3.0 </li><li style="flex:1">4.0 </li></ul>푤 = 푤0 1 + Beam profile at beam waist • Gaussian beam diffracting from the beam waist Far-field divergence half-angle 푤0 휆 휃 = = 푧푅 휋푤0 휆 휃푤0 = 휋푧푅 24 Radiation Beam FWHM, Radius and Emittance Gaussian beam radial FWHM 퐼0 ( 푤0 = 1/e2 radius) 훿ꢓ퐹푊퐻푀 = 2ꢎ푛2 푤0 Gaussian beam angular divergence FWHM ꢌ푟 ꢌ푟′ rms 퐼 = .606퐼0 (휃 = 1/e2 half-angle) 퐼 ꢓ 훿ꢓꢔ퐹푊퐻푀 = 2ꢎ푛2 휃 FHWM 퐼 ꢓ′ 푤0 ꢌ푟 = rms beam radius 1ꢃꢒ 2푤0 휃 퐼 = .135퐼0 휃2rms angular divergence ꢌ푟′ = <ul style="display: flex;"><li style="flex:1">-4.0 </li><li style="flex:1">-3.0 </li><li style="flex:1">-2.0 </li><li style="flex:1">-1.0 </li><li style="flex:1">0.0 </li><li style="flex:1">1.0 </li><li style="flex:1">2.0 </li><li style="flex:1">3.0 </li><li style="flex:1">4.0 </li></ul>휖푟 = ꢌ푟ꢌ푟′ Radial dimension (or angle) Gaussian beam emittance Photon beam emittance for transversely incoherent (not diffraction limited) radiation 휆 휆ꢕꢒ > 1

Undulator Radiation & FEL

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