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 푒푢퐵0 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 푒𝑖 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−𝑖 휔−휔푟 푡푒 2휎푡 푑푡 is a Gaussian spectrum centered at ± 휔푟 2휋

2 2 푡 휔−휔푟 − 2 퐸0 − 2 −𝑖휔푟푡 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

−𝑖휔 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 푒𝑖풌·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 − 𝑖 푘 퐸(푟, 푧) = 퐸 0 푒 푤2 푧 푒𝑖 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 1.0 2.0 3.0 4.0 푧R Beam profile at beam waist • Gaussian beam diffracting from the beam waist Far-field divergence half-angle 푤 휆 휃 = 0 = 푧푅 휋푤0 휆 휃푤0 = 푧푅 휋 24 Radiation Beam FWHM, Radius and Emittance

Gaussian beam radial FWHM 퐼0

2 훿푟퐹푊퐻푀 = 2푙푛2 푤0 ( 푤0 = 1/e radius) Gaussian beam angular divergence FWHM 휎푟 rms 퐼 = .606퐼0 휎 ′ 훿푟′ = 2푙푛2 휃 (휃 = 1/e2 half-angle) 퐼 푟 푟 퐹푊퐻푀 FHWM 퐼 푟′ 푤0 rms beam radius 휎 = 푟 1 2 푤 퐼 = .135퐼 0 푒2 0 휃 휃 rms angular divergence 휎푟′ = 2 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 ′ Gaussian beam emittance 휖푟 = 휎푟휎푟 Radial dimension (or angle)

휆 Photon beam emittance for 휆 휖 = 2 2 푟 transversely incoherent (not 휖푟 = 푀 푀 > 1 4휋 diffraction limited) radiation 4휋

25 Photon Beam Transverse Coherence

2a Screen

Incoherent x' Narrow slit Young’s monochromatic diameter = 2a double slit diffuse source (e.g., Na lamp) x Condition for interference Coherent phase-space area 푎 pattern to appear on screen Slope = a 퐷 휆 퐴 ≤ 퐴 = 2휋휖푟 Radiation phase-space 2 26 Spontaneous vs Stimulated Emission

Spontaneous emission Pump (source of energy) 3 2 pump out

Gain Medium 1

Stimulated emission Pump (source of energy)

Partial 3 Total 2 reflector reflector in out

Gain Medium 1 Population inversion 27 How does an FEL differ from a laser?

▪ FEL gain medium is free electrons in vacuum (lasers have bound electrons)

▪ FEL have broad wavelength tunability (lasers have no or limited tunability)

▪ FEL beams are distortion-free (laser gain media have optical distortions)

▪ FEL work at x-ray wavelengths (x-ray laser upper state lifetimes are too short)

▪ The coherence length of a SASE FEL is much shorter than that of a typical laser.

28 Electron Motions in an Undulator

29 Lorentz Force

Lorentz force governs the rate of change in the electron beam energy and momentum 퐅 = −푒 퐄 + 훖 × 퐁

Force caused by an electric field acts along the electron beam propagation direction, thus changing the beam energy Δ푊 = න퐅 ⋅ 푑퐬 = − න푒퐄 ⋅ 푑퐬

Force caused by a magnetic field is perpendicular to the beam propagation direction, thus changing the beam momentum by Dp and the beam direction by Dp/p0. Magnetic force does not change the beam energy. Δ퐩 = න퐅 푑푡 = − න푒 훖 × 퐁 푑푡

30 Planar Hybrid Permanent Magnet Undulators Steel poles Magnets Electrons travel mainly in the z direction x y Electrons also have a small initial velocity in z The on-axis ( 푦 = 0 ) magnetic field is x sinusoidal with z and points in the y direction

B = 퐵0푠푖푛 푘푢푧 푦ො  2휋 B u Undulator wavenumber 푘푢 = 푢 Lorentz force F = −e v × B

The Lorentz force imparts a force in the x direction that is sinusoidal with z and opposes the electrons’ motion (electrons going into the page experience a force pointing out of the page, and vice versa). 31 Transverse Motion in a Planar Undulator Electrons enter the undulator with a small initial Transverse acceleration velocity 푣 푥 . Lorentz force is the restoring force 푑풑푥 푑퐯푥 that brings them back to the equilibrium position, = 훾푚푒 = −푒푣푧 퐵0푠푖푛 푘푢푧 푥ො similar to an oscillating mass on a spring. 푑푡 푑푡

푑푣푥 푑푣푥 푒퐵0 = = − sin(푘푢푧) 푣푧푑푡 푑푧 훾푚푒

Integrate with respect to z 푒퐵0 푣푥 = c cos(푘푢푧) 훾푚푒푘푢c 퐾 Transverse velocity 푣 = 푐 cos(푘 푧) 푥 훾 푢 퐾 Transverse displacement 푥 = sin(푘푢푧) 훾푘푢 32 Longitudinal Motion in a Planar Undulator

푥 훽 푣푥 = 0 훽푥 휃푚푎푥 푣푥 = 푚푎푥 훽푧 Transverse velocity in x relative to c Longitudinal velocity in z relative to c

2 2 훽푥 = 훽푠푖푛휃 훽푧 = 훽 − 훽푥

Average transverse velocity squared 2 훽푥 2 휃 훽푧 = 훽 1 − 2 훽2 ≈ 훽2 푚푎푥 훽 푥 2 퐾 휃 = Average longitudinal velocity along the undulator 푚푎푥 훾 2 2 2 2 퐾 1 퐾 훽푥 ≈ 훽 훽ҧ ≈ 1 − − 2훾2 푧 2 2 2훾 4훾 33 Moving with the Electrons 푢 • In the laboratory frame, the electrons are moving in the z direction at the average longitudinal velocity 1 y 푣푧 1 퐾2 2 푣ഥ = 푐 1 − 1 + 푧 훾2 2 x

z • In the frame moving with the electrons, the “beam Laboratory frame frame,” the undulator period is shortened by g* 푢 ∗ 1 ′ = 훾 = 푢 훾∗ 2 푣ҧ 푣′ 1 − 푧 y' 푧 푐2 훾 x' 훾∗ = 퐾2 1 + z' Beam frame 2 34 Oscillations in Electron Beam Frame

′ In the beam frame, the undulator is a traveling EM wave with the Lorentz contracted period  푢 ′ ′ The electron oscillates at a wavelength equal to the contracted undulator period,  = 푢 E ′ ′ 푢 푣푧 B

FEL wavelength in beam frame   퐾2 ′ = 푢 = 푢 1 + 훾∗ 훾 2

In the beam frame, the electron oscillates transversely (along the x' axis) and also longitudinally (along the z' axis) at twice the frequency of the transverse oscillations. This figure-8 motion gives rise to radiation at the harmonics of the fundamental frequency. 35 Fast and Slow Electron Motions Motion What causes the motion

▪ Fast transverse motion in x Lorentz force due to 푣푧 cross By ▪ Once every undulator period

▪ Fast longitudinal motion in z Modulations of 푣푥 in a planar undulator ▪ Twice every undulator period (Helical undulators do not have this motion)

▪ Slow transverse motion Weak focusing due to transverse field gradient ▪ Occurring over many undulator periods Strong focusing due to external quadrupoles

▪ Slow longitudinal motion Microbunching due to FEL interaction ▪ Occurring over the entire undulator length

36 Undulator Radiation

37 Synchrotron Radiation (SR) Synchrotron Radiation Facility Spectral property

Broadband

Sharp lines

38 Electrons radiate when they are accelerated

Electron trajectory

In the electron rest frame, their motions are non-relativistic. We can calculate the total power radiated by an oscillating dipole.

Larmor formula

6 2 2 훾1 푒 22 퐯 × 퐯ሶ 푃 = 3 퐯퐯ሶ ሶ − 2 6휋휀0 푐 푐

In the Lab frame, electrons’ motions are relativistic and we need to use the relativistic Larmor formula. The electrons emit the highest radiation power where they experience the greatest acceleration. 39 Relativistic Doppler Shift

In the beam frame, the electron oscillates like a dipole antenna and emits radiation in a dipole radiation pattern with two spherical lobes propagating in opposite directions.

Transforming back to the laboratory frame, the lobes get Doppler shifted and turn into a narrow x-ray beam with a half-cone angle of 1/g* 1/g* ′  = 2훾∗ Length Contraction & Doppler Shift 2 1 1 1 푢 퐾 x =  = 1 + 훾∗ 2훾∗ 2훾∗2 2훾2 2

40 Undulator Radiation Wavelength Consider an observer 푡 =푡 =0.5ΔΔ푡푡 휆 looking at the electron at 휆푢푐표푠휃 an angle q w.r.t. the z axis. z 휃 휆 In the time Dt the electron 푣푧ҧ 푢 travels one period, 휆푢

휆푢 휆푢푐표푠휃 + 휆 Average velocity along z = 푣푧ҧ 푐 휆푢 휆푢푐표푠휃 1 퐾2 − = 휆 푣ҧ = 푐 1 − 1 + 푣푧ҧ 푐 푧 2훾2 2 Fundamental undulator wavelength Small-angle approximation 2 2 휆푢 퐾 휆푢 휃 휆 = 1 + + 훾2휃2 휆 ≈ −휆 1 − 2 1 퐾2 푢 2 2훾 2 1 − 1 + 2훾2 2 41 Undulator Radiation Harmonics Radiation beam transverse profile Harmonics wavelength Harmonics number 휆 퐾2 휆 = 푢 1 + + 훾2휃2 m = 1 푚 푚 2훾2 2

m = 2 m=1

m = 3 m=3

m=5 m = 4

42 SR Brilliance and Photon Energy Spectra

Undulator radiation spectrum with odd harmonics. The gray lines denote energy tuning curves toward higher energy by opening up the magnet gap.

1.4E+15

1.2E+15 m=1

1E+15

8E+14

6E+14 m=3

4E+14

m=5 2E+14 m=7 m=9 0 100 1,000 1,900 2,800 3,700 4,600 5,500 6,400 7,300 8,200 9,100 10,000

Photon Energy (eV) 43 Undulator Radiation Spectral Property

2 Peak ∝ 푁푢

∆휔 1 ∝ 휔0 푁푢

A single electron traversing an undulator with Nu periods will produce a constant amplitude train of 2 ∆휔 electromagnetic waves with Nu wavelengths. 2 푠푖푛 휋푁푢 푑 푈 휔0 The Fourier Transform of a constant amplitude 2 2 ∝ 푁푢 퐸푏 2 푑Ω푑휔 ∆휔 wave train with Nu wavelengths is a sinc function 휋푁푢 휔0 with a spectral FWHM of approximately 1/Nu The number of photons within the coherent angular 퐾 2 and spectral bandwidths is proportional to the number 푁 = 휋훼푁 푐표ℎ푒푟푒푛푡 푏 1 + 퐾2 44 of electrons and the fine-structure constant, 훼 = 1/137 Undulator Radiation and FEL Brilliance

Peak Brilliance # 푝ℎ표푡표푛푠 (Spectral Brightness) 퐵 = 푝 ∆퐸 퐴 퐴 휏 푥 푦 퐸 Phase-space areas in x and y Relative x-ray energy bandwidth Pulse length

Und. Radiation FEL # photons/pulse ~ 108 ~ 1012

Phase space area 퐴푥 > 100 퐴ℎ휈 퐴ℎ휈 Relative BW ~ 1% ~ 0.1% Pulse length ~ ps 10s of fs Total increase 1 108 - 1010

45 Particle Transverse Positions and Angles

Consider a single electron (red) in an ensemble of billions of particles co-traveling with the reference particle (blue)

푥 푥′

푥 is the transverse position of the particle relative to the reference particle 푥′ is the angle the particle makes with respect to the reference particle’s trajectory 푑푥 푝 푣 푥′ = = 푥 ≈ 푥 푑푧 푝푧 푐 Similarly, the particle is also described by its transverse position 푦 and angle 푦′ Paraxial approximation: transverse velocities are much smaller than c so the angles 푥′ and 푦′ ≪ 1

46 Ensemble Averages and rms Values

Ensemble average value of 푥2 푥2 = න푥2푓 푥, 푥′, 푦, 푦′ 푑푥 푑푥′푑푦 푑푦′

휎푥′ Ensemble average value of 푥′2 푥′2 = න푥′2푓 푥, 푥′, 푦, 푦′ 푑푥 푑푥′푑푦 푑푦′

휎 Ensemble average value of 푥푥′ 푥 For cases where f is a Gaussian distribution, the ′ ′ ′ 푥푥 = න푥푥 푓 푥, 푥′, 푦, 푦′ 푑푥 푑푥 푑푦 푑푦′ ellipse corresponding to rms values, 휎푥 and 휎푥′ (blue) with a phase-space area equal to 휋휀푥,푟푚푠 encompasses only 39% of the particles. Going to 푥푥′ is the correlation between the particle’s position 2휎푥 and 2휎푥′ increases the phase-space area to and angle (푥푥′ is also known as the flow). 4휋휀푥,푟푚푠 (black) and raises the fraction of particles enclosed in the ellipse to 87%. 47 Normalized and Un-normalized Emittance Un-normalized rms emittance in x

2 ′2 ′ 2 휀푥,푟푚푠 = 푥 푥 − 푥푥

Un-normalized emittance is larger at low energy

푥′ Un-normalized emittance decreases as the beams are accelerated to higher energy (adiabatic damping) 푥′ To compare emittance of particle beams with different energy, we “normalize” the emittance by multiplying it by 훽훾 (or 훾 since 훽~1 ). The normalized emittance is 2 ′2 ′ 2 conserved in the absence of non-linear forces. 휀푛,푟푚푠 = 훽훾 푥 푥 − 푥푥

48 Photon Beam Emittance 푤0 1 • Consider the TEM00 of a Gaussian beam at the beam waist Τ푒 휃 푟2 푟2 퐸(푟) = 퐸0푒푥푝 − 2 = 퐸0푒푥푝 − 2 푤0 4휎푟

푤0 rms beam radius 휎푟 = 푧푅 2 퐸(푟) 휋푤2 휆 푧 = 0 휃푤 = 푅 휆 0 휋 • We also write the electric field as a function of beam divergence

푟′2 푟′2 ℇ(푟′) = ℇ 푒푥푝 − = ℰ 푒푥푝 − Photon beam rms emittance 0 휃2 0 4휎2 푟′  휃 휀푟 = 휎푟휎푟′ = rms angular divergence 휎 ′ = 4휋 푟 2

49 Beam-limited Radiation Brightness

Phase space area in x, y 퐴푥 = 2휋푥푥′ 퐴푦 = 2휋푦푦′

2 2 Source size Σ푥 = 휎푥 + 휎푟

Angular divergence 2 2 Σ푥′ = 휎푥′ + 휎푟′

Third generation undulator radiation is e-beam emittance dominated

휎푥 ≫ 휎푟 휎푥′ ≫ 휎푟′ 휀푥 = 휎푥휎푥′ ≫ 휀푟

# 푝ℎ표푡표푛푠 Undulator radiation brightness ℱ ℱ ≡ Spectral photon flux 0.1% 퐵푊 ∙ 푠 ℬ푈푅 = 2 4휋 휀푥휀푦

50 Diffraction-limited Radiation Brightness If the electron beam emittance is less than or equal to the radiation beam emittance, the output radiation is considered diffraction limited 휆 휀 ≤ 푥,푦 4휋 Diffraction-limited phase-space area

퐴푟 = 2휋휎푟휎푟′ = 2휋휀푟 휆 퐴 = 푟 2 Diffraction-limited radiation brightness 4ℱ ℬ = 퐷퐿 휆2

This is true for both FEL and diffraction-limited synchrotron radiation 51 Benefits of FEL over Undulator Radiation

• Small beam size Full transverse coherence Diffraction limited radiation • Low angular divergence Coherent diffractive imaging

• Femtosecond pulses Time-resolved studies of physical, chemical, biological and materials science dynamics

• Narrow spectral BW High-resolution spectroscopy

X-ray diffraction of small crystals, single viruses, etc. • Higher photon flux Better signal-to-noise ratios

52 Introduction to FEL

53 Electron Position & Velocity in the Undulator

Electron position and velocity in one undulator period 1 퐾 푥 = sin(푘 푧) 푧 = 0 푢 푧 = 푢 푘푢 훾 x = 0

퐾 푣푥 = 푐 cos(푘푢푧) Example:  = 5.6 cm, K = 3 and E = 8 푣 훾 푢 b x v GeV, the maximum transverse position 휃 and angle of the trajectory are: 푣z 1 푥푚푎푥 = 1.7 휇푚 2 2 1 퐾 2 푣 = 푐 1 − − 푐표푠 푘 푧 휃푚푎푥 = 192 휇푟푎푑 푧 훾2 훾2 푢 54 Lorentz Force in a Planar Undulator

The v x B force produces the transverse acceleration, i.e., rate of change in the electrons’ transverse momentum. By integrating the rate of change in the relativistic transverse momentum, we obtain the electrons’ sinusoidal velocity along the x direction. 푑풑 = −푒흼 × 퐁 푑푡 푑 훾푚푒푣푥 = −푒푣푧퐵푦 푒퐵0 푑푡 푣푥 = 푐 cos(푘푢푧) 훾푚푒푐푘푢 The rate of change in the electron energy (W) is proportional to the dot product of the transverse electron current (A-m) and radiation beam transverse electric field (V/m). v x 푑푊 푑훾 = 풋 ∙ 퐄 푚 푐2 = −푒푣 퐸 t 푑푡 푒 푑푡 푥 푥 j E

55 Resonant Wavelength

푢 + 푟



푢 In the time the electrons travel one undulator period (blue), the optical wave (red) has traveled one undulator period plus one wavelength. The wave slips ahead of the electron one wavelength. This special wavelength is called the Resonant Wavelength.

  +   푐 푟 1 푢 푢 푟 푟 = − 1 = = − 1  1 퐾2 푣ഥ푧 푐 푢 푣ഥ푧 푢 1 − 1 + 2훾2 2  퐾2  = 푢 1 + 푟 2훾2 2

56 Electron-Wave Energy Exchange

Transverse electron Rate of energy exchange Transverse electric field current density 푑푊 of the optical radiation = 퐣 ∙ 퐄 푗푥 = −푒푣푥 푑푡 퐸푥 = 퐸0 푐표푠 푘푧 − 휔푡 + 휑 Energy exchange occurs via the vector dot product between the transverse electron current density and the transverse electric field of the radiation 푐퐾 푗푥 = −푒 푐표푠 푘푢푧 푑 푐퐾퐸 훾 훾푚 푐2 = −푒 0 푐표푠 푘 푧 푐표푠 푘푧 − 휔푡 + 휑 푑푡 푒 훾 푢 푑훾 푐퐾퐸 0 푐표푠 푘 + 푘 푧 + 휑 − 휔푡 + 푐표푠 푘 − 푘 푧 + 휑 − 휔푡 = −푒 2 푢 푢 푑푧 2훾푚푒푐 휓 This phase can be made constant with a judicious choice of k 57 Resonant Wavenumber

Ponderomotive phase The ponderomotive phase remains constant with time as the electrons travel along z for a special 휓 = 푘푢 + 푘푟 푧 + 휑 − 휔푟푡 wavenumber called the resonant wavenumber 푘푟 Differentiate with respect to t and set it to zero 푑휓 = 푘 + 푘 푣ഥ − 휔 = 0 Average axial velocity of the nth electron 푑푡 푢 푟 푧 푟 1 퐾2 푣ഥ푧푛 ≈ 푐 1 − 2 1 + Divide both sides by c 2훾푛 2 1 퐾2 푘푢 + 푘푟 1 − 2 1 + − 푘푟 = 0 Resonant wavenumber 2훾푛 2 2 2훾푟 2 푘 = 푘 푘푟 퐾 푟 푢 퐾2 푘푢 = 2 1 + 1 + 2훾푛 2 2

58 Resonance Condition Energy Gain

Snap shots of an optical wave (red) traveling collinearly with an electron (black circle) that follows a sinusoidal trajectory (blue) at three different points along an undulator period from top to bottom.

The ponderomotive phase is equal to -p/2. The wave electric field vector points in the opposite direction of the transverse electron velocity. The electron is accelerated by the optical electric field.

The rate of energy exchange is positive, i.e., the electron gains energy from the optical wave.

59 Resonance Condition Energy Loss

The ponderomotive phase is equal to p/2. The wave electric field vector points in the same direction with the transverse electron velocity. The electron is decelerated by the optical electric field.

The rate of energy exchange is negative, i.e., the electron loses energy to the optical wave.

60 FEL Energy-Phase & Pendulum Equations

Untrapped

Trapped l Separatrix q 휂

−휋 휓 휋 푑휓 푑휃 FEL energy-phase equations =2푘푢휂 Pendulum equations = 휔 푑푧 푑푡

푒퐸0퐾෡ 푑휂 푑휔 푔 푎 = = − 푎 푠푖푛휓 =− 푠푖푛휃 2푚 푐2훾2 푙 푒 0 푑푧 푑푡 61 Electrons Gain/Lose Energy & Bunch Up

Hamiltonian (kinetic energy + potential energy) of the pendulum

2 푚푙2 휃ሶ 휔 퐻 = − 푚푔푙 1 − cos 휃 2

Pendulum potential energy 휃 푉 = 푔푙 1 − cos 휃 POTENTIAL

Half of the electrons (ponderomotive phase from -p to 0) gain energy and move up in the bucket. The other half of the electrons (phase from 0 to p) lose energy and move down in the bucket. In term of pendulum potential energy, they all fall down to the bottom of the potential well and bunch up near phase = 0.

-2휋 -휋 0 휋 2휋62 Energy Modulations & Density Modulations

Electrons interacting with the ponderomotive waves develop energy and density modulations. 63 Radiation from a Bunch of Electrons

Electrons are randomly distributed along z Incoherent undulator radiation

FEL interaction induces density modulations with period equal to the radiation wavelength. 2 The emitted fields are in phase and add coherently. Coherent intensity scales with N

Electrons are bunched with period of a radiation  Coherent FEL radiation



Ratio of coherent power to incoherent power is N (the number of electrons in one ) 64 Bunched Beams Emit Coherent FEL Radiation

Bunching factor 푁휆 Radiation from an ensemble of 푁 electrons 1 휆 푏 = ෍ 푒𝑖휓푛 푧 푁휆 is number of electrons in one wavelength 푁휆 Unbunched beam 푛 2 2 2 퐸 = 휖 푁휆 + 푁휆 푁휆 − 1 푏

휖 2 = power emitted by one electron

Bunching factor Incoherent undulator radiation

2 2 푏 = 0 퐸 푈푅 = 휖 푁휆 Bunched beam Bunching factor Coherent FEL emission

2 2 2 푏 ~ 1 퐸 퐹퐸퐿 = 휖 푁휆 + 푁휆

65 Segmented Undulators in a FODO Lattice

Undulator Undulator Undulator Undulator Undulator

QF QD QF QD QF QD

xrms

rms radius (m)

yrms

Distance along undulator beamline (m) 66 FEL Animation

Courtesy of Gabriel Marcus 67 Summary of FEL Radiation Properties

• FELs are tunable sources of coherent radiation based on the same principle of operation, i.e., resonant wavelength, energy and density modulations followed by coherent bunched beam radiation, over the entire electromagnetic spectrum. • FEL radiation, similar to undulator radiation, originates from the sinusoidal motions of electrons in undulators. However, the FEL beams have full transverse coherence, large numbers of photons per pulse and peak brightness several orders of magnitude above the peak brightness of undulator radiation. • X-ray FELs produce nearly Gaussian coherent beams similar to a high-quality conventional laser beam but with very small angular divergence. • The radiation generation process in an FEL is completely classical. The motions of electrons in energy-phase space can be described by two coupled differential equations similar to those describing the motions of a classical pendulum.

68 References

1. “An Introduction to Synchrotron Radiation: Techniques and Applications” by Philip Willmott, John Wiley & Sons (2019).

2. “Free-Electron Lasers in the Ultraviolet and X-ray Regime” by Peter Schmüser, Martin Dohlus, Jorg Rössbach and Christopher Behrens, Second Edition, Springer Tract in Modern Physics, Volume 258.

3. “Synchrotron Radiation and Free-Electron Lasers: Principles of Coherent X-ray Generation” by Kwang-Je Kim, Zhirong Huang and Ryan Lindberg, Cambridge University Press (2017).

4. “Review of Free-Electron Laser Theory” by Zhirong Huang and Kwang-Je Kim, Phys. Rev. Spec. Topics in Accel. Beams, 10, 034801 (2007). 69