Advanced Research on Free Electron Laser Technology

Free electron laser 25.05.23 Group meeting Jang Won

Schematic diagram of PAL-XFEL Reproduced from H.-S. Kang et al., Hard X-ray free-electron laser with femtosecond-scale timing jitter, Nature Photonics, vol. 11, pp. 708 713, 2017.

Undulator radiation Undulator radiation ??? ? ??= ? ? ? ? ? ,??? ,??(?) ?? ? ?? ??(?) ? = ? (constructive interference condition) 2??2(1 +?2 ?? ? = (0, ??sin2?? ?= 2+ ?2?2) ,0) ?? Undulator radiation Condition ??? =?? ??????? 1 2?21 +?2 2??? :Undulator parameter ? =??0?? ?2 4?2cos4? ??? = 1 ??? 2

Undulator radiation Radiation by a moving charge, energy radiated frequency distribution (by Jackson Ch.11) 2 ?? ? ? ? ??? ? ? ? ? ?2? ? ??= ?2?2 4?2? 1 1 2? ? 4??0 2 ???? ? ? ? ??? ? ? ? ? ? ?2?? 4?2? 2?2 1 ?(?? ? = ?1) ? ? 4??0 ?? Line-shape function sin2??? ? ?? ? ?1 ?? ? ??????? ?1 ? = 2sin2? ? ?? ?1

Energy Transfer between Electron beam and light wave Basic concept of FEL ??? =?? ???????: electron ??= E0cos(klz ??? + ?0) :incident light

Low gain The electric field of the light wave exerts a force on the electron, ? ?? ??= ? ? = ????? = ??? ? = ????0 2? ???(???)E0cos(klz ??? + ?0) cos(?) ????0 cos(?) 2? Ponderomotive phase ? = ??+ ??? ??? + ?0 ? = ?? ??? ??? + ?0(Phase velocity>c), neglected term (? = ???) ??= E0cos(klz ??? + ?0) :incident light ??? =?? ??????? Condition for stationary phase ?? ??= ??+ ?? ??? ??? + ?0= 0 ?? 2?2(1 +?2 radiation condition! 2)same as the undulator ? =

Low gain ?? ??= ????0 cos(?) 2? ??= E0cos(klz ??? + ?0) :incident light -:light wave -:electron trajectory

FEL Pendulum Equations ? =? ?? =? ?? :energy deviation ?? ?? ?? ??= ????0 ?? ??= cos(?) ?? ??= ??+ ?? ??? ??? + ?0 2???? ?? ??= 2??? Solving the equation using the perturbation method ? = ?0+ ??1? + ?2?2? + ? = ?0+ ??1? + ?2?2? + 2? ??0? 2??2??2???? ??0? 2??2??2) (? = =< ? > ??2 ???2 sin2? ?2 ? ? second order eq ? ? ?? = ??2 ?2?? 4?0??2??3 3?? 2?? ? ?? ?? ,(? = ??? ) ?? ? ??????? ? Derivative of line shape function Gain undulator Microbunching, ?0 increasing

High gain(1D) Microbunching 1 +?2 ?2 1 4?2cos4? Incident light ??? = 1 ? 2?2 2 ?? ??(?,?) = ? > ?? energy transfer from electrons to light ? ?? ? < ?? energy transfer from light to electron ? ?? ??? exp[?(??? ???)] Electron-light interaction ?1? ??? ? ?,? = ?0+ ???,? = ?0+ ?1? ??? ? = ??+ ??? ??? + ?0 Modulation effect Energy modulation & Density modulation

High gain(1D) Radiation Field ?2 ??2? = ?0 2 1 ? ? ??+1 ?1? ??? ? ?,? = ?0+ ???,? = ?0+ ?1? ??? ?0 ? ?2 ? ?? ?? ?2 ??2 1 ?2 ??2 ? ?? ??= ?0?? For x component: ??= ?0 ?1(?) ?2 4?? ???,? = ???) ??? exp[?(??? ???)] ? Slowly varying amplitude(???) approximation | ?? (?)|?? | | ?? (?)|?? | ??(?)| ?? (?)| ? ?? ??= ??0 ? ?? ?? exp[ ? ??? ??? ] ??= ???, ??= ??? 2??

High gain(1D) Space charge field ? =? ?1??? generates a periodic longitudinal field ?0 ? ??(?,?) ?? ??(?) = ??0?2 = ?1(?) exp[?( ??+ ??? ???)] ?1(?) ?0 ?? ???,? = ??? exp[?(??? ???)] Slowly varying amplitude(???) approximation | ?? (?)|?? | | ?? (?)|?? | ??(?)| ?? (?)|

Coupled First-Order Equations Result of Maxwell equation (under modulation) ? ?? ??= ?0?? ?? ??= 2??? Radiation field ?1(?) From phase equation: 4?? ??(?) = ??0?2 ?? ??= ??0? 2??2??2???? Space charge field ?1(?) From ?? ??= ? ?: ?? ? ? ?? ??? ?? ??= ????: 2??2??2??{ ?????} = ?? ???? ? ?? ??= ????: ??2??2??{ ?????}Energy deviation due to interaction between space charge field and electron = ? ?? 2??+ ?? ??= ? ?? ???} ?1? , modulated current ! ??2????{

Coupled First-Order equations Longitudinal distribution of electron in microbunch ? ? ? = ?(? ??) ?=1 ? ? can be expanded in a real Fourier series ? ? =?0 2+ ??{ ??exp(???)} First at the dc current density, ?0 ?=1 2? ????) ,(?0 ? ? 2?) ?0= ?? 2? ???? ?0 2 ??=1 ?0= ????,(??= 2= ? ?(?)exp ??? ?? 0 Modulation current, ?1(first harmonic term) ? 2? ???? 2 ? ?=1 ?1= ?? ?1= ?0 exp( ???)

Coupled First-Order equations The complete set of coupled first-order equations ??? ?? = 2???? Describe time evolution of the phase(?), energy deviation(??), modulated current density(?1), amplitude of the light wave( ? ?? 2?? ??0?2 ??? ??= ? ?1 ????} ??2????{ ? ?? ??) 2 ? ?=1 ?1= ?0 exp( ???) ? ?? ??= ?0?? ?1(?) 4?? 2N+2 coupled differential, algebraic equations

Third-Order equations Under mono-energetic condition, ?1= ?2= = ?? assume periodic initial phases: ??0 ??= 2?? ? ??? = ??+ 2???? + ??{ ? ? ????} ??? = ? + ??{ ? ? ????} 1 3: gain parameter ?0 ??2???? 4??3? = ?? ?: space charge parameter 2?? ?? + ? 4??? + ?? ? 3 kp= ?? 2?2 2 4?? ?? ?? ??= 0 ???2 ???0?: plasma frequency = ?? (If the electron beam is on resonance and space charge force is small,) (? = 0,??= 0) ? 3 ?? ??= 0

Third-Order equations ? 3 ?? ?? 3 trial solution ??= 0 ??? = ?1?1? + ?2?2? + ?3?3? , ??? = exp(???) ? + 3 2 ? 3 2 (?1= ,?2= ,?3= ? ) Initial condition(z=0) ??0 ?? 0 ?? 0 1 ?1 ?1 1 ?2 ?2 1 ?3 ?3 ?1 ?2 ?3 = 2 2 2 FEL process incident plane light wave initial modulation , Exz,t = Eincos(klz ???) (0) ?10 = 0,?? ?? ??0 ?? 0 ?? 0 ??? 0 0 ??? =??? i + 3 2 i 3 2 ?1= ?2= ?3=??? = 3(exp ? + exp ? + exp( i ?)) 3

Third-Order equations ??? =??? i + 3 2 i 3 2 3(exp ? + exp ? + exp( i ?)) Exponential growth! Dominating term, when ? 1 ??? 1 ??? 3exp 2??0? , 1 3 : power gain length ??0 Gain function low gain 2 ???,? ??? ? ?,? = 1

Third-Order-equations FEL process starts with initial periodic charge density modulation (There is no external Lightwave) ???,? = ?0+ ?1? ??? ??0 ?? 0 ?? 0 0 ?0? ? 4?? ?1(0) 1 = Exponential growth ? 2??? ??? = ?1?1? + ?2?2? + ?3?3? , ??? = exp(???) Periodic initial density modulation

Saturation

SASE Incident light (seed) or initial modulation leads to an exponential growth process. ??0 ?? 0 ?? 0 0 Seed ??0 ?? 0 ?? 0 0 ??? 0 ?0? ? 4?? ?1(0) 1 = = ? 2??? Initial modulation SASE means Self-Amplified Spontaneous Emission(no seed) Shot noise(white noise) makes initial modulation. ??0 ? ? 1 ?10 = ??

SASE-shot noise Consider relativistic electrons that are randomly distributed along the bunch and call N numbers of electrons in the time interval T. Then ? =? ? is the average number of electrons per unit time. ?0= ? ?: absolute magnitude of dc beam current ? ?,?? ? 2,? ? ? = ? ?(? ??), 2,(? 1) ? ?=1 The Fourier transform of ? ? < ? ? >=?? iT? = ? ??? (????) ?0 ? ?=1 1 2=??0 S ? = lim Spectral density function ????? Constant Independent of frequency (Characteristic feature of white noise) ? ? ? 2 ?0+ ? =< ? ?2>=1 1 ? 2?? = ? ? ? =??0 2 2 2?? = lim ???? ? ? ? ??? ? ? ? ?0 ? ? 2 2 2 ???? ?? ??0 ? ? 1 Random distribution makes initial modulation ?1? = 0 = = ??

SASE-shot noise A long bunch can be subdivided into sections that are each one coherence length long. In the SASE process the microbunching takes place independently in different sections which then radiate incoherently. The longitudinal coherence of SASE radiation is poor. The straightforward solution is seeding external laser(coherent source).

High-Gain Harmonic Generation(HGHG) ?1 The precondition is a seed beam of sufficient power to generate an energy modulation well above the energy spread of the electron beam. Density modulation is formed where high/low energy electrons take different length of path in chicane. The Fourier expansion of ? contains higher harmonics with angular frequency ??= ??1

Echo-Enabled Harmonic Generation(EEHG) Generating microbunches with a considerable substructure Can generate much higher harmonic numbers than HGHG, ? = ?1?1+ ?2?2

Self-Seeding(X-ray region) For X-rays, suitable seed lasers are presently not available. 1. 2. 3. SASE radiation (undulator 1) Radiation passes through a monochromator, electron beam passes a magnetic bypass. The magnetic bypass is adjusted such that the emerging electron and radiation pulse have a good temporal overlap. Microbunching from first undulator is washed out in the magnetic bypass.

Reference 1. Peter Schm ser, Martin Dohlus, J rg Rossbach, Christopher Behrens , Free- Electron Lasers in the Ultraviolet and X-Ray Regime, 2nd ed. (Springer International Publishing, Switzerland, 2014). 2. Moohyun Yoon, Lecture Notes on PHYS683, DIVISION OF ADVANCED NUCLEAR ENGINEERING, POSTECH.