Lecture: LWFA electrons: staged acceleration 1/2 Masaki Kando kando.masaki@qst.go.jp Kansai Photon Science Institute QST, Japan Advanced Summer School on “Laser-Driven Sources of High Energy Particles and Radiation” 9-16 July 2017, CNR Conference Center, Anacapri, Capri, Italy This work was partially funded by ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan).
Contents of Lecture 1 • Introduction of staged acceleration • Accelerator physics basics • Beam dynamics • Transverse motion • Longitudinal motion • Transfer matrix • Comparison of RF accelerators and LWFA Advanced Summer School, 9-16 July 2017, Capri, Italy 2
What is “staging”? • Implement several stages of acceleration and/or functions in particle accelerators. Example: Spring-8 Angstrom Compact Free-electron Laser (SACLA, Japan) T. Hara et al., PRST-AB 19 , 020703 (2016) • Particle accelerators are composed of many stages . • Injector, Sub-Harmonic Buncher, S-band, C-band • Bunch compressor (BC) Advanced Summer School, 9-16 July 2017, Capri, Italy 3
Novel schemes http://www.eupraxia-project.eu P. A. Walker et al., TUOBB3, IPAC 2017 Advanced Summer School, 9-16 July 2017, Capri, Italy 4
SPring-8/SACLA Road Map up to 2041 2041 Major Upgrade to SPring-8-III Overall Configuration (Generation of X-ray beams through Major Upgrade to XFEL-II 2033 Next Next Generation SR Elements Development laser acceleration) Generalization Return XFEL Research Outcome to the Society � Next Generation XFEL Development X-ray beam Sub-PW Laser � Electron beam Generalization (1 keV) � (> 1 GeV) � New Generation SR Use Industrial Use Return XFEL Research Outcome to the Society � SPring-8 Major Upgrade to SPring-8-II Industrial Use 2020 Academic Use of XFEL � XFEL SACLA Plasma devices � Microundulator � XFEL Use � 3 rd Generation-like Use Next Generation SR Elements Development Impulsing Paradigm Change through Disruptive Technologies Program (ImPACT Program) XFEL Inauguration 2011 Cabinet Office, Government of Japan Generalization Next Generation SR Conceptual Development XFEL Project � 2009 “Ubiquitous Power Laser for Achieving 2006-2010 a Safe, Secure and Longevity Society” Return SR Research Outcome to the Society � LWFA Platform ~20M € / 5 Years (2014.10~) Groundbreaking for the Next Generation Industrial Use Project Manager: Yuji SANO 3rd Generation SR Use 2nd Generation-like Use SPring-8 Inauguration � 1997 Photon Pioneers Center, Osaka University �� Presentation Material of Prof. T. ISHIKAWA, Director general of SACLA(XFEL Facility, Japan)
������������� MP2 amplifier Booster XPW Compressor3 Compressor2 Compressor1 2 x 20 J at 527 nm 2 x 4 J at 532 nm 2 x 2.5 J at 532 nm TWIN amplifier MP3 amplifier Ti:Sa ultrafast oscillator multi-pass amplifier MP1 amplifier MP0 amplifier 125 mJ at 532 nm 30 mJ at 527 nm 30 mJ at 527 nm Stretcher + AOPDF Regenerative amplifier + AOPGCF + 2.5 J at 532 nm Laser Platform@HARIMA (SACLA) ������������������������� ������������������������� 30 µJ at 100 Hz 3 mJ at 100 Hz 25 mJ at 10 Hz 200 mJ 100 mJ 350 mJ at 10 Hz at 10 Hz at 10 Hz � ����� � ������������ ���������� ��� ����� �� �� 1 J at 10 Hz, 20 fs 2 J at 5 Hz, 50 fs 10 J at 0.1 Hz, 100 fs ���������� � ����������� Injector � Phase rotator � Booster � ���������� Accelerator Tunnel (SCSS) @ SPring8 Platform laser installation started!! � ���������� �� ���� ���������� Prototype XFEL (SCSS) 1 st ! 2016 Apr. 2017 2 nd Laser � 3 rd Laser � Laser � 2005 ������������ E > 1 GeV ! E ~ a few - 10s MeV ! E ~ 10s MeV ! Δ E/E ~ 1 % ! Δ E/E = 10 ~ 100% ! Δ E/E < 1 % !
Why staging in laser-electron accelerators? • To overcome length limitation • Pump depletion • Dephasing • To manipulate electron beams • Bunch compression (temporal compression) • Energy compression • Still there is a room for discussion… • “to be staged or not to be..” • It depends on what you need. • If a single stage satisfies your requirement, no need to stage. Advanced Summer School, 9-16 July 2017, Capri, Italy 7
Preparation: my lecture focuses on staging http://www.eupraxia-project.eu P. A. Walker et al., TUOBB3, IPAC 2017 Advanced Summer School, 9-16 July 2017, Capri, Italy 8
Motion of particles in a magnetic field Assume that the displacement x is small: Coordinate with a moving particle in a horizontal plane � � ρ 0 + x ≈ 1 1 1 − x ρ 0 ρ 0 ρ 0 B z ( x ) ≈ B z (0) + dB z dx x z x In the central orbit the forces are qev × B z balanced: centrifugal force s m γ v 2 B + qevB z (0) = 0 x ρ 0 The equation of motion in a moving Eq. (1.1) can be rewritten as particle coordinate is m γ v 2 d 2 x ds 2 = − m γ v 2 x + qevdB z (1.2) dx x m γ d 2 x dt 2 = m γ v 2 ρ 2 (1.1) ρ 0 + x + qevB z ( x ) 0 Here we define the field gradient index We rewrite this by using dB z n : = − ρ 0 (1.3) B z (0) dx d 2 x dx dt = dx ds dt = vdx dt 2 = v 2 dx Eq. (1.2) is simplified to ds ds ds d 2 x ds 2 = − 1 − n x (1.4) Horizontal direction ρ 2 0 Advanced Summer School, 9-16 July 2017, Capri, Italy 9
Motion of particles in a magnetic field dB x dz � dB z Coordinate with a moving particle in a rot � Using dx = 0 B = 0 vertical plane Here we use the field gradient index z dB z n : = � ρ 0 s qev × B x B z (0) dx B The equation of motion is x d 2 z ds 2 = � n z (1.6) Vertical direction ρ 2 The equation of motion in the vertical 0 (perpendicular to the bending plane) is d 2 x ds 2 = � 1 � n m γ d 2 z (1.4) x Horizontal direction (1.5) dt 2 = � qevB x ( z ) ρ 2 0 Similarly we use the trans. t->x and The solutions are stable if 0<n<1 . Weak focusing condition � + dB x ( z ) � � B x ( z ) ≈ B x (0) z � dz � z = 0 ���� = 0 In the median place, Bx is zero. Advanced Summer School, 9-16 July 2017, Capri, Italy 10
Transverse dynamics Suppose that a solution can be written as d 2 x ds 2 = � 1 � n (1.4) x Horizontal direction ρ 2 y ( s ) = w ( s ) e i ψ ( s ) (1.8) 0 d 2 z Then from (1.7) the two equations are ds 2 = � n (1.6) Vertical direction z obtained. ρ 2 0 1 (1.9) w �� ( s ) + K ( s ) w ( s ) � w 3 ( s ) = 0 1 � n n The terms and express 1 ρ 2 ρ 2 ψ � ( s ) = (1.10) 0 0 w 2 ( s ) focusing forces. Generally it is C/w^2 but here we choose C=1. Let us write (1.4) and (1.6) in the general The general solution of (1.7) can be form: expressed by a linear combination of two special solutions: d 2 y ds 2 = � K ( S ) y (1.7) (1.11) y ( s ) = c 1 w ( s ) e i ψ ( s ) + c 2 w ( s ) e � i ψ ( s ) Hill’s equation N.B. The solution is expressed in the form according to Floquet theory. Here we assume a periodicity such a ring accelerator K ( s + C ) = K ( s ) Advanced Summer School, 9-16 July 2017, Capri, Italy 11
Transverse dynamics Let the solutions at s=s 1 (s 2 ) y 1 , w 1 , ψ 1 ( y 2 , w 2 , ψ 2 ) The y2 and y1 are expressed by the following relationship. w 2 � � w 1 cos( ψ 2 � ψ 1 ) � w 2 w � 1 sin( ψ 2 � ψ 1 ) w 1 w 2 sin( ψ 2 � ψ 1 ) � y 2 � � y 1 � � � � � � � � � = 1 + w 1 w � 1 w 2 w � w � w � � � w 1 y � � � y � 2 sin( ψ 2 � ψ 1 ) � 1 2 cos( ψ 2 � ψ 1 ) w 2 cos( ψ 2 � ψ 1 ) + w 1 w � 2 sin( ψ 2 � ψ 1 ) � � � w 2 � � � 2 1 w 1 w 2 w 1 (1.12) This matrix is called “ Transfer matrix ”. Later, we will see transfer matrices are very useful to analyze the motion of particles in the orbit. Let us introduce parameters as following. β : = w 2 α : = � ww � Twiss parameters γ : = 1 + ( ww � ) 2 (1.13) w 2 µ : = ψ 2 � ψ 1 1 + α 2 Twiss parameters are NOT independent. (1.14) = γ β Advanced Summer School, 9-16 July 2017, Capri, Italy 12
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