DESIGN OF AN ULTIMATE STORAGE RING FOR FUTURE LIGHT SOURCE

Transcription

1 DESIGN OF AN ULTIMATE STORAGE RING FOR FUTURE LIGHT SOURCE Yichao Jing Submitted to the faculty of the University Graduate School in partial fulfillment of the requirement for the degree Doctor of Philosophy in the Department of Physics, Indiana University August, 2011

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3 Accepted by the Graduate Faculty, Indiana Univeristy, in partial fulfillment of the requirement for the degree of Doctor of Philosophy. iii Shyh-Yuan Lee, Ph.D. Doctoral Committee Chen-Yu Liu, Ph.D. Paul E. Sokol, Ph.D. August 1st, 2011 Rex Tayloe, Ph.D.

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5 v Copyright c 2011 by Yichao Jing ALL RIGHTS RESERVED

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7 To my parents. vii

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9 ix Acknowledgments I would like thank all those who have provided precious help during my graduate study to make this thesis possible. I would like to give my sincere gratitude to my advisor Prof. Shyh-Yuan Lee, who led me into accelerator physics. His deep understanding of physics has shown me the path through my graduate study. I have benefited a lot from his wide vision and innovative ideas. He is very supportive and always ready to help in every aspect of life which makes my living at Indiana University very comfortable and enjoyable. I was so lucky to meet such a knowledgeable mentor, a good friend and a respectable senior when I first came to a new country and a new environment. I would like to thank my committee members, Prof. Paul Sokol, Prof. Chen-Yu Liu and Prof. Rex Tayloe for their great help on my Ph.D. thesis study. I really want to thank Prof. Paul Sokol for providing me the great opportunity to work for ALPHA project where I gained a lot of practical experience. I want to thank Prof. Chen-Yu Liu and Prof. Rex Tayloe for providing many great help and advices during my Ph.D. thesis work. I sincerely thank Dr. Kingyuen Ng, who has provided a lot of useful discussion and deep insights in many research topics during the past few years. I wish to thank Dr. Xiaoying Pang and Dr. Xin Wang who graduated from our group recently. They helped me a lot in every aspect and the time we were working together will always be a precious memory for me. Also, I am grateful to my colleagues: Tianhuan Luo, Honghuan Liu, Alfonse Pham, Hung-Chun Chao, Kun Fang, Zhenghao Gu, Ao Liu and Xiaozhe Shen for their wonderful friendship and creating such a fun environment for me to live and work.

10 x My biggest thank goes to my parents. Without their support and love, none of my accomplishments would have been possible. I can only express my gratitude by dedicating this thesis to them.

11 xi Yichao Jing DESIGN OF AN ULTIMATE STORAGE RING FOR FUTURE LIGHT SOURCE Electron storage rings are the main sources of very bright photon beams. They are driving the majority of condensed matter material science and biology experiments in the world today. There has been remarkable progress in developing these light sources over the last few decades. Existing third generation light sources continue to upgrade their capabilities to reach higher quality photon beam while new light sources are being planned and designed with ever improving performance. Idea of ultimate storage rings (USR) has recently been proposed to have beam emittance down to few tens of pico-meters, reaching diffractive limit of hard X-ray. This theses work is dedicated to designing a storage ring with ultra-small beam emittance using n-bend achromat (n-ba) structure. For ultimate storage rings, large natural chromaticities require strong sextupoles to correct. Strong non-linear effect requires the study of dynamic aperture (DA). We calculate and optimize the DA to achieve a 1.5 mm by 1.5 mm aperture size. Other instabilities such as intra-beam scattering (IBS) and microwave instabilities (MI) are evaluated self-consistently. Possible free electron laser (FEL) scheme has been proposed to facilitate the implementation of this ultimate storage ring design.

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13 CONTENTS xiii Contents Acceptance Acknowledgments Abstract iii ix xi 1 Introduction Motivation of study Introduction to Accelerator Physics Frenet-Serret coordinates and Hill s equation Floquet theorem and betatron oscillation Synchrotron motion Free Electron Laser Angular-Modulated Harmonic Generation (AMHG) and Echo Enable Harmonic Generation (EEHG) Optics Free FEL Oscillator (OFFELO) Linear Lattice for 10 pm Storage Ring pm storage ring and n-ba structure Theoretical Minimum Emittance (TME) Effort in shortening the circumference

14 xiv CONTENTS 2.2 Combined function magnet lattics Nonlinear Lattice and Dynamic Aperture (DA) optimization Positive chromaticities and sextupole correction Dynamic aperture and tune shift with amplitude Injection issues Intra-beam Scattering (IBS) and Microwave Instability (MI) Single bunch collective instability Intra-beam scattering (IBS) and its effect Comparison between microwave instability and IBS effect SASE FEL performance study under microwave instability Conclusions 63 Appendix 65 A Undulator theory and laser study 65 A.1 Background A.2 Elastic photon-electron collision A.3 Klein-Nishina formula A.3.1 Total cross section and differential cross section A.4 Laser-Beam Interaction A.5 Laser induced damping B Multipole effect on higher order momentum compaction factor 81 B.1 Multipoles effect B.1.1 Momentum compaction factor B.1.2 Higher order dispersion

15 CONTENTS xv B.1.3 Multipole effect using Hamiltonian expansion Bibliography 95

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17 LIST OF TABLES xvii List of Tables 2.1 Parameters for 10pm storage ring Parameters for 10pm storage ring with 25BA

18 xviii LIST OF TABLES

19 LIST OF FIGURES xix List of Figures 1.1 Frenet-Serret coordinate system Schematic drawing of SASE FEL and undulators Oscillator FEL involves a shorter undulator and two reflective mirrors to confine the optical wave. Laser reaches saturation in N turns Initial beam longitudinal phase space (a) with vertical axis the particle momentum. Different colors are used to depict different regions in initial particle distribution. Energy modulation is observed after modulator(b). Microbunching forms after chicane with condition R 56 δ = λ/4(c) Particles over microbunch when passing through a large dispersive chicane with R 56 δ = λ(a) and R 56 δ = 2λ(b) Particles form very fine energy strips after the large dispersive chicane(a).particles further experience energy modulation in second modulator(b). At the end of 2nd chicane, a density modulation with ultrashort period is formed(c). Microbunching is observed in current distribution at the end of 2nd chicane Optics Free FEL Oscillator requires two circulating beams. Low energy beam is the information carrier from modulator to radiator. Radiation reaches saturation in a few turns

20 xx LIST OF FIGURES 2.1 Plot of TWISS parameters for 11BA structure. Horizontal dispersion is magnified by 100 times Plot of tune space with up to 8th order resonance lines. Red square is the location for 10 pm storage ring s tunes Plot of TWISS parameters for 25BA structure. Horizontal dispersion is magnified by 100 times TWISS parameters for a superperiod of the combined function magnet lattice Dispersion vs matching quadrupole strength for the combined function magnet lattice. The different colors represent different drift space lengths. Longer drift space requires weaker matching quadrupole strength. Boundary reaches stability limits Beta funtion vs matching quadrupole strength for the combined function magnet lattice. The different colors represent different drift space lengths. Solution found from Fig.2.5 does match to theoretical value. K c = 0.5(1/m 2 ) is too small for this case. Plot s boundary reaches stability limits Beta tune space plot for the combined function magnet lattice. The different colors represent different drift space lengths. Longer drift space results in a larger ratio between β x and β y thus a change in the quadrupole strength is not sensitive in changing β y. Thus vertical betatron tune is not changed much. Plot s boundary reaches stability limits turn dynamic aperture for 10 pm storage ring. 1.5 mm aperture is obtained after correcting the large tune shift with amplitude Quadratic tune amplitude dependence

21 LIST OF FIGURES xxi 3.3 ICA analysis for a particle in DA with a small initial offset. It experiences mostly betatron oscillation although some of the instability induced by nonlinear effect can be observed in temporal wave function(up-right plot) ICA analysis for a particle in DA with a large initial offset. This mode shows frequency spectrum with noisy peaks which indicates the particle experiences many different resonances at the boundary of DA Schematic drawing of a longitudinal impedance Bunch length vs beam current for ALS. Bunch lengthening is observed due to single bunch microwave instability Rms energy spread vs beam current for ALS. Energy spread can be blown up by a few times under single bunch microwave instability Bunch length vs beam current for SPEAR3. It has similar performance as ALS due to the similar parameters of the storage ring Rms energy spread vs beam current for SPEAR3. The calculated FEL parameter is much lower than rms energy spread so SASE FEL is not possible Bunch length vs beam current for 10 pm storage ring. Bunching factor is very big that peak current can reach few ka when beam current is high Rms energy spread vs beam current for 10 pm storage ring. FEL parameter is closer to rms energy spread when beam current is low Rms energy spread vs beam current for 10 pm storage ring. FEL parameter is closer to rms energy spread when beam current is low Rms energy spread vs beam current for 10 pm storage ring. FEL parameter is closer to rms energy spread when beam current is low.. 55

22 xxii LIST OF FIGURES 4.10 Rms energy spread vs beam current for 10 pm storage ring. FEL parameter is closer to rms energy spread when beam current is low Rms energy spread vs beam current for 10 pm storage ring. FEL parameter is closer to rms energy spread when beam current is low Rms energy spread vs beam current for 10 pm storage ring. FEL parameter is closer to rms energy spread when beam current is low Rms energy spread vs beam current for 10 pm storage ring. FEL parameter is closer to rms energy spread when beam current is low Rms energy spread vs beam current for 10 pm storage ring. FEL parameter is closer to rms energy spread when beam current is low Rms energy spread vs beam current for 10 pm storage ring. FEL parameter is closer to rms energy spread when beam current is low.. 61 A.1 A 2D plot of differential cross section vs photon energy according to Klein-Nishina formula. At half of the peak energy, the differential cross section is half of the value of its peak cross section A.2 A 3D plot of differential cross section vs photon energy and emission angle θ according to Klein-Nishina formula A.3 Ratio between damping times induced by laser and dipoles under different laser cross section and laser power A.4 Horizontal damping time induced by laser under different laser cross section and laser power A.5 Vertical damping time induced by laser under different laser cross section and laser power A.6 Longitudinal damping time induced by laser under different laser cross section and laser power

23 Introduction 1 Chapter 1 Introduction 1.1 Motivation of study Storage rings are the main sources of high-brightness photon beams. They are driving the majority of condensed matter material science and biology experiments in the world today. There has been remarkable progress in developing these light sources over the last few decades. Existing third generation light sources continue to upgrade their capabilities to reach lower emittances and smaller energy spreads, while new light sources are being planned and designed with ever improving performance [1, 2]. As light sources, storage rings have many attractive features. They provide a wide, easily tunable energy spectrum from infrared to hard X-ray with high repetition rates thus high average flux and brightness. The beams are very stable in energy, intensity, position, and size. Storage rings usually have many beamlines which can serve many experiments simultaneously and reliably. The cost for each user is also considerably low. Besides this combination of properties, storage rings can be designed to implement other advanced techniques such as free electron laser (FEL), which offers extremely high peak brightness in much shorter pulse durations but with typically

24 2 1. Introduction lower repetition rate. Although FELs are playing more and more prominent role in biology science and material science which require ultra short pulses with high instantaneous brightness for tissue tomography, a broad class of X-ray science still relies on the low peak brightness (to avoid damaging samples) and high photon pulse repetition rates (to reach sufficient flux) provided by storage rings. Such experiments simple cannot be conducted using the ultra-high peak brightness from FEL sources. Storage rings will continue to be the important sources for a large user community for the indefinite future. While storage rings are a mature technology, they have the potential for significantly enhanced performance. One can imagine an ultimate storage ring that produces high- brightness, transversely coherent X-rays while simultaneously serves dozens of beamlines and thousands of users annually. For such a source to maximize transverse photon coherence, the beam emittance must be extremely small in both transverse planes, approaching and even exceeding the wavelength-dependent diffraction limit. Storage ring sources have achieved diffraction limited emittances for hard X-rays in the vertical plane by minimizing horizontal-vertical beam coupling, but horizontal emittance must be reduced by a factor of 100 or more from the lowest emittance values achieved today to reach that limit. On the other hand, possible designs for ultimate rings would necessarily have large circumference and large number of magnets. Given present day s technology, there is no difficulties in reaching such low emittances. However, the cost of such big rings would be considerably high due to the large vacuum system and magnet construction. An ultimate storage ring would retain all the general strengths of today s storage rings mentioned above while delivering high transverse coherence up to the hard X-ray ( 10 kev) regime.ultimate storage rings would have brightnesses and coherent flux one or two orders of magnitude higher than the highest performance ring-based light sources in operation or presently being constructed.

25 1.2 Introduction to Accelerator Physics 3 In this thesis, the author intends to present the current progress of designing an ultimate storage ring including linear lattice design and nonlinear properties study, while instabilities study will also be discussed. The following 4 chapters are organized in following orders. First chapter gives an introduction of basic accelerator physics including Hill s equation, Floquet transformation, betatron motion and synchrotron motion. There will also be an introduction about Free Electron Laser(FEL) which we want to implement on this ultimate storage ring. In the second chapter, we report the linear lattice design which achieves Theoretical Minimum Emittance (TME) and study the possibility of using combined function magnets to shorten the circumference. In the third chapter, the calculation and optimization of dynamic aperture (DA) and effort of understanding the resonances in DA are presented. In the fourth chapter, the effects of beam instabilities as Microwave Instability (MI) and Intra Beam Scattering (IBS) are analyzed and calculated. Possible FEL performance is also evaluated. Chapter five will be the conclusion. 1.2 Introduction to Accelerator Physics Frenet-Serret coordinates and Hill s equation In accelerator, a reference orbit (or designed orbit) is formed once bending magnets are in place. Under perfect conditions, particles will follow the reference orbit when circulating in the accelerator. In reality, particles have small amplitude oscillation around the reference orbit which we call betatron oscillation. To discuss particle motion with respect to the reference orbit, we use Frenet-Serret coordinate system as is shown in Fig In the Frenet-Serret coordinate system, particle position can be expressed as r = r0 + xˆx + zẑ (1.1)

26 4 1. Introduction s Particle Position v z r x r 0 Reference Orbit Figure 1.1: Frenet-Serret coordinate system. where ˆx is the radial (horizontal) unit vector, ẑ is the normal (vertical) unit vector and ŝ is the tangential (longitudinal) unit vector. The three unit vectors ˆx, ŝ and ẑ form the basis of the curvilinear coordinate system. In this new coordinate system, particle motion can be described by a new Hamiltonian H = (1 + x eφ)2 )[(H m 2 c 2 (p ρ c 2 x ea x ) 2 (p z ea z ) 2 ] 1/2 ea s, (1.2) where p x, p z are transverse momenta, A x, A z, A s are the vector potentials, H = ps is the new Hamiltonian and (x, p x, z, p z, t, H) are the new phase space coordinates. We have corresponding Hamilton s equations t = H H, H = H t ; = H H, p x x = p x x ; = H H, p z z = p z z. (1.3) where the apostrophe indicates differentiation with respect to s and we use s instead of t as the new independent variable. If we look into the Hamiltonian described in

27 1.2 Introduction to Accelerator Physics 5 Eq.(1.2), the first two terms in the middle parentheses is particle s total momentum p squared. Typically, the transverse momenta p x and p z are much smaller than total momentum p. We can expand the Hamiltonian up to second order in p x and p z H p(1 + x ρ ) x/ρ [(p x ea x ) 2 + (p z ea z ) 2 ] ea s. (1.4) 2p Applying Hamilton s equations in transverse directions to Eq.(1.4), we end up with betatron equations of motion x ρ + x ρ 2 = ± Bz p 0 Bρ p (1 + x ρ )2, (1.5) and z = Bx p 0 Bρ p (1 + x ρ )2, (1.6) where the upper and lower signs are correspondent to positive and negative charged particles respectively, Bρ = p 0 /e is the beam rigidity for a reference particle. With the magnetic field expansion B z = B 0 + B 1 x, B x = B 1 z, (1.7) with B 1 = Bz, we can get Hill s equations x x + K x (s)x = ± B z Bρ, z + K z (s)z = B x Bρ, (1.8) with K x (s) = 1/ρ 2 B 1 /Bρ and K z (s) = ±B 1 /Bρ being the horizontal and vertical focusing functions. The inhomogeneous term on the equation s right illustrates the field imperfections and higher order magnet components. For an ideal accelerator with pure dipole and quadrupole fields, the Hill s equations become homogeneous and focusing functions K x (s), K z (s) are periodic with a period of accelerator circumference.

28 6 1. Introduction Floquet theorem and betatron oscillation As we discuss above, the focusing functions satisfies relation K y (s+c) = K y (s) with subscript y denoting both horizontal and vertical directions. Thus the solution of Hill s equation can be written in such a form y = aw(s)e jψ(s), (1.9) where oscillation amplitude satisfies w(s+c) = w(s) and phase ψ(s+c) = ψ(s)+φ with Φ the phase advance in one revolution. If the accelerator has a symmetric structure and, is composed of superperiods with period of L, then we can impose stronger requirement and ask for a periodic solution over superperiods. Plugging Eq.(1.9) into Eq.(1.8) results in the differential equations: w + K(s)w 1 = 0, (1.10) w3 ψ = 1 w2. (1.11) The Courant-Snyder parameters are related to the amplitude function by: β = w 2, α = ww, γ = 1 + α2. (1.12) β Hence, the betatron function and phase advance can be written in forms of: 1 2 β + K(s)β 1 [1 + (β β 2 )2 ] = 0, (1.13) ψ(s) = s Thus a general solution of Hill s equation is: 0 ds β(s). (1.14) y(s) = a β(s)cos (ψ(s) + χ) (1.15) where a β is the oscillation amplitude for a single particle and ν y = (ψ(s + C) ψ(s))/2π is the betatron tune depicting the number of betatron oscillations in one

29 1.2 Introduction to Accelerator Physics 7 revolution. For a beam which is a cluster of particles, a β is the rms size of the beam and ɛ u = a 2 is the unnormalized rms emittance of the beam which is the phase space area divided by π. When there is no acceleration, ɛ u is an invariant quantity. When beam acceleration happens, normalized emittance given by ɛ n = βγɛ u is invariant. The phase space coordinates (y, y ) depict the particle s position and deflection angle (velocity) and can be transported in an accelerator by transfer map (linear map is 6D matrix including 4 dimensions in transverse and 2 dimensions in longitudinal direction) from point to point. For a linear system, the phase space coordinates at any particular position can be obtained by propagating from any initial position. y = M(s, s 0 ) y 0, (1.16) y with M(s, s 0 ) the transfer matrix from initial s 0 to final s. In any beam transport line it can be expressed by Courant-Snyder parameters as: β(s) 0 M(s, s 0 ) = cosφ sin φ α(s) 1 β(s) β(s) sin φ cosφ y 0 1 β0 0 β0 α 0 β0 where φ = φ(s) φ 0 is the phase advance from initial position to final position. For one complete revolution, the transfer matrix can be simplified to: M(s 0, s 0 ) = cos Φ + α 0 sin Φ β 0 sin Φ, (1.17) γ 0 sin Φ cos Φ α 0 sin Φ where Φ represents the phase advance for one complete revolution., Synchrotron motion We have discussed transverse betatron oscillation in an accelerator. Particles also experience longitudinal oscillation which is called synchrotron oscillation. With longitudinal electric field provided by rf cavity, particles gain energy or lose energy in

30 8 1. Introduction longitudinal direction. We can derive conjugate equations of longitudinal motion: and d dt ( E ω 0 ) = 1 2π ev (sin φ sin φ s), (1.18) dφ dt = hω2 0 η β 2 E ( E ω 0 ), (1.19) where V is the voltage across rf cavity, φ and φ s are the rf phases for off momentum particle and synchronous particle respectively, h is the harmonic number of rf cavity and η is the phase slip factor. Thus we can write the Hamiltonian for synchrotron motion H = 1 hηω0 2 2 β 2 E ( E ) 2 + ev ω 0 2π [cosφ cos φ s + (φ φ s ) sin φ s ]. (1.20) Using definition of fractional momentum spread we can express the Hamiltonian in a new form δ = p = ω 0 E, (1.21) p 0 β 2 E ω 0 H = 1 2 hηω 0δ 2 + ω 0eV 2πβ 2 E [cosφ cosφ s + (φ φ s ) sin φ s ]. (1.22) When particle is experiencing slow acceleration, one should use the Hamiltonian given by Eq.(1.20) for phase space tracking. On the other hand, if there is no acceleration (storage mode), one should use the Hamiltonian given by Eq.(1.22) for turn by turn phase space tracking. Two fixed points (φ s, 0) and (π φ s, 0) can be easily found for the Hamiltonian. Around the stable fixed point (φ s, 0), the particle s motion is elliptical, while being hyperbolic around unstable fixed point (π φ s, 0). Starting with Hamiltonian given by Eq.(1.22), for a small amplitude oscillation around stable fixed point, particle s motion becomes simple harmonic oscillation d 2 dt 2(φ φ s) = hω 0eV η cosφ s (φ φ 2πβ 2 s ). (1.23) E

31 1.3 Free Electron Laser 9 The stability condition requires η cos φ s 0. So below transition energy η < 0, the synchronous phase should satisfy 0 cosφ s π/2. On the other hand, when above transition energy η > 0, the synchronous phase should satisfy π/2 cos φ s π. Thus we can define the synchrotron tune (number of synchrotron oscillations per revolution) to be Q s = ω s ω 0 = hev η cosφ s 2πβ 2 E. (1.24) Typically synchrotron tune is a small number of the order of As we can see from above equation, when the phase slip factor η goes to zero, the synchrotron tune becomes zero thus the longitudinal phase space freezes and we call this isochronous condition. This would have some special applications in FEL when we need to preserve longitudinal microbunching structure. 1.3 Free Electron Laser Free Electron Laser (FEL) is a technique first invented in 1976 by John Madey involving coherent addition of synchrotron radiation emitted by electrons passing through periodic structure like alternating magnetic fields undulators as is shown in Fig It usually requires long undulator before the radiation can reach saturation. During this process, radiation spectrum starts out to be noise like with all wavelengths and a single wavelength λ determined by the period of undulator, electron beam energy and undulator parameter K = 0.94B 0 [Tesla]λ u [cm] as λ = λ u K2 2γ2(1 + 2 ) (1.25) will experience a power growth and peak out with very high intensity. Usually the peak brightness of FEL is at least 10 orders of magnitude higher than its peer from a 3rd generation light source. This process is also called Self-Amplified Spontaneous

32 10 1. Introduction Figure 1.2: Schematic drawing of SASE FEL and undulators. Emission (SASE). The power growth process requires the FEL parameter, given by ( ) ρ FEL = 1 1/3 Î λ 2 w K2 w 1, (1.26) 2 I A 2πγ 3 4πσ x σ y larger than the rms energy spread σ E. Thus one needs to have a very good quality beam with high peak current, low sliced emittance and low sliced energy spread (most important!). Electron beam properties in Linacs are determined by the injector. According to current technology, injectors can be designed to have ultra small emittance and low energy spread and bunch current can be made very high with bunch compressors so Linacs are good candidates for FEL. On the other hand, properties of the electron beams in storage rings are equilibrium values which involve radiation damping, quantum excitation, intra beam scattering and many other factors. The equilibrium rms energy spread is usually high when comparing with Linacs due to the quantum excitation. Also the bunch current is kept low so that many of

33 1.3 Free Electron Laser 11 the instabilities (intensity dependent) don t appear to destroy the storage of electron beam. A FEL technique called oscillator FEL (shown in Fig. 1.3) was invented to overcome these weaknesses of storage rings. It requires the implementation of a relatively shorter and weaker undulator together with optical cavities to store the radiation waves. During every revolution, the electron beam interacts with optical wave in undulator and losses a little fraction of its energy to the radiation. The radiation keeps growing and reaches saturation in many interactions with the electron beam. The growth of the power stored in the cavities needs to be larger than the losses at the optical mirrors so that the growing process can be continued. The gain of FEL usually does not have to be high as long as a mirror with high reflectivity is chosen so that it does not require such a high current electron beam. Because the oscillator FEL reaches saturation rather slowly comparing with SASE FEL and requires many interactions between electron beam and optical laser, it requires very fine alignment of the optical mirrors for the least degradation of FEL performance. If one is targeting hard X-ray which can be used for biology material science experiments and crystallography experiments, one would have difficulties in finding proper materials for reflection mirrors. Recently, some mirrors made of diamonds have been proposed but none experimental data has proven its validity. Some innovative ideas have come out in the effort of solving this problem Angular-Modulated Harmonic Generation (AMHG) and Echo Enable Harmonic Generation (EEHG) In the standard High-Gain Harmonic Generation (HGHG) scheme [3], two stages of undulators are implemented. A seed laser with wavelength λ is first used to generate energy modulation in the electron beam in the first undulator modulator. After

34 12 1. Introduction Figure 1.3: Oscillator FEL involves a shorter undulator and two reflective mirrors to confine the optical wave. Laser reaches saturation in N turns. passing through a chicane satisfying constraint R 56 δ = λ/4, the energy modulation is converted to density modulation with the information of high harmonics of wavelength λ. Then the density modulated beam is sent into the second undulator radiator to generate coherent radiation at wavelength λ/n. The phase space evolution of this process is shown in Fig Typically generating the nth harmonic of the seed laser requires the energy modulation amplitude to be approximately n times larger than the beam energy spread. Because of the inherent large energy spread of the beam in storage rings, the harmonic number is limited to about 3 to 5, good for ultraviolet but still one order of magnitude lower than soft X-ray. One may think about making use of the tiny vertical emittance of the beam in storage rings and propose an angular modulation instead of energy modulation to achieve higher harmonics. A vertical wiggling motion of the electron beam in modulator will introduce angular modulation

35 1.3 Free Electron Laser 13 Figure 1.4: Initial beam longitudinal phase space (a) with vertical axis the particle momentum. Different colors are used to depict different regions in initial particle distribution. Energy modulation is observed after modulator(b). Microbunching forms after chicane with condition R 56 δ = λ/4(c). to the beam. A chicane between modulator and radiator with nonzero R 54 with proper focusing magnets will transfer the angular modulation to phase modulation. The electron beam will further generate coherent radiation at nth harmonic. The bunching factor is crucial during this process. For the nth harmonic, it scales with b n e 1 2 n2 r 2, (1.27) where n is the harmonic number and r = kσ yr 54 with k the wave number of seeded laser and σ y the rms angular width for the electron beam. In order to make the bunching factor for the nth harmonic large enough the modulated angular or energy amplitude should be n times larger than beam divergence or energy spread respectively. For HGHG, due to the large energy spread in storage rings, laser power needed for generating such a large energy modulation is high. Also the energy spread after

36 14 1. Introduction lasing is high so may affect beam stability. If one uses angular modulation, with very tiny vertical emittance, laser power needed to generate the angular modulation is significantly lowered and the energy spread growth during lasing is very low given by γ = 2Bγɛ y /λ, (1.28) where B is the angular modulation amplitude and ɛ y is the vertical emittance. Typically, the harmonic number using angular modulation can be up to around 50 which reaches soft X-ray regime. Going back to HGHG, when two stages of modulators and chicanes are implemented, so called EEHG [4] can also further extend to higher harmonic number. The electron beam in the first modulator develops energy modulation before entering a strong dispersive chicane with large R 56. As shown in Fig. 1.5, the longitudinal phase space will be over rotated so a fine strip pattern forms in energy. After it experiences further energy modulation in the second modulator and a bunch compressor(2nd chicane), a very fine microbunching exists in the electron beam, which can radiate ultrashort X-ray in radiator. The longitudinal phase space evolution during the whole process is shown in Fig The bunching factor of EEHG decays much slower over harmonics comparing with HGHG or AMHG b n n 1/3. (1.29) For EEHG, harmonic number can go up to a few hundred. But the phase space control is very constringent considering multiple stages are being implemented. Also the coherent synchrotron radiation (CSR) and incoherent synchrotron radiation (ISR) introduces a large energy spread which will smear out the fine energy bands required for EEHG. Thus the beam current cannot be high as CSR scales with the beam current. The beam energy also cannot be very high as ISR scales with the beam energy to the 7th power. Many other practical issues such as rf power jittering will severely affect the performance of EEHG and experimental proof is yet to come.

37 1.3 Free Electron Laser 15 Figure 1.5: Particles over microbunch when passing through a large dispersive chicane with R 56 δ = λ(a) and R 56 δ = 2λ(b) Optics Free FEL Oscillator (OFFELO) Recently, people have invented a new method Optics Free FEL Oscillator (OFFELO) to overcome the shortage of proper mirrors working in X-ray regime. OFFELO requires two circulating beams (storage rings or Energy Recoverty Linacs) with one being low energy and the other being high energy. Schematic drawing is shown in Fig At first, a fresh low energy beam goes into modulator and develops a little bit of the microbunching structure. Then it circulates one turn and enters radiator with microbunching structure preserved. In radiator, the low energy beam generates a light at the wavelength of its microbunching period and goes to dump after the radiator. This generated light acts as a seed and interacts with the high energy circulating beam in a long undulator and enters high gain regime thus power of the light grows up fast. After the interaction with high energy beam, the light with high power goes into modulator as a seed and interacts with another fresh beam with low energy in the modulator. Due to the high power the light carries, this low energy beam starts out in its 2nd run with stronger microbunching structure thus radiates

38 16 1. Introduction Figure 1.6: Particles form very fine energy strips after the large dispersive chicane(a).particles further experience energy modulation in second modulator(b). At the end of 2nd chicane, a density modulation with ultrashort period is formed(c). Microbunching is observed in current distribution at the end of 2nd chicane. a light with higher power in the radiator. After a few iterations, the power of the light radiated will reach saturation and FEL is done without using any of the optical mirrors. One important and most difficult problem for this process to realize is to preserve the longitudinal phase space structure for low energy beam when it circulates one turn. As we have discussed before, particles are doing oscillation (rotation) in longitudinal phase space and their time structure will be completely destroyed after 1 of the synchrotron period. However, if phase slip factor is zero(up to a few orders of 4

39 1.3 Free Electron Laser 17 Figure 1.7: Optics Free FEL Oscillator requires two circulating beams. Low energy beam is the information carrier from modulator to radiator. Radiation reaches saturation in a few turns. δ), the synchrotron motion is very slow. Thus the particle distribution in longitudinal direction freezes during the transportation from modulator to radiator so that any of the microbunching structure (starts out to be small) developed in modulator can be preserved. Zeroth order of η can be made zero with stronger focusing quadrupoles so that particles with different energies are strongly focused to have the same pass length. Higher orders of η are also important in this sense and they can be tuned to zero with the help of higher order magnets such as sextupoles, octupoles, etc. A systematic study of how magnets affect phase slip factor or momentum compaction factor will be discussed in the appendix of this thesis. Another important issue is the repetition rate. If the high energy beam s current is very high, although the power of the light grows up fast and can reach saturation in fewer iterations, the energy spread growth in high energy beam induced by this FEL interaction is very large. If high energy beam is a storage ring beam, it usually take

40 18 1. Introduction a few thousand turns of revolutions to damp the energy spread back to a small value. This would greatly reduce the reptition rate of the lasing. If both beams are from ERL and the energy spread blown up by this process is still within the acceptance of energy recovery to work, then it requires superconducting cryomodules and does not have advantage in rep rate comparing with what is now achieved in ERL based FEL. It only has advantage in extending the radiation spectrum to X-ray or even hard X-ray which cannot be possible with the current ERL based FEL where reflective cavities are used. This is being achieved at a very high cost (construction of two ERLs). A technical breakthrough in mirror design is probably more economical and affordable.

41 Linear Lattice for 10 pm Storage Ring 19 Chapter 2 Linear Lattice for 10 pm Storage Ring Most electron storage rings in the world are designed to make use of synchrotron radiation. The brightness of a storage ring is given by B = F photon (2πɛ x)(2πɛ y) where ɛ x, ɛ y are the transverse electron beam emittances. The photon flux F photon is the number of photons per unit time in a given bandwidth ω/ω which by convention is usually chosen to be 0.1%. For a typical 3rd generation light source, to achieve high beam brightness, the transverse emittances are usually very small ( 1 nm for horizontal emittance and less than 1% for vertical emittance) and beam current is high. The beam brightness usually ranges from to photons/(s mm 2 -mrad 2 0.1% of bandwidth). When a storage ring has a small emittance that reaches the diffractive limit of the radiation, the brightness will greatly increase not only because the transverse beam size is small but also because the photons add up coherently. This condition is satisfied when ɛ y λ/4π, where y stands for both horizontal and vertical directions. Due to the lack of vertical bending magnets, the vertical emittance is usually induced

42 20 2. Linear Lattice for 10 pm Storage Ring by field errors or misalignments in magnets. Skew quadrupoles can be implemented to couple the horizontal emittance to vertical direction. The vertical emittance for a typical 3rd generation light source can reach diffractive limit by a very small coupling coefficient ( 1%). To have both planes reaching diffractive limit, we need to design a storage ring with ultimate low horizontal emittance. In this chapter, we will describe the linear lattice design of an ultimate storage ring using n-ba structure pm storage ring and n-ba structure In order to maximize the number of straight sections, achromat structure with n bending magnets (n-ba) is used. Dispersion function outside of the achromat structure is zero so insertion devices or user beamlines can be implemented. Due to the break of symmetry, its theoretical minimum emittance (TME) is larger than nonachromat lattice which can be purely symmetric and have nonzero dispersion everywhere. Typically, the natural emittance for a storage ring is given by ɛ x = F lattice C q γ 2 θ 3, (2.1) where C q = m is a radiation constant, and θ is the total bending angle in each dipole. The scaling factor F lattice is F lattice = H dip J x ρ dip θ3, (2.2) where H dip is the average H-function over all the dipoles and J x is the horizontal damping partition number. F lattice is a quantity that depends on the design of the storage ring lattice. For a Theoretical Minimum Emittance (TME) lattice with non zero disperion, it is 1/(12 15J x ). From Eq.(2.1), we know that the most efficient way of making natural emittance small is to reduce θ. Thus we need to increase the total number of dipoles. For 10

43 pm storage ring and n-ba structure 21 pm storage ring, we use 11BA structure with total of 440 dipoles separated into 40 superperiods. Each superperiod has 11 dipoles with non zero dispersion inside the superperiod. Dispersion is closed between superperiods so a 10 meter long straight section can be used for insertion devices and user beamlines Theoretical Minimum Emittance (TME) To match to the TME lattice, we need to minimize the H dip. Starting from transport matrix theory, the dispersion function and its first derivative in dipoles are given by D = ρ(1 cosφ) + D 0 cosφ + ρd 0 sin φ, (2.3) and where D 0 and D 0 D = (1 D 0 ρ ) sinφ + D 0 cos φ, (2.4) are the dispersion function and its first derivative at the entrance of the dipole and φ is the phase advance along the dipole. Thus H-function which is defined as H = γd 2 + 2αDD + βd 2, (2.5) can be expressed as H(φ) = H 0 + 2(α 0 D 0 + β 0 D 0) sin φ 2(γ 0 D 0 + α 0 D 0)ρ(1 cos φ) + β 0 sin 2 φ + γ 0 ρ 2 (1 cosφ) 2 2α 0 ρ sin φ(1 cosφ). After averaging the phase advance in the dipole, we arrive at H = H 0 + (α 0 D 0 + β 0 D 0)θ 2 E(θ) 1 3 (γ 0D 0 + α 0 D 0)ρθ 2 F(θ) + β 0 3 θ2 A(θ) α 0 4 ρθ3 B(θ) + γ 0 20 ρ2 θ 4 C(θ), where E(θ) = 2(1 cosθ)/θ 2, F(θ) = 6(θ sin θ)/θ 3, A(θ) = (6θ 3 sin 2θ)/(4θ 3 ), B(θ) = (6 8 cosθ + 2 cos2θ)/θ 4, C(θ) = (30θ 40 sinθ + 5 sin2θ)/θ 5.

44 22 2. Linear Lattice for 10 pm Storage Ring Under small angle approximation, we have A = B = C = D = E = F = 1. We note that θ here is the bending angle in a dipole. To find out the minimum H-function without the achromat condition, we can simply take derivatives with respect to initial D 0 and D 0 H D 0 = H D 0 = 0, then we finally arrive at the matching conditions D0 = 1 Lθ, (2.6) 12 and β 0 = L 60, (2.7) where the L is the total length of the dipole. Both dispersion and beta function have waists at the center of dipoles. In the 10 pm lattice, we use quadrupole triplets in between two dipoles to match the optics to the TME conditions listed above. In order to make beta function and dispersion minimum at the center of dipoles, we choose a layout of QD-QF-QD for the quadrupole triplets. Quadrupoles with lengths of 25 cm are used and a drift space of 40 cm between quadrupoles is kept to accommodate sextupoles and avoid collision of magnets. The quadrupole field gradients are /m 2, /m 2 and /m 2 respectively. In 10 pm storage ring, we have both achromatic region and non-achromatic region. We need a good transition between these two regions. In other words, we need to match the H-functions in both region. For an isomagnetic storage ring, the lengths of the center dipole and outer dipole should satisfy the condition L 2 = 3 1/3 L 1 (2.8) with L 2 the length of center dipoles and L 1 the length of the edge ones. For 10 pm storage ring, we choose middle dipoles to be 1.95 m and edge ones 1.3 m to satisfy this

45 pm storage ring and n-ba structure 23 Table 2.1: Parameters for 10pm storage ring. Parameter Beam energy Ring circumference Value 5 GeV 2663 m Equilibrium energy spread E/E (rms) % Natural emittance (rms) 9.1 nm-mrad Natural horizontal chromaticity Natural vertical chromaticity Horizontal betatron tune Vertical betatron tune Momentum compaction factor 1.223e-5 requirement. In a storage ring design, another important property is the fractional energy spread δ = de. Derived from the equilibrium longitudinal emittance, the E fractional energy spread is given by ( σ E E )2 = C q γ 2 J E ρ, (2.9) with J E the longitudinal damping partition number. And we have relations J x = 1 D, J z = 1, J E = 2 + D. (2.10) For isomagnetic storage rings with separated function magnet, D = αcr ρ is a small number. For 10 pm storage ring, we have J x = and J E = We choose the bending radius of the dipoles to be 78 m and the length of the edge dipoles to be 1.3 m so the rms energy spread is %. The total circumference of the ring is 2663m. Table. 2.1 shows the main parameters for this design. Since

46 24 2. Linear Lattice for 10 pm Storage Ring β (m), DX MINIMUM EMITTANCE FBA1 LATTICE Linux version 8.23/08 03/05/ β x β y DX s (m) δ E/ p 0c = 0. Table name = TWISS Figure 2.1: Plot of TWISS parameters for 11BA structure. Horizontal dispersion is magnified by 100 times. in vacuum undulator with period of 1 cm is available and we are aiming at hard X-ray lasing so we choose nominal beam energy to be 5 GeV and the storage ring can accommodate energy from 4 GeV to 7 GeV. The optics for one superperiod is shown in Fig The beta-function at the center of middle dipoles is matched to L dip / 60 and dispersion matched to L dip θ dipole /12 and natural emittance is 9.1 pico-meters for this lattice. We note that this is still 3.3 times larger than the TME predicted emittance 2.77 pm because of the breaking of symmetry. The betatron tunes are chosen to be ν x = and ν y = respectively so the zeroth order tunes stay relatively far away from lower order resonances. In order to move tunes

47 pm storage ring and n-ba structure 25 Figure 2.2: Plot of tune space with up to 8th order resonance lines. Red square is the location for 10 pm storage ring s tunes. to a safe location without changing optics and lattice properties much, we vary the quadrupole triplet in the non-dispersive region. The beta functions and dispersions in the central dipole regions are not changed thus the achieved minimum emittance is not affected. Fig. 2.2 shows the tune space with up to the 8th order resonance lines Effort in shortening the circumference As we can see, 10 pm storage ring design has a large circumference due to the large number of dipoles. This is common for ultimate storage ring designs because tiny

48 26 2. Linear Lattice for 10 pm Storage Ring β (m), DX MINIMUM EMITTANCE FBA1 LATTICE Linux version 8.23/08 03/05/ β x β y DX s (m) δ E/ p 0c = 0. Table name = TWISS Figure 2.3: Plot of TWISS parameters for 25BA structure. Horizontal dispersion is magnified by 100 times. dispersion is required to reach low emittance. Large circumference is very costly especially for the construction of beam tunnel and vacuum system. An alternative design with shorter circumference, on the other hand, is preferable if the natural emittance could be maintained at the same level. As we have discussed above, the emittance of a lattice is determined by the bending angle in the dipole ɛ x = F lattice C q γ 2 θ 3. (2.11) In order to get the same emittance, we need to keep bending angle in each dipole the same. While keeping the n-ba structure, we change the number of dipoles in each superperiod and number of superperiods to get the same bending angle. To match the

49 pm storage ring and n-ba structure 27 Table 2.2: Parameters for 10pm storage ring with 25BA. Parameter Beam energy Ring circumference Value 5 GeV 2334 m Equilibrium energy spread E/E (rms) % Natural emittance (rms) 9.5 nm-mrad Natural horizontal chromaticity Natural vertical chromaticity Horizontal betatron tune Vertical betatron tune Momentum compaction factor 1.4e-5 H-function between dispersive region and non-dispersive region, a ratio of 1/3 3 should be maintained between the length of the central dipoles and edge dipoles. A simple calculation of two cases, 11BA with 40 superperiods and 25BA with 17 superperiods is shown to have same bending angles 2π 17 (1.5 (25 2) + 2) 1.5 = = 2π 40 (1.5 (11 2) + 2) 1.5. (2.12) The optics of this new layout with 25BA structure is shown in Fig Beta function and dispersion remain the same as previous 11BA structure thus the emittance is matched to theoretical minimum. Main parameters for 25BA structure is shown in Table The natural chromaticities remain huge because the total number of magnets and minimum beta functions are similar to the 11BA design. The circumference of the 25BA lattice is reduced to 2334 m. This seems to be the limit of what we

50 28 2. Linear Lattice for 10 pm Storage Ring can achieve unless we implement combined function magnets. The momentum compaction factor, α c L dipθ dip R D min R with R the average radius of the circumference, is slightly larger than 11BA structure due to the reduction of ring circumference. 2.2 Combined function magnet lattics β (m), DX MINIMUM EMITTANCE CFM LATTICE Linux version 8.23/08 25/08/ β x β y DX s (m) δ E/ p 0c = 0. Table name = TWISS Figure 2.4: TWISS parameters for a superperiod of the combined function magnet lattice. A straightforward way of thinking to make a storage ring with shorter circumference is to use combined function magnets. By making dipoles with defocusing gradient, we can absorb two quadrupoles from the quadrupole triplet into the central dipole and shorten the circumference. The new lattice will be composed of dipoles with defocusing gradient and single quadrupole in between two dipoles to match the

51 2.2 Combined function magnet lattics 29 optics to TME. A simple lattice is made in the effort of achieving this purpose. Figure. 2.4 shows the optics for a central dipole cell in the combined function magnet lattice. The entire ring is composed of identical structures as previous 11BA lattice except the dipoles have gradients. The whole lattice resembles FODO lattice. We use D x (m) Kq(1/m 2 ) Figure 2.5: Dispersion vs matching quadrupole strength for the combined function magnet lattice. The different colors represent different drift space lengths. Longer drift space requires weaker matching quadrupole strength. Boundary reaches stability limits. a sort of analytical way to search for all possible solutions. The parametric space for this structure is relatively simple the central dipole s bending angle and bend radius are determined by the emittance and rms energy spread that we want to achieve with