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Simulations of Optical Stochastic Cooling with Elegant∗ 9th International Particle Accelerator Conference IPAC2018, Vancouver, BC, Canada JACoW Publishing ISBN: 978-3-95450-184-7 doi:10.18429/JACoW-IPAC2018-THPAK057 SIMULATIONS OF OPTICAL STOCHASTIC COOLING WITH ELEGANT∗ M. B. Andorf1, P. Piot1;2, and V. A. Lebedev2 1 Department of Physics and Northern Illinois Center for Accelerator and Detector Development, Northern Illinois University DeKalb, IL 60115, USA 2 Fermi National Accelerator Laboratory, Batavia IL 60510, USA Abstract Fermilab is pursuing a proof-of-principle test of the Opti- cal Stochastic Cooling (OSC) of 100 MeV electrons in the Integrable Optics Test Accelerator. In support of this we present simulations of horizontal damping with OSC. We find excellent agreement with theory on the amplitude depen- dent damping rates. Additionally particle tracking is used to confirm the necessity and effectiveness of sextupoles used Figure 1: Overview of the OSC beamline (the beam propa- i j to correct non-linear path lengthening in the OSC chicane. gates from left to right). The letters B and Q respectively refer to quadrupole and dipole magnets. INTRODUCTION The Optical Stochastic Cooling (OSC) is a beam cooling we report on simulations of the OSC in the IOTA ring using method that holds promise for the cooling of dense particle the particle tracking code elegant [4] and focus on the beams. It is similar in concept to the widely used stochastic horizontal phase-space damping. cooling with microwaves but by utilizing optical frequencies enables cooling to be done with bandwidth 3-4 orders of Table 1: Undulator Parameters for the OSC Test in IOTA magnitude larger than any current stochastic cooling sys- with 100 MeV Electrons tem [1] [2]. At optimum gain, this increase in bandwidth Parameter Value Unit reduces damping time by that same amount. The transition to optical frequencies is done by replacing the traditional undulator parameter, K 1.038 - ’pickup’ and ’kicker’ with two identical undulators. A par- length, Lu 77.4 cm ticle radiates a short wave-pulse in the pickup and then, by undulator period, λu 11.06 cm interacting with its own radiation in the kicker, receives a number of periods, Nu 7 - 2018). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI. © corrective kick. A magnetic chicane between the undulators on-axis wavelength, λo 2.2 µm simultaneously gives space for the required focusing optics electron Lorentz factor, γ 195.69 - and optical amplifier while also providing the appropriate de- lay (with respect to the particles wave-packet) necessary for cooling(see Fig.1). Currently, Fermilab is pursuing a proof- HORIZONTAL COOLING RATE of-principle test of the OSC using 100 MeV electrons in the The horizontal damping rate from the OSC can be found Integrable Optics Test Accelerator (IOTA) [3]. The expected in a very similar way as that of damping of betatron os- undulator and beam parameters are given in Table 1. cillations from the emission of synchrotron radiation [5]. The energy kick, δu, a particle will receive in the kicker Coupling between longitudinal and horizontal planes occurs is determined by the particles longitudinal displacement, where dispersion in the ring is non-zero. In this case the par- s, relative to the reference particle upon traveling between ticles position is given as x = xβ + ¹u/UsºD and its angle by undulator centers 0 0 0 x = xβ + ¹u/UsºD . Here the β subscript is used to denote δu displacement from betatron motion, D and D0 is dispersion = κ sin¹kosº; (1) Us and its derivative and u is the particles difference in energy with respect to the reference particle. The longitudinal dis- κ where is the kick amplitude normalized by particles energy placement of the particle due to its horizontal coordinate in U k ≡ π/λ λ s, and o 2 o where o is the undulator-radiation on- the pickup center is axis wavelength. Although the kick is purely in energy, 0 coupling between longitudinal and horizontal planes in the s = M51 x + M52 x (2) cooling insertion results in horizontal cooling. In this paper M ∗ where 5n are the elements of the transfer matrix from Work supported by the US Department of Energy (DOE) contract DE- pickup to kicker centers. At first lets assume k s 1 so that SC0013761 to Northern Illinois University. Fermilab is operated by o the Fermi research alliance LLC under US DOE contract DE-AC02- Eq. 1 can be Taylor expanded. During the kick the particles 07CH11359. position and angle do not change appreciably, and yet clearly Content fromTHPAK057 this work may be used under the terms of the CC BY 3.0 licence ( 05 Beam Dynamics and EM Fields 3354 D09 Emittance Manipulation, Bunch Compression and Cooling 9th International Particle Accelerator Conference IPAC2018, Vancouver, BC, Canada JACoW Publishing ISBN: 978-3-95450-184-7 doi:10.18429/JACoW-IPAC2018-THPAK057 its energy has; hence there must be a corresponding change where x is the betatron phase and ax is the amplitude of the 0 to xβ and xβ: particles longitudinal displacement over the course of a beta- tron oscillation expressed in units of phase of the undulator ∆xβ = − κDM51ko x central wavelength [6]: (3) ∆ 0 − 0 0 xβ = κD M52ko x : q 2 2 2 ax = ko βM − 2αM51 M52 + ¹1 + α ºM /β : (7) From these equations a corresponding decrease in the par- 51 52 02 0 ticles Courant-Synder invariant, = βxβ + αxβ xβ + ¹1 + The damping time is long compared to the period of the 2 2 α ºxβ/β is found. Dividing this change by the revolution betatron oscillation and so the cooling rate can be averaged period τs, and averaging over betatron oscillations yields over the period yielding the amplitude dependent damping rate: d ko 0 = κ ¹M51D + M52D º: (4) λx¹axº = 2λx J1¹ax)/ax (8) dt 2τs and the rate of change of ax is and so ¹tº = o exp(−tλxº where λx is the horizontal damp- ing rate: dax = −2J1¹axºλx : (9) ko 0 dt λx = κ ¹M51D + M52D º: (5) 2τs In elegant the kick is computed by recording the arrival Table 2: IOTA Parameters Extracted From or Piven to Ele- For the more general case of arbitrary particle amplitude gant we re-express Eq. 1 as Parameter Value Unit δu = κ sin ax sin¹ xº (6) Revolution period, τs 133 ns Us emittance (without cooling), σ 2.67 nm Normalized Kick Amplitude, κ 2.0×10−6 - OSC damping time, 1/λx 22 µs −4 M51 0.6×10 - M52 4.0 mm M56 3.8 mm times of the particle in the pickup and kicker, tp and tk re- spectively. During the first pass through the insertion the ht − t i average time difference k p of the bunch is computed 2018). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI. and used to compute the arrival phase of the reference par- © ticle. In subsequent passes through the cooling insertion a particles time of flight tk −tp is used to find s and compute the corresponding change in the particles momentum. The top pane in Fig. 2 shows the damping of ax¹tº as computed from elegant using particle positions and angles and Eq. 7. It is in excellent agreement with the damping curved obtained from numerical integration of Eq. 9. Table 2 summarizes lattice and cooling parameters used in simulations1. Our model neglects kicks by neighboring particles since we are interested in the dynamics of a single classical particle in the accelerator. This choice is further justified considering that the OSC test in IOTA will first be done passively where the kick amplitude is far from the optimal gain when the incoherent kicks from neighboring particles become non- negligible. We can account for separation that occurs between the par- ticle and its radiation for a particle that is off of the optical Figure 2: The amplitude of longitudinal displacement a ¹tº x axis. This is particularly important in active OSC. In order for different initial amplitudes. Dots are computed from ele- to suppress depth of the field effects associated with a finite gant outputs while solid lines are from numeric integration of Eq. 9. The top pane assumes no separation occurs be- 1 In our simulations κ and consequently the OSC damping time can be tween particle and radiation while the bottom pane accounts set arbitrarily. The values given in the Table 2 were chosen to limit the number of turns needed to observe damping. The OSC damping rate in separation for the case of the +I telescope. IOTA is expected to be more than three orders of magnitude slower. 05 Beam Dynamics and EM Fields THPAK057 Content from this work may be used under the terms of the CC BY 3.0 licence ( D09 Emittance Manipulation, Bunch Compression and Cooling 3355 9th International Particle Accelerator Conference IPAC2018, Vancouver, BC, Canada JACoW Publishing ISBN: 978-3-95450-184-7 doi:10.18429/JACoW-IPAC2018-THPAK057 Figure 3: Same setup as Figure 2 but without sextupole to correct the non-linear path lengthening. Separation between light and particle is not accounted. undulator length the focusing light optics are made to be a telescope with a transfer matrix ±I where I is the identity matrix. For an active test of the OSC in IOTA the positive case must be chosen so that radiation from the pickup can be focused to a reasonable size in the OA crystal making the pump laser power and associated heating manageable. And so if a particle radiates at position xp the center of its imaged radiation will appear at xp in the kicker.
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