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Optically Based Diagnostics for Optical Stochastic Cooling⇤ M ISBN 978-3-95450-180-9 Proceedings of NAPAC2016, Chicago, IL, USA WEPOA38 OPTICALLY BASED DIAGNOSTICS FOR OPTICAL STOCHASTIC COOLING⇤ M. B. Andorf1, V. A. Lebedev2, P. Piot1,2, J. Ruan2 1 Department of Physics and Northern Illinois Center for Accelerator & Detector Development, Northern Illinois University DeKalb, IL, USA 2 Fermi National Accelerator Laboratory, Batavia, IL, USA Abstract the two transverse phase spaces provides a path for cooling An Optical Stochastic Cooling (OSC) experiment with along all three degrees of freedom. electrons is planned in the Integrable Optics Test Accelera- Table 1: Undulator Parameters for the OSC in IOTA tor (IOTA) ring currently in construction at Fermilab. OSC requires timing the arrival of an electron and its radiation parameter value unit generated from the upstream pickup undulator into the down- stream kicker undulator to a precision on the order of less Uo 100 MeV than a fs. The interference of the pickup and kicker radiation K 1.04 - suggests a way to diagnose the arrival time to the required period 11.06 cm precision. λ (at zero angle) 2.2 µm Total length 77.4 cm INTRODUCTION In IOTA, the OSC proof-of-principle experiment will be Fermilab is currently preparing a proof-of-principle of accomplished in two phases. In a first stage, we will not Optical Stochastic Cooling (OSC) using 100 MeV electrons have an amplifier and rely on passive cooling: the optical at the Integrable Optics Test Accelerator (IOTA). In OSC system will consists of a telescope composed of three lenses a particle emits radiation in a pickup undulator. The light chosen so that the optical transfer matrix between the pickup from the pickup is then transported, and in the presence of and kicker undulators is I where I is identity matrix. The an optical amplifier, amplified, to a kicker undulator located − telescope can transport 90% of the energy from the pickup downstream and identical to the pickup undulator. There- ⇠ to the kicker. Parameters for the undulators can be found fore the particle follows a curved path in a magnetic bypass in Table 1 and yield a value for the kick amplitude, ⇠ = chicane such that it arrives at the entrance of the kicker 0 8.5 10 10. Simulations show that the kick is further reduced just as the light from the pickup is also arriving [1,2]. For ⇥ − by roughly 10 % due to dispersion in telescope [3]. In the the reference particle the arrival time of radiation and par- second stage, a 7-dB Cr:ZnSe optical amplifier (with an ticle in the kicker is such that it receives no kick. Parti- operating band from 2.2 2.9 µm) will be installed and is cles with a nonzero synchrotron or betatron coordinate are − expected to double the cooling rates over the passive cooling displaced relative to the reference particle by an amount scheme. ∆s = M51 x + M52✓ + M56(∆p/p) and receive a longitudinal kick given as In this paper we explore how to measure the timing error δp between the arrival of the pickup signal and the particle in = ⇠ sin(k∆s). (1) the kicker. The timing precision needs to be < 1 fs in order p − 0 to fully optimize the cooling rates. We propose a technique In the above Mij are the elements of the 6 6 transfer matrix that uses optical interference between the pickup and kicker ⇥ from pickup to kicker, k 2⇡/λ is the radiation wave- radiation signals. Before detailing the measurement tech- ⌘ number, and (δp)/p is the relative momentum change of the nique we first consider in the impact of arbitrary timing error particle after a single pass through the cooling section. The on the OSC process. kick normalized amplitude is ⇠0 = 2E0pG/(pc) where E0 corresponds to, in the case of a sufficiently small undulator LONGITUDINAL TIMING parameter, the energy radiated in a single undulator, G is the gain in power from the optical amplifier (if present), p Cooling Rates and Transit Time Error the design momentum and c the velocity of light. When the The damping decrements λi (i [x, s]) for small betatron 2 dispersion function is non vanishing at the kicker undulator and synchrotron amplitudes are given by [4] location a horizontal kick is also applied thereby resulting in horizontal cooling. Likewise, introducing coupling between λx k⇠o M SP = 56 − , (2) "λs # 2 " SP # ⇤ Work supported by the by the US Department of Energy (DOE) contract DE-SC0013761 to Northern Illinois University. Fermilab is operated 2016 CC-BY-3.0 and by the respective authors where Sp = M Dp + M D0 + M is the partial slip factor by the Fermi research alliance LLC under US DOE contract DE-AC02- 51 52 P 56 © 07CH11359. such that a particle with a momentum deviation ∆p/p will be 3: Advanced Acceleration Techniques and Alternative Particle Sources 779 Copyright WEPOA38 Proceedings of NAPAC2016, Chicago, IL, USA ISBN 978-3-95450-180-9 damping and excitation dp"˜ p"˜ = λx (ax, ap ) dt ⌧s λx,damp + (p"˜ p"o) ⌧s − d∆p/p ∆p/p = λp (ax, ap ) dt ⌧s λp,damp + (∆p/p σp ), (5) ⌧s − Figure 1: Trajectories in ax,ap space. Top figures are for where ⌧s is the revolution period of the accelerator, λx,damp, ksd = 0. The left shows trajectories from OSC without λp,damp are the synchrotron cooling decrements in ampli- diffusion from synchrotron damping, while the right figure tude per turn, "0 is the equilibrium beam emittance and σp includes both. The bottom figure is set for ksd = ⇡ with is equilibrium momentum spread, both without OSC. again the left figure showing trajectories from OSC and the The above equations exclude fluctuations. Thus, they right including diffusion from synchrotron radiation. describe an evolution of average amplitudes for an ensemble of particles having the same initial amplitudes. They do not imply, in the absence of OSC, that all particles are driven longitudinally displaced by the amount Sp∆p/p while travel- to have the values "˜ = "0 and ∆p/p = σp, which is only ing from the pickup to kicker undulators (without accounting statistically true for the whole beam. In reality particles make the additional displacement due to betatron motion). The random walks about these values, reflecting the random 0 dDP quantities DP is the dispersion in the pickup and DP = ds . process of photon emission. Table 2 summarizes the OSC As can be seen from Eq. (1) the corrective kick becomes chicane and beam parameters used in this contribution. nonlinear for large particle amplitudes and so the damping decrements depend on amplitude. This is accounted for in [4] Table 2: OSC and Beam Parameters by averaging kicks over betatron and synchrotron oscillations. Here we follow what was done in the latter reference but parameter value unit delay 2 mm include an additional phase term in Eq. (1), ksd, where sd/c is the difference between between the arrival times measured Sp 1.38 mm at the kicker between the reference particle and its radiation Sx 2.24 µm 4 σp 1.1 10 - emitted (at a retarded time) from the pickup. Including this ⇥ − additional term yields the damping decrements "o 9.1 nm 7 λp,damp 1.2 10− - ⇥ 8 λx (ax, ap ) Lp sd λx,damp 7.2 10− - = − | | cos(ksd ) ⇥ "λp (ax, ap )# Lp λ 2J (a )J (a )/a Momentarily neglecting the synchrotron radiation terms x o p 1 x x , (3) ⇥ "λp2Jo (ax )J1(ap )/ap # we look for the fixed points of Eq. (5). It is easily seen that such points occur only when both J0(ax ) and J0(ap ) are where the left most factor has been introduced to account for zero, or when J1(ax ) and J1(ap ) are zero. Computation the decrease in the kick amplitude due to the finite length of of the eigenvalues of the Jacobian reveals that the J0 fixed the undulator pulse, Lp Nundλ, and Nund is the number points correspond to saddle points while the J fixed points ⇡ 1 of undulator periods. The parameters ax and ap are the am- are either stable or unstable nodes. plitudes of longitudinal displacement from pickup to kicker, There are three particularly important fixed points expressed in units of the wave phase, respectively due to [0,µ, ], [µ , ,µ, ],and [µ , , 0] where µ , 2.405 is the 1 1 0 1 0 1 1 1 0 1 ⇡ betatron and synchrotron oscillations first zero of Jo and µ , 3.832 is the second zero for J . 1 1 ⇡ 1 These three points roughly map out the cooling boundary ax = k "˜Sx, and enclose most particles. When cos(ksd ) > 0 all three ap = kpSp (∆p/p), (4) points are unstable and all particles within this region are attracted to the stable node [0, 0],aJ1 = 0 fixed point. where "˜ is the particles Courant-Snyder invariant, Sx = When cos(ksd ) < 0 the stability of the nodes are switched 2 2 2 ( βp M 2↵p M M + (1 + ↵ )M )), and ↵p and βp are while the saddle points remain unstable. Thus all particles 51 − 51 52 p 52 the Courant-Snyder parameters in the pickup. within the cooling range are attracted to either [0,µ1,1] or We can now write a system of coupled rate equations for [µ1,1, 0] depending on their initial coordinate. Therefore 2016 CC-BY-3.0 and"˜ by the respectiveand authors ∆p/p, where in addition to the effects of OSC we longitudinal timing errors correspond to only three modes © include a term to approximately describe both synchrotron of behavior, the two just described which are determined Copyright 780 3: Advanced Acceleration Techniques and Alternative Particle Sources ISBN 978-3-95450-180-9 Proceedings of NAPAC2016, Chicago, IL, USA WEPOA38 only by the sign of cos(ksd ).
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