Interferometer design for optical stochastic cooling ...

Interferometer design for optical stochastic cooling demonstration at Bates James Hays-Wehlea † , Wilbur Franklina , Franz X. Kärtnerb , Jan van der Laana , Richard Milnera , Aleem Siddiquib , Christoph Tschalära , Fuhua Wanga a

Massachusetts Institute of Technology Laboratory for Nuclear Science, Cambridge, MA 02139 and MIT-Bates Accelerator Center, Middleton, MA 01949; b MIT Research Laboratory of Electronics, Cambridge, MA 02139 ABSTRACT

Optical stochastic cooling (OSC) holds significant promise for enhancing the luminosity of high energy colliders by cooling charged particle beams. This paper describes the conceptual design and requirements for an interferometer to be built for an OSC demonstration experiment with stored electrons at the MIT-Bates South Hall Ring. The paper presents an overview of the optical and charged particle beamlines, the intensity detection system and the phase stabilizing feedback loop. Keywords: Accelerator physics, optical stochastic cooling, phase lock

1. INTRODUCTION 1.1 Accelerator application High energy collider applications benefit greatly from small “emittance,” or phase space distribution of the charged particle beams. The collision rate in an accelerator experiment varies inversely with the emittance, and additionally, low emittance allows more current to be stored, and is in general a measure of beam quality. Most of the particle beam’s evolution is governed by Hamiltonian dynamics, where Liouville’s Theorem holds that the phase space distribution can be manipulated and distorted, but never compressed. Where compression corresponds to a decrease in emittance. Conventional beam manipulations by magnetic fields use static potentials and conservative forces, such that phase space distribution is conserved. Cooling, on the other hand, requires a non-symplectic transformation, engendered either by a non conservative force (as in synchrotron damping), or a time dependent Hamiltonian, as is the case with stochastic cooling. Stochastic cooling is a revolutionary technique for damping a beam to a smaller emittance. It is termed “stochastic” as the method repeatedly addresses small random samples of the beam (not the beam as an unambiguous whole), and “cooling” as it reduces random motion within the beam as would cooling a hot gas. Stochastic cooling was designed in 1972 by Simon van der Meer1 using two pairs of electrostatic strips in a storage ring. The first pair picked up a signal proportional to the vertical position of the beam, and sent this signal amplified to intercept the beam on the other side of the ring, where the second strip produced an electric field, giving a correction impulse, (or “kick”) to steer the same section of beam back towards the central orbit, thereby creating a negative feedback loop. This first use of the technique allowed the Super Proton Synchrotron at CERN to store anti-protons and collide them with protons to discover the W and Z bosons. It is currently in use at the anti-proton source at Fermilab and RHIC at Brookhaven National Laboratory. The technique hinges on sampling the beam and correcting based on the average phase location of that sample, where the sample size is determined by the pickups and the bandwidth of the information carrier. Therefore, it is vitally important to have a method which is fast enough to resolve the beam on a scale where inhomogeneity is manifest. To deal with denser beams, a new approach to this technique moves the sampling frequency to the faster, broadband regime of optical light. Proposed in 1994 by Zholents and Zolotorev,2 OSC works by passing the beam through a “pickup” undulator, producing synchrotron light (electromagnetic radiation from ultrarelativistic † Send correspondence to: [email protected]

Interferometry XIV: Techniques and Analysis, edited by Joanna Schmit, Katherine Creath, Catherine E. Towers, Proc. of SPIE Vol. 7063, 706319, (2008) · 0277-786X/08/$18 · doi: 10.1117/12.795064

Proc. of SPIE Vol. 7063 706319-1 2008 SPIE Digital Library -- Subscriber Archive Copy

Chicano dipolo magnots

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liii.

Optical Parametric Amplifier

Upstream pickup undulator

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Downstream kicker undulator

Figure 1. OSC device layout, showing the path of the particle beam (dot), and light (wave) through the undulators and bypass chicane.

particles accelerated in magnetic fields) which is amplified and returned to the beam at the location of a “kicker” undulator. Figure 1 shows a schematic of the particle and light trajectories through the undulators, and where the particle beam is split off from the light to travel through a bypass chicane. The larger chicane dipole magnets steer the particles around the optical parametric amplifier, but also alter the bunch distribution through Hamiltonian dynamics. Just like white light spreading out through a prism, the more energetic particles are less deflected by the magnets and fly a straighter, shorter path to the other end. Thus, when the bunch reaches the other end of the chicane, it has acquired a phase space correlation: the highest energy particles lead the bunch. If the phase relation of the amplified light with respect to particles is properly maintained, then the kick given to the particle will be proportional to its offset from this reference trajectory (the reference trajectory is that of an ideal particle exactly at the center of the bunch, with exactly the right momentum), thereby decelerating the higher momentum particles and accelerating the lower ones. The net effect is to bring particles closer to a central momentum, “cooling” the beam. A proposal has been submitted to test this technique at the William H. Bates linear accelerator laboratory in Middleton, MA.

1.2 Interferometry and phase control Moving to this new optical regime, however, requires that phase stability be maintained over an optical path length of many meters. This phase relation is critical because the ‘kick’ is produced by coinciding the light’s transverse electric field with the particles sideways momentum in the undulator. If this coincidence were to be off by half a wavelength, the effective kick would be opposite to what was intended. The second undulator will also produce light (light not shown in Figure 1). When the phase is correct for cooling, the light from the second undulator will lag behind the amplified pulse by a quarter wave. The phase relation can be controlled by delaying, with refractive material, the amplified light. Then the interference between these two pulses can be used to measure this phase relation. Direct interference of two pulses propagating along the same axis is the basis of a Mach-Zehnder device, and laser feedback interferometry has been explored thoroughly in recent years. However, neither of these techniques have been widely used in accelerator applications, nor has the interference between pulses of different magnitudes been approached. This paper will demonstrate the feasibility of using this technique in the OSC setup.

2. VISIBILITY The light created by a particle deflected in a magnetic field is referred to as synchrotron radiation, and has been well employed by condensed matter physicists experimenting with soft X-rays. An undulator (or “wiggler” or “insertion device”) is a series of alternating magnets designed to maximize this light and its coherence. The high coherence of synchrotron light allows us to treat the pulses as classical waves. In this analysis, we will ignore higher order effects from the amplifier setup, and treat as identical the two electric fields of the two undulator

Proc. of SPIE Vol. 7063 706319-2

x

Px

(a)

(b)

(c)

Figure 2. The phase distribution for the beam before it enters the chicane (a), and after,with arrows indicating the correction applied by OSC, in the cases of cooling (b) when in phase and heating (c)when out of phase.

light pulses, though one amplified by an amplitude gain factor g and the other delayed by some time τ . Then the amplitude of the electric field incident on the detector is simply the sum of the two fields: Etotal (ω) = gE(ω) + eiωτ E(ω)

(1)

I(ω) = |g + eiωτ |2 E 2 (ω)

(2)

The particles execute a sinusoidal trajectory though the undulator, from which the Liénard-Wiechert potentials can be calculated. From those, the undulator light can usually be characterized as a plane wave in a boxcar envelope:4 ω−ω E0 sin(πN ω0 0 ) , (3) E(ω) = √ 0 2π πN ω−ω ω0 where ω0 is the central frequency, in the case of the Bates experiment, 2 µm, chosen to suit the bandwidth of the optical amplifier. But in general it is determined by the magnetic field and the beam energy. Integrating over frequencies from equations (2) and (3), we have the power, E02 dω(g 2 + 1 + 2g cos ωτ ) sinc2 (N ∆ω) U= (4) 2π N ω0 τ (5) U ∝ (g 2 + 1) + 2g cos ω0 τ tri 2π This curve, U (τ ), is shown as part of Figure 3. The figure also shows the correspondence to light wave overlap (with maximum interference at full coincidence), as well as the target delays for cooling. Note that optimal cooling occurs on the nodes of the curve, where no net power is put in or extracted from the bunch. However, on the nodes π later, the phase is such that the correction will be applied backwards: the high energy head of the bunch will be accelerated while the lower energy tail will be decelerated, thereby increasing the energy spread such that the beam will in fact be heated (see Figure 2.) When the delay lands on the curves trough or crests, however, the whole bunch is being accelerated or decelerated respectively. The effective signal to noise ratio of the phase measurement will be heavily influenced by the difference in max−min the amplitudes of the two pulses. Interferometric visibility of a fringe is defined as the amplitude average or max+min of the power. Thus, interference between two identical coherent plane waves will have a visibility of 1. We can then see from equation (5) that the background goes as g 2 + 1 and the amplitude of the modulation as 2g. Thus, the visibility is 2g (6) ν= 2 g +1


fl—fl 2

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2NAr4Cj.tm Figure 3. Graphical interpretation of power as a function of delay. The top of the figure shows the overlap of the two pulses, while the bottom shows the result of their interference.

before slight corrections due to angular spread. This curve is graphed as Figure 4. From this we can see that there is a tradeoff between amplifier gain and visibility. However, while running at unity gain could be of use, measurements done at half gain are not appreciably easier than those done at full gain. But even at full gain (g 2 ≈ 103 ), the visibility is reduced to 1/20, which, while small, can be resolved with the signal-to-noise ratios available in the detectors of interest. Future analysis will include the gain factor as a complex function of frequency, but given that the optical parametric amplifier has sufficiently flat response and minimal dispersion, we can treat the amplification factor simply as a real number.

3. DETECTOR To determine the phase, we simply need to measure the incident power, which can be done by placing a detector at the next synchrotron port ten meters downstream, where the particle and light beams part ways. See Figure 5 for layout. There are a wide variety of detectors available for the infrared range that our experiment is using. Those ideally suited to our application are the InGaAs or PbS photoresistive type such as the Hamamatsu P9217,5 whose peak wavelength sits at 2.2 µm and has a rise time of 200 µs, allowing plenty of measurement within both the cooling time and the stability of the chicane. Furthermore, the small detector surface means that we can measure only the central, most coherent core of the beam. It also means that the steepest angle ω0 4 involved is 3 mm/10m = .3 mr, and the corresponding central frequency ω = 1+γ 2 θ 2 , due to doppler effects, ω is less than 10% off from its value on axis. Thus, because ω−ω > N , we can use a more divergent lens post amplifier to improve visibility without worrying that the different frequencies will no longer interfere. 2 With current parameters, the undulator alone will produce a peak power intensity3 dP dΩ ≈ .6[mW/mr ] meaning that at full amplifier gain, the power received by the detector would be a background average of 30 mW with an varying signal amplitude of 2 mW. The Hamamatsu detector of choice is ideally suited to much lower power than will be incident upon it at full gain. But this can be remedied easily with a neutral density filter such as Newport’s FIR-ND30.6 Further power will be lost in the use of an optical chopper to give the detector recovery time between measurements. The detector is responsive over 3 orders of magnitude in power, so the same filter can be fitted and left in place even if the amplifier gain is to be varied over its full range.


Visibility 1

0.75

0.5

0.25

0

4

8

12

16

20

24

28

32

Gain

Figure 4. Visibility as a function of gain. Eq. (6)

A

IDDDD 'I'I

IDDDD j\ Wiggler I

Wiggler 2

Amp

U _____I Filters

Phase control Figure 5. Locations of the detector and phase control.


Detector

4. FINDING THE ZERO CROSSING The phase can be controlled by varying the thickness of a refractive material in the optical beamline. See Figure 5. For the Bates experiment, this material is barium flouride, with a refractive index of 1.45. By adjusting these BaF2 delay wedges, we can easily scan through the range of delay τ to map out the curve shown in Figure 3, and then identify the zero crossing where maximal cooling will occur. Keeping the phase locked at this point requires a feedback loop of some sort. The most obvious method is to determine the power associated with the zero-level, and simply feed this level along with the actual detector output through a differential amplifier and out to the actuators which drive the wedges. This is susceptible to systematics inherent in determining the target power level. Another possibility is that if we can precisely vary the gain of the amplifier without altering the phase, we can observe the response of the detected power, which will be proportional to g 2 + 1 at the nodes and nowhere else, thus if and only if the phase is correct. A third approach is to drive the wedges, and thus the delay, sinusoidally, and to measure the response of the detector. On the zero crossing, the second derivative is also zero, and small oscillations about it will be purely harmonic, while oscillations about some other power level will produce higher harmonics. A feedback loop that alters the phase to minimize these harmonics could keep the device at the correct delay. To wit, if we let τ = 0 define the zero crossing, then the power goes as P = A sin ω0 τ + B. If we then vary the delay as τ = τ0 + τ1 sin ω1 t, where τ0 is the offset we are looking to minimize, we have P (t) = A sin ω0 (τ0 + τ1 sin ω1 t) + B

(7)

If we only use small oscillations (ω0 τ1 ≡ < 1), which are preferable, as they remain closest to the target, we have 2 P (t) = A{ cos ω0 τ0 sin ω1 t − sin ω0 τ0 sin2 ω1 t} + B (8) 2 clearly showing that the amplitude of the second harmonic goes monatonically with the offset. We can then use a heterodyne mixer to remove the central frequency from the detector power output using the same local oscillator that drives the delay wedge actuator, leaving the second harmonic signal. Using the local oscillator and a voltage controlled oscillator, we can phase detect the remainder of the signal, returning a DC signal proportional to the offset from the zero crossing delay, which can then be fed back to the actuator, so as to make the necessary adjustment. If faster response is needed, the option to run at higher frequency (up to the speed of the wedge actuators) remains, and if better precision is needed, we could also phase detect on higher harmonics as well. A bench test of the feedback system can be achieved easily by using beam splitters and a laser to simulate the wiggler light. By building all the optics in this way, we can test optics and feedback scenarios without using valuable beam time, and possibly before insertion devices are even installed in the ring.

5. CONCLUSIONS Optical stochastic cooling is a promising new technique in accelerator physics for improving luminosity, but it depends heavily on optical path length stability. However, this stability can be achieved through the use of direct interferometry. Measuring interference between the amplified light from the pickup undulator with the light from the kicker undulator produces a signal with adequate visibility for resolution with available detector technology, even at full optical amplifier gain. This signal and a simple feedback loop will be able to compensate for instability in the optical path.


REFERENCES [1] van der Meer, Simon, Nobel Lecture (1984) [2] Zolotorev, M. S. and Zholents, A.A; “Transit-time method of optical stochastic cooling” Phys Rev E 50, 4 3089-3091 (1994) [3] Kim, K. J., “Angular Distribution of Undulator Power for an Arbitrary Deflection Parameter K*” Nuc. Inst. Meth. A246 67-70 (1986) [4] Hofmann, A. [The Physics of Synchrotron Radiation] Cambridge University Press, 154-167 (2004). [5] Hamamatsu catalog, http://www.datasheetcatalog.org/datasheet/hamamatsu/P9217.pdf [6] Newport catalog, http://www.newport.com/file store/PDFs/tempPDFse3568 Absorptive-Neutral-DensityFilters.pdf
