1. Toward a proof-of-principle experiment of Optical Stochastic Cooling
Northern Illinois University
Toward a proof-of-principle experiment of Optical Stochastic Cooling
Matthew Andorf1 On the Behalf of V. Lebedev2, A. Romanov2, J. Ruan2, P. Piot1,2
1Northern Illinois University
2Fermi National Accelerator Laboratory

2. Introduction to Optical Stochastic Cooling (OSC)
First proposed in mid 1990’s as an improvement over stochastic cooling. Working principles are the same as stochastic cooling but movement to much shorter wavelengths increases the bandwidth (W) of the cooling system by ~104.
Ordinary pickup/kicker are replaced by undulators.
Since damping time is inversely related to the bandwidth OSC has the potential to decrease damping times by 3-4 orders of magnitude. However a major challenge to achieving this is the design of the optical amplifier.
The OSC kick is always longitudinal (energy kick). Presence of dispersion in the kicker allows for horizontal cooling. Coupling of x and y outside of cooling insertion allows for full 6D particle beam cooling.
Energy kick is determined by phase between transverse particle motion and pickup radiation inside kicker.

3. Experiment broken into two phases:
OSC in IOTA
Integrable Optics Test Accelerator (IOTA) at Fermilab has a ~6 m straight section reserved for proof-of-principle demonstration the OSC.
Cooling will be demonstrated on 100 MeV electrons.
Identical 7 period undulators will be used for kicker/pickup. Base wavelength 2.2 μm.
Experiment broken into two phases:
Phase 1 (passive OSC): Bunched cooling using only lenses to focus pickup radiation into kicker. Optics designed to suppress depth of field.
Phase 2 (active OSC): An OA based on Cr:ZnSe is used to amplify pickup signal. Expected to increase damping rates by a factor of 2.
-Additionally experiments with single electron are being considered.
OSC damping times assumes x-y coupling outside cooling insertion.

4. Damping Rates for small particle amplitude
The kick depends on longitudinal particle displacement relative to the reference particle as:
For small particle amplitude this becomes
where
Is the kick amplitude. In the absence of an OA and sufficiently small undulator parameter K, Eo is the energy loss from one undulator. G is energy gain in amplifier not accounting pulse broadening.
Damping decrements (amplitude per turn) are found to be
The sum of the damping decrements is determined by M56. Dispersion merely redistributes cooling between longitudinal and horizontal planes.
V. Lebedev, “OPTICAL STOCHASTIC COOLING” in Beam Dynamics Newsletter, No. 65, Issue Editor: Y. Zhang, Editor in Chief: W. Chou, pp , (2014);

5. Large Particle Amplitude and the Cooling Range I
Sinusoidal behavior of kick implies nonlinearity and even the “wrong” sign for large particle amplitude. However a particle will undergo many cycles of betatron and synchrotron oscillations in the course of damping. To account this let:
Averaging over oscillations yields
For damping in both planes at all times we need ap , ax <μ01=2.405 corresponding to the first zero of J0. This yields the cooling boundaries:

6. Large Particle Amplitude and the Cooling Range II
The square cooling boundary ax,p <μ01 ≈ assures particles are always damping in both planes. It is possible for a particle to anti-damp momentarily in 1-plane before eventually being cooled in both.
Three fixed saddle points in ax,p phase space sketch out the cooling boundary. [0, μ11≈ 3.832] ,[μ01 , μ01 ], [μ11,0]. The actual contour is dependent on the ratio ofλx /λp.

7. Large Particle Amplitude and the Cooling Range III
Cooling ranges can be defined as the ratio of cooling boundaries to rms momentum spread and horizontal emittance:
In the case of equal damping between transverse and longitudinal planes both cooling ranges are linear in wavelength and inversely proportional to total delay of the chicane Δso.
In order to obtain sufficient cooling we choose Δso =2 mm and the base wavelength of 2.2 μm. This seriously constrains the design of an OA and any diagnostics utilizing undulator radiation.

8. Determining number of Undulator periods
Length of cooling insertion is constrained by available space in IOTA, wavelength is determined by the amplifier and cooling range requirements. This leaves the number of periods (or undulator parameter) as a free variable.
A trade off is made between increasing number of periods to increase kick amplitude
without also causing significant increase in the equilibrium emittance that results from the necessary dispersion in the kicker. Emittance also determined OSC experiment will be done at 100 MeV instead of IOTA’s nominal 150 MeV.
An additional consideration is the tune shift caused by turning undulators off and on. Being able to do so will be useful for transversely aligning pickup and kicker radiation.
A. Romanov
A. Romanov
Undulator length (m)
0.774
Number of Periods 7
Period length (cm)
11.05 K
1.03

9. Light Optics for passive OSC
V. Lebedev
Light Optics for passive OSC
Optic design accounts for depth of field. Transfer matrix from pickup to kicker center should be +/-1, the identity matrix.
The -1 case is a closer match to beam optics transfer and so was selected to prevent separation from radiation and particle in the kicker in the transverse plane. Lens focal lengths and positions are given by
Lenses are made from BaF2, chosen for its small chromaticity/GVD in OSC band. Pulse lengthening from GVD expected to reduce kick ~6%
Total telescope length (m)
3.3
F1 L1 (cm) 32
F2 (cm)
-46
Outer/Inner lens thickness (mm)
1.24, 0.76
γϴmax
0.8
GVD fs2/mm
-9.654
Error in lens focal length ~2% can be compensated by slight (~mm) adjustment in lens position. Alignment done on optic bench.
Long term Stability requirement of lenses determined by ρmax ≈ γλ=430 μm.
-Transverse <6 μm (displacement ~ ρmax /10)
-Longitudinally ~0.2 mm

10. Simulations with Synchrotron Radiation Workshop (SRW)
SRW (O.Chubar) can be used to calculate electric field from pickup, propagated through the telescope and computed along the entire length of kicker.
Presently working towards verifying semi-analytic treatment of OSC kick. Should be able to account effects (lens dispersion, transverse field dependence) not accounted in theory.

12. An amplifier for OSC: General remarks
Although passive OSC works for a proof-of-principle demonstration, any real application of the OSC will require an amplifier capable of delivering 20-30dB of gain in a single pass.
For sufficient cooling ranges the amplifier must operate in the mid-IR where, compared to visible wavelengths, lasing crystals are less capable.
For example no mid-IR crystals can match Ti:Sapphire in performance (gain, amplification bandwidth, thermal load handling)
Optical Parametric Amplification (OPA) would make a superb amplifier for OSC (high single pass gain, large bandwidth, virtually no thermal effects). However in an OPA gain only occurs when signal (pickup radiation) and pump (laser pulse) are overlapping. Which implies a pump pulse on the order of nanoseconds and pump energy of a few mJ. At MHz rep rate this brings the pump laser power to several kW.
One solution to this problem would be to recycle the pump pulse. This is possible since the signal (even at 30 dB gain) is a very small fraction (<0.1%) of pump energy.
Hypothetical Example!

13. An amplifier for OSC: In IOTA
Courtesy IPG photonics
An amplifier for OSC: In IOTA
Based on a highly doped Cr:ZnSe crystal pumped with a CW thulium laser with ~kHz frequency modulation.
Major limitation of the crystal is saturable absorption of the pump laser. Problem exacerbated by short crystal requirement.
Gain Medium
Cr:ZnSe
Max gain (dB) 7
Operating Band (μm)
Thickness (mm) 1
Required pump intensity (MW/cm2 )
0.1
Pump Power (W) 50
Optical Delay (mm)
1.45
Gain in power does not directly determine increase in the kick amplitude since pulse is distorted by i) GVD in the crystal ii) Finite amplifier bandwidth iii) nonlinear phase shift in the amplifier. By far GVD has the greatest effect (pulse lengthening).
Using SRW to compute the focused electric field entering the amplifier, these effects can be accounted analytically. The result is 7dB of gain increases kick amplitude by a factor of 2.

14. Closing remarks
-Main parameters of test have stabilized.
-Our group currently writing a Conceptual Design Report for OSC in IOTA.
-Construction of IOTA is well underway.