Progress towards testing non-linear integrable optics in a ... · 05/11/2019 Lucy Martin 1 Progress towards testing non-linear integrable optics in a Paul trap Lucy Martin 4th ICFA

05/11/2019 Lucy Martin 1

Progress towards testing non-linear integrable optics in a Paul trap

Lucy Martin

4th ICFA mini workshop on space charge – 05/11/19


Introduction

• What is a Paul trap?

• How can it be used to study accelerator physics?

• How does it work?

• What is Nonlinear Integrable Optics (NIO)

• Why do we want to test NIO on a Paul trap

• Designing the experiment on IBEX

• Can it be done?


Motion in a Paul trap

• Hamiltonian of a Paul trap :

where

• Hamiltonian of a conventional accelerator:

where


IBEX and SPOD: Linear Paul Traps

IBEX at the Rutherford Appleton Lab, UKS-POD at Hiroshima University, JapanPTSX at Princeton Plasma Physics lab, US

• What are the advantages of a LPT?

• What are the limitations?• No longitudinal effects = coasting beam


Past experiments on LPTs

• Coherent motion

• Excitation of dipole and quadrupole resonances

• Study of how resonances interact at half integer tunes

• Studying resonance crossing for novel FFA accelerators

• Beam Halo and beam size growth

• Cm factor studies8


Linear Paul Trap

4. Ions confined longitudinally with end caps

1. Argon gas introduced to vessel at ~10-7 mbar

2. Electron gun ionises Ar in trapping region

3. Ions confined transversely via 4 cylindrical rods


Experimental procedure

• A simple FODO cell is created using a sine wave• 1 MHz

• System scans across a large range of tunes

• Ions are created at the optimal tune

• Tune is then changed to the desired operating point by changing voltage

• At this tune ions are stored for a given time (eg. 100ms)

• Ions are then extracted onto an Micro Channel Plate (MCP) or a Faraday cup


Diagnostics and CalibrationMCP image

Faraday cup signal

Cell tune

17 mm

Tune scan takes ~ 2 hours

200 ms

[Taken on S-POD, Hiroshima]


Future accelerators (Nonlinear Integrable Optics)

• In linear accelerators the motion is integrable – it is known to be bounded

• This is exactly what we want, the beam won’t be lost!

• Susceptible to resonances

• Realistically an accelerator can never be totally linear (errors + space charge)

• Nonlinearities -> no longer integrable due to coupling between x and y

• Require integrable system where small perturbations are allowed

• At Fermilab they found such a system2

• Unfortunately it requires a complicated potential


Non-linear integrable optics

• Requires round beams and np phase advance• This is called a “T-insert”

• Quasi-integrable version involves only octupoles

• Octupole must vary in strength proportional to 1/b3

Image from [3]

To test in a Paul trap we require 2 things:1. To be able to create a good enough T-insert with the Paul trap2. Create correct octupole potential independent of linear focusing


Why bother testing NIO in a Paul trap?• No dispersion

• No chromatic effects

• Can create a continuous octupole, or simulate a stepping field

• Octupole strength is easily variable

• T-insert parameters are easily variable

• Space charge effects easily included

• Can sit on resonance to study stability and excite resonances at arbitrary frequencies

• Already testing at IOTA and UMER – each facility has different advantages


1. T-insert

Constraints:• np phase advance in x and y• bx = by in centre of drift• ax = ay = 0 in centre of drift

Solution (first attempt): • Simplify situation by only varying voltage pulse length, not amplitude• 6 constraints -> 6 voltage pulses• Basin hopping algorithm with low amplitude, 25V• At this point did not have a clear idea of the maximum allowable beta function

• Max bx and by• Phase advance close to 0.3 over drift• Bandwidth of amplifiers – pulses can’t be too short

or too large• Try to avoid too much asymmetry – more

susceptible to resonance at high intensity


2. Quadrupole + Octupole trap• Building heavily on work from Hiroshima4

Vo

ltag

e (V

)


Quality of octupole

Blue = perfect rod and plate placementRed = example with 20mm error on rod and

plate locationRmeas = 3mm (from simulation)

• Plot shows sum of the harmonic coefficients of higher order fields (from simulation) in an attempt to compare to IOTA and UMER magnets.

• For octupole only desired phase advance tolerance in IOTA7 is 2p * 10-2.

• Magnet tolerance is stated in [8] to be an integrated strength of harmonics relative to octupole field is less than 1% (100 units).

UMER: From [6]IOTA: From [8]


Dynamic Ap

= 2.6 mm

Dynamic Ap

= 2.1 mm

• Dynamic aperture and

tune spread over 1000

periods.

• The T-insert is applied

using a single matrix

transformation.

• The colour bar shows

the log of the change in

tune over the simulation

for particles within the

dynamic aperture.

• Grey regions are

particles that survived

but are outside of the

dynamic aperture10.

Perfect

octupole

Realistic

Octupole

Log(Du

)

Log(Du

)

Log(Du

)

Log(Du

)


Tested T-insert on IBEX in Jan 2019…

No IONs

• Trying to run before we can walk, this was before the installation of the MCP• Concluded that we probably were trapping ions but not enough to see• Received custom built amplifiers from Oxford electronics

• Improved band width – however only on central rods.• Installed the MCP, giving us:

• Improved calibration of trap • Ability to see down to 103 ions


Simple simulationImages from MCP

(rotated through 45o as trap is 45 degrees from accelerator axis)

Cell tune = 0.1 Cell tune = 0.2




Varied the emittance in simulation, fitting to the rms beam size and matching the result to the images on the MCP


1. T-insert, again• This time we varied pulse strength and length• Created a shorter T-insert that was easier to correct • Looked at 3 different T-inserts scaling voltage and length• To minimise beta function operated at lower tune (0.5 + 0.135)

• UMER also looking to operate at this phase advance6

• For lattice with larger beta function still saw no trapping• Decided to match into the T-insert region (automated)• Read waveforms from scope -> apply calibration factor and calculate optics -> call MAD-

X to calculate matching -> create waveforms to be applied -> send waveforms to AWGs

Length2 = Length1 * 1

𝑘

Voltage2 = Voltage1 * k


• For the waveforms with larger voltages ran into problems with the amplifiers, especially those on the endcaps, with the lower band width:

• Decided to not apply the alternating voltage to the end caps and to match back to original waveform before extracting:

Central rods

End caps

05/11/2019 Lucy Martin 21B

eta

funct

ion (

m)

0 3.02.01.0

Time (ms)

Designed T-insert, MAD-X:

Best beta function reconstruction:

For the shortest (highest voltage/ lowest beta function) lattice studied the optics are reconstructed by taking each voltage point from an oscilloscope trace and

transforming it into a transfer matrix:


• For lower beta functions more ions were stored• Matching reduced ion loss


Testing the T-insert• How can we prove that we have the correct phase advance?• How can we prove that we have the correct tune?

Cartoon showing the perturbation applied to the rods (perturbation voltage enlarged)

• Method using a dipole kick5

• Potential to use high speed camera for time resolved imaging

Quadrupole perturbation: See tune

control of ~10-2


From [8]

• The upper estimate that we have on emittance is 2.16*10-9.

• The largest beta function in the region where the octupole

will be on is 229.89.

• Using the emittance estimate this gives 0. 70467mm as an

rms beam size, v. small due to small beta function.

• Can then use 2 * this as reference radius = 1.4093 mm

Octupole quality


Simulation of new setup

• Want to vary octupole strength in simulation - can also do this within reasonable bounds experimentally.

• Further simulations of the new T-insert required• Full lattice• Space charge


Conclusions and further work• IBEX is a useful tool for accelerator physics studies

• Commission of IBEX has ended and we’re now capable of a range of interesting physics

• To test quasi-NIO a number of often competing constraints should be met

• However, it should be possible.

• Further simulations of the new T-insert required• Full lattice• Space charge

• Octupole upgrade to the trap needed – must be well designed

• Experimental testing of quasi integrable NIO

• Potential to vary a wide range of experimental parameters.


Thank you!


References1. Martin, L. K. et al., “Can a Paul Ion Trap Be Used to Investigate Nonlinear asi-Integrable Optics?”, 2019, J. Phys.: Conf. Ser. (In Press)

2. Danilov V and Nagaitsev S, “Nonlinear accelerator lattices with one and two analytic invariants”, PRAB, 2010

3. Antipov S et al, JINST 12 P04008, 2017

4. FUKUSHIMA K and OKAMOTO H, “Design Study of a Multipole Ion Trap for Beam Physics Applications”, Volume 10, 1401081, 2015

5. Martin, L. K. et al., “A new method to measure the beta function in a Paul trap”, 2018, J. Phys.: Conf. Ser. 1067 062016

6. Ruisard, K. et al, “Single-invariant nonlinear optics for a small electron recirculator”, 2019, Phys. Rev. Accel. Beams, 10.1103/PhysRevAccelBeams.22.041601

7. S. Antipov, S. Nagaitsev, and A. Valishev, “Single-particle dynamics in a nonlinear accelerator lattice: attaining a large tune spread with octupoles in IOTA”, 2017, J. Instrum. 12, P04008

8. Martin, L. K. et al., “A study of coherent and incoherent resonances in high intensity beams using a linear Paul trap” New J. Phys. 21 053023


Matching code:

First estimate of matching waveforms applied to the rods

Waveforms from oscilloscopes saved

Code analyses save waveforms calculating the

optics from each

Twiss parameters at the end of the square wave

section and start of the T-insert are found

Start and end parameters for the match are written

into a MADX file

MADX run from python code to calculate correct

matching

Voltage pulse heights extracted from MADX

code

New waveforms to be applied to the rods are created with the new

matching section

New waveforms are applied to the rods via the Labview (requires human

input)

Waveforms from oscilloscope are saved

Progress towards testing non-linear integrable optics in a ... · 05/11/2019 Lucy Martin 1 Progress towards testing non-linear integrable optics in a Paul trap Lucy Martin 4th ICFA

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