1. AN UNDULATOR BASED ON BEAM DRIVEN
WAKEFIELDS IN A DIELECTRIC TUBE
Josh Cutler
UCLA Department of Physics & Astronomy, Los Angeles, CA 90095
March 2012
1 INTRODUCTION
Light sources are devices which produce brilliant X-rays used in researching topics from
medicine to geology. Free electron lasers (FELs) are now providing the latest generation of light
sources. The free electron laser is composed of an electron source, an accelerator, and an
undulator. Our focus is on the undulator. The oscillation of an electron bunch through the
undulator generates photons. The wavelength of released photons can be decreased with a
shortened undulator period. The magnetic undulators, which are the most common, are difficult
to produce with periods shorter than the order of cm.
We are focusing on a new approach in producing the undulator field. Instead of using the typical
magnetic undulator, we are using a dielectric tube to simulate the same wakefield pattern seen in
traveling wave undulators. With the proper tube structure, a single mode can be generated from
an electron bunch and create wakefields in the THz range (infrared) using coherent Cherenkov
radiation as shown below in Figure 1. The lowest transverse electric mode is preferred because
the electron beam would follow a helical path due to the wakefield. Such a beam-driven
structure could be used to produce an undulator-like field with periods less than 1 mm and
frequencies on the order of terahertz. [1]

2. Figure 1: The nature of coherence. We want the wakefields left by the first electron bunch to
make the next bunch coherent.
2 THEORY
2.1 FEL AND UNDULATOR DESIGN
In order to generate infrared wakefield radiation in the FEL, we needed to alter the undulator
period and electron energies for the following formula to work:
𝜆 𝑟 =
𝜆 𝑢
2𝛾2 (1 +
𝑎 𝑢
2
2
) (1)
where 𝜆 𝑢 is the undulator period, 𝛾 is the Lorentz factor, and 𝑎 𝑢 is the dimensionless normalized
field strength. To find these variables, we started with previously determined values that were
useful to us.
We chose 𝜆 𝑟 to be 10-10 m because of its previous FEL use in the LCLS x-rays. This wavelength
of x-rays is also the approximate radius of an atom. [2]
Because of its success in previous dielectric wakefield accelerator experiments on the order of 1
THz, we chose 𝜆 𝑢 =300 μm.
We chose 𝑎 𝑢 based on:
𝑎 𝑢 =
𝑒𝜆 𝑢 𝐵 𝑢
2𝜋𝑚 𝑒 𝑐
≈ 0.94𝜆 𝑢 𝐵𝑢 (2)
Since we are only approximating on orders at this stage and would look at ranges later, we put au
on the order of 1. When we substitute these parameters back into equation (1), γ is
approximately 1500, which signifies that the beam has energy of 0.75 GeV.

3. Table 1: FEL parameters
Radiation wavelength, Å 1.0
Undulator period, μm 300
Undulator parameter 1
Beam energy, GeV 0.75
To find the peak electric field at the inner surface of the dielectric, we can substitute Bu=
𝐸 𝑟,𝑠𝑢𝑟𝑓
𝑐
,
and we will get 𝐸𝑟,𝑠𝑢𝑟𝑓 ≈ 1010
V/m = 10 GV/m. The wakefield driven dielectric breakdown is
about13.8 GV/m, so this result is within breakdown level. [3]
Using the above electric field, we can find approximations for inner radius a, bunch length 𝜎𝑧,
and electron amount 𝑁 𝑏 as shown below:
𝐸𝑟,𝑠𝑢𝑟𝑓 =
4𝑁 𝑏 𝑟 𝑒 𝑚 𝑒 𝑐2
𝑎𝑒(√
8𝜋
𝜀 𝑟−1
𝜀 𝑟 𝜎𝑧+𝑎)
(3)
where re is the classical electron radius, me is the mass of the electron, e is the charge of the
electron, and εr is the dielectric constant. The dielectric is made of fused silica due to its
availability, so εr=3.8. [4]
The data combination of a=200 μm, 𝜎𝑧=100 μm, and 𝑁 𝑏=109 electrons worked best for a radial
electric field on the order of 107 V/m on the inner radius, which is far below 𝐸𝑟,𝑠𝑢𝑟𝑓 above. Our
next goal was to modify these values to get a TE01 mode frequency of 1 THz in a tube
waveguide.
2.2 RESONANT FREQUENCIES OF DIELECTRIC TUBE
The TE01 frequencies are given by:
𝑓01 =
1
2𝜋
𝛽𝑐𝜅01
√𝜀 𝑟 𝛽2−1
(4)
where κ is the wavenumber determined by the energy of the electron and 𝛾 =
1
√1−𝛽2 . The tube
radii are given by inner radius a and outer radius b. [5]
We want to generate TE01 frequencies close to 1 THz with an appropriate tube waveguide based
on the energy of the electron bunch and radii of the tube as shown above in Equation (4). This
would generate wakefields that can accelerate the next electron bunch if they are coherent. We
desire a transverse electric mode from this bunch to amplify the photon emissions through self-
amplified stimulated emission. Because of this process, SASE could negate the need for an
external power source in an undulator.

4. The parameters and process leading to Equation (4) were evaluated on a Mathematica file based
in Alan Cook’s longitudinal wakefield analysis.
2.3 WAKEFIELD EXCITATION OF TUBE (OOPIC)
An OOPIC simulation was used to display the wakefield excitation of the tube. The OOPIC
simulations need to be consistent with the generated electric field in Mathematica to prove that
these parameters will generate one transverse mode in a simulation. The required frequency of 1
THz must also be generated to verify that the intended design could work with an actual
dielectric tube modeled after the undulator.
3 NUMERICAL RESULTS
In Mathematica and OOPIC, we tested different undulator radii and bunch lengths to see if we
can yield a wakefield with a 1 THz frequency.
When we tested each value of inner radius a for certain ratios of b:a (εr=3.8, γ=1500), we got
these THz values for f01:
Table 2: Radii ratios and wakefield frequencies
a b:a=1.5:1 b:a=1.75:1 b:a=2:1 b:a=2.25:1 b:a=2.5:1
50 2.9 2.07 1.61 1.33 1.13
75 1.93 1.38 1.08 0.89 0.76
100 1.45 1.03 0.81 0.67 0.57
125 1.16 0.83 0.65 0.53 0.45
150 0.97 0.69 0.54 0.44 0.38

5. Figure 2: Radii Ratio Chart
Next, we tested the values of σz and Nb for the radii ratios that had frequencies within 0.1 THz of
1 THz to see which ones yielded a sinusoidal wakefield with a sufficiently strong amplitude
(Nb=109):
Table 3: Mathematica results
a (μm) b (μm) σz (μm) THz Frequency E amplitude (V/m)
150 225 75 0.97 1.22x107
100 175 70 1.03 1x107
75 150 60 1.08 8.71x106
0
0.5
1
1.5
2
2.5
3
3.5
1.5 1.7 1.9 2.1 2.3 2.5
Wakefieldfrequency(THz)
Outer:Inner radius ratio
Radii ratios
50 micron
radius
75 micron
radius
100 micron
radius
125 micron
radius
150 micron
radius

6. Figure 3: Mathematica Longitudinal Wakefield for a=150 μm, b=225 μm
Figure 4: Mathematica Longitudinal Wakefield for a=100 μm, b=175 μm
Figure 5: Mathematica Longitudinal Wakefield for a=75 μm, b=150 μm
0.0002 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010
1 107
5 106
0
5 106
1 107
s m
EzVm
Longitudinal Wakefield Ez
0.0002 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010
1 107
5 106
0
5 106
1 107
s m
EzVm
Longitudinal Wakefield Ez
0.0002 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010
5 106
0
5 106
s m
EzVm
Longitudinal Wakefield Ez

7. All of these wakefields had a magnitude near 107 V/m: the desired straw-man wakefield
amplitude.
In OOPIC, we tested the parameters that worked to see which longitudinal wakefield graph
would resemble the Mathematica plot and also give a 1 THz frequency. An axial field was
calculated at r=0 and the radial field was calculated at r=a. Here are the results:
Table 4: OOPIC Simulation results
a (μm) b (μm) σz (μm) Beam radius
(μm)
THz
Frequency
Longitudinal
E amplitude
(V/m)
Radial E
amplitude
(V/m)
150 225 75 91 0.53 7.13x107
4.02x107
100 175 70 91 0.59 1.08x108
6.66 x107
75 150 60 61 0.57 1.8x108
8.48 x107
Figure 6: OOPIC Longitudinal Wakefield for a=150 μm, b=225 μm
Figure 7: OOPIC Longitudinal Wakefield for a=100 μm, b=175 μm
0.000 0.002 0.004 0.006 0.008
6 107
4 107
2 107
0
2 107
4 107
6 107
z m
EzVm
Longitudinal Wakefield Ez
0.000 0.001 0.002 0.003 0.004 0.005 0.006
1 108
5 107
0
5 107
1 108
z m
EzVm
Longitudinal Wakefield Ez

8. Figure 8: OOPIC Longitudinal Wakefield for a=75 μm, b=150 μm
Figure 9: OOPIC Radial Wakefield for a=150 μm, b=225 μm
Figure 10: OOPIC Radial Wakefield for a=100 μm, b=175 μm
0.000 0.001 0.002 0.003 0.004 0.005
1.5 108
1.0 108
5.0 107
0
5.0 107
1.0 108
1.5 108
z m
EzVm
Longitudinal Wakefield Ez
0.000 0.002 0.004 0.006 0.008
8 107
6 107
4 107
2 107
0
2 107
4 107
z m
ErVm
Radial Wakefield Er
0.000 0.001 0.002 0.003 0.004 0.005 0.006
1.5 108
1.0 108
5.0 107
0
5.0 107
z m
ErVm
Radial Wakefield Er

9. Figure 11: OOPIC Radial Wakefield for a=75 μm, b=150 μm
4 DISCUSSION
As we can see from Figures 6-11, the problem with the OOPIC simulations is that we have had
trouble creating a sinusoidal wakefield due to the interference of other modes. This could be
why the wakefield measurements are so erratic and the frequency only came within half of its
expected value.
When the straw-man design was tested on OOPIC, the resonant frequency did not come close to
1 THz. However, these figures were approximations to be modified for the simulations above.
This tube is possible to engineer if we can build one on the order of hundreds of μm.
The best measurements of frequency and an almost sinusoidal wakefield come from inner radius
100 μm, outer radius 175 μm, and bunch length 70 μm. However, simulation results indicate that
the tube structure is still flawed, perhaps due to the beam radius or number of electrons.
5 CONCLUSIONS
After reanalyzing Cook’s Mathematica file and his dissertation, we believe the reason that the
frequencies haven’t come close to 1 THz is because we were generating a TM mode, and not a
TE mode. The field at the boundary of the dielectric resembles the TM mode, despite changing
the TE01 frequency in Mathematica.
In order to achieve a wakefield frequency of 1 THz, we plan to generate a dipole mode like
HEM11 or TE11.
0.000 0.001 0.002 0.003 0.004 0.005
1.5 108
1.0 108
5.0 107
0
5.0 107
z m
ErVm
Radial Wakefield Er

10. 1 C. Pellegrini:“X-BandMicrowave UndulatorsforShortWavelength Free-ElectronLasers”
2 LCLS Glossary:http://www-ssrl.slac.stanford.edu/lcls/glossary.html
3 M.C. Thompson et al.,Phys.Rev.Letter: “BreakdownLimitsonGigavolt-per-meterElectron-beam-
drivenwakefieldinDielectricStructures”
4 A.M. Cook et al.: “Beam-drivendielectricwakefieldacceleratingstructure asaTHz radiationsource”
5 A.M. Cook:“Generationof Narrow-BandTerahertzCoherentCherenkovRadiationinaDielectric
WakefieldStructure”