Analysis of images of a self-modulated proton bunch exiting a plasma in AWAKE∗
Paul Andreini†
University of California at Santa Cruz, Santa Cruz, CA, 95064, USA
Patric Muggli‡
(Dated: August 23, 2018)
We have analyzed data recorded by the streak cameras at the CERN AWAKE experiment on 10 September 2017. We have discriminated between streak plots of proton bunches propagating through plasma and those that propagate through neutral Rubidium vapor. We have also removed from consideration misleading data. We have aligned the 73ps streak plots to produce 375ps plots. We have divided these element-wise to learn how the charge distribution varies when protons propagate through plasma, versus a neutral gas.
I. INTRODUCTION
Accelerator and beam scientists continue to demand lepton accelerators capable of ever-higher energies. As the energy of an accelerator increases, so does the cost, since length of the accelerator must grow, in order to house more accelerating cavities and bending magnets. We require a new accelerator technology to lower the size and cost of future high-energy accelerators. Fortunately, the Plasma-based Accelerator (PBA), promises far larger accelerating gradients, on the order of GeV/m, as com- pared to the ubiquitous accelerating cavities.
PBAs ultimately transfer energy from a driver to a witness, namely leptons. Accelerator scientists use either an intense laser pulse or a relativistic particle bunch as a PBA driver. [6] The driver donates energy to wakefields, sustained by plasma electrons, which in turn energize the witness. Hence the driver energy places an upper bound on the energy transferable to the witness.
The Advanced Proton-driven Plasma Wakefield Ac- celeration Experiment (AWAKE) at CERN uses proton bunches from the Super Proton Synchrotron (SPS) as a driver. SPS proton bunches consist of ∼ 3 · 1011 protons, each with energy Ep+ = 400 GeV, thus each bunch car- ries kilojoules of energy. This energetic driver will allow AWAKE to accelerate leptons to very high energies with only one application of plasma-wakefield acceleration.
∗ This research experience was funded by the NSF’s International Research Experience for Students (IRES) program. Professor Philip Rubin (George Mason University) helped significantly to arrange this experience. † pandrein@ucsc.edu ‡ muggli@mpp.mpg.de
A. Plasma Wakefields
Consider a “cold” plasma at equilibrium – a gas of positive-ions and electrons at a sufficiently-low tempera- ture to ignore the thermal motion of the electrons. Dis- placing an electron (or group of electrons) slightly will in- duce oscillations at the plasma’s natural frequency1, ωpe, given by the following formula:
ωpe =
√ nee2
meε0 (1)
where ne is the number density of electrons in the plasma, e is the electron charge, me is the electron mass, and ε0
is the permittivity of free space. [5] To resonantly drive a wakefield to a large amplitude
requires a particle bunch to have both transverse and longitudinal dimensions on the order of the skin depth of the cold plasma, δpe: [3]
δpe = c
ωpe (2)
We often express the constraints on rms longitudinal and transverse bunch size in terms of the wavenumber,
kpe = ωpe
c (3)
kpeσz ≤ 1 (4)
kpeσr ≤ 1 (5)
where c is the speed of light in a vacuum, σz is the lon- gitudinal, and σr the transverse rms bunch size.
The longitudinal plasma wakefield amplitude can ap- proach the wave-breaking field [3],
EWB = mecωpe
e ≈ 1
2
FIG. 1. A schematic of the AWAKE experiment. [1]
where ne is the plasma electron number denisty, in inverse cubic centimeters (cm−3).
If we enforce the plasma density constraint that
kpeσz ≈ 1 (7)
(substitute Eq. 4 into Eq. 6), then we see that the mag- nitude of the wave-breaking field is inversely proportional to the bunch length:
EWB = mec
σz = 10− 12 cm.
Using σz = 12 cm, we find a wave-breaking field of
EWB ≈ 27 MV/m
with ne = 8 · 1010 cm−3. In this instance, the PBA would not significantly outperform accelerating cavities.
To attain EWB > 1 GV/m, we require that
ne > 1014 cm−3 ;
σz < 5 mm,
the latter of which proves experimentally difficult with a proton driver. We must also ensure that the beam remains focused. AWAKE can typically focus a beam to a transverse dimension of
σr = 2 · 10−4 m
Enforcing the wavenumber constraint in Eq. 5, we re- quire a plasma density
ne ≤ 7 · 1014 cm−3
These conditions create a bunch many factors of λpe in length2; such a long bunch cannot drive large amplitude wakefields effectively. [4] AWAKE requires shorter pro- ton bunches to effectively and resonantly excite wake- fields.
B. Seeded Self-Modulation
From linear plasma wakefield theory, even weak trans- verse wakefields, driven initially by the presence of the long proton bunch, can modulate (increase or decrease) the bunch density over the length scale of the bunch. Furthermore, the wakefield amplitude is directly propor- tional to the local bunch density. [4] The stronger wake- fields further compress the protons locally, thereby in- creasing the density. In other regions, the protons are de- focused. We call this feedback loop self-modulation (SM), since this favorable process will automatically modu- late the proton density, creating the short bunches that AWAKE requires.
Self-modulation results in a chain of micro-bunches of protons, each shorter than λpe. Further, this oc- curs periodically, with wavelength λpe. Since the micro- bunches automatically satsify our size constraints, these resonantly drive wakefields in plasma to very large am- plitudes. However, we would prefer to control this mod- ulation, rather than rely upon noise within the plasma electrons or the bunch protons to begin the modulation process. This way, we would retain the ability to in- ject the witness electrons in the accelerating and focusing phase of the wakefield. [4]
We control the modulation through a process called seeding. Seeding the self-modulation process involves ion- izing the plasma approximately halfway through the pro- ton bunch. Protons thus fall incident on the plasma sud- denly, and this provides an initial wakefield amplitude
2 λpe is the plasma wavelength; recall that λpe = 2π/kpe = 2πc/ωpe.
3
that exceeds the noise level. [2] Although seeding meth- ods can take many forms3, AWAKE chooses the rela- tivistic ionization front method to ensure that the seeded self-modulation (SSM) excites the initial wakefields.
Fig. 2 shows a computer simulation of the structure of a proton bunch undergoing SSM. This consists of proton micro-bunches, regions of the proton bunch focused by the wakefields, starting behind the ionizing laser, and regions of defocused protons. We note that Fig. 2 is an idealization; actual streak plots appear different.
FIG. 2. A computer-generated image displaying seeded self- modulation. The pink surroundings represent neutral gas; the red points and clumps represent protons; the green line represents an ionizing laser pulse, behind which the yellow stripe represents plasma. Image courtesy of Fabian Bastch.
II. STREAK CAMERA AND STREAK PLOTS
In order to analyze the SSM of proton bunches within AWAKE, we require a side-on image of the proton bunch traversing the plasma. AWAKE records such images us- ing a device known as a streak camera, which gives im- ages known as streak plots. In this section, we explain the streak camera, and we discuss key graphical features of the streak plots.
A. Streak Camera
AWAKE scientists collect raw data using a tool called a “streak camera,” designated “OTR” in Fig. 1. This ap- paratus consists of a 150µm-thick silicon screen, placed ∼ 3m downstream of the plasma exit, situated at a 45
angle with respect to the beam axis. The streak camera streaks in time the optical transition radiation (OTR)
3 Seeding methods include: a laser pulse preceding the drive bunch, a sharp increase in number density in the drive bunch, a bunch of oppositely-charged particles traveling within the drive bunch, a relativistic ionization front (created by a laser pulse within the drive bunch), pre-modulation of the drive bunch.
that each proton emits when entering the metallic screen. Time-resolved images of the OTR signal provide informa- tion on both the seeding of the self-modulation (nanosec- ond scale), as well as the period of the modulation (pi- cosecond scale). [1]
Longitudinal Window Size [ps] Temporal Resolution [ps] 73 ∼ 1 200 ∼ 3 500 ∼ 6
TABLE I. The resolution of a streak plot decreases as the length of the plot increases.
Much like a wide-angle lens with digital zoom, the streak camera’s resolution decreases as we increase the time window over which we record a streak plot. In Ta- ble I, we show the resolution of the streak camera for various plot lengths.
Since we require the optimal resolution for accurate analysis of SSM, we elected to take data with a 73 ps window. This gives streak plots with resolution ∼ 1 ps.
B. Streak Plots
We analyzed streak plots, recorded by the streak cam- era, which took the form of matrices with dimensions (512, 672). The longitudinal dimension (512 px) corre- sponds to exactly 73 ps, or 2.2 cm; the transverse mea- surement (672 px) thus equals 2.8 cm. The numerical value within a given matrix cell is proportional to the amount of light (charge) in that position in the plasma.
Along the vertical (longitudinal) axis, early time is low and late time is high; the protons propagate upwards in the streak plots.
The raw data streak plots presented in this paper fea- ture the event number and the ionizing laser energy meter in the title, as well as a colorbar on the right-hand side of the plot. These features prove crucial for analysis, which we explain in greater detail throughout Section III.
We show examples of streak plots (raw data) in Figs. 3, 4, 5, and 6. We explain the significance of each of these in Section III A.
III. COMPUTATIONAL ALGORITHM
Since SPS Proton bunches have rms length of order 200ps, we require several 73ps plots to image the entire bunch. Moreover, streak plots have a jitter on the order of 20ps, much greater than the modulation period of
T = λpe
c ≈ 8 ps.
This long period is obtained with a low plasma density (ne = 2 · 1014 cm−3), in order for the streak camera to resolve the image properly (see Table I). We use a
4
low-power replica of the ionizing laser pulse, called the fiducial laser pulse, that we delay in steps of 50ps, as a time fiducial on each image. To improve image quality, we acquire multiple images (10 with plasma) for each 50ps delay, and two images without plasma. We align these, and stitch together the delays to form two final images (see Figs. 7 and 8).
In this Section, we outline the computational algo- rithm by which we analyzed the streak plots, described above. First, we sorted the streak plots (visually) into three categories. After this, we added the streak plots to- gether to construct two longer4 plots of proton bunches in AWAKE: in one of which, the protons propagate through plasma; in the other of which, the protons travel through neutral Rb gas. Finally we divided the long plot by the number of images that contributed to each pixel.
A. Sorting Streak Plots
Our data set consisted of 89 streak plots that depicted different types of events. Our first task involved sorting these into three categories: “plasma-on,” “plasma-off,” and “rejects.”
We sorted plasma-on and plasma-off plots using the ionizing laser energy, recorded by an energy meter. Of the plasma-on plots, we sorted the good data from the bad data manually by looking carefully at every streak plot, and deciding which category best described the plot. Below, we outline the visual cues in the streak plots that led us to our categorization.
1. Plasma-on
“Plasma-on” plots depict a proton bunch propagating through Rb1+ plasma. The ionizing laser energy must significantly exceed 100mJ in order to ionize the Rubid- ium, thereby creating a plasma.5
Elaser on 100 mJ
Plasma-on plots must also include a visible fiducial laser, in the bottom left-hand corner of the plot. The time fiducial allowed us to align several streak plots, which we describe in Section III B.
Since these plots feature plasma, we expect to see the protons self-modulating to form micro-bunches in these plots. We also require that the proton beam in plasma- on plots appears very symmetric about the beam axis. For example, if a streak plot features protons moving off- axis asymmetrically, slanted micro-bunches, or otherwise unsightly behavior, we would not classify this plot as
4 Each individual streak plot is 73ps long; the two long streak plots are ∼ 400ps long.
5 Usually, the value of Elaser on varies between 150 - 200 mJ.
plasma-on, since this could potentially lead our analysis (charge comparison) and conclusions astray.
We show an example of a “plasma-on” streak plot in Fig. 3.
FIG. 3. A “plasma-on” streak plot. Notice the laser energy in the plot title, the presence of the fiducial laser in the bottom- left corner, and the self-modulated proton beam. The clumps of higher charge density represent the micro-bunches; the lines of lower charge density between successive micro-bunches rep- resent the vacuities, present in Fig. 2.
2. Plasma-off
“Plasma-off” streak plots depict a proton bunch propa- gating through Rb0 gas. There is no ionizing laser present in plasma-off experimental runs, and thus we expect the ionizing laser energy to be much lower than 100mJ.6
Elaser off 100 mJ
We show an example of a “plasma-off” streak plot in Fig. 4.
Since no plasma is present in these events, we do not expect the proton bunches in these plots to self-modulate. In fact, one can detect very little picosecond-scale struc- ture in plasma-off plots; these should appear like a high- density stripe of protons.
3. Rejects
We rejected some of the plasma-on plots as “bad data” for several reasons, which we outline here.
6 Although the ionizing laser plays no role in the plasma-off plots, the energy meter seldom registers “0 mJ,” due to the effects of noise.
5
FIG. 4. A “plasma-off” streak plot. Notice the laser energy in the plot title, the presence of the fiducial laser in the bottom- left corner, and the self-modulated proton beam. The clumps of higher charge density represent the micro-bunches; the lines of lower charge density between successive micro-bunches rep- resent the vacuities, present in Fig. 2.
If a proton bunch veered from the beam axis in any measure, then we classified the streak plot in question as a reject. These plots exhibit a well-documented ad- verse effect, known as hosing, which transpires due to a mismatch between the centroids of the proton bunch and the plasma wave. [7] This causes the unwanted off-axis protons in the streak plots.
In a select few streak plots, the longitudinal coordi- nate of the fiducial laser was ∼ 400px. Streak data was recorded so that the fiducial laser would be placed within the bottom 200 pixels of the streak plot. These plots were acquired when the fiducial laser pulse delay was be- ing changed, hence the unreliability of this data. The odd location of the fiducial laser would have confused the plot-adding algorithm that we describe in detail in Section III B.
In very few of the streak plots, the proton bunches were especially difficult to detect visually, by looking at the streak plots. Moreover, neighboring streak plots in the data set did not exhibit this strange dearth of protons. In these streak plots, the proton bunch was not extracted from the SPS. We thus excluded these plots from our analysis.
We show two examples of rejected streak plots in Figs. 5 and 6.
B. Adding Streak Plots
We would like to use several streak plots from our data set to construct a longer plot, that depicts an entire pro- ton bunch in the plasma chamber. We will show visually that such a streak image has length ` = 375ps, so we need to add several shorter (` = 73ps) plots end-to-end to create this longer plot.
FIG. 5. Example of a rejected streak plot. Notice, first, the protons that drift to the left-hand side of the plot, but do not drift symmetrically to the right. Furthermore, the micro- bunches in this plot are slanted with respect to the beam axis.
FIG. 6. Example of a rejected streak plot. Notice the fiducial laser around pixel z = 425. Any streak plot with a fiducial laser at z 6∈ {0, 200} will mislead our results.
It is worth noting here that the fiducial laser in the plasma-on plot serves another purpose than merely mark- ing which plots feature plasma: the fiducial laser flashes every t = 50 ps. This knowledge permits us to align plasma-on plots constructively, in the proper temporal sequence so that the micro-bunches in one plot overlap with the micro-bunches in a later plot.
Dissimilarly, a time fiducial is not important for constructing long plasma-off plots, since these feature little picosecond-scale structure.7
7 It is not possible to visually detect the Gaussian number density of the proton bunches by looking at a “plasma-off” streak plot.
6
1. Addition process: Plasma-on
When we add plasma-on plots, it is imperative that we pay close attention to the fiducial laser in each plot. We must ensure that the fiducial lasers align properly in plots with the same delay. For two successive delays, e.g. 0 ps and 50 ps, we must make certain that the successive laser pulses appear delayed by 50 ps, or 351 px.
The fiducial laser is found in columns
{40, 41, 42, ..., 170}
of the streak plot. The proton beam axis is situated in the middle of the streak plot, in column 336. In order to find the fiducial laser, we run a “center of mass” algorithm that finds the brightest row, considering only pixels in the columns where the fiducial laser shines.
We start by creating a “mass” vector of length 512, and we load the vector with the sum of the values in columns 40-170 for every row in the streak plot. Next, we take the dot product of the mass vector and the row number (position vector, np.arange(512)), and we divide by the sum of every element in the mass vector. As shown below, this gives the weighted average row, considering columns 40-170.
~m · ~x∑511 i=0mi
=
≡ x (9)
In Eq. 9, ~m is a 512-vector, containing the sums of the values in columns 40-170 in each row of one streak plot; ~x is a similar vector that labels each row; x is the average longitudinal position of all of the pixels in the relevant columns. We repeat this process for every streak plot in our plama-on dataset.
We know, from the dataset, which streak plots show the self-modulating proton bunch at which delay, i.e.
δt = {0 ps, 50 ps, 100 ps, ... , 300 ps}
For each of the delays above, we align each streak plot so that the fiducial lasers fall atop one another (for each 50 ps event), through the following method.
We first create a long streak plot of zeros, to be filled with plasma-on streak data, using np.zeros. Then we add one streak plot from the first delay, δt = 0 ps. After this, we add all of the streak plots with this “zeroth” de- lay, but we shift them longitudinally (using the weighted averages that we determined for each streak plot) so that the fiducial laser pulses all coincide.
Once we have finished adding the streak plots from the zeroth delay, we are ready to do the same for the remain- ing six delays. In our streak plots, 50 ps corresponds to 351 px, so we repeat the process above, but we introduce a longitudinal shift of 351 px to each plot. We present the relatively-simple delay function below:
d(n) = (351 px) · n (10)
However, this data would not accurately represent the spatial charge distribution, because the number of plots
that comprise a given row in the long streak plot is pro- portional to the number of events that contribute to each row. It is quite possible that row a of the long plot may contain twice as many streak inputs as row b, potentially leading us to the nave conclusion that row a contains twice as much charge as row b. In order to account for this, we must create a second “index” array (same size as the long streak plot) that tracks how many streak plots comprise each cell of the long streak array.
Each time we add a streak plot to the long streak ar- ray, we add a brick of ones (np.ones(512, 672)) to the index array. Once we have finished seeding both arrays, each cell of the index array will contain the number of streak plots that contributed to the same pixel in the long streak array. We set all zeros in the index array to 1, before dividing (np.divide) the long streak array by the index array element-wise. Through this addition and division process, we have essentially averaged the pixel value of many individual streak plots. This process greatly reduces the noise in the long plot. We include the resultant long streak plot in Fig. 7.
2. Addition process: Plasma-off
Due to the lack of picosecond-scale structure in plasma- off plots, creating a similar long streak plot for plasma-off data involves a less-complicated algorithm. We followed a similar procedure to the plasma-on plot, without align- ing the fiducial lasers, since plasma-off plots feature no fiducials.
First, we created an empty array, using np.zeros. We filled this with plasma-off streak data from each delay, using the longitudinal-shift function described in Eq. 10.
As in the plasma-on case, we created and seeded an index array, containing the number of streak plots that comprise each pixel of the long streak array; we set the zeros to 1, and we divide the long streak array element- wise by this index array. We include the resultant plot in Fig. 8.
C. Comparing Charge
Our goal throughout this endeavor is to compare the charge in the plasma-on plot to that in the plasma-off plot. We now have both long streak plots to compare, via element-wise division of both arrays, as we outline below.
Since we aim to divide plasma-on by plasma-off, there can exist no zeros in the plasma-off array. We fix this by setting all zeros in the plasma-off array to 1. We then use np.divide to divide the two arrays element-wise. We include the final result in Fig. 9.
7
FIG. 9. Final charge comparison plot (t = 375 ps). [1]
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IV. ANALYSIS
We discuss some key features of these long plots (see Figs. 7, 8, and 9), below.
We would like to comment on three important fea- tures of this plasma-on long streak plot. Firstly, the self-modulation is immediately apparent in this longer streak plot, signified by the bright vertical lines (proton micro-bunches) followed by dark vertical lines (vacuities between micro-bunches). Secondly, notice that the fidu- cial lasers are evenly spaced by 351px = 50ps. Finally, the proton density decreases over time. This is a mani- festation of the defocusing E acting on the proton micro- bunches.
Figure 7 shows that the protons self-modulate over more than 2500 pixels, corresponding to more than 356 ps. The stitching procedure produces a long bunch with the modulation of each plot adding to that of the others to form a regular pattern. This shows that the SM process is consistent and reproducible. Otherwise, the modulation pattern would wash out upon addition of the many images.
Similarly to the short (73ps) plasma-off streak plots, the long plasma-off plot reveals little picosecond-scale structure. We expect such behavior, since these protons do not interact with any plasma, and there is no credible reason for the protons to fly off-axis. Further, the proton bunch number density is Gaussian in time.
In order to compare the charge contained within the respective bunches, we divided element-wise the plasma- on plot by the plasma-off plot. This gave the final image (see Fig. 9). Since both the dividend- and the divisor- plots were 375 ps long, so was the final charge comparison plot. We discuss four key features of this final plot, below.
Firstly, note that the color bar scale ranges from [0, 2]. A yellow pixel in this plot signifies that this pixel in the plasma-on plot contains at least twice as much charge as the corresponding pixel in the plasma-off plot.
Secondly, the self-modulation is visible here, as well. Note the bright vertical lines (slightly higher than 1; slightly more charge in plasma-on than in plasma-off) fol- lowed by dark vertical lines (slightly lower than 1; slightly less charge in plasma-on than in plasma-off).
Additionally, the cells surrounding the beam axis (rows {0, 1, 2, ... , 250} and rows {422, 423, 424, ... , 672} in Fig. 9) get progressively brighter as time progresses. This evidences the defocusing effect, and reflects a similar trend in the plasma-on plot.
Finally, the random noise off-axis, especially earlier along the time scale reflects actual noise present in the plasma-on and -off plots. We expect this behavior; we do not find it worrisome.
V. CONCLUSION
We set out to analyze image data from AWAKE to compare the charge distribution between plasma-on and plasma-off events. We stitched together streak plots of proton bunches traversing the plasma (or gas) to generate long plots, which show the entire proton bunch, and we divided these element-wise to create a com- parison long streak plot. Through the long plasma-on streak plot, we confirmed the reproducibility of SM. The long plasma-off plot confirmed our assumption that, absent a plasma, the SPS proton bunch has very little picosecond-scale structure. Finally, both the long plasma-on plot and the comparison plot prove the viability of future electron acceleration using a proton-driven plasma wakefield accelerator.
ACKNOWLEDGEMENTS
Firstly, we would like to acknowledge the support of the NSF, who funded this research endeavor. Many thanks to Dr. Edda Gschwendtner and Dr. Patric Muggli, who served as mentors for this summer research project. Finally, thanks to Prof. Philip Rubin, who or- ganized the summer research experience from his office at George Mason University.
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