Towards a compact 0.1-10 MeV broadband betatron photon source

Towards a compact 0.1-10 MeV broadband betatron photon source C.E. Clayton*a, S. F. Martinsd, J. L. Martinsd, D. K. Johnsona, S. Wanga, K. A. Marsha, P. Mugglib,

M. J. Hoganb, D. Walzb, R. A. Fonsecad, E. Ozb, C. D. Barnesb, C. L. O’Connellb, I. Blumenfeldb, N. Kirbyb, R. Ischebeckb, C. Huanga, M. Zhoua, W. Lua, S. Dengb, T. Katsouleasb, W. B. Moria, R. H.

Siemannb, L. O. Silvad, C. Joshia aUniversity of California, Los Angeles, California 90095, USA

bUniversity of Southern California, Los Angeles, California 90089, USA cStanford Linear Accelerator Center, Stanford, California 94309, USA

dInstituto Superior Téchico, Lisbon, Portugal

ABSTRACT

When a highly relativistic electron is injected off-axis into an ion channel, the restoring force of the radial field of the ions will cause the electron to accelerate towards the axis, overshoot, and begin to undergo oscillations about the ion-column axis at a characteristic frequency; the betatron frequency. This so-called betatron motion will cause the electron to radiate hard x-rays in the forward direction. In two recent experiments at the Stanford Linear Accelerator Center (SLAC), betatron x-rays in the 1-20kV range and in the 1-50MV range were produced with an electron beam with an energy of 28.5 GeV for ion densities of about 1 x 1014 cm-3 and 1 x 1017cm-3, respectively. To make such an x-ray source more compact, the 3km long SLAC linac would be replaced by a source of electrons from a Laser Wakefield accelerator (LWFA). To increase the efficiency of converting laser into photons at high photon energies, we propose adding a second stage where the LWFA electrons radiate via a second ion channel, independent of the accelerating process. This two stage concept allows one to control the critical frequency of the emitted radiation as well as the efficiency of the process.

Keywords: relativistic electron, betatron oscillations, betatron x-ray photons, laser wakefield accelerator, plasma wakefield accelerator

1. INTRODUCTION Broadband, incoherent photon sources at high photon energies have several potential uses. For example, photons in the 100 kV range would be useful for spectroscopy in the very high-density plasmas relevant to laser fusion or high energy-density plasma physics. Photons in the 1 MV range could be useful in nuclear physics while photons in the 10 MV range could be used to induce fission at the so-called giant resonance. X-rays or gamma rays from betatron oscillations in plasmas can cover this range. For example, when a highly relativistic electron (Lorentz factor γ >> 1) is injected off-axis (initial radius r0 > 0) into an ion channel, the restoring force of the radial field of the ions will cause the electron to accelerate towards the axis, overshoot, and begin to undergo oscillations about the ion-column axis at a characteristic frequency; the betatron frequency. This so-called betatron motion will cause the electron to radiate hard x-rays in the forward direction. In two recent experiments at the Stanford Linear Accelerator Center (SLAC), betatron x-rays in the 1-20kV range and in the 1-50MV range were produced with an electron beam with γ = 56000 for ion densities ni0 of about 1 x 1014 cm-3 and 1 x 1017cm-3, respectively. These experiments were in connection to a electron beam driven plasma accelerator; i.e., the Plasma Wakefield accelerator (PWFA) concept. To make such an x-ray photon source more compact, the 3km long SLAC linac would be replaced by a source of electrons from a Laser Wakefield accelerator (LWFA); that is, where the longitudinal fields of an electron plasma wave driven by an intense laser become a miniature linac. With today’s high-intensity lasers, the laser-driven plasma wave can be very nonlinear. In this case, the ion channel occurs naturally since the charge separation producing the plasma wave occurs with the plasma electrons blown out radially by the nonlinear force of the laser pulse. This is the so-called bubble or blowout regime and the electron acceleration takes place in an ion channel. The characteristic frequency of the resulting photon spectrum scales as γ2 x r0

*[email protected]; phone +01 310 206-9139; fax +01 310 206-8220.

Invited Paper

Harnessing Relativistic Plasma Waves as Novel Radiation Sources from Terahertz to X-Rays and Beyond, edited by Dino A. Jaroszynski, Antoine Rousse, Proc. of SPIE Vol. 7359, 735902 · © 2009 SPIE

CCC code: 0277-786X/09/$18 · doi: 10.1117/12.820782

Proc. of SPIE Vol. 7359 735902-1

Orbit of an individual electron

e Orbit of Centroid

e

C) Envelope oscillations

Orbit of head of bunch

() - Tail sloshing

rbit of tail of bunch

x ni0, but in a LWFA that is one dephasing-length long, the high-γ contribution to the spectrum occurs only at the end of the acceleration process. To increase the efficiency of converting laser into photons at high photon energies, we propose adding a second stage where the LWFA electrons radiate via a second ion channel, independent of the accelerating process. This two-stage concept allows one to control the cut-off frequency of the emitted radiation as well as the efficiency of the process. The challenges are to get a high charge of accelerated electrons at γ well over 2000. The ongoing development of the simulation tools at UCLA, QuickPIC and OSIRIS, will be used to help plan a roadmap to the desired photon source.

In Sec. 2 we introduce some concepts of betatron motion and its radiation. Section 3 summarizes some of the observations from the experiments at SLAC. In Sec. 4, the two-stage concept is tested via three dimensional (3D) particle-in-cell (PIC) computer simulations and we conclude in Sec. 5.

2. BETATRON CONCEPTS An exact definition of betatron motion may depend on the accelerating system for which it is observed. Betatron motion in cyclotrons, storage rings, linacs, and plasma accelerators may have different implications. Here, we will simply refer to this motion as that where the orbits (or trajectories) of the charged particles oscillate about an ideal central orbit. For example, in the (now) 30 GeV linac at SLAC, beam position monitors typically show the centroid of the electron bunch slowly oscillating down the machine. Here, the focusing system is a lattice of quadrapole lenses (thin lenses). Corrector dipole magnets are used to correct this wayward orbit since, without correction, the beam can come close to irises in the accelerating structure and pick up a transverse tail or tilt that increases the emittance of the beam.

In a single stage of a plasma accelerator, the focusing force of the ions is continuous (a thick lens). Again, the centroid of the bunch can oscillate but now the transverse size (or envelope) σr of the bunch will oscillate even if the centroid is not. This is illustrated in Figs. 1(a) and (b). Electrons at, say, the top and bottom of the bunch will oscillate about the central orbit as illustrated by the sine-like curves in Fig. 1(a) which illustrates a little more than one half of a betatron wavelength λβ. Since the transverse spacing of these two orbits is proportional to σr, contours of constant beam density would appear as shown in Fig. 1(b). Note that a full oscillation period in the envelope size as given by σr corresponds to half of a single-electron betatron period. Thus betatron motion leads to “envelope oscillations”.

If the electron bunch is injected into the ion channel with a tilt, as illustrated in Fig. 1(c), both characteristic periods are apparent: the tail oscillates as if it were a single electron while the envelope varies as in Fig. 1(b). In the case where the electron bunch itself is driving the wake, the axis of the ion channel is defined by the head of the bunch and travels in a straight line while the tail oscillates about this axis. This we call “tail sloshing”.

Fig. 1. Betatron motion of electrons in a ion channel (or thick lens). (a) The orbits of two electrons. (b) The envelope of a

beam containing the two electrons in (a). (c) Envelope oscillations of a tilted beam. In addition to the variation in the transverse size of the beam, the tail sloshes up and down at half the period of the envelope period.

Let γz0 be the Lorentz factor associated with the longitudinal velocity of the electrons. For K << γ and γ >> 1, γz0 is approximately γ. The motion of an electron in the ion column is given approximately byi

d2rdt 2 + ωβ

2 r = 0 (1)


which has the solution

r(t) =K

γ z0kβ

cos(ωβ t) (2)

where kβ = ωb /c , ωb = ω p 2γ is the betatron frequency in terms of the electron-plasma frequency ωp associated with nio, r0 is the initial radius of the electrons, and the wiggler strength parameter K = γ z0kβ r0 ≈ γkβ r0 is introduced. Note that here r and r0 are actually a measure in Cartesian coordinates since they can take negative values in Eq. 2. However, σr is calculated from the envelope equation

′ ′ σ r z,ne( )+ kβ

2 −εN

2

γ 2σ r2 z,ne( )

⎡

⎣ ⎢

⎤

⎦ ⎥ σ r z,ne( )= 0

(3)

and is positive definite. Here εN is the normalized emittance of the electron beam. This accounts for difference in periodicity of the envelope oscillations (period = π/kβ) and the single-electron or tail sloshing (period = λβ = 2π/kβ). Note also that the term in the brackets of Eq. 3 can be made zero in which case the electron beam is “matched” to the plasma; the restoring force of the ions is balanced by the “emittance force” or the natural tendency of the beam to diverge. However, for all this simple analysis so far, we have neglected any change in the energy of the beam.

There are several things to observe at this point. First, the electrons are executing curved trajectories and are highly relativistic and will therefore radiate in an instantaneous angle of 1/γ. As in an free electron laser, γz0 < γ due to the finite radial excursion of the electrons and thus the curved trajectories will not be purely harmonic. For K >> 1, the on-axis harmonics ωn occur at intervals of the harmonic number n, ωn = n4γ z0ωβ K 2 and gives a critical harmonic number

nc = 3K 3 /8 >>1, corresponding to a critical frequency ωc of (dropping the distinction between γz0 and γ)

ωc =3ωβ

2γ 3

2cr0

(4)

for which half the total radiated energy lies above this frequency. In practice, each electron has a different r0, γ, and injection time while any detector has a finite angular acceptance. Thus all the harmonic structure is washed out and the radiation will appear to be continuous in frequency having, not surprisingly, the characteristic shape of synchrotron radiation from electrons executing circular motion, as will be discussed further in Sec. 3. The spectrum will be less hard off axis. With large K, the characteristic radiation angle is now K/γ.

From the Larmor formula, the radiated power Pe(t) is

Pe =2c 2

3m2c 3 γ 2 dr p dt

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2

− m2c 2 dγdt

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2⎡

⎣ ⎢

⎤

⎦ ⎥ . (5)

For our assumptions on K and γ, the second term in Eq. 5 is negligible and dr p dt ≈ −γωβ

2 r so that

Pe ≈23

e2cγ 4kβ2 r2 cos2 ωβ t( ) (6)

which peaks twice per each cycle of r(t), as expected. Averaging over a betatron period gives <Pe> and converting average power per oscillation to an average rate of energy loss Wloss/dz from the electrons into radiations; i.e., Wloss/dz = <Pe>/c, we get

dWloss

dz=

13

remc 2γ 2kβ2K 2 (7)

where the classical electron radius re = e2/mc2. For efficient conversion of the electron betatron motion into photons, we want the dz∫ of the right-hand side of Eq. 7 to be as high as possible. As for the spectrum of the betatron radiation, Eq.


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4 says that each burst of on-axis light will be characterized by the quantities nio, r0, and γ at that instant. This is summarized in Fig. 2. Plotted is hνc vs. γ for a pure ion channel for various values of ambient electron density ne (ne = ni0). Clearly, obtaining spectra in the 0.1–10 MeV range is likely in the near term as the electron energies from LWFA go beyond 1 GeV.

Fig.2. From Eq. 4: ωc (expressed in MeV) vs. γ (expressed in GeV) for various plasma densities ranging from 1 x 1017 cm-3

(bottom curve) up to 64 x 1017 cm-3 (top curve) in steps of a factor of 21/2. The initial offset r0 is taken as 10 μm.

3. REVIEW OF SLAC RESULTS 3.1 Physical picture and experimental setup

A cartoon of the betatron x-ray source investigated in two experiments at SLAC is shown in Fig 3. The space charge of the incoming electron bunch blows out the plasma electrons which return to the axis roughly one plasma wavelength behind the head of the bunch. For this near-light-speed plasma wave, the wavenumber is approximately kp = ωp/c. (Note that this cartoon of a PWFA looks very much like a LWFA.) The plasma density was adjusted such that bi-Gaussian electron bunch with bunch length σz (and initial spot size σr0) occupied the full longitudinal extent of the resulting ion channel. If the bunch was too short, Eq. 7 says that Wloss would be small since all the electrons would be in the decelerating portion of the wake and γ would drop too fast with propagation direction z.

Fig.3. Cartoon of an electron-beam driven wakefield (PWFA). Gaussian curve is longitudinal bunch current; two circles are

electrons orbiting ion-column axis; arrows represent betatron x-ray emission. The longitudinal field changes sign slightly behind the peak of the bunch current.

A schematic of the experimental setup is shown in Fig. 4. Two regimes were studied. In the first, low-density regime, the bunch duration τb = σz/c was about 700 fsec, the density ne,low was about 1–2 x 1014 cm-3, and the length of the plasma Lp was 1.5 m and was produce by single-photon ionization of the lithium vapor by an eximer laser (not shown). In the second, high-density regime, τb = 60–120 fsec, ne,high = 1–3 x 1017 cm-3, and Lp = 8–30 cm. Here the plasma was produced by field ionization from the transverse fields of the electron bunch itself.


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Fig.4. Schematic of the PWFA experimental setup at SLAC. The incoming electron bunch is focused at the entrance to the

Li vapor column which is either 1.5 m long and laser-ionized (ne,low experiments) or 8–30 cm long and field ionized by the electron beam itself (ne,high experiments). For the low (high) density experiments, σz was 700 μm (20–40 μm).

The x-ray diagnostics for the two density regimes were very different. For both density regimes, the photon-spectrum measurements were made 40 m from the plasma exit at the x-ray diagnostic area (see Fig. 4), along the photon beam-line, after the electrons beam was diverted downward. For the ne,low conditions (see Fig. 6), the spectrum was sampled in two spectral bands by inserting a 1 mm thick silicon crystal and having two absolutely calibrated, 1 mm thick silicon surface barrier detectors (SBD). One SBD (narrowband SBD) was placed at the Bragg angle for 14 keV photons for an absolute spectral brightness measurement while the other SBD (broadband SBD) was placed in the same plane, but not k-matched for any particular photon energy. This latter detector measured the relative yield of photons in the 5 to 30 keV range via incoherent Thomson scattering of the photons from the electrons within the silicon crystal.

For the ne,high conditions, the expected critical photon energy was 50 MeV making it impossible to disperse the photons. Instead, we used the photons to create electron-positron pairs and dispersed them. This setup is shown in Fig. 5. Figure 5(a) is a schematic showing the plasma, the main electron spectrometer, two collimators, the converter target and the positron (e+) spectrometer (typically, the magnet polarity was set to measure e+ but the e- spectra looked the same). More detail of the converter target and e+ spectrometer is shown in Fig. 5(b). For each experimental condition (ne, Lp, σz, σr0, etc.), the magnetic field of the e+ spectrometer was ramped up and then ramped back down over at least 400 linac shots (1 Hz). Each of the three SBD’s thus recorded spectra that would slightly overlap from which a composite spectrum was constructed.

Fig.5. (a) Schematic of the high density experiment where the critical photon energy was 50 MeV. (b) Detail of the positron

spectrometer that follows the gamma-pair conversion target.

3.2 Observations of betatron motion in SLAC experiments

Evidence for betatron oscillations in the ne,low experiments is shown in Fig. 6. A beam position monitor (BPM) was located Lbpm downstream of the plasma (and beyond the downstream optical transition radiator (DSOTR) of Fig. 4). The signal from the BPM reveals the position of the centroid of the bunch. For a perfectly axial beam, for every electron at an initial r0, there is one at –r0. However, this beam is more like that shown in Fig. 1(c) so that the centroid oscillates like the motion of a single electron–tail sloshing. Plotted in Fig. 6(a) is this centroid position vs. ne,low. From Eq. 2, we expect the exit angle θexit of the tail at the end of the plasma to follow


E400

00)E 000cc

-400

Plasma density (1014 cm3)0 1 2

2-t 3tPhase advance W

θexit =drdz

(z = Lp ) = −r0kβ sin(kβ Lp ) (8)

As ne,low is increased from zero to 1.7 x 1014 cm-3, kβ increases and, from Eq. 8, both the period and maximum amplitude

of θexit should increase. The “phase advance” Ψ = kβ dz0

L p∫ is the integrated argument of the sine term in Eq. 8 and we

calculate a maximum Ψ of about 3π. The centroid displacement should be proportional to θexit Lbpm and for 0 < Ψ < 3π there should be about 1.5 oscillations of the centroid and this is what we see in Fig. 6(a)ii.

Fig.6. Two views of betatron oscillations for the low density experiment. (a) Tail-sloshing induced centroid displacement as

seen by a beam position monitor downstream of the plasma. (b) Envelope oscillations as measured of images of the beam taken in optical transition radiation 1 m downstream of the plasma. Both are plotted versus density (upper scale), which is related to phase advance of the beam envelope (lower scale). The incoming beam parameters were not the same for the two data sets. The arrows in (a) point to “transparency points” where the beam downstream of the plasma is roughly the same size as the beam size at zero density.

From Eq. 3, the envelope of the beam should also oscillate. This is shown in Fig. 5(b) where the transverse size of the beam as measured on the DSOTR is plotted vs. density. Recall that σr is positive definite so the solution to Eq. 3 would look like a rectified version of Eq. 2; i.e., the absolute value of the right-hand-side of Eq. 2. Thus, a phase advance of 3π should correspond to about 3 oscillations of the envelope size and that is what we see in Fig. 5(b). Note the clear increase in period and amplitude, as expectediii. These two data sets were for different incoming beam parameters and thus do not line up exactly, although surprisingly close.

Generally, the incoming electron beam had an energy chirp, dropping in energy from the head to the tail of the bunch at a known rate. Thus an imaging electron energy spectrometer can act as a “streak camera” for the transverse dynamics of the bunch exiting the plasma. For the data in the ne,low regime, the spatial resolution of the spectrometer was insufficient for this purpose. However, for the ne,high data, the spectrometer had excellent imaging properties and could be used as a single-shot streak camera. Some interesting manifestations of betatron motion are shown in Fig. 6 using this technique.

Figure 6(a) shows such a single-shot streak-camera-like image of the electron spot size at the exit of the plasma. The initial γ, γ0, is at the bottom of this image and the electrons above have gained energy. The maximum energy gain here is 5-10 GeV. The exit spot size σexit varies according to the solution of the envelope equation, Eq. 3. The parameters ne,high and σ0 are fixed and only the variation of the final γ makes the envelope oscillations visible; i.e., Ψ = Ψ(γ) so that σexit = σexit(γ). Apparently, a Δγ of about 1 GeV corresponds to a ΔΨ of π. No tail sloshing is apparent here. However, tail sloshing is clear in Fig. 6(b) for similar plasma conditions (the linac was poorly tuned for this data). The size of the image is different than Fig. 6(a), but it is clear that the sloshing of the tail reduced the maximum energy gain. This is due to the fact that the peak-accelerating field is near the axis so that the average field seen in the tail is reduced. Also visible in Fig. 6(b) is the decelerated portion of the bunch. This is double-valued since, along the bunch, the decelerating field


Low

er?

Hig

her

00

10

10 20 30

(b)

40

2n.=1x1017cm3

5 10 15 20Energy (MeV)

(c)

grows to a maximum and then returns to zero. Beyond this is the accelerating field (see Fig. 3). If one looks carefully, the head of the bunch drops down in energy with very little transverse oscillation. Moving further back in the bunch, the oscillations grow as γ approaches γ0 and continues to grow for γ > γ0. This is not a hosing instability. It is simply a tilted beam and the initial offset r0 was increasing along the bunch. The fact that the head of the bunch does not oscillate confirms that the ion channel axis is defined by the head of the bunch, as was illustrated in Fig. 1(c).

Fig.7. Interesting images of beam energy vs. transverse position from the imaging electron spectrometer. The image plane is

the plasma exit. (a) Beaded look for the spectrum above γ0 shows the energy-dependency of the betatron phase advance. (b) Energy-dependent variation of the tail sloshing of an accelerated beam that was tilted as it entered the plasma. (See text.)

3.3 Measurements of betatron x-rays at SLAC

In Sec. 3.1 we described the diagnostics for the two density regimes. In both cases, if we can have knowledge of the betatron orbits, as illustrated in Fig. 8(a), we can calculate the photon spectrum and compare it with that inferred from experiment. Since the photons are emitted tangentially to the orbit of the electrons, we need only calculate the contribution when the momentum vector points to the target. For example, the particle with momentum p1 in Fig. 8(a) will not contribute high-energy photons (occurring only in a 1/γ cone) to the target at that instant. However, the same

Fig.8. (a) Illustration of the method of calculating the photon spectrum once the betatron orbit is known. (b) Image of 5–20

keV photons (bright spot) on a phosphor 40 m downstream of the plasma for the low-density experiments. The vertical stripe is synchrotron radiation from the dipole magnet that dumped the beam (imperfect reference image subtraction). (c) Measured (open circles) and calculated (solid line) positron spectra.


particle later in time and now having momentum p2 will contribute for a brief period. At a distance of 40 m to the target, this will correspond to the moment of maximum excursion of the electron. This is also the moment of maximum radiated power according to Eq. 6. Finally, since the instantaneous radius of curvature of the trajectory is smallest here, this is where the highest energy photons are emitted. The curved orbit is Taylor expanded to second order about this point of maximum contribution approximating circular motion. The resulting synchrotron-like spectrum is then added to that part of the target. For the ne,high experiments, the photons are not measured but rather the positrons that come off the target. In this case the target is subdivided into many sub-regions or pixels and many betatron orbits are followed in order to have a large number of photons per photon-energy bin in each pixel. The reason for this is that these pixilated photon spectra will be the input deck for a Monte Carlo simulation of the pair production and transport to the SBD’s in the e+ spectrometer.

Some raw data from the ne,low experiments is shown in Fig. 8(b). This image was taken at Ψ = π where K/γ = 0.9 x 10-4. The width of the betatron spot is 4 mm at 40 m corresponding to an angle of 10-4, in good agreement with the expected angle. The density scaling of the signal on the broadband SBD followed roughly a ne

2 scaling, as expected from Eq. 6. Finally, the absolute energy captured on the narrowband SBD was in reasonable agreement with that expected for the beam conditions and the resulting photon spectrum as calculated via the method described above.

For the ne,high experiments, a simple ne2 scaling was never observed. In fact none of the “simple” scaling laws implied by

the equations in this paper were observed. In the ne,low experiments, we have: (1) a preionized plasma; (2) an initial bunch density 10 times larger than the plasma density; (3) a relatively long betatron period; and (4) very little change in the γ of the beam. All this leads to a very good knowledge of the betatron orbits. Contrast this to the ne,high experiments where we have: (1) a plasma produced by the bunch itself; (2) an initial bunch density only a few times the plasma density; (3) a very short betatron period; and (4) a large change in the γ of the beam. This leads to a situation in which knowledge of the betatron orbits requires an estimate of the actual number of electrons residing in the ion column, their energy variation, and the highly evolving transverse size of the beam.

The measured e+ spectrum for ne = 1 x 1017 cm-3 and Lp = 11 cm is shown in Fig. 8(c) along with predicted spectrum from the Monte Carlo code EGS4 (Electron-Gamma Shower version 4)iv. As mentioned above, many betatron orbits are needed to obtain enough statistics to run the code. Also, as listed above, there are four aspects to the ne,high experiments that make the estimates of the betatron orbits experimentally challenging. The details are given in Ref. 2 and the excellent agreement between the measured and simulated e+ spectra indicate that we were able to make very good estimates of the betatron motion and therefore of the resultant photon spectrum. The thick line in Fig. 9(b) shows the calculated photon spectrum for the data of Fig. 8(c).

Although we had many experimental knobs for optimizing the photon yield at SLAC, in the end we were limited by the quantity Ib/σr where Ib is the drive beam current and is proportional to Nb/σz. Here, Nb is the number of electrons in the bunch. The transverse field of the beam, that ionizes the lithium vapor and makes our plasma, is proportional to Ib/σr. If we could place more electrons where the longitudinal field of the plasma wave is near zero or even accelerating, the drop in γ could be minimized for the beam as a whole. However, when we increase σz for this purpose, eventually Ib/σr drops to the point where the ionization threshold occurs later and later into the bunch. For lithium, this threshold is roughly 5 kA for a σr of 10 μm. In principal, an electron beam generated in a LWFA could have a Ib/σr much higher than in the SLAC experiments. In this case, there could be more flexibility in parameter space for this PWFA betatron x-ray source. This will be discussed in more detail in the next Section.

4. TWO-STAGE CONCEPT The concept is illustrated in Fig. 9. Figure 9(a) shows a simulation image from the 3D particle in cell code OSIRIS and the resulting energy spectrum of the self-trapped electron beam at about 1.5 GeV. Figure 9(b), for illustration purposes only, a desired photon spectrum. These are the spectra calculated for the SLAC data of Fig. 8(c), not from this simulation. Figure 9(c) shows how all this is put together. A laser drives a wakefield in a He plasma producing a 1.5 GeV beam of electrons. These electrons are allowed to freely expand until they enter a gas vapor with a low ionization threshold such as cesium. Optionally, the free expansion could be replace by magnetic optics to controllably expand the beam (larger r0 is good for x-ray conversion efficiency and x-ray photon energy). The optics could also stretch the beam to scale with the plasma wavelength of the variable-density Cs vapor. A collimator and beam dump follows allowing only the most energetic photons to exit through the Be window. Note that the various curves in Fig. 9(b) are for different observation angles.


CPAlaser

system

Pinhole,L/ i

hv

- Cs plasma

He plasma

Dipolemagnet

'1Collimated

BearWdump X-rays (C)and collimator

Trapped e- trajectories (y-projected), color-coded for Energy

VacuumTrapping20begins

40 x 1019 -3

m

1000I

x-20U

8 9 10 11 12 13

z (mm)

Fig.9. Illustration of the two-stage concept for betatron x-ray production. (a) The input; an electron beam (1.5 GeV)

produced from self-trapping in a LWFA driven by a intense laser (simulation result). (b) The output; a photon spectrum (curves taken for SLAC-like parameters, not for 1.5 GeV). (c) How it all fits together (see text).

Recently, the Callisto laser at the Jupiter Laser Facility at LLNL has been upgraded to a sub-100 fsec pulse at an energy of nearly 10 J. Simulations using the 3D code OSIRIS have been performed for parameters which may be available soon on the Callisto laser; namely, 9.5 J, 47 fsec focused at z = 0 to a 20 μm spot size into a gas cell of 12.3 mm length and a plasma density of 1.5 x 1018 cm-3. (These parameters are close to that used in Fig. 9(a).) Trajectories of selected electrons are shown in Fig. 10. In this particular simulation, the density was increased by about 27 times for the last 2 mm to study one possible way to increase the laser to x-ray conversion efficiency; i.e., let the last 2 mm be a PWFA betatron x-ray source as in the SLAC experiments. A density of 4 x 1019 cm-3 was chosen mainly to push the x-ray energies up into the MeV range.

Fig.10. Orbits of selected electrons from a 3D simulation. The laser was focused at z = 0 and was self guided over at least 10

mm. Self trapping begins around 3 mm into the 10.3 mm long lower density part of the plasma. The beam is slowed down within 2 mm inside the higher density plasma (see annotations on the figure).

Note that the period of the betatron oscillations is substantially reduced in the high-density portion of the plasma. This is adventitious since the critical frequency of the x-rays is proportional to the instantaneous radius of curvature of the orbits (for a fixed γ). Thus, a shorter betatron period should yield photons out to a higher energy—at least early on before the electrons lose too much energy. This is the purpose of the “two-stage” idea. Note: this is a first result and the density of the second stage has not been optimized. Nor has σz or σr been allowed to change as might be desirable in experiments.

The data in Fig. 10 was made possible due to the new features that have been added to OSIRIS 2.0. First, the simulation in Fig. 10 was performed in the “boosted frame” which allowed the simulation to run to completion 25 times faster than usual (see L. Silva et al., these proceedings). Another of these new features of particular interest is described here. To calculate the betatron x-ray spectrum in post-processing, the entire 3D trajectories of representative electrons are needed.


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Given that there are on the order of 108 electrons in the simulation, many of which could be trapped, it was important to limit the number of orbits to be post-processed. The code is run until self-trapping and acceleration begins. Using a cut in longitudinal momentum, a subset of the trapped electrons is tagged. The code is then re-run to completion and the trajectories of the tagged electrons are saved. The resulting space- and time-resolved photon spectra are then calculated in post-processing.

To allow for many iterations of the post-processing calculation of the photon spectrum, the electron beam at z ~ 10 mm from the simulation of Fig. 10 was approximated by a bi-gaussian bunch with a similar phase space to the tracked electrons. The bi-gaussian which best represented the electrons had a σz of about 1.2 μm and a σr of about 0.3 μm. Even if the charge is 10 times smaller than the 1 nC given by simulation, this is still a very dense beam. Apparently, this beam drove its own wakefield in the high-density portion of the simulation box in Fig. 10. Furthermore, the strongly curved orbits and the rapid deceleration of these electrons, apparent in Fig. 10, suggests that this was a PWFA in the blowout regime. However, a σz of 1.2 μm is about 0.25 of a plasma wavelength in the high density and is not optimum for conversion efficiency to high-energy x-rays–γ drops too fast.

Fig.11. (a) Electron betatron radiation; energy deposition on a virtual detector 2.3 mm away from the end of the plasma. The

dashed line shows the row of pixels selected for the plot in (b) which is the x2-resoved spectrum (see text). The horizontal bar in (b) represents 4.6 μm at 2.3 mm or 2 mrad of angle in x2.

Some results from the post-processing of selected trajectories are shown in Fig. 11 (see J. Martins et al., these proceedings for details on the spectral calculations). Figure 11(a) is a virtual energy-deposition detector located 2.3 mm from the end of the high-density plasma or at z = 14.6 in Fig. 10. The synthetic bi-gaussian beam was launched into the high-density plasma so the contribution from the acceleration stage (0 < z < 10.3 mm in Fig. 10) is not included. So far, the number of electrons that have been post-processed is too small to produce a smooth “spot” on the detector although nearly all the photon-hits fit nicely on this 97.6 μm square detector and tend to add up in the center.

The individual “streaks” apparent in Fig. 11(a) are due to this lack of statistics (the statistics can easily be increased). These streaks are roughly 40-50 μm long. The reason that they are of “finite” length is that the off-axis contributions to the streaks have much lower photon energies and thus fall off the color scale (see Fig. 9(b)). The highest-energy photons are on axis. Each pixel in this detector not only records the total energy deposited, but also has the full spectrum of the photons that struck that pixel. The horizontal dashed line in Fig. 11(a) was chosen to illustrate the x2-resolved spectrum of the photons along this row of pixels. This is shown (zoomed in x2) in Fig. 11(b). To emphasize the spectrum from the high-γ electrons, only the contribution from the orbits between 10.3 < z < 11.1 mm in Fig. 10 are included in Fig. 11(b) (the full spectrum looks very similar since beyond z = 11 mm, γ has dropped substantially). We see that the photon spectrum near x2 = 10 μm has a critical energy close to 1 MeV. This is roughly the critical energy calculated for electrons of 1.3 GeV and an electron density of 4 x 1019 cm-3. This peak is more than 10 times higher in photon energy that that obtained from the 10.3 mm of the accelerating portion of the LWFA. Although the post-processing code produces data with absolute scales, we need the more statistics and the implantation of magnetic beam manipulation in the code before we can optimize the photon yield and quantify the photon-to-photon (laser energy to x-ray energy within some bandwidth) conversion efficiency.

Proc. of SPIE Vol. 7359 735902-10

5. CONCLUSIONS In this paper we have provided an introduction to betatron motion of electrons within the ion channel of a Laser Wakefield accelerator (LWFA) or a Plasma Wakefield accelerator (PWFA) in the bubble or blowout regime. Experimental results on betatron motion and the resulting x-ray spectra from a PWFA at the Stanford Linear Accelerator Center (SLAC) are reviewed. We believe that the betatron x-ray experiments at SLAC benefitted from the independence of the electron accelerator (the SLAC linac) and the ion-channel wiggler (a PWFA driven in beam-ionized lithium vapor). Three-dimensional simulations of the LWFA in the self-guided and self-trapping regime suggest that very high-current electron bunches can be produced above 1 GeV. However, without a secondary wiggler, the photon spectrum peaks at a relatively low energy (< 100 keV). By sending this bunch into a secondary plasma, a sort of two-stage process occurs where, after being accelerated in the LWFA stage, the electrons slow down in the second, PWFA stage and give some of their energy into x-rays. This 2-stage process was simulated in the 3D PIC code OSIRIS 2.0 and the peak photon energy became ~ 1 MeV, more than 10 times higher than that of the accelerator alone. However, much more needs to be done to optimize this concept—perhaps with magnetic optics between the stages—and to quantify the laser-to-x-ray conversion efficiency.

Acknowledgements: This work was funded with DOE Grants No. DE-FG02-92ER40727, No. DE-FG03-92ER40745, No. DE-AC02-76SF00515 (SLAC), NSF Grant: No. PHY-0406758, Fundação para a Ciência e Tecnologia, Portugal No. PTDC/FIS/66823/2006 and NGST-M173-30.

REFERENCES

i Esarey, E. et al., Phys. Rev. E 65, 056505 (2002).

ii Wang, S. et al., Phys. Rev. Lett. 88, 135004 (2002).

iii Clayton, C. E. et al., Phys. Rev. Lett. 88, 154801 (2002)

iv Johnson, D. K. et al., Phys. Rev. Lett. 97, 175003 (2006).

Proc. of SPIE Vol. 7359 735902-11

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