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Particle accelerators are the key tools to accelerating charged particles to high speed by using electromagnetic field. In 1928, the worlds first accelerator was built in Germany . By considering the evolution of accelerators and colliders, the energy needed for two nuclei which are positively charged and they repel each other to interact between a close enough distance has to be sufficient enough. In quantum mechanics, particles are considered not only by their physical trajectory, but also through a wave that gives the probabilities of a particle can be localised at a given point. Furthermore, this relates a increase of the energy, a decrease of the wavelength of the colliding particles. There is a purpose for using high energy technique, a majority of the interest of the particles do not exist naturally, and therefore they will need to be created in laboratory. The relationship of indicates that energy is required to produce particle with mass m during the collision. Unfortunately, most of interesting particles are extremely heavy and energies needed to produce them will be up to GeV scale. Hence, many physicists were committed to construct of bigger and better accelerators.
Fig . Livingston plot .
The graph in fig.1 shows that acceleration energy against time scale and clearly to notice that energy of accelerators has been grown exponentially these few decades and increased by a factor of 10 approximately every six years. The Livingston plot was produced firstly by M. Stanley Livingston in 1954, and this plot was updated by putting all the modern and advanced machines. From the graph above, noted that the largest and highest-energy particle accelerator is the Large Hadrons Collider (LHC), which is 27km long circular colliders and achieved to operate 4 TeV per beam in the end of 2012 and recently discovered the Higgs boson. LHC will be shut down and engineers will upgrade the machine to be able to attain 7 or 7.5 TeV per beam during the next two years . After LHC, linear colliders over 30km will be needed. The International Linear Collider (ILC) and The Compact Linear Collider (CLIC) are the next big machines afterwards and developed at CERN. Compare to circular accelerators (e.g. LHC), which has a big advantage of the higher efficiency of accelerating particles over long distances, especially for heavy particles, such as LHC for proton-proton collision. However, an electron-positron linear collider will provide more precisely measurements, even though they cannot achieve the same energies for collisions. As the accelerating gradient in radio frequency linear accelerators are limited around about 100 MV/m, which due to breakdown occurs, and the expensive cost of the big and high-energy machines with the increase rate of beam energy, therefore the new high-gradient accelerators are required to need in high-energy physics for future, that leads to introduce the concept of plasma-based accelerators . The laser- and beam driven plasma accelerators can undertake considerable great amount of electric fields. During last few decades, the development of plasma accelerators was really fast. For instance, the most well known investigated accelerators are the laser wakefield accelerator (LWFA), the plasma wakefield accelerator (PWFA), the plasma beat-wave accelerator (PBWA) and self-modulated laser wakefield accelerator (SM-LWFA), etc. The aim of these plasma-based accelerators that reduce the cost and minimise the size of machines and high gradients roughly in 50-100 GV/m have been achieved in labs.
The idea of laser wakefield accelerator is using a short ultrahigh intensity laser pulse to drive a plasma wave, which is approximately less than 1 ps. The most efficiently way to drive plasma wave is when the laser pulse length equals to the plasma wavelength. Laser wakefield accelerator (LWFA) was first proposed by T. Tajima and J. M. Dawson since 1979. A laser technology can supply an electric field of V/cm and a high power density of W/. Therefore, Tajima and Dawson have used the technique that electrons were accelerated by this high-power radiation in a short distance. The use of this mechanism was stimulated by computer in 1988 by Sullivan and Godfrey and Mori. In the meanwhile, only the PBWA concept is accepted because of the technology of ultrahigh intensities and picoseconds short laser pulse did not exist. So the LWFA was reconsidered by Gorbunov and Kirsanov later. The first experimental work for generating plasma wave by the LWFA regime was probably by Hamster. The results of these experiments have shown that when the laser pulse length by the driven plasma wave approximately equal to the plasma wavelength at the plasma frequency of terahertz radiation was emitted . The method of using probe pulses and optical interferometer have been obtained some measurements of generating plasma wave recently at Ecole Polytechnique and University of Texas at Austin groups in 1996. This technique allows a comparison between the particle-in-cell stimulations and experiments to help the understanding of future developments.
The self-modulated laser wakefield accelerator (SMLWFA) uses the same short ultrahigh intensity laser pulse as in LWFA, but operates at even higher densities than the LWFA. Therefore, the length of laser pulse is longer than the plasma wavelength, i.e. . The power P is larger than a critical power . At the beginning of 1990s, the "chirped pulse amplification (CPA)" has been invented and for the use of storing large energy. For example, the generation of subpicosecond pulses by using the Nd laser pulse produced a few terawatts of power. These pulses have enough intensity to stimulate a plasma wave via the forward Raman scattering (RFS), where the beam decays into Stokes wave, anti Stokes wave and relativistic plasma wave . When these waves become high intense appropriately, they will produce large amplitude of electric field.
The plasma wakefield accelerator (PWFA) has proposed by electron beams which is driven by the plasma wave with a high phase velocity. In 1956, Fainberg proposed that the use of beam driven plasma accelerator. Chen has analyzed that the PWFA in the linear regime in 1985 . Ruth has shown that a limit of transformer ratio for a symmetric driving beam, where is the ratio of energy obtained by beam energy . Later, Rosenzweig illustrated that a two-dimensional nonlinear regime has even better benefit by using electron bunches on the PWFA .
For the PBWA regime, in order to excite a plasma wave at resonance, so use two low power and long pulse laser beams with frequencies and respectively and satisfy the condition that . It has been analysed by several researchers after first used by Tajima and Dawson. One of the attractive feature of PBWA is the potential obtain the beam energy doubly in a single stage of the accelerator only few tens meters long. Katsouleas and Dawson brought out a transverse magnetic field into the PBWA for conquering the phase detuning between the plasma wave and electrons. After some early work, the PBWA is simulated, for instance, the relativistic plasma wave for the self-focusing laser beams. The observation of the PBWA was firstly used by two lasers in a plasma density and accelerated in energy of more than 10 MeV. More recently, the Nd lasers have been used in a PBWA experiment.
2. Physical content of plasma accelerators in different regime
2.1 Basic concepts for plasma acceleration
The capability of sustaining the high electric field for plasma makes itself to one of the most attractive medium for particle acceleration. Since Tajima and Dawson have proposed the plasma based accelerator in 1979, a longitudinal plasma wave has been used to provide the energy for particles. Hence, with the purpose of producing relativistic particle beams, a phase speed has to be close to the speed of light in vacuum for plasma wave to achieve enough intense . Moreover, the accelerating electrons can be trapped by injection of a relativistic plasma wave were shown by Clayton (1993) and Everett (1994). There are a large number of experiments about this topic and recently results obtained high accelerating fields about 1 in 2004. In beat wave experiments, the limitation of maximum accelerating field gradient is due to the wavebreaking, which occurs at nonlinear distortion of a sinusoidal wave and the gradient becomes to infinity. Except for the accelerating gradient, luminosity and emittance are also the parameters for building more advanced conventional accelerators. Unlike the conventional accelerators, plasma accelerators have no electrical limit, which cannot excess maximum field strength of around 1. The plasma waves oscillate at the plasma frequency which defined as, where is the plasma number density and is the mass of electron. Then, these waves have the phase speed and the electric field E of relativistic plasma waves can be estimated by , where is the amplitude of plasma wave and n is the plasma number density. For a density of can possibly achieves high gradients of 1 .
2.2 Laser wakefield accelerator
In the LWFA regime, a short laser pulse and high intensity drives a plasma wake. The plasma wave in an electric field can trap and accelerate electrons. The maximum energy on the order of 100 MeV was gained in early results in 2002. a short laser pulse has been used and the frequency of the pulse is much greater than the plasma frequency, which creates the blowout or bubble regime due to the ponderomotive force, .
Fig 2. One-dimensional graph of the excitation by (a) a short laser pulse or (b) a short electron beam pulse which propagates through the plasma .
The laser case shows that the ponderomotive force of the pulse is proportional to the intensity and the beam case shows that the space-charge force of the beam pushes away the plasma electrons. Furthermore, the phase velocity of the wake is equal to the group velocity for the laser case, while in the beam case the phase velocity is the same as beam velocity . But one of the common between these two cases is that the plasma electrons snap back quickly as the restoring force and the pulse length is estimated around half of a plasma wavelength. For the nonlinear plasma waves, they are generated by the leading edge of the intense laser pulse. If the length of laser pulse is longer than the plasma wavelength, then the plasma wave energy is absorbed by trailing beam of the laser pulse. In contrast, if the length of laser pulse is equal or shorter than the plasma wavelength, a wakefield will be excited by the ponderomotive force with a phase velocity, which is equal to the group velocity. Such lasers can have intensities greater than and in the strong nonlinear regime.
The bubble or blowout regime was considered by an electron beam driven accelerator firstly, but it applied for a laser driven accelerator as well. In the electron beam case, it is the beam pulse of the space-charge force. While in the laser case, the radiation pressure pushes all the plasma electrons away radially and leaving a bubble of ions (See Fig.3). The plasma electrons in both cases form a sheath around the bubble of ions and return to the beam axis, overshoot and form a three- dimensional oscillation.
Fig.3 This diagram describes a short laser pulse or short electron beam driven plasma. In both cases, all the plasma electrons have been blown out. A bubble is created and surrounds the drive beam and the left plasma ions. The wakefield can trap some electrons and can accelerates the trailing beam .
Although the phase velocity of the bubble regime is relativistic, the accelerating particles can still escape in a specific distance called dephasing distance. This limitation have been gained the maximum energy, but within a narrower energy spread. Therefore, a quasimonoenergetic bunch is observed and to produce them, the drive pulse blows out some of the first trapped electrons by the spike of the accelerating field. As a large amount of electrons are trapped, therefore the wake becomes beam loaded and the amplitude of the accelerating field decreases, then there is no further trapping. And a quasimonoenergetic bunch is generated by the phase-space rotation, because the electrons in the front diphase and lose energy gradually and the electrons behind keep gaining the energy. In order to generate such bunches process has to be close to the dephasing distance within a plasma-vacuum condition. Otherwise, the trapped electrons will lose energy and the monoenergetic beam will lose either.
The principle of relativistic optical guiding in the LWFA is a short (1), high power () and single frequency laser pulse which can accelerating the wakefield by through long distances . The plasma wave is not resonantly excited in the LWFA compare to that in the PBWA and which means the plasma density does not have to achieve large enough amplitude of accelerating fields.
2.3 Self-modulated laser wakefield accelerator
A large amplitude plasma wave with long laser pulse can separate into a train of short pulses with separation equals to approximately and each short pulse have a width on the order of . These occur via Raman forward scattering in one-dimensional and via an envelope self-modulation instability in two-dimensional. One of the most remarkable results observed from a group at RAL. They have reported that the electrons at high-energy about 120 MeV were correlated with radiation by RFS. These experiments used a 25 TW laser with intensities and pulse length. Then, the acceleration gradient of from greater than 100 MeV electrons observed over a 600 interaction length approximately. Also, this experiment observed that there are 12 Rayleigh lengths for laser self-channelling. Furthermore, the short pulse wakefield have achieved the largest acceleration gradients which was nearly equals to 200 . The limitation of extending the acceleration length is usually depends on the diffraction length or as known as Rayleigh length , where is the laser frequency and is the spot size. A useful accelerator must propagate through the plasma stably over much larger distance than the Rayleigh length.
2.4 Plasma wakefield accelerator
The perturbations for plasma electrons are the space charge of the beams and produced the plasma oscillations at the plasma frequency . The phase velocity is equal to the velocity of the driving beam, while the group velocity is almost zero. Chen has been studied the PWFA with a model contains of a number of short driving bunches. These short bunches limit the trailing electrons gain the energy to , where represents the energy of the driving electrons. The maximum energy gained can be considered by R times the energy of driving beam. The transformer ratio R of PWFA is defined as the ration of energy gained to the initial energy, i.e. . This gives that an electric field causes a deceleration from initial energy to zero in the length . On the other hand, the driving electrons gain energy ,where is the maximum accelerating field of the wake.
As the electron bunch enters into a given region, the plasma sees much more negative charge. The plasma moves to neutralise the field while the charge slowly builds up, i.e. , where is the maximum density and is the background plasma density. After the plasma was nearly neutral, a non-neutral space charge of amplitude equal to . Unlike the PBWA, one of the advantages of PWFA is that it does not need to satisfy the resonance condition. From Poisson's equation, the amplitude of the plasma wave electric field , where. Hence, we find and .
For a driving bunch, the transverse instabilities will cause distortion of the shape of the bunch. For example, the self-focusing result a transverse wake by the driving beam. The transverse instability is depending mainly on the radius of beam. For a narrow beam of order with radius "a", the beam is self-focused by its own wake. On the contrast, wide beams are subject to the Weibel instability, such as filamentation.
2.5 Plasma beat wave accelerator
In the PBWA regime, the energy and momentum conservation give that and , where is the frequencies of the two lasers, is the wavenumbers of the two lasers and is the plasma wave wavenumber. If , therefore the phase velocity is equal to the group velocity . Since the frequency for two laser pulses are very close to each other and much greater than , then the Lorentz factor becomes to . If the electromagnetic wave scattered was travelling the same direction as the incident wave, then called it as forward scattering. Hence, the equation of the plasma number density perturbation will be,
Where is the density and is the normalised quiver velocity for .
In 1972, Rosenbluth and Liu have solved the equation abouv by using the limit of zero pump depletion, so the equation becomes
It shows that the amplitude of plasma wave has a linearly relationship with time. However, the amplitude will be saturated as the increase of the relativistic electron mass. They also demonstrated that the wave saturated before it reaches the breaking limit, which is . Thus,
A theoretical maximum amplitude for the longitudinal field was obtained from Poisson equation and gives that , where is the saturation value. The diffraction of the laser beams limits the depth of the laser. The pump depletion limits to avoid by using more intense lasers. The experiment at UCLA injected a 2MeV electron beam by using two frequencies of laser. This experiment shows a result of nearly electrons are accelerated within a 1cm diffraction length from 2 to 30 MeV, which related of a gradient of . The most important point of view of this experiment is that wave potential trapped the electrons. In this case, there are above 16 MeV trapped electrons were observed and propagated in the forward direction of the wave .
2.6 Proton driven plasma wakefield accelerator
Recently, some experiments have already achieved a range of gradient . Therefore, need to improve and reach even higher energy regime of teraelectron volt scheme. Caldwell has proposed a new regime called "proton-driven plasma wakefield acceleration" . Similar to the beam-driven PWFA, a motion of plasma electrons is set by the space charge of the driving beam. Nowadays, the benefit of using a proton drive beam instead of electron beam is that there are several TeV synchrotron facilities exist in the world, such as Tevatron, LHC etc. Moreover, a typical TeV proton bunch can store two or three orders magnitude higher than that of the electron bunch from Stanfor Linear Collider (SLC). In the PDPWA regime, energy transfer from high-energy protons to the plasma wave, then transfer to the electrons.
Fig. 4 A quadrupole magnets surround a thin tube within Li gas.
The structure of PDPWA is shown by Caldwell in 2009(See Fig.4). In the linear regime, the maximum gradient can achieve is . Plasma electrons oscillate with frequency for a relativistic driving bunch, where is the density of plasma electrons and is the permittivity of free space. The electrons oscillate towards the beam axis and then create a cavity after they pass though each other with strong electric fields. So a witness bunch accelerated and placed in the electric field varying with time. Also, a radial force is provided by the plasma for the purpose of keeping witness bunch and the drive bunch .
3. Some experiments and properties for different regime accelerators.
3.1 The LWFA applications and experiment with few-cycle pulses.
In September 2002, the ALPHA-X (Advanced Laser-Plasma High-energy Accelerators towards X-rays) project has begun. The application of this project is to produce short wavelength radiation coherently by using a free-electron laser. Moreover, it can help to develop the laser-plasma accelerators as well .
Fig. 5 Schematic overview of the ALPHA-X set-up.
In order to have the short-wavelength regime, the electron energy must be high enough and plasma-based accelerator is used for producing ultra-short electron bunches with high charge, low energy spread, and low transverse emittance by driving a free-electron laser.
3.2 The PWFA experiments at FACET.
Facilities for Accelerator science and Experimental Test beams (FACET) at Stanford Linear Accelerator Center (SLAC) will provide electron and positron beams with high-evergy, high-peak-current and low-transverse-emittance. The aim of the experiments at FACET are provide the evidence that in a single and high-gradient PWFA stage can obtain high-energy by bunches of electron and positron under the conditions of preserving the beam emittance and small spread of momentum, and illustrating the energy transfer efficiency.
Fig. 6 FACET modifications to the linear accelerator systems marked in red. The positron bunch compressor is at Sector 10 while the experimental area is at Sector 20 along the system.
FACET| beams drives the plasma wakes over long distances to obtain energy of the order of 25 GeV. The facility is placing at the upstream of the LCLS injector (see Fig.6). In order to gain about 23 GeV beams of electron and positron with small emittances, FACET uses the first 2km of SLAC linac to produce it. A high peak current and a small transverse size are the two key factors to excite large amplitude wakes with the significant high intensities. To achieve high peak current in FACET, use a threefold compression process. An electron gun produced and accelerated the electron bunch to 1.19 GeV. The long bunch travels into the radio frequency cavity and leaves the unchanged energy. Then, the bunch travels through the ring and compressed by the momentum compression factor to 1.5 mm. However, the bunch is accelerated to 9 GeV in the first km of the linac. Finally, the bunch compresses to a minimum length of 14 and a peak current which is above 20 kA.
3.3 Energy doubling of 42 GeV in PWFA of 85 cm length.
An excellent result of a plasma wakefield accelerator of 85 cm length to gain the energy of more than 42 GeV has been found at the SLAC. A schematic set-up of this experiment is shown by Fig.7. And these are the perfect agreement of the predictions of 3D particle-in-cell simulations.
Fig. 7 Schematic of the experiment set-up.
In the early work at the Final Focus Test Beam facility, usually the 50-femtosecond-long electron beam contains particles which is focused to a spot size of about with the density . Therefore, the 42 GeV beam has an energy spread roughly about 1.5 GeV and electrons in the front have higher energies than in the back of the beam. The electrons beam disperses after exiting the plasma traverses a dipole magnet.
Fig. 8 Energy spectrum 
Figure 8 shows that a energy distribution between 35 and 100 GeV for the electrons after traversing the plasma. The angle can be neglected at the plasma exit in this case as it was smaller than 100 by calculation. Thus, energy is relating to the position only. Clearly, some electrons with an initial energy of 41 GeV have more than twice of their initial value as the highest energy of the electron is . This implied the maximum accelerating field of approximately states that the energy gain of the plasma extends to at least 85 cm long. If the length extended from 85 cm to 113 cm, the maximum energy will be measured to be with a similar incoming current. The saturation of energy gain for this experiment might have three possible reasons. Firstly, the acceleration is stopped in the last 28 cm of the plasma as the energy of producing the wake has been depleted to about zero. Secondly, the hosing instability breaks up the beam. The last possible reason is head erosion, i.e. the front of the beam expands. This caused a decrease of the beam density and moves the ionisation front backward in the beam frame.
3.4 PDPWA based on CERN SPS.
The Super Proton Synchrotron (SPS) tunnel has a length of 600 m and can provide high intense and high-energy proton beams for the LHC and other experiments. The maximum beam energy can achieve by SPS is 450 GeV. In the first experiment, the externally injected electrons can be accelerated to 1 GeV with a proton bunch of 5-10 m plasma. A plan for reaching 100 GeV within 100 m plasma will be developed later.
Fig. 9 Schematic of the beam line.
Nowadays, with an increase of beam energy, the size and cost of high-energy particle accelerators reached the limit, so conventional accelerators are no longer satisfy our needs. Therefore, scientists decide to build up plasma-based accelerators. One of the most important features of plasma is that it can sustain very large electric fields. The plasma accelerators both of the laser driven-wakefield and beam driven-wakefield have developed rapidly in last few decades, the accelerating gradients of 50-100 GV/m have been demonstrated in labs. For future laser experiments, physicists are expecting something like particles per laser pulse accelerated to TeV energies.