# Looking At Compton Backscattering Biology Essay

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Compton Backscattering has been understood for many years, but with the emergence of quasi-monoenergetic electron bunch sources more opportunities have arisen to exploit this phenomenon. By utilising laser-plasma mechanisms a tuneable monochromatic (very little beam and energy divergence, in the region of a few mrads) radiation source can be developed for many purposes across many disciplines.

Laser wakefield acceleration has been a major development in the laser-plasma physics field within the last decade. Now relativistic electron bunches can be formed over extremely short scales; compared to massive conventional particle accelerators used at particle physics labs such as CERN in Switzerland. This mechanism creates betatron radiation due to interaction with the ion cavity created by the laser pulse propagating in the plasma. When the accelerated electrons 'collide' with a counter propagating laser, massive upshifts in frequency can be accomplished into the X-ray part of the electromagnetic spectrum.

Understandably, with the above being applied to create a tabletop short pulse X-ray radiation source, the topic has become attractive to research groups globally. A novel x-ray source would become a valuable tool in molecular dynamics, biomedical imaging, medicine and material science to name but a few.

Groups have recently been able to demonstrate high charge (1 nC), high energy (>50MeV), almost monokinetic laser-plasma produced electron bunches using terawatt class, small Joule level, femtosecond chirped pulse systems [1].

A further justification for the amount of research into the topic was on the back of successful Thomson scattering experiments, where photons of energy 0.4 - 2 keV were produced by MeV electrons. So it is a natural progression to study the higher energy Compton scattering interactions. Also with constant advances in laser technology, the boundaries are constantly being extended.

In this thesis we will discuss the theory behind laser-plasma acceleration and Compton backscattering. We will analyse the profiles of the radiation emitted as several key parameters are changed to 'tune' the output, which is both the betatron component and the scattered radiation. Advances will also be discussed, such as the possibility of the creation of a 'gamma laser' using the technologies and theories included in this report. We will discuss conclusions and future work to be carried out on the topic.

## 2 Background Theory

## 2.1 Plasma Physics

A plasma is a gas where certain amounts of its particles are ionized, where the potential energy of a particle due to its nearest neighbour is much smaller than its kinetic energy. Plasma is the fourth form of matter. The presence of non-negligible amount of charge carriers in plasmas makes them electrically conductive and highly interactive with electromagnetic fields. Like a gas, a plasma does not have specific dimensions unless it is contained; unlike a gas, in the presence of magnetic fields the plasma can change to form structures [2].

## 2.1.1 Plasma Frequency

The plasma frequency is the most fundamental timescale in plasma physics. It is given by:

(1)

Where n is the number density, e is the charge of an electron, Ïµ0 is the permittivity of free space and m is the mass of the electron. This shows that a unique plasma frequency exists for every different mix of plasma. It can be seen that the plasma frequency, Ï‰p, relates to the typical electrostatic oscillation frequency in a species due to charge separation. Let us consider a one-dimensional plasma regime consisting of only one charge species being displaced from its quasi-neutral state by some infinitesimal amount, Î´x. This gives rise to a charge density on the "leading face" of the plasma of

(2)

An equal and opposite charge density is formed on the opposing face of the plasma. The electric field in the x-direction generated within the one-dimensional regime is of magnitude

(3)

Applying Newton's Law to an individual particle yields

(4)

giving Î´x = Î´x0 cos(Ï‰pt) [2].

It should be noted that plasma oscillations would only be observed if the observation time is longer than the plasma period

(5)

Likewise, in the observation over a length scale less than vtÏ„p, which is the distance travelled by a particle during a singular plasma period, will not detect plasma behaviour. In this case, the particle will leave the system before the completion of a plasma oscillation. This distance is the spatial equivalent of Ï„p, called the Debye Length [3]. This takes the form

(6)

Noting that

(7)

Is independent of mass and is therefore broadly comparable across different species of plasma. Considering the above conditions, the idealised parameters would be:

## ,

## 2.1.2 Debye Shielding

In a plasma there are many particles travelling at high velocity. Consider a test particle of charge qT>0 and infinite mass located at the origin of a three-dimensional Cartesian coordinate reference frame containing an infinite, uniform plasma. In this case, the test charge will repel ions and attract electrons, the electron density (ne) around the test charge will therefore increase and the ion density (ni) will decrease. The test charge gathers a shielding cloud, cancelling its own charge. Using Poisson's equation relating the electric potential (Ï†) to the charge density (Ï) due to charged particles and test charge

(8)

where Î´(r) â‰¡ Î´(x) Î´(y) Î´(z) is the product of 3 Dirac delta functions [3].

## 2.2 Plasma Based Charged Particle Acceleration

## 2.2.1 Information and Innovations

The large magnetic fields that plasmas can accommodate make them a valuable resource , with methods of particle acceleration remaining an important research topic internationally. In this section, the fundamental physics of the process will be discussed. In a plasma-based accelerator, particles are accelerated by relativistic plasma waves that can be created by interaction with a sufficiently intense laser or particle beam. In these accelerators, the particles gain energy from longitudinal plasma waves. To accelerate electrons charged particles to relativistic energies the plasma waves must be sufficiently intense and must have a phase speed close to the speed of light, c.

Both of the above methods have been accomplished in a number of laboratories. An extremely successful example of intense plasma wave generation is show by Tajima and Dawson. Particle acceleration by relativistic electron plasma waves was demonstrated to produce electrons of energy greater than 100 MeV over distances in the order of a millimetre. These experiments yield accelerating fields of 1 GV cm-1, it is worth noting that this is much greater than the fields created in high-energy accelerators that are of the order of 20 MVcm-1.

The 1980s developments in laser technology greatly increased the possibilities of laser-plasma interaction acceleration. It was in this decade that femtosecond pulse amplification was developed which allowed the construction of compact terawatt and petawatt lasers, with ultra-high intensities (â‰¥ 1018 W cm-2) and small energies (<100 J). Using the new tabletop-terawatt (T3) lasers, the laser-plasma interaction becomes highly nonlinear and relativistic that leads to a number of interesting phenomena, such as laser wakefield acceleration [4].

This report will be focussed on high power laser-plasma acceleration techniques.

## 2.2.2 Laser Wakefield Acceleration

In laser wakefield accelerator schemes, an extremely short laser pulse of frequency (Ï‰0) much greater than the plasma frequency (Ï‰p) excites a wake of plasma oscillations (at Ï‰p) due to the pondermotive force created. In this method of plasma wave generation, the density of the plasma is not required to be of high uniformity to produce large amplitude waves. As the laser pulse propagates through an underdense plasma ( where) the pondermotive force associated with the laser envelope expels electrons from the region of the laser pulse and thus excites electron plasma waves. If the laser pulse length (cÏ„L) is large relative to the electron plasma wavelength then the energy in the plasma wave will be reabsorbed by the trailing section of the incident laser pulse. If the pulse length is comparative to or shorter than plasma wavelength, , the pondermotive force excites plasma waves (or wakefields) with a phase velocity equal to the laser group velocity, the energy is not reabsorbed. Thus, a pulse with a sharp rise or fall on the scale of c/Ï‰p will excite a wake.

To understand the laser wakefield mechanism, the model used considers one fluid, cold relativistic hydrodynamics and Maxwell equations with a quasistatic approximation, a set of two coupled nonlinear equations describing the self consistent evolution in one-dimension of the laser pulse vector potential envelope and the scalar potential of the excited wakefield. Starting with the relativistic electron momentum equation

(9)

where the relativistic momentum is

(10)

where v is the electron velocity. Equation (9) assumes that all variables depend on z and t; where the laser pulse propagates in the z direction. Therefore, the electromagnetic fields can be calculated by

(11)

Here Aâ”´ is the vector potential of the electromagnetic pulse and Ï•, the ambipolar potential due to separation of charge within the plasma. The perpendicular component of the electron momentum can be found using equations (9) and (11). These yield

(12)

therefore, we can express Î³ in the form

(13)

where

. (14)

From the analysis above of equation (9) we can now express; its longitudinal component, the electron continuity equation, Poisson's equation and the wave equation for a(z, t) (respectively) as follows:

(15)

(16)

(17)

and finally

. (18)

The above assumes a driving pulse of form

(19)

where Î¸ = Ï‰0 - k0z, Ï‰0 and k0 are the central frequency and the wave number (respectively), Î¾ = z - vgt, where vg is the group velocity and Ï„ is a slow timescale such that

and c.c. is the complex conjugate [4].

## 3 Experimental Setup

The basic experimental setup consists of a high-powered laser interacting with an underdense plasma. This creates a beam of accelerated electrons by the laser wakefield acceleration mechanism. A counter-propagating laser then strikes this electron beam with causes the electrons to change trajectory (and velocity) and emit radiation. This is shown in the Fig 1 below:

Figure 1: Schematic of a Compton scattering x-ray experiment using laser wakefield accelerated electrons. The energy of the emitted x-rays depends on the angle between the laser and electron beam as well as the angle of observation and the electron energy [5].

The first high powered laser is a Ti:Sapph laser for wavelength 800nm. It has a femtosecond pulse duration delivering a 1J energy pulse to the underdense plasma. This corresponds to an output power of approximately 30 Terawatts. The profile of the laser pulse is Gaussian in form.

It has an electron number density of between 1018 and 1019 cm-3. This interaction gives rise to accelerated electrons of approximately 200 MeV over an acceleration length of 2mm. The electron bunches have a charge of a few Pico coulombs.

The counter-propagating laser is also a Ti:Sapph as above, but will be varied in this report for analytical purposes.

## 4 Betatron Radiation

## 4.1 Derivation of Betatron Radiation Equation

The betatron radiation from the accelerated electrons beam is a result of interaction with the ion cavity created by the accelerating laser pulse can be derived using the general form of power radiated per unit solid angle

(20)

where

(21)

and E is the electric field. In equation (20), the power is expressed in the observer's time frame since we are considering the frequency spectrum with regards to the observer. To achieve a finite total energy radiated we assume that any acceleration occurring does so over a finite time. The total energy radiated per unit solid angle is the time integral of equation (20):

(22)

This can also be expressed as an integral over all frequencies by introducing A(Ï‰), the Fourier transform of A(t) such that

(23)

and its inverse then becomes

(24)

Using these, equation (22) can now be expressed as

(25)

By interchanging the order of integration, we can represent the total energy radiated per unit solid angle as

(26)

The equality of equations (22) and (26) along with suitable constraints on the function A(t) is a special case of Parseval's theorem. As the sign of the frequency has no physical meaning, it is now accepted that one need only integrate over positive frequencies. Then

(27)

defines the energy radiated per unit solid angle per unit frequency:

(28)

If A(t) is a real function, from equation (24) we can see that A(-Ï‰) = A*(Ï‰). This then becomes

(29)

Equation (29) relates the behaviour of the radiated power as a function of time to the frequency spectrum of the radiated energy. By combining the results above with the electric field created by an accelerated charge:

(30)

Where Î² and are the charged particle velocity and acceleration respectively. We now must calculate the Fourier transform, equation (23), of A(t), given by equation (21). This yields

(31)

where ret means is should be evaluated at t' + [R(t')/c] = t, the retarded time. Using this change of variable of integration and the assumption that the observation point is far away from the acceleration (the unit vector n stays constant in time) and the approximation

(32)

where x is the distance from an origin to the observer and r(t') is the particle position in relation to the same origin, equation (31) becomes

. (33)

Therefore, from equation (33) we can find that the energy radiated per unit solid angle per unit frequency [6,7] is

. (34)

## 4.2 Electron Motion

When considering the electron motion it will be in the case of an idealised model of wakefield acceleration. The wake of the laser pulse propagates through the underdense plasma and expels highly energetic electrons; this leaves an ion cavity behind. This cavity is assumed to be of radius , where a0 is the laser field strength parameter and Î»p the plasma wave period. The phase velocity of the cavity is assumed equal to the laser group velocity. Within the cavity, a strong electromagnetic field is created due to charge separation that drives the electron. The cavity accelerates the electrons longitudinally in its back half and decelerates them in its front half. The transverse element of the field focuses electrons toward the axis. In the model used, the electron motion is expressed as

(35)

where p is the electron momentum, m is the electron mass and Ï‰p is the plasma frequency. In the above expression, Fâˆ¥ is responsible for longitudinal acceleration along the path of laser propagation. The Fâ”´ term is responsible for the transverse oscillation across the cavity axis at the Betatron frequency

(36)

where Î³ is the Lorentz factor [8].

## 4.3 Analysis of Betatron Radiation Emission

Using codes supplied by Dr E. Brunetti, the betatron radiation could be analysed to assess the effect of changing fundamental parameters. We will consider the difference made to energy, electron spectra, electron trajectory and the angular distribution of radiation. The electron trajectory for the following analysis I given in Fig 2, from this we can see that there is no movement in the y-plane; the electron is propagating in the z-direction while oscillating in the x-plane.

C:\Users\Scott\Desktop\Project\thomson\betatron_basic_traj.png

Figure 2: Electron trajectory as set in program thb.cpp

As can be seen in Fig. 3 a very clear quasi-monoenergetic structure can be noticed with energies less than approximately 4 keV. Some energy spread can be seen as the energy increases towards 5 keV.

C:\Users\Scott\Desktop\Project\thomson\thb_basic.png

Figure 3: Betatron Emission spectrum from driving laser of intensity Iw=1x1018 Wcm-2 and wavelength, Î»= 800nm

The first parameter to be changed is the intensity of the driving laser. Fig 4 shows the spectrum for a driving laser of intensity of 1x1020 Wcm-2. We would expect this change to create a higher energy spike.

C:\Users\Scott\Desktop\Project\thomson\thb_10^20.png

Figure 4: Betatron Emission spectrum from driving laser of intensity Iw=1x1020 Wcm-2 and wavelength, Î»= 800nm

By comparing the spectrum for the two different intensities we can notice a much wider energy spectrum, but with a much higher peak in the spectrum. We can see the peak in the energy spectrum at approximately 2 keV compared to <1 keV as in fig 3. The next step is to decrease the intensity of the laser to 1x1016 Wm-2. Fig 4 has its peak in the soft x-ray region but has some divergence into the hard x-ray part of the electromagnetic spectrum.

Fig 5 shows a similar monoenergetic radiation spectrum with a peak 2 orders of magnitude lower than that of the spectrum generated by the 1x1018 Wcm-2 intensity laser-plasma interaction. It should also be noted that the energy spread of the spectrum spike has also decreased. From these graphs we can see that a higher intensity greatly increases the energy density of the spectrum but increases the energy spread.

C:\Users\Scott\Desktop\Project\thomson\thb_10^16.png

Figure 5: Betatron Emission spectrum from driving laser of intensity Iw=1x1016 Wcm-2 and wavelength, Î»= 800nm

The step in this reports analysis is to vary the wavelength of the driving laser, Î»w, to evaluate its effect on the radiation profile from the betatron mechanism. The intensity will remain constant at 1x1018 Wcm-2 to allow comparison with Fig 3. Fig 6 shows the radiation spectrum for a laser wavelength of 950 nm. As can be seen when comparing the graphs, the emission spectrum profile is extremely similar for both cases, except for a small increase in the peak in the spectrum to just over 1.4 x10-40 from 1.3 x10-40. Fig 7 shows the emission spectrum for a driving laser wavelength of 650nm, again with the same intensity of driving laser as in Fig 6 and Fig 3. Fig 7 shows a small decrease in the peak of the radiation spectrum as would be expected; also, the energy spread has stayed fairly constant throughout all changes in the driving laser wavelength.

C:\Users\Scott\Desktop\Project\thomson\thb_950.png

Figure 6: Betatron Emission spectrum from driving laser of intensity Iw=1x1018 Wcm-2 and wavelength, Î»= 950nm

C:\Users\Scott\Desktop\Project\thomson\thb_650.png

Figure 7: Betatron Emission spectrum from driving laser of intensity Iw=1x1018 Wcm-2 and wavelength, Î»= 650nm

From the above parameter changes, we can conclude that the intensity of the driving laser has a strong affect on the spectrum peak and spread, and the wavelength of the laser has a small effect on the peak of the spectrum but not on the energy spread.

Now that we have learned the effects of the laser parameters on the betatron emission spectrum, we can analyse the angular distribution. The original settings for the variables of the electron trajectory are the initial x position, x0=2, and both y0 and z0 are both zero (these are the original y and z positions respectively). The graph is created after the electrons have been through an accelerating potential over a length of approximately 0.5mm. Fig 8 shows the angular distribution of radiation for these initial conditions.

C:\Users\Scott\Desktop\Project\thomson\thb_basic_ad.png

Figure 8: Betatron Emission Angular Distribution from driving laser of intensity Iw=1x1018 Wcm-2 and wavelength, Î»= 800nm

We can observe from Fig 8 that the radiation is emitted of the axis of propagation. The most intense areas or radiation distribution are within a few mrad of the electron trajectory. These maximums are also located where the greatest changes in acceleration take place in the electrons motion. Now we will analyse the angular distribution of radiation from a source which initial position is on the origin, i.e. x0, y0, z0 = 0. This is shown in Fig 9.

C:\Users\Scott\Desktop\Project\thomson\thb_000_initpos.png

Figure 9: Betatron Emission Angular Distribution from driving laser of intensity Iw=1x1018 Wcm-2 and wavelength, Î»= 800nm and initial position x0, y0, z0 = 0.

The radiation in the above case is emitted in a cone shape around the electrons trajectory. This shows a maximum in the angular distribution at approximately 0.01 rad in a circular pattern. In addition, we can observe a drop of in radiation the more that the angle of observation diverges from the electron trajectory.

Now let us experiment with a more exotic electron trajectory. Initial parameters have been set to cause the electron to travel in a helical orbit. these initial parameters are x0=2Î¼m, y0=0, px0=0, py0=2, pz0=20 and an increased acceleration length of approximately 1mm. The trajectory is shown in Fig 10 as a three-dimensional plot.

C:\Users\Scott\Desktop\Project\thomson\traj_g_1e18_6.png

Figure 10: Electron trajectory from driving laser of intensity Iw=1x1018 Wcm-2 and wavelength, Î»= 800nm and initial parameters x0=2Î¼m, y0=0, px0=0, py0=2, pz0=20 and an increased acceleration length of approximately 1mm.

The radiation spectrum for this trajectory cannot be viewed properly in graph form due to its extremely close proximity to the spectrum axis. It has a peak of 1.15x10-38 on the spectrum axis at energy of 0.01 keV. This may be the case, or it is possible that the program did not analyse a wide enough frequency range to pick up all emissions. The angular Distribution for this exotic trajectory is in Fig 11.

C:\Users\Scott\Desktop\Project\thomson\traj_g_1e18_6.png

Figure 11: Betatron Emission angular distribution from exotic electron trajectory as outlined above.

From Fig 11 we can see that the radiation distribution follows the electron trajectory in a spiralling orbit. We can also notice that the radiation density changes with the path of the accelerated electron. The radiation becomes very intense as the electron gains energy, as the acceleration length was increased its impact can be readily noticed on the angular distribution map.

## 5 Compton Backscattered Radiation

## 5.1 Thomson Scattering as a Prelude to Compton Backscattering

To understand Compton Backscattering it is useful if we first discuss the Thomson Scattering phenomenon. In the Thomson case, a plane wave of monochromatic light is incident on a free particle of charge e and mass m. This interaction changes the acceleration of the free particle and therefore causes it to emit radiation. It should be noted that the emitted radiation would be in directions other than that of the incident wave, for nonrelativistic particle motion it will have the same frequency as the incident radiation.

The instantaneous power radiated into a polarized state Ïµ by a nonrelativistic particle of charge e is

(37)

The acceleration is caused by the interaction with the incident plane wave. The plane waves propagation vector is k0 and the polarization vector is Ïµ0, therefore the electric field can be written

Using the force equation of nonrelativistic mechanics, we have acceleration

(38)

Using the assumption that the charge moves a negligible part of a wavelength during a single cycle of oscillation, then the average power per unit solid angle can be expressed by

(39)

This process is viewed as simple scattering and therefore it is convenient to introduce a scattering cross-section of

(40)

The incident energy flux in equation (40) is the time averaged Poynting vector for the plane wave; therefore, we can derive the differential scattering cross-section as

(41)

The classical Thomson formulae are only valid for low frequency interactions, where the momentum of the incident photon is negligible. When the photons momentum becomes comparable to mc modifications occur. These are quantum-mechanical effects from the concept of photons have zero mass but have a momentum, though many of the modifications are kinematical. Compton observed the most important change, the energy or momentum of the scattered photon is less than the incident energy due to the recoiling of the charged particle in the collision. Using a two-body relativistic kinematical approach, we find that the ratio of the emitted to incident plane wave number is given by the Compton formula

where Î¸ is the scattering angle in the rest frame of the target [9].

In Compton Scattering, the basic effect relies on a double Doppler upshift of incident photons by relativistic electrons. For a head on collision the energy of scattered photons is

(42)

where is the relativistic factor of the electrons that are accelerated through some potential V, and Ï‰0 is the frequency of the incident photon. Here we can see the extreme frequency upshifts that are possible. For example, consider an incident photon from an 800nm Ti:Al2O3 laser which has an energy of 1.55 eV which is incident on a 50 MeV electron beam. This interaction would create 60 keV for head on collisions [1].

## 5.2 Analysis of Compton Backscattered Radiation Emission

As in the section on the analysis of betatron radiation, in this section we will alter key parameters to assess their effect on the profile of the radiation emitted. First, we will analyse a non-accelerating case for Compton Backscattering. Fig 12 shows the linear constant velocity electron trajectory propagating in the z-direction.

C:\Users\Scott\Desktop\Project\thomson\traj_g_1e18_6.png

Figure 12: Non-accelerating Electron trajectory from ths.cpp

Using the standard values in the program and setting the z component of the momentum, pz0=20, we constructed Fig 13 which has the radiation spectrum profile.

C:\Users\Scott\Desktop\Project\thomson\ths_10^14_pz.png

Figure 13: Compton Emission spectrum from scattering laser of intensity Is=1x1014 Wcm-2 and wavelength, Î»= 800nm

It can be seen in Fig 13 that the emission is extremely monoenergetic; the very fine peak of the spectrum displays this. In Fig 13, the counter propagating scattering laser has an intensity of 1x1014 Wcm-2 and a wavelength of 800 nm. Now we will study the effects that the changing the scattering parameters has on the spectrum.

Fig 14 shows a scenario where the scattering laser is of intensity 1x1020 Wcm-2. The energy remains constant between these two plots, but the spectrum increases by several orders of magnitude. It should be noted, that the width of the peak has remain incredibly small, showing excellent monoenergetic characteristics. But we also notice a second peak rising at a higher energy. This could be noise or the emergence of some sort of harmonic structure.

C:\Users\Scott\Desktop\Project\thomson\ths 10^20 SB.png

Figure 14: Compton Emission spectrum from scattering laser of intensity Is=1x1020 Wcm-2 and wavelength, Î»= 800nm

Now we will observe what a decrease in scattering laser intensity will do to the radiation spectrum profile. The intensity has been set to 1x1012 Wcm-2 and the resulting spectrum plotted in Fig 15. As is consistent with the result above, we observe a decrease in the intensity of the spectrum at the peak, but the energy has again remained constant. The next stage is to vary the wavelength of the scattering laser and to compare it with Fig 13.

Fig 16 shows the effect that an increase of wavelength to 950nm has on the spectrum profile. We can observe that the peak has remained at the same energy but the peak has increased by 2 orders of magnitude in the spectrum.

C:\Users\Scott\Desktop\Project\thomson\rad1e18_offa_6.png

Figure 15: Compton Emission spectrum from scattering laser of intensity Is=1x1012 Wcm-2 and wavelength, Î»= 800nm

C:\Users\Scott\Desktop\Project\thomson\ths_950 SB.png

Figure 16: Compton Emission spectrum from scattering laser of intensity Is=1x1014 Wcm-2 and wavelength, Î»= 950nm

The radiation spectrum was also plotted for a scattering laser wavelength of 650nm. to be compared to Fig 13 and 16. This is contained in Fig 17.

C:\Users\Scott\Desktop\Project\thomson\ths_650nm_pz.png

Figure 17: Compton Emission spectrum from scattering laser of intensity Is=1x1014 Wcm-2 and wavelength, Î»= 650nm

The peak in the spectrum is also 2 orders of magnitude greater than in the graph, Fig 13. However, it is also less than the peak in the case with a 950 nm scattering laser wavelength. As expected the energy of the peak has not changed and has remained the same through all changes of scattering laser wavelength. So it seems that there is not a directly proportional relationship between the wavelength and the spectrum. The emissions fit with the soft x-ray part of the spectrum.

The angular distribution of this Compton Backscattering is shown in Fig 18.

C:\Users\Scott\Desktop\Project\thomson\ad1e18_6.png

Figure 18: Compton emission angular distribution for initial parameters set out above.

We can observe here that there is a strong on axis radiation density; this would infer that the electron emitted the radiation in the direction of its propagation to a maximum on its own path. The plot quite similar to the pattern created by a radiating dipole system.

We will now consider an accelerating electron; the radiation spectrum is shown in Fig 19. We can observe a very low peak at a very low energy, also a gradual increase as the energy rise. However, this gradual increase could be due to or exaggerated by noise in the dataset.

C:\Users\Scott\Desktop\Project\thomson\rad1e18_offa_6.png

Figure 19: Compton Emission spectrum from scattering laser of intensity Is=1x1014 Wcm-2 and wavelength, Î»= 800nm

Fig 20 shows the angular distribution of the emitted radiation. It is of a similar pattern to that of the betatron discussed earlier. It shows a similar 'doughnut' formation around the axis of propagation and interaction with the scattering laser. An angular distribution plot was also taken with the initial conditions being given a small momentum in the y-component. This plot is in Fig 21. We can see from this plot that the emission distribution has completely moved away from the origin of the plot, also we can observe that the angular distribution of radiation is not uniform around the ring, but it is in fact more intense toward the origin.

C:\Users\Scott\Desktop\Project\thomson\ad1e18_6.pngFigure 20: Compton emission angular distribution for accelerating electron.

C:\Users\Scott\Desktop\Project\thomson\ad1e18_6.png

Figure 21: Compton emission angular distribution for accelerating electron with initial momentum in y direction.

## 6 Gamma Laser

With the increasing possible energies being created by laser-plasma acceleration techniques, it will soon be possible to create monochromatic light sources in the gamma section of the electromagnetic spectrum. Gamma rays generally have frequencies of >1019 Hz. Using equations already laid out in this report we will analyse the critical electron energy needed to create the gamma beam by Compton Backscattering.

By rearranging equation (42) we find that

We know the wavelength of the incident photons are 800nm in the experimental set up used for this report. By using the relation Ï‰0=2Ï€c/Î». We can derive

Now we substitute for known quantities

We can therefore solve for the gamma factor. This yields a gamma factor of

## .

To accomplish the idea of a gamma laser it would seem that electrons would have to be accelerated to energies of multiple GeV. This research is currently being carried out in several groups worldwide [10,11].

## 7 Conclusions and Future Work

The radiation mechanisms discussed in this thesis are both extremely promising for the creation of a highly tuneable, tabletop, x-ray and gamma source. As laser physics progresses, higher energies of accelerated electrons will be generated and used to create higher energy (and frequency) monochromatic light sources.

Through the above research, it is shown that the betatron radiation mechanism is much stronger than the Compton backscattering mechanism. When both are considered together, the betatron emission would swamp the Compton emission until it was lost or only resembled noise in the data analysis. The factor of 4Î³2 in the Compton backscattering regime is incredibly useful for massive upshifts in the frequency of emitted photons.

These techniques will be further developed and refined to create a useful tool in many branches of science and medicine as discussed in the introduction of this report.

Future work that can be carried out on this topic is the inclusion of different laser innovations, such as chirped pulses and Ramaan lasers, in an attempt to increase the energy of the emitted photons and radiation. The use of more powerful lasers will also make a huge difference to power achieved though the discussed methods. In addition, the pursuit of longer acceleration lengths without dephasing taking place would be a major area of research. This again would help to create higher energy interactions and improve the usefulness of such monochromatic light sources to industry.