Laser Interaction and Plasma Based Accelerators

Transcription

1 Laser Interaction and Plasma Based Accelerators - ١ -

2 Abstract Plasma is an attractive medium for particle acceleration because of the high electric field can be sustained by Plasma. Our objective in this thesis concentrate mainly to study the physics of particle acceleration by different methods like microwave radiation propagates in the waveguides and also like beating two intense lasers in plasma based accelerators. So, it has been of great interest to consider the following subjects: 1- The dynamics of an electron in the fields associated with transverse magnetic (TM) wave propagating inside rectangular waveguide is studied analytically. We have solved exactly the relativistic momentum and energy equations of a single electron which injected initially along the propagation of microwave. Expressions for the acceleration gradient and deflection angle are obtained. - The dynamics of an electron in the fields associated with TEelectromagnetic wave propagating inside a circular waveguide is analytically studied. The motion of this electron along the axis of the waveguide is investigated in the existence of a helical magnet (in which the field is perpendicular to the axis of waveguide and rotating as a function of position along the magnet). 3- The study of the beat wave plasma accelerator due to the interaction of two linearly polarized Bessel laser beams is investigated. The electron acceleration which driven by the generated longitudinal plasma waves with phase velocities near the speed of the light is studied. The wave equation descried the fields of this beat wave is obtained. - ٢ -

3 I. I.1. History of Plasma Physics: At the beginning of 0 th century, American scientists Irving Langmuir proposed that the electrons, ions and neutrals in an ionized gas could similarly be considered as corpuscular material entrained in some kind of fluid medium and called this entraining medium plasma. In the 190 s and 1930 s a few isolated researchers, each motivated by a specific practical problem, began the study of what is now called plasma physics. This work was mainly directed towards understanding (i) the effect of ionospheric plasma on long distance shortwave radio propagation and (ii) gaseous electron tubes used for rectification, switching and voltage regulation in the pre-semiconductor era of electronics [1]. In the 1940 s Hanns Alfven developed theory [] of hydromagnetic waves (now called Alfven waves) which proposed that these waves would be important in astrophysics plasmas. In the early 1950 s large-scale plasma physics based magnetic fusion energy research started simultaneously in the USA, Britain and the then Soviet Union where this work was an offshoot of thermonuclear weapon research. Fusion progress was slow through most of the 1960 s, but by the end of that decade the empirically developed Russian tokamak configuration began producing plasmas with parameters far better than the lackluster results of the previous two decades. International agreement was reached in the early 1 st century to build the international thermonuclear experiment reactor (ITER) designed to produce 500 Megawatts of fusion output power. Besides, inertial confinement schemes were also developed in which high power laser bombard millimeter diameter pallets of thermonuclear fuel. However, starting in the late 1980 s an important effort has been directed towards using plasmas in many practical applications. Plasma thrusters are now in use on some spacecraft and are under serious consideration for new and more ambitions spacecraft designs. Plasma processing a critical aspect of the fabrication of the tiny, complex integrated circuits used in modern electronic devices. Besides, plasma is an attractive medium for particle acceleration (Dawson 1989) [3] because of the high electric field it can sustain in a plasma based accelerator, particles gain energy from longitudinal plasma waves, with a phase speed close to the speed of light c in vacuum. - ٣ -

4 I.. Plasma Parameters: First and foremost, plasma is ionized gas. When a gas is heated enough that the atoms collides with each other and knock their electron off, plasma is formed (so called "fourth state of matter"). Three fundamental parameters characterize plasma: 1- The particle density n (measured in particles per cubic meter). - The temperature T of each species (measured in ev, where 1ev= 11,605 K). 3- The steady state magnetic field B (measured in tesla). A host of subsidiary parameters (e.g., Deby length, Larmor radius, plasma frequency, cyclotron frequency, thermal velocity) can be derived from these three fundamental parameters [4]. Exactly, the transition between a very weakly ionized gas and plasma occurs when the following conditions are satisfied: 1- The Debye shielding length λ D λ ( KT /4 e n) 1/ D = π = 6.9 (T/n) 1/ cm ( T in Kelvin) - The no. of charged particles inside the specific region must be more than one, i.e., N D >>1, where: 4 3 N D = π nλ D 3 3 / T = 1380 (T inκ ) n 1 / 3- The frequency of the plasma oscillations (ω) must be more than collision frequency with neutral atom (υ, ω > υ - ٤ -

5 Generally, it could be noticed that there are some typically parameters density and temperature of naturally occurring and laboratory plasmas are illustrated in figure (I-1). Figure (I-1). Naturally occurring and laboratory plasmas. - ٥ -

6 I.3. Wave Plasma Interaction and Generation A large effort made to study plasma waves and generations are mainly due to their importance for the development of plasma dynamics and plasma confinement in fusion research. Generally speaking, by means of waves and generation, we can feed plasma with energy, and to study waves gives us important diagnostic tools for determining plasma composition, studying the atomic processes of ionization, excitation and recombination phenomena for plasma heating. The interaction of electromagnetic waves with plasma is very important when considering the reflection or absorption and amplification of radio waves and stability of the device based on the use of the charged particle beam. In addition, this interaction plays also an important role for other effects such as breakdown, self focusing, trapping, instabilities, etc, and various nonlinear effect. Recently the study of nonlinear interaction of powerful electromagnetic waves with fully ionized plasma becomes very urgent. First of all, because it is connected with some new method of particle acceleration [5] and also with the possibility of heating [6,7] and compression of plasma in laser driven fusion devices [8]. At the interaction of the intense electromagnetic waves with medium, the latter become inhomogeneous optically, the polarization vector, dielectric constant, the index of reflection and other values, characterizing physical properties of the medium, become dependent of the amplitude of the incident wave. Powerful radiation changes the medium physical properties, that in its turn affect the wave propagation, i.e., there is self-effect of the wave. Until other nonlinear effects such as harmonic generation, the stimulated (Raman and brillioun) scattering and parametric process where the interaction take place at slightly different frequency does not change in the process of self effect, and the effect is observed in the change of its amplitude, polarization, shape of the angular and frequency spectrum. In spite of that technique for study of nonlinear wave interaction and wave generation in the plasma is too complex, large numbers of investigation and research have been devoted to these studies [3, 4, 9, 10]. Linear theory shows that electromagnetic waves with frequency less than the electron wave plasma cannot propagate in unmagnetized plasma. However, for large field intensities, nonlinear effects [11] such as the relativistic electron mass variation [1] and the ponderomotive force [11] can lead to a downshift of the local electron plasma frequency. This result in the possibility of the electromagnetic wave energy penetration into the overdense region as defined classically. Phenomena such as this are very important in the studies of laser pellet interaction, the RF heating of magnetically confined fusion plasmas, the laser induced modification of the ionosphere as well as some wave-driven particle accelerators. - ٦ -

7 Also, it is well known that a plasma is bounded leads to a series of specific properties connected with the propagation of the waves; discrete frequency spectrum for the natural modes of the system and the possibility of surface waves. The radiated modes emitted from a plane plasma layer in the absence of external magnetic field have been considered by T.A. Davydova and A. A. Zhmudskii [13]. As a result of the energy loss by radiation the frequency are complex and it means that the waves are damped in the layer. - ٧ -

8 I.4. Microwave Waveguides In general, waveguide is a structure of a special form of transmission line consisting of a hollow metallic tube (rectangular or circular shape) used to guide an electromagnetic wave. Waveguides are used principally at frequencies in the microwave range; inconveniently large guide would be required to transmit radio- frequency power at longer wavelengths. Figure (I.): Wave guides (rectangular or circular shape) conduct microwave energy at lower loss than coaxial cables Waveguides are practical only for signals of extremely high frequency, where the wavelength approaches the cross-sectional dimensions of the waveguide. Below such frequencies, waveguides are useless as electrical transmission lines. Waveguides differ in their geometry which can confine energy in one dimension such as in slab waveguides or two dimensions as in fiber or channel waveguides. In addition, different waveguides are needed to guide different frequencies: an optical fiber guiding light (high frequency) and will not guide microwaves (which have a much lower frequency). As a rule of thumb, the width of a waveguide needs to be of the same order of magnitude as the wavelength of the guided wave. There are structures in nature which act as waveguides, for example, a layer in the ocean can guide whale song to enormous distances. Also, there is a plasma waveguides which used to guide laser beams through the plasma in order to achieve a lot of applications like electron acceleration. It is possible to propagate several modes of electromagnetic waves within a waveguide. These modes correspond to solutions of Maxwell s equations for particular waveguide. A given waveguide has a definite cutoff frequency for each allowed mode. If the frequency of the impressed signal is above the cutoff frequency for given mode, the electromagnetic energy can be transmitted through the guide without attenuation. Otherwise the electromagnetic energy with a frequency below the cutoff frequency for that particular mode will be attenuated to a negligible value in a relatively short distance. The dominant mode in a particular guide is the mode having the lowest cutoff frequency. So, it - ٨ -

9 is advisable to choose the dimensions of a guide in such a way that, for a given input signals, only the energy of the dominant mode can be transmitted through the guide. I.4.1. Waveguide Classification: Waveguides are considered essentially coaxial lines without center conductors. They are constructed from conductive material and may be rectangular, circular or elliptical cross section, as shown in Fig. (I-3).The conducting walls of the guide confine the electromagnetic fields and thereby guide the electromagnetic wave. So, a number of distinct field configurations or modes can exist in waveguides. They can be constructed to carry waves over a wide portion of the electromagnetic spectrum, but are especially useful in the microwave frequency ranges (microwave waveguide) and optical frequencies (optical waveguide) depending on its dielectric materials. Also, an acoustic waveguide is a physically structure for guiding sound waves. A plane wave in a waveguide resolves into two components one standing wave in the direction normal to the reflecting walls of the guide or at one traveling wave in the direction parallel to the reflecting walls. In lossless waveguides the modes may be also classified as either transverse electric (TE) mode or transverse magnetic(tm) mode. Figure (I.3): Waveguide shapes I.4.. Microwave: - ٩ -

10 Microwave is a descriptive term used to identify electromagnetic waves in the frequency spectrum ranging approximately from 1 Gigahertz (10 9 hertz) to 30 Gigahertz. This corresponds to wavelength from 30 cm to 1cm. Sometimes higher frequencies (expanding up to 600 GHz) are also called (Microwaves). These waves present several interesting and unusual features not found in order portion of electromagnetic frequency spectrum. These features make microwaves uniquely suitable for several useful applications. The main characteristic features of microwaves originate from the small size of wavelengths (1 cm to 30 cm) in relation to the sizes of components or devices commonly used. Since the wavelengths are small, the phase varies rapidly with distance; consequently the technique of circuit analysis and design, of measurements and of power generation, and amplification at these frequencies are distinct from those at lower frequencies. Study and research in microwave hasn t been as interesting and challenging academic. It has led to several useful applications in communication in radar, in physical research, in medicine and in industrial measurement and also for heating and drying of agricultural and food products. A significant advantage associated with the use of microwaves for communications is their large bandwidth. A ten per cent bandwidth at 3 GHz implies availability of 300MHz spectrum. This means all the radio, television and other communications that are transmitted in frequency spectrum from DC to 300 MHz can be accommodated in a ten per cent bandwidth around 3GHz (say from 8.50 to 31.50MHz). Since the lower frequency part of radio spectrum is getting crowded, there is a trend to use more and more of microwave region (and beyond) for varies different services. Presently, microwave communications are widely used for telephone networks, in broadcast and television systems and in several other communication applications by services, railways,..etc. - ١٠ -

11 I.5. Particle Acceleration: The charged particle acceleration has a subject of great interest to the research community due to its diverse applications in the field of nuclear physics, thermonuclear fusion research, coherent harmonic generation and high-energy particle physics. All over the world various efforts [14] have been made for achieving higher acceleration gradients for the particle acceleration, plasma beatwave acceleration, laser wakefield acceleration etc. The researchers have made theoretical as well experimental attempts for the particle acceleration [14-5]. In 1950 McMillan [15] had used the electromagnetic wave field for explaining the possible origin of cosmic rays. Later Chan [16] showed that an abnormally large amount of energy can be transferred from the radiation to relativistic charged particle when it interacts with a laser beam moving almost together in same direction. In direct acceleration scheme, McKinstrieand Startsev [17] have shown that the pre-accelerated electron can be accelerated significantly, but they neglected the effect of longitudinal field of the laser pulse. Lu et al. [18] have studied the electron motion in electromagnetic field of a short pulse high- intensity laser in the vacuum for the electron acceleration. Taking the effects of longitudinal and transverse laser field, Du and Xu [19] have studied the direct acceleration of positron by the laser, and significant acceleration has been noted even for the particle at rest. The properties of the waveguides that they can guide the electromagnetic fields for very long distances have been extensively used for the purpose of particle acceleration. However, in view of low cost structures and significant acceleration, it is prefer to use microwave fields for the particle acceleration in place of ultrahigh lasers. r Electron acceleration has been realized via v B r process [6,7] where an electrostatic wave (e.g., electron plasma wave) propagates in direction perpendicular to a magnetic field B r r and an electron that traps in the wave trough gets accelerated in the vp B r direction. In these experiments, the electrons could be accelerated up to 400 ev. Plasma as a medium for particle acceleration has a number of advantages. It has no electrical breakdown limit like convential accelerating structures which are limited to maximum field strength of less than 1 MV cm -1. A plasma supports longitudinal plasma 1 / waves which oscillates at the plasma frequency ω = 4πn e / m ) where n o and m e are p p ( o e the electron number density and the mass, respectively. In these waves the plasma electrons oscillate back and forth at ω irrespective of the wavelength. Therefore, these waves can p have arbitrary phase speed, ν ph ; relativistic plasma waves have ν ph c. Plasmas have been known for a long time to support space charge waves, waves in which the electron and ions densities are not equal. The electrons oscillate back and forth about the neutrality condition (the ion motion is negligibly small). These oscillations have the remarkable property that they all have essentially the same frequency ( ω = 4πe n / m ) independent of their wavelength. Thus the phase velocity p o e ( = λυ = ω / k) can take on any desired value by making an appropriate choice of the V ph wavelength. In particular the phase velocity can be made an equal to the speed of light which is ideal for accelerating particles to high energy. The fact that the frequency is - ١١ -

12 independent of wavelength lead to another interesting aspects, the group velocity ( = ω / k) of the wave is zero. This implies that once the wave is established, it will V g not propagate away; it remains where produced until it damps or some nonlinear process destroys it. Thus, a relativistic electron injected in the traveling wave will stay in phase with an accelerating field for a long distance as in Fig. (I.4) Figure 4. An electron plasma wave. Upper curves: representation of the electron density, ion density, the electric field E and the restoring force F Lower curves: the electric field at three successive times showing the propagation at the phase velocity. The space charge electric field E is responsible for the electron oscillation at the plasma frequency.depending on the wavelength of the plasma wave; the phase velocity can be relativistic. - ١٢ -

13 I.6- Plasma Based Accelerators: Laser-driven plasma-based accelerators were originally proposed three decades ago by Tajima and Dawson 1979 [8] where particle gain energy from a longitudinal plasma wave. To produce relativistic particle beams, the plasma waves have to be sufficiently intense, with a phase speed close to speed of the light in vacuum. Dawson [9] made many of the early developments in this field beside he was one of the early pioneers of particle-in-cell (PIC) simulation of plasmas which is now a widely used tool in the study of plasma-based accelerators [30, 3]. Particle acceleration by relativistic electron plasma waves has been demonstrated in a number of experiments (Clayton et al. (1993) [33] and Everett et al. (1994) [34] ). In these experiments, the maximum accelerating field gradient E was limited by wave breaking, which occurs for a cold plasma when the plasma wave density perturbation δn equals the mean plasma number density n o. Recently, three groups (Faure et al. 004 [35] ; Geddes et al. 004 [36] ; Mangles et al. 004 [37] ) reported the production of high quality electron bunches characterized by significant charge (> 100pC) at high energy (100 MeV) with small energy spread (approximately few percent) and low divergence (approximately a few milliradians). These high-quality electron bunches were a result of a higher degree of control of the laser and plasma parameters, an improvement of diagnostic techniques, an extension of the laser propagation through the plasma, and a greater understanding of the underlying physics, in particular, the importance of matching the acceleration length to dephasing length. Much of this growth is also due to the rapid development of chirped-pulse amplification (CPA) laser technology. Laser-driven plasma based accelerators, which are capable of supporting fields in excess of 100 GV/m, are reviewed by Esarey E. and et al. [38]. They gave an overview of some of the concepts and schemes relevant to the most widely investigated plasma based accelerators, namely the plasma wakefield accelerator (PWFA), the plasma beat- wave accelerator (PBWA) and the laser wakefield accelerator (LWFA). Accelerating gradient is an important parameter but is not the only parameter needed to make a successful acceleration; luminosity and emittance are two other parameters that have to match or better conventional accelerators. For example the next linear colliders being planned will have Luminosity in the region cm s which is beyond the present capability of plasma accelerators. Plasma accelerators are ideal at providing a compact, short pulses accelerators, they may also be useful at increasing the energy of conventional accelerators using the afterburner concept (Joshi and Katsouleas 003) [39] ). - ١٣ -

14 Here, we intend to give an overview of some of the concepts and schemes of the plasma wakefield accelerator (PWFA), the plasma beat- wave accelerator (PBWA) and the laser wakefield accelerator (LWFA): I.6.1- Laser Wakefield Accelerator: As it was mentioned above, plasmas are capable of supporting large amplitude electromagnetic waves with phase velocities near the speed of light. Such waves can be used to accelerate electrons. In the laser Wakefield accelerator (LWFA)[18], a 18 single short ( 1 β s) ultrahigh intensity ( 10 W / cm ) laser pulse drives a plasma wave. Wakefield is driven most efficiently when the laser pulse length L= cγ L is approximately the plasma wavelength λp = πc / ωp, i.e., L λp. (LWFA) was first proposed by Tajma and Dawson [5] and stimulated one dimensional (1-D) (particle in cell) by Siullivan and Godfreq [18] and by Mori [19]. Prior to 1988, the technology for generating ultra-intense picosecond laser pulses did not exist and only (PBWA) concept appeared feasible (which relied on long pulses of modest intensity). (LWFA) was later reinvented independent by Gorbnnov and Kirsanov [0] and by Sprangle et al [1]. This roughly coincides with the time when chirped pulse amplification (CPA) was applied to compact solid state lasers and a table top Terawatt laser system. In laser driven plasma based accelerators, Wakefield are driven via the pondermotive force. The pondermotive force can be derived by considering the electron momentum equation in the cold fluid limit [] r p r r r r r + ( ν. ) p= e[ E+ ( ν B) / c]. t The electric and magnetic fields of the laser can be written as : r r A E= c t and B r = A r where the vector potential of the laser is polarized pre-dominately in the transverse direction (linearly polarized), e.g., A r = A cos( kz ωt) e 0. Defining the normalized vector potential as a = ea mec in the linear limit a = e A m c e << 1, the leading order electron motion is the quiver velocity V g = c am e as indicated by ( / t) = ee. V g Let v= v g + δv the second- order of motion is given by : - ١٤ -

15 dδp dt = m [( ν. ) ν + cν ( a)] e q q q = m c ( a e ) Hence, F m c a p = e ( / ) is the pondermotive force in linear limit ( a << 1). So, as an intense laser pulse propagates through an underdense plasma, λ / << 1. The pondermotive force associated with the laser pulse envelope, is equal λ p Fp a expels electrons from the region of the laser pulse. If the length scale L z of the axial gradient in pulse profile is approximately equal to the plasma wavelength, L ~ λ, the pondermotive force excites a large amplitude plasma wave (Wakefield) z p with a phase velocity approximately equal to the laser pulse group velocity [1, ]. The laser strength parameter is given by: e λoi 1 / 9 1 / ao = ( ) I [ w/ cm ] λ [ m] 5 o µ πme c Where, I is the laser peak intensity and λ o = πc/ω o is the laser wavelength with frequency ω o.the amplitude of the transverse electric field of linearly polarized laser is:, E TV m a λ µ m I W cm e.g., I =10 18 W/cm, gives E L =.7 TV/m 9 1/ L[ / ] 3. o / [ ].7 10 [ / ] Hence, the wakefield amplitude can be increased by operating at high densities and shorter pulse length. I.6.. Plasma Wakefield Accelerator: Plasma based accelerators in which the plasma wave is driven by one or more electron beams are referred to as plasma wakefield accelerator (PWFA). A series of experiments at Stanford linear accelerator centre (SLAC) has demonstrated successfully the beam driven PWFA (Lee et al. (001); Blue et al. (003); Hogan et al. (005)). In the plasma wakefield accelerator (PWFA), plasma waves are excited by the space charge force of the drive electron beam. An electron beam creates a wake like displacing water. In the plasma case, the massive ions are relatively immobile and provide the restoring force (a space charge force of a beam ) to the displaced electrons. Provided that the electron beam terminates in a time τ f short compared to the plasma period, τ fω f < 1, the electron beam will excite a plasma wave. The wake phase velocity (ω/k) is the speed of the driver approximately equal c. Thus a - ١٥ -

16 second group of relativity particles can surface on the wake and remain in phase with it for some distance. It has been investigate that the physical origin of the plasma wave in the PWFA is the space charge force associated with the drive electron beam. When the electron beam propagates into a uniform plasma, n = n o, where n is the plasma electron density, the beam density n b generates a space charge potential via Poisson s equation, E max = mc ωδ p / e, where K p = ωp / c the resulting space charge force F03 = me c ϕ can drive a plasma wakefield. The plasma electron will respond to cancel a space charge potential of the beam. Hogan et al. (005) have demonstrated multi-gev energy gain in a 10 cm long plasma using electron beams. In the beam driven wakefield experiment the beam energy is transferred from a large number of particles in the core of the bunch to a fewer number of particles in the back of the same bunch. The wakefield thus acts like a transformer with a ratio of accelerating field to decelerating field of about 1.3 in the experiment. I.6.3. Plasma Beat Wave Accelerator The most mature plasma acceleration scheme is the beat-wave accelerator, first proposed by Tajima and Dawson [6]. In the (PBWA) a relativistic plasma wave is generated by pondermotive force of two lasers separated in frequency by amount equal to the plasma frequency, that the energy and momentum conservation relations are satisfied, viz. ω= ω1 ω = ωpe and k k 1 k, where ( ω 1,, k1,) are the frequencies and wave numbers of the two lasers, respectively, and k p is the plasma wave number. The beat pattern can be viewed as a series of short light pulses each πcω p long moving through the plasma at the group velocity of light which for ω1, ω p is close to c. the plasma electrons feel the periodic ponderomotive force of these pulses. Since this frequency difference matches the natural oscillation frequency of the electron plasma waveω, the plasma responds resonantly to the ponderomotive p force and large amplitude plasma waves would be build up. υ ph If ω1, ω p then the phase velocity of the plasma waves = ω = ( ω ω ) /( k k ) = / k equals the group velocity of the laser p / k p 1 1 ω 1 / beams ν g = c( 1 ωp / ω1, ) which is almost equal to c in an under dense plasma. Particles that are injected into the beat wave region with a velocity comparable to the phase velocity of the electron plasma waves, can gain more energy from the - ١٦ -

17 longitudinal electric field. Since ω 1 is close to ω and much larger than ω p, the Lorentz factor δ p associated with the beat waves is δ p 1/ ν ph ω1, 1 1. c ω p = = The beat wave process is related to stimulate Raman forward scattering (SRFS). Stimulated Raman scattering is the terminology used in plasma physics for the scattered electromagnetic wave propagates in the same direction as the incident electromagnetic wave we refer to this as forward scattering. Electron plasma waves are also sometimes referred to as Langmuir waves after E. Langmuir, who was the first to discover them. The general equations describing the beat wave and SRFS are similar. It is sufficient to analyze the problem of plasma wave growth and saturation using the relativistic fluid equations for electrons, and Maxwell and Poisson equations. Experiments have been conducted using microwaves, CO lasers and glass lasers as the drive beams. In the experiments the plasma wave was not driven to its limiting relativistic saturation level due to the growth of modulation instabilities which have a growth rate determined by the ion plasma frequency. It was pointed out by Tang et al. (1984) that by deliberately allowing for the relativistic mass variation effect [49] and having a denser plasma such that the plasma frequency was initially larger than the laser frequency difference the plasma wave would come into resonance as it grew, allowing a larger maximum saturation value to be attained. An increase of about 50% in the saturated wave amplitude can be achieved by this technique. The longitudinal field amplitude of these relativistic plasma waves can be extremely large with a theoretical maximum obtained from Poisson equation, and it is given by An important consideration in the beat wave scheme is to have sufficiently intense lasers such that the time to reach saturation is short compared to the ion plasma period. When the time-scale is longer than the latter, the ion dynamics becomes important and the electron plasma wave becomes modulationally unstable by coupling to low-frequency ion density perturbations (Amiranoff et al. 1995) [50]. - ١٧ -

18 I.7. Previous Work: The dynamics of charged particle (an electron, for example) in the fields of the electromagnetic wave is one of the basic problems in plasma physics. This subject has great interest due to its applications in the fields of laser particle interactions, thermonuclear fusion, high energy particle physics, etc. Here we presented a number of studies were dedicated to the electron acceleration by the interaction of the laser or microwave field. There are some attempts in waveguides for the wakefield excitation by electron bunches [51-53] the lasers have been extensively used for the purpose of particle acceleration, and noticeable amount of energy gain has been achieved for the electrons and protons. These types of systems cost very high and very sophisticated instrumentation is also required for the specific/accurate measurement on ultra fast time scale. However, in view of low cost structureand significant acceleration by moderate intensity microwaves, the wakefield excited by Gaussian-like microwave pulse in a rectangular waveguide filled with homogeneous plasma is investigated by Aria A. K. and Malik H. K. [54]. They showed the effects of microwave frequency, waveguide width and microwave intensity on the wakefield. The amplitude of the wakefield is found to increase with microwave pulse duration and its intensity, but it gets decreased with microwave frequency and waveguide width. By optimizing various parameters, they have achieved wakefield of appreciable strength for the purpose of particle acceleration. Pantell and Smith [55] have analyzed the interaction of a slow microwave signal and a laser light beam. They calculated the energy gradient for an electron as a function of fields of both the electromagnetic waves. Electron acceleration under the collisionless condition by the oscillating electric field is investigated in an experimental system [56], where electrons were injected from a tip of electron source along the magnetic field between two electrostatic potentials. The energy gain by an electron during its motion in the field of a fundamental TE 10 mode excited by a microwave in a rectangular wave-guide is analyzed by Malik [57]. He also studied the particle acceleration due to the two fundamental TE 10 modes interfere obliquely and obtained the expressions for the energy gain and the acceleration gradient [58]. Besides, the microwave breakdown threshold in a circular waveguide excited in the lowest order TE 11 - mode has been investigated by Tomala et al. [59]. Jawla et al. [60] have also studied the electron acceleration and evaluated the fields for fundamental mode in a waveguide filled with plasma under the effect of external magnetic field. However, some of the researchers have made theoretical as well experimental attempts for the particle acceleration by using microwave radiation [19, 55,61, 6 ]. The description of the electron dynamics in a high frequency field is complicated - ١٨ -

19 because of the large number of oscillations. The problem becomes more intricate analytically with the addition of an extra- electric or magnetic field. Pioneer experiments are reported by R. B. Yoder, T.C. Marshall, and J. L. Hirshfield [6] on inverse free-electron-laser acceleration, including for the first time observations of the energy change as a function of relative injection phase of the electron bunches. The microwave accelerating structure consists of a uniform circular waveguide with a helical wiggler and an axial magnetic field. Acceleration of the entire beam by 6% is seen for 6 MeV electron bunches at optimum relative phase. Experimental results compare favorably, for accelerating phases, with predictions of a three-dimensional simulation that includes large-orbit effects. Chunguang Du [63] has analytically studied the motion of charged particle in the fields associated with a circularly polarized laser pulse propagating in a plasma. In contrast to an electron, a positron can be accelerated by the leading edge of the pulse to very high energy, even if it is initially at rest. Since the beat-wave scheme was first proposed as a possible Electron accelerator, it has generated much interest. The usual configuration of a plasma beat-wave accelerator (PBWA) consists of two parallel laser beams that resonantly excite a plasma wave in a background plasma. The plasma density must be precisely controlled in order to fulfill the resonant condition. J. Faure et al. [36] have presented experimental results on electron acceleration using two counter propagating ultrashort and ultraintense laser pulses. At the collision, the two pulses drive a standing wave which is able to pre- accelerate plasma electrons which can then be trapped in plasma wave. Zhiguo Zhao and Baida Lu have studied the direct acceleration of electrons by using two crossed linearly polarized Bessel beams with equal frequency and amplitude in vacuum is studied and compared with the case of single linearly polarized Bessel beam. It is found that two zeroth- and firstorder Bessel beams with p-rad phase difference have a nonvanishing longitudinal electric field on the z-axis, which can be maximized under certain conditions and used to accelerate electrons. Changbum Kim, et al. [ ] have studied two dimensional simulation are performed for modified laser wakefield acceleration. After one laser pulse, another identical laser pulse is sent to the plasma to amplify the wake wave resonantly. The simulation results show that the number of injected electrons is bigger than that of the single pulse case and the beam energy is higher as well. In addiation, increase of the transverse amplitude is noticed in the wake wave after the second laser pulse. - ١٩ -

20 This shows that the transverse motion of the wake wave enhances the wave breaking for strong injection and acceleration of the electron beams. Salamin Y. I. [ ] has studied the energy gain for single electron injected sideways into the focal point of a tightly focused laser beat wave, all results are obtained from solving the relativistic equations of motion numerically. Through numerical modeling of the test particle motion of an ensemble of relativistic electrons in the combined longitudinal plasma wave and transverse waves of the two PBWA laser beams, Petrzilka V. and Krlin L. [ ] show in the present contribution that a strong enhancement of electron acceleration arises if an additional perpendicularly propagating transverse laser beam is present. They show that electrons are also strongly accelerated in such PBWA configurations, where a significant acceleration by the plasma wave would not occur without the presence of the additional laser beam. In 004, a modified version of the plasma beat-wave accelerator scheme is proposed by Lindberg R.R..Charman A.E., and Wurtele J.S., based on autoresonant phase locking of the Langmuir wave to the slowly chirped beat frequency of the driving lasers by passage through resonance. Peak electric fields above standard detuning limits seem readily attainable, and the plasma wave excitation is robust to large variations in plasma density or chirp rate. This scheme might be implemented in existing chirped pulse amplification or CO laser systems. References Petrzilka V. and Krlin L., 5th Workshop on fast Ignition of Fusion Targets Funchal,, 18- (001) R. R. Lindberg,* A. E. Charman, and J. S.Wurtele, Physical Review Letters,V. 93, No 5 (004) - ٢٠ -

21 I.7. Present Work: The present work gives, in chapters II IV, a detailed analysis of the dynamics of an electron inside the waveguide and the possibility to accelerate it. Besieds, the acceleration of the electron depending on plasma beat-wave scheme (PBWA) is investigated as follow: In chapter II: The dynamics of an electron in the fields associated with transverse magnetic (TM) wave propagating inside rectangular waveguide is studied analytically. We have solved exactly the relativistic momentum and energy equations of a single electron which injected initially along the propagation of microwave. Expressions for the acceleration gradient and deflection angle are obtained. In principle, it is shown that the energy of the electron can be accelerated in this environment and there is no deflection when the electron is injected from the centre of waveguide front. However, it is found that, the acceleration gradient and deflection angle are strongly depended on the parameters of microwave (intensity, frequency etc) and the dimensions of waveguide. In chapter III: The dynamics of an electron in the fields associated with TEelectromagnetic wave propagating inside a circular waveguide is analytically studied. The motion of this electron along the axis of the waveguide is investigated in the existence of a helical magnet (in which the field is perpendicular to the axis of waveguide and rotating as a function of position along the magnet). It is shown that it can be accelerated due to its interaction with polarized fields of microwave radiation propagating along the waveguide. The fields for the lowest order TE 11 -modes and the deflection angle of electron trajectory, due to these fields, are obtained. Also, An expression of the acceleration gradient of the electron and its energy gain are evaluated for different intensities and frequencies of the microwave. In chapter IV: By using MHD model, the generation of longitudinal plasma waves in relativistic underdense plasma between accelerators is studied. - ٢١ -

22 The relativistic electron fluid eqns describing driven electron acceleration with phase velocities near the speed of the light in cold, collisionless plasma are reduced to single, approximate ordinary differential equations of a parametrically excited nonlinear oscillator. The model has used two linearly polarized Bessel laser beams to describe the generation of the langmuir wave due to the interaction of the beams. Chapter V : The results and conclusions of chapters II IV are summarized. The major part of the work included in this thesis has published in the following: 1- Electron Acceleration by Microwave Radiation Inside Rectangular Waveguide B. F. Mohamed and A. M. Gouda Sent to be published in J. Plasma Science and Technology - Electron Dynamics in Presence of Static Helical Magnet inside Circular Waveguide B. F. Mohamed, A. M. Gouda and L. Z. Ismail Accepted to published in IEEE Transactions on Plasma Science. - ٢٢ -

23 CHAPTER II ELECTRON ACCELERATION BY MICROWAVE RADIATION INSIDE RECTANGULAR WAVEGUIDE - ٢٣ -

24 CHAPTER II ELECTRON ACCELERATION BY MICROWAVE RADIATION INSIDE RECTANGULAR WAVEGUIDE The dynamics of a charged particle (an electron, for example) in the fields of the electromagnetic wave is a basic problem in plasma physics. This subject has been a great interest due to its applications in the fields of laser particle interactions, thermonuclear fusion, high energy particle physics, etc. Recently, considerable progress has been achieved to the problem of acceleration of the charged particle to the high energy [1]. Different mechanics could be used for achieving high energy gain through the direct acceleration by the fields of electromagnetic waves, plasma or laser wakefield acceleration and beat wave acceleration. The direct acceleration scheme, as in microwave plasma interaction experiments, has been realized by the Lorentz force. A number of studies were dedicated to the electron acceleration by the interaction of the laser and microwave fields [-5]. However, the properties of the waveguides that they can guide the electromagnetic fields for very long distances have been extensively used for purpose of the particle acceleration. Also, in view of low cost structure and significant particle acceleration, it is better to use microwaves in place of ultrahigh lasers. So, in the present chapter, we have investigated the dynamics and the acceleration of an electron in the fields associated with TM 11 modes which excited by the microwave propagates along a rectangular waveguide. The expressions for the acceleration gradient and the energy gain are obtained which are affected by the dimensions of the waveguide and the parameters of the microwave. - ٢٤ -

25 II.1. The Dynamics of an Electron in the Fields of TM 11 Modes A hollow metallic tube (waveguide) with rectangular cross section of the width a along x-axis and height b along y-axis is assumed. The propagation of the microwave radiation of the frequency (ω) is made use to excite TM mn modes inside the waveguide. The field components associated with the mode can be obtained -iwt from Maxwell equations (with time dependence as e ): uv ω uv + = c E 0 (II- 1) uv ω uuv + = c H 0 (II- ) However, the TM mn modes in rectangular waveguide are characterized by H z = 0 and the energy transmission in the guide is done by z- component of the electric field. Consequently, equations (II-1) and (II-) are solved by considering the boundary conditions that the tangential component of the electric field E 0 and the normal component of the magnetic field must vanish on the conducting surface. So, the field components of TM 11 are: iπβ g Ex = E cos ( / )sin( / ) o π x a π y b e akc β i g z iπβ g E y = E sin ( / )cos( / ) o π x a π y b e bkc iβ g z E = E sin ( π x / a)sin( π y / b) e z o iβ g z (II- 3) iπωε H x = E sin( / )cos( / ) o π x a π y b e b kc iβ g z iπωε H y = E cos( / )sin( / ) o π x a π y b e akc iβ g z In this situation, ω = π f and f > f c with k c and f c are the wave number and frequency corresponding to the cutoff condition, f = + = ω µ ε (II- 4) 1/ c ( ), k c c µ a b 0ε 0-٢٥ -

26 Also, β g is the propagation constant which given by amplitude constant associated with the field mode. ω π π β g = and E 0 is the c a b It can be seen from equation (II-4) that the cutoff frequency f c depends on the waveguide width a and height b. This shows that the frequency of the mode could be propagated in a waveguide of dimensions, for example a= 4 cm and b=.5 cm should be bigger than 7.15 GHz. Previous theoretical studies of the electron accelerations in waveguide have considered the fundamental TE 01 mode excited by a microwave either in an evacuated rectangular waveguide [6] or in a waveguide filled with plasma [3]. Now we are going to investigate the dynamics of an electron in the fields of another mode (TM 11 mode). Let the electron is injected along the direction of the propagation of the mode inside the waveguide. In the first, we calculate resultant electric and magnetic fields in the plane perpendicular to the direction of propagation of the mode. The resultant electric field is obtained as: 1/ g iβ tan ( / ) tan ( / ) g z x y y b x a cos cos o c πβ π π π π E = i E e + k a b a b (II- 5) And also the resultant magnetic field in the transverse plane is 1/ iβ tan ( / ) tan ( / ) g z x y y b x a cos cos o c πωε π π π π H = i E e + k a b a b (II- 6) These resultant fields made angles θ 1 and θ respectively with the x-axis: a π x π y tanθ 1= tan cotan b a b (II- 7) b π x π y tanθ = cotan tan a a b (II- 8) The angle (α), at which the electron gets deflected from the z-axis, can be evaluated from the following electron momentum equation: r dp r r r = e( E+ µ 0 v H ) d t (II- 9) - ٢٦ -

27 Taking into account the resultant field components in the transverse component of the electron equation of motion, d p d z= ( e / ν z )[ E + µ o( v H) ], it has been integrated (by considering ν to be slowly varying quantity) and finally we obtain: z p tanα = p z 1/ eπ π z cos ( π x / a)sin ( π y / b) sin ( π x / a)cos ( π y / b) E cos( ) o e cν z λg = + m k a b ωεν zµ o sin( π x / a)cos( π y / b) cos( π x / a)sin( π y / b) + k g b a (II- 10) From this expression, it can been see that the deflection angle strongly depends on the injection point of the electron at the entrance of waveguide along x-y plane. In addition, it is affected by initial energy of the injected electron (or velocity ν z ), and frequency f and intensity E o of microwave. Here E o is the maximum electric field of the mode (corresponding to the intensity of the microwave radiation) and ν z is the velocity of the electron (corresponding to its energy) along the z-axis. It can be also seen from equation (II- 10) that the angle α change from maximum to minimum along z- axis at the range (0 / ) and after that the electron executes sinusoidal oscillation. In addition, if λ g the electron is injected at the centre of the waveguide (x = a /, y = b / ), it will not be suffered any deflection at all. However, for simplicity, let us assume that the electron is injected at x = a /. So, we have: 1 eπ π z ωεν zµ o α = tan E ( )cos( / ) 1 ocos π y b bmekcν z λ g k g (II- 11) (11) Deflection Angle (degree ) - ٢٧ -

28 z-axis (m) (a) 40 Deflection Angle (degree ) z-axis (m) (b) Fig. (1): Change of angle of deflection (by degree) along the Z-axis (m) in rectangle waveguide of width 4.0 cm and height.5 cm for microwave intensity (a) 10 7 W/m and (b) W/m when the electron is injected with energy 50 kev. Therefore, the distance traveled by the electron along the waveguide can be obtained from equation (11) which is calculated in terms of λ g. It is clear from this equation that the angle of deflection is directly proportional to E o, (i.e., to the microwave intensity I o ) and it has also dependence on waveguide width b and microwave frequency ω ( = π f ). It is clear that the angle of deflection will be maximum in the front of the waveguide (z = 0) and it decreases along z-axis to becomes zero at z = λ g / 4. However, the angle of deflection can be analyzed by giving typical values to the waveguide dimensions, frequency and intensity of the microwave radiation. So, the results can be discussed graphically from the following figures. From figure (II-1), it can be seen how the deflection angle change along the z- axis in rectangular waveguide of width 4.0 cm and height.5 cm for different microwave intensities (a) 10 7 W/m and (b) W/m. Here, it is considered that an electron of energy 50 kev is injected at y = 0.65 cm. Also, figure (II-) displays the variation of the angle of deflection with microwave intensity along the waveguide. It is clear that the angle increases for larger microwave intensity and decreases with the distance. - ٢٨ -

29 Deflection Angle (degree ) z-axis Microwave Intensity (W/m ) Fig. (II-): Variation of the angle of deflection with the intensity of microwave along the axis of waveguide. Other parameters are the same as in Fig. (II-1) Deflection Angle (degree ) z-axis (m) y-axis (m) Fig. (II-3): The variation of deflection angle with the injection point of the electron on y-axis of waveguide along the z-axis for microwave intensity 10 7 W/m and the other parameters are the same as in Fig. (II-1). Fig. (II-3) shows the variation of injection point of the electron on y-axis of waveguide with the angle of deflection along the axis of waveguide for microwave intensity 10 7 W/m. It is clear that the angle of deflection gets smaller as the injection point near to y = b /. Also it gets smaller when the width of waveguide becomes wider. - ٢٩ -

30 Deflection Angle (degree ) z-axis (m) Initial Energy (kev) Fig. (II-4): The effect of initial energy of the injected electron on the deflection angle during its motion along the z-axis. The microwave intensity is 10 7 W/m and the other parameters are the same as in figure (II-1).. Deflection Angle (degree ) Frequency (Hz) z-axis Fig. (II-5): The effect of the frequency of the incident microwave on the deflection angle of the electron during its motion along the z-axis. The microwave intensity is 10 7 W/m and the other parameters are the same as in figure (II-1). Finally, the effects of the initial energy of the electron and the frequency of microwave on the deflection of the electron during its motion along the z-axis are displayed in figures (II-4) and (II-5) respectively. They show that the energy of the injected electron has small effect on the deflection of the electron while the frequency has important effect to decrease this deflection with the other parameters are the same in the figure (II-1). - ٣٠ -

31 II.. Electron Acceleration and Energy Gain In this section, the electron acceleration during its motion in the field of the TM 11 mode of microwave radiation along the direction of its propagation has been studied. To obtain the acceleration gradient and the energy gain by the electron, the following momentum equation and energy equation are used: d r ur r uur m e dt ( eγν ) = [ E + µ o ( ν H )] E H (II- 1) d r r ( m c eγ ) = e v. E (II- 13) dt Here, the energy U = meγ c and m e is the rest mass of the electron. As we can see that the velocity ( ν z ) is the dominant, (i.e. the propagation of the electron is solely specified in the z-direction), we can transform the coordinates to ζ =ν g t z to find: m ( ν v )( γ v ) = e ( E µ v H ) (II- 14) e g z x ζ x 0 z y m ( ν v )( γ v ) = e ( E + µ v H ) (II- 15) e g z y ζ y 0 z x me ( ν g vz )( γ vz ) ζ = e E z + µ 0 ( vx H y v y H x ) (II- 16) m ec ( ν g vz ) γ ζ = e[ vxex + v y E y + vz Ez ] (II- 17) where, the subscript (ζ ) denote the differentiation with respect to (ζ ). The equations of the momentum and energy (13-16) can be solved to obtain the following relation: e me c γ ( vg vz ) γ = ( E x+ E y ) ζ ζ m ( v v ) e g z µ v ( E H E H ee ( γ v ) 0 z x y y x z z ζ (II- 18) For the sake of simplicity, we assume that γ and ν z (in the case of the microwave) are slowly varying (i.e., γ << γ << γ, ( v ) << ( v ) << v ) [7]. Then, ζζ ζ z ζζ z ζ z after making the derivatives involved in equation (17), it becomes: - ٣١ -

32 e γγ ζζ = + µ m c v v ( E ) x Ey 0 vz ( ExH y EyH x e ( g z ) (II- 19) By putting the values of the various field components of the mode (equation (II-3)) in the above differential equation (18), one can integrate it, (by considering ζ as dummy variable), to obtain: d γ d ζ = R l n γ + C (II- 0) where, C is the constant of the integration and e π k E R = (1 µν ο zωε / k g )cos( π y / b)sin( π z / λg ) m c v v k g o e ( g z ) c (II- 1) The factor γ is related to the deflection angle α through the following expression: γ = c c v tan (II- ) / z ( α +1) When the electron is injected along the z-axis at the entrance (z = 0), v z = vzo and γ = which gives the constant, C = R lnγ. So, equation (II-19) becomes: γ o o dγ γ eπ kg E o µ ο vzωε = ln 1 cos ( π y / b)sin ( π z / λg ) d z γ o mec( vg vz ) kcb kg Here, it has been used d dζ = d d z 1/ (II- 3) Therefore, the change in energy gain ( U the energy gradient) has been obtained as: γ = me c ) per unit distance ( ev / m ) (i.e. d U π c E o γ = (ln )( µ oεω vz kg ) cos( π y / b)sin( π z / λg ) d z ( vg vz ) b k c γ o ev / m 1/ (II- 4) It is clear from expression (II-3) that the acceleration gradient depends on the parameters of the microwave where it increases for example with the microwave intensity. It is also clear that no acceleration will be done when the electron is injected at y = b/, while the acceleration gradient has maximum value when the electron completes the distance ( nλ g / 4 ) ( n = 1, 3, 5, ). - ٣٢ -

33 Acceleration Gradient (Ev/M) z-axis Microwave Intensity (W/m ) Fig. (II-6): The dependence of acceleration gradient on the microwave intensity along the axis of waveguide (z) when the frequency of microwave is 7.5 GH and the other parameters are the same as in Fig. (II-1). Acceleration Gradient (ev/m) z-axis Frequency (Hz) Fig. (II-7): Variation of the acceleration gradient as a function of microwave frequency along the z-axis of the waveguide for intensity 10 7 W/m and the other parameters are taken as in figure (II-1). - ٣٣ -

34 Again, we can give the typical values of the waveguide dimensions and microwave parameters in order to analyze the energy gain of the electron during its motion. Figure (II-6) displays how the energy gain (acceleration gradient) acquired by the electron is changed with microwave intensity along the z-axis of waveguide for microwave frequency 7.5 GHz when the electron is injected with energy 50 kev. While figure (II-7) shows the changes of acceleration gradient with frequency of microwave radiation for intensity 10 7 W/m. CHAPTER III Electron Dynamics in Presence of Static Helical Magnet inside Circular Waveguide - ٣٤ -

35 CHAPTER III Electron Dynamics in Presence of Static Helical Magnet inside Circular Waveguide In recent years, as we mentioned in chapter II, a number of studies have been dedicated to the dynamics of electrons in electromagnetic fields depending on the basis of the Newton equation with the Lorentz force [1, ]. This subject has a great interest due to its diverse applications to particle acceleration in the field of nuclear physics, thermonuclear fusion research and high energy particle physics. Also another mean for coupling electromagnetic energy to particle has discussed as a possible origin of cosmic rays [3]. The same mechanism has been proposed for the most of the investigations including the direct acceleration scheme make use of short-pulse high- intensity lasers [4, 5]. However, some of the researchers have made theoretical as well experimental attempts for the particle acceleration by using microwave radiation [6-9]. The description of the electron dynamics in a high frequency field is complicated because of the large number of oscillations. The problem becomes more intricate analytically with the addition of an extra- electric or magnetic field. Besides, the microwave breakdown threshold in a circular waveguide excited in the lowest order TE 11 -mode has been investigated by Tomala et al. [1]. Pioneer experiments have been reported on inverse free-electron laser acceleration depending on microwave accelerating structure consists of TE 11 rotating waveguide mode and an axial magnetic field by Yoder et al. [9]. In the present chapter we study the dynamics of an electron in the fields associated with TE-electromagnetic wave propagating inside a circular waveguide. It has been investigated the possibility of acceleration for this electron inside the waveguide when it is injected along the direction of propagation of the TE 11 -mode excited by microwave radiation. It is also considered that the motion of this electron along the axis of a helical magnet (in which the field is perpendicular to the axis of the waveguide and rotating as a function of position along the magnet). - ٣٥ -

36 III.1. Electron Trajectory and Energy Gain III.1-A Electron Dynamics A high intensity microwave of frequency ωis used to excite the lowest order TE 11 mode (with E z = 0 ) in an evacuated circular waveguide of radius "a". Therefore, one can obtain the field components of this mode from the following Maxwell s equations[13] (with Time dependence as -iωt e ): ω ρ + = 0 E c ω ϖ + = 0 H c (III-1) (III-) So that, inside a waveguide of ideally conducting material, the fields of the excited TE 11 mode can be obtained under the condition that the tangential component of the electric field must vanish on the boundary as follows: iω H o Er = J1( kcr) sinφ e r kc iω H o Eϕ = J 1( kcr) cosφ e kc i kg Ho Hr = J 1( kcr) cosφ e kc i kg Ho Hϕ = J1( kcr) sinφ e r kc H = H J ( k r) cosφ e z o 1 c iβ z g iβ z g iβ z g g iβ z g iβ z (III-3) Here, H o is the amplitude of the magnetic field intensity which is related to the intensity of the microwave radiation and k c is the wave number corresponding to the cutoff condition (i.e., J1 ( k c a) = 0 where J 1 is the first order cylindrical Bessel function) whileβ g is the propagation constant with kc = ω µε β g. It is noticed that the strength of the fields depends on both of radius r and azimuthal angle ϕ, such - ٣٦ -

37 that its magnitude is largest in the centre of the waveguide and decreases radially outwards. Now, we discuss the dynamics of an electron in the fields of this mode when the electron is injected along the propagation mode inside the waveguide. The electron deflected from the z-axis due to the force of the field E and also the force due to the field H which also acts on it at the same time. Besides, we consider an external magnetic field that is always perpendicular to the mean particle motion, but whose direction rotates about that axis as a function of position along that axis. This field can be produced by a magnet referred as "helical magnet" and given by B= Bo exp (π i z / Λ) where, Λ is the length of one turn of the helix and B o is the helical magnetic field induction [6]. Under the force of all these fields, the deflection angle η of the electron during its motion along z-axis can be calculated from the following electron equation of motion: r d p d t r r r r = e E+ v ( µ oh + B) (III-4) The rate of change for transverse momentum P of an electron injected with momentum P z in z-direction is calculated from the following: where, d P e = E + v z ( µ o H ϕ + B ) d z v (III-5) z iω iβ 1 g z E = Hoe J1 sin φ+ J 1 cos φ kc rkc The equation (III-5) has been obtained under P << Pz (which means that the propagation of the electron is solely specified in the z-direction), then one can put the values of the field components and integrate it to find P at ϕ = π /. Therefore, the angle of deflection can be evaluated as: 1/ P eµ oh o tanη = = J 1( kcr)( ω β gvz ) cos β g z P z me vz kc β g r e BoΛ π + sin z π m v Λ e z ( ) (III-6) - ٣٧ -

38 III. 1-B. Electron Acceleration Now, we analyze the electron acceleration during its motion in the field of the TE11 -mode. The acceleration gradient and energy gains by the electron can be obtained by considering the following momentum equation and energy equation: d r r r r r ( m v eγ ) = e E + v ( µ H B o + ) d t d r r ( meγ c ) = e v. E d t (III-7) (III-8) We transform the coordinates to ξ = v g t z in equations (7) and (8) and then they are simplified under the above condition ( P ) P z to obtain the following relations: d ( γ vr ) e = Er vz ( µ ohφ + B0 ) dξ m ( v v ) e g z (III-9) v r = m e ( v g v z ) d ( γ v z ) e ( µ H + B ) dξ o φ o (III-10) dγ e Er = v dξ m c ( v v ) r e g z (III-11) Equations (10) and (11) give the following relation: d ( ) d c H B d γ Er d γ vz = ξ ( µ o ϕ + 0 ) ξ (III-1) So, the radial component of the electron momentum equation (9) with (10) give: d ( γ vz ) dγ d ( γ vz ) d( γ vz ) ( vg vz ) γ + v g dξ dξ dξ dξ ( µ ) = Er vz ( µ ohϕ + B0 ) e ohϕ + B0 me ( vg vz ) (III-13) - ٣٨ -

39 Now, from (1), (13) and for the sake of simplicity, γ and vz (in the case of the microwave [11] ) can be considered as slowly varying. So, the first term in (13) is dominated over the other two terms. Then (13) can be integrated with considering ξ, as the dummy variable [9], and putting the field components in it, we have: 1/ dγ γ = R ln dξ γ o where, e ωµ o H o R = J 4 1 ( kcr) sin φ{ ω β gv z sin β g z m c ( v v ) r k e g z c r v k B ( ) z c o sin( ( β g π Λ) z) µ o H o J1( kcr) sinφ (III-14) Multiplication (14) with ( m e c / e), one can obtain the energy ( U = meγ c ) gradient (in ev/m) achieved by the electron as follows: d U c ωµ oh o J 1( kc r) sin( g z) ln γ = β d z ev r kc ( vg vz ) γ o m 1/ (( β g π Λ) z) β g vz r vz kc B sin o 1 ω ωµ o H o J1( kcr) sin ( β g z) 1/ (III-15) It is worth mention that for deducing (6) and (15) we considered that the electron is injected at angle ϕ = π /. This expression shows that the acceleration gradient depends on the parameters of waveguide and the properties of microwave. III.. NUMERICAL RESULTS The motion of the injected electron is investigated by studying the expressions of deflection angle and energy gradient It can be seen, from (III-6), that the deflection angle depends on the amplitude of the magnetic field H o (which related to the intensity of incident microwave). It also changes sinusoidally along the z-axis and is a maximum at the entrance of the waveguide. In addition, equation (III-15) illustrates that the energy gradient of the electron could be increased rapidly along the z-axis depending on different parameters. - ٣٩ -

40 In this section, the numerical evaluation of the formulas of the deflection angle and the energy gradient derived in the above sections is employed to further explore the behavior of motion of the electron. Deflection Angle (degrees) (d) (c) (b) (a) z-axis (m) Fig. (III-1). The variation of deflection angle along z-axis for different microwave intensities ( a) = 10, ( b) = *10, ( c ) = 3*10, ( d) = 4*10 W / m with microwave frequency 140 kev f = GHz, helical magnetic field 0.1 Tesla and electron kinetic energy A numerical example is taken to investigate the propagation of the dominant TE 11 mode in a circular waveguide of radius r= 5 cm is considered. It can be found 9 that the microwave frequency f should be bigger than Hz which corresponding to cutoff wave number kc = m -1. The other used parameters, in our case, the initial energy of the injected electron (related to the initial velocity) and the helical magnetic field, are taken to be 140 kev and 0.1 Tesla, respectively. Deflection Angle (degrees) GH 5GH 4GH z-axis 3GH Fig. (III-). The variation of deflection angle along z-axis for different microwave 4 frequencies with microwave intensity 3 10 W / m and the other parameters are the same as Fig. (III-1). - ٤٠ -

41 Fig. (III-1) displays the variation of deflection angle of the electron during its motion along z-axis for different microwave intensities. It is noted that the deflection angle is directly proportional to the intensity of the microwave and the angle is maximum at the entrance of the waveguide and decreases along its axis attains zero at = / 4. z λ g Deflection Angle (degree) (b) (c) (a) z-axis(m) Fig. (III-3). The deflection angle along z-axis at different points of injections 4 ( a ) = 3a / 4, ( b) = a /, ( c) = a / 4 with microwave intensity 3 10 W / m, and the other parameters are the same as in Fig.(III-1). Deflection Angle (degree) z-axis Initial electron velocity(m/s)) Fig. (III-4). The effect of the initial kinetic energy of the electron on the deflection 4 angle along z-axis for microwave intensity 3 10 W / m and the other parameters are the same as in Fig.(III-1). The dependence of the deflection angle on the frequency of the microwave along the z-axis is also investigated in Fig. (III-) for microwave intensity W / m. It can be noted that the angle of deflection is decreasing with increasing of the microwave frequency and it is going to zero at smaller distance with greater frequency along z-axis. This is due to that λ g (the guide wavelength) is depending on the microwave frequency. - ٤١ -

42 Fig. (III-3) shows that the dynamics of the electron and deflection of its motion is strongly depend on the point of injection for this electron at entrance of the waveguide. It displays that the angle of deflection is decreasing when the electron is injected at longer distance from the centre of the entrance of waveguide. One can note, from Fig. (III-4), that the deflection angle changes inversely proportional with the kinetic energy of injected electron in the front of the waveguide along its axis. Besides, the effect of the helical magnetic field on the deflection angle of the electron can also be investigated in Fig. (III-5). It displays that the deflection is increased with the strong magnetic field.. Deflection Angle (degree) z-axis (m) Helical magnetic field (Tesla) Fig. (III-5). The effect of the helical magnetic field on the deflection angle along z- 4 axis with microwave intensity 3 10 W / m and the other parameters are the same as figure (III-1). On the other hand, the acceleration gradient of the electron during its motion along the axis of waveguide has also been studied with different parameters. In the first, figure (III-6) shows output the effect of different intensities of microwave on the energy gradient along the z-axis for microwave frequency f = GHz and electron kinetic energy140 kev. It can be seen that the energy gradient is directly proportional to the microwave intensity. Also, the gradient started from a point closer to the front of waveguide with higher intensity. - ٤٢ -

43 d U d z ev / m z-axis (m) Fig. (III-6). The variation of energy gradient along the z-axis for different intensities of microwave ( a ) = 10, ( b) = *10, ( c) = 4 *10, and the other parameters are the same as figure (III-1). The effect of microwave frequency on the maximum acceleration gradient is also shown in Fig. (III-7). It is clear that the acceleration gradient increases with microwave frequency and makes to decrease the acceleration length (the distance on which the energy gradients become maximum). d U d z ev / m z-axis Fig. (III-7). The variation of energy gradient along z-axis for different microwave frequencies ( a) = 3GH, ( b) = 4GH, ( c) = 5GH, with microwave intensity W / m and the other parameters are the same as figure (III-1). - ٤٣ -

44 Here, it can be seen that the values of λ g and vg decrease as the microwave frequency is increased. These results on the variation of β g (orλ g ) are similar to the ones obtained by Shenggang et al. [14] and Xiao et al. [15] in a circular waveguide for the similar types of the modes. du dz ev / m z-axis(m) Initial Velocity (m/s) Fig. (III-8). The effect of the initial energy of the electron on the energy gradient 4 along z-axis with microwave intensity 3 10 W / m and the other parameters are the same as Fig. (III-1). The plot in Fig. (III-8) illustrates the dependence of energy gradient on the electron initial energy of the electron. The injected electron with higher kinetic energy also increases the energy gradient. In summary, it can be deduced that the acceleration gradient and hence the energy gain attained by the electron could be enhanced by optimizing the parameters of microwave and the initial energy. So, it must be taken into account that when we increase the intensity of microwave it must increase the microwave frequency to keep the amplitude of oscillation (through deflection angle) less than the radius of waveguide. - ٤٤ -