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1

Impact of the mass and other parameters of charged particles on the results of laser resonance acceleration

Dubik A., Małachowski M. J.

Biuletyn Wojskowej Akademii Technicznej

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2014

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Vol. 63, nr 1

13-42

EN

Theoretical and numerical analyses are presented concerning the conditions at which the charged particles of different masses can be accelerated to significant kinetic energy in the circularly polarized laser or maser beams and a static magnetic field. The studies are carried out using the analytical derivations of the particles dynamics and theirs kinetic energy. The presented illustrations enabled interpretation of the complex motion of particles and the possibilities of their acceleration. At the examples of an electron, proton and deuteron, the velocity, kinetic energy and trajectory as a function of the acceleration time at the resonance condition are illustrated in the appropriate graphs. The particles with larger masses require the application of enhanced magnetic field intensity at the resonance condition. However, this field intensity can be significantly reduced if the particles are preaccelerated.

PL

Stosując metody teoretyczną i numeryczną przebadano warunki, w których naładowane cząstki o rożnych masach można przyspieszać do znacznej energii w kołowo spolaryzowanej laserowej bądź maserowej wiązce z dodatkowym statycznym polem magnetycznym. Badania przeprowadzono za pomocą wyprowadzonych analitycznych relacji dotyczących dynamiki i kinetycznej energii cząstek. Dzięki stosunkowo licznym wykresom stała się możliwa interpretacja dość złożonego ruchu cząstek oraz przebiegu ich akceleracji. Na przykładach elektronu, protonu i deuteronu zostały zilustrowane zależności od czasu trwania akceleracji takich wielkości jak kształt trajektorii oraz kinetyczna energia. Wszystkie ilustracje dotyczą warunku rezonansu, czyli synchronizacji ruchów obrotowych cząstki i wektora natężenia pola elektrycznego. Czym większa masa cząstki, tym większe natężenie stałego pola magnetycznego jest niezbędne do uzyskania warunku synchronizacji. Jednak to natężenie można znacznie zredukować, jeśli cząstka będzie posiadała prędkość początkową.

2

Impact of the chirping effect on charged particle acceleration in laser radiation

Małachowski M. J., Dubik A.

Journal of Achievements in Materials and Manufacturing Engineering

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2011

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Vol. 48, nr 1

87--96

EN

Purpose: The aim of this paper is to find in a numerical way the trajectories and kinetic energies gained by electrons, protons and deuterons accelerated in the laser or maser chirped radiation propagating in a vacuum, with an additionally applied external static axial magnetic field. The accelerated particles to the well defined energies are of interest in many applications, among others in medicine or in processing of different materials. Design/methodology/approach: The acceleration processes of electrons, protons and deuterons were found to be strongly depending on the way the frequency of the laser or maser radiation changes in time. In order to design the realistic acceleration processes the appropriate parameters of a laser or maser and a static magnetic field were used. Findings: The quantitative illustrations of the calculation results in a graphical form enable to discuss the impacts of the chirping effect on the acceleration process of electrons, protons and deuterons. It was found that the rate at which a particle gains the energy depends not only on the particle’s mass but also on the laser radiation frequency variation rate. Due to the different rate at which a relativistic mass of an electron, proton or deuteron increases during the acceleration process the rate at which chirped frequency decreases in time should be different. Research limitations/implications: Limits in the gained energy by the accelerated particles are a consequence of the limits in the available at present the laser or maser beam energy and the static magnetic field intensity. Originality/value: The authors have found, in an exact numerical way, the values of the acceleration equipment parameters which should be applied to obtain the desired energy of the accelerated particles. It is explained why the rate at which a particle gains the energy depends on the way the radiation frequency varies in time.

3

Basic features of the laser acceleration of charged particles

Małachowski M. J., Dubik A.

Journal of Achievements in Materials and Manufacturing Engineering

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2011

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Vol. 49, nr 2

412--420

EN

Purpose: of this paper is to study the acceleration of the charged particles by the laser beam in the range outside the resonance conditions. The studies have been limited in the subresonance region since in order to achieve the resonance acceleration a very high constant magnetic field is needed. Design/methodology/approach: The studies are carried out using the analytical derivations of the particles dynamics and its kinetic energy. The evolution of the acceleration process in time has been studied. The presented illustrations enabled interpretation of the obtained equations. Findings: The kinetic energy of the particle periodically achieves the maximal energy. Its value and the distance between the subsequent maxima rise with the increasing magnetic field or the laser beam intensity. However, these parameters of oscillating energy decrease with the decreasing wavelength. Research limitations/implications: Limits in the energy of accelerated particles are caused by the limits of the available at present the laser beam energy and the static magnetic field intensity. Practical implications: The authors of this paper believe that the presented results of the studies will help the designing of the experimental studies. It has been shown the way of achieving the high energy particles without the application of a very high magnetic field. Originality/value: The value of the paper is the analytical derivation of the parameters describing the oscillatory shape of the particles energy and numerical analysis its course. According to the authors best knowledge there are no performed such analysis of the acceleration process.

4

Difference in acceleration of electrons, protons and deuterons in a laser beam

Małachowski M. J., Dubik A.

Journal of Achievements in Materials and Manufacturing Engineering

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2010

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Vol. 41, nr 1-2

82-90

EN

Purpose: The aim of this paper is to find in a numerical way the main differences in the trajectories and kinetic energies of electrons, protons and deuterons accelerated in the laser or maser beams propagating in a vacuum, with an additionally applied external static co-axial magnetic field. The accelerated particles to the well defined energies are of interest in many applications, among others in medicine or in processing of different materials. Design/methodology/approach: Due to differences in masses the comparison between the acceleration processes of electrons, protons and deuterons is possible to perform after appropriate parameters of radiation of a laser, maser and a static magnetic field have been designed. Findings: The quantitative illustrations of the calculation results in a graphical form enable to discuss the main differences in the acceleration process of electrons, protons and deuterons. It was found that the rate at which a particle gains the energy depends not only on the particle’s mass but also on the stage of the process. Due to the mass differences, in order to keep a particle inside the radiation beam, significantly different static magnetic fields should be used to each kind of a particle. The authors have found an answer to the question why the rate at which particles energy increases in time, is different for different particles and why the difference depends on a stage of the acceleration process. Research limitations/implications: Limits in the energy of accelerated particles are caused by the limits of laser or maser beam energy or power available at present and the static magnetic fields. Originality/value: The authors show, in an exact numerical way, the values of the acceleration equipment parameters which should be selected to obtain the desired energy of the accelerated particles. It is explained why the rate at which a particle gains the energy depends on the stage of the process and on the particle’s mass.

5

Basic features of a charged particle dynamics in a laser beam with static axial magnetic field

Dubik A., Małachowski M. J.

Opto-Electronics Review

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2009

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Vol. 17, No. 4

275-286

EN

In this paper, the trajectory and kinetic energy of a charged particle, subjected to interaction from a laser beam containing an additionally applied external static axial magnetic field, have been analyzed. We give the rigorous analytical solutions of the dynamic equations. The obtained analytical solutions have been verified by performing calculations using the derived solutions and the well known Runge-Kutta procedure for solving original dynamic equations. Both methods gave the same results. The simulation results have been obtained and presented in graphical form using the derived solutions. Apart from the laser beam, we show the results for a maser beam. The obtained analytical solutions enabled us to perform a quantitative illustration, in a graphical form of the impact of many parameters on the shape, dimensions and the motion direction along a trajectory. The kinetic energy of electrons has also been studied and the energy oscillations in time with a period equal to the one of a particle rotation have been found. We show the appearance of, so-called, stationary trajectories (hypocycloid or epicycloid) which are the projections of the real trajectory onto the (x, y) plane. Increase in laser or maser beam intensity results in the increase in particle's trajectory dimension which was found to be proportional to the amplitude of the electric field of the electromagnetic wave. However, external magnetic field increases the results in shrinking of the trajectories. Performed studies show that not only amplitude of the electric field but also the static axial magnetic field plays a crucial role in the acceleration process of a charged particle. At the authors of this paper best knowledge, the precise analytical solutions and theoretical analysis of the trajectories and energy gains by the charged particles accelerated in the laser beam and magnetic field are lacking in up to date publications. The authors have an intention to clarify partly some important aspects connected with this process. The presented theoretical studies apply for arbitrary charged particle and the attached figures-for electrons only.

6

Resonance acceleration of a charged particle in a laser beam and static magnetic field

Dubik A., Małachowski M. J.

Journal of Technical Physics

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2009

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Vol. 50, no 2

97-120

EN

Conditions at which the charged particles can be accelerated to the significant energies under interaction with the laser or maser radiation and the static magnetic field have been studied. The studies have been performed on the basis of the derived by the author's analytical solutions. With the example of an electron, the trajectory, velocity and kinetic energy as a function of the acceleration time have been shown. We have defined the maximal static magnetic field which was found to be appropriate to achieve the resonance condition in the acceleration process. At certain values of the magnetic field, the projections of the 3-dimensional trajectory onto the plane perpendicular to the axis show quite regular curves in the shape of epicycloids or hypocycloids with convex and concave bows, respectively. In the resonance region, there appears an increase in effectiveness of the energy transfer from the electromagnetic field to the accelerated particle and it can be realized simply through a very small increase of magnetic induction. The significant gain of the energy by the accelerated electron occurs only in the case of inversely directed laser beam velocity and the static magnetic field. The maximal energy can be increased by rising the laser beam or the magnetic field intensities. The energy gained by an electron depends on the acceleration time and it increases until the maximum energy is reached. The distance the electron covers to gain the maximal energy, depends on the intensities of the laser beam and the static magnetic field, and on the acceleration time. The analysis of acceleration process in the maser beam shows many advantages arising from the longer wavelength radiation emitted. The essential advantage is connected with the reduction of the static axial magnetic field intensity.

7

Ścisłe rozwiązanie relatywistycznych równań ruchu naładowanej cząstki w wiązce laserowej w obecności stałego, osiowego pola magnetycznego

Dubik A., Małachowski M. J.

Biuletyn Wojskowej Akademii Technicznej

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2009

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Vol. 58, nr 1

7-32

PL

W pracy przedstawiono wyniki teoretycznej analizy zachowania się naładowanej cząstki w polu elektromagnetycznym w wiązce laserowej oraz w stałym polu magnetycznym skierowanym osiowo względem wiązki laserowej. Wyniki w postaci graficznej uzyskano, korzystając z wyprowadzonych oryginalnych analitycznych wyrażeń. Weryfikację otrzymanych równań analitycznych przeprowadzono, rozwiązując wyjściowe równania różniczkowe metodą Runge-Kutta. Otrzymane analityczne równania umożliwiły ilościowe zobrazowanie za pomocą wykresów wpływu różnych parametrów na kształt, rozmiary, kierunek ruchu elektronu wzdłuż trajektorii, a także na jego energię kinetyczną. Przez zwiększanie natężenia pola elektrycznego wiązki laserowej można zwiększać rozmiary trajektorii zarówno w przypadku hipocykloid, jak i w przypadku epicykloid tzn. krzywych będących rzutem trajektorii na płaszczyznę (x, y). Ta zmiana jest proporcjonalna do zmiany amplitudy natężenia pola elektrycznego. Natomiast zwiększanie indukcji stałego, wzdłużnego pola magnetycznego prowadzi do zmniejszania rozmiarów trajektorii i zmian ich kształtów.

EN

The results of theoretical analysis of behaviour of charged particle in the electromagnetic field of laser beam in the presence of the axially directed static magnetic field are presented in the graphical form. The results have been obtained using the originally derived analytical equations. The equations have been verified by numerical solution of differential equations of motion using the Runge-Kutta method. The analytical equations with the aid of the proper curves enabled a quantitative illustration of the impact of different parameters on the shape, dimensions and the electron motion direction along the trajectory. By changing the intensity of the laser beam it is possible to change the dimension of the trajectories of hypocycloids as well as epicycloids which present the projection of the trajectories on the (x, y) plane perpendicular to the axis direction. This change was found to be proportional to the change of the electric field intensity. However, the increase in the static axial magnetic field leads to the decrease in the trajectory dimensions and the change of theirs shape.

8

Acceleration of charged particles in laser beam

Małachowski M., Dubik A.

Archives of Materials Science and Engineering

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2009

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Vol. 40, nr 2

98-103

EN

Purpose: The aim of this paper was to find parameters of the laser and maser beams in numerical ways with additionally applied external static axial magnetic field which satisfies the proper conditions for charged particle acceleration. Design/methodology/approach: The set acceleration was designed in order to obtain the possible high kinetic energy of the charged particles in the controllable manner. This was achieved applying a circularly polarized high intensity laser beam and a static axial magnetic field, both acting on the particle during the proper period. Findings: The quantitative illustrations of the calculation results, in a graphical form enabled to discuss the impact of many parameters on the acceleration process of the electrons and protons. We have found the impact of the Doppler Effect on the acceleration process to be significant. Increase in laser or maser beam intensity results in particle's energy increase and its trajectory dimension. However, increase in external magnetic field results in shrinking of the helical trajectories. It enables to keep the particle inside the laser beam. Research limitations/implications: Limits in the energy of accelerated particles arise from the limits in up-to-date available laser beam energy and the beam diameters. Originality/value: The authors show the parameters of the circularly polarized laser beam which should be satisfied in order to obtain the desired energy of the accelerated particles. The influence of the magnetic field strength is also shown.