شتابدهی لیزری

1. Acceleration سیدمحسن میرجلیلی

2. Contents Single Electron Motion in a Laser Wave Laser Wake-Field Acceleration Conventional Acceleration

3. Single electron When a charged particle enters a uniform magnetic field with its velocity at right angles to the magnetic field B,The speed of the particle will remain constant and the particle moves in a circular pathin a plane perpendicular to B. From balancing between the centripetal magnetic force and the centrifugal force we can find: radius of gyration R=

4. Single electron Helical motion A charged particle having a velocity vector that has a component parallel to a uniform magnetic field moves in a helical path

5. Single electron particles drift due to a gradient of magnetic field. When a plasma is subject to a magnetic field gradient, positive and negative charged particles drift in opposite directions, thus leading to plasma polarization.

6. Single electron Crossed – Field (magnetron) The motion of a charge in crossed electric and magnetic fields, yielding the average drift velocity E/B. average value of is 0 The average velocity in the x direction is the crossed-field drift velocity )

7. Single electron Non-relativistic Motion in a Plane Wave SingleElectron Motion in EM field: Relativistic Motion in a Plane Wave Non-relativistic : Equations of motion for an electron : Plane wave : For weak field, |V|in the linear approximation we neglect the V×B r = E V=- Ethere is no force in the longitudinal direction So: In lowest order the particle just oscillates around its current position due to the electric field The trajectory is a line for linear polarization, and a circle for circular polarization.

8. Single electron higher order: Since the peak amplitude is , the dimensionless parameter is defined as , so the linear solution is adequate as long as 1 If is not so small the effects of the magnetic force should be estimated : , to the lowest order in , the electron velocity is given by the linear solution: () we use this solution to evaluate the next order term: V= + of order of order

9. Single electron For linear polarization along y, the electron oscillates along the x direction with frequency 2ω is obtained for the trajectory the trajectory of an electron in a monochromatic plane wave, in the reference frame where there is no average drift along the propagation direction

10. Single electron Relativistic: the “relativistic regime” of laser-plasma interaction is defined by the condition 1 in practical units =0.85 The electron is accelerated during the interaction with the laser pulse, but after the pulse has passed the electron is again stationary with no net acceleration So: In an infinite planegeometry, even with a strong electromagnetic wave, an electron has no energy gain according to the Woodward–Lawson theorem. Lawson-Woodward theorem: A charge cannot be accelerated by a radiation field in vacuum extending over an infinite region The consequence of using a plane wave laser pulse without including the radiation damping

11. Single electron • Lawson–Woodward theorem: There is no net energy gain for an electron when interacting with a laser field • Laser field in vacuum, no boundaries or walls • The electron is highly relativistic, β →1 • No static E or B field in presence • Region of interaction is infinite For an electron propagating in z is accelerated by in vacuum and assuming linear polarization along x : If there is no boundary and the integration for z is on infinity =0 • How to break the Lawson-Woodward theorem for energy gain? • focused laser pulse: electron acceleration in vacuum is possible by violating one or more hypotheses of the “no acceleration” theorem, for example use a focused laser pulse (ponderomotive force). • Bound electron: For a 1D laser pulse, if an electron starts from rest and encounters both the front and back of the pulse, the net force on the electron is zero. However, if the electron is initially bound in an atom, there is a possibility for net acceleration.

12. Single electron • Experiments(1999): laser pulse of , 340keV • Simulations(2002, 2006): GeV electron energies are possible • Chirp pulse: Another way for net acceleration is by having a frequency variation (chirp) in the laser pulse • Semi Vacuum energy gain: • multiple laser pulses • a combination of external static magnetic fields and a laser pulse • …

13. Single electron a monochromatic wave of frequency ω (the laser pulse has been ramped up adiabatically): constant drift of the electron The “universal” trajectory of an electron in a monochromatic, linearly polarized wave plane wave of dimensionless amplitude The trajectory in a circularly polarized plane wave for = 2 ( helicoidal) the laser pulse is adiabatically ramped up and of constant amplitude afterwards.

14. Single electron Radiation Friction: at ultrahigh laser irradiances radiation reaction must be taken into account: -e (E + the figure-of-eight opens up and the electron is accelerated. radiationfrictionforce The drifting “figure of eight” trajectory (black line) for an electron in a plane wave when the radiation friction force is included. The red line gives the usual closed trajectory obtained when is neglected.

15. Single electron • Ponderomotive force: • For normal laser light incidence on plasma: • Independent of the sign of the charge • The immediate effect on electrons is much larger than on ions, thus in general the ponderomotiveeffect on ions is negligible with respect to that on free electrons. • Ponderomotivesignificance increases with increasing laser wavelength. • The force is positive on the front side of the pulse and negative on the backside This nonlinear force is very important phenomenon in view of harmonic generation, beat wave excitation, wakefieldexcitation for particle acceleration, self-focusing of laser beam, filamentation of laser beam and…

16. Laser-Plasma Progress in laser technology New subject of physics New theoretical model and new experimental setup • higher laser pulse peak intensities • shorter pulses sun strong laser field strength [V/m]: intensity [W/] : 1000 In recent years using laser–plasma interactions for acceleration has been brought about by the development of ultrashort high-power lasers via the CPA method.

17. introduction • Why laser plasma interactions? • To achieve higher energies using conventional techniques, larger and larger accelerator systems are required. So, huge cost and constructions many years are necessary. • Traditional accelerators: • gradient: <100 MV/m limited by material break down • largefacilities (To shrink the facility one needs higher gradient) • Plasmas are not limited by breakdown as they are already ionized and indeed can support electric fields of the order 10 – 100 GeV/m • A laser beam propagating in a plasma can excite electron plasma wave, which being longitudinal can be used to accelerate electrons. (Tajima and Dawson 1979) Accelerating Into the Future: formZero to 1 GeV in a Few Centimeters

18. Laser-Plasma laser-plasma interactions: Generation of high-energy electrons Wake waves In underdense plasmas overdenseplasma Acceleration at the interface Underdense Plasmas: Laser Wake - field Acceleration: Ashort laser pulse interacts with a plasma generating a strong oscillating electrostatic field in its wake (wake-field) much in the same manner as a boat generates a wake when it travels through water. WakefieldGeneration: An ultrashortlaser pulse can induce large electrostatic fields in plasma. Displacement of electrons due to ponderomotive force, creates large amplitude plasma wave, which is called the wake.

19. Laser-Plasma electron – laser pulse interaction in vacuum Pushing of plasma electrons by a laser pulse • In the vacuum case, the electron returns to zero momentum after interacting with the laser pulse. However, in the plasma case the electrons have net momentum after the laser pulse has passed.

20. Laser-Plasma Density perturbations (dotted line) and wake-field (bold line) induced by the laser pulse (solid line). Expanded view of plasma electrons behind the laser pulse. As opposed to the case in vacuum, the laser pulse induces charge density perturbations. As can be seen, the charge perturbations are in the form of a sinusoidal type of wave. These charge perturbations result in an electrostatic field or wake-field(bold line)

21. Laser-Plasma • Wake-Field Acceleration: • After exciting a wake-field in plasma by a short laser pulse or multiple pulses, how electrons can achieve net acceleration? • In a standard linear wake-field there is no net acceleration of electrons in the plasma. After interacting with the laser pulse the electrons simply oscillate in time. • Net acceleration occurs when the electrons are injected into the wake-field in some manner with velocities comparable to the phase velocity of the wave. A surfers must attain some minimum velocity to be “injected” into an ocean wave. Start to surf: Injection of electrons

22. Laser-Plasma • Injection of electrons into the wake wave can occur by several mechanisms: • inject electrons externally with velocities comparable to the phase velocity of the wake-field. • self-injection of background plasma electrons: • Inhomogeneous plasmas One way of controlling the injection of the electrons into the wake-field is by using multiple laser pulses. A primary laser pulse: to generate the wake-field secondary laser pulses: to ponderomotively kick some of the background electrons into the appropriate accelerating phase of the wake wave. After the collision with the second laser pulse. before

23. Conventional accelerators Linear accelerator Conventional particle accelerators: Cyclotron Betatron Cynchrotron Circular Accelerator most common Linac (straight particle accelerator):

24. Conventional accelerators Cyclotron: Two hollow RF electrodes, the dees, are connected to an RF power supply oscillating at the gyrofrequencyof the ions in the magnetic induction B. gyrofrequency of the ion in the magnetic field: The driving RF electric field should have a frequency = resonance condition)

25. Conventional accelerators Because of the relativistic mass increase, the RF driving frequency must be reduced to keep pace with the gyrofrequency: The principle of phase stability is very important to the operation of the relativistic cyclotron: T(period of revolution(kinetic energy of the particle) The principle of phase stability illustrates how relativistic effects, bunch particles in a stable phase in a relativistic cyclotron. In operation, these devices produce protons up to at least 450 MeV, with a pulsed repetition rate that is typically 60 Hz

26. Conventional accelerators Betatron: To keep the radius constant, the magnetic induction B must increase as p increases Betatron can produce electrons with energies up to approximately 300 MeV

27. Conventional accelerators Synchrotron: • An electric field with a variable RF frequency is imposed across this gap • In the synchrotron, it is not necessary to fill the area enclosed by the accelerating tube with magnetic induction, resulting in a significant saving in the cost of these devices, relative to the betatron or the cyclotron • centrifugal acceleration synchrotron radiation

28. Conventional accelerators Circular -Particles pass RF-cavities several times. -No space problem for RF-cavities. -Smaller in length than a Linac of comparable energy. -Particles loose energy by synchrotron radiation (especially low mass particles). Although in some cases it can be intended. -High magnetic fields are needed to turn high mass particles (protons) Linear -No energy loss due to synchrotron radiation. -High gradient: high energy gain in a short length. Handle high currents. -Particles cross only once the RF-cavities. -RF-cavities have to work at high frequencies (or small wavelengths) to reduce space. • In big projects, it is used a combination of both types.

29. References: • Andrea Macchi.2013. A Super intense Laser–Plasma Interaction • Shalom Eliezer.2009. Applications of Laser–Plasma Interactions • J Reece Roth.1995. Industrial Plasma Engineering

30. Do your best , Allah will do the rest That is all.