Laser Pulse Evolution In Plasma

1. Laser Pulse Evolution In Plasma سیدمحسن میرجلیلی

2. Contents Basic Concepts Self-Focusing Self-Compression Variational Analysis Reviewing Papers

3. concepts Linear and Nonlinear Optics • :consequences of linearity of the medium • The optical properties are independent of light intensity. • The principle of superposition • The frequency of light cannot be altered by its passage through the medium. • Light cannot interact with light, thus light cannot control light. • The invention of the laser in 1960 possibility of light propagation in optical materials at higher intensities optical media exhibit nonlinear behavior in many experiments: • The refractive index changes with the light intensity • The principle of superposition is violated • Light can alter its frequency as it passes through a nonlinear optical material (for example from red to blue) • Light can control light; photons do interact. • Linearity or nonlinearity is a property of the medium through which light travels, rather than a property of the light itself.

4. concepts If the polarization is not linearly proportional to the electric field E, the medium is “nonlinear” and is described by the “nonolinear optics” Linear optics: P = E = applied optical field (input) P = polarization(output of a system) = linear susceptibility = a dimensionless proportionality constant that indicates the degree of polarization of a material in response to an applied electric field nonLinear optics: P = = + Nonlinear optics: laser-matter interactions

5. concepts The Nonlinear Wave Equation:E - = ( basic equation of NL optics) Second harmonic generation Frequency conversion Parametric amplification Parametric oscillation Second order nonlinear relation between P and E: Third-harmonic generation Self-phase modulation Self –focusing Four-wave mixing …. Third order nonlinear P-E relation: (Kerr medium) Kerr medium: n = = linear = intensity of beam Kerr Effect : refractive index (Intensity-Dependent Refractive Index)

6. concepts Lasers : Pulse: is made of sum of many monochromatic waves, so in dispersive medium each of which is modified differently is characterized by a frequency-dependent refractive index Pulse duration or pulse width: the temporal length of laser pulse, in fact it is the time during which the laser actually emits energy Pulsed laser: the laser only oscillates for a short time span(e.g, some fs) Pulse Power = Intensity = • higher laser pulse peak intensities • shorter pulses Ultrashortlaser pulses enable the study of ultrafast processes

7. concepts Laser pulse generation is achieved by Q-switch and mode-locking methods For high peak power production For short pulse production(femto) and more PRR than Q-switching Intensity: mode-locking (GW) >Q-switching(MW) CPA(1985): Power of the femtosecond laser pulses is increased Chirp: time dependence of the instantaneous frequency of an optical pulse, so in a chirped pulse the frequency of the electromagnetic wave varies with time. time-bandwidth product: to generate a short pulse (Δt), a broad spectral bandwidth (Δω) is required.

8. concepts CPA scheme: the output femtosecond laser pulse is first stretched in time (chirped in frequency), then amplified and finally recompressed Intensity [W/] : Order of Pulse length :

9. concepts Gaussian beam : light beam where the E field profile in a plane perpendicular to the beam can be described with a Gaussian function The wavefronts are approximately planar near the beam waist, but they gradually curve and become approximately spherical far from the waist. Beam waist: location with min beam radius Rayleigh Length: the distance from a beam waist where the radius increased by 2 The depth of focus of a Gaussian beam

10. concepts The width of this function is minimum at the beam waist () and grows gradually in both directions.

11. concepts • Ponderomotive force ( nonlinear force): • For normal laser light incidence on plasma: • Independent of the sign of the charge and this force acts to expel the electron from the laser focus (regions of high intensity) • 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. • It is due to the spatial gradient in intensity across the focused laser pulse 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…

12. concepts Self Actions : occur when an electromagnetic field induces a refractive index change in the medium through which the field propagates. The change in index then exhibits a back-action on the field so its propagation characteristics is influencedA beam of light modifies its own propagation by means of the nonlinear response of a material medium Self-action effectsemerge for high-power light waves because a significant contribution to the response of a medium, along with linear polarization, is made by nonlinear polarization, which is proportional to odd powers of the electric field strength. Self-action effects arise from the 3rd nonlinear polarization

13. focusing • Self- Focusing • The term self-focusing was introduced by Askar'yan (1962) in his paper “Effect of the field gradient of a high-intensity electromagnetic beam on electrons and atoms‘’ • focusing of a beam in a transparent medium, caused by the beam itself through a nonlinear process in the medium. It is produced by the optical Kerr effect • It can occur when a beam of light have a nonuniform transverse intensity distribution • The physical mechanism is based on a Kerr nonlinearity with positive (positive • The medium acts as a positive lens.

14. focusing laser beam with an intensity profile peakedon axis index of refraction, η(r) have a maximum on axis the laser phase velocity depends on the index of refraction depends on the laser intensity Local variation in the phase velocity will modify the shape of the laser pulse (phase velocity lower in the vicinity of the axis, max on axis) So: Wavefronts curve inward and the laser beam converges.

15. focusing Importance of self focusing: In the absence of any external guiding, the laser beam diffracts in vacuum in one Rayleigh length 5μm spot size 0.8μm wavelength the Rayleigh length is ∼ 100μm For a Gaussian laser pulse: the ionization of the gaseous medium local electron density maximum along the axis of propagation is increased The refractive index acts as a diverging lens and defocusses the non-linearity of laser plasma interaction comes at the rescue Plasma Self Focusing is an effective solution to the problem of long-distance transfer of laser energy

16. focusing Self focusing in plasmas can occur through thermal, relativistic and ponderomotive effects. Thermal : collisional heating of a plasma exposed to EM radiation T hydrodynamic expansion index of refractions Relativistic : mass of electrons = Ponderomotive : push electrons away from region of high intensity Plasma refractive index is modified (increased) charge displacement Inducing focusing effect

17. focusing When the effect of self-focusing precisely compensates the diffraction beam spreading, this regime is referred to as self-channeling (self – trap) A beam of light propagates with a constant diameter as a consequence of an exact balance between self – focusing and diffraction effects

18. SPM /Compression Self-Phase Modulation: Due to the optical Kerr effect, an optical wave traveling in a third-order nonlinear medium undergoes self-phase modulation(change in the phase of an optical pulse ). SPM was first observed in 1967 Kerr medium: n(t) =(t) effect of the medium on transmitted pulse :change the phase of the pulse ) (t)=+) )=(t) I (t) Leading edge of the pulse: I (t)>0 Instantaneous frequency is red shifted trailing edge of the pulse: I (t)<0 Instantaneous frequency is blue shifted

19. SPM /Compression Qualitative picture Normal dispersion: high frequency components of an optical pulse travel slower than low-frequency Anomalous dispersion: shorter wavelength components are faster

20. Laser Plasma Interactions: New gate of physics as a further promise for a bright future • Application of Laser_ Plasma interactions • Inertial Confinement Fusion (ICF) • Accelerators: A plasma accelerator can cut down significantly the acceleration distance to boost particles from rest to several MeV over a short distance (less than the millimeter range) and still provide high quality electron • X -Ray sources and X-Ray Lasers • Material Processing with Femtosecond Lasers • Magnetic field generation (mega gauss range) • …

21. Variational Variational Principles have been used widely throughout physics, both as a foundation for fundamental theory, and as an approximation method. • Pierre de Fermat: Nature always acts by the shortest paths • Fermat: principle of least time/ principle of natural economy • In mechanics: the least amount of action, in other word action is minimal • Variational principles: • present natural phenomena as problems of optimization under constraints • Express physical laws in a global manner • Are the mathematical formulation of the superlative • Variational calculus: finding a function(s) that minimizes some integral = ) (x), (x), x To minimize or maximize Lagrange-Euler equation = Lagrange function.

22. Variational , This method has been applied in the field of Laser-Plasma interactions by Anderson and Bonnedal . .

23. Variational . The Euler-Lagrange equation for various laser parameters

24. Variational simultaneous evolution of laser spot-size and pulse length

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