Laser Pulse Evolution In Plasma

سیدمحسن میرجلیلی

Contents Basic Concepts

Self-Focusing Self-Compression

Variational Analysis Reviewing Papers

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.

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

nonLinear optics: P == +

= a dimensionless proportionality constant that indicates the degree of polarization of a material in response to an applied electric field

Nonlinear optics: laser-matter interactions

concepts

Second order nonlinear relation between P and E:

Second harmonic generationFrequency conversionParametric amplificationParametric oscillationThird order nonlinear P-E relation:(Kerr medium)

Third-harmonic generation Self-phase modulationSelf –focusingFour-wave mixing….

conceptsThe Nonlinear Wave Equation:E - = ( basic equation of NL optics)

= linear = intensity of beam

Kerr medium: n =

Kerr Effect : refractive index (Intensity-Dependent Refractive Index)

Pulse Power = Intensity =

concepts

• higher laser pulse peak intensities• shorter pulses

Lasers :Pulse: is made of sum of many monochromatic waves, so in dispersive medium each of which is modified differentlyis characterized by a frequency-dependent refractive index

Pulsed laser: the laser only oscillates for a short time span(e.g, some fs)Pulse duration or pulse width: the temporal length of laser pulse, in fact it is the time during which the laser actually emits energy

Ultrashort laser pulses enable the study of ultrafast processes

Laser pulse generation is achieved by Q-switch and mode-locking methodsFor high peak power production For short pulse production(femto)

and more PRR than Q-switching

Intensity: mode-locking (GW) > Q-switching(MW)

time-bandwidth product: to generate a short pulse (Δt), a broad spectral bandwidth (Δω) is required.

concepts

CPA(1985): Power of the femtosecond laser pulses is increasedChirp: time dependence of the instantaneous frequency of an optical pulse, so in a chirped pulse the frequency of the electromagnetic wave varies with time.

CPA scheme: the output femtosecond laser pulse is first stretched in time (chirped in frequency), then amplified and finally recompressed

Intensity [W/] :1 𝑓𝑠

1𝑠𝑒𝑐𝑜𝑛𝑑=

5𝑚𝑖𝑛𝑢𝑡𝑒𝐴𝑔𝑒𝑜𝑓 𝐸𝑎𝑟𝑡 h

Order of Pulse length :

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.

The depth of focus of a Gaussian beam

Rayleigh Length: the distance from a beam waist where the radius increased by

2

Beam waist: location with min beam radius

concepts

concepts

𝑤 (𝑧 )=𝑤0 ¿¿

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

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 ponderomotive effect on ions is negligible with respect to that on free electrons.This nonlinear force is very important phenomenon in view of harmonic generation, beat wave excitation, wakefield excitation for particle acceleration, self-focusing of laser beam, filamentation of laser beam and…

𝑓 𝑝=−𝜔𝑝❑2

8𝜋𝜔𝐿2𝛻𝐸

2

• It is due to the spatial gradient in intensity across the focused laser pulse

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 influenced

A beam of light modifies its own propagation by means of the nonlinear response of a material mediumSelf-action effects emerge 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.

concepts

Self-action effects arise from the 3rd nonlinear polarization

Self- FocusingThe 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.

focusing

focusing𝒗𝒑𝒉=𝒄/𝜼

laser beam with an intensity profile peaked on 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

.

Importance of self focusing:In the absence of any external guiding, the laser beam diffracts in vacuum in one Rayleigh length

5μm spot size0.8μm wavelength the Rayleigh length is 100μm ∼

focusing

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

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 refractionsRelativistic : mass of electrons =

Ponderomotive : push electrons away from region of high intensity Plasma refractive index is modified (increased)

Inducing focusing effect

focusing

charge displacement𝜔𝑝 ∈𝑎𝑛𝑑𝜂

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

focusing

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 shiftedtrailing edge of the pulse: I (t)<0 Instantaneous frequency is blue shifted

SPM /Compression

Qualitative picture

Normal dispersion: high frequency components of an optical pulse travel slower than low-frequencyAnomalous dispersion: shorter wavelength components are faster

SPM /Compression

Application of Laser_ Plasma interactions Inertial Confinement Fusion (ICF) Accelerators:

Laser Plasma Interactions: New gate of physics as a further promise for a bright future

X -Ray sources and X-Ray Lasers Material Processing with Femtosecond Lasers Magnetic field generation (mega gauss range) …

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

Pierre de Fermat: Nature always acts by the shortest paths Fermat: principle of least time/ principle of natural economyIn 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

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To minimize or maximizeLagrange function.

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Lagrange-Euler equation

Variational Principles have been used widely throughout physics, both as a foundation for fundamental theory, and as an approximation method.

Variational

𝛿 I=0

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

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