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بسم الله الرحمن الرحیم

Self-focusing of Gaussian laser beam in relativistic cold quantum plasma

سیدمحسن میرجلیلی93آذر

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Contents

Basic Concepts1

Laser – Plasma Interactions2

Self-focusing3

Article and Results4

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What is Plasma?

Plasma/Laser Relativistic nonlinearity

Laser interactions with Plasma

Classical approximation andQuantum plasma

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Definition of plasma:A plasma is a quasi-neutral gas consisting of positively and negatively charged particles (usually ions and electrons) which exhibit collective behaviour.Remarkable difference fromneutral gas for plasmas is collective behavior

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Quasineutrality means almost neutral, for example, a system containing 100 protons (positively charged) and 101 electrons (negatively charged) is a quasineutral system. Plasmas resulting from ionization of neutral gases generally contain equal numbers of positive and negative charge carriers. In this situation, the oppositely charged fluids are strongly coupled, and tend to electrically neutralize one another on macroscopic length-scales. Such plasmas are termed quasi-neutral.

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Collective behaviour: Charged particles in plasma are by no means free. in fact, they are strongly affected by each others’ electromagnetic fields. Nevertheless, because the charges are no longer bound, their assemblage becomes capable of collective motions. A plasma has interesting properties because the electrostatic force is a long range force and every charged particle interacts with many of its neighbors. (collective behavior).

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Plasma frequency: = (SI ) = (c.g.s) Plasma frequency is the most fundamental time-scale in plasma physics.

In fact plasma oscillations try to preserve quasineutrality.

Charge separation

E is created (restoring force)

simple harmonic motion

if electrons were displaced from their equilibrium positions

The electrostatic force due to the ions would pull electrons back

the electrons wouldovershoot and ocillations would ensue

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The most important feature of a plasma is its ability to reduce electric fields very effectively. In plasma the Coulomb interaction range is reduced due to the screening effect. Imagine a positive ion in a plasma. A cloud of electrons is attracted around this positive ion due to the Coulomb law. Therefore, the force of attraction of a positive ion does not extend to infinity but to a finite distance. This distance is called the Debye radius or Debye length and the shielding is called Debye shielding. in other words, the electric field from the test charge is effectively shielded at distances larger than the Debye length.

Debye length : a fundamental length scale in plasma

𝜆𝐷=( 𝜀0𝑘𝐵𝑻

𝑛0𝑒2 )

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A plasma of size L must be sufficiently large, i.e., L >> to behave in a collective manner. A plasma must also exist for a period of time larger then the response time, T>> to behave in a collective manner.

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Plasma as a classical system obeys Boltzmann statistics, and degeneracy is ignored.Quantum effects come into play when the interparticle distance of the electrons becomes comparable with their thermal de Broglie wavelength =

In quantum limit we must use Fermi-Dirac statistics.Since the de Broglie wavelength of an electron is much larger than that of an ion, quantum effects occur for electrons before they do for ions. Therefore, to determine the boundary between classical and quantum regimes, it is sufficient to consider only electrons.

=

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It is interesting to note that quantum effects become more important as the temperature decreases. This occurs because the de Broglie wavelength increases as the electron velocity (temperature) decreases.(cold) On the other hand, the conditions for the quantum regime are met at very high densities, as in metals and various astrophysical objects such as white dwarfs.(dense)

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The Maxwell-Boltzmann statistics is appropriated for dilute, or non-degenerate plasmas, for which T>> , = .

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PLASMA

Classical

system

Quantum

system

cold

dense

hot

Low density

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Shielding∈quantum plasm : 𝜆𝐹=( 2𝜀0𝐸𝐹

3𝑛0𝑒2 )

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L>>

so: if T=0 = 0

Condition for quantum plasma𝑏𝑢𝑡 𝜆𝐹≠0

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This happens because of the exclusion principle which prevents theaccumulation of electrons in the same place as the test charge

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Plasmas play an important role in nonlinear optics in two different ways: (1) Nonlinear optical processes such as multiphoton ionization can create a plasma. (2) A plasma can respond in an intrinsically nonlinear manner to an applied optical field.There are two primary mechanisms of nonlinearity: (1)ponderomotive effects and (2) relativistic effects.In a sufficiently intense laser beam (I W/cm2) a free electron can be accelerated to relativistic velocities

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Gaussian beam: is usually considered in situations where the beam divergence is relatively small, so that paraxial approximation can be applied.In Gaussian beam transverse electric Field and intensity (irradiance) distributions are well approximated by Gaussian function. The width of thisfunction is minimum at the beam waistand grows gradually in both directions.

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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 electron17

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X -Ray sources

X-Ray Lasers

Material Processing with Femtosecond Lasers

Magnetic field generation (mega gauss range)

…

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Self actions : self-focusing, self defocusing, self-trapping, and filamentation are the major self-action effects in which a beam of light modifies its own propagation by means of the nonlinear response of a material medium.Self focusing : Let's consider a beam of light propagating into a material with a positive nonlinear refractive index. If the intensity distribution in the beam is higher in the center than at the edge, the material that is near the center of the beam, will have a higher refractive index than others, so the different portions of the beam will be refractive to different degrees. Thus NL material behaves like a positive lens that focuses light that propagates through it.

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Self focusing 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 n =

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

Inducing focusing effect

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Introductions and analysis : applications of laser-plasma interactions requires the beam to propagate over several Rayleigh lengths without loss of energy Quantum plasmas are characterized by low-temperature and high density and can be studied via the Wigner formalism

Quantum effects play a role when the de Broglie wavelength becomes of the order (or larger than) the average interparticle distance Quantum effects are modeled by the Bohm potential = - ()

Equivalent to the classical Vlasov equation

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the dielectric constant for cold quantum plasma : ( 1−𝛿𝑞𝛿

¿¿−1

if =0 classical relativistic dielectric constant ignoring quantum effectUsing WKB and paraxial approximation one can derive a differential equation for beam-width that governing the behavior of beam width parameter as a function of distance of propagation and it can be solved numerically.

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Result of article

Self-focusing

Interactions

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Results:By comparing the self focusing effect in cold quantum plasma with classical relativistic case, we see that quantum effects significantly adds self focusing than classical relativistic case.Variation of beam width parameter f with dimensionless distance of propagation in cold quantum plasma with value of intensity parameter =0.1 Solid curve: relativisticDashed curve: cold quantum

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Result of article

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It is observed that quantum contribution decreases the focusing length with additional self focusing effect in comparison to that of classical relativistic case. This is because of stronger beam localization due to the quantum effects which adds the self focusing.

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References:1. S.D. patil,  M.V. Takale, S.T. Navare, M.B. Dongare: Self-focusing of Gaussian laser beam in relativistic cold quantum plasma 2. A. Krall: Principles of Plasma Physics3. D. A. GURNETT: INTRODUCTION TO PLASMA PHYSICS4. Shalom Eliezer: The Interaction of High-Power Lasers with Plasmas5. J.A. Bittencourt: Fundamentals of Plasma Physics6. Paul M. Bellan: Fundamentals of Plasma Physics

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Do your best ,

That is all.

Allah will do the rest

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