Nonlinear interaction of intense laser beams with magnetized plasma Rohit Kumar Mishra Department of Physics, University of Lucknow Lucknow

Interaction of intense lasers with plasma involves a number of interesting nonlinear physical phenomenon including self-focusing, wakefield generation and quasi-static magnetic field generation. Experiments report that quasi-static magnetic fields (both axial and azimuthal) of the order of MG are generated when intense laser beams interact with underdense plasma. These fields affect the propagation characteristics of the laser pulses and hence play vital role in fast ignition schemes of inertial confinement fusion, charged particle acceleration and harmonic generation.

In the present thesis a theoretical analysis of intense laser plasma interaction, in the presence of a uniform magnetic field has been presented. The effect of magnetic field on (a) self-focusing property, (b) modulation instability and (c) possible generation of second harmonic frequencies have been shown.

Self-focusing of intense laser beams propagating in magnetized plasma For a laser beam having Gaussian radial profile, the intensity is peaked on axis causing the plasma electrons to be expelled away from the axis. Therefore, the refractive index tends to maximize along the axis. Due to this refractive index gradient the phase velocity of the laser wavefront increases with the radial distance, causing the wavefronts to curve inwards and the laser beam to converge

 Consider a linearly polarized laser pulse propagating in a uniform plasma embedded in a constant external magnetic field  Basic equations describing the evolution of laser beam in magnetized plasma are  Wave equation AmplitudeWave numberFrequency Jha et al Phys. Plas. 13, (2006) (a) Linearly polarized laser beam propagating in transversely magnetized plasma

 Current density equation  Lorentz force equation and  Continuity equation is the relativistic factor and is the magnetic vector of the radiation field. Jha et al Phys. Plas. 13, (2006)

 Using perturbative technique all quantities are simultaneously expanded in orders of the radiation field. Using Eq. (4) first order velocities are given by  Presence of magnetic field increases the transverse quiver velocity of plasma electrons and also leads to the generation of a longitudinal velocity component due to force acting on plasma electrons.  is the cyclotron frequency and is the normalized field amplitude. Jha et al Phys. Plas. 13, (2006)

 The second order high frequency x-component of velocity is generated due to uniform magnetic field and reduces to zero in its absence. However z- component and third order tranverse velocities are modified due to external magnetic field.  Second and third order velocities are Jha et al Phys. Plas. 13, (2006)

 Density perturbations introduced in the plasma due to interaction with the laser beam can be obtained by expanding the continuity Eq. (5). Thus first order density perturbation is given by  The first order density perturbation arises due to the presence of external magnetic field and reduces to zero in its absence. The second order density perturbation is given by Jha et al Phys. Plas. 13, (2006)

 Perturbed velocities and densities are used to obtain the transverse current density [Eq. (3)].  Using the value of current density obtained with the help of Eq.(8) and using it in the wave equation and assuming the radiation amplitude to be slowly varying function of z, the paraxial wave equation is given by Linear current density Nonlinear current density terms external magnetic field Relativistic mass correction Jha et al Phys. Plas. 13, (2006)

 N includes nonlinear perturbations due to  Using source dependent expansion (SDE) method the equation for laser spot-size is obtained as  Relativistic effects  Density fluctuations  Coupling of radiation field with magnetic field Jha et al Phys. Plas. 13, (2006)

 Here is the normalized laser power and is the ‘Rayleigh Length’. defines the critical laser power for nonlinear self-focusing of a laser beam in magnetized plasma and its value is  A graphical analysis of the normalized laser spot size variation with propagation distance and magnetic field; and the variation of critical power with magnetic field is presented. Jha et al Phys. Plas. 13, (2006)

Fig. 1 Variation of r s /r 0 with z/Z R for (a) unmagnetized plasma (b) = 0.2 and (c) = 0.4, with,, and = 0.1.

Fig. 2 Variation of with at = 0.3 for 0.271, s -1 and = 0.1.

Fig. 3 : Variation of with for, s -1 and = 0.1.

(b) Circularly polarized laser beam propagating in axially magnetized plasma A circularly polarized laser beam propagating in plasma is embedded in a uniform, axial magnetic field. The normalized electric field vector of the radiation field propagating along the z- direction is represented by where k 0 and ω 0 are the normalized amplitude, wave number and frequency of the radiation field, respectively. σ takes values ±1for right or left circularly polarized radiation, respectively. Wave equation governing the propagation of a circularly polarized laser beam in presence of axial magnetic field is given by

Proceeding in the same manner as in the case of linearly polarized laser beam the spot-size of the circularly polarized laser beam is given by where S is given by

Fig. 4 Variation of r s /r 0 with z/Z R for (a) ω c /ω 0 = 0, (b) ω c /ω 0 =0.15, σ = -1 and (c) ω c /ω 0 =0.15; σ =+1with a 0 = and ω 0 = 1.88×10 15 s -1.

Fig. 5 Variation of r s /r 0 with ω c /ω 0 for right circularly polarized laser beam having z/Z R = 0.3, a 0 = 0.271, ω 0 = 1.88×10 15 s -1 and ω c /ω 0 = 0.1.

Fig. 6 Variation of r s /r 0 with ω c /ω 0 for left circularly polarized laser beam having z/Z R = 0.3, a 0 = 0.271, ω 0 = 1.88×10 15 s -1 and ω c /ω 0 = 0.1.

Modulation instability of laser pulses in axially magnetized plasma Modulation instability is the process in which the pump wave amplitude gets modulated in space or time. Modulation occurs due to the interplay between the nonlinearity and dispersive effects Due to this instability the actual wave number (k 0 ) of laser beam change into k 0 ±K (where K is modulation wave number). Modulation instability of a circularly polarized laser beam propagating through axially magnetized, cold and underdense plasma has been studied. The governing wave equation is

Considering only linear source term and taking the Fourier Transform of wave equation gives where is the Fourier Transform of slowly varying amplitude a 0 (,t) and is the linear part of the total refractive index, having contributions due to vacuum, finite spot-size of the laser radiation and presence of magnetized plasma respectively. Defining mode propagation constant and considering the limit that mode propagation constant is close to the unperturbed wave number (k 0 ), Eq. (15) may be written as

Using Taylor series expansion the frequency dependent function β m (ω) may be expanded about ω 0 as where. In Eq. (17) is related to the group velocity dispersion (GVD). Substituting Eq. (17) in Eq. (16), retaining terms up to β 2m (ω) and introducing nonlinear current source term on the right hand side gives the nonlinear non-paraxial wave equation as

In order to study the spatial modulation instability, transformations are carried out from spatial and temporal coordinates (z, t) in the laboratory frame to the spatial coordinates (z, ξ)in the pulse frame. The transformation is achieved by substituting ξ=z – v g t and z = z. Substituting the nonlinear parameter, setting β 0 = k 0, and neglecting in comparison to 2 Eq. (18) may be written in the 1-D limit as Solution of Eq.(19) may be written as where a 10 (z, ξ) is the perturbed beam amplitude and is the normalized laser power in presence of axial magnetic field.

The exponentially varying perturbed amplitude may be taken to be of the form where k is the propagation wave number of the perturbed wave amplitude. Taking to vary with z as exp(±Kz), where K is the modulation wave number, the dispersion relation for one-dimensional modulation instability is written as where, and are normalized dimensionless quantities.

Modulation instability is excited provided is sufficiently negative,, so that can be complex. Consequently the range of unstable wave numbers for which the instability exists is given by The growth rate of modulation instability for the laser beam propagating through transversely magnetized plasma is given by

Fig. 7 Variation of modulation instability growth rate for right (curve a), and left (curve c) circularly polarized laser beam propagating in magnetized plasma and for laser beam propagating in unmagnetized (curve b) plasma, with normalized wave number with r 0 =15μm, a 0 =0.271, ω 0 =1.88×10 15 s -1, ω p /ω 0 =0.1 and ω c /ω 0 =0.05 (curves a and b).

Fig. 8 Stability boundry curves showing the variation of normalized laser power with for right (curve a), left (curve c) circularly polarized laser beam propagating in magnetized plasma and unmagnetized case (curve b). The parameters used a 0 =0.271, ω 0 =1.88×10 15 s -1, ω p /ω 0 =0.1 and ω c /ω 0 =0.05.

Second harmonic generation in laser magnetized plasma interaction I t has been shown that when an intense laser beam interacts with homogeneous plasma embedded in a transverse magnetic field, second order transverse plasma electron velocity oscillating with frequency twice that of the laser field is set up This plasma electron velocity couples with the ambient plasma density leading to a transverse plasma current density oscillating at the second harmonic frequency. Also first order density perturbation oscillating at the laser frequency arises due to the presence of the magnetic field. This density perturbation couples with the fundamental transverse quiver velocity to give transverse plasma current density oscillating at twice the laser frequency.

 Consider a linearly polarized laser beam propagating along the z-direction as  As the beam propagates through transversely magnetized plasma, transverse current density at twice the laser frequency arises and acts as a source of second harmonic generation.  Corresponding to the frequencies and the electric fields are assumed to be given by Propagation constant Laser frequency Amplitude

 Here and. and are wave refractive indices corresponding to the frequencies and.  The equation governing the propagation of the laser pulse through plasma is given by where.  The plasma electron density is given by Plasma electron velocity Plasma electron density

 Relativistic interaction between the electromagnetic field and plasma electron is governed by  Lorentz force equation  Continuity equation Magnetic vector of radiation field Transverse magnetic field

 Using perturbative technique all quantities can be expanded in orders of the radiation field. Using Lorentz force equation the first and second order longitudinal velocities and second order transverse velocity of the plasma electrons is given by where is the cyclotron frequency of plasma electrons and and are normalized amplitudes.

 The first order plasma electron density is obtained by using continuity equation as The transverse current density can now be written as  From the current density equation it is observed that current density at second harmonic arises via  Transverse plasma electron velocity oscillations at second harmonic frequency.  Coupling of electron density oscillations at fundamental frequency and electron quiver velocity also oscillating at fundamental frequency. This contribution is attributed to the external magnetic field and provides source for the generation of second harmonic radiation.

 Linear fundamental and second harmonic dispersion relations are given by  For obtaining second harmonic amplitude the value of the current density is substituted in the wave equation and it is assumed that the distance over which changes appreciably is large compared with the wave length and that depletes (with z) very slightly so that quantity can be assumed to be independent of z.  The evolution of the second harmonic amplitude is given by where.

 The second harmonic conversion efficiency is defined as For a given conversion efficiency is periodic in z.  The minimum value of z for which is maximum is given by The length represents the maximum plasma length up to which the second harmonic power increases. For z > the second harmonic power reduces again.

 The maximum second harmonic efficiency, after traversing a distance is given by  The maximum conversion efficiency is zero in the absence of magnetic field and increases in with increase in magnetic field. However, near the electron cyclotron resonance the theory breaks down. The conversion efficiency also increases with the increase in the intensity of the laser beam.

Fig. 9 Variation of conversion efficiency (η) with the propagation distance z forω c /ω 0 =ω p / ω 0 =0.1, a 1 2 =0.09 and ω 0 =1.88×10 15 s -1.

Fig. 10 Variation of maximum conversion efficiency (η max ) with ω c /ω 0 for ω p /ω 0 =0.1, a 1 2 =0.09 and ω 0 =1.88×10 15 s -1.

Conclusions Transverse magnetization of plasma enhances the self-focusing property of the laser beam and the critical power required to self-focus the linearly polarized laser beam propagating in transversely magnetized plasma is reduced. This above explanation is also valid for a left circularly polarized laser beam propagating in axially magnetized plasma If the laser beam is right circularly polarized, the beam will be defocused. Focusing of the right circularly polarized beam can be brought about by reversing the direction of the external magnetic field. Magnetic fields alter the growth rate of modulation instability. The peak growth rate of modulation instability in the presence of the magnetic field for a left circularly polarized laser beam is found to increase while for right circularly polarized beam the spatial growth rate reduces as compared to the absence of magnetic field. The stability boundary curve shows that for left circularly polarized beam, the area representing the unstable interaction is increased while that for left circularly polarized laser beam it reduces.

It is seen that second harmonic conversion efficiency oscillates as the wave propagates along the z-direction. It is found that maximum conversion efficiency is zero in the absence of magnetic field and increases as the magnetic field is increased. The conversion efficiency also increases with increase in intensity of the laser beam. observation of second harmonics in homogeneous plasma could point towards the possibility of presence of a magnetic field, since second harmonics have so far been generated by the passage of linearly polarized laser beams through inhomogeneous plasma.

Journal Publications 1.Self focusing of intense laser beam in magnetized plasma Pallavi Jha, Rohit K. Mishra, Ajay K. Upadhyay and Gaurav Raj, Physics of Plasmas, 13, (2006). Also published in ‘Virtual Journal of Ultrafast Science’, 5, Issue 10 (2006) 2.Second harmonic generation in laser magnetized-plasma interaction Pallavi Jha, Rohit K. Mishra, Gaurav Raj and Ajay K. Upadhyay, Physics of Plasmas, 14, (2007). Also published in ‘Virtual Journal of Ultrfast Science’, 6, Issue 5, (2007) 3.Spot-size evolution of laser beam propagating in plasma embedded in axially magnetic field. Pallavi Jha, Rohit K. Mishra, Ajay K. Upadhyay and Gaurav Raj, Physics of Plasmas, 13, (2006).

Conference Proceedings 1.Interaction of laser pulses with magnetized plasma Rohit K. Mishra, Ajay K. Upadhyay, Gaurav Raj and Pallavi Jha Presented at ’20 th National Symposium on Plasma Science and Technology’ Cochin (2005). 2.Modulation instability of a laser beam in a transversely magnetized plasma Rohit K. Mishra, Ajay K. Upadhyay, Gaurav Raj and Pallavi Jha Presented at ’21 st National Symposium on Plasma Science and Technology’ Jaipur (2006). 3.Spot-size evolution in axially magnetized plasma Rohit K. Mishra, Ajay K. Upadhyay, Gaurav Raj and Pallavi Jha Presented at ’6 th National Laser Symposium’ Indore (2007). 4.Magnetic field detection via second harmonic generation Rohit K. Mishra, Ram G. Singh and Pallavi Jha Presented at ’22 nd National Symposium on Plasma Science and Technology’ Ahmedabad (2007).