Density Transition Based Self-Focusing of cosh-gaussian Laser Beam in Plasma with Linear Absorption
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1 Commun. Theor. Phys. 64 ( Vol. 64, No. 1, July 1, 2015 Density Transition Based Self-Focusing of cosh-gaussian Laser Beam in Plasma with Linear Absorption Niti Kant and Manzoor Ahmad Wani Department of Physics, Lovely Professional University, Phagwara , Punjab (Received February 27, 2015; revised manuscript received April 20, 2015 Abstract Density transition based self-focusing of cosh-gaussian laser beam in plasma with linear absorption has been studied. The field distribution in the plasma is expressed in terms of beam width parameter, decentered parameter, and linear absorption coefficient. The differential equation for the beam width parameter is solved by following Wentzel Kramers Brillouin (WKB and paraxial approximation through parabolic wave equation approach. The behaviour of beam width parameter with dimensionless distance of propagation is studied at optimum values of plasma density, decentered parameter and with different absorption levels in the medium. The results reveal that these parameters can affect the self-focusing significantly. PACS numbers: Hb, Dx Key words: self-focusing, cosh-gaussian beam, plasma density ramp, nonlinearity, linear absorption 1 Introduction The interaction of intense laser beams with plasmas has been a fascinating field of research due to various applications like laser electron acceleration, 1 3] inertial confinement fusion 4 6] and ionospheric modification 7 10] etc. These applications require large interaction region up to several Rayleigh lengths without loss of energy. When a high power laser beam interacts with the plasma, it provides an oscillatory velocity to the electron, which modifies the dielectric constant of the plasma, which leads to the relativistic self-focusing of the laser beam. 11] The steady-state self-focusing/defocusing of a laser beam in a medium is characterized by the real part of dielectric constant having saturating nonlinearity and the imaginary part being determined by multiphoton absorption under paraxial approximation. 12] Takale et al. 13] investigated the self-focusing and defocusing of first six TEM op Hermite Gaussian laser beams in collision-less plasma and found that modes with odd p-values defocus and that with even p-values exhibit oscillatory as well as defocusing character of the beam width parameter variation during laser propagation in collision-less plasma. However, it has been observed that a plasma density ramp of suitable length can reduce these oscillations. 14] The propagation of Hermite-cosh Gaussian (HchG laser beams in n-insb was studied for 0, 1 and 2 mode indices and incorporates the desirability of self-focusing effect in a particular application by exploiting the decentered parameter of the beam. 15] The relativistic selffocusing of cosh-gaussian laser beams in a plasma illustrates that oscillatory self-focusing takes place for decentered parameter; b = 0 & 1 and sharp self-focusing occurs for b = 2. 16] Nanda and Kant et al. 17] have observed that decentered parameter and ramp density profile are important for stronger self-focusing of laser beam. However, by considering the magnetic field and plasma density ramp for Hermite-cosh Gaussian laser beams, it has been found that the presence of plasma density ramp and magnetic field enhances the self-focusing effect to a greater extent. 18] The proper selection of decentered parameter is very much sensitive to self-focusing. 19] However, Kant et al. 20] have observed the effect of plasma density ramp and initial intensity of the laser beam on self-focusing of laser beam. Again, the parameters like density profile, intensity parameter, and decentered parameter play an important role in the improvement of self-focusing ability of laser beam. Further, the upward plasma density ramp in weakly relativistic and ponderomotive regime can accelerate the electron to higher energy over a long propagation distance as compared with uniform density relativistic plasma. 21] The oscillatory self-focusing takes place for different values of intensity parameter and with increase in intensity parameter, the distance between two consecutive points of intersection of two beams increases. 22] The laser beam is compressed and amplified in an enhanced manner by the combined effect of magnetic field, relativistic nonlinearity, and negative initial chirp. 23] It is to be noted that strong self-focusing is obtained by optimizing wavelength and intensity parameters of cosh-gaussian laser beams. 24] In the investigation of self-focusing and frequency broadening of laser pulse in water, the laser beam initially undergoes self-focusing due to Kerr nonlinearity and then nonlinear refraction takes place which causes the laser beam to defocus. 25] Also, it has been found that Supported by a Financial Grant from CSIR, New Delhi, India, under Project No. 03(1277/13/EMR-II nitikant@yahoo.com c 2015 Chinese Physical Society and IOP Publishing Ltd
2 104 Communications in Theoretical Physics Vol. 64 with increase of power density and control parameters lead to trapping of particle in the potential and hence strong focusing. 26] Laser pulse propagating in a plasma under plasma density ramp tends to become more focused. Further, if there is no density ramp, the laser pulse is defocused due to the dominance of the diffraction effect. As the plasma density increases, self-focusing effect becomes stronger. 27] Furthermore, the quantum effects play a vital role in laser-plasma interaction and significantly adds self-focusing in plasma as compared to that of classical relativistic case. 28] However, in addition to quantum effects the upward ramp density profile causes much higher oscillation and better focusing of laser beam in cold quantum plasma. 29] In the present communication, authors have studied the self-focusing of cosh-gaussian laser beam in plasma by taking into account the effect of plasma density ramp and linear absorption through parabolic equation approach under paraxial approximation. The second order differential equation governing the nature of self-focusing of the beam in plasma is obtained. The results are presented graphically and are discussed. Finally, a conclusion is drawn in the last section of the paper. 2 Field Distribution of cosh-gaussian Beams The field distribution of cosh-gaussian laser beam at the plane of z = 0 is described by 30 32] E(r, 0 = E 0 exp ( r2 cosh(ω 0 r, (1 where r 0 is the waist width of the Gaussian amplitude distribution, r is the radial coordinate of the cylindrical coordinate system, E 0 is the amplitude of the electric field at the centre position of r = z = 0 and Ω 0 is the parameter associated with the hyperbolic cosine function also called the cosh factor. On the other hand Eq. (1 can be expressed as follows: E(r, 0 = E ( 0 b 2 2 exp 4 + exp r 2 0 { ( r exp + b 2 ] r 0 2 ( r r 0 b 2 2 ]}, (2 where b = r 0 Ω 0 is the decentered parameter also called the normalized modal parameter. Now, in the presence of absorption alone, the energy concentration of the laser beam decreases by a factor of exp( 2 k i dz which weakens the nonlinear effect. Therefore, in compliance with Eq. (2, we can construct the following ansatz for the field distribution of cosh-gaussian laser beam propagating along the z-axis in a parabolic medium. E(r, 0 = E ( 0 b 2 2f exp 4 { ( r exp( 2k i z exp 2 ] r 0 f + b 2 2 ]}, (3 ( r + exp r 0 f b 2 where k i is the absorption coefficient and f = f(r, z is the dimensionless beam-width parameter of the laser beam in medium, which measures axial intensity and width of the beam. 3 Non-Linear Dielectric Constant The propagation of cosh-gaussian laser beam in a plasma is characterized by a dielectric constant of the form ε = ε 0 + Φ(EE. (4 With ε 0 = 1 ω 2 p/ω 2, ω 2 p = 4πn(ξe 2 /m and n(ξ = n 0 tan (ξ/d, where ε and Φ represent the linear and nonlinear parts of dielectric constant respectively, ω p is plasma frequency, e and m are the electronic charge and rest mass respectively, n 0 is the equilibrium electron density, ξ = z/r d is the normalized propagation distance, R d is the diffraction length and d is a dimensionless adjustable parameter. 4 Self-Focusing The wave equation governing the propagation of laser beam may be written as ( ω 2 2 ( E ε E + c 2 εe + = 0. (5 ε The last term of Eq. (5 on left hand side can be neglected provided that k 2 2 (ln ε 1 where k represents the wave number. Thus, ( ω 2 2 E + c 2 εe = 0. (6 This equation is solved by employing Wentzel Kramers Brillouin (WKB approximation. Under WKB approximation, one of the coupled equations, for intensity A 2 0 and eikonal S(r, z 33 35] of a laser beam in a nonlinear medium is obtained as: 1 ω 2 c 2 ω 2 ωp 2 0 tan(z/dr d ω2 p 0 zsec 2 (z/dr d 2dR d + 1 ( s 2 ω 2 c 2 ω2 ωp 2 0 tan(z/dr d ] r ωp sec 2 (z/dr d 4ω 2 c 2 d 2 Rd 2(ω2 ωp 2 0 tan(z/dr d ω 2 p 0 z(z + ssec 2 (z/dr d 2dR d (s + 2z ]( s z (ω 2 ωp 2 0 tan(z/dr d ] = 1 2 A 0 ω 2 A 0 r A ] 0 + Φ(A2 0 r r c 2. (7 The eikonal S = S(r, z which determines the convergence or divergence of the beam is S(r, z = r 2 /2β(z + ϕ(z. Here, ϕ is the phase factor and β(z = 1/f(z( f/ z represents the curvature of the wavefront. For a parabolic medium, i.e. the nonlinearity factor is proportional to A 2 0, we write Φ(A 2 0 = 1 2 ε 2A 2 0, (8 where ε 2 is the nonlinear coefficient. Employing the field distribution of Eq. (3 and following the procedure of Akhmanov et al. 33] and its extension by Sodha et al., 34] the general differential equation for the propagation of
3 No. 1 Communications in Theoretical Physics 105 cosh-gaussian laser beam in a parabolic medium with linear absorption is formulated as: 1 ω2 p 0 ω 2 tan(ξ/d ω2 ( p 0 ξ ] ω 2 sec 2 2 f (ξ/d d ξ 2 where + 1 ω2 p 0 ω 2 tan(ξ/d + ω2 ( p 0 ξ ] 1 ω 2 sec 2 (ξ/d d f = 2 ( f 3 1 ω2 ( ( p 0 ω 2 tan(ξ/d 5 + 3b2 1 + b2 2 4 ( r0 ω c ( f 2 ξ 2ε2 ( b E0 2 2 ] exp 2 k iξ, (9 k i = k i R d (r 0 ω/c 1 (ωp0 2 /ω2 tan(ξ/d is the normalized absorption coefficient. Equation (9 is the required equation for the beam width parameter and can be solved for f as a function of ξ for various k i levels, from which the variation of beam width parameter f with the dimensionless distance of propagation ξ for cosh-gaussian beams can be studied. 5 Results and Discussion Equation (9 is the second order nonlinear differential equation governing beam width parameter of cosh- Gaussian laser beam in plasma with density ramp and linear absorption. The self-focusing (convergence or defocusing (divergence of the laser beam is determined by the relative magnitude of nonlinear and diffraction terms of Eq. (9. The numerical solution of this equation is possible by using Runge Kutta method with following set of parameters; ω = rad/s, r 0 = 253 µm, d = 8, n 0 = cm 3 and the value of intensity is I 0 = W/cm 2. Further, by choosing suitable laser and plasma parameters, we investigate the focusing/defocusing of cosh-gaussian laser beam in plasma with density ramp profile. Figure 1 shows the variation of beam-width parameter f with normalized propagation distance ξ for decentered parameter b = 0 with ω p0 /ω = 0.2, 0.3, 0.4 & 0.5 for different absorption levels k i = 0.5, 0.6, 0.7 & 0.8. It is clear from the figure that the beam-width parameter first decreases, attains a minimum value and then increases due to the dominance of diffraction term. Strong selffocusing occurs at ω p0 /ω = 0.5, k i = 0.8 and then defocusing takes place as absorption weakens self-focusing effect. Density transition plays an important role to make the self-focusing early and stronger. The variation of beamwidth parameter f with normalized propagation distance ξ for b = 1 is depicted in Fig. 2. It is clear from the figure that sharp self-focusing is observed for ω p0 /ω = 0.5, k i = 1.3. So, with further increase in absorption level, the laser beam is defocused. This is because, the parameters like decentered parameter, plasma density ramp and absorption coefficient are such that they change the nature of self-focusing/defocusing of the laser beam significantly. Fig. 2 Variation of f(ξ with the normalised propagation distance (ξ for different values of k i and ω p0/ω at b = 1. Fig. 1 Variation of f(ξ with the normalised propagation distance (ξ for different values of k i and ω p0/ω at b = 0. Fig. 3 Variation of f(ξ with the normalised propagation distance (ξ for different values of k i and ω p0/ω at b = 2.
4 106 Communications in Theoretical Physics Vol. 64 that the absorption plays a vital role in the self-focusing effect and destroys the oscillatory self-focusing character of laser beam during propagation. Hence, in comparison to Refs. 36] and 37], by applying the density ramp and taking into account the effect of linear absorption, we observe that self-focusing occurs earlier even at ξ = Hence self-focusing length increases with absorption level under the influence of plasma density ramp. Further, we found that study of cosh-gaussian beams can be analyzed in a nonlinear medium like plasma, but the important thing is that the decentered parameter, absorption coefficient, and plasma density ramp are found to change the nature of self-focusing/defocusing of the laser beam significantly. Fig. 4 Variation of f(ξ with the normalised propagation distance (ξ for ω p0/ω = 0.2, k i = 2 and for different values of decentered parameter b. Figure 3 shows the variation of beam width parameter with normalized propagation distance for different combinations of ω p0 /ω and k i. Beam width parameter attains a minimum value at ω p0 /ω = 0.2 with k i = 2 and ω p0 /ω = 0.3 with k i = 3 for decentered parameter b = 2 which leads to strong self-focusing in plasma. Thereafter as soon as the values of ω p0 /ω and k i are increased, defocusing of laser beam takes place and beam width parameter decreases slowly. But, the self-focusing length increases with absorption level. However, in Fig. 4, the variation of beam width parameter with normalized propagation distance ξ is shown for decentered parameter b = 0, 1, 2 and keeping ω p0 /ω and absorption coefficient k i constant at 0.2 and 2 respectively. It is clear from Fig. 4 that sharp self-focusing occurs for b = 2 and for b = 0 and 1, the beam width parameter first decreases and then increases very slowly for lower values of b. Our results support the results obtained with different approach by Gill et al. 38] Figure 5 shows the variation of beam width parameter f with normalized propagation distance ξ for ω p0 /ω = 0.5 and b = 1 with different values of absorption level k i. It is important to notice that early and strong self-focusing occurs for k i < 1.2. After k i > 1.2, the beam width parameter increases slowly and self-defocusing takes place. However, Patil et al. 36] have studied the self-focusing of cosh-gaussian beams in a parabolic medium at various values of linear absorption (k i and decentered parameter (b and found that the self-focusing length increases with absorption level. Further, in the work of Navare et al., 37] while considering the collisional nonlinearity, they found Fig. 5 Variation of f(ξ with the normalised propagation distance (ξ for ω p0/ω = 0.5, b = 1 and for different values of absorption coefficient k i. 6 Conclusion This communication presents an analysis of the propagation of cosh-gaussian laser beam in plasma with density ramp and linear absorption using paraxial approximation. The effect of density ramp on the self-focusing of laser has been analyzed at different values of absorption level and decentered parameter. By optimizing laser and plasma parameters, the combined effect of plasma density ramp and linear absorption on self-focusing has been observed. The results show that self-focusing occurs earlier and becomes stronger under the influence of plasma density ramp. It is noticed that the decentered parameter, absorption coefficient, and plasma density ramp are found to affect the nature of self-focusing/defocusing of the laser beam significantly. References 1] H.Y. Niu, X.T. He, B. Qiao, and C.T. Zhou, Laser Part. Beams 26 ( ] J.X. Lee, W.P. Zang, Y.D. Li, and J.G. Tian, Opt. Exp. 17 ( ] S. Lourenco, N. Kowarsch, W. Scheid, and P.X. Wang,
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