Research Article Relativistic Propagation of Linearly/Circularly Polarized Laser Radiation in Plasmas

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1 ISRN Optics Volume 213, Article ID , 8 pages Research Article Relativistic Propagation of Linearly/Circularly Polarize Laser Raiation in Plasmas Sonu Sen, 1 Meenu Asthana Varshney, 2 an Dinesh Varshney 3 1 Department of Engineering Physics, Inore Institute of Science an Technology, Inore , Inia 2 Department of Physics, M. B. Khalsa College, Inore 4522, Inia 3 School of Physics, Vigyan Bhawan, Devi Ahilya University, Khanwa Roa Campus, Inore 4521, Inia Corresponence shoul be aresse to Sonu Sen; ssen.plasma@gmail.com Receive 27 May 213; Accepte 25 July 213 Acaemic Eitors: M. Hu, V. Matejec, an J. Shibayama Copyright 213 Sonu Sen et al. This is an open access article istribute uner the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite. Paraxial theory of relativistic self-focusing of Gaussian laser beams in plasmas for arbitrary magnitue of intensity of the beam has been presente in this paper. The nonlinearity in the ielectric constant arises on account of relativistic variation of mass. An appropriate expression for the nonlinear ielectric constant has been use to stuy laser beam propagation for linearly/circularly polarize wave. The variation of beamwith parameter with istance of propagation, self-trapping conition, an critical power has been evaluate. The saturating nature of nonlinearity yiels two values of critical power of the beam (P cr1 an P cr2 )forself-focusing. When P<P cr1 <P cr2 the beam iverges. When P cr1 <P<P cr2 the beam first converges then iverges an so on. When P>P cr2 the beam first iverges an then converges an so on. Numerical estimates are mae for linearly/circularly polarize wave applicable for typical values of relativistic laser-plasma interaction process in unerense an overense plasmas. Since the relativistic mechanism is instantaneous, this theory is applicable to unerstaning of self-focusing of laser pulses. 1. Introuction The interaction of ultrahigh-power laser beams with plasmas is not only of technological importance but also rich in a variety of nonlinear phenomena. These phenomena become particularly interesting an involve when the laser power is sufficiently intense to cause the electron oscillation (quiver) velocity to become relativistic [1]. An important process that can affect the size of the focuse spot of the raiation is selffocusing. In recent years the fiel of laser plasma interaction in the relativistic regime has been ientifie as an emerging area an is often referre to as high-fiel science. Several complex phenomena are inclue in the area of high-fiel science. These phenomena (on account of their nonlinear nature) are all significantly affecte by the fiel istribution in thebeam,anhence,self-focusingoccupiesauniqueposition inthefielasitaffectsallotherphenomena. Relativistic electron motion in a plasma ue to an intense laser pulse moifies the refractive inex an leas to two effects: relativistic inuce transparency an relativistic selffocusing. In ense plasma with ω < ω p, light cannot propagateanisreflectefromthesurface.however,for relativistic intensities generating large γ factors, the plasma becomes transparent. The epenence of the electron mass on intensity causes significant change of characteristic properties of the plasma an its nonlinear processes; the increase in effective electron mass ecreases the electron plasma frequency an makes the penetration of electromagnetic waves into the overense area of plasma, which is known as self-inuce transparency or electromagnetically inuce transparency, an is the cause of relativistic self-focusing. A combination of the relativistic self-focusing an relativistic inuce transparency enables transmission of laser energy eep into plasmas which is useful for fast ignition of inertial fusion [2]. This so-calle penetration sensitively epens on the focal position of the laser intensity ue to the inhomogeneous ensity profile of the plasma an convergence of the laser pulse by final focusing optics. For ultrafast laser pulses lasting the orer of a picosecon or less, the rift velocity of electrons in a plasma can be comparable to the velocity of light, causing a significant increase in the mass of the electron an consequently in the

2 2 ISRN Optics effective ielectric constant of the plasma. This leas to selffocusing of the laser beam, as pointe out [3]. The fact that the relativistic mechanism is the only mechanism of selffocusing that can manifest itself for subpicosecon pulses makes the unerstaning of this mechanism very important [4, 5]. The earlier analysis concerne with relativistic selffocusing of laser beams in plasmas ignore the effect of the saturating nature of the nonlinear ielectric constant an was essentially a perturbation treatment, base on the quaratic epenence of the ielectric constant on the electric fiel of the beam [6]. Hence, such an analysis has limite applicability to unerstan the self-focusing of beams with arbitrarily large electric fiel an consequently arbitrarily large nonlinear part of the ielectric constant. Relativistic focusing of raiation beams in plasma is applicable to pulse length τ R in the range 1/ω < τ R <r s /C s, where C s is the ion-acoustic spee [7], an can arise owing to the creation of a ensity epression in the plasma as well as to an increase in the electron mass ue to relativistic effects. The time scale for self-focusing ue to a ensity epression is longer than that for self-focusing ue to relativistic mass increase. The latter time scale is τ R ω 1,whereω is the raiation frequency an is assume to be much greater than the electron plasma frequency. The analysis presente here will be concerne only with relativistic self-focusing on a time scale sufficiently short that the plasma ensity profile oes not evolve significantly uner the influence of the raiation beam. This implies that the pulse length τ R of the raiation beam must be short compare with τ s =r s /C s (the time scale for the ensity epression to occur) an, of course, long compare with a raiation perio τ R.Wemakeuseofthesteay-state paraxial-ray theory, as evelope by Akhamanov et al. [8]an Soha et al. [9]. The organization of the paper is as follows. In Section 2 the nonlinear ielectric constant ue to relativistic variation of mass for a time-harmonic plane wave (i.e., a linearly/circularly polarize wave) is foun. Section 3 is concerne with the relativistic self-focusing equation relating the variation of beamwith parameter with istance of propagation. The self-trapping conition an the critical power are evaluate in Section 4. Results an iscussion are mae in Section Nonlinear Dielectric Constant of the Plasma The relativistic equation of motion for a point charge is t [ m k ]=e [E + (k B)], (1) (1 V 2 /c 2 1/2 ) where m is the rest mass of a point charge, c is the velocity of light in vacuum, E is the electric fiel, an B is the wave magnetic fiel. For a homogeneous plane wave traveling in the irection of a unit vector n, theelectricanmagnetic vectors are perpenicular to n an are relate by B = n ( E ). (2) c Substituting for B on the right-han sie of the equation of motion, (1) gives t [ k (1 V 2 /c 2 ) ]= e 1/2 m (1 n k c ) E + e mc (E k) n. (3) Consiering the scalar prouct of (3)with n yiels t [ k (1 V 2 /c 2 ) ]= e (E k). 1/2 (4) mc Equation (4) together with the energy equation establishing that t [ mc 2 ]=ee k (5) (1 V 2 /c 2 1/2 ) 1 n k/c λ [ ] (6) (1 V 2 /c 2 1/2 ) is constant. We now eliminate t in favour of the phase variable φ=ω(t n r ). (7) c Here ω is the angular frequency of the wave, an r is the positive vector of the particle. Then k = r t =ω(1 n k c ) r, (8) where the prime enotes ifferentiation with respect to φ,an using (6) t [ k (1 V 2 /c 2 ) ]=λ 1/2 t ( k 1 n k/c ) =ω 2 λ(1 n k c ) r. Substitution of (8)an(9)into(3) enables the factor (1 n k/c) to be cleare an leaves r e = λmω 2 [E + ω c (E r ) n] (1) which is a secon-orer linear ifferential equation for r as a function of φ. The ielectric constant of the plasma is given by (9) ε=1 ( ω2 p ω 2 ) ( m m r ), (11) where m r = γ mis the relativistic mass, γ = (1 V 2 /c 2 ) 1/2 is the relativistic factor, an ω p =(4πn e 2 /m) 1/2 is the plasma

3 ISRN Optics 3 frequency in the absence of the beam. For a circularly polarizewave,thecomponentsof(1)are x = y = ee cos φ λmω 2, ee sin φ λmω 2, z = eex cos φ 8m 2 λ 2 cω 3, (12) where φ = (ωt kz) is the phase, an ω an k approximately satisfy the linear ispersion relation ω 2 =ω 2 p +c2 k 2. (13) Using (12), theaverage relativisticfactoris where γ = [1 + e2 E 2 1/2 m 2 ω 2 c 2 C 1], (14) C 1 =1+ e 2 E 2 16λ 2 m 2 ω 2 c 2 (15) for self-focusing consiering the orer of electric fiel up to secon orer an that the value of C 1 is unity for circularly polarize wave [1] an half for linearly polarize wave. The ielectric constant can be written as [9] where is the linear part of ielectric constant, ε=ε +φ(e E ), (16) ε =1 ( ω2 p ω 2 ) (17) φ(e E )=ω2 p ω 2 [1 + (1 1 2 αe E 1/2 C 1) ] (18) is the nonlinear relativistic term, an α= 2e 2 m 2 ω 2 c 2. (19) The effective ielectric constant for arbitrary large nonlinearity can be written as ε=ε ε 1 (f) r 2, (2) where f(z) is the imensionless beamwith parameter an ε an ε 1 represent, respectively, the linear an nonlinear ielectric constant for arbitrary nonlinearity. 3. Relativistic Self-Focusing Equation Consier the propagation of Gaussian laser beam of frequency ω alongz-irection; at z = theintensity istribution of the beam is given by EE =E 2 exp ( r2 r 2 ), (21) an here, r is the raial coorinate of the cylinrical coorinate system, r is the initial beamwith, an E is the axial amplitue. The wave equation governing the electric vector of beam in plasmas with the effective ielectric constant (2)is 2 E ( E) + ω2 c 2 εe =. (22) To solve the prevoius equation, the Wentzel-Kramers- Brillouin (WKB) approximation has been use. In the WKB approximation, the secon term ( E) of (22) hasbeen neglecte, which is justifie when (c 2 /ω 2 ) 1/ε 2 ln ε 1. The electric vector of the main beam can be expresse as 2 E + ω2 c 2 εe =. (23) In the presence of the moifie backgroun electron concentration ue to relativistic nonlinearity, the intensity istribution of the beam insie the plasma can be obtaine by using the WKB an paraxial ray approximations. Following Akhmanov et al. [8] ansohaetal.[9] thesolutionfore can be written as 1/2 k () E (r, z) =A(r, z) [ k(f) ] exp ( i k(f)z), (24) where k(f)= ω c [ε (f)]1/2, k () = ω c [ε (f = 1)]1/2. (25) Substituting for E an ε from (24) an(2) in(23), one obtains 2ik (f) A z + 2 A r A r r ω2 c 2 ε 1 (f) r 2 A=. (26) Putting A = A (r, z) exp [ iks(r, Z)] in (26) anseparating real an imaginary parts we get 2( S 2 z )+( S r ) + ω2 c 2 ε 1 (f) k 2 (f) r2 1 ( 2 A k 2 (f) A r A r r )=, (27) A 2 z 2 + S A 2 S r r +A2 ( 2 r S ) =, (28) r r

4 4 ISRN Optics where S(r, z) is calle the eikonal an relate to the curvature of the wavefront. For a slightly converging/iverging beam the solution for a Gaussian laser beam (21)canbewrittenas S= r2 2 β (z) +φ(z), (29) A 2 = E2 r2 exp ( f2 r 2 ), (3) f2 where β= 1 f f z. (31) Itcanbeseenpreviouslythatβis the inverse of raius of curvature of the wavefront an r f is the with of the beam. In the geometrical optics approximation, r=r f(z) represents arayinaplanecontainingthez-axis. On substituting for S into (27), using the paraxial ray approximation, such that (r/r f) 1, equating the coefficients of r 2 on both sies of the resulting equations an substituting for β into (29), the imensionless beamwith parameter is given by k 2 (f) 2 f z 2 = 1 ω2 r 4 f3 c 2 ε 1 (f) f. (32) Transforming the coorinate z an the initial beamwith r to imensionless forms ξ=( zc r 2ω), ρ =( r ω c ), we get the characteristic beam propagation equation as (33) 2 f ξ 2 = 1 ε (f) ρ2 r2 ε 1 (f) f3 ε (f) f. (34) Physically, (34) governs the variation of beamwith parameter f with istance of propagation. Substituting for E from (24)anS, A,anβ from (29) (31)into(2)anusingthe paraxial ray approximation, φ canbewrittenas φ(ee ) E 2 k () φ( { k(f) ) r2 { { α ω2 p k () ω 2 k(f) [1+ (1+ E 2 4r 2f4 k () k(f) α E2 k () k(f) α E2 3/2 8λ 2 )] [1+ 1 4λ 2 k () k(f) α E2 ] (35) correct to terms in r 2,where E 2 k () φ( k(f) ) { k () ω 2 { 1 [1 + k(f) α E2 { = ω2 p (1+ k () k(f) α E 2 1/2 16λ 2 )]. (36) Equation (35) can put into the convenient form of (2), with [11] ε =ε + ω2 p { k () ω 2 { 1 [1 + k(f) α E2 { ε 1 =α ω2 p k () ω 2 k(f) E 2 4r 2f4 (1+ k() k(f) α E 2 1/2 16λ 2 )], { {[1 + k() k(f) α E2 k() (1 + 2f2 { k(f) α E2 [1+ 1 k () 4λ 2 k(f) α E2 ]. 3/2 8λ 2 )] (37) Consier secon orer of nonlinearity, for circularly polarize wave; we can rewrite the previous set of equations as ε (f) = 1 ω2 p k () (1 + ω2 k(f) ε 1 (f) = ω2 p αe 2 k () 4ω 2 (1 + f4 k(f) r 2 αe 2 1/2 ), αe 2 3/2 ). (38) On substituting values of ε (f) an ε 1(f) from (38) in(34), the equation governing the beamwith parameter is 2 f ξ 2 = 1 [1 ω2 p (1 + k() αe ω 2 2 ) 1/2 ]f k(f) 3 ω 2 p ρ 2 4ω 2 r2 αe 2 r 2f4 [1 ω2 p ω 2 (1 + k() k(f) αe 2 αe 2 ) 3/2 (1 + k(f) k() ) 1/2 ] f. (39)

5 ISRN Optics D.7 D 1.2 Relativisitc factor (γ) Dielectric function Axial intensity 1 17 Figure 1: Depenence of relativistic factor on axial intensity for linearly an circularly polarize laser beam. In (39) the first term on right-han sie is responsible for iffraction ivergence of the beam, an the secon term correspons to convergence, arising ue to nonlinearity through therelativisticfactoroflinearlyorcircularlypolarizebeam, which contributes to focusing. 4. Self-Trapping Conition When the two terms on the right-han sie of (39) balance each other the beam propagates without convergence or ivergence, which is referre to as uniform waveguie propagation or self-trapping of the beam. For an initial plane wavefront of the beam the initial conitions on f are f(ξ = ) = 1 an (f/ξ) ξ= =.Asboththetermsonright-han sie cancel each other hence at z=, ( 2 f/ξ 2 ) ξ= =;since (f/ξ) is also zero an f=1at z=, f=1for all values of z; in other wors beam can propagate without convergence or ivergence. Therefore, the conition for self-trapping is Axial intensity 1 17 Figure 2: Variation of ielectric function ε as function of axial intensity for linearly (soli line) an circularly (ot line) polarize laser beam. Curves D.7 an D 1.2 correspon to (ω p /ω) 2 =.7 an 1.2, respectively. Dimensionless beam raius ρ Axial intensity D.7 D Figure 3: Critical curves, that is, the relation between imensionless beam raius ρ =(r ω/c) an axial intensity for linearly (soli line) an circularly (ot line) polarize laser beam. Curves D.7 an D 1.2 correspon to (ω p /ω) 2 =.7 an 1.2, respectively. ρ 2 Lin = 4ω2 (1 + αe 2 /2)3/2 ωp 2 αe 2/2 for linearly polarize beam an (4a) ρ 2 Cir = 2ω2 (1 + αe 2 )3/2 (4b) ωp 2 αe 2 for circularly polarize beam. The critical power of the beam is P= c 8π ε 1/2 E 2 cr exp ( r2 r 2 )2πrr, P= c 8π r2 E2 cr [ε (f = 1)]1/2. (41) 5. Results an Discussion The ifferential equation (39) is to stuy the propagation of linearly an circularly polarize beams through plasma. We have chosen parameters accessible for interaction of such beams with moerately unerense plasma to slightly overense plasma. The numerical values are chosen for laser intensity ranging between I =1 16 to 1 18 W/cm 2, electron ensity between.1 to 1.5 of critical ensity, laser frequency ω = sec 1 aninitialraiusofthebeamr laying between 1 to 3 μm. The ielectric function given by (2) an beam propagation equation (39)togetherwithbounary conitions, are numerically solve. Further we consier the propagation of the beam in axially inhomogeneous plasma by incluing linear ensity variation in plasma.

6 6 ISRN Optics Beamwith parameter (f) I VI V VII IV Unerense plasma II III (a) Beamwith parameter (f) II III I Overense plasma IV (b) Figure 4: (a) Variation of beamwith parameter (f) with imensionless istance of propagation (ξ) for linearly (soli line) an circularly (ot line) polarize beams. Curves I, II, III, IV, V, VI, an VII correspon to (P/P cr1 ) =.5, 5, 1, 2, 6, 1, an 2, respectively. Here (ω p /ω) 2 =.7, ρ =4,an (P cr2 /P cr1 ) = (for linearly polarize laser beam) an (for circularly polarize laser beam). (b) Variation of beamwith parameter (f ) with imensionless istance of propagation (ξ) for linearly (soli line) an circularly (ot line) polarize beams. Curves I, II, III, an IV correspon to (P/P cr1 ) = 1, 6, 1, an 2, respectively. Here (ω p /ω) 2 = 1.2, ρ =3,an(P cr2 /P cr1 ) = 99.6 (for linearly polarize laser beam) an (for circularly polarize laser beam). Figure 1 shows the variation of relativistic factor γ versus intensity for linearly an circularly polarize laser beam. From the graph it is evient that the growth in relativistic factor for circularly polarize laser beam is more as compare to linearly polarize beam. Figure 2 shows plot of ielectric function versus axial intensity for linearly an circularly polarize beams in both unerense an overense plasmas corresponing to ifferent concentration of electron ensity. It is seen that initially there is an increase in ielectric function an after a certain value it attains a constant value. Becauseofverystrongfiels,thenonlinearitysaturates.Asa result the electromagnetic wave rives all of the plasma out of the regions of large fiel intensity an establishes equilibrium with raiation insie a vacuum channel surroune by plasma. As we move from unerense region to overense region the saturation effect slows own. The critical curve (i.e., a plot of normalize raius of the beam ρ against the axial intensity proportional to critical beam power) (4a) an(4b) isrepresentebyfigure 3 for linearly an circularly polarize beams at Ω 2 p = (ω p/ω) 2 =.7 an 1.2. Equations(4a) an(4b) havetwocritical powers P cr1 an P cr2, corresponing to two ifferent values of electric fiel strength (or intensity of electric fiel), say E ocr1 (or I ocr1 ) an E ocr2 (or I ocr2 ) such that (E ocr1 < E ocr2 ). Thebeamcanbeself-focusewhenitspowerP lies between the two critical values (P cr1 < P < P cr2 ),antherange (P cr2 P cr1 ) increases rapily with increasing normalize raius. The region above the critical curve (i.e., P cr1 < P < P cr2 ) is the self-focusing region, an region below the critical curve is known as the efocusing region. The effect of ifferent concentration of electron ensity can be clearly seen from these critical curves. As concentration of electron ensity increases, that is, if we move from unerense region to overense region the critical curve shifts ownwar. Figures 4(a) an 4(b) show the variations of imensionless beamwith parameter f with imensionless istance of propagation ξ at relativistic intensity in plasma for ifferent values of electron ensity in unerense an overense plasmas. From the critical curves for Ω 2 p =.7(D.7)anρ = 4, the value (P cr2 /P cr1 ) = for linearly/plane polarize beam, an (P cr2 /P cr1 ) = for circularly polarize beam. At Ω 2 p = 1.2 (D 1.2) anρ =3, (P cr2 /P cr1 ) = 99.6 for linearly/plane polarize beam, an (P cr2 /P cr1 ) = for circularly polarize beam. The beam shows focusing/efocusing for ifferent values of P/P cr1.thisisbecauseaswechange the strength of the electric fiel, the maxima an minima get shifte. For a given value of ρ when (P cr1 <P<P cr2 ),the secon term on the RHS of (39) ominates the first term at z=,an( 2 f/ξ 2 ) is negative; hence, f ecreases with the istance of propagation. For certain value of ξ, the two terms on RHS of (39) get cancelle resulting in ( 2 f/ξ 2 ) =,anforlargervaluesofξ, RHS becomes positive. However, the beam still continues to converge on account of the curvature it has alreay gaine, though (f/ξ) becomes less an less negative. At certain ξ, ( 2 f/ξ 2 ) = an f = f min. Beyon this point, the iffraction term ominates over the nonlinear term, an hence f exhibits the oscillatory behavior. From the figures it is evient that ue to relativistic mass effect, increase in effective electron mass ecreases the electron plasma frequency an makes the penetration of the beam into the

7 ISRN Optics Dimensionless axial intensity (q /f 2 ) V I IV III II Dimensionless axial intensity (q /f 2 ) I II (a) (b) Figure 5: (a) Variation of imensionless axial intensity (q /f 2 ) with imensionless istance of propagation (ξ) for linearly/plane (soli line) an circularly (ot line) polarize beams. Curves I, II, III, IV, an V correspon to (P/P cr1 ) = 5, 1, 2, 6, an 1, respectively. Remaining parameters are as mentione in Figure 4(a). (b) Variation of imensionless axial intensity (q /f 2 ) with imensionless istance of propagation (ξ) for linearly (soli line) an circularly (ot line) polarize beams. Curves I an II to (P/P cr1 ) = 1 an 6, respectively. Remaining parameters areasmentione infigure 4(b). Beamwith parameter (f) (I, D.7 ) (II, D 1.2 ) Dimensionless axial intensity (q /f 2 ) (II, D 1.2 ) (I, D.7 ) (a) (b) Figure 6: (a) Variation of beamwith parameter (f)with imensionless istance of propagation (ξ) for linearly (soli line) an circularly (ot line) polarize beams. Curves (I, D.7 ) correspon to (ω p /ω) 2 =.7(1+2.5ξ), ρ =4,an(P cr2 /P cr1 ) = (for linearly polarize laser beam) an (for circularly polarize laser beam), an curves (II, D 1.2 ) correspon to (ω p /ω) 2 = 1.2( ξ), ρ =3,an(P cr2 /P cr1 ) = (for linearly polarize laser beam) an (for circularly polarize laser beam). (b) Variation of imensionless axial intensity (q /f 2 )with imensionless istance of propagation (ξ) for linearly (soli line) an circularly (ot line) polarize beams. Curves (I, D.7 ) correspon to (ω p /ω) 2 =.7( ξ), ρ =4,an(P cr2 /P cr1 ) = (for linearly polarize laser beam) an (for circularly polarize laser beam) an curves (II, D 1.2 ) correspon to (ω p /ω) 2 = 1.2( ξ), ρ =3,an(P cr2 /P cr1 ) = (for linearly polarize laser beam) an (for circularly polarize laser beam).

8 8 ISRN Optics overense region of plasma, which is known as self-inuce transparency. At higher concentration of electron ensity beam gets more focuse as compare to low concentration of electron ensity. The variation of imensionless axial intensity (αe 2 /f2 ) = (q /f 2 ) of the beam with istance of propagation ξ corresponing to Figures 4(a) an 4(b) for self-focusing region is epicte in Figures 5(a) an 5(b).The intensity peaks for higher concentration of electron ensities are higher with a reuce istance between two successive peaks. Figure 6(a) illustrates the variations of imensionless beamwith parameter f with istance of propagation ξ for ifferent concentrations of electron ensities in self-focusing region with linear increase in electron ensity. Careful observation of the results from figure inicates that for linearly increasing ensity plasma, minimum value of the normalize beamwith parameter (f min ) for the secon an higher orers ecreases continuously in unerense an overense plasmas. In unerense plasma it propagates long istance as compare to overense plasma, but its focusing is less. The corresponing graph for variation of imensionless axial intensity (q /f 2 ) of the beam versus istance of propagation ξ is shown in Figure 6(b). From the numerical analysis an graphs it is evient that both linearly polarize an circularly polarize beams can propagate through unerense plasma as well as they penetrate overense plasma or say they exhibit self-inuce transparency effect. In unerense plasma there is not much ifference in propagation of linearly polarize an circularly polarize beams, while the propagation is affecte in case of overense plasma an focal point changes. A threshol exists for onset of self-focusing, as this effect must overcome the spreaing of the beam ue to iffraction. A light beam is focuse to be self-trappe at any arbitrary iameter an will thus not sprea. Further, self-trapping occurs at a critical power level, which is inepenent of the beam iameter. The theory presente here has a much wie range of application than the previous one, because it is not a perturbation treatment. Furthermore, it is applicable at all intensities high an low. [7] E. Esarey, A. Ting, an P. Sprangle, Relativistic focusing an beat wave phase velocity control in the plasma beat wave accelerator, Applie Physics Letters, vol. 53, no. 14, pp , [8] S.A.Akhmanov,A.P.Sukhorukov,anR.V.Khokhlov, Selffocusing an iffraction of light in a nonlinear meium, Soviet Physics Uspekhi,vol.1,no.5,article69,1968. [9] M.S.Soha,A.K.Ghatak,anV.K.Tripathi,Self-Focusing of Laser Beams in Dielectrics, Semiconuctors an Plasmas, Tata McGraw-Hill, New Delhi, Inia, [1] M. Asthana, Relativistic self-focusing of laser beams of arbitrary intensity in plasmas, Inian Pure an Applie Physics,vol.31,p.564,1993. [11] M. Asthana, M. S. Soha, an K. P. Maheshwari, Relativistic self-focusing of laser beams in time-harmonic plane waves: arbitrary intensity, Plasma Physics, vol.51,no.1,pp , References [1] D. Umstater, Relativistic laser-plasma interactions, Physics D, vol. 36, Article ID R151, 23. [2] A. L. Lei, K. A. Tanaka, R. Koama et al., Stuy of ultraintense laser propagation in overense plasmas for fast ignition, Physics of Plasmas, vol. 16, no. 5, Article ID 5637, 29. [3] H. Hora, Theory of relativistic self-focusing of laser raiation in plasmas, JournaloftheOpticalSocietyofAmerica, vol. 65, no. 8, pp , [4] G. Schmit an W. Horton, Self-focusing of laser beams in the beatwave accelerator, Comments on Plasma Physics an Controlle Fusion,vol.9,p.85,1985. [5] G.Z.Sun,E.Ott,Y.C.Lee,anP.Guzar, Self-focusingofshort intense pulses in plasmas, Physics of Fluis, vol. 3, p. 526, [6] M. R. Siegrist, Self-focusing in a plasma ue to poneromotive forces an relativistic effects, Optics Communications, vol. 16, no. 3, pp , 1976.

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