Sharma, Ashutosh

Sodha, M P

Verma, M P

IACS Institutional Repository

Indian J. Phys. 79(4), 393-399 (2005).KtlOA,4i5 UP\Self-focussing o f  TEM io m ode laser beam s in p lasm asAshutosh Sharma, M P Verma and M S Sodha*Institute o f  Advanced Studies in Education, Univcrsit|| o f Lucknow, Lucknow-226 U07. Uttar Pradesh, IndiaV K ifipathiDepartment o f  Physics, Indian Institute ^ T ech n ology , New D elh i-110 016, India E-mail : insodha|l>rediffmail.comd eceived  13 July 2004, a tcepred  2 February 2005Abstract : The nonlinear propagation o f  a laser beam in the TEM lo mode, through a plasma, has been studied by expanding the eikonal around the axis and an intensity maximum. Three kinds o f nonlineamies viz ponderomoiive, collisional and relativistic, have been considered. It has been  observed that uneven refraction o f  the beam in a- and y directions, rules out the possibility o f u uniform waveguide mode propagation (Le. without convergence or divergence). The beam width parameters characterising convergence/di vcrgence in x  and y directions respectively, have been obtained by numerically integrating the corresponding differential equations. In the axial region, the beam suffers steady divergence, enhanced by nonlinearity. Por regions close to irradiance maxima, two coupled differential equations for beam widths in x  and y directions have been obtained. These have been solved for a set o f  typical parameters and a discussion o f  the results is presented.Keywords : Laser, plasma, self-focussingPACS Nos. : 52.38.H b, 52.35.M w1. In tro d u c tio nThe self-focussing o f laser beams in plasm as has been extensively investigated in the last three decades [1-12]. Most of these investigations consider beams with Gaussian distribution o f  intensity along the wavefront implying that the laser is operated in the TEMoo mode. However, there are situations where higher order modes (e,g. the TEMio mode) also becom e important. W hen lasers are operated with m irrors having annular output coupling apertures, the low est order m ode (TEMoo) is suppressed and higher order m odes, such as the TEMio mode are excited. Sodha e t al  [13] have studied the asymmetric focussing o f  this m ode in a dielectric with quadratic nonlinearity, for regions around the axis and around the maximum o f irradiance, the analysis was limited to the geometrical optics approxim ation. This treatm ent is valid for modest intensities as at higher intensities, the dielectric ^ s ta n t  tends to  saturate. The saturating nonlinearity C<>rTesponding Authormodifies the nature o f .self-focussing of Gaussian beam s. One expects similar effects in the case o f non-Gaussian beams.There are two regions o f the TEMio m ode laser beam, which are o f interest v/z. (i) around the axis jr = 0, y = 0 and (ii) around points o f m axim um  irradiance. Since a solution applicable to the whole wave front o f  the beam is not obtainable, the authors have adopted an approach [13], similar to that adopted in the paraxial approximation [1-3]. In this approxim ation [1—3, 13], the nature o f intensity distribution o f the beam  rem ains unchanged and beam width param eters f\  and f i  define the intensity distribution. It is well known that the results of paraxial approximation are not valid for points far from the axis, since the expansion is made as a series in (jc^  and y^). Hence, the intensity distributions for the paraxial (x = 0, y = 0) region and regions around the m ax im um  in te n s ity  a re  o f  d if fe re n t n a tu re ; th e<S) 2005  lA C S3 9 4 A sh u to sh  Sharm a, M  P  Verma, M  S  S o d h a  a n d  V  K  T ripa th im athem atical basis for this difference is the fact that the expansion is made in term s o f  and i.x—x\lf\jr ,respectively. In the approxim ations used, the beam  retains TEM io m ode structure with different values o f  f \  and /2  in the tw o regions o f interesl.In this paper, we study the self-focussing o f  a laser beam  in the TEMu) mode propagating in plasm a. Three ty p es o f  n o n lin e a r itie s  have been co n s id e re d  viz. ponderom otive, collisional and relativistic, w hich conform  to the saturating nature o f the dielectric constant & with increasing EE^. As expected, it is seen that the focussing o f  the beam  is asym m etric. Case (i) corresponds to steady divergence, because EE* —> 0 as .-v, y go to zero and consequently, the nonlinearity  is negligible. H ow ever for case (ii), depending on the initial values o f  the beam  p o w e r  an d  b e a m  w id th , (a ) s te a d y  d iv e rg e n c e , (b) oscillatory divergence and (c) self-focussing in x  and y  d irections are predicted. This result is qualitatively sim ilar to  the results o f  the recent analysis, related to the G aussian (TEM(X)) m ode [12].2. Self-focussingC onsider the propagation o f  a TEM io m ode laser beam in a plasm a along the ^-direction,£ (.x ,y ,z) = A (.r,y ,z )ex p (-/(< w /-/:z )) .A t z = 0AA*  = E ^ ix ^ /r o )e x p { - ( ^ ^  + r ^ )  (1)w here Eo and ro are constants; the intensity m axim a in the z = 0 plane lie at (± 0). Follow ing Sharm a et al[12] the d ielectric constan t ^  o f  the p lasm a can be expressed ase{x. y. z) -  eo(z) + eiix,y,z). (2)The JW K B  solution o f the one-dim ensional wave equation suggests that for a slow ly d iverging o r converging beam, one may w riteE C x,y ,z)  = > '»^)expx i ^ c o t - j ' ^ J e ^ J ^ d z - ^ J e ^ T ^ S ( x . y , z ) ^ ,  (3)w here the te rm ^ Q  (0 ) /^ o  included in A q and 5is an eikonal. Substitu ting  for E  in to  the w ave equation and separating the real and im aginary parts o f  the resulting equation, one obtainsdA(y ds BA^  ds 2Bz dx dx dy By * (4}and2 d S j d S ^ ^  f d S~dz V ax  J [ 3 VI . , 1(<.) A) Bx^ By^Wc solve these equations in tw o different limits.3, Axial beam propagationFor the portion o f  the beam  close to the z-axis.(• . (5)I y l« /"o , eq. (4) is satisfied byS (x ,y .Z )  = ^ P i ( z ) x ■ v u )and A ^ ( x ,y ,z )  =f i f i  r^A‘explr i f iIX i.( 6)(7)w here (p iz) is an arbitrary function o f z, w hich does not affect focussing, and/^ i(z)  = 1dzP A z ) (8»i _  ^f ,    ^ '  / 2 * dz  ‘and f i 2iz)  represent the curvatures o f  the wavefroni in the respective d irections and vanish for a plane wave front.The d ielectric constant can in general, be expressed as [2]e = £ o  = E o + 0 (a J ) . (4| IThe functional forni o f 0  depends on the specific operative j nonlinearity. For ponderom otive nonlinearity  [2,3]<!>(EE') = a ^ \ \ - e x p \  - ^ a ^ E E " ( 10)w here i2  = cop/oj  ^ cOp is the plasm a frequency corresponding to unm odified electron density  and electron rest maSwS m. M  is the ion m ass, and a  -  e^M f6m ^co^kffTQ , -e is the electron charge, ks  is the B oltzm ann constant and To the p lasm a tem perature.Substitu ting  EE* from  eq. (7) in eq. (1^ ’^expanding the function in pow ers o f  the coordinates i retaining term s upto squares o f  x  and y, one finally j obtainsSelf-focussing ofTEM io mode laser beams in plasmas 3954 w  / , / ,  mV,"The dimensionless quantity _ 2  m Eq _  Po^ " 4  A / / . / 2  / , /2 ( i l),s proportional to £ ^  and represents dimensionless beam power. Pit is the dimensionless beam power at z = 0. Thus,e ( i, y, c) = (1Q ^p'b J\The lowest order term in y  involves and is therefore neglected. Comparison with eq. (2) givesandr i pSubstituting for S, A q , ^o(z) and € i(a, y, z) in eq. (5) and equating the coefficients of and y^ separately to zero, one obtains( i - o = )2W V 1 _ 1ac / i{i + £ ? V p } ( 12)and (13)\_f i  'where P -  Pof\  ^ The param eters f  = rz/r(fa> and /!,) -  rQco/c represent dimensionless distance of propa­gation and dim ensionless beam width at e = 0, respectively. The boundary conditions corresponding to a plane wave front at ^  = 0, are^  = ^  = 0 and / ,  = / 2  = 1: ^  = 0. df d<f •' ^ ^(14)4. Propagation close to irradiance maximaThe propagation of the beam around the intensity maxima can be analyzed in a way similar to that in Section 3. One can use an approximation akin to the paraxial approximation by .substituting a = /o / i  + Ai and requiring 1 X\ l« ro . Eqs. (4) and (5) remain unchanged except that ^ d x  is replaced by d/dA,. The eikonal may be expressed a&t 2 2■■liicn A,-, satisfying eq. (4) can be written as(15)i J\J21 +rof{expl - |  1 +rof{expl -ro /z"where P'iU) = - 4 —f(U )  dz '  n i z )  dz(16)(17)The eikonal expansion may have a term linear in Ai. However, it does not contribute to the curvature of the wavefront and hence, self-focussing.4.1, Fonderomotive nonlinearity (collisionless case) : The dielectric constant e(x,y,z) for the ponderomotive nonlinearity may be expressed up to terms in x f  andas€ (A ,y .2 )  =  GoU) +  €i(x ,y ,z\where Gq (z) = 1 “-^2" exp(~ ;?.exp(~l)) andGi (A,y.z) = ~ i2^cxp(~ p .exp (-l))pSince the terms on RHS of eqs. (12) and (13) are always positive / ,  and fz  keep on increasing with increasing ^  implying steady divergence.It is also instructive to visualize the physics behind the steady divergence. Since near a = 0, y = 0, the electric field increases with increasing a, the refractive index also increases with increasing a; this augments the diffraction divergence. For the case of collisional and t^lativistic nonlinearities, qualitatively similar results are obtained.2a^xexp(“ l)j -V775 'o / i  ‘( 18)'b J2Substituting for S{x, y  z \  A l( x ,y ,z ) ,  S q(z) and G i(A,y,z) in eq. (5) and equating the coefficients of x^ and y^ separately to zero, one obtains1d ^ f { ^e „  (z)/,'^ 32Q^ p f  p .cxp{-\)  1 exp(p.cxp(-l))flfVz _____ l_  ^ £ > V |p .ex p (-l)eo  \ exp(p .exp(-l))(19)(20)3 9 6 A sh u to sh  Sharm a , M  P  Verma, M  S  S o d h a  a n d  V  K  T ripa th iw here p , =  P o /i ', P 2 = P 0/ 2 ' and the other sym bols are the sam e as in Section 3, At  ^ = ()(^ = 0). one has / r  — h*  — I and if in addition, the w avefront is plane, ~ ‘ T h u s , th e  b o u n d a ryc o n d itio n s  fo r in teg ra tin g  cqs. (19 ) and (20 ) are/ , '  = / 2 =  1. ■?= 0.•^ ..2. CoUisinnal notilinearily :In a collisional plasm a, ohm ic nonlinearity arises through the heating o f electrons and subsequent redistribution o f plasm a density. In this case, the dielectric constant can be written ase = \ - a - \  1 + — EE* (21 )w here a  and have the sam e m eaning as in collisionicss case, w hile s is a param eter characterizing the nature o f  collisions. Proceeding as before, one obtains for the portion o f  the beam  around the intensity  m axim a, the split parts o f  the d ielectric constant asande o { 2 )  = I - i 2 2 ( ,  + , , .e x p ( -D ) ^ )(l + p .e x p (- l) )^(22)2 * r'bV,'^ 2 f  >2'() J2(23)where p  represents the dim ensionless quantity ^  L fi f i   ^The differential equations for / , '  a n d c a n  then be obtained in the form :5. D iscussionThe behaviour o f  the portion o f  the beam  close to the axis is governed by eqs. (12) and (13). Eq. (13) directly  integrated for initially  plane w ave front (df/dq~^ 0 and /2  = 1 at = O) to get/z  -  a/ )Eq. (12) involves /2  and can be integrated numericallv after substituting this expression f o r / 2. The parameters/ and f 2 as functions o f  ^  have been graphically  repre.somcd in F igure 1 for su itable values o f  param eters 12, p„ and Piu (subscript zero indicates values at ^  = 0). The grapns are clearly as expected.Figure 1. Beam width parameters / ,  a n d /2 corrc.spcmding to x and v axes respectively, as functions o f  ^ fo r  £2^ = 0 .2 , Po^  * (axial beampropagation in co llision less plasma).d ^ / :  Id i^  e.)d ^ f i2, ~ .di'^  e o  (^)/2-(2 -  s ) f i “/>,’ p .e x p (- l)  (l + /j.e x p (- l) )^ " i2(l + p .exp{-I))^  2(24)(2 - .y) Q  p .o xp {-D" (25)s  is equal to 1 or —3 corresponding to collisions with neutrals and ions respectively. T he expression for the relativistic nonlinearity  (as pointed out by Sharm a et a l f l2 ] )  can be obtained from  the expressions given above by putting  s  =  1 and  ~  f  .From  eq. (19) (collisionless p lasm a), one observes that for d '^ f\!d ^ ^  = 0, to vanish at = 0, the initial beam  width /7o and initial beam  pow er po must satisfy2 _  e x p (p o e x p (- l) )” 2 £ ? ^ P (,e x p (-l)  ■(26)A correspond ing  relation for d '^ fy d ^ '^  to  vanish at ^  = 0, is2 _  e x p (p o e x p (- l) )  " £ /^ p o e x p ( - l )(27)TTie tw o conditions are different and can  not simul' taneously be satisfied fo r any choice o f  initial beamSelf-focussing o f  TEM,o m ode laser beam s in p lasm as 397v^ idtti and beam power. Since the beam power p  is equal even if one started with cE f\/d ^^  = 0, / ',  will ,t remain unchanged because / i  will be changing. One .^^ n^clucles that a uniform wave guide mode of propagation ,s just not possible for the TEMjo mode. Eqs. (26) and 1^ 7) represent relations between the initial beam width initial beam power shown by curves C (l)  and C(2) respectively in Figure 2. These curves are not like thepropagation. Obviously, if the point (/>o,/>o) lies in between the two curves or below C( 1) or much below C( I), the variation of /,' and yV will be very different. I ’hesc variations can be obtained by integrating eqs. (19) and(20) numerically. J'hc two equations arc not independent and must be integrated simultaneously. This has been done choosing different pairs of initial values (/?(>,/>(>) and the corresponding plots of /,' and / '^ as functions of ^ are given in Figures 3 to 6. Figure 3 corresponds to the pewnt {p(u/h)) lying above C(2) in Figure 2.Figure 2. Initial beam width Po versus initial beam power near irradiance f!K!xima for (collisionless plasma) with = 0 2. iP j indicated byi('( I j) andcritical power curve obtained for the Gaussian beam in so far as they are relevant only at ^  = 0; yet they are significant in as much as they determine the sign of and for any chosen initial values ofbeam width po  and beam power po. thus d - f \ /d ^ -  at f = 0 will be positive, zero or negative according as the point (poiA)) ties on the same side of the curve C (l)  as the origin, on the curve and on the other side of the curve, respectively. Similar conclusion can be drawn for at ^  = 0. The signs o f these quantities indicate whether / /  a n d / 2 ' will initially decrease or increase; negative sign indicates decrease viz* self-focussing, while the positive sign indicates initial divergence.If now, the initial point (po.Po) ties above the curve ^(2) and hence also above C (l)  (see Figure 2), then both d ^ f\fd ^^  and d ^ fy d ^ ^  arc initially = 0) negative, ^ is  implies that f ' and f y  both start falling below their initial values unity. In other words, the beam gets self- focussed in both directions in the early part of itsFigure 3. Beam width parameters / and f \  corresponding to a and y directions respectively, and their product /',./?  as functions o f dimensionless distance of propagation ^f(^r the point a(2, 5) ot FigureFigure 4. Beam width parameters/' , a n d /;  corresponding to the x  and y  directions respectively, function.? o f dimensionless distance o f  propagation^for the point b(3, 3) o f  Figure 2.398 A sh u to sh  Sharm a , M  P  Verma^ M  S  S o d h a  a n d  V  K  T ripa th iF igure  5. Beam width parameters f \  and corresponding to the x  and v directions respectively, as functions o f  dimensionless distance o f propagation f  for the point c(4, 2.5) o f Figure 2.Figure 6. Beam width parameters f \  and f '2 corresponding to the x  and y  directions respectively, as functions of dimensionless distance of propagation f  for the point d(l, I) of Figure 2.This figure also shows the variation of fifz^  its reciprocal determining the irradiance of the beam at Figure 4 corresponds to a point lying between the curves C(l) and C(2) and shows that f i '  experience's oscillatorydivergence w hile f \  undergoes an oscillatory  convergence in o ther w ords, gets self^focussed. F igure 5 depicts the results fo r a point (po,Po) lying below  the curve C (l) and Figure 6 plots f \  and f 2 for a point m uch below Q i)Figure 5 corresponds to  oscillatory  divergence, while F igure 6 corresponds to  steady divergence o f the bean both in jc and y  d irections.The above analysis and calculations can be conduciei for the collisional p lasm a (and relativistic plasm a) stanm< w ith eqs. (24) and (25). T he results so obtained an sim ilar to  those reported  fo r the co llisionless plasma discussed above.6. ConclusionsN ear the axis (jc = 0, y = 0), the electric  field and hence the refractive index increases w ith increasing x\ thi augm ents the diffraction divergence o f  the beam. Ho finite values o f x  (near the axis), the field  decreases wu increasing y; thus the nonlinearity  tends to converge ili beam  in the y direction effectively reducing the diffraciio divergence. This is obvious from  the slow er rise of /;. i F igure 1 com pared to that o f  f \ .For regions close to the irradiance m axim a, one obtain tw o coupled equations for beam  w idths f \  and in and y directions. T he critical curves for x  and y direction are d ifferent and so are the curves for the two bear w idth param eters /] ' and f 2 . Specific typies o f  nonlineariiie have been considered and dependence o f  f {  and j \ ’ n d istance o f  propagation, has been investigated  for typia values o f  the param eters.AcknowledgmentThe authors are grateful to  the D epartm ent o f  Scienc and T echnology, G ov ern m en t o f  Ind ia  fo r financii support.References[1] S A Akhraanov, A P Sukhorukov and R V Khokhlov Sov. Phv Usp. 10 609 (1968)(23 M S Sodha, A K Ghatak and V K Tripathi S e l f  focusing of Ust Beams in Dielectrics, Plasmas and Semiconductors (Delhi : Tai McGraw Hill) (1974)[31 M S Sodha. A K Ghatak and V KTripathi Prog. Opt. 13 169[4] E Esarey, P Sprangle. J Krall and A Ting JEEE J. Quant. E lectr 3 1879(1997)(51 T M Antonsen (Jr.) and P Mora Phys. Rev. Lett. 69  2204 (1992)[6] P Sprangle. E Esarey. J Krall and G Joyee Phys. Rev. Lett. 69 220 (1997)S e lf- fo cu ss in g  o j T E M k, m o d e  la se r  b ea m s in p la sm a s 399,7, D Subbarao. R Um a and H Singh P h ys. P la s m a s  S 3440 (1998) 1111 u  Garm.rc. R Y Chmo and C: H Townes P h ss . R ev. L e u .  16 347|8] Piero Chessa, Patrick M ora and T  M Anion,scn (Jr.) P hys, P la sm a s  < 1966)5 3451 (1998) H 21 A Sharma, G Prakash, M P Vcrma and M S Sodha P h ys . P la s m a s  10jt>] Li-M ing Chen and Hai Lin P h y s . P la s m a s  8 2974 (2001) 4079 (2(X)3)[JO] B Hafizi, A Ting, R F Hubbard, P Sprangle and .1 R Pcnano Pliws, M S  Scxlha, V P Nayvar and V K IVipalhi J. O p t.S a c . A m - 64 941P la sm a s  10 1483 (2003) ( 1974)

Self-Focussing of TEM10 Mode Laser Beams in Plasmas

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Self-Focussing of TEM10 Mode Laser Beams in Plasmas

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