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Gupta et al. Vol. 24, No. 5 /May 2007/J. Opt. Soc. Am. B 1155

Plasma density ramp for relativistic self-focusingof an intense laser

Devki N. Gupta, Min S. Hur, Ilmoon Hwang, and Hyyong Suk

Center for Advanced Accelerators, Korea Electrotechnology Research Institute, Changwon 641-120, Korea

Ashok K. Sharma

Center for Energy Studies, Indian Institute of Technology, New Delhi 110-016, India

Received August 28, 2006; revised December 22, 2006; accepted December 26, 2006;posted January 12, 2007 (Doc. ID 74334); published April 17, 2007

It is known that a high-power laser propagating through an underdense plasma can acquire a minimum spotsize due to relativistic self-focusing. Beyond the focus, the nonlinear refraction starts weakening, and the spotsize of the laser increases, showing periodic self-focusing/ defocusing behavior with the distance of propagation.To overcome the defocusing, we propose the introduction of a localized upward plasma density ramp. In thepresence of an upward ramp of plasma density, the laser beam obtains a minimum spot size and maintains itwith only a mild ripple. For suitable parameters of the laser and the plasma, we have deduced conditions forthe self-focusing. This kind of plasma density ramp may be observed in a gas-jet plasma experiment and re-sembles a plasma lens. © 2007 Optical Society of America

OCIS codes: 260.5950, 190.4410, 190.3270.

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. INTRODUCTIONnteraction of an intense laser (intensity of �1019 W/cm2

r higher) with an underdense plasma is a major area ofxperimental and theoretical study due to its relevance toaser-driven fusion, laser-driven accelerators, x-ray la-ers, and other areas.1–6 In these applications one needshe laser pulse to propagate several Rayleigh lengthshile preserving an efficient interaction with the plasma.ith the advent of a very high-power source of electro-agnetic radiation, the electron velocity in a plasma may

ecome quite large (comparable to the light velocity inree space). Hence, the effect of relativistic mass variationust be taken into account. The relativistic effect of an

ntense laser propagation in a plasma leads to self-ocusing because the dielectric constant of a plasma is anncreasing function of the intensity. The ponderomotiveorce of the focused laser beam pushes the electrons out ofhe region of high intensity, reduces the local electronensity, and increases the plasma dielectric function fur-her, leading to even more self-focusing of the laser.7–15

Relativistic self-focusing has been observed in many ex-eriments and has been proved to be an efficient way touide a laser pulse over distances much longer than theayleigh length. Fedosejevs et al.16 have reported the ex-erimental observations on relativistic self-focusing forydrogen gas. They employed a 0.3 TW, 250 fs laserulse, which gave an axial intensity of 3�1017 W cm−2.arkisov et al.17 have observed relativistic self-focusingnd channel formation by using an intense laser pulse ofn axial intensity of 6�1018 W cm−2. Hora and others18–20

ave presented numerous theoretical discussions on theoncept of relativistic self-focusing of a high-power lasernd its applications for fast ignition. Esarey et al.21 pre-ented an extensive review of the self-focusing and self-

0740-3224/07/051155-5/$15.00 © 2

uiding of short laser pulses in ionized gases and plas-as. Hafizi et al.22 studied the propagation of an intense

aser beam in a plasma, including the relativistic andonderomotive effects. They described the oscillations ofhe laser spot size in terms of an effective potential. Liund Tripathi23 have observed the effect of a self-generatedzimuthal magnetic field on the relativistic self-focusingf an intense laser in a plasma. In the previous studies, itas observed that the laser can acquire a minimum spot

ize due to the relativistic self-focusing in a plasma. Be-ond the focus, the nonlinear refraction starts weakeningnd the spot size of the laser increases, showing oscilla-ory behavior with the distance of propagation. To over-ome the diffraction and the successive high-amplitudescillation of the spot size, we propose the introduction ofslowly increasing plasma density gradient.The plasma density transition plays an important role

uring laser–plasma interaction. Suk et al.24 proposed acheme for plasma electron trapping by using a densityransition. The density ramp could be important for theelf-focusing of a high-power laser by choosing the lasernd plasma parameters. The motivation of the presentork is to study the nonlinear propagation of a high-ower laser during a slowly varying upward plasma den-ity ramp, when the relativistic effect is operative. As wenow, an intense laser becomes self-focusing in under-ense plasma because of the relativistic mass and pon-eromotive effects. Generally, the self-focused laser dif-racts and focuses periodically because of the mismatch inhannel and spot size. For a given laser spot size, the os-illation amplitude becomes larger for a higher plasmaensity due to the enhanced relativistic effect. However,y slowly (i.e., adiabatically) increasing the density, thescillation amplitude of the laser spot size can be signifi-

007 Optical Society of America

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1156 J. Opt. Soc. Am. B/Vol. 24, No. 5 /May 2007 Gupta et al.

antly reduced. As the laser propagates through the den-ity ramp region, it sees a slowly narrowing channel. Inuch an environment, the oscillation amplitude of thepot size shrinks, while its frequency increases, which is aesult of the adiabatic invariance theorem. Therefore, theaser tends to become more focused during propagation in

plasma density ramp. Second, as the equilibrium elec-ron density is an increasing function of the distance ofropagation of the laser, the diffraction length decreasesapidly as the beam penetrates deeper and deeper intohe plasma. Consequently, the diffraction effect reducesnd the laser focuses more. Hence, the upward plasmaensity ramp plays an important role in enhancing laserocusing. The proposed scheme is similar to a thin plasmaens,25 where a properly shaped plasma slab may be usedo focus the incident beam on a target. This work has di-ect application to plasma-based accelerators where alasma density transition is very important for plasmalectron injection into the acceleration phase.

In Section 2, we calculate the nonlinearity due to theelativistic effect and the wake field generation. The con-ition for laser self-focusing is found in Section 3, wherehe evolution of the beam-width parameter is discussed.he results are discussed in Section 4, and the conclusion

s presented in Section 5.

. NONLINEARITYet us consider the propagation of a Gaussian laser beamhrough an unmagnetized cold plasma of electron density

with a density gradient (ramp) along the z direction,

E� L = x̂E0�r,z,t�exp�− i��0t − k0z��, �1�

here �E02�z=0=A0

2g�t�exp�−r2 /r02�, k0�z�= ��0 /c��0

1/2, �p�z��4�n�z�e2 /m�1/2 is the electron plasma frequency, �0 is

he plasma dielectric constant, g�t� characterizes the tem-oral shape of the pulse, g�t�=1 for t�0 and g�t�=0 for t0, and −e and m are the electron’s charge and mass, re-

pectively. Here we use the step-pulse profile of the laseror simplicity; however, the Gaussian temporal profile of aaser pulse gives a profound effect on self-focusing. Theroup velocity of the laser increases with laser intensity;ence, the latter portion of the pulse reinforces with theront, causing stronger focusing and sharpening of theulse as investigated by Upadhyay et al.26,27 For z�0, weay write, A2= �A0

2 / f2�exp�−r2 /r02f2�, where f�z� is known

s the beam-width parameter, A0 is the axial amplitude ofhe laser, r0 is the half-width of the laser, c is the velocityf light, and r is the radial coordinate of the cylindricalystem. Here we assume that the pulse propagates with-ut changing shape. Also, we use a linearly polarized la-er pulse. A linearly polarized pulse is more complex totudy because the analytic simplifications that are pos-ible in the case of circularly polarized laser pulses, whichack harmonic content, do not apply. In addition, for aransverse circularly polarized electromagnetic wave, thelectron moves along a circular trajectory. Its longitudinalomentum is equal to zero, and the transverse momen-

um is finite. In a linearly polarized electromagnetic waven plasmas, the transverse and longitudinal motions oflectrons are always coupled and have a finite value.28

lso, an intense circularly polarized laser light is un-table in Kerr media, as investigated in an experiment bylose et al.29 As a result, to date, there is some confusion

n the literature with regard to circular polarization sta-ility. On behalf of the above discussion, we consider a lin-arly polarized laser to study the self-focusing needed toake this model simple and convenient.The upward plasma density ramp profile can be mod-

led by a simple expression n�z�=n0+tan�z /d�, as shownn Fig. 1, where n0 is the initial electron density and d isconstant, which is adjustable. We take this constant fac-

or d=10 so that the electron density approaches 2.5-foldf the initial electron density during the ramp lengthRd0, where Rd0=�0r0

2 /c and �0 is the laser frequency.his kind of density profile can be achieved in an experi-ent by using transient laser drilling on the gas jet that

s relevant to laser wake field acceleration (LWFA).30

The radial component of the ponderomotive forceushes the electrons radially outward on the time scale ofplasma period �p

−1, creating a radial space–charge field�

s=−��s. From the equation of motion and Poisson’squation, one may obtain the modified electron densityssociated with the wake field as

n1 = n�z���1 +1

m�p2�z���mc2�21 +

a2

2 1/2�� . �2�

Here we neglect the effect of the self-generated mag-etic field in the context of a linearly polarized laserulse. Although it is not definite that linearly polarizedulses do not generate a magnetic field in plasmas, it isikely that the effect may be small. However, in the case of

circularly polarized laser, the self-generated magneticeld will be quite significant, as predicted earlier. At high

aser intensity, the effect of the magnetic field on self-ocusing is indefinite because it signs in the radial direc-ion owing to electron cavitation. Also, at high laser inten-ity, the ponderomotive force dominates the electric fieldroduced by charge separation in a macroscopic regionnd the density given by Eq. (2) can take nonphysical val-es. This negative density can be avoided by putting thelectron density to zero in the entire spatial region. Inhis paper, we consider the numerical parameters wherehe condition n1�0 holds, i.e., where the electron cavita-ion does not take place. The argument behind this condi-ion in the context of our model will be elaborated onater.

ig. 1. Plasma density ��p /�0� variation with normalized dis-ance ���. The normalized parameters are d=10 and � /� =0.2.

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Gupta et al. Vol. 24, No. 5 /May 2007/J. Opt. Soc. Am. B 1157

Following Tripathi et al.,31 the modified electron den-ity can be rewritten as

n1 = n�z���1 −c2

�p2�z��r0

2f2

a2

�1 + a2/2�1/2

+c2

�p2�z��

r2

r04f4

a2�1 + a2/4�

�1 + a2/2�3/2� . �3�

Here we have approximated the normalized laser in-ensity distribution, conserving power in any cross sections the beam propagates, leading to a2= �a0

2 / f2�exp�r2 /r0

2f2�, where a0=eA0 /m�0c is the laser intensity pa-ameter.

By using the Taylor expansion of the dielectric constantf the plasma in the paraxial region �r2r0

2f2�, we obtainhe dielectric constant of the plasma as

� = �0 + r2, �4�

here

=�p

2�z�

4�02

a02f4

�1 + a02/2f2�3/2�1 +

c2

r02�p

2�z��f2

8 + a02/f2

�1 + a02/2f2�1/2� ,

�5�

�0 = 1 −�p

2�z�/�02

�1 + a02/2f2�1/2

, �6�

here the first term in Eq. (8) is due to the relativisticass nonlinearity and the second is due to the density de-

ression associated with the wake field. The density inho-ogeneity comes with a z dependence through �p in the

bove equations.

. SELF-FOCUSINGhe wave equation governing the propagation of the laser

s given below:

�2E� +�0

2

c2 �E� = 0. �7�

Even if E� has a longitudinal component,32 the term��E� � can be neglected provided �c2 /�0

2���1/�0��2 ln �0�1. Substituting for E� from Eq. (1) into Eq. (7), one ob-

ains, in the Wentzel–Kramers–Brillouin (WKB) approxi-ation

k02A + 2ik0A − ��

2 A =�0

2

c2 �A. �8�

Following the approach given by Sodha et al.,33 we in-roduce an eikonal, A=A00�z ,r�exp�ik0S�z ,r��, where00�z ,r� and S�z ,r� are the real functions of space. Weubstitute the value of A and separate the real and imagi-ary parts of the resulting equation. The coupled set ofquations for A00

2 and S are solved in the paraxial ap-roximation by expanding S as S=S0+ �r2 /2��1/ f��df /dz�,here S0�z� is the axial phase. By following the above

teps, we obtain A002 = �A0

2 / f2�exp�−r2 /r02f2� and the result-

nt equations governing the spot size of the laser

�2f

��2 =1

f3 −1

2 ��0

�� �f

��−

Rd02 f

r02 1

�0− 1, �9�

here �=z /Rd0.Here can be represented by Eq. (5), where the first

erm on the right-hand side is a nonlinear term on the ac-ount of relativistic nonlinearity and the second term is aonlinear term due to the density depression associatedith a wake field in a plasma. In Eq. (9), the first term isue to the diffraction effect, the second term is due to thelasma inhomogeneities, and the last one is the nonlinearerm that is responsible for laser self-focusing. In an un-erdense plasma the second term on the right-hand sidean be neglected. Equation (9) is a second-order differen-ial equation, and we solve it by choosing the suitable la-er and plasma parameters. In this way one can find theuitable condition for self-focusing of the laser duringropagation through the density ramp. In the generalase, we use the boundary conditions f=1 and �f /��=0.

. COMPUTATIONAL RESULTSo check the validity of the above analysis, we conductomputational simulations for solving the beam-width pa-ameter equation. All the numerical parameters are cho-en for a Ti:sapphire–Nd:glass laser of intensity I 5.71017 W cm−2. The laser spot size and wavelength are 40

nd 1 �m, respectively. The normalized minimum plasmaensity is �p0 /�0=0.2 (where �p0 represents the mini-um plasma electron density at �=0). The plasma den-

ity is increasing linearly with longitudinal distance andttains a maximum value �p0 /�0=0.5 in the distance ofRd0 (cf. Fig. 1).Figure 2 shows the variation of the beam-width param-

ter f as a function of the normalized propagation dis-ance �=z /Rd0 in an underdense plasma with an upwardensity ramp (gray curve) and without plasma densityamp (black curve). In the region of low plasma density,he electrons are expelled from the high-intensity regiony a ponderomtive force; the nonlinearity in a plasmaomes by electron mass variation, which is due to the in-ense laser and to the change in electron density on ac-ount of wake field generation. If there is no density

ig. 2. Beam-width parameter �f� with the normalized propaga-ion distance ��� with a plasma density ramp (gray curve) andithout a plasma density ramp (black curve). The used numeri-

al parameters are given in the text.

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1158 J. Opt. Soc. Am. B/Vol. 24, No. 5 /May 2007 Gupta et al.

amp, the beam-width parameter decreases monotoni-ally up to a Rayleigh length because of the nonlinear ef-ects. As the diffraction effects become predominant, theeam-width parameter increases after attaining a mini-um value, and the laser beam starts diverging owing to

aturation of nonlinearity. Hence, the laser becomes fo-used and defocused and shows an oscillatory behavior. Tovoid laser defocusing, we introduce an upward plasmaensity ramp. If there is a density ramp, the beam-widtharameter decreases up to a Rayleigh length and does notncrease much, as in the case with no ramp, for the suit-ble parameters given above. After a couple of Rayleighengths, the beam-width parameter attains a minimumalue and maintains it for a longer distance. The satura-ion behavior of the beam-width parameter shows thetrong self-focusing of the laser in a plasma with a den-ity ramp. The physics behind the laser self-focusing dur-ng propagation in an upward plasma density ramp cane understood as follows: After initial focusing of the la-er, the relativistic mass effect will be much more pro-ounced in the region of increasing plasma density.herefore, the laser focuses more during propagation in alasma density ramp. On the other hand, as the equilib-ium electron density is an increasing function of the dis-ance of propagation of the laser, the plasma dielectriconstant decreases rapidly as the beam penetrates deepernd deeper into the plasma. Consequently, the self-ocusing effect is enhanced and the laser is more focused.ence, the upward plasma density ramp plays an impor-

ant role in enhancing laser focusing.The minimum plasma density is chosen in the assump-

ion of an underdense plasma. The length of a plasmaensity ramp is considered to avoid the maximum defo-using of the laser. The laser focuses up to 0.5Rd0 in bothases (with or without density ramp). So it is necessary tose a density ramp longer than 0.5Rd0. If we choose the

ength of density ramp Rd0, there is only about 20% focus-ng. Better focusing is observed by increasing the lengthf the density ramp. But the plasma density should not beuch larger; otherwise, the laser can be reflected because

f the overdense plasma effect. For a typical scale lengthf axial plasma density variation �3Rd0�, the laser spotize reduces about 60%. Durfee and Milchberg34 have ob-erved optical guiding of a laser over a distance of morehan 20Rd0 in their experiment. To achieve laser guidingver this distance, one must have a plasma density rampith a typical scale length of 3Rd0.In this paper, we consider the numerical parameters,

.e., where electron cavitation does not take place. The ar-ument behind this condition can be explained from theesults of Fig. 2. In both cases (with and without densityamp), the laser spot size shows an oscillatory behaviorith propagation distance. In the case of electron cavita-

ion, the electron density will be zero inside the cavitationone, and the plasma effect will not be dominant in thatase. Hence, the laser would be defocused during propa-ation inside the channel. In this paper, we observed thathe laser becomes more focused (not much defocusing)uring propagation in the density ramp. Hence, it is ob-ious that the plasma effect is operative and that electronavitation is avoided in this case. Charge nonconservationtarts just after the appearance of the cavitation.35 How-

ver, if cavitation does not take place, the total charge wille conserved. This can be proved by using the approachsed in Ref. 35.

. CONCLUSIONSn many applications one needs the laser pulse to propa-ate several Rayleigh lengths while preserving an effi-ient interaction with the plasma. To achieve this goal, wentroduced an upward density ramp in an underdenselasma. As a result of relativistic mass nonlinearity andake field generation, the laser becomes self-focused in

he underdense plasma and attains a minimum spot size.f there is no density ramp, the laser is defocused beyondhis distance due to the dominance of the diffraction ef-ect. To reduce this defocusing, an upward density ramp isntroduced. As the plasma density increases, the self-ocusing effect becomes stronger and, as in the case of noamp, the beam-width parameter does not increase much.fter several Rayleigh lengths, the beam-width param-ter attains a minimum value and maintains it for a longistance. Hence, an upward density ramp is very impor-ant for self-focusing of a laser in a plasma. This workould be applicable to plasma-based accelerators andaser-driven fusion.

CKNOWLEDGMENTShis work was supported by the Korean Ministry of Sci-nce and Technology through the Creative Research Ini-iative Program/Korean Science and Engineering Foun-ation. The authors would like to thank V. K. TripathiIndian Institute of Technology, New Delhi, India) andictor Kulagin in our group for helpful discussions.Corresponding author H. Suk can be reached by e-mail

t hysuk@keri.re.kr.

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1

1

1

1

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11

2

2

2

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