Propagation of an intense laser beam in a tapered plasma channelPallavi Jha, Ram Gopal Singh, Ajay K. Upadhyaya, and Rohit K. Mishra Citation: Phys. Plasmas 15, 033101 (2008); doi: 10.1063/1.2841018 View online: http://dx.doi.org/10.1063/1.2841018 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v15/i3 Published by the American Institute of Physics. Related ArticlesCompetition of circularly polarized laser modes in the modulation instability of hot magnetoplasma Phys. Plasmas 20, 012120 (2013) Propagation of femtosecond terawatt laser pulses in N2 gas including higher-order Kerr effects AIP Advances 2, 042190 (2012) A polar-drive–ignition design for the National Ignition Facility Phys. Plasmas 19, 056308 (2012) Laser produced streams of Ge ions accelerated and optimized in the electric fields for implantation into SiO2substrates Rev. Sci. Instrum. 83, 02B305 (2012) Feasibility study of broadband efficient “water window” source Appl. Phys. Lett. 100, 014103 (2012) Additional information on Phys. PlasmasJournal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors

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Propagation of an intense laser beam in a tapered plasma channelPallavi Jha, Ram Gopal Singh, Ajay K. Upadhyaya, and Rohit K. MishraDepartment of Physics, University of Lucknow, Lucknow 226007, India

�Received 3 October 2007; accepted 17 January 2008; published online 11 March 2008�

Propagation characteristics and modulation instability of an intense laser beam propagating in anaxially tapered plasma channel, having a parabolic radial density profile, are studied. Using thesource-dependent expansion technique, the evolution equation for the laser spot is set up andconditions for propagation of the laser beam with a constant spot size �matched beam� are obtained.Further, the dispersion relation and growth rate of modulation instability of the laser pulse as itpropagates through linearly and quadratically tapered plasma channels, have been obtained. © 2008American Institute of Physics. �DOI: 10.1063/1.2841018�

I. INTRODUCTION

Nonlinear interaction of a laser beam with underdenseplasma leads to self-focusing1,2 and self-channeling.3 Plasmachannels are required to further guide the laser beam beyondthe Rayleigh length4 ZR �=�r0

2 /�; r0 and � are the beamwaist and wavelength, respectively�. Optical guiding of in-tense laser beams in plasma channels finds application inlaser driven accelerators,5 x-ray sources,6 and the fast igni-tion concept.7,8 A plasma channel with a radially parabolicdensity profile allows the propagation of a laser beam withconstant spot size, over large distances, if the channel depth�n is equal to a certain critical value �nc.

Using segmented capillary discharge9,10 the density of aplasma channel can be tapered such that the channel has aparabolic radial density profile and an arbitrary axial densityvariation. Tapered plasma channels are important for improv-ing the performance of laser wakefield accelerators�LWFAs�.10 When untapered channels are used for LWFAs,the acceleration distance is limited by dephasing length11

while in a tapered channel, the dephasing distance is in-creased relative to an untapered channel and large energygains are achieved.10 Hence, it is important to study thepropagation of laser beams in tapered channels. Effects oftapering the plasma channel on forward Raman and self-modulation instabilities have been studied.12 Propagationcharacteristics of intense laser beams in tapered plasma chan-nels have also been reported.13

In this paper, a detailed analysis of evolution of the en-velope of a laser beam propagating in a parabolic plasmachannel, tapered along the axis, is presented. The transverseplasma density gradient and the channel radius have beenconsidered to be functions of the propagation distance. Con-ditions for propagation of the laser beam with a constant spotsize �matched beam� have been obtained. Since the plasmadensity �frequency� varies along the axis in a tapered chan-nel, the wave number of the laser beam will also vary withpropagation distance. In addition, the modulationinstability14–17 of the laser beam as it propagates in linearlyand quadratically tapered channels, has been studied. Thegrowth rate and the range of frequencies over which the in-stability occurs have been obtained. The organization of thepaper is as follows. In Sec. II, a nonlinear wave equation for

a laser beam propagating in an axially tapered plasma chan-nel is set up. In Sec. III, using the source-dependent expan-sion �SDE� method, the evolution equation for the laser spotis obtained. Conditions for matched beam propagation in atapered plasma channel have been derived and numerical so-lutions of the envelope equation for the laser spot propagat-ing in linearly and quadratically tapered plasma channels aregraphically depicted. In Sec. IV, the dispersion relation formodulation instability of a laser beam propagating in a ta-pered plasma channel is derived and temporal growth ratesare obtained. Summary and conclusions are presented in Sec.V.

II. FORMULATION

Consider a linearly polarized laser beam propagatingalong the z direction in a plasma channel, spatially taperedalong the axis. The profile of the plasma density is of theform n�r ,z�=n�z�+�n�z�r2 /rch

2 �z�, where n�z� is the longitu-dinally varying plasma density, while rch�z� and �n�z� are thechannel radius and channel depth, respectively. The electricfield of the laser beam is given by

E�r,z,t� =1

2E�r,z,t�exp�i�

0

z

k0�z�dz − �0t�x̂ + c.c., �1�

where E�r ,z , t�, �0, and k0�z� are the laser field envelope,frequency, and wave number, respectively. The wave numberis axially varying due to the tapered plasma density. Thenonlinear wave equation describing the evolution of the laserfield in a tapered plasma channel �in the mildly relativisticregime�, is given by

��2 −1

c2

�2

�t2a =�p

2�z�

c2 �1 +�n�z�

n�z�

r2

rch2 �z�

−a2

4

−�n�z�

4

a2

n�z�

r2

rch�z�a , �2�

where a�r ,z , t��=�eE /m�0c���1 is the normalized electricfield, �p�z��=4�e2n�z� /m� is the plasma frequency, and

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Coloumb gauge �� ·E=0� has been used. The source termson the right side of Eq. �2� arise due to the plasma channel,relativistic nonlinearity, and coupling of this nonlinearitywith the channel.18 Introducing independent variables z and ��=t− 0

zdz /vg�z�, where vg�z��=c2k0�z� /�0� is the group ve-locity�, the wave equation �2� becomes

���2 − k0

2�z� +�0

2

c2 + 2ik0�z��1 +i

k0�z�vg�z�

�

��� �

�z

+1

c2� 1

�g2�z�

− 1� �2

��2 + i�k0�z�

�z �1 −i

k0�z�vg�z�

�

���

+ 2i��0

c2 −k0�z�

vg�z�� �

��+

�2

�z2�a

=�p

2�z�

c2 �1 −a2

4+

�n�z�r2

n�z�rch2 �z�

−a2�n�z�r2

4n�z�rch2 �z� �a , �3�

where �g�z��=vg�z� /c� is the normalized group velocity. Thelowest order dispersion relation gives the wave number

k0�z�= �1 /c���02−�p

2�z�−4c2 /rs2�z�, where rs�z� is the laser

spot size. Substituting the value of vg�z� in Eq. �3� and con-sidering paraxial propagation of the laser beam gives

���2 + 2ik0�z�

�

�za�r,z,t� = k0

2�z��1 − �2�r,z��a�r,z,t� ,

�4a�

where the z-dependent, nonlinear refractive index is given by

��r,z� =� �02

c2k02�z�

+i

k02�z�

�k0

�z−

�p2�z�

c2k02�z�

�1 +�n�z�

n�z�

r2

rch2 �z�

−a2

4−

r2a2�n�z�

4rch2 �z�n�z� ��1/2

.

�4b�

In deriving Eq. �4a�, group velocity dispersion and finitepulse length effects have been neglected.

III. ENVELOPE EVOLUTION AND MATCHED BEAMPROPAGATION

In order to solve Eq. �4a�, the source-dependent expan-sion �SDE� technique19 is used. The laser field amplitude isexpanded as a series of orthogonal Laguerre–Gaussianmodes given by

a�r,z,t� = �m=0

am�z,��Lm���exp�− �1 − i�s�z���

2� , �5�

where m=0,1 ,2 . . . , �=2r2 /rs2�z�, �s�z�=krs

2�z� /2RC, Lm isthe Laguerre polynomial of order m, and RC is the radius of

curvature associated with the wave front. Substituting Eq. �5�in Eq. �4a� and following the standard SDE procedure leadsto

�s�z� =k0�z�rs�z�

2

�rs�z��z

, �6a�

��s�z�2�z

−�s�z�rs�z�

�rs�z��z

−�1 − �s

2�z��k0�z�rs

2�z�

=k0�z�

2�

0

�1 − �2��1 − ��exp�− ��d� . �6b�

In deriving Eqs. �6�, a0 am�0 is assumed, where a0= �a00r0 /rs�z���k00 /k0�z�; a00, k00 are vacuum amplitude andwave number, respectively, and r0 is the minimum spot size.

Differentiating Eq. �6a� with respect to z and substitutinginto Eq. �6b� along with Eq. �4b�, the evolution equation forthe laser spot size is given by

�2rs�z��z2 =

4

k02�z�rs

3�z�−

�p2�z�a00

2 r02k00

8c2rs3�z�k0

3�z�−

4��n�z�e2rs�z�mc2k0

2�z�rch2 �z�

+ ��p�z��0

2

��p�z��z

�� �rs�z��z

� −4c2

rs3�z��0

2� �rs�z��z

�2

.

�7�

The first three terms on the right side of Eq. �7� are respon-sible for diffractive effect, relativistic nonlinear effect, andthe contribution due to parabolic channel, respectively, whilethe last two terms are present due to variation of wave num-ber with propagation distance ��k0�z� /�z�.

Propagation of a matched laser beam �rs=r0� in a taperedplasma channel can be obtained by substituting �rs /�z=�2rs /�z2=0 and rs=r0. With the help of Eq. �7�, the beammatching condition is given by

�n�z� = rch2 G�1 − a00

2 r02�p

2�z�k00/32c2k0�z�� , �8�

where G=mc2 /�e2r02. In deriving Eq. �8�, the channel radius

is considered to be constant. The laser beam can also bematched by varying the channel radius assuming the channeldepth to be constant. In this case the beam matching condi-tion is given by

rch�z� = ��n�1/2/�G�1 − a002 r0

2�p2�z�k00/32c2k0�z���1/2. �9�

The laser beam can therefore propagate with a constant spotsize in an axially tapered channel by tapering either the chan-nel depth or its radius. An appropriate longitudinal variationof the channel depth �radius�, depending on the axial plasmadensity profile, would be required for satisfying beam match-ing condition.

Matched beam propagation for two typical axial densityprofiles �linear and quadratic� has been considered. Linear

033101-2 Jha et al. Phys. Plasmas 15, 033101 �2008�

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density taper �plasma channel density increases linearlyalong the propagation direction� may be represented byn�z�=n0�1+z /L�. Here, L is the scale length for tapering ofdensity along the z direction and n0 is the plasma electrondensity at z=0. If the density increases quadratically with z,the tapering function is given by n�z�=n0�1+z2 /L2�. It maybe noted that if z�L, linear taper will be more effective incomparison to quadratic taper and vice versa. Figure 1 showsthe variation of channel depth with normalized distance forpropagation of a matched laser beam in a linearly tapered�curve �a�� and a quadratically tapered �curve �b�� channelfor r0=15 �m, �0=1.881015 s−1, a00

2 =0.1, rch=20 �m,�p=10 �m, n0=4.91018 cm−3, and ZR /L=0.2. It is seen

that the axial density taper and the channel depth shouldhave inverse profiles. Similarly, Fig. 2 shows the variation ofchannel radius with normalized distance for propagation of amatched laser beam in a linearly tapered �curve �a�� and aquadratically tapered �curve �b�� channel for the same pa-rameters as in Fig. 1, with �n=3.961018 cm−3. It may benoted that in order to obtain matched beam propagation, theaxial density taper and the channel radius should have simi-lar profiles.

Numerical solutions to the envelope equation �7� for lin-early and quadratically tapered channels have been obtainedusing fourth-order Runge–Kutta method. In Fig. 3, variationof the normalized spot size �rs /r0� of a laser beam is plottedagainst the normalized propagation distance �z /ZR� for unta-pered �curve �a��, linearly tapered �curve �b��, and quadrati-cally tapered �curve �c�� density channels, for normalizedscale length ZR /L=0.2. The laser and plasma parametersused are �0=1.881015 s−1, a00

2 =0.1 �intensity 1.31017 W /cm2�, r0=15 �m, �p=10 �m, n0=4.91018 cm−3, rch=20 �m, and �n=3.961018 cm−3. It isseen that the laser spot size undergoes betatron oscillationsdue to the radial density channel.4 For the untapered channel,the spot size oscillates about the z axis with a constant meanvalue. However, when the channel is longitudinally tapered,the mean value of the spot size decreases with z. It has beenobserved that the decrease in mean value of the laser spotsize is larger for the linearly tapered channel as compared tothe quadratically tapered channel when propagation distancein the channel is less than five Rayleigh lengths �z /ZR�5�,while beyond five Rayleigh lengths, the reverse phenomenonis observed. Thus, a tapered plasma channel can be used forfocusing the laser beam with enhanced intensity.

If the density of the plasma channel decreases along theaxial direction �negative taper� the laser spot will increase,thereby reducing the intensity. Thus, a negatively taperedchannel acts as an inverse lens.20

FIG. 2. Variation of channel radius �rch� with normalized propagation dis-tance �z /ZR� for linearly tapered channel �curve �a��, and for quadraticallytapered channel �curve �b�� for r0=15 �m, �0=1.881015 s−1, a00

2 =0.1,�p=10 �m, �n=3.961018 cm−3, n0=4.91018 cm−3, and ZR /L=0.2.

FIG. 1. Variation of channel depth ��n� with normalized propagation dis-tance �z /ZR� for linearly tapered �curve �a�� and quadratically tapered �curve�b�� channels for r0=15 �m, �0=1.881015 s−1, a00

2 =0.1, �p=10 �m, n0

=4.91018 cm−3, rch=20 �m, and ZR /L=0.2.FIG. 3. Variation of normalized spot size �rs /r0� with normalized propaga-tion distance �z /ZR� for untapered �curve �a��, linearly tapered �curve �b��and for quadratically tapered �curve �c�� channels for r0=15 �m, �0=1.881015 s−1, a00

2 =0.1, �p=10 �m, rch=20 �m, n0=4.91018 cm−3, �n=3.961018 cm−3, ZR /L=0.2.

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IV. ANALYSIS OF MODULATION INSTABILITY

In this section, the modulation instability of a laser beampropagating in a tapered plasma channel is studied. Consid-ering only the first-order source terms in the wave equationand taking its Fourier transform, we get

��2 +4

r02 +

�2

c2 �L2�z,���E�r,z,� − �0�exp ik0�z�z = 0,

�10�

where E�r ,z ,�−�0� is the Fourier transform of the slowlyvarying laser field amplitude and �L�z ,���=�1−�p

2�z� /�2

− �4c2 /�2r02��1/2� is the linear refractive index having contri-

bution due to vacuum, free plasma electron, and finite spotsize of the laser radiation respectively. Defining the modepropagation21 constant ��z ,���=��L

2�z ,�� /c� and substitut-ing it into Eq. �10� gives

��2 + 2k0�z��i�

�z+

�2�z,�� − k02�z�

2k0�z�+

2

k0�z�r02�

E�r,z,� − �0� = 0. �11�

If the mode propagation constant is considered to beclose to the unperturbed wave number �k0�z��, we can write��2�z ,��−k0

2�z� /2k0�z�����z ,��−k0�z�. Substituting theTaylor series expansion of ��z ,�� about �0 �in terms of vari-ous order dispersion parameters� in Eq. �11� and taking theinverse Fourier transform, we obtain

��2 + 2k0�z��i�

�z+ �0�z� − k0�z� +

2

k0�z�r02 + i�1�z�

�

�t

−�2�z�

2

�2

�t2��E�r,z,t� = 0, �12�

where �0�z�=k0�z� and �1�z�=1 /vg�z�, while �2�z��=−1 /vg

2�z���vg�z� /��0�� is the group velocity dispersion �GVD�parameter. Now transforming the variable �z , t� to �z ,�� andretaining dispersion terms up to the second order, Eq. �12�becomes

���2 + 2k0�z��i

�

�z−

�2�z�

2

�2

��2 −2

vg�z�

�2

�z���a�r,z,��

= −�p

2�z�

4c2 �1 + R�r,z��a2a�r,z,�� , �13�

where R�r ,z��=��n�z�r2 /n�z�rch2 �− �4�n�z�r2 /n�z�rch

2 a2��represent effects due to channel and coupling of channel withnonlinearity. In deriving Eq. �13�, 1 /k0�z� has been neglectedin comparison to �2�z� and �2 /�z2�2k0�z�� /�z.

In order to study the longitudinal modulation instabilityof a laser beam propagating in a tapered channel, we con-sider a simple one-dimensional model. Neglecting transversevariation of the laser field amplitude, the plane wave limit ofEq. �13� is given by

�2k0�z��i�

�z−

�2�z�

2

�2

��2� −2

vg�z�

�2

�z��+

�p2�z�a2

4c2 �a�r,z,�� = 0. �14�

In the long pulse limit, the variation in laser pulse amplitudewith respect to � may be considered to be a perturbation thatresults in modulation instability. Thus, the zeroth-order ��independent� solution of Eq. �14� is given by

a0�z� = a00 exp�4iP0�z�k0�z�r0

2 z� , �15�

where P0�z�=�p2�z�a00

2 r02 /32c2. Assuming the first-order con-

tribution to the pulse amplitude obtained due to � variation tobe of the same form as that of the unperturbed amplitude�Eq. �15��, the total amplitude may be written as

a0�z� = a00 exp�4iP0�z�k0�z�r0

2 z� + a01�z,��exp�4iP0�z�k0�z�r0

2 z� ,

�16�

where a01�z ,�� is the complex perturbed beam amplitude.Substituting Eq. �16� in Eq. �14� and considering a01� a00, we get

i�a01

�z−

�2�z�2

�2a01

��2 −1

k0�z�vg�z��2a01

���z

−4iP0�z�

vg�z�r02k0

2�z��a01

��+

8P0�z�k0�z�r0

2a01 = 0. �17�

The perturbed wave amplitude may be considered to bean exponentially varying function of z and �. Thus, the as-sumed solution of Eq. �17� may be written as

a01�z,�� = exp�i�Kz − ���� + exp�− i�Kz − ���� , �18�

where K and � are wave number and frequency of the per-turbed wave amplitude, respectively. If K is complex, itsimaginary part will lead to amplitude modulation thus excit-ing modulation instability. Now substituting Eq. �18� in Eq.�17�, we get the dispersion relation for one dimensionalmodulation instability as

�1 −�̂2

S4�z��K̂2 + �P̂0X�z� + �̂2�̂2�

�̂

S3�z�K̂

− 16� �̂22�̂2

S2�z�+

P̂0�̂2X�z�S2�z�

−P̂0

2X2�z�4S6�z�

��̂2 = 0, �19�

where K̂=KZR0, �̂=� /k00vg0, and �̂2=vg02 k00

2 ZR0�2 /8 aredimensionless quantities.vg0 is the z-independent group ve-locity �at z=0�. The terms S�z� and X�z� depend on the ta-pering configuration. For linear taper S�z�= �1−�p0

2 z /2c2k002 L� and X�z�= �1+z /L�, while for quadratic

taper S�z�= �1−�p02 z2 /2c2k00

2 L2� and X�z�= �1+z2 /L2�. Equa-

tion �15� represents a quadratic in K̂, having roots

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K̂ = − 4� P̂0X�Z�S3�z�

+�̂2�̂2

S3�z���1 −

�̂2

S4�z�

�

4�� P̂0X�z�S3�z�

+�̂2�̂2

S3�z�2

�̂2 + �1 −�̂2

S4�z�� �̂2

2�̂2

S2�z�+

P̂0�̂2X�z�S2�z�

−P̂0

2X2�z�4S6�z�

�̂2

�1 −�̂2

S4�z� . �20�

Since �̂2 is negative, K̂ can be complex, leading to modulation instability. The condition for K̂ to be complex in Eq. �16�defines the range of unstable frequencies for which the instability exists as

�̂2�z� �

P̂0� 3P̂0X2�z�

4S6�z�+

�̂2X�z�

S2�z��

� �̂2

S+

P̂0X�z�

2S6�z��2

. �21�

It may be noted that the critical value of �̂ for which the instability exists, is a function of z, due to tapered plasma. Themodulation instability growth rate for the laser beam propagating through the tapered plasma channel �given by the imaginarypart of Eq. �20�� is

��z� =

4�3P̂02�̂2X2�z�4S6�z�

+P̂0�̂2X�z�

S6�z�+

�̂22�̂4

S2�z�+

P̂0�̂2�̂2X�z�S2�z�

+P̂0

2�̂4X2�z�4S10�z�

�1 −�̂2

S4�z� . �22�

The variation of growth rate of modulation instability

��� with �̂ is shown in Fig 4 for r0=15 �m, �0=1.881015 s−1, a00

2 =0.1, and �p=10 �m. Curve �a� shows thegrowth rate for a laser beam propagating in untaperedplasma. The set of curves �bq ,bl� and �cq ,cl� depict growthrate variations for quadratically and linearly tapered plasmafor normalized distance z /ZR=3 and 10, respectively. It isseen that axially tapering the plasma leads to enhanced in-stability. It may be noted that for z /ZR=3, the instabilitygrowth for linear taper predominates over quadratic taper,whereas for z /ZR=10 enhancement of growth rate for qua-dratic taper is larger than that obtained for linear taper. Theenhancement of modulation instability occurs due to in-creased focusing of the laser beam in tapered channels.

V. SUMMARY AND CONCLUSIONS

In this paper, nonlinear propagation of a laser beam in atapered plasma channel is studied. The source-dependent ex-pansion �SDE� method is used to obtain the evolution equa-tion for the laser spot. Matched propagation of the laserbeam in an axially tapered channel can be obtained by taper-ing the channel density gradient, while keeping the channelradius constant. The profile of the channel density gradientshould be inverse of the axial density tapering profile. Inaddition, the beam matching condition can be satisfied by

varying the channel radius with a profile similar to that of theaxial density variation.

Numerical methods are used to solve the equation fortypical laser and plasma parameters for linearly as well asquadratically tapered plasma channels. The dependence of

FIG. 4. Modulation instability growth rate variation with normalized fre-

quencies �̂ for untapered �curve �a�� and tapered plasma density �curves �bq,bl� and �cq, cl�, the subscripts q and l represent quadratically and linearlytapered channels, respectively. Here the set of curves bq, bl and cq, cl areplotted at normalized propagation distance z /ZR=3,10, respectively, with�0=1.881015 s−1, r0=15 �m, a00

2 =0.1, and �p=10 �m.

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the laser wave number on propagation distance has beentaken into account. It is seen that for a positively taperedchannel the laser spot undergoes betatron oscillations withcontinuously decreasing radius as the beam propagates alongthe z direction. This allows the tapered plasma channel to beused as a lens for focusing the laser beam.

The dispersion relation of perturbed laser beam ampli-tude propagating in tapered density plasma �in the long pulselimit� is established and the growth rate for modulation in-stability is evaluated for different propagation distances andis compared with untapered plasma case. The present analy-sis shows that in an axially tapered plasma, growth rate ofmodulation instability of the laser beam as well as the rangeof unstable frequencies increases with normalized propaga-tion distance.

ACKNOWLEDGMENTS

This work has been done with the financial support ofthe Science and Engineering Research Council, Departmentof Science and Technology, Government of India. The au-thors thank the organization for funding the research project.

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