Three-halves harmonic emission from a laser filament in a plasma channel

Three-Halves Harmonic Emission From a Laser Filament in a Plasma Channel

GANESWAR MISHRA, INDRANIL TALUKDAR, VIJAYSHRI, AND V. K. TRIPATHI

Abstract—A self-trapped laser filament is susceptible to decay-pro- ducing radially localized Langmuir waves. A nonlinear interaction of the pump wave with the density oscillation at the Langmuir frequency gives rise to three-halves harmonic emissions. Using a basis-function expansion technique, the emitted power in the backward direction is obtained. It decreases with the increasing size of the filament.

I. INTRODUCTION

I

N RECENT years investigations on long scale-length laser-produced plasmas have become important. In many experiments, filamentation instability has been ob- served [l]-[5]. It is an important process, producing lo- calized density depressions with large laser power densi- ties, and may have a strong influence on parametric processes [6], [7]. Langdon and Lasinski [8], Sharma and Tripathi [9], and Short et al. [10] have examined the pro- cess of two plasmon decay in a plasma channel induced by a cylindrical laser filament and have demonstrated that the filamentation should enhance the growth rate. This process is important near the quarter critical density layer [11], [12] and is responsible for 3wo/2 generation ob- served in many experiments [13], [14]. Information about the plasma waves at frequency coo/2 is largely gathered from 3o)0/2 emissions in laser-plasma interactions.

In this paper, we study three-halves harmonic emission from a self-trapped laser filament in a plasma channel. We consider the situation in which the nonlinear refraction- induced self-convergence effect is balanced by the dif- fraction-induced divergence. In Section II, we obtain nonlinear three-halves harmonic current density by using the fluid approach. In Section III, the wave equation is solved for the back-scattered wave. The power density of the three-halves harmonic is obtained in Section IV. A brief discussion of results follows in Section V.

II. NONLINEAR CURRENT DENSITY

Consider the propagation of a self-trapped laser pump wave (i.e., a filament) in a plasma:

Manuscript received May 6, 1987; revised July-43, 1989. This work was supported by the Department of Science and Technology (DST) and by the CSIR, Government of India.

The authors are with the Department of Physics, Indian Institute of Technology, New Delhi 110016, India.

IEEE Log Number 8931370.

Eo = $E0(r) exp [i(

El = Elo exp - -2

2 _

2 C00

(1) where a is the radial width of the beam, copO is the electron plasma frequency at r = 0, and ktfi » 1. The beam exerts a radial ponderomotive force on the electrons, causing radial outward ambipolar diffusion of the plasma on a time scale a/cs, where cs is the sound speed. The density variation near the axis of the filament is given by [9]

n - "o 1 + 72 (2)

with

where vtb is the electron thermal speed, and t>00 and n0 are the electron oscillatory velocity and electron density at r

= 0. The self-made plasma duct tends to focus the laser beam. This tendency is opposed by the diffraction diver- gence. The two effects are balanced when a = v2 cfth/ojpo^oo and the laser pump propagates in the self- made waveguide without undergoing any divergence or convergence [15].

The laser beam-induced oscillatory velocity can be written as

V0 =

eE0

(3) where e and m are the electronic charge and mass, re- spectively. We assume that the laser undergoes two plas- mon decay near the quarter critical layer, and that the electrostatic potential of one of the decay waves at satu- ration is </>i/2- The electron response to <j>i/2 can be writ- ten as

(4)

850 IEEE TRANSACTIONS ON PLASMA SCIENCE. VOL. 17. NO. 6. DECEMBER 1989

4en0 2

n , /2 = 2 v <

mw0

( 5 ) The pump and the half-harmonic waves produce a non- linear current density at 3 / 2 harmonics,

~ 3 (6)

where

v3/2 = 2eE3/2/3miu0

and £3/2 is the self-consistent field. Equation (6) can be rewritten as

- —2n0e2 -> 2noe3' -. 2

^ = ^ 3 / 2 2~3 £(>V 01/2- (7) For resonant 3 O J0/ 2 emission the phases of the two terms in (7) must be the same; i.e., &3/2 = k0 + k\/2. For backward emission, jfc3/2 ~ —3/2(coo/c), hence kx/2 ~

— 5 / 2 ( wo/ c ) . Since kl/2vlh/w0/2 « 1 to ensure weak Landau damping of the co0/2 Langmuir wave, one re- quires 5vlh/c « 1 or Te < 1 keV. In fact, the other Langmuir wave produced in the decay process would have k\/2 « - 7 / 2 ( wo/ c ) , thereby putting a little more severe restriction on the electron temperature to Te < 0.5 keV.

However, it is possible in hot plasmas that the nonlinear mixing of the laser and a forward-propagating Langmuir wave produce a forward-propagating 3 O J0/ 2 wave which would suffer reflection in the overdense region to give a backward 3a>0/2 emission. This process requires kx/2 ~ wo/2c, at which the decay wave damping is small.

III. BACKWARD EMISSION

The wave equation for the three-halves harmonic can be written as

• £3 / 2) = ~ -h/2-

( 8 ) Substituting the value of J3/2 from (7) and neglecting the second term on the LHS of (8), we obtain in the cy- lindrical coordinate system,

Id2 d2 ,2 .

= 2"^ £0V me WQ

(9) with

Ac c

Using (2), we may write,

(10) with

Equation (9) can be rewritten as

1 d 1 d2

rdrVdr + r2 dd2 + dz2

where

5NL _ r — me co^{2 ^}Q

2 .

<p\/2. (11) The 0 dependence of £3/2 is the same as that of <j>{/2

and may be taken as exp (imd). For r dependence we expand Ei/2( 7) in terms of the basis functions <t>j(r):

£ 3 / 2 ( r ) = y S <t>j(r)tj(z)

j = i

where 0 j ( r ) satisfies the equation, 1 d

r dr

(12)

= 0 . ( 1 3 )

Equation (13) has well-behaved solutions when

a2 = —— ( 2 / + \m\ - 1)

7 = 1 , 2 , 3 , . . . . m = 0 , ± 1 , ±2, . . . . The solutions are

(14)

(15) where

c - 2 ( 7 " 1)!

7 + w - 1)!

and L]'"\ are the associate Laguerre polynomials. 4>j(r) satisfies the ortho-normality condition

(16) Using (12) in (11), multiplying the resulting equation by <f>*r dr and integrating over r, we get

(17) where

/»00

/ = PNL4>*(r)rdr Jo

and

(3

2

= k

2/2

(0) - a?.

To evaluate /;, we need to know the r dependence of <j>\ /2. The mode structure equation for </>,/2 can be written as

with

(2n'

2T

th

2 rl ~ 2 21.2 22 ^1/2 = 0 ( 1 8 )

n' = 0, 1, 2, . . . and #„• as Hermite polynomials.

We choose n' = 0 in (23) to model a gaussian intensity profile of the laser filament in the axial direction:

where

We have artificially introduced a z dependence of elec- tron density, although much weaker, but important to ax- ially localize the Langmuir wave and hence the region of 3coo/2 emission; n0 = «o(l + r 2/ ^2 + z2/b2). As far as the propagation of w0 and 3wo/2 waves are concerned, the effect of axial inhomogeneity is negligible.

Equation (18) can be solved using the method of the separation of the variable. Let

<$>\/i{r, 6, z) = Al/2R(r)Z(z) exp (imd) (19) where R(r) and Z(z) satisfy the differential equations

.2 J.

\ '/4 r

o \— exp . (24)

and where 7 and r\ are constants, such that y2 + 172 = k2/2(0).

Equations (20) and (21) have well-defined solutions:

(n 1)!

( | m | + 2 ) / 2

with

« = 1, 2, 3, . . . . m = 0, ± 1 , ± 2 , . . . and

ZB.(z) = exp

(23)

Substituting the value of V2<£i/2 from (18) and employ- ing (1) and (24) in (11), we get

tip0 j

^ ^ C n n / l l r t t - n | — K\I2\V) "I T7l ~ COQ

• exp 1

+ —52a2

and

mc2wo

'

n

cf exp

^{ai*_ 2}

2y/2b vth

2

(26) where

and

d _ 1 , * * > , ^ 2a2 2cLn 2*/thLn

( 2 7 )

852 IEEE TRANSACTIONS ON PLASMA SCIENCE. VOL. 17. NO. 6. DECEMBER 1989

Defining

(28) we may write (26) as

h =

_E

°°

^C

'"

^C

*

^exp

1

^ik

°

^Z

~~

-*?*«>)

(29) // can be solved by using the standard integration for- mulae,

1 {n+j - 2 + 2 (n-V

m

d -

Fig. 1. Ratio of the total 3w^o/2 power radiated in the backward direction to the total fundamental laser power [ | D,, t• ,„ \2/EIQ-KO2] as a function of j, the radial mode number of 3 uo/ 2 wave, for parameters OJ0 = 2 x 10¹⁵ rad/s, u^p0 = u^o/ 2 , a - 2 ftm, f^th = 1.7 x 10⁹ cm/s, r^lh/i>⁰ = 4.7, amplitude /!,/, = 0.1 ESU and radial mode number n = 1 for the Langmuir wave, for different values of the azimuthal mode number m:

m = 0 ( ); m = 1 ( ); and m = 2 ( ).

d - ~ ) cLj - 1)!

2FA -n + 1, -j + 1; -n - j - \m + 2 ; (30)

for Re | m | > - 1 and Re d > 0.

Using the expression for /,, (17) can be solved to ob-

tain, IV. RADIATED POWER

The radiated power for three-halves harmonic emission (31) may now be calculated as

where

1/2

= £- S DfD}, exp (-10,-*) ( 4>f{r)4>y(r)rdr

sir j . / J

8TT (34)

exp ( 3 2 )

Hence

• exp^I

\ 2cLⁿ

exp ( —i@jZ + imd).

(33)

where Dj is given by (32). To have a numerical appreci- ation of the emitted power, we consider a typical set of parameters w0 — 2 X 1015 rad/s, wp0 = wo/ 2 , a = 2 /xm, Tf = 500 eV,vo/vth = 0.2, L,, « 15 ^im, the radial width of the Langmuir wave ( = vft hLn/wpo) — 0-5 /^m, the z width of the interaction region ( = (v2feft h/w/ j 0)l / / 2) « 2, and the amplitude /4,/2 of the Langmuir wave = 0 . 1 ESU. We have plotted in Fig. 1 (for n = 1, m = 0, 1, 2) the backward 3wo/2 power as a function of j, the radial mode number. As the azimuthal mode number increases, the radiated power decreases for m = 0, whereas it in- creases for m > 1 modes.

V. RESULTS AND DISCUSSION

The Langmuir waves generated via the 2wp decay are localized near the density minimum of the laser filament.

For azimuthally symmetric modes (m = 0 ) , the Lang- muir wave amplitude maximizes near the density bottom, whereas for higher azimuthal mode numbers it peaks slightly away from the density bottom of the filament.

Since the intensity of the filament peaks at its density bot- tom, the nonlinear current density generating 3wo/2 wave is largest when the Langmuir waves have m = 0. The source region of 3wo/2 emissions is also limited by the axial localization of the Langmuir wave. In actual exper- iments the localization is due to the density gradient and the wave number mismatch thus introduced. However, here we have modeled it by a slowly varying parabolic density profile, so as to obtain localized analytical solu- tions. Thus, the 3wo/2 power radiated in the backward direction is maximum for the azimuthally symmetric mode, showing a slow dependence on j, the radial mode number of 3o>0/2 emission. For higher azimuthal mode numbers m = 1, 2, it shows a significant increase with the radial mode number j. For a given pump power, an increase in the beam cross-section leads to a decrease in the emitted 3wo/2 power by almost the same factor.

The power density 3 O J0/ 2 radiation in our model should be taken as an upperbound for the following reasons: 1) We consider the filament channel to localize 3wo/2 radia- tion, giving a very strongly peaked angular distribution, whereas in an actual experiment the diffraction effects in the underdense region may reduce the on-axis intensity considerably; and 2) we have assumed some artificial value of the saturation amplitude of the Langmuir wave which might be smaller than 0.1 ESU.

In small filaments, lower azimuthal mode numbers of decay waves are preferred; higher ones would suffer stronger Landau damping. Hence one should expect 3wo/2 emissions largely in the m = 0, 1, 2 modes.

REFERENCES

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Ganeswar Mishra was born on May 3, 1962, in Orissa, India. He received the Master's degree in physics from Ravenshaw College in 1985. He is currently working towards the Ph.D. degree at the Department of Physics, Indian Institute of Tech- nology, Delhi. His research interests include un- dulator Cerenkov free-electron lasers, gyrotrons, and laser-plasma interaction.

Indranil Talukdar was born in 1960 in West Bengal, India. He received the Master's degree in physics from the Indian Institute of Technology, Kharagpur, in 1985. Since then he has been work- ing towards the Ph.D. degree at the Department of Physics, Indian Institute of Technology, Delhi.

His research interests include whistler instabili- ties, RF heating, and the current drive in toka- maks.

Vijayshri, photograph and biography not available at the time of publica- tion.

V. K. Tripathi was born on March 11, 1948, in India. He obtained the Master's degree in physics from Agra University in 1967 and the Ph.D. de- gree in plasma physics from the Indian Institute of Technology, Delhi, in 1971.

In 1971 he joined the faculty of the Indian In- stitute of Technology, Delhi, and started working on laser-plasma interaction. In 1976 he moved to the University of Maryland, College Park, where he worked on radio-frequency heating of fusion devices. In 1983 he rejoined the Indian Institute of Technology, Delhi, as Professor of Physics, where he initiated experi- mental and theoretical research in the areas of beam-plasma instabilities, gyrotron, free-electron lasers, and RF heating of plasmas.