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Effect of cross focusing of two laser beams on the growth of laser ripple in plasma

Published online by Cambridge University Press:  25 March 2004

GUNJAN PUROHIT ,
H.D. PANDEY  and
R.P. SHARMA
GUNJAN PUROHIT
Affiliation:
Centre for Energy Studies, Indian Institute of Technology, New Delhi 1, India
H.D. PANDEY
Affiliation:
Centre for Energy Studies, Indian Institute of Technology, New Delhi 1, India
R.P. SHARMA
Affiliation:
Centre for Energy Studies, Indian Institute of Technology, New Delhi 1, India
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Abstract

This article presents an effect of cross focusing of two laser beams on the growth of a laser ripple in laser-produced plasmas. The mechanism of nonlinearity is assumed to be ponderomotive force, arising because of the Gaussian intensity distribution of the laser beams. The dynamical equation governing the laser ripple intensity has been set up and a numerical solution has been presented for typical laser plasma parameters. It is found that the change in the intensity of the second laser beam can affect the growth of the laser ripple significantly.

Keywords

LasersNonlinearityPonderomotiveRipple
Type
Research Article
Information
Laser and Particle Beams , Volume 21 , Issue 4 , October 2003 , pp. 567 - 572
Copyright
© 2003 Cambridge University Press

1. INTRODUCTION

There has been considerable interest in the interaction of intense laser beams with plasmas because of its relevance to laser fusion and charged particle acceleration. In laser-induced fusion, the most important problem is the efficient coupling of the energy of the laser beam to plasma to heat the latter. In this coupling process, many nonlinear phenomena such as self-focusing, filamentation instabilities, stimulated Raman scattering (SRS), stimulated Brillouin scattering (SBS) and so forth (Kruer, 1988) play a crucial role.

Filamentation instability or hot spot formation by laser beam in plasmas has been studied in detail (Kruer, 1988). This instability may also arise because of small-scale fluctuations in the intensity distribution of the pump laser beam. This perturbation could grow inside the plasma, leading to the phenomena of filamentation.

In the past decade, several new laser beam smoothing techniques, namely, random phase plates (RPP; Kato et al., 1984), smoothing by spectral dispersion (SSD; Lehmberg et al., 1987), and induced spatial incoherence (ISI; Skupsky et al., 1989), have been used for controlling natural intensity nonuniformities in laser beams. These techniques introduce randomness into the laser beam through spatial and/or temporal incoherence to produce a smooth laser intensity distribution in a focal spot region. This, in turn, improves the efficiency of laser energy coupling to the plasma and uniformity of fuel compression in inertial confinement fusion (ICF) targets.

In the present article, we have investigated the growth of the laser ripple when a second laser beam is also present. To keep the mathematics simple but to understand the nonlinear mechanism, we have considered the ripple superimposed on one laser beam only. Because of the ponderomotive nonlinearity, the dynamical equation governing the intensity of the laser ripple depends on the total intensity of both lasers. Therefore, by changing the intensity of the second laser, one can control the growth of the ripple in the plasmas.

In Section 2, we present an analysis of the nonlinear effective dielectric constant of the plasma and derive the differential equations governing the nature of the laser ripple intensity in plasma. In Section 3, a brief discussion and conclusions of the numerical results of the present investigation are presented.

2. MODEL EQUATIONS

Consider the propagation of two coaxial Gaussian laser beams of frequencies ω1 and ω2 along the z direction. The initial intensity distributions of the beams are given by

where r is the radial coordinate of the cylindrical coordinate system and r10 and r20 are their initial beamwidths. The expression for ponderomotive force in the presence of two beams can be written as (Schmidt, 1973)

and the modified electron concentration because of ponderomotive force can be written as

where

e and m are the electronic charge and mass, respectively, M is the mass of ion, kB is Boltzmann constant, T0 is the equilibrium temperature of the plasma, and N0 is the electron concentration in the absence of the beams. The effective dielectric constant of the plasma at frequencies ω1 and ω2 can be written as

where

where φ1,2 is the nonlinear part of the dielectric constant and ωp02 = (4πN0 e2)/m is the electron plasma frequency. By using the Taylor expansion of the dielectric constant around r = 0, the equation can be rewritten as

The wave equation governing the electric vectors of the two beams in plasma can be written as

In writing Eq. (6), we have neglected the ∇(∇·E) term, which is justified as long as

The solution for E1,2 can be written as (Akhmanov et al., 1968; Sodha et al., 1976, 1979)

where the dimensionless beam width parameter equation f1,2 is given by

For initial plane wavefronts of the beams, the initial conditions on f1,2 are

Let a perturbation be superimposed on the first beam such that its initial intensity distribution is given by

where n is a positive number and r100 is the width of the ripple. By changing the value of n, the position of the ripple changes. The total electric vector of the laser having ripple superimposed on it can be written as

where E01 is the ripple superimposed on the first laser beam. The total electric vector E satisfies the wave equation

In the WKB approximation, the second term of Eq. (11) can be neglected. This is justified because one has (ωp02/ω12)(1/ε1)In ε1 ≤ 1, and the electric vectors of the main beam E1 and the ripple E01 satisfy the equations

respectively. To obtain the solution of Eq. (13), we express

where A01(r, z) is a complex function of its argument. Substituting for E01 from Eq. (14) into Eq. (13) we get the following equation within the WKB approximation:

Further, substituting for A01 in Eq. (15)

where A010(r, z) is a real function and S10 is the eikonal; we obtain the following equations after separating the real and imaginary parts

where

where φp is the angle between the electric vectors of the main laser beam (one) and the ripple. In writing Eq. (16) we have expanded

Following Akhmanov et al. (1968), the solutions of Eqs. (16a) and (16b) can be written as

where f is the dimensionless beam width parameter of the ripple and φ(z) is a constant. For an initially plane wave front, df/dz = 0 and f = 1 at z = 0. Substituting for A010 and S10 from Eq. (17) into Eq. (16a) and expanding around r = r100 fn1/2 by Taylor expansion, we get

where

Using Eqs. (18a), (18b), and (16a), we get the following equation for f after equating the coefficient of r2:

Equation (19) determines the focusing/defocusing of a ripple. It is apparent from Eq. (17) for A0102 that the ripple grows/decays inside the plasma and the growth rate is ki. The growth rate depends on the intensity of the main beams, the phase angle φp, and parameters of the pump wave and plasma. The following set of parameters has been used in the numerical calculations: r10 = 15 μm, r20 = 20 μm, r100, = 10 μm, ω1 = 1.778 × 1015 rad/s, ω2 = 1.778 × 1014rad/s, ωpo = 0.95 ω2, 2φp = 3π/2, n = 2.8 and 3.

3. DISCUSSION AND CONCLUSIONS

It is obvious from the present analysis that because of the coupling between the main beams and the ripple, the ripple can grow in the plasma. The growth rate of the ripple (ki) is given by Eq. (17), which depends on the effective intensity of the main beam in the plasma, electron density in the plasma, frequency of the laser beams, and the phase angle φp. It is clear from the equation that, when sin 2φp is positive, the ripple will not grow and it will be attenuated at a distance of the order of 1/ki. The ripple will grow only when sin 2φp is negative. From Figure 1, we observe the variation of exp ki(z) of the ripple with the normalized distance of propagation for different powers of the second laser beam. It is obvious from the graph that the growth rate of the ripple increases with the distance of propagation. When the power of the second beam is increased, a similar effect has been found, but the growth rate is decreased.

Variation of exp(ki z) for the two beams with the normalized distance of propagation ξ(= zc/ω12r102) for a fixed power of the first beam [3/4 (m/M)α1 E102 = 3] and different powers of the second beam. Solid line, dotted line, and semi-dotted line are for 3/4 (m/M)α2 E202 = 3, 4, and 5, respectively.

Equation (8) is the fundamental equation for cross focusing of two laser beams when the ripple is not present on the first laser beam. On neglecting the contribution of the second beam in Eq. (8), one can obtain the usual ponderomotive self-focusing of the first beam. With the simultaneous propagation of two laser beams, the ponderomotive nonlinearity introduced in the plasma depends on the total intensity of the two beams, and the behavior of f1 is also governed by f2, and vice versa. In other words, the self-focusing of one beam is affected by the presence of another beam; this is referred to as cross focusing. The first term on the right-hand side of Eq. (19) represents the diffraction phenomenon of the ripple. The second term, which arises from the ponderomotive nonlinearity, describes nonlinear refraction. The relative magnitude of these terms determines the focusing/defocusing behavior of the ripple. If the first term is large in comparison to the second term, the diffraction dominates over the nonlinear phenomena, leading to defocusing of the ripple. When second term is larger than the first term, self-focusing of the ripple is observed. The growth rate (ki) contributes significantly to focusing/defocusing of the ripple. The variations of the beam width parameter f of the ripple with the distance of propagation is illustrated in Figures 2a,b for fixed 2φp = 3π/2 but for different powers of the second laser beam in two cases when n = 2.8 (Fig. 2a) and n = 3 (Fig. 2b). When the power of the second laser beam is increased, the ripple shows diverging behavior continuously. The focusing/defocusing of the ripple is found to be considerably affected by the power of the main beams, phase angle between the electric vectors of the first laser beam, but not much affected by the parameter n.

a: Variation of dimensionless beam width parameter (f) of the ripple with the normalized distance of propagation ξ(= zc/ω12r102) for a fixed power of the first beam [3/4 (m/M)α1 E102 = 3, n = 2.8, and 2φp = 3π/2] and different powers of the second beam. b: Variation of dimensionless beam width parameter (f) of the ripple with the normalized distance of propagation ξ(= zc/ω12r102) for a fixed power of the first beam [3/4 (m/M)α1 E102 = 3, n = 3.0, and 2φp = 3π/2] and different powers of the second beam. Solid line, dotted line, and semi-dotted line are for 3/4 (m/M)α2 E202 = 3, 4, and 5, respectively.

Figure 3a,b depicts the variation of normalized intensity of the ripple at r = r100 fn1/2 in the plasma with the normalized distance of propagation for fixed 2φp = 3π/2, but for different values of the ripple position parameter n. For fixed power of two laser beams, intensity of the ripple increases with the distance of propagation because growth rate (ki) affects significantly the intensity dynamics of the ripple. It is obvious from the graphs that the intensity of the ripple is decreased by increasing the power of the second laser beam. When the power of the second laser beam is increased, the nonlinearity, which governs the growth of the laser ripple, becomes affected [see Eq. (17)]; therefore, the ripple intensity behaves as shown in Figures 3a,b.

a: Variation in intensity of the ripple with the normalized distance of propagation ξ(= zc/ω12r102) for a fixed power of the first beam [3/4 (m/M)α1 E102 = 3, n = 2.8, and 2φp = 3π/2] and different powers of the second beam. b: Variation in intensity of the ripple with the normalized distance of propagation ξ(= zc/ω12r102) for a fixed power of the first beam [3/4 (m/M)α1 E102 = 3, n = 3.0, and 2φp = 3π/2] and different powers of the second beam. Solid line, dotted line, and semi-dotted line are for 3/4 (m/M)α2 E202 = 3, 4, and 5, respectively.

In conclusion, two laser beams copropagating in plasma modify each other's characteristics of ponderomotive self-focusing. One important result that comes from the present analysis is that we can control the fluctuations on the first laser beam by increasing the power of the second laser beam. This study may be useful in beat wave excitation, laser plasma coupling where the filamentation process plays a very important role. This will be a part of a future investigation.

ACKNOWLEDGMENT

This research work is supported by the Council of Scientific and Industrial Research (CSIR), India.

References

REFERENCES

Akhmanov, S.A., Sukhorukov, A.P. & Khokhlov, R.V. (1968). Soviet Phys. Usp. 10, 609.
Kato, Y., Mima, K., Miyaraga, N., Arinaga, S., Kitagawa, Y., Nakatsuka, M. & Yamanaka, C. (1984). Phys. Rev. Lett. 53, 1057.
Kruer, W.L. (1988). The Physics of Laser Plasma Interactions. New York: Addison-Wesley Publishing Company.
Lehmberg, R.H., Schmitt, A.J. & Bodner, S. (1987). J. Appl. Phys. 62, 2680.
Schmidt, G. (1973). Phys. Fluids 16, 1676.
Skupsky, S., Short, R.W., Kessler, T., Craxton, R.S., Letzring, S. & Soures, J.M. (1989). J. Appl. Phys. 66, 3456.
Sodha, M.S., Ghatak, A.K. & Tripathi, V.K. (1976). Prog. Optics 13, 171.
Sodha, M.S., Govind., Tewari, D.P., Sharma, R.P., &Kaushik, S.C. (1979). J. Appl. Phys. 50, 158.
Figure 0

Variation of exp(ki z) for the two beams with the normalized distance of propagation ξ(= zc/ω12r102) for a fixed power of the first beam [3/4 (m/M)α1 E102 = 3] and different powers of the second beam. Solid line, dotted line, and semi-dotted line are for 3/4 (m/M)α2 E202 = 3, 4, and 5, respectively.

Figure 1

a: Variation of dimensionless beam width parameter (f) of the ripple with the normalized distance of propagation ξ(= zc/ω12r102) for a fixed power of the first beam [3/4 (m/M)α1 E102 = 3, n = 2.8, and 2φp = 3π/2] and different powers of the second beam. b: Variation of dimensionless beam width parameter (f) of the ripple with the normalized distance of propagation ξ(= zc/ω12r102) for a fixed power of the first beam [3/4 (m/M)α1 E102 = 3, n = 3.0, and 2φp = 3π/2] and different powers of the second beam. Solid line, dotted line, and semi-dotted line are for 3/4 (m/M)α2 E202 = 3, 4, and 5, respectively.

Figure 2

a: Variation in intensity of the ripple with the normalized distance of propagation ξ(= zc/ω12r102) for a fixed power of the first beam [3/4 (m/M)α1 E102 = 3, n = 2.8, and 2φp = 3π/2] and different powers of the second beam. b: Variation in intensity of the ripple with the normalized distance of propagation ξ(= zc/ω12r102) for a fixed power of the first beam [3/4 (m/M)α1 E102 = 3, n = 3.0, and 2φp = 3π/2] and different powers of the second beam. Solid line, dotted line, and semi-dotted line are for 3/4 (m/M)α2 E202 = 3, 4, and 5, respectively.