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UCRL-JC-121196 PREPRINT

Measurement of the Bespalov-Talanov Gain Spectrum in a Dispersive Medium with Large N2

P. J. Wegner M. D. Feit

A. J.lFleck, Jr. D. Eimerl

This paper was prepared for submittal to the 1st Annual Solid-state Lasers for Application

to Inertial Confinement Fusion Monterey, CA

May 30 -June 2, 1995

June 15,1995

~ ~ a p ~ ~ a ~ ~ r ~ ~ ~ r ~ b t i ~ ~ ~ ~ a joPmrlorploceedings. Since changes may be made before publication, this preprint is nude available with the understanding that it will not be ated or reproduced without the permission of the author.

DISCLAIMER

This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor the University of California nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Referens herein to any specific commeraal product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or the University of California, and shall not be used for advertising or product endorsement purposes.

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DISCLAIMER

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Measurement of the Bespalov-Talanov gain spectrum in a dispersive medium with large nz

P. Wegner, M. D. Feit, J. A. Fleck, Jr., D. Eimerl

Lawrence Livermore National Laboratory PO Box 5508, Livermore. CA. 94550

ABSTRACT

Conditions which seed the self focussing of high-powex broadband laser beams are determined by examining growth rates for planewave perturbations on a strong pump field as a €unction of frequency and angle. Measurements verifying predictions of growth based on the linearized stability analysis of BespaIov and Talanov extended to broadband fields are reported.

1. INTRODUCTION

In the last five to ten years reseatchers working on laser drivers for inertial confinement fusion have recognized the importance of using small amounts of bandwidth- on the order of a few tenths of a percent- to achieve smoother, more uniform laset irradiance profiles at the target. Beam smoothing techniques which utilize controlled temporal or angular dispersion of laser bandwidth to produce target plane irradiance which is spatially smooth in a timeaveraged sense have been developed and demonstrated in a number of laboratories. With the development and implementarion of such smoothing schemes on glass ICF lasers has come an increased interest in the propagation characteristics of high-power broadband beams, and in @cular the self focussing of such beams in solid-state laser materials with the potential limitations to systean performance that this pight pose.

* .

Smoothing schemes implemented on the the Nova N&:glass laser at Lawrence Livermore National Laboratory have typi- -tally incorporated a grating at the front end of the laser e m , which causes the spectral comp6nents of the laser field to propagate through the ampIifier chain at different angles. As fm pointed out by Bespalov and Talanov and subsequently verified by Bliss et at. ', the initial growth rate of pert;abgtions on a strong pump field (ko, 00) which chara- the onset of self focussing is a function of the transverse wave vector kA, or angle of the peatubtion. The Bespalov-Talanov (BT) stabiity argument was based on a l i i analysis of the nonlinear paraxiat wave equation applicable to monochromatic fields which has recently been extended to the case of fields with bandwidths of up to a few percent by authors Feit and Fleck. The genemlhd BT analysis indicates that growth rate depends on the 6requency offset of the perturbation A- co-610 as well as kL and that the interplay betweem difitiaction and group-velocity dispersion determines the initial behavior (oscillation, stability or growth). In the foliowing Seaion we derive the mnditionS on Aa and kl that are conducive for nonlinear ripple growth in a dispeFsive medium. Measurements verifying the predictions of this theory are described in Section 3.

'2.THEORY

A laser pulse having moderate spatial and femporal bandwidth, propagating through a transparent dielectric with an intensitydependent refractive index can be shown to evolve according to the nonlinear paraxial wave equation:

where A represents the slowly-varying amplitude of the electric field:

Equation (1) is second-order in both space and time to account for diffraction and dispwsion effects; V12 represents the trans-

verse Laplacian, Y, and v:= dv,ldo are the group velocity and group-velocity dispersion at a,,. In defining an intensity- dependent refiactive index n = &+ y1 (or n = &+ n ~ l A I ’), it is assumed that the dispersion of x”) over the relatively narrow temporal bandwidth of the pulse can be neglected.

To examine the nonlinear growth of ripples on the laser pulse we consider the case when the field is initially comprised of a strong plant warn Eo at &,%) and a weak perturbation IC at (IC,,+&, a+&). Energy and momentum conservation dic- tate that the nonlinear coupling of 4 with E, and Eo* must generate an additional field component I at &-&, c&-Aa), which has tfaditionaly been r e f d to as the conjugate of E, (see discussion in the Appendix). When conditions are optimal for growth, these two fsld components grow togethw. E+ mix- ing with the pump to form the conjugate which then mixes with the pump to generate % and so on. As the process continues, higher-ordercomponents06fmhk,~f~o)proportionalto %’” also build up, with the additional bandwidth leading to a steepening of the ripple intensity profile that is characteristic of self focussing. Such highex-order effects are neglected in the linearized analysis which follows.

-0.5 0.0 0.5 1.0 1.5 a

Figure 1. Plot of the square of the perturbation eigen- valueXvasusthe~hasanatchingparamevra~NLB.. Exponedad growth of the pexturwon OCMS when 1 is imaginary. indicated by region n.

With the goal of Backing &+ and I of arbitrary relative initial magnitnde we adopt an amplitude for the input field of the following form:

* ‘ . I

~ ( z , t ) = [A. + .s+(z)ei(k”-h) + tz- (z)e- i (kz-h) 1 eiB.2 , 7 = t -z /vg (3)

where we have anticipated the nonlinear phase shift per unit length B e (kJnJn2 I A,, I that is incurred by the field as a result of the htenSitydependent refractive index. Substituting (3) into (1) and keeping terms linear in Q results in the following coupled equations for the evolution of the perturbations:

when e+ = G this is just the familiar BT result, cxcept with R now given by (l&)[k? + (~,V;/V:)&~]. Whereas the ariginaI analysis predicted growth or Oscillation depending on the magnitudes of k, and B, this result indicates that the fresuency of the perturbation and the material dispersion will be determining factors as as well:Equations (4) are readily solved and have the following general solutions:

where 5 = and the eigenvalue X = QE,. As shown in Figure 1 there exist three distinct regions of solution, depending on the value of the phasematching parameter QLB, = a. In regions I (a < 0) and III (a > 1) h is real and the solutions are

osciliatory. Physically, region I corresponds to the case in which the firequency shift of the perturbation is so large that the dispersion of the material prohibits efficient phasematching of the four-wave mixing process (see appbdix). In Region III it is excessive angle that inhibits efficient nonlinear coupling, al- though in this case the phasematching can be improved by increasing B, Only in region I1 (0 e a c 1) will the perturbation actually grow.

A special &e of interest is that in which the conjugate field L is initially zero, which would be the situation for ex- ample in a pump&obe "two-beam mixing" experiment like the one discussed in the next section. Solving for the resulting field intensities we find I+(z) = Z+(O) + I&), with the intensity of the conjugate field given by:

. <

where B= BJ. Ratios of 1-11, versus B for several different val- ues of the phamwhln * gparameter a are shown in Figure 2

0.0 0.5 1 .o 1.5 2.0 8 (radians)

Figure 2. Calculated ratios of I J + versus B g for phase- matchkg parametexs u of 5 (a), .2 and .8 @), 0 and 1 (c),-.2sndl2(d),-5and15(e),-l and2(f),-2and 3 (g). Calculations assume L(0) = 0.

With perfed phasematching (a = 5) the conjugate will grow firom zero to -60% of I+ at a B of 1 radian, beyond which the . ,ydidity of the linearized theory should probably be questioned Note that even when a lies outside of Re 'on II the intensity of

demonskated in Figures 3 (a) and @) where we pick a Be of .025 radianskm and plot the expected ratio of I J + as a function -of W@ aM tk kJk0 for two different glasses- f t k d silica and SF 6- at a wavelength of 1.053 p. The growth region (region

tI& conjugate field for a given B may be a m b l e as it oscjllates over a whemce length z, = n/2B, P- Ma-1). This is further

0.0 0.5 1 .o 1.5 0.0 0.5 1 .o 1 .El W @ O @I w o o (XI

. .. - . . . . . . - . -._ . . - - . :::figure:3(a),BT-gain spectrum of fused silica at B, of .025 raditm/cm. Plot shows calculated ratios of ZJI+ versus frequency and angle of &.. Contour values are B-dependent as per Figure 2.

Figure 3@). Corresponding spectrum of lead-silicate glass SF 6 (MIL# 805254). which has approximately five times the dispersion of fused silica.

1I) is shaded. Maximum growth occurs at the angles and frequencies indicated by the solid line e,.,= J(2EJh - &'(Wq,)*) where the groupvelocity dispersion v i is -72 cm/s/rad-GHz for Si& and -364 crn/irad-GHz for SF 6. The dependence of L, on frequency is seen to increase with the dispersion of the material.

3. EXPERIMENT

To verify the growth spectrum depicted in Figure 3 we configured an experiment to mix tunable-frequency probe puls- es and 0 5 GW/cm2 pump pulses at various angles inside a u)an long sample of SF 6 glass. The glass was chosen for its high dispersion content and its large nonlinearity: the nonlinear coefficient y is 21 x IO-' cm2/cW, of which the instantaneous electronic component is 81%? As discussed above, at probe angles and frequencies (kx, Aw) which satisfy optimal growth conditions, conjugate fields (-kx,-Aco) build up which are measurable. The experimental layout for this measurement is shown in Figure 4.

temporal and spectral

The pump and probe pulses are derived from the output pulse train of an actively modelocked and Q-switched NdYLF oscillator, that is amplified to - 35 rnJ and directed to a 50/% split In one leg of the split, single 85-ps pulses sliced from the cenm of the train are amplified to -10 mJ in a double-pass glass amplifier for use as experimental pump pulses. The pump pulses are temporally gaussian wirh a content that is . .lpoadened slighuy beyond the transform limit to -6 GHz as a result qf tow-level self-phase modulation in the oscilhtor: Ca+g wy$tgand ga@Ilf~tedng ensure a w U i spatiaUy smooth 5.65-mm &uneter (l/e intensity) gaussian beam at the SF 6 sample. In the other leg of the split, the pulse train is frequencyconverted to the third hannonic and used to synchro- nously pump an optical parametric oscillator (OPO). The OPO employs a 12-mm Brewstereut crystal of Lithium tribome, an intracavity etalon, and cavily dumping to generate single 80 to 85-ps 0.5-MW probe pulses with bandwidths of 9 GHz and wavelmg!h tunable by BO nm about 1.053 pm! The size of the probe beam at the Sample is 3.8 mm and its spatial quality is excellent.Relative W i g of thetwopulsesisbetter than 10psasmeasured with a streak camera

Figure 4. Experimental layout used to measure the growth- spectrum depicted in Figure 3. Paths of the pump p b e (...) and conjugate C 3 beams arc mdicated Energy diodes are denoted "d".

Ihe 'p'-polarized beams ate c o m b i i at ti 15% spIitter situated just upstream of the glass sample. Relative pointing adjutmen& ae accomplished to within f 35 pad (imtemal angle) as determined by viewing the first surface return 6rom the sample in the far field with a CIDcamera. "be same camera is &to view the image of a cross-hait positioned in the reflected beam path at a plane equivalent to the sample center, permitting evaluation of beam centration to better than J nun. At the sample output, the probe beam and its conjugate am separated fiom the pump in a double momchromator, consisting of two 1480-gpmm gratings positioned at the relay planes of a 1:l telescope. Neutral density fdtexs placed at the focal plane of the telescope attenuate the pump and probe beams to a level that permits the imaging of the fxlds in this plane onto a CCD camem. The camem is situated at the output of the monochromator and has an angular fE1d of view off 3.5 nuad, referred back to inside the sample. A slight tilt of the second grating relative to the first provides residual dispersion of - 30 p d / A m the horizontal axis for wavelength discrimination at the camera. Adjustment to the pointing of the probe is performed in the vertical axis, so that wavelength and angular offsets ate recorded in orthogonal axes (see Figure 5).

The measurements wnsist of angular linescans through the spectrum of Figure 3. The OPO is fm tuned to a specific p b e wavelength; three wavelengths were chosen, shifted by 40.40, and +120 angstroms relalive to the 1.053-p pump. A series of shots are then taken at each wavelength during which the probe angk is scanned in increments of - 250 pad to cover the angular range of the Figure. Tuxmaround time between shots is - 2 minutes, limited by thermal effects in the glass amplifier. Because multiple data points are required at each angle for statistical purposes, a scan from 0 to 35 mrad and back takes upwards of four hours to complete, during which time the laser parameters and system calibrations must remain stable. Shot to

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