Ž .Fusion Engineering and Design 44 1999 427]430

Temporal waveform control of laser pulse by frequencychirping

Yoshinori KatoU, Hidetsugu Yoshida, Hisanori Fujita, Sadao Nakai

Institute of Laser Engineering, Osaka Uni ersity, 2]6 Yamada-oka, Suita, Osaka, 565 Japan

Abstract

Arbitrary temporal waveform control can be achieved by means of amplifier gain control with chirped laser pulse.As lasers for fusion research are broad-band glass lasers, a chirping method of the THz region is required to obtainarbitrary temporal waveform control. In order to precisely simulate amplification properties of chirped pulse, wehave developed the simulation code for an inhomogeneous glass laser amplifier. Included physics are as follows:

Žtemporal chirping; relaxation of lower level; and change of distribution density between sub-levels. Chirping 0.9.THz applied to a Gaussian pulse increased the peak intensity of output in 20% to the case without chirping.

Q 1999 Elsevier Science S.A. All rights reserved.

Keywords: Laser pulse; Frequency chirping; Gaussian pulse; Sub-levels

1. Introduction

The Laser fusion experiment requires high en-ergy and a precisely shaped optical pulse toachieve high compression of fusion targets. Now,

Ž ( ) 2.tailored pulse quadratic curve: I t A t is con-sidered the most desired temporal waveform. Asa frequency of laser light is changed by chirping,and amplifier gain of glass laser is a function ofinput frequency, desired temporal waveform canbe made by using chirped laser pulse. In addition,

U Corresponding author. Tel.: q81 687 98761; fax: q81 68770900; e-mail: yosinori@ile.osaka-u.ac.jp

if the chirped pulse is focused by a lens, theposition of the focal point changes quickly due todispersion of the refractive index of the lensmedium. As a result, pulse intensity on the targetcan be increased during the laser pulse.

In this report, we present a simulation modelfor chirped pulse amplification on a gain mediumŽ .LHG-8 with inhomogeneous broadening.

2. Simulation model

Previously, we have performed basic experi-ments using a Nd:YAG laser and a fast EOmodulator and developed a simulation model fortemporal waveform control of laser pulse by fre-

0920-3796r99r$ - see front matter Q 1999 Elsevier Science S.A. All rights reserved.Ž .P I I: S 0 9 2 0 - 3 7 9 6 9 8 0 0 2 9 4 - 4

( )Y. Kato et al. r Fusion Engineering and Design 44 1999 427]430428

quency chirping. Experimental results were com-pared with simulation results of chirped pulseamplification based on the Frantz]Nodvik theoryw x1 . Experimental results showed good agreementwith simulation results. It was indicated that de-sired temporal waveform can be achieved by fre-quency chirping.

As lasers used in fusion research are broad-band Nd:glass lasers, THz chirping is required toobtain desired temporal waveform control. In or-der to generate optimum tailored pulses, chirpingof approx. 3 THz is required by the two-levelsimulation. The fluorescence of neodymium inlaser glass consists of 12 overlapping transitions.As the gain of amplification is proportional to theproduct of cross-section and population inversion,it is necessary to consider the product for individ-ual transitions. The fluorescence spectrum wasobtained by the direct measurement using a fluo-

Ž .rescence spectrometer SS-25: JASCO Co. Ltd.and characterized and overlapping by the summa-tion of the products of cross-sections and popula-tions of upper sub-levels. In order to preciselysimulate the chirped pulse amplification, thesimulation for the two-level model should be ex-tended to the multi-level.

At first, the spectrum of neodymium phosphateŽ .laser glass LHG-8 was divided into 12 sub-tran-

sitions by a curve fitting method. Fitting parame-ters were energy levels, relative intensifies and

line widths. The individual line shape was as-sumed by convolution of the Lorentzian andGaussian shapes with constant line widths. Theindividual cross-sections were calculated by

2t

h ?exp yln2 ?k ,i ½ 5ž /v` gŽ .s n s d t ,Hk ,i 2Ž .y` nyu y tk ,i1q ½ 5v l

Ž .ks1,2,3; is1 to 6, 11 Ž . Ž .u s E yE , 2k ,i k ih

where h and u are line strengths and centerk ,i k ,ifrequencies of individual transitions, n is the fre-

Ž .quency, v and v , are line widths HWHM ofg lGaussian and Lorentzian, respectively, h is thePlank constant and E , E are energy of upperk iand lower sub-levels. The fluorescence spectrumcan be described by summation of products ofcross-section and population of upper levels.

N2 qN21 2Ž . Ž . Ž .s n s s n ? , 3Ý k ,i N2 kk ,i

where N2 is the populations of upper levels.kAs a result, the frequencies and line strengths

of the individual transitions were determined asshown in Fig. 1. The Lorentzian and Gaussian

Ž .Fig. 1. Measured and calculated fluorescence line shapes of LHG-8. Spectrum of neodymium phosphate laser glass LHG-8 wasdivided into 12 sub-transitions by a curve fitting method. As a result, the frequencies and line strengths of the individual transitionswere determined. The Lorentzian and Gaussian line widths were determined to be 0.75 and 2.4 THz, respectively.

( )Y. Kato et al. r Fusion Engineering and Design 44 1999 427]430 429

line widths were determined to be 0.75 and 2.4THz, respectively.

The number densities of active atoms in thelower and upper levels are denoted, respectively,N1 and N2 and photon density by n. Ratei kequations for number densities and photons areas follows.

1 ­ n ­ n Ž . Ž .q ss n N2 yN1 , 4k ,i k iž /c ­ t ­ x

­ N1 1i Ž . Ž .s s cn N2 yN1 ? 1y , 5Ý k ,i k i ž /­ t trk

­ N2 k Ž . Ž .sy s cn N2 yN1 , 6Ý k ,i k i­ ti

where c is the velocity of light in the medium, sk ,iis the cross-section, t is relaxation time from therlower level to the ground level.

The multi-level rate equation code includes thefollowing effects: temporal chirping of the inci-dent laser; relaxation process of lower level; andredistribution of population density betweenStark-levels. Recently, Bibeau et al. reported thatthe relaxation time of the lower level was 250]450

w xps for the phosphate laser glass 2 . The relax-ation time of 300 ps was used in the simulationcode. Since thermalization of closely spaced en-ergy levels within the Stark manifold is extremely

Ž . w xrapid -10 ps 3]5 , the Boltzmann equilibriumis assumed in both upper and lower laser levels by

Ž .simulation mesh time 20 ps . The temperature of300 K was used in the simulation code.

3. Simulation results

Amplification properties were simulated by thetwo-level model and the multi-level model. Theresults are shown in Fig. 2. The simulationparameters were as follows: the peak stimulatedemission cross-section was 4.0=10y20 cm2; inputpulse width was 5 ns; amplifier length was 100 cm;and the small-signal gain coefficient was 6%rcmwhich corresponded with upper level populationof 1.5=1018 cmy3.

The relaxation process from 4 I to 4 I11r2 9r2increased the extraction energy significantly. Theextraction efficiency was reached at 82% in thecase of the multi-level model, and at 81% in thecase of the two-level model. When the lower levelrelaxation is not included in the two-level model,the extraction efficiency saturated at 46% for thesame parameters.

Typical simulation results with and withoutchirping are shown in Fig. 3. The maximum chirpwas 0.9 THz as shown in the figure and inputpulse intensity was 20 MWrcm2. The otherparameters were the same as mentioned above.

Although the output pulse energy was lower,the maximum intensity was 20% higher for thecase with chirping. Because the energy extractionwas suppressed due to lower gain in front of thepulse, rapid energy extraction was achieved at thepeak of the incident pulse. Maximum intensityincreased more effectively with higher small-sig-nal gain and broader frequency chirping. In thecase where maximum chirp was 3 THz, maximumintensity was 100% higher. Waveform control for

Ž .Fig. 2. Amplification properties are simulated for LHG-8 by simulation codes of the two-level and multilevel. a Output fluence vs.Ž . Ž 2 .input fluence and b extraction efficiency vs. gain distance input fluence 0.1 Jrcm .

( )Y. Kato et al. r Fusion Engineering and Design 44 1999 427]430430

Fig. 3. Typical simulation results of amplification propertieswith and without chirping. The maximum chirp was 0.9 THz asshown in the figure and the input peak intensity was 20MWrcm2.

high energy is easier and is sufficient at lowerchirping.

4. Conclusion

The simulation code for YAG laser system wasextended to an inhomogeneous glass laser system.The code developed can precisely simulate tem-poral waveform of optical pulse propagating inthe amplifier. We will be examining the devel-

oped code by comparing it with experimentalresults.

As the laser fusion driver is a broad-band glasslaser, a chirping method of the THz region isrequired to obtain an arbitrary shaped pulse. Weare considering a chirping method of the THzregion by using a fiber or a grating or a highspeed EOM or mixing these.

Furthermore, we must consider temporal focus-ing intensity on target. In implosion experiments,tangential irradiation is adopted for uniform irra-diation. When a chirped pulse of 3 THz is focusedon target of 1 mm in diameter by using a lensŽ .BK-7 with F number of 3 and the focal length is1000 mm, the laser peak intensity on target isestimated 1.5 times higher due to dynamic focus-ing. Laser intensity on target is four times higherby using LaF9 with larger dispersion instead ofBK-7.

References

w x Ž .1 L.M. Frantz, et al., J. Appl. Phys. 34 1963 2346]2349.w x Ž .2 C. Bibeau, et al., J. Opt. Soc. Am. B 12 1995 1981]1992.w x3 J.D. Zuegel, et al., IEEE J. Quantum Electron. 31

Ž .1995 1742]1746.w x Ž .4 T. Kushida, et al., Phys. Rev. 167 1968 289]291.w x Ž .5 T. Kushida, et al., Phys. Rev. 185 1969 500]508.