Modeling the Temporal-Pulse-Shape Dynamics of an Actively Stabilized Regenerative Amplifier for OMEGA Pulse-Shaping Applications

Transcription

1 Modelig the Temporal-Pulse-Shape Dyamics of a Actively Stabilized Regeerative Amplifier for OMEGA Pulse-Shapig Applicatios Advaces i laser-fusio techology idicate that the temporal profile of the laser pulse applied to laser-fusio targets is importat for improvig the performace of these targets. 1 The OMEGA laser is a 60-beam laser-fusio system capable of producig a total of 30 kj of ultraviolet (351-m) eergy o target, where the temporal profile of the optical pulse applied to a laser-fusio target ca be specified i advace. This is accomplished by a pulse-shapig system that produces a optical pulse with a specific temporal pulse shape at the aojoule eergy level. 2 This pulse seeds a actively stabilized Nd:YLF regeerative amplifier 3 (rege) followed by ad wavelegth matched (1053 m) to a series of Nd:glass amplifiers. The beams are the frequecy tripled to the third harmoic usig KDP oliear crystals. To achieve the desired o-target optical pulse shape, the temporal dyamics of the etire OMEGA laser system must be accurately modeled to determie the specific temporal profile of the seed pulse required from the pulse-shapig system at the begiig of the laser. The temporal profile of this low-eergy seed pulse, whe amplified ad frequecy tripled by the laser system, will the compesate for the temporal distortios caused by gai saturatio i the rege ad amplifiers ad by the triplig process, ad will produce the desired pulse shape o target. To determie the required temporal profile of the optical pulse at the begiig of the system, all sources of temporal distortios i the system must be uderstood ad compesated for. The temporal distortio due to the frequecy-triplig process is modeled with a time-depedet simulatio of the appropriate oliear equatios for this process. 4 The temporal-pulse distortios i the system s Nd:glass amplifiers are easily modeled with our beam code RAINBOW. Modelig the actively stabilized Nd:YLF rege at the begiig of the system is the topic of this article. With the rege model described here, OMEGA s temporal dyamics ca ow be completely modeled. Pulse distortios i the system ca be easily compesated for by proper choice of the seed temporal profile determied from the overall model. The gai of the OMEGA system from the pulse-shapig modulator to the target is approximately ; a gai of 10 7 i the actively stabilized rege is icluded i this overall gai. Modelig the actively stabilized rege is complicated by may factors. The rege must be treated as a multipass amplifier, the last few passes of which experiece sigificat gai saturatio i the Nd:YLF laser rod. The lifetime of the lower-laser-level maifold i Nd:YLF has bee measured to be 21 s, 5 ad the roud-trip time i the cavity is 26 s. A Fratz-Nodvik type solutio 6 for the gai i the Nd:YLF medium caot accout for this fiite lower-laser-level lifetime ad, hece, is iappropriate; the rate equatios must be used to describe the siglepass gai i the Nd:YLF medium. Fially, the rege icorporates a feedback mechaism that measures the circulatig pulse eergy each roud-trip. 3 Whe the circulatig pulse eergy exceeds a threshold (~25 µj), a feedback mechaism is activated. The feedback mechaism itroduces appropriate losses ito the cavity each roud-trip i order to stabilize the circulatig pulse eergy to a fixed but low-eergy level. Durig this prelase stabilizatio phase the rege is operatig with a et gai (roud-trip gai/loss) approximately equal to uity ad establishes a costat ad stable pulse-eergy output from the rege. After this prelase stabilizatio is achieved, the laser ca be Q-switched by elimiatig the feedback losses from the cavity. The rege will the emit a Q-switched evelope of pulses. The pulse at the peak of the Q-switched evelope is switched out ad set to the OMEGA amplifiers. This stabilized rege produces pulses with a log-term shot-to-shot eergy stability of approximately %, despite the fluctuatios itroduced by the flash-lamp pumpig, ad is isesitive to ijected-pulse eergy variatios of more tha two orders of magitude. 3 The rege is modeled by umerical itegratio of the rate equatios ad careful cosideratio of the rege dyamics. The rege model described here icludes gai saturatio i the Nd:YLF laser rod, itracavity losses, lower-laser-level lifetimes, ad the active losses itroduced by the stabilizer-feed- 18 LLE Review, Volume 69

2 back circuit. Careful measuremets of the iput ad output shaped optical pulses from the rege have bee made ad will be discussed. The calculatios o this rege agree well with the measured output of the rege ad serve as a model for this importat OMEGA compoet. With this rege model, the etire temporal-pulse-shapig dyamics of OMEGA ca ow be modeled from the pulse-shapig system to the fial otarget pulse shape. This modelig provides us with the capability to accurately produce ay desired temporally shaped optical pulse o target for laser-fusio experimets. Rate Equatios OMEGA s pulse-shapig system produces a shaped optical pulse that is ijected ito the actively stabilized regeerative amplifier. The output-pulse shape of the rege is determied by gai saturatio i the active medium ad by the cavity dyamics. The rege model cosists of ijectig a temporally shaped pulse ito the cavity ad calculatig the ew shape after every pass through the cavity. The effect o the pulse shape due to each compoet is treated separately i the calculatio. I this sectio we discuss the temporal distortio due to a sigle pass through the gai medium ad i the ext sectio icorporate this ito the calculatio of the overall rege dyamics. Gai saturatio due to a sigle pass through a gai medium is calculated by solvig the laser rate equatios 7 ad lower laser maifolds. I these equatios c is the speed of light i vacuum; is the idex of refractio of the gai medium; σ is the stimulated emissio cross sectio ( cm 2 ); ad τ i,j is the relaxatio time of the trasitio from maifold i to maifold j (here, level 0 represets the groud state). The upper laser maifold N 2 cosists of two sublevels labeled with eergies E 2, ( = 1,2), ad the lower laser maifold N 1 cosists of six sublevels labeled with eergies E 1,m (m = 1 to 6), two of which are degeerate i eergy (sublevels 2 ad 3) as show i Fig The stimulated emissio terms i the rate equatios [Eqs. (1)] (first terms o the right side i each equatio) ivolve trasitios betwee the sublevels E 21 ad E 12 as show i Fig ; hece, the thermal occupatio f i N i of these laser-active sublevels is used i these terms. The thermal occupatio of these sublevels is calculated by ad e E12 ktn fn 1 1 = 1 m= 6 e E1m kt = N1 (2a) m= 1 e E21 ktn f2n2 = 2 e E21 kt + e E22 kt = N2, (2b) where E i,j is the eergy of level i sublevel j relative to the lowest ϕ( zt, ) c ϕ( zt, ) + t z cσϕ( z, = [ f N ( zt, ) fn( zt, )], (1a) N 2 E 22 E 21 11,597 cm 1 11,538 cm 1 N1( z, cσϕ( z, = [ f2n2( zt, ) fn 1 1( zt, )] t + N2( z, N1( z,, τ τ 21 N2( z, cσϕ( z, = t N z t f2n2 zt, fn 1 1 zt, 2(, ) [ ( ) ( )], τ (1b) (1c) E cm 1 N 1 E cm 1 E cm 1 E 12, cm 1 E cm 1 Groud level 0 E8269 which describe the evolutio of the cavity photo flux ϕ, the atomic populatio N 1 of the lower laser maifold, ad the atomic populatio N 2 of the upper laser maifold. These rate equatios explicitly accout for the lifetimes of these upper Figure The eergy levels ivolved i the 53-µm-Nd:YLF laser trasitio. The arrow shows the laser trasitio betwee sublevels withi the maifolds show. LLE Review, Volume 69 19

3 eergy level i the maifold, k is the Boltzma costat, ad T is the temperature (assumed to be room temperature). The rate equatios [Eqs. (1)] ca be solved umerically. We trasform these equatios alog their characteristics i the time-distace plae with the trasformatio equatios ct z x z+ τ t. (3) c If we use the chai rule ad the substitutio σϕ x N = 2 ( ) [ ( ) ( )] f2n2 zt, fn 1 1 zt, dx (4) with cdt = dz = dx 2, we get the set of fiite-differece equatios dϕ = N, (5a) dn N N N 1 = dt 1 dt, (5b) τ21 τ10 dn N N 2 = 2 2 dt τ21 (5c) for the trasformed rate equatios. Here we have trasformed Eq. (1a) usig the trasformatio equatios [Eqs. (3)], ad we have left Eqs. (1b) ad (1c) utrasformed sice the photo flux evolves i both space ad time, whereas the populatios evolve i time oly. These equatios ca be solved umerically give appropriate boudary coditios. I the model, the photo flux is specified at the etrace face of the laser rod ad is give by the temporal profile of the pulse eterig the rod. The iitial upper-laser-level populatio is determied from measuremets of the laser rod small-sigal gai, ad for simplicity the iitial lower-laser-level populatio is assumed to be zero. With these boudary coditios, Eqs. (5) ca be umerically itegrated to yield the photo flux at ay time ad for ay positio i the laser rod. Of iterest for our calculatios is the output-pulse shape specified at the output face of the laser rod. These equatios with these boudary coditios, alog with the rege dyamics discussed below, have bee solved umerically, ad the results are preseted below. Regeerative Amplifier Model Modelig the rege cosists of ijectig a pulse with a give pulse shape ad eergy ito the rege ad calculatig the ew pulse shape ad eergy after each roud-trip through the cavity. A sigle roud-trip through the regeerative amplifier is depicted i Fig The pulse first experieces gai through the laser rod followed by propagatio to the outputcouplig mirror ad back. The pulse the experieces gai agai followed by propagatio to the ed mirror ad back. Losses due to the output-couplig mirror ad the feedback stabilizer (discussed below) are icluded i the calculatio. Durig propagatio of the pulse i the cavity, the upper- ad lower-laser-level maifolds are allowed to decay with their respective lifetimes. This calculatio for a sigle pass through the cavity gives the output-pulse shape ad eergy, give the iput-pulse shape ad eergy for the pass. The output pulse for each pass is used as the iput pulse for the ext pass through the cavity, ad the procedure is repeated for a give umber of roud-trips through the cavity. The loss due to the feedback stabilizer depeds o several factors. The cavity icorporates two Pockels cells, oe of which is feedback cotrolled. Specific voltages are applied to all four electrodes of the two Pockels cells at specific times. 3 Durig the begiig of the flash-lamp cycle, high losses are itroduced ito the cavity to allow the gai to build up i the rod. At the peak of the gai, a pulse is ijected ito the cavity at time t 1, ad all losses are removed from the cavity (with the exceptio of the static losses here assumed to be 55% i our laser, which icludes the 50% output coupler loss) allowig the circulatig-pulse eergy to icrease. The applied voltages after time t 1 are show schematically i Fig (a) (however, ot to scale). Whe the circulatig-pulse eergy reaches a threshold value (adjusted to ~25 µj), the feedback stabilizer is activated. At this time (t 2 ) a dc voltage V dc is applied to oe electrode of the first Pockels cell, which itroduces a dc loss ito the cavity. Simultaeously, a modulated feedback-co- E8313 Feedback loss Feedback PC Propagatio Gai Nd:YLF Iject pulse Propagatio Mirror loss Figure The rege model calculates the pulse shape ad eergy after a roud-trip i the cavity, the iterates for may roud-trips. The calculatio icludes the effects of gai saturatio, propagatio, ad static ad feedback losses. 20 LLE Review, Volume 69

4 Feedback voltage Measured rege output E8270 V π 0 t 1 (a) (b) V( V dc t 2 t 3 t 2 t Time (µs) Figure Rege temporal dyamics showig (a) feedback-cotrolled Pockels cell voltages (ot to scale), ad (b) measured rege-output evelope filtered to remove idividual pulses i the trai. trolled voltage V( is applied to a electrode of the feedbackcotrolled Pockels cell, which itroduces a feedback-cotrolled modulated loss i the cavity. The fuctio of the feedbackcotrolled modulated loss is to stabilize the circulatig pulse eergy to a specified costat low value. If the pulse eergy falls below (above) this eergy, loss is removed (added) to maitai the specified output-pulse eergy. Specially desiged circuitry for this modulated feedback-cotrolled voltage 3 elimiates pulse-shape distortios caused by fast feedback-voltage chages durig pulse propagatio through the Pockels cell, as i the previous desig. 8 This esures that pulse-shape distortios i the rege are due maily to gai saturatio. Fially, after the output-pulse eergy is stabilized by the feedback mechaism durig this prelase phase, the laser is Q-switched, at which time (t 3 ) all feedback loss is removed ad a Q- switched pulse evelope is allowed to build up. (Durig this time a adjustable low-level dc loss is left i the cavity to cotrol the fial output-pulse eergy; however, this loss is ot icluded i the model.) The measured output-pulse trai evelope from the rege is show i Fig (b). The voltage applied to the feedback-cotrolled Pockels cell durig the prelase stabilizatio is modulated every roud-trip so that the Pockels cell trasmissio is give by Vt T = cos 2 π V, 2 π where V( is the istataeous value of the modulated voltage differece betwee the electrodes ad V π is the quarter-wave voltage of the Pockels cell. The modulated voltage for a particular pass whe the feedback circuitry is active is modeled by [ ] (6) Vi+ = Vi + Vi e 1 τrt τfb, (7) where V i is the value of the modulated voltage at the begiig of the pass, V i is the icrease i voltage due to the feedback circuitry, ad V i+1 is the value of the modulated voltage after the pass. The chage i voltage V i is give by V i = pulse eergy J feedback gai ( V J ), (8) where the feedback gai is determied by the feedback circuitry. I Eq. (7), the fial voltage is allowed to decay every roud-trip (roud-trip time τ rt = 26 s) with the expoetially decreasig feedback decay time τ fb = 35 s. Whe the laser is Q-switched at time t 3, all feedback loss is removed from the cavity allowig the free buildup of the Q-switched pulse trai. The above model describes how to calculate the outputpulse shape from the rege give the iput-pulse shape. Ofte it is ecessary to calculate the iverse, that is, calculate the required iput-pulse shape to the rege that will produce a desired output-pulse shape. A good approximatio for this iput-pulse shape ca be gotte from the output-pulse shape with a simple procedure. A trasfer fuctio for the rege ca be calculated by usig the desired rege-output-pulse shape Iout [ t ] as iput to the calculatio to obtai a ew outputpulse shape e.g., Iew [ t ]. The trasfer fuctio T( for the rege is obtaied by dividig these two fuctios to get Tt = The required iput-pulse shape Ii t with this trasfer fuctio ad is give by Iew t. Iout (9) t [ ] ca ow be calculated Iout t Ii= t. (10) Tt LLE Review, Volume 69 21

5 This simple procedure is used to obtai the required regeiput-pulse shape that will produce the desired rege-outputpulse shape. More importatly, this procedure is useful i producig the desired OMEGA o-target pulse shape. Experimets The rege i OMEGA uses a Nd:YLF laser rod pumped to a sigle-pass, small-sigal gai of approximately 2.9. The laser uses a 50% reflectig output coupler, the cavity-roud-trip time is 26 s, ad the laser operates at 5 Hz. Typical output eergies of the pulse switched out at the peak of the Q- switched evelope are approximately mj. The measured output-pulse trai from the rege is show i Fig (b). The output has bee filtered to show oly the evelope of the pulse-trai output from the rege. It ca be see that the feedback is activated at time t 2 approximately 600 s after the pulse is ijected ito the cavity at time t 1 = 0. At t 3 = 2.9 µs, the laser is Q-switched ad a pulse trai builds up ad decays as the gai is depleted. Figure shows the calculated-output-pulse trai from the rege for the above case. Idividual pulses withi the trai are show. The calculatio is based o the model described above with typical values for the rege parameters. Note the good agreemet betwee the measured-output-pulse trai i Fig (b) ad the predictios show i Fig Figure shows rege iput/output-pulse shapes for a square pulse ijected ito the rege. The output-pulse shape is the pulse that is switched out at the peak of the Q-switched evelope. The iput square pulse (curve plotted with log dashed lies) ad measured-rege-output pulse (curve plotted with short dashed lies) are show i Fig , alog with the calculated-output-pulse shape (curve plotted with solid lie) obtaied with the above umerical method usig the measured-square-pulse shape as iput to the calculatio. The rege parameters used i the calculatio correspod to the measured rege parameters with slight adjustmets to obtai good agreemet with the data. By adjustig the rege parameters i this way, the model is calibrated to the data. Oce this calibratio procedure is performed, the parameters i the model are left uchaged ad other shaped pulses ca be calculated ad compared to measuremets. Figure shows the same iformatio as Fig , but for a shaped optical pulse ijected ito the rege. The rege parameters were idetical to those used for the calculatio i Fig This pulse shape, whe ijected ito OMEGA, will produce a square pulse shape at 351-m wavelegth at the output of OMEGA. I summary, we have modeled the temporal evolutio of a shaped optical pulse ijected ito our feedback-stabilized rege to a high degree of accuracy. We have solved the rate Calculated rege output t 2 t 3 Amplitude Measured iput Calculated output Measured output E Time (µs) E Time (s) Figure Calculated rege-output evelope correspodig to the case measured i Fig (b). Idividual pulses are show. Figure Square-pulse distortio from the rege showig the iput-pulse shape (log dashed lies), the measured-output-pulse shape (short dashed lies), ad the calculated-output-pulse shape (solid lie). 22 LLE Review, Volume 69

6 Amplitude Measured iput Calculated output Measured output ACKNOWLEDGMENT This work was supported by the U.S. Departmet of Eergy Office of Iertial Cofiemet Fusio uder Cooperative Agreemet No. DE-FC03-92SF19460, the Uiversity of Rochester, ad the New York State Eergy Research ad Developmet Authority. The support of DOE does ot costitute a edorsemet by DOE of the views expressed i this article. REFERENCES E Time (s) Figure Shaped pulse from the rege showig the iput-pulse shape (log dashed lies), the measured-output-pulse shape (short dashed lies), ad the calculated-output-pulse shape (solid lie). equatios icludig upper- ad lower-laser-level lifetimes explicitly. We provide a prescriptio for determiig the ijectio-pulse shape required to produce a give output-pulse shape from this rege. Fially, with this model of the rege, the etire OMEGA laser system ca be modeled, ad o-target pulse shapes ca be specified i advace by OMEGA users. 1. J. D. Lidl, Phys. Plasmas 2, 3933 (1995). 2. A. Okishev, M. D. Skeldo, S. A. Letzrig, W. R. Doaldso, A. Babushki, ad W. Seka, i Superitese Laser Fields, edited by A. A. Adreev ad V. M. Gordieko (SPIE, Belligham, WA, 1996), Vol. 2770, pp A. Babushki, W. Bittle, S. A. Letzrig, A. Okishev, M. D. Skeldo, ad W. Seka, Stable, Reproducible, ad Exterally Sychroizable Regeerative Amplifier for Shaped Optical Pulses for the OMEGA Laser System, to be preseted at Advaced Solid-State Lasers, Orlado, FL, Jauary 1997, paper ME9. 4. R. S. Craxto, IEEE J. Quatum Electro. QE-17, 1771 (1981). 5. J. D. Zuegel ad W. Seka, IEEE J. Quatum Electro. 31, 1742 (1995). 6. L. M. Fratz ad J. S. Nodvik, J. Appl. Phys. 34, 2346 (1963). 7. A. E. Siegma, Lasers (Uiversity Sciece Books, Mill Valley, CA, 1986). 8. A. Okishev, M. D. Skeldo, S. A. Letzrig, W. Seka, ad I. Will, i OSA Proceedigs o Advaced Solid-State Lasers, edited by B. H. T. Chai ad S. A. Paye (Optical Society of America, Washigto, DC, 1995), Vol. 24, pp LLE Review, Volume 69 23