We report on the modeling of transient stimulated rotational Raman scattering in H2 gas. We predict a multifrequency output, spanning a bandwidth greater than the pump frequency, that may be generated without any significant delay with respect to the pump pulses. The roles of dispersion and transiency are quantified

A. P. Lutsenko

Eimerl

Flusberg

G. H. C. New

G. S. McDonald

Hickman

L. L. Losev

Losev

M. Shaw

Reiser

Shen

English

We report on the modeling of transient stimulated rotational Raman scattering in H2 gas. We predict a multifrequency output, spanning a bandwidth greater than the pump frequency, that may be generated without any significant delay with respect to the pump pulses. The roles of dispersion and transiency are quantified. Multiple high Stokes and anti-Stokes orders are fre-quently observed in stimulated Raman scattering (SRS). Usually, higher orders are generated under phase-matched conditions and thus are at widely differing angles. This results in a multifrequency beam that is not focusable to a single spot. The much more interesting case is that in which higher orders are generated with a phase mismatch but are collinear. Such a beam may have application to laser fusion.&quot;2 Recently multiple orders of compa-rable amplitude spanning a broad bandwidth were generated with both H2 and N2 gas.3&apos;4 Previously i

G. S. Mcdonald

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Ultrabroad-bandwidth multifrequency Raman generation

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Mcdonald, GS

New, GHC

Losev, LL

Lutsenko, AP

Shaw, M

University of Salford Institutional Repository

1400 OPTICS LETTERS / Vol. 19, No. 18 / September 15, 1994Ultrabroad-bandwidth multifrequency Raman generationG. S. McDonald and G. H. C. NewLaser Optics and Spectroscopy Group, The Blackett Laboratory, Imperial College of Science, Technology and Medicine,Prince Consort Road, London SW7 2BZ, UKL. L. Losev and A. P. LutsenkoP. N. Lebedev Physical Institute, Leninsky Prospekt 53, 117924 Moscow, RussiaM. ShawRutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 OQX, UKReceived March 1, 1994We report on the modeling of transient stimulated rotational Raman scattering in H2 gas. We predict amultifrequency output, spanning a bandwidth greater than the pump frequency, that may be generated withoutany significant delay with respect to the pump pulses. The roles of dispersion and transiency are quantified.Multiple high Stokes and anti-Stokes orders are fre-quently observed in stimulated Raman scattering(SRS). Usually, higher orders are generated underphase-matched conditions and thus are at widelydiffering angles. This results in a multifrequencybeam that is not focusable to a single spot. Themuch more interesting case is that in which higherorders are generated with a phase mismatch butare collinear. Such a beam may have application tolaser fusion."2 Recently multiple orders of compa-rable amplitude spanning a broad bandwidth weregenerated with both H2 and N2 gas.3'4 Previously itwas thought that long delays arising from the T2 timeof the medium would severely limit the usefulness ofthe generated spectra.12'4 SRS with identical pumpand first-Stokes pulse envelopes (symmetric pump-ing) has not, to our knowledge, been investigated inthe transient regime, in which turn-on effects are im-portant. It is precisely in this regime that we predictthe generation of more than 40 orders of compara-ble magnitude during the local rise time of the pumppulse. The central motivation for this research is theoptimization of collisional absorption in high-energylaser-target coupling experiments.' The benefitsof increased bandwidth are multifold and may im-prove the efficiency of the laser fusion process andsignificantly increase the number of alternative ex-perimental configurations.4 Since the underlyingmechanisms involved in this work are fundamental,there may be a much wider range of applications.To model SRS, we expand the total electric fieldE in terms of constituent plane waves (the pumpand Raman sidebands):E(z, t) = 2 Fn(z, t)exp[i(o,,t-k kz)] + c.c. (1)Each wave has a complex envelope Fn varying in bothlongitudinal space z and time t and has a carrier fre-quency given by con = co + nfOR (n = 0 ±1, ±2,...),where coo and cOR are the pump frequency and Stokesshift, respectively. For collinear waves the propa-gation constants kn are scalar and given by ncn/c,where Iin (Ref. 5) is the refractive index at On and cis the speed of light. To study the essential physicsof the multiline generation process, we recast thestandard model6'7 into a dimensionless form by defin-ing the following: An = Fn/Fo(0,0), Z = gIo(O)z,T = t/tp, and yn = An/[gho(O)], where gIo(O) is theRaman gain coefficient, In(O) = IFn(0, 0)12, n=(Akn - AknI1) - (Ako - Akl), AIkn = (An - 1)On/c,and tp is the input pulse width. In any one con-figuration the set of normalized mistunings Yn areparameterized by yi. With respect to the resonantdriving of a single transition the model equations areaAZ = con [p* - PAnI, exp(iynZ)] I(T ) P= -P + >AjAj-A exp(-iyjZ),O r(2)(3)where T2 is the dephasing time of the polarization P.Previously attention had centered on the injectionof a small Stokes seed at an angle to the pump.However, recent analysis and simulation of thesteady-state equations8 investigated the dependencyof bandwidth on the initial Stokes wave and pre-dicted that a cw Stokes seed of amplitude equal tothe pump would maximize the generated bandwidth.Here we consider symmetric pumping at Z = 0,Ao(r) = A-1(X) = exp(-r 2 ) and set all other compo-nents initially to zero. Collinear symmetric pump-ing permits maximum spatiotemporal overlap in theterm driving the polarization grating; regions withoutsuch overlap could also lead to noise amplification.In this Letter we model rotational SRS in H2 gas,coR/(2 1TC) = 587 cm-', pumped by a second harmonic0146-9592/94/181400-03$6.00/0 © 1994 Optical Society of AmericaSeptember 15, 1994 / Vol. 19, No. 18 / OPTICS LETTERS 1401persion also suppresses the generation of Stokes or-___________11______ l _ ___ ders, resulting in a relatively monotonic decrease inv - .I I I I I I I output bandwidth.-30 -20 -10 0 10 20 30 Instantaneous bandwidths (at particular values ofComponent umber r) may exceed those of time-integrated spectra andIo - __1 Ply _1___I1_ I - I ___ ]LI can be greater that 5OcoR. In Fig. 3(b) the pointsF ig. l. Logaritnmic plot of lignt pulses in the generatedRaman spectrum. The component number n gives thefrequency of each line through (In = co + nfWR. The timescale r is in units of the input pulse width tp. Otherparameters are tpIT2 = 1, Z = 200, and yj ' 0.0018.of Nd:YAG, oj0/(27rc) = 18900 cm-'. We solveEqs. (2) and (3) numerically, using a Runge-Kuttaalgorithm in Z and a predictor-corrector method inr. Figure 1 shows the temporal distributions of in-tensity, IA.(-r)12, of the generated Raman lines forZ = 200. Although the logarithmic scale exagger-ates pulse widths, the high rate of energy transferinto the higher orders may still be inferred. Closerexamination reveals that output spectra are modu-lated with a rapidly oscillating envelope.28 In thesteady-state limit, and neglecting dispersion, we seethat this envelope slowly converges to a fixed pat-tern after Z = 100, whereas for finite dispersionthe envelope continues to evolve, in both space andtime, for Z values as high as several hundred. Fortransient SRS, time integration of the output pulsesyields a clear quantitative picture of the distributionof energy density and also smooths out any tempo-ral modulations. This is shown in Fig. 2, in whichlog 0Io J JAI(r),12)2dr is plotted. To demonstrate themagnitude of this bandwidth, one may calculate thatthe visible section of the spectrum spans approxi-mately 18wR.Output bandwidth depends on the amplitude ofthe grating and also on the phase matching of theinteracting waves. Figure 3 illustrates the role ofdispersion (each data point corresponds to a singlesimulation). Correlating a number of simulationsresults in apparently noisy data since small changesat the start of a calculation can lead to large quanti-tative changes for a fixed Z. For each t,/T 2 consid-ered, a large Z is specified to minimize Z-dependentvariation of the bandwidth. In Fig. 3(a) 10% band-widths of the time-integrated spectra are shown.This definition of bandwidth reflects the spectralwidth within which the orders have comparable en-ergy. Choosing gIo(0) = 1 cm-', one can interpretZ as a cell length in centimeters and y, to be pro-portional to the gas pressure (y, = 0.008 for 3-atmH2). For t,/T 2 2 1 and y' 0, anti-Stokes ordersdominate the output spectrum. Here Stokes ordersexperience gain suppression,7 9 while the explicit fre-quency dependence coI,/coo in Eq. (2) enhances thegeneration of higher frequencies. For increased dis-persion the bandwidth reaches a maximum as a re-sult of a rapid increase in the number of the Stokesorders. This peak coincides with an optimal phasemismatch for the parametric interactions, which re-sults in a large number of Stokes and anti-Stokesorders. For large mistunings the parametric inter-actions are suppressed and generate only weak seedsfor the Stokes cascade process. For t,/T 2 << 1, dis-1.20.8Time-integratedoutput0.4Fig. 2. Time-integratedshown in Fig. 1.5OWR4OWR -GeneratedBandwidth3 OWR20WR10WR0tptp/2Local timeof peakbandwidth0-tp-20 -10 0 10 20Component numberintensity of the Raman lines0 2 4 6 8ayl x 103 (Dispersion)(a)0 2 4 6 8al x l03 (Dispersion)(b)Fig. 3. Characteristics of the Raman spectra as a func-tion of dispersion for (a) 10% bandwidths and (b) points inlocal time for maximum bandwidths [values of t,/T 2 areas indicated in (a)]. Z = 200 for t,/T 2 Ž 1 and Z = 500for t,/T 2 = 0.125.IlO L-30-1402 OPTICS LETTERS / Vol. 19, No. 18 / September 15, 199450WR4OWR FGeneratedBandwidth3 OWR L ....20WR1OWR F0 _0.01tptp/2Local timeof peakbandwidth00.1 1tp/T2(a)0.01 0.1 1 1tp/T2(b)Fig. 4. Variation of output characteristifor Z = 500 and y, - 0.0018: (a) 10% bemagnitude of the pump frequency in unitsshift and (b) local times for maximum (ibandwidth.in time for maximim instantaneous baare shown. Bandwidth tends to saturawhereas Tmax is local and remains morZ. We can interpret the overall trendsof this sensitivity by correlating these rerally, with decreasing t,/T 2, Tmax is seeHowever, the delays shown are still 1input pulse width tp. For t,/T 2 = 0.1persion suppresses bandwidth, and consdoes not increase greatly. When T2 < tbandwidth may be switched on even durise time of the input pulses.Figure 4 shows the effect of transie:width and Tmax for a value of yV that isnite. In the direction of the extreme ti(tp << T2) there is a reduction in banddue to the simultaneous suppression canti-Stokes orders. Toward the steady(tp >> T2) bandwidth can be maximal ,imal. Even though tp/T2 varies overmagnitude, symmetric pumping leads to a signifi-cant bandwidth that coincides (temporally) with thepump frequency to within a pulse width. It shouldbe stressed that bandwidths of the same order are.......... . . ... also predicted for lower Z values. For example, theI simulations of Fig. 4 predict a bandwidth of approxi-mp mately the pump frequency for t,/T 2 2 1 and Z 2 60.iency In summary, modeling of transient SRS in H2 gaspredicts output bandwidths greater than the pumpfrequency when symmetric pumping is implemented.Previously, long transients were predicted beforebandwidths of just several components switch on.We have presented results that suggest that ultra-10 100 broad bandwidths may be quickly generated. Theexperimental realization of just a fraction of the pre-dicted bandwidth would be a significant result. Theconsideration of additional orders (which are dueto a vibrational transition) are not expected to al-ter our overall conclusions significantly since onlythe rotational transition is resonantly driven. Fur-thermore, vibrational orders are more sensitive todispersion-a sensitivity that is heightened by tran-sient effects. Any vibrational orders that do growfrom noise are also expected to form cones that leavethe space-time interaction region. Research is inprogress to generalize these results through the sys-tematic inclusion of asymmetries and spatiotemporalinhomogeneities5"10 in the pumping scheme. Stud-ies with more complex models to determine limitingfactors that could suppress the generation of suchultrabroad bandwidths need to be undertaken..0 100This research was supported in part by UK Scienceand Engineering Research Council grant GR/J04746.L. Losev and A. Lutsenko acknowledge financial sup-cs with tp/T2 port under a Royal Society Joint Project on RamanLndwidths and Lasers.3 of the Stokesnstantaneous) References1. D. Eimerl, W. L. Kruer, and E. M. Campbell, Comm.ndwidth rmax Plasma Phys. Controlled Fusion 15,85 (1992).te at large Z, 2. D. Eimerl, R. S. Hargrove, and J. A. Paisner, Phys.e sensitive to Rev. Lett. 46, 651 (1981).and the level 3. M. Hermann, M. Norton, L. Hackel, and D. 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Ultrabroad-bandwidth multifrequency Raman generation

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