We show that under certain conditions the time evolution of the conjugate reflectivity for degenerate four wave mixing in saturable absorbers can exhibit a resonant spike after a time equal to the molecular relaxation time of the system

Heffernan, Daniel M.

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.SST(-Transi ent Resonant Spiking in DegenerateFour Wave Mixing in Saturable AbsorbersDaniel N. HeffernanDepartment of Physics, Dublin Institute of TechnolocyKevin Street, Dublin 8, IrelandandSchool of Theoretical Physics, Dublin Institute for Advanced Studies10 Burlington Road, Dublin 4, Ireland.Abstract.We show that under certain conditions the time evolution of the conjugatereflectivity for degenerate four wave mixing in saturable absorbers can exhibita resonant spike after a time equal to the molecular relaxation time of thesystem.1.Silverberg and Bar-Joseph have developed a model for transientdegenerate four wave mixing (DFWM) in saturable absorbers [1]. Themodel is an extension of the steady state model of Abrams and Lind [2]and it gives the conjugate reflectivity as a function of time when thepump beams consists of pulses. of width much longer than the moleculardecay time of the system. The model has.recently being extended to thecase where the pump beams consist of pulses of arbitrary width, inparticular comparable to or much shorter than the molecular decaytime [3]. The model is in good agreement with experiment [1]. Itpredicts an enhanced transient reflectivity which has been observedexperimentally [1,4]. We now show that this model also predicts atransient resonant spiking in the time evolution of the conjUatereflectivity. The difficulties that would be faced in observing thistransient spiking experimentally are also pointed out.We consider degenerate four wave mixing in a three level saturableabsorber. The saturable absorber has a fast 2-3 transition and arelatively long lifetime T of level 3. This includes most of the organcdye saturable absorbers and many solid state systems [5]. If no is thetotal population, then during the pulse the gound state population, n,evolves according to [1]L (,_M.)—— (1)where I is the intensity and is the frequency of the radiation and athe absorptive cross section for the transition 1 -2.With the usual DFWM notation we consider four plane waves of theform [1,3]C.C1 (2)where E1 and E2 are two strong counter-propagating pump fields and E3and E4 are two weak fields also counter-propagating but in a different= (;y oL) (4)and the absorption coefficient a and coupling coefficient > are given byand —p(H-P)]Here Qç,P-where I s thesaturation intensity and the angular brackets < > denote averaging overthe phase term Q 4 In obtaining the above results it isassumed that (a) the transverse relaxation time of the medium is muchless than the pulse duration, (b) it is an optically thin saturableabsorber and Cc) the transition is on resonance [1,3].The above model has been found to be in good agreement withexperiment [1]. It predicts an enhanced transient conjugate reflectivitywhich has been observed and found to be in good agreement with experiment [1,4)çC\(2)direction from E1 and E2. These fields interact through thesusceptibility of the medium. In the case where the pumpof rectangular pulses of width -un, Silverberg and Bar-Josephthat, in the non-depleted pump approximation and for r >>t,reflectivity is given by [1]non—linearbeams consisthave shownthe conjugate(3)wheret2.I x—W CtD-: w L + a, L: p(i÷P)]j&C (-i-))Cpft+P)_____(I +.5))‘•= (6)(3)We will now show that this model also predicts the possibility oftransient resonant spiking in the time evolutionof the conjugatereflectivity after a time equa) to the molecular relaxation timeofthe absorber.Suppose that the intensity of the pump beams is chosen so thatP = 1. From equations (5) and (6) we see that after a time t== a i.e.’W = 0 instantaneously at t = r. Hence R = at t =(equation (n)). This transient resonant spiking is seen clearly infigure (1) where R is shown as a function of time for P = 1, a0L = 05and Tp/T = 1000. Note that the resonance is trudytransient since itis only after a time -r that j can equal a and from equations (5) and(6) this can only occur when P = 1.Great care would have to be taken in order to observethis spikingexperimentally. Apart from having to ensure that P =1, thermal andoff—resonant effects would tend to broaden and flattenthe resonant spike.Furthermore, any experimental apparatus measures theaverage value ofthe conjugate field over some time interval and as a result the spikecouldbe lost in the transient enhancement on which it sits.Any experimentdesigned to see transient resonant spiking in the conjugate reflectivitywould have to ensure that the measuring apparatus averages over timeintervals much shorter than the width of the spike.In conclusion we predict the possibility of transient resonant spikingin the time evolution of the conjugate reflectivity for DFWM in saturableabsorbers. This spike occurs after a time equal to themolecularrelaxation time of the absorber.Acknowledgements.I wish to acknowledge many fruitful discussions withProfessor D.J. Bradley F.RS.(4)Figure Captjoi15Figure 1 Time evolutioti of the conjugate reflectivity when P 1,= 0’5 and T/t boo.References111 Y. Silverberg and I. Bar-Joseph, iEEE 3. Quantni ElectronQE-17 (1981) 1967.[2] R.L. Abrams and R.C. Lind, Opt. Lett. 2 (1978) 94.[31 D.M. Heffernan Phys. Lett. 103A (1984) 286.[41 S. Boothroyd and C. Grey Morgan, 3. Phy. 0: Appl. Phys. 16 (1983) L165.[5] N. Hercher App. Opt. 6 (1967) 974.•03rn•oIi7Pt‘04•00 2 53 4

Transient Resonant Spiking in Degenerate

Four Wave Mixing in Saturable Absorbers

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Transient Resonant Spiking in Degenerate Four Wave Mixing in Saturable Absorbers

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