Theoretical analysis of instabilities in optically pumped molecular lasers

Volume 60, number 4

OPTICS COMMUNICATIONS

15 November 1986

THEORETICAL ANALYSIS OF INSTABILITIES IN OPTICALLY P U M P E D MOLECULAR LASERS S.C. MEHENDALE ~and R.G. HARRISON Department of Physics, Heriot- Watt University, Edinburgh EH14 4AS, UK Received 16 July 1986

Results of a theoretical investigation of instabilities in a resonant, coherently pumped, homogeneously broadened three-level system are presented. It is found that the behaviour of this system is significantly different and apparently much less complex than that of an incoherently pumped two-level system under similar conditions. In the absence of pump-field induced Rabi splitting, the emission is stable even when relative excitation is very high while presence of such splitting is found to lead only to regular pulsations.

Optically pumped molecular lasers (OPML's) have recently emerged [ 1-3] as particularly attractive systems for experimental investigation of laser instabilities. Of special interest, in this context, is the possibility that the somewhat prohibitive conditions necessary for occurence of chaos in the Lorenz-Haken model [4 ] may be met in OPML's operating in the far infrared region [ 5 ]. However, while the LorenzHaken model describes an incoherently pumped twolevel system, an OPML involves a coherently pumped three-level system and, as shown recently by Dupertuis et al. [ 6 ], the two are equivalent only under certain restrictive conditions which may not be satisfied in a practical OPML system. While there exists a considerable amount of theoretical work concerning instabilities in the conventional two-level laser systems under various conditions of broadenings, detunings, etc., to our knowledge a theoretical analysis of instabilities in coherently pumped laser systems has so far not been considered. With this in view, we have studied the time-dependent behaviour of OPML emission under various conditions and the results are presented in this communication. To highlight the basic features of OPML behaviour, we consider a simple resonant, homogeneously broadened system. A rather intriguing result of our analyS.C. Mehendale is now with the Laser Division, Bhabha Atomic Research Centre, Bombay 400 085, India.

0 030-401/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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sis is that the behaviour of this system appears to be very much less complicated compared to that of its two-level counterpart. In the absence of pump-field induced Rabi splitting the emission has a steady-state solution even when relative excitation is quite high, while presence of such splitting has been found to lead to pulsating instabilities only. Consider a resonantly pumped homogeneously broadened three-level system shown schematically in the inset of fig. 1. Let F and 7 be respectively the population and polarisation rates, assumed to be the 257

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same for the three levels. We define dimensionless quantities a = c~'/y and fl =fl'/y where a ' and fl' are the Rabi frequencies o f the p u m p and the generated fields respectively. Let N O be the initial population density in level 1; and D21, D23 the inversion densities, normalised to N °, corresponding to the transitions 2--, 1 and 2 ~ 3 respectively. We shall assume this simplified three-level system to be the active medium in a uni-directional ring cavity with co-propagating p u m p and generated waves. In the absence o f appreciable p u m p depletion the field and matter equations [6 ] governing the dynamics o f this system can be shown to be

j~=o'[ -fl-k iGp23 ] , P21 = - P 2 1 +iaD21--i~P31 , b23 = - - P 2 3 "~- iflD23 - -

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/~31 =--P3J --ifl*p2t q-iap*3 , O21 = -b(D21 - D ° l ) - 4 Im(ot*P21) - 2 Im(fl*P23) , D23 = - b (D23 -- D°3 ) -- 4 Im (r'P23 ) - 2 Im(ot*p21). Here p;j is the density matrix element, normalised to N °, corresponding to transition between levels i and j; tr = x/y, where x is the cavity decay constant; b=F/y; G is the ratio o f coupling constant COs1/12312N~1/2h%y2 to a; and D°l, D°3 are the initial equilibrium values o f D2~ and D23. The derivatives on lhs are with respect to the normalised time z = yt. We assume a cavity mode to be resonant with the centre o f the transition between levels 2 and 3. The unsaturated gain per unit length, g ( 0 ) for the generated wave is determined by the parameters G, tr and or. F o r given G and tr it is m a x i m u m for a 2 = ½ [7] and given by gmax=2yGtr/3c. The variation of the ratio g(0)gmax with ot2 is shown in fig. 1. The decrease in gain for a 2 > ½ is caused by a splitting of the gain feature due to the ac Stark effect [ 7 ]. The parameter G can be shown to correspond to relative excitation, being proportional to the ratio o f small signal gain to the cavity losses. Thus for o~2=½, threshold corresponds to G = 3 for all values o f a. For 258

15 November 1986 10.0

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Fig. 2. Dependence of the steady state emission intensity Ifll 2on the relative excitation parameter G. The three curves marked a, b and c correspond to a2= 0.5, 0.4 and 0.3 respectively. other values o f a 2, threshold is reached for larger values of G, in accordance with the decreased gain. Since fl and pCs are complex, the six equations given above correspond to ten real coupled differential equations. In view o f the difficulty o f applying techniques of linear stability analysis to these equations, we have chosen to study the temporal dependence of the generated emission under various conditions by a direct numerical integration. A standard subroutine based on the Runge-Kutta-Merson method was used and integration was performed over a sufficiently long interval to allow initial transients to die out. In the following we restrict to the case for which b = 1 and initially only the level l is populated, i.e. D°3 = 0 and D°l = -- I. Let us first consider the situation when a 2 ~ ½ SO that there is no ac-Stark splitting and the gain profile is smooth with a single peak at the centre o f the transition from level 2 to level 3 [ 7 ]. As mentioned earlier, in this case for a significant range o f values of G, ¢t and tr explored by us, it was found that Ifll 2, which is proportional to the intensity of generated emission, always reached a steady-state value. This is illustrated in fig. 2, where curves a, b and c give steady-state values of Ifl 12 as a function of G for tr = 5 and a2= 0.5, 0.4 and 0.3 respectively. We note that since G corresponds to relative excitation, identical curves were obtained for other values o f the cavity decay parameter or. Although not apparent from the figure, the three curves have slightly different threshold values o f G at which lasing starts ( ~ 3.00, 3.03 and 3.18 respectively) as is to be expected from the

Volume 60, number 4

OPTICS C O M M U N I C A T I O N S

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15 November 1986

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G Fig. 3. Results o f numerical integration for parameter values a2 = 2 and a = 10. For G values between Gm~. and G . . . . an oscillatory solution was obtained and the dashed curve shows variation o f the normalised period 7T within this region. For larger G the smooth curve gives variation o f the steady state intensity 1ill 2 with G.

different small-signal gains. For curve a, G = 300 corresponds to an unsaturated gain 100 times above threshold, but we have found no evidence o f any instability, even with a varied from 1 to 20. Computations with even larger gains have also given essentially the same result. We note that a large O P M L gain at a small value of ot can be realised in an O P M L for which oscillator strength for the emission transition is much larger than that for the p u m p transition. Next we consider the case when ot is sufficiently large to cause Rabi splitting of the gain profile. In this case, a pulsating instability occurred in the O P M L emission under certain conditions. Typically, for suitable values o f a 2 and a, u n d a m p e d oscillations were observed for G greater than a certain value Gmin which was lower than Gthr, the value of G necessary for making gain at centre equal to the cavity losses. The oscillations persisted for some range o f G values above Gtnr up to a value G . . . . after which a steady state solution resulted. This is illustrated in fig. 3 which shows results o f calculations with a 2 = 4, a = 10 and G varying from 5 to 15. In this c a s e Gthr was ~ 7.08. The dotted curve shows variation of the normalised oscillation period 7 Twith G in the instability region while the smooth curve shows the steady state values o f I#12 . As an illustration of the nature of instability, a projection o f the phase-space trajectory - a limit cycle - in the O23-fll plane is shown in fig. 4 for G = 8. It was found that

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Fig. 4. A projection of the phase space trajectory in the D23-fll plane. Parameter values G = 8, a 2 = 40 and a = 10.

for a given ol2 there was a m i n i m u m value o f a below which there was no instability (e.g. with or2= 4, only stable solutions resulted for tr = 5 ) and the instability r e g i o n , Gmax - Gmin, increased with a. Similary, for a given a, the instability region increased with increasing values of a 2. In our calculations so far, no instability was observed for a ~<2, implying necessity of a bad cavity. The stability of O P M L emission in the absence o f Rabi splitting even under very large gain conditions is somewhat unexpected. For comparison, we note that for a resonant, homogeneously broadened twolevel system the m i n i m u m instability threshold corresponds to an excitation ~ 9 times above threshold for a cavity with a ~ 5 [ 8 ]. This difference in the behaviour o f the two systems could be due to different effects produced by a strong oscillating mode in the two cases. In a two-level system, the associated dynamic Stark effect produces a distortion in the gain and dispersion o f a nearly-resonant probe field leading to creation of side-modes with net gain [ 9 ]. In a three-level system, on the other hand, such Rabi splitting would influence not only the probe field but also the p u m p field leading to an effective detuning from resonance and hence reduced excitation. It is conceivable that the net result could be a lower probegain preventing side-mode build up. Such a mechanism could also be responsible for another significant difference in the dynamics o f the two systems. 259

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While in a two-level system the emission intensity increases linearly with relative excitation after threshold, the dependence is much slower in a threelevel system as can be seen from the curves in fig. 2. An analysis of the probe-field gain in the presence of a strong mode is necessary for a clarification of the possible role of this mechanism in preventing instabilities. The regular pulsations in the presence of pump field induced Rabi splitting have their origin in mode splitting caused by anomalous dispersion associated with the split gain profile [ 10]. The behaviour in this case is somewhat similar to that of a system of two groups of two-level atoms with slightly different resonant frequencies [l l]. However while regions of chaotic dynamics have been identified in the latter case, our results so far have shown only oscillatory instabilities. A more extensive and finer scan of the parameter space is currently under progress to verify if a transition to chaos can occur. Finally it is necessary to comment on the apparent contradiction between our theoretical results and the experimental results in OPML's [ 1-3 ] which have shown existence of a rich variety of instabilities. In this context we note that while we have considered an idealised system to highlight the basic features of a three-level system, many other factors neglected here such as off-resonant pumping, spatially nonuniform gain, unequal F and ~, M-degeneracy of molecular levels, etc. are likely to play an important role under typical experimental conditions. To conclude, we have presented results of a theo-

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15 November 1986

retical analysis of instabilities in a simple resonantly pumped, homogeneously broadened three-level system which show that the behaviour in this case is significantly different from that of incoherently pumped two-level systems. Instabilities were observed only when the pump-field was strong enough to induce Rabi splitting of the emission gain profile. The research work was supported by the Science and Engineering Research Council (U.K.) and in part by the Procurement Executive, U.K. Ministry of Defence.

References [ 1 ] R.G. Harrison and D.J. Biswas, Phys. Rev. Lett. 55 (1985) 63. [2] N.M. Lawandy, J. Opt. Soc. Am. B2 (1985) 108. [3] E.H.M. Hogenboom, W. Klische, C.O. Weiss and R. Godone, Phys. Rev. Lett. 55 (1985) 2571. [4] H. Haken, Phys. Lett. 53A 0975) 77. [5] C.O. Weiss and W. Klische, Optics Comm. 51 (1984) 47. [ 6 ] M.A. Dupertuis, R.R.E. Salomaa and M.R. Siegrist, Optics Comm. 57 (1986) 410. [7] R.L. Panock and R.J. Temkin, IEEE J. Quantum Electron. QE-13 (1977) 425. [8] M.L. Minden and L.W. Casperson, IEEE J. Quantum Elec-. tron. QE-18 (1982) 1952. [9] S.T. Hendow and M. Sargent III, J. Opt. Soc. Am. B2 (1985) 84. [ 10 ] S.C. Mehendale and R.G. Harrison, Phys. Rev. A., to appear. [ l 1 ] N.B. Abraham, D. Dangoisse, P. Glorieux and P. Mandel, J. Opt. Soc. Am. B2 (1985) 23.