Volume 79A, number 5,6 PHYSICS LETTERS 27 October 1980 HARD-MODE INSTABILITY IN A SIX-STATE MODEL OF A DYE LASER S.T. DEMBINSKI’ , A. KOSSAKOWSKI and K. STEFANSK1 Institute of Physics, Nicholas Copernicus University, Toruñ, Poland Received 23 July 1980 It is shown that without external saturable absorber but due to dynamic internal triplet losses single mode theory of a six-state model of a dye laser predicts the occurrence of undamped spiking. The onset of undamped periodic spiking in lasers “fully-adiabatic” approximation, i.e.,in the one-dimen- may be considered as an example of a hard-mode in- sional (number of photons) space. Some of the far reach- stability in systems far from thermal equilibrium. The ing similarities between such a description of a dye pulsation can occur in multi-mode lasers as a result of laser and theories of laser with saturable absorber [15— mode—mode coupling which leads to phase locking 17] have recently been discussed [18]. The similarities but can also be understood in single mode lasers. In have their origin in the absorbing role being played by the multimode case the basic physical mechanism re- the triplet—-triplet transition in dye molecule. Among sponsible for locking was found to be the nonlinear problems which stem from the above analogy and are saturation of the laser transition (self mode locking) stifi open the following one is of interest to us: are or the saturation of the absorbing transition in an ab- there, as in the case of a laser with saturable absorber sorber (passive Q-switched mode locking) [1—4].The [51 ,stable time-dependent solutions in single mode single mode mechanism was explained on the basis of theory of dye lasers? the dynamic saturation of a resonant absorber (j~assive Starting from the generalized von Neumann equa- Q-switching) [5] and was recently the subject of a sto- tion for the field-dye molecules density matrix, the chastic description [6,7]. Due to their broad emission semiclassical equations of motion (thermodynamic band dye lasers are particularly attractive to produce limit) for the dye laser were derived in ref. [12]. We ultra-short optical pulses and therefore most of the ef- will follow the formalism and notation adopted there- fort was devoted to obtain and explain undamped in. The energy diagram of a dye molecule and the de- spiking in Q-switched mode locked dye lasers (see [8] scription of rate constants of radiationless transitions and references therein, as well as [9] on pulses in lasers are given in the upper part of fig. 1. The semiclassical with distributed feedback). Relaxation oscillation equation of motion form a set of 12 ordinary first have also been studied [10]. On the other hand, how- order nonlinear differential equations for mean values ever, in order to describe extraordinary threshold be- of complex field amplitude ct, complex transition cli- haviour of dye lasers (possible first order-like phase pole moments r 1 and r2 between the pair (2,3) of sin- transition analogy) single mode quantum theories glet states and the pair (5,6) of triplet states and six based on the six-state model of a dye molecule have probabilities p~, i = 1 ... 6, of populations of respective been developed [11—13] (see however also [14]). energy levels. Using the normalisation condition ~p Semiclassical and quantum stationary solutions were = 1 we eliminate one variable namely p1 . We further found and their stability was examined within the assume that the inverse relaxation times ~ i and ~ 2 of polarisations r1 and r2 as well as inverse relaxation 1 Temporary address: II. Institut für Theoretische Physik, times a1 2’ a34,a16 and a15 of P2’ p4, P6 and p~ are Universität Stuttgart, and Max-Planck-Institut für Fest- greater than ic (inverse life-time of passive cavity), a23 kdrperforschung, Stuttgart, Fed. Rep. Germany. and a53 so that the adiabatic elimination of r1 , 383