VOLUME 62, NUMBER 11 PHYSICAL REVIEW LETTERS Spatial and Temporal Instabilities in a CO2 Laser 13 MARcH 1989 J. R. Tredicce, E. J. Quel, t'~ A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato Department of Physics and Atmospheric Sciences, Drexel University, Philadelphia, Pennsylvania 19104 (Received 11 August 1988) We show experimentally the appearance of time-dependent and stationary complex spatial structures in a C02 laser. A comparison of the data with a theoretical model supports the notion that the observed phenomena result from the nonlinear interaction of transverse cavity modes with the active medium. The simultaneous appearance of spatial and temporal instabilities hints to the interesting prospect that one may be able to study turbulence in a laser system. PACS numbers: 42. 60.3f, 42. 50.Tj Deterministic time-dependent intensity fluctuations, both of periodic and chaotic type, have been observed in a variety of free-running and driven laser systems. ' Many other nonlinear systems, such as fluids and chemi- cal oscillators, commonly display a richer phenomenolo- gy because of the appearance of spatial in addition to temporal complexity. "' When dealing with a laser, one is faced with unique constraints imposed not only by the geometry and the finite transverse dimensions of the ac- tive medium, but also, and especially, by the configura- tion of the resonator. The laser cavity, which acts as an optical filter, imposes a significant reduction in the num- ber of dynamical degrees of freedom. In general, this favors the development of ordered spatial structures, the so-called transverse cavity modes. Passive optical sys- tems driven by a Gaussian field have shown recently the ability to generate complex space-time patterns. Free-running lasers, of course, differ from passive sys- tems because the resonant medium is active and because of the absence of an injected field. In Ref. 9 we developed a laser model that accounts for the interaction of the transverse modes of the resonator and the active medium. Our calculations show that, of the many geometrical and physical parameters of the system, the radius of curvature of the output mirrors plays an especially critical role in affecting the strength and the nature of the modal coupling. Over a range of experimentally accessible parameter values our analysis predicts the existence of instabilities and cooperative fre- quency locking; the latter is a multimode regime of oper- ation where the field modes oscillate in synchrony and the output intensity is time independent. In this Letter we present experimental evidence for self-pulsing, cooperative frequency locking and the devel- opment of complex stationary spatial structures at the output of a CO2 laser when the cavity is swept slowly and continuously from a quasiconfocal to a nearly planar configuration. Our experiments are carried out with a polarized CO2 laser of the Fabry-Perot type with fat-end reflectors. The laser is designed to operate with two in- tracavity ZnSe lenses whose separation is controlled by a stepper motor and whose role is to produce a variable eff'ective radius of curvature of the output couplers. The tube is designed with special care in order to maintain the cylindrical symmetry of the output beam over most of the interesting parameter range, even in the presence of the Brewster windows. Its diameter is suKciently large to allow the operation of the second-order (p =2) Gauss-Laguerre mode without excessive diffraction losses. The output beam is detected by a HgCdTe detector with a bandwidth of 100 MHz after reflection from a ro- tating polygonal mirror whose function is to sweep the beam across a small opening placed in front of the detec- tor. First the detector is adjusted in such a way as to maximize the diameter of the beam cross section; the detected signal is then analyzed with a digital oscillo- scope and processed by a spectrum analyzer. The separation between the intracavity lenses controls the eff'ective configuration of the cavity; this can be varied continuously in the experiments from nearly pla- nar to nearly hemispherical. Varying the distance be- tween the lenses also causes a change in the frequency spacing between the transverse cavity modes; this can be made as large as about 4 of the longitudinal free spec- tral range before transverse modes characterized by a lower longitudinal index also fall under the gain curve. The longitudinal free spectral range, instead, remains constant because of the fixed length of the cavity. For values of the effective radius of curvature of the mirrors that produce the largest separation between con- secutive transverse modes, a combination of the natural diffraction losses of the cavity and the transverse profile of the atomic inversion can easily lead to single-Gaus- sian-mode operation. The transverse intensity profile of the beam in this case takes the form shown in Fig. 1(a). Upon increasing the effective radius of curvature of the mirrors, a small but detectable modification of the beam profile begins to develop [Fig. 1(b)]. The output field can no longer be described in terms of a single cavity mode, and even if the profile is still time independent, the contribution of the first higher-order mode (p =1) to the 1274 1989 The American Physical Society