N O NLINEAR DYNAMICS OF BROAD-BAND LASERS

A number of advances were made in improving the understanding of multimode dye laser dynamics. The source of mode-intensity fluctuations in a three-mirror, multimode dye laser has been i dentified in the typical operating regime well above threshold, lt was found experimentally that deterministic four-wave mixing interactions are the underlying cause. We have also made new theoretical studies concerning the regime just above threshold and predicted that the mode fluctuations will be dominated by quantum noise in this regime. We have constructed a short-cavity dye laser in order to enhance this quantum effect, and have made measurements of the turn-on time statistics of individual modes . Effects of coherent, four-wave mixing interactions on these statistics have been identified.

ABSTRACT A number of advances were made in improving the understanding of multimode dye laser dynamics.
The source of mode-intensity fluctuations in a three-mirror, multimode dye laser has been identified in the typical operating regime well above threshold, lt was found experimentally that deterministic four-wave mixing interactions are the underlying cause. We have also made new theoretical studies concerning the regime just above threshold and predicted that the mode fluctuations will be dominated by quantum noise in this regime. We have constructed a short-cavity dye laser

NONLINEAR DYNAMICS OF BROAD-BAND LASERS
Following the invention of the laser, much of the theoretical foundation of quantum-noise effects was laid down in the 1960's, by Haken [1], Lamb [2], and Lax [3]. The two-mode ring dye laser is well described by an equation of motion for the slowly varying electric-field amplitudes E. for each mode, ie where N=2 (two modes), 7. is the cavity-decay rate for the n-th mode, g. is the unsaturated gain coefficient for the n-th mode, the sum includes self gain saturatio'l and cross saturation (gain competition) by the modes, and F.(t) is a delta-correlated random noise (Langevin term) representing spontaneous emission into the n-th mode. Any noise present in the pumping rate will appear as a modulation ,3f the gain coefficient. The switching between modes can be pictured as a random walk between two stable minima in a potential function, separated by a barrier.
Multimode cw dye lasers were studied by our group to determine if quantum-induced switching plays a role in these more complex systems. [4,7] With a typical three-mirror, folded cavity with length 50cm we found, in agreement with previous studies, [8,9,10] that the modec nave no quasistable behavior--rather they continuously exchange energy in a seemingly  (1) to the case of many modes, as done in [11,12]. The reason, we pointed out, [7] is the absence in Eq. (1) Fig.2. Furthermore, we found that the trend of decreasing correlation time is properly described by this theory. Evidently the four-wave mixing leads to a continual exchange of energy between the modes, the time scale of which decreases when the intensities are higher as expected for a nonlinear optical process.
As a further confirmation that the dynamics are deterministic, we employed the data-analysis algorithm of Grassberger and Procaccia [13] to find the "correlation dimension" of the strange attractor underlying the dynamics. We found, as did the group of Atmanspacher, [14] a low dimension, indicating that the typical cw, multimode dye laser displays deterministic evolution, possibly chaos.
However, this leaves a gap in our understanding of Class A multimode lasers--that is, where are the effects of quantum noise in these systems?
To answer this question we solved Eq.(2) numerically with quantum noise, [15] and the results are shown in Fig.2 Mode turn-on statistics have; been measured by our group for a short-cavity dye laser (length 2.5cm) following sudden turn on of the pump power. [16] This provides a way to observe the effects of quantum noise in the presence of pump fluctuations since the turn-on dynamics occur on a time scale short compared to the pump variations. [17] To attempt to unravel the effects of coherent mode coupling from quantum noise, we studied both a laser with "small" mode spacing (30cm-1) and a laser with larger spacing (300cm -1) determined by intracavity etalons.
For the large mode spacing we obtained the data shown as points in  qualitative behavior of the modes were observed (overshoot, very slow turn on), as shown in Fig.4, which shows 256 realizations of turn-on behavior, As will be discussed below in the theory section, we believe that these effects are caused by coherent mode coupling.

C. Theory of Class B lasers
An important element in the success of this program is the development of a predictive theory of multimode, Class B lasers, in which the cavity lifetime is comparable to or less than the population lifetime.
As mentioned above, in this case the population cannot be adiabatically where 7_ is the population decay rate and c_is the linewidth enhancement factor arising from amplitude-phase coupling.II8] These equations are similar to Eq.(2) except that the population inversion is now a dynamical variable. This allows for relaxation oscillations, in addition to coherent mode coupling.
Numerical solutions of Eq.(4) show behavior qualitatively similar to that found in our experiments. Figure 5 shows typical solutions for parameters similar to those in the short-cavity dye laser whose dynamics i ,J are shown in Fig.4, The slow turn-on behavior seen for some of the shots is seen in both the theory and the experiment in the case of the smaller mode spacing. We conclude that this behavior is due to the coherent fourwave mixing among modes, which is enhanced in this case,

D. Conical emission from short-cavity lasers
We have designed and constructed a dye laser with cavity length that is adjustable from 501_m to a few mm, In this laser an unexpected spatial distribution of the output of the short-cavity laser has been observed, Intense, sharp rings were found surrounding the usual lasing spot. In the case of a cavity with plane-parallel mirrors, the rings were founa to be at