We analyze the conditions under which the Risken-Nummedal-Graham-Haken multimode instability can be observed in ring-cavity class-B three- and four-level lasers. Relevant examples of these are rare-earth doped fiber lasers. We show that the standard two-level laser model can be applied to three- and four-level laser systems under quite general conditions. We take advantage of this isomorphism to derive the conditions for the observation of the instability. The analysis is extended to the case in which the uniform field limit does not apply in order to cover more realistic situations.
We analyze the conditions under which the Risken-Nummedal-Graham-Haken multimode instability can be observed in ring-cavity class-B three- and four-level lasers. Relevant examples of these are rare-earth doped fiber lasers. We show that the standard two-level laser model can be applied to three- and four-level laser systems under quite general conditions. We take advantage of this isomorphism to derive the conditions for the observation of the instability. The analysis is extended to the case in which the uniform field limit does not apply in order to cover more realistic situations.
In 1966 Haken and Risken, Schmid and Weidlich(1) showed that under proper conditions the stationary state of a homogenously broadened, single mode ring laser can become unstable. The instability requires a "bad cavity" situation, i.e. the cavity damping constant must be larger than the atomic decay rates. Later, Haken(2) proved that such an instability coincides with the well known Lorenz instability, which leads to chaotic behaviour. On the other hand, Risken and Nummedal and independently Graham and Haken(3) analyzed the homogeneously broadened ring laser taking into account all the longitudinal cavity modes. They showed that under appropriate conditions in the good cavity case some cavity modes, different from the one resonant with the atomic system, can become unstable. In this case, one has the formation of a pulse that travels in the cavity. In 1978, Bonifacio and Lugiato(4) showed that a similar self-pulsing behaviour in good cavity conditions occurs in optical bistability in a ring cavity filled with a homogeneously broadened atomic sample.
This paper contains a description of the behavior of a multimode unidirectional ring laser with a homogeneously broadened active medium. Our formulation is based on the conventional Maxwell-Bloch (MB) equations, but is distinguished from other treatments by the inclusion of a finite mirror reflectivity and an arbitrary value of the gain parameter. We review the steady-state behavior of the system and analyze the longitudinal profile of the field and of the atomic variables. With an appropriate transformation of variables, we transform the boundary conditions of the ring cavity into standard periodicity type, even in the presence of a finite reflectivity, and derive an infinite hierarchy of coupled mode equations. We analyze exactly the linear stability of the system, and investigate the dependence of the instability domain on the reflectivity and gain parameters. A numerical study of the full MB equations for a parameter range of the type explored in the recent experiments by Hillman et al. [Phys. Rev. Lett. 52, 1605 (1984)] reveals similarities, but also considerable differences between the results of the theory and the main experimental signatures of their instability. However, the injection of numerical noise shows the presence of numerous coexisting basins of attraction which are likely to play a significant role in the dynamics of a noisy laser.
An analysis is made of the operation of a laser with a saturable absorber and dominant inhomogeneous line broadening. A four-level model is used to determine the characteristics of mode intensities and conditions for the appearance of an optical bistability. By analogy with a two-level model, the results obtained demonstrate that in single-mode operation this bistability appears only if the relaxation coefficients of the upper levels of atomic or molecular systems are different in the amplifying and absorbing parts of a laser. Moreover, a change in the inhomogeneous broadening alters greatly the optical bistability interval. The calculation method used in the present study can be readily extended to analyze a multimode laser.
We analyze the behavior of a laser upon injection of a weak-field resonant with a cavity mode adjacent to the lasing mode; the four-wave-mixing coupling generates also a field with the frequency of the symmetrical adjacent mode. Using the Maxwell-Bloch equations, we show that under appropriate conditions there is a spectacular amount of frequency conversion, with a peak when the free spectral range is equal to the Rabi frequency of the laser field multiplied by \ensuremath{\surd}2 . The origin of this effect is identified with the help of the linear stability analysis of the free-running laser, which shows also the deep connections with the classical laser instability, predicted by Risken and Nummedal [J. Appl. Phys. 39, 4662 (1968)] and Graham and Haken [Z. Phys. 213, 420 (1968)]. The Rabi peak is a precursor of the multimode laser instability, but it arises for smaller values of the pump parameter and is only weakly sensitive to the transverse shape of the laser beam.
The instabilities in homogeneously broadened lasers, which we described in the previous chapter, have a paradigmatic importance because they arise in models which allow one to perform the linear-stability analysis in a simple analytic form. The treatment of the corresponding instabilities in the case of in homogeneously broadened lasers is substantially more complex in general. However, it is very interesting because, as we shall see, the instabilities arise in parametric domains that are much more accessible from the experimental viewpoint. A related issue is that of the possibility of multimodal emission in a laser. It is common knowledge that in order to attain multimode emission it is necessary to have a sort of in homogeneity in the gain medium, because in a completely homogeneous configuration all modes interact with all atoms, and the arising competition among modes implies that only one of them survives. There are basically two main kinds of in homogeneity, the first is of spatial nature and the other is of spectral nature. The first kind is realized in Fabry–Perot lasers, because different cavity modes interact with atoms at different spatial positions in the cavity thanks to the standing-wave configuration of the field and to the consequent phenomenon of spatial hole- burning (see Section 14.2 and Fig. 14.1). This circumstance is capable of weakening the mode–mode competition and allowing multimode operation. In unidirectional ring lasers, on the other hand, there is an “inhomogeneous” configuration in the case of inhomogeneous broadening, because different groups of atoms have different transition frequencies and therefore interact with different modes thanks to the spectral hole-burning (see Section 15.2 and Fig. 15.3). Again, this weakens the mode–mode competition and allows multimode operation. However, these are qualitative considerations, and the only reliable and quantitative method to determine the possibility of multimodal operation is the linear-stability analysis of the single-mode stationary solutions.
A homogeneously broadened two-level ring laser with an intracavity parametric amplifier is studied. The presence of the parametric amplifier is shown to lead to two distinct nontrivial cw solutions compared to the one in its absence. The conditions for the stability of these cw solutions are obtained. It is shown that in the context of the resonant modes one cw solution is always phase stable, whereas the other is always phase unstable. For pump parameters beyond a critical value (the second threshold), both these cw solutions become amplitude unstable, and self-pulsing can set in. The shape and the velocity of these pulses are evaluated numerically. It is shown that two distinct types of steady-state pulses beyond the second laser threshold are possible due to the phase sensitivity introduced by the parametric amplifier. This is in sharp contrast to the case without the parametric amplifier, where only Risken-Nummedal pulses [see H. Risken and K. Nummedal, J. Appl. Phys. 39, 4662 (1968)] are possible. It is shown that the parametric amplifier can lead to the narrowing or broadening of the Risken-Nummedal pulses. Moreover, it is shown that the ring laser with or without the parametric amplifier allows for a pulsed solution distinct from the above-mentioned one, which, however, turns out to be unstable.