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Topical Meeting on Optical Bistability (OB)

Optical Bistability due to mode-mode competition in a CO2 laser

Lorenzo M. Narducci, Jorge R. Tredicce, N. B. Abraham, Donna Kay Bandy

Published Jan 1, 1985• 0 citations
Publisher

Abstract

Homogeneously broadened lasers are considered as intrinsically single mode monostable devices [1] where intensity and frequency are defined by the cavity mode that lies nearest to the center of the atomic line. Here we show theoretically and experimentally that this heuristic picture falls when the interaction between different cavity modes becomes relevant. In fact, for sufficiently high detuning between cavity and atomic resonance frequencies (δAC), self pulsing of the laser arise from the simultaneous operation in two modes (phase instability) even if it is operating near threshold (C ⪯ 1) and in the so-called good cavity limit (k ≪ γ⊥, γ‖). Using parameter values which are typical of many solid state and molecular gas lasers (γ‖, k ≪ γ⊥), the theory predicts a bistable behavior in both the output intensity and frequency, as δAC is varied (Fig. 1). These theoretical results have been confirmed by experiments carried out on a CO2 laser where the distance between modes was controlled by changing the losses inside the cavity and the detuning δAC was varied by a fine tuning of the distance between mirrors. The output intensity versus δAC is shown in Fig. 2 for two different values of losses and gas pressure. Direct comparison between Fig. 2b) and Fig. 1 shows a good qualitative agreement. Measurements of the transients during the jump of Fig.2b) confirmed the mode-mode competition interpretation.

DOI
10.1364/obi.1985.wc5

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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.

Topical Meeting on Optical Bistability (OB)•Jan 1, 1983•0 citations
Homogeneously broadened lasers are considered as intrinsically single mode monostable devices [1] where intensity and frequency are defined by the cavity mode that lies nearest to the center of the atomic line. Here we show theoretically and experimentally that this heuristic picture falls when the interaction between different cavity modes becomes relevant. In fact, for sufficiently high detuning between cavity and atomic resonance frequencies (δAC), self pulsing of the laser arise from the simultaneous operation in two modes (phase instability) even if it is operating near threshold (C ⪯ 1) and in the so-called good cavity limit (k ≪ γ⊥, γ‖). Using parameter values which are typical of many solid state and molecular gas lasers (γ‖, k ≪ γ⊥), the theory predicts a bistable behavior in both the output intensity and frequency, as δAC is varied (Fig. 1). These theoretical results have been confirmed by experiments carried out on a CO2 laser where the distance between modes was controlled by changing the losses inside the cavity and the detuning δAC was varied by a fine tuning of the distance between mirrors. The output intensity versus δAC is shown in Fig. 2 for two different values of losses and gas pressure. Direct comparison between Fig. 2b) and Fig. 1 shows a good qualitative agreement. Measurements of the transients during the jump of Fig.2b) confirmed the mode-mode competition interpretation.
Unstable and bistable behavior has been predicted recently in multimode homogeneously broadened lasers as a function of the detuning between the cavity and the atomic frequencies. As an example of a device for which most required physical conditions for the appearance of this instability are satisfied, we have analyzed the behavior of a low-pressure (10–20 Torr) CO2 laser. The operating parameters (relaxation rates for the population inversion, 104s−1, and for the polarization, 108s−1) restrict the accessible range of theoretically expected phenomena to a stable intensity output without evidence of stable self-pulsing. However, mode–mode competition is responsible for bistability in both the intensity and the operating frequency on varying the detuning parameter. This behavior is confirmed by additional observations of the transient that accompanies the jump between the two stable branches. We have also studied the effect of varying the cavity losses and discovered bistability also in the vicinity of the threshold for laser action. This kind of behavior is not predicted by the plane wave Maxwell-Bloch equations but can be explained in terms of a radial gain profile. Both effects—mode–mode competition and transverse—lead to a variety of bistable behaviors depending on gas pressure and discharge current.
Beams of homogeneously broadened two-state sodium atoms in a confocal ring resonator have been employed to investigate the steady-state characteristics of absorptive [1] and dispersive optical bistability. Also studied in the same system is the dynamical behavior due to the intrinsic single-longitudinal-mode instability [2] originating in the alteration of the atomic absorption profile by a strong field as in the Lorenz-type instability in the homogeneously broadened laser. All experimental parameters are carefully measured and corrected for transit broadening, finite absorption, and residual Doppler broadening in order to be able to make an absolute comparison with the single-Gaussian-transverse-mode theory [3-6]. Despite the frequency degeneracy of the transverse modes of the resonator, the agreement with this theory for absorptive bistability is remarkably good. The dispersive bistability data are more difficult to interpret, but they seem to be in reasonable agreement with the theory; the same can be said of the results for the domain and oscillation frequency of the instability, at least within a certain region of parameter space. Outside this region, multiple unstable states appear for various intracavity intensities; recent observations suggest that these may be due in part to instabilities involving higher-order transverse modes of the resonator.
Self-pulsing of a single-mode laser has been studied with the use of the inhomogeneously broadened 3.51-\ensuremath{\mu}m transition in xenon. Time-averaged intensity power spectra, average power output, and real-time observations of intensity fluctuations have been recorded for various detunings of the Fabry-Perot cavity. Two forms of self-pulsing have been observed, both of which can be explained in terms of the phase locking of several frequencies which simultaneously exist in the same spatial mode, made possible by the rapid variations of the index of refraction in the high-gain medium. One form of such mode splitting occurs in the wings of the atomic response due to the variations in the inhomogeneously broadened anomalous dispersion. The second form can occur anywhere in the tuning range of the laser as it is caused by distortions induced in the dispersion by the hole-burning effects of the oscillating mode. The pulsation frequency of this induced mode splitting varies with cavity tuning, shows a dip at the atomic line center, and may be absent for detuning of one or two homogeneous linewidths away from center. The instabilities have shown subharmonic bifurcations of the pulsation frequencies. Both high-QQQ pulsation frequencies and broad bands of frequencies in the power spectra have been observed for various operating conditions.
We review and extend the theory of instabilities in absorptive optical bistability for ring cavities containing homogeneously broadened two-level media. The instabilities occur in the high-transmission branch and result in either a precipitation to the low-transmission branch or in pulsed multimode operation. Two numerical approaches are used to predict operation away from the simple mean-field limit. First, an iterative method is used to solve the general field eigenvalue equation for cavity modes with frequencies displaced from the input frequency. This work shows that the instability region decreases in size as the mirror transmission increases. These calculations are confirmed by the second numerical technique, consisting of direct integrations of the coupled Maxwell-Bloch equations. This second approach also provides time histories of the instability evolutions, and divides the instability range into a precipitation regime and a self-pulsing regime. The results are discussed in terms of first- and second-order phase transitions, and agree with the analytical results obtained within the dressed-mode description of optical bistability. They show that when the incident field is adiabatically decreased along the high-transmission branch, the spiking behavior always appears abruptly. By further decreasing the incident intensity, the self-pulsing disappears either continuously if the system remains in the high-transmission branch, or discontinuously if the system precipitates to the low-transmission branch. Connection is made with induced probe gain known in the saturation spectroscopy of absorbers (uninverted media), in which population pulsations transfer energy from a saturating wave to the probe waves. The very close relationship with multimode operation in homogeneously broadened unidirectional ring lasers is also established.
This paper gives a detailed description of the plane-wave Gaussian models of optical bistability and of the experimental investigations of the single-mode instability induced on a beam of sodium atoms in a cavity by an external driving field. Our goal is to provide a careful comparison of the theoretical and experimental results. We focus on the shape of the instability domain, the nature and behavior of the spontaneous output oscillations produced by the instability, and their frequency dependence on the control parameters. We carry out this comparison using both the plane-wave and Gaussian models. Our analysis shows that the latter is in good quantitative agreement with the experimental results. We investigate in detail the main assumption of the Gaussian model, namely, that the internal cavity field retains the same radial profile as the input field.
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.