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Journal of the Optical Society of America B

Single- and two-mode theory for coupled lasers with Doppler-broadened gain media

Mark P. Jelonek, Won B. Roh

Published Feb 1, 1990• 0 citations
Publisher

Abstract

Existing coupled-laser theories have been extended to analyze the case of two coupled lasers supporting two modes in each cavity. A set of eight coupled equations describing the amplitudes and phases of the two modes oscillating in each cavity has been solved numerically to evaluate the operating characteristics of the coupled-laser system. Locking ranges were computed for both single- and two-mode coupled-laser systems, and it was found that the presence of the additional mode reduces the locking range of the two-mode system to approximately one half that of the single-mode coupled lasers. A number of other oddities in behavior have been found, including a mutually reinforced hole-burning minimum in intensity and a phase-dependent shift of the laser frequency from the bare-cavity frequency.

DOI
10.1364/josab.7.000193

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Existing coupled-laser theories have been extended to analyze the case of two coupled lasers supporting two modes in each cavity. A set of eight coupled equations describing the amplitudes and phases of the two modes oscillating in each cavity has been solved numerically to evaluate the operating characteristics of the coupled-laser system. Locking ranges were computed for both single- and two-mode coupled-laser systems, and it was found that the presence of the additional mode reduces the locking range of the two-mode system to approximately one half that of the single-mode coupled lasers. A number of other oddities in behavior have been found, including a mutually reinforced hole-burning minimum in intensity and a phase-dependent shift of the laser frequency from the bare-cavity frequency.
We propose a simple and convenient analytical model for the description of coherent passive mode locking in a two-section ring-cavity laser based on the generalized area theorem for the coherent pulse propagation in a resonant medium with inhomogeneous line broadening. This model is applied to theoretically analyze the possible regimes of the spatiotemporal dynamics of such a laser together with their stability properties. We examined the laser dynamics, when varying the energy and phase relaxation times in the media, the gain and absorption rate in the respective laser sections, as well as the ratio of the transition dipole moments of the absorber and gain media. As the result, the stability of the coherent mode locking as well as the large flexibility of the arising laser behavior were shown in wide parameter ranges.
The multimode strong-signal theory of the preceding paper is applied to a number of two- and three-mode cases. A two-mode stability analysis yields strong-signal counterparts to the third-order coupling parameter. Two-mode homogeneous-broadened operation leads to the discussion of population inversion gratings and associated Bragg scattering. Symmetry properties of the equations for the bidirectional ring are examined. Results for Doppler-broadened two-mode special cases and the three-mode locked unidirectional-ring operation are presented. The strong-signal results are compared to those of the third-order and rate-equation-approximation theories. For the bidirectional ring the historical third-order domain of relative excitation from 1.0 to 1.2 is divided into a valid third-order region and a less accurate region. The condition of bistable operation vs monostable operation is seen to be dependent on the relative excitation and the amount of homogeneous broadening. Two types of bistable operation are seen to exist, one where one mode is completely suppressed and another where both modes may have nonzero intensities.
Aid or competition between two laser modes oscillating on different transitions of a three-level medium are studied following the method of Lamb. An arbitrary amount of Doppler broadening is allowed for. The "electric-quadrupole" term generated by back-to-back transitions of the two modes but having vanishing electric dipole moment is found to have little effect on the oscillation intensities in typically observed circumstances. This fact justifies, in part, the rate-equation approximation commonly used in multilevel analysis. For a bidirectional ring cascade configuration, however, this coherence term can be important because of constructive interference familiar from two-photon, Doppler-free spectroscopy. The ordinary (i.e., hole-burning) cross-saturation terms for cascade transitions produce mutual mode aid, rather than competition as in the competitive cases. A curious consequence of this difference is greater accuracy in the third-order theory of cascade transitions than in that of competitive transitions, as a result of partial cancellation of errors incurred through neglect of higher orders. The treatment is generalized to multilevel media (four or more levels) with numerical illustration and discussion of the method's relationship to laser oscillation involving diatomic molecules.
We use the semiclassical strong-signal theory of the laser to predict and explain the onset of side-mode buildup in lasers with one oscillating mode. Two general categories are considered: one for which the side modes and the oscillating mode all have the same wavelength and the other for which they have different wavelengths. The treatments include an arbitrary amount of inhomogeneous broadening. Our approach unifies the treatments of the side-mode instabilities presented earlier and extends them to handle standing waves in addition to the previously treated running waves. We write the field and the population matrix elements as Fourier series in the adjacent-mode beat frequency. This approach has been used extensively in both multimode laser theory and saturation spectroscopy. This technique coincides with linear stability analyses used by others, provided that our beat frequency includes a contribution that is proportional to the complex side-mode gain. We give a solution that allows for detuned operation along with its simpler, centrally tuned special case. The connection with saturation spectroscopy clearly reveals that the side-mode instabilities require side-mode gain. For the single-wavelength case, nonlinear anomalous dispersion is also required. The side-mode gain and dispersion result from both inhomogeneous broadening and population pulsations. The lowest instability thresholds occur when both of these mechanisms play a role. The approach can also be used to treat instabilities in optical bistability by substituting the appropriate equation of state for the strong-mode intensity and by changing the sign of the absorption coefficient. In homogeneously broadened, standing-wave lasers, we show that multiwavelength instabilities depend strongly on the position of the medium in the cavity. We illustrate the theory by giving numerical results for the output pulsation frequency and for the instability threshold by using parameters that are appropriate for the He–Xe laser. These results correlate well with experimental observations.
A gas laser oscillating steadily in a single mode is described theoretically using the "generalized Bloch equations" as the point of departure. The nonlinear susceptibility is calculated without utilizing perturbation techniques, so that the theory retains validity for laser fields of arbitrary intensity. Expressions are presented for (1) the "gain profile" which exhibits the phenomenon of dual "hole burning," (2) the powertuning characteristics, including the conditions for the appearance of the central tuning dip, and (3) the frequency of oscillation, in which are displayed the effects of mode pulling and pushing. Emphasis is on a strongly inhomogeneous line and a high-QQQ cavity, although some more general results are included as well. For excitations only slightly above threshold, a comparison is made with the Lamb single-mode theory, and in all cases agreement is found.
A transmitting window is used to weakly couple two resonant optical cavities. When active medium is placed in one cavity, the system describes a laser coupled to a resonant cavity, while active medium in both cavities describes two coupled lasers. Equations of motion for the fields in the two cavities are found. These equations are solved in the steady state when the cavities are sufficiently closely tuned that the lasers in the two cavities oscillate with the same frequency. In the locked situation, the second cavity tends to stabilize the frequency of operation against changes in length of the first cavity. When there is no active medium in the second cavity, and that cavity becomes very long, the laser acts as if it were looking into free space.