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Physical Review A

Model of coherent passive mode locking in a two-section ring-cavity laser

A. V. Pakhomov, Rostislav M. Arkhipov

Published Mar 18, 2024• 2 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.109.033519

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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.
Generation of single-cycle pulses from a laser passively mode locked by the technique of coherent mode locking is theoretically demonstrated for the case when resonant lines of both the gain and absorbing media are inhomogeneously broadened. In contrast to conventional mode-locked lasers, for which the inhomogeneous nature of line broadening sets a severe limitation on their performance, here, the stable single-cycle pulse operation is shown to persist even when the spectral widths of the inhomogeneously broadened lines of the amplifier and the absorber become nearly as wide as the central resonance frequency of their two-level transitions. This study is aimed at showing the potential for application of the coherent-mode-locking technique for quantum-dot lasers, which are typically characterized by a great amount of inhomogeneous broadening.
Coherent mode-locking (CML) represents a method for ultrashort pulse generation in lasers based on the self-induced transparency (SIT) phenomena, which allows us to overcome the principal limitations on the pulse duration imposed by the active medium's gain bandwidth in standard passively mode-locked lasers. So far, this regime was only studied in lasers with transition dipole moments in the absorber medium being twice larger than in the gain medium. This fact ensures that the $2\ensuremath{\pi}$-SIT-soliton in the absorber also forms the stable $\ensuremath{\pi}$-soliton in the gain but can be barely realized in experiments. In this paper we demonstrate theoretically that the self-starting stable coherent mode-locking regime can also arise if the same medium is used both in the gain and absorber laser sections, i.e., the transition dipole moments are equal. These results are both obtained analytically using the area theorem and confirmed by the numerical solution of the Maxwell-Bloch equations. Our findings can enable new opportunities in the area of ultrafast lasers.
Stable coherent mode-locking based on π\piπ pulse formation in single-section lasers
Here we consider coherent mode-locking (CML) regimes in single-section cavity lasers, taking place for pulse durations less than atomic population and phase relaxation times, which arise due to coherent Rabi oscillations of the atomic inversion. Typically, CML is introduced for lasers with two sections, the gain and absorber ones. Here we show that, for certain combination of the cavity length and relaxation parameters, a very stable CML in a laser, containing only gain section, may arise. The mode-locking is unconditionally self-starting and appears due to balance of intra-pulse de-excitation and slow interpulse-scale pump-induced relaxation processes. We also discuss the scaling of the system to shorter pulse durations, showing a possibility of mode-locking for few-cycle pulses.
We provide a comprehensive analysis of the standard model of the bidirectional ring laser in which only one mode can be supported in each direction and in the limit that the polarization can be eliminated adiabatically. The interaction between the two counterpropagating modes can be derived and it is most naturally viewed as a coupling between them via scattering from a spatial grating formed in the population inversion. If the grating is a sufficiently small modulation of the spatial average of the population inversion, it can be approximated by a sinusoidal function. A systematic derivation of the model with only a sinusoidal grating for the homogeneously broadened case is presented that reveals and corrects errors in several previously published analyses. The stability of the steady-state solution is analyzed. The bidirectional steady-state solution is unstable and the unidirectional steady-state solutions may be stable or unstable depending on the parameters. The well-established result of bistability between the two modes when the cavity is tuned to resonance is recovered, a result that persists, in part, even when the losses between the two modes are different.With detuning, the unidirectional steady-state solutions can be destabilized, creating regions in the parameter space where only time-dependent solutions are found. For parameters characteristic of solid-state or molecular gas lasers, the instability can occur for very small detunings and even very close to the lasing threshold. Asymptotic limits of the instability boundaries for these parameters are presented. Additional results are derived formally for inhomogeneous broadening with a Lorentzian line shape and for arbitrary inhomogeneous linewidths. Explicit analytic results are presented in the limits of very small and very large inhomogeneous linewidths compared with the homogeneous linewidth. Time-dependent solutions in the homogeneously broadened case for relaxation rates appropriate to CO2{\mathrm{CO}}_{2}CO2​ lasers show that there are broad regions in the parameter space of gain and detuning for which the behavior is dynamically chaotic in the form of nearly ``square-wave'' alternation between the counterpropagating modes of the laser with irregular switching times. In other regions of that parameter space we find more irregular pulsations and pulse alternations. Regions of periodic pulsations also have been found.
We solve numerically the time-dependent equations of motion for an inhomogeneously broadened single-mode ring laser. As expected from the linear stability analysis, pulsations develop in the output of the laser for appropriate values of the gain parameter. Under resonant conditions, the tendency is for the system to slip from a periodic regime into irregular oscillations; out of resonance, however, periodic oscillations and period-doubling bifurcations are more typical over a large range of gain values. We find that, as the ratio of the population-to-polarization decay rates varies from a value of two (radiative limit) to smaller values, the periodic oscillations turn into a train of well-separated pulses whose peak intensity scales approximately as the square of the atomic density.
The coupling of an electric field to an atomic beam prepared in a phase-locked coherent superposition of energy eigenstates is examined for a ring-laser cavity. The active medium is assumed to be a collection of noninteracting homogeneously broadened two-level atoms whose transition is electric-dipole allowed. The atom-field interaction is treated semiclassically and the atomic coherence injected into the cavity is assumed to be phase matched to the unidirectional plane-wave cavity mode. The polarization is derived for the case in which the electric-field-mode envelope has at most linear phase variations in time and is applied to the problem of frequency locking in a cavity. The presence of the atomic coherence causes the active medium to possess an inherent polarization, i.e., a field-independent contribution. This enables an optical cavity to support laser oscillation over the entire range of relative excitations, in particular, even for strongly absorbing media. A simple linear stability analysis is performed on the electric-field envelope by describing the medium with a slowly varying polarization valid when the time scales of the atomic response are much smaller than those of the variations in the envelope. The stability of frequency-locked oscillation depends on the detuning of the cavity from line center and the strength of the injected coherence relative to that of the injected population difference.