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Physical Review A

Transverse patterns in a laser with an injected signal

Stefano Longhi

Published Sep 1, 1997• 12 citations
Publisher

Abstract

Pattern formation near threshold in large-aspect-ratio, single-longitudinal-mode, two-level lasers with plane mirrors under the action of a weak injected plane-wave field resonant with the atomic transition frequency and tilted with respect to the laser cavity axis is analytically investigated in terms of the amplitude equations for the system. It is shown that, when the laser emission occurs off-axis, the appearance of resonances among the injected signal field and the unstable emerging tilted waves drastically changes the pattern selection properties of the free-running laser. As a result new transverse patterns, which include singly and doubly periodic stripes and undulating hexagons, may be stabilized by the injected signal. These patterns arise from the superposition of the forced mode with either one, two, or three traveling waves on the critical circle of the free-running laser. This is a distinctive feature from similar structures more commonly found in passive optical systems, where complex patterns always arise as a superposition of standing waves. The stability properties of the various patterns and their selection rules are analytically investigated within the amplitude equations.

DOI
10.1103/physreva.56.2397

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Conference on Lasers and Electro-Optics•Aug 29, 2005•0 citations
Pattern formation near threshold in large-aspect-ratio, single-longitudinal-mode, two-level lasers with plane mirrors under the action of a weak injected plane-wave field resonant with the atomic transition frequency and tilted with respect to the laser cavity axis is analytically investigated in terms of the amplitude equations for the system. It is shown that, when the laser emission occurs off-axis, the appearance of resonances among the injected signal field and the unstable emerging tilted waves drastically changes the pattern selection properties of the free-running laser. As a result new transverse patterns, which include singly and doubly periodic stripes and undulating hexagons, may be stabilized by the injected signal. These patterns arise from the superposition of the forced mode with either one, two, or three traveling waves on the critical circle of the free-running laser. This is a distinctive feature from similar structures more commonly found in passive optical systems, where complex patterns always arise as a superposition of standing waves. The stability properties of the various patterns and their selection rules are analytically investigated within the amplitude equations.
A complex order parameter description of the spatiotemporal dynamics close to threshold for a single-longitudinal-mode, large-aspect-ratio laser with a weak injected signal is presented in one transverse spatial dimension and for a longitudinal cavity frequency smaller than the atomic resonance frequency. It is shown that, when the frequency of the external signal is tuned close to the atomic resonance frequency, the spatiotemporal dynamics of the laser may be described by two coupled Ginzburg-Landau equations with parametric terms analogous to those derived in hydrodynamics to study excitation of parametric waves in oscillatory convection. As a result of the parametric terms, which physically arise by means of a four-wave-intercation process mediated by the external signal, the bifurcating modes which are superimposed to the forced (homogeneous) mode may be either traveling or standing waves. In particular, it is shown that for small frequency detunings of the injected signal standing waves are preferred to traveling waves close to threshold. Stability analysis of traveling and standing waves beyond threshold within the amplitude equations reveals the emergence of both phase (Eckhaus) and amplitude instabilities.
Transverse effects in the laser dynamics due to near dipole-dipole interactions are studied considering the presence of permanent electric dipole moments. The semiclassical two-level Maxwell-Bloch equations are used and a single longitudinal mode is assumed. A traveling wave is selected at threshold when the sum of the cavity detuning and the near dipole-dipole parameter is larger than zero. As a consequence of this, transverse pattern can occur even when the laser frequency is larger than the frequency of the atomic transition. Also, a cutoff in the laser field spectrum arises. We found that the near dipole-dipole interactions significantly modify the stability picture of the traveling waves. Numerical simulations have been carried out and the effect of the near dipole-dipole on the pattern formation is addressed.
We analyze the effect of injected atomic coherence on transverse patterns of a broad area laser by means of the semiclassical two-level Maxwell-Bloch equations. A single longitudinal mode is considered. The injected atomic coherence forces a spatially homogeneous profile to appear and locks the field phase to a single value. Above a pump threshold value a very rich scenario of patterns is developed. Near threshold we find stationary patterns such as rhombic and hexagonal lattices. Well above threshold nonstationary patterns such as complex highly ordered vortex lattices traveling along the cross section, and nearly traveling waves appear.
In this paper we analyze the steady-state and stability properties of bistable optical systems, lasers with an injected signal, and ordinary free-running lasers. The starting point of our study is a unified model of a ring cavity containing a finite-size cylindrical sample of homogeneously broadened two-level atoms, and capable of supporting a single field mode with a Gaussian transverse profile. After solving the equations of motion in the steady state, we carry out a linear stability analysis of the stationary solutions, identify the domains of unstable operation of each of the three systems, and compare in detail the results of this work with those of earlier plane-wave uniform-field calculations. In the case of the laser with an injected signal, we find a significant enhancement of the instability domain relative to the plane-wave limit. A similar conclusion holds for mixed absorptive and dispersive optical bistability, although the enhancement of the domain of instability is less pronounced. Unstable behavior is predicted to occur for experimentally accessible values of the bistability parameter, and to be favored by the absence of bistability and by the selection of atomic and cavity detunings having opposite signs. This configuration ensures that the instability will be responsible for the emergence of undamped output pulsations of the type that will make the bistable system behave as an optical clock.
Pattern formation in large-aspect-ratio single-mode inhomogeneously broadened lasers is studied by means of the integro-differential Maxwell-Bloch equations. As the inhomogeneous linewidth increases, the neutral stability curve shows a relaxation in the traveling wave selection allowing structures with different sizes to grow. We have performed numerical simulations with a simplified model based on a few discrete groups of atoms at different resonant frequencies to observe transverse dynamics above threshold for good and bad cavity configurations. We obtain, in general, that close to threshold the inhomogeneous broadening leads to a more complex pattern in comparison to the homogeneous broadening case. At higher pumpings, a stabilization in the number of spatial frequencies taking part in the transverse pattern is found. The influence of the laser aperture on temporal dynamics is also studied. It is shown how the typical self-pulsing regime present in single-mode inhomogeneously broadened lasers is destroyed for a wide enough aperture.
Summary form only given. Stationary localized structures (LS) and localized patterns have been predicted in passive resonators in regimes where the homogeneous steady states exhibit a subcritical Turing (or modulational) instability. These solutions consist of bright or dark spots in the transverse profile of the intracavity field amplitude. They can be either spatially independent and randomly distributed or clusters forming well-defined spatial patterns in the transverse plane orthogonal to the propagation axis. We consider an optical cavity with plane mirrors filled with resonant two-level atoms and driven by a coherent plane-wave injected signal. We focus on the purely absorptive limit where the two-level atoms are in resonance with the injected field. In this system, two type of LS may be generated, depending on the magnitude of the critical wave number associated with the modulational instability: circular localized bright spots or a stationary localized single stripe. Both types of LS can form for the same value of parameters.