The objective of this presentation is to discuss a simple model of laser in which the atoms can undergo diffusive motion and to characterize the main dynamic effects associated with this often ignored feature of theoretical models. For this purpose we reconsider a laser cavity with flat mirrors1 and special geometry that allows diffraction along only one of the transverse directions to the resonator axis. As already found in an earlier model, here we also show that low threshold instabilities can lead to a departure from the homogeneous stationary state, to the creation of complex spatial patterns, and to cooperative frequency locking, a state of operation in which several transverse modes can oscillate synchronously. There are significant differences, however, in the shape of the domains of instability and in details of the laser dynamics following the destabilization of the homogeneous state. In addition, for selected parameter values the operation of the laser may actually become more stable in the presence of atomic diffusion.
The objective of this presentation is to discuss a simple model of laser in which the atoms can undergo diffusive motion and to characterize the main dynamic effects associated with this often ignored feature of theoretical models. For this purpose we reconsider a laser cavity with flat mirrors1 and special geometry that allows diffraction along only one of the transverse directions to the resonator axis. As already found in an earlier model, here we also show that low threshold instabilities can lead to a departure from the homogeneous stationary state, to the creation of complex spatial patterns, and to cooperative frequency locking, a state of operation in which several transverse modes can oscillate synchronously. There are significant differences, however, in the shape of the domains of instability and in details of the laser dynamics following the destabilization of the homogeneous state. In addition, for selected parameter values the operation of the laser may actually become more stable in the presence of atomic diffusion.
With the help of a microscopic model we investigate the effects of collisions and atomic recoil on the behavior of absorbing atoms placed inside a bidirectional resonator and driven by an external injected field. This model complements and generalizes earlier studies of the collective atomic recoil laser and of optical bistability. According to our model, even in the presence of collisions, the resonator can support bidirectional propagation. In particular, for appropriate selection of the parameters, we predict the existence of stationary solutions such that the cavity field that copropagates with the injected signal is locked in frequency with the external source, while the counterpropagating field is frequency shifted from both, even in steady state. The early stage of growth of the counterpropagating field is accompanied by a spatial modulation (grating structure) in the density of the medium, but this modulation decays away in a time roughly of the order of the average interval between collisions.
The dynamics of a semiconductor laser with an intracavity atomic absorber is studied numerically. The study is motivated by the experiments of Barbosa et al. [Opt. Lett. 32, 1869 (2007)], using a semiconductor junction as an active medium, with its output face being antireflection coated, and a cell containing cesium vapor placed in a cavity that was closed by a diffraction grating (DG). The DG allowed scanning the lasing frequency across the D2 line in the Cs spectrum, and different regimes such as frequency bistability or dynamic instability were observed depending on the operating conditions. Here we propose a rate-equation model that takes into account the dispersive losses and the dispersive refractive index change in the laser cavity caused by the presence of the Cs vapor cell. These effects are described through a modification of the complex susceptibility. The numerical results are found to be in qualitative good agreement with some of the observations; however, some discrepancies are also noticed, which can be attributed to multi-longitudinal-mode emission in the experiments. The simulations clearly show the relevant role of the Lamb dips and crossover resonances, which arise on top of the Doppler-broadened D2 line in the Cs spectrum, and are due to the forward and backward intracavity fields interacting resonantly with the Cs atoms. When the laser frequency is locked in a dip, a reduction in the frequency noise and of the intensity noise is demonstrated.
Up to this point we have always assumed that all the atoms which interact with the radiation field have the same atomic transition frequency, with an atomic linewidth given by γ ⊥ . This configuration is called homogeneous broadening. However, in the great majority of lasers the linewidth is inhomogeneously broadened , because the atomic frequency follows a distribution. For example, in solid-state lasers such as the ruby laser and fiber lasers the inhomogeneity is due to the imperfections in the crystalline structure of the material. In gas lasers, where the atoms or molecules are free to move, the inhomogeneous broadening is due to the Doppler effect, for which a molecule moving with velocity v in a field of frequency ω and wave vector k experiences an effective frequency ω ′ = ω − k · v [8]. Since what matters is the difference between the atomic transition frequency and the field frequency, we can keep the field frequency fixed and imagine that the field interacts with different groups of atoms that move with different velocities and therefore have different transition frequencies. The realistic velocity distribution is a Gaussian, for which the corresponding distribution for the modulus of the velocity is Maxwellian. However, in the following we will mainly analyze the case of a Lorentzian distribution for the atomic transition frequency, because this allows an analytical treatment. Again for the sake of simplicity, we will limit our analysis to the case of a ring cavity. In Section 15.1 we will generalize to the inhomogeneous-broadening configuration the modal equations for the case of homogeneous broadening, which were formulated in Chapter 12. In Section 15.2 we will specialize to the single-mode model and discuss its stationary solutions for the laser case. Multimode dynamical equations The atomic transition frequency of the generic group of atoms is given by ω a + δ D , where ω a is the central atomic frequency and δ D takes positive and negative values. We call ω c the frequency of the cavity mode nearest to ω a and denote by δ ac the difference ω a − ω c .
It is well known that transverse intensity profiles of lasers may show a nontrivial spatial structure. Usually, depending on the degree of symmetry, the spatial patterns have been associated with modes of a passive resonator or combinations of them which cause oscillations in the laser intensity (mode beating).1 It was recently shown2 that the fluctuations may be due, instead, to the nonlinear coupling between atoms and the field. Here we show that a spontaneous symmetry-breaking process is also at the origin of the formation of complex spatiotemporal dynamics in which oscillations at a single frequency, quasiperiodicity, and chaos may coexist within in the same spatial structure. We compare our experimental results with those obtained from bifurcation theory and numerical integration of the Maxwell–Bloch equations. The transition from a simple (Gaussian) profile to a complex (turbulent) profile is studied by means of the average intensity patterns, frequencies, and phases of the local intensity and cross-correlation functions.
In Chapters 27 and 28 we discussed the topic of optical spatial pattern formation in planar cavities. In this chapter, instead, we consider the case of ring cavities with spherical mirrors, on the basis of the discussion of transverse modes in Chapter 26. In the first section we derive, first of all, a set of modal equations to be coupled with the atomic equations, with a configuration similar to that of the modal equations of Chapter 12, with the difference that now transverse modes are included in addition to longitudinal modes. This provides an extremely complex context, and therefore in the following we focus on the case of quasi-planar cavities, in which it is possible, by introducing the low-transmission and single-longitudinal-mode approximations, to derive a single-longitudinal-mode model that represents, in the framework of cavities with spherical mirrors, the counterpart of the single-longitudinal-mode model for planar cavities discussed in Chapter 27. In Section 29.2 we consider the simplest configuration, in which only the fundamental Gaussian transverse mode is assumed to have negligible losses, so that it is possible to consider a single-mode model that includes only the Gaussian mode. This is precisely the model considered in Section 24.3.1 for the comparison with experimental data concerning the single-mode instability of optical bistability. We calculate the stationary curve, and in the laser case we demonstrate the validity of the mode-pulling formula also in this configuration. In the following section we consider, instead, the general case in which several transverse modes of a laser are involved and demonstrate that the modal equations admit multimodal stationary intensity states, a configuration that is not possible in the case of longitudinal modes, at least in the low-transmission approximation. In this phenomenon, which we call cooperative frequency locking , the modes agree a common oscillation frequency that is given by the average of the (mode-pulled) modal frequencies, with weights given by the modal intensities. Numerical and experimental evidence of cooperative frequency locking is provided.
An ideal atom laser would produce an atomic beam with highly stable flux and energy. In practice, the stability is likely to be limited by technical noise and nonlinear dynamical effects. We investigate the dynamics of an atom laser using a comprehensive one-dimensional, mean-field numerical model. We fully model the output beam and experimentally important physics such as three-body recombination. We find that at high-pump rates, the latter plays a role in suppressing the high-frequency dynamics, which would otherwise limit the stability of the output beam.