Bob Nagler

Minimal rate equations describing polarization switching in vertical-cavity surface-emitting lasers

Intensity rate equations for two nearly degenerate modes are derived for current driven polarization switching (PS) in vertical-cavity surface-emitting lasers (VCSELs). We show that the introduction of a current-dependent gain that saturates with increasing optical power leads to a minimal description of a PS experiment. The main advantage of these rate equations compared to more sophisticated model equations is the possibility to describe analytically the PS dynamical phenomenon. We obtain a simple expression for the switching time which agrees with experimental observations.

Frequency response of polarization switching in vertical-cavity surface-emitting lasers

We present an experimental study of the current-driven polarization modulation properties of VCSELs. In some VCSELs, abrupt polarization switching (PS) between two polarization modes is observed at a particular value of the pump current. We investigate the dynamics and the associated dominating time scales of PS as these features are strongly linked with the underlying physical mechanism causing the PS. To this end, we measure both for gain- and index-guided VCSELs the critical modulation amplitude necessary to steadily force PS back and forth across the PS point as a function of the modulation frequency. This yields the current-driven polarization modulation frequency response, which we compare with the thermal frequency response of the studied devices. The dynamic behavior turns out to be strikingly different for the different VCSEL types. Thermal effects only play a minor role in the PS in our index-guided VCSELs, while they really seem to lie at the origin of PS in the gain-guided VCSELs. By implementing this in a rate-equation based theoretical model, we are able to explain the peculiarities of the measured response curves and to reproduce the experimental findings.

Stochastic polarization switching dynamics in vertical-cavity surface-emitting lasers: theory and experiment

We present an analytical, numerical, and experimental study of the switching time and jitter of current-induced polarization switching in vertical-cavity surface-emitting lasers (VCSELs) as an example of switching in a nonlinear system in the presence of noise. Assuming that the switching is induced by changes in the dichroism, the problem can be reduced to the first-passage-time problem in gain-switched Class-A lasers. The theoretical results show excellent agreement both with numerical simulations based on the full-rate equations model and with experiments performed on oxide-confined VCSELs.

Polarization behavior of vertical-cavity surface-emitting lasers under the influence of in-plane anisotropic strain

It is well known that vertical-cavity surface-emitting lasers (VCSELs) can abruptly switch between two orthogonal linear polarization states if the current is changed. The impact of externally induced in-plane anisotropic strain on this switching was experimentally demonstrated in proton-implanted devices. In this contribution we present a further and thorough experimental investigation of the polarization behavior of different types of VCSELs (proton-implanted, air-post and oxide-confined), under varying strain conditions. We first measure the influence of the strain on the orientation of the axes of the linear polarization states. These axes can be rotated from the crystallographic direction [110] over [100] to [110]. At the same time, we monitor the exact birefringence. From the combination of these two measurements the amount of residual strain in these devices is deduced. Applying strain not only changes the frequency splitting between the two modes (due to birefringence) and their orientation, but also lifts the degeneracy in the gain of the polarization modes. We therefore also measure the gain difference (dichroism) as a function of the applied strain, via the mode suppression ratio and the optical spectrum. Due to the effect on both the birefringence and the dichroism, strain also changes the position of the polarization switching point as a function of current and can lead to the observation of double (consecutive) polarization switching. All this experimental evidence will help to build up a better understanding of the physics of polarization switching in VCSELs.

Polarization switching dynamics in single-mode VCSELs

We present an accurate experimental characterization of the dynamical properties of polarization switching (PS) in single transverse and longitudinal mode vertical-cavity surface-emitting lasers (VCSELs). When a VCSEL is driven with a constant current at its polarization switching point, it makes random jumps between its two linear polarization states. This phenomenon is called mode-hopping. The permanence times in the two polarization states show an exponentially decreasing distribution, according to Arrhenius? law. The average permanence time varies over several orders of magnitude depending on the relative difference between threshold and switching current. We have performed a statistical experimental characterization of the residence times of mode hopping VCSELs for both proton implanted and oxide confined samples, and find our results to be in excellent agreement with the theoretical predictions from a novel intensity rate equation model.

In-plane strain modification of polarization behavior of vertical-cavity surface-emitting lasers

We demonstrate experimentally how an external mechanical stress applied to a VCSEL wafer results in an in-plane anisotropic strain, which dramatically alters the polarization behavior of our VCSELs. In the presence of in-plane strain, the VCSEL still emits linearly polarized light but its direction strongly depends on the magnitude and the orientation of the strain (with respect to the crystal co-ordinate system). The latter behavior can be understood by taking into account the elasto-optic effect and the proper strain distribution. Furthermore, for a specific range in magnitude and orientation of the externally induced strain, current induced polarization switching between the two fundamental modes (with orthogonal linear polarization) is present in a reproducible way. The current at which switching occurs strongly depends on the magnitude of the external stress and can be tuned in the whole region of single-mode operation. These effects can be explained by accounting for the anisotropy of the valence band. The latter is induced by the in-plane uniaxial strain, leading to a modification of density of states and effective masses for different directions in the plane of the QW, and henceforth to a gain anisotropy and a different gain curve for each of the two polarization modes. Furthermore, the frequency splitting between the two cavity eigenmodes (also a result of the stress via the elasto-optic effect) has to be taken into account. We will discuss how the gain anisotropy changes with current, lattice temperature and carrier density, and how all these determine the polarization behaviour of VCSELs.

Stochastic polarization dynamics in vertical-cavity surface-emitting lasers described by simple rate equations

Current driven switching between two orthogonal linear polarizations in the fundamental mode of VCSELs is the object of intensive experimental and theoretical work. We developed a model based on the experimental evidence that the gain/loss difference between the two modes is an important factor in polarization switching. We analytically and numerically study a nearly-degenerate two-mode intensity rate equation model. The gain coefficients we use are current dependent and gain saturation with increasing optical power is also taken into account. Two types of stationary solutions emerge: pure-mode solutions (one mode lases) and mixed mode solutions (both modes lase). Stability analysis shows that when the gains equalize, switching between the two pure-mode solutions occurs. The nonlinear gain induces a region of bistability around the switching point. Taking into account the different time scales present in the model and using asymptotic techniques, we can further reduce the model to a single dynamical equation, which can be solved analytically. Stochastic effects (e.g. due to spontaneous emission) can be incorporated in this 1D Langevin-type equation. This allows us to explain the mode hopping in terms of a First Passage Time over the potential barrier in a double potential well.

Time scales of polarization switching in different types of VCSELs

We present a study of the time-scale at which current induced polarization switching (PS) in VCSELs takes place. To this end, we measure the step and frequency response in three different types of PS VCSELs, showing that the dominating time-scales differ strongly from one VCSEL structure to another. We characterize the current-driven polarization modulation frequency response by measuring the critical modulation amplitude necessary to steadily force PS back and forth across the PS point as a function of the modulation frequency. The polarization step response is obtained by measuring the stochastic properties of the delay between the applied current step and the resulting change in the polarization, for various values of the initial and final current. For the studied proton-implanted VCSEL the polarization response is characterized by the thermal relaxation time. The measured polarization response of the air-post VCSEL also shows a clear signature of thermal effects, however PS is not at all inhibited at higher frequencies. In the oxide-confined device studied, there seems to be no thermal influence on the PS at all. Comparing the frequency response and the step response measurements done on the same device leads to similar conclusions and allows us to crosscheck our results. In all cases, we are able to reproduce our experimental findings using a rate-equation model, where PS is supposed to be induced by changes in the gain balance between the two polarization modes.

Frequency response of current modulation induced polarization switching in VCSELs

We present an experimental and rate-equation based theoretical study of the current-driven polarization modulation properties of VCSELs. In such lasers a high-contrast polarization flip is often observed at a particular value of the pump current. When modulating the current around the polarization switching value, we measure the critical modulation amplitude necessary to force synchronized back-and-forward polarization flips, as a function of the modulation frequency. This yields the polarization modulation frequency response. For a proton-implanted VCSEL the shape of the measured response curve is characterized by time constants that are very long compared with the usual time scales of laser dynamics (such as photon and carrier lifetimes), and compatible with the measured thermal relaxation time. Indeed, both the polarization modulation and the thermal frequency response curves show a cut-off frequency of about 90kHz, independent of the particular value of the switching current. In the frequency response curve of an air-post VCSEL one clearly sees remnants of the thermal influence on the switching. However, one cannot say that a thermal cut-off inhibits polarization switching above a certain modulation frequency. Notwithstanding the difference in impact of thermal effects depending on the type of device under study, our results indicate that it is necessary to incorporate a temperature-dependent variable in realistic models describing the dynamical polarization properties of VCSELs.

Stochastic current-driven polarisation switching in VCSELs: theory and experiment

We investigate the dynamics of current driven polarisation switching in VCSELs, taking stochastic effects into account. From a physical point of view, such a study allows to gain insight in the physical mechanism behind polarisation switching. Moreover, it is also of practical importance if one wishes to exploit the VCSELs polarisation properties for applications where controlled PS is required.