class-B Laser Research Articles

The impact of delayed feedback on the pulsating oscillations of class-B lasers

Class-B laser systems can be described by a set of slow–fast differential equations with a small parameter, and they exhibit pulsating oscillations in common. In this paper, the impact of the delayed feedback on the pulsating solutions is investigated. At first, a careful analysis of the local stability and bifurcation shows the existence of a series of Hopf bifurcations and double Hopf bifurcations when the strength of the delayed feedback or the delay increases, by means of stability switches. Then, an application of the geometric singular perturbation theory reveals the evolutional mechanism of the pulsating solutions from transients to stationary states, and the effect of the delayed feedback upon the pulsating solutions is examined.

3D-vortex labyrinths in the near field of solid-state microchip laser

The usage of vortex-labyrinth fields and Talbot lattices as optical dipole traps for neutral atoms is considered for the wavelength of trapping radiation in the range 0.98–2.79 µm. The square vortex lattices generated in high Fresnel number solid-state microchip lasers are studied as a possible realization. The distribution of light field is obtained via a nonstationary computational model based on Maxwell–Bloch equations for a class-B laser, discrete Fox–Lee map with relaxation of inversion and a static model based on superposition of copropagating Gaussian beams. The spatial patterns obtained numerically and observed experimentally previously are interpreted as nonlinear superposition of vortices with helicoidal phase dislocations. The distribution of light field is approximated analytically by a sum of array of vortex lines and an additional parabolic subtrap. The separable optical trapping potential is proposed with similar intensity distribution. The factorization of the macroscopic wavefunction has led to the solution of the Gross–Pitaevsky equation for an ensemble of quantum particles trapped in a vortex labyrinth formed by a spatially periodic array of Laguerre–Gaussian beams.

Single–Frequency operation of External–Cavity VCSELs: Non-linear multimode temporal dynamics and quantum limit.

We present an experimental and theoretical investigation of the non-linear multimode dynamics of external-cavity VCSELs emitting at 1 and 2.3mm. We account for the stable single-frequency and linearly polarized emission by these laser sources, even in the presence of quantum noise and non-linear mode interactions originating from Four-Wave-Mixing via population pulsations in the quantum-wells. This fact is a consequence of the mode antiphase dynamics. Thanks to the high-Q external cavity configuration, the laser dynamics fall into the oscillation-relaxation-free class-A regime. The characteristic time to achieve single mode emission is ~ 1ms for a 15mm long cavity with an antireflection coated structure and no spectral filter, as for an "ideal" homogeneous gain laser. The side mode suppression ratio is as high as 40 dB, close to the quantum limit. The laser linewidth is at the quantum limit, and is ~ 1Hz at 1mW output. An experimental value <20 kHz has been established. Under standard conditions, without spectral filtering, the optimum cavity length for highly coherent single mode operation is expected in the range 5 to 30mm. Finally, for cavity lengths typically shorter than 5mm, we rather have an "ideal" homogeneous gain class-B laser, exhibiting oscillation-relaxation of the intensity in the 0.1GHz range. These properties contrast with the intrinsic strongly non-linear dynamics of conventional semiconductor lasers.

Development of spatial turbulence from boundary-controlled patterns in class-B lasers

We report time-resolved measurements of transverse dynamics in a pulsed custom-made Nd:YAG (yttrium aluminum garnet) broad-area laser, where a dynamic transition from order to turbulentlike patterns appears, without the modification of the Fresnel number. To our knowledge, it is the first time this dynamical transition is observed in a class-B laser. Higher values of the pumping and a higher ratio between the electric field and the inversion population decay rates diminish the time at which the transition takes place. A theoretical model based on the adiabatic approximation allows us to explain the main features of the observed dynamics and to compare the results with those previously seen, especially in broad-area $\mathrm{C}{\mathrm{O}}_{2}$ lasers.

Dynamical Control: Comparison of Map and Continuous-Flow Approaches

Continuous and pulsed forms of control of a multistable system are compared directly, both theoretically and numerically, taking as an example the switching of a periodically driven class-B laser between its stable and unstable pulsing regimes. It is shown that continuous control is the more energy efficient. This result is illuminated by making use of the close correspondence that exists between the problems of energy-optimal control and the stability of a steady state.

Broad-bandwidth shot-noise-limited class-A operation of a monomode semiconductor fiber-based ring laser

We demonstrate the operation of a monomode semiconductor laser with a relative intensity noise limited by the shot-noise floor, -156 dB/Hz for a typical detected photocurrent of 1 mA, over a large frequency range from 50 MHz to 18 GHz. We achieve direct control of photon lifetime to turn an initially class-B laser into a relaxation-oscillation-free class-A one while preserving strict single-mode operation. Finally, we confirm experimentally that the laser operation in the desired class-A regime allows a dramatic filtering out of the relative intensity noise.

Stability theory of class-C (far-infrared) lasers with an injected signal

The behavior of a single-mode class-C laser in the presence of an injected signal is investigated theoretically in detail. We have determined the ranges of input signal strength and the frequency detuning for which the oscillations of the cavity electric field and the atomic induced dipole moment are simultaneously stable and have only one oscillation component with the frequency equal to that of the input signal (injection-locked state). One can reproduce all the previous results corresponding to class-A and class-B lasers from our results by using their relevant conditions. Finally, the energy conservation law is demonstrated for both the below-threshold and the injection-locked states of class-C laser amplifiers.

Problems of excess noise and adequate laser models

We discuss the problem of excess noise in lasers of different dynamical classes, as well as adequate models for the description of such noise. It is shown that the excess noise in lasers is caused by mode-mode coupling. The linear coupling leads to the phenomenon of excess noise in the narrow sense (Petermann’s excess noise), while the nonlinear mode coupling is the universal mechanism for excess noise of dynamical nature. It is shown that the theory of excess noise in class-B lasers does not require new approaches and can be developed on the basis of models with adiabatically eliminated polarization obtained in a standard way.

Poincar� maps for studying relaxation oscillations in periodically pumped solid-state lasers

We present an asymptotic analysis of relaxation oscillations in periodically pumped single- and multimode class-B lasers. Discrete maps which allow one to describe the hierarchy of coexisting periodic attractors are obtained and their bifurcations leading to period-doubling regimes and quasi-periodic and chaotic oscillations are studied analytically. For systems of coupled longitudinal modes, the maps determine conditions for antiphase dynamics.

Coherent effects in the multimode dynamics of inhomogeneously broadened ring lasers

We investigate under which conditions coherent effects manifest in the multimode dynamics of inhomogeneously broadened ring lasers. In particular, we demonstrate that for long enough cavities standard rate equations for class-B lasers fail in describing the multimode dynamics.

Secondary bifurcations and transverse standing-wave patterns in anisotropic microcavity lasers close to the first laser threshold

It is well known that in a laser--in the limit of an infinite extent of the transverse aperture--traveling tilted waves are excited at the laser threshold for positive detuning between the frequency of the gain maximum of the active medium and the cavity frequency [Phys. Rev. A 45, 8129 (1992)]. However, a transverse standing wave is unstable. In this paper, it is shown that in anisotropic lasers there can be a chain of secondary bifurcations very close to threshold, which stabilizes the standing wave and then destabilizes it again through a supercritical Hopf bifurcation. The parameter dependence of these bifurcations is discussed. The investigations are motivated by interest in pattern formation in vertical-cavity surface-emitting lasers in which the rotational symmetry is broken due to the dependence of the reflectance of the Bragg reflectors on the polarization vector of the field. The applicability of the results to other class-B lasers is discussed.

Spontaneous transverse pattern formation in a microchip laser excited by a doughnut pump profile

A transition from a pure Laguerre–Gaussian (LG) mode to a pattern of optical vortex lattices in a large-Fresnel-number microchip laser is experimentally demonstrated by controlling the cavity Q-factor. The cooperative frequency locking of nearly degenerate modes is found to be a primary process for the generation of the optical vortex lattices in a class-B laser. When the cavity Q-factor is high enough, a LG-like mode and a structure of optical vortex lattices are found to coexist. Competition between coexisting transverse patterns of different symmetry gives rise to chaotic fluctuations.

Synchronization effects in a dual-wavelength class-B laser with modulated losses.

Different types of synchronization: in-phase, antiphase, phase, and lag synchronization, as well as amplitude death have been found theoretically in a dual-wavelength class-B laser with modulated losses in one of the channels. Depending on the laser parameters, oscillations in master and slave channels can be either completely or partially synchronized. The conditions for the dual-wavelength regime have been established. The analysis has been performed on the basis of transfer functions of the master and slave channels.

Coherent laser detection by frequency-shifted optical feedback

The dynamical response of a laser to frequency-shifted optical feedback is investigated both theoretically and experimentally. The ultrahigh sensitivity of a class-B laser (i.e., a laser with a cavity damping rate ${\ensuremath{\gamma}}_{c}$ higher than the population damping rate ${\ensuremath{\gamma}}_{1})$ to external light injection is demonstrated by the optical detection of weak optical feedback. Compared to a conventional optical beating, the intensity modulation induced by the coherent interaction between the laser electric field and the frequency-shifted reinjected electric field can be several orders of magnitude higher. This method permits high sensitive interferometry and hence imaging. We call this laser detection technique laser optical feedback imaging (LOFI). When the optical frequency shift is resonant with the laser relaxation frequency, the intracavity amplification of the beating is maximum and the enhancement is given by the laser damping rate ratio ${\ensuremath{\gamma}}_{c}/{\ensuremath{\gamma}}_{1}.$ This amplification is of the order of ${10}^{6}$ for a microchip laser. We also show that without optical feedback the strong fluctuations of the laser output power are well described by the Langevin noise process. In a broad range around the laser relaxation frequency the laser quantum noise is also resonantly amplified and is then several orders of magnitude higher than the detector noise. In these conditions, the LOFI is a shot noise limited detection technique. Reflectivity as low as ${10}^{\ensuremath{-}13}$ is then easily detectable with a laser output power of a few milliwatts with a detection bandwidth of 1 kHz. Experimentally, for weak optical feedback the laser fluctuations are principally composed of the LOFI modulation signal at the shifted frequency and of the laser quantum noise amplified at the relaxation frequency. For strong optical feedback, nonlinear effects appear in the laser dynamics. In these conditions, harmonics and parametrics peaks appear in the power spectrum. The LOFI detection system is then saturated.

Chaotic antiphase dynamics and synchronization in multimode semiconductor lasers

The dynamics of chaotic multimode semiconductor lasers is currently of considerable interest from the viewpoints of both fundamental physics and applications. Experimental observations and numerical simulations have been reported which show that multimode dynamics can play a significant role in a variety of complex dynamical phenomena, including low-frequency fluctuations ~LFF! and high-frequency chaotic oscillations @1‐6 #. Also, the synchronization of multimode dynamics in lasers has been considered as a mechanism for multiwavelength transmission of encoded data @7‐9#. A key issue in the behavior of multimode lasers is how the oscillations of different modes are related to each other. One general behavior that has been observed in multimode lasers is antiphase dynamics @10‐17#. Antiphase is a property related to coherence of the phases of the oscillations of the mode intensities rather than the optical phases. Alternating pulsations in two modes due to mode coupling is a simple example of antiphase dynamics. Antiphase dynamics have been shown to occur in the relaxation in a multimode class-B laser with cross saturation, where the lower-frequency components of oscillation of different longtitudinal modes cancel each other completely, so the total laser intensity oscillates with just a single oscillation frequency, resembling the relaxation of a single-mode laser @12,13#. Antiphase dynamics of multilongitudinal modes have been experimentally observed in various laser systems, including microchip lasers @12#, solid-state lasers @14,15#, fiber lasers @16#, and CO2 lasers @17#.

Interacting pairs of periodic solutions lead to tori in lasers subject to delayed feedback.

Models of class-B lasers subject to either an optoelectronic or an optical feedback are investigated analytically and numerically. We derive slow time amplitude equations from the laser delay differential equations and find multiple bifurcating and isolated branches of periodic solutions. We then show that secondary bifurcations to tori result from the interaction of pairs of Hopf bifurcations. The branches emerging from these bifurcations are followed numerically using a continuation method developed for delay differential equations.

Cooperative parametric instability of natural modes and coherent mechanism of self-mode-locking in a ring class-B laser

Based on recent experimental observations and numerical simulations [1-4], we develop an analytical theory of spontaneous low-threshold mode-locking in a long homogeneously-broadened fiber laser with slow relaxation of inversion. We show that the phenomenon originates from the self-consistent parametric resonance and beat-frequency-locking of paired “hot” modes taking place due to coherent nonlinear effects in the system of active two-level centers. An important role of the Risken-Nummedal-Graham-Haken [5-7] multimode instability is discussed.

Bi-instability and the global role of unstable resonant orbits in a driven laser

Driven class-B lasers are devices which possess quadratic nonlinearities and are known to exhibit chaotic behavior. We describe the onset of global heteroclinic connections which give rise to chaotic saddles. These form the precursor topology which creates both localized homoclinic chaos, as well as global mixed-mode heteroclinic chaos. To locate the relevant periodic orbits creating the precursor topology, approximate maps are derived using matched asymptotic expansions and subharmonic Melnikov theory. Locating the relevant unstable fixed points of the maps provides an organizing framework to understand the global dynamics and chaos exhibited by the laser.

Order parameters for class-B lasers with a long time delayed feedback

A local dynamics for class-B laser systems with optoelectronic feedback is considered in the case of a very long time delay. We determine the conditions for the bifurcation of infinite dimension. It is shown that the (ordinary) delay system can be described by a systematically derived complex Ginzburg–Landau equation without delay terms. The complicated quasi-periodic dynamics near equilibrium can be demonstrated analytically on the basis of space–time representation.

FM lasing phenomena in external cavity lasers

Numerical simulations of frequency modulated external cavity class-B lasers have been performed and a wide range of dynamic and spectral phenomena observed. It is shown that FM, pulsed or coherence collapse like behaviour can be obtained depending on the relative magnitudes of the mirror reflectivities and cavity round-trip times.