Spatial localization and pattern formation in discrete optomechanical cavities and arrays

We investigate theoretically the generation of nonlinear dissipative structures in optomechanical (OM) systems containing discrete arrays of mechanical resonators. We consider both hybrid models in which the optical system is a continuous multimode field, as it would happen in an OM cavity containing an array of micro-mirrors, and also fully discrete models in which each mechanical resonator interacts with a single optical mode, making contact with Ludwig & Marquardt [Phys. Rev. Lett. 101, 073603 (2013)]. Also, we study the connections between both types of models and continuous OM models. While all three types of models merge naturally in the limit of a large number of densely distribu…

Electric quantum walks in two dimensions

We study electric quantum walks in two dimensions considering Grover, Alternate, Hadamard, and DFT quantum walks. In the Grover walk the behaviour under an electric field is easy to summarize: when the field direction coincides with the x or y axes, it produces a transient trapping of the probability distribution along the direction of the field, while when it is directed along the diagonals, a perfect 2D trapping is frustrated. The analysis of the alternate walk helps to understand the behaviour of the Grover walk as both walks are partially equivalent; in particular, it helps to understand the role played by the existence of conical intersections in the dispersion relations, as we show th…

Creating highly squeezed vacua in hybrid Laguerre-Gauss modes

In this communication we study the above threshold quantum properties of a degenerate optical parametric oscillator (DOPO) tuned to a given transverse mode family at the signal frequency. We will show that under this configuration DOPOs are versatile sources of nonclassical light, in which one could be able to generate highly squeezed vacua with the non trivial shapes of Hybrid Laguerre-Gauss modes.

Experimental approach to transverse wave-number selection in cavity nonlinear optics

Spontaneous transverse pattern formation is experimentally studied in a ${\text{BaTiO}}_{3}$ photorefractive oscillator under degenerate four-wave mixing conditions. A near self-imaging resonator of high Fresnel number and quasi-one-dimensional in the transverse plane is used. A fine control technique of the cavity detuning, $\ensuremath{\Omega}$, is described. It allows a precise study of the relation of $\ensuremath{\Omega}$ with the transverse wave number ${k}_{\ensuremath{\perp}}$ of the roll patterns selected by the system. The law ${k}_{\ensuremath{\perp}}^{2}=\ensuremath{-}\ensuremath{\Omega}∕a$ is verified, which evidences that wave-number selection is mainly dictated by the cavity …

Generalized complex Swift-Hohenberg equation for optical parametric oscillators

A generalized complex Swift-Hohenberg equation including diffraction and nonlinear resonance terms is derived for spatially extended nondegenerate optical parametric oscillators (OPOs) with flat end mirrors. For vanishing pump detuning this equation becomes the complex Swift-Hohenberg (SH) equation valid also for lasers. Nevertheless the similarities between OPOs and lasers are limited, since the diffractive character of OPOs is lost when the diffraction coefficients of signal and idler fields are equal. This manifests, e.g., in the absence of advection by traveling waves (TWs), a clear difference with lasers. When pump detuning is nonzero a nonlinear resonance develops, as it occurs in deg…

Transverse effects in a thin slab of material with local-field induced intrinsic optical bistability

We consider a thin slab of dense material exhibiting local-field induced intrinsic optical bistability irradiated by a transversely uniform optical field (holding beam). We study the transverse effects that can arise when local excitations are created by means of a narrow optical beam (writing beam). We show that whereas diffraction effects are negligible, diffusion effects make the excitation-domain walls to move inward or outward in the transverse direction, with a speed that depends on the holding-beam intensity and the diffusion coefficient. Conditions can be found, however, for which the wall movement is counterbalanced by the field transverse gradient so that stable narrow excitation …

The dynamics of optically pumped molecular lasers. On its relation with the Lorenz - Haken model

In this paper we review the work on dynamical instabilities in optically pumped molecular lasers (OPLs) that has been carried out during the last 15 years. The main purpose of this review article is to survey and extend the authors' work on optically pumped molecular lasers and to place it in context with other research done in this area, without being a comprehensive review of all previous work done on this topic. In particular, we concentrate on the theoretical interpretation of the Lorenz dynamics observed in the far-infrared ammonia laser by reviewing the results obtained with different models of OPLs. New results corresponding to the dynamics obtained with the Doppler-broadened OPL mod…

Quantum walk on a cylinder

We consider the 2D alternate quantum walk on a cylinder. We concentrate on the study of the motion along the open dimension, in the spirit of looking at the closed coordinate as a small or "hidden" extra dimension. If one starts from localized initial conditions on the lattice, the dynamics of the quantum walk that is obtained after tracing out the small dimension shows the contribution of several components, which can be understood from the study of the dispersion relations for this problem. In fact, these components originate from the contribution of the possible values of the quasi-momentum in the closed dimension. In the continuous space-time limit, the different components manifest as …

Hysteretic nonequilibrium Ising-Bloch transition

We show that a parametrically driven cubic-quintic complex Ginzburg-Landau equation exhibits a hysteretic nonequilibrium Ising-Bloch transition for large enough quintic nonlinearity. These results help to understand the recent experimental observation of this pheomenon [A. Esteban-Martin et al., Phys. Rev. Lett. 94, 223903 (2005)].

Rocking bidirectional lasers

Abstract We study the emission properties of a class A bidirectional laser under the action of an amplitude modulated injected signal, i.e. a rocked bidirectional laser. We derive two coupled autonomous amplitude equations valid close to the emission threshold and study their solutions. The most relevant result is that while in the absence of rocking the laser can only emit in either of the two unidirectional solutions, under suitable rocking conditions cw bidirectional emission appears and, moreover, it coexist bistably with unidirectional emission.

Nonlinear dynamics of a two-photon Fabry–Pérot laser

Abstract The steady-state emission, stability and temporal dynamics of a single-mode two-photon laser with a Fabry–Perot cavity is investigated and compared with that of a ring-cavity laser. It is found that the Fabry–Perot cavity makes the laser less efficient than the ring cavity because of spatial hole burning, but the domain of stability is larger for the Fabry–Perot laser. The intensity and phase dynamics are numerically investigated and distinctive features are found in the phase dynamics as compared with one-photon lasers.

The Ising–Bloch transition in degenerate optical parametric oscillators

Domain walls in type I degenerate optical parametric oscillators are numerically investigated. Both steady Ising and moving Bloch walls are found, bifurcating one into another through a nonequilibrium Ising--Bloch transition. Bloch walls are found that connect either homogeneous or roll planforms. Secondary bifurcations affecting Bloch wall movement are characterized that lead to a transition from a steady drift state to a temporal chaotic movement as the system is moved far from the primary, Ising--Bloch bifurcation. Two kinds of routes to chaos are found, both involving tori: a usual Ruelle-Takens and an intermittent scenarios.

Type-II intermittency in a cascade laser model

Quantum noise properties of cavity solitons

General method for studying quantum fluctuations of dissipative structures formed in nonlinear optical cavities is presented. Application to cavity soliton supported by degenerate optical parametric oscillator is presented. Squeezing and intensity fluctuations spectra are discussed.

Generating highly squeezed Hybrid Laguerre-Gauss modes in large-Fresnel-number Degenerate Optical Parametric Oscillators

We theoretically describe the quantum properties of a large Fresnel number degenerate optical parametric oscillator with spherical mirrors that is pumped by a Gaussian beam. The resonator is tuned so that the resonance frequency of a given transverse mode family coincides with the down-converted frequency. After demonstrating that only the lower orbital angular momentum (OAM) Laguerre-Gauss modes are amplified above threshold, we focus on the quantum properties of the rest of (classically empty) modes. We find that combinations of opposite OAM (Hybrid Laguerre-Gauss modes) can exhibit arbitrary large quadrature squeezing for the lower OAM non amplified modes.

Domain wall dynamics in an optical Kerr cavity

An anisotropic (dichroic) optical cavity containing a self-focusing Kerr medium is shown to display a bifurcation between static --Ising-- and moving --Bloch-- domain walls, the so-called nonequilibrium Ising-Bloch transition (NIB). Bloch walls can show regular or irregular temporal behaviour, in particular, bursting and spiking. These phenomena are interpreted in terms of the spatio-temporal dynamics of the extended patterns connected by the wall, which display complex dynamical behaviour as well. Domain wall interaction, including the formation of bound states is also addressed.

Controlled Observation of a Nonequilibrium Ising-Bloch Transition in a Nonlinear Optical Cavity

We describe the controlled observation of the nonequilibrium Ising-Bloch transition in a broad area nonlinear optical cavity, namely, a quasi-1D single longitudinal-mode photorefractive oscilator in a degenerate four-wave mixing configuration. Our experimental technique allows for the controlled injection of the domain walls. We use cavity detuning as control parameter and find that both Ising and Bloch walls can exist for the same detuning values within a certain interval of detunings, i.e., the Ising-Bloch transition is hysteretic in our case. A complex Ginzburg-Landau model is used for supporting the observations.

Cavity solitons in bidirectional lasers.

We show theoretically that a broad area bidirectional laser with slightly different cavity losses for the two counterpropagating fields sustains cavity solitons (CSs). These structures are complementary, i.e., there is a bright (dark) CS in the field with more (less) losses. Interestingly, the CSs can be written/erased by injecting suitable pulses in any of the two counterpropagating fields.

Noncritical quadrature squeezing in two-transverse-mode optical parametric oscillators

In this article we explore the quantum properties of a degenerate optical parametric oscillator when it is tuned to the first family of transverse modes at the down-converted frequency. Recently we found [C. Navarrete-Benlloch et al., Phys. Rev. Lett. 100, 203601 (2008)] that above threshold a TEM${}_{10}$ mode following a random rotation in the transverse plane emerges in this system (we denote it as the bright mode), breaking thus its rotational invariance. Then, owing to the mode orientation being undetermined, we showed that the phase quadrature of the transverse mode orthogonal to this one (denoted as the dark mode) is perfectly squeezed at any pump level and without an increase in the…

Types I and II intermittencies in a cascade laser model

Abstract We report on types I and II intermittencies found in a cascade laser model. A continuous transition from one to another type of intermittency, which involves the coexistence of both types of laminar phases within the same time series, is found. Type II intermittency has special characteristics such as its origin at a frequency locked two-torus. When frequency unlocked this torus bifurcates to a three-torus, further giving rise to a type II intermittent like behaviour with new features during the laminar phases.

Transverse effects in ring fiber laser multimode instabilities

We study the influence of the transverse structure of pump and lasing fields and of the width of the doped region on the conditions for the appearance of the multimode instability in an ${\mathrm{Er}}^{3+}$-doped ring fiber laser. We show that the instability can be inhibited while maintaining a large output power when the radius of the doped region is a fraction of the core radius.

Cavity solitons in nondegenerate optical parametric oscillation

Abstract We find analytically cavity solitons in nondegenerate optical parametric oscillators. These solitons are exact localised solutions of a pair of coupled parametrically driven Ginzburg–Landau equations describing the system for large pump detuning. We predict the existence of a Hopf bifurcation of the soliton resulting in a periodically pulsing localised structure. We give numerical evidence of the analytical results and address the problem of cavity soliton interaction.

Models, predictions, and experimental measurements of far-infrared NH3-laser dynamics and comparisons with the Lorenz-Haken model

Dynamics of the intensity and optical field amplitude of a coherently pumped far-infrared NH3-laser are measured and characterized. The experimental findings in certain parameter ranges closely follow the dynamics of the Lorenz model and its generalization for laser systems. Similarities and some specific differences are found in geometrical or statistical characterizations of the attractors. The experimental results are also consistent with the results of a model of optically pumped three-level lasers which takes into account the presence of a multiplicity of velocity groups as well as three-level coherence effects. For a certain region in parameter space, this far more complex model with …

Stabilizing and controlling domain walls and dark-ring cavity solitons.

We demonstrate two alternative techniques for controlling and stabilizing domain walls (DW) in phase-sensitive, nonlinear optical resonators. The first of them uses input pumps with spatially modulated phase and can be applied also to dark-ring cavity solitons. An optical memory based on the latter is demonstrated. Here the physical mechanism of control relies on the advection caused to any feature by the phase gradients. The second technique uses a plane wave input pump with holes of null intensity across its transverse plane, which are able to capture DWs. Here the physical mechanism of control is of topological nature. When distributed as a regular array, these holes delimit spatial opti…

Squeezing induced by spontaneous rotational symmetry breaking

In this communication we study in depth the phenomenon of quadrature squeezing generated via spontaneous rotational symmetry breaking discussed for the first time in [1]. The idea can be put in short as follows. Consider a degenerate optical parametric oscillator (DOPO) tuned to the first family of transverse modes at the signal frequency, and having perfectly spherical mirrors. When pumped above threshold with a Gaussian beam and within a classical description, it is easy to show that a TEM 10 mode with an arbitrary orientation (measured by θ at Fig. 1) emerges at the subharmonic, hence breaking the rotational symmetry of the system in the transverse plane. Quantum effects are then quite i…

Lorenz character of the Doppler-broadened far-infrared laser

The dynamic behavior of an optically pumped Doppler-broadened single-mode far-infrared laser is theoretically investigated in detail and compared with that of the simpler Lorenz–Haken laser. Through the analysis of phase diagrams, three-dimensional attractor’s projections, intensity maps, and the different terms of the laser equations, the analogies and the differences between the two models are determined. Optical pumping and Doppler broadening, present in this far-infrared laser model, can be approximately incorporated into a Lorenz–Haken model with effective parameters. These results represent a further step toward the understanding of the Lorenz-like behavior observed in recent years in…

Theory of quantum fluctuations of optical dissipative structures - Application to the study of squeezing and intensity fluctuations of DOPO cavity solitons

We present a general theory of quantum fluctuations of dissipative structures in nonlinear optical cavities with transverse translation invariance. Perfect squeezing of the transverse momentum, detectable under homodyning, occurs irrespectively of the system parameters.

Noncritically squeezed light via spontaneous rotational symmetry breaking.

We theoretically address squeezed light generation through the spontaneous breaking of the rotational invariance occuring in a type I degenerate optical parametric oscillator (DOPO) pumped above threshold. We show that a DOPO with spherical mirrors, in which the signal and idler fields correspond to first order Laguerre-Gauss modes, produces a perfectly squeezed vacuum with the shape of a Hermite-Gauss mode, within the linearized theory. This occurs at any pumping level above threshold, hence the phenomenon is non-critical. Imperfections of the rotational symmetry, due e.g. to cavity anisotropy, are shown to have a small impact, hence the result is not singular.

Class-B two-photon Fabry–Pérot laser

Abstract We study the stationary operation and stability properties of a class-B two-photon Fabry–Perot laser. We show that, differently from the one-photon laser, the intensity emitted by the two-photon laser is larger in a Fabry–Perot than in a ring cavity. The lasing solution loses stability through a subcritical Hopf bifurcation, as it occurs in the unidirectional ring laser. The stability domain in the parameter space is larger in the Fabry–Perot than in the ring cavity configuration.

Role of field losses on the Risken?Nummedal?Graham?Haken laser instability: application to erbium-doped fibre lasers

We analyse the effect of both distributed and localised losses in a laser cavity on the Risken–Nummedal–Graham–Haken multimode instability. For two-level lasers, distributed losses are found to have a negligible influence on the instability conditions as long as they remain below 10 dB, a value hardly ever exceeded under common experimental conditions. If one keeps raising the distributed loss above that value, finally the uniform-field-limit results are recovered: localised loss becomes less and less important, and in the limit does not enter at all. In contrast, for three-level lasers – in particular for erbium-doped fibre lasers – distributed losses are found to have a profound quantitat…

Vectorial Kerr-cavity solitons.

It is shown that a Kerr cavity with different losses for the two polarization components of the field can support both dark and bright cavity solitons (CS’s). A parametrically driven Ginzburg–Landau equation is shown to describe the system for large-cavity anisotropy. In one transverse dimension the nonlinear dynamics of the bright CS’s is numerically investigated.

Theory of quantum fluctuations of optical dissipative structures and its application to the squeezing properties of bright cavity solitons

We present a method for the study of quantum fluctuations of dissipative structures forming in nonlinear optical cavities, which we illustrate in the case of a degenerate, type I optical parametric oscillator. The method consists in (i) taking into account explicitly, through a collective variable description, the drift of the dissipative structure caused by the quantum noise, and (ii) expanding the remaining -internal- fluctuations in the biorthonormal basis associated to the linear operator governing the evolution of fluctuations in the linearized Langevin equations. We obtain general expressions for the squeezing and intensity fluctuations spectra. Then we theoretically study the squeezi…

Laser instabilities in a Gaussian cavity mode with Gaussian pump profile

We analytically demonstrate that both single-mode and multimode instabilities may occur in a Gaussian-cavity-mode laser model with Gaussian pump profile. As a necessary condition, the ratio of the beam waist to the pump waist must exceed a given limiting value, which depends on the population decay rate. For an infinitely concentrated pump the plane-wave model instability thresholds are recovered, and there exists an optimum value of the waists ratio for which the second laser threshold is minimum.

Spontaneous symmetry breaking as a resource for noncritically squeezed light

[EN] In the last years we have proposed the use of the mechanism of spontaneous symmetry breaking with the purpose of generating perfect quadrature squeezing. Here we review previous work dealing with spatial (translational and rotational) symmetries, both on optical parametric oscillators and four-wave mixing cavities, as well as present new results. We then extend the phenomenon to the polarization state of the signal field, hence introducing spontaneous polarization symmetry breaking. Finally we propose a Jaynes-Cummings model in which the phenomenon can be investigated at the singlephoton-pair level in a non-dissipative case, with the purpose of understanding it from a most fundamental …

Four-phase patterns in a forced nonlinear optical oscillator

We present preliminary theoretical and experimental results indicating that a high Fresnel number nonlinear optical oscillator with planar mirrors can display four-phase multistability, eventually leading to four-phase patterns. Such situation is similar to that emerging in extended oscillatory systems forced within a 4:1 resonance and, to the best of our knowledge, has not been predicted nor observed previously in an optical system.

Generalized rate equations for multimode lasers

Abstract A generalized rate equations model for class B lasers outside the uniform field limit is presented. This model allows to correctly describe the Risken–Nummedal–Graham–Haken instability and its associated multimode dynamics. When parameters suitable for fiber lasers – where the instability has been observed – are adopted, the computation time is shown to be greatly reduced with respect to the complete model based on the full set of Maxwell–Bloch (MB) equations.

Multimode emission in inhomogeneously broadened ring lasers

The threshold for multilongitudinal-mode emission in inhomogeneously broadened ring lasers is analytically investigated. In the homogeneous limit the multimode instability corresponds to the classical Risken–Nummedal–Graham–Haken instability. It is found that by increasing the inhomogeneous linewidth, the instability threshold is decreased and the growth of high-frequency side modes is favored. In the limit where the population-inversion decay rate γ‖ is much smaller than the polarization decay rate γ⊥ (class B lasers), analytical expressions for the instability threshold are found, which are then generalized to three-level lasers for a comparison with experimental results obtained with erb…

Physical interpretation of laser phase dynamics

The basic features characterizing the dynamical evolution of the phase of a detuned-laser field under an unstable regime are physically interpreted in terms of dispersive and dynamical effects. A general method for obtaining any attractor projection containing the phase information is established, which provides evidence for the heteroclinic character of the attractor in the presence of cavity detuning for any emission regime.

Bistable phase locking in a low fresnel number nondegenerate optical oscillator with injected signal

Degenerate four-wave mixing oscillators are phase-bistable cavities. In such systems, above the oscillation threshold, two equivalent states, of equal intensities but opposite phases are generated. This phase bistability extends over the whole range of stable emission, unlike the intensity bistability (in, e.g. a saturable absorber cavity) that exits in a limited range of injection. When the cavity Fresnel number is large different patches of the beam transverse section can have different phases and a pattern forms. Basic patterns here are phase fronts (or domain walls), which are 1D structures separating regions with opposite phase that manifest as dark lines (as the phase jumps by p acros…

Diffusion stabilizes cavity solitons in bidirectional lasers

We study the influence of field diffusion on the spatial localized structures (cavity solitons) recently predicted in bidirectional lasers. We find twofold positive role of the diffusion: 1) it increases the stability range of the individual (isolated) solitons; 2) it reduces the long-range interaction between the cavity solitons. Latter allows the independent manipulation (writing and erasing) of individual cavity solitons.

Domain walls and ising-BLOCH transitions in parametrically driven systems.

Parametrically driven systems sustaining sech solitons are shown to support a new kind of localized state. These structures are walls connecting two regions oscillating in antiphase that form in the parameter domain where the sech soliton is unstable. Depending on the parameter set the oppositely phased domains can be either spatially uniform or patterned. Both chiral (Bloch) and nonchiral (Ising) walls are found, which bifurcate one into the other via an Ising-Bloch transition. While Ising walls are at rest Bloch walls move and may display secondary bifurcations leading to chaotic wall motion.

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.

Risken–Nummedal–Graham–Haken instability in class-B lasers

We determine analytical expressions for the Risken-Nummedal-Graham-Haken multimode laser instability outside the uniform field limit in the case of very fast polarization decay (class-B laser). A new condition for the observability of that instability, concerning the value of the cavity mirrors reflectivity, is predicted.

Deviation from Lorenz-type dynamics of an NH3 ring laser

Abstract We show that the differences of the intensity spiral dynamics of an optically pumped NH 3 ring laser from that of the Lorenz model are caused by counterpropagating emission and by three-level coherence effects. In particular we find that under appropriate conditions the differences disappear and the laser emits purely according to the Lorenz model.

Multiphase patterns in a degenerate nonlinear oscillator

Degenerate four-wave mixing (DFWM) oscillators are phase-bistable devices. In such systems, two equivalent states, of equal intensities but opposite phases can be generated. When the cavity Fresnel number is large, different regions of the beam transverse section can have different phases, leading to phase patterns like phase fronts (or domain walls), i.e. ID structures separating regions with opposite phase that manifest as dark lines (as the phase jumps by π across the wall), phase domains, and phase solitons, among others.

Bistable phase locking in a laser with injected signal

Intermittent and quasiperiodic behavior in a Zeeman laser model with large cavity anisotropy

Optical implementability of the two-dimensional Quantum Walk

We propose an optical cavity implementation of the two-dimensional coined quantum walk on the line. The implementation makes use of only classical resources, and is tunable in the sense that a large number of different unitary transformations can be implemented by tuning some parameters of the device.

Polarization-sensitive population trapping in an optically pumped laser

Bidirectional laser cavity solitons

Cavity solitons in optical systems have been studied for two decades in a large variety of optical systems. In principle, bidirectional lasers can emit only unidirectionally in steady state, as the two-mode solution (non null steady state in the two possible emission directions) is an unstable solution (winner-takes-all competition prevents the bidirectional cw emission). But for a wide aperture cavity this is not necessarily true as, in different regions of the transverse (with respect to the propagation axis) direction, emission in different propagation directions could occur if the fronts separating these domains are stable. In fact this is exactly what happens when the cavity losses for…

Type I optical parametric oscillators above threshold are perfect squeezers for empty gauss-hermite modes at any pumping level

A type I optical parametric oscillator pumped by a Gaussian beam above threshold and tuned to its first transverse mode family is shown to yield a perfectly squeezed, empty Gauss-Hermite mode at any pumping level.

Propagating quantum walks: The origin of interference structures

We analyze the solution of the coined quantum walk on a line. First, we derive the full solution, for arbitrary unitary transformations, by using a new approach based on the four "walk fields" which we show determine the dynamics. The particular way of deriving the solution allows a rigorous derivation of a long wavelength approximation. This long wavelength approximation is useful as it provides an approximate analytical expression that captures the basics of the quantum walk and allows us to gain insight into the physics of the process.

Multimode instability in ring fiber lasers

Very low instability threshold in a three-level laser model with incoherent optical pumping

Abstract The stability properties of a laser model based on a closed three-level atomic scheme with incoherent optical pumping are studied. Unexpectedly, the instability threshold can be very low approaching the lasing threshold for large unsaturated gain values.

Unveiling two-dimensional discrete quantum walks dynamics via dispersion relations

The discrete, or coined, quantum walk (QW) [1] is a process originally introduced as the quantum counterpart of the classical random walk (RW). In both cases there is a walker and a coin: at every time step the coin is tossed and the walker moves depending on the toss output. Unlike the RW, in the QW the walker and coin are quantum in nature what allows the coherent superpositions right/left and head/tail happen. This feature endows the QW with outstanding properties, such as making the standard deviation of the position of an initially localized walker grow linearly with time t, unlike the RW in which this growth goes as t1/2. This has strong consequences in algorithmics and is one of the …

Observability of the Risken–Nummedal–Graham–Haken instability in Nd:YAG lasers

Multilongitudinal mode instability in ring Nd:YAG lasers is theoretically analyzed. After we review the way in which the standard two-level laser theory applies to this laser we extend the theoretical treatment to include transverse effects. We do this by taking into account the finite transverse section of the active medium and by assuming a Gaussian transverse distribution for the intracavity field. Finally we demonstrate that multimode emission develops whenever the intracavity field waist diameter is almost equal to the active rod diameter. We conclude that continuous-wave diode-pumped Nd:YAG lasers with low cavity losses are good candidates for the observation of the Risken–Nummedal–Gr…

Addressing optical pixel bits in a slab of dense optical material via intrinsic optical bistability

It is well known that dense materials with local-field effects can show "intrinsic" optical bistability when they are directly irradiated by a light beam. This has been shown theoretically in a number of works and also experimentally in several cases, in gas media and also in doped solid-state materials where nonlinearities based on standard local-field effects can be reinforced with other ion interaction effects. Although from the point of view of applications nonlinearities stronger than those found so far would be desirable, the fact that no optical resonator is needed to achieve bistability makes these materials potentially interesting for applications in optical information storage and…

Understanding and controlling N-dimensional quantum walks via dispersion relations: application to the two-dimensional and three-dimensional Grover walks-diabolical points and more

The discrete quantum walk in N dimensions is analyzed from the perspective of its dispersion relations. This allows understanding known properties, as well as designing new ones when spatially extended initial conditions are considered. This is done by deriving wave equations in the continuum, which are generically of the Schrodinger type, and allows devising interesting behavior, such as ballistic propagation without deformation, or the generation of almost flat probability distributions, which is corroborated numerically. There are however special points where the energy surfaces display intersections and, near them, the dynamics is entirely different. Applications to the two- and three-d…

Nonlinear optical Galton board

We generalize the concept of optical Galton board (OGB), first proposed by Bouwmeester et al. {[}Phys. Rev. A \textbf{61}, 013410 (2000)], by introducing the possibility of nonlinear self--phase modulation on the wavefunction during the walker evolution. If the original Galton board illustrates classical diffusion, the OGB, which can be understood as a grid of Landau--Zener crossings, illustrates the influence of interference on diffusion, and is closely connected with the quantum walk. Our nonlinear generalization of the OGB shows new phenomena, the most striking of which is the formation of non-dispersive pulses in the field distribution (soliton--like structures). These exhibit a variety…

Coexistence of single-mode and multi-longitudinal mode emission in the ring laser model

A homogeneously broadened unidirectonal ring laser can emit in several longitudinal modes for large enough pump and cavity length because of Rabi splitting induced gain. This is the so called Risken-Nummedal-Graham-Haken (RNGH) instability. We investigate numerically the properties of the multi-mode solution. We show that this solution can coexist with the single-mode one, and its stability domain can extend to pump values smaller than the critical pump of the RNGH instability. Morevoer, we show that the multi-mode solution for large pump values is affected by two different instabilities: a pitchfork bifurcation, which preserves phase-locking, and a Hopf bifurcation, which destroys it.

Multimode instability in inhomogeneously broadened class-Bring lasers: Beyond the uniform-field limit

The multimode emission threshold of class-$B$ ring lasers is analytically investigated taking into account the localized nature of the cavity losses and the inhomogeneous broadening of the amplifying medium. This analysis finds a relevant application to erbium-doped fiber lasers (EDFL's). The main conclusion is that the predictions of the simplest models are deeply modified both quantitatively and qualitatively. Thus, any attempt to interpret multimode emission in EDFL's must incorporate the two considered factors. Two main results are: (i) While in homogeneously broadened lasers instabilities are inhibited for values of the mirrors' reflectivity $\mathcal{R}l0.54ca,$ this limitation disapp…

Control and steering of phase domain walls

We show experimentally the feasibility of optically controlled location, individual addressing/erasure and steering of phase domain walls by injection of coherent addressing pulses into a phase-locked four-wave-mixing photorefractive oscillator.

Dissipative structures in optomechanical cavities

Motivated by the increasing interest in the properties of multimode optomechanical devices, here we study a system in which a driven mode of a large-area optical cavity is despersively coupled to a deformable mechanical element. Two different models naturally appear in such scenario, for which we predict the formation of periodic patterns, localized structures (cavity solitons), and domain walls, among other complex nonlinear phenomena. Further, we propose a realistic design based on intracavity membranes where our models can be studied experimentally. Apart from its relevance to the field of nonlinear optics, the results put forward here are a necessary step towards understanding the quant…

Transverse patterns in degenerate optical parametric oscillation and degenerate four-wave mixing.

Transverse pattern formation in both degenerate optical parametric oscillation and degenerate four-wave mixing is considered both theoretically and numerically. In the limit of small signal detuning both systems are shown to be described by the real Swift-Hohenberg equation. Contrarily, for small signal and large pump detunings the Swift-Hohenberg equation is modified differently in both systems, by the appearance of additional nonlinear terms, which signal the existence of nonlinear resonances that are theoretically studied through the derivation of the amplitude equation for the roll pattern in both systems. Numerical analysis supports the theoretical predictions. \textcopyright{} 1996 Th…

Polarization instability in anisotropic-cavity degenerate four-wave mixing

Abstract The emission and stability properties of a plane-wave model of intracavity degenerate four-wave mixing including self- and cross-phase modulation are studied. A Kerr medium inside an anisotropic cavity in which a linearly polarized field is injected is considered. Cavity anisotropy leads to qualitative new phenomena such as a subcritical polarization instability.

Generalization of the Lorenz-Haken model to atomic systems with different relaxation rates for the two laser levels

Abstract The fundamental Lorenz-Haken laser model is generalized to the case of a two-level amplifying medium with different external relaxation rates for the two levels and with internal relaxation. This represents one further degree of freedom, and important quantitative differences in the laser dynamics. i.e., in the stationary solutions, linear stability analysis, and timedependent solutions, are found. No significant qualitative differences, however, are observed.

Experimental demonstration of phase bistability in a broad-area optical oscillator with injected signal

We demonstrate experimentally that a broad-area laserlike optical oscillator (a nondegenerate photorefractive oscillator) with structured injected signal displays two-phase patterns. The technique [de Valc\'arcel and Staliunas, Phys. Rev. Lett. 105, 054101 (2010)] consists in spatially modulating the injection, so that its phase alternates periodically between two opposite values, i.e., differing by $\ensuremath{\pi}$.

Polarization phenomena in a laser coherently pumped by a linearly polarized field

The field intensity and polarization behaviour of an optically pumped laser is investigated in different operating conditions. For a linearly polarized pump field, a strong gain anisotropy is induced which favours generation of light with a polarization parallel to that of the pump field. Thus gain anisotropy can be counterbalanced by cavity-loss anisotropy only at low pumping field intensities, and the interplay between both types of anisotropy leads to polarization switching phenomena. In contrast to the case of the incoherently pumped laser, the decay rate for the magnetic dipole induced on the J = 1 level plays a minor role in determining the polarization dynamics. The influence of a lo…

A Theoretical Approach to the Risken-Nummedal Instability in Erbium-Doped Fibre Lasers

Multidimensional quantum walks: Diabolical points, optical wave-like propagation, and multipartite entanglement

Quantum walks (QWs) are important for quantum information science, but are becoming also interesting for other fields of research as this simple quantum diffusion model finds analogues in diverse physical systems, optical ones in particular. The experimental capabilities regarding QWs have remarkably increased along recent years and several aspects of QWs are now open to experimental research, multidimensional QWs in particular [1].

One- and two-photon lasers with injected signal in a high-Q fabry-Pérot cavity

Explicit models are derived for good cavity one- and two-photon lasers with an injected signal in a Fabry-Perot cavity. The steady solutions and their stability properties are obtained analytically and compared with the corresponding ring cavity model ones. Only quantitative differences between both types of cavities are found. In particular we show that (i) the Fabry-Perot cavity reduces significantly the domain of self-pulsing with respect to the ring cavity, and for the two-photon laser case (ii) larger output can be extracted from a Fabry-Perot cavity than from a ring cavity under certain conditions, something impossible in free-running lasers. We conclude that ring cavity models are se…

Experimental demonstration of bistable phase locking in a photorefractive oscillator

We report experimental evidence of bistable phase locking in nonlinear optics, in particular, in a photorefractive oscillator emitting in few transverse modes. Bistable phase locking is a recently proposed method for converting a laserlike system, which is phase invariant, into a phase-bistable one by injecting a suitable spatially modulated monochromatic beam, resonant with the laser emission, into the optical cavity. We experimentally demonstrate that the emission on the fundamental TEM00 mode becomes phase bistable by injection of a beam with the shape of the TEM10 mode with appropriate frequency, in accordance with recent theoretical predictions [K. Staliunas et al., Phys. Rev. A 80, 02…

Two-photon laser dynamics.

Degenerate as well as nondegenerate three-level two-photon laser (TPL) models are derived. In the limit of equal cavity losses for both fields, it is shown that the nondegenerate model reduces to the degenerate one. We also demonstrate the isomorphism existing between our degenerate TPL model and that of a dressed-state TPL. All these models contain ac-Stark and population-induced shifts at difference from effective Hamiltonian models. The influence of the parameters that control these shifts on the nonlinear dynamics of a TPL is investigated. In particular, the stability of the periodic orbits that arise at the Hopf bifurcation of the system and the extension of the self-pulsing domains of…

Quadrature and polarization squeezing in a dispersive optical bistability model

We theoretically study quadrature and polarization squeezing in dispersive optical bistability through a vectorial Kerr cavity model describing a nonlinear cavity filled with an isotropic chi(3) medium in which self-phase and cross-phase modulation, as well as four--wave mixing, occur. We derive expressions for the quantum fluctuations of the output field quadratures as a function of which we express the spectrum of fluctuations of the output field Stokes parameters. We pay particular attention to study how the bifurcations affecting the non-null linearly polarized output mode squeezes the orthogonally polarized vacuum mode, and show how this produces polarization squeezing.

Noncritical quadrature squeezing through spontaneous polarization symmetry breaking

We discuss the possibility of generating noncritical quadrature squeezing by spontaneous polarization symmetry breaking. We first consider Type II frequency-degenerate optical parametric oscillators but discard them for a number of reasons. Then we propose a four-wave-mixing cavity, in which the polarization of the output mode is always linear but has an arbitrary orientation. We show that in such a cavity, complete noise suppression in a quadrature of the output field occurs, irrespective of the parameter values.

Lorenz–Haken instability in a laser with arbitrary mirrors reflectivity

Abstract We study the Lorenz–Haken instability in a laser with arbitrary mirrors reflectivity R . In the limit of slow population difference we demonstrate that in order to observe the instability the reflectivity must exceed the critical value R c ≃0.5379, which coincides with that found recently for the multimode (Risken–Nummedal–Graham–Haken) instability. Several other differences with respect to the uniform field limit R →1 are presented.

Modal expansions in lasers outside the uniform-field limit

We show that, in lasers characterized by a slow population dynamics, the expansion of the electric field on longitudinal modes is useful even beyond the uniform-field limit. The dynamical behavior of the laser above the second threshold can be well reproduced by a set of ordinary differential equations, whose integration is much faster than that of the complete Maxwell–Bloch equations. The conditions for the uniform-field limit are also clarified.

Multi-longitudinal mode emission in a bidirectional laser model

Multi-longitudinal mode emission is a fundamental issue in laser physics. Interestingly enough, the mechanisms responsible for the transition from single- to multi-longitudinal mode emission have not been completely clarified yet. For example, it is well known that in unidirectional ring lasers the Rabi splitting of the lasing transition can lead to multimode emission even in a homogeneously broadened medium, the so called Risken-Nummedal—Graham-Haken instability (RNGHI) [1]. In spite of being known since the late sixties, only in the recent years a couple of experiments have demonstrated “dressed” versions of the RNGHI [2], i.e., up to day there are not clear demonstrations of this basic m…

Bistable phase locking of a laser via monochromatic signal injection

In free running lasers the field phase is not fixed and any value possible value is equally likely (invariant), but can be locked to an external reference by injecting a monochromatic signal field into the cavity. In this way the phase of the slave laser locks to a single value resulting in a monostable phase locking. It could be however of practical interest that the laser field be locked not to a single value but to two different possible values, hence the name bistable phase locking.

Quantum walk with a time-dependent coin

We introduce quantum walks with a time-dependent coin, and show how they include, as a particular case, the generalized quantum walk recently studied by Wojcik et al. {[}Phys. Rev. Lett. \textbf{93}, 180601(2004){]} which exhibits interesting dynamical localization and quasiperiodic dynamics. Our proposal allows for a much easier implementation of this particular rich dynamics than the original one. Moreover, it allows for an additional control on the walk, which can be used to compensate for phases appearing due to external interactions. To illustrate its feasibility, we discuss an example using an optical cavity. We also derive an approximated solution in the continuous limit (long--wavel…

Quantum fluctuations in cavity solitons

Quantum fluctuations of degenerate optical parametric oscillators' cavity solitons (CS) are studied. We show that CSs are sources of perfectly squeezed light that exhibit photon fluctuations below the shot-noise level as well.

N-dimensional alternate coined quantum walks from a dispersion-relation perspective

We propose an alternative definition of an N-dimensional coined quantum walk by generalizing a recent proposal [Di Franco et al., Phys. Rev. Lett. 106, 080502 (2011)]. This N-dimensional alternate quantum walk, AQW_N, in contrast with the standard definition of the N-dimensional quantum walk, QW_N, requires only a coin-qubit. We discuss the quantum diffusion properties of AQW_2 and AQW_3 by analyzing their dispersion relations that reveal, in particular, the existence of diabolical points. This allows us to highlight interesting similarities with other well known physical phenomena. We also demonstrate that AQW_3 generates genuine multipartite entanglement. Finally we discuss the implementa…

Operation conditions and stability of a degenerate two-photon laser

Abstract The presence of a far off-resonant intermediate level in a degenerate three-level two-photon laser model manifests through a steady pump-dependent and a dynamic frequency shift on the laser equations. The steady shift strongly affects the pump threshold necessary for emission, preventing it in certain cases. The dynamic shift makes the frequency pulling/pushing to be intensity-dependent and substantially modify the domains of existence and stability of the lasing solution permitting stable operation for large cavity losses in certain cases.

Bistable phase locking of a nonlinear optical cavity via rocking: Transmuting vortices into phase patterns.

We report experimental observation of the conversion of a phase-invariant nonlinear system into a phase-locked one via the mechanism of rocking [G. J. de Valcarcel and K. Staliunas, Phys. Rev. E 67, 026604 (2003)]. This conversion results in that vortices of the phase-invariant system are being replaced by phase patterns such as domain walls. The experiment is carried out on a photorefractive oscillator in two-wave mixing configuration.A model for the experimental device is given that reproduces the observed behavior.