Unifying Frequency Combs in Active and Passive Cavities: Temporal Solitons in Externally Driven Ring Lasers.

Frequency combs have become a prominent research area in optics. Of particular interest as integrated comb technology are chip-scale sources, such as semiconductor lasers and microresonators, which consist of resonators embedding a nonlinear medium either with or without population inversion. Such active and passive cavities were so far treated distinctly. Here we propose a formal unification by introducing a general equation that describes both types of cavities. The equation also captures the physics of a hybrid device-a semiconductor ring laser with an external optical drive-in which we show the existence of temporal solitons, previously identified only in microresonators, thanks to symmetry breaking and self-localization phenomena typical of spatially extended dissipative systems.

From the Lugiato-Lefever equation to microresonator-based soliton Kerr frequency combs.

The model, that is usually called the Lugiato-Lefever equation (LLE), was introduced in 1987 with the aim of providing a paradigm for dissipative structure and pattern formation in nonlinear optics. This model, describing a driven, detuned and damped nonlinear Schroedinger equation, gives rise to dissipative spatial and temporal solitons. Recently, the rather idealized conditions, assumed in the LLE, have materialized in the form of continuous wave driven optical microresonators, with the discovery of temporal dissipative Kerr solitons (DKS). These experiments have revealed that the LLE is a perfect and exact description of Kerr frequency combs-first observed in 2007, i.e. 20 years after the original formulation of the LLE-and in particular describe soliton states. Observed to spontaneously form in Kerr frequency combs in crystalline microresonators in 2013, such DKS are preferred state of operation, offering coherent and broadband optical frequency combs, whose bandwidth can be extended exploiting soliton-induced broadening phenomena. Combined with the ability to miniaturize and integrate on-chip, microresonator-based soliton Kerr frequency combs have already found applications in self-referenced frequency combs, dual-comb spectroscopy, frequency synthesis, low noise microwave generation, laser frequency ranging, and astrophysical spectrometer calibration, and have the potential to make comb technology ubiquitous. As such, pattern formation in driven, dissipative nonlinear optical systems is becoming the central Physics of soliton micro-comb technology.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)'.

Backscattering differential ghost imaging in turbid media.

In this Letter we present experimental results concerning the retrieval of images of absorbing objects immersed in turbid media via differential ghost imaging (DGI) in a backscattering configuration. The method has been applied, for the first time to our knowledge, to the imaging of thin black objects located inside a turbid solution in proximity of its surface. We show that it recovers images with a contrast better than standard noncorrelated direct imaging, but equivalent to noncorrelated diffusive imaging. A simple theoretical model capable of describing the basic optics of DGI in turbid media is proposed.

Difference differential equations for a resonator with a very thin nonlinear medium.

We derive from the classic Maxwell-Bloch equations a set of difference-differential equations valid, in general, when the length of the nonlinear medium in the optical cavity is much smaller than a wavelength. Such equations provide an elegant and simple framework in which the case of Fabry-Perot and ring cavity can be discussed in a unified way. We outline a complete scenario for the multimode laser instability in the Fabry-Perot case, illustrating the results for parameter values appropriate to quantum cascade lasers. Our approach can have a relevant impact also on the study of dynamical instabilities in external cavity semiconductor lasers, including multiple quantum well or quantum-dot structures.

Differential ghost imaging.

We present a new technique, differential ghost imaging (DGI), which dramatically enhances the signal-to-noise ratio (SNR) of imaging methods based on spatially correlated beams. DGI can measure the transmission function of an object in absolute units, with a SNR that can be orders of magnitude higher than the one achievable with the conventional ghost imaging (GI) analysis. This feature allows for the first time, to our knowledge, the imaging of weakly absorbing objects, which represents a breakthrough for GI applications. Theoretical analysis and experimental and numerical data assessing the performances of the technique are presented.

X entanglement: the nonfactorable spatiotemporal structure of biphoton correlation.

We investigate the spatiotemporal structure of the biphoton entanglement in parametric down-conversion (PDC) and we demonstrate its nonfactorable X-shaped geometry. Such a structure gives access to the ultrabroad bandwidth of PDC, and can be exploited to achieve a biphoton temporal localization in the femtosecond range. This extreme localization is connected to our ability to resolve the photon positions in the source near field. The nonfactorability opens the possibility of tailoring the temporal entanglement by acting on the spatial degrees of freedom of twin photons.

Microresonator defects as sources of drifting cavity solitons.

Cavity solitons (CS) are localized structures appearing as single intensity peaks in the homogeneous background of the field emitted by a nonlinear (micro)resonator. In real devices, their position is strongly influenced by the presence of defects in the device structure. In this Letter we show that the interplay between these defects and a phase gradient in the driving field induces the spontaneous formation of a regular sequence of CSs moving in the gradient direction. Hence, defects behave as a device built-in CS source, where the CS generation rate can be set by controlling the system parameters.

High-resolution ghost image and ghost diffraction experiments with thermal light.

High-resolution ghost image and ghost diffraction experiments are performed by using a single classical source of pseudothermal speckle light divided by a beam splitter. Passing from the image to the diffraction result solely relies on changing the optical setup in the reference arm, while leaving the object arm untouched. The product of spatial resolutions of the ghost image and ghost diffraction experiments is shown to overcome a limit which seemed to be achievable only with entangled photons.

Ghost imaging with thermal light: comparing entanglement and classical correlation.

We consider a scheme for coherent imaging that exploits the classical correlation of two beams obtained by splitting incoherent thermal radiation. This case is analyzed in parallel with the configuration based on two entangled beams produced by parametric down-conversion, and a precise formal analogy is pointed out. This analogy opens the possibility of using classical beams from thermal radiation for ghost imaging schemes in the same way as entangled beams.

Ghost imaging schemes: fast and broadband.

In ghost imaging schemes information about an object is extracted by measuring the correlation between a beam that passed the object and a reference beam. We present a spatial averaging technique that substantially improves the imaging bandwidth of such schemes, which implies that information about high-frequency Fourier components can be observed in the reconstructed diffraction pattern. In the many-photon regime the averaging can be done in parallel and we show that this leads to a much faster convergence of the correlations. We also consider the reconstruction of the object image, and discuss the differences between a pixel-like detector and a bucket detector in the object arm. Finally, it is shown how to non-locally make spatial filtering of a reconstructed image. The results are presented using entangled beams created by parametric down-conversion, but they are general and can be extended also to the important case of using classically correlated thermal-like beams.

Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion.

Using a 1 GW, 1 ps pump laser pulse in high-gain parametric down conversion allows us to detect sub-shot-noise spatial quantum correlation with up to 100 photoelectrons per mode by means of a high efficiency charge coupled device. The statistics is performed in single shot over independent spatial replica of the system. Evident quantum correlations were observed between symmetrical signal and idler spatial areas in the far field. In accordance with the predictions of numerical calculations, the observed transition from the quantum to the classical regime is interpreted as a consequence of the narrowing of the down-converted beams in the very high-gain regime.

Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm.

We formulate a theory for entangled imaging, which includes also the case of a large number of photons in the two entangled beams. We show that the results for imaging and for the wave-particle duality features, which have been demonstrated in the microscopic case, persist in the macroscopic domain. We show that the quantum character of the imaging phenomena is guaranteed by the simultaneous spatial entanglement in the near and in the far field.

Rotating and Fugitive Cavity Solitons in semiconductor microresonators.

We describe two different methods that exploit the intrinsic mobility properties of cavity solitons to realize periodic motion, suitable in principle to provide soliton-based, all-optical clocking or synchronization. The first method relies on the drift of solitons in phase gradients: when the holding beam corresponds to a doughnut mode (instead of a Gaussian as usually) cavity solitons undergo a rotational motion along the annulus of the doughnut. The second makes additional use of the recently discovered spontaneous motion of cavity solitons induced by the thermal dynamics, it demonstrates that it can be controlled by introducing phase or amplitude modulations in the holding beam. Finally, we show that in presence of a weak 2D phase modulation, the cavity soliton, under the thermally induced motion, performs a random walk from one maximum of the phase profile to another, always escaping from the temperature minimum generated by the soliton itself (Fugitive Soliton).

Introduction.

Homogeneous, periodic, quasiperiodic, irregular and disordered spatial structures have been at the heart of scientific interest and research for a long time and in many disciplines as diverse as chemistry, physics, biology and morphology. Some recent, spectacular achievements in experimental quantum optics have, however, made it possible to study quantum effects in a variety of spatial structures. The main aim of this Focus Issue is to present some of the most recent theoretical, numerical and experimental progresses in the area of Quantum Structures, i.e. spatial structures that display quantum features. A short background overview of Quantum Structures is provided at the beginning of this Focus Issue, in which individual authors' works are specifically commented. In brief, we can say that the articles can be catalogued into two main areas of research interest in Quantum Structures: quantum effects in spatial structures in nonlinear optics, and in atomic physics.

Quantum structures in nonlinear optics and atomic physics: a background overview.

A brief overview of quantum effects in spatial structures such as nonlinear optical patterns, chains of trapped ions and atoms in optical lattices is presented. Some of the main results of the contributions to this Focus Issue are also briefly described.

Spatial patterns in optical parametric oscillators with spherical mirrors: classical and quantum effects.

We investigate the formation of transverse patterns in a doubly resonant degenerate optical parametric oscillator. Extending previous work, we treat the more realistic case of a spherical mirror cavity with a finite-sized input pump field. Using numerical simulations in real space, we determine the conditions on the cavity geometry, pump size and detunings for which pattern formation occurs; we find multistability of different types of optical patterns. Below threshold, we analyze the dependence of the quantum image on the width of the input field, in the near and in the far field.

Spatial patterns in optical parametric oscillators with spherical mirrors: classical and quantum effects: errata.

The correct author affiliation for K. I. Petsas, A. Gatti and L. A. Lugiato is: Dipartimento di Fisica dell'Universita di Milano, Istituto Nazionale per la Fisica della Materia, Via Celoria 16, Milano 20133, Italy.

From quantum to classical images.

We display the results of the numerical simulations of a set of Langevin equations, which describe the dynamics of a degenerate optical parametric oscillator in the Wigner representation. The scan of the threshold region shows the gradual transformation of a quantum image into a classical roll pattern. An experiment on parametric down-conversion in lithium triborate shows strikingly similar results in both the near and the far field, displaying qualitatively the classical features of quantum images. c 1997 Optical Society of America.

Formation and control of localized structures in nonlinear optical systems.

Diffractive effects in passive nonlinear optical resonators can lead to pattern-forming instabilities. When the pattern (in our case, a regular hexagonal lattice of intensity peaks) coexists with the homogeneous solution, soliton-like intensity peaks in the transverse plane can be excited. These solutions have the characteristics of localized structures and are highly degenerate with respect to the peak location. By injecting narrow laser pulses, it is possible to turn on such peaks at desired locations and to turn them off selectively. The conditions to ensure independence among the peaks are described as well. These features suggest the possibility of encoding optical information in the structure of the field profile. (c) 1996 American Institute of Physics.