We use a spatially resolved cavity ring-down technique to show that the 2D eigenmode of an unstable optical cavity has a fractal pattern, i.e. it looks the same at different length scales. In agreement with theory, we find that this pattern has the maximum conceivable roughness, i.e., its fractal dimension is 3.01 plus\minus 0.04. This insight in the nature of unstable cavity eigenmodes may lead to better understanding of wave dynamics in open systems, for both light and matter waves.
Codes were written to simulate the propagation of monochromatic light through a bare optical resonator, using a computational Fourier method to solve the Huygens-Fresnel integral. This was used, in the Fox-Li method, to find the lowest-loss eigenmodes of arbitrary cavity designs. An implicit shift `hopping' method was employed to allow a series of increasingly higher-loss eigenmodes to be found, limited in number by computational time. Codes were confirmed in their accuracy against the literature, and were used to investigate a number of different cavity configurations. In addition to confirmi…
Materials which possess a high local density of states varying at a subwavelength scale theoretically permit to focus waves onto focal spots much smaller than the free space wavelength. To do so metamaterials -manmade composite media exhibiting properties not available in nature- are usually considered. However this approach is limited to narrow bandwidths due to their resonant nature. Here, we prove that it is possible to use a fractal resonator alongside time reversal to focus microwaves onto $λ/15$ subwavelength focal spots from the far field, on extremely wide bandwidths. We first numerica…
The second part of the article is devoted to field transfers by diffraction that are represented by fractional Fourier transformations whose orders are complex numbers. The corresponding effects on the Wigner distributions associated with optical fields are still represented by $4\times4$ matrices operating on the scaled phase-space, but unlike matrices involved in the first part, those matrices decompose into two matrices that essentially represent 2--dimensional hyperbolic rotations, not elliptical rotations. The result is applied to the theory of unstable resonators.
We find that in nonhermitian PT-symmetric systems (as realized in resonators with balanced absorption and amplification), a mechanism based on quantum-to-classical correspondence reduces the occurrence of strongly amplified states. The reduction arises from semiclassically emerging hierarchical phase-space structures that are associated with the coupling of the amplifying and absorbing regions (forward and backward-trapped sets and their complements), and amounts to a generalization of the fractal Weyl law, earlier proposed for ballistically open systems. In the context of the recently introdu…
The eigenmodes of unstable canonical optical resonators possess fractal structure in their transverse intensity cross-sections [Karman et al., Nature 402, 138 (1999)]. In one particular plane, the magnified self-conjugate plane, this structure can be explained in terms of a combination of imaging and diffraction [Courtial and Padgett, PRL 85, 5320 (2000)]. Here we show that this combination of imaging and diffraction simultaneously occurs in the longitudinal direction, resulting in three-dimensional self-similar fractal structure around the centre of the magnified self-conjugate plane.
Fractals, complex shapes with structure at multiple scales, have long been observed in Nature: as symmetric fractals in plants and sea shells, and as statistical fractals in clouds, mountains and coastlines. With their highly polished spherical mirrors, laser resonators are almost the precise opposite of Nature, and so it came as a surprise when, in 1998, transverse intensity cross-sections of the eigenmodes of unstable canonical resonators were predicted to be fractals [Karman et al., Nature 402, 138 (1999)]. Experimental verification has so far remained elusive. Here we observe a variety of …
The third part of the paper is devoted to ray tracing in optical resonators. The employed method for dealing with the issue uses the elliptical or hyperbolic rotations that Wigner distributions associated with optical fields undergo during propagation from one spherical mirror of a resonator to the other. It is illustrated by various examples concerning meridional or skew rays, reentrant in stable resonators or propagating in unstable resonators. A classification of optical resonators is eventually deduced.
The collective response of closely spaced metal particles in non-periodic arrangements has the potential to provide a beneficial angular and frequency dependence in sensing applications. In this paper, we investigate the optical response of a Sierpinski fractal arrangement of gold nanoparticles and show that it supports a collective resonance similar to the surface lattice resonances that exist in periodic arrangements of plasmonic resonators. Using back focal plane microscopy, we observe the leakage of radiation out of a surface lattice resonance that is efficiently excited when the wavenumbe…
We demonstrate a nonlinear optical resonator that tunes itself onto resonance with an input beam. In a monolithic Fabry-Perot cavity implemented in rubidium-doped periodically-poled potassium titanyl phosphate, an intensity-dependent refractive index produces line-pulling by multiple free-spectral ranges (FSRs). In this condition, the cavity passively maintains optical resonance in the face of FSR-scale excursions of the drive laser frequency: when one resonant operating-point becomes unstable, the resonator rapidly transitions to another resonant operating point. We demonstrate stable second-…
A path within the parameter space of detuning and pump power is demonstrated in order to obtain a single cavity soliton (CS) with certainty in SiN microring resonators in the anomalous dispersion regime. Once the single CS state is reached, it is possible to continue a path to compress it, broadening the corresponding single free spectral range (FSR) Kerr frequency comb. The first step to achieve this goal is to identify the stable regions in the parameter space via numerical simulations of the Lugiato-Lefever equation (LLE). Later, using this identification, we define a path from the stable m…
We demonstrate lasing in arrays of microring resonators with underlying para-Fermi symmetry. The properties of the algebra allow the analytic prediction of propagation constants and normal modes of an array of passive resonators that show the optical analogue of a zero-energy mode that is its own chiral pair. The rest of the normal modes in the linear model form chiral pairs that are evenly distributed around the pseudo zero-energy mode. We use this information to construct a class-B laser model where even and odd sites are driven with different strength in a pattern following para-Fermi algeb…
We present a numerical investigation of the ray dynamics in a paraxial optical cavity when a ray splitting mechanism is present. The cavity is a conventional two-mirror stable resonator and the ray splitting is achieved by inserting an optical beam splitter perpendicular to the cavity axis. We show that depending on the position of the beam splitter the optical resonator can become unstable and the ray dynamics displays a positive Lyapunov exponent.
We study lasing emission from asymmetric resonant cavity (ARC) GaN micro-lasers. By comparing far-field intensity patterns with images of the micro-laser we find that the lasing modes are concentrated on three-bounce unstable periodic ray orbits, i.e. the modes are scarred. The high-intensity emission directions of these scarred modes are completely different from those predicted by applying Snell's law to the ray orbit. This effect is due to the process of ``Fresnel filtering'' which occurs when a beam of finite angular spread is incident at the critical angle for total internal reflection.
The dynamics of Fabry-Perot cavity with suspended mirrors is described. The suspended mirrors are nonlinear oscillators interacting with each other through the laser circulating in the cavity. The degrees of freedom decouple in normal coordinates, which are the position of the center of mass and the length of the cavity. We introduce two parameters and study how the dynamics changes with respect to these parameters. The first parameter specifies how strong the radiation pressure is. It determines whether the cavity is multistable or not. The second parameter is the control parameter, which det…
Complex systems are a proving ground for fundamental interactions between components and their collective emergent phenomena. Through intricate design, integrated photonics offers intriguing nonlinear interactions that create new patterns of light. In particular, the canonical Kerr-nonlinear resonator becomes unstable with a sufficiently intense traveling-wave excitation, yielding instead a Turing pattern composed of a few interfering waves. These resonators also support the localized soliton pulse as a separate nonlinear stationary state. Kerr solitons are remarkably versatile for application…
We study the spontaneous temporal symmetry breaking instability in a coherently-driven passive optical Kerr resonator observed experimentally by Xu and Coen in Opt.~Lett.~{\bf 39}, 3492 (2014). We perform a detailed stability analysis of the Lugiato-Lefever model for the optical Kerr resonators and analyze the temporal bifurcation structure of stationary symmetric and the emerging asymmetric states as a function of the pump power. For intermediate pump powers a pitchfork loop is responsible for the destabilization of symmetric states towards stationary asymmetric ones while at large pump power…
Dissipative solitons can emerge in a wide variety of dissipative nonlinear systems throughout the fields of optics, medicine or biology. Dissipative solitons can also exist in Kerr-nonlinear optical resonators and rely on the double balance between parametric gain and resonator loss on the one hand and nonlinearity and diffraction or dispersion on the other hand. Mathematically these solitons are solution to the Lugiato-Lefever equation and exist on top of a continuous wave (cw) background. Here we report the observation of temporal dissipative solitons in a high-Q optical microresonator. The …
Physikalisches Institut, Universität Bonn（波恩大学物理研究所）；Research Unit Satellite Geodesy, Technical University of Munich（慕尼黑工业大学卫星大地测量研究部）；School of Physical Sciences, University of Canterbury（坎特伯雷大学物理科学学院）
The Eckhaus instability is a secondary instability of nonlinear spatiotemporal patterns in which high-wavenumber periodic solutions become unstable against small-wavenumber perturbations. We show in this letter that this instability can take place in Kerr combs generated with ultra-high $Q$ whispering-gallery mode resonators. In our experiment, sub-critical Turing patterns (rolls) undergo Eckhaus instabilities upon changes in the laser detuning leading to cracking patterns with long-lived transients. In the spectral domain, this results in a metastable Kerr comb dynamics with a timescale that …
We discuss one possible method for suppression of the nonlinear effect of a parametric oscillatory instability in a Fabry-Perot cavity of the next generation of gravitational wave detectors by determination of optimal radii of curvature of interferometers' mirrors. Varying the radii of curvature of the mirrors we can change the frequencies of the optical modes of the Fabry-Perot resonators and distributions of optical fields in optical modes, which, in turn, can change the number of unstable modes, leading to parametric oscillatory instability. The determination of the optimal values of the ra…
The microwave phonon stimulated emission (SE) has been experimentally and numerically investigated in a nonautonomous microwave acoustic quantum generator, called also microwave phonon laser or phaser (see previous works arXiv:cond-mat/0303188 ; arXiv:cond-mat/0402640 ; arXiv:nlin.CG/0703050) Phenomena of branching and long-time refractority (absence of the reaction on the external pulses) for deterministic chaotic and regular processes of SE were observed in experiments with various levels of electromagnetic pumping. At the pumping level growth, the clearly depined increasing of the number of…
Topological photonics has recently emerged as a route to realize robust optical circuitry, and nonlinear effects are expected to enable tunability of topological states with the light intensity. Here we realize experimentally nonlinear self-induced spectral tuning of the electromagnetic topological edge states in an array of coupled nonlinear resonators in a pump-probe experiment. In a weakly nonlinear regime, we observe that resonators frequencies exhibit spectral shifts, that are concentrated mainly at the edge mode affecting only weakly the bulk modes. For a strong pumping, we describe seve…
Photothermal effects can alter the response of an optical cavity, for example, by inducing self-locking behavior or unstable anomalies. The consequences of these effects are often regarded as parasitic and generally cause limited operational performance of the cavity. Despite their importance, however, photothermal parameters are usually hard to characterize precisely. In this work we use an optical cavity strongly coupled to photothermal effects to experimentally observe an optical back-action on the photothermal relaxation rate. This effect, reminiscent of the radiation-pressure-induced opti…
Ray optics has proven to be an effcient and versatile tool to describe dielectric optical microcavities and their far-field emission based on the principle of ray-wave correspondence. Whereas often the well-known ray-optics at planar interfaces yields reasonable results, semiclassically and boundary-curvature induced corrections will become more important as the cavity size is further reduced. In this paper we summarize the various ray optics descriptions of optical microresonators, paying in addition special attention to the differences that arise between chaotic vs. non-chaotic (integrable o…
Standard optomechanical cooling methods ideally require weak coupling and cavity damping rates which enable the motional sidebands to be well resolved. If the coupling is too large then sideband-resolved cooling is unstable or the rotating wave approximation can become invalid. In this work we describe a protocol to cool a mechanical resonator coupled to a driven optical mode in an optomechanical cavity, which is also coupled to an optical mode in another auxiliary optical cavity, and both the cavities are frequency-modulated. We show that by modulating the amplitude of the drive as well, one …
A deformed dielectric microcavity is used as an experimental platform for the analysis of the statistics of chaotic resonances, in the perspective of testing fractal Weyl laws at optical frequencies. In order to surmount the difficulties that arise from reading strongly overlapping spectra, we exploit the mixed nature of the phase space at hand, and only count the high-Q whispering-gallery modes (WGMs) directly. That enables us to draw statistical information on the more lossy chaotic resonances, coupled to the high-Q regular modes via dynamical tunneling. Three different models [classical, Ra…
