Exceptional points enhance sensing in an optical microcavity.

Sensors play an important part in many aspects of daily life such as infrared sensors in home security systems, particle sensors for environmental monitoring and motion sensors in mobile phones. High-quality optical microcavities are prime candidates for sensing applications because of their ability to enhance light-matter interactions in a very confined volume. Examples of such devices include mechanical transducers, magnetometers, single-particle absorption spectrometers, and microcavity sensors for sizing single particles and detecting nanometre-scale objects such as single nanoparticles and atomic ions. Traditionally, a very small perturbation near an optical microcavity introduces either a change in the linewidth or a frequency shift or splitting of a resonance that is proportional to the strength of the perturbation. Here we demonstrate an alternative sensing scheme, by which the sensitivity of microcavities can be enhanced when operated at non-Hermitian spectral degeneracies known as exceptional points. In our experiments, we use two nanoscale scatterers to tune a whispering-gallery-mode micro-toroid cavity, in which light propagates along a concave surface by continuous total internal reflection, in a precise and controlled manner to exceptional points. A target nanoscale object that subsequently enters the evanescent field of the cavity perturbs the system from its exceptional point, leading to frequency splitting. Owing to the complex-square-root topology near an exceptional point, this frequency splitting scales as the square root of the perturbation strength and is therefore larger (for sufficiently small perturbations) than the splitting observed in traditional non-exceptional-point sensing schemes. Our demonstration of exceptional-point-enhanced sensitivity paves the way for sensors with unprecedented sensitivity.

Ray-Wave Correspondence in Microstar Cavities.

In a previous work by the authors (Phys. Rev. Research 2, 012072(R) (2020)) a novel concept of light confinement in a microcavity was introduced which is based on successive perfect transmissions at Brewster's angle. Hence, a new class of open billiards was designed with star-shaped microcavities where rays propagate on orbits that leave and re-enter the cavity. In this article, we investigate the ray-wave correspondence in microstar cavities. An unintuitive difference between clockwise and counterclockwise propagation is revealed which is traced back to nonlinear resonance chains in phase space.

Regulated Photon Transport in Chaotic Microcavities by Tailoring Phase Space.

Manipulating light dynamics in optical microcavities has been made mainly either in real or momentum space. Here we report a phase-space tailoring scheme, simultaneously incorporating spatial and momentum dimensions, to enable deterministic and in situ regulation of photon transport in a chaotic microcavity. In the time domain, the chaotic photon transport to the leaky region can be suppressed, and the cavity resonant modes show stronger temporal confinement with quality factors being improved by more than 1 order of magnitude. In the spatial domain, the emission direction of the cavity field is controlled on demand through rerouting chaotic photons to a desired channel, which is verified experimentally by the far-field pattern of a quantum-dot microlaser. This work paves a way to in situ study of chaotic physics and promoting advanced applications such as arbitrary light routing, ultrafast random bit generation, and multifunctional on-chip lasers.

Regular-Orbit-Engineered Chaotic Photon Transport in Mixed Phase Space.

The dynamical evolution of light in asymmetric microcavities is of primary interest for broadband optical coupling and enhanced light-matter interaction. Here, we propose and demonstrate that the chaos-assisted photon transport can be engineered by regular periodic orbits in the momentum-position phase space of an asymmetric microcavity. Remarkably, light at different initial states experiences different evolution pathways, following either regular-chaotic channels or pure chaotic channels. Experimentally, we develop a nanofiber technique to accurately control the excitation position of light in the phase space. We find that the coupling to high-Q whispering gallery modes depends strongly on excitation in islands or chaotic sea, showing a good agreement with the theoretical prediction. The engineered chaotic photon transport has potential in light manipulation, broadband photonic devices, and phase-space reconstruction.

Chiral modes and directional lasing at exceptional points.

Controlling the emission and the flow of light in micro- and nanostructures is crucial for on-chip information processing. Here we show how to impose a strong chirality and a switchable direction of light propagation in an optical system by steering it to an exceptional point (EP)-a degeneracy universally occurring in all open physical systems when two eigenvalues and the corresponding eigenstates coalesce. In our experiments with a fiber-coupled whispering-gallery-mode (WGM) resonator, we dynamically control the chirality of resonator modes and the emission direction of a WGM microlaser in the vicinity of an EP: Away from the EPs, the resonator modes are nonchiral and laser emission is bidirectional. As the system approaches an EP, the modes become chiral and allow unidirectional emission such that by transiting from one EP to another one the direction of emission can be completely reversed. Our results exemplify a very counterintuitive feature of non-Hermitian physics that paves the way to chiral photonics on a chip.

Pair of Exceptional Points in a Microdisk Cavity under an Extremely Weak Deformation.

One of the interesting features of open quantum and wave systems is the non-Hermitian degeneracy called an exceptional point, where not only energy levels but also the corresponding eigenstates coalesce. We demonstrate that such a degeneracy can appear in optical microdisk cavities by deforming the boundary extremely weakly. This surprising finding is explained by a semiclassical theory of dynamical tunneling. It is shown that the exceptional points come in nearly degenerated pairs, originating from the different symmetry classes of modes. A spatially local chirality of modes at the exceptional point is related to vortex structures of the Poynting vector.

Separatrix modes in weakly deformed microdisk cavities.

Optical modes in deformed dielectric microdisk cavities often show an unexpected localization along unstable periodic ray orbits. We reveal a new mechanism for this kind of localization in weakly deformed cavities. In such systems the ray dynamics is nearly integrable and its phase space contains small island chains. When increasing the deformation the enlarging islands incorporate more and more modes. Each time a mode comes close to the border of an island chain (separatrix) the mode exhibits a strong localization near the corresponding unstable periodic orbit. Using an EBK quantization scheme taking into account the Fresnel coefficients we derive a frequency condition for the localization. Observing far field intensity patterns and tunneling distances, reveals small differences in the emission properties.

Rotating optical microcavities with broken chiral symmetry.

We demonstrate in open microcavities with broken chiral symmetry that quasidegenerate pairs of copropagating-wave resonances are transformed by rotation to counterpropagating ones, leading to a striking change of emission directions. The rotation-induced relative change in output intensity increases exponentially with cavity size, in contrast to the linear scaling of the Sagnac effect. By tuning the degree of spatial chirality with cavity shape, we are able to maximize the emission sensitivity to rotation without spoiling the quality factor.

Semiclassical evaluation of frequency splittings in coupled optical microdisks.

When two microdisks are placed close to each other and the evanescent fields of their whispering gallery modes are overlapped, a strong coupling can be induced in the modes and lead to a doublet state. We attempt to evaluate the frequency splittings of the doublets by applying a semiclassical analysis in the regime of small wavelengths. Since a whispering gallery mode in a microdisk is a leaky mode, an established semiclassical method that deals with coupled closed systems is modified. As a result, we attain an analytic formula which can conveniently compute the frequency splittings of coupled whispering gallery modes. The derived formula is verified by demostrating a perfect agreement with numerical solutions of Maxwell's equations.

Mode selection in electrically driven quantum dot microring cavities.

Within this paper a novel method for selecting certain lasing modes from a whispering gallery mode (WGM) spectrum of electrically pumped microrings is presented. Selection is achieved by introducing sub-wavelength sized notches of about 50 nm width and 500 nm depth to the sidewalls of ring shaped quantum dot micro cavities with 80 µm diameter and ridge widths below 2 µm. It is shown that the notches act as scattering centers, suppressing modes that have maxima in intensity at the notch position. By a variation of the angle between the notches, different repetitive patterns of lasing modes and suppressed modes are conceivable.

Whispering-gallery mode resonators for highly unidirectional laser action.

Optical microcavities can be designed to take advantage of total internal reflection, which results in resonators supporting whispering-gallery modes (WGMs) with a high-quality factor (Q factor). One of the crucial problems of these devices for practical applications such as designing microcavity lasers, however, is that their emission is nondirectional due to their radial symmetry, in addition to their inefficient power output coupling. Here we report the design of elliptical resonators with a wavelength-size notch at the boundary, which support in-plane highly unidirectional laser emission from WGMs. The notch acts as a small scatterer such that the Q factor of the WGMs is still very high. Using midinfrared (λ ∼ 10 µm) injection quantum cascade lasers as a model system, an in-plane beam divergence as small as 6 deg with a peak optical power of ∼5 mW at room temperature has been demonstrated. The beam divergence is insensitive to the pumping current and to the notch geometry, demonstrating the robustness of this resonator design. The latter is scalable to the visible and the near infrared, thus opening the door to very low-threshold, highly unidirectional microcavity diode lasers.

Local chirality of optical resonances in ultrasmall resonators.

In wavelength-scale cavities with chiral-symmetric geometry, wave optical effects can introduce local chirality, that is, a spatial separation of the clockwise and counterclockwise propagating resonant modes. We show that this local chirality results in unidirectional lasing emission in the far field. In the presence of a waveguide, the local chirality also allows for directional evanescent coupling of the lasing modes, and the output direction can be varied by selecting the coupling position along the cavity boundary. Our results demonstrate that the local chirality of optical resonances can be utilized to control the output directionality and enhance the collection efficiency of emission from ultrasmall resonators.

Morphology of wetting-layer states in a simple quantum-dot wetting-layer model.

The excitation of semiconductor quantum dots often involves an attached wetting layer with delocalized single-particle energy eigenstates. These wetting-layer states are usually approximated by (orthogonalized) plane waves. We show that this approach is too crude. Even for a simple model based on the effective-mass approximation and containing one or a few lens-shaped quantum dots on a rectangular wetting layer, the wetting-layer states typically show a substantially irregular and complex morphology. To quantify this complexity we use concepts from the field of quantum chaos such as spectral analysis of energy levels, amplitude distributions, and localization of energy eigenstates.

Marginally unstable periodic orbits in semiclassical mushroom billiards.

Optical mushroom-shaped billiards offer a unique opportunity to isolate and study semiclassical modes concentrated on nondispersive, marginally unstable periodic orbits. Here we show that the openness of the cavity to external electromagnetic fields leads to unanticipated consequences for the far-field radiation pattern, including directional emission. This is mediated by interactions of marginally unstable periodic orbits with chaotic modes. We also show that the semiclassical modes are robust against perturbations to the shape of the cavity, despite the lack of structural stability of the corresponding classical orbits.

Reciprocal transmissions and asymmetric modal distributions in waveguide-coupled spiral-shaped microdisk resonators: comment.

In a recent article [Opt. Express 15, 14650 (2007)] Lee et al. claimed that optical modes in spiral-shaped microcavities come in pairs of clockwise and counterclockwise traveling-wave modes having the same frequencies and Q-factors but different modal distributions. In this comment, we show that the opposite is true: the modes are in general nondegenerate in terms of frequencies and Q-factors and the modal distributions are similar.

Fractal Weyl law for chaotic microcavities: Fresnel's laws imply multifractal scattering.

We demonstrate that the harmonic inversion technique is a powerful tool to analyze the spectral properties of optical microcavities. As an interesting example we study the statistical properties of complex frequencies of the fully chaotic microstadium. We show that the conjectured fractal Weyl law for open chaotic systems [Lu, Phys. Rev. Lett. 91, 154101 (2003)] is valid for dielectric microcavities only if the concept of the chaotic repeller is extended to a multifractal by incorporating Fresnel's laws.

Goos-Hänchen shift and localization of optical modes in deformed microcavities.

Recently, an interesting phenomenon of spatial localization of optical modes along periodic ray trajectories near avoided resonance crossings has been observed [Wiersig, Phys. Rev. Lett. 97, 253901 (2006)]. For the case of a microdisk cavity with elliptical cross section, we use the Husimi function to analyze this localization in phase space. Moreover, we present a semiclassical explanation of this phenomenon in terms of the Goos-Hänchen shift, which works very well even deep in the wave regime. This semiclassical correction to the ray dynamics modifies the phase-space structure such that modes can localize either on stable islands or along unstable periodic ray trajectories.

Spontaneous T-symmetry breaking and exceptional points in cavity quantum electrodynamics systems.

Spontaneous symmetry breaking has revolutionized the understanding in numerous fields of modern physics. Here, we theoretically demonstrate the spontaneous time-reversal symmetry breaking in a cavity quantum electrodynamics system in which an atomic ensemble interacts coherently with a single resonant cavity mode. The interacting system can be effectively described by two coupled oscillators with positive and negative mass, when the two-level atoms are prepared in their excited states. The occurrence of symmetry breaking is controlled by the atomic detuning and the coupling to the cavity mode, which naturally divides the parameter space into the symmetry broken and symmetry unbroken phases. The two phases are separated by a spectral singularity, a so-called exceptional point, where the eigenstates of the Hamiltonian coalesce. When encircling the singularity in the parameter space, the quasi-adiabatic dynamics shows chiral mode switching which enables topological manipulation of quantum states.

Leaking billiards.

Billiards are idealizations for systems where particles or waves are confined to cavities, or to other homogeneous regions. In billiard systems a point particle moves freely except for specular reflections from rigid walls. However, billiard walls are not always completely reflective and measurements inside can also open the billiard. Since boundary openings have been studied extensively in the literature, we rather model leakages inside the billiard. In particular, we investigate the classical dynamics of a leakage for a continuous family of billiard systems, that is, the stadium-lemon-billiard family. With a single parameter the geometry of the billiard can be tuned from stadium (being fully hyperbolic) over circle (integrable) to the lemon-shaped billiard (mixed chaotic). For the stadium billiard we found an algebraically decaying mean escape time with the linear size of the leakage n(esc) approximately epsilon-1 together with an exponential decay of the survival probability distribution. The finding is nearly independent of the position and size of the leakage, as long as the leakage is much smaller than the system size, and it is in good agreement with a stochastic map approximation of the dynamics. Due to the mixed phase space for lemon billiards, the mean escape time depends both on the position and geometry of the leakage. For systems where quasiregular motion dominates, we found a linear dependence of the mean escape time, n(esc) approximately 1-epsilon, which we refer to as flooding law. Our findings are helpful in understanding dynamics of leaking Hamiltonian systems.

Chaos-assisted broadband momentum transformation in optical microresonators.

The law of momentum conservation rules out many desired processes in optical microresonators. We report broadband momentum transformations of light in asymmetric whispering gallery microresonators. Assisted by chaotic motions, broadband light can travel between optical modes with different angular momenta within a few picoseconds. Efficient coupling from visible to near-infrared bands is demonstrated between a nanowaveguide and whispering gallery modes with quality factors exceeding 10 million. The broadband momentum transformation enhances the device conversion efficiency of the third-harmonic generation by greater than three orders of magnitude over the conventional evanescent-wave coupling. The observed broadband and fast momentum transformation could promote applications such as multicolor lasers, broadband memories, and multiwavelength optical networks.