Research: Modeling in Micro- and Nano-photonics
This research is concerned in part with nonlinear dynamics in light confining – closed and open – systems, a set of themes that we loosely classify as cavity-based photonics, and more generally on theoretical and numerical modeling in photonics.
Cavity-based photonics: the fundamentals
To understand the ray-wave dynamics of closed and open planar electromagnetic resonators.
From the point of view of dynamics, in the short wavelength limit (geometrical or classical), we refer to closed systems as billiards, and to open systems as dielectric (or Fresnel) billiards. The corresponding ray-dynamics can be regular (e.g. for a circle or an ellipse), chaotic (e.g. for a stadium) or mixed (e.g. for the quadrupole, the annular disk) as portrayed in the phase space of positions of impact and angles of incidence on the outer-boundary. In the long wavelength limit (wave) on the other hand, we generically speak of optical cavities whether the boundary conditions are of the Dirichlet-type (closed systems) or of the dissipative-type (open systems). The wave dynamics is obtained from the appropriate solutions of Maxwell equations or more often from the associated Helmholtz equation.
Our fundamental interests lie in the characterisation of classical chaos and its counterpart - wave chaos, in the clarification of the ray-wave dualism (the optical version of classical-quantum dualism), and in a deeper understanding of the mesoscopic region where semi-classical physics reigns. A wealth of fascinating phenomena has emerged over the years as the optical analogue of existing processes in other fields (e.g. dynamical and chaos-assisted tunnelling, dynamical localisation); this makes these studies of fundamental importance and of ever challenging research. What happens when the objects considered approach the wavelength scale, from microscopic to nanoscopic size, is also a source of ongoing questioning.
Cavity-based photonics: microlasers without mirrors, sensors without labels
To enhance, confine, and channel light in optical structures for lasing and sensing applications.
Confinement and manipulation of light in optical microcavities has been a subject of intense fundamental and applied research for more than a decade [Vahala04]. Recent advances in micro- and nano-technology and in the field of optoelectronics have indeed allowed the realization of small micron size cavities capable, among other things, of low-threshold coherent light emission [Spillane02]. In fact, dielectric optical microcavities (for recent reviews, see [Vahala04, Astratov07]) are already used in various applications such as filters [Bergeron08], single-molecule sensors [Armani07, Vollmer08], tunable optical frequency combs [Savchenkov08], and optomechanical oscillators [Kippenberg07] to name a few.
The operating principle of these small resonators is based on the concept of planar dielectric cavities, owing their interest to the high-quality factor Q of the supported modes. In this type of cavity, light is trapped within the cavity by total internal reflection (TIR) as showed in Fig. 1b. In analogy to acoustic modes, the modes with the highest Q of these two dimensional dielectric resonators are of the whispering gallery (WG) type, confined in the vicinity of the cavity boundary (left panel of Fig. 1b) and candidate for an easy coupling to the outside world.
The resonance spectrum, the free spectral range (FSR), the potentially high quality factor Q (proportional to the photon residence time inside the cavity), the (ultra-small) modal volume Vm and the spatial distribution of the electromagnetic field E (x,y,z) are the principal characteristics of these resonators. In turn, these properties are highly sensitive to the medium composing the cavity, to its geometry and to its immediate surroundings. It is precisely this sensitivity that makes these resonators such versatile candidates for a number of applications as noted above
Depending on the symmetry of the cavity, two classes of WG resonators exist. The first class, having rotational or axial symmetry, is characterized by low radiation loss, therefore high quality factor, and isotropic emission patterns. To this category belong for instance the geometry of the sphere [Chiasera10], the cylinder [Kippenberg03] and the torus [Armani03]. It is the very long circulation time of the light inside the cavity that allows for superlative performances for applications ranging from sensors, to cavity quantum electrodynamics and multifunction optical circuit devices [Malekin06].
However, despite these impressive results, the highly symmetrical cavities present a serious disadvantage in terms of emission properties. The rotational symmetry imposes an isotropic field distribution, a major drawback for their use as microlasers, single-photon sources or efficient interacting devices with other optical components. To overcome this problem, the concept of two-dimensional asymmetric resonant cavities (ARCs) has been proposed [Nöckel97, for an early description]. This second class of resonators has a broken rotational symmetry induced by a smooth deformation of the circular cavity leading to directional emission modes. This has led to the fabrication of new types of microlasers supporting specific modes with very interesting properties – ultralow excitation threshold, high power output and emission in well defined directions [Gmachl98, for the first experimental realization] and [Harayama11, for a recent review].
Ever since the demonstration of a directional emission of coherent light from microcavities, the Grail of this line of research has been the production and control of the laser directionality through various combinations of geometry and medium optimization. Our recent efforts (theory and experiment, see below) have been directed towards the development of a new directional microlaser operating configuration based on the coupling between an annular cavity microlaser and an optical wave guide.
Among the vast possibilities of geometry/medium for
microresonators, our choice has fallen on a special class of
cavities called annular cavities [Hentschel02]. Although,
different geometries can lead to a directional character of the
optical modes, and even unidirectional emission ([Kneissl04],
spriral-shaped cavity; [Shang09], peanut-shaped cavity; [Song09],
limaçon of Pascal; [Wang11], notched-ellipse), they are
usually associated with a strong degradation of the cavity quality
factor Q compared to the circular case.
This degradation is unfortunately generic to most other geometries.
One of the few systems that offer a solution - directionality
without Q spoiling - is precisely the annular cavity.
In this system we break the rotational symmetry by introducing an off-center defect (a circular inclusion, a noble metal nanoparticle or a quantum dot) inside the circular cavity. This kind of cavity supports non-directional long lived modes (high Q), similar to those of a circular one and confined close to the boundary, and directional short lived modes (low Q) that can be affected by a step index perturbation of the cavity medium. Our recent theoretical studies (that we have coined phase space engineering) [Painchaud-Poirier10] have revealed 3 regimes of operation displayed in Figs 3a-c as the inclusion radius and cavity-hole centers distance are varied. The hybridized modes can therefore, under appropriate circumstances, display strong directionality and high Q (Fig. 3b). Even more, we have already demonstrated that one can induce, predict and control the FF and the NF emission.
For optimal delivery of the laser light, the evanescent coupling with a waveguide addresses the NF intensity. Concentrating this field in the vicinity of the waveguide for maximal output efficiency, while keeping a large enough Q factor, means to operate between regimes 2 (Fig. 3b) and 3 (Fig. 3c). The experimental verification of these expectations will be in itself a major advance in our understanding of microlasers.
One of the advantages of these studies is that the same system can be the basis of two independent applications: whereas shaping the FF turns the microcavity into a directional microlaser,exploiting the NF can make it a highly sensitive bio- or chemical-sensor. With this in mind, we have embarked in an ambitious combined theory-experiment research for the principal goal of developing optical microcavity array sensors for an integrated, multiplexed, label-free, sensitive detection of viral/bacterial agents. In this respect, it is noteworthy that our present configuration allows for plasmonic enhancement of the NF by judicious placement of a metallic nanoparticle on the surface of the cavity to create hot-spots where the field intensity is concentrated, the net effect being an enhanced sensitivity of detection of selected targets (virus, bacteria).
Modeling in photonics: optimization of photonics structures
The design of on-chip optical elements is paramount to many applications such as micro-manipulation, sensing and optical communications. Moreover, engineering such planar micro-structures with great precision is now possible due to advances in photo-lithographic techniques. Prior to the experimental step, it is often necessary to model the interplay of light with in-plane scattering elements in order to harness the full potential of integrated optical devices. Our group has been interested lately in the design of integrated devices based on cylindrical scatterers, as shown on Fig. 4. Since our goal is to enable new device functionalities such as beam shaping and lensing, a large number of adjustable geometric parameters have to be considered, for instance the position of scatterers on a given cylinder lattice. This means a large number of lattice configurations must be tested before converging to acceptable design. Therefore, the speed of the algorithms used to solve Maxwell's equations is critical. Keeping this speed requirement in mind, we advocate the use of algorithms exploiting the circular symmetry of scatterers, loosely called 2D-GLMT (two-dimensional generalized Lorenz-Mie theories).
Using our implementation of 2D-GLMT, we have proposed integrated optical elements tailored for the conversion of a Gaussian beam into Hermite-Gauss profiles (see Fig. 5). These devices are designed without making any a priori assumptions on the scatterer arrangement, and exhibit reasonable backscattering losses and good tolerance to variations in the input beam parameters, refractive index and operating wavelength. This inverse design problem has led us to embrace metaheuristics, optimization algorithms based on empirical rules for exploring solution spaces. More specifically, we have developed expertise in the use of genetic [Gagnon12] and tabu search algorithms [Gagnon13].
Our group has developed (and is continuing development of) versatile numerical algorithms for computing resonances of closed and open dielectric resonators/cavities. The emphasis is on the generality of the system's configuration, i.e. the geometry of the (main) cavity (and possible inclusions) and the internal and external dielectric media (homogeneous and inhomogeneous). Two complementary techniques have now reached maturity for 2D resonators, and are currently being extended to photonic molecules (combination of a few cavities), and more generally, to photonic complexes (periodic or aperiodic arrangements of cavities).
The first method is applicable to cavities of arbitrary shape and arbitrary inhomogeneities of the medium (continuous or discontinuous) and is based on a scattering formalism to obtain the position and width of the (quasi)-eigenmodes. The calculated S-matrix contains all the relevant information of the corresponding scattering experiment, and the electromagnetic near- and far-fields are readily extracted (see for instance Fig 3).
The second method is a generalization to multiply connected regions of the boundary integral equation (BIE), implemented as a boundary element method (BEM) with discretisation of the different interfaces. It was originally designed with the annular cavity in mind, but is more general in that the shapes of the connected regions can be arbitrary. However, within each region, the refractive index is presently limited to be a constant.
Full finite-element algorithms are also available to model 3D resonators. At present, tests are underway to validate the methods for geometries of interest: torus, ring, (thick) disk, and sphere. We are also working on semi-analytic and perturbative methods to complete our 3D arsenal.
Theoretical & Experimental Collaborations
The reality check is provided by collaborations with 2 experimental groups: the Microphotonics Laboratory of Prof. Y.-A. Peter (Ecole Polytechnique de Montréal) and the Biomedical Engineering in Advanced Applications - Quantum, Oscillatory, and Nanotechnological Systems – (BEAAQONS) of Prof. J. L. Nadeau (McGill University). Some of the related activities are funded (or have been funded) under the following specific projects:
A new class of optical microcavities – inhomogeneous dielectric resonators –
optical microcavities, wave and classical chaos, microsystems,
optical communications, optical sensors, nanotechnology
Label free high sensitivity detection of bacteria by phages using functionalized optical microcavities
biotechnology, microsystems, lab on chip, optical microcavity,
nanotechnology, bacteria, label free detection, high quality factor
In addition, a further in situ collaboration is developing with the Chair of Excellence for Innovations in Photonics of Prof. Y. Messaddeq.Funding for this research is provided in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada and the Fonds de Recherche du Québec - Nature et Technologies (FRQ-NT).