Michael A. Byrne

Modes of random lasers

In conventional lasers, the optical cavity that confines the photons also determines essential characteristics of the lasing modes such as wavelength, emission pattern, directivity, and polarization. In random lasers, which do not have mirrors or a well-defined cavity, light is confined within the gain medium by means of multiple scattering. The sharp peaks in the emission spectra of semiconductor powders, first observed in 1999, has therefore lead to an intense debate about the nature of the lasing modes in these so-called lasers with resonant feedback. We review numerical and theoretical studies aimed at clarifying the nature of the lasing modes in disordered scattering systems with gain. The past decade has witnessed the emergence of the idea that even the low-Q resonances of such open systems could play a role similar to the cavity modes of a conventional laser and produce sharp lasing peaks. We focus here on the near-threshold single-mode lasing regime where nonlinear effects associated with gain saturation and mode competition can be neglected. We discuss in particular the link between random laser modes near threshold and the resonances or quasi-bound (QB) states of the passive system without gain. For random lasers in the localized (strong scattering) regime, QB states and threshold lasing modes were found to be nearly identical within the scattering medium. These studies were later extended to the case of more lossy systems such as random systems in the diffusive regime, where it was observed that increasing the openness of such systems eventually resulted in measurable and increasing differences between quasi-bound states and lasing modes. Very recently, a theory able to treat lasers with arbitrarily complex and open cavities such as random lasers established that the threshold lasing modes are in fact distinct from QB states of the passive system and are better described in terms of a new class of states, the so-called constant-flux states. The correspondence between QB states and lasing modes is found to improve in the strong scattering limit, confirming the validity of initial work in the strong scattering limit.

Finite element computation of grating scattering matrices and application to photonic crystal band calculations

We consider the calculation of the band structure and Bloch mode basis of two-dimensional photonic crystals, modelled as stacks of one-dimensional diffraction gratings. The scattering properties of each grating are calculated using an efficient finite element method (FEM) and allow the complete mode structure to be derived from a transfer matrix method. A range of numerical examples showing the accuracy, flexibility and utility of the method is presented.

Modal formulation for diffraction by absorbing photonic crystal slabs

A finite element-based modal formulation of diffraction of a plane wave by an absorbing photonic crystal slab of arbitrary geometry is developed for photovoltaic applications. The semianalytic approach allows efficient and accurate calculation of the absorption of an array with a complex unit cell. This approach gives direct physical insight into the absorption mechanism in such structures, which can be used to enhance the absorption. The verification and validation of this approach is applied to a silicon nanowire array, and the efficiency and accuracy of the method is demonstrated. The method is ideally suited to studying the manner in which spectral properties (e.g., absorption) vary with the thickness of the array, and we demonstrate this with efficient calculations that can identify an optimal geometry.

Effects of polarization on the transmission and localization of classical waves in weakly scattering metamaterials

We summarize the results of our comprehensive analytical and numerical studies of the effects of polarization on the Anderson localization of classical waves in one-dimensional random stacks. We consider homogeneous stacks composed entirely of normal materials or metamaterials, and also mixed stacks composed of alternating layers of a normal material and a metamaterial. We extend the theoretical study developed earlier for the case of normal incidence A. A. Asatryan et al., Phys. Rev. B 81, 075124 2010 to the case of off-axis incidence. For the general case where both the refractive indices and layer thicknesses are random, we obtain the long-wave and short-wave asymptotics of the localization length over a wide range of incidence angles including the Brewster “anomaly” angle. At the Brewster angle, we show that the long-wave localization length is proportional to the square of the wavelength, as for the case of normal incidence, but with a proportionality coefficient substantially larger than that for normal incidence. In mixed stacks with only refractive-index disorder, we demonstrate that p-polarized waves are strongly localized, while for s polarization the localization is substantially suppressed, as in the case of normal incidence. In the case of only thickness disorder, we study also the transition from localization to delocalization at the Brewster angle.

Fano resonances of photonic crystal slabs

A Bloch mode theory for diffraction of plane waves by planar PC slabs is developed. The theory provides physical insight into the origin of Fano resonances, allowing a simple pole model to be deduced rigorously.

Plane-wave scattering by a photonic crystal slab: Multipole modal formulation and accuracy

The optical properties of photonic crystal slabs are often simulated with general three-dimensional methods such as finite-difference time-domain. Here we develop a multipole modal method, which is specialized to exploit two symmetries of the photonic crystal slab: the slabs vertically invariant nature allows the field to be expressed in Bloch modes, while the cylindrical inclusions allow the Bloch modes themselves to be expressed in the multipole basis. We find the multipole method to be fast and efficient in finding the Bloch modes, with convergence approaching the numerical accuracy possible. By matching the Bloch modes to plane waves at the top and bottom interfaces of the photonic crystal, the field scattered by the slab is calculated. Values of transmittance and reflectance accurate to 2–3 digits are easily and quickly achieved, whereas 5–6 digits are possible with greater numbers of modes and plane waves in field expansions. Higher accuracy is limited by Gibbs-related phenomena arising from the matching at the interface of necessarily discontinuous Bloch modes to necessarily continuous plane waves. We believe this limit may be present in all modal methods that use Bloch modes to expand the field within the photonic crystal.

Novel modeling techniques for photonic devices

The combination of purely numerical methods, such as the finite element method, with an analytical treatment can lead to a powerful semi-analytical technique. We present such a technique, which combines the finite element method with a modal approach, with a focus on the modeling three-dimensional photonic structures.

Dispersion effects on the Anderson localization in disordered one dimensional metamaterial stacks

We have carried out a comprehensive study of dispersion and absorption effects on Anderson localization in one-dimensional metamaterial stacks and have shown that the field is delocalized in μ or ∊-near-zero media at normal incidence.

Threshold lasing modes of a random laser: from the localised to the ballistic regime

We use the rigorous multipole method to calculate lasing and quasi-bound states from localised to diffusive regimes and show these coincide in the former regime but differ in the latter.

Polarisation effects on Anderson localisation in the presence of metamaterials

We have derived an elegant and effective equation to characterise the Anderson localisation length in one-dimensional systems that may contain metamaterials and have undertaken a comprehensive study, uncovering a number of striking polarisation effects.