'Parabolic' trapped modes and steered Dirac cones in platonic crystals.

This paper discusses the properties of flexural waves governed by the biharmonic operator, and propagating in a thin plate pinned at doubly periodic sets of points. The emphases are on the design of dispersion surfaces having the Dirac cone topology, and on the related topic of trapped modes in plates for a finite set (cluster) of pinned points. The Dirac cone topologies we exhibit have at least two cones touching at a point in the reciprocal lattice, augmented by another band passing through the point. We show that these Dirac cones can be steered along symmetry lines in the Brillouin zone by varying the aspect ratio of rectangular lattices of pins, and that, as the cones are moved, the involved band surfaces tilt. We link Dirac points with a parabolic profile in their neighbourhood, and the characteristic of this parabolic profile decides the direction of propagation of the trapped mode in finite clusters.

Making waves round a structured cloak: lattices, negative refraction and fringes.

Using the framework of transformation optics, this paper presents a detailed analysis of a non-singular square cloak for acoustic, out-of-plane shear elastic and electromagnetic waves. Analysis of wave propagation through the cloak is presented and accompanied by numerical illustrations. The efficacy of the regularized cloak is demonstrated and an objective numerical measure of the quality of the cloaking effect is provided. It is demonstrated that the cloaking effect persists over a wide range of frequencies. As a demonstration of the effectiveness of the regularized cloak, a Young's double slit experiment is presented. The stability of the interference pattern is examined when a cloaked and uncloaked obstacle are successively placed in front of one of the apertures. This novel link with a well-known quantum mechanical experiment provides an additional method through which the quality of cloaks may be examined. In the second half of the paper, it is shown that an approximate cloak may be constructed using a discrete lattice structure. The efficiency of the approximate lattice cloak is analysed and a series of illustrative simulations presented. It is demonstrated that effective cloaking may be obtained by using a relatively simple lattice structure, particularly, in the low-frequency regime.

Blazing evanescent grating orders: a spectral approach to beating the Rayleigh limit.

We develop a way to enhance the amplitudes of the nonpropagating evanescent orders of resonant dielectric gratings. We use this blazing to design gratings with spectra tailored to generate steerable sub-Rayleigh field concentrations on a surface. We investigate the enhancement and customization of evanescent fields necessary to create a virtual and passive scanning probe with no moving parts. Spot size can be decreased 1 order of magnitude below the free-space Rayleigh limit.

Modes of shallow photonic crystal waveguides: semi-analytic treatment.

We investigate the formation of photonic crystal waveguide (PCW) modes within the framework of perturbation theory. We derive a differential equation governing the envelope of PCW modes constructed from weak perturbations using an effective mass formulation based on the Luttinger-Kohn method from solid-state physics. The solution of this equation gives the frequency of the mode and its field. The differential equation lends itself to simple analytic approximations which reduce the problem to that of solving slab waveguide modes. By using this model, we demonstrate that the nature of the projected band structure and corresponding Bloch functions are central to the behaviour of PCW modes. With this understanding, we explain why the odd mode in a hexagonal PCW spans the entire Brillouin zone while the even mode is cut off.

Efficient coupling into slow light photonic crystal waveguide without transition region: role of evanescent modes.

We show that efficient coupling between fast and slow photonic crystal waveguide modes is possible, provided that there exist strong evanescent modes to match the waveguide fields across the interface. Evanescent modes are required when the propagating modes have substantially different modal fields, which occurs, for example, when coupling an index-guided mode and a gap-guided mode.

Modal method for conical diffraction by slanted lamellar gratings.

Slanted lamellar gratings made of dielectric materials are considered, used in conical diffraction mounts. We extend the modal method for slanted lamellar gratings from classical to conical incidence, develop fully generalized Fresnel matrices, and derive energy conservation relations for these matrices. Using the method, we verified a uniaxial crystal model for slanted lamellar gratings in a homogenization regime, examined the effects of grating symmetry on the maximum reflectance of Fano resonances, and showed that slanted lamellar gratings support Fano resonances despite the homogenization of their other optical properties.

Efficient slow-light coupling in a photonic crystal waveguide without transition region.

We consider the coupling into a slow mode that appears near an inflection point in the band structure of a photonic crystal waveguide. Remarkably, the coupling into this slow mode, which has a group index ng>1000, can be essentially perfect without any transition region. We show that this efficient coupling occurs thanks to an evanescent mode in the slow medium, which has appreciable amplitude and helps satisfy the boundary conditions but does not transport any energy.

Total absorption of unpolarized light by crossed gratings.

We present both experimental and numerical data showing the absorption of unpolarized, normally incident light by a gold crossed grating having a shallow sinusoidal profile. We show furthermore that the total absorption of unpolarized light can be achieved for an angle of incidence of 30 degrees with a crossed grating having its period adjusted appropriately from the normal incidence case to preserve the plasmonic resonance responsible for the enhanced absorptance. We contrast the process for achieving high absorptance in the principal plane of incidence aligned with the grooves of one of the gratings, with that for the principal plane at 45 degrees to each grating.

Evidence of a mobility edge for photons in two dimensions.

A scaling analysis of conductance for photons in two dimensions is carried out and, contrary to widely held belief, we find strong evidence of a mobility edge. Such behavior is compatible with the existence of an Anderson transition for electronic systems under symplectic symmetry, and indeed we show that the transfer matrix in the photonic system we have modelled has such a symmetry. We verify single parameter scaling of the conductance and demonstrate the transition from the metallic phase to localization. Key parameters, including the critical disorder, the conductance, and the critical exponent of the localization length are calculated, and it is shown that the value of the critical exponent is similar to that for electronic systems with symplectic symmetry.

Gap-edge Asymptotics of defect modes in 2D Photonic Crystals.

We consider defect modes created in complete gaps of 2D photonic crystals by perturbing the dielectric constant in some region. We study their evolution from a band edge with increasing perturbation using an asymptotic method that approximates the Green function by its dominant component which is associated with the bulk mode at the band edge. From this, we derive a simple exponential law which links the frequency difference between the defect mode and the band edge to the relative change in the electric energy. We present numerical results which demonstrate the accuracy of the exponential law, for TE and TM polarizations, hexagonal and square arrays, and in each of the first and second band gaps.

Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance.

Discrete systems of infinitely long polarizable line dipoles are considered in the quasistatic limit, interacting with a two-dimensional cloaking system consisting of a hollow plasmonic cylindrical shell. A numerical procedure is described for accurately calculating electromagnetic fields arising in the quasistatic limit, for the case when the relative permittivity of the cloaking shell has a very small imaginary part. Animations are given which illustrate cloaking of discrete systems, both for the case of induced dipoles and induced quadrupoles on the interacting particles. The simulations clarify the physical mechanism for the cloaking.

Wide-angle coupling into rod-type photonic crystals with ultralow reflectance.

We describe the surprising phenomenon of near-perfect coupling from free space into uniform two-dimensional rod-type photonic crystals over a wide range of incident angles. This behavior is shown to be a generic feature of many rod-type photonic crystal structures that is related to strong forward scattering resonances of the individual cylinders. We explain these results using both semianalytic analysis and two-dimensional numerical calculations and identify the conditions under which efficient, wide-angle coupling can occur. The results may lead to more efficient designs for in-band photonic crystal devices such as superprisms and self-collimation based photonic circuits.

Spontaneous emission suppression via quantum path interference in coupled microcavities.

We examine theoretically the spontaneous emission rate in optical microstructures with cavity resonances that overlap in both position and frequency. Using projection techniques, we show that the spontaneous emission in such structures can be accurately described by the direct emission and quantum path interference of emission into a few discrete resonant modes, even though the exact infinite-dimensional problem involves a coupling to the continuum of radiation states. Moreover, we obtain an efficient numerical time-domain method for determining the spontaneous emission rate that incorporates these effects, including the suppression of spontaneous emission into some modes.

Manipulation of spontaneous emission in a tapered photonic crystal fibre.

We characterize the spontaneous emission of dye that is introduced into the central core of a tapered photonic crystal fiber. Since the photonic crystal period in the fibre cladding varies along the taper, the transmission and spontaneous emission spectra over a wide range of relative frequencies can be observed. The spontaneous emission spectra of the fibre transverse to the fiber axis show suppression due to partial band-gaps of the structure, and also enhancement of spontaneous emission near the band edges. We associate these with van Hove features, as well as finite cluster size effects.

Two-dimensional treatment of the level shift and decay rate in photonic crystals.

We present a comprehensive treatment of the level shift and decay rate of a model line source in a two-dimensional photonic crystal (2D PC) composed of circular cylinders. The quantities in this strictly two-dimensional system are determined by the two-dimensional local density of states (2D LDOS), which we compute using Rayleigh-multipole methods. We extend the critical point analysis that is traditionally applied to the 2D DOS (or decay rate) to the level shift. With this, we unify the crucial quantity for experiment--the 2D LDOS in a finite PC--with the band structure and the 2D DOS, 2D LDOS, and level shift in infinite PC's. Consistent with critical point analysis, large variations in the level shift are associated with large variations in the 2D DOS (and 2D LDOS), corroborating a giant anomalous Lamb shift. The boundary of a finite 2D PC can produce resonances that cause the 2D LDOS in a finite 2D PC to differ markedly from the 2D LDOS in an infinite 2D PC.

Modeling of defect modes in photonic crystals using the fictitious source superposition method.

We present an exact theory for modeling defect modes in two-dimensional photonic crystals having an infinite cladding. The method is based on three key concepts, namely, the use of fictitious sources to modify response fields that allow defects to be introduced, the representation of the defect mode field as a superposition of solutions of quasiperiodic field problems, and the simplification of the two-dimensional superposition to a more efficient, one-dimensional average using Bloch mode methods. We demonstrate the accuracy and efficiency of the method, comparing results obtained using alternative techniques, and then concentrate on its strengths, particularly in handling difficult problems, such as where a mode is highly extended near cutoff, that cannot be dealt with in other ways.

Analytic properties of photonic crystal superprism parameters.

We study the analytic properties of the photonic crystal superprism resolution parameters p , q , and r introduced previously by Baba and Matsumoto [Appl. Phys. Lett. 81, 2325 (2002)], which characterize the potential dispersive power of a superprism. We find closed form expressions for these quantities that greatly simplify their accurate evaluation and reveal significant insights about their behavior. The expressions imply general properties of the parameters which are true for all bands and all photonic crystals. In particular, we demonstrate that all photonic crystals exhibit infinite resolution as measured by the parameter r along particular contours in any photonic band.

Conductance of photons in disordered photonic crystals.

The conductance of photons in two-dimensional disordered photonic crystals is calculated using an exact multipole-plane wave method that includes all multiple scattering processes. Conductance fluctuations, the universal nature of which has been established for electrons in the diffusive regime, are studied for photons, in both principal polarizations and for varying disorder. Our simulations show that universal conductance fluctuations can be observed in H(||) (TE) polarization for weak and intermediate disorder while, for E(||) (TM) polarization, we show that the conductance variance is essentially independent of sample size but strongly dependent on disorder. The probability distribution of the conductance is also calculated in the diffusive and localized regimes, and also at their transition, for which the distributions for both polarizations are seen to be very similar.

Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory.

We present a rigorous Bloch mode scattering matrix method for modeling two-dimensional photonic crystal structures and discuss the formal properties of the formulation. Reciprocity and energy conservation considerations lead to modal orthogonality relations and normalization, both of which are required for mode calculations in inhomogeneous media. Relations are derived for studying the propagation of Bloch modes through photonic crystal structures, and for the reflection and transmission of these modes at interfaces with other photonic crystal structures.

Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications.

The Bloch mode scattering matrix method is applied to several photonic crystal waveguide structures and devices, including waveguide dislocations, a Fabry-Pérot resonator, a folded directional coupler, and a Y-junction design. The method is an efficient tool for calculating the properties of extended photonic crystal (PC) devices, in particular when the device consists of a small number of distinct photonic crystal structures, or for long propagation lengths through uniform PC waveguides. The physical insight provided by the method is used to derive simple, semianalytic models that allow fast and efficient calculations of complex photonic crystal structures. We discuss the situations in which such simplifications can be made and provide examples.