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.

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.

Supermodes in multiple coupled photonic crystal waveguides.

We analyze the supermodes in multiple coupled photonic crystal waveguides for long-wavelengths. In the tight-binding limit we obtain analytic results that agree with fully numerical calculations. We find that when the field flips sign after a single photonic crystal period, and there is an odd number of periods between adjacent waveguides, the supermode order is reversed, compared to that in conventional coupled waveguides, generalizing earlier results obtained for two coupled waveguides.

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.

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.

Modes of coupled photonic crystal waveguides.

We consider the modes of coupled photonic crystal waveguides. We find that the fundamental modes of these structures can be either even or odd, in contrast with the behavior in coupled conventional waveguides, in which the fundamental mode is always even. We explain this finding using an asymptotic model that is valid for long wavelengths.

Density of states functions for photonic crystals.

We discuss density of states functions for photonic crystals, in the context of the two-dimensional problem for arrays of cylinders of arbitrary cross section. We introduce the mutual density of states (MDOS), and show that this function can be used to calculate both the local density of states (LDOS), which gives position information for emission of radiation from photonic crystals, and the spectral density of states (SDOS), which gives angular information. We establish the connection between MDOS, LDOS, SDOS and the conventional density of states, which depends only on frequency. We relate all four functions to the band structure and propagating states within the crystal, and give numerical examples of the relation between band structure and density of states functions.

Ultracompact resonant filters in photonic crystals.

A novel design for an ultracompact, high-Q notch-rejection filter is presented, and an analytic expression for the transmission properties is derived. This folded directional coupler shares the properties of a Fabry-Perot resonator and a directional coupler. We compare and contrast the device to high-Q Fabry-Perot cavities in photonic crystal waveguides.

Effects of disorder in two-dimensional photonic crystal waveguides.

The effects of randomness on the guiding properties of waveguides embedded in disordered two-dimensional photonic crystals composed of a finite cluster of circular cylinders of infinite length are investigated for TM-polarized radiation. Different degrees of disorder in the radius, filling fraction, refractive index, and position are considered for both straight and 90 degrees bent guides. The crystals exhibit similar sensitivity to refractive index and radius disorder, with a degree of disorder from 15%-20% yielding little substantial change in the guiding properties. A smaller range of position disorder is also considered. For strong disorder in radius and refractive index, the guide effectively closes. These results were obtained by a Monte Carlo simulation method, and the performance of this method is analyzed. The method requires at least ten realizations in some cases for convergence to commence; substantially more realizations are required for moderate and strong disorder to achieve accurate results.

Cylinder gratings in conical incidence with applications to woodpile structures.

We use our previous formulation for cylinder gratings in conical incidence to discuss the photonic band gap properties of woodpile structures. We study scattering matrices and Bloch modes of the woodpile, and use these to investigate the dependence of the optical properties on the number of layers. We give data on reflectance, transmittance and absorptance of metallic woodpiles as a function of wavelength and number of layers, using both the measured optical constants of tungsten and using a perfect conductivity idealization to characterize the metal. For semi-infinite metallic woodpiles, we show that polarization of the incident field is important, highlighting the role played by surface effects as opposed to lattice effects.

Semianalytic treatment for propagation in finite photonic crystal waveguides.

We present a semianalytic theory for the properties of two-dimensional photonic crystal waveguides of finite length. For single-mode guides, the transmission spectrum and field intensity can be accurately described by a simple two-parameter model. Analogies are drawn with Fabry-Perot interferometers, and generalized Fresnel coefficients for the interfaces are calculated.