Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders

Plane wave scattering from a plasmonic nanowire-film system with the inclusion of non-local effects

In this paper we present a theoretical analysis of the electromagnetic response of a plasmonic nanowire-film system. The analytical solution accounts for both the dispersive as well as non-local nature of the plasmonic media. The physical structure comprises of a plasmonic nanowire made of a plasmonic metal such as gold or silver placed over a plasmonic film of the same material. Such a nanostructure exhibits a spectrum that is extremely sensitive to various geometric parameters such as spacer thickness and nanowire radius, which makes it favorable for various sensing applications. The non-locality of the plasmonic medium, which can be captured using the hydrodynamic model, significantly affects the resonant wavelength of this system for structures of small dimensions (~ less than 5 nm gap between the nanowire and the film). We present an analytical method that can be used to predict the effect of non-locality on the resonances of the system. To validate the analytical method, we also report a comparison of our analytical solution with a numerical Finite Difference Time Domain analysis (FDTD) of the same structure with the plasmonic medium being treated as local in nature.

Backscatter analysis based algorithms for increasing transmission through highly scattering random media using phase-only-modulated wavefronts

Recent theoretical and experimental advances have shed light on the existence of so-called "perfectly transmitting" wavefronts with transmission coefficients close to 1 in strongly backscattering random media. These perfectly transmitting eigen-wavefronts can be synthesized by spatial amplitude and phase modulation. Here, we consider the problem of transmission enhancement using phase-only-modulated wavefronts. Motivated by biomedical applications, in which it is not possible to measure the transmitted fields, we develop physically realizable iterative and non-iterative algorithms for increasing the transmission through such random media using backscatter analysis. We theoretically show that, despite the phase-only modulation constraint, the non-iterative algorithms will achieve at least about 25π%≈78.5% transmission with very high probability, assuming that there is at least one perfectly transmitting eigen-wavefront and that the singular vectors of the transmission matrix obey the maximum entropy principle such that they are isotropically random. We numerically analyze the limits of phase-only-modulated transmission in 2D with fully spectrally accurate simulators and provide rigorous numerical evidence confirming our theoretical prediction in random media, with periodic boundary conditions, that is composed of hundreds of thousands of non-absorbing scatterers. We show via numerical simulations that the iterative algorithms we have developed converge rapidly, yielding highly transmitting wavefronts while using relatively few measurements of the backscatter field. Specifically, the best performing iterative algorithm yields ≈70% transmission using just 15-20 measurements in the regime, where the non-iterative algorithms yield ≈78.5% transmission, but require measuring the entire modal reflection matrix. Our theoretical analysis and rigorous numerical results validate our prediction that phase-only modulation with a given number of spatial modes will yield higher transmission than amplitude and phase modulation with half as many modes.

Iterative, backscatter-analysis algorithms for increasing transmission and focusing light through highly scattering random media

Scattering hinders the passage of light through random media and consequently limits the usefulness of optical techniques for sensing and imaging. Thus, methods for increasing the transmission of light through such random media are of interest. Against this backdrop, recent theoretical and experimental advances have suggested the existence of a few highly transmitting eigen-wavefronts with transmission coefficients close to 1 in strongly backscattering random media. Here, we numerically analyze this phenomenon in 2D with fully spectrally accurate simulators and provide rigorous numerical evidence confirming the existence of these highly transmitting eigen-wavefronts in random media with periodic boundary conditions that are composed of hundreds of thousands of nonabsorbing scatterers. Motivated by bio-imaging applications in which it is not possible to measure the transmitted fields, we develop physically realizable algorithms for increasing the transmission through such random media using backscatter analysis. We show via numerical simulations that the algorithms converge rapidly, yielding a near-optimum wavefront in just a few iterations. We also develop an algorithm that combines the knowledge of these highly transmitting eigen-wavefronts obtained from backscatter analysis with intensity measurements at a point to produce a near-optimal focus with significantly fewer measurements than a method that does not utilize this information.

Analyzing diffraction gratings by a boundary integral equation Neumann-to-Dirichlet map method

For analyzing diffraction gratings, a new method is developed based on dividing one period of the grating into homogeneous subdomains and computing the Neumann-to-Dirichlet (NtD) maps for these subdomains by boundary integral equations. For a subdomain, the NtD operator maps the normal derivative of the wave field to the wave field on its boundary. The integral operators used in this method are simple to approximate, since they involve only the standard Green's function of the Helmholtz equation in homogeneous media. The method retains the advantages of existing boundary integral equation methods for diffraction gratings but avoids the quasi-periodic Green's functions that are expensive to evaluate.

Visual gamma oscillations: waves, correlations, and other phenomena, including comparison with experimental data

Mean-field theory of brain dynamics is applied to explain the properties of gamma (> or approximately 30 Hz) oscillations of cortical activity often seen during vision experiments. It is shown that mm-scale patchy connections in the primary visual cortex can support collective gamma oscillations with the correct frequencies and spatial structure, even when driven by uncorrelated inputs. This occurs via resonances associated with the the periodic modulation of the network connections, rather than being due to single-cell properties alone. Near-resonant gamma waves are shown to obey the Schrödinger equation, which enables techniques and insights from quantum theory to be used in exploring these classical oscillations. Resulting predictions for gamma responses to stimuli account in a unified way for a wide range of experimental results, including why oscillations and zero-lag synchrony are associated, and variations in correlation functions with time delay, intercellular distance, and stimulus features. They also imply that gamma oscillations may enable a form of frequency multiplexing of neural signals. Most importantly, it is shown that correlations reproduce experimental results that show maximal correlations between cells that respond to related features, but little correlation with other cells, an effect that has been argued to be associated with segmentation of a scene into separate objects. Consistency with infill of missing contours and increase in response with length of bar-shaped stimuli are discussed. Background correlations expected in the absence of stimulation are also calculated and shown to be consistent in form with experimental measurements and similar to stimulus-induced correlations in structure. Finally, possible links of gamma instabilities to certain classes of photically induced seizures and visual hallucinations are discussed.

Multipole Dirichlet-to-Neumann map method for photonic crystals with complex unit cells

The periodicity of photonic crystals can be utilized to develop efficient numerical methods for analyzing light waves propagating in these structures. The Dirichlet-to-Neumann (DtN) operator of a unit cell maps the wave field on the boundary of the unit cell to its normal derivative, and it can be used to reduce the computation to the edges of the unit cells. For two-dimensional photonic crystals with complex unit cells, each containing a number of possibly different circular cylinders, we develop an efficient multipole method for constructing the DtN maps. The DtN maps are used to calculate the transmission and reflection spectra for finite photonic crystals with complex unit cells.

Three-color photonic crystal demultiplexer based on ultralow-refractive-index metamaterial technology

We numerically demonstrate the operation of a novel class of wavelength-division demultiplexing circuit based on photonic crystal waveguides that are entirely synthesized by ultralow-refractive-index metallic nanopillars. The operational principle of the newly proposed device is based on the phenomenon of total external reflections in ultralow-refractive-index metallic photonic crystal structures (metamaterials). In addition we provide detailed design guidelines for optimum device performance. The low propagation losses and compact size, as well as temperature-insensitive operation over a wide temperature range, are only a few of the advantages of the proposed technology, making this new type of demultiplexer an excellent candidate for applications in the visible spectrum.

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.

Cylinder gratings in conical incidence with applications to modes of air-cored photonic crystal fibers

We develop a formulation for cylinder gratings in conical incidence, using a multipole method. The theory, and its numerical implementation, is applied to two-dimensional photonic crystals consisting of a stack of one-dimensional gratings, each characterized by its plane wave scattering matrix. These matrices are used in combination with Bloch's theorem to determine the band structure of the photonic crystal from the solution of an eigenvalue problem. We show that the theory is well adapted to the difficult task of locating the complete band gaps needed to support air-guided modes in microstructured optical fibers, that is, optical fibers in which the confinement of light in a central air hole is achieved by photonic band-gap effects in a periodic cladding comprising a lattice of air holes in a glass matrix.

From scattering or impedance matrices to Bloch modes of photonic crystals

The dispersion relation of Bloch waves is derived from the properties of a single grating layer. A straightforward way to impose the Bloch condition leads to the calculation of the eigenvalues of the transfer matrix through the single grating layer. Unfortunately, the transfer-matrix algorithm is known to be unstable as a result of the growing evanescent waves. This problem appears again in the calculation of the eigenvalues, making unusable the transfer matrix in numerous practical problems. We propose two different algorithms to circumvent this problem. The first one takes advantage of scattering matrices, while the second one takes advantage of impedance matrices. Numerical evidence of the efficiency of the algorithms is given. Dispersion diagrams of simple cubic and woodpile photonic crystals are obtained by using, respectively, the scattering and impedance matrices.

Numerical computation of the scattering matrix of an electromagnetic resonator

A method is presented to investigate diffraction of an electromagnetic plane wave by an infinitely thin infinitely conducting circular cylinder with longitudinal slots. It is based on the use of the combined boundary conditions method that consists of expressing the continuity of the tangential components of both the electric and the magnetic fields in a single equation. This method proves to be very efficient for this kind of problem and leads to fast numerical codes. The scattering matrix that is obtained from this theory can then be used in a multiscattering method to study wave propagation in square arrays of such resonators with an emphasis on the low-frequency behavior.

Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method

We present a formulation for wave propagation and scattering through stacked gratings comprising metallic and dielectric cylinders. By modeling a photonic crystal as a grating stack of this type, we thus formulate an efficient and accurate method for photonic crystal calculations that allows us to calculate reflection and transmission matrices. The stack may contain an arbitrary number of gratings, provided that each has a common period. The formulation uses a Green's function approach based on lattice sums to obtain the scattering matrices of each layer, and it couples these layers through recurrence relations. In a companion paper [J. Opt Soc. Am. A 17, 2177 (2000)] we discuss the numerical implementation of the method and give a comprehensive treatment of its conservation properties.

Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation

A numerical implementation and generalized conservation properties of a formulation for calculating wave propagation through stacked gratings comprising metallic and dielectric cylinders are presented. The basic formulation of the method was given in a companion paper [J. Opt. Soc. Am. A. 17, 2165 (2000)]. Here, details of the numerical implementation of the method are discussed and are illustrated for the ensemble average of a strongly scattering structure with refractive index and radius disorder. Also presented are a comprehensive treatment of energy conservation and generalized phase relations, as well as reciprocity.

Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals

The effects of disorder in the geometry and refractive index on the transmittance of two-dimensional photonic crystals composed of dielectric circular cylinders are considered, including randomness of radii, positions of the cylinder centers, and thickness of each layer of the photonic crystal. The effects of combinations of different types of strong disorder are also considered. The localization and homogenization properties of disordered photonic crystals are investigated. Analytical expressions for the two-dimensional localization length in the form of integrals are presented for both polarizations. It is shown numerically that the slope of the exponential divergence of the localization length in two dimensions is proportional to the inverse of the square of randomness for strong disorder and proportional to the inverse of the randomness for weak disorder. The effective dielectric constants for both polarizations in the case of strong disorder are also found. The transition from localization to homogenization is discussed and the terms responsible for this transition are identified and investigated.