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We present a novel approach for achieving tightly concentrated optical field by a hybrid photonic-plasmonic device in an integrated platform, which is a triangle-shaped metal taper mounted on top of a dielectric waveguide. This device, which we call a plasmomic light concentrator (PLC), can achieve side-coupling of light energy from the dielectric waveguide to the plasmonic region and light focusing into the apex of the metal taper(at the scale ~10nm) at the same time. For demonstration, we numerically investigate a PLC, which is a metal (Au) taper on a dielectric (Si3N4) waveguide at working wavelengths around 800nm. We show that three major effects (mode beat, nanofocusing, and weak resonance) interplay to generate this light concentration phenomenon and govern the performance of the device. By coordinating these effects, the PLC can be designed to be super compact while maintaining high efficiency over a wide band. In particular, we demonstrate that under optimized size parameters and wavelength a field concentration factor (FCF), which is the ratio of the norm of the electric field at the apex over the average norm of the electric field in the inputting waveguide, of about 13 can be achieved with the length of the device less than 1um for a moderate tip radius 20nm. Moreover, we show that a FCF of 5-10 is achievable over a wavelength range 700-1100nm when the length of the device is further reduced to about 400nm., Comment: 14 pages, 9 figures

We report a new analytical method for exact solution of homogeneous linear ordinary differential equations with arbitrary order and variable coefficients. The method is based on the definition of jump transfer matrices and their extension into limiting differential form. The approach reduces the $n$th-order differential equation to a system of $n$ linear differential equations with unity order. The full analytical solution is then found by the perturbation technique. The important feature of the presented method is that it deals with the evolution of independent solutions, rather than its derivatives. We prove the validity of method by direct substitution of the solution in the original differential equation. We discuss the general properties of differential transfer matrices and present several analytical examples, showing the applicability of the method. We show that the Abel-Liouville-Ostogradski theorem can be easily recovered through this approach.

In this paper, we present a new approach for the exact calculation of band structure in one-dimensional periodic media, such as photonic crystals and superlattices, based on the recently reported differential transfer matrix method (DTMM). The media analyzed in this paper can have arbitrary profile of refractive index. We derive a closed form dispersion equation using differential transfer matrix formalism, and simplify it under the assumptions of even symmetry and real-valued wavenumber. We also show that under normal incidence both TE and TM modes must have the same band structure. Several numerical test cases are also studied and discussed.