FINITE-DIFFERENCE FREQUENCY-DOMAIN ANALYSIS OF NOVEL PHOTONIC WAVEGUIDES

This paper concerns the study of novel waveguide structures in photonics, in particular, photonic crystal fibers (PCFs) and photonic wire waveguides. One common feature among these guides is that they are not perfect waveguiding structures in that they possess leakage or confinement losses in the propagation. We present efficient and accurate analysis of these waveguides using a finite-difference frequency-domain (FDFD) method with perfectly matched layer (PML) absorbing boundary conditions. We formulate the eigenvalue problem for waveguide mode solutions using the finite difference method but based on the Yee mesh. To obtain high-accuracy results, proper matching of interface conditions near the dielectric interface is performed. INTRODUCTION We present an efficient and accurate method for studying two classes of photonic waveguides which are of much current interest: the photonic crystal fibers (PCFs) [1] and the photonic wire waveguides [2]. Both of them possess leakage or confinement losses in the propagation due to their waveguiding structures being not perfect. Their propagation constants are thus complex and accurate determination of these complex values is essential in designing and applying these waveguides. The method we employ is our recently developed finite-difference frequency-domain (FDFD) method with perfectly matched layer (PML) absorbing boundary conditions [3]. The formulation of the eigenvalue problem for waveguide mode solutions using the finite difference method is relatively simple compared to the popularly employed finite element methods and others. In our method, the finite difference scheme is based on the Yee mesh, which is the well-known mesh used in the finite-difference time-domain (FDTD) simulation [4]. Most importantly, the PMLs are employed to surround the domain of computation so that the confinement losses of leaky waveguides can be calculated. The FDFD eigenvalue problem formulation is derived directly from Maxwell’s equations. It is attractive in that the obtained mode fields can easily be incorporated into the FDTD computation. To obtain high-accuracy results, proper matching of interface conditions near the dielectric interface through the Taylor series expansion of the fields is performed [3]. PCFs are one successful application of the photonic crystal concept and have been intensively studied in the past few years. The technology of fabricating PCFs has also advanced rapidly. The family of PCFs include general holey fibers composed of array of air holes running down the fiber length [1], the photonic-band-gap (PBG) fibers making use of the PBG effect in the cladding [5], and the air-core fibers in which the light is guided in air region surrounded by a PBG cladding [6]. We are able to efficiently determine the guided modes and their confinement losses of these PCFs using the FDFD method with accuracy comparable to other more sophisticated methods such as the multipole method and the finite element method [3]. In this paper, we will show the analysis results of PBG fibers with different numbers of air-hole rings and coupling characteristics of two-core PCFs. Photonic wire waveguides, or photonic wires, are high index contrast optical waveguides with core region of submicron cross-sectional sizes. Currently, silicon-on-insulator (SOI) structures are most often considered, in which the silicon core is situated on a SiO2 layer. Such structure is promising for achieving future high-density integrated photonic circuits. Light can be strongly confined within the core region due to the high index contrast and lowradiation-loss bends with radius in the micron range are possible. However, sidewall surface roughness due to the fabrication process in such sub-micron structures becomes significant and may cause unacceptably high propagation losses. Many efforts have been made in reducing such sidewall roughness. For example, propagation loss as low as 2.4 dB/cm has recently been achieved based on fabrication with deep ultraviolet lithography and dry etching [2]. In addition, there is intrinsic substrate leakage loss because the substrate below the SiO2 layer and the core have the same refractive index since they are both made of silicon. Therefore, the propagating modes are leaky ones with complex propagation constant even in the absence of surface roughness scattering loss. According to the calculation conducted by Dumon et al. [2] based on an eigenmode expansion solver with PML boundary conditions, 1–1.5 dB/cm of the 2.4 dB/cm loss was due to substrate leakage. In this paper, we analyze the same structure and calculate the leakage loss using our FDFD method. Waveguides with different values of core width and SiO2 layer thickness are studied. Our calculation predicts smaller leakage losses compared with the 1–1.5 dB/cm value. Our results have been carefully checked with those calculated using a mode solver based on the finite-element imaginary-distance beam propagation method (FE-ID-BPM) with PMLs [7] and very good agreement has been obtained. THE FDFD METHOD The cross-section of a general waveguide problem we consider is shown in Fig. 1 with the computational window surrounded by PML regions I, II, and III, each having thickness of d, with x and y being the transverse directions and z being the direction of propagation. Regions I and II have the normal vectors parallel to the x-axis and y-axis, respectively, and regions III are the four corner regions. In the PML region, using the stretched coordinate transform [8], Maxwell’s equations can be written as