# Designing a Novel High-Performance FBG-OADM Based on Finite Element and Eigenmode Expansion Methods
## 1 Introduction  
Fiber gratings refer to periodic structures that change according to the refractive index (RI) of the core of photosensitive optical fibers. According to the periodic length of fiber gratings, they can be categorized as short- or long-period fiber gratings. Short-period fiber gratings, also known as fiber Bragg gratings (FBGs), were first proposed by [PERSON] et al. at the Canadian Communication Center in 1978 [1; 2; 3]. Of all related methods currently used to produce FBGs, the interferometric and phase mask methods are the most common ones. The interferometric method involves irradiating a photosensitive optical fiber with two interleaved beams of UV light; the UV light wavelength or angle between the interleaved beams is adjusted to change the RI of the optical fiber, thereby producing the desired fiber grating. The phase mask method involves irradiating UV light onto a phase mask, which creates constructive and destructive interference in the core of the photosensitive optical fiber. Changes in energy intensity then cause the RI of the optical fiber to display a periodic distribution [4; 5; 6]. Optical add/drop multiplexers (OADMs) are key components for creating wavelength-division multiplexers (WDMs) in fiber-optic communication networks. The main function of an OADM is todrop or add a client signal in the fiber-optic communication network. Specifically, OADMs employ wavelength division multiplexing techniques in the frequency domain to emulate the time division multiplexing capabilities of the traditional synchronous digital hierarchy standard in the time domain. In the past several decades, numerous types of OADM have been proposed, many of which are widely used in various fiber-optic communication systems [7; 8; 9; 10; 11; 12]. Of all of the currently available OADMs, the most common one is composed of an FBG and two optical circulators. However, this particular OADM is disadvantageous because of its large size and high-cost, complex production process. To overcome the disadvantages of current OADMs, the present study referenced optical coupling theory to develop a novel high-performance FBG-OADM model comprising two single mode fibers (SMFs), which are placed side by side so that they work in the same manner as a \(2\times 2\) optical fiber coupler, and two FBGs, which have identical parameters so that the they work in the same manner as a mode coupler. In contrast to traditional OADMs, the proposed FBG-OADM is advantageous because it is a miniature-sized all-optical fiber-based multiplexer that can be fabricated through a low-cost and simple manufacturing process [12]. In researching and designing the proposed FBG-OADM, this study combined the finite element method (FEM) and eigenmode expansion method (EEM) to perform numerical simulations for calculation and analysis. The remainder of this paper is organized as follows: Section 2 details the geometric structure, parameters, and working principle of the proposed FBG-OADM. Optical coupling theory is used to explain how the FBG-OADM achieves outstanding bandwidth performance (\(-3\) dB). In addition, to minimize discrepancies between simulations and actual performance results, a perfectly matched layer (PML) and perfectly reflecting boundary (PRB) were integrated into the FBG-OADM design. Section 3 explains how the FEM technique was used to solve and analyze the FBG-OADM mode. To obtain suitable mesh-cutting resolutions, the object meshing method (OMM) and boundary meshing method (BMM)--two methods that are currently used in the FEM--were adopted to determine the optimal tradeoff between computational performance and cost. Mathematically, solutions derived from partial differential equations (PDEs) with boundary conditions must be pairwise orthogonal; in other words, their orthogonal value must be equal to zero. However, in numerical simulations, because of memory and computational time constraints, achieving pairwise orthogonality between modes is impossible. In other words, errors are inevitable.