# Comparative Analysis of the Methods for Fiber Bragg Structures Spectrum Modeling
## 1 Introduction  
Mathematical modeling of the reflection and/or transmission spectra of passive and active elements of fiber-optic circuits is one of the main approaches used in the design of both fiber communication lines and fiber-optic sensor systems. Mathematical modeling allows not only to predict the results of a certain impact on a fiber-optic system, but also to optimize individual elements of such systems. One of the most commonly used elements of fiber-optic sensor systems, as well as data transmission networks, is a periodic or quasi-periodic structure formed in the core of an optical fiber by changing the refractive index. In particular, a wide class of passive fiber-optic structures, such as fiber Bragg gratings (FBGs), belongs to this type of elements. Over the past decade, in addition to classical FBGs, numerous fiber-optic structures have become widespread: FBGs with one or more phase shifts, apodized and chirped FBGs, superstructured FBGs, FBGs combined with open or closed Fabry-Perot resonators (FPRs), etc. The basic components of the abovementioned complex or combined fiber-optic structures are FBGs and FPRs. Combining these basic elements in different configurations makes it possible to obtain various sensor designs with specific reflection spectra and different properties and characteristics (Figure 1). An internal (or external) FPR (Figure 1a) can be used as a high-precision point temperature sensor [1]. A conventional uniform FBG (Figure 1b) is also used as a sensing element for temperature, deformation, vibration, etc. measurements [2]. Two FBGs recorded sequentially (Figure 1c) or one over the other [3,4] (Figure 1d) have found applications as addressed fiber Bragg structures [5]. An FBG-basedstructure with a discrete phase shift (Figure 1e) is used as a high-precision Bragg-type sensor due to the presence of an ultra-narrow transparency window in its reflectance spectrum, the wavelength position of which can be measured with higher accuracy than the shift of the entire FBG spectrum [6]. The FBG structures with two or more phase shifts can also be used as addressed [5] (Figure 1f) or even multi-addressed [7] structures. The sensing elements based on the combination of FBG and FPR (Figure 1g) represent a promising solution for gas concentration measurement systems [8; 9; 10]. To enhance the performance of the designed elements of the fiber-optic data transmission or sensor systems, it is necessary to have reliable methods and tools that enable spectral characteristics modeling of fiber-optic structures with high accuracy. The basic element of such tools, which is of particular interest, is the mathematical apparatus that makes it possible to reconstruct the FBG reflection spectrum. The current paper investigated and compared various methods of FBG spectral response modeling. The first method is based on determination of the reflectance and the transmittance of plane waves propagating through a layered structure similarly to the layer peeling algorithm [11], which results in the solution of a system of linear equations for the transmittance and reflectance of each layer using the sweep method (further referred to as the Layer Sweep (LS) method). The second technique is based on the formulation of transfer matrices for the entire FBG structure [12; 13] (Transfer Matrix (TM) method). The FBG transfer matrix is often reduced to a matrix with Chebyshev polynomials of the second kind [14], which requires their calculation for each wavelength. Modeling techniques based on recurrence relations, including the ones using Chebyshev polynomials, were deliberately excluded from consideration, since they do not provide performance benefits in comparison with the LS method. Figure 1: Several types of layered structures formed in the optical fiber core: (**a**) Fabry–Perot resonator (FPR); (**b**) fiber Bragg grating (FBG); (**c**) two sequential FBGs; (**d**) two FBGs recorded one over the other; (**e**) FBG with \(\pi\) phase shift; (**f**) FBG with two \(\pi\) phase shifts; (**g**) FBG and FPR combination.