# Fiber Bragg Grating-Based Optical Signal Processing: Review and Survey
## 1 Introduction  
The discovery of fiber Bragg gratings (FBGs) about four decades ago entailed a revolution in the field of telecommunications [1]. FBGs appeared as an all-optical device component capable of performing signal processing with low loss, relatively low cost and full-compatibility with fiber optic systems. FBGs immediately attracted much of researchers' attention, being a fruitful field of research with widespread application in a number of scientific and industrial fields [2, 3, 4]. FBGs are considered as basic building blocks in photonic circuits aimed at ultrafast information transmission and computing, as they deliver broadband operation (even greater than 1 THz) while avoiding inefficient electro-optical and opto-electronic (EO/OE) conversions [5].  
A Bragg grating in a light-guiding medium such as an optical fiber is a periodic perturbation of its refractive index, causing certain reflectance and phase change in a wavelength range nearby that accomplishing the Bragg condition, i.e., \(\lambda_{B}=2\pi\Lambda\), where \(\lambda_{B}\) is the Bragg wavelength and \(\Lambda\) is the period of the refractive index perturbation [6].  
Nowadays, two of the more prominent areas of application of FBGs are optical sensing [7, 8] and optical signal processing [9, 10]. In optical sensing, the Bragg wavelength of FBGs is highly sensitive to variations in temperature and strain, offering a precise transducer to these effects. This fact, together with their low weight, low size, easiness to multiplex and easiness to embed fiber in different materials have made FBG a widespread photonic sensing solution of high interest in an increasing number of fields such as civil engineering, medicine, aerospace, and more [11, 12, 13, 14, 15, 16]. As optical signal processors, FBGs operate as broadband, linear filtering devices. In the 2000s, the versatility of FBGs to implement any desired linear filtering operation (bandpass, bandstop, customized phase filtering, etc.) was unveiled [6]. For this purpose, FBG design techniques consisting in engineering the apodization profile (i.e., the envelope of the periodic refractive index modulation of the fiber) and/or the grating period were developed. Hence, different types of optical filters and optical signal processing units (such as differentiators, integrators, pulse shapers,etc.) were implemented with bandwidths ranging from tens of GHz to THz [17]. Recently, the rapid growth of semiconductor technologies and silicon-based photonics integrated circuits (PICs) has speeded up the interest in translating the functionalities attained by FBGs to integrated platforms (i.e., the so-called waveguide Bragg gratings), aimed at the realization of entire functionalities on a chip [18; 19; 20]. Besides, recent prospects foster a low cost for massive production using CMOS technology.  
In this review, we focus on the application of FBG as signal processing units. First, we review the algorithms employed to analyze and synthesize FBG as linear filters, from approximate methods to \"exact\" analytical solutions. Then, we revisit the research on signal processors based on FBG carried out in the past two decades, discussing the design methodology employed and their applications. Finally, we provide insight for potential future lines of work in this area.