# Fiber Bragg Grating-Based Optical Signal Processing: Review and Survey
## 2 Review of Design Methods for Fiber Bragg Gratings  
Fiber Bragg Gratings (also known as short-period gratings) are distributed reflectors that couple light from a forward-propagating core-mode (\(E^{+}(z,f)\)), to the same counter-propagating mode (\(E^{-}(z,f)\)) in an optical fiber, as represented in Figure 1. For simplicity, single mode operation can be generally assumed, since a single co- and counter-propagating mode intervenes in the coupling process [6]. This coupling dominates at a particular wavelength, specified by the grating period (typically in the sub-micrometer) via the Bragg phase-matching condition [6]. Mathematically, an FBG can be modeled as the modulation of the effective refractive index of the guided mode of interest along the fiber length \(z\) as:  
\[n_{FBG}(z)=n_{eff}(z)+\Delta n(z)\cdot\cos\left(\int_{0}^{z}\frac{2\pi}{\Lambda (z^{\prime})}dz^{\prime}\right) \tag{1}\]  
where \(n_{eff}(z)\) is the effective refractive index for the fundamental mode, \(\Delta n(z)\) is the apodization profile, i.e., the envelope of the refractive index modulation, \(\Lambda(z)\) is the period variation (also defined as the chirp function) along the grating length, which is assumed to be \(L\) (Figure 1). As previously mentioned, FBGs behave as linear, passive filters. They have a linear and time invariant (LTI) spectral response and hence, diverse signal processing functions can be generated by using techniques from Fourier analysis [21]. FBGs can work in both reflection and transmission, depending on whether the target output is attained at the same fiber end where the input is launched or in the opposite end. Their frequency responses (or  
Figure 1: Nomenclature for the parameters of an arbitrary FBG of length \(L\) and for the electromagnetic optical fields at the two extremes (the grating period of the perturbation has been increased for illustrative purposes). spectral transfer functions) in reflection and transmission, \(H_{R}(f)\) and \(H_{T}(f)\), respectively, and the corresponding temporal impulse responses, \(h_{R}(t)\) and \(h_{T}(t)\) are defined as:  
\[H_{R}(f)=\left.\frac{E^{-}(z=0,f)}{E^{+}(z=0,f)}\right|_{E^{-}(z=L,f)=0};h_{R}(t )=\Im^{-1}[H_{R}(f)] \tag{2}\]  
\[H_{T}(f)=\left.\frac{E^{+}(z=L,f)}{E^{+}(z=0,f)}\right|_{E^{-}(z=L,f)=0};h_{T}(t )=\Im^{-1}[H_{T}(f)] \tag{3}\]  
where \(\Im^{-1}\) denotes inverse Fourier transformation, \(t\) is the time variable, and \(f\) is the optical frequency variable. The spectral response of the grating in reflection and transmission, also defined as reflective and transmissive field coefficients, are generally complex functions. Along this work, we will define these spectral responses as \(H_{R}(f)=|H_{R}(f)|\cdot\exp\{i\phi_{R}(f)\}\) and \(H_{T}(f)=|H_{T}(f)|\cdot\exp\{i\phi_{T}(f)\}\), respectively. The reflectivity of the FBG is then obtained as \(R(f)=|H_{R}(f)|^{2}\), while the transmissivity is \(T(f)=|H_{T}(f)|^{2}\). In linear regime, the reflective coefficient of an FBG acts as an optical band-pass filter, while the transmissive coefficient defines a band-stop filter. When operating in reflection, FBGs offer an extraordinary flexibility to achieve almost any desired complex-valued spectral filtering response, fundamentally constrained by practical fabrication limitations. In transmission, however, their response is minimum-phase [22], i.e., the imaginary part, \(\text{Im}[H_{T}(f)]\) of the FBG's spectral response in transmission is uniquely determined by the real part, \(\text{Re}[H_{T}(f)]\) through the Kramers-Kronig relationship [23,24]. To achieve a user-defined spectral response, the Bragg grating can be engineered, e.g., by changing its length \(L\), modulating its envelope, \(\Delta n(z)\), or modulating its grating period, \(\Lambda(z)\)[6].