# Fiber Bragg Grating-Based Optical Signal Processing: Review and Survey
## 3 Survey of Signal Processing Units based on FBGs
### Optical Differentiators  
An optical differentiator obtains the derivative of the complex envelope of an input optical signal [52]. Therefore, the transfer function of an arbitrary order differentiator can be written as  
\[y(t)\propto\frac{d^{N}x(t)}{dt^{N}}\to H_{diff}(f)\propto\left(j\cdot(f-f_{0} )\right)^{N} \tag{42}\]  
where \(j\) is the imaginary unit and \(N\) is the differentiator order. Applications of optical differentiators in the literature include the generation of arbitrary-order Hermite-Gaussian (HG) optical pulses [53], which can be employed as advanced coding for network access applications. Besides, optical differentiators can be applied for solving differential equations (ODEs) in analog computing systems. These equations play a fundamental role in practically any field of science or engineering, and the possibility of performing these computations all-optically implies potential processing speeds well beyond the reach of present electronic computing systems [5]. Several approaches for first- and high-order optical differentiators have been proposed in the literature based on FBGs, providing either different operation bandwidth or spectral resolution, still within the practical limitations of the technology [52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63]. First approaches proposed multiple-phase-shifts FBGs operating in reflection for first- and high-order differentiation, but with a limited bandwidth around 20 GHz [54; 55]. Afterwards, an alternative design approach based on the use of especially apodized linearly-chirped FBG operated  
Figure 6: Measured reflection spectral characteristic of a typical linearly-chirped FBG: (**a**) Reflectivity versus optical frequency, and (**b**) reflection group delay with a linear slope of 1100 ps\({}^{2}\). in transmission (following the space-to-frequency-to-time mapping design method) were proposed and experimentally demonstrated increasing the operation bandwidths to a few hundreds of GHz [56; 57; 60]. Finally, optical differentiators have been also proposed in a transmissive configuration based on synthesis algorithms, as \(H_{diff}(f)\) is a minimum-phase function. A first-order all-optical differentiator with a 2 THz bandwidth (full-width at 0.1% of the maximum amplitude) has been designed using a linearly chirped FBG and following the synthesis algorithm described in Section 2.4.1[41]. The obtained apodization profile is plotted in Figure 7a. The synthesized grating device was readily feasible with existing fabrication methods, in terms of effective length, maximum refractive index modulation and average spatial resolution of the ripples in the apodization profile. A comparison between the fabrication-constrained spectral response in amplitude and phase with the originally defined response is shown in Figure 7b. Figure 8 presents the output temporal characterization for the first-order optical differentiator as well as their performance based on the correlation coefficient and the TBP. Thus, Figure 8a shows the comparison between the ideally expected output waveform and the output from an FBG tailored to typical fabrication constraints. The two curves show an excellent match. Besides, Figure 8b shows the comparison in performance between the original designed spectral response and the fabrication-constrained spectral response. Figure 8: **(a)** Input Gaussian pulse (dotted black line), output of an ideal first-order differentiator (dashed blue line) and output of a first-order differentiator based on a fabrication-constrained FBG in transmission (red solid line); **(b)** Cross-correlation coefficient as a function of the 3 dB-bandwidth of the input pulse for the original design (blue dashed line) and the design tailored to fabrication constraints (red solid line).