# Fiber Bragg Grating-Based Optical Signal Processing: Review and Survey
## 3 Survey of Signal Processing Units based on FBGs
### Photonic Hilbert Transformers  
FBG-based photonic Hilbert transformers (PHT), also known as phase shifters, have been also proposed and experimentally demonstrated based on FBGs working in reflection [72; 73; 74; 75; 76; 77] and transmission [78; 79]. A PHT is a pulse processor that delivers the Hilbert transform of an input optical pulse. The transfer function of a general PHT is defined as  
\[y(t)\propto H^{(P)}[x(t)]\to H_{PHT}(f)\propto\cos(\varphi)+\sin(\varphi) \times(-j\cdot sign(f-f_{0})) \tag{44}\]  
where \(H\) represents Hilbert transformation, \(\varphi=P\cdot\pi/2\), and \(P\) is the fractional order, being \(P\) a real number. The PHT is called integer when \(P=1\). PHTs are important components for a high number of applications in the fields of computing and communications. As optical signal processors, PHTs have been employed in the generation of phase-shifted pulse doublets, where the PHT order allows one to define the amplitude ratio between the pulse lobes. They have been also proposed to implement single side-band modulation from amplitude modulation formats [76].  
The first approach to the design of an FBG-based integer PHT was based on a weak-coupling uniform FBG (using the first-order Born approximation) with a properly designed amplitude-only grating apodization profile incorporating a single phase shift in the middle of the grating length [72]. The generalization for higher- and fractional-order PHTs were proposed based on strong-coupling uniform FBGs using grating synthesis algorithms [74; 75]. Practical fabrication constraints imposed operation bandwidths typically in the order of 200 GHz.  
To overcome this bandwidth limitation, FBG-based PHTs in transmission mode have been proposed taking into account that the PHT transfer function in not minimum phase [42]. By following the design approach presented in Section 2.4.2, two Hilbert transformers with an operation bandwidth of 3 THz, i.e., an integer, i.e., \(P=1\) and a fractional one, \(P=0.81\), were experimentally demonstrated [78]. As in previous processors based on FBGs in transmission, the grating profile (apodization and phase) were readily feasible, even for a relatively strong-coupling grating. Figure 10a,b shows the reflectivity and group delay in reflection of the integer HT, comparing the ideal and the experimentally measured curves. The reflectivity follows the anticipated interferogram-like profile described in Section 2.4.2. In particular, the target discrete shift in the phase spectral response of the PHT is encoded as the phase shift in the middle of the sinusoidal interferogram profile, while the all-pass PHT filter response imposes the nearly uniform interferogram envelope.  
The signal at the output of the PHTs are represented in Figure 10c,d. The experimentally obtained Hilbert-transformed output component of the fabricated devices when the input pulse is Gaussian-like and has an FWHM (Full-Width Half-Maximum) of 0.88 ps are presented in Figure 10c,d, with the corresponding simulated outputs for comparative purposes, showing a fairly good match in both cases. Some deviations are attributed to the low signal-to-noise ratio of the measured signals as a result of the application of a Fourier transform spectral interferometry procedure for the pulse characterization [42].