## 2 Distributed Optical Fibre Sensors
### Rayleigh-Based Distributed Sensing
#### 2.1.2 Optical Frequency Domain Reflectometry  
As discussed above, optical frequency domain reflectometry aims at characterising the frequency response of the fibre under roundtrip propagation. Instead of using a pulse signal, the source is modulated either in intensity or frequency. Optical frequency domain reflectometry comes in two variants:  
* Incoherent OFDR or I-OFDR, obtained by modulating the optical intensity with radio frequency (RF) signals;
* Coherent OFDR, obtained by sweeping the optical frequency. In these two variants, the sensing information is encoded on an RF or an optical carrier, respectively. The basic setup of an I-OFDR is similar to that of a conventional OTDR (Figure 1), where the pulsed source is replaced by a continuous wave light modulated in amplitude by an RF signal. The RF signal frequency is linearly swept in a given bandwidth to probe the frequency response of the fibre [35; 36]. Similar to conventional OTDR, I-OFDR spatial resolution is inversely proportional to the system bandwidth, here corresponding to the bandwidth swept by RF modulation. Nonetheless, owing to the RF modulation, this approach can benefit from electrical heterodyne detection, in general resulting in better performance compared to OTDR [37; 38]. Despite that, at the moment there are no commercial implementations of Rayleigh-based I-OFDR, probably due to the outstanding performance achieved by the coherent counterpart. The basic setup of coherent OFDR is shown in Figure 2. The light from the source--which is optical frequency-modulated--interferes coherently with the light that was emitted a short time before and scattered back from a certain distance along the sensing fibre [39]. If one assumes that the sweep is linear, the frequency difference between these two light signals is proportional to the propagation time delay along the sensing fibre, and therefore, to distance. This means that the instantaneous beat frequency measured at the detector is mapped to a specific position along the fibre. Figure 1: The principle of operation of the OTDR and I-OFDR schemes (CW: continuous wave; OTDR: optical time domain reflectometer; I-OFDR: incoherent optical frequency domain reflectometer; RF: radio frequency). Modified from [9]. The spatial resolution \(\delta z\) is in this case determined by the wavelength scanning range \(\Delta\lambda\) (or, equivalently, by the frequency scanning range, \(\Delta f\)). It reads \(\delta z=c/(2n\Delta f)=\lambda_{s}\lambda_{f}/(2n\Delta\lambda)\), where \(c\), \(n\), \(\lambda_{s}\), and \(\lambda_{f}\) are the speed of light, the effective refractive index of the fibre, and lower and upper wavelengths of the frequency scan, respectively [40]. At the same time, if the sweep time is slow enough, the electrical bandwidth needed to manage the frequency sweep can be reasonably modest, down to some MHz. Indeed, the expression of maximum beat frequency, \(f_{max}\), is given by \(f_{max}=L/(\delta z\,t_{sweep})=\gamma\tau_{max}\), where \(L\), \(t_{sweep}\), \(\gamma\), and \(\tau_{max}\) are the length of the fibre, the frequency sweep duration, the slope of the frequency sweep (Hz/s), and the maximum fibre delay, respectively. This means that a 1 km-long fibre can be sampled with 10 cm of spatial resolution by sweeping the frequency over 1 GHz in 1 ms (\(\gamma=1\) THz/s) and with a maximum beat frequency of 10 MHz. These values show the advantage of coherent OFDR over other time-domain techniques: high spatial resolution is achieved with a wavelength sweep of some tens of nanometres and a relatively small electric bandwidth. From early OFDR implementations over short fibres more than 30 years ago [41], it took a while to reach longer distances, mainly due to the need for coherent sources capable of very linear frequency sweep [42]. To circumvent the issue of the nonlinearity of the frequency sweep, schemes integrating an additional reference interferometer were proposed [43].