## 3 Synthesis of Optical Coherence Function
### FBG-Based Multiplexed Fiber Optic Sensing with SOCF  
Fiber Bragg grating (FBG) is a useful point sensor for strain and/or temperature measurement. As for FBG multiplexing schemes, wavelength division techniques have been mainly developed and used [1]. However, in the techniques, FBGs should have different Bragg wavelengths from each other [1]. On the contrary, the SOCF technology provides a multiplexing system of FBGs with identical Bragg wavelengths [25; 27; 28]. Figure 10 is an example of proposed systems, in which a SOCF with simple sinusoidal FM is used [28]. The Bragg wavelengths of the FBGs are identical, but their reflectivity is small. By changing the SOCF frequency \(f_{2}\), we can select one FBG along the fiber. Additionally, the LD frequency is scanned with a saw-tooth waveform, which gives us an FBG reflection spectrum shape that has the applied strain or temperature from the spectral peak. In this system, the LD intensity modulation associated with the FM is compensated by an intensity modulator driven by another saw-tooth waveform, as shown in Figure 10.  
When the FBGs are set along a long-length fiber, for example, of 1000-m length, as shown in Figure 10, an optical path-length difference (OPD) takes place between the sensing lightwave reflected at the FBG and the reference lightwave. In this case, the LD frequency scan with the saw-tooth waveform makes an additional frequency difference between the two lightwaves due to the OPD. Therefore, the beat frequency between the two lightwaves, which is the heterodyne output frequency of the interferometer system, changes for each FBG to be measured. Therefore, we need to add some scheme to manage the heterodyne frequency change. In Figure 10, the amount of the frequency shift set at the acoustic optical modulator (AOM), which is an optical frequency shifter, is changed for each FBG, so that the heterodyne frequency at the system output becomes constant. Figure 11a shows the experimental results of the detection of reflection spectra for the three FBGs with adopting the random accessibility function [28] Here, FBG1 and FBG2 are set after a delay fiber with a length of 1010 m, which have a meter-order separation from each other, and FBG3 is set after another delay fiber with a length of 1000 m, as shown in Figure 10. In this case, the AOM driving frequency, which corresponds to the optical frequency shift by this device, is 66.98 MHz for FBG1 and FBG2, and 99.84 MHz for FBG3, respectively. This makes the heterodyne frequency of the system constant: that is, 135 MHz. In this experiment, the total random access speed was 60 Hz with a sequence of FBG1, FBG2, FBG3, FBG1, FGB2, FBG3, etc. Then, each FBG spectral shape was measured  
Figure 9: Delta function-like optical coherence function with sinusoidal FM waveform [14]. **(a)** FM waveform; **(b)** power spectrum shape of the light source modulated with the sinusoidal FM waveform, and **(c)** synthesized delta function-like optical coherence function. 20 times/s, as shown in Figure 11a. A time-division multiplexed measurement of the dynamic strain change at the three FBGs can be realized. In this experimental setup, the number of the FBGs can be increased more, while keeping the total measurement speed. The random access speed in this experiment was restricted, in practice, by the limitation in the driving frequency change of the AOM. The additional heterodyne frequency change due to the OPD is proportional to the OPD length and the saw-tooth frequency for the LD frequency sweep. In this experiment, a measurement fiber length of 1000 m and a width of the AOM driving frequency change of 60-120 MHz restricted the measurement speed to 60 Hz. This is not the inherent limitation. If a wider frequency change could be applied to the AOM and/or to the center frequency of the BPF, we have the possibility of enhancing the random access measurement speed to a greater degree.