# Long-Period Grating with Asymmetrical Modulation for Curvature Sensing
## 2 Sensor Fabrication and Principle  
To construct the proposed sensor, a standard single-mode fiber is used. The fibers we used in the experiment are SMF-28 (Yangtze Optical Fibre and Cable Joint Stock Limited Company, YOFC, Wuhan, China) with the core diameter of 8.2 \(\upmu\)m and the nominal effective index of 1.4682 (@ 1550 nm). The fiber is stripped before being fixed on a 3D translation stage for the laser micromachining. A femtosecond laser (Spectra Physics, Solstice, Andover, MA, USA) with repetition rate of 1 kHz and wavelength of 800 nm is focused into the fiber with a 100 X oil objective, as shown in Figure 1a. The position of the focusing spot is monitored with a CCD camera. The pulse energy for the inscribing of the LPFG is 200 nJ, and the exposure time for each point is 1 s. To make a comparison, we first focus the laser into the center of the fiber core to inscribing a normal LPFG. The microscope image of the fabricated LPFG is shown in Figure 1b. Figure 2a shows the micrograph of the laser-ablated spot in the fiber core. The period of LPFG is 500 \(\upmu\)m, and the transmission spectrum of the grating is monitored during the fabrication process. The period of 500 \(\upmu\)m is selected so that the resonant wavelength of the LPFG can be expected to be located within the C band. The transmission spectrum is collected by a broadband source (BBS) and an optical spectrum analyzer (OSA) (Yokogawa AQ6370C). The evolution of the grating spectrum is shown in Figure 3a, where the spectra of the LPFG at different numbers of periods are plotted. From Figure 3a, we can see that three resonant dips appear as the number of periods increases. The dips are deepest when the number of periods is 20. After that number, those resonant dips start to degenerate, which corresponds to the case of overcoupling. Figure 1: (**a**) The schematic diagram of the fabrication of the LPFG. (**b**) Microscope photograph of the LPFG. Figure 2: Grating modulation with offset of (**a**) 0 \(\upmu\)m, (**b**) 3 \(\upmu\)m, and (**c**) 6 \(\upmu\)m.  
Then, LPFG samples with different offsets are fabricated. Figure 2b,c show the micrographics of grating modulations with offsets of 3 \(\upmu\)m and 6 \(\upmu\)m, respectively. The laser pulse energy used in the fabrication is the same as that used for the normal LPFG fabrication. The size of the laser-ablated spots are the same, while the locations of the modulation are different. Figure 3b shows the grating spectrum of the sample with an offset of 3 \(\upmu\)m at different numbers of grating periods during the fabrication. The wavelength of the resonant dips are a little different from that of the normal LPFG. Such difference in the resonant wavelength is due to the change of cladding modes that are involved in the grating resonance. For the sample with an offset of 6 \(\upmu\)m, the effective index modulation is much weaker compared with the former two cases, and the grating spectrum grows relatively slower during the fabrication process, as seen from Figure 3c. Thirty periods are introduced to fabricate the grating, and the resonant dip is not as deep as the former two. The amounts of index change introduced to the silica material in different samples are close to each other, since the laser energy deposited is the same. However, since the majority of light is propagating in the central part of the fiber, a smaller offset will have a larger influence on the mode effective index. That is why the sample with an offset of 6 \(\upmu\)m has the weakest effective index modulation. Considering that nonuniformities are introduced to fabricated the LPFG, conventional coupled mode theory is not applicable here. On the other hand, the coupled local mode theory can be adopted to deal with such a case.