We have investigated experimentally and numerically the temperature and refractive index (RI) sensitivity characteristics of excessively tilted fiber gratings (Ex-TFGs) in detail. Both results have shown that the temperature and RI sensitivities of Ex-TFGs are mode order dependent. For temperature sensitivity, the higher order cladding mode of Ex-TFG exhibited lower temperature sensitivity, quantitatively, the temperature sensitivities of TM cladding modes at the resonance wavelength around 1550nm are 9pm/°C, 6.8pm/°C, 5.6pm/°C and, 4pm/°C for cladding mode 28th, 31st, 35th, 40th, respectively, indicating the overall temperature sensitivity of Ex-TFGs were lower than that of normal FBGs. The SRI sensing results have shown that the RI sensitivity of Ex-TFG at the special index value could be improved by choosing the cladding mode with effective index close to the refractive index of the detecting medium. The SRI sensitivities at the effective mode index were 2250nm/RIU at 1.408, 864nm/RIU at 1.395, 1536nm/RIU at 1.380 and 1360nm/RIU at 1.355, for the cladding mode of 28th, 31st, 35th, 43rd, respectively. The experimental results have also shown the SRI sensitivity of Ex-TFG was increasing with increasing of the resonance wavelength.
© 2017 Optical Society of America
Different from fiber Bragg gratings (FBGs) and long period gratings (LPGs), tilted fiber gratings (TFGs) have a pile of periodically slanted fringe plates with respect to the normal of optical fiber axis. Such tilted structure has broken the circular symmetry of fiber and induced the birefringence of the fiber, then fostering interesting polarization dependent mode coupling behavior, which has initiated many applications, including biosensors [1–4], accelerator , in-fiber polarizer , twist sensor [7, 8], loading sensor , water level sensor  et al.. According to different mode coupling behaviors, TFGs could be sorted into three different types: (1) tilted fiber Bragg gratings (TFBGs), which couple light from the forward propagating core mode into the backward propagating core/cladding modes; (2) radially tilted fiber gratings (RTFGs), which couple light from the forward propagating core mode into radiation modes; (3) excessively tilted fiber gratings (Ex-TFGs), which couple light from the forward propagating core mode into the forward propagating core/cladding modes . The TFBGs were firstly reported and proposed as an in-fiber tap by G. Meltz in 1990 . Since that, many relative works have been reported [13, 14], among them, T. Erdogan et al have contributed the main works in the theory of the TFBGs , and Jacque Albert’s group has contributed lots of TFBGs based sensing application , and this year, Christophe Caucheteur et al. reported a TFBGs based ultrasensitive plasmonic sensors in Nature Communication . RTFGs were firstly proposed by Westbrook and demonstrated as an in-line polarimeter . In 2005, the Photonics Research Group (now AIPT) from Aston University has proposed to use RTFGs as an in-fiber polarizer . A detailed characterization about RTFGs has been reported in  by the same group, and later developed a commercial in-fiber linear polarizer by inscribing RTFGs into PM fiber in 2012 . The main applications of RTFGs were mostly in all fiber ultrafast laser systems [20–22]. In 1997, J. L. Wagener et al. have firstly proposed a tilted chirped fibre grating as optical spectrum analyzer tap . Recently, our collaborators from Bern University of Applied Sciences in Switzerland and Kent University in UK have introduced the RTFGs as an in-fiber diffractive grating into spectroscopy and imaging system [24, 25]. In the first tilted long period grating as mode convertor was reported by K. O. Hill in 1990 . Lately, Zhou et al. reported an >45° TFBG with tens micrometers grating period as a low thermal cross sensitivity RI sensor [11, 27], however, such gratings do not have any Bragg reflection, so in later research, they were renamed as Ex-TFGs as their structures are excessively tilted. Comparing with TFBGs and RTFGs, Ex-TFGs have a relative larger period, which have similar mode coupling behavior with the LPGs, the only difference was that Ex-TFGs induce polarization dependent mode coupling. In our previous papers, we have explored and verified that Ex-TFGs have a series of dual peaks with orthogonal polarization in their transmission spectra , and also demonstrated a highly sensitive RI sensor based on Ex-TFG structure in thin cladding fiber , and successfully applied as a glucose sensor . The Ex-TFGs have also been utilized in fiber laser system to generate orthogonally polarized bright–dark pulse pair mode locking . Although many papers have been published about Ex-TFGs, there was no report on detailed discussion about the RI and temperature sensitivity characteristics. In this paper, we would present a detailed analysis of Ex-TFGs with results from experiment and simulation.
2. Theoretical analysis of Ex-TFG
The phase matching condition (PMC) is the most important mathematical formula to describe the mode coupling behavior and the sensing function of fiber gratings. Ex-TFGs have the similar phase matching condition as the LPGs . However, for the Ex-TFGs, the phase matching condition not only includes the mode information, but also contains the polarization states. The PMC of an Ex-TFG could be expressed as:
where λ is the resonance wavelength, neffco(λ) is the effective index of the core mode at the wavelength λ; is the effective index of mth TE/TM cladding mode at the wavelength λ; is the normal period of grating; is the tilt angle of the grating.
The resonance wavelength of an Ex-TFG was determined by the grating period, tilt angle and the effective refractive indexes of the core and cladding modes. These parameters could be also affected by environmental condition, such as temperature, loading, strain and surrounding-medium RI (SRI) experienced by the fiber. The resonance wavelength of an Ex-TFG after environmental perturbation could be expressed by re-writing Eq. (1) as:Eq. (5) is the waveguide dispersion (usually, known as γ factor), the second one is the environment dependence of waveguide dispersion (Г) and the third one is the material expansion caused by the changing of environment (α). Then, Eq. (5) could be written as:
2.1 The temperature sensitivity
When an Ex-TFG was subjected to the temperature change, the refractive index of fiber would change because of the thermo-optic effect, and the period of grating would also change due to the temperature expansion of material. In Eq. (6), the second term as the temperature dependence of waveguide dispersion could be expressed as:32]. In general, for the core composited with 4.1m% GeO2 and 95.9m% SiO2, ξco is about 7.97 × 10−6/°C, and for the core doped by 9.7m% B203 and 4.03m% GeO2, ξco is about 7.3 × 10−6/°C . The ξco of SM-28 telecom fiber we used in the experiment was around 7.07 × 10−6/°C, which was obtained by measuring the temperature sensitivity (~11.6 pm/°C) of FBG UV-inscribed in SM-28 fiber at 1550nm. The third term of Eq. (6) is the coefficient of temperature expansion, which is around 4.1 × 10−7 for silica based material.
In our previous work, we have found the TE and TM cladding mode have the similar variation trends as the environmental change, and the TM cladding mode was a bit higher sensitive than the TE cladding mode. So, in this paper, we would only analyze the temperature and SRI sensitivity of TM cladding mode. Generally, it was thought that because ГTemp>>α, the temperature sensitivity was determined predominantly by the product of γ, ГTemp and λ. Figure 1(a) showed the simulation results of (a) temperature dependence of waveguide dispersion, in which ГTemp depended on ξco and mode order, and ξco depended on the dopant of fiber core. Our previous works have concluded that for ξco>ξcl, ГTemp was always a positive value and approaching to zero with increasing mode order; while for ξco<ξcl, there was a switchover point at which ГTemp was equal to zero and after which it changed from negative to positive . Figure 1(b) showed the temperature sensitivity of TM resonance cladding mode against the mode order at 1550nm for different ξco values. As it shown in Fig. 1(b), cladding mode at the turning point was ultra-sensitive to the temperature, which was because the γ factor trended to unlimited at that point. And in our experiment, the cladding mode order was far away from the turning point, which showed very low temperature sensitivity. Overall, for a specific cladding mode, the temperature sensitivity was a constant, and decreasing as the increasing of mode order. For example, for SM-28 fiber ξco = 7.07 × 10−6, the calculated results showed the temperature sensitivity are 21pm/°C, 11pm/°C, and 5pm/°C for 30th, 35th, and 40th cladding mode at 1550nm, respectively, which would be verified in our lateral experiment.
2.2 The SRI sensitivity of Ex-TFG
Ex-TFGs have been proposed and demonstrated as optical biosensors as they are intrinsically sensitive to SRI changes, due to the effective indexes of cladding modes depend on the SRI. In general, the effective index of core mode was only determined by the indexes of core and cladding materials, thus not directly affected by the surrounding medium, whereas only the effective index of cladding mode was changing with the SRI. We could regard that in Eq. (5), the changing of effective index of core (δnco) and the material expansion (α) induced by the changing of SRI were zero. Then, SRI dependence of waveguide dispersion could be written as:32, 33], the effective mode index of optical fiber waveguide could be approximately expressed as:
From Eq. (9), the differential of cladding mode index about SRI is:
According to Eq. (12), the RI sensitivity depended on γ factor, RI dependence of waveguide dispersion, and wavelength. Among of them, γ factor could be dramatically increased by designing the cladding mode closing to the turning point. However, with increasing γ factor, the temperature and strain cross sensitivity were also increased as well, which is not good for bio/chemistry sensing application. In previous section, our analysis resulted showed that the higher order cladding mode had lower temperature sensitivity. And from Eq. (11), the RI dependence of waveguide dispersion would increase as the cladding mode order moving to higher. Figure 2(a) plotted the calculated results of ГSRI values for the first 50 TM resonant cladding modes at 1300nm, 1550nm and 1700nm for SRI = 1.345 (the reason for choosing SRI = 1.345 was that the RI of most bio/chemical solution is water based and around 1.345). As it shown in Fig. 2(a), ГSRI rapidly increased with increasing of mode order before reaching the dispersion turning point, and then slowly increased and almost reached a saturation value after the turning point (The turning point is the point of inflexion of the γ factor, at that point, γ has the maximum value. So, the sensors designed at that point is ultra-sensitive to temperature, RI, strain et al.). From Fig. 2(a), we could conclude that the absolute value of ГSRI was proportional to the operation mode order under the same wavelength, and proportional to the operation wavelength at the same mode order. The general trend was the longer the resonance wavelength, the higher the SRI sensitivity was. We have also analyzed the SRI sensitivity of 28th, 31st, 35th, 40th and 43rd TM cladding mode coupled by the grating with axial period of 35.1µm, 28.8µm 22.6µm, 17.4µm and 15µm, and they showed SRI sensitivities of 140nm/RIU, 150nm/RIU, 170nm/RIU, 240nm/RIU and 480nm/RIU at SRI = 1.345, respectively (see in Fig. 2(b)).
Figure 2(c) showed the SRI sensitivities of the 33rd, 35th and 37th TM cladding modes of Ex-TFG with 23.2 µm axial period. The resonances of these three cladding modes were at 1343nm, 1508nm and 1702nm, and the SRI sensitivities are 140nm/RIU, 190nm/RIU and 320nm/RIU at SRI = 1.345, respectively. The simulated results in Fig. 2(c) have shown the longer resonance wavelength was the higher SRI sensitivity. Moreover, it was we clearly shown that the higher order cladding mode was more sensitive under the same resonance wavelength range, and the SRI sensitivity increases for longer resonance wavelength range.
3. Fabrication of Ex-TFG
The Ex-TFGs had relatively large periods compared with FBGs, however, because of their slanted fringes, they couldn’t be fabricated by point-by-point inscription method. A more realistic method would be using tilted amplitude mask. The detailed fabrication method has been described in . To verify our simulation, 5 Ex-TFGs were fabricated by UV-inscription technique using a 6.6μm amplitude mask with different tilt angles, and the detailed fabrication parameters were listed in Table 1. The relationship between the tilt angle, period of mask and grating structure inside the fiber have been discussed in several papers [19, 34]. We have summarized in Eqs. (13) and (14).
The axial grating period was defined as:
According to the previous calculation results, it could be identified that the TM resonance wavelength, mode order and mode index of Ex-TFGs with tilt angles at 72°, 75°, 79°, 81° and 83°, and these parameters were listed in Table 2.
4. Experimental evaluation on temperature and SRI sensitivity of Ex-TFGs
The detailed theoretical analysis and numerical simulation on temperature and SRI sensitivity of Ex-TFGs have been discussed in previous section. In this section, we would present the experimental evaluation results on the temperature and SRI sensing characterization of Ex-TFGs.
4.1 Temperature sensing experiment using Ex-TFGs
Four Ex-TFGs with structures tilted at 75°, 79°, 81° and 83° were selected for temperature sensitivity evaluation. Each grating was subjected to the temperature elevation. The temperature was tuned from 10°C to 60°C with an increment of 10°C. As we known, the Ex-TFGs have two orthogonal polarization state transmission peaks. During the experiment, a polarizer and polarization controller were employed to make sure only one of the dual peaks selected for the characterization, which the detailed measuring method has been described in . Figure 3(a) shows the temperature sensitivity of TM (at 1560nm) and TE (at 1567nm) peak of the 81°-TFG. The fitted results indicate the temperature sensitivities of TE and TM cladding mode are 5.6pm/°C and 6.8pm/°C; the latter was slightly higher than the former.
In the experiment, we also investigated the temperature sensitivity of TM peak at around 1550nm for Ex-TFGs tilted at 83°, 81°, 79° and 75°, in which the cladding modes of 83°-, 81°-, 79°- and 75°-TFG at around 1550nm should be 28th, 32nd, 35th, and 40th, respectively, which has been listed in Table 2. Thus, before reaching the switchover point, we only see that the temperature sensitivity was decreasing with increasing of the mode order. Figure 3(b) plots the wavelength shift of TM cladding mode against the temperature for these four Ex-TFGs. From the linear fitting, we can obtain the temperature sensitivities are 9pm/°C, 6.8pm/°C, 5.6pm/°C, 4pm/°C, respectively, for the four gratings. All these values are lower than the sensitivity of normal FBGs at ~1550nm. The fiber used to inscribe the Ex-TFGs is SM-28, which has a temperature optical coefficient of 7.07 × 10−6. The previous simulated results show the switchover point of temperature sensitivity was at 47th cladding mode for ξco = 7.07 × 10−6.
4.2 SRI sensing experiment using Ex-TFGs
To evaluate SRI sensing capability of the Ex-TFGs, we have applied a series of index oil (from Cargille laboratory) with different RIs from 1.305 to 1.408 (measured at 532nm at 25°C) to the gratings and measured their spectral evolution. To avoid the wavelength shift induced by the bending and axial strain, the grating was straightly clamped between two translation stages with the same height. The index oil was placed on a flat glass substrate and the glass substrate was left up to immerge the grating fiber in the index oil, which the experiment setup was shown in Fig. 4. During the RI measuring process, the temperature was kept at room temperature (around 25°C). After each index oil measurement, the grating was rinsed with methanol to remove the residual oil until the original spectrum in air was restored on the optical spectra analyzer.
The wavelength shift against SRI variation from 1.305 to 1.408 for the 35th TE and TM cladding modes from 79° Ex-TFG was plotted in Fig. 5(a). The results have shown that the SRI sensitivity of TM mode was slightly higher than the TE mode. Unlike the linear trend of the temperature sensitivity, Ex-TFG exhibited a nonlinear SRI sensitivity. The general trend for SRI response of Ex-TFG was that the SRI sensitivity increased with increasing SRI value and reached the maximum when the SRI approaching to the effective cladding mode index. For 79° Ex-TFG, we could see from Fig. 5(a), the wavelength shifted of TM and TE mode were around 44.04nm and 37.44nm when SRI was changing from 1.305 to 1.408. When SRI > 1.408, the 35th cladding mode resonances of 79°Ex-TFG completely disappeared as it was larger than the effective mode index. Figure 5(b) showed the comparison of SRI responses of the 33rd (1710nm), 35th (1550nm) and 37th (1320nm) TM cladding modes of 79° Ex-TFG. As it shown in the Fig. 5(b), each cladding mode had a different SRI sensing range. The maximum detectable SRI was 1.375 for the 33rd cladding mode at 1710nm, 1.395 for the 35th cladding mode at 1550nm and 1.408 for the 37th cladding mode at 1320nm, and the sensitivities at SRI = 1.345 were 232nm/RIU, 200nm/RIU and 132nm/RIU, respectively. All these results were in very good agreement with the calculation results shown in Fig. 2(c). As discussed before and shown in Fig. 2, the wavelength shift against SRI was not linear, but exponentially increasing with the SRI and reaching the maximum when approaching to the effective mode index. Table 3 listed the calculated results of the effective index and measured SRI sensitivity of the TM modes of the 79°-TFG at 1320nm, 1550nm and 1710nm.
So, in order to achieve higher sensitivity at a special SRI index or an SRI range, we could design the grating structure to make sure the effective mode index was close to the index of sensing medium. In the experiment, we have evaluated SRI response for the four different cladding modes at the resonance wavelength around 1550nm Ex-TFGs. Figure 6(a) plotted the wavelength shift of the TM peaks at around 1550nm region against SRI for these four Ex-TFGs. From the measurement results, we estimated that the SRI sensitivities at the effective mode index are 2250nm/RIU at 1.408, 864nm/RIU at 1.395, 1536nm/RIU at 1.380 and 1360nm/RIU at 1.355, for the four Ex-TFGs respectively. As we could find that the effective mode index measured from experiment was a bit higher than the simulation results, which was because the index oil we used was calibrated by 532nm light, and the simulation results was calculated at 1550nm, and the simulation model of cladding mode was also an approximated model. These experiment results matched very well with our theoretical analysis discussed in previous sections. Finally, we demonstrated a 72° Ex-TFG as the milk fat concentration sensing experiment, in which we tested milks with different fat concentrations 4%, 2% and 0% (whole, semi-skimmed and skimmed milk), respectively. The experimental results showed the wavelength shifts are 21.4nm, 20.8nm and 20.2nm for these three milk samples comparing with the peak in air (see in Fig. 6(b)). Also, in comparison with our reference data, we could conclude that the 4%, 2% and 0% fat concentrating milk samples had the index values around 1.344, 1.342, and 1.340, respectively (Table 4).
We have presented the detailed analysis of SRI and temperature sensing characterization in theory and experiment. Both the theoretical and experimental results showed that the TM transmission cladding peak was linearly responding to the temperature, and the temperature sensitivity was decreasing when the cladding mode towards to higher order. The SRI sensing results have shown the RI sensitivity of cladding mode of the Ex-TFGs could be achieved high sensitivity at the special RI sensing range by design the cladding mode index close to the sensing index. Both the theoretical and experimental results were agreed very well with each other. Finally, we have applied a 72° Ex-TFG to measure fat concentration of milk samples, the results showed a linear wavelength shift versus the fat concentration.
National Natural Science Foundation of China (No. 61505244). The sub-project of the National Natural Science Foundation of China (N0.61290315).
We thank Dr Guolu Yin from Aston Institute of Photonic Technologies, Aston University who provided insight and expertise that greatly assisted the research.
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