The measurement of reflective index (RI) has played an important role in physical, chemical, and biomedical areas. The optical fiber-based RI sensors have been intensively investigated and widely applied due to their unique advantages such as simplicity, immunity to electromagnetic interference, corrosion resistance, and remote sensing capability. Most of them are based on fiber gratings, Fabry–Pérot (FP) or Mach–Zehnder (MZ) interferometer, micro fiber, selectively infiltrated photonic crystal fiber (PCF) coupler, deposition of a thin film over the sensing element, and many other structures^{[1–11" target="_self" style="display: inline;">–11]}. The FP interferometer (FPI)-based sensors have been much more extensively studied owing to their characteristics of simple configurations, compactness, stable performances, and endurance for high temperature and high pressure environments, etc. RI sensing cannot be carried out reliably without the simultaneous measurement of temperature. In recent years, optical fiber-based RI sensors with temperature compensation have been extensively investigated^{[5,7,8,1214" target="_self" style="display: inline;">–14]}.

In this Letter, an FPI is presented. The sensing head of the interferometer consists of a closed air cavity and a section of PCF. The former responds to the envelope of the reflective spectrum, and the latter contributes to the high-frequency fringe spectrum. The spectral response of the interferometer with different lengths of PCF is analyzed. The frequency characteristics of reflective spectra when the interferometer is dipped into solutions with different RIs are studied. The influence of temperature on reflective spectra is also studied. Taking interaction into consideration, a matrix equation is used to demodulate RI and temperature simultaneously.

The FPI was made by splicing a standard single mode fiber (SMF) with a solid-core PCF whose cross section is shown in Fig. 1(a) (core diameter: 7.0 μm, cladding diameter: 125 μm). The splicing was performed on a fusion splicer (Fujikura 80S) and the discharge power and time were −40dB and 1000 ms, respectively. The discharge position was on the SMF, about 100 μm from the end, in order not to destroy the end face of the PCF. Such fusion parameters ensure complete collapsing of the air holes in a limited region. After three discharge times, air holes in the cladding of the PCF collapsed and an elliptical air cavity was formed in the fiber. The collapsed region can be regarded as pure silica. Subsequently, the PCF was cut off with a small section remaining, as shown in Fig. 1(b). An optical sensing analyzer si720 provided by Micron Optics was used to act as both the light source and the spectral receiver. The scanning laser ranging from 1510 to 1590 nm was launched to the sensor from the SMF. The reflective spectra before cleaving (depicted with a black solid line), and when the remaining PCF are 1 and 9 mm after cleaving (dotted in blue and red, respectively), are shown in Fig. 2.

Fig. 1. Microscopic image of (a) cross-sectional image of PCF and (b) sensing head.

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Fig. 2. Reflective spectra of sensors with different lengths of PCF.

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We found out that before cleaving the reflective spectrum is influenced only by the closed air cavity. Taking both dips as λ1 and λ2, we calculated that the length of the air cavity is about 19.1 μm by using L=λ1λ22n(λ2−λ1),(1)which is almost the same as the measured value (18.85 μm). However, the other cavity was formed when a certain length of PCF remained after cleaving. The reflective spectra consist of an envelope and a high-frequency fringe spectrum whose frequency varies with the length of the remaining PCF. In order to find out where the fringe spectrum comes from, we analyzed the schematic diagram of the sensing head and the propagation of the light.

As shown in Fig. 3, after being reflected by the air cavity, the rest of the incident light is coupled to the PCF through the collapsed region. Lots of higher modes are excited at the beginning of the PCF, including the cladding modes. But the cladding modes vanish gradually through the propagation distance due to the power dissipation. As a result, the guided modes propagating in the PCF are the core modes, including the high-order modes. They are reflected by the PCF end face cleavage, and are coupled to the SMF again through the collapsed region. Taking the interference between the fundamental mode (LP01) and the high-order mode (LP31)^[15] into consideration, we calculated that the FP cavity should be 412 mm (n=3.51×10−2^[15]) when the interference spectrum is the same as the one dotted in red in Fig. 2. But it was not coincident with the experimental result (9 mm). So we inferred that the fringe spectrum was caused by the silica cavity (L2), which was composed of the collapsed region and the PCF, as illustrated in Fig. 3. With n=nsilica=1.457, we calculated that the cavity length should be 9.99 mm, which was well coincident with L2 in the experiment (9.78 mm).

Fig. 3. Schematic diagram of the sensing head.

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Fig. 4. Simulated reflective spectrum of the interferometer.

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There are three reflection surfaces in the sensing head, as follows: SMF-air, air-silica, silica-surrounding with power reflection coefficients of R1, R2, and R3, respectively. R1 and R2 are equal to (n0−1)2/(n0+1)2=0.034, R3=(n0−nLiq)2/(n0+nLiq)2, with n0 being the RI of pure silica and nLiq being the RI of the measured solution. The total reflected field from the sensor is given approximately by the sum of the first-order reflected fields from the three surfaces. The total contribution from the high-order reflections is less than 0.1% because of the low reflection coefficients and therefore can be neglected^[6]. So the normalized reflection spectrum R(λ) is obtained as follows^[6,16]: $$\begin{gathered} R(λ)=|Er/Ei|2=R1+(1−α)2(1−R1)2R2+(1−α)2(1−γ)2(1−R1)2(1−R2)2R3 \\ −2R1R2(1−α)(1−R1)cos(4πL1/λ) \\ −2R1R3(1−α)(1−γ)(1−R1)(1−R2)⁢cos[4π(L1+n0L2)/λ)] \\ +2R2R3(1−α)2(1−γ)(1−R1)2(1−R2)⁢cos(4πn0L2/λ), \end{gathered}$$(2)where Ei is the incident wave field, Er is the reflected wave field, α and γ are the diffraction loss factors of air cavity and silica cavity, respectively. A phase shift of π takes place at the reflection surface of air and fiber because light is reflected from an optically dense medium to a thinner medium. As a result, if the refractive index of the solution to be measured (nLiq) is bigger than n0, there is also a π phase shift at the reflection surface of the PCF. From Eq. (2), we find out that the reflection coefficient R3, which depends on nLiq, only affects the amplitude of R(λ). So when the sensor is dipped into different solutions, the fringe contrast of the reflective spectrum will change, whereas the phase will remain unchanged.

The reflective interference spectrum is simulated with the parameters L1=18.75μm, L2=1531μm, n0=1.457, R1=R2=0.034, and R3=0.002 with nLiq=1.33, α=0.2, γ=0.32. The simulated result in wavelength range from 1510 to 1590 nm is depicted in Fig. 4, from which we can see that the reflective spectrum mainly consists of a low-frequency envelope and a high-frequency waveform. To analyze the characteristics of the interference pattern, the wavelength spectrum was Fourier transformed to a spatial frequency. The reflective spectra with nLiq in the range from 1.33 to 1.45 were analyzed after a fast Fourier transform (FFT).

Taking the parameter “Amplitude” as the Y axis, the FFT results in the spatial frequency domain are shown in Fig. 5. Amplitude is defined as Re2+Im2/n, where Re and Im are the real and imaginary part of the FFT data, and n is the number of sample points, respectively.

Fig. 5. Simulated FFT results of spectral responses with different nLiq.

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From Fig. 5, we can see that the first several peaks in the spatial frequency domain, which correspond to interferences in the air cavity, do not change with nLiq. But the peak near the frequency of about 1.8nm−1, which corresponds to interferences in the cavity of silica, decreases when nLiq increases. The relationship between the amplitude peak (P) near the frequency of about 1.8nm−1 and the RI of the solution is depicted in Fig. 6, from which we can see that the simulated P has a linear relationship with the RI.

Fig. 6. Simulated relationship between the P and RI of the solution.

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Given that the RI of a material itself exhibits a dependence on temperature, RI sensing cannot be carried out reliably without the simultaneous measurement of temperature. When temperature changes, the dip wavelength (λdip) of the reflective optical spectrum will shift and the amplitude peaks of the frequency spectrum will also change. If both parameters are perturbed simultaneously by the RI and temperature, the variation of the dip wavelength Δλdip caused by the temperature and RI can be expressed as^[12]Δλdip=Kλ,TΔT+Kλ,RIΔRI,(3)where Kλ,T is the sensitivity coefficient of the temperature and the unit is nm/°C, and Kλ,RI is the sensitivity coefficient of RI and the unit is nm/RIU. The variation of the amplitude peak (ΔP) of the FFT results caused by the variation in temperature and RI can be expressed as ΔP=KP,TΔT+KP,RIΔRI,(4)where KP,T is the sensitivity coefficient of the temperature with units of /°C, and KP,RI is the sensitivity coefficient of RI with units of /RIU. The equation of the sensitivity matrix can be written as^[12,13][ΔλdipΔP]=[Kλ,TKλ,RIKP,TKP,RI][ΔTΔRI].(5)The equation of the sensitivity matrix can be transposed as [ΔTΔRI]=1D[KP,RI−Kλ,RI−KP,TKλ,T][ΔλdipΔP],(6)where D=Kλ,TKP,RI−KP,TKλ,RI, and Kλ,T, Kλ,RI, KP,T, and KP,RI are regarded as constant values and can be determined by measuring the temperature and wavelength response separately for Δλdip and ΔP. The temperature and RI variations can thus be evaluated by Eq. (6).

Several all-fiber optical FPIs with different lengths of air cavity and PCF were fabricated. Two reflective spectra in the air at 10°C are shown in Fig. 7, with parameters as follows: for sensor1, L1=15.6μm, L2=925μm; for sensor2, L1=6.2μm, L2=1400μm. The solid line represents the reflective spectrum of sensor1 and the dotted line represents sensor2. To calibrate the sensor, we immersed sensor1 into the prepared sugar solution with different RI, and we found that the fringe contrast of the reflective spectrum decreases with RI. The FFT results of the reflective spectra with different RIs are shown in Fig. 8.

Fig. 7. Reflective spectra of two sensors with different L1 and L2.

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Fig. 8. FFT results of spectral responses with different solution RIs.

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Subsequently, the sensor was placed in an oven and the temperature was twice increased from 15°C to 75°C and then decreased to 15°C.

Assuming ΔPRI is the difference between the measured P in the air and in different solutions, ΔRI is the difference between the RI of the solution and the air, the experimental ΔPRI were dotted in Fig. 9. Assuming ΔPT is the difference of the measured P at different temperatures and at 10°C, the experimental ΔPT values were marked with square points in Fig. 9. After linear fitting, we found out that the parameter ΔP increases almost linearly with ΔRI, with the sensitivity of 5.30/RIU, and ΔP also increases linearly with ΔT, with the sensitivity of 6.80×10−4/°C. Consequently, the parameters KP,T and KP,RI in Eq. (5) are 6.80×10−4/°C and 5.30/RIU, respectively. In this Letter, the solution samples were characterized through an Abbe refractometer.

Fig. 9. Comparison of ΔP caused by ΔRI and ΔT.

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The relationship between Δλdip and ΔRI, together with the relationship between Δλdip and ΔT were explored in this Letter. The reflective spectra of the sensor after a low-pass filter with a cutoff frequency of 0.15 Hz at temperatures of 15°C, 20°C, and 25°C were shown in Fig. 10. The dip wavelength shifts to short wavelength with the increase of temperature. Assuming ΔλRI is the wavelength shift caused by ΔRI, the experimental ΔλRI were dotted in Fig. 11. Assuming ΔλT is the wavelength shift caused by ΔT, the experimental ΔλT were marked with square points in Fig. 11. After linear fitting, we found out that the parameter Δλdip increases almost linearly with ΔRI, with a sensitivity of 8.46×10−1nm/RIU and Δλdip also increases linearly with ΔT with a sensitivity of 2.48×10−3nm/°C. Consequently, the parameters Kλ,T and Kλ,RI in Eq. (5) are 2.48×10−3nm/°C and 8.46×10−1nm/RIU, respectively, and Eq. (5) can be written as follows: [ΔλdipΔP]=[2.48×10−38.46×10−16.80×10−45.30][ΔTΔRI].(7)

Equation (6) can be written as [ΔTΔRI]=[4.05×102−6.55×10−4−5.25×10−32.00×10−6][ΔλdipΔP].(8)

Fig. 10. Reflective spectra at different temperatures.

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Fig. 11. Comparison of Δλdip caused by ΔRI and ΔT.

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Equation (8) can be applied to simultaneously determine the surrounding RI and temperature, as long as the wavelength shift Δλdip and ΔP are measured.

In conclusion, a simple optical sensor is proposed and demonstrated that can be used to simultaneously measure RI and temperature. The sensing head is made by splicing a small section of PCF with SMF, with an ellipsoid air cavity reserved. The interference model is analyzed theoretically and verified experimentally. ΔP from the FFT analysis and Δλdip of the envelope after the FFT low-pass filter can be measured. An amplitude sensitivity of 5.30/RIU after the FFT and a wavelength sensitivity of 8.46×10−1nm/RIU are experimentally achieved over the RI range from 1.34 to 1.43. For temperature measurements, an amplitude sensitivity of 6.80×10−4/°C and a wavelength sensitivity of 2.48×10−3nm/°C are achieved. Easy fabrication, low cost, good temperature compensation, and dual-parameter measurement make it appropriate for practical applications.