A highly sensitive sensor of methane and hydrogen in tellurite photonic crystal fiber based on four-wave mixing

A tellurite photonic crystal fiber (PCF) sensor structure is proposed for simultaneous measurements of methane and hydrogen. The structure is a simple hexagonal three-cladding structure, and six air holes in the inner cladding are coated with methane sensitive film and hydrogen sensitive film respectively. Based on the degenerate four-wave mixing (DFWM) theory, the direct relationship between the wavelength shifts of the Stokes spectrum or anti-Stokes spectrum and the variations in gas concentration can be established to realize the accurate detection of gas concentration. The influences of the pump wavelength and gas-sensitive film thickness on the gas sensitivity are investigated, and the maximum sensitivities of methane and hydrogen after parameter optimization are − 2.052 nm/% and − 0.236 nm/%, respectively. The linearity of the fitting can reach up to 99.95%, and the low detection limit of methane is 450 ppm and hydrogen is 2500 ppm. The sensing method based on four-wave mixing in non-silica photonic crystal fiber can also be extended to other detections of gas mixtures in the mid-infrared field.


Introduction
Due to the particularity of structure, photonic crystal fibers (PCF) have been widely used in the sensing field because of their various advantages such as endlessly singlemode (Baselt et al. 2019), controllable dispersion characteristic (Cheng et al. 2014), large mode field area (Petersen and Alkeskjold 2015), and high nonlinearity (Krishna, et al. 2020). Many gas-sensing methods based on PCFs have been reported, including interference (Ma et al. 2019), grating (Yan et al. 2019), functional filling , and surface plasmon resonance (Lu et al. 2014). However, most of the above sensors can only detect a single component gas and have complicated structures. The four-wave mixing (FWM) effect can solve these problems and obtain high sensitivity, the Stokes and anti-Stokes peak shifts can provide two independent sensing channels 1 3 215 Page 2 of 17 to realize double parameter sensing. Since 2011, Frosz (Frosz and Stefani 2011) first demonstrated the feasibility of using the FWM effect for the measurement of refractive index, and many researchers have conducted related investigations. In 2012, Gu et al. (Gu et al. 2012) proposed a nonlinear optical fiber strain sensor by measuring the displacements of the Stokes peak and anti-Stokes peak which originated from the refractive index (RI) changes caused by strain. In 2018, N. Nallusamy et al. (Nallusamy and Amunadevi 2018) proposed a highly sensitive nonlinear fiber RI sensor using the DFWM technique. The proposed sensing mechanism was analysed in both anomalous and normal dispersion regimes. That same year, they designed a D-Shaped plasmonic PCF sensor that simultaneously measures both seawater temperature and salinity with the salinity sensitivity of 1.6 nm/%(kg/kg) and temperature sensitivity of 12.31 nm/•C . In 2019, Geng et al. (Geng et al. 2019) designed a micro-fluid refractive index sensor with a fiber length of only 60 mm, RI sensitivity of 6238.9 nm / RIU, and RI sensor resolution of 2.2 × 10 −5 RIU. All of the above studies adopt quartz fibers, and the transmission window of quartz fibers is 0.38-2.3 um. When the wavelength of the transmitted light wave exceeds 2.3 μm, the transmission loss will increase sharply (Steinmeyer and Skibina 2014). Therefore, such quartz material cannot be used in mid-infrared applications.
Compared with quartz materials, tellurite glass has attracted wide attention due to its unique characteristics, including a wider infrared transmission range, greater nonlinear refractive index, better insulation constant, damage threshold, and better third-order nonlinear optical performance (Wang and Vogel 1994;Sun et al. 2020). The refractive index (RI) of tellurite glass is about 2.0, and its nonlinear coefficient is one order of magnitude larger than that of quartz glass, which is beneficial for enhancing the nonlinearity. The zero-dispersion wavelength (ZDW) of tellurite glass is approximately 1.85 μm, and fibers with the ZDW within the range of 1.5-2.0 μm can be obtained by adjusting the fiber structure, which more easily meets the phase matching conditions to generate the FWM. That is to say, to detect Stokes and anti-Stokes signals, most siliconbased sensors require a very high pump peak power of 1-10 kW, while tellurite glass only requires much lower pump power to excite the FWM under the phase-matching condition. In 2020, Sun et al. (Sun et al. 2020) realized temperature sensing by using FWM in tellurite photonic crystal fibers, and the temperature sensitivity of the sensor was 0.7 nm/′C at the pumped wavelength of 3550 nm. Moreover, due to the RI of quartz material (1.44) and hydrogen sensitive film (1.99), ordinary filled quartz fiber cannot be used for hydrogen detection, unless using surface plasma resonance (Tabassum and Gupta 2015;Liu et al. 2018) or Bragg grating (Masuzawa et al. 2015;Fisser et al. 2019). Therefore, the PCFs made of tellurite glass can easily realize the detection of hydrogen and methane concentrations by functional filling method without considering special structures, avoiding leakage loss caused by fiber post-processing.
In this paper, a new PCF sensor based on the DFWM effect is proposed by tellurite glass. The methane and hydrogen gas-sensitive film are plated into the air hole of the inner cladding to detect the concentrations of methane and hydrogen simultaneously. When the gas mixture passes through, the refractive index of the gas-sensitive film changes, which affects the dispersion and nonlinear coefficient of the propagation mode of the PCF and causes the gain peak of the Stokes and anti-Stokes spectrum to shift. The final sensitivity of methane is -2.052 nm/% and that of hydrogen can reach -0.236 nm/%. The sensor has a simple structure and high sensitivity, which applies to the detection of other gas mixtures.

The theory of DFWM
Four-wave mixing is caused by the third-order nonlinear polarizability (3) of the medium. Only when the phase mismatch is almost zero can arise significant FWM effect, namely, phase matching. Under normal circumstances, two new photons will be produced when two photons are annihilated in the FWM process: In the case of 1 ≠ 2 , it is necessary to use two beams of pump light to generate an FWM. However, if 1 = 2 , only one pump beam is needed to excite the FWM, and the FWM in optical fibers usually adopts this degenerate case (DFWM). To understand the sensing mechanism of DFWM, we can use linear stability analysis, ignoring the Raman effect, self-steepening effect, and linear loss to obtain the following nonlinear Schrödinger equation (MNLSE) (Ott et al. 2008), where A is the normalized light pulse amplitude, z is the propagation distance, t is time, and = 2 n 2 A eff is the nonlinear coefficient. Here, A eff is the effective mode field area, which can be expressed by Formula (3).
In Formula (2), n is the nth order dispersion parameter, which is the Taylor series expansion coefficient of the propagation constant ( ) at the central angular frequency, and its definition is: By applying linear stability analysis to the nonlinear Schrodinger equation for the long picosecond pulse, the formula for Stokes and anti-Stokes optical parameter signal gains in nondestructive media can be obtained. Considering the loss of light wave in the propagation process, the influence of loss on gain needs to be considered, so the gain formula is shown as Formula (5) (Agrawal 2006), where is the limit loss; p 0 is the peak power of the pump; L is the length of optical fiber; L eff is the effective length, and the formula is calculated by L eff = 1 − exp (− L) ∕ . The loss is given by Formula (6), where Im n eff is the imaginary part of the effective index of refraction. The gain coefficient g is ln 10 Im n eff × 10 6 (dB∕m) nonlinear phase mismatch 2 P 0 . The linear phase mismatch can be represented by the Taylor expansion at the pump frequency as is shown in Formula (7), where Δ s is the frequency shift between pump frequency and Stokes light frequency or anti-Stokes light frequency; 2 and 4 are the dispersion coefficients of the second-order and the fourth-order at the pump wavelength, respectively, which can be calculated by formula(4).
It can be seen from the above formulas that if the gas concentration varies, it will affect the effective refractive index ( n eff ) of the PCF, and then change the , 2 and 4 , finally leading to the peak shifts of the Stokes and anti-Stokes spectra. The gas concentration can be measured by detecting the peak movements of the Stokes and anti-Stokes spectra.

Simulation results and analysis
The PCF is a conventional hexagonal structure with three cladding layers, which is simple and easy to implement, as is shown in Fig. 1. The air hole distance is Λ=3 m , air hole diameter is d = 1.45 m , the thickness of methane gas-sensitive film is t 1 , and the thickness of hydrogen gas-sensitive film is t 2 .
The substrate is made of tellurite glass TeO 2 -ZnO -Na 2 O -Bi 2 O 3 (TZNB), with a wider middle-infrared transmission window and larger nonlinear coefficient. Its nonlinear refractive index n 2 = 5.9 × 10 −19 m 2 ∕W (Zhao et al. 2016), and the wavelength-dependent refractive index of the material can be described by Sellmeier Formula (8) (Zhao et al. 2016), The material refractive index is 2.005 at 1.55 μm and the ZDW is at 1.857 μm, but the ZDW can be manipulated to the short wavelength by fiber structure adjustment ). In addition, Formulas (9) (Liu et al. 2018) and (10) (Zhao et al. 2017) illustrate the linear relationship between the refractive index of gas-sensitive film and the concentrations of methane and hydrogen, respectively. It should be noted that all the linear relationships here are obtained at 1.55 μm, and only their linear relationships are selected to verify Cross-section diagram of the PCF sensor for methane and hydrogen the feasibility of the FWM sensing method in the mid-infrared region. Next, COMSOL software will be used to analyse the influence of structural parameters on the sensor.
The specific parameters of the PCF sensor are selected as Λ = 3 m,d = 1.45 m ,t 1 = 360nm,t 2 = 360nm , and fiber length L = 0.4m . After calculations, the ZDW of the proposed PCF is near 1705 nm. The pumping wavelength is selected as 1705 nm and the pump peak power is set as 20 W. Except for the length of PCF and the parameters of pumping source, the gain spectrum of the FWM also needs to know the phase mismatch =Δ +2 P 0 and loss . The final gain spectrum is described by Formula (5). Figure 2a-c show the curves of 2 , 4 and as the methane concentration increases from 0% to 3.5%.
As shown in Fig. 2a-c, 2 varies from − 3.25001 × 10 -4 ps 2 /m to − 4.77584 × 10 -4 ps 2 /m, 4 ranges from − 1.41702 × 10 -6 to − 1.41844 × 10 -6 ps 4 /m and changes from 0.140387 to 0.140522 W −1 ·m −1 , which is ten times of the nonlinear coefficient of silicon-based PCFs. The variation of these parameters will eventually affect the movement of Stokes and anti-Stokes peaks, thus achieving the measurement of methane concentration. Figure 2d, e show the phase mismatch curves and FWM gain curves as the methane concentration varies from 0 to 3.5%. As is can be seen from Fig. 2e, with increasing methane concentration, the Stokes spectrum has a blue-shift while the anti-Stokes spectrum has a red-shift. The Similarly, we change the hydrogen concentration from 0 to 3% without methane. Then, the parameters 2 , 4 and related to the hydrogen concentration can be obtained, 2 goes from − 3.25001 × 10 -4 to − 3.44014 × 10 -4 ps 2 /m, 4 shifts from − 1.41702 × 10 -6 to -1.41830 × 10 -6 ps 4 /m, and varies from 0.1403866 to 1,408,409 W −1 ·m −1 . To avoid repetition, we describe the corresponding phase mismatch and gain curves directly in Fig. 4a, b.
From Fig. 4b, we can draw the same conclusion as Fig. 2e. The Stokes peak blueshifts and anti-Stokes peak redshifts, despite the movement being slightly smaller. The wavelength variations and hydrogen sensitivity fitting of the gain peak are shown in Fig. 4c, d. The fitting degree of Stokes peak and anti-Stokes peak can also reach 99.9%, and the sensitivity of hydrogen is − 0.22 nm/% and 0.166 nm/%, respectively. Ultimately, the maximum sensitivities of methane and hydrogen under this set of parameters are − 1.685 nm/% and − 0.22 nm/%, respectively.

Parameter optimization
In this section, we will study the influence of various parameters of the PCF and pump light on gas sensitivity to select the optimal parameters. First, we investigate the effect of the pump wavelength. Supposing that methane concentration increases from 0 to 2% while the hydrogen concentration remains at 0%, and we select three pump wavelengths of 1705, 1706, and 1707 nm near the zero-dispersion wavelength for example with a pump peak power of 20 W. Figure 5a shows that the variation of Stokes and anti-Stokes peak decreases with each 1% change in methane concentration as the pump wavelength increases. The tendency can be more intuitively seen in Fig. 6a, b. Meanwhile, the gain peak decreases as the pump wavelength moves away from the zero-dispersion wavelength, and the wavelength positions of the two gain peaks gradually converge to the pump wavelength. The  same tendency occurs with the variation of hydrogen concentration. Obviously, the closer the pump wavelength is to the zero-dispersion wavelength, the higher the gas sensitivity is.
Second, we study the effect of pump peak powers on sensitivity. Three pump peak powers of 15 W, 20 W, and 25 W are selected with the pump wavelength of 1705 nm. From  Fig. 5b, the gain peak value increases, and the two gain peaks gradually move away from the pump wavelength with the increase of pump peak power. As shown in Fig. 6c, d, the wavelength offsets and sensitivity decrease. Considering the gain peak value and sensitivity, we choose the pump power of 20 W. At present, the peak power of 1.7um pulse laser can reach the level of several hundred watts (Khegai, et al. 2018), but further developments are needed.
Next, we investigate the influence of the thickness of two gas-sensitive films on the sensitivity. Here, the thickness of the hydrog film should keep unchanged ( t 2 = 360nm ), and the methane film varies from 360 to 400 nm. Figure 7a shows the movement of gain peaks corresponding to different thicknesses of methane film when methane concentration varies from 0 to 2%. As can be seen from Fig. 7a, under the condition of constant hydrogen film and methane concentration, the Stokes peaks of each group move to long wave direction and the anti-Stokes peaks move to short wave direction with increasing methane film thickness. When methane concentration changes, it can be clearly seen in Fig. 8a, that Stokes peaks blueshift and anti-Stokes peaks redshift. The shifts of the Stokes and anti-Stokes peaks all increase with the augment of methane film thickness. When the methane film thickness is 400 nm and the hydrogen film thickness is 360 nm, the offset of the gain peak is maximum.
Under the same conditions, the variations in the gain peak of hydrogen concentration can be obtained by changing the hydrogen concentration directly. Figure 7b shows the movements of gain peaks corresponding to different thicknesses of methane film as the hydrogen concentration increases from 0 to 2%. The Stokes and anti-Stokes peaks gradually move away from the pump wavelength as the thickness of methane film increases. Figure 8c, d indicate the same tendency as Fig. 8a, The offsets of the gain peak are in direct propotion to the methane film thickness and hydrogen concentration, but the increments are small. Similarly, when the thickness of methane film is 400 nm and the thickness of hydrogen film is 360 nm, the migration of the gain peak is the largest.
Next, we keep the thickness of the methane film unchanged at 400 nm, and the thickness of the hydrogen film varies from 360 to 400 nm. First, we keep the hydrogen concentration constant. The variation of degenerate FWM gain spectra under different methane concentrations (0-2%) are plotted in Fig. 9a. With the augment of hydrogen film thickness, the pump wavelength increases while the gain peak decreases. In addition, the Stokes and anti-Stokes wavelength shifts decrease with increasing methane concentration, as shown in Fig. 9b.
Then, we change the hydrogen concentration from 0 to 2% with the same thickness of the gas-sensitive film. It can be inferred from Fig. 10a that the Stokes and anti-Stokes wavelength shifts will be the same as Fig. 9a. Figure 10b shows that the Stokes and anti-Stokes wavelength shifts decrease with the augment of hydrogen concentration. This is because the refractive index of hydrogen gas-sensitive film is closer to that of the core material (2.0). Therefore, the hydrogen gas-sensitive film thickness has a great influence on the effective refractive index of the fundamental mode. With the augment of hydrogen gas-sensitive film thickness, ZDW moves to the long wave direction, and the pump wavelength should also redshift. The conclusion is obtained according to Fig. 5(a), the pump wavelength redshift will cause a decline in sensitivity, which cancels out the part of sensitivity that increases with the augment of gas-sensitive film thickness. Overall, the gas sensitivity is in inverse proportion to the hydrogen gas-sensitive film thickness.
The specific sensitivity changes are listed in Table 1. Taking the above factors into consideration, the numerical values shown in Table 2 are selected as the optimal structural parameters of the sensor structure.
According to the above parameters, we can obtain 2 , 4 and of the varying methane concentration after parameter optimization. These variation curves are shown in Fig. 11a-c. With different methane concentrations, the calculated PCF parameters 2 , 4 and at  1705 nm vary linearly from − 9.9641 × 10 -5 to − 2.67071 × 10 -4 ps 2 /m, from − 1.41166 × 10 -6 to − 1.41329 × 10 -6 ps 4 /m and from 0.14033 to 0.14049 W −1 m −1 , respectively. It is can be observed from Fig. 12a, at the wavelength of phase mismatch = 0 corresponds to the peak wavelength of Stokes lines and anti-Stokes lines under different gas concentrations. In addition, as is shown in Fig. 12(b), the Stokes lines have a blue-shift and anti-Stokes lines have a red-shift with the augment of methane concentration, and both of the peak intensities increase. The specific movements are described in Fig. 12c, d, which indicate the peak wavelength of gain signal and methane sensitivity fitting curves when the methane concentration varies from 0 to 3.5% without hydrogen gas at the pumping wavelength of 1705 nm. The Stokes and the anti-Stokes lines shift from 1839.09 to 1831.91 nm and 1589.14 to 1594.53 nm, respectively. The final results indicate that the linear fitting degree of the Stokes and anti-Stokes peaks can reach 99.95%, and the sensitivity coefficients for methane are k 1 = − 2.052 nm/% and k 3 = 1.542 nm/%, respectively. Similar to the impact of methane concentration, we can also obtain the variations of PCF parameters under different hydrogen concentrations, as is shown in Fig. 13. As shown in Fig. 11a-c, 2 shifts from − 9.9641 × 10 -5 to − 1.17787 × 10 -4 ps 2 /m, 4 varies from − 1.41166 × 10 -6 to − 1.4129 × 10 -6 ps 4 /m and changes from 0.14033 to 0.14078 W −1 m −1 . Then, we can obtain the phase mismatch curve and gain spectra. From Fig. 13a, b, the Stokes lines blueshift from 1839.09 to 1838.38 nm and the anti-Stokes lines redshift from 1589.14 to 1589.67 nm. The linear variations are represented in Fig. 13c, and Fig. 13d shows the hydrogen sensitivity fitting curves of Stokes lines and anti-Stokes lines when the hydrogen concentration varies from 0 to 3% without methane at the pumping wavelength of 1705 nm. The linearity can also reach up to 99.95%. The sensitivity coefficients for hydrogen are k 2 = − 0.236 nm/% and k 4 = 0.175 nm/%, respectively.
In addition, the variations of methane and hydrogen concentrations can be measured simultaneously using a 2 × 2 sensitivity matrix. According to the double-parameter demodulation method, matrix (11) is a coefficient matrix composed of the sensitivity values of methane and hydrogen. We can rewrite matrix (11) as matrix (12) by inverse transformation (Liu, et al. 2020), where Δ 1 and Δ 2 are the peak movements of Stokes and anti-Stokes gains, respectively. When the concentrations of methane and hydrogen change, the variations can be calculated by substituting the wavelength shifts Δ 1 and Δ 2 into matrix (12). Where, k 1 = −2.052 , k 2 = −0.236 , k 3 = 1.542 , k 4 = 0.175. Taking a two-dimensional parameter C 1 ∕%, C 2 ∕% as an example, we select (1% , 0.5%) as the initial state and (3% , 2.5% ) as the final state to verify whether the measurement method of the above sensor is feasible. From Fig. 14, we can obtain Δ 1 = − 4.51 nm, Δ 2 = 3.4 nm. After substituting it into matrix (12), we can determine that the result is (3.09%, 2.57%) within the margin of error. It is vigorously proved that the proposed structure can accurately measure methane and hydrogen concentrations simultaneously.
In addition, the resolution of the dual-parameter sensor proposed in this paper can be defined as Eq. 13 (Wu and Guan 2011), where ΔC expresses the variation of methane/hydrogen concentration, Δ min is 0.2 nm, Δ peak is the wavelength shift. The calculated methane and hydrogen resolutions are 450 ppm and 2500 ppm, respectively.
13) R = ΔC × Δ min Δ peak Fig. 12 a Curves of the phase mismatch at different methane concentrations; b Stokes and anti-Stokes spectrum gain curves (the methane concentration is 0-3.5%); c Peak wavelength shift curves; d Sensitivity fitting curve at the pump wavelength of 1705 nm  Table 3 shows some reported sensors based on photonic crystal fiber published. After comparison, the proposed dual-parameter sensor based on the DFWM effect has obvious advantages in terms of both sensitivity and linearity. Meanwhile, such gas sensing method using FWM in non-silica PCF can be extended to other gas detections.

Conclusion
A methane and hydrogen sensor based on Tellurite PCF is proposed and optimized. The structure is a simple three-clad hexagonal structure, and methane and hydrogen sensitive films are coated onto the six air holes of the inner cladding. By adopting the DFWM effect to measure the peak movements of Stokes spectrum and anti-Stokes spectrum caused by the change of gas concentration, the concentrations of methane and hydrogen can be accurately measured. After parameter optimization, the sensitivity of methane can reach up to − 2.052 nm/% and that of hydrogen is − 0.236 nm/%. The measurement method can be further extended to the detection of some specific gas concentration in other gas mixtures.