## Abstract

A dual-mode long period fiber grating (DM-LPFG) sensor for simultaneous measurement of the surrounding refractive index (SRI) and temperature is proposed and demonstrated theoretically. It is made by inscribing a LPFG based on the dual-peak resonance in a dual-mode fiber. Known from coupled mode theory, the sensing principle of the sensor is originated from the sensibility of the dual peaks to the SRI. There are three peaks corresponding to the dual-peak resonance and single-peak resonance in different core modes in the transmission spectrum, respectively. And the three peaks have a different response to the SRI and temperature. Through the monitoring of the distance change between dual peaks and the wavelength shift of single peak, the dual-wavelength matrix method can be used to simultaneously measure the SRI and temperature. As the results show, the SRI sensitivities corresponding to the distance between the dual peaks and the resonant wavelength of the single peak are 1940.0nm/RIU and 590.0nm/RIU in the SRI range of 1.33 to 1.37, respectively. Besides, the temperature sensitivities corresponding to the distance between the dual peaks and the wavelength of the single peak are -0.42nm/°C and 0.23nm/°C with the temperature changing from 20°C to 120°C, respectively. It has great potential application in chemical and biological sensing fields, owing to its advantages of high sensitivity and multi-parameter measurement.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Optical fiber grating sensors have many unique advantages and have been extensively investigated in measuring various parameters [1–2], such as temperature, surrounding refractive index (SRI), curvature, strain, pressure, pH and so on. Long period fiber grating (LPFG) is a kind of grating with a period from tens to hundreds of micrometers, which can couple the core modes to the cladding modes [3]. Because of the strong interaction between the cladding modes and the surrounding medium, various LPFG-based devices have high sensitivity to the change of surrounding medium. For a LPFG, improving sensitivity and multi-parameter simultaneous measurement are two important research topics.

There are many schemes to improve the sensitivity of LPFG sensors. The dual-peak resonance based on higher order cladding mode in a LPFG is one of the most effective methods, and has attracted researchers’ attention. D. Villar et al. [4] proposed that the dual-peak resonance between the core mode and a higher order cladding mode, and the sensitivity of this LPFG sensor for strain, SRI [5], and temperature is higher than that of the sensor based on single peak. M. Smietana et al. [6] reported the LPFG based on the dual-peak resonance near PMTP has high SRI sensitivity. In addition, such LPFG sensor based on dual-peak resonance can also be used to detect film parameters [7], bacteria [8], solution concentration [9] and so on.

Simultaneous measurement has become a hot issue, because the cross sensitivity should not be neglected for LPFG sensor. In the past few years, there are various kinds of LPFG-based schemes for simultaneous measurement, such as cascaded LPFG [10], ultralong-period fiber grating [11] and so on. W. Zhang et al. [12] proposed the cascaded LPFG sensor for dual-parameter measurement, and the sensor consists of two LPFGs with different fiber types and different grating periods. Experiments show that the two peaks respond to the change of strain and temperature. By building the sensitivity matrix, the proposed sensor can be efficiently used for dual-parameter measurement. A. Zhang et al. [13] presented a simultaneous measurement of SRI and temperature by using a sandwiched structure of LPFGs, which consists of three gratings with different periods. And the measurement sensitivity of the sensor to the SRI and temperature are -27.77nm/RIU and -34nm/°C, respectively. J. Li [14] et al. demonstrated an efficient fiber-optic sensor based on cascaded LPFG and S-fiber taper for simultaneous measurement of SRI and temperature. There are different resonant peaks corresponding to the coupling of LPFG and Mach-Zehnder interference (MZI). The SRI and temperature sensitivities of LPFG resonant peaks are -52.57nm/RIU and 45.87pm/°C, and the SRI and temperature sensitivities of the MZI resonant wavelengths are 311.48nm/RIU and 12.87pm/°C. Different SRI and temperature sensitivities are helpful to build a sensitivity matrix to simultaneously measure SRI and temperature. In conclusion, these multi-parameter measurement schemes can be used to realize the double-parameter measurement, but most of them are based on the cascaded structure or special type of fiber structure, which requires complex fabrication process, and generally has low sensitivity.

LPFG can be fabricated both in a single-mode fiber and multi-mode fiber (MMF). Although the MMF offers the advantage of low fiber nonlinearity, it is not suitable for sensing because of the influence of intermodal crosstalk [15]. Few-mode fiber (FMF) has been recently presented as the number of guided modes is much smaller than MMF for alleviating the intermodal crosstalk problem. At present, there are few reports about the sensitivity characteristics for few-mode LPFG. G. Chen et al. [16] analyzed the sensing characteristics of the four-mode LPFG by the coupled-mode theory. The four-mode LPFG was fabricated by the CO_{2} laser, and the temperature characteristics were measured in water. The temperature sensitivity is −43.98pm/°C. Q. Wang et al. [17] presented a four-mode LPFG sensor, which is sensitive to SRI and the SRI sensitivity is 863nm/RIU. And J. Dong et al. [18] reported that the LPFG with a period of 145µm has the temperature sensitivity of 115.1pm/°C, which is inscribed in FMF by CO_{2} laser-irradiation method. The reported few-mode LPFG sensors are mostly about single environmental parameters rather than multi-parameter measurement, besides, the sensitivity is still low. The FM-LPFG can be used not only in single parameter sensing field, but also in multi-parameter measurement field. L. Wang et al. [19] proposed a sensor by cascading a LPFG in FMF and SMF for simultaneous detection of strain and temperature. The strain and temperature sensitivity of the few-mode LPFG sensor are −2.9pm/µɛ and −17.6pm∕°C, respectively. Q. Liu et al. [20] pointed out a superimposed FM-LPFG sensor, which is sensitive to SRI but insensitive to temperature. In general, the sensitivity of the above-proposed FM-LPFG sensor is not high enough. In order to improve the sensitivity, the FM-LPFG sensor based on the dual-peak resonance is presented in our paper.

In our paper, a dual-parameter sensor for simultaneous measurement of SRI and temperature has been proposed. The device is based on the DM-LPFG and the dual-peak resonance. Firstly, the coupling-mode theory of the DM-LPFG is introduced, and the phase matching equation is given. Secondly, the dual-peak resonance of core mode LP_{01} and the transmission characteristics are analyzed. The coupling between the core mode LP_{11} and the cladding modes show single-peak resonance when the dual peaks corresponding to LP_{01}-EH_{1,22} occur in the wavelength range of 1200-1800nm. In addition, the dual peaks and single peak have disparate spectral response to the change of SRI and temperature. Furthermore, by analyzing the sensitivity differences between the resonant peaks, the scheme to simultaneously measure SRI and temperature is proposed. The proposed sensor has the advantages of high sensitivity, good linearity and simplified structure, which can be extended to multi-parameter sensing.

## 2. Principle

#### 2.1 Mode coupling theory for DM-LPFG

The DM-LPFG has a similar grating structure to that of single mode long period fiber grating (SM-LPFG). Therefore, the two LPFGs have the similar numerical solution method [16]. The coupled-mode theory is also applicable to the transmission characteristics of DM-LPFG. For a DM-LPFG, the coupled equations that describe the coupling between the core modes $L{P_{mn}}$ ($L{P_{01}}$ and $L{P_{11}}$) and the $({1\nu } )$ cladding mode can be written as:

By solving the coupled-mode Eqs. (1) and (2), a numerical value of transmission can be calculated according to Eq. (5).

where is the amplitude of the core mode $L{P_{mn}}$ on the starting point of the fiber grating along the*z*axis, and

*L*is the length of the grating in fiber.

#### 2.2 Dual-peak resonance

Based on the coupled-mode theory, the coupling characteristics of DM-LPFG are analyzed. The coupling-mode equations are solved by the following parameters: the radii of core and cladding are 6µm and 62.${A^{L{P_{mn}}}}(0 )= 1$5µm, respectively; the refractive indices of core, cladding and surrounding are 1.458, 1.45 and 1.33, respectively. The average index change of the grating is 4×10-4, and the length of grating is 2.5cm. Under such fiber parameters, the normalized frequency $V = 3.7097$ at wavelength 1550nm [22]. There are two mode groups in the FMF.

Within a certain wavelength range, the coupling between a core mode and a higher-order cladding mode possibly generate dual peaks in the transmission spectrum. According to Eq. (4), the phase-matching curves (PMCs) of core modes and cladding modes (LP_{01}-HE_{1,21}, LP_{01}-EH_{1,22}, LP_{11}-HE_{1,25}, LP_{11}-EH_{1,26}) can be calculated and demonstrated in Fig. 1(a). There is only one intersection (1446nm) between the PMC of LP_{01}-HE_{1,21} cross coupling and the vertical line corresponding to the period 108.6µm in the wavelength range of 1200nm-1800nm. It means that one resonant peak corresponding to LP_{01}-HE_{1,21} would appear in the wavelength range 1200-1800nm, as shown in Fig. 1(b). The nearby PMC of LP_{01}-HE_{1,22} has two intersections with the vertical line, corresponding to the two peaks at 1530nm and 1740nm in the transmission spectrum. In general, the coupling between the fundamental mode LP_{01} and the even-order cladding mode of SM-LPFG is often ignored because of the small coupling coefficient. However, the coupling coefficients between the core mode LP_{01} and the even-order cladding mode could not be ignored [23]. In addition, the coupling coefficients between the higher-order core mode LP_{11} and odd-order cladding modes are much smaller than the coupling coefficients between the LP01 and even-order cladding modes. Therefore, there exists a resonant peak near 1644nm, which corresponds to the cross-coupling between LP_{11}-EH_{1,26}. And the two resonant peaks near 1284nm and 1692nm of LP_{11}-HE_{1,25} are larger. The coupling between LP_{01}-HE_{1,21}, LP_{01}-EH_{1,22}, LP_{11}-HE_{1,25} and LP_{11}-EH_{1,26} exhibits a quadratic PMC with a turning point where the slope of the PMC turns from positive to negative. And the turning point is phase-matching turning point (PMTP). And this PMTP corresponds to a specific grating period ${\Lambda _T}$ and resonant wavelength ${\lambda _T}$. To study the dual-peak sensing properties of LP_{01}-EH_{1,22} and other adjacent peaks, we define the peaks at 1446nm, 1530nm, 1692nm and 1740nm as left single peak P_{1}, left peak of dual peaks P_{2}, right single peak P_{3} and right peak of dual peaks P_{4}, respectively.

## 3. Sensing characteristics

For DM-LPFG, the resonant peaks show different sensing characteristics with the change of SRI and temperature. Combined with the difference, the dual-parameter detection method of SRI and temperature is proposed to solve the cross-sensitivity problem.

#### 3.1 SRI sensitivity

To study the response of DM-LPFG to SRI, the transmission spectra with different SRIs are illustrated in Fig. 2. As the SRI increases from 1.33 to 1.37, the left single peak P_{1} and the left peak of dual peaks P_{2} shift toward the shorter wavelengths, and the right single peak P_{3} and the right peak of dual peaks P_{4} shift toward longer-wave direction. For the resonant peak with the resonant wavelength less than ${\lambda _T}$ [24], it moves to the short-wave direction with the increase of SRI, as peaks P_{1} and P_{2}. And if the resonant peak with the resonant wavelength greater than ${\lambda _T}$, it moves to the long-wave direction with the increase of SRI, as P_{3} and P_{4}. The wavelength shifts of the four peaks are calculated to judge the SRI sensitivities, as shown in Fig. 3. In Fig. 3(a), the shifts of P_{3} and P_{4} are both positive, while the shifts of P_{1} and P_{2} are both negative. The resonant wavelength changes of P_{1}, P_{2}, P_{3} and P_{4} are -14nm, -30nm, 24nm and 48nm, respectively. Therefore, we can calculate the sensitivity by comparing the resonant wavelength distance between peak P_{2} and peak P_{4} and the resonant wavelength of peak P_{3}, which is shown in Fig. 3(b). It can be clearly seen that the SRI sensitivities of the resonant wavelength distance between P_{2} and P_{4} and the resonant wavelength of P_{3} are 1940.0nm/RIU and 590.0nm/RIU, respectively, and a linear fit with a high R^{2} of 0.998 and 0.985, respectively, are obtained.

#### 3.2 Temperature sensitivity

When the FM-LPFG is affected by temperature, the grating period and the radii of core and cladding will be changed by the thermal expansion effect. And the refractive index of core and cladding will change by the thermo-optic effect. As a result, the change of these parameters e cause the changes of period and effective refractive index, which will bring the change of resonant wavelength. To evaluate the sensitivity of the DM-LPFG sensor, Fig. 4(a) shows the transmission spectra with temperature range from 20°C to 120°C at the SRI 1.33. As the temperature increases, peak P_{2} moves toward a longer wavelength linearly and peak P_{3} and peak P_{4} move toward a shorter wavelength linearly. Figure 4(b) shows the resonant wavelength of P_{3} and the resonant wavelength distance between P_{2} and P_{4} versus temperature. The temperature sensitivity corresponding to the resonant wavelength of P_{3} and the temperature sensitivity of the resonant wavelength distance between P_{2} and P_{4} are 0.23nm/°C and -0.42nm/°C, respectively, and a linear fit with a high R^{2} value of 0.999 and 0.999, respectively, are obtained.

#### 3.3 Multi-parameter sensitivity

According to the different responses of resonant peaks to the measurands change, the measurands can be easily identified by the dual-wavelength matrix method [22]. For the proposed sensor, both the resonant wavelength distance between P_{2} and P_{4} (Δλ_{24}) and resonant wavelength of P_{3} (λ_{3}) almost change linearly with the temperature and SRI. The sensing matrix equation can be expressed as:

The matrix equation of the proposed sensor can be rewritten as:

_{3}, and $\Delta {\lambda _{24}}$ is the wavelength distance between peak P

_{2}and P

_{4}.

For the sensitivity matrix, the larger the condition number is, the greater the relative error of numerical solution will be. To reduce the condition number of the sensitivity matrix, the matrix can be written as follows:

## 4. Conclusions

Our paper has proposed a novel DM-LPFG sensor based on dual-peak resonance for simultaneous detection of SRI and temperature by theoretical simulation. As the SRI increases, the single-peak wavelength increases linearly and the distance between the dual peaks enlarges. And the SRI sensitivities corresponding to the distance between the dual peaks and the wavelength of the single peak are 1940.0nm/RIU and 590.0nm/RIU, respectively. As the temperature increases, the single-peak wavelength increases linearly and the distance between the dual peaks reduces. And the temperature sensitivities corresponding to the distance between the dual peaks and the wavelength of the single peak are -0.42nm/°C and 0.23nm/°C, respectively. The results show that the proposed sensor has enormous potential in some applications that require sensors with high sensitivities for dual-parameter monitoring, such as biochemical sensing and food safety monitoring.

## Disclosures

The authors declare no conflicts of interest.

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