Open Access
Add to BibSonomy
Abstract
In this paper, high-order LP modes based Sagnac interference for temperature sensing are proposed and investigated theoretically. Based on the specific high-order LP modes excited through the mode selective couplers (MSCs), we design a stress-induced Panda-type few-mode fiber (FMF) supporting 4 LP modes and construct a Sagnac interferometer to achieve a highly sensitive temperature sensor. The performances of different LP modes (LP01, LP11, LP21, and LP02) are explored under a single Sagnac interferometer and paralleled Sagnac interferometers, respectively. LP21 mode has the highest temperature sensitivity. Compared with fundamental mode (LP01), the temperature sensitivity based on LP21 mode improved by 18.2% at least. In addition, a way to achieve the enhanced optical Vernier effect is proposed. It should be noted that two Sagnac loops are located in two temperature boxes of opposite variation trends, respectively. Both two Sagnac interferometers act as the sensing element, which is different from the traditional optical Vernier effect. The temperature sensitivity of novel enhanced optical Vernier effect is magnified by 8 times, which is larger than 5 times the traditional Vernier effect. The novel approach avoids measurement errors and improves the stability of the sensing system. The focus of this research is on high-order mode interference, which has important guiding significance for the development of highly sensitive Sagnac sensors.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical fiber sensors [1–4] have attracted significant research owing to their prominent advantages including high sensitivity, compact size, immunity to electromagnetic interference, and stability in hostile environments. Driven by continuous advancements of optical fibers and related sensor devices, they have been applied in diverse fields and achieved a variety of physical quantity sensing and measurement, such as temperature [5], strain [6], liquid level [7], refractive index [8], transversal force [9], biomedicine [10], high-energy physics [11], gas detection [12,13], structural health monitoring [14], and marine environmental pollution tracking [15,16].
In the past decades, temperature sensors have been widely studied and applied. The temperature sensitivity of conventional high birefringent fiber as the Sagnac loops was reported approximately 0.94 nm/$^{\circ }$C [17]. Due to limitations in the inherent characteristics of silica, the sensitivity improvement of optical fiber sensors was restricted. There were three main approaches for enhancing the temperature sensitivity in optical fiber interferometric sensors. The first approach involved the introduction of a high thermo-optic or a high thermal expansion coefficient material into the fiber [18]. The second approach involved using the unique transmission mechanisms of special optical fibers [19]. The third approach combined the optical Vernier effect [20]. It is significant to explore other ways to improve the sensitivity of Sagnac interference sensors.
Bucaro and Carome first proposed the idea of mode-mode interference in the late 1970s [21]. In 2021, Dawei Du $et$ $al$. fabricated an isopropanol-sealed modal interferometer with LP$_{01}$ and LP$_{11}$ modes by a tapered two-mode fiber, which provides a temperature sensitivity of up to -140.5 nm/$^{\circ }$C [22]. Junlong Kou $et$ $al$. used focused ion beam (FIB) technology to process a micro-notch cavity on single-mode fiber (SMF) taper tips, where SMF taper tips are composed of an SMF and a multimode fiber (MMF) in essence. A reflective Fabry-Perot interferometer with a high sensitivity temperature sensor was developed using modal interference between LP$_{01}$ and high-order LP$_{0m}$. Its temperature sensitivity was approximately 20 pm/$^{\circ }$C [23]. Ruihang Wang $et$ $al$. introduced an axial offset at the FMF cross-section to excite the modal interference between LP$_{01}$ and LP$_{11}$, The optical respiratory monitoring sensor based on the Mach-Zehnder interferometer (MZI) can monitor important respiratory parameters [24]. Chenxu Lu $et$ $al$. used the single mode fiber-few-mode fiber-single mode fiber structure to fabricate a modal interferometer. The LP$_{01}$ and LP$_{02}$ modes within the FMF were excited by the fundamental LP$_{01}$ mode in the input SMF [25]. The humidity sensitivity of MZI up to 153.5 pm/${\% }$ relative humidity (RH). The above-mentioned interferometers do not include the Sagnac interferometer, they mainly utilize interferences between the fundamental mode and the higher-order LP modes. In theory, any type of interferometer can be used for temperature sensing, but there are differences in sensitivity. Sagnac interference also belongs to modal interference in nature. Generally, the birefringence between fast and slow axes (two orthogonal polarization fundamental modes) leads to a phase difference during transmission [13,26]. Because the fundamental mode is the easily transmitted mode in optical fibers. However, each mode in FMF has distinct propagation constants, resulting in different birefringence between two orthogonal polarization states. Thus, high-order LP modes are introduced in Sagnac interference to improve the sensitivity of optical fibers.
High-order LP modes with high mode purity selectively excited and steadily transmitted in the FMF were challenging. With the development of mode-division-multiplexed systems [27], many research works on fiber lasers that generate various LP modes have been published, such as spatial light modulators (SLMs) [28], fiber Bragg gratings (FBGs) [29–31], and photonic lanterns [32,33] for generating specific LP modes. Compared to reported fiber lasers, ring type core fiber was made into mode selective couplers (MSCs) [34–36], and they were integrated into spatial-mode switchable lasers [37]. The spatial-mode switchable lasers based ring fibers exhibited superior characteristics, including low crosstalk, low insertion loss, and high mode conversion efficiency. In addition, MSCs, photonic lanterns, or mode excitation technology were used to excite different LP modes for optical fiber sensing experiments [38–40]. To some extent, it provides a certain reference for the feasibility of the theory in this paper. Therefore, we decided to use MSCs to excite specific high-order modes in FMF.
In this paper, high-order LP modes Sagnac interference sensors are theoretically proposed. GeO$_2$ and B$_2$O$_3$ with a high coefficient of thermal expansion are considered to manufacture a stress-induced Panda-type FMF as sensing units in Sagnac interference. MSCs are used to excite specific high-order LP modes in the FMF. By utilizing high-order LP modes instead of fundamental modes, high temperature sensitivity Sagnac interference sensors are achieved. In addition, the application of the traditional and a novel enhanced optical vernier effect in temperature measurement is explored.
2. Design of the stress-induced PANDA-type few-mode fiber
The schematic cross-section and initial refractive index profile of the designed PANDA-type FMF are shown in Fig. 1(a). The inner and outer diameters of the PANDA-type FMF ring-type core are $D_1$ = 6.5 µm and $D_2$ = 8.5 µm, respectively. The stress rods and cladding diameter are $D_3$ = 33 µm and $D_4$=125.6 µm, respectively. The ring-type core is divided into two parts: one part is the yellow region, which is composed of SiO$_2$ doped a with 19% molecular fraction of GeO$_2$. The other part is the red region, which is composed of SiO$_2$ doped with a 20% molecular fraction of GeO$_2$. The stress rod is composed of SiO$_2$ doped with a 30% molecular fraction of B$_2$O$_3$ and the cladding is composed of silica. The distance between stress rods and the core center is 27.8 µm. The refractive indexes at 1550 nm of the ring-type core, stress rods, and cladding are $n_1$=1.4723, $n_2$=1.4738, $n_3$=1.4361, and $n_4$=1.444, respectively. The detailed refractive indexes of materials are referred to the following formula:
where $n_G$ is the refractive index of GeO$_2$ and $\lambda$ is the wavelength of the incident light, and its unit is µm. The constants values of GeO$_2$ [41] dispersion equation are $G_{A_1}$ = 0.80686642, $G_{A_2}$ = 0.71815848, $G_{A_3}$ = 0.85416831, $G_{l_1}$ = 0.06897261 µm$^2$, $G_{l_2}$ = 0.15396605 µm$^2$, $G_{l_3}$ = 0.15396605 µm$^2$. The constants values of B$_2$O$_3$ dispersion equation are $B_{A_1}$ = 0.690618, $B_{A_2}$ = 0.341996, $B_{A_3}$ = 0.898817, $B_{l_1}$ = 0.898817 µm$^2$, $B_{l_2}$ = 0.123662 µm$^2$, $B_{l_3}$ = 9.098960 µm$^2$. Silica dispersion [41] equation constants values are $S_{A_1}$ = 0.6961663, $S_{A_2}$ = 0.4079426, $S_{A_3}$ = 0.8974794, $S_{l_1}$ = 0.0684043 µm$^2$, $S_{l_2}$ = 0.1162414 µm$^2$, $S_{l_3}$ = 9.896161 µm$^2$. The above dispersion equations are applicable only when the three materials are undoped. However, if the material is doped, the dispersion equation of dopants is determined following the formula [41]:
Fig. 1. (a) Schematic cross-section and initial refractive index profile of the PANDA-type FMF (b) Von Mises stress of the designed Panda-type FMF.
Table 1. Doped material parameters used for numerical simulation.
The designed PANDA-type FMF is numerically simulated under geometry-induced birefringence and stress-induced birefringence. Geometry-induced birefringence is related to the geometrical structure of PANDA-type FMF, two stress rods are introduced on the x-axis. The difference in geometry between the x and y axes causes geometry-induced birefringence. Only geometry-induced birefringence is considered, and the magnitude of birefringence in the PANDA-type FMF is 10$^{-8}$. Generally speaking, the birefringence of polarization-maintaining fibers is on the order of 10$^{-4}$. Therefore, the effect of geometry-induced birefringence and stress-induced birefringence must be taken into account, which is more consistent with practical situations. Stress-induced birefringence is attributed to the utilization of doped materials with different thermal expansion coefficients in optical fiber manufacturing. After high-temperature annealing, thermal stress is generated inside the Panda-type FMF. Thermal stress transformed the initial isotropic effective refractive index distribution into an anisotropic distribution under the influence of the stress-optic effect. When SiO$_2$ is doped with GeO$_2$ and B$_2$O$_3$ in the PANDA-type FMF. The refractive index change of the PANDA-type FMF is calculated by the following equations [43].
At different operating temperatures, thermal stress leads to a birefringence change in the designed PANDA-type FMF, causing a shift in the interference dip. Figure 1(b) displays the Von Mises stress of the designed Panda-type FMF. This difference in thermal expansion coefficients of the ring core, stress rods, and cladding leads to the generation of thermal stress within the fiber. Thermal stress is mainly concentrated around the ring core and stress rods. In this paper, we studied temperature ranges between 20-25 $^{\circ }$C. However, for a clear understanding of Von Mises stress change with temperature, the color legend displays the magnitude of Von Mises stress from 20 to 60$^{\circ }$C sequentially from left to right. Panda-type fiber Von Mises stress decreased as temperature increased.
3. Results and disscussion
3.1 Single Sagnac interferometer
3.1.1 Operation principles
Figure 2 displays the device diagram for the PANDA-type FMF temperature measurement system by utilizing the between high-order LP modes interference and detailed MSC device structure diagram. The output light passes through an optical switching into a multiplexer (MUX). The multiplexer (MUX) integrates four MSCs with different LP modes, as is shown in Fig. 2(b), an MSC device made of an SMF and an FMF, which can convert the fundamental mode into higher-order LP modes. The output light passes through different MSC devices, and specific LP modes are selectively excited. All four modes (LP$_{01}$, LP$_{11}$, LP$_{21}$, and LP$_{02}$) exceed 90% coupling efficiency in the experiment at present [45]. The fundamental modes through MUX are almost completely converted into specific LP modes. MCSs with higher mode conversion efficiency can be manufactured with improved manufacturing technology. At this time, the energy of other modes is extremely small and negligible. The designed PANDA-type FMF supports multiple LP modes. The specific LP mode output light is evenly divided into two beams of equal energy by the few-mode (FM) coupler, which moves clockwise and counterclockwise along the optical path respectively. The polarization controller (PC) or lobe orientation controller (LOC) is integrated into the demultiplexer (DEMUX) to adjust the relative polarization state of the interference light in the Sagnac interferometer to optimize the transmission spectrum. Particularly spatial lobe orientation of the asymmetric modes such as LP$_{11}$ and LP$_{21}$ [36]. The PANDA-type FMF has different birefringence between its fast and slow axes under different modes, which results in a phase difference during light transmission. When the two beams of light are returned to the FM coupler, they are synthesized into a DEMUX, and high-order LP modes are reconverted into the fundamental mode and transmitted in SMF. The isolator (ISO) ensures one-way light transmission in the optical path. Finally, the light enters optical spectrum analyzers (OSA), forming a periodic interference dip. When the surrounding temperature changes, the interference dip shifts. The temperature is measured by detecting the interference dip movement. The designed PANDA-type FMF as the Sagnac loop is placed on a temperature controller box, which adjusts the surrounding temperature around the Sagnac loop.
The normalized transmittance ($T$) of the transmission spectrum of the Sagnac interferometer is described as follows:
where $\varphi$ represents the optical transmission phase difference between the fast and the slow axis under different LP modes.$B$ and $L$ are the birefringence and fiber lengths of the designed PANDA-type FMF, respectively.
where $n_{x}$ and $n_{y}$ are the refractive index at fast and slow axes of the designed PANDA-type FMF under different LP modes. When the phase difference meets the phase matching conditions (2$\pi \,m$=2$\pi \,B\,L$/$\lambda$). The maximum and minimum values of transmission spectrum appear, indicating the presence of an interference dip. where $m$ is an integer in the above equation. When the external temperature changes, the PANDA-type FMF thermal stress also changes. Due to the stress-optic effect, the birefringence of the PANDA-type FMF changes. Therefore, the interference dip will shift. In addition, the PADA-type FMF is also affected by the thermal expansion effect. As a result, fiber length also changes slightly.Differentiate Eq. (8) by $\Delta$ $T$,
The change of length $\Delta$ $L$ is caused by the thermal expansion effect, and the thermal expansion coefficient of silica is reported to be $0.54 \times 10^{-6}$/$^{\circ }$C. When the temperature rises uniformly by 1 $^{\circ }$C, the $\Delta \,L$ is extremely small and negligible. Therefore, Eq. (9) simplifies to the following equation.
Fig. 2. (a) Diagram of the PANDA-type FMF temperature measurement in a single Sagnac interferometer. (b) Detailed MSC device structure diagram.
3.1.2 Sensing characteristics and results discussion
Figure 3 shows the function between birefringence and wavelength at different temperatures under multiple LP modes. The birefringence of Panda-type fiber decreases as temperature increases. Combined with Fig. 1(b), it is observed that there exists a causal relationship between birefringence and thermal stress. Specifically, the birefringence of modes increases as the thermal stress increases. In addition, when the temperature is 20 $^{\circ }$C, the linear fitting function between birefringence and wavelength is also written out under different LP modes, which can be used to estimate the sensitivity magnitude.
Fig. 3. The function between birefringence and wavelength at different temperatures. (a) LP$_{01}$; (b) LP$_{11}$; (c) LP$_{21}$ (d) LP$_{02}$.
When lengths of the Panda-type FMF are 40 or 50 cm respectively, the change of the interference dip in the wavelength range of 1500-1630 nm under different modes is illustrated in Figs. 4 and 5. However, two Panda-type FMFs as sensing units are placed in temperature control boxes with opposite temperature variation trends, respectively. The interference dip of the 40 cm Panda-type FMF is a red shift as temperature decreases, while the interference dip of the 50 cm Panda-type FMF is a blue shift as temperature increases.
Fig. 4. Transmission spectrum at different temperatures when fiber length ($L$) = 40 cm. (a) LP$_{01}$; (b) LP$_{11}$; (d) LP$_{21}$ (e) LP$_{02}$; (c) Temperature sensitivity of different modes in a single Sagnac interferometer; (f) $B$ as a function of temperature under different modes.
Fig. 5. Transmission spectrum at different temperatures when fiber length ($L$) = 50 cm; (a) LP$_{01}$; (b) LP$_{11}$; (d) LP$_{21}$ (e) LP$_{02}$; (c) Temperature sensitivity of different modes in a single Sagnac interferometer; (f) $B$ as a function of temperature under different modes.
Figures 4(c) and 5(c) show the temperature sensitivity of different modes by using linear fitting, when the fiber length is 40 cm and 50 cm, respectively. When the fiber length is 40 cm, the temperature sensitivity of LP$_{01}$, LP$_{11}$, LP$_{21}$, and LP$_{02}$ are found to be 1.62857 nm/$^{\circ }$C, 1.69143 nm/$^{\circ }$C, 1.92571 nm/$^{\circ }$C, and 1.75429 nm/$^{\circ }$C, respectively. When the fiber length is 50 cm, the temperature sensitivity of LP$_{01}$, LP$_{11}$, LP$_{21}$, and LP$_{02}$ are found to be 1.62857 nm/$^{\circ }$C, 1.67429 nm/$^{\circ }$C, 1.92571 nm/$^{\circ }$C, and 1.8 nm/$^{\circ }$C, respectively. The LP$_{21}$ mode of the designed Panda-type FMF has the highest temperature sensitivity. Compared with the fundamental mode, the temperature sensitivity of the LP$_{21}$ mode is improved by 18.2%. Higher LP modes have larger effective mode field areas, which have stronger interaction with the surrounding environment [39,46,47].
The temperature sensitivity of sensors under different modes can be roughly estimated by Eq. (10). According to the Eq. (10), we will find that the sensitivity is related to $\frac {\lambda }{B}$ and $\frac {\Delta B}{\Delta T}$. We use linear fitting functions to describe these two quantities and draw Fig. 3, 4(f) and 5(f). Therefore, the Eq. (10) is transformed into the following equation. Figures 4(f) and 5(f) show the linear fitting function between birefringence and temperature under different LP modes. It is observed that birefringence decreases as the temperature increases. In addition, the slope of linear fitting functions is equal to the $\Delta \,B$/$\Delta \,T$.
3.2 Parallel Sagnac interferometers
3.2.1 Operation principles of traditional optical Vernier effect
The implementation of the Vernier effect [48] involves the integration of two interferometers, where the interferometric signals from interferometers are regarded as Vernier scales. To generate an envelope modulation when these interferometric signals overlap, two interferometric signals need to be slightly detuned. This envelope modulation, known as the Vernier envelope, provides enhanced sensing capabilities compared to the individual interferometer performance. In the traditional optical Vernier effect, one interferometer is placed in the temperature controller box as the sensing arm, and the other interferometer is placed in a fixed temperature as the reference arm, as shown in Fig. 6. The two interferometers are named "Sensor 1" and "Sensor 2", respectively. The fiber length of "Sensor 1" and "Sensor 2" were 50 cm and 40 cm, respectively. Temperature increases around "Sensor 1" while the temperature of "Sensor 2" is maintained at 20 $^{\circ }$C.
Fig. 6. Diagram of the PANDA-type FMF temperature measurement based on the traditional optical Vernier effect in parallel Sagnac interferometers.
The intensity of the superposed transmission spectrum of the sensing arm and reference arm by the parallel Sagnac loops can be expressed as:
where $I$ is the intensity of the superposed transmission spectrum. $I_s$ and $I_R$ are the the intensity of the sensing arm and reference arm, respectively. The transmitted intensity of the superposed transmission spectrum can be calculated by the following equation. where $I_{out}$ and $I_{in}$ are the intensities of the output light and the input light, respectively. The free spectrum range ($FSR$) of a single Sagnac interference spectrum is shown in the following equation:The length of fiber affects the $FSR$ size, and the $FSR$ size decides the detection limit. The detuning ($\delta$) represents the difference in optical path lengths between the two interferometers introduces the optical Vernier effect can be expressed as follows [49]:
where $L_1$ and $L_2$ are the lengths of the sensing arm and the reference arm, respectively.The $FSR$ of the Vernier envelope is expressed as a function of the $FSRs$ of two interferometers that comprise the system.
The magnification factor (M-factor) describes the optical Vernier effect. It establishes a relationship between the Vernier envelope and the interference signal of the sensing interferometer. The M-factor is defined as the ratio between the Vernier envelope $FSR$ and the sensing interferometer $FSR$. The M-factor is expressed as follows [50]:
3.2.2 Sensing characteristics and results discussion
The temperature sensitivity based on the traditional Vernier effect is shown in Fig. 7(e). The temperature sensitivity of LP$_{01}$, LP$_{11}$, LP$_{21}$, and LP$_{02}$ are 8.36 nm/$^\circ \text{C}$, 8.6685 nm/$^\circ \text{C}$, 9.88571 nm/$^\circ \text{C}$ and 9.33714 nm/$^\circ \text{C}$, respectively. LP$_{21}$ mode has the highest temperature sensitivity. Compared with the fundamental mode, the temperature sensitivity of the LP$_{21}$ mode is improved by 18.3%. The theoretical M-factor of the traditional optical Vernier effect is approximately 5 times, that is, compared with a single Sagnac interferometer, the temperature sensitivity is approximately improved 5 times. Results demonstrate the temperature sensitivity of LP$_{01}$, LP$_{11}$, LP$_{21}$, and LP$_{02}$ is improved 5.1 times, 5.2 times, 5.1 times, and 5.2 times, respectively.
Fig. 7. Transmission spectrum at different temperature based traditional optical Vernier effect in parallel Sagnac interferometers. (a) LP$_{01}$; (b) LP$_{11}$; (c) LP$_{21}$ (d) LP$_{02}$; (e) Temperature sensitivity of different modes based traditional optical Vernier effect in parallel Sagnac interferometers.
3.2.3 Operation principles of enhanced optical Vernier effect
The enhanced optical Vernier effect is a complex optical Vernier effect. In the enhanced optical Vernier effect, both interferometers are considered as a combined sensing structure. The experimental device diagram of multiple LP mode temperature sensing is based on the enhanced optical Vernier effect, as shown in Fig. 8. Both interferometers are placed simultaneously in the temperature controller boxes with opposite variation trends. The sensitivity of the Vernier envelope is defined as [51]:
Fig. 8. Diagram of the PANDA-type FMF temperature measurement based on the enhanced optical Vernier effect in parallel Sagnac interferometers.
3.2.4 Sensing characteristics and results discussion
When the interference dip of two interferometers shift in opposite directions, the difference of Eq. (18) is transformed into a sum, so that the enhanced Vernier effect is realized. Figure 9(a), (b), (c), and (d) show transmission spectrum at different temperatures based on the novel enhanced optical Vernier effect. The Vernier envelope wavelength shift is a blue shift as temperature increases. The temperature sensitivity of multiple LP modes is shown in Fig. 9(e). The temperature sensitivity of LP$_{01}$, LP$_{11}$, LP$_{21}$, and LP$_{02}$ are 13.72 nm/$^\circ \text{C}$, 14.77143 nm/$^\circ \text{C}$, 16.44571 nm/$^\circ \text{C}$ and 14.15429 nm/$^\circ \text{C}$, respectively. LP$_{21}$ mode has the highest temperature sensitivity. Compared with the fundamental mode, the temperature sensitivity of the LP$_{21}$ mode is improved by 19.9%. The temperature sensitivity based on the enhanced optical Vernier effect of LP$_{01}$, LP$_{11}$, LP$_{21}$, and LP$_{02}$ is improved 8.4 times, 8.2 times, 8.5 times, and 7.9 times, respectively. Therefore, by utilizing the enhanced optical Vernier effect, a higher sensitivity temperature sensor is achieved.
Fig. 9. Transmission spectrum at different temperature based enhanced optical Vernier effect in parallel Sagnac interferometers. (a) LP$_{01}$; (b) LP$_{11}$; (c) LP$_{21}$ (e) LP$_{02}$; (d) Temperature sensitivity of different modes based enhanced optical Vernier effect in parallel Sagnac interferometers.
4. Discussion
In the references that have been reported so far [52–54], the enhanced optical vernier effect is achieved by using the opposite response of two interferometers to temperature in the same environment. The illustration of the traditional enhanced optical effect in practical application is shown in Fig. 10(a), (b) and (c). In the laboratory, two interferometers are placed in a narrow, enclosed temperature controlled chamber, which can easily achieve a constant temperature field. However, a constant temperature field is difficult to achieve in the practical application. The velocity of heat transfer is generally in the order of mm/s. The sensor requires rapid measurement of temperature changes, and it is not possible to heat continuously or heat dissipation to obtain a constant temperature field. In addition, in real life, the sensing system is in a large space, and the temperature is generally different in different spaces. In actual production, many factors will produce temperature changes, we assume the simplest case, there is a heat source in the external environment. (a) When the distance between two interferometers is far, and there is a heat source that is the same distance from the two interferometers, the heat source temperature changes, and after the same heat transfer distance, the temperature field around the two interferometers is the same. The temperature of the heat source is constant, the closer the interferometer is to the heat source, the higher the temperature. The red line indicates the accurate temperature measurement range, while the yellow line indicates the inaccurate temperature measurement range. Therefore, there is a large difference between the temperature around the interferometer and the temperature further away from the interferometer. The distance between the heat source and the two interferometers is the same, which is an ideal situation in real life. (b) The distance between two interferometers is far, and the distance between the heat source and the two interferometers is different, the actual temperature of the two interferometers is considerably large temperature difference when the heat source temperature changes. The above-mentioned situation is very common in real life. (c) The distance between the two interferometers is far, and the two interferometers are extremely close, and the heat source is almost the same distance from the two interferometers. Therefore, the temperature difference between the two interferometers is extremely small. However, when the two interferometers are fused, two interferometers must be kept at a distance. In conclusion, the above three cases are either in an ideal environment or there are large or small measurement errors in the practical application process. Figure 10(d) shows the illustration of novel enhanced optical vernier effect. The traditional optical vernier effect is to place the reference arm at room temperature. However, in practical applications, if the reference arm is exposed to the external environment, the outside temperature fluctuates more or less slightly. The addition of a narrow, enclosed temperature-controlled chamber can not only keep the temperature constant but also avoid the influence of external temperature changes on the reference arm. The novel approach avoids measurement errors and improves the stability of the sensing system.
However, the same response of two interferometers to temperature, but they are placed in two environments with opposite temperature variation trends, respectively. Therefore, the enhanced optical vernier effect also can realized. In some environments where the temperature environment changes dramatically or maintains a constant temperature, it is necessary to realize high-sensitivity continuous monitoring of temperature, and the principle of enhanced optical vernier effect in this paper can also be applied. Two interferometers are employed in this system: one placed in the external environment and the other in a temperature control box. In normal circumstances, as long as the temperature of the external environment does not fluctuate rapidly and randomly up and down, that is, the external temperature is in a relatively stable state for a certain period of time. Subsequently, the temperature control box is adjusted accordingly. Otherwise, the temperature control box maintains a constant temperature. This allows for relatively highly sensitive measurements using the traditional optical vernier effect. When the temperature control box automatically adjusts towards the opposite trend of the temperature variation. As a result, an enhanced optical vernier effect is achieved, enabling precise monitoring of small change temperature changes. In addition, the novel enhanced optical vernier effect can be directly applied to artificially changing the external temperature to make it monotonically variable. The novel method in the paper shows great potential for application in agricultural production. When sensing systems use both traditional and enhanced optical vernier effects, higher requirements are put forward for optical signal processing technology, which needs to identify the temperature corresponding to the spectrum. At present, some references use artificial intelligence techniques to identify spectrum data using full spectrum features that can solve this problem [55–59].
Fig. 10. Illustration of the novel enhanced optical vernier effect. (a) The distance between two interferometers is far, and a heat source is the same distance from the two interferometers. (b) The distance between two interferometers is far, and the distance between the heat source and the two interferometers is different. (c) The distance between two interferometers is far, and two interferometers are extremely close. (d) Novel optical enhanced vernier effect.
5. Conclusion
In this study, high-order LP modes based Sagnac interference for temperature sensing were proposed and investigated theoretically. MSCs were added to excite specific higher-order modes. A stress-induced Panda-type few-mode fiber (FMF) supporting 4 LP modes was designed by numerical simulation method, which was used to act as sensing units. When the external temperature increased, the thermal stress inside the Panda-type FMF decreased, resulting in the birefringence decreased. Therefore, the interference dip shifted as the temperature changed. LP$_{21}$ mode has the highest temperature sensitivity of 1.92571 nm/$^\circ \text{C}$, 9.88571 nm/$^\circ \text{C}$, and 16.44571 nm/$^\circ \text{C}$ in a single Sagnac interferometer, paralleled Sagnac interferometers based on traditional and enhanced optical Vernier effect, respectively. Compared with fundamental mode (LP$_{01}$), the temperature sensitivity of LP$_{21}$ mode is improved by 18.2%, 18.3%, and 19.9% in above three different conditions, respectively. Due to the opposite spectrum shift direction of two interferometers that as sensing arms, the enhanced Vernier effect provided a higher sensing amplification capability and an M-factor of approximately 8 times rather than 5 times in the traditional optical Vernier effect. The novel approach of the enhanced Vernier effect avoids measurement errors and improves the stability of the sensing system. The findings in this paper hold significant guidance for the advancement of highly sensitive sensors by utilizing high-order mode interference.
Funding
National Natural Science Foundation of China (12074331); Natural Science Foundation of Hebei Province (F2021203112); Yangtze Optical Fibre and Cable Joint Stock Limited Company (x2022084).
Acknowledgments
This work was supported by the Natural Science Foundation of Hebei Province, China (Grant No. F2021203112), National Natural Science Foundation of China (Grant No. 12074331), and Yangtze Optical Fibre and Cable Joint Stock Limited Company (Grant No. x2022084).
Disclosures
The authors declare that they have no competitive economic interests or personal relationships that may affect the work reported in this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. Y. Liu, H. Chen, Q. Chen, et al., “Experimental Study on Dual-Parameter Sensing Based on Cascaded Sagnac Interferometers With Two PANDA Fibers,” J. Lightwave Technol. 40(9), 3090–3097 (2022). [CrossRef]
2. Y. Ran, L. Xia, D. Niu, et al., “Design and demonstration of a liquid level fiber sensor based on self-imaging effect,” Sens. Actuators, A 237, 41–46 (2016). [CrossRef]
3. L. Wei, H. Li, T. Hu, et al., “Magnetic Field and Temperature Two-Parameter Sensor Based on Mach–Zehnder Interferometer and Faraday Rotation Effect,” IEEE Sens. J. 23(13), 14041–14048 (2023). [CrossRef]
4. K. Wang, X. Dong, M. H. Köhler, et al., “Advances in optical fiber sensors based on multimode interference (MMI): a review,” IEEE Sens. J. 21(1), 132–142 (2020). [CrossRef]
5. X. Fan, H. Chen, Y. Zheng, et al., “Ultrasensitive fiber-optic temperature sensor based on cascaded Sagnac interferometers with a nematic liquid crystal film,” Opt. Laser Technol. 152, 108169 (2022). [CrossRef]
6. Q. Chen, H. Chen, Y. Liu, et al., “An optical fiber sensor for the detections of liquid level and strain through cascading Sagnac interference and modal interference,” Infrared Phys. Technol. 127, 104387 (2022). [CrossRef]
7. Z. Rui, Z. Xiang, F. Zeng, et al., “Liquid Level Sensor With High Sensitivity Based on Hetero Core Structure,” IEEE Sens. J. 22(14), 14051–14057 (2022). [CrossRef]
8. H. Y. Choi, G. Mudhana, K. S. Park, et al., “Cross-talk free and ultra-compact fiber optic sensor for simultaneous measurement of temperature and refractive index,” Opt. Express 18(1), 141–149 (2010). [CrossRef]
9. Z. Wang, Z. Tan, R. Xing, et al., “Liquid level sensor based on fiber ring laser with single-mode-offset coreless-single-mode fiber structure,” Opt. Laser Technol. 84, 59–63 (2016). [CrossRef]
10. T. Dong, B. Gao, B. Yan, et al., “Label-Free Tapered Fiber Optic Sensor for Real-Time In Situ Detection of Cell Activity,” IEEE Sens. J. 23(14), 15622–15627 (2023). [CrossRef]
11. M. Consales, A. Buosciolo, A. Cutolo, et al., “Fiber optic humidity sensors for high-energy physics applications at CERN,” Sens. Actuators, B 159(1), 66–74 (2011). [CrossRef]
12. S. Narasimman, L. Balakrishnan, and Z. C. Alex, “Highly Selective and Rapid Detection of Ethanol Using Ni-Functionalized ZnO Nanoflakes Coated Fiber Optic Sensor,” IEEE Sens. J. 23(1), 344–352 (2023). [CrossRef]
13. C. Liu, H. Chen, Q. Chen, et al., “An ultra-high sensitivity methane gas sensor based on Vernier effect in two parallel optical fiber Sagnac loops,” Opt. Commun. 540, 129509 (2023). [CrossRef]
14. S. Mustapha, A. Kassir, K. Hassoun, et al., “Joint Crowd Management and Structural Health Monitoring Using Fiber Optic and Wearable Sensing,” IEEE Commun. Mag. 57(4), 62–67 (2019). [CrossRef]
15. J. Lu, Y. Yu, S. Qin, et al., “High-performance temperature and pressure dual-parameter sensor based on a polymer-coated tapered optical fiber,” Opt. Express 30(6), 9714–9726 (2022). [CrossRef]
16. S. Chen, K. Zhu, J. Han, et al., “Photonic Integrated Sensing and Communication System Harnessing Submarine Fiber Optic Cables for Coastal Event Monitoring,” IEEE Commun. Mag. 60(12), 110–116 (2022). [CrossRef]
17. Y. Liu, B. Liu, X. Feng, et al., “High-birefringence fiber loop mirrors and their applications as sensors,” Appl. Opt. 44(12), 2382–2390 (2005). [CrossRef]
18. B. H. Kim, S. H. Lee, A. Lin, et al., “Large temperature sensitivity of Sagnac loop interferometer based on the birefringent holey fiber filled with metal indium,” Opt. Express 17(3), 1789–1794 (2009). [CrossRef]
19. Y. Xin, X. Dong, Q. Meng, et al., “Alcohol-filled side-hole fiber Sagnac interferometer for temperature measurement,” Sens. Actuators, A 193, 182–185 (2013). [CrossRef]
20. W. Yang, R. Pan, L. Zhang, et al., “Highly sensitive fiber-optic temperature sensor with compact hybrid interferometers enhanced by the harmonic Vernier effect,” Opt. Express 31(9), 14570–14582 (2023). [CrossRef]
21. J. A. Bucaro and E. F. Carome, “Single fiber interferometric acoustic sensor,” Appl. Opt. 17(3), 330–331 (1978). [CrossRef]
22. D. Du, C. Xu, Z. Yang, et al., “Ultrasensitive temperature sensor and mode converter based on a modal interferometer in a two-mode fiber,” Opt. Express 29(20), 32135–32148 (2021). [CrossRef]
23. J.-L. Kou, J. Feng, L. Ye, et al., “Miniaturized fiber taper reflective interferometer for high temperature measurement,” Opt. Express 18(13), 14245–14250 (2010). [CrossRef]
24. R. Wang, J. Zhao, Y. Sun, et al., “Wearable respiration monitoring using an in-line few-mode fiber Mach-Zehnder interferometric sensor,” Biomed. Opt. Express 11(1), 316–329 (2020). [CrossRef]
25. X. Lei, X. Dong, and C. Lu, “Sensitive Humidity Sensor Based on a Special Dual-Mode Fiber,” IEEE Sens. J. 19(7), 2587–2591 (2019). [CrossRef]
26. Q. Yu, X. Li, X. Zhou, et al., “Temperature compensated fiber optic magnetic sensor based on the combination interference principle,” Opt. Lett. 47(10), 2558–2561 (2022). [CrossRef]
27. C. Caucheteur, J. Villatoro, F. Liu, et al., “Mode-division and spatial-division optical fiber sensors,” Adv. Opt. Photonics 14(1), 1–86 (2022). [CrossRef]
28. S. Ngcobo, I. Litvin, L. Burger, et al., “A digital laser for on-demand laser modes,” Nat. Commun. 4(1), 2289 (2013). [CrossRef]
29. T. Liu, S.-P. Chen, and J. Hou, “Selective transverse mode operation of an all-fiber laser with a mode-selective fiber Bragg grating pair,” Opt. Lett. 41(24), 5692–5695 (2016). [CrossRef]
30. B. Sun, A. Wang, L. Xu, et al., “Low-threshold single-wavelength all-fiber laser generating cylindrical vector beams using a few-mode fiber Bragg grating,” Opt. Lett. 37(4), 464–466 (2012). [CrossRef]
31. Y. Zhou, A. Wang, C. Gu, et al., “Actively mode-locked all fiber laser with cylindrical vector beam output,” Opt. Lett. 41(3), 548–550 (2016). [CrossRef]
32. N. Wang, J. C. A. Zacarias, J. E. Antonio-Lopez, et al., “Transverse mode-switchable fiber laser based on a photonic lantern,” Opt. Express 26(25), 32777–32787 (2018). [CrossRef]
33. N. Wang, J. C. Alvarado-Zacarias, M. S. Habib, et al., “Mode-selective few-mode Brillouin fiber lasers based on intramodal and intermodal SBS,” Opt. Lett. 45(8), 2323–2326 (2020). [CrossRef]
34. H. Wan, J. Wang, Z. Zhang, et al., “Passively Mode-Locked Ytterbium-Doped Fiber Laser With Cylindrical Vector Beam Generation Based on Mode Selective Coupler,” J. Lightwave Technol. 36(16), 3403–3407 (2018). [CrossRef]
35. Y. Gao, J. Cui, D. Ge, et al., “A Degenerate-Mode-Selective Coupler for Stable DSP-free MDM Transmission,” J. Lightwave Technol. 37(17), 4410–4420 (2019). [CrossRef]
36. S. H. Chang, H. S. Chung, N. K. Fontaine, et al., “Mode division multiplexed optical transmission enabled by all–fiber mode multiplexer,” Opt. Express 22(12), 14229–14236 (2014). [CrossRef]
37. F. Ren, J. Yu, and J. Wang, “Spatial-mode switchable ring fiber laser based on low mode-crosstalk all-fiber mode MUX/DEMUX,” Opt. Laser Technol. 101, 21–24 (2018). [CrossRef]
38. J. Jia, Y. Yang, M. Zuo, et al., “Distributed Transverse Stress Sensor Based on Mode Coupling in Weakly-Coupled FMF,” IEEE Photonics J. 14(1), 1–6 (2022). [CrossRef]
39. C. Song, X. Liu, F. Liu, et al., “Fault detection of few-mode fiber based on high-order mode with high fault detection sensitivity,” Opt. Lett. 44(18), 4487–4490 (2019). [CrossRef]
40. Z. Luo, H. Yu, Y. Zheng, et al., “Selective Mode Excitation in a Few-Mode Photonic Crystal Fiber for Strain Sensing With Restrained Temperature Response,” J. Lightwave Technol. 38(16), 4560–4571 (2020). [CrossRef]
41. J. W. Fleming, “Dispersion in GeO2–SiO2 glasses,” Appl. Opt. 23(24), 4486–4493 (1984). [CrossRef]
42. H. Yan, S. Li, Z. Xie, et al., “Design of PANDA ring-core fiber with 10 polarization-maintaining modes,” Photonics Res. 5(1), 1–5 (2017). [CrossRef]
43. S. Zhang, Z. Cao, T. Xia, et al., “Lateral Force Sensing Based on Sagnac Interferometry Realized by a High-Birefringence Suspended-Core Fiber,” J. Lightwave Technol. 40(12), 3935–3941 (2022). [CrossRef]
44. W. Urbanczyk, T. Martynkien, and W. J. Bock, “Dispersion effects in elliptical-core highly birefringent fibers,” Appl. Opt. 40(12), 1911–1920 (2001). [CrossRef]
45. R. Ismaeel, T. Lee, B. Oduro, et al., “All-fiber fused directional coupler for highly efficient spatial mode conversion,” Opt. Express 22(10), 11610–11619 (2014). [CrossRef]
46. H. Z. Yang, M. M. Ali, M. R. Islam, et al., “Cladless few mode fiber grating sensor for simultaneous refractive index and temperature measurement,” Sens. Actuators, A 228, 62–68 (2015). [CrossRef]
47. G. Bai, S. Li, X. Fan, et al., “Highly sensitive refractive index sensor based on a D-shaped few-mode fiber with silver and graphene film,” Optik 267, 169653 (2022). [CrossRef]
48. A. D. Gomes, H. Bartelt, and O. Fraz ao, “Optical Vernier Effect: Recent Advances and Developments,” Laser Photonics Rev. 15(7), 2000588 (2021). [CrossRef]
49. A. D. Gomes, M. S. Ferreira, J. Bierlich, et al., “Optical Harmonic Vernier Effect: A New Tool for High Performance Interferometric Fiber Sensors,” Sensors 19(24), 5431 (2019). [CrossRef]
50. Y. Li, C. Zhao, B. Xu, et al., “Optical cascaded Fabry–Perot interferometer hydrogen sensor based on vernier effect,” Opt. Commun. 414, 166–171 (2018). [CrossRef]
51. A. D. Gomes, M. S. Ferreira, J. Bierlich, et al., “Hollow microsphere combined with optical harmonic Vernier effect for strain and temperature discrimination,” Opt. Laser Technol. 127, 106198 (2020). [CrossRef]
52. J. Li, M. Zhang, M. Wan, et al., “Ultrasensitive refractive index sensor based on enhanced Vernier effect through cascaded fiber core-offset pairs,” Opt. Express 28(3), 4145–4155 (2020). [CrossRef]
53. W. Luo, Z. Cao, G. Zhang, et al., “A highly sensitive optical fiber temperature sensor based on the enhanced Vernier effect,” Opt. Fiber Technol. 67, 102702 (2021). [CrossRef]
54. J. Liang, J. Qu, J. Ye, et al., “Ultra-Sensitive Temperature Sensor of Cascaded Dual PDMS-Cavity Based on Enhanced Vernier Effect,” IEEE Sens. J. 23(3), 2264–2269 (2023). [CrossRef]
55. A. G. Leal-Junior, V. Campos, C. Díaz, et al., “A machine learning approach for simultaneous measurement of magnetic field position and intensity with fiber Bragg grating and magnetorheological fluid,” Opt. Fiber Technol. 56, 102184 (2020). [CrossRef]
56. C. Zhu, Y. Zhuang, and J. Huang, “Machine Learning Assisted High-Sensitivity and Large-Dynamic-Range Curvature Sensor Based on No-Core Fiber and Hollow-Core Fiber,” J. Lightwave Technol. 40(16), 5762–5767 (2022). [CrossRef]
57. C. Zhu, O. Alsalman, and W. Naku, “Machine learning for a Vernier-effect-based optical fiber sensor,” Opt. Lett. 48(9), 2488–2491 (2023). [CrossRef]
58. C. Zhu and O. Alsalman, “Vernier effect-based optical fiber sensor for dynamic sensing using a coarsely resolved spectrometer,” Opt. Express 31(13), 22250–22259 (2023). [CrossRef]
59. Y. Mei, S. Zhang, Z. Cao, et al., “Deep Learning Assisted Pressure Sensing Based on Sagnac Interferometry Realized by Side-Hole Fiber,” J. Lightwave Technol. 41(2), 784–793 (2023). [CrossRef]
