A Review of Sensitivity Enhancement in Interferometer-Based Fiber Sensors

Optical fiber sensors based on an interferometer structure play a significant role in monitoring physical, chemical, and biological parameters in natural environments. However, sensors with high-sensitivity measurement still present their own challenges. This paper deduces and summarizes the methods of sensitivity enhancement in interferometer based fiber optical sensors, including the derivation of the sensing principles, key characteristics, and recently-reported applications.The modal coupling interferometer is taken as an example to derive the five terms related to the sensitivity: (1) the wavelength-dependent difference of phase between two modes/arms ∂ϕd/∂λ, (2) the sensor length Lw,A, (3) refractive index difference between two modes/arms Δneff,A, (4) sensing parameter dependent length change α, and (5) sensing parameter dependent refractive index change γ. The research papers in the literature that modulate these terms to enhance the sensing sensitivity are reviewed in the paper.

Among manifold fiber-optic sensors, the interferometer-based fiber optic sensor is one of the most important devices since it offers high sensitivity, high resolution, easy operation, and multiplexing capabilities. Although Fabry-Perot interferometry-based sensors can react to changing physical environments through changes as visibility of fringes and finesse of cavity [31], monitoring the wavelength shift of the interference spectrum is a common way for interferometry-based sensors when the surrounding environment changes [32][33][34][35][36]. To date, a variety of reviews aim to categorize fiber optic interferometric sensors according to their operating principles, fabrication methods, and application fields [31,37]. For example, the performances of the modal-interferometer fiber optical sensor have been reviewed based on the standard fibers, polarization maintaining fiber, photonic crystal fiber, and fiber Bragg grating [38]; and a review of the basic sensing platforms implemented using tapered optical is presented exploited to facilitate the development of fiber-optic physical, chemical, and bio-sensors [39]. In this paper, we aim to offer a summary of the interferometer-based fiber-optic sensors including mode-coupling-based sensors, Mach-Zehnder-interferometerbased sensors, Sagnac-interferometer-based sensors, and birefringence-coupling-based sensors. Moreover, the sensing sensitivity is numerically analyzed, which shows that the sensitivity is related to the terms of (1) the wavelength-dependent difference of phase between two modes/arms ∂φ d /∂λ, (2) the sensor length L w,A , (3) refractive index difference between two modes/arms ∆n e f f ,A , (4) sensing parameter dependent length change α, and (5) sensing parameter dependent refractive index change γ. If a high sensitivity is obtained, at least one of these terms should be enhanced. It is likely that this review of the sensitivity enhancement of the interferometer-based fiber-optic sensors is helpful for readers' understanding of their state-of-the-art applications.

Sensing Principle of Interferometer-Based Sensors
The interferometer-based sensors monitor the wavelength shift of the interference spectrum when the surrounding environment changes. Then, the key point is to find the relationship between the wavelength shift and the changing parameters. There are many types of interferometers, such as mode-coupling-based sensors, Mach-Zehnderinterferometer-based sensors, Sagnac-interferometer-based sensors, and birefringencecoupling-based sensors. Here, an interferometer with a coupling of two modes is taken as an example to derive the theoretical sensing principle and other techniques can be applied to a similar analysis.
In a mode-coupling inteferometer, let us assume that a fiber sensor structure sustains two main modes (mode e and mode o). When light is launched into the fiber sensor, the mode e and mode o are excited and propagate along the sensor. The output pattern is a superposition of the fields of the mode e and mode o and is given by The intensity at the output port is given by , ∆n e f f = n e e f f − n o e f f , a i is the complex amplitude with i being e or o, θ i is the phase of a i , n i e f f is the effective refractive index of mode i, L w is the length of the sensor, and A is the parameter characterizing the change of the surrounding environment.
Troughs are observed in the transmission spectrum when the difference between the phases of mode e and mode o satisfies the condition φ d (λ, A) = (2m + 1)π, where m is an integer. Changes in the surrounding environment A from A to A + ∆A lead to a change in the phase-difference φ d given by The phase-difference at the mth trough in the transmission spectrum always satisfies φ d (λ, A) = (2m + 1)π; as the surrounding environment changes from A to A + ∆A, the mth trough shifts from wavelength λ m,A to wavelength λ m,A+∆A = λ m,A + ∆λ m,A such that φ d (λ m,A , A) = φ d (λ m,A + ∆λ m,A , A + ∆A) = (2m + 1)π leading to ∆φ d (λ, A) = 0. In addition, Equation (1) becomes Due to φ d (λ, A) = 2π∆n e f f L w /λ + θ e − θ o , the term of ∂φ d /∂A becomes ∂φ d /∂A = (2π/λ) L w,A ∂∆n e f f ,A /∂A + ∆n e f f ,A ∂L w,A /∂A and then the wavelength-shift ∆λ m,A becomes where ∆n e f f ,A is the refractive indices difference between mode e and mode o, L w,A is the taper length, α = (1/L w,A )∂L w,A /∂A, and γ = 1/∆n e f f ,A ∂∆n e f f ,A /∂A. To achieve a high sensitivity to the change of the surrounding environment ∆A, the value of ∆λ m,A should be large, which requires that the terms of (1) the wavelengthdependent difference of phase between two modes/arms ∂φ d /∂λ, (2) the sensor length L w,A , (3) refractive index difference between two modes/arms ∆n e f f ,A , (4) sensing parameter dependent length change α, and (5) sensing parameter dependent refractive index change γ should be enhanced. In the next section, the research about sensitivity enhancement utilized the terms of the (∂φ d /∂λ) −1 , L w,A , ∆n e f f ,A , α and γ are reviewed, respectively. Table 1 presents the sensing characteristics of different interferometer-based sensors with different modulated terms.

Interferometer-Based Sensors
All the terms in Equation (2) play a role in sensitivity enhancement of interferometerbased sensors. The sensing sensitivity is usually a combined effect of many terms and here we review the most significant one determining the enhanced sensitivity. For example, many sensors based on Mach-Zehnder interferometers utilize the coupling between the fundamental mode and a high-order mode and the high-sensitive sensing performance is achieved by monitoring the shift of the transmission spectrum of the two modes. The highorder mode is usually more sensitive to the surrounding environment change, indicating a large value of γ; the refractive index difference ∆n e f f ,A between the fundamental mode and the high-order mode is large, which also leads to a large wavelength shift in the transmission spectrum based on Equation (2). There are many research papers showing that multiple terms dominate the sensitivity [51][52][53][54][55][56] and various environmental parameters are sensed, such as magnetic field [57] and chemical vapor [58].
Recently, a label-free DNA biosensor based on an exposed core microstructured optical fiber has been demonstrated to utilize a multimode Mach-Zehnder interferometer by splicing a section of tapered exposed core fiber between two single-mode fibers [40]. The binding of biomolecules on the surface of the optical fiber induces the change of the refractive index of the surrounding environment and the tapered design provides the evanescent field with the ability of refractive index detection. Two terms (∆n e f f ,A and γ) are modulated to enhance the measurement sensitivity. Figure 1 shows that, as the refractive index of the external environment increases, the effective refractive index of the high-order mode is more affected by the external refractive index than the fundamental mode, indicating a large value of γ using this structure. Figure 2 presents relationship between the effective refractive index difference (∆n e f f ,A ) between the fundamental mode and the high-order mode and the external refractive index. The sensor design can be optimized depending on the values of γ and ∆n e f f ,A calculated in Figures 1 and 2.  and ∆n e f f ,A (between the fundamental mode and high-order mode) for three fiber diameters [40]. Reprinted with permission from Ref. [40]. Copyright 2022 Elsvier Science and Technology Journals.

Dual-Core Tapers with Large Diameters
A high-sensitivity temperature and strain sensor based on a dual-core As 2 Se 3 -PMMA taper was proposed [41], which modulates the value of ∂φ d (λ)/∂λ by increasing the As 2 Se 3 core diameter. The values of (∂φ d /∂λ) −1 can be calculated as a function of diameter of As 2 Se 3 core using Comsol Multiphysics with parameters found in [59]. Based on Equation (2), a large value of (∂φ d /∂λ) −1 induces a large wavelength shift of the transmission spectrum when the surrounding temperature and imposed longitudinal strain changes, which enhances the measurement sensitivity with the temperature sensitivity of 572 pm/ • C and strain sensitivity of −6.23 pm/µε.

Effective Group-Velocity Matching Induced by an Antisymmetric Long-Period Grating
Another way to enhance the value of (∂φ d /∂λ) −1 is based on effective group-velocity matching between the even mode and odd mode in a dual-core taper with an antisymmetric long-period grating [60]. An antisymmetric long-period grating is inscribed by launching optical pulses into a dual-core As 2 Se 3 -PMMA taper, which causes the effective group-velocity matching between the even and odd modes propagating around the central wavelength of the input pulses [61]. The value of ∂φ d /∂λ tends to 0 near the resonance wavelength of the antisymmetric long-period grating indicating that, near the resonance wavelength, the large value of (∂φ d /∂λ) −1 leads to a large wavelength shift of the transmission spectrum and then the measurement sensitivity is enhanced.
The sensitivity enhancement based on a dual-core taper with an antisymmetic longperiod grating is achieved and temperature measurement sensitivity is of 0.121 nm/°C that is enhanced by a factor of 4.0 due to the large value of (∂φ d /∂λ) −1 induced by an antisymmetric long-period grating.

Cut-Off Wavelength of High-Order Modes
The value of (∂φ d /∂λ) −1 can also be modulated near the cut-off wavelength of highorder modes in a optical waveguide. An optical fiber refractive index sensor consisting of an optical microfiber Mach-Zehnder interferometer was demonstrated near the cut-off wavelength of mode HE 12 [42]. Around the cut-off wavelength, the mode HE 12 ceases to propagate and only the fundamental mode HE 11 propagates, leading to a constant value of φ d and then a large value of (∂φ d /∂λ) −1 . Figure 3 shows the evolution of the transmission spectrum of the microfiber MZI with the variation of refractive index, showing that the refractive index sensitivity around the cut-off wavelength of HE 12 (up to 44,271 nm/RIU) is much higher than that far away from the cut-off wavelength.  [42]. Reprinted with permission from Ref. [42]. Copyright 2020 Elsvier Science and Technology Journals.

Modulation of ∆n e f f to Enhance the Sensor Sensitivity
There are many ways to enhance ∆n e f f in the interferometer-based sensors, such as utilizing the coupling between the fundamental mode and high-order modes, inducing a waveguide with a high refractive index in one arm of an MZI, and using a large refractive index difference between the two polarization axes in polarization-maintaining fibers.

The Coupling between Fundamental Mode and High-Order Modes
A fiber strain sensor was reported by inducing higher-order interference modes using a torsional multimode fiber [62]. As is shown in Figure 4a, the procedure of single mode-twisted-multimode-single mode fiber structure (STMS) is presented, and the twist multimode-fiber region is able to couple more power of light into the cladding and introduce high-order LP modes coupling. Figure 4b presents the transmittance spectrum of SMS and STMS, respectively, which shows that the STMS strucure gives a trough with a higher contrast in the transmission spectrum. The coupling between the fundamental mode and high-order modes has a large ∆n e f f , which leads to a high strain sensitivity of −42.5 pm/µε based on Equation (2).
A curvature sensor based on a thin-core fiber Mach-Zehnder interferometer was proposed with the sensor head composed by a short section of thin-core fiber embedded between two single-mode fibers [43]. The high-order fiber cladding mode LP 018 is excited because of the dramatic structural discontinuity at the splicing point of the single-mode fiber and the thin-core fiber. The Mach-Zehnder interferometer works on the basis of interference between the core mode LP 01 and the excited cladding mode LP 018 in the thincore fiber, and the large refractive index difference ∆n e f f between the fundamental mode LP 01 and high-order mode LP 018 achieves a sensitivity of −13.53 nm/m −1 . Similarly, a high-sensitivity temperature sensor was also proposed based on a Mach-Zehnder interferometer fabricated by concatenating two microcavities separated by a middle section [63]. High-order cladding modes are excited when light propagates the first microcavity and the interference pattern of the fundamental mode and cladding modes are formed when light propagates the second microcavity. The large refractive index ∆n e f f between the fundamental mode and high-order cladding mode induces a large wavelength shift when the temperature changes in the range of n the range of 500-1200°C with a high sensitivity of 109 pm/°C.

Inducing a Waveguide with a High Refractive Index in One Arm of a MZI
The way of increasing the value of ∆n e f f in a Mach-Zehnder interferometer is usually to form the Mach-Zehnder interferometer using two different waveguides as the two arms, respectively [4,64]. For example, a fiber in-line Mach-Zehnder interferometer fabricated by femtosecond laser micromachining for refractive index measurement was achieved [65]. As is shown in Figure 5, one of the interferometer arms in the air contains a microcavity created by removing part of the fiber core near the core and cladding interface, and the other arm lies in the remaining part of the fiber core, which induces a large effective index difference between the two arms of the MZI to achieve high sensitivity. Figure 5. Schematic of the MZI structure fabricated by the use of femtosecond laser pulse irradiation [65]. Reprinted with permission from Ref. [65]. Copyright 2010 Optica Society.

Using the Large Refractive Index Difference of the Two Polarization Axes in Polarization-Maintaining Fibers
To obtain a large value of ∆n e f f , the large refractive index difference of the two polarization axes in polarization-maintaining fibers can be utilized [66]. A Sagnac-interferometer biosensor was reported based on an exposed core microstructured optical fiber that has noncircular symmetry and thus exhibits birefringence [44]. Figure 6 shows a schematic diagram of the proposed Sagnac interferometer. Light from a broadband source is split into clockwise and counterclockwise arms with two polarization states as it enters the Sagnac loop. The two polarization states experience different optical paths and interfere with each other when the two beams are recombined at the output port. In addition, then the large wavelength shift is achieved by a large refractive index difference ∆n e f f between the two polarization states. Figure 6. Schematic diagram of the proposed sensing system [44]. Reprinted with permission from Ref. [44]. Copyright 2018 Elsvier Science and Technology Journals.

Modulation of L w,A to Enhance the Sensor Sensitivity
Longer sensors have been used for many applications, such as salinity sensing [67]. In addition, it has been demonstrated that the longer interference length of L w,A leads to a larger wavelength shift. A dual-taper-based Mach-Zehnder interferometer was fabricated and applied for refractive index sensing demonstrating that the Mach-Zehnder interferometer with a longer interferometer length has a higher sensitivity [45]. A humidity sensor based on an in-fiber Mach-Zehnder interferometer was proposed, which demonstrated that the high sensitivity is achieved by increasing the length of the L w,A of the Mach-Zehnder interferometer [46].

Specialty Fiber That Is Sensitive to the Surrounding Environment
As shown in Figure 7a, a temperature sensor consisting of a segment of multimode fiber with a polymer cladding spliced between two single-mode fibers was proposed. The packaging material employed in the experiments was an aluminum tube and the structure was bonded to the aluminum tube by using epoxy. The measured temperature sensitivity is 3.195 nm/ • C, which is shown in Figure 7b [47]. The high sensitivity of temperature measurement is mainly attributed to the high thermo-optic coefficient (a large value of γ) induced by the polymer cladding. The core material with a large thermal-optic coefficient can also be utilized to enhance γ [68].

Coating Sensitive Materials on the Surface of the Sensors
Depending on the environmental parameter that needs to be sensed, different materials can be coated on the surface of the sensors. For example, a humidity sensor was demonstrated utilizing an interferometer structure that is coated by gelatin [69], gold nanoparticles [70], or a thin film of tin oxide [71] working as sensitive materials to improve the value of γ. In addition, a high-sensitive temperature sensor can be coated with a material with a high thermal-optic coefficient such as a film of polydimethylsiloxane [72] or deionized water [73]. Figure 8 shows the experimental setup of coated polydimethylsiloxane-based MZI temperature sensor with the advantage of the low Young's modulus, good linear thermal expansion effect, and thermo-optic effect from the polydimethylsiloxane material.  [72]. Reprinted with permission from Ref. [72]. Copyright 2019 Elsvier Science and Technology Journals.

Filling the Air Holes in Photonic Crystal Fibers Using a High Sensitive Solution
A temperature sensor based on a fiber loop with an alcohol-filled photonic crystal fiber was demonstrated [48]. The alcohol has a large thermo-optic coefficient (a large γ), which leads to a large wavelength shift of the interference spectrum of the fiber loop when the surrounding temperature changes. Figure 9a shows the SEM image of the used photonic crystal fiber, and the alcohol is filled in the air holes in the PCF structure. Figure 9b shows the transmission spectra of the alcohol-filled HiBi-PCF FLM at a temperature range of 20°C to 34°C and the sensitivity is as high as 6.6 nm/°C, which is 660 and seven times higher than that of an FBG and that of the FLM made of a conventional HiBi fiber.  [48]. Reprinted with permission from Ref. [48]. Copyright 2011 The Optical Society.

High-Order Modes
High-order modes are usually more sensitive to the change of the surrounding environment and the excitation of high-order cladding modes can be controlled by changing the waveguide structure such as tapering single-mode fibers and coreless fibers [74]. A fiber refractometer based on a tapered multimode fiber between two single-mode fibers was proposed with a sensitivity better than 1900 nm/RIU [75]. The tapered multimode fiber enables high-order cladding modes, which provides a large overlap with the surrounding solutions. High order modes are more sensitive to the surrounding environment leading to high sensitivity of the refractive index measurement [76].

Modulation of α to Enhance the Sensor Sensitivity
Polymer fibers usually have a large α that is related to the length change when the sensing parameter changes. An ultrasonic sensor based on polymer optical fibers in a Mach-Zehnder interferometer has been proposed [49]. The sensitivities of two Mach-Zehnder interferometric sensors under the same conditions were investigated and compared: One is based on a polymethylmethacrylate fiber and the other based a silica fiber. The sensitivity of the polymer optical fiber is more than 12 times larger than that of the fused silica fiber due to the low Young's modulus of polymer fibers since polymer fibers have a large deformation under the same strain leading to a large value of α.
Similarly, a high-sensitive temperature and strain measurement has been demonstrated in a microwire with an As 2 Se 3 core and a PMMA cladding [50]. This structure combines the large thermal-expansion coefficient of the PMMA cladding (α) and the large difference between the refractive-indices of As 2 Se 3 core and PMMA cladding (∆n e f f ) to achieve a temperature sensitivity of −115 pm/°C and strain sensitivity of −4.21 pm/µε.

Discussion
In this review, the methods of sensitivity enhancement are summarized based on the five terms derived by monitoring the wavelength shift of the interferometer-based sensors, and there are a few points that need to be noted. Firstly, wavelength is not the only parameter to monitor in the interferometer-based fiber sensors and there are other parameters, such as intensity change that can be linked to the environmental changes. Secondly, as described in Section 3, all the terms in Equation (2) play a role in sensitivity enhancement of interferometer-based sensors and sometimes it is very difficult to distinguish which term is the most significant oneFinally, there are limitations for the presented methods: (1) Modulation of the term (∂φ d /∂λ) −1 always requires changing the waveguide structure or choosing high-order modes, which makes it difficult to achieve; however, once the conditions are met, the sensitivity is much enhanced; (2) Modulation of the term ∆n e f f is often together with the modulation of another term; (3) A larger value of L w,A provides a higher sensitivity, but the modulation of the term L w,A is impractical, since the size of the point sensor should be as small as possible; (4) Modulation of the term γ is a common way for the sensitivity enhancement; as the development of material fabrication technique and fiber fabrication technique, it is easy to find a material that has a large γ to the sensing parameters; (5) Modulation of the term α is mostly associated with the length-related sensing parameters such as temperature and vibration.

Conclusions
Although interferometer-based fiber optic sensors have been widely applied for various parameter sensing in a variety of applications, there remains a limited knowledge of methods for sensitivity enhancement. This paper reviews various structures and methods that have been used for enhancing the sensitivity, including the derivation of the sensing principles, key characteristics, and recently-reported applications. Five terms related to the sensitivity enhancement of the interferometer-based fiber sensors are reviewed: ∂φ d /∂λ, L w,A , ∆n e f f ,A , α, and γ . Author Contributions: Conceptualization, S.G. and Z.W.; methodology, S.G., Z.W., Z.G., J.D. and H.L.; validation, S.G., Z.W. and Y.C.; formal analysis, S.G. and Z.W.; investigation, Z.W., Z.G., J.D. and H.L.; resources, S.G.; data curation, S.G. and Z.W.; writing-original draft preparation, S.G.; writing-review and editing, S.G. and Z.W.; visualization, S.G., Z.W., Z.G., J.D. and H.L.; supervision, S.G. and Y.C.; project administration, S.G.; funding acquisition, S.G. All authors have read and agreed to the published version of the manuscript.
Funding: This work was funded by the National Natural Science Foundation of China (12104265, 12104272) and Shandong Provincial Natural Science Foundation (ZR2021QA019).