Characteristics of Critical-Wavelength-Existed Fiber-Optic Mach–Zehnder Interferometers and Their Sensing Applications

: In this paper, we review the characteristics of critical wavelength (CWL)-existed ﬁber-optic Mach–Zehnder interferometers (MZIs), including special few-mode ﬁbers and microﬁbers, and their sensing applications in physical, chemical, and marine ﬁelds. Owing to the existence of CWL in the transmission spectra, the in-line MZIs show some speciﬁc characteristics. The closer the peak/dip wavelength to the CWL, the larger the wavelength shift or the related sensitivity when the interferometer is under testing. Meanwhile, CWL shifts monotonically with the variations in measurands, such as temperature (in the air or seawater), axial strain, water pressure, surrounding refractive index, etc., when they are applied to the sensing ﬁbers. These characteristics of the CWL-existed in-line MZIs make them appealing solutions for fabricating various interferometric sensors, with the advantages of large measurement range, high sensitivity, multiparameter sensing, etc. Theoretical and experimental studies on the properties of the CWL-existed in-line MZIs are reviewed and discussed in this paper.

A typical in-line MZI shows periodic interference fringes.The interference peaks/dips in the periodically changed transmission spectrum exhibit the same behavior, i.e., shifting in the same direction, with the same sensitivity, when monitoring sensing parameters.In this case, when such a sensor system is being applied to detect any particular parameter through monitoring the wavelength variations of a specific interference peak/dip or intensity variations at a particular wavelength, some critical issues will limit the veracity of the sensor performance: (1) multivalue outputs and limited measurement range induced by overlapping of adjacent peaks/dips; (2) limitations in multiparameter applications caused by the uniformity of the interference fringes and their wavelength sensitivity.
However, in the transmission spectra of some in-line MZIs, a critical wavelength (CWL) may appear in the operating wavelength range.At the CWL, the wavelength response period seen is infinite in the transmission spectrum.This interference fringe, which shows the least dependence on wavelength, is defined as the achromatic fringe [26].In fact, the CWL is the wavelength at which the two interference modes' group velocities equal each other.Therefore, the propagation constant difference in the two interference modes changes nonlinearly with wavelength and turns around at CWL.In this way, the CWL is also called the dispersion turning point (DTP).
The CWL-existed in-line MZIs exhibit intriguing characteristics that researchers have thoroughly studied and applied in a wide range of sensing fields: (1) The peak lying closest to CWL at each side has its maximum spacing from the adjacent peak.Thus, compared with periodically changed interference fringes, the CWL is easy to identify in the transmission spectrum.Moreover, the CWL is exclusive at the operating wavelength range, which becomes an excellent candidate for measuring various parameters, especially in large-range measurement.(2) The interference peak/dip closest to the CWL has the maximum wavelength sensitivity among all the interference peaks/dips.(3) The interference peaks/dips on each side of the CWL shift to opposite directions under sensing parameter variations.The characteristics of the CWL and the interference peaks/dips have been proved with competitive advantages in large-range measurement [7,8,30], high wavelength sensitivities of interference peaks/dips [10,[28][29][30][31][32][33]35,36], and co-located multiparameter sensing [9,11].With these advantages, the CWL-existed in-line MZIs have been reported in sensing parameters of axial strain, curvature, displacement, surrounding refractive index, temperature, relative humidity, static pressure, chemical solution concentration, etc., according to the particular circumstances.Difference fibers such as liquid-core fiber, MMF, FMF, and few-mode microfibers are employed in these CWL-existed in-line MZIs.
In this paper, we provide a simple but overall review of the developments and recent progress on CWL-existed in-line MZIs and their sensing applications.In Section 2, the generation of the CWL is explained.In Section 3, we review the wavelength sensitivity characteristics of the interference peaks/dips lying near the CWL and their applications in sensors.The large-range measurement sensors employing CWL detection are reviewed in Section 4. Lastly, in Section 5, some conclusions and prospects are given.
Photonics 2022, 9, x FOR PEER REVIEW 3 of 15 planned, the FMF supports only the LP01 and LP02 modes propagating in the fiber under well-centered splicing [44].Figure 2 shows the cross-section structure of the FMF and the schematic diagram of the SFS structure.The relative refractive index difference is defined as ∆n co/cli = (n co/cli − n 0 )/n 0 , where the n co/cli and n 0 are the refractive indices of the core/i th inner cladding and pure silica of the FMF, respectively.FMF is specially designed with five layers and a W-shaped refractive index difference profile.Therefore, at the operational wavelength planned, the FMF supports only the LP 01 and LP 02 modes propagating in the fiber under well-centered splicing [44].The LP01 and LP02 modes propagating in the FMF are excited by the fundament mode LP01 in input SMF.The interference between the LP01 and LP02 modes is transmitte by output SMF.The ratios of optical power transferred to the LP01 and LP02 modes in th FMF from input SMF are  01 =  01 /  and  02 =  02 /  , and calculated as 81% an 11.5%, respectively.The transmission, T, through the SFS structure with  FMF -length FM is given by [15] as follows: where () = Δ •  FMF is the phase difference between LP01 and LP02 modes.Δ is th propagation constant difference between LP01 and LP02 modes.
With the parameters of FMF given in Figure 1, a finite element model is establishe to simulate the relationship between the Δ and wavelength, which is depicted as dashed line in Figure 3.The calculated transmission spectrum combining Equation (1) an the simulated value of Δ is shown as a solid line in Figure 3, where the physical lengt of the employed FMF is 50 cm.Because the dispersion of Δ exhibits nonlinear behavio and has a maximum, corresponding to the value of CWL in the transmission spectrum the operational wavelength, the wavelength response period at CWL is infinite.Moreove the periods of interference fringes closest to CWL from both sides reach a maximum.A shown in Figure 3, numbering from the peak that lies closest to CWL from each side, th peaks that are located on the higher wavelength side of CWL (i.e., within the region fro CWL to 1580 nm) are denoted as PH, 1, PH, 2,..., respectively.Peaks located on the low wavelength side of CWL (i.e., within the region from 1530 nm to CWL) are denoted as P The LP 01 and LP 02 modes propagating in the FMF are excited by the fundamental mode LP 01 in input SMF.The interference between the LP 01 and LP 02 modes is transmitted by output SMF.The ratios of optical power transferred to the LP 01 and LP 02 modes in the FMF from input SMF are t 01 = P 01 /P in and t 02 = P 02 /P in , and calculated as 81% and 11.5%, respectively.The transmission, T, through the SFS structure with L FMF -length FMF is given by [15] as follows: where ϕ(λ) = ∆β•L FMF is the phase difference between LP 01 and LP 02 modes.∆β is the propagation constant difference between LP 01 and LP 02 modes.
With the parameters of FMF given in Figure 1, a finite element model is established to simulate the relationship between the ∆β and wavelength, which is depicted as a dashed line in Figure 3.The calculated transmission spectrum combining Equation (1) and the simulated value of ∆β is shown as a solid line in Figure 3, where the physical length of the employed FMF is 50 cm.Because the dispersion of ∆β exhibits nonlinear behavior and has a maximum, corresponding to the value of CWL in the transmission spectrum at the operational wavelength, the wavelength response period at CWL is infinite.Moreover, the periods of interference fringes closest to CWL from both sides reach a maximum.As shown in Figure 3, numbering from the peak that lies closest to CWL from each side, the peaks that are located on the higher wavelength side of CWL (i.e., within the region from CWL to 1580 nm) are denoted as P H, 1 , P H, 2 ,. . ., respectively.Peaks located on the lower wavelength side of CWL (i.e., within the region from 1530 nm to CWL) are denoted as P L, 1 , P L, 2 ,. . ., respectively.The interference dips are also numbered in the same way.With the achromatic fringes in the transmission spectrum, multiple sensing indicators are provided in the sensors employing CWL-existed in-line MZIs, including monitoring wavelength shifts in CWL [6][7][8]18,40], PH, 1/DH, 1, PL, 1/DL, 1 [9][10][11]15,17,[28][29][30][31][32][33][34][35][36], and intensity variations [12] at a specific wavelength or wavelength ranges.Studies in CWLexisted in-line MZIs show that utilizing CWL shifts or other monitoring parameters listed above, the CWL-existed in-line MZIs can provide competitive advantages over conventional periodically changed interferometers in the applications of large-range measurement, co-located multiparameter sensing, and high-sensitivity sensing.Moreover, the theoretical study of wavelength sensitivities of interference fringes offers a direction for mode interferometers to maximize their peak wavelength sensitivities.

Characteristics of Interference Peaks/Dips in the CWL-Existed in-Line MZIs and Their Applications
In a CWL-existed transmission spectrum, the interference peaks/dips, especially those lying closest to CWL on both sides, exhibit essential characteristics different from the typical periodically changing fringes.These differences were experimentally observed by Salik et al. [15] and Zhou et al. [17].However, theoretical analysis was not presented.The wavelength sensitivities of the interference fringes under temperature, axial strain, static pressure, and surrounding refractive index (SRI) variations were further theoretically and experimentally studied in Refs.[10,11,[28][29][30]32,36], which are reviewed in this section.

Characteristics of Interference Peaks/Dips in the CWL-Existed SFS Structures
For the CWL-existed SFS structure employing the FMF shown in Figure 2, because the geometrical structure and relative refractive index difference profile of the FMF are determined, the temperature, axial strain, and static pressure wavelength sensitivities of the interference peaks/dips are governed by the wavelength separations between the measured wavelengths and the CWL.The temperature (T), axial strain (), and static pressure (P) wavelength sensitivities can be written as follows [11,13]: With the achromatic fringes in the transmission spectrum, multiple sensing indicators are provided in the sensors employing CWL-existed in-line MZIs, including monitoring wavelength shifts in CWL [6][7][8]18,40], P H, 1 /D H, 1 , P L, 1 /D L, 1 [9][10][11]15,17,[28][29][30][31][32][33][34][35][36], and intensity variations [12] at a specific wavelength or wavelength ranges.Studies in CWL-existed in-line MZIs show that utilizing CWL shifts or other monitoring parameters listed above, the CWL-existed in-line MZIs can provide competitive advantages over conventional periodically changed interferometers in the applications of large-range measurement, co-located multiparameter sensing, and high-sensitivity sensing.Moreover, the theoretical study of wavelength sensitivities of interference fringes offers a direction for mode interferometers to maximize their peak wavelength sensitivities.

Characteristics of Interference Peaks/Dips in the CWL-Existed in-Line MZIs and Their Applications
In a CWL-existed transmission spectrum, the interference peaks/dips, especially those lying closest to CWL on both sides, exhibit essential characteristics different from the typical periodically changing fringes.These differences were experimentally observed by Salik et al. [15] and Zhou et al. [17].However, theoretical analysis was not presented.The wavelength sensitivities of the interference fringes under temperature, axial strain, static pressure, and surrounding refractive index (SRI) variations were further theoretically and experimentally studied in Refs.[10,11,[28][29][30]32,36], which are reviewed in this section.

Characteristics of Interference Peaks/Dips in the CWL-Existed SFS Structures
For the CWL-existed SFS structure employing the FMF shown in Figure 2, because the geometrical structure and relative refractive index difference profile of the FMF are determined, the temperature, axial strain, and static pressure wavelength sensitivities of the interference peaks/dips are governed by the wavelength separations between the measured wavelengths and the CWL.The temperature (T), axial strain (ε), and static pressure (P) wavelength sensitivities can be written as follows [11,13]: Photonics 2022, 9, 378 5 of 15 where β 01 and β 02 are the propagation constants of the LP 01 and LP 02 modes, respectively.α = (1/L FMF )(∂L FMF /∂T) = 5.1 × 10 −7 / • C is the thermal expansion coefficient for pure silica [20].
12, ρ 12 = 0.27 are strain-optic coefficients for fused silica, respectively [45].ν = 0.17 and E = 78 GPa is the Poisson ratio and Young's modulus for pure silica, respectively [46].k 0 = 2π/λ is the free space wave vector.r co is the core radius of the FMF.n 1 and n 2 is the effective refractive index of the LP 01 and LP 02 modes propagating in the FMF, respectively.
is the normalized frequency of the FMF.β i and b i (i = 1, 2) is the propagation constant and normalized propagation constant of LP 0i mode, respectively, and can be represented as The wavelength sensitivity curves under temperature, axial strain, and static pressure are calculated and depicted in solid curves in Figure 4a-c, together with the experimental results.From Equations ( 2)-( 5) and Figure 4, the wavelength sensitivities of interference fringes in the spectrum of the SFS structure with CWL can be summarized as follows: (1) The interference peaks/dips lying on the different sides of the CWL shift to opposite directions under temperature, axial strain, and static pressure variations.For example, when temperature increases, there is a blueshift for peaks on the lower wavelength side of the CWL (i.e., within the region from 1500 nm to the CWL), while for peaks on the longer wavelength side of the CWL (i.e., within the region from the CWL to 1580 nm), there is a redshift.The experimental transmission spectra of the SFS structure under temperature variations are shown in Figure 4d, which agrees well with the calculation results.(2) The wavelength sensitivity of the interference peak is only related to the wavelength spacing between the measured wavelength and the CWL.The L FMF used in the SFS sensor and the sensing parameters pre-applied to the FMF section may change the wavelength separation between the peak wavelength and the CWL.Thus, the variation in wavelength separation will affect the corresponding wavelength sensitivity of the measured peaks, as shown in Figure 4a-c.The experimental results show close agreement with the simulated curves.(3) As the peak wavelength value approaches the CWL value, i.e., λ/λ CW L → 1 , the wavelength sensitivity increases significantly, as shown in Figure 4a-c, from both the theoretical and experimental points of view.However, the mathematically derived infinitely high sensitivity at the CWL has no practical meaning because ∂(∆β)/∂λ = 0 at CWL from the theoretical analysis, and an infinite length of FMF is needed in experiments to minimize the wavelength spacing between the wavelength of P L, 1 /P H, 1, and the CWL, which is unpractical in applications.
Because the wavelength sensitivities of interference peaks and CWL are different under sensing parameter variations, the SFS structure can be applied in multiparameter sensing by simultaneously measuring the shift in CWL and interference peaks.Simultaneous measurements of temperature and axial strain [9] have been studied by detecting wavelength shifts in P H, 1 and P L, 1 , taking advantage of their highest wavelength sensitivities among all of the peaks on the right and left sides of the CWL, respectively.
To further improve the sensitivity performance, a cascaded CWL-existed SFS structure was proposed by our group, which has been applied in static water pressure and temperature sensing [11].Figure 5 shows the schematic diagram and transmission spectra under different static water pressures of the cascaded SFS structure.Vernier effect was adopted to magnify the wavelength sensitivities of interference peaks.The free spectrum range of the envelope in the superimposed transmission spectra varies with wavelength and reaches the maximum when approaching the envelope critical wavelength (CWL E ).Therefore, the envelop peaks, P EL, 1 and P EL, 2 , are easy to identify.A maximum static water pressure sensitivity of 4.072 nm/MPa and temperature sensitivity of 1.753 nm/ • C is achieved using this sensing approach, which is more than 7 times larger in terms of the wavelength sensitivity than those reported in sensors using a single CWL-existed SFS structure [14].This cascaded CWL-existed SFS static-pressure sensor can be applied in marine pressure sensing with temperature compensation by simultaneously measuring the envelope peaks P EL, 1 and P EL, 2 .
Photonics 2022, 9, x FOR PEER REVIEW 6 of 15 Because the wavelength sensitivities of interference peaks and CWL are different under sensing parameter variations, the SFS structure can be applied in multiparameter sensing by simultaneously measuring the shift in CWL and interference peaks.Simultaneous measurements of temperature and axial strain [9] have been studied by detecting wavelength shifts in PH, 1 and PL, 1, taking advantage of their highest wavelength sensitivities among all of the peaks on the right and left sides of the CWL, respectively.
To further improve the sensitivity performance, a cascaded CWL-existed SFS structure was proposed by our group, which has been applied in static water pressure and temperature sensing [11].Figure 5 shows the schematic diagram and transmission spectra under different static water pressures of the cascaded SFS structure.Vernier effect was adopted to magnify the wavelength sensitivities of interference peaks.The free spectrum range of the envelope in the superimposed transmission spectra varies with wavelength and reaches the maximum when approaching the envelope critical wavelength (CWLE).Therefore, the envelop peaks, PEL, 1 and PEL, 2, are easy to identify.A maximum static water pressure sensitivity of 4.072 nm/MPa and temperature sensitivity of 1.753 nm/°C is achieved using this sensing approach, which is more than 7 times larger in terms of the wavelength sensitivity than those reported in sensors using a single CWLexisted SFS structure [14].This cascaded CWL-existed SFS static-pressure sensor can be applied in marine pressure sensing with temperature compensation by simultaneously measuring the envelope peaks PEL, 1 and PEL, 2. with permission, from [11].

Characteristics of Interference Peaks/Dips in the CWL-Existed Few-Mode Microfiber in-Line MZIs
The few-mode microfiber interferometer fabricated by tapering a normal-sized optical fiber to the diameter of several micrometers is widely applied in SRI sensing using the evanescent waves propagating in the microfiber section.Wei et al. [28] proposed a few-

Characteristics of Interference Peaks/Dips in the CWL-Existed Few-Mode Microfiber in-Line MZIs
The few-mode microfiber interferometer fabricated by tapering a normal-sized optical fiber to the diameter of several micrometers is widely applied in SRI sensing using the evanescent waves propagating in the microfiber section.Wei et al. [28] proposed a fewmode microfiber interferometer designed at the CWL.They explored the SRI sensing properties of tapered microfiber interferometers in a gaseous medium, where the SRI changes around 1. The nonadiabatic few-mode microfiber is tapered from a standard SMF.The fundamental HE 11 mode and the high-order HE 12 mode are excited and propagate in the uniform waist region, as shown in Figure 6. with permission, from [11].

Characteristics of Interference Peaks/Dips in the CWL-Existed Few-Mode Microfiber in-Line MZIs
The few-mode microfiber interferometer fabricated by tapering a normal-sized optical fiber to the diameter of several micrometers is widely applied in SRI sensing using the evanescent waves propagating in the microfiber section.Wei et al. [28] proposed a fewmode microfiber interferometer designed at the CWL.They explored the SRI sensing properties of tapered microfiber interferometers in a gaseous medium, where the SRI changes around 1. The nonadiabatic few-mode microfiber is tapered from a standard SMF.The fundamental HE11 mode and the high-order HE12 mode are excited and propagate in the uniform waist region, as shown in Figure 6.For a particular interference peak/dip with the peak/dip wavelength of   , the SRI wavelength sensitivity is written as follows [28,47]: where  =    11 −    12 is the difference between the group effective index of HE11 and HE12 modes.The group effective index can be calculated by   =   −   (  )/  .Therefore, with known material and diameter parameters of the uniform microfiber section, the term Δ  /Δ increases dramatically as the value of G approaches zero, which For a particular interference peak/dip with the peak/dip wavelength of λ N , the SRI wavelength sensitivity is written as follows [28,47]: where is the difference between the group effective index of HE 11 and HE 12 modes.The group effective index can be calculated by Therefore, with known material and diameter parameters of the uniform microfiber section, the term ∆λ N /∆SRI increases dramatically as the value of G approaches zero, which corresponds to the CWL point.Moreover, the interference peaks/dips lying on each side of the CWL shift to opposite wavelength directions under SRI variations.These conclusions agree well with the wavelength sensitivity study of CWL-existed SFS structure.The calculated wavelength sensitivity curves of SRI under different microfiber diameters are shown in Figure 7.By measuring the peak wavelength separation of D L, 1 and D H, 1 , a maximum sensitivity of −69,984.3 nm/RIU ("−" denotes blue wavelength shifts) around the air refractive index is obtained in experiments.The temperature influence on the gas RI sensor is also reported, and the measured temperature sensitivities of D L, 1 and D H, 1 are −2.134nm/ • C and 1.899 nm/ • C, respectively.Thus, the cross-sensitivity of the gas refractive index sensor is 5.763 ×10 −5 RIU/ • C.
sions agree well with the wavelength sensitivity study of CWL-existed SFS structure.The calculated wavelength sensitivity curves of SRI under different microfiber diameters are shown in Figure 7.By measuring the peak wavelength separation of DL, 1 and DH, 1, a maximum sensitivity of −69,984.3 nm/RIU ("−" denotes blue wavelength shifts) around the air refractive index is obtained in experiments.The temperature influence on the gas RI sensor is also reported, and the measured temperature sensitivities of DL, 1 and DH, 1 are −2.134nm/°C and 1.899 nm/°C , respectively.Thus, the cross-sensitivity of the gas refractive index sensor is 5.763 × 10 −5 RIU/℃.With the merits of excellent sensitivity performances and simple fabrication processes, few-mode microfibers have been further studied and applied in various SRI-related sensing parameters.Guan et al. [29] thoroughly investigated the sensitivity of a twomode microfiber MZI as a function of SRI.Since Equation ( 7) can be rewritten as where Γ = 1 −   /∆  • ∆  / is a dispersion factor that characterizes the effect of index difference variations with wavelength, ∆  / .Since the whole term in the bracket was negative [48], the sign of Γ is crucial in determining the SRI sensitivity sign.CWL exists at the point where the sign of the SRI sensitivity changes.The simulated dispersion factors Γ and Δ  /Δ  as functions of the fiber diameter is shown in Figure 8.
The numerical simulation revealed that two CWLs exist as the microfiber diameter decreases.Experiments were conducted to study the wavelength sensitivities of interference fringes around both CWLs.At the second CWL, an ultrahigh SRI sensitivity of approximately 95,836 nm/RIU with the microfiber diameter of 1.87 µ m was achieved around the SRI of air.Wang et al. [33] applied the few-mode microfiber MZI in marine fields by measuring sodium nitrate in seawater.They achieved the wavelength sensitivity of 5.98 pm/ppm using DL, 1, corresponding to the SRI sensitivity of 50396.09nm/RIU.CWL was also observed in Z-shaped few-mode microfibers between the interference of HE11 and HE21 modes [34].The SRI sensitivity of 1.46 × 10 5 nm/RIU was experimentally achieved around the refractive index of water.Zhao et al. [32] demonstrated an ultrasensitive sensor for SRI and temperature measurement by employing a CWL-existed two-mode microfiber.An ultrahigh SRI sensitivity of 24,209 nm/RIU around 1.3320 and a high-temperature sensitivity of −2.47 nm/°C were experimentally achieved by a microfiber with a waist diameter of ~4.8 μm.Meanwhile, the CWL-existed microfiber was coated with a polydimethylsiloxane (PDMS) layer to increase the temperature sensitivity to about 8.33 nm/°C .To further increase the measurement range of the interference dips, they [30] proposed a fiber Bragg grating (FBG) cascaded microfiber in-line MZI operating at the CWL and applied it in seawater temperature With the merits of excellent sensitivity performances and simple fabrication processes, few-mode microfibers have been further studied and applied in various SRI-related sensing parameters.Guan et al. [29] thoroughly investigated the sensitivity of a two-mode microfiber MZI as a function of SRI.Since Equation ( 7) can be rewritten as where Γ = 1 − λ N /∆n e f f •∆n e f f /∂λ is a dispersion factor that characterizes the effect of index difference variations with wavelength, ∆n e f f /∂λ.Since the whole term in the bracket was negative [48], the sign of Γ is crucial in determining the SRI sensitivity sign.CWL exists at the point where the sign of the SRI sensitivity changes.The simulated dispersion factors Γ and ∆λ N /∆n SRI as functions of the fiber diameter is shown in Figure 8.The numerical simulation revealed that two CWLs exist as the microfiber diameter decreases.
Experiments were conducted to study the wavelength sensitivities of interference fringes around both CWLs.At the second CWL, an ultrahigh SRI sensitivity of approximately 95,836 nm/RIU with the microfiber diameter of 1.87 µm was achieved around the SRI of air.Wang et al. [33] applied the few-mode microfiber MZI in marine fields by measuring sodium nitrate in seawater.They achieved the wavelength sensitivity of 5.98 pm/ppm using D L, 1 , corresponding to the SRI sensitivity of 50,396.09nm/RIU.CWL was also observed in Z-shaped few-mode microfibers between the interference of HE 11 and HE 21 modes [34].The SRI sensitivity of 1.46 × 10 5 nm/RIU was experimentally achieved around the refractive index of water.
Photonics 2022, 9, x FOR PEER REVIEW 9 of 15 measurement, as shown in Figure 9a.The FBG resonant dip, located in the achromatic fringe region, works as a rough measurement tool to identify the temperature measurement ranges.Meanwhile, the interference dip closest to the CWL with the highest sensitivity among all the interference dips was employed to measure the water temperature once the rough range was determined.The two-mode microfiber MZI was packaged in PDMS to increase temperature sensitivity.The transmission spectrum of the FBG-cascaded two-mode microfiber MZI in PDMS package is shown in Figure 9b.With the aid of FBG, the temperature measurement range could be enlarged from ~2 °C to ~36 °C, with a temperature sensitivity of ~38 nm/℃, as shown in Figure 9c.Zhao et al. [32] demonstrated an ultrasensitive sensor for SRI and temperature measurement by employing a CWL-existed two-mode microfiber.An ultrahigh SRI sensitivity of 24,209 nm/RIU around 1.3320 and a high-temperature sensitivity of −2.47 nm/ • C were experimentally achieved by a microfiber with a waist diameter of ~4.8 µm.Meanwhile, the CWL-existed microfiber was coated with a polydimethylsiloxane (PDMS) layer to increase the temperature sensitivity to about 8.33 nm/ • C. To further increase the measurement range of the interference dips, they [30] proposed a fiber Bragg grating (FBG) cascaded microfiber in-line MZI operating at the CWL and applied it in seawater temperature measurement, as shown in Figure 9a.The FBG resonant dip, located in the achromatic fringe region, works as a rough measurement tool to identify the temperature measurement ranges.Meanwhile, the interference dip closest to the CWL with the highest sensitivity among all the interference dips was employed to measure the water temperature once the rough range was determined.The two-mode microfiber MZI was packaged in PDMS to increase temperature sensitivity.The transmission spectrum of the FBG-cascaded two-mode microfiber MZI in PDMS package is shown in Figure 9b.With the aid of FBG, the temperature measurement range could be enlarged from ~2 • C to ~36 • C, with a temperature sensitivity of ~38 nm/ • C, as shown in Figure 9c.
PDMS to increase temperature sensitivity.The transmission spectrum of the FBG-cascaded two-mode microfiber MZI in PDMS package is shown in Figure 9b.With the aid of FBG, the temperature measurement range could be enlarged from ~2 °C to ~36 °C, with a temperature sensitivity of ~38 nm/℃, as shown in Figure 9c.Apart from using nonadiabatic tapers to fabricate the few-mode microfibers by tapering a standard SMF, other special fibers were employed to increase the sensor performance and redshift the cutoff wavelength of the high-order mode in the microfiber.In 2020, Wang et al. [31] proposed a CWL-existed in-line MZI by tapering a ring-core twomode fiber.Quasi-adiabatic transition tapers were fabricated to provide HE11 and HE21 modes with continuous mode evolutions to the microfiber section.The two modes launched into the fiber were collected at the fiber output.A CWL was observed when the Apart from using nonadiabatic tapers to fabricate the few-mode microfibers by tapering a standard SMF, other special fibers were employed to increase the sensor performance and redshift the cutoff wavelength of the high-order mode in the microfiber.In 2020, Wang et al. [31] proposed a CWL-existed in-line MZI by tapering a ring-core two-mode fiber.Quasi-adiabatic transition tapers were fabricated to provide HE 11 and HE 21 modes with continuous mode evolutions to the microfiber section.The two modes launched into the fiber were collected at the fiber output.A CWL was observed when the tapered two-mode microfiber was immersed in a liquid environment.The SRI sensitivities of 79,283.7 nm/RIU and −48,772 nm/RIU were reported, respectively, by detecting the right and left interference dip closest to CWL, i.e., D L, 1 and D H, 1 .Thus, a maximum sensitivity of 128,055.7 nm/RIU was achieved by measuring the wavelength separation variations between these two interference dips.This CWL-based two-mode microfiber was further applied in an immunoglobulin G (IgG) level measurement.The limits of detection of 10 fg/mL of IgG in phosphate-buffered saline solution were achieved in the experiments.Luo et al. [36] reported a refractive index sensor fabricated by tapering a single stressapplying fiber (SSAF), where a stress-applying part was introduced into the fiber cladding to engineer the cladding material index.The SSAF-based two-mode microfiber could significantly reduce the sensitivity of the group effective refractive index difference between the two interference modes-HE 11 and TE 01 .Therefore, the operation bandwidth of the two-mode microfiber can be broadened to 500 nm, compared with that of conventional SMF-based two-mode microfiber with the same diameter operating at CWL.By detecting the wavelength separation changes between D L, 1 and D H, 1 , a maximum sensitivity of 30,563 nm/RIU was reported in the refractive index measurement range of 1.3212~1.3216.
The interference fringes in the transmission spectrum of a CWL-existed few-mode microfiber structure can also be applied in vibration measurement.Luo et al. [35] demonstrated a highly sensitive vibration sensor using a CWL-existed two-mode microfiber MZI and packaging it in a rectangular through-hole cantilever beam.By detecting the interference dip near the CWL, the maximum axial-strain sensitivity reached −45.55 pm/µε.This structure was also applied in acceleration sensing experiments, where the acceleration sensitivity of the sensor was 0.764 V/g at 45 Hz.
The sensors employing the CWL-based in-line MZI structures and measuring the wavelength shifts in the interference peaks/dips closest to the CWL on both sides are listed in Table 1.Compared with SFS structures, the few-mode microfibers exhibit higher sensitivity in temperature, axial strain, acceleration, and surrounding refractive index measurements.However, the taper structures and ultrasmall waists of the microfibers may cause structural fragility.Therefore, the microfibers and the taper regions should be packaged carefully to guarantee robustness in applications.During experiments, the noise induced by temperature fluctuations and axial strain in the packaging process should also be considered.The CWL-based in-line MZIs have advantages in that the CWL and interference fringes near the CWL exhibit different wavelength sensitivities under the same measurement and variation.Thus, the temperature-and strain-induced noises can be compensated by simultaneously detecting two or three peaks/dips.Few-mode fiber Axial Strain 43 pm/µε [10] Few-mode fiber Temperature 704 pm/°C [10] Cascaded few-mode fiber temperature 1.753 nm/°C [11] Cascaded few-mode fiber Static pressure 4.072 nm/MPa [11] Few − mode microfiber with a diameter of 2 µm Refractive index around 1 −69,984.3nm/RIU [28] Few − mode microfiber with a diameter of 1.87 µm Refractive index around 1 95,836 nm/RIU [29] Few − mode microfiber with a diameter of 3.57 µm Sodium nitrate in seawater 5.98 pm/ppm [33] Z-shaped few-mode microfiber Refractive index around 1.333 1.46 × 105 nm/RIU [34] Few-mode microfiber coated with PDMS Temperature 38 nm/°C [30] Few-mode microfiber tapered by a ring-core two-mode fiber Refractive index around 1.333 128,055.7 nm/RIU [31] Few-mode microfiber tapered by a single stress-applying fiber Refractive index around 1.3212 30,563 nm/RIU [36] Few − mode microfiber with a diameter of 2.2 µm Axial strain −45.55 pm/µε [35] Few − mode microfiber with a diameter of 2.2 µm Acceleration 764 mV/g [35] 4. Characteristics of the Achromatic Fringe in CWL-Existed in-Line MZIs and Their Applications In the transmission spectrum of a CWL-existed in-line MZI, achromatic fringes are different from the periodically changing interference fringes and, therefore, are easy to identify.Because the achromatic fringe shifts monotonously with sensing parameters, CWL detection sensors can be applied in large-range measurements.This section reviews a demodulation method of the CWL and sensing applications employing CWL detection.

Detection Scheme of CWL in the SFS Sensing Structure
The demodulation method of the CWL in the transmission spectrum and its advantages, compared with the traditional interference peak detection, is explained in Ref. [8], using an etched SFS structure in SRI detection.Figure 10 shows the simulated transmission spectra, the shift in CWL, and the wavelength shift in peak/dip located closest to the CWL on each side under different SRIs.The peaks/dips in both sides of CWL will move toward the CWL with increasing SRI.If SRI is measured by monitoring the wavelength shift in peaks/dips lying around the CWL, the peaks/dips will end at CWL with only a small measurement range of SRI, as shown in Figure 10.Therefore, traditional interference peak/dip wavelength detection methods limit the measurement range significantly.
CWL detection sensors can be applied in large-range measurements.This section reviews a demodulation method of the CWL and sensing applications employing CWL detection.

Detection Scheme of CWL in the SFS Sensing Structure
The demodulation method of the CWL in the transmission spectrum and its advantages, compared with the traditional interference peak detection, is explained in Ref. [8], using an etched SFS structure in SRI detection.Figure 10 shows the simulated transmission spectra, the shift in CWL, and the wavelength shift in peak/dip located closest to the CWL on each side under different SRIs.The peaks/dips in both sides of CWL will move toward the CWL with increasing SRI.If SRI is measured by monitoring the wavelength shift in peaks/dips lying around the CWL, the peaks/dips will end at CWL with only a small measurement range of SRI, as shown in Figure 10.Therefore, traditional interference peak/dip wavelength detection methods limit the measurement range significantly.Since the peaks and dips in the transmission spectrum are nearly symmetrically distributed on both sides of CWL, to avoid difficulties in identifying CWL from achromatic Since the peaks and dips in the transmission spectrum are nearly symmetrically distributed on both sides of CWL, to avoid difficulties in identifying CWL from achromatic fringes, we can obtain CWL from Equation ( 8) by averaging the wavelength of P H, 1 , D H, 1 , P L, 1 , and D L, 1 , written as The shift in CWL with SRI is calculated from Equation (8) above, and the results are given in Figure 10b.Compared with the change in first peak or dip wavelengths simultaneously plotted in Figure 10b, the CWL changes monotonically with SRI in a large measurement range, with unambiguous demodulation of SRIs.Thus, the CWL detection sensors can effectively solve the overlapping spectrum problem and provide large sensing ranges in applications.

Sensing Applications of in-Line MZIs Based on CWL Detection
The advantages of the CWL detection method in large-range measurement are first experimentally observed in temperature detection.Tripathi et al. [18] spliced a section of MMF between two sections of SMF (SMS) and observed a CWL in the transmission spectrum at 1318 nm.The experiments showed that the CWL shifted monotonously with temperature from 25 • C to 1100 • C. Martincek et al. [40] employed a liquid-core optical fiber, which consisted of a fused silica cladding and a core with medicinal oil, to increase the temperature sensitivity of the CWL.The maximum temperature sensitivity of the measured CWL was −56 nm/ • C, around 24 • C. The transmission spectra of the liquid-core fiber with temperature variations are shown in Figure 11.
trum at 1318 nm.The experiments showed that the CWL shifted monotonously w perature from 25 ℃ to 1100 ℃.Martincek et al. [40] employed a liquid-core opti which consisted of a fused silica cladding and a core with medicinal oil, to incr temperature sensitivity of the CWL.The maximum temperature sensitivity of th ured CWL was −56 nm/°C, around 24 °C .The transmission spectra of the liquid-c with temperature variations are shown in Figure 11.Our group theoretically studied the characteristics of the CWL in the tran spectrum of an SFS structure and applied the SFS structure in the large-range m ment of SRI [8], curvature [6], and displacement measurements [7]; the sensin achieved 1.316~1.44 at 1550 nm, 0~125 m −1 , and 0~350 mm, respectively, as show ure 12.Our group theoretically studied the characteristics of the CWL in the transmission spectrum of an SFS structure and applied the SFS structure in the large-range measurement of SRI [8], curvature [6], and displacement measurements [7]; the sensing ranges achieved 1.316~1.44 at 1550 nm, 0~125 m −1 , and 0~350 mm, respectively, as shown in Figure 12.

Conclusions
In this paper, we reviewed the transmission characteristics of CWL-existed in-line MZIs.The detection of the CWL in the transmission spectrum provides a solution to the overlapping problem in conventional MZIs with periodically changing interference fringes.Moreover, the interference peaks/dips on each side of the CWL shift in opposite directions.Their wavelength sensitivities increase significantly when peak wavelengths approach the CWL, which provides a solution for fiber-optic MZIs to enhance their sensi-

Conclusions
In this paper, we reviewed the transmission characteristics of CWL-existed in-line MZIs.The detection of the CWL in the transmission spectrum provides a solution to the overlapping problem in conventional MZIs with periodically changing interference fringes.Moreover, the interference peaks/dips on each side of the CWL shift in opposite directions.Their wavelength sensitivities increase significantly when peak wavelengths approach the CWL, which provides a solution for fiber-optic MZIs to enhance their sensitives by designing the operational wavelength near the CWL.Therefore, taking full advantage of the CWL-existed transmission spectrum, these in-line MZIs can be developed into several different sensors, including large-range measurement sensors, multiparameter sensors, and high-sensitivity sensors, to satisfy the requirements in practical applications.

Figure 1 .
Figure 1.Schematic of the critical-wavelength-existed in-line Mach-Zehnder interferometer measurement system.

Figure 1 .
Figure 1.Schematic of the critical-wavelength-existed in-line Mach-Zehnder interferometer measurement system.

Figure 2 .
Figure 2. (a) Geometrical structure and relative refractive index difference profile of the few-mod fiber; (b) the SMF-FMF-SMF (SFS) structure.The insets are simulated mode field distributions the LP01 mode propagating in SMF and the LP01 and LP02 modes propagating in FMF at 1550 n wavelength using the finite element model.SMF: single-mode fiber; FMF: few-mode fiber.

Figure 2 .
Figure 2. (a) Geometrical structure and relative refractive index difference profile of the few-mode fiber; (b) the SMF-FMF-SMF (SFS) structure.The insets are simulated mode field distributions of the LP 01 mode propagating in SMF and the LP 01 and LP 02 modes propagating in FMF at 1550 nm wavelength using the finite element model.SMF: single-mode fiber; FMF: few-mode fiber.

Figure 3 .
Figure 3.The simulated curve of the propagation constant difference Δ of LP01 and LP02 modes propagating in the few-mode fiber, and the transmission spectrum of the SFS structure ( FMF = 50 cm) under the temperature of 25 ℃ versus wavelength.

Figure 3 .
Figure 3.The simulated curve of the propagation constant difference ∆β of LP 01 and LP 02 modes propagating in the few-mode fiber, and the transmission spectrum of the SFS structure (L FMF = 50 cm) under the temperature of 25 • C versus wavelength.

Figure 4 .
Figure 4. Characteristics of interference fringes in the transmission spectra of the CWL-existed SFS structure: (a) temperature wavelength sensitivities of interference fringes; (b) axial strain wavelength sensitivities of interference fringes; (c) static pressure wavelength sensitivities of interference fringes; (d) transmission spectra of the SFS structure (LFMF = 50 cm) under temperature variations.

Figure 4 . 15 Figure 5 .
Figure 4. Characteristics of interference fringes in the transmission spectra of the CWL-existed SFS structure: (a) temperature wavelength sensitivities of interference fringes; (b) axial strain wavelength sensitivities of interference fringes; (c) static pressure wavelength sensitivities of interference fringes; (d) transmission spectra of the SFS structure (L FMF = 50 cm) under temperature variations.s 2022, 9, x FOR PEER REVIEW 7 of 15

Figure 5 .
Figure 5. (a) Diagram of the cascaded SMF-FMF-SMF(SFS) structure; (b) experimental transmission spectra of the cascaded SFS structure under static water pressure variations; (c) wavelength shifts in PE L. 1 and PE L. 2 under static water pressure and temperature variations.Reprinted with permission from ref. [11].Copyright 2020 IEEE.

Figure 5 .
Figure 5. (a) Diagram of the cascaded SMF-FMF-SMF(SFS) structure; (b) experimental transmission spectra of the cascaded SFS structure under static water pressure variations; (c) wavelength shifts in PEL. 1 and PEL. 2 under static water pressure and temperature variations.@ (2020) IEEE.Reprinted,

Figure 6 .
Figure 6.(a) The microscopic view of the tapered optical microfiber; (b) the origin (black) and the fast Fourier transform (FFT)-filtered (red) transmission spectra of the tapered few-mode microfiber surrounded by air; (c) the simulated mode profiles of HE11 and HE12 modes.Adapted with permission from Ref. [28].

Figure 6 .
Figure 6.(a) The microscopic view of the tapered optical microfiber; (b) the origin (black) and the fast Fourier transform (FFT)-filtered (red) transmission spectra of the tapered few-mode microfiber surrounded by air; (c) the simulated mode profiles of HE 11 and HE 12 modes.Adapted with permission from [28].

Figure 7 .
Figure 7.The surrounding refractive index sensitivities of interference fringes around CWLs at different waist diameters of microfibers.Adapted with permission from Ref. [28].

Figure 7 .
Figure 7.The surrounding refractive index sensitivities of interference fringes around CWLs at different waist diameters of microfibers.Adapted with permission from Ref. [28].

Figure 8 .
Figure 8.(a) Schematic diagram of the taper-based in-line MZI; the inset presents calculated intensity profiles of the interference two modes, HE11 and HE12, respectively; (b) simulated dispersion factor Γ and (c) RI sensitivity as functions of the fiber diameter.Adapted with permission from Ref. [29].

Figure 8 .
Figure 8.(a) Schematic diagram of the taper-based in-line MZI; the inset presents calculated intensity profiles of the interference two modes, HE 11 and HE 12 , respectively; (b) simulated dispersion factor Γ and (c) RI sensitivity as functions of the fiber diameter.Adapted with permission from Ref. [29].

Figure 8 .
Figure 8.(a) Schematic diagram of the taper-based in-line MZI; the inset presents calculated intensity profiles of the interference two modes, HE11 and HE12, respectively; (b) simulated dispersion factor Γ and (c) RI sensitivity as functions of the fiber diameter.Adapted with permission from Ref. [29].

Figure 9 .
Figure 9. (a) Schematic of the polydimethylsiloxane (PDMS)-packaged two-mode microfiber for temperature measurement; (b) the transmission spectrum of the PDMS-packaged two-mode microfiber MZI cascaded with FBG; (c) cooperation of the FBG resonant dip and the interference dips (DL,

Figure 9 .
Figure 9. (a) Schematic of the polydimethylsiloxane (PDMS)-packaged two-mode microfiber for temperature measurement; (b) the transmission spectrum of the PDMS-packaged two-mode microfiber MZI cascaded with FBG; (c) cooperation of the FBG resonant dip and the interference dips (D L, 1 and D H, 1 ) lying on each side of the CWL in temperature measurement.Reprinted with permission from Ref. [30].Copyright 2020 Elsevier.

Figure 10 .
Figure 10.The simulated results of the etched SFS structure (  = 21.3 μm,   = 20 cm): (a) the transmission spectra under the SRIs of 1.350, 1.355, and 1.360; (b) wavelength shifts in the critical wavelength and interference peaks/dips on each side of the critical wavelength.

Figure 10 .
Figure 10.The simulated results of the etched SFS structure (d FMF = 21.3 µm, L FMF = 20 cm): (a) the transmission spectra under the SRIs of 1.350, 1.355, and 1.360; (b) wavelength shifts in the critical wavelength and interference peaks/dips on each side of the critical wavelength.

Figure 11 .
Figure 11.Shift in the CWL in MZI employing the liquid-core fiber shown on measured spectra for temperatures of 28.2, 27.5, and 26.2 • C. Reprinted with permission from Ref. [40].Copyright 2011 Elsevier.

Figure 12 .
Figure 12.The wavelength shifts in CWL and sensor structures employing the few-mode fiber in the SFS structure; (a) experimental (marks with error bars) and calculated results of CWL shift with SRI variations.@ (2017) IEEE.Reprinted, with permission, from [8]; (b) calculated and experimental results of CWL shift with equivalent curvature.@ (2016) IEEE.Reprinted, with permission, from [6]; (c) schematic diagram of the experiment setup of the displacement sensor employing SFS structure; (d) the experimental and the theoretical results of the CWL changes in the sensor and the theoretical equivalent curvature of FMF in the helix coil (dashed line) versus displacement.@ (2016) IEEE.Reprinted, with permission, from [7].

Figure 12 .
Figure 12.The wavelength shifts in CWL and sensor structures employing the few-mode fiber in the SFS structure; (a) experimental (marks with error bars) and calculated results of CWL shift with SRI variations.Reprinted with permission from ref. [8].Copyright 2017 IEEE.(b) calculated and experimental results of CWL shift with equivalent curvature.Reprinted with permission from ref. [6].Copyright 2016 IEEE.; (c) schematic diagram of the experiment setup of the displacement sensor employing SFS structure; (d) the experimental and the theoretical results of the CWL changes in the sensor and the theoretical equivalent curvature of FMF in the helix coil (dashed line) versus displacement.Reprinted with permission from ref. [7].Copyright 2016 IEEE.

Table 1 .
Performance comparisons of different CWL-existed in-line MZI sensors.