Modal interference discrepancy and its application to a modified fiber Mach-Zehnder Vernier interferometer

Xiaoyan Wen; Guohui Lin; Xinao Jia; Min Li; Ming-Yu Li; Haifei Lu; Jiafu Wang

doi:10.1364/OE.474302

1. Introduction

As a kind of excellent fiber interferometers, all-fiber Mach-Zehnder interferometers (MZI) have received considerable attention in recent years due to their high sensitivity, easy fabrication, compactness, etc. [1–6] Various all-fiber MZI have been developed, for which the common background is modal interference in the fiber. For example, the most studied all-fiber MZIs are based on a multiple mode fiber (MMF) fused with a lead-in and a lead-out single mode fiber (SMF) [7]. Due to core diameter mismatch, several cladding modes are excited together with core mode. All the excited modes interference with each other and the output spectrum is a superimposition of modal interferences. Other novel all-fiber MZIs based on thin-core fiber [8], hollow-core fiber [9], no-core fiber [10,11], tapered few modes fiber [12], dispersion compensation fiber [13] and PDMS-coated S-tapered fiber [14], etc., have also been developed. In these structures, different kinds of cladding modes were excited and involved in the output spectrum. The variation in output spectrum was directly related to the fluctuation of certain ambient parameter such as refractive index (RI) [8], curvature [9], liquid level [10], lateral stress [11], magnetic field [12], temperature [13], and transverse load [14]. However, discrepancies in modal interference involved in these sensors are neglected.

It has been reported that effective refractive index of higher modes in fiber are more sensitive to ambient media compared with lower modes [15]. In addition, mode dispersion, thoroughly studied by theoretical calculation and simulation [16,17], is another feature that differ in different modes. In-consistence in mode sensitivity and dispersion characteristic will bring about discrepancy in modal interference, which will further affect the profile, dip wavelength and fringe visibility of the output spectrum. For example, spectral profile is important to the choice of a dip that is usually used for sensing tests. Dip wavelength is critical to the response sensitivity and range of the sensor, especially for a wavelength demodulation system. Fringe visibility affects the signal intensity and accuracy of the sensor. Ming-Jun Li and Costas Saravanos demonstrated that modal interference can be controlled by optimizing both the fiber design and its surrounding material [18]. However, to the best of our knowledge, study on the discrepancy in different modal interference involved in an all-fiber MZI is far from enough.

In addition, optical Vernier effect is an efficient method for enhancing the sensitivity of an all-fiber MZI. Most of the Vernier effect-based sensors consist of two cascaded MZI interferometers, one acts as a reference arm and the other acts as a sensing arm [19–21]. The two arms are designed to possess of spectra with slightly different free spectrum ranges (FSRs) to work as Vernier calipers. As the two unit MZIs are separately fabricated and then cascaded, modal interferences involved in the output spectrum are more complicated. Research on the influence of modal interference discrepancy on Vernier effect is also limited.

In this paper, modal interference discrepancy in an all-fiber MZI is theoretically analyzed and experimentally verified. Theoretical calculation suggests that ambient RI response trend of core-cladding modal interference is opposite to that of cladding-cladding modal interference, while temperature response trends of the two kinds of modal interference are consistent. Based on the discrepancy, an improved Vernier sensor cascaded by two unit MZIs with slight/obvious core-offset welding are designed, which respectively offered core-cladding modal interference and cladding-cladding modal interference. Experimental verification is carried out, and results in good agreement with the theoretical analysis are obtained. Due to the opposite ambient RI and consistent temperature response trend of the two unit MZIs, the cascaded sensor possesses of an improved ambient RI sensitivity with a restrained temperature cross-talk.

The rest of the paper is organized as follows: Section 2 theoretically analyzes the discrepancy between different modes and modal interferences. Based on the discrepancy a modified Vernier effect is obtained by cascading two unit interferometers. One unit interferometer is dominated by core-cladding modal interference and the other one is dominated by cladding-cladding modal interference. Section 3 introduces the sensor fabrication and sensing system design. Section 4 gives experimental results and discussion. Finally, a summary is given in Section 5.

2. Principle and simulation

2.1 Modal interference in fiber

Generally, the output spectrum of an all-fiber MZI is a superposition of multiple modal interferences. The output light intensity can be expressed as [19]:

(1)$$I\textrm{ = }\sum\limits_{i = 1,N} {{I_{\textrm{mode}i}}} + 2\sum\limits_{i< j = 1,N} {\sqrt {{I_{\textrm{mode}i}}{I_{\textrm{mode}j}}} } \cos (\Delta {\varphi _{ij}}),$$

where I_modei and I_modej are the intensities of the i-th and j-th mode, respectively, Δφ_ij is the phase difference between them. Taking two-mode interference as an example, the output light intensity could be rewritten as:

(2)$$I = {I_{\textrm{mode}1}} + {I_{\textrm{mode}\textrm{2}}} + 2\sqrt {{I_{\textrm{mode}1}}{I_{\textrm{mode}2}}} \cos [\frac{{2\pi L(n_{\textrm{eff}}^{\bmod \textrm{e}1} - n_{\textrm{eff}}^{\bmod \textrm{e}2})}}{\lambda }],$$

where I_mode1 and I_mode2 represent the mode intensity involved in the interference, and n_mode1-n_mode2 is the effective refractive index difference between them. L is interference arm length and λ is incident light wavelength. Usually, a wide spectrum light source is used for sensing test so that an intensity minimum will occur when the following condition is satisfied:

(3)$${\lambda _{\textrm{dip,}i}} = \frac{{2L[n_{\textrm{eff}}^{\textrm{mode1}}(\lambda ,{n_{\textrm{ex}}}) - n_{\textrm{eff}}^{\bmod \textrm{e}2}(\lambda ,{n_{\textrm{ex}}})]}}{{2i + 1}}.$$

Thus, response of the dip wavelength to ambient RI could be obtained by:

(4)$$\frac{{d{\lambda _{\textrm{dip},i}}}}{{d{n_{ex}}}} = \frac{{{\lambda _{\textrm{dip},i}}\left( {\frac{{\partial n_{\textrm{eff}}^{\bmod \textrm{e}1}}}{{\partial {n_{ex}}}} - \frac{{\partial n_{\textrm{eff}}^{\bmod \textrm{e}2}}}{{\partial {n_{ex}}}}} \right)}}{{\Delta {n_{\textrm{eff}}} - {\lambda _{\textrm{dip},i}}\left( {\frac{{\partial n_{\textrm{eff}}^{\bmod \textrm{e}1}}}{{\partial \lambda }} - \frac{{\partial n_{\textrm{eff}}^{\bmod \textrm{e}2}}}{{\partial \lambda }}} \right)}},$$

where the bracket term in numerator depends effective refractive index response of the two modes, and that in denominator depends on dispersion of mode effective index [17].

Both the effective refractive index response and dispersion characteristic are inconsistent among different modes. For example, it has been reported that effective refractive index of higher modes in fiber are more sensitive to ambient media compared with lower modes [15]. Thus, for different modal interference, for example core-cladding modal interference and cladding-cladding modal interference, both the bracket terms in numerator and denominator of Eq. (4) are different, and will have outstanding influence on the wavelength response of an MZI sensor.

Similarly, response of the dip wavelength to ambient temperature is:

(5)$$\frac{{d{\lambda _{\textrm{dip},i}}}}{{dT}} = \frac{{{\lambda _{\textrm{dip},i}}\left( {\frac{{\partial n_{\textrm{eff}}^{\bmod \textrm{e}1}}}{{\partial T}} - \frac{{\partial n_{\textrm{eff}}^{\bmod \textrm{e}2}}}{{\partial T}}} \right)}}{{\Delta {n_{\textrm{eff}}} - {\lambda _{\textrm{dip},i}}\left( {\frac{{\partial n_{\textrm{eff}}^{\bmod \textrm{e}1}}}{{\partial \lambda }} - \frac{{\partial n_{\textrm{eff}}^{\bmod \textrm{e}2}}}{{\partial \lambda }}} \right)}},$$

in which the bracket term in numerator depends on temperature response of the two modes and that in denominator represents mode dispersion. Equation (5) indicates that discrepancy also exists in the temperature response of different modal interference in an MZI sensor. In the following part detailed calculation of discrepancy in different modes and modal interferences will be presented. Mode dispersion discrepancy is demonstrated together for the convenience of analysis.

2.2 Discrepancy in different modes

Firstly, effective refractive index (n_eff) of those typical modes under different ambient RI were calculated with COMSOL software. Fiber parameters are as follows: effect refractive index and fiber radius is 1.4682, 4.1 µm for core and 1.4628, 62.5 µm for cladding, respectively. Incident light wavelength is 1550 nm. The results are shown in Fig. 1(a) and (b). It can be seen that n_eff of core mode (LP₀₁) keeps a constant when ambient RI is increased from 1.34 to 1.38, while n_eff of all the cladding modes change along with ambient RI and the slopes are much different. Thus, effective refractive index difference between LP₀₁ and cladding modes are much different as shown in Fig. 1(c) and (d). The discrepancy would induce inconsistency in ambient RI response of modal interference as illustrated by Eq. (4).

Fig. 1. Variation of effect refractive index of different modes versus ambient RI (a), (b). Modal effect refractive index difference under different ambient RI (c), (d) and temperature (e), (f).

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Then effective refractive index of those typical modes under different ambient temperature were summarized in Table 1. Fiber parameters are the same as those described above. Thermal-optic coefficient of fiber core and cladding is 8.6 × 10⁻⁶K^-1 and 2.2 × 10⁻⁶K^-1, respectively. It can be seen that for each mode its effective refractive index increases along with ambient temperature to a different extent. LP₀₁ presents the largest increase compared with other modes. Effective refractive index differences between LP₀₁ and cladding modes at each temperature are unequal as shown in Fig. 1(e) and (f). The discrepancy would induce inconsistency in temperature response of different modal interference as illustrated by Eq. (5).

Table 1. Mode effective refractive index under different ambient temperature

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Discrepancy in mode dispersion characteristic is calculated by solving dispersion equation and the results are shown in Fig. 2(a). It can be seen that all the effective refractive indexes decrease along with the increase of incident light wavelength, but the slopes are slightly different. Based on the data in Fig. 2(a), dispersion discrepancy in different modal pairs were calculated as shown in Fig. 2(b) and (c). Effective refractive index difference between LP₀₁ and several cladding modes, LP₀₃, LP₀₅, LP₁₁, LP₁₃ and LP₁₅, decreases when wavelength increases from 1500 to 1600 nm. The slopes are different (Fig. 2(b)). However, effective refractive index difference between low order cladding mode (LP₀₂) and the same cladding modes involved above, LP₀₃, LP₀₅, LP₁₁, LP₁₃ and LP₁₅, increases along with the increase of wavelength in the same range (Fig. 2(c)). The slopes are different. According to Eqs. (4) and (5), the opposite trends in Fig. 2(b) and (c) would lead to discrepancy in ambient RI and temperature response of different modal interferences. Detailed demonstration is given below.

Fig. 2. Dispersion characteristic of different modes (a). Effective refractive index difference between core-cladding mode pairs (b) and cladding-cladding mode pairs (c).

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2.3 Discrepancy in different modal interferences

2.3.1 Ambient RI response

Ambient refractive index response of modal interference can be simulated according to Eq. (4) based on effective refractive index and dispersion results described above. As an example, Fig. 3(a) shows the spectral shift of a typical core-cladding modal interference, LP₀₁-LP₀₅ interference, when ambient RI is increased from 1.34 to 1.36 and 1.38. The spectrum is blue-shifted with a sensitivity of -23.610 nm/RIU (negative response). Spectra of other core-cladding modal interferences were also simulated. The wavelength shifts are shown in Table 2 (without underline), all of which present blue-shifts (negative shifts).

Fig. 3. Response of core-cladding modal interference (LP₀₁-LP₀₅) (a, c) and cladding-cladding modal interference (LP₀₂-LP₀₅) (b, d) to ambient RI and temperature.

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Table 2. Wavelength shift of modal interference at different ambient RI

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Similarly, spectral shift of a typical cladding-cladding modal interference, LP₀₂-LP₀₅ interference, was simulated and shown in Fig. 3(b). A red-shift (positive response) with a sensitivity of 104.250 nm/RIU was obtained. Spectral shifts of other cladding-cladding modal interferences were also simulated as shown in Table 2, all of which presented red shifts (underlined, positive response). Thus Fig. 3(a), (b) and Table 2 indicates that ambient RI response of cladding-cladding modal interference is opposite to that of core-cladding modal interference.

2.3.2 Ambient temperature response

In this section temperature response of different modal interference were simulated. Figure 3(c) shows the series spectra of LP₀₁-LP₀₅, which is a core-cladding modal interference, at ambient temperature of 25, 45 and 65°C. The spectrum is red-shifted (positive response) when temperature is increased. The sensitivity is 0.026 nm/°C. Responses of other core-cladding modal interference were also calculated as shown in Table 3 (without underline), all of which present red shift responses.

Table 3. Wavelength shift of modal interference at different temperature

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Similarly, spectrum of a typical cladding-cladding modal interference (LP₀₂-LP₀₅) at ambient temperature of 25, 45 and 65°C were calculated and shown in Fig. 3(d). The spectrum red-shifts along with the increase of temperature with a sensitivity of 0.010 nm/°C (positive response). Response of other cladding-cladding modal interferences were also calculated and shown in Table 3 (underlined), all of which present red shift responses (positive response). Thus, temperature response trend of cladding-cladding modal interference is the same as that of core-cladding modal interference, which is much different from the reverse ambient RI response trend described above.

2.4 Inverse Vernier effect based on modal interference discrepancy

It is well known that Vernier enhancement could be obtained by cascading two interferometers with close free spectral ranges [19]. A traditional Vernier effect-based sensor usually consists of a reference interferometer and a sensing one with the intensity respectively described by:

(6)$${I_\textrm{R}} = {I_{\textrm{R1}}} + {I_{\textrm{R2}}} + 2\sqrt {{I_{\textrm{R1}}}{I_{\textrm{R2}}}} \cos (\frac{{2\pi {L_\textrm{R}}}}{\lambda }\Delta {n_{\textrm{eff}}}) = {A_\textrm{R}} + {B_\textrm{R}}\cos {\varphi _\textrm{R}},$$

(7)$${I_\textrm{S}} = {I_{\textrm{S1}}} + {I_{\textrm{S2}}} + 2\sqrt {{I_{\textrm{S1}}}{I_{\textrm{S2}}}} \cos (\frac{{2\pi {L_S}}}{\lambda }\Delta {n_{\textrm{eff}}}) = {A_\textrm{S}} + {B_\textrm{S}}\cos {\varphi _\textrm{S}},$$

where λ is the incident light wavelength, L_R, A_R, A_S and L_S, B_R, B_S represent the interference arm length, the DC component and the spectral fringe contrast of the reference and sensing interferometer, respectively. FSR of each interference spectrum is given by [19,22]:

(8)$$FS{R_R} = \frac{{{\lambda ^2}}}{{\Delta {m_R}{L_R}}},$$

(9)$$FS{R_S} = \frac{{{\lambda ^2}}}{{\Delta {m_S}{L_S}}},$$

where Δm_R and Δm_s is the effective RI difference between the two modes involved in interference with mode dispersion considered [19]. Light intensity of the cascaded interferometer could be expressed as:

(10)$$\begin{array}{l} I = {A_\textrm{R}}{A_\textrm{S}} + {A_\textrm{S}}{B_\textrm{R}}\cos {\varphi _\textrm{R}} + {A_\textrm{R}}{B_\textrm{S}}\cos {\varphi _\textrm{S}}\\ \textrm{ } + 0.5{B_\textrm{R}}{B_\textrm{S}}\cos ({\varphi _\textrm{S}} + {\varphi _\textrm{R}}) + 0.5{B_\textrm{R}}{B_\textrm{S}}\cos ({\varphi _\textrm{S}} - {\varphi _\textrm{R}}). \end{array}$$

FSR of the envelope of the superimposed fringe is:

(11)$$FS{R_e} = \frac{{FS{R_\textrm{R}}FS{R_\textrm{S}}}}{{FS{R_\textrm{S}} - FS{R_\textrm{R}}}}.$$

Vernier magnification folds, M, is determined by:

(12)$$M = \frac{{FS{R_\textrm{R}}}}{{FS{R_\textrm{S}} - FS{R_\textrm{R}}}}.$$

Schematic diagram of the Vernier effect could be illustrated by Fig. 4(a). For the convenience of description, we rewrite Eq. (12) as Eq. (13) if unit interferometer 1 is used as sensing interferometer (with a free spectral range of FSR₁) and unit interferometer 2 is used as reference interferometer (with a free spectral range of FSR₂):

(13)$${M_1} = \frac{{FS{R_\textrm{2}}}}{{FS{R_\textrm{1}} - FS{R_2}}}.$$

When the opposite is the case, we can get Eq. (14):

(14)$${M_2} = \frac{{FS{R_1}}}{{FS{R_2} - FS{R_1}}}.$$

Thus, the difference between FSR₁ and FSR₂ has an outstanding influence on the magnification folds of a traditional Vernier effect-based sensor. It is well known that FSR₁ and FSR₂ is determined by interference arm length of the interferometer. Theoretically FSR₁ and FSR₂ could be as close as possible so that magnification folds could be infinitely large. However, due to the arm length error caused by inaccurate fiber cutting and welding, the reported magnification folds of a traditional Vernier sensor is limited [19–21].

Fig. 4. Schematic diagram of a traditional (a), an improved (b) and an decayed (c) Vernier effect.

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Considering that both of the two unit interferometers are used as sensing units, in such a condition sensitivity of the cascaded interferometer, S, is determined by [19]:

(15)$$S = {S_1}{M_1} + {S_2}{M_2},$$

where S₁ and S₂ is the sensitivity of unit interferometer 1 and 2, respectively. Based on Eq. (13)–(15), different Vernier effect could be obtained as shown in Fig. 4. Detailed analysis is given below:

(1) According to Eqs. (13) and (14), the mark of M₁ and M₂ must be opposite, that is, M₁*M₂< 0;
(2) According to Eq. (15), S will be increased to a large extent if S₁*S₂< 0, that is, the response trends of the two unit interferometers are opposite (one presents positive response and the other negative response). In this case, an improved Vernier effect will be obtained (Fig. 4(b));
(3) On the contrary, S will be decreased obviously if S₁*S₂> 0, that is, the response trends of the two unit interferometers are consistent (both are positive or negative response). In this case, an attenuated Vernier effect is formed (Fig. 4(c)).
(4) The improved Vernier effect can be used to increase the sensitivity of desired sensing parameter (for example, ambient RI measurement), while the attenuated Vernier effect can be used to restrain the crosstalk from the undesired parameter (for example, temperature disturbance on ambient RI measurement).

To obtain an improved Vernier effect, an opposite response trend in the two unit interferometers should be ensured. As described in Section 2.3, ambient RI response trend of core-cladding modal interference is opposite to that of cladding-cladding modal interference in an SMF. Thus, an improved Vernier effect for ambient RI measurement could be obtained if we fabricate a Vernier interferometer with two sensing unit interferometers respectively dominated by core-cladding and cladding-cladding modal interference. At the same time, in Section 2.3 we also illustrated that ambient temperature response trend of core-cladding modal interference is the same as that of the cladding-cladding modal interference. Based on this characteristic, the decayed Vernier effect on temperature sensitivity could be obtained in the same structure mentioned above. As a result, the proposed Vernier interferometer with dual-sensing units will simultaneously possesses of an improved Vernier effect for ambient RI measurement and a decayed Vernier effect for temperature crosstalk. Sensor fabrication, ambient RI and temperature sensing performance test will be demonstrated in the followed section.

3. Experiment and discussion

3.1 Design and fabrication of the sensor

To design and fabricate two unit interferometers respectively dominated by core-cladding modal interference and cladding-cladding modal interference, excitation of specific mode should be realized in advance. It is well known that offset fusion will induce asymmetrical optical mode field distribution in fiber and excite a specific mode. Excitation efficiency is determined by fusion offset. Assume field from a lead-in SMF with offset is as follows [23]:

(16)$${E_{\textrm{SMF}}} = Exp( - \frac{{{r^2} + 2r\cos \theta \cdot \Delta x + \Delta {x^2}}}{{\omega _g^2}}) \cdot Exp( - i{\beta _{01}}z) \cdot \hat{x},$$

where Δx is the axis offset, β₀₁ is the propagation constant of LP₀₁ mode and ω_g is the Gaussian half height width of the incident light beam:

(17)$${\omega _\textrm{g}} = \frac{r}{{\sqrt {\ln 2} }}\left( {0.65 + \frac{{1.619}}{{{V^{1.5}}}} + \frac{{2.879}}{{{V^6}}}} \right),$$

in which r and V respectively represents the core radius and normalized frequency. Excitation efficiency of a specific mode, LP_v,µ, could be calculated as [24]:

(18)$${\eta _{v,u}} = \frac{{{{\left|{\int\!\!\!\int\limits_\infty {{E_{\textrm{SMF}}} \cdot {E_{\mathrm{SMF^{\prime}}}}dA} } \right|}^2}}}{{\left( {\int\!\!\!\int\limits_\infty {|{E_{\textrm{SMF}}^2} |\textrm{dA}} } \right) \cdot \left( {\int\!\!\!\int\limits_\infty {|{E_{\mathrm{SMF^{\prime}}}^\textrm{2}} |\textrm{dA}} } \right)}},$$

in which E_SMF’ is the electric field of LP_v,µ mode in a single-mode fiber.

When light is transmitted from the lead-in SMF into a sensing SMF by offset fusing, the excitation efficiency of LP₀₁ mode in the sensing SMF is [24]:

(19)$${\eta _{0,1}}\textrm{ = }\frac{{8{{\left|{\int\limits_0^\infty {Exp(\frac{{ - {r^2}}}{{{\omega_g}}}){I_0}(\frac{{2r\Delta x}}{{\omega_g^2}}){J_0}({u_{0,1}})rdr} } \right|}^2}}}{{{r^2}\omega _\textrm{g}^2Exp(\frac{{2\Delta {x^2}}}{{\omega _g^2}})\left[ {J_0^2({u_{0,1}})\textrm{ + }J_1^2({u_{0,1}})\textrm{ + }\frac{{J_0^2({u_{0,1}})}}{{K_0^2({w_{0,1}})}}({K_1^2({w_{0,1}}) - K_0^2({w_{0,1}})} )} \right]}},$$

where, J, K and I represent Bessel J function, Bessel K function and Bessel I function, respectively, r is the radius of the single-mode fiber. u_0,1 and w₀_,1 are the normalized mode parameters of LP₀₁ [24]:

(20)$${u_{0,1}} = r\sqrt {k_0^2n_{\textrm{co}}^2 - \beta _{0,1}^2} ,\textrm{ }{w_{0,1}} = r\sqrt {\beta _{0,1}^2 - k_0^2n_{\textrm{cl}}^2} ,$$

where k₀ is the wave vector, β_0,1 is the propagation constant of LP₀₁ mode. n_co and n_cl is the core and cladding refractive index of the SMF, respectively.

According to above formula, excitation efficiency of LP₀₁, LP₁₁, LP₂₁ and LP₀₂ at the offset point between lead-in SMF and sensing SMF could be calculated. The result is shown in Fig. 5(a). Simulation parameter is the same as those described in Section 2.2. At a small offset (<6 µm), LP₀₁ and LP₁₁ mode could be excited with moderate excitation efficiencies, while LP₀₂ and LP₂₁ could only be slightly excited. Excitation efficiency of LP₀₁ mode drops rapidly and becomes smaller than that of LP₁₁ mode when offset is further increased. In this case most of the light energy will enter into fiber cladding for transmission. Energy in LP₀₂ and LP₂₁ modes remain to be low and could be neglected.

Fig. 5. (a) Mode excitation efficiency under different fiber core-offset. (b) Mode field distributions of LP₀₁, LP₁₁, LP₂₁ and LP₀₂ modes. (c) Simulation of light propagation in a 6 µm offset fused SMF unit MZI. (d): Schematic diagram of the cascaded sensor consisting two unit interferometers respectively with a 6 µm and a 30 µm core-offset. Photos of two fused points in the fabricated sensor are shown in the insects. Schematic diagram of the ambient RI (e) and temperature (f) sensing system. OSA: Optical spectrum analyzer; BOA: Booster optical amplifiers.

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Electric field distributions of LP₀₁, LP₁₁, LP₂₁ and LP₀₂ modes have been given in Fig. 5(b). Figure 5(c) gives the simulation of light propagation in a 6 µm offset fused single mode fiber unit MZI obtained by Rsoft software. Simulation parameter is the same as above. At the first bias point, considerable amount of energy is confined in the core mode while a small part of energy enters into the cladding due to offset. Energy in core and cladding modes re-coupled at the second bias point. As a result, core-cladding modal interference is the major composition in the spectrum. As for an obvious offset, most of the light energy enter into cladding for transmission, and cladding-cladding modal interference is the main component (not shown). Thus, modal interference could be controlled by core offset adjustment during fiber fusing.

Structure schematic of the sensor cascaded by two unit interferometers is shown in Fig. 5(b). Both of the two unit interferometers are based on an offset spliced SMF-SMF-SMF structure. Core and cladding diameter of SMF is 8.2 and 125 µm, respectively. offset is optimized by taking mode excitation efficiency and interference fringe visibility into account. As a result, a slight offset of 6 µm is adopted to obtain core-cladding modal interference (named by unit interferometer 1), and an obvious offset of 30 µm is used to get cladding-cladding modal interference (unit interferometer 2).

The sensing arm length will affect the free spectral range (FSR) of the interference spectrum as demonstrated by Eqs. (8), (9) and Vernier magnification folds (Eqs. (12)–(14)). To increase magnification folds, sensing arm lengths of the two unit interferometers should be close to each other. In addition, arm length should be carefully optimized so that more than one dip could be observed by the optical spectrum analyzer. Besides, the sensing arm should not be too long for fear that cladding modes be decayed severely. Based on these considerations, sensing arm length is optimized to be 6 cm for unit interferometer 1 and 8 cm for unit interferometer 2, respectively.

As for the fabrication of unit interferometer 1, a lead-in SMF and a 6 cm sensing fiber segment were cleaved and put on a fusion splicer (Fujikura FSM-60S) working in manual mode. The two fiber segments were carefully adjusted to make them aligned along the X axis and 6 µm offset along the Y axis. After that, the two segments were fused with a 2000ms discharge time and a standard discharge intensity. The other end of the 6 cm sensing fiber was offset fused with a connecting SMF in the same way. Before that the upper part of the sensing fiber on the welding machine was marked so that the offset direction of the second offset point was in the same plane. Unit interferometer 2 was fabricated in the same way. Finally, the two units were cascaded through the connecting SMF. Ambient RI and temperature sensing system is shown in Fig. 5(c) and (d). Incident light from a Booster optical amplifier (BOA) was emitted into the sensor, and the output interference spectrum was recorded by an optical spectrum analyzer (OSA).

3.2 Ambient RI response

Ambient RI and temperature sensing performance were experimentally tested. Firstly, refractive index response of each unit interferometer was measured by immersing the unit interferometer in NaCl solutions with different concentrations at room temperature. RI of each solution was measured to be 1.3362, 1.3540, 1.3629, 1.3740 and 1.3811 by an Abbe refractometer. Before each test the senor was cleaned with deionized water and dried in air. Figure 6(a) and (c) shows the refractive index responses of unit interferometer 1 and 2, respectively. For unit interferometer 1, the spectrum presents a blue shift when ambient RI is increased (Fig. 6(b)). The slope is measured to be -54.009 nm/RIU. As for unit interferometer 2, the spectrum presents a red shift with a slope of 142.581 nm/RIU (Fig. 6(d)). The opposite response trend is consistent with the theoretical result demonstrated by Fig. 3(a) and (b).

Fig. 6. Refractive index response characteristics of unit interferometer with a slight (a), an obvious (c) offset and the cascaded interferometer (e).

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Then the two unit interferometers were cascaded together and simultaneously immersed in NaCl solution to repeat the test. The ambient RI sensitivity of the cascaded sensor is measured to be -1788.160 nm/RIU (Fig. 6(e)), which is improved by 33.1 folds in comparison with unit interferometer 1 and improved by 12.5 folds when compared with unit interferometer 2, respectively. Thus, an improved Vernier effect for ambient RI measurement is obtained due to cascading of the two unit interferometers with different modal interferences.

In Fig. 6 the waveform diagram of the refractive index response shows a certain degree of distortion in addition to spectral shift. The reason is that, in a practical sensor more than one cladding mode are excited with different excitation efficiencies as demonstrated by Fig. 5(a). These cladding modes interference with the core mode. Thus, the measured spectrum is a superposition of multiple modal interference, and will be more complicated than the simulated two-mode interference spectrum. Detailed simulation analysis about the effect of modal interference quantity on the spectrum has been demonstrated in our previous work [25]. An increase in interference components could cause wavelength shift even no surrounding medium fluctuation occurs. The spectrum of one core-cladding modal interference is a regular sine curve. The curve becomes irregular and shows off certain spectral shift when another kind of core-cladding modal interference is added. Irregularity and shift are more obvious when a third modal interference is involved. Spectral irregularity is common in the experimental results of fiber MZIs [26–33].

3.3 Temperature cross talk

Temperature response characteristic of the two unit interferometers and the cascaded interferometer were tested. For each test the sensor was placed in an incubator with temperature ranging from 27∼67 °C with a step of 10°C. The results are shown in Fig. 7. For each of the two unit interferometers, the spectrum present a slight red shift. The slope is fitted to be 0.042 nm/°C for unit interferometer 1 and 0.025 nm/°C for unit interferometer 2, respectively. The consistent response trend agrees well with the theoretical results demonstrated by Fig. 3(c) and (d). Ambient RI measurement error induced by temperature cross-talk could be calculated as [26]:

(21)$${E_\textrm{r}}\textrm{ = }\left|{\frac{{{S_\textrm{T}}}}{{{S_\textrm{R}}}}} \right|,$$

where S_T and S_R represent the temperature and ambient RI sensitivity of the sensor, respectively. According to Eq. (21) temperature cross-sensitivity is calculated to be 7.776 × 10⁻⁴ RIU/°C for unit interferometer 1 and 1.753 × 10⁻⁴ RIU/°C for interferometer 2, respectively. As for the cascaded interferometer the temperature sensitivity is measured to be 0.167 nm/°C (Fig. 7(c)). Ambient RI error caused by temperature cross-talk in the cascaded interferometer is as low as 9.339 × 10⁻⁵ RIU/°C, which is 12 and 53 folds reduced compared with unit interferometer 1 and 2, respectively. Therefore, due to decayed Vernier effect temperature crosstalk on ambient RI measurement was obviously restrained. The result is consistent with the analysis demonstrated in Section 2.4. Table 4 lists the performance comparison of the proposed sensor with similar sensors based on fiber interferometers or Vernier effect. Mechanism of the increased temperature sensitivity in the cascaded sensor needs further research. The probable reason is that elasto-optical effect caused by fiber thermal expansion brings about an outstanding influence on temperature sensitivity of sensor.

Fig. 7. Temperature response of unit interferometer with a slight (a), an obvious (b) offset and the cascaded interferometer (c) and (d).

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Table 4. RI sensitivity and temperature cross-talk data of the proposed sensor and other similar sensors

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4. Conclusion

In this paper, modal interference discrepancy in an all-fiber MZI has been theoretically analyzed and experimentally verified. Theoretical analysis demonstrated that core-cladding modal interference induced a blue-shift spectral response to the increase of ambient RI, while cladding-cladding modal interference resulted in a red-shift response. However, temperature response trends of the two kinds of modal interferences were consistently along with a red-shift. Based on the discrepancy, an improved Vernier sensor cascaded by two unit MZIs with slight/obvious core-offset welding, respectively offering core-cladding modal interference (unit interferometer 1) and cladding-cladding modal interference (unit interferometer 2) was designed. Due to the opposite ambient RI response trend of the two unit MZIs, ambient RI sensitivity of the cascaded sensor was improved to a large extent. However, temperature sensitivity of the cascaded sensor was restrained because of the consistent temperature response trend of the two unit MZIs. Sensing tests verified that Ambient RI sensitivity of the cascaded interferometer reaches -1788.160 nm/RIU within RI range of 1.3362-1.3811, which was 33.1 folds improved over unit interferometer 1 (-54.009 nm/RIU) and 12.5 folds improved over unit interferometer 2 (142.581 nm/RIU), respectively. Temperature sensitivity of the cascaded interferometer was as low as 0.167 nm/°C and only caused a slight RI sensing error of 9.339 × 10⁻⁵ RIU/°C. Benefiting from the characteristic of simple structure, ease of fabrication and low temperature cross talk, the modal interference discrepancy based dual-sensing unit Vernier sensor was believed to have potential application prospects in biochemical sensing fields.

Funding

National Natural Science Foundation of China (11974266, 62075174).

Acknowledgments

We thank the funding from National Natural Science Foundation of China (62075174, 11974266).

Disclosures

The authors declare no conflicts of interest.

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.

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Mode	Effective refractive index under different ambient temperature
Mode	25 °C	35 °C	45 °C	55 °C	65 °C
LP₀₁	1.4652044	1.4652903	1.4653762	1.4654621	1.4655479
LP₀₂	1.4627461	1.4628311	1.4629162	1.4630012	1.4630861
LP₀₃	1.4625675	1.4626525	1.4627375	1.4628225	1.4629076
LP₀₄	1.4622679	1.4623530	1.4624380	1.4625230	1.4626081
LP₀₅	1.4618506	1.4619357	1.4620207	1.4621058	1.4621909
LP₁₁	1.4627271	1.4628120	1.4628971	1.4629821	1.4630671
LP₁₂	1.4625692	1.4626543	1.4627393	1.4628244	1.4629094
LP₁₃	1.4623352	1.4624203	1.4625054	1.4625905	1.4626756
LP₁₄	1.4620087	1.4620939	1.4621790	1.4622642	1.4623493
LP₁₅	1.4615711	1.4616563	1.4617414	1.4618266	1.4619118

Modal interference	Wavelength shift at different ambient RI (nm)					Shift rate (nm/RIU)
Modal interference	1.34	1.35	1.36	1.37	1.38	Shift rate (nm/RIU)
LP₀₁-LP₀₂	0.000	-0.009	-0.020	-0.031	-0.041	-1.040
LP₀₁-LP₀₃	0.000	-0.043	-0.084	-0.142	-0.197	-4.930
LP₀₁-LP₀₄	0.000	-0.090	-0.203	-0.331	-0.476	-11.930
LP₀₁-LP₀₅	0.000	-0.204	-0.408	-0.651	-0.957	-23.610
LP₀₁-LP₁₁	0.000	-0.023	-0.047	-0.052	-0.073	-1.750
LP₀₁-LP₁₂	0.000	-0.039	-0.081	-0.132	-0.178	-4.490
LP₀₁-LP₁₃	0.000	-0.092	-0.179	-0.271	-0.378	-9.350
LP₀₁-LP₁₄	0.000	-0.157	-0.326	-0.539	-0.775	-19.320
LP₀₁-LP₁₅	0.000	-0.271	-0.595	-0.934	-1.363	-33.890
LP₀₂-LP₀₃	0.000	0.891	1.773	2.942	4.474	109.990
LP₀₂-LP₀₄	0.000	0.870	1.739	2.811	4.185	103.110
LP₀₂-LP₀₅	0.000	0.823	1.745	2.886	4.181	104.250
LP₀₂-LP₁₂	0.000	0.344	0.775	1.191	1.913	46.730
LP₀₂-LP₁₃	0.000	0.358	0.743	1.196	1.817	44.720
LP₀₂-LP₁₄	0.000	0.385	0.867	1.402	2.071	51.590
LP₀₂-LP₁₅	0.000	0.510	1.091	1.773	2.604	64.710

Modal interference	Wavelength shift at different temperature (nm)					Shift rate (nm/°C)
Modal interference	25 °C	35 °C	45 °C	55 °C	65 °C	Shift rate (nm/°C)
LP₀₁-LP₀₂	0.000	0.238	0.471	0.690	0.919	0.023
LP₀₁-LP₀₃	0.000	0.239	0.472	0.713	0.952	0.024
LP₀₁-LP₀₄	0.000	0.255	0.507	0.747	0.985	0.025
LP₀₁-LP₀₅	0.000	0.264	0.533	0.771	1.032	0.026
LP₀₁-LP₁₁	0.000	0.233	0.455	0.679	0.915	0.023
LP₀₁-LP₁₂	0.000	0.227	0.471	0.708	0.950	0.024
LP₀₁-LP₁₃	0.000	0.264	0.507	0.762	1.003	0.025
LP₀₁-LP₁₄	0.000	0.251	0.532	0.782	1.055	0.026
LP₀₁-LP₁₅	0.000	0.269	0.561	0.838	1.119	0.028
LP₀₂-LP₀₃	0.000	0.091	0.198	0.312	0.439	0.011
LP₀₂-LP₀₄	0.000	0.105	0.204	0.292	0.406	0.010
LP₀₂-LP₀₅	0.000	0.103	0.201	0.294	0.395	0.010
LP₀₂-LP₁₂	0.000	0.162	0.310	0.454	0.587	0.015
LP₀₂-LP₁₃	0.000	0.164	0.323	0.451	0.597	0.015
LP₀₂-LP₁₄	0.000	0.151	0.290	0.425	0.553	0.014
LP₀₂-LP₁₅	0.000	0.139	0.258	0.387	0.508	0.013

Reference	RI sensitivity	Temperature sensitivity	Temperature cross-talk
[27]	13.7592 nm/RIU	0.0462 nm/°C	3.36 × 10⁻³ RIU/°C
[28]	-1364.343 nm/RIU	0.033 nm/°C	2.419 × 10⁻⁵ RIU/°C
[8]	-25.2935 nm/RIU	0.0615 nm/°C	0.00243 RIU/°C
[29]	158.4 nm/RIU	not mentioned	not mentioned
[30]	123.40 nm/RIU	not mentioned	not mentioned
[31]	1191.5 nm/RIU	0.0648 nm/°C	5.439 × 10⁻⁵ RIU/°C
[32]	2460.07 nm/RIU	not mentioned	not mentioned
[33]	-26.22 nm/RI	46.5 pm/°C	1.773 × 10⁻³ RIU/°C
Our work	-1788.160 nm/RIU	0.167 nm/°C	9.339 × 10⁻⁵ RIU/°C

Mode	Effective refractive index under different ambient temperature
Mode	25 °C	35 °C	45 °C	55 °C	65 °C
LP₀₁	1.4652044	1.4652903	1.4653762	1.4654621	1.4655479
LP₀₂	1.4627461	1.4628311	1.4629162	1.4630012	1.4630861
LP₀₃	1.4625675	1.4626525	1.4627375	1.4628225	1.4629076
LP₀₄	1.4622679	1.4623530	1.4624380	1.4625230	1.4626081
LP₀₅	1.4618506	1.4619357	1.4620207	1.4621058	1.4621909
LP₁₁	1.4627271	1.4628120	1.4628971	1.4629821	1.4630671
LP₁₂	1.4625692	1.4626543	1.4627393	1.4628244	1.4629094
LP₁₃	1.4623352	1.4624203	1.4625054	1.4625905	1.4626756
LP₁₄	1.4620087	1.4620939	1.4621790	1.4622642	1.4623493
LP₁₅	1.4615711	1.4616563	1.4617414	1.4618266	1.4619118

Modal interference discrepancy and its application to a modified fiber Mach-Zehnder Vernier interferometer

Abstract

1. Introduction

2. Principle and simulation

2.1 Modal interference in fiber

2.2 Discrepancy in different modes

2.3 Discrepancy in different modal interferences

2.3.1 Ambient RI response

2.3.2 Ambient temperature response

2.4 Inverse Vernier effect based on modal interference discrepancy

3. Experiment and discussion

3.1 Design and fabrication of the sensor

3.2 Ambient RI response

3.3 Temperature cross talk

4. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Tables (4)

Equations (21)

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