## Abstract

A Mach-Zehnder interferometer (MZI) composed by a pair of long period gratings (LPGs) fabricated in silica microfiber for sensing applications is demonstrated. Each LPG is fabricated with a pulsed CO_{2} laser by creating six periodical deformations along fiber length with only one scanning cycle. The length of the MZI can reach as short as 8.84 mm when the diameter of the microfiber is 9.5 μm. Compared with the ones fabricated in single-mode fibers, the present MZI is much shorter owing to the large effective-index difference between the fundamental and higher order modes. The microfiber MZI exhibits a sensitivity to surrounding refractive index (RI) of 2225 nm per refractive index unit and the temperature sensitivity of only 11.7 pm/°C. Theoretical analysis suggests that the performances of the MZI sensor can be improved by using thinner microfibers with a diameter down to 3.5 μm: The sensitivity can be greatly enhanced due to the stronger evanescent-field interaction and reduced dispersion factor; the transmission dips become narrower which benefits high-resolution measurement; the thinner fiber also allows further reduction in device length. The present device has great potential in biochemical and medical sensing due to the advantages including easy fabrication, excellent compactness and high sensitivity.

© 2013 OSA

## 1. Introduction

In-fiber Mach-Zehnder interferometers (MZIs) have received great interest for a variety of sensing applications [1–4]. A number of methods have been proposed to fabricate a MZI, by forming abrupt fiber tapers [2, 3] or introducing core mismatch [4]. Alternatively, two wavelength-matched long period gratings (LPGs) can be cascaded to form a MZI [5]. In this configuration, the first grating couples a fraction of the fundamental mode into a co-propagating higher order mode (HOM) and then the two modes are combined by the second grating, resulting in sharp interference fringes. The LPG-based MZIs have been used for refractive-index sensing by tracking the shift of the interference fringes [1, 2]. However, the previous LPG-based MZI formed in conventional single-mode fibers usually has a typical length of up to tens of centimeters due to a relative small index difference between the fundamental and higher-order modes (10^{−3} order), restricting its application as a sensor when RI variation needs to be precisely located [1, 6–8].

Recently, the micron-scaled microfibers are increasingly used for fabrication of various optical devices and sensors [9–11]. The sensors have presented high sensitivity for the refractive-index measurement of surrounding media due to the strong evanescent-field interaction. A MZI has been reported by assembling two microfibers with a dual-path geometry as an add-drop filter but is not suitable for sensing applications [12]. In this paper, we demonstrate an alternative to fabricate a compact LPG-based MZI formed in microfiber for sensing applications by using a pulsed CO_{2} laser. Due to a larger effective-index difference between the fundamental and higher order modes of the microfiber, the LPG-based MZI can be less than 1cm in length, much shorter than those fabricated in conventional single-mode fibers. The microfiber LPG-based MZI with a diameter of 9.5 μm presents a RI sensitivity of as high as 2225 nm/RIU and a temperature sensitivity of only 11.7 pm/°C. Theoretical analysis suggests that the RI sensitivity can be further enhanced by using thinner microfibers, due to the effect of mode dispersion [13]. The present device is promising for sensing applications due to the high RI sensitivity and the ability of highly localized RI measurement enabled by the device compactness.

## 2. Fabrication and working principle

The microfibers are fabricated from standard single-mode fibers (SMFs) by flame-heated taper drawing approach [11]. The SMF is stretched by use of two linear stages while a flame moves back and forth to heat the fiber. The geometry of the fiber taper is determined by parameters including the moving ranges and speeds. By use of this method, a microfiber with a diameter of 9.5 μm is fabricated. The uniform region of the microfiber is 7.5 cm in length. Figure 1(a)
shows of the schematic of the setup for the fabrication of the microfiber MZI. The fabrication process is similar with the LPG fabrication as described in [14]. A pulsed CO_{2} laser (SYNRAD 48-5) is used for the fabrication. Its maximum output power is 50 W. The laser beam is focused by a ZnSe lens to a spot of a diameter of ~50 μm. The scanning path, speed, and the number of scanning cycles are controlled by a computer. In the experiment, the scanning speed, output power and repetition frequency are 230 mm/s, 3W and 5 kHz, respectively. Figure 1(b) shows the microscopic image of a deformation after the laser beam scans across the microfiber. The silica glass absorbs light at 10.6 μm from the CO_{2} laser during the irradiation and produces a great amount of heat. The local temperature reaches above the softening temperature of silica glass and then the treated region rapidly cools down to room temperature. As a result, a microtaper-like deformation is created, with a length of 73.5 μm and a waist diameter of 5.85μm. A mass of 0.7 g has been attached on one of the fiber pigtails to keep the fiber straight during the tapering. Such deformations are produced at desired positions along the fiber to form two identical long period gratings, as shown in Fig. 1(c). Each LPG contains six periodic deformations. The separation between two adjacent deformations, i. e., the grating pitch Λ is 170 μm and the separation of the two gratings is 6.8 mm, which equals 40 pitches. The total length of the interferometer is 8.84 mm. The MZI fabrication is completed with only one laser-scanning cycle. The two SMF pigtails of the microfiber are connected to a broadband light source (BBS) and an optical spectrum analyzer (OSA), respectively, to record the transmission spectrum.

Figure 2(a) shows the recorded transmission spectrum of a single LPG which contains six deformations. Three resonant dips located at 817.8, 1061.2 and 1252.7nm are observed. The depths of the dips are 5.6, 4.5, and 7.3 dB, respectively. The mode coupling process in the grating can be explained based on the coupled local-mode theory as described in [15], since strong perturbations are introduced. According to the theory, each deformation greatly changes the local-mode fields and causes fractional energy coupling from the fundamental mode to higher-order modes (HOMs). When periodic deformations are formed along the fiber length, the fundamental mode is resonantly coupled to co-propagating phase-matched HOMs. The phase-matching condition can be expressed by

where $\overline{\Delta n}=\frac{1}{\Lambda}{\displaystyle \underset{{z}_{0}}{\overset{{z}_{0}+\Lambda}{\int}}\Delta n(z)\text{d}z}$ is the average index difference between the fundamental mode and HOM, and Δ*n*(

*z*) represents the longitudinal index-difference variation within a period. Equation (1) becomes the well-known equation

*λ*

_{LPG}= Δ

*n*·Λ for weak-perturbation approximation. Figure 2(b) shows the calculated phase-matched curves (PMCs), i. e., the relations between the resonant wavelengths and grating pitch, for unperturbed microfiber with a diameter of 9.5 μm. We found that the observed dips are most possibly caused by resonant couplings to the LP

_{21}mode group, which contains degenerate EH

_{11}and HE

_{31}modes. Figure 2(c) shows their mode energy and electric fields. The PMCs for other order modes are not shown here.

The local coupling coefficient can be expressed by [15]

*j*and

*l*denote the two coupled modes. Equation (2) suggests that the coupling strength is determined by how the mode field is changed within a certain length. The electric fields and mode energy profiles of the LP

_{21}modes are shown in Fig. 2(c). We assume the cross section of the deformed microfiber is circular and these modes present anti-symmetrical electric field distribution. The anti-symmetrical electrical fields result in zero overlap integrals with unperturbed fundamental modes, which indicates that the geometrical deformations cannot effectively drive the mode couplings. The asymmetrical refractive-index modification, on the other hand, can shift the mode fields (as shown in Fig. 2(c)) and causes a non-zero overlap integral. As a result, the index modification dominates in the grating couplings. Compared with the HE

_{31}modes, the EH

_{11}mode profile changes significantly with the perturbation so that the degeneracy cannot be well maintained, i. e., the effective indexes of the two EH

_{11}modes are considerably different. The PMC for the coupling to EH

_{11}mode splits into two curves and three resonant dips can be observed, instead of two. The discrepancy between the calculated and measured resonant wavelengths is a result of the additional change in effective-index difference induced by the CO

_{2}-laser treatment. The calculated PMCs can be modified to match with the measured wavelengths provided that the exact spatial index change is known. The LP

_{02}modes have close effective indexes but can hardly be excited by the grating, due to their cylindrically symmetrical electric field profiles. The resultant overlap integral is much smaller than those for the LP

_{21}modes.

Figure 2(a) also shows the transmission spectrum of the MZI. A zoomed-in spectrum at around 1050 nm is shown in Fig. 3(a) . A series of interferometric fringes can be observed within the envelope of the LPG dips. The wavelength spacings between adjacent transmission dips are 14.3, 15.2 16.1 nm at around the individual resonant wavelengths, respectively. The transmission of the MZI can be described by [16]

*r*and

*t*denote the complex amplitudes of the fundamental and higher-order modes, ${M}_{1}=\left[\begin{array}{cc}{t}_{1}& {r}_{1}\\ {r}_{1}& {t}_{1}^{*}\end{array}\right]$ and ${M}_{2}=\left[\begin{array}{cc}{t}_{2}& {r}_{2}\\ {r}_{2}& {t}_{2}^{*}\end{array}\right]$ are the transmission matrices for the two gratings, respectively. ${M}_{p}=\left[\begin{array}{cc}\mathrm{exp}(i\frac{\Delta \phi}{2})& 0\\ 0& \mathrm{exp}(-i\frac{\Delta \phi}{2})\end{array}\right]$ represents the phase shifts over the unperturbed microfiber in between the two gratings, where Δ

*φ*= 2πΔ

*nL*/

*λ*. The matrix $\left[\begin{array}{c}1\\ 0\end{array}\right]$ represents the boundary condition: the amplitudes of the incident fundamental and higher-order modes are 1 and 0, respectively. The transmission power of the MZI can be expressed byFigure 3(b) shows the calculated transmission spectrum at around 1050 nm, based on the above theory. In the calculation, the effective indexes at individual wavelengths are calculated by numerically solving the wave equation at individual wavelengths to obtain the corresponding phase shifts. For the two identical gratings, the matrices

*M*

_{1}and

*M*

_{2}can be naturally expressed by ${M}_{1}={M}_{2}={M}_{0}^{6}={\left[\begin{array}{cc}{t}_{0}& {r}_{0}\\ {r}_{0}& {t}_{0}^{*}\end{array}\right]}^{6}$, where

*M*

_{0}is the transmission matrix for each period. The components

*r*

_{0}and

*t*

_{0}contain the intercoupling efficiency and phase changes over a single period. In the calculation, the values of coupling coefficients and grating pitch are set to approximate the measured spectrum. The difference in wavelength spacing is a result of the discrepancy between the calculated and actual mode dispersions. For comparison, we have calculated the transmission property of a LPG-based MZI formed in standard SMF. LPGs which induce resonant coupling to LP

_{06}mode are selected for the comparison, since the corresponding coupling strength is higher than other cladding modes. We found that the MZI needs to be 25.8 mm in length, not taking into account the grating lengths, three times longer than the present microfiber MZI. That is because the index differences between the fundamental and higher order modes are much larger than the SMF, which results in larger phase differences.

## 3. Sensitivity characteristics

The response to refractive index is measured by immersing the MZI into ethyl alcohol with a purity of 99.8%. The refractive index is controlled by changing the temperature of alcohol by use of an oven. The alcohol has a thermo-optic coefficient of −4 × 10^{−4} /°C. We changed the alcohol RI by varying the temperature from 45° to 0°, with the corresponding RI from 1.352 to 1.37. Figure 4
shows the measured shifts of four selected dips as a function of the surrounding RI. All the dips red shift with increasing refractive index. The measured sensitivities are estimated as 1155.5, 1025.8, 1882.4, and 2225.2nm/RIU, respectively, by linearly fitting with the external RI range from 1.352 to 1.370. The sensitivity is much higher than the previously reported MZI sensors in SMFs due to the much stronger evanescent-field interaction in microfibers [2, 3, 17]. Note that the thermal effect of the silica glass can also affect the wavelength shift when the temperature changes, but this effect is much lower than that of the RI change of the liquid due to the evanescent-field interaction. The testing method should be further improved for accurate measurement of the RI response.

For an immersed microfiber, the phase difference between the fundamental and higher-order modes changes with external refractive index, and therefore the shift of interferometric fringes can be observed. At the transmission dips, the phase difference satisfies $\Delta \phi =2(k+1)\pi $, where *k* denotes the interference order. Derived from this condition, the dependence of dip wavelength on external refractive index can be expressed by

The RI responses have been calculated based on Eq. (5), as shown in Fig. 4. In the calculation, the dispersion factor is obtained from the calculated index curves. The index-difference change with external RI is calculated by repeatedly solving the wave equation with the corresponding dielectric structure. The calculated results for the four dips are 1217.8, 1266.2, 2098.5, 2230.4 nm/RIU, respectively, in good agreement with the measured results. For a wider RI range, the dip wavelength shifts cannot be considered as linear due to the change in dispersion factor and the strength of the evanescent-field interaction. More accurate calculation should be performed by plotting the phase-matching curves for the individual RIs.

The temperature response in air is measured by placing the MZI into a resistance furnace. The temperature changes from 30 to 110 °C. Figure 5(a) shows the recorded spectrum of a single transmission dip at around 1239.4nm at different temperatures. Figure 5(b) shows the measured wavelength shift with temperature. The temperature sensitivity is about 11.7 pm/°C.

Similarly with the characterization of RI response, the temperature sensitivity can be expressed by

*α*= 5 × 10

^{−7}/°C is the thermal-expansion coefficient and

*β*

_{si}= d

*n*

_{si}/d

*T*= 6 × 10

^{−6}/°C is the thermo-optic coefficient of silica glass. We have calculated the variation of index-difference as

*n*

_{si}changes and found that the raise of

*n*

_{si}results in a reduction of index difference Δ

*n*. Therefore, the value of the second term in the bracket is about −1.306 × 10

^{−5}/°C and much larger than the expansion coefficient. Considering the dispersion factor

*Γ*is also negative, the fringes red shift with temperature. Substituting dispersion factor into Eq. (6), the calculated temperature sensitivity around 1240 nm is about 16.6 pm/°C, which is close to the measured result. The low temperature sensitivity greatly reduces the effect of cross-sensitivity in the RI measurement.

## 4. Discussion

In this section, we intend to carry out a further investigation on how to enhance the RI sensitivity and determine whether there is a limit of sensitivity for the microfiber MZI sensor. The aim is to propose a proper strategy to enhance the RI sensitivity for the detection of slight componential change of the surrounding medium.

Equation (5) suggests that the RI sensitivity is mainly determined by two factors: the dispersion factor *Γ* and the dependence of index difference on external RI $\text{d}\Delta n/\text{d}n\text{si}$ which measures the strength of evanescent-field interaction. Both the factors depend on fiber diameter for circular microfibers. Figure 6
shows the calculated index differences between the fundamental mode and EH_{11} mode, and the corresponding dispersion factor, RI sensitivities and wavelength spacing for different fiber diameters, respectively. The index differences increase with wavelength and thinner microfibers have higher slopes. For microfibers with a diameter of 9.5 μm in the experiment, the resultant dispersion factor *Γ* is around −1 and hardly affects the sensitivities. When the diameter is down to 3.5 μm, the dispersion factor can be about −0.85, which means that its contribution to the sensitivity can be increased by ~17%. Figure 6(c) shows that the RI sensitivity can be greatly enhanced by using thinner microfibers. Considering the limited effect of the mode dispersion, the high sensitivity is mainly attributed to the much stronger evanescent-field interaction. When further reducing the fiber diameter, the EH_{11} mode becomes radiative and most of its mode energy spread into the surrounding medium. As a result, the grating coupling is greatly weakened and the interferometric fringes cannot be seen. This indicates that the maximum RI sensitivity that a microfiber MZI can achieve is about 40000 nm/RIU when the fiber diameter is 3.5 μm at around 1450 nm. However, in the experiment we attempted to fabricate MZI sensors in such thin microfibers but failed because the laser treatment is intense and can easily break the fiber. In addition, the grating pitch should be less than 20 μm for such thin microfibers due to the large index difference. However, the spot of the focused CO_{2} laser beam is typically 50 μm. As a result, the fabrication of LPGs with such extremely high sensitivity requires further selection of micromachining technique.

The dependence of wavelength spacing on fiber diameter is also investigated. The dip wavelengths of the interference fringes can be expressed by

Where*L*is the length between the center of the first grating and the center of the second grating, thus

*L*is 7.82 mm. Equation (7) suggests that the peak wavelengths are determined by the interference order

*k*and the optical path difference Δ

*n·L*. Derived from Eq. (7), the dependence of peak wavelength

*λ*on interference order

*k*can be expressed byThe fringe spacing is the wavelength variation when d

*k*= 1 and can be expressed byEquation (9) suggests that for an interferometer with a certain length, the wavelength spacing is not only determined by the index difference, but also depends on the mode dispersion property of the microfiber. Figure 6(d) shows the calculated wavelength spacing for microfiber MZIs with different diameters. Thinner microfiber results in sharper interferometric fringes and narrower transmission dips, as a result of the rapid increment of Δ

*nΓ*.

The dispersion effect on the sensitivity characteristics for the MZI sensors is quite similar with the LPG sensors, as described in [18]. Here we consider the extreme situation when *Γ* approaches zero. For a random index-difference curve, i. e., the variation of index difference with wavelength, if we can find a straight line Δ*n* = *qλ* which is tangent to the curve, as shown in Fig. 7
, zero dispersion factor *Γ* can be found at the point of tangency because $\frac{\text{d}\Delta n}{\text{d}\lambda}=\frac{\Delta n}{\lambda}$. According to our calculation, *Γ* = 0 can be hardly achieved in circular microfibers. However, the physical picture in Fig. 7 provides a guideline to attain higher sensitivity. This can probably be realized by tailoring the transverse geometry of microfibers.

The *Γ* = 0 point in Fig. 7 corresponds to the “turning point” of the phase matching curve for LPGs in [18] (the λ-Λ relation). However, the “turning point” of the LPGs causes broadening of the resonant dips as well as high sensitivity. The dips can be so broad that the RI measurement cannot be properly performed. In contrast, by using thinner microfibers, the present MZI sensors can present high sensitivity and sharp fringes at the same time, which benefits the RI measurement.

In this section, we described the interferometric fringes as a result of the phase shift induced over the unperturbed region in between the gratings. In fact, the gratings introduce additional phase shifts at the off-resonance wavelengths due to the detuning. Its effect on the transmission property has been included in the transmission matrices in Sec. 3. Its contribution to the transmission property is relatively small for the dips within the bands and thus ignored here [19].

## 5. Conclusion

In this paper, we have preliminarily demonstrated the MZI sensors based on LPG pairs fabricated in micro-scaled optical fibers for sensing applications. The sensor with a diameter of 9.5 μm presents a RI sensitivity as high as ~2225 nm/RIU and a temperature sensitivity of only 11.7 pm/°C. Compared with the conventional MZI sensors, the present sensor can be much shorter and much more sensitive to surrounding RI due to stronger evanescent-field interaction. We further perform a theoretical investigation and predict that the maximum RI sensitivity that can be achieved is about 40000 nm/RIU, when the fiber is about 3.5 μm in diameter at around 1450 nm. Its sensitivity dependence is quite similar to the LPG sensors, considering the effect of mode dispersion. However, the LPG sensors sacrifice the spectral width of the resonant dips to achieve high sensitivity. In contrast, the high sensitivity and sharper fringes can be obtained at the same time. This feature also enables further reduction of the device length. Our next step is to realize higher RI sensitivity by tailoring the transverse geometry of the microfiber to approach zero dispersion factor.

## Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant No. 61225023, 11104117, and 11004085), the Research Fund for the Doctoral Program of Higher Education (Grant No. 20114401110006), and the Fundamental Research Funds for the Central Universities (Grant No. 21609102).

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