## Abstract

In-line fiber interferometers based on abrupt tapers have been shown as a promising low-cost platform for various sensing applications. Many variations have been demonstrated experimentally using combinations and permutations of novel photonic devices such as photonic crystal fibers or multimode fibers. In this manuscript, an analytical model for light propagation in an abrupt taper Mach-Zehnder Interferometer in commercial single-mode fiber based on coupled-mode theory is detailed. The model calculates the mode propagation process, and the result is compared with previous numerical simulations. For experimental verification, an in-line Mach-Zehnder interferometer based on an abruptly tapered fiber is fabricated and tested. The spectrum calculated by the model matches well with the measured spectrum. To reduce the computational complexity, a method for step size estimation and the corresponding error accumulation is discussed and verified. This can serve as a basis to estimate the optical responses of this class of abrupt taper based fiber sensors. With the right packaging, these sensors can play a vital role in rugged and hostile environments.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Optical fiber sensors based on in-line interferometer has been investigated and realized in recent years. The in-line fiber interferometric sensor eliminates the need to have two physical fiber branches by confining the fundamental mode in the core and the higher-order modes in the cladding of the bare fiber. The in-line fiber interferometers are used in such applications as sensing humidity [1], refractive index [2], gas [3], temperature [4], strain [5], and curvature [6]. Various structures of in-line interferometers have been investigated using different types of fibers. In [6], multiple cascaded tapers fused on multimode fiber (MMF) is shown to improve curvature sensing. In [7], a photonic crystal fiber (PCF) is used in the tapered section as a biosensing unit. Among all various implementation methods, the use of abrupt tapers fused on single-mode fiber (SMF) delivers the least fabrication complexity and cost.

Different in-line fiber sensor designs with abrupt SMF taper have been proposed: e.g., Mach-Zehnder interferometer (MZI) [8], Michelson interferometer [9] and long tapered fiber [10]. Numerical simulation using the beam propagation method (BPM) has been applied in the simulation of these devices to calculate the interferometer spectrum [11]. In [12], the mode coupling method is compared versus the BPM for a down taper. However, there is no analytical model for mode propagation in abrupt tapered fiber MZI, and the simulation based on BPM is time-consuming and lends little insight into the mode propagation dynamics. In this study, we detail an analytical model based on coupled-mode theory to describe the mode propagation in tapered fiber MZI [13] and provide a method to estimate the step size in the calculation based on the taper profile. The error estimation corresponding to the different step sizes will be discussed. Although our model is based on the SMF, this model can shine a light on the modeling of most fibers with a step-index profile such as MMF. Also, by studying the abrupt taper as the most severe case of tapering slope, our error estimation can be considered as a bound for various lengths and slopes of tapered devices. This is while our model has been shown to predict the long taper profile spectral response [10].

The paper is organized as follows. Section 2 provides an overview and theoretical analysis of the abrupt taper as the primary mode-coupling mechanism between the fundamental mode and the cladding modes. Section 3 details the formulation of the step size investigation. Section 4 details the modeling results and discusses the effect of the step size on the error accumulation. Section 5 verifies the modeling computation with experimental results in section 6.

## 2. Principle of mode coupling mechanisms

The modal fields in an optical fiber, a circular symmetric dielectric waveguide under the weakly guiding condition, can be described as a combination of linearly polarized (LP) modes. While the single-mode fiber (SMF) provides the confinement of the fundamental (LP_{01}) mode, an abruptly tapered fiber can perform mode conversion efficiently between the fundamental mode and higher-order modes [14]. As a result of tapering, the uniformity of the SMF in the longitudinal direction (z) is perturbed, and higher-order cladding modes (e.g., LP_{02}, LP_{03}…) confined by the air-cladding boundary, are excited by the partial coupling of LP_{01} mode energy into cladding modes. The mode profiles and their associated effective refractive indices can be calculated through a double-clad waveguide model at each point in the tapered SMF [15]. The geometrical profile of an [16] tapered SMF is depicted in Fig. 1. Fabrication details can be found in [17]. The choice of the taper geometry is driven by the ease of fabrication and robustness, which eliminates nanofiber waist taper from practical, low-cost packaging and disposable applications.

#### 2.1 Coupling coefficients

Approximating by LP modes, the scalar modal fields can be calculated. Then, the bounded field intensities can be applied to the modal coupling coefficients [18]:

_{0m}and LP

_{0n}modes are noted as ${\beta _m}$ and ${\beta _n}$.

*k*is the propagation constant in free space. $\varepsilon $ is the relative permittivity. ${\varphi _m}$ and ${\varphi _n}$ are describing the normalized modal fields of LP

_{0}_{0m}and LP

_{0n}(i.e., the time-averaged power of the mode $\frac{{{\varphi _n}}}{{N_n^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}} \right.}\!\lower0.7ex\hbox{$2$}}}}}$ is 1, where $N_n^{} = \frac{1}{2}\sqrt {\frac{{{\varepsilon _0}}}{{{\mu _0}}}} \frac{{{\beta _n}}}{{{k_0}}}\int\!\!\!\int\limits_S {{\varphi _n}\varphi _n^\ast } dS$ and

*S*is the fiber cross-section). The coupling coefficient, formulated in Eq. (1), depends upon not only the overlap of two field profiles but also the phase mismatch of these two modes. At the same time, the rate of change in relative permittivity with respect to the axis of propagation plays an important role. As a result of the mode coupling both on the core-cladding and cladding-surrounding boundaries, representing the mode conversion in a tapered SMF, the coupling coefficients (

*C*) between LP

_{mn}_{0m}and LP

_{0n}modes ($m \ne n$) can be calculated by combining the coefficients on each boundary [18]:

*n*

_{1},

*n*

_{2}, and

*n*

_{3}are the refractive indices of the core, cladding, and surrounding medium, respectively. The effective refractive indices of local modes LP

_{0n}and LP

_{0m}are noted as

*n*and

_{eff,n,}*n*. Considering the change of

_{eff,m}*n*through the down and up-taper, the modified normalized local mode ${\hat{\varphi }_m}$ is defined as:

_{eff}_{0m}mode. In Eq. (3), the integral in the denominator represents the mode power in the calculation of which the upper limit of infinity can be superseded with the cladding’s local radius. These local modes are orthogonal to each other. The radius of core and cladding are denoted by

*a*and

*b*as functions of

*z*along the fiber. So, the two differential operators account for the slope of tapered fiber.

*r*is the radial position of the fiber taper. As formulated in Eq. (4), the overlap integral of normalized mode profiles is denoted by ${[{{{\hat{\varphi }}_m}{{\hat{\varphi }}_n}} ]_{r = a(z )}}$ at the boundary of core-cladding, similarly, ${[{{{\hat{\varphi }}_m}{{\hat{\varphi }}_n}} ]_{r = b(z )}}$ accounts for the overlap at the boundary of cladding-air:

#### 2.2 Mode coupling equations

Utilizing the coupling coefficients between every two modes, the mode coupling equations can be completed towards the calculation of the field intensity and phase of different modes:

*A*is the complex amplitude of LP

_{n}_{0n}mode, comprising the field intensity and phase of the mode. $\lambda $ stands for the wavelength. The first summation term in the right-hand side of the equation denotes the mode conversion between LP

_{0n}and other modes. The second term represents the mode propagation along the fiber taper with effective index that varies along the taper. Given the parameters are all functions of z, this equation can be treated as a set of nonlinear differential equations.

## 3. Parameter calculation and analysis

Matching our SMF model with the characteristics of Corning SMF-28, the refractive indices of core and cladding are set as 1.455 and 1.450 at 1570 nm, with core and cladding’s diameters given as 8.2$\mathrm{\mu }\textrm{m}$, and 125$\mathrm{\mu }\textrm{m}$. The taper’s length is about 700 microns, while its waist’s diameter is about 40$\mathrm{\mu }\textrm{m}$. In this section, we set the surrounding medium to be air (*n*_{3 }= 1.000). The local V factor mainly determines the effective indices of local modes:

When V>0.807, the LP_{01} mode acts as a core mode, however, it converts to a cladding mode once the condition is not met where the effective index of the fundamental mode is less than the cladding’s index, i.e., n_{3}< n_{eff}_{,1}< n_{2}. As it is shown in Fig. 2(b), the LP modes’ powers are no longer confined within the core-cladding boundary; instead, it is contained by the cladding-air interface. From Eq. (2), we notice that only the field at the boundaries makes a contribution to the coupling coefficients, so we calculate the normalized mode power at the core-cladding and cladding-air boundaries as a function of the core radius, which is a proxy for the fiber tapering process. This is shown in Fig. 3.

For higher-order modes, the power intensities at the boundaries increase when the core radius decreases due to the unit power after normalization. This also applies to the fundamental mode intensity at the cladding-air boundary. The fundamental mode intensity, however, decreases after the core radius becomes smaller than the criteria of fundamental mode cut-off at the core-cladding interface, as shown in Fig. 3(a). This is because the power of fundamental mode after sufficient tapering is confined by the cladding instead of the core. This can also be seen from Fig. 3(b), where the power of LP_{01} mode increases after the cut-off point.

In order to calculate the coupling coefficients, we apply the effective index and the normalized modes at the boundaries into Eq. (1), which vary along the tapered region. Also, we calculated the effective indices along the tapered region to illustrate their influence along with fundamental mode’s cut-off. This is illustrated in Figs. 4(a) and 4(b) where the dependance between the coupling coefficient of two modes and their inverse difference of ERI is magnified; i.e. the smaller the difference between the modes’ ERI the greater their coupling coefficient.

## 4. Step size estimation

To calculate the mode-coupling equations, the tapered region of fiber needs to be segmented so that we can assume that the coupling coefficients remain unchanged in each segment. However, the segmentation of the taper will induce computation error, which we call a segmentation error. To determine the maximum step size in segmentation, we need to find a method to estimate the segmentation error with given step size before model calculation, based on the shape of tapered fiber and other given parameters (e.g., refractive index). We apply derivative on both sides of Eq. (5):

*C*, we can estimate the limit of segmentation error as: As seen in Fig. 3, the field intensities at boundaries are functions of the core radius, so we apply simple functions to estimate the upper limits of the field intensities at both core and cladding boundaries:

_{12}*B*and

_{m1}*B*are constants. The sign of

_{m2}*B*can be positive and negative, according to the order of the mode. Then we can use these functions in the analysis of coupling coefficients.

_{m2}*D*is a constant concerning the fiber parameters, (of the magnitude10

_{mn}^{−2}), which is estimated based on the refractive index of the material and the parameters in Eq. (9). There are two main parts in the final result of Eq. (11), one is related to the effective index mismatch between two modes, and the other is related to the shape of tapered fiber. However, the first part is slowly varying; hence, the second part mainly determines the change of coupling coefficients along the tapered fiber. The effective index mismatch between the two modes has a magnitude of 10

^{−2}. So the ratio between

*D*and the effective index mismatch in Eq. (11) is roughly ±1. (

_{mn}*D*can take on positive or negative values based on the sign of

_{mn}*B*

_{m2}_{.}) If we regard the ratio as a constant, we can perform the derivative on the coupling coefficient, which is proportional to the derivative of “da/dz(1/a)”:

*dC*. This is further illustrated in the following.

_{mn}/dzFor verification, we calculate the numerical derivative of coupling coefficients (C_{12}, among others), which are shown in Fig. 4(b) to compare with the estimation based on the longitudinal core radius profile. In Figs. 5(b) and Fig. 5(c), we can see that the trends of the values are similar to that in Fig. 5(a). The difference of the sign in the center parts of (a) and (b) is due to the fundamental mode’s cut-off, as mentioned in Fig. 3(a), which has a subsequent impact on the coupling coefficients C_{12}, considering the sign and magnitude of the overlap integral between LP_{01} and LP_{02} modes at the cladding-air interface. The sign is determined by order of the LP_{0m} mode: negative for even (e.g., LP_{02}) and positive for odd (e.g., LP_{03}) m. An example is shown in Fig. 6, where the relative amplitude of the cross-sectional electric field is plotted as a function of the fiber radius (normalized by a) around the LP_{01} cut-off point (V < 0.807). At the cladding-air interface, the relative field amplitude of the LP_{02} (even) mode is negative, and that of the LP_{03} (odd) mode is positive, justifying the sign selection in Eq. (11).

As seen in Fig. 5, the three figures share a similar order of magnitudes, which means the function in Fig. 5(a) can estimate the derivatives of the coupling coefficients. The difference in sign and value between Figs. 5(a) and 5(b), as previously explained, depends on the numerical value of the coupling coefficients.

In Eq. (8), e accounts for the error in each segment, but we need to consider the accumulated segmentation error in modeling. To estimate it, we will numerically integrate the function shown in Fig. 5(a) with a different step size. The result is shown in Fig. 7 with highlighted maximum values, which is regarded as the maximum accumulated segmentation error.

The integral over the function estimates the maximum segmentation error in the mode coupling procedure calculation, which is proportional to the step size in the calculation. We provide an estimation table based on the shape of the abruptly tapered fiber in our paper:

As shown in Table 1, the accumulated error is proportional to the step size in the simulation. To estimate the error in the simulation, we need to follow these steps: (1) calculate the estimation function in Fig. 5(a) according to the taper profile; (2) integrate over the tapered region with given step size and get the maximum value as the estimated error; (3) adjust the step size to reach the desired error limit and apply the step size in the calculation. We will verify this method in the following section.

## 5. Model calculation

In our simulation, the baseline step size is set as 0.005 microns, which, according to the model, will confine the segmentation error at 10^{−5}. After calculating the mode coupling equations step by step, we will get the mode power distribution at the end of the tapered fiber, as shown in Fig. 8.

The abrupt taper performs as a 3-dB attenuator between the fundamental mode and cladding modes. The output power is confined in the fundamental mode and the first ten cladding modes (LP_{02}∼LP_{011}). This result agrees with the previous simulation using BPM [10]. We can also get the mode power distribution (Fig. 9) along the tapered fiber after the calculation, which cannot be acquired directly using BPM. On accuracy and speed, the proposed analytical model requires only half of the time in BPM simulation for the same result.

As shown in Fig. 9, during the down-taper region, where the boundary’s slope is negative, the fundamental mode couples mostly to the cladding mode LP_{02}. Concerning the phase of the modes, the LP_{01} mode has a proclivity to couple with a mode that meets the smallest phase mismatch, i.e., the smallest effective index difference, namely LP_{02}, as presented in Fig. 4(a). Around $z ={-} 100\mathrm{\mu }\textrm{m}$, ${n_{eff,1}} - {n_{eff,2}}$ approaches the minimum value, whilst the magnitude of coupling coefficient between LP_{01} and LP_{02} reaches the maximum. After the LP_{01} transfers its power to the primary cladding mode (LP_{02}), its intensity at cladding-air boundary increases. As shown in Fig. 4(b), the field’s sign of those modes with an even order (e.g., LP_{02}) at the core-cladding boundary and cladding-air boundary are opposite, and the sign of coupling coefficient between LP_{02} mode and LP_{01} mode changes near the core to the cladding transfer point. The rate of power conversion at the waist of the taper reduces to zero since the gradient of the tapered geometry at the waist is zero. Conversely, in the up-taper region, where the gradient of the tapering is positive, the power in LP_{01} and LP_{02} is transferred into other higher-order cladding modes amongst which LP_{05} mode encompasses the highest power.

Despite the taper profile being symmetric, throughout the down and up-taper, the mode coupling process results in an asymmetric power evolution. This is because the signs of the coupling coefficient, appearing as the first term at the right-hand side of the coupling equations in Eq. (5), for every two symmetric positions are opposite, as indicated in Fig. 4(b), while the mode propagation term in Eq. (5) (the second term at the right-hand side) is not changed. Hence, the coupling process in these two symmetric sections cannot be treated as the converse of each other. Nevertheless, in the center of the fiber taper, the process looks symmetric (as shown in Fig. 9) since there is only a slight phase change between two symmetric points near the center of the fiber taper. While current modeling literature to date has offered results only in the downward tapering region [12,18,19], the current work emphasizes the ability of abrupt fiber taper in exciting higher order LP_{0m} modes.

This simulation result is set as our reference. To verify the step size estimation method, we applied different step sizes and compared the calculation result with this reference. By examining the difference in the magnitude of the fundamental mode between these results, we can get the maximum simulation error for the step size and verify the estimation method, as shown in Fig. 10.

In Fig. 10, we show the regions where the maximum accumulated error occurs as well as the point where the accumulated error is reduced to zero, which is −100$\mu m$ and −30$\mu m$, respectively. The reason for the occurrence of maximum and zero error points are the shape of the function in Fig. 5(a), which is similar to the shape of a normal sinc function.

We calculated the difference of power in fundamental mode between the approximate and reference calculations and compared it with the estimation (Fig. 7) in the previous section in Fig. 11. The maximum value in calculation and estimation matches well, which suggests that the estimation method is valid.

## 6. Experimental verification

An MZ interferometer is fabricated on a bare fiber between two abrupt fiber tapers to confirm the modeling result. The scheme of the experiment is shown in Fig. 12(a), which has been commonly used in many in-line fiber optic sensing systems. Using a broadband source (BBS), we launched light in the wavelength range of 1520∼1570 nm and measured the MZ interferometer attenuation spectrum with an optical spectrum analyzer (OSA) by deducting the output spectrum from the input spectrum. Modeling results are compared with the measured spectrum in Fig. 12(b).

In our measurement setup, the MZ interferometer is immersed in pure water, hence the surrounding’s refractive index in the model (*n _{3)}* is 1.31500, considering the wavelength range of 1520∼1570 nm. Each mode’s complex amplitude at the beginning of the second taper is estimated based on the modal amplitudes, gained at the end of the first taper, given the parameters of the bare fiber:

_{0m}mode at the end of the first taper is denoted by

*A*, and

_{m}*n*is the effective index of LP

_{eff,m}_{0m}mode in the bare fiber between two tapers.

*L*is the length of the bare fiber. The theoretical attenuation spectrum of the MZ interferometer is estimated via Eq. (13), once we calculate the mode coupling equations in the second taper within a range of wavelength (1520 nm to 1570 nm).

From Fig. 12, the attenuation spectrum is well-matched not only at the peaks and dips but also around the flat top between 1540 nm and 1545 nm. However, due to fabrication imperfection, bending along the taper profile, and connector losses in the setup, the measured attenuation in the experiment is more significant than the simulation result.

## 7. Conclusion

We have detailed an analytical model for an abrupt tapered MZ interferometer. The result of the model matches with the experimental result. The error of the proposed model is proportional to the step size in segmentation. Based on the shape of taper and fiber parameters, we can estimate the error induced by segmentation for different step sizes. By calculating the integral of the function related to the longitudinal taper profile, we can estimate the maximum accumulated error in the simulation and use it to determine the appropriate step size. Compared with other numerical approaches, the proposed formulation allows increased computation efficiency. The method of segmentation error estimation in this paper can be applied to the modeling of other tapered fiber MZ interferometer.

## Funding

Natural Sciences and Engineering Research Council of Canada (RPGIN/311-817-2012).

## Disclosures

The authors declare no conflicts of interest.

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