Analytical and experimental study on a bent abrupt taper

Pourya Ghasemi; Scott S.-H. Yam

doi:10.1364/OE.413139

1. Introduction

Single-mode fiber (SMF-28 by Corning) is an essential component of the fiber optics industry. Due to its low cost and versatility, it has been pervasively deployed in many applications, from communication to sensing. Although SMF-28 fiber supports a single transverse mode in the 1550 nm range, there are procedures under which it can perform as a multimode device. These procedures are associated with those that change the boundary conditions of the fiber. Alternations in the boundary conditions are introduced either geometrically along the axis of propagation, such as Fiber Taper (FT), or by engraving a periodic refractive index pattern, namely Fiber Bragg Grating (FBG) and Long Period Grating (LPG). Amongst these resultant multimode devices, abrupt fiber tapers can be quickly fabricated using the most commercially available fusion splicers at the lowest cost possible.

The multimode performance of the single-mode fiber’s down-taper has been investigated through experimental observations, and the modal coupling coefficients were studied [1,2]. In [3], the power evolution of the first four modes (LP$_{0m}$; m=1$\ldots$4) for a down-taper have been studied using the coupled-mode theory (CMT), indicating the successive excitation of the higher-order modes (HOMs) on a down-taper. Investigation on the modal power transfer in the subsequent up-taper region, after a down-taper, is developed in [4]. In [5], the modal excitation of the LP$_{0m}$ modes (for m = 1$\ldots$10) in an abrupt taper is studied using the CMT method. Accumulation of phase and coupling results in a continuum of LP$_{0m}$ HOMs, amongst all, LP$_{05}$ is ranked with the highest power [4,5]. It has also been shown that by concatenation of two tapers, HOMs can form a modal Mach-Zehnder interferometer [6]. Another type of in-line interferometry using abrupt tapers has been investigated in [7], forming a Michelson interferometer employing a gold plate at the end of the taper HOMs will bounce back, causing interference between the HOMs and the fundamental mode through the bouncing length. These modal interferometers have shown their application in refractive index (RI) sensing of solutions, as dipping them into a solvent changes the difference of Effective RI (ERI) between the fundamental mode, and the HOMs [8,9].

There have been reports on modal interferometry of the bent tapers (namely S-Tapers) along with their sensing capabilities [10 –13]. In these studies, an approximation of a two-mode interference is studied. Also, it has been shown that with a single S-Taper, one can achieve higher degrees of sensitivity while sensing strain and RI, compared with conventional tapers. Nevertheless, a coupling model interpreting the experimental observations and the physical behavior of the device is missing.

Aside from the tapering effect, there have been earlier analytical and numerical studies regarding fibers’ bending effect. Field deformations and losses in a double-clad fiber medium due to curved perturbation in the RI layers have been studied using bending models in [14,15]. In the case of slight perturbations, the overlap coupling integral between LP$_{01}$ and LP$_{11}$ mode has been expressed in [16], providing an approximation of phase distortion for small bend angles. For more severe bending cases, it is shown in [17] that the boundary conditions can be modified through the conformal mapping technique such that the field distribution can be resolved more efficiently. However, this approach relies on finite difference methods, and compared with the analytical model, is computationally costlier and less intuitive of the underlying physical phenomena. In any case, none of these techniques are addressing a mode coupling scheme comprising both tapering and bending at the same time.

In this paper, we introduce an analytical coupling approximation for a perturbed abrupt taper along with the definition of a perturbation function. We can treat the perturbations on the RI boundary conditions and propagation axis throughout this model, namely tapering and bending, separately. The core and cladding modes’ power evolution is then studied utilizing the modified coupling model for a specific perturbation function and its other variations. A comparison between our model and the experimental results shows our estimate’s agreement with the spectral measurements. The emergence of the LP$_{11}$ and LP$_{12}$ modes is verified numerically and experimentally by beam profile observations.

2. Theoretical study of bent taper profiles

The measured outlines of straight and bent taper are shown in Fig. 1. The device’s tapering and bending outline measurements are done via image processing over the microscope images. Initially, Sobel’s edge detection technique is used with a tuned threshold to avoid background noise [18]. Then, a moving average filter is applied upon the extracted edges to form a smooth outline. The taper’s bending is calculated as the average point between the cladding’s upper and lower extracted edges. To minimize the measurement error, we have chosen the window size of the moving average filter so that the initial value of the bending is set to zero. Also, we have imposed the symmetry with respect to the taper’s waist. Once the cladding is estimated, the core’s outline is calculated, maintaining the original core-cladding fraction of $\frac {8.4}{125}$ along the taper.

Fig. 1. (a) Edge detection and filtering results on microscope image of a bend taper, (b) The measured outline of tapered fiber for symmetrical and asymmetrical fabrication outcomes is plotted versus the propagation axis. The bending outline p(z) is estimated along z. The maximum bending angle $\theta _{max}$ is denoted as the angle between the primary axis of propagation and the bent fiber.

Download Full Size | PDF

In the straight taper, variations in the boundary conditions along the z-axis induce the fiber field profiles to change, resulting in energy coupling from the fundamental to the HOMs. The circular symmetry of the RI layers induces just the LP$_{0m}$ modes. However, introducing a bend across the taper breaks this symmetry and causes further coupling to other HOMs (e.g., LP$_{1m}$ modes).

Under the slight bending condition, where the maximum bending angle ($\theta _{max}$) is less than $3^{\circ}$, maintaining the linear approximation over the sine function, tapering and bending can be treated separately. Thus, the field profiles at each z-position along the taper can be calculated regardless of bending, and then the bending outline effects the coupling process.

2.1 Analytical model

The field profile can be spanned by a set of orthonormal modes $\left \{{\ \Psi }_\nu \right \}$ resulting from the scalar’s solution wave equations. Given the fiber’s boundary conditions, $\left \{{\ \Psi }_\nu \right \}$ satisfy the continuity condition up to the first derivative [19,20]. The total field can be represented as follows:

(1)$$\Psi_{total}(r,\phi,z)=\sum_{\nu}{a_\nu(z)\Psi_\nu\left(r,\phi\right)e^{i\int_{0}^{z}{\beta_\nu(z^\prime)dz^\prime}}},$$

where $\Psi _{\nu }\left (r,\phi \right )$, is the $\nu$th mode’s field distribution, $\left (r,\phi \right )$ are representing the radial and angular dependencies of the field function, $a_{\nu }$ is a z-dependent function denoting the amplitude of the mode, $\beta _{\nu }$ is the mode’s propagation constant, and the axis of propagations lies in the z-direction. The coupling equation can be compactly described as:

(2)$$\frac{da_\mu}{dz}=\sum_{\nu\neq\mu}{\mathfrak{C}_{\mu\nu}a_\nu},$$

where $a_\nu$ and $a_\mu$ are the amplitudes of the $\mu$th and $\nu$th mode’s power, and, $\mathfrak {C}_{\mu \nu }$ is the modified coupling coefficient, between the modes of order $\mu$ and $\nu$, defined as:

(3)$$\mathfrak{C}_{\mu\nu}={\widetilde{C}}_{\mu\nu}e^{\left(i\Phi_{\mu\nu}\left(z\right)\right)},$$

within which, $\Phi _{\mu \nu }$ accounts for the differential phase accumulation between the modes $\mu$ and $\nu$ across taper:

(4)$$\Phi_{\mu\nu}\left(z\right)=k_0\int_{0}^{z}\left(n_\mu\left(z^\prime\right)-n_\nu\left(z^\prime\right)\right)dz^\prime.$$

The integral is between the starting point of the down-taper ($z^\prime =0$), up to the point z along the taper’s propagation axis. The z-dependent functions, $n_\mu$ and $n_\nu$, are the effective refractive indices of modes $\mu$ and $\nu$, and $k_0=\frac {2\pi }{\lambda }$ with $\lambda$ as wavelength.

The coupling coefficient ${\widetilde {C}}_{\nu \mu }$ describes the change in the overlap integral between two modes through the taper, meaning a differential overlap integral. As perturbations are introducing slight bends over the RI’s profile, a phase change is introduced along with the field’s profile, causing the phase of the profile to face less change in the compressed section (inside of the bend) and more in the extended section (outside of the bend). Although the field profile is prone to displacement for more significant degrees of agitation, for minor values of perturbation (slight bend angle smaller than 0.055 radians), we can assume that the field’s profile stands the same while it is masked with a distorted phase front [16]. This distortion appears as an exponential term in the coupling coefficient integral as:

(5)$${\widetilde{C}}_{\mu\nu}=\iint{\frac{\partial\Psi_\mu}{\partial z}\Psi_\nu^\ast e^{{ik}_0n\left(1-\chi\right)r\theta\cos{\left(\phi\right)}}}rdrd\phi$$

The differential term in the integrand $(\frac {\partial \Psi _\nu }{\partial z})$ relates to the modulation of the mode’s profile along the propagation axis due to the modulation of the refractive index profile of the fiber along the same axis caused by topological variations. The term $\chi$($\approx 0.22$ for fused silica [16]) accounts for the stress-related changes in the refractive index profile, and $\theta$ stands for the average curvature against the dynamic axis of propagation, which can be mathematically defined as the absolute moving-average over the second derivative of the fiber’s perturbation outline:

(6)$$\theta\left(z_l\right)=\left|\frac{\Delta_z^2}{W}\ \sum_{i=l-(W-1)/2}^{l+(W-1)/2}D\left(z_i\right)\right|$$

Considering the z direction as a discrete set of L points:$\left \{z_l\right \}_{l=1..L}$, with $\Delta _z$ as the spacing grid, W is the window size of the moving-average filter; given zero-padding for the truncated endpoints. $D\left (z\right )$ is the second derivative of the perturbation outline ($D\left (z\right )=d^2p(z)/dz^2$; with p(z) as the perturbation pattern, shown in Fig. 1.) Equation (5) can be further simplified and expanded as the following:

(7)$${\widetilde{C}}_{\mu\nu}=Re\left\{{\widetilde{C}}_{\mu\nu}\right\}+iIm\left\{{\widetilde{C}}_{\mu\nu}\right\},$$

where the real part matches the conventional coupling coefficient as [3]:

(8)$$Re\left\{{\widetilde{C}}_{\mu\nu}\right\}\approx\frac{1}{2n_{co}\left(n_\mu-n_\nu\right)}\iint{\frac{dn^2}{dz}\Psi_\mu\Psi_\nu^\ast}rdrd\phi\ ,$$

the term $dn^2/dz$ directly points to the variations of the refractive index profile $n(r,\phi ,z)$ of the fiber along the z-axis of propagation. It is to note that the coupling coefficient, appearing in (8), stands nonzero if two modes in the integrant are coming from the same family of modes, e.g., both from LP$_{0m}$ or LP$_{1m}$. The imaginary term in (7) is nonzero when the modes are attained from two different families of modes. This term can be simplified for the case of coupling between LP$_{0m}$ and LP$_{1m}$ modes as:

(9)$$Im\left\{{\widetilde{C}}_{\mu\nu}\right\}\approx\frac{\pi^2\left(1-\chi\right)\theta}{\lambda\left(n_\mu-n_\nu\right)}\int{\frac{dn^2}{dz}\psi_\mu\psi_\nu^\ast}r^2dr,$$

where $\psi _\nu$, is the mode’s radial function; for LP$_{0m}$ modes $\Psi _\nu (r,\phi )=\psi _\nu \left (r\right )$, and $\Psi _\nu (r,\phi )=\psi _\nu \left (r\right )\cos (\phi )$ for LP$_{1m}$ modes. Utilizing the perturbed coupling coefficient, introduced in Eq. (5), the integral is no longer symmetrical between the LP$_{0m}$ and LP$_{1m}$ set of modes as a result of which the energy transfer is no longer confined solely within the LP$_{0m}$ modes. Hence, the LP$_{1m}$ modes are excited throughout the perturbed taper.

2.2 Matrix implementation of the coupling equations

The coupling equation has been simplified to (2), where $\mathfrak {C}_{\mu \nu }$ is the modified coupling coefficient accounting for the phase accumulation within itself. Now we discretize the first derivative on the left-hand side of the equation as:

(10)$$\frac{da_\mu}{dz}\overset{\Delta}{=}\frac{a_\mu^{l+1}-a_\mu^l}{\Delta_z},$$

where $l$ accounts for the layer index in the z-direction, and $\Delta _z$ is the discrete differential of the propagation axis. In the same way, each layer’s coupling coefficient can be noted as $\mathfrak {C}_{\mu \nu }^l$ accounting for the coupling between the modes $\mu$ and $\nu$ at layer l, which can be calculated from (3). Then, the recursive coupling equation can be written in the matrix format as:

(11)$$A^{l+1}=\left(I+\Delta_z\mathfrak{C}^l\right)A^l,$$

and

(12)$$\mathfrak{C}^l=\left[\begin{matrix}\mathfrak{C}_{11}^l & \begin{matrix}\mathfrak{C}_{12}^l & \ldots\\\end{matrix}&\mathfrak{C}_{1m}^l\\\begin{matrix}\mathfrak{C}_{21}^l\\\vdots\\\end{matrix}&\begin{matrix}\begin{matrix}\mathfrak{C}_{22}^l\\\vdots\\\end{matrix}&\begin{matrix}\ldots\\\ddots\ \\\end{matrix}\\\end{matrix}&\begin{matrix}\mathfrak{C}_{2m}^l\\\vdots\\\end{matrix}\\\mathfrak{C}_{m1}^l&\begin{matrix}\mathfrak{C}_{m2}^l & \ldots\\\end{matrix}&\mathfrak{C}_{mm}^l\\\end{matrix}\right],$$

with m as the total number of modes under study, and I as an m-by-m the identity matrix. In order to further simplify the notation, we express $M^l=\left (I+\Delta _z\mathfrak {C}^l\right )$. Thus, the modes amplitude vector at each layer can be calculated as the following product:

(13)$$A^l=\left(\prod_{j=0}^{l-1}M^j\right)A^0,$$

where $A^0$ stands for the initial excitation amplitude vector. In the next section, we will use this matrix method to calculate the power evolution of the modes along the taper assuming the excitation with LP$_{01}$ mode as the first mode, i.e., $A^0=\left [1\ 0\ldots 0\right ]^T$.

3. Modeling and simulation results

3.1 Simulation parameters and assumptions

The modeling parameters are set to simulate a tapered SMF-28 fiber with 8.4 $\mu$m and 125 $\mu$m for diameters of core and cladding. The cores refractive index (RI) is taken to be 1.455, and the RI difference between the core and cladding is taken to be 0.0036. As the fiber is stripped, the outer layer’s RI equates to that of air. For typically fabricated samples, the fiber’s diameter at the tapered section is set to 50 $\mu$m, and the total length of the tapered section is taken to be 700 $\mu$m. The wavelength ($\lambda$) is 1550 nm. LP$_{01}$ is the excitation mode conditioned to carry the SMF’s energy before the tapered section. This modal study is limited to the first ten modes on the LP$_{0m}$ family as they carry 98% of the total energy for a typical abrupt taper [4]. Also, amongst the LP$_{1m}$ modes, we decided to limit our study to the first two modes provided that we are within the low energy transfer regime, and higher ERI’s are having a remarkable proclivity to carry the energy. The tapering geometry profile, regarding the core’s radius along the z-axis, is formulated as:

(14)$$r_{co}\left(z\right)=\frac{1}{2}\left[\left(r_1+r_2\right)-\left(r_1-r_2\right)cos{\left(2\pi\frac{z}{L_t}\right)}\right],$$

where $r_1$ and $r_2$ are the core’s radius at the taper’s waist and plain area, respectively; $r_1=1.7 \mu$m and $r_2=4.2 \mu$m. $L_t$ is the taper length (700 $\mu$m), and z is associated with the positions along the propagation axis. The spacing grid in the z-direction is taken to be 1 micron ($\Delta _z=1 \mu$m). We maintain the assumption that the fraction of the core-cladding radius remains the same throughout the taper geometry, i.e. $r_{cl}\left (z\right )=14.88\times \ r_{co}(z)$. Figure 2(a) plots the outcome of (14).Aside from the tapering profile, the final result of the power evolution depends on the bending perturbation function p(z) and its associated absolute curvature $\theta (z)$. In order to reach a generalized fitting model for p(z) that is measured via fabrication (Fig. 1), we first introduced the curvature function $\theta (z)$ utilizing our fabrication observations. Throughout the tapering process, the fiber will be stretched as soon as the electric arc is applied. As a result of off-axis force at the jigs, the sections at the vicinity of the fixtures are curved. However, most of the tapered region that has been stretched stands uncurved. There are three characteristics to define an initial curvature function $\theta (z)$: first, the maximum amount of bend angle ($\theta _{max}$), next, the curvature linewidth ($\Delta z_\theta$), which is set to 150 $\mu$m for this case. Last, the start point of bending in the first section (offset). These features can be extracted from the measurement results, as shown in Fig. 1. Moreover, developing this function, we assumed the symmetry in perturbation such that p(z) is odd with respect to the tapered waist. Once the curvature function is estimated, the perturbation outline p(z) is calculated using the reverse averaging process described in (6) with W=9, addressing to an average curvature in 9 $\mu$m as the minimum window length to achieve smoothness on p(z). In Fig. 2(c), the curvature and bending functions are plotted. In the same figure, Fig. 2(b), a microscope view of a fabricated taper is shown, scaled with the z-axis, which supports the perturbation outline shown in Fig. 2(c). The bending of the geometric profile starts at the beginning of the taper, and then the fiber stands straight before the next bending at the end of the tapered section. All in all, three sections of Fig. 2 depict the separation between the bending and tapering in the modal analysis. The modal content is being calculated using the tapered boundary condition, while the bending characteristics appear to modify the modal coupling integrant with a phase mask, manifesting in terms of an exponential described in (5).

Fig. 2. (a) the core’s tapering outline formulated in (14) as a function of z ($r_{co}(z)$), and the estimated core’s radii from the edge detection process, (b) typical fabrication result, in-line with (c) the absolute curvature function: $\theta (z)$ (right axis) and the plot of the extracted bending outline p(z) (left axis) versus the taper length.

Download Full Size | PDF

3.2 Power evolution and modified coupling coefficients

In this paper, the study of power evolution and modal coupling is carried upon twelve modes, comprising the first ten modes of the LP$_{0m}$ class and the first two modes of the LP$_{1m}$ class. The collected LP$_{0m}$ modes have been previously considered as the most dominant modes in the study of abrupt tapers [6]. We have included the first two modes from the LP$_{1m}$ family of modes as their ERIs are in close vicinity and even intersect with LP$_{02}$ and LP$_{03}$, respectively, causing them to highly couple and gain power through the power evolution.

Using the matrix formulation introduced in (14), we can calculate the amplitude of modes at each layer along z. Then, the power values are then extracted as the absolute squared amplitudes. In Fig. 3, the power evolution of modes for the case of $\theta _{max}=3^{\circ}$ is depicted. It is shown that LP$_{04}$ has higher power than LP$_{05}$ at the end of the taper in comparison with an unperturbed power evolution studied in [8]. The LP$_{12}$ mode, a cladding mode, comes as the second most significant HOM after LP$_{04}$. The LP$_{11}$ mode is carrying 4% of the total power. It is to note that LP$_{07}$ to LP$_{010}$ modes are excluded from Fig. 3 and Fig. 4, as they together carry less than 1% of the total power, and their coupling contribution is negligible.The direction of the power flow has a dependence on the slope of the taper, the inverse difference between ERIs of the modes, and the minimum amount of change in the phase accumulation term. Thus, starting from the excitation mode (LP$_{01}$), the first mode to ignite is LP$_{02}$, then the LP$_{03}$, and the list goes on sequentially, within the down taper section (from 0 to 350 $\mu$m). Notwithstanding, the power transfer through the up-taper section (from 350 to 700 $\mu$m) is no longer ordered due to the phase accumulation term introduced in (4). Hence, the study of the modified coupling coefficient formulated in (3) containing the coupling and phase accumulation can bring us insight into the effective coupling through the whole device. The absolute value over the averaged modified coupling values throughout the perturbed taper is shown in Fig. 4 and expressed as follows:

(15)$$\frac{\left|\sum_{l=1}^{L}{\mathfrak{C}^l}\right|}{L},$$

where L is the total number of layers, and $\mathfrak {C}^{\mathfrak {l}}$ is the modified coupling coefficient matrix at layer $l$ formulated in (12). The zero values at the diagonal are associated with the self-coupling of each mode. The power coupling shows the highest value of coupling between LP$_{05}$ and LP$_{04}$ amongst the cladding modes. This translates the mode swapping between LP$_{05}$ and LP$_{04}$, presented before in Fig. 3, and the excitation of the LP$_{1m}$ cladding modes. Also, the coupling between LP$_{0m}$ and LP$_{1m}$ modes shows its peak for LP$_{05}$ and LP$_{12}$, i.e., power transfer from LP$_{05}$ to LP$_{12}$ mode. This will be further investigated in the next section.

Fig. 3. Power evolution of the significant modes throughout an abrupt bent taper with a maximum bending angle ($\theta _{max}$) of three degrees. The left axis associates with the fundamental mode, while the right axis shows the relative power of other HOMs.

Download Full Size | PDF

Fig. 4. Absolute mean value of the modified coupling coefficients throughout the taper. LP$_{05}$ shows to have the maximum coupling to LP$_{12}$ and LP$_{04}$.

Download Full Size | PDF

3.3 Perturbation function variations

The three elements of the bend function $\theta \left (z\right )$ being the linewidth ($\Delta z_{\theta }$), the start point of the bent section (offset), and the maximum bend ($\theta _{max}$), will phenomenologically interpret most of the cases of S-taper fabrication. In this section, we limit our investigation to 0.035 radians translating a bend that causes the taper’s tail to dislocate from its primary axis by 24 $\mu$m. This curvature extent can occur at the beginning of the taper, where the offset feature is set to zero. Thus, we have investigated the output power of significant modes at the end of the taper via different values of $\theta _{max}$ up to 0.035 rad, assuming a gentle curve introduced by a relatively broad linewidth of $\Delta z_{\theta }=200 \mu$m, the result of which is shown in Fig. 5(a). The fundamental mode’s power drops from 0.5 to 0.3, coupling its energy to LP$_{04}$, while LP$_{05}$ surrenders its energy to cladding modes (LP$_{12}$).We then studied the effect of HOM harmonics at the spectrum utilizing the Fourier transform of the simulated spectrum, a method that has been extensively used for modal decomposition [21]. Figure 5(b) depicts the outcome of applying the FFT algorithm on the resultant spectrums through rising values of $\theta _{max}$. The horizontal axis interprets the time delay between the HOMs and the fundamental mode in picoseconds. The lobe peaking around 1ps, conjoining LP$_{04}$ and LP$_{05}$, has the highest energy amongst other lobes and HOMs at zero bends. However, as $\theta _{max}$ increases, the width and zenith of this lobe decay moderately, reflecting the gain and loss of power in LP$_{04}$ and LP$_{05}$, respectively. At the same time, LP$_{11}$ and LP$_{12}$ are being triggered as they gain in width and height around 0.2ps and 0.5ps. Meanwhile, the LP$_{06}$’s peak at 1.4ps remains relatively steady. In the next section, our model will be compared with the experimental results.

Fig. 5. (a) the power of the significant modes LP$_{01}$, LP$_{04}$, LP$_{05}$, LP$_{06}$, LP$_{11}$, LP$_{12}$ at the end of the taper versus the maximum bending angle ($\theta _{max}$) and (b) FFT of the simulated spectral outcomes of modal amplitudes against $\theta _{max}$.

Download Full Size | PDF

4. Experimental results and model validation

As shown in Fig. 6, the experimental setups comprise a Broadband Source (BBS) in the transmitter side, and either an Optical Spectrum Analyzer (OSA) or a Beam Profiler (BP) on the receiver’s end. The fiber’s tail at OSA is connected through a Bare Fiber Adaptor (BFA), which avails the multimode spectral reading within OSA.A formulation that best describes the OSA’s spectral response is noted in (16), showing the summation of all the modes on a specific wavelength span:

(16)$$\widetilde{S}\left(\lambda\right)=\left|\sum_{\nu}{a_\nu\exp{\left(i\left(\frac{2\pi}{\lambda}n_\nu\left(\lambda\right)L_d+\phi_\nu\right)\right)}}\right|^2$$

Using our coupling model, we can calculate the complex amplitude of each mode ($A_\nu =a_\nu e^{i\phi _\nu }$) and its related ERI ($n_\nu$). The change of ERIs over the wavelengths is non-negligible since the order of inverted wavelength will magnify the modal dispersion. Likewise, the spectrum highly depends on the interferometry length ($L_d$), from the end of taper to OSA, since lambda is in the order of microns, and $L_d$ stands in the order of centimeters. The proportion of interferometry length over the wavelength magnifies the fabrication imperfections over $L_d$ so that a small change over the value $L_d$ causes a drastic change over the total spectrum. We used a simulated annealing technique to optimize our modeling estimation with the experimental data. This optimization technique returned us a set of modal amplitudes for each spectrum, meeting our mean square error criteria:

(17)$$\underset{a_\nu,\phi_\nu,L_d}{\mathrm{argmin}}{\left|\widetilde{S}\left(\lambda\right)-S\left(\lambda\right)\right|^2},$$

where $\widetilde {S}\left (\lambda \right )$ is our spectral estimate, and $S\left (\lambda \right )$ is the spectral reading from OSA.

Fig. 6. (a) spectral and (b) profile examination setups. Broad Band Source (BBS) is utilized as the input source in both setups. The spectral response is measured via an Optical Spectrum Analyzer (OSA). The taper is connceted to the OSA via a Bare Fiber Adaptor (BFA). The fiber’s length after the tapered section is noted with $L_d$. A Beam Profiler (BP) is used to capture the taper’s output intensity profile. The gap between the taper’s tail and BP is noted as $l_g$.

Download Full Size | PDF

The spectral response of the spectrum can be studied through the Fourier transform formalism. After factorization with respect to the fundamental mode, Eq. (16) can be simplified to the following expansion:

(18)$$\begin{aligned} \widetilde{S}\left(\omega\right)=\left|a_{01}\right|^2\Bigg(1&+\sum_{\nu}\frac{\left|a_\nu\right|^2}{\left|a_{01}\right|^2}+\sum_{\nu}{\frac{2a_\nu}{a_{01}}\cos{\left(\omega\tau_\nu+\Delta\phi_{\nu,01}\right)}}\\ &+ \sum_{\nu>\mu}{2\frac{a_\nu a_\mu}{\left|a_{01}\right|^2}\cos{\left(\omega\Delta\tau_{\nu,\mu}+\Delta\phi_{\nu,\mu}\right)}}\Bigg), \end{aligned}$$

with $\Delta \phi _{\nu ,\mu }=\phi _\nu -\phi _\mu$, $\Delta \tau _{\nu ,\mu }=\tau _\nu -\tau _\mu$, and $\omega =2\pi c/\lambda$. Where $\tau _\nu =\frac {(n_\nu -n_{01})L_d}{c}$ is the relative group-delay difference between the mode $\nu$ and LP$_{01}$. The modeled spectrum’s dependency on the difference between the ERIs is better reflected in (18), whilst the time-delay between the HOMs and fundamental mode is emphasized. The relative time-delays are initially estimated using the ERIs’ modeling values and the measured interferometry length of $L_d$. As the model matches through the optimization, the FFT of the spectrum reveals the delay between the modal content. The Fourier transform of (18) results in a function depending on relative delay differences between the HOMs and the LP$_{01}$.

A typical spectral response and their associated Fourier transform are compared with our modeling results in Fig. 7 and Fig. 8, respectively. The interferometry pattern results from the different time delays between the fundamental mode and the HOM’s. The dominant time delay associates with the most dominant HOM, which is particularly marked as LP$_{04}$ with 1ps delay with respect to the LP$_{01}$ mode. In Fig. 8, the peaks appearing below the 1ps are related to HOMs with ERIs higher than LP$_{04}$’s, such as LP$_{02}$, LP$_{03}$, LP$_{11}$, and LP$_{12}$. Amongst all, the LP$_{12}$ acts as the most dominant at 0.5ps. To further justify our estimate, we have gone through a mode tagging and decomposition procedure through a set of spectral responses. Using the resultant modal amplitudes from Eq. (17). Next, the mode extraction results, using the simulated annealing method, are averaged over a total number of ten spectral outcomes for different fabricated tapers. Our initial guess over the model amplitudes and the optimization process’s averaged outcome is depicted in Fig. 9.The excitation of LP$_{12}$ and the LP$_{04}$’s priority in power, as predicted by our model, matches the mode extraction over experimental data. The deviation of the estimates from the experimental data is due to minor multi-axis fabrication imperfections that are not considered in our model. Also, the BFA’s connection mismatch prompts mode excitations assumed negligible in our modeling.

Fig. 7. Spectral response of a fabricated result compared with our modeled spectrum.

Download Full Size | PDF

Fig. 8. Fourier transform of the experimental and modeling spectrum versus the delayed time with respect to the fundamental mode.

Download Full Size | PDF

Fig. 9. The initial guess and averaged optimization outcome over ten different fabricated tapers considering the first ten modes from LP$_{0m}$ modes and the first two modes from the LP$_{1m}$ family.

Download Full Size | PDF

Furthermore, utilizing a beam profiler, we have made observations over the spatial profile of an SMF with-and-without its polymer coating as a reference for LP$_{01}$. Next, a series of straight and bent tapers are fabricated, and their beam profiles are measured before and after tapering. The differential mode profiles are studied for the modal residue between the straight tapers, the bend tapers, and the SMF. We have used the optimized features as the phase information to decompose the modal content.

Also, a Fresnel Transformation technique to account for the free-space propagation of the beam from the fiber’s tip to the profiler’s capturing screen [22]. The Fresnel transformation is formulated as the following:

(19)$$H\left(u,v\right)=\frac{e^{i\pi\left(u^2+v^2\right)}}{\left(\lambda l_g\right)^3}\mathcal{F}^{2\mathcal{D}}\left\{h\left(x,y\right)e^{i\pi\lambda l_g\left(x^2+y^2\right)}\right\},$$

where $\mathcal {F}^{2\mathcal {D}}\left \{.\right \}$ is the two-dimensional Fourier transform. $h\left (x,y\right )$ is the intensity profile. $l_g$ is the gap between the fiber’s tip and the profiler; $l_g\approx 0.5$ mm. Throughout the transformation, the initial coordinates (x,y) at the magnified distance from the fiber’s tip will be converted to a new coordinate (u,v) at the fiber’s tip. The result of the transformations translates the effect of the free space as a lens with its focal fixed at infinity and demagnifies the profiles in Fig. 10(a) and (c) to the fiber’s original dimensions resulting in Fig. 10(b) and (d), respectively. Nevertheless, by tuning the transformation parameters, we have achieved higher spatial resolution describing the beam profiles. Also, the LP$_{12}$’s profile deformation (Fig. 10(a) and (b)) interprets the bending effect on the tapered fiber.

Fig. 10. Intensity difference between the tapered fiber and a stripped SMF for two instances of fabrication (a and c) with their associated two-dimensional Fresnel transformation (b and d); the presence of LP$_{12}$ is confirmed.

Download Full Size | PDF

5. Conclusion

An estimated model of perturbed coupling for slightly bent tapers is presented. The effect of mode coupling on an off-axis fiber taper is analyzed and formulated, representing the separation between the tapering and bending on the coupling process. The modified coupling coefficient and power evolution of the core and cladding modes is studied against a distinct perturbation function. The HOMs time delay difference is studied for variations of the defined perturbation function. The emergence of LP$_{11}$ and LP$_{12}$ modes is verified by matching our estimate with experimental data. The experimental observation has been made on the differential intensity profiles of a series of tapered fibers and a stripped SMF to detect the excitation of the LP$_{12}$ mode as a dominant cladding mode.

Funding

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

Disclosures

The authors declare no conflicts of interest related to this work.

References

1. A. J. Fielding, K. Edinger, and C. C. Davis, “Experimental observation of mode evolution in single-mode tapered optical fibers,” J. Lightwave Technol. 17(9), 1649–1656 (1999). [CrossRef]

2. D. Marcuse, “Mode conversion in optical fibers with monotonically increasing core radius,” J. Lightwave Technol. 5(1), 125–133 (1987). [CrossRef]

3. F. Gonthier, A. Hénault, S. Lacroix, R. J. Black, and J. Bures, “Mode coupling in nonuniform fibers: comparison between coupled-mode theory and finite-difference beam-propagation method simulations,” J. Opt. Soc. Am. B 8(2), 416–421 (1991). [CrossRef]

4. Z. Tian and S. S.-H. Yam, “In-line abrupt taper optical fiber mach–zehnder interferometric strain sensor,” IEEE Photonics Technol. Lett. 21(3), 161–163 (2009). [CrossRef]

5. X. Leng and S. S.-H. Yam, “Analytical model for abrupt tapered mach–zehnder interferometer based on coupled mode theory,” IEEE Photonics Technol. Lett. 31(19), 1600–1603 (2019). [CrossRef]

6. X. Leng, S. S.-H. Yam, and P. Ghasemi, “Error estimation in the analytical modeling of abrupt taper mach-zehnder interferometers,” OSA Continuum 3(11), 3048–3060 (2020). [CrossRef]

7. Z. Tian, S. S. Yam, and H.-P. Loock, “Refractive index sensor based on an abrupt taper michelson interferometer in a single-mode fiber,” Opt. Lett. 33(10), 1105–1107 (2008). [CrossRef]

8. Z. Tian and S. S.-H. Yam, “In-line single-mode optical fiber interferometric refractive index sensors,” J. Lightwave Technol. 27(13), 2296–2306 (2009). [CrossRef]

9. Z. Tian, S. S.-H. Yam, J. Barnes, W. Bock, P. Greig, J. M. Fraser, H.-P. Loock, and R. D. Oleschuk, “Refractive index sensing with mach–zehnder interferometer based on concatenating two single-mode fiber tapers,” IEEE Photonics Technol. Lett. 20(8), 626–628 (2008). [CrossRef]

10. R. Yang, Y.-S. Yu, Y. Xue, C. Chen, Q.-D. Chen, and H.-B. Sun, “Single s-tapered fiber mach–zehnder interferometers,” Opt. Lett. 36(23), 4482–4484 (2011). [CrossRef]

11. R. Yang, Y.-S. Yu, C. Chen, Y. Xue, X.-L. Zhang, J.-C. Guo, C. Wang, F. Zhu, B.-L. Zhang, Q.-D. Chen, and H.-B. Sun, “S-tapered fiber sensors for highly sensitive measurement of refractive index and axial strain,” J. Lightwave Technol. 30(19), 3126–3132 (2012). [CrossRef]

12. H. Liu, Y. Miao, B. Liu, W. Lin, H. Zhang, B. Song, M. Huang, and L. Lin, “Relative humidity sensor based on s-taper fiber coated with sio2 nanoparticles,” IEEE Sensors J. 15(6), 3424–3428 (2015). [CrossRef]

13. W. Wu, Y. Cao, H. Zhang, B. Liu, X. Zhang, S. Duan, and Y. Liu, “Compact magnetic field sensor based on a magnetic-fluid-integrated fiber interferometer,” IEEE Magn. Lett. 10, 1–5 (2019). [CrossRef]

14. D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. 21(23), 4208–4213 (1982). [CrossRef]

15. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66(4), 311–320 (1976). [CrossRef]

16. J. N. Blake, B. Y. Kim, H. E. Engan, and H. J. Shaw, “Analysis of intermodal coupling in a two-mode fiber with periodic microbends,” Opt. Lett. 12(4), 281–283 (1987). [CrossRef]

17. R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007). [CrossRef]

18. R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital image processing using MATLAB (Pearson Education India, 2004).

19. Z.-J. Zhang and W.-K. Shi, “Eigenvalue and field equations of three-layered uniaxial fibers and their applications to the characteristics of long-period fiber gratings with applied axial strain,” J. Opt. Soc. Am. A 22(11), 2516–2526 (2005). [CrossRef]

20. M. Monerie, “Propagation in doubly clad single-mode fibers,” IEEE Trans. Microwave Theory Techn. 30(4), 381–388 (1982). [CrossRef]

21. D. M. Nguyen, S. Blin, T. N. Nguyen, S. D. Le, L. Provino, M. Thual, and T. Chartier, “Modal decomposition technique for multimode fibers,” Appl. Opt. 51(4), 450–456 (2012). [CrossRef]

22. T. Alieva, M. J. Bastiaans, and M. L. Calvo, “Fractional transforms in optical information processing,” EURASIP J. Adv. Signal Process. 2005(10), 920687 (2005). [CrossRef]

Analytical and experimental study on a bent abrupt taper

Abstract

1. Introduction

2. Theoretical study of bent taper profiles

2.1 Analytical model

2.2 Matrix implementation of the coupling equations

3. Modeling and simulation results

3.1 Simulation parameters and assumptions

3.2 Power evolution and modified coupling coefficients

3.3 Perturbation function variations

4. Experimental results and model validation

5. Conclusion

Funding

Disclosures

References

Cited By

Figures (10)

Equations (19)

Optics Express