1. Introduction

Few-mode fibers have been studied for several years. The first applications were based on using one of the guided higher-order modes, which can have properties that are not obtainable for the fundamental mode, for applications such as dispersion compensation [1], large effective area [2], and anomalous dispersion below 1200 nm [3]. Recently, mode-division-multiplexed transmission in few-mode fibers has attracted a lot of attention as a means of increasing the transmission capacity of a single fiber [4]. Methods for characterization of few-mode fibers are therefore of interest. A method for measuring the difference between effective group indices based on measuring interference pattern in short lengths of fiber has been known for long [5]. In this paper we show a new method for measuring difference in effective phase indices based on measuring the change in interference pattern when the fiber is stretched. The difference in effective phase indices is an interesting property for few-mode fibers because it governs the amount of coupling between the modes.

2. Experimental setup and measurements

The experimental setup for measuring the effective refractive-index difference is shown in Fig. 1 . It consists of a broadband light source and an optical spectrum analyzer (OSA). Both ends of the few-mode fiber under test (FUT) are fusion spliced together with single-mode pigtails. The splices are made with large offset in order to excite the higher order modes considerably. The fiber used in the experiments was a fiber designed for support of LP₀₁ and LP₁₁. It had a simple step-index profile with a core diameter of 19 μm, and an NA of 0.12 [6]. The theoretical effective refractive indices are obtained by solving the scalar wave equation using a finite element mode solver for the fiber index profile and are shown in Fig. 2 .

 

Fig. 1 The experimental setup. The fiber under test is fusion spliced to single-mode pigtails, and glued on two places to a translation stage and a force sensor. Optical power spectrums are recorded as a function of axial fiber stretch induced by the stage.

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Fig. 2 The numerically calculated effective refractive indices found by solving the scalar wave equation for the specific fiber profile.

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The few-mode fiber is glued at two points, where the polymer coating has been removed; one point to a micrometer-controlled translational stage and the other on a stationary stage equipped with a force sensor (model: Thorlabs FSC102). The force sensor is used to record the stress-strain curve, which is used for two purposes. The first is to verify that the stress-strain curve is linear, proving that the fiber does not slip, i.e. that the fiber stretches by the same amount as the translation stage is moved. Secondly, we also compare the slope of this curve to the expected value calculated from the Young’s moduli of the fiber, including the primary and secondary coatings. These two values agree implying that the exact, unknown elasticity of the glue is insignificant.

The translational stage is moved in 0.05 mm steps, and for each step the OSA is scanned from 1530 nm to 1570 nm in 0.04 nm steps, using 0.2 nm resolution. The data presented in this communication corresponds to two separate pieces of the same fiber, which are 46 cm and 52 cm long, respectively, and are from now on referred to as measurement 1 and measurement 2. The stress is applied for fiber lengths 21 cm and 42 cm in the two cases. The total length increases in the two measurements are 1.9 mm and 3.3 mm, corresponding to a strain of 0.9% and 0.8%.

The broadband light source emits light in a well-defined polarization state. The polarization was varied with polarization control and observed to modify the optical spectrum. However, during the experiments the polarization was kept fixed and the optical spectrum did not change between subsequent scans, implying that the polarization was stable throughout the experiment.

3. Results and discussion

Figure 3 shows the transmitted power as a function of both wavelength and strain for measurement 1. The oscillations are due to interference between propagating modes. For example, summing two fields with different propagation constants gives for a specific wavelength a power fluctuation proportional to$2A_1{A}_2\cos (\Delta \beta L)$, where $A_i$ are the field amplitudes, $\Delta \beta =2\pi \Delta n_{eff}/{\lambda }_0$is the difference in propagation constants, and $L$is the fiber length.

 

Fig. 3 The transmitted power as a function of wavelength and axial fiber stretch. The intensity is oscillating both as a function of wavelength and fiber stretch.

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As the fiber is stretched, the relative phase $\Delta \beta \cdot L$ between the two modes will also change. This is mainly due to the direct length increase; however$\Delta \beta =\Delta \beta (L)$depends also on $L$ due to strain-induced refractive-index changes. Assuming the length increase $\Delta L$to be small, the Taylor expansion of the phase to the first order gives

Alternatively, considering $\Delta \beta =\Delta \beta (\lambda )$as a function of wavelength for a specific strain and Taylor expanding $\Delta \beta (\lambda )$ around ${\lambda }_0$ gives the period of the oscillation as a function of wavelength [5]

These considerations explain the qualitative behavior of the transmitted intensity in Fig. 3, showing oscillations both as a function of fiber stretch and wavelength. Quantitatively, the measured $\Lambda$(Fig. 3) agrees well with Eq. (4) when using the numerically calculated group delay, which is also obtained by solving the scalar wave equation.

As we are interested in $\Delta n_{eff}$, we fit a sine function to the transmitted power vs. $\Delta L$in order to extract the effective refractive-index difference at a given wavelength with the help of Eq. (3). The $\Delta n_{eff}$obtained in this way is shown in Fig. 4(a) as a function of wavelength together with the numerical solution obtained from the fiber index profile. The measured effective refractive-index difference agrees well with the LP₀₁-LP₁₁ interference. In order to reduce the small fluctuations in the signal, a cubic spline smoothing of the experimental data is performed. Comparison with the model at an expanded vertical scale is shown in Fig. 4(b) as the solid line. The dark shaded area corresponds to the estimated error (one times standard deviation).

 

Fig. 4 (a) The measured effective refractive-index difference as a function of wavelength for the first FUT shown as the solid curve. Dashed lines correspond to the numerically solved differences in effective refractive-index between different modes. LP₀₁-LP₂₁ and LP₀₁-LP₀₂ are also calculated but they are outside the figure limits. (b) A close-up of the data in (a) (solid curve), and additionally the result for the second FUT with two distinct interferences (dotted and dash-dotted curves, corresponding to LP₀₁-LP₁₁ and LP₁₁-LP₂₁, respectively). The shaded regions represent the estimated one standard deviation error.

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For the second measurement, the transmitted power shows more complicated oscillations. As shown in Fig. 5 , a sum of two sine functions fits the observed data well. The additional frequency is probably due to excitation of the mode with the third largest effective refractive index, LP₂₁, in addition to the LP₀₁ and LP₁₁ modes (see Fig. 2). This is assigned to a difference in launch conditions due to difficulty in reproducing the misaligned fusion splices exactly. Comparison of the results (dashed and dash-dotted curves in Fig. 4(b) with the model shows that the extra oscillations might indeed originate from the LP₁₁-LP₂₁ coupling. On average the results deviate from the theoretical curves by 2.6% for the first measurement and 4.9% and 1.8% for the second measurement.

 

Fig. 5 The transmitted power as a function of the fiber strain for the second measurement at 1537 nm. The solid line shows the fit by a sum of two sine functions.

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Previous experiments close to cutoff in two-moded fibers, where the transmitted power has been measured as a function of fiber strain, have shown deviations from the approximation that $\Delta \beta$ does not depend on strain [7–9]. These nonlinear effects should, however, be small in the present investigation as the relevant wavelengths used here are quite far from the cut off wavelengths, which are 2713 nm and 1714 nm for LP₁₁ and LP₂₁, respectively. In accordance, no sign of such effects are seen in the observed data. It is interesting to note that in [9], it was found that the observed period was two times larger than predicted by Eq. (3) even when operating far from the cut off wavelength. We do not observe a similar discrepancy between the theory and measurements.

There are several possible future improvements to the measurement technique presented here. First, using a different force sensor would allow a measurement with higher loads and thus provide data for longer absolute length increases. This would allow better accuracy in the fitting, but eventually the possible nonlinear effects discussed above might become problematic. Using a longer fiber would also allow larger absolute length differences; however, the oscillation period as a function of wavelength must be kept so large that the oscillations can be clearly resolved at the available resolution of the OSA. An alternative approach would be to use a (narrow band) laser at the wavelength of interest or a tunable laser to probe different wavelengths, both together with a power meter or a photo diode.

The method will work well for fibers, where only few modes propagate. When the number of modes increases, the analysis becomes more challenging. The maximum number of solvable modes depends on the experimental parameters, such as wavelength range and maximum displacement, and also on the actual effective refractive-index differences between the modes. For example, if two interferences have the same beat length at a given wavelength, the analysis is not simple. Furthermore, the method cannot alone label, which modes the measured interferences correspond to. In order to achieve this, a comparison to modeling as done in the present investigation, is needed. Alternatively, the modes in play can be identified by combining the current method with techniques for mode analysis, such as spatially and spectrally resolved imaging [10].

4. Conclusion

We have shown that the effective refractive-index difference of a few-mode fiber can be determined by measuring the transmitted power as a function of axial fiber stretching. The first measurement showed the interference between LP₀₁ and LP₁₁, whereas both this and also LP₁₁-LP₂₁ were observed in the second measurement. This difference is assigned to variations in the launch conditions in the misaligned fusion splices. In both cases, the mean deviation from the calculated effective refractive-index difference was, however, only a few percent. This shows that this technique can be used to test the quality of the modeling and/or to reliably measure the effective refractive-index difference, at least when operating far from the cut off wavelengths of the relevant modes. In addition, several improvements for the measuring technique were discussed.