1. Introduction

Acousto-optic (AO) interaction in optical fibers, whose operation principles are known since the 90s, has been exploited over the years for the development of dynamic and reconfigurable all-fiber devices. In the literature, one can find filters, equalizers, frequency shifters and switches based on the interaction between an acoustic mode –either flexural, longitudinal or torsional mode–, and optical modes propagating in the fiber [1–3].

AO coupling has also been exploited as a technique to investigate the properties of optical fibers [4, 5]. Recently, time-resolved in-fiber acousto-optics was utilized for the characterization of axial diameter variations –in the order of nm– along an optical fiber [6], and to analyze refractive index changes produced by an optical pump in doped fibers [7].

With the advent of new strategies to develop optical transmission systems with larger capacity based on space-division multiplexing (SDM), few-mode fibers have gained relevancy again. A two-mode optical fiber provides six spatial channels in a single fiber [8]. Accurate characterization of the propagation properties of the different fiber modes is essential for the design of SDM schemes. Several methods for the characterization of few-mode fibers have been proposed. Most of them give information about the difference in group index or chromatic dispersion between the guided modes [9, 10]. Very few methods for measuring difference in modal effective (phase) indices have been reported so far [11, 12]. Differences in modal effective index is a key property for few-mode fiber systems because it governs the intermodal coupling. In this work, we explore the use of AO in optical fibers as a non-interferometric technique for a complete characterization of two-mode optical fibers.

2. Basis

An acoustic wave that propagates along an optical fiber creates a travelling periodic refractive-index perturbation that can cause coupling between co-propagating modes. When the acoustic wave is a flexural mode, the coupling occurs between modes with symmetric and antisymmetric field distribution across the fiber: coupling between the fundamental LP₀₁ mode and modes of the series LP_1m (m = 1, 2,...) is enabled. Maximum coupling between two modes is achieved at the optical wavelength at which the phase-matching condition is satisfied, i.e., when the acoustic wavelength matches the beatlength between the two optical modes. The resonance wavelength, λ_R, is given by,

Figure 1 shows the LP₀₁-LP₁₁ modal index difference as a function of wavelength, calculated for a silica optical fiber with 125 μm cladding diameter, and different values of NA and cut-off wavelength (λ_c). We assumed here a simple, step-index fiber geometry surrounded by air. The exact modes for such a three-layer fiber have been detailed in [13]. In all cases, Δn shows a maximum centred close to the cut-off wavelength. Obviously, for wavelengths below cut-off the LP₁₁ mode is guided by the core, while it has evolved to a cladding mode for wavelengths above cut-off. The two guiding regimes for the LP₁₁ mode are at the origin of the different slope sign of Δn at both sides of the maximum. It can remarked that the two characteristic parameters of the fiber, i.e. NA and λ_c, produce changes in the curve Δn(λ) in different ways, at least within a given range. While a change of NA introduces a vertical shift, increasing Δn for all wavelengths, a change of λ_c shifts the wavelength at which Δn is maximum. It is worth to note that a small change of NA leads to a remarkable increase of Δn. Therefore, NA and λ_c of a two-mode optical fiber can be determined accurately and without ambiguity from measurements of Δn performed in a broadband wavelength range around the LP₁₁ cutoff wavelength.

 

Fig. 1 Theoretical calculation of the LP₀₁-LP₁₁ modal effective index difference, Δn, for different step-index silica optical fibers. (a) Fibers with 1.28 μm cutoff wavelength, and different NA (indicated in the figure), and (b) fibers with NA = 0.12, and different cutoff wavelengths (indicated in the figure). In all cases, the cladding diameter was 125 μm.

Download Full Size | PDF

On the other hand, if the dependence of Δn with the optical wavelength is known with accuracy, it is then a simple matter to obtain numerical derivatives of Δn with respect to wavelength. Hence, the group index difference, Δn_g, and the chromatic dispersion difference, ΔD, of the fiber modes can be investigated since,

3. Experimental results and discussion

3.1 Experimental arrangement

Figure 2 shows a diagram of the experimental setup. Flexural elastic waves were generated by a piezoelectric transducer (provided by Morgan Advanced Ceramic), and were launched along an uncoated section of the fiber under test via an aluminium horn. The piezoelectric transducer was able to operate up to 10 MHz, and it was driven by an RF signal generator. The fiber was illuminated using an unpolarized broadband light source. The spectrum of the light transmitted through the fiber was recorded with an OSA. At the input of the fiber section, a mode stripper was inserted to ensure that only light carried by the LP₀₁ mode enters the interaction region. A second mode stripper was inserted at the end of the interaction region to remove the light carried by the LP₁₁ mode that was transferred from the input LP₀₁ mode by the elastic wave. In this way, coupling between the two modes leads to a notch in the spectrum of the light that remains guided by the core. The notch can be tuned by changing the wavelength of the acoustic wave, Λ, which can be done by simply changing the frequency of the RF signal generator, Ω. The relationship between the frequency Ω and wavelength of the elastic wave Λ is determined by the dispersion relation of the flexural mode. In a first experiment, we investigated the dispersion properties of the elastic waves propagating along the fiber. Experimental measurements were performed using an optical vibrometer. We also calculated theoretically the dispersion of the fundamental flexural elastic mode propagating in a silica optical fiber following the methods described in [14]. Both results are shown in Fig. 2(b), with good agreement between theory and experimental data.

 

Fig. 2 (a) Experimental arrangement: BLS: broadband light source; MS: mode stripper; FUT: fiber under test; T: piezoelectric transducer. (b) Wavelength of the flexural elastic wave on the fiber as a function of its frequency. Dots are experimental measurements. Solid line is the theoretical calculation for the fundamental flexural elastic mode propagating in a 125 μm diameter silica optical fiber

Download Full Size | PDF

3.2 Modal effective index difference

We investigated the properties of two commercially available optical fibers: Corning SMF-28e and SM2000 (commercialized by Thorlabs). The nominal values of the fiber characteristic parameters are included in Table 1. Figures 3(a)-3(b) show the resonance wavelength as a function of frequency for different couplings between the fundamental LP₀₁ mode and LP_1m modes.

Table 1. Nominal characteristics of the fibers and best-fit parameters

View Table

 

Fig. 3 (top) Resonance wavelength vs. frequency of the elastic wave, for different resonances. Dots are experimental data and solid line is a guide to the eye. (bottom) Modal indices difference vs. wavelength. Solid lines show the results obtained from the experiment. Dashed lines are theoretical calculations assuming a step-index profile. FUT: (a), (c) SMF-28e; (b), (d) SM2000.

Download Full Size | PDF

Notice that all the LP_1m modes, except the LP₁₁ mode for wavelengths below its cutoff, are cladding modes within the wavelength range of the experiment. For a given frequency, the lower is the order of the LP_1m mode, the shorter is the resonance wavelength. When coupling to cladding modes, the resonance wavelength decreases as the frequency of the elastic wave increases, as expected [2]. However, the trend is more complex for the LP₀₁-LP₁₁ coupling as a result of the two guiding regimes of the LP₁₁ mode, which depends on the optical wavelength. For some frequencies, a particular feature is observed experimentally: the LP₀₁-LP₁₁ coupling gives rise to two notches in the spectrum. This happens because phase-matching is satisfied simultaneously at two different wavelengths.

Using Eq. (1) and the dispersion relation of the flexural elastic mode, the modal phase index difference was obtained as a function of wavelength from the experimental data shown in Figs. 3(a)-3(b). The results are shown in Figs. 3(c)-3(d). Additionally, theoretical calculations of Δn were performed for a 125μm fiber with a step refractive index profile. By varying NA and λ_c in the model, numerical calculations were fitted to experimental data. The numerical results that best fit the experimental data are included in Figs. 3(c)-3(d). Table 1 gives the resulting best-fit parameters. In general terms, the step-index model describes properly the experimental results of Δn. The fiber parameters provided by the model for the SM2000 fiber are in good agreement with the nominal data provided by the manufacturer. However, for the SMF-28e the agreement in not so good, the NA differs in about 13% and the cut-off wavelength obtained experimentally is remarkably larger than the nominal value. We investigated the origin of this apparent disagreement in the cut-off wavelength. We performed an auxiliary experiment to investigate the cutoff wavelength of the fiber using the conventional method [15]. We used an halogen lamp and an OSA to measure the transmission spectrum. This experiment was done with sections of fiber of 1 m in length. We found that the LP₁₁ mode was guided up to 1.4 μm when the fiber was held straight. However, we observed that a small fiber curvature shifted the cutoff wavelength down to the nominal value, which indicates that the nominal value provided by the manufacturer is a practical parameter, but not the strict cutoff wavelength.

3.2 Group index and chromatic dispersion difference

The group index difference, Δn_g, and the chromatic dispersion difference, ΔD, between the LP₀₁ and LP_1m modes were obtained from the modal effective index difference, Δn, using Eq. (2). The 1st and 2nd derivatives of Δn with respect to λ were found by fitting a proper function to the experimental measuremets of Δn, that was subsequently derived (the experimental data were fitted to polynomial functions by segments). The results corresponding to LP₀₁-LP₁₁ for the two fibers are shown in Fig. 4. For both fibers, there is a wavelength range far below cutoff, where Δn_g is negative, meaning that the group index of the LP₁₁ mode is larger than that of the LP₀₁. On the other hand, the dispersion rises as the wavelength approaches the cutoff of the LP₁₁ mode. Theoretical calculations of Δn_g and ΔD for a 125 μm diameter step-index fiber with the fiber parameters that resulted from the best-fit of Δn (see Table 1) are also included. In general terms, the agreement between experimental and modelling results is rather good. However, there is a remarkable disagreement in the chromatic dispersion difference near the strict cutoff wavelength, where the theory predicts a sharper peak. We believe that the origin of such disagreement lies in the model that has been assumed for the fiber that involves a perfect step-index profile.

 

Fig. 4 (a) LP01-LP₁₁ group index difference and (b) chromatic dispersion difference vs. wavelength. Solid lines are obtained from experimental measurements. Dashed line are theoretical calculations. Vertical dotted lines indicate the cut-off wavelength values obtained in section 4.1.

Download Full Size | PDF

The group index difference between the LP₀₁ and LP_1m modes with m = 2 - 5 is shown in Fig. 5 for the SMF-28e fiber. Within the wavelength range of our experimental arrangement, the LP_1m (m > 1) modes are cladding modes. It is known that the cut-off of a guided mode affects the guiding properties of higher order modes whose fields have the same angular order. Using the acousto-optic method, we were able to investigate the perturbation of the group index in LP_1m modes (with m > 1) as a result of the cut-off of the LP₁₁ mode. Figure 5(a) shows the experimental result. For comparison, the corresponding theoretical calculation is shown in Fig. 5(b). Δn_g shows a bent at wavelengths near the LP₁₁ cut-off wavelength, which is more clear for LP₁₂ and LP₁₃ modes.

 

Fig. 5 LP01-LP_1m (m = 2 - 5) group index difference vs. wavelength. (a) Experimental result, and (b) theoretical calculations. FUT: SMF-28e.

Download Full Size | PDF

3.3 Vector modes: TE, TM, and hybrid

The fine mode structure of LP_1m modes can also be investigated. It is known that the LP₁₁ mode results from the superposition of three vector modes: TE₀₁, TM₀₁, and two HE₂₁ modes with orthogonal polarizations. In conventional fibers, the triplet of modes are nearly degenerated, so they propagate with very similar phase velocity. As a result, when a LP₁₁ mode is excited by the elastic wave, the three notches caused by the coupling from the fundamental HE₁₁ mode are usually overlapped, giving rise to just one notch. However, the bandwidth of the notches depends on the length of the elastic grating: the longer is the elastic grating, the narrower are the notches. Therefore, the resolution of the characterization technique can be adjusted and the three notches can be observed separately provided that interaction length and uniformity of the fiber, are sufficient. So, we repeated the experiments with 50 cm of SM2000 fiber. Inset of Fig. 6(a) shows an example of the spectrum recorded when the elastic wave was adjusted to excite the LP₁₁ mode. Now, three notches that result from the coupling to the triplet of modes modes are observed. Figure 6 shows Δn as a function of wavelength for the three modes. As expected, the splitting is larger as the wavelength approaches the cut-off wavelength. Far from cut-off, the modes are nearly degenerated. Even so, the AO technique can distinguish the three notches up to the point where the difference in modal effective index between the vector modes was as small as 3 × 10⁻⁶.

 

Fig. 6 Modal index difference vs. wavelength. (a) For wavelengths near cut-off and (b) for wavelengths well below cutoff. Inset shows a transmission spectrum. FUT: SM2000. Interaction length: 50 cm

Download Full Size | PDF

4. Conclusions

Accurate mode characterization of two-mode optical fibers can be performed using in-fiber acousto-optic interaction. The effective index difference between modes was measured in a broad wavelength range around the LP₁₁ cut-off wavelength. NA and λ_c were determined precisely and without ambiguity by fitting the experimental data to theoretical calculations based on a step-index model. Modal group index difference and modal chromatic dispersion difference were also obtained from the experiment, and were compared to theoretical calculations showing good agreement. Additionally, the fine mode structure of LP_1m modes, and other interesting features were also investigated by means of this technique.