1. Introduction

Over the last decade, microstructured optical fibers (MOFs) have been a subject of many theoretical and experimental investigations. The interest in the fibers is mainly motivated by flexibility of engineering their structures enabling control over their optical properties. Until now, tailoring of such properties as dispersion [1], nonlinearity [2], beam shape [3], and birefringence [4] was demonstrated and specific fiber structures optimized for particular applications, e.g., endlessly single-mode transmission [5], generation of ultra-high power light [6], and chemical sensing [7] were elaborated.

In this paper, we focus on a particular class of MOFs – the suspended-core fibers (SCF). Such fiber consists of three air holes/channels surrounding the core suspended on three thin struts. Typically, a diameter of the light-guiding core is of the order of a micrometer, which makes light to penetrate the channels in a form of the evanescent field [8]. The channels may be filled by a foreign medium: gas or liquid. This enables studies of the interaction between light and the medium under specific experimental conditions and opens up an avenue for applications of this type of fibers in chemical and biological sensing [8]. However, in order to obtain quantitative information about the filling medium, e.g., concentration of a specific compound, a detailed knowledge of the fiber properties is required. Recently, we have investigated a number of optical properties of SCFs such as mode structure, numerical amplitude, etc [9]. Herein, we concentrate on another important property of the fiber, the birefringence.

Breaking rotation symmetry of a fiber structure is one of the techniques for introducing birefringence in optical fibers. However, it was demonstrated theoretically that only fibers with a rotational symmetry of the order lower than three may reveal such structural birefringence [10]. Thus, a fiber like SCF should not possess any birefringence. Nonetheless, in reality, deviations from the perfect structures are always observed and result in residual birefringence of optical fibers.

In optical fibers two parameters may be used to describe birefringence: the phase and the group birefringence. The two properties are closely related to each other and it is generally possible to calculate one of them based on spectral dependence of the other [11]. The group birefringence is also related to other important characteristics of a fiber – the differential group delay and polarization mode dispersion [11].

There are many techniques used to determine birefringence and/or polarization-mode dispersion of optical fibers. One may, for instance, investigate the change of a polarization state of light departing a fiber as a function of its length (the cutting-end method) [12]. This approach, however, requires destruction of the fiber. A different technique is based on mechanical coupling of different polarization modes. In such an approach the state of light at the fiber end is monitored as a function of the position of a mechanical stress-point. i.e., the length of co-propagation of orthogonal polarization modes [13]. Both these techniques enable to determine phase birefringence of a fiber. Also the white-light (low-coherence) techniques can be applied for measurements of the polarization-mode dispersion [14]. The most common approach utilizes an interferometer which compensates the optical-path difference introduced by the fiber at certain wavelengths and enables to determine the dispersion [15].

In this paper we use another approach to analyze birefringence, - the wavelength-scanning method (WSM) [16]. It combines different aspects of all the above mentioned techniques. In the method a fiber placed between two crossed polarizers is illuminated by a broadband light source. The first polarizer enables to excite orthogonal polarization modes in the fiber, while the second polarizer allows one to observe interference between the modes. Application of a broadband light makes it possible to observe the interference in a spectral domain based on relation [17] for the group birefringence:

Herein, in order to retrieve information about birefringence of SCF, data recorded using WSM are analyzed using the continuous wavelet transform. It allows us to determine the birefringence of the fundamental mode but also to determine the birefringence of higher-order modes, which, to our best knowledge, was done for the first time. We show that the higher-order mode-birefringence strongly depends on the wavelength, which is an obstacle in its determination using traditional approaches, e.g. the Fourier transform. A possibility of measuring birefringence of many modes is important in analysis of few-mode optical fibers, in particular SCFs.

2. Experimental apparatus and results

Our experimental setup is shown in Fig. 1 . A halogen quartz lamp (StallarNet SL-1) serves as a broadband light source and a fiber optic spectrometer (AvaSpec-3648) is used for transmission measurements in the 500-1000-nm range with a resolution of 0.5 nm. To couple broadband light into the fiber, we use a system of several spherical mirrors and optical fibers denoted in Fig. 1 as a beam-shaping system. The investigated fiber is situated between two linear polarizers. The first linearly polarizes the beam and excites different polarization modes in the fiber (incident polarization at 45° with respect to the optical axis of the fiber), while the second is used to observe the mode interference. The axes of both polarizers are crossed, which provides the best signal-to-noise ratio, note, however, that other angles are also possible and, in some situations, are worth considering (see below).

 

Fig. 1 Experimental setup for birefringence measurements using the wavelength scanning method.

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The SCF we examine in this work (Fig. 2 ) was fabricated in the Laboratory of Optical Fibre Technology at the Marie Curie-Sklodowska University in Lublin, Poland [19]. This silica fiber possess a small, triangular-like silica core (800 nm diameter of the inscribed circle) attached to three thin struts (length of 9 µm and a thickness of 70 nm) to the solid fiber cladding. The fiber supports propagation of a number of modes within a range from 500 nm to 1000 nm. The SCF structures which have three-fold rotation symmetry are not birefringent and all their modes either have the same symmetry or exist in degenerated doublets. Thus any imperfection of the structure which breaks the symmetry will lead to a very high birefringence between higher-order modes [10].

 

Fig. 2 Scanning electron microscope images of the cross section of the examined fiber: a) whole fiber (80 μm diameter), b) core region (800 nm diameter).

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A typical signal measured with the WSM is shown in Fig. 3 . It was obtained with the spectrometer detecting the light exiting the 290-mm long SCF and interfering at the output polarizer. Normalization of the spectra eliminates influence of spectral properties of the fiber, halogen lamp, spectrometer and other optical devices, hence the recorded spectra demonstrate only the mode interference. Besides the clearly visible oscillations of about 25-nm periodicity, the spectrum shown in Fig. 3 reveals higher-frequency components of the signal, most distinctly at longer wavelengths. It is important to note that this high frequency modulation is stable in time, hence is not noise but, as discussed in Sec. 3.3, it originates from group birefringence of the fiber.

 

Fig. 3 Typical transmission spectrum obtained for a 290 mm long piece of a suspended-core fiber.

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3. Data analysis

3.1. Continuous wavelet transform

A successful technique that typically enables to distinguish different components of multi-frequency signal is the Fourier transform. However, application of the technique in the considered case is not possible due to the strong dependence of the oscillations on the wavelength of light (see Fig. 3). The case, however, may be conveniently analyzed with the help of the Continuous Wavelet Transform (CWT) [20]. Note that CWT has already proved its usefulness for analysis of liquid crystal birefringence [21], but to our best knowledge, has never been applied for investigation of optical fibers.

Continuous wavelet transform of a given function $f\left(x\right)$ is expressed as:

Function φ(x) is the, so-called, mother wavelet which is the basis for constructing the dilated and translated wavelets by using parameters a and b corresponding to the frequency and the central position of the wavelet, respectively. Choosing the appropriate wavelet for a given application is not trivial. In our case of fiber birefringence we apply the Morlet wavelet, which is a Gaussian function modulated by a plane wave [20]:

Figure 4(a) presents the two-dimensional wavelet spectrum of the data shown in Fig. 3 (note that the plots were adapted for the considered case, i.e., wavelet frequency was replaced by its period and position of the wavelet is labeled as a wavelength). The spectrum reveals three well visible traces marked as A, B, and C. The strongest trace corresponds to the oscillation period of about 25 nm that does not significantly vary with the wavelength. This result may be almost completely reproduced by application of the Fourier transform [Fig. 4(b)]. However, Fig. 4(a) reveals other important features. For example, the other trace marked as B is visible in the range of 750-900 nm but its period decreases from ~10 nm for shorter wavelengths to ~2 nm for longer wavelengths. The last trace, marked as C, is observed for the 600-750 nm range. Its period begins at ~3 nm at 550 nm and rises until ~10 nm at 650 nm where it start decreasing back to ~2 nm at 750 nm. Since none of these traces is observed using the Fourier transform, Fig. 4(a) demonstrates well the potential of the CWT method.

 

Fig. 4 (a) Real part of the wavelet spectrum of the signal shown in Fig. 3. Three distinct visible traces are marked as A, B, C, (b) FFT spectrum of the same signal. Besides of a large zero-frequency component, only the primary birefringence mode is visible (at frequency close to 0.04 nm⁻¹). Other features relevant for CWT trances B and C should be located in the 0.1-0.5 nm⁻¹ frequency range, but are not visible.

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3.2. Data filtering

In order to identify the signal associated with the fiber from the overall background, the spectrum shown in Fig. 4(a) is analyzed by searching for local extremes. It allows one to obtain two-dimensional point plot [Fig. 5(a) ], which, however, due to simplicity of the applied identification technique, contains many points not associated with the fiber properties. The signal is hence processed further by application of three main criteria. The first two are set by the resolution of the spectrometer which limits the lowest and the highest periods that can be determined. Somehow arbitrary we assume that no less than four points per period are required to detect the oscillations (in fact, two points per period are enough, but applying a more restrictive criterion increases credibility of the results). Thus the applied limits are: 2 nm (four times the interferometer resolution of 0.5 nm) and ~100 nm (about one quarter of the 450 nm interferometer bandwidth). The third criterion is related to the threshold value of $\mathrm{Re}\left[CWT\left(a,b\right)\right],$ set to 10% of the maximum value available with the normalized spectrum. It needs to be stressed that the threshold has to be set very carefully as too high threshold causes loss of data points, while too low threshold level leaves too much noise.

 

Fig. 5 (a) Local extremes of the real part of the wavelet spectrum without filtering, (b) Signal filtered out using the criteria described in the text. Each point of the plots is related to an extremum in the CWT spectrum in Fig. 4(a) and defines a pair (λ, Δλ). The pairs group into bent lines which reflect the systematic wavelength dependence of the group birefringence of the resolved modes

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The raw data and the data processed using the criteria discussed above are presented in Fig. 5. Since the applied filtering technique is rather simple, some points originating from the fiber properties may be lost. Nonetheless even such rough filtering yields valuable information about fiber (see below). In particular, the points depicted in Fig. 5 group in a way which reveals dispersion of the resolved modes. Such analysis is was not possible with the regular standard FFT method (compare with Fig. 4(b)).

3.3. Group birefringence

The data shown in Fig. 5(b) may be recalculated into the group birefringence based on Eq. (1). In order to verify that they are associated with the fiber properties and are not caused by any apparatus artifacts, additional cross-checks are performed. The most basic test is to measure pieces of the fiber of different lengths. If certain points are related to the fiber properties, they should lay along a given curve after length normalization. At the same time the points which are not associated with the birefringence are just randomly scattered over the whole plot area and represent a uniform background. The purpose of this procedure is to show that obtained points are in fact originated by the fiber birefringence. In case of typical birefringence measurement, such step is not required and analysis of single interferogram is sufficient.

Other test consisted in modification of the measurement geometry, i.e. rotation of the optical axes of the polarizers. Such change deteriorated the contrast of the observed signal, but also allowed us to observe signals for different light coupling conditions. This test was very important, since it revealed birefringence of the higher-order modes where proper polarizer arrangement was not easy to accomplish.

The data taken with different fiber lengths and polarization arrangements are brought together in Fig. 6 . It is clearly visible that most of the collected points form four strongly wavelength-dependent traces. Points originating from the birefringence of the fundamental modes lay on a curve which rises monotonously with $\lambda$ from about _{$5\times {10}^{-4}$} at 600 nm to _${10}^{-3}$ at 900 nm. This is a very high group birefringence for the fiber which was fabricated without intentionally introduced birefringence. Other three traces are connected with higher-order modes. Two of them are rising with _$\lambda$ until ~800 nm where they reach their maximum values, $6\times {10}^{-4}$ and _${10}^{-3},$ respectively. For longer wavelength their birefringence decreases. The behavior of the curves around 600-650 nm and 900-950 nm may suggest that the group birefringence changes its sign in these regions and for certain wavelengths is equal zero. However, since in such measurements only the absolute value of the group birefringence is measured, the sign of birefringence is not known. The fourth trace starts at about 725 nm and ranges as far as near 950 nm.

 

Fig. 6 Group birefringence values collected from measurements done for different arrangement of polarizers and different fiber lengths. Dashed lines limit the birefringence range accessible in measurements using our spectrometer.

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It is worth noting that the resolution of a spectrometer limits the range of the birefringence that can be measured. High birefringence yields oscillations with a short period [see Eq. (1)] and requires a spectrometer with high resolution. The range of birefringence that can be measured using a spectrometer of a given resolution changes with the wavelength of light and the length of a fiber. The limits imposed by the spectrometer in presented measurements are marked in Fig. 6 by two dashed lines.

4. Conclusions

We described the approach for determination of the birefringence dispersion of few-mode optical fibers. It is based on the wavelength scanning technique and continuous wavelet transform. Application of these techniques allowed us to measure group-birefringence dispersion of the suspended-core fiber in a wide spectral range. This was done despite of a very strong wavelength dependence of the birefringence signals. This approach can be improved by more careful data filtering for better compromise between the number of eliminated data points and noise. The range of birefringence values that can be measured is limited by the range and accuracy of the spectrometer.

To the best to our knowledge, group birefringence of the higher-order modes has been measured for the first time. We expect the described method will prove useful for determination of polarization modes associated with the measured signals and for verification of various models of the suspended-core fibers that we planned to used for chemical sensing.