## Abstract

We show that the spectral interferometry method and the lateral point-force method used up to now to measure spectral dependence of the group and the phase modal birefringence in highly birefringent fibers with linearly polarized eigenmodes, can be after some modifications extended for the class of spun highly birefringent fibers with elliptically polarized modes. By combining the two methods, it is possible to determine spectral dependence of the group and phase elliptical birefringence in spun highly birefringent fibers. Moreover, if the fiber spin pitch is independently measured, the spectral dependence of ellipticity angle of polarization eigenmodes as well as the built-in linear phase and group birefringence, can be also obtained using the analytical relations between the parameters of spun and non-spun fibers. We demonstrate the effectiveness of the proposed approach in spectral measurements (700-1600 nm) of the spun side-hole and microstructured highly birefringent fibers with different birefringence dispersion and spin pitches ranging from 4.1 to 200 mm.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Spun fibers have been known since early 1980s [1,2] and originally attracted attention due to twist induced circular birefringence and possible applications in magnetic field and electric current sensing [3]. Since recently the field has been gaining increasing interest as fabrication of fibers with very small spin pitches became technologically feasible [4–6]. Moreover, new numerical methods have been developed for modelling spun fibers and analyzing complex twist-induced effects [6–8]. Spun fibers have already found several interesting applications which include generation of modes with orbital angular momentum [6,9,10], fiber lasers with efficient single mode operation [11,12] and reduced polarization mode dispersion [13]. Furthermore, resonant coupling between core and cladding modes was observed in conventional and microstructured spun fibers for selected combination of wavelengths and spin pitches [14–20]. This effect may be exploited for building novel fiber optic devices, such as filters, polarizers, sensors or couplers.

Spun highly birefringent fibers (SHBFs) [3] are fabricated by spinning the fiber preform during the drawing process. The circular birefringence induced by fiber twist adds to intrinsic linear birefringence induced by core ellipticity [21], stress applying elements [22] or asymmetry of microstructured cladding [23]. This results in helical elliptically-polarized eigenmodes of the SHBF, which have orthogonal azimuths and opposite handedness. Axes of the polarization ellipses rotate along the fiber length and overlap with its symmetry axes (axes of linear built-in birefringence), while their ellipticity is determined by the ratio of linear birefringence beat length (*L _{l}*) and spin pitch (

*L*). The latter might be directly determined by microscope examination of the SHBFs in transverse illumination or calculated from the drawing parameters, namely the preform spinning velocity and the drawing speed.

_{T}There has been a number of papers presenting different approaches to characterization of the SHBFs. The most straightforward way is to measure the evolution of the polarization state of light as it propagates through the fiber [1], which may be represented as cycloidal curve on the Poincare sphere [1,24]. Its characteristics may be used to calculate the value of total elliptical birefringence as well as its linear and circular components [1]. This method requires the use of electrooptic or magnetooptic polarization modulator moved down the fiber length and is therefore applicable to fibers with relatively long beat lengths. Moreover, the measurements are conducted for a single wavelength. Other techniques for characterization of the SHBFs with small built-in linear birefringence are based on wavelength scanning [25] or the use of binary polarization rotators [26], but they are both limited to a narrow wavelength range. In [27] analytical expressions were derived which allow for determining beat length of the linear birefringence at specific wavelength from periodic variations of the output polarization state induced by a fiber elastic twist. On the other hand, the ellipticity angle of polarization eigenmodes for specific wavelength can be measured directly with the use of linear polarizers and a quarter-wave retardation plate, as it was shown in [28,29]. Then, knowing the spin pitch, it is possible to calculate the linear birefringence as well as the total elliptical birefringence. In recent work [30], the method of measurement of the linear and elliptical phase birefringence in spun birefringent fibers has been proposed using short-length fiber Bragg gratings.

To sum up, there exist numerous techniques proposed in literature for characterization of the SHBFs, but none of them provides full information on fiber birefringence, which includes the phase and group elliptical birefringence, the phase and group linear birefringence, and the ellipticity angle of polarization eigenmodes measured in a wide spectral range. In this work we demonstrate that all these parameters can be measured in the SHBFs by combining the spectral interferometry method and the lateral point-force method used up to now for characterization of highly birefringent fibers with linearly polarized modes [31–33]. We demonstrate the effectiveness of the proposed approach in measurements of the spun side-hole and microstructured highly birefringent fibers conducted in the spectral range from 700 to 1600 nm. The two fibers have different elliptical birefringence dispersion induced by their geometry and spin pitches ranging from 4.1 to 200 mm. We have also derived analytical relations linking the elliptical group modal birefringence with the linear group modal birefringence and fiber spin pitch. Moreover, we show analytically and confirm by the measurement results that in the SHBFs the ellipticity angle of the polarization eigenmodes reaches the maximum value at zero-crossing wavelength of the linear group modal birefringence.

## 2. Theoretical background

In this section we define the parameters characterizing fibers with linear and elliptical birefringence as well as the relations between them further used in our experimental approach. We start with linearly birefringent fibers, which support two orthogonally polarized fundamental modes propagating with different phase effective indices, respectively *n _{x}* and

*n*. For linearly birefringent fibers, the coordinate system is typically chosen in such a way that

_{y}*x*-polarized mode has greater effective index and therefore the linear phase modal birefringence defined in the following way

*n*significantly varies with wavelength and therefore the linear group modal birefringence ∆

_{l}*N*understood as the difference of the group effective indicescan be significantly different than ∆

_{l}*n*. The linear phase birefringence can be also characterized by a linear beat length

_{l}*L*representing the distance over which the phase shift between orthogonally polarized modes increases by 2π:

_{l}When optical fiber of perfect cylindrical symmetry (no birefringence) and length *L* is non-elastically twisted by angle *α* around its symmetry axis (during the drawing process), the linear polarization state introduced at the fiber input rotates in the helicoidal coordinate system related to the fiber with the rate *α/L,* when the light propagates down the fiber length. One can treat this rotation as the result of twist-induced circular birefringence (existing only in the coordinate system related to the fiber), which can be expressed in the following way:

*α*depending on its direction and

*L*is the spin pitch. Because of combined effects of circular and linear birefringence, the twisted fibers with initially linear birefringence support helical elliptically-polarized eigenmodes, having orthogonal azimuths and opposite handedness. Axes of the polarization ellipses rotate along the fiber length and overlap with the axes of linear built-in birefringence, while their ellipticity angle is determined by the ratio of linear birefringence beat length (

_{T}*L*) and spin pitch (

_{l}*L*):

_{T}The evolution of an arbitrary polarization state of light propagating in spun highly birefringent fiber depends on the choice of coordinate system. In the local coordinate system rotating with the fiber symmetry axes, the polarization state variation along the fiber length is represented by a circle on a Poincare sphere. The rotation axis is defined by the vector representing the elliptical eigenstate of lower effective index while the angular rotation velocity is given by:

where ∆*n’*represents the elliptical phase birefringence of the SHBF in the rotating coordinate system related to the fiber axes and is expressed by:The initial state of polarization is reproduced in the SHBF after light travels the elliptical beat length

_{e}*L’*, which in the rotating coordinate system is expressed in the following way:

_{e}For the coordinate system associated with the laboratory (not rotating with the fiber axes) the evolution of the polarization state on the Poincare sphere becomes more complex as the rotation around the polarization eigenstate vector must be combined with the rotation of the Poincare sphere itself. As a result, the evolution of the polarization state along the fiber length is represented on the Poincare sphere as a cycloidal curve [1,24] and the expression for elliptical beat length takes the form [29]:

The above relation may be inverted to obtain the linear beat length*L*:The elliptical phase birefringence Δ

_{l}*n*of the SHBF in the laboratory coordinate system may be obtained from the relation Δ

_{e}*n*=

_{e}*λ/L*, which yields:Similarly as for the linearly birefringent fibers, one can define the group modal birefringence for the elliptically birefringent fibers:It is worth to notice that in contrast to the phase birefringence ∆

_{e}*n*, the group birefringence ∆

_{e}*N*does not depend on the choice of the reference system. After performing differentiation of ∆

_{e}*n*with respect to

_{e}*λ*, a simple relation between linear and elliptical group birefringence can be derived:

*n*experiences high chromatic dispersion and as a result, the group birefringence ∆

_{l}*N*may strongly differ from ∆

_{l}*n*. In some cases ∆

_{l}*N*has different sign than ∆

_{l}*n*and crosses zero value at certain wavelength [34,35]. The above relation proves that spinning such a fiber does not change the zero-crossing wavelength of the group birefringence. Moreover, it can be shown that the ellipticity angle of the polarization eigenmodes in the SHBFs reaches the maximum value for the zero-crossing wavelength of the linear group birefringence ∆

_{l}*N*. Differentiation of Eq. (5) over

_{l}*λ*yields:Since the derivative

*dL*can be expressed as:the ellipticity angle takes the maximum value at wavelength at which the linear group birefringence Δ

_{l}/dλ*N*(

_{l}*λ*) crosses zero. This general observation is confirmed by our experimental results obtained for the spun side-hole fiber, which are presented in the following sections.

## 3. Measurement method

In this section, we propose an approach for determining spectral dependences of all relevant parameters characterizing the SHBFs, such as the phase and group elliptical birefringence, the phase and group linear birefringence and the ellipticity angle of the helical polarization eigenmodes. Using the proposed method, we have characterized two spun highly birefringent fibers whose cross sections are shown in Fig. 1. The first one was the side-hole fiber with elliptical core and two side-holes located in a direct vicinity of the core. As it was shown for the non-spun side hole fibers [36], the holes are responsible for the increase in the linear phase modal birefringence in the long wavelength range and as a result, the linear group modal birefringence changes its sign from positive to negative at certain wavelength. The second investigated fiber was the microstructured fiber with birefringence induced by a pair of large holes. It is known from earlier publications [37,38] that in the non-spun fibers of this type the phase modal birefringence increases against wavelength according to the power law function and as a result, the group modal birefringence has negative sign. In Fig. 2 we show the side views of the two investigated fibers drawn without preform spinning and with spin pitches equal to *L _{T}* = 5.0 mm, (side-hole) and

*L*= 4.1 mm, (microstructured).

_{T}To directly measure the spectral dependences of the elliptical group and phase birefringence in a broad spectral range, we have combined the spectral interferometry method [32] and lateral point-force method [31], previously used to characterize the linearly birefringent fibers. A broadband light from supercontinuum source (NKT Photonics SuperK) was linearly polarized at 45° with respect to the symmetry axes of the SHBF at its input and coupled to the fiber. A transmission azimuth of the polarizer placed at the fiber output was aligned at 45° to the local symmetry axes of the SHBF. Because the axes of the ellipses representing the polarization states of the eigenmodes are parallel to the symmetry axes of the SHBF, such alignment of the polarizers ensures the maximum contrast of the spectral interference fringes visible in the transmission spectrum. In this experimental setup, we registered the transmission spectra of the spun highly birefringent fibers. In Fig. 3 we show an exemplary spectrum acquired with the optical spectrum analyzer (OSA) for the spun SHF of length *L* = 10.0 m, with the spin pitch *L _{T}* = 10.0 mm.

Similarly as it is for linearly birefringent fibers, the density of interference fringes in the output spectrum is related to the group elliptical birefringence ∆*N _{e}*(

*λ*) of the investigated spun fiber:

*L*is the fiber length,

*∆λ*is the spectral width of interference fringe and

*λ*is the central wavelength for a given fringe. The characteristic wide interference fringe centered at around 980 nm indicates that the elliptical group birefringence ∆

*N*(

_{e}*λ*) crosses zero at this wavelength.

Instead of using Eq. (16) to determine the ∆*N _{e}*(

*λ*), one can reconstruct from the registered interferogram the spectral dependence of the phase difference

*∆φ*(

*λ*) between the elliptically polarized eigenmodes given by:

*m*-th order

*∆φ*is an odd multiplicity of π, one obtains:Because it is not possible to identify the absolute value of the fringe order

*m,*the reconstructed phase difference

*∆φ(λ)*is known with a precision to a constant value

*φ*. Nonetheless, by performing differentiation of the

_{0}*∆φ(λ)*versus wavelength, it is possible to determine the spectral dependence of the elliptical group modal birefringence according to the following relation:

*∆φ(λ),*one may calculate the elliptical phase modal birefringence

*∆n*with precision up to a constant value

_{e}(λ)*∆n*

_{e}^{0}:

*∆n*

_{e}^{0}can be determined by directly measuring the elliptical beat length

*L*for chosen wavelengths using the lateral point-force method. Then an appropriate constant

_{e}*∆n*

_{e}^{0}is set to achieve agreement between ∆

*n*(

_{e}*λ*) calculated according to Eq. (20) and the values measured directly for a set of selected wavelengths. In this way, one can determine the spectral dependence of the following parameters

*∆φ(λ)*, ∆

*n*(λ),

_{e}*L*(λ) and ∆

_{e}*N*(λ). Finally, if the twist period

_{e}*L*is independently measured in the transverse illumination, it is possible to calculate also the linear beat length

_{T}*L*(

_{l}*λ*), the linear phase birefringence ∆

*n*(

_{l}*λ*) and the ellipticity angle

*ϑ(λ)*of the polarization eigenstates by using the relations (10), (3) and (5), respectively.

The lateral point-force method was up to now used for measuring beat length in fibers with linear birefringence [31–33]. The principle of measurement in case of the SHBFs with elliptically polarized eigenmodes is similar as for linearly birefringent fibers. In Fig. 4 we show the scheme of the measurement setup allowing to excite only one elliptically polarized eigenmode in the SHBF. At the fiber input the rotatable polarizer *P _{1}* and achromatic quarter wave plate

*λ/4*with polarization axes aligned in parallel to the fiber symmetry axes are placed. The polarizer

*P*aligned at the fiber output at 45° to its symmetry axes allows for interference of the elliptically polarized eigenmodes, which results in the spectral interference fringes shown in Fig. 3. By rotating the polarizer

_{2}*P*it is possible to tune the ellipticity angle of the input polarization state while keeping its azimuth parallel to the fiber symmetry axis until it matches one of the polarization eigenmodes of the SHBF. This happens for the transmission azimuth of the polarizer set at

_{1}*ϑ*with respect to one of the fiber symmetry axes and manifests itself by extinguishing the interference fringes in certain spectral range. The black curve in Fig. 3 shows the best possible suppression of interference fringes in the spectral range from 0.8 to 1.4 μm for the side-hole fiber of total length

*L*= 10.0 m and spin pitch

*L*= 10.0 mm. As it is clearly visible in Fig. 3, full suppression of the interference fringes in a wide spectral range is not possible since the polarization eigenstates of the spun side-hole fiber are wavelength dependent.

_{T}Pressing the fiber at some point causes that the power from the initially excited polarization mode is partially coupled to the non-excited mode. Therefore, the phase delay between the two modes accumulates when they propagate over the distance *L _{F}* from the coupling point to the fiber output, which results in the appearance of the spectral interference fringes. In Fig. 5 we show such force-induced interference fringes for the spun side-hole fiber with

*L*= 10.0 mm and the distance

_{T}*L*9.80 m and for the spun microstructured fiber with

_{F}=*L*= 8.2 mm and distance

_{T}*L*0.20 m. For better visibility of the interference fringes, we have subtracted the reference spectrum registered for non-pressed fiber from the spectrum registered after pressing the fiber (black curve in Fig. 3).

_{F}=When the coupling point is moved down the fiber length, it causes the displacement of interference fringes observed in the output spectrum. If for the initial position of the coupling point at *L _{F}* from the fiber end, one of the interference minima arises at the wavelength

*λ*

_{0}, then by moving the coupling point by

*L*(

_{e}*λ*

_{0}) towards the fiber end, we shift the interference pattern by the full fringe width and replicate another interference minimum of order differing by 1 at this specific wavelength. One has to be aware that random factors, like the amount and direction of applied force, may influence the fringe position. It is therefore important to shift the coupling point by a multiple of

*L*(

_{e}*λ*

_{0}), while observing the displacement of subsequent interference fringes at the wavelength

*λ*

_{0}. The value

*L*(

_{e}*λ*

_{0}) is then obtained as a slope of the straight line ∆

*L*, where ∆

_{F}= f(∆M)*L*is the shift of the coupling point inducing the displacement of

_{F}*∆M*interference fringes at the wavelength

*λ*

_{0}. In Fig. 6(a) we show the enlarged fraction of the force-induced fringe pattern for different displacements ∆

*L*of the coupling point. The distance ∆

_{F-n}*L*is approximately equal to half of the elliptical beat length for the wavelength 1290 nm, which resulted in the shift of the fringe by half of its width. In Fig. 6(b) we show the linearly approximated dependence ∆

_{F-5}*L*for

_{F}= f(∆M)*λ*= 1290 nm, indicating that the elliptical beat length at this wavelength equals

*L*= 200 ± 4 mm. In Fig. 7(a) and Fig. 7(b) analogical experimental results are shown for the spun birefringent microstructured fiber with spin pitch

_{e}*L*= 8.2 mm.

_{T}One can see in Fig. 6(b) and Fig. 7(b) a random spread of measurement points around the linear dependence ∆*L _{F} = f(∆M),* which results in uncertainty in determining the elliptical beat length

*L*of about 2%. Consequently, the phase elliptical birefringence being inversely proportional to

_{e}*L*is charged with the same uncertainty. The precision of determining the phase and group linear birefringences as well as the ellipticity angle of the polarization eigenmodes is related to precision of measurements of not only

_{e}*L*but also

_{e}*L*. In our experiments, we have determined the

_{T}*L*parameter by microscopic examination of the spun fibers in transverse illumination and the accuracy of these measurements was estimated at 1%. This implies that the uncertainty of determining the phase and group linear birefringences, and the ellipticity angle is of about 3%.

_{T}## 4. Experimental results

Using the proposed method, we have characterized seven side-hole highly birefringent fibers denoted in Fig. 8 as SHF1-7. The fiber denoted as SHF1 was a non-spun side-hole fiber whereas the fibers SHF2-7 had spin pitches equal to 200 mm, 100 mm, 50 mm, 30 mm, 10 mm and 5.0 mm, respectively. In Fig. 8(a), the measured group elliptical birefringence ∆*N _{e}*(

*λ*) is shown for all the examined side-hole fibers. The absolute value of the ∆

*N*(

_{e}*λ*) increases with the fiber spin pitch and is the highest for the non-spun side-hole fiber SHF1. The spectral dependences of the ∆

*N*(

_{e}*λ*) for six out of seven examined side-hole fibers cross zero level in the range between 940 and 980 nm. Some deviation was observed for the fiber SHF7 with the shortest spin pitch

*L*= 5.0 mm, as in this case the zero-crossing wavelength equals 916 nm. According to Eq. (13), the zero-crossing wavelength for the group elliptical birefringence is not affected by the fiber twist and depends only on the fiber geometrical parameters defining the linear group birefringence ∆

_{T}*N*(

_{l}*λ*). The observed differences in the zero-crossing wavelengths of the ∆

*N*(

_{e}*λ*) imply that the geometrical parameters of the examined fibers were affected by the preform spinning during the drawing process.

In Fig. 8(b) we show the obtained spectral dependences of the phase elliptical birefringence ∆*n _{e}*(

*λ*) for all seven side-hole fibers. In accordance with Eqs. (9), (11) also in this case the highest value of the phase birefringence

*Δn*was observed for the non-spun fiber SHF1, whereas the lowest for the SHF7 fiber with the shortest spin pitch

_{e}*L*= 5.0 mm.

_{T}The linear phase birefringences ∆*n _{l}*(

*λ*) shown in Fig. 8(c) for all the seven examined side-hole fibers differ by up to 30% depending on the spin pitch

*L*, which confirms again that the preform spinning during the drawing process affects the fiber geometry. Knowing the spectral dependences of the phase linear phase birefringence ∆

_{T}*n*(

_{l}*λ*), we calculated the group linear birefringences ∆

*N*(

_{l}*λ*) according to Eq. (2), which are shown in Fig. 8(d) for all the examined side-hole fibers. Similarly as for the linear phase birefringences ∆

*n*(

_{l}*λ*), small differences in the ∆

*N*(

_{l}*λ*) are visible depending on the spin pitch

*L*. One should note, however, that according to Eq. (13) the zero-crossing wavelengths for the linear ∆

_{T}*N*(

_{l}*λ*) and elliptical ∆

*N*(

_{e}*λ*) group birefringences are the practically same. The largest mismatch of the zero-crossing wavelengths was only 2 nm and was most probably caused by numerical errors.

The spectral dependences of the ellipticity angle *ϑ(λ)* of the polarization eigenmodes determined according to Eq. (5) for the spun side-hole fibers are shown in Fig. 8(e). The relation between the *ϑ(λ)* and the fiber spin pitch *L _{T}* is clearly visible. Moreover, the maxima of the

*ϑ(λ)*functions arise at the wavelengths at which the group elliptical (and linear) birefringence crosses zero level, as it was predicted by theoretical analysis in Section 2. It is generally noted that the ellipticity angles

*ϑ(λ)*determined for the spun side-hole fibers are weakly dispersive. For the SHF7 fiber, the

*ϑ(λ)*changes between 39.6° and 41.4° in the full spectral range, so the corresponding polarization eigenstates are nearly circular. On the other hand, the smallest ellipticity angle in the SHF2 fiber with the largest spin pitch ranges between 4.1° and 6°.

In Fig. 9 we show the measured characteristics for five birefringent microstructured fibers denoted as BMF1-5. The fiber BMF1 was non-spun, whereas the fibers BMF2-5 had the spin pitches equal respectively to 16.4 mm, 8.2 mm, 5.5 mm, and 4.1 mm.

In Fig. 9(a), 9(b) we show the results of measurements of the group and phase elliptical birefringences ∆*N _{e}*(

*λ*) and ∆

*n*(

_{e}*λ*). Similarly as it was for the side-hole fibers, the non-spun fiber BMF1 shows the greatest birefringences ∆

*N*(

_{e}*λ*) and ∆

*n*(

_{e}*λ*). For the spun fibers BMF2-5 there is no clear relation between the spin pitch

*L*and the value of elliptical group and phase birefringences ∆

_{T}*N*(

_{e}*λ*) and ∆

*n*(

_{e}*λ*) (Fig. 9(a), Fig. 9(b)). There are two reasons responsible for this effect. The first one is much lower differentiation in spin pitches of the microstructured fibers (16.4 ÷ 4.1 mm) compared to the side-hole fibers (200 ÷ 5.0 mm), whereas the second one is more complex structure of the BMFs causing that it is more likely affected by different velocities of the preform spinning during the drawing process. This hypothesis is supported by clearly visible differences in geometry (holes size) of the microstructured fibers drawn with different spin pitches shown in Fig. 1 and by high variations in the linear birefringences ∆

*n*(

_{l}*λ*) and ∆

*N*(

_{l}*λ*) between the fibers with different spin pitches

*L*Fig. 9(c), 9(d).

_{T,}For the investigated BMFs, the group elliptical (and linear) birefringence does not cross zero level in the analyzed spectral range and consequently the ellipticity angle *ϑ(λ)* does not reach the maximum value. Simultaneously, the ellipticity angle *ϑ(λ)* for the microstructured fibers is much more dispersive than for the side-hole fibers, which is related to much stronger dispersion of the linear phase birefringence ∆*n _{l}*(

*λ*). The most dispersive dependence of the

*ϑ(λ)*was observed for the fiber BMF2 with the largest spin pitch

*L*= 16.4 mm, for which the measured ellipticity angle

_{T}*ϑ(λ)*varied between 28° and 6.8° in the analyzed spectral range.

## 5. Conclusions

In this paper, we have proposed a novel approach for characterizing highly birefringent spun fibers, which is based on combination of the spectral interferometry method and lateral point-force method. Our approach allowed to determine in a wide spectral range all relevant parameters of the spun highly birefringent fibers such as the group and phase elliptical birefringence, the group and phase linear birefringence and the ellipticity angle of the polarization eigenmodes. To our best knowledge, such detailed characterization of the spun highly birefringent fibers was conducted for the first time. In addition, we have derived analytical relations between the fiber parameters to support and explain our measurement results. In particular, we derived the relation between the group linear and elliptical birefringence in spun highly birefringent fibers and the condition for the maximum of ellipticity angle of polarization eigenmodes.

Using the proposed approach, we have examined two families of the spun highly birefringent fibers with different spin pitch *L _{T}* and dispersion characteristics. The experimental results obtained for the fibers drawn from the same preform with different spin pitches

*L*were not always in agreement with the analytical predictions. This suggests that the preform spinning during the fiber drawing process affects the fiber geometry and therefore its linear birefringence. Our results show that this effect is particularly well pronounced in the birefringent microstructured fibers of more complex geometry and therefore more sensitive to the drawing conditions.

_{T}## Funding

National Science Centre of Poland, grant Maestro 8, DEC- 2016/22/A/ST7/00089.

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