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Spectral polarization characteristics of short-length fiber Bragg gratings UV-written in a highly-birefringent spun-fiber have been investigated. Based on the analysis of the characteristics the technique for measuring the built-in linear phase birefringence as well as the spin period in this fiber type has been suggested. In this method the birefringence dispersion is excluded and therefore the built-in linear phase birefringence can be measured with an improved accuracy.
© 2016 Optical Society of America
1. Introduction
Spinning of optical fibers provides an additional degree of freedom, which allows modifying the fiber waveguiding properties. A helical structure can be created in optical fibers both twisting as-drawn fibers (twisted fibers) [1] and by drawing the fiber from a rotating fiber preform (spun fibers) [2].
Useful applications of helical fiber structures include circular polarizing fibers created by spinning linear polarizing fibers [3], fiber transformers of polarization state (phase plates) [4], helical fiber gratings with different spin periods [5, 6], long-period fiber gratings with the coupling coefficient depending on the twist rate [7], and some others. Axial rotation of microstructured optical fibers can also lead to novel fiber-optic components [8, 9].
Spun fibers drawn from preforms with a low birefringence (LoBi spun fiber) are attractive for high speed telecommunication as this fiber type possesses a reduced polarization mode dispersion (PMD). Recall that PMD is the main limiting factor in modern optical communication systems [10, 11].
Singlemode fibers drawn from rotating preforms with a high linear birefringence (HiBi spun fiber) are widely used in fiber-optic current sensors, because circular polarized waves propagating along such a fiber accumulate a phase shift induced by an external magnetic field [12, 13]. The polarization state of HiBi spun fiber eigenmodes is elliptical, and the ellipse axes rotate on rotating the birefringence axes. If the spin period is sufficiently short, the polarization eigenstates are almost circular and the polarization state evolution becomes similar to that in circularly birefringent media [12].
The polarization properties of a HiBi spun fiber are defined by the built-in linear phase birefringence Δβ and the spin rate of the birefringence axes 2ξ, the latter being two times higher than the spin rate of the fiber itself ξ = 2π/Ls. Here the fiber spin period Ls = V/fs is defined by the fiber drawing speed V and the preform rotation frequency fs. The dimensionless ratio of the two mentioned values σ = Δβ/2ξ = Ls/2Lb (where Lb = 2π/Δβ is the built-in linear birefringence beatlength) is the most important parameter of a HiBi spun fiber. This ratio defines the degree of ellipticity of the polarization eigenstates, the polarization evolution and, as a consequence, the magneto-optical sensitivity of the fiber S = (1 + σ2)-1/2 [14]. Thus, the practically important problem of determining the polarization properties of a HiBi spun fiber is essentially reduced to measuring parameters Δβ and 2ξ.
Spin rate ξ of a HiBi spun fiber can be calculated from the known fiber drawing conditions or, in some cases, measured using an optical microscope through the lateral fiber surface. The value of the built-in linear phase birefringence Δβ can be determined by measuring the beatlength of the linear polarizations (as is done for HiBi unspun fibers [15]). The beatlength is recalculated taking into account the ξ-value and the built-in linear group birefringence Δβgr = Δβ - λdΔβ/dλ [16]. At the same time, the birefringence dispersion dΔβ/dλ is often neglected, which may decrease the accuracy of the thus obtained Δβ-value [17].
Fiber Bragg gratings (FBG) provide additional possibilities for measuring the fiber parameters under discussion, as the spectral polarization characteristics of the FBG written in a HiBi spun fiber are essentially modified by the periodic modulation of the fiber properties. A coupled mode theory suitable for calculation of the FBG transmission/reflection spectra as well as their polarization dependence has been suggested in [18, 19]. It was shown both theoretically and experimentally that additional sidelobes appear in the spectra of long-length FBGs (Lg > Ls), the sidelobes frequency detuning from the resonance wavelength being defined by the fiber spin rate. The intensity of the sidelobes proved to strongly depend on the polarization state of the propagating radiation, whereas the position and the shape of the main FBG resonance proved to be polarization independent.
This work is devoted to the study of the spectral polarization properties of short-length FBGs (Lg ≤ Ls) written in HiBi spun fibers. A simplified approach for calculating the FBG spectra for the linear polarization states directed along the birefringence axes is described. The analysis of the measured FBG spectra based on this approach allowed us to propose a useful technique for measuring the built-in linear phase birefringence and the spin rate in a HiBi spun fiber. The technique does not require any additional information on the birefringence dispersion and on the parameters of the fiber drawing process.
2. Short-length FBGs in HiBi spun fiber
The starting idea of the measuring technique of the built-in linear phase birefringence in a HiBi spun fiber with the help of short-length FBGs was as follows. If the length of an FBG written in a HiBi spun fiber is sufficiently short, i.e. much shorter than the spin period Lg << Ls, the FBG-containing fiber section can be considered to be free of rotation of the birefringence axes. If so, as in the case of a HiBi unspun fiber, the value of the linear phase birefringence B = Δβ/k (where k = 2π/λ is the wavevector in vacuum) can be determined by measuring the difference between the resonance wavelengths Δλ for the modes with linear polarizations directed along the slow (s) and fast (f) birefringence axes.
where ns and nf are the effective refractive indices of these modes, n and ngr are the effective phase and group indices, respectively. Note that the group index in Eq. (1) means that the Bragg resonance condition is satisfied at different wavelengths for the orthogonal linear polarization states and, therefore, the index dispersion should be taken into consideration. At the same time, for all-solid silica-based fibers the dispersion effect is rather small and ngr with an acceptable accuracy (1 - 2%) can be considered to be equal to n and to the refractive index of undoped silica glass [20]. Thus, the described procedure allows measuring the built-in linear phase birefringence in a rather simple and accurate way.Shortening the FBG leads to broadening of the FBG spectrum and to a decrease of the reflection coefficient; therefore, in practice the FBG length should not be shorter than Lg = 0.3 - 0.5 mm. In this case, the desired measurement accuracy of linear phase birefringence δB ~ 10−5 can be achieved. A typical value of the spin period in HiBi spun fibers is about 3 mm, which means that condition Lg << Ls may not be satisfied, and rotation of the birefringence axes along the FBG length should not be neglected with. The possible way to overcome this problem is to analyze the dependence of the spectral shift Δλ on the FBG length and to extrapolate it to an extremely short FBG length. To do this, the polarization state evolution along the FBG length should be taken into consideration, because, generally speaking, the input polarization state is not conserved, in case the FBG length is comparable to the spin period (Lg ~Ls).
The polarization state evolution in birefringent optical fibers of various types, including HiBi spun fibers, has already been analyzed in detail [1, 21]. Let us consider the Poincaré sphere in laboratory coordinate system (x, y, z) with z-axis directed along the fiber axis while x- and y-axes lie in the perpendicular plane [Fig. 1]. Suppose that points H and V on the sphere correspond to the linear horizontal (x) and vertical (y) polarization states, P and Q, to the linear polarization states with + 45° and −45° azimuth, respectively, and L and R correspond to left- and right-circular polarization state. Suppose also that the geometrical center of the FBG is located at z = 0. At this point, the axes of birefringence coincide with directions x and y.
Fig. 1 Evolution of orthogonal linear polarization states in a HiBi spun fiber with σ = 0.2 (see text for details).
With these assumptions, points H and V on the sphere indicate two orthogonal linear polarization states directed along the birefringence axes of the fiber at the FBG center. In addition, these two points on the sphere correspond to polarization states discussed above for extremely short FBG (see Eq. (1)). If so, trajectories CH(z) and CV(z) (- Lg/2 < z < Lg/2) going through these points correspond to the polarization state evolution along an FBG of arbitrary length. The trajectories shown in Fig. 1 were calculated by numerical integration of the differential Jones matrix [13] for a HiBi spun fiber with σ = 0.2, the value of σ being close to that of the fiber used in our experiments. Trajectories CH(z) and CV(z) oscillate with a spatial period nearly equal to the spin period of the fiber birefringence axes, Ls/2. This means that for condition Lg ~Ls, despite the rotation of the birefringence axes in a HiBi spun fiber, trajectories CH(z) and CV(z) deviate only slightly from the exact linear polarization states (points H and V).
Note that a more general statement is valid for HiBi spun fibers with small σ: an arbitrary polarization state deviates only slightly from its initial state at a fiber distance of about Ls. In other words, the rotation of the axes of the built-in linear birefringence in this case can be considered as a weak perturbation of the initial polarization state. The drift of trajectories CH(z) and CV(z) along the equator is associated with the rotation of polarization state axes, as it occurs in media with circular birefringence [16].
Thus, for the modes with the orthogonal linear polarization states, more strictly for the modes which polarization states within the FBG correspond to trajectories CH(z) and CV(z) [Fig. 1], the effective refractive index can be approximately written as:
where n0 = (ns + nf)/2 is the mean index of the linearly polarized modes. Dependences (2) are schematically shown in Fig. 2 together with the FBG and fiber parameters involved in the consideration. The FBG reflection spectra calculated using these dependencies were compared with the measured spectra. The calculations were performed numerically by solving the coupled-mode equations, with the FBG being divided into short sections with a constant refractive index (transfer matrix method [22]).Fig. 2 Spatial distributions of effective refractive indices for the orthogonal linear polarization modes in a HiBi spun fiber.
3. Experiments
A singlemode HiBi spun fiber drawn from a HiBi fiber preform with borosilicate stress-applying rods (PANDA-type spun fiber) has been used in our experiments. The fiber was fabricated by JSC Perm Scientific-Industrial Instrument-Making Company (PNPPK, Russia). The fiber had a step-index profile with a core/cladding difference of about 0.01. The diameters of the fiber core and cladding were 6.2 and 125 μm respectively, the cut-off wavelength being 1.35 μm. The fiber was drawn with a spin period Ls of about 3.2 mm, which corresponds to the period of the built-in linear birefringence axes rotation of about 1.6 mm. The beatlength in the fiber drawn from the initial fiber preform was measured to be Lb ≈7 mm in the 1.55 μm spectral range. This value corresponded to the built-in linear birefringence B ≈2.2 × 10−4.
Thus, the expected value of the σ-parameter of the HiBi spun fiber used in our experiments was about 0.23, the value typical for magneto-sensitive fibers in current sensors. With this σ-value the eigen polarization states of the fiber are almost circular and the linear polarization states evolve along the fiber as shown in Fig. 1.
FBGs were written in the core of the HiBi spun fiber through its lateral surface (the acrylate coating was preliminarily removed) using a continuous-wave frequency-doubled Ar-ion laser radiation (244 nm) in a Lloyd interferometer scheme. High photosensitivity of the fiber (at least 10−3) is required to write even weakly reflecting FBGs (R ∼10 - 20%) if the FBGs length is 0.5 mm or less. The intrinsic photosensitivity of fibers with a relatively low GeO2 concentration in the core (in our case, this value was about 6 mol.%) does not reach the required level. To increase photosensitivity, the fiber was subjected to low temperature hydrogen loading in a 100-atm-pressure chamber during 12 hours at a temperature of 100°С. A UV-induced index change as high as 1.8 × 10−3 was achieved in the hydrogen-loaded fiber during 30-min irradiation (maximum exposure time used in our experiments). The remaining molecular hydrogen was outdiffused from the glass network after the FBG fabrication using heat treatment in air at the same temperature.
Six uniform FBGs of different length (Lg = 0.4, 0.8, 1.2, 1.9, 2.3, 2.9 mm) separated by 10-cm fiber sections were written in the HiBi spun fiber, the Bragg wavelengths of the FBGs being different. The transmission spectrum of the fabricated FBGs measured with depolarized light is shown in Fig. 3. The FBGs spectra proved to be rather uniform even for the gratings longer than Ls/2 = 1.6 mm (the writing UV radiation evidently crossed the borosilicate rods in writing such FBGs). The spectral widths of all FBGs were in a good agreement with the calculated values for a specific FBG length.
The scheme of the experimental setup used for the investigation of the spectral polarization properties of the FBGs is shown in Fig. 4. Broadband polarized radiation of an erbium-doped fiber source passed through a fiber polarizer and a polarization controller and then was directed to the fiber length containing the FBGs by a fiber circulator. The transmission or reflection spectrum of a certain FBG was measured by an optical spectrum analyzer ANDO-6317B, the polarization state of the light coupled to the HiBi spun fiber being set by the polarization controller. The specified polarization state at any point of the HiBi spun fiber was supposed to be unchanged during the measuring time (several seconds) despite the fact that all the elements of the optical scheme were pigtailed with an SMF-28 fiber, which is not polarization-maintaining.
Fig. 4 Optical scheme of the experimental setup used for the investigation of the spectral polarization properties of the FBGs written in the HiBi spun fiber. P – polarizer, PC – polarization controller, C – fiber circulator, BBS – broad band erbium-doped fiber source, OSA – optical spectrum analyzer.
Thus, we supposed that the proper adjustment of the polarization controller allowed us to ensure linear polarization states (points H or V on the Poincare sphere) at the geometrical center of the tested FBG so that the polarization state evolution inside the grating coincides with trajectories CH(z) or CV(z) in Fig. 1.
4. Experimental results and analysis
Preliminary calculations of the FBG reflection spectra made for the FBGs of different length using Eq. (2) showed that the difference between the wavelengths of the first spectrum minimum of the orthogonal polarization states (Δλm in Fig. 4(b)) is the most convenient parameter to monitor the polarization state at the FBG center. As followed from the calculations, the extreme spectral positions of the first spectrum minimum corresponded to the linear polarization states. Therefore, before measuring the FBG reflection spectra, the polarization state was adjusted by the polarization controller so as to provide these extreme spectral positions.
Figure 5 shows the experimental (top) and the calculated (bottom) reflection spectra obtained for FBGs of three lengths, the experimental and calculated spectra being separated by an arbitrary offset along the ordinate axis, the offset value being different for each subfigure. Variations of the effective mode index along the FBG length for the orthogonal linear polarization states are illustrated schematically in the insets of each figure.
Fig. 5 Experimental (top) and calculated (bottom) reflection spectra obtained for FBGs with length of 0.4 (a), 1.9 (b) and 2.9 (c) mm. Solid (dotted) curves correspond to the linear polarization state directed along the slow (fast) birefringence axis at the FBG center.
As can be seen, if Lg << Ls/2, the polarization dependence of the FBG spectrum is similar to that observed for HiBi unspun fibers [Fig. 4(a)] [23]: for linear polarization states directed along the built-in birefringence axes the spectra are of identical shape, but shifted by Δλ = Bλ/ngr with respect to each other. When the grating length is increased, the position of the main resonance becomes less sensitive to the polarization state, while the spectral changes of the sidelobes become more pronounced [Fig. 5(b), (c)]. Note that the good agreement between the calculated and experimental spectra for all the FBG lengths testifies to the validity of the proposed simplified approach to the calculation of the spectra of short-length FBGs.
Note that for FBG with Lg ≈Ls/2 the values of the effective mode index averaged along the FBG length are equal for the orthogonal linear polarization states. Nevertheless, the reflection spectra in the region of the sidelobes are asymmetric [Fig. 4(b)]. The observed asymmetry is due to the difference in the phase matching conditions between different parts of the FBG. In fact, the pitches at the FBG center are better phase matched than the pitches located at the FBG ends, which are separated by a fiber section with a different effective index [Fig. 1].
The measured spectral difference Δλm is shown by symbols in Fig. 6 as a function of the FBG length. The solid curve presents the calculated dependence obtained using approximation (2). In the calculations Ls was equal to 3.2 mm and B was varied to achieve the best coincidence with the experimental data. The determined built-in linear phase birefringence B = 1.7 ± 0.1 × 10−4 proved to be 30% less than the measured one by the spectral beating technique mentioned above. Such a reduction has been already noticed in [24] and can be explained as follows. The rotation of the fiber preform during the drawing process can cause a reduction of the elastic stresses in the fiber core. Another reason of the discrepancy can be associated with the birefringence dispersion effect which was ignored in the spectral beating measurements.
Fig. 6 The experimental (symbols) and calculated (solid curve) dependences of the polarization sensitivity of the first minimum in the FBG reflection spectrum on the FBG length.
It should be noted that the approach proposed allows one to determine the B-value of the HiBi spun fiber, even if the spin period is unknown. Indeed, the above one-dimensional procedure to find the B-value can be extended to a two-dimensional one to find simultaneously B and Ls values.
It can be shown that the polarization sensitivity of the first minimum Δλm is reduced two-fold with respect to its maximum value, when the FBG length is equal to the period of the birefringence axes rotation, i.e. relationship Δλm(Ls/2) = Δλm(0)/2 shown by a dashed line in Fig. 6 is satisfied. In this case, doubled polarization sensitivity of the first minimum Δλm of an FBG of length Ls/2 allows one to obtain the B-value directly from Eq. (1). Similarly, to estimate the spin rate of the HiBi spun fiber, it is sufficient to find the FBG length at which Δλm decreases by a factor of 2 with respect to the value measured for the shortest FBG.
Note also that an additional linear birefringence may be induced by UV radiation in the fiber core during the FBG fabrication procedure. The value of this birefringence is typically small (BUV ~10−5 [25]), and, therefore, has not been taken into our consideration.
5. Conclusion
We have investigated the spectral polarization characteristics of short-length FBGs UV-written in a PANDA-type HiBi spun fiber with the σ-parameter less than unity. It was shown that in such a fiber an arbitrary polarization state deviates only slightly from its initial state along the FBG length. This allowed us to suggest a simplified approach to the calculation of the short-length FBG transmission/reflection spectra, if light at the FBG center possesses linear polarization states directed along the birefringence axes. A good agreement between the experimental and calculated spectra has been observed for both linear polarization states for all FBGs of different length used in our experiments.
It has been shown that the difference between the wavelengths of the first minimum in the FBG reflection spectra measured for the orthogonal linear polarization states can be considered as a useful tool to determine the unknown HiBi spun fiber parameters. By measuring this difference in a series of FBGs of different length, it is possible to determine both the built-in linear phase birefringence Δβ and the spin rate of the birefringence axes 2ξ. It is important that the built-in linear phase birefringence is determined in this way with a better accuracy, because the value of the birefringence dispersion is not utilized in the calculations.
Acknowledgments
The authors are grateful to JSC Perm Scientific-Industrial Instrument Making Company (PNPPK) for providing the HiBi spun fiber samples.
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