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Effects of spin process on birefringence strength of single-mode fibers |
Optics Express, Vol. 20, Issue 1, pp. 1-6 (2012)
http://dx.doi.org/10.1364/OE.20.000001
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– Abstract
1. Introduction
2. Measurement procedure
3. Experimental results
4. Conclusion
– References and links
– Abstract
1. Introduction
2. Measurement procedure
3. Experimental results
4. Conclusion
– References and links
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Abstract
Spin process is the most effective and diffused way to reduce polarization mode dispersion in single-mode optical fibers. All theoretical models adopted so far to describe spun fibers assume that the only effect of spin is to rotate fiber birefringence, without affecting its strength. Yet, experimental analyses of this hypothesis are controversial. In this paper, we report on an extensive experimental characterization of birefringence in spun and unspun fibers. Results indicate that the spinning process has no instantaneous effect on birefringence strength, regardless of the kind of fiber; nevertheless, there might be a small average effect on G.652 fibers.
© 2011 OSA
1. Introduction
Since the earliest days of fiber optic communications, it was clear that fiber asymmetry and anisotropy could have been sources of impairment. Indeed, polarization mode dispersion (PMD) confirmed to be the black beast of high bit-rate, long haul, optical communication systems, only partially tamed by the new polarization-diversity coherent systems [1–3]. The vector equalization of optical channel, exploited in such systems, allows to mitigate linear impairments due to PMD. On the other hand, however, the steady increase of aggregate bit-rate transmitted through fibers has enhanced non-linear effect and their interaction with PMD, hindering equalization [4, 5]. In this context, to reduce PMD ab ovo is still an advisable approach to the problem.
A. Galtarossa and C. R. Menyuk, Polarization Mode Dispersion (Springer, New York, 2005). [CrossRef]
L. Nelson, S. Woodward, S. Foo, X. Zhou, M. Feuer, D. Hanson, D. McGhan, H. Sun, M. Moyer, M. O. Sullivan, and P. Magill, “Performance of a 46-Gb/s dual-polarization QPSK transceiver with real-time coherent equalization over high PMD fiber,” J. Lightwave Technol. 27, 158–167 (2009). [CrossRef]
O. Bertran-Pardo, J. Renaudier, G. Charlet, P. Tran, H. Mardoyan, M. Salsi, and S. Bigo, “Experimental assessment of interactions between nonlinear impairments and polarization-mode dispersion in 100-Gb/s coherent systems versus receiver complexity,” IEEE Photon. Technol. Lett. 21, 51–53 (2009). [CrossRef]
P. Serena, N. Rossi, O. Bertran-Pardo, J. Renaudier, A. Vannucci, and A. Bononi, “Intra- versus inter-channel PMD in linearly compensated coherent PDM-PSK nonlinear transmissions,” J. Lightwave Technol. 29, 1691–1700 (2011). [CrossRef]
The most effective way to reduce PMD in single-mode fibers consists in spinning them while they are drawn. In fact, spun fibers are widely diffused and have been studied in deep details by many research groups [6–15]. Despite all these efforts, however, one aspect still needs clarification. Actually, all theoretical models adopted so far assume that spin just rotates fiber birefringence, without affecting its strength. Nonetheless, experimental analyses of this hypothesis provided controversial results. Ferrario, Pietralunga and co-workers compared the birefringence of constantly-spun fibers with that of unspun fibers drawn from the same preforms under the same conditions, finding that spin may reduce birefringence by about 33–40% [11, 12]. The variation was confirmed by two different measurement techniques (although results were not in complete agreement), and qualitatively supported by a theoretical model. Recently, however, we performed an analogous analysis on periodically-spun non-G.652 fibers, without finding any significant evidence of birefringence reduction induced by the spin process [15].
A. J. Barlow, J. J. Ramskov-Hansen, and D. N. Payne, “Birefringence and polarization mode-dispersion in spun single-mode fibers,” Appl. Opt. 20, 2962–2968 (1981). [CrossRef] [PubMed]
M. Ferrario, S. M. Pietralunga, M. Torregiani, and M. Martinelli, “Modification of local stress-induced birefringence in low-PMD spun fibers evaluated by high-resolution optical tomography,” IEEE Photon. Technol. Lett. 16, 2634–2636 (2004). [CrossRef]
S. M. Pietralunga, M. Ferrario, M. Tacca, and M. Martinelli, “Local birefringence in unidirectionally spun fibers,” J. Lightwave Technol. 24, 4030–4038 (2006). [CrossRef]
In this paper, we extend our analysis including G.652 fibers and quasi-periodically spun fibers. Results confirm that in non-G.652 fibers spin process does not have a significant effect on birefringence strength (neither instantly nor on average). On the contrary, we have found a weak experimental evidence suggesting that spin process might have a small average (not instantaneous) effect on birefringence strength of G.652 fiber.
2. Measurement procedure
The aim of the experiment is to verify if spin process may reduce fiber birefringence strength. Specifically, we have addressed two possible effects. The first one is an instantaneous effect in which birefringence strength possibly decreases with local spin rate. If this effect exists, it should be clearly visible in periodically spun fibers, where absolute spin rate varies cyclically from 0 to a maximum value. On the contrary, it cannot be clearly assessed in unidirectionally spun fibers, where the spin rate is constant. The second effect is an average effect, in which spin possibly causes an average reduction of birefringence strength. This average effect is the one addressed in [12], and it can be assessed by comparing the birefringence of spun and unspun fibers drawn from the same preform. The evaluation of the average effect may be hindered by longitudinal homogeneity of the fiber (hence of the preform), a parameter not easy to quantify.
S. M. Pietralunga, M. Ferrario, M. Tacca, and M. Martinelli, “Local birefringence in unidirectionally spun fibers,” J. Lightwave Technol. 24, 4030–4038 (2006). [CrossRef]
We have analyzed a set of 23 fiber samples, drawn from 13 different preforms, listed in Table 1. The set of preforms includes the most common telecommunication fibers. From some of these preforms (F to K) more than one sample has been drawn, with varying spin properties among unspun, periodically spun, and quasi-unidirectionally spun. Quasi-unidirectional spin, in particular, has been obtained by applying triangular spin profiles with periods exceeding 200 m, and putting special care in removing the torsion unavoidably accumulated by the fiber during the drawing. From these quasi-unidirectional samples we have selected both spans with unidirectional spin and spans comprising the spin-rate inversion. As an example, Fig. 1(a) shows the physical angle of rotation of birefringence measured (as described below) on a quasi-unidirectionally spun fiber. Dashed curve refers to a sample comprising only unidirectional spin (namely, sample G2; see Table 2); solid curve refers to a sample comprising also spin-rate inversion (namely, sample G3). Note that spin properties of quasi-unidirectionally spun fibers are quite different from those of periodically spun ones. In particular, the spin inversion (where the spin-rate is close to zero and its effects should be minimum) lasts several meters in the former case, but is much shorter in the latter one.
Table 1 List of Preforms Used to Draw Fiber Samples and Their General Properties
Fig. 1 (a) Measured angle of rotation of birefringence measured on quasi-unidirectionally spun fiber samples without (dashed curve, sample G2) and with (solid curve, sample G3) spin inversion. (b)–(c) Example of analysis of instantaneous effect (data refer to sample B; see Table 2): (b) measured absolute spin rate; (c) measured birefringence strength (blue curve) and the fitting curves given by the linear model (black) and the polynomial one (red).
Table 2 Experimental Results
Birefringence and spin properties of the collected samples have been measured by means of the polarization-sensitive optical frequency reflectometer (POFDR) described in ref. [14], with a spatial resolution ranging from 5 mm to 50 mm, depending on the fiber sample. In order to avoid twist and to minimize external perturbations, samples have been laid as straight as possible on the laboratory floor, for lengths ranging from 15 to 25 m. The state of polarization (SOP) of the backscattered field has been measured with the POFDR as a function of the scattering position z, for 5 different input SOPs. These 5 measurements have been grouped in combinations of 3, each of which was used to calculate the birefringence vector as described in [16]. As a result, this procedure yields 10 different estimates of the birefringence vector for each fiber sample. Birefringence strength, β(z), has been determined as the average over the 10 estimates. Spin rate, α(z), has been calculated as the derivative of the birefringence angle, averaged over the 10 estimate and filtered with a fourth-degree, 5-cm-long, Savitzky-Golay filter [17].
A. Galtarossa, D. Grosso, L. Palmieri, and M. Rizzo, “Spin-profile characterization in randomly birefringent spun fibers by means of frequency-domain reflectometry,” Opt. Lett. 34, 1078–1080 (2009). [CrossRef] [PubMed]
A. Galtarossa, D. Grosso, L. Palmieri, and L. Schenato, “Reflectometric measurement of birefringence rotation in single-mode optical fibers,” Opt. Lett. 33, 2284–2286 (2008). [CrossRef] [PubMed]
A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964). [CrossRef]
2.1. Data analysis
Figure 1(b) shows, as an example, the modulus of the spin rate measured on a fiber sample (namely, sample B). The birefringence strength measured on the same sample is shown in Fig. 1(c) (blue curve). A qualitative comparison of the two curves may suggest that when the spin rate is at its maximum, the birefringence strength decreases. Yet, it is important to quantify this effect. To this aim, we have already proposed a method in [15], which applies, however, only to periodically spun fibers. Here, we introduce a new and more general data analysis algorithm, which does not rely on assumptions about the spin.
If the birefringence strength, β(z), is influenced by the modulus of spin rate, then we may model it as β(z) = 〈β〉 + η(z) + Δβa(z), where 〈β〉 is the mean birefringence strength, η(z) is a zero-mean random process describing intrinsic randomness and measurement noise, and a(z) = |α(z)|/max{|α(z)|} is the normalized modulus of spin rate. Parameter Δβ quantifies what we call the instantaneous effect of spin rate on birefringence strength. In particular, if the birefringence decreases as the spin-rate increases, Δβ should be negative. The quantities 〈β〉 and Δβ are the free parameters of the proposed model and they can be estimated by fitting experimental data in the least square error (LSE) sense, while treating the random term η(z) as a noise. Let b be the column vector build stacking the values of β(z) sampled along z; similarly, let a be the column vector build from a(z). Then, the LSE estimate of 〈β〉 and Δβ can be obtained as v = (MTM)−1MTb, where v = (〈β〉, Δβ)T and M is the matrix (u, a), u being a vector with the same size as a and with all the elements equal to 1. The application of this procedure to the data shown in figs. 1(b) and 1(c) yields 〈β〉 ≃ 21.42 rad/m and Δβ ≃ −1.9 rad/m. The resulting fitting curve, 〈β〉+Δβa(z), is shown in Fig. 1(c) by the black curve. Clearly, the model is able to extract the sought information from data. Note that 〈β〉 corresponds also to the ergodic average of birefringence strength, evaluated along z.
One might object that the proposed model is too simple, for it assumes a linear influence of spin rate on birefringence strength. Indeed, the model can be generalized as β(z) = η(z) + q[a(z)], where q(·) is a polynomial of arbitrary degree, whose coefficients should be estimated by LSE fit. Note, however, that the starting hypothesis states that birefringence decreases as the spin rate increases; thus q(·) must be monotonically decreasing. The problem of LSE fitting under the constraint that q(·) is monotonic, can be cast in terms of a semidefinite programming problem and solved with standard algorithms [18]. Yet, the task is quite involved. Moreover, the solutions found by this more complex model are comparable with those provided by the other one, as shown in Fig. 1(c) for a polynomial of 10th degree (red curve, almost overlapped with the black one). Therefore, hereinafter we use, for simplicity and clarity, only the linear model.
L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Rev . 38, 49–95 (1996). [CrossRef]
Before concluding this section, we remark that the average effect introduced in sec. 2 can be quantified as the variation of the parameter 〈β〉 among a set of fibers samples drawn from the same preform.
3. Experimental results
The procedure described above has been applied to a set of 23 fiber samples, as shown in Table 2. The first column of the table indicates the fiber sample, the letter being the same of the corresponding preform. Second and third column report spin period and maximum spin rate. The last four columns report average birefringence 〈β〉, relative instantaneous effect, Δβ/〈β〉, relative average effect, δβ/〈β〉, and relative measurement uncertainty, σβ/〈β〉. In particular, the average effect δβ/〈β〉 has been calculated as the difference between 〈β〉 measured on a spun sample and 〈β〉 measured on the unspun sample drawn from the same preform, normalized to this latter value (except preform J, where the reference is the least spun sample J3). Measurement uncertainty, σβ, is the standard deviation of birefringence strength, calculate over the 10 estimates and averaged along z. If we consider samples A-E, the apparent correlation between 〈β〉 and spin rate is only due to the fact that in these samples spin rate and birefringence were selected so to spatially resolve the fast birefringence rotations and to minimize PMD [19].
L. Palmieri, T. Geisler, and A. Galtarossa, “Limits of applicability of polarization sensitive reflectometry,” Opt. Express 19, 10874–10879 (2011). [CrossRef] [PubMed]
Regarding the instantaneous effect, Δβ, we may note that the relative variation induced by the spin is always comparable with, or even smaller than, the relative uncertainty of measurement. This means that the observed variations are not significant; most important, it also means that if any instantaneous effect actually exists, it has to be no more than some percents of mean birefringence, thus quite smaller than the variation of 33–40% reported in [12]. We deem worthwhile remarking that this conclusion apply to both periodically and quasi-unidirectionally spun fibers samples, which correspond, as noted in sec. 2, to two rather different spin conditions. It is also evident from Table 2 that in all cases Δβ is negative. Apparently, and despite measurement uncertainty, this may suggest that spin induces a birefringence reduction; on the contrary, this feature is just a consequence related to measurement uncertainty. In fact, when the spin rate is at its maximum, the backscattered SOP undergoes fast and faint oscillations, which may be blurred by measurement noise, resulting in an apparent reduction of birefringence strength [19].
S. M. Pietralunga, M. Ferrario, M. Tacca, and M. Martinelli, “Local birefringence in unidirectionally spun fibers,” J. Lightwave Technol. 24, 4030–4038 (2006). [CrossRef]
L. Palmieri, T. Geisler, and A. Galtarossa, “Limits of applicability of polarization sensitive reflectometry,” Opt. Express 19, 10874–10879 (2011). [CrossRef] [PubMed]
Coming to the average effect, δβ, we may note that both positive and negative variations have been observed. Positive variations contradict the starting assumption that spin might reduce birefringence strength. Rather, they may be attributed to fiber inhomogeneity. Negative variations that are consistent with the starting hypothesis have been observed only on samples drawn from preforms G and K. In the first case, however, δβ is small and we cannot exclude that also these reductions are due to fiber inhomogeneity. Differently, in the case of preform K the observed reduction may be considered significant, despite smaller than the variations reported in ref. [12]. Although not conclusive, this result may still lead to think that spin could actually cause an average reduction of birefringence strength in G.652 fibers.
S. M. Pietralunga, M. Ferrario, M. Tacca, and M. Martinelli, “Local birefringence in unidirectionally spun fibers,” J. Lightwave Technol. 24, 4030–4038 (2006). [CrossRef]
A possible explanation of this effect can be given considering that birefringence in optical fibers is caused by stress-induced anisotropy and geometrical asymmetries [20]. As stated in [11,12], the spin process may reduce stress anisotropy. At the same time, the analysis performed in ref. [21] shows that material anisotropy is the dominant source of birefringence in SSMF (i.e. G.652) fibers, while NZDF fibers are affected most by geometrical birefringence. On the ground of this argumentation, it is reasonable to state that spin process may reduce birefringence strength in G.652 fibers, where stress effects overwhelm geometrical ones, whereas it has a negligible effect on non-G.652 fibers, where the opposite occurs. Indeed, this conclusion is consistent with the results reported in Table 2. Finally, we remark that the instantaneous effect (differently from the average one) is negligible also in G.652 fibers. This may suggest that (if any) the reduction action of spin occurs only on average.
R. Ulrich and A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt. 18, 2241–2251 (1979). [CrossRef] [PubMed]
M. Ferrario, S. M. Pietralunga, M. Torregiani, and M. Martinelli, “Modification of local stress-induced birefringence in low-PMD spun fibers evaluated by high-resolution optical tomography,” IEEE Photon. Technol. Lett. 16, 2634–2636 (2004). [CrossRef]
S. M. Pietralunga, M. Ferrario, M. Tacca, and M. Martinelli, “Local birefringence in unidirectionally spun fibers,” J. Lightwave Technol. 24, 4030–4038 (2006). [CrossRef]
D. Chowdhury and D. Wilcox, “Comparison between optical fiber birefringence induced by stress anisotropy and geometric deformation,” IEEE J. Sel. Top. Quantum Electron. 6, 227–232 (2000). [CrossRef]
4. Conclusion
In this paper we have experimentally analyzed the birefringence of a wide set of spun (spin rates up to 275 rad/m) and unspun fibers of various kind, in order to see if spin process might induce some birefringence reduction. Actually, while spin-induced PMD reduction is a fact, it is not clear if this PMD reduction is just due to birefringence rotation or is due also to a spin-induced local reduction of birefringence strength.
Our results show clearly that spin does not induce any significant reduction of the birefringence in non-G.652 fibers. Differently, we have found weak evidence that the spin might actually decrease on average the birefringence strength in G.652 fibers. This result, in agreement with previous analysis [11], may be explained considering that, first, spin can reduce stress and its related anisotropy [11, 12], and, second, G.652 fibers are more affected by stress birefringence, whereas birefringence in non-G.652 fibers is mainly due to geometrical asymmetry [21].
M. Ferrario, S. M. Pietralunga, M. Torregiani, and M. Martinelli, “Modification of local stress-induced birefringence in low-PMD spun fibers evaluated by high-resolution optical tomography,” IEEE Photon. Technol. Lett. 16, 2634–2636 (2004). [CrossRef]
M. Ferrario, S. M. Pietralunga, M. Torregiani, and M. Martinelli, “Modification of local stress-induced birefringence in low-PMD spun fibers evaluated by high-resolution optical tomography,” IEEE Photon. Technol. Lett. 16, 2634–2636 (2004). [CrossRef]
S. M. Pietralunga, M. Ferrario, M. Tacca, and M. Martinelli, “Local birefringence in unidirectionally spun fibers,” J. Lightwave Technol. 24, 4030–4038 (2006). [CrossRef]
D. Chowdhury and D. Wilcox, “Comparison between optical fiber birefringence induced by stress anisotropy and geometric deformation,” IEEE J. Sel. Top. Quantum Electron. 6, 227–232 (2000). [CrossRef]
Acknowledgments
This work has been performed under the agreement with ISCTI, Rome, Italy. Partial support from the Italian Ministry of Foreign Affairs (Direzione Generale per la Promozione del Sistema Paese), by the
Italian Ministry of Instruction, University and Research (project PRIN
2008MPSSNX), and by
University of Padova (project
NINFO) is acknowledged.
References and links
A. Galtarossa and C. R. Menyuk, Polarization Mode Dispersion (Springer, New York, 2005). [CrossRef] | |
S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16, 804–817 (2008). [CrossRef] [PubMed] | |
L. Nelson, S. Woodward, S. Foo, X. Zhou, M. Feuer, D. Hanson, D. McGhan, H. Sun, M. Moyer, M. O. Sullivan, and P. Magill, “Performance of a 46-Gb/s dual-polarization QPSK transceiver with real-time coherent equalization over high PMD fiber,” J. Lightwave Technol. 27, 158–167 (2009). [CrossRef] | |
O. Bertran-Pardo, J. Renaudier, G. Charlet, P. Tran, H. Mardoyan, M. Salsi, and S. Bigo, “Experimental assessment of interactions between nonlinear impairments and polarization-mode dispersion in 100-Gb/s coherent systems versus receiver complexity,” IEEE Photon. Technol. Lett. 21, 51–53 (2009). [CrossRef] | |
P. Serena, N. Rossi, O. Bertran-Pardo, J. Renaudier, A. Vannucci, and A. Bononi, “Intra- versus inter-channel PMD in linearly compensated coherent PDM-PSK nonlinear transmissions,” J. Lightwave Technol. 29, 1691–1700 (2011). [CrossRef] | |
A. J. Barlow, J. J. Ramskov-Hansen, and D. N. Payne, “Birefringence and polarization mode-dispersion in spun single-mode fibers,” Appl. Opt. 20, 2962–2968 (1981). [CrossRef] [PubMed] | |
M. J. Li and D. A. Nolan, “Fiber spin-profile designs for producing fibers with low polarization mode dispersion,” Opt. Lett. 23, 1659–1661 (1998). [CrossRef] | |
A. Galtarossa, L. Palmieri, and A. Pizzinat, “Optimized spinning design for low PMD fibers: an analytical approach,” J. Lightwave Technol. 19, 1502–1512 (2001). [CrossRef] | |
A. Galtarossa, L. Palmieri, and D. Sarchi, “Measure of spin period in randomly birefringent low-PMD fibers,” IEEE Photon. Technol. Lett. 16, 1131–1133 (2004). [CrossRef] | |
G. Bouquet, L. A. de Montmorillon, and P. Nouchi, “Analytical solution of polarization mode dispersion for triangular spun fibers,” Opt. Lett. 29, 2118–2120 (2004). [CrossRef] [PubMed] | |
M. Ferrario, S. M. Pietralunga, M. Torregiani, and M. Martinelli, “Modification of local stress-induced birefringence in low-PMD spun fibers evaluated by high-resolution optical tomography,” IEEE Photon. Technol. Lett. 16, 2634–2636 (2004). [CrossRef] | |
S. M. Pietralunga, M. Ferrario, M. Tacca, and M. Martinelli, “Local birefringence in unidirectionally spun fibers,” J. Lightwave Technol. 24, 4030–4038 (2006). [CrossRef] | |
L. Palmieri, “Polarization properties of spun single-mode fibers,” J. Lightwave Technol. 24, 4075–4088 (2006). [CrossRef] | |
A. Galtarossa, D. Grosso, L. Palmieri, and M. Rizzo, “Spin-profile characterization in randomly birefringent spun fibers by means of frequency-domain reflectometry,” Opt. Lett. 34, 1078–1080 (2009). [CrossRef] [PubMed] | |
L. Palmieri, T. Geisler, and A. Galtarossa, “Experimental evidences of independence between birefringence modulus and spin rate in periodically spun fibers,” in 37th European Conference on Optical Communication (ECOC) (2011), paper Th.12.LeCervin.3. | |
A. Galtarossa, D. Grosso, L. Palmieri, and L. Schenato, “Reflectometric measurement of birefringence rotation in single-mode optical fibers,” Opt. Lett. 33, 2284–2286 (2008). [CrossRef] [PubMed] | |
A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964). [CrossRef] | |
L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Rev . 38, 49–95 (1996). [CrossRef] | |
L. Palmieri, T. Geisler, and A. Galtarossa, “Limits of applicability of polarization sensitive reflectometry,” Opt. Express 19, 10874–10879 (2011). [CrossRef] [PubMed] | |
R. Ulrich and A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt. 18, 2241–2251 (1979). [CrossRef] [PubMed] | |
D. Chowdhury and D. Wilcox, “Comparison between optical fiber birefringence induced by stress anisotropy and geometric deformation,” IEEE J. Sel. Top. Quantum Electron. 6, 227–232 (2000). [CrossRef] |
OCIS Codes
(060.2270) Fiber optics and optical communications : Fiber characterization
(060.2300) Fiber optics and optical communications : Fiber measurements
ToC Category:
Fibers, Fiber Devices, and Amplifiers
History
Original Manuscript: October 3, 2011
Revised Manuscript: December 2, 2011
Manuscript Accepted: December 7, 2011
Published: December 19, 2011
Virtual Issues
European Conference on Optical Communication 2011 (2011) Optics Express
Citation
Luca Palmieri, Tommy Geisler, and Andrea Galtarossa, "Effects of spin process on birefringence strength of single-mode fibers," Opt. Express 20, 1-6 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-1-1
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References
- A. Galtarossa and C. R. Menyuk, Polarization Mode Dispersion (Springer, New York, 2005). [CrossRef]
- S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express16, 804–817 (2008). [CrossRef] [PubMed]
- L. Nelson, S. Woodward, S. Foo, X. Zhou, M. Feuer, D. Hanson, D. McGhan, H. Sun, M. Moyer, M. O. Sullivan, and P. Magill, “Performance of a 46-Gb/s dual-polarization QPSK transceiver with real-time coherent equalization over high PMD fiber,” J. Lightwave Technol.27, 158–167 (2009). [CrossRef]
- O. Bertran-Pardo, J. Renaudier, G. Charlet, P. Tran, H. Mardoyan, M. Salsi, and S. Bigo, “Experimental assessment of interactions between nonlinear impairments and polarization-mode dispersion in 100-Gb/s coherent systems versus receiver complexity,” IEEE Photon. Technol. Lett.21, 51–53 (2009). [CrossRef]
- P. Serena, N. Rossi, O. Bertran-Pardo, J. Renaudier, A. Vannucci, and A. Bononi, “Intra- versus inter-channel PMD in linearly compensated coherent PDM-PSK nonlinear transmissions,” J. Lightwave Technol.29, 1691–1700 (2011). [CrossRef]
- A. J. Barlow, J. J. Ramskov-Hansen, and D. N. Payne, “Birefringence and polarization mode-dispersion in spun single-mode fibers,” Appl. Opt.20, 2962–2968 (1981). [CrossRef] [PubMed]
- M. J. Li and D. A. Nolan, “Fiber spin-profile designs for producing fibers with low polarization mode dispersion,” Opt. Lett.23, 1659–1661 (1998). [CrossRef]
- A. Galtarossa, L. Palmieri, and A. Pizzinat, “Optimized spinning design for low PMD fibers: an analytical approach,” J. Lightwave Technol.19, 1502–1512 (2001). [CrossRef]
- A. Galtarossa, L. Palmieri, and D. Sarchi, “Measure of spin period in randomly birefringent low-PMD fibers,” IEEE Photon. Technol. Lett.16, 1131–1133 (2004). [CrossRef]
- G. Bouquet, L. A. de Montmorillon, and P. Nouchi, “Analytical solution of polarization mode dispersion for triangular spun fibers,” Opt. Lett.29, 2118–2120 (2004). [CrossRef] [PubMed]
- M. Ferrario, S. M. Pietralunga, M. Torregiani, and M. Martinelli, “Modification of local stress-induced birefringence in low-PMD spun fibers evaluated by high-resolution optical tomography,” IEEE Photon. Technol. Lett.16, 2634–2636 (2004). [CrossRef]
- S. M. Pietralunga, M. Ferrario, M. Tacca, and M. Martinelli, “Local birefringence in unidirectionally spun fibers,” J. Lightwave Technol.24, 4030–4038 (2006). [CrossRef]
- L. Palmieri, “Polarization properties of spun single-mode fibers,” J. Lightwave Technol.24, 4075–4088 (2006). [CrossRef]
- A. Galtarossa, D. Grosso, L. Palmieri, and M. Rizzo, “Spin-profile characterization in randomly birefringent spun fibers by means of frequency-domain reflectometry,” Opt. Lett.34, 1078–1080 (2009). [CrossRef] [PubMed]
- L. Palmieri, T. Geisler, and A. Galtarossa, “Experimental evidences of independence between birefringence modulus and spin rate in periodically spun fibers,” in 37th European Conference on Optical Communication (ECOC) (2011), paper Th.12.LeCervin.3.
- A. Galtarossa, D. Grosso, L. Palmieri, and L. Schenato, “Reflectometric measurement of birefringence rotation in single-mode optical fibers,” Opt. Lett.33, 2284–2286 (2008). [CrossRef] [PubMed]
- A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem.36, 1627–1639 (1964). [CrossRef]
- L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Rev. 38, 49–95 (1996). [CrossRef]
- L. Palmieri, T. Geisler, and A. Galtarossa, “Limits of applicability of polarization sensitive reflectometry,” Opt. Express19, 10874–10879 (2011). [CrossRef] [PubMed]
- R. Ulrich and A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt.18, 2241–2251 (1979). [CrossRef] [PubMed]
- D. Chowdhury and D. Wilcox, “Comparison between optical fiber birefringence induced by stress anisotropy and geometric deformation,” IEEE J. Sel. Top. Quantum Electron.6, 227–232 (2000). [CrossRef]
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