Abstract

The concurrent realization of multiple polarization-based link impairments such as polarization-mode dispersion, polarization-dependent loss, and differential-attenuation slope creates a nontrivial measurement and analysis problem. The difficulties concerning the robustness of raw data, of minimizing drift artifacts experienced during multiple wavelength measurements, and the analysis methods that lead to physical significant interpretations are addressed. Measurements of an in-service wavelength-division-multiplexed metro-area network are presented that explicitly illustrate the limitations when using industry-standard commercial test equipment.

© 2004 Optical Society of America

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References

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  1. P. Hernday, “Measurement of polarization effects in lightwave systems,” presented at the International Conference on Integrated Optics and Optical Fiber Communication, San Diego, Calif., February 21, 1999.
  2. D. Derickson, ed., Fiber Optic Test and Measurement (Prentice-Hall, New York, 1998).
  3. C. J. K. Richardson, R. J. Runser, M. Goodman, and L. Mercer, “Statistical evaluation of polarization-dependent losses and polarization-mode dispersion in an installed fiber networkD,” NIST Spec. Publ. 988 (2002).
  4. D. S. Waddy, C. Liang, and B. Xiaoyi, “Theoretical and experimental study of the dynamics of polarization-mode dispersion,” IEEE Photonics Technol. Lett. 14, 468–470 (2002).
    [CrossRef]
  5. T. Takahashi, T. Imai, and M. Iki, “Time evolution of pmd in 120 km installed optical submarine cable,” Electron. Lett. 29, 1605 (1993).
    [CrossRef]
  6. H. Bulow and G. Veith, “Temporal dynamics of error-rate degradation induced by polarisation mode dispersion fluctuation of a field fiber link,” presented at the 23rd European Conference on Optical Communications, Edinburgh, UK, September 22–25 (1997).
  7. M. Brodsky, P. Maggill, and N. Frigo, “Evidence of parametric dependence of pmd on temperature in installed 0.05 ps/km1/2,” presented at the European Conference on Optical Communications, Copenhagen, Denmark, September 2002.
  8. R. Caponi, B. Riposati, A. Rossaro, and M. Schiano, “WDM system impairments due to highly-correlated pmd spectra of buried optical cables,” Electron. Lett. 38, 737–738 (2002).
    [CrossRef]
  9. B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent loss,” IEEE J. Sel. Top. Quantum Electron. 6, 317–329 (2000).
    [CrossRef]
  10. B. Huttner, C. DeBarros, B. Gisin, and N. Gisin, “Polarization-induced pulse spreading in birefringent optical fibers with zero differential group delay,” Opt. Lett. 24, 370–372 (1999).
    [CrossRef]
  11. R. Doverspike, M. Maeda, S. Narain, J. Pastor, Chien-Chung Shen, N. Stoffel, Yukun Tsai, and B. Wilson, “Network management research in atdnet,” IEEE Network 10, 30–41 (1996).
    [CrossRef]
  12. B. L. Heffner, “Automated measurements of polarization-mode dispersion using Jones matrix eigenanalysis,” IEEE Photonics Technol. Lett. 5, 814 (1993).
    [CrossRef]
  13. P. A. Williams, A. J. Barlow, C. Mackechnie, and J. B. Schlager, “Narrowband measurements of polarization-mode dispersion using the modulation phase shift technique,” NIST Spec. Publ. 930, 23–26 (1998).
  14. C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization-mode dispersion in long single-mode fibers,” Electron. Lett. 22, 1029–1030 (1986).
    [CrossRef]
  15. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. (U.S.A.) 97, 4541–4550 (2000).
    [CrossRef]
  16. B. Nyman, “Automated system for measuring polarization-dependent loss,” Conference on Optical Fiber Communication, Vol. 4 of 1994 OSA OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 230.
  17. R. Craig, S. Gilbert, and P. Hale, “Accurate polarization dependent loss measurement and calibration standard development,” NIST Spec. Publ. 930, 5–8 (1998).
  18. B. L. Heffner, “Deterministic, analytically complete measurement of polarization-dependent transmission through optical devices,” IEEE Photonics Technol. Lett. 4, 451–454 (1992).
    [CrossRef]
  19. Y. Li and A. Yariv, “Solutions to the dynamical equation of polarization-mode dispersion and polarization-dependent losses,” J. Opt. Soc. Am. B 17, 1821–1827 (2000).
    [CrossRef]
  20. P. Andrekson, M. Karlsson, and J. Brentel, “Long-term measurement of pmd and polarization drift in installed fibers,” J. Lightwave Technol. 18, 941–951 (2000).
    [CrossRef]
  21. D. A. Holmes, “Exact theory of retardation plates,” J. Opt. Soc. Am. 54, 1115 (1964).
    [CrossRef]
  22. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998).
  23. R. C. Jones, “A new calculus for the treatment of optical systems. VI. Experimental determination of the matrix,” J. Opt. Soc. Am. 37, 110–112 (1947).
    [CrossRef]
  24. P. J. Leo, G. R. Gray, G. J. Simer, and K. B. Rochford, “State of polarization changes: classification and measurement,” J. Lightwave Technol. 21, 2189–2193 (2003).
    [CrossRef]
  25. Hewlett-Packard Company, HP 8509A/B Lightwave Polarization Analyzer User’s/Reference Guide (Hewlett-Packard Company, Santa Rosa, Calif., 1994).

2003 (1)

2002 (2)

R. Caponi, B. Riposati, A. Rossaro, and M. Schiano, “WDM system impairments due to highly-correlated pmd spectra of buried optical cables,” Electron. Lett. 38, 737–738 (2002).
[CrossRef]

D. S. Waddy, C. Liang, and B. Xiaoyi, “Theoretical and experimental study of the dynamics of polarization-mode dispersion,” IEEE Photonics Technol. Lett. 14, 468–470 (2002).
[CrossRef]

2000 (4)

B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent loss,” IEEE J. Sel. Top. Quantum Electron. 6, 317–329 (2000).
[CrossRef]

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. (U.S.A.) 97, 4541–4550 (2000).
[CrossRef]

Y. Li and A. Yariv, “Solutions to the dynamical equation of polarization-mode dispersion and polarization-dependent losses,” J. Opt. Soc. Am. B 17, 1821–1827 (2000).
[CrossRef]

P. Andrekson, M. Karlsson, and J. Brentel, “Long-term measurement of pmd and polarization drift in installed fibers,” J. Lightwave Technol. 18, 941–951 (2000).
[CrossRef]

1999 (1)

1998 (2)

P. A. Williams, A. J. Barlow, C. Mackechnie, and J. B. Schlager, “Narrowband measurements of polarization-mode dispersion using the modulation phase shift technique,” NIST Spec. Publ. 930, 23–26 (1998).

R. Craig, S. Gilbert, and P. Hale, “Accurate polarization dependent loss measurement and calibration standard development,” NIST Spec. Publ. 930, 5–8 (1998).

1996 (1)

R. Doverspike, M. Maeda, S. Narain, J. Pastor, Chien-Chung Shen, N. Stoffel, Yukun Tsai, and B. Wilson, “Network management research in atdnet,” IEEE Network 10, 30–41 (1996).
[CrossRef]

1993 (2)

B. L. Heffner, “Automated measurements of polarization-mode dispersion using Jones matrix eigenanalysis,” IEEE Photonics Technol. Lett. 5, 814 (1993).
[CrossRef]

T. Takahashi, T. Imai, and M. Iki, “Time evolution of pmd in 120 km installed optical submarine cable,” Electron. Lett. 29, 1605 (1993).
[CrossRef]

1992 (1)

B. L. Heffner, “Deterministic, analytically complete measurement of polarization-dependent transmission through optical devices,” IEEE Photonics Technol. Lett. 4, 451–454 (1992).
[CrossRef]

1986 (1)

C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization-mode dispersion in long single-mode fibers,” Electron. Lett. 22, 1029–1030 (1986).
[CrossRef]

1964 (1)

1947 (1)

Andrekson, P.

Barlow, A. J.

P. A. Williams, A. J. Barlow, C. Mackechnie, and J. B. Schlager, “Narrowband measurements of polarization-mode dispersion using the modulation phase shift technique,” NIST Spec. Publ. 930, 23–26 (1998).

Brentel, J.

Caponi, R.

R. Caponi, B. Riposati, A. Rossaro, and M. Schiano, “WDM system impairments due to highly-correlated pmd spectra of buried optical cables,” Electron. Lett. 38, 737–738 (2002).
[CrossRef]

Craig, R.

R. Craig, S. Gilbert, and P. Hale, “Accurate polarization dependent loss measurement and calibration standard development,” NIST Spec. Publ. 930, 5–8 (1998).

DeBarros, C.

Doverspike, R.

R. Doverspike, M. Maeda, S. Narain, J. Pastor, Chien-Chung Shen, N. Stoffel, Yukun Tsai, and B. Wilson, “Network management research in atdnet,” IEEE Network 10, 30–41 (1996).
[CrossRef]

Geiser, C.

B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent loss,” IEEE J. Sel. Top. Quantum Electron. 6, 317–329 (2000).
[CrossRef]

Gilbert, S.

R. Craig, S. Gilbert, and P. Hale, “Accurate polarization dependent loss measurement and calibration standard development,” NIST Spec. Publ. 930, 5–8 (1998).

Gisin, B.

Gisin, N.

B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent loss,” IEEE J. Sel. Top. Quantum Electron. 6, 317–329 (2000).
[CrossRef]

B. Huttner, C. DeBarros, B. Gisin, and N. Gisin, “Polarization-induced pulse spreading in birefringent optical fibers with zero differential group delay,” Opt. Lett. 24, 370–372 (1999).
[CrossRef]

Gordon, J. P.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. (U.S.A.) 97, 4541–4550 (2000).
[CrossRef]

Gray, G. R.

Hale, P.

R. Craig, S. Gilbert, and P. Hale, “Accurate polarization dependent loss measurement and calibration standard development,” NIST Spec. Publ. 930, 5–8 (1998).

Heffner, B. L.

B. L. Heffner, “Automated measurements of polarization-mode dispersion using Jones matrix eigenanalysis,” IEEE Photonics Technol. Lett. 5, 814 (1993).
[CrossRef]

B. L. Heffner, “Deterministic, analytically complete measurement of polarization-dependent transmission through optical devices,” IEEE Photonics Technol. Lett. 4, 451–454 (1992).
[CrossRef]

Holmes, D. A.

Huttner, B.

B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent loss,” IEEE J. Sel. Top. Quantum Electron. 6, 317–329 (2000).
[CrossRef]

B. Huttner, C. DeBarros, B. Gisin, and N. Gisin, “Polarization-induced pulse spreading in birefringent optical fibers with zero differential group delay,” Opt. Lett. 24, 370–372 (1999).
[CrossRef]

Iki, M.

T. Takahashi, T. Imai, and M. Iki, “Time evolution of pmd in 120 km installed optical submarine cable,” Electron. Lett. 29, 1605 (1993).
[CrossRef]

Imai, T.

T. Takahashi, T. Imai, and M. Iki, “Time evolution of pmd in 120 km installed optical submarine cable,” Electron. Lett. 29, 1605 (1993).
[CrossRef]

Jones, R. C.

Karlsson, M.

Kogelnik, H.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. (U.S.A.) 97, 4541–4550 (2000).
[CrossRef]

Leo, P. J.

Li, Y.

Liang, C.

D. S. Waddy, C. Liang, and B. Xiaoyi, “Theoretical and experimental study of the dynamics of polarization-mode dispersion,” IEEE Photonics Technol. Lett. 14, 468–470 (2002).
[CrossRef]

Mackechnie, C.

P. A. Williams, A. J. Barlow, C. Mackechnie, and J. B. Schlager, “Narrowband measurements of polarization-mode dispersion using the modulation phase shift technique,” NIST Spec. Publ. 930, 23–26 (1998).

Maeda, M.

R. Doverspike, M. Maeda, S. Narain, J. Pastor, Chien-Chung Shen, N. Stoffel, Yukun Tsai, and B. Wilson, “Network management research in atdnet,” IEEE Network 10, 30–41 (1996).
[CrossRef]

Narain, S.

R. Doverspike, M. Maeda, S. Narain, J. Pastor, Chien-Chung Shen, N. Stoffel, Yukun Tsai, and B. Wilson, “Network management research in atdnet,” IEEE Network 10, 30–41 (1996).
[CrossRef]

Pastor, J.

R. Doverspike, M. Maeda, S. Narain, J. Pastor, Chien-Chung Shen, N. Stoffel, Yukun Tsai, and B. Wilson, “Network management research in atdnet,” IEEE Network 10, 30–41 (1996).
[CrossRef]

Poole, C. D.

C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization-mode dispersion in long single-mode fibers,” Electron. Lett. 22, 1029–1030 (1986).
[CrossRef]

Riposati, B.

R. Caponi, B. Riposati, A. Rossaro, and M. Schiano, “WDM system impairments due to highly-correlated pmd spectra of buried optical cables,” Electron. Lett. 38, 737–738 (2002).
[CrossRef]

Rochford, K. B.

Rossaro, A.

R. Caponi, B. Riposati, A. Rossaro, and M. Schiano, “WDM system impairments due to highly-correlated pmd spectra of buried optical cables,” Electron. Lett. 38, 737–738 (2002).
[CrossRef]

Schiano, M.

R. Caponi, B. Riposati, A. Rossaro, and M. Schiano, “WDM system impairments due to highly-correlated pmd spectra of buried optical cables,” Electron. Lett. 38, 737–738 (2002).
[CrossRef]

Schlager, J. B.

P. A. Williams, A. J. Barlow, C. Mackechnie, and J. B. Schlager, “Narrowband measurements of polarization-mode dispersion using the modulation phase shift technique,” NIST Spec. Publ. 930, 23–26 (1998).

Shen, Chien-Chung

R. Doverspike, M. Maeda, S. Narain, J. Pastor, Chien-Chung Shen, N. Stoffel, Yukun Tsai, and B. Wilson, “Network management research in atdnet,” IEEE Network 10, 30–41 (1996).
[CrossRef]

Simer, G. J.

Stoffel, N.

R. Doverspike, M. Maeda, S. Narain, J. Pastor, Chien-Chung Shen, N. Stoffel, Yukun Tsai, and B. Wilson, “Network management research in atdnet,” IEEE Network 10, 30–41 (1996).
[CrossRef]

Takahashi, T.

T. Takahashi, T. Imai, and M. Iki, “Time evolution of pmd in 120 km installed optical submarine cable,” Electron. Lett. 29, 1605 (1993).
[CrossRef]

Tsai, Yukun

R. Doverspike, M. Maeda, S. Narain, J. Pastor, Chien-Chung Shen, N. Stoffel, Yukun Tsai, and B. Wilson, “Network management research in atdnet,” IEEE Network 10, 30–41 (1996).
[CrossRef]

Waddy, D. S.

D. S. Waddy, C. Liang, and B. Xiaoyi, “Theoretical and experimental study of the dynamics of polarization-mode dispersion,” IEEE Photonics Technol. Lett. 14, 468–470 (2002).
[CrossRef]

Wagner, R. E.

C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization-mode dispersion in long single-mode fibers,” Electron. Lett. 22, 1029–1030 (1986).
[CrossRef]

Williams, P. A.

P. A. Williams, A. J. Barlow, C. Mackechnie, and J. B. Schlager, “Narrowband measurements of polarization-mode dispersion using the modulation phase shift technique,” NIST Spec. Publ. 930, 23–26 (1998).

Wilson, B.

R. Doverspike, M. Maeda, S. Narain, J. Pastor, Chien-Chung Shen, N. Stoffel, Yukun Tsai, and B. Wilson, “Network management research in atdnet,” IEEE Network 10, 30–41 (1996).
[CrossRef]

Xiaoyi, B.

D. S. Waddy, C. Liang, and B. Xiaoyi, “Theoretical and experimental study of the dynamics of polarization-mode dispersion,” IEEE Photonics Technol. Lett. 14, 468–470 (2002).
[CrossRef]

Yariv, A.

Electron. Lett. (3)

T. Takahashi, T. Imai, and M. Iki, “Time evolution of pmd in 120 km installed optical submarine cable,” Electron. Lett. 29, 1605 (1993).
[CrossRef]

R. Caponi, B. Riposati, A. Rossaro, and M. Schiano, “WDM system impairments due to highly-correlated pmd spectra of buried optical cables,” Electron. Lett. 38, 737–738 (2002).
[CrossRef]

C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization-mode dispersion in long single-mode fibers,” Electron. Lett. 22, 1029–1030 (1986).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent loss,” IEEE J. Sel. Top. Quantum Electron. 6, 317–329 (2000).
[CrossRef]

IEEE Network (1)

R. Doverspike, M. Maeda, S. Narain, J. Pastor, Chien-Chung Shen, N. Stoffel, Yukun Tsai, and B. Wilson, “Network management research in atdnet,” IEEE Network 10, 30–41 (1996).
[CrossRef]

IEEE Photonics Technol. Lett. (3)

B. L. Heffner, “Automated measurements of polarization-mode dispersion using Jones matrix eigenanalysis,” IEEE Photonics Technol. Lett. 5, 814 (1993).
[CrossRef]

D. S. Waddy, C. Liang, and B. Xiaoyi, “Theoretical and experimental study of the dynamics of polarization-mode dispersion,” IEEE Photonics Technol. Lett. 14, 468–470 (2002).
[CrossRef]

B. L. Heffner, “Deterministic, analytically complete measurement of polarization-dependent transmission through optical devices,” IEEE Photonics Technol. Lett. 4, 451–454 (1992).
[CrossRef]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (1)

NIST Spec. Publ. (2)

R. Craig, S. Gilbert, and P. Hale, “Accurate polarization dependent loss measurement and calibration standard development,” NIST Spec. Publ. 930, 5–8 (1998).

P. A. Williams, A. J. Barlow, C. Mackechnie, and J. B. Schlager, “Narrowband measurements of polarization-mode dispersion using the modulation phase shift technique,” NIST Spec. Publ. 930, 23–26 (1998).

Opt. Lett. (1)

Proc. Natl. Acad. Sci. (U.S.A.) (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. (U.S.A.) 97, 4541–4550 (2000).
[CrossRef]

Other (8)

B. Nyman, “Automated system for measuring polarization-dependent loss,” Conference on Optical Fiber Communication, Vol. 4 of 1994 OSA OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 230.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998).

H. Bulow and G. Veith, “Temporal dynamics of error-rate degradation induced by polarisation mode dispersion fluctuation of a field fiber link,” presented at the 23rd European Conference on Optical Communications, Edinburgh, UK, September 22–25 (1997).

M. Brodsky, P. Maggill, and N. Frigo, “Evidence of parametric dependence of pmd on temperature in installed 0.05 ps/km1/2,” presented at the European Conference on Optical Communications, Copenhagen, Denmark, September 2002.

P. Hernday, “Measurement of polarization effects in lightwave systems,” presented at the International Conference on Integrated Optics and Optical Fiber Communication, San Diego, Calif., February 21, 1999.

D. Derickson, ed., Fiber Optic Test and Measurement (Prentice-Hall, New York, 1998).

C. J. K. Richardson, R. J. Runser, M. Goodman, and L. Mercer, “Statistical evaluation of polarization-dependent losses and polarization-mode dispersion in an installed fiber networkD,” NIST Spec. Publ. 988 (2002).

Hewlett-Packard Company, HP 8509A/B Lightwave Polarization Analyzer User’s/Reference Guide (Hewlett-Packard Company, Santa Rosa, Calif., 1994).

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Figures (11)

Fig. 1
Fig. 1

Comparison of pulse distortion when passing through a PMD and DAS components that are oriented at 45° with respect to each other. The Gaussian pulse is introduced at the median angle between the components. The output when τ0=30 ps and χ0=30 ps (solid), when τ0=30 ps and χ0=0 ps (dashed), and when τ0=0 ps and χ0=30 ps (dotted). Even though the JME yields zero impairments for the solid curve, it is apparent that the pulses propagating through this simple system are degraded more than the case of only DGD.

Fig. 2
Fig. 2

Comparison between analysis techniques for a simple first-order transmission matrix composed of DGD and DAS that are rotated with respect to each other. The results are plotted for the JME of the measured Jones matrix (dashed), JME following polar decomposition (dotted), and Pauli-matrix equivalence (solid).

Fig. 3
Fig. 3

Measurement scheme typically followed while making wavelength-dependent measurements of Jones matrices. Mij represent measurements, and Iij indicate interpolated Jones matrices that are used for isochronal analysis.

Fig. 4
Fig. 4

Reduction in the DGD error and DAS error associated with dynamic rotation of the SOP during measurement τ=1 ps and χ=1 ps. Raw data (gray) and the same analysis using interpolated values for isochronal analysis of the Jones matrices (black) show significant reduction in the aliasing caused by a simple rotation of the SOP.

Fig. 5
Fig. 5

Reduction in the DGD error and DAS error associated with dynamic rotation of the SOP during measurement, for a Jones matrix with τ=10 ps and χ=1 ps. Raw data (gray) and the same analysis using interpolated values for isochronal analysis of the Jones matrices (black) show significant reduction in the aliasing caused by a simple rotation of the SOP.

Fig. 6
Fig. 6

Scatter plot of the simulated measurement of the PDL determined from the dynamic Monte Carlo simulation. The increased scatter at higher rotations of the SOP demonstrate the dynamic artifact resulting when the Jones-matrix measurement time is the same order of magnitude as the dynamic effects.

Fig. 7
Fig. 7

Scatter plot of the averaged instantaneous PDL determined from the dynamic Monte Carlo simulation. The independence of the output state of rotation demonstrates the independence of the dynamic evolution of the fiber realization and the actual PDL.

Fig. 8
Fig. 8

Configuration of the WDM metro network that was analyzed for polarization-based impairments. The analysis system was at one node, passively multiplexed into the WDM trunk, separated again at the return node, and returned through a separate fiber in the same fiber bundle.

Fig. 9
Fig. 9

Magnitudes of the DGD, DAS, and PDL vectors are plotted along with the region’s wind speed. Measurements of all quantities are collected with a resolution of approximately one minute. The correlation between the wind and the polarization-based impairments is a clear indication of the need to account for all possible complications when conducting link-assessment experiments.

Fig. 10
Fig. 10

Statistics of the SOP rotation occurring in 1-min intervals during the course of the measurement window. This amount of rotation should be easily handled using an isochronal analysis and data interperpolation.

Fig. 11
Fig. 11

Scatter plot of the measured PDL and rotation of the SOP showing a correlation that indicates the influence of the average wind speed on measurements of the Jones matrices.

Tables (1)

Tables Icon

Table 1 Linear Regression of the Polarization-Dependent Loss and Rotation in the Output State of Polarization

Equations (42)

Equations on this page are rendered with MathJax. Learn more.

Sω=Ω×S,
Sω=Ω×S-(Λ×S)×S,
J=VRLR-1,
V=exp(-iτ0ω/2)00exp(iτ0ω/2),
R=cos(ϕ)-sin(ϕ)sin(ϕ)cos(ϕ),
L=exp(-χ0ω/2)00exp(χ0ω/2).
A(ω)=cos(θ)sin(θ)exp[-(ω/ωτ)2],
J=HU,orJ=VL.
jJωJ-1=jVωV-1+jVLωL-1V-1.
jLωL-1=jRωR-1+jRAωA-1R-1+jRARω-1RA-1R-1.
jVωV-1=-12 Ω·σ,
jVLωL-1V-1=-j2 Λ·σ.
σ=(σ1, σ2, σ3).
σ1=100-1,σ2=0110,σ3=0-ii0.
jJωJ-1=jHωH-1+jHUωU-1H-1
Jm(ω)=C(ω)a(ω)+ib(ω)c(ω)+id(ω)e(ω)+if(ω)1,
J(ω)=expk=03[ak(ω)+iϕk(ω)]σk.
ξ1(ω),ξ2(ω)=exp12 [a0(ω)+iϕ0(ω)]±k=13[ak(ω)+iϕk(ω)]2.
J(ω)=a(ω)Jm(ω)=exp-ln[ξ1(ω)]+ln[ξ2(ω)]2Jm(ω).
JωJ-1=k=13 12 ω[ak(ω)+iϕk(ω)]σk.
Ω1=-2I([JωJ-1]11),
Ω2=-I([JωJ-1]21+[JωJ-1]12),
Ω3=-R([JωJ-1]21-[JωJ-1]12).
Λ1=2R([JωJ-1]11),
Λ2=R([JωJ-1]21+[JωJ-1]12),
Λ3=I([JωJ-1]21-[JωJ-1]12),
τ=|Ω|,
χ=|Λ|.
dS=Sω dω+Stdt,
dS=s|σ|ss|sωdω+s|σ|ss|stdt.
M=m0I+m·σ,
s|a×σ|s=a×s|σ|s=a×σ,
s|a|s=a×s|s=a|s|.
dS=[Ω×S-(Λ×S)×S]dω+[Ψ×S-(Υ×S)×S]dt.
τerror=J(ωm, tn+1)-J(ωm, tn)δω × J(ωm, tn+1)+J(ωm, tn)2-1,
J(m+δt, n)=J(m, n)+δtJc+δt22Jc+[δt(δt2-1)]6Jc,
PDL=10 logβ0+β12+β22+β32β0-β12+β22+β32,
β0=12(|J11|2+|J12|2+|J21|2+|J22|2),
β1=12(|J11|2-|J12|2+|J21|2-|J22|2),
β2=12 R(J11*J12+J21*J22),
β3=12 I(J11*J12+J21*J22),
βˆ=(β1, β2, β3)β12+β22+β32.

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