## Measurement of polarization mode dispersion vectors using the polarization-dependent signal delay method

Optics Express, Vol. 6, Issue 8, pp. 158-167 (2000)

http://dx.doi.org/10.1364/OE.6.000158

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### Abstract

We describe a new time-domain method for determining the vector components of polarization-mode dispersion from measurements of the mean signal delays for four polarization launches. Using sinusoidal amplitude modulation and sensitive phase detection, we demonstrate that the PMD vector components measured with the new method agree with results obtained from the more traditional Müller Matrix Method.

© Optical Society of America

## 1. Introduction

3. B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Technol. Lett. **4**, 1066–1069 (1992). [CrossRef]

5. R. M. Jopson, L. E. Nelson, and H. Kogelnik, “Measurement of second-order polarization-mode dispersion vectors in optical fibers,” IEEE Photon. Technol. Lett. **11**, 1153–1155 (1999). [CrossRef]

6. P. A. Williams, “Modulation phase-shift measurement of PMD using only four launched polarisation states: a new algorithm,” Electron. Lett. **35**, 1578–1579 (1999). [CrossRef]

## 2. Theory

### 2.1 Moments

_{g}, is defined by the first moment of the pulse envelope in the time domain [8–10

10. W. Shieh, “Principal states of polarization for an optical pulse,” IEEE Photon. Technol. Lett. **11**, 677–679 (1999). [CrossRef]

_{o}is the polarization-independent delay component. More precisely, τ

_{g}is expressed as the difference between the normalized first moments at the fiber output and input,

*W*=∫

*dt*

^{†}

*d*ω

*Ẽ*

^{†}

*Ẽ*is the energy of the signal pulse represented by the complex field vector

*z*,

*t*) with Fourier transform

*Ẽ*(

*z*,ω), and

*W*

_{1}(

*z*)=∫

*dtt*

^{†}

*j*∫

*d*ω

*Ẽ*

^{†}

*Ẽ*

_{ω}is the first moment. Eq. (1) assumes that τ

_{o}and

*ω*[9

9. M. Karlsson, “Polarization mode dispersion-induced pulse broadening in optical fibers,” Optics Lett. **23**, 688–690 (1998). [CrossRef]

11. J. P. Gordon and H. Kogelnik, “PMD Fundamentals: Polarization mode dispersion in optical fibers,” Proceedings of the National Academy of Sciences, Vol. 97, April 25, 2000. [CrossRef]

_{o}and the vector

*Ŝ*

_{1}, -

*Ŝ*

_{1},

*Ŝ*

_{2}, and

*Ŝ*

_{3}, with corresponding mean signal delay measurements τ

_{g1}, τ

_{g(-1)}, τ

_{g2}, and τ

_{g3}. Then τ

_{o}and the vector components of

*i*=1,2,3. Eq. (3) can be generalized to any four non-degenerate input polarization launches

_{i}(

*i*=1,2,3,4) that span Stokes space. The corresponding signal delay measurements,

*τ*

_{gi}, can be grouped into a four-dimensional (4-D) vector

_{g}=(τ

_{g1},τ

_{g2},τ

_{g3},τ

_{g4}), and similarly, the desired delay components can be grouped into a 4-D vector

_{4D}=(τ

_{0},τ

_{1},τ

_{2},τ

_{3}). The four launches together with Eq. (1) generate the required four linear equations for the four unknowns represented by

_{4D}. Thus,

_{g}=

*X*

_{4D}, where

*X*is a 4×4 matrix containing the components of the launched polarizations,

*s*

_{ij}is the jth component of polarization launch

_{i}. The equivalent of Eq. (3) is then

_{4D}=

*X*

^{-1}

_{g}, and therefore τ

_{o}and the other vector components of

_{4D}can be determined by inverting the matrix

*X*at each frequency of interest.

### 2.2 Sinusiodal modulation

_{gi}, can also be obtained by observing phase shifts of transmitted sinusoidal signals and benefiting from the precision of sensitive phase-detection techniques. (Note that Williams [6

6. P. A. Williams, “Modulation phase-shift measurement of PMD using only four launched polarisation states: a new algorithm,” Electron. Lett. **35**, 1578–1579 (1999). [CrossRef]

_{gi}with polarization and optical frequency and to not resolve the ambiguity presented by the use of a signal with periodic modulation. For a sinusoidal signal, with the assumption of frequency-independent PSP’s and DGD, we can use the dot-product rule [11

11. J. P. Gordon and H. Kogelnik, “PMD Fundamentals: Polarization mode dispersion in optical fibers,” Proceedings of the National Academy of Sciences, Vol. 97, April 25, 2000. [CrossRef]

_{out}=ω

_{m}τ

_{ϕ}, defines a signal delay τ

_{ϕ}that obeys the relation

_{m}is the angular modulation frequency. For small ω

_{m}, the signal delays τ

_{ϕ}(defined by sinuosidal phase) and τ

_{g}(defined by momenta) are approximately equal. When the four launch polarizations coincide with the Poincaré sphere axes,

*Ŝ*

_{1}, -

*Ŝ*

_{1},

*Ŝ*, and

*Ŝ*

_{3}, Eq. (3) is still valid for sinusoidal modulation:

*Δ*(the magnitude of

*p*

_{i}, of the unit vector pointing in the direction of

_{i}(

*i*=1,

*a*,

*b*,

*c*) that span Stokes space. We orient the input Stokes space so that

_{1}=

*Ŝ*

_{1}. Then, with

_{a}=(

*a*

_{1},

*a*

_{2},

*a*

_{3}),

_{b}=(

*b*

_{1},

*b*

_{2},

*b*

_{3}), and

_{c}=(

*c*

_{1},

*c*

_{2},

*c*

_{3}), we measure the corresponding delays τ

_{ϕ1}, τ

_{ϕa}, τ

_{ϕb}, and τ

_{ϕc}. We first express

*Ŝ*

_{1},

*Ŝ*

_{2}, and

*Ŝ*

_{3}, in terms of

_{a},

_{b}, and

_{c},

*α*

_{i},

*β*

_{i}, and

*γ*

_{i}are obtained from

_{a},

_{b}, and

_{c}should not lie in a common plane. If

_{a},

_{b}, and

_{c}are coplanar, for purposes of analysis, rotate to a different Stokes space with

*Ŝ*

_{1}’ aligned with one of the other launch polarizations.

_{o}:

_{0}can be obtained by linearizing Eq. (9),

_{0}and using

*p*

_{i}, of the PMD vector,

_{m}, we can approximate tan (

*x*)⋍

*x*in Eq. (5) and use the procedure outlined above (i.e. Eq. (3–4)). These linear expressions are valid for sinusoidal modulation to within 6% as long as ω

_{m}Δτ<π/4. For instance, for the peak DGD we observed here, 88 ps, better than 6% accuracy will be obtained for modulation frequencies, f

_{m}=ω

_{m}/2π, less than 1.5 GHz.

## 3. Experiment

_{3}modulator typically operating at 1 GHz with an extinction ratio of 1/3. A wavemeter monitored the carrier wavelength. Three of the four launch polarizations were sequentially provided by a portion of a commercial polarization analyzer instrument containing three linear polarizers that could be switched into the beam. The fourth polarization was set by polarization controller 2 (PC2) and obtained by switching all three polarizers out of the signal path. In principle, the fourth polarization could have been provided by a polarizer as well. The phase of the modulation was measured with a network analyzer. The apparatus contained a polarimeter to allow PMD measurements by the MMM to verify the accuracy of the PSD method. The polarimeter also provided a convenient monitor when adjusting PC2.

*Ŝ*

_{1}), 60.4°, and 120.6°, and right-circular polarization,

*Ŝ*

_{3}. Figure 3 shows the wavelength dependence of the signal delays for these four inputs in a fiber span with 35-ps mean DGD. The span included an 8-km length of high-PMD dispersion compensating fiber and a 54-km length of standard fiber to compensate the chromatic dispersion to a residual dispersion of +124 ps/nm at 1542 nm. The fiber’s PMD causes the polarization dependence of the signal delays, and the variations in these relative signal delay curves originate from the change of direction and magnitude of

*Ŝ*

_{1}, 60.4°, and 120.6° delay curves shown in Fig. 3 have been corrected for the ≈2 ps additional delay caused by the thickness of the polarizers. The data in Fig. 3, however, have not been corrected for drift caused by changes in the optical length of the fiber during the course of the measurement.

## 4. Results and Discussion

_{0R}, for the fiber span, obtained from the data of Fig. 3. We used the linear approximation to the tangent, i.e. Eq. (4) and

_{ϕ}=(τ

_{ϕ1},τ

_{ϕ2},τ

_{ϕ3},τ

_{ϕ4}). Here, τ

_{0R}is the polarization-independent delay through the fiber span at each wavelength compared to the delay at the center wavelength, 1542.0 nm. The modulation frequency was 1.0 GHz, providing ω

_{m}Δτ=0.55 at the maximum DGD. The measurement points were separated by 0.02 nm. We show a scan of only 1 nm so that the magnitude and components of

_{0R}computed from the 3-GHz data using the exact expression and the linear approximation. For this case, ω

_{m}Δτ=1.66 at the maximum DGD. The largest difference between the two DGD curves was 2.79 ps at 1542.14 nm, corresponding to a 5.3% difference. Note that with a 3-GHz modulation frequency, the modulation sidebands are spread out over 48 pm, so there is averaging of the signal delay over this bandwidth.

## 5. Conclusion

## 6. Appendix

*S*

_{3}=+1 for right-circular polarization), conforming with the traditional optics literature and the available measurement instrumentation. The discussion of the Müller Matrix Method (MMM) in Ref. [5

5. R. M. Jopson, L. E. Nelson, and H. Kogelnik, “Measurement of second-order polarization-mode dispersion vectors in optical fibers,” IEEE Photon. Technol. Lett. **11**, 1153–1155 (1999). [CrossRef]

*S*

_{3}=+1 for left-circular polarization) and provides an algorithm for the PMD vector at the fiber output. This appendix provides a bridge between the two.

11. J. P. Gordon and H. Kogelnik, “PMD Fundamentals: Polarization mode dispersion in optical fibers,” Proceedings of the National Academy of Sciences, Vol. 97, April 25, 2000. [CrossRef]

*R*

_{0}and

*R*

_{+}at two adjacent optical frequencies ω

_{0}and ω

_{+}=ω

_{0}+Δω [5

5. R. M. Jopson, L. E. Nelson, and H. Kogelnik, “Measurement of second-order polarization-mode dispersion vectors in optical fibers,” IEEE Photon. Technol. Lett. **11**, 1153–1155 (1999). [CrossRef]

*R̃*denotes the transpose of

*R*. From this

*R*

_{Δ}the MMM extracts the output PMD vector

_{0}+Δω/2.

_{s}, as

*R*(ω) is the fiber’s rotation matrix at ω given by

*R*

_{Δs}at the fiber input is the matrix transform of

*R*

_{Δ},

*R*

_{Δ}and

*R*

_{Δs}.) The PMD vectors

_{s}are extracted from Eqs. (A2) and (A6) following the MMM procedure sketched in the following. Note that the expressions for

*R*

_{Δ}and

*R*

_{Δs}have the same form for both right- and left-circular Stokes space, while their actual components are different for the two spaces.

*R*

_{Δ}, which is different for the two spaces,

*I*is the 3×3 unit matrix,

**11**, 1153–1155 (1999). [CrossRef]

## References and links

1. | C. D. Poole and J. A. Nagel, “Polarization effects in lightwave systems,” in |

2. | F. Heismann, “Polarization mode dispersion: fundamentals and impact on optical communication systems,” ECOC’98 Digest , |

3. | B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Technol. Lett. |

4. | L. E. Nelson, R. M. Jopson, and H. Kogelnik, “Müller matrix method for determining polarization mode dispersion vectors,” ECOC ’99 Digest , |

5. | R. M. Jopson, L. E. Nelson, and H. Kogelnik, “Measurement of second-order polarization-mode dispersion vectors in optical fibers,” IEEE Photon. Technol. Lett. |

6. | P. A. Williams, “Modulation phase-shift measurement of PMD using only four launched polarisation states: a new algorithm,” Electron. Lett. |

7. | R. M. Jopson, L. E. Nelson, H. Kogelnik, and J. P. Gordon, “Polarization-dependent signal delay method for measuring polarization mode dispersion vectors,” LEOS’99 Postdeadline paper, PD1.1, San Francisco, CA (1999). |

8. | L. F. Mollenauer and J. P. Gordon, “Birefringence-mediated timing jitter in soliton transmission,” Optics Lett. |

9. | M. Karlsson, “Polarization mode dispersion-induced pulse broadening in optical fibers,” Optics Lett. |

10. | W. Shieh, “Principal states of polarization for an optical pulse,” IEEE Photon. Technol. Lett. |

11. | J. P. Gordon and H. Kogelnik, “PMD Fundamentals: Polarization mode dispersion in optical fibers,” Proceedings of the National Academy of Sciences, Vol. 97, April 25, 2000. [CrossRef] |

**OCIS Codes**

(060.2270) Fiber optics and optical communications : Fiber characterization

(060.2300) Fiber optics and optical communications : Fiber measurements

(060.4510) Fiber optics and optical communications : Optical communications

(260.5430) Physical optics : Polarization

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 31, 2000

Published: April 10, 2000

**Citation**

Lynn Nelson, Robert Jopson, Herwig Kogelnik, and James Gordon, "Measurement of polarization mode dispersion vectors using the polarization-dependent signal delay method," Opt. Express **6**, 158-167 (2000)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-6-8-158

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### References

- C. D. Poole and J. A. Nagel, "Polarization effects in lightwave systems," in Optical Fiber Telecommunications IIIA, I. P. Kaminow and T. L. Koch, eds. (Academic Press, San Diego, 1997), pp. 114-161.
- F. Heismann, "Polarization mode dispersion: fundamentals and impact on optical communication systems," ECOC'98 Digest, Vol. 2, Tutorials, pp. 51-79, Madrid, (1998).
- B. L. Heffner, "Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis," IEEE Photon. Technol. Lett. 4, 1066-1069 (1992). [CrossRef]
- L. E. Nelson, R. M. Jopson, and H. Kogelnik, "M�matrix method for determining polarization mode dispersion vectors," ECOC '99 Digest, Vol. II, pp. 10-11, Nice, (1999).
- R. M. Jopson, L. E. Nelson, and H. Kogelnik, "Measurement of second-order polarization-mode dispersion vectors in optical fibers," IEEE Photon. Technol. Lett. 11, 1153-1155 (1999). [CrossRef]
- P. A. Williams, "Modulation phase-shift measurement of PMD using only four launched polarisation states: a new algorithm," Electron. Lett. 35, 1578-1579 (1999). [CrossRef]
- R. M. Jopson, L. E. Nelson, H. Kogelnik, and J. P. Gordon, "Polarization-dependent signal delay method for measuring polarization mode dispersion vectors," LEOS'99 Postdeadline paper, PD1.1, San Francisco, CA (1999).
- L. F. Mollenauer, and J. P. Gordon, "Birefringence-mediated timing jitter in soliton transmission," Optics Lett. 19, 375-377 (1994).
- M. Karlsson, "Polarization mode dispersion-induced pulse broadening in optical fibers," Optics Lett. 23, 688-690 (1998). [CrossRef]
- W. Shieh, "Principal states of polarization for an optical pulse," IEEE Photon. Technol. Lett. 11, 677-679 (1999). [CrossRef]
- J. P. Gordon and H. Kogelnik, "PMD Fundamentals: Polarization mode dispersion in optical fibers," Proceedings of the National Academy of Sciences, Vol. 97, April 25, 2000. [CrossRef]

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