## Limits of applicability of polarization sensitive reflectometry |

Optics Express, Vol. 19, Issue 11, pp. 10874-10879 (2011)

http://dx.doi.org/10.1364/OE.19.010874

Acrobat PDF (973 KB)

### Abstract

We present a detailed theoretical analysis of the measurement limits of polarization sensitive reflectometry, imposed by spatial resolution and measurement accuracy. The limits are conveniently represented in a map of constraints. We also describe and experimentally verify a procedure that allows to measure spin profiles of single-mode fibers with spin rates exceeding the measurable range of the reflectometer. The technique consists in twisting the fiber to locally unwind the spin.

© 2011 OSA

## 1. Introduction

1. A. J. Rogers, “Polarization-optical time domain reflectometry: a technique for the measurement of field distributions,” Appl. Opt. **20**, 1060–1074 (1981). [CrossRef] [PubMed]

2. A. Galtarossa and C. R. Menyuk, eds., *Polarization Mode Dispersion* (Springer, 2005). [CrossRef]

11. A. Galtarossa, D. Grosso, and L. Palmieri, “Accurate characterization of twist-induced optical activity in single-mode fibers by means of polarization-sensitive reflectometry,” IEEE Photon. Technol. Lett. **21**, 1713–1715 (2009). [CrossRef]

12. L. Palmieri, “Polarization properties of spun single-mode fibers,” J. Lightwave Technol. **24**, 4075–4088 (2006). [CrossRef]

10. 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]

## 2. Analysis of the constraints

3. R. E. Schuh, X. Shan, and A. S. Siddiqui, “Polarization mode dispersion in spun fibers with different linear birefringence and spinning parameters,” J. Lightwave Technol. **16**, 1583–1588 (1998). [CrossRef]

14. 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]

**R**(

*z*) representing forward propagation as a function of

*z*is described by the differential equation ∂

**R**

*=*

_{z}**b**×

**R**, where

**b**= (

*β*, 0, −2

*α*)

*is the*

^{T}*equivalent birefringence vector*in the rotated frame,

*β*is the constant birefringence and

*α*is the constant spin rate (i.e. physical angle of rotation per unit length). Accordingly,

**R**(

*z*) can be written as

**R**(

*z*) =

**I**+ (

**b**×)(sin

*γz*)/

*γ*+ (

**b**×)

^{2}(1 − cos

*γz*)/

*γ*

^{2}, where

*γ*= (

*β*

^{2}+ 4

*α*

^{2})

^{1/2}is the

*equivalent birefringence strength*. The Mueller matrix that represents round-trip propagation is invariant with respect to a rotation of the reference frame around the fiber longitudinal axis [9

9. 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]

**B**(

*z*) =

**MR**(

*z*)

^{T}**MR**(

*z*), where

**M**= diag(1, 1, −1) is a diagonal matrix. Explicit calculations reveal that

**B**(

*z*) can be written as

**B**(

*z*) =

**C**

_{0}(

*ε*) +

**C**

_{1}(

*ε*) cos

*γz*+

**C**

_{2}(

*ε*) cos 2

*γz*+

**S**

_{1}(

*ε*) sin

*γz*+

**S**

_{2}(

*ε*) sin 2

*γz*, where

*ε*= arctan(2

*α*/

*β*) is the

*equivalent birefringence ellipticity*,

**C**

_{0}=

**I**–

**C**

_{1}–

**C**

_{2},

**I**is the 3 × 3 identity matrix and where

*c*= cos

_{n}*nε*and

*s*= sin

_{n}*nε*. We now use these results to assess the constraint on spatial resolution and SOP accuracy.

### 2.1. SOP accuracy

*ε*→

*π*/2,

**C**

_{1},

**C**

_{2},

**S**

_{1}and

**S**

_{2}approach zero, hence

**B**(

*z*) →

**I**. This means that as the spin rate increases

*with respect to the linear birefringence*, the amplitude of the oscillation of the backscattered SOP decreases and the measurement becomes more critical. To quantify this effect, we should compare the amplitude of the SOP oscillation with the uncertainty of SOP measurement. This amplitude may be estimated as 2〈|[

**B**(

*z*) –

**C**

_{0}]

**s**

_{0}|〉, which is twice the mean modulus of the oscillating component of the backscattered SOP,

**s**

*(*

_{B}*z*) =

**B**(

*z*)

**s**

_{0}, where

**s**

_{0}is the input SOP. The average 〈·〉 is evaluated with respect to

**s**

_{0}, which we assume to be uniformly distributed on the Poincaré sphere. Indeed, this assumption is justified by the fact that, in practice, the actual SOP at the input of each highly spun fiber section is randomly oriented. As a result, the mean amplitude reads Δ = 2(5/3 – cos 2

*ε*)

^{1/2}cos

*ε*, hence we may conclude that the oscillations of the SOP can be accurately detected only if Δ >

*ρσ*, where

*σ*is the SOP uncertainty and

*ρ*> 1 is an arbitrary constant representing the least acceptable SNR. Typically,

*σ*is less than 10% and an SNR of 10 dB is more than enough; hence,

*ρσ*≲ 1. Recalling that tan

*ε*= 2

*α*/

*β*, the constraint Δ >

*ρσ*may finally be expressed as

### 2.2. Spatial resolution

**s**

*(*

_{B}*z*). Since this frequency is equal to 2

*γ*/(2

*π*), the spatial sampling frequency should be at least the double and hence the spatial resolution

*δz*must respect the constraint

*δz*<

*π*/(2

*γ*), which yields Note, however, that as the spin rate increases with respect to the linear birefringence (

*ε*→

*π*/2), all the elements of

**C**

_{2}and

**S**

_{2}approach zero faster than the elements of

**C**

_{1}and

**S**

_{1}. Consequently, in the case of highly spun fibers the terms of angular frequency 2

*γ*may be neglected and the constraint on the spatial resolution

*δz*can be relaxed by a factor 2.

*γ*be at least

*κ*times larger that the amplitude of the terms at 2

*γ*. As we have done for Δ, we may estimate these amplitudes as Δ

*= 〈|*

_{n}**C**

_{n}**s**

_{0}cos

*nγz*+

**S**

_{n}**s**

_{0}sin

*nγz*|〉, (

*n*= 1, 2), which yields

*α*>

*κβ*/4, where

*κ*≃ 10 is a reasonable choice.

### 2.3. Range of applicability

*L*of such a section, however, should be at least 2

*π*/

*γ*, so to enable the observation of a complete period of oscillation of

**s**

*(*

_{B}*z*). Following these considerations, we may show that the constraints calculated above may be applied to a randomly birefringent, periodically spun fiber if where

*L*is the birefringence correlation length [2

_{F}2. A. Galtarossa and C. R. Menyuk, eds., *Polarization Mode Dispersion* (Springer, 2005). [CrossRef]

*p*is the period of the spin profile. The first of (3) is an immediate consequence of the condition 2

*π*/

*γ*≪

*L*. The second of (3) is obtained by assuming, for simplicity, that the spin profile is sinusoidal and by requesting that 2

_{F}*π*/

*γ*is much larger than the distance over which the spin is approximately linear.

*τ*, sets a limit to PSR, because it may distort the probe signal and hinder the measurement [15]. As a rule of thumb, it should be Δ

*τ*Δ

*f*≪ 1, where Δ

*f*is the bandwidth of the probe signal. However, this limit has to be assessed for each specific fiber sample, because Δ

*τ*is not a local parameter and depends on fiber length, on birefringence and (in a non trivial way) on spin profile [12

12. L. Palmieri, “Polarization properties of spun single-mode fibers,” J. Lightwave Technol. **24**, 4075–4088 (2006). [CrossRef]

### 2.4. The map of constraints

_{1}represents the constraint (2) given by the spatial resolution, whereas line ℒ

_{1}represents the constraint (1) given by the SOP accuracy. Curve 𝒞

_{3}represents the tightest of the constraints imposed by (3). All these constraints define the area of measurable parameters, represented in yellow. Also shown in the figure is the area of the relaxed constraints, colored in orange. This area is defined by the curve 𝒞

_{2}, which is the spatial resolution constraint relaxed by a factor 2, and by the line ℒ

_{2}, which represents the lower range,

*α*>

*κβ*/4, for the application of the relaxed constraint. Note that this area exists if and only if ℒ

_{1}is steeper than ℒ

_{2}, which means

*σ*≲ 6.5/(

*ρκ*) — i.e. the SOP uncertainty should be sufficiently small.

_{1}and 𝒞

_{2}tend to shrink, but at the same time ℒ

_{1}becomes steeper.

_{1}may seem of poor interest; but actually this is the range of medium-to-low birefringence, typically populated by telecommunication fibers. Therefore, it is convenient to represent the map of constraints as a function of beat length,

*L*= 2

_{B}*π*/

*β*, as shown in Fig. 1(b). The markers represent a broad selection of various types of fibers measured with the POFDR described in refs. [10

10. 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]

11. A. Galtarossa, D. Grosso, and L. Palmieri, “Accurate characterization of twist-induced optical activity in single-mode fibers by means of polarization-sensitive reflectometry,” IEEE Photon. Technol. Lett. **21**, 1713–1715 (2009). [CrossRef]

*p*and

*L*were both larger than 8 m, therefore the constraints (3) fall outside the shown axes. Note also that some results lay outside the achievable area; how those measurements have been possible is described in the next section.

_{F}## 3. Measurements beyond the measurable range

*z*-derivative are shown. The curves clearly have two portions, highlighted in orange, that are much smoother than the rest and correspond to the sites where the local spin rate falls within the measurable range. On the contrary, the rest of the curve corresponds to fiber sections where the spin rate is too large for the SOP to be accurately measured. We remark that these differences in the portions are due to the limits of the PSR. Data refer to the G.655 fiber represented by the star in Fig. 1(b); the fiber is sinusoidally spun with period 8.85 m and amplitude 20 turns (which corresponds to a maximum spin rate of 89.2 rad/m) and its mean birefringence is about 1.2 rad/m. This measurement (as well as the other ones described below) has been performed with the POFDR described in refs. [10

10. 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]

11. A. Galtarossa, D. Grosso, and L. Palmieri, “Accurate characterization of twist-induced optical activity in single-mode fibers by means of polarization-sensitive reflectometry,” IEEE Photon. Technol. Lett. **21**, 1713–1715 (2009). [CrossRef]

*ψ*

_{i}_{,}

*(*

_{m}*z*) be the

*m*th good portion of the physical angle of the birefringence measured when the fiber is twisted at a constant rate

*τ*. Then we may write

_{i}*ψ*

_{i}_{,}

*(*

_{m}*z*) =

*A*(

*z*) + (1 −

*g*/2)

*τ*+

_{i}z*ϕ*

_{i}_{,}

*+*

_{m}*η*(

*z*), where

*A*(

*z*) is the spin profile,

*g*≃ 0.147 is the elasto-optic rotation coefficient [11

**21**, 1713–1715 (2009). [CrossRef]

*ϕ*

_{i}_{,}

*is the unknown offset that affects the portion and*

_{m}*η*(

*z*) accounts for both the intrinsic random rotation of birefringence and the measurement noise. We may get rid of the offsets

*ϕ*

_{i}_{,}

*simply by taking the*

_{m}*z*-derivative of

*ψ*

_{i}_{,}

*(*

_{m}*z*). Note, however, that this is basically equivalent to perform a second order differentiation on experimental data—a task that requires some care. Therefore, we have calculated

*dψ*

_{i}_{,}

*/*

_{m}*dz*by applying a Savitzky-Golay filter of forth degree, on an 11-point-long (5 cm) sliding window [16

16. A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. **36**, 1627–1639 (1964). [CrossRef]

*g*/2)

*τ*to each curve, using the corresponding value of

_{i}*τ*. In this way the spin rate of the fiber is reconstructed as shown in Fig. 3(c).

_{i}*dA*/

*dz*, can be obtained upon averaging the overlapping portions of the curves

*dψ*

_{i}_{,}

*/*

_{m}*dz*; the result is shown in Fig. 3(d) (black, solid curve). This curve can be numerically integrated to achieve the estimate of the spin profile,

*A*(

*z*), which is shown on the same figure (black, dashed curve). Note that this is in very good agreement with the estimate provided by the method described in [13] and shown by the green curve. Finally, the orange curve in Fig. 3(d) represents the best fitting sinusoid; the resulting period and amplitude are 8.9 m and 21.6 turns, respectively, in good agreement with the nominal values.

## 4. Conclusion

## Acknowledgments

## References and links

1. | A. J. Rogers, “Polarization-optical time domain reflectometry: a technique for the measurement of field distributions,” Appl. Opt. |

2. | A. Galtarossa and C. R. Menyuk, eds., |

3. | R. E. Schuh, X. Shan, and A. S. Siddiqui, “Polarization mode dispersion in spun fibers with different linear birefringence and spinning parameters,” J. Lightwave Technol. |

4. | H. Sunnerud, B. E. Olsson, M. Karlsson, P. A. Andrekson, and J. Brentel, “Polarization-mode dispersion measurements along installed optical fibers using gated backscattered light and a polarimeter,” J. Lightwave Technol. |

5. | M. Wuilpart, P. Megret, M. Blondel, A. Rogers, and Y. Defosse, “Measurement of the spatial distribution of birefringence in optical fibers,” IEEE Photon. Technol. Lett. |

6. | M. Wegmuller, M. Legré, and N. Gisin, “Distributed beatlength measurement in single-mode fibers with optical frequency-domain reflectometry,” J. Lightwave Technol. |

7. | H. Dong, P. Shum, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Spectral-resolved backreflection measurement of polarization mode dispersion in optical fibers,” Opt. Lett. |

8. | T. Geisler, P. Kristensen, and O. Knop, “New details of spun fibers measured with an OFDR,” |

9. | A. Galtarossa, D. Grosso, L. Palmieri, and L. Schenato, “Reflectometric measurement of birefringence rotation in single-mode optical fibers,” Opt. Lett. |

10. | 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. |

11. | A. Galtarossa, D. Grosso, and L. Palmieri, “Accurate characterization of twist-induced optical activity in single-mode fibers by means of polarization-sensitive reflectometry,” IEEE Photon. Technol. Lett. |

12. | L. Palmieri, “Polarization properties of spun single-mode fibers,” J. Lightwave Technol. |

13. | L. Palmieri, T. Geisler, and A. Galtarossa, “Characterization of strongly spun fibers with spin rate exceeding OFDR spatial resolution,” |

14. | A. Galtarossa, L. Palmieri, and D. Sarchi, “Measure of spin period in randomly-birefringent low-PMD fibers,” IEEE Photon. Technol. Lett. |

15. | D. K. Gifford, M. E. Froggatt, S. T. Kreger, M. S. Wolfe, and B. J. T. Soller, “Polarization echoes based on scatter de-correlation in polarization maintaining fiber,” |

16. | A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. |

**OCIS Codes**

(060.2270) Fiber optics and optical communications : Fiber characterization

(060.2300) Fiber optics and optical communications : Fiber measurements

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: March 16, 2011

Revised Manuscript: May 9, 2011

Manuscript Accepted: May 10, 2011

Published: May 19, 2011

**Citation**

Luca Palmieri, Tommy Geisler, and Andrea Galtarossa, "Limits of applicability of polarization sensitive reflectometry," Opt. Express **19**, 10874-10879 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-11-10874

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### References

- A. J. Rogers, “Polarization-optical time domain reflectometry: a technique for the measurement of field distributions,” Appl. Opt. 20, 1060–1074 (1981). [CrossRef] [PubMed]
- A. Galtarossa and C. R. Menyuk, eds., Polarization Mode Dispersion (Springer, 2005). [CrossRef]
- R. E. Schuh, X. Shan, and A. S. Siddiqui, “Polarization mode dispersion in spun fibers with different linear birefringence and spinning parameters,” J. Lightwave Technol. 16, 1583–1588 (1998). [CrossRef]
- H. Sunnerud, B. E. Olsson, M. Karlsson, P. A. Andrekson, and J. Brentel, “Polarization-mode dispersion measurements along installed optical fibers using gated backscattered light and a polarimeter,” J. Lightwave Technol. 18, 897–904 (2000). [CrossRef]
- M. Wuilpart, P. Megret, M. Blondel, A. Rogers, and Y. Defosse, “Measurement of the spatial distribution of birefringence in optical fibers,” IEEE Photon. Technol. Lett. 13, 836–838 (2001). [CrossRef]
- M. Wegmuller, M. Legré, and N. Gisin, “Distributed beatlength measurement in single-mode fibers with optical frequency-domain reflectometry,” J. Lightwave Technol. 20, 828–835 (2002). [CrossRef]
- H. Dong, P. Shum, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Spectral-resolved backreflection measurement of polarization mode dispersion in optical fibers,” Opt. Lett. 32, 1665–1667 (2007). [CrossRef] [PubMed]
- T. Geisler, P. Kristensen, and O. Knop, “New details of spun fibers measured with an OFDR,” Tech Digest. OFC (paper OWA3) (Anaheim (CA), 2006).
- 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. 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, and L. Palmieri, “Accurate characterization of twist-induced optical activity in single-mode fibers by means of polarization-sensitive reflectometry,” IEEE Photon. Technol. Lett. 21, 1713–1715 (2009). [CrossRef]
- L. Palmieri, “Polarization properties of spun single-mode fibers,” J. Lightwave Technol. 24, 4075–4088 (2006). [CrossRef]
- L. Palmieri, T. Geisler, and A. Galtarossa, “Characterization of strongly spun fibers with spin rate exceeding OFDR spatial resolution,” Tech. Digest OFC (paper OMF2) (Los Angeles (CA), 2011).
- 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]
- D. K. Gifford, M. E. Froggatt, S. T. Kreger, M. S. Wolfe, and B. J. T. Soller, “Polarization echoes based on scatter de-correlation in polarization maintaining fiber,” Tech Digest. OFC (paper JWA8) (Anaheim (CA), 2007).
- A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964). [CrossRef]

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