Abstract

We present a detailed derivation of the locus of Rayleigh backscattered states of polarization for polarization optical time domain reflectometry in uniformly twisted optical fiber with intrinsic linear birefringence. The locus is algebraically a quartic whose topology is determined by the relative orientation on the Poincaré sphere of the input SOP and the effective birefringence vector. We present an analysis that indicates how experimental data may be interpreted, through geometric parameters of the locus, for evaluating the fiber parameters. The analysis also indicates the minimum number of experimental data points required for meaningful values for the fiber parameters to be obtained.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. E. Schuh, J. G. Ellison, A. S. Siddiqui, D. H. O. Bebbington, “Polarization OTDR measurements and theoretical analysis on fibres with twist and their implications for estimation of PMD,” Electron. Lett. 32, 387–388 (1996).
    [CrossRef]
  2. M. O. van Deventer, “Polarization properties of Rayleigh backscattering in single mode fibers,” J. Lightwave Technol. 11, 1895–1899 (1993).
    [CrossRef]
  3. C. D. Graves, “Radar polarization power scattering matrix,” Proc. IEEE 44, 248–252 (1956).
  4. D. H. O. Bebbington, “Target vectors—spinorial concepts and geometry,” presented at the Second International Workshop on Radar Polarimetry, Nantes, France, September 8–11, 1992.
  5. R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, D. H. O. Bebbington, “Theoretical analysis and measurement of the effects of fibre twist on the differential group delay of optical fibres,” Electron. Lett. 31, 1772–1773 (1995).
    [CrossRef]
  6. W. V. D. Hodge, D. Pedoe, Methods of Algebraic Geometry (Cambridge U. Press, Cambridge, UK, 1954), Chap. XIII.
  7. R. E. Schuh, A. S. Siddiqui, X. Shan, J. G. Ellison, D. H. O. Bebbington, “A novel simple polarization OTDR technique to detect fibre sections with and without twist,” Int. J. Optoelectron. 11, 321–324 (1997).
  8. D. H. O. Bebbington, J. G. Ellison, R. E. Schuh, X. Shan, A. S. Siddiqui, S. D. Walker, “Fully polarimetric optical time domain reflectometer with one metere spatial resolution,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper WL24.
  9. J. G. Ellison, A. S. Siddiqui, “A fully polarimetric optical time domain reflectometer,” IEEE Photonics Technol. Lett. 10, 246–248 (1998).
    [CrossRef]

1998 (1)

J. G. Ellison, A. S. Siddiqui, “A fully polarimetric optical time domain reflectometer,” IEEE Photonics Technol. Lett. 10, 246–248 (1998).
[CrossRef]

1997 (1)

R. E. Schuh, A. S. Siddiqui, X. Shan, J. G. Ellison, D. H. O. Bebbington, “A novel simple polarization OTDR technique to detect fibre sections with and without twist,” Int. J. Optoelectron. 11, 321–324 (1997).

1996 (1)

R. E. Schuh, J. G. Ellison, A. S. Siddiqui, D. H. O. Bebbington, “Polarization OTDR measurements and theoretical analysis on fibres with twist and their implications for estimation of PMD,” Electron. Lett. 32, 387–388 (1996).
[CrossRef]

1995 (1)

R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, D. H. O. Bebbington, “Theoretical analysis and measurement of the effects of fibre twist on the differential group delay of optical fibres,” Electron. Lett. 31, 1772–1773 (1995).
[CrossRef]

1993 (1)

M. O. van Deventer, “Polarization properties of Rayleigh backscattering in single mode fibers,” J. Lightwave Technol. 11, 1895–1899 (1993).
[CrossRef]

1956 (1)

C. D. Graves, “Radar polarization power scattering matrix,” Proc. IEEE 44, 248–252 (1956).

Bebbington, D. H. O.

R. E. Schuh, A. S. Siddiqui, X. Shan, J. G. Ellison, D. H. O. Bebbington, “A novel simple polarization OTDR technique to detect fibre sections with and without twist,” Int. J. Optoelectron. 11, 321–324 (1997).

R. E. Schuh, J. G. Ellison, A. S. Siddiqui, D. H. O. Bebbington, “Polarization OTDR measurements and theoretical analysis on fibres with twist and their implications for estimation of PMD,” Electron. Lett. 32, 387–388 (1996).
[CrossRef]

R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, D. H. O. Bebbington, “Theoretical analysis and measurement of the effects of fibre twist on the differential group delay of optical fibres,” Electron. Lett. 31, 1772–1773 (1995).
[CrossRef]

D. H. O. Bebbington, J. G. Ellison, R. E. Schuh, X. Shan, A. S. Siddiqui, S. D. Walker, “Fully polarimetric optical time domain reflectometer with one metere spatial resolution,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper WL24.

D. H. O. Bebbington, “Target vectors—spinorial concepts and geometry,” presented at the Second International Workshop on Radar Polarimetry, Nantes, France, September 8–11, 1992.

Ellison, J. G.

J. G. Ellison, A. S. Siddiqui, “A fully polarimetric optical time domain reflectometer,” IEEE Photonics Technol. Lett. 10, 246–248 (1998).
[CrossRef]

R. E. Schuh, A. S. Siddiqui, X. Shan, J. G. Ellison, D. H. O. Bebbington, “A novel simple polarization OTDR technique to detect fibre sections with and without twist,” Int. J. Optoelectron. 11, 321–324 (1997).

R. E. Schuh, J. G. Ellison, A. S. Siddiqui, D. H. O. Bebbington, “Polarization OTDR measurements and theoretical analysis on fibres with twist and their implications for estimation of PMD,” Electron. Lett. 32, 387–388 (1996).
[CrossRef]

D. H. O. Bebbington, J. G. Ellison, R. E. Schuh, X. Shan, A. S. Siddiqui, S. D. Walker, “Fully polarimetric optical time domain reflectometer with one metere spatial resolution,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper WL24.

Gleeson, L. M.

R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, D. H. O. Bebbington, “Theoretical analysis and measurement of the effects of fibre twist on the differential group delay of optical fibres,” Electron. Lett. 31, 1772–1773 (1995).
[CrossRef]

Graves, C. D.

C. D. Graves, “Radar polarization power scattering matrix,” Proc. IEEE 44, 248–252 (1956).

Hodge, W. V. D.

W. V. D. Hodge, D. Pedoe, Methods of Algebraic Geometry (Cambridge U. Press, Cambridge, UK, 1954), Chap. XIII.

Pedoe, D.

W. V. D. Hodge, D. Pedoe, Methods of Algebraic Geometry (Cambridge U. Press, Cambridge, UK, 1954), Chap. XIII.

Schuh, R. E.

R. E. Schuh, A. S. Siddiqui, X. Shan, J. G. Ellison, D. H. O. Bebbington, “A novel simple polarization OTDR technique to detect fibre sections with and without twist,” Int. J. Optoelectron. 11, 321–324 (1997).

R. E. Schuh, J. G. Ellison, A. S. Siddiqui, D. H. O. Bebbington, “Polarization OTDR measurements and theoretical analysis on fibres with twist and their implications for estimation of PMD,” Electron. Lett. 32, 387–388 (1996).
[CrossRef]

R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, D. H. O. Bebbington, “Theoretical analysis and measurement of the effects of fibre twist on the differential group delay of optical fibres,” Electron. Lett. 31, 1772–1773 (1995).
[CrossRef]

D. H. O. Bebbington, J. G. Ellison, R. E. Schuh, X. Shan, A. S. Siddiqui, S. D. Walker, “Fully polarimetric optical time domain reflectometer with one metere spatial resolution,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper WL24.

Shan, X.

R. E. Schuh, A. S. Siddiqui, X. Shan, J. G. Ellison, D. H. O. Bebbington, “A novel simple polarization OTDR technique to detect fibre sections with and without twist,” Int. J. Optoelectron. 11, 321–324 (1997).

D. H. O. Bebbington, J. G. Ellison, R. E. Schuh, X. Shan, A. S. Siddiqui, S. D. Walker, “Fully polarimetric optical time domain reflectometer with one metere spatial resolution,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper WL24.

Siddiqui, A. S.

J. G. Ellison, A. S. Siddiqui, “A fully polarimetric optical time domain reflectometer,” IEEE Photonics Technol. Lett. 10, 246–248 (1998).
[CrossRef]

R. E. Schuh, A. S. Siddiqui, X. Shan, J. G. Ellison, D. H. O. Bebbington, “A novel simple polarization OTDR technique to detect fibre sections with and without twist,” Int. J. Optoelectron. 11, 321–324 (1997).

R. E. Schuh, J. G. Ellison, A. S. Siddiqui, D. H. O. Bebbington, “Polarization OTDR measurements and theoretical analysis on fibres with twist and their implications for estimation of PMD,” Electron. Lett. 32, 387–388 (1996).
[CrossRef]

R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, D. H. O. Bebbington, “Theoretical analysis and measurement of the effects of fibre twist on the differential group delay of optical fibres,” Electron. Lett. 31, 1772–1773 (1995).
[CrossRef]

D. H. O. Bebbington, J. G. Ellison, R. E. Schuh, X. Shan, A. S. Siddiqui, S. D. Walker, “Fully polarimetric optical time domain reflectometer with one metere spatial resolution,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper WL24.

Sikora, E. S. R.

R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, D. H. O. Bebbington, “Theoretical analysis and measurement of the effects of fibre twist on the differential group delay of optical fibres,” Electron. Lett. 31, 1772–1773 (1995).
[CrossRef]

van Deventer, M. O.

M. O. van Deventer, “Polarization properties of Rayleigh backscattering in single mode fibers,” J. Lightwave Technol. 11, 1895–1899 (1993).
[CrossRef]

Walker, N. G.

R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, D. H. O. Bebbington, “Theoretical analysis and measurement of the effects of fibre twist on the differential group delay of optical fibres,” Electron. Lett. 31, 1772–1773 (1995).
[CrossRef]

Walker, S. D.

D. H. O. Bebbington, J. G. Ellison, R. E. Schuh, X. Shan, A. S. Siddiqui, S. D. Walker, “Fully polarimetric optical time domain reflectometer with one metere spatial resolution,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper WL24.

Electron. Lett. (2)

R. E. Schuh, J. G. Ellison, A. S. Siddiqui, D. H. O. Bebbington, “Polarization OTDR measurements and theoretical analysis on fibres with twist and their implications for estimation of PMD,” Electron. Lett. 32, 387–388 (1996).
[CrossRef]

R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, D. H. O. Bebbington, “Theoretical analysis and measurement of the effects of fibre twist on the differential group delay of optical fibres,” Electron. Lett. 31, 1772–1773 (1995).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

J. G. Ellison, A. S. Siddiqui, “A fully polarimetric optical time domain reflectometer,” IEEE Photonics Technol. Lett. 10, 246–248 (1998).
[CrossRef]

Int. J. Optoelectron. (1)

R. E. Schuh, A. S. Siddiqui, X. Shan, J. G. Ellison, D. H. O. Bebbington, “A novel simple polarization OTDR technique to detect fibre sections with and without twist,” Int. J. Optoelectron. 11, 321–324 (1997).

J. Lightwave Technol. (1)

M. O. van Deventer, “Polarization properties of Rayleigh backscattering in single mode fibers,” J. Lightwave Technol. 11, 1895–1899 (1993).
[CrossRef]

Proc. IEEE (1)

C. D. Graves, “Radar polarization power scattering matrix,” Proc. IEEE 44, 248–252 (1956).

Other (3)

D. H. O. Bebbington, “Target vectors—spinorial concepts and geometry,” presented at the Second International Workshop on Radar Polarimetry, Nantes, France, September 8–11, 1992.

D. H. O. Bebbington, J. G. Ellison, R. E. Schuh, X. Shan, A. S. Siddiqui, S. D. Walker, “Fully polarimetric optical time domain reflectometer with one metere spatial resolution,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper WL24.

W. V. D. Hodge, D. Pedoe, Methods of Algebraic Geometry (Cambridge U. Press, Cambridge, UK, 1954), Chap. XIII.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Poincaré sphere from two viewpoints, (a) and (b), showing the double cone on which the reflection plane, Mq, relating input and output SOP rolls, and its complementary cone swept by the axes q that are obtained by rotating the circular polarization axis γˆ about the effective birefringence axis β.

Fig. 2
Fig. 2

Locus of the SOP considered as the intersection of the Poincaré sphere with a circular cone whose vertex is the reflection in the equatorial plane of the input SOP and whose axis is parallel to the effective birefringence vector β.

Fig. 3
Fig. 3

Four configurations of the intersection of the sphere and the cone with vertex at the input SOP, according to the location of the input SOP relative to the rolling cone: (a) above the equatorial plane, outside the cone; (b) as for (a) but on the equator; (c) as for (a) but below the equatorial plane; (d) below equatorial plane, inside the rolling cone.

Fig. 4
Fig. 4

Experimental setup, comprising a fully polarimetric optical time-domain reflectometer and the fiber under test.

Fig. 5
Fig. 5

Experimental points for a section of moderately high linear birefringence fiber subjected to a uniform rate of twist. The dashed curve is the fit to the measured data with use of the methodology described.

Fig. 6
Fig. 6

As for Fig. 5 but for a section with lower linear birefringence relative to the twist-induced circular birefringence, leading to a narrower vertex angle for the intersecting cone.

Equations (24)

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

S=MγS=100001000010000-1S0S1S2S3,
ST10000-10000-10000-1S=0.
S(l)=Rz(2lγ)RF(lβ)S(0),
SR(l)=RFT(lβ)RzT(2lγ)MγRz(2lγ)RF(lβ)ST.
SR(l)=RFT(lβ)MγRF(lβ)ST.
θ=2 sin-1sin ψ sinβl2,
ϕ=cot-1cos ψ tanβl2,
S=100001-2q12-2qˆ1qˆ2-2qˆ1qˆ30-2qˆ2qˆ11-2q22-2qˆ2qˆ30-2qˆ3qˆ1-2qˆ3qˆ21-2q32S=MqS,
Pcopol=½SMqS=(1-2qˆ·sˆ)2,
Γ0=10000-10000-10000-1,Γ1=10000-10000100001,
Γ2=1000010000-100001,Γ3=100001000010000-1,
Γ4=0100100000000000,Γ5=0010000010000000,
Γ6=0001000000001000,Γ7=0000001001000000,
Γ8=0000000100000100,Γ9=0000000000010010.
A=i=09aiΓi.
{xk}TAxk=0.
ξk·a=0,
ξik={xk}TΓixk.
Ξai=09ξik ai=0,
{xk}TΓ0xk=0,
a=a0+λb,
A=Γ0+λB
|Γ0+λB|=0,
aTΞTΞa,

Metrics