Abstract

In this work we analyze the birefringence matrix developed for a twisted fiber in order to identify the basic optical effects that define its birefringence. The study was performed using differential Jones calculus. The resultant differential matrix showed three independent types of birefringence: circular, linear at 0 degrees and linear at 45 degrees (Jones birefringence). We applied this birefringence matrix to the description of the output state of polarization measured for three commercial fibers that due to its higher rigidity present stronger birefringence changes when twisted. The torsion applied to the erbium-doped fiber samples varied from 0 to 1440 degrees.

© 2013 Optical Society of America

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References

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  1. R. Ulrich and A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt.18(13), 2241–2251 (1979).
    [CrossRef] [PubMed]
  2. D. Tentori, C. Ayala-Díaz, E. Ledezma-Sillas, F. Treviño-Martínez, and A. García-Weidner, “Birefringence matrix for a twisted single-mode fiber: Geometrical contribution,” Opt. Commun.282(5), 830–834 (2009).
    [CrossRef]
  3. D. Tentori, A. García-Weidner, and C. Ayala-Díaz, “Birefringence matrix for a twisted single-mode fiber: Photoelastic and geometrical contributions,” Opt. Fiber Technol.18(1), 14–20 (2012).
    [CrossRef]
  4. D. Tentori and A. Garcia-Weidner, “Right- and left-handed twist in optical fibers,” submitted for publication.
  5. R. C. Jones, “A new calculus for the treatment of optical systems. VII properties of the N-matrices,” J. Opt. Soc. Am.38(8), 671–685 (1948).
    [CrossRef]
  6. C. Tsao, Optical Fibre Waveguide Analysis (Oxford University, 1992).
  7. P. Hauge, R. Muller, and C. Smith, “Conventions and formulas for using the Mueller-Stokes calculus in ellipsometry,” Surf. Sci.96(1-3), 81–107 (1980).
    [CrossRef]
  8. D. Tentori, C. Ayala-Díaz, F. Treviño-Martínez, and F. J. Mendieta-Jiménez, “Evaluation of the residual birefringence of single-mode erbium-doped silica fibers,” Opt. Commun.271(1), 73–80 (2007).
    [CrossRef]
  9. A. Rizzo and S. Coriani, “Birefringences: A challenge for both theory and experiment,” Adv. Quantum Chem.50, 143–184 (2005).
    [CrossRef]
  10. V. A. De Lorenci and G. P. Goulart, “Magnetoelectric birefringence revisited,” Phys. Rev. D Part. Fields Gravit. Cosmol.78(4), 045015 (2008).
    [CrossRef]
  11. D. Shcherbin, A. J. Thorvaldsen, D. Jonsson, and K. Ruud, “Gauge-origin independent calculations of Jones birefringence,” J. Chem. Phys.135(13), 134114 (2011).
    [CrossRef] [PubMed]
  12. M. Izdebski, W. Kucharczyk, and R. E. Raab, “Effect of beam divergence from the optic axis in an electro-optic experiment to measure an induced Jones birefringence,” J. Opt. Soc. Am. A18(6), 1393–1398 (2001).
    [CrossRef] [PubMed]

2012

D. Tentori, A. García-Weidner, and C. Ayala-Díaz, “Birefringence matrix for a twisted single-mode fiber: Photoelastic and geometrical contributions,” Opt. Fiber Technol.18(1), 14–20 (2012).
[CrossRef]

2011

D. Shcherbin, A. J. Thorvaldsen, D. Jonsson, and K. Ruud, “Gauge-origin independent calculations of Jones birefringence,” J. Chem. Phys.135(13), 134114 (2011).
[CrossRef] [PubMed]

2009

D. Tentori, C. Ayala-Díaz, E. Ledezma-Sillas, F. Treviño-Martínez, and A. García-Weidner, “Birefringence matrix for a twisted single-mode fiber: Geometrical contribution,” Opt. Commun.282(5), 830–834 (2009).
[CrossRef]

2008

V. A. De Lorenci and G. P. Goulart, “Magnetoelectric birefringence revisited,” Phys. Rev. D Part. Fields Gravit. Cosmol.78(4), 045015 (2008).
[CrossRef]

2007

D. Tentori, C. Ayala-Díaz, F. Treviño-Martínez, and F. J. Mendieta-Jiménez, “Evaluation of the residual birefringence of single-mode erbium-doped silica fibers,” Opt. Commun.271(1), 73–80 (2007).
[CrossRef]

2005

A. Rizzo and S. Coriani, “Birefringences: A challenge for both theory and experiment,” Adv. Quantum Chem.50, 143–184 (2005).
[CrossRef]

2001

1980

P. Hauge, R. Muller, and C. Smith, “Conventions and formulas for using the Mueller-Stokes calculus in ellipsometry,” Surf. Sci.96(1-3), 81–107 (1980).
[CrossRef]

1979

1948

Ayala-Díaz, C.

D. Tentori, A. García-Weidner, and C. Ayala-Díaz, “Birefringence matrix for a twisted single-mode fiber: Photoelastic and geometrical contributions,” Opt. Fiber Technol.18(1), 14–20 (2012).
[CrossRef]

D. Tentori, C. Ayala-Díaz, E. Ledezma-Sillas, F. Treviño-Martínez, and A. García-Weidner, “Birefringence matrix for a twisted single-mode fiber: Geometrical contribution,” Opt. Commun.282(5), 830–834 (2009).
[CrossRef]

D. Tentori, C. Ayala-Díaz, F. Treviño-Martínez, and F. J. Mendieta-Jiménez, “Evaluation of the residual birefringence of single-mode erbium-doped silica fibers,” Opt. Commun.271(1), 73–80 (2007).
[CrossRef]

Coriani, S.

A. Rizzo and S. Coriani, “Birefringences: A challenge for both theory and experiment,” Adv. Quantum Chem.50, 143–184 (2005).
[CrossRef]

De Lorenci, V. A.

V. A. De Lorenci and G. P. Goulart, “Magnetoelectric birefringence revisited,” Phys. Rev. D Part. Fields Gravit. Cosmol.78(4), 045015 (2008).
[CrossRef]

García-Weidner, A.

D. Tentori, A. García-Weidner, and C. Ayala-Díaz, “Birefringence matrix for a twisted single-mode fiber: Photoelastic and geometrical contributions,” Opt. Fiber Technol.18(1), 14–20 (2012).
[CrossRef]

D. Tentori, C. Ayala-Díaz, E. Ledezma-Sillas, F. Treviño-Martínez, and A. García-Weidner, “Birefringence matrix for a twisted single-mode fiber: Geometrical contribution,” Opt. Commun.282(5), 830–834 (2009).
[CrossRef]

Goulart, G. P.

V. A. De Lorenci and G. P. Goulart, “Magnetoelectric birefringence revisited,” Phys. Rev. D Part. Fields Gravit. Cosmol.78(4), 045015 (2008).
[CrossRef]

Hauge, P.

P. Hauge, R. Muller, and C. Smith, “Conventions and formulas for using the Mueller-Stokes calculus in ellipsometry,” Surf. Sci.96(1-3), 81–107 (1980).
[CrossRef]

Izdebski, M.

Jones, R. C.

Jonsson, D.

D. Shcherbin, A. J. Thorvaldsen, D. Jonsson, and K. Ruud, “Gauge-origin independent calculations of Jones birefringence,” J. Chem. Phys.135(13), 134114 (2011).
[CrossRef] [PubMed]

Kucharczyk, W.

Ledezma-Sillas, E.

D. Tentori, C. Ayala-Díaz, E. Ledezma-Sillas, F. Treviño-Martínez, and A. García-Weidner, “Birefringence matrix for a twisted single-mode fiber: Geometrical contribution,” Opt. Commun.282(5), 830–834 (2009).
[CrossRef]

Mendieta-Jiménez, F. J.

D. Tentori, C. Ayala-Díaz, F. Treviño-Martínez, and F. J. Mendieta-Jiménez, “Evaluation of the residual birefringence of single-mode erbium-doped silica fibers,” Opt. Commun.271(1), 73–80 (2007).
[CrossRef]

Muller, R.

P. Hauge, R. Muller, and C. Smith, “Conventions and formulas for using the Mueller-Stokes calculus in ellipsometry,” Surf. Sci.96(1-3), 81–107 (1980).
[CrossRef]

Raab, R. E.

Rizzo, A.

A. Rizzo and S. Coriani, “Birefringences: A challenge for both theory and experiment,” Adv. Quantum Chem.50, 143–184 (2005).
[CrossRef]

Ruud, K.

D. Shcherbin, A. J. Thorvaldsen, D. Jonsson, and K. Ruud, “Gauge-origin independent calculations of Jones birefringence,” J. Chem. Phys.135(13), 134114 (2011).
[CrossRef] [PubMed]

Shcherbin, D.

D. Shcherbin, A. J. Thorvaldsen, D. Jonsson, and K. Ruud, “Gauge-origin independent calculations of Jones birefringence,” J. Chem. Phys.135(13), 134114 (2011).
[CrossRef] [PubMed]

Simon, A.

Smith, C.

P. Hauge, R. Muller, and C. Smith, “Conventions and formulas for using the Mueller-Stokes calculus in ellipsometry,” Surf. Sci.96(1-3), 81–107 (1980).
[CrossRef]

Tentori, D.

D. Tentori, A. García-Weidner, and C. Ayala-Díaz, “Birefringence matrix for a twisted single-mode fiber: Photoelastic and geometrical contributions,” Opt. Fiber Technol.18(1), 14–20 (2012).
[CrossRef]

D. Tentori, C. Ayala-Díaz, E. Ledezma-Sillas, F. Treviño-Martínez, and A. García-Weidner, “Birefringence matrix for a twisted single-mode fiber: Geometrical contribution,” Opt. Commun.282(5), 830–834 (2009).
[CrossRef]

D. Tentori, C. Ayala-Díaz, F. Treviño-Martínez, and F. J. Mendieta-Jiménez, “Evaluation of the residual birefringence of single-mode erbium-doped silica fibers,” Opt. Commun.271(1), 73–80 (2007).
[CrossRef]

Thorvaldsen, A. J.

D. Shcherbin, A. J. Thorvaldsen, D. Jonsson, and K. Ruud, “Gauge-origin independent calculations of Jones birefringence,” J. Chem. Phys.135(13), 134114 (2011).
[CrossRef] [PubMed]

Treviño-Martínez, F.

D. Tentori, C. Ayala-Díaz, E. Ledezma-Sillas, F. Treviño-Martínez, and A. García-Weidner, “Birefringence matrix for a twisted single-mode fiber: Geometrical contribution,” Opt. Commun.282(5), 830–834 (2009).
[CrossRef]

D. Tentori, C. Ayala-Díaz, F. Treviño-Martínez, and F. J. Mendieta-Jiménez, “Evaluation of the residual birefringence of single-mode erbium-doped silica fibers,” Opt. Commun.271(1), 73–80 (2007).
[CrossRef]

Ulrich, R.

Adv. Quantum Chem.

A. Rizzo and S. Coriani, “Birefringences: A challenge for both theory and experiment,” Adv. Quantum Chem.50, 143–184 (2005).
[CrossRef]

Appl. Opt.

J. Chem. Phys.

D. Shcherbin, A. J. Thorvaldsen, D. Jonsson, and K. Ruud, “Gauge-origin independent calculations of Jones birefringence,” J. Chem. Phys.135(13), 134114 (2011).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

D. Tentori, C. Ayala-Díaz, F. Treviño-Martínez, and F. J. Mendieta-Jiménez, “Evaluation of the residual birefringence of single-mode erbium-doped silica fibers,” Opt. Commun.271(1), 73–80 (2007).
[CrossRef]

D. Tentori, C. Ayala-Díaz, E. Ledezma-Sillas, F. Treviño-Martínez, and A. García-Weidner, “Birefringence matrix for a twisted single-mode fiber: Geometrical contribution,” Opt. Commun.282(5), 830–834 (2009).
[CrossRef]

Opt. Fiber Technol.

D. Tentori, A. García-Weidner, and C. Ayala-Díaz, “Birefringence matrix for a twisted single-mode fiber: Photoelastic and geometrical contributions,” Opt. Fiber Technol.18(1), 14–20 (2012).
[CrossRef]

Phys. Rev. D Part. Fields Gravit. Cosmol.

V. A. De Lorenci and G. P. Goulart, “Magnetoelectric birefringence revisited,” Phys. Rev. D Part. Fields Gravit. Cosmol.78(4), 045015 (2008).
[CrossRef]

Surf. Sci.

P. Hauge, R. Muller, and C. Smith, “Conventions and formulas for using the Mueller-Stokes calculus in ellipsometry,” Surf. Sci.96(1-3), 81–107 (1980).
[CrossRef]

Other

D. Tentori and A. Garcia-Weidner, “Right- and left-handed twist in optical fibers,” submitted for publication.

C. Tsao, Optical Fibre Waveguide Analysis (Oxford University, 1992).

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Figures (6)

Fig. 1
Fig. 1

Geometrical relations between total birefringence coefficient Γ and the circular and linear (0 and 45°) birefringence coefficients. This representation holds in the space of the Poincaré sphere.

Fig. 2
Fig. 2

Evolution of the output state of polarization for a 1540 nm linearly polarized input signal when the absolute value of the twist applied to the EDF varied from 0 to 1440°. In this figure red is used for those sections of the curve located at front. (a) F1, φ = 30°, right twist; (b) F3, φ = 60°, left twist.

Fig. 3
Fig. 3

Numerical fitting of Stokes parameters (a) and (c), and SOP evolution depicted on the Poincaré sphere (b) and (d) obtained for fiber F1 for two different signal wavelengths (1530 and 1560 nm). The azimuth angle of the input linear polarization φ is shown for each case. Purple line on the Poincaré sphere corresponds to theoretical prediction and the experimental data are presented with a blue line (the absolute value of the twist applied to the EDF varied from 600 to 1440°). The variation of Stokes parameters is shown for applied torsions τa between 0 and 1440°.

Fig. 4
Fig. 4

Numerical fitting and experimental data obtained for fiber F2. These graphs show the evolution of the Stokes parameters of the output state of polarization of a 1530 and a 1550 nm linearly polarized input signal (φ = 30 and 120°, respectively), when the value of the twist applied to the EDF varied from 0 to −1440°. Figures 4(a), 4(b), 4(d), and 4(e) show the Stokes parameters evolution and Figs. 4(c) and 4(f), the output SOP evolution on the Poincaré sphere for applied torsions τa between 700 and 1440° (purple line = theory, blue line = experimental data).

Fig. 5
Fig. 5

Numerical fitting and experimental data obtained for fiber F3. These graphs show the evolution of the Stokes parameters of the output state of polarization of a 1530 and a 1550 nm linearly polarized input signal (φ = 30 and 120°, respectively), when the value of the twist applied to the EDF varied from 0 to −1440°. Figures 5(a), 5(b), 5(d), and 5(e) show the Stokes parameters evolution and Figs. 5(c) and 5(f) the output SOP evolution on the Poincaré sphere for applied torsions τa between 700 and 1440° (purple line = theory, blue line = experimental data).

Fig. 6
Fig. 6

Evolution of the output state of polarization of a 1530 nm linearly polarized signal when the twist applied to the fiber varies from: (a) 600 to 1400° for erbium-doped fiber F2 (Fibercore 1500E, 1.51 m), (b) 0 to 1440° for a standard fiber (SMF-28e, 1.75 m).

Tables (5)

Tables Icon

Table 1 Fiber’s parameters.

Tables Icon

Table 2 Fitting parameters.

Tables Icon

Table 3 Geometrical parameters

Tables Icon

Table 4 Erbium-doped fibers wavelength dependent parameters.

Tables Icon

Table 5 Parameters related with fiber’s anisotropy.

Equations (4)

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N T =A[ j c 2  cos2( β+bτ ) cos2ε b+ c 2  sin2εj c 2  sin2( β+bτ ) cos2ε b c 2  sin2εj c 2  sin2( β+bτ ) cos2ε j c 2  cos2( β+bτ ) cos2ε ],
β 0 ( c,β,b,ε,τ )=A c cos2( β+bτ ) cos2ε 2  =  η y η x 2 ,linear birefringence coefficient at 0°
M R =[ cos2δ+2 sin 2 σ cos 2 σ l sin 2 δ cos σsin2δsin2 σ l sin 2 σ sin 2 δ sin σsin σ l sin2δsin2σcos σ l sin 2 δ cos σsin2δsin2 σ l sin 2 σ sin 2 δ cos2δ+2 sin 2 σ l sin 2 σ sin 2 δ sin σcos σ l sin2δ+sin2σsin σ l sin 2 δ sinσsin σ l sin2δsin2σcos σ l sin 2 δ sin σcos σ l sin2δ+sin2σsin σ l sin 2 δ cos2δ+2 cos 2 σ sin 2 δ ]
S 1 out =cos2( β+bτ )[ cos2φ( cos2δ+2 sin 2 σ  cos 2 σ l   sin 2 δ )sin2φ( cosσsin2δ+sin2 σ l sin 2 σ sin 2 δ ) ] +sin2( β+bτ ) [ cos2φ( cosσsin2δsin2 σ l sin 2 σ sin 2 δ ) +sin2φ( cos2δ+2 sin 2 σ l sin 2 σ sin 2 δ ) ] S 2 out =sin2( β+bτ )[ cos2φ( cos2δ+2 sin 2 σ  cos 2 σ l   sin 2 δ )    sin2φ( cosσsin2δ+sin2 σ l sin 2 σ sin 2 δ ) ] +cos2( β+bτ ) [ cos2φ( cosσsin2δsin2 σ l sin 2 σ sin 2 δ ) +sin2φ( cos2δ+2 sin 2 σ l sin 2 σ sin 2 δ ) ] S 3 out =cos2φ( sinσsin σ l sin2δ+sin2σcos σ l sin 2 δ )+sin2φ( sinσcos σ l sin2δ+sin2σsin σ l sin 2 δ ).

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