Abstract

A coupled-mode equation (CME) of twisted birefringent fiber is presented in this paper, which uses the degenerate polarization modes of single-mode fibers as eigenmodes in a unified coordinate. The inconsistency between the coordinate and the rotating principal axis, existing in the previous CME, is solved by conversion to the lab coordinate. The CME gives self-consistent results for fibers with high birefringence or low birefringence and for single-mode fibers as well. Analyses and simulations show the CME gives characteristics of twisted birefringent fiber coincident with the property of polarization-maintaining fibers.

© 2013 Optical Society of America

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References

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  1. J. I. Sakai and T. Kimura, “Polarization behavior in multiply perturbed single-mode fibers,” IEEE J. Quantum Electron. 18, 59–65 (1982).
    [CrossRef]
  2. S. C. Rashleigh, “Origins and control of polarization effects in single-mode fibers,” J. Lightwave Technol. 1, 312–331 (1983).
    [CrossRef]
  3. H. C. Huang, “Fiber-optic analogs of bulk-optic wave plates,” Appl. Opt. 36, 4241–4258 (1997).
    [CrossRef]
  4. H. C. Huang, “Practical circular-polarization-maintaining optical fiber,” Appl. Opt. 36, 6968–6975 (1997).
    [CrossRef]
  5. A. H. Rose, N. Feat, and S. M. Etzel, “Wavelength and temperature performance of polarization-transforming fiber,” Appl. Opt. 42, 6897–6904 (2003).
    [CrossRef]
  6. D. Tentori, C. Ayala-Diaz, F. Trevino-Martinez, and J. Mendieta-Jimenez, “Spectral birefringence characterization of erbium-doped monomode optical fibers: relative values of the linear and the circular birefringence,” Proc. SPIE 5622, 868–872 (2004).
    [CrossRef]
  7. R. Ulrich, and A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt. 18, 2241–2251 (1979).
    [CrossRef]
  8. A. J. Barlow, and D. N. Payne, “Polarization maintenance in circularly birefringent fibers,” Electron. Lett. 17, 388–389 (1981).
    [CrossRef]
  9. P. Shen, and J. C. Palais, “Passive single-mode fiber depolarizer,” Appl. Opt. 38, 1686–1691 (1999).
    [CrossRef]
  10. H. J. El-Khozondar, M. S. Muller, R. J. El-Khozondar, and A. W. Koch, “Polarization rotation in twisted polarization maintaining fibers using a fixed reference frame,” J. Lightwave Technol. 27, 5590–5596 (2009).
    [CrossRef]
  11. H. Y. Kim, E. H. Lee, and B. Y. Kim, “Polarization properties of fiber lasers with twist-induced circular birefringence,” Appl. Opt. 36, 6764–6769 (1997).
    [CrossRef]
  12. M. S. Müller, H. J. El-Khozondar, A. Bernardini, and A. W. Koch, “Transfer matrix approach to four mode coupling in fiber Bragg gratings,” IEEE J. Quantum Electron. 45, 1142–1148 (2009).
    [CrossRef]
  13. J. Dacles-Mariani, and G. Rodrigue, “Study of optically induced effects due to bending and twisting using the vector finite-element method,” J. Opt. Soc. Am. B 23, 1743–1751 (2006).
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  14. R. Ulrich, S. C. Rashleigh, and W. Eickhoff, “Bending-induced birefringence in single-mode fibers,” Opt. Lett. 5, 273–275 (1980).
    [CrossRef]
  15. N. G. Walker, and G. R. Walker, “Polarization control for coherent communications,” J. Lightwave Technol. 8, 438–458 (1990).
    [CrossRef]
  16. M. Tsubokawa, T. Higashi, and Y. Negishi, “Mode couplings due to external forces distributed along a polarization-maintaining fiber—an evaluation,” Appl. Opt. 27, 166–173 (1988).
    [CrossRef]

2009 (2)

M. S. Müller, H. J. El-Khozondar, A. Bernardini, and A. W. Koch, “Transfer matrix approach to four mode coupling in fiber Bragg gratings,” IEEE J. Quantum Electron. 45, 1142–1148 (2009).
[CrossRef]

H. J. El-Khozondar, M. S. Muller, R. J. El-Khozondar, and A. W. Koch, “Polarization rotation in twisted polarization maintaining fibers using a fixed reference frame,” J. Lightwave Technol. 27, 5590–5596 (2009).
[CrossRef]

2006 (1)

2004 (1)

D. Tentori, C. Ayala-Diaz, F. Trevino-Martinez, and J. Mendieta-Jimenez, “Spectral birefringence characterization of erbium-doped monomode optical fibers: relative values of the linear and the circular birefringence,” Proc. SPIE 5622, 868–872 (2004).
[CrossRef]

2003 (1)

1999 (1)

1997 (3)

1990 (1)

N. G. Walker, and G. R. Walker, “Polarization control for coherent communications,” J. Lightwave Technol. 8, 438–458 (1990).
[CrossRef]

1988 (1)

1983 (1)

S. C. Rashleigh, “Origins and control of polarization effects in single-mode fibers,” J. Lightwave Technol. 1, 312–331 (1983).
[CrossRef]

1982 (1)

J. I. Sakai and T. Kimura, “Polarization behavior in multiply perturbed single-mode fibers,” IEEE J. Quantum Electron. 18, 59–65 (1982).
[CrossRef]

1981 (1)

A. J. Barlow, and D. N. Payne, “Polarization maintenance in circularly birefringent fibers,” Electron. Lett. 17, 388–389 (1981).
[CrossRef]

1980 (1)

1979 (1)

Ayala-Diaz, C.

D. Tentori, C. Ayala-Diaz, F. Trevino-Martinez, and J. Mendieta-Jimenez, “Spectral birefringence characterization of erbium-doped monomode optical fibers: relative values of the linear and the circular birefringence,” Proc. SPIE 5622, 868–872 (2004).
[CrossRef]

Barlow, A. J.

A. J. Barlow, and D. N. Payne, “Polarization maintenance in circularly birefringent fibers,” Electron. Lett. 17, 388–389 (1981).
[CrossRef]

Bernardini, A.

M. S. Müller, H. J. El-Khozondar, A. Bernardini, and A. W. Koch, “Transfer matrix approach to four mode coupling in fiber Bragg gratings,” IEEE J. Quantum Electron. 45, 1142–1148 (2009).
[CrossRef]

Dacles-Mariani, J.

Eickhoff, W.

El-Khozondar, H. J.

M. S. Müller, H. J. El-Khozondar, A. Bernardini, and A. W. Koch, “Transfer matrix approach to four mode coupling in fiber Bragg gratings,” IEEE J. Quantum Electron. 45, 1142–1148 (2009).
[CrossRef]

H. J. El-Khozondar, M. S. Muller, R. J. El-Khozondar, and A. W. Koch, “Polarization rotation in twisted polarization maintaining fibers using a fixed reference frame,” J. Lightwave Technol. 27, 5590–5596 (2009).
[CrossRef]

El-Khozondar, R. J.

Etzel, S. M.

Feat, N.

Higashi, T.

Huang, H. C.

Kim, B. Y.

Kim, H. Y.

Kimura, T.

J. I. Sakai and T. Kimura, “Polarization behavior in multiply perturbed single-mode fibers,” IEEE J. Quantum Electron. 18, 59–65 (1982).
[CrossRef]

Koch, A. W.

H. J. El-Khozondar, M. S. Muller, R. J. El-Khozondar, and A. W. Koch, “Polarization rotation in twisted polarization maintaining fibers using a fixed reference frame,” J. Lightwave Technol. 27, 5590–5596 (2009).
[CrossRef]

M. S. Müller, H. J. El-Khozondar, A. Bernardini, and A. W. Koch, “Transfer matrix approach to four mode coupling in fiber Bragg gratings,” IEEE J. Quantum Electron. 45, 1142–1148 (2009).
[CrossRef]

Lee, E. H.

Mendieta-Jimenez, J.

D. Tentori, C. Ayala-Diaz, F. Trevino-Martinez, and J. Mendieta-Jimenez, “Spectral birefringence characterization of erbium-doped monomode optical fibers: relative values of the linear and the circular birefringence,” Proc. SPIE 5622, 868–872 (2004).
[CrossRef]

Muller, M. S.

Müller, M. S.

M. S. Müller, H. J. El-Khozondar, A. Bernardini, and A. W. Koch, “Transfer matrix approach to four mode coupling in fiber Bragg gratings,” IEEE J. Quantum Electron. 45, 1142–1148 (2009).
[CrossRef]

Negishi, Y.

Palais, J. C.

Payne, D. N.

A. J. Barlow, and D. N. Payne, “Polarization maintenance in circularly birefringent fibers,” Electron. Lett. 17, 388–389 (1981).
[CrossRef]

Rashleigh, S. C.

S. C. Rashleigh, “Origins and control of polarization effects in single-mode fibers,” J. Lightwave Technol. 1, 312–331 (1983).
[CrossRef]

R. Ulrich, S. C. Rashleigh, and W. Eickhoff, “Bending-induced birefringence in single-mode fibers,” Opt. Lett. 5, 273–275 (1980).
[CrossRef]

Rodrigue, G.

Rose, A. H.

Sakai, J. I.

J. I. Sakai and T. Kimura, “Polarization behavior in multiply perturbed single-mode fibers,” IEEE J. Quantum Electron. 18, 59–65 (1982).
[CrossRef]

Shen, P.

Simon, A.

Tentori, D.

D. Tentori, C. Ayala-Diaz, F. Trevino-Martinez, and J. Mendieta-Jimenez, “Spectral birefringence characterization of erbium-doped monomode optical fibers: relative values of the linear and the circular birefringence,” Proc. SPIE 5622, 868–872 (2004).
[CrossRef]

Trevino-Martinez, F.

D. Tentori, C. Ayala-Diaz, F. Trevino-Martinez, and J. Mendieta-Jimenez, “Spectral birefringence characterization of erbium-doped monomode optical fibers: relative values of the linear and the circular birefringence,” Proc. SPIE 5622, 868–872 (2004).
[CrossRef]

Tsubokawa, M.

Ulrich, R.

Walker, G. R.

N. G. Walker, and G. R. Walker, “Polarization control for coherent communications,” J. Lightwave Technol. 8, 438–458 (1990).
[CrossRef]

Walker, N. G.

N. G. Walker, and G. R. Walker, “Polarization control for coherent communications,” J. Lightwave Technol. 8, 438–458 (1990).
[CrossRef]

Appl. Opt. (7)

Electron. Lett. (1)

A. J. Barlow, and D. N. Payne, “Polarization maintenance in circularly birefringent fibers,” Electron. Lett. 17, 388–389 (1981).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. S. Müller, H. J. El-Khozondar, A. Bernardini, and A. W. Koch, “Transfer matrix approach to four mode coupling in fiber Bragg gratings,” IEEE J. Quantum Electron. 45, 1142–1148 (2009).
[CrossRef]

J. I. Sakai and T. Kimura, “Polarization behavior in multiply perturbed single-mode fibers,” IEEE J. Quantum Electron. 18, 59–65 (1982).
[CrossRef]

J. Lightwave Technol. (3)

S. C. Rashleigh, “Origins and control of polarization effects in single-mode fibers,” J. Lightwave Technol. 1, 312–331 (1983).
[CrossRef]

H. J. El-Khozondar, M. S. Muller, R. J. El-Khozondar, and A. W. Koch, “Polarization rotation in twisted polarization maintaining fibers using a fixed reference frame,” J. Lightwave Technol. 27, 5590–5596 (2009).
[CrossRef]

N. G. Walker, and G. R. Walker, “Polarization control for coherent communications,” J. Lightwave Technol. 8, 438–458 (1990).
[CrossRef]

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

Opt. Lett. (1)

Proc. SPIE (1)

D. Tentori, C. Ayala-Diaz, F. Trevino-Martinez, and J. Mendieta-Jimenez, “Spectral birefringence characterization of erbium-doped monomode optical fibers: relative values of the linear and the circular birefringence,” Proc. SPIE 5622, 868–872 (2004).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Schematic diagram of (a) twisted fiber under lateral pressure and (b) twisted birefringent fiber.

Fig. 2.
Fig. 2.

Output polarization rotation versus the twist angle of the output end for LB=30m. α1: calculated by Eq. (28); α2: calculated by Eq. (30); α3: calculated by Eq. (31).

Fig. 3.
Fig. 3.

Output polarization rotation versus the twist angle of the output end for LB=3mm. α1: calculated by Eq. (28); α2: calculated by Eq. (30); α3: calculated by Eq. (31).

Fig. 4.
Fig. 4.

(a) Polarization rotation α and (b) intensities Ix, Iy, and S3 varied with the twisting angle of PMF with LB=3mm and input polarization θ0=5°.

Fig. 5.
Fig. 5.

Polarization rotation α and output Stokes components varied with the propagation distance in PMF with LB=3mm and input polarization θ0=5°.

Fig. 6.
Fig. 6.

Polarization rotation α and Stokes components varied with the propagation distance in PMF with LB=0.02m and torsion of LT=0.2m.

Fig. 7.
Fig. 7.

(a) Bent and twisted PMF. (b) Strain state composed of internal strain and lateral pressure-induced strain.

Equations (40)

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(ab)=(jΔβ/2κτκτjΔβ/2)(ab),
(a(z)b(z))=(pqqp*)(a(0)b(0)),
(ab)=(jΔβ/2κτ+τκττjΔβ/2)(ab).
a=(τκτ)b,b=(κττ)a,
a=κτb,b=κτa.
Δ(1ε)=1ε2(ΔεxΔεyΔεzΔεyzΔεzxΔεxy)=(p11p12p12000p12p11p12000p12p12p11000000p44000000p44000000p44)(exeyezeyzezxexy),
ε=(εxεxyεxzεxyεyεyzεxzεyzεz)=ε0ε02(p11ex+p12eyp44exyp44τyp44exyp12ex+p11eyp44τxp44τyp44τxp12(ex+ey))=ε0(p11+p12)e¯ε02+ε02p44(Δe0τy0Δeτxτyτx2e¯)=ε¯+ε˜,
x=x1cosτz+y1sinτzy=x1sinτz+y1cosτz.
ex=ex1cos2τz+ey1sin2τzexy1sinτzcosτz,ey=ex1sin2τz+ey1cos2τz+exy1sinτzcosτz,exy=2exy1cos2τz+(ex1ey1)sin2τz,exz=exz1cosτz+eyz1sinτz,eyz=eyz1cosτzexz1sinτz.
ex=ex1cos2τz+ey1sin2τz,ey=ex1sin2τz+ey1cos2τz,exy=(ex1ey1)sin2τz.
ε=ε0(p11+p12)e¯ε02ε02p44(ΔecosθΔesinθτyΔesinθΔecosθτxτyτx2e¯)=ε¯+ε˜,
2E+k2εE(·E)=0.
·E=·(ε˜E)/ε¯,
Φ^E=ε˜k2E(·E)=ε˜k2E+[·(ε˜E)]/ε¯.
E=a(z)E1(x,y)+b(z)E2(x,y).
E1=[J0(βtr)0jhcosϕJ1(βtr)]Tejβz[J0(βtr)00]Tejβz,E2=[0J0(βtr)jhsinϕJ1(βtr)]Tejβz[0J0(βtr)0]Tejβz,
ε˜Ep44ε02(ΔecosθExΔesinθEyΔesinθEx+ΔecosθEyτ(yExxEy)).
a=jE1*Φ^E/2β,b=jE2*Φ^E/2β,
E2*ε˜E2=E1*ε˜E1=ε¯2p44Δecosθ,
E1*ε˜E2=E2*ε˜E1=ε¯2p44Δesinθ.
·(ε˜E1)=ε02p44[Δe(ysinθxcosθ)1rJr+jβτyJ]ejβz,
·(ε˜E2)=ε02p44[Δe(ycosθ+xsinθ)1rJrjβτxJ]ejβz.
|E1*·[·(ε˜E1)]dS|=|E2*·[·(ε˜E2)]dS|0,
E2*·[·(ε˜E1)]dS=E1*·[·(ε˜E2)]dSjβε02p44τ.
(a1a2)=(jκbcos2τzκτ+jκbsin2τzκτ+jκbsin2τzjκbcos2τz)(a1a2),
(ab)=jκb(ab).
E1=a(z)ejβz=a(0)ej(β+κb)z,E2=b(z)ejβz=b(0)ej(βκb)z.
β1,2=β±κb,
(ab)=(j(Δβ/2)cos2τzgτ+j(Δβ/2)sin2τzgτ+j(Δβ/2)sin2τzj(Δβ/2)cos2τz)(ab).
(S1S2S3)=2(0κτκbsin2τzκτ0κbcos2τzκbsin2τzκbcos2τz0)(S1S2S3),
ddz(S12+S22+S32)=0,
(S1S2S3)=2(0gτ0gτ0Δβ/20Δβ/20)(S1S2S3),
(S1S2S3)=2(0(g1)τ0(g1)τ0Δβ/20Δβ/20)(S1S2S3).
(a(z)b(z))=(t1t2*t2t1*)(a(z1)b(z1)),
a(z)=a0cosρzb0κτsinρzρ+ja0κbρsinρz,b(z)=b0cosρz+a0κτsinρzρjb0κbρsinρz.
tan2θ=S2S1=ρ2tan2θ0cos2ρz+ρκτsin2ρzκb2+κτ2cos2ρz+κτρsin2ρztan2θ0.
tan2θ=ρκτsin2ρzκb2+κτ2cos2ρz.
ex=ex0+ex1cos2ϕ+ey1sin2ϕ,ey=ey0+ex1sin2ϕ+ey1cos2ϕ,exy=(ex1ey1)sin2ϕ.
tan2χ=2exyexey=2Δe1sin2ϕΔe0+Δe1cos2ϕ.
Δe2=ex2ey2=(exey)2+4exy2=(Δe0+Δe1cos2ϕ)2+4(Δe1)2sin22ϕ.

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