Abstract

We analyze propagation in a nonlinear, birefringent optical fiber with twist. The results show that the polarization evolution is periodic, and they are applied to the analysis of a Sagnac interferometer. The period is calculated by using perturbation theory, and we find a condition for it to be independent of the initial polarization state. We derive a simplified set of equations to describe the nonlinear evolution of the phase. We give a useful way to visualize the behavior of the nonlinear optical loop mirror (as a function of birefringence, twist, length, and input polarization) in terms of the Poincaré sphere.

© 2001 Optical Society of America

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References

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  1. J. D. Moores, K. Bergman, H. A. Haus, and E. P. Ippen, “Optical switching using fiber ring reflectors,” J. Opt. Soc. Am. B 8, 594–601 (1991).
    [CrossRef]
  2. H. C. Lim, F. Futami, and K. Kikuchi, “Polarization independent wavelength-shift-free optical phase conjugator using a nonlinear fiber sagnac interferometer,” IEEE Photon. Technol. Lett. 11, 578–580 (1999).
    [CrossRef]
  3. N. Finlayson, B. K. Nayar, and N. J. Doran, “Switch inversion and polarization sensitivity of the nonlinear-optical loop mirror,” Opt. Lett. 17, 112–114 (1992).
    [CrossRef] [PubMed]
  4. C. Baskev Clausen, J. H. Povlsen, and K. Rottwitt, “Polarization sensitivity of the nonlinear amplifying loop mirror,” Opt. Lett. 21, 1535–1537 (1996).
    [CrossRef] [PubMed]
  5. E. A. Kuzin, J. A. Andrade-Lucio, B. Ibarra-Escamilla, R. Rojas-Laguna, and J. Sanchez-Mondragon, “Nonlinear optical loop mirror using the nonlinear polarization rotation effect,” Opt. Commun. 144, 60–64 (1997).
    [CrossRef]
  6. E. A. Kuzin, B. Ibarra-Escamilla, R. Rojas-Laguna, and J. Sanchez-Mondragon, “Nonlinear optical loop mirror with low birefringence twisted fiber in the loop,” Opt. Commun. 149, 73–76 (1998).
    [CrossRef]
  7. J. W. Haus, G. Shaulov, E. A. Kuzin, and J. Sanchez-Mondragon, “Vector soliton fiber lasers,” Opt. Lett. 24, 376–378 (1999).
    [CrossRef]
  8. F. Matera and S. Wabnitz, “Nonlinear polarization evolution and instability in a twisted birefringent fiber,” Opt. Lett. 11, 467–469 (1986).
    [CrossRef] [PubMed]
  9. N. Akhmediev and A. Ankiewicz, Solitons—Nonlinear Pulses and Beams (Chapman & Hall, London, 1997).
  10. E. A. Kuzin, N. Korneev, J. W. Haus, and B. Ibarra-Escamilla, “Polarization independent nonlinear fiber Sagnac interferometer,” Opt. Commun. 183, 389–393 (2000).
    [CrossRef]
  11. Y. Liang, J. W. Low, J. K. Andersen, J. C. Stocker, O. Boyraz, M. N. Islam, and D. A. Nolan, “Polarization-insensitive nonlinear optical loop mirror demultiplexer with twisted fiber,” Opt. Lett. 24, 726–728 (1999).
    [CrossRef]
  12. S. F. Feldman, D. A. Weinberger, and H. G. Winful, “Polarization instability in a twisted birefringent optical-fiber,” J. Opt. Soc. Am. B 10, 1191–1201 (1993).
    [CrossRef]
  13. V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer, New York, 1989).
  14. C. R. Menyuk and P. K. A. Wai, “Elimination of nonlinear polarization rotation in twisted fibers,” J. Opt. Soc. Am. B 11, 1305–1309 (1994).
    [CrossRef]
  15. E. A. Kuzin, B. Ibarra-Escamilla, and J. M. Estudillo-Ayala, “Low birefringence measurement in optical fiber,” Electron. Lett. 35, 332–333 (1999).
    [CrossRef]
  16. A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Measurements of beat length and perturbation length in long single-mode fibers,” Opt. Lett. 25, 384–386 (2000).
    [CrossRef]

2000 (2)

E. A. Kuzin, N. Korneev, J. W. Haus, and B. Ibarra-Escamilla, “Polarization independent nonlinear fiber Sagnac interferometer,” Opt. Commun. 183, 389–393 (2000).
[CrossRef]

A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Measurements of beat length and perturbation length in long single-mode fibers,” Opt. Lett. 25, 384–386 (2000).
[CrossRef]

1999 (4)

E. A. Kuzin, B. Ibarra-Escamilla, and J. M. Estudillo-Ayala, “Low birefringence measurement in optical fiber,” Electron. Lett. 35, 332–333 (1999).
[CrossRef]

H. C. Lim, F. Futami, and K. Kikuchi, “Polarization independent wavelength-shift-free optical phase conjugator using a nonlinear fiber sagnac interferometer,” IEEE Photon. Technol. Lett. 11, 578–580 (1999).
[CrossRef]

J. W. Haus, G. Shaulov, E. A. Kuzin, and J. Sanchez-Mondragon, “Vector soliton fiber lasers,” Opt. Lett. 24, 376–378 (1999).
[CrossRef]

Y. Liang, J. W. Low, J. K. Andersen, J. C. Stocker, O. Boyraz, M. N. Islam, and D. A. Nolan, “Polarization-insensitive nonlinear optical loop mirror demultiplexer with twisted fiber,” Opt. Lett. 24, 726–728 (1999).
[CrossRef]

1998 (1)

E. A. Kuzin, B. Ibarra-Escamilla, R. Rojas-Laguna, and J. Sanchez-Mondragon, “Nonlinear optical loop mirror with low birefringence twisted fiber in the loop,” Opt. Commun. 149, 73–76 (1998).
[CrossRef]

1997 (1)

E. A. Kuzin, J. A. Andrade-Lucio, B. Ibarra-Escamilla, R. Rojas-Laguna, and J. Sanchez-Mondragon, “Nonlinear optical loop mirror using the nonlinear polarization rotation effect,” Opt. Commun. 144, 60–64 (1997).
[CrossRef]

1996 (1)

1994 (1)

1993 (1)

1992 (1)

1991 (1)

1986 (1)

Andersen, J. K.

Andrade-Lucio, J. A.

E. A. Kuzin, J. A. Andrade-Lucio, B. Ibarra-Escamilla, R. Rojas-Laguna, and J. Sanchez-Mondragon, “Nonlinear optical loop mirror using the nonlinear polarization rotation effect,” Opt. Commun. 144, 60–64 (1997).
[CrossRef]

Bergman, K.

Boyraz, O.

Clausen, C. Baskev

Doran, N. J.

Estudillo-Ayala, J. M.

E. A. Kuzin, B. Ibarra-Escamilla, and J. M. Estudillo-Ayala, “Low birefringence measurement in optical fiber,” Electron. Lett. 35, 332–333 (1999).
[CrossRef]

Feldman, S. F.

Finlayson, N.

Futami, F.

H. C. Lim, F. Futami, and K. Kikuchi, “Polarization independent wavelength-shift-free optical phase conjugator using a nonlinear fiber sagnac interferometer,” IEEE Photon. Technol. Lett. 11, 578–580 (1999).
[CrossRef]

Galtarossa, A.

Haus, H. A.

Haus, J. W.

E. A. Kuzin, N. Korneev, J. W. Haus, and B. Ibarra-Escamilla, “Polarization independent nonlinear fiber Sagnac interferometer,” Opt. Commun. 183, 389–393 (2000).
[CrossRef]

J. W. Haus, G. Shaulov, E. A. Kuzin, and J. Sanchez-Mondragon, “Vector soliton fiber lasers,” Opt. Lett. 24, 376–378 (1999).
[CrossRef]

Ibarra-Escamilla, B.

E. A. Kuzin, N. Korneev, J. W. Haus, and B. Ibarra-Escamilla, “Polarization independent nonlinear fiber Sagnac interferometer,” Opt. Commun. 183, 389–393 (2000).
[CrossRef]

E. A. Kuzin, B. Ibarra-Escamilla, and J. M. Estudillo-Ayala, “Low birefringence measurement in optical fiber,” Electron. Lett. 35, 332–333 (1999).
[CrossRef]

E. A. Kuzin, B. Ibarra-Escamilla, R. Rojas-Laguna, and J. Sanchez-Mondragon, “Nonlinear optical loop mirror with low birefringence twisted fiber in the loop,” Opt. Commun. 149, 73–76 (1998).
[CrossRef]

E. A. Kuzin, J. A. Andrade-Lucio, B. Ibarra-Escamilla, R. Rojas-Laguna, and J. Sanchez-Mondragon, “Nonlinear optical loop mirror using the nonlinear polarization rotation effect,” Opt. Commun. 144, 60–64 (1997).
[CrossRef]

Ippen, E. P.

Islam, M. N.

Kikuchi, K.

H. C. Lim, F. Futami, and K. Kikuchi, “Polarization independent wavelength-shift-free optical phase conjugator using a nonlinear fiber sagnac interferometer,” IEEE Photon. Technol. Lett. 11, 578–580 (1999).
[CrossRef]

Korneev, N.

E. A. Kuzin, N. Korneev, J. W. Haus, and B. Ibarra-Escamilla, “Polarization independent nonlinear fiber Sagnac interferometer,” Opt. Commun. 183, 389–393 (2000).
[CrossRef]

Kuzin, E. A.

E. A. Kuzin, N. Korneev, J. W. Haus, and B. Ibarra-Escamilla, “Polarization independent nonlinear fiber Sagnac interferometer,” Opt. Commun. 183, 389–393 (2000).
[CrossRef]

J. W. Haus, G. Shaulov, E. A. Kuzin, and J. Sanchez-Mondragon, “Vector soliton fiber lasers,” Opt. Lett. 24, 376–378 (1999).
[CrossRef]

E. A. Kuzin, B. Ibarra-Escamilla, and J. M. Estudillo-Ayala, “Low birefringence measurement in optical fiber,” Electron. Lett. 35, 332–333 (1999).
[CrossRef]

E. A. Kuzin, B. Ibarra-Escamilla, R. Rojas-Laguna, and J. Sanchez-Mondragon, “Nonlinear optical loop mirror with low birefringence twisted fiber in the loop,” Opt. Commun. 149, 73–76 (1998).
[CrossRef]

E. A. Kuzin, J. A. Andrade-Lucio, B. Ibarra-Escamilla, R. Rojas-Laguna, and J. Sanchez-Mondragon, “Nonlinear optical loop mirror using the nonlinear polarization rotation effect,” Opt. Commun. 144, 60–64 (1997).
[CrossRef]

Liang, Y.

Lim, H. C.

H. C. Lim, F. Futami, and K. Kikuchi, “Polarization independent wavelength-shift-free optical phase conjugator using a nonlinear fiber sagnac interferometer,” IEEE Photon. Technol. Lett. 11, 578–580 (1999).
[CrossRef]

Low, J. W.

Matera, F.

Menyuk, C. R.

Moores, J. D.

Nayar, B. K.

Nolan, D. A.

Palmieri, L.

Povlsen, J. H.

Rojas-Laguna, R.

E. A. Kuzin, B. Ibarra-Escamilla, R. Rojas-Laguna, and J. Sanchez-Mondragon, “Nonlinear optical loop mirror with low birefringence twisted fiber in the loop,” Opt. Commun. 149, 73–76 (1998).
[CrossRef]

E. A. Kuzin, J. A. Andrade-Lucio, B. Ibarra-Escamilla, R. Rojas-Laguna, and J. Sanchez-Mondragon, “Nonlinear optical loop mirror using the nonlinear polarization rotation effect,” Opt. Commun. 144, 60–64 (1997).
[CrossRef]

Rottwitt, K.

Sanchez-Mondragon, J.

J. W. Haus, G. Shaulov, E. A. Kuzin, and J. Sanchez-Mondragon, “Vector soliton fiber lasers,” Opt. Lett. 24, 376–378 (1999).
[CrossRef]

E. A. Kuzin, B. Ibarra-Escamilla, R. Rojas-Laguna, and J. Sanchez-Mondragon, “Nonlinear optical loop mirror with low birefringence twisted fiber in the loop,” Opt. Commun. 149, 73–76 (1998).
[CrossRef]

E. A. Kuzin, J. A. Andrade-Lucio, B. Ibarra-Escamilla, R. Rojas-Laguna, and J. Sanchez-Mondragon, “Nonlinear optical loop mirror using the nonlinear polarization rotation effect,” Opt. Commun. 144, 60–64 (1997).
[CrossRef]

Schiano, M.

Shaulov, G.

Stocker, J. C.

Tambosso, T.

Wabnitz, S.

Wai, P. K. A.

Weinberger, D. A.

Winful, H. G.

Electron. Lett. (1)

E. A. Kuzin, B. Ibarra-Escamilla, and J. M. Estudillo-Ayala, “Low birefringence measurement in optical fiber,” Electron. Lett. 35, 332–333 (1999).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

H. C. Lim, F. Futami, and K. Kikuchi, “Polarization independent wavelength-shift-free optical phase conjugator using a nonlinear fiber sagnac interferometer,” IEEE Photon. Technol. Lett. 11, 578–580 (1999).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (3)

E. A. Kuzin, J. A. Andrade-Lucio, B. Ibarra-Escamilla, R. Rojas-Laguna, and J. Sanchez-Mondragon, “Nonlinear optical loop mirror using the nonlinear polarization rotation effect,” Opt. Commun. 144, 60–64 (1997).
[CrossRef]

E. A. Kuzin, B. Ibarra-Escamilla, R. Rojas-Laguna, and J. Sanchez-Mondragon, “Nonlinear optical loop mirror with low birefringence twisted fiber in the loop,” Opt. Commun. 149, 73–76 (1998).
[CrossRef]

E. A. Kuzin, N. Korneev, J. W. Haus, and B. Ibarra-Escamilla, “Polarization independent nonlinear fiber Sagnac interferometer,” Opt. Commun. 183, 389–393 (2000).
[CrossRef]

Opt. Lett. (6)

Other (2)

V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer, New York, 1989).

N. Akhmediev and A. Ankiewicz, Solitons—Nonlinear Pulses and Beams (Chapman & Hall, London, 1997).

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Figures (13)

Fig. 1
Fig. 1

Projection of trajectories on the AC plane of the Poincaré sphere at high power (a projection of the Poincaré sphere on the AC plane). 1a and 2a are eigenmodes for nonzero twist. They are rotated from the principal axes of the birefringent fiber by Φ0. At low intensities the trajectories are circles on the Poincaré sphere orthogonal to the 1a2a axis (i.e., straight lines in the AC plane projection), but at high intensities the eigenmodes are displaced from their low-power positions as shown by 1b and 2b. The trajectories are no longer circles and the trajectories are no longer straight lines on the AC plane projection.

Fig. 2
Fig. 2

Dependence of phase shift of eigenmodes. The lines are linear and the results from analytical and numerical solutions are indistinguishable.

Fig. 3
Fig. 3

Analytical and numerical nonlinear phase shifts versus the initial polarization state. For a power equal to 1 the results are the same to within a few percent.

Fig. 4
Fig. 4

Schematic drawing of the polarization-insensitive NOLM.

Fig. 5
Fig. 5

Transmission of the polarization-insensitive NOLM versus angle on the Poincaré sphere and input power. We use a 0.4/0.6 coupler, critical twist rate, a NOLM with 10 beat lengths, and the adjusted polarization controller.

Fig. 6
Fig. 6

Polarization controller angle is misaligned by θ. Except for the controller adjustment angle, all parameters are as in Fig. 5.

Fig. 7
Fig. 7

Transmission of polarization-insensitive NOLM with the controller misaligned by π/4. Except for the misaligned polarization controller, the parameters are the same as in Fig. 5.

Fig. 8
Fig. 8

Transmission of the polarization-insensitive NOLM versus angle on the Poincaré sphere and input power. We use a 0.4/0.6 coupler, critical twist rate, a NOLM with 50 beat lengths, and the adjusted polarization controller.

Fig. 9
Fig. 9

AC plane of the Poincaré sphere for highly twisted fiber. The eigenstate is depicted by the black dot and is nearly circularly polarized. The quarter-wave polarization controller is designed to change the trajectory from the point C+ to the line of linear polarizations.

Fig. 10
Fig. 10

Schematic of the NOLM with a 0.5/0.5 coupler and the quarter-wave (QW) plate.

Fig. 11
Fig. 11

Transmission versus twist rate and input power of the NOLM with a symmetric coupler. The quarter-wave polarization controller is used and the NOLM is 10 beat lengths long. The input polarization is C=0, A=1.

Fig. 12
Fig. 12

Transmission versus twist rate and input power with a symmetrical coupler. The parameters are the same as in Fig. 11, except that C=0.05, A=0.

Fig. 13
Fig. 13

Polarization dependence of the NOLM’s transmission with a symmetrical coupler. In this case g=14.1.

Equations (24)

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

zC+=iα2nC++ik exp(2iqz)C-+iβ(|C+|2+2|C-|2)C+,
zC-=-iα2nC-+ik exp(-2iqz)C++iβ(|C-|2+2|C+|2)C-.
C±=C± exp(±iqz)PIN,
zC+=igC++iπC-+iPN(|C+|2+2|C-|2)C+,
zC-=-igC-+iπC++iPN(|C-|2+2|C+|2)C-,
izC±=H/C±*,
H(C±, C±*)=-gA-πC-3/4PN+1/4PNA2,
A=|C+|2-|C-|2,
B=i(C+*C--C+C-*),
C=C+*C-+C+C-*.
A2+B2+C2=1,
zA=±2π1-A2-(C0-gA/π+PNA2/4π)2,
C=C0-(g/π)A+(PN/4π)A2,
C-C+=C-iB1+A=1-AC+iB,
C±(z)=C±(0)exp0zdz±ig+iπC(z)iB(z)1±A(z)+iPN[3/21/2A(z)],
S+=cos αC++sin αC-,
S-=-sin αC++cos αC-.
izS+=-μ-3/2PN-[1/2PN(As cos Φ0-CS sin Φ0)cos Φ0]S+
+[1/2PN(AS cos Φ0-CS sin Φ0)sin Φ0]S-,
izS-=-μ-3/2PN+[1/2PN(AS cos Φ0-CS sin Φ0)cos Φ0]S-
+[1/2PN(AS cos Φ0-CS sin Φ0)sin Φ0]S+.
izS+=[-μ-3/2PN+1/4PN sin2Φ0+1/2PNAS(cos2 Φ0-1/2 sin2 Φ0)]S+,
izS-=[μ-3/2PN+1/4PN sin2 Φ0-1/2PNAS(cos2 Φ0-1/2 sin2 Φ0)]S-.
T=2π2µ-PNAS(cos2 Φ0-1/2 sin2 Φ0).

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