Abstract

The fast-axis polarization instability arises in a weakly birefringent fiber as a result of competition between the natural fiber birefringence and the nonlinear ellipse rotation. Direct observation of the fast-axis polarization instability is reported. A full theoretical development of the polarization instability in a twisted, birefringent optical fiber is presented. The theory includes the derivation of and full solutions for the evolution of light in a twisted fiber as well as stability analysis and phase-plane representation of the solutions. The experiment is described in detail; good agreement is obtained between theory and experiment. As a result of the instability, very small variations in either the input power or the input polarization to the fiber result in large changes in the output polarization. A crossed polarizer at the fiber end converts the polarization variation into intensity information. Thus the modulation depth of an input pulse has been increased from 15% to 100%. Modulation gains of as much as 10 times are theoretically possible.

© 1993 Optical Society of America

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References

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  1. R. H. Stolen, J. Botineau, and A. Ashkin, “Intensity discrimination of optical pulses with birefringent fibers,” Opt. Lett. 7, 512–514 (1982).
    [Crossref] [PubMed]
  2. N. J. Halas and D. Grischkowsky, “Simultaneous optical pump compression and wing reduction,” Appl. Phys. Lett. 48, 823–825 (1986).
    [Crossref]
  3. B. Nikolaus, D. Grischkowsky, and A. C. Balant, “Optical pulse reshaping based on the nonlinear birefringence of single-mode optical fibers,” Opt. Lett. 8, 189–191 (1983).
    [Crossref] [PubMed]
  4. K.–I. Kitayama, Y. Kimura, and S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
    [Crossref]
  5. J. M. Dziedzic, R. H. Stolen, and A. Ashkin, “Optical Kerr effect in long fibers,” Appl. Opt. 20, 1403–1406 (1981).
    [Crossref] [PubMed]
  6. B. Daino, G. Gregori, and S. Wabnitz, “New all-optical devices based on third-order noninearity of birefringent fibers,” Opt. Lett. 11, 42–44 (1986).
    [Crossref]
  7. H. G. Winful, “Polarization instabilities in birefringent nonlinear media: application to fiber-optic devices,” Opt. Lett. 11, 33–35 (1986).
    [Crossref] [PubMed]
  8. Related theoretical work was also done by K. L. Sala, “Nonlinear refractive-index phenomena in isotropic media subjected to a dc electric field: exact solutions,” Phys. Rev. A 29, 1944–1956 (1984).
    [Crossref]
  9. Recently researchers have theoretically shown that an analogous polarization instability may be expected for spatial solitons in a waveguide: C. M. de Sterke and J. E. Sipe, “Polarization instability in a waveguide geometry,” Opt. Lett. 16, 202–204 (1991).
    [Crossref] [PubMed]
  10. R. Ulrich and A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt. 18, 2241–2251 (1979).
    [Crossref] [PubMed]
  11. H. G. Winful and A. Hu, “Intensity discrimination with twisted birefringent optical fibers,” Opt. Lett. 11, 668–670 (1986).
    [Crossref] [PubMed]
  12. F. Matera and S. Wabnitz, “Nonlinear polarization evolution and instability in a twisted birefringent fiber,” Opt. Lett. 11, 467–469 (1986).
    [Crossref] [PubMed]
  13. S. F. Feldman, D. A. Weinberger, and H. G. Winful, “Observation of polarization instabilities and modulational gain in a low-birefringence optical fiber,” Opt. Lett. 15, 311–313 (1990).
    [Crossref] [PubMed]
  14. S. F. Feldman, D. A. Weinberger, and H. G. Winful, “Polarization instability in a twisted optical fiber,” in Nonlinear Dynamics in Optical Systems, N. B. Abraham, E. M. Garmire, and P. Mandel, eds., Vol. 7 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1990), pp. 471–474.
  15. S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber”, Appl. Phys. Lett. 49, 1224–1226 (1986).
    [Crossref]
  16. C. R. Menyuk, “Stability of solitons in birefringent optical fibers. II. Arbitrary amplitudes,” J. Opt. Soc. Am. B 5, 392–402 (1988).
    [Crossref]
  17. K. J. Blow, N. J. Doran, and D. Wood, “Polarization instabilities for solitons in birefringent fibers,” Opt. Lett. 12, 202–204 (1987).
    [Crossref] [PubMed]
  18. P. McIntyre and A. W. Snyder, “Light propagation in twisted anisotropic media: application to photoreceptors,” J. Opt. Soc. Am. 68, 149–157 (1978).
    [Crossref] [PubMed]
  19. H. G. Winful, “Self-induced polarization changes in birefringent fibers,” Appl. Phys. Lett. 47, 213–215 (1985).
    [Crossref]
  20. P. D. Maker, R. W. Terhune, and C. M. Savage, “Intensity-dependent changes in the refractive index of liquids,” Phys. Rev. Lett. 12, 507–509 (1964).
    [Crossref]
  21. S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
    [Crossref] [PubMed]
  22. P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer-Verlag, New York, 1971).
    [Crossref]
  23. R. Seydel, From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis (Elsevier, New York, 1988).
  24. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1983).
  25. A. Papp and H. Harms, “Polarization optics of index-gradient optical waveguide fibers,” Appl. Opt. 14, 2406–2411 (1975).
    [Crossref] [PubMed]
  26. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  27. N. Finlayson, B. K. Nayar, and N. J. Doran, “An ultrafast multibeatlength all-optical fibre switch,” Electron. Lett. 27, 1209–1210 (1991).
    [Crossref]
  28. P. Ferro, M. Haelterman, S. Trillo, S. Wabnitz, and B. Daino, “All-optical polarisation switch with long low-birefringence fibre,” Electron. Lett. 27, 1407–1408 (1991).
    [Crossref]

1991 (3)

N. Finlayson, B. K. Nayar, and N. J. Doran, “An ultrafast multibeatlength all-optical fibre switch,” Electron. Lett. 27, 1209–1210 (1991).
[Crossref]

P. Ferro, M. Haelterman, S. Trillo, S. Wabnitz, and B. Daino, “All-optical polarisation switch with long low-birefringence fibre,” Electron. Lett. 27, 1407–1408 (1991).
[Crossref]

Recently researchers have theoretically shown that an analogous polarization instability may be expected for spatial solitons in a waveguide: C. M. de Sterke and J. E. Sipe, “Polarization instability in a waveguide geometry,” Opt. Lett. 16, 202–204 (1991).
[Crossref] [PubMed]

1990 (1)

1988 (2)

S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
[Crossref] [PubMed]

C. R. Menyuk, “Stability of solitons in birefringent optical fibers. II. Arbitrary amplitudes,” J. Opt. Soc. Am. B 5, 392–402 (1988).
[Crossref]

1987 (1)

1986 (6)

1985 (2)

H. G. Winful, “Self-induced polarization changes in birefringent fibers,” Appl. Phys. Lett. 47, 213–215 (1985).
[Crossref]

K.–I. Kitayama, Y. Kimura, and S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[Crossref]

1984 (1)

Related theoretical work was also done by K. L. Sala, “Nonlinear refractive-index phenomena in isotropic media subjected to a dc electric field: exact solutions,” Phys. Rev. A 29, 1944–1956 (1984).
[Crossref]

1983 (1)

1982 (1)

1981 (1)

1979 (1)

1978 (1)

1975 (1)

1964 (1)

P. D. Maker, R. W. Terhune, and C. M. Savage, “Intensity-dependent changes in the refractive index of liquids,” Phys. Rev. Lett. 12, 507–509 (1964).
[Crossref]

Ashkin, A.

Assanto, G.

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber”, Appl. Phys. Lett. 49, 1224–1226 (1986).
[Crossref]

Balant, A. C.

Blow, K. J.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1983).

Botineau, J.

Byrd, P. F.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer-Verlag, New York, 1971).
[Crossref]

Daino, B.

P. Ferro, M. Haelterman, S. Trillo, S. Wabnitz, and B. Daino, “All-optical polarisation switch with long low-birefringence fibre,” Electron. Lett. 27, 1407–1408 (1991).
[Crossref]

B. Daino, G. Gregori, and S. Wabnitz, “New all-optical devices based on third-order noninearity of birefringent fibers,” Opt. Lett. 11, 42–44 (1986).
[Crossref]

de Sterke, C. M.

Doran, N. J.

N. Finlayson, B. K. Nayar, and N. J. Doran, “An ultrafast multibeatlength all-optical fibre switch,” Electron. Lett. 27, 1209–1210 (1991).
[Crossref]

K. J. Blow, N. J. Doran, and D. Wood, “Polarization instabilities for solitons in birefringent fibers,” Opt. Lett. 12, 202–204 (1987).
[Crossref] [PubMed]

Dziedzic, J. M.

Feldman, S. F.

S. F. Feldman, D. A. Weinberger, and H. G. Winful, “Observation of polarization instabilities and modulational gain in a low-birefringence optical fiber,” Opt. Lett. 15, 311–313 (1990).
[Crossref] [PubMed]

S. F. Feldman, D. A. Weinberger, and H. G. Winful, “Polarization instability in a twisted optical fiber,” in Nonlinear Dynamics in Optical Systems, N. B. Abraham, E. M. Garmire, and P. Mandel, eds., Vol. 7 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1990), pp. 471–474.

Ferro, P.

P. Ferro, M. Haelterman, S. Trillo, S. Wabnitz, and B. Daino, “All-optical polarisation switch with long low-birefringence fibre,” Electron. Lett. 27, 1407–1408 (1991).
[Crossref]

Finlayson, N.

N. Finlayson, B. K. Nayar, and N. J. Doran, “An ultrafast multibeatlength all-optical fibre switch,” Electron. Lett. 27, 1209–1210 (1991).
[Crossref]

Friedman, M. D.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer-Verlag, New York, 1971).
[Crossref]

Gregori, G.

Grischkowsky, D.

N. J. Halas and D. Grischkowsky, “Simultaneous optical pump compression and wing reduction,” Appl. Phys. Lett. 48, 823–825 (1986).
[Crossref]

B. Nikolaus, D. Grischkowsky, and A. C. Balant, “Optical pulse reshaping based on the nonlinear birefringence of single-mode optical fibers,” Opt. Lett. 8, 189–191 (1983).
[Crossref] [PubMed]

Haelterman, M.

P. Ferro, M. Haelterman, S. Trillo, S. Wabnitz, and B. Daino, “All-optical polarisation switch with long low-birefringence fibre,” Electron. Lett. 27, 1407–1408 (1991).
[Crossref]

Halas, N. J.

N. J. Halas and D. Grischkowsky, “Simultaneous optical pump compression and wing reduction,” Appl. Phys. Lett. 48, 823–825 (1986).
[Crossref]

Harms, H.

Hu, A.

Kimura, Y.

K.–I. Kitayama, Y. Kimura, and S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[Crossref]

Kitayama, K.–I.

K.–I. Kitayama, Y. Kimura, and S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[Crossref]

Maker, P. D.

P. D. Maker, R. W. Terhune, and C. M. Savage, “Intensity-dependent changes in the refractive index of liquids,” Phys. Rev. Lett. 12, 507–509 (1964).
[Crossref]

Matera, F.

McIntyre, P.

Menyuk, C. R.

Nayar, B. K.

N. Finlayson, B. K. Nayar, and N. J. Doran, “An ultrafast multibeatlength all-optical fibre switch,” Electron. Lett. 27, 1209–1210 (1991).
[Crossref]

Nikolaus, B.

Papp, A.

Sala, K. L.

Related theoretical work was also done by K. L. Sala, “Nonlinear refractive-index phenomena in isotropic media subjected to a dc electric field: exact solutions,” Phys. Rev. A 29, 1944–1956 (1984).
[Crossref]

Savage, C. M.

P. D. Maker, R. W. Terhune, and C. M. Savage, “Intensity-dependent changes in the refractive index of liquids,” Phys. Rev. Lett. 12, 507–509 (1964).
[Crossref]

Seaton, C. T.

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber”, Appl. Phys. Lett. 49, 1224–1226 (1986).
[Crossref]

Seikai, S.

K.–I. Kitayama, Y. Kimura, and S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[Crossref]

Seydel, R.

R. Seydel, From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis (Elsevier, New York, 1988).

Simon, A.

Sipe, J. E.

Snyder, A. W.

Stegeman, G. I.

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber”, Appl. Phys. Lett. 49, 1224–1226 (1986).
[Crossref]

Stolen, R. H.

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber”, Appl. Phys. Lett. 49, 1224–1226 (1986).
[Crossref]

R. H. Stolen, J. Botineau, and A. Ashkin, “Intensity discrimination of optical pulses with birefringent fibers,” Opt. Lett. 7, 512–514 (1982).
[Crossref] [PubMed]

J. M. Dziedzic, R. H. Stolen, and A. Ashkin, “Optical Kerr effect in long fibers,” Appl. Opt. 20, 1403–1406 (1981).
[Crossref] [PubMed]

Terhune, R. W.

P. D. Maker, R. W. Terhune, and C. M. Savage, “Intensity-dependent changes in the refractive index of liquids,” Phys. Rev. Lett. 12, 507–509 (1964).
[Crossref]

Trillo, S.

P. Ferro, M. Haelterman, S. Trillo, S. Wabnitz, and B. Daino, “All-optical polarisation switch with long low-birefringence fibre,” Electron. Lett. 27, 1407–1408 (1991).
[Crossref]

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber”, Appl. Phys. Lett. 49, 1224–1226 (1986).
[Crossref]

Ulrich, R.

Wabnitz, S.

P. Ferro, M. Haelterman, S. Trillo, S. Wabnitz, and B. Daino, “All-optical polarisation switch with long low-birefringence fibre,” Electron. Lett. 27, 1407–1408 (1991).
[Crossref]

S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
[Crossref] [PubMed]

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber”, Appl. Phys. Lett. 49, 1224–1226 (1986).
[Crossref]

B. Daino, G. Gregori, and S. Wabnitz, “New all-optical devices based on third-order noninearity of birefringent fibers,” Opt. Lett. 11, 42–44 (1986).
[Crossref]

F. Matera and S. Wabnitz, “Nonlinear polarization evolution and instability in a twisted birefringent fiber,” Opt. Lett. 11, 467–469 (1986).
[Crossref] [PubMed]

Weinberger, D. A.

S. F. Feldman, D. A. Weinberger, and H. G. Winful, “Observation of polarization instabilities and modulational gain in a low-birefringence optical fiber,” Opt. Lett. 15, 311–313 (1990).
[Crossref] [PubMed]

S. F. Feldman, D. A. Weinberger, and H. G. Winful, “Polarization instability in a twisted optical fiber,” in Nonlinear Dynamics in Optical Systems, N. B. Abraham, E. M. Garmire, and P. Mandel, eds., Vol. 7 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1990), pp. 471–474.

Winful, H. G.

S. F. Feldman, D. A. Weinberger, and H. G. Winful, “Observation of polarization instabilities and modulational gain in a low-birefringence optical fiber,” Opt. Lett. 15, 311–313 (1990).
[Crossref] [PubMed]

H. G. Winful and A. Hu, “Intensity discrimination with twisted birefringent optical fibers,” Opt. Lett. 11, 668–670 (1986).
[Crossref] [PubMed]

H. G. Winful, “Polarization instabilities in birefringent nonlinear media: application to fiber-optic devices,” Opt. Lett. 11, 33–35 (1986).
[Crossref] [PubMed]

H. G. Winful, “Self-induced polarization changes in birefringent fibers,” Appl. Phys. Lett. 47, 213–215 (1985).
[Crossref]

S. F. Feldman, D. A. Weinberger, and H. G. Winful, “Polarization instability in a twisted optical fiber,” in Nonlinear Dynamics in Optical Systems, N. B. Abraham, E. M. Garmire, and P. Mandel, eds., Vol. 7 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1990), pp. 471–474.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1983).

Wood, D.

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Appl. Opt. (3)

Appl. Phys. Lett. (4)

H. G. Winful, “Self-induced polarization changes in birefringent fibers,” Appl. Phys. Lett. 47, 213–215 (1985).
[Crossref]

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber”, Appl. Phys. Lett. 49, 1224–1226 (1986).
[Crossref]

N. J. Halas and D. Grischkowsky, “Simultaneous optical pump compression and wing reduction,” Appl. Phys. Lett. 48, 823–825 (1986).
[Crossref]

K.–I. Kitayama, Y. Kimura, and S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[Crossref]

Electron. Lett. (2)

N. Finlayson, B. K. Nayar, and N. J. Doran, “An ultrafast multibeatlength all-optical fibre switch,” Electron. Lett. 27, 1209–1210 (1991).
[Crossref]

P. Ferro, M. Haelterman, S. Trillo, S. Wabnitz, and B. Daino, “All-optical polarisation switch with long low-birefringence fibre,” Electron. Lett. 27, 1407–1408 (1991).
[Crossref]

J. Opt. Soc. Am. (1)

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

Opt. Lett. (9)

K. J. Blow, N. J. Doran, and D. Wood, “Polarization instabilities for solitons in birefringent fibers,” Opt. Lett. 12, 202–204 (1987).
[Crossref] [PubMed]

Recently researchers have theoretically shown that an analogous polarization instability may be expected for spatial solitons in a waveguide: C. M. de Sterke and J. E. Sipe, “Polarization instability in a waveguide geometry,” Opt. Lett. 16, 202–204 (1991).
[Crossref] [PubMed]

H. G. Winful and A. Hu, “Intensity discrimination with twisted birefringent optical fibers,” Opt. Lett. 11, 668–670 (1986).
[Crossref] [PubMed]

F. Matera and S. Wabnitz, “Nonlinear polarization evolution and instability in a twisted birefringent fiber,” Opt. Lett. 11, 467–469 (1986).
[Crossref] [PubMed]

S. F. Feldman, D. A. Weinberger, and H. G. Winful, “Observation of polarization instabilities and modulational gain in a low-birefringence optical fiber,” Opt. Lett. 15, 311–313 (1990).
[Crossref] [PubMed]

R. H. Stolen, J. Botineau, and A. Ashkin, “Intensity discrimination of optical pulses with birefringent fibers,” Opt. Lett. 7, 512–514 (1982).
[Crossref] [PubMed]

B. Nikolaus, D. Grischkowsky, and A. C. Balant, “Optical pulse reshaping based on the nonlinear birefringence of single-mode optical fibers,” Opt. Lett. 8, 189–191 (1983).
[Crossref] [PubMed]

B. Daino, G. Gregori, and S. Wabnitz, “New all-optical devices based on third-order noninearity of birefringent fibers,” Opt. Lett. 11, 42–44 (1986).
[Crossref]

H. G. Winful, “Polarization instabilities in birefringent nonlinear media: application to fiber-optic devices,” Opt. Lett. 11, 33–35 (1986).
[Crossref] [PubMed]

Phys. Rev. A (2)

Related theoretical work was also done by K. L. Sala, “Nonlinear refractive-index phenomena in isotropic media subjected to a dc electric field: exact solutions,” Phys. Rev. A 29, 1944–1956 (1984).
[Crossref]

S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

P. D. Maker, R. W. Terhune, and C. M. Savage, “Intensity-dependent changes in the refractive index of liquids,” Phys. Rev. Lett. 12, 507–509 (1964).
[Crossref]

Other (5)

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer-Verlag, New York, 1971).
[Crossref]

R. Seydel, From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis (Elsevier, New York, 1988).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1983).

S. F. Feldman, D. A. Weinberger, and H. G. Winful, “Polarization instability in a twisted optical fiber,” in Nonlinear Dynamics in Optical Systems, N. B. Abraham, E. M. Garmire, and P. Mandel, eds., Vol. 7 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1990), pp. 471–474.

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Figures (8)

Fig. 1
Fig. 1

Roots and allowed values of the quartic Q(u). The quartics are calculated for circularly polarized light in the same fiber (a) with no twist and (b) with a twist rate of one-half twist per beat length. The normalized power P = 2.2 in both cases. The roots are indicated by filled circles. The value of u is constrained to lie within one of the regions bounded by arrows.

Fig. 2
Fig. 2

Bifurcation diagrams showing the stable (solid curves) and unstable (dotted curves) eigenmodes for both (a) an untwisted fiber and (b) a fiber with a twist rate of 0.3 twist per beat length (after Ref. 14).

Fig. 3
Fig. 3

Phase portraits displaying the state of polarization of the light as it propagates in both untwisted and twisted fibers at powers less than and greater than the critical power (after Ref. 14). (a) Low-power phase portrait for an untwisted fiber (P = 0.5). Note the two regions of oscillatory motion centered on linearly polarized light aligned with the principal axes. (b) High-power phase plane for an untwisted fiber (P = 3.0). Note the separatrix orbit (drawn as a heavy curve) separating the regions of oscillatory and rotatory motion. (c) Low-power phase plane for a weakly twisted fiber (P = 0.01, twist rate = 0.3). The phase plane now consists of two regions of oscillatory motion as well as a region of rotatory motion. (d) High-power phase plane for a weakly twisted fiber (P = 4.0, twist rate = 0.3). The phase plane is once again divided by a separatrix orbit. Note that the saddle point has moved from the linear polarization to a slightly elliptical polarization.

Fig. 4
Fig. 4

Determination of the beat length of a weakly birefringent fiber by means of a generalized Jones matrix technique. The measurements required in Eq. (19) are indicated.

Fig. 5
Fig. 5

Schematic of the experimental layout. The attenuator is composed of a half-wave plate followed by a polarizer. The wave plate and the polarizer marked with an asterisk form a second attenuator, permitting more accurate control over the pulse power. The fiber is held in vacuum chucks and supported in an aluminum bar to minimize stress-induced birefringence. Neutral-density (ND) filters are used as needed to prevent damage to the photodiodes. 10× microscope objectives (MO’s) are used to couple light into and out of the fiber.

Fig. 6
Fig. 6

Comparison of the oscilloscope traces from the experiment with the theoretical calculations for a weakly twisted fiber. The upper half of each part of the figure shows the theoretically predicted pulse shape transmitted through the crossed polarizer. The upper trace in the oscilloscope photographs corresponds to P, while the lower trace corresponds to P. The lower oscilloscope trace should be compared with the theoretical prediction. Note that the vertical scale on the oscilloscope plots is in arbitrary units, and therefore the transmittance near the slow and fast axes may not be directly compared. In addition, after data were taken for (e), the neutral-density filters in front of the photodiode were changed from 1.9 to 3.1 for (f). (a) Theoretical approximation and oscilloscope trace of the Q-switched pulse input into the fiber. The modulation is due to longitudinal mode beating in the laser. (b) Light at the critical power input at ≈3° to the slow axis. Little light leaks through the crossed polarizer. (c) Intense light (P = 2.5) input to the fiber at ≈3° to the slow axis. Again, little pulse shaping is observed. (d) Low power light (P = 0.45) is input to the fiber at ≈10° to the fast axis. Nonlinear pulse-shaping effects begin to appear at a light intensity of only one half of the critical power. (e) Light at the critical power is input to the fiber at ≈10° from the fast axis. Significant pulse shaping is observed at the critical power. The fiber-crossed polarizer system is far more transmitting for the central region of the pulse than in the wings. (f) Intense light (P = 2.5) input at ≈10° from the fast axis. Significant pulse shaping is observed. The pulse appears narrower since the power in the wings is so strongly suppressed relative to the peak.

Fig. 7
Fig. 7

AM gain as a function of normalized input power. No AM gain is expected below the critical power. AM gain as high as 10 may be realized.

Fig. 8
Fig. 8

Oscilloscope trace showing a nearly fully modulated output pulse for an AM gain of ≈6. This expanded view was taken at the same time as the pulses in Figs. 6(d)–6(f) (after Ref. 13).

Tables (2)

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Table 1 Solutions for the Nonlinear Polarization Evolution in a Twisted Optical Fiber

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Table 2 Representative Data Used to Calculate Fiber Beat Lengtha

Equations (28)

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twist = [ 1 1 2 i η - 1 2 i η 2 ] .
R ( q z ) = [ cos ( q z ) sin ( q z ) - sin ( q z ) cos ( q z ) ] .
= R ( - q z ) twist R ( z ) = [ 0 + a cos ( 2 q z ) a sin ( 2 q z ) + i η a sin ( 2 q z ) - i η 0 - a cos ( 2 q z ) ] ,
d 2 E i d z 2 + ( ω c ) 2 i j E j = - μ 0 ω 2 P i NL ,
P NL = ( χ 0 / 2 ) [ ( E · E * ) E + 1 / 2 ( E · E ) E * ] ,
c ± ( z ) exp ( i k 0 z ) = 1 / 2 [ E x ( z ) ± i E y ( z ) ] .
d d z c + = i α 2 n c + + i κ exp ( i 2 q z ) c - + i β ( c + 2 + 2 c - 2 ) c + ,
d d z c - = i κ exp ( - i 2 q z ) c + - i α 2 n c - + i β ( c - 2 + 2 c + 2 ) c - ,
d d z c + = κ c - sin Ψ ,
d d z c - = - κ c + sin Ψ ,
d d z Ψ = α n - 2 q + κ ( cos ψ ) ( c - c + - c + c - ) + β ( c - 2 - c + 2 ) .
p = u + v ,
d d z ( c + c - cos Ψ ) = cos Ψ d d z ( c + c - ) - c + c - sin Ψ d Ψ d z .
Γ = u v cos Ψ - u ( u + μ - p ) ,
d u d z = 2 κ u v sin Ψ ,
d v d z = - 2 κ u v sin Ψ ,
d Ψ d z = κ ( v u - u v ) cos Ψ - 2 κ μ + 2 κ ( v - u ) .
u 0 u d u Q ( u ) = ± 0 z 2 κ d z = ± 2 κ z ,
d u d z = f 1 ( u , v , Ψ ) = k v sin Ψ 0 ,
d v d z = f 2 ( u , v , Ψ ) = - k u sin Ψ 0 ,
d Ψ d z = f 3 ( u , v , Ψ ) = κ ( v u - u v ) ( cos Ψ ) - 2 μ + 2 κ ( v 2 - u 2 ) 0 ,
16 ( u 2 ) 4 + 6 ( μ - 2 p ) ( u 2 ) 3 + 4 ( 1 - 6 μ p + 5 p 2 + μ 2 ) ( u 2 ) 2 + 4 ( - p - p μ 2 + 2 p 2 μ - p 3 ) ( u 2 ) + p 2 = 0.
J [ f 1 u f 1 v f 1 Ψ f 2 u f 2 v f 2 Ψ f 3 u f 3 v f 3 Ψ ] ,
J [ 0 0 κ v cos Ψ 0 0 - κ u cos Ψ - κ v ξ ( cos Ψ ) - 4 κ u κ u ξ ( cos Ψ ) + 4 κ v 0 ] .
exp ( i θ ) ( P ± i P ) = P TOT [ cos R out sin R out - sin R out cos R out ] × [ exp ( - i Γ / 2 ) 0 0 exp ( i Γ / 2 ) ] [ cos R in - sin R in sin R in cos R in ] [ 1 0 ] ,
cos 2 Γ 2 = P / P TOT - sin 2 ( R in + R out ) sin 2 ( R in - R out ) - sin 2 ( R in + R out ) .
L b = fiber length × 2 π Γ ( rad ) = fiber length × 360 Γ ( deg ) .
g AM = Δ P / P Δ P in / P in = g D T ,

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