Abstract

The author describes a new idea for making circular-polarization-maintaining optical fiber with an existing fabrication technique. The method simply requires one to spin at a constant rate a special preform consisting of only one off-axis stress-applying element in addition to the on-axis core. Measurements taken with such a fiber specimen verify the existence of circular eigenmodes, the ease of joining or splicing two fiber segments, the tolerance to macrobending with a small radius, etc. Good agreement exists between the experimental data and the theoretical analysis. Prospective applications are discussed.

© 1997 Optical Society of America

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References

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  1. R. Ulrich, A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt. 18, 2241–2251 (1979).
    [CrossRef] [PubMed]
  2. L. Jeunhomme, M. Monerie, “Polarization maintaining single-mode fiber cable design,” Electron. Lett. 16, 921–922 (1980).
    [CrossRef]
  3. J-i. Sakai, S. Machida, T. Kimura, “Existence of eigen polarization modes in anisotropic single-mode optical fibers,” Opt. Lett. 6, 496–498 (1981).
    [CrossRef] [PubMed]
  4. S. Machida, J. Sakai, T. Kimura, “Polarization preservation in long-length twisted single-mode optical fibers,” Trans. IECE Jpn. E 65, 642–647 (1982).
  5. A. J. Barlow, D. N. Payne, “Polarization maintenance in circularly birefringent fibres,” Electron. Lett. 17, 388–389 (1981).
    [CrossRef]
  6. F. Gauthier, J. Dubos, S. Blaison, Ph. Graindorge, H. J. Arditty, “Attempt to draw a circular polarization conserving fiber,” in Proceedings of the International Congress on Fiberoptic Rotation Sensors (MIT Press, Cambridge, Mass., 1981), p. 196.
  7. M. P. Varnham, R. D. Birch, D. N. Payne “Helical-core circularly birefringent fibres,” in Technical Digest of Fifth International Conference on Integrated Optics and Optical Fiber Communication/Eleventh European Conference on Optical Communication (Compagnia dei Librai Editrice, Genova, Italy, 1985), p. 135.
  8. M. P. Varnham, R. D. Birch, D. N. Payne, J. D. Love, “Design of helical core circularly birefringent fibres,” in Digest of Meeting on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1986), p. 68.
  9. R. D. Birch, “Fabrication and characterization of circularly birefringent helical fibres,” Electron. Lett. 23, 50–51 (1987).
    [CrossRef]
  10. Fang Xi-sheng, Lin Zong-qi, “Field in single-mode helically wound optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-33, 1150–1154 (1985).
    [CrossRef]
  11. F. Maystre, A. Bertholds, “Magneto-optic current sensor using a helical fiber Fabry–Perot resonator,” Opt. Lett. 14, 587–589 (1989).
    [CrossRef] [PubMed]
  12. A. J. Rogers, Jincheng Xu, Jialing Yao, “Vibration immunity for optical-fiber current measurement,” J. Lightwave Technol. 13, 1371–1377 (1995).
    [CrossRef]
  13. R. Dandliker, “Rotational effects of polarization in optical fibers,” in Anisotropic and Nonlinear Optical Waveguides, C. G. Someda, G. Stegeman, eds., Vol. 2 of Optical Wave Sciences and Technology (Elsevier, Amsterdam, 1992), pp. 39–76.
  14. C. D. Hussey, R. D. Birch, Y. Fujii, “Circularly birefringent single-mode optical fibres,” Electron. Lett. 22, 129–130 (1986).
    [CrossRef]
  15. Y. Fujii, C. D. Hussey, “Design considerations for circularly form-birefringent optical fibres,” Proc. Inst. Electr. Eng. Part J 133, 249–255 (1986).
  16. C. G. Someda, “Circularly birefringent dielectric waveguides,” Italian Patent41584A/85 (July1985).
  17. C. G. Someda, “Circularly birefringent dielectric waveguides,” Italian Patent41638A/86 (December1986).
  18. R. Castelli, F. Irrera, C. G. Someda, “Circularly birefringent optical fibres: new proposals Part I. Field analysis,” Opt. Quantum Electron. 21, 35–46 (1989).
    [CrossRef]
  19. C. G. Someda, “Circularly birefringent optical fibres: new proposals Part II. Birefringence and coupling loss,” Opt. Quantum. Electron. 23, 713–725 (1991).
  20. V. Ramaswamy, W. G. French, R. D. Standley, “Polarization characteristics of noncircular core single-mode fibers,” Appl. Opt. 17, 3014–3017 (1978).
    [CrossRef] [PubMed]
  21. Huang Hung-chia, “Practical circular-polarization-maintaining optical fiber,” U.S. Patent5,452,394 (September1995).
  22. Huang Hung-chia, “Generalized theory of coupled local normal modes in multi-wave guides,” Sci. Sin. 9, 142–154 (1960).
  23. Huang Hung-chia, He Ying-chun, “Polarization behavior of spun fiber versus conventional fiber under strong and slight twisting,” Microwave Opt. Technol. Lett. 9, 37–45 (1995).
    [CrossRef]
  24. Huang Hung-chia, Microwave Approach to Highly Irregular Fiber Optics (Wiley, New York, to be published).
  25. Huang Hung-chia, “Passive fiber-optic polarization control,” U.S. Patent4,943,132 (July1990).
  26. Huang Hung-chia, “Passive fiber-optic polarization control element,” U.S. Patent5,096,312 (March1992).
  27. Huang Hung-chia, “Fiber-optic analogs of bulk-optic wave plates,” Appl. Opt. 36, 4241–4258 (1997).
    [CrossRef]

1997 (1)

1995 (2)

Huang Hung-chia, He Ying-chun, “Polarization behavior of spun fiber versus conventional fiber under strong and slight twisting,” Microwave Opt. Technol. Lett. 9, 37–45 (1995).
[CrossRef]

A. J. Rogers, Jincheng Xu, Jialing Yao, “Vibration immunity for optical-fiber current measurement,” J. Lightwave Technol. 13, 1371–1377 (1995).
[CrossRef]

1991 (1)

C. G. Someda, “Circularly birefringent optical fibres: new proposals Part II. Birefringence and coupling loss,” Opt. Quantum. Electron. 23, 713–725 (1991).

1989 (2)

R. Castelli, F. Irrera, C. G. Someda, “Circularly birefringent optical fibres: new proposals Part I. Field analysis,” Opt. Quantum Electron. 21, 35–46 (1989).
[CrossRef]

F. Maystre, A. Bertholds, “Magneto-optic current sensor using a helical fiber Fabry–Perot resonator,” Opt. Lett. 14, 587–589 (1989).
[CrossRef] [PubMed]

1987 (1)

R. D. Birch, “Fabrication and characterization of circularly birefringent helical fibres,” Electron. Lett. 23, 50–51 (1987).
[CrossRef]

1986 (2)

C. D. Hussey, R. D. Birch, Y. Fujii, “Circularly birefringent single-mode optical fibres,” Electron. Lett. 22, 129–130 (1986).
[CrossRef]

Y. Fujii, C. D. Hussey, “Design considerations for circularly form-birefringent optical fibres,” Proc. Inst. Electr. Eng. Part J 133, 249–255 (1986).

1985 (1)

Fang Xi-sheng, Lin Zong-qi, “Field in single-mode helically wound optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-33, 1150–1154 (1985).
[CrossRef]

1982 (1)

S. Machida, J. Sakai, T. Kimura, “Polarization preservation in long-length twisted single-mode optical fibers,” Trans. IECE Jpn. E 65, 642–647 (1982).

1981 (2)

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

J-i. Sakai, S. Machida, T. Kimura, “Existence of eigen polarization modes in anisotropic single-mode optical fibers,” Opt. Lett. 6, 496–498 (1981).
[CrossRef] [PubMed]

1980 (1)

L. Jeunhomme, M. Monerie, “Polarization maintaining single-mode fiber cable design,” Electron. Lett. 16, 921–922 (1980).
[CrossRef]

1979 (1)

1978 (1)

1960 (1)

Huang Hung-chia, “Generalized theory of coupled local normal modes in multi-wave guides,” Sci. Sin. 9, 142–154 (1960).

Arditty, H. J.

F. Gauthier, J. Dubos, S. Blaison, Ph. Graindorge, H. J. Arditty, “Attempt to draw a circular polarization conserving fiber,” in Proceedings of the International Congress on Fiberoptic Rotation Sensors (MIT Press, Cambridge, Mass., 1981), p. 196.

Barlow, A. J.

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

Bertholds, A.

Birch, R. D.

R. D. Birch, “Fabrication and characterization of circularly birefringent helical fibres,” Electron. Lett. 23, 50–51 (1987).
[CrossRef]

C. D. Hussey, R. D. Birch, Y. Fujii, “Circularly birefringent single-mode optical fibres,” Electron. Lett. 22, 129–130 (1986).
[CrossRef]

M. P. Varnham, R. D. Birch, D. N. Payne “Helical-core circularly birefringent fibres,” in Technical Digest of Fifth International Conference on Integrated Optics and Optical Fiber Communication/Eleventh European Conference on Optical Communication (Compagnia dei Librai Editrice, Genova, Italy, 1985), p. 135.

M. P. Varnham, R. D. Birch, D. N. Payne, J. D. Love, “Design of helical core circularly birefringent fibres,” in Digest of Meeting on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1986), p. 68.

Blaison, S.

F. Gauthier, J. Dubos, S. Blaison, Ph. Graindorge, H. J. Arditty, “Attempt to draw a circular polarization conserving fiber,” in Proceedings of the International Congress on Fiberoptic Rotation Sensors (MIT Press, Cambridge, Mass., 1981), p. 196.

Castelli, R.

R. Castelli, F. Irrera, C. G. Someda, “Circularly birefringent optical fibres: new proposals Part I. Field analysis,” Opt. Quantum Electron. 21, 35–46 (1989).
[CrossRef]

Dandliker, R.

R. Dandliker, “Rotational effects of polarization in optical fibers,” in Anisotropic and Nonlinear Optical Waveguides, C. G. Someda, G. Stegeman, eds., Vol. 2 of Optical Wave Sciences and Technology (Elsevier, Amsterdam, 1992), pp. 39–76.

Dubos, J.

F. Gauthier, J. Dubos, S. Blaison, Ph. Graindorge, H. J. Arditty, “Attempt to draw a circular polarization conserving fiber,” in Proceedings of the International Congress on Fiberoptic Rotation Sensors (MIT Press, Cambridge, Mass., 1981), p. 196.

French, W. G.

Fujii, Y.

C. D. Hussey, R. D. Birch, Y. Fujii, “Circularly birefringent single-mode optical fibres,” Electron. Lett. 22, 129–130 (1986).
[CrossRef]

Y. Fujii, C. D. Hussey, “Design considerations for circularly form-birefringent optical fibres,” Proc. Inst. Electr. Eng. Part J 133, 249–255 (1986).

Gauthier, F.

F. Gauthier, J. Dubos, S. Blaison, Ph. Graindorge, H. J. Arditty, “Attempt to draw a circular polarization conserving fiber,” in Proceedings of the International Congress on Fiberoptic Rotation Sensors (MIT Press, Cambridge, Mass., 1981), p. 196.

Graindorge, Ph.

F. Gauthier, J. Dubos, S. Blaison, Ph. Graindorge, H. J. Arditty, “Attempt to draw a circular polarization conserving fiber,” in Proceedings of the International Congress on Fiberoptic Rotation Sensors (MIT Press, Cambridge, Mass., 1981), p. 196.

Hung-chia, Huang

Huang Hung-chia, “Fiber-optic analogs of bulk-optic wave plates,” Appl. Opt. 36, 4241–4258 (1997).
[CrossRef]

Huang Hung-chia, He Ying-chun, “Polarization behavior of spun fiber versus conventional fiber under strong and slight twisting,” Microwave Opt. Technol. Lett. 9, 37–45 (1995).
[CrossRef]

Huang Hung-chia, “Generalized theory of coupled local normal modes in multi-wave guides,” Sci. Sin. 9, 142–154 (1960).

Huang Hung-chia, “Practical circular-polarization-maintaining optical fiber,” U.S. Patent5,452,394 (September1995).

Huang Hung-chia, Microwave Approach to Highly Irregular Fiber Optics (Wiley, New York, to be published).

Huang Hung-chia, “Passive fiber-optic polarization control,” U.S. Patent4,943,132 (July1990).

Huang Hung-chia, “Passive fiber-optic polarization control element,” U.S. Patent5,096,312 (March1992).

Hussey, C. D.

C. D. Hussey, R. D. Birch, Y. Fujii, “Circularly birefringent single-mode optical fibres,” Electron. Lett. 22, 129–130 (1986).
[CrossRef]

Y. Fujii, C. D. Hussey, “Design considerations for circularly form-birefringent optical fibres,” Proc. Inst. Electr. Eng. Part J 133, 249–255 (1986).

Irrera, F.

R. Castelli, F. Irrera, C. G. Someda, “Circularly birefringent optical fibres: new proposals Part I. Field analysis,” Opt. Quantum Electron. 21, 35–46 (1989).
[CrossRef]

Jeunhomme, L.

L. Jeunhomme, M. Monerie, “Polarization maintaining single-mode fiber cable design,” Electron. Lett. 16, 921–922 (1980).
[CrossRef]

Kimura, T.

S. Machida, J. Sakai, T. Kimura, “Polarization preservation in long-length twisted single-mode optical fibers,” Trans. IECE Jpn. E 65, 642–647 (1982).

J-i. Sakai, S. Machida, T. Kimura, “Existence of eigen polarization modes in anisotropic single-mode optical fibers,” Opt. Lett. 6, 496–498 (1981).
[CrossRef] [PubMed]

Love, J. D.

M. P. Varnham, R. D. Birch, D. N. Payne, J. D. Love, “Design of helical core circularly birefringent fibres,” in Digest of Meeting on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1986), p. 68.

Machida, S.

S. Machida, J. Sakai, T. Kimura, “Polarization preservation in long-length twisted single-mode optical fibers,” Trans. IECE Jpn. E 65, 642–647 (1982).

J-i. Sakai, S. Machida, T. Kimura, “Existence of eigen polarization modes in anisotropic single-mode optical fibers,” Opt. Lett. 6, 496–498 (1981).
[CrossRef] [PubMed]

Maystre, F.

Monerie, M.

L. Jeunhomme, M. Monerie, “Polarization maintaining single-mode fiber cable design,” Electron. Lett. 16, 921–922 (1980).
[CrossRef]

Payne, D. N.

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

M. P. Varnham, R. D. Birch, D. N. Payne, J. D. Love, “Design of helical core circularly birefringent fibres,” in Digest of Meeting on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1986), p. 68.

M. P. Varnham, R. D. Birch, D. N. Payne “Helical-core circularly birefringent fibres,” in Technical Digest of Fifth International Conference on Integrated Optics and Optical Fiber Communication/Eleventh European Conference on Optical Communication (Compagnia dei Librai Editrice, Genova, Italy, 1985), p. 135.

Ramaswamy, V.

Rogers, A. J.

A. J. Rogers, Jincheng Xu, Jialing Yao, “Vibration immunity for optical-fiber current measurement,” J. Lightwave Technol. 13, 1371–1377 (1995).
[CrossRef]

Sakai, J.

S. Machida, J. Sakai, T. Kimura, “Polarization preservation in long-length twisted single-mode optical fibers,” Trans. IECE Jpn. E 65, 642–647 (1982).

Sakai, J-i.

Simon, A.

Someda, C. G.

C. G. Someda, “Circularly birefringent optical fibres: new proposals Part II. Birefringence and coupling loss,” Opt. Quantum. Electron. 23, 713–725 (1991).

R. Castelli, F. Irrera, C. G. Someda, “Circularly birefringent optical fibres: new proposals Part I. Field analysis,” Opt. Quantum Electron. 21, 35–46 (1989).
[CrossRef]

C. G. Someda, “Circularly birefringent dielectric waveguides,” Italian Patent41584A/85 (July1985).

C. G. Someda, “Circularly birefringent dielectric waveguides,” Italian Patent41638A/86 (December1986).

Standley, R. D.

Ulrich, R.

Varnham, M. P.

M. P. Varnham, R. D. Birch, D. N. Payne “Helical-core circularly birefringent fibres,” in Technical Digest of Fifth International Conference on Integrated Optics and Optical Fiber Communication/Eleventh European Conference on Optical Communication (Compagnia dei Librai Editrice, Genova, Italy, 1985), p. 135.

M. P. Varnham, R. D. Birch, D. N. Payne, J. D. Love, “Design of helical core circularly birefringent fibres,” in Digest of Meeting on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1986), p. 68.

Xi-sheng, Fang

Fang Xi-sheng, Lin Zong-qi, “Field in single-mode helically wound optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-33, 1150–1154 (1985).
[CrossRef]

Xu, Jincheng

A. J. Rogers, Jincheng Xu, Jialing Yao, “Vibration immunity for optical-fiber current measurement,” J. Lightwave Technol. 13, 1371–1377 (1995).
[CrossRef]

Yao, Jialing

A. J. Rogers, Jincheng Xu, Jialing Yao, “Vibration immunity for optical-fiber current measurement,” J. Lightwave Technol. 13, 1371–1377 (1995).
[CrossRef]

Ying-chun, He

Huang Hung-chia, He Ying-chun, “Polarization behavior of spun fiber versus conventional fiber under strong and slight twisting,” Microwave Opt. Technol. Lett. 9, 37–45 (1995).
[CrossRef]

Zong-qi, Lin

Fang Xi-sheng, Lin Zong-qi, “Field in single-mode helically wound optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-33, 1150–1154 (1985).
[CrossRef]

Appl. Opt. (3)

Electron. Lett. (4)

L. Jeunhomme, M. Monerie, “Polarization maintaining single-mode fiber cable design,” Electron. Lett. 16, 921–922 (1980).
[CrossRef]

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

R. D. Birch, “Fabrication and characterization of circularly birefringent helical fibres,” Electron. Lett. 23, 50–51 (1987).
[CrossRef]

C. D. Hussey, R. D. Birch, Y. Fujii, “Circularly birefringent single-mode optical fibres,” Electron. Lett. 22, 129–130 (1986).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

Fang Xi-sheng, Lin Zong-qi, “Field in single-mode helically wound optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-33, 1150–1154 (1985).
[CrossRef]

J. Lightwave Technol. (1)

A. J. Rogers, Jincheng Xu, Jialing Yao, “Vibration immunity for optical-fiber current measurement,” J. Lightwave Technol. 13, 1371–1377 (1995).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

Huang Hung-chia, He Ying-chun, “Polarization behavior of spun fiber versus conventional fiber under strong and slight twisting,” Microwave Opt. Technol. Lett. 9, 37–45 (1995).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

R. Castelli, F. Irrera, C. G. Someda, “Circularly birefringent optical fibres: new proposals Part I. Field analysis,” Opt. Quantum Electron. 21, 35–46 (1989).
[CrossRef]

Opt. Quantum. Electron. (1)

C. G. Someda, “Circularly birefringent optical fibres: new proposals Part II. Birefringence and coupling loss,” Opt. Quantum. Electron. 23, 713–725 (1991).

Proc. Inst. Electr. Eng. Part J (1)

Y. Fujii, C. D. Hussey, “Design considerations for circularly form-birefringent optical fibres,” Proc. Inst. Electr. Eng. Part J 133, 249–255 (1986).

Sci. Sin. (1)

Huang Hung-chia, “Generalized theory of coupled local normal modes in multi-wave guides,” Sci. Sin. 9, 142–154 (1960).

Trans. IECE Jpn. E (1)

S. Machida, J. Sakai, T. Kimura, “Polarization preservation in long-length twisted single-mode optical fibers,” Trans. IECE Jpn. E 65, 642–647 (1982).

Other (10)

F. Gauthier, J. Dubos, S. Blaison, Ph. Graindorge, H. J. Arditty, “Attempt to draw a circular polarization conserving fiber,” in Proceedings of the International Congress on Fiberoptic Rotation Sensors (MIT Press, Cambridge, Mass., 1981), p. 196.

M. P. Varnham, R. D. Birch, D. N. Payne “Helical-core circularly birefringent fibres,” in Technical Digest of Fifth International Conference on Integrated Optics and Optical Fiber Communication/Eleventh European Conference on Optical Communication (Compagnia dei Librai Editrice, Genova, Italy, 1985), p. 135.

M. P. Varnham, R. D. Birch, D. N. Payne, J. D. Love, “Design of helical core circularly birefringent fibres,” in Digest of Meeting on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1986), p. 68.

C. G. Someda, “Circularly birefringent dielectric waveguides,” Italian Patent41584A/85 (July1985).

C. G. Someda, “Circularly birefringent dielectric waveguides,” Italian Patent41638A/86 (December1986).

R. Dandliker, “Rotational effects of polarization in optical fibers,” in Anisotropic and Nonlinear Optical Waveguides, C. G. Someda, G. Stegeman, eds., Vol. 2 of Optical Wave Sciences and Technology (Elsevier, Amsterdam, 1992), pp. 39–76.

Huang Hung-chia, “Practical circular-polarization-maintaining optical fiber,” U.S. Patent5,452,394 (September1995).

Huang Hung-chia, Microwave Approach to Highly Irregular Fiber Optics (Wiley, New York, to be published).

Huang Hung-chia, “Passive fiber-optic polarization control,” U.S. Patent4,943,132 (July1990).

Huang Hung-chia, “Passive fiber-optic polarization control element,” U.S. Patent5,096,312 (March1992).

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Figures (8)

Fig. 1
Fig. 1

Fiber and preform: (a) structure of the Screw fiber with an on-axis straight core and an off-axis helical stress filament, (b) preform with an on-axis straight core rod and an off-axis straight stress rod.

Fig. 2
Fig. 2

Microscopic photograph of a cross section of the Screw fiber (initial specimen fabricated in 1994).

Fig. 3
Fig. 3

Screw fiber circular eigenmodes independent of fiber length (numbers underneath the curves indicate the number of fiber end cuttings). The values for 10 log10(Imax/Imin) were ≈0.05-dB incident light and <0.2-dB output light.

Fig. 4
Fig. 4

Jointing and splicing of two screw fiber segments with the same sense of spin and only the cores aligned: J, fiber segments butt-jointed; S, fiber segments spliced; N, fiber placed in its natural course. The values for 10 log10(Imax/Imin) were ≈0.05-dB incident light and <0.2-dB output light.

Fig. 5
Fig. 5

Jointing and splicing of two screw fiber segments with opposite senses of spin and only the cores aligned: J, fiber segments butt-jointed; S, fiber segments spliced. The values for 10 log10(Imax/Imin) were ≈0.05-dB incident light and <0.2-dB output lights.

Fig. 6
Fig. 6

Macroscopic bending of the Screw fiber with bending diameters of ϕ3 = 3 cm and ϕ5 = 5 cm. The values for 10 log10(Imax/Imin) were ≈0.05-dB incident light and <0.2-dB output light.

Fig. 7
Fig. 7

Twisting experiment that discerns a circular hi-bi Screw fiber: (a) Screw fiber immune to slight twists (numbers on top of curve represent the number of revolutions of the twist), (b) conventional fiber strongly affected by a slight twist [numbers mean the same as in (a)].

Fig. 8
Fig. 8

All-fiber transmission link structured with the circular hi-bi Screw fiber and a fiber-optic PPT. The four minicircles represent splices, the two vertical arrows represent linear light, and the three circles with arrows represent circular light.

Equations (23)

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

dA/dz=KA,  A=AxAy,  K=jΔβ/2κξ-κξ-jΔβ/2,
κ=2π/ξLb,  Lb=2π/κξ.
A=OW,
Q=cos ϕj sin ϕj sin ϕcos ϕ,
ϕ=1/2arctan2κξ/Δβ.
dW/dz=ΛW,
Λ=O-1KO=jg00-jg,
g=Δβ/22+κξ21/2.
Wz=Λ˜W0,  Λ˜=expjgz00exp-jgz,
Az=OΛ˜O-1A0,
Q=κξ/Δβ1,
O121jj1,
O-1121-j-11.
gκξ.
W0=10.
A0=121j,
Az=OΛ˜O-1A0=121jexpjκξz.
Az=121-jexpjκξz.
A0=cos θsin θ,
W0=O-1A0=exp-jθ-j expjθ,  Wz=Λ˜W0=exp-jθ-κξz-j expjθ-κξz,  Az=OWz=cosθ-κξzsinθ-κξz.
A0=cos θsin θ expjδ.
Az=OΛ˜O-1A0=cosκξzsinκξz-sinκξzcosκξzcos θsin θ expjδ.
κξκξ+VH,

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