Abstract

We investigate the nonlinear dynamic behavior of an optical ray in a focusing fiber when curving of the fiber occurs. The study indicates that, when the fiber curves, the behavior of the ray in the fiber evolves from regular to chaotic. When chaos appears, the ray’s behavior in the focusing fiber becomes irregular; then self-focusing becomes difficult, dispersion is enlarged, and the loss of light energy is increased.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. R. Hsing, “Imaging system using gradient index fibers,” in Fiber Optics, J. C. Daly, ed. (CRC Press, Boca Raton, Fla., 1984), pp. 195–236.
  2. S. D. Personick, Fiber Optics: Technology and Applications (Plenum, New York, 1985).
  3. D. D. Holm and G. Kovačič, “Homoclinic chaos for ray optics in a fiber,” Physica D 51, 177–188 (1991).
    [Crossref]
  4. X. Li, Y. Zhang, and G. Du, “Influence of perturbation on chaotic behavior of the parabolic ray system” J. Acoust. Soc. Am. 105, 2142–2148 (1999).
    [Crossref]
  5. X. Li, Y. Zhang, Z. Ping, and G. Du, “An investigation of ray chaos in underwater acoustics,” [Acta Phys. Sin. 8, S304–S307 (1999); overseas edition].
  6. K. B. Smith, M. G. Brown, and F. D. Tappert, “Ray chaos in underwater acoustics,” J. Acoust. Soc. Am. 91, 1939–1949 (1992).
    [Crossref]
  7. S. S. Abdullaev and G. M. Zaslavskii, “Classical nonlinear dynamics and chaos of rays in wave propagation problems in inhomogeneous media,” Usp. Fiz. Nauk 161, 1–43 (1991).
    [Crossref]
  8. G. M. Zaslavsky and S. S. Abdullaev, “Chaotic transmission of waves and ‘cooling’ of signals,” Chaos 7, 182–186 (1997).
    [Crossref] [PubMed]
  9. A. L. Virovlyansky and G. M. Zaslavsky, “Wave chaos in terms of normal modes,” Phys. Rev. E 59, 1656–1668 (1999).
    [Crossref]
  10. A. A. Tsonis, Chaos: From Theory to Applications (Plenum, New York, 1992).
  11. F. P. Kapron, “Geometrical optics of parabolic index-gradient cylindrical lenses,” J. Opt. Soc. Am. 60, 1433–1435 (1970).
    [Crossref]

1999 (3)

X. Li, Y. Zhang, and G. Du, “Influence of perturbation on chaotic behavior of the parabolic ray system” J. Acoust. Soc. Am. 105, 2142–2148 (1999).
[Crossref]

X. Li, Y. Zhang, Z. Ping, and G. Du, “An investigation of ray chaos in underwater acoustics,” [Acta Phys. Sin. 8, S304–S307 (1999); overseas edition].

A. L. Virovlyansky and G. M. Zaslavsky, “Wave chaos in terms of normal modes,” Phys. Rev. E 59, 1656–1668 (1999).
[Crossref]

1997 (1)

G. M. Zaslavsky and S. S. Abdullaev, “Chaotic transmission of waves and ‘cooling’ of signals,” Chaos 7, 182–186 (1997).
[Crossref] [PubMed]

1992 (1)

K. B. Smith, M. G. Brown, and F. D. Tappert, “Ray chaos in underwater acoustics,” J. Acoust. Soc. Am. 91, 1939–1949 (1992).
[Crossref]

1991 (2)

S. S. Abdullaev and G. M. Zaslavskii, “Classical nonlinear dynamics and chaos of rays in wave propagation problems in inhomogeneous media,” Usp. Fiz. Nauk 161, 1–43 (1991).
[Crossref]

D. D. Holm and G. Kovačič, “Homoclinic chaos for ray optics in a fiber,” Physica D 51, 177–188 (1991).
[Crossref]

1970 (1)

Abdullaev, S. S.

G. M. Zaslavsky and S. S. Abdullaev, “Chaotic transmission of waves and ‘cooling’ of signals,” Chaos 7, 182–186 (1997).
[Crossref] [PubMed]

S. S. Abdullaev and G. M. Zaslavskii, “Classical nonlinear dynamics and chaos of rays in wave propagation problems in inhomogeneous media,” Usp. Fiz. Nauk 161, 1–43 (1991).
[Crossref]

Brown, M. G.

K. B. Smith, M. G. Brown, and F. D. Tappert, “Ray chaos in underwater acoustics,” J. Acoust. Soc. Am. 91, 1939–1949 (1992).
[Crossref]

Du, G.

X. Li, Y. Zhang, Z. Ping, and G. Du, “An investigation of ray chaos in underwater acoustics,” [Acta Phys. Sin. 8, S304–S307 (1999); overseas edition].

X. Li, Y. Zhang, and G. Du, “Influence of perturbation on chaotic behavior of the parabolic ray system” J. Acoust. Soc. Am. 105, 2142–2148 (1999).
[Crossref]

Holm, D. D.

D. D. Holm and G. Kovačič, “Homoclinic chaos for ray optics in a fiber,” Physica D 51, 177–188 (1991).
[Crossref]

Hsing, T. R.

T. R. Hsing, “Imaging system using gradient index fibers,” in Fiber Optics, J. C. Daly, ed. (CRC Press, Boca Raton, Fla., 1984), pp. 195–236.

Kapron, F. P.

Kovacic, G.

D. D. Holm and G. Kovačič, “Homoclinic chaos for ray optics in a fiber,” Physica D 51, 177–188 (1991).
[Crossref]

Li, X.

X. Li, Y. Zhang, and G. Du, “Influence of perturbation on chaotic behavior of the parabolic ray system” J. Acoust. Soc. Am. 105, 2142–2148 (1999).
[Crossref]

X. Li, Y. Zhang, Z. Ping, and G. Du, “An investigation of ray chaos in underwater acoustics,” [Acta Phys. Sin. 8, S304–S307 (1999); overseas edition].

Personick, S. D.

S. D. Personick, Fiber Optics: Technology and Applications (Plenum, New York, 1985).

Ping, Z.

X. Li, Y. Zhang, Z. Ping, and G. Du, “An investigation of ray chaos in underwater acoustics,” [Acta Phys. Sin. 8, S304–S307 (1999); overseas edition].

Smith, K. B.

K. B. Smith, M. G. Brown, and F. D. Tappert, “Ray chaos in underwater acoustics,” J. Acoust. Soc. Am. 91, 1939–1949 (1992).
[Crossref]

Tappert, F. D.

K. B. Smith, M. G. Brown, and F. D. Tappert, “Ray chaos in underwater acoustics,” J. Acoust. Soc. Am. 91, 1939–1949 (1992).
[Crossref]

Tsonis, A. A.

A. A. Tsonis, Chaos: From Theory to Applications (Plenum, New York, 1992).

Virovlyansky, A. L.

A. L. Virovlyansky and G. M. Zaslavsky, “Wave chaos in terms of normal modes,” Phys. Rev. E 59, 1656–1668 (1999).
[Crossref]

Zaslavskii, G. M.

S. S. Abdullaev and G. M. Zaslavskii, “Classical nonlinear dynamics and chaos of rays in wave propagation problems in inhomogeneous media,” Usp. Fiz. Nauk 161, 1–43 (1991).
[Crossref]

Zaslavsky, G. M.

A. L. Virovlyansky and G. M. Zaslavsky, “Wave chaos in terms of normal modes,” Phys. Rev. E 59, 1656–1668 (1999).
[Crossref]

G. M. Zaslavsky and S. S. Abdullaev, “Chaotic transmission of waves and ‘cooling’ of signals,” Chaos 7, 182–186 (1997).
[Crossref] [PubMed]

Zhang, Y.

X. Li, Y. Zhang, Z. Ping, and G. Du, “An investigation of ray chaos in underwater acoustics,” [Acta Phys. Sin. 8, S304–S307 (1999); overseas edition].

X. Li, Y. Zhang, and G. Du, “Influence of perturbation on chaotic behavior of the parabolic ray system” J. Acoust. Soc. Am. 105, 2142–2148 (1999).
[Crossref]

Acta Phys. Sin. (1)

X. Li, Y. Zhang, Z. Ping, and G. Du, “An investigation of ray chaos in underwater acoustics,” [Acta Phys. Sin. 8, S304–S307 (1999); overseas edition].

Chaos (1)

G. M. Zaslavsky and S. S. Abdullaev, “Chaotic transmission of waves and ‘cooling’ of signals,” Chaos 7, 182–186 (1997).
[Crossref] [PubMed]

J. Acoust. Soc. Am. (2)

K. B. Smith, M. G. Brown, and F. D. Tappert, “Ray chaos in underwater acoustics,” J. Acoust. Soc. Am. 91, 1939–1949 (1992).
[Crossref]

X. Li, Y. Zhang, and G. Du, “Influence of perturbation on chaotic behavior of the parabolic ray system” J. Acoust. Soc. Am. 105, 2142–2148 (1999).
[Crossref]

J. Opt. Soc. Am. (1)

Phys. Rev. E (1)

A. L. Virovlyansky and G. M. Zaslavsky, “Wave chaos in terms of normal modes,” Phys. Rev. E 59, 1656–1668 (1999).
[Crossref]

Physica D (1)

D. D. Holm and G. Kovačič, “Homoclinic chaos for ray optics in a fiber,” Physica D 51, 177–188 (1991).
[Crossref]

Usp. Fiz. Nauk (1)

S. S. Abdullaev and G. M. Zaslavskii, “Classical nonlinear dynamics and chaos of rays in wave propagation problems in inhomogeneous media,” Usp. Fiz. Nauk 161, 1–43 (1991).
[Crossref]

Other (3)

A. A. Tsonis, Chaos: From Theory to Applications (Plenum, New York, 1992).

T. R. Hsing, “Imaging system using gradient index fibers,” in Fiber Optics, J. C. Daly, ed. (CRC Press, Boca Raton, Fla., 1984), pp. 195–236.

S. D. Personick, Fiber Optics: Technology and Applications (Plenum, New York, 1985).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Geometrical relation among variable. OZ is the optical axis, OX is the polar axis, plane Θ is the plane considered at range z, and M is the point considered. OM is the position vector q(r, φ). Vector n(z, r, φ) represents the direction and velocity of the optical ray at point M. Vector p is a projection of the vector n onto the plane.

Fig. 2
Fig. 2

Planes at range z. OX is the polar axis. The original optical axis OZ is vertical to the page surface and points outward. O is the new position of the real optical axis after the fiber has been curved. (a) Sinusoidal curvature. The fiber bends within plane XOZ. D(z) is the deviation. (b) Helical curvature. The fiber appears as a helix with radius R and pitch H about axis OZ.

Fig. 3
Fig. 3

Result for double-parabolic distribution of the refractive index.

Fig. 4
Fig. 4

Skew ray under sinusoidal curvature (r0=10 µm, φ0=0,α0=5°,β0=10°). Poincaré section when (a) Dm=0.1 µm, (b) Dm=1 µm, (c) Dm=10 µm, and (d) Dm=30 µm. (e) Phase portrait when Dm=30 µm. (f) Path when Dm=30 µm. (d), and (f) Same as for ray ❺ in Fig. 3.  

Fig. 5
Fig. 5

Lyapunov exponent on the degree of curvature Dm. Here only the maximum Lyapunov exponent is shown.

Fig. 6
Fig. 6

Poincaré sections of rays with φ0=0,r0=10 µm, β0=10°,Dm=25 µm, and α0=0°, 1° ,, 9°, a coexistence of chaotic and regular ray trajectories. The closed curve corresponds to α0=0° and the speckled regions correspond to α0=1°, 2° ,, 9°.

Fig. 7
Fig. 7

Result for a parabolic distribution of the refractive index.

Tables (1)

Tables Icon

Table 1 Expressions of the Refractive Index

Equations (11)

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

r˙=Hpr,
p˙r=-Hr,
φ˙=Hpφ,
p˙φ=-Hφ,
H=-[n2(z, r, φ)-p2]1/2,
n2(r)=λ2+(μ-νr2)2,
n(r)=n01-A2r2,
D(z)=Dm sin2πTz,
r2-2D(z)r cos φ+D2(z)=r2.
θ(z)=(2π/H)z,
r2-2Rr cos[φ-θ(z)]+R2=r2

Metrics