Abstract

A novel and simpler method to calculate the main parameters in fiber optics is presented. This method is based in a planar dielectric waveguide in rotation and, as an example, it is applied to calculate the turning points and the inner caustic in an optical fiber with a parabolic refractive index. It is shown that the solution found using this method agrees with the standard (and more complex) method, whose solutions for these points are also summarized in this paper.

©2008 Optical Society of America

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References

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  1. M. Eve and J. H. Hannay, “Ray theory and random mode coupling in an optical fibre waveguide I,” Opt. Quantum Electron. 8, 503–508 (1976).
    [Crossref]
  2. M. Eve and J. H. Hannay, “Ray theory and random mode coupling in an optical fibre waveguide II,” Opt. Quantum Electron. 8, 509–512 (1976).
    [Crossref]
  3. T. Kudou, M. Yokota, and O. Fukumitsu, “Complex Ray analysis of reflection and transmission of Hermite Gaussian beam at curved interface,” Electron. Lett. 24, 1519–1520 (1988).
    [Crossref]
  4. S. Guo and S. Albin, “Transmission property and evanescent wave absorption of cladded multimode fiber tapers,” Opt. Express 11, 215–223 (2003).
    [Crossref] [PubMed]
  5. K. Iwai, Y. Abe, Y. Matsuura, and M. Miyagi, “Equivalent complex refractive indices for ray-tracing evaluation of dielectric-coated hollow waveguides,” Opt. Eng. 41, 1471–1472 (2002).
    [Crossref]
  6. F. Pérez-Ocón, A. Peña, J. R. Jiménez, and J. A. Díaz “A simple model for fibre optics: planar dielectric waveguides in rotation,” Eur. J. Phys. 27, 657–665 (2006).
    [Crossref]
  7. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).
  8. F. Pérez-Ocón and J. R. Jiménez, 2003 Fibras Ópticas: Estudio Geométrico 1st edn (FPO)
  9. E. M. Dianov, A. B. Grudinin, A. N. Gurjanov, D. D. Gusovsky, Z. E. Harutjunian, S. V. Ignatjev, and O. B. Smirnov, “Circular core polarization-maintaining optical fibers with elliptical stress-induced cladding,” J. Lightwave Technol. 10, 118–124 (1992).
    [Crossref]
  10. H. Hung-chia, “Practical circular-polarization-maintaining optical fiber,” Appl. Opt. 36, 6968–6975 (1997).
    [Crossref]

2006 (1)

F. Pérez-Ocón, A. Peña, J. R. Jiménez, and J. A. Díaz “A simple model for fibre optics: planar dielectric waveguides in rotation,” Eur. J. Phys. 27, 657–665 (2006).
[Crossref]

2003 (1)

2002 (1)

K. Iwai, Y. Abe, Y. Matsuura, and M. Miyagi, “Equivalent complex refractive indices for ray-tracing evaluation of dielectric-coated hollow waveguides,” Opt. Eng. 41, 1471–1472 (2002).
[Crossref]

1997 (1)

1992 (1)

E. M. Dianov, A. B. Grudinin, A. N. Gurjanov, D. D. Gusovsky, Z. E. Harutjunian, S. V. Ignatjev, and O. B. Smirnov, “Circular core polarization-maintaining optical fibers with elliptical stress-induced cladding,” J. Lightwave Technol. 10, 118–124 (1992).
[Crossref]

1988 (1)

T. Kudou, M. Yokota, and O. Fukumitsu, “Complex Ray analysis of reflection and transmission of Hermite Gaussian beam at curved interface,” Electron. Lett. 24, 1519–1520 (1988).
[Crossref]

1976 (2)

M. Eve and J. H. Hannay, “Ray theory and random mode coupling in an optical fibre waveguide I,” Opt. Quantum Electron. 8, 503–508 (1976).
[Crossref]

M. Eve and J. H. Hannay, “Ray theory and random mode coupling in an optical fibre waveguide II,” Opt. Quantum Electron. 8, 509–512 (1976).
[Crossref]

Abe, Y.

K. Iwai, Y. Abe, Y. Matsuura, and M. Miyagi, “Equivalent complex refractive indices for ray-tracing evaluation of dielectric-coated hollow waveguides,” Opt. Eng. 41, 1471–1472 (2002).
[Crossref]

Albin, S.

Dianov, E. M.

E. M. Dianov, A. B. Grudinin, A. N. Gurjanov, D. D. Gusovsky, Z. E. Harutjunian, S. V. Ignatjev, and O. B. Smirnov, “Circular core polarization-maintaining optical fibers with elliptical stress-induced cladding,” J. Lightwave Technol. 10, 118–124 (1992).
[Crossref]

Díaz, J. A.

F. Pérez-Ocón, A. Peña, J. R. Jiménez, and J. A. Díaz “A simple model for fibre optics: planar dielectric waveguides in rotation,” Eur. J. Phys. 27, 657–665 (2006).
[Crossref]

Eve, M.

M. Eve and J. H. Hannay, “Ray theory and random mode coupling in an optical fibre waveguide II,” Opt. Quantum Electron. 8, 509–512 (1976).
[Crossref]

M. Eve and J. H. Hannay, “Ray theory and random mode coupling in an optical fibre waveguide I,” Opt. Quantum Electron. 8, 503–508 (1976).
[Crossref]

Fukumitsu, O.

T. Kudou, M. Yokota, and O. Fukumitsu, “Complex Ray analysis of reflection and transmission of Hermite Gaussian beam at curved interface,” Electron. Lett. 24, 1519–1520 (1988).
[Crossref]

Grudinin, A. B.

E. M. Dianov, A. B. Grudinin, A. N. Gurjanov, D. D. Gusovsky, Z. E. Harutjunian, S. V. Ignatjev, and O. B. Smirnov, “Circular core polarization-maintaining optical fibers with elliptical stress-induced cladding,” J. Lightwave Technol. 10, 118–124 (1992).
[Crossref]

Guo, S.

Gurjanov, A. N.

E. M. Dianov, A. B. Grudinin, A. N. Gurjanov, D. D. Gusovsky, Z. E. Harutjunian, S. V. Ignatjev, and O. B. Smirnov, “Circular core polarization-maintaining optical fibers with elliptical stress-induced cladding,” J. Lightwave Technol. 10, 118–124 (1992).
[Crossref]

Gusovsky, D. D.

E. M. Dianov, A. B. Grudinin, A. N. Gurjanov, D. D. Gusovsky, Z. E. Harutjunian, S. V. Ignatjev, and O. B. Smirnov, “Circular core polarization-maintaining optical fibers with elliptical stress-induced cladding,” J. Lightwave Technol. 10, 118–124 (1992).
[Crossref]

Hannay, J. H.

M. Eve and J. H. Hannay, “Ray theory and random mode coupling in an optical fibre waveguide I,” Opt. Quantum Electron. 8, 503–508 (1976).
[Crossref]

M. Eve and J. H. Hannay, “Ray theory and random mode coupling in an optical fibre waveguide II,” Opt. Quantum Electron. 8, 509–512 (1976).
[Crossref]

Harutjunian, Z. E.

E. M. Dianov, A. B. Grudinin, A. N. Gurjanov, D. D. Gusovsky, Z. E. Harutjunian, S. V. Ignatjev, and O. B. Smirnov, “Circular core polarization-maintaining optical fibers with elliptical stress-induced cladding,” J. Lightwave Technol. 10, 118–124 (1992).
[Crossref]

Hung-chia, H.

Ignatjev, S. V.

E. M. Dianov, A. B. Grudinin, A. N. Gurjanov, D. D. Gusovsky, Z. E. Harutjunian, S. V. Ignatjev, and O. B. Smirnov, “Circular core polarization-maintaining optical fibers with elliptical stress-induced cladding,” J. Lightwave Technol. 10, 118–124 (1992).
[Crossref]

Iwai, K.

K. Iwai, Y. Abe, Y. Matsuura, and M. Miyagi, “Equivalent complex refractive indices for ray-tracing evaluation of dielectric-coated hollow waveguides,” Opt. Eng. 41, 1471–1472 (2002).
[Crossref]

Jiménez, J. R.

F. Pérez-Ocón, A. Peña, J. R. Jiménez, and J. A. Díaz “A simple model for fibre optics: planar dielectric waveguides in rotation,” Eur. J. Phys. 27, 657–665 (2006).
[Crossref]

F. Pérez-Ocón and J. R. Jiménez, 2003 Fibras Ópticas: Estudio Geométrico 1st edn (FPO)

Kudou, T.

T. Kudou, M. Yokota, and O. Fukumitsu, “Complex Ray analysis of reflection and transmission of Hermite Gaussian beam at curved interface,” Electron. Lett. 24, 1519–1520 (1988).
[Crossref]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

Matsuura, Y.

K. Iwai, Y. Abe, Y. Matsuura, and M. Miyagi, “Equivalent complex refractive indices for ray-tracing evaluation of dielectric-coated hollow waveguides,” Opt. Eng. 41, 1471–1472 (2002).
[Crossref]

Miyagi, M.

K. Iwai, Y. Abe, Y. Matsuura, and M. Miyagi, “Equivalent complex refractive indices for ray-tracing evaluation of dielectric-coated hollow waveguides,” Opt. Eng. 41, 1471–1472 (2002).
[Crossref]

Peña, A.

F. Pérez-Ocón, A. Peña, J. R. Jiménez, and J. A. Díaz “A simple model for fibre optics: planar dielectric waveguides in rotation,” Eur. J. Phys. 27, 657–665 (2006).
[Crossref]

Pérez-Ocón, F.

F. Pérez-Ocón, A. Peña, J. R. Jiménez, and J. A. Díaz “A simple model for fibre optics: planar dielectric waveguides in rotation,” Eur. J. Phys. 27, 657–665 (2006).
[Crossref]

F. Pérez-Ocón and J. R. Jiménez, 2003 Fibras Ópticas: Estudio Geométrico 1st edn (FPO)

Smirnov, O. B.

E. M. Dianov, A. B. Grudinin, A. N. Gurjanov, D. D. Gusovsky, Z. E. Harutjunian, S. V. Ignatjev, and O. B. Smirnov, “Circular core polarization-maintaining optical fibers with elliptical stress-induced cladding,” J. Lightwave Technol. 10, 118–124 (1992).
[Crossref]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

Yokota, M.

T. Kudou, M. Yokota, and O. Fukumitsu, “Complex Ray analysis of reflection and transmission of Hermite Gaussian beam at curved interface,” Electron. Lett. 24, 1519–1520 (1988).
[Crossref]

Appl. Opt. (1)

Electron. Lett. (1)

T. Kudou, M. Yokota, and O. Fukumitsu, “Complex Ray analysis of reflection and transmission of Hermite Gaussian beam at curved interface,” Electron. Lett. 24, 1519–1520 (1988).
[Crossref]

Eur. J. Phys. (1)

F. Pérez-Ocón, A. Peña, J. R. Jiménez, and J. A. Díaz “A simple model for fibre optics: planar dielectric waveguides in rotation,” Eur. J. Phys. 27, 657–665 (2006).
[Crossref]

J. Lightwave Technol. (1)

E. M. Dianov, A. B. Grudinin, A. N. Gurjanov, D. D. Gusovsky, Z. E. Harutjunian, S. V. Ignatjev, and O. B. Smirnov, “Circular core polarization-maintaining optical fibers with elliptical stress-induced cladding,” J. Lightwave Technol. 10, 118–124 (1992).
[Crossref]

Opt. Eng. (1)

K. Iwai, Y. Abe, Y. Matsuura, and M. Miyagi, “Equivalent complex refractive indices for ray-tracing evaluation of dielectric-coated hollow waveguides,” Opt. Eng. 41, 1471–1472 (2002).
[Crossref]

Opt. Express (1)

Opt. Quantum Electron. (2)

M. Eve and J. H. Hannay, “Ray theory and random mode coupling in an optical fibre waveguide I,” Opt. Quantum Electron. 8, 503–508 (1976).
[Crossref]

M. Eve and J. H. Hannay, “Ray theory and random mode coupling in an optical fibre waveguide II,” Opt. Quantum Electron. 8, 509–512 (1976).
[Crossref]

Other (2)

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

F. Pérez-Ocón and J. R. Jiménez, 2003 Fibras Ópticas: Estudio Geométrico 1st edn (FPO)

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Figures (3)

Fig. 1.
Fig. 1. Trajectory of the rays within the core of a gradual-index fiber with the position of the turning points; (a) shows the trajectory of a meridional ray and (b) an oblique one. In (c), the angle (r) is represented and the azimuthal direction; θz and θϕ are projections of the path on a plane perpendicular to the axis of the fiber.
Fig. 2.
Fig. 2. Path of a meridional ray moving along a graded-index profile fiber. Detail of one layer magnified.
Fig. 3.
Fig. 3. Section of the fiber on a plane perpendicular to the axis. Projection of the path traveled within each layer over a section perpendicular to the axis.

Equations (25)

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

dL i = dx sin θ zi sin θ ϕ i
dt i = dL i v i = n i dx c sin θ Zi sin θ ϕ i
ω i ω ( x ) = d θ ϕ ( x ) dt
ω ( x ) = d θ ϕ ( x ) dx dx dt = d θ ϕ ( x ) dx c sin θ z ( x ) n ( x ) sin θ ϕ ( x ) = { 0 if x > dx π c sin θ z ( x ) n ( x ) 2 dx if x < dx
d θ ϕ ( x ) dx = π 2 dx
ω = π c sin θ z ( x ) n ( x ) 2 dx
dL i = sin θ ϕ i dx sin θ Zi
dt i = dL i v i = n i sin θ ϕ i dx c sin θ Zi
ω ( x ) = d θ ϕ ( x ) dx dx dt = d θ ϕ ( x ) dx c sin θ Z ( x ) n ( x ) sin θ ϕ ( x )
α = d ω ( x ) dt = 0
α = d ω ( x ) dt = d ω ( x ) dx dx dt = c sin θ Z ( x ) n ( x ) sin θ ϕ ( x ) d ω ( x ) dx = 0 d ω ( x ) dx = 0
d ω ( x ) dx = c sin θ Z ( x ) n ( x ) sin θ ϕ ( x ) d 2 θ ϕ ( x ) dx 2 +
c d θ ϕ ( x ) dx [ n ( x ) sin θ ϕ ( x ) cos θ Z ( x ) d θ Z ( x ) dx sin θ Z ( x ) [ sin θ ϕ ( x ) dn ( x ) dx + n ( x ) cos θ ϕ ( x ) d θ ϕ ( x ) dx ] n 2 ( x ) sin 2 θ ϕ ( x ) ] = 0
d 2 θ ϕ ( x ) dx 2 + d θ ϕ ( x ) dx [ cot g θ Z ( x ) d θ Z ( x ) dx cot g θ ϕ ( x ) d θ ϕ ( x ) dx 1 n ( x ) dn ( x ) dx ] = 0
cot g θ ϕ ( x ) d θ ϕ ( x ) dx + 1 n ( x ) dn ( x ) dx = 0
n ( x ) sin θ ϕ ( x ) = 1
n 2 ( x ) = n nu 2 [ ( 1 2 Δ x ρ ) 2 ]
n 2 ( x ) sin 2 θ ϕ ( x ) = 1 ( x ρ ) 2 = 1 2 Δ ( 1 1 n nu 2 sin 2 θ ϕ ( x ) )
x = ρ n nu 2 Δ n nu 2 1 sin 2 θ ϕ ( x )
r ci = ρ 2 n nu Δ ( n nu 2 β - 2 ) ( n nu 2 β 2 ¯ ) 2 8 Δ l - 2 n nu 2
r pl = ρ 2 n nu Δ ( n nu 2 β - 2 ) ( n nu 2 β 2 ¯ ) 2 8 Δ l - 2 n nu 2
r pl = ρ 2 n nu Δ n nu 2 β - 2 + n nu 2 β - = ρ n nu 2 Δ n nu 2 1 2 ( β - 2 β - )
r ci = ρ 2 n nu Δ n nu 2 β - 2 n nu 2 + β 2 - = 0
First: n nu > 1 | sin θ ϕ ( x ) |
Second : 1 sin 2 θ ϕ ( x ) = 1 2 ( β - 2 β - )

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