Abstract

We demonstrate that the beam-propagation method can be used to calculate accurately both the pure bending loss and the transition loss of bent single-mode optical waveguides and fibers. Our results allow us to establish the accuracy of several commonly used theories of bending loss and to investigate the degree to which theories of step-index monomode fiber losses can be used to predict the losses of graded-index monomode fibers.

© 1983 Optical Society of America

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References

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  1. W. A. Gambling, H. Matsumura, and C. M. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–58 (1979).
    [CrossRef]
  2. W. A. Gambling, H. Matsumura, and C. M. Ragdale, “Field deformation in curved single-mode fibers,” Electron. Lett. 14, 130–132 (1978).
    [CrossRef]
  3. R. A. Sammut, “Discrete radiation from curved single-mode fibers,” Electron. Lett. 13, 418–419 (1977).
    [CrossRef]
  4. D. Marcuse, “Radiation losses of parabolic-index slabs and fibers with bent axes,” Appl. Opt. 17, 755–768 (1978).
    [CrossRef] [PubMed]
  5. K. Petermann, “Microbending loss in monomode fibers,” Electron. Lett. 12, 107–109 (1976).
    [CrossRef]
  6. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66, 216–220 (1976).
    [CrossRef]
  7. L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
    [CrossRef]
  8. W. A. Gambling and H. Matsumura, “Propagation in radially-inhomogeneous single-mode fibre,” Opt. Quantum Electron. 10, 31 (1978).
    [CrossRef]
  9. R. Baets and P. E. Lagasse, “Loss calculation and design of arbitrary curved integrated-optic waveguide,” J. Opt. Soc. Am. 73, 177–182 (1983).
    [CrossRef]
  10. M. D. Fleit and J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef]
  11. M. D. Feit and J. A. Fleck, “Computation of mode eigenfunctions in graded-index optical fibers by the propagating beam method,” Appl. Opt. 19, 2240–2246 (1980).
    [CrossRef] [PubMed]
  12. A. W. Snyder and R. A. Sammut, “Fundamental (HE11) modes of graded optical fibers,” J. Opt. Soc. Am. 69, 1663–1670 (1979).
    [CrossRef]
  13. M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), p. 132.
  14. E. Brinkmeyer, “Spot size of graded-index single-mode fibers: profile-independent representation and new determination method,” Appl. Opt. 18, 932–937 (1979).
    [CrossRef] [PubMed]
  15. H. Matsumura and T. Suganuma, “Normalization of single-mode fibers having an arbitrary index profile,” Appl. Opt. 19, 3151–3158 (1980).
    [CrossRef] [PubMed]
  16. C. D. Hussey and C. Pask, “Characterizations and design of single-mode optical fibers,” Opt. Quantum Electron. 14, 347–358 (1982).
    [CrossRef]

1983 (1)

1982 (1)

C. D. Hussey and C. Pask, “Characterizations and design of single-mode optical fibers,” Opt. Quantum Electron. 14, 347–358 (1982).
[CrossRef]

1980 (2)

1979 (3)

1978 (4)

W. A. Gambling, H. Matsumura, and C. M. Ragdale, “Field deformation in curved single-mode fibers,” Electron. Lett. 14, 130–132 (1978).
[CrossRef]

W. A. Gambling and H. Matsumura, “Propagation in radially-inhomogeneous single-mode fibre,” Opt. Quantum Electron. 10, 31 (1978).
[CrossRef]

D. Marcuse, “Radiation losses of parabolic-index slabs and fibers with bent axes,” Appl. Opt. 17, 755–768 (1978).
[CrossRef] [PubMed]

M. D. Fleit and J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
[CrossRef]

1977 (1)

R. A. Sammut, “Discrete radiation from curved single-mode fibers,” Electron. Lett. 13, 418–419 (1977).
[CrossRef]

1976 (2)

K. Petermann, “Microbending loss in monomode fibers,” Electron. Lett. 12, 107–109 (1976).
[CrossRef]

D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66, 216–220 (1976).
[CrossRef]

1974 (1)

L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
[CrossRef]

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), p. 132.

Baets, R.

Brinkmeyer, E.

Feit, M. D.

Fleck, J. A.

Fleit, M. D.

Gambling, W. A.

W. A. Gambling, H. Matsumura, and C. M. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–58 (1979).
[CrossRef]

W. A. Gambling and H. Matsumura, “Propagation in radially-inhomogeneous single-mode fibre,” Opt. Quantum Electron. 10, 31 (1978).
[CrossRef]

W. A. Gambling, H. Matsumura, and C. M. Ragdale, “Field deformation in curved single-mode fibers,” Electron. Lett. 14, 130–132 (1978).
[CrossRef]

Hussey, C. D.

C. D. Hussey and C. Pask, “Characterizations and design of single-mode optical fibers,” Opt. Quantum Electron. 14, 347–358 (1982).
[CrossRef]

Lagasse, P. E.

Lewin, L.

L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
[CrossRef]

Marcuse, D.

Matsumura, H.

H. Matsumura and T. Suganuma, “Normalization of single-mode fibers having an arbitrary index profile,” Appl. Opt. 19, 3151–3158 (1980).
[CrossRef] [PubMed]

W. A. Gambling, H. Matsumura, and C. M. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–58 (1979).
[CrossRef]

W. A. Gambling and H. Matsumura, “Propagation in radially-inhomogeneous single-mode fibre,” Opt. Quantum Electron. 10, 31 (1978).
[CrossRef]

W. A. Gambling, H. Matsumura, and C. M. Ragdale, “Field deformation in curved single-mode fibers,” Electron. Lett. 14, 130–132 (1978).
[CrossRef]

Pask, C.

C. D. Hussey and C. Pask, “Characterizations and design of single-mode optical fibers,” Opt. Quantum Electron. 14, 347–358 (1982).
[CrossRef]

Petermann, K.

K. Petermann, “Microbending loss in monomode fibers,” Electron. Lett. 12, 107–109 (1976).
[CrossRef]

Ragdale, C. M.

W. A. Gambling, H. Matsumura, and C. M. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–58 (1979).
[CrossRef]

W. A. Gambling, H. Matsumura, and C. M. Ragdale, “Field deformation in curved single-mode fibers,” Electron. Lett. 14, 130–132 (1978).
[CrossRef]

Sammut, R. A.

A. W. Snyder and R. A. Sammut, “Fundamental (HE11) modes of graded optical fibers,” J. Opt. Soc. Am. 69, 1663–1670 (1979).
[CrossRef]

R. A. Sammut, “Discrete radiation from curved single-mode fibers,” Electron. Lett. 13, 418–419 (1977).
[CrossRef]

Snyder, A. W.

Suganuma, T.

Appl. Opt. (5)

Electron. Lett. (3)

W. A. Gambling, H. Matsumura, and C. M. Ragdale, “Field deformation in curved single-mode fibers,” Electron. Lett. 14, 130–132 (1978).
[CrossRef]

R. A. Sammut, “Discrete radiation from curved single-mode fibers,” Electron. Lett. 13, 418–419 (1977).
[CrossRef]

K. Petermann, “Microbending loss in monomode fibers,” Electron. Lett. 12, 107–109 (1976).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Quantum Electron. (3)

W. A. Gambling and H. Matsumura, “Propagation in radially-inhomogeneous single-mode fibre,” Opt. Quantum Electron. 10, 31 (1978).
[CrossRef]

W. A. Gambling, H. Matsumura, and C. M. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–58 (1979).
[CrossRef]

C. D. Hussey and C. Pask, “Characterizations and design of single-mode optical fibers,” Opt. Quantum Electron. 14, 347–358 (1982).
[CrossRef]

Other (1)

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), p. 132.

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Figures (6)

Fig. 1
Fig. 1

(a) Electric field distributions in a bent two-dimensional cosh profile at axial distances 0 to 1500 μm from the start of the bend in a fixed coordinate system, computed with a 160-μm-wide window and 256 grid points, V = 1.4, a = 3 μm, and R = 1.5 cm. (b) Electric field distributions as in (a) in a coordinate system moving with the fiber axis [Eq. (5)] at axial distances 0 to 2000 μm from the start of the bend; 20-μm-thick absorbers are used at window distance 60–80 μm from the origin of the window (middle point). (c) Same as (b) but with 3-μm-thick absorbers (at 77–80 μm from the origin). (d) Asymptotic bent waveguide field distributions (z = 3000 μm) of (b) and (c) computed with thick (solid line) and thin (dotted solid line) absorbers. The dashed-line field distribution corresponds to a thick absorber placed 30–50 μm from the waveguide axis, computed with 32 grid points. (e) Asymptotic field distribution of (d) computed by direct integration of the wave equation in a bent waveguide. (f) Differential power loss from the computational window as a function of propagation distance from the start of the bend expressed in decibels per meter for the waveguide computed with the 20-μm absorber (dashed line) and the 3-μm absorber (solid line) of (d).

Fig. 2
Fig. 2

Comparison of the normalized propagation constant of a parabolic single-mode fiber, computed with the BPM, with that computed by Gambling.4

Fig. 3
Fig. 3

(a) Field distributions of a bent (three-dimensional) fiber computed at axial distances from 0 to 2000 μm from the start of the bend using a 20-μm-wide absorber and a 32 × 32 computational window. Also shown are the effective refractive-index distribution (n2k2) along with the propagation constant β and turning point xp. The fiber is a parabolic single-mode fiber with V = 2, a = 3 μm, and R = 2.5 cm. (b) Asymptotic field distribution of (a) shown as a contour plot in the transverse fiber plane.

Fig. 4
Fig. 4

Differential power coupled into higher-order radiated modes in the waveguide of Fig. 1(a), computed with perturbation theory. The dashed curve corresponds to approximating the fundamental and radiated modal fields by simple analytical functions (see text).

Fig. 5
Fig. 5

Differential bending loss as a function of propagation distance in a parabolic fiber with (a) V = 2 and a = 3 μm and (b) V = 3 and a = 4.5 μm with various bending radii R from 0.8 to 5 cm.

Fig. 6
Fig. 6

Comparison of the BPM pure bending loss (circles) in the two parabolic fibers of Fig. 5 with that of ESI profiles, computed by the methods of Matsumara (solid lines) and Brinkmeyer (dashed lines).

Tables (1)

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Table 1 Transition Loss in a Bent Parabolic Single-Mode Fiber

Equations (8)

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E ( x , y , z + d z ) = exp ( - i d z 4 n 0 k 2 ) × exp { - i n 0 k 2 z z + d z [ n ( x , y , z ) 2 / n 0 2 - 1 ] d z } × exp ( - i d z 4 n 0 k 2 ) E ( x , y , z ) exp ( - i k n 0 d z ) + O ( d z ) 3 .
C ( z ) = ɛ ( x , y , 0 ) * ɛ ( x , y , z ) d x d y , ɛ ( x , y , z ) = exp ( - i k n 0 z ) E ( x , y , z ) ,
absorb ( x ) = { 1 ,             x < x a , 1 / 2 { 1 + cos [ π ( x - x b ) / ( x a - x b ) ] γ } ,             x a < x < x b , 0 ,             x b < x < x R ,
n ( x ) 2 = n cl 2 + 2 Δ n cl 2 / cosh 2 ( x / a ) ,
n eq 2 = n 2 + 2 x n cl 2 / R ,
K ( β ) = ω ɛ 0 4 i - + δ n 2 E 0 ( β 0 ) E β ( β ) * d x ,
P ( z ) = 0 k n cl K ( β ) 2 | ( 1 / z 1 / 2 ) 0 z exp [ - i ( β - β 0 ) z ] d z | 2 d β .
α = ( R / a ) - 2 V 4 32 Δ 2 ( 0.65 + 1.62 V - 1.5 + 2.88 V - 6 ) 6 .