Abstract

Curvature loss formulas of optical fibers usually ignore the effect of field deformation caused by the curved axis of the fiber. Contrary to naive intuition, this field deformation may substantially decrease the radiation losses of modes with low mode number. Losses of modes with high mode numbers are, however, increased. We present a theoretical evaluation of curvature losses of the modes of a step-index fiber with a bent axis incorporating the influence of field deformation on the loss coefficient. The limitations of the simple loss formula in case of a sharply bent overmoded waveguide are pointed out.

© 1976 Optical Society of America

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References

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  1. M. A. Miller and V. I. Talanov, “Electromagnetic Surface Waves Guided by a Boundary with Small Curvature,” Zh. Tekh. Fiz. 26, 2755 (1956).
  2. E. A. J. Marcatili, “Bends in Optical Dielectric Guides,” Bell Syst. Tech. J.,  48, 2013–2132 (1969).
    [Crossref]
  3. L. Lewin, “Radiation from Curved Dielectric Slabs and Fibers,” IEEE Trans. Microwave Theory Tech., Vol.  MTT-22, No, 7, 718–727 (1974).
    [Crossref]
  4. J. A. Arnaud, “Transverse Coupling in Fiber Optics Part III: Bending Losses,” Bell Syst. Tech. J.,  53, 1379–1394 (1974).
    [Crossref]
  5. A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from Bent Optical Waveguides,” Electron. Lett.,  11, 332–333 (1975).
    [Crossref]
  6. V. V. Shevchenko, “Radiation Losses in Bent Waveguides for Surface Waves,” Radiophys. Quantum Electron. 14, 607–614 (1973) (Russian original 1971).
    [Crossref]
  7. D. C. Chang and E. F. Kuester, “General Theory of Surface-Wave Propagation on a Curved Optical Waveguide of Arbitrary Cross Section,” , Electromagnetics Laboratory, Dept. Electr. Eng., University of Colorado, Boulder, Colorado; also IEEE J. Quantum Electron. QE-11, 903–907 (1975).
  8. D. Marcuse, Light Transmission Optics (Van Nostrand, Reinhold, New York, 1972), pp. 398–406.
  9. D. Marcuse, “Curvature Loss Formula for Optical Fibers,” J. Opt. Soc. Am. 66, 216–220 (1976).
    [Crossref]
  10. S. E. Miller, “Directional Control in Lightwave Guidance,” Bell Syst. Tech. J.,  43, 1727–1739 (1964).
    [Crossref]
  11. E. A. J. Marcatili and S. E. Miller, “Improved Relations Describing Directional Control in Electromagnetic Wave Guidance,” Bell Syst. Tech. J.,  48, 2161–2188 (1969).
    [Crossref]
  12. D. Gloge, “Weakly Guiding Fibers,” Appl. Opt.,  10, 2252–2258 (1971).
    [Crossref] [PubMed]
  13. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  14. M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Eq. 9.2.4, p. 364. U. S. Dept. of Commerce, Natl. Bur. Stds., Appl. Math. Ser. Vol. 55 (U. S. GPO, Washington, D. C., 1964).
  15. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).
  16. D. Marcuse, “Bent Optical Waveguide with Lossy Jacket,” Bell Syst. Tech. J.,  53, 1079–1101 (1974).
    [Crossref]
  17. D. Gloge, “Bending Losses in Multimode Fibers with Graded and Ungraded Core Index,” Appl. Opt. 11, 2506–2513 (1972).
    [Crossref] [PubMed]

1976 (1)

1975 (1)

A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from Bent Optical Waveguides,” Electron. Lett.,  11, 332–333 (1975).
[Crossref]

1974 (3)

L. Lewin, “Radiation from Curved Dielectric Slabs and Fibers,” IEEE Trans. Microwave Theory Tech., Vol.  MTT-22, No, 7, 718–727 (1974).
[Crossref]

J. A. Arnaud, “Transverse Coupling in Fiber Optics Part III: Bending Losses,” Bell Syst. Tech. J.,  53, 1379–1394 (1974).
[Crossref]

D. Marcuse, “Bent Optical Waveguide with Lossy Jacket,” Bell Syst. Tech. J.,  53, 1079–1101 (1974).
[Crossref]

1973 (1)

V. V. Shevchenko, “Radiation Losses in Bent Waveguides for Surface Waves,” Radiophys. Quantum Electron. 14, 607–614 (1973) (Russian original 1971).
[Crossref]

1972 (1)

1971 (1)

1969 (2)

E. A. J. Marcatili and S. E. Miller, “Improved Relations Describing Directional Control in Electromagnetic Wave Guidance,” Bell Syst. Tech. J.,  48, 2161–2188 (1969).
[Crossref]

E. A. J. Marcatili, “Bends in Optical Dielectric Guides,” Bell Syst. Tech. J.,  48, 2013–2132 (1969).
[Crossref]

1964 (1)

S. E. Miller, “Directional Control in Lightwave Guidance,” Bell Syst. Tech. J.,  43, 1727–1739 (1964).
[Crossref]

1956 (1)

M. A. Miller and V. I. Talanov, “Electromagnetic Surface Waves Guided by a Boundary with Small Curvature,” Zh. Tekh. Fiz. 26, 2755 (1956).

Abramovitz, M.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Eq. 9.2.4, p. 364. U. S. Dept. of Commerce, Natl. Bur. Stds., Appl. Math. Ser. Vol. 55 (U. S. GPO, Washington, D. C., 1964).

Arnaud, J. A.

J. A. Arnaud, “Transverse Coupling in Fiber Optics Part III: Bending Losses,” Bell Syst. Tech. J.,  53, 1379–1394 (1974).
[Crossref]

Chang, D. C.

D. C. Chang and E. F. Kuester, “General Theory of Surface-Wave Propagation on a Curved Optical Waveguide of Arbitrary Cross Section,” , Electromagnetics Laboratory, Dept. Electr. Eng., University of Colorado, Boulder, Colorado; also IEEE J. Quantum Electron. QE-11, 903–907 (1975).

Gloge, D.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

Kuester, E. F.

D. C. Chang and E. F. Kuester, “General Theory of Surface-Wave Propagation on a Curved Optical Waveguide of Arbitrary Cross Section,” , Electromagnetics Laboratory, Dept. Electr. Eng., University of Colorado, Boulder, Colorado; also IEEE J. Quantum Electron. QE-11, 903–907 (1975).

Lewin, L.

L. Lewin, “Radiation from Curved Dielectric Slabs and Fibers,” IEEE Trans. Microwave Theory Tech., Vol.  MTT-22, No, 7, 718–727 (1974).
[Crossref]

Marcatili, E. A. J.

E. A. J. Marcatili, “Bends in Optical Dielectric Guides,” Bell Syst. Tech. J.,  48, 2013–2132 (1969).
[Crossref]

E. A. J. Marcatili and S. E. Miller, “Improved Relations Describing Directional Control in Electromagnetic Wave Guidance,” Bell Syst. Tech. J.,  48, 2161–2188 (1969).
[Crossref]

Marcuse, D.

D. Marcuse, “Curvature Loss Formula for Optical Fibers,” J. Opt. Soc. Am. 66, 216–220 (1976).
[Crossref]

D. Marcuse, “Bent Optical Waveguide with Lossy Jacket,” Bell Syst. Tech. J.,  53, 1079–1101 (1974).
[Crossref]

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

D. Marcuse, Light Transmission Optics (Van Nostrand, Reinhold, New York, 1972), pp. 398–406.

Miller, M. A.

M. A. Miller and V. I. Talanov, “Electromagnetic Surface Waves Guided by a Boundary with Small Curvature,” Zh. Tekh. Fiz. 26, 2755 (1956).

Miller, S. E.

E. A. J. Marcatili and S. E. Miller, “Improved Relations Describing Directional Control in Electromagnetic Wave Guidance,” Bell Syst. Tech. J.,  48, 2161–2188 (1969).
[Crossref]

S. E. Miller, “Directional Control in Lightwave Guidance,” Bell Syst. Tech. J.,  43, 1727–1739 (1964).
[Crossref]

Mitchell, D. J.

A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from Bent Optical Waveguides,” Electron. Lett.,  11, 332–333 (1975).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

Shevchenko, V. V.

V. V. Shevchenko, “Radiation Losses in Bent Waveguides for Surface Waves,” Radiophys. Quantum Electron. 14, 607–614 (1973) (Russian original 1971).
[Crossref]

Snyder, A. W.

A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from Bent Optical Waveguides,” Electron. Lett.,  11, 332–333 (1975).
[Crossref]

Stegun, I. A.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Eq. 9.2.4, p. 364. U. S. Dept. of Commerce, Natl. Bur. Stds., Appl. Math. Ser. Vol. 55 (U. S. GPO, Washington, D. C., 1964).

Talanov, V. I.

M. A. Miller and V. I. Talanov, “Electromagnetic Surface Waves Guided by a Boundary with Small Curvature,” Zh. Tekh. Fiz. 26, 2755 (1956).

White, I.

A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from Bent Optical Waveguides,” Electron. Lett.,  11, 332–333 (1975).
[Crossref]

Appl. Opt. (2)

Bell Syst. Tech. J. (5)

D. Marcuse, “Bent Optical Waveguide with Lossy Jacket,” Bell Syst. Tech. J.,  53, 1079–1101 (1974).
[Crossref]

S. E. Miller, “Directional Control in Lightwave Guidance,” Bell Syst. Tech. J.,  43, 1727–1739 (1964).
[Crossref]

E. A. J. Marcatili and S. E. Miller, “Improved Relations Describing Directional Control in Electromagnetic Wave Guidance,” Bell Syst. Tech. J.,  48, 2161–2188 (1969).
[Crossref]

J. A. Arnaud, “Transverse Coupling in Fiber Optics Part III: Bending Losses,” Bell Syst. Tech. J.,  53, 1379–1394 (1974).
[Crossref]

E. A. J. Marcatili, “Bends in Optical Dielectric Guides,” Bell Syst. Tech. J.,  48, 2013–2132 (1969).
[Crossref]

Electron. Lett. (1)

A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from Bent Optical Waveguides,” Electron. Lett.,  11, 332–333 (1975).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

L. Lewin, “Radiation from Curved Dielectric Slabs and Fibers,” IEEE Trans. Microwave Theory Tech., Vol.  MTT-22, No, 7, 718–727 (1974).
[Crossref]

J. Opt. Soc. Am. (1)

Radiophys. Quantum Electron. (1)

V. V. Shevchenko, “Radiation Losses in Bent Waveguides for Surface Waves,” Radiophys. Quantum Electron. 14, 607–614 (1973) (Russian original 1971).
[Crossref]

Zh. Tekh. Fiz. (1)

M. A. Miller and V. I. Talanov, “Electromagnetic Surface Waves Guided by a Boundary with Small Curvature,” Zh. Tekh. Fiz. 26, 2755 (1956).

Other (5)

D. C. Chang and E. F. Kuester, “General Theory of Surface-Wave Propagation on a Curved Optical Waveguide of Arbitrary Cross Section,” , Electromagnetics Laboratory, Dept. Electr. Eng., University of Colorado, Boulder, Colorado; also IEEE J. Quantum Electron. QE-11, 903–907 (1975).

D. Marcuse, Light Transmission Optics (Van Nostrand, Reinhold, New York, 1972), pp. 398–406.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Eq. 9.2.4, p. 364. U. S. Dept. of Commerce, Natl. Bur. Stds., Appl. Math. Ser. Vol. 55 (U. S. GPO, Washington, D. C., 1964).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

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Figures (10)

FIG. 1
FIG. 1

Fiber core of refractive index n1 is shown embedded in an infinite medium of refractive index n2, This figure defines the radius of curvature of the fiber and the mathematical matching surface (n1 = 1.515, n2 = 1.5 was used throughout).

FIG. 2
FIG. 2

Field distribution E as a function of normalized radius r/a in the plane of curvature. The dotted line represents the field of the straight, single mode fiber, the solid lines represent the corresponding mode for two different values of R/a. V = 2.4, ν = 0, p = 1, n1 = 1.515, n2 = 1.5.

FIG. 3
FIG. 3

Field distribution E as a function of normalized radius r/a for the HE11 mode in a multimode fiber. V = 12.76, ν = 0, p = 1.

FIG. 4
FIG. 4

Angular field distribution, E as a function of ϕ, at r/a = 0.8 for the HE11 mode with V = 12.76, ν = 0, p = 1.

FIG. 5
FIG. 5

Field distribution of the mode ν = 1, p = 1, for V = 12.76.

FIG. 6
FIG. 6

Field distribution of the mode ν = 1, p = 3, as a function of normalized radius r/a for V = 12.76.

FIG. 7
FIG. 7

Angular distribution of the mode shown in Fig. 6 at r/a = 0.8.

FIG. 8
FIG. 8

Radial field distribution of the mode ν = 7, p = 1 for V =12.76.

FIG. 9
FIG. 9

Angular field distribution of the mode shown in Fig. 8.

FIG. 10
FIG. 10

Field distribution E of the mode ν = 7, p = 1 along the matching surface shown as the dotted line in Fig. 1.

Tables (3)

Tables Icon

TABLE I Conventional curvature loss (normalized with respect to the fiber core radius) and the ratio of the actual loss to the loss calculated from the conventional formula are listed for several values of the normalized radius of curvature for a single mode fiber. γa = 1.7461359, V = 2.4.

Tables Icon

TABLE II Eigenvalues and conventional curvature losses for all modes of a multimode fiber. The loss values of modes with ν ≥ 3 are of questionable accuracy. V = 12.759702.

Tables Icon

TABLE III Ratios of the losses calculated from Eq. (1) to the conventional curvature losses calculated from (11) for the modes of the multimode fiber listed in Table II. The loss ratios for modes with ν ≥ 3 are of questionable accuracy. V = 12.759702.

Equations (37)

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2 α = 4 I 1 2 π 3 / 2 γ 1 / 2 R 3 / 2 I 2 H μ ( 2 ) ( ξ ) 2 .
I 1 = 0 r 0 E ( z ) d z
I 2 = 0 2 π d ϕ 0 r 0 E 2 r d r .
I 1 ( β ) = 1 2 - E ( z ) e i β z d z .
γ = ( β g 2 - n 2 2 k 2 ) 1 / 2 ,
μ = β g R ,
ξ = n 2 k ( R + a ) .
H μ ( 2 ) ( ξ ) = i e μ ( θ - tanh θ ) ( 1 2 π μ tanh θ ) 1 / 2 ,
cosh θ = μ / ξ .
H μ ( 2 ) ( ξ ) = i e - γ a ( 1 2 π γ R ) 1 / 2 exp ( γ 3 3 β g 2 R ) .
2 α = 2 κ 2 e - 2 γ a e ν π γ 5 / 2 R 3 / 2 V 2 H μ ( 2 ) ( ξ ) 2 K ν - 1 ( γ a ) K ν + 1 ( γ a ) ,
e ν = { 2 for ν = 0 , 1 for ν 1.
κ 2 = n 1 2 k 2 - β g 2 ,
V 2 = ( n 1 2 - n 2 2 ) k 2 a 2 .
2 E r 2 + 1 r E r + 1 r 2 2 E ϕ 2 + 2 E z 2 + n 2 [ 1 + 2 ( r / R ) cos ϕ ] k 2 E = 0.
n eff = n [ 1 + ( r / R ) cos ϕ ] .
E = e i β g z ν , p = 1 A ν p J ν ( σ ν p r ) cos ν ϕ .
J ν ( σ ν p r 0 ) = 0.
M A = ( γ a ) 2 D A .
M ¯ A ¯ = ( γ a ) 2 A ¯ .
M ¯ = D - 1 / 2 M D - 1 / 2 ,
A ¯ = D 1 / 2 A .
I 2 = π ν , p = 1 e ν N ν p A ν p 2 .
q = ( γ R / n 2 2 k 2 ) β 2 ,
[ γ a / ( n 2 k a ) 2 ] ( R / a ) 1.23
β g R = n 2 k ( R + a ) .
β g = n 1 k cos θ n 1 k ( 1 - 1 2 θ 2 ) .
θ eff = θ c [ 1 - 2 ( n 2 - n 1 ) a / R θ c 2 ] 1 / 2 .
θ c = [ 1 - ( n 2 / n 1 ) 2 ] 1 / 2 [ 2 ( 1 - n 2 / n 1 ) ] 1 / 2 .
N s = 1 2 ( a k n 1 θ c ) 2 .
N = N s [ 1 - 2 ( n 2 / n 1 ) a / R θ c 2 ] .
M ν p , ν p = ( V 2 P ν p p - a 2 σ ν p 2 N ν p δ p p ) e ν δ ν ν + ( 2 n 2 2 k 2 a 3 / R ) Q ν p , ν p e ¯ ν ν , D ν p , ν p = N ν p e ν δ ν ν δ p p ,
P ν p p = 1 2 [ J ν 2 ( σ ν p a ) - J ν - 1 ( σ ν p a ) J ν + 1 ( σ ν p a ) ] ,
P ν p p = P ν p p = σ ν p a J ν ( σ ν p a ) J ν + 1 ( σ ν p a ) - σ ν p a J ν + 1 ( σ ν p a ) J ν - 1 ( σ ν p a ) ( a σ ν p ) 2 - ( a σ ν p ) 2             for p p ,
N ν p = ( r 0 2 / 2 a 2 ) J ν + 1 2 ( σ ν p r 0 ) ,
Q ν + 1 , p , ν p = Q ν p , ν + 1 , p = V 2 n 2 2 k 2 a 4 1 σ ν + 1 , p 2 - σ ν p 2 { σ ν + 1 , p a J ν + 2 ( σ ν + 1 , p a ) J ν ( σ ν p a ) - σ ν p a J ν + 1 ( σ ν + 1 , p a ) J ν + 1 ( σ ν p a ) + 2 σ ν p σ ν + 1 , p 2 - σ ν p 2 [ σ ν p J ν + 1 ( σ ν + 1 , p a ) J ν ( σ ν p a ) - 2 a ( ν + 1 ) J ν + 1 ( σ ν + 1 , p a ) J ν + 1 ( σ ν p a ) + σ ν + 1 , p J ν + 2 ( σ ν + 1 , p a ) J ν + 1 ( σ ν p a ) ] } + 2 σ ν p σ ν + 1 , p r 0 a 3 ( σ ν + 1 , p 2 - σ ν p 2 ) 2 J ν + 2 ( σ ν + 1 , p r 0 ) J ν + 1 ( σ ν p r 0 ) .
e ¯ ν v = { 1 for ν = 0 ,             ν = 1 , 1 for ν = 1 ,             ν = 0 , 1 2 for ν = ν + 1 ,             ν 0 , 1 2 for ν = ν - 1 ,             ν 0 , 0 for all other combinations of ν and ν .