Abstract

An extension is derived of a bending-loss-coefficient equation derived by Marcuse [ J. Opt. Soc. Am. 66, 216 ( 1976)]. The original derivation employed approximations that limited the loss formula to low-order modes. We develop a closed-form integral and an algorithm for an approximate integral that allow loss coefficients to be computed for high-order modes. The overall loss of a multimode fiber is then analyzed by making simple assumptions about the power distribution among the modes of a fiber.

© 1981 Optical Society of America

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References

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  1. M. A. Miller and V. I. Talanow, “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. MTT-22, 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).
    [CrossRef]
  7. D. C. Chang and E. F. Kuester, “General theory of surface-wave propagation on a curved optical waveguide of arbitrary cross-section,” IEEE J. Quantum Electron. QE-11, 903–907 (1975).
  8. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).
  9. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66, 216–220 (1976).
    [CrossRef]
  10. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66, 311–320 (1976).
    [CrossRef]
  11. D. Gloge, “Weakly guided fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [CrossRef] [PubMed]
  12. R. Terras, “A Miller algorithm for an incomplete Bessel function,” J. Comput. Phys. 39, 233–240 (1981).
    [CrossRef]
  13. R. Terras, “Algorithms for integrals of Bessel functions and multivariate Gaussian integrals,” J. Comput. Phys. 41, 192–199 (1981).
    [CrossRef]
  14. M. Lebedev, Special Functions (Dover, New York, 1965).
  15. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Nat. Bur. of Stand. (U.S.) Applied Mathematics Service 55 (U.S. Government Printing Office, Washington, D.C., 1964).

1981 (2)

R. Terras, “A Miller algorithm for an incomplete Bessel function,” J. Comput. Phys. 39, 233–240 (1981).
[CrossRef]

R. Terras, “Algorithms for integrals of Bessel functions and multivariate Gaussian integrals,” J. Comput. Phys. 41, 192–199 (1981).
[CrossRef]

1976 (2)

1975 (2)

A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975).
[CrossRef]

D. C. Chang and E. F. Kuester, “General theory of surface-wave propagation on a curved optical waveguide of arbitrary cross-section,” IEEE J. Quantum Electron. QE-11, 903–907 (1975).

1974 (2)

L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 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]

1973 (1)

V. V. Shevchenko, “Radiation losses in bent waveguides for surface waves,” Radiophys. Quantum Electron. 14, 607–614 (1973).
[CrossRef]

1971 (1)

1969 (1)

E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2013–2132 (1969).
[CrossRef]

1956 (1)

M. A. Miller and V. I. Talanow, “Electromagnetic surface waves guided by a boundary with small curvature,” Zh. Tekh. Fiz. 26, 2755 (1956).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Nat. Bur. of Stand. (U.S.) Applied Mathematics Service 55 (U.S. Government Printing Office, 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,” IEEE J. Quantum Electron. QE-11, 903–907 (1975).

Gloge, D.

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,” IEEE J. Quantum Electron. QE-11, 903–907 (1975).

Lebedev, M.

M. Lebedev, Special Functions (Dover, New York, 1965).

Lewin, L.

L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 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]

Marcuse, D.

Miller, M. A.

M. A. Miller and V. I. Talanow, “Electromagnetic surface waves guided by a boundary with small curvature,” Zh. Tekh. Fiz. 26, 2755 (1956).

Mitchell, D. J.

A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975).
[CrossRef]

Shevchenko, V. V.

V. V. Shevchenko, “Radiation losses in bent waveguides for surface waves,” Radiophys. Quantum Electron. 14, 607–614 (1973).
[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. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Nat. Bur. of Stand. (U.S.) Applied Mathematics Service 55 (U.S. Government Printing Office, Washington, D.C., 1964).

Talanow, V. I.

M. A. Miller and V. I. Talanow, “Electromagnetic surface waves guided by a boundary with small curvature,” Zh. Tekh. Fiz. 26, 2755 (1956).

Terras, R.

R. Terras, “A Miller algorithm for an incomplete Bessel function,” J. Comput. Phys. 39, 233–240 (1981).
[CrossRef]

R. Terras, “Algorithms for integrals of Bessel functions and multivariate Gaussian integrals,” J. Comput. Phys. 41, 192–199 (1981).
[CrossRef]

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. (1)

Bell Syst. Tech. J. (2)

E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2013–2132 (1969).
[CrossRef]

J. A. Arnaud, “Transverse coupling in fiber optics. Part III: bending losses,” Bell Syst. Tech. J. 53, 1379–1394 (1974).
[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 J. Quantum Electron. (1)

D. C. Chang and E. F. Kuester, “General theory of surface-wave propagation on a curved optical waveguide of arbitrary cross-section,” IEEE J. Quantum Electron. QE-11, 903–907 (1975).

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. Comput. Phys. (2)

R. Terras, “A Miller algorithm for an incomplete Bessel function,” J. Comput. Phys. 39, 233–240 (1981).
[CrossRef]

R. Terras, “Algorithms for integrals of Bessel functions and multivariate Gaussian integrals,” J. Comput. Phys. 41, 192–199 (1981).
[CrossRef]

J. Opt. Soc. Am. (2)

Radiophys. Quantum Electron. (1)

V. V. Shevchenko, “Radiation losses in bent waveguides for surface waves,” Radiophys. Quantum Electron. 14, 607–614 (1973).
[CrossRef]

Zh. Tekh. Fiz. (1)

M. A. Miller and V. I. Talanow, “Electromagnetic surface waves guided by a boundary with small curvature,” Zh. Tekh. Fiz. 26, 2755 (1956).

Other (3)

M. Lebedev, Special Functions (Dover, New York, 1965).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Nat. Bur. of Stand. (U.S.) Applied Mathematics Service 55 (U.S. Government Printing Office, Washington, D.C., 1964).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

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Figures (6)

Fig. 1
Fig. 1

The integrand of Ib, f ( β ) = I ν β 2 exp ( - γ R n 2 - 2 k - 2 β 2 ), versus x = βa. In(a) ν =0, = 219, and R/a = 1000. In (b) ν =130, = 144, and R/a = 100.

Fig. 2
Fig. 2

Curvature loss as a function of the mode numbers p and ν. The fiber has a = 62.5 μm, n1 = 1.619, and n2 = 1.547, and the wavelength of the light is λ = 0.86 μm. In (a) R = 50a, and in (b) R = 2260a.

Fig. 3
Fig. 3

The number of propagating modes versus the radius of curvature. The fiber has a = 62.5 μm, n1 = 1.619, and n2 = 1.547 and carries light of λ = 0.86 μm. In (a) log10(R/a) varies from 1.5 to 5.5, but (b) takes a more detailed look at the region from 3.5 to 5.5.

Fig. 4
Fig. 4

A three-dimensional plot of the mode numbers present versus the radius of curvature for a fiber with a = 62.5 μm, n1 = 1.619, and n2 = 1.547, carrying light of λ = 0.86 μm. In (a) 0 ≦ ν ≦ 160 and 1.0 ≦ log10(R/a) ≦ 5.5. In (b) is a closeup with 0 ≦ ν ≦ 20 and 3.5 ≦ log10(R/a) ≦ 5.5.

Fig. 5
Fig. 5

Curvature loss at 1 m versus the radius of curvature. The fiber has a = 62.5 μm, n1 = 1.619, n2 = 1.547 and carries light with λ = 0.86 μm. In (a) one sees the fractional loss q′ occurring in the propagating modes, and in (b) one sees the overall fractional loss q.

Fig. 6
Fig. 6

Curvature losses of the lossiest individual modes versus the radius of curvature for a fiber with a = 62.5 μm, n1 = 1.619, and n2 = 1.547, carrying light with λ = 0.86 μm.

Equations (43)

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2 α = 2 κ 2 exp ( - 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 ) ,
γ = ( β g 2 - n 2 2 k 2 ) 1 / 2 ,             κ = ( n 1 2 k 2 - β g 2 ) 1 / 2 , e ν = { 2 ; ν = 0 1 ; ν 0 ,             V 2 = k 2 a 2 ( n 1 2 - n 2 2 ) , k = 2 π / λ = ( 0 μ 0 ) 1 / 2 ω ,             ξ = n 2 k ( R + a ) ,             μ = β g R ,
I b = - I ν β 2 exp ( - γ R n 2 - 2 k - 2 β 2 ) d β ,
I ν β = - H ν ( 1 ) [ i γ ( a 2 + z 2 ) 1 / 2 ] × cos [ ν arctan ( z / a ) ] exp ( i β z ) d z ,
I ν β = i - ν - 1 γ - ν exp [ - a ( γ 2 + β 2 ) 1 / 2 ] ( γ 2 + β 2 ) - 1 / 2 × { [ ( γ 2 + β 2 ) 1 / 2 - β ] ν + [ ( γ 2 + β 2 ) 1 / 2 + β ] ν } .
I b = 2 γ { exp ( - 2 a γ ) ( π G γ 2 + a γ ) 1 / 2 + exp [ h ( η ) ] ( 1 + η 2 ) - 1 × [ π ( 1 + η 2 ) G γ 2 ( 1 + 2 η 2 ) + a γ ( 1 + η 2 ) 1 / 2 ] 1 / 2 } ,
h ( β ) = G γ 2 β 2 - 2 a γ ( 1 + β 2 ) 1 / 2 + 2 ν ln [ ( 1 + β 2 ) 1 / 2 + β ] ,
2 α = κ 2 I b 2 π e ν V 2 n 2 k R H μ ( 2 ) ( ξ ) 2 K ν - 1 ( γ a ) K ν + 1 ( γ a ) .
I b = 4 γ 2 exp ( - 2 a γ ) π G - 1 / 2 ,
q = 1 - 1 P 0 P ν , p exp ( - 2 α ν , p d ) ,
q = 1 - 1 N exp ( - 2 α ν , p d ) ,
I ν β = 2 a π i - ν - 1 - K ν [ γ a ( 1 + z 2 ) 1 / 2 ] × cos ( ν arctan z ) exp ( i a β z ) d z ,
I ν β = a π i - ν - 1 - K ν [ γ a ( 1 + z 2 ) 1 / 2 ] ( 1 + z 2 ) - ν / 2 × [ ( 1 + i z ) ν + ( 1 - i z ) ν ] exp ( i a β z ) d z .
K ν ( x ) = 1 2 ( x 2 ) ν 0 exp [ - t - ( x / 2 ) 2 t - 1 ] t - ν - 1 d t ,
K ν ( x ) = ( x 2 ) ν 0 exp [ - t 2 - ( x / 2 ) 2 t - 2 ] t - 2 ν - 1 d t .
I ν β = a π i - ν - 1 ( a γ 2 ) ν 0 exp [ - t 2 - ( a γ / 2 ) 2 t - 2 ] t - 2 ν - 1 d t × - exp [ - ( a γ / 2 ) 2 z 2 t - 2 ] × [ ( 1 + i z ) ν + ( 1 - i z ) ν ] exp ( i a β z ) d z .
- exp [ - ( a γ 2 ) 2 z 2 t - 2 ] ( 1 + i z ) ν exp ( i a β z ) d z = 2 π ( t a γ ) ν + 1 H ν ( a γ 2 t - t β γ ) exp ( - t 2 β 2 γ - 2 ) ,
I ν β = 2 - ν + 1 i - ν - 1 γ π 0 exp [ - t 2 ( γ 2 + β 2 γ 2 ) - a 2 γ 2 4 t 2 ] t - ν × [ H ν ( a γ 2 t - t β γ ) + H ν ( a γ 2 t + t β γ ) ] d t ,
I ν β = i - ν - 1 ( a γ ) - ν + 1 γ π 0 exp [ - a 2 ( γ 2 + β 2 ) 4 t 2 - t 2 ] × t ν - 2 [ H ν ( t - a β 2 t ) + H ν ( t + a β 2 t ) ] d t ,
I ν β = i - ν - 1 ( a γ ) - ν + 1 γ π { L [ a ( γ 2 + β 2 ) 1 / 2 , a β / 2 ] + L [ a ( γ 2 + β 2 ) 1 / 2 , - a β / 2 ] } ,
L ( z , w ) = 0 exp ( - t 2 - z 2 4 t 2 ) t ν - 2 H ν ( t - w t ) d t .
H ν ( t - w t ) = λ = 0 ν ν ! λ ! ( ν - λ ) ! ( - 2 t ) λ w λ H ν - λ ( t ) .
H n ( x ) = k = 0 [ n / 2 ] ( - 1 ) k n ! k ! ( n - 2 k ) ! ( 2 x ) n - 2 k ,
[ n / 2 ] = { n / 2 n even ( n - 1 ) / 2 n odd ,
L ( z , w ) = λ = 0 ν ( ν λ ) ( - 2 w ) λ k = 0 [ ( ν - λ ) / 2 ] ( - 1 ) k ( ν - λ ) ! 2 ν - λ - 2 k k ! ( ν - λ - 2 k ) ! × 0 exp ( - t 2 - z 2 4 t 2 ) t 2 ν - 2 λ - 2 k - 2 d t .
L ( z , w ) = λ = 0 ν ( ν λ ) ( - 2 w ) λ k = 0 [ ( ν - λ ) / 2 ] ( - 1 ) k ( ν - λ ) ! 2 ν - λ - 2 k k ! ( ν - λ - 2 k ) ! × ( z 2 ) ν - λ - k - 1 / 2 K - ν + λ + k + 1 / 2 ( z ) .
L ( z , w ) = π λ = 0 ν ( ν λ ) ( - 2 w ) λ z ν - λ - 1 e - z ,
L ( z , w ) = π z - 1 e - z ( z - 2 w ) ν .
I ν β = i - ν - 1 γ - ν exp [ - a ( γ 2 + β 2 ) 1 / 2 ] ( γ 2 + β 2 ) - 1 / 2 × { [ ( γ 2 + β 2 ) 1 / 2 - β ] ν + [ ( γ 2 + β 2 ) 1 / 2 + β ] ν } .
I b = - exp ( - G β 2 ) I ν β 2 d β ,
I b = 1 γ - exp ( - G γ 2 β 2 ) ( 1 + β 2 ) - 1 exp [ - 2 a γ ( 1 + β 2 ) 1 / 2 ] × { 2 + [ ( 1 + β 2 ) 1 / 2 - β ] 2 ν + [ ( 1 + β 2 ) 1 / 2 + β ] 2 ν } d β ,
I 1 = 2 γ - exp ( - G γ 2 β 2 ) × exp [ - 2 a γ ( 1 + β 2 ) 1 / 2 ] ( 1 + β 2 ) - 1 d β
I 2 = 2 γ - exp ( - G γ 2 β 2 ) exp [ - 2 a γ ( 1 + β 2 ) 1 / 2 ] × [ ( 1 + β 2 ) 1 / 2 + β ] 2 ν ( 1 + β 2 ) - 1 d β .
exp [ - 2 a γ ( 1 + β 2 ) 1 / 2 ] = exp ( - 2 a γ ) exp ( - a γ β 2 ) exp [ 0 ( β 4 ) ] .
I 1 = exp ( - 2 a γ ) ( π G γ 2 + a γ ) 1 / 2
I 2 = - e h ( β ) ( 1 + β 2 ) - 1 d β .
h ( β ) = - 2 A β + 2 ( ν - λ β ) ( 1 + β 2 ) - 1 / 2 ,
ν = β [ λ + A ( 1 + β 2 ) 1 / 2 ] .
β n + 1 = ν [ λ + A ( 1 + β n 2 ) 1 / 2 ] - 1 ,             β 0 = 0.
β n = ν ( λ + A x n ) - 1 , x n + 1 = [ x n + ( 1 + β n 2 ) x n - 1 ] / 2 ,             x 0 = 1 ,
h ( β ) = h ( η ) + h ( η ) 2 ( β - η ) 2 + ,
h ( η ) = - 2 A ( 1 + 2 η 2 ) + λ ( 1 + η 2 ) 1 / 2 1 + η 2 .
I 2 = e h ( η ) ( 1 + η 2 ) - 1 ( π ( 1 + η 2 ) G γ 2 ( 1 + 2 η 2 ) + a γ ( 1 + η 2 ) 1 / 2 ) 1 / 2 .