Abstract

Excess losses of single-mode and multimode cabled optical fibers due to random bends of the Gaussian correlation function are discussed theoretically. By comparing theoretical results of various correlation functions to measured excess losses of the cabled fibers, it is verified that the correlation function of random bends of single-mode and multimode cabled fibers is estimated to be Gaussian. Using the estimated Gaussian correlation function, random-bend loss formulas of the single-mode and the graded-index multimode fiber cables are given. To use a cabled single-mode fiber over a wider wavelength region intended for a wavelength-division-multiplexing system, the relative refractive-index difference between core and cladding should be made larger to reduce the random-bend loss substantially. Cabling losses of graded-index multimode fibers depend strongly on the relative refractive-index difference and the diameter of the core.

© 1980 Optical Society of America

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References

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  1. T. Miyashita, A. Kawana, M. Nakahara, M. Kawachi, T. Hosaka, Trans. Inst. Electron. Commun. Eng. Jpn. 61-E, 891 (Nov.1978).
  2. Y. Katsuyama, Y. Ishida, K. Ishihara, T. Miyashita, H. Tsuchiya, Electron. Lett. 15, 94 (1979).
    [CrossRef]
  3. D. Marcuse, Bell Syst. Tech. J. 55, 937 (1976).
  4. K. Petermann, Electron. Lett. 12, 107 (1976).
    [CrossRef]
  5. S. Kawakami, Appl. Opt. 15, 2778 (1976).
    [CrossRef] [PubMed]
  6. R. Olshansky, Appl. Opt. 14, 935 (1975).
    [PubMed]
  7. K. Inada, Y. Sugawara, T. Kobayashi, T. Yamada, M. Tokuda, H. Ishihara, Natl. Conv. Rec. Inst. Electron. Commun. Eng. Jpn. Commun. Sec. S3-2 (Oct.1978) (in Japanese).
  8. D. Marcuse, J. Opt. Soc. Am. 66, 311 (1976).
    [CrossRef]
  9. A. W. Snyder, IEEE Trans. Microwave Theory Tech. MTT-17, 1138 (1969).
    [CrossRef]
  10. D. Gloge, Appl. Opt. 10, 2252 (1971).
    [CrossRef] [PubMed]
  11. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  12. H. Beteman, Table of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2.
  13. M. Kubota, K. Furuya, Y. Suematsu, “Random-Bend Loss of Single-Mode Fiber with Various Index Profiles,” unpublished.
  14. Y. Katsuyama, Y. Mitsunaga, S. Mochizuki, A. Kawana, K. Ishihara, H. Tsuchiya, Papers on Technical Group Meeting of Inst. Electron Commun. Eng. Jpn. OQ378-43 (June1978) (in Japanese).
  15. K. Furuya, Y. Suematsu, H. Tokiwa, Opt. Quantum Electron. 10, 323 (1978).
    [CrossRef]
  16. Y. Suematsu, H. Tokiwa, Trans. Inst. Electron. Commun. Eng. Jpn. 61-E, 355 (May1978).

1979 (1)

Y. Katsuyama, Y. Ishida, K. Ishihara, T. Miyashita, H. Tsuchiya, Electron. Lett. 15, 94 (1979).
[CrossRef]

1978 (5)

K. Inada, Y. Sugawara, T. Kobayashi, T. Yamada, M. Tokuda, H. Ishihara, Natl. Conv. Rec. Inst. Electron. Commun. Eng. Jpn. Commun. Sec. S3-2 (Oct.1978) (in Japanese).

Y. Katsuyama, Y. Mitsunaga, S. Mochizuki, A. Kawana, K. Ishihara, H. Tsuchiya, Papers on Technical Group Meeting of Inst. Electron Commun. Eng. Jpn. OQ378-43 (June1978) (in Japanese).

K. Furuya, Y. Suematsu, H. Tokiwa, Opt. Quantum Electron. 10, 323 (1978).
[CrossRef]

Y. Suematsu, H. Tokiwa, Trans. Inst. Electron. Commun. Eng. Jpn. 61-E, 355 (May1978).

T. Miyashita, A. Kawana, M. Nakahara, M. Kawachi, T. Hosaka, Trans. Inst. Electron. Commun. Eng. Jpn. 61-E, 891 (Nov.1978).

1976 (4)

S. Kawakami, Appl. Opt. 15, 2778 (1976).
[CrossRef] [PubMed]

D. Marcuse, J. Opt. Soc. Am. 66, 311 (1976).
[CrossRef]

D. Marcuse, Bell Syst. Tech. J. 55, 937 (1976).

K. Petermann, Electron. Lett. 12, 107 (1976).
[CrossRef]

1975 (1)

1971 (1)

1969 (1)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. MTT-17, 1138 (1969).
[CrossRef]

Beteman, H.

H. Beteman, Table of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2.

Furuya, K.

K. Furuya, Y. Suematsu, H. Tokiwa, Opt. Quantum Electron. 10, 323 (1978).
[CrossRef]

M. Kubota, K. Furuya, Y. Suematsu, “Random-Bend Loss of Single-Mode Fiber with Various Index Profiles,” unpublished.

Gloge, D.

Hosaka, T.

T. Miyashita, A. Kawana, M. Nakahara, M. Kawachi, T. Hosaka, Trans. Inst. Electron. Commun. Eng. Jpn. 61-E, 891 (Nov.1978).

Inada, K.

K. Inada, Y. Sugawara, T. Kobayashi, T. Yamada, M. Tokuda, H. Ishihara, Natl. Conv. Rec. Inst. Electron. Commun. Eng. Jpn. Commun. Sec. S3-2 (Oct.1978) (in Japanese).

Ishida, Y.

Y. Katsuyama, Y. Ishida, K. Ishihara, T. Miyashita, H. Tsuchiya, Electron. Lett. 15, 94 (1979).
[CrossRef]

Ishihara, H.

K. Inada, Y. Sugawara, T. Kobayashi, T. Yamada, M. Tokuda, H. Ishihara, Natl. Conv. Rec. Inst. Electron. Commun. Eng. Jpn. Commun. Sec. S3-2 (Oct.1978) (in Japanese).

Ishihara, K.

Y. Katsuyama, Y. Ishida, K. Ishihara, T. Miyashita, H. Tsuchiya, Electron. Lett. 15, 94 (1979).
[CrossRef]

Y. Katsuyama, Y. Mitsunaga, S. Mochizuki, A. Kawana, K. Ishihara, H. Tsuchiya, Papers on Technical Group Meeting of Inst. Electron Commun. Eng. Jpn. OQ378-43 (June1978) (in Japanese).

Katsuyama, Y.

Y. Katsuyama, Y. Ishida, K. Ishihara, T. Miyashita, H. Tsuchiya, Electron. Lett. 15, 94 (1979).
[CrossRef]

Y. Katsuyama, Y. Mitsunaga, S. Mochizuki, A. Kawana, K. Ishihara, H. Tsuchiya, Papers on Technical Group Meeting of Inst. Electron Commun. Eng. Jpn. OQ378-43 (June1978) (in Japanese).

Kawachi, M.

T. Miyashita, A. Kawana, M. Nakahara, M. Kawachi, T. Hosaka, Trans. Inst. Electron. Commun. Eng. Jpn. 61-E, 891 (Nov.1978).

Kawakami, S.

Kawana, A.

T. Miyashita, A. Kawana, M. Nakahara, M. Kawachi, T. Hosaka, Trans. Inst. Electron. Commun. Eng. Jpn. 61-E, 891 (Nov.1978).

Y. Katsuyama, Y. Mitsunaga, S. Mochizuki, A. Kawana, K. Ishihara, H. Tsuchiya, Papers on Technical Group Meeting of Inst. Electron Commun. Eng. Jpn. OQ378-43 (June1978) (in Japanese).

Kobayashi, T.

K. Inada, Y. Sugawara, T. Kobayashi, T. Yamada, M. Tokuda, H. Ishihara, Natl. Conv. Rec. Inst. Electron. Commun. Eng. Jpn. Commun. Sec. S3-2 (Oct.1978) (in Japanese).

Kubota, M.

M. Kubota, K. Furuya, Y. Suematsu, “Random-Bend Loss of Single-Mode Fiber with Various Index Profiles,” unpublished.

Marcuse, D.

D. Marcuse, J. Opt. Soc. Am. 66, 311 (1976).
[CrossRef]

D. Marcuse, Bell Syst. Tech. J. 55, 937 (1976).

Mitsunaga, Y.

Y. Katsuyama, Y. Mitsunaga, S. Mochizuki, A. Kawana, K. Ishihara, H. Tsuchiya, Papers on Technical Group Meeting of Inst. Electron Commun. Eng. Jpn. OQ378-43 (June1978) (in Japanese).

Miyashita, T.

Y. Katsuyama, Y. Ishida, K. Ishihara, T. Miyashita, H. Tsuchiya, Electron. Lett. 15, 94 (1979).
[CrossRef]

T. Miyashita, A. Kawana, M. Nakahara, M. Kawachi, T. Hosaka, Trans. Inst. Electron. Commun. Eng. Jpn. 61-E, 891 (Nov.1978).

Mochizuki, S.

Y. Katsuyama, Y. Mitsunaga, S. Mochizuki, A. Kawana, K. Ishihara, H. Tsuchiya, Papers on Technical Group Meeting of Inst. Electron Commun. Eng. Jpn. OQ378-43 (June1978) (in Japanese).

Nakahara, M.

T. Miyashita, A. Kawana, M. Nakahara, M. Kawachi, T. Hosaka, Trans. Inst. Electron. Commun. Eng. Jpn. 61-E, 891 (Nov.1978).

Olshansky, R.

Petermann, K.

K. Petermann, Electron. Lett. 12, 107 (1976).
[CrossRef]

Snyder, A. W.

A. W. Snyder, IEEE Trans. Microwave Theory Tech. MTT-17, 1138 (1969).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Suematsu, Y.

Y. Suematsu, H. Tokiwa, Trans. Inst. Electron. Commun. Eng. Jpn. 61-E, 355 (May1978).

K. Furuya, Y. Suematsu, H. Tokiwa, Opt. Quantum Electron. 10, 323 (1978).
[CrossRef]

M. Kubota, K. Furuya, Y. Suematsu, “Random-Bend Loss of Single-Mode Fiber with Various Index Profiles,” unpublished.

Sugawara, Y.

K. Inada, Y. Sugawara, T. Kobayashi, T. Yamada, M. Tokuda, H. Ishihara, Natl. Conv. Rec. Inst. Electron. Commun. Eng. Jpn. Commun. Sec. S3-2 (Oct.1978) (in Japanese).

Tokiwa, H.

Y. Suematsu, H. Tokiwa, Trans. Inst. Electron. Commun. Eng. Jpn. 61-E, 355 (May1978).

K. Furuya, Y. Suematsu, H. Tokiwa, Opt. Quantum Electron. 10, 323 (1978).
[CrossRef]

Tokuda, M.

K. Inada, Y. Sugawara, T. Kobayashi, T. Yamada, M. Tokuda, H. Ishihara, Natl. Conv. Rec. Inst. Electron. Commun. Eng. Jpn. Commun. Sec. S3-2 (Oct.1978) (in Japanese).

Tsuchiya, H.

Y. Katsuyama, Y. Ishida, K. Ishihara, T. Miyashita, H. Tsuchiya, Electron. Lett. 15, 94 (1979).
[CrossRef]

Y. Katsuyama, Y. Mitsunaga, S. Mochizuki, A. Kawana, K. Ishihara, H. Tsuchiya, Papers on Technical Group Meeting of Inst. Electron Commun. Eng. Jpn. OQ378-43 (June1978) (in Japanese).

Yamada, T.

K. Inada, Y. Sugawara, T. Kobayashi, T. Yamada, M. Tokuda, H. Ishihara, Natl. Conv. Rec. Inst. Electron. Commun. Eng. Jpn. Commun. Sec. S3-2 (Oct.1978) (in Japanese).

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

D. Marcuse, Bell Syst. Tech. J. 55, 937 (1976).

Electron. Lett. (2)

K. Petermann, Electron. Lett. 12, 107 (1976).
[CrossRef]

Y. Katsuyama, Y. Ishida, K. Ishihara, T. Miyashita, H. Tsuchiya, Electron. Lett. 15, 94 (1979).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. MTT-17, 1138 (1969).
[CrossRef]

J. Opt. Soc. Am. (1)

Natl. Conv. Rec. Inst. Electron. Commun. Eng. Jpn. Commun. Sec. (1)

K. Inada, Y. Sugawara, T. Kobayashi, T. Yamada, M. Tokuda, H. Ishihara, Natl. Conv. Rec. Inst. Electron. Commun. Eng. Jpn. Commun. Sec. S3-2 (Oct.1978) (in Japanese).

Opt. Quantum Electron. (1)

K. Furuya, Y. Suematsu, H. Tokiwa, Opt. Quantum Electron. 10, 323 (1978).
[CrossRef]

Papers on Technical Group Meeting of Inst. Electron Commun. Eng. Jpn. (1)

Y. Katsuyama, Y. Mitsunaga, S. Mochizuki, A. Kawana, K. Ishihara, H. Tsuchiya, Papers on Technical Group Meeting of Inst. Electron Commun. Eng. Jpn. OQ378-43 (June1978) (in Japanese).

Trans. Inst. Electron. Commun. Eng. Jpn. (2)

Y. Suematsu, H. Tokiwa, Trans. Inst. Electron. Commun. Eng. Jpn. 61-E, 355 (May1978).

T. Miyashita, A. Kawana, M. Nakahara, M. Kawachi, T. Hosaka, Trans. Inst. Electron. Commun. Eng. Jpn. 61-E, 891 (Nov.1978).

Other (3)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

H. Beteman, Table of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2.

M. Kubota, K. Furuya, Y. Suematsu, “Random-Bend Loss of Single-Mode Fiber with Various Index Profiles,” unpublished.

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Figures (10)

Fig. 1
Fig. 1

Coordinate system of a bending fiber.

Fig. 2
Fig. 2

Coordinate system of radiation.

Fig. 3
Fig. 3

Random-curvature function. Random function of the curvature 1/R is assumed as the random sequence of pulses C1, C2,…, Ci.

Fig. 4
Fig. 4

Dependence of random-bending loss on wavelength. Curves depend remarkably on the power spectrum. λ c is the cutoff wavelength of higher-order modes.

Fig. 5
Fig. 5

Theoretical and measured cabling loss of a single-mode fiber. Measured data are based on those of Katsuyama et al.14 From fitting a theoretical curve to the measured data, the power spectrum of random bends is estimated as Gaussian.

Cabling: Δ = 0.12%; Theory: Gaussian W ¯ = 1 . 2 mm ; N ( 1 / R 1 ) 2 ¯ = 92 m 3 ;
Jacketing:Δ0.12%0.17%
Measured

TheoryGaussian W ¯ = 1 mm , N ( 1 / R 1 ) 2 ¯ = 8.4 m 3
- - - and – – are theoretical curves corresponding to different spectra from Gaussian.
Fig. 6
Fig. 6

Factors x and y in Eqs. (31), (40), and (44).

Fig. 7
Fig. 7

Cabling loss vs wavelength. Fiber loss is based on Ref. 1: λ c = 1 μm, W ¯ = 1 . 2 mm , N ( 1 / R 1 ) 2 ¯ = 92 m 3 and n1 = 1.46.

Fig. 8
Fig. 8

Theoretical and measured cabling loss of a parabolic multimode fiber. Measured data are based on those of Inada et al.7 Solid lines are theoretical losses corresponding to a Gaussian power spectrum.

Fig. 9
Fig. 9

Normalized losses of single-mode and parabolic-index multimode fibers vs index difference, cutoff wavelength, core diameter, and correlation length. Parameter is the wavelength normalized by the cutoff wavelength for a single-mode fiber. Ls and Lm are in dB/km, N is in 1/m. x is given in Fig. 6 and Table II.

Fig. 10
Fig. 10

(a) Exponents s, and (b) exponent m in Eqs. (42) and (43).

Tables (2)

Tables Icon

Table I Factors C and F and Exponent ν in Eq. (30)

Tables Icon

Table. II Values x and y in Eqs. (31), (40), and (44)

Equations (52)

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δ n 2 ( r , θ ) = n 2 ( r ) 2 r R cos ( θ ψ ) ,
J ( e ) ( r , θ , z ) = j ω ɛ 0 δ n 2 ( r , θ ) E ( r , θ , z ) ,
E y ( r , θ , z ) = AD ( r ) exp ( j β z ) ,
D ( r ) = { J 0 ( u r a ) r a J 0 ( u ) K 0 ( υ ) K 0 ( υ r a ) r > a ,
υ [ V 1 2 5.784 exp ( 1.797 / V 1 ) ] 1 / 2 ,
u = ( V 1 2 υ 2 ) 1 / 2 .
β = [ ω 2 ɛ 0 n 1 2 μ ( u / a ) 2 ] 1 / 2 ,
V 1 = 2 π n 1 ( 2 Δ ) 1 / 2 a / λ ,
Δ ( n 1 1 n 2 2 ) / ( 2 n 1 2 ) ,
A = [ 2 π ( μ ɛ ) 1 / 2 ] 1 / 2 υ V 1 J 1 ( u ) a .
Π ( R , Θ , Φ ) = j 1 4 π ɛ 0 ω n 2 2 0 L dz 0 rdr 0 2 π d θ × 1 r J ( e ) ( r , θ , z ; t r ¯ c ) ,
r ¯ R z cos Φ r sin Φ cos ( θ Θ ) .
E k 2 2 [ R 0 × ( R 0 × Π ) ] ,
k i ω n i ( ɛ 0 μ 0 ) 1 / 2 , ( i = 1 , 2 ) .
| E | = | Π y | k 2 2 ( 1 sin 2 Φ sin 2 Θ ) 1 / 2 .
P = 0 2 π d Θ 0 π R 2 sin Φ d Φ 1 2 n 2 ( ɛ 0 μ 0 ) 1 / 2 | E | 2 .
α = P L ( neper / unit length ) .
α = 1 L k 2 4 4 π 3 υ 2 V 1 2 J 1 2 ( u ) a 2 × 0 2 π d Θ 0 π sin Φ ( 1 sin 2 Φ sin 2 θ ) d Φ × | 0 L dz 0 rdr 0 2 π d θ r R cos ( θ ψ ) D ( r ) × exp [ j k 2 r sin Φ cos ( Θ θ ) j ( β k 2 cos Φ ) z ] | 2 .
α = 2 ( k 2 a ) 4 u 2 V 1 2 υ 6 K 1 2 ( υ ) 0 π d Φ sin Φ ( 1 + cos 2 Φ ) × ( k 2 a sin Φ υ ) 2 [ 1 + ( k 2 a sin Φ υ ) 2 ] 4 G R ( β k 2 cos Φ ) ,
G R ( x ) = 1 L | 0 L dz 1 R exp ( jxz ) | 2 .
α = 1 6 1 Δ u 2 K 1 2 ( υ ) υ 4 G R [ 4 / 3 k 1 Δ ( υ V 1 ) 2 ] .
G R ( x ) = 1 L | i c i 1 R exp ( jxz ) dz | 2 = 1 L { i | c i 1 R exp ( jxz ) dz | 2 + Σ i Σ j ( j j ) [ c i 1 R exp ( jxz ) dz ] [ c j 1 R exp ( jxz ) dz ] } = N | c 1 R exp ( jxz ) dz | 2 ¯
1 R = { 1 R 1 W 2 < z < W 2 0 z < W 2 , W 2 < z ,
G R ( x ) = 2 N x 2 ( 1 R 1 ) 2 | cos xW | ¯ .
G R ( x ) = 2 N ( 1 R 1 ) 2 1 x 2 .
1 R = { 1 R 1 exp ( 2 z / W ) z 0 1 R 1 exp ( 2 z / W ) z > 0
G R ( x ) = 16 N ( 1 R 1 W ) 2 1 x 4 [ 1 1 + ( 2 xW ) 2 ] 2 16 N ( 1 R 1 W ) 2 1 x 4 ( xW ) 2 1 .
1 R = 1 R 1 exp [ ( 2 Z W ) 2 ] ,
G R ( x ) = π 4 N ( W R 1 ) 2 exp [ ( xW ) 2 / 8 ] ¯ .
α = NC [ ( 1 / R 1 ) 2 ¯ , W ¯ , λ c ] F ( λ λ c , Δ , W ¯ , λ c ) Δ ν ,
L s = xN ( 1 / R 1 ) 2 ¯ W ¯ 2 × 1 Δ exp [ y ( W ¯ n 1 λ c ) 2 Δ 2 ] ( dB / km ) ,
C i , i + 1 = π 2 n 1 d λ ( 2 Δ ) 1 / 2 ( i + 1 ) G R ( β i + 1 β i ) ,
β i + 1 β i = 2 ( 2 Δ ) 1 / 2 / d .
h = ( i + 1 ) / i m = 2 λ ( i + 1 ) / [ π d n 1 ( 2 Δ ) 1 / 2 ] ,
i m = π d n 1 ( 2 Δ ) 1 / 2 / ( 2 λ ) ,
C ( h ) = π 3 16 N ( W R ) 2 n 1 2 d 2 λ 2 h exp [ Δ ( W d ) 2 ] .
z p ( h , z ) = 1 i m 2 h C ( h ) h [ hp ( h , z ) ] ,
p ( h , z ) = h J 0 [ j q ( 0 ) ( h ) 1 / 2 ] exp ( κ q z ) , κ q = ( π / 32 ) [ j q ( 0 ) ] 2 N ( 1 / R ) 2 W 2 ¯ 1 Δ × exp [ ( W ¯ d ) 2 Δ ] ( neper / unit length ) ,
L m = 2500 N ( 1 / R 1 ) 2 W 2 ¯ 1 Δ exp [ ( W ¯ d ) 2 Δ ] ( dB / km ) ,
L s Δ / [ xN ( 1 R ) 2 W ¯ 2 ] = exp [ y ( W ¯ n 1 λ c Δ ) 2 ] [ y ( W ¯ n 1 λ c Δ ) 2 ] [ y ( W n 1 / λ c Δ ) 2 ] ,
L m Δ / [ 2500 N ( 1 R ) 2 W ¯ 2 ] = exp [ ( W ¯ d ) 2 Δ ] [ ( W ¯ d ) 2 Δ ] [ ( W / d ) 2 Δ ]
L s Δ ( s + 1 ) λ c s W ¯ ( s 2 ) ,
L m Δ ( m + 1 ) d 2 m W ¯ ( 2 m 2 ) ,
s = 2 [ y ( W ¯ n 1 λ c Δ ) 2 ] , m = ( W ¯ d ) 2 Δ ,
W ¯ = 1 mm , N ( 1 / R 1 ) 2 ¯ = 8.4 m 3
1 12 π 2 ( λ c n 1 R 1 ) 2 ¯
1 6 π 4 ( λ C 2 n 1 2 R 1 W ) 2 ¯
A X 4
π 24 ( W R 1 ) 2 ¯
A ( V 2 v 2 ) / ( K 1 2 ( v ) v 4 ) , X ( 4 / 3 ) ( v / V ) 2 ( λ C / λ )
P = π ( ɛ μ ) 1 / 2 A 2 { 0 a J 0 2 ( u r a ) rdr + [ J 0 ( u ) K 0 ( υ ) ] 2 0 a K 0 2 ( υ r a ) rdr } = π ( ɛ μ ) 1 / 2 A 2 a 2 2 [ J 1 2 ( u ) + J 0 2 ( u ) K 1 2 ( υ ) K 0 2 ( υ ) ] .
u J 1 ( u ) J 0 ( u ) = υ K 1 ( υ ) K 0 ( υ ) .

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