Abstract

In this paper, we consider the scattering losses of single-mode fibers that are caused by microdeformations such as microbends of the fiber axis and random fluctuations of the fiber core diameter. Since very little is known about the statistics of microdeformations of actual fibers, we assume that the autocorrelation functions of random bends and random core diameter fluctuations are Gaussian, characterized by the rms deviation and the correlation length of the random function. We consider single-mode fibers with step, parabolic, and triangular (linear) refractive-index profiles and reach the following conclusions: (1) Whereas for equal (large) mode radii the microbending losses of all three fiber types are the same, losses due to random core diameter fluctuations can be three times as high in step-index fibers as in triangular-index fibers. Since triangular-index fibers have sometimes been observed to have lower scattering losses than step-index fibers, one might conclude that, in these cases, excess losses may be caused by random radius fluctuations rather than by microbends. (2) Radial refractive-index ripples, which tend to be present in the deposited claddings of single-mode fibers, seem unlikely to be a major source of microdeformation losses. (3) The wavelength dependence of microdeformation losses depends strongly on the value of the correlation length of the Gaussian autocorrelation function of the fiber deformations. If the correlation length is of the same order of magnitude as the fiber radius, the losses are only slightly wavelength dependent. For very long correlation lengths the losses are very much smaller (for the same rms variation of the random functions), but they become strongly wavelength dependent, increasing sharply with increasing wavelength.

© 1984 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. B. Gardner, Bell. Syst. Tech. J. 54, 457 (1975).
  2. K. Petermann, AEU Arch. Elektron. Uebertragungstech. Electron. Commun. 30, 337 (1976).
  3. S. Kawakami, Appl. Opt. 15, 2778 (1976).
    [CrossRef] [PubMed]
  4. D. Marcuse, Bell Syst. Tech. J. 55, 937 (1976).
  5. T. Tanaka, S. Yamada, M. Sumi, K. Mikoshiba, Appl. Opt. 16, 2391 (1977).
    [CrossRef] [PubMed]
  6. W. A. Gambling, H. Matsumura, C. M. Ragdae, Opt. Quantum Electron. 11, 43 (1979).
    [CrossRef]
  7. K. Furuya, S. Suematsu, Appl. Opt. 19, 1493 (1980).
    [CrossRef] [PubMed]
  8. D. Gloge, Appl. Opt. 10, 2252 (1971).
    [CrossRef] [PubMed]
  9. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  10. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), pp. 377–378.
  11. D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).
  12. Ref. 10, Sec. 8.7.
  13. M. Abromowitz, I. Stegun, Eds., Handbook on Mathematical Functions, Vol. 55 (Dover, New York, 1965).
  14. M. A. Saifi, S. J. Jang, L. G. Cohen, J. Stone, Opt. Lett. 7, 43 (1982).
    [CrossRef] [PubMed]
  15. K. I. White, Electron. Lett. 18, 725 (1982).
    [CrossRef]
  16. B. J. Ainslie, K. J. Beales, D. M. Cooper, C. R. Day, J. D. Rush, Electron. Lett. 18, 842 (1982).
    [CrossRef]

1982

M. A. Saifi, S. J. Jang, L. G. Cohen, J. Stone, Opt. Lett. 7, 43 (1982).
[CrossRef] [PubMed]

K. I. White, Electron. Lett. 18, 725 (1982).
[CrossRef]

B. J. Ainslie, K. J. Beales, D. M. Cooper, C. R. Day, J. D. Rush, Electron. Lett. 18, 842 (1982).
[CrossRef]

1980

1979

W. A. Gambling, H. Matsumura, C. M. Ragdae, Opt. Quantum Electron. 11, 43 (1979).
[CrossRef]

1977

1976

K. Petermann, AEU Arch. Elektron. Uebertragungstech. Electron. Commun. 30, 337 (1976).

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

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

1975

W. B. Gardner, Bell. Syst. Tech. J. 54, 457 (1975).

1973

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

1971

Ainslie, B. J.

B. J. Ainslie, K. J. Beales, D. M. Cooper, C. R. Day, J. D. Rush, Electron. Lett. 18, 842 (1982).
[CrossRef]

Beales, K. J.

B. J. Ainslie, K. J. Beales, D. M. Cooper, C. R. Day, J. D. Rush, Electron. Lett. 18, 842 (1982).
[CrossRef]

Cohen, L. G.

Cooper, D. M.

B. J. Ainslie, K. J. Beales, D. M. Cooper, C. R. Day, J. D. Rush, Electron. Lett. 18, 842 (1982).
[CrossRef]

Day, C. R.

B. J. Ainslie, K. J. Beales, D. M. Cooper, C. R. Day, J. D. Rush, Electron. Lett. 18, 842 (1982).
[CrossRef]

Furuya, K.

Gambling, W. A.

W. A. Gambling, H. Matsumura, C. M. Ragdae, Opt. Quantum Electron. 11, 43 (1979).
[CrossRef]

Gardner, W. B.

W. B. Gardner, Bell. Syst. Tech. J. 54, 457 (1975).

Gloge, D.

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

D. Gloge, Appl. Opt. 10, 2252 (1971).
[CrossRef] [PubMed]

Jang, S. J.

Kawakami, S.

Marcatili, E. A. J.

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

Marcuse, D.

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

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

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), pp. 377–378.

Matsumura, H.

W. A. Gambling, H. Matsumura, C. M. Ragdae, Opt. Quantum Electron. 11, 43 (1979).
[CrossRef]

Mikoshiba, K.

Petermann, K.

K. Petermann, AEU Arch. Elektron. Uebertragungstech. Electron. Commun. 30, 337 (1976).

Ragdae, C. M.

W. A. Gambling, H. Matsumura, C. M. Ragdae, Opt. Quantum Electron. 11, 43 (1979).
[CrossRef]

Rush, J. D.

B. J. Ainslie, K. J. Beales, D. M. Cooper, C. R. Day, J. D. Rush, Electron. Lett. 18, 842 (1982).
[CrossRef]

Saifi, M. A.

Stone, J.

Suematsu, S.

Sumi, M.

Tanaka, T.

White, K. I.

K. I. White, Electron. Lett. 18, 725 (1982).
[CrossRef]

Yamada, S.

AEU Arch. Elektron. Uebertragungstech. Electron. Commun.

K. Petermann, AEU Arch. Elektron. Uebertragungstech. Electron. Commun. 30, 337 (1976).

Appl. Opt.

Bell Syst. Tech. J.

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

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

Bell. Syst. Tech. J.

W. B. Gardner, Bell. Syst. Tech. J. 54, 457 (1975).

Electron. Lett.

K. I. White, Electron. Lett. 18, 725 (1982).
[CrossRef]

B. J. Ainslie, K. J. Beales, D. M. Cooper, C. R. Day, J. D. Rush, Electron. Lett. 18, 842 (1982).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

W. A. Gambling, H. Matsumura, C. M. Ragdae, Opt. Quantum Electron. 11, 43 (1979).
[CrossRef]

Other

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

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), pp. 377–378.

Ref. 10, Sec. 8.7.

M. Abromowitz, I. Stegun, Eds., Handbook on Mathematical Functions, Vol. 55 (Dover, New York, 1965).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Microbending losses of single-mode step-index fibers as functions of wavelength for several values of the core radius a. The relative index difference Δ is adjusted so that V = 2.4 at λ = 1.3 μm. The rms deviation from straightness is σ = 1 nm. The correlation length is (a) L c = 1 μm; (b) L c = 5 μm; (c) L c = 10 μm; (d) L c = 50 μm; (e) L c = 100 μm; (f) L c = 500 μm.

Fig. 2
Fig. 2

(a)–(f) are similar to Figs. 1(a)–(f) but apply to losses caused by random radius fluctuations.

Fig. 3
Fig. 3

(a) Microbending loss as function of wavelength for a step-index fiber with a = 5 μm, Δ = 0.003, rms deviation σ = 1 nm for several values of the correlation length L c . (b) Similar to Fig. 3(a) but applying to random radius fluctuations.

Fig. 4
Fig. 4

Ratios of loss coefficients for triangular (linear)-index and parabolic-index fibers relative to step-index fibers as functions of wavelength for several values of the correlation length L c . All fibers have the same cutoff for LP11.

Fig. 5
Fig. 5

(a) Ratio of microbending losses of graded-index fibers relative to step-index fibers as functions of the relative mode radius w/a for two different correlation lengths. All fibers have the same core radius a = 5 μm and the same mode radius w but different Δ values. The wavelength is λ = 1.5 μm. (b) Same as Fig. 5(a) but for random radius fluctuations. (c) V values for the fibers in Figs. 5(a) and (b).

Fig. 6
Fig. 6

(a) Microbending loss ratios of triangular (linear)-index (solid lines) and parabolic-index (dotted lines) fibers relative to the step-index fiber loss as functions of the relative mode radius w/alin. All fibers have the same mode radius w. The radius of the triangular fiber is alin = 5 μm, the radii of the other fibers are given by Eq. (35). Δ is nominally the same (see text). (b) Same as Fig. 6(a) but for random radius fluctuations. (c) V values for the fibers in Figs. 6(a) and (b). (d) Δ values of the fibers in Figs. 6(a) and (b).

Equations (35)

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

n ( r , φ , z ) = n 0 [ x 0 + f ( z ) , y 0 ] = n 0 ( x 0 , y 0 ) + n 0 x f ( z ) = n 0 ( r ) + n 0 r f ( z ) cos φ ,
n ( r , z ) = n 0 [ r , a + f ( z ) ] = n 0 ( r , a ) + n 0 a f ( z ) ,
ψ g = u ( r ) exp ( - i β g z ) .
ψ ν s = A ν s J ν ( J ν s r b ) cos ( ν φ ) exp ( - i β ν s z )
J ν ( j ν s ) = 0.
2 α = k 2 b 2 s F ( β g - β 1 s ) 2 [ 0 b r n 0 r J 1 ( j 1 s r b ) U ( r ) d r ] J 0 2 ( j 1 s ) 0 r U 2 ( r ) d r .
R ( u ) = σ 2 f ( z ) f ( z + u ) ,
F ( θ ) 2 = - R ( u ) cos ( θ u ) d u .
R ( u ) = σ 2 exp [ - ( u L c ) 2 ] .
F ( β g - β ν s ) 2 = π σ 2 L c exp { - [ 1 2 ( β g - β ν s ) L c ] 2 } .
2 α = 2 k 2 b 2 s F ( β g - β 0 s ) 2 × [ 0 b r n 0 a J 0 ( j 0 s r b ) U ( r ) d r ] 2 J 1 2 ( j 0 s ) 0 r U 2 ( r ) d r .
n 0 ( r ) = { n 1 [ 1 - ( r a ) g Δ ] for 0 < r < a , n 2 for r > a ,
Δ = ( n 1 - n 2 ) / n 1 .
U ( r ) = exp [ - ( r w ) 2 ] .
2 α = π σ 2 L c ( 2 n 1 k a g Δ b w ) 2 · s exp { - [ ( β g - β 1 s ) L c 2 ] 2 } × { 0 1 x g J 1 ( j 1 s a b x ) exp [ - ( a x w ) 2 ] d x } 2 J 0 2 ( j 1 s ) .
2 α = π σ 2 L c ( 2 n 1 k a Δ b w ) 2 - s exp { - [ ( β g - β 1 s ) L c 2 ] 2 } J 1 2 ( j 1 s a b ) J 0 2 ( j 1 s ) exp ( - 2 a 2 w 2 ) .
2 α = 2 π σ 2 L c ( 2 n 1 k a g Δ b w ) 2 · s exp { - [ ( β g - β 0 s ) L c 2 ] 2 } × { 0 1 x g + 1 J 0 ( j 0 s a b x ) exp [ - ( a x w ) 2 ] d x } 2 J 1 2 ( j 0 s ) ,
2 α = 2 π σ 2 L c ( 2 n 1 k a Δ b w ) 2 · s exp { - [ ( β g - β 0 s ) L c 2 ] 2 } J 0 2 ( j 0 s a b ) J 1 2 ( j 0 s ) exp ( - 2 a 2 w 2 ) .
β g = [ n 1 2 k 2 - κ 2 ] 1 / 2 .
κ 2 = 2 w 2 + 4 k 2 w 2 0 r [ n 1 2 - n 2 ( r ) ] exp ( - 2 r 2 w 2 ) d r .
1 + 2 k 2 0 r ( 2 r 2 w 2 - 1 ) n 2 ( r ) exp ( - 2 r 2 w 2 ) d r = 0.
1 Δ = 2 [ 1 + n 2 2 k 2 a 2 exp ( - 2 a 2 w 2 ) + 2 n 2 2 k 2 r a r ( r a ) g ( 2 r 2 w 2 - 1 ) exp ( - 2 r 2 w 2 ) d r ] .
0 r ( 2 r 2 w 2 - 1 ) exp ( - 2 r 2 w 2 ) d r = 0.
κ 2 = 1 a 2 + 2 w 2 ,
1 Δ = 2 [ 1 + n 2 2 k 2 a 2 exp ( - 2 a 2 w 2 ) ]
κ 2 = n 1 2 k 2 Δ w 2 a 2 [ 1 - exp ( 2 a 2 w 2 ) ] + 2 w 2 ,
1 Δ = 2 { 1 + ( n 2 k w 2 2 a ) 2 [ 1 - ( 1 + 2 a 2 w 2 ) exp ( - 2 a 2 w 2 ) ] } ,
κ 2 = 2 w 2 + 2 n 1 2 k 2 Δ [ 1 2 w a γ ( 3 2 , 2 a 2 w 2 ) + exp ( - 2 a 2 w 2 ) ] ,
1 Δ = 2 ( 1 + n 2 2 k 2 a 2 { exp ( - 2 a 2 w 2 ) + w - 3 / 2 w 3 a 3 [ γ ( 5 2 , 2 a 2 w 2 ) - γ ( 3 2 , 2 a 2 w 2 ) ] } ) .
γ ( a , x ) = 0 x exp ( - t ) t a - 1 d t .
V = n 1 k a 2 Δ ,
w = a ln V .
β ν s = [ n 2 2 k 2 - ( j ν s / b ) 2 ] 1 / 2 ,
I = 0 a r n ( r ) d r
a step a lin = 1 3 ;             a parab a lin = 2 3 .

Metrics