Abstract

This paper reports some theoretical and experimental investigations on the propagation of a pulse in long step-index optical fibers where mode conversion is present. We derive simple analytical expressions for the frequency response and the radiation pattern as a function of the fiber’s length provided the launching conditions are known. A comparison with experimental observations made on a 3-km long Corning fiber shows reasonable agreement. We conclude that the approximations used to obtain the simple analytical solutions are satisfactory for predicting the transmission characteristics of the fiber.

© 1976 Optical Society of America

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References

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  1. S. D. Personnick, Bell Syst. Tech. J. 50, 843 (1971).
  2. D. Gloge, Bell Syst. Tech. J. 51, 1767 (1972).
  3. D. Gloge, Bell Syst. Tech. J. 52, 801 (1973).
  4. R. Olshansky, Appl. Opt. 14, 935 (1975).
    [PubMed]
  5. D. B. Keck, Appl. Opt. 13, 1882 (1974).
    [CrossRef] [PubMed]
  6. L. Jeunhomme, J. P. Pocholle, Opt. Commun. 12, 89 (1974).
    [CrossRef]
  7. L. G. Cohen, in Digest of OSA Topical Meeting on Optical Fiber Transmission, Williamsburg (Optical Society of America, Washington, D.C., 1975).
  8. W. A. Gambling, D. N. Payne, H. Matsumura, Appl. Opt. 14, 1538 (1975).
    [CrossRef] [PubMed]
  9. L. Jeunhomme, J. P. Pocholle, Electron. Lett. 11, 425 (1975).
    [CrossRef]
  10. L. Jeunhomme, J. P. Pocholle, Electron. Lett. 12, 63 (1976).
    [CrossRef]
  11. W. B. Gardner, Bell Syst. Tech. J. 54, 457 (1975).

1976 (1)

L. Jeunhomme, J. P. Pocholle, Electron. Lett. 12, 63 (1976).
[CrossRef]

1975 (4)

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

R. Olshansky, Appl. Opt. 14, 935 (1975).
[PubMed]

W. A. Gambling, D. N. Payne, H. Matsumura, Appl. Opt. 14, 1538 (1975).
[CrossRef] [PubMed]

L. Jeunhomme, J. P. Pocholle, Electron. Lett. 11, 425 (1975).
[CrossRef]

1974 (2)

D. B. Keck, Appl. Opt. 13, 1882 (1974).
[CrossRef] [PubMed]

L. Jeunhomme, J. P. Pocholle, Opt. Commun. 12, 89 (1974).
[CrossRef]

1973 (1)

D. Gloge, Bell Syst. Tech. J. 52, 801 (1973).

1972 (1)

D. Gloge, Bell Syst. Tech. J. 51, 1767 (1972).

1971 (1)

S. D. Personnick, Bell Syst. Tech. J. 50, 843 (1971).

Cohen, L. G.

L. G. Cohen, in Digest of OSA Topical Meeting on Optical Fiber Transmission, Williamsburg (Optical Society of America, Washington, D.C., 1975).

Gambling, W. A.

Gardner, W. B.

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

Gloge, D.

D. Gloge, Bell Syst. Tech. J. 52, 801 (1973).

D. Gloge, Bell Syst. Tech. J. 51, 1767 (1972).

Jeunhomme, L.

L. Jeunhomme, J. P. Pocholle, Electron. Lett. 12, 63 (1976).
[CrossRef]

L. Jeunhomme, J. P. Pocholle, Electron. Lett. 11, 425 (1975).
[CrossRef]

L. Jeunhomme, J. P. Pocholle, Opt. Commun. 12, 89 (1974).
[CrossRef]

Keck, D. B.

Matsumura, H.

Olshansky, R.

Payne, D. N.

Personnick, S. D.

S. D. Personnick, Bell Syst. Tech. J. 50, 843 (1971).

Pocholle, J. P.

L. Jeunhomme, J. P. Pocholle, Electron. Lett. 12, 63 (1976).
[CrossRef]

L. Jeunhomme, J. P. Pocholle, Electron. Lett. 11, 425 (1975).
[CrossRef]

L. Jeunhomme, J. P. Pocholle, Opt. Commun. 12, 89 (1974).
[CrossRef]

Appl. Opt. (3)

Bell Syst. Tech. J. (4)

S. D. Personnick, Bell Syst. Tech. J. 50, 843 (1971).

D. Gloge, Bell Syst. Tech. J. 51, 1767 (1972).

D. Gloge, Bell Syst. Tech. J. 52, 801 (1973).

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

Electron. Lett. (2)

L. Jeunhomme, J. P. Pocholle, Electron. Lett. 11, 425 (1975).
[CrossRef]

L. Jeunhomme, J. P. Pocholle, Electron. Lett. 12, 63 (1976).
[CrossRef]

Opt. Commun. (1)

L. Jeunhomme, J. P. Pocholle, Opt. Commun. 12, 89 (1974).
[CrossRef]

Other (1)

L. G. Cohen, in Digest of OSA Topical Meeting on Optical Fiber Transmission, Williamsburg (Optical Society of America, Washington, D.C., 1975).

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Figures (9)

Fig. 1
Fig. 1

Differential attenuation as a function of propagation angle. Present assumption and measurements on the CGW-CGE 68 fiber.

Fig. 2
Fig. 2

Far-field radiation pattern of the fundamental θ mode for different values of V.

Fig. 3
Fig. 3

Steady-state loss γ1 as a function of A (a) and C0 (b).

Fig. 4
Fig. 4

Effective angular width of the steady-state radiation pattern as a function of V.

Fig. 5
Fig. 5

Electrical 6-dB bandwidth as a function of length for different launching conditions and given values of γ and θeff(∞). The dashed lines of slope 1 indicate the values obtained without mode conversion.

Fig. 6
Fig. 6

Far-field radiation patterns of the 3-km long CGW-CGE 68 fiber: (a) He–Ne laser (b) L.E.D.; (c) fundamental quasi-mode Q1(θ) with V = 9.4 (same value of θeff).

Fig. 7
Fig. 7

Effective width of the far-field radiation pattern as a function of length for CGW-CGE 68 fiber with different launching condition. Theory and experiment.

Fig. 8
Fig. 8

Electrical 6-dB bandwidth of the CGW-CGE 68 fiber as a function of length. Theory and experiment.

Fig. 9
Fig. 9

Frequency response and group delay for the 3-km long CGW-CGE 68 fiber. Theory and experiment.

Tables (1)

Tables Icon

Table I Spectral Attenuation Dependence on Launching Conditions

Equations (32)

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β = k 1 cos θ ,
k 2 = k 1 cos θ m cos θ M = ( n 2 ) / ( n 1 ) .
Δ θ = λ / ( 4 a n 1 ) ,
2 P ( θ , z ) θ 2 + 1 θ P ( θ , z ) θ - 1 C 0 ( Δ θ ) 2 P ( θ , z ) z - α ( θ ) C 0 ( Δ θ ) 2 P ( θ , z ) = 0 ,
P ( θ , z ) = Q ( θ ) exp ( - γ z ) .
d 2 Q d θ 2 + 1 θ d Q d θ + γ - α ( θ ) C 0 ( Δ θ ) 2 Q ( θ ) = 0.
d 2 E z d r 2 + 1 r dEz d r + [ ( 2 π λ ) 2 n 2 ( r ) - β 2 - ν 2 r 2 ] E z = 0 ,
θ r ,             Q ( θ ) E z ( r ) , ( 2 π λ ) 2 n 2 ( r ) - α ( θ ) C 0 ( Δ θ ) 2 ,             β 2 - γ C 0 ( Δ θ ) 2 .
Q ( θ ) and dQ d θ ( θ )
α ( θ ) = 0 0 θ < θ M α ( θ ) = A θ θ M ( A 0 ) { .
u 2 = a 2 [ ( 2 π λ ) 2 n 2 ( 0 ) - β 2 ] U 2 = γ C 0 ( θ M Δ θ ) 2 ,
w 2 = a 2 [ β 2 - ( 2 π λ ) 2 n 2 ( a ) ] W 2 = A - γ C 0 ( θ M Δ θ ) 2 ,
v 2 = u 2 + w 2 V 2 = U 2 + W 2 = A C 0 ( θ M Δ θ ) 2 .
U n J 1 ( U n ) J 0 ( U n ) = - W n K 1 ( W n ) K 0 ( W n ) U n 2 + W n 2 = V 2 } .
Q n ( θ ) = J 0 ( U n θ θ M ) , 0 θ θ M ,
Q n ( θ ) = J 0 ( U n ) K 0 ( W n ) K 0 ( W n θ θ M ) , θ θ M .
γ 1 = U 1 2 C 0 ( Δ θ θ M ) 2 = ( U 1 V ) 2 A .
E ( z ) = 0 π / 2 2 π sin θ P ( θ , z ) d θ .
0 θ eff ( z ) 2 π sin θ P ( θ , z ) d θ 0 π / 2 2 π sin θ P ( θ , z ) d θ = 0.9.
θ eff 2 ( z ) θ eff 2 ( ) = θ eff 2 ( 0 ) / θ eff 2 ( ) + t h γ z 1 + θ eff 2 ( 0 ) θ eff 2 ( ) t h γ z .
γ = ( γ n - γ n - 1 ) / 2 = γ 1 .
E ( z ) E ( 0 ) = 1 c h γ 1 z + θ eff 2 ( 0 ) θ eff 2 ( ) s h γ 1 z
E ( z ) E ( 0 ) = θ eff 2 ( ) θ eff 2 ( 0 ) 1 c h γ 1 z + θ eff 2 ( 0 ) θ eff 2 ( ) s h γ 1 z + [ 1 - θ eff 2 ( ) θ eff 2 ( 0 ) ] exp ( - A z )
P z ( θ , z , t ) + t g ( θ ) P r ( θ , z , t )
t g ( θ ) n 1 c ( 1 + θ 2 2 ) ,
P ( θ , z , ω ) = - + exp ( - i ω t ) P ( θ , z , t ) d t ,
2 P ( θ , z , ω ) θ 2 + 1 θ P ( θ , z , ω ) θ - 1 C 0 ( Δ θ ) 2 P ( θ , z , ω ) z - α ( θ ) C 0 ( Δ θ ) 2 P ( θ , z , ω ) - i ω C 0 ( Δ θ ) 2 n 1 c P ( θ , z , ω ) - i ω θ 2 C 0 ( Δ θ ) 2 n 1 2 c P ( θ , z , ω ) = 0.
i ω C 0 ( Δ θ ) 2 n 1 c P
H ( z , ω ) = c h γ z + θ eff 2 ( 0 ) θ eff 2 ( ) s h γ z c h σ γ z + σ θ eff 2 ( 0 ) θ eff 2 ( ) s h σ γ z ,
σ = [ 1 + i n 1 2 c ω θ eff 2 ( ) γ ln ( 10 ) ] 1 / 2 .
C 0 ( Δ θ ) 2 = 7.9 × 10 - 7 r d 2 m - 1 C 0 = 0.115 m - 1 ,
γ 1 = ( U 1 V ) 2 A = 1 dB / km .

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