Abstract

The WKB analysis is extended to take into account the discontinuity in index profile shape that occurs at the core-cladding boundary of graded-index optical waveguides. The presence of the cladding introduces a large correction to the delay times of the highest 5% of the guided modes and greatly increases the predicted rms pulse broadening of a parabolic profile fiber. Using a model of differential mode attenuation, it is shown that these highest order modes are very susceptible to attenuation. A small degree of waveguide irregularity restores the low pulse broadening originally predicted for a parabolic fiber.

© 1977 Optical Society of America

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References

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  1. D. Gloge, U.S. Patent3,785,718.
  2. K. Okamoto, T. Okoshi, IEEE Trans. Microwave Theory Tech. MTT-24, 416 (1976).
    [CrossRef]
  3. K. Okamoto, J. Okoshi, IEEE Trans. Microwave Theory Tech.MMT-24 (to be published).
  4. D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).
  5. R. Olshansky, D. B. Keck, in Digest of Topical Meeting on Optical Fiber Transmission (Optical Society of America, Washington, D.C., 1975).
  6. R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
    [CrossRef] [PubMed]
  7. R. Olshansky, D. A. Nolan, Appl. Opt. 15, 1045 (1976).
    [CrossRef] [PubMed]
  8. See, for example, L. G. Cohen, Appl. Opt. 15, 1808 (1976).
    [CrossRef] [PubMed]
  9. M. Hashimoto, IEEE Trans. Microwave Theory Tech. MTT-24, 404 (1976).
    [CrossRef]

1976 (5)

1973 (1)

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

Cohen, L. G.

Gloge, D.

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

D. Gloge, U.S. Patent3,785,718.

Hashimoto, M.

M. Hashimoto, IEEE Trans. Microwave Theory Tech. MTT-24, 404 (1976).
[CrossRef]

Keck, D. B.

R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
[CrossRef] [PubMed]

R. Olshansky, D. B. Keck, in Digest of Topical Meeting on Optical Fiber Transmission (Optical Society of America, Washington, D.C., 1975).

Marcatili, E. A. J.

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

Nolan, D. A.

Okamoto, K.

K. Okamoto, T. Okoshi, IEEE Trans. Microwave Theory Tech. MTT-24, 416 (1976).
[CrossRef]

K. Okamoto, J. Okoshi, IEEE Trans. Microwave Theory Tech.MMT-24 (to be published).

Okoshi, J.

K. Okamoto, J. Okoshi, IEEE Trans. Microwave Theory Tech.MMT-24 (to be published).

Okoshi, T.

K. Okamoto, T. Okoshi, IEEE Trans. Microwave Theory Tech. MTT-24, 416 (1976).
[CrossRef]

Olshansky, R.

R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
[CrossRef] [PubMed]

R. Olshansky, D. A. Nolan, Appl. Opt. 15, 1045 (1976).
[CrossRef] [PubMed]

R. Olshansky, D. B. Keck, in Digest of Topical Meeting on Optical Fiber Transmission (Optical Society of America, Washington, D.C., 1975).

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

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

IEEE Trans. Microwave Theory Tech. (2)

K. Okamoto, T. Okoshi, IEEE Trans. Microwave Theory Tech. MTT-24, 416 (1976).
[CrossRef]

M. Hashimoto, IEEE Trans. Microwave Theory Tech. MTT-24, 404 (1976).
[CrossRef]

Other (3)

K. Okamoto, J. Okoshi, IEEE Trans. Microwave Theory Tech.MMT-24 (to be published).

R. Olshansky, D. B. Keck, in Digest of Topical Meeting on Optical Fiber Transmission (Optical Society of America, Washington, D.C., 1975).

D. Gloge, U.S. Patent3,785,718.

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Figures (2)

Fig. 1
Fig. 1

Delay times for the ν = 0 and ν = 1 modes of a parabolic fiber are shown as a function of the principal mode number. The solid curve includes the effect of the uniform cladding, while the dashed curve shows the previous result obtained when this correction was neglected.

Fig. 2
Fig. 2

The rms pulse width is shown as a function of the degree of random diameter variation. A small amount of perturbation causes severe loss to the high order modes and sharply decreases the rms pulse width.

Equations (36)

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n 2 ( x ) = n 1 2 [ 1 - 2 Δ f ( x ) ] x 1 , n 2 ( x ) = n 2 2 = n 1 2 ( 1 - 2 Δ ) x 1 ,
W = a ( n 1 2 k 2 - β 2 ) 1 / 2 ,
Q = a ( β 2 - n 2 2 k 2 ) 1 / 2 ,
V = n 1 k a ( 2 Δ ) 1 / 2 ,
W 2 + Q 2 = V 2 .
R ( x ) = 1 ( x u ¯ ) 1 / 2 exp ( - x x 1 d x u ¯ )             0 x < x 1 ,
R ( x ) = 2 ( x u ) 1 / 2 cos ( x 1 x d x u - π / 4 ) x 1 < x < x 2 ,
R ( x ) = 1 ( x u ¯ ) 1 / 2 [ sin ϕ 1 exp ( - x 2 x d x u ¯ ) + 2 cos ϕ 1 exp ( x 2 x d x u ¯ ) ]             x 2 < x 1.
R ( x ) = 1 ( x v ) 1 / 2 exp ( - 1 x d x v ) .
u 2 = - u ¯ 2 = W 2 - V 2 f ( x ) - ν 2 / x 2 , v 2 = Q 2 + ν 2 / x 2 ,
ϕ 1 = x 1 x 2 d x u .
u ( x 1 ) = u ( x 2 ) = 0
cot ϕ 1 = δ ,
δ = 4 - 2 exp ( - 2 ϕ 2 ) ,
= V 2 f ( 1 ) ( Q 2 + ν 2 ) - 3 / 2 / 4 ,
ϕ 2 = x 2 1 d x v .
ϕ 1 = ( μ + 1 2 ) π ,
τ μ ν = 1 c d β μ ν d k .
ϕ 1 k + ϕ 1 β β k + 1 1 + δ 2 ( δ k + δ β β k ) = 0.
τ μ ν = τ μ ν C ,
τ μ ν = - 1 c ϕ 1 k / ϕ 1 β ,
C = ( 1 + 1 1 + δ 2 δ k / ϕ 1 k ) / ( 1 + 1 1 + δ 2 δ β / ϕ 1 β ) .
k ϕ 1 k = n 1 2 k 2 a 2 x 1 x 2 d x u [ 1 - 2 Δ f ( x ) ] ,
ϕ 1 β = - β a 2 x 1 x 2 d x u ,
k δ δ k = 2 n 2 2 k 2 a 2 x 2 1 d x v + 2 2 - [ 2 + 3 n 2 2 k 2 a 2 / ( Q 2 + ν 2 ) ] ,
1 δ δ β = - 2 β a 2 x 2 1 d x v - 2 2 - ( 3 β a 2 ) / Q 2 + ν 2 ) .
f ( x ) = x 2 ,
ϕ 1 = ( - ν 2 + W 2 4 V ) π ,
ϕ 2 = 1 2 ( ν 2 + Q 2 ) 1 / 2 - W 2 4 V log | 2 V ( ν 2 + Q 2 ) 1 / 2 + V 2 + Q 2 D | - ν 2 log [ 2 ν ( ν 2 + Q 2 ) 1 / 2 + 2 ν 2 - W 2 D ] ,
x 1 x 2 d x u = π 2 V ,
x 1 x 2 d x u x 2 = W 2 4 V 3 ,
x 2 1 d x v = 1 2 V log [ V 2 + Q 2 + 2 V ( ν 2 + Q 2 ) 1 / 2 D ] ,
D = ( W 4 - 4 ν 2 V 2 ) 1 / 2 .
τ μ ν = n 1 2 c ( n 1 k / β + β / n 1 k ) .
β ° = n 1 k [ 1 - 2 Δ ( m / M ) ] 1 / 2 ,
τ μ ν = n 1 c [ 1 + Δ 2 2 ( m / M ) 2 ] + 0 ( Δ 3 ) .

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