Abstract

Pulse spreading in a single-mode optical fiber is discussed taking into account the third-order dispersion term of the waveguide when an optical source with a finite spectral width is modulated by a Gaussian-shaped pulse. Explicit forms for group velocity and pulse width are obtained taking into account both the optical source bandwidth and modulation bandwidth, and these are compared with those obtained by a simple theory. A condition is also made clear how small the second-order dispersion term has to be before the third-order dispersion term becomes important.

© 1979 Optical Society of America

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References

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  1. D. N. Payne, W. A. Gambling, Electron. Lett. 11, 176 (1975).
    [CrossRef]
  2. D. N. Payne, A. H. Hartog, Electron. Lett. 13, 627 (1977).
    [CrossRef]
  3. M. Miyagi, S. Nishida, Appl. Opt. 18, 678 (1979).
    [CrossRef] [PubMed]
  4. F. P. Kapron, Electron. Lett. 13, 96 (1977).
    [CrossRef]
  5. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 446.
  6. S. Kawakami (Tohoku University), private communication.
  7. K. Okoshi, K. Okamoto, K. Hotate, Fundamentals of Optical Fibers (Ohm, Tokyo, 1977), p. 80 (in Japanese).

1979 (1)

1977 (2)

F. P. Kapron, Electron. Lett. 13, 96 (1977).
[CrossRef]

D. N. Payne, A. H. Hartog, Electron. Lett. 13, 627 (1977).
[CrossRef]

1975 (1)

D. N. Payne, W. A. Gambling, Electron. Lett. 11, 176 (1975).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 446.

Gambling, W. A.

D. N. Payne, W. A. Gambling, Electron. Lett. 11, 176 (1975).
[CrossRef]

Hartog, A. H.

D. N. Payne, A. H. Hartog, Electron. Lett. 13, 627 (1977).
[CrossRef]

Hotate, K.

K. Okoshi, K. Okamoto, K. Hotate, Fundamentals of Optical Fibers (Ohm, Tokyo, 1977), p. 80 (in Japanese).

Kapron, F. P.

F. P. Kapron, Electron. Lett. 13, 96 (1977).
[CrossRef]

Kawakami, S.

S. Kawakami (Tohoku University), private communication.

Miyagi, M.

Nishida, S.

Okamoto, K.

K. Okoshi, K. Okamoto, K. Hotate, Fundamentals of Optical Fibers (Ohm, Tokyo, 1977), p. 80 (in Japanese).

Okoshi, K.

K. Okoshi, K. Okamoto, K. Hotate, Fundamentals of Optical Fibers (Ohm, Tokyo, 1977), p. 80 (in Japanese).

Payne, D. N.

D. N. Payne, A. H. Hartog, Electron. Lett. 13, 627 (1977).
[CrossRef]

D. N. Payne, W. A. Gambling, Electron. Lett. 11, 176 (1975).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 446.

Appl. Opt. (1)

Electron. Lett. (3)

F. P. Kapron, Electron. Lett. 13, 96 (1977).
[CrossRef]

D. N. Payne, W. A. Gambling, Electron. Lett. 11, 176 (1975).
[CrossRef]

D. N. Payne, A. H. Hartog, Electron. Lett. 13, 627 (1977).
[CrossRef]

Other (3)

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 446.

S. Kawakami (Tohoku University), private communication.

K. Okoshi, K. Okamoto, K. Hotate, Fundamentals of Optical Fibers (Ohm, Tokyo, 1977), p. 80 (in Japanese).

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Figures (2)

Fig. 1
Fig. 1

A model system for analysis.

Fig. 2
Fig. 2

Waveforms of pulses for several values of τ0Δω and B. For a comparison a pulse shape for τ0Δω = B = 0 is also shown by a dashed line.

Tables (2)

Tables Icon

Table I Values of B for Fused Silica Fiber at Wavelength near 1.3 μm

Tables Icon

Table II Relative Increment of Pulse Width Δτ/τ0

Equations (36)

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β ( ω ) = β ( ω 0 ) + β ( ω 0 ) ( ω - ω 0 ) + ½ β ( ω 0 ) ( ω - ω 0 ) 2 + 1 6 β ( ω 0 ) ( ω - ω 0 ) 3 ,
r ( z , t ) = - R ( ω ) exp j [ ω t - β ( ω ) z ] d ω ,
R ( ω ) = - N ( u ) S ( ω - u ) d u .
r ( z , t ) 2 = - N ( ω ) N * ( μ ) S ( ω - ω ) S * ( μ - μ ) × exp j { ( ω - μ ) t - [ β ( ω ) - β ( μ ) ] z } d ω d ω d μ d μ ,
N ( ω ) N * ( μ ) = P ( ω ) δ ( ω - μ ) ,
r ( z , t ) 2 = - P ( ω ) S ( ω - ω ) S * ( μ - ω ) × exp j { ( ω - μ ) t - [ β ( ω ) - β ( μ ) ] z } d ω d ω d μ ,
P ( ω ) = 2 Δ ω ( π ) 1 / 2 exp [ - ( ω - ω 0 ) 2 / ( Δ ω 2 ) 2 ] .
S ( ω ) = ( 2 ) 1 / 2 τ 0 4 ( π ) 1 / 2 exp ( - τ 0 2 8 ω 2 ) .
r ( z , t ) 2 = 2 Δ ω ( π ) 1 / 2 - exp { - [ ( 2 Δ ω ) 2 + ( τ 0 2 ) 2 ] ( ω - ω 0 ) 2 } × I ( ω ) I * ( ω ) d ω ,
I ( ω ) = 1 ( π ) 1 / 2 - exp { [ - τ 0 ( 2 ) 1 / 2 ( ω - ω 0 ) + j A ] u - ( 1 + j C ) u 2 - j 1 3 B u 3 } d u ,
A = 4 ( 2 )  1 / 2 τ 0 [ t - β ( ω 0 ) z ] ,
B = 16 ( 2 ) 1 / 2 τ 0 3 β ( ω 0 ) z ,
C = 4 τ 0 2 β ( ω 0 ) z .
r ( z , t ) 2 = 8 ( π ) 1 / 2 Δ ω B - 2 / 3 exp [ 2 ( 2 - 3 A B - 6 C 2 ) / 3 B 2 ] × - A i { [ ( 1 + j C ) 2 - ( A - j ( 2 ) 1 / 2 τ 0 ω ) B ] B - 4 / 3 } 2 × exp { - [ ( τ 0 2 ) 2 + ( 2 Δ ω ) 2 ] ω 2 - ( 2 ) 1 / 2 C B τ 0 ω } d ω ,
r ( z , t ) 2 = 4 π B - 2 / 3 exp [ 2 ( 2 - 3 A B - 6 C 2 ) / 3 B 2 ] × A i { [ ( 1 + j C ) 2 - A B ] B - 4 / 3 } 2 .
r ( z , t ) 2 = 1 ( T ) 1 / 2 exp ( - 1 2 T A 2 ) ,
T = 1 + [ 1 + ( τ 0 Δ ω 4 ) 2 ] C 2 .
τ = τ 0 { 1 + [ 1 + ( τ 0 Δ ω 4 ) 2 ] ( 2 τ 0 ) 4 β ( ω 0 ) 2 z 2 } 1 / 2 ,
τ = β ( ω 0 ) Δ ω z
τ = [ β ( ω 0 + 1 2 Δ ω ) - β ( ω 0 - 1 2 Δ ω ) ] z = β ( ω 0 ) Δ ω z .
r ( z , t ) 2 = 1 ( T ) 1 / 2 exp [ - A 2 2 T + Q 24 ( 1 + C 2 ) 3 A B ] ,
Q = 12 ( 1 - C 4 ) - 2 ( 1 - 3 C 2 ) A 2 + 3 8 ( τ 0 Δ ω ) 2 1 T [ 1 - 8 C 2 + 7 C 4 + C 2 ( 3 - C 2 ) A 2 ] - 3 16 ( τ 0 Δ ω ) 4 C 2 T × [ 1 + 2 C 2 - C 4 - ( 1 - 3 C 2 ) A 2 ] - 1 32 ( τ 0 Δ ω ) 6 C 4 T ( 3 - C 2 ) A 2 .
r ( z , t ) 2 = exp { - 1 2 A 2 + 1 24 [ 2 ( 6 - A 2 ) + 3 8 ( τ 0 Δ ω ) 2 ] A B } .
1 v g = 1 v g 0 + 2 [ 1 + 1 32 ( τ 0 Δ ω ) 2 ] β ( ω 0 ) τ 0 2 ,
1 v g = 1 v g 0 + ( Δ ω 4 ) 2 β ( ω 0 ) ,
1 v g = 1 v g 0 + 2 ( Δ ω 4 ) 2 β ( ω 0 ) ,
1 v g = β ( ω ) = β ( ω 0 ) + β ( ω 0 ) ( ω - ω 0 ) + 1 2 β ( ω 0 ) ( ω - ω 0 ) 2 ,
r ( z , t ) 2 = exp ( - F ) ,
F = 1 2 A 2 - 1 12 [ 6 - A 2 + 3 16 ( Δ ω τ 0 ) 2 ] A B + 1 32 { [ A 2 - 8 - 3 8 ( Δ ω τ 0 ) 2 - 1 128 ( Δ ω τ 0 ) 4 ] A 2 + 1 2 ( Δ ω τ 0 ) 2 + 3 256 ( Δ ω τ 0 ) 4 } B 2 .
τ = τ 0 { 1 + 1 72 [ 5 + 9 ( Δ ω τ 0 4 ) 4 + 9 2 ( Δ ω τ 0 4 ) 4 ] B 2 } .
Δ τ τ 0 ) C = 8 τ 0 4 [ 1 + ( τ 0 Δ ω 4 ) 2 ] β ( ω 0 ) 2 z 2 ,
Δ τ τ 0 ) B = 16 9 τ 0 6 [ 5 + 9 ( τ 0 Δ ω 4 ) 2 + 9 2 ( τ 0 Δ ω 4 ) 4 ] β ( ω 0 ) 2 z 2 ,
Δ τ τ 0 ) C Δ τ τ 0 ) B ,
β ( ω 0 ) 2 10 9 · 1 + 9 5 ( τ 0 Δ ω 4 ) 2 + 9 10 ( τ 0 Δ ω 4 ) 4 1 + ( τ 0 Δ ω 4 ) 2 [ β ( ω 0 ) τ 0 ] 2 .
β ( ω 0 ) | ( 10 9 ) 1 / 2 | β ( ω 0 ) τ 0 | 1 τ 0 | β ( ω 0 ) | ;             Δ ω τ 0 1 ,
β ( ω 0 ) Δ ω 4 | β ( ω 0 ) | ;             Δ ω τ 0 1.

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