Abstract

A theory is presented of the propagation of Gaussian pulses in single-mode optical fibers by expanding the propagation constant in a Taylor series that includes the third derivative with respect to frequency. The light source is assumed to have a Gaussian spectral distribution whose width relative to the width of the Gaussian signal pulse is arbitrary. Formulas are derived for the spectrum of the ensemble average of the optical pulse, from which the shape of the average pulse itself is obtained by the fast Fourier transform. Also derived is an expression for the rms pulse width. The theory is applicable at all wavelengths including the vicinity of the zero first-order dispersion point.

© 1980 Optical Society of America

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References

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  1. H. G. Unger, AEU Arch. fuer Electron. und Uebertragungstech. Electron. and Commun. 31 (12), 518 (1977).
  2. D. N. Payne, W. A. Gambling, Electron. Lett. 12, 549 (1976).
    [CrossRef]
  3. F. P. Kapron, Electron. Lett. 13, 96 (1977);in Digest of Topical Meeting on Optical Fiber Communication (Optical Society of American, Washington, D.C., 1979), p. 104.
    [CrossRef]
  4. M. Miyagi, S. Nishida, Appl. Opt. 18, 678 (1979).
    [CrossRef] [PubMed]
  5. M. Miyagi, S. Nishida, Appl. Opt. 18, 2237 (1979).
    [CrossRef] [PubMed]
  6. S. H. Wemple, Appl. Opt. 18, 31 (1979).
    [CrossRef] [PubMed]
  7. D. Gloge, Electron. Lett. 15, 686 (1979).
    [CrossRef]
  8. S. Kobayashi, S. Shibata, N. Shibata, T. Izawa, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo (1977), paper B8-3.

1979 (4)

1977 (2)

H. G. Unger, AEU Arch. fuer Electron. und Uebertragungstech. Electron. and Commun. 31 (12), 518 (1977).

F. P. Kapron, Electron. Lett. 13, 96 (1977);in Digest of Topical Meeting on Optical Fiber Communication (Optical Society of American, Washington, D.C., 1979), p. 104.
[CrossRef]

1976 (1)

D. N. Payne, W. A. Gambling, Electron. Lett. 12, 549 (1976).
[CrossRef]

Gambling, W. A.

D. N. Payne, W. A. Gambling, Electron. Lett. 12, 549 (1976).
[CrossRef]

Gloge, D.

D. Gloge, Electron. Lett. 15, 686 (1979).
[CrossRef]

Izawa, T.

S. Kobayashi, S. Shibata, N. Shibata, T. Izawa, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo (1977), paper B8-3.

Kapron, F. P.

F. P. Kapron, Electron. Lett. 13, 96 (1977);in Digest of Topical Meeting on Optical Fiber Communication (Optical Society of American, Washington, D.C., 1979), p. 104.
[CrossRef]

Kobayashi, S.

S. Kobayashi, S. Shibata, N. Shibata, T. Izawa, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo (1977), paper B8-3.

Miyagi, M.

Nishida, S.

Payne, D. N.

D. N. Payne, W. A. Gambling, Electron. Lett. 12, 549 (1976).
[CrossRef]

Shibata, N.

S. Kobayashi, S. Shibata, N. Shibata, T. Izawa, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo (1977), paper B8-3.

Shibata, S.

S. Kobayashi, S. Shibata, N. Shibata, T. Izawa, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo (1977), paper B8-3.

Unger, H. G.

H. G. Unger, AEU Arch. fuer Electron. und Uebertragungstech. Electron. and Commun. 31 (12), 518 (1977).

Wemple, S. H.

AEU Arch. fuer Electron. und Uebertragungstech. Electron. and Commun. (1)

H. G. Unger, AEU Arch. fuer Electron. und Uebertragungstech. Electron. and Commun. 31 (12), 518 (1977).

Appl. Opt. (3)

Electron. Lett. (3)

D. N. Payne, W. A. Gambling, Electron. Lett. 12, 549 (1976).
[CrossRef]

F. P. Kapron, Electron. Lett. 13, 96 (1977);in Digest of Topical Meeting on Optical Fiber Communication (Optical Society of American, Washington, D.C., 1979), p. 104.
[CrossRef]

D. Gloge, Electron. Lett. 15, 686 (1979).
[CrossRef]

Other (1)

S. Kobayashi, S. Shibata, N. Shibata, T. Izawa, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo (1977), paper B8-3.

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Figures (15)

Fig. 1
Fig. 1

Gaussian input pulse.

Fig. 2
Fig. 2

Pulse is broadened by traveling through a fiber characterized by the normalized dispersion parameters B = 0, D = 3. Light source is assumed to be monochromatic, V = 0.

Fig. 3
Fig. 3

Pulse at zero first-order dispersion wavelength, D = 0, and monochromatic source, V = 0. The second-order dispersion parameter is B = 1. Changing B to B = −1 would convert the pulse tail to a precursor.

Fig. 4
Fig. 4

Compared with Fig. 3, the operating wavelength has been shifted off the zero first-order dispersion point so that now D = 3, while 5 = 1 and V = 0.

Fig. 5
Fig. 5

Operation at zero first-order dispersion wavelength, D = 0, with broader source spectrum, V = 5. It is again B = 1.

Fig. 6
Fig. 6

Similar to Fig. 5 but with D = 3.

Fig. 7
Fig. 7

More second-order dispersion with B = 5 for monochromatic source, V = 0, at zero (first-order) dispersion wavelength D = 0. Note the pronounced pulse distortion with oscillating pulse tail. Changing B = 5 to B = −5 would convert the pulse tail to a precursor.

Fig. 8
Fig. 8

Similar to Fig. 7 but with a slight shift off the zero (first-order) dispersion wavelength to make D = 7.5.

Fig. 9
Fig. 9

Similar to Fig. 8 with D = 15.

Fig. 10
Fig. 10

Similar to Fig. 8 with D = 30.

Fig. 11
Fig. 11

Similar to Fig. 7 with D = 0 but finite source spectral width so that V = 1.

Fig. 12
Fig. 12

Compared with Fig. 11 the source spectral width is further increased to V = 5.

Fig. 13
Fig. 13

Similar to Fig. 12 but off the zero (first-order) dispersion wavelength, causing D = 15.

Fig. 14
Fig. 14

Plot of the first-order dispersion parameter D as a function of wavelength for an input pulse of width 2T = 1 psec and a fiber of 1-km length.

Fig. 15
Fig. 15

Plot of the second-order dispersion parameter B as a function of wavelength for an input pulse of width 2T = 1 psec and a fiber of 1-km length.

Equations (61)

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ϕ 0 ( ω ) = 1 2 π ψ 0 ( t ) exp ( i ω t ) d t .
ψ 0 ( t ) = A ( t ) exp ( i ω 0 t ) .
ϕ 0 ( ω ) ϕ 0 * ( ω ) = 1 ( 2 π ) 2 exp [ i ( ω ω ) t ] d t × R ( t t ) exp [ i ( ω 0 ω ) ( t t ) ] d t .
A ( t ) A * ( t ) = R ( t t ) .
ϕ 0 ( ω ) ϕ 0 * ( ω ) = | ϕ ̂ 0 ( ω 0 ω ) | 2 δ ( ω ω ) ,
| ϕ ̂ 0 ( ω ω 0 ) | 2 = 1 2 π R ( u ) exp [ i ( ω 0 ω ) u ] d u .
P ( t ) = s ( t ) P 0 ( t ) ,
s ( t ) = S exp [ ( t / T ) 2 ] ,
P ( t ) = | ψ ( t ) | 2 .
ψ ( t ) = exp [ i g ( t ) ] s 1 / 2 ( t ) ψ 0 ( t ) .
ψ ( z , t ) = ϕ ( ω ) exp [ i ( ω t β z ) ] d ω ,
P ( z , t ) = ϕ ( ω ) ϕ * ( ω ) × exp { i [ ( ω ω ) t ( β β ) z ] } d ω d ω .
ϕ ( ω ) = 1 2 π s 1 / 2 ( t ) ψ 0 ( t ) exp ( i ω t ) d t = ϕ 0 ( ω ) F ( ω ω ) d ω ,
F ( ω ) = 1 2 π s 1 / 2 ( t ) exp ( i ω t ) d t = ( S 2 π ) 1 / 2 T exp [ 1 2 ( T ω ) 2 ] .
ϕ ( ω ) ϕ * ( ω ) = | ϕ ̂ 0 ( ω 0 ω ) | 2 × F ( ω ω ) F * ( ω ω ) d ω .
P ( z , t ) = | ϕ ̂ 0 ( ω 0 ω ) | 2 × | F ( ω ω ) × exp { i [ ( ω ω ) t ( β β ) z ] } d ω | 2 d ω .
β = β 0 + β ˙ 0 ( ω ω 0 ) + 1 2 β ¨ 0 ( ω ω 0 ) 2 + 1 6 β ( ω ω 0 ) 3 .
β β = { β ˙ 0 + 1 2 β ¨ 0 [ ( ω ω 0 ) + ( ω ω 0 ) ] + 1 6 β 0 [ ( ω ω 0 ) 2 + ( ω ω 0 ) 2 + ( ω ω 0 ) ( ω ω 0 ) ] } ( ω ω ) .
| ϕ ̂ 0 ( ω 0 ω ) | 2 = P 0 π 1 / 2 W exp [ ( ω ω 0 ) 2 / W 2 ] ,
P ( z , t ) = G ( z , x ) exp ( i t β ˙ 0 z T x ) d x ,
G ( z , x ) = P 0 S 2 π 3 / 2 V exp ( 1 2 x 2 ) exp ( i D x 2 ) exp ( i B x 3 ) × d η exp [ ( η / V ) 2 ] exp ( 2 i D x η ) × exp [ ξ ( ξ + x ) 2 i D x ξ ] × exp [ 3 i B x η ( x + η ) ] × exp [ 3 i B x ξ ( ξ + 2 η + x ) ] d ξ ,
V = T W ,
D = β ¨ 0 z / ( 2 T 2 ) ,
B = β 0 z / ( 6 T 3 ) = 1 3 T d D d ω .
G ( z , x ) = P 0 S 2 π 1 / 2 × exp ( x 2 / 4 ) exp ( i B x 3 / 4 ) [ 1 + 3 i B x ( 1 + V 2 ) ] 1 / 2 × exp [ D 2 x 2 ( 1 + V 2 ) 1 + 3 i B x ( 1 + V 2 ) ] ,
x = ω T .
P ( z , t ) = S P 0 [ 1 + 4 D 2 ( 1 + V 2 ) ] 1 / 2 exp { [ ( t β ˙ 0 z ) / T ] 2 1 + 4 D 2 ( 1 + V 2 ) } .
t ¯ = [ t P ( z , t ) d t ] / P ( z , t ) d t ,
σ = ( t 2 ¯ t ¯ 2 ) 1 / 2 ,
t 2 ¯ = [ t 2 P ( z , t ) d t ] / P ( z , t ) d t .
G ( z , x ) = 1 2 π P ( z , t ) exp ( i u x ) d u ,
σ = [ ( T 2 d 2 G d x 2 G 1 ) x = 0 ( i T d G d x G 1 ) x = 0 2 ] 1 / 2 .
σ = T 2 { 1 + 4 D 2 ( 1 + V 2 ) + 9 B 2 ( 1 + V 2 ) 2 } 1 / 2 ,
σ = T 2 { 1 + ( β ¨ 0 z ) 2 T 4 ( 1 + T 2 W 2 ) + 1 4 ( β 0 z ) 2 T 6 ( 1 + T 2 W 2 ) 2 ) 1 / 2 .
σ 0 = T / 2 .
σ σ 0 = β ¨ 0 z T 2 ( 1 + T 2 W 2 ) 1 / 2 .
σ σ 0 = β ¨ 0 z T W .
σ σ 0 = β 0 z 2 T 3 ( 1 + T 2 W 2 ) ,
σ σ 0 = β 0 z 2 T W 2 .
σ 0 = R σ ,
σ 0 = ( 0.5 R β ¨ 0 z ) 1 / 2 .
σ 0 = R β ¨ 0 W z / 2 .
σ 0 = ( R β 0 z / 2 5 / 2 ) 1 / 3 ,
σ 0 = R β 0 W 2 z / 2 3 / 2 ,
β ˙ = d β d ω = τ / z .
β ¨ = d 2 β d ω 2 = λ 2 2 π z c d τ d λ ,
β = d 3 β d ω 3 = λ 2 ( 2 π c ) 2 z ( λ 2 d 2 τ d λ 2 + 2 λ d τ d λ ) .
β = n k = ( 2 π n ) / λ .
β ˙ = 1 c ( n λ d n d λ ) ,
β ¨ = λ 3 2 π c 2 d 2 n d λ 2 ,
β = λ 2 ( 2 π ) 2 c 3 ( 3 λ 2 d 2 n d λ 2 + λ 3 d 3 n d λ 3 ) .
D = z λ 3 4 π c 2 T 2 d 2 n d λ 2 ,
B = z λ 2 24 π 2 c 3 T 3 ( 3 λ 2 d 2 n d λ 2 + λ 3 d 3 n d λ 3 ) = λ 2 6 π c T d D d λ .
n 2 1 = j = 1 N λ 2 B j λ 2 λ j 2 .
d n d λ = λ n j = 1 N λ j 2 B ( λ 2 λ j 2 ) 2 ,
d 2 n d λ 2 = 1 n j = 1 N λ i 2 ( 3 λ 2 + λ j 2 ) B j ( λ 2 λ j 2 ) 3 1 n ( d n d λ ) 2 ,
d 3 n d λ 3 = 12 λ n j = 1 N λ j 2 ( λ 2 + λ j 2 ) B j ( λ 2 λ j 2 ) 4 3 n d n d λ d 2 n d λ 2 .
B 1 = 0.6961663 λ 1 2 = 0.004679148 μ m 2 B 2 = 0.4079426 λ 2 2 = 0.01351206 μ m 2 B 3 = 0.8974994 λ 3 2 = 97.934002 μ m 2 } .
2 D = 3 B ( 1 + V 2 ) 1 / 2 .
D = d D d λ Δ λ ,
( Δ λ ) s / λ = λ ( 1 + V 2 ) 1 / 2 4 π c T .

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