Abstract

The theory of pulse distortion in single-mode fibers is extended to include laser sources that suffer a linear wavelength sweep (chirp) during the duration of the pulse. The transmitted pulse is expressed as a Fourier integral whose spectral function is given by an analytical expression in closed form. The rms width of the transmitted pulse is also expressed in closed form. Numerical examples illustrate the influence of the chirp on the shape and rms width of the pulse. A somewhat paradoxical situation exists. A given input pulse can be made arbitrarily short by a sufficiently large amount of chirping, and, after a given fiber length, this chirped pulse returns to its original width. But at this particular distance an unchirped pulse would be only 2 times longer. Thus chirping can improve the rate of data transmission by only 40%.

© 1981 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Marcuse, Appl. Opt. 19, 1653 (1980).
    [CrossRef] [PubMed]
  2. D. Marcuse, Appl. Opt. 20, 2969 (1981).
    [CrossRef] [PubMed]
  3. J. V. Wright, B. P. Nelson, Electron. Lett. 13, (1977).
    [CrossRef]
  4. D. L. Franzen, G. W. Day, “Measurement of Optical Fiber Bandwidth in the Time Domain,” Natl. Bur. Stand. U.S. Tech. Note 1019 (Feb.1980).
  5. T. Suzuki, Electron. Commun. Jpn. 59-C, 117 (1981).
  6. D. Grishkowsky, Appl. Phys. Lett. 25, 566 (1974).
    [CrossRef]
  7. M. M. T. Loy, Appl. Phys. Lett. 26, 99 (1975).
    [CrossRef]
  8. D. Grischkowsky, M. M. T. Loy, Appl. Phys. Lett. 26, 156 (1975).
    [CrossRef]
  9. B. Steverding, Appl. Phys. Lett. 30, 231 (1977).
    [CrossRef]
  10. D. Marcuse, C. Lin, IEEE J. Quantum Electron QE-17, 869 (1981).
    [CrossRef]
  11. Since only average pulses are being considered in this paper, conclusions about error rates cannot immediately be drawn.

1981

T. Suzuki, Electron. Commun. Jpn. 59-C, 117 (1981).

D. Marcuse, C. Lin, IEEE J. Quantum Electron QE-17, 869 (1981).
[CrossRef]

D. Marcuse, Appl. Opt. 20, 2969 (1981).
[CrossRef] [PubMed]

1980

D. Marcuse, Appl. Opt. 19, 1653 (1980).
[CrossRef] [PubMed]

D. L. Franzen, G. W. Day, “Measurement of Optical Fiber Bandwidth in the Time Domain,” Natl. Bur. Stand. U.S. Tech. Note 1019 (Feb.1980).

1977

J. V. Wright, B. P. Nelson, Electron. Lett. 13, (1977).
[CrossRef]

B. Steverding, Appl. Phys. Lett. 30, 231 (1977).
[CrossRef]

1975

M. M. T. Loy, Appl. Phys. Lett. 26, 99 (1975).
[CrossRef]

D. Grischkowsky, M. M. T. Loy, Appl. Phys. Lett. 26, 156 (1975).
[CrossRef]

1974

D. Grishkowsky, Appl. Phys. Lett. 25, 566 (1974).
[CrossRef]

Day, G. W.

D. L. Franzen, G. W. Day, “Measurement of Optical Fiber Bandwidth in the Time Domain,” Natl. Bur. Stand. U.S. Tech. Note 1019 (Feb.1980).

Franzen, D. L.

D. L. Franzen, G. W. Day, “Measurement of Optical Fiber Bandwidth in the Time Domain,” Natl. Bur. Stand. U.S. Tech. Note 1019 (Feb.1980).

Grischkowsky, D.

D. Grischkowsky, M. M. T. Loy, Appl. Phys. Lett. 26, 156 (1975).
[CrossRef]

Grishkowsky, D.

D. Grishkowsky, Appl. Phys. Lett. 25, 566 (1974).
[CrossRef]

Lin, C.

D. Marcuse, C. Lin, IEEE J. Quantum Electron QE-17, 869 (1981).
[CrossRef]

Loy, M. M. T.

D. Grischkowsky, M. M. T. Loy, Appl. Phys. Lett. 26, 156 (1975).
[CrossRef]

M. M. T. Loy, Appl. Phys. Lett. 26, 99 (1975).
[CrossRef]

Marcuse, D.

Nelson, B. P.

J. V. Wright, B. P. Nelson, Electron. Lett. 13, (1977).
[CrossRef]

Steverding, B.

B. Steverding, Appl. Phys. Lett. 30, 231 (1977).
[CrossRef]

Suzuki, T.

T. Suzuki, Electron. Commun. Jpn. 59-C, 117 (1981).

Wright, J. V.

J. V. Wright, B. P. Nelson, Electron. Lett. 13, (1977).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

D. Grishkowsky, Appl. Phys. Lett. 25, 566 (1974).
[CrossRef]

M. M. T. Loy, Appl. Phys. Lett. 26, 99 (1975).
[CrossRef]

D. Grischkowsky, M. M. T. Loy, Appl. Phys. Lett. 26, 156 (1975).
[CrossRef]

B. Steverding, Appl. Phys. Lett. 30, 231 (1977).
[CrossRef]

Electron. Commun. Jpn.

T. Suzuki, Electron. Commun. Jpn. 59-C, 117 (1981).

Electron. Lett.

J. V. Wright, B. P. Nelson, Electron. Lett. 13, (1977).
[CrossRef]

IEEE J. Quantum Electron

D. Marcuse, C. Lin, IEEE J. Quantum Electron QE-17, 869 (1981).
[CrossRef]

Natl. Bur. Stand. U.S. Tech. Note 1019

D. L. Franzen, G. W. Day, “Measurement of Optical Fiber Bandwidth in the Time Domain,” Natl. Bur. Stand. U.S. Tech. Note 1019 (Feb.1980).

Other

Since only average pulses are being considered in this paper, conclusions about error rates cannot immediately be drawn.

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 (9)

Fig. 1
Fig. 1

Width reduction of a chirped pulse. Input pulse is shown on the left; central figure shows the minimum width of the pulse (after 5.5 km of travel in the fiber) due to a wavelength sweep that is equal to the inherent spectral width of the source. Pulse on the right is also shown at its minimum width at L = 2.2 km with a wavelength sweep that is 5 times larger than the inherent source bandwidth. Pulse has central wavelength λ = 1.3 μm and a spectral width Δλ = 10 Å; rms width of the input pulse is σ0 = 10 psec.

Fig. 2
Fig. 2

Dispersion of a chirped pulse at the minimum dispersion wavelength, D = 0. Input pulse is shown on the left. Central figure shows a pulse without chirp after traveling through a fiber of 100-km length with a source spectral width Δλ = 30 Å. Figure on the right shows a chirped pulse after traversing 100 km of fiber. In this case the source is assumed to be inherently monochromatic but with a linear wavelength sweep that equals the source bandwidth of the case shown in the central figure; rms input pulse width is σ0 = 1 psec in all cases.

Fig. 3
Fig. 3

Pulse dispersion at the minimum dispersion wavelength. Three pulses all have traversed 100 km of fiber. Chirp is increasing from zero on the left to δλ = Δλ in the center and δλ = 3Δλ at the right-hand side. D = 0, Δλ = 10 Å, rms input pulse width σ0 = 1 psec.

Fig. 4
Fig. 4

Dependence of rms width of a chirped pulse on fiber length. Curve C = −V belongs to a pulse whose chirp causes its width to grow monotically. Curve C = V belongs to a chirped pulse whose width decreases initially. Source wavelength λ = 1.5 μm, inherent spectral width Δλ = 10 Å, wavelength sweep δλ = 10 Å. Input rms pulse width σ0 = 10 psec.

Fig. 5
Fig. 5

Same as Fig. 4 with σ0 = 20 psec.

Fig. 6
Fig. 6

Same as Fig. 4 with σ0 = 50 psec.

Fig. 7
Fig. 7

Left: ratio of minimum rms pulse width to input pulse width as a function of the relative wavelength sweep. Right: fiber length at which minimum pulse width is reached as a function of relative wavelength sweep. Source wavelength λ = 1.5 μm, source width Δλ = 10 Å, input pulse width σ0 = 10 psec.

Fig. 8
Fig. 8

Same as Fig. 4 for λ = 1.3 μm.

Fig. 9
Fig. 9

Same as Fig. 7 for λ = 1.3 μm.

Equations (35)

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

ψ 0 ( t ) = A ( t ) exp [ i ( Δ ω 2 T t 2 + ω 0 t ) ] .
ω i ( t ) = Δ ω T t + ω 0 .
ϕ 0 ( ω ) = 1 2 π - A ( t ) × exp [ i ( Δ ω 2 T t 2 + ω 0 t ) ] exp ( - i ω t ) d t ,
ϕ 0 ( ω ) ϕ 0 * ( ω ) = 1 ( 2 π ) 2 - d t - d t R ( t - t ) × exp { i Δ ω 2 T ( t 2 - t 2 ) + i [ ω 0 ( t - t ) - ω t + ω t ] }
R ( t - t ) = A ( t ) A * ( t ) .
ϕ 0 ( ω ) ϕ 0 * ( ω ) = T 2 π Δ ω R ( ω - ω Δ ω T ) × exp { i [ ω 0 - / 2 1 ( ω + ω ) ] ω - ω Δ ω T } .
ϕ ^ 0 ( ω - ω 0 ) 2 = 1 2 π - R ( u ) exp [ i ( ω - ω 0 ) u ] d u .
ϕ ^ 0 ( ω - ω 0 ) 2 P 0 W π exp [ - ( ω - ω 0 ) 2 / W 2 ]
R ( u ) = P 0 exp [ - ( u W / 2 ) 2 ] .
s ( t ) = S exp [ - ( t / T ) 2 ]
P ( 0 , t ) = ψ ( 0 , t ) 2 = s ( t ) ψ 0 ( t ) 2 .
ψ ( z , t ) = - ϕ ( ω ) exp [ i ( ω t - β z ) ] 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 [ - / 2 1 ( T ω ) 2 ] .
P ( z , t ) = ψ ( z , t ) 2 ,
P ( z , t ) = - - ϕ ( ω ) ϕ * ( ω ) · exp { i [ ( ω - ω ) t - ( β - β ) ] } d ω d ω .
ϕ ( ω ) ϕ * ( ω ) = - - ϕ 0 ( ω ) ϕ 0 * ( ω ) × F ( ω - ω ) F * ( ω - ω ) d ω d ω .
ϕ ( ω ) ϕ * ( ω ) = P 0 S 2 π T 2 { 1 + T 2 [ W 2 + ( Δ ω ) 2 ] } 1 / 2 · exp { - T 2 [ ( 1 + 1 2 T 2 W 2 ) ( ω - ω ) 2 2 + ( ω - ω 0 ) ( ω - ω 0 ) + i ( ω + ω ' 2 - ω 0 ) ( ω - ω ) Δ ω T ] 1 + T 2 [ W 2 + ( Δ ω ) 2 ] } .
P ( z , t ) = - G ( z , x ) exp ( i t - β ˙ z T x ) d x ,
G ( z , x ) = P 0 S 2 π exp { - 1 + V 2 4 ( 1 + V 2 + C 2 ) x 2 } exp ( - i B x 3 / 4 ) [ 1 + 3 i B x ( 1 + V 2 + C 2 ) ] 1 / 2 · exp { [ 1 2 C + D ( 1 + V 2 + C 2 ) ] 2 x 2 [ 1 + 3 i B x ( 1 + V 2 + C 2 ) ] ( 1 + V 2 + C 2 ) }
V = W T ,             x = T ω ,
C = T Δ ω ,             D = β ¨ z 2 T 2 ,             B = β z 6 T 3 .
P ( z , t ) = S P 0 [ ( 1 + 2 D C ) 2 + 4 D 2 ( 1 + V 2 ) ] 1 / 2 · exp [ - ( t - β ˙ z T ) 2 ( 1 + 2 D C ) 2 + 4 D 2 ( 1 + V 2 ) ] .
σ = { - t 2 P ( z , t ) d t / - P ( z , t ) d t - [ - t P ( z , t ) d t / - P ( z , t ) d t ] 2 } 1 / 2 .
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 = σ 0 [ ( 1 + 2 D C ) 2 + 4 D 2 ( 1 + V 2 ) + 9 B 2 ( 1 + V 2 + C 2 ) 2 ] 1 / 2 .
σ 0 = T / 2 .
D min = - C 2 ( 1 + V 2 + C 2 ) .
σ min = σ 0 ( 1 + V 2 1 + V 2 + C 2 ) 1 / 2 .
z min = - C T 2 β ¨ ( 1 + V 2 + C 2 ) .
D 1 = C 1 + V 2 + C 2 .
( D 1 ) max = - 1 2 ( 1 + V 2 ) 1 / 2 ,
C = ( 1 + V 2 ) 1 / 2 .
σ / ( σ 0 ) = 2 .
σ σ 0 2 D ( 1 + V 2 + C 2 ) 1 / 2 .

Metrics