Abstract

The theory of pulse distortion in single-mode fibers is extended to include laser sources such as injection lasers operating simultaneously at several distinct wavelengths. 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 spectral width of the source and of its asymmetry on the shape and rms width of the pulse.

© 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, C. Lin, IEEE J. Quantum Electron. 17, 869 (1981).
    [CrossRef]
  3. L. G. Cohen, W. L. Mammel, H. M. Presby, Appl. Opt. 19, 2007 (1980).
    [CrossRef] [PubMed]
  4. C. Lin, L. G. Cohen, Electron. Lett. 14, 170 (1978).
    [CrossRef]
  5. C. Lin, D. Marcuse, Electron. Lett. 17, 54 (1981).
    [CrossRef]

1981 (2)

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

C. Lin, D. Marcuse, Electron. Lett. 17, 54 (1981).
[CrossRef]

1980 (2)

1978 (1)

C. Lin, L. G. Cohen, Electron. Lett. 14, 170 (1978).
[CrossRef]

Cohen, L. G.

Lin, C.

C. Lin, D. Marcuse, Electron. Lett. 17, 54 (1981).
[CrossRef]

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

C. Lin, L. G. Cohen, Electron. Lett. 14, 170 (1978).
[CrossRef]

Mammel, W. L.

Marcuse, D.

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

C. Lin, D. Marcuse, Electron. Lett. 17, 54 (1981).
[CrossRef]

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

Presby, H. M.

Appl. Opt. (2)

Electron. Lett. (2)

C. Lin, L. G. Cohen, Electron. Lett. 14, 170 (1978).
[CrossRef]

C. Lin, D. Marcuse, Electron. Lett. 17, 54 (1981).
[CrossRef]

IEEE J. Quantum Electron. (1)

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

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 (10)

Fig. 1
Fig. 1

Source spectra showing the effect of the skew parameter λs = 1.5 μm.

Fig. 2
Fig. 2

Output to input rms pulse width ratio σ/σ0 as a function of the skew parameter R for L = 10 km, λs = 1.5 μm for the source of Fig. 1(b).

Fig. 3
Fig. 3

Output to input rms pulse width ratio as a function of R for three different wavelengths, L = 100 km, σ0 = 50 psec.

Fig. 4
Fig. 4

Output rms pulse width as a function of input pulse width for two different wavelengths and fiber lengths for the source shown in Fig. 1(b).

Fig. 5
Fig. 5

Input pulse with σ0 = 50 psec.

Fig. 6
Fig. 6

Output pulse for three different values of R and L = 10 km, λs = 1.5 μm, and σ0 = 50 psec.

Fig. 7
Fig. 7

Output pulse for two different values of R for L = 10 km, λs = 1.5 μm, and σ0 = 10 psec.

Fig. 8
Fig. 8

This figure corresponds to Figs. 6(a) and (b) but applies to a source with finite linewidth of its wavelength components, Vj = 20, L = 10 km, λs = 1.5 μm, and σ0 = 50 psec.

Fig. 9
Fig. 9

Output pulses for different values of the skew parameter R for λs = λ0 = 1.273 μm, L = 100 km, σ0 = 1 psec, and Vj = 0.

Fig. 10
Fig. 10

Output pulse for R = 1 and an infinitely narrow source spectrum.

Equations (35)

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

ψ 0 ( t ) = j = 1 N A j ( t ) exp ( i ω j t ) ,
A j ( t ) A k * ( t ) = A j ( t ) A j * ( t ) δ j k ,
ϕ 0 ( ω ) = 1 2 π - j = 1 N A j ( t ) exp [ - i ( ω - ω j ) t ] d t .
ϕ 0 ( ω ) ϕ 0 * ( ω ) = 1 ( 2 π ) 2 · j , k = 1 N - A j ( t ) A k * ( t ) × exp [ - i ( ω - ω j ) t + i ( ω - ω k ) t ] d t d t .
R j ( t - t ) = A j ( t ) A j * ( t )
ϕ ^ 0 j ( ω - ω j ) 2 = 1 2 π - R j ( x ) exp [ - i ( ω - ω j ) x ] d x .
ϕ 0 ( ω ) ϕ 0 * ( ω ) = j = 1 N ϕ ^ 0 j ( ω - ω j ) 2 δ ( ω - ω j ) .
s ( t ) = S exp [ - ( t / T ) 2 ]
ψ ( t ) = s 1 / 2 ψ 0 ( t )
ϕ ( ω ) = 1 2 π - s 1 / 2 ψ 0 ( t ) exp ( - i ω ) 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 [ - ½ ( T ω ) 2 ] .
P ( z , t ) = j = 1 N - ϕ ^ 0 j ( ω - ω j ) 2 · | - F ( ω - ω ) exp [ i ( ω - ω ) t - i ( β - β ) z ] d ω | 2 d ω .
β = β j + β ˙ j ( ω - ω j ) + ½ β ¨ j ( ω - ω j ) 2 + / 6 1 β j ( ω - ω j ) 3 .
P ( z , t ) = j = 1 N - G j ( z , x ) exp ( i t - β ˙ j z T x ) d x ,
G j ( z , x ) = P j S 2 π exp ( - x 2 / 4 ) exp ( - i B j x 3 / 4 ) [ 1 + 3 i B j x ( 1 + V j 2 ) ] 1 / 2 · exp [ - D j 2 x 2 ( 1 + V j 2 ) 1 + 3 i B j x ( 1 + V j 2 ) ] .
ϕ ^ 0 ( ω - ω j ) 2 = P j W j π exp [ - ( ω - ω j ) 2 / W j 2 ] ,
V j = T W j .
D j = β ¨ j z / ( 2 T 2 ) ,
B j = β j z / ( 6 T 3 ) .
σ 2 = - t 2 P ( z , t ) d t - P ( z , t ) d t - { - t P ( t , z ) d t - P ( z , t ) d t } 2 ,
σ 2 = - T 2 { d 2 d x 2 [ j G j ( z , x ) exp ( - i β ˙ j z T x ) ] } x = 0 j G j ( z , 0 ) + { T d d x [ j G j ( z , x ) exp ( - i β ˙ j z T x ) ] j G j ( z , x ) } x = 0 2 .
σ 2 / T 2 = j { 1 + 2 ( β ˙ j z T ) 2 + 4 D j 2 [ 1 + 6 β ˙ j z T B j + 27 2 B j 2 ( 1 + V j 2 ) ] ( 1 + V j 2 ) } P j 2 j P j - { j [ β ˙ j z T + 3 2 B j ( 1 + V j 2 ) ] j P j } 2 .
β j = β j + β ˙ s ( ω j - ω s ) + ½ β ¨ s ( ω j - ω s ) 2 + / 6 1 β s ( ω j - ω s ) 3 .
σ 2 T 2 = ½ + 2 D s 2 { 2 T 2 [ ( ω j - ω s ) ¯ 2 - ( ω j - ω s ) 2 ¯ ] + 1 + V j 2 ¯ } + 6 D s B s { 2 T 3 [ ( ω j - ω s ) 3 ¯ - ( ω j - ω s ) ¯ ( ω j - ω s ) 2 ¯ ] + T [ 3 ( ω j - ω s ) ( 1 + V j 2 ) ¯ - ( ω j - ω s ) ¯ ( 1 + V j 2 ¯ ) ] } + 9 B s 2 { T 4 [ ( ω j - ω s ) 4 ¯ - ( ω j - ω s ) 2 ¯ 2 ] + / 4 1 [ 3 ( 1 + V j 2 ) 2 ¯ - ( 1 + V j 2 ) ¯ 2 ] + T 2 [ 3 ( ω j - ω s ) 2 ( 1 + V j 2 ) ¯ - ( ω j - ω s ) 2 ¯ ( 1 + V j 2 ¯ ) ] .
F j ¯ = j = 1 N F j P j j = 1 N P j .
σ 2 / T 2 = ½ + 2 D s 2 [ 1 + V 2 + 2 T 2 ( ω j - ω s ) 2 ¯ ] + 9 B s 2 { ½ ( 1 + V 2 ) 2 + 2 T 2 ( ω j - ω s ) 2 ¯ ( 1 + V 2 ) + T 4 [ ( ω j - ω s ) 4 ¯ - ( ω j - ω s ) 2 ¯ 2 ] } .
V e = T Δ ω
σ 2 / T 2 = ½ { 1 + 4 D s 2 ( 1 + V 2 + V e 2 ) + 9 B s 2 [ 1 + V 2 + V e 2 ] 2 } .
σ = T 2 { 1 + 4 D s 2 ( 1 + V c 2 ) + 9 B s 2 ( 1 + V c 2 ) 2 } 1 / 2 .
ϕ ^ 0 ( ω j - ω s ) 2 = { A exp [ - ( ω j - ω s ) 2 / w 1 2 ] for ω j < ω s , A exp [ - ( ω j - ω s ) 2 / w 2 2 ] for ω j > ω s .
P = - ϕ ^ 0 ( ω - ω s ) 2 d ω = ½ A π ( w 1 + w 2 ) .
R 2 = 0 ( ω - ω s ) 2 ϕ ^ 0 ( ω - ω s ) 2 d ω - 0 ( ω - ω s ) 2 ϕ ^ 0 ( ω - ω s ) 2 d ω = w 2 3 w 1 3
w 1 = K 1 + R 2 / 3 ,
w 2 = K 1 + R 2 / 3 R 2 / 3 = w 1 R 2 / 3 ,
σ 0 = T / 2 ,

Metrics