Abstract

An earlier paper [ Applied Optics 19, 1653 ( 1980)] dealt with the ensemble averages of pulses propagating in single-mode fibers. In this paper we discuss pulse fluctuations. The light pulses are generated by modulation of the power of a continuously operating light source consisting of N discrete sinusoidal frequencies randomly phased relative to each other. The fixed amplitudes of the sinusoidal frequency components of the source are adjusted to fit into a Gaussian envelope, and the modulating pulse has a Gaussian distribution in time. This mathematical model approximates a laser light source operating in several free-running longitudinal modes. We find that the fluctuations of the modulated light pulses can die out if the pulses travel a long distance in a dispersive fiber, provided the spacings between the sinusoidal frequency components of the light source are larger than the spectral width of the modulating signal. If the source frequency components are spaced more closely than the spectral width of the modulating pulse, fluctuations persist indefinitely independent of fiber length. However, in a practical system, whose input pulse is only about half as short as the output pulse, fluctuations are practically unaffected by transmission through a fiber.

© 1980 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Marcuse, Appl. Opt. 19, 1653 (1980).
    [CrossRef] [PubMed]
  2. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).
  3. T. Ito, S. Machida, K. Nawata, T. Ikegami, IEEE J. Quantum Electron. QE-13, 574 (1977).
    [CrossRef]
  4. K. Juergensen, Appl. Opt. 17, 2412 (1978).
    [CrossRef]
  5. K. Furuya, M. Miyamoto, Y. Suematsu, Trans. Inst. Electron. Commun. Eng. Jpn. E62(5), 305 (1979).

1980 (1)

1979 (1)

K. Furuya, M. Miyamoto, Y. Suematsu, Trans. Inst. Electron. Commun. Eng. Jpn. E62(5), 305 (1979).

1978 (1)

1977 (1)

T. Ito, S. Machida, K. Nawata, T. Ikegami, IEEE J. Quantum Electron. QE-13, 574 (1977).
[CrossRef]

Furuya, K.

K. Furuya, M. Miyamoto, Y. Suematsu, Trans. Inst. Electron. Commun. Eng. Jpn. E62(5), 305 (1979).

Ikegami, T.

T. Ito, S. Machida, K. Nawata, T. Ikegami, IEEE J. Quantum Electron. QE-13, 574 (1977).
[CrossRef]

Ito, T.

T. Ito, S. Machida, K. Nawata, T. Ikegami, IEEE J. Quantum Electron. QE-13, 574 (1977).
[CrossRef]

Juergensen, K.

Machida, S.

T. Ito, S. Machida, K. Nawata, T. Ikegami, IEEE J. Quantum Electron. QE-13, 574 (1977).
[CrossRef]

Marcuse, D.

Miyamoto, M.

K. Furuya, M. Miyamoto, Y. Suematsu, Trans. Inst. Electron. Commun. Eng. Jpn. E62(5), 305 (1979).

Nawata, K.

T. Ito, S. Machida, K. Nawata, T. Ikegami, IEEE J. Quantum Electron. QE-13, 574 (1977).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).

Suematsu, Y.

K. Furuya, M. Miyamoto, Y. Suematsu, Trans. Inst. Electron. Commun. Eng. Jpn. E62(5), 305 (1979).

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

T. Ito, S. Machida, K. Nawata, T. Ikegami, IEEE J. Quantum Electron. QE-13, 574 (1977).
[CrossRef]

Trans. Inst. Electron. Commun. Eng. Jpn. (1)

K. Furuya, M. Miyamoto, Y. Suematsu, Trans. Inst. Electron. Commun. Eng. Jpn. E62(5), 305 (1979).

Other (1)

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).

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 (8)

Fig. 1
Fig. 1

Input pulse produced by a light source consisting of eleven sinusoidal components with fixed amplitudes and random phases, modulated by a Gaussian-shaped pulse; (a) and (b) were obtained by using different sets of random phases. 2TΔF = 15 with 2T = width of Gaussian pulse; Δf = frequency spacing of sinusoidal source components.

Fig. 2
Fig. 2

Pulse of Fig. 1 after transmission through a dispersive fiber with D = 2, B = 0.

Fig. 3
Fig. 3

Same as Fig. 1 with fifty-one sinusoidal frequency components. The pulses of (a) and (b) have different sets of random phases.

Fig. 4
Fig. 4

Pulse of Fig. 3 after transmission through dispersive fiber with D = 2, B = 0.

Fig 5
Fig 5

Pulse of Fig. 1 after transmission through dispersive fiber with D = 0, 2TΔf = 1.5. The following parameters apply: (a) B = 0.051; (b) B = 0.1; (c) B = 0.2.

Fig. 6
Fig. 6

Same as Fig. 5(b) with 2TΔf = 3.

Fig. 7
Fig. 7

Relative pulse fluctuation ΔP/P as function of D for several values of 2TΔf.

Fig. 8
Fig. 8

Same as Fig. 7 with larger values of 2TΔf.

Equations (41)

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

ψ 0 ( t ) = j = 1 N A j exp [ i ( ω j t + ϕ j ) ] ,
P 0 ( t ) = | ψ 0 ( t ) | 2 .
P ( t ) = s ( t ) P 0 ( t ) .
ψ ( t ) = s 1 / 2 ( t ) ψ 0 ( t )
ϕ ( ω ) = 1 2 π s 1 / 2 ( t ) ψ 0 ( t ) exp ( i ω t ) d t .
s ( t ) = S exp [ ( t / T ) 2 ]
ϕ ( ω ) = ( S 2 π ) 1 / 2 T j = 1 N A j exp ( i ϕ j ) exp [ 1 2 ( ω j ω ) 2 T 2 ] .
ψ ( z , t ) = ϕ ( ω ) exp [ i ( ω t β z ) ] d ω ,
ψ ( z , t ) = ( S 2 π ) 1 / 2 T j = 1 N A j exp ( i ϕ j ) × exp [ 1 2 ( ω j ω ) 2 T 2 ] exp [ i ( ω t β z ) ] d ω .
β = β 0 + β ˙ 0 ( ω ω 0 ) + 1 2 β ¨ 0 ( ω ω 0 ) 2 + 1 6 β 0 ( ω ω 0 ) 3 ,
x = ( ω ω 0 ) T ,
x j = ( ω j ω 0 ) T ,
u = ( t β ˙ 0 z ) / T ,
D = 1 2 [ ( β ¨ 0 z ) / T 2 ] ,
B = 1 6 [ ( β 0 z ) / T 3 ] ,
ψ ( z , t ) = ( S 2 π ) 1 / 2 exp [ i ( ω 0 t β 0 z ) ] exp ( iux ) { j = 1 N A j exp ( i ϕ j ) exp [ 1 2 ( x j x ) 2 ] } exp [ i ( D x 2 + B x 3 ) ] d x .
ψ ( z , t ) = ( S 1 + 2 i D ) 1 / 2 exp [ i ( ω 0 t β 0 z ) ] j = 1 N A j exp ( i ϕ j ) × exp ( 1 2 x j 2 ) exp [ ( u i x j ) 2 2 ( 1 + 2 i D ) ] .
A j = ( P 0 V π 1 / 2 ) 1 / 2 exp ( x j 2 2 V 2 ) ( Δ x j ) 1 / 2 .
Δ x j = x j + 1 x j .
V = W T .
lim N j = 1 N A j 2 = P 0 .
P ( z , t ) = | ψ ( z , t ) | 2 ,
( Δ P ) 2 = P 2 ( z , t ) P ( z , t ) 2
P = S P 0 ( π ) 1 / 2 V ( 1 + 4 D 2 ) 1 / 2 j = 1 N exp ( x i 2 V 2 ) × exp [ ( u 2 D x j ) 2 1 + 4 D 2 ] Δ x j .
( Δ P ) 2 = S 2 P 0 2 π V 2 ( 1 + 4 D 2 ) ( [ j = 1 N exp ( x j 2 V 2 ) exp [ ( u 2 D x j ) 2 1 + 4 D 2 ] Δ x j ] 2 j = 1 N { exp ( x j 2 V 2 ) exp [ ( u 2 D x j ) 1 + 4 D 2 ] } 2 ( Δ x j ) 2 ) .
[ Δ P ( z , t ) ] 2 = P ( z , t ) 2
Δ P ( z , t ) = P ( z , t ) .
( 2 D Δ x j ) 2 1 + 4 D 2 K ,
D = 1 2 ( K Δ x j 2 K ) 1 / 2 .
Δ x j > K 1 / 2 .
Δ x j = 2 π Δ f T ,
Δ f ( 2 T ) K 1 / 2 π .
σ σ 0 = [ 1 + 4 D 2 ( 1 + V 2 ) + 9 B 2 ( 1 + V 2 ) 2 ] 1 / 2 .
1 + 4 D 2 ( 1 + V 2 ) = 4
D 2 ( 1 + V 2 ) = ¾ .
V = M Δ x j .
D 2 ( 1 + M 2 Δ x j 2 ) = ¾ .
1 M 2 ( 3 4 D 2 1 ) > 10 .
4 D 2 Δ x j 2 > 10 ,
4 D 2 Δ x j 2 = 3 / M 2 .
3 / M 2 > 10 .

Metrics