Abstract

Fiber four-wave mixing in transmission lines with multistage optical amplifiers is studied. A theoretical model is described, and a fundamental experiment is carried out. It is shown that the phase-mismatching characteristic differs in transmission lines with and without amplifiers.

© 1992 Optical Society of America

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  1. K. O. Hill, D. C. Johnson, B. S. Kawasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1978).
    [CrossRef]
  2. M. W. Maeda, W. B. Sessa, W. I. Way, A. Yi-Yan, L. Curtis, R. Spicer, R. I. Laming, IEEE J. Lightwave Technol. 8, 1402 (1990).
    [CrossRef]
  3. E. Lichtman, J. Opt. Commun. 12, 53 (1991).
    [CrossRef]
  4. K. Inoue, H. Toba, K. Oda, IEEE J. Lightwave Technol. 10, 350 (1992).
    [CrossRef]
  5. K. Inoue, H. Toba, IEEE J. Lightwave Technol. 10, 361 (1992).
    [CrossRef]
  6. N. Edagawa, H. Taga, J. Yoshida, M. Suzuki, S. Yamamoto, H. Wakabayashi, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP3.
  7. P. M. Gabla, O. Scaramucci, J. O. Frorud, G. Bassier, L. B. Leroy, V. Havard, E. Leclerc, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP4.
  8. S. Saito, M. Murakami, A. Naka, Y. Fukuda, T. Imai, M. Aiki, T. Ito, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP5.
  9. R. Welter, R. I. Laming, W. B. Sessa, R. S. Vodhanel, M. W. Maeda, R. E. Wagner, Electron. Lett. 25, 1333 (1989).
    [CrossRef]
  10. K. Inoue, H. Toba, K. Nosu, IEEE J. Lightwave Technol. 9, 368 (1991).
    [CrossRef]
  11. D. G. Schadt, Electron. Lett. 27, 1805 (1991).
    [CrossRef]
  12. N. Shibata, R. P. Braun, R. G. Waarts, IEEE J. Quantum Electron. QE-23, 1205 (1987).
    [CrossRef]

1992 (2)

K. Inoue, H. Toba, K. Oda, IEEE J. Lightwave Technol. 10, 350 (1992).
[CrossRef]

K. Inoue, H. Toba, IEEE J. Lightwave Technol. 10, 361 (1992).
[CrossRef]

1991 (3)

K. Inoue, H. Toba, K. Nosu, IEEE J. Lightwave Technol. 9, 368 (1991).
[CrossRef]

D. G. Schadt, Electron. Lett. 27, 1805 (1991).
[CrossRef]

E. Lichtman, J. Opt. Commun. 12, 53 (1991).
[CrossRef]

1990 (1)

M. W. Maeda, W. B. Sessa, W. I. Way, A. Yi-Yan, L. Curtis, R. Spicer, R. I. Laming, IEEE J. Lightwave Technol. 8, 1402 (1990).
[CrossRef]

1989 (1)

R. Welter, R. I. Laming, W. B. Sessa, R. S. Vodhanel, M. W. Maeda, R. E. Wagner, Electron. Lett. 25, 1333 (1989).
[CrossRef]

1987 (1)

N. Shibata, R. P. Braun, R. G. Waarts, IEEE J. Quantum Electron. QE-23, 1205 (1987).
[CrossRef]

1978 (1)

K. O. Hill, D. C. Johnson, B. S. Kawasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1978).
[CrossRef]

Aiki, M.

S. Saito, M. Murakami, A. Naka, Y. Fukuda, T. Imai, M. Aiki, T. Ito, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP5.

Bassier, G.

P. M. Gabla, O. Scaramucci, J. O. Frorud, G. Bassier, L. B. Leroy, V. Havard, E. Leclerc, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP4.

Braun, R. P.

N. Shibata, R. P. Braun, R. G. Waarts, IEEE J. Quantum Electron. QE-23, 1205 (1987).
[CrossRef]

Curtis, L.

M. W. Maeda, W. B. Sessa, W. I. Way, A. Yi-Yan, L. Curtis, R. Spicer, R. I. Laming, IEEE J. Lightwave Technol. 8, 1402 (1990).
[CrossRef]

Edagawa, N.

N. Edagawa, H. Taga, J. Yoshida, M. Suzuki, S. Yamamoto, H. Wakabayashi, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP3.

Frorud, J. O.

P. M. Gabla, O. Scaramucci, J. O. Frorud, G. Bassier, L. B. Leroy, V. Havard, E. Leclerc, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP4.

Fukuda, Y.

S. Saito, M. Murakami, A. Naka, Y. Fukuda, T. Imai, M. Aiki, T. Ito, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP5.

Gabla, P. M.

P. M. Gabla, O. Scaramucci, J. O. Frorud, G. Bassier, L. B. Leroy, V. Havard, E. Leclerc, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP4.

Havard, V.

P. M. Gabla, O. Scaramucci, J. O. Frorud, G. Bassier, L. B. Leroy, V. Havard, E. Leclerc, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP4.

Hill, K. O.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1978).
[CrossRef]

Imai, T.

S. Saito, M. Murakami, A. Naka, Y. Fukuda, T. Imai, M. Aiki, T. Ito, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP5.

Inoue, K.

K. Inoue, H. Toba, K. Oda, IEEE J. Lightwave Technol. 10, 350 (1992).
[CrossRef]

K. Inoue, H. Toba, IEEE J. Lightwave Technol. 10, 361 (1992).
[CrossRef]

K. Inoue, H. Toba, K. Nosu, IEEE J. Lightwave Technol. 9, 368 (1991).
[CrossRef]

Ito, T.

S. Saito, M. Murakami, A. Naka, Y. Fukuda, T. Imai, M. Aiki, T. Ito, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP5.

Johnson, D. C.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1978).
[CrossRef]

Kawasaki, B. S.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1978).
[CrossRef]

Laming, R. I.

M. W. Maeda, W. B. Sessa, W. I. Way, A. Yi-Yan, L. Curtis, R. Spicer, R. I. Laming, IEEE J. Lightwave Technol. 8, 1402 (1990).
[CrossRef]

R. Welter, R. I. Laming, W. B. Sessa, R. S. Vodhanel, M. W. Maeda, R. E. Wagner, Electron. Lett. 25, 1333 (1989).
[CrossRef]

Leclerc, E.

P. M. Gabla, O. Scaramucci, J. O. Frorud, G. Bassier, L. B. Leroy, V. Havard, E. Leclerc, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP4.

Leroy, L. B.

P. M. Gabla, O. Scaramucci, J. O. Frorud, G. Bassier, L. B. Leroy, V. Havard, E. Leclerc, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP4.

Lichtman, E.

E. Lichtman, J. Opt. Commun. 12, 53 (1991).
[CrossRef]

MacDonald, R. I.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1978).
[CrossRef]

Maeda, M. W.

M. W. Maeda, W. B. Sessa, W. I. Way, A. Yi-Yan, L. Curtis, R. Spicer, R. I. Laming, IEEE J. Lightwave Technol. 8, 1402 (1990).
[CrossRef]

R. Welter, R. I. Laming, W. B. Sessa, R. S. Vodhanel, M. W. Maeda, R. E. Wagner, Electron. Lett. 25, 1333 (1989).
[CrossRef]

Murakami, M.

S. Saito, M. Murakami, A. Naka, Y. Fukuda, T. Imai, M. Aiki, T. Ito, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP5.

Naka, A.

S. Saito, M. Murakami, A. Naka, Y. Fukuda, T. Imai, M. Aiki, T. Ito, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP5.

Nosu, K.

K. Inoue, H. Toba, K. Nosu, IEEE J. Lightwave Technol. 9, 368 (1991).
[CrossRef]

Oda, K.

K. Inoue, H. Toba, K. Oda, IEEE J. Lightwave Technol. 10, 350 (1992).
[CrossRef]

Saito, S.

S. Saito, M. Murakami, A. Naka, Y. Fukuda, T. Imai, M. Aiki, T. Ito, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP5.

Scaramucci, O.

P. M. Gabla, O. Scaramucci, J. O. Frorud, G. Bassier, L. B. Leroy, V. Havard, E. Leclerc, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP4.

Schadt, D. G.

D. G. Schadt, Electron. Lett. 27, 1805 (1991).
[CrossRef]

Sessa, W. B.

M. W. Maeda, W. B. Sessa, W. I. Way, A. Yi-Yan, L. Curtis, R. Spicer, R. I. Laming, IEEE J. Lightwave Technol. 8, 1402 (1990).
[CrossRef]

R. Welter, R. I. Laming, W. B. Sessa, R. S. Vodhanel, M. W. Maeda, R. E. Wagner, Electron. Lett. 25, 1333 (1989).
[CrossRef]

Shibata, N.

N. Shibata, R. P. Braun, R. G. Waarts, IEEE J. Quantum Electron. QE-23, 1205 (1987).
[CrossRef]

Spicer, R.

M. W. Maeda, W. B. Sessa, W. I. Way, A. Yi-Yan, L. Curtis, R. Spicer, R. I. Laming, IEEE J. Lightwave Technol. 8, 1402 (1990).
[CrossRef]

Suzuki, M.

N. Edagawa, H. Taga, J. Yoshida, M. Suzuki, S. Yamamoto, H. Wakabayashi, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP3.

Taga, H.

N. Edagawa, H. Taga, J. Yoshida, M. Suzuki, S. Yamamoto, H. Wakabayashi, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP3.

Toba, H.

K. Inoue, H. Toba, IEEE J. Lightwave Technol. 10, 361 (1992).
[CrossRef]

K. Inoue, H. Toba, K. Oda, IEEE J. Lightwave Technol. 10, 350 (1992).
[CrossRef]

K. Inoue, H. Toba, K. Nosu, IEEE J. Lightwave Technol. 9, 368 (1991).
[CrossRef]

Vodhanel, R. S.

R. Welter, R. I. Laming, W. B. Sessa, R. S. Vodhanel, M. W. Maeda, R. E. Wagner, Electron. Lett. 25, 1333 (1989).
[CrossRef]

Waarts, R. G.

N. Shibata, R. P. Braun, R. G. Waarts, IEEE J. Quantum Electron. QE-23, 1205 (1987).
[CrossRef]

Wagner, R. E.

R. Welter, R. I. Laming, W. B. Sessa, R. S. Vodhanel, M. W. Maeda, R. E. Wagner, Electron. Lett. 25, 1333 (1989).
[CrossRef]

Wakabayashi, H.

N. Edagawa, H. Taga, J. Yoshida, M. Suzuki, S. Yamamoto, H. Wakabayashi, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP3.

Way, W. I.

M. W. Maeda, W. B. Sessa, W. I. Way, A. Yi-Yan, L. Curtis, R. Spicer, R. I. Laming, IEEE J. Lightwave Technol. 8, 1402 (1990).
[CrossRef]

Welter, R.

R. Welter, R. I. Laming, W. B. Sessa, R. S. Vodhanel, M. W. Maeda, R. E. Wagner, Electron. Lett. 25, 1333 (1989).
[CrossRef]

Yamamoto, S.

N. Edagawa, H. Taga, J. Yoshida, M. Suzuki, S. Yamamoto, H. Wakabayashi, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP3.

Yi-Yan, A.

M. W. Maeda, W. B. Sessa, W. I. Way, A. Yi-Yan, L. Curtis, R. Spicer, R. I. Laming, IEEE J. Lightwave Technol. 8, 1402 (1990).
[CrossRef]

Yoshida, J.

N. Edagawa, H. Taga, J. Yoshida, M. Suzuki, S. Yamamoto, H. Wakabayashi, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP3.

Electron. Lett. (2)

R. Welter, R. I. Laming, W. B. Sessa, R. S. Vodhanel, M. W. Maeda, R. E. Wagner, Electron. Lett. 25, 1333 (1989).
[CrossRef]

D. G. Schadt, Electron. Lett. 27, 1805 (1991).
[CrossRef]

IEEE J. Lightwave Technol. (4)

K. Inoue, H. Toba, K. Nosu, IEEE J. Lightwave Technol. 9, 368 (1991).
[CrossRef]

M. W. Maeda, W. B. Sessa, W. I. Way, A. Yi-Yan, L. Curtis, R. Spicer, R. I. Laming, IEEE J. Lightwave Technol. 8, 1402 (1990).
[CrossRef]

K. Inoue, H. Toba, K. Oda, IEEE J. Lightwave Technol. 10, 350 (1992).
[CrossRef]

K. Inoue, H. Toba, IEEE J. Lightwave Technol. 10, 361 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

N. Shibata, R. P. Braun, R. G. Waarts, IEEE J. Quantum Electron. QE-23, 1205 (1987).
[CrossRef]

J. Appl. Phys. (1)

K. O. Hill, D. C. Johnson, B. S. Kawasaki, R. I. MacDonald, J. Appl. Phys. 49, 5098 (1978).
[CrossRef]

J. Opt. Commun. (1)

E. Lichtman, J. Opt. Commun. 12, 53 (1991).
[CrossRef]

Other (3)

N. Edagawa, H. Taga, J. Yoshida, M. Suzuki, S. Yamamoto, H. Wakabayashi, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP3.

P. M. Gabla, O. Scaramucci, J. O. Frorud, G. Bassier, L. B. Leroy, V. Havard, E. Leclerc, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP4.

S. Saito, M. Murakami, A. Naka, Y. Fukuda, T. Imai, M. Aiki, T. Ito, in Digest of European Conference on Optical Communication (Société des Electriciens et des Electroniciens, Paris, 1991), paper PDP5.

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Figures (4)

Fig. 1
Fig. 1

Configuration of the transmission line.

Fig. 2
Fig. 2

FWM efficiency η as a function of the equivalent frequency separation in the transmission line consisting of 10 sections with a section length of 50 km (solid curve). The dashed curve is for the same total length without the amplifier.

Fig. 3
Fig. 3

Experimental setup. PC’s, polarization controllers.

Fig. 4
Fig. 4

Experimental result (points). The calculation result is shown by the solid curve.

Equations (12)

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2 z 2 E ( ω F , z ) - n 2 c 2 2 t 2 E ( ω F , z ) - α n c t E ( ω F , z ) = 4 π c 2 2 z 2 ( D χ ) E ( ω p , z ) E ( ω q , z ) E ( ω r , z ) * ,
E ( ω F , z ) = E F ( z ) exp ( - i ω F t ) exp [ i β F ( z - j = 1 k - 1 l j ) ] ,
E ( ω i , z ) = E i ( k ) exp ( - i ω i t ) × exp [ ( - α 2 + i β i ) ( z - j = 1 k - 1 l j ) ] ,             i = p , q , r ,
E i ( k ) = E i ( 1 ) exp [ ( - α 2 + i β i ) j = 1 k - 1 l j ] ( j k - 1 G j ) ,
d d z E F ( z ) = - α 2 E F ( z ) + i 2 π ω n c ( D χ ) E p ( k ) E q ( k ) E r ( k ) * × exp [ ( - 3 2 α + i Δ β ) ( z - j = 1 k - 1 l j ) ] ,
E F ( k ) = i 2 π ω n c ( D χ ) E p ( 1 ) E q ( 1 ) E r ( 1 ) * × exp [ ( - 3 2 α + i β F ) j = 1 k - 1 l j ] [ j k - 1 ( G j ) 3 / 2 ] × exp ( i Δ β j = 1 k - 1 l j ) exp [ ( - 1 2 α + i β F ) l k ] × exp [ ( - α + i Δ β ) l k ] - 1 i Δ β - α ,
E F ( k end ) = E F ( k ) exp [ ( - α 2 + i β F ) j = k + 1 N l j ] × ( j = k N - 1 G j ) .
E F ( total ) = k = 1 N - 1 E F ( k end ) + E F ( N ) = i 2 π ω n c ( D χ ) E p ( 1 ) E q ( 1 ) E r ( 1 ) * × exp [ ( - α 2 + i β F ) j = 1 N l j ] ( j = 1 N - 1 G j ) × 1 i Δ β - α k = 1 N exp ( - α j = 1 k - 1 l j ) ( j = 1 k - 1 G j ) × exp ( i Δ β j = 1 k - 1 l j ) { exp [ ( - α + i Δ β ) l k ] - 1 } ,
E F ( total ) = i 2 π ω n c ( D χ ) E p ( 1 ) E q ( 1 ) E r ( 1 ) * × exp ( - α 2 + l N ) exp ( i β F j = 1 N l j ) × 1 i Δ β - α k = 1 N exp ( i Δ β j = 1 k - 1 l j ) × { exp [ ( - α + i Δ β ) l k ] - 1 } .
P F = 1024 π 6 n 4 λ 2 c 2 ( D χ ) 2 P p ( 0 ) P q ( 0 ) P r ( 0 ) A eff 2 exp ( - α l N ) × 1 α 2 + Δ β 2 | k = 1 N exp ( i Δ β j = 1 k - 1 l j ) × { exp [ ( - α + i Δ β ) l k ] - 1 } | 2 ,
P F = 1024 π 6 n 4 λ 2 c 2 ( D χ ) 2 P p ( 0 ) P q ( 0 ) P r ( 0 ) A eff 2 × exp ( - α l 0 ) [ 1 - exp ( - α l 0 ) ] 2 α 2 N 2 η ,
η = α 2 N 2 ( α 2 + Δ β 2 ) sin 2 ( N Δ β l 0 / 2 ) sin 2 ( Δ β l 0 / 2 ) × { 1 + 4 exp ( - α l 0 ) sin 2 ( Δ β l 0 / 2 ) [ 1 - exp ( - α l 0 ) ] 2 } .

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