Abstract

Exact formulas are obtained for the amplitudes of light waves involved in four-wave-mixing cascades near the zero-dispersion frequency of a fiber. The cascade that is initiated by two strong pump waves is phase insensitive, whereas the cascade that is initiated by two strong pump waves and a weak signal wave is phase sensitive. In both cascades, the number of waves that have significant power increases with distance.

© 2006 Optical Society of America

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  1. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li and P. O. Hedekvist, "Fiber-based optical parametric amplifiers and their applications," IEEE J. Sel. Top. Quantum Electron. 8, 506-520 (2002).
    [CrossRef]
  2. <jrn>. S. Radic and C. J. McKinstrie, "Optical amplification and signal processing in highly-nonlinear optical fiber," IEICE Trans. Electron. E88C, 859-869 (2005).</jrn>
    [CrossRef]
  3. C. J. McKinstrie, S. Radic and A. R. Chraplyvy, "Parametric amplifiers driven by two pump waves," IEEE J. Sel. Top. Quantum. Electron. 8, 538-547 and 956 (2002).
    [CrossRef]
  4. C. J. McKinstrie and S. Radic, "Phase-sensitive amplification in a fiber," Opt. Express 12, 4973-4979 (2004).
    [CrossRef] [PubMed]
  5. C. J. McKinstrie, M. Yu, M. G. Raymer and S. Radic, "Quantum noise properties of parametric processes," Opt. Express 13, 4986-5012 (2005).
    [CrossRef] [PubMed]
  6. C. J. McKinstrie, M. G. Raymer, S. Radic and M. V. Vasilyev, "Quantum mechanics of phase-sensitive amplification in a fiber," Opt. Commun. 257, 146-163 (2006).
    [CrossRef]
  7. I. P. Kaminow and T. L. Koch, Editors, Optical Fiber Telecommunications IIIA and IIIB (Academic Press, 1997).
  8. S. T. Cundiff, "Phase stabilization of ultrashort optical pulses," J. Phys. D: Appl. Phys. 35, R43-R59 (2002).
    [CrossRef]
  9. I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, 1994), pp. 987 and 994.
  10. G. Cappellini and S. Trillo, "Third-order three-wave mixing in single-mode fibers: Exact solutions and spatial instability effects," J. Opt. Soc. Am. B 8, 824-838 (1991).
    [CrossRef]
  11. C. J. McKinstrie, X. D. Cao and J. S. Li, "Nonlinear detuning of four-wave interactions," J. Opt. Soc. Am. B 10,1856-1869 (1993).
    [CrossRef]
  12. B. I. Cohen, A. N. Kaufman and K. M. Watson, "Beat heating of a plasma," Phys. Rev. Lett. 29, 581-584 (1972).
    [CrossRef]
  13. S. J. Karttunen and R. R. E. Salomaa, "Electromagnetic field cascading in the beat-wave generation of plasma waves," Phys. Rev. Lett. 56, 604-607 (1986).
    [CrossRef] [PubMed]
  14. K. Inoue, K. Nakanishi, K. Oda and H. Toba, "Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions," J. Lightwave Technol. 12, 1423-1439 (1994).
    [CrossRef]
  15. R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck and R. M. Derosier, "Four-photon mixing and highspeed WDM systems," J. Lightwave Technol. 13, 841-849 (1995).
    [CrossRef]
  16. R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan and P. Kumar, "In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier," IEEE Photon. Technol. Lett. 17, 1845-1847 (2005).
    [CrossRef]

2006

C. J. McKinstrie, M. G. Raymer, S. Radic and M. V. Vasilyev, "Quantum mechanics of phase-sensitive amplification in a fiber," Opt. Commun. 257, 146-163 (2006).
[CrossRef]

2005

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan and P. Kumar, "In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier," IEEE Photon. Technol. Lett. 17, 1845-1847 (2005).
[CrossRef]

C. J. McKinstrie, M. Yu, M. G. Raymer and S. Radic, "Quantum noise properties of parametric processes," Opt. Express 13, 4986-5012 (2005).
[CrossRef] [PubMed]

2004

2002

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li and P. O. Hedekvist, "Fiber-based optical parametric amplifiers and their applications," IEEE J. Sel. Top. Quantum Electron. 8, 506-520 (2002).
[CrossRef]

S. T. Cundiff, "Phase stabilization of ultrashort optical pulses," J. Phys. D: Appl. Phys. 35, R43-R59 (2002).
[CrossRef]

1995

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck and R. M. Derosier, "Four-photon mixing and highspeed WDM systems," J. Lightwave Technol. 13, 841-849 (1995).
[CrossRef]

1994

K. Inoue, K. Nakanishi, K. Oda and H. Toba, "Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions," J. Lightwave Technol. 12, 1423-1439 (1994).
[CrossRef]

1993

1991

1986

S. J. Karttunen and R. R. E. Salomaa, "Electromagnetic field cascading in the beat-wave generation of plasma waves," Phys. Rev. Lett. 56, 604-607 (1986).
[CrossRef] [PubMed]

1972

B. I. Cohen, A. N. Kaufman and K. M. Watson, "Beat heating of a plasma," Phys. Rev. Lett. 29, 581-584 (1972).
[CrossRef]

Andrekson, P. A.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li and P. O. Hedekvist, "Fiber-based optical parametric amplifiers and their applications," IEEE J. Sel. Top. Quantum Electron. 8, 506-520 (2002).
[CrossRef]

Cao, X. D.

Cappellini, G.

Chraplyvy, A. R.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck and R. M. Derosier, "Four-photon mixing and highspeed WDM systems," J. Lightwave Technol. 13, 841-849 (1995).
[CrossRef]

Cohen, B. I.

B. I. Cohen, A. N. Kaufman and K. M. Watson, "Beat heating of a plasma," Phys. Rev. Lett. 29, 581-584 (1972).
[CrossRef]

Cundiff, S. T.

S. T. Cundiff, "Phase stabilization of ultrashort optical pulses," J. Phys. D: Appl. Phys. 35, R43-R59 (2002).
[CrossRef]

Derosier, R. M.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck and R. M. Derosier, "Four-photon mixing and highspeed WDM systems," J. Lightwave Technol. 13, 841-849 (1995).
[CrossRef]

Devgan, P.

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan and P. Kumar, "In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier," IEEE Photon. Technol. Lett. 17, 1845-1847 (2005).
[CrossRef]

Forghieri, F.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck and R. M. Derosier, "Four-photon mixing and highspeed WDM systems," J. Lightwave Technol. 13, 841-849 (1995).
[CrossRef]

Gnauck, A. H.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck and R. M. Derosier, "Four-photon mixing and highspeed WDM systems," J. Lightwave Technol. 13, 841-849 (1995).
[CrossRef]

Grigoryan, V. S.

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan and P. Kumar, "In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier," IEEE Photon. Technol. Lett. 17, 1845-1847 (2005).
[CrossRef]

Hansryd, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li and P. O. Hedekvist, "Fiber-based optical parametric amplifiers and their applications," IEEE J. Sel. Top. Quantum Electron. 8, 506-520 (2002).
[CrossRef]

Hedekvist, P. O.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li and P. O. Hedekvist, "Fiber-based optical parametric amplifiers and their applications," IEEE J. Sel. Top. Quantum Electron. 8, 506-520 (2002).
[CrossRef]

Inoue, K.

K. Inoue, K. Nakanishi, K. Oda and H. Toba, "Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions," J. Lightwave Technol. 12, 1423-1439 (1994).
[CrossRef]

Karttunen, S. J.

S. J. Karttunen and R. R. E. Salomaa, "Electromagnetic field cascading in the beat-wave generation of plasma waves," Phys. Rev. Lett. 56, 604-607 (1986).
[CrossRef] [PubMed]

Kaufman, A. N.

B. I. Cohen, A. N. Kaufman and K. M. Watson, "Beat heating of a plasma," Phys. Rev. Lett. 29, 581-584 (1972).
[CrossRef]

Kumar, P.

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan and P. Kumar, "In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier," IEEE Photon. Technol. Lett. 17, 1845-1847 (2005).
[CrossRef]

Li, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li and P. O. Hedekvist, "Fiber-based optical parametric amplifiers and their applications," IEEE J. Sel. Top. Quantum Electron. 8, 506-520 (2002).
[CrossRef]

Li, J. S.

McKinstrie, C. J.

Nakanishi, K.

K. Inoue, K. Nakanishi, K. Oda and H. Toba, "Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions," J. Lightwave Technol. 12, 1423-1439 (1994).
[CrossRef]

Oda, K.

K. Inoue, K. Nakanishi, K. Oda and H. Toba, "Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions," J. Lightwave Technol. 12, 1423-1439 (1994).
[CrossRef]

Radic, S.

Raymer, M. G.

C. J. McKinstrie, M. G. Raymer, S. Radic and M. V. Vasilyev, "Quantum mechanics of phase-sensitive amplification in a fiber," Opt. Commun. 257, 146-163 (2006).
[CrossRef]

C. J. McKinstrie, M. Yu, M. G. Raymer and S. Radic, "Quantum noise properties of parametric processes," Opt. Express 13, 4986-5012 (2005).
[CrossRef] [PubMed]

Salomaa, R. R. E.

S. J. Karttunen and R. R. E. Salomaa, "Electromagnetic field cascading in the beat-wave generation of plasma waves," Phys. Rev. Lett. 56, 604-607 (1986).
[CrossRef] [PubMed]

Tang, R.

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan and P. Kumar, "In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier," IEEE Photon. Technol. Lett. 17, 1845-1847 (2005).
[CrossRef]

Tkach, R. W.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck and R. M. Derosier, "Four-photon mixing and highspeed WDM systems," J. Lightwave Technol. 13, 841-849 (1995).
[CrossRef]

Toba, H.

K. Inoue, K. Nakanishi, K. Oda and H. Toba, "Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions," J. Lightwave Technol. 12, 1423-1439 (1994).
[CrossRef]

Trillo, S.

Vasilyev, M. V.

C. J. McKinstrie, M. G. Raymer, S. Radic and M. V. Vasilyev, "Quantum mechanics of phase-sensitive amplification in a fiber," Opt. Commun. 257, 146-163 (2006).
[CrossRef]

Voss, P. L.

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan and P. Kumar, "In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier," IEEE Photon. Technol. Lett. 17, 1845-1847 (2005).
[CrossRef]

Watson, K. M.

B. I. Cohen, A. N. Kaufman and K. M. Watson, "Beat heating of a plasma," Phys. Rev. Lett. 29, 581-584 (1972).
[CrossRef]

Westlund, M.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li and P. O. Hedekvist, "Fiber-based optical parametric amplifiers and their applications," IEEE J. Sel. Top. Quantum Electron. 8, 506-520 (2002).
[CrossRef]

Yu, M.

IEEE J. Sel. Top. Quantum Electron.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li and P. O. Hedekvist, "Fiber-based optical parametric amplifiers and their applications," IEEE J. Sel. Top. Quantum Electron. 8, 506-520 (2002).
[CrossRef]

IEEE Photon. Technol. Lett.

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan and P. Kumar, "In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier," IEEE Photon. Technol. Lett. 17, 1845-1847 (2005).
[CrossRef]

J. Lightwave Technol.

K. Inoue, K. Nakanishi, K. Oda and H. Toba, "Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions," J. Lightwave Technol. 12, 1423-1439 (1994).
[CrossRef]

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck and R. M. Derosier, "Four-photon mixing and highspeed WDM systems," J. Lightwave Technol. 13, 841-849 (1995).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. D: Appl. Phys.

S. T. Cundiff, "Phase stabilization of ultrashort optical pulses," J. Phys. D: Appl. Phys. 35, R43-R59 (2002).
[CrossRef]

Opt. Commun.

C. J. McKinstrie, M. G. Raymer, S. Radic and M. V. Vasilyev, "Quantum mechanics of phase-sensitive amplification in a fiber," Opt. Commun. 257, 146-163 (2006).
[CrossRef]

Opt. Express

Phys. Rev. Lett.

B. I. Cohen, A. N. Kaufman and K. M. Watson, "Beat heating of a plasma," Phys. Rev. Lett. 29, 581-584 (1972).
[CrossRef]

S. J. Karttunen and R. R. E. Salomaa, "Electromagnetic field cascading in the beat-wave generation of plasma waves," Phys. Rev. Lett. 56, 604-607 (1986).
[CrossRef] [PubMed]

Other

I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, 1994), pp. 987 and 994.

I. P. Kaminow and T. L. Koch, Editors, Optical Fiber Telecommunications IIIA and IIIB (Academic Press, 1997).

<jrn>. S. Radic and C. J. McKinstrie, "Optical amplification and signal processing in highly-nonlinear optical fiber," IEICE Trans. Electron. E88C, 859-869 (2005).</jrn>
[CrossRef]

C. J. McKinstrie, S. Radic and A. R. Chraplyvy, "Parametric amplifiers driven by two pump waves," IEEE J. Sel. Top. Quantum. Electron. 8, 538-547 and 956 (2002).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Mode powers plotted as functions of mode number for (a) x=1 (left) and (b) x=3 (right).

Fig. 2.
Fig. 2.

(a) Mode powers plotted as functions of the distance parameter x. The solid, dot-dashed and dashed curves represent modes -1, -3 and -5, respectively. (b) Power asymmetry plotted as a function of distance.

Fig. 3.
Fig. 3.

Mode powers plotted as functions of mode number for (a) x=1 and (b) x=3. Red bars denote pumps, whereas blue bars denote signals. The input-signal phase ϕ0=0.

Fig. 4.
Fig. 4.

(a) Mode powers plotted as functions of the distance parameter x. The solid, dot-dashed and dashed curves represent modes 0, -2 and -4, respectively. (b) Total pump and signal powers plotted as functions of distance. The solid and dashed curves represent the pumps and signals, respectively. The input-signal phase ϕ0=0.

Fig. 5.
Fig. 5.

Mode powers plotted as functions of mode number for (a) x=1 and (b) x=3. Red bars denote pumps, whereas blue bars denote signals. The input-signal phase ϕ 0=π/2.

Fig. 6.
Fig. 6.

(a) Mode powers plotted as functions of the distance parameter x. The solid, dot-dashed and dashed curves represent modes 0, -2 and -4, respectively. (b) Total pump and signal powers plotted as functions of distance. The solid and dashed curves represent the pumps and signals, respectively. The input-signal phase ϕ0=π/2.

Equations (38)

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

i z A = β ( i τ ) A + γ A 2 A ,
A ( τ , z ) = A ( τ , 0 ) exp [ i γ A ( τ , 0 ) 2 z ] .
A ( τ , 0 ) = ρ + exp ( i ϕ ) + ρ exp ( i ϕ ) ,
exp ( ix cos θ ) = m i m J m ( x ) exp ( im θ ) ,
A ( τ , z ) = n A n ( x ) exp ( in ϕ ) ,
A n ( x ) = i ( n 1 ) 2 ρ + J ( n 1 ) 2 ( x ) + i ( n + 1 ) 2 ρ J ( n + 1 ) 2 ( x ) ,
n = J n 2 ( x ) = 1 .
n A n ( x ) 2 = ρ + 2 + ρ 2 ,
n = 1 [ A n ( x ) 2 A n ( x ) 2 ] = ( ρ + 2 ρ 2 ) J 0 2 ( x ) .
n = 1 A n ( x ) A n ( x ) constant .
A ( τ , 0 ) = ρ exp ( i ϕ ) + ρ 0 exp ( i ϕ 0 ) + ρ exp ( i ϕ ) ,
A n ( x ) = m i n m J m ( x ) [ ρ 0 exp ( i ϕ 0 ) J n 2 m ( ε 0 x ) 2 i ρ J n 2 m ( ε 0 x ) ] ,
A n ( x ) i n 2 ρ 0 { J n 2 ( x ) [ exp ( i ϕ 0 ) + 2 ix cos ϕ 0 ] + J n 2 ( x ) 2 x cos ϕ 0 } ,
d z B 3 i 1 c B 3 + i 1 d + i 2 n B 3 * ,
B 3 ( z ) i 1 d z + 1 d ( 2 n 1 c ) z 2 2 .
d z B 1 i 1 d B 3 B 1 B 1 * + i 2 n B 3 B 3 B 1 * ,
B 1 ( z ) 1 1 d 2 z 2 2 .
A 1 ( z ) exp ( iz ) 1 z 2 2 ,
A 3 ( z ) exp ( iz ) iz + z 2 2 ,
A 0 ( z ) exp ( iz ) ( c 0 + i s 0 ) + ( s 0 + 3 i c 0 ) z 3 ( c 0 + i s 0 ) z 2 2 ,
A 2 ( z ) exp ( iz ) ( 3 i c 0 s 0 ) z + ( i s 0 c 0 ) z 2 ,
d z B 0 = i 1 c B 0 + i 2 m B 0 * ,
B 0 ( z ) = ( c 0 + i s 0 ) + [ ( 2 m 1 c ) s 0 + i ( 2 m + 1 c ) c 0 ] z
+ ( 2 m 2 1 c 2 ) ( c 0 + i s 0 ) z 2 2 .
d z B 2 i 1 c B 2 + i 1 m B 0 * + i 2 b B 0 + i 2 p B 2 * ,
B 2 [ i ( 1 m + 2 b ) c 0 + ( 1 m 2 b ) s 0 ] z .
d z B 2 i 2 b ( i 1 d z ) B 0 + i 2 p ( i 1 d z ) B 0 * + i 2 b ( i 1 d z ) B 0 ,
B 2 1 d 2 p ( i s 0 c 0 ) z 2 2 .
d z B 0 i 1 m B 2 * + i 2 b B 2 + i 1 m B 2 * + i 2 b B 2 ,
B 0 ( 2 b 2 1 m 2 ) ( c 0 + i s 0 ) z 2 .
d z B 0 = i δ B 0 + i 2 γ B 1 B 1 B 0 * ,
d z B 0 = i δ B 0 + i 2 ρ 2 B 0 * ,
B 0 ( z ) = μ ( z ) B 0 ( 0 ) + ν ( z ) B 0 * ( 0 ) ,
μ ( z ) = cosh ( κ z ) + i δ sinh ( κ z ) κ ,
ν ( z ) = i 2 ρ 2 sinh ( κ z ) κ ,
B 0 ( z ) 2 = ρ 0 2 [ μ 2 + ν i 2 + 2 μ r ν i sin ( 2 ϕ 0 ) + 2 μ i ν i cos ( 2 ϕ 0 ) ] ,
2 ϕ m = tan 1 ( μ r μ i ) .
B 0 ( z ) ( c 0 + i s 0 ) + ( s 0 + 3 i c 0 ) ( ρ 2 z ) + 3 ( c 0 + i s 0 ) ( ρ 2 z ) 2 2 ,

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