Abstract

We analyze modulational instability (MI) of light waves in fiber systems with periodically varying dispersion. The dispersion fluctuation generates special waves, called nonconventional MI sidebands, which are shown to be highly sensitive to two fundamental system parameters. The first one is the average dispersion of the system. Surprisingly, the second parameter turns out to be the mean value of the dispersion coefficients of the two types of fibers of the system, which is then called “central dispersion.” These two parameters are used to control and optimize the MI process. In particular, we establish the existence of a critical region of the central dispersion at which the power gain of the nonconventional sidebands undergoes a dramatic enhancement.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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  21. N. J. Smith and N. J. Doran, “Modulation instability in fibers with periodic dispersion management,” Opt. Lett. 21, 570-572 (1996).
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    [CrossRef]
  23. F. Kh. Abdullaev and J. Garnier, “Modulational instability of electromagnetic waves in birefringent fibers with periodic and random dispersion,” Phys. Rev. E 60, 1042-1050 (1999).
    [CrossRef]
  24. F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, and M. P. Sorensen, “Modulational instability of electromagnetic waves in media with varying nonlinearity,” J. Opt. Soc. Am. B 14, 27-33 (1997).
    [CrossRef]
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  26. F. Matera, A. Mecozzi, M. Romagnoli, and M. Settembre, “Sideband instability induced by periodic power variation in long-distance fiber links,” Opt. Lett. 18, 1499-1501 (1993).
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  27. K. Kikuchi, C. Lorattanasane, F. Futami, and S. Kaneko, “Observation of quasiphase matched four-wave mixing assisted by periodic power variation in a long-distance optical amplifier chain,” IEEE Photon. Technol. Lett. 7, 1378-1380 (1995).
    [CrossRef]
  28. S. Trillo and S. Wabnitz, “Bloch wave theory of modulational polarization instabilities in birefringent optical fibers,” Phys. Rev. E 56, 1048-1058 (1997).
    [CrossRef]
  29. G. Millot, P. Tchofo Dinda, E. Seve, and S. Wabnitz, “Modulational instability and stimulated Raman scattering in normally dispersive highly birefringent fibers,” Opt. Fiber Technol. 7, 170-205 (2001).
    [CrossRef]
  30. R. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1-68 (1977).
    [CrossRef]
  31. C. Lin, “Designing optical fibers for frequency conversion and optical amplification by stimulated Raman scattering and phase matched four-photon mixing,” J. Opt. Commun. 4, 2-9 (1983).
    [CrossRef]
  32. P. Tchofo Dinda, G. Millot, and S. Wabnitz, “Polarization switching and suppression of stimulated Raman scattering in birefringent optical fibers,” J. Opt. Soc. Am. B 15, 1433-1441 (1998).
    [CrossRef]

2007 (1)

2006 (1)

2005 (1)

2003 (2)

J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. 28, 2225-2227 (2003).
[CrossRef] [PubMed]

A. Kumar, A. Labruyére, and P. Tchofo Dinda, “Modulational instability in fiber systems with periodic loss compensation and dispersion management,” Opt. Commun. 219, 221-232 (2003).
[CrossRef]

2002 (2)

2001 (2)

J. Garnier, F. Kh. Abdullaev, E. Seve, and S. Wabnitz, “Role of polarization mode dispersion on modulational instability in optical fibers,” Phys. Rev. E 63, 066616 (2001).
[CrossRef]

G. Millot, P. Tchofo Dinda, E. Seve, and S. Wabnitz, “Modulational instability and stimulated Raman scattering in normally dispersive highly birefringent fibers,” Opt. Fiber Technol. 7, 170-205 (2001).
[CrossRef]

1999 (1)

F. Kh. Abdullaev and J. Garnier, “Modulational instability of electromagnetic waves in birefringent fibers with periodic and random dispersion,” Phys. Rev. E 60, 1042-1050 (1999).
[CrossRef]

1998 (1)

1997 (2)

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, and M. P. Sorensen, “Modulational instability of electromagnetic waves in media with varying nonlinearity,” J. Opt. Soc. Am. B 14, 27-33 (1997).
[CrossRef]

S. Trillo and S. Wabnitz, “Bloch wave theory of modulational polarization instabilities in birefringent optical fibers,” Phys. Rev. E 56, 1048-1058 (1997).
[CrossRef]

1996 (3)

1995 (1)

K. Kikuchi, C. Lorattanasane, F. Futami, and S. Kaneko, “Observation of quasiphase matched four-wave mixing assisted by periodic power variation in a long-distance optical amplifier chain,” IEEE Photon. Technol. Lett. 7, 1378-1380 (1995).
[CrossRef]

1993 (1)

1990 (2)

J. E. Rothenberg, “Modulation instability for normal dispersion,” Phys. Rev. A 42, 682-685 (1990).
[CrossRef] [PubMed]

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137-142 (1990).
[CrossRef]

1989 (1)

S. Sudo, H. Itoh, K. Okamoto, and K. Kubodera, “Generation of 5 THz repetition optical pulses by modulation instability in optical fibers,” Appl. Phys. Lett. 54, 993-994 (1989).
[CrossRef]

1988 (1)

S. Wabnitz, “Modulation polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018-2021 (1988).
[CrossRef] [PubMed]

1987 (1)

G. P. Agrawal, “Modulational instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880-883 (1987).
[CrossRef] [PubMed]

1986 (2)

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135-138 (1986).
[CrossRef] [PubMed]

N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. 69, 1089-1092 (1986).
[CrossRef]

1984 (1)

1983 (1)

C. Lin, “Designing optical fibers for frequency conversion and optical amplification by stimulated Raman scattering and phase matched four-photon mixing,” J. Opt. Commun. 4, 2-9 (1983).
[CrossRef]

1977 (1)

R. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1-68 (1977).
[CrossRef]

1970 (1)

A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486-490 (1970).

Appl. Phys. Lett. (1)

S. Sudo, H. Itoh, K. Okamoto, and K. Kubodera, “Generation of 5 THz repetition optical pulses by modulation instability in optical fibers,” Appl. Phys. Lett. 54, 993-994 (1989).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

K. Kikuchi, C. Lorattanasane, F. Futami, and S. Kaneko, “Observation of quasiphase matched four-wave mixing assisted by periodic power variation in a long-distance optical amplifier chain,” IEEE Photon. Technol. Lett. 7, 1378-1380 (1995).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Commun. (1)

C. Lin, “Designing optical fibers for frequency conversion and optical amplification by stimulated Raman scattering and phase matched four-photon mixing,” J. Opt. Commun. 4, 2-9 (1983).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (3)

F. Consolandi, C. De Angelis, A. D. Capobianco, G. Nalesso, and A. Tonello, “Parametric gain in fiber systems with periodic dispersion management,” Opt. Commun. 208, 309-320 (2002).
[CrossRef]

A. Kumar, A. Labruyére, and P. Tchofo Dinda, “Modulational instability in fiber systems with periodic loss compensation and dispersion management,” Opt. Commun. 219, 221-232 (2003).
[CrossRef]

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137-142 (1990).
[CrossRef]

Opt. Express (1)

Opt. Fiber Technol. (1)

G. Millot, P. Tchofo Dinda, E. Seve, and S. Wabnitz, “Modulational instability and stimulated Raman scattering in normally dispersive highly birefringent fibers,” Opt. Fiber Technol. 7, 170-205 (2001).
[CrossRef]

Opt. Lett. (6)

Phys. Lett. A (1)

F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers with variable dispersion,” Phys. Lett. A 220, 213-218 (1996).
[CrossRef]

Phys. Rev. A (2)

S. Wabnitz, “Modulation polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018-2021 (1988).
[CrossRef] [PubMed]

J. E. Rothenberg, “Modulation instability for normal dispersion,” Phys. Rev. A 42, 682-685 (1990).
[CrossRef] [PubMed]

Phys. Rev. E (3)

J. Garnier, F. Kh. Abdullaev, E. Seve, and S. Wabnitz, “Role of polarization mode dispersion on modulational instability in optical fibers,” Phys. Rev. E 63, 066616 (2001).
[CrossRef]

F. Kh. Abdullaev and J. Garnier, “Modulational instability of electromagnetic waves in birefringent fibers with periodic and random dispersion,” Phys. Rev. E 60, 1042-1050 (1999).
[CrossRef]

S. Trillo and S. Wabnitz, “Bloch wave theory of modulational polarization instabilities in birefringent optical fibers,” Phys. Rev. E 56, 1048-1058 (1997).
[CrossRef]

Phys. Rev. Lett. (2)

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135-138 (1986).
[CrossRef] [PubMed]

G. P. Agrawal, “Modulational instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880-883 (1987).
[CrossRef] [PubMed]

Prog. Quantum Electron. (1)

R. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1-68 (1977).
[CrossRef]

Sov. Phys. JETP (1)

A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486-490 (1970).

Theor. Math. Phys. (1)

N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. 69, 1089-1092 (1986).
[CrossRef]

Other (3)

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).

K. Pasu and A. Tuptim, “Sideband instability in the presence of periodic power variation and periodic dispersion management,” in Proceedings of the Optical Fiber Communication Conference (OFC2001) (IEEE, 2001), paper WDD32.

F. Kh. Abdullaev, S. A. Darmanyan, and J. Garnier, “Modulational instability of electromagnetic waves in inhomogeneous and in discrete media,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2002), Vol. 44, pp. 303-365.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic configuration of the dispersion map of the system.

Fig. 2
Fig. 2

MI gain as a function of the average dispersion β av , obtained from the LSA, for β ̃ = 0 ps 2 km , Δ β = 4 ps 2 km , β + = 2 ps 2 km , and Z d = 30 m . (a) Gain spectra. (b) Optimum modulation frequency. (c) Optimum gain.

Fig. 3
Fig. 3

MI gain as a function of the average dispersion β av , obtained from the LSA, for the same parameter as for Fig. 2 but with Δ β = 8 ps 2 km and β + = 4 ps 2 km . (a) Gain spectra. (b) Optimum modulation frequency. (c) Optimum gain.

Fig. 4
Fig. 4

MI gain as a function of the average dispersion β av , obtained from the LSA, for β ̃ = 0.5 ps 2 km , Δ β = 7 ps 2 km , β + = 4 ps 2 km , and Z d = 30 m . (a) Gain spectra. (b) Optimum modulation frequency. (c) Optimum gain.

Fig. 5
Fig. 5

MI gain as a function of the average dispersion β av , obtained from the LSA, for the same parameter as for Fig. 4 but with β ̃ = 2 ps 2 km and Δ β = 4 ps 2 km . (a) Gain spectra. (b) Optimum modulation frequency. (c) Optimum gain.

Fig. 6
Fig. 6

MI gain as a function of the average dispersion β av , obtained from the LSA, for the same parameter as for Fig. 4 but with β ̃ = 3.5 ps 2 km and Δ β = 1 ps 2 km . (a) Gain spectra. (b) Optimum modulation frequency. (c) Optimum gain.

Fig. 7
Fig. 7

MI obtained from the LSA. (a) Optimum gain, (b) optimum modulation frequency, and (c) optimum average dispersion as a function of the central dispersion β ̃ .

Fig. 8
Fig. 8

MI gain spectrum [(a1), (b1), and (c1)] are obtained from the NLSE. [(a2), (b2), and (c2)] are obtained from the LSA. Z d = 30 m , β + = 4 ps 2 km , and ( β ̃ = 1 ps 2 km , β av = 1.84.6 ps 2 km ), ( β ̃ = 2 ps 2 km , β av = 0.07 ps 2 km ), and ( β ̃ = 3 ps 2 km , β av = 2.8 ps 2 km ), respectively. L T = 25 Z d .

Fig. 9
Fig. 9

MI gain spectrum. (a1) and (b1) are obtained from the NLSE. (a2) and (b2) are obtained from the LSA. Z d = 30 m , β + = 4 ps 2 km , and ( β ̃ = 1.9 ps 2 km , β av = 0.59 ps 2 km ) and ( β ̃ = 2.1 ps 2 km , β av = 0.7 ps 2 km ), respectively. Propagation distance L T = 25 Z d .

Fig. 10
Fig. 10

MI gain spectrum. (a) and (b) are obtained from the NLSE. Z d = 30 m , β + = 4 ps 2 km , and ( β ̃ = 1.98 ps 2 km , β av = 0.2 ps 2 km ) and ( β ̃ = 2 ps 2 km , β av = 0.07 ps 2 km ), respectively. ρ = 0.36 . Propagation distance L T = 10 Z d .

Tables (1)

Tables Icon

Table 1 Critical Parameter Regions

Equations (30)

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Ω opt = 2 γ P 0 β ,
q z + i 2 [ β av + β ̃ ( z ) ] 2 q t 2 i γ ( 1 ρ ) q 2 q + α 2 q i γ ρ q 0 q ( z , t s ) 2 χ R ( s ) d s = 0 ,
q ( z , t ) = Q ( z , t ) exp ( α 2 z ) ,
Q z + i 2 [ β av + β ̃ ( z ) ] 2 Q t 2 i γ f ( z ) ( 1 ρ ) Q 2 Q i γ ρ f ( z ) Q 0 Q ( z , t s ) 2 χ R ( s ) d s = 0 ,
f ( z ) = exp ( α z ) .
β av ( β + L + + β L ) Z d ,
Z d L + + L .
Q ( z , t ) = P 0 exp [ i ϕ ( z ) ] ,
ϕ ( z ) = γ P 0 [ 1 exp ( α z ) ] α .
Q ( z , t ) = [ P 0 + a ( z , t ) ] exp [ i ϕ ( z ) ] ,
a ( z , t ) = a ( Ω , z ) exp ( i Ω t ) + a ( Ω , z ) exp ( i Ω t ) ,
d a ( Ω , z ) d z = i M ( Ω , z ) a ( Ω , z ) + i W ( Ω , z ) a * ( Ω , z ) ,
d a * ( Ω , z ) d z = i W ( Ω , z ) a ( Ω , z ) i M ( Ω , z ) a * ( Ω , z ) ,
M ( Ω , z ) η ( Ω , z ) + W ( Ω , z ) ,
W ( Ω , z ) γ P 0 f ( z ) [ 1 ρ + ρ χ ̃ R ( Ω ) ] ,
η ( Ω , z ) η av ( Ω ) + η ̃ ( Ω , z ) = 1 2 [ β av + β ̃ ( z ) ] Ω 2 .
[ b ( Ω , z ) b ( Ω , z ) ] = [ a ( Ω , z ) a ( Ω , z ) ] exp ( i 2 Ω 2 0 z β ̃ ( z ) d z ) .
d d z [ b ( Ω , z ) b * ( Ω , z ) ] = i [ η av ( Ω ) + W ( Ω , z ) W ( Ω , z ) g ( Ω , z ) W ( Ω , z ) g * ( Ω , z ) η av ( Ω ) W ( Ω , z ) ] [ b ( Ω , z ) b * ( Ω , z ) ] ,
g ( Ω , z ) = exp ( i Ω 2 0 z β ̃ ( z ) d z ) .
g ( Ω , z ) = n = + g n ( Ω ) exp ( i n k d z ) ,
g n ( Ω ) = exp [ i ( n k d Ω 2 β ̃ + ) L + 2 ] i Z d ( n k d Ω 2 β ̃ ) { exp [ i ( n k d Ω 2 β ̃ ) L ] 1 } + 1 i Z d ( n k d Ω 2 β ̃ + ) [ { exp [ i ( n k d Ω 2 β ̃ + ) L + 2 ] 1 } × { exp [ i ( n k d Ω 2 β ̃ ) L ] exp [ i ( n k d + Ω 2 β ̃ + ) L + 2 ] + 1 } ] ,
k d = 2 π Z d ,
β ̃ ± = ± Δ β L Z d ,
Δ β = β + β .
[ b ( Ω , z ) b ( Ω , z ) ] = [ v ( Ω , z ) v ( Ω , z ) ] exp ( i 2 p k d z ) ,
d d z [ v ( Ω , z ) v * ( Ω , z ) ] = i M [ v ( Ω , z ) v * ( Ω , z ) ] ,
M = [ η av ( Ω ) + W ( Ω , 0 ) + p k d 2 W ( Ω , 0 ) g p ( Ω ) e i p k d z W ( Ω , 0 ) g p * ( Ω ) e i p k d z η av ( Ω ) W ( Ω , 0 ) p k d 2 ]
K ± = ± [ ( β av Ω 2 + p k d 2 + γ P 0 [ 1 ρ + ρ χ R ( Ω ) ] ) 2 γ 2 P 0 2 [ 1 ρ + ρ χ R ( Ω ) ] 2 g p ( Ω ) 2 ] 1 2 .
G ( Ω ) = 2 I ( K ± ) .
β ̃ = ( β + + β ) 2 ,

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