Abstract

Optical phase conjugation is demonstrated to enable simultaneous wide-band compensation of the residual dispersion and the fiber nonlinearities in dispersion-managed fiber transmission lines employing slope-compensating fibers. When the dispersion slope of transmission fibers is equalized by slope-compensating fibers, the residual dispersion and the slope of dispersion slope are compensated by middle-span optical phase conjugation. More importantly, fiber nonlinearity may be largely suppressed by arranging the fibers into conjugate pairs about the phase conjugator, where the two fibers of each pair are in scaled translational symmetry. The translational symmetry is responsible for cancelling optical nonlinearities of the two fibers up to the first-order perturbation, then a mirror-symmetric ordering of the fiber pairs about the conjugator linearizes a long transmission line effectively.

© 2004 Optical Society of America

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References

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    [CrossRef]
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  13. M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, �??1.5- µm-band wavelength conversion based on cascaded second-order nonlinearity in LiNbO3 waveguides,�?? IEEE Photon. Technol. Lett. 11, 653-655 (1999).
    [CrossRef]
  14. S. Radic, R. M. Jopson, C. J. McKinstrie, A. H. Gnauck, S. Chandrasekhar, and J. C. Centanni, �??Wavelength division multiplexed transmission over standard single mode fiber using polarization insensitive signal conjugation in highly nonlinear optical fiber,�?? OFC 2003, paper PD12.
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  18. K. Rottwitt and A. J. Stentz, �??Raman amplification in lightwave communication systems,�?? in Optical Fiber Telecommunications IV A: Components, I. P. Kaminow and T. Li, eds. (Academic Press, San Diego, 2002).
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    [CrossRef]
  21. J.-C. Bouteiller, K. Brar, and C. Headley, �??Quasi-constant signal power transmission,�?? European Conference on Optical Communication 2002, paper S3.04.
  22. M. Vasilyev, �??Raman-assisted transmission: toward ideal distributed amplification,�?? OFC 2003, paper WB1.
  23. M. E. Marhic, N. Kagi, T.-K. Chiang, and L. G. Kazovsky, �??Cancellation of third-order nonlinear effects in amplified fiber links by dispersion compensation, phase conjugation, and alternating dispersion,�?? Opt. Lett. 20, no. 8, 863-865 (1995).
    [CrossRef] [PubMed]
  24. A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University Press, 1997), Chapter 3.
  25. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic Press, San Diego, 1995), Chapter 2.
  26. The use of βk�??s defined by Eq. (4) in the NLSE is connected to an approximation β2(w)-β2�?? 2β[β((w)-βо] with sacrificed accuracy. An alternative definition in Ref. [17] may be used for better accruacy.
  27. K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations (Springer-Verlag, New York, 2000).
  28. E. E. Narimanov and P. Mitra, �??The channel capacity of a fiber optics communication system: perturbation theory,�?? J. Lightwave Technol. 20, 530-537 (2002).
    [CrossRef]
  29. J. A. Buck, Fundamentals of Optical Fibers (Wiley, New York, 1995), Chapter 4.
  30. C. Rasmussen, T. Fjelde, J. Bennike, F. Liu, S. Dey, B. Mikkelsen, P. Mamyshev, P. Serbe, P. van der Wagt, Y. Akasaka, D. Harris, D. Gapontsev, V. Ivshin, P. Reeves-Hall, �??DWDM 40G transmission over trans-Pacific distance (10,000 km) using CSRZ-DPSK, enhanced FEC and all-Raman amplified 100 km UltraWaveTM fiber spans,�?? OFC 2003, paper PD18.
  31. L. Gruner-Nielsen, Y. Qian, B. Palsdottir, P. B. Gaarde, S. Dyrbol, T. Veng, and Y. Qian, �??Module for simultaneous C + L-band dispersion compensation and Raman amplification,�?? OFC 2002, paper TuJ6.
  32. T. Miyamoto, T. Tsuzaki, T. Okuno, M. Kakui, M. Hirano, M. Onishi, and M. Shigematsu, �??Raman amplification over 100 nm-bandwidth with dispersion and dispersion slope compensation for conventional single mode fiber,�?? OFC 2002, paper TuJ7.
  33. M. Eiselt, M. Shtaif, R. W. Tkach, F. A. Flood, S. Ten, and D. Butler, �??Cross-phase modulation in an L-band EDFA,�?? IEEE Photon. Technol. Lett. 11, 1575-1577 (1999).
    [CrossRef]
  34. H. S. Chung, S. K. Shin, D. W. Lee, D. W. Kim, and Y. C. Chung, �??640Gbit/s (32�?20Gbit/s) WDM transmission with 0.4(bit/s)/Hz spectral efficiency using short-period dispersion-managed fiber,�?? Elec. Lett. 37, 618-620 (2001).
    [CrossRef]
  35. R.-J. Essiambre, G. Raybon, and B. Mikkelson, �??Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s,�?? in Optical Fiber Telecommunications IV B: Systems and Impairments, I. P. Kaminow and T. Li, eds. (Academic Press, San Diego, 2002).
  36. P. Kaewplung, T. Angkaew, and K. Kikuchi, �??Simultaneous suppression of third-order dispersion and sideband instability in single-channel optical fiber transmission by midway optical phase conjugation employing higher order dispersion management,�?? J. Lightwave Technol. 21, 1465-1473 (2003).
    [CrossRef]
  37. For example, a nearly perfect translational symmetry may be formed between Corning�??s LEAF, a +NZDSF with D�??4 ps/nm/km, S�??0.1 ps/nm2/km, and its Vascade LEAF, a -NZDSF with D�??-4 ps/nm/km, S�??0.1 ps/nm2/km in the C band. The fiber parameters are available at <a href="http://www.corning.com/opticalfiber">http://www.corning.com/opticalfiber<a>.
  38. F. Forghieri, R.W. Tkach, A. R. Chraplyvy, and D. Marcuse, �??Reduction of four-wave mixing crosstalk in WDM systems using unequally spaced channels,�?? IEEE Photon. Technol. Lett. 6, 754-756 (1994).
    [CrossRef]
  39. F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, �??WDM systems with unequally spaced channels,�?? J. Lightwave Technol. 13, 889-897 (1995).
    [CrossRef]

arXiv:physics/0307020 (1)

H.Wei and D. V. Plant, �??On the capacity of nonlinear fiber channels,�?? arXiv:physics/0307020 at, <a href= " http://arxiv.org/">http://arxiv.org</a>

Elec. Lett. (1)

H. S. Chung, S. K. Shin, D. W. Lee, D. W. Kim, and Y. C. Chung, �??640Gbit/s (32�?20Gbit/s) WDM transmission with 0.4(bit/s)/Hz spectral efficiency using short-period dispersion-managed fiber,�?? Elec. Lett. 37, 618-620 (2001).
[CrossRef]

Electron. Lett. (1)

M. Vasilyev, B. Szalabofka, S. Tsuda, J. M. Grochocinski, and A. F. Evans, �??Reduction of Raman MPI and noise figure in dispersion-managed fiber,�?? Electron. Lett. 38, no. 6, 271-272 (2002).
[CrossRef]

Eur. Conf. on Optical Commun. 2002 (1)

J.-C. Bouteiller, K. Brar, and C. Headley, �??Quasi-constant signal power transmission,�?? European Conference on Optical Communication 2002, paper S3.04.

Eur. Conf. on Optical Communic. 2001 (1)

M. J. Li, �??Recent progress in fiber dispersion compensators,�?? European Conference on Optical Communication 2001, paper Th.M.1.1.

IEEE Photon. Technol. Lett. (3)

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, �??1.5- µm-band wavelength conversion based on cascaded second-order nonlinearity in LiNbO3 waveguides,�?? IEEE Photon. Technol. Lett. 11, 653-655 (1999).
[CrossRef]

F. Forghieri, R.W. Tkach, A. R. Chraplyvy, and D. Marcuse, �??Reduction of four-wave mixing crosstalk in WDM systems using unequally spaced channels,�?? IEEE Photon. Technol. Lett. 6, 754-756 (1994).
[CrossRef]

M. Eiselt, M. Shtaif, R. W. Tkach, F. A. Flood, S. Ten, and D. Butler, �??Cross-phase modulation in an L-band EDFA,�?? IEEE Photon. Technol. Lett. 11, 1575-1577 (1999).
[CrossRef]

J. Lightwave Technol. (4)

F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, �??WDM systems with unequally spaced channels,�?? J. Lightwave Technol. 13, 889-897 (1995).
[CrossRef]

E. E. Narimanov and P. Mitra, �??The channel capacity of a fiber optics communication system: perturbation theory,�?? J. Lightwave Technol. 20, 530-537 (2002).
[CrossRef]

P. Kaewplung, T. Angkaew, and K. Kikuchi, �??Simultaneous suppression of third-order dispersion and sideband instability in single-channel optical fiber transmission by midway optical phase conjugation employing higher order dispersion management,�?? J. Lightwave Technol. 21, 1465-1473 (2003).
[CrossRef]

S. Watanabe and M. Shirasaki, �??Exact compensation for both chromatic dispersion and Kerr effect in a transmission fiber using optical phase conjugation,�?? J. Lightwave Technol. 14, 243-248 (1996).
[CrossRef]

OFC 2000 (2)

I. Brener, B. Mikkelsen, K. Rottwitt, W. Burkett, G. Raybon, J. B. Stark, K. Parameswaran, M. H. Chou, M. M. Fejer, E. E. Chaban, R. Harel, D. L. Philen, and S. Kosinski, �??Cancellation of all Kerr nonlinearities in long fiber spans using a LiNbO3 phase conjugator and Raman amplification,�?? OFC 2000, paper PD33.

S. N. Knudsen and T. Veng, �??Large effective area dispersion compensating fiber for cabled compensation of standard single mode fiber,�?? OFC 2000, paper TuG5.

OFC 2001 (2)

Q. L. N.T., T. Veng, and L. Gruner-Nielsen, �??New dispersion compensating module for compensation of dispersion and dispersion slope of non-zero dispersion fibres in the C-band,�?? OFC 2001, paper TuH5.

V. Srikant, �??Broadband dispersion and dispersion slope compensation in high bit rate and ultra long haul systems,�?? OFC 2001, paper TuH1.

OFC 2002 (4)

L. Gruner-Nielsen, Y. Qian, B. Palsdottir, P. B. Gaarde, S. Dyrbol, T. Veng, and Y. Qian, �??Module for simultaneous C + L-band dispersion compensation and Raman amplification,�?? OFC 2002, paper TuJ6.

T. Miyamoto, T. Tsuzaki, T. Okuno, M. Kakui, M. Hirano, M. Onishi, and M. Shigematsu, �??Raman amplification over 100 nm-bandwidth with dispersion and dispersion slope compensation for conventional single mode fiber,�?? OFC 2002, paper TuJ7.

K. Mukasa, H. Moridaira, T. Yagi, and K. Kokura, �??New type of dispersion management transmission line with MDFSD for long-haul 40 Gb/s transmission,�?? OFC 2002, paper ThGG2.

M. Gorlier, P. Sillard, F. Beaumont, L.-A. de Montmorillon, L. Fleury, Ph. Guenot, A. Bertaina, and P. Nouchi, �??Optimized NZDSF-based link for wide-band seamless terrestrial transmissions,�?? OFC 2002, paper ThGG7.

OFC 2003 (3)

M. Vasilyev, �??Raman-assisted transmission: toward ideal distributed amplification,�?? OFC 2003, paper WB1.

C. Rasmussen, T. Fjelde, J. Bennike, F. Liu, S. Dey, B. Mikkelsen, P. Mamyshev, P. Serbe, P. van der Wagt, Y. Akasaka, D. Harris, D. Gapontsev, V. Ivshin, P. Reeves-Hall, �??DWDM 40G transmission over trans-Pacific distance (10,000 km) using CSRZ-DPSK, enhanced FEC and all-Raman amplified 100 km UltraWaveTM fiber spans,�?? OFC 2003, paper PD18.

S. Radic, R. M. Jopson, C. J. McKinstrie, A. H. Gnauck, S. Chandrasekhar, and J. C. Centanni, �??Wavelength division multiplexed transmission over standard single mode fiber using polarization insensitive signal conjugation in highly nonlinear optical fiber,�?? OFC 2003, paper PD12.

Opt. Lett. (3)

Proc. SPIE (1)

H.Wei and D. V. Plant, �??Fundamental equations of nonlinear fiber optics,�?? in Optical Modeling and Performance Predictions, M. A. Kahan, ed., Proc. SPIE 5178, 255-266 (2003).

Other (12)

K. Rottwitt and A. J. Stentz, �??Raman amplification in lightwave communication systems,�?? in Optical Fiber Telecommunications IV A: Components, I. P. Kaminow and T. Li, eds. (Academic Press, San Diego, 2002).

E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (John Wiley & Sons, New York, 1994).

J. A. Buck, Fundamentals of Optical Fibers (Wiley, New York, 1995), Chapter 4.

H. Wei and D. V. Plant, �??Two means of compensating fiber nonlinearity using optical phase conjugation arXiv:physics/03070202 at <a href="http://arxiv.org/">http://arxiv.org/<a>

For example, a nearly perfect translational symmetry may be formed between Corning�??s LEAF, a +NZDSF with D�??4 ps/nm/km, S�??0.1 ps/nm2/km, and its Vascade LEAF, a -NZDSF with D�??-4 ps/nm/km, S�??0.1 ps/nm2/km in the C band. The fiber parameters are available at <a href="http://www.corning.com/opticalfiber">http://www.corning.com/opticalfiber<a>.

R.-J. Essiambre, G. Raybon, and B. Mikkelson, �??Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s,�?? in Optical Fiber Telecommunications IV B: Systems and Impairments, I. P. Kaminow and T. Li, eds. (Academic Press, San Diego, 2002).

A. H. Gnauck and R. M. Jopson, �??Dispersion compensation for optical fiber systems,�?? in Optical Fiber Telecommunications III A, I. P. Kaminow and T. L. Koch, eds. (Academic Press, San Diego, 1997).

F. Forghieri, R. W. Tkach and A. R. Chraplyvy, �??Fiber nonlinearities and their impact on transmission systems,�?? in Optical Fiber Telecommunications III A, I. P. Kaminow and T. L. Koch, eds. (Academic Press, San Diego, 1997).

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University Press, 1997), Chapter 3.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic Press, San Diego, 1995), Chapter 2.

The use of βk�??s defined by Eq. (4) in the NLSE is connected to an approximation β2(w)-β2�?? 2β[β((w)-βо] with sacrificed accuracy. An alternative definition in Ref. [17] may be used for better accruacy.

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations (Springer-Verlag, New York, 2000).

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Figures (11)

Fig. 1.
Fig. 1.

Two fiber spans in translational symmetry about an optical phase conjugator. The shaded areas represent two typical fiber segments that are in scaled translational symmetry about the conjugator.

Fig. 2.
Fig. 2.

A mirror-symmetric configuration of pairs of fiber spans in scaled translational symmetry, with the dispersion in each span compensated to zero. Top: schematic arrangement of fibers and amplifiers with respect to OPC. Middle: map of signal power P(z) along the propagation distance z. Bottom: map of accumulated dispersion b 2(z) along the propagation distance z.

Fig. 3.
Fig. 3.

A mirror-symmetric configuration of pairs of fiber spans in scaled translational symmetry, with non-zero residual dispersion in the spans. There are pre- and post-dispersion compensators (DCs), as well as a dispersion conditioner immediately after OPC. Top: schematic arrangement of fibers and amplifiers with respect to OPC. Middle: map of signal power P(z) along the propagation distance z. Bottom: map of accumulated dispersion b 2(z) along the propagation distance z.

Fig. 4.
Fig. 4.

A mirror-symmetric configuration of pairs of fiber spans in scaled translational symmetry, with non-zero residual dispersion in the spans. There are pre- and post-dispersion compensators (DCs) but no dispersion conditioner at the site of OPC. Top: schematic arrangement of fibers and amplifiers with respect to OPC. Middle: map of signal power P(z) along the propagation distance z. Bottom: map of accumulated dispersion b 2(z) along the propagation distance z.

Fig. 5.
Fig. 5.

A transmission line consisting of SMFs and slope-matching DCFs.

Fig. 6.
Fig. 6.

Received eye diagrams of the 2nd DEMUX channel. Top row: transmission results of the setup in Fig. 5. Top-left: fiber nonlinearity is OFF, the signal is only impaired by amplifier noise. Top-right: fiber nonlinearity is ON, the signal distortion is only increased slightly. Bottom row: transmission results when the setup is modified, and the fiber non-linearity is always ON. Bottom-left: fiber lengths of and input powers to the two types of spans are exactly the same. Bottom-right: all fiber spans are identical in length and input signal power as well as the ordering of fibers (SMF followed by DCF).

Fig. 7.
Fig. 7.

A transmission line consisting of +NZDSFs, -NZDSFs, and DCFs compensating the dispersion slope.

Fig. 8.
Fig. 8.

Received eye diagrams of the 2nd DEMUX channel. Top row: transmission results of the setup in Fig. 7. Top-left: fiber nonlinearity is OFF, the signal is only impaired by amplifier noise. Top-right: fiber nonlinearity is ON, no extra penalty is visible. Bottom row: degraded transmission results when all -NZDSFs are replaced by +NZDSFs. Bottom-left: with OPC. Bottom-right: without OPC, of the 3rd MUX/DEMUX channel.

Fig. 9.
Fig. 9.

Scalability and cascadability of the nonlinearity-suppressed NZDSF transmission line in Fig. 7. Left: the number of circulations on each side of OPC is doubled to ten times and the signal power is increased by 3 dB. Right: two identical transmission lines as in Fig. 7 are in cascade all-optically and the signal power is increased by 3 dB. The eye diagrams are still of the 2nd DEMUX channel.

Fig. 10.
Fig. 10.

A transmission line consisting of ten fiber spans on each side of OPC, each span has 50 km DSF and a slope-compensating DCF.

Fig. 11.
Fig. 11.

Received eye diagrams of the 2nd DEMUX channel. Top row: transmission results of the setup in Fig. 10. Top-left: fiber nonlinearity is OFF, the signal is only impaired by amplifier noise. Top-right: fiber nonlinearity is ON. Bottom row: transmission results when the setup in Fig. 10 is modified by setting D = 0 ps/nm/km for the DCFs while keeping the dispersion slope. Bottom-left: with OPC in the middle of the link. Bottom-right: when OPC is removed.

Equations (26)

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

A ˜ ( ω ) = def A ( t ) = E ( t ) exp [ i ( ω 0 + ω ) t ] dt ,
H ( ω ) = exp ( i k = 2 + b k ω k k ! ) ,
b k = β k ( z ) dz , k 2 ,
β k ( z ) = def k β z ω ω k ω = ω 0 , k 2 ,
b k R = ( 1 ) k b k L , k 2 ,
H R ( ω ) OPC [ H L ( ω ) A ˜ ( ω ) ] = H R ( ω ) H L * ( ω ) A ˜ * ( ω ) = A ˜ * ( ω ) .
E z t = A z t exp [ i z β 0 ( ζ ) i ω 0 t ] ,
A z t z + k = 2 + i k 1 β k ( z ) k ! ( t ) k A z t + α ( z ) 2 A z t =
( z ) A z t 2 A z t + i [ g z t A z t 2 ] A z t ,
A z t z + k = 2 + i k 1 β k ( z ) k ! ( t ) k A z t + α ( z ) 2 A z t = 0 ,
H ( z 1 , z 2 , ω ) = def exp [ i k = 2 + ω k k ! z 1 z 2 β k ( z ) dz 1 2 z 1 z 2 α ( z ) dz ] .
P ( z 1 , z 2 ) = def 1 H ( z 1 , z 2 , ω ) ,
h ( z 1 , z 2 , t ) = def 1 [ H ( z 1 , z 2 , ω ) ] ,
P ( z 1 , z 2 ) = h ( z 1 , z 2 , t ) ,
A 0 ( z 2 , t ) = P ( z 1 , z 2 ) A ( z 1 , t ) ,
A 1 ( z 2 , t ) = z 1 z 2 P ( z , z 2 ) { ( z ) A 0 ( z , t ) 2 A 0 ( z , t )
+ i [ g ( z , t ) A 0 ( z , t ) 2 ] A 0 ( z , t ) } dz ,
α ( z ) = ( Rz ) ,
β k ( z ) = ( 1 ) k 1 R β k ( Rz ) , k 2 ,
γ ( z ) = ( Rz ) ,
g ( z , t ) = Qg ( Rz , t ) , t ( , + ) ,
A ( z , t ) z + k = 2 + i k 1 β k ( z ) k ! ( t ) k A ( z , t ) + α ( z ) 2 A ( z , t ) =
( z ) A ( z , t ) 2 A ( z , t ) + i [ g ( z , t ) A ( z , t ) 2 ] A ( z , t ) ,
A ( z , t ) R z + k = 2 + ( i ) k 1 β k ( Rz ) k ! ( t ) k A ( z , t ) + α ( Rz ) 2 A ( z , t ) =
iQ R 1 γ ( Rz ) A ( z , t ) 2 A ( z , t ) iQ R 1 [ g ( Rz , t ) A ( z , t ) 2 ] A ( z , t ) ,
A ( z , t ) = R / Q A * ( Rz , t ) ,

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