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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 21 — Oct. 17, 2005
  • pp: 8460–8468
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Unifying theory of compensation techniques for intrachannel nonlinear effects

Paolo Minzioni and Alessandro Schiffini  »View Author Affiliations


Optics Express, Vol. 13, Issue 21, pp. 8460-8468 (2005)
http://dx.doi.org/10.1364/OPEX.13.008460


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Abstract

We show a new graphical method to identify and create configurations yielding to nonlinearity compensation in a fiber transmission system. Method validity is shown with regards to different link configurations and different compensation techniques. It is demonstrated that a unifying principle can always be applied, because only one physical effect is involved, even if different practical arrangements are proposed. Disclosed method allows gaining physical insight and can be applied to derive new compensation techniques; two examples of configurations derived using the proposed technique are also reported.

© 2005 Optical Society of America

1. Introduction

2. Mid Span Spectral Inversion and Graphical Approach

Mid Span Spectral Inversion (MSSI) states that it is possible to compensate nonlinear effects completely, by placing an Optical Phase Conjugator (OPC) in the middle of a system that presents a symmetrical distribution of both dispersive and nonlinear effects. If this condition is satisfied the pulse evolution after the OPC is symmetric to that experienced before the OPC and thus all the distortions caused by nonlinearity, and odd-order dispersion, can be compensated. This happens because the nonlinearity produced on a pulse is perfectly compensated by that produced on its “conjugated copy”, an exact copy of the original pulse spectrally-inverted and with an opposite value of accumulated dispersion.

In the particular situation of high bit-rate transmission systems, the nonlinear distortion can be generally modeled as a perturbation to the dominant dispersive effects. Thus pulse changes are mainly due to temporal broadening (or narrowing), more than to spectral modifications. It is important to underline that this allows mitigating the nonlinear effect produced on a pulse by the nonlinearity produced on its temporally-inverted (but not spectrally-inverted) copy, because the spectral modifications of pulses are, at a first order approximation, generally negligible. This requirement on the temporally-inverted pulses can be viewed as a graphical-symmetry condition, by reporting on a diagram the value of the optical power as a function of the dispersion accumulated by pulses during propagation (in the following PADD: Power - Accumulated Dispersion Diagram). Nonlinearity compensation can be achieved, using this graph, when two regions of “nonlinear propagation” are obtained for symmetrical values of pulse’s accumulated dispersion, because the previously found condition (generating nonlinear effect on temporally-inverted pulses) is then satisfied.

Fig. 1. Left: power and accumulated dispersion (solid and dotted line) required for MSSI. Right: derived PADD shows the perfectly symmetric distribution of nonlinear regions with respect to the zero of accumulated dispersion; DLamp is the dispersion accumulated by signal during propagation on one span. The nonlinear (grey) regions A and B are symmetrical to regions D and C respectively.

In a general situation nonlinearity will be completely compensated if two conditions are satisfied: the entire power profile curve is symmetrical and pulses are both temporally and spectrally inverted; anyway, even if both these conditions are not verified, but a substantial symmetry of highly nonlinear regions is present, a partial nonlinearity compensation will be obtained. It is worth noting that this graphical approach is, in principle, less effective than the MSSI (no “perfect compensation” is guaranteed), but it’s more viable, because it doesn’t require unpractical constraints on power profiles, or the use of nonlinear devices like the optical phase conjugator (OPC). Moreover this approach may be viewed as a generalized version of the MSSI: if the system symmetries required for MSSI are satisfied the same result is obtained applying both techniques.

3. Analysis of Known Techniques

In a realistic link, like that illustrated in Fig. 2, MSSI can’t be applied, because the nonlinearity distribution, in a typical optical system, is strongly asymmetrical. Applying the graphical approach a well defined OPC position, creating a symmetrical distribution of nonlinear regions in the PADD, can be found. This position, obtained by simple graphical considerations, is exactly coincident with the one that in [8

8 . P. Minzioni , F. Alberti , and A. Schiffini , “ Optimized Link Design for Non Linearity Cancellation by Optical Phase Conjugation ” IEEE Photonics Technol. Lett , 16 , 813 – 815 , March 2004 . [CrossRef]

] has been mathematically demonstrated to minimize pulse distortion.

Fig. 2. Left: evolution of power (solid) and accumulated dispersion (dotted) considering the optimal, OPC position. Right: derived PADD highlights the symmetry of nonlinear regions

In a similar way any configuration, not only those reported in Fig. 1 and Fig. 2, producing INLE compensation in systems including an OPC can [4

4 . I. Brener et al., “ Cancellation of all Kerr nonlinearities in long fiber spans using a LiNbO3 phase conjugator and Raman amplification ”, Optical Fiber Communication Conf. 2000, vol.4 , 266 – 268 , 2000

]–[7

7 . S. Watanabe , et al., “ Generation of optical phase-conjugate waves and compensation for pulse shape distortion in a single-mode fiber ” IEEE J. Lightwave Technol 12 , 2139 – 2146 , Dec 1994 [CrossRef]

] be easily reviewed as a technique to produce symmetrical nonlinear regions.

Once observed that the PADD could describe several techniques for nonlinearity compensation by optical phase conjugation it is worth noticing that it can successfully explain also techniques for nonlinearity compensation in systems without an OPC. A well known solution to achieve nonlinearity mitigation is to carefully design link’s dispersion map. Different maps optimizations reducing INLE have been proposed in literature, but always applied to particular configurations; thus no unifying explanation of their behaviour is known.

Three main classes of dispersion maps, depending on the amount of dispersion compensation inserted in each span, can be identified: no in-span compensation [9

9 . A. Mecozzi , C. B. Clausen , and M. Shtaif , “ System impact of intra-channel nonlinear effects in highly dispersed optical pulse transmission ”, IEEE Photonics Technol. Lett. , 12 , 1633 – 1635 , Dec. 2000 [CrossRef]

], partial span compensation [10

10 . S. Kumar , J.C. Mauro , S. Raghavan , and D.Q. Chowdhury , “ Intrachannel Nonlinear penalties in Dispersion-Managed Transmission Systems ” IEEE J. Sel. Top. Quantum Electron. 8 , 626 – 631 ( 2002 ). [CrossRef]

], and complete span compensation [11

11 . A. Sano , et al. “ A 40-Gb/s/ch WDM Transmission with SPM/XPM Suppression Through Prechirping and Dispersion management ”, IEEE J. Lightwave Technol. 18 , 1519 – 1527 ( 2000 ). [CrossRef]

]. It must be underlined that in the three situations three diverse solutions for nonlinearity compensation have been identified, all coincident with those that can be obtained analyzing the corresponding PADD. As an example let’s consider the configuration including dispersion compensation only at the transmitter and receiver site. The mathematical analysis in [9

9 . A. Mecozzi , C. B. Clausen , and M. Shtaif , “ System impact of intra-channel nonlinear effects in highly dispersed optical pulse transmission ”, IEEE Photonics Technol. Lett. , 12 , 1633 – 1635 , Dec. 2000 [CrossRef]

] shows that the optimum distance to be pre-compensated, if lumped amplification is considered, is given by the following equation (where Lamp is the amplifier spacing, α the fiber losses and n the number of spans).

zpres,opt=nLamp2n(αLamp1)+(n1)eαLamp2α[n(n1)eαLamp]
(1)

Being zmsa the position of mid-span amplifier and Leff the effective length, defined in [14

14 . G.P. Agrawal Nonlinear Fiber Optics , (Academic Press 90-91)

] as the length over which nonlinearity is significant, (1) can be reduced, considering long amplifier spacing (exp(-αLamp)≪1) and an odd number of spans, to a form similar to that reported in [8

8 . P. Minzioni , F. Alberti , and A. Schiffini , “ Optimized Link Design for Non Linearity Cancellation by Optical Phase Conjugation ” IEEE Photonics Technol. Lett , 16 , 813 – 815 , March 2004 . [CrossRef]

]:

zpres,opt=zmsa+Leff2Lamp2
(2)

zpres,opt=n2Lamp+Leff2
(3)

which matches solutions found in [8

8 . P. Minzioni , F. Alberti , and A. Schiffini , “ Optimized Link Design for Non Linearity Cancellation by Optical Phase Conjugation ” IEEE Photonics Technol. Lett , 16 , 813 – 815 , March 2004 . [CrossRef]

], where two equations have been provided in the case of an odd or even value of n, and include (1).

Fig. 3. Left: optimal positioning if n=2. Right: obtained PADD.

Similarly it can be demonstrated that all the configurations proposed in literature, and relating to different dispersion maps, act in the same way: making nonlinear region positions to be symmetrical in the PADD.

4. Configurations for Nonlinearity Compensation Derived by Graphical Analysis

Considering first the opportunity to achieve nonlinearity compensation by using an OPC, two preliminary considerations have to be done: generally MSSI can’t be applied because system power profile is strongly asymmetrical, and the “optimal OPC positioning” solution proposed in [8

8 . P. Minzioni , F. Alberti , and A. Schiffini , “ Optimized Link Design for Non Linearity Cancellation by Optical Phase Conjugation ” IEEE Photonics Technol. Lett , 16 , 813 – 815 , March 2004 . [CrossRef]

], useful for newly-designed transmission systems, seems not easily employable into embedded links, because an intra-span access point to fiber cable is required.

The PADD can thus be used to determine new system configurations satisfying the three following requirements: identified solutions must provide a sensible compensation of nonlinear effects, they must be applicable to real systems (where power profiles are generally asymmetrical, using EDFA), and no in-line access-point to fiber cable must be needed. These three conditions can be easily translated into three constraints regarding the PADD: nonlinear regions must be symmetrically placed, asymmetrical power profiles must be considered, and changes to system configuration can only be applied in a position corresponding to an amplifier site (not in the middle of the span between two amplifiers). Moreover to gain the maximum advantage from the presence of an OPC it is useful to focus on configurations employing one single OPC, and producing at the same time a substantial compensation of nonlinear effects as well as of chromatic dispersion.

Let’s consider a system composed of four spans and including lumped amplification. With these conditions, several different approaches can be considered, depending on the kind (and number) of changes introduced along the system. The more interesting solution for practical implementation can be derived observing the PADD diagram of the original system configuration, shown in the following figure.

Fig. 4. Left: evolution, along the link, of optical power (solid) and pulse’s accumulated dispersion (dotted). Right: The related PADD shows that nonlinear regions (areas A, B, C, D) are not symmetrical. In this example dispersion compensation is included only at the receiver.

Since four nonlinear regions are distributed along the system, the OPC has to be positioned between second and third nonlinear region; moreover the only access point to fiber cable available in that section of the system is given by the third amplification stage, at a distance 2Lamp from the transmitter. With these simple considerations the position of the OPC is then defined, but this positioning does not create a symmetrical distribution of nonlinear regions inside the PADD, and some system modification must thus be included.

The simplest way to achieve symmetry is to modify system dispersion map introducing an adequate displacement of nonlinear regions; practically this can be obtained adding, at the OPC site, an adequate dispersive element. According to the sign of introduced dispersion two different embodiments of this solution can be defined, as reported in Fig. 5.

If the dispersion introduced by added element has the same (or opposite) sign of the dispersion introduced by transmission fiber, it will be positioned after (or before) the OPC respectively; thus producing, in the PADD, an adequate translation of nonlinear regions. The absolute value of dispersion that has to be introduced (Delement expressed as ps nm-1) is, in both situations, coincident with the dispersion accumulated by pulses during propagation along last “linear region”. Indicating the dispersion of transmission fiber with Df (ps nm-1 km-1), it is possible to observe that the value of Delement producing the maximum compensation of nonlinear effects is defined by the following equation:

Delement=Df(LampLeff)
(4)

The evolution of power and accumulated dispersion, together with derived PADDs, obtained considering both possible system implementations are reported in Fig. 5. It must be underlined that adding an appropriate dispersive element is a not critical operation: it can be easily obtained by adding an appropriate span of fiber, or a chirped Bragg grating.

Fig. 5. PADD relatives to the two proposed configurations. In both cases nonlinear regions are symmetrical with respect to the axis of zero accumulated dispersion. Left: Added element (positioned upstream the OPC) has a dispersion sign opposite to that of transmission fiber. Right: The dispersion of fiber and of added element have the same sign, the element is thus placed downstream the OPC

Table 1. fiber parameters considered for simulations

table-icon
View This Table

Three systems implementations have been simulated using parameters of fibers F1 and F2; moreover RZ pulses, with TFWHM=5ps (duty-cycle=0.2), have been considered to propagate on both the F1 and the F2 link, while NRZ pulses have been considered on the F2 link.

Dispersion compensation, at both the OPC site and the receiver site, has been supposed as ideal: no distortion effect or insertion loss has been included in simulations, thus making equivalent to analyze the two techniques reported in Fig. 5. In the following figure the obtained EOP, as a function of the dispersion compensated at the OPC site, is reported. To simplify the analysis, compensated dispersion has been indicated as the “compensated length” (L comp):

Lcomp.=DelementDf
(5)

Combining (4) and (5) the optimum value of L comp is equivalent to (Lamp-Leff); thus, with given data, it is approximately equal to 79km. It must be noted that considering the MSSI technique (compensated length equal to 0km in Fig. 6) a worst EOP is obtained; while through appropriate dispersion compensation an EOP of about 1dB, or lower, can be achieved.

Fig. 6. Left: EOP as a function of dispersion compensated before the OPC.. Right: eye diagrams (RZ propagation on F1) when 0km and 79km (top and bottom) are compensated

The fourth system implementation considered was based on RZ pulse propagation on a system (18 × 100 km) of F3 fiber, to underline the absence of restriction on fiber dispersion. In Fig. 7 the evolution of the EOP and of the timing jitter accumulated by pulses during propagation on this system is reported. It can be seen that, by adding an appropriate Dispersion Compensating Module (DCM), a noticeable distortion compensation is produced while pulses propagate along the second half of the link.

Fig. 7. The evolution of EOP (left) and timing jitter (right) is almost symmetrical if an appropriate DCM is added together with the OPC. Conversely both EOP and timing jitter grow in the second part of the link if no DCM is added, thus showing that the MSSI is not effective in this configuration.

In the second configuration proposed, the PADD can also be applied to identify nonlinearity compensation techniques based on the use of proper dispersion maps. To design a dispersion map suitable for practical implementation, and producing INLE compensation, two requirements must be satisfied: dispersion map must use low number of fiber types and “nonlinear regions” must be symmetrical in the PADD. Moreover, to allow simple path-reconfiguration, accumulated dispersion must be compensated on a span-by-span basis.

This can be achieved using a dispersion-map (Fig. 8) including two fibers in each span, and producing INLE compensation between nonlinear regions of subsequent spans; this solution, graphically identified in 2003 [13

13 . P. Minzioni , A. Schiffini , and A. Paoletti “ Patent Application WO2003IT00455 20030724 ” ( 2003 )

], has recently been mathematically demonstrated in [12

12 . A.G. Striegler and B. Schmauss , “ Compensation of Intrachannel Effects in Symmetric Dispersion-Managed Transmission Systems ” IEEE J. Lightwave Technol , 22 , 1877 – 1882 , Aug. 2004 [CrossRef]

].

Fig. 8. Proposed dispersion map (left) and related PADD (right). Intra-span dispersion compensation, and inter-span nonlinearity compensation are obtained.

Fig. 9. Left: the identified dispersion map employ two fibers in each span and its period is twice the span length. This fiber arrangement allows producing a symmetrical PADD. Right: commonly used map yielding periodical dispersion compensation.

Fig. 10. Evolution, during propagation, of EOP and timing jitter in a standard map and in the map identified using the PADD

5. Conclusion

A new, graphical approach able to explain all known techniques for nonlinearity compensation has been disclosed. This method, derived from a generalization of the mid-span-spectral-inversion, is based on mid-nonlinearity-temporal-inversion and allows a fast identification of systems configuration producing nonlinearity compensation. Moreover it allows to overcome various implementation constraints (like unequal amplifier spacing or non periodic dispersion maps), thus proving useful also for the analysis of non-ideal systems.

References and links

1 .

A. Mecozzi , C.B. Clausen , and M. Shtaif , “ System impact of intra-channel nonlinear effects in highly dispersed optical pulse transmission ” IEEE Photonics Technol. Lett. 12 , 1633 – 1635 ( 2000 ). [CrossRef]

2 .

P.V. Mamyshev and N.A. Mamysheva , “ Pulse-overlapped dispersion-managed data transmission and intrachannel four-wave mixing ” Opt. Lett. 24 , 1454 – 1456 ( 1999 ). [CrossRef]

3 .

D. Kunimatsu , et al., “ Subpicosecond Pulse Transmission over 144 km Using Midway Optical Phase Conjugation via a Cascaded Second-Order Process in a LiNbO3 Waveguide ” IEEE Photonics. Technol. Lett. 12 , 1621 – 1623 ( 2000 ). [CrossRef]

4 .

I. Brener et al., “ Cancellation of all Kerr nonlinearities in long fiber spans using a LiNbO3 phase conjugator and Raman amplification ”, Optical Fiber Communication Conf. 2000, vol.4 , 266 – 268 , 2000

5 .

W. Pieper , et al., “ Nonlinearity-insensitive standard-fibre transmission based on optical-phase conjugation in a semiconductor-laser amplifier ” Electron. Lett. 30 , 724 – 726 , Apr. 1994 [CrossRef]

6 .

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 ” IEEE J. Lightwave Technol. 21 , 1465 – 1473 ( 2003 ) [CrossRef]

7 .

S. Watanabe , et al., “ Generation of optical phase-conjugate waves and compensation for pulse shape distortion in a single-mode fiber ” IEEE J. Lightwave Technol 12 , 2139 – 2146 , Dec 1994 [CrossRef]

8 .

P. Minzioni , F. Alberti , and A. Schiffini , “ Optimized Link Design for Non Linearity Cancellation by Optical Phase Conjugation ” IEEE Photonics Technol. Lett , 16 , 813 – 815 , March 2004 . [CrossRef]

9 .

A. Mecozzi , C. B. Clausen , and M. Shtaif , “ System impact of intra-channel nonlinear effects in highly dispersed optical pulse transmission ”, IEEE Photonics Technol. Lett. , 12 , 1633 – 1635 , Dec. 2000 [CrossRef]

10 .

S. Kumar , J.C. Mauro , S. Raghavan , and D.Q. Chowdhury , “ Intrachannel Nonlinear penalties in Dispersion-Managed Transmission Systems ” IEEE J. Sel. Top. Quantum Electron. 8 , 626 – 631 ( 2002 ). [CrossRef]

11 .

A. Sano , et al. “ A 40-Gb/s/ch WDM Transmission with SPM/XPM Suppression Through Prechirping and Dispersion management ”, IEEE J. Lightwave Technol. 18 , 1519 – 1527 ( 2000 ). [CrossRef]

12 .

A.G. Striegler and B. Schmauss , “ Compensation of Intrachannel Effects in Symmetric Dispersion-Managed Transmission Systems ” IEEE J. Lightwave Technol , 22 , 1877 – 1882 , Aug. 2004 [CrossRef]

13 .

P. Minzioni , A. Schiffini , and A. Paoletti “ Patent Application WO2003IT00455 20030724 ” ( 2003 )

14 .

G.P. Agrawal Nonlinear Fiber Optics , (Academic Press 90-91)

OCIS Codes
(060.2330) Fiber optics and optical communications : Fiber optics communications
(060.2360) Fiber optics and optical communications : Fiber optics links and subsystems
(190.3270) Nonlinear optics : Kerr effect
(190.4370) Nonlinear optics : Nonlinear optics, fibers
(190.5040) Nonlinear optics : Phase conjugation

ToC Category:
Research Papers

History
Original Manuscript: July 18, 2005
Revised Manuscript: September 23, 2005
Published: October 17, 2005

Citation
Paolo Minzioni and Alessandro Schiffini, "Unifying theory of compensation techniques for intrachannel nonlinear effects," Opt. Express 13, 8460-8468 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-21-8460


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References

  1. A. Mecozzi, C.B. Clausen, and M. Shtaif, �??System impact of intra-channel nonlinear effects in highly dispersed optical pulse transmission�?? IEEE Photonics Technol. Lett. 12, 1633�??1635 (2000). [CrossRef]
  2. P.V. Mamyshev, N.A. Mamysheva, �??Pulse-overlapped dispersion-managed data transmission and intrachannel four-wave mixing�?? Opt. Lett. 24, 1454�??1456 (1999). [CrossRef]
  3. D. Kunimatsu, et al., �??Subpicosecond Pulse Transmission over 144 km Using Midway Optical Phase Conjugation via a Cascaded Second-Order Process in a LiNbO3 Waveguide�?? IEEE Photonics. Technol. Lett. 12, 1621�??1623 (2000). [CrossRef]
  4. I. Brener et al., �??Cancellation of all Kerr nonlinearities in long fiber spans using a LiNbO3 phase conjugator and Raman amplification�??, Optical Fiber Communication Conf. 2000, vol.4, 266-268, 2000
  5. W. Pieper, et al., �??Nonlinearity-insensitive standard-fibre transmission based on optical-phase conjugation in a semiconductor-laser amplifier�?? Electron. Lett. 30, 724�??726, Apr. 1994 [CrossRef]
  6. P. Kaewplung, T. Angkaew, 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�?? IEEE J. Lightwave Technol. 21, 1465-1473 (2003) [CrossRef]
  7. S.Watanabe, et al., �??Generation of optical phase-conjugate waves and compensation for pulse shape distortion in a single-mode fiber�??IEEE J. Lightwave Technol 12, 2139-2146, Dec 1994 [CrossRef]
  8. P. Minzioni, F. Alberti, A. Schiffini, �??Optimized Link Design for Non Linearity Cancellation by Optical Phase Conjugation�?? IEEE Photonics Technol. Lett, 16, 813�??815, March 2004. [CrossRef]
  9. A. Mecozzi, C. B. Clausen, and M. Shtaif, �??System impact of intra-channel nonlinear effects in highly dispersed optical pulse transmission�??, IEEE Photonics Technol. Lett., 12, 1633-1635, Dec. 2000 [CrossRef]
  10. S. Kumar, J.C. Mauro, S. Raghavan, D.Q. Chowdhury, �??Intrachannel Nonlinear penalties in Dispersion- Managed Transmission Systems�?? IEEE J. Sel. Top. Quantum Electron. 8, 626-631 (2002). [CrossRef]
  11. A. Sano, et al. �??A 40-Gb/s/ch WDM Transmission with SPM/XPM Suppression Through Prechirping and Dispersion management�??, IEEE J. Lightwave Technol. 18, 1519-1527 (2000). [CrossRef]
  12. A.G. Striegler, and B.Schmauss, �??Compensation of Intrachannel Effects in Symmetric Dispersion-Managed Transmission Systems�?? IEEE J. Lightwave Technol, 22, 1877-1882, Aug. 2004 [CrossRef]
  13. P. Minzioni, A. Schiffini, A. Paoletti �??Patent Application WO2003IT00455 20030724�?? (2003)
  14. G.P. Agrawal Nonlinear Fiber Optics, (Academic Press 90-91)

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