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IPO Working Group                                 Dimitri Papadimitriou
Category: Informational Draft                           Jean-Paul Faure
Expiration Date: May 2002                               Olivier Audouin

                                                           Roy Appelman

                                                          November 2001

                     Non-linear Routing Impairments
                in Wavelength Switched Optical Networks


Status of this Memo

   This document is an Internet-Draft and is in full conformance with
      all provisions of Section 10 of RFC2026 [1].

   Internet-Drafts are working documents of the Internet Engineering
   Task Force (IETF), its areas, and its working groups. Note that
   other groups may also distribute working documents as Internet-
   Drafts. Internet-Drafts are draft documents valid for a maximum of
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   documents at any time. It is inappropriate to use Internet- Drafts
   as reference material or to cite them other than as "work in

   The list of current Internet-Drafts can be accessed at
   The list of Internet-Draft Shadow Directories can be accessed at

1. Abstract

   Today, in transparent optical networks, the increasing bit-rate (10
   Gbit/s and up to 40 Gbit/s in the future), combined with the
   increasing number of wavelengths (16 and higher up to 320) and a
   narrowing of the channels spacing, enhance the impact of non-linear
   effects on optical signal quality.

   Thus, non-linear effects like Self-Phase Modulation (SPM), Cross-
   Phase Modulation (XPM), Four-Wave Mixing (FWM) as well as Stimulated
   Raman Scattering (SRS) and Brillouin scattering have to be examined
   in order to evaluate their impacts on the transmission quality. If
   these effects appear to be significant, they have to be taken into
   account in the routing of a wavelength throughout a transparent
   optical network.

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   The aim of this draft is to extend the previous works dedicated to
   routing impairments ([IPO-IMP] and [IPO-ORI]) in order to determine
   which are the non-linear effects that must be considered and which
   kind of engineering rules may be used to take these effects into
   account in constraint-based optical routing.

   Moreover, we propose to introduce IGP routing protocol extensions to
   transport information related to non-linear impairments relevant for
   wavelength routing decisions.

2. Conventions used in this document

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   this document are to be interpreted as described in RFC-2119 [2].

3. Introduction

   Non-linear effects are due to the fact that the optical properties
   of the medium (refractive index, loss, etc.) become dependent of the
   signal power of the optical channels present in this medium. As a
   consequence, this power dependency tends to modify the propagation
   of the optical waves and also lead to interactions between these
   waves. Non-linear interactions depend on the transmission length
   (distance between the transmitter and the receiver), the type of
   fiber, the cross-sectional area of the fiber, the wavelength and the
   power level. Basically, these effects become more intensive when the
   optical power or the transmission length increase or when the
   channel spacing becomes narrower (the different wavelengths tend to
   interact more each other). As a consequence, non-linearities can
   impose significant limitations on high bit-rates (10 Gbit/s and
   higher), Long Haul (LH) and Ultra-Long Haul (ULH) systems, or high
   capacity DWDM systems.

   Linear impairments are extensively addressed in [IPO-IMP] and
   corresponding IGP routing protocol extensions in [IPO-ORI]. In these
   previous works, the approach for non-linear impairments was to
   consider that: ôOne could assume that non-linear impairments are
   bounded and increase the required OSNR level by X dB, where X for
   performance reasons would be limited to 1 or 2 dB, consequently
   setting a limit on the maximum number of spans. For the approach
   described here to be useful, it is desirable for this span limit to
   be longer than that imposed by the constraints which can be treated

   However, this approximation may lead to both an over or an under
   estimation of the real impact of non-linear effects. If the actual
   impact is less than 2 dB on the OSNR, then, the margin taken will
   forbid some feasible path(s) (as it limits the maximum number of
   spans). On the contrary, if the real impact is over 2 dB, the
   corresponding route(s) may be chosen despite they are not feasible
   from the optical transmission point of view.

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   Therefore, the objective of this document is first to determine
   which kind of non-linear effects must be taken into account, and to
   give some simple engineering rules to determine their maximum
   tolerable value, second, to propose IGP routing protocol extensions
   in order to cover non-linear optical routing impairments.

   Additional complexity may arise from the fact that for instance when
   minimizing degradations through Self Phase Modulation (SPM) after
   setting a distinct Lambda LSP (L-LSP) or optical channel in the
   network, this L-LSP will suffer a changing degradation by Cross-
   Phase Modulation (XPM) through the changing number of concurrent
   optical channels on the fiber links. Thus, as we will point out,
   cross channels effects should be minimized at the system design in
   order to be compatible with SPM and other impairments.

4. Non-Linear Impairments

   Non-linear optical impairments can be classified into two
   categories. The first category consists of effects occurring due to
   the dependence of the refractive index on the optical signal power
   (generally called Kerr effect). This category includes Self-Phase
   Modulation (SPM), Cross-Phase Modulation (XPM) and Four-Wave Mixing
   (FWM). The second category of effects consists of inelastic
   scattering effects in the fiber medium and are due to the
   interaction of the light waves with the optical phonons of the fiber
   medium leading to Stimulated Raman Scattering (SRS) or with the
   acoustic phonons (sound waves) of the medium leading to Stimulated
   Brillouin Scattering (SBS).

   SPM and XPM essentially affect the phase of the signals and cause
   its spectral broadening which lead to temporal distorsions because
   of dispersion. FWM lead to energy exchange between signals that
   induces in-band crosstalk, whereas SBS and SRS provide gain or loss
   to the light waves. Nevertheless, the actual impact of these non-
   linear effects on transmission quality depends strongly on
   dispersion management.

4.1 Refractive Index

   The general equation for the refractive index of the core in an
   optical fiber is given by:

   n = n(0) + [n(2) x P / A(eff)]

   - n(0)   = the refractive index of the fiber core at low optical
              power level (no unit)
   - n(2)   = the non-linear refractive index coefficient (for
              instance, 2.35 x 10^(-20) m^2/W for silica)
   - P      = optical signal power in Watts (W)
   - A(eff) = the effective area of the core in square meters (m^2)

   Clearly, this equation indicates that two strategies for minimizing

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   non-linearities due to refractive index power dependence are to
   minimize the launched optical power P and/or to maximize the
   effective area of the fiber A(eff).

   Minimizing P is limited by the fact that during network design,
   there is a strong trade-off between non-linear effects and optical
   signal to noise ratio (decreasing P will decrease non-linear effects
   but also the OSNR). On the other hand, augmenting A(eff) is being
   targeted by some fiber vendors while keeping other non-linear
   effects unchanged.

   This optical power dependence of the refractive index introduces the
   following non-linear effects: SPM, XPM, FWM, SBS and SRS as
   explained here below.

4.1.1 Self-Phase Modulation (SPM)

   Self-Phase Modulation (SPM) arises from the power dependency of the
   refractive index of the fiber core. Fluctuations in the optical
   signal power cause changes in the phase of the signal referred to as
   a non-linear phase shift. This induces an additional frequency chirp
   on the spectrum of the optical pulse which interacts with the
   fiberÆs dispersion to broaden the pulse and lead to intensity
   fluctuations. Therefore, this effect leads to higher penalties due
   to Inter-Symbol Interference (ISI). This chirping effect affects
   each channel independently of the other and is proportional to the
   optical channel power. Therefore SPM effects are more pronounced in
   systems using higher transmitted signal power. Moreover because the
   SPM effect leads to extra ISI, higher bit-rate systems will be more

   It is important to point out that in a DWDM transmission system at
   10 Gbit/s with 100 GHz channel spacing, SPM is generally considered
   as a significant non-linear effect (except may be when low
   dispersion fibers are used).

4.1.2 Cross-Phase Modulation (XPM)

   For a given optical signal, Cross-Phase Modulation (XPM) is a
   consequence of a modification of the refractive index of the medium
   due to the optical power of the closest neighboring channels present
   in the fiber. As for SMP, the induced phase shift lead to intensity
   fluctuations after interaction with dispersion. XPM increases when
   optical channel spacing becomes narrower as long as the adjacent
   channels are closer in the spectral domain, so that they travel
   roughly at the same velocity and interact over a longer time period.
   As XPM effect depends on channel spacing, it can be a significant
   problem for high capacity DWDM system with channel spacing of 50 GHz
   or lower.

   On the other hand, when the bit rate increases (from 10 Gbit/s to
   40 Gbit/s), the impact of XPM decrease as long as the time during
   which the interacting channels temporally overlap is considerably

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   For moderate bit rate systems (10 Gbit/s), XPM effect generally
   becomes significant compared to SPM when channel spacing is lower
   than 100 GHz, and when the local dispersion is low. This means that
   XPM should be taken into account for 50 GHz spacing DWDM systems.
   Here, we want to point out that solutions demonstrated in laboratory
   environments may be implemented to decrease the effect of XPM by
   introducing a phase-mismatch between optical channels at the link
   input by using interleaved polarization or using a suitable
   dispersion management.

4.1.3 Four-Wave Mixing (FWM)

   In a (high capacity) DWDM system, based on different optical
   channels at different wavelengths (i.e. frequencies), the power
   dependence of the refractive index of the fiber core also gives rise
   to the generation of new frequencies (i.e. new optical signal).

   Practically, there is an interaction between the different channels,
   leading to energy transfer between these channels. This effect is
   called Four-Wave Mixing (FWM) because if three optical channels with
   frequencies f1, f2 and f3 propagates simultaneously within the same
   fiber, a fourth optical channel is generated and frequency f4 which
   is related to the other frequencies by the following relation: f4 =
   f1 (+ or -) f2 (+ or -) f3. In theory, several frequencies
   corresponding to different combinations are possible. However, in
   practice only the frequency combinations of the form f4 = f1 + f2 û
   f3 are the most troublesome for (high capacity) DWDM systems. These
   fourth optical channels can become even nearly phase-matched when
   optical channel wavelengths are close to the zero-dispersion point.

   As a consequence, significant optical power can be transferred
   between neighboring optical channels through the FWM effect. Though,
   in contrast to SPM and XPM, which are bit-rate dependent, the FWM
   effect is not really dependent of the bit-rate. Nevertheless, like
   XPM, it depends strongly on the optical channels spacing and the
   fiber dispersion. Clearly, FWM becomes significant only at narrow
   channel spacing (50 GHz or lower) or when the local dispersion is

   As a consequence, significant optical power can be transferred
   between neighboring optical channels through the FWM effect. Though,
   in contrast to SPM and XPM, which are bit-rate dependent, the FWM
   effect is not really dependent of the bit-rate. Nevertheless, like
   XPM, it depends strongly on the optical channels spacing and the
   fiber dispersion. Clearly, FWM becomes significant and should be
   taken into account for DWDM systems with narrow channel spacing (50
   GHz or lower) or when the local dispersion is low.

4.2 Scattering Effects

   Scattering effects, the second set of mechanisms generating non-

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   linearities give rise to SRS and SBS.

4.2.1 Stimulated Raman Scattering (SRS)

   In DWDM systems, the fiber acts as a Raman amplifier such that the
   longer wavelengths channel are amplified by the shorter wavelengths
   channels as long as the wavelength difference is within the Raman
   Gain spectrum.

   Therefore, if two or more signals at different wavelengths are
   injected into a fiber, the SRS effect causes optical signal power to
   be transferred from the lower wavelength optical channels to the
   higher wavelength optical channels. The gain coefficient increases
   with increasing channel spacing up to 125 nm. This amplification
   leads to increase power fluctuations, which add to receiver and
   degrade receiver performance. This phenomenon known as Raman inter-
   channel crosstalk can be avoided if channel powers are made so small
   that Raman amplification is negligible over the fiber length.

   However, the coupling between wavelengths occurs only if both
   optical channels are launched simultaneously so that the impact of
   the SRS is reduced by the dispersion introduced by the silica
   medium. Basically, SRS induces a gain tilt over the whole bandwidth
   of the fiber, which is proportional to the total power of all
   channels present in the fiber.

   Moreover, periodic amplification of the DWDM signal in ULH fiber
   links can also increase the impact of the SRS-induced degradation.
   This phenomenon occurs because in-line amplifiers add noise which
   experiences less Raman loss than the signal itself, resulting in
   degradation of the SNR. In brief, the total capacity of DWDM systems
   is then limited to below 100 Gbps for a transmission distance of
   5000 km or more.

   As a matter of fact, SRS should not be considered for impairment
   based optical routing, as long as its induced Raman tilt will be
   managed link by link during the network design.

4.2.2 Stimulated Brillouin Scattering (SBS)

   Scattering effects in the optical fiber occur due to the interaction
   of the optical channels with the sound waves (acoustic phonons)
   present in the silica medium. In SBS, the scattering process is
   stimulated by photons with a wavelength higher than the wavelength
   of the incident signal. This interaction takes place over a very
   narrow band of 20 MHz at 1550nm. The scattered waves and the
   incident optical light waves propagate in opposite directions. Thus,
   SBS produces an additional loss in the propagating signal but does
   not induce any interaction between different optical channels.

   In practical implemented systems, the Brillouin inter-channel
   crosstalk phenomenon can be easily avoided by always keeping the SBS
   threshold power higher than the optical signal power. Moreover, the

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   probability for SRS to occur is much higher than that for SBS
   because the gain bandwidth for SRS is ~5 THz, while the gain
   bandwidth for SBS is ~0.05 GHz. Consequently, the losses induced by
   the SBS effect are not a real problem when considering impairment-
   based optical routing.

5. Fiber and Optical Amplifiers

   In the course of moving from pure optical centrally managed
   transmission to flexible wavelength switched networks many problems
   have to be solved. One of these problems is the fiber diversity
   among the different links, and the impact of non-linear effects
   inside the different transmission fibers.

5.1 Influence of the fiber medium

   An optical transparent network is composed of many nodes (optical
   LSR) connected by links (a link is a transmission system between two
   nodes). Each link is composed of transmission spans of identical
   fiber, whereas in the most general case, different types of fibers
   may be deployed among the different links. The main types of fibers
   that are generally deployed in today optical networks are mainly
   based on G.652 as well as G.655 and G.653 ITU-T Recommendations.
   Also and Dispersion Compensating Fibers (DCF) used to compensate the
   dispersion is present inside the network element.

   The intensity of non-linear effects is also dependent on the
   intrinsic optical properties of each fiber, thus, fiber diversity in
   optical networks must be taken into account when regarding the non-
   linear impairments. In particular, non-linear effects arise not only
   in the transmission fiber but also inside the Dispersion
   Compensating Fiber (DCF) present in the network elements.

   An additional important point is that in today optical networks, the
   dispersion of the transmission fiber is compensated for each link by
   a specific dispersion map. The residual dispersion after cascading a
   certain number of links must be compatible with the cumulated impact
   of non-linear effect such as SPM or XPM.

   In most common optical networks, of moderate bit-rate (10 Gbit/s)
   and channel spacing (100 GHz), Self-phase modulation (SPM) is the
   strongest non-linear degradation effect. Changing the path length in
   the network will then increase the SPM contribution when it
   increases the total path length, or when the optical path crosses
   highly non-linear fiber links.

   Only for highly dense DWDM systems (channel spacing of 50 GHz or
   less), XPM at its turn will take more and more importance, as well
   as FWM (even if this latter effect can be reduced with specific
   design in the link). In that case, after setting a distinct path in
   the network, this path may suffer a changing degradation by XPM and
   eventually FWM through the changing number of concurrent channels on
   a same fiber link. Thus, XPM and FWM must be minimized at system

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   design but even then their impact must be taken into account for
   impairment-based optical routing.

5.2 Optical Amplifiers

   Potentially, non-linear effects may also occur in the fiber
   amplifiers, but they can be considerably reduced provided a specific
   design will be done. Then, it is not worth to take them into
   account, as long as they are kept under control at system design.

6. Non-linear Phase

   The intensity of the SPM, XPM and FWM non-linear effects can be
   quantified through the Non-Linear Phase shift NLP (see [AGR-FOCS])
   induced in the fiber by the Kerr effect. As a matter of fact, the
   NLP was shown to be a robust empiric parameter able to evaluate the
   impact of non-linear effects as described in [OFC00-NLP] and [OFC02-
   NLP] while related considerations can be found in [ELEC-ODS].

   In this section we propose to use the Non-Linear Phase (NLP) as an
   empiric criterion to correlate the cumulated effects of SPM, XPM and
   FWM with a given penalty. This penalty corresponds to an upper bound
   value of the NLP (NLPmax) which depends on the bit-rate, the channel
   spacing and the fiber type.

   Since the NLP is additive along an optical path (including several
   links and spans), the cumulated NLP value (NLPcum) can be compared
   to the maximum tolerated value of the NLP (NLPmax). Consequently,
   this method ensures that an optical channel is not affected by non-
   linear effects when NLPcum < NLPmax.

6.1 Definition

   The Non-Linear Phase (NLP) for a given transmission span NLP(span)
   is given by the following formula (see [AGR-NFO]):

   NLP(span) = P(in) x F(span)

   - P(in)   = optical power in Watts (W) at the span input
   - F(span) = function assumed to be constant for a given span.

   The F(span) function is directly proportional to:

   F(span) ~ [n(2) x L(eff)] / [w x A(eff)]

   - n(2)    = non-linear refractive index coefficient (m^2/W)
   - L(eff)  = effective interaction length (m)
   - w       = wavelength of the optical channel (m)
   - A(eff)  = effective area of the fiber core in square meters (m^2)

   While the effective interaction length L(eff) is defined as:

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   L(eff) = [1 û exp(-aL)] / a

   - a  = linear absorption coefficient of the fiber (m^-1)
   - L  = fiber length (m)

   It is important to point out that the function F is defined as an
   integral of a rational function, which can be easily calculated and
   leads to an analytical formula. Then, for each span, the NLP can be
   easily computed using the above coefficients: w, n(2), A(eff) and

   For a transmission span, one must take into account the NLP due to
   the transmission fiber and Dispersion Compensating Fiber (DCF).
   Therefore, the total NLP for a whole span (NLP(span)) is then given
   by the following formula:

   NLP(span) = NLP(fiber) + NLP(DCF) = P(in) x F(span)

   - P(in)   = optical power at the span input
   - F(span) = global function containing the parameters of the span
               including transmission fiber and inline DCF

   Notice that the only varying parameter in the formula described in
   this section is the optical power at the span input P(in).

6.1.1 Calculation of the NLP for a link

   A link between two optical LSR is constituted by N transmission
   spans. Then, the cumulated NLP for a given link (NLP(link)) is equal
   to the sum of the different NLP due to each span and is given by:

   NLP(link) = Sum(NLP[i]) = Sum(P[i] x F[i])

   - NLP[i]  = NLP due to span ôiö
   - P[i]    = power at the input of span ôiö
   - F[i]    = F function of span ôiö

   This formula allows the calculation of the cumulated NLP for any
   link, provided one knows the optical power at each span input, and
   the F function for each span.

   An interesting case is when the N spans (including the same fiber
   type) can be considered as roughly identical, so that we can
   simplify the above formula to:

   NLP(link) = N x NLP(span) = N x P x F(span)

   - NLP(span) = NLP of the spans

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   - P         = power at the input of the link
   - F(span)   = F function of the spans

   It is important to note that the NLP(link) is a static information
   which characterizes the link, and is calculated locally. Therefore,
   it is the only required parameter to be flooded by IGP protocols.

6.1.2 Calculation of the NLP for an optical path

   Considering the establishment of an optical path within a network,
   this path will be a succession of ôjö different links, each link
   being composed of a specific fiber type. For instance, consider an
   optical path going from ingress node S to egress node D, via node A
   and B where the link between S and A is includes SMF fiber, the link
   between A and B, E-Leaf fiber and the link between B and D, True-
   Wave Fiber.

   Then, the total cumulated NLP over the whole optical path NLP(path)
   is given by:

   NLP(path) = Sum(NLP(link)[j])

   where NLP(link)[j] is the NLP due to link ôjö.

   With this simple formula, it is possible to calculate the total
   cumulated NLP (referred to as NLPcum) for any optical path, using
   the previously calculated NLP of the different links.

6.2 NLP Constraint

   For a given optical path, the total cumulated NLP due to SPM, XPM
   and FWM non-linear effects can be computed according to the previous

   In Section 6.1, we have demonstrated that the NLP is an additive
   variable along an optical path (including several links and spans)
   which depends on the bit-rate, the channel spacing and the fiber
   type. Therefore, the cumulated NLP value (NLP(path)) for a given
   optical path can be compared to the maximum tolerated value of the
   NLP (NLPmax). The latter is used as empiric criterion to correlate
   the SPM, XPM and FWM non-linear effects leading to the NLPmax upper
   bound value of the NLP.

   Consequently, the non-linear optical routing cumulative constraint
   including the SPM, XPM and FWM effects can be expressed as follow:
   for a given residual dispersion after crossing an optical path, the
   total cumulated dispersion NLP(path) must be lower than NLP(max),
   the maximum tolerable value for the NLP:

   NLP(path) < NLPmax

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   When the NLP(path) fulfills this constraint, the corresponding
   optical channel is not limited by the SPM, XPM and FWM non-linear
   effects do not limit a given optical system.

   For example, simulations have shown that the non-linear constraint
   NLPmax can be expressed as follows (assuming an accurate dispersion
   compensation management):

   - NLPmax < 0.45 pi at 10Gbit/s
   - NLPmax < 0.3 pi  at 40Gbit/s

   It is important to point out that we assume in this approach that
   the dispersion is managed at the link level using available
   technology being developed, so that for each link, the residual
   dispersion is compatible with the NLP of the link.

   Moreover, in a high density DWDM system, the NLP shift per span for
   a given optical channel does not only depend on the optical power of
   that channel and the fiber type. The NLP depends also on the power
   of the closest neighboring optical channels typically (8 in
   practical applications), as well as on the channel spacing. This
   implies that one have to consider the NLPmax constraint with respect
   to the channel spacing. Consequently, the NLP value per link
   (NLP(link)) must be flooded by the IGP routing protocol to take the
   channel spacing effect into account.

7. Traffic-Engineering Routing Protocol Extension

   As mentioned here above, the NLP parameter must be flooded per
   optical channel spacing (i.e. 100 GHz, 50 GHz and 25 GHz) using a
   dedicated extension to the IGP TE-Routing protocol.

   In OSPF, these NLP parameters are included in a common sub-TLV of
   the Link TLV in the Traffic Engineering LSA. The Type value of this
   sub-TLV is to be attributed (TBA). The length of this sub-TLV is 12
   octets and the corresponding value specifies the NLP value (in IEEE
   floating point format) per channel spacing. The format of the NLP
   sub-TLV is as shown:

      0                   1                   2                   3
      0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
     |  Type = TBA                   |         Length = 12           |
     |                         NLP at 100GHz                         |
     |                         NLP at 50GHz                          |
     |                         NLP at 25GHz                          |

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   In IS-IS, we propose to enhance the sub-TLVs for the extended IS-IS
   reachability TLV. The length of the NLP sub-TLV is 12 octets and
   specifies the NLP value (in IEEE floating point format) per channel
   spacing (in IEEE floating point format). Specifically, we add the
   following sub-TLV:

   - Sub-TLV type: TBA
   - Length(in bytes): 12
   - Name: NLP

8. Security Considerations

   There are no additional security considerations than the ones
   already covered in OSPF and IS-IS.

9. Reference

   1. Bradner, S., "The Internet Standards Process -- Revision 3", BCP
      9, RFC 2026, October 1996.

   2. Bradner, S., "Key words for use in RFCs to Indicate Requirement
      Levels", BCP 14, RFC 2119, March 1997

   3. [AGR-NFO] Govind P. Agrawal, ôNonlinear Fiber Opticsö, (section
      2.6.2: ôNonlinear refractionö), Academic Press, 1995.

   4. [AGR-FOCS] Govind P. Agrawal, ôFiber-Optic Communication
      Systemsö, Second Edition, Wiley Series in Microwave and Optical
      Engineering, March 1997.

   5. [ELEC-ODS] A. F„rbet et al., ôOptimised dispersion scheme for
      long-haul optical communication systemsö, Electronic Letters 14,
      October 1999, Vol.35, No.21.

   6. [GYS-XT] T. Gyselings, ôInvestigation and Reduction of CrossTalk
      in Wavelength Division Multiplexed All-Optical Cross-Connectsö,
      PhD Thesis, INTEC, Universiteit Gent.

   7. [IPO-IMP] A. Chiu et al., ôImpairments And Other Constraints On
      Optical Layer Routingö, Internet Draft, Work in progress, draft-
      ietf-ipo-impairments-00.txt, May 2001.

   8. [IPO-ORI] A. Banerjee et al., ôImpairment Constraints for Routing
      in All-Optical Networksö, Internet Draft, Work in progress,
      draft-banerjee-routing-impairments-00.txt, May 2001.

   9. [OFC02-NLP] J.-C. Antona et al. ôNonlinear cumulated phase as a
      criterion to assess performance of terrestrial WDM systemsö,
      Technical paper submitted to OFCÆ02.

   10. [OFC00-NLP] Y. Frignac and S. Bigo, ôNumerical optimization of
       residual dispersion in dispersion-managed systems at 40 Gbit/sö,
       Paper TuD3, OFCÆ00, Baltimore.

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draft-papadimitriou-ipo-non-linear-routing-impairm-01    November 2001

10. Acknowledgments

   The authors would like to thank B. Sales, E. Desmet, J.C. Antona, S.
   Bigo and A. Jourdan for their constructive comments and inputs.

11. Author's Addresses

   Dimitri Papadimitriou
   Francis Wellesplein 1,
   B-2018 Antwerpen, Belgium
   Phone: +32 3 240-8491
   Email: dimitri.papadimitriou@alcatel.be

   Jean-Paul Faure
   Route de Nozay
   91461 Marcoussis Cedex, France
   Phone: +33 1 6963-1307
   Email: jean-paul.faure@ms.alcatel.fr

   Olivier Audouin
   Route de Nozay
   91461 Marcoussis Cedex, France
   Phone: +33 1 6963-2365
   Email: olivier.audouin@ms.alcatel.fr

   Roy Appelman
   Phone: +1 972 3 922-9229
   Email: roy.a@civcom.com

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draft-papadimitriou-ipo-non-linear-routing-impairm-01    November 2001

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