## Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links

Optics Express, Vol. 16, Issue 20, pp. 15777-15810 (2008)

http://dx.doi.org/10.1364/OE.16.015777

Acrobat PDF (1328 KB)

### Abstract

We develop an analytic model of *Coherent Optical Orthogonal Frequency Division
Multiplexing* (OFDM) propagation and detection over multi-span long-haul fiber links,
comprehensively and rigorously analyzing the impairments due the combined effects of
*FWM, Dispersion and ASE* noise. Consistent with prior work of Innoe and Schadt
in the WDM context, our new closed-form expressions for the total FWM received power
fluctuations in the wake of dispersive phase mismatch in OFDM transmission, indicate that the
FWM contributions of the multitude of spans build-up *on a phased-array basis*.
For particular ultra-long haul link designs, the effectiveness of dispersion in reducing FWM is
far greater than previously assumed in OFDM system analysis. The key is having the dominant FWM
intermodulation products due to the multiple spans, destructively interfere, mutually
cancelling their FWM intermodulation products, analogous to operating at the null of a
phased-array antenna system. By applying the new analysis tools, this mode of effectively
mitigating the FWM impairment, is shown under specific dispersion and spectral management
conditions, to substantially suppress the FWM power fluctuations. Accounting for the
phased-array concept and applying the compact OFDM design formulas developed here, we analyzed
system performance of a 40 Gbps coherent OFDM system, over standard G.652 fiber, with cyclic
prefix based electronic dispersion compensation but no optical compensation along the link. The
transmission range for 10^{-3} target BER is almost tripled from 2560 km to 6960 km,
relative to a reference system performing optical dispersion compensation in every span
(ideally accounting for FWM and ASE noise and the cyclic prefix overhead, but excluding
additional impairments).

© 2008 Optical Society of America

## 1. Introduction

*Coherent Optical Orthogonal Frequency Division Multiplexing*(CO-OFDM) is emerging as a key high-performance optical transmission technique [1

1. W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division
multiplexing,” Electron. Lett. **42**, 587–589 (2006) [CrossRef]

9. K. Inoue, “Phase mismatching characteristic of four-wave
mixing in fiber lines with multistage optical amplifiers”
Opt. Lett **17**, 801 (1992). [CrossRef] [PubMed]

*Chromatic Dispersion*(CD) and PMD impairments. Notice that there are numerous direct detection studies of optical OFDM, however our exclusive interest in this paper is in OFDM systems using coherent detection, henceforth referred to as OFDM for brevity. The resilience of such systems to CD and PMD is due to the long symbol length of the individual data tributaries carried over multiple low-rate orthogonal sub-channels. In particular, residual CD affecting the low-rate multiple received sub-channels is simply suppressed by applying digital

*Dispersion Compensation*(DC) in the frequency domain, consisting of simple one-tap multiplications of the FFT outputs.

*Four-Wave-Mixing*(FWM) that ultimately sets the limit to OFDM transmission performance.

2. J. Lowery, S. Wang, M. Premaratne, and J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM” Opt. Express15, 12965–12970, (2007), http://www.opticsexpress.org/abstract.cfm?id=125469. [CrossRef] [PubMed]

*mixing products, intermods*or

*beats*) corresponding to all possible triplets of OFDM subcarrier frequencies (excluding those corresponding to SPM and XPM, but including degenerate FWM pairs, wherein two of the three frequencies coincide).

3. W. Shieh, X. Yi, and Y. Tang, “Transmission experiment of multi-gigabit coherent
optical OFDM systems over 1000 km SSMF fiber,” Electron.
Lett. **43**, 183–185 (2007). [CrossRef]

9. K. Inoue, “Phase mismatching characteristic of four-wave
mixing in fiber lines with multistage optical amplifiers”
Opt. Lett **17**, 801 (1992). [CrossRef] [PubMed]

14. T.-K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple
optical amplifiers and dispersion compensation,” J.
Lightwave Technol. **14**249–260 (1996). [CrossRef]

2. J. Lowery, S. Wang, M. Premaratne, and J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM” Opt. Express15, 12965–12970, (2007), http://www.opticsexpress.org/abstract.cfm?id=125469. [CrossRef] [PubMed]

*NonLinear Schroedinger Equation*(NLSE), proceeding to detailed modeling of multichannel propagation in terms of the

*Non-Depleted Pump Approximation*[11], yielding high accuracy for the case at hand. The FWM dispersive build-up over a single span as well as over multiple spans is initially thoroughly analyzed.

*multiple spans*in a

*dispersive*fiber transmission link, based on methods first introduced in [9

9. K. Inoue, “Phase mismatching characteristic of four-wave
mixing in fiber lines with multistage optical amplifiers”
Opt. Lett **17**, 801 (1992). [CrossRef] [PubMed]

10. D. G. Schadt, “Effect of amplifier spacing on four wave mixing in
multichannel coherent communications,” Electron. Lett **27**, 805–7 (1991). [CrossRef]

2. J. Lowery, S. Wang, M. Premaratne, and J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM” Opt. Express15, 12965–12970, (2007), http://www.opticsexpress.org/abstract.cfm?id=125469. [CrossRef] [PubMed]

**17**, 801 (1992). [CrossRef] [PubMed]

10. D. G. Schadt, “Effect of amplifier spacing on four wave mixing in
multichannel coherent communications,” Electron. Lett **27**, 805–7 (1991). [CrossRef]

*N*

*transmission spans in a multi-span OFDM long-haul link do*

_{span}*not*add up in-phase. The compounding law for the FWM contributions of multiple spans in any multi-channel system, as briefly previewed in [12,13] for an OFDM system, and as first modeled by Innoe [9

**17**, 801 (1992). [CrossRef] [PubMed]

10. D. G. Schadt, “Effect of amplifier spacing on four wave mixing in
multichannel coherent communications,” Electron. Lett **27**, 805–7 (1991). [CrossRef]

*the individual spans effectively act as antennas in a distributed one-dimensional Phased-Array (PA)*. Hence, the FWM terms, due to each of the individual spans, do add up on a field basis with definite phases, nevertheless these phases are not all equal (i.e. the contributions are not in-phase). Rather, the optical field FWM contributions from successive spans (for any given triplet of subcarriers) are regularly de-phased, analogous to the fields generated by the antenna elements of a radio-frequency phased-array, hence the e Δ

*β*dependence of any FWM mixing product at the link end is given by that of a single span multiplied by a phased-array factor, accounting for the interference between the multiplicity of spans. This is also analogous to the linear transfer function (vs. k-vector) of a short fiber grating (with the number of grating periods matching the number of spans in the OFDM system). Moreover, a different PA-factor is applicable to each of the mixing products. As a result, for certain OFDM multi-span link configurations, it is even possible to have the dominant FWM mixing products interfere destructively and nearly cancel. Akin to the generation of nulls in the radiation pattern of a wireless PA, or the formation of bandgaps in the transmission pattern of a fiber grating, destructive interference may set in between the FWM contributions of the individual fiber spans, providing substantial FWM reduction effect, for specific link configurations. The cumulative FWM build-up effect over thousands of frequency triplets (intermods) is analytically formulated here, enabling accurate prediction of the overall FWM cancellation attained by the PA effect.

**17**, 801 (1992). [CrossRef] [PubMed]

**27**, 805–7 (1991). [CrossRef]

**17**, 801 (1992). [CrossRef] [PubMed]

**27**, 805–7 (1991). [CrossRef]

*Cross-Phase Modulation*(XPM) generated in an ASK-WDM multi-span system [14

14. T.-K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple
optical amplifiers and dispersion compensation,” J.
Lightwave Technol. **14**249–260 (1996). [CrossRef]

15. M. E. Marhic, N. Kagi, T.-K. Chiang, and L. G. Kazovsky, “Optimizing the location of dispersion compensators
in periodically amplified fiber links in the presence of third-order nonlinear
effects” Photon Technol. Lett. **8**, 145–147 (1996). [CrossRef]

16. M. Eiselt, “Limits on WDM Systems Due to Four-Wave Mixing: A
Statistical Approach,” J. Lightwave Technol. **17**, 2261–2267 (1999). [CrossRef]

16. M. Eiselt, “Limits on WDM Systems Due to Four-Wave Mixing: A
Statistical Approach,” J. Lightwave Technol. **17**, 2261–2267 (1999). [CrossRef]

**17**, 801 (1992). [CrossRef] [PubMed]

**27**, 805–7 (1991). [CrossRef]

**17**, 801 (1992). [CrossRef] [PubMed]

**27**, 805–7 (1991). [CrossRef]

**17**, 801 (1992). [CrossRef] [PubMed]

## 2. System description

*u*(

*t*;

*z*) be the real-valued optical field at time

*t*and position

*z*along the fiber,

*u*̰(

*t*;

*z*) its

*Complex-Envelope*(CE), and

*ŭ*(

*t*;

*z*) its

*SpatioTemporal CE*(STCE):

*i*-th channel is at frequency

*ω*

*=Ω*

_{i}*+*

_{i}*ω*

_{0}, i.e. it deviates Ω

*away from a reference freq*

_{i}*ω*

_{0}. This formulation also applies to OFDM+DWDM, in which case there are multiple OFDM signals further multiplexed in wavelength. In this paper we focus on a single OFDM channel, however, multiple OFDM WDM channels, each carrying its own OFDM channel, may be similarly treated using the formulation of this paper.

*t*=

*n*

*T*where

*n*is an integer denoting discrete-time. Assume that a lone OFDM block is transmitted at discrete-time

*n*=0, simultaneously exciting all

*M*OFDM sub-channels with independent PSK symbols (equivalently if there are multiple OFDM blocks transmitted, we ignore inter-block interference, as ensured by the cyclic prefix technique). The signal STCE transmitted at the i-th subcarrier radian frequency

*ω*

_{i}=

*ω*

_{0}+Ω

*is assumed to consist of a modulated rectangular pulse (where*

_{i}*1*[

*,*

_{a}*](*

_{b}*t*)≡1 if

*t*∈[

*a*,

*b*],

*1*[

*a*,

*b*](

*t*)≡0, otherwise):

*L*=

*N*

_{span}*L*

*, typically consisting of*

_{span}*N*

*identical spans, each initiated and terminated in an*

_{span}*Optical Amplifier*(OA) perfectly compensating the power loss

*j*Ω

*t*] corresponding to the

*i*-th sub-channel), and applying integrate-and-dump (I&D) filtering onto each of the downconverted signals.

*T*

^{-1}scaling factor, amounting to a time-average:

*T*

^{-1}, then one-tap-equalized (i.e. multiplied by a complex weight) cancelling the dispersion, i.e. realigning the received constellation axes and normalizing the magnitude. Each of the equalized sub-channel constellations is input into its own PSK decision device.

## 3. Coupled-mode equations for the OFDM sub-channel complex amplitudes

*Non-linear Schroedinger’s Equation*(NLSE) [11].

**NonLinear Schroedinger Equation:**The STCE

*ŭ*(

*t*,

*z*) at position

*z*along the fiber and at time

*t*, satisfies the NLSE

*t*→

*t*-

*β*

^{′}

*z*is assumed,

*α*is the loss coefficient,

*γ*is the non-linear coefficient [11],

*∂*

*is the derivative with respect to the variable*

_{t}*t*,

*∂*

^{2}

*the second derivative,*

_{t}*β*

^{′}≡

*∂*

_{ω}*β*(

*ω*) and

*β*

^{″}≡

*∂*

^{2}

_{ω}*β*(

*ω*).

17. G.P. Agrawal, *Lightwave Technology: Telecommunication Systems*
(Wiley2005). [CrossRef]

*S*[

*i*]of indexes of frequency triplets corresponding to FWM

*Intermodulation*(intermod) products falling on

*ω*

*, namely the set of “proper FWM” intermods, excluding the SPM and XPM “coherent” intermods (for which*

_{i}*j*=

*i*or

*k*=

*i*):

*β*

^{T}*, also features in Eq. (8), describing the spatial rates of phase-change due to Dispersion, SPM, XPM, and representing loss as imaginary*

_{i}*β*:

*P*

*≡∑*

^{T}

^{M}*=*

_{k}_{1}|

*ŭ*

*|*

_{k}^{2},

*p*

*≡*

_{i}*|*ŭ

*|*

_{i}^{2}, (all functions of

*t*,

*z*), and

*Dispersion*and

*Loss*, with

**An alternative point of view regards the composite OFDM waveform Eq. (3) as generating its own SPM, as reflected in the RSH of NLSE Eq. (7) prior to substituting into it the superposition Eq. (3) of multiple OFDM tones. Indeed, the RSH of Eq. (7) may be rewritten as -**

*SPM and XPM:**jγ*|

*ŭ*|

^{2}ŭ=-

*jβ*

*ŭ, with*

_{NL}*β*

*≡*

_{NL}*γ*|

*ŭ*|

^{2}a phase constant proportional to the composite signal intensity, which modulates its own phase, i.e. the nonlinear distortion generated by the OFDM signal is actually an SPM effect. While in principle this SPM description would capture the full nonlinear effects treated here, this point of view is not actually useful in moving further. Indeed the composite OFDM waveform

*ŭ*(

*t*) would look erratic on a scope (as it is the speckle-like addition of a large number of randomly phased phasors), and its further analysis in the time-domain would be intractable without taking advantage of its frequency domain structure as the superposition of randomly phased subcarriers (Eq. (3)). To this end we must actually perform the substitution of Eq. (3) into Eq. (7) and simplify the resulting expression, leading to Eq. (8) - this amounts to working out the details of the “SPM” effect by a Fourier analysis of the composite OFDM waveform, breaking it into its individual single frequency ingredients, the substitution of which into the triple product in the RSH of Eq. (7) would yield a sum over triplets of OFDM subcarriers as in the RSH of Eq. (8), eventually leading to FWM view, which is formally equivalent to the abstract SPM top view, but is more operationally more amenable. Another relevant remark is that distortions of the SPM/XPM/FWM type should be specified relative to a signal set. In our case the SPM of

*ŭ*(

*t*) happens to coincide with the FWM+XPM/FWM of the spectral constituents {

*ŭ*

*(*

_{i}*;*

**t***z*)

*e*

^{jΩ}

^{i}*} of*

^{t}*ŭ*(

*t*).

*j≠i≠k*onto the frequency triplets considered for OFDM modeling. The excluded triplets are of the form [

*j*,

*k*,

*l*]=[

*i*,

*k*,-

*k*] with

*k*≠

*i*(XPM terms) and [

*j*,

*k*,

*l*]=[

*i*,

*i*,-

*i*] (SPM terms). For these triplets,

*ŭ*

_{j}*ŭ*

_{k}*ŭ**

*reduces to*

_{l}*ŭ*

*|*

_{i}*ŭ*

*|*

_{k}^{2}and the summations over all such terms effectively represents the intensity-dependent term in the propagation constant

*β*

^{T}*in Eq. (10), describing an overall frequency-dependent phaseshift of each received OFDM sub-channel, say the*

_{i}*i*-th one. In QPSK-modulated OFDM all the amplitudes of the OFDM sub-channels are constant (just the phases are modulated), hence the mixing products associated with XPM/SPM of the subcarriers do not contribute to statistical fluctuations of the received angle, but they rather contribute to a bias in the mean of the received angle — a fixed constellation rotation due to the intensity- dependent term in the propagation constant, which may be calibrated out at the receiver, without closing the reception eye (unlike FWM which does close the OFDM eye, due to random fluctuations in the received angle due to the buildup of the FWM distortion. Hence, unlike in ASK multichannel transmission, the XPM between the subcarriers (and the SPM of each subcarrier) does not contribute to BER degradation. The only nonlinear source to BER degradation in OFDM is “proper” FWM (i.e. triplets with

*j*≠

*i*≠

*k*as accounted for in the

*S*[

*i*] set of Eq. (9)).

**We now set**

*Undepleted pump approximation:*^{(1)}refers to the effective linear propagation of the “pumps”, i.e. the launched channels (propagating with modified refractive index induced by SPM and XPM). The index

^{(3)}refers to the total FWM intermods generated by the pumps at the

*i*-th channel.

*ŭ*

^{(1)}

*of Eq. (15) is used to formulate a non-linear equation Eq. (16) for the perturbation*

_{i}*ŭ*

^{(3)}

*≡*

_{i}*ŭ*

*-*

_{i}*ŭ*

^{(1)}

*. Formally, Eq. (16) is obtained by setting*

_{i}*ŭ*

*=*

_{i}*ŭ*

^{(1)}

*+*

_{i}*ŭ*

^{(3)}

*into (8), and applying the*

_{i}*Undepleted Pumps Approximation*(UPA) [11]. Since

*ŭ*

^{(1)}

*satisfies the homogeneous equation Eq. (15), the LSH operator of Eq. (8) acting on*

_{i}*ŭ*

^{(1)}

*+*

_{i}*ŭ*

^{(3)}

*nulls out the*

_{i}*ŭ*

^{(1)}

*component, leaving just*

_{i}*ŭ*

^{(3)}

*in the LSH of Eq. (16). Adopting the UPA, we set*

_{i}*ŭ*

^{(1)}

*+*

_{i}*ŭ*

^{(3)}

*≈*

_{i}*ŭ*

^{(1)}

*, into RSH of Eq. (16), as well as into the*

_{i}*β*

^{T}*term Eq. (17). The physical significance of the UPA is as follows: the third-order field is driven by the polarization currents generated by the undepleted pumps alone, neglecting, for the purpose of further evaluation of the intermods, small corrections due to the already generated intermods superposing onto the pumps.*

_{i}*ŭ*

*=*

_{i}*ŭ*

^{(1)}

*,*

_{i}*ŭ*

^{(3)}

*=0. Then Eq. (15) is solved. Its solution*

_{i}*ŭ*

^{(1)}

*is inserted into Eq. (16) to calculate the FWM-induced perturbation*

_{i}*ŭ*

^{(3)}

*. At the end*

_{i}*ŭ*

*=*

_{i}*ŭ*

^{(1)}

*+*

_{i}*ŭ*

^{(3)}

*is formed as the full solution.*

_{i}**We next consider the**

*Quasi-CW solution:**quasi-CW*or

*weakly-dispersive*case, wherein the durations of the transmitted pulses are long relative to the dispersion delay spread (corresponding to a large number of OFDM sub-channels each carrying a low data rate), such that the waveform distortion due to dispersion may be neglected in the LHS of Eq. (15) and Eq. (16). Notice that the impact of dispersion on the phase mismatch efficiency of FWM is still accounted for in the RHS of Eq. (16). Discarding the time-derivatives in the LSH then yields the following set of coupled differential equations, one for each observation frequency:

*ŭ*

*(*

_{i}*t*;

*z*) over a single span:

*α*(

*z*)=

*α*

_{0}=

*const*.)

17. G.P. Agrawal, *Lightwave Technology: Telecommunication Systems*
(Wiley2005). [CrossRef]

*ŭ*

*(*

_{i}*t*;

*z*), introducing (in our notation) a modified SCTE

*z*-dependent

*β*

^{T}*(*

_{i}*z*). We then generalize Agrawal’s normalization to the following formulation:

*p*

*(*

_{i}*t*,

*z*)≡

*|*ŭ

*(*

_{i}*t*,

*z*)|

^{2}.

*z*-dependent loss,

*α*(

*z*). Typically the loss is modeled as constant along each of the equal fiber spans, but it may differ from span to span, and so may the lengths of the spans differ. Moreover, we find it useful to model the optical amplifiers gains as a negative impulsive losses. The typical loss profile assumed in this paper corresponds to a regular multi-span system with identical spans:

*α*(

*z*)=

*α*

_{0}, while the amplifiers are modeled as negative loss spatial impulses at the span boundaries. Notice that Eq. (30) excludes the initial transmitter post-amplifier (considered part of the optical source) and the last receiver pre-amplifier (separately treated). It follows that the “pumps”

*v*̆

^{(1)}

*are constant over each fiber span, expressible in terms of the initial power*

_{i}*p*

_{0}(

*t*) per sub-channel (assumed identical over all sub-channels) launched at the beginning of the span, and the transmitted phase,

*ϕ*

*(*

_{i}*t*) of the

*i*-th channel:

*ω*, such that the

*i*-th output frequency is

*ω*

*=*

_{i}*ω*

_{0}+

*i*Δ

*ω*, i.e. Ω

*i*=

*ω*

*-*

_{i}*ω*

_{0}=

*i*Δ

*ω*, and similarly for

*ω*

_{j}*ω*

_{k}*ω*

*, now represented by their indexes*

_{l}*j*,

*k*,

*l*.

*ω*

*+*

_{j}*ω*

*+-*

_{k}*ω*

*=*

_{l}*ω*

*then reduces to*

_{i}*j*+

*k*-

*l*=

*i*or equivalently

*l*=

*j*+

*k*-

*i*,i.e. a given output index,

*i*along with the pair of indexes

*j*,

*k*, determine the 4th index,

*l*.

*double*sum over

*j*,

*k*:

*M*, the intermods (mixing products) summation domain may be compactly re-expressed as

*j*+

*k*-

*i*≤

*M*stems from the requirement that

*l*be in-band as well: 1≤

*l*=

*j*+

*k*-

*i*≤

*M*. Figure. 1 plots the set

*S*[

*i*] in the (

*j*,

*k*) plane. Henceforth the [

*j*,

*k*] pairs, will be interchangeably referred to as

*intermods*(recalling that the [

*j*,

*k*] indexes determine the

*l*, the intermods actually refer to [

*j*,

*k*,

*l*] triplets of frequencies, generating the fourth

*i*-th frequency.

*i*=

*i*Δ

*ω*, and similarly for the other indexes, the mismatch Eq. (11) is compactly expressed (and relabeled) in terms of three (rather than four) indexes,

*j*

^{′}≡

*j*-

*i*,

*k*

^{′}≡

*k*-

*i*) to:

*p*

*rigorously emerging here in our derivation of total effective phase mismatch coefficient Δ*

_{ijk}*β*

*, was previously derived in 18,19*

_{ijk}19. S. Song, C. T. Allen, K. R. Demarest, and R. Hui, “Intensity-Dependent Phase-Matching Effects on
Four-Wave Mixing in Optical Fibers,” J. Lightwave Technol. **17**, 2285–2290 (1999). [CrossRef]

*β*mismatch coefficient. Such correction terms were not included in [9

**17**, 801 (1992). [CrossRef] [PubMed]

**27**, 805–7 (1991). [CrossRef]

**17**, 801 (1992). [CrossRef] [PubMed]

**27**, 805–7 (1991). [CrossRef]

*Equi-power sub-channels:*Notice that when the observation coincides with an OFDM sub-channel frequency,

*i*=1,2,…,

*M*, the index pairs [

*j*,

*k*] belonging to the summation domain,

*S*[

*i*], together with the index

*i*, satisfy Δ

*p*

*=0, provided that all (sub)channels are launched with equal power, (as applicable to*

_{ijk}*m*-ary PSK):

*p*

^{(1)}

*(*

_{j}*t*,0)=

*p*

^{(1)}

*(*

_{k}*t*,0)=

*p*

^{(1)}

*l*(

*t*,0)=

*p*

^{(1)}

*(*

_{i}*t*,0)≡

*p*

_{0}(

*t*). Indeed, as the initial conditions for the launched powers are identical, and all four signals

*ŭ*

^{(1)}

*(*

_{i}*t*,

*z*),

*ŭ*

^{(1)}

*(*

_{j}*t*,

*z*),

*ŭ*

^{(1)}

*(*

_{k}*t*,

*z*),

*ŭ*

^{(1)}

*(*

_{l}*t*,

*z*), satisfy the same differential equation, at any point along the link, irrespective of the multispan structure of the link equality of powers also holds:

*p*

*Eq. (29) to the FWM mismatch, Δ*

_{ijk}*β*

^{T}*, nulls out, yielding Δ*

_{ijk}*β*

^{T}*=Δ*

_{ijk}*β*

^{DL}*. It is only upon observing an out-of-band intermod, not falling onto any of the OFDM sub-channels, that we would have Δ*

_{ijk}*p*

*≠0. E.g., this is the case for a 3-tone test with three pumps not equally spaced, such that their intermods do not coincide in frequency with one of the pump channels. However, out-of-band intermods are not of interest to us, as they do not overlap with existing channels, hence their FWM fluctuations may be filtered out. Conveniently, at OFDM sub-carrier frequencies (i.e. points of the frequency grid occupied by actual OFDM channels) the power correction to the phase-matching condition vanishes, hence it is sufficient to account for wave-vector mismatch and loss, ignoring the power-dependent SPM/XPM induced corrections to the propagation constant.*

_{ijk}*Setting Δ*

**Propagation equations:***β*

^{T}*=Δ*

_{ijk}*β*

^{DL}*in (37) yields*

_{tjk}*final coupled-mode equations for FWM build-up at the carrier frequencies of the OFDM sub-channels over a single span:*

## 4. FWM build-up for an OFDM signal over a single dispersive span

*L*=

*L*we have Δ

*β*

^{DL}*(*

_{ijk}*t*,

*z*)=Δ

*β*

*-*

_{ijk}*jα*=

*const.*.

*Field propagation:*The integral Eq. (42) then yields ΔΦ

*(*

^{DL}*t*,

*z*)=Δ

*β*

^{DL}

_{ijk}*z*, hence Eq. (41) reduces to

*v*̆

^{(1)}

*, for any index,*

_{n}*n*(standing for

*i*,

*j*,

*k*,

*l*) are constant along

*z*. As it is only the exponent in the RSH of Eq. (43) that is z-dependent, this differential equation is readily integrated over the segment[0,

*L*], assuming the boundary condition

*v*̆

^{(3)}

*=0, yielding the STCE of the FWM fluctuation affecting the*

_{i}*i*-th OFDM channel (with its time-dependence not explicitly indicated):

*single-span Nonlinear Effective Length*(NEL) parameter, describing the phase matching efficiency of FWM generation:

*β*

*=0, one must still account for loss, hence the DNEL Eq. (45) reduces to the well-known*

_{ijk}*Effective Length*parameter,

*L*

*, appearing in the description of SPM/XPM generation [11,17*

_{eff}17. G.P. Agrawal, *Lightwave Technology: Telecommunication Systems*
(Wiley2005). [CrossRef]

*L*,

*e*-

*αL*≪1, the DNEL simplifies to

*L*

^{FWM}*≈(*

_{ijk}*j*Δ

*β*

*+*

_{ijk}*α*)

^{-1}.

*Power propagation:*Substituting Eq. (31) into Eq. (44), then Eq. (44) into Eq. (23) and squaring, yields the FWM power at the end of a single-span link (

*L*=

*L*

*):*

_{span}*S*[

*i*] into two sets: a

*degenerate*(DG) subset (the points in the hexagonal domain along the bisector of the [

*j*,

*k*] plane in Fig. 1):

*non-degenerate*(NDG) subset (

*j*≠

*k*), in turn expressed as the union of two subsets:

*S*

*<[*

^{NDG}*i*], is obtained by transposition of an element in

*S*

*>[*

^{NDG}*i*] and viceversa. The summation in Eq. (47) then breaks into two subsums:

*j*,

*k*], [

*k*,

*j*] are indistinguishable, yielding identical phases in their respective intermods, hence these two pairs add up coherently, on an amplitude basis. However, whenever [

*j*,

*k*]≠[

*j*

^{′},

*k*

^{′}](meaning at least one index is different), then the random phase factors

*j*>

*k*, and

*j*

^{′}>

*k*

^{′}contain at least one different phase out of the three phases, hence the correlation of these two terms comes out zero, since the

*m*-ary PSK angles are equiprobable over the

*M*-ary PSK set, and independent over the distinct indexes:

*S*[

*i*]over all intermods, and the NEL was normalized (as denoted by a hat) by dividing through the effective length,

*β*

*is explicitly given by Eq. (36). The modulus of the normalized NEL, |*

_{ijk}*L*̂

^{FWM}*|, is referred to as*

_{ijk}*single-span FWM Attenuation*for the particular

*ijk*beat. Now

*root-mean-square*(rms) average Eq. (53) over all intermods,

*denotes rms averaging (here over all pairs of the set*

_{rms}*S*[

*i*]), and

*j*,

*k*] pairs belonging to the hexagonal domain of beat points in Fig. 1. Notice that for for

*M*≫1, out of this large number of intermods, just a relatively small number, namely

*N*

^{DG}*[*

_{beats}*i*,

*M*]=

*M*/2-1, are degenerate.

*L*̂

^{FWM}*≤1 for each of the intermods, hence*

_{ijk}*L*̂

^{FWM}*[*

_{rms}*i*,

*M*]≤1, with equality attained in the dispersion-free case. Similarly, rms-average over the degenerate intermods, yielding

*Effective FWM Supression*(EFWMS), defined as:

*M*≫1, we have

*N*

^{DG}*[*

_{beats}*i*,

*M*]≪

*N*

*[*

_{beats}*i*,

*M*]. The fact that the DG intermods form an infraction of the total number of intermods has also been noticed in [2

*FWM power:*Using Eq. (57) and Eq. (56) we note that (

*L*̂

^{FWM}*[*

_{eff}*i*,

*M*])

^{2}

*N*

*reduces to the braces in the last expression in Eq. (52). The FWM power Eq. (52) is then compactly expressed as follows:*

_{beats}*received FWM power*(for a single span system) then compactly expressed as:

## 5. FWM compounds over multiple spans as radiation from a phased-array

*L*=

*N*

_{span}*L*

*the intermods coherently add up in-phase, i.e. the complex amplitude of each intermod is increased by a factor of*

_{span}*N*

*, hence the overall FWM power at the end of an*

_{span}*N*

*-link increases by a factor of*

_{span}*N*

^{2}

*relative to the FWM power Eq. (60) generated at the end of a single span. This statement is only correct in the absence of dispersion. The rule for the compounding of dispersive spans, turns out to be quite different, as briefly previewed in [12]. We proceed to develop the precise model of FWM generation over a link comprising multiple identical spans in the presence of dispersion. Similarly to [9*

_{span}**17**, 801 (1992). [CrossRef] [PubMed]

**27**, 805–7 (1991). [CrossRef]

*Inhomogeneous loss/gain:*Assuming Δ

*β*

^{DL}*(*

_{ijk}*z*)=Δ

*β*

^{DL}*=*

_{ijk}*const*., but allowing for an inhomogeneous loss profile (possibly including negative loss, modeling of the OAs as negative loss spatial impulses Eq. (30)) it is apparent that Eq. (42) reduces to:

*α*(

*z*) per unit length (effectively representing the optical amplifiers as regularly spaced negative loss impulses) and introduced the notation

*N*

*periods of a sawtooth waveform, with a discontinuity at each optical amplifier restoring it to its zero peak value. Using Eq. (61), we have*

_{span}*G*(

*z*)=

*G*(

*z*)

**1**[

_{0},

*](*

_{L}*z*), windowed over the domain [0,

*L*] of interest:

*G*(

*z*) is

*finite periodic*, i.e. it is expressible as the superposition of

*N*

*replicas of a waveform of duration*

_{span}*L*

*, shifted by integer multiples of*

_{span}*L*

*:*

_{span}*D*

^{FWM}*≡*

_{ijk}*L*

_{eff}*N*

_{span}*D*̂

^{FWM}*(in units of distance) is the*

_{ijk}*multi-span NEL*, providing a generalization of the single-span expression

*L*

^{FWM}*. The multi-span NEL is given by*

_{ijk}^{nd}equality we used G(

*z*)=G(

*z*)

**1**[

_{0},

*](*

_{L}*z*), and the last is identified as a spatial

*Fourier Transform*(FT):

*D*

^{FWM}*:*

_{eff}*β*

^{T}

*=*

_{i}*β*

_{i}-

*β*

_{0}-

*jα*(

*z*)/2, yielding:

*̰, as before, we have*

**r***D*

^{FWM}*plays for a multi-span link a role analogous to that of*

_{ijk}*L*

^{FWM}*for a single-span link.*

_{ijk}*We now consider a “regular” multi-span link with dispersion, wherein all the spans are identical in fiber losses, propagation constants and lengths, proceeding to evaluate the FT Eq. (69). For such a system, the gain profile is finite-periodic, as described in Eq. (66), which expression is readily Fourier transformed as the product of the FTs of two convolutional terms,*

**Array of spans as phased array:***F*

*is called*

_{ijk}*array factor*, readily evaluated as a finite geometric series,

*dinc*function in terms of the

*FWM Coherence Length*, (CL)

*j*,

*k*substantially deviating from

*i*.

*Coherence Number*, (CN), given by

*F*

*|≤1. In the absence of dispersion we have*

_{ijk}*F*

*=1=*

_{ijk}*F*

*and Eq. (77) reduces to*

_{ijk}*D*

^{FWM}*=*

_{ijk}*N*

*, i.e. the spans add up coherently, in phase, consistent with [2*

_{span}*D*̂

^{FWM}*|=|*

_{ijk}*F*

*|*

_{ijk}*L*̂

^{FWM}*| represents the*

_{ijk}*multi-span FWM attenuation*, equal to the single-span FWM attenuation multiplied by the modulus of the array factor.

**Our phased-array-like analysis was formulated for homogenous links wherein all spans were assumed identical in length, loss and propagation constant. Inspecting the proof of our key result Eq. (69), it is apparent that it did not make use of the specific periodic form assumed for the gain profile**

*Irregular Array:**G*(

*z*). The result Eq. (69) then more generally applies to multi-span structures with arbitrary fiber losses, lengths and amplifier gains, varying from span to span, provided that the gain profile

*G*(

*z*) is redefined accordingly:

*Multi-Span DNEL :*

*L*

^{FWM}*[*

_{ijk}*s*] is the FWM effective length of the

*s*-th span which has loss

*α*

*and length*

_{s}*L*

*[*

_{span}*s*]=

*z*

*+1-*

_{s}*z*

*, where*

_{s}*z*

_{0}=0,

*z*

_{1},

*z*

_{2},…,

*z*

_{N}*are the span boundaries. This is the most general description corresponding to an irregular phased-array, wherein the “antennas” have variable strengths and are not regularly spaced. The conventional regular array formulation [9*

_{span}**17**, 801 (1992). [CrossRef] [PubMed]

**27**, 805–7 (1991). [CrossRef]

*z*

*form an arithmetic sequence, and setting the FWM effective lengths of the individual spans all equal). Notice that throughout the paper it was assumed that the propagation constant is spatially uniform along the fiber link and each span loss is compensated by an equal OA gain. The most general formulation [21*

_{s}21. W. Zeiler, F. D. Pasquale, P. Bayvl, and J. E. Midwinter, “Modeling of four-wave mixng and gain peaking in
amplified WDM optical communication systems and networks,”
J. Lightwave Technol. **14**, 1933–1942 (1996). [CrossRef]

*In order to significantly reduce FWM, i.e. have |*

**Large FWM suppression via the PA effect:***D*̂

^{FWM}*|≪|*

_{ijk}*L*̂

^{FWM}*|≤1, we must have |*

_{ijk}*F*

*|≪1, i.e. the operating point should be rolled into sidelobes of the dinc function; as illustrated in Fig. 1(b,c) for 83 spans, for large*

_{ijk}*N*

*(i.e. long links), the sidelobes tend to become very low. Having |*

_{span}*u*|>1 places the operating point in the sidelobes (barring the periodicity of the

*dinc*). Inspecting (84), the

*dinc*argument is identified as

*u*=

*N*

*/*

_{span}*N*

^{ijk}*, hence the condition that a particular FWM beat attain substantial attenuation (be placed in the dinc sidelobes) is reformulated as:*

_{coh}*N*

^{ijk}*,*

_{coh}*L*

^{ijk}*are inverse measures of dispersion, made small by having sufficient dispersion. We conclude that the more dispersive intermods (those having coherence lengths less than the total length of the link) experience substantial FWM attenuations. Notice that substantial overall (average) FWM suppression may still be obtained even when there are intermods with coherence lengths exceeding the link length, provided that those intermods form a small fraction of the full*

_{coh}*S*[

*i*]set of intermods. Using Eq. (81), condition Eq. (88) is equivalently formulated as

*Hyperbolic Distance*(HD)

*critical HD*, to be exceeded in order to attain high FWM attenuation:

*W*=

*M*Δ

*ν*is the total OFDM bandwidth over the

*M*subcarriers.

*j*,

*k*deviates a lot from

*i*(i.e. [

*j*,

*k*]is far away from [

*i*,

*i*] in HD) that potentially experience high FWM attenuation. The distribution of FWM attenuations is next investigated for specific example addressed in Fig. 1, showing the points corresponding to the

*S*[

*i*], further superposing contour lines of the HD between each of the intermods ([

*j*,

*k*] pairs) in

*S*[

*i*] and the [

*i*,

*i*] pair, with all hyperbolic distances normalized by

*d*

^{crit}*. In the (*

_{h}*j*,

*k*) plane, the “truth set” of condition Eq. (89) corresponds to the exterior of a “hyperbolic circle” (actually a star-shape in the Euclidean plane) of “radius”

*d*

^{crit}*(i.e.*

_{h}*d*̂

^{crit}*=1). The intermods belonging to this truth set all experience high FWM attenuation. To maximize the size of this desirable set, one should make the “hyperbolic radius”*

_{h}*d*

^{crit}*as small as possible, so that its exterior cover more and more of the*

_{h}*S*[

*i*] domain. The interior of the “critical circle” then mainly contains low FWM attenuation intermods (in fact, not all of the intermods in the “circle” interior entail low FWM attenuation. As these intermods correspond to the dinc mainlobe, towards its fringes there is sufficient rolloff to provide high attenuation. Hence our count of high FWM attenuation intermods, as just those in the sidelobes - critical circle exterior - is a conservative one).

*M*=128 subcarriers transmitted over

*N*

*=83 spans, it turns out that the*

_{span}*β*

*=-21.7psec*

^{′}^{2}/Kmcoefficient of the G.652 fiber,

*M*=128 channels at inter-carrier-spacing Δ

*v*=200

*MHz*) suffice to reduce the critical HD Eq. (91), to the low level

*d*

^{crit}*≡27.6. These are the parameters actually assumed in Fig. 1. The 12033 intermods in the*

_{h}*S*[

*i*] set are then partitioned into two subsets, the “truth set” of condition Eq. (89), and its complement. It is the “truth set” that provides high sidelobes attenuation. For the system in [12], this high FWM attenuation ”sidelobes” set contains 11653/12033=96.8% of the intermods. Its complement, namely the low FWM attenuation set (the “mainlobe”), depicted in Fig. 1 as the black cross in the center, then contains just 3.2% of the intermods, hence contributes very little to the average FWM suppression, which is predominantly determined by the vast majority of highly attenuated intermods in the truth set (the orange colored area exterior to the cross).

*u*Eq. (82) of the array factor Eq. (84), next reformulated by making use of Eqs. (81)–(83) and Eq. (89) to show that

*u*is actually given by normalizing the HD of the [

*j*,

*k*] beat from [

*i*,

*i*], through the critical HD, expressed using Eq. (91) in the last equality:

*S*[

*i*] domain in the (

*j*,

*k*) plane, the arguments

*u*of the array factor (the values whereat the dinc plot of Fig. 2(a) is sampled) may be directly read off as the labels of the constant HD contours, provided that the “hyperbolic radii” are first normalized by division through

*d*

^{crit}*. In particular the label*

_{h}*u*=1 labeling the boundary of the high attenuation region (the center “cross” of dark points), corresponds to the first zero crossing of the dinc, namely the transition point from the mainlobe to the first sidelobe.

*L*̂

^{FWM}*, the single-span FWM attenuation, and Fig. 3(b–e) plotting*

_{ijk}*D*̂

_{FWM}*=*

_{ijk}*F*

_{ijk}*L*̂

^{FWM}*, the multispan FWM attenuation (obtained from the single-span attenuation by multiplying by the array factor*

_{ijk}*F*

*). Notice that*

_{ijk}*L*̂

^{FWM}*is a mildly varying function over all intermods, hovering above -0.5 dB for the majority of the intermods — rolling off substantially just in the fringes of the*

_{ijk}*S*[

*i*] domain. Accordingly, the average single-span attenuation turns out to be as little as

*L*̂

^{FWM}*≈1*

_{rms}*dB*. For a multi-span system, transitioning to

*D*̂

^{FWM}*by multiplying*

_{ijk}*L*̂

^{FWM}*through*

_{ijk}*F*

*, the shape of*

_{ijk}*D*̂

^{FWM}*is then essentially determined by*

_{ijk}*F*

*. However, as already seen above, the vast majority of the intermods fortunately sample*

_{ijk}*F*

*into its sidelobes, yielding substantial attenuation for the vast majority of the*

_{ijk}*D*̂

^{FWM}*terms, as plotted in Fig. 3(b-e). Substituting our design parameters into the EFWMS formula Eq. (95) (with*

_{ijk}*F*

*,*

_{ijk}*L*̂

^{FWM}*evaluated in turn according to Eq. (93) and Eq. (60)), then yields the overall FWM suppression of*

_{ijk}*D*̂

^{FWM}*=18.5*

_{eff}*dB*for 83 spans. For the 94 spans system yielding 10

^{-3}BER, the EFWMS is 19.2 dB. For a 61 spans system attaining 10

^{-4}BER, the suppression is 17.1 dB.

**FWM suppression goes as the***bandwidth*

^{2}×length×GVD*product, nearly independent of M:*Throughout the discussion below take

*i*=

*M*/2, i.e. observe the FWM mid-band.

*S*[

*i*] domain (the orange hexagon in Fig. 1). In turn, the

*S*[

*i*] domain area is proportional to the number of [

*j*,

*k*] points falling within the domain, as given Eq. (55), which is nearly linear in

*M*

^{2}. We next show that the mainlobe area is also linear in

*M*

^{2}. It then follows that the FWM suppression is independent of the sub-channel count

*M*(as both the mainlobe and total areas are linear in

*M*

^{2}, which cancels in their ratio) and we are left with a

*bandwidth*dependence for the FWM suppression.

^{2}×length×GVD*u*

*=1 in (92). The mainlobe area is proportional to its characteristic linear dimension, which is taken as half the side,*

^{ijk}*j*-

*i*, of a square centered on [

*i*,

*i*] inscribed into the mainlobe, with the [

*j*,

*j*] vertex of the square touching the mainlobe boundary in the first quadrant of a “straight” coordinate system centered on [

*i*,

*i*]. Half the side of this inscribed square,

*l*

_{0}≡

*j*-

*i*, is then adopted as the characteristic length dimension of the mainlobe. The hyperbolic distance of the [

*j*,

*j*] vertex from the square center [

*i*,

*i*] is then given by

*j*=

*k*, the

*u*

*=1 condition for the mainlobe boundary, as well as (93), yielding the following relation:*

^{ijk}*l*

^{2}

_{0}:

*A*{

*mainlobe*} ∝

*l*

^{2}

_{0}=

*M*

^{2}(2

*πLW*

^{2}

*β*

^{′})

^{-1}

*S*[

*i*] domain is

*A*{

*domain*}∝

*M*

^{2}. Taking the ratios of the two areas yields

*A*{

*mainlobe*}/

*A*{

*domain*}∝(2

*πLW*

^{2}

*β*

*)*

^{′}^{-1}which indicates that the FWM supression is independent of

*M*, and is inversely proportional to the

*bandwidth*

^{2}×

*length*×

*GVD product*. Actually, as

*M*gets very low, the density of [

*j*,

*k*] grid points sampling the mainlobe is reduced to a level where the mainlobe area ceases being a good linear predictor of the number of included [

*j*,

*k*] points, hence at low

*M*values the FWM suppression starts exhibiting some

*M*dependence, improving relative to its nearly constant level predicted by the analysis above. However, there is little value in practicing OFDM with very low

*M*, hence barring this end effect the FWM may be taken as independent of

*M*, essentially set by the

*bandwidth*

^{2}×

*length*×

*GVD product*.

*W*=

*M*Δ

*ν*, that sets the level of FWM suppression. The longer and more dispersive the fiber, and the wider band the OFDM system, the better its FWM suppression, which makes sense, as the three factors — bandwidth, length, GVD — are measures of increased dispersion, mitigating FWM via the phase mismatch.

*ν*=Δ

*ω*/(2

*π*), as it is seen in (36) that it is solely Δ

*ω*that sets the Δ

*β*mismatch. In light of this, the independence of

*M*is surprising at first sight, as for fixed bandwidth

*W*, increasing

*M*decreases the inter-subcarrier spacing Δ

*ν*=

*W*/

*M*, hence the Δ

*β*mismatch is reduced, which would seem to degrade the FWM suppression. Nevertheless, the argument just made ignores the distribution of mixing products: when

*M*is increased there are quadratically more mixing products overall, yet the percentage of the degraded mixing products out of the total number remains nearly constant, as there are quadratically more mixing products in the mainlobe, as demonstrated above. We note that 12] erroneously assumed Δ

*ν*to be the key predictor of FWM performance, not realizing the inverse impact of

*M*, such that it is the aggregate bandwith,

*W*, that is the main factor setting the level of FWM suppression.

*β*

^{′a′},

*W*,

*N*

*, i.e.*

_{span}*use higher dispersion fiber*(

*exclude dispersion-shifted fibers, use standard G.652 fiber to provide ample GVD), design the system to occupy as large overall bandwidth as possible, and remove all or nearly all dispersion compensation, while taking as large as possible a number of spans*

*N*

*. As for selection of*

_{span}*M*, this is determined not from FWM considerations, but rather from its interplay with the overall bandwidth via the cyclic prefix overhead inthe presence of dispersive temporal spread, as exemplified in section 8.

*N*

*, the FWM and ASE induced impairments start taking their toll. Nevertheless, the higher the FWM suppression, the higher the number of spans supported for a specified target BER.*

_{span}## 6. FWM build-up over multiple dispersive spans — OFDM performance analysis

*dispersive nonlinear*interaction among tens or hundreds of thousands of nonlinear inter-modulation products generated by triplets of subcarriers in each of the spans of the OFDM communication link, would be a daunting task, quite intractable analytically, to be relegated to numerical evaluation. Nevertheless, building on the methods and results of the previous sections, we develop a simple yet accurate design formula for the Q-factor applicable to the Bit Error Rate (BER) performance of an

*M*-ary PSK optical OFDM link. Our results generalize the dispersion-free treatment [2

*multi-span Effective FWM Suppression*(EFWMS) parameter, representing the FWM generation efficiency rms-averaged over all possible intermodulation products (while accounting for NDG/DG effects).

*D*̂

^{FWM}*=*

_{ijk}*F*

_{ijk}*L*̂

^{FWM}*, it is apparent that the*

_{ijk}*multi-span FWM suppression*factors (95) differ from their

*single-span*counterparts Eqs. (54),(56), (57) by applying multiplications through the array factors

*F*

*before rms-averaging.*

_{ijk}*N*

*[*

_{beas}*i*,

*M*](

*D*̂

^{FWM}*[*

_{eff}*i*,

*M*,

*N*

*])*

_{span}^{2}, hence (94) is compactly reformulated as

*N*

*spans (dispersion compensation is only applied at the end of the link). These results for the FWM-induced phase noise variance in the multi-span dispersive case are seen similar to those for the single-span case Eq. (60), the differences being that there now appears a factor*

_{span}*N*

^{2}

*, and the singles-pan parameter*

_{span}*L*̂

^{FWM}*[*

_{eff}*i*,

*M*] in Eq. (60) is now replaced by

*D*̂

^{FWM}*[*

_{eff}*i*,

*M*,

*N*

*] in Eq. (96), accounting for the interplay of FWM, dispersion and the phased-array effect over the multiple spans.*

_{span}*D*̂

^{FWM}*on*

_{eff}*N*

*is due to the array factor*

_{span}*F*

*[*

_{ijk}*N*

*].*

_{spans}*To evaluate the BER degradation due to the FWM (temporarily ignoring all other noise sources) we work out the e variance*

**Angular variance:***σ*

^{2}

_{∠}of the phase noise induced by FWM in the angular decision variable

*φ*

*≡∠*

_{i}*r*̰

*. Here*

_{i}*r*̰

*is a circular gaussian RV with equal variance of its real and imaginary parts. We assume that the FWM-induced phase noise is small relative to the angular distance of the noiseless angle to the decision boundary,*

_{i}*π*/

*m*, for

*m*-ary PSK. In this case, the phase noise,

*φ*

*is essentially determined by the variance of the fluctuations in the imaginary part*

_{i}*r*

^{im}*of*

_{i}*r*̰

*(equal to half the variance of*

_{i}*r*̰

*), normalized by the signal power (i.e. the inverse of the signal to intermodulation ratio):*

_{i}*p*

_{0}, which in turn equals a fraction 1/

*M*of the total power

*P*

*transmitted over all*

_{T}*M*sub-channels:

*N*

*(55):*

_{beats}*N*

*[*

_{beats}*i*,

*M*] Eq. (55) has a quadratic dependence on

*M*, then, for large

*M*, its normalized version is weakly dependent on

*M*, as seen in Eq. (102). In particular, for the carrier at the mid-band frequency,

*i*=

*M*/2 (assuming even

*M*), we obtain the numerical value 0.734:

*N*̂

*≈0.734 for other values of*

_{beats}*M*(≠128) as well, since

*N*̂

*is weakly dependent on*

_{beats}*M*. Considering now the dispersion-free special case, we use this approximation for

*N*̂

*, and also set*

_{beats}*D*̂

^{FWM}*[*

_{eff}*i*,

*M*,

*N*

*]=1 in Eq. (101), yielding:*

_{span}*In the absence of dispersion, the FWM-induced phase noise power is proportional to the total power of all OFDM sub-channels, nearly independent of the number of sub-channels*. In the absence of dispersion the EFWMS reduces to unity (0 dB) to high accuracy, and our formulation reduces to that of Lowery’s.

*dispersive FWM*effects, compactly described in terms of the key multi-span EFWMS parameter,

*D*̂

^{FWM}*[*

_{eff}*i*,

*M*,

*N*

*] Eq. (95), representing the FWM attenuation rms-averaged over all intermods. Unlike the dispersion-free result Eq. (104), the FWM power Eq. (101) in the presence of dispersion may exhibit non-negligible dependence on*

_{span}*M*(via

*D*̂

^{FWM}*[*

_{eff}*i*,

*M*,

*N*

*], which depends on the array factor). Now, using square root of the approximate value*

_{span}*N*̂

*≈0.734 as coefficient in Eq. (101), we approximate that expression for the dispersive case as*

_{beats}**As the FWM-induced phase noise distribution is approximately gaussian (to the extent the approximation**

*Q-factor and BER:**φ*≈

*r*

*/*

^{im}*A*holds) we may compactly describe the

*Bit Error Rate*(BER), induced by FWM (assuming it to be the only impairment for now),

*Q*[

*u*]=(2

*π*)

^{-1/2}∫∞

*exp[-*

_{u}*x*

^{2}/2]

*dx*with a Q-factor (argument)

23. C. Hiew, F. M. Abbou, H. T. Chuah, S. P. Majumder, and A. A. R. Hairul, “BER estimation of optical WDM RZ-DPSK systems
through the differential phase Q,” IEEE Photon. Technol.
Lett. **16**, 2619–2621 (2004). [CrossRef]

*κ*

*was applied to the Q-factor, to better fit for the tails of the actual distribution, yielding improved accuracy of the phase noise model. The modified Q-factor applicable to our case is then corrected by a fit-factor*

_{m}*κ*

*(e.g.*

_{m}*κ*

_{4}=1.11):

*D*̂

^{FWM}*≈1), assuming*

_{eff}*M*≫1 holds, the Q-factor is approximately independent of the number of sub-channels. For QPSK and fiber parameters

*m*=4,

*κ*

*=1.11,*

_{m}*N*̂

*≈,*

_{beats}*γ*=1.3/

*W*/

*Km*,

*α*

_{0}=0.22

*dB*/

*Km*

*L*

*=19.38*

_{eff}*Km*

*q*

*≡20log*

^{dB}_{10}

*q*=log

_{10}

*q*

^{2}, we have a linear relation:

*decreases 6 dB per octave of the spans number, and 2 dB per dB of optical power increase*.

*D*̂

^{FWM}*[*

_{eff}*i*,

*M*,

*N*

*]|<1. Hence the*

_{span}*Q*

^{2}-factor (in dBE units) is increased above its dispersion-free value by the positive factor -20log

_{10}|

*D*̂

^{FWM}*[*

_{eff}*i*,

*M*,

*N*

*].*

_{span}*N*

*via the key EFWMS factor Eq. (95), tending to offset the nominal degradation of the Q-factor with*

_{span}*N*

*.*

_{span}*UPA revisited:*We finally briefly consider the validity of the

*undepleted pump approximation*, assumed throughout the paper. In a refined analysis, the 3

^{rd}order generated field must be included in the RSH Eq. (16) alongside the 1

^{st}order field (the solution of Eq. (15)). The summand in the triple summation in the RSH of Eq. (16) must then be replaced by

*ŭ*

^{(1)}

_{j}*ŭ*

^{(1)}

_{k}*ŭ*

^{(1)}*

_{j}_{+}

*-*

_{k}*, mixing the*

_{i}^{(3)}field with a pair of

^{(1)}fields (with the

^{(3)}field evaluated using the UPA). It would be of interest to evaluate the power fluctuations contributed by these “next-order perturbation” terms to the

*i*-th sub-channel, comparing them with the power of the FWM fluctuations as evaluated under the UPA in Eq. (96). Such analysis, to be reported in a future publication, establishes that the next-order perturbation correction to the UPA is negligible: the UPA error is merely of the order of one thousandth of the UPA-evaluated FWM power Eq. (96). Therefore, our closed-form UPA-based approach and resulting OFDM design formulas are highly accurate.

## 7. Adding-in linear ASE noise

*Q*

^{2}-factors), compounding to a total

*Q*

^{2}-factor expressed as the harmonic mean of the individual

*Q*

^{2}-factors:

*q*

_{∠}

*the Q-factor of*

_{LN}*the linear ASE-induced phase noise*to be derived in this section.

*q*

*. Our treatment in this section essentially follows the principles outlined in [9*

_{LN}**17**, 801 (1992). [CrossRef] [PubMed]

*t*≤

*T*, the received signal in the

*i*-th sub-channel prior to demodulation is:

*n*̰

*|*

_{s}^{2}〉≡

*N*

_{0}/

*T*, where

*N*

_{0}is the

*Power Spectral Density*(PSD) of

*N*̰

*(*

_{s}*t*), expressed in terms of the OA parameters as

*N*

_{0}=(

*F*

*/2)(*

_{N}*G*

*-1)*

_{OA}*hν*

_{0}, hence

*N*

_{0}/2

*T*is the PSD of each of the real and imaginary parts of

*n*̰

*, with*

_{s}*F*

*the OA noise figure (on a linear scale, i.e. 10log*

_{N}*F*

*=*

_{N}*F*

^{dB}*).*

_{N}*A*|

^{2}=

*p*

_{0}=

*P*

*/*

_{T}*M*, as well as the relation

*MT*

^{-1}=

*W*between the symbol rate

*T*

^{-1}of each of the

*M*sub-channels and the total bandwidth,

*W*.

*N*

_{0}represents the one-sided PSD of each of the I and Q quadrature components of the OA ASE in each polarization,

*W*

*≡12.5*

_{ref}*GHz*is the 0.1nm bandwidth, and the factor of two is due to the requirement to include the two noise polarizations in the standard definition. Comparing Eq. (117) with the last equation, it is apparent that

*P*

*/*

_{T}*W*[Watt/Hz] of the OFDM signal, and inversely proportional to the number of OAs,

*N*

*+1.*

_{span}*BER*

_{∠}

*≅2*

_{T}*Q*[

*q*

_{∠}

*], where, consistent with (112), with the two variances given by Eqs. (105) and (119), the total Q-factor is:*

_{T}## 8. Application — designing a 40 Gb/s per *λ* OFDM
system with the phased-array effect

*Cyclic Prefix*(CP) [7

7. W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and
design,” Opt. Express **16**, 841–859 (2008),
http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-2-841. [CrossRef] [PubMed]

*β*

^{′}=-21.7psec

^{2}/Km. Additional assumed parameters are fiber differential loss of

*α*

_{0}=0.22

*dB*/

*Km*, nonlinear coefficient

*γ*=1.3/

*W*/

*Km*, comprising

*N*

*spans, each of length*

_{span}*L*

*=80Km terminated and initiated in OAs, all with gain*

_{span}*F*

*=6.5*

_{N}*dB*.

*Cyclic Prefix*(CP) extension of the transmitted OFDM block [7

7. W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and
design,” Opt. Express **16**, 841–859 (2008),
http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-2-841. [CrossRef] [PubMed]

*QPSK*modulate each OFDM sub-channel at 21.7

*Msym*/

*sec*, in order to attain the target 40 Gb/s aggregate bit-rate:

*sym*/

*Hz*. The 21.7

*Msym*/

*s*data rate per subcarrier would then translate into a subcarrier separation of Δ

*v*=21.7

*MHz*, and since there are

*M*=512 channels, the aggregate bandwidth would be

*W*=

*M*Δ

*v*=11.11

*GHz*. Such CP-free design applies to the reference system of Fig. 4(a) (the dotted lines), wherein dispersion is optically corrected at the end of every span. As this reference system requires no electronic compensation and no cyclic prefix, its spectral-efficiency is very high:

*D*̂

^{FWM}*[*

_{eff}*i*,

*M*,

*N*

*] in (108) is replaced here by*

_{span}*L*̂

^{FWM}*[*

_{eff}*i*,

*M*]. Indeed, recall that

*L*̂

^{FWM}*[*

_{eff}*i*,

*M*] describes the (relatively small) FWM degradation over a

*single span*due to dispersive phase mismatch, whereas

*D*̂

^{FWM}*[*

_{eff}*i*,

*M*,

*N*

*] is the*

_{span}*multi-span*expression, comprising the array factor. Both formulas (108) and Eq. (124) feature a

*N*

*term in the denominator. In the current Eq. (124), this is accounted for by the quadratic phases being reset by the dispersion compensation at the end of every span, hence the FWM contributions of the various spans add up in phase, i.e. the total FWM variance grows quadratically in*

_{span}*N*

*, yielding an inverse linear dependence on*

_{span}*N*

*for the Q-factor. Using this formula for the FWM contribution to the Q-factor, as well as Eq. (120) for the ASE noise contribution to the Q-factor, and combining the two Q-factor contributions according to Eq. (112), we obtain an expression for the total Q-factor. Then*

_{span}*N*is successively increased until the Q-factor Eq. (124) becomes 3.29 corresponding to the target BER of 10

_{span}^{-3}, which occurs at 32 spans (notice that the dotted total BER curve line in Fig. 4(a), also plotted for 33 spans, is slightly worse than 10

^{-3}; reducing the number of spans to 32 would improve BER to 10

^{-3}). In fact, for the relatively small frequency separation of 21.7 MHz between adjacent subcarriers, and with the large number

*M*=512 of subcarriers, it turns out that

*L*̂

^{FWM}*[*

_{eff}*i*,

*M*] in Eq. (124) is nearly unity, hence the performance of the reference system in this weakly dispersive case is nearly indistinguishable from that attainable in the dispersion-free case. Subsequently, we consider a second version of the reference system in which the subcarriers are almost three times more spaced out in frequency, ameliorating the FWM impairment by almost 1 dB, enabling to extend the reach by one span, to 33 spans (still for 10

^{-3}BER) — this is actually the reference case described by the solid curves in Fig. 4(a), to be compared with Fig. 4(b) describing the performance of our proposed OFDM system transmitting 40 Gb/s per

*λ*without in-line dispersion compensation. We next describe how the reach of the system under study has been determined to be 87 spans (Fig. 4(b)).

*v*constrained by the CP overhead, as a function of the range

*L*=

*N*

_{span}*L*

*. A basic equation governing the relations between the OFDM parameters in the presence of the CP overhead is*

_{span}*R*

*the aggregate bitrate (40Gbps),*

_{b}*ρ*the fraction of subcarriers used to carry useful data (0.9 — as 51 carriers out of 512 are used as pilots),

*η*[b/sym] per subcarrier

*T*

*the CP guard band time, set in our case equal to the dispersive delay spread, expressed in terms of signal bandwidth and link length 11,17*

_{CP}17. G.P. Agrawal, *Lightwave Technology: Telecommunication Systems*
(Wiley2005). [CrossRef]

*ν*yields an expression (not explicitly reproduced here) of the form Δ

*ν*(

*N*

*;*

_{span}*R*

*,*

_{b}*M*

*ρ*,

*η*,

*L*

*) for the frequency separation as a function of*

_{span}*N*

*, with the other quantities viewed as parameters. The resulting system performance is readily obtained using the suite of Q-factor analytic formulas derived here for the first time in sections 6 and 7: specifically, Eq. (108) for*

_{span}*q*

_{∠}

*, Eq. (121) for*

_{FWM}*q*

_{∠}

*, and Eq. (112) for the composition*

_{LN}*q*

_{∠}

*of Q-factors. In turn, Eq. (108) for*

_{T}*q*

_{∠}

*depends on*

_{FWM}*D*̂

^{FWM}*, which (although not explicitly labeled as such in Eq. (108) is a decreasing function of Δ*

_{eff}*ν*, i.e. ultimately

*D*̂

^{FWM}*=*

^{eff}*D*̂

^{FWM}*[*

_{eff}*i*,

*M*,

*N*

*,Δ*

_{span}*ν*(

*N*

*) is a function of*

_{span}*N*

*both directly, and via the CP-constrained Δ*

_{span}*ϛ*(

*N*

*) dependence. The maximal*

_{span}*N*

*subject to the target BER constraint was then determined by repeatedly evaluating*

_{span}*q*

_{∠}

*for successively larger*

_{T}*N*

*values, until the target Q-factor of 3.27 was met, yielding the extended reach of*

_{span}*N*

*=87 for the OFDM system under study. Having determined*

_{span}*N*

*also sets the inter-subcarrier frequency Δ*

_{span}*ν*(

*N*

*)=Δ*

_{span}*ν*(87)61.33

*MHz*, as well as the aggregate bandwidth

*W*=512Δ

*ν*(87)=31.4

*GHz*, which in turn determines the spectral efficiency

*R*

*/*

_{b}*W*=(40

*Gb*/

*s*)/(31.4

*GHz*)=1.27

*b*/

*s*/

*Hz*of the system under study, seen to be a factor of 2.83 worse than the spectral efficiency of 3.6

*b*/

*s*/

*Hz*of the first version of the reference system, which used an initial frequency separation of 21.7 MHz.

*MHz*, providing almost 1 dB of FWM dispersive mismatch, allowing to increase the reach of the reference system from the initially evaluated 32 spans to 33 spans (Fig. 4(a)).

*P*

*, (nearly) independent of the spectral distribution, as indicated in our Eq. (104) and previously shown in [2*

_{T}*ν*=21.7

*MHz*(both with per-span dispersion compensation) were both seen to essentially attain the same performance (this is variant (i)), whereas upon expanding the bandwidth by a factor of 2.83, (with the subcarrier spacing increased to Δ

*ν*=61.33

*MHz*), the reference system reach hardly changed from 32 to 33 spans (this is variant (ii) in Fig. 4(a)). Hence we conclude that the lack of CP in the reference system is not helpful in increasing the reach — however it would significantly improve the spectral efficiency relative to a CP-based system. Conversely, spectral efficiency may be traded off for range: The OFDM technique may attain ultra-long-haul transmission over 87 spans, free of dispersion compensation, while incurring a spectral efficiency reduction by a factor of 2.83, down to 1.27

*b*/

*s*/

*Hz*.

**17**, 801 (1992). [CrossRef] [PubMed]

**27**, 805–7 (1991). [CrossRef]

## 9. Conclusions

**17**, 801 (1992). [CrossRef] [PubMed]

**27**, 805–7 (1991). [CrossRef]

^{-3}to 10

^{-4}target BERs, relative to conventionally designed systems, with dispersion compensation applied in each span.

*dynamically-reconfigurable*networks, providing full flexibility in system provisioning as the link configuration is changed. Unfortunately, legacy submarine links wherein physical access to the nodes is precluded, may not be readily upgraded by removal of their compensation, as terrestrial networks would.

3. W. Shieh, X. Yi, and Y. Tang, “Transmission experiment of multi-gigabit coherent
optical OFDM systems over 1000 km SSMF fiber,” Electron.
Lett. **43**, 183–185 (2007). [CrossRef]

8. J. Lowery, “Amplified-spontaneous noise limit of optical OFDM
lightwave systems,” Opt. Express **16**, 860–865, (2008),
http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-2-860. [CrossRef] [PubMed]

## References and links

1. | W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division
multiplexing,” Electron. Lett. |

2. | J. Lowery, S. Wang, M. Premaratne, and J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM” Opt. Express15, 12965–12970, (2007), http://www.opticsexpress.org/abstract.cfm?id=125469. [CrossRef] [PubMed] |

3. | W. Shieh, X. Yi, and Y. Tang, “Transmission experiment of multi-gigabit coherent
optical OFDM systems over 1000 km SSMF fiber,” Electron.
Lett. |

4. | W. Shieh, X. Yi, Y. Ma, and Y. Tang, “Theoretical and experimental study on PMD-supported
transmission using polarization diversity in coherent optical OFDM
systems,” Opt. Express |

5. | W. Shieh, R. S. Tucker, W. Chen, X. Yi, and G. Pendock, “Optical performance monitoring in coherent optical
OFDM systems,” Opt. Express |

6. | H. Bao and W. Shieh, “Transmission simulation of coherent optical OFDM
signals in WDM systems,” Opt. Express |

7. | W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and
design,” Opt. Express |

8. | J. Lowery, “Amplified-spontaneous noise limit of optical OFDM
lightwave systems,” Opt. Express |

9. | K. Inoue, “Phase mismatching characteristic of four-wave
mixing in fiber lines with multistage optical amplifiers”
Opt. Lett |

10. | D. G. Schadt, “Effect of amplifier spacing on four wave mixing in
multichannel coherent communications,” Electron. Lett |

11. | T. Schneider, |

12. | M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “The FWM Impairment in Coherent OFDM Compounds on a Phased-Array Basis over Dispersive Multi-Span Links” COTA’08, Boston, July 13–16, (2008). |

13. | Please notice that in the published proceedings of [12] an error fell in the stated system performance. The early simulations did not account for the coherent addition of transposed pairs of non-degenerate intermods, as properly carried out in section 6 of this paper. |

14. | T.-K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple
optical amplifiers and dispersion compensation,” J.
Lightwave Technol. |

15. | M. E. Marhic, N. Kagi, T.-K. Chiang, and L. G. Kazovsky, “Optimizing the location of dispersion compensators
in periodically amplified fiber links in the presence of third-order nonlinear
effects” Photon Technol. Lett. |

16. | M. Eiselt, “Limits on WDM Systems Due to Four-Wave Mixing: A
Statistical Approach,” J. Lightwave Technol. |

17. | G.P. Agrawal, |

18. | T. Yamamoto and M. Nakazawa “Highly efficient four-wave mixing in an optical
fiber with intensity dependent phase matching,” J.
Lightwave Technol. |

19. | S. Song, C. T. Allen, K. R. Demarest, and R. Hui, “Intensity-Dependent Phase-Matching Effects on
Four-Wave Mixing in Optical Fibers,” J. Lightwave Technol. |

20. | R. P. Feynman, R. B. Leighton, and M. Sands, |

21. | W. Zeiler, F. D. Pasquale, P. Bayvl, and J. E. Midwinter, “Modeling of four-wave mixng and gain peaking in
amplified WDM optical communication systems and networks,”
J. Lightwave Technol. |

22. | Y. Atzmon and M. Nazarathy, “A Gaussian Polar Model for Error Rates of Differential Phase Detection Impaired by Linear, Non-Linear and Laser Phase Noises,” J. Lightwave Technol. (acccepted for publication). |

23. | C. Hiew, F. M. Abbou, H. T. Chuah, S. P. Majumder, and A. A. R. Hairul, “BER estimation of optical WDM RZ-DPSK systems
through the differential phase Q,” IEEE Photon. Technol.
Lett. |

**OCIS Codes**

(060.2330) Fiber optics and optical communications : Fiber optics communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: April 25, 2008

Revised Manuscript: June 17, 2008

Manuscript Accepted: July 23, 2008

Published: September 22, 2008

**Citation**

Moshe Nazarathy, Jacob Khurgin, Rakefet Weidenfeld, Yehuda Meiman, Pak Cho, Reinhold Noe, Isaac Shpantzer, and Vadim Karagodsky, "Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links," Opt. Express **16**, 15777-15810 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15777

Sort: Year | Journal | Reset

### References

- W. Shieh and C. Athaudage, "Coherent optical orthogonal frequency division multiplexing," Electron. Lett. 42, 587-589 (2006) [CrossRef]
- J. Lowery, S. Wang, and M. Premaratne, "Calculation of power limit due to fiber nonlinearity in optical OFDM systems," Opt. Express 15, 13282-13287 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-20-13282 [CrossRef] [PubMed]
- J. Lowery, "Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM" Opt. Express 15, 12965-12970, (2007), http://www.opticsexpress.org/abstract.cfm?id=125469. [CrossRef]
- W. Shieh, X. Yi, and Y. Tang, "Transmission experiment of multi-gigabit coherent optical OFDM systems over 1000 km SSMF fiber," Electron. Lett. 43, 183-185 (2007). [CrossRef] [PubMed]
- W. Shieh, X. Yi, Y. Ma, and Y. Tang, "Theoretical and experimental study on PMD-supported transmission using polarization diversity in coherent optical OFDM systems," Opt. Express 15, 9936-9947 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-16-9936. [CrossRef] [PubMed]
- W. Shieh, R. S. Tucker, W. Chen, X. Yi, and G. Pendock, "Optical performance monitoring in coherent optical OFDM systems," Opt. Express 15, 350-356 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-2-350 [CrossRef] [PubMed]
- H. Bao and W. Shieh, "Transmission simulation of coherent optical OFDM signals in WDM systems," Opt. Express 15, 4410-4418 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-8-4410 [CrossRef] [PubMed]
- W. Shieh, H. Bao, and Y. Tang, "Coherent optical OFDM: theory and design," Opt. Express 16, 841-859 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-2-841. [CrossRef] [PubMed]
- J. Lowery, "Amplified-spontaneous noise limit of optical OFDM lightwave systems," Opt. Express 16, 860-865, (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-2-860. [CrossRef] [PubMed]
- K. Inoue, "Phase mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers" Opt. Lett 17, 801 (1992). [CrossRef]
- D. G. Schadt, "Effect of amplifier spacing on four wave mixing in multichannel coherent communications," Electron. Lett 27, 805-7 (1991).
- T. Schneider, Nonlinear Optics in Telecommunications (Springer 2004).
- M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, "The FWM Impairment in Coherent OFDM Compounds on a Phased-Array Basis over Dispersive Multi-Span Links" COTA???08, Boston, July 13-16, (2008).
- Please notice that in the published proceedings of [12] an error fell in the stated system performance. The early simulations did not account for the coherent addition of transposed pairs of non-degenerate intermods, as properly carried out in section 6 of this paper. [CrossRef]
- T.-K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, "Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensation," J. Lightwave Technol. 14249-260 (1996). [CrossRef]
- M. E. Marhic, N. Kagi, T.-K. Chiang, and L. G. Kazovsky, "Optimizing the location of dispersion compensators in periodically amplified fiber links in the presence of third-order nonlinear effects" Photon Technol. Lett. 8, 145-147 (1996). [CrossRef]
- M. Eiselt, "Limits on WDM Systems Due to Four-Wave Mixing: A Statistical Approach," J. Lightwave Technol. 17, 2261-2267 (1999). [CrossRef]
- G.P. Agrawal, Lightwave Technology: Telecommunication Systems (Wiley 2005).
- T. Yamamoto and M. Nakazawa "Highly efficient four-wave mixing in an optical fiber with intensity dependent phase matching," J. Lightwave Technol. 9, 327-329 (1997). [CrossRef]
- S. Song, C. T. Allen, K. R. Demarest, and R. Hui, "Intensity-Dependent Phase-Matching Effects on Four-Wave Mixing in Optical Fibers," J. Lightwave Technol. 17, 2285-2290 (1999).
- R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. I, (Addison-Wesley 1965). [CrossRef]
- W. Zeiler, F. D. Pasquale, P. Bayvl, and J. E. Midwinter, "Modeling of four-wave mixng and gain peaking in amplified WDM optical communication systems and networks," J. Lightwave Technol. 14, 1933-1942 (1996).
- Y. Atzmon and M. Nazarathy, "A Gaussian Polar Model for Error Rates of Differential Phase Detection Impaired by Linear, Non-Linear and Laser Phase Noises," J. Lightwave Technol. (acccepted for publication). [CrossRef]
- C. Hiew, F. M. Abbou, H. T. Chuah, S. P. Majumder, and A. A. R. Hairul, "BER estimation of optical WDM RZ-DPSK systems through the differential phase Q," IEEE Photon. Technol. Lett. 16, 2619-2621 (2004).

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