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

Moshe Nazarathy; Jacob Khurgin; Rakefet Weidenfeld; Yehuda Meiman; Pak Cho; Reinhold Noe; Isaac Shpantzer; Vadim Karagodsky

doi:10.1364/OE.16.015777

1. Introduction

Coherent Optical Orthogonal Frequency Division Multiplexing (CO-OFDM) is emerging as a key high-performance optical transmission technique [1–9] providing major advantages in spectral efficiency and in mitigating the 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.

In contrast, CD is more difficult to mitigate in single channel ultra-high-speed transmission, requiring far more complex equalization techniques, whereas in multichannel OFDM CD is more readily manageable. It is rather the nonlinear Four-Wave-Mixing (FWM) that ultimately sets the limit to OFDM transmission performance.

It is well known that dispersion acts to somewhat mitigate the FWM impairment, making the non-linear interactions less efficient, by phase mismatching the various subcarrier triplets interacting through the fiber third-order nonlinearity. Therefore, for the purpose of analysis and design of OFDM transmission performance, it is imperative to properly model FWM, including dispersive phase mismatch effects in OFDM transmission, which are not wellunderstood.

FWM generation in OFDM transmission is quite difficult to model due to the non-linear nature of this impairment, and the large number of subcarriers. Lowery [2] provided an analytic estimate of FWM build-up in OFDM transmission, assuming a dispersion-free system. The overall FWM performance is set by the cumulative effect of the incoherent addition of thousands of intermodulation products (interchangeably referred to as 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).

The question arises whether or not dispersion is a significant factor in determining the FWM impairment. In a recent treatment of non-linear compensation of OFDM [3], the argument was advanced that the dispersive phase mismatch between closely packed subcarriers is very low, hence dispersion was dismissed as affecting OFDM negligibly. Here we show that the overall effect of CD on FWM, over the full frequency band, may be quite significant for a high-speed OFDM system operating at ultra-high bitrates, occupying an extended bandwidth of tens of GHz. Granted, the dispersive phase mismatch between adjacent carriers is very small, yet among subcarriers well-separated in the band there is substantial phase mismatch, leading to sizable FWM mutual interference effects first analyzed by Innoe [9] and Schadt 14] in the WDM context, ported here to OFDM for the first time. Once this effect is properly accounted for, the CD-free analysis in [2] is seen to have merely provided a loose upper bound on the FWM impairment.

An understanding of the interaction between CD and FWM, including the accrual of nonlinear contributions from the multiple transmission spans, is then critical to advanced OFDM system analysis and design. This paper provides for the first time a comprehensive analytic treatment of the combined effect of dispersion and FWM nonlinearity in OFDM transmission.

The analysis rigorously starts from the 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.

A key contribution of our new model is its proper modeling of the overall FWM contributed by the multiple spans in a dispersive fiber transmission link, based on methods first introduced in [9,10] for the modeling of the FWM impairment in WDM systems, as further reviewed in the textbook [11].

Notice that Lowery [2] has assumed coherent addition for the FWM contributions of multiple spans. Here we establish that such in-phase multi-span addition as assumed in [2], would be strictly correct only in the absence of dispersion. Extending the treatments in [9,10] we introduce in the context of OFDM modeling a new mechanism of FWM cancellation, in the wake of CD, whereby, unlike predicted in [2] for the CD-free case, the FWM contributions of the individual N _span transmission spans in a multi-span OFDM long-haul link do 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] and Schadt [10] for WDM systems, is neither strictly coherent (in-phase addition), nor incoherent (addition with random phases): Remarkably, 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 Δβ 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.

Accounting for the PA effect, under certain system configurations, the effectiveness of dispersion in reducing the amount of FWM is then far greater than previously assumed in the OFDM literature. The key is having the dominant FWM intermodulation products due to the multiple spans, destructively interfere, capitalizing on the PA effect. The non-linear analysis capability is put to work to realize an advanced 40 Gb/s OFDM systems design with range exceeding 6000 km, void of optical dispersion compensation along the link. Initial OFDM system design consequences of the very substantial FWM cancellationwere briefly previewed in [12]. The PA effect was introduced there unaware of the precedence of [9,10] who first discovered the PA effect in the WDM context, without referring to it as such. Here we rederive the PA effect by a different method than used in [9,10], and extend its applicability to non-periodic optically amplified links, then explore the consequences of this effect to QPSK-OFDM transmission. Similar PA-like expressions arise in the analysis of Cross-Phase Modulation (XPM) generated in an ASK-WDM multi-span system [14,15], however it turns out that XPM does not degrade the BER of DPSK-OFDM transmission (provided the fixed XPM-induced constellation rotations are calibrated out, and neglecting temporal dispersive walk-off effects which may result in time-varying XPM as the symbol intervals run out of synchonism). It should be mentioned that our QPSK-OFDM model differs from that introduced in [16] to model ASK-WDM, whereby incoherent addition of FWM mixing products from multiple spans was assumed. In our opinion, the incoherent addition assumption in [16] is not accurate, as it does not account for the PA-effects first pointed out in [9,10].

The paper is structured as follows: Section 2 briefly introduces the notation and OFDM system description. Section 3 adapts the NLSE to multichannel transmission over nonlinear fiber, formulating coupled mode equations. Section 4 evaluates FWM build-up over a single dispersive span, in the field and power domains. Section 5 rederives and extends the key PA effect first introduced in [9,10] proving that FWM compounds over multiple spans as radiation from a phased-array. Section 6 works out compact formulas for evaluating the Q-factors and BER performance for a multi-span link, in the wake of the PA effect. In Section 7 the ASE limit for OFDM systems is worked out, consistent with [9]. Section 8 develops the guidelines for substantial FWM suppression via the PA effect, working the details of a specific 40 Gb/s system example. Section 9 concludes the paper, discussing outstanding limitations and potential future work.

2. System description

Stating our notation, let u(t;z) be the real-valued optical field at time t and position z along the fiber, ṵ(t;z) its Complex-Envelope (CE), and ŭ(t;z) its SpatioTemporal CE (STCE):

u (t; z) = Re {\underset{\sim}{u} (t; z) e^{j ω_{0} t}} = Re {\overset{˘}{u} (t; z) e^{- j (β_{0} z - ω_{0} t)}}

The relation between the (temporal) CE and the STCE is then

{\underset{\sim}{u}}_{i} (t, z) = {\overset{˘}{u}}_{i} (t, z) e^{- j β_{0} z}

The OFDM-DWDM signal is expressed as a superposition of multiple sub-channels:

\overset{˘}{u} (t; z) = \sum_{i = 1}^{M} {\overset{˘}{u}}_{i} (t; z) e^{j Ω_{i} t}

where the i-th channel is at frequency ω _i=Ω_i+ω ₀, i.e. it deviates Ω_i away from a reference freq 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.

Symbols are launched at times t=n Twhere 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_i is assumed to consist of a modulated rectangular pulse (where 1[_a,_b](t)≡1 if t∈[a,b], 1[a,b](t)≡0, otherwise):

{\underset{\sim}{S}}_{i} (t) = {\overset{˘}{u}}_{i} (t, 0) e^{j Ω_{i} t} = A_{c} e^{j Ω_{i} t} e^{j μ_{0}^{(i)}} 1_{[- \frac{T}{2}, \frac{T}{2}]} (t)

This launched signal propagates along the fiber link of length L=N _span L _span, typically consisting of N _span identical spans, each initiated and terminated in an Optical Amplifier (OA) perfectly compensating the power loss $e^{- α L_{span}}$ by providing power gain

G_{OA} = e^{α L_{span}} .

The OFDM receiver is modeled as a band-pass correlator bank, splitting the signal to multiple parallel paths, down-converting each path to baseband, in effect frequency demultiplexing the signal by demodulating each path according to its subcarrier frequency (cancelling the modulation factor exp[jΩt] corresponding to the i-th sub-channel), and applying integrate-and-dump (I&D) filtering onto each of the downconverted signals.

Since receiver filtering gain is irrelevant (equally affecting signal and noise), the I&D is most conveniently modeled by applying a T ^-1 scaling factor, amounting to a time-average:

y (t) = \frac{1}{T} \int_{- \frac{T}{2}}^{\frac{T}{2}} x (t) dt .

The complex-valued output of each I&D filter is sampled at the OFDM symbol rate 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

In this section we derive coupled mode equations for the fields of each of the OFDM sub-channels, starting from the 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

\partial_{z} \overset{˘}{u} - j (\frac{β ″}{2}) \partial_{t}^{2} \overset{˘}{u} + (\frac{α}{2}) \overset{˘}{u} = - j γ {∣ \overset{˘}{u} ∣}^{2} \overset{˘}{u} .

In this equation and hereafter, t denotes retarded time, i.e. the substitution t→t-β ^′ z is assumed, α is the loss coefficient, γ is the non-linear coefficient [11], ∂ _t is the derivative with respect to the variable t, ∂ ² _t the second derivative, β ^′≡∂ _ω β(ω) and β ^″≡∂ ² _ω β(ω).

Following [17, sec. 4.3.1] we substitute the OFDM signal Eq. (3) into the NLSE Eq. (7) and simplify, yielding a set of coupled mode equations, equivalently written in our notation as

\partial_{z} {\overset{˘}{u}}_{i} + β ″ Ω_{i} \partial_{t} {\overset{˘}{u}}_{i} - j \frac{β ″}{2} \partial_{t}^{2} {\overset{˘}{u}}_{i} = - j β_{i}^{T} {\overset{˘}{u}}_{i} - j γ \sum_{[j, k, l] \in S [i]} {\overset{˘}{u}}_{j} {\overset{˘}{u}}_{k} {\overset{˘}{u}}_{l}^{*}

with the triple summation running over the set S[i]of indexes of frequency triplets corresponding to FWM Intermodulation (intermod) products falling on ω _i, namely the set of “proper FWM” intermods, excluding the SPM and XPM “coherent” intermods (for which j=iork=i):

S [i] \equiv {[j, k, - l] : ω_{j} + ω_{k} - ω_{l} = ω_{i}, j \neq i \neq k}, i = 1, 2, \dots, M,

A power-dependent “total” propagation constant β ^T _i, also features in Eq. (8), describing the spatial rates of phase-change due to Dispersion, SPM, XPM, and representing loss as imaginary β :

β_{i}^{T} \equiv \frac{β ″}{2} Ω_{i}^{2} - j \frac{α}{2} + γ ({∣ {\overset{˘}{u}}_{i} ∣}^{2} + 2 \sum_{\underset{k \neq i}{k = 1}}^{M} {∣ {\overset{˘}{u}}_{k} ∣}^{2}) = β_{i}^{DL} - β_{0} + 2 γ P^{T} - γ p_{i}

with P ^T≡∑^M _k=₁|ŭ _k|², p _i≡|ŭ_i|², (all functions of t,z), and

β_{i}^{DL} \equiv β_{0} + (\frac{β ″}{2}) Ω_{i}^{2} - \frac{j α (z)}{2} = β_{i} - \frac{j α (z)}{2}

a complex-valued propagation constant associated with just Dispersion and Loss, with

β_{i} \equiv β_{0} + (\frac{β ″}{2}) Ω_{i}^{2}

Finally,

β_{i}^{T} = β_{i} - β_{0} - j \frac{α}{2} + 2 γ P^{T} - γ p_{i}

SPM and XPM: 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 -jγ|ŭ|²ŭ=-jβ _NLŭ, with 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 ^t} of ŭ(t).

Finally, we clarify why the XPM between the subcarriers (and likewise the SPM of each subcarrier) have been excluded from the summation of Eq. (9), by imposing the criterion 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*_l reduces to ŭ _i|ŭ _k|² and the summations over all such terms effectively represents the intensity-dependent term in the propagation constant β ^T _i in Eq. (10), describing an overall frequency-dependent phaseshift of each received OFDM sub-channel, say the 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)).

Undepleted pump approximation: We now set

{\overset{˘}{u}}_{i} = {\overset{˘}{u}}_{i}^{(1)} + {\overset{˘}{u}}_{i}^{(3)},

where the superscripts indicate a perturbation approach to solving the set of coupled differential equations Eq. (8). The index⁽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.

Substituting Eq. (14) into Eq. (8) yields two sets of coupled equations:

\partial_{z} {\overset{˘}{u}}_{i}^{(1)} + β ″ Ω_{i} \partial_{t} {\overset{˘}{u}}_{i}^{(1)} - j \frac{β ″}{2} \partial_{t}^{2} {\overset{˘}{u}}_{i}^{(1)} + j β_{i}^{T} {\overset{˘}{u}}_{i}^{(1)} = 0

\partial_{z} {\overset{˘}{u}}_{i}^{(3)} + β ″ Ω_{i} \partial_{t} {\overset{˘}{u}}_{i}^{(3)} - j \frac{β ″}{2} \partial_{t}^{2} {\overset{˘}{u}}_{i}^{(3)} + j β_{i}^{T} {\overset{˘}{u}}_{i}^{(3)} = - j γ \sum_{[j, k, l] \in S [i]} {\overset{˘}{u}}_{j}^{(1)} {\overset{˘}{u}}_{k}^{(1)} {\overset{˘}{u}}_{l}^{{(1)}^{*}}

with

β_{i}^{T} (t, z) \equiv \frac{β ″}{2} Ω_{i}^{2} - j \frac{α}{2} + γ ({∣ {\overset{˘}{u}}_{i}^{(1)} ∣}^{2} + 2 \sum_{\underset{k \neq i}{k = 1}}^{M} {∣ {\overset{˘}{u}}_{k}^{(1)} ∣}^{2}) = β_{i}^{DL} - β_{0} + 2 γ P^{T} (t, z) - γ p_{i}^{(1)} (t, z)

To first-order in the perturbation, neglecting the FWM generation, we set the FWM triple sum term on the RHS of Eq. (8) to zero, yielding the homogeneous equation Eq. (15). The linear solution ŭ ⁽¹⁾ _i of Eq. (15) is used to formulate a non-linear equation Eq. (16) for the perturbation ŭ ⁽³⁾ _i≡ŭ _i-ŭ ⁽¹⁾ _i. Formally, Eq. (16) is obtained by setting ŭ _i=ŭ ⁽¹⁾ _i+ŭ ⁽³⁾ _i into (8), and applying the Undepleted Pumps Approximation (UPA) [11]. Since ŭ ⁽¹⁾ _i satisfies the homogeneous equation Eq. (15), the LSH operator of Eq. (8) acting on ŭ ⁽¹⁾ _i+ŭ ⁽³⁾ _i nulls out the ŭ ⁽¹⁾ _i component, leaving just ŭ ⁽³⁾ _i in the LSH of Eq. (16). Adopting the UPA, we set ŭ ⁽¹⁾ _i+ŭ ⁽³⁾ _i≈ŭ ⁽¹⁾ _i, into RSH of Eq. (16), as well as into the β ^T _i 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.

To summarize the UPA-based solution procedure, in the first stage ŭ _i=ŭ ⁽¹⁾ _i,ŭ ⁽³⁾ _i =0. Then Eq. (15) is solved. Its solution ŭ ⁽¹⁾ _i is inserted into Eq. (16) to calculate the FWM-induced perturbation ŭ ⁽³⁾ _i. At the end ŭ _i=ŭ ⁽¹⁾ _i+ŭ ⁽³⁾ _i is formed as the full solution.

Quasi-CW solution: We next consider the 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:

\partial_{z} {\overset{˘}{u}}_{i}^{(1)} = - j β_{i}^{T} {\overset{˘}{u}}_{i}^{(1)}

\partial_{z} {\overset{˘}{u}}_{i}^{(3)} = - j β_{i}^{T} {\overset{˘}{u}}_{i}^{(3)} - j γ \sum_{[j, k, l] \in S [i]} {\overset{˘}{u}}_{j}^{(1)} {\overset{˘}{u}}_{k}^{(1)} {\overset{˘}{u}}_{l}^{{(1)}^{*}}

readily yielding the first order solution of ŭ _i(t;z) over a single span:

{\overset{˘}{u}}_{i}^{(1)} (t, z) = {\overset{˘}{u}}_{i}^{(1)} (t, 0) e^{- j \int_{0}^{z} β_{i}^{T} (t, z') dz'} = A_{c} e^{j μ_{0}^{(i)}} 1_{[- \frac{T}{2}, \frac{T}{2}]} (t) e^{- (\frac{α_{0}}{2}) z} e^{- j (\frac{β ″}{2}) Ω_{i}^{2} z} e^{- j γ (2 M - 1) P_{0} Z_{eff}}

where (assuming α(z)=α ₀=const.)

Z_{eff} = \int_{0}^{z} e^{- α_{0} z'} dz' = \frac{(1 - e^{- α_{0} z})}{α_{0}}

Agrawal [17] normalized the SCTE, ŭ _i(t;z), introducing (in our notation) a modified SCTE ${\overset{˘}{ν}}_{i} (t; z) \equiv {\overset{˘}{u}}_{i} (t; z) e^{j β_{i}^{T} z}$ . For our purposes we must treat a z-dependent β ^T _i(z). We then generalize Agrawal’s normalization to the following formulation:

{\overset{˘}{ν}}_{i} (t; z) = {\overset{˘}{u}}_{i} (t; z) e^{j \int_{0}^{z} β_{i}^{T} (t, z') dz′}

Substituting the inverse transformation

{\overset{˘}{u}}_{i} (t; z) = {\overset{˘}{ν}}_{i} (t; z) e^{- j \int_{0}^{z} β_{i}^{T} (t, z') dz′}

into the i-th coupled NLSEs Eq. (18) and Eq. (19), yields after some simplification

\partial_{z} {\overset{˘}{ν}}_{i}^{(1)} = 0

\partial_{z} {\overset{˘}{ν}}_{i}^{(3)} = - j γ \sum_{[j, k, l] \in S [i]} {\overset{˘}{ν}}_{j}^{(1)} {\overset{˘}{ν}}_{k}^{(1)} {\overset{˘}{ν}}_{l}^{{(1)}^{*}} e^{- j Δ Φ^{T} (t, z)}

where

Δ Φ^{T} (t, z) = \int_{0}^{z} Δ β_{ijkl}^{T} (t, z') dz' = \int_{0}^{z} Δ β_{ijkl}^{DL} (t, z') dz' - γ \int_{0}^{z} Δ p_{ijkl} (t, z') dz′

Δ β_{ijkl}^{DL} \equiv β_{j}^{DL} + β_{k}^{DL} - {(β_{l}^{DL})}^{*} - β_{i}^{DL}

is a complex-valued phase mismatch associated with dispersion and loss, and

Δ β_{ijkl} \equiv \frac{β ″}{2} [(Ω_{j}^{2} + Ω_{k}^{2} - Ω_{l}^{2}) - Ω_{i}^{2}]

is the phase mismatch associated with dispersion alone, and

Δ p_{ijkl} (t, z) \equiv p_{j} (t, z) + p_{k} (t, z) - p_{l} (t, z) - p_{i} (t, z) .

is a power imbalance term, with p _i(t,z)≡|ŭ_i(t,z)|².

Notice that Eq. (27) was formulated in terms of a 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} - α_{0} L_{span} Σ_{s = 1}^{N_{span} - 1} δ (z - s L_{span})

Over any single span 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” v̆⁽¹⁾ _i are constant over each fiber span, expressible in terms of the initial power 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:

\forall z : {\overset{˘}{ν}}_{i}^{(1)} (t, z) = {\overset{˘}{u}}_{i}^{(1)} (t, 0) = \sqrt{p_{0} (t)} e^{j ϕ_{i} (t)},

or

\forall z : p_{i} (t, z) \equiv {∣ {\overset{˘}{ν}}_{i}^{(1)} (t, z) ∣}^{2} = p_{0} (t)

Hence, using Eq. (23) yields v̆⁽¹⁾ _i≡v̆⁽¹⁾ _i(t;0)=ŭ ⁽¹⁾ _i(t;0) ∀z.

Frequency grid: Henceforth assume that the set of OFDM-DWDM frequencies resides on a frequency grid with spacing Δω, such that the i-th output frequency is ω _i=ω ₀+iΔω, i.e. Ωi=ω _i-ω ₀=iΔω, and similarly for ω _j ω _k ω _l, now represented by their indexes j,k,l.

The condition ω _j+ω _k+-ω _l=ω _i then reduces to 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.

Each DWDM channel contains an OFDM signal with multiple sub-channels, all assumed on a common grid. While such synchronization of multiple OFDM grids for the various wavelengths is not necessary in practice, it simplifies the analysis. The common grid seems to provide the worst overlap of the intermods with the sub-carriers.

The triple sum over frequencies now reduces to a double sum over j, k :

\partial_{z} {\overset{˘}{ν}}_{i}^{(3)} = - j γ \sum_{[j, k] \in S [i]} {\overset{˘}{ν}}_{j}^{(1)} {\overset{˘}{ν}}_{k}^{(1)} {\overset{˘}{ν}}_{j + k - i}^{{(1)}^{*}} e^{- j Δ Φ^{T} (t, z)}

Further assuming a contiguous set of sub-channels, labeled 1,2,…,M, the intermods (mixing products) summation domain may be compactly re-expressed as

S [i] = {[j, k] ∣ 1 \leq j, k \leq M & 0 < j + k - i \leq M & j \neq i \neq k},

where the condition 0<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.

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

Δ β_{ijk} \equiv β_{j} + β_{k} - β_{j + k - i} - β_{i} = \frac{{β ″ (Δ ω)}^{2}}{2} [(j^{2} + k^{2} - {(j + k - i)}^{2}) - i^{2}]

which simplifies (using j ^′≡j-i,k ^′≡k-i) to:

Δ β_{ijk} = - β ″ {(Δ ω)}^{2} (j - i) (k - i) = - β ″ {(Δ ω)}^{2} j' k'

The various propagation constants are redefined to depend on three rather than four indexes:

Δ Φ^{T} (t, z) = \int_{0}^{z} Δ β_{ijk}^{T} (t, z') dz' = \int_{0}^{z} Δ β_{ijk}^{DL} (t, z') dz' - γ \int_{0}^{z} Δ p_{ijk} (t, z') dz′

Δ β_{ijk}^{T} \equiv β_{j}^{T} + β_{k}^{T} - β_{j + k - i}^{T} - β_{i}^{T} = Δ β_{ijk}^{DL} - γ Δ p_{ijk}, with

Δ p_{ijk} (t, z) \equiv p_{j} (t, z) + p_{k} (t, z) - p_{j + k - i} (t, z) - p_{i} (t, z) .

The power-dependent term Δp _ijk rigorously emerging here in our derivation of total effective phase mismatch coefficient Δβ _ijk, was previously derived in 18,19] extending the analysis of FWM generation for WDM signals to account for phase-shifts due to self-phase-modulation of the “pump” signals, affecting Δβ mismatch coefficient. Such correction terms were not included in [9,10], however a more general extension of [9,10], as carried out here, accounts for this term. Fortunately, as shown next, in the OFDM context the correction terms Eq. (39) null out when the OFDM channels are transmitted with equal power.

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 Δp _ijk=0, provided that all (sub)channels are launched with equal power, (as applicable to 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_{j}^{(1)} (t, z) = p_{k}^{(1)} (t, z) = p_{j + k - i}^{(1)} (t, z) = p_{i}^{(1)} (t, z) \equiv p^{(1)} (t, z) = p_{0} (t) \int_{0}^{z} e^{- α (z') dz'} .

In this case the SPM/XPM-induced power-dependent modification Δp _ijk Eq. (29) to the FWM mismatch, Δβ ^T _ijk, nulls out, yielding Δβ ^T _ijk=Δβ ^DL _ijk. It is only upon observing an out-of-band intermod, not falling onto any of the OFDM sub-channels, that we would have Δp _ijk≠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.

Propagation equations: Setting Δβ ^T _ijk=Δβ ^DL _tjk in (37) yields final coupled-mode equations for FWM build-up at the carrier frequencies of the OFDM sub-channels over a single span:

\partial_{z} {\overset{˘}{ν}}_{i}^{(3)} = - j γ \sum_{[j, k] \in S [i]} {\overset{˘}{ν}}_{j}^{(1)} {\overset{˘}{ν}}_{k}^{(1)} ν_{j + k - i}^{{(1)}^{*}} e^{- j Δ Φ^{DL} (t, z)}

where (using (38)):

Δ Φ^{DL} (t, z) \equiv \int_{0}^{z} Δ β_{ijk}^{DL} (t, z') dz' = \int_{0}^{z} Δ β_{ijk} (t, z') dz' - j \int_{0}^{z} α (z') dz′

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

Over a single uniform fiber span, span L=L we have Δβ ^DL _ijk(t,z)=Δβ _ijk-jα=const..

Field propagation: The integral Eq. (42) then yields ΔΦ^DL(t,z)=Δβ ^DL _ijk z, hence Eq. (41) reduces to

\partial_{z} {\overset{˘}{ν}}_{i}^{(3)} = - j γ \sum_{[j, k] \in S [i]} {\overset{˘}{ν}}_{j}^{(1)} {\overset{˘}{ν}}_{k}^{(1)} {\overset{˘}{ν}}_{j + k - i}^{{(1)}^{*}} e^{- j Δ β_{ijk}^{DL} z} single-span equation

As seen in Eq. (31) the “pumps”, v̆⁽¹⁾ _n, for any index, 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 v̆⁽³⁾ _i=0, yielding the STCE of the FWM fluctuation affecting the i-th OFDM channel (with its time-dependence not explicitly indicated):

{\overset{˘}{ν}}_{i}^{(3)} (L) = - j γ \sum_{[j, k] \in S [i]} {\overset{˘}{ν}}_{j}^{(1)} {\overset{˘}{ν}}_{k}^{(1)} {\overset{˘}{ν}}_{j + k - i}^{{(1)}^{*}} L_{ijk}^{FWM} single-span solution

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

L_{ijk}^{FWM} \equiv \int_{0}^{L} e^{- j Δ β_{ijk}^{DL} z} dz = \frac{e^{- j Δ β_{ijk}^{DL} L} - 1}{- j Δ β_{ijk}^{DL}} = \frac{1 - e^{- α L} e^{- j Δ β_{ijk} L}}{j Δ β_{ijk}^{DL}} = \frac{1 - e^{- α L} e^{- j Δ β_{ijk} L}}{j Δ β_{ijk} + α}

In particular, in the dispersion-free case, Δβ _ijk=0, one must still account for loss, hence the DNEL Eq. (45) reduces to the well-known Effective Length parameter, L _eff, appearing in the description of SPM/XPM generation [11,17]:

L_{ijk}^{FWM} ∣_{Δ β_{ijk} = 0} = \frac{(1 - e^{- α L})}{α} \equiv L_{eff} .

Finally, for appreciable loss over the distance L, e-αL≪1, the DNEL simplifies to L ^FWM _ijk≈(jΔβ _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):

{∣ {\overset{˘}{u}}_{i}^{(3)} (L) ∣}^{2} = {∣ {\overset{˘}{ν}}_{i}^{(3)} (L) ∣}^{2} e^{- α L} = γ^{2} e^{- α L} p_{0}^{3} {∣ \sum_{[j, k] \in S [i]} ∣ L_{ijk}^{FWM} ∣ e^{j (ϕ_{j} + ϕ_{k} - ϕ_{j + k - i} + ∠ L_{ijk}^{FWM} (L))} ∣}^{2}

We now partition 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):

S^{DG} [i] \equiv {[j, k] \in S [i] | j = k} = {[j, j] | j \leq M & 0 < 2 j - i \leq M & j \neq i}

and a non-degenerate (NDG) subset (j≠k), in turn expressed as the union of two subsets:

S^{NDG} [i] \equiv {[j, k] \in S [i] ∣ j \neq k} = S_{>}^{NDG} [i] \cup S_{<}^{NDG} [i]

Notice that each element of the one-sided set, 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:

Σ [i] \equiv \sum_{[j, k] \in S [i]} ∣ L_{ijk}^{FWM} ∣ e^{j (ϕ_{j} + ϕ_{k} - ϕ_{j + k - i} + ∠ L_{ijk}^{FWM} (L))} = Σ^{DG} [i] + Σ^{NDG} [i]

where in the last equality we notice that the transposed pairs [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 $e^{j (ϕ_{j} + ϕ_{k} - ϕ_{j + k - i})}$ with j>k, and $e^{j (ϕ_{j'} + ϕ_{k'} - ϕ_{j' + k' - i'})}$ with 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:

〈 e^{j (ϕ_{j} + ϕ_{k} - ϕ_{j + k - i} + ∠ L_{ijk}^{FWM})} e^{- j (ϕ_{j'} + ϕ_{k'} - ϕ_{j' + k' - i} + ∠_{ij′ k'}^{FWM})} 〉 = δ_{jj'} δ_{kk'}

Indeed, if j=j ^′ and k=k ^′ we get unity, else either j≠j ^′ or k≠k ^′ (or both), and we also have j≠k and j ^′≠k ^′ (as these are NDG intermods) hence in the last expectation in Eq. (51), at least one of the phase terms (say ϕ _j) is independent of the other, hence the expectation breaks into a product of expectations, say $〈 e^{j ϕ_{j}} 〉〈 e^{- j ϕ_{j'}} e^{j (ϕ_{k} - ϕ_{k'})} e^{- j (ϕ_{j + k - i} - ϕ_{j' + k' - i})} 〉$ , and as $〈 e^{j ϕ_{j}} 〉 = 0$ , then the terms in the NDG summation in Eq. (50) are uncorrelated, adding up on a power basis (since the cross-terms cancel upon expanding the square and taking the expectation). Similarly the terms in the DG summation in Eq. (50) are uncorrelated, also adding up on a power basis. The total power of the overall summation of fields Eq. (47) is then the sum of the individual powers, itemized according to NDG or DG (with an amplitude factor of 2 squared, i.e. 4, in the one-sided NDG, or equivalently a factor of 2 in the two-sided NDG):

σ_{FWM}^{2} \equiv σ_{{\overset{˘}{u}}_{i} (L)}^{2} = 〈 {∣ {\overset{˘}{u}}_{i}^{(3)} (L) ∣}^{2} 〉 = γ^{2} e^{- α L} p_{0}^{3} {4 \sum_{[j, k] \in S_{>}^{NDG} [i]} {∣ L_{ijk}^{FWM} ∣}^{2} + \sum_{[j, j] \in S^{DG} [i]} {∣ L_{ijj}^{FWM} ∣}^{2}}

In the last form we found it convenient to reintroduce the summation S[i]over all intermods, and the NEL was normalized (as denoted by a hat) by dividing through the effective length,

{\hat{L}}_{ijk}^{FWM} \equiv \frac{L_{ijk}^{FWM}}{L_{eff}} = \frac{\frac{1 - e^{- α L} e^{- j Δ β_{ijk} L}}{j Δ β_{ijk} + α}}{\frac{1 - e^{- α L}}{α}} = \frac{\frac{(1 - e^{- α L} e^{- j Δ β_{ijk} L})}{1 - e^{- α L}}}{\frac{j Δ β_{ijk}}{α + 1}}

recalling that Δβ _ijk is explicitly given by Eq. (36). The modulus of the normalized NEL, |L̂^FWM _ijk|, is referred to as single-span FWM Attenuation for the particular ijk beat. Now root-mean-square (rms) average Eq. (53) over all intermods,

{\hat{L}}_{rms}^{FWM} [i, M] \equiv {〈 L_{ijk}^{FWM} 〉}_{rms} = \sqrt{N_{beats}^{- 1} [i, M] \sum_{[j, k] \in S [i]} {∣ {\hat{L}}_{ijk}^{FWM} ∣}^{2}}

where 〈·〉_rms denotes rms averaging (here over all pairs of the set S[i]), and

N_{beats} [i, M] \equiv ∣ S [i] ∣ = \frac{(M^{2} - 5 M + 2)}{2 + (M + 1) i - i^{2}}

is the number of FWM intermods, evaluated by counting the [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.

We generally have L̂^FWM _ijk≤1 for each of the intermods, hence L̂^FWM _rms[i,M]≤1, with equality attained in the dispersion-free case. Similarly, rms-average over the degenerate intermods, yielding

{\hat{L}}_{rms}^{DG - FWM} [i, M] \equiv {〈 L_{ijj}^{FWM} 〉}_{rms} = \sqrt{\frac{1}{N_{beats}^{DG} [i, M]} \sum_{[j, j] \in S^{DG} [i]} {∣ {\hat{L}}_{ijj}^{FWM} ∣}^{2}}

The degenerate contribution Eq. (56) turns out introduce a small negative correction to the overall single-span Effective FWM Supression (EFWMS), defined as:

{\hat{L}}_{eff}^{FWM} [i, M] \equiv \sqrt{\frac{1}{N_{beats} [i, M]} {\sum_{[j, k] \in S [i]} {∣ {\hat{L}}_{ijk}^{FWM} ∣}^{2} - \frac{1}{2} \sum_{[j, j] \in S^{DG} [i]} {∣ {\hat{L}}_{ijj}^{FWM} ∣}^{2}}}

We notice that the second subtractive term, associated with the degenerate intermods, is very small, since for 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].

Fig. 1. (a): Graphical (j,k)-plane representation of the “FWM set” S[i] of 12033 intermods (FWM mixing products) falling onto sub-channel i=64, for an OFDM system with M=128 sub-channels on a regular frequency grid. (b),(c): Array Factor Dinc function parameterized by N _span=83 (section 5). (b) is a zoomed-in version of (c). (a): The points corresponding to the intermods in S[64], reside on a dense 2D grid (the individual “beat” points may be resolved in the e-version by sufficiently zooming in), forming an elongated tilted hexagon with two right angles, in turn partitioned into two subsets: the star-shaped “mainlobe” region consisting of the black points, and its complement, the “sidelobes” region, containing the orange points (appearing gray in colorless print). The excluded SPM/XPM intermods are the missing points in the white crosshairs at j=64 or k=64. A point at [j,k] then represents the triplet of sub-channels indexed [j,k,j+k-64], generating a FWM beat at sub-channel 64. The total FWM power at sub-channel 64 is then the sum of 12033 contributions from each of the intermods. The relative strength of the FWM contribution of each beat (the amount of FWM attenuation relative to the dispersionless case) is indirectly represented by the contour map blue overlay. The rest of this caption describing (b),(c) and the blue overlay, refers to sections 5,8. It is shown there that the points of the mainlobe/sidelobe regions (j,k)-plane in the respectively map into the first mainlobe (|u|<1)/sidelobes (|u|>1) of the dinc function representing the phased-array FWM cancellation factor. Accordingly, the orange (gray) intermods in the “sidelobe” set experience high FWM attenuation, when accounting for the phased-array effect, making very weak contributions to the overall buildup of FWM at sub-channel i=64 (for an 83-spans system, the peaks of the dinc sidelobes range from -13.3 dB to -38.4 dB). Most of the black points in the star-shaped “mainlobe” make strong FWM contributions, however there are just 320 such points (or 3.2% of the 12033 intermods), hence, fortunately, the overall (rms-average) FWM suppression is dominated by the very weak contributions (high FWM attenuation) of the vastly larger sidelobes region occupying 96.8%. The overall FWM suppression attained with this configuration turns out to be 18.4 dB. The blue contour lines are hyperbolas, with their labels, u, representing the normalized hyperbolic distances between each point and the [64, 64] center point (normalized by the critical HD, d _h ^crit for 200 MHz sub-channel separation and standard G.652 fiber). Six such contours are actually plotted with labels u=1,5,35,82,84,125, however one should visualize the infinite family of contour lines passing through every point. Consider the contour line passing through a given beat point. As indicated by the elbowed arrows connecting between the graphs, the contour line labels u represent the argument values whereat the dinc function (b) or (c) is sampled in order to obtain the corresponding array factor (FWM attenuation) for the given beat. E.g., any beat point on the displayed integer-labeled contour lines generates no FWM (as the dinc is sampled at its zero crossings). The thicker contours lines labeled 1, 82, 84 have special significance. The boundary between the main mainlobe (black) and sidelobes (orange) region is the thick blue contour line with label u=1. As seen in (b) the abscissa u=1 corresponds to the first zero-crossing separating the mainlobe of the dinc function from its first sidelobe. The narrow stripe between the two thick contour lines labeled {82,84}=83±1 represents the second mainlobe, |u-83|<1, for a 83-spans system treated here. Most intermods within this secondary mainlobe, yields high FWM, but there are very few of these intermods. Indeed, this region, consisting of a very thin, curved stripe, captures just a few grid points (in a zoom-in not reproduced here, just 23 beat points were counted within either of the two thin curved stripes in the second and fourth quadrants, with many of these point landing quite close to the stripe boundaries (the contour lines labeled 82 and 84), hence potentially experiencing higher attenuation even though these are mainlobe points. This effect will be confirmed in Fig. 3, plotting the amount of FWM attenuation experienced by all intermods.

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The EFWMS is just slightly smaller than the rms-averaged quantity: L̂^FWM _eff[i,M]~>L̂^FWM _rms[i,M]≤1E.g., in the dispersion-free case we have L̂^FWM _rms[i,M]=1=L̂^DG-^FWM _rms[i,M], then Eq. (57) yields L̂^FWM _eff[i,M]=0.999. Generally, the DG correction is negligible and one may use L̂^FWM _rms in lieu of the EFWMS, L̂^FWM _eff, with high accuracy. It turns out that the EFWMS (in its single-span version above, as well as its multi-span version to be introduced in the sequel) is the key parameter representing the averaged reduction in the FWM generation due to dispersive phase mismatch detuning over all the sub-channel triplets. The details of FWM generation in the presence of dispersion are complicated, but the complexity is hidden in the EFWMS.

FWM power: Using Eq. (57) and Eq. (56) we note that (L̂^FWM _eff[i,M])² N _beats reduces to the braces in the last expression in Eq. (52). The FWM power Eq. (52) is then compactly expressed as follows:

σ_{{\overset{˘}{u}}_{i} (L_{span})}^{2} = 2 {(γ L_{eff} {\hat{L}}_{eff}^{FWM} [i, M])}^{2} N_{beats} [i, M] p_{0}^{3} e^{- α_{0} L_{span}} single-span FWM power

Reviewing receiver operation (section 2), flat NRZ pulses were assumed to be transmitted over each OFDM sub-channel. As each sub-channel pulse is assumed to experience negligible dispersion, then these flat transmitted pulses are also received flat over each symbol interval in each sub-channel path. The complex-valued decision variable in the absence of non-linear distortion is then

{\underset{\sim}{r}}_{i}^{(1)} = e^{(\frac{α_{0}}{2}) L_{span}} {\overset{˘}{u}}_{i}^{(1)} (L_{span}) .

where labeling by the discrete-time at which this sample occurs was omitted, merely indicating the sub-channel index. Likewise, the I&D filtering of the FWM intermods also amounts to just passing their constant value to the output.

The FWM signals obtained by triple products of flat sub-channel waveforms over symbol-intervals, may also be well approximated as constant over symbol-intervals. Hence the I&D time-averaging Eq. (6) of both the sub-channel linearly propagated signals, and of the FWM intermods, reduces to a null operation (the time-average of a constant waveform is the same constant), i.e. the decision sample at the I&D output simply equals the input.

The received FWM power (for a single span system) then compactly expressed as:

σ_{FWM}^{2} [i, M] \equiv σ_{\underset{\sim}{r_{i}}}^{2} = e^{α_{0} L_{span}} σ_{{\overset{˘}{u}}_{i} (L)}^{2} = 2 {(γ L_{eff} {\hat{L}}_{eff}^{FWM} [i, M])}^{2} N_{beats} [i, M] p_{0}^{3}

Notice that the attenuation factor $e^{- α_{0} L_{span}}$ was compensated for by the power gain (5) of optical pre-amplifier placed ahead of the receiver.

Gaussian distribution: The key result Eq. (60) provides the second moment of r̰_i (its first moment is 〈r̰_i〉=0), however the question arises what the distribution of this complex-valued random variable is. We have already seen that the phasors in the summation Eq. (47) or Eq. (44) are uncorrelated. Each of these FWM intermods phasors is randomly phased. Their random sum Eq. (44) is speckle-like over the ensemble of equi-probable multiple m-ary PSK transmission symbols. The isotropic nature of the distribution of the double-sum is evident (multiplying it by e ^jθ preserves the distribution). It is conjectured by virtue of the Central Limit Theorem that the sum ŭ ⁽³⁾ _i(L)of the FWM phasors is complex circular gaussian distributed, as widely held in the optical OFDM literature (e.g. [8]). Hence, for each OFDM sub-channel, the received carrier phasor becomes the center of an FWM-induced circular noise cloud, the quadrature component of which causes phase-noise, degrading the BER.

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

In the previous section we analyzed FWM generation over a fiber-optic link consisting of a single-span. We now extend the FWM model to address the effect of propagation over multiple optically amplified spans. In the dispersion free-case, it was implicitly assumed in [2] that the FWM build-up over multiple spans is coherent: For a link of length L=N _span L _span the intermods coherently add up in-phase, i.e. the complex amplitude of each intermod is increased by a factor of N _span, hence the overall FWM power at the end of an N _span-link increases by a factor of N ² _span 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 [9,10], and generalizing those results to arbitrary non-periodic gain/loss profiles, we show that unlike the dispersion-free case [2], wherein the FWM from multiple spans coherently accumulates in-phase, the FWM from the multiple dispersive spans builds-up as the radiation from a phased-array would. The role of phased-array antenna elements is played here by the fiber span. More precisely the compounding of FWM in a multispan fiber link with periodic amplification is analogous to the linear buildup of radiation from an end-fire antenna array, wherein the antenna elements are all aligned onto a single line, and the observation point is also on this line.

Inhomogeneous loss/gain: Assuming Δβ ^DL _ijk(z)=Δβ ^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:

Δ Φ^{DL} (t, z) = \int_{0}^{z} Δ β_{ijkl}^{DL} (t, z) dz = Δ β_{ijkl} z - j \int_{0}^{z} α (z') dz′ = Δ β_{ijkl} z + j G_{\log} (z)

where we defined the log-gain,

G_{\log} (z) \equiv - \int_{0}^{z} α (z') dz' = - α_{0} {[z]}_{L_{span}}, 0 \leq z \leq L

used Eq. (30) for the differential loss α(z) per unit length (effectively representing the optical amplifiers as regularly spaced negative loss impulses) and introduced the notation

{[z]}_{L_{span}} \equiv z \mod L_{span} .

The log-gain function Eq. (62) then consists of N _span periods of a sawtooth waveform, with a discontinuity at each optical amplifier restoring it to its zero peak value. Using Eq. (61), we have

1_{[0, L]} (z) e^{- j Δ Φ^{DL} (t, z)} = 1_{[0, L]} (z) e^{G_{\log} (z)} e^{- j Δ β_{ijkl} z} \equiv G (z) e^{- j Δ β_{ijkl} z}

where we introduced a gain function, G(z)=G(z)1[₀,_L](z), windowed over the domain [0,L] of interest:

G (z) \equiv e^{G_{\log} (z)} = e^{- α_{0} {[z]}_{L_{span}}} 1_{[0, L]} (z) = Σ_{S = 0}^{N_{span} - 1} e^{- α_{0} (z - s L_{span})} 1_{[0, L_{span}]} (z - s L_{span}) .

Notice that G(z) is finite periodic, i.e. it is expressible as the superposition of N _span replicas of a waveform of duration L _span, shifted by integer multiples of L _span:

G (z) = Σ_{S = 0}^{N_{span} - 1} δ (z - s L_{span}) \otimes e^{- α_{0} z} 1_{[0, L_{span}]} (z)

where ⊗ denotes convolution. We next replace $e^{- j Δ Φ^{DL} (t, z)}$ in Eq. (41) by Eq. (64), yielding:

\partial_{z} {\overset{˘}{ν}}_{i}^{(3)} = - j γ \sum_{[j, k] \in S [i]} {\overset{˘}{ν}}_{j}^{(1)} {\overset{˘}{ν}}_{k}^{(1)} {\overset{˘}{ν}}_{j + k - i}^{{(1)}^{*}} G (z) e^{- j Δ β_{ijk} z}

FWM efficiency as FT of gain profile: Integrating Eq. (67) over the interval [0,L] yields

{\overset{˘}{ν}}_{i}^{(3)} (L) = - j γ \sum_{[j, k] \in S [i]} {\overset{˘}{ν}}_{j}^{(1)} {\overset{˘}{ν}}_{k}^{(1)} {\overset{˘}{ν}}_{j + k - i}^{{(1)}^{*}} D_{ijk}^{FWM} = - j γ L_{eff} N_{span} \sum_{[j, k] \in S [i]} {\overset{˘}{ν}}_{j}^{(1)} {\overset{˘}{ν}}_{k}^{(1)} {\overset{˘}{ν}}_{j + k - i}^{{(1)}^{*}} {\hat{D}}_{ijk}^{FWM}

where D ^FWM _ijk≡L _eff N _span D̂^FWM _ijk (in units of distance) is the multi-span NEL, providing a generalization of the single-span expression L ^FWM _ijk. The multi-span NEL is given by

D_{ijk}^{FWM} \equiv \int_{0}^{L} G (z) e^{- j Δ β_{ijkl} z} dz = \int_{- \infty}^{\infty} G (z) e^{- j Δ β_{ijkl} z} dz = {F {G (z)} ∣}_{k = Δ β_{ijk}}

where in the 2^nd equality we used G(z)=G(z)1[₀,_L](z), and the last is identified as a spatial Fourier Transform (FT):

F {G (z)} \equiv \int_{- \infty}^{\infty} G (z) e^{- jkz} dz

The elegant key result Eq. (69) states that the multi-span NEL, describing the FWM generation efficiency of each beat, is given by the spatial Fourier Transform of the gain profile, evaluated at a spatial frequency equal to the Δβ mismatch of that particular beat. Hence, the relative strengths of the FWM terms are obtained as (non-uniform) samples of the spectrum of the gain profile at various spatial frequencies corresponding to the Δβ mismatches. A related result was derived in [14,15] in the context of XPM, where it was shown that it is the Fourier Transform of the dispersion map (rather than the gain profile) that yields the XPM response, though our interest here is in FWM rather than XPM.

In the last expression in Eq. (68) we conveniently introduced a normalized version of D ^FWM _eff :

{\hat{D}}_{ijk}^{FWM} \equiv \frac{D_{ijk}^{FWM}}{(L_{eff} N_{span})} .

Considering the field at the output of the last optical amplifier (receiver pre-amplifier), it is useful to transition from the normalized SCTE Eq. (68) to a CE formulation. To this end we first transition from Eq. (68) at the input of the receiver pre-amp, to the corresponding modified SCTE at the same point. The integral in the exponent in Eq. (23) is evaluated using Eq. (13), while excluding the SPM/XPM induced phase, The modeling of the SPM/XPM induced term $\exp [- j γ \int_{z_{1}}^{z_{2}} (2 P^{T} - p_{i} (t, z')) dz']$ is outside the scope of this analysis, and will be addressed in a future paper, hence we discard this term, taking β ^T _i=β _i-β ₀-jα(z)/2, yielding:

\int_{0}^{L} β_{i}^{T} (t, z') dz' = (β_{i} - β_{0}) L - j \frac{1}{2} \int_{0}^{L} α (z) dz' = (β_{i} - β_{0}) L - j \frac{1}{2} \int_{L - L_{span}}^{L} α_{0} dz'

Then Eq. (23) reduces to:

{\overset{˘}{u}}_{i}^{(3)} (L) = {\overset{˘}{v}}_{i}^{(3)} (L) e^{- j (β_{i} - β_{0}) L} e^{(\frac{- α_{0}}{2}) L_{span}}

Accounting for the last optical amplifier along the line (the receiver pre-amplifier) denoting its output as r̰, as before, we have

{\underset{\sim}{r}}_{i} = {\overset{˘}{u}}_{i}^{(3)} (L) e^{(\frac{α_{0}}{2}) L_{span}} = {\overset{˘}{v}}_{i}^{(3)} (L) e^{- j (β_{i} - β_{0}) L}

Substituting (68) in the last equation, yields (with D ^FWM _ijk given by Eq. (69)):

{\underset{\sim}{r}}_{i} = - j γ e^{- j (β_{i} - β_{0}) L} \sum_{[j, k] \in S [i]} {\overset{˘}{v}}_{j}^{(1)} {\overset{˘}{v}}_{k}^{(1)} {\overset{˘}{v}}_{j + k - i}^{(1) *} D_{ijk}^{FWM}

This is our final expression for the decision variable. Comparing Eq. (75) with Eq. (44), it is apparent that the key parameter D ^FWM _ijk plays for a multi-span link a role analogous to that of L ^FWM _ijk for a single-span link.

Array of spans as phased array: 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,

F {\sum_{s = 0}^{N_{span} - 1} δ (z - s L_{span})} = \sum_{s = 0}^{N_{span} - 1} e^{- j Δ β_{ijkl} s L_{span}} \equiv N_{span} F_{ijk}^{FWM}

yielding:

D_{ijk}^{FWM} = F_{ijk} N_{span} \cdot L_{ijk}^{FWM}

Here F _ijk is called array factor, readily evaluated as a finite geometric series,

F_{ijk} \equiv \frac{1}{N_{span}} \sum_{s = 0}^{N_{span} - 1} e^{- j Δ β_{ijkl} s L_{span}} = e^{- \frac{j Δ β_{ijk} (L - L_{span})}{2}} \frac{\sin (\frac{Δ β_{ijk} L}{2})}{N_{span} \sin (\frac{Δ β_{ijk} L_{span}}{2})}

further expressed in terms of the “digital sinc” function (Fig. 1(b,c)),

{dinc}_{N} [u] \equiv \frac{\sin (π u)}{N \sin (\frac{π u}{N})} = N^{- 1} (1 + e^{- j θ} + e^{- j 2 θ} + \dots + {e^{- j (D - 1) θ} ∣}_{θ \to 2 \frac{π u}{N}})

in the following compact form:

F_{ijk} [N_{span}] = e^{- \frac{j Δ β_{ijk} (L - L_{span})}{2}} {dinc}_{N_{span}} [\frac{L Δ β_{ijk}}{2 π}]

More formally, in signal processing theory, the dinc function Eq. (79), describing the Discrete-Time FT of a finite sequence {1,1,…,1} of N ones (or rather a closely related form), is referred to as the Dirichlet kernel. In antenna theory, a phased array is a group of antennas in which the relative phases of the respective signals feeding the antennas are varied in such a way that the effective radiation pattern of the array is reinforced in a desired direction and suppressed in undesired directions. The array factor describes the geometrical structure - the relative positioning of the antennas - independent of the common radiation pattern of the individual antennas. In an optical context (but equally applicable to the RF context), the dinc function is shown in Feynman’s textbook [20, sec. 30-1], to describe the resultant amplitude due to N equal oscillators subsequently applied in the textbook to model the field of a diffraction grating. In fact a direct analogy may be set between the buildup of the FWM field from N spans of fiber in an amplified fiber link, and the linear transfer function of a short (sub-mm) fiber grating with N periods. In our case the oscillators are the FWM sources of each fiber span. Eq. (77) is interpreted as the product of the frequency dependence of the distributed FWM generated in a single fiber span (corresponding to the radiation pattern of a single “antenna”) and a array factor corresponding to placing point oscillators at the center of each fiber span, irrespective of the details of the distributed radiation from each of the identical spans. The generation of the dinc function by adding up N _span regularly dephased phasors each of length 1/N _span, as indicated in the summation in Eq. (79), graphically forming a (partial) polygon, is illustrated in Fig. 2. Physically, the summed-up phasors correspond to the normalized FWM contributions due to the individual spans, superposing to the total FWM at freq. i. The possibility of getting a very small resultant for the phasors addition is apparent. Indeed, the successively dephased phasor contributions from the individual spans form a polygon which may close upon itself, yielding zero net FWM resultant. However, each triplet of subcarrier frequencies is described by its own partial polygon with a different pitch, hence the various FWM triplets experience various degrees of cancellation. A key objective of this paper is to work out the distribution of FWM suppression over all subcarrier triplets.

Fig. 2. Array Factor (dinc function) as resultant (red arrow) adding up N _span phasors (black arrows), each of length 1/N _span,, regularly dephased by an angle θ=β _ijk L _span=2π/N _coh=2πu ^ijk N _span between successive phasors. (a): N _span=12, θ=10°, 14°, 18°, 26°, 30°. In the last case the polygon closes upon itself (12·30°=360°) corresponding to the first zero crossing of the dinc. The other five points sample the dinc in its mainlobe. (b): θ=18°, N _span=1,4,8,16,32,64. In last two cases the polygon retraces itself, making several revolutions. In fact N _span=20 accomplishes one full revolution. 64 modulo 12=4, hence the resultants for N _span=4,64 are parallel. The condition for making one complete revolution (which yields zero resultant) is N _span θ=2π or N _span=N _coh. The condition for zero resultant (possibly making multiple complete revolutions) is that N _coh divide N _span. When N _span<N _coh (N _span>N _coh) the dinc is sampled in its mainlobe (sidelobes). In the dinc sidelobes, the polygon curls up upon itself, completing at least one full revolution, while becoming quite small.

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Our dinc function Eq. (77), was first shown in [9,10] to describe the compounding of FWM along a regular multi-span fiber link, in the context of WDM, rather than OFDM transmission, and a phased-array interpretation is offered here for the first time. More significantly, Eq. (77) coinciding with the results of [9,10], has been obtained here as a special case of the generalized formulation Eq. (70), which is new to the best of our knowledge. Further below we apply our more general result Eq. (70), to treat irregular arrays, extending the regular array treatment of [9,10]. We further note that a similar phased-array factor arises in the evaluation of XPM in a multi-span system [14,15], which is expected as the FWM and XPM are related types of third-order mixing products as explained in section 3. In fact, the strength of the total XPM distortion along the link experienced by subcarrier i due to subcarrier k, is precisely given by the array factor Eq. (80), with Δβ _ijk→Δβ _iik, since as shown in section 3, XPM is obtained as [j,k,l]→[i,k,-k]. Again, we remark that XPM does not degrade BER in OFDM transmission (provided the fixed XPM-induced constellation rotations are calibrated out) hence the array factor arising in XPM [14,15] is irrelevant for OFDM transmission.

It is convenient to re-express the argument of the dinc function in terms of the FWM Coherence Length, (CL)

L_{coh}^{ijk} \equiv \frac{2 π}{∣ Δ β_{ijk} ∣} = \frac{2 π}{β ″ {(Δ ω)}^{2} ∣ j - i ∣ \cdot ∣ k - i ∣},

where Eq. (36) was used in the second equality. The CL provides an equivalent measure of the amount of phase-mismatch (the higher the dispersion, the lower the coherence length).

Notice that the CL becomes low for intermods with j, k substantially deviating from i.

The argument of the dinc function in the array factor is given by:

u \equiv \frac{L Δ β_{ijk}}{2 π} = \frac{L}{L_{coh}^{ijk}} = \frac{N_{span} L_{span}}{L_{coh}^{ijk}} = \frac{N_{span}}{N_{coh}^{ijk}}

where in the last equality we introduced a FWM Coherence Number, (CN), given by

N_{coh}^{ijk} \equiv \frac{L_{coh}^{ijk}}{L_{span}}

essentially normalizing the coherence length by the length of a single span.

It is the modulus of the array factor Eq. (80) that matters most, expressed in terms of Eq. (82) as follows:

∣ F_{ijk} [N_{span}] ∣ = ∣ {dinc}_{N_{span}} [\frac{L}{L_{coh}^{ijk}}] ∣ = ∣ {dinc}_{N_{span}} [\frac{N_{span}}{N_{coh}^{ijk}}] ∣

We note that the dinc mainlobe peaks at unity, hence |F _ijk|≤1. In the absence of dispersion we have F _ijk=1=F _ijk and Eq. (77) reduces to D ^FWM _ijk=N _span, i.e. the spans add up coherently, in phase, consistent with [2], yielding the worst (highest) FWM.

Next, Eq. (71) is expressed using Eq. (77), yielding a key relation as follows:

{\hat{D}}_{ijk}^{FWM} = F_{ijk} {\hat{L}}_{ijk}^{FWM}

Irregular Array: 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 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:

G (z) = \sum_{s = 0}^{N_{span} - 1} δ (z - z_{s}) \otimes e^{- α_{s} z} 1_{[z_{s} - z_{s - 1}]} (z)

The spatial FT (69) of this expression is readily evaluated to yield the Multi-Span DNEL :

D_{ijk}^{FWM} = \sum_{s = 0}^{N_{span} - 1} e^{- j Δ β_{ijkl} z_{s}} L_{ijk}^{FWM} [s] = \sum_{s = 0}^{N_{span} - 1} e^{- j (Δ β_{ijkl} z_{s} + ∠ L_{ijk}^{FWM} [s])} ∣ L_{ijk}^{FWM} [s] ∣

where L ^FWM _ijk[s] is the FWM effective length of the s-th span which has loss α _s and length L _span[s]=z _s+1-z _s, where z ₀=0,z ₁,z ₂,…,z _N _span 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 [9,10], leading to the “dinc” array-factor, is readily seen to be a special case of this more general formulation (obtained by having the z _s 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] allowing varying propagation constants from span to span as well as arbitrary amplifier gains, fiber losses and span lengths, may also be rederived based on our formalism.

Large FWM suppression via the PA effect: In order to significantly reduce FWM, i.e. have |D̂^FWM _ijk|≪|L̂^FWM _ijk|≤1, we must have |F _ijk|≪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 N _span (i.e. long links), the sidelobes tend to become very low. Having |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 _coh, hence the condition that a particular FWM beat attain substantial attenuation (be placed in the dinc sidelobes) is reformulated as:

N_{coh}^{ijk} < N_{span} or L_{coh}^{ijk} < L Large FWM attenuation condition

Recall that N ^ijk _coh,L ^ijk _coh 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 S[i]set of intermods. Using Eq. (81), condition Eq. (88) is equivalently formulated as

d_{h}^{crit} < ∣ j - i ∣ \cdot ∣ k - i ∣ \equiv d_{h} {[j, k], [i, i]} High-FWM Mattenuation condition

where the RSH in the last inequality is defined as a Hyperbolic Distance (HD)

d_{h} {[x_{1}, x_{2}], [x_{2}, x_{2}]} \equiv ∣ x_{1} - x_{2} ∣ \cdot ∣ y_{1} - y_{2} ∣ .

and we introduced a critical HD, to be exceeded in order to attain high FWM attenuation:

d_{h}^{crit} \equiv \frac{2 π}{L ∣ β ″ ∣ {(Δ ω)}^{2}} = {[2 π L β ″ {(Δ ν)}^{2}]}^{- 1} = M^{2} {[2 π β ″ L_{span} N_{span} W^{2}]}^{- 1}

where W=MΔν is the total OFDM bandwidth over the M subcarriers.

It is apparent that it is those intermods for which 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 _h. In the (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 _h (i.e. d̂^crit _h=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” d ^crit _h as small as possible, so that its exterior cover more and more of the 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).

In the 100Gb/s OFDM system treated in [12] with M=128 subcarriers transmitted over N _span=83 spans, it turns out that the β ^′=-21.7psec²/Kmcoefficient of the G.652 fiber, $(D = 17 \frac{p^{\sec}}{nm \cdot Km})$ along with the assumed relatively large bandwidth (M=128 channels at inter-carrier-spacing Δv=200 MHz) suffice to reduce the critical HD Eq. (91), to the low level d ^crit _h≡27.6. These are the parameters actually assumed in Fig. 1. The 12033 intermods in the 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).

The amount of FWM attenuation for each individual beat is determined by the argument 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:

u^{ijk} \equiv \frac{N_{span}}{N_{coh}^{ijk}} = \frac{d_{h} {[j, k], [i, i]}}{d_{h}^{crit}} = 2 π β ″ \frac{{LW}^{2}}{M^{2}} d_{h} {[j, k], [i, i]}

Then, plotting concentric “hyperbolic circles” (contours of constant HD) over the 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 _h. In particular the label 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.

The distribution of FWM attenuations for each of the 12033 intermods is shown in Fig. 3(a), plotting L̂^FWM _ijk, the single-span FWM attenuation, and Fig. 3(b–e) plotting D̂_FWM _ijk=F _ijk L̂^FWM _ijk, the multispan FWM attenuation (obtained from the single-span attenuation by multiplying by the array factor F _ijk). Notice that L̂^FWM _ijk 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 S[i] domain. Accordingly, the average single-span attenuation turns out to be as little as L̂^FWM _rms≈1 dB. For a multi-span system, transitioning to D̂^FWM _ijk by multiplying L̂^FWM _ijk through F _ijk, the shape of D̂^FWM _ijk is then essentially determined by F _ijk. However, as already seen above, the vast majority of the intermods fortunately sample F _ijk into its sidelobes, yielding substantial attenuation for the vast majority of the D̂^FWM _ijk terms, as plotted in Fig. 3(b-e). Substituting our design parameters into the EFWMS formula Eq. (95) (with F _ijk,L̂^FWM _ijk evaluated in turn according to Eq. (93) and Eq. (60)), then yields the overall FWM suppression of D̂^FWM _eff=18.5 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.

In contrast, if it were not for the PA effect (i.e. if in-line DC were to be applied the end of each fiber span), then the FWM reduction due to the phase mismatch over the whole band would be merely ~1 dB, consistent with Fig. 3(a).

By now, the substantial FWM supression (attainable in case all in-line dispersion is removed) should appear more plausible - the analytical/graphical analysis carried out above enables visualizing the distribution of FWM attenuations experienced by each individual intermods of the full set, providing insight into the accumulation of FWM supression over all intermods.

FWM suppression goes as the bandwidth²×length×GVD product, nearly independent of M: Throughout the discussion below take i=M/2, i.e. observe the FWM mid-band.

In light of the discussion above, it is intuitively evident that the FWM suppression (further quantified in (95)a below) is essentially determined by the area of the mainlobe (black cross in Fig. 1), relative to the area of the full 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 bandwidth²×length×GVD dependence for the FWM suppression.

Fig. 3. (a): Single-span FWM attenuation L̂^FWM _ijk (contour plot labels are in dB units) for the [j,k] intermods falling onto sub-channel i=64, over the elongated tilted hexagonal-shaped S[64] domain. The relevant link parameters are stated in the text. A key parameter is the 200 MHz OFDM subcarrier separation. Even at the dispersive phase phase mismatch corresponding to this large frequency separation, most of the mixing products experience less than 0.5 dB FWM attenuation. The average single-span FWM suppression over all intermods is just 1 dB (relative to a dispersion-free system). Figs. (b)–(e) address an 83-span system with the same parameters, but with the phased-array effect at work, plotting the D̂^FWM _ijk=F _ijk L̂^FWM _ijkFWM efficiency (attenuation) over all intermods. In particular, Figs. (b),(c) provide respective 3D and 2D top views (with contour lines at 2 dB intervals) over the [-18.4,0] dB vertical range, whereas (d),(e) provide a deeper cross-section of the FWM attenuation function over the [-30.4,-18.4] dB range. (b) should be visualized laid on top of (d). The cross-sectional level of -18.4 dB coincides with the average FWM suppression over all intermods. Consistent with this average value, the efficiencies of most intermods are seen to fall under the -18.4 dB floor in (b) or (c) (most of the area is white in (c) and most of the area of the bottom facet of the box is visible in (b); conversely the cross-shaped mainlobe occupies a relatively small area; Notice that (b),(c) also feature the first sidelobe barely sticking up through the bottom, in the four quadrants). Further inspecting (d), (e) it is apparent that many of the intermods have FWM efficiencies falling even lower than -30.4 dB. All this is indicative of the very favorable averaging of FWM attenuation, making it plausible that the large overall 18.4 dB suppression be attained. Figs. (b)–(e) also display “topographic features” consisting of “ridges/spikes/thin sheets” sticking out in the second and fourth quadrant. The origin of this effect is the sparse second-mainlobe region, |u-83|<1, as detailed in the caption of Fig. 1.

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It remains to derive the mainlobe area dependence. The mainlobe boundary corresponds to setting u ^ijk=1 in (92). The mainlobe area is proportional to its characteristic linear dimension, which is taken as half the side, 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

d_{h} {[j, j], [i, i]} = {(j - i)}^{2} = l_{0}^{2} .

Substitute into (92) the following: j=k, the u ^ijk=1 condition for the mainlobe boundary, as well as (93), yielding the following relation:

1 = 2 π β ″ (\frac{{LW}^{2}}{M^{2}}) d_{h} {[j, j], [i, i]} = 2 π (\frac{{LW}^{2} β ″}{M^{2}}) l_{0}^{2}

The mainlobe area is proportional to l ² ₀ : A{mainlobe} ∝l ² ₀=M ²(2πLW ² β ^′)^-1

As remarked above, the area of the full S[i] domain is A{domain}∝M ². Taking the ratios of the two areas yields A{mainlobe}/A{domain}∝(2πLW 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.

Therefore, for a given length of fiber, it is the overall OFDM bandwidth W=MΔν, 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.

However, for the purposes of FWM suppression, “bandwidth” could have been interpreted as Δν=Δω/(2π), as it is seen in (36) that it is solely Δω that sets the Δβ 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 Δν=W/M, hence the Δβ 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 Δν 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.

The design guidelines for OFDM systems to take advantage of the PA effect for cancelling FWM are then simple: strive to increase some or all of the three parameters β ^′a′, W, N _span, i.e. 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 _span. As for selection of 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.

Evidently, the number of spans may only be increased up to a point in order to improve FWM suppresion. Taking excessive N _span, 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.

6. FWM build-up over multiple dispersive spans — OFDM performance analysis

At first sight, it would seem that evaluating the 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] to the dispersive non-linear case at hand. The complexity of the non-linear interaction in the presence of dispersion is encapsulated in a 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).

FWM power: Again, we realize that the phasors describing transposed NDG intermods, [j,k],[k,j], bear identical phases, summing up on an amplitude basis, whereas the various pairs of NDG intermods are uncorrelated, adding up on a power basis. We then evaluate the power of (75), by summing up the powers of uncorrelated terms, similarly to the single span case (52):

σ_{FWM}^{2} [i, M] = 〈 {∣ {\underset{\sim}{r}}_{i} ∣}^{2} 〉 = {(γ L_{eff} N_{span})}^{2} p_{0}^{3} \cdot 2 {\sum_{[j, k] \in S [i]} {∣ {\hat{D}}_{ijk}^{FWM} ∣}^{2} - \frac{1}{2} \sum_{[j, j] \in S^{DG} [i]} {∣ {\hat{D}}_{ijj}^{FWM} ∣}^{2}}

We now introduce FWM suppression factors for the multi-span case at hand, respectively defined as rms-averages over the full intermods set, as well as over the DG intermods:

{\hat{D}}_{eff}^{FWM} [i, M, N_{span}] \equiv \sqrt{{({\hat{D}}_{rms}^{FWM} [i, M, N_{span}])}^{2} - \frac{N_{beats}^{DG} [i, M]}{2 N_{beats} [i, M]} {({\hat{D}}_{rms}^{DG - FWM} [i, M, N_{span}])}^{2}}

{\hat{D}}_{rms}^{FWM} [i, M, N_{span}] \equiv \sqrt{\frac{1}{N_{beats} [i, M]} \sum_{[j, k] \in S [i]} {∣ {\hat{D}}_{ijk}^{FWM} ∣}^{2}}

{\hat{D}}_{rms}^{DG - FWM} [i, M, N_{span}] \equiv \sqrt{\frac{1}{N_{beats}^{DG} [i, M]} \sum_{[j, j] \in S^{DG} [i]} {∣ {\hat{D}}_{ijj}^{FWM} ∣}^{2}}

Recalling Eq. (85), repeated here for convenience, D̂^FWM _ijk=F _ijk L̂^FWM _ijk, it is apparent that the multi-span FWM suppression factors (95) differ from their single-span counterparts Eqs. (54),(56), (57) by applying multiplications through the array factors F _ijk before rms-averaging.

Using these definitions, the term in braces in Eq. (94) is compactly expressed as N _beas[i,M](D̂^FWM _eff[i,M,N _span])², hence (94) is compactly reformulated as

σ_{FWM}^{2} = 2 {(γ L_{eff} N_{span} {\hat{D}}_{eff}^{FWM} [i, M, N_{span}])}^{2} N_{beats} [i, M] p_{0}^{3}

The last formula refers to an uncompensated link containing N _span 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 N ² _span, and the singles-pan parameter L̂^FWM _eff[i,M] in Eq. (60) is now replaced by D̂^FWM _eff[i,M,N _span] in Eq. (96), accounting for the interplay of FWM, dispersion and the phased-array effect over the multiple spans.

Notice that the dependence of D̂^FWM _eff on N _span is due to the array factor F _ijk[N _spans].

Angular variance: To evaluate the BER degradation due to the FWM (temporarily ignoring all other noise sources) we work out the e variance σ ² _∠ of the phase noise induced by FWM in the angular decision variable φ _i≡∠r̰_i. Here r̰_i 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, π/m, for m-ary PSK. In this case, the phase noise, φ _i is essentially determined by the variance of the fluctuations in the imaginary part r ^im _i of r̰_i (equal to half the variance of r̰_i), normalized by the signal power (i.e. the inverse of the signal to intermodulation ratio):

σ_{∠ FWM}^{2} [i, M] \equiv var {φ_{i}} \approx var {\frac{r_{i}^{im}}{A}} = \frac{var {{\underset{\sim}{r}}_{i}}}{(2 A^{2})} = \frac{σ_{FWM}^{2} [i, M]}{2 p_{0}}

where, due to the optical amplification, the received power was set equal to the transmitted power per sub-channel, p ₀, which in turn equals a fraction 1/M of the total power P _T transmitted over all M sub-channels:

p_{0} = A^{2} = \frac{P_{T}}{M}

Substituting Eq. (96) into Eq. (97) yields

σ_{∠ FWM}^{2} [i, M, N_{span}] = {(γ L_{eff} N_{span} {\hat{D}}_{eff}^{FWM} [i, M, N_{span}])}^{2} N_{beats} [i, M] p_{0}^{2}

Finally, substituting Eq. (98) into Eq. (99), yields our final form for the angular variance

σ_{∠ FWM}^{2} [i, M, N_{span}] = {(γ L_{eff} N_{span} {\hat{D}}_{eff}^{FWM} [i, M, N_{span}])}^{2} {\hat{N}}_{beats} [i, M] \cdot P_{T}^{2}

and taking the square root yields its standard deviation,

σ_{∠ FWM} [i, M, N_{span}] = γ L_{eff} N_{span} {\hat{D}}_{eff}^{FWM} [i, M, N_{span}] \sqrt{{\hat{N}}_{beats} [i, M]} \cdot P_{T} dispersive

In the last two formulas we introduced a normalized version of N _beats (55):

{\hat{N}}_{beats} [i, M] \equiv \frac{N_{beats} [i, M]}{M^{2}} = 0.5 + \frac{(i - 2.5)}{M} + \frac{(1 + i - i^{2})}{M^{2}} .

Since 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:

{\hat{N}}_{beats} [\frac{M}{2}, M] = \frac{3}{4} - \frac{2}{M} + \frac{1}{M^{2}}; {\hat{N}}_{beats} [64, 128] = 0.734 \approx {\hat{N}}_{beats} [\frac{M}{2}, M]

We shall approximate N̂_beats≈0.734 for other values of M (≠128) as well, since N̂_beats is weakly dependent on M. Considering now the dispersion-free special case, we use this approximation for N̂_beats, and also set D̂^FWM _eff[i,M,N _span]=1 in Eq. (101), yielding:

σ_{∠ FWM}^{2} [\frac{M}{2}, M] \approx 0.734 {(γ L_{eff} N_{span} P_{T})}^{2} dispersion-free

reproducing a feature already stated in [2]: 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.

However, well beyond the dispersion-free approximation Eq. (104), our more general expression Eq. (101) accounts for dispersive FWM effects, compactly described in terms of the key multi-span EFWMS parameter, D̂^FWM _eff[i,M,N _span] 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 M (via D̂^FWM _eff[i,M,N _span], which depends on the array factor). Now, using square root of the approximate value N̂_beats≈0.734 as coefficient in Eq. (101), we approximate that expression for the dispersive case as

σ_{∠ FWM} [\frac{M}{2}, M, N_{span}] \approx 0.857 γ L_{eff} \cdot N_{span} {\hat{D}}_{eff}^{FWM} [i, M, N_{span}] P_{T} dispersive

Q-factor and BER: As the FWM-induced phase noise distribution is approximately gaussian (to the extent the approximation 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),

{BER}_{FWM} [i, M] ≅ 2 Q [q_{FWM} [i, M, N_{span}]],

using the gaussian-Q function Q[u]=(2π)^-1/2∫∞_uexp[-x ²/2]dx with a Q-factor (argument)

q_{FWM} [i, M, N_{span}] \equiv \frac{π}{m σ_{∠ FWM} [i, M, N_{span}]}

where m is the number of phase states (m-ary PSK) and σ _∠ _FWM is given by Eq. (101).

In a phase-noise context, such approximation to linear phase-noise mechanisms induced by circular Gaussian noise fluctuations, was shown in [22,23] to underestimate the BER of DPSK detection, and a correction factor, κ _m 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 κ _m (e.g. κ ₄=1.11):

q_{∠ FWM} [i, M, N_{span}] \equiv \frac{\frac{π}{m}}{\sqrt{κ_{m}} σ_{∠ FWM}} = \frac{π}{m γ L_{eff} \sqrt{κ_{m} {\hat{N}}_{beats} [i, M]} {\hat{D}}_{eff}^{FWM} [i, M, N_{span}] N_{span} P_{T}}

It follows that the Q-factor is inversely proportional to the total power carried by all OFDM channels. In the dispersion-free case (D̂^FWM _eff≈1), assuming M≫1 holds, the Q-factor is approximately independent of the number of sub-channels. For QPSK and fiber parameters m=4, κ _m=1.11, N̂_beats≈, γ=1.3/W/Km, α ₀=0.22dB/Km L _eff=19.38Km

Eq. (108) yields the following Q-factor for the FWM induced angular fluctuations:

q_{∠ FWM}^{QPSK} [i, M, N_{span}] \approx 0.0332 {({\hat{D}}_{eff}^{FWM} [i, M, N_{span}] N_{span} P_{T})}^{- 1}

On an “electrical dB” scale, q ^dB≡20log₁₀ q=log₁₀ q ², we have a linear relation:

q_{∠ FWM}^{QPSK} {[\frac{M}{2}, M]}^{dBE} \approx - 29.6 - N_{span}^{dBE} - 2 [P_{T}^{dBm} - 30] - {∣ {\hat{D}}_{eff}^{FWM} [i, M, N_{span}] ∣}^{dBE}

These are our key results for a multi-span link. The nominal effect is a linear degradation of the Q-factor in the number of spans, and in the optical power. In electrical dB, the Q-factor decreases 6 dB per octave of the spans number, and 2 dB per dB of optical power increase.

In the presence of dispersion, the FWM suppression term acts to improve the Q-factor, since |D̂^FWM _eff[i,M,N _span]|<1. Hence the Q ²-factor (in dBE units) is increased above its dispersion-free value by the positive factor -20log₁₀|D̂^FWM _eff[i,M,N _span].

To summarize the operational formulas, the Q-factor Eq. (108) for a dispersive multispan link (determining the BER as in Eq. (106)) nominally exhibits an inverse linear dependence on the number of spans and the optical power, precisely as a dispersion-free link would. Moreover, accounting for dispersion, the Q-factor exhibits additional dependence (typically monotonically increasing) of Q-factor on N _span via the key EFWMS factor Eq. (95), tending to offset the nominal degradation of the Q-factor with 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

({\overset{˘}{u}}_{j}^{(1)} + {\overset{˘}{u}}_{j}^{(3)}) ({\overset{˘}{u}}_{k}^{(1)} + {\overset{˘}{u}}_{k}^{(3)}) ({\overset{˘}{u}}_{l}^{(1) *} + {\overset{˘}{u}}_{l}^{(3) *}) \approx {\overset{˘}{u}}_{j}^{(1)} {\overset{˘}{u}}_{k}^{(1)} {\overset{˘}{u}}_{l}^{(1) *} +

The three terms in the second line in Eq. (111) describe the main perturbation to the UPA term ŭ ⁽¹⁾ _j ŭ ⁽¹⁾ _k ŭ ⁽¹⁾*_j ₊ _k-_i, mixing the ⁽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

In this section we consider the joint impact of FWM+CD and the linear ASE-induced phase noise, developing a combined model incorporating both noise sources. Granted, there are additional phase noise sources affecting the BER performance: ASE-induced linear phase noise, SPM/XPM induced non-linear phase noise (the Gordon-Mollenauer effect) and laser phase noise. Treatment of the additional noise sources (in particular those nonlinearly generated by the ASE fluctuations) is deferred to future study.

Focusing here on FWM and ASE, these two independent gaussian noise sources are described by their respective Q-factors (or equivalently Q ²-factors), compounding to a total Q ²-factor expressed as the harmonic mean of the individual Q ²-factors:

\frac{1}{q_{∠ T}^{2}} = \frac{1}{q_{∠ FWM}^{2}} + \frac{1}{q_{∠ LN}^{2}} .

with q _∠ _LN the Q-factor of the linear ASE-induced phase noise to be derived in this section.

The last equation is equivalent to the additivity of the noise variances of these two effects, due to their statistical independence. It now remains to derive q _LN. Our treatment in this section essentially follows the principles outlined in [9], differing only in notation and presentation.

We derive the angle of the noisy received phasor, assuming linear addition of the ASE noise contributions. Ignoring non-linear distortion, and assuming a lone transmitted symbol over the interval 0≤t≤T, the received signal in the i-th sub-channel prior to demodulation is:

A e^{j ϕ_{i}} e^{j Ω_{i} t} + {\underset{\sim}{N}}_{1} (t) + {\underset{\sim}{N}}_{2} (t) + \dots + {\underset{\sim}{N}}_{N_{span} + 1} (t),

The demodulation removes the factor $e^{j Ω_{i} t}$ it off the signal component, leaving the noises statistically equivalent to the original ones, hence we denote the demodulated signal as

ρ (t) = A + {\underset{\sim}{N}}_{1} (t) + {\underset{\sim}{N}}_{2} (t) + \dots + {\underset{\sim}{N}}_{N_{span} + 1} (t)

Finally, applying the I&D averaging operation Eq. (6) yields

\underset{\sim}{r} = A + {\underset{\sim}{n}}_{1} + {\underset{\sim}{n}}_{2} + \dots + {\underset{\sim}{n}}_{N_{span} + 1}, {\underset{\sim}{n}}_{s} \equiv \frac{1}{T} \int_{- \frac{T}{2}}^{\frac{T}{2}} {\underset{\sim}{N}}_{s} (t) dt, s = 1, 2, \dots, N_{span} + 1

The variances of these RVs are readily evaluated as 〈|n̰_s|²〉≡N ₀/T, where N ₀ is the Power Spectral Density (PSD) of N̰_s(t), expressed in terms of the OA parameters as N ₀=(F _N/2)(G _OA-1)hν ₀, hence N ₀/2T is the PSD of each of the real and imaginary parts of n̰_s, with F _N the OA noise figure (on a linear scale, i.e. 10log F _N=F ^dB _N).

The phase angle of the decision statistic Eq. (115) is

ϕ_{LN} \equiv ∠ \underset{\sim}{r} = ∠ {A + {\underset{\sim}{n}}_{1} + {\underset{\sim}{n}}_{2} + \dots + {\underset{\sim}{n}}_{N_{span} + 1}} \approx \sum_{s = 1}^{N_{span} + 1} \frac{{\underset{\sim}{n}}_{s}^{im}}{A}

Its variance (the linear phase noise power) is

σ_{∠ LN}^{2} \equiv Var ϕ_{LN} = (N_{span} + 1) \frac{\frac{1}{2} N_{0} / T}{{∣ A ∣}^{2}} = (N_{span} + 1) \frac{\frac{1}{2} N_{0} / T}{P_{T} / M} = \frac{\frac{1}{2} N_{0} (N_{span} + 1) W}{P_{T}}

where we used |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.

It is seen that the numerator in Eq. (117) is the total ASE noise power (in one quadrature) over the overall bandwidth of the OFDM signal, while the denominator is the total signal power. It follows that the variance of the angular fluctuations induced by ASE is proportional to the inverse of the optical signal to noise ratio (OSNR), defined in a standard way as

{OSNR}_{OFDM} \equiv \frac{P_{T}}{2 (N_{span} + 1) N_{0} W_{ref}} = \frac{P_{T}}{(N_{span} + 1) F_{N} (G_{OA} - 1) h ν_{0} W_{ref}}

where N ₀ represents the one-sided PSD of each of the I and Q quadrature components of the OA ASE in each polarization, W _ref≡12.5GHz 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

σ_{∠ LN}^{2} = {OSNR}_{OFDM}^{- 1} \cdot \frac{B}{(4 W_{ref})} = \frac{1}{4} (N_{span} + 1) F_{N} (G_{OA} - 1) h ν_{0} W \cdot P_{T}^{- 1}

We then have the following Q-factor associated with the linear phase noise impairment:

q_{∠ LN} \equiv \frac{\frac{π}{m}}{\sqrt{κ_{m}} σ_{∠ NL}} = \frac{π}{m} \sqrt{\frac{{OSNR}_{OFDM}}{\frac{κ_{m} W}{(4 W_{ref})}}}

with related Q ²-factor

q_{∠ LN}^{2} \equiv \frac{{(\frac{π}{m})}^{2}}{κ_{m}} σ_{∠ NL}^{- 2} = \frac{{(\frac{2 π}{m})}^{2}}{κ_{m}} {[F_{N} (G_{OA} - 1) h ν_{0} (N_{span} + 1)]}^{- 1} \frac{P_{T}}{W}

proportional to the PSD P _T/ W [Watt/Hz] of the OFDM signal, and inversely proportional to the number of OAs, N _span+1.

The total BER incurred in the M-ary PSK detection of each OFDM sub-channel, due to the combination of FWM and linear ASE induced noise is given by BER _∠ _T≅2Q[q _∠ _T], where, consistent with (112), with the two variances given by Eqs. (105) and (119), the total Q-factor is:

q_{∠ T} = \frac{π}{(m \sqrt{κ_{m} (σ_{∠ FWM}^{2} + σ_{∠ NL}^{2})})}

8. Application — designing a 40 Gb/s per λ OFDM system with the phased-array effect

Section 5 numerically demonstrated that substantial FWM suppression levels are attainable for systems engineered to take advantage of the phased-array cancellation effect. In this section we put this advantage to work in a reasonably realistic design scenario, assessing the overall system performance tradeoffs between the BER, throughput and reach (i.e. distance) parameters of an ultra-long-haul 40 Gb/s OFDM system, while taking into account the overhead due to the Cyclic Prefix (CP) [7].

As already stated in section 5, it is advantageous to discontinue usage of dispersion-shifted fiber for OFDM transmission, and just use G.652 standard fiber with dispersion parameter $D = 17 \frac{p^{\sec}}{nm \cdot Km}$ , corresponding to β ^′=-21.7psec²/Km. Additional assumed parameters are fiber differential loss of α ₀=0.22 dB/Km, nonlinear coefficient γ=1.3/W/Km, comprising N _span spans, each of length L _span=80Km terminated and initiated in OAs, all with gain $G_{0} = e^{α_{0} L_{span}} = 17.6 dB$ , and noise figure F _N=6.5dB.

In the proposed OFDM system under study, a 40Gbps OFDM signal is carried over 461 out of 512 QPSK modulated subcarriers provided by 512-point (I)FFTs. The remaining 51 sub-channels are assumed to be used as pilot tones for phase noise tracking (impairments due to phase noise are not considered here). The system uses polarization multiplexing, i.e. two independent sets of 512 subcarriers are transmitted over two orthogonal polarizations. In principle this design may be duplicated for each of the wavelengths of a WDM system, although some extra performance degradation (not modeled) may result from FWM cross-talk between WDM channels. The proposed system is assumed to include no in-line optical dispersion-compensation modules. Instead, dispersion is electronically compensated in the receiver, based on Cyclic Prefix (CP) extension of the transmitted OFDM block [7].

If no Cyclic Prefix were used, it would suffice to QPSK modulate each OFDM sub-channel at 21.7 Msym/sec, in order to attain the target 40 Gb/s aggregate bit-rate:

R_{b} = 40 \frac{Gb}{s} = 461 Ch \cdot 21.7 \frac{Msym / s .}{Ch \cdot Pol} \cdot 2 \frac{b}{sym} \cdot 2 pol

The assumed NRZ QPSK modulation of each sub-channel exhibits OFDM spectral multiplexing efficiency of 1 sym/Hz. The 21.7 Msym/s data rate per subcarrier would then translate into a subcarrier separation of Δv=21.7 MHz, and since there are M=512 channels, the aggregate bandwidth would be W=M Δv=11.11GHz. 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: $R_{b} / W = 40 \frac{Gb}{s} / 11.11 GHz = 3.6 b / s / Hz$ . However, the per-span compensation in this reference system precludes taking advantage of the phased-array effect, hence its FWM impairment is quite sizable, and its attainable range will be limited, as next determined using a variant of Eq. (108):

q_{∠ FWM} {[i, M, N_{span}]}_{DC after every span} = \frac{π}{m γ L_{eff} \sqrt{κ_{m} {\hat{N}}_{beats} [i, M]} {\hat{L}}_{eff}^{FWM} [i, M] N_{span} P_{T}}

The small yet significant modification in this formula is that D̂^FWM _eff[i,M,N _span] in (108) is replaced here by L̂^FWM _eff[i,M]. Indeed, recall that L̂^FWM _eff[i,M] describes the (relatively small) FWM degradation over a single span due to dispersive phase mismatch, whereas D̂^FWM _eff[i,M,N _span] is the multi-span expression, comprising the array factor. Both formulas (108) and Eq. (124) feature a N _span 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 N _span, yielding an inverse linear dependence on N _span 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 N_span is successively increased until the Q-factor Eq. (124) becomes 3.29 corresponding to the target BER of 10^-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 L̂^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)).

As our proposed system requires a CP, its induced overhead will translate into an enhanced bandwidth requirement, penalizing spectral efficiency, but the enhanced bandwidth actually turns out to be beneficial for FWM suppression via the phased-array effect (enabled by having no in-line optical compensation), as shown at the end of section 6.

To determine the bandwidth, we express the frequency separation Δv constrained by the CP overhead, as a function of the range L=N _span L _span. A basic equation governing the relations between the OFDM parameters in the presence of the CP overhead is

\frac{R_{b}}{(M ρ η)} = {(\frac{1}{Δ ν} + T_{CP})}^{- 1}

with R _b the aggregate bitrate (40Gbps), 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 $2 \frac{b}{sym} \cdot 2 pol = \frac{4 b}{sym}$ , and T _CP 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] as:

T_{CP} = β ″ L (2 π W) = 2 π β ″ N_{span} L_{span} M Δ ν

Combining the last two equations, and solving for Δν yields an expression (not explicitly reproduced here) of the form Δν(N _span;R _b,M ρ,η,L _span) for the frequency separation as a function of N _span, 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 q _∠ _FWM, Eq. (121) for q _∠ _LN, and Eq. (112) for the composition q _∠ _T of Q-factors. In turn, Eq. (108) for q _∠ _FWM depends on D̂^FWM _eff, which (although not explicitly labeled as such in Eq. (108) is a decreasing function of Δν, i.e. ultimately D̂^FWM ^eff=D̂^FWM _eff[i,M,N _span,Δν(N _span) is a function of N _span both directly, and via the CP-constrained Δϛ(N _span) dependence. The maximal N _span subject to the target BER constraint was then determined by repeatedly evaluating q _∠ _T for successively larger N _span values, until the target Q-factor of 3.27 was met, yielding the extended reach of N _span=87 for the OFDM system under study. Having determined N _span also sets the inter-subcarrier frequency Δν(N _span)=Δν(87)61.33MHz, as well as the aggregate bandwidth W=512Δν(87)=31.4 GHz, which in turn determines the spectral efficiency R _b/W=(40Gb/s)/(31.4GHz)=1.27 b/s/Hz of the system under study, seen to be a factor of 2.83 worse than the spectral efficiency of 3.6b/s/Hz of the first version of the reference system, which used an initial frequency separation of 21.7 MHz.

As the spectral efficiencies of the two systems are disparate, in order to make the comparison as fair as possible, we considered a variant of the reference system. In fact Fig. 4(a) displays both variants (dotted vs. solid curves): (i): the reference system also carries the same 40Gb/s as the system under study does (being 2.8 times more spectrally efficient, the reference system then requires 2.83 times less bandwidth). (ii): the reference system is allowed to occupy the same bandwidth as the system under study (hence it carries 2.83 times the bitrate — namely 40Gb/s×2.83=113 Gb/s). This second sub-case actually slightly improves the range performance of the reference system, since the subcarrier separation is now increased to 61.33MHz, 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)).

We remark that for a non-dispersive system (or for a weakly dispersive system, such as in scenario (i) of the reference system wherein dispersion is compensated span-by-span), the main parameter essentially setting performance is the total OFDM power P _T, (nearly) independent of the spectral distribution, as indicated in our Eq. (104) and previously shown in [2]. Therefore, upon varying the bandwidth of the reference system while attempting to formulate a fair comparison of the two systems, the resulting variation in reference system performance would be minute. Indeed, a non-dispersive system and a weakly dispersive system with Δν =21.7MHz (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 Δν=61.33MHz), 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.

In summary, the operational insight attained with the concepts and analytical tools developed here indicates that the phased-array effect (first discovered by [9,10]) yields a dramatic performance difference for OFDM systems properly designed to take advantage of this effect. Our comparison indicates that the transmission distance of the proposed OFDM system is almost tripled to 87 spans (6960 km), relative to the reference 33-span (2560 km) system, wherein dispersion is compensated in every span, with both systems transmitting 40Gbps at identical BER. The dramatically improved reach is seen to stem from engineering dispersion to attain large FWM suppression via the phased-array effect. A similar ultra-longhaul OFDM system was first briefly considered in [12] (please notice the erratum correction [13]).

Fig. 4. BER performance of two OFDM links both links attaining BER=10^-3, a BER level still amenable to Forward Error Correction, compatible with OFDM: (a): 33-span link carrying 40 Gb/s, with dispersion compensated at the end of every span, hence not using a cyclic prefix - does not display the phased-array effect, as its fully compensated spans add up coherently. (b): 87-span link with no in-line dispersion compensation — dispersion is electronically compensated in the receiver, based on the cyclic prefix inserted at the transmitter. The solid curves in (a) and (b) are the BERs related to FWM, ASE-induced phase noise, and both effects combined (TOTAL). The dotted curves in (a) are the FWM-induced BER, and total (FWM+ASE) BER for a 40 Gb/s system either in the absence of dispersion or for weak dispersion, e.g. when M=512 sub-channels are used in the system (a) to carry 40 Gb/s over 11.11 GHz such that the subcarrier separation is 11.11 GHz/512=21.7 MHz, in which case the effect of dispersion is unnoticeable with the densely packed subcarriers. The cyclic-prefix-free design (a) is superior in its spectral efficiency, relative to the ultra-long-haul system (b), which has its spectral efficiency reduced by a factor of 0.35, relative to that of system (a), as explained in the text. Therefore the total bandwidth required by system (a) is 1/0.35=2.83 times smaller. To make a fair comparison between the two systems, the spectral efficiency advantage of the cyclic-prefix-free system (a) is utilized to carry 2.83 times more data over this system, 40 Gb/s×2.83=113 Gb/s, while bringing the bandwidths of the two uneven-spectralefficiency systems to be equal (both 31.4 GHz), by increasing the spectral separation between the subcarriers of system (a) by a factor of 2.83 to 21.7 MHz×2.83=61.33 MHz (to coincide with that of system (b)) while retaining the 512-point FFT. Having increased spectral separation between subcarriers — see solid FWM curve in (a) — actually helps system (a) providing some dispersive phase mismatch even over a single span, ameliorating the FWM impairment by almost 1 dB in this span-by-span configuration —with no phased-array effect. The resulting TOTAL solid curve there is slightly better than the dispersion-free or weak dispersion case (the dotted lines), accounting for an increase in reach by one span (from 32 to 33 spans). Once all in-line dispersion compensation is removed (b), the phased-array effect is seen to provide dramatically enhanced reach (87 spans), almost tripling transmission range relative to (a), by virtue of the large FWM suppression generated via the phased-array effect. We conclude that a conventional design (a), with span-by-span dispersion compensation, is very inefficient in reach, and lacks flexibility in provisioning, however such cyclic-prefix-free design is superior in its spectral efficiency, relative to the ultra-long-haul system (b), which has its spectral efficiency reduced by a factor of 0.35, relative to that of system (a), as explained in the text.

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9. Conclusions

This paper was devoted to the rigorous development of an analytic model of the major FWM non-linear impairment affecting coherent optical OFDM transmission, incorporating into the OFDM model, for the first time, the beneficial impact of a phased-array effect first discovered in [9,10]. It was demonstrated in the OFDM context that multiple transmission spans may yield large (17–19 dB) mutual FWM cancellation, enabling to almost triple long-haul transmission ranges for 10^-3 to 10^-4 target BERs, relative to conventionally designed systems, with dispersion compensation applied in each span.

Ideally, for optimal OFDM performance, the link should be “dispersion managed” in an unconventional way: dispersion compensation should only be electronically applied at the receiver end of the link, removing the in-line dispersion compensation altogether, enabling the phased-array effect to run its course. Such dispersion compensation-free design would be advantageous for 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.

We reiterate that further to removing or at least spacing out the optical dispersion compensation, and using conventional (rather than low-dispersion) fiber, a key enabler is the usage of very large total OFDM bandwidth. The surprisingly large amount of total FWM cancellation was justified by visualizing the levels of FWM attenuation experienced by the multitude of individual intermods, and working out their distribution.

The limitations of the current model are reiterated: The quoted performance parameters represent upper bounds, since the only phase-noise sources considered were FWM and linear-ASE induced. A host of additional noise sources and impairments affecting practical OFDM systems were excluded from the analysis and will be addressed in future work: laser phase noise, non-linear phase noise sources (e.g. the Gordon-Mollenauer effect), polarization and PMD effects, clipping due to the large peak to average ratio of the OFDM signal, the nonlinearity of the optical modulator, the number of bits and non-ideality of the Analog to Digital conversion, and XPM effects. Therefore, the long ranges attained in the current analysis, should be merely viewed as upper bounds on performance. The main effect of the OFDM cyclic prefix overhead, which becomes appreciable, of the order of the OFDM symbol length, at the long ranges afforded by efficiently suppressing the FWM by means of the PA effect, was seen to substantially curb the system spectral efficiency. The CP overhead also turns out to limit the attainable aggregate bitrates for ultra-long-haul transmission, hence future research should be aimed at devising new methods to circumvent the CP limitation.

Yet another effect not addressed in this paper pertains to DWDM transmission. While the current formulation is in principle applicable to OFDM-DWDM systems, we have not actually considered multiple WDM channels, along with the mixing products between triplets of frequencies not all belonging to the same single wavelength OFDM channel. As the frequency separations are typically larger between DWDM channels than between OFDM sub-channels, this effect is conjectured to be small. Future work will further evaluate the performance resulting from the presence of dispersion compensation every few spans, and the interaction between the PA effect and various proposals for non-linear compensation [3,8].

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

Abstract

1. Introduction

2. System description

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

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

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

6. FWM build-up over multiple dispersive spans — OFDM performance analysis

7. Adding-in linear ASE noise

8. Application — designing a 40 Gb/s per λ OFDM system with the phased-array effect

9. Conclusions

References and links

Cited By

Figures (4)

Equations (142)

Optics Express