1. Introduction

Group-velocity dispersion and optical nonlinearity are the major limiting factors in high-speed long-distance fiber-optic transmissions [1, 2]. Dispersion-compensating fibers (DCFs) have been developed to offset the dispersion effects of transmission fibers over a wide frequency band. The most advanced DCFs are even capable of slope-matching compensation, namely, compensating the dispersion and the dispersion slope of the transmission fiber simultaneously [3, 4]. By contrast, it proves more difficult to compensate the nonlinear effects of optical fibers because of the lack of materials with negative nonlinearity and high group-velocity dispersion simultaneously [5]. Optical phase conjugation (OPC) in the middle of a transmission line may compensate the nonlinear effects between fibers on the two sides of the phase conjugator [6], especially when the two sides are configured into a mirror [7, 8, 9] or translational [9, 10, 11] symmetry in a scaled sense, although the benefit of OPC may still be appreciable in the absence of such scaled symmetry [12]. However, wide-band optical phase conjugation exchanges the channel wavelengths, so to complicate the design and operation of wavelength-division multiplexed (WDM) networks. Also, the performance and reliability of prototype conjugators are not yet sufficient for field deployment. Fortunately, it has been found that ordinary fibers could compensate each other for the intra-channel Kerr nonlinear effects without the help of OPC. The intra-channel nonlinear effects, namely, nonlinear interactions among optical pulses within the same wavelength channel, are the dominating nonlinearities in systems with high modulation speeds of 40 Gb/s and above [13], where the nonlinear interactions among different wavelength channels become less-limiting factors. As a result of the short pulse width and high data rate, optical pulses within one channel are quickly dispersed and overlap significantly so to interact through the Kerr nonlinearity. In the past a few years, intra-channel nonlinearities have been extensively investigated by several research groups [14, 15, 16, 17, 18, 19, 20, 21, 22]. A method has been identified for suppressing the intra-channel nonlinearity-induced jitters in pulse amplitude and timing, using Raman-pumped transmission lines manifesting a lossless or mirror-symmetric map of signal power [16, 22]. However, there is a problem with such mirror-symmetric power map. Namely, the loss of pump power makes it difficult to maintain a constant gain in a long transmission fiber. Consequently, the significant deviation of signal power profile from a desired mirror-symmetric map degrades the result of intra-channel nonlinear compensation using mirror symmetry [23]. By contrast, we shall demonstrate here that two fiber spans in a scaled translational symmetry [9, 11] could cancel out their intra-channel nonlinear effects to a large extent without resorting to OPC, and a significant reduction of intra-channel nonlinear effects may be achieved in a multi-span system with scaled translationally symmetric spans suitably arranged. The results shall be derived analytically from the nonlinear Schrödinger equation and verified by numerical simulations using commercial software.

2. Basics of nonlinear wave propagation in fibers

The eigenvalue solution of Maxwell’s equations in a single-mode fiber determines its transverse model function and propagation constant β(ω) as a function of the optical frequency ω[24, 25]. When a fiber transmission line is heterogeneous along its length, the propagation constant could also depend on the longitudinal position z in the line, and may be denoted as β(z,ω). The slow-varying envelope form,

with ${\beta }_0\left(z\right)\stackrel{\mathrm{def}}{=}\beta ({\omega }_0,z)$, is often employed to represent an optical signal, which may be of a single time-division multiplexed channel or a superposition of multiple WDM channels. The evolution of the envelope A(z,t) in an optical fiber of length L is governed by the nonlinear Schrödinger equation (NLSE) [9, 25],

∀ z ∈ [0,L], in the retarded reference frame with the origin z = 0 moving along the fiber at the signal group-velocity. In the above equation, α(z) is the loss/gain coefficient,

are the z-dependent dispersion coefficients of various orders, γ(z) is the Kerr nonlinear coefficient of the fiber, g(z,t) is the impulse response of the Raman gain spectrum, and ⊗ denotes the convolution operation [9]. Note that all fiber parameters are allowed to be z-dependent, that is, they may vary along the length of the fiber. Because of the definition in terms of derivatives, β ₂ may be called the second-order dispersion (often simply dispersion in short), while β ₃ may be called the third-order dispersion, so on and so forth. The engineering community has used the term dispersion for the parameter D = ${\mathit{\text{dv}}}_g^{-1}$/dλ, namely, the derivative of the inverse of group-velocity with respect to the optical wavelength, and dispersion slope for S = dD/dλ [1]. Although β ₂ and D are directly proportional to each other, the relationship between β ₃ and S is more complicated. To avoid confusion, this paper adopts the convention that dispersion and second-order dispersion are synonyms for the β ₂ parameter, while dispersion slope and third-order dispersion refer to the same β ₃ parameter, and similarly the slope of dispersion slope is the same thing as the fourth-order dispersion β ₄.

Had there been no nonlinearity, namely γ(z) = g(z, t) ≡ 0, Eq. (2) would reduce to,

which could be solved analytically using, for example, the method of Fourier transform. Let F denote the linear operator of Fourier transform, a signal A(z, t) in the time domain can be represented equivalently in the frequency domain by $\mathit{Ã}(z,\omega )\stackrel{\mathrm{def}}{=}\mathcal{F}\left[A(z,t)\right]$. Through a linear fiber, a signal Ã(z ₁,ω) at z = z ₁ would be transformed into Ã(z ₂, ω) = H(z ₁,z ₂,ω)Ã(z ₁,ω) at z ₂ ≤ z ₁, where the transfer function H(z ₁,z ₂, ω) is defined as,

In the time domain, the signals are related linearly as A(z ₂,t) = P(z ₁,z ₂)A(z ₁,t), with the linear operator P(z ₁,z ₂) given by,

Namely, P(z ₁,z ₂) is the concatenation of three linear operations: firstly Fourier transform is applied to convert a temporal signal into a frequency signal, which is then multiplied by the transfer function H(z ₁,z ₂,ω), finally the resulted signal is inverse Fourier transformed back into the time domain. In terms of the impulse response,

P(z ₁,z ₂) may also be represented as,

That is, the action of P(z ₁, z ₂) on a time-dependent function is to convolve the function with the impulse response. All linear operators P(z ₁,z ₂) with z ₁ ≤ z ₂, also known as propagators, form a semigroup [26] for the linear evolution governed by Eq. (4).

However, the existence of nonlinear terms in Eq. (2) makes the equation much more difficult to solve. Fortunately, when the signal power is not very high so that the nonlinearity is weak and may be treated as perturbation, the output from a nonlinear fiber line may be represented by a linearly dispersed version of the input, plus nonlinear distortions expanded in power series of the nonlinear coefficients [27]. In practical transmission lines, although the end-to-end response of a long link may be highly nonlinear due to the accumulation of nonlinearity through many fiber spans, the nonlinear perturbation terms of higher orders than the first are usually negligibly small within each fiber span. Up to the first-order perturbation, the signal A(z ₂,t) as a result of nonlinear propagation of a signal A(z ₁,t) from z ₁ to z ₂ ≥ z ₁, may be approximated using,

where A(z ₂,t)≈A ₀(z ₂,t) amounts to the zeroth-order approximation which neglects the fiber nonlinearity completely, whereas the result of first-order approximation A(z ₂,t)≈A ₀(z ₂,t) + A ₁(z ₂, t) accounts in addition for the lowest-order nonlinear products integrated over the fiber length. The term A ₁(·, t) is called the first-order perturbation because it is linearly proportional to the nonlinear coefficients γ(·) and g(·,t).

3. Theory of intra-channel nonlinearity compensation using scaled translational symmetry

Within one wavelength channel, it is only necessary to consider the Kerr nonlinearity, while the Raman effect may be neglected. The translational symmetry [9, 11] requires that the corresponding fiber segments have the same sign for the loss/gain coefficients but opposite second-and higher-order dispersions, which are naturally satisfied conditions in conventional fiber transmission systems, where, for example, a transmission fiber may be paired with a DCF as symmetric counterparts. The scaled translational symmetry further requires that the fiber parameters should be scaled in proportion and the signal amplitudes should be adjusted to satisfy [9, 11],

∀ z ∈ [0,L] and ∈ t ∈ (-∞, +∞), where α(z), β ₂(z), β ₃(z), and γ(z) denote the loss coefficient, second-order dispersion, third-order dispersion, and Kerr nonlinear coefficient respectively for one fiber stretching from z = 0 to z = L > 0, while the primed parameters are for the other fiber stretching from z′ = 0 to z′ = L/R, R > 0 is the scaling ratio, A(z,t) and A′(z′,t) are the envelopes of optical amplitude in the two fiber segments respectively. Even though the effect of dispersion slope may be neglected within a single wavelength channel, the inclusion of the β ₃-parameters in the scaling rules of Eq. (11) ensures that good dispersion and nonlinearity compensation is achieved for each wavelength channel across a wide optical band. When a pair of such fiber segments in scaled translational symmetry are cascaded, and the signal power levels are adjusted in accordance with Eq. (11), it may be analytically proved that both the timing jitter and the amplitude fluctuation due to intra-channel nonlinear interactions among overlapping pulses are compensated up to the first-order perturbation of fiber nonlinearity, namely, up to the linear terms of the nonlinear coefficient. Since the dispersive and nonlinear transmission response is invariant under the scaling of fiber parameters and signal amplitudes as in Eq. (11) [9, 11], it is without loss of generality to consider two spans that are in translational symmetry with the ratio R = 1. The cascade of such two spans would constitute a transmission line stretching from z = 0 to z = 2L, with the fiber parameters satisfying,

∀ z ∈ [0,L] and ∀ t ∈ (-∞, +∞). The translational symmetry is illustrated in Fig. 1 with plots of signal power and accumulated dispersion along the propagation distance.

 

Fig. 1. The signal power and dispersion maps for a cascade of two fiber spans in scaled translational symmetry with scaling ratio R = 1. Top: the variation of signal power along the propagation distance. Bottom: the dispersion map, namely, the variation of accumulated dispersion along the propagation distance.

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The amplitude envelope of a single channel may be represented by a sum of optical pulses, namely, A(z,t) = Σ_k u_k (z,t), where u_k (z,t) denotes the pulse in the kth bit slot and centered at time t = kT, with k ∈ Z and T > 0 being the bit duration. The following NLSE describes the propagation and nonlinear interactions among the pulses [13],

where the right-hand side keeps only those nonlinear products that satisfy the phase-matching condition. The nonlinear mixing terms with either m = k or n = k contribute to self-phase modulation and intra-channel cross-phase modulation (XPM), while the rest with both m ≠ k and n = k are responsible for intra-channel four-wave mixing (FWM) [13]. It is assumed that all pulses are initially chirp-free or they may be made so by a dispersion compensator, and when chirp-free the pulses u_k , ∀ k ∈ Z should all be real-valued. This includes the modulation schemes of conventional on-off keying as well as binary phase-shift keying, where the relative phases between adjacent pulses are either 0 or π. It is only slightly more general to allow the pulses being modified by arithmetically progressive phase shifts ϕ_k = ϕ ₀ + k∆ϕ, ∀k∈Z, with ϕ ₀,∆ϕ ∈ [0,2π),because Eq. (13) is invariant under the multiplication of phase factors exp(iϕ_k ) to u_k , ∀ k ∈ Z. The linear dependence of ϕ_k on k is in fact equivalent to a readjustment of the frequency and phase of the optical carrier. The pulses may be return-to-zero (RZ) and nonreturn-to-zero (NRZ) modulated as well, for an NRZ signal train may be viewed the same as a stream of wide RZ pulses with the half-amplitude points (with respect to the peak amplitude) on the rising and falling edges separated by one bit duration.

Were there no nonlinearity in the fibers, the signal propagation would by fully described by the dispersive transfer function,

with z ₁,z ₂ ∈ [0,2L], and,

or equivalently the corresponding impulse response,

which is calculated from F^-1 [H(z ₁,z ₂, ω)] up to a constant phase factor. The impulse response defines a linear propagator P(z ₁,z ₂) as in Eq. (8). In reality, the signal evolution is complicated by the Kerr nonlinear effects. Nevertheless, the nonlinearity within each fiber span may be sufficiently weak to justify the application of the first-order perturbation theory:

∀ k ∈ Z, where u_k (z,t) ≈ v_k (z,t) is the zeroth-order approximation which neglects the fiber nonlinearity completely, whereas the result of first-order perturbation u_k (z,t) ≈ v_k (z,t)+$v_k^{\mathit{\prime }}$ (z,t) accounts in addition for the nonlinear products integrated over the fiber length. For the moment, it may be assumed that both fiber spans are fully dispersion- and loss-compensated to simplify the mathematics. It then follows from the translational symmetry of Eq. (12) that b ₂(0,z+L) = -b ₂(0,z), ${\int }_0^{z+L}$ α(s)ds = ${\int }_0^z$ α(s)ds, γ(z + L) = γ(z), ∀ z ∈ [0,L], and v_k (2L,t) = v_k (L,t) = v_k (0,t) = u_k (0,t), which is real-valued by assumption, ∀ k ∈ Z. It further follows that h(0,z + L,t) = h ^*(0,z,t) and h(z + L,2L,t) = h ^*(z,2L,t), hence,

∀ z ∈ [0,L]. Consequently, the pulses at z and z+L are complex conjugate,namely,v_k (z+L,t) = $v_k^{*}$(z,t), ∀ k ∈ Z, ∀ z ∈ [0,L]. At the end z = 2L, a typical term of nonlinear mixing reads,

which is therefore real-valued. It follows immediately that the first-order nonlinear perturbation $v_k^{\mathit{\prime }}$ (2L,t) is purely imaginary-valued, which is in quadrature phase with respect to the zeroth-order approximation v_k (2L,t) = u_k (0,t), ∀ k ∈ Z. When the span dispersion is not fully compensated, namely, b ₂(0,L) ≠ 0, the input pulses to the first span at z = 0 should be pre-chirped by an amount of dispersion equal to - ½b ₂(0,L), so that the input pulses to the second span at z = L are pre-chirped by ½b ₂(0,L) as a consequence. In other words, the input signals to the two spans should be oppositely chirped. Under such condition, the equation v_k(z + L,t) = $v_k^{\mathit{*}}$ (z,t), ∀ k ∈ [0,L],∀k ∈ Z, are still valid, so are the above argument and the conclusion that v_k and $v_k^{\mathit{\prime }}$ are real- and imaginary-valued respectively when brought chirp-free.

Mathematically, that v_k and $v_k^{\mathit{\prime }}$ are in quadrature phase implies |u_k |²= |v_k + $v_k^{\mathit{\prime }}$ |² = |v_k |² + |$v_k^{\mathit{\prime }}$ |², where |$v_k^{\mathit{\prime }}$ |² is quadratic, or of the second-order, in terms of the Kerr nonlinear coefficient, ∀ k ∈ Z. This fact has significant implications to the performance of a transmission line. Firstly, it avoids pulse amplitude fluctuations due to the in-phase beating between signal pulses and nonlinear products of intra-channel FWM, which could seriously degrade the signal quality if not controlled [13, 15, 16, 21]. The quadrature-phased nonlinear products due to intra-channel FWM lead to the generation of “ghost” pulses in the “ZERO”-slots [14,18,19] and the addition of noise power to the “ONE”-bits. As second-order nonlinear perturbations, these effects are less detrimental. Secondly, it eliminates pulse timing jitter due to intra-channel XPM up to the first-order nonlinear perturbation. Using the moment method [15, 16], the time of arrival for the center of the kth pulse may be calculated as,

which is clearly jitterless up to the first-order perturbation, ∀ k ∈ Z. In the above calculation, the |$v_k^{\mathit{\prime }}$ |² terms are simply neglected as they represent second-order nonlinear perturbations. It may be noted that our mathematical formulation and derivation are straightforwardly applicable to transmission lines with scaled mirror symmetry for compensating intra-channel nonlinear effects without using OPC, and provide a theoretical framework of intra-channel nonlinearity that is more general than previous discussions [14, 15, 16, 17, 18, 19, 20, 21, 22]. No matter which type is the scaled symmetry, the essence of intra-channel nonlinear compensation is to annihilate the in-phase components of the nonlinear mixing terms, with respect to the unperturbed signals. How well the nonlinear effects are suppressed in a fiber transmission line depends largely upon how clean the in-phase nonlinear components are removed.

4. Optimal setups of fiber-optic transmission lines

A transmission fiber, either standard single-mode fiber (SMF) or non-zero dispersion-shifted fiber (NZDSF), and its corresponding slope-matching DCF [3, 4] are a perfect pair for compensating intra-channel nonlinearities, as their dispersions and slopes of dispersion satisfy the scaling rules in Eq. (11) perfectly, and the signal amplitudes may be easily adjusted to fulfil the corresponding scaling rule. The so-called reverse-dispersion fibers (RDFs) [28, 29], as a special type of DCFs, may be suitably cabled into the transmission line and contribute to the transmission distance, since the absolute dispersion value and loss coefficient of RDFs are both comparable to those of the conventional transmission fiber. Only the smaller modal area requires a lower level of signal power for an RDF to compensate the nonlinearity of a conventional transmission fiber. Otherwise a “one-for-many” compensation scheme may be employed, where the signal power may be slightly adjusted for an RDF to compensate the nonlinearity of multiple conventional transmission fibers [11]. There is usually no power repeater between the conventional transmission fiber and the cabled RDF within one span, so that the signal power decreases monotonically in each fiber span, as shown in Fig. 1. Note that one fiber span has a conventional transmission fiber followed by an RDF, while the other span has an RDF followed by a conventional transmission fiber, in accordance with the scaling rules in Eq. (11) for non-linearity compensation. Alternatively, if distributive Raman amplification [30, 31, 32, 33, 34], especially backward Raman pumping, is used to repeat the signal power, then one span should have the conventional transmission fiber Raman pumped in accordance with the RDF being Raman pumped in the other span. The signal power variation in each span may no longer be monotonic, but the power profiles in two compensating spans should still be similar and obey the scaling rules of Eq. (11), especially in portions of fibers that experience high signal power.

For DCFs having absolute dispersion values much higher than the transmission fiber, it is suitable to coil the DCF into a lumped dispersion-compensating module (DCM) and integrate the module with a multi-stage optical amplifier at each repeater site. Two fiber spans in scaled translational symmetry for intra-channel nonlinearity compensation should have oppositely ordered transmission fibers and DCFs. As shown in Fig. 2, one span has a piece of transmission fiber from A to B, in which the signal power decreases exponentially, and an optical repeater at the end, in which one stage of a multi-stage optical amplifier boosts the signal power up to a suitable level and feeds the signal into a lumped DCM, where the signal power also decreases exponentially along the length of the DCF from B to C, finally the signal power is boosted by another stage of the optical amplifier. The other span has the same transmission fiber and the same DCM, with the signal power in the DCF from C to D tracing the same decreasing curve. However, this span has the DCM placed before the transmission fiber. Ironically, the efforts of improving the so-called figure-of-merit [1, 4] by DCF manufacturers have already rendered the loss coefficients of DCFs too low to comply with the scaling rules of Eq. (11). To benefit from nonlinearity compensation enabled by scaled translational symmetries, DCFs, at least parts of them carrying high signal power, may be intentionally made more lossy during manufacturing or by means of special packaging to introduce bending losses. As illustrated in Fig. 2, the DCFs from B to C and from C to D are arranged in scaled translational symmetry to the transmission fibers from D to E and from A to B respectively, such that the transmission fiber from A to B is compensated by the DCF from C to D, and the DCF from B to C compensates the transmission fiber from D to E, for the most detrimental effects of jittering in pulse amplitude and timing due to intra-channel FWM and XPM. In practice, the DCMs from B to D and the multi-stage optical amplifiers may be integrated into one signal repeater, and the same super-span from A to E may be repeated many times to reach a long-distance, with the resulting transmission line enjoying the effective suppression of intra-channel nonlinear impairments. In case distributive Raman pumping in the transmission fibers [30, 31, 32, 33, 34] is employed to repeat the signal power, the DCFs may also be Raman pumped [35, 36] or erbium-doped for distributive amplification [37] to have similar (scaled) power profiles as that in the transmission fibers for optimal nonlinearity compensation.

It should be noted that in regions of fibers carrying lower optical power, the scaling rules of fiber parameters in Eq. (11) may be relaxed without sacrificing the performance of nonlinearity compensation, both for systems using cabled DCFs into the transmission lines and for systems using lumped DCMs at the repeater sites. Such relaxation may be done for practical convenience, or to control the accumulated dispersion in a span to a desired value, as well as to reduce the DCF loss so to reduce the penalty due to optical noise. As an example and a potentially important method in its own right, a DCM compensating the dispersion and nonlinearity of transmission fibers may be so packaged that the first part of DCF experiencing a high level of signal power may have a higher loss coefficient satisfying the scaling rule in Eq. (11), whereas the second part of DCF may ignore the scaling rule and become less lossy such that the signal power at the end of the DCM is not too low to be significantly impaired by the amplifier noise. In fact, the low-loss part of the DCM may even use optical filters other than DCFs, such as fiber Bragg gratings and photonic integrated circuits. This method of packaging DCMs achieves the capability of nonlinearity compensation and good signal-to-noise ratio performance simultaneously. For instance, it takes 10 km DCF with D′ = -80 ps/nm/km to compensate 100 km NZDSF with dispersion D = 8 ps/nm/km and loss α = 0.2 dB/km. The first 4 km of the DCF may be made highly lossy by a special treatment in manufacturing or packaging, with a loss coefficient α′ = 2 dB/km to form a scaled translational symmetry with respect to the first 40 km NZDSF for optimal nonlinearity compensation. However, the remaining 6 km DCF may ignore the scaling rules and have a much lower nominal loss α′ = 0.6 dB/km. The total loss is reduced by 8.4 dB as compared to a DCM that complies strictly with the scaling rules throughout the length of the DCF. Another important parameter of DCFs is the effective modal area, or more directly the nonlinear coefficient. Traditional designs of DCFs have always strived to enlarge the modal area so to reduce the nonlinear effects of DCFs. However, for DCFs used in our method of nonlinearity compensation, there exists an optimal range of modal area which should be neither too large nor too small. According to the scaling rules in Eq. (11), a DCF with a large modal area may require too much signal power to generate sufficient nonlinearity to compensate the nonlinear effects of a transmission fiber, when optical amplifiers may have difficulty to produce that much signal power. On the other hand, when the effective modal area is too small, the scaling rules of Eq. (11) dictate a reduced power level for the optical signal in the DCF, which may be more seriously degraded by optical noise, such as the amplified-spontaneous-emission noise from an amplifier at the end of the DCF.

 

Fig. 2. The signal power and dispersion maps for a cascade of two fiber spans in scaled translational symmetry with lumped dispersion compensators. Top: the variation of signal power along the propagation distance. Bottom: the dispersion map, namely, the variation of accumulated dispersion along the propagation distance.

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It is further noted that the nonlinear responses of fiber spans of different lengths may be approximately the same so long as each of them is much longer than the effective length L _eff = 1/α. This makes nonlinearity compensation possible among spans with different lengths, which are commonly seen in terrestrial and festoon systems, where the span-distance between repeaters may vary according to the geographical conditions. The dispersion of each fiber span may not be always fully compensated, in which case it is desirable to fine-tune the fiber lengths such that any pair of compensating spans have the same amount of residual dispersion. A final note is that two compensating fiber spans are not necessarily located immediately next to each other as drawn in Figs. 1 and 2. Sometimes, it may be advantageous to order pairs of compensating fiber spans in a mirror-symmetric manner [11], especially when all spans are not compensated to zero dispersion. Indeed, it is convenient to have the two spans of any compensating pair accumulating the same amount of total dispersion including the sign. This would be achieved naturally if the two compensating spans consist of exactly the same DCF and transmission fiber of exactly the same lengths, with the only difference being the ordering of the fibers. When a pair of compensating spans are not the same in span distance, the length of either a DCF or a transmission fiber may be fine-tuned, namely slightly elongated or shortened, to make sure that the two spans have the same accumulated dispersion. If the spans of a long-distance transmission line are labelled by -N, - N + 1, ⋯, -2, - 1 and 1,2, ⋯ ,N - 1,N from one end to the other, N > 1, a mirror-symmetric arrangement of spans requires that spans -n and n, ∀ n ∈ [1,N], should be paired for nonlinearity compensation. That is, their fiber parameters should satisfy the scaling rules in Eq. (11) approximately and their accumulated dispersions should be the same. Note that the scaling rules may only be fulfilled approximately if the two spans have the same non-zero accumulated dispersion. Then pre- and post-dispersion compensators may be employed at the two ends of the transmission line to equalize the total dispersion and importantly, to make sure that the accumulated dispersion from the transmitter to the beginning of span -n is opposite to the accumulated dispersion from the transmitter to the beginning of span n, ∀ n ∈ [1,N], such that the input signals to spans -n and n are complex conjugate, that is oppositely chirped, as required for compensating intra-channel nonlinearities. As an example, when all spans have the same accumulated dispersion b ₂, the pre-dispersion compensator should provide - (N - ½) b ₂, while the post-dispersion compensator should contribute - (N + ½) b ₂. Or the amount of post-dispersion compensation may be slightly different from - (N + ½)b ₂, such that the overall dispersion of the transmission line is not zero but within the tolerance of the transmitted pulses. It is worth pointing out that the single-channel nature of intra-channel nonlinearity compensation permits the use of channelized pre- and post-dispersion compensators. Namely, at each end of the transmission line, apart from a common pre- or post-dispersion compensator shared by all channels, each individual channel may have a channelized dispersive element, such as a short piece of fiber with the length fine-tuned, to compensate the channel-dependence of dispersion if any. Finally, it should be mentioned that a recent paper [38] proposes to compensate the timing jitter due to intra-channel XPM in a transmission fiber using the nonlinearity of a DCF, which is similar in spirit to our method of intra-channel nonlinearity compensation using scaled translational symmetry. However, the proposal in [38] is limited to the compensation of timing jitter of RZ pulses that are Gaussian-shaped, whereas our method could compensate both the amplitude fluctuation and timing jitter due to intra-channel nonlinear interactions of arbitrarily shaped pulses, with the only condition for suppressing intra-channel FWM that the signal pulses (when chirp-free) should be all real-valued upon a suitable choice of frequency and phase for the optical carrier. More importantly, the work presented in [38] did not recognize the significance of scaling the dispersion, loss and nonlinear coefficients of the DCF with respect to the transmission fiber, which is a necessary condition for optimal compensation of nonlinear effects. On the practical side, the proposal in [38] requires fiber Bragg grating dispersion compensators, which are limited in operating bandwidth and may suffer problems as thermal instability and group-delay ripples.

 

Fig. 3. A transmission line consists of 6 pairs of fiber spans, with the first span in each pair having 50 km SMF followed by 50 km RDF then 15.74 dB EDFA gain, and the second span having 39.35 km RDF followed by 39.35 km SMF then 20 dB EDFA gain.

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5. Simulation results and discussions

Numerical simulations using commercial software are carried out to support our theoretical analysis and verify the effectiveness of our method of suppressing intra-channel nonlinearity using scaled translational symmetry. In one test system, as depicted in Fig. 3, the transmission line consists of 6 pairs of compensating fiber spans totaling a transmission distance of 1072.2 km. The first span in each pair has 50 km SMF followed by 50 km RDF then an erbium-doped fiber amplifier (EDFA) with gain 15.74 dB, the second span has 39.35 km RDF followed by 39.35 km SMF then an EDFA with gain 20 dB. The other test system consists of the same number of spans with the same span lengths, which are constructed using the same fibers and EDFAs as the first system except that the second span in each span-pair has the 39.35-km SMF placed before the 39.35-km RDF, as shown in Fig. 4. The EDFA noise figure is 4 dB. The SMF has loss α = 0.2 dB/km, dispersion D = 16 + δD ps/nm/km, and dispersion slope S = 0.055 ps/nm²/km, effective modal area A _eff = 80 μm², while the RDF has α = 0.2 dB/km, D = -16 ps/nm/km, S = -0.055 ps/nm²/km, and A _eff = 30 μm². Fiber-based pre- and post-dispersion compensators equalize 11/24 and 13/24 respectively of the total dispersion accumulated in the transmission line. Both the SMF and the RDF have the same nonlinear index of silica n ₂ = 2.6 × 10^-20 m²/W. The transmitter has four 40 Gb/s WDM channels. The center frequency is 193.1 THz, and the channel spacing is 200 GHz. All four channels are co-polarized and RZ-modulated with 33% duty cycle and peak power of 15 mW for the RZ pulses. The MUX/DEMUX filters are Bessel of the 7th order with 3dB-bandwidth 80 GHz. The electrical filter is third-order Bessel with 3dB-bandwidth 28 GHz. The results of four-channel WDM transmissions have been compared with that of single-channel transmissions, with no clearly visible difference observed, which indicates the dominance of intra-channel nonlinearity and the negligibility of inter-channel nonlinear effects. Several trials with various values for δD have been simulated for each test system. The following figures present the eye diagrams of optical pulses after wavelength DEMUX, in order to signify the nonlinear deformation (timing and amplitude jitters) of optical pulses and the generation of ghost pulses. Fig. 5 shows the received optical pulses of δD = 0 for the two test systems, with the amplifier noise being turned off to signify the nonlinear impairments (right diagram) and the effectiveness of nonlinearity compensation (left diagram). Clearly shown is the suppression of nonlinear impairments by using scaled translational symmetry, and especially visible is the reduction of pulse timing jitter, as seen from the thickness of the rising and falling edges as well as the timing of pulse peaks. In both eye diagrams, there are optical pulses with small but discernable amplitudes above the floor of zero signal power, which could be attributed to ghost-pulse generation [14, 18, 19] due to the uncompensated and quadrature-phased components of intra-channel FWM. When the amplifier noise is turned back on, as shown in Fig. 6, the received signals become slightly more noisy, but the suppression of nonlinear distortions is still remarkable when there is scaled translational symmetry. Then δD = 0.2 ps/nm/km was set for the two test systems of Fig. 3 and Fig. 4 respectively, in order to showcase that a mirror-symmetric ordering of pairwise translationally symmetric fiber spans is fairly tolerant to the residual dispersions in individual fiber spans. In this setting, each fiber span has 10 or 7.87 ps/nm/km worth of residual dispersion, and the accumulated dispersion totals 107.22 ps/nm/km for the entire transmission line. Importantly, the pre- and post-dispersion compensators are set to compensate 11/24 and 13/24 respectively of the total dispersion, ensuring at least approximately the complex conjugation between the input signals to each pair of spans in scaled translational symmetry. The amplifier noise is also turned on. The transmission results, as shown in Fig. 7, are very similar to that with δD = 0, which demonstrates the dispersion tolerance nicely. In a better optimized design to tolerate higher dispersion mismatch |δD|, either SMFs or RDFs may be slightly elongated or shortened in accordance with the value of δD, such that the same residual dispersion is accumulated in all spans. As an example, δD is set to 0.6 ps/nm/km and each 39.35-km SMF is elongated by 0.385 km, so that all spans have the same residual dispersion of 30 ps/nm/km, and the whole transmission line accumulates 360 ps/nm/km worth of dispersion. The pre- and post-dispersion compensators equalize 360×11/24= 165 and 360×13/24 = 195 ps/nm/km worth of dispersion respectively. The gain of each 15.74-dB EDFA is increased to 15.817 dB in correspondence to the elongation of the 39.35-km SMF. The amplifier noise is still on. The transmission results are shown in Fig. 8.

 

Fig. 4. A transmission line consists of 6 pairs of fiber spans, with the first span in each pair having 50 km SMF followed by 50 km RDF then 15.74 dB EDFA gain, and the second span having 39.35 km SMF followed by 39.35 km RDF then 20 dB EDFA gain.

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Fig. 5. The transmission results with δD = 0 and amplifier noise turned off to signify the nonlinear effects. Left: received optical eye diagram of the scaled translationally symmetric setup in Fig. 3. Right: received optical eye diagram of the setup in Fig. 4 without scaled translational symmetry.

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Fig. 6. The transmission results with δD = 0 and amplifier noise turned on. Left: received optical eye diagram of the scaled translationally symmetric setup in Fig. 3. Right: received optical eye diagram of the setup in Fig. 4 without scaled translational symmetry.

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6. Conclusion

In conclusion, we have demonstrated through analytical derivation and numerical simulations that two fiber spans in a scaled translational symmetry could cancel out their intra-channel nonlinear effects to a large extent. And a significant reduction of intra-channel nonlinear effects may be achieved in a long-distance transmission line consisting of multiple pairs of translationally symmetric spans. We have also discussed a method of packaging dispersion-compensating fibers to optimally compensate the nonlinear effects of transmission fibers and to minimize the signal power loss at the same time.

 

Fig. 7. The transmission results with δD = 0.2 ps/nm/km and amplifier noise turned on. Left: received optical eye diagram of the scaled translationally symmetric setup in Fig. 3. Right: received optical eye diagram of the setup in Fig. 4 without scaled translational symmetry.

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Fig. 8. The transmission results with δD = 0.6 ps/nm/km and amplifier noise turned on. Left: received optical eye diagram of the scaled translationally symmetric setup in Fig. 3. Right: received optical eye diagram of the setup in Fig. 4 without scaled translational symmetry.

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