Phase jitter control of ultrashort solitons by use of Butterworth filters and nonlinear gain

Optical Fiber Technology 13 (2007) 67–71 www.elsevier.com/locate/yofte

Phase jitter control of ultrashort solitons by use of Butterworth filters and nonlinear gain Y.J. He, H.Z. Wang ∗ State Key Laboratory of Optoelectronic Materials and Technologies, Zhongshan (Sun Yat-Sen) University, Guangzhou 510275, People’s Republic of China Received 7 December 2005; revised 29 May 2006 Available online 10 August 2006

Abstract We demonstrate that soliton phase jitter can be efficiently suppressed by both Butterworth filters and nonlinear gain for ultrashort solitons with higher-order effects, as well as for long duration solitons. The nonlinear gain added to the system is to suppress no only background instability but also phase jitter, whereas the Butterworth filters are used to reduce both self-frequency shift and phase jitter. This scheme especially exploits a possibility for achieving a higher-speed soliton communication system using ultrashort solitons based on differential phase-shift keying. © 2006 Elsevier Inc. All rights reserved. Keywords: Differential phase-shift keying; Nonlinear gain; Phase jitter; Ultrashort soliton

1. Introduction There has recently been a renewed effort to develop coherent optical communication systems, particularly differential phase-shift keying (DPSK) [1,2], which dose not require a local oscillator to perform decoding. And because of its potential to eliminate the cross-phase modulation penalty, DPSK has attracted much attention. The use of DPSK has recently allowed the demonstration of impressive transmission capacities in the context of long-haul fiber-optic communication systems [3]. In such systems the essential parameter to be retrieved at the receiving end is the soliton optical phase. The error-free distance is mainly limited by random fluctuations of the phase caused by the amplified spontaneous emission that occurs in optical amplifiers. Nonlinear phase-shift compensation [1,2] and in-line phase conjugation [4] can reduce soliton phase jitter. In-line filtering was also shown to reduce phase jitter [5–8] (including Butterworth filters [6]) through the damping of amplifiers noise which results in a linear increase of the phase jitter with distance. Butterworth filters are suggested to replace conventional Gaussian filters in control phase jitter [6] because Butterworth filters have larger damping coefficient than that * Corresponding author.

E-mail address: [email protected] (H.Z. Wang). 1068-5200/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.yofte.2006.07.001

of Gaussian filters. However, when the filters are used, an excess gain must be provided to compensate for the loss induced by filters. The excess gain amplifiers linear waves coexistent with the soliton trains, leading to instability of the background. Hanna et al. [5] have confirmed that solitons phase jitter cannot be effectively controlled exceeding a distance of 8 Mm using Gaussian filters due to instability of the background. These phenomena lead to a trade-off involving the strength of the filter and limit the achievable length of the communication channel. The use of a sliding-frequency filter whose center frequency moves with distance along the transmission line can reduce continuum wave. However, the sliding action of filter increases phase jitter [9]. Another method to suppress the linear-wave growth is to use nonlinear gain (the amplitude-dependent gain). The nonlinear gain preferentially amplifies the soliton with large amplitudes while the linear waves with small amplitudes are unamplified or attenuated, thereby the background instability is suppressed [10], which leads to solitons stably transmission in a ultralong distance. This method of nonlinear gain may be particularly useful for transmission of solitons with subpicosecond or femtosecond durations [10]. In soliton communication systems, the transmission capacity can be improved by using ultrashort soliton (width < 5 ps). However, it is well-known that the use of ultrashort soliton in designing high-speed fiber-optic communication system

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(bit rate > 20 Gb/s) is bringing new challenges for system and fiber designs, because as the soliton width decreases, the higher-order effects such as third-order dispersion (TOD), selffrequency shift (SFS), and self-steepening become important and ultimately limit the bit rate. An example to stabilize ultrashort soliton propagation is the combination of filters and nonlinear gain in nonlinearity Schrödinger equation [11,12]. To date, main efforts that have been developed in controlling phase jitter in other authors’ studies were based on long duration solitons. In this work, by use of the perturbation theory used in [13,14] where filters are used to control timing jitter, we demonstrate that the phase jitter can be efficiently suppressed by both Butterworth filters and nonlinear gain for ultrashort solitons with higher-order effects, as well as for long duration solitons. Numerical simulations show an agreement with theoretical analysis. 2. Theoretical analysis The propagation equation for ultrashort solitons transmission with higher-order effects, Butterworth filters and nonlinear gain is described by a generalized nonlinear Schrödinger equation   2n  ∂u ∂ i ∂ 2u 1 = + i|u|2 u + u + γ1 |u|2 u α − ηn i ∂z 2 ∂t 2 2 ∂t + γ2 |u|4 u + s1

∂ 3u ∂(|u|2 u) ∂|u|2 − s , + is u 2 3 ∂t ∂t ∂t 3

(1)

where α > 0 (α < 0) is excess gain (loss), ηn are filters strength, n is the order of filters, γ1 and γ2 are the nonlinear gain coefficients. The parameter s1 = β3 /(6T0 |β2 |) is responsible for the TOD, here β2 and β3 are the group-velocity dispersion coefficient and third order dispersion coefficient, respectively; s2 = −TR /T0 (here TR is the Raman gain curve parameter) is the SFS parameter and s3 = 1/(ω0 T0 ) is self-steepening parameter, here T0 is soliton initial duration and ω0 is center frequency. Without nonlinear gain and higher-order effects, Eq. (1) has been shown in [15]. Introducing the usual ansatz for the soliton u = A sech(At − q) exp(−iΩt + iσ ) (notice that for nonsmall values of η1 and γ1 , Eq. (1) has other types of stable stationary solutions, namely the “composite pulses” [12]), and solving Eq. (1), we get the following evolution equations for the soliton amplitude A, frequency Ω, and phase σ :  n   dA 2n Mn−j Ω 2j A2(n−j )+1 = αA − ηn 2j dz j =0

4 16 + γ1 A3 + γ2 A5 , (2a) 3 15  n−1   dΩ 8 2n Mn−j Ω 2j +1 A2(n−j ) + s2 A4 , = −ηn 2j + 1 dz 15 j =0

(2b) dσ = dz

A2

− Ω2 2

1 + s1 (Ω 3 − 3ΩA2 ). 6

(2c)

Here Mn are the 2nth derivatives of the generating function f (s) = s/ sin(s) calculated at s = 0 (see [15]). Without nonlinear gain and higher-order effects, Eqs. (2a) and (2b) have been shown in [15]. The stable fixed points for the amplitude are given by minimums of the potential φ defined by dA/dz = −dφ/dA [10]. By solving dφ/dA = 0 and d 2 φ/dA2 > 0, we get double minimum potentials at A = 1 and A = 0 when α < 0, γ2 < 0, 0 < ηn < 4γ1 /(3Mn ), ηn Mn > 4/3γ1 +32/15γ2 , and α = ηn Mn − 4/3γ1 − 16/15γ2 . Here we neglect the higher order terms containing Ω0i (i > 1), because from Eq. (2b) Ω0 = 4s2 /(15nηn Mn ) is a very small value (it is same below). Introducing the first-order quantities A → 1 + a, Ω → Ω0 + δ, and σ → σ + σ , and linearizing Eqs. (2) about the steady state, we obtain   8 da (3a) = − e1 + (5γ1 + 8γ2 ) a − e2 δ + Fa (z, t), dz 15   16 dδ = −e1 δ − e3 − s2 a + Fδ (z, t), (3b) dz 15 d σ = c1 a − c2 δ + Fσ (z, t), (3c) dz where e1 = 2nηn Mn , e2 = 8s2 Mn−1 /(15Mn ), e3 = 16ns2 /15, c1 = 1 − s1 Ω0 , and c2 = Ω0 − 1/2s1 (1 + Ω02 ). On the righthand side of Eqs. (3), we add three delta-correlated noise factors of zero mean and correlation functions to account for the amplified spontaneous emission noise added by the in-line amplifiers [13,14]. They satisfy the relation [16]

1 Fi (z, t)Fj (z, t  ) = nsp Ni δi,j δ(z − z )δ(t − t  ), 2 i, j = a, δ, σ, (G − 1) (G − 1) , Nδ = , Na = N0 Za 3N0 Za   2 (G − 1) π +1 , Nσ = 12 3N0 Za



(4)

(5)

where N0 = P0 T0 / hν = Aeff Dλ4 /(4π 2 n2 hT0 c2 ) is the ratio of the number of photons to the unit energy, nsp is the spontaneous emission factor, G is the gain of the amplifier, Za is amplifier spacing and h is the Planck constant. In addition, P0 , Aeff , and D are the input power, effective fiber cross-section area and group delay (or time-of-flight dispersion) parameter of the fiber, respectively. The normal modes of Eqs. (3a) and (3b) have two eigenvalues (damping constants):  (6a) l1 = e1 − d1 − d12 − d2 ,  (6b) l2 = e1 − d1 + d12 − d2 , where d1 = 4(5γ1 + 8γ2 )/15 and d2 = e2 (16s2 /15 − e3 ). The steady state of soliton transmission systems need Re(l1 ) = Re(l2 ) > 0. Solving Eqs. (3), we obtain the variance of phase jitter  σ 2 (z) with distance. The result is

Na a0 + a1 z + a2 exp(−l1 z) + a3 exp(−l2 z) σ 2 (z) = 12 + a4 exp(−2l1 z) + a5 exp(−2l2 z)



Y.J. He, H.Z. Wang / Optical Fiber Technology 13 (2007) 67–71

 

Na π 2 + 1 z, + a6 exp[−(l1 + l2 )z] + 3 12

(7)

where a0 =

a2 = a3 =

− 3f22 ) + l23 (g12 2l13 l23 (l1 + l2 )

3l1 l2 (l12

+ l1 l2 + l22 )(4f1 f2 2l13 l23 (l1 + l2 )

+ 3f12 )]

+ g1 g2 )

3l2 (f12 + 2g12 ) + 3l1 (f1 f2 + g1 g2 ) l13 l2 3l1 (f22 + 2g22 ) + 3l2 (f1 f2 + g1 g2 )

a5 = −

γ2 < 0,

l23 l1 3f12 + g12 2l13

long solitons: 5γ1 + 8γ2 < 0,

,

3(f1 l2 + f2 l1 )2 + (g1 l2 + g2 l1 )2 , l12 l22

a4 = −

ultrashort solitons:  10γ1 + 16γ2  − e2 (240s2 − 225e3 ), ηn < 4γ1 /(3Mn ),

3(l1 + l2 )[l13 (g22

+ a1 =

69

(8a) (8b)

,

(8c)

,

(8d)

,

3f22 + g22

, 2l23 2(3f1 f2 + g1 g2 ) , a6 = l2 l1 (l1 + l2 )

(8e) (8f) (8g)

and

   2 f1 = c1 1 + d1 / d1 − d2 − 16s2 c2 / d12 − d2 ,     f2 = c1 1 − d1 / d12 − d2 + 16s2 c2 / d12 − d2 ,     g1 = c2 1 − d1 / d12 − d2 + 4ηn Ω0 c1 / d12 − d2 ,     g2 = c2 1 + d1 / d12 − d2 − 4ηn Ω0 c1 / d12 − d2 . 

(9a) (9b) (9c) (9d)

On the right-hand side of Eq. (7), the first term shows that amplitude-to-phase noise conversion occurs through the selfphase modulation and frequency-to-phase noise conversion occurs through the SFS and TOD [also see Eqs. (3)], i.e., the higher-order effects increase phase jitter. In the absence of higher-order effects, only amplitude-to-phase noise conversion occurs but not frequency-to-phase noise conversion occurs. The last term is a small contribution of direct coupling of the noise on phase. For large value of z and l1 z  1 and l2 z  1, all the exponentials in Eq. (7) are negligible, the phase jitter becomes (in real units)

2 (10) σ (Z) = M(a1 + π 2 /3 + 4)Z, which means that the phase jitter grows only in linear proportion to Z, here M = π 2 hc2 n2 nsp f (G)Γ T0 /(6λ4 Aeff D) and Z = zLD (LD is dispersion length), where f (G) = F (G)/ ln(G), F (G) = (G − 1)2 /[G(ln G)] is the ratio of the soliton peak power at the amplifier output to the peak power of the average soliton. From Eqs. (6) and (10), we find that the nonlinear gain plays a helpful role to suppress soliton phase jitter when the nonlinear gain coefficients satisfy following conditions:

γ2 < 0,

ηn < 4γ1 /(3Mn ).

(11)

This is one of important results in this paper. The proper coefficients of nonlinear gain γ1 and γ2 (γ2 < 0) are necessary to have the double-minimum potential [see the earlier analytical result following Eqs. (2)], i.e., the introduction of nonlinear gain inhibits the background instability. In addition, under these conditions with proper coefficients of nonlinear gain, two damping constants l1 and l2 in Eqs. (6) increase due to the introduction of nonlinear gain, so that a1 in Eq. (10) becomes small, which leads to phase jitter decrease. Thus the nonlinear gain added to the system can also suppress phase jitter for both long solitons and ultrashort solitons. We stress that the conditions of Eq. (11) are included in the steady-state conditions mentioned above, i.e., under these conditions the soliton transmission systems are stable. For long solitons, the higher-order effects are negligible, we get a1 = 12/ l12 and l1 = e1 − 2d1 . For ultrashort solitons, from Eqs. (8b) and (9), the value of a1 increases due to higherorder effects, accordingly phase jitter is increased. The reasons are (i) the second terms on the right of Eqs. (3a) and (3b) show that the amplitude and frequency produce extra couple terms through SFS, which significantly enhances phase jitter, this is the main influence, and (ii) the expressions of c1 and c2 in Eq. (3c) show that three factors increase the phase jitter, including SFS [through Ω0 = 4s2 /(15nηn Mn )], TOD and the couple terms between SFS and TOD. Among higher-order effects, the most important effect increasing phase jitter is SFS, while the least effect is self-steepening. However, without in-line control (ηn = γ1 = γ2 = 0), for ultrashort solitons, we have  σ 2 (Z) = 2M/75[(50 + 25s1 )Z 3 − 8s1 s2 Z 5 ] + (π 2 /3 + 4)MZ, i.e., the phase jitter obtains a quintic growth with distance due to higher-order effects. Whereas for long solitons, the phase jitter obtains a cubic growth with distance [5,8]. This shows the need which Butterworth filters and nonlinear gain suppress the growth of phase jitter. We must point out that the suppression of phase jitter by Butterworth filters is larger than that by nonlinear gain. The physical mechanism for nonlinear gain controlling phase jitter is that the nonlinear gain described here is amplitude-dependent gain which directly modulates the soliton amplitude [also see Eq. (2a)]. In addition, it is well-known that phase jitter increases arising from amplitude fluctuation due to the conversion of amplitude-to-phase noise. So through selecting proper coefficients of nonlinear gain, nonlinear gain can efficiently suppress phase jitter by overcoming the amplitude fluctuation. 3. Numerical example As a numerical example for Butterworth filters with n = 2, we used lumped filters with the equivalent distributed fil-

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 Fig. 1. Phase jitter  σ 2 /π versus propagation distance with Butterworth filters n = 2, for long soliton (s1 = s2 = s3 = 0). Theory (• • • curve) and simulations (+ + + curve) with both nonlinear gain and filters, theory (— curve) with filters only, theory (∗ ∗ ∗ curve) without in-line control, here 1.763T0 = 10 ps.

press not only phase jitter but also the background instability when nonlinear gain coefficients satisfy Eq. (11). The phase jitter is also suppressed by nonlinear gain for ultrashort solitons with higher-order effects, which has been described above. The variance of phase jitter versus propagation distance with filters alone (i.e., in the absence of nonlinear gain) is not shown in Fig. 2 because such system, whose background instability arising from the higher-order effects cannot be fully suppressed, is a quasi-steady system. Actually in this case the two damping constants shown in Eqs. (6) are complex values. In addition, the variance of phase jitter versus propagation distance with nonlinear gain alone is also not shown in Fig. 2 because both most phase jitter and SFS cannot be decreased due to the absence of filters. Therefore, using ultrashort solitons accompanied with higher-order effects, higher-speed soliton communication systems based on DPSK must simultaneously employ both the filters and nonlinear gain, and otherwise it is impossible to effectively communicate. 4. Conclusion

 Fig. 2. Phase jitter  σ 2 /π versus propagation distance with Butterworth filters n = 2 for ultrashort soliton. Theory (• • • curve) and simulations (+ + + curve) with nonlinear gain and Butterworth filters, theory (∗ ∗ ∗ curve) without in-line control, here 1.763T0 = 1 ps, s1 = 0.0022, s2 = −0.0106, and s3 = 0.0014. The inserts: 1 (input), 2 (output with Butterworth filters and nonlinear gain), and 3 (output with Butterworth filters only).

ter strength η2 = 0.123 [5,8]. The amplifier spacing Za = 45 km. The nonlinear gain coefficients are γ1 = 0.5 and γ2 = −0.34375 [10,11]. Other typical parameters: λ = 1.55 µm, β2 = −0.32 ps2 /km, nsp = 1.5, Γ = 0.046 km−1 , n2 = 3 × 10−16 cm2 /W, Aeff = 50 µm2 , β3 = 0.0024 ps3 /km, TR = 6 fs. Averaging over 500 realizations by a slip-step Fourier method, we perform the computer simulations. Figures 1 and 2 show the variance of phase jitter with distance based on Eq. (10) and the direct simulations of Eq. (1). Figure 1 uses long duration solitons 1.763T0 = 10 ps. Figure 2 uses ultrashort solitons 1.763T0 = 1 ps (corresponding to a 100-Gbit/s system). In Figs. 1 and 2, the computer simulations show a good agreement with theories. In Fig. 2, as a result of our computer limit, our results are shown only at a distance of 500 km, but we convince our results can be achieved at larger distance. In Figs. 1 and 2, there are large discrepancies between theories and simulations in the initial stage, which result from the neglect of the exponentials in Eq. (7). In Fig. 2, the insert output pulse 3 show that the linear waves lead to the background instability with filters only at Z = 80 km, whereas the SFS and linear waves are completely suppressed by, respectively, filters and nonlinear gain (see output pulse 2) at Z = 500 km. Note that in the plots of pulses 2 and 3 we remove an unessential common time shift that result from the TOD and self-steepening. Comparing the cases of with nonlinear gain and without nonlinear gain in Fig. 1, we demonstrate that nonlinear gain added to the system is to sup-

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