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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 18 — Aug. 30, 2010
  • pp: 19039–19054
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Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems

Xi Chen and William Shieh  »View Author Affiliations


Optics Express, Vol. 18, Issue 18, pp. 19039-19054 (2010)
http://dx.doi.org/10.1364/OE.18.019039


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Abstract

There has been a trend of migration to high spectral efficiency transmission in optical fiber communications for which the frequency guard band between neighboring wavelength channels continues to shrink. In this paper, we derive closed-form analytical expressions for nonlinear system performance of densely spaced coherent optical OFDM (CO-OFDM) systems. The closed-form solutions include the results for the achievable Q factor, optimum launch power density, nonlinear threshold of launch power density, and information spectral efficiency limit. These analytical results clearly identify their dependence on system parameters including fiber dispersion, number of spans, dispersion compensation ratio, and overall bandwidth. The closed-form solution is further substantiated by numerical simulations using distributed nonlinear Schrödinger equation.

© 2010 OSA

1. Introduction

High spectral efficiency transmission can be readily achieved with the concept of coherent optical OFDM (CO-OFDM) [1

1. W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–589 (2006). [CrossRef]

]. In such systems, the CO-OFDM wavelength channels can be either continuously spaced without frequency guard band [2

2. E. Yamada, A. Sano, H. Masuda, E. Yamazaki, T. Kobayashi, E. Yoshida, K. Yonenaga, Y. Miyamoto, K. Ishihara, Y. Takatori, T. Yamada, and H. Yamazaki, “1Tb/s (111Gb/s/ch × 10ch) no-guard-interval CO-OFDM transmission over 2100 km DSF,” Opto-Electronics Communications Conference/Australian Conference on Optical Fiber Technology, paper PDP6, Sydney, Australia (2008).

5

5. G. Goldfarb, G. F. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. 19(24), 2015–2017 (2007). [CrossRef]

], or densely spaced with extremely small frequency guard band [6

6. R. Dischler, and F. Buchali, “Transmission of 1.2 Tb/s continuous waveband PDM-OFDM-FDM signal with spectral efficiency of 3.3 bit/s/Hz over 400 km of SSMF,” Optical Fiber Communication Conference, paper PDP C2, San Diego, USA (2009).

,7

7. H. Takahashi, A. Al Amin, S. L. Jansen, I. Morita, and H. Tanaka, “Highly spectrally efficient DWDM transmission at 7.0 b/s/Hz using 8 x 65.1-Gb/s coherent PDM-OFDM,” J. Lightwave Technol. 28(4), 406–414 (2010). [CrossRef]

]. These densely spaced systems present the ultimate limit of achieving high spectral efficiency by allowing very little or no frequency guard band. Most recently, nonlinear transmission performance of CO-OFDM systems have attracted much attention [8

8. A. J. Lowery, S. Wang, and M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Opt. Express 15(20), 13282–13287 (2007). [CrossRef] [PubMed]

12

12. M. Mayrock and H. Haunstein, “Monitoring of linear and nonlinear signal distortion in coherent optical OFDM transmission,” J. Lightwave Technol. 27(16), 3560–3566 (2009). [CrossRef]

]. In particular, analytical results are shown in [8

8. A. J. Lowery, S. Wang, and M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Opt. Express 15(20), 13282–13287 (2007). [CrossRef] [PubMed]

] for single-channel transmission without consideration of chromatic dispersion; complete analytical expressions are presented in [9

9. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

] involving summation of a large number of terms for practical OFDM systems; system performances via numerical simulation is reported in [10

10. W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16(2), 841–859 (2008). [CrossRef] [PubMed]

12

12. M. Mayrock and H. Haunstein, “Monitoring of linear and nonlinear signal distortion in coherent optical OFDM transmission,” J. Lightwave Technol. 27(16), 3560–3566 (2009). [CrossRef]

]. It would be of great interests to derive concise closed-form solutions that capture the dependence of the nonlinear performance on some major system parameters such as chromatic dispersion and dispersion compensation ratio. Similar analytical work was pioneered in [13

13. P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001). [CrossRef] [PubMed]

] where nonlinear launch power and information capacity are derived in closed-form. However, there are two limitations for the report [13

13. P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001). [CrossRef] [PubMed]

]: (i) it only includes the cross phase modulation (XPM) as the dominant effect ignoring four-wave-mixing (FWM) and self-phase-modulation (SPM). This only applies to sparsely spaced WDM systems and would not apply to the densely spaced CO-OFDM systems where XPM, FWM, and SPM are all important, and very often indistinguishable [8

8. A. J. Lowery, S. Wang, and M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Opt. Express 15(20), 13282–13287 (2007). [CrossRef] [PubMed]

11

11. X. Liu, F. Buchali, and R. W. Tkach, “Improving the nonlinear tolerance of polarization-division-multiplexed CO-OFDM in long-haul fiber transmission,” J. Lightwave Technol. 27(16), 3632–3640 (2009). [CrossRef]

]; (ii) it assumes that nonlinear phase noise is generated independently in different spans, ignoring an important phase array effect of the FWM products that accounts for the interference among multiple spans [9

9. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

]. In this paper, we derive closed-form analytical expressions for nonlinear system performance of densely spaced CO-OFDM systems. The closed-form solution entails the results for achievable Q factor, optimum launch power density, nonlinear threshold of launch power density, and information spectral efficiency limit. These analytical results clearly identify the nonlinear performance dependence on system parameters including fiber dispersion, number of spans, dispersion compensation ratio, and overall bandwidth. The closed-form solutions are further substantiated by numerical simulations using distributed nonlinear Schrödinger equation.

2. Theoretical derivation and analysis

η=η1η2
(2)
η1=|1eαLejΔβijkLjΔβijk+α|21(Δβijk)2+α2
(3)
η2=sin2{NsΔβ˜ijk/2}sin2Δβ˜ijk/2,         Δβ˜ijk=ΔβijkL+Δβijk,1L1
(4)

In Eq. (2) the overall FWM efficiency is decomposed into two separate contributions: (i) η1, the FWM efficiency coefficient for single span (for simplicity, the contribution from dispersion compensation fiber is omitted), and (ii) η2, the interference effect between Ns spans of FWM products, also known as phase array effect [9

9. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

,14

14. K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett. 17(11), 801–803 (1992). [CrossRef] [PubMed]

]. Δβijkβi+βjβkβgis the phase mismatch in the transmission fiber. In Eq. (3), we have assumed that the span loss eαLis much larger than 1, eαLejΔβijkLis removed from the nominator. In Eq. (4), the subscript ‘1’ stand for the parameters associated with the dispersion compensation fiber (DCF)). Assuming mth subcarrier frequency has the form of fm=mΔf, the phase mismatch terms Δβijk and Δβ˜ijkin Eq. (4) and Eq. (5) can be rewritten as
Δβijk=2πλ2c(fifk)(fjfk)D=4π2β2(fifk)(fjfk)           =4π2β2Δf2(ik)(jk),             β2=λ22πcD
(5)
Δβ˜ijk=2πλ2c(fifk)(fjfk)Dr=4π2β2(fifk)(fjfk)L(1ρ)         =4π2β2Δf2L(1ρ)(ik)(jk)Dr=DL+D1L1=DL(1ρ)=DLζ
(6)
where D (orD1) is the chromatic dispersion of the transmission fiber (or DCF), ρ (or ζ) is the dispersion compensation (or residual dispersion) ratio, Dris the residual chromatic dispersion per span accounting for both transmission fiber and DCF. At the end of each span, the FWM product Pgalong with the signal will be amplified by a gain of G equal to the loss of each span eαLand the FWM product becomes

Pg=Dx29γ2PiPjPkη
(7)

We adopt the approach used in [13

13. P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001). [CrossRef] [PubMed]

] where the nonlinear effect is considered as the multiplicative noise to the signal. In essence, we will consider ith subcarrier as the reference frequency, and j and k frequencies as the interferers, namely, frequency j and frequency k generates a beating frequency component at (fjfk), which in turn modulates the subcarrier i, creating fourth components of fg. It has been shown for large number of subcarriers, the non-degenerate FWM is the dominate effects [9

9. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

], and Dx is set to 6 in Eq. (7). Consequently, the nonlinearity impinging on subcarrier i, PNLiis given as

PNLi=2γ2Pik=N/2N/2   j=N/2N/2PjPkη
(8)

A factor of one half is added in Eq. (7) because of the double counting in the dual summation. Equation (8) can be understood as the number of photons or amount of energy scattered off the subcarrier i, and should be equivalent to the photons scattered into this subcarrier i with large bandwidth assumption which we will clarify later. From now on, we drop index i and set it to zero, or equivalently, we are investigating the performance of center wavelength channels in broad bandwidth DS-OFDM systems. We also assume all the subcarriers have the same power of P for the sake of simplicity. The FWM power at the center subcarriers becomes
                                                         PNL=2γ2P3k=N/2N/2   j=N/2N/2η1η2η1=|sin(Nsj(kj)Δf2/(2fPA2))sin(j(kj)Δf2/(2fPA2))|2,     η2=1β22(2π)41Δf4j2(kj)2+fW4                                           fPA12π1|β2|Lζ,             fW12πα|β2|
(9)
where fPAis defined as the phase array bandwidth indicating frequency range of the effectiveness of phase array effects, fWis the defined as the walk-off bandwidth indicating the frequency range of the effectiveness of FWM nonlinearity in the presence of the dispersion. Substituting a new variablem=kj, Eq. (9) becomes
PNL=2γ2P3β22(2π)4m=N/2jN/2j   j=N/2N/2|sin(NsjmΔf2/(2fPA2))sin(jmΔf2/(2fPA2))|21Δf4j2m2+fW4
(10)
Equation (10) is similar to the result in [9

9. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

] where the FWM products scattering into each subcarrier is formulated. To understand the complexity of Eq. (10), we use a typical system where subcarrier frequency spacing Δfof 100 MHz, and a total bandwidth of 400 GHz, which gives Ns of 4000 subcarriers. This implies that in order to compute the FWM effect, it requires a summation in the order of 16 millions (4000x4000) of FWM terms in Eq. (10). Aside from the apparent mathematical complexity, the physical interpretation of FWM dependence on various key system parameters is difficult to ascertain. It is highly desirable to have a concise closed-form solution to the nonlinearity products in Eq. (10).

Although corroborating by the numerical simulation using distributed Schrödinger equation is the ultimate validation of the closed-form solutions, we would like to go through the derivation step by step below stating our assumptions and intermediate procedures towards final analytical solutions, in order to ensure certain degree of mathematical rigorousness is enforced. In each step we first summarize the main task, lay out the assumption and its justification, and then present the operations as a result of the simplification.

2.1 Conversion from discrete summation to integration

We observe that the FWM coefficient η1 represents the phase array effects and the major contribution of the summation takes place where jmΔf2/fPA21. This implies that dominate contribution is coming from the terms where j=m=int(fPA/Δf) where ‘int’ is the integer round off function. In this work, we assume that phase array effect bandwidth fPAis much larger than Δf, namely
fPA>>Δf
(11)
which is generally true in CO-OFDM systems. It can be easily shown around j=m=int(fPA/Δf), the phase jmΔf2/fPA2 inside sine function is changing slowly each time when j or m is changed by 1, therefore the conversion from discrete summation is justified as far as the η1is concerned. Similarly if we assume the walk-off bandwidth fWis much larger than the subcarrier spacing, namely
fW>>Δf
(12)
conversion from discrete to integration can be also justified in relation to η2. We call the conditions of Eqs. (11) and (12) as ‘dense subcarrier’ assumption. Under the assumptions of Eqs. (11) and (12), substituting the continuous integral variable f for mΔf, f1 for jΔf, the FWM power is transformed into

PNL=2γ2β22(2π)4P3Δf2B/2f1B/2f1B/2B/2η1(f,f1)η2(f,f1)df1dfη1(f,f1)=|sin(Nsf1f/(2fPA2))sin(f1f/(2fPA2))|2,       η2(f,f1)=1(f1f)2+fW4
(13)

According to the definition of m in Eq. (10), the variable f represents the frequency of the multiplicative noise impairing the channel. We now introduce more convenient and also fundamentally more important terms, power (spectral) densities given by
INLPNLΔf,               IPΔf
(14)
where INLand I are respectively FWM noise power (spectral) density and launch power (spectral) density. Substituting Eq. (14) into Eq. (13), we arrive at the FWM noise density
INL=2γ2β22(2π)4I3B/2f1B/2f1B/2B/2η1(f,f1)η2(f,f1)df1df
(15)
where B=NΔfis the total bandwidth of the DS-OFDM systems. The important conclusion is that under ‘dense subcarrier’ assumption, the result of the nonlinearity is independent of the subcarrier spacing.

2.2 Conversion of the integration range to more manageable forms

The proof is given in the Appendix A. The FWM power density becomes
INL=8γ2β22(2π)4I3B0/2B/20η1(f,f1)η2(f,f1)df1df
(16)
B0=2fW2/B
(17)
B>>fW
(18)
Equation (18) is another important assumption that is used for deriving Eq. (16), which states that the overall bandwidth is much larger than the walk-off bandwidth. We call this condition ‘large bandwidth’ assumption which will be repeatedly used in the remainder of the paper.

2.3 Closed-form expressions for nonlinear noise density

Since f is the nonlinearity noise frequency, the integration over f1 in Eq. (16) would generate the nonlinear noise spectral density. We rewrite Eq. (16) in terms of the one-sided nonlinear multiplicative noise spectral density iNL(f) given by

INL=IB0B/2iNL(f)df,     iNL(f)=8γ2β22(2π)4I20η1(f,f1)η2(f,f1)df1
(19)

INL=(II0)2I,         I01γπα|β2|Nsheln(B/B0)
(23)

2.4 Signal-to-noise ratio and spectral efficiency limit in the presence of nonlinearity

The signal power in presence of the nonlinear interference can be expressed as [13

13. P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001). [CrossRef] [PubMed]

]

I=Iexp((I/I0)2)I
(24)

The noise can be considered as the summation of the white optical amplified-spontaneous-noise (ASE), n0 and the FWM noise, and is shown given by [13

13. P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001). [CrossRef] [PubMed]

]
n=n0+I(1exp((I/I0)2)),       n0=Ns(G1)nsphυ0.5NseαLhυNF
(25)
where nspis the spontaneous noise factor equal to half of the noise figure of the optical amplifier NF, h is the Planck constant, and υ is the light frequency. The signal-to-noise is thus given by

SNR=Iexp((I/I0)2)n0+I(1exp((I/I0)2))
(26)

For the SNR larger than 10, Eq. (26) can be approximated as

SNRIn0+I(I/I0)2
(27)

The simplification is generally valid for the case of interests where the signal power density is much smaller than I0.

We have verified through our simulation under ‘dense subcarrier’ and ‘large bandwidth’ assumptions of Eqs. (11), (12), and (18), the FWM noise is of Gaussian distribution. This is also verified previously in [16

16. X. Liu and F. Buchali, “Intra-symbol frequency-domain averaging based channel estimation for coherent optical OFDM,” Opt. Express 16(26), 21944–21957 (2008). [CrossRef] [PubMed]

]. Under the assumption of Gaussian noise distribution, the information spectral efficiency (defined as the maximum information capacity C normalized to bandwidth B) for single-polarization is readily given by [17

17. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948).

]

S=log2(1+SNR)=log2(1+Iexp((I/I0)2)n0+I(1exp((I/I0)2)))     log2(1+In0+I(I/I0)2)
(28)

From Eq. (28), the maximum spectral efficiency Sopt in the presence of fiber nonlinearity can be easily shown as

Sopt=log2(1+13(2I0/n0)2/3)
(29)

2.5 Optimal launch power density, maximum Q, and nonlinear threshold of launch power density

In Eq. (28), the ultimate spectral efficiency is obtained. However in practice, the performance is always lower because of the practical implementation of modulation and coding. We therefore derive a few important parameters that are commonly used in the optical communications community. The first one is the maximum achievable Q factor. Under the Gaussian noise assumption and QPSK modulation, the Q factor is equal to the SNR given by

Q=SNR=Iexp((I/I0)2)n0+I(1exp((I/I0)2))In0+I(I/I0)2
(30)

The optimum launch power density is another important parameter and is defined as the launch power density where the maximum Q takes place. By simply differentiating Q of Eq. (30) over I, and setting it to zero, we obtain the optimum launch power density and the optimal Q given by

Iopt=(n0I02/2)1/3=(n0πα|β2|2γ2Nsheln(B/B0))1/3
(31)
Qmax=13(2I0/n0)2/3=(4πα|β2|)1/33(n02γ2Nsheln(B/B0))1/3
(32)

One of the inconveniences of using the optimum launch power expression in Eq. (31) is that it is dependent on the amplifier noise figure. Another commonly used term is nonlinear threshold launch power density that is defined as the maximum launch power density at which the BER due to the nonlinear noise can no longer be corrected by a certain type of forward-error-correction (FEC) code. For standard Reed-Solomon code RS(255, 239), the threshold Q is 9.8 (dB), or linear q0 of 3.09. In Eq. (30), setting n0 to zero and Q to q02, we arrive at the nonlinear threshold of power density
Ith=I0q0=1q0γπα|β2|Nsheln(B/B0)
(33)
where q0 is the correctable linear Q for a specific FEC.

The closed-form expressions for nonlinear noise spectral density iNL(f) in Eq. (20), nonlinear noise power density INLin Eq. (21), nonlinear multi-span noise enhancement factor hein Eq. (22), nonlinear characteristic power density I0in Eq. (23), information spectral efficiency S in Eq. (28), system Q factor and its optimal value in Eq. (30) and Eq. (32), optimal launch power density in Eq. (31), and nonlinear threshold of launch power density in Eq. (33) comprise the major findings in this work.

3. Corroboration of the theories with numerical simulation

During the derivation in Section 2, there are quite many assumptions and approximations made in order to arrive at concise closed-form expressions. The concern is that whether several approximations may accumulate and make the closed-form solutions inaccurate. In this section, we intend to corroborate the theoretical results with the numerical simulations. Among all the analytical results, the FWM noise density INLof Eq. (21) is the most fundamental one, and other expressions can be considered as the derivatives of the INL. We have conducted simulation to validate the expression for INL.

The parameters for the simulated systems are as follows: 16 wavelength channels, each covering 31-GHz bandwidth, giving total bandwidth B of 496 GHz; OFDM subcarrier frequency spacing of 85 MHz; QPSK modulation for each subcarrier; no frequency guard band between wavelength channels; 10-span of 100 km fiber link; Fiber loss α of 0.2 dB/km; nonlinear coefficientγ=1.22W1km1; noise figure of the amplifier of 6 dB; The FWM noise density is simulated by using an perfect optical notch filter to notch out 100 MHz gap at the center of the input signal spectrum, and the power density is measured at the output after 1000-km transmission. Figure 2(a)
Fig. 2 Comparison of closed-form theory and simulation results for (a) FWM power density, and (b) Q factor as a function of the launch power density. Theo.: Theory; Simu.: Simulation; CD: Chromatic dispersion with a unit of ps/nm/km; CR: (CD) compensation ratio. Both (a) and (b) assume 10x100 km single-polarization transmission systems.
shows the simulated nonlinear noise density compared with the computed nonlinear density using the closed-form expression of Eq. (21). Three transmission systems are investigated: (i) SSMF type system with CD of 16 ps/nm/km without any dispersion compensation, abbreviated as ‘system I’, (ii) CD of 16 ps/nm/km, but with dispersion 95% compensated per span, abbreviated as system II, and (iii) non-zero dispersion-shifted type fiber with CD of 4 ps/nm/km, abbreviated as ‘system III’. For systems I, II, and III, the average difference of FWM density is 14%, 12%, and 17%. This shows excellent match between the closed-form formula and simulation, considering the extreme sensitivity of the FWM density as a function of launch power density (cubic dependence). We also perform the simulation of the system Q factors with the above-described three systems, the results of which is shown in Fig. 2(b). We can see a good match between theoretical expressions based on Eq. (23) and Eq. (30) and simulation results. For instance, the difference between the optimal Q from theory and simulation is within 0.15 dB for all simulated dispersion maps. The difference of launch power between the simulation and closed-form theory for the same Q factor is less than 0.4 dB for wide range of launch power density of −28 to −16 dBm/GHz. All these confirm the excellent match between the simulation and the closed-form expression of Q factors in (30).

3.1 Discussion about the closed-form expression

Because the concise closed-form expressions are available, we are ready to quickly identify their dependence on system parameters including fiber dispersion, number of spans, dispersion compensation ratio, and overall bandwidth. In this section, we will discuss in detail the achieved system Q factor, optimum launch power, information spectral efficiency, and multi-span noise enhancement factor

3.2 System Q factor and optimum launch power

The immediate benefits of having closed-form formulas of Eq. (31) and Eq. (32) for system Q factor and optimum launch power density are their scaling over the underlying parameters. From Eq. (31) and Eq. (32), it follows that for every 3 dB increase in fiber dispersion, there is 1 dB increase in the optimal launch power density and the achievable Q; for every 3 dB increase in fiber nonlinear coefficient γ, there is 2 dB increase in the optimal launch power and achievable Q. We can quickly generate the optimum launch power and achievable Q for variety of dispersion maps. We use the three systems, systems I, II and III studied in Figs. 2 and Fig. 3
Fig. 3 (a) The maximum Q factor, and (b) the optimal launch power density versus number of spans with various dispersion maps. CD: chromatic dispersion. CR: (CD) compensation ratio
as an example. As shown in Fig. 3(a), system I has the best performance due to large local dispersion and no per-span dispersion compensation. The advantage of system I over system II increases with the increase of the number of spans, for instance from 0 dB to 2.4 dB when the reach increases from single-span to 10 spans. The advantage of system I over system III is maintained at 1.7 dB independent of the number of spans. However, Fig. 3(b) shows the optimal launch power versus number of spans. The optimum launch powers for non-compensated systems, systems I and III are constant. This is because both the linear and nonlinear noises increase linearly with the number of the spans that leads to the optimum power independent of the number of spans. However, for the dispersion compensated system II, the optimum launch power density decreases with number of spans due to the multi-span noise enhancement effect. Another interesting observation from Eq. (31) and Eq. (32) is that both the optimal Q factor and launch power has very week dependence on the overall system bandwidth: proportional to 1/3 power of logarithm of the overall bandwidth. It can be easily shown that for both systems I and II, the Q is decreased by only about 0.7 dB with the 10-fold increase of the overall system bandwidth from 400 to 4000 GHz whereas system II incurs a larger decrease of the Q factor of 0.84 dB with the same bandwidth increase.

3.3 Information spectral efficiency

3.4 Multi-span noise enhancement factor

4. Conclusion

In this paper, we have derived closed-form analytical expressions for nonlinear system performance for densely spaced CO-OFDM systems. The closed-form solutions include the results for the achievable Q factor, optimum launch power density, and nonlinear threshold of launch power density, and information spectral efficiency limit. These analytical results clearly identify their dependence on system parameters including fiber dispersion, number of spans, dispersion compensation ratio, and overall bandwidth. The closed-form solution is further substantiated by numerical simulations using distributed nonlinear Schrödinger equation.

Appendix A: Change of the integration range.

We first argue that we can move the integration boundary for f from [-B/2-f 1, B/2-f 1] to more tidy form of [-B/2, B/2]. This is reasonable simplification as the major contribution of the integral is when f is aroundfW. In essence, this fringe effect of subtracting f1 can be ignored when B >> fW. This corresponds to the ‘large bandwidth’ assumption in Eq. (18). Under the large bandwidth assumption, Eq. (15) can be rewritten as
INL=8γ2I3β22(2π)4INL1,         INL1=14B/2B/2B/2B/2η1(f,f1)η2(f,f1)df1dfη1(f,f1)=|sin(Nsf1f/(2fPA2))sin(f1f/(2fPA2))|2,       η2(f,f1)=1(f1f)2+fW4
(35)
where INLand INL1differs merely by a constant factor. We will focus on our simplifying effort on INL1. Using symmetry over zero for the integral variables f1 and f, Eq. (35) can be expressed as

INL1=0B/20B/2η1(f,f1)η2(f,f1)df1df
(36)

We now partition INL1into three regions of integration as follows:

INL1=A1A2+A3
(37)

where

A1=B0/2B/20η1η2df1df,     A2=B0/2B/2B/2η1η2df1df,     A3=0B0/20B/2η1η2df1dfB0=2fW2/B
(38)

We will show below that A1 is dominant over A2 and A3, and therefore the contribution of A2 and A3 can be omitted as a good approximation.

We first study the special case of Ns of 1. It is one of the most important cases as it corresponds to the system scenarios for which nonlinear noises originated from different spans are uncorrelated, and the overall noise is simply Ns times of the noise of one span. As we show later, an uncompensated fiber system is almost a perfect example of independent span noises. Substituting N s = 1 into Eq. (38), it can be easily shown that A1 can be completely integrated in a closed-form as

A1=π2fW2ln(B/B0)
(39)

We now carry out the integration for A2 over f1, we have

A2=B0/2B/21ffW2arctan(fW2fB/2)dfB0/2B/21ffW2fW2fB/2df       =1B/2B0/2B/21f2df=1B/21B0/2=2fW2
(40)

In Eq. (40), we have used the inequality of x>argtan(x) for x>0, and B>>B0, which can be considered as part of the ‘large bandwidth’ assumption. It can be seen that A2 is upper bounded by 2/fW2, the ratio between A1 /A2 is upper bounded by π4ln(B/B0). Under large bandwidth assumption of B>>fW, we consider this is a large number and thus A2 in general can be ignored. The upper bound for the term A3 can be found as follows

A3=0B00B/21(f1f)2+fW4df1df       <0B0/20B/21fW4df1df=1fW4(B0/2)B/2=12fW2
(41)

Therefore the ratio between A1/A3 is upper bounded by πln(B/B0). Consequently under the large bandwidth assumption, A1 is the dominant contribution to INL1, and can be used as an good approximation to INL1.

We now move to the multi-span scenario for which there is an additional factor of η1in the integrand of Eq. (38). We show that by introducing this factor, the conclusion that A1 being a dominate component in PNL1still stands. η1generates a interference pattern with a main lobe in the low frequency region. When considering the ratio between A1 and A2, the weight will shift toward A1 as A1 is integrated at the low frequency of f1, [0, B/2], and A2 is integrated at the high frequency of f1, [B/2, ]. Therefore the ratio of A1/A2 should be no less than π4ln(B/B0) we derived for η1= 1 for the single-span case, and subsequently A2 can be ignored to a good degree of approximation. Regarding the ratio of A1/A3, the integration variable ‘f’ is over low frequency region of [0, B 0/2] for ‘A3 whereas over [B 0/2, B/2] for A1. So the existence of the phase array pattern η1will shift weight toward A3. However, we will show the ratio of A1/A3 is still upper bounder by a factor in the order of πln(B/B0) to a good approximation. The phase array effects dependent on the phase difference between each span largely determined by fPA. When fPA>>fW, for instance, the dispersion is almost completely compensated, η1remains a constant over a large range frequency range. This is closed to the scenario for the single-span case for which we have shown A3 can be ignored. Therefore we only need to study the ratio of A1/A3 when fPA<<fW as this is the scenario for which the main lob shifts close to the low frequency, favoring A3. We would like to derive the ratio of A1/A3 whenfPA<<fW. η1can be also expressed as

η1=|1+ejθ+ej2θ+...++ej(N1)θ|2=N+2n=1N1(Nn)cos(nθ)                                                                     θ=ff1/fW2
(42)

Substituting Eq. (42) into Eq. (38), we have

A3=0B0/20B/2η11(f1f)2+fW4df1df1fW40B0/20B/2η1df1df     =1fW40B0/20B/2[N+2n=1NS1(NSn)cos(nff1/fPA2)]df1df       =Ns2fW2+21fW4n=1N1(NSn)0B0/20B/2cos(nff1/fPA2)df1df       =Ns2fW2+21fW4n=1N1(NSn)0B0/2fPA2nfsin(nfB/(2fPA2))df       =Ns2fW2+21fW4n=1N1(NSn)fPA2n0nfW24fPA21fsin(f)df
(43)

As the upper boundary of integration for variable f in Eq. (43), nfW2/(4fPA2) is much larger than 1 under the assumption offPA<<fW, and can be considered approximated as infinity. Using formula0sin(f)/fdf=π/2 and after some simple arrangement, Eq. (43) can be transformed into

       A3NsfW2+πfPA2fW4n=1N1(NSn)nfPA2         Ns2fW2+πfPA2(NslogNsNs+1)fW4Ns2fW2
(44)

In Eq. (44), we have used the assumption of fPA<< fW to justify omitting the second term. It can be also shown that using the closed-form developed for A1 in Appendix B, under assumption of fPA<< fW, A1 can be approximated as

A1=πNs2fW2ln(B/B0)
(45)

Combing Eq. (44) and Eq. (45), we conclude that A 1/A 3 is also upper bounded in the order of πln(B/B0). In summary, we conclude that A 1 can be used as an approximation for INL1 for a wide range of the system parameters. Unless specifically mentioned in this paper, we consider

INL=8γ2I2β22(2π)4INL1,           INL1=B0/2B/20η1(f,f1)η2(f,f1)df1df
(46)

Using the definition of iNL(f)in Eq. (19), we have

iNL(f)=8γ2I2β22(2π)4iNL1(f),   iNL1(f)=0η1(f,f1)η2(f,f1)df1
(47)

There are two other important factors that make Eq. (46) a very good approximation: First, in Eq. (37), not only are A 2 and A 3 small compared to A 1, but they also have opposite signs and their combined effects tend to cancel each other, which further improves the accuracy of using A 1 as an approximation to INL1. Second, the phase array effect will tend to deemphasize the contribution of A 2 and add more weight to A 3. However, because A 3 is integrated in the low frequency of f, in practical application, the low frequency noise including DC components can be estimated and removed. Therefore the lower integration boundary for f, B 0/2 in Eq. (46) can be also interpreted as that the low frequency noise from [0, B 0/2] is being removed due to the phase estimation. Therefore, to be complete, we redefine the B0 as

B0=max(2fW2/B,   2BPE)
(48)

where BPEis the phase estimation bandwidth which is equal to half of the subcarrier channel spacing Δfin CO-OFDM systems, or the phase locked loop (PLL) bandwidth in single-carrier systems. This is to accommodate the scenario for which the phase estimation bandwidth is high enough such that the lower integration boundary for f in Eq. (46) should be BPE instead of fW2/B.

Appendix B: Closed-form solution for FWM noise spectral density

In deriving close-form for INL1 we will repeatedly employ a useful formula of complex functional analysis as follows:

f(x)x2+a2dx=f(x)x2+a2dx=πaf(ia)
(49)

wheref(x)is an analytical function over the upper half of the complex plane. Substituting Eq. (42) into Eq. (47), we have

iNL1(f)=1f0[N+2n=1NS1(NSn)cos(nf2/fPA2)]1f22+fW4df2                   =12fRe{[N+2n=1NS1(NSn)exp(jnf2/fPA2)]1f22+fW4df2}
(50)

where ‘Re’ stands for operation of extracting the real component. We have exchanged variable f1f to f2 in Eq. (50). Applying formula of Eq. (49) into Eq. (50), we obtain

iNL1(f)=12f{Ns+2n=1NS1(NSn)exp(nfW2/fPA2)}                   =π2ffW2{2(Ns1+eNsfW2/fPA2NsefW2/fPA2)efW2/fPA2(efW2/fPA21)2+Ns}
(51)

Substituting the definitions for fWand fPAin Eq. (9) into Eq. (51), we have

iNL1(f)=2π3|β2|fα{2(Ns1+eNsαLζNseαLζ)eαLζ(eαLζ1)2+Ns}
(52)

Substituting Eq. (52) into Eq. (47), we arrive at the FWM noise spectral density

iNL(f)=γ2πI2α|β2|{2(Ns1+eNsαLζNseαLζ)eαLζ(eαLζ1)2+Ns}1f
(53)

References and links

1.

W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–589 (2006). [CrossRef]

2.

E. Yamada, A. Sano, H. Masuda, E. Yamazaki, T. Kobayashi, E. Yoshida, K. Yonenaga, Y. Miyamoto, K. Ishihara, Y. Takatori, T. Yamada, and H. Yamazaki, “1Tb/s (111Gb/s/ch × 10ch) no-guard-interval CO-OFDM transmission over 2100 km DSF,” Opto-Electronics Communications Conference/Australian Conference on Optical Fiber Technology, paper PDP6, Sydney, Australia (2008).

3.

Y. Ma, Q. Yang, Y. Tang, S. Chen, and W. Shieh, “1-Tb/s single-channel coherent optical OFDM transmission over 600-km SSMF fiber with subwavelength bandwidth access,” Opt. Express 17, 9421–9427 (2009). [CrossRef] [PubMed]

4.

S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “Transmission of a 1.2-Tb/s 24-carrier no-guard-interval coherent OFDM superchannel over 7200-km of ultra-large-area fiber,” Eur. Conf. Optical Commun.,Vienna, Austria (2009), post-deadline Paper PD2.6.

5.

G. Goldfarb, G. F. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. 19(24), 2015–2017 (2007). [CrossRef]

6.

R. Dischler, and F. Buchali, “Transmission of 1.2 Tb/s continuous waveband PDM-OFDM-FDM signal with spectral efficiency of 3.3 bit/s/Hz over 400 km of SSMF,” Optical Fiber Communication Conference, paper PDP C2, San Diego, USA (2009).

7.

H. Takahashi, A. Al Amin, S. L. Jansen, I. Morita, and H. Tanaka, “Highly spectrally efficient DWDM transmission at 7.0 b/s/Hz using 8 x 65.1-Gb/s coherent PDM-OFDM,” J. Lightwave Technol. 28(4), 406–414 (2010). [CrossRef]

8.

A. J. Lowery, S. Wang, and M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Opt. Express 15(20), 13282–13287 (2007). [CrossRef] [PubMed]

9.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

10.

W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16(2), 841–859 (2008). [CrossRef] [PubMed]

11.

X. Liu, F. Buchali, and R. W. Tkach, “Improving the nonlinear tolerance of polarization-division-multiplexed CO-OFDM in long-haul fiber transmission,” J. Lightwave Technol. 27(16), 3632–3640 (2009). [CrossRef]

12.

M. Mayrock and H. Haunstein, “Monitoring of linear and nonlinear signal distortion in coherent optical OFDM transmission,” J. Lightwave Technol. 27(16), 3560–3566 (2009). [CrossRef]

13.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001). [CrossRef] [PubMed]

14.

K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett. 17(11), 801–803 (1992). [CrossRef] [PubMed]

15.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13(5), 841–849 (1995). [CrossRef]

16.

X. Liu and F. Buchali, “Intra-symbol frequency-domain averaging based channel estimation for coherent optical OFDM,” Opt. Express 16(26), 21944–21957 (2008). [CrossRef] [PubMed]

17.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948).

OCIS Codes
(060.1660) Fiber optics and optical communications : Coherent communications
(060.2330) Fiber optics and optical communications : Fiber optics communications

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: June 10, 2010
Revised Manuscript: August 6, 2010
Manuscript Accepted: August 7, 2010
Published: August 23, 2010

Citation
Xi Chen and William Shieh, "Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems," Opt. Express 18, 19039-19054 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-18-19039


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References

  1. W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–589 (2006). [CrossRef]
  2. E. Yamada, A. Sano, H. Masuda, E. Yamazaki, T. Kobayashi, E. Yoshida, K. Yonenaga, Y. Miyamoto, K. Ishihara, Y. Takatori, T. Yamada, and H. Yamazaki, “1Tb/s (111Gb/s/ch × 10ch) no-guard-interval CO-OFDM transmission over 2100 km DSF,” Opto-Electronics Communications Conference/Australian Conference on Optical Fiber Technology, paper PDP6, Sydney, Australia (2008).
  3. Y. Ma, Q. Yang, Y. Tang, S. Chen, and W. Shieh, “1-Tb/s single-channel coherent optical OFDM transmission over 600-km SSMF fiber with subwavelength bandwidth access,” Opt. Express 17, 9421–9427 (2009). [CrossRef] [PubMed]
  4. S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “Transmission of a 1.2-Tb/s 24-carrier no-guard-interval coherent OFDM superchannel over 7200-km of ultra-large-area fiber,” Eur. Conf. Optical Commun.,Vienna, Austria (2009), post-deadline Paper PD2.6.
  5. G. Goldfarb, G. F. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. 19(24), 2015–2017 (2007). [CrossRef]
  6. R. Dischler, and F. Buchali, “Transmission of 1.2 Tb/s continuous waveband PDM-OFDM-FDM signal with spectral efficiency of 3.3 bit/s/Hz over 400 km of SSMF,” Optical Fiber Communication Conference, paper PDP C2, San Diego, USA (2009).
  7. H. Takahashi, A. Al Amin, S. L. Jansen, I. Morita, and H. Tanaka, “Highly spectrally efficient DWDM transmission at 7.0 b/s/Hz using 8 x 65.1-Gb/s coherent PDM-OFDM,” J. Lightwave Technol. 28(4), 406–414 (2010). [CrossRef]
  8. A. J. Lowery, S. Wang, and M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Opt. Express 15(20), 13282–13287 (2007). [CrossRef] [PubMed]
  9. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]
  10. W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16(2), 841–859 (2008). [CrossRef] [PubMed]
  11. X. Liu, F. Buchali, and R. W. Tkach, “Improving the nonlinear tolerance of polarization-division-multiplexed CO-OFDM in long-haul fiber transmission,” J. Lightwave Technol. 27(16), 3632–3640 (2009). [CrossRef]
  12. M. Mayrock and H. Haunstein, “Monitoring of linear and nonlinear signal distortion in coherent optical OFDM transmission,” J. Lightwave Technol. 27(16), 3560–3566 (2009). [CrossRef]
  13. P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001). [CrossRef] [PubMed]
  14. K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett. 17(11), 801–803 (1992). [CrossRef] [PubMed]
  15. R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13(5), 841–849 (1995). [CrossRef]
  16. X. Liu and F. Buchali, “Intra-symbol frequency-domain averaging based channel estimation for coherent optical OFDM,” Opt. Express 16(26), 21944–21957 (2008). [CrossRef] [PubMed]
  17. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948).

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