國 立 交 通 大 學
電信工程學系
博 士 論 文
超密波長分工及次載波有線電視系統的
基本限制
Fundamental Transmission Limitations on both
Ultra-Dense WDM and
Subcarrier Multiplexed CATV Systems
研究生：吳明佳
指導教授：尉應時
超密波長分工及次載波有線電視系統的基本限制
Fundamental Transmission Limitations on both
Ultra-Dense WDM and Subcarrier Multiplexed CATV Systems
研 究 生：吳明佳 Student：Ming-Chia Wu
指導教授：尉應時 Advisor：Winston I. Way
國 立 交 通 大 學
電 信 工 程 學 系
博 士 論 文
A Dissertation
Submitted to Department of Communications Engineering
College of Electrical Engineering and Computer Science
National Chiao Tung University
in partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
in
Communications Engineering
July 2004
Hsinchu, Taiwan, Republic of China
超密波長分工及次載波有線電視系統的基本限制
學生：吳明佳 指導教授：尉應時
國立交通大學電信工程學系
摘
要
本論文研究超密波長分工系統及次載波有線電視系統的基本限制. 主要分為四個部份:
在第一部分中, 我們分析了 2.5 及 10 Gb/s 的超密波長分工系統在不同單模光纖中的傳
輸限制並分析了週期性放大的系統在 C 波帶的各項基本限制及最佳色散補償比例. 第
二部分, 我們發現在 1550nm 外調式類比有線電視系統中, 自發性及外加相位調變必須
同時考慮才能精準地預測二階拍差, 特別是當外加相位調變的調變深度或頻率高的時侯.
這個結果對長距離 1550nm 外調式光發射機的設計有很重要的影響. 第三部分, 我們證
明了用多個 CW 載波來模擬測試 M-QAM 次載波有線電視系統的準確度是在量測誤差的範
圍內. 最後, 我們架設了世界第一個類比的光纖循環迴圈並創造了 78 個頻道 64-QAM
1550nm 次載波系統的最長傳輸記錄-740km.
Fundamental Transmission Limitations on both
Ultra-Dense WDM and Subcarrier Multiplexed CATV Systems
Student：Ming-Chia Wu
Advisors：Dr.Winston I. Way
Department of Communications Engineering
National Chiao Tung University
ABSTRACT
This thesis investigates fiber nonlinearity limitation in both ultra dense wavelength division
multiplexing (U-DWDM) digital systems and subcarrier multiplexing analog lightwave
transmission systems. The main research results are organized into four parts.
First of all, transmission performance of ultra-dense 2.5 and 10 Gbps NRZ IM/DD
wavelength division multiplexing systems in various single-mode fibers is investigated.
Fundamental limiting factors and their remedies by using optimum dispersion compensation
for periodically amplified systems in C-band are presented.
Second, we found that the combined effects of self- and external-phase modulations must be
considered in order to precisely predict the CSO distortions in a long-distance 1550 nm
externally-modulated AM-CATV system, especially when the applied phase modulation
index and modulating tone frequency to the integrated phase modulator are high. This result
has important implications to the optimum design of 1550 nm transmitter for long-distance
AM- and QAM-CATV systems.
Third, the validity of using multiple CW tones as the signal source to test the linearity of a
multichannel M-ary quadrature-amplitude-modulation (M-QAM) subcarrier multiplexed
(SCM) lightwave system was investigated. We consider the following representative optical
fiber system nonlinearities: (1) laser clipping, and (2) the combined effect of laser frequency
chirp and fiber dispersion. The results show that, if all orders of nonlinear distortions (NLDs)
in a signal bandwidth are included in the total NLD power, the error caused by replacing
M-QAM signals with CW tones can be within measurement uncertainty.
Finally, a long-distance 1550 nm subcarrier multiplexed lightwave trunk system which
transported 78 channels of 64-QAM signals was demonstrated in a recirculating loop
experiment. Each channel can achieve a carrier-to-(noise + nonlinear distortion) ratio of 30
dB after 740 km transmission through conventional single-mode fiber without dispersion
compensation.
誌 謝
時光飛逝, 我在交大已過了十幾個年頭, 這一路走來要感謝的人很多, 首先感謝系
上的師長及口試委員給我很多意見及指導, 也非常感謝我的老師尉應時及其家人, 在國
內外都關心我的生活, 讓我有機會學習社會的經驗．感謝實驗室的學長及學弟帶給我許
多歡樂及幫助, 在此也感謝系上行政人員給我相當大幫助． 最後, 將此論文獻給我的
家人, 因為有你們的支持才能讓我無後顧之憂的完成學業．
Index
1. Introduction ...1
2. Fiber Nonlinearity Limitations in Ultra-Dense WDM Systems ...4
2.1. Fundamental Limiting Factors in U-DWDM Systems...4
2.1.1 Four Wave Mixing ...10
2.1.2 Cross Phase Modulation ...13
2.1.3 Self Phase Modulation and Residual Linear Dispersion...15
2.2. Overall System Limitations...16
2.3. Discussions...29
2.4. Conclusions ...30
3. CSO Distortions due to the Combined Effects of Self- and External-Phase Modulations
in Long-distance 1550 nm AM-CATV Systems ...33
3.1. Experimental setup:...33
3.2. Experimental, numerical, and analytical results...34
3.3. Conclusion...37
4. On the Validity of Using CW Tones to Test the Linearity of Multichannel M-QAM
Subcarrier Multiplexed Lightwave Systems ...38
4.1. Analysis and numerical simulation ...38
4.2. Experiment ...40
4.3. Discussion ...42
4.4. Conclusions ...43
5. 740 km Transmission of 78-Channel 64-QAM Signals (2.34 Gb/s) Without Dispersion
Compensation by Using a Recirculating Loop ...44
5.1. Experimental setup ...44
5.2. Results and discussion...46
5.3. Conclusion...47
Table Index
Table 1 Fiber parameter assumptions...5 Table 2 Summary of optimum launch power per channel under different system conditions ...31 Table 3 Summary of maximum transmission distance under different system conditions ...32
Figure Index
Fig. 1 Illustrations of typical NLDs generated from (a) CW tones and (b) QAM. Signal level were normalized to 0 dB for comparison...3 Fig. 2 (a) A general U-DWDM system model; (b) A periodically amplified U-DWDM system model
with post dispersion compensation in every amplifier stage...6 Fig. 3 Cumulative waveform amplitude distribution of simulated nonlinear interference/distortion
obtained at an optimum sampling point, for the central 4 channels of a U-DWDM system. The result is a distribution of 50 independent simulations. Dashed curves are Gaussian Approximation. ...8 Fig. 4 FWM and ASE limited maximum fiber input power per channel of 2.5Gbps/6.25GHz systems as
a function of amplifier stages for different DCR’s in (a) SMF and (b) NZDSF. Assume 40km per amplifier stage. Dotted curve is ASE noise-limited minimum input power. Symbols are simulation results. ...17 Fig. 5 Q2 as a function of average launched optical power of a 2.5Gbps/6.25GHz system after 4640km
NZDSF, which corresponds to the maximum transmission point of Fig. 6(a). Solid curve and circles are calculated and simulated received Q2, respectively. Dotted line represents the
optical nonlinearity-limited Q2. Dashed line is ASE limited Q2. Solid triangles are the
combined results of ASE and fiber nonlinearity...19 Fig. 6 (a) Calculated and simulated ASE-, fiber linear dispersion- and nonlinearity-limited maximum
transmission distance of 2.5Gbps/6.25GHz-spaced systems (Q2≧15.6dB) as a function of
in-line DCR in NZDSF. Results for both 40 and 80 km fiber spans are shown. Dashed and dotted curves were calculated based on D = 2 and 6 ps/nm/km, respectively, while solid curves represent the worse of the two. ● and ▲ are simulation results. In-line DCFs are used for dispersion compensation and the input power into DCF is 3dB lower than that into transmission fiber. A tunable PDC (up to -2,000 ps/nm) was used to optimize the individual channel performance. The dashed-dotted curve is the linear dispersion limitation at D=6 ps/nm/km. (b)&(c) Calculated individual Q2 for 40km fiber span...20 Fig. 7 (a) Calculated and simulated ASE-, fiber linear dispersion- and nonlinearity-limited maximum
transmission distance of 2.5Gbps/6.25GHz-spaced systems (Q2≧15.6dB) as a function of
in-line DCR in SMF. Results for both 40 and 80 km fiber spans are shown. ● and ▲ are simulation results. In-line DCFs are used for dispersion compensation and the input power into DCF is 3dB lower than that into transmission fiber. A tunable PDC (up to -2,000 ps/nm)
was used to optimize the individual channel performance. (b) Calculated individual Q2 for
40km fiber span...22 Fig. 8 Optimum launched power per channel as a function of DCR in 2.5Gbps/6.25GHz systems. The
symbols correspond to the maximum transmission distances at different DCR in Fig.6(a) and Fig.7(a)...23 Fig. 9 Fiber input power per channel as a function of amplifier stages for different DCRs in a 10Gbps/
25 GHz system, with 40km per amplifier stage. Curves are calculation results, and symbols are numerical simulation results. (a) NZDSF: ▲: DCR=100%; ●: DCR= 90%; (b) SMF: ▲: DCR=100%; ●: DCR= 99% ...24 Fig. 10 Calculated and simulated Eye-opening penalty due to SPM and linear dispersion as a function
of PDC. Assume a single channel 10Gb/sec system over 3600km of NZDSF fiber (D=6ps/nm/km). DCR=90%. (a) Solid and dashed curves are calculated eye-opening penalties due to SPM+linear dispersion and linear dispersion only, respectively. ▲ and △ are simulated eye-opening penalties due to SPM+ linear dispersion and linear dispersion only, respectively. (b) Eye diagrams obtained via calculations ((i) and (iii)) and simulations ((ii) and (iv))...25 Fig. 11 (a) Calculated and simulated fiber linear-dispersion- and fiber nonlinearity-limited maximum
transmission distance of a 10Gbps/25GHz system (Q2=15.6dB) as a function of in-line DCR in
NZDSF. Solid curves are analytical results for 40 and 80km fiber spans with dashed and dotted curves for D = 2 and 6 ps/nm/km, respectively. ● and ▲ are simulation results with 40 and 80km fiber span, respectively. In-line DCFs are used for dispersion compensation and the input power into DCF is 3dB lower than that into transmission fiber. A tunable PDC (up to -2,000 ps/nm) was used to optimize the individual channel performance. ...26 Fig. 12 (a) Calculated and simulated fiber linear dispersion- and nonlinearity-limited maximum
transmission distance of 10Gbps/25GHz systems (Q2≧15.6dB) as a function of in-line DCR in
SMF. Solid and dashed curves are analytical results for 40 and 80km, respectively. ● and ▲ are simulation results with 40 and 80km fiber spans, respectively. In-line DCFs are used for dispersion compensation and the input power into DCF is 3dB lower than that into transmission fiber. A tunable PDC (up to -2,000 ps/nm) was used to optimum the individual channel performance. (b)&(c) Calculated corresponding individual Q2 of 40&80 km fiber
span, respectively. ...27 Fig. 13 Calculated optimum PDC as a function of dispersion compensation ratio at the optimum
Fig. 14 Optimum launched power per channel as a function of DCR, corresponding to the maximum transmission distances at the various DCR’s in Fig.10(a) and Fig.11(a) ...28 Fig. 15 Q2 as a function of transmission distance when FWM and ASE dominate. Assume a
2.5Gbps/6.25GHz system in SMF with span=80km and DCR=91%. Curves are analytical results for Q2
FWM (dashed), Q2ASE (Dash-dotted) and Q2FWM+ASE (solid). ▲ and ○ are
simulation results with split-step FFT method and with commercial simulation software- VPI-TransmissionMaker, respectively. ...30 Fig. 16 Experimental Setup...33 Fig. 17 CSO @ channel 78 as a function of launched optical power into a repeaterless AM-CATV
system with three different transmission distances: 64, 74, and 87 km. Numerical results are for CSOs caused by both SPM and EPM effects, while analytical results are based Eq.(1). Key parameters include: OMI/ch = 2.8%, number of AM channels = 78, λ0 = 1551.7 nm, D =
17ps/nm/km, n2 = 2×10-20 m2/W, fiber loss = 0.2 dB/km, and Aeff = 90µm2. β’s for the three
tones at 1.9, 3.8, 5.7 GHz are 3.9, 3.9, and 1.3, respectively. Soild lines, open symbols, and solid symbols represent analytical, numerical, and measured results, respectively. ...35 Fig. 18 Measured, numerically calculated, and analytical CSOs @ channel 78 as a function of the total
fiber length in an equal-span, multi-stage-repeatered AM-CATV system. Inter-stage fiber span is 60 km. The launched optical power from each EDFA (Pout,i, i ≥1) was 12 dBm. (OMI/ch =
3%, λ0 = 1561.1 nm. Single tone phase modulation at 1.9GHz. Other parameters are the same as
those given in Fig. 16)...36 Fig. 19 Numerically calculated and analytical CSOs @ channel 78 as a function of the total fiber length
in an equal-span, multi-stage repeatered AM-CATV system. (β = 2.5 for all PM modulating tone frequencies. Other parameters are the same as those given in Fig. 18)...37 Fig. 20 Spectral analysis (solid and dotted lines) and numerical (solid and open symbols) results of LD
clipping induced SNLDs. 74 channels of CW tones or 64-QAM signals were used. Number of averages were 1000 and 150 for CW and QAM, respectively. RMS OMI/ch=3.9% ...39 Fig. 21 Analytical (solid lines) and numerical results (open and solid symbols) of signal to second order
nonlinear distortions ratio due to laser frequency chirping and fiber dispersion. Three different fiber lengths were considered. Solid lines are calculated results based on [40]. Solid and open symbols are resulted from 74-channel CW tones and 64-QAM, respectively...40 Fig. 22 Measured SNLD results for laser clipping. Results for 16 channels of CW tones (solid symbols)
or 64-QAMs (open symbols) ranging from 258 to 354 MHz are shown. Solid lines are calculated results based on spectral analysis...41
Fig. 23 Measure results of SNLD due to fiber dispersion and laser chirping for both 16-channel CW tones (solid symbols) and 64-QAM (open symbols). Solid lines are calculated results from [40]. Total laser frequency chirp is 3.6 GHz, and the fiber dispersion is 17 ps/nm/km...42 Fig. 24 Experimental setup of an SCM recirculating loop...44 Fig. 25 Captured 78-channel CW tones (a) and 64-QAM signals (b) after 0, 4, 8 times of recirculating
loops. (c) is the measured 64-QAM signal constellation diagram of channel 78 after 7 times of the recirculating loop (714 km)...46 Fig. 26 (a) Measured CNR of Ch.3 (■), Ch.39(◆) and Ch.78(▲) in a 6MHz bandwidth, and the worst
case CSO @Ch.78(○), for the case of transmitting 78 CW tones. (b) Measured C/(NLD+N) of Ch.3 (□) and Ch.78(△) when transmitting 78 CW tones, and the measured SNR of Ch.3 (■) and Ch.78(▲) when transmitting 78 64-QAM signals. Solid lines in (a) and (b) are the calculated results. Launched optical power was +6 dBm. ...47
1. Introduction
This thesis investigates fundamental transmission limitations in both ultra dense wavelength division multiplexing (U-DWDM) digital systems and subcarrier multiplexing lightwave systems. The main research results are organized in four parts as follows.
First of all, transmission performance of ultra-dense 2.5 and 10 Gbps NRZ IM/DD wavelength division multiplexing systems in various single-mode fibers is investigated. Fundamental limiting factors and their remedies by using optimum dispersion compensation for periodically amplified systems in C-band are presented.
In order to increase the transmission capacity of a DWDM optical system, one can either increase the transmission data rate per wavelength, or increase the number of wavelengths while keeping proper transmission granularity. The first approach can be illustrated by the recent increase in data rate per wavelength from 2.5 Gbps to 10 and 40 Gbps. The second approach is to significantly increase the number of wavelengths in a fixed optical spectrum (e.g., C-band) by decreasing the spacing between neighboring wavelengths. By using this approach, capacity increase can be achieved without resorting to high-speed (e.g., > 40 Gbps) electronics, while keeping compatibility with existing 2.5/10 Gbps SONET/SDH equipment. Along this line, the focus of this paper is on ultra-dense wavelength division multiplexing (U-DWDM) transmission systems. Examples of U-DWDM systems include 25 GHz-spaced 10 Gbps and 6.25 GHz-spaced 2.5 Gbps transmission systems.
It should be noted that, even though a few U-DWDM system experiments have been carried out recently [1-5], the fundamental limiting factors and their remedies in such systems remain unclear. It is obvious that there are different transmission issues to be dealt with in the above-mentioned two distinct approaches. When the transmitting data rate is higher than 40 Gbps, severe chromatic dispersion and polarization mode dispersion problems will have to be resolved even before dealing with optical nononlinearity-induced penalties. On the other hand, U-DWDM systems intuitively should have optical nonlinearity-induced system limitations such as four-wave mixing (FWM) and cross-phase modulation (XPM) penalties. The first part of this thesis considers various nonlinear distortions/interferences to determine the fiber nonlinearity limited maximum transmission distances in U-DWDM systems. Optimum launched power and dispersion compensation ratio (DCR), and the dominant optical nonlinearities in different systems are
also discussed.
In Section 2.1, we provide an overview of the main nonlinear distortions/interferences in U-DWDM systems. In Section 2.2, we analytically calculate and numerically simulate the capacity and distance limitations of U-DWDM systems. Discussions and conclusions are provided in Sections 2.3 and 2.4, respectively.
Second, we found that the combined effects of self- and external-phase modulations must be considered in order to precisely predict the CSO distortions in a long-distance 1550 nm externally-modulated AM-CATV system, especially when the applied phase modulation index and modulating tone frequency to the integrated phase modulator are high. This result has important implications to the optimum design of 1550 nm transmitter for long-distance AM- and QAM-CATV systems.
Recently, there has been intense interests in long-distance 1550 nm external-modulation AM-CATV systems based on conventional single-mode fibers [6-13]. When the optical power launched into these long-distance systems is below the stimulated Brillouin scattering (SBS) threshold, it has been found that the transmission distance is mainly limited by the fiber dispersion-induced composite second order (CSO) distortions. These CSO distortions in turn were believed to be caused by self-phase modulation (SPM) [8-10], and by the residual intensity modulation from an imperfect phase modulator [11]. However, analyses developed for either of the above mechanisms were not accurate in predicting the resultant CSO values in long-distance AM-CATV transmission systems [10-12].
It is noted that all commercially available 1550 nm CATV transmitters have an integrated phase modulator and a Mach-Zehnder interferometer (MZI) modulator. In order to increase the SBS threshold, a ~2 GHz tone (or a few tones > ~2GHz) is usually applied to the phase modulator, and a high SBS threshold can be obtained by using a high phase modulation (PM) index (β) [13]. However, the resultant high launched optical power into the system can unavoidably increase the SPM effects. In addition, when the applied PM index β or the PM modulating tone frequency are high, or when the transmission distance is long, both SPM and external phase modulation (EPM) can be mixed with intensity modulation in a nonlinear dispersive optical fiber system. The resultant CSO distortions due to their combined effects, however, have not yet been thoroughly investigated. Section 3 of this thesis presents both experimental and numerical results on the above subject.
Third, the validity of using multiple CW tones as the signal source to test the linearity of a multichannel M-ary quadrature-amplitude-modulation (M-QAM) subcarrier multiplexed (SCM) lightwave system was investigated. We consider the following representative optical fiber system nonlinearities: (1) laser clipping, and (2) the combined effect of laser frequency chirp and fiber dispersion. The results show that, if all orders of nonlinear distortions (NLDs) in a signal bandwidth are included in the total NLD power, the error caused by replacing M-QAM signals with CW tones can be within measurement uncertainty.
It is believed that SCM lightwave systems can be used to transport multichannel M-QAM signals to provide broadband digital services such as Internet access, digital video, IP telephony, etc. [14,15]. In the past, the linearity characteristics of such systems were investigated by using multiple CW tones [16-18], mainly because of the practical difficulty in generating multiple distinct M-QAM channels. However, the spectral distributions of nonlinear distortions (NLDs) caused by multiple CW tones and multiple wideband M-QAM signals are quite different. In the former case, NLDs consist of various distinct beats such as composite second orders (CSOs) and composite triple beats (CTB’s). In the latter case, NLDs are spread over several channels and are like white noise. Fig.1 (a) and (b) illustrate the spectra of the two cases. To our knowledge, there is no report discussing the validity of using CW tones to replace the actual M-QAM signals. In section 4 of this thesis, we study this validity by performing spectral analysis, numerical simulation and experimental verification. We chose two representative optical fiber nonlinearities in SCM lightwave systems to study: (1) laser clipping, and (2) the combined effect of laser chirping and optical fiber dispersion.
Fig. 1 Illustrations of typical NLDs generated from (a) CW tones and (b) QAM. Signal level were normalized to 0 dB for comparison.
Finally, a long-distance 1550 nm subcarrier multiplexed lightwave trunk system which transported 78 channels of 64-QAM signals was demonstrated in a recirculating loop experiment. Each channel can
achieve a carrier-to-(noise + nonlinear distortion) ratio of 30 dB after 740 km transmission through conventional single-mode fiber without dispersion compensation.
A subcarrier multiplexed (SCM) lightwave system transporting multi-channel M-ary quadrature-amplitude-modulation (M-QAM) signals can have transmission features such as high system capacity and long transmission distance [19-20]. This is due to the fact that the carrier-to-noise ratio (CNR) and carrier-to-nonlinear distortion ratio (CNLD) requirements of M-QAM signals are lower than those of AM-VSB signals. In addition, M-QAM signals have a high spectral efficiency, which makes multi-gigabit/sec data transmission feasible when using conventional CATV optical transceivers. Therefore, multi-channel M-QAM SCM trunk systems have a great potential to be used for interconnecting CATV headends and delivering various digital communication services.
It was found that the fundamental M-QAM system capacity of either a laser diode- or a linearized external modulator-based transmitter could be as high as tens of gigabit/sec [22, 23]. However, the transmission distances of all reported M-QAM SCM systems are still rather limited. In section 5 of this thesis, we experimentally demonstrated that the transmission distance of an M-QAM external modulation SCM system carrying an equivalent data capacity of 2.34 Gb/s could exceed 740 km. In addition, for the first time, an optical fiber recirculating loop was implemented in an SCM system experiment.
2. Fiber Nonlinearity Limitations in Ultra-Dense WDM Systems
2.1. Fundamental Limiting Factors in U-DWDM Systems
The three fundamental limiting factors in U-DWDM systems include (1) various random noise terms, (2) fiber chromatic dispersion-induced inter-symbol interference (ISI), and (3) optical-nonlinearity-induced distortion and interference. All three factors are common to conventional DWDM systems, although the third factor can be unique in the case of extremely close channel spacing.
We assume a multi-segmented, optically amplified U-DWDM system as shown in Fig. 2(a) to derive the general forms of fiber nonlinearity induced interferences/distortions. The transmitter side contains M external-modulated light sources at wavelength
λ
i , where i=1, 2,..., M. The M wavelengths are Non-Return to ZERO (NRZ) modulated with independent data pattern and are multiplexed and demultiplexed by either a pair of ideal multiplexer and demultiplexer, or by a pair of broadband couplers(with an ideal filter in each receiver). A tunable post dispersion compensator (PDC), which can be tuned up to –2000 ps/nm, is located right before each optical receiver to optimize the received signal performance of each individual channel. The nth segment contains a span L(n) of fiber and an erbium-doped optical
amplifier with a noise figure of 5dB and a power gain g( )n =exp(α( ) ( )nLn) to compensate the fiber loss. The fiber can be either SMF, or NZDSF, or dispersion compensation fibers (DCFs) used for broadband dispersion compensation. Fiber parameters assumed for analysis and numerical simulations throughout this paper are summarized in Table 1. When dispersion compensation fibers are used in a transmission link, their additional loss must be compensated by additional stages of optical amplifiers. In this case, we use a system configuration shown in Fig.2(b), which can be considered a special case of Fig.2(a). Note that in Fig. 2(b), each amplifier stage comprises a span of SMF (or NZDSF), an optical amplifier to compensate the fiber loss, a span of DCF whose span length is determined by the designed DCR, and another optical amplifier to compensate for the DCF loss. Therefore, it is equivalent to having a total of 2N stages in the
general system model shown in Fig. 2(a). We will use this periodically amplified and
dispersion-compensated system model to analyze the optimum design of U-DWDM systems in Section III.
Fig. 2 (a) A general U-DWDM system model; (b) A periodically amplified U-DWDM system model with post dispersion compensation in every amplifier stage.
Throughout this paper, we will use Q2 = 15.6 dB (which gives BER=10−9under the assumption of Gaussian distributed noise) as the minimum system performance requirement. Q2 follows the conventional Q2
definition and is given by
2 1 0 10 1 0 ( ) 20 log m m Q dB σ σ − = + (1)
where m and i σi are the mean and standard deviation values for mark (i=1) and space (i=0), respectively.
encoders/decoders are added. It should be noted that although many published analytical results showed that nonlinear distortions or interference are usually non-Gaussian distributed, they did show that under small optical nonlinearity penalties, Gaussian approximation can serve as an upper bound and can be viewed as a good approximation [24-25]. As will be seen later in our analysis, fiber nonlinearity-induced distortions/interference is assumed to be of the same order of magnitude as amplified spontaneous emission (ASE) noise, and at Q2 = 15.6 dB it can be considered small enough to be approximated as random noise just
as ASE noise. Note that optical-signal-to-noise ratio (OSNR) [26-27] is not used as a performance index in our study. This is because XPM and self-phase-modulation (SPM) are not measurable in optical domain unless they are converted to electrical signal at a photo-detector.
The nonlinear interferences/distortions under consideration typically occur at mark, and thus
σ
12>>
σ
02. The standard deviationσ
1 in (1) is modeled as2 1 , 2 1 , 2 1
σ
ASEσ
NLσ
= + (2)where
σ
ASE2 ,1 is the noise variance due to optical amplifiers, and is dominated by signal-spontaneous emission beat noise at the presence of mark.σ
NL2 ,1 =σ
FWM2 ,1+σ
XPM2 ,1+σ
SPM2 ,1 is the interference/distortion variance due to fiber nonlinearities, including FWM, XPM and SPM. Note that although SPM induces deterministic waveform distortion, we approximate it as a normal variance termσ
SPM2 ,1because (1) we want to compare the effect of SPM with that of FWM and XPM across various DCR, distances, fibers, and data rates by using the same parameter, and (2) in cases where SPM distortion has the same order of magnitude as FWM/XPM interference, it is difficult to separate deterministic distortion from random interference in numerical simulations. The range of validity for this approximation will be shown in Fig.3.Fig. 3 Cumulative waveform amplitude distribution of simulated nonlinear interference/distortion obtained at an optimum sampling point, for the central 4 channels of a U-DWDM system. The result is a distribution of 50 independent simulations. Dashed curves are Gaussian Approximation.
(a) (FWM-dominant) 2.5Gbps/6.25GHz system after 5400km SMF transmission, corresponding to the 40km-span, DCR=93% point in Fig. 7(a). 51,200 symbols, Q2=21.5dB.
(b) (SPM, XPM-dominant) 10Gbps/25GHz system after 4,000km NZDSF (D=6psec/nm/km), corresponding to the 40km-span, DCR=90% point in Fig. 11(a). 204,800 symbols, Q2=21dB.
(c) (SPM-dominant) Comparison between Gaussian approximation and the true distribution. The true BER’s (solid curves) are obtained by using the method given in Appendix.
Note that space level impairment is dominated by deterministic distortions due to linear dispersion. Spontaneous-spontaneous emission beat noise and nonlinearity are negligible at space level. Space-level deterministic distortion is taken into account in our calculations and simulations by replacing m0 in (1) with
the highest distorted space amplitude. Therefore, the effect of liner dispersion-induced ISI on (1) is to effectively introduce additional eye-opening penalty.
To verify the validity of treating optical nonlinearities as random noise in (2), under the conditions of a relatively large Q2 and the fact that the variance of these impairments are smaller than that of ASE, Fig.3
shows the amplitude distributions of mark of simulation data at the optimum decision point of received eye diagrams. Fig. 3(a) shows the amplitude distribution in a 40-channel 2.5Gbps/6.25GHz system transmitted over 5400km SMF with DCR=93% (corresponding to the point shown in Fig. 7(a)). The result is a distribution of 51,200 symbols, which were obtained from the central 4 channels (256 symbols per channel) and 50 independent simulations. The average Q2 is 21.5dB and is dominated by FWM. Dashed curves are
Gaussian approximation with the same Q2. We can see that the FWM amplitude distribution matches very
well with the Gaussian approximation. This is because the dominant interference is contributed from many different independent channels via FWM. Because of finite U-DWDM channels and simulation symbols, the cumulative distribution of the simulated waveform (excluding ASE noise) deviates away from Gaussian approximation at the tail. However, when ASE noise is added, the true BER (solid curve, the cumulative distribution calculated based on a method detailed in Appendix A [25, 28]) and BER with Gaussian approximation (dashed curve) match each other extremely well. Fig. 3(b) shows the amplitude distribution in a 40-channel 10Gbps/25GHz system transmitted over 4000km NZDSF with 6 ps/nm/km fiber dispersion and DCR=93% (corresponding to the maximum transmission distance in Fig. 11(a)). The result is a distribution of 204,800 symbols, which were obtained from the central 4 channels (1,024 symbols per channel) and 50 independent simulations. The average Q2 is 21dB and is dominated by SPM and XPM (see
Fig.11(b)). Although a slight deviation from Gaussian distribution is observed, the distribution can be brought closer to Gaussian when ASE noise is added (see the cumulative distribution (BER) curve with ASE noise included). Noise and nonlinearities on space level are neglected and not shown in the above figures. Fig. 3(c) shows the amplitude distribution when SPM-induced waveform distortion is the dominant impairment. We checked an extreme case, in which the mark level is split into 3 levels due to SPM (see the eye diagram shown in the inset of Fig.3(c)). We assume the three levels are binomially distributed with probabilities of 1/4, 1/2, and 1/4, representing the probabilities of bit sequences “00”, “10” or ”01”, and “11”, respectively. Because SPM results in deterministic waveform distortion, they cannot be approximated as Gaussian noise. However, when ASE noise with the same order of magnitude is added, the situation is different. The true BER (solid curves) are obtained with a method detailed in Appendix A. The Gaussian approximation by treating SPM as a normal variance in (2) is shown as dotted curve. Three different values of Q2
SPM, 18, 20 and 25dB are shown in the figure. The corresponding Q2ASE are chosen so that the
combined Q2
SPM+ASE = 15.6dB. We can see that, although a large deviation between Gaussian
approximation and the true BER occurs when Q2
SPM <Q2ASE,Gaussian approximation matches very well
with the corresponding true BER when Q2
Q2
SPM > 25dB is generally satisfied in our calculations/simulations, especially in the region near the
optimum DCR.
In the followings, we will concentrate on analytical tools for each individual fiber-nonlinearity-induced interference or distortion, and linear-dispersion induced ISI. In Section III, we will add amplifier noise in the overall system performance evaluation.
2.1.1 Four Wave Mixing
In an amplified and dispersion compensated DWDM system, FWM terms generated at every amplifier stage are added according to their phase relations. Starting from the coupled nonlinear Schrödinger’s equations (NLSE) with three CW wavelengths located in fi, fj and fk and applying the general system model as shown in Fig.2(a), we can get a general form of FWM in E-field at a frequency fFWM = +fi fj− fk and a
distance ( ) 1 N n T n L =
∑
=L [24, 29-31].( )
(
)
(
)
( ) 3 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 * , ( ) ( ) ( ) ( ) 1 ( ) ( ) exp exp 2 exp 1 n n n n N FWM ijk T i j k n n n n n n n L g L i L E L d E E E L i L i α γ α β α β α β − = = = = − ⋅ − + ∆ = ⋅ ⋅ − + ∆ − ⋅ − + ∆ ∑
∏
∑
∑
A A A A A A A A A A (3) where( )
2( )
, , FWM ijk T FWM ijk T E L =P L is the FWM power, 2 , , , , i j k i j kE =P is the launched CW signal power at frequenciesfi j k, , ; d is the degeneracy factor ( d = 1 for i = j, and d = 2 for i ≠ j). ( ) 2( )
( ) 2 n n i n eff n A π γ λ = is the
fiber nonlinearity parameter in the nth fiber section and ∆β( )n represents the phase mismatch at nth fiber section and may be expressed in terms of signal frequency differences as
(
)
2 2 ( ) ( ) 2 ( ) 2 n n n i k j k i k j k dD f f f f D f f f f c c d πλ λ β λ ∆ = − − ⋅ + ⋅ − + − (4)In general, the dispersion slope term can be neglected in non-zero dispersion regions- for example, the second term due to dispersion slope in C-band is only 0.022 ps/nm/km for 25GHz spacing in NZDSF, which
is much smaller than typical D = 2 to 6ps/nm/km. For periodically amplified systems without dispersion compensation, (3) can be simplified to the result given in [29] by settingα( )n =α, L( )n =L, g( )n =eαL and
( )n
β β
∆ = ∆ for all n, such that
2 2 2 2 , 2 sin ( / 2) ( ) sin ( / 2) ijk L
FWM ijk ijk eff i j k
ijk N L P N L d L PP P e L α β η γ β − ∆ ∆ ⋅ = ⋅ ⋅ ∆ ∆ (5)
where Leff = {1 − exp(−αL)}/α is the effective length, and ηijk is the mixing efficiency given by
(
)
2 2 2 2 2 sin ( / 2) 1 4 1 L ijk L L e e α α α β η α β − − ∆ = + ⋅ + ∆ − (6)For systems with dispersion compensation, the accumulated phase term 1
(
( ) ( ))
1 exp n− i β L = ∆ ∑
A A A in (3) represents the phase relation among FWM terms generated in each amplifier stage, and is proportional to fiber dispersion, channel spacing and fiber length. For conventional DWDM systems in non-ZERO dispersion fibers, where ∆β( )A is large (e.g. ∆β( )A ≈2 /rad km for a 100GHz spaced DWDM system in NZDSF), small variation in span length ( )LA can result in large phase variation in FWM terms. Therefore,
(
)
1 ( ) ( ) 1 exp n− i β L = ∆ ∑
A A Awas sometimes assumed to be randomly distributed over [0, 2π] and the FWM terms generated in different fiber spans are statistically power added regardless of the phase relation among stages, and the resultant FWM power is proportional to N [32]. This assumption, however, is not valid for U-DWDM systems, especially in the case of analyzing optimum DCR. For example, consider a 6.25GHz spaced U-DWDM systems in NZDSF, ∆β( )A ≈0.008 /rad km, therefore small variations on L( )A have negligible effect on 1
(
( ) ( ))
1 exp n− i β L = ∆ ∑
A A A .As a result, random variable assumption is not valid, and we need to use the exact form in (3), where the FWM terms generated in different fiber spans are added in E-field. Note that the above analysis is based on three CW optical carriers without modulations. In an M-channel NRZ modulated U-DWDM system, the total FWM at a certain channel with frequency fS = fFWM is
expressed as the sum of all the FWM terms with fFWM = +fi fj− fk in E-field [24]
∑
= − +=
FWM k j i ijk f f f f j ijk FWM k j i FWMb
b
b
P
e
E
, θ (7)where bi, j, k = 1 or 0 depending on whether the i, j, kth channel is mark or space. θijk is the random phase of
FWM terms. If the wavelengths in a U-DWDM system are equally spaced, some FWM terms will fall right below other signal bands. At the receiver, the interference on mark is originated from the beating between signal and FWM after a photodiode. The interference on space is originated from the beating between FWM terms (similar to spontaneous-spontaneous emission beat noise) and can usually be neglected in the SNR range of interest. The exact probability density functions of the FWM interference on both mark and space are detailed in [24]. It was shown in [24] that the Gaussian distribution can serve as a good approximation and the equivalent Q-factor due to FWM can be written as:
( )
= FWM S FWM P P dB Q 2 1 log 10 10 2 (8)where PS is the peak received signal power per channel. PFWM = EFWM 2 can be written as [24]
, , ,
1 1 1
8 i j k FWM 4 i j k FWM 4 i j k
FWM FWM ijk FWM ijk FWM ijk
f f f f f f f f f f f
P P P P
≠ ≠ ≠ ≠ ≠ = = ≠
=
∑
+∑
+∑
(9)where the three fractional numbers represent the probabilities of coexisting-marks among the four wavelengths (including the signal channel itself). Note that in a U-DWDM system with equal launched power per channel,
P
FWM∝
P
S3. Therefore, according to (8),Q
FWM2∝
1
/
P
S2.It is a time-consuming task to calculate all the FWM terms in (9) because the number of FWM terms is proportional to the cubic of channel numbers M3. However, a strong phase matching among the four
wavelengths (fi, fj, fj and fFWM) is required for the new FWM term to build up and was reflected in the FWM
efficiency, (6). Note that (6) has a low-pass characteristics with a -3dB point at
2 2 c f D α πλ ∆ = , e.g. the
3dB points are about 7.3GHz and 21.2GHz for D = 17 and 2 ps/nm/km , respectively. This means that, first, for a U-DWDM with channel spacing close to 7.3/21.2GHz in SMF/NZDSF fibers, FWM would become very severe. Secondly, when calculating or simulating the penalty due to FWM in a U-DWDM system with a large channel count, a smaller number of closely spaced channels can possibly used to obtain the final results (with the contribution from farther away channels neglected). Using (9) we find that for a 6.25GHz channel spacing in a transmission fiber with D as low as 2 ps/nm/km, the difference of FWM power is increased by less than 1 dB when the number of channels under consideration is increased from 40 to 640 channels (fully loaded C-band). The difference is even smaller when considering SMF and larger channel spacing. Therefore, in the following analysis and numerical simulations, only up to 40 channels are used. This could save considerable computer time in Section III.
2.1.2 Cross Phase Modulation
The XPM-induced PM-to-IM interference was analyzed in frequency domain by using a modulated pump channel, k, and a CW probe channel, s, and can be written as [33-34]
(
ω
)
ω
ω
)
(
)
~
(
0
,
)
,
,
(
~
, T XPM sk k T s T sk XPML
P
L
P
H
L
P
=
⋅
⋅
(10)where
P
s(
L
T)
is the optical power of a CW channel at a distance LT, P~k(0,ω
) is the Fourier transform ofthe modulated pump channel,
∫
∞∞ − − ⋅ = P t e dt P j t k k(0,
ω
) (0, ) ω ~ , and XPM(
T,
ω
)
skL
H
is the normalizedfrequency response of the XPM induced intensity modulation from channel k to channel s and can be written as [33]
(
)
( ) ( )
[
(
(
)
)
(
( )
)
]
(
)
∑
∏
∑
= − − − = − = − − − + − + − + ⋅ = N n n n sk n s n s n s n sk n s n s n s n sk n s n sk n sk n s n s T XPM sk L a B b B a B b B a b a L jd L g L H 1 ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) 1 ( ) ( 2 ) ( 2 ) ( 1 1 ) ( ) ( ) ( ) ( 1 1 ) ( ) ( exp cos sin cos sin 4 exp ) , ( A A A A A A A α ω γ ω (11)where N is the stage number in Fig. 2(a),
a
(skn)=
α
(n)−
j
ω
d
sk(n) ,c
D
b
s n s n sπ
λ
ω
4
2 ) ( 2 ) (=
, and∑
+ = = N n s s n s D L c B 1 ) ( ) ( 2 2 ) ( 4 A A Aπ
λ
ω
.( ) ( )
s
k
f
c
D
v
v
d
n gk n gs n sk=
−
≈
⋅
⋅
−
⋅
∆
− −1 ( ) 1 2 ) ( )(
λ
is the walk-off parameterbetween channels s and k, v(ngs) is the group velocity of channel s in the nth fiber span, and ∆f is the frequency spacing between adjacent channels. The XPM induced interference in channel s in a digitally modulated U-DWDM system can be written as:
∑∫
≠ − ⋅ ⋅ = s k B B t j XPM sk k T s T s XPM e e d e H P t L P t L P ω ω ω π ω ) ( ) , 0 ( ~ 2 1 ) , ( ˆ ) , ( ˆ , (12)where Pˆs(LT,t) is the optical waveform due to residual linear dispersion at a distance LT and is given by [34]:
∫
−∞∞∑
= − ⋅ =ω
π
λ
ω
ω
π
P c D L PDC e ωd t L P N j t n n n s T s 1 ) ( ) ( 2 2 4 cos ) , 0 ( ~ 2 1 ) , ( ˆ (13)where P~s(0,
ω
)is the Fourier transform of the launched intensity waveform of channel s. This general form can be used to analyze arbitrary modulation format, DCR and PDC. We can see that the DC term of XPM induced interference in (11) is zero and the total variance of XPM induced interference in channel s of a U-DWDM system can be obtained by summing all the variances as∫ ∑
∑
− ≠ ≠ ⋅ = = e e B B s k XPM sk k T s s k sk XPM s XPM σ π P L P ω H ω dω σ 2 2 2 2 , 2 , (0, ) ( ) ~ ) ( 2 1 2 1 (Mark) (14) 0 ≈ (Space)where Be is the signal bandwidth and ( )
2 1
T s L
P is the average received optical power of channel s after a transmission distance LT . Accordingly, the equivalent Q2 due to XPM only can be written as:
∫ ∑
− ≠ = e e B B s k XPM sk k XPM d H P Q ω ω ω π 2 2 2 ) ( ) , 0 ( ~ 2 1 4 (15) We note that XPM(ω) skH is a high pass transfer function, which means the variance of XPM interference in (14) is higher if the modulation signal has higher bandwidth. Therefore, modulation schemes with smaller bandwidths (e.g., 2.5 Gbps) are preferred to larger bandwidths (e.g., 10 Gbps) from the viewpoint of minimizing XPM interference.
In conventional DWDM systems, one can assume that exp(-αL) « 1 and the modulation bandwidth is much smaller than the channel spacing, i.e.,
c
D
d
s n s n skπ
λ
ω
ω
4
2 ) ( 2 ) (>>
, then (n) sb
can be neglected and (11) can be simplified to[35](
) (
)
∑
∏
∑
= − − = − = − + ⋅ = N n skn n s n sk n n k n s T XPM sk a B L jd L g L H 1 ( ) ) 1 ( 1 1 ) ( ) ( ) ( ) ( 1 1 ) ( ) ( exp sin 4 ) , ( A A A A A A ω α γ ω (16)However, this assumption is not valid in U-DWDM systems and the exact form (11) must be used in the following analysis.
2.1.3 Self Phase Modulation and Residual Linear Dispersion
The analytical form for XPM interference can also be used to analyze SPM distortion. In (11), in the limiting case when k=s, the normalized frequency response of SPM distortion can be written as:
(
)
(
)
( ) ( )
[
(
(
)
)
(
( )
)
]
(
)
∑
∏
∑
= − − − = − = = − − − + − + − ⋅ = = N n n n n s n s n s n n s n s n s n n s n n n s n s s k T XPM sk T SPM s L B b B B b B b L g L H L H 1 ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) 1 ( ) ( 2 ) ( 2 ) ( 1 1 ) ( ) ( 1 1 ) ( ) ( exp cos sin cos sin 2 exp , 2 1 ) , ( α α α α α γ ω ω A A A A A (17)The distorted pulse waveform due to ISI and SPM can be approximated as
(
)
+ ⋅ ⋅ = ∆ + ⋅ =∫
− e e B B t j SPM s s T s t s SPM T s T s SPM d e H P t L P t L P t L P t L Pω
ω
ω
π
(0, ) ( ) ω ~ 2 1 1 ) , ( ˆ ) , ( ˆ 1 ) , ( ˆ ) , ( ˆ , , (18)In (18),
∆
P
ˆ
SPM,s(
L
T,
t
)
is waveform distortion due to the combination of SPM and linear dispersion (similar to what was derived for XPM in (12)). In our calculations in Sec.III, SPM-limited Q2 (Eq.(1)) willbe obtained by using the mean (m1) and variance (σ1) at the mark level of distorted waveform (obtained
from (18) at the optimum sampling point) and neglect the contribution from space level (i.e., σ0 ≈0). The
waveform distortion at space due to linear dispersion is taken into account by replacing m0 with the highest
level at space, which causes eye–opening penalty.
2.2. Overall System Limitations
In this Section, we will calculate the overall U-DWDM system limitations by considering ISI, optical amplifier noise and all the optical nonlinearities discussed above. A generalized periodically amplified and dispersion compensated U-DWDM system shown in Fig. 2(b) is used. We assume optical amplifier gain tilts are perfectly equalized and DCFs are used for broadband dispersion and dispersion slope compensation. A tunable PDC, with continuous tuning range up to -2000ps/nm is used for individual channel performance optimization between linear dispersion and SPM/XPM. For simplicity, periodically amplified systems are assumed which have the same fiber length and DCR per span. We assume that the launched optical power into each transmission fiber span is equal, and the optical power into each DCF is 3 dB lower than the launched power to avoid additional nonlinearities generated in DCF. To focus on the fundamental system limitation due to optical fiber nonlinearities, we assumed an ideal rectangular optical filter whose bandwidth
equals to channel spacing and a receiver whose electrical bandwidth equals to 0.8 times data rate. Numerical results are obtained by solving NLSE directly by using the Split-step Fourier (SSF) method with a sampling rate of 2.56THz and 218 sample points (256 and 1024 symbols per channel per simulation for 2.5 and 10 Gbps, respectively). The accuracy of the SSF method was confirmed by gradually reducing the step size. A maximum nonlinear phase change of 0.05 degree per step was used in the numerical simulation. Q2 is calculated directly from the sample mean and standard deviation of the simulated
waveform at an optimum sampling point.
Fig. 4 FWM and ASE limited maximum fiber input power per channel of 2.5Gbps/6.25GHz systems as a function of amplifier stages for different DCR’s in (a) SMF and (b) NZDSF. Assume 40km per amplifier stage. Dotted curve is ASE noise-limited minimum input power. Symbols are simulation results.
Solid and dotted curves in Figs. 3 show the calculated FWM and ASE limited fiber input powers per channel (at Q2(dB)≥15.6dB) as a function of cascaded amplifier stages N, for 2.5Gbps/6.25GHz systems in both NZDSF and SMF. An amplifier span length of 40 km was assumed. Three different DCRs-100, 90 and 60% are shown to illustrate the effect of DCR on FWM. All symbols were obtained by choosing a specific input power/channel, and run a simulation to obtain the maximum amplifier stages which can be cascaded to reach Q2(dB)≥15.6dB. Simulation results include all fiber nonlinearity, linear dispersion and ASE noise. Symbols near the interception points between ASE- and FWM-limited curves deviate away from their corresponding analytical FWM/ASE curves because in that region both FWM and ASE contribute to the combined Q2 value (therefore Q2 due to individual FWM or ASE may be a few dB greater than 15.6 dB).
2.5Gbps/6.25GHz systems in both types of fiber (Fig.4(a) for SMF and 3(b) for NZDSF). Since numerical simulation results include all degradation factors, the results indicate that FWM is the dominant impairment in such systems. From Fig. 4, we can see that with DCR=100%, FWM-limited launched power in NZDSF is about 5dB lower than that in SMF due to the lower local dispersion in NZDSF. We also observe that DCR=60% offers the maximum transmission distance (= 40 km/stage × 115 stages) among the three cases in NZDSF. The transmission distances and optimum launched powers of 2.5Gbps/6.25GHz systems in NZDSF (D=2ps/nm/km) are about 4600, 1600, 900 km and -15, -20, -22dBm for 60, 90, 100% DCR, respectively. Similarly, the numbers for 2.5Gbps/6.25GHz systems in SMF are about 5200, 1400 km and -13, -18.5dBm for 90%, 100% DCR, respectively.
From Fig. 4, we can see that for DCR=100%, FWM-limited launched power is inversely proportional to N for both NZDSF and SMF. This is because with DCR=100%, FWM terms generated at every amplifier stage are added in-phase according to (3). Owing to the fact that phase matching is critical to the generation of FWM, maximum input power levels strongly depend on DCR. For DCR other than 100%, the relation between FWM interference and stage number is not simply amplitude addition (∝N2) or power addition (∝N), but depends on the dispersion map of the system. The resultant FWM limited maximum input power at a non-100% DCR can be much higher than that with 100% DCR, especially for large number of cascaded amplifier stages. However, even though DCR other than 100% can effectively cause the residual dispersion to suppress FWM, it can also cause pulse broadening and enhance the PM-IM conversion via SPM and XPM (which is especially critical in 10 Gbps systems). Therefore, there exists an optimum DCR for a U-DWDM system in which FWM is dominant. In contrast, 100% DCR is always the optimum point for a conventional DWDM system in which FWM is not the limiting factor.
From Fig. 4, we can see that as the number of cascaded amplifiers increases, the maximum FWM-limited input power decreases while the minimum ASE-limited input power increases. Therefore, the optimum fiber input power is a balance between FWM and ASE. With a launched power P per channel, we know from (8) that
Q
FWM2∝
1
/
P
2. LetQ
FWM2=
K
FWM/
P
2 where KFWM is a constant for a fixed system at acertain transmission distance and can be calculated by (8). Similarly, we have
Q
ASE2=
K
ASE⋅
P
for ASE noise (assuming signal-spontaneous beat noise dominate). In an U-DWDM system, where FWM is the dominant fiber nonlinearity, we have(
2) (
1 2) (
1 2)
1total ASE FWM
Q − = Q − + Q − and the optimum launched powerP at a opt certain distance can be found by
(
) (
)
(
1 1)
1 2 2 2 1 0 ASE FWM ASE FWM d d P Q Q dP dP K P K − − − + = + = (19) Therefore, 31/ 2 opt eqP = ⋅P or P dBopt( )P dBeq( ) 1− dB , where Peq= 3KFWM /KASE is the launched
power level at which 2 2
ASE FWM
Q =Q . The relation between 2 and 2
ASE FWM
Q Q when a system is optimized is
2 ( ) 2 ( ) 3
ASE FWM
Q dB Q dB − dB. Furthermore, if we assume 2 15.6( )
total
Q = dB at the maximum transmission
distance, we find 2 ( ) 17.5
ASE
Q dB dB and 2 ( ) 20.5
FWM
Q dB dB at the optimum launched power. Note
that this general rule holds for any
Q
NLD2∝
1
/
P
s2 (e.g., FWM, XPM, SPM, etc.). To show how the optimum launched power is found in a particular long-haul U-DWDM system, an example is illustrated in Fig.5. We plot the Q2 as a function of launched optical power in a 4640 km 2.5Gbps/6.25GHz system usingNZDSF fibers (D=2 ps/nm/km), with 40km per span and DCR=60%. In this case FWM is the dominant nonlinearity and the dotted line represents analytical results based on (8). As expected, the optimum launched power is a balance between ASE and FWM. The optimum launched power (Popt = -15dBm) is
about 1dB lower than the power (Peq = -14dBm) which gives QASE2 =QNL2 . Also note that at Popt = -15dBm,
2 ( ) 17.5
ASE
Q dB dB and 2 ( ) 20.5
NL
Q dB dB , as expected.
Fig. 5 Q2 as a function of average launched optical power of a 2.5Gbps/6.25GHz system after 4640km NZDSF, which corresponds to the maximum transmission point of Fig. 6(a). Solid curve and circles are calculated and simulated received Q2, respectively. Dotted line represents the optical nonlinearity-limited Q2. Dashed line is ASE limited Q2. Solid triangles are the combined results of ASE and fiber nonlinearity.
Fig.4 can be plotted in a different way, as shown in Fig.6(a), to explicitly show that the maximum transmission distance in an NRZ modulated 2.5Gbps/6.25GHz NZDSF system is obtained at optimum DCR’s of about 40-60%. The maximum distance was calculated based on a received Q2 of 15.6dB. For
every DCR, the input power per channel was swept from 0 to –25dBm with a 0.5dB step size to find the maximum achievable transmission distance. Because a typical NZDSF has a dispersion value ranging from 2-6 ps/nm/km, we analyzed both the upper limit (D = 6 ps/nm/km, dotted curves) and lower limit (D = 2ps/nm/km, dashed curves), and use a solid curve to represent the worse of the two. The analytical and numerical results of maximum transmission distance for 80km-span as a function of DCR are also shown in Fig. 6(a). We can see that there exists an optimum DCR of about 40-60% and 20-40% for 40 and 80km spans, respectively, rather than ~ 100% in conventional DWDM systems. The maximum distances are about 4500 and 2300km for 40 and 80km spans, respectively. Also indicated in the same figure is the linear-dispersion limitation (dash-dotted curve) at long wavelength (D = 6ps/nm/km) and low DCR (DCR<50%) region, and is given by [36]:
(
)
2 ( ) ( ) 2 1 104000 (Gb/s) ( / ) N n n n R∑
= D ⋅L −PDC < ⋅ ps nm (20)where R is the data rate in Gbps, D(n) is the fiber dispersion (ps/nm/km) of stage n, L(n) is the fiber span length
(km) of stage n, N is the total number of stages, and PDC represents the dispersion of a post dispersion compensator (PDC = -2000ps/nm in Fig.6(a)).
Fig. 6 (a) Calculated and simulated ASE-, fiber linear dispersion- and nonlinearity-limited maximum transmission distance of 2.5Gbps/6.25GHz-spaced systems (Q2≧15.6dB) as a function of in-line DCR in NZDSF. Results for both 40 and 80 km fiber spans are shown. Dashed and dotted curves were calculated based on D = 2 and 6 ps/nm/km, respectively, while solid curves represent the worse of the two. ● and ▲ are simulation results. In-line DCFs are used for dispersion compensation and the input power into DCF is 3dB lower than that into
transmission fiber. A tunable PDC (up to -2,000 ps/nm) was used to optimize the individual channel performance. The dashed-dotted curve is the linear dispersion limitation at D=6 ps/nm/km. (b)&(c) Calculated individual Q2 for 40km fiber span.
Figs. 5(b) and 5(c) show the corresponding calculated Q2 of individual noise and interference terms. From
Figs 5(a), (b), and (c), we can see that in a 2.5Gbps/6.25GHz system, the optimum DCR is resulted from the tradeoff between linear dispersion and FWM. FWM is the dominant optical nonlinearity for all DCRs especially in the short wavelength region (D = 2ps/nm/km). Note that
Q
ASE2≈
17
.
5
dB
anddB
Q
NLD2≈
20
.
5
for all DCR>50%. The dash-dotted curve shown in Fig 5(b) is the sum of all nonlinearity and ASE noise. The difference between the dash-dotted and dotted curves is due to linear-dispersion-induced eye-opening penalty. Because we use (20) as linear dispersion limitation, eye-opening penalty is kept below 1dB. The dash-dotted curve is not shown in Fig.6(c), because in this region linear dispersion effect can be neglected.Having discussed the maximum transmission distances of 2.5Gbps/6.25GHz systems in NZDSF, we now turn to the cases of SMF, as shown in Fig.7. Fig.7(a) shows the linear dispersion limitation with and without PDC- we can see that the effect of PDC on the optimum DCR and maximum distance is small. This can be understood by the fact that linear dispersion limited transmission distances (dotted curves) are much longer than the maximum transmission distance at the optimum DCR. The maximum transmission distance in SMF is about 900 and 2300km longer than that in NZDSF for 40 and 80km span length, respectively. Fig. 7(b) shows the calculated Q2 of individual nonlinearity, linear dispersion and ASE. The optimum DCR is around 85-93% (Fig.7(a)), which is a trade-off among FWM, SPM/XPM and linear dispersion (Fig.7(b)). We can see that FWM dominates in the range DCR>90%; SPM, XPM and FWM must all be considered for DCR between 80% and 90%; linear dispersion dominates for DCR<80%.
Fig. 7 (a) Calculated and simulated ASE-, fiber linear dispersion- and nonlinearity-limited maximum transmission distance of 2.5Gbps/6.25GHz-spaced systems (Q2≧15.6dB) as a function of in-line DCR in SMF. Results for both 40 and 80 km fiber spans are shown. ● and ▲ are simulation results. In-line DCFs are used for dispersion compensation and the input power into DCF is 3dB lower than that into transmission fiber. A tunable PDC (up to -2,000 ps/nm) was used to optimize the individual channel performance. (b) Calculated individual Q2 for 40km fiber span.
Fig. 8 shows the optimum launched powers for different DCR’s to achieve the maximum transmission distances given previously in Fig. 6 and 6 for NZDSF and SMF, respectively. Note that because of the stronger FWM in NZDSF, the optimum launched power levels in NZDSF are lower than those in SMF. The optimum launched power levels in NZDSF are -15 and -12dBm at DCR = 40-60% (40km-span) and 20-40% (80km-span), respectively. The optimum launched power levels in SMF are about -13 and -10dBm at about 85-93% DCR for 40 and 80km spans, respectively.
Fig. 8 Optimum launched power per channel as a function of DCR in 2.5Gbps/6.25GHz systems. The symbols correspond to the maximum transmission distances at different DCR in Fig.6(a) and Fig.7(a)
Figs. 9 (a) and (b) show fiber nonlinearities (including FWM, SPM and XPM) and ASE limited optical launched powers per channel for 10Gbps/25GHz systems as a function of cascaded amplifier stages (Fig. 2(b)) in NZDSF and SMF, respectively. Curves are calculated results based on (8), (15) and (18) for FWM, XPM and SPM, respectively; discrete symbols are numerical simulation results, which include all fiber nonlinearities, linear dispersion and ASE. Similar to Fig.4, near the interception points of nonlinearity- and ASE-limited curves, symbols deviate away from calculated results because in these regions symbols include the contributions from both nonlinearity and ASE, therefore the input power for ASE-limited case needs to be higher than that at Q2=15.6 dB, and the input power for nonlinearity-limited case needs to be lower than
that at Q2=15.6 dB. For ideal DCR=100%, it is always FWM limited, and XPM and SPM effects can be
neglected. This is clear from the good match between simulation data (▲) and calculated FWM limitations with DCR=100% in Figs. 9(a) and (b). We can see in Fig. 9(a) and (b) that XPM and SPM start to dominate over FWM after about 20 stages (800km) for DCR≠100% in both NZDSF and SMF. As opposed to a 2.5Gbps/6.25GHz system in which FWM is almost always the dominant limiting factor at optimum DCR, a 10 Gbps/25GHz system at an optimum DCR (e.g., 90% for NZDSF) must consider SPM, XPM and FWM altogether.
