長距離次載波/分波多工光纖系統與混合光纖同軸擷取網路

全文

(1)國立交通大學電信工程研究所博士論文. 長距離次載波/分波多工光纖系統與混合光纖同軸擷取網路 Long-Distance DWDM/Subcarrier Multiplexed Lightwave Systems and HFC Access Network Architecture 研究生：陳威宏指導教授：尉應時教授. 中. 華. 民. 國. 九. 十 I. 三. 年. 七. 月.

(2) 長距離次載波/分波多工光纖系統與混合光纖同軸擷取網路 Long-Distance DWDM/Subcarrier Multiplexed Lightwave Systems and HFC Access Network Architecture 研究生：陳威宏指導教授：尉應時教授. Student: Wei-Hung Chen Advisor: Dr. Winston I. Way. 國立交通大學電信工程研究所博士論文 A Dissertation Submitted to the Institude of Communication 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 Communication Engineering July 2004 Hsinchu, Taiwan, Republic of China. 中. 華. 民. 國. 九. 十 II. 三. 年. 七. 月.

(3) 長距離次載波/分波多工光纖系統與混合光纖同軸擷取網路研究生：陳威宏. 指導教授：尉應時教授國立交通大學電信工程研究所中. 文. 摘. 要. 在本論文中，我們研究將單頻帶/次載波（SSB/SCM）技術結合分波多工（DWDM）架構應用於都會型光網（Metropolitan Optical Network）的可行性。都會型光網不同一般長距離光纖通訊網路的特點在於：它要求高頻寬效益、高容量、高適應性（也就是說，每個次載波頻道裡的資料，彼此間是與通訊協定或資料速率無關的。）、且低成本。為了了解此系統在傳輸上的限制，我們在論文中推導出，由於（1）Mach-Zehnder. 光. 調變器與（2）光纖色散所導致的二階和三階拍差失真（CSO、CTB）的理論公式。另為了與分波多工的架構相結合，我們也同時推導在分波多工的架構下，由於光纖非線性所導致之物理現象（Cross-Phase Modulation）因而產生的光波長間干擾（Crosstalk）的理論與數值分析上的結果。此理論與數值分析並不受限於光波長間距的寬窄、每波長入射光的功率大小、調變訊號頻率的高低乃至光訊號的調變方式。以上所有的理論與數值分析結果都經由電腦模擬予以一一驗證。我們亦實際討論了兩個 SSB/SCM/DWDM 的系統以了解它們各自的基本限制與未來衍生的研究方向。在本論文中，為了提供窄頻服務到用戶端，我們也提出了一個全新的，具有漸進式演化發展特性的混合光纖與同軸（HFC）擷取網路的架構，並以實驗驗證了它的可行性。此架構在演化的最初期並不要求一開始就舖設昂貴的分波多工的許多主被動元件，例如 ITU 規格的雷射、分波多工器與解分波多工器等等。雖然如此，然而當每個光節點用戶的頻寬要求持續增加時，它卻也可以逐漸升等為多波長的分波多工架構。此外，此架構的特點是每個波長均可載多個次載波頻道以滿足節點用戶高容量的要求。 i.

(4) Long-Distance DWDM/Subcarrier Multiplexed Lightwave Systems and HFC Access Network Architecture Student: Wei-Hung Chen. Advisor: Dr. Winston I. Way. Institute of Communication Engineering, National Chiao-Tung University. Abstract In. this. dissertation,. subcarrier-multiplexed. we. investigate. (SSB/SCM). the. applicability. technology. of. combined. single-sideband with. dense. wavelength-division-multiplexing (DWDM) scheme in the metropolitan optical network, which requires significantly more bandwidth-efficiency and high capacity with high granularity (protocol-and rate-independent at the same time) at low cost. To understand the transmission limitation of the SSB/SCM signals, we present a close-form analysis to predict the Mach-Zehnder intensity (MZI) modulator-induced composite triple beat (CTB), and linear-fiber-dispersion-induced composite second-order (CSO) and CTB distortions. To combine SSB/SCM with DWDM systems, analytical and numerical tools, which are not constrained by any wavelength spacing and modulation frequencies, are used to analyze cross-phase modulation (XPM) -induced crosstalk. are verified by computer simulations.. All the analytical and numerical results. Several multichannel SSB/SCM/DWDM systems. with transport capacities of 10 or 20 Gb/s per wavelength, with a wavelength spacing of 25,50,100 GHz, are also studied in this dissertation to understand the fundamental transmission limitations.. We also proposed an evolutionary hybrid-fiber-coax (HFC) network architecture, which does not require DWDM in the beginning stage of adding narrowcast services, yet can be eventually upgraded using multiple DWDM wavelengths with a large channel loading per wavelength, is proposed and experimentally verified. ii.

(5) 誌. 謝. 能夠順利畢業，最要感謝的是尉應時教授。尉老師不但在學術研究上具有先知卓見，指引我們正確而前膽的研究方向；而老師嚴謹治學的態度，也使得我個人所獲甚多，使我們實驗室的研究成果，可以毫不畏懼地站上世界舞台，與全世界首屈一指的頂級實驗室分庭抗禮。這些參加大型國際學術會議的經驗，對於在台灣求學的我來說，實在是非常寶貴的學習機會。而在待人處世方面，老師更是以身作則，身教重於言教，對我們後輩有很深的啟發。此外，老師與師母對於我們，乃至於推己及人地，對於我們家人的貼心關懷與關心，亦使我銘記在心。同時也要感謝祁甡教授、高銘盛教授、陳智弘助理教授、溤開明助理教授、張清添教授與王惠民學長等口試委員的不吝指正。尤其是高銘盛老師，不但在學術研究上常常給我新的啟發，且在我遇到問題與困難時，適時地給予鼓勵並提供我一個澄清的人生方向，繁此種種都令我深為感動。而在實驗室八年當中，能與實驗室的學長、學弟妹們共同研究、互相扶持、一起生活則將會是我永遠難以忘懷的愉快回憶。同樣地，曾同我走過這一階段的交大學弟妹們與朋友們，不論以後如何地物換星移，我亦會永遠記得發生在我們之間的點點滴滴。最後，我要深深感謝這一路陪我走過來的家人—爸爸、媽媽、姊姊和最最最可愛的外甥們—怡慧和冠穎。謝謝你們的支持與陪伴！ iii.

(6) Table of Contents 中. 文. 摘. 要.........................................................................................................................i. Abstract ......................................................................................................................................ii 誌. 謝...................................................................................................................................... iii. Table of Contents ......................................................................................................................iv List of Tables.............................................................................................................................vi List of Figures ..........................................................................................................................vii Chapter 1. Introduction (Overview) ....................................................................................1. 1.1. Metropolitan Network........................................................................................1. 1.2. Why using Subcarrier Multiplexing (SCM)?.....................................................1. Chapter 2. Multi-Channel Single-Sideband SCM/DWDM Transmission Systems ............4. 2.1. System Configuration ........................................................................................5. 2.2. Nonlinear Distortions in an SSB/SCM Transmission System — Single Subcarrier Channel.............................................................................................6. 2.3. Nonlinear Distortions in an SSB/SCM Transmission System — Multiple Subcarrier Channels ......................................................................................... 11. 2.4. SPM-Induced Impairment................................................................................18. 2.5. DWDM Configuration – XPM Limitations.....................................................19. 2.6. Complete System Performance Considering All Transmission Impairments..30. 2.7. Summary ..........................................................................................................36. Chapter 3. An Evolutionary HFC Network Architecture for Adding Narrowcast Services39. 3.1. Architectural Concept ......................................................................................40. 3.2. Experiment and Analysis .................................................................................41. 3.3. Summary ..........................................................................................................44. Chapter 4. Conclusions......................................................................................................45 iv.

(7) References................................................................................................................................47 Tables .......................................................................................................................................50 Figures......................................................................................................................................55 Publication List ........................................................................................................................76 簡. 歷......................................................................................................................................77. v.

(8) List of Tables. Table 2.1. Analytic expressions for the individual third order distortions due to SSB/SCM Modulation, Ω i , Ω j and Ω k , are three arbitrarily chosen distinct subcarrier frequencies. ...........................................................................................................50. Table 2.2. Analytic expressions for SSB/SCM 2nd order NLDs due to linear fiber dispersion.. ρ = 1 2 ⋅ β 2 ⋅ z ...................................................................................51. Table 2.3. Parameters for a 20 Gb/s SSB/SCM systems .......................................................52. Table 2.4. For the system given in Table 2.3, a summary of the optimum optical power, RMS OMI/ch and the corresponding worst-case NSR in a transmission system with four multiplexed wavelengths. Three wavelength spacings are given: 25, 50 and 100 GHz.. The required coding gains to reach a BER of 10-12 are also. shown. ...................................................................................................................53 Table 2.5. For 32 channels of 16-QAM signals equally distributed between 6.882 and 12.648 GHz (same transport capacity as that given in Table 2.3), a summary of the optimum optical power, RMS OMI/ch and the corresponding worst-case NSR in a transmission system with four multiplexed wavelengths. wavelength spacings are studied: 25, 50 and 100 GHz.. Three. The required coding. gains to reach a BER of 10-12 for QPSK and 16-QAM are also shown................54. vi.

(9) List of Figures Fig. 2.1. An SSB/SCM/DWDM system..............................................................................55. Fig. 2.2. Optical spectrum of a single channel SSB/SCM signal with m=0.1 (note that ω0 is subtracted from all frequency components). .....................................................56. Fig. 2.3. Optical spectrum of a 2-ch SSB/SCM signal with m=0.1 (note that ω0 is subtracted from all frequency components)..........................................................57. Fig. 2.4. Noise-to-signal ratio (NSR) versus frequency in an SSB/SCM optical transmission system with 64 QPSK channels (ranging from 0.93 to 12.648 GHz), for fiber lengths of 0 and 80 km.. Dashed line and upper triangular symbols are. the theoretical and simulation results for the case of no fiber transmission, respectively.. Solid lines and circular symbols are the theoretical and. simulation results after 80 km fiber transmission, respectively.. RMS OMI per. channel is around 4%. ...........................................................................................58 Fig. 2.5. Linear fiber dispersion-induced CSO versus transmission distance in a SSB/SCM system with 64 QPSK channels.. Dashed and solid lines are the. results of channel 32 and worst-case channel, respectively.. RMS OMI per. channel is 4%. .......................................................................................................59 Fig. 2.6. Noise-to-signal ratio (NSR) versus frequency in an SSB/SCM optical transmission system with 32 16-QAM channels (ranged from 6.882 to 12.648 GHz), for fiber lengths of 0, 30 and 80 km.. Dashed line and upper triangular. symbols are the theoretical and simulation results for the case of no fiber transmission, respectively.. Square and circle symbols are the simulation. results after 30 and 80km fiber transmissions, respectively. Solid lines are the vii.

(10) theoretical results. Fig. 2.7. RMS OMI per channel is around 1.8%. ..............................60. Linear fiber dispersion-induced CTB versus transmission distance in an SSB/SCM system with 32 16-QAM channels. Dashed and solid lines are the results of channel 16 and worst-case channel, respectively. Dotted line is the intrinsic MZI-induced CTB at channel 16.. Fig. 2.8. RMS OMI per channel is 1.8%. ....61. Two wavelengths XT versus pump modulation frequency for an average optical power per wavelength of 0, 6 or 10 dBm.. Circular ( ∆λbp = –0.2 nm) and. triangular ( ∆λbp = +0.2 nm) symbols are simulation results; dashed ( Θ p = 0) and dotted ( Θ p ≠ 0) lines show the theoretical predictions; solid lines are numerical results.. The pump wave is modulated with 64 QPSK channels. between 0.93 and 12.648 GHz using USB/SCM....................................................62 Fig. 2.9. Four wavelengths XT versus pump modulation frequency for wavelength separations of 25, 50 and 100 GHz (with the probe channel in the center ( λ3 ). Circular, dotted and solid lines show the simulation, theoretical and numerical results, respectively. Three pump wavelengths are simultaneously modulated by three uncorrelated 64 QPSK channels (between 0.93 and 12.648 GHz) using USB/SCM.. Fig. 2.10. The average optical power per wavelength is 6 dBm.....................63. Theoretical results of the worst channel NSR versus the RMS OMI/ch for an average optical power per wavelength of -2, 4 and 10 dBm, for the system given in Table 2.3.. Dotted lines are for single wavelength, while solid lines are for. four wavelengths. The wavelength separation is 50 GHz.. Dashed line is due. to linear fiber dispersion-induced CSO.................................................................64 Fig. 2.11. NSR versus channel frequency in the case of four wavelengths for the system viii.

(11) given in Table 2.3.. Average optical power per wavelength is -2, 4 or 10 dBm.. Circular symbols are simulation results; dotted lines are theoretical results; and solid lines are numerical results. The wavelength separation is 50 GHz and RMS OMI per channel is 1.8%.............................................................................65 Fig. 2.12. Theoretical results of the worst channel NSR versus the RMS OMI/ch for an average optical power per wavelength of -6, 0 and 6 dBm.. Dotted lines are for. the case of single wavelength, while solid lines are for the case of four wavelengths.. The wavelength separation is 50 GHz.. Dashed line is due to. linear fiber dispersion-induced CTB.....................................................................66 Fig. 2.13. NSR versus the channel frequency in the case of four wavelengths. optical power per wavelength is -6, 0 or 6 dBm.. Average. Circular symbols are. simulation results; dotted lines are theoretical results; and solid lines are numerical results.. The wavelength separation is 50 GHz and RMS OMI per. channel is 5.5%. ....................................................................................................67 Fig. 3.1. Proposed HFC Network architecture for adding narrowcast services to broadcast services..................................................................................................................68. Fig. 3.2. Measured SBS threshold versus OMI/ch for 81 km conventional SMF. ..............69. Fig. 3.3. Measured C/CTB (●) and C/(CTB+N) (■) of Ch.50 (centered@ f = 745.25 MHz, Bandwidth = 6 MHz) versus OMI/ch.. Dashed and solid lines are. analytical results....................................................................................................70 Fig. 3.4. Received RF spectrum after 102.5 km conventional SMF transmission and 14 dB splitting loss.....................................................................................................71. Fig. 3.5. C/CTB (●) and C/(CTB+N) (■) as a function of channel frequency. The ix.

(12) dashed and solid lines are analytical results..........................................................72 Fig. 3.6. 4-wavelength. experimental. setup. for. measuring. DWDM. crosstalk.. Wavelength spacing was 3.2 nm. ..........................................................................73 Fig. 3.7. Measured crosstalk in the probe wavelength due to λ1 (▼), λ2 (■), λ3 (▲) and total crosstalk (●) as a function of channel frequency.. All broken lines. are analytical results. Solid line is the predicted crosstalk at the middle (worst case) wavelength. Fig. 3.8. Transmission distance is 80 km............................................74. Predicted maximum transmission distance between primary and secondary hub versus wavelength spacing for 4 (■), 8 (●) and 16 (▲) wavelengths. The transmission distance between secondary hub and optical nodes is fixed at 20 km. ........................................................................................................................75. x.

(13) Chapter 1 Introduction (Overview) 1.1. Metropolitan Network. Metropolitan optical networks are characterized by densely populated areas, a large number of central offices, short distances between these interconnecting central offices, and lower data rates than long distance networks (typically less than 2.5 Gbps).. In metropolitan. networks, the traffic patterns change rapidly and require dynamic interconnections at these numerous central offices.. Capacity constraints in metropolitan optical networks are becoming frequent and common as data transmission consumes all available metropolitan fiber optical bandwidth. Capacity demand is being driven principally by the growth of the Internet, home office usage, e-commerce, and other broadband services.. Moreover, enterprises that currently. access the WAN through T1 circuits will require DS-3 and OC-3 connections in the near future in order to support virtual private networks and high-speed Internet access. Heavy and unpredictable data traffic, a condition present in many metropolitan areas, is driving an intense need for a metropolitan network solution that will optimize the installed fiber infrastructure and provide flexible capacity cost-effectively.. 1.2. Why using Subcarrier Multiplexing (SCM)?. Demand for more flexible capacity creation at lower cost is causing service providers to look at alternative and complementary technologies to traditional time division multiplexing (TDM) systems.. Subcarrier multiplexing (SCM) provides an additional dimension of. multiplexing to increase the bandwidth-efficiency, overall capacity, and flexibility of optical 1.

(14) transport networks.. TDM is the other technology usually employed to increase capacity in metropolitan networks. TDM equipment offers flexible add/drop multiplexing, remote management, and performance monitoring.. However, TDM solutions do not offer the level of. granularity required on the tributary inputs and outputs. In order to serve the OC-3 or lower speed services required in the metropolitan market, OC-192 systems require external multiplexers, which adds cost and consumes valuable space.. Higher data rate TDM is. therefore not cost-effective in many metropolitan applications.. Furthermore, TDM system. is not particularly scalable, since the common equipment high-speed components represent high first cost.. Finally, TDM dose not offer the protocol and data rate independence. obtained with DWDM technology.. SCM has been used in radio, satellite, and hybrid fiber coax (cable-TV, cable-modem, etc.) applications at much lower data rates for many years, providing low-cost, highly bandwidth-efficient physical-layer transport.. SCM is a scheme where multiple. bandwidth-efficient modulated signals, translating to lower speed (and cost) electronics and to an ability to transmit more information in a given bandwidth, are multiplexed in the radio-frequency (RF) domain and transmitted by a single wavelength.. For optical. modulation, a scheme of optical single sideband (OSSB) is used to eliminate the need for chromatic dispersion compensation in all but long-distance optical networks. Such an SCM system described above achieves the features desired for metropolitan networks.. It. provides significantly more bandwidth-efficiency and high capacity with high granularity at low cost. Significantly more bandwidth-efficiency and high capacity are attributed to the combined technologies of SCM and OSSB in the system.. Low cost is achieved by using. commercially available, standard components developed and proved for other high-volume 2.

(15) industries. In the meanwhile, multiple advanced Forward-error-correction (FEC) coding algorithms were exploited so that performance comparable to or better than alternative approaches can be achieved with similar power.. The SCM channels can be. protocol-independent and rate-independent, since no synchronization is required among the channels. Finally, SCM can be combined with TDM on the tributaries and DWDM on the high-speed side to achieve even greater granularity and higher overall capacity.. 3.

(16) Chapter 2 Multi-Channel Single-Sideband SCM/DWDM Transmission Systems In a conventional SCM transmission system, lower and upper subcarrier sidebands appear on both sides of an optical carrier, and their phases tend to rotate with respect to each other due to the chromatic dispersion in a standard single-mode fiber (SMF). At a certain transmission distance, the two sidebands could be 180o out of phase, and after down-conversion through a square-law photo-detector, the signal is cancelled out. This kind of “fading” phenomenon could periodically occur at different transmission distances along a fiber link.. Single sideband SCM (SSB/SCM) modulation eliminates the energy on. one side of the optical carrier, and consequently its chromatic dispersion tolerance is significantly enhanced – limited only by the symbol rate of the individual subcarriers not by the aggregate signal bandwidth while an optical filter before the photodiode is adopted to select a certain sub-channel; or the usage of the bandwidth is not limited any more by the first-order RF power degradation while utilizing broadband detection at the receiver end. As a result, SSB/SCM has been used in wireless-optical fiber links operating at millimeter-wave subcarriers to carry a narrowband signal, such as 51.8 Mb/s at 12 GHz over 80 km of SMF [1], and 155 Mb/s at 38 GHz over 50 km of SMF [2].. SSB/SCM has. also been applied to a transmission system carrying multiple channels, for example, a system with 70 channels of 6 MHz AM-VSB at 28 GHz over 23.8 km of SMF was studied [3].. However, only simulation results were provided in [3], and no clear analytical. guideline was given.. In this chapter, we first study the fundamental transmission limitations in a single wavelength, multi-channel SSB/SCM system. 4. These include fiber dispersion-induced.

(17) second- and third-order nonlinear distortions, and the effect of self-phase modulation (SPM). We then investigate whether DWDM can be used in combination with multi-channel SSB/SCM modulation technique.. Computer simulations, numerical calculations, and. fundamental analytical tools are provided to understand the effect of cross-phase modulation (XPM) [4] and XPM-to-IM conversion via group velocity dispersion, for various wavelength spacing and modulation frequencies.. We also present a few metro. optical network case studies to understand the applicability of SSB/SCM/DWDM.. Our. fundamental analysis and numerical calculations can be easily extended to multi-channel wireless-optical fiber links.. 2.1. System Configuration. Fig. 2.1 illustrates the basic SSB/SCM/DWDM system configuration.. Multiple SCM. channels from digital modems at a single intermediate frequency, with or without FEC codecs, are frequency converted and filtered to a series of equally spaced subcarrier frequencies (f1, f2, …fk).. The subcarrier channels are then combined together, creating a. single composite RF signal consisting of k independent subcarrier channels.. Subsequently,. the composite RF signal is used to modulate an ITU wavelength via a dual-drive Mach-Zehnder (MZI) electro-optical modulator [1].. The composite modulating signal from an SCM block (see Fig. 2.1) was applied to both electrodes of the MZI modulator, with a π/2 relative phase shift between the arms.. A DC. bias sets the modulator at a quadrature point. Note that if a π/2 phase shift cannot be met over a wide frequency range, multiple phase shifters will have to be built.. Thereafter, a. number of wavelengths (λ1, λ2, …λN), each carrying plural single-sided SCM signals, are combined by a conventional DWDM multiplexer onto the same fiber. At the receiving 5.

(18) terminal, these wavelengths are demultiplexed and a high-speed photo-detector converts the composite SCM signals in a single wavelength back to their original RF frequencies. The composite SCM signal is then separated into constituent signals by a sequence of electrical frequency converters, filters, demodulators, and decoders.. 2.2. Nonlinear. Distortions. in. an. SSB/SCM. Transmission. System — Single Subcarrier Channel 2.2.1 Third Order Harmonic Due to MZI Modulator The output electrical field of a differential dual-electrode MZI modulator can be written as. Eout =. Ein. =. Ein. 2. 2. [cos(ω o t + φ1 ) + cos(ω o t − φ 2 )]. (2.1). [(cos φ1 + cos φ 2 ) ⋅ cos ω o t − (sin φ1 − sin φ 2 ) ⋅ sin ω o t ]. (2.2). where ω o is the lightwave carrier angular frequency; Ein and Eout are the input and output electrical fields of the MZI modulator, respectively; φ1 and φ 2 are the voltage-induced phase shifts of the two arms of the MZI modulator, both of which can be represented in the form of. π ⋅V dc Vπ , dc. +. π ⋅V ac (t ) Vπ , ac. (where Vπ ,dc and Vπ ,ac are the voltages. required at DC bias port and RF input ports, respectively, to cause a 180-degree phase shift between the two interferometric arms).. Vdc and Vac (t ) are the DC bias voltage and RF. drive signal, respectively. The opposite signs of φ1 and φ 2 in the two cosine terms of (2.1) are attributed to the “differential” electrode configuration of the MZI modulator.. 6.

(19) For SSB/SCM modulation, the RF signals at the two electrodes of a modulator must be a Hilbert transform pair, which means that one of the two RF signals has to pass through a 90° phase shifter.. Furthermore, the modulator is DC biased at a quadrature point of the. nonlinear E/O transfer curve of the modulator to avoid second-order nonlinear distortions. Therefore, if the modulator is modulated by cos(Ωt ) , where Ω is a modulating RF frequency, φ1 and φ 2 can be respectively represented as π 2 + m ⋅ cos(Ωt ) and m ⋅ sin (Ωt ) , where m = π ⋅ Vac Vπ ,ac is the root-mean-square optical modulation index. (RMS OMI).. Note that m 2 2 is the normalized electrical power of the subcarrier. channel at an RF input port of the MZI modulator. The output electrical field of the single RF-frequency modulated optical signal can then be obtained by rewriting (2.1) as. Eout =. Ein ⎧⎡ ⎛ π ⎤ ⎞ ⎨⎢cos⎜ + m cos Ωt ⎟ + cos(m sin Ωt )⎥ ⋅ cos ω o t 2 ⎩⎣ ⎝ 2 ⎠ ⎦ ⎫ ⎛ ⎛π ⎞ ⎞ − ⎜⎜ sin ⎜ + m cos Ωt ⎟ − sin (m sin Ωt )⎟⎟ ⋅ sin ω o t ⎬ ⎠ ⎝ ⎝2 ⎠ ⎭. =. Ein ⎧ π⎞ ⎛ ⎨ 2 J 0 (m) ⋅ cos⎜ ω o t + ⎟ − 2 J 1 (m) ⋅ cos[(ω o + Ω ) ⋅ t ] 4⎠ 2⎩ ⎝ ⎧ ⎡ π⎤ π ⎤⎫ ⎡ + 2 J 2 (m) ⋅ ⎨cos ⎢(ω o + 2Ω ) ⋅ t − ⎥ + cos ⎢(ω o − 2Ω ) ⋅ t − ⎥ ⎬ 4⎦ 4 ⎦⎭ ⎣ ⎩ ⎣ + 2 J 3 (m) ⋅ cos[(ω o − 3Ω ) ⋅ t ]. ⎧ ⎡ π⎤ π ⎤⎫ ⎡ + 2 J 4 (m) ⋅ ⎨cos ⎢(ω o + 4Ω ) ⋅ t + ⎥ + cos ⎢(ω o − 4Ω ) ⋅ t + ⎥ ⎬ 4⎦ 4 ⎦⎭ ⎣ ⎩ ⎣. }. − 2 J 5 (m) ⋅ cos[(ω o + 5Ω ) ⋅ t ] + ". (2.3). where we have used Bessel function expansion to obtain the Bessel function of the first kind, k th order. 7. (2.3), and J k (⋅) (k=1,2,3…) is. As expected, the electrical field is.

(20) composed of, in addition to the single-sided fundamental modulating frequency, an infinite number of harmonics of the modulating frequency.. The even harmonics are double-sided,. whereas the odd harmonics consist of alternative upper or lower sidebands at different orders.. Fig. 2.2 shows the optical spectrum of a single channel SSB/SCM signal.. If only transmission loss is considered, the detected photocurrent is given by. ℜη Ein ⎧ 2 π⎞ ⎛ ≈ ⎨ J 0 (m) − 2 2 J 0 ( m) J1 (m) ⋅ cos ⎜ Ωt − ⎟ 2 4⎠ ⎝ ⎩ 2. I = ℜ ⋅η ⋅ Eout. 2. ⎫ π⎞ ⎛ + 2 2 [J 0 (m) J 3 (m) − J 1 (m) J 2 (m)]⋅ cos⎜ 3Ωt + ⎟ + "⎬ 4⎠ ⎝ ⎭ where ℜ is the photodiode responsivity, and η is the optical power attenuation.. (2.4) The. first term ℜη Ein J 0 (m) 2 = Pr ℜ is the average photocurrent, and Pr is the received 2. 2. optical power. Note that the result contains no even order harmonics, as can be seen in (2.4).. The third order harmonic distortion (HD3) due to the intrinsic nonlinear E/O. “cosine” transfer curve of an MZI modulator, while using SSB/SCM modulation without considering fiber dispersions, is given by. HD3SSB / SCM , MZI. ⎡ J ( m ) J 3 ( m ) − J1 ( m ) J 2 ( m ) ⎤ =⎢ 0 ⎥ J 0 ( m ) J1 ( m ) ⎣ ⎦. 2. (2.5). 2.2.2 Second and Third Order Harmonic Distortions due to Linear Fiber Dispersion Eq.(2.3) can be modified to include the effect of linear fiber dispersion in a transmission link. The electrical field of an SSB/SCM signal after propagating through a SMF with a 8.

(21) length z is given by. Eout , D =. [. ]. Ein ⎧ π ⎛ ⎞ ⎨ 2 J 0 (m) ⋅ cos⎜ ω o t + − β ωo z ⎟ − 2 J 1 (m) ⋅ cos (ω o + Ω ) ⋅ t − β ωo +Ω z 4 2⎩ ⎝ ⎠ ⎧ ⎡ π π ⎤ ⎡ ⎤⎫ + 2 J 2 (m) ⋅ ⎨cos ⎢(ω o + 2Ω ) ⋅ t − − β ωo + 2Ω z ⎥ + cos ⎢(ω o − 2Ω ) ⋅ t − − β ωo −2 Ω z ⎥ ⎬ 4 4 ⎦ ⎣ ⎦⎭ ⎩ ⎣. [. + 2 J 3 (m) ⋅ cos (ω o − 3Ω ) ⋅ t − β ωo −3Ω z. ]. ⎧ ⎡ π π ⎤ ⎡ ⎤⎫ + 2 J 4 (m) ⋅ ⎨cos ⎢(ω o + 4Ω ) ⋅ t + − β ωo + 4Ω z ⎥ + cos ⎢(ω o − 4Ω ) ⋅ t + − β ωo − 4 Ω z ⎥ ⎬ 4 4 ⎦ ⎣ ⎦⎭ ⎩ ⎣. [. ]. }. − 2 J 5 (m) ⋅ cos (ω o + 5Ω ) ⋅ t − β ωo +5Ω z + ". (2.6). where βω is the propagation constant at a frequency ω , which can be expanded around an optical carrier frequency ω o as. 1 2. βω = β 0 + β1 (ω − ω o ) + β 2 (ω − ω o ). 2. (2.7). where the higher order dispersion terms are neglected.. β 1 and β 2 denote the first and. second derivatives of the propagation constant with respect to the optical carrier frequency, The propagation phase shift, βω z , for each optical subcarrier term is. respectively.. different due to the change in the refractive index as a function of the frequency. The SSB/SCM signal propagates in a fiber link with a group velocity 1 β 1 , and the fiber dispersion. D. is related to the group velocity dispersion parameter. β2. by. D = − 2πcβ 2 λ2 where λ is the operating wavelength and c the velocity of light in 9.

(22) vacuum.. According to (2.6), the detected photocurrent can be written as. ℜη Ein ⎧ 2 1 π⎞ ⎛ 2 ≈ ⎨ J 0 (m) − 2 2 J 0 ( m) J1 (m) ⋅ cos ⎜ Ωt − β1Ωz − β 2Ω z − ⎟ 2 2 4⎠ ⎝ ⎩ 2. I = ℜ ⋅η ⋅ Eout , D. 2. ⎛1 ⎞ − 4 J 0 (m) J 2 (m) sin ⎜ β 2 (2Ω) 2 z ⎟ ⋅ cos(2Ωt − β 1 (2Ω) z ) ⎝2 ⎠. π⎞ ⎛ +2 2C1 ⋅ cos ⎜ 3Ωt − β1 (3Ω) z + ⎟ 4⎠ ⎝ ⎫ π⎞ ⎛ −2 2C2 ⋅ sin ⎜ 3Ωt − β1 (3Ω) z + ⎟ + "⎬ 4⎠ ⎝ ⎭. (2.8). where ⎛1 ⎞ ⎛1 ⎞ C1 = J 0 (m) J 3 (m) cos⎜ β 2 (3Ω) 2 z ⎟ − J 1 (m) J 2 (m) cos⎜ β 2 3Ω 2 z ⎟ ⎝2 ⎠ ⎝2 ⎠ ⎛1 ⎞ ⎛1 ⎞ C 2 = J 0 (m) J 3 (m) sin ⎜ β 2 (3Ω) 2 z ⎟ − J 1 (m) J 2 (m) sin ⎜ β 2 3Ω 2 z ⎟ ⎝2 ⎠ ⎝2 ⎠. Note that higher-order harmonics have been neglected in (2.8). Based on (2.8), we can see that when SSB/SCM modulation is used and a fiber transmission link with a distance z is considered, the normalized second and third order harmonic distortions are given by. HD 2 SSB / SCM , D. ⎡ ⎛1 2 ⎞⎤ ⎢ J 2 (m) sin ⎜ 2 β 2 (2Ω) z ⎟ ⎥ ⎝ ⎠⎥ = 2⎢ J m ( ) ⎢ ⎥ 1 ⎢⎣ ⎥⎦. HD3SSB / SCM , D. C + C2 = 21 2 J 0 ( m ) J1 ( m ) 2. 2. (2.9). 2. (2.10). 10.

(23) 2.3. Nonlinear. Distortions. in. an. SSB/SCM. Transmission. System — Multiple Subcarrier Channels 2.3.1 CTB Distortions Due to MZI Modulator The derivations in Sec. 2.2 can be extended to the case of multiple subcarrier channels, i.e.,. φ1 and φ 2 in (2.1) can be replaced by π 2 + m ⋅ ∑ cos(Ω i t ) and m ⋅ ∑ sin (Ω i t ) , i. i. respectively, Ω i is the radian carrier frequency of the subcarrier channel i , and all channels have the same RMS OMI value, m. Fig. 2.3 shows the optical spectrum of a 2-ch SSB/SCM signal. The resultant SSB/SCM-modulation 3rd order nonlinear distortions (NLDs) are summarized in Table 2.1.. It is well known that when the number of subcarrier channels is large, the dominant 3rd order NLD is the triple beat product, which is expressed in decibels as. CTBSSB / SCM , MZI. ⎧⎪ ⎡ J (m) ⎤ 4 ⎫⎪ N = 10 ⋅ log ⎨4 ⋅ ⎢ 1 ⋅ CTB ⎬ ⎥ ⎪⎩ ⎣ J 0 (m) ⎦ ⎪⎭. (2.11). where N CTB is the triple-beat product count at a particular subcarrier channel of interest, whose center frequency Ω d is equal to Ω i ± Ω j ± Ω k .. 2.3.2 CSO due to Linear Fiber Dispersion The derivations in Sec. 2.2.2can be extended to the case of multiple subcarrier channels, and we summarize the results of various 2nd order NLDs in a transmission system using conventional single-mode fiber with a length of z in Table 2.2. 11.

(24) The second order intermodulation distortion terms include two categories, IM i + j and IM i − j , as a result of SSB/SCM modulation. The reason why the amplitudes of IM i + j and IM i − j are different can be explained as follows.. IM i + j results from two types of. beatings, i.e., ωo beats with ⎡⎣ω o + ( Ω1 + Ω 2 ) ⎤⎦ or ⎡⎣ω o − ( Ω1 + Ω 2 ) ⎤⎦ ; while IM i − j results from three types of beatings, i.e., in addition to ωo beating with ⎡⎣ω o + ( Ω1 − Ω 2 ) ⎤⎦ or ⎡⎣ω o − ( Ω1 − Ω 2 ) ⎤⎦ , there is a third contributing term from the beating between (ω o − Ω1 ) and (ωo − Ω 2 ) .. Note that IM i − j in Table 2.2 is not only distortion frequency ( Ω d ) dependent, but also is source frequency ( Ωi and Ω j ) dependent.. When the number of subcarrier channels is large, the dominant 2nd order NLD is the combination of IM i + j and IM i − j , and is expressed in decibels as. ⎧⎪ ⎫⎪ CSOSSB / SCM , D = 10 ⋅ log ⎨ IM i + j ⋅ N Ωi +Ω j =Ωd + ∑ IM i − j ⎬ Ωi −Ω j =Ω d ⎪⎩ ⎭⎪. (2.12). where Ω i and Ω j should be arbitrarily chosen from the frequencies of multiple subcarrier channels and N Ωi +Ω j =Ω d. is the number of second order intermodulation. components whose source beating-frequencies satisfy Ω i + Ω j = Ω d . On the other hand, 12.

(25) when the distortion is source beating-frequency dependent, the accumulation of the second order intermodulation distortions should be rewritten as the summation form such as the second term in (2.12).. 2.3.3 CTB due to Linear Fiber Dispersion The derivations on linear fiber dispersion-induced harmonic distortions in Sec. 2.2 can be extended to third-order intermodulation distortions as follows:. 4. IM i ± j ± k. ⎡ J (m) ⎤ 2 2 =⎢ 1 ⎥ ⋅ C1i ± j ± k + C 2i ± j ± k ⎣ J 0 (m) ⎦. (. ). (2.13). where C1i ± j ± k and C 2i ± j ± k are given by (2.14a)-(2.14h) shown below (note that ρ in these equations is again given by 1 2 ⋅ β 2 ⋅ z ):. (. ). 2 C1i + j + k = cos ⎡⎣ ρ ⋅ (Ωi + Ω j + Ωk )2 ⎤⎦ − cos ⎡ ρ ⋅ Ωi − (Ω j + Ωk )2 ⎤ ⎣ ⎦ 2 2 − cos ⎡ ρ ⋅ Ω j − (Ωi + Ωk )2 ⎤ − cos ⎡ ρ ⋅ Ωk − (Ωi + Ω j )2 ⎤ ⎣ ⎦ ⎣ ⎦. (. ). (. (. ). ). C 2i + j + k = sin ⎡⎣ ρ ⋅ (Ωi + Ω j + Ωk )2 ⎤⎦ + sin ⎡ ρ ⋅ Ωi − (Ω j + Ωk )2 ⎤ ⎣ ⎦ 2 2 + sin ⎡ ρ ⋅ Ω j − (Ωi + Ωk )2 ⎤ + sin ⎡ ρ ⋅ Ωk − (Ωi + Ω j )2 ⎤ ⎣ ⎦ ⎣ ⎦. (. ). 2. (. 13. ). (2.14a). (2.14b).

(26) (. ). C1i + j −k = − cos ⎡⎣ ρ ⋅ (Ωi + Ω j − Ωk )2 ⎤⎦ − cos ⎡ ρ ⋅ Ωi − (Ω j − Ωk )2 ⎤ ⎣ ⎦ 2 2 2 2 − cos ⎡ ρ ⋅ Ω j − (Ωi − Ωk ) ⎤ + cos ⎡ ρ ⋅ Ωk − (Ωi + Ω j ) ⎤ ⎣ ⎦ ⎣ ⎦. (. ). 2. (. ). (. ). 2 C 2i + j −k = sin ⎡⎣ ρ ⋅ (Ωi + Ω j − Ωk )2 ⎤⎦ + sin ⎡ ρ ⋅ Ωi − (Ω j − Ωk )2 ⎤ ⎣ ⎦ 2 2 + sin ⎡ ρ ⋅ Ω j − (Ωi − Ωk )2 ⎤ + sin ⎡ ρ ⋅ Ωk − (Ωi + Ω j )2 ⎤ ⎣ ⎦ ⎣ ⎦. (. ). (. (. ). (2.14c). (2.14d). )⎤⎦ )⎤⎦. (2.14e). C 2i − j + k = sin ⎡⎣ ρ ⋅ (Ωi − Ω j + Ωk )2 ⎤⎦ + sin ⎡ ρ ⋅ Ωi − (−Ω j + Ωk )2 ⎤ ⎣ ⎦ 2 2 + sin ⎡ ρ ⋅ Ω j − (Ωi + Ωk )2 ⎤ + sin ⎡ ρ ⋅ Ωk − (Ωi − Ω j )2 ⎤ ⎣ ⎦ ⎣ ⎦. (2.14f). C1i − j +k = − cos ⎡⎣ ρ ⋅ (Ωi − Ω j + Ωk )2 ⎤⎦ − cos ⎡ ρ ⋅ Ωi − (−Ω j + Ωk )2 ⎣ 2 2 + cos ⎡ ρ ⋅ Ω j − (Ωi + Ωk )2 ⎤ − cos ⎡ ρ ⋅ Ωk − (Ωi − Ω j )2 ⎣ ⎦ ⎣. (. ). (. (. (. 2. ). 2. ). (. (. ). ). C1i − j −k = cos ⎡⎣ ρ ⋅ (Ωi − Ω j − Ωk )2 ⎤⎦ − cos ⎡ ρ ⋅ Ωi − (−Ω j − Ωk )2 ⎤ ⎣ ⎦ 2 2 2 2 + cos ⎡ ρ ⋅ Ω j − (Ωi − Ωk ) ⎤ + cos ⎡ ρ ⋅ Ωk − (Ωi − Ω j ) ⎤ ⎣ ⎦ ⎣ ⎦. (. ). 2. (. (. ). ). 2 C 2i − j −k = sin ⎡⎣ ρ ⋅ (Ωi − Ω j − Ωk )2 ⎤⎦ + sin ⎡ ρ ⋅ Ωi − (−Ω j − Ωk )2 ⎤ ⎣ ⎦ 2 2 + sin ⎡ ρ ⋅ Ω j − (Ωi − Ωk )2 ⎤ + sin ⎡ ρ ⋅ Ωk − (Ωi − Ω j )2 ⎤ ⎣ ⎦ ⎣ ⎦. (. ). (. 14. ). (2.14g). (2.14h).

(27) We can write the composite triple beat (CTB) in decibel as:. ⎛ ⎞ CTBSSB / SCM , D = 10 ⋅ log ⎜ IM i ± j ± k ⎟ ∑ ⎜ Ω ±Ω ±Ω =Ω ⎟ ⎝ i j k d ⎠. (2.15). where Ω i , Ω j and Ω k could be chosen arbitrarily from the frequencies of multiple subcarrier channels and Ω d is the distortion frequency.. 2.3.4 Theoretical Predictions and Simulation Results Computer simulations were carried out to verify the analytical results derived in Sec. 2.3.1 to 2.3.3, and the results are given below. Note that we used a time window (the reciprocal of "frequency resolution") of 128/155e6 sec, and a sample rate of 8×64×155e6 Hz (79.36GHz).. To verify the results derived in Sec. 2.3.1 and 2.3.2, we used 64 channels of QPSK signals in the frequency range from 0.93 to 12.648 GHz. RMS OMI per channel was assumed to be 4%. Each QPSK signal has a symbol rate of 155 Msps and an excess bandwidth factor of 0.2; and the channel spacing is 186 MHz (1.2×155 MHz). Bit-streams among those 64 channels were assumed to be uncorrelated.. To study the fundamental nonlinear distortions due to intrinsic MZ modulator nonlinearity and fiber dispersion, thermal and shot noise are not taken into account. At the receiver end, noise-to-signal ratio (NSR) is measured to estimate the impact of NLDs.. 15. CW.

(28) RF/microwave tones are used to represent the modulated signals [5].. Two cases were studied: back-to-back and 80 km SMF transmission (with a fiber dispersion of 17 ps/nm/km at 1.55 µm). In both cases, after all subcarrier channels were converted to electrical signals by a photodiode, one of the 64 channels was sampled and electrically filtered to measure its NSR performance.. The sampled channel should be turned off while. measuring the total noise-like NLD in its bandwidth.. Since different orders of NLD. falling into a particular channel cannot be separated in the simulation, we need to know the dominant NLD in each case — for the case of back-to-back, the dominant NLD is CTB due to MZI modulator, and it is the fundamental system limit even when all fiber dispersions are compensated; for the case of 80 km transmission, the dominant NLD is CSO.. Fig. 2.4 shows the frequency distribution of the resultant NSR for transmission lengths of 0 and 80 km, respectively. In the case of 0 km, the simulation result agrees well with the analytical CTB approximation obtained from (2.11). However, in the case of 80 km SMF fiber transmission, although the simulation result agrees quite well with the analytical CSO approximation obtained from (2.12) in most cases, there does exist some large discrepancy at those frequencies close to dips. This is because at those dips, CTBs due to linear fiber dispersion are not negligible.. Note that in general CSO is proportional to m2 while CTB is proportional to m4, as can be observed from Eqs. (2.11) and (2.12) by letting J 0 (m) ≈ 1 , J 1 (m) ≈ m , J 2 (m) ≈ m 2 2 and J 3 (m) ∝ m3 for RMS OMI/ch << 1. Therefore, in a multi-channel, dispersive system. with small RMS OMI/ch, CSOs are the dominant nonlinear terms.. 16.

(29) Fig. 2.5 shows the linear fiber dispersion-induced CSO as a function of transmission distance. At an OMI/ch of 4%, the CSO in channel 32 (dashed line) first monotonously increases with transmission distance until at about 50 km and then fluctuates around ~ -14 dB with a peak-to-peak variation of ~ 2.5 dB.. This phenomenon may be explained from. IM i − j in (2.12), where each contribution has distance-dependent cosine or sine terms and. with different period along the transmission distance z.. The solid line in Fig. 2.5. represents the CSO in a worst-case channel at different distances. Note that the worst-case channel is not a fixed channel, rather, it is distance-dependent, i.e., for each distance, the worst CSO may occur in different channels. It can be seen that the worst-case CSO increases with distance initially, but saturates at ~ -10 dB beyond about 100 km.. It is of interest to compare SSB/SCM system to a conventional SCM system, in which two sidebands exist on each side of an optical carrier. The transmission distance of the latter is known to be bounded by the dispersion-limited bandwidth [6], rather than by CSOs.. The. limited transmission bandwidth is caused by the periodic cancellation of the two sidebands. For example, when a conventional SCM system has the same maximum microwave subcarrier frequency of 12.648 GHz as the current SSB/SCM system, its transmission distance is limited to only ~ 23 km.. To verify the results derived in Sec. 2.3.3, a computer simulation on fiber dispersion-induced CTB was carried out for the case of 32 16-QAM signals between 6.882 and 12.648 GHz (within an octave), with an RMS OMI per channel of 1.8%. Fig. 2.6 shows the frequency distribution of the calculated and simulated NSR for transmission lengths of 0, 30 and 80 km, respectively. In the case of 0 km, the dominant NLD is the intrinsic MZI-induced CTB (see (2.11)). However, with fiber transmissions, the dominant 17.

(30) NLD becomes the fiber dispersion-induced CTB. It is clear that the simulation results agree well with the analytical CTB given in (2.15). Note the dispersion-induced CTB at 30 km is more severe than that at 80 km. This is because the dispersion-induced CTBs oscillate around the MZI-induced CTB, as shown in Fig. 2.7, and is due to the cosine and sine terms in (2.14a)-(2.14h).. Fig. 2.7 shows the linear fiber dispersion-induced CTB as a function of transmission distance. The intrinsic MZI-induced CTB of channel 16 (dotted line) obtained from (2.11) is shown as a distance-independent baseline. It is shown that the CTB in channel 16 (dashed line) irregularly oscillates around the baseline, and the deviation from the baseline becomes smaller as the distance increases.. This may be attributed to the. distance-dependent cosine and sine terms in (2.14a)-(2.14h). Note that the worst-case CTB which occurs in channel 16 has a maximum value of ~ -45 dB after a fiber transmission of ~ 30 km.. Furthermore, the worst-case CTB approaches the baseline. beyond a certain distance, i.e., 150 km in the current QAM/SSB/SCM system. This result has an important implication to future metro and wireless optical networks in that the transmission distance of a high-capacity SSB/SCM system can be essentially distance-independent, provided that all SCM channels can be kept within one octave.. 2.4. SPM-Induced Impairment. The effects of SPM in an SSB/SCM system include (1) SPM-induced subcarrier amplitude fluctuation which varies among multiple modulation frequencies (considering a fixed distance); (2) SPM and chromatic dispersion induced residual optical single sideband [14]; and (3) SPM-induced nonlinear distortions.. We have found that the third factor is. negligible when compared with the nonlinear distortions caused by linear fiber dispersion 18.

(31) and nonlinear cross-phase modulation (in the case of multiple wavelengths). The first and second factors can also be ignored if we limit the launched power per wavelength to below ~ 6 dBm, which gives a subcarrier to subcarrier amplitude variation of less than 1 dB [14], [19].. 2.5. DWDM Configuration – XPM Limitations. In this section, the effect of XPM on SSB/SCM/DWDM optical fiber links is investigated theoretically and numerically. A generalized theoretical model of crosstalk (XT) induced by an arbitrary number of wavelength channels through XPM is derived, and is verified by computer simulations.. 2.5.1 XPM-induced XT Considering Two Wavelengths Here we generalize the analysis presented in [7] and [8] to obtain the results for SSB/SCM systems. Note that the small-signal perturbation approach shown in [4] cannot be applied to subcarrier frequencies higher than about 1 GHz, and is not applicable to the microwave SSB/SCM system under consideration.. We first consider a CW probe wavelength and a. modulated pump wavelength with the same polarization co-propagating in a single-mode optical fiber. At the fiber input, the expression for a sine-modulated pump channel is given by. _ ~ Pp (t ) = m Pp (0) ⋅ cos(Ωt + θ ) , (input pump). ~ whose complex amplitude is given by Pp (Ω ) = mPp (0) ⋅ e jθ , where Pp (0 ) is the average pump power at the fiber input. The change of the pump signal power due to both GVD 19.

(32) and fiber nonlinearity can modulate the phase of a probe channel through XPM. Let us assume the normalized complex amplitude of the pump signal at a distance z can be expressed as [8]. − jΩz / vgp Pp ( z , Ω ) = Pp ( Ω ) ⋅ H Fp ( z , Ω ) ⋅ e −α z e , (pump at a distance z). where e −αz e. − jΩz / v gp. accounts for the attenuation and propagation delay at a distance z from. the fiber input. Note that v gp and α are the group velocity and attenuation coefficient, respectively. The function H F x ( z , Ω ) (x = p for a pump channel, and x = b for a probe channel) is defined as the ratio between the intensity modulation at the fiber output (at a distance z) and fiber input (z = 0) [9], and is called the intensity modulation-fiber transfer function (IM-FTF), which takes into account the interaction of fiber nonlinearity and dispersion.. To separate the effect of IM-IM and PM-IM conversions, we can. express H F x ( z , Ω ) as [9]–[15],. H Fx ( z , Ω ) = C IM − IM ( z , Ω ) + H PM (Ω ) ⋅ C PM − IM ( z , Ω ). where C IM − IM ( z , Ω ) = ~ p N (z, Ω ) ~ p N (0, Ω ) φ~ (0,Ω )=0 is a conversion function representing IM-IM conversion, CPM − IM ( z , Ω ) = p N ( z , Ω ) ⎡⎣ 2 P ( 0 ) ⋅ φ ( 0, Ω ) ⎤⎦ is a conversion p N ( 0, Ω ) = 0 ~ ~ function representing PM-IM conversion, and H PM (Ω ) = 2 P (0 ) ⋅ φ (0, Ω ) PN (0, Ω ) gives. the relation between IM and PM for a given optical transmitter. Note that ~ p N ( z , Ω ) and. φ ( z, Ω ) are the Fourier transforms of the normalized signal power and phase terms ~. ~ ~ p N (z , t ) and φ (z, t ) (normalized by P (0) ) of the pump channel (or probe channel), 20.

(33) respectively.. In a directly modulated semiconductor laser [10]–[12], H PM (Ω ) = α c (1 − j f c f ) , where. α c is the chirp parameter and f c is the frequency at which the adiabatic chirp equals to the transient chirp. In an externally modulated IM-DD system [15], H PM (Ω ) = α c . In an SSB/SCM modulation system with ideal dual-electrode MZI modulator [14], i.e., the case we are interested in, H PM (Ω ) = ± j ( + j represents optical lower side band (LSB), whereas − j represents optical upper side band (USB)).. For a given H PM (Ω ) , one can numerically solve CIM − IM ( z , Ω ) , CPM − IM ( z , Ω ) and H F x ( z , Ω ) (by using the fourth-fifth-order Runge-Kutta method) from the following coupled linear differential equations, which depicts the evolution of small-signal power ~ ~ p N ( z , Ω ) and phase φ ( z , Ω ) along a transmission link by considering both GVD and fiber nonlinearities [16]:. p N (z, Ω ) ∂~ ~ = β 2 Ω 2 P (0) ⋅ φ (z , Ω ) ∂z. (2.16). ~ ⎡ β Ω2 ⎤ ∂φ (z , Ω ) p N (z, Ω ) = −⎢ 2 + γ ⋅ exp(− αz )⎥ ⋅ ~ ∂z ⎣ 4 P (0 ) ⎦. (2.17). where β 2 is the first-order group velocity dispersion parameter presented in section 2.2,. γ = 2πn2 (λAeff ) is the nonlinearity coefficient accounting for the nonlinear Kerr effect, n2 is the nonlinear refractive index, and Aeff the effective core area of the optical fiber.. 21.

(34) Once H Fp ( z , Ω ) is obtained, the XPM-induced complex sinusoidal PM in the probe channel (originated in an infinitesimal section of length dz at a distance z from the fiber input) is given by [8]. − jΩz / vgp dφXPM ( z , Ω ) = −2γ b Pp ( Ω ) ⋅ H Fp ( z , Ω ) ⋅ e −α z e dz. (XPM-induced PM in a probe. channel),. (2.18). where γ b is the nonlinear coupling coefficient of a probe channel.. XPM-induced PM in a probe channel is converted to IM at the output of a transmission link through PM-IM conversion. The total XPM-induced IM of a probe channel at the output of a transmission link with a length L is given by. L ~ ~ − jΩ ( L − z ) / v gb PXPM ( z , Ω ) = ∫ 2 Pb ( z ) ⋅ e −α ( L − z )e ⋅ C PM − IM ,b (L − z , Ω ) ⋅ dφ XPM ( z , Ω ) 0. (PM-to-IM. induced IM in a probe channel),. (2.19). where Pb ( z ) = Pb (0) ⋅ e −αz is the average power of the probe channel at a distance z, and Pb (0 ) is the average power of a probe channel at the fiber input; e −α ( L − z )e. − jΩ ( L − z ) / v gb. represents the attenuation and propagation delay of the probe channel from z to L; and C PM − IM ,b (L − z , Ω ) is the PM-IM conversion of the probe channel from z to L.. Substituting Eqs. (2.18) into (2.19) yields. 22.

(35) − jΩL vgb PXPM ( Ω ) = −4γ b Pb ( 0 ) ⋅ Pp ( Ω ) ⋅ e −α L e. ⋅∫ exp ( −σ z ) ⋅ H Fp ( z , Ω ) ⋅ CPM − IM ,b ( L − z , Ω ) ⋅ dz L. (2.20). 0. where σ = α − jΩd bp and d bp ≡ (v gb ) − (v gp ) = ∫ D(λ )dλ is the walk-off parameter −1. −1. λb. λp. between. pump. and. probe. wavelengths.. In. a. non-zero. dispersion. region,. d bp ≈ Db ⋅ ∆λbp − S b ⋅ (∆λbp ) , where ∆λbp ≡ λb − λ p is the separation between probe and 2. pump wavelengths, and S b ≡ dD dλ λ is the dispersion slope of the probe channel. b. Note that H Fp ( z , Ω ) in (2.20), instead of being numerically obtained via (2.16) and (2.17), can possibly be approximated by using a closed-form obtained from a linear dispersive system.. In the next Section, we will point out the limitation of this approach, despite that. it could save significant computing time and help us gain more physical insights. The IM-IM and PM-IM conversions of a pump channel can be expressed by [9] C IM − IM , p ( z , Ω ) = cos(Θ p z ). Θ p = Ω2 Dpλp. 2. (4πc ). and. C PM − IM , p (z , Ω ) = − sin (Θ p z ) ,. respectively,. where. with D p being the dispersion coefficient of the pump channel.. Consequently, the IM-FTF of a pump channel is given by a simplified expression: H Fp ( z , Ω ) = cos ( Θ p z ) − H PM ( Ω ) ⋅ sin ( Θ p z ) .. In. an. USB/SCM. system,. H Fp ( z , Ω ) = H Fp ,USB ( z , Ω ) = cos ( Θ p z ) + j ⋅ sin ( Θ p z ). and. CPM − IM ,b ( L − z , Ω ) = − sin ⎡⎣Θb ( L − z ) ⎤⎦ can be used in (2.20) to obtain a closed-form. solution as 23.

(36) − jΩL vgb PXPM ,USB ( Ω ) = 4γ b Pb ( 0 ) ⋅ Pp ( Ω ) ⋅ e −α L e. [. ⎧⎪ 1 (σ − jΘ p )⋅ sin (Θ b L ) − Θ b ⋅ cos(Θ b L ) + Θ b ⋅ e −(σ − jΘ p )L ⋅⎨ 2 2 ⎪⎩ (σ − jΘ p ) + Θ b. ]⎫⎪⎬ (2.21) ⎪⎭. For an LSB/SCM system, the result in (2.21) can be modified by replacing jΘ p with − jΘ p .. The. XT. (. induced. ~ ~ XT XPM = PXPM (Ω ) Pp (Ω ) ⋅ e −αL. ). 2. by. XPM. in. a. probe. channel. is. given. by. ~ ,where Pp (Ω ) is the complex amplitude of a pump. channel.. (2.21) shows that the strength of the XPM-induced power fluctuation in a probe channel depends on the average power of the pump and probe channels, the modulation frequency Ω , the walk-off and dispersion parameters.. (2.21) also shows that in general the major. contribution to the XPM-induced XT is generated in the front part of the fiber as a result of the optical power attenuation e −α L . (2.21) further shows that only when there is a local zero fiber dispersion at the probe wavelength (i.e., Θb = 0 ) can the XPM-induced XT be eliminated.. If we rewrite the term σ − jΘ p in (2.21) as α − j ( Ωdbp ± Θ p ) , we can see that when Θ p and the walk-off parameter Ωdbp have the same sign, Θ p increases the walk-off; whereas when they have different signs, Θ p decreases the walk-off.. It is interesting to see that. the phase change effect due to GVD in SSB/SCM modulation is significantly comparable to. 24.

(37) the walk-off effect at a small channel separation and at a high modulation frequency. This can be easily observed from the ratio of Θ p and Ωd bp , given by. Ω ωp 2 ∆λbp λ p. , where ω p. is the optical angular frequency of the pump channel. For example, when the wavelength separation is 0.2 nm and the modulation frequency is 1 GHz, the ratio of Θ p and Ωd bp is about 1/50, and therefore Θ p is negligible. Following the same logic, this is also true when the channel spacing is more than 0.2 nm.. However, when the modulation frequency. is increased to > ~10 GHz, Θ p becomes comparable to Ωd bp and cannot be neglected.. From the above discussion, it is possible to further simplify (2.21) by letting Θ p = 0 when the WDM channel spacing is larger than 0.2 nm and when the RF modulation frequency is lower than a few GHz.. Under this condition, we have H Fp ( z , Ω ) ≈ 1 , cos(Θ b L ) ≈ 1 , and. sin (Θ b L ) ≈ Ω 2 Db λb L (4πc ) . 2. Consequently, XT based on (2.21) can be approximated as. [4], [8]. ⎡ 1 − e −(α − jΩdbp ) L ⎤ λb Ω ⎥ ⋅ ⎢1 − ( Ω ) = γ b Pp ( 0 ) ⋅ Db L πc α − jΩdbp ⎢ α − jΩdbp ⎥ ⎣ ⎦ 2. XTXPM. 2. 2. (2.22). It can be seen from (2.22) that for a weak walk-off and small modulation frequencies, Ωd dp << α , the XPM-induced XT increases linearly with Ω 4 ; while for a strong walk-off and higher modulation frequencies, Ωd dp >> α , the XPM-induced XT is linearly proportional to Ω 2 , and inversely proportional to the square of channel spacing. Also. 25.

(38) note that XPM-induced XT is independent of OMI/ch.. Under the conditions of large wavelength spacing and small modulation frequencies, both an SSB/SCM system and a dispersion-compensated optical double sideband (ODSB) system have neither dispersion-induced fading nor pump envelope change, i.e., H Fp ( z , Ω ) = 1 (note that an ODSB system without dispersion compensation has H Fp ( z , Ω ) = cos ( Θ p z ) ).. It should be noted, however, that in an SSB/SCM. system H Fp ( z , Ω ) = cos ( Θ p z ) ± j ⋅ sin ( Θ p z ) = exp ( ± jΘ p z ) ( ＋ : USB; － : LSB), hence H Fp ( z , Ω ) = 1 does not imply Θ p = 0 .. Consequently, the assumption of Θ p = 0 may. cause errors, which we will discuss subsequently.. 2.5.2 Multiple Wavelengths (2.18) can be easily extended to cases when there are more than two wavelengths co-propagating in a fiber link. In those cases, each pump wavelength k modulates the phase of a probe wavelength independently [17], [18] and therefore (2.18) can be modified as. ⎡ − jΩz / vgk ⎤ −α z dφXPM ,total ( z , Ω ) = −2γ b ⎢ ∑ Pk ( Ω ) ⋅ H Fk ( z , Ω ) ⋅ e ⎥ ⋅ e dz ⎣ k ≠b ⎦. (2.23). ~ where Pk (Ω ) and H Fk ( z , Ω ) are the complex amplitude and the IM-FTF of pump channel. k , respectively.. The total power fluctuation induced by XPM in a probe channel at the. output of a fiber link can be written as PXPM ,total ( Ω ) = ∑ PXPM ,bk ( Ω ) with each individual k ≠b. 26.

(39) ~ PXPM ,bk (Ω ) given by (2.20) or (2.21). When random phase relationship of the modulating signals in each wavelength is assumed, the total XPM-induced XT in a probe channel can be expressed by. (. XTXPM ,total = ∑ PXPM ,bk ( Ω ) Pb ( Ω ) ⋅ e −α L k ≠b. ). 2. (2.24). 2.5.3 Transmission Link Performance due to XPM: Simulation, Numerical, and Theoretical Results To validate the theoretical results on XPM-induced XT, we carried out numerical calculations and computer simulations on a system with parameters shown in Table 2.3. The system is a 64-channel 155 Msps QPSK USB/SCM transmission system over 80 km of conventional SMF, and has a theoretical transport capacity of 20 Gbps per wavelength.. 1) Two Wavelengths: Assuming a pump wavelength is modulated by 64-channel. QPSK signals occupying a frequency range beyond an octave, the distribution of XT at different RF frequencies in a co-propagating probe wavelength is shown in Fig. 2.8. Three average optical power levels per wavelength were studied: 0, 6 and 10 dBm.. Simulation results are shown by circles ( ∆λbp = –0.2 nm, i.e., λb − λ p = –0.2 nm) or triangles ( ∆λbp = +0.2 nm, but only for input power of 0 dBm per wavelength). Note that in the simulation, the digital modulation data among different RF channels were carefully kept independent. Also, we used a time window of 128/155e6 sec, and a sample rate of 8×8×64×155e6 Hz (634.88 GHz).. 27.

(40) Numerical results obtained from (2.20) and the coupled nonlinear Schrödinger equations (2.16) and (2.17) are shown by solid lines.. Theoretical results obtained from (2.21) ( Θ p ≠0, dotted line; and Θ p = 0, dashed line) are also shown in Fig. 2.8.. Several observations from Fig. 2.8 are summarized below:. The numerical results show an excellent agreement with those of simulations, even in the case of a high optical power level of 10 dBm per wavelength. In general, The XT is nearly proportional to Ω 4 at low modulation frequencies, and nearly proportional to the square of Pb (0 ) , as predicted by (2.22).. Theoretical results based on (2.21) ( Θ p ≠0) agree very well with simulation results (within ~2 dB).. The ~2 dB discrepancy occurs only at the case of 10. dBm/λ with a modulation frequency > ~8 GHz. attributed to two factors. wavelength,. the. This discrepancy may be. The first is that at a very high input power per linear-system-approximated. H Fp ( z , Ω ) = cos ( Θ p z ) ± j ⋅ sin ( Θ p z ) is no longer valid.. IM-FTF. This is because the. interaction between SPM and chromatic dispersion could generate a second residual optical single-sideband [14], causing two optical sidebands to interfere with each other at the receiver. The second is that a high input power could cause enhanced IM-IM and PM-IM conversions, and in addition, the 28.

(41) fading-induced dip frequency could shift toward slightly higher frequencies [13], [14], [19]. The theoretical approximation based on (2.21) ( Θ p = 0) is seen to cause more errors in the higher modulation frequency range when compared to the more exact theoretical estimation based on Θ p ≠0. At a launched power of 0 dBm per wavelength, the XT of ∆λbp = –0.2 nm is always greater than that of ∆λbp = +0.2 nm by up to ~5 dB. The difference is more pronounced at higher modulation frequencies. This confirms what we said earlier about the pump phase change along the fiber (caused by the combined effects of SSB/SCM modulation and GVD) could enhance or mitigate the walk-off effect.. Since we used USB/SCM here, the walk-off effect will be. affected via α − j (Ωd bp + Θ p ) (see (2.21)). Considering 1550 nm wavelength propagating in a conventional SMF, the dispersion is positive, and therefore Θ p is positive.. Consequently, a negative ∆λbp ( Ωd bp < 0) will generate more. XPM-induced XT than a positive ∆λbp ( Ωd bp > 0), because the latter enhances the walk-off effect. The CNR requirement of a QPSK signal is 17 and 20 dB in theory and practice, respectively. Therefore, the power level per wavelength should be kept below ~3 dBm in this transmission system. A dip frequency occurs at somewhere between 9.6 to 9.8 GHz under different conditions. This is due to the fact that PM-IM conversion vanishes around 9.6 GHz. for. a. SMF. fiber. length. 29. of. 80. km,. i.e.,. Θb z = nπ. in.

(42) CPM − IM ,b ( z , Ω ) = − sin ( Θb z ) , and therefore the XT performance is improved.. 2) Four Wavelengths: Fig. 2.9 shows the XPM-induced XT due to four adjacent. wavelengths (spaced at 25, 50, or 100 GHz) for the system given in Table 2.3. average optical power per wavelength is 6 dBm. center ( λ3 in the inset of Fig. 2.9).. The. The probe channel is located at the. Simulation results are shown in circles, numerical. results (using (2.16), (2.17) (2.20) and (2.24)) are shown in solid lines, and theoretical results (using (2.21) and (2.24)) are shown in dotted lines. The followings can be observed in Fig. 2.9:. All results from simulations, numerical calculations, and theory match very well. However, it should be noted that at higher power levels (e.g., 10 dBm), the influence of fiber nonlinearity on the PM-IM conversion becomes non-negligible and the small-signal analysis results used to derive (2.21) may no longer be valid. The XT is nearly inversely proportional to the square of wavelength separation for moderate to high frequencies, as predicted by (2.22). It can be predicted that the XT estimates based on either Θ p ≠0 or Θ p = 0 (not shown in figure) will have a negligible difference (within 1 dB for 0.2 nm wavelength separation). This is because, from the viewpoint of a central probe channel, that the mitigated or enhanced XTs due to left- or right-hand side neighbor channels are averaged out.. 2.6. Complete System Performance Considering All Transmission Impairments 30.

(43) In this section, we shall include all possible transmission impairments, in addition to the XPM-induced XT described in the last section.. These impairments include receiver. thermal noise, signal-spontaneous and spontaneous-spontaneous beat noise due to optical amplifiers, shot noise, and linear fiber dispersion induced nonlinear distortions (CSOs and CTBs which were discussed in Sec. 2.3). Note that in the 64 channels QPSK system example shown in Table 2.3, only CSOs need to be considered in the transmission system when OMI per channel is <<1 (this is because CSO ~ m2 and CTB ~ m4). However, if we consider a system with modulation frequencies distributed within an octave, e.g., 32-channel 155 Msps 16-QAM or QPSK equally distributed between 6.882 and 12.648 GHz, then only CTB needs to be considered. As we will see, keeping all modulation channels within an octave is a way to achieve high capacity, long distance transmission without dispersion compensation.. We used an EDFA pre-amplifier in the system so that the receiver thermal noise is not dominant [20].. The pre-amplifier has an unsaturated gain of 30 dB, a 3-dB saturation. output power of 10 dBm, and a noise figure of 5 dB.. 1) 64 channels of QPSK: Four wavelengths, each carrying the same 64-channel QPSK. signals (see Table 2.3), are considered in an 80 km transmission system. No dispersion compensation is used.. The probe wavelength again is λ3 .. Fig. 2.10 shows the theoretical results of the worst-case noise-to-signal-ratio (NSR) (the worst among 64 subcarrier channels) as a function of RMS OMI/ch. Note that in the case of four wavelengths, the worst-case channel frequency depends on the combination of CSO and XPM-induced XT.. The dashed line represents the NSR. caused by linear fiber dispersion-induced CSOs, which was obtained from (2.12). It 31.

(44) shows that the NSR is proportional to the square of RMS OMI/ch.. Due to this. linear-dispersion-limited NSR, we must have a low OMI/ch in order to obtain a low NSR, e.g., OMI/ch less than 1.5% to obtain NSR < –20 dB. However, to reach a long transmission distance with such a small OMI/ch, the launched optical power per wavelength has to be high.. When there is only a single wavelength, we see from the. three dotted lines in Fig. 2.10 that the launched optical power has to be as high as 10 dBm so that NSR can reach its best value of –20 dB to meat the requirement of a QPSK signal.. We also notice that there is an optimum OMI/ch, i.e., 2.2%, 1.6%, and. 1.1%, for the three power levels, respectively. When the OMI/ch is low, e.g., lower than 1%, the dominant noise is the signal-spontaneous beat noise, which is linearly proportional to the signal power. When the OMI/ch is high, e.g., higher than 2%, the dominant noise is the linear-dispersion-induced CSO.. In the case of four wavelengths, XPM-induced XT has to be included, and the theoretical results based on (2.21) and (2.24) are shown by solid lines in Fig. 2.10. Note that the wavelength separation is 50GHz, and the probe wavelength is at the center of the four wavelengths, i.e., λ3 . As shown in (2.20), XPM-induced XT is proportional to the square of optical power per wavelength, thus a 10 dBm input power per wavelength causes a significant degradation to the single-wavelength NSR. When the input optical power is -2 dBm per wavelength, the theoretical results are the same for both single and four wavelengths. This is because the XPM-induced XT is negligible when compared to the combination of fiber dispersion-induced CSOs and signal-spontaneous beat noise. When the input power is 4 dBm per wavelength, the NSR is only slightly degraded by the XPM-induced XTs.. Fig. 2.11 shows the distribution of NSR in all the RF channels (OMI/ch= 1.8%) given 32.

(45) in Table 2.3, when there are four equal-power wavelengths with per wavelength power of –2, 4, or 10 dBm and a channel spacing of 50 GHz. The theoretical results (dotted lines) are based on (2.12), (2.21), (2.24) and various receiver noise terms. Again, the NSR limited by the linear fiber dispersion-induced CSOs (see Eq.(2.12)) serves as the fundamental limitation of the given transmission system despite of the wavelength power level.. We see from Fig. 2.11 that it is impossible for all QPSK channels to. reach the minimum practical requirement of –20 dB NSR after 80 km of transmission without dispersion compensation, unless a forward error correction (FEC) coding gain is available.. Also shown in the figure are the simulation (circle) and numerical (solid line) results, which exhibit excellent agreement over the entire frequency range of the subcarrier channels for all optical powers. Theoretical results at 10 dBm, however, show slight discrepancy (~2 dB) from the simulation results.. This is because the. high-power-induced fiber nonlinearity affects the applicability of the linear-system IM-FTF, as was discussed in section 2.5.1.. When the input power per wavelength is 4. dBm, we can see in Fig. 2.11 that the worst-channel NSR is about –16 dB, and is consistent with what was presented in Fig. 2.10.. Table 2.4 summarizes the theoretical results based on the optimum input optical power per wavelength, RMS OMI/ch and the corresponding worst-case NSR for a wavelength spacing of 25, 50 and 100 GHz, respectively. It can be seen that the wider the wavelength spacing, the higher input optical power per wavelength can be launched into the system, the lower OMI per channel can be used, and a better NSR performance can be obtained.. 33.

(46) The coding gains required to reach a BER of 10-12 in practice for all wavelength spacings are also shown.. These coding gains should be within the reach of. conventional Reed-Solomen FEC codecs.. For longer transmission distance, more. optical amplifier-induced noise will have to be added (even though CSO is already close to its highest value, similar to what was shown in Fig. 2.5), and the coding gain may not be enough to cope with this additional noise due to longer transmission. Setting the total modulation bandwidth to within an octave may be a good solution to this problem.. The following example confirms the feasibility of this approach.. 2) 32 channels of 16-QAM or QPSK: Four wavelengths, each carrying 32-channel. 155 Msps 16-QAM (or QPSK) signals equally distributed between 6.882 and 12.648 GHz, are considered in an 80 km transmission system.. No dispersion compensation. is used. The probe wavelength again is λ3 .. Fig. 2.12 shows the theoretical results of the worst-case NSR as a function of RMS OMI/ch.. The dashed line represents the NSR caused by linear fiber. dispersion-induced third order nonlinear distortions, which were obtained from (2.15). It shows that the NSR is proportional to m4. It is obvious that we can now work in a region with much higher OMI/ch than that in Fig. 2.10. To reach an NSR < -27 dB for practical 16-QAM signals without coding gain, OMI/ch needs to be less than 6.5%. When there is only a single wavelength, we see from the three dotted lines in Fig. 2.12 (for a launched power of –6, 0, and 6 dBm per wavelength) that the launched optical power has to be as high as 6 dBm such that NSR can reach its best value of –29 dB to meet the requirement of a 16-QAM signal. We also notice that there is an optimum OMI/ch, i.e., 7.1%, 5.6%, and 4.5%, for the three power levels, respectively. When the OMI/ch is low, e.g., lower than 4%, the dominant noise is the signal-spontaneous 34.

(47) beat noise, which is linearly proportional to the signal power. When the OMI/ch is high, e.g., higher than 7%, the dominant noise is the linear-dispersion-induced CTB.. In the case of four wavelengths, XPM-induced XT is included, and the theoretical results based on (2.21) and (2.24) are shown by solid lines in Fig. 2.12.. We can. observe: (1) when the input optical power is as low as -6dBm per wavelength, the theoretical results are the same for both cases of single and four wavelengths, because the XPM-induced XT is negligible; (2) when the input optical power is as high as 6 dBm per wavelength, the XPM-induced XT degrades the single-wavelength NSR severely; and (3) when the input power is 0 dBm per wavelength, the NSR is only slightly degraded by the XPM-induced XTs.. Fig. 2.13 shows the distribution of NSR in all the RF channels (OMI/ch= 5.5%), when there are four equal-power wavelengths with per wavelength power of –6, 0, or 6 dBm and a channel spacing of 50 GHz.. The theoretical results (dotted lines) are based on. (2.15), (2.21), (2.24) and additive receiver noise terms.. The NSR limited by the. linear fiber dispersion-induced CTBs (see Eq.(2.15)) serves as the fundamental limitation of the transmission system, which gives a NSR limit of about –30 dB after 80 km transmission. If we carefully choose 0 dBm as the per wavelength power level to avoid XPM-induced XT, the resultant NSR is about –24 dB and is limited by the amplifier signal-spontaneous beat noise.. This NSR is more than sufficient for QPSK. signals, but requires > 3 dB FEC gain for 16-QAM signals.. Also shown in the figure are the simulation (circle) and numerical (solid line) results, which exhibit excellent agreement over the entire frequency range of the subcarrier channels for all optical powers. Theoretical results at 6 dBm, however, show slight 35.