國 立 交 通 大 學
光電工程研究所
博士論文
光通訊系統中的全光訊號處理
All-Optical Signal Processing in Optical
Communication Systems
研
究
生 ： 魏 嘉 建
指 導 教 授 ： 陳 智 弘 老師
光通訊系統中的全光訊號處理
All-Optical Signal Processing in Optical Communication
Systems
研 究 生 : 魏嘉建
Student : Chia-Chien Wei
指導教授: 陳智弘 老師
Advisor : Associate Prof. Jyehong Chen
國 立 交 通 大 學
光 電 工 程 研 究 所
博 士 論 文
A Disseration
Submitted to Institute of Electro-Optical Engineering
College of Electrical and Computer Engineering
National Chiao Tung University
In Partial Fulfillment of the Requirements
For the Degree of Philosophy
In
Institute of Electro-Optical Engineering
October 2008
摘要
光通訊系統中的全光訊號處理
研究生： 魏嘉建 指導教授： 陳智弘 老師 本論文將介紹及分析數種新穎的全光訊號處理技術，包含了全光的波長轉換、全光 的開關移鍵到差分相位移鍵的格式轉換、全光的差分相位移鍵的再生。 為了提升在半導體光放大器中的交叉極化調變之波長轉換頻寬，我們提出了一種稱 作差分交叉極化調變的方法，它與傳統的交叉極化調變的不同僅在於一段額外的雙折射 延遲線。在差分交叉極化調變的分析部份，包括訊號的數值模擬以及小訊號模型的推導， 兩者都能和實驗結果互相對應。其中數值模擬清楚地顯示出波長轉換的效能提昇，而小 訊號模型則提供了更多對差分交叉極化調變直觀且深入的理解。 在全光調變格轉換中，我們使用光子晶體光纖中的交叉相位調變，來達成 40 Gb/s 的開關移鍵到差分相位移鍵的訊號格式轉換。因為我們所使用的光子晶體光纖具有高度 的線性雙折射特性，並且其雙折射光軸在整段光纖中是固定的，將受感應光的偏振態控 制在相對雙折射光軸 45o 輸入光纖時，只要受感應光與調變光的波長差超過約 6 nm 就能 明顯降低該全光調變格式轉換效率受調變光偏振態的影響程度，並且偏振態與波長差對 於格式轉換的影響有理論的定量分析。由於該格式轉換基於以調變光光強度調變受感應 光相位，當我們需要的相位變化不足、或調變光強度雜訊引起相位雜訊，都會影響差分相位移鍵的訊號性能，該問題的造成的損耗也有理論的分析。 關於差分相位移鍵的訊號再生，因為差分相位移鍵訊號是將訊號載在光相位上，其 訊號表現會同時受到振幅及相位雜訊影響。因此，除了振幅的再生，差分相位移鍵的再 生必須要保留原有相位，甚至要降低相位雜訊。我們提出了一個新的概念：相位雜訊平 均器，來降低差分相位移鍵的系統中的相位雜訊。它能夠有效地增加相鄰位元間相位雜 訊的相關性，並因此減少了系統的差分相位雜訊。考慮各種不同特性的相位雜訊，我們 發現多次的相位雜訊平均皆能使差分相位雜訊收斂，而且無關於系統傳輸長度。最後， 對於不同的差分相位移鍵的再生，我們也理論分析了它們的差異。
ABSTRACT
Title of dissertation:
ALL-OPTICAL SIGNAL PROCESSING
IN OPTICAL COMMUNICATION SYSTEMS
Chia-Chien Wei, Doctor of Philosophy, 2008
Dissertation directed by:
Associate Professor Jason (Jyehong) Chen
Department of Photonics
National Chiao Tung University, Taiwan
In this dissertation, new approaches for all-optical wavelength conversion, all-optical on-off keying (OOK) to differential phase-shift keying (DPSK) format conversion and all-optical DPSK regeneration are proposed and analyzed theoretically.
In order to improve the wavelength conversion bandwidth of cross polarization mod-ulation (XPoM) in a semiconductor optical amplifier (SOA), a novel conversion scheme named differential cross-polarization conversion (DXPoM) is proposed, and it is realized by simply adding an extra birefringence delay line in the conventional XPoM. In this part, both the large-signal simulation and small-signal model of DXPoM are developed, and they successfully predict results that agree with experimental outcomes. The large-signal simulation of DXPoM evidently shows the performance improvement and match well with experimental results. The small-signal model gives more comprehensible and intuitive physical insight of DXPoM.
For format conversion, using the cross-phase modulation (XPM) effect in a nonlinear photonic crystal fiber (PCF), the conversion of OOK-to-DPSK is achieved at 40 Gb/s for the first time. Because the PCF has high linear birefringence, and its birefringent axes remain for entire length, launching the probe at 45◦ relative to the birefringence axes of
the PCF and having the pump-probe detuning (PPD) greater than ∼ 6 nm can realize polarization-insensitive XPM-based conversion. The polarization- and PPD-dependent nonlinear phase shift is investigated theoretically. It indicates improper nonlinear strength could generate insufficient phase shift (< π) in XPM-based conversion, and the amplitude noise (AN) of an OOK pump will also be converted into the phase noise (PN) of the converted DPSK signal, the penalties induced by these issues are discussed analytically as well.
Because the DPSK format carries information on the phase domain, it is influenced by both AN and PN. Therefore, except amplitude regeneration, the regeneration of DPSK signals has to preserve phase information or eliminate PN. A novel all-optical phase noise averager (PNA) is proposed to reduce residual PN in the DPSK transmission system with phase-preserving amplitude regenerators. It can increase the correlation between the PNs of neighboring bits and greatly reduce the differential PN in the transmission system. Considering various PN, the multiple PN averaging effect is investigated, and the effec-tive PN is convergent regardless of the transmission distance. Furthermore, theoretical bit error rates of the DPSK format with various regeneration schemes is presented for comparison.
誌 謝
能夠完成博士的學業，我最要感謝的就是我的指導教授，陳智弘老師。除了專業領 域的幫助，陳老師生活態度給我的影響，也是我研究生涯的一個重大收穫。另外，要感 謝美國馬里蘭大學的指導教授，陳永睿老師，以及 Prof. Gary M. Carter 和 Dr. William Astar 的幫助，沒有他們我就不能在美國順利度過一年、並拿到馬里蘭大學的博士學位。 還要感謝實驗室成員，彭煒仁、林俊廷、黃明芳、施伯宗，以及我最好的朋友，紀 建宇，他們的陪伴使我的研究生涯更添色彩，並且在互相的討論中，讓我學習到更多。 最後，感謝于珊一路的關懷與鼓勵；以及我最重要的家人，我的父、母親、大姐、 二姐以及哥哥，感謝他們的包容，並讓我有一個溫暖的家。 嘉建 2008/11/24
Table of Contents
List of Tables viii
List of Figures ix
List of Abbreviations xiii
1 Introduction 1
1.1 All-optical signal processing . . . 1
1.2 Speed enhancement of semiconductor optical amplifier-based wavelength conversion . . . 4
1.3 All-optical OOK-to-Binary phase-shift keying conversion . . . 6
1.4 All-optical phase noise suppression of DPSK signals . . . 8
1.5 Organization of the dissertation . . . 10
2 Differential cross-polarization modulation in an SOA 11 2.1 Cross-polarization modulation in an SOA . . . 11
2.1.1 Concept of XPoM . . . 11
2.1.2 Concept of DXPoM . . . 11
2.2 Large-signal model . . . 13
2.2.1 Simulation model of SOA-based wavelength converters . . . 13
2.2.2 Simulation results of DXPoM . . . 17
2.3 Experimental results . . . 21
2.4 Small-signal model . . . 24
2.4.1 Small-signal model of XPoM . . . 24
2.4.2 Small-signal model of DXPoM . . . 29
2.4.3 Discussion . . . 36
3 All-optical OOK-to-PSK conversion 38 3.1 Introduction to the OOK and DPSK formats . . . 38
3.2 Polarization-insensitive OOK-to-DPSK conversion . . . 40
3.2.1 Concept of XPM-based format conversion . . . 40
3.2.2 XPM effect in a birefringent fiber . . . 41
3.2.3 PPD-dependent XPM effect . . . 43
3.2.4 Experimental RZ-OOK-to-RZ-DPSK conversion in a PCF . . . 49
3.3 Theoretical analyses of the penalty arising from OOK-to-DPSK conversion 56 3.3.1 Insufficient phase shift . . . 57
3.3.2 Finite SNR of OOK . . . 65
4 All-optical phase noise suppression of DPSK signals 74 4.1 Introduction to the regeneration of DPSK signals . . . 74
4.2 Concept and realization of phase noise averaging . . . 75
4.3 Capacity of phase noise averaging . . . 80
4.3.1 Nonlinear phase noise . . . 80
4.4 Analysis of error probability . . . 97 5 Conclusions 105 5.1 Wavelength conversion . . . 105 5.2 Format conversion . . . 106 5.3 Phase regeneration . . . 107 5.4 Future works . . . 108 A Error probabilities of the DPSK format with nonzero threshold 110
B Differential correlative phase noise 113
Publication List 114
List of Tables
List of Figures
1.1 The evolution of WDM networks . . . 3
2.1 Concept of an XPoM wavelength converter. (PC: polarization controller, OBF: optical bandpass filter, and TP: tunable polarizer) . . . 12
2.2 Concept of a DXPoM wavelength converter. (PC: polarization controller, OBF: optical bandpass filter, BDL: birefringent delay line, and TP: tunable polarizer) . . . 13
2.3 Illustration of a differential sinusoidal wave . . . 13
2.4 Schematic transfer matrix . . . 16
2.5 Simulated eye-diagram of converted signal based on XPoM . . . 19
2.6 Simulated eye-diagram of converted signal based on DXPoM with TM delay 19 2.7 Simulated eye-diagram of converted signal based on DXPoM with TE delay 20 2.8 The output phase of TM and TE modes . . . 20
2.9 The phase difference between TM and TE modes . . . 21
2.10 Experimental Setup of DXPoM. (TL: tunable laser; IM: intensity modula-tor; PG: pattern generamodula-tor; PC: polarization controller; OBF: optical band-pass filter; BDL: birefringent delay line; PBS: polarization beam splitter; TP: tunalbe polarizer; Att: optical attenuator; BERT: bit error rate tester) 23 2.11 Measured eye-diagrams of 10 Gb/s converted signal based on (a) XPoM, (b) DXPoM, and (c) XGM . . . 23
2.12 BER curves of wavelength conversion at 10 Gb/s . . . 24
2.13 Frequency response of XPoM and DPoM with different delay time . . . 31
2.14 Amplitude (solid curves) and delay (dashed curves) responses of XPoM, DXPoM with TM delay and DXPoM with TE delay . . . 32
2.15 Simulated eye-diagrams derived from the transfer functions of small-signal model based on (a) XPoM, (b) DXPoM with TM delay, and (c) DXPoM with TE delay . . . 33
2.16 Improvement of 3-dB bandwidth with different γχ . . . 34
2.17 Amplitude (solid curves) and delay (dashed curves) responses with delays of ∆t0 opt, 0.95 × ∆t0opt and 1.05 × ∆t0opt . . . 35
3.1 (a) The transmitter and (b) receiver of the DPSK format . . . 39 3.2 Constellation diagrams of the OOK and DPSK formats . . . 40 3.3 Schematic OOK-to-BPSK format conversion . . . 41 3.4 (a) All SOP of the pump beam used in simulation, and (b) the effective
phase shift with ∆BL À 2π as the function of ψ2 . . . 47
3.5 With 2γP1L = π, the effective phase shift of the best scenario, ψ2 = π/4,
as the function of PPD. . . 48 3.6 Maximum and minimum effective phase shifts as functions of nonlinear
effects and PPD . . . 49 3.7 Experimental setup of 40 Gbps RZ-OOK-to-RZ-DPSK format converter
(TL: tunable laser; SMLL: semiconductor mode-locked laser; EAM: electro-absorption modulator; PS: polarization scrambler; OA: optical amplifier; PC: polarization controller; ATT: optical attenuator; CR: clock recovery; BERT: bit error rate tester) . . . 50 3.8 The experimental and theoretical power differential of XPM-induced
spec-tral pedestal as a function of PPD . . . 52 3.9 The eye-diagrams of converted DPSK signals with the pump at 1547 nm,
while (a) the pump and probe beams were launched at the same birefringent axis; (b) the pump and probe beams were launched at different birefrin-gent axes; (c) the pump was scrambled and the probe was launched at a birefringent axis; (d) the probe was launched at 45◦ relative to birefringent
axes and the pump was adjusted to optimize the signal; (e) the probe was launched at 45◦ relative to birefringent axes and the pump was adjusted
to get the worst signal; (f) the pump was scrambled and the probe was launched at 45◦ relative to birefringent axes. . . 53
3.10 The BER measurement of the converted DPSK signals. (Scr.: scrambled; 45◦: 45◦ relative to the birefringent axes; 0◦: one of the birefringent axes) . 54
3.11 The optical spectra obtained with a resolution bandwidth of 0.05 nm and a PPD of 10 nm . . . 55 3.12 The error probabilities of DPSK signals with various phase shifts . . . 62 3.13 Normalized optimal threshold . . . 63 3.14 The sensitivity of DPSK signals as a function of effective phase shift . . . . 64 3.15 The phase distributions with different SNR of OOK signals . . . 68
3.17 Error probability as a funciton of the finite SNR of the OOK signal . . . . 71
3.18 Sensitivity as a function of ρ1 . . . 73
4.1 Concept of a PNA . . . 77
4.2 Setup of a PNA . . . 78
4.3 The output power and phase of a PSA . . . 79
4.4 (a) Normalized output power, (b) output phase and (c) output phase slope of a PNA . . . 81
4.5 Classification of phase impairments . . . 82
4.6 Optical DPSK transmission system with amplitude regenerators and PNAs inserted every K spans. . . 83
4.7 Multiple PN averaging effect in the transmission system . . . 83
4.8 The variances of differential linear PN as functions of the number of PNAs 86 4.9 The distributions of IFWM-induced PN, ϑn and ϑn−1, generated in (a) SSMF and DCF (b) only SSMF, and (c) only DCF . . . 90
4.10 The correlation coefficients of the IFWM-induced PN . . . 91
4.11 The variances of differential PN as functions of the number of spans . . . . 92
4.12 The curves of error probability as functions of the SNR and nonlinear impairment . . . 93
4.13 The simulation setup of 40 Gbps 33% RZ-DPSK . . . 94
4.14 (a) Normalized IN and (b) differential PN versus transmission distance for 40 Gbps 33% RZ-DPSK . . . 95
4.15 Differential phasor diagrams for 40 Gbps 33% RZ-DPSK after 2800 km transmission. For (a), FWM-based regenerators are inserted, and for (b), FWM-based regenerators and PSA-based PNA are both inserted every 400 km. (c) and (d) are the differential phase eye-diagrams corresponding to (a) and (b), respectively. . . 96
4.16 Schematic of regeneration configuration . . . 97
4.17 Setup of the PN-averaged regenerator. PPAR: phase-preserving amplitude regenerator. . . 99
4.18 (a) pdf of phase distributions with ρ1 = 14 dB, and (b) variance of phase distributions as a function of ρ1 . . . 103
4.19 Error probabilities of DPSK signals with various regeneration schemes (curves: analytical results and markers: Monte Carlo method) . . . 104 4.20 SNR at the error probability of 10−9 . . . 104
List of Abbreviations
AMI alternate-mark inversion
AN amplitude noise
ASE amplified spontaneous emission
BER bit error rate
BPSK binary phase-shift keying
CW continuous wave
DB duobinary
DCF dispersion compensating fiber
DI delay interferometer
DPSK differential phase-shift keying
DQPSK differential quadrature phase-shift keying DXPoM differential cross-polarization modulation EAM electro-absorption modulator
EDFA Erbium doped fiber amplifier
ER extinction ratio
FWHM full width at half maximum
FWM four wave mixing
GVD group-velocity dispersion HNLF highly nonlinear fiber
IFWM intra-channel four wave mixing iid independent identically distributed
IN intensity noise
ISPM isolated pulse self-phase modulation IXPM intra-channel cross-phase modulation LN-MZM LiNbO3 Mach-Zehnder modulator
MAN metropolitan area network
MPS minimum phase system
MZI Mach-Zehnder interferometer
NF noise figure
NLS nonlinear Schr¨odinger
NOLM nonlinear optical loop mirror
NRZ non-return-to-zero
OADM optical add-drop multiplexer
OOK on-off keying
OSA optical spectral analyzer OSNR optical signal-to-noise ratio
OXC optical cross-connect
PC polarization controller
PCF photonics crystal fiber pdf probability density functions
PLL phase-lock loop
PN phase noise
PNA phase noise averager
PPAR phase-preserving amplitude regenerator
PPD pump-probe detuning
PRBS pseudo random binary sequence
PS polarization scrambler
PSA phase-sensitive amplifier
QAM quadrature-amplitude modulation
RMS root mean square
RZ return-to-zero
SBS stimulated Brillouin scattering
SNR signal-to-noise ratio
SOA semiconductor optical amplifier
SOP state of polarization
SPM self-phase modulation
SSMF standard single mode fiber
STD standard deviation
TE transverse electronic
TIA transimpedance amplifier
TM transverse magnetic
TMM transfer matrix method
WDM wavelength division multiplexing
XGM cross-gain modulation
XPM cross-phase modulation
Chapter 1
Introduction
1.1 All-optical signal processing
Due to the advancement of optical communications since 1980s [1], the main function of public communications networks has been modified from voice-only telephony to the data transfer created by internet applications. In fact, while the huge growing requirement of communication bandwidth drives the evolution of optical communication networks at an accelerating rate, the new era for communication networks has come and moved this world toward a global village [2].
Owing to the requirement of network capacity, the line bit rate and the channel num-ber of wavelength division multiplexing (WDM) systems are increasing. Furthermore, the need to enhance network efficiency is also driving traditional point-to-point WDM networks towards meshed flexible WDM networks [3], as shown in Fig. 1.1. Accordingly, the capacity and speed of signal processing in optical networks are increasing dramati-cally. If the processing is carried out in electrical domain, expensive high-speed electronic components, such as electrical amplifiers, optical modulators and photodetectors, must be provided. In order to lower the cost of optical networks, the number of optoelectronic interfaces has to be minimized. Consequently, all-optical signal processing which can realize transparent optical networks without using O/E/O conversion are required in fu-ture cost-effective networks. Various all-optical signal processing have been investigated, such as packet switching [4], logic gates [5], demultiplexing [6][7], wavelength conversion
[8]-[11], format conversion [12]-[14] and regeneration [15]-[20].
To enable a flexible meshed network topology, the functionality and design of optical cross-connects (OXC) and optical add-drop multiplexer (OADM) are key issues. The basic functionalities of these new network elements are switching and routing, which provide the connectivity between the channels on the input ports and those on the output ports [21][22]. Actually, switching and routing in WDM networks can be realized through controlling and converting signal wavelength. Furthermore, the number of wavelength in WDM networks which corresponds to the number of independent wavelength addresses is generally not enough to support all the nodes of a huge network. While two channels with the same wavelength might be routed to the same output port, the blocking probability would rise due to time-slot contention. Wavelength conversion can also overcome this limitation. Therefore, all-optical wavelength converters can increase the flexibility and reduce blocking probability of networks, and have been treated as key components in WDM systems.
In long-distance fiber transmission, optical noise from in-line optical amplifiers and other active optical devices and fiber impairments, such as dispersion and nonlinear effects, can degrade signal performance significantly in transmission. Actually, even in a small-size network, multiple filtering in OXC or OADM can also degrade optical signals. All-optical signal regenerators could be inserted to regenerate optical signals and improve the quality of optical signals, and therefore, they can increase transmission distance and relax the limit of multiple filtering and launch power [17]. Generally speaking, regeneration can be only 1R (amplifying), 2R (amplifying + shaping) or 3R (amplifying + re-shaping + re-timing). Among them, the basic function, re-amplifying, can be simply
OADM OADM OADM OADM OADM OADM OADM OADM OADM OADM OADM OADM OXC OXC OXC OXC OXC WDM transmission
WDM transmission with add/drop
WDM rings with node addressing
Mesh networks with full connectivity
Interconnected multi-ring topology
T ec h n o lo g y e v o lu ti o n OXC OXC
the first thing in transparent regenerators.
Recently, advanced modulation formats have attracted much attention due to their various advantages [23], such as better spectral efficiency, chromatic dispersion tolerance, nonlinear tolerance, and sensitivity to achieve error free, compared with traditional on-off keying (OOK). However, to obtain these advantages, one has to pay the price of a more complex and expensive transmitter or receiver. Considering both cost and system re-quirement, the most suitable modulation format varies in different network architectures, and therefore, all-optical format converters at the node between different networks could connect different modulation formats transparently [12]. Otherwise, an optical network might carry mixed-format signals, and cannot achieve the optimal performance [24].
In the dissertation research, the all-optical wavelength conversion, format conversion and regeneration are covered. To increase the cascadability of wavelength conversion, the modified approach of wavelength conversion is proposed to improve conversion speed. Because the advanced format of differential phase-shift keying (DPSK) carries information on the optical phase domain, a novel method is proposed to manage the phase of DPSK signals. Moreover, a polarization-insensitive OOK-to-DPSK conversion was carried out in a birefringent fiber to avoid mixed-format transmission. In the following sections, the dissertation works with regard to these three parts are introduced further.
1.2 Speed enhancement of semiconductor optical amplifier-based
wave-length conversion
[8], cross-phase modulation (XPM) [8], cross-polarization modulation (XPoM) [9][10], and four wave mixing (FWM) [11]. Each scheme has its own advantages and disadvantages. For example, FWM has low conversion efficiency, and XGM, due to the limitation of carrier’s recovery time, has lower conversion speed and higher frequency chirping. Among all these parameters, conversion speed is considered to be one of the most important factor. This is because insufficient speed response causes larger timing jitter and thus limits cascadability [25].
XPoM is operated on the interferometric principle similar to the Mach-Zehnder inter-ferometer (MZI), given that it exploits the varied phase difference between the transverse electronic (TE) and transverse magnetic (TM) modes of the CW beam induced by the signal beam as the probe beam passes through an SOA. Therefore, similar to XPM which also bases on MZI, XPoM has several advantages, such as pulse reshaping and unre-strained conversion logics [10]. Nonetheless, XPoM is a more economic conversion scheme because only one SOA is needed compared with XPM which typically needs two SOAs. Furthermore, adding an extra delay line to one arm of either XPM or XPoM scheme [26][27] can realize high speed wavelength conversion of return-to-zero (RZ) format, and this kind of modified scheme is unrestricted by the relaxation tail of carrier depletion and recovery in SOAs. Actually, it has been proposed that the similar XPM scheme can also improve the conversion speed of non-return-to-zero (NRZ) format [28]. By adding an extra birefringence delay line in the conventional XPoM, a novel conversion scheme is pro-posed to improve the speed performance, named differential cross-polarization modulation (DXPoM).
To investigate DXPoM, both the large-signal model and small-signal model of DXPoM are developed and successfully predict results that agree with experimental outcomes.
The large-signal model, based on time dependent transfer matrix method (TMM) [9], of DXPoM evidently shows the improvements of rise time, timing jitter and extinction ratio (ER). The small-signal model, based on frequency domain Fourier transform approach, gives more comprehensible physical insight of DXPoM. From the small-signal model, analytic expressions are elaborated to show that the modulation bandwidths of XPoM and XGM are identical and limited by the carrier’s recovery time. From the transfer function of DXPoM, the relations between the modulation bandwidth, the delay time, the operating points, the parameters of an SOA and the conversion logics, are clearly described in both frequency and time domains.
1.3 All-optical OOK-to-Binary phase-shift keying conversion
In DPSK format, an optical pulse appears in each bit slot with the binary data encoded as either a zero or π phase shift between adjacent bits. This format emerged as an alternative to OOK format, especially for long-haul transmission [29]. The most obvious advantage of DPSK format with balanced detection over OOK is that its optical signal-to-noise ratio (OSNR) required to reach a given bit error rate (BER) is approximately 3 dB lower, which implies that the maximum transmission distance could be approximated double if a linear system is considered. However, DPSK format requires more complex transmitter and receiver. Accordingly, DPSK and OOK formats are suitable for networks with different sizes. While the size of networks is large and the transmission distance is long, such as long-hual backbone networks, the DPSK format could be the better one. Oppositely, OOK format should be adopted in a network with smaller size, such as metropolitan area networks (MAN). However, without format conversion in the gateway
be mixed in a WDM system, and it has been found that the phase information of DPSK signals would be seriously distorted by OOK signals through the inter-channel XPM effect [24]. As a result, an OOK-to-DPSK modulation format converter is needed in the nodes to avoid mixed-format transmission.
Intuitively, converting intensity-modulated format to phase-modulated format, such as OOK to binary phase-shift keying (BPSK), can be achieved by using power-dependent XPM effect in a nonlinear medium, such as highly nonlinear fiber (HNLF) and an SOA. While the space of OOK signals do not induce nonlinear phase shift to another probe beam, the mark with specific power is supposed to generate π phase shift to encode binary data on the phase of the probe beam. However, to avoid error propagation at the receiver side while BPSK signals are demodulated by a delay interferometer (DI), the binary data have to be precoded in advance. Basically, this precoding is not a problem, and it can be simply done before generating OOK signals. Therefore, the issue about precoding or how to convert BPSK to DPSK will not be discussed in this dissertation. Moreover, although SOAs could realize polarization insensitive format conversion [13], DPSK signals would be degraded by the XGM effect and slow response time of SOAs, which can be avoided by using ultra-fast (∼fs) χ3 effect in HNLF [14]. Nevertheless, this
fiber-based format converter is polarization sensitive. Recently, a novel photonics crystal fiber (PCF) has been discovered that its birefringent axes remain fixed over the entire length, and this characteristic can perform independent [7] or polarization-insensitive [30] XPM effect. Using the similar approach, a polarization-polarization-insensitive 40 Gb/s RZ-OOK-to-RZ-BPSK format conversion can be realized in a highly nonlinear PCF, and the experimental and theoretical results are shown in this dissertation. The theoretical part also analyzes the penalty of DPSK signals induced by insufficient phase shift, which
originates from improper nonlinear strength or polarization mismatching. Furthermore, as using amplitude-to-phase conversion, the amplitude noise (AN) of OOK signals will also be transferred to the phase noise (PN) of converted DPSK signals. How the OSNR of OOK signals influence the performance of DPSK signals will be carried out theoretically.
1.4 All-optical phase noise suppression of DPSK signals
Compared with OOK signals, besides ∼3-dB lower OSNR required, DPSK signals are also less sensitive to nonlinear effects due to lower peak power and the same power of each pulse, particularly those of self-phase modulation (SPM) and inter-channel XPM [31], improved dispersion tolerance and high spectral efficiency [32]. However, without considering timing jitter, unlike OOK systems that are limited only by AN, DPSK systems are affected by both AN and PN. The PN in DPSK systems will be converted to AN in the receiver through a DI. Furthermore, the PN contains linear PN and nonlinear PN. In dispersion-managed systems, AN and linear PN are generated mainly from the amplified spontaneous emission (ASE) noise of optical amplifiers. The nonlinear PN is translated from AN through the fiber Kerr effect, often called the Gordon-Mollenauer effect [33], and dispersion-induced pattern effects through XPM in WDM systems [34].
Either PN or AN must be prevented from being accumulated to expand the reach of DPSK systems. Unfortunately, most of the all-optical regeneration schemes for OOK format [15]-[17] do not fit the DPSK format, because the phase information is distorted by these regenerators. Theoretically, a phase-sensitive amplifier (PSA) can simultane-ously reduce both AN and PN, and the regeneration was experimentally demonstrated by pumping a PSA with an original undistorted DPSK signal [18]. Even so, the coherent
owing to the requirement of optical-carrier phase-locking between the pump beam and the signal beam. Additionally, several phase-preserving amplitude regeneration approaches for DPSK format have been proposed [19][20]. The reduction of AN is such that the nonlinear PN caused by the Gordon-Mollenauer effect will be reduced [20], and therefore, the transmission distance will be extended due to the reduction of both AN and nonlinear PN. Nevertheless, these regenerators can constrain only some of the nonlinear PN and preserve the original linear PN, and the accumulated PN still distorts DPSK signals and limits the transmission distance.
In this dissertation, a novel all-optical phase noise averager (PNA) based on a PSA is proposed. This PNA can average the PN of one bit with that of its neighboring bit coherently and an extra phase-locking pump beam is not required. These PNAs can increase the correlation between neighboring PNs, effectively diminish differential PN, and greatly extend the reach of DPSK signals. The most important and appealing feature of the proposed PNA is that, when cascaded, the chain of PNAs results in the convergent variance of PN. Particularly, in amplitude-managed linear systems with repeated PNAs, the total differential PN is always less than that before the first averager and is irrelevant to the number of spans.
Recently, a nonlinear optical loop mirror (NOLM) with an SOA [35][36] has been found to yield PN suppression of DPSK signals. Later, this PN suppression has been shown to be achieved by increasing the correlation between neighboring PNs, similar to PN averaging [37]. In this approach, the residual AN after PN averaging could be simply neglected, and it is easier to be analyzed. Hence, to further investigate the efficacy of PN averaging based on the other approach [37], the theoretical analysis of sensitivity improvement by PN averaging is also carried out.
1.5 Organization of the dissertation
In chapter 1, the introduction of all-optical signal processing and the subjects covered in the dissertation are included. In chapter 2, after introducing the concept of XPoM and DXPoM, large-signal simulation, experimental realization and theoretical small-signal analysis are given. In chapter 3, the properties of OOK and DPSK formats are discussed in advance. The concept, theory and experiment of polarization-insensitive OOK-to-DPSK conversion are provided, and then, the impairment due to OOK-to-DPSK conversion is analyzed. In chapter 4, the concept and realization of a PNA are given, as well as the effects of PN averaging on various kinds of PN. Additionally, the PN averaging is examined by the theoretical analysis of error probability. Finally, the conclusions are give in chapter 5.
Chapter 2
Differential cross-polarization modulation in an SOA
2.1 Cross-polarization modulation in an SOA
2.1.1 Concept of XPoM
Figure 2.1 illustrates the conept of XPoM. When a signal pump beam at wavelength λpump is fed into an SOA, the signal light will modulate the carrier density of the SOA.
In addition, a continuous wave (CW) beam (called the probe) which is placed at the desired output wavelength λprobeis transmitted into the SOA simultaneously. Owing to the
asymmetric waveguide geometry, the confinement factors, effective guide refractive indices and carrier distribution of an SOA are not identical at TE and TM orientations, and these differences correlate monotonously with the input signal power through the SOA. When the pump power changes, the nonlinear polarization rotation of the probe beam is induced through carrier density modulation. Moreover, this polarization modulation can be converted into intensity modulation by a polarizer after the SOA, and the ER and the conversion logic of the converted signal can be selected by properly controlling the polarizer in the XPoM scheme. However, the conversion speed of XPoM is limited by the carrier’s response.
2.1.2 Concept of DXPoM
Figure 2.2 presents the configuration of DXPoM. As in a typical XPoM, properly con-trolling the states of polarization (SOP) of λpump and λprobeallows the injected pump light
Ȝprobe Ȝpump Ȝprobe TP OBF PC PC
Figure 2.1: Concept of an XPoM wavelength converter. (PC: polarization controller, OBF: optical bandpass filter, and TP: tunable polarizer)
to introduce additional birefringence in the SOA, resulting in a change in the difference between the refractive indices of the TE and TM modes of the probe beam. At the polar-izer, these two orthogonal modes are partially combined coherently. Namely, the XPoM exploits the phase difference between the TE and TM modes, when the probe beam passes through the SOA, to rotate the polarization state. This phase difference, controlled by the signal power, determines the output power of the CW beam after it has passed through a polarizer. While an extra birefringent delay line is added in front of the polarizer in the DXPoM, the conversion speed of the traditional XPoM could be improved. A simplified and intuitive example illustrated in Fig. 2.3 can help to imagine why this extra delay line improves the conversion speed. Figure 2.3 shows two sinusoidal waves with the same angular frequency, Ω, but different amplitudes, φa and φb. The difference between these
two waves is also a sinusoidal wave with frequency, Ω, and amplitude, φc. If the time
offset, ∆t, as presented in Fig. 2.3, is applied, then the differential amplitude becomes φ2
c = φ2a+ φ2b − 2φaφbcos(Ω∆t). For Ω < π/∆t, the differential amplitude increases with
frequency. Namely, properly selecting ∆t increases the differential amplitudes at some high frequencies. Accordingly, adding an extra delay between the TE and TM modes may amplify some high-frequency components of the phase difference in DXPoM, to
com-in an SOA. As a result, the SOP of the output CW beam after the delay lcom-ine is rotated more rapidly as the signal power varies. In other words, DXPoM has a higher conversion speed and a better performance, compared with XPoM.
Ȝprobe Ȝpump Ȝprobe TP OBF PC PC BDL
Figure 2.2: Concept of a DXPoM wavelength converter. (PC: polarization controller, OBF: optical bandpass filter, BDL: birefringent delay line, and TP: tunable polarizer)
Figure 2.3: Illustration of a differential sinusoidal wave
2.2 Large-signal model
2.2.1 Simulation model of SOA-based wavelength converters
An SOA is generally treated as a two-level system, which is an approach suitable for gaseous and solid state amplifiers. It can be extended for SOAs, if the active region is modeled as a collection of non-interacting two-level systems with transition energies extending over the whole range of the conduction and valence bands. By assuming the
input signal pulse width to be much larger than the intraband relaxation time which governs the dynamics of the induced polarization, the considerable simplification which neglects the intraband processes could be held. In general, the condition for simplification is typically held when signal pulse width is larger than 1 ps. In this approximation, the rate equation of the carrier density in an SOA, N, which responds to the optical power, P , are described by [38], ∂N ∂t = I qV − N τs −X i X k gkPik ~ωiAeff , (2.1)
where i = 1, 2 represent the pump and the probe beam, respectively; k = e, m represent TE and TM component; I is the injection current; q is the elementary charge, V is the active volume; g is the material gain; ~ is the reduced Planck constant; ωi is the optical
angular frequency; Aeff is the effective area of the waveguide, and τs is the lifetime of
the carriers governed by spontaneous emission and nonradiative recombination. In large-signal model, τs is dependent on the carrier density,
τs =
1
c1+ c2N + c3N2
, (2.2)
where c1, c2 and c3 are the coefficients of nonradiative recombination rate, radiative
re-combination rate and Auger rere-combination rate, respectively. Moreover, the rate equation of slow-varying optical envelope, Aik =
√
Pikexp(jφik), is given by,
∂Aik ∂z + 1 vg ∂Aik ∂t = 1 2Γk(1 − jαH)gkAik− 1 2αwAik, (2.3) where φ is the phase of optical field; vg is the group velocity; Γ is the confinement factor;
αw is the waveguide loss, and αH is the linewidth enhancement factor. To consider two
where σg is the differential gain coefficient; N0 is the carrier density at transparency; ² is
the gain compression factor; Sk is the total photon density in k mode,
Sk = X i Pik vg~ωiAeff ,
and ηk is the modified imbalance factor used to describe the asymmetry of optical
tran-sitions between TE and TM modes, when extra strain is built into the active layer of an SOA in order to make an SOA polarization insensitive. This factor is accounted for by an imbalance factor f [9], ηe = 2 3 µ 1 + 2f 1 + f ¶ , ηm = 2 3 µ 2 + f 1 + f ¶ . (2.5)
If extra strain is not applied, then both f and ηk are unity, and the optical transition is
almost isotropic.
Based on Eqs. (2.1) and (2.3), the large signal model is performed by the time-dependent TMM [40][41]. The basis of the TMM is to divide a laser structure longi-tudinally into a number of sections where the structural and material parameters are assumed to be homogeneous throughout each section. However, the parameters may vary between sections to allow longitudinal inhomogeneities, such as those produced by lon-gitudinal spatial hole burning and nonlinear gain, to be incorporated into the model. Each of the sections is characterized by a transfer matrix which modifies the forward and backward traveling waves as they propagate though the section. The use of the TMM here is different from that in the steady-state model where the objective is to obtain the overall transfer matrix for the structure from which the oscillation characteristics are de-termined. In this application, the parameters of the individual section are renovated in
In general, the transfer matrix, Tlof the lthsection expresses the following relationship: Af,l+1 Ab,l+1 = Tl Af,l Ab,l , (2.6)
where Af and Ab are the forward and backward fields, as shown in Fig. 2.4. However,
Eq. (2.6) implies steady-state operation, and to develop a time-dependent implementation of TMM, the transfer matrix should be expressed as Tl(t). Moreover, if the input fields
are Af,l(t) and Ab,l+1(t), the output fields of the section should be Af,l+1(t + ∆t) and
Ab,l(t + ∆t), where ∆t is the transit time of each section. Assuming that Tl(t) remains
unchanged over the interval, t ∼ t + ∆t, Eq. (2.6) can be written as, Af,l+1(t + ∆t) Ab,l+1(t) = Tl11(t) Tl12(t) Tl21(t) Tl22(t) Af,l(t) Ab,l(t + ∆t) . (2.7)
Rearranging Eq. (2.7) produces the expression for the updated fields in terms of the past fields and the transfer matrix elements, it becomes,
Af,l+1(t + ∆t) Ab,l(t + ∆t) = Tl11(t) − Tl12(t)Tl21(t) Tl22(t) Tl12(t) Tl22(t) −Tl21(t) Tl22(t) 1 Tl22(t) Af,l(t) Ab,l+1(t) . (2.8)
2.2.2 Simulation results of DXPoM
The parameters used in the simulation are summarized in Table 2.1. The OOK pump signal is simulated by super Gaussian pulses with the order of 3, and the pattern is the pseudo random binary sequence (PRBS) of 27− 1. The input powers of the pump and
probe beams are 2 and 5 dBm, respectively.
Using time-dependent TMM and the rate equations, the simulation results are shown in Figs. 2.5-2.9. Figs. 2.5-2.7 compare the simulated eye diagrams of 10 Gb/s wavelength conversion employing XPoM, DXPoM with TM delay, and DXPoM with TE delay, re-spectively. In Fig. 2.6, the extra delay added on the TM mode was 13 ps, and in Fig. 2.7, the extra delay added on the TE mode was also 13 ps. These figures illustrate that DX-PoM with TM delay performs much better than the other schemes. The ER is better, the timing jitter is lower and the rise time is markedly improved. However, if the extra delay is added on the TE mode, the conversion performs is even worse than that of XPoM. Because the power gain at the TE mode is 1 dB higher than that at the TM mode in the simulation model, the TE mode induces larger phase changes than the TM mode, as shown in Fig. 2.8. Fig. 2.9 compares the phase difference between the TE and TM modes corresponding to Figs. 2.5-2.7. Due to the asymmetric shape of the phase varia-tion, the phase difference introduced by adding TM delay shows a square-wave type of sharp transition and flatness. Conversely, the phase difference by adding TE delay show continuously slow rising and falling bit patterns. Due to the interferometric principles of XPoM, the phase difference between the TE and TM modes principally controls the output power of the CW beam after the polarizer. Therefore, a square wave style phase difference translates into a square wave style intensity pattern. Furthermore, the slow and irregular rising and falling differential phase bit patterns cause a larger timing jitter
Table 2.1: Device parameters
Symbol Description Value Unit
Γe Confinement factor in TE mode 0.2
Γm Confinement factor in TM mode 0.15
L SOA length 5 × 10−4 m
c1 Nonradiative recombination rate constant 1 × 108 s−1
c2 Radiative recombination rate constant 2.5 × 10−17 m3s−1
c3 Auger recombination rate constant 9.4 × 10−41 m6s−1
σge Differential gain coefficient in TE mode 2 × 10−20 m2
σgm Differential gain coefficient in TM mode 2 × 10−20 m2
N0 Carrier density at transparency 1.1 × 1024 m−3
vg Group velocity 7.5 × 107 m s−1
² Gain compression factor 1.3 × 10−23 m3
αH Linewidth enhancement factor 5
αw Waveguide loss 1.5 × 104 m−1
and eye closure. This roughly explains why the TM delay outperforms the TE delay. 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (ns) Power
Figure 2.5: Simulated eye-diagram of converted signal based on XPoM
0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (ns) Power
0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (ns) Power
Figure 2.7: Simulated eye-diagram of converted signal based on DXPoM with TE delay
0 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 Time (ns) Phase ( π ) TM mode TE mode
0 0.2 0.4 0.6 0.8 1 1.2 Phase difference (0.1 π /div) Time (ns) w/o delay TM delay TE delay
Figure 2.9: The phase difference between TM and TE modes
2.3 Experimental results
Fig. 2.10 shows the experimental setup of a DXPoM wavelength converter. A DFB signal pump laser at 1554.9 nm was intensity modulated at 10 Gb/s with a PRBS length of 27 − 1. A tunable CW probe laser at 1548.5 nm was combined with the signal and
injected into the SOA. The bias current of the SOA (JDSU CQF872) was set at 300mA. The average power of the DFB and the tunable laser was 1.5 dBm and 5.5 dBm, respec-tively. After the SOA, the signal pump beam was filtered out by an optical bandpass filter. The combination of the polarization controller (PC) and the Panda polarization maintained (PM) fiber acted as a tunable birefringence delay line. The maximum differ-ential delay introduced by the PM fiber was approximative 14 ps, which was optimized by using PM fibers of different lengths. Proper control of PC3 allowed the relative delay
between the TE and TM modes to be adjusted, and this produced a desired DXPoM function. The TE and TM modes were then coherently combined at the polarization beam splitter. Figs. 2.11(a) and (b) display the 10 Gb/s eye diagrams of the measured XPoM and DXPoM, respectively. Comparing Figs. 2.11(a) and (b) revealed that the rise time was improved by more than 300%, from 74 ps to 23 ps, the extinction ratio showed a 9% enhancement, from 11 dB to 12 dB, and the root mean square (RMS) timing jitter decreased by 50%, from 5.4 ps to 2.7 ps. In the configuration of Figs. 2.11(a) and (b), PC4 was adjusted to form a destructive interference between the TE and TM modes when only the CW beam was present, thus non-inverted conversion was obtained. As a comparison, Fig. 2.11(c) shows the eye diagram of XGM with the CW beam of -7 dBm and the signal beam of 2.3 dBm. This demonstrated that the used SOA can not support 10 Gb/s XGM due to the constraint of the carrier’s slow recovery time. Fig. 2.12 shows the bit error rate measurement results of source (back-to-back), XPoM and DXPoM. According to this figure, the DXPoM scheme improved the sensitivity by more than 7 dB compared with XPoM. Furthermore, the wavelength conversion itself had a conversion penalty of roughly 1.5 dB compared with the back to back result. The power penalty was contributed mainly by the ASE noise of the SOA by lowering the OSNR. The experiment confirms that the DXPoM is feasible and the performance of the converted signal is improved significantly. However, because the remote pump signal could be arbitrarily polarized after transmitting through fibers, PC1 in Fig. 2.10 has to be an automatic PC to optimize the wavelength converter in practice.
PG
DFB
TL
Oscilloscope
Rx
BERT
PBS
Isolator
Att
OBF
PC1
PC4
PC2
PC3
PandafiberIM
BDL
TP
Figure 2.10: Experimental Setup of DXPoM. (TL: tunable laser; IM: intensity modu-lator; PG: pattern generator; PC: polarization controller; OBF: optical bandpass filter; BDL: birefringent delay line; PBS: polarization beam splitter; TP: tunalbe polarizer; Att: optical attenuator; BERT: bit error rate tester)
˄ˈˁˈʳ̃̆˂˷˼̉
˅˃ʳ̃̆˂˷˼̉
˄ˈˁˈʳ̃̆˂˷˼̉
ʻ˵ʼ
ʻ˴ʼ
ʻ˶ʼ
Figure 2.11: Measured eye-diagrams of 10 Gb/s converted signal based on (a) XPoM, (b) DXPoM, and (c) XGM
−22 −20 −18 −16 −14 −12 −10 10−12 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 Received Power (dBm) BER Back−to−back XPoM DXPoM
Figure 2.12: BER curves of wavelength conversion at 10 Gb/s
2.4 Small-signal model
Since the time-consuming large-signal simulations and experiments with some invari-able parameters are difficult to look into the connections between the conversion perfor-mance and other parameters, based on frequency domain Fourier transform approach, the small-signal model was developed to obtain more comprehensible and intuitive physical insight of DXPoM.
2.4.1 Small-signal model of XPoM
agation functions of the optical fields in different polarization modes in an SOA can be described as, dN dt = I qV − N τs0 −X i X k · σgk(ηkN − N0) Pik ~ωiAeff ¸ , (2.9) dPik dz = Γkσgk(ηkN − N0)Pik, (2.10)
where the retarded time frame, t − z/vg → t, is adopted, and the waveguide loss and the
gain compression effect described by ² are neglected. By defining the integrated carrier density as,
N(t, z) = Z z
0
N(t, z0) dz0, (2.11)
the solution of Eq. (2.10) is,
Pik(t, z) = Pik(t, 0) exp[Γkσgk(ηkN − N0z)] , (2.12)
and the phase of the probe beam is, φ2k = −
αH
2 Γkσgk(ηkN − N0z) . (2.13) Using Eq. (2.10) and integrating both sides of Eq. (2.9) with respect to z yield,
dN dt = − N − N0 τs0 −X i X k Pik(t, z) − Pik(t, 0) Γk~ωiAeff , or, dN dt = − N − N0 τs0 −X i X k Pik(t, 0) Γk~ωiAeff n exp[Γkσgk(ηkN − N0z)] − 1 o , (2.14) where N0 = Izτs0 qV .
While the input pump signal is modulated harmonically, the input powers can be repre-sented as,
because P2k(t, 0) is the CW probe. Assuming ∆P0,1k ¿ P0,1k, the integrated carrier
density, the output probe power, and the output probe phase can be written as, N(t, L) = N + ∆NejΩt+ c.c. ,
P2k(t, L) = P2k+ ∆P2kejΩt+ c.c. , (2.16)
φ2k(t, L) = φ2k+ ∆φ2kejΩt+ c.c. ,
where L is the length of the SOA. Using Eq. (2.14) and considering only the first-order terms, the small-signal response of the integrated carrier density can be written as,
∆N = − X k G1k− 1 ΓkσgkηkPsat,1k ∆P0,1k jΩτs0+ 1 + X i X k GikP0,ik Psat,ik , (2.17) where Gik = exp £ Γkσgk(ηkN − N0L) ¤ , Psat,ik = ~ωiAeff τs0σgkηk ,
are the bias gain and the saturation power, respectively. Similarly, the small-signal response of the intensity and phase of the probe beam are,
∆P2k = G2kP0,2kΓkσgkηk∆N , (2.18)
∆φ2k = −
αH
2 Γkσgkηk∆N . (2.19)
Since the TE and TM modes of the probe beam interfere at the tunable polarizer which consists of a PC and a polarizer, the relation between the fields before and after the tunable polarizer has to be derived in advance. Using Jone’s matrix, the field after the tunable polarizer can derived by,
Eout 1 0 e jπ/2 0 cos θ1 sin θ1
× ejπ/4 0 0 e−jπ/4 cos θ2 sin θ2 − sin θ2 cos θ2 | {z } quarter-wave plate Ex Ey , (2.20)
where Ex and Ey are the input field at orthogonal modes, and θ1 and θ2 are the angles of
the half-wave and quarter-wave plates, respectively. Accordingly, the output field is, Eout = ej
3π
4 [(cos θ1cos θ2+ j sin θ1sin θ2)Ex
+(cos θ1sin θ2− j sin θ1cos θ2)Ey] . (2.21)
Owing to
|(cos θ1cos θ2+ j sin θ1sin θ2)|2+ |(cos θ1sin θ2− j sin θ1cos θ2)|2 = 1 ,
Eq. (2.21) can be simplified by defining,
(cos θ1cos θ2+ j sin θ1sin θ2)Ex = cos θ · ejϑx|Ex| ,
(cos θ1sin θ2− j sin θ1cos θ2)Ey = − sin θ · ejϑy|Ey| . (2.22)
Since θ and ϑy− ϑx are the functions of two variables, θ1 and θ2, they could be adjusted
independently. If Ex and Ey correspond to the TE and TM modes of the probe beam,
the power of the probe beam after the tunable polarizer can be represented as, PT P(t, L) = 1 2 ¯ ¯ ¯cos θpP2e(t, L) − sin θ p P2m(t, L) · ej(ϑ+∆φ) ¯ ¯ ¯2 , (2.23) where ϑy− ϑx = ϑ + ∆φ, and ∆φ = ∆φ2m− ∆φ2e is the phase difference between TE and
TM modes. In order to obtain maximum ER after the interference, θ needs to be chosen to make sure that the TE ad TM modes beyond the polarizer have equal DC intensity, i.e., cos2θ · P
2e = sin2θ · P2m= Pθ, and the selection of ϑ is to ensure that PT P2 (t, L) has a
operation, where ∆φoff is the phase difference when the input pump signal is logical zero. Thus, under this condition, the small-signal response of XPoM, ∆PT P, given by Eq. (2.23)
could be written as, ∆PT P = 1 2(1 − cos ϑ) ¡ cos2θ · ∆P 2e+ sin2θ · ∆P2m ¢ + Pθsin ϑ · ∆φ . (2.24)
The first and second terms of the right hand side of Eq. (2.24) are associated with XGM and XPM effects, respectively. From Eq. (2.19), if there is no polarization dependent gain, i.e. ∆φ2m= ∆φ2e, the remaining contribution of Eq. (2.24) is only the XGM effect.
By defining the lifetimes of stimulated recombination due to the pump and probe beams as, 1 τstim,i =X k GikP0,1k τs0Psat,ik , (2.25)
Eqs. (2.17)-(2.19), (2.24) and (2.25) yield,
∆PT P = [χ (1 + rme) − (1 − rme)] αHsin ϑ 2τs0 X k · ΓeσgeηePθ ΓkσgkηkPsat,1k (G1k− 1) ∆P0,1k ¸ jΩ + 1 τs0 + 1 τstim,1 + 1 τstim,2 , (2.26) where χ = − 1 αH tan ϑ 2 , rme = Γmσgmηm Γeσgeηe ,
are the factors associated to the operating point and the polarization dependence of an SOA. Restated, the first and second terms in the first bracket of Eq. (2.26) repre-sent the contributions from XGM and XPM effects, respectively. Moreover, the ratio of XGM to XPM parts contributing to the output small-signal response can be
writ-χ < 0 represent that XPoM works with non-inverted and inverted conversion scheme, respectively.
Furthermore, from Eqs. (2.17) and (2.18), the small-signal response of XGM is, X k ∆P2k = − 1 τs0 X k X k0 ΓkσgkηkP2k Γk0σgk0ηk0Psat,1k0 (G1k 0 − 1) ∆P0,1k0 jΩ + 1 τs0 + 1 τstim,1 + 1 τstim,2 . (2.27)
It is evident from Eqs. (2.26) and (2.27) that the modulation bandwidths of XPoM and XGM are the same and both are limited by the carrier’s recovery time.
2.4.2 Small-signal model of DXPoM
In this section, the analytical small-signal response of DXPoM is carried out to explain the relations between conversion performance and various operating parameters. If the delay time, ∆t, is added on TM modes, the intensity and phase responses of TM mode described in Eqs. (2.18) and (2.19) would become:
∆P2k = G2kP0,2kΓkσgkηk∆N · e−jΩ∆t, (2.28)
∆φ2k = −
αH
2 Γkσgkηk∆N · e
−jΩ∆t. (2.29)
Using Eqs. (2.24), (2.28) and (2.29), the small-signal response of DXPoM is, ∆PT P = χ¡1 + rmee−jΩ∆t ¢ −¡1 − rmee−jΩ∆t ¢ jΩ + 1 τs0 + 1 τstim,1 + 1 τstim,2 ×αHsin ϑ 2τs0 X k · ΓeσgeηePθ ΓkσgkηkPsat,1k (G1k− 1) ∆P0,1k ¸ . (2.30)
It is clear that the modulation bandwidth is determined by the first part in Eq. (2.30). Therefore, by dividing this part by (χ − 1)τT, it can be difined as a transfer function,
T (Ω),
T (Ω) = 1 − γχe
−jΩ∆t
where 1 τT = 1 τs0 + 1 τstim,1 + 1 τstim,2 , γχ = rme 1 + χ 1 − χ.
Unfortunately, in Eq. (2.31), the analytical expression of 3-dB bandwidth cannot be obtained after the time delay, ∆t, is applied. Therefore, as shown in Fig. 2.13, several numerical examples are given to illustrate the substantial improvement of the conversion bandwidth. With τT = 5 × 10−11s and γχ = 0.8, the 3-dB bandwidths of XPoM and
DXPoM under 2.5, 5, 10, 20, and 40 ps delay are 5.5, 7.0, 69.9, 67.3, 38.4, and 41.3 GHz, respectively. With proper delay, the conversion bandwidth can be improved by more than 1000%.
When ∆t is smaller than zero, the TM mode will get ahead of the TE mode and the DXPoM is operated with TE delay. By defining T (Ω) = |T (Ω)| exp (jΘ(Ω)), the group delay of the transfer function is −dΘ/dΩ. With the same operating parameters, the conversion bandwidth and the group delay of the TE and TM delay are shown in Fig. 2.14. Although the amplitude responses of DXPoM with TM and TE delay are identical, the TE delay has the largest phase variation, which will induce to the worst timing delay disparity and will cause the biggest converted signal distortion compared with XPoM or DXPoM with TM delay. Actually, the same conclusions are reached in large-signal simulation as well, as shown in Figs. 2.6 and 2.7. Compared with XPoM, DXPoM with TM delay has better conversion bandwidth and lower timing jitter due to flatter delay response. In order to portray the effects of phase response to the signal distortion, the PRBS input signal spectrum is directly multiplied by the transfer functions
100 101 102 −20 −15 −10 −5 0 5 −3
Frequency (GHz)
Response (dB)
∆
t = 0
∆
t = 2.5 ps
∆
t = 5 ps
∆
t = 10 ps
∆
t = 20 ps
∆
t = 40 ps
Figure 2.13: Frequency response of XPoM and DPoM with different delay time back to time domain. One can clearly see the eye closure with TE delay due to the worse phase response.
Furthermore, normalizing Ω and ∆t by τT, Eq. (2.31) can be re-written as,
T0(Ω0) = 1 1 − γχ
×1 − γχe
−jΩ0∆t0
1 + jΩ0 , (2.32)
where T0, Ω0 and ∆t0 are the normalized transfer function, angular frequency and delay
time, respectively. Thus, γχis the only variable which will affect the conversion bandwidth
under different normalized delay times. Figure 2.16 illustrates the relationships between normalized 3 dB conversion bandwidth and time delay under different γχ. The maximum
100 101 102 −25 −20 −15 −10 −5 0 5
Frequency (GHz)
Amplitude response (dB)
100 101 102 −50 0 50 100 150 200 250Delay (ps)
∆
t = 0
∆
t = 10 ps
∆
t = −10 ps
Figure 2.14: Amplitude (solid curves) and delay (dashed curves) responses of XPoM, DXPoM with TM delay and DXPoM with TE delay
factor of the bandwidth, and the bandwidth will be decreased with any extra birefringence delay.
Accordingly, the conversion performance is determined by not only the bandwidth of amplitude response but also the flatness of group delay from phase response. In other words, for a transfer function, the flat and large pass band and the linear phase response corresponding to |T0(Ω0)| and Θ0(Ω0), respectively, are required. For example, the
DX-PoM with TM and TE delay have the same bandwidth improvement but they perform differently due to the unlike group delays. Therefore, the delay response with ∆t0
corre-0.05 0.1 0.15 0.2 0 0.5 1 1.5 2 2.5 Time (ns) Power 0.05 0.1 0.15 0.2 0 0.5 1 1.5 2 2.5 Time (ns) Power 0.05 0.1 0.15 0.2 0 0.5 1 1.5 2 2.5 Time (ns) Power
Figure 2.15: Simulated eye-diagrams derived from the transfer functions of small-signal model based on (a) XPoM, (b) DXPoM with TM delay, and (c) DXPoM with TE delay s = jΩ0 in the normalized transfer function in Eq. (2.32), the transfer function becomes,
H(s) = 1 1 − γχ
× 1 − γχe
−s∆t0
1 + s . (2.33)
Then, T0(Ω0) can be derived by evaluating H(s) on the imaginary axis in the complex
s plane. The poles and zeros of Eq. (2.33) are pH = −1, −∞ + j2nπ/∆t0 and zH =
ln γχ/∆t0 + j2nπ/∆t0, respectively, where n are all integrals. When all poles of a causal
system are in the left half of the s plane, this system is stable. If all zeros are also in the left half of the s plane simultaneously, this system is said to be a minimum phase system (MPS) [43]. An important property of MPS is that Θ0(Ω0) can be got by applying
Hilbert transform to ln |T0(Ω0)|. If ∆t0 is positive and γ
χ lies in between 0 and 1 which
correspond to the condition of DXPoM with TM delay, the whole zeros of Eq. (2.33) are in the left half of the s plane. Namely, DXPoM with TM delay is an MPS and its amplitude and phase responses are not independent. It has been shown that a constant logarithmic amplitude response implies linear phase response as they are a Hilbert transform pair [43].
0 0.5 1 1.5 −2 0 2 4 6 8 10 12
Normalized delay time
Improvement of 3−dB bandwidth (dB)
γ
χ= 0.8
γ
χ= 0.6
γ
χ= 0.4
γ
χ= 0.2
γ
χ= −0.2
Figure 2.16: Improvement of 3-dB bandwidth with different γχ
is getting flatter until the largest bandwidth is approximately achieved. Consequently, relying on the properties of MPS and Hilbert transform, when ∆t is selected to get the largest bandwidth, the delay response is almost constant in the pass band and the best performance of DXPoM with TM delay is obtained simultaneously. Using second-order approximation and forcing |T0(Ω0)| = 1, yield the optimum delay,
∆t0opt = 1 − γ√ χ γχ
, (2.34)
tions of the amplitude responses. The good match at low frequency indicates that this approximation can be applied to determine the optimum delay. Furthermore, a nearly constant delay response and the flattest amplitude response are achieved simultaneously by applying ∆t0 opt. 10−1 100 101 −2 −1.5 −1 −0.5 0 0.5
Normalized angular frequency
Amplitude response (dB)
10−1 100 101 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2Delay (ps)
∆
t’
opt0.95
⋅
∆
t’
opt1.05
⋅
∆
t’
opt2
nd−order approx.
Figure 2.17: Amplitude (solid curves) and delay (dashed curves) responses with delays of ∆t0
opt, 0.95 × ∆t0opt and 1.05 × ∆t0opt
The information concerning the time domain can be obtained from the impulse re-sponse, which is the inverse Fourier transform of the frequency response of the small-signal model. Taking inverse Fourier transform of Eq. (2.32), the impulse response is,
h(t0) = F−1{T0(Ω0)} = √ 2π 1 − γχ h e−t0us(t0) − γχ· e−(t 0−∆t0) us(t0 − ∆t0) i , (2.35) Rt
Figure 2.18 illustrates several impulse responses with different γχ and ∆t0. Fig. 2.18(a) is
XPoM without an extra delay, and Fig. 2.18(b) is DXPoM with the TM delay of 0.35. It’s obvious that Fig. 2.18(b) has narrower impulse response due to the extra delay canceling the relaxation tail. However, increasing the extra delay may enlarge the response time, as shown in Fig. 2.18(c), which is still better than Fig. 2.18(a). This also explains the change of the bandwidths in Fig. 2.13. While γχ is closer to 1, as shown in Fig. 2.18(d), smaller
time delay is needed to reach the maximum bandwidth which comes to the same conclusion as in Fig. 2.16. Furthermore, with negative γχ as in Fig. 2.18(e), the impulse response
is distorted by the extra time delay and causes the bandwidth decrease as depicted in Fig. 2.16. Lastly, Fig. 2.18(f) plots the the case with TE delay. Although the TE delay and the TM delay perform the same bandwidth, as shown in Fig. 2.14, due to distinct phase performance, the impulse responses of Figs. 2.18(b) and (f) are so different that the TE delay couldn’t improve the conversion performances.
2.4.3 Discussion
From the small-signal model, the optimized delay is a constant, if the operation con-ditions are fixed, such as the injection current, the injection powers, the SOP of both the pump and probe beams, and the tunable polarizer. Namely, once the operation condi-tions are selected to maximize the conversion bandwidth, the desired delay is irrelative to the bit rate of incoming signals. However, the remote pump beam could be arbitrarily polarized after transmitting through fibers, and the SOP could vary significantly in time periods of about 1 ms [44]. Therefore, a polarization stabilizer is required to realize and optimize DXPoM in practice. Actually, both XPoM and DXPoM need the automatic
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Chapter 3
All-optical OOK-to-PSK conversion
3.1 Introduction to the OOK and DPSK formats
Considering both transmitter and receiver, the OOK format, which carries binary data by turning on and off the optical power, is the simplest format to be carried out. At the transmitter side, besides using an external modulator, such as a LiNbO3 Mach-Zehnder
modulator (LN-MZM) or an electro-absorption modulator (EAM), even a direct modula-tion laser can be adopted to generate OOK signals economically. On the receiver side, only an optical detector with suitable bandwidth is needed. However, as transmission distance increases and channel spacings in WDM systems decrease, the conventional OOK format is not suitable for all the application anymore. Accordingly, various modulation formats which show different benefits, such as higher dispersion tolerance, lower sensitivity, higher nonlinear tolerance, and higher spectral efficiency, have been proposed [23]. Actually, the coherently detected BPSK format had attracted a lot of attention before the success-ful introduction of Erbium doped fiber amplifiers (EDFA). To avoid the requirement of phase-lock loops (PLL) in coherent BPSK receivers, the DPSK format carries information by differential phase between two neighboring pulses. The transmitter of DPSK signals is almost identical to that of BPSK signals, but the self interference of DPSK signals by a DI makes its receiver to be implemented much easier. Figure 3.1(a) shows the transmitter of the DPSK format. To avoid error propagation at the receiver, an electrical precoder is
