A WAVEFORM DIVISION MULTIPLEXING SCHEME FOR FIBER- OPTIC COMMUNICATION SYSTEM

Transcription

1 A WAVEFORM DIVISION MULTIPLEXING SCHEME FOR FIBER- OPTIC COMMUNICATION SYSTEM i

2 A WAVEFORM DIVISION MULTIPLEXING SCHEME FOR FIBER- OPTIC COMMUNICATION SYSTEM BY ZEYU HU, B.ENG. (HUAZHONG UNIV. OF SCI. & TECH.) A Thesis Submitted to the School of Graduate Studies in Partial Fulfillment of the Requirements for the Degree Master of Applied Science in Electrical Engineering McMaster University Zeyu Hu, 2014 ii

3 McMaster University MASTER OF APPLIED SCIENCES (2014) (Electrical and Computer Engineering) TITLE: A New Waveform Multiplexing for Fiber-Optic Communication AUTHOR: Zeyu Hu, B.Eng. Huazhong Univ. of Sci. & Tech. SUPERVISOR: Prof. Xun Li, Department of Electrical and Computer Engineering NUMBER OF PAGES: xiv, 106 iii

4 Abstract The increasing demand for large capacity of the fiber-optic communication system leads people to look at a variety of possibilities for the multiplexing technique. A multiplexing fiber-optic communication system based on the orthogonality of the waveform, Orthogonal Waveform Division Multiplexing (OWFDM), is proposed to increase the capacity of fiber-optic communications systems. The OWFDM system is designed and analyzed in this work. The generation of signals, the modulation process, the wave propagation in fiber and the technique of signals extraction are presented in this paper. Simulations in each section have been conducted by numerical algorithms on Matlab. The Chebyshev polynomials of the second kind are selected as the pulse shape. The reason of waveform selection is also given. To ensure that a set of the orthogonal Chebyshev polynomials of the second kind can be obtained at the receiver, the inverse process is performed from the receiver to the transmitter. Based on the inverse process, simulation results of inputs and outputs of sections including the receiver, fiber, modulator and digital filters are presented. Verification through the whole process is given in this work. Simulation results of the whole process from the initial input end to the output end are presented. Finally, the orthogonality of the waveform at the receiver has been tested. The test result confirms that OWFDM can increase the capacity of the fiber-optic communication in the future. iv

5 Acknowledgements First of all, I would like to sincerely express my thanks to my supervisor Professor Xun Li for his inspiring mentorship, spirit of freedom to explore and create in science, patience, and his financial support. These years with his effective mentoring, are the most unforgettable time of my academic career. He gives me a lot of encouragement and guidance throughout my research process, and helps me to establish self-confidence. Secondly, I feel I am very lucky and grateful to have Professor Shiva Kumar, Professor James P. (Jim) Reilly, Professor Jiankang Zhang and Professor T. Kirubarajan. I have learned a lot through their courses. Then I want to thank Cheryl Gies who is always willing to help me and answer questions from us, as well as my classmates Zhao Sangzhi. Finally I want to express my thanks to my parents for their unconditional love during these years. v

6 Table of Contents Chapter 1 Introduction Background Motivation Thesis Organization... 9 Chapter 2 Configuration of OWFDM System OWFDM System Configuration Signal Waveform Selection Chapter 3 Receiver Design Design scheme of receiver Receiver Simulation Chapter 4 Optical Waves Propagation in Fiber Maxwell s Equations Pulse Propagation Equation Split-Step Method Backward Propagation Chapter 5 Transmitter Design Transmitter Structure Modulator Design Generation of Electric Driving Signals Chapter 6 System Simulation Chapter 7 Conclusion Bibliography vi

7 List of Figures and Tables Fig. 1.1 Basic Fiber Optic Communication System... 7 Fig. 2.1 OWFDM System Diagram Fig. 2.2 The first few Chebyshev polynomials of the second kind Fig. 3.1 Diagram of the receiver structure of the OWFDM system Fig. 3.2 Expected electrical signals from each channel Fig. 3.3 The optical wave amplitudes at the end of the fiber Fig. 4.1 Expected field amplitude of the first order wave before fiber transmission (the first set of fiber specifications) Fig. 4.2 Expected phase of the first order wave before fiber transmission (the first set of fiber specifications) Fig. 4.3 Expected field amplitude of the second order wave before fiber transmission... (the first set of fiber specifications) Fig. 4.4 Expected phase of the second order wave before fiber transmission (the first set of fiber specifications) Fig. 4.5 Expected field amplitude of the third order wave before fiber transmission (the first set of fiber specifications) Fig. 4.6 Expected phase of the third order wave before fiber transmission (the first set of fiber specifications) Fig. 4.7 Expected field amplitude of the fourth order wave before fiber transmission (the first set of fiber specifications) Fig. 4.8 Expected phase of the fourth order wave before fiber transmission (the first set of fiber specifications) Fig. 4.9 The field amplitude of the first set at end of fiber (the second set of fiber specifications) Fig Expected field amplitude of the first order wave before fiber transmission (the second set of fiber specifications) Fig Expected phase of the first order wave before fiber transmission (the second set of fiber specifications) Fig Expected field amplitude of the second order wave before fiber transmission (the second set of fiber specifications) vii

8 Fig Expected phase of the second order wave before fiber transmission (the second set of fiber specifications) Fig Expected field amplitude of the third order wave before fiber transmission (the second set of fiber specifications) Fig Expected phase of the third order wave before fiber transmission (the second set of fiber specifications) Fig Expected field amplitude of the fourth order wave before fiber transmission (the second set of fiber specifications) Fig Expected phase of the fourth order wave before fiber transmission (the second set of fiber specifications) Fig The field amplitude at end of fiber (the second set of fiber specifications). 38 Fig. 5.1 The transmitter structure of the OWFDM system Fig. 5.2 Mach-Zehnder modulator Fig. 5.3 Expected modulator's input voltage V 1 of the first order (the first set of fiber specifications) Fig. 5.4 Expected modulator's input voltage V 2 of the first order (the first set of fiber specifications) Fig. 5.5 Expected modulator's input voltage V 1 of the second order (the first set of fiber specifications) Fig. 5.6 Expected modulator's input voltage V 2 of the second order (the first set of fiber specifications) Fig. 5.7 Expected modulator's input voltage V 1 of the third order (the first set of fiber specifications) Fig. 5.8 Expected modulator's input voltage V 2 of the third order (the first set of fiber specifications) Fig. 5.9 Expected modulator's input voltage V 1 of the fourth order (the first set of fiber specifications) Fig Expected modulator's input voltage V 2 of the fourth order (the first set of fiber specifications) Fig Expected modulator's input voltage V 1 of the first order (the second set of fiber specifications) Fig Expected modulator's input voltage V 2 of the first order (the second set of fiber specifications) viii

9 Fig Expected modulator's input voltage V 1 of the second order (the second set of fiber specifications) Fig Expected modulator's input voltage V 2 of the second order (the second set of fiber specifications) Fig Expected modulator's input voltage V 1 of the third order (the second set of fiber specifications) Fig Expected modulator's input voltage V 2 of the third order (the second set of fiber specifications) Fig Expected modulator's input voltage V 1 of the fourth order (the second set of fiber specifications) Fig Expected modulator's input voltage V 2 of the fourth order (the second set of fiber specifications) Fig The block diagram of the digital filter Fig FIR filter parameters h(n) for V 1 of the first order (the first set of fiber specifications) Fig FIR filter parameters h(n) for V 2 of the first order (the first set of fiber specifications) Fig FIR filter parameters h(n) for V 1 of the second order (the first set of fiber specifications) Fig FIR filter parameters h(n) for V 2 of the second order (the first set of fiber specifications) Fig FIR filter parameters h(n) for V 1 of the third order (the first set of fiber specifications) Fig FIR filter parameters h(n) for V 2 of the third order (the first set of fiber specifications) Fig FIR filter parameters h(n) for V 1 of the fourth order (the first set of fiber specifications) Fig FIR filter parameters h(n) for V 2 of the fourth order (the first set of fiber specifications) Fig FIR filter parameters h(n) for V 1 of the first order (the second set of fiber specifications) Fig FIR filter parameters h(n) for V 2 of the first order (the second set of fiber specifications) ix

10 Fig FIR filter parameters h(n) for V 1 of the second order (the second set of fiber specifications) Fig FIR filter parameters h(n) for V 2 of the second order (the second set of fiber specifications) Fig FIR filter parameters h(n) for V 1 of the third order (the second set of fiber specifications) Fig FIR filter parameters h(n) for V 2 of the third order (the second set of fiber specifications) Fig FIR filter parameters h(n) for V 1 of the fourth order (the second set of fiber specifications) Fig FIR filter parameters h(n) for V 2 of the fourth order (the second set of fiber specifications) Fig Amplitude of the modulator's output of the first order (the first set of fiber specifications) Fig Phase of the modulator's output of the first order (the first set of fiber specifications) Fig Amplitude of the modulator's output of the second order (the first set of fiber specifications) Fig Phase of the modulator's output of the second order (the first set of fiber specifications) Fig Amplitude of the modulator's output of the third order (the first set of fiber specifications) Fig Phase of the modulator's output of the third order (the first set of fiber specifications) Fig Amplitude of the modulator's output of the fourth order (the first set of fiber specifications) Fig Phase of the modulator's output of the fourth order (the first set of fiber specifications) Fig Amplitude of the modulator's output of the first order (the second set of fiber specifications) Fig Phase of the modulator's output of the first order (the second set of fiber specifications) Fig Amplitude of the modulator's output of the second order (the second set of fiber specifications) x

11 Fig Phase of the modulator's output of the second order (the second set of fiber specifications) Fig Amplitude of the modulator's output of the third order (the second set of fiber specifications) Fig Phase of the modulator's output of the third order (the second set of fiber specifications) Fig Amplitude of the modulator's output of the fourth order (the second set of fiber specifications) Fig Phase of the modulator's output of the fourth order (the second set of fiber specifications) Fig. 6.1 Input binary signal of the first order (the first set of fiber specifications) Fig. 6.2 Input binary signal of the second order (the first set of fiber specifications).. 72 Fig. 6.3 The modulator's input voltage V 1 of the first order (the first set of fiber specifications) Fig. 6.4 The modulator's input voltage V 2 of the first order (the first set of fiber specifications) Fig. 6.5 The modulator's input voltage V 1 of the first order (the first set of fiber specifications) Fig. 6.6 The modulator's input voltage V 2 of the first order (the first set of fiber specifications) Fig. 6.7 The amplitude of the modulator's output field of the first order (the first set of fiber specifications) Fig. 6.8 The phase of the modulator's output field of the first order (the first set of fiber specifications) Fig. 6.9 The amplitude of the modulator's output field of the second order (the first set of fiber specifications) Fig The phase of the modulator's output field of the second order (the first set of fiber specifications) Fig Fiber's output power of the first order (the first set of fiber specifications).. 77 Fig Eye-diagram of fiber's output power of the first order (the first set of fiber specifications) Fig Fiber's output power of the second order (the first set of fiber specifications) xi

12 Fig Eye-diagram of fiber's output power of the second order (the first set of fiber specifications) Fig The input mixed signal of the first order and the second order (the first set of fiber specifications) Fig The received mixed signal of the first order and the second order (the first set of fiber specifications) Fig The output of ideal integrator of the first order (the first set of fiber specifications) Fig The output of ideal integrator of the second order (the first set of fiber specifications) Fig Input binary signal of the first order (the second set of fiber specifications) 82 Fig Input binary signal of the second order (the second set of fiber specifications) Fig The modulator's input voltage V 1 of the first order (the second set of fiber specifications) Fig The modulator's input voltage V 2 of the first order (the second set of fiber specifications) Fig The modulator's input voltage V 1 of the second order (the second set of fiber specifications) Fig The modulator's input voltage V 2 of the second order (the second set of fiber specifications) Fig The amplitude of the modulator's output field of the first order (the second set of fiber specifications) Fig The phase of the modulator's output field of the first order (the second set of fiber specifications) Fig The amplitude of the modulator's output field of the second order (the second set of fiber specifications) Fig The phase of the modulator's output field of the second order (the second set of fiber specifications) Fig Fiber's output power of the first order (the second set of fiber specifications) Fig Eye-diagram of fiber's output power of the first order (the second set of fiber specifications) xii

13 Fig Fiber's output power of the second order (the second set of fiber specifications) Fig Eye-diagram of fiber's output power of the second order (the second set of fiber specifications) Fig The input mixed signal of the first order and the second order (the second set of fiber specifications) Fig The received mixed signal of the first order and the second order (the second set of fiber specifications) Fig The output of ideal integrator of the first order (the second set of fiber specifications) Fig The output of ideal integrator of the second order (the second set of fiber specifications) Table 1.1 Overview of recent achievements in high-capacity fiber-optic communication systems... 3 Table 4.1 Fiber specifications: SMF indicates single mode fiber Table 4.2 The second set of fiber specifications Table 5.1 Laser specifications Table 5.2 Modulator Specifications xiii

14 List of Abbreviations and Symbols DFB FIR FPGA MZM OFDM OWFDM PDM QCSE SDM SMF TDM WDM Distributed Feedback Laser Finite Impulse Response Field Programmable Gate Array Mach-Zehnder Modulator Optical Frequency Division Multiplexing Orthogonal Waveform Division Multiplexing Polarization Division Multiplexing Quantum Confined Stark Effect Space Division Multiplexing Single-Mode Fiber Time-Division Multiplexing Wavelength Division Multiplexing xiv

15 Master Thesis - Zeyu Hu Chapter 1 Introduction 1.1 Background Over the past few decades, there has been explosive growth in data transmission and signal processing using optical communication systems. The research on fiber-optic communication systems started around After several years of research, the first commercial fiber-optic communication system was developed, which operated at a wavelength of approximately 0.8 μm and using GaAs semiconductor lasers. The small repeater spacing ( 1 km), which was a drawback of this system, made it relatively expensive to operate. In the early 1980s, following the development of InGaAsP semiconductor lasers, the repeater spacing increased significantly [1]. These fiber-optic communication systems operated at a wavelength of 1.3 µm [2]. However, the bit rates of early systems were limited to less than 100 Mb/s because of the dispersion in multimode optical fiber [3]. Even though the application of single-mode fiber was developed to overcome the limitation, dispersion and loss remained as obstacles to the enhancement of properties of fiber-optic communication systems. Worldwide efforts were made to reduce dispersion and loss 1

16 Master Thesis - Zeyu Hu for the development of a new fiber-optic communication system. Researchers found that both dispersion and loss reached minimum near the wavelength of 1.55 μm, which was attained using the dispersion-shifted fiber [4]. By 1985, laboratory-based experimental systems could operate at 2.5 Gb/s with repeater spacing in excess of 100 km and losses of 0.2 db/km [5]. The late 1990s saw a dramatic increase in optical communication system capacity through the utilization of various division multiplexing technologies. In 1996, the first 1 Tb/s system with tens of channels was demonstrated. Today, the application of space-division multiplexing (SDM) [6], timedivision multiplexing (TDM) [7], and wavelength-division multiplexing (WDM) with various other division multiplexing-based techniques [8 10] are prevalent in fiberoptic communication systems. For example, the frequency-division multiplexing (FDM) technique was combined with mode-division multiplexing (MDM) [11 14], and polarization-division-multiplexed quadrature phase-shift keying (PDM-QPSK) has been used to increase the capacity of the system [15]. Fiber-optic communication systems have also made major advances in other techniques in recent years [16], including advanced optical modulation techniques, coherent detection combined with the adoption of high speed electronics (i.e., digital signal processors), advanced preparatory techniques for materials and devices, and 2

17 Master Thesis - Zeyu Hu photonic integrated circuit technologies. As a result of these advancements, several other important aspects of fiber-optic communication systems have improved, including spectral efficiency and transmission distance. Table 1.1 shows an overview of properties of recent experimental systems with high transmission capacities and high spectral efficiencies. The highest demonstrated total capacity listed in Table 1.1 is Tb/s with a spectral efficiency of 4.0, which was achieved adopting a WDM transmission technique. Transmission technique Capacity Spectral Efficiency WDM [17] (2014) Tb/s 4.0 (bit/s)/hz SDM [18] (2013) Tb/s 3.2 (bit/s)/hz PDM [19] (2010) 64 Tb/s 8.0 (bit/s)/hz Table 1.1 Overview of recent achievements in high-capacity fiber-optic communication systems The principle of the multiplexing techniques used to increase the capacities of fiber-optic communication systems are briefly described as follows: (a) SDM: The SDM technique enables the creation of parallel spatial channels to enhance the communication capacity through spatial multiplexing [20, 21]. In other words, SDM takes advantage of the space dimension on the optic waveguide to send 3

18 Master Thesis - Zeyu Hu independent signals at the same time. Three types of fiber for SDM transmission applications have been reported: weakly coupled multi-core fiber, strongly coupled multi-core fiber, and multimode fiber (MMF). In principle, the SDM transmission capacity scales linearly with the number of parallel channels, i.e., the total capacity is proportional to the number of modes per core multiplied by the number of cores in fiber [22, 23]. (b) TDM: TDM is a technique for transmitting and receiving independent signals over a common signal path using synchronized switches at each end of the transmission channel, so that each signal only appears on the channel for a fraction of the total time in an alternating pattern. Because TDM utilizes time segmentation in a physical connection, the different signals cannot overlap with each other in the time domain [24]. TDM is a very powerful technique for ultra-high-speed communication systems. However, as the transmission rate in TDM systems increases, higher-order fiber dispersion and coherent cross-talk make long haul transmission difficult. In recent years, some theoretical and experimental works have been conducted to overcome these obstacles [25-28]. (c) WDM: Because a single fiber can transmit optical carriers of different wavelengths at the same time, WDM technology takes this advantage and multiplexes a number of optical carrier signals on a single optical fiber adopting different laser 4

19 Master Thesis - Zeyu Hu light wavelengths. This technique enables bidirectional communication over a single strand of fiber, and leads to multiplication of the system capacity [29]. A WDM system uses a number of multiplexers at the transmitter end, which multiplexes more than one optical signal onto a single fiber. At the receiver end, demultiplexers separate them apart. WDM network splits this into a number of small bandwidths optical channels. As optical network supports huge bandwidth; the entire wavelength range is divided into several bands. Each wavelength band is an independent transmission channel of predetermined frequency for optical signals [30]. The conventional wavelength window, which is known as the C band, covers the wavelength range μm. After the year of 2000, the wavelength range was extended to the L and S bands, which stand for the long and short wavelength regions, respectively. There are different types WDM network, including Coarse Wavelength Division Multiplexing (CWDM) [31] and Dense Wavelength Division Multiplexing (DWDM) [32], etc. In recent years, more complex methods for data encoding on multiple carrier frequencies have been used, including ultra-dense FDM [33] and orthogonal FDM (OFDM) [34]. OFDM is an FDM scheme that is adopted as a digital multi-carrier modulation method. A large number of closely spaced orthogonal subcarrier signals are utilized to carry the data on several parallel data streams or 5

20 Master Thesis - Zeyu Hu channels [35-37]. Generally speaking, WDM is an important technology which has been widely adopted in today s telecommunication systems. (d) MDM: Mode-division multiplexing is utilizing degrees of freedom in the space of the multi-core fiber or multimode fiber to reuse multipath channels[38]. It takes advantage of the orthogonality between every mode in fiber. Every spatial mode is characterized by a so called mode function, which determines the spatial distribution of its field in the transverse plane. All different mode functions are orthogonal to each other and so the modes can theoretically be multiplexed and demultiplexed without any loss of information [39]. Each mode is regarded as independent load signal in different channel. Recently, people are interested in technique of MDM over few-mode fiber (FMF). They think this technique is a promising approach to extend system transmission capacity and is worth for further study [40-45]. Compared with the single mode fiber, the MDM over few-mode fiber technology can expand the transmission capacity; compared with the multimode fiber, it can control the number of modes and reduce dispersion and crosstalk. (e) PDM: The polarization division multiplexing is one of the ways to improve the system transmission capacity. It is relatively new in fiber-optic communication. In this technique, two independent and mutually orthogonal polarization states are regarded as different channels [46-49]. It transmits two signals at the same time, so 6

21 Master Thesis - Zeyu Hu that the transmission capacity of fiber is doubled and additional bandwidth is not required. Recently, in order to improve the transmission capacity of the fiber-optic communication system, offset-quadrature phase-shift keying (QPSK) modulation technique has been developed. More researchers tend to adopt this modulation technique and combine it with PDM. Nowadays, PDM-QPSK has been widely used to increase the capacity of the fiber optic communication system [50]. So far, various existing multiplexing techniques have been introduced. No matter what kinds of multiplexing techniques are adopted, the basic configurations of the fiber-optic communication system are the same. The typical configuration of the fiber-optic communication system mainly consists of three components: transmitter, optical fiber, and receiver [1]. Figure 1.1 illustrates this system structure. Figure 1.1 Configuration of a typical fiber-optic communication system. The meanings of the abbreviated symbols are as follows: PA indicates pre-amplifier; MA, main amplifier; AGC, automatic gain control unit; EQU, equalizer. 7

22 Master Thesis - Zeyu Hu 1.2. Motivation The increase in data traffic has resulted in enormous demand for high-capacity communication systems with low loss and dispersion. Multiplexing techniques will lead to the next generation of ultra-high-capacity fiber-optic communication systems. In this thesis, a new multiplexing technique is proposed, which is a promising alternative technique for next-generation high-capacity fiber-optic communication system. The concept of this technique is based on the orthogonality of the signal waveform. A number of signals with orthogonal waveforms are multiplexed into a single optical fiber. In this scheme, the user s information is encoded into the amplitude and the phase of the signal waveform. Then multiplexed signals can be transmitted through the fiber-optic communication system. At the receiver end, the orthogonal signals can be separated using decision circuits. Thus this technique improves the system efficiency and increases the transmission capacity. We call this technique as the orthogonal waveform division multiplexing (OWFDM). The OWFDM technique can be combined with the other multiplexing techniques described in the previous section to increase transmission capacity. To realize the OWFDM system, in this work, the mathematical expression of the orthogonal waveform for the system is carefully selected. And the feasibility of 8

23 Master Thesis - Zeyu Hu OWFDM is studied through simulations of the transmission processes in transmitter, optical fiber, and receiver of the fiber-optic communication system. The final simulation results show that this technique is applicable to fiber-optic communication systems. 1.3 Thesis Organization This thesis is organized as follows. Chapter 2 discusses the theoretical framework of OWFDM and presents the rationale for waveform selection. In Chapter 3, the design and simulation of the receiver for the OWFDM system are presented. A physical model of optical signal transmission in the optical fiber is established in Chapter 4, and the corresponding simulation of the backward-tracing propagation of the optical signal is described. In Chapter 5, the design and simulation of the transmitter for the OWFDM system are given. To verify the orthogonality at the final output, system simulation is performed from the initial input to the final output in Chapter 6. Finally, Chapter 7 gives conclusions for this thesis. 9

24 Master Thesis - Zeyu Hu Chapter 2 Configuration of the OWFDM System Similar to conventional fiber-optic telecommunication systems, the OWFDM system is also composed of an optical transmitter, an optical fiber, and an optical receiver. The only difference between the OWFDM and other multiplexing system lies in that the OWFDM system sends a number of optical signals with mutually orthogonal waveforms simultaneously over a single fiber for raising the transmission capacity. To describe the working principle of the OWFDM, we must firstly define the orthogonality of the signal waveform. If we use a set of orthogonal functions {f i (x): i = 1, 2, 3...} to represent a set of orthogonal signal waveforms, the orthogonality of two signal waveforms can be defined as the validity of the following formula on the integral of their inner product f n ( x) fm ( x) w( x) dx n, m (2.1) with w (x) denoting the weight function, and δ the Kronecker symbol, respectively. To ensure that the multiplexed signal in each channel can be correctly extracted, the orthogonality of signal waveforms must be guaranteed right before the de-multiplexing circuit in the receiver end. In design of such a system, waveform distortion and other impairments in signal transmission such as attenuation and noise 10

25 Master Thesis - Zeyu Hu must be taken into account. Since the waveform orthorgonality is only required right before the demultiplexing circuit at the receiver end, a backward-tracing algorithm must be followed to find a set of signal waveforms at the transmitter end to ensure that, after the modulation (i.e., electro-opto conversion), fiber transmission, and detection (i.e., opto-electro conversion), the waveform orthogonality will be restored. Namely, we need to adopt a pre-distortion scheme for achieving minimum crosstalk from channel to channel after demultiplexing. Under the pre-distortion scheme, signal original waveforms in all channels will firstly be converted into a set of desired shapes through, e.g., a set of filters, respectively, after modulation, transmission and detection, these shapes become orthogonal as illustrated by Fig. 2.1 shown below. 2.1 OWFDM System Configuration A block diagram of the OWFDM system is shown in Figure

26 Master Thesis - Zeyu Hu 12

27 Master Thesis - Zeyu Hu As shown in the diagram, the transmitter of the system is composed of a laser, digital filters, modulators, and a combiner. A laser provides a steady optical power for modulators. Various user signals are sent into a number of digital filters to produce input signals for the modulators. Modulators will generate specific signal waveforms which are not orthogonal at the moment, but will be orthogonal to each other before they reach the de-multiplexing circuit. Optical signals from various modulators are added together by a combiner. Signals from the combiner are then launched into fiber. Because of dispersion and nonlinear effects of fiber, signals in different channels will experience distortion and attenuation when transmitting through fiber, and will finally be detected by a p-in photodiode, which converts optical signals into electrical signals. These signals are sent into PA, MA, and EQU before being divided equally into n parts which are then dispersed among the n decision circuits respectively. In each channel, a decision circuit contains a multiplier unit, an integrator, and a digital filter. All "1" square wave signals are sent into a digital filter to produce the signal with orthogonal waveform. The signal will be multiplied by the input signal of the decision circuit, and then integrated over a certain period of time. Because of the orthogonality of the waveform in each channel, the integration result will be "1" if that channel has sent a "1" from the transmitter; otherwise, the result will be "0". Also, 13

28 Master Thesis - Zeyu Hu no crosstalk emerges in this system. Thus, the signal from each individual channel can be extracted. 2.2 Signal Waveform Selection The signal waveform adopted in the OWFDM communication system must satisfy two conditions. First, the signal must be causal. Second, the set of waveforms must be orthogonal to each other. Various mutually-orthogonal signal waveforms have been taken into consideration and discussed as follows: Soliton: The mathematical expressions for soliton of different orders are overly complicated for computation and analysis. Consequently, soliton has not been considered for the simulation. Gauss-Hermite polynomials: These waveforms encapsulate the entire time domain. In fiber-optic communication, a pulse must be confined within a certain finite time domain. If a pulse in the waveform of a Gauss-Hermite polynomial is truncated into a specific finite time domain, the orthogonality of the waveform cannot be guaranteed. As a result, Gauss-Hermite polynomials have been excluded from the simulation. Legendre polynomials: These polynomials are divergent in the time domain, which makes it impossible for them to be of the required pulse waveform. 14

29 Master Thesis - Zeyu Hu Chebyshev polynomials of the second kind: These polynomials with a weight function have simple mathematical expressions, and corresponding waveforms are finite and convergent in the time domain. As noted from the analysis of different waveforms, Chebyshev polynomials of the second kind with weight a function are the most appropriate to provide the distinct choice of waveform required for our study. Therefore, Chebyshev polynomials of the second kind with weight a function are adopted for further examination of the communication system simulation. Chebyshev polynomials of second kind are defined by the following recurrence relation [51-55]: U n 1( n n 1 x x) 2xU ( x) U ( ) (2.2) The first few Chebyshev polynomials of the second kind are: U 0( x) 1 U ( x) 2x 1 U2( x) 4x U ( x) 8x - 4x U4( x) 16x -12x 1 (2.3) A graph of these Chebyshev polynomials of the second kind is shown in Figure

30 Master Thesis - Zeyu Hu Figure 2.2 the first six Chebyshev polynomials of the second kind in the domain To satisfy the orthogonality condition, each function must multiply itself by 1 4 the weight function ( 1 x 2 ), The orthogonality of the Chebyshev polynomials of the second kind is shown as follows: 1 1 U ( x) U n m ( x) 1 x 2 0 n m dx (2.4) n m 2 as a result, values of these functions are 0 at x = ±1. This indicates that one can use the period -1 to 1 as the pulse region. Thus, the pulse is power-limited. Because of their underlying characteristics, Chebyshev polynomials of the second kind with a weight function are consistent with our requirements and befit the mathematical expression of the waveforms. S ( x) U ( x)(1 x n n ) (2.5) 16

31 Master Thesis - Zeyu Hu where S n (x) represents the waveform of the current signal. Equation 2.4 then becomes: 1 1 S ( x) S n m 0 n m ( x) dx (2.6) n m 2 17

32 Master Thesis - Zeyu Hu 3.1 Design scheme of receiver Chapter 3 Receiver Design A diagram of the receiver is shown in Figure 3.2. Figure 3.1 Diagram of the receiver structure of the OWFDM system Optical signals from the fiber are converted into electrical signals by a photodetector. Specific requirements such as high sensitivity, reliability, fast response, low noise and low cost should be met by the photodetector. The most commonly used photodetectors are photoelectric diodes, which include both p-n photodiodes and p-i-n photodiodes. When incident photons with energy h reach the photodiode's surface, a 18

33 Master Thesis - Zeyu Hu photocurrent I p is emitted under the influence of an electric field produced by an applied voltage. The amplitude of this photocurrent is proportional to the incident optical power P in, I p RP in (3.1) where the incident optical power P in is proportional to the square of the amplitude of the electric field. R is the photodetector responsivity, which can be expressed in terms of the quantum efficiency [56]: R (3.2) 1.24 After the photodiode, received signals are divided into n equal parts. These n equal parts of the signals are then sent to decision circuits. In the receiver end, right before the de-multiplexing circuit, electrical signals received from each channel are expected to be with a set of smooth and ideal waveforms of Chebyshev polynomials of the second kind with their weight functions, as shown in Figure 3.2. Only the first four order signal waveforms are illustrated here as examples: 19

34 Master Thesis - Zeyu Hu Figure 3.2 Expected electrical signals from each channel Right before the de-multiplexing circuit in the receiver end, signal waveforms should be the same as shown in Figure 3.2. Therefore, signals waveforms in different channels with different orders can be extracted readily in the de-multiplexing circuit. Because of the orthogonality, they have no crosstalk to each other, despite the fact that they share the same optical and electric path. Each decision circuit contains a multiplying unit, an integrator and a digital filter. All "1" square wave signals are sent into digital filters to produce the signal with orthogonal waveform. The signal is multiplied by the input signal of the decision circuit, and then integrated over a certain period of time. Because of the orthogonality of the waveform in each channel, the integration result will be "1" if that channel has 20

35 Master Thesis - Zeyu Hu sent a "1" from the transmitter; otherwise, the result will be "0". Also, no crosstalk emerges in this system. Thus, the signal from each individual channel can be extracted. 3.2 Simulation of receiver A backward process simulation is performed to obtain the optical wave amplitude at the end of the fiber. In this part, we assume that it is the simplest case to obtain optical signals from electrical signals using Equation 3.1, and we assume that the value of responsivity is 0.9. The simulation results are shown in Figure 3.3. These results will be used as the inputs for the next section of the inverse simulation. Figure 3.3 The ideal optical wave amplitudes at the end of the fiber 21

36 Master Thesis - Zeyu Hu Chapter 4 Optical Waves Propagation in Fiber In this Chapter, to describe the wave propagation in an optical fiber clearly, a physical model is set up first, which is based on Maxwell's electromagnetic theory. 4.1 Maxwell s Equations In this chapter, the wave propagation process in optical fiber will be described. This physical phenomenon is governed by Maxwell s equations: B E t D H J t D f B 0 (4.1) where E and H are electric and magnetic field vectors respectively; D and B are corresponding flux densities. J and f are the current density vector and the charge density respectively. In the absence of free charges in a medium such as an optical fiber, J= 0 and 0. D and B arise in response to the propagation of the electric f and magnetic fields E and H inside the medium and their relationships are given by: D E P B H M 0 0 (4.2) where 0 is the vacuum dielectric constant, 0 is the vacuum permeability, and P and M are the induced electric and magnetic polarizations. In a nonmagnetic medium such as an optical fiber, M=0. 22

37 Master Thesis - Zeyu Hu 4.2 Pulse Propagation Equation From the equations given in the last section, we obtain 1 E 2 c 2 2 E P t t (4.3) where P is the induced electric polarization. When an optical pulse propagates in a fiber, nonlinear effect influences its waveform and spectrum. Then, the electric polarization can be written as [1, 56 59] P( r, t) P ( r, t) P ( r, t) (4.4) L NL 2 2 Using E ( E) E E, one can obtain: 2 1 E 2 c 2 E 2 t 2 2 PL PNL (4.5) 2 t t To obtain the wave equation for the slowly varying amplitude E(r; t), it is convenient to work in the Fourier domain. The general processing method is, taking the P NL as perturbation. By applying the Fourier transform to (4.5), one can obtain: E ~ ~ ( r, ) E( r, t) exp[ i( ) t] dt 0 0 (4.6) Then, one obtains the Helmholtz equation in the frequency domain. 2 ~ E ~ ( ) k 2 E 0 (4.7) 0 Equation (4.7) can be solved utilizing the method of separation of variables. One can assume a solution has the form: ~ ~ E( r, ) F( x, y) A( z, ) exp( i ) (4.8) 0 0 0z ~ where A ( z, ) is a slowly varying function of z, and 0 is the wave number 23

38 Master Thesis - Zeyu Hu corresponding to 0. Equation (4.7) leads to the following two equations for F ( x, y) ~ and A ( z, ) : 2 2 F F 2 ~ 2 [ ( ) k0 ] F 0 (4.9) 2 2 x y ~ A ~ 2 2 ~ 2i 0 ( 0 ) A 0 (4.10) z The wave number ~ is determined by solving the eigenvalue equation (4.9) for F ( x, y), and the dielectric constant ( ) in Equation (4.9) can be approximated by [60]: 2 2 ( n n) n 2n n (4.11) where n is a small perturbation factor, which does not affect F ( x, y). However, the eigenvalue ~ becomes: ~ ( ) ( ) (4.12) ~ The Fourier transform of A ( z; t) satisfies Equation (4.10), which can be written as: A ~ ~ i[ A ~ ( ) 0) z (4.13) One can expand ( ) into a Taylor series about the carrier frequency 0 as follows: ( ) 0 ( 0) 1 ( 0) 2 ( 0) 3 (4.14) 2 6 M m d ( ) m 0 d m 1,2... (4.15) Based on the works of Hardin and Tappert [61], in the time domain one can 24

39 Master Thesis - Zeyu Hu find that the Equation (4.10) becomes [60]: 2 3 A( z, T) A( z, T) 3 A( z, T) 2 z 2 i i e A ( z, T) A( z, ) 2 3 z 2 T 6 T T (4.16) where 2 is the second order dispersion and 3 is the third order dispersion. is the fiber nonlinear parameter. T is a new reference time to exclude the time spent on propagating at a speed of the group velocity v g from the real time ( T 1 t z ). 4.3 Split-Step Method The split-step method is adopted to simulate the propagation process of the optical wave [62]. According to the split-step method, Equation (4.16) can be rewritten as: A ( D N) A z (4.17) where D is a linear operator that represents the dispersion term, and N is a nonlinear operator that counts the nonlinear term: D i T 6 T (4.18) N 2 i e z A (4.19) Assuming that these two operators interact alternately over a small propagation distance z, the dispersion and the nonlinear effect could be treated independently. This calculation is carried out in two steps. In the first step, only the dispersion is considered (i.e., N = 0). In the second step, only the nonlinear effect is 25

40 Master Thesis - Zeyu Hu considered (i.e., D = 0). The solution of Equation (4.16) is therefore given by [60, 63]: ~ A( z z, ) H D ( ) F[ A( z, T )] (4.20) ~ ~ (, ) exp( z A z z T i z e F [ A( z z, )] ) F [ A( z z, )] (4.21) where H D ( ) e i( 2 z 3 z ) 2 6, with F[ ] and F -1 [ ] denoting the Fourier and inverse Fourier transforms, respectively. As it can be seen from Equations (4.20) and (4.21), the algorithm is straightforward: by converting the envelope function in the time domain at position z to the frequency domain and applying Equation (4.20), the function envelope in the frequency domain at position z z is first obtained where only the linear effect is considered. Then one can convert the envelope in the frequency domain function back into the time domain and use Equation (4.21) to evaluate the nonlinear effect at position z z. As indicated above, through stepwise calculations for Equations (4.20) and (4.21), final result can be obtained when a certain distance is reached. This is the process of wave propagation adopting the split-step method. 4.4 Backward Propagation A backward-tracing algorithm must be followed to find a set of signal waveforms at the transmitter end to ensure that, after the fiber transmission, and the detection, the waveform orthogonality will be restored. The algorithm of the 26

41 Master Thesis - Zeyu Hu backward propagation is also based on Equations (4.18) and (4.19). However, different from the algorithm of the forward propagation, the algorithm of the backward propagation starts at the end of fiber, using - z instead of z as the step: A ~ ( z z, ) H ( ) F[ A( z, T )] (4.22) D ~ ~ (, ) exp( z A z z T i z e F [ A( z z, )] ) F [ A( z z, )] (4.23) where H D ( ) e i( 2 z - 3 z ) 2 6. One can get input waveforms of fiber by calculating Equations (4.22) and (4.23) repeatedly until the position is 0. Expected signals at the output end of fiber are shown in Figure 3.3. They are also input signals of the simulation of the backward propagation. The simulation is conducted on MATLAB. Two sets of fiber specifications are applied to the simulation. The first set of fiber specifications are shown in Table 4.1. It is a simple case working at 1310 nm with very small dispersion. Fiber Parameter SMF Second order dispersion (ps 2 /km) Loss (db/km) 0.35 Length (km) 40 Nonlinear coefficient (W-1/km) 2.4 Table 4.1 Fiber specifications: SMF indicates single mode fiber Results of the first set are as shown in Figure , which contain both amplitude and phase information. They are expected signals at the input end of fiber. 27

42 phase (rads) Amplitude of field (V/m) Master Thesis - Zeyu Hu x Figure 4.1 Expected field amplitude of the first order wave before fiber transmission (the first set of specifications) shifted time T (s) x Figure 4.2 Expected phase of the first order wave before fiber transmission (the first set of specifications) 28

43 phase (rads) Amplitude of field (V/m) Master Thesis - Zeyu Hu x Figure 4.3 Expected field amplitude of the second order wave before fiber transmission (the first set of specifications) shifted time T (s) x Figure 4.4 Expected phase of the second order wave before fiber transmission(the first set of specifications) 29

44 phase (rads) Amplitude of field (V/m) Master Thesis - Zeyu Hu x Figure 4.5 Expected field amplitude of the third order wave before fiber transmission 0.06 (the first set of specifications) shifted time T (s) x Figure 4.6 Expected phase of the third order wave before fiber transmission(the first set of specifications) 30

45 phase (rads) Amplitude of field (V/m) Master Thesis - Zeyu Hu x Figure 4.7 Expected field amplitude of the fourth order wave before fiber 0.4 transmission (the first set of specifications) shifted time T (s) x Figure 4.8 Expected phase of the fourth order wave before fiber transmission (the first set of specifications) 31

46 Master Thesis - Zeyu Hu Then, above signals are adopted as input signals of fiber and are transmitted through fiber. After the propagation, waveforms of the first set at the end of fiber become: Figure 4.9 The received field amplitude at end of fiber (the first set of specifications) The figure given above shows that, expected fiber's inputs which have been calculated using backward-tracing algorithm can lead to expected fiber' outputs when noise has not been considered in the simulation of the first set. The second case is working at 1550nm. It has SMF and dispersioncompensating fiber (DCF). The dispersion is different. The second set of fiber specifications are shown in Table

47 Amplitude of field (V/m) Master Thesis - Zeyu Hu Fiber Parameter SMF DCF second order dispersion (ps 2 /km) third order dispersion (ps 3 /km) Nonlinear coefficient (W-1/km) Loss (db/km) Length (km) Table 4.2 The second set of fiber specifications After adopting the backward-tracing algorithm in fiber transmission simulation, both amplitude and phase information of the second set have been obtained. They are shown in Figure , which are expected signals at the input end of fiber x Figure 4.10 Expected field amplitude of the first order wave before fiber transmission (the second set of fiber specifications) 33

48 Amplitude of field (V/m) phase (rads) Master Thesis - Zeyu Hu shifted time T (s) x Figure 4.11 Expected phase of the first order wave before fiber transmission (the second set of fiber specifications) x Figure 4.12 Expected field amplitude of the second order wave before fiber transmission (the second set of fiber specifications) 34

49 Amplitude of field (V/m) phase (rads) Master Thesis - Zeyu Hu shifted time T (s) x Figure 4.13 Expected phase of the second order wave before fiber transmission (the second set of fiber specifications) x Figure 4.14 Expected field amplitude of the third order wave before fiber transmission (the second set of fiber specifications) 35

50 Amplitude of field (V/m) phase (rads) Master Thesis - Zeyu Hu shifted time T (s) x Figure 4.15 Expected phase of the third order wave before fiber transmission (the second set of fiber specifications) x Figure 4.16 Expected field amplitude of the fourth order wave before fiber transmission (the second set of fiber specifications) 36

51 phase (rads) Master Thesis - Zeyu Hu shifted time T (s) x Figure 4.17 Expected phase of the fourth order wave before fiber transmission (the second set of fiber specifications) Then, above signals are adopted as input signals of fiber and transmitted through fiber. After the propagation in fiber, waveforms of the second set at the end of fiber are shown in Figure

52 Master Thesis - Zeyu Hu Figure 4.18 The received field amplitude at end of fiber (the second set of specifications) The above result shows that, in the simulation of the second set, expected fiber's inputs which have been calculated using backward-tracing algorithm can also lead to expected fiber' output when noise has not been considered. As can be seen from above results, the algorithm of the backward-tracing in fiber is correct. 38

53 Master Thesis - Zeyu Hu Chapter 5 Transmitter Design 5.1 Transmitter Structure The transmitter structure is shown in Figure 5.1. Figure 5.1 The transmitter structure of the OWFDM system In the transmitter, the laser provides steady optical output power for each channel. In each channel, the input optical wave is modulated by the external 39

54 Master Thesis - Zeyu Hu modulator. In this work, the dual-drive structure of Mach-Zehnder modulator (MZM) will be adopted as the modulator structure [64, 65]. The pair of driving voltages of the modulator carry user s message of each channel. The phase and the amplitude of the modulator's output wave vary in accordance with the modulator's driving voltages. Moreover, finite impulse response (FIR) filters are needed to convert user s message signals into input signals of modulators. Then, output signals of modulators are expected signals which will be launched into optical fiber at the end of the transmitter. 5.2 Modulator Design The modulator design weighs more than other parts in design the transmitter. Commonly used external modulators include phase modulators, Mach-Zehnder modulators (MZM), and electro-absorption modulators [66, 67]. One can assume that the input optical wave has the following form: A ( t,0) A0 exp( 2i f t) (5.1) in c where A0 is the amplitude of the optical field, f c is the carrier's frequency. After passing through the arm, the output optical wave will be [68,69]: A ( t, L) A0 exp[ i(2 f c t )] (5.2) 2 L 2 L and n( V ) [ neff n( V )] (5.3) 0 where is the phase shift. 0 is the carrier's wavelength. n (V ) is the refractive index. 0 40

55 Master Thesis - Zeyu Hu L is the arm length. The working principle of a dual-drive MZM is briefly described as follows. The MZM consists of one optical input port, two arms and one optical output port [70]. An illustration of the MZM is shown in Figure 5.2: Figure 5.2 Mach-Zehnder modulator Voltages V 1 and V 2 are applied to the upper arm and the lower arm respectively. Each arm has an output wave as follows: A j ( 0 j c j t, L) A exp[ i(2 f t )], j=1, 2 (5.4) 2 2 where 1 and 2 denote the ratio splits to each arm, where After adding them together, the output optical wave will be: A out A0 exp( i2 f ct) [ 1 exp( i 1) 2 exp( i 2)] (5.5) 1 2 and the output power will be: P0 2 2 P out [ cos( 1 2)] 2 (5.6) ( )