Transmission impairments mitigation in next generation coherent optical access networks

Transmission Impairments Mitigation in Next-Generation Coherent Optical Access Networks Author Jeison Alejandro Tabares Giraldo Advisor Josep Prat Gomà A thesis submitted in fulfillment for the degree of Doctor of Philosophy At the Optical Communications Group Department of Signal theory and Communications Universitat Politècnica de Catalunya July 2021 2 Abstract Worldwide, the coherent technologies have revolutionized the optical communication systems, significantly increasing the capacity of the fiber channel owing to transmission of advanced modulation formats and effective mitigation of propagation impairments. However, the actual commercial solutions for long-haul core/backbone networks are still complex and costly, and therefore hardly feasible for deployment in optical access networks. In particular, the main limitations arise from the customer premises equipment whose cost, footprint and power consumption may be kept down. Thus, the optimal solutions for next generation coherent optical access are required to achieve high performance but at lower complexity and cost, since in the access scenario the cost-effectiveness takes more relevance over achieving the best system performance. The research described in this thesis primarily aims at the development of the customer equipment namely the coherent transceiver for a passive optical access network that implements the novel wavelength-to-the-user concept by serving hundreds of users (e.g., 256 users) with dedicated wavelengths allocated in ultra-narrow optical grid. The proposed access network features complexity-reduced coherent technologies by leveraging photonic integration, commercial lowcost lasers and optics, and consumer electronics. To this end, the thesis investigates on the main transmission impairments that affect signal integrity from source to destination in the access network, and proposes novel and enhanced mitigation strategies by either low-complexity digital signal processing or analog hardware design. The covered topics spread over both the optical transmitter and the coherent receiver subsystems. Simplified optical modulation is addressed by direct phase modulation of semiconductor lasers profiting from the laser chirp. Digital pre-equalization of non-ideal frequency response from electronic/photonic devices such as lasers, amplifiers and data converters is investigated, focusing on the tolerance to quantization noise from digital-to-analog converters with limited resolution. Hardware-efficient strategies for optical carrier recovery based on differential phase detection are explored in two scenarios: homodyne receivers aided by digital signal processing, or fully analog heterodyne receivers. Finally, to deal with the critical polarization matching in coherent systems, simplified architectures for polarization-independent coherent receivers using low-cost optics and simpler receiver front-end are investigated. 3 Table of Contents 1. Introduction…………………………………………………………………………………… 1.1. Optical access networks…………………………………………………………………... 1.2. Motivation and objectives………………………………………………………………… 1.3. Research scenario…………………………………………………………………………. 1.4. Thesis outline……………………………………………………………………………... 1.5. List of publications………………………………………………………………………... 1.5.1. Patents……………………………………………………………………………… 1.5.2. Publications in international, peer-reviewed Journals……………………………… 1.5.3. Publications in scientific conferences………………………………………………. 2. State of the Art………………………………………………………………………………… 2.1. Fundamentals of coherent systems...…………………………………………………….... 2.1.1. Coherent receiver analysis………………………………………………………….. 2.1.2. Key architectures in coherent receivers…………………………………………….. 2.1.2.1. 90° hybrid phase-diversity homodyne……………………………………. 2.1.2.2. 3x3 coupler phase-diversity homodyne…………………………………… 2.1.2.3. 2x2 coupler heterodyne…………………………………………………... 2.2. State-of-polarization of lightwaves……………………………………………………….. 2.2.1. Stokes space representation………………………………………………………… 2.2.2. Polarization-diversity coherent receivers…………………………………………... 2.2.2.1. Phase-and-polarization diversity 90° hybrid / 3x3 coupler homodyne…… 2.2.2.2. Polarization-diversity 2x2 coupler heterodyne……………………………. 2.3. Modulation formats……………………………………………………………………….. 2.3.1. Binary phase shift keying…………………………………………………………... 2.3.2. Differential phase shift keying……………………………………………………… 2.3.3. Amplitude shift keying……………………………………………………………... 2.3.4. Complex quadrature amplitude modulation………………………………………... 2.3.4.1. Differential detection of m-ary PSK………………………………………. 2.4. Direct modulation of semiconductor lasers……………………………………………….. 2.4.1. Laser rate equations………………………………………………………………… 2.4.2. Direct phase modulation……………………………………………………………. 7 7 10 11 12 13 13 13 14 16 16 18 21 21 22 24 25 26 27 28 29 30 30 31 33 34 37 38 39 41 4 3. Transmission impairments in coherent access networks…………………………………… 3.1. Design of coherent transceivers for PON…………………………………………………. 3.1.1. 3x3 homodyne transceiver………………………………………………………….. 3.1.2. 2x2 heterodyne transceiver…………………………………………………………. 3.2. Coherent receiver sensitivity in the udWDM-PON……………………………………….. 3.2.1. Additive noise at the receiver………………………………………………………. 3.3. Laser phase and frequency error………………………………………………………….. 3.3.1. Impact of phase noise………………………………………………………………. 3.3.2. Impact of frequency detuning………………………………………………………. 3.4. Interand intra-channel crosstalk…………………………………………………………. 3.4.1. Crosstalk by adjacent udWDM channels…………………………………………… 3.4.1.1. Raised cosine pulse-shaping……………………………………………… 3.4.1.2. Gaussian pulse-shaping…………………………………………………… 3.4.1.3. Minimum channel spacing in udWDM-PON……………………………... 3.4.1.4. Laser side-modes…………………………………………………………. 3.4.1.5. Experimental assessment of channel spacing in udWDM-PON…………... 3.4.2. Crosstalk by non-ideal isolation and optical reflections…………………………….. 3.5. Non-flat electro-optical frequency response……………………………………………… 3.5.1. Frequency dip in laser FM response……………………………………………….. 3.6. Coherent ONU specifications for udWDM-PON…………………………………………. 3.6.1. Summary of transmission impairments and mitigation strategies…………………... 3.6.2. Performance estimate of coherent receivers for PON………………………………. 3.7. Chapter summary…………………………………………………………………………. 4. The COCONUT PON: next generation wavelength-to-the-user access……………………. 4.1. Network architecture…………………………………………………………………….... 4.2. Coherent transceivers technology………………………………………………………… 4.2.1. Digital FPGA homodyne ONU…………………………………………………….. 4.2.1.1. DSP subsystems…………………………………………………………... 4.2.2. Analog DPSK heterodyne ONU……………………………………………………. 4.3. COCONUT field-trial…………………………………………………………………….. 4.4. Results and discussion……………………………………………………………………. 44 45 47 49 50 53 56 56 59 61 62 62 63 64 69 71 74 77 80 82 82 86 87 88 88 90 91 77 95 96 98 5 4.5. Chapter summary…………………………………………………………………………. 5. Optical carrier recovery strategies…………………………………………………………... 5.1. Algorithms for frequency detuning estimation in homodyne receivers…………………… 5.1.1. Differential mth-power estimator…………………………………………………... 5.1.2. IQ cross-correlation estimator……………………………………………………… 5.1.3. Experimental validation……………………………………………………………. 5.2. Analog frequency detuning estimation in heterodyne receivers…………………………... 5.2.1. Analytical model…………………………………………………………………… 5.2.2. Simulation and implementation in hardware……………………………………….. 5.3. Differential carrier recovery for homodyne receivers…………………………………….. 5.3.1. Hardware optimization……………………………………………………………... 5.3.2. Experimental validation real-time………………………………………………….. 5.3.3. Results and discussion……………………………………………………………… 5.3.3.1. Tolerance to optical frequency dithering………………………………….. 5.4. Chapter summary…………………………………………………………………………. 6. Novel architectures for polarization-independent coherent receivers……………………... 6.1. Polarization-independent 3x3 heterodyne receiver……………………………………….. 6.1.1. Receiver architecture……………………………………………………………….. 6.1.2. Analytical model…………………………………………………………………… 6.1.3. Experimental validation……………………………………………………………. 6.2. Polarization-diversity 3x3 homodyne receiver……………………………………………. 6.2.1. Receiver architecture……………………………………………………………….. 6.2.2. Symbol-rate digital signal processing………………………………………………. 6.2.3. Experimental validation real-time………………………………………………….. 6.3. Chapter summary…………………………………………………………………………. 7. Digital pre-emphasis for optical transmitters……………………………………………….. 7.1. Directly modulated lasers for coherent transmitters………………………………………. 7.1.1. Linear pre-emphasis filter design…………………………………………………... 7.1.2. Impact of quantization noise from DACs…………………………………………... 7.1.3. Simulation results…………………………………………………………………... 7.2. Dual-EML coherent transmitter…………………………………………………………... 7.2.1. Characterization of the photonic integrated DEML chip…………………………… 103 104 105 105 106 107 111 112 115 118 120 122 123 128 129 131 131 132 133 137 140 141 142 146 149 150 150 152 156 158 160 161 6 7.2.2. Experimental validation……………………………………………………………. 7.3. Chapter summary…………………………………………………………………………. 8. Conclusions and future work………………………………………………………………… 8.1. Future research…………………………………………………………………………… References……………………………………………………………………………………... List of acronyms………………………………………………………………………………. 163 166 168 171 172 183 7 Chapter 1 Introduction 1.1 Optical Access Networks The global telecommunication infrastructure needs to continuously evolve facing the upcoming bandwidth-hungry multimedia services including ultra-high definition video, electronic commerce, Internet of Things (IoT) and business connectivity among others. In this regard, the optical communication networks have demonstrated that they can satisfy the requirements for broadband data services in a cost-effective manner, leveraging the vast bandwidth (BW) of the optical fibers and devices. The optical fiber exhibits the desired characteristics in a transmission medium: large BW, low attenuation, and immunity to electromagnetic interference. Moreover, the capital and operational expenditures (CAPEX and OPEX) for optical fiber deployment are lower than that of the traditional copper lines. For these reasons, optical fibers are now the preferred media to effectively transmit high speed data through long, medium, or even short distances. Depending on the topology and the transmission link distance, optical networks are hierarchically classified as: core/backbone, metropolitan, and access. The access network is the last segment that reaches the final subscribers. Nowadays, access networks are becoming optical everywhere, substituting the legacy copper lines all the way from the central office (CO) up to the final user premise. These technologies are collectively known as fiber-to-the-x (FTTx), where x denotes the final optical fiber destination, i.e., home/building, business or radio antenna. Worldwide, the FTTx technologies experience an exponential growth driven by the Internet market demands and mobile connectivity. According to the number of residential fiber subscribers forecast, in 2022 most of the population will be reached by fiber-to-the-home (FTTH), at least in urban areas [Eff17]. The passive optical network (PON) is a type of access network intended for FTTx distribution that only employs passive elements such as wavelength selective devices and power splitters for the optical transport through the optical distribution network (ODN); thus, the overall PON only requires power at the transceiver (TRX) sides yielding high cost-savings in both implementation and maintenance. Since no active devices are placed within the ODN, the link power budget, i.e., the total power that can be assigned for a transmission path from the optical line terminal (OLT) to the optical network unit (ONU) by considering power splitting and insertion losses, should be carefully dimensioned to ensure that the receiver (RX) reaches the minimum optical signal power to operate error-free after forward error correction (FEC). Therefore, the RX sensitivity becomes a critical factor to be addressed in next generation PONs. Currently deployed PON systems, or being under development as for next generation PON, are based on ITU/FSAN and IEEE international standards. Fig. 1.1 shows three different point-tomultipoint (P2MP) architectures for optical access networks. By considering the downstream 8 (a) (b) (c) Figure 1.1. Network configuration for PON systems: (a) TDM-PON, (b) WDM-PON and (c) udWDM-PON. (DS) direction, in the time division multiplexing PON (TDM-PON) in Fig. 1.1(a), all the users share the same wavelength (𝜆) and data transfer is achieved in the time domain by the assignment of time slots for each ONU. In this configuration, the aggregate PON capacity does not increase when new users are added since all users are served by the same 𝜆. The user equipment implements direct detection (DD) RXs that are advantageous in terms of simplicity, but they are required to operate at the very high aggregate bit rate in both the opto-electronic components and in a relevant part of the media access control (MAC) layer, while the final user only benefits from a small fraction of it. For example, the XG-PON recommendation for current 10G optical access networks rely on intensity modulation and DD (IM-DD) TRXs at 10 Gb/s and lower, shared by TDM among 32/64 users [ITU17]; in these systems, the guaranteed effective BW per user is about 150/300 Mb/s while the ONU operates at the full 10 Gb/s. Taking into account that in fast Fiber TDM OLT DD ONU DD ONU Power splitter CO ODN Fiber WDM OLT DD ONU DD ONU CO AWG Fiber udWDM OLT Coherent ONU Coherent ONU Power splitter CO 9 electronics (e.g., in CMOS) the power consumption scales with the clock (CLK) speed, there is a huge power and BW inefficiency in such a pure TDM distribution. A step forward from TDM-PON consist of exploiting the optical frequency domain. The wavelength division multiplexing PON (WDM-PON) in Fig. 1.1(b) employs different 𝜆s for data transport that are distributed through the ODN by 𝜆-selective optical components such as arrayed waveguide grating (AWG) behaving as optical channel selector. Now, each DD ONU detects a dedicated single 𝜆 already separated from the others by the AWG and operating only at the user BW, relaxing the hardware (HW) requirements and power consumption of the ONUs. However, the 𝜆-selective characteristic of the ODN is the main drawback in this architecture because of the incompatibility with currently deployed TDM-PON ODN, and also because of the extra challenges and constraints for ONU provisioning and reconfiguration since each ONU needs to be matched in 𝜆 to that the assigned port of the AWG. In the case of PON users requiring lower data rates, the WDM is also compatible with TDM by sharing some 𝜆s among several ONUs. Precisely, the latest ITU-T standard NG-PON2 [ITU19] exploits the WDM by stacking 4/8 XG-PON 𝜆s increasing the aggregate PON capacity up to 80 Gb/s. To keep up with increased BW demand and support future mobile traffic, next generation PONs target over 100 Gb/s aggregate capacity [Hou19] while ensuring coexistence with legacy access systems. However, scaling the actual time-and-wavelength division multiplexing (TWDM) technologies to higher data rates in a cost-effective way might not be compatible with low-cost commercial photonic and electronic devices, mandatory for affordable PON implementation. Specifically, increasing the bit rate per single 𝜆 constitutes a high cost and power consuming solution due to the large required BW of the HW devices such as lasers/modulators, photodiodes (PDs), and amplifiers. Moreover, transmission impairments like chromatic dispersion (CD) become more apparent on large BW signals, eventually requiring digital signal processing (DSP) techniques for their mitigation, with necessary data converters that are expensive and power-hungry if the required BW and sampling rate are high. As a rule of thumb, the novel solutions for next generation PON must be: (I) affordable in cost by the final users, (II) fully compatible with legacy access systems based on power splitting at the outside plant, and (III) flexible in terms of user BW and channel allocation, with reconfigurable TRXs enabling convergence of residential, business and mobile services. A key technology fulfilling these requirements is a new class of coherent TRX implementing the 𝜆-to-the-user concept, where each ONU owns a dedicated 𝜆 in a cost-effective manner by multiplexing in ultradense WDM (udWDM), as depicted in Fig. 1.1(c). The channel selection is done by 𝜆-tuning of the local oscillator (LO) laser at the coherent RX; it increases the available optical BW by allowing very narrow channel spacing to accommodate hundreds of 𝜆s on a single fiber, in contrast with the sparse optical grid of WDM-PON in Fig. 1.1(b). Consequently, 𝜆-selective devices are not required at the ODN that can be implemented with power splitters, achieving colorless (no pre-selected 𝜆s) PON operation fully compatible with legacy access systems. The use of coherent detection further enhances the RX sensitivity with respect to DD. Such extra power budget allows to either increase the maximum reach of the PON, or to allocate more users by adding more power splitting. 16 Chapter 2 State of the Art 2.1 Fundamentals of Coherent Systems Coherent systems present many advantages with respect to the conventional DD systems mainly due to its high selectivity, enhanced sensitivity, and ability for linear detection of the electrical field. First, in the WDM environment with coherent detection, channel selection is done by 𝜆tuning of the LO combined with sharp electrical filtering after photodetection, much better discriminating the 𝜆 channels compared with optical filters whose selectivity is in the order of nanometers (hundreds of GHz). Second, the use of a LO further improves the sensitivity when compared with DD because of the boosting effect of the high LO power on the received optical carrier after mixing. Third, the DD is only sensitive to changes in the optical power thus only detects intensity modulation formats; instead, coherent detection linearly maps all the information carried by the received electrical field into a photocurrent, enabling transmission of advanced modulation formats and effective mitigation of transmission impairments with complex dynamics. In terms of implementation, the main difference between DD and coherent detection is that the received signal is mixed with the LO in an optical hybrid or coupler, as depicted in Fig. 2.1. Afterwards the resulting combination is photodetected, then low-pass filtered for noise suppression. Figure 2.1. Ideal coherent receiver. Depending on the use of an intermediate frequency (IF) stage, defined as the difference between the signal and LO optical frequencies, the coherent detection can be classified into homodyne, intradyne and heterodyne. Table 2.1 provides the electrical spectra after photodetection comparing the coherent detection schemes, as well as the main requirements for each. In homodyne systems, the received signal and the LO are matched in frequency and phase, then the data-carrying electrical field is directly downconverted into base-band at the photodetection stage. The intradyne detection differs from homodyne detection in that the LO is free running, i.e., not matched in phase with the received signal. Such a phase mismatch is alleviated by the use of optical phase diversity along with electrical signal processing, that will be treated in Section 2.1.2. In addition, the IF in intradyne systems is set to be lower than the data BW, and as close to zero as possible in many applications. Last, in heterodyne detection the received optical signal is downconverted into a high frequency electrical carrier with IF larger than the data BW. Due to 17 Table 2.1. Photodetected electrical spectra and main requirement for coherent detection schemes. B: data BW; IF: intermediate frequency. System Photodetected spectrum IF Requirements Homodyne IF = 0 Frequency + phase locking. Baseband processing. Intradyne IF < B Frequency locking + optical phase diversity. ~Baseband processing. Heterodyne IF > B Only frequency locking. Large receiver BW the fast oscillations of the IF carrier, the absence of signalLO phase matching has no effect on the photodetection and phase-looking is not required. As a particular feature, the intradyne detection can be considered either a “near-zero” IF heterodyne detection, or a homodyne detection without phase locking. Each of the three coherent detection schemes exhibit some advantages and drawbacks that must be considered when dimensioning and designing the coherent PON. As a brief comparison, the homodyne RX requires only base-band signal processing thus featuring the highest spectral efficiency; however, the need of optical phase-and-frequency locking between TX and LO leads to problems with the phase and frequency noise of commercial low-cost lasers. On the other hand, the intradyne RX replaces the phase-locking loop (PLL) system by simpler feedforward electrical processing, at the cost of extra HW at the RX front-end for the optical phase diversity. Moreover, the use of free-running LO in intradyne detectors slightly spreads the photodetected spectrum with respect to pure homodyne detection (see Table 2.1), requiring extra BW for the residual IF yet neglictible in most of the applications. In the case of heterodyne RX, as no optical phase matching is needed for correct photodetection, the RX implementation is the simplest; nevertheless, the use of an IF carrier, usually larger than twice the data BW, imposes extra challenges as limited electronic functionalities and larger channel spacing between users, lowering the spectral efficiency. In all cases, frequency locking is required at the coherent RX to maintain the LO 𝜆 at the selected value, maximizing the detection quality but also avoiding interference from adjacent channels in the tight udWDM optical grid. Next sections provide a more detailed comparison among the coherent detection architectures, evaluating their performance, complexity and robustness against transmission impairments. 18 2.1.1 Coherent receiver analysis The basic model for an ideal coherent receiver is shown in Fig. 2.2. The incoming optical signal 𝑒𝑆(𝑡) is ideally mixed with the continuous wave (CW) LO laser beam 𝑒𝐿𝑂(𝑡) resulting in a beating electrical field 𝑒𝐶(𝑡) that is next mapped into the electrical domain by an ideal photodetector. Figure 2.2. Ideal coherent receiver. The electrical field for the received signal and the LO can be modelled respectively as 𝑒𝑆(𝑡)=𝐸𝑆(𝑡)𝑒𝑗(𝜔𝑆𝑡+𝜙𝑆(𝑡)) 𝑒𝐿𝑂(𝑡)=𝐸𝐿𝑂𝑒𝑗(𝜔𝐿𝑂𝑡+𝜙𝐿𝑂(𝑡)) (2.1) Where 𝜔𝑆 and 𝜔𝐿𝑂 are the angular optical frequencies, 𝜙𝑆(𝑡) and 𝜙𝐿𝑂(𝑡) the relative optical phase of each signal. 𝐸𝑆(𝑡) is the envelope of the received electrical field, that is a constant (𝐸𝐿𝑂) for the LO as it emits in CW. The optical signal after ideal coherent mixing becomes 𝑒𝐶(𝑡)=𝐸𝑆(𝑡)𝑒𝑗(𝜔𝑆𝑡+𝜙𝑆(𝑡))+𝐸𝐿𝑂𝑒𝑗(𝜔𝐿𝑂𝑡+𝜙𝐿𝑂(𝑡)) (2.2) In the photoconversion process, the output current from a photodetector is proportional to the incident optical power. Therefore, without BW limitations it can be calculated as: 𝑖𝑃𝐷(𝑡)= ℜ|𝑒𝐶(𝑡)|2, with ℜ the responsivity of the photodiode, yielding 𝑖𝑃𝐷(𝑡)=ℜ[𝐸𝑆(𝑡)𝐸𝑆∗(𝑡)+𝐸𝐿𝑂2+𝐸𝑆(𝑡)𝐸𝐿𝑂𝑒𝑗((𝜔𝑆−𝜔𝐿𝑂)𝑡+𝜙𝑆(𝑡)−𝜙𝐿𝑂(𝑡)) +𝐸𝑆∗(𝑡)𝐸𝐿𝑂𝑒−𝑗((𝜔𝑆−𝜔𝐿𝑂)𝑡+𝜙𝑆(𝑡)−𝜙𝐿𝑂(𝑡))] (2.3) Here we define the intermediate frequency 𝜔𝐼𝐹=𝜔𝑆−𝜔𝐿𝑂 and the instantaneous phase difference ∆𝜙(𝑡)=𝜙𝑆(𝑡)−𝜙𝐿𝑂(𝑡). The (∙)∗ denotes the complex conjugate. The photocurrent can be rewritten as 𝑖𝑃𝐷(𝑡)=ℜ[|𝐸𝑆(𝑡)|2+𝐸𝐿𝑂2+𝐸𝑆(𝑡)𝐸𝐿𝑂(𝑒𝑗(𝜔𝐼𝐹𝑡+∆𝜙(𝑡))+𝑒−𝑗(𝜔𝐼𝐹𝑡+∆𝜙(𝑡)))] (2.4) where |𝐸𝑆(𝑡)|2=𝑃𝑆(𝑡) and 𝐸𝐿𝑂2=𝑃𝐿𝑂 represent the signal and LO power respectively. Finally, the photocurrent assumes the form 𝑖𝑃𝐷(𝑡)=ℜ[𝑃𝑆(𝑡)+𝑃𝐿𝑂+2√𝑃𝑆(𝑡)𝑃𝐿𝑂𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+∆𝜙(𝑡))] (2.5) which is the noiseless general expression for ideal coherent detection. It is composed by the DD terms 𝑃𝑆(𝑡) and 𝑃𝐿𝑂, and the third coherent beating term. Afterwards, the signal 𝑖𝑃𝐷(𝑡) is lowpass filtered by the convolution in time with the filter impulse response ℎ𝐼𝐹(𝑡) with equivalent BW 𝐵, that limits the noise and crosstalk components before the decision stage of the RX. For the upcoming analysis of the coherent detection performance, let us focus on the beating term by 19 neglecting the DD terms 𝑃𝐿𝑂 and 𝑃𝑆(𝑡) that are the constant LO power the DC level of the photocurrent, and the received signal power 𝑃𝑆(𝑡) being much lower than the coherent term. Moreover, both terms can be removed by balanced photodetection as explained in next section. Hence, the variable for data decision at the filter output can be expressed as: 𝑖𝑑(𝑡)=2ℜ√𝑃𝑆(𝑡)𝑃𝐿𝑂𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+∆𝜙(𝑡))∗ℎ𝐼𝐹(𝑡) (2.6) To calculate the signal-to-noise ratio (SNR) and the detection performance, initially the considered noise sources are the shot noise produced in the photoconversion process due to the quantum nature of light, and the thermal noise generated at the electronic part of the RX, both being white Gaussian random process with variance 𝜎2. The SNR, defined as the ratio between the average signal power and the average noise power, can be calculated as 𝑆𝑁𝑅= 〈𝑖𝑑2(𝑡)〉 〈𝜎𝑠ℎ2〉+〈𝜎𝑡ℎ2〉 (2.7) where 〈∙〉 stands for averaging and 𝜎𝑠ℎ2 and 𝜎𝑡ℎ2 represent the variance of the shot noise and thermal noise respectively, which are defined as [Kaz96]: 𝜎𝑠ℎ2=2𝑞ℜ(𝑃𝑆+𝑃𝐿𝑂)𝐵 𝜎𝑡ℎ2=4𝑘𝑇𝐵/𝑅𝐿 (2.8) (2.9) with 𝑞 the electron charge, 𝐵 the BW of the RX, 𝑘 the Boltzmann constant, 𝑇 the absolute temperature of the RX and 𝑅𝐿 the load resistance. Let us first assume the case of homodyne detection for binary 0−180° phase modulation. In that case, in Eq. (2.6) the ∆𝜙(𝑡) is the information-bearing variable and the signal power 𝑃𝑆(𝑡) becomes a constant. For homodyne detection (𝜔𝐼𝐹=0), the signal power 𝑖𝑑2(𝑡) within the RX BW (𝐵) yields 𝑖𝑑2(𝑡)=4ℜ2𝑃𝑆𝑃𝐿𝑂𝑐𝑜𝑠2∆𝜙(𝑡) (2.10) where the term 𝑐𝑜𝑠2∆𝜙(𝑡)=1 for 0−180° phase modulation. Thus, by replacing Eq. (2.10) into Eq. (2.7) the SNR is calculated as 𝑆𝑁𝑅= 4ℜ2𝑃𝑆𝑃𝐿𝑂 (2𝑞ℜ(𝑃𝑆+𝑃𝐿𝑂)+4𝑘𝑇/𝑅𝐿)𝐵 (2.11) Taking into account that 𝑃𝐿𝑂≫𝑃𝑆, there exist a 𝑃𝐿𝑂 value that makes the left term in the denominator of Eq. (2.11) dominant, simplifying the SNR to: 𝑆𝑁𝑅=2ℜ𝑃𝑆 𝑞𝐵 (2.12) which is defined as the quantum-limit SNR. Here one can see one of the main advantages of coherent detection; by controlling the LO power the shot-noise SNR limit can be achieved even for electronic RXs whose performance is generally dominated by thermal noise. For the case of heterodyne detection, the power of the signal in Eq. (2.6) becomes 20 𝑖𝑑2(𝑡)=2ℜ2𝑃𝑆𝑃𝐿𝑂(1+𝑐𝑜𝑠(2𝜔𝐼𝐹𝑡+2∆𝜙(𝑡))) (2.13) The second term in Eq. (2.13) oscillates at twice the 𝜔𝐼𝐹 and completely falls out of the RX BW, being suppressed by filtering. Thus, the SNR for heterodyne detection reads as 𝑆𝑁𝑅= 2ℜ2𝑃𝑆𝑃𝐿𝑂 (2𝑞ℜ(𝑃𝑆+𝑃𝐿𝑂)+4𝑘𝑇/𝑅𝐿)𝐵 (2.14) Note that it only differs from the calculated homodyne SNR in Eq. (2.11) by a factor of two, that corresponds to 3dB SNR penalty for heterodyne detection against homodyne (or intradyne with neglictible IF). In order to estimate the LO power that reaches the quantum-limit SNR in Eq. (2.11), the SNR in terms of the received and LO optical power are plotted in Fig. 2.3 for the case of homodyne detection with binary 0−180° phase modulation, assuming 1 GHz RX BW capable for gigabit data transmission. (a) (b) Figure 2.3. SNR as a function of received and LO optical power for homodyne detection: (a) contour plot; (b) surface plot. Parameters: ℜ=1𝐴𝑤 ⁄,𝐵=1 𝐺𝐻𝑧, 𝑅𝐿=50 Ω,𝑇= 300 𝐾 As observed in both Fig. 2.3a and Fig. 2.3b, for a fixed LO power the SNR is linearly proportional to the received optical power. Conversely, for a constant received power the shot noise contribution in equation (2.11) becomes dominant for a LO power higher than ~0dBm, the limit for linear SNRLO power variations. The quantum-limit SNR is achieved for LO power higher than ~15dBm; beyond this limit, the SNR is totally independent from the LO power and only depends on the signal power. Otherwise, if the LO power is not strong enough, the SNR is linearly reduced. The heterodyne detection performs similar, except for the 3dB penalty in the SNR values. In summary, the foregoing analyses emphasize the three main advantages of coherent detection when compared with DD systems:  Enhanced sensitivity. Due to the boosting effect of the LO laser after mixing, the coherent RX reaches the shot-noise SNR limit overcoming the thermal noise effect that generally dominates the electronic RXs. 21  Linear electrical-field detection. Coherent detection is sensitive to the incident optical power but, more important, also sensitive to the frequency and phase differences between signalLO after optical mixing, enabling full reconstruction of the complex transmitted electrical field.  High selectivity. In a WDM scenario, the coherent RX is able to discriminate among the set of optical frequencies by tuning the LO 𝜆, which in combination with sharp electrical filtering after photodetection allows for ultra-narrow channel spacing. 2.1.2 Key architectures in coherent receivers The coherent RX needs to combine the received signal containing data, with the LO laser before photodetection. Such a mixing is realized by optical hybrids composed by directional couplers, or by simple bi-directional fused-fiber couplers, depending on the RX configuration. This section provides an overview of the key architectures to implement the three aforementioned coherent detection schemes, i.e., homodyne, intradyne and heterodyne. As mentioned in Section 2.1, intradyne detection with zero IF is similar to homodyne detection except for the LO that is freerunning without PLL for the case of intradyne. Therefore, in the remainder of this thesis the terms homodyne and intradyne indistinctly refer to RXs that perform base-band detection, whereas heterodyne is used for RXs that perform IF detection. 2.1.2.1 90° hybrid phase-diversity homodyne The first approach for homodyne coherent RX uses an optical 90° hybrid whose internal structure is shown Fig. 2.4. The upper and lower arms of the optical hybrid differ in that one is 90º phase shifted against the other, leading to the upper arm detects the in-phase (I) beat product while the lower arm detects the quadrature (Q) beat product, composing a phase-diversity homodyne architecture. The four PDs are connected in pairs for balanced photodetection. Figure 2.4. Structure of an optical 90º hybrid with balanced photodetection for phasediversity coherent homodyne RX. In practice, the optical 90° hybrid can be implemented in a waveguide platform by polarization beam-splitters (PBS) setting appropriately the state-of-polarization (SOP) of signal and LO, or by 90˚ optical hybrid 22 passive couplers with correctly dimensioned coupling region [Kik16]. In this phase-diversity homodyne architecture, the I and Q components of the coherent beating have to be present to correctly track the random optical phase from transmitter (TX) and LO lasers without PLL. The output of the optical 90° hybrid can be obtained through the following scatter matrix: [𝑒1(𝑡) 𝑒2(𝑡) 𝑒3(𝑡) 𝑒4(𝑡)]=12[1 1 1 −1 1 𝑗 1 −𝑗][𝑒𝑆(𝑡) 𝑒𝐿𝑂(𝑡)] (2.15) The resulting photocurrents at each of the PD outputs is calculated by: 𝑖𝑘(𝑡)=ℜ|𝑒𝑘(𝑡)|2, being ℜ the photodiode responsivity and 𝑘∈{1,2,3,4} the photocurrent index. Hence, the four photocurrents turn to be [𝑖1(𝑡) 𝑖2(𝑡) 𝑖3(𝑡) 𝑖4(𝑡)]=ℜ 4 [ 𝑃𝑆(𝑡)+𝑃𝐿𝑂 𝑃𝑆(𝑡)+𝑃𝐿𝑂 𝑃𝑆(𝑡)+𝑃𝐿𝑂 𝑃𝑆(𝑡)+𝑃𝐿𝑂 ] +ℜ 2 [ √𝑃𝑆(𝑡)𝑃𝐿𝑂 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+∆𝜙(𝑡)) −√𝑃𝑆(𝑡)𝑃𝐿𝑂 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+∆𝜙(𝑡)) √𝑃𝑆(𝑡)𝑃𝐿𝑂 𝑠𝑖𝑛(𝜔𝐼𝐹𝑡+∆𝜙(𝑡)) −√𝑃𝑆(𝑡)𝑃𝐿𝑂 𝑠𝑖𝑛(𝜔𝐼𝐹𝑡+∆𝜙(𝑡)) ] (2.16) which are composed by the DD terms and the coherent beating terms. Here, 𝜔𝐼𝐹 represents the residual frequency mismatch between signalLO that may be kept as low as possible for homodyne detection (ideally 𝜔𝐼𝐹=0), and ∆𝜙(𝑡) contains the phase noise contribution from the lasers. Note that the coherent mixing in the 90° hybrid produces four photocurrents with relative 𝜋/2 phase difference between consecutive index currents; therefore, those photocurrents with relative 𝜋 phase difference between them are later connected in pairs for balanced photodetection, as seen in Fig. 2.4. It cancels the DD terms that contains the relative intensity noise (RIN) from the lasers, and the strong LO power term 𝑃𝐿𝑂 that may cause amplifiers saturation. After balanced photodetection, the extracted I and Q components read as 𝑖𝐼(𝑡)=ℜ√𝑃𝑆(𝑡)𝑃𝐿𝑂 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+∆𝜙(𝑡)) 𝑖𝑄(𝑡)=ℜ√𝑃𝑆(𝑡)𝑃𝐿𝑂 𝑠𝑖𝑛(𝜔𝐼𝐹𝑡+∆𝜙(𝑡)) (2.17) (2.18) This phase-diversity homodyne RX is usually the reference for implementation in long-haul core networks with coherent detection. The main drawback for real deployments in access networks is the optical 90° hybrid that is still a complex and expensive device. Then, simplified architectures with less complex optical couplers are also explored for homodyne RX implementation. 2.1.2.2 3x3 coupler phase-diversity homodyne A much simpler alternative for the optical 90° hybrid at the homodyne RX is the use of symmetric 3x3 fused-fiber couplers. The main advantage is that the critical 𝜋/2 phase shift between the 2 branches of the 90° hybrid is not required thus the fabrication complexity is significantly lower. To achieve the phase diversity characteristic, the relative phase difference at the 3x3 coupler outputs is adjusted by a proper control of the coupling region length of the three fibers. When 23 symmetric 33% power splitting is observed at the coupler outputs, 120° phase difference among outputs is naturally obtained. The simplified architecture for phase-diversity homodyne detection implementing the 3x3 coupler is depicted in Fig. 2.5. This architecture also saves one PD, lowering the overall RX complexity and power consumption. Figure 2.5. Simplified phase-diversity homodyne RX based on 3x3 fiber coupler. The three electrical fields at the 3x3 coupler outputs are calculated by [Pie89]: [𝑒1(𝑡) 𝑒2(𝑡) 𝑒3(𝑡)]=13[𝑎 𝑏 𝑏 𝑏 𝑎 𝑏 𝑏 𝑏 𝑎][𝑒𝑆(𝑡) 0 𝑒𝐿𝑂(𝑡)] (2.19) where the scatter matrix coefficients 𝑎,𝑏 from the symmetric 3x3 coupler are respectively 𝑎=𝑒−𝑗4𝜋9 ⁄+2𝑒𝑗2𝜋9 ⁄ 𝑏=𝑒−𝑗4𝜋9 ⁄−𝑒𝑗2𝜋9 ⁄ (2.20) (2.21) After photodetection, the three photocurrents read as [𝑖1(𝑡) 𝑖2(𝑡) 𝑖3(𝑡)]=ℜ 3[𝑃𝑆(𝑡)+𝑃𝐿𝑂 𝑃𝑆(𝑡)+𝑃𝐿𝑂 𝑃𝑆(𝑡)+𝑃𝐿𝑂]+2ℜ 3[√𝑃𝑆(𝑡)𝑃𝐿𝑂 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+∆𝜙(𝑡)+2𝜋3 ⁄) √𝑃𝑆(𝑡)𝑃𝐿𝑂 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+∆𝜙(𝑡)) √𝑃𝑆(𝑡)𝑃𝐿𝑂 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+∆𝜙(𝑡)−2𝜋3 ⁄)] (2.22) In contrast with the 90° hybrid RX, here single-ended photodetection is carried out and the three photocurrents have a relative 2𝜋/3 phase shift among them. To extract the orthogonal IQ signals, as well as to cancel the DD terms and common mode noise, the three photocurrents are linearly combined according to the following expressions [Nic89], [Xie12]: 𝑖𝐼(𝑡)=𝑖2(𝑡)−12(𝑖3(𝑡)+𝑖1(𝑡)) 𝑖𝑄(𝑡)=√3 2 (𝑖3(𝑡)−𝑖1(𝑡)) (2.23) (2.24) that can be realized by passive hardware for analog RX implementation, or into the digital domain after analog-to-digital (A/D) conversion for digital RXs. Finally, the IQ recovery yields 𝑖𝐼(𝑡)=ℜ√𝑃𝑆(𝑡)𝑃𝐿𝑂 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+∆𝜙(𝑡)) 𝑖𝑄(𝑡)=ℜ√𝑃𝑆(𝑡)𝑃𝐿𝑂 𝑠𝑖𝑛(𝜔𝐼𝐹𝑡+∆𝜙(𝑡)) (2.25) (2.26) IQ recovery 24 This simpler architecture enables phase-diversity homodyne detection with only three PDs, employing conventional and low-cost 3x3 couplers, becoming an attractive alternative for coherent homodyne RXs for access networks deployments. 2.1.2.3 2x2 coupler heterodyne In the coherent heterodyne RX the beating signal after photodetection is not directly downconverted to base-band (𝜔𝐼𝐹=0) as in homodyne detection, but downconverted into an IF carrier with a frequency larger than the data BW, as pointed out in Table 2.1. Owing to the fast oscillations of such IF carrier, Iand Qphotodetection namely phase-diversity is not required. Therefore, optical mixing between signal and LO is made by common 2x2 optical couplers, followed by balanced photodetection with two PDs, as represented in Fig. 2.6. Figure 2.6. 2x2 coupler heterodyne RX architecture. The two coherent beating components are obtained by the scatter matrix of a symmetric 3dB coupler, as follows: [𝑒1(𝑡) 𝑒2(𝑡)]=1 √2[1 𝑗 𝑗 1][𝑒𝑆(𝑡) 𝑒𝐿𝑂(𝑡)] (2.27) After photodetection, the two photocurrents are found to be: [𝑖1(𝑡) 𝑖2(𝑡)]=ℜ 2[𝑃𝑆(𝑡)+𝑃𝐿𝑂 𝑃𝑆(𝑡)+𝑃𝐿𝑂]+ℜ[√𝑃𝑆(𝑡)𝑃𝐿𝑂 𝑠𝑖𝑛(𝜔𝐼𝐹𝑡+∆𝜙(𝑡)) −√𝑃𝑆(𝑡)𝑃𝐿𝑂 𝑠𝑖𝑛(𝜔𝐼𝐹𝑡+∆𝜙(𝑡))] (2.28) which correspond to two similar signals but shifted 180° between them. After balancing the photocurrents, the heterodyne RX output becomes 𝑖𝑃𝐷(𝑡)=2ℜ√𝑃𝑆(𝑡)𝑃𝐿𝑂 𝑠𝑖𝑛(𝜔𝐼𝐹𝑡+∆𝜙(𝑡)) (2.29) The frequency 𝑓𝐼𝐹 (𝜔𝐼𝐹=2𝜋𝑓𝐼𝐹) of the IF carrier in Eq. (2.29) is larger than the data BW, and usually set to two or three times. Depending on the modulation format, data can be demodulated by squaring the signal for envelope detection, or by delay-and-multiply for differential demodulation (see Section 2.3). In both cases, the demodulator performs as an RF down-converter by extracting the base-band data from the IF carrier without the need of extra processing. The heterodyne RX is also attractive for access networks due to its simplicity, requiring only two PDs and a conventional 2x2 optical coupler. However, the simplicity comes at the cost of larger electronics BW and larger spacing with adjacent WDM channels compared with homodyne detection. 25 2.2 State-of-Polarization of Lightwaves In coherent systems, the detection process is based on the non-linear mixing between the received signal and the LO that occurs within the photodiodes. This holds for any type of coherent RX: homodyne, intradyne or heterodyne. The full mixing is only possible if photons from the signal and LO reach the PDs with the same SOP; otherwise, the coherent mixing suffers a power loss due to destructive interference between waves. When signal and LO have orthogonal SOP there is no coherent mixing between both ligthwaves, as the worst case. The SOP of lightwaves defines the geometry of the electrical field propagation through the waveguide or the transmission medium. Let us consider that the electrical field can be decomposed into two orthogonal components 𝑒𝕏, 𝑒𝕐 which in turn are orthogonal to the propagation direction of the ligthwave, as depicted in Fig. 2.7(a). The complex field propagation geometry will depend on the interaction between the two oscillating waves 𝑒𝕏, 𝑒𝕐 having different amplitudes −represented by the polarization splitting ratio 𝛿− but also having a relative phase shift 𝜓 between them, as shown in Fig. 2.7(b). As a result, one may have three fundamental SOP: (I) Linear: when one of the components vanishes, or when 𝜓= 0 or 𝜋. (II) Circular: when both components have the same amplitude and 𝜓=±𝜋2 ⁄ (III) Elliptical: in the rest of cases (a) (b) (c) Figure 2.7. Orthogonal decomposition of ligthwaves: (a) complex electrical field propagation and (b) the individual orthogonal plane waves. (c) Three different polarization states (from left to rigth): linear, circular and elliptical. 32 demodulation. An alternative consist of recovering the transmitted information from the phase difference between two consecutive symbols. The binary modulation obtained from this encoding technique is called differential-PSK (DPSK), having the great advantage of relaxing the use of synchronization systems. To correctly decode the optical phase changes at the coherent RX, also avoiding error propagation, the binary data must be encoded at the TX side according to the structure depicted in Fig. 2.12(a), that implements the expression: 𝑚[𝑘]=𝑑[𝑘] ⨁ 𝑚[𝑘−1] (2.51) At the RX side, the photodetected current 𝑖𝑃𝐷(𝑡) after filtering is multiplied by a version of itself delayed by one symbol time 𝑇𝑏 to extract the phase difference between two consecutive symbols, as shown in Fig. 2.12(b). Afterwards, the product is low-pass filtered by the base-band filter to get rid of the second order harmonic from the electrical mixing. (a) (b) Figure 2.12. DPSK system: (a) differential encoding at the TX; (b) differential detection at the RX. IF: intermediate frequency; BB: base-band. From Eq. (2.48), the photocurrent at the input of the differential demodulator in Fig. 2.12(b), after being filtered by the IF filter matched to the BW of the information signal 𝑎(𝑡), reads as 𝑖𝑃𝐷(𝑡)=2ℜ𝑎(𝑡)√𝑃𝑆(𝑡)𝑃𝐿𝑂cos(𝜔𝐼𝐹𝑡+∆𝜙(𝑡))+𝑛(𝑡) (2.52) In this case, the carrier phase term 𝜔𝐼𝐹𝑡+∆𝜙(𝑡) related to the optical frequency-and-phase mismatch between signal and LO is also included due to the absence of PLL synchronization. After the demodulation process, the performance of the DPSK modulation in presence of AWGN is calculated by computing the statistics of the decision variable 𝑖𝑑(𝑡). Nevertheless, analytical derivation of the error probability for DPSK is not a straightforward task especially for higher order modulation due to the statistics of the non-linear term 𝑛(𝑡)𝑛(𝑡−𝑇𝑏) that appears in 𝑖𝑑(𝑡) after the delay and multiply operation. By following the analytical derivation in [Pro02] with respect to Eq. (2.52), the error probability for DPSK with homodyne detection is found to be 𝑃𝑒=12𝑒−𝑆𝑁𝑅 (2.53) It only exhibits a small SNR penalty compared with synchronous detection of BPSK, with less than ~1 dB penalty for error probabilities below 10-3. Such behavior is plotted in Fig. 2.13. It is worth emphasizing that, for the case of heterodyne detection, the photocurrent in Eq. (2.52) is a 33 Figure 2.13. Error probability for homodyne detection of DPSK as a function of SNR and received optical power. 𝑅𝑏= 1 Gb/s for received power calculation. high frequency carrier centered at 𝜔𝐼𝐹, that carries the base-band data 𝑎(𝑡). The delay-andmultiply operation of the differential demodulator realizes the frequency down-conversion by extracting the base-band data from the IF carrier. This is verified by making 𝑖𝑑(𝑡) explicit, after the base-band filtering and without considering the noise terms, yielding 𝑖𝑑(𝑡)=2ℜ2𝑎(𝑡)𝑎(𝑡−𝑇𝑏)𝑃𝑆𝑃𝐿𝑂𝑐𝑜𝑠(𝜔𝐼𝐹𝑇𝑏) (2.54) where 𝑎(𝑡)𝑎(𝑡−𝑇𝑏) are the differentially-demodulated symbols. In order to maximize the decision variable 𝑖𝑑(𝑡), the condition 𝜔𝐼𝐹𝑇𝑏=2𝑛𝜋, with 𝑛=1,2,3,…, must be satisfied, being fulfilled when 𝑓𝐼𝐹=𝑛𝑅𝑏. Hence, for optimal decision in heterodyne DPSK detection the IF should be set as an integer multiple of the symbol rate 𝑅𝑏. Although 𝑓𝐼𝐹=𝑅𝑏 is the lowest IF satisfying the optimal condition, in practice the IF is usually set to two or three times 𝑅𝑏 to avoid low-frequency issues near DC. As a final remark, the DPSK modulation format is very attractive for cost-effective implementation in access networks, as the phase-encoded information can be demodulated in asynchronous way without any PLL system. 2.3.3 Amplitude shift keying The foregoing phase modulation formats transport information as the variations in the optical carrier phase, thus they have a constant envelope characteristic. Conversely, in amplitude shift keying (ASK) the binary data are carried on the amplitude of the optical carrier by switching it on and off, in the ideal case. By following the same procedure than the BPSK modulation in Eq. (2.47), for ASK one can define an information-bearing random variable 𝑎(𝑡) taking the values 0 and 1 when the carrier amplitude is switched off and on respectively. Thus the ideal noiseless ASK-modulated optical signal becomes 𝑒𝑆(𝑡)=𝑎(𝑡)𝐸𝑠𝑒𝑗𝜔𝑆𝑡 (2.55) 34 If synchronous homodyne detection is assumed at the RX, with ideal phase matching with the LO, the decision variable for ASK in presence of AWGN is given by 𝑖𝑑(𝑡)=2ℜ𝑎(𝑡)√𝑃𝑆𝑃𝐿𝑂+𝑛(𝑡) (2.56) which is exactly the same than the BPSK decision variable in Eq. (2.48), except for the random variable 𝑎(𝑡) that takes the values 1 and −1 for BPSK, and now 1 and 0 for ASK. The latter implies that, for a given noise variance, the two ASK symbols have half the Euclidean distance compared with the BPSK symbols, thus the SNR is halved. Accordingly, the error probability for ASK derived from Eq. (2.50) is 𝑃𝑒=12𝑒𝑟𝑓𝑐(√𝑆𝑁𝑅 2) (2.57) In Fig. 2.14, the performance of ASK is compared with the phase-modulated formats, showing 3dB SNR penalty for synchronous ASK with respect to BPSK. Although practical implementations of ASK are, in general, simpler than the PSK systems, this thesis primarily focuses on the phase modulated formats as they exhibit better performance, and are assumed to be more robust against the intensity noise from lasers and the optical power fluctuations through the transmission path. Figure 2.14. Error probability for ideal homodyne detection, as a function of SNR and received optical power. 𝑅𝑏= 1 Gb/s for received power calculation. 2.3.4 Complex quadrature amplitude modulation To fully harness the capabilities of the coherent detection, and more specifically, the ability for full electrical field reconstruction at the RX, binary data are mapped into 𝑚-ary complex constellations by simultaneous amplitude and phase modulation of the complex electrical field. It composes the so-called quadrature amplitude modulation (QAM), where 𝑚 represents the number of constellation points in the complex IQ plane. 35 In the previous real-valued modulations, i.e., BPSK, DPSK, and ASK, the two-level electrical data directly drives the optical modulation either by directly modulated laser or by external modulator. Now, for 𝑚-QAM modulation, the bit-to-symbol mapping is required to translate each group of 𝑙𝑜𝑔2(𝑚) bits into a constellation symbol, according to a predefined modulation alphabet. Rectangular 𝑚-QAM constellations are usually generated for convenient optical modulation employing external IQ modulators. Alternatively, 𝑚-QAM constellations in polar coordinates 𝑟,𝜙 can also be generated by directly modulating intensity and phase respectively. It is advantageous, for instance, when cascading AM/PM external modulators, or when semiconductor lasers are directly phase-modulated by exploiting the laser chirp, as explained in Section 2.4.2. These kind of polar coordinates-like constellations are also known as amplitude-and-phase shift keying (APSK). Since this thesis primarily targets access networks applications with low-cost coherent technologies, here we focus on APSK modulation rather than rectangular QAM due to the possibility of simple and cheaper direct laser modulation. For 𝑚-ary phase modulation only, namely 𝑚-PSK, the constellation symbols are 𝑠𝑘=𝑅𝑒𝑗𝜙𝑘, with 𝑅 the constant radius and 𝜙𝑘 the phase-encoded data defined by 𝜙𝑘={2𝜋𝑛𝑚; ⁄𝑛= 0,1,…,𝑚−1}. The mean symbol power is given by Γ=𝑅2. Table 2.1 depicts the constellation diagrams for 𝑚-PSK, with 𝑚={1,2,3} corresponding to BPSK, QPSK and 8-PSK respectively. As expected, the higher is the constellation order the smaller is the Euclidean distance 𝑑 between symbols for the same average power Γ, thus more difficult the symbol decision when the constellation is corrupted by noise. Table 2.2. Constellation diagrams for 𝑚-ary PSK. BPSK QPSK 8-PSK Constellation Euclidean distance 𝑑=2𝑅 𝑑=√2𝑅 ≈1.41𝑅 𝑑=(2−√2)12 ⁄𝑅 ≈0.76𝑅 In the case of amplitude and phase modulation, consecutive symbols in the 𝑚-APSK constellation are set to have symmetric distance 𝑑 among them to optimize the decision thresholds. For instance, Table 2.3 shows the constellation diagram for 8-APSK, in two different configurations. In conventional 8-APSK (Table 2.3, left), the outer ring is rotated 45° with respect to the inner ring. The symbols are defined by 𝑠𝑘=𝑟𝑘𝑒𝑗𝜙𝑘, with 𝑟𝑘={𝑅1,𝑅2} the two radii, and 𝜙𝑘= {𝑛𝜋4; ⁄𝑛=0,1,…,𝑚−1} the phase information similar to 8-PSK. The ratio between radii for symmetric 𝑑 is found to be 𝑅2𝑅1 ⁄=(1+√3)√2 ⁄. In the other configuration for 8-APSK in Table 2.3(right), referred as star 8-APSK, the constellation points are disposed with equal phase angles for both rings. The symbols are also defined by 𝑠𝑘=𝑟𝑘𝑒𝑗𝜙𝑘, with 𝑟𝑘={𝑅1,𝑅2} the two radii, but the phase information is 𝜙𝑘= 36 {0,±𝜋2, ⁄𝜋}, equivalent to QPSK. The ratio between radii for symmetric 𝑑 becomes 𝑅2𝑅1 ⁄= 1+√2. Table 2.3. Constellation diagram for 8-APSK in two configurations. 8-APSK Star 8-APSK Constellation Euclidean distance 𝑑= 2 (3+√3)12 ⁄𝑅 ≈0.92𝑅 𝑑=(2−√2)12 ⁄𝑅 ≈0.76𝑅 In both cases, the mean symbol power is given by Γ=(𝑅12+𝑅22)2 ⁄. By setting the same average power Γ for all the constellations in Table 2.2 and 2.3, the Star 8-APSK exhibits smaller Euclidean distance between points compared with the two rotated-rings 8-APSK constellation, thus the SNR requirement is higher for the same error probability. Here, 𝑅 is the radius of the benchmark 𝑚PSK constellation in Table 2.2. In order to assess the performance of the complex QAM modulation, binary data were mapped into different 𝑚-(A)PSK constellations, spanning from BPSK up to 16-APSK, then corrupted by AWGN to set the SNR. The bit error ratio (BER) was computed by direct error counting over 107 bits. For comparison, rectangular 16-QAM was also considered. Results are plotted in Fig. 2.15. The constellation diagrams at BER = 103 are also shown in Fig. 2.16. Figure 2.15. Error probability for 𝑚-(A)PSK as a function of the SNR, in presence of AWGN. As expected, the different modulation formats perform according to their Euclidean distance 𝑑. BPSK and QPSK show the lowest SNR requirement respectively, as they have the largest 𝑑 for 37 the same average symbol power. For the case of modulation formats with 3-bits/symbol (𝑚 = 8), 8-APSK shows 1.2 dB SNR improvement compared with 8-PSK and Star 8-APSK, for BER = 10-3. Moreover, 8-PSK and Star 8-APSK constellations have similar 𝑑 (See Table 2.2 and 2.3) thus perform almost the same. However, Star 8-APSK is expected to be more robust against the phase noise due to the underpinning QPSK phase modulation that has larger angle separation than 8-PSK. For 𝑚 = 16, i.e. 4-bit/symbol, 16-APSK performs worse than 16-QAM, with ~1 dB SNR penalty for error probability of 10-3. Nevertheless, it is worth noting that 16-APSK only has two different amplitudes or rings, in contrast with the three different symbol amplitudes of rectangular 16-QAM, as observed in Fig. 2.16(f) and 2.16(g) respectively. This feature translates into less (a) (b) (c) (d) (e) (f) (g) Figure 2.16. Constellation diagrams at error probability of 10-3 for (a) BPSK, (b) QPSK, (c) 8-PSK, (d) 8-APSK, (e) Star 8-APSK, (f) 16-APSK, (g) 16-QAM. tolerance of 16-QAM against nonlinearities of amplifiers and/or modulators. In general, APSK is preferred over rectangular QAM, for the same constellation order 𝑚, when nonlinearities of the TX HW dominate [Häg13]. 2.3.4.1 Differential detection of 𝑚-ary PSK The differential PSK, introduced in Section 2.3.2 for binary signals, can also be applied to 𝑚-ary PSK constellations. In binary DPSK the encoding process at the TX is realized on the bits, as depicted in Fig. 2.12(a). Nonetheless, for 𝑚-ary PSK where the constellation symbols are composed by groups of 𝑙𝑜𝑔2(𝑚) bits, the encoding process in the basis of bits becomes a very complex task as the constellation order increases. A more practical solution consists of differentially encoding the 𝑚-PSK symbols 𝑠[𝑘] after bitto-symbol mapping [Gin98], as represented in Fig. 2.17(a). At the RX, the differential demodulator shown in Fig. 2.17(b) is fed by the complex signal 𝑐𝑑(𝑡)=𝐼(𝑡)+𝑗𝑄(𝑡) arising from the phase-diversity coherent detection process. Here, (∙)∗ stands for the complex conjugate. Note that the TX encoding and the RX demodulation in Fig. 2.17 are complementary operations, by integrating and differentiating the phase between consecutive symbols respectively. In terms 38 of performance, differential 𝑚-PSK detection yields 3 dB SNR penalty with respect to synchronous 𝑚-PSK, for 𝑚≥ 4 [Pro02]. In the case of 𝑚-APSK, the differential detection can be easily implemented when the symbols have equal phase angles among rings, like in Star 8-APSK in Fig. 2.16(e). In such case, the phasors 𝑠[𝑘] and 𝑐𝑑(𝑡) entering the differential encoder and decoder respectively, must be normalized in amplitude to only operate on the symbol phase. Otherwise, if the 𝑚-APSK constellation has rotated rings, like in 8-APSK and 16-APSK in Fig 2.16(d) and 2.16(f) respectively, symbol-phase ambiguity occurs after the encoding process [Fis00]. (a) (b) Figure 2.17. Differential (a) encoding at the TX and (b) demodulation at the RX for 𝑚-ary PSK. 2.4 Direct Modulation of Semiconductor Lasers Injection-current modulation of semiconductor lasers is a widespread technology in optical communication networks, with the benefit of less complexity, lower footprint and higher power efficiency when compared with external modulation. In particular, direct laser modulation dominates the PON and intra-data center interconnect (DCI) market segments, where simple and low-cost TRXs are mandatory. In these scenarios, intensity modulation (IM) formats are usually implemented by either binary or multilevel signaling (e.g., pulse-amplitude modulation (PAM)). Nevertheless, the modulation information could potentially be carried on the optical phase of the laser output field through modulation of the frequency chirp. It exploits the fact that when a semiconductor laser is under injection-current modulation ∆𝐼, the emission optical frequency ∆𝜈 varies according to variations in the optical power ∆𝑃, due to the chirp parameter of the laser. This behavior is represented in Fig. 2.18. Figure 2.18. Direct laser modulation characteristics. Such a frequency chirp –or dynamic change in the optical frequency when the laser is under injection current modulation obeys two reasons. First, changes in the laser injection current produce variations in the density of excited carriers (electrons), which in turn induce changes in the semiconductor refractive index and, therefore, in the optical frequency. Secondly, temperature variations modify the relative permittivity of the semiconductor, also varying the semiconductor Frequency chirp 39 refractive index. Since the temperature variations have large time constants, those thermal effects are only relevant for modulation frequencies lower than few MHz (e.g., < 10 MHz). Hence, for the upcoming analysis let us assume modulation frequencies larger than 10 MHz, so that only carrier density effects are considered. In this section, the dynamics of the direct semiconductor laser modulation are briefly examined, giving special attention to the frequency chirp parameter. Moreover, simple strategies to generate direct optical phase modulation through the laser chirp are discussed. 2.4.1 Laser rate equations The laser rate equations are a set of differential equations that describe the dynamic characteristics of the laser diode operation. From the general expression for the complex laser field output 𝑒(𝑡)=√𝑆(𝑡)𝑒𝑗𝜙(𝑡) (2.58) let us define 𝑆(𝑡) as the photons density inside the optical cavity, related to the optical power, and being calculated as 𝑆(𝑡)=|𝑒(𝑡)|2. Using Eq. (2.58), the dynamics of both 𝑆(𝑡) and the instantaneous optical phase 𝜙(𝑡) are derived from 𝑑𝑆(𝑡) 𝑑𝑡 =𝑒∗(𝑡)𝑑𝑒(𝑡) 𝑑𝑡 +𝑒(𝑡)𝑑𝑒∗(𝑡) 𝑑𝑡 𝑑𝜙(𝑡) 𝑑𝑡 =1 𝑆(𝑡)Im{𝑒∗(𝑡)𝑑𝑒(𝑡) 𝑑𝑡 } (2.59) (2.60) If the electrical field 𝑒(𝑡) at the laser output is expressed in terms of the physical parameters of the semiconductor optical cavity, as in [Pet88] and [Bje96], the solutions of Eq. (2.59) and (2.60) yield the standard set of rate equations for single-mode laser emission, for the photon density 𝑆(𝑡), the carrier density 𝑁(𝑡), and the optical phase 𝜙(𝑡) respectively, as 𝑑𝑆(𝑡) 𝑑𝑡 =𝑆(𝑡)[𝑅𝑠𝑡−1 𝜏𝑝ℎ]+𝑅𝑠𝑝 𝑑𝑁(𝑡) 𝑑𝑡 =𝐼(𝑡)−𝐼𝑡ℎ 𝑒𝑉 −𝑅𝑠𝑡 𝑉𝑆(𝑡)−𝑅(𝑁) 𝑑𝜙(𝑡) 𝑑𝑡 =12𝛼𝐻𝑣𝑔𝜕𝑔 𝜕𝑁[𝑁(𝑡)−𝑁𝑡ℎ] (2.61) (2.62) (2.63) In Eq. (2.61), 𝑅𝑠𝑡 and 𝑅𝑠𝑝 are the effective stimulated/spontaneous emission coefficient respectively, related to the optical cavity gain and the confinement factor, and 𝜏𝑝ℎ is the photon lifetime. This rate equation for the photons density may be simply interpreted as follows: 𝑆(𝑡)𝑅𝑠𝑡 photons are generated per unit of time due to stimulated emission, and 𝑆(𝑡)𝜏𝑝ℎ ⁄ photons are lost per unit of time due to stimulated absorption subject to the photon lifetime. Also, 𝑅𝑠𝑝 photons are generated per unit of time due to spontaneous emission. Eq. (2.62) describes the dynamics of the carrier density. Here, 𝐼(𝑡) is the injection current, 𝐼𝑡ℎ is the threshold current, 𝑒 stands for the electron charge, 𝑉 is the volume of the active region, and 40 𝑅(𝑁) is the carrier recombination with respect to spontaneous emission. This rate equation is explained as follows: the carrier density generated by the continuous pumping of the injection current (first term of Eq. (2.62)) decays 𝑅(𝑁) per unit of time due to spontaneous recombination, and (𝑅𝑠𝑡 𝑉 ⁄ )𝑆(𝑡) per unit of time due to stimulated recombination, i.e., the carrier consumption of the stimulated emission. The last Eq. (2.63) is the rate equation for the optical phase. Note that the time derivative of the optical phase is proportional to the optical frequency variation the laser chirp , given that Δ𝜐(𝑡)=1 2𝜋𝑑𝜙(𝑡) 𝑑𝑡 (2.64) Hence, it can be derived from Eq. (2.63) that the optical frequency varies proportional to the carrier density variations as a result of modulating the injection current. This explains the underlying mechanism for the laser FM modulation. In Eq. (2.63), 𝑣𝑔 is the group velocity of the lightwave, 𝑁𝑡ℎ the carrier density at the threshold, 𝜕𝑔𝜕𝑁 ⁄ is the slope of the stimulated emission gain, and 𝛼𝐻 is the linewidth enhancement factor or Henry coefficient. The pair of rate equations (2.61) and (2.62) mathematically model the three different interactions that occur between photons and electrons inside the lasing medium (i.e., the optical cavity), namely the spontaneous emission, the stimulated emission and the stimulated absorption. If the pumping rate, the spontaneous emission effects and the loss terms are neglected in Eqs. (2.61) and (2.62), one obtains 𝑑𝑆(𝑡) 𝑑𝑡 =𝑅𝑠𝑡𝑆(𝑡) 𝑑𝑁(𝑡) 𝑑𝑡 =−𝑅𝑠𝑡 𝑉𝑆(𝑡) (2.65) (2.66) Now, Eq. (2.65) is the stimulated emission contribution and Eq. (2.66) is the stimulated recombination contribution. They clearly show that an increase in the number of generated photons directly relates to a decrease in the number of carriers (excited electrons). These phenomena constitutes the operating principle of lasers: light amplification inside the semiconductor optical cavity by stimulated emission of photons, that is graphically summarized in Fig. 2.19. Figure 2.19. Laser principle: light amplification by stimulated emission of radiation Carrier density Photons density Chirp Optical power Injection current Stimulated emission Stimulated absorption 41 2.4.2 Direct phase modulation The inherent frequency chirp of semiconductor lasers under direct current-modulation can be exploited for generation of phase-modulated optical signals, without the need of external modulators. This is particularly advantageous for coherent PON that may benefit from the high performance, capacity and robustness of advanced 𝑚-PSK modulation formats. In such case, the coherent TX is now conceptually more sophisticated than the employed in conventional IM-DD systems, yet simple and low-cost. Using Eqs. (2.61) and (2.63), one can set up an expression that relates the photons density (i.e., the optical power ∆𝑃) with the frequency chirp Δ𝜐, yielding [Hen82], [Tuc85] Δ𝜐(𝑡)=𝛼𝐻 4𝜋(𝜅𝑃(𝑡)+1 𝑃(𝑡)𝑑𝑃(𝑡) 𝑑𝑡 ) (2.67) where 𝜅 is the adiabatic chirp coefficient. The first term in Eq. (2.67) is the adiabatic chirp that is dominant at mid frequencies (from MHz up to several GHz [Pro11]), whereas the second term is the transient chirp that is more apparent for high frequencies with fast symbol transitions. It is worth mentioning that Eq. (2.67) lacks from the thermal chirp contribution since only carrier density effects are considered. Nevertheless, the optical frequency variations due to lowfrequency thermal effects and their impact on the laser FM response for broadband modulation, are examined in Section 3.3.3.1. Also, the detailed technique for laser chirp measurement is presented in [Pro11] and [Sat05]. The best configuration to directly modulate the optical phase through the frequency chirp consist of setting the laser bias current to large values, while applying a small current swing for modulation. This produces variations of the optical power that are significantly lower than the mean emitted power. Under this small-signal condition and considering the modulation BWs for the proposed PON scenario, < 10 GBd with 𝑚-(A)PSK formats (see Fig 1.2), the adiabatic chirp term in Eq. (2.67) prevails; then ∆𝜐(𝑡)∝𝛼𝐻𝜅𝑃(𝑡), and the semiconductor laser can be modelled as an optical FM modulator with residual IM. Accordingly, one can infer that lasers with large 𝛼𝐻𝜅 value are optimal for direct chirp modulation to lower the modulation current swing thus the residual IM. Next step is to translate the chirp FM modulation into phase modulation (PM). It is a matter of observing the relationship between frequency and phase given by Eq. (2.64), that can be rewritten as 𝜙(𝑡)=2𝜋∫Δ𝜐(𝜏)𝑑𝜏 𝑡 0 ∝2𝜋∫𝛼𝐻𝜅𝑃(𝑡)𝑑𝜏 𝑡 0 (2.68) or expressed in the frequency domain as ∆𝜙(𝜔)=2𝜋 𝑗𝜔Δ𝜐(𝜔) 48 (a) (b) Figure 3.2. Coherent homodyne TRX for udWDM-PON with simplified architecture based on (a) DEML at the TX, and (b) 3x3 optical front-end at the RX. chirp. Chapter 7 provides technical details for the DEML characterization and operation with 𝑚APSK. Moreover, optical amplification is not necessary since no external power-losing devices are attached to the DEML output, lowering the power consumption of the coherent TX. At the RX in Fig. 3.2(b), optical 90° hybrids are replaced by the much simpler 3x3 coupler frontend presented in Section 2.2.2.1. This allows for phase-and-polarization diversity with six singleended PDs, two less than the eight PDs of the conventional balanced homodyne RX. The linear combination of the photocurrents for IQ recovery is carried out by passive HW. Note that although the IQ recovery may be implemented into the DSP of the RX, this simple analog pre-processing saves two ADC channels that are critical components of the RX. The DSP part, the algorithms inherited from the coherent TRX for core networks in Fig. 3.1 can also be adapted and further simplified for the udWDM access scenario having different requirements and constraints. Specifically, some of the DSP algorithms for equalization and impairments mitigation play a secondary role in the access scenario with lower data rates, simpler modulation formats, and shorter fiber spans. For example, at the simplified TX the EAM integrated into the DEML has a nonlinear electro-optic response (see Fig. 7.10). Nevertheless, the EAM bias and the extinction ratio (ER) can be adjusted to operate in the linear region, and the nonlinear pre-distortion by DSP might be optional or discarded. Therefore, the key DSP subsystems for the TX are: symbol mapping for 𝑚-(A)PSK (Section 2.3.4), pulse shaping for both spectral compactness (Section 3.4.1.1) and direct phase modulation (Section 2.4.2), and linear pre-emphasis (Chapter 7). Similarly, the DSP of the RX is further reduced by considering that the impairments from the optical fiber channel, like CD and PMD, have minor impact for the signal BWs and fiber spans ASIC Symbol Mapping DAC DAC Integrated Dual-EML EAM Optical output Linear Pre-Emphasis Pulse Shaping PBS PBS Optical input ADC ADC ADC ADC IH QH IV QV Phase & Pol. diversity front-end Equalization CLK recovery Carrier frequency & phase recovery Decision & demapping ASIC IQ recovery IQ recovery 49 of the udWDM-PON scenario. Also, the front-end correction and orthonormalization for imbalanced optical hybrids are more critical for dense QAM constellations that are not feasible for PONs. Hence, the key DSP subsystems of the RX are: equalization for residual ISI, CLK recovery (Section 6.2.2), CR based on differential phase detection (Chapter 5), and constellation demapping. The DSP of the coherent TRX can be implemented either in an application-specific integrated circuit (ASIC) or in a FPGA. Whereas, in general, the FPGA is more versatile in terms of reconfiguration and synthesis of the DSP algorithms, the ASIC is more suitable for mass production and lower cost, including embedded data converters. As a final remark, the DSP-based coherent TRX reconfigures easily, dynamically adjusting its parameters such as modulation format and data BW to adapt to the type of service and user. This is advantageous for network flexibility compared with analog systems that require HW changes. 3.1.2 2x2 heterodyne transceiver The heterodyne TRX in Fig. 3.3, intended for residential services that require mass ONUs deployment, may benefit from the remarkable lower cost and simpler HW complexity offered by the fully analog implementation, without DACs or ADCs. The heterodyne detection has the drawback of extra BW requirement at the RX to process the broadband detected spectrum centered at IF. However, in the particular case of residential user at 1.25 – 2.5 Gb/s, this has very limited practical impact for state-of-art electronics. (a) (b) Figure 3.3. Coherent heterodyne TRX for udWDM-PON with fully analog transmission and detection of DPSK signals: (a) TX and (b) RX. The TX in Fig 3.3(a) uses direct phase modulation of a DFB laser. Here, DPSK modulation is adopted to avoid the use of DSP for data processing at both the TX and RX side. The binary NRZ data are first differentially encoded, then followed by the high pass pre-equalization, similar to Fig. 2.21(a), to proper adapt the pulse shape of the modulating current for direct phase modulation. The RX in Fig. 3.3(b) implements the polarization-diversity heterodyne front-end introduced in Section 2.2.2.2. All the electrical signal processing is carried out by analog HW. Band-pass filtering (BPF) centered at IF (~2 or 3 times 𝑅𝑏) and matched in BW to that of the transmitted signal is applied. The automatic frequency control (AFC) system owns an analog frequency discriminator (see Section 5.2) to drive the optical frequency tuning of the LO. The binary data Analog HW NRZ data Optical output Pulse shaping Differential encoding PBS PBS H V LO Optical Input H V H V Automatic frequency control Differential demodulation BPF Analog HW Pol. diversity front-end 50 encoded in the DPSK signal are recovered by differential demodulation, then thresholded for data decision and bit recovery. 3.2 Coherent Receiver Sensitivity in the udWDM-PON One of the great advantages of coherent detection is the high RX sensitivity, which eventually reaches the quantum-limit for strong LO power as analyzed in Section 2.1.1. Nonetheless, the final RX sensitivity in the PON is determined by the external sources of noise, interferences and distortions, as well as by the specific implementation of the coherent RX. In this section, the impact of the udWDM-PON scenario on the final coherent RX sensitivity, employing the RX architectures described in previous Section 3.1, is assessed by numerical simulation with MonteCarlo method, using VPITransmissionMaker® and MATLAB®. For the sake of brevity, here and in the remainder of the chapter, the analysis and key conclusions are restricted to the FTTH use case, i.e., residential users with dedicated 𝜆 at 1.25 Gb/s with DPSK. Next chapters, though, cover the rest of scenarios including higher data rates with complex modulation formats. The general simulation setup is represented in Fig. 3.4. The 1.25 Gb/s DPSK signal generated at the TX is sent through 25km of standard SMF. The ODN losses from power splitting are emulated by a variable optical attenuator (VOA) at the coherent RX input, that also adjust the received optical power for the target BER. The simulation results are mostly referred to BER = 10-3, the FEC limit given in ITU-T Rec. 975.1. This FEC limit is assumed as the basis for sensitivity and penalty definition. Figure 3.4. General setup for numerical simulation with DPSK at 1.25 Gb/s. The direct DPSK TX in Fig. 3.5 emulates the direct phase modulation of a DFB. The CW optical carrier from the laser is modulated in phase by an ideal PM modulator, followed by an AM modulator driven by the high-pass equalized data needed for the direct phase modulation (see Section 2.4.2). It mimics the simultaneous intensity-chirp variations of the laser under injection current modulation. Data consist of pseudo-random binary sequences (PRBS) that are mapped into the electrical domain by a non-return-to-zero (NRZ) pulse shaper with 0.1𝑇𝑏 raise time. The mean emitted power is set to 0dBm. It is worth mentioning that for PRBS the differential encoding at the TX is not necessary because the differentially demodulated data are a time-delayed version of the transmitted PRBS. At the RX side, the coherent front-end, i.e., the section of the RX that realizes the optical beating with the LO and the photodetection, is implemented by the two different architectures analyzed in Section 3.1 for PON: the 3x3 homodyne and the 2x2 heterodyne. Also, the conventional 90° 1: Power splitter 25 km SMF VOA DPSK Tx ONU 1 ONU 2 Coherent RX ONU 51 Figure 3.5. DPSK TX for simulation of the direct phase modulation. The inset shows the residual IM on the complex field envelope. (a) (b) (c) Figure 3.6. RX architecture for numerical simulation of the (a) 90º homodyne, (b) 3x3 homodyne, and (c) 2x2 heterodyne coherent DPSK RXs. hybrid homodyne front-end is included as benchmark for performance evaluation. In the simulations, only one SOP is considered as the polarization matching is assumed to be addressed by polarization-diversity at the RX duplicating its architecture. For the homodyne RXs, depicted in Fig. 3.6(a) and 3.6(b), the two photodetected currents corresponding to the I and Q components, are first low-pass filtered by the RF filter for out-band noise suppression. Next, the photocurrents are differentially demodulated each to extract the data PSK with residual IM PM CW AM PRBS G 0180° Direct DPSK TX Equalizer Optical 90˚ hybrid LO RF filter BB filter LO IQ recovery RF filter BB filter LO BB filter RF filter 52 encoded in the phase of the optical carrier, then combined to overcome the phase mismatch due to the free-running LO, without PLL. Finally, the base-band (BB) signal is low-pass filtered for further noise suppression before the decision part of the RX. The RF and BB filters are 4th-order Bessel filters because of their desirable linear-phase characteristics. In the heterodyne RX in Fig.3.6(c), the single photocurrent after balancing is first band-pass filtered by the RF filter, whose center frequency is set to the IF value. Then, the signal is differentially demodulated and lowpass filtered for noise suppression and rejection of second-order harmonics after the frequency down-conversion carried out by the differential demodulator. The LO is separated in 𝜆 from the Tx to obtain IF = 2𝑅𝑏. The performance of each coherent RX is evaluated by computing the BER over the recovered data 𝑑𝑟(𝑡) by direct error counting. Table 3.1 summarizes the main simulation parameters; here, Δ𝜐 represents the effective laser linewidth and RIN is the relative intensity noise. Table 3.1. Summary of main parameters for Monte-Carlo simulation. Device Parameter Value General Bit rate 1.25 Gb/s Sample rate 20 GSa/s Number of bits 218 Modulation DPSK Simulation method Monte-Carlo Reference λ 1553 nm TX Laser Output power 0 dBm RIN -150 dB/Hz ∆𝜐 100 kHz Residual IM 1 dB Data generator Data sequence PRBS-7 Pulse shaper Format NRZ Rise time 0.1𝑇𝑏 ODN Optical fiber Type SMF Length 25 km Attenuation 0.2 dB/km Dispersion 16 ps/(nm·km) RX LO Output power 0 dBm RIN -150 dB/Hz ∆𝜐 100 kHz IF 0 (homodyne) 2𝑅𝑏 (heterodyne) SOP Adjusted to be matched Photodiode Type PIN Responsivity 1 A/W 53 First, the optimal cut-off frequency of the RF and BB filters for each RX is identified. Results are plotted in Fig. 3.7 in terms of the measured BER as a function of the filter BW normalized to the symbol time 𝑇𝑏. (a) (b) (c) Figure 3.7. Filter BW optimization for the (a) 90º homodyne, (b) 3x3 homodyne and (c) 2x2 heterodyne coherent RXs. For the 90° homodyne, plotted in Fig. 3.7(a), the optimal RF filter BW is found to be 0.6/𝑇𝑏, close to the conventional 0.75/𝑇𝑏 filtering, while the optimal BB filter BW ranges from 0.8/𝑇𝑏 to 1.1/𝑇𝑏. The 3x3 homodyne in Fig. 3.7(b) performs very similar to the previous one, with optimal RF filter BW = 0.55/𝑇𝑏, and BB filter BW ranging from 0.6/𝑇𝑏 to 1.1/𝑇𝑏. In the 2x2 heterodyne in Fig. 3.7(c), the optimal RF filter BW is about twice the homodyne RF filter BW, because of the two modulation side-lobes of the IF spectrum after heterodyning. The BB filtering behaves similar than that of homodyne RXs. For 1.25 Gb/s data rate, the optimal filter values are summarized in Table 3.2. These values are obtained from simulation with ideal frequency responses and absence of distortions; in practice, however, these values must be optimized in order not to incur on extra symbol distortion and ISI. Table 3.2. Optimal 3dB filter BW for 1.25 Gb/s DPSK. Filter BW Receiver RF filter BB filter 90º homodyne 750 MHz 1.25 GHz 3x3 homodyne 700 MHz 1.25 GHz 2x2 heterodyne 1.5 GHz 1 GHz After filter optimization, the RX sensitivity with DPSK at 1.25 Gb/s is evaluated. Results plotted in Fig. 3.8 indicate that, for a FEC limit of BER = 10-3, the RX sensitivity reaches -58.5 dBm for both the 90° and the 3x3 homodyne architectures, that perform very similar. The 2x2 heterodyne RX exhibits 2.6 dB power penalty at FEC level when compared with the homodyne RXs, close to the expected 3 dB penalty from the heterodyne detection. These values are taken as a reference in next sections to evaluate de performance degradation due to transmission impairments. 3.2.1 Additive noise at the receiver The high sensitivity achieved by the coherent RXs is ascribed to the boosting effect of the strong LO power. The limit sensitivity is determined by the amount noise generated at the optical-to- 54 Figure 3.8. Coherent RXs sensitivity for 1.25 Gb/s DPSK, assessed by Monte-Carlo simulation. electrical conversion in the photodetection stage. On the one hand, the shot noise originated by the quantum-nature of light, is linearly dependent on the incident optical power to the PDs after coherent beating, and it establishes the quantum-limit sensitivity as analyzed in Section 2.1.1. On the other hand, the most relevant thermal noise contribution is the noise generated by the transimpedance amplifiers (TIA) after photodetection [Zha12], employed to convert the photodetected current into voltage variations. The thermal noise, expressed in terms of the equivalent noise spectral density 𝑆𝑓 in pA/√Hz, depends on the quality of the TIAs and the architecture of the coherent Rx, and specifically, in the number of photocurrent branches, as can be appreciated in Fig. 3.9 for the 90º homodyne, 3x3 homodyne and 2x2 heterodyne RX frontends respectively. To evaluate the impact of the thermal noise, isolating from other effects, let us first focus on the homodyne detection. Numerical simulation was done by using the coherent homodyne front-ends depicted in Figs. 3.9(a) and 3.9(b). The LO power was initially set to 0 dBm, in the region where (a) (b) (c) Figure 3.9. Coherent RX front-end for (a) 90º homodyne, (b) 3x3 homodyne and (c) 2x2 heterodyne. Optical 90˚ hybrid PIN TIA TIA PIN + TIA PIN TIA 55 the thermal noise still has influence on the SNR of the RX (see Fig. 2.3). The results are plotted in Fig. 3.10, in terms of the sensitivity penalty at BER = 10-3 as a function of the spectral noise density 𝑆𝑓. The 3x3 homodyne RX with three branches is slightly more sensitive to increments of the thermal noise factor than the 90° homodyne RX with two branches. For such constant LO power (0 dBm), the thermal noise factor increase that yields 1dB power penalty at BER = 10-3 is nearby 5 pA/√Hz for both RXs. Note that the calculated penalty in Fig. 3.10 is relative to the case when the LO power is limited, but the thermal noise factor is 0 pA/√Hz, thus reaching the quantum-limit sensitivity in Fig. 3.8. Otherwise, as the thermal noise factor increases the power penalty increases as well. This dictates the RX sensitivity penalty when, in practice, the LO power is not strong enough due to HW constraints, with thermal noise factor spanning from 10 to 18 pA/√Hz for commercial TIAs. Figure 3.10. Sensitivity penalty at BER = 10-3 vs. thermal noise spectral density for homodyne RXs. As stated in Section 2.1.1, for a given received optical power, with other parameters unchanged, the LO power can be adjusted to reach the quantum-limit sensitivity, limited only by the shot noise. Hence, a further simulation was carried out to establish the optimal LO power that minimizes the thermal noise effect, thus maximizing the SNR, for the coherent RXs in Fig. 3.9. The thermal noise spectral density was set to 𝑆𝑓 = 12 pA/√Hz. Results are plotted in Fig. 3.11. (a) (b) Figure 3.11. (a) BER and (b) sensitivity penalty at BER = 10-3 against LO power, for thermal noise spectral density 𝑆𝑓=12 pA/√Hz. 56 As observed in Fig. 3.11(a), the BER varies when sweeping the LO power, even if the received signal power keeps constant, as a consequence of the noise generated inside the coherent RX that dominates the ultimate RX sensitivity, reached for high LO power. In Fig. 3.11(b) the detriment of the BER is translated into sensitivity penalty at BER = 10-3, showing that LO power >10 dBm ensures sensitivity penalty less than 1dB with respect to the quantum-limit sensitivity, for the three RX types. These results are in good agreement with theoretical values obtained from Fig. 2.3. The LO power, however, must be carefully adjusted in practice according to the HW parameters. When the LO power increases, the undesired RIN from semiconductor lasers makes impact on the RX performance if the balancing of photocurrents is not ideal. The RIN relates to intensity fluctuations from spontaneous emission even if the bias current of the laser keeps constant. For ideal balanced detection, these power fluctuations are completely suppressed from the photodetected signal. In contrast, the RIN can be only partially suppressed when imperfect photocurrent balancing, and it becomes the dominant noise source at the RX for large LO power. The suppression ratio is determined by the common-mode rejection ratio (CMRR) of the balanced RX. For example, results in [Zha12] for a RIN value of 150 dB/Hz, the typical of DFBs, indicate that for CMRR > 25 dB the RIN has minimum impact and the system performance is close to that the ideal balanced RX in Fig. 3.11. Otherwise, for CMRR < 25 dB there is an optimum LO power between 8 and 12 dBm. These results very well match the simulations here. 3.3 Laser Phase and Frequency Error Having in mind the strict requirements for PON in terms of complexity and cost, DFBs or verticalcavity surface-emitting lasers (VCSELs) are the best solution as for optical sources in the udWDM scenario because of their low fabrication cost and mass production. Moreover, they allow for direct intensity/phase modulation, and thermal 𝜆-tuning. Nevertheless, those benefits come at the cost of wide spectral linewidth and optical frequency drifts. This section evaluates the impact of the carrier phase fluctuations on phase modulated data, employing differential phase detection. 3.3.1 Impact of phase noise Phase noise from a laser relates to its 3dB power spectrum bandwidth so-called laser linewidth Δ𝜐. It becomes a serious problem in coherent detection of phase modulated data, and eventually, the laser linewidth-symbol time product (Δ𝜐𝑇𝑏) determines the feasibility and accuracy to recover the phase encoded symbols. To analytically determine the impact of phase noise on the PSK signals, let us consider the case of BPSK with homodyne detection. Here, the phase noise is represented by the random variable 𝜙 and modeled as a Wiener-Lévy process [Kik12], i.e., a Gaussian process with zero-mean and variance 𝜎2=2𝜋𝛥𝜐𝑇𝑏 (3.1) 57 From Eq. (2.48), the decision variable at the RX for homodyne detection in presence of the phase error 𝜙 turns out to be 𝑖𝑑(𝑡)=2ℜ𝑎(𝑡)√𝑃𝑆𝑃𝐿𝑂cos𝜙+𝑛(𝑡) (3.2) and the error probability in Eq. (2.50), assuming AWGN channel, becomes [Kaz96] 𝑃𝑒′=12𝑒𝑟𝑓𝑐(√𝑆𝑁𝑅cos𝜙) (3.3) Now, the error probability is also dependent from the random process 𝜙; therefore, the expression in Eq. (3.3) must be averaged over the statistics of 𝜙, assumed to be Gaussian, yielding 𝑃𝑒=1 2√2𝜋𝜎∫ 𝑒𝑟𝑓𝑐(√𝑆𝑁𝑅cos𝜙)𝑒−𝜙2 2𝜎2 ∞ −∞ 𝑑𝜙 (3.4) The integral in Eq. (3.4) was numerically evaluated by Gauss-Hermite quadrature rules [Ben02], and the results plotted Fig. 3.12 in terms of the error probability as a function of the SNR, for several phase-error standard deviation values. As observed, the fact of having a phase error different from zero leads to an error-floor, thus the error probability limit is a finite value independent from the SNR. Figure 3.12. Error probability of homodyne BPSK for several phase-noise standard deviation values. The phase-noise standard deviation (𝜎) values shown in Fig. 3.12 are useful to evaluate the limits of the BPSK modulation corrupted by phase noise. From Eq. (3.1) one can calculate the product 𝛥𝜐𝑇𝑏, which denotes the maximum laser linewidth tolerance, for a given symbol rate and an error probability. Table 3.3 summarizes the 𝜎 values and their corresponding 𝛥𝜐𝑇𝑏 values to produce 1 dB SNR penalty at several error probabilities. The equivalent error-floor is also computed for each case. For FEC limit (𝑃𝑒=10−3), the phase noise tolerance at 1dB SNR penalty is found to be 𝛥𝜐𝑇𝑏=0.018, which means that, for 1.25 Gb/s bit rate, maximum 22.5 MHz total laser linewidth is tolerated. For 𝑚-ary PSK, with shorter distance between constellation symbols as 𝑚 increases, the linewidth tolerance is correspondingly lower. Table 1 of [Pfa09] summarizes the maximum 𝛥𝜐𝑇𝑏 for 1dB SNR penalty at BER = 10-3 with QAM constellations, indicating 𝛥𝜐𝑇𝑏= 4.1×10−4 for QPSK as the best case. 64 PRBS data sequences with NRZ pulse-shape passed through the Gaussian filter prior to the optical modulation. The generated Gaussian PSK signal was detected by the 90º homodyne RX, as reference. The input power was adjusted to obtain BER = 10-3 for the case of NRZ data without Gaussian filtering, then the power penalty due to the BW of the Gaussian filter was computed and plotted in Fig. 3.18(b). The results indicate that the sensitivity penalty raises to 1 dB for BW= 1.2/𝑇𝑏, which in terms of the rise time yields 𝑡𝑟=0.29𝑇𝑏. 3.4.1.3 Minimum channel spacing in udWDM-PON To evaluate the benefit of electrical pulse-shaping at the TX on the mitigation of crosstalk by adjacent channels, numerical simulation was carried out using the setup depicted in Fig. 3.19(a) to determine the minimum channel separation in the udWDM grid. The setup consisted of two users, one of them at a fixed 𝜆1, and the other passing through by sweeping ±7.5 GHz with respect to 𝜆1, as illustrated in Fig. 3.19(b). The channel to be detected modulated PRBS-7, whereas the interference channel modulated PRBS-15 for uncorrelated data. The emission power was set to 0 dBm for both users, that were later combined in a 3 dB optical coupler, propagated through 25 km of SMF, and detected by a 90º homodyne Rx. The LO of the RX was tuned to detect 𝜆1, and the received optical power adjusted to obtain BER = 10−3 for the case of single channel. (a) (b) Figure 3.19. (a) Simulation setup to evaluate the crosstalk by adjacent channel in the udWDMPON; (b) Optical spectra for the 𝜆-sweep of User 2 with respect to User 1. At the TX, bits from PRBS sequences are mapped into the electrical domain by the pulse-shaping filter, and the resulting electrical signal drives the optical DPSK modulation at 1.25 Gb/s. Three pulse shapes were considered for the simulation: pure NRZ, Gaussian and raised cosine. The BW of the Gaussian filter was set to BW=1.2/𝑇𝑏, corresponding to 𝑡𝑟=0.29𝑇𝑏, and derived from Fig. 3.18(b) for maximum 1 dB sensitivity penalty. For the raised cosine filter, the roll-off was first set to 𝛽=1, then the BW =1/𝑇𝑏. In both cases, the BW of the pulse-shaping filter was properly selected to reduce the secondary modulation lobes at most, without producing excessive BER degradation. To only translate the filter complexity to the TX side, the RX implements standard low-pass Bessel filtering for noise suppression. Table 3.4 summarizes the main pulseshaping filter parameters, and depicts the electrical spectrum and the eye diagram of the electrical signal at the filter output. nm 90˚ homodyne Rx LO 25km SMF VOA DPSK Tx1 PC DPSK Tx2 65 Table 3.4. Electrical spectra and eye diagram for PRBS data at 1.25 Gb/s, for three different pulse-shaping filters implemented at the TX. Pulse-shaping filter Electrical spectrum Eye diagram NRZ 𝑡𝑟=0.1𝑇𝑏 Gaussian BW = 1.2/𝑇𝑏 𝑡𝑟=0.29𝑇𝑏 Raised cosine BW = 1/𝑇𝑏 𝛽=1 The main conclusions of Table 3.4 are as follows. In the case of NRZ data, with 𝑡𝑟=0.1𝑇𝑏, the sharp symbol transitions resulted in high-frequency harmonics of the spectrum, with significant power beyond the 1.25 GHz data BW, and 14 dB suppression of the main modulation lobe with respect to the secondary lobe. By applying Gaussian filtering the sharp symbol transitions are smoothed and more of the power is now close to the data BW, enhancing the secondary lobe suppression up to 18 dB, and reducing the power of the rest of harmonics. When raised cosine filtering is applied, the electrical pulses in the time domain adopt the sinc (cardinal sine) shape, and all the signal power is now within the data BW, with more than 60 dB suppression over the residual harmonics of the spectrum. Next, optical DPSK modulation at 1.25 Gb/s takes place. After modulation, though, the same spectral distribution and secondary-lobe suppression obtained in the electrical domain in Table 3.4, are not expected to be achieved in the optical domain due to the non-linear characteristic of the phase modulation, which exhibits strong Bessel harmonic dynamics influencing on the modulated optical BW [Hay01]. Table 3.5 shows the obtained optical DPSK spectra for two users separated by 6.25 GHz. First, the optical DPSK modulation with NRZ data reveals 12 dB 66 suppression over the secondary lobes that potentially lead to interchannel crosstalk in the udWDM grid. Second, DPSK modulation by Gaussian-shaped signals increases by 1 dB the secondary lobe suppression, and the total spectral width per user is lowered. Last, the raised cosine filtering enhances the secondary-lobe suppression of the DPSK spectrum up to more than 19 dB, and the harmonics at higher frequencies have been completely suppressed. Table 3.5. Optical spectra of two DPSK users at 1.25 Gb/s separated by 6.25GHz, for three different pulse-shaping filters implemented at the TX. Pulse-shaping filter Two-channel optical spectra NRZ 𝑡𝑟=0.1𝑇𝑏 Gaussian BW = 1.2/𝑇𝑏 𝑡𝑟=0.29𝑇𝑏 Raised cosine BW = 1/𝑇𝑏 𝛽=1 It is worth mentioning that reducing the roll-off parameter (𝛽) of the raised cosine filter does not increase the secondary-lobe suppression nor lower the modulated optical BW achieved in Table 3.5, because of the non-linear phase modulation. Further details and experimental verification are provided in Section 3.4.1.5. Next, the minimum channel spacing between udWDM users was evaluated by computing the sensitivity penalty at BER = 10-3 as a function of channel separation, for the three different pulseshaping filters at the TX and homodyne detection at the RX. Results are presented in Fig. 3.20. Not surprisingly the raised cosine pulse-shape outperforms the NRZ and Gaussian filtering as the 67 Figure 3.20. Sensitivity penalty at BER = 10-3 against channel separation between udWDM users at 1.25 Gb/s with homodyne detection. channel spacing decreases, because of the lowest modulated optical BW and the highest secondary-lobe suppression achieved in Table 3.5. Notably, the minimum channel spacing for a sensitivity penalty of 0.5 dB only, is found to be 4 GHz, 3.1 GHz and 2.6 GHz for the NRZ, Gaussian and raised cosine pulse-shape respectively, validating for the implementation of the udWDM grid with ultra-narrow channel spacing. Note that the power penalty in Fig 3.20 was computed from a single interfering channel; in practice, however, every single channel in the udWDM experiences crosstalk from two adjacent users at each side, as illustrated in Fig. 3.21 for three channels at the same power. 𝑃𝐼 denotes the crosstalk power. Figure 3.21. Crosstalk from two adjacent users in the udWDM-PON with narrow channel spacing. In order to relate the power penalty found for a single interferer to that of two interfering users, let us define ∆𝑃(1), ∆𝑃(2) as the power penalty for one and two adjacent channels respectively, calculated as ∆𝑃(𝑘)𝑑𝐵=10𝑙𝑜𝑔10(𝑐𝑘) (3.9) for 𝑘=1,2. Here 𝑐𝑘 represents the power scaling factor translating into the power penalty. In the absence of adjacent channels, 𝑆𝑁𝑅=𝑃𝑆𝑃𝑁 ⁄ is the basic definition SNR at the RX that dictates the BER of the system, with 𝑃𝑆 and 𝑃𝑁 the received signal power and the noise power respectively. On the other hand, the SNR at the RX in presence of crosstalk from one or two adjacent channels becomes signalchann2 chann1 68 𝑆𝑁𝑅(𝑘)=𝑐𝑘𝑃𝑆 𝑃𝑁+𝑐𝑘(𝑘𝑃𝐼) (3.10) The power penalty factor 𝑐𝑘 scales the optical power reaching the RX thus affects 𝑃𝑆 and 𝑃𝐼, but not 𝑃𝑁 since in coherent detection the dominating noise source at the RX is the shot noise from the strong LO power, and 𝑃𝑁 is independent from the received power, as examined in Section 2.1.1. For a given BER, one may find the power penalty factor 𝑐2 for two interfering users that keeps constant the reference SNR, by equating the Eq. (3.10) for 𝑘=2 to the single-channel 𝑆𝑁𝑅=𝑃𝑆𝑃𝑁 ⁄, yielding 𝑐2=𝑃𝑁 𝑃𝑁−2𝑃𝐼 (3.11) and expressed in dBs from Eq. (3.9) as ∆𝑃(2)𝑑𝐵=10𝑙𝑜𝑔10(𝑃𝑁 𝑃𝑁−2𝑃𝐼) (3.12) Similarly, the crosstalk power 𝑃𝐼 can be expressed in terms of the power penalty factor 𝑐1 for one adjacent user, by equating the Eq. (3.10) for 𝑘=1 to the reference single-channel 𝑆𝑁𝑅=𝑃𝑆𝑃𝑁 ⁄, obtaining 𝑃𝐼=𝑃𝑁(𝑐1−1) 𝑐1 (3.13) By replacing Eq. (3.13) into Eq (3.12), one gets ∆𝑃(2)𝑑𝐵=𝑃𝑁𝑑𝐵−10𝑙𝑜𝑔10(𝑃𝑁−2𝑃𝑁(𝑐1−1) 𝑐1) (3.14) that yields ∆𝑃(2)𝑑𝐵=∆𝑃(1)𝑑𝐵−10𝑙𝑜𝑔10(2−10(∆𝑃(1)𝑑𝐵 10 ⁄)) (3.15) Eq. (3.15) is a useful expression to determine the power penalty from two adjacent channels emitting at the same power, using the power penalty calculated from only one adjacent channel, and independently from the SNR or the modulation format. Graphically, the relationship is depicted in Fig. 3.22. As observed, for single-interferer power penalties lower than 1dB, there is a linear relationship (slope = 2.14) with the penalty for two adjacent users. This means that the calculated channel spacing in Fig. 3.20 for 0.5 dB sensitivity penalty and a single interferer, translate into about 1 dB penalty when two users generate crosstalk at each side. Table 3.6 summarizes the minimum channel spacing found. Taking as reference the 50/100 GHz spaced WDM grid defined in NG-PON2 [ITU19], the channelization of the proposed udWDM-PON is found to be as low as 6.25 GHz for residential users at 1.25 Gb/s. This value was selected as the lowest integer divisor of the standardized 50 GHz NG-PON2 grid that satisfies the required minimum channel spacing in Table 3.6. In the case 69 Figure 3.22. Relationship between the power penalty for two interfering users and the power penalty for a single interferer. of heterodyne RXs, the defined channel spacing is twofold, i.e., 12.5 GHz, in order to allocate for the image frequency of the heterodyne detection located at 2×IF. The minimum 6.25 GHz channel spacing found for the udWDM-PON provides for full backward compatibility with legacy PON systems (G-PON, XG-PON, TWDM-PON) and allows to exploit narrow spectral windows where no other communication systems are allocated. Table 3.6. Summary of pulse-shaping filters performance for DPSK at 1.25 Gb/s, with closely spaced udWDM channels and homodyne detection. Pulse shaping Optical secondary lobe suppression Minimum channel spacing for 1dB power penalty tolerance NRZ 12 dB 4 GHz Gaussian 13 dB 3.1 GHz Raised cosine 19 dB 2.6 GHz 3.4.1.4 Laser side-modes Another highly relevant characteristic of semiconductor lasers based on optical cavities is that they generate light at several spurious 𝜆s around the central 𝜆. Typical lasers for optical communication systems exhibit side-modes separated by ~150 GHz (1.2 nm) with respect to the central 𝜆. Also, the power difference between the main and side modes of the laser is over 30 dB, denoted by the side-mode suppression ratio (SMSR) parameter. These side-modes are close enough to generate crosstalk on other users in the udWDM-PON, that implements up to 256 𝜆𝑠 spaced by minimum 6.25GHz, for a total optical band covering 12.8nm. The optical spectrum in Fig. 3.23 illustrates the potential interference of the laser side-modes on the other PON users. The power difference between one channel and the overlapped spurious 𝜆 is represented by the signalto-interference ratio (SIR). 70 To determine how variations in the SMSR of one of the channels affect the performance of other channel located at the same 𝜆 then the interfering side-mode, further simulation was carried out using the same setup and parameters in Fig. 3.19(a), but in this case, the two DPSK TXs emit at the same 𝜆 for total spectral overlap (𝜆1=𝜆2), as the worst case. It is worth mentioning that the interfering 𝜆2 was also modulated because, in practice, the side-modes exhibit approximately the same spectral shape than the main laser mode [Pet88]. The output power of the transmitted channel 𝜆1 was set to 0dBm, whereas the interfering 𝜆2 power was variable to adjust the SIR. Figure 3.23. Optical spectrum of a single-mode laser with spurious side-modes generating crosstalk on adjacent udWDM channels. First, the BER against the received optical power was computed for several SIR values, and plotted in Fig. 3.24(a). As the SIR decreases, the laser side-modes crosstalk causes BER degradation and produces error-floor on the channel located at the same 𝜆 than the spurious side mode. Fig. 3.24(b) depicts the sensitivity penalty at BER = 10-3 in terms of the SIR, revealing that the crosstalk power must be at least 18 dB lower than the transmitted power to avoid penalties larger than 1 dB. This value serves as reference to determine the required SMSR for the lasers in the udWDM-PON, to avoid severe interference on adjacent channels. Note that in the udWDM scenario, the side-modes crosstalk originates from not only one but two far channels located at the same 𝜆-difference from the user 𝜆, but in opposite sides. Hence, extra 3 dB tolerance should be added to the SMSR limit, as the worst case. Furthermore, all the udWDM channels do not emit at the same power, being allowed 15 dB maximum power difference between 𝜆s [ITU19] that should be considered as well. (a) (b) Figure 3.24. (a) BER as a function of the received power for several SIR, and (b) sensitivity penalty at BER = 10-3 for the same test. Power SIR SMSR Side-mode separation nth-channel 71 Accordingly, SMSR = 36 dB is considered an appropriate value to mitigate the crosstalk by laser side-modes, in the margin of the available technology for optics. Another interesting remark that can be derived from the simulations is that, during the activation process of new ONUs connecting to the PON, it is necessary to attenuate the output power of the TX during its 𝜆 transition, in at least 18 dB with respect to its nominal emission power, to avoid BER degradation on the already active channels in the PON. 3.4.1.5 Experimental assessment of channel spacing in udWDM-PON The udWDM system was implemented to experimentally verify the pulse shaping effect on the mitigation of adjacent channel crosstalk, and the minimum channel separation in the udWDMPON found by numerical simulation. The experimental setup is depicted in Fig. 3.25, and consisted of two identical TXs (TX1 and TX2) based on directly-modulated DFBs, with 4 MHz and 3 MHz linewidth respectively, and modulated by two uncorrelated PRBS sequences. The RZshapers perform the time-derivative of the NRZ data for direct phase modulation, as pointed out in Section 2.4.2, followed by the raised cosine filters for Nyquist spectral shaping. After 25 km of SMF, the signal was detected by the 3x3 homodyne RX introduced in Fig. 3.6(b), that implemented a 100 kHz linewidth ECL as LO, and emitting at 0 dBm. The LO 𝜆 was tuned to detect TX1. No matched filtering at the RX is considered to lower the overall ONU complexity. Figure 3.25. Experimental setup to evaluate the pulse-shaping impact on the minimum channel spacing between udWDM users. AWG: arbitrary waveform generator; RTO: real-time oscilloscope. The eye diagram and time-domain waveforms of the direct-DPSK modulating current are plotted in Fig. 3.26, for 50% duty-cycle RZ signal with and without Nyquist filtering. It is worth noting that in direct phase modulation, data symbols with zero phase-shift apply no current swing to the laser during the symbol time 𝑇𝑏 producing zero-crossing in the eye diagram of the laser driving current, as appreciated in Fig. 3.26(a). An important outcome is that the Nyquist filtering eliminates the modulation side-lobes, but the filter roll-off has no impact on the width of the main modulation-lobe because of the non-linear PM modulation, as stated before. This phenomena can be appreciated in Fig. 3.27 that plots the experimental photodetected spectra for DPSK at 2.5 Gb/s, centered at 10 GHz, for two roll-off values of the Nyquist filter. Observe that the width of the main PM modulation-lobe remains IQ recovery LPF LPF LO 25km SMF VOA PC PRBS RZshaper DAC Nyquist shaper TX 1 PRBS RZshaper DAC Nyquist shaper TX 2 AWG DFB DFB ECL BER RTO TX RX 72 (a) (b) (c) (d) Figure 3.26. Eye diagram of the 50% duty-cycle RZ signal for direct DPSK modulation: (a) without Nyquist pulse-shaping, and with Nyquist shaping for (b) 𝛽=1 and (c) 𝛽=0.25; (d) time domain waveforms of (a) and (c). (a) (b) Figure 3.27. Photodetected spectra for DPSK at 2.5 Gb/s with Nyquist filtering, for (a) 𝛽=1 and (b) 𝛽=0.25. unchanged, regardless the filter roll-off, while the width of the residual IM in the base-band is narrowed, because the IM is generated by a linear modulation. Accordingly, 𝛽=1 was selected for the following tests as the effect of lower 𝛽 is barely noticed. First, the DFB laser were directly modulated at 1.25 and 2.5 Gb/s DPSK. The BER curves in Fig. 3.28(a) indicate that, without Nyquist shaping, the Rx sensitivities were -52 and -49.5 dBm respectively, measured at a FEC limit of BER = 10-3. These values were approximately 3 and 5.5 dB better with respect to the case with Nyquist shaping, (-49 and -44 dBm for 1.25 and 2.5 Gb/s respectively). The sensitivity penalty can be ascribed to strong Nyquist filtering in combination with non-flat RF response of the electro-optical components that introduce distortion and extra ISI, thus reducing the eye-opening at the RX. This effect is clearly observed in Fig. 3.28(b) by comparing the received eye diagrams w/. and w/o. Nyquist shaping. The use of matched filtering at the RX with further equalization to flatten the channel response may lead to ISI reduction and improvement of the eye-opening. Next, the minimum channel spacing between users was assessed for the highest 𝑅𝑏, i.e., 2.5 Gb/s. The received optical power was adjusted for BER = 3×10-4 when detecting TX1, and the TX2 𝜆 73 (a) (b) Figure 3.28. (a) BER against Rx optical power for Tx1 and eye diagrams at BER = 3 · 10–6 without and with Nyquist shaping at 1.25 Gb/s (b,c) and 2.5 Gb/s (d,e) respectively. (a) (b) Figure 3.29. BER against channel spacing for 2.5 Gb/s DPSK with and without Nyquist shaping: (a) both users at the same optical power; (b) the interfering TX2 emitting 15 dB higher in power than TX1. was swept ±12.5 GHz with respect to TX1. The results plotted in Fig. 3.29(a) show no meaningful channel spacing reduction after applying Nyquist filtering (𝛽=1), mostly due to (I) the unchanged main PM modulation-lobe width, and (II) the significant suppression of the other harmonics of the spectrum w/o. Nyquist filtering in Fig. 3.27, due to band limited electronics. Nevertheless, the benefit of the Nyquist shaping becomes more apparent when maximum 15 dB differential optical path losses are allowed in the PON, as specified in [ITU19] for NG-PON2, due to the strong interference of modulation side-lobes on weaker adjacent channels. The BER curves in Fig. 3.29(b) were computed from TX1 emitting 15 dB lower than the interfering TX2. The results indicate that the Nyquist shaping reduces in 25% the required channel spacing for BER = 10-3, allowing that 2.5 Gb/s DPSK users having 15 dB power difference between adjacent channels can be fitted in a 6.25 GHz spaced optical grid. As a final remark, the benefits of spectral shaping by Nyquist filtering are not exclusive for ONUs with DSP systems. The analog ONUs can also leverage the analog Nyquist shaping by means of sharp electrical brick-wall microstrip filters [Sch15]. w/o. Nyquist w/. Nyquist 1.25 Gb/s 2.5 Gb/s 80 of directly phase-modulated DFBs, and non-ideal and band-limited frequency response of RF amplifiers and DACs. (a) (b) Figure 3.35. BER dependence on the distortion delay (𝜏), for a magnitude ripple of (a) 1.75 dB and (b) 3.5 dB. 3.5.1 Frequency dip in laser FM response Besides the ripple in the magnitude, the non-flat FM response of semiconductor lasers also include a frequency dip at low modulation frequencies. This obeys the changes in both the temperature and the carrier density under injection current modulation, which in turn causes variations in the refractive index of the semiconductor active layer, as explained in Section 2.4. As a result, the overall FM response of the laser is the combined effect of thermal modulation and carrier density modulation, which may either produce a dip or an enhancement in the FM response at the frequencies where both effects converge [Vod90]. The thermal modulation, with large time constants, is only effective at low frequencies below few MHz (usually < 10 MHz), whereas the carrier density modulation produces a nearly flat response extending from MHz up to about several GHz, with a resonance peak at higher frequencies. This behavior is illustrated in Fig. 3.36. Figure 3.36. General FM response of semiconductor lasers. Experimental characterization of the laser FM response at lower frequencies is shown in Fig 3.37, using a commercial DFB with sinusoidal current modulation. The laser emitted at 𝜆 = 1550.6 nm when biased at Ibias = 82 mA. The result is a wideband FM modulation that prevails over the AM modulation, as appreciated in the photodetected spectrum after heterodyning in Fig. 3.37(inset). The laser FM response was measured in terms of the FM efficiency for a modulation current 81 swing of 3 mApp, revealing a frequency dip clearly visible whose minimum is located about 400 kHz. Figure 3.37. Experimental FM response at lower frequencies for a DFB with 3 mApp sinusoidal current modulation. The inset shows the electrical spectrum for 𝑓𝑚 = 10 KHz. The frequency dip may induce system degradation and pattern-dependent performance, because of the pass-band characteristic of the effective laser BW. The tolerable cut-off frequency imposed by the dip, and its impact on the low frequencies of the data pattern, were evaluated by numerical simulation. The simulation setup was simply implemented by high-pass filtering the NRZ-PRBS data at the TX in Fig. 3.5, to suppress the frequency components falling into the dip of the laser. The high-pass filter (HPF) consisted of a 4th order Bessel whose 3dB cut-off frequency 𝑓𝑐 was adjusted for the different values of the laser dip. The optical signal was detected by a 90º homodyne RX. The received optical power was swept and the BER computed for several cut-off frequencies of the HPF, and for two different data sequences: PRBS-7 and PRBS-15, to evaluate the pattern dependence. Higher order PRBS were not considered due to the computational limit of 218 total bits for Monte-Carlo simulation. Results are plotted in Fig. 3.38. (a) (b) Figure 3.38. BER as a function of Rx power for several HPF cut-off frequencies, and two different data sequences: (a) PRBS-7 and (b) PRBS-15. 870 MHz 82 As expected, the frequency dip leads to pattern-dependent performance. For PRBS-7 in Fig. 3.38(a), the maximum cut-off frequency for 1 dB maximum penalty at BER = 10-3 is 6 MHz, whereas for PRBS-15 in Fig. 3.38(b) the maximum tolerance is 1.5 MHz, and the performance degradation is much more severe with apparent error-floor. This is ascribed to the repetition rate of the PRBS, that is 9.8 MHz and 38 kHz for PRBS-7 and PRBS-15 respectively, at 1.25 Gb/s. The repetition rate represents the minimum frequency component of data, and dictates the sensitivity penalty at the detection because of the high-pass characteristics of the frequency dip. Several methods have been reported to counteract the low-frequency dip of semiconductor lasers, including pre-equalization at the TX [Ale89] or post-equalization at the RX [Enn90] of the laser FM response, or the use of line coding like the 8B10B code, used to transport Gigabit Ethernet traffic, that removes the low frequency part of the baseband data by adding a codified redundancy [Bar08]. Although the direct-phase modulation of broadband data only leverages the carrier density modulation region of the laser FM response, there is not necessarily a waste of BW in direct laser modulation. Indeed, for an efficient use of the available BW, the thermal modulation region below the laser dip can be exploited for low-frequency FSK signaling [Hon15], [Nak14], [Tan12] useful for implementation of the auxiliary management and control channel (AMCC), mandatory in WDM-PON deployments for network control and optical grid management [ITU19]. 3.6 Coherent ONU Specifications for udWDM-PON The previous sections carried out a comprehensive analysis, through numerical simulation and preliminary experiments, on the techno-economic factors and the transmission impairments that have the highest relevance for the design and operation of the coherent ONUs for PON. The main conclusions drawn and the maximum tolerance found for each of the impairments serve as reference for ONU specifications and PON dimensioning, in terms of number of users and maximum fiber reach ‒given in this scenario by the final RX sensitivity‒, channel spacing, data rate, among others. In what follows, the identified transmission impairments in udWDM-PON are summarized, highlighting their impact on both the coherent TRX design and the final system performance. 3.6.1 Summary of transmission impairments and mitigation strategies The list of the main impairments identified here, which affect the performance of the udWDMPON is presented in Table 3.8. As stated before, this chapter primarily aims at the residential users segment, with dedicated 𝜆 at 1.25 – 2.5 Gb/s. For each of the impairments, it is provided: the physical origin, the effect relevance, the key parameters defining the phenomena, the typical expected value, the possible mitigation strategies, and the maximum tolerance for a given RX sensitivity penalty. 83 Table 3.8. Summary of the most relevant transmission impairments in udWDM-PON. Laser phase and frequency error Phase noise  Origin: Internal laser-cavity phase fluctuations.  Typical expected value: 1 – 10 MHz, for commercial DFBs.  Mitigation: Phase-diversity coherent detection, differential demodulation, DSP phase estimation.  Maximum tolerance at BER = 10-3: 1dB power penalty for 𝛥𝜐𝑇𝑏=1.6% Frequency drift  Origin: Thermal fluctuations, noisy current sources.  Typical expected value: ±100 MHz, for thermally-controlled DFBs  Mitigation: LO automatic frequency control, DSP frequency estimation.  Maximum tolerance at BER = 10-3: 1dB power penalty for 𝛥𝑓𝐼𝐹𝑇𝑏=6% Additive noise at the RX Shot and thermal noise  Origin: Thermal noise originates from the electronic part of the RX, mostly from TIAs. Shot noise originates from the photodetection due to the quantum-nature of light.  Typical expected value: Thermal noise spectral density 𝑆𝑓=10−18 pA/√Hz, for commercial TIAs.  Mitigation: LO power increase. Shot noise establishes the quantum-limit RX sensitivity.  Maximum tolerance at BER = 10-3: LO power higher than 10dBm for maximum 1 dB power penalty, with respect to the quantum-limit sensitivity 84 Additive noise at the RX RIN  Origin: Intensity noise contribution from the strong LO  Typical expected value: -150 dB/Hz.  Mitigation: Balanced photodetection, CMRR improvement.  Maximum tolerance: CMRR > 25 dB for no meaningful RIN impact. Optimum 8 – 12 dBm LO power for CMRR < 25 dB Interand intrachannel crosstalk Crosstalk by adjacent udWDM channels  Origin: Modulated spectrum side-lobes interfering adjacent channels in the ultranarrow optical udWDM grid.  Typical expected value: >10dB modulated side-lobes suppression.  Mitigation: Electrical pulse-shaping, accurate 𝜆-control of lasers.  Maximum tolerance at BER = 10-3: <1 dB power penalty for channel spacing larger than 4 GHz, 3.1 GHz, and 2.6 GHz for NRZ, Gaussian, and raised cosine pulseshape respectively. Up to 15 dB power difference tolerance with adjacent channels, for 6.25 GHz spaced udWDM optical grid with raise-cosine pulse shaping. Laser sidemodes interference  Origin: Laser cavities coupling, modepartitioning noise.  Typical expected value: SMSR > 30 dB.  Mitigation: Spectral management, improvement of laser quality.  Maximum tolerance at BER = 10-3: 1dB power penalty for SMSR = 21 dB, when all the udWDM channels emit at the same power. 85 Interand intrachannel crosstalk Laser sidemodes interference 1dB power penalty for SMSR = 36 dB, when 15 dB power difference between udWDM channels are considered. Crosstalk by optical reflections  Origin: Limited return loss of connectors and optical devices, non-ideal isolation. Relevant when the crosstalk has the same 𝜆 than the signal (HOC).  Typical expected value: Return loss >50 dB  Mitigation: Optical isolators, APC connectors.  Maximum tolerance at BER = 10-3: Optical reflections lower than -30 dB have no impact. Optical reflections lower than -18 dB for 1dB power penalty, with total laser 𝛥𝜐= 8 MHz (TX+LO), the typical for DFBs. Non-flat electrooptical frequency response Non-flat RF frequency response  Origin: Limited RF BW, electrical distortions, impedance mismatching.  Typical expected value: magnitude spectrum ripple < 2 dB  Mitigation: RF design improvement, electronic and digital equalization.  Maximum tolerance at BER = 10-3: < 2 dB magnitude spectrum ripple for 1dB power penalty. Laser frequency dip  Origin: Thermal and carrier density effects frontier in the laser FM response.  Typical expected value: frequency dip nearby 1 MHz  Mitigation: preand post-equalization, data encoding (e.g., 8B10B).  Maximum tolerance at BER = 10-3: Pattern dependent performance. 1dB power penalty for 1.5 MHz frequency dip, with PRBS-15, as the worst simulated case. 86 3.6.2 Performance estimate of coherent receivers for PON With all these impairments present, if all those sensitivity penalties for each are allowed, one may end up with the total sensitivity penalty summarized in Table 3.9 over the benchmark RX sensitivity▲, for homodyne and heterodyne detection. Yet, the accumulation of impairment effects may not be linear. The values reported in Table 3.9 are first purely orientative, may vary depending on the implemented devices, and will be experimentally assessed in subsequent chapters. Thus, a conservative estimate of the final RX sensitivity for 1.25 Gb/s DPSK users, is of -52 and -49 dBm for homodyne and heterodyne RXs respectively. These coherent RX sensitivities are substantially better than the IM-DD counterpart, that offer -32 dBm sensitivity at the same 1.25 Gb/s bit rate [ITU19]. Table 3.9. Expected performance of coherent RXs in udWDM-PON . Power penalty [dB] @ 10-3 BER Remarks Residual IM 1 Direct phase modulation Thermal noise 1 LO power limited by RIN Linewidth 1 Commercial DFBs Frequency drifts 1 Thermally-tuned DFBs Differential detection 0.5 - Optical front-end 1 + 0.5 Excess loss + connectors IF to baseband 3 Heterodyne detection only RX sensitivity [dBm] @ 10-3 BER Remarks Benchmark▲ -58.5 Homodyne, simulation Expected in real scenario ~ -52 Homodyne ~ -49 Heterodyne ▲Sensitivity curves in Fig. 3.8 from numerical simulation with the best parameter configuration. Moreover, the theoretical analysis and numerical simulations carried out in this chapter do provide sufficient evidence to support the main claims of the proposed udWDM-PON scenario. More concretely, cost-effective 𝜆-to-the-user enabled by a novel class of coherent TRXs for PON, with superior RX sensitivity. The optical spectrum is efficiently utilized by allocating the 𝜆s in ultranarrow channel spacing, as low as 6.25 GHz in between 1.25 – 2.5 Gb/s users. The coherent ONUs can leverage commercial DFBs, low-cost optics, and simple electrical signal processing ‒ either analog or digital‒, but still retaining high the performance. Among the three options evaluated for coherent RX, the 90º homodyne RX with 8 PDs (for full polarization) and two optical 90º hybrids, is not a feasible candidate for PON, and only served as benchmark. The other two architectures, 3x3 homodyne and 2x2 heterodyne, are the most promising solutions as they exhibit the best trade-off in terms of performance, power consumption, and cost per user. Accordingly, they were selected for implementation in the COCONUT network to demonstrate the complete functionalities of the coherent udWDM-PON. The following Chapter 4 covers this topic. 87 3.7 Chapter summary This chapter provided an evaluation of the main transmission impairments that were identified as the main limitations in the udWDM-PON, enabled by simplified coherent systems. The impairments assessments was realized through numerical simulations and preliminary experiments. At the fiber channel, both linear and non-linear propagation effects can degrade the PON performance, thus limiting on the maximum symbol rate and launch power. Yet, impairments from the optical fiber are not considered here as the main limitation for the PON scenario, and the dominant transmission impairment mostly originate from the TX and RX subsystems of the coherent ONU, as the phase noise, frequency drifts, LO-RIN, crosstalk by adjacent channels, non-flat electro-optical frequency response, among others. Possible solutions were discussed to mitigate the impact of such impairments, drawing design rules to implement the PON and optimizing network parameters. The results indicate the feasibility of using commercial DFBs and simpler optics to implement the coherent ONU, but still achieving high RX sensitivity and transmission robustness. At the TX, bulk IQ modulators are replaced by direct amplitude-and-phase modulation of semiconductor lasers. At the coherent RX, optical 90º hybrids are replaced by much simpler 2x2 and 3x3 fusedfiber couplers, also reducing the number of PD thus lowering the power consumption of the ONU. With this novel class of simplified coherent systems, two application scenarios within the PON, , were proposed: for enterprises and mobile services at 10 – 20 Gb/s, 𝑚-(A)PSK optical TXs with 3x3 homodyne RXs; and for residential users at 1.25 – 2.5 Gb/s, DPSK TXs with 2x2 heterodyne RXs. This allows for convergence of services within the same deployed optical fiber. 88 Chapter 4 The COCONUT PON: Next Generation Wavelength-tothe-User Access The overview of the PON technologies evolution provided in Section 1.1, emphasized the fundamental idea underpinning the next generation access networks deployment: enhancing the PON performance and capabilities, but retaining the cost down, as it lies on the final PON subscribers. This translates into reducing the complexity, cost and energy consumption of the ONU equipment at most without excessively sacrificing the performance. To fulfill these strict requirements of next generation access, efforts in research are being taken to develop the Gigabitto-the-user concept, towards user BWs in excess of 1 Gb/s, but enhancing the spectral efficiency by allocating hundreds of 𝜆s in ultra-dense grid configuration [Pra16], [Agr16], [Kaz17]. Concerning this, the research involved in the COCONUT ‒Cost-effective Coherent Ultra-dense WDM-PON for Lambda-to-the-user Access‒ project (http://www.ict-coconut.eu) aimed at the development of cost-effective solutions covering both photonic technologies and system design, that will enable the significant capacity increase in the throughput of an OLT, from several Gb/s offered today by TWDM-PON systems like XG/NG-PON systems, to hundreds of Gb/s supported by the use of coherent technologies and novel signal processing techniques, yet simple and HWefficient. Precisely, the main novelty of the COCONUT PON is the use of coherent detection in an optical access network, to realize cost-effective WDM in ultra-dense optical spectrum configuration. The coherent udWDM-PON solution enables, as analyzed in Chapter 3, a large number of 𝜆-channels to be densely allocated in the optical spectrum, while the RX sensitivity is significantly improved, nearby 15 – 20 dB, with respect to conventional DD. This results in connectivity to a larger number of end users (up to 256) compared to standardized PON solutions. The primary application aimed in COCONUT-PON is next generation FTTH networks, since they constitute a major driving force worldwide. Nevertheless, the different capabilities of the two types of coherent TRXs for PON reported in Section 3.1, potentially suggest the convergence of heterogeneous applications/services within the PON, such as residential, business connectivity and mobile antenna. Hence, the COCONUT-PON uses dedicated 𝜆-channel per user, with channel spacing from ultra-dense 6.25 GHz to conventional 100 GHz ITU-grid, for bit rates ranging from 1.25 to 10 Gb/s respectively, depending on the coherent TRX type and the application scenario. This chapter reports on the main research outcomes of the European project COCONUT, whose complete functionalities were demonstrated in a field trial open to public visitors, using coherent TRXs prototypes operating in real-time with multimedia traffic, and transmitting trough dark fiber installed across the city center of Pisa, Italy. 4.1 Network Architecture 89 Fig. 4.1 illustrates the architecture of the COCONUT udWDM-PON, consisting of a standard tree-based PON with pure power splitting, where the drop fiber to every coherent ONU branches from the feeder fiber or from a previous drop. Each user connected to the PON is served by a dedicated 𝜆, efficiently subdividing the optical spectrum into many ultra-densely spaced channels, as low as 6.25 GHz separation between adjacent users, and individually detected by high-selective coherent ONUs without optical filters. The bit rates per 𝜆 span from 1.25 or 2.5 Gb/s for the case of residential subscribers, up to 10 or 20 Gb/s for enterprises and mobile services. The bidirectional streams are symmetric, with the upstream (US) and the downstream allocated in different 𝜆s and flexibly spaced, according to the coherent detection type as will be explained in upcoming sections. Figure 4.1. Network architecture of the COCONUT udWDM-PON; OLT: optical line terminal, ONU: optical network unit, DS: downstream, US: upstream, FEC: forward error correction, DE: differential encoding, OSA: optical spectrum analyzer. The splitting ratio of the udWDM-PON has flexibility depending on the balance between fiber distance reach, number of users served and its geographical distribution. The high sensitivity of the coherent RXs enables large power splitting without the need of optical amplification, maintaining the ODN totally passive and compatible with legacy optical access systems. This allows to the PON serving up to 256 users, in contrast with the 32/64 users of typical TWDMPON [ITU17], [ITU19]. At the customer premises, the ONUs are based on the maximally simplified coherent TRXs architectures described in Section 3.1, replacing the optical 90° hybrids by low-cost fused-fiber couplers (2x2 for heterodyne and 3x3 for homodyne) at the coherent RX front-end, also with less number of photodiodes. The use of directly modulated DFBs at the TX instead of external modulators contributes to reduce the form factor and power consumption of the coherent TRX. Next Section 4.2 outlines the technical details of the coherent ONUs, for both homodyne and heterodyne detection. The udWDM OLT comprises the coherent TRX array for the virtual point-to-point (P2P) connectivity with the PON users, referring to the fact that each users owns a dedicated 𝜆-channel. The OLT also concentrates the more complex elements, which can be shared, like the highDFB Tx1 TX array CoherentRx 1 RXarray udWDM OLT Coherent ONU 1 1: DS US hr-OSA -control Eth. switch FEC FEC US data DS data USDS PON Coherent ONU 2 DE DFBTX Equalizer Direct-DFB TX Coherent frontend Signal processing FEC CoherentRX FEC -control Ethernet Coherent ONU DS data US data MUX 96 decoded the signal real-time, and subsequently both demodulated data streams were added to overcome SOP mismatching. The baseband electrical data at 1.25 Gb/s were finally low-pass filtered and amplified by a 1 GHz low-noise amplifier, then delivered to a media converter (MC) module connected to multimedia streaming servers through a RJ-45 connector, or to a BER tester for sensitivity measurements. The direct DPSK TX is based on reflective semiconductor optical amplifier (RSOA) to reuse the same emission 𝜆 of the LO for the US link, i.e., shifted by 2.5 GHz from the DS. The RSOA operated in the saturation gain region that, along with the high input power from the DFB (~1 dBm), optimized the PM modulation of the RSOA and lowered the residual IM [Chu15]. The 1.25 Gb/s NRZ data for the US were differentially encoded, then phase-modulated the RSOA. The generated DPSK signal passed through a circulator to be coupled into the COCONUT ODN with 0 dBm optical power. Note that the optical front-end of the heterodyne ONU implements two optical circulators: one for US/DS separation, and other for feeding the RSOA with the LO lightwave. The optical spectrum at the heterodyne ONU is illustrated in Fig. 4.7. In contrast with the homodyne ONU in previous section, the heterodyne ONU features the closest separation between the US and DS links, set to 2.5 GHz by reusing the DFB LO as TX for the US. Nevertheless, the adjacent udWDM users are separated by 12.5 GHz, twice the homodyne channel spacing, to allocate for the image frequency characteristic of the heterodyne detection, as studied in Section 3.4.1.3. Figure 4.7. Optical spectrum at the heterodyne ONU for bidirectional transmission. 4.3 COCONUT Field-Trial The udWDM-PON field-trial carried out for the public demonstration of the COCONUT project, was implemented in the access network laboratories of the Scuola Superiore Sant'Anna in Pisa, Italy. The overall scheme of the network is illustrated in Fig. 4.8. The udWDM COCONUT PON consisted of a set of coherent TRXs operating in real-time at 1.25 Gb/s (the 10 Gb/s TRX implemented off-line processing), using various modulation formats (ASK or DPSK) and coherent detection schemes (homodyne, intradyne or heterodyne). The coherent TRX prototypes were implemented by either analog processing with electronic hardware or DSP with FPGA. At the OLT side, 14 WDM signals performing as DS were multiplexed by an arrayed waveguide . . . 2.5 GHz 12.5 GHz DS+US 97 Figure 4.8. COCONUT udWDM-PON field-trial, using real-time coherent ONU prototypes with multimedia services. The inset shows the optical bands used for bidirectional transmission in the trial. CO: central office; CE: coexistence element. Figure 4.9. Optical distribution network: 10 km external trunk fiber. grating (AWG) with 100 GHz channel spacing and 100 GHz full-width at half maximum (FWHM) on each channel. The DS signals were connected to seven input ports of the 100 GHz AWG in the following configuration: 1.25 Gb/s DPSK for the homodyne ONU; 1.25 Gb/s DPSK for the heterodyne ONU; 8× 1.25 Gb/s ASK, on a 6.25 GHz grid, for the intradyne ONU; and 4× 10 Gb/s ASK, using four individual 100 GHz AWG channels, also for intradyne ONU. The channel selection at the ONUs was done by tuning the LO to the assigned transmitted 𝜆. In addition, a coexistence element (CE) inserted an E-PON signal into the COCONUT PON to assess the coexistence and seamless integration of the udWDM network with legacy PON systems. The CE consisted of a commercially available optical filter originally designed for NG-PON2 applications, that guaranteed error-free operation of the E-PON system. The COCONUT OLT also implemented the hr-OSA for monitoring the DS and US channels. The ODN exploited a 10 km G.652 SMF dark-fiber installed across the city center of Pisa, Italy, Pisa external trunkfiber COCONUT OLT DS US hr-OSA GbE Switch 100 GHz AWG ASK Intradyne 8 1.25 Gb/s US DS WiFi ASK Intradyne 4 10 Gb/s DPSK Homodyne 1.25 Gb/s DPSK Heterodyne 1.25 Gb/s CE E-PON OLT IP-TV Broadcast CO 10 km IntradyneONU 1.25G ASK IntradyneONU 10G ASK HomodyneONU 1.25G DPSK HeterodyneONU 1.25G DPSK GbE Switch 25 km CE 10 km E-PON ONU IP-TV interface HD TV US/DS US DS O-band S-band C-band COCONUT E-PON 98 as shown in Fig. 4.9, and made available by the local company AGESTEL. It was followed by a cascade of optical power splitters and extra SMF spools, connecting a total of 5 ONUs. With this configuration, the fiber reach ranged from 10 to 35 km. At the OLT, the launched power per udWDM channel was 0 and 3 dBm for the DPSK and ASK systems respectively. At the ONU side, 3 different options for US allocation were considered: the ASK intradyne systems used two different 100 GHz ITU bands for US and DS, the DPSK homodyne system employed ultra-dense 6.25 GHz spacing between US and DS, and the DPSK heterodyne system featured ultra-narrow 2.5 GHz spacing between US and DS by reusing the LO laser as TX for the US (see Fig. 4.7). Both at the OLT and the ONU sides, some of the coherent TRXs were connected to Ethernet switches or multimedia streaming servers to demonstrate network operations in a heterogeneous distribution of real-time traffic. The COCONUT TRXs have been operated at the C-band, with 𝜆s ranging from 1546.12 to 1551.72 nm, as depicted in Fig. 4.10(a), that also shows the 𝜆 allocation for the E-PON system hosted in the testbed. Figs. 4.10(b) and 4.10(c) report on the optical spectrum at the OLT, showing the 14 udWDM channels for DS plus the E-PON signal. (a) (b) (c) Figure 4.10. (a) Wavelength allocation plan used for the COCONUT field-trial; (b) optical spectrum at the OLT; (c) zoom into the 14 COCONUT DS channels at the C-band: (a) pilot tone, (b) 8× 1.25G ASK, (c)-(e) 10G ASK, (f) 1.25G DPSK, (g) 10G ASK, (h) 1.25G DPSK. 4.4 Results and Discussion The coherent ONU prototypes were evaluated in the COCONUT testbed in terms of RX sensitivity, ultra-dense WDM multiplexing with narrow channel spacing, and coexistence with heterogeneous real-time traffic and legacy PON systems. Due to the fact that each COCONUT user owns a dedicated 𝜆, the udWDM-PON behaves as a virtual point-to-point network with both DS and US operating full-duplex at different 𝜆s; therefore, the results and analysis provided in this section are reasonably valid for both streams. For BER measurements, PRBS-31 digital 8 1.25G ASK DS (nm) E-PON DS 1290 1330 E-PON US 1480 1500 1546.12 Intradyne 310G ASK DS 1546.92 1548.51 Intradyne 1.25G DPSK US/DS 1549.32 Heterodyne 10G ASK DS 1550.12 Intradyne 1.25G DPSK US/DS 1550.92 Homodyne 1.25G ASK US 1551.72 Intradyne COCONUT -70 -60 -50 -40 -30 -20 -10 0 10 1488 1504 1520 1536 1552 Power (dBm) Wavelength (nm) COCONUT E-PON -70 -60 -50 -40 -30 -20 -10 0 10 1542 1546 1550 Power (dBm) Wavelength (nm) (a) (b)(c)(d)(e)(f)(g)(h) 99 sequences at 1.25 or 10 Gb/s were used as data, then detected and processed real-time by the analog/digital coherent TRXs, and the recovered data were finally delivered to BER testers for error computing. Let us first consider the homodyne ONU with DPSK at 1.25 Gb/s. This system was initially tested in optical back-to-back (btb) as benchmark, and the results are reported in Fig. 4.11(a) in terms of the BER against the received optical power. As observed, the homodyne RX achieved a high sensitivity of -53 and -46 dBm for BER = 10-3 and 10-9 respectively, that matches well with the performance estimate in Table 3.9. The recovered eye diagrams after differential demodulation and CLK recovery (i.e., data are already at one sample per bit) are depicted in Fig. 4.11(b), for the two BER values in the btb test. (a) (b) (c) (d) Figure 4.11. (a) Real-time BER vs received power for 1.25 Gb/s DPSK detected with homodyne ONU in the COCONUT PON; (b) normalized eye diagram at the RX after demodulation for btb test, with BER = 10-3 (upper) and BER = 10-9 (lower). Electrical spectrum at the RX after photodetection for (c) only US direction in btb, and (d) bidirectional transmission through the ODN. Next, the system was connected to the COCONUT testbed with all the other systems disabled, and the US was detected. As observed in Figure 4.11(a), the RX sensitivity penalty at FEC threshold of BER = 10-3 is found to be 1.5 dB compared with btb transmission due to impairments that arise from the ODN, mostly optical reflections. The US direction was detected again with all the systems connected to the PON for the coexistence test, with no apparent penalty as observed 1.25 GHz US Back-scattered DS1.25 GHz US 6.25 GHz 100 in the curves, thus confirming the filterless operation of the PON driven by the superior selectivity of coherent detection, which uses 𝜆-tuning of the LO in combination with sharp electrical filtering to reject the out-band interference from the other systems in the PON. Last, the DS was activated to assess the bidirectional udWDM operation. The DFB TX of the OLT modulated PRBS data uncorrelated with the US, to generate the 1.25 Gb/s DPSK signal for DS. The US and DS were separated in 𝜆 by 6.25 GHz. The BER curve in Fig. 4.11(a) for the US detection indicates 2.5 dB penalty at FEC threshold for bidirectional operation when compared with only US transmission through the ODN, and 4 dB penalty with respect to the btb test. The sensitivity penalty originates from the back-reflections in the external trunk section of ODN, which included several connections (SC/PC connectors) in outside cabinets. This phenomena can be clearly appreciated when comparing the electrical spectrum at the OLT Rx after photodetection for the btb test, in Fig. 4.11(c), with that of the bidirectional transmission through the ODN, in Fig. 4.11(d). Note that the strong back-reflection of the counter propagating DS channel is comparable in power to the weak detected US channel; although US and DS are separated by 6.25 GHz, the back-scattered spectral tails DS fall into the US BW, producing penalty at the detection. The BER performance of the heterodyne ONU with 1.25 Gb/s DPSK data is reported in Fig. 4.12(a). The first test, optical btb, differed from the homodyne btb test in that bidirectional US and DS were active, because the single DFB laser in the heterodyne ONU lighted up both the TX and the RX. The BER corresponds to DS detection by the ONU RX, and the achieved sensitivities were nearby -38 and -32 dBm for 10-3 and 10-9 BER respectively. In this case, the RX sensitivity is lower than the estimated in Table 3.9 due to several reasons: (I) the single-ended detection added 3 dB penalty because one branch of the 2x2 coupler is unused (see Fig. 4.6); (II) the LO power was divided with a 3 dB optical coupler to also feed the RSOA, thus the RX sensitivity was accordingly reduced by 3.5 dB approximately including the coupler excess loss; (III) the common-mode noise and the LO RIN that are not cancelled due to the single-ended detection. The eye diagram in btb after differential demodulation is reported in Fig. 4.12(b) for two BER. (a) (b) Figure 4.12. (a) Real-time BER vs received power for 1.25 Gb/s DPSK detected with heterodyne ONU in the COCONUT PON; (b) eye diagram at the RX after demodulation for btb test, with BER = 10-5 (upper) and BER = 10-9 (lower). 101 The heterodyne system was later connected to the COCONUT ODN, with and without the other systems. The results in Fig. 4.12(a) indicate that, at FEC threshold, the power penalty after transmitting through the ODN is negligible, but arises up to 1.5 dB when coexisting with all the other COCONUT systems. It is worth noting that, for bidirectional transmission, the heterodyne ONU suffers less sensitivity penalty (1.5 dB) than the homodyne ONU (4 dB), with respect to the b2b test. This can be ascribed to the better sensitivity of the homodyne RX at FEC level: the received signal is very weak (-53 dBm), thus more affected by the strong back-reflections from the ODN that are aliased into the signal BW (see Fig. 4.11(d)). The functioning of the real-time coherent ONU prototypes was also demonstrated in the trial with multimedia services. For the DPSK prototypes, GbE frames were used to simultaneously transport four data streams, consisting of three multi-service video and one high-definition (HD) video broadcasted by a PROMAX HD video player module. In the case of the ASK systems, several 4K video were transmitted from a multimedia server, as well as WiFi internet connectivity through an ERICSSON access point. In all cases for multimedia real traffic demonstration, the COCONUT ONUs were set to operate error-free (i.e., the received power was adjusted to BER = 10-9), since none of the prototypes incorporated FEC encoders/decoders. Table 4.1 summarizes the main features and performances of the coherent ONUs for bidirectional transmission in the COCONUT testbed. The results include the ASK systems at 1.25 and 10 Gb/s. For the 10G system, BER values smaller than 10-6 were not measured due to the off-line processing. In general, all the systems show excellent RX sensitivities both at BER of 10-3 (where 2nd generation FEC can be implemented) and 10-9 (used for real-time GbE and video streaming tests). The homodyne ONU achieved the best sensitivity in its polarization-dependent configuration and digital processing with FPGA. Conversely, the heterodyne ONU shows the lowest RX sensitivity but features the simplest architecture and the highest power efficiency by owing single-laser at the ONU. Table 4.1. Summary of performance and main characteristics of the coherent ONU prototypes operating bidirectional real-time in the COCONUT testbed. System Receiver type Intermediate frequency Channel spacing [GHz] Signal processing Testbed sensitivity [dBm] BER = 10-3 2nd generation FEC BER = 10-9 Real-time GbE 1.25 Gb/s DPSK 2x2 heterodyne 2.5 GHz 12.5 Real-time analog -36.5 -31 3x3 Homodyne < 50 MHz 6.25 Real-time digital with FPGA -49 -42 1.25 Gb/s ASK 3x3 Intradyne 900 MHz 6.25 Real-time analog -47 -39 10 Gb/s ASK 3x3 Intradyne 3 GHz 100 Off-line with RTO -35 − 102 Overall, the coherent ONUs outperform conventional IM-DD ONUs by improving the RX sensitivity, reported to be -36 (the lowest, for E2 class ODN) and -28 dBm for 1.25 and 10 Gb/s respectively [ITU19]. Furthermore, the trial successfully confirmed that the TRXs preserve their performance also when operated in the testbed, where the systems are affected by reflections in the ODN, and a significant number of channels are lighted up: 17 WDM channels in total, 14 channels as DS and 3 channels as US. Based on the results from the field-trial in Table 4.1, the COCONUT udWDM-PON can be dimensioned and compared with XG-PON / NG-PON2 standards. Depending on the ODN class, optical power budget (PB) from 30 up to 35 dB are targeted in next-generation PON [ITU19], to coexist with legacy PON systems. The PB is a crucial parameter dictating the choice of technology to deploy next-generation optical access. In udWDM-PON, for instance, larger PB directly translates into larger splitting ratio serving more users, and/or extended fiber reach. Owing to the superior sensitivity of the coherent RXs, even larger PB > 40 dB can be achieved in the COCONUT PON. Table 4.2 details the PON dimensioning when taking as reference a target PB of 40 dB. Table 4.2. COCONUT udWDM-PON dimensioning and comparison with standards for PON. TX power per channel for 40 dB PB1 [dBm] Maximum number of channels2 Bit rate per 𝝀 [Gb/s] Aggregated PON capacity [Gb/s] Channel spacing [GHz] XG-PON [ITU17] 12 − 10 10 − NG-PON2 [ITU19] 12 8 10 80 50/100 COCONUT udWDMPON -3 256 1.25 320 6.25 1Referred to -28 dBm RX sensitivity at 10G for XG-PON and NG-PON2. 2Below the eye safety threshold of 21 dBm. By considering 21 dBm emitted optical power as the eye safety frontier, and having in mind 256 udWDM users, the emitted power per channel in COCONUT PON must be reduced to -3 dBm, which represents not any limitation by virtue of the high sensitivity of the coherent RXs (regardless the heterodyne DPSK RX sensitivity in the trial, that obeys a particular implementation and can be improved). Hence, the total number of ONUs that can be served by dedicated 𝜆s at 1.25 Gb/s is 256, and the aggregated PON capacity reaches 320 Gb/s, four times larger than NG-PON2. The spectral efficiency is further improved by reducing the channel spacing from 50/100 GHz down to ultra-dense 6.25 GHz, and the power consumption of the ONU is lowered as the TRXs operate at lower data rate and lower emitted optical power. Note that the TWDM-PON (NG-PON2) improves the optical efficiency with respect to TDMPON (XG-PON) by multiplexing several TDM 𝜆s. Alternative techniques for optical access like 103 radio-over-fiber (RoF) or orthogonal frequency division multiplexing (OFDM) aim at improving the electrical efficiency of the ONU [Cvi12], [Can15a]. Notably, the great improvement in both spectral efficiency domains is achieved with the COCONUT udWDM-PON, which can translate into relevant energy consumption and cost savings. This is particularly relevant in access networks and other optical networks with high terminals density, because any increase in efficiency scales rapidly to significant quantities. 4.5 Chapter summary This chapter reported on the results and main outcomes of the European COCONUT project, aimed at the development of cost-effective optical access by implementing the 𝜆-to-the-user concept, where hundreds of dedicated 𝜆s for each user are efficiently allocated in the optical spectrum in ultra-dense WDM grid. The main enabling technology is the use coherent detection with simplified TRX architecture, thus filling the gap between the high-performance yet costly commercial coherent TRXs, and the simple IM-DD TRXs that dominate the PON market segment. Prototypes of the novel class of low-cost coherent TRXs developed in COCONUT, were built and tested in real-time in a complete udWDM-PON field-trial. The system testbed has been implemented to demonstrate the most relevant COCONUT-PON concepts: network capacity upgrade by reusing legacy infrastructure, coexistence with legacy PON systems, ultra-dense WDM multiplexing with statistical 𝜆 allocation, and support for long reach and/or large splitting ratio thanks to enhanced optical PB. The trial achieved the successful transmission of multiple PON signals and real-time service distribution over a real optical fiber network deployed in the city center of Pisa, Italy. The results of the COCONUT field-trial indicate that the udWDM-PON can effectively implement dedicated 𝜆-to-the-user at 1.25 Gb/s (or higher, when required) in a filterless PON with channel spacing as low as 6.25 GHz, serving up to 256 users. Optical PBs in excess of 40 dB are obtained by virtue of the high sensitivity of the coherent RXs, implemented with low-cost optics and simple electrical signal processing, either analog or digital. In this regard, the trial demonstrated that the coherent TRX can be assisted by simple DSP optimized for use in PONs, exploiting simpler and cheaper FPGAs at the ONUs rather than custom expensive ASICs, thus revealing their potential for the future coherent optical access networks. More generally, the proposed COCONUT PON is flexible in 𝜆 allocation, and can conveniently be adapted to operate within the C-band, where low-cost DFBs are available. In terms of overall PON consumption, the optimality of the COCONUT udWDM approach with respect to current TWDM-PON systems, is to reduce the electrical BW to match that the demanded user traffic BW, supported by efficient exploitation of the optical domain, therefore reducing the power consumption of the ONUs. 104 Chapter 5 Optical Carrier Recovery Strategies Up until now, this thesis focused on investigating the enabling technologies, key design parameters and challenges to implement the next generation 𝜆-to-the-user access networks with low complexity and cost coherent TRXs. The main transmission impairments were identified and addressed in Chapter 3, pointing out their impact and the maximum tolerance against each of them to operate the coherent TRXs reliably below the BER threshold for FEC. The foregoing investigation contributed to the implementation of the COCONUT testbed to demonstrate the performance and functionalities of the coherent udWDM-PON in a field-trial, whose results were reported in Chapter 4. Hereafter, the remainder of the thesis takes a step forward towards the improvement of the udWDM-PON in terms of performance, complexity, cost, and power consumption, by introducing novel and/or enhanced mitigation strategies for the main transmission impairments employing either low-complexity DSP or analog HW design. To begin with, one of the major challenges in coherent systems is the synchronization of the received optical carrier with the LO. To avoid the use of PLL systems in homodyne RXs, the phase-diversity architectures can operate with free-running LO and overcome phase synchronization problems by further processing the I and Q signals, as it was derived in Section 2.1.2. In such phase-diversity schemes, the optical frequency of TX and LO lasers must remain as close as possible to each other, with an error less than 6% of the bit rate for DPSK (see Section 3.3.2), to avoid penalty at the detection. This contrast with typical low-cost DFB lasers that exhibit severe 𝜆 variability due to thermal environment changes and current supply fluctuations. Laser aging is another factor contributing to 𝜆 instability. For instance, tunable lasers integrated with external modulator for use in telecommunications equipment operating in the C or L band, exhibit frequency accuracy of ±2.5 GHz over lifetime [OIF08]. This also becomes a major issue in udWDM-PON as the ODN is filterless, and the channel selection is done at the coherent RX by tuning the 𝜆 of the LO. Hence, accurate LO 𝜆-control is mandatory for correct data detection and channel stability in such a closely-spaced udWDM grid. The LO 𝜆-control system is driven by the CR subsystem of the RX, in which this chapter focuses on. The block diagram of a conventional digital coherent RX implementing the complete CR subsystem is illustrated in Fig. 5.1. After coherent beating at the RX front-end, the ADCs gather digital samples of the received I+𝑗Q signal to be processed by the DSP stage, which performs frequency estimation (FE), LO tuning, and optical phase recovery (PR). Usually FE takes place prior PR, although they can be swapped depending on the implemented algorithms. The estimated detuning Δ𝜆 drives both the feedforward correction of the I+𝑗Q signal, and the LO feedback for 𝜆-tuning. Afterwards, the signal I’+𝑗Q’ already in baseband is further processed to extract the data encoded in the optical phase, getting rid of the harmful phase noise. This chapter presents the design and implementation of AFC systems, also known as automatic wavelength control (AWC), for coherent RXs either digital or analog. In the case of digital RXs, 105 Figure 5.1. Conventional carrier recovery for digital coherent RXs. simultaneous frequency detuning compensation is achieved by correcting data and tuning the 𝜆 of the DFB LO, using the same estimation block, thus lowering the HW complexity. The reported results include frequency noise characterization and DPSK transmission tests. The chapter also present an optimized CR architecture based on differential field detection for digital homodyne RXs, that substantially reduces the required DSP HW resources by reusing the 1-symbol complex correlation required for frequency estimation and differential phase detection. The results indicate improvement in the energy consumption of the digital processor and enhanced tolerance against fast 𝜆-drifts of lasers. 5.1 Algorithms For Frequency Detuning Estimation In Homodyne Receivers There exist several algorithms reported in literature to estimate the frequency error in digital homodyne RXs. Some of them are based on time domain estimation, while others rely on spectral analysis using fast Fourier transform (FFT) [Sav10]. Alternative strategies have also been proposed like remote 𝜆-control from the OLT by a centralized 𝜆-locker [Rop13]. The goal of the time-domain FE algorithm is to estimate the phase drift ∆𝜑=2𝜋Δ𝑓𝑇𝑠 between consecutive samples of the detected signal, produced by the frequency mismatch ∆𝑓=𝑓𝑆−𝑓𝐿𝑂 between TX (𝑓𝑆) and LO (𝑓𝐿𝑂) optical frequencies. Here, 𝑇𝑠 represents the sampling time. Note that if the 𝑚-PSK modulation is removed from the detected I+𝑗Q signal, there is no phase drift caused by the laser phase noise over a certain number of consecutive samples, whereas the phase drift caused by frequency mismatch increases a fixed value every 𝑇𝑠. This section considers two algorithms from the variety of available algorithms for time-domain FE in coherent RXs. One approach is the differential 𝑚th-power estimator, and the other is based on the cross-correlation between I and Q signals. These algorithms were selected due to their convenient implementation in DSP units, achieving good performance with relatively simple design [Nat84]. 5.1.1 Differential mth-power estimator The first approach consists of a phase-increment estimation algorithm based on the feedforward Viterbi&Viterbi estimation model for 𝑚-PSK [Lev07]. The FE architecture is shown in Fig. 5.2. Optical input I+jQ I’+jQ’ Carrier recovery ADCs LO Coherent RX front-end -control DAC Frequency estimation Phase recovery Baseband data 112 of RF detectors convert the power of the filtered RF signals into a DC level, followed by ACblocking ‒i.e., low-pass filtering with narrow BW, in the order of kHz‒ to reject the residual high frequency components after DC conversion. Finally, the two branches are either subtracted to generate the frequency error signal, and added to obtain the received optical power thus eliminate the dependency of the frequency discriminator on it by means of an automatic gain control (AGC) circuit. Figure 5.10. Frequency discriminator for AWC system in analog heterodyne RXs. The output signal 𝑦𝑒(𝑡) is a low frequency signal proportional to the frequency variations of the 𝑖𝑅𝐹(𝑡) spectrum with respect to IF. Note that the BW of the signal 𝑦𝑒(𝑡) to make the 𝜆-tracking does not need to be very high because the characterization of the frequency drift rate for freerunning TX−LO lasers, as calculated from Fig. 5.8, was about 3.6 MHz/s. It is worth also mentioning that the frequency discriminator in Fig. 5.10 implements 1st-order RC filters because they are the simplest solution; however, the AWC system design can be also extended to higher order filters. 5.2.1 Analytical model The frequency discriminator implements two complementary filters (LPF and HPF) with cut-off frequency 𝑓𝑐=𝑓𝐼𝐹 each. Without loss of generality, these RF filters are linear time-invariant (LTI) whose outputs are the convolution between the input signal and the filter impulse response ℎ(𝑡), as shown in Fig. 5.11. The result at the filter output is the same photocurrent 𝑖𝑅𝐹(𝑡) at the input but modified in amplitude and phase by the transfer function of the filters 𝐻(𝑓)=ℱ{ℎ(𝑡)}, with ℱ{∙} the Fourier transform. Figure 5.11. Analytical model of the analog frequency discriminator, with sinusoidal carrier at the input. The filter response 𝐻(𝑓) is a complex number that can be expressed in magnitude and phase as AC-block AC-block LPF HPF R R C C RF detector AGC Gain HPF RF detector LPFs LPF 113 𝐻(𝑓)=|𝐻(𝑓)|𝑒𝑗arg{𝐻(𝑓)} (5.8) From Eq. (2.29), the balanced photodetected current in the heterodyne RX writes as 𝑖𝑅𝐹(𝑡)=𝐴sin(2𝜋𝑓𝐼𝐹𝑡+∆𝜙(𝑡)) (5.9) with 𝐴=2ℜ√𝑃𝑆(𝑡)𝑃𝐿𝑂. Let us first consider the case where Eq. (5.9) is unmodulated, therefore it corresponds to a sinusoidal carrier oscillating at 𝑓𝐼𝐹 with Lorentzian-shape spectrum due to the phase noise ∆𝜙(𝑡). By using Eq. (5.8), it can be easily proven that both the low-pass and the highpass filter outputs assume the form 𝑖𝐿𝑃/𝐻𝑃(𝑡)=|𝐻(𝑓𝐼𝐹)|𝐴sin(2𝜋𝑓𝐼𝐹𝑡+∆𝜙(𝑡)+arg{𝐻(𝑓𝐼𝐹)}) (5.10) This notation is assumed for clarity, instead of the time convolution with the filter impulse response ℎ(𝑡). Let us define ℎ𝐿𝑃𝐹(𝑡) and ℎ𝐻𝑃𝐹(𝑡) for the 1st-order LPF and HPF respectively, as ℎ𝐿𝑃𝐹(𝑡)=1 𝑅𝐶𝑒−𝑡 𝑅𝐶 𝑢(𝑡) ℎ𝐻𝑃𝐹(𝑡)=𝛿(𝑡)−1 𝑅𝐶𝑒−𝑡 𝑅𝐶 𝑢(𝑡) (5.11) (5.12) with 𝛿(𝑡) the Dirac delta function and 𝑢(𝑡) the Heaviside function. By applying Fourier transform, the frequency response of the filters are respectively 𝐻𝐿𝑃𝐹(𝑓)=ℱ{ℎ𝐿𝑃𝐹(𝑡)}=1 1+𝑗2𝜋𝑅𝐶𝑓 𝐻𝐻𝑃𝐹(𝑓)=ℱ{ℎ𝐻𝑃𝐹(𝑡)}=𝑗2𝜋𝑅𝐶𝑓 1+𝑗2𝜋𝑅𝐶𝑓 (5.13) (5.14) Since the LPF and HPF are followed by RF detectors to calculate the mean power of the filtered RF signals, in this analysis only the magnitude response of the filters |𝐻(𝑓)| is considered, and the phase response arg{𝐻(𝑓)} is of no interest. The magnitude responses read as |𝐻𝐿𝑃𝐹(𝑓)|=1 √1+(2𝜋𝑅𝐶𝑓)2 |𝐻𝐻𝑃𝐹(𝑓)|=2𝜋𝑅𝐶𝑓 √1+(2𝜋𝑅𝐶𝑓)2 (5.15) (5.16) whose behavior is depicted in Fig. 5.12 with respect to the normalized frequency 𝑓𝑓𝑐 ⁄, with 𝑓𝑐= 12𝜋𝑅𝐶 ⁄ the 3 dB cut-off frequency of both LPF and HPF filters. By replacing Eqs. (5.15), (5.16) into Eq. (5.10), and normalizing with respect to 𝑓𝑐, the filtered photocurrents are expressed as 𝑖𝐿𝑃(𝑡)=𝐴 √1+(𝑓𝐼𝐹 𝑓𝑐 ⁄ )2sin(2𝜋𝑓𝐼𝐹𝑡+∆𝜙(𝑡)+arg{𝐻𝐿𝑃𝐹(𝑓𝐼𝐹)}) (5.17) 114 Figure 5.12. Magnitude response vs. normalized frequency for the 1st-order RC filters. 𝑖𝐻𝑃(𝑡)=𝐴(𝑓𝐼𝐹 𝑓𝑐 ⁄ ) √1+(𝑓𝐼𝐹 𝑓𝑐 ⁄ )2sin(2𝜋𝑓𝐼𝐹𝑡+∆𝜙(𝑡)+arg{𝐻𝐻𝑃𝐹(𝑓𝐼𝐹)}) (5.18) Next, the RF detectors square the signals to calculate the mean power, and the narrow LPFs reject the frequency-doubled sinusoidal term after RF to DC conversion, yielding 𝑥𝐿𝑃=𝐴2 2(1+(𝑓𝐼𝐹 𝑓𝑐 ⁄ )2) 𝑥𝐻𝑃=𝐴2(𝑓𝐼𝐹 𝑓𝑐 ⁄ )2 2(1+(𝑓𝐼𝐹 𝑓𝑐 ⁄ )2) (5.19) (5.20) Finally, these two expressions are added and subtracted to get the received optical power and the frequency error signal respectively. First, by adding Eqs. (5.19) and (5.20), one obtains 𝑥𝐻𝑃+𝑥𝐿𝑃=𝐴2 2 (5.21) corresponding to the received optical power. Conversely, subtraction of Eqs. (5.19) and (5.20) produces 𝑥𝐻𝑃−𝑥𝐿𝑃=𝐴2((𝑓𝐼𝐹 𝑓𝑐 ⁄ )2−1) 2((𝑓𝐼𝐹 𝑓𝑐 ⁄ )2+1) (5.22) which is a frequency dependent signal that serves as frequency error monitor. The dependency on the carrier amplitude 𝐴2 can be cancelled by using Eq. 5.21 to drive the AGC circuit, as in Fig. 5.10. The behavior of Eq. (5.22) by varying the carrier frequency is plotted in Fig. 5.13(a). As observed, the frequency discriminator output is a bipolar signal with zero-crossing at the cut-off frequency of the filters. The error estimation is not linear as expected, because of the frequency response characteristics of the RC filters (see Fig. 5.12); however, Fig. 5.13(b) indicates that there exists a good approximation to a linear estimation across ±0.3𝑓𝑐 with respect to the zero-crossing point. Replacing the 1st-order RC filters by higher order filters, or different RF filter technology, can modify the output characteristics of the frequency error estimator, and even increase the liner region. Nevertheless, by considering the typical 𝑓𝐼𝐹=2𝑅𝑏 in optical heterodyne RXs, the ±0.3𝑓𝑐 115 linear region in Fig. 5.13(b) translates into ±60% of the bit rate, enough for practical applications with low HW complexity. Furthermore, frequency errors beyond ±0.3𝑓𝑐 can effectively be corrected under LO feedback tuning as the polarity of the frequency error does not change. (a) (b) Figure 5.13. (a) Characteristics of the frequency error estimated by the analog frequency discriminator, and (b) zoom into the linear region around the filter’s cut-off frequency 𝑓𝑐. 5.2.2 Simulation and implementation in hardware In order to validate the operation of the frequency discriminator within the analog heterodyne RX, including the phase-modulated data, numerical simulation was carried out with VPI using the setup shown in Fig. 5.14. The data consisted of PRBS-7 sequences running at 1.25 Gb/s, that directly phase-modulated the TX laser. The 𝑓𝐼𝐹 was set to 2.5 GHz. After photodetection the RF signal was filtered by a Bessel BPF with 1.5 GHz BW at 3 dB, and centered at 𝑓𝐼𝐹, and then passed on to the differential demodulator for data recovery. At the same time, the RF photocurrent for the frequency discriminator input was tapped from two different points, before and after the BPF; this helps to figure out what might be, in practice, the effect of the BW limitations of the implemented HW. In the frequency discriminator, the values of R and C were adjusted for 𝑓𝑐 = 2.5 GHz, matched with that the IF of the receiver. Figure 5.14. Simulation setup for heterodyne RX with analog frequency discriminator. In Fig. 5.15(a), the LO was swept ±7.5 GHz with respect to the TX, to visualize the characteristics of the frequency error estimation. The results are plotted with and without the Bessel BPF. First, for the case of no BPF the obtained curve is symmetrical against 𝑓𝐼𝐹 = 0, with the zero-crossing Frequency error discriminator (i) = 2.5 GHz LO LPFBPF 25km SMF VOA PC Direct DPSK TX PRBS-7 1.25 Gb/s data (ii) Analog heterodyne RX 116 at IF = ±2.5 GHz. This matches well with the theoretical result in Fig. 5.13 without modulation. On the other hand, the Bessel BPF folds to zero the frequency error estimation beyond the linear region. As stated before, this represents the maximum BW limitation that the system imposes on the range of the frequency discriminator. Nevertheless, Fig. 5.15(b) zooms into the linear region around 𝑓𝐼𝐹 = 2.5 GHz and shows that ±500 MHz of linear error estimation are still available, even if the system is BW limited. (a) (b) Figure 5.15. (a) Simulation results for the analog frequency estimation in 1.25 Gb/s heterodyne RX with 𝑓𝐼𝐹 = 2.5 GHz; (b) zoom into the linear region of the error estimation. Last, the frequency discriminator was assessed in terms of the tolerance to the phase noise from lasers. The RF photocurrent was tapped after the Bessel BPF, and the results are plotted in Fig. 5.16 in terms of the linear range of the frequency discriminator output against the total linewidth (TX + LO). As expected, the system is practically immune to the laser linewidth because the frequency discrimination is based on the power difference between the two filtered RF spectra, being the phase contribution negligible. Moreover, the detrimental effect of the phase noise becomes apparent at linewidths nearby 1 GHz, close to the bit rate, far from realistic values for typical DFBs. Figure 5.16. Linewidth tolerance of the analog frequency discriminator. After performance assessment by numerical simulation, a prototype of the frequency discriminator was implemented in a PCB with standard FR-4 substrate and commercial electronic components. Fig. 5.17(a) shows the circuitry of the implemented prototype. The frequency 117 responses of the two complementary HPF and LPF are depicted in Fig. 5.17(b), showing the crossing-frequency at ~2.5 GHz at -6 dB (calculated as 20log (∙)). A low-barrier Schottky diode can be used as RF power detector. In this particular implementation, a HSMS-286Y microwave Schottky diode was configured as zero-bias detector. It operates at RF frequencies from 900 MHz up to 5.8 GHz. To operate the diode with zero-bias, the RF power at the diode input must be higher than -30 dBm to guarantee that the root mean square (RMS) voltage of the RF signal is high enough to polarize the Schottky barrier, without external DC bias, as plotted in Fig. 5.17(c). The negative or positive half-wave rectification can be selected by simply inverting the polarization of the Schottky diode. The rectified signals are then low-pass filtered at 10 kHz to obtain 𝑥𝐿𝑃 and 𝑥𝐻𝑃, that are later delivered to the electronic 𝜆-control system of the DFB LO. The addition and subtraction of 𝑥𝐿𝑃 and 𝑥𝐻𝑃 are done in the LO laser board with operational amplifiers. (a) (b) (c) Figure 5.17. Analog circuitry of the frequency error discriminator prototype; (b) frequency responses of both 2.5 GHz low-pass and high-pass 1st-order RC filters; (c) output voltage vs. RF input power for the zero-bias Schottky detector diode. The frequency discriminator prototype was implemented and tested real-time within the analog DPSK heterodyne ONU for the COCONUT PON, discussed in Section 4.2.2. Note that in such polarization diversity RX architecture, there are not one but two photodetected currents, one per each SOP. Hence, two parallel frequency discriminators are needed, and one will end up with four outputs from the frequency discriminator system: 𝑥𝐿𝑃𝕏, 𝑥𝐿𝑃𝕐 and 𝑥𝐻𝑃𝕏, 𝑥𝐻𝑃𝕐. On the one hand, the addition of the four signals yields the total received power independent from the received HSMS 286-Y HSMS 286-Y RF input 2.5 GHz LPF/HPF RF detector 10 KHz LPF 2.5 GHz 6 dB 118 SOP; this signal drives the AGC circuit for power normalization. On the other hand, the frequency error signal 𝑦𝑒(𝑡) is given by 𝑦𝑒(𝑡)=(𝑥𝐿𝑃𝕏+𝑥𝐿𝑃𝕐)−(𝑥𝐻𝑃𝕏+𝑥𝐻𝑃𝕐) (5.23) In Fig. 5.18 the DFB LO of the heterodyne RX was swept in 𝜆 to produce a frequency drift of ±1.25 GHz with respect to 𝑓𝐼𝐹 = 2.5 GHz. The TX laser modulated PRBS-7 at 1.25 Gb/s for direct DPSK. The resulting error signal from the frequency discriminator prototype was evaluated with and without the broadband modulated data. As observed, the analog frequency discriminator correctly tracked the 𝑓𝐼𝐹 mismatch across the entire LO sweep, exhibiting a good approximation to a linear error signal in the range ±1.25 GHz from 𝑓𝐼𝐹, for the case of PRBS data. Interestingly, the frequency discriminator can also operate with unmodulated optical signals, even though the quality of the frequency error estimation is worse because the power of the photodetected RF signal is now concentrated within a narrow frequency band of some MHz ‒the Lorentzian-shaped spectrum of the phase noise‒ and the frequency response non idealities of the LPF and HPF filters become more apparent. Figure 5.18. Output signal of the analog frequency discriminator prototype, for heterodyne RX with 𝑓𝐼𝐹 = 2.5 GHz and 1.25 Gb/s DPSK data. 5.3 Differential Carrier Recovery For Homodyne Receivers One of the key aspects of the DSP implemented in the homodyne RX for the COCONUT-PON was the use of differential field detection to recover the phase-encoded data (see Chapter 4). Its implementation is easy, and shows high robustness against the phase noise from low-cost lasers as studied in Section 3.3.1. Indeed, if the phase noise is dominant at the decision part of the RX, e.g., at lower data rates and/or employing common DFBs or VCSELs with large linewidth, the performance of differential phase detection is comparable to that of the synchronous detection using carrier phase estimation algorithms [Fat13], but with a remarkable lower implementation complexity. To visualize the latter, let us consider the well-known Viterbi & Viterbi feedforward phase estimator [Vit83] illustrated in Fig. 5.19(a), compared with the differential field detector for 𝑚-PSK in Fig. 5.19(b). 119 (a) (b) Figure 5.19. (a) Viterbi & Viterbi feedforward phase estimation and (b) differential field detection for optical PR. In order to calculate the complexity and the required HW resources of both PR algorithms, let us examine the case of QPSK modulation. The HW resources estimate was done by decomposing the mathematical operations with complex numbers into the required number of real multipliers and adders. As an example, raising the complex number 𝐼+𝑗𝑄 to the power of two yields: (𝐼+𝑗𝑄)2=(𝐼2−𝑄2)+𝑗(2𝐼𝑄); therefore, four real multipliers and two adders are needed. Other operations like the trigonometrical functions to calculate arg (∙) are more efficiently implemented by look-up tables (LUTs). The HW resources estimate for QPSK, as summarized in Table 5.1, indicates that the differential demodulation requires 50% less multipliers, 56% less adders, and no LUTs. Here, 𝑀 represents the number of parallel bits processed by the digital processor. This directly translates into less complexity for HW prototyping, improved performances, and lower energy consumption of the digital RX. For a fair comparison, it should be mentioned that the differential detection also requires symbol encoding at the TX (see Section 2.3.5.1); however, such differential encoding is widely applied and recommended, even for synchronous detection with phase estimation algorithms, to overcome the phase-noise induced cycle-slips that produce phase ambiguity and error propagation [Kik06], and therefore, represents no significant extra complexity. An alternative to deal with the cycle slips is the use of data-aided DSP by transmitting pilot symbols for carrier synchronization and further channel equalization [Zha10], at the expenses of lowering the net data rate. Table 5.1. Hardware resources for QPSK phase recovery. PR algorithm Multipliers Adders LUTs Viterbi & Viterbi 8×𝑀 7×𝑀−2 𝑀+1 Differential demodulation 4×𝑀 3×𝑀 0 Reduction 50% 56% 100% Owing to the advantages mentioned above, namely simplicity and robustness, this thesis promotes the use of differential phase detection for optical CR in coherent TRXs for access networks, where the cost-effectiveness takes the primary attention. Hence, by selecting the differential 𝑚𝑡ℎ-power estimator (Section 5.1.1) for FE, and the differential demodulator for PR, the complete CR architecture is as represented in Fig. 5.20. Here, 𝑚 is the number of constellation points of the 𝑚PSK (e.g., 𝑚=2 for DPSK), 𝑁 is the block length for averaging, 𝑘 is the symbol index within the estimation block, and 𝐷 is the process delay of the FE algorithm. In this CR algorithm, the phase shift ∆𝜑 induced by the frequency detuning between Tx and LO lasers is estimated from each sample of the received signal 𝑟[𝑛]=𝐼[𝑛]+𝑗𝑄[𝑛], then cancelled by the phase rotator j I[n] Q[n] c[n] + j I[n] Q[n] [n] 120 𝑒−𝑗𝑘(∙) before PR. Afterwards, the corrected signal 𝑟′[𝑛] is differentially demodulated by a 1symbol complex correlation (i.e., 1-symbol delay-and-multiply) to recover the phase-encoded data 𝑑[𝑛]. Note that both FE and PR stages implement a complex correlation with different signal delay to extract the phase difference between samples (𝑧−1) in FE, or between symbols (𝑧−𝑃) in PR, where 𝑃=𝑅𝑠𝑅𝐵 ⁄ represents the oversampling factor and is defined as the ratio between the sample rate (𝑅𝑠) and the symbol rate (𝑅𝑏). Usually, 𝑅𝑠≥2𝑅𝐵 at the ADCs for mapping into the digital domain, to satisfy the Nyquist sampling criterion. Figure 5.20. CR algorithm based on differential 𝑚𝑡ℎ-power for FE and differential demodulation for PR. 5.3.1 Hardware optimization The CR architecture in Fig. 5.20 can be further optimized for HW prototyping based on the fact that the differential 𝑚𝑡ℎ-power algorithm for FE cancels the 𝑚-PSK modulation by rising to the power of 𝑚, then the frequency detuning can be correctly estimated either between consecutive samples or symbols. This contrasts with the IQ cross-correlation algorithm in Section 5.1.2 that was specifically conceived to estimate the frequency detuning between samples, and requires a signal delay for the cross-correlation shorter than the symbol duration to avoid the influence of the phase-modulated data. In consequence, by going back to Fig. 5.20, if the CLK recovery at the RX DSP is carried out prior the CR (i.e., the I and Q signals for CR were already downsampled to one sample per symbol), then 𝑃=1 and both FE and PR algorithms need the same 1-symbol complex correlation for differential phase detection; otherwise, for unrecovered CLK (𝑃≥2) the signal delays for correlation are different. Therefore, for the case of 𝑃=1 (recovered CLK) the architecture in Fig. 5.20 can be rearranged to now calculate the 1-symbol correlation for PR only once per symbol, then reuse it for the next feedforward FE algorithm, as illustrated in Fig. 5.21(a), thus optimizing the implementation and saving the HW resources. A related idea was proposed and tested in [Fer16b], working with differential 𝑚𝑡ℎ-power estimator for FE and Viterbi & Viterbi phase estimator for PR, both requiring the calculation of the 𝑚𝑡ℎ-power. In such architecture, the HW optimization was achieved by calculating the 𝑚𝑡ℎ-power only once, then sharing it to the FE and the PR algorithms. Fig. 5.21(b) shows the evolution of the signal constellation within the simplified CR algorithm, for the case of DPSK. First, the complex received signal 𝑟[𝑛] ((i), left handed) completely rotates over the complex plane due to the phase noise and the frequency error. Next, the informationbearing optical phase is recovered by differential demodulation, getting rid of the phase noise effect. However, each symbol of 𝑤[𝑛] ((ii), center) has a phase shift ∆𝜑=2𝜋∆𝑓𝑅𝑏 ⁄ due to the 121 (a) (b) Figure 5.21. (a) Simplified CR architecture that reuses the 1-symbol complex correlation required for both the FE and the PR algorithms; (b) evolution of the DPSK constellation within the simplified CR. frequency detuning, denoted by ∆𝑓, as analyzed in Section 3.3.2. This produces a fixed incremental rotation of the constellation in the complex plane. Finally, the residual phase shift ∆𝜑 is estimated and corrected from the phase of each symbol, yielding a phase/frequency-noise-free signal 𝑑[𝑛] ((iii), right handed) that goes on to the decision part of the RX. The estimated ∆𝜑 simultaneously drives the feedforward symbol phase correction 𝑤[𝑛]×exp[−𝑗∆𝜑], and the LO feedback for automatic 𝜆-tuning, as demonstrated in Section 5.1. Note that the phase rotator exp[−𝑗∆𝜑] in the simplified CR in Fig. 5.21(a) does not include the symbol index 𝑘, as in Fig. 5.20 for the conventional CR, because the frequency error of the signal 𝑤[𝑛] after differential demodulation translates into a constant phase shift for all the symbols within the estimation window of length 𝑁, with time-dependency already suppressed. The proposed CR algorithm in Fig. 5.21(a) with optimized architecture requires less operations than the conventional CR in Fig. 5.20, thus relaxing the HW resources and the energy consumption per processed symbol when the coherent RX is prototyped in a FPGA for real-time operation. Table 5.2 summarizes the required HW resources for DPSK, indicating 23% and 25% of savings in the number of real adders and multipliers respectively with the simplified architecture. Table 5.2. Hardware resources for DPSK carrier recovery. CR architecture Multipliers Adders Process delay [symbols] Conventional 16×𝑀 13×𝑀−2 4×𝑀 Proposed 12×𝑀 10×𝑀−2 3×𝑀 Reduction 25% 23% 25% 128 These results were obtained from direct laser modulation and band limited electronics that influenced the spectral width of the DPSK modulation. Further DSP at the transmitter for spectral shaping may lower the required channel spacing, as examined in Section 3.4.1. 5.3.3.1 Tolerance to optical frequency dithering Motivated by the reduction in the process delay of the proposed CR algorithm, by 25% (see Table 5.2), a further experiment was designed to evaluate how fast the CR, prototyped in the FPGA, can effectively estimate and correct the frequency detuning. The test consisted of applying a lowfrequency triangle current waveform to the bias of the DFB TX, as illustrated in Fig. 5.31(a), to produce an optical frequency dithering in the frequency band where the thermal chirp is dominant, and below the laser dip, up to few MHz (see Sections 2.4 and 3.5.1). The amplitude of the dithering (i.e., the maximum deviation with respect to its central frequency) is controlled by the amplitude of the bias current swing, and was set to ±250 MHz to operate in the region of Fig. 5.27(b) where the sensitivity penalty is almost negligible after detuning compensation. The frequency of the dithering varied from 1 Hz up to 1 MHz to determine the impact on the RX sensitivity and the maximum tolerance. Fig. 5.31(b) reports on the electrical spectra at the RX after photodetection, for DPSK at 1.25 Gb/s without dithering (upper) and with ±250 MHz dithering amplitude (lower). (a) (b) Figure 5.31. (a) Implementation of the optical frequency dithering in the DFB TX; (b) DPSK photodetected spectra at 1.25 Gb/s without dithering (upper), and with ±250 MHz dithering amplitude (lower). The results plotted in Fig. 5.32, in terms of the BER as a function of the frequency of the optical dithering, show that the conventional CR can tolerate up to 70 kHz dithering frequency for 1 dB sensitivity penalty, whereas the proposed CR tolerance extends up to 350 kHz thanks to the substantial reduction in the number of operations and process delay of the CR algorithm, which translates into faster tracking of the frequency error. This confirms the feasibility of inserting the low frequency FSK signaling among the OLT and the ONUs for network control and management, as discussed in Chapter 4, without interfering the broadband data modulation. DFB TX No dithering 500MHz dithering amplitude 129 Figure 5.32. BER against frequency of the optical dithering, with ±250 MHz amplitude of the dithering, for conventional and proposed CR algorithms. 5.4 Chapter summary This chapter explored the strategies for optical carrier synchronization at the coherent RX, dealing with the phase noise and the frequency drifts of lasers, which are more notorious when working with low-cost DFBs or VCSELs. First, for optical frequency matching between TX and LO, a complete DSP-aided AWC system for homodyne RXs was demonstrated with dual correction strategies: feedforward error tracking and feedback stabilization of the LO. The two simultaneous strategies can effectively deal with strong environment temperature changes and laser frequency noise, as the feed-forward DSP corrects any residual frequency error after LO tuning. Accurate performance in closed-loop was achieved, with ±500 MHz detuning correction in 600ms within 2 dB maximum penalty in sensitivity. The proposed AWC system is simple enough to be implemented in a cost-effective ONU for udWDM-PON, using common DFBs. For the case of fully analog ONUs without DSP, an analog frequency discriminator for AWC was designed and implemented in a 1.25 Gb/s heterodyne RX. The frequency error is calculated from the power of the photodetected current, properly filtered by two complementary LPF/HPF filters. The system is robust and requires simple electronics with few complexity. The experimental tests included characterization of the frequency drift of DFB lasers free-running and under environment temperature conditions. The maximum measured drift rate was about 3.6 MHz/s, and indicates that the thermal tuning of conventional DFBs suffices for the coarse correction of the LO detuning, despite the slow nature of the thermal tuning with time constants in the order of seconds. On the other hand, to recover the phase-encoded data in presence of optical phase noise, a simplified CR algorithm based on differential field detection for homodyne 𝑚-PSK RXs was proposed and tested real-time. The proposed method shares the 1-symbol complex correlation required for the FE and the PR algorithms, thus lowers the required HW resources, the power consumption, and the process delay of the RX DSP. The CR was prototyped in FPGA and tested with DPSK at 1.25 Gb/s, achieving a high sensitivity of -55 dBm for BER = 10-3, with robustness against the frequency drifts of lasers. Notably, the proposed CR with simplified architecture increases by a factor of five the tolerance against fast frequency drifts of lasers, compared with 130 the conventional CR architecture. The good results obtained with direct laser modulation, phase noise tolerance of 𝛥𝜐Tb = 6.4·10-3 with common DFBs, and 6.25 GHz spaced udWDM grid, demonstrates that a cost-effective PON can be implemented with user terminals based on lowcost components and simple techniques, but still maintaining high the performance. 131 Chapter 6 Novel Architectures for Polarization-Independent Coherent Receivers Polarization handling has been a major concern in coherent optical communications because in the optical link the transmitted signal reaches the coherent RX at the destination with random SOP, that evolves unpredictably along the birefringent fiber channel. This fact is particularly critical in coherent detection because incorrect SOP matching with the LO can even completely prevent the coherent mixing in the PDs, as analyzed in Section 2.2. The conventional methods to cope with the signal SOP variations are: (I) the use of polarization-maintaining fibers, not feasible for coherent PON that is planned to reuse the fiber infrastructure already deployed for legacy optical access systems; (II) polarization-diversity RX architectures that duplicate the optical frontend to process separately each SOP, widely used in commercial coherent RXs for transport networks; or (III) fast polarization tracking systems [Koc13], [Li19], that add the corresponding extra HW complexity. Since the adoption of coherent systems for PON and other short reach applications with high terminal density, like DCI, is mainly limited by the complexity of the coherent RX, efforts in research have been conducted in pursuing novel polarization-independent coherent detection techniques. One of them is polarization switching in time [Can14c], [Erk16], in which the RX front-end reduces to an optical 3dB coupler and one or two PDs, for single-ended or balanced photodetection respectively. This solution, however, requires polarization modulation at the TX with external modulators. Another attractive alternative ‒outcome of the research in the COCONUT project‒ is a novel front-end configuration based on simple 3x3 coupler and PBS [Cia14], [Bot15], intended for ASK signals with envelope demodulation. This chapter presents two different approaches for low complexity polarization-independent coherent detection: one for analog heterodyne RXs with DPSK signals, exploiting the simple front-end composed by 3x3 coupler + PBS; and the other for digital homodyne RXs leveraging low-complexity DSP that uses only 1 sample per symbol, thus lowering the ADCs requirements for implementation of the polarization-diversity front-end. 6.1 Polarization-Independent 3x3 Heterodyne Receiver In previous chapters, the optical front-end for polarization-diversity heterodyne RXs was implemented with 2x2 couplers, in two different configurations. The first option is the conventional 2x2 heterodyne RX presented in Fig. 3.3(b) and hereafter named “4-PD balanced”, that implements a pair of PBSs and 2x2 couplers to separate the orthogonal SOP components of the received signal and the LO, later photodetected and processed separately. The four PDs are connected in pairs for balanced photodetection, as illustrated in Fig. 6.1(a). The analytical model can be found in Section 2.2.2.2. 132 (a) (b) Figure 6.1. Optical front-end for (a) 4-PD balanced and (b) 2-PD single-ended heterodyne RX. (a) (b) Figure 6.2. Novel 3x3 optical front-end with the PBS attached (a) to the LO, and (b) to the input signal path. The second option is the simplification of the previous front-end, which was implemented in the heterodyne ONU for the COCONUT-PON (Fig. 4.6). It consists of a 2x2 coupler that mixes signal and LO, followed by a PBS that separates the SOP components of the coherent mixing to be photodetected by a single PD each, as shown in Fig. 6.1(b). This scheme is hereafter named “2PD single-ended”. It pretty simplifies the complexity of the RX front-end, but suffers from the influence of the DD terms and the common-mode noise due to the single-ended detection; it also has an inherent 3 dB sensitivity penalty as only one arm of the coherent beating is processed. The 2x2 coupler with 50:50 coupling ratio could be replaced by, e.g., a 90:10 coupler such that the TX input loss is 0.5 dB only, much lower than 3 dB, but at the expenses of increasing the LO input loss to 10dB. Now, let us consider the 3x3 homodyne RX introduced in Fig. 3.6(b). If the LO is first attached to a PBS, and the second PBS output is connected to the unused input port of the 3x3 coupler, the result is the novel and simple front-end depicted in Fig. 6.2(a) for polarization-independent coherent detection. Note that the PBS can also be attached to the signal path, as seen in Fig. 6.2(b), but the first connection to the LO is preferred in order not to add extra insertion loss from the PBS to the already weak received signal. With this novel front-end, Ciaramella analytically demonstrated in [Cia14] that by applying a large frequency detuning between signal and LO (larger than 0.75𝑅𝑏), the random SOP components affecting data fall beyond the RX BW and can be suppressed by electrical filtering. This holds for ASK modulation that is quite insensitive to the frequency detuning. Nevertheless, it does not suit for PSK because the phase modulated signals are highly affected by the frequency detuning, as reported in Chapter 5. This motivated to investigate the heterodyne detection of DPSK signals using this new 3x3 optical front-end. 6.1.1 Receiver architecture PBS PBS 45⁰ Signal LO Signal PBS 45⁰ LO PBS Signal 45⁰ LO PBS Signal 45⁰ LO 133 The proposed RX is depicted in Fig. 6.3, and implements the 3x3 optical front-end of Fig. 6.2(a) to beat the incoming signal with the 𝕏 and 𝕐 components of the LO SOP, previously separated by the PBS. For optimality of the coherent mixing, the LO is polarized linearly at 45° so that the 𝕏 and 𝕐 components are the same. Three PDs perform single-ended detection, but the photocurrents are later balanced when they are linearly combined for IQ recovery, similar than the 3x3 homodyne RX in Section 2.1.2.2. This cancels the DD terms, which is of particular interest in udWMD-PON with hundreds of users, as outlined below. The IQ recovery can be realized by passive hardware for fully analog RXs, or inside the DSP unit if the preferred RX implementation is digital. Figure 6.3. Polarization-independent 3x3 balanced heterodyne RX for DPSK. At the electrical processing part of the RX, the recovered 𝐼𝕏𝕐(𝑡) and 𝑄𝕏𝕐(𝑡) signals enter the frequency control systems that, for analog heterodyne RX, consists of the frequency discriminator analyzed in Section 5.2. Later on the two signals are band-pass filtered and differentially demodulated to recover the DPSK data, then combined and low-pass filtered to overcome the polarization effects before the decision part of the RX. Indeed, the complete RX in Fig. 6.3 is quite similar than that of Fig. 3.6(b) for phase-diversity homodyne, but replacing the LPFs by BPFs and adding the PBS at the input. Here, though, the detection is heterodyne with the 𝑓𝐼𝐹 properly selected as an integer multiple of the bit rate to maximize the amplitude of the decision variable (see Section 2.3.2). It is worth mentioning that the proposed RX achieves the polarization-independent detection of DPSK signals with a remarkably simple front-end, and without adding extra complexity to the electrical differential detection process nor the optical TX, compared with the schemes based on 2x2 coupler. 6.1.2 Analytical model For the system model, let us consider the Jones notation in Eq. (2.31) for the arbitrary-polarized received signal, and given by 𝒥𝑆(𝑡)=[𝑒𝑆,𝕏(𝑡) 𝑒𝑆,𝕐(𝑡)]=[cos𝜑𝕏 sin𝜑𝑒𝑗𝜓 𝕐][𝐸𝑆(𝑡)𝑒𝑗(𝜔𝑆𝑡+𝜙𝑆(𝑡))] (6.1) where 𝜑,𝜓 are the azimuth and ellipticity of the signal SOP respectively. The LO is linearly polarized at 45° (𝜑=45°,𝜓=0) at the input of the PBS to ensure optical beating with LO components at the same power; then, its Jones vector reads as 𝒥𝐿𝑂(𝑡)=[𝑒𝐿𝑂,𝕏(𝑡) 𝑒𝐿𝑂,𝕐(𝑡)]=1 √2[𝕏 𝕐][𝐸𝐿𝑂(𝑡)𝑒𝑗(𝜔𝐿𝑂𝑡+𝜙𝐿𝑂(𝑡))] (6.2) PBS Signal 45⁰ LO IQ recovery BPF LPF Freq. control 134 The optical mixing between signal and LO in the 3x3 coupler is described by the following scatter matrix [𝑒1(𝑡) 𝑒2(𝑡) 𝑒3(𝑡)]=13[𝑎 𝑏 𝑏 𝑏 𝑎 𝑏 𝑏 𝑏 𝑎][𝒥𝑆(𝑡) 𝑒𝐿𝑂𝕏(𝑡) 𝑒𝐿𝑂𝕐(𝑡)] (6.3) with 𝑎, 𝑏 the 3x3 scatter matrix coefficients given by Eqs. (2.20) and (2.21). If ideal photodetection is assumed, the three photocurrents are derived by 𝑖𝑘(𝑡)=ℜ(|𝑒𝑘,𝕏(𝑡)|2+ |𝑒𝑘,𝕐(𝑡)|2), with 𝑘= {1, 2, 3} and ℜ the responsivity of the PD. The result is the following set of photocurrents 𝑖1(𝑡)=ℜ 3[𝑃𝑆(𝑡)+2𝑃𝐿𝑂+2√𝑃𝑆(𝑡)𝑃𝐿𝑂(𝑐𝑜𝑠𝜑 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡)+2𝜋 3)+ 𝑠𝑖𝑛𝜑 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡)+𝜓+2𝜋 3))] 𝑖2(𝑡)=ℜ 3[𝑃𝑆(𝑡)+2𝑃𝐿𝑂+2√𝑃𝑆(𝑡)𝑃𝐿𝑂(𝑐𝑜𝑠𝜑 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡)−2𝜋 3)+ 𝑠𝑖𝑛𝜑 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡)+𝜓))] 𝑖3(𝑡)=ℜ 3[𝑃𝑆(𝑡)+2𝑃𝐿𝑂+2√𝑃𝑆(𝑡)𝑃𝐿𝑂(𝑐𝑜𝑠𝜑 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡))+ 𝑠𝑖𝑛𝜑 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡)+𝜓−2𝜋 3))] (6.4) (6.5) (6.6) Here, 𝑃𝑆(𝑡) and 𝑃𝐿𝑂 are the signal and LO power respectively, 𝜔𝐼𝐹=2𝜋𝑓𝐼𝐹 is the intermediate frequency, and Δ𝜙(𝑡)=𝜙𝑑(𝑡)+𝜙𝑛(𝑡) is the phase difference between signal and LO that carries the phase modulated data 𝜙𝑑(𝑡), along with the random phase term 𝜙𝑛(𝑡). Each photocurrent is composed by the DD terms in baseband and the coherent terms oscillating at 𝜔𝐼𝐹. Afterwards, the three signals are linearly combined according to Eqs. (2.23) and (2.24), to obtain two orthogonal I and Q signals and also cancel the DD terms, yielding 𝐼𝕏𝕐(𝑡)=ℜ√𝑃𝑆(𝑡)𝑃𝐿𝑂[𝑐𝑜𝑠𝜑 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡)−2𝜋 3)+𝑠𝑖𝑛𝜑 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡)+𝜓)] 𝑄𝕏𝕐(𝑡)=ℜ√𝑃𝑆(𝑡)𝑃𝐿𝑂[−𝑐𝑜𝑠𝜑 𝑠𝑖𝑛(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡)−2𝜋 3)+𝑠𝑖𝑛𝜑 𝑠𝑖𝑛(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡)+𝜓)] (6.7) (6.8) Note that the first term of each of the equations above is the 𝕏polarization contribution, whereas the second term is the 𝕐polarization, and that is precisely the reason why the notation 𝕏𝕐 appears in the subindexes. In practice this means that there is spectral overlap between the SOP components, thus potential destructive interference depending on the values of 𝜑,𝜓. This contrasts with the 2x2 heterodyne RXs in Fig 6.1 that produce one photocurrent for each SOP. However, a key point worth noting here is the inverted sign in the two terms of 𝑄𝕏𝕐(𝑡) originated at the optical mixing in the 3x3 coupler because the 𝕏and 𝕐polarizations of the LO are shifted in phase by relative 2𝜋3 ⁄, and the quadrature signal is sensitive to that change of quadrant. 135 Next, both signals are filtered by two BPFs centered at 𝑓𝐼𝐹 and properly adjusted in BW to be matched with the data BW. Differential demodulation is carried out to extract the DPSK data Δ𝜙𝑑(𝑡)=𝜙𝑑(𝑡)−𝜙𝑑(𝑡−𝑇𝑏), then the signals are added into 𝑚(𝑡) to obtain 𝑚(𝑡)=ℜ2𝑃𝑆(𝑡)𝑃𝐿𝑂[𝑐𝑜𝑠(𝜔𝐼𝐹𝑇𝑏+Δ𝜙𝑑(𝑡)) +𝑠𝑖𝑛(2𝜑)𝑐𝑜𝑠(2𝜔𝐼𝐹𝑡−𝜔𝐼𝐹𝑇𝑏+Δ𝜙(𝑡)+Δ𝜙(𝑡−𝑇𝑏)+𝜓−2𝜋 3)] (6.9) This key equation shows a very clean term in baseband that contains the demodulated data Δ𝜙𝑑(𝑡), and a polarization-dependent term at high frequency ‒oscillating at 2𝜔𝐼𝐹‒ that contains the random SOP coordinates 𝜑,𝜓. Since in heterodyne detection the 𝑓𝐼𝐹 is usually two or three times the data rate 𝑅𝐵, low-pass filtering of 𝑚(𝑡) completely filters out the polarization-dependent term at 2𝑓𝐼𝐹 in Eq. (6.9), thus canceling the SOP overlap. This is illustrated in Fig. 6.4 that shows frequency domain representation of 𝑚(𝑡). Figure 6.4. Frequency domain representation of 𝑚(𝑡) after adding the differentially demodulated signals. At the baseband data (first term of Eq. (6.9)), in order to maximize the variable for data decision, the 𝑓𝐼𝐹 is set to be 𝑛𝑅𝐵, with 𝑛={ 2,3,…}; then 𝜔𝐼𝐹𝑇𝑏=2𝑛𝜋 and has no impact on the recovered data Δ𝜙𝑑(𝑡). Finally, the recovered DPSK signal after the LPF is found to be 𝑑(𝑡)=ℜ2𝑃𝑆(𝑡)𝑃𝐿𝑂cosΔ𝜙𝑑(𝑡) (6.10) demonstrating the polarization-independent detection of DPSK signal. Therefore, the main outcome is that in the coherent RX with 3x3 optical front-end, the polarization independent detection is achieved by the joint effect of the intermediate frequency that eliminates the spectral overlap, and the subsequent sharp electrical filtering that suppress the random SOP components. If the 3x3 front-end is configured with the PBS in the signal path, as in Fig. 6.2(b), the three photodetected currents are correspondingly written as 𝑖1(𝑡)=ℜ 3[𝑃𝑆(𝑡)+2𝑃𝐿𝑂+2√𝑃𝑆(𝑡)𝑃𝐿𝑂(𝑐𝑜𝑠𝜑 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡)+2𝜋 3)+ 𝑠𝑖𝑛𝜑 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡)+𝜓))] 𝑖2(𝑡)=ℜ 3[𝑃𝑆(𝑡)+2𝑃𝐿𝑂+2√𝑃𝑆(𝑡)𝑃𝐿𝑂(𝑐𝑜𝑠𝜑 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡))+ 𝑠𝑖𝑛𝜑 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡)+𝜓+2𝜋 3))] (6.11) (6.12) LPF Baseband data Polarization-dependent data 136 𝑖3(𝑡)=ℜ 3[𝑃𝑆(𝑡)+2𝑃𝐿𝑂+2√𝑃𝑆(𝑡)𝑃𝐿𝑂(𝑐𝑜𝑠𝜑 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡)−2𝜋 3)+ 𝑠𝑖𝑛𝜑 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡)+𝜓−2𝜋 3))] (6.13) which only differ from Eqs. (6.4)‒(6.6) in the distribution of the phase shift ±2𝜋3 ⁄ and the SOP ellipticity 𝜓 among the three photocurrents. The recovered 𝐼𝕏𝕐(𝑡) and 𝑄𝕏𝕐(𝑡) after cancelling the DD terms are now 𝐼𝕏𝕐(𝑡)=ℜ√𝑃𝑆(𝑡)𝑃𝐿𝑂[𝑐𝑜𝑠𝜑 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡))+𝑠𝑖𝑛𝜑 𝑐𝑜𝑠(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡)+𝜓+2𝜋 3)] 𝑄𝕏𝕐(𝑡)=ℜ√𝑃𝑆(𝑡)𝑃𝐿𝑂[𝑐𝑜𝑠𝜑 𝑠𝑖𝑛(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡))−𝑠𝑖𝑛𝜑 𝑠𝑖𝑛(𝜔𝐼𝐹𝑡+Δ𝜙(𝑡)+𝜓+2𝜋 3)] (6.14) (6.15) This time, the sign inversion in 𝑄𝕏𝕐(𝑡) occurs for the 𝕐polarization term. The signal 𝑚(𝑡) after differential demodulation and signals addition becomes 𝑚(𝑡)=ℜ2𝑃𝑆(𝑡)𝑃𝐿𝑂[𝑐𝑜𝑠(𝜔𝐼𝐹𝑇𝑏+Δ𝜙𝑑(𝑡)) +𝑠𝑖𝑛(2𝜑)𝑐𝑜𝑠(2𝜔𝐼𝐹𝑡−𝜔𝐼𝐹𝑇𝑏+Δ𝜙(𝑡)+Δ𝜙(𝑡−𝑇𝑏)+𝜓+2𝜋 3)] (6.16) that produces identical result after the LPF than the configuration with the PBS attached to the LO, preferred in this thesis to minimize the power loss of the received signal. Another important characteristic of the 3x3 heterodyne RX is the cancellation of the DD terms. By looking at the three photocurrents in Eqs. (6.4)‒(6.6) ‒or equivalently (6.11)‒(6.13)‒ after the single-ended detection, the DD terms are composed by the strong DC component from the LO due to the high LO power 𝑃𝐿𝑂, and by the received signal power 𝑃𝑆(𝑡)=|𝐸𝑆(𝑡)|2 that normally is much smaller than the coherent term. However, in the udWDM-PON scenario with hundreds of users, the DD term 𝑃𝑆(𝑡) accumulates the power of all the individual udWDM channels, being expressed as 𝑃𝑆(𝑡)=∑|𝐸𝑘(𝑡)|2 𝑁 𝑘=1 (6.17) with 𝑁 the number of users. Taking the example of the COCONUT PON in Chapter 4, with equal 256 users spaced by 12.5 GHz for heterodyne ONUs, the total optical power is 10×𝑙𝑜𝑔10(256) = +24 dB at the PDs input, thus the DD term might become comparable in power to the coherent term. This scenario is illustrated in Fig. 6.5, and makes us to conclude that for a large number of users the DD terms can cause interference and negative impact on the detection if they are not cancelled correctly. Figure 6.5. Frequency domain representation of the single-ended heterodyne detection. 256 udWDM users DD Coherent term Single-user DD 137 6.1.3 Experimental validation To validate correct DPSK detection with the 3x3 heterodyne RX, independent from the received signal SOP, the experimental setup depicted in Fig. 6.6 was implemented. At the TX, PRBS data from a PPG running at 1.25 Gb/s were properly amplified and equalized to drive the direct modulation of a 4 MHz linewidth DFB (𝜆 = 1549 nm) and emitting at -10 dBm. The signal was transmitted through 50 km of SMF, and the VOA emulated for the ODN losses. At the RX, the received signal was mixed with LO using the three optical front-ends mentioned above, namely 4-PD balanced, 2-PD single-ended and 3x3 balanced. The LO was an ECL with 100 kHz linewidth and emitting at 3dBm, that entered linearly polarized at 45º to the RX front-end. The 𝑓𝐼𝐹 was set to 2.5 GHz. After photodetection, the two photocurrents from the 2x2 coupler-based front-ends were digitalized by a 50 GSa/s RTO for differential demodulation and BER computing. In the case of 3x3 coupler front-end, the balancing of the photocurrents was carried out by DSP, thus a third ADC channel sampled the output of the third photodiode, as shown in Fig. 6.6 (dotted line). Within the DSP, the BPF was achieved by combination of two 4th order 1.5 GHz high-pass and 3.5 GHz low-pass Bessel filters. After the differential demodulation both signals were added, then low-pass filtered at 1 GHz and thresholded for data decision. The BER was calculated by direct error counting over 218 bits. Figure 6.6. Experimental setup with 1.25 Gb/s DPSK transmission. The inset shows the DPSK spectrum after heterodyne detection at 𝑓𝐼𝐹= 2.5 GHz. First, the BER was evaluated in terms of the received optical power for 0 (b2b) and 50km SMF. The results in Fig. 6.7(a) reveal that, at BER = 10-3 FEC threshold, the 3x3 balanced RX achieved a sensitivity of -49 dBm, 0.7 dB worse than the 4-PD balanced RX mostly due to the excess losses from the 3x3 coupler, that ranged from 0.11 to 0.59 dB between ports. For the 2-PD single-ended RX, the power penalty rises to 2.6 dB, close to the 3 dB expected from theory. No meaningful difference was observed between b2b and 50km tests. Fig. 6.7(b) reports the eye diagram after differential demodulation for the 3x3 balanced RX. Next, the RX tolerance to the frequency detuning was measured by sweeping the LO 𝜆. The results are plotted in Fig. 6.8 in terms of the power penalty at BER = 10-4 against the LO detuning. As observed, the detuning must be kept lower than 60 MHz in the three RXs to avoid penalties larger 50 km SMF VOA DPSK TX ECL LO 50 GSa/s RTO DFB TX PPG Equalizer BPF LPF Coherent RX front-end BER ADC ADC ADC Polarization-independent heterodyne RX 45⁰ 1.25 Gb/s 144 phase 𝜙𝜏 progressively in steps of ∆𝜏, and within −𝑇𝑏2 ⁄<𝜏<𝑇𝑏2 ⁄, until converge to the optimal sampling point. This timing adjustment process is graphically illustrated in Fig. 6.15. Let us assume that there is a positive timing delay +∆𝜏 between 𝑃𝑘−1       and 𝑃𝑘    , which means that the last calculated ∆𝜏𝑘−1> 0. If the sampling is early (Fig. 6.15 left hand), 𝜀𝑘=𝑃𝑘    −𝑃𝑘−1       is positive, ∆𝜏𝑘−1 is also positive, then the TED output ∆𝜏𝑘 indicates that a timing delay by +∆𝜏 is required for the next symbols. On the other hand, if the sampling is late (Fig. 6.15 right hand), and assuming the same +∆𝜏 between 𝑃𝑘−1       and 𝑃𝑘    , then 𝜀𝑘<0, ∆𝜏𝑘−1>0, and ∆𝜏𝑘 indicates that a timing advance by −∆𝜏 is required for the next symbols. Within the loop, the sampling phase 𝜙𝜏 is updated every 𝑁 symbols. Figure 6.15. Graphic representation of the ADC sampling phase tuning by maximum finding algorithm over the total received signal power 𝑃𝕏+𝑃𝕐. An immediate conclusion from Fig. 6.15 is that the performance of the proposed CLK recovery and the ADC sampling tuning process highly depends on the shape of the received signal power, which is mainly determined by the LPF filter response ℎ(𝑡) if other BW limitations and signal distortions are not considered, in the ideal case. Accordingly, the influence of ℎ(𝑡) on the powerbased TED was assessed by numerical simulation using standard low-pass Bessel filters of the 4th order, with variable BW. The results are plotted in terms of the timing error 𝜀𝑘 in open loop, as a function of the sampling phase deviation with respect to the optimal sampling point, in Fig. 6.16(a) for DPSK at 1 Gb/s, and in Fig. 6.17(a) for QPSK at 2 Gb/s. Without filtering, 𝜀𝑘=0 as expected because of the constant envelope of the 𝑚-PSK modulation. Conversely, after low-pass filtering 𝜀𝑘 draws the typical S-curve of the TED, which is smoother and steeper for filter BWs between 0.75𝑅𝑏 and 𝑅𝑏. This matches well with the BWs used in homodyne detection. Additionally, Figs. 6.16(b) and 6.17(b) show the impact of the frequency detuning Δ𝑓 on the TED for DPSK and QPSK respectively. As Δ𝑓 increases, most of the received signal power falls beyond the BW of the LPFs at the RX input, and is filtered out, then the slope of the S-curve reduces accordingly. The results indicate that for Δ𝑓= 400 MHz (40% of the baud rate) the Scurve slope reduces about 55% with respect to Δ𝑓= 0. Despite the slope reduction, the tolerance against the frequency detuning is much higher than conventional TED based on signal amplitude; for instance, Fig. 2(a) of [Tan08] shows that the Gardner TED fails for Δ𝑓 larger than 50% of the Optimal sampling Earlysampling Optimal sampling Late sampling 145 (a) (b) Figure 6.16. S-curves of the power-based TED for DPSK at 1 Gb/s, for various (a) LPF filter BWs and (b) frequency detuning of the homodyne detection. (a) (b) Figure 6.17. S-curves of the power-based TED for QPSK at 2 Gb/s, for various (a) LPF filter BWs and (b) frequency detuning of the homodyne detection. baud rate, because the slope of the S-curve is almost cero. Without loss of generality, the S-curves in Figs. 6.16 and 6.17 were computed by assuming positive sampling phase delay +∆𝜏 between 𝑃𝑘−1       and 𝑃𝑘    , similar to Fig. 6.15, with sampling phase tuning resolution of ∆𝜏× 0.05𝑇𝑏. It is also worth noting that DPSK and QPSK perform the same, thus confirming the independency of the power-based TED on the optical phase content. The other subsystem of the symbol-rate DSP in Fig. 6.13 is the CR algorithm. It implements the simplified differential CR studied in Section 5.3, that was specifically conceived to operate at one sample per symbol. The differential CR algorithm in its polarization-diversity configuration is depicted in Fig. 6.18. The digital samples of the received complex signal 𝐼[𝑛]+𝑗𝑄[𝑛] are differentially demodulated individually for each SOP to recover the m-PSK information from the phase difference between consecutive symbols; here m is the order of the PSK modulation (e.g., m = 4 for QPSK). This operation cancels the phase noise effect, as long as the symbol rate is several orders of magnitude larger than the total laser linewidth (TX + LO), and depending on the 146 modulation format (see Section 3.3.1). Since the direct laser modulation at the TX proposed in this thesis is single-polarization, the demodulated signals for each received SOP 𝑑𝕏[𝑛] and 𝑑𝕐[𝑛] are added to counteract variations of the transmitted SOP after propagating through the fiber channel. Last, the FE compensates for the frequency detuning of the non-ideal homodyne detection, and the recovered PSK symbols 𝑠[𝑛] are finally delivered to the decision and demapping subsystem of the DSP for symbol-to-bit recovery. Figure 6.18. Differential CR architecture for symbol-rate polarization-diversity m-PSK receivers. 6.2.3 Experimental validation real-time The symbol-rate DSP described in previous section was prototyped in a commercial FPGA for real-time assessment, using the FPGA homodyne RX for COCONUT presented in Section 4.2.1. This time, however, the optical front-end was polarization-diversity 3x3 homodyne, as depicted in Fig. 6.19. The experiment featured direct-DPSK modulation at the TX, generated from a PPG running at 1.25 Gb/s. Figure 6.19. Real-time prototype of the polarization-diversity homodyne RX for 1.25 Gb/s DPSK, with the ADCs and the DSP operating at 1 sample per symbol. LO feedback j FE PR j PPG IQ Recovery BERT PIN + TIA DFB LO ML605 Virtex-6 FPGA Pol. diversityfront-end HomodynecoherentRX ADC 25 km SMF VOA 1-bit DAC DFB TX 1.25 Gb/s 1.25 GSa/s Equalizer ADC CLK recovery1.25 GSa/s Carrier recovery 1.25 GSa/s Direct-DPSK TX PBS PBS 45⁰ IQ Recovery 1GHz LPF PIN + TIA ADC 1.25 GSa/s ADC MMCM 1.25 GSa/s 147 Two DFBs with linewidth Δ𝜈 = 4 MHz each were used as TX and LO. The DPSK signal was transmitted through 25 km of SMF, then coherently detected by the polarization-diversity homodyne RX. The LO SOP was adjusted to 45⁰ for optimality of the detection. The four photodetected signals 𝐼𝕏, 𝑄𝕏, 𝐼𝕐 and 𝑄𝕐 were low-pass filtered by standard 4th-order Bessel filters with 1 GHz cut-off frequency, then mapped into the digital domain. The ADC was the FMC125 from 4DSP vendor that has four high-speed channels at 1.25 GSa/s each. In the COCONUT testbed in Section 4.2.1, the four ADC channels were interleaved in pairs to configure 2× 2.5 GSa/s ADC channels to sample the 𝐼 and 𝑄 signals at 2 samples per symbol. This work instead, uses all four ADC channels independently at 1.25 GSa/s to sample the four polarization-diversity photocurrents at 1 sample per symbol. At the DSP of the RX, the CLK recovery and the carrier recovery subsystems implemented the algorithms reported in Figs. 6.14 and 6.18 respectively. In this FPGA prototyping, the computed timing delay ∆𝜏 from the CLK recovery drives a mixed-mode CLK manager (MMCM) that dynamically shifts the phase of a 1.25 GHz reference CLK embedded in the FPGA board. This in turn drives the ADC sampling instants with a sampling phase tuning resolution of ∆𝜏× 23 ps, representing 2.8% of 𝑇𝑏. The other RX functionalities are similar to that of the Section 4.2.1. To experimentally validate the functioning of the proposed DSP, the sensitivity of the coherent RX operating at 1 sps was evaluated in two different configurations: single polarization and polarization diversity. First, the single polarization test used only the upper branch of the optical front-end in Fig. 6.19, without PBS. Then, only 𝐼 and 𝑄 were detected and processed by the FPGA at 1 sps; the received signal SOP was manually adjusted at the RX input. Besides the DFB LO with Δ𝜈 = 4 MHz, other ECL with ∆𝜈 = 100 kHz also performed as LO to evaluate the impact of the phase noise. The results in Fig. 6.20 show RX sensitivity of -54 dBm for BER = 10-3 FEC threshold, with no meaningful difference in the performance between 4 MHz and 8 MHz total Figure 6.20. BER vs. received power for 1.25 Gb/s DPSK, detected by 3x3 homodyne RX operating at 1 sample per symbol in single polarization (Single pol.) and polarization diversity (Pol. divers.) configuration. 148 linewidth. The extra phase noise had no impact on the CLK recovery performance, as expected, due to the timing error estimation based on the received signal power thus immune to the carrier phase, as stated before. Notably, the measured BER curves are roughly similar to Figs. 4.11(a) and 5.25(a) that utilized conventional ADC sampling and CLK recovery at 2 sps. After successfully validating the symbol-rate DSP in single polarization, the system was evaluated in polarization-diversity configuration with the two DFBs as TX and LO (total Δ𝜈 = 8 MHz ). The BER curves in Fig 6.20 were computed for three cases of interest: adjusting the received signal SOP to (I) 𝕏-polarization, (II) 𝕐-polarization, and (III) random. At FEC threshold, less than 1 dB penalty in sensitivity was observed among the different received signal SOP, and 8 dB penalty compared with the single polarization test due to the extra excess losses from the more complex optical front-end with the PBSs. The tolerance of the CLK recovery algorithm against CLK detuning between TX and RX was also evaluated in real time. The CLK frequency of the PPG was swept ±4 kHz with respect to 𝑅𝐵 (1.25 Gb/s), and the BER was measured for three different received optical powers. The block length for averaging in the CLK recovery was 𝑁 = 2048 bits. The results are reported in Fig. 6.21 and reveal that the CLK recovery has non-symmetric tolerance to the CLK mismatch as the received power decreases, with the lowest BER at ~1 kHz CLK detuning for the lowest received power (-43 dBm). This behavior is ascribed in part to accuracy errors in the implemented system for tuning the sampling phase of the ADCs with the FPGA, and in part to non-flat frequency response of the electronic HW that causes non-ideal pulse shape of the received signal at the ADCs input; this might produce different slopes for the rising edge and the falling edge of the calculated power 𝑃𝕏+𝑃𝕐 in Fig. 6.15. The test also shows that, when the optical power is as high as -34 dBm, the CLK recovery at 1 sps tolerates up to ±2.5 kHz CLK detuning in a constant BER regime. For -41 dBm and -43 dBm received power the tolerance lowers to ±1.5 kHz and ±1 kHz respectively (with respect to the optimal), which represents 40% and 60% reduction compared with high received power. Figure 6.21. BER vs. CLK detuning between TX and RX, for real-time DPSK at 1.25 Gb/s sampled by 1.25 GSa/s ADCs (1 sample per symbol). 149 The good performance achieved in real-time demonstrates the feasibility of the proposed lite DSP at 1 sample per symbol, leveraging low-speed ADCs and cheap FPGAs. However, the main challenge of this technique is that the accuracy of the CLK recovery highly depends on the quality of the detected pulses, as the maximum finding algorithm establishes the optimal sampling point at the maximum of the detected signal power, determined by the pulse shape. Therefore, preequalization at the TX of the linear channel response seems to be necessary for optimal RX performance. This topic is addressed in the next chapter. 6.3 Chapter Summary This chapter reported on a novel architecture for polarization-independent heterodyne detection, based on 3x3 optical coupler and differential field detection. The proposed RX exploits a simple front-end consisted of 3x3 coupler, PBS and three single-ended PDs. The electrical signal processing requires no extra complexity compared with standard heterodyne RXs with differential demodulation, and makes the proposed RX simple enough to be implemented with analog HW. The concept was successfully demonstrated with DPSK at 1.25 Gb/s, achieving -49 dBm sensitivity at BER = 10-3, after 50 km SMF transmission. Notably, the RX exhibits less than 1dB power penalty for a reference BER of 10-4 when the SOP of the received signal is randomly varied. The RX can also tolerate up to 25 dB power difference with adjacent channels by featuring cancellation of the direct-detection terms, thus outperforming by 7 dB the single-ended detection in the multiuser udWDM-PON scenario. The high performance along with the low-complexity of the RX, with common components and simple electronics, constitutes an attractive technology for cost-effective next generation udWDM-PON. Besides the heterodyne RX with analog HW, this chapter also explored solutions for the digital homodyne RX counterpart, aiming at simplifying the complexity of the polarization-diversity detection. In this regard, a complete DSP for polarization-diversity 𝑚-PSK RXs that operates at the symbol rate has been successfully implemented in FPGA and evaluated in real-time for shortreach coherent applications. The CLK recovery algorithm estimates the timing error from the detected signal power, and takes profit from a maximum finding algorithm for continuous tuning of the sampling phase of the ADCs, which operate at 1 sample per symbol. The proposed DSP is simple and robust. It can withstand to ±2.5 kHz detuning between TX and RX CLK for 1.25 Gb/s DPSK data, if the received optical power is high, and to ±1 kHz CLK detuning if the received power is near to the BER threshold for FEC. The homodyne RX achieved a high sensitivity of - 46 dBm at BER = 10-3 using low-complexity DSP at 1 sps, commercial DFB lasers, and directDPSK modulation. 150 Chapter 7 Digital Pre-Emphasis for Optical Transmitters As it has been stressed throughout the different chapters of this thesis, the next generation PONs and other optical networks with high terminal density like DCI, present strict requirements in terms of power budget, cost, and footprint of the photonic devices [Agr16], [Nes17]. Therefore, the adoption of coherent technologies for those systems will depend heavily on the overall cost, complexity and footprint reduction that can be achieved on the coherent TRX. To this aim, the previous chapters explored and proposed effective solutions to lower the complexity of the coherent RX. This chapter, now, focuses on the optical TX and particularly in direct laser modulation as an alternative to the IQ modulators employed in commercial coherent TXs for core networks, which encompass optical losses and substantial footprint. In order to keep on using the very low-cost direct laser modulation technology of actual commercial TDM-PON systems ‒that is limited to binary intensity modulation‒, optical 𝑚-PSK TXs based on direct modulation of DFB lasers have been proposed and successfully demonstrated in [Hu15], [Can16b] using the laser chirp to achieve the optical phase modulation, as studied in Section 2.4.2. This technique is particularly attractive in the udWDM-PON scenario, and has been adopted in this thesis, because it employs commercial low-cost lasers and reduces the form factor and power consumption of the coherent TX. This TX simplification though, does not imply sacrificing the capacity and spectral efficiency of the coherent PON because complex modulation formats can still be generated, thus fully exploiting the quadrature detection capabilities of the coherent RX. Nonetheless, commercial DFBs present different challenges for an effective implementation, as they exhibit non-flat frequency response and suffer from severe BW limitation. Moreover, to scale from binary to higher modulation formats, a DAC is required to drive the injection current modulation of the laser. To keep the cost of the coherent TX down, a low-cost DAC might present narrow BW and/or low resolution, quantified through the effective number of bits (ENOB). The last chapter of this thesis explores digital pre-emphasis (DPE) linear filtering at the TX to jointly mitigate the BW limitations and flatten the frequency responses of the laser and DAC. The design of the DPE filter also takes into account the quantization noise introduced by a DAC with low-resolution. The chapter also presents a novel and simple 8-ary coherent TX by simultaneous amplitude-and-phase modulation of an integrated DEML. Bit rates up to 10 Gb/s are experimentally achieved by applying DPE at the TX to mitigate the non-ideal DEML response. 7.1 Directly Modulated Lasers For Coherent Transmitters The high level block diagram of the considered optical transmission system with directly modulated lasers (DMLs) is illustrated in Fig. 7.1. The optical TX does not employ external IQ modulators to generate complex modulation formats; instead, they are replaced by simpler and cheaper direct laser chirp modulation for 𝑚-PSK, or alternatively, by integrated DEML for 𝑚- 151 APSK constellations, as examined in Sections 2.3.4 and 3.1.1. Referring to Fig. 7.1, let us initially consider the case of DML at the TX. First, the binary NRZ data are mapped into the constellation symbols 𝑠[𝑛𝑇𝑠], which consist of a 𝑚-level signal needed for direct 𝑚-PSK; here, 𝑇𝑠 is the sampling time. The 𝑚-PSK symbols can be differentially-encoded or not, depending on the detection scheme. Next, the signal 𝑠[𝑛𝑇𝑠] passes through a linear DPE filter designed to mitigate the non-ideal frequency responses of the TX HW, including the DAC and the DML. After the DPE the signal 𝑟[𝑛𝑇𝑠] enters a differentiator to perform the time-derivative of the modulating signal, as required to modulate the optical phase of the DML by its inherent frequency chirp. Figure 7.1. Considered optical transmission system for 𝑚-PSK, with DPE, DAC and DML at the TX, and coherent detection at the RX. For the ideal case, i.e., ideal frequency responses and no DPE, the differentiator output is ∆𝑇𝑠𝑟[𝑛𝑇𝑠], where ∆𝑇𝑠 is the discrete time difference operator with respect to the sampling time, as given by Eq. (2.70). Afterwards, the DAC performs two operations. First, the digital quantization of ∆𝑇𝑠𝑟[𝑛𝑇𝑠] as a function of the ENOB. Then, the sample-and-track circuit of the DAC reconstructs in the analog domain the output signal 𝑥(𝑡) that drives the direct laser chirp modulation, according to 𝑒𝑃𝑆𝐾(𝑡)=𝐸𝑐(𝑡)𝑒𝑗[2𝜋𝑓𝑐𝑡+2𝜋𝑘𝑓∫𝑥(𝜏)𝑑𝜏+𝜑𝑐(𝑡) 𝑡 0] (7.1) This corresponds to the small-signal approach to the laser rate equations examined in Section 2.4, in which the DML behaves as an optical FM modulator with residual IM. Here, 𝐸𝑐(𝑡) identifies the residual IM, 𝑓𝑐 and 𝜑𝑐(𝑡) are the frequency and the random phase of the optical carrier respectively, while 𝑘𝑓 denotes the frequency modulation depth, derived from Eq. (2.67) and expressed as 𝑘𝑓=𝛼𝐻𝜅𝜂 4𝜋 (7.2) with 𝛼𝐻 the Henry coefficient, 𝜅 the adiabatic chirp parameter, and 𝜂 the current-to-power efficiency of the laser. This holds for modulation frequencies from few MHz up to several GHz, the region where the adiabatic chirp is dominant. After propagating through the optical fiber channel, the transmitted 𝑚-PSK signal finally reaches the coherent RX that first recovers the information-bearing optical phase thanks to the linear field detection properties of the coherent RX, and afterwards recovers the transmitted 𝑚-PSK symbols 𝑠[𝑛𝑇𝑠] free from phase noise by utilizing either synchronous detection with phase estimation algorithms or differential field detection. 152 7.1.1 Linear pre-emphasis filter design The linear DPE filter with transfer function 𝑃(𝑓) is designed in the frequency domain with respect to the equivalent system model of the optical TX with DML illustrated in Fig. 7.2. The 𝑚-PSK symbols 𝑠(𝑓) in frequency domain first pass through the DPE filter 𝑃(𝑓); then, to convert from digital to analog domain, the DPE output 𝑟(𝑓) is converted by a DAC with transfer function 𝐷(𝑓), that also introduces the quantization noise 𝑛𝑞(𝑓) as a function of the ENOB. Note that the transfer function of the differentiator depicted in Fig. 7.1 is cancelled by the integral term of the optical phase modulation in the DML (compare with Eq. (2.69)), and therefore, does not impair the signal neither is considered in Fig. 7.2. The resulting signal 𝑥(𝑓) after the DAC drives the modulation of the DML that behaves as an optical FM modulator, and whose transfer function 𝐿(𝑓) is derived from Eq. (2.67) in the adiabatic regime, as 𝐿(𝑓)=Δ𝜈(𝑓) Δ𝑥(𝑓)=𝛼𝐻𝜅𝜂 4𝜋 (7.3) where 𝜈(𝑓) is the optical frequency, and the rest of terms follow the definition of 𝑘𝑓 in Eq. (7.2). Theoretically, 𝐿(𝑓) draws a totally flat frequency response if the DML operates in the adiabatic chirp regime only (from MHZ up to several GHz). In practice, however, commercial low-cost DMLs suffer from BW limitation and ripple in the passband of 𝐿(𝑓) that can be ascribed to constraints in the RF circuitry and package parasitic effects. This, in turn, limits the maximum bit rates that can be achieved with the DMLs. The optical channel is modeled with transfer function 𝐻(𝑓), while 𝑛𝑐ℎ(𝑓) stands for the additive channel noise. Last, the transmitted 𝑚-PSK symbols are recovered at the coherent RX output, denoted by 𝑦(𝑓). Figure 7.2. Frequency-domain equivalent system model of the optical TX with DML depicted in Fig. 7.1. The objective of the DPE filter 𝑃(𝑓) is to jointly mitigate the BW limitation and flatten the frequency response 𝐿(𝑓) of the DML (that might include the diver amplifier response), and 𝐷(𝑓) of the DAC. To this aim, the most conventional method consists of assigning to 𝑃(𝑓) the inverse frequency responses of the DML and DAC, as 𝑃(𝑓)= 1 𝐷(𝑓)𝐿(𝑓) (7.4) known in literature as zero-forcing (ZF) equalization, which can cancel all linear distortion and ISI if extra sources of noise are not considered [Don12], [Zho11]. This technique, however, presents some disadvantages since it might over-amplify the noise greatly in the spectral region where 𝐷(𝑓)𝐿(𝑓) is more attenuated. It might also produce large peaks in the time domain signal that could lead to quantization noise enhancement if the DAC has low resolution, as well as driver amplifier saturation and clipping. 153 A more advanced solution, yet simple, for pre-equalization at the TX consist of calculating the linear filter 𝑃(𝑓) through the minimum mean squared error (MMSE) between the desired and the real system output. This does not eliminate the distortion and ISI completely, but minimizes the total power of the noise and the interference components [Che12], [Zha14]. Precisely, it is optimal in presence of significant quantization noise due to ADCs/DACs with low ENOB [Nap16], as one could expect from the low-cost converters intended for deployment in PONs. For the analytical derivation of 𝑃(𝑓), this work aims at optimizing the TX only, then the channel and the coherent RX are assumed ideal and noiseless (i.e., 𝐻(𝑓)= 1 and 𝑛𝑐ℎ(𝑓)= 0). This assumption is correct for the transmission rates and fiber link lengths considered in this thesis for the access scenario, of <10 GBd and <60 km respectively, where the impairments from the fiber are not dominant, as analyzed in Chapter 3. Let 𝑦𝑖𝑑(𝑓)=𝑠(𝑓) be the ideal output signal from the coherent RX, i.e., the transmitted 𝑚-PSK symbols are ideally recovered. On the other hand, the real system output in Fig. 7.2 is given by 𝑦(𝑓)=𝑠(𝑓)𝑃(𝑓)𝐷(𝑓)𝐿(𝑓)+𝑛𝑞(𝑓)𝐷(𝑓)𝐿(𝑓) (7.5) Here, 𝑠(𝑓) and 𝑛𝑞(𝑓) are stochastic processes, whereas 𝑃(𝑓), 𝐷(𝑓) and 𝐿(𝑓) are linear transfer functions. The DPE filter 𝑃(𝑓) is obtained through the mean squared error (MSE) between 𝑦(𝑓) and 𝑦𝑖𝑑(𝑓), calculated as 𝑀𝑆𝐸(𝑃(𝑓))=∫ E[|𝑦(𝑓)−𝑦𝑖𝑑(𝑓)|2]𝑑𝑓 𝑅𝑏 −𝑅𝑏 (7.6) where E[∙] is the expectation operator with respect to the stochastic processes 𝑠(𝑓) and 𝑛𝑞(𝑓), and 𝑅𝑏 is the symbol rate. By replacing Eq. (7.5) into Eq. (7.6) one obtains 𝑀𝑆𝐸(𝑃(𝑓))=∫ E[|𝑠(𝑓)[𝑃(𝑓)𝐷(𝑓)𝐿(𝑓)−1]+𝑛𝑞(𝑓)𝐷(𝑓)𝐿(𝑓)|2]𝑑𝑓 𝑅𝑏 −𝑅𝑏 (7.7) that can be solved using the triangle inequality for complex numbers: |𝑧1+𝑧2|2≤(|𝑧1|+|𝑧2|)2. Accordingly, the upper bound (considering the equality) of the MSE in Eq. (7.7) is found to be: 𝑀𝑆𝐸(𝑃(𝑓))=∫ E[|𝑠(𝑓)[𝑃(𝑓)𝐷(𝑓)𝐿(𝑓)−1]|2+|𝑛𝑞(𝑓)𝐷(𝑓)𝐿(𝑓)|2 𝑅𝑏 −𝑅𝑏 +2|𝑠(𝑓)𝑛𝑞(𝑓)𝐷(𝑓)𝐿(𝑓)[𝑃(𝑓)𝐷(𝑓)𝐿(𝑓)−1]|]𝑑𝑓 (7.8) which can be rewritten as 𝑀𝑆𝐸(𝑃(𝑓))=∫ |𝑃(𝑓)𝐷(𝑓)𝐿(𝑓)−1|2E[|𝑠(𝑓)|2]𝑑𝑓 𝑅𝑏 −𝑅𝑏 +∫ |𝐷(𝑓)𝐿(𝑓)|2E[|𝑛𝑞(𝑓)|2]𝑑𝑓 𝑅𝑏 −𝑅𝑏 +2∫ |𝐷(𝑓)𝐿(𝑓)[𝑃(𝑓)𝐷(𝑓)𝐿(𝑓)−1]|E[|𝑠(𝑓)𝑛𝑞(𝑓)|]𝑑𝑓 𝑅𝑏 −𝑅𝑏 (7.9)