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Integrated-optic electrorefraction modulators Ristic, Sasa 2007

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Integrated-Optic Electrorefraction Modulators by Sasa Ristic B.A.Sc, The University of British Columbia, 2001 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor in Philosophy in The Faculty of Graduate Studies (Electrical and Computer Engineering) The University of British Columbia April 2007 © Sasa Ristic 2007 Abstract The large electroabsorption (EA) and electrorefraction (ER) effects that can be achieved in coupled quantum well (CQW) structures give these structures great potential for use in intensity, phase, and polarization modulators for future optical links. This thesis focuses on designing structures that will result in state-of-the-art performance of CQW-based modulators. In this thesis, a design for a push-pull polarization-modulation scheme is presented, and it requires placing multiple repetitions of two different CQW structures in the optical waveguide of the proposed polarization modulator. The main function of one of the two structures is to provide a large increase of the refractive index for the TM polarization, without a proportionate increase of the refractive index for the TE polarization. It is shown that this effect can be achieved by anticrossing light holes. The main function of the other structure is to provide a large decrease of the refractive index for the TE polarization, without a proportionate decrease of the refractive index for the TM polarization. It is shown that, although this effect can be efficiently achieved by anticrossing heavy holes, the CQW structures that achieve this effect by anticrossing electrons are less sensitive to the layer thickness and compositional variations that may be expected during the growth process. The large EA and ER effects in CQW structures are typically seriously compromised by the layer variations, and, it is shown in this thesis that decreasing the confinement of the anticrossing wave functions can decrease the sensitivities of the CQW structures to these variations. In addition, the very small confinement of light holes in the InGaAlAs-based material systems together with the small sensitivity of the light-hole band edge to the compositional variations, make the anticrossing of the light holes the most robust-to-growth and, therefore, the most promising mode of operation for CQW-based ER intensity, phase, and polarization modulators. The absorption spectra of the novel CQW structures designed in this thesis are obtained using the variational method for solving the 1S exciton equation in the effective-mass, envelope-function approximation, and the electric-field-induced refractive index changes are obtained using the Kramers-Krbnig relations. iii Table of Contents Abstract ii Table of Contents iii List of Figures v List of Abbreviations vii Acknowledgments viii Co-Authorship Statement x Chapter 1. Introduction 1 1.1 Optical Fiber Communications 4 1.1.1 Digital Communications 5 1.1.2 Analog Communications 8 1.2 Optical Transmitters 10 1.3 Overview of External Modulators 15 1.3.1 Linear Electro-Optic Effect in Semiconductors 16 1.3.2 Semiconductor Polarization Modulators 17 1.3.3 Semiconductor Mach-Zehnder Modulators 23 1.3.4 Semiconductor Mode-Conversion Modulators 26 1.3.5 Lithium Niobate Modulators 28 1.3.6 Polymer Modulators 32 1.3.7 Comparison of Linear Electro-Optic Modulators 34 1.4 Multiple Quantum Well Modulators 44 1.4.1 Electroabsorption in SQWs 46 1.4.2 Electrorefraction in SQWs 57 1.4.3 Electroabsorption vs. Electrorefraction in M Q W Modulators 60 1.5 Coupled Quantum Wel l Structures 66 1.5.1 Modeling of C Q W Structures 67 1.5.2 Electrorefraction in C Q W Structures 77 1.5.3 Strain in C Q W Structures Grown on (001) InP 99 1.6 Thesis Outline 104 References 106 Chapter 2. Push-Pull CQW Polarization Modulator 118 2.1 Introduction 118 iv 2.2 Device Basics 120 2.2.1 TS Structure 121 2.2.2 CS Structure 126 2.3 Low-Chirp Push-Pull Q W P C Modulator 127 References 128 Chapter 3. Robustness of Push-Pull CQW Polarization Modulator... 130 3.1 Introduction 130 3.2 Influence of Layer Variations 133 3.2.1 TS Structure 135 3.2.2 CS Structure 138 3.3 Discussion ••• 140 References ; 141 Chapter 4. Robust CQW Electrorefraction Modulator 143 4.1 Introduction 143 4.2 Influence of Layer Variations 145 References 153 Chapter 5. Improving Robustness in CQW Structures 155 5.1 Introduction 155 5.2 Robustness Improvement Approach 156 5.3 Discussion 162 References 162 Chapter 6. Improved Push-Pull CQW Polarization Modulator 164 6.1 Introduction 164 6.2 Improved Structure #1 and Structure #2 166 References 174 Chapter 7. Summary, Conclusions, and Suggestions for Future Work 1 7 5 7.1 Summary 175 7.2 Conclusions 184 7.3 Suggestions for Future Work : . . . . 188 References • 191 V List of Figures 1.1 Directly-modulated and externally-modulated transmitters 6 1.2 Wavelength-division-multiplex digital fiber-optic link 8 1.3 Sub-carrier-multiplexed analog fiber-optic link 10 1.4 Return-to-zero optical modulation scheme 15 1.5 Semiconductor polarization modulator 18 1.6 Intensity modulator transfer function 22 1.7 Semiconductor Mach-Zehnder intensity modulator 24 1.8 Semiconductor mode-conversion modulator 26 1.9 Lithium Niobate modulator 30 1.10 Polymer push-pull Mach-Zehnder intensity modulator 34 1.11 Traveling-wave electrode with T-shaped capacitive loads 37 1.12 States in two uncoupled InGaAs-InAlAs SQWs 45 1.13 Effect of the electric field on the lowest energy states in a S Q W 48 1.14 Schematic representation of SQW optical absorption for T M light 50 1.15 Experimental TE and T M absorption spectra in G a A s - A l G a A s SQWs 54 1.16 T E absorption and refractive-index changes in G a A s - A l G a A s SQWs 59 1.17 InGaAs-InAlAs p-i-n diode Mach-Zehnder M Q W modulator 60 1.18 Band-edge diagram and states of an InGaAs-InAlAs C Q W structure 80 1.19 Overlap integrals for states in an InGaAs-InAlAs C Q W structure 81 1.20 Binding energies for excitons in an InGaAs-InAlAs C Q W structure 84 1.21 T M absorption spectra in an InGaAs-InAlAs C Q W structure 86 1.22 T E absorption spectra in an InGaAs-InAlAs C Q W structure 88 1.23 Exciton transition energies in an InGaAs-InAlAs C Q W structure 89 1.24 Wavelength dependence of T M electroabsorption and electrorefraction 90 1.25 Wavelength dependence of TE electroabsorption and electrorefraction 91 1.26 Electric-field dependence of the T E and T M refractive index changes 92 1.27 Electrorefraction dependence on broadening functions 94 1.28 Experimental optical loss and phase shift in InGaAsP-InP SQWs 96 1.29 Effect of a thickness variation on an InGaAs-InAlAs C Q W structure 98 1.30 Effects of biaxial strain on the band edges in InGaAs and InAlAs 101 2.1 Schematic of layers for TS and CS structures 120 2.2 Band-edge diagram and wave functions for the TS structure 122 vi 2.3 Refractive index changes in TS and CS structures for T E and T M light 125 2.4 Band-edge diagram and wave functions for the CS structure 126 3.1 Schematic of layers for TS and CS structures 132 3.2 Effects of layer variations on the T M light for the T S structure 135 3.3 Effects of layer variations on the TE light for the TS structure 136 3.4 Effects of layer variations on the TE light for the CS structure 138 3.5 Effects of layer variations on the T M light for the CS structure 139 4.1 Band-edge diagrams and wave functions for structures #1 and #2 146 4.2 T E and T M refractive-index changes for structure #1 and structure #2 147 4.3 Effects of layer variations on electrorefraction in structures # 1 and #2...... 150 5.1 Band-edge diagrams and ground state wave functions for S A and S B 157 5.2 Sensitivities of S A and S B to layer variations in the left-hand side well 159 5.3 Sensitivities of S A and S B to layer variations in the middle barrier 160 5.4 Sensitivities of S A and S B to layer variations in the left-hand side barrier... 161 6.1 Schematic of layers for structure #1 and structure #2 166 6.2 Band-edge diagrams and wave functions for structures #1 and #2 167 6.3 Effects of layer variations on refractive indices in structures #1 and #2 168 6.4 Effects of layer variations on a ID polarization modulator 173 List of Abbreviations ACQW Asymmetric Coupled Quantum Well AWG Arrayed Waveguide Grating CATV Cable Television CQW Coupled Quantum Well DFB Distributed Feedback DGD Differential Group Delay EA Electroabsorption EDFA Earbium-Doped Fiber Amplifier ER Electrorefraction HWHM Half-Width-at-Half-Maximum ITU International Telecommunication Union L A N Local-Area Network MQW Multiple Quantum Well NRZ Non-Return-to-Zero OOK On-Off Keying PCS Personal Communications Services QCSE Quantum-Confined Stark Effect QWPC Quantum Well Polarization Conversion RF Radio Frequency RZ Return-to-Zero RZ-DPSK Return-to-Zero Differential-Phase-Shift-Keyed SCM Sub-Carriers Multiplexing SNR Signal-to-Noise Ratio SQW Square Quantum Well TE Transverse-Electric TDM Time-Division Multiplexing T M Transverse-Magnetic VCSEL Vertical-Cavity Surface-Emitting WDM Wavelength-Division Multiplexing Acknowledgments Throughout my PhD studies, I have received great love and support from my wife, Dr. Jelena Ristic, and for this, I am very thankful to her. I am also very tharikful to my parents, Nada and Djordjo Ristic, and my sister, Sanja Ristic, for their understanding and emotional support. Many thanks to my supervisor Dr. Nicolas Jaeger. Our seven-year relationship started when he accepted me in his laboratory as an undergraduate student who wanted to gain experimental experience. From the moment he took me in his laboratory, I was immediately sold on the field of Photonics before I even knew what exactly it was. It is difficult not to be affected by Dr. Jaeger's drive and passion for the field. His multifaceted mentorship guidance and support, which also meant generous financial support, help me develop into an independent and confident researcher. I would also like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC), the University of British Columbia, and the British Columbia Advanced Systems Institute for their financial support. Thanks are due to several faculty members, primarily Dr. David Pulfrey. His support has helped me secure both external funding from the N S E R C and internal funding from University Graduate Fellowship (UGF). I am very thankful for all the help and encouragement he has given me throughout my graduate and undergraduate studies and for making me feel that I could rely on his help in the future as well. The same is true for Dr. John Madden and Dr. Lukas Chrostowski, who have also been supportive of my work. Dr. Mario Beaudoin, who is in charge of the Nanofabrication Facility in the A M P E L building, has been very helpful and extremely approachable whenever I had any growth-related questions. I am grateful to him and also Dr. Tom Tiedje, from the Physics Department, for growth-related discussions. Although, I have enjoyed many interesting research-related discussions with the following graduate U B C students from Dr. Pulfrey's group, I also cannot thank them enough for all the good times we have had. Thank you Dr. Leonardo Castro, Dylan McGuire, Dr. David John, and Paul Pereira. Also, I have been lucky to be surrounded by a great group of people in Dr. Jaeger's laboratory. Thank you Dr. Jeffrey Bull , Pejman Ghanipour, Michael Manarovici, Kyle MacDonald, Sameer Chandani, and Ryan Kisch. Also, thanks are due to our dear friends who made Vancouver feel like home for me and my wife for the past 12 years. Thank you Nebojsa and Jelena Plavsic, Aleksandar and Vanja Vitkovic, Zanka and Aleksandar Lizdek, Jasna and Svetlana Sokolovic, Pedja Kesic, Sinisa and Ljubica Predojevic, Igor Rudic and Snezana Slavic, and, particularly, Ljubomir Kaludjercic and Dijana Mandic with their children, our nieces, Ksenija and Katarina Kaludjercic. Co-Authorship Statement The work presented in this thesis was a collaboration between me and my supervisor, which I took the leading role. Chapter 1 Introduction 1 High-speed optical fiber communication networks play a vital role in the ongoing information revolution. Consequently, there is a huge demand for improving performance of the optical networks and their components, such as: semiconductor lasers, photodetectors, and electro-optic modulators. The research objective of this thesis is to make a contribution to the area of optical communications by pursuing the development of InGaAlAs-based electro-optic modulators using coupled quantum well (CQW) structures [ l ] - [ 3 ] . Electro-optic modulators based on C Q W structures have the potential to become a leading technology for efficient high-speed conversion of electrical into optical signals, not only in optical communications but also in signal processing and instrumentation. This is mainly because C Q W structures provide very large electroabsorption effect (electric-field-induced change of optical absorption) and electrorefraction effect (electric-field-induced change of refractive index) and should enable modulators based on them to have shorter lengths and lower drive voltages than modulators based on square quantum wells or on the linear electro-optic effect. Both the short lengths and the low drive voltages possible with C Q W modulators wi l l be essential requirements for next-generation high-speed optical networks, which wi l l , in turn, require electro-optic modulators having bandwidths of 100 G H z and above. [4] In this thesis, a design for the first push-pull polarization modulator that is based on any type of quantum well structure is presented [1]. The effect is achieved by placing multiple repetitions of two types of C Q W structure in the optical waveguide of the modulator. Upon the application of an electric field normal to the planes of the quantum wells, one of the structures mainly provides a positive change of the refractive index for the T M polarization, while the other structure mainly provides a similar negative change of the refractive index for the T E polarization. As a result, an input polarization state containing equal powers in both the TE and T M fundamental modes of the waveguide can be switched between orthogonal states. Polarization modulation has been traditionally associated with linear electro-optic modulators, and it has been used in an increasing number of modulation schemes, which have provided considerable improvements to the systems using them [5]. The C Q W polarization modulator design presented in this thesis wi l l enable these modulation schemes to be applied in next-generation high-speed optical networks as well. 3 Although they provide large electroabsorption and electrorefraction effects, C Q W structures are quite sensitive to the layer thickness and compositional variations that are likely to occur in their quantum well layers during growth processes [2], [3]. The layer variations severely affect the properties of C Q W structures, reducing the magnitude of the electroabsorption and electrorefraction effects that can be achieved. This thesis also presents a method that can be used to decrease the sensitivities of C Q W structures to layer variations and, thus, significantly improve the magnitude of the electroabsorption and electrorefraction effects achievable with these structures. The method is mainly based on decreasing confinement of the electronic states in the C Q W structures to their quantum well layers [3], and it can be used to improve not only the performance of the polarization modulator, but also the performance of electroabsorption and electrorefraction modulators in general. The large improvement achievable is expected to make a considerable impact on the state-of-the-art of electro-optic modulators. The discussion of C Q W modulators begins in Section 1.5 of this Introduction, Chapter 1, and continues throughout Chapters 2-6, which contain manuscripts that have been published, have been submitted for publication, or wi l l be submitted for publication in journals. The last chapter of this thesis, Chapter 7, presents a discussions that relates Chapters 2-6 to each other, and to the field of optical fiber communications in general, and it also gives recommendations for future work on C Q W modulators. The remainder of the Introduction starts with a discussion of analog and digital optical links (Section 1.1), which is followed by a comparison of directly modulated transmitters (Section 1.2) and linear electro-optic effect based external modulators (Section 1.3) for use in these optical links. Section 1.3 presents an overview of the most common linear electro-optic effect based external modulators: semiconductor polarization modulators, semiconductor Mach-Zehnder modulators, semiconductor mode converters, LiNb03 -based modulators, and polymer modulators. More specifically, Section 1.3.7 presents a comparison of the linear electro-optic effect based external modulators and gives a motivation for using quantum well based modulators in modern optical links. In Section 1.4, a discussion of the electroabsorption effect in square quantum wells (Section 1.4.1) is followed by a discussion of the electrorefraction effect in them (Section 1.4.2) and by a comparison of these two types of modulator (Section 1.4.2). Although the discussion of square quantum well modulators in Section 1.4 points to the qualities that make them very popular in today's optical links, it also reveals the shortcomings that motivate research into C Q W modulators. 1.1 Optical Fiber Communications Today's global need for greater telecommunication network capacities seems insatiable. In the 1970s, the transmission loss of optical fibers was reduced to only -0.5 dB/km within a 12 THz spectral region near 1310 nm and to only ~0.2 dB/km within a 15 THz spectral region near 1550 nm [6]. It was clear that data transmission capacities of optical 5 fibers would eventually overshadow those achievable with the copper wire [7]. Subsequently, a global research effort was initiated in order to develop optical transmitters and receivers operating around both 1310 nm and 1550 nm. Although the spectral region around 1550 nm has smaller optical loss than that around 1310 nm, and is dominant in long-haul communications, the region near 1310 nm has also been widely used in fiber-optic communications as the group-velocity dispersion is zero, i.e., the dispersion changes sign, close to 1310 nm. Besides optical loss, group-velocity dispersion in fibers is one of the most important properties limiting the performance of a communication system [8]. It is also worth mentioning that the first-generation fiber-optic systems operated in the region near 800 nm using multimode fiber. The relatively large loss in this region (~2.5 dB/km) and the intermodal dispersion [8] limited digital systems to bit rates of only -45 Mbps and repeater spacing of only -10 km [7]. Due to its cost-effectiveness and maturity, this short-wavelength technology has reemerged recently in short-haul applications, such as in local-area networks (LANs) [9]. 1.1.1 Digital Communications The majority of fiber-optic systems are digital and operate in the spectral range around 1550 nm. In these systems, data are typically encoded in binary representation. Among the various characteristics of lightwaves that can be modulated, e.g., intensity, phase, frequency, and polarization, intensity modulation has been most commonly used mainly because of the simplicity of the envelope photodetection [10]. Therefore, as will be explained in more detail below, optical transmitters are realized either by direct modulation or by external modulation of the intensity of light coming out of a semiconductor laser. In direct-modulation transmitters, the current of a semiconductor laser is directly modulated by the binary electrical data stream, which switches the intensity of the laser's output from the "on" state (maximum light intensity at the output) to the "off state (almost no light at the output), and vice versa [11], see Fig. 1.1(a). In external-modulation transmitters, the intensity of light from a semiconductor laser is switched on and off by an external optical modulator, which has its electrical port driven by the digital data stream [10], see Fig. 1.1(b). The on-off modulation in digital optical systems is also referred to as on-off keying (OOK). 0 10010 optical data output Laser Modulator T DCi D C 2 0 10010 •_n_n_ optical data output electrical data input (a) 0 10010 electrical data input (b) Figure 1.1: Schematics of an optical transmitter for digital communications: (a) directly-modulated semiconductor laser and (b) externally-modulated semiconductor laser. In order to maximize the transmission capacity of a digital fiber-optic network, many different data streams at lower bit rates are commonly multiplexed to achieve higher bit rates in a single optical channel operating at a single wavelength. Typically, the 7 interleaving is done electronically, and is referred to as time-division multiplexing (TDM). The highest transmission rate in commercially available T D M systems is around 10 Gbps for each channel [12]. Many researchers are working on the next generation of T D M systems that wi l l operate at 40 Gbps data rates. Traditionally, when planning future networks, service providers expect a fourfold increase in a network capacity at about 2.5 times the cost of the network equipment [13]. In combination with the T D M , commercial digital fiber-optic networks employ wavelength-division multiplexing (WDM), increasing the transmission capacities to about 1 Tbps [12]. In W D M links, many 10 Gbps (or lower bit rate) channels, each at a different carrier wavelength (X), are multiplexed onto the same single-mode fiber using, for example, an arrayed waveguide grating (AWG) [14], as shown in Fig. 1.2. The channels form an infinite frequency grid centered at 193.1 THz, and are typically separated by 50 or 100 GHz, as regulated by the International Telecommunication Union (ITU) [12]. As also shown in Fig. 1.2, the multiplexed light is amplified by a cascade of erbium-doped fiber amplifiers (EDFAs), with about 100 km spacing between them [7], before entering the A W G demultiplexer at the other side of the link. Due to their ability to simultaneously amplify, in the optical domain, many W D M channels in the low-loss 1550 nm spectral region, E D F A s play a very important role in fiber-optic communications [15]. The gains of E D F A s cover the whole C-band (1530 nm-1565 nm) and a large part of the L-band (1565 nm-1625 nm), i.e., the spectrum between 1565 nm and 1610 nm. For amplification of wavelengths shorter than these, including those around 1310 nm, Raman amplifiers are commonly 8 used [14] (Raman amplifiers are also used to complement E D F A s in ultra-long-haul systems). As shown in Fig. 1.2, at the end of a W D M link, which can be several thousand kilometers long [7], each channel is demultiplexed and passed to a separate receiver, which is a fast photodetector, for a simple envelope demodulation [14]. Transmitter #1 h Transmitter #2 • • • Transmitter #N 0) X w 3 ED FA EDFA w X JD ~B E i) Q A.1 Receiver #1 ^2 Receiver #2 • • • Receiver #N Figure 1.2: Schematics of a wavelength-division-multiplexed digital fiber-optic link. 1.1.2 Analog Communications Although the vast majority of fiber-optic links are digital, there are an increasing number of applications for analog fiber-optic links. They are used for faithful transmission of RF signals over distances that are typically not as long as the distances covered by most digital links but are still too long for the use of coaxial cables. For example, the standard 3 /8" coaxial cable exhibits a transmission loss of over 50 dB/km at the frequency of 1 G H z [16]. The fact that analog links mainly span relatively short distances makes the zero-dispersion 1310 nm spectral region very popular for these applications because the relatively high loss in the spectral region is not a limiting factor for short distances [11]. 9 Similar to digital links, in analog links, transmitters are based on direct or external intensity modulation of semiconductor lasers, and the receivers are simply high-speed photodetectors. However, in analog links, the light intensity of the carrier is not switched on-off, but, rather, small-signal modulation is used in order to faithfully transmit the RF signal over the link. Small-signal modulation in analog links is more bandwidth-efficient than the O O K in digital links, and it also circumvents the costly analog-to-digital and digital-to-analog data conversion. Some of the applications of analog links include the up-link (down-link) for cellular/PCS antenna remoting (here, PCS stands for Personal Communications Services), and transmit (receive) function in radar systems. Also, one of the first, and perhaps most commercial, applications of analog links was the distribution of cable television ( C A T V ) signals [17]. Depending on the application, analog links can use only a single optical carrier, or they can employ wavelength-division multiplexing, similar to the digital links [18]. Also, similar to the T D M used in digital links for combining multiple signals into a single channel at a certain wavelength, sub-carrier multiplexing (SCM) is used in C A T V systems to electronically combine multiple low-bandwidth signals onto the same carrier wavelength, as shown in Fig. 1.3 [19]. In an S C M link, N different signals (S t to Sn), each having a different RF carrier frequency (fi to fw), are added into a larger bandwidth signal that modulates a semiconductor laser. At the other end of the fiber, the optical 10 signal enters the high-speed photodetector (PD), where it is converted back into an electrical signal. At the end of the link, N band pass filters (BPFi to BPFN) are used to separate the N input signals. Also, it is usual that some of the signals in an S C M link are digital. Laser 1 1 p n T D C i t D C 2 BPFi B P F 2 B P F N S 2 S n Figure 1.3: Schematics of a sub-carrier-multiplexed analog fiber-optic link. 1.2 Optical Transmitters As mentioned above, optical transmitters used in both digital and analog fiber-optic links are based on either direct or external modulation of the light coming out of a semiconductor laser. The performance requirements for these transmitters depend on the application and whether the optical link in question is digital or analog. A n overview of the performance requirements for optical transmitters, with an emphasis on external modulators, is given in [10]; the most important of these requirements are summarized below. One of the basic requirements for optical transmitters used in most digital links is, again, operation near 1550 nm. In addition, the transmitters are required to operate at high data rates, i.e., have wide modulation bandwidths. In modern W D M links, data transmission capacities could be further increased by designing the links to have even larger numbers of low bit rate channels. However, such designs are difficult to implement because the optical multiplexing and demultiplexing circuits become very complex for large channel counts. Therefore, wide-modulation-bandwidth transmitters are very important for future high-capacity digital networks. A n optical transmitter for digital links is also required to have large signal-to-noise ratio (SNR) and large extinction ratio. For example, an O O K transmitter should introduce as little noise into the system as possible. The device should leak as little light as possible when in its " o f f state, and it should put out as much power as possible when in its "on" state. Also, as electronic drives that output more than 3.5 V become very expensive for data rates of 40 Gbps and higher, optical transmitters for the next-generation digital networks must have very low drive voltages. Finally, the last requirement for optical transmitters used in digital links, that wi l l be mentioned here, is low chirp. Both directly modulated lasers and external intensity modulators introduce a certain amount of undesirable phase modulation in addition to the intensity modulation of the lightwave carrier. This effect results in an optical frequency (wavelength) chirp. In 12 optical waveguides that have nonzero group velocity dispersion, such as the common single-mode fibers used in the low loss 1550 nm spectral region, chirped optical pulses suffer excessive broadening and reduction of their peak powers. This effect limits the bit-rate-distance product (BL) in 1550 nm digital links, making them dispersion-limited, rather than loss-limited systems. One way of quantifying chirp in external modulators is using Henry's alpha parameter (aH) [10], which is defined as the ratio between the change of the real part of the refractive index (Sn) and the imaginary part of the refractive index (dk) in the optical waveguide of the modulator. A n equivalent to an in directly modulated lasers is the "linewidth enhancement factor" and is defined in the same way. It can also be shown that a H = — - - 2 / ( 0 , (1.1) " 5k yjdi(t)/dt ' where <J>(0 and I(t) are time-dependent phase and intensity of light exiting the external modulator. In analog links, low frequency chirp is not such an important requirement for optical transmitters, as analog links are typically shorter than digital links and many of them operate near 1310 nm, where common single-mode fiber has zero group velocity dispersion. Also, small-signal modulation, which is characteristic of analog links, typically introduces less frequency chirp than O O K in digital links. Similar to digital links, many analog links require very fast transmitters that introduce little noise into the ! 3 system, and output as much power as possible. Similar to digital transmitters, analog transmitters are required to achieve large changes in the intensity of light upon only small changes in the drive voltage, which is directly related to the so called small-signal modulation efficiency. Maybe the greatest difference between analog and digital transmitters is that the response of analog transmitters should be as linear as possible because in analog links data signals are supposed to be transmitted over the fiber link in a faithful manner without significant nonlinear distortions. There are several methods that can be used to compensate, to a certain degree, for nonlinear transmitter response, both in the electrical domain (e.g., using pre-distortion circuitry) and in the optical domain (e.g., cascading two external modulators in a way that their nonlinear responses cancel each other) [17]. Nonetheless, for these schemes to be effective, the transfer function of the transmitter should be as linear as possible. Finally, it is very important for analog transmitters, as well as the digital transmitters, to be affordable, to be insensitive to changes in temperature, and to have long lifetimes. Having summarized the basic requirements for digital and analog transmitters, a , performance comparison between the directly modulated transmitters and externally modulated transmitters wi l l now be presented. The simplest and most economical way to implement an optical transmitter is to directly modulate the current of a semiconductor laser and, thus, modulate the intensity of its output light. Directly modulated lasers have small footprints and low power consumption. However, they have quite limited 14 frequency responses. The resonance frequencies of distributed feedback (DFB) lasers or vertical-cavity surface-emitting lasers (VCSELs) are typically lower than about 30 GHz [20]. As the modulation frequency is increased toward the resonance frequency, noise and non-linear distortions rapidly increase. In addition, directly modulated lasers generate significantly more chirp than most external modulators do. Although the performance of directly modulated lasers is being continuously improved using innovative methods, such as injection-locking [20], it is a widely accepted opinion that externally modulated lasers wi l l continue to dominate in high-frequency (>20 GHz) links [10]. Furthermore, external modulators are essential for some modulation schemes, such as return-to-zero (RZ) modulation [14]. Unlike non-return-to-zero modulation (NRZ), where optical pulses completely fi l l the bit interval, in R Z modulation, optical pulses are much narrower than the bit interval. The N R Z modulation scheme was implied in Fig. 1.1. The R Z modulation scheme is illustrated in Fig. 1.4 below. Here, a pulse generator outputs a train of optical pulses that are significantly narrower than the bit interval, which is determined by the N R Z data stream driving the optical modulator. The optical modulator gates the pulses corresponding to "bit 0," outputting R Z optical data stream. Using a mode-locked laser as a pulse generator is not cost effective, and the pulses are typically generated by another external modulator that modulates the output of a D F B laser [14]. The major disadvantage of R Z modulation is that it requires modulators that have higher bandwidths than those used for N R Z modulation. Nonetheless, R Z modulation offers important 15 benefits, such as clock transitions for the receiver even when the data contains a long string of Is [21]. Pulse Generator optical pulse train A A A A A 1 1 0 0 >AA A optical RZ data stream 1 1 0 0 1 i—I n electrical NRZ data stream Figure 1.4: Return-to-zero optical modulation scheme. As wi l l be explained in more detail in the following discussion on external modulators, unlike directly modulated lasers, some external modulators are capable of modulating the polarization of light. Polarization-modulation schemes have provided great improvements to both the analog and digital links that use them. 1.3 Overview of External Modulators In this section, an overview of most common external modulation technologies is presented. First, attention is paid to the technologies that use the linear electro-optic effect in crystals and do not need any quantum wells for optical modulation. Subsequently, the main limitations of these technologies are discussed, giving a 16 motivation for modulators based on quantum well structures, particularly those based on coupled quantum well structures, as they are the main focus of this thesis. 1.3.1 Linear Electro-Optic Effect in Semiconductors In GaAs and other crystals having the zincblende structure (e.g., InP), the change in the refractive index of the bulk crystal upon the application of the electric field ( F) (after relatively simple tensor mathematics) can be shown to be one of the following three cases depending on the field direction [22]: Case I: electric field applied in [001] direction rc[001] = « 0 «[Tl0] = n 0 + A « (1.2) n[l 10] = n0 - An Case II: electric field applied in [110] direction n[\l0] = n0 n[l\y[2] = n0+An (1.3) n[l lV2 ] = n0 - An Case III: electric field applied in [111] direction An n[\ 1 0] = n0 + V3" r i , ^rn An n[U2] = n0 +-j= (1.4) n 1 n 2 A " n[l 11] = n0 - -j^ V 3 17 In all of three cases, n0 stands for the refractive index of the unperturbed, isotropic crystal, and 3 M = ^ -r4lF, (1.5) where r 4 1 (negative constant) is the electro-optic coefficient of the crystal. However, because [11 l]-cut substrates and electric fields in [111] direction are not very common, the first two cases (Case I and Case II) are typically used for electro-optic modulators, resulting in three different modulation schemes: polarization modulation, intensity modulation (using, for example, Mach-Zehnder interferometer), and mode conversion. 1.3.2 Semiconductor Polarization Modulators A GaAs polarization modulator is shown in Fig. 1.5(a). Several micrometers of an A f G a ^ A s epitaxial layer are first grown on top of (001) GaAs. Typically, in order to provide the vertical confinement of light, during the growth of the ALGari .^As epitaxial layer, the A l mole fraction, x, is decreased in the region where the waveguide core is intended to be because the refractive index of the core needs to be higher than that of the cladding, and the refractive index of A L - G a ^ A s increases as x is decreased. When the top cladding layer is grown, x is increased to the same value it was for the bottom cladding layer. In order to provide the horizontal confinement of light, the AlGaAs is then etched to obtain a narrow, about 3 um wide, optical single-mode ridge waveguide incident light T M >• TE (a) (b) Figure 1.5: Semiconductor polarization modulator: (a) device schematics and (b) refractive index change directions (Case I). parallel to the [110] direction. Once the optical waveguide has been fabricated, electrodes, which are about 1 um thick and 100 um wide, are formed, and the sample is cleaved along (110) plane in order to get smooth input and output facets of the waveguide. As it is only the vertical component of the applied electrical field that is utilized for the polarization modulation scheme, one of the electrodes is evaporated directly above the optical waveguide in order to maximize the field "seen" by the light traveling through the waveguide. It is also a common practice to deposit about 0.2 um of an optical buffer layer on the substrate before the electrodes are formed. This decreases the optical loss in the waveguide due to the free electrons in the metal. 19 For these particular crystal and field orientations, the refractive index change wi l l be as shown in the Case I: Refractive index in the vertical direction (refractive index "seen" by the transverse-magnetic (TM) polarized light, i.e., the fundamental T M optical mode of the waveguide) wi l l not change upon an application of the electric field, and the refractive index in the horizontal direction (refractive index "seen" by the transverse-electric (TE) polarized light, i.e., the fundamental TE optical mode) wil l change by a small amount. The change of the refractive index along the direction of the waveguide, [110] direction, is unimportant because there is no appreciable longitudinal component of the optical field. Consequently, i f the incident light is polarized at 45° with respect to [001] direction, which can be achieved by passing unpolarized light through a polarizer rotated at the same 45° with respect to [001] direction [see Fig. 1.5(a)], or by aligning highly polarized laser light with the 45° direction, the light couples equal amounts of power into the TE and T M modes, which "see" two different effective refractive indices (t t^and n™, respectively). This is illustrated in Fig. 1.5(b). Here, it is important to note that a plane wave polarized, for example, in the [001] direction and propagating in bulk semiconductor crystal would "see" only the refractive index of the bulk crystal, i.e., n0. This is different from waveguide modes, such as the TM-polarized fundamental mode propagating in an optical waveguide, which wi l l "see" an average refractive index that depends on the waveguide material and geometry as well as surrounding materials, i.e., the effective refractive index for the fundamental T M mode(n™ ). 20 As the TE and T M modes "see" different effective refractive indices, they wi l l undergo different phase shifts after propagating the length of the waveguide (L ), as given in the following two equations: <PTM=--n™-L (1.6) *TEiF)---»#{F)-L, (1.7) c where n%(F)=rt%(F = 0)+An% (1.8) and (1.9) 2 Also, here, a> is the frequency, and c is the speed of the light. The magnitude of r 4 1 (1.4 12 x 10" m/V for wavelength near 1550 nm) is very nearly the same for GaAs and AlGaAs of the waveguide. The phase shift of the T M mode is just a passive phase shift, and it is not affected at all by the presence of the electrical field. The TE mode, however, in addition to the passive phase shift, experiences a phase shift proportional to the electric field. The phase difference between the two components at the output of the waveguide is often called "phase retardation." It can be easily shown that for a phase retardation of 180°, the polarization modulator can switch polarization of light between two orthogonal states using simple phase modulation of the fundamental T E mode [23]. 21 If the waveguide was designed in such a way that n]^ (F = fj) and n™ had the same magnitudes, and i f no voltage (electric field) was applied, the TE and T M modes would undergo the same passive phase shifts. There would be no phase difference between them at the output of the device, where they would combine to form light polarized at the same angle as the incident light. If a polarizer is placed at the output of the device (such polarizer is often called the analyzer) and rotated 90° relative to the polarization of the incident light, as shown in Fig. 1.5(a), no light wi l l go through the analyzer. However, i f enough voltage is applied {Vn), the phase difference between the two modes at the output of the waveguide wi l l be 180°, and this means that the polarization of the incoming light is rotated by 90°, so that all of the light wi l l pass through the analyzer. Therefore, simply by adding the analyzer to a polarization modulator, the device can be used to convert polarization modulation into intensity modulation. The transfer function of the intensity modulator is described by the following expression [23] 7 ° " ' =s in 2 ( U 0 ) 2 V I where Iin and Iout are intensity of light at the input and the output of the waveguide, respectively, V is the applied voltage, and Vn (half-wave voltage) is the voltage needed to induce 180° of phase retardation between the two modes. The transfer function, which has the same functional form for all intensity modulators based on linear electro-optic effect, is also plotted in Fig. 1.6. If nTejf (F = u) is not matched to n™, the transfer 22 function would have an extra phase term in the argument of the sinusoidal, which wil l move the function left or right on the horizontal axis. 1.5 2 vivn Figure 1.6: Intensity modulator transfer function. 2.5 If used as an intensity modulator for digital applications, this device would switch intensity of light from " o f f state (V = 0) to "on" state (V = Vn ). The transfer function given in Eq. (1.10) does not include the waveguide loss for the two modes. If the waveguide loss was included in Eq. (1.10), which is straightforward to do following the approach given in [23], it would be obvious that the waveguide loss is detrimental to the maximum optical power available from the device, and that differential loss between the two modes is detrimental to the extinctio ration of the device, which is defined as (L...\ Extinction Ratio = 101og10 (1.11) Also, due to the fact that the effective refractive index changes for the two modes are not of the same magnitude and opposite sign, there wil l be a net phase change during the 23 operation of the device, and it wi l l suffer from large frequency chirp. This is one of the main reasons why the polarization modulator is not widely used in digital communications. If used in analog intensity-modulation applications, the device would be biased at V = Vn 12, where the transfer function is most linear, and only small-signal modulation would be applied. Applications of polarization modulation, rather than intensity modulation, wi l l be discussed below following the summary of the other most common polarization and intensity modulators. 1.3.3 Semiconductor Mach-Zehnder Modulators A GaAs Mach-Zehnder intensity modulator is shown in Fig. 1.7. In this case, the incident TE light is split between two different waveguides, which in turn merge into a single output waveguide. The electrodes are designed and positioned in such a way that light traveling in one waveguide "sees" a vertical electric field pointing up, and the light in the other waveguide "sees" the vertical electric field pointing down. The orientation of the crystal is the same as for the polarization converter, so that the refractive index change is that of the Case I. G N D V G N D s Figure 1.7: Semiconductor Mach-Zehnder intensity modulator (Case I). 24 In the Mach-Zehnder case, however, only TE light is launched into the device, and only the refractive index change in the horizontal direction matters. The refractive index changes in the two waveguides wi l l have the same magnitude but opposite sign because of the opposite polarity of the field applied to each of the branches. If no field is applied to the device, the incident TE light splits into two equal-power components that undergo the same (only passive) phase shifts and add up in phase as the two branches merge into the single output waveguide. This is the "on" state of the modulator. When an electric field is applied to the device, the phase shift in one branch wi l l be -90° and in the other +90°, so that the total phase retardation wil l be 180°. The two components of light wi l l add up destructively upon merging into the single output waveguide, and the analyzer is 25 not needed. This is the " o f f state of the modulator. This type of a Mach-Zehnder intensity modulator is called a "push-pull" Mach-Zehnder modulator [10], and its transfer function has the same sinusoidal form as shown in Fig. 1.6. In theory, push-pull Mach-Zehnder modulators introduce no net phase change and do not suffer from the problem of chirp. In practice, it is impossible to fabricate the two branches of the modulator to have the same geometry and provide exactly the same magnitude of the refractive index change, which is required for "zero-chirp" operation. The problem of the two arms not being perfectly symmetric can be solved by driving the push-pull device differentially, i.e., applying different voltages to the two electrodes of the device, which allows for the chirp to be fine-tuned to the desired value. However, i f chirp is not an issue at all for a particular application, voltage can be applied to only one arm of the modulator and the intensity modulation would still be possible. When the two arms of the push-pull Mach-Zehnder modulator are not perfectly balanced, they wi l l also have different waveguide losses and optical mode shapes, preventing the light signals from the two branches to cancel completely in the " o f f state. Similarly, the splitting ratio of the input Y-splitter is never ideally 50:50. The imbalance between the two arms is the limiting factor for the extinction ratio in push-pull Mach-Zehnder modulators [10]. 1.3.4 Semiconductor Mode-Conversion Modulators 26 A n illustration of a mode converter is shown in Fig. 1.8 (a). As can be seen from the figure, the electric field is applied in the horizontal, [110], direction, and the refractive > TE Figure 1.8: Semiconductor mode-conversion modulator: (a) device schematics and (b) refractive index change directions (Case II). indices in [1 lV2] and [1l>/2] directions (oriented ±45° with respect to the horizontal) wi l l change by the same amount, although with opposite signs, corresponding to the Case II [see Fig. 1.8.(b)]. Horizontally polarized incident light (TE mode) can be thought of as being effectively split into the two equal-power "hybrid modes" components polarized along these two directions, as shown is Fig. 1.8(b). If no voltage is applied to the 27 electrodes, and i f the effective refractive indices are the same for T E and T M modes, these two components wi l l add up in phase at the output and combine into the TE mode again. If, however, enough voltage is applied to the electrodes (V K ) , one of the vectors wi l l undergo -90° phase shift and the other one will undergo +90° phase shift. The two components of light wi l l then recombine at the output to form a T M mode. By definition, optical waveguide modes are states of light that do not change, unless somehow perturbed, as they propagate down a longitudinally invariant waveguide. The ability of the device to transfer power from one fundamental mode of the waveguide into another (TE into T M , or vice versa) is the reason the device is called a mode converter, not simply a polarization converter. It is important to emphasize that complete mode conversion, which means that all power from one mode is transferred into the other, is possible only i f the unperturbed (no external field condition) effective refractive indices of the TE and the T M mode are the same. This condition is also called the zero modal birefringence condition. In order to fully understand this, one can use the coupled-mode theory, rather than the simplistic explanation given above [24]. The effective refractive indices are inherently similar in I I I -V semiconductor optical waveguides. However, in order to achieve the low modal birefringence required for mode conversion, the geometry of the optical waveguide needs to be carefully designed. [4]. 28 Similar to the polarization modulator, the 90° polarization rotation in a mode converter can be turned into intensity modulation simply by adding a cross-polarized analyzer. Its transfer function can be represented by Eq. (1.10) and illustrated by Fig. 1.6. Following the same logic as in the two previous modulation schemes, it is obvious that this is a push-pull device capable of zero-chirp operation. In comparison with the polarization modulator, the mode converter is twice as efficient because the material refractive index changes for two axes, not just one. Also, i f the two modes experience different amounts of loss in the waveguide, the extinction ratio could be severely affected. Fortunately, the problem could be easily solved by rotating the input polarizer in order to put more power into the mode that suffers greater loss, matching the powers of the two modes at the output. In Mach-Zehnder modulators, the differential optical loss due to the imbalance of the two arms cannot be compensated so easily after the device has been fabricated. 1.3.5 Lithium Niobate Modulators Modulators based on the linear electro-optic effect in Lithium Niobate (LiNbC^) represent the most mature external optical modulator technology used in optical telecommunications. In them, optical waveguides are most often fabricated by diffusing titanium (Ti) into the crystal [10]. After the optical waveguide has been fabricated, its facets are polished, not simply cleaved as in I I I -V semiconductor modulators. Also, a thin optical buffer layer is often deposited on top of the waveguide (particularly for z-cut 29 modulators introduced below) to decrease the optical loss due to the metallic electrodes. Unlike the isotropic I I I -V semiconductors, LiNb03 is an anisotropic, uniaxial crystal. Its material refractive index in one direction is different from those in the other two directions. This special direction is called the extraordinary optical axis, and the refractive index corresponding to it is called the extraordinary refractive index (ne). The other two axes are called the ordinary axes, and the refractive indices are called the ordinary refractive indices (n0). When the crystal is used for its electro-optic effect, its extraordinary axis is called z-axis, and the other two axes are called x-axis and y-axis. Unlike, for example, the I I I -V semiconductor GaAs crystal that has only one nonzero electro-optic coefficient (r 4 1 = -1 .4x 10"1 2 m/V), LiNb03 has 4 different nonzero electro-optic coefficients: r 3 3 = 30 .8x l0" 1 2 , r 1 3 = 8 .6xl0~ 1 2 , r22 = 3 .4x lO" 1 2 , and r 5 1 = 28.0 x 10"1 2 m/V [10]. The electro-optic effect in LiNbC>3 is typically used in three different modulation schemes, as shown in Fig. 1.9. Upon the application of an electric field in the z direction (Fz), z-cut and y-propagating (waveguide is parallel to the y-axis) LiNbC*3 crystals wi l l "see" the refractive index in the x direction change from n0 (-2.29) to 3 nx=n0-^-ruF:, (1.12) and the refractive index in the z direction change from ne (~2.2) to 33-* z J 30 (1.13) as shown in Fig. 1.9(a). Because, r 3 3 » r 1 3 , the phase of the vertical, T M , polarization is much more efficiently modulated than the phase of the horizontal, TE, polarization. This G N D V G N D V G N D I n j = V 1HH G N D y © — (a) (b) - * y 22J y (C) Figure 1.9: Lithium Niobate modulator: (a) z-cut, y-propagating device, (b) x-cut, y-propagating device, and (c) x-cut, z-propagating device. type of refractive index change provides for an efficient TM-polarization phase/frequency modulator, TM-polarization Mach-Zehnder modulator, as well as a polarization converter, because the differential phase change for the TE and T M modes is quite large. The fact that the largest electro-optic coefficient for LiNbCh, r 3 3 , is more than twenty times larger than r 4 l for GaAs is somewhat deceiving, as it may seem that the refractive index change achievable in LiNb03 is more than twenty times larger than the one achievable in GaAs. However, the refractive index change in LiNb03 is only 3-4 times larger because it also strongly depends on the material refractive index itself, and the 31 isotropic material refractive index for GaAs (-3.4) is considerably larger than either the ordinary or extraordinary index of LiNbCh. Fig. 1.9(b) illustrates x-cut, y-propagating LiNb03. The refractive indices in the transverse x and z directions change in the same way as for z-cut, y-propagating LiNb03, except that here the z-axis and Fz are in the horizontal direction, so that x-cut, y-propagating LiNbCh is most appropriately used for TE-polarization phase/frequency/intensity modulation, or again, in polarization modulators. Although in z-cut, y-propagating LiNbCh and x-cut, y-propagating LiNbCh efficient polarization modulation is possible, the polarization modulation is not push-pull. In addition, unlike the isotropic I I I -V semiconductors, y-propagating LiNbCh has large material birefringence (n0 » ne), which, in turn, results in large modal birefringence that cannot be simply overcome with any particular waveguide design. Consequently y-propagating LiNb03 polarization modulators suffer from large (10-15 ps) differential group delay (DGD) between the two modes, which is significantly larger than the D G D for GaAs mode converters (<0.5 fs). Such large D G D precludes y-propagating LiNbCh polarization modulators from being used in very important polarization-modulation applications [25]. 32 Fig. 1.9(c) shows another typical LiNb03 modulation scheme. Here, LiNb03 is x-cut and z-propagating, and the electric field (Fv) is applied in the horizontal, y-direction. Consequently, the field-dependent refractive index in the x-direction, 3 » ; = " 0 + y ^ o (1-14) and the field-dependent refractive index in the y-direction, '3 n y - n 0 - ^ - r 2 2 F y , (1.15) experience same-magnitude, opposite-sign changes from the unperturbed ordinary refractive index. The push-pull, low-modal-birefringence polarization modulation is ideal for certain long-haul applications [25], [26]. Unfortunately, r22 is quite a bit smaller than r 3 3 , and the drive voltages in z-propagating LiNb03 modulators are larger than those in y-propagating LiNb03 modulators. 1.3.6 Polymer Modulators r V ; Electro-optic polymers are an immature modulator technology [10], [27]. The polymers are synthetic organic materials obtained by embedding active electro-optic polymers, i.e., nonlinear optical chromophores, into a matrix of a standard polymer host, such as P M M A . It is very easy to form multilayer polymer stacks using spin-coating, which makes polymers very easy to integrate with various optoelectronic circuits. Unlike 33 metallic electrodes on semiconductor epitaxial layers, metallic electrodes can be easily sandwiched between polymer layers. In order to obtain a macroscopic electro-optic effect from the microscopic electro-optic effect that is characteristic of chromophore molecules, the molecules need to be aligned. This is achieved by heating (in the range between 100 and 200 °C) and cooling the polymer in a strong electric field (100-200 V/um). This process is called polling. The electrodes used to apply the polling field are often removed and replaced by more appropriate, typically faster, microwave electrodes. After polling, the isotropic electro-optic polymer becomes uniaxial. Its extraordinary axis is aligned with the direction of the field, and its ordinary axis is normal to the direction of the applied field. The extraordinary material refractive index is ne = n + 2 A , and the ordinary material refractive index is n0 = n - A , where n »-l .6 and A » 0.1. If the z axis is aligned with the extraordinary axis, and only electric field in z direction is applied, the refractive index change for electro-optic polymers wi l l be similar to the one for z-cut, y-propagating LiNb03, as shown in Fig. 1.9(a), and Eqs. (1.12) and (1.13). r 3 3 is about three times larger than r 1 3 , and it could be increased i f a greater polling field is applied. Unfortunately, the greater the polling field, the greater the material loss and birefringence. r 3 3 can vary from several pm/V to -100 pm/V. In electro-optic polymers, due to the low refractive index, r 3 3 needs to be -80 pm/V in order for the refractive index change of the polymer to be similar to that of L i N b 0 3 for the same electric-field change. 34 Fig. 1.10 illustrates a push-pull Mach-Zehnder polymer modulator. During the polling process, the electric field is applied in opposite directions in two neighboring top electrode optical waveguide core/cladding bottom electrode silica substrate its—-*-* r ,F GND Figure 1.10: Cross-section of push-pull polymer Mach-Zehnder modulator, regions containing the electro-optic polymer, providing for the push-pull effect. The two regions form the cores of the optical waveguides in the two branches of a Mach-Zehnder modulator. The opposite alignment of the molecules (white arrows) with respect to the applied electric field (black arrows) is indicated in the figure. The waveguide cores in the horizontal direction are defined by decreasing the refractive index of the surrounding material (cladding) using either dry etching or photobleaching techniques [10]. Light confinement in the vertical direction is achieved by spin-coating materials with lower refractive indices than that of the core, above and below the waveguide. 1.3.7 Comparison of Linear Electro-Optic Modulators The preceding sections presented only the basics of some of the best known linear electro-optic modulation technologies. Here, a more detailed performance comparison of 35 the technologies is given. One of the most important performance requirements for optical modulators is wide electro-optic bandwidth. For example, future W D M links wi l l require optical modulators to operate at data rates of 40 Gbps and higher. The optical modulators that wi l l be used in future W D M links wi l l , therefore, need to have electro-optic bandwidths greater than 30 GHz, depending on the spectral efficiency of the digital signal (ratio of the bit rate to the bandwidth), which depends on the modulation and coding schemes used in the links [21]. In addition, high-frequency operation must be achieved without sacrificing modulator performance, e.g., the R F drive voltage should be less than 3.5 V [10]. Most high-frequency modulators, particularly those intended for operation at 40 Gbps and higher data rates, employ some type of traveling-wave electrodes, rather than lumped electrodes [28]. A lump electrode structure is basically a capacitor, such as shown in Fig. 1.8, connected in parallel with an impedance-matching resistor (typically, a 50 Q resistor for broadband matching is used). In lumped electrode structures, the electrode length is short compared to the RF wavelength. The bandwidth of a lumped electrode structure is determined by the RC time constant, and the electro-optic bandwidth of the modulator is the smaller of the inverse of the R C time constant or the inverse of the time the light takes to propagate along the length of the capacitor, between its plates. The former is usually more limiting. In order to increase the electro-optic effect and lower the drive voltage, it is desirable to make the electrodes as long as possible, increasing the interaction between 36 the light and the modulating electric field. However, the capacitance increases with the electrode length, decreasing the bandwidth. As the linear electro-optic effect requires the electrodes of electro-optic modulators to be several centimeters long, traveling-wave electrodes need to be used. Even the short electroabsorption modulators (L < 1 mm), which wi l l be described in more detail in the following section, have shown better performance using traveling-wave electrodes [29]. In traveling-wave electro-optic modulators, the modulating signal is applied to one end of the electrodes, and the other end of the electrodes is terminated with a resistive load. The impedance of both the electrode and the load are matched to the impedance of the source, which is typically 50 Q, so the electrode can be considered to be an extension of the transmission line and its bandwidth is not limited by the electrode charging time. Here, the basic idea is that the modulating electric field launched at one end of the electrodes propagates down the traveling-wave electrode at the same speed as the light, which is launched in the same direction in the optical waveguide. If the two speeds are not matched, the light-electric-field interaction wi l l not integrate effectively as the two waves propagate. In semiconductor modulators, the electrical signal travels faster than the optical signal and is relatively easily slowed down by capacitive loading, as illustrated in Fig. 1.11 [4], [30], [31]. 37 incident electric signal Q incident light Figure 1.11: Top view of a mode converter with coplanar traveling-wave electrode with periodic T-shaped capacitive loads. shown in the following equation fida = — i — r. 0-16] where n^c and n"^ are the effective refractive index of the electrode structures and the effective refractive index of the optical mode, respectively, and c is the speed of light [4]. In LiNbCb modulators, unlike in I I I -V semiconductor modulators, the material refractive index for electromagnetic waves is significantly larger for the microwave/millimeterwave frequencies than it is for optical frequencies. Therefore, the velocity of the electrical signal in the traveling-wave electrode needs to be increased considerably for fine velocity matching, which is challenging to do without increasing the drive voltage [27]. The electro-optic 3 dB bandwidth due to the velocity mismatch between the modulating signal and the optical mode is inversely proportional to the electrode length (L ), as 38 The electro-optic bandwidth of modulators is also limited by the electrode microwave loss, which is usually more dominant than the velocity mismatch effect for very high frequencies. The electro-optic 3dB bandwidth due to electrode loss is inversely proportional to the square of the length of the electrodes, as shown below 2 54 A * = 7 ^ , 0 -17) (a0L) where a 0 is a constant dependent on the dimensions and the resistivity of the electrodes, and L is, as before, the length of the electrode [4]. In addition, a very important design constraint for high-performance electro-optic modulators is impedance matching. If the impedance of an electrode is not closely matched to the impedance of the source and/or the termination, the modulator wi l l suffer from large frequency-dependent reflection loss and resonance issues. The large electro-optic effect in LiNbCh modulators (again, the effect is 3-4 times larger than that in GaAs) does not necessarily provide for a device that is strikingly shorter and, thus, faster than other electro-optic modulators, because of constraints such as impedance matching, velocity matching, and low drive voltages, as mentioned above. To the contrary, LiNbC>3 Mach-Zehnder modulators typically have the largest footprints of all the available modulator technologies, which is perceived as one of their major 39 disadvantages [10]. Although a LiNb03 Mach-Zehnder modulator with bandwidth greater than 75 GHz , having 2 cm long electrode, has been demonstrated, the drive voltage for the device was large (Vn = 5.1 V ) [32]. The length of the electrode had to be extended to 3 cm and its bandwidth consequently lowered to 30 GHz, in order for the modulator, with otherwise the same design, to have a drive voltage of only 3.5 V . Besides their large footprint, bias instability in LiNbO"3 Mach-Zehnder modulators is also perceived as one of their major disadvantages. A sudden change of the bias voltage can trigger a slow bias drift with a time constant of several hours to several days. The bias drift is due to slowly moving charge that is either trapped in LiNb03 and SiC>2 buffer layers, or is trapped at their surfaces and interfaces. The charge can be process related or it can be generated during the operation of the device by thermal or optical excitation. To minimize the drift problem, LiNbCh surfaces should be treated, and, also, the conductivities of the layers should be matched [10]. Nevertheless, due to large investments in LiNb03, it is still considered the benchmark technology for the linear electro-optic modulators [10]. LiNbCh optical waveguides exhibit low loss (-0.2 dB/cm), they can handle the largest optical powers (10s to 100s of mW), and they can be designed for very efficient coupling with the standard single-mode fibers (5 dB fiber-to-fiber insertion loss) [27]. In contrast with LiNbCh modulators, polymer modulators are a very immature technology, which is not yet very popular in the marketplace [10]. The technology has a 40 major advantage of being inexpensive and easy to integrate with various devices. The small refractive indices of polymers (~1.6) make them well matched to optical fibers (refractive index of silica is -1.46). Also, the small refractive index and its small change from optical frequencies to microwave frequencies, provides for easy fabrication of 50 Q traveling-wave electrodes, and excellent microwave-lightwave velocity match. Polymer Mach-Zehnder modulators with bandwidths of over 40 GHz, and drive voltages of about 10 V , have been reported [27]. The optical loss in polymer waveguides is typically -3 dB/cm and the total, fiber-to-fiber loss is about 10-12 dB. The intense research in the field is continuously improving the properties of electro-optic polymers. One of the major challenges that the technology has to overcome is the instability of the electro-optic chromophores. If the optical power is large enough (-10 mW), the chromophores become excited and misaligned, diminishing the macroscopic electro-optic effect. For example, a 30 % decrease in the electro-optic efficiency has been estimated after a 104 h operation for a device with a 10 um 2 optical waveguide cross-sectional area and 10 mW of light power at 1320 nm [27]. The misalignment of the chromophores can also be induced by high-electric-field or thermal excitation. Further improvements to the stability of electro-optic polymers and to room-temperature deposition techniques may make electro-optic polymer modulators into a very desirable technology. GaAs Mach-Zehnder electro-optic modulators have benefited immensely from the growth and fabrication processes that have been developed for telecommunication lasers 41 and photo-detectors. Being a GaAs technology, they can, potentially, be monolithically integrated with lasers, photodetectors, and other GaAs-based devices. Many devices can be obtained from a 6" GaAs wafer, and their waveguide facets are easily cleaved (LiNb03 waveguides must be polished). Their typical chip length of ~3 cm is smaller than that of LiNbCb Mach-Zehnder devices, which is ~5 cm [10]. They can also handle large optical powers and their optical waveguide loss is ~ ldB/cm, and the total, fiber-to-fiber loss is about 10-15 dB [27]. The GaAs dielectric properties provide for good microwave electrodes, and 40 Gbps devices with drive voltages of about 5 V are commercially available [10]. GaAs polarization converters also benefit from a mature GaAs growth and fabrication process, inheriting many properties that are similar to those of GaAs Mach-Zehnder modulators. Versawave Technology Inc. (formerly J G K B Photonics Inc.) has been a leader in developing ultrahigh-speed mode-conversion modulators [33]. Although still a young technology, Versawave's mode converters exhibit remarkable properties when packaged as intensity polarization modulators as well as polarization modulators. Versawave's mode converter, being a push-pull device, inherently has low chirp (±0.1) when packaged as an intensity modulator [34]. It also has the benefit that the chirp can be adjusted simply by rotating the input polarizer and/or the analyzer. Its package size (2-2.5 inches long) is the shortest of all the available linear electro-optic modulators [33]. Compared to GaAs Mach-Zehnder modulators, the mode converter is single-optical-waveguide device and does not suffer from additional optical losses due to the Y -splitters. Fiber-to-fiber optical loss is less than 7 dB. Having a bandwidth of ~33 G H z and drive voltage of 3.5-4 V , the device has a great potential for future 40 Gbps optical networks [33], [34]. When packaged as a polarization modulator, a mode converter has very similar properties to the intensity modulator [35]. Due to its low birefringence, it has very low differential group delay (10s of 10"15s), which is much smaller than that for y-propagating LiNb03 polarization modulators [25], as mentioned in the section on LiNbCb modulators. Ever since it was introduced, Versawave's polarization modulator has attracted considerable attention for being a fast device, with many desirable properties, which enable an ever-increasing number of polarization-modulation applications to be used at ultrahigh speeds [5]. As explained in [5], Versawave's polarization modulator was recently used at Bel l Laboratories for a 10"2 reduction (i.e., 100 times improvement) in uncorrected bit error rate of a W D M transmission using in a novel polarization-alternating scheme. Here, the polarization modulator was used to switch the polarization of alternating pulses in a return-to-zero differential-phase-shift-keyed (RZ-DPSK) data stream in order to reduce nonlinear degradation in a communication system, providing for a record-breaking reach. The same modulator was also used in an experiment at the Naval Research Laboratory, where, in an analog optical link, polarization modulation provided a 35 dB improvement in inter-channel crosstalk compared to amplitude modulation. Also, it has been shown 43 that high speed polarization modulators can be used for scrambling the polarization of a forward pumped Raman amplifier in order to reduce the effect of the pump's polarization dependent gain on the data. In conclusion, the outstanding performance of the Versawave's mode converter as an intensity modulator, as well as a polarization modulator, that can be used in an ever-increasing number of applications is very inspiring. If a similar but shorter device could be made, it would enable these important modulation schemes to be used in still faster, future optical networks. For it to be significantly shorter, the device must be based on an effect that is significantly greater than the linear electro-optic effect used in the devices mentioned above. Various quantum well structures have been studied over the years in order to realize large electrorefraction as well as electroabsorption effects in semiconductors, both of which are essential for future electro-optic modulators and related electro-optic devices. The main focus of the thesis is improving the-state-of-the-art in polarization and intensity modulators. In the thesis, coupled quantum well structures, rather than the more common square quantum well structures, are recognized as the most promising technology to achieve this goal in the foreseeable future. The rest of the Introduction chapter presents insights into the optical processes in square and coupled quantum well structures and their application to electro-optic modulators. 1.4 Multiple Quantum Well Modulators 44 Multiple quantum well (MQW) modulators are based on an array of identical, uncoupled square quantum wells (SQWs). Typically, 10-20 repetitions of a SQW are placed in the optical waveguide of an M Q W modulator in order to enhance the interaction with the optical mode [36], [37]. However, as the SQWs are uncoupled, it is sufficient to focus on the optical processes for only one SQW in order to understand the basic operation of the M Q W modulator. Fig. 1.12 illustrates a short-period M Q W structure, with only two quantum wells, that has Ino.53Gao.47As wells and Ino.52Alo.48As barriers. In the left-hand side quantum well, the lowest energy electron wave function (WEI) and the second lowest energy electron wave function Q¥E2) are shown in the conduction band and have " L " (for left-hand side) superscripts. Also, the two lowest energy heavy-hole wave functions OPHHI and WHH2) and the two lowest energy light-hole wave functions OPLHI and WLm) are shown in the valence band and have the " L " superscripts. The corresponding wave functions in the right-hand side well have " R " superscripts. The wave functions in Fig. 1.12 are obtained separately for each well, without a possibility of coupling with the corresponding wave functions in the other well, i.e., assuming that the two wells are infinitely spaced. When the SQWs are brought together, as long as they remain sufficiently far apart, the coupling between the wave functions, i.e., their overlap integrals, wi l l remain small. The overlap integral for two 1-dimensional 45 T E 2 T E 2 ^ E 1 L ^ T E 1 Figure 1.12: A n array of two uncoupled SQWs, where, in each well, the two lowest energy electron wave functions (WEI and WE2), heavy-hole wave functions (WHHI and W H H 2 ) , and light-hole wave functions ( W L H I and WLH2) are shown. The wave functions in the left-hand side well (those with superscript " L " ) are obtained separately from those in the right-hand side well (those with superscript "R"), i.e., without a possibility of coupling. The quantum wells are Ino .53Gao.47As, and the barriers are Ino.52Alo.48As. wave functions fA (z) and fB (z) is defined as lAB=ffA(z)fB(z)dz- (1 .18) -00 As SQWs are brought together, the high energy wave functions start to couple before the low energy wave functions do. It is clearly shown in Fig. 1.12 that * P L H 2 L and W L H 2 R exhibit the deepest penetrations into the middle barrier and, thus, wi l l have the 46 highest degree of coupling. Consequently, the energies and distributions of the two wave functions wi l l change. However, it is often necessary to keep the barriers as thin as possible in order to increase the number of SQWs in the optical waveguide and increase their interaction with the optical mode. Fortunately, as explained below, it is the lowest energy wave functions (WEI, ^ H H I , and JPLHI), which are more strongly confined to the quantum well layers than the respective second lowest energy wave functions Q^E2, WHH2, and ^ua), that dominate the optical processes in M Q W modulators, and the barriers are typically as thin as 50 to 100 A . The characteristic thicknesses for the quantum wells are also typically smaller than 100 A , depending on the specific application and the M Q W modulator design. Also, the choice of material for the quantum wells depends on the specific application and intended wavelength of operation. 1.4.1 Electroabsorption in SQWs Optical absorption in semiconductor structures of various dimensions (e.g., ID, 2D, or 3D) strongly depends on the density of states in these structures. This is also true for quantum well structures, which have dimensionality between 2D and 3D [38]. In order for light to excite an electron from a valence-band state into a conduction band state, and become absorbed in the process, it needs to have the energy equal to the energy difference between these two states, assuming that the final state is not already occupied. The energy of light (E) is inversely proportional to its wavelength in vacuum (Xa), i.e., 47 E = hx clXo, where h is Planck's constant and c is speed of light in vacuum. The two electron states need to be at the same point in space, i.e., they need to have a significant spatial overlap, and, because light adds negligible momentum to the system, they must have the same value for the wave vector k. Such electron transitions are called direct interband transitions. In addition, the ability of light to induce the direct interband transition between these two states depends on its polarization and on the symmetry of the unit cells corresponding to the states [39] , which is referred to as the "polarization selection rule." Fig. 1.13 illustrates how the lowest energy electron wave function OPEI) and either of the lowest energy hole wave functions QVm\\ or WLHI) in a SQW react to an electric field applied in the direction normal to the plane of the quantum well [40]. In order to achieve the electroabsorption and the electrorefraction effects that are used in optical modulators, the electric field must be applied in the direction normal to the planes of the quantum well structures regardless whether they are uncoupled SQWs, coupled quantum wells, or some other type of quantum well structure. The effective mass of the heavy holes is much larger than the effective mass of the light holes, and, consequently, the heavy-hole wave functions and the light-hole wave functions have different energies in a quantum well, they experience different degrees of confinement by a quantum well, and they exhibit different degrees of sensitivity to an applied electric field. However, the nature of their 48 response to an electric field is the same, and the hole wave function in Fig. 1.13 can be used to represent either W H H I or W L H i . Figure 1.13: Effect of the electric field (a) on the lowest energy electron wave function, W E I , and on either of the lowest energy hole wave functions, W L H I or W H m : (a) zero-electric-field case, (b) nonzero-electric-field case [40]. © 1988 IEEE As shown in Fig. 1.13, upon the application of an electric field, the electron and the hole wave functions wil l be pulled to opposite sides of the quantum well, where they "see" the lowest potential energies. Their movement to the opposite sides of the quantum well is not surprising considering that electrons and holes have charges of opposite signs. The field, therefore, reduces the overlap integral [Eq. (1.18)] between these two states. In general, when no electric field is applied to a symmetric quantum well, such as those in Figs. 1.12 and 1.13(a), only those electron and hole wave functions corresponding to VALENCE BAND CONDUCTION BAND \ (a) (b) 49 states that have the same quantum number wi l l have appreciable overlaps (WEI and WHHI, W E I and W L H i , W E 2 and W H H 2 , W E 2 and W L H 2 , W E 3 and W H H 3 , W E 3 and W L H 3 , etc.) [41]. Light-induced transitions between these states are called "allowed" transitions. As the electric field is increased, the nominally "allowed" transitions become less possible and the nominally "forbidden" transitions (WEI and W H H 2 , W E I and W L H 2 , etc.) start to dominate. Such behaviour can be easily proved mathematically, using simple geometrical arguments and the orthogonality of wave functions [41]. Fig. 1.13 illustrates another important effect that the electric field has on these states. As shown in the figure, the electric field reduces the energy separation (E a) between these two states. Conversely, the field increases the corresponding wavelength separation between these two states, and this phenomenon is, therefore, commonly called "red shifting." Similar to [10], Fig. 1.14 shows a rough sketch of zero-electric-field optical absorption in SQWs corresponding to electron-light-hole transitions. In the literature, optical absorption is represented by the absorption coefficient a (1/cm), which determines the fraction of photons absorbed over a given distance [39]. As mentioned above, the optical absorption is directly related to the density of states. The two steps in Fig. 1.14 are related to the joint density of states [38] for electrons and light holes, assuming that there is no coulombic interaction between them, and assuming that the SQW confines only the two 5 0 k c u 'o m , o c _o & 1 o < I 1—I Optical Wavelength Figure 1.14: Schematic representation of S Q W optical absorption for electron-light-hole transitions. lowest energy electron and the two lowest energy light-hole wave functions. The lower step starts at the wavelength corresponding to the energy separation between WEI and W L H I , and its height is proportional to the overlap integral for these two wave functions. Similarly, the second step starts at the wavelength corresponding to the energy separation between and WLH2, and, also, its height is proportional to the overlap integral for these two wave functions. As explained above, when an electric field is applied to the SQW, the edges of these steps wil l shift on the horizontal axis and their heights wi l l start diminishing, while the heights of the nominally "forbidden" transitions wi l l start appearing. In order to fully describe optical absorption in SQWs and explain the origin of the two vertical lines in Fig 1.14, the electron and the light-hole states cannot be treated as independent, and the coulombic attraction between them must be included. As light excites an electron from a state in the valence band into an available state in the 51 conduction band, an oppositely charged light-hole (or heavy hole) is created in the valence band. The coulombic attraction forces the electron to spend more time in the vicinity of the light hole, and the two-particle system resembles that of the hydrogen atom. The more time the electron spends in the vicinity of the light-hole, the greater the probability of finding the electron and hole in the same unit cell and, thus, the greater the strength of the optical absorption [38]. This two-particle system is called an exciton. The vertical line sitting at a higher wavelength corresponds to the lowest energy state of the exciton associated with the W E I - W L H I transition (E1-LH1 exciton), while the other line corresponds to the lowest energy state of the exciton associated with the W E 2 — W L H 2 transition (E2-LH2 exciton). These states are similar to the lowest energy, radially symmetric 1S states of the 2-dimensional hydrogen atom. In this thesis, for the purpose of modeling electroabsorption and electrorefraction, the relative motion of an electron around a hole in the plane of the quantum well wi l l be restricted to radially symmetric IS orbitals [42]. The energy separation of the 2D IS exciton state from the edge of the corresponding free-particle density of states step is called the exciton binding energy (EB) [42]. In the literature, EB is given as a negative energy measured relative to the free-particle density of states step, i.e., the energy separation between the electron and hole states when the coulombic coupling is ignored. This is analogous to the 2D hydrogen atom, where the energies of the bound states are negative and those of the unbound states are positive (the same is true for the 3D hydrogen atom). Also, similar to the 2D 52 hydrogen atom, there are an infinite number of bound exciton states, where the energy separation between the states becomes progressively smaller for high-energy states. However, these states make smaller contributions to the electroabsorption and electrorefraction effects and are commonly neglected in the numerical modeling of these effects [38], [42], again, as is done in this thesis. It is also worth mentioning that the coulombic force enhances the free-particle density of states in the vicinity of the step edge, increasing the absorption coefficient for the corresponding energies. This is also analogous to the unbound positive-energy hydrogen atom states. Therefore, in the following discussions, these states wi l l be referred to as exciton continuum states. It has already been mentioned above that the electric field pulls electrons and holes in opposite directions and decreases the absorption due to the continuum. This is also true for the excitonic absorption. As an electric field applied in the direction normal to the quantum well separates an electron and a hole in the same direction, the separation between the two particles also increases in the two directions parallel to the plane of the quantum well. The in-plane increase of the exciton occurs because it is energetically favorable for the two-particle system to be nearly spherical. Stretching an exciton into a "pancake" would imply mixing in some highly non-spherical orbitals that have high energies. Conversely, squeezing the exciton by narrowing the quantum well, for example, makes it smaller in all three dimensions. Therefore, the electric field wi l l separate the electron and hole in an exciton and decrease the time they spend in the vicinity of one 53 another, dramatically decreasing their probability of being in the same unit cell and, in turn, decreasing the excitonic absorption coefficient. As the in-plane motion of an electron around a hole is assumed to be restricted to the radially symmetric 1S orbital, the application of an electric field implies an increase of the orbital's radius and a decrease of EB, which is small compared to the "red shift" of the absorption edge. It is interesting to note that excitons also contribute to the absorption in bulk, 3D semiconductors, however, the contribution is much smaller because the exciton in the bulk material is considerably larger. Also, the potential confinement due to the quantum well prevents the quantum well excitons from being easily destroyed by the electric field, and the excitonic absorption features can be observed at electric fields higher than 100 kV/cm, which is a very useful effect for quantum well optical modulators. Fig. 1.15 shows the effect that the electric field has on the absorption spectra for a M Q W modulator containing only two 94 A wide GaAs-AlGaAs SQWs for the polarization of light that is parallel to the planes of the quantum wells (TE polarization) and for the polarization of light that is normal to the planes of the quantum wells ( T M polarization) [43]. According to the polarization selection rules for interband transitions in I I I -V semiconductors [39], only the light holes contribute to the absorption of T M polarization (i.e., there is no heavy-hole contribution), and the basic features of the absorption spectra in Fig. 1.15(b) are similar to those of the absorption spectrum in Fig. 1.14. However, Fig. 1.15(b) shows realistic, experimental absorption spectra where the contributions of the discrete IS exciton state and the continuum exciton states are not separate functions (delta function and step-like function, respectively), but rather, they form smooth, continuous curves that arise due to several broadening mechanisms. 1-42 146 1.5 PHOTON ENERGY (eV) Figure 1.15: Experimental absorption spectra of a waveguide modulator containing only 2 G a A s - A l G a A s SQWs (94 A wide wells) for: (a) TE polarization and (b) T M polarization. The curves (i)-(v) correspond to electric fields of 16, 100, 130, 180, and 220 kV/cm [43]. As wi l l be explained below, the lowest energy transitions dominate in the absorption processes for M Q W s , and only the exciton and the beginning of the continuum states 55 corresponding to the lowest energy W E I - W L H I transition are shown in Fig. 1.15(b). The linewidth function for the broadening evident in the figure is usually well described by a Lorentzian function (for homogeneous broadening mechanisms), a Gaussian function (for inhomogeneous broadening mechanisms), or a convolution of the two (for simultaneous homogeneous and inhomogeneous broadening mechanisms). The energy width of an absorption peak reflects the resonant state of the corresponding exciton. By the uncertainty principle, the longer the lifetime of the exciton, the better defined the energy, and consequently, the corresponding absorption peak wi l l be sharper. The absorption peak wil l be broadened (or the exciton lifetime wil l be shortened) by the following five major mechanisms. First, optical phonon scattering efficiently separates an electron from the corresponding hole (homogeneous broadening). Second, the external electric field wi l l pull the electron and hole in opposite directions (homogeneous broadening). This effect is clearly shown in Fig. 1.15 (b). Third, in M Q W modulators, any variations in the sizes of the quantum wells wi l l contribute to the total absorption peak broadening, as each distinct quantum well wi l l have its absorption peak at a distinct energy (inhomogeneous broadening). Fourth, i f the electric-field varies across the region containing the quantum wells, each quantum well wi l l have a slightly different response (inhomogeneous broadening). Fifth, particles have finite probabilities of tunnelling out of quantum wells, which means that many states close in energy to each bound state are possible (homogeneous broadening). 56 Considering the points made above, it is simple to understand the basic principle behind electroabsorption modulators. The wavelength of operation is chosen so that the corresponding energy is -20-30 meV below the lowest energy exciton, and the electric field F is switched from a low value of F~ 0 kV/cm to F~ 100 kV/cm. Comparing the curve (i) in Fig. 1.15(b) to, for example, the curve (iv), it is clear that the absorption of TM-polarized light increases from a small value (the "on" state of the electroabsorption modulator) to a much larger value (the " o f f state of the electroabsorption modulator). Large "red shifting" of the lowest energy exciton and its persistence for large fields needed to achieve the "red shift," provide for large extinction ratios in electroabsorption modulators. This phenomenon is called the Quantum-Confined Stark Effect (QCSE) [44]. Most common electroabsorption modulators, however, modulate the T E light. The polarization selection rules [39], state that both the heavy holes and the light holes participate in electroabsorption for the TE polarization, where the contribution of the heavy holes dominates. Fig. 1.15(a) illustrates electroabsorption for TE polarization, which is very similar to that for T M polarization except that the lowest energy exciton peak corresponds to the W E I - W H H I transition and the second lowest peak corresponds to the W E I - V L H I transition. The light holes have smaller effective masses than the heavy holes, so that the W E I - ^ L H I energy separation is greater than the ^ E I - ^ H H I energy separation. 1.4.2 Electrorefraction in SQWs 57 Beside the possibility of quite efficient modulation of the optical absorption in SQWs, the refractive indices for the T E and T M polarizations in SQWs also depend on the applied electric field and can be efficiently modulated. The real and imaginary parts of a linear, passive system's response function are related, and, therefore, the same is also true for the complex refractive index response function, n + j k. The absorption coefficient is related to the imaginary part of the complex refractive index by and where a> and c are the radian frequency and speed of light, respectively. The radian frequency of light is of course related to its energy by a simple formula E = hco, whereh = hi 2JI is the reduced Planck's constant. The refractive index at frequencya> is a weighted integral of absorption coefficients for all frequencies except for a;, and it can be obtained from the Kramers-Krbnig relation [44] 2a> , a = —k , c (1.19) n (a, ) - l -£/ f«W n J Q2 -(1.20) Here, the symbol P stands for the Cauchy principal value of the integral [10] (1.21) 58 The Kramers-Kronig relation thus states that electroabsorption is always accompanied by electrorefraction. However, using the Kramers-Kronig relation in the form given in Eq. (1.20) is impractical because the equation requires a knowledge of the absorption coefficients for all frequencies of light. Fortunately, the spectrum of the electric-field-induced change of the absorption coefficient in SQWs is well localized around the frequencies corresponding to the lowest energy transitions. Consequently, it is possible to calculate the magnitude of the electrorefraction close to the lowest energy transitions using the following equation [44] whereo)l ando;2 are the lower and upper frequency limits confining the localized change of the absorption coefficient. Fig. 1.16 illustrates the application of the Kramers-Kronig relation, given in Eq. (1.22), for calculating the refractive index change spectra (An) from the simulated absorption coefficient change spectra (Aa ) in a G a A s - A l G a A s S Q W for T E polarization around 850 nm [45]. Here, the electric field is switched from 0 to 25 kV/cm, and the figure clearly shows a positive change in A a for wavelengths longer than the wavelength corresponding to the W E I - ^ H H I transition (labeled as ehhl 1 in the figure). The electroabsorption is similar to that in Fig. 1.15 (a). Fig. 1.16 shows that An reaches a maximum or a minimum close to the wavelength where A a changes sign, and vice versa. (1.22) 0.05 0 -0.05 f-T—j—x j « j 1 , , r F» 0—25 fcV/cm Alo.3Goo.7As "lGoAsf lOnm e h h l 2 eShM e l i h l l fc_x L ( b ) 1 820 840 860 880 WAVELENGTH (nm) 10000 5000 0 •5000 -10000 E o Figure 1.16: Simulated TE absorption change ( A a ) and refractive index change ( Aw) in a G a A s - A l G a A s SQW for a field change from 0 to 25kV/cm [45]. This behavior is expected because A a and A M are actually a Hilbert-transform pair [46]. For the wavelengths just above the W E I - W H H I transition, it is clear that there is a nonzero change in the refractive index. The refractive index change is quite large, and it causes electroabsorption modulators to have significant chirp [10]. Although not clearly visible in Fig. 1.16, A a decreases exponentially for energies below the lowest energy exciton peak, changing much more rapidly than An, which exhibits approximately a quadratic decrease for these energies [44]. Therefore, i f the amount of detuning from the exciton peak is ~60-80 meV, rather than -20-30 meV that is typical for electroabsorption modulators, An wi l l still be appreciable although A a anda wil l be small. This is the basic idea behind the use of SQWs for electrorefraction modulators. 60 1.4.3 Electroabsorption vs. Electrorefraction M Q W Modulators For electrorefraction (ER) M Q W modulators, SQWs are commonly placed in the waveguide of a Mach-Zehnder interferometer in order to convert the refractive index modulation into intensity modulation. This is different from electroabsorption (EA) modulators, where the intensity of light is modulated in a single, straight waveguide by simply modulating the absorption coefficient in the M Q W region. TE light is typically modulated in both E R and E A modulators. Also, both E R and E A modulators are typically implemented in a p-i-n diode configuration, as shown in Fig. 1.17. The p-i-n diode configuration is used for providing an electric field that is Mach-Zehnder (top view) single branch (3D view) Figure 1.17: InGaAs-InAlAs p-i-n diode Mach-Zehnder modulator. 61 normal to the planes of quantum wells, and for increasing the overlap between the electric field and the thin (<0.5 um) M Q W region. The average refractive index of the M Q W region is higher than that of the surrounding semiconductor for either polarization and provides for confinement of light in the vertical direction. However, although it can be quite short, the p-i-n diode structure is not inherently fast, and there has been considerable research dedicated to designing fast M Q W modulators to fully realize their potential for high-frequency operation [47], [48]. One of the most attractive properties of E A M Q W modulators is their small size. They are typically 80-300 um long [10]. Such a small size provides for high yields per wafer, and, therefore, a low cost of fabrication. Similar to E R M Q W modulators, the choice of the substrate and the material for the quantum well structures greatly affects the wavelength of operation of these devices. The large band gap in GaAs constrains GaAs / Al x Gai_ x As quantum well structures on GaAs substrates to be used at wavelengths close to 850 nm. Quantum well structures on InP substrates are used at telecommunication wavelengths. There are two main material technologies grown on InP. These are the Ini. x iGa x iASyiPi.yi / In1.x2Gax2ASy2P1.y2 and I n X | G a y i A l i . x i . y i A s / Inx2Gay2Ah.x2.y2As material systems. The first material technology typically consists of Ino.53Gao.47As (x l = 0.47 and y l = 1) quantum wells lattice-matched to InP (x2 = 0 and y2 = 0) barriers and InP substrate. The second material technology typically consists of Ino.53Gao.47As (x l = 0.53 and y l = 0.47) quantum wells lattice-matched to Ino.52Alo.48As (x2 = 0.52 and 62 y2 = 0) barriers and, also, to the InP substrate. The composition of the material used for the quantum wells and barriers is chosen depending on the desired potential energies for these layers, which, in turn, determine the energies of the bound states and the wavelength of operation. As wi l l be shown in more detail below, the compositions of the epitaxial layers can also be chosen to introduce strain in the quantum well structures and to manipulate the relative magnitude of the T E and T M polarization responses. Tensile strain has been used in many E A modulators in order to make them polarization-insensitive [10]. The In x iGa y iAl i_ x i_ y iAs / I n X 2 G a y 2 A l i . X 2 - y 2 A s material system has proven to be very beneficial for E A modulators because it has a very small valence band discontinuity ratio. The valence band discontinuity ratio is defined as the ratio of the band-edge discontinuity for the valence band (AEV) to the band-edge discontinuity for the conduction band (AE C), i.e., AEVIAEC. For the In x iGa y iAl i_ x i_ y iAs / I n x 2 G a y 2 A l i . X 2 - y 2 A s material system, A£ v / AEC ~ 30/70 %, which is considerably smaller than the valence band discontinuity.ratio for the Ini_ x iGa x iAs y iPi_ y i / In i_ X 2Ga X 2As y 2Pi- y 2 material system (A£ v / AEC ~ 60/40 %) [49], [50]. Consequently, due to the small valence band discontinuity ratio, the quantum wells in the valence band are quite shallow and give the heavy holes a good chance of tunneling out of the M Q W structure. Due to their large effective masses, the heavy holes tend to be very strongly confined to the quantum wells, and their accumulation due to high input optical powers screens the applied electric field and saturates the electroabsorption effect. Typically, the maximum input optical power 63 that can be launched into the waveguide of a M Q W E A modulator is between 10 and 20 mW [10]. The large absorption in the " o f f state of E A modulators also creates considerable current, which is detrimental to the operation of the high-frequency electrodes. Still, 40 Gbps devices with drive voltages lower than 3 V have been demonstrated [27]. O f all the modulator technologies, E A modulators for 40 Gbps links have the lowest drive voltages. However, for long-haul applications, E A modulators are not the strongest candidates because they have the largest chirp. The chirp parameter, aw, for E A modulators can be greater than 2 [10]. Although, the E A modulators have large inherent optical losses in the "on" state (15-20 dB/mm) [27], they are very short devices and their total fiber-to-fiber loss can be made lower than 10 dB (typically between 6 and 8 dB) using additional optics to match the small, highly elliptical mode of the E A modulator to the large, round mode of standard single-mode fiber [51]. On the other hand, the large inherent optical loss prevents the length of the E A modulator from being significantly increased for an efficient use of traveling-wave electrodes, and E A modulators have largely been used in lumped-electrode configurations. Also, for the traveling-wave electrode configurations, E A modulators tend to have low impedances (20-30 Q). O f course, being very short, they do not suffer from the microwave-lightwave velocity mismatch. The footprint and cost of packaged E A modulators can be further reduced by integrating them with D F B lasers [10]. Lastly, as they are operated very close to the semiconductor band edge, E A 64 modulators tend to be very sensitive to the temperature and wavelength of operation. Depending on the specific design and optimization of the E A modulator, its properties can change significantly across the C-band [27], [51], [52]. This is unlike the mode converter, as well as other linear electro-optic modulators, which operate very far from the band edge and have flat wavelength responses over the combined C and L bands [35]. As mentioned above, the wavelength of operation for E R M Q W modulators is more detuned from the lowest energy exciton than that for E A M Q W modulators. Consequently, they have smaller optical loss in the active region than E A modulators. However, because they require Y-splitters, the E R Mach-Zehnder modulators have considerably larger footprints and tend to have similar or even greater fiber-to-fiber optical loss than E A modulators. For example, the active region of a 10 Gbps lumped-electrode Mach-Zehnder modulator (InGaAsP wells and InP barriers) in [53] is only 600 um long, whereas its total length including the Y-splitters and tapered-waveguide spot-size converters is 3 mm. The fiber-to-fiber loss for this device is ~6 dB. Having smaller absorption in the active region than the E A M Q W modulators, E R M Q W modulators have larger optical saturation powers. Similar to other Mach-Zehnder modulators, E R M Q W modulators can be driven differentially and have small and adjustable chirp. Also, the imbalance between the two arms typically restricts their extinction ratio to about 15 dB (for a drive voltage of ~4V) [53], whereas -20 dB extinction ratio is not typical for E A modulators [27]. As E R modulators absorb much less optical power than E A 65 modulators, they exhibit lower temperature sensitivity due to heating effects. The optically generated currents, therefore, interfere less with the operation of fast electrodes [27]. Although the development of high-frequency electrodes for EA and ER modulators is an active area of research, modern ER MQW modulators are mostly lumped-electrode p-i-n devices limited to 10 Gbps applications [10]. Recently, in an on-wafer experiment, an ER MQW modulator having travelling-wave p-i-n electrodes and a modulation bandwidth of 28 GHz has been demonstrated [54]. The drive voltage of this device was about 3 V, and its extinction ratio and fiber-to-fiber loss were >14 dB and ~9 dB, respectively. Also, recent modelling of ER MQW modulators having travelling-wave p-i-n electrodes has shown that modulation bandwidths exceeding 40 GHz could be achieved, although the drive voltages would be larger than 4 V and additional optical loss due to the electrode configuration would be inevitable [48]. The p-doped region in p-i-n devices introduces significant optical loss and microwave loss. The microwave loss is the main bandwidth-limiting factor of these p-i-n devices. The bandwidth could be improved with better electrode designs, which require larger thicknesses of the intrinsic regions containing the quantum wells and, therefore, provide lower electric fields to the quantum wells. These devices are likely to have Schottky-contact coplanar or microstrip electrodes [55], [56], [57]. Therefore, improving the ER effect of quantum well structures is an important requirement for future ER modulators. ER modulators based on coupled quantum wells, 66 rather than SQWs, are capable of providing this important benefit. The remainder of this thesis is dedicated to provide a background for the operation of coupled quantum well modulators (Section 1.5) and to provide the convincing arguments that the coupled quantum well modulators developed in this thesis wi l l play an important role in future optical links (Chapters 2-7). 1.5 Coupled Quantum Well Structures Coupled quantum well (CQW) structures consist of multiple quantum wells that are separated by thin barriers, which permit strong coupling between the states of the wells. In this thesis, we wil l be looking at C Q W structures containing only two quantum wells separated by a thin barrier. The coupling creates new states, and these states exhibit electric-field dependences that are qualitatively different from those of the states in SQWs. C Q W structures offer refractive index changes that have similar magnitudes to those typically obtainable in SQWs. The magnitude of the electric-field-induced refractive index change in SQWs is An ~ 10"3 for 100 kV/cm change in the electric field [58]. In C Q W structures, however, similar changes in the refractive index can be obtained for ~20 kV/cm change in the electric field, which is a significant improvement. Unlike in SQWs, in C Q W structures, the electroabsorption is based on the modulation of the absorption strength associated with the exciton peaks, rather than on their "red shifting." As C Q W structures are the main focus of this thesis, the electroabsorption and electrorefraction in these structures wi l l be discussed in detail. However, a brief description of the numerical models used to simulate electroabsorption and electrorefraction in these structures wi l l be given first. 67 1.5.1 Modeling of C Q W Structures In this section, the model used for simulating electroabsorption and electrorefraction in the C Q W structures studied in the thesis is described. This model is commonly used for simulation of C Q W structures as well as SQWs [38], [42], [60]. In the model, the exction equation is used within the effective-mass, envelope-function approximation, and it is solved using the variational method. The effective mass exciton equation can be written as e \ He-Hh+Eg- ®(re,rh)=E<$>{re,rh), (1.23) where H - - — v 2 +V (z ) (1.24) and -Hh=-^V\+Vh{zh) 2m, (1.25) 68 are the electron Hamiltonian and the hole Hamiltonian, respectively. Here, m*e and m*h are the effective masses for electrons and either light or heavy holes. It is assumed that the electric field is applied in the z direction and its effect is included in the potential energy for both electrons and holes (Ve and Vh, respectively). The third term on the left-hand side of Eq. (1.23) is the band gap energy, and the fourth term is the Coulomb potential energy, where e is the elementary charge, and es is the permittivity of the quantum well material. The coulombic interaction in the above equation couples the electron and hole coordinates (re and rh, respectively) in all three dimensions. If it were not for this coupling, the two-particle wave function in Eq. (1.23) would be a simple product of the electron and hole wave functions and the equation would be separable. Neglecting the quantum well potential (Ve and Vh) in Eq. (1.23) would reduce it to the well-known hydrogen atom equation [61]. Similar to the hydrogen atom problem, Eq. (1.23) is more easily solved when recast in the center-of-mass coordinates in the plane of the quantum wells Rt=Xx + Yy = meP'+m*»P» M (1.26) and the difference coordinates in the plane of the quantum wells p = xx + yy = pe-ph, (1.27) where M = m*e+m*h (1.28) 69 is the total mass corresponding the electron-hole system. Performing the transformation, and defining the reduced mass of the electron-hole system as — = — + — m , m* ml (1.29) make Eq. (1.23) become. h2 -*(re,rh)- 0,(1.30) where and 2ra dz" 2mh dzh (1.31) (1.32) are the usual 1-dimensional electron Hamiltonian and the 1-dimensional hole Hamiltonian, respectively, from which the energies and 1-dimensional wave functions are readily obtained by solving H{Ze)fn{Ze)=Eenfn{Ze) (1.33) and H{zh)gm{zh) = Ehmgm(zh), using the transfer matrix method [62]. (1.34) 70 The first term in Eq. (1.30) is the only term describing the center of mass of the system, which can be regarded as a freely moving particle of mass M a n d momentum (1.35) which is negligible for direct interband transitions. The wave function can, therefore, be written as frfc>rj= ^ - F(p,ze,zh). (1.36) Using the completeness property of the free-electron and free-hole wave functions, the exciton envelope function can be written as F{P^zh) = Yl^Mfn{ze)gm{zh). (1-37) n m After some further manipulations of the equation, it can be shown that it satisfies HB<P„m{p)=EB<PnM, (1-38) where ' • £ V J - K „ ( p f (1.39) and Kmiphjdze\fn{ze)2fdzh\gm(zhf Ajtes^pA +\ze -zh (1.40) Eq. (1.40) is a simpler version of a more general equation [38] and does not allow distinct subbands in the conduction band and distinct subbands in the valence band to be coupled by the coulombic interaction. In Eq. (1.38), 71 EB = E-(Eg + Een-Ehm) ( 1 . 4 1 ) is the exciton binding energy, which is the negative quantity that was already mentioned in the discussion of electroabsorption in SQWs. In order to o b t a i n ^ and the wave function from Eq. (1.38), an additional approximation is made. It is assumed that the relative motion of an electron around a hole in the plane of the quantum well is restricted to the lowest energy 1S orbital, which is a reasonable assumption considering that the higher energy states make smaller contributions to the absorption coefficient, as mentioned in the discussion of electroabsorption in SQWs. The IS state is radially symmetric in the plane of the quantum well structure < ^ ( P ) = A P } A (1-42) and it has an effective radius A . This radius is found, and the problem is solved, using the variational method. In this method, the binding energy as a function of the in-plane radius as an adjustable parameter is minimized. The in-plane extension of the exciton is comparable to the length of the quantum well structure in the z direction. In [38], it is shown how £ 5 can be analytically obtained from the variational method for a simple case in which no coulombic coupling of distinct subband states is allowed. 72 However, this minimization procedure needs to be done numerically if the coulombic coupling of distinct subband states is allowed. For example, in C Q W structures, the energy separation between the two lowest energy electron wave functions, / , (ze) and f2 {ze), and the energy separation between the two lowest energy hole wave functions, g] {ze) and g2 (ze), can be on the order of EB (<10 meV). In such a case, the coulombic coupling of these states cannot be neglected, and the variational wave function becomes F{p,ze,zll) = [afl{ze) + ^[^^f2{ze)] [bg^ + ^ ^ 2 g2{zh)]^i{^, (144) as explained in [63]. In this case, A , a, and b have to be varied independently in order to find EB for the 1S excitons. As wi l l be shown below, /j (ze) and f2 (ze) or g, (ze) and Si ize) a r e confined to opposite wells of a C Q W structure and wi l l switch wells when the applied electric field reaches a certain value, which is called the resonant electric field, FR. One of the main effects of the coupling is to increase FR by several kV/cm. The absorption spectra consist of the discrete and continuum exciton state contributions corresponding to both the electron-heavy-hole transitions and the electron-light-hole transitions [38]. The discrete exciton states contribution to the absorption coefficient is given as a 0 ( H = C0^-\tm{P-0) \2\lnm\2\e;~pjB{E - too), (1.45) qw 73 where a transition between an n electron state and an m hole state (light-hole or heavy-hole state) is considered. In C Q W structures, it is typically sufficient to consider only the two lowest energy electron, heavy-hole, and light-hole states, and the absorption coefficients corresponding to the transitions between these conduction-band and valence-band states are simply added together. The first term in Eq. (1.45), depends on some of the already defined material parameters and fundamental constants, as well as the refractive index of the quantum well material (nr), the free-space permittivity (e 0 ) , and the free electron mass (m 0). The second term in Eq. (1.45) is inversely proportional to Lqw, which is the length of one period of the quantum well structure. The third term is the in-plane exciton envelope function of Eq. (1.42) evaluated for zero relative displacement between the electron and the hole, which is proportional to the probability of finding the electron and the hole in the same unit cell. The fourth term is the overlap integral between the 1-dimensional electron and hole wave functions, as defined in Eq. (1.18). The fifth term is determined by the polarization selection rules, given below. The vector e is the unit vector describing the polarization of light, and the vector C n = 2 ' (1.46) nrceQa>m0 (1.47) 74 which is called the momentum matrix element. Here, p is the momentum operator and uc and uv are the conduction-band and valence-band unit-cell wave functions, respectively. For T E polarization (e = x or e = y), f 3 e-p„ =M2b\ heavy - hole exciton 1 — light - hole exciton (1.48) and for T M polarization (e = £ ), , „ , 2 i fO heavv - hole exciton \e-pJ =Ml\ , , (1-49) 2 light - hole exciton where / ™2 MI* (1.50) {6me(Eg+2A/3)j is the bulk matrix element (squared). In Eq. (1.50), A is the spin-orbit splitting energy. Eqs. (1.48) and (1.49) state, as was already mentioned, that the electron-heavy-hole transitions do not contribute to absorption for T M polarization, and they dominate the absorption for TE polarization. Finally, the last term in Eq. (1.45) is the exciton broadening function, which is represented by the Lorentzian function B(E-hco)- y—. _ (1.51) v ' 7i[(E-hco)2 +y2] for homogeneous broadening mechanisms and by the Gaussian function 5 ( ^ - n w ) = - ^ = ^ e 2 a l (1.52) 75 for inhomogeneous broadening mechanisms [10]. In Eq. (1.51), y is the half-width-at-half-maximum (HWHM) . In Eq. (1.52), the H W H M corresponds to -1.175o. In both of these functions E stands for the total exciton energy, as given in Eq. (1.41). In reality, the broadening function is a convolution of the broadening functions corresponding to each of the broadening mechanisms that are relevant in a specific device. In this thesis, a Lorentzian broadening function that has 7 = 6 meV is used. The semi-empirical formula used to obtain this value is given in [64]. It is also stated in [64], and confirmed by many simulations in this thesis, that the choice of the broadening function for C Q W modulators has no significant effect on the electrorefraction process in these devices. In the formula for the continuum states contribution to the absorption coefficient 00 ac{hm) = C,^^Ml\lnn^dE,M{Et)S{E,)B{E-noy), (1.53) some of the terms that appear in Eq. (1.45) can be recognized. However, because the exciton states form a continuum here, Eq. (1.53) is an integral with respect to the transverse energy h2k2 E , - ¥ - 0-54) of the electron-hole two-body system. Also, the momentum matrix element in Eq. (1.53) is different from that in Eq. (1.45) because the momentum matrix element in Eq. (1.45) is approximated with kt = 0, which is a commonly used approximation. In Eq. (1.53), the 76 momentum matrix element in the integral depends on E,, and for TE polarization becomes M{E,) = + cos 2 6nm) heavy - hole exciton -h- 3cos 2 Qnm j light - hole exciton (1.55) while for T M polarization, it becomes M(E,)-In these two equations, cos 2 0N_ = Also, in Eq. (1.53) S(E,)~ heavy - hole exciton - ( l + 3cos 2 dnm) light - hole exciton (1.56) Ehm Een + Ef,m (1.57) In (1.58) which is the Sommerfeld enhancement factor for a 2D exciton. The Sommerfeld enhancement factor comes from the solution of0„ m (p = 0) for the continuum states of a 2D exciton. A detailed derivation of the eigenfunctions and eigenenergies of the continuum states, as well as bound states of the 3D and the 2D hydrogen atom, is given in [61]. Typically, 1 =s so ^  2, and so = 2 for a purely 2D exciton. Ry is the Ryberg energy. The total exciton energy for the continuum states in B{E - hco) is given as E = Eg+Een-Ehm+E,. (1.59) 77 Once the absorption spectra for either polarization are obtained by adding the discrete and continuum components, the corresponding refractive index changes can be obtained using the Kramers-Kronig relation given in Eq. (1.22). In the following section, the electrorefraction of a particular C Q W structure is simulated using this model. 1.5.2 Electrorefraction in C Q W Structures Although the research on C Q W structures has not been as intense as the research on SQWs, there has been considerable research effort dedicated to C Q W structures in the last two decades. However, a large part of this effort was focused on studying the basic physics and optical phenomena in G a A s - A l G a A s C Q W structures, both experimentally and theoretically [63], [65]—[71]. Compared to SQWs, there have been few attempts to develop and optimize devices based on C Q W structures for E A and E R modulators, especially for 1550 nm operation. Regarding E A modulators, some of the research that did include specific device considerations, such as chirp, focused on GaAs material systems [72], [73]. A rare example of a 1550 nm C Q W E A modulator, that demonstrated an improved effect compared to SQW E A modulators, was presented only recently [74] (in 2004). As regards E R modulators, the C Q W structures studied have also been mainly based on GaAs material systems and have demonstrated significant improvements over SQW E R modulators, both theoretically [75] and, recently, experimentally [76], [77] (in 78 2001 and 2004, respectively). A 50 GHz modulator based on G a A s - A l G a A s quantum well structures studied in [77] has already been demonstrated in [78]. The drive voltage of the device was 3 V , which was unexpectedly large considering the theoretical predictions of a very large electrorefraction effect. Subsequently, it was shown that the electrorefraction effect in these structures is very sensitive to the layer thickness and compositional variations that occur during the growth process [79], making it challenging to realize the large electrorefraction effect predicted by theory. This is true for C Q W structures in general, and it is probably the main factor impeding the development of ER, as well as E A , C Q W modulators. In general, the C Q W structures intended for 1550 nm operation are expected to be even more sensitive to the variations than GaAs-based structures because they contain ternary and/or quaternary epitaxial layers. So far, research on C Q W structures in these material systems has only been theoretical [45], [60], [l]-[3], and no fabricated 1550 nm E R C Q W modulators have been reported. As wi l l be shown in the following chapters, the two main objectives of this thesis are the improvement of the robustness to growth of C Q W structures for E R as well as E A modulators and the facilitation of polarization modulation using these structures. However, before addressing these two objectives, basics of electrorefraction in C Q W structures are presented. Fig. 1.18 shows the energy-band diagram for an InGaAs-InAlAs C Q W structure as well as the two lowest energy electron wave functions ( W E | and W E 2 ) , the two lowest energy light-hole wave functions (WLHI and WLTO), and the two lowest energy heavy-hole wave functions ( W H H i and WHM) for an electric field, F, of 0, 19, and 35 kV/cm. The C Q W structure also confines some other, higher energy Wave functions, which are not shown in Fig. 1.18 because they do not make significant contributions to the electrorefraction effect. The C Q W structure has Ino.53Gao.47As wells and Ino.52Alo.48As barriers, so that all of the epitaxial layers are lattice-matched to (001) InP. The left-hand side well (QW1) is 48 A thick, and the right-hand side well (QW2) is 54 A thick. The middle barrier is 36 A thick, and each of the external barriers is 90 A thick. For F = 0 kV/cm, W E i , W H H I , and W L H I are localized in QW2 because it is thicker than QW1 and provides lower potential energy to these wave functions, as shown in Fig. 1.18(a). QW2 would have the same effect i f it was made deeper than QW1. Compared to the other two wave functions, WHHI is the most strongly localized because it has the largest effective mass. The second lowest energy wave functions, i.e., WE2, WHH2, and WLH2, are localized in QW1 for F= 0 kV/cm. When the electric field is increased to about 19 kV/cm, as shown in Fig. 1.18(b), WEI and WE2 become approximately equally distributed between the two wells. For this electric field, the energy separation between the two wave functions becomes the smallest, and the electric field is called the resonant electric field for electrons, F E . When the electric field is increased above FE, as shown in Fig. 1.18(c), WEI and W E 2 become strongly QWl (a> (b) (c) Figure 1 . 1 8 : Energy-band diagram of a InGaAs-InAlAs C Q W structure and W E I and W E 2 , W L H I and W L H 2 , W H H i and W H H 2 for (a) F= 0 , (b) 1 9 , and (c) 3 5 kV/cm. W E i and W L H I are shown as full thick lines, while WE2 and W L H 2 are shown as full thin lines. WH H I is shown as dashed thick line, while WHH2 is shown as dashed thin line. localized in Q W l and Q W 2 , respectively. No similar anticrossing of W L H I and W L H 2 or WH H I and W H H 2 occurs in Fig. 1 .18 . These wave functions simply become more strongly localized to their quantum wells as the electric field is increased. These wave functions would anticross i f the electric field was applied in the opposite direction, or i f Q W l was wider than Q W 2 and W L H I and WHHI were localized in it for F = 0 kV/cm. The electric-field-induced anticrossing of only one type of wave function, e.g., the electron wave functions, without the anticrossing of the other types, e.g., the light-hole and the heavy-81 hole wave functions, gives rise to a special electrorefraction effect largely exploited in this thesis. Figs. 1.19(a) and (b) show the electric-field dependence of the squared overlap integrals between the conduction and valence band wave functions of Fig. 1.18. The overlap integrals are evaluated using Eq. (1.18), and, for example, the curve Electric Field (kV/cm) Electric Field (kV/cm) (a) (b) Figure 1.19: Squared overlap integrals of the wave functions corresponding to the transitions (a) between the two lowest energy electron and heavy holes and (b) between the two lowest energy electron and light holes. corresponding to the overlap integral for the W E I - W H H I transition is labeled as " E 1 - H H 1 " in Fig. 1.19(a). In the literature, the transitions between W E I and W H H I , W E 2 and WHH2, W E I and W L m , and W E 2 and W L H 2 are sometimes referred to as the "symmetric-mode transitions" [80]. Similarly, the transitions between W E I and WHH2, W E 2 and W H H I , W E I 82 and WLH2, and WE2 and W L H I are sometimes referred to as the "asymmetric-mode transitions." In Fig. 1.19(a), the squared overlap integrals corresponding to the electron-heavy-hole symmetric-mode transitions decrease with F, while the squared overlap integrals corresponding to the electron-heavy-hole asymmetric-mode transitions increase with F. This behavior results from the electric-field-dependent redistribution of W E I and W E 2 only, as the distributions of WH H I and WHH2 barely depend on F in the region of interest (0 <. F <35 kV/cm). Also, the shapes of the curves in Fig. 1.19(a) suggest that the distributions of W E i and WE2 are most sensitive to changes in F when these wave functions are approximately equally distributed between QW1 and QW2, i.e., when F ~ FE. In general, a distribution of any wave function in Fig. 1.18 is most sensitive to both changes in F and relative changes in the thicknesses and compositions of the two wells when the wave function is approximately equally distributed between the two wells, and, conversely, it is least sensitive to these changes when the wave function is strongly confined to one of the wells. Later in this thesis, this phenomenon wi l l be used to design C Q W structures which are quite robust to the growth process. Fig. 19(b) shows that the squared overlap integrals corresponding to the electron-light-hole transitions exhibit similar dependence on F to that of the electron-heavy-hole transitions. However, the squared overlap integrals for the electron-light-hole symmetric-mode transitions first increase as F is changed from 0 to about 7 kV/cm, and then decrease for higher F. The reason for this behavior is that W L H I and WLH2 are less strongly confined to their respective wells than W E i and W E 2 are [see Fig. 1.18(a)], and it takes a small change in F (again, about 7 kV/cm) to maximize the spatial overlaps between the wave functions for the symmetric-mode transitions before they start decreasing toward very small values. Fig. 19(b) also shows that the opposite is true for the electron-light-hole asymmetric-mode transitions. In addition, the weak confinement of W L H I and WL H2 for small F makes the value of F for which the squared overlap integrals of the symmetric-mode and asymmetric-mode transitions are equal be somewhat larger than the one for the electron-heavy-hole transitions in Fig. 19(a). Besides obtaining the electric-field dependence of the squared overlap integrals associated with the wave functions confined by C Q W structures, it is essential to obtain the electric-field dependence of the exciton binding energies associated with these wave functions. This is because the contributions of the discrete exciton states to the absorption spectra [Eq. (1.45)] dominate the electroabsorption and, in turn, electrorefraction in C Q W structures, and these contributions strongly depend on both the squared overlap integrals and binding energies associated with the wave functions. In Eq. (1.45), the discrete exciton state absorption spectrum, a c , is proportional to the probability of an electron in state n and a hole in state m being in the same unit cell in the plane of the quantum well, '•E-> |#NM (P = 0)( , which, in turn, is proportional to the exciton binding energy. Again, 84 according to Eq. (1.42), the exciton binding energy is inversely proportional to the exciton in-plane radius, A . Figs. 1.20(a) and (b) show how the binding energies of the heavy-hole and the light-hole excitons depend on F . -2.5 -3 5 10 15 20 25 30 35 0 Electric Field (kV/cm) ( a ) E l - L H l \ . / . E2-LH2 y E2-LH1 / \ \ ' — E 1 - L H 2 / \ ^ 5 10 15 20 25 30 35 Electric Field (kV/cm) (b) Figure 1.20: Binding energies of the excitons corresponding to the transitions (a) between the two lowest energy electron and heavy holes and (b) between the two lowest energy electron and light holes. For small values of F, those below FE, the binding energies corresponding to the symmetric-mode transitions are large, and the binding energies corresponding to the asymmetric-mode transitions are small. For large values of F, those above F E , the binding energies corresponding to the symmetric-mode transitions are small, and the binding energies corresponding to the asymmetric-mode transitions are large. This behavior of the binding energies in Fig. 1.20 is similar to the behavior of the squared overlap integrals in Fig. 1.19 because, as mentioned above, separating the electron and hole wave functions 85 of an exciton in the z direction (direction normal to the planes of the quantum wells) increases the in-plane exciton size. Now, having obtained the electric-field dependence of the squared overlap integrals and the binding energies in C Q W structures, it is easy to describe electroabsorption in these structures. Fig. 1.21 shows the absorption spectra for the T M polarization for F= 0, 20, and 35 kV/cm. According to the polarization selection rules given in Eqs. (1.48) and (1.49), heavy-hole excitons do not contribute to the T M absorption spectra, and only the light-hole exciton peaks are present in Fig. 1.21. For F - 0 kV/cm, the two exciton peaks in the figure are dominated by the symmetric-mode transitions, which have large squared overlap integrals and large binding energies for small F (F<FE). In the absorption spectrum, the symmetric-mode transitions are indistinguishable from the asymmetric-mode transitions, not only because the asymmetric-mode transitions have small squared overlap integrals and small binding energies for small F, but also because the energy separations between the symmetric-mode transitions and asymmetric-mode transitions are of the same order of magnitude as the H W H M of the broadening function. F o r i 7 = 35 kV/cm, the heights and the energy/wavelength positions of the two peaks are similar to those for F = 0 kV/cm, however, for F = 35 kV/cm, the peaks are dominated by asymmetric-mode transitions, as they have large squared overlap integrals and large binding energies for large F(F> F E ) . For F = 20 kV/cm, however, the symmetric-mode transitions and the asymmetric-mode transitions make similar contributions to the absorptions peaks, which are considerably lower than the absorption peaks for F = 35 kV/cm. 86 Oand S u a u B o U c o t o < 1033 8000 7000 6000 H 5000 4000 3000 2000 h 1000 1200 992 954 Energy (meV) 918 886 855 826 800 F = 0 kV/cm: E2-LH2 F = 20 kV/cm: E2-LH2 and E1-LH2 F = 35 kV/cm: E1-LH2 F = 0 kV/cm: E1-LH.1 F = 20 kV/cm: E1-LH1 and E2-LH1 F = 35 kV/cm: E2-LH1 775 - F = 0 k V / c m - - F = 2 0 kV/cm - F = 3 5 kV/cm 1300 1350 1400 1450 1500 Wavelength (nm) 1550 1600 Figure 1.21: Absorption spectra for the T M polarization. The fact that the absorption peaks are smaller when the symmetric-mode transitions and asymmetric-mode transitions make similar contributions, as compared to when one type of transition dominates, cannot be solely explained by the squared overlap integrals in Fig. 1.19. This is because there is no significant electric-field-induced change in the energies of the transitions and the total sum of the squared overlap integrals. For 87 example, in Fig. 1.19(b), the squared overlap integrals for the symmetric-mode transitions are ~1, and those for the asymmetric-mode transitions are ~0 for F~ 7 kV/cm. Therefore, the total sum of the squared overlap integrals is ~1. When F~22 kV/cm, the squared overlap integrals for both the symmetric-mode and asymmetric mode transitions are ~0.5, so that the total sum of the squared overlap integrals is, again, ~1. The squared overlap integrals, alone, therefore, conserve the absorption strength in Eq. (1.45). However, the squared overlap integrals in Eq. (1.45) are scaled by the corresponding values of|0nm (p = o)| , which are, again, proportional to the binding energies. For small F, the absorption peaks are large because they are dominated by the symmetric-mode transitions, which have their squared overlap integrals (~1) scaled by large |< n^m (p = 0)j calculated for large binding energies of about -5.5 meV. For F ~ 20 kV/cm, however, the absorption peaks are small because the squared overlap integrals for the symmetric-mode and asymmetric-mode transitions (their sum is ~1) are scaled by small |0„m (p = 0)j2 calculated for small binding energies of about -4.5 meV. For F ~ 35 kV/cm, the absorption peaks are large again because they are now dominated by the asymmetric-mode transitions, which have their squared overlap integrals (~1) scaled by large \(j)nm (p = o)j calculated for small binding energies of about -6 meV. Similar to Fig. 1.21, Fig. 1.22 shows the absorption spectra for the TE polarization. The electroabsorption process for the TE polarization is analogous to that for the T M 88 1033 992 Energy (meV) 918 886 1350 1400 1450 Wavelength (nm) 1600 Figure 1.22: Absorption spectra for the TE polarization, polarization, except that the polarization selection rules allow for the participation of the heavy-hole excitons as well. Because the heavy holes have larger masses than the light holes, the heavy-hole excitons typically have lower transition energies than the light-hole excitons. This point is also illustrated in Fig. 1.23, which shows the transition energies/wavelength for the heavy-hole and light-hole excitons. For F < F E , the absorption spectra for both polarizations are dominated by the symmetric-mode transitions, and the energies of these transitions change only slightly with F. As F is increased above FE, the symmetric-mode transitions show a certain amount of red shift 89 (E1-HH1 and E 1 - L H 1 transitions) or blue shift (E2-HH2 and E 2 - L H 2 transitions). Here, however, the absorption spectra are dominated by the asymmetric-mode transitions, which do not display significant red or blue shifts. Consequently, the absorption peaks do not move left or right on the horizontal axis in response to changes in F. Rather, the absorption peaks only move up and down, creating large changes in the areas below the absorption spectra. Such behavior is very desirable for obtaining large electrorefraction effects in C Q W structures [60]. > to 6, >•> 00 L000 980 960 F 940 c a o •~ 920 900 880 h 860 E l - H H l E1-HH2 E2-HH1 E2-.H.H2 E1-LH1 E1-LH2 E2-LH1 E2-LH2 10 15 20 25 Electric Field (kV/cm) 30 1240 1265 1291 s <L> 1319 "S 1348 :•= C 1378 1409 1442 35 Figure 1.23: Transition energies and wavelengths for the heavy-hole and light-hole excitons. 90 Figs. 1.24(a) and (b) show the change of the absorption coefficient and the refractive index for the T M polarization when F is switched between 0 and 20 kV/cm and between 20 and 35 kV/cm, respectively. Because there is no significant blue or red shifting of the Energy (meV) Energy (meV) 1033 954 886 826 775 1033 954 886 826 775 Figure 1.24: T M polarization absorption coefficient change and refractive index change when F is switched between (a) 0 and 20 kV/cm and (b) 20 and 35 kV/cm. absorption peaks, the absorption coefficient changes in Figs. 1.24(a) and (b) are well localized in wavelength. For the same reason, the absorption coefficient changes in Figs. 1.24(a) and (b) are almost entirely negative and positive, respectively. Consequently, for the wavelengths that are longer than the wavelength corresponding to the lowest energy exciton peak, the Kramers-Kronig relation in Eq. (1.22) predicts large refractive index 91 changes that are respectively also negative and positive, as shown in Figs. 1.24(a) and (b). The same is true for the absorption coefficient change and the refractive index corresponding to the T E polarization, as shown in Figs. 1.25(a) and (b). 1033 Energy (meV) 954 886 826 775 U o u U c _o f o JD < 1 1 r— 1 F: 0 to 20 kV/cm "—\r'\ V V 1 — 1 i i 00 c CO JS u 0 « c 1200 1300 1400 1500 Wavelength (nm) (a) 1600 1033 Energy (meV) 954 886 826 1200 1300 1400 Wavelength (nm) (b) -2 500 1600 Figure 1.25: TE polarization absorption coefficient change and refractive index change when F is switched between (a) 0 and 20 kV/cm and (b) 20 and 35 kV/cm. Unlike in C Q W structures, in SQWs, an application of the electric field causes large red shifts of the absorption peaks, so that both large positive and large negative changes of the absorption spectra are created (see Fig. 1.15). The opposite-sign changes of the absorption coefficients in SQWs create opposite-sign changes in the refractive indices at higher wavelengths, which is highly undesirable [60]. 92 Fig. 1.26 shows the electric-field-dependent refractive index changes for three different wavelengths across the C-band. Disregarding the increase of the refractive indices for the 0 5 10 15 20 25 30 35 Electric Field (kV/cm) Figure 1.26: Electric field dependence of the refractive index changes for the TE and T M polarizations across the C-band. T M polarization around F ~ 7 kV/cm, which are due to the increase of the light-hole absorption peaks mentioned above, the refractive indices decrease with F in the region below FE and increase with F in the region above FE for all curves in Fig. 1.26. The " V " shapes of the curves are characteristic for C Q W structures. For any wavelength in Fig. 1.26, the electrorefractive effect is larger for the TE polarization than for the T M 93 polarization because the heavy-hole excitons participate in the electrorefraction for the T E polarization and are closer to the wavelength of operation than the light-hole excitons. Consequently, the C Q W structures are typically intended for TE Mach-Zehnder modulators, where either the negative slope of the refractive index change below FE or the positive slope of the refractive index change above FE can be used [60], [80]. In Fig. 1.26, the refractive index change shows relatively small dependence on the wavelength of operation across the C-band. If this dependence was required to be decreased even further, detuning of the lowest energy excitons from the wavelength of operation would need to be increased. The exciton energies could simply be increased by decreasing the thicknesses of the quantum wells in the C Q W structure. Unfortunately, the larger the detuning, the smaller the electrorefraction effect. Both of these trends are evident from Fig. 1.26, where the T M polarization exhibits a smaller dependence on the wavelength of operation as well as a smaller electrorefraction effect. Also, due to the larger detuning of the corresponding excitons, the T M polarization exhibits smaller optical loss and modulation of the optical loss at the wavelength of operation than the TE polarization does. However, the magnitude of the optical loss is not easily modeled as it strongly depends on the broadening function used in the model. The Gaussian broadening functions tend to underestimate the loss, while the Lorentzian broadening functions tend to overestimate the loss. For example, the semi-empirical Lorentzian broadening function used for modeling of the C Q W structure analyzed above, and all of the C Q W structures 94 in this thesis, has a H W H M of about 6 meV [64] and predicts the absorption coefficients for the two polarizations to be in the 10s of cm"1 at 1550 nm. However, i f a Gaussian broadening function with the same value for the H W H M is used, the corresponding absorption coefficients are several orders of magnitude smaller for both polarizations, again at 1550 nm. Nevertheless, the choice of the broadening function has little (practically no) effect on the refractive index change at the wavelength of operation. Fig. 1.27 compares the refractive index changes obtained using the Gaussian and the Lorentzian broadening functions at 1550 nm, showing no appreciable difference between the two. It turns out that no appreciable difference between the two broadening functions Electric Field (kV/cm) Figure 1.27: Comparison of the 1550 nm refractive index changes obtained using the Gaussian and the Lorentzian broadening functions. is seen even i f the H W H M s of the broadening functions are changed. The refractive 95 index changes are insensitive to the choice of the broadening function because the choice of the broadening function does not significantly affect the areas under the absorption spectra for relatively large wavelength detunings. According to the Kramers-Kronig relation given in Eq. (1.22), it is these areas that affect the magnitudes of the refractive index changes. Unfortunately, due to the lack of experimental reports on the absorption spectra in InGaAlAs-based C Q W structures, choosing the appropriate broadening function for the modeling of these structures is a somewhat challenging task. One way of ensuring small optical losses at the wavelengths of operation is to allow for large wavelength detunings that are comparable to those used in typical 1550 nm E R M Q W modulators. For example, in [81], an optical waveguide of a p-i-n diode contained multiple repetitions of 85 A thick InGaAsP quantum wells and 85 A thick InP barriers in the 0.4 um thick intrinsic region. Detuning of the lowest energy exciton in the SQWs from the 1550 nm wavelength of operation was 133 nm. Optical waveguide loss for the TE polarization was measured to be ~1 cm"1 (~0.4 dB/mm) for zero applied voltage. Due to the large red shift of the lowest energy exciton, which is characteristic for SQWs, a 4 V change in the applied voltage induced more than a three-fold increase in the measured waveguide loss and less than 4x 10"4 increase in the measured effective refractive index. A similar behavior was observed in [82] for a p-i-n diode containing twenty 103 A InGaAsP wells and 80 A InP barriers in the intrinsic region, which formed -60 % overlap with the optical mode. A 96 voltage change from 0 to 5 V induced a red shift of the lowest energy exciton from -1440 nm to -1480 nm, resulting in a large refractive index change (i.e., large phase change across the device) as well as a large optical loss increase for 1557 nm wavelength in a 0.6 mm long waveguide, as shown in Fig. 1.28. As can be seen in the figure, optical loss for 4.71 Figure 1.28: Optical loss and phase shift for 1557 nm wavelength in a 0.6 mm long section of an optical waveguide containing twenty InGaAsP-InP SQWs. The squares and triangles are experimentally obtained data points, while the lines come for theoretical models that use fitting parameters [82]. small voltages is only a fraction of a decibel and increases to -1.5 dB for a half-wave voltage of-4 .5 V . As the excitons in C Q W structures do not exhibit significant red shifts for the small changes in F that are typically applied to them, the optical loss at the wavelength of operation stays relatively independent of F. For example, let's look at a half-wave phase change 97 2JI = — x AnxF x L = JI (1.60) A in an optical modulator at X = 1550 nm, i f we assume that the optical mode has an overlap factor of T = 0.60 for the C Q W region, which provides a refractive index change of An = 10 x 10"4, the interaction length would need to be L - 1.3 mm. Consequently, the optical loss of the C Q W modulator can be estimated to be below 1 dB for a wavelength detuning that is somewhat larger than 100 nm. Even smaller losses can be expected in C Q W modulators that are not based on p-i-n diode structures. The p-doped regions are lossy not only for the electrical signal, but also for the optical mode. The optical loss due to the p-doped region, a p - (25 x 10"18 cm 2) x T p x p, depends on the overlap that the region makes with the optical mode, T p , as well as the amount of doping, p ( -10 1 8 cm"3) [52]. Besides a p , another major contribution to the residual, wavelength-independent loss is the photon scattering due to the imperfections of the optical waveguide geometry. For example, in a p-i-n diode having a reactive-ion-etched optical waveguide and p = 2 x 10 1 7 cm" ), the residual loss for the T E polarization was measured to be -0.3 dB/mm [83]. The contribution of the wavelength-dependent loss due to the 0.4 um thick InGaAlAs- InAlAs M Q W region was -0.2 dB/mm for -130 nm wavelength detuning. In addition to the above discussion on optical loss in C Q W structures, there are two remaining aspects of them that should be addressed in this chapter before proceeding to the following chapters, where the contributions of this thesis to the state-of-the-art in C Q W modulators are given. 98 First, as wi l l be explained in more detail in Chapters 4 and 5, increasing the thickness of the middle barrier decreases the magnitude of F needed for switching in C Q W structures. Unfortunately, increasing the thickness of the middle barrier also increases the sensitivity to the thickness (-3 A) and compositional variations (1 -2 %) that can be expected to occur in the C Q W epitaxial layers during the growth process. In the C Q W structures that have switching fields smaller than -20 kV/cm, the layer thickness and compositional variations can be very detrimental to the electrorefraction effect. For example, Fig. 1.29 illustrates the effect of a 3 A increase in the thickness of QW2 for the C Q W structure 1 1 1 s ' \ *• \ *™ —""--w. x \ \ \ t \ * / * / \ * ' • \ 4 ' \ \ / \ \ If \ \ If \ \ if \ \ * I ' \ \ * 1 \ * ' / A«TM A « T E 5 10 15 20 25 30 35 0 Electric Field (kV/cm) (a) 10 15 20 25 30 35 Electric Field (kV/cm) (b) Figure 1.29: (a) 1550 nm refractive index changes for the T E and T M polarizations in the C Q W structure of Fig. 1.18 (same as Fig. 1.26) and (b) the same for a 3 A increase in the thickness of QW2. shown in Fig. 1.18. The 3 A thickness variation increases FE and places the negative 99 slope of the refractive index change in the electric-field region in which it is supposed to be positive. Randomly distributed thickness variations (of approximately ±3 A) are unavoidable between the epitaxial layers in C Q W structures [58], and therefore, the total change in the refractive index that the optical mode experiences wi l l be an average of the contributions to the refractive index changes that correspond to each variation. Many of these contributions wil l have opposite signs and diminish the total, average refractive index change in C Q W structures. Again, methods for decreasing the sensitivities of C Q W structures to thickness and compositional variations, without compromising the drive electric fields, wi l l be presented in Chapters 4 and 5. Second, in order to increase the difference between the electrorefraction effects for the TE and T M polarizations, and, thus, achieve efficient polarization modulation, as wil l be discussed in Chapters 2, 3, and 6, the epitaxial layers in C Q W structures need to be slightly strained. 1.5.3 Strain in C Q W Structures grown on (001) InP The lattice constant of InP is ao = 5.869 A [84], [85]. Therefore, unless the epitaxial layers in C Q W structures are intended to be strained, their compositions have to be chosen so that their lattice constants match that of the (001) InP substrate. For example, Ino.53Gao.47As and Ino.52Alo.48As are both lattice-matched to (001) InP. If an epitaxial layer 100 having a lattice constant a (^«o) is grown on top of a (001) InP substrate, or some other epitaxial layer (or a stack of layers) having lattice constant aa, the new epitaxial layer wi l l stretch to accommodate to the lattice constant of the thick bottom layer. Strictly speaking, this is true only i f the thickness of the top layer is smaller than the critical thickness, hc, which is the maximum allowed thickness of the epitaxial layer that can be accommodated by the system without generating any material dislocations and defects [86]. The in-plane strain, i.e., in-plane fractional distortion of the lattice, * u = ^ = ^ = ^ - l (1-61) a is the same in both x and y directions for cubic semiconductors (e.g., InP and GaAs) that have their lattice constants the same in all directions [85]. The strain in Eq. (1.61) is called biaxial tension or biaxial compression depending whether a is smaller or larger than ao, respectively. The biaxial strain is accompanied by strain in the growth direction £ x = ^ = - 2 — ( 1 - 6 2 ) where cn and c\ \ are elastic stiffness constants [85]. One of the main properties of strained epitaxial layers is energy splitting of otherwise degenerate heavy-hole and light-hole band edges. Fig. 1.30 illustrates this effect for InGaAs and InAlAs layers grown on (001) InP. In both cases, the light-hole band edge is above the heavy-hole band edge for tensile strain, and it is below the heavy-hole band edge for compressive strain. Consequently, it is possible to design the depths of Q W 1 and 101 QW2 in C Q W structures to be different for light holes and heavy holes, and thereby it is possible to manipulate the relative magnitudes of the electrorefraction effect for the TE and T M polarizations, as wil l be shown in Chapters 2, 3, and 6. The nature of the band-edge splitting is the same for the quaternary layers studied in these chapters. Also, opposite-sign biaxial strains in some of the layers in these C Q W structures are used to balance the strains in neighboring layers and increase hc of the overall structure. > £? u c LU K OO -a to I 1000 800 600 400 200 0 -200 -400 compressive strain (x < 0.47) E C E H H E L H tensile strain (x > 0.47) ^ 2500 2000 1500 In(i_ x)Ga xAs 0 0.2 0.4 0.6 0.1 Ga Mole Fraction (a) 1000 500 0 -500 -1000 compressive strain (z < 0.48) ' E C E H H — — E L H tensile strain (z>0.48) In( i . z )Al z As 0 0.2 0.4 0.6 0.8 Al Mole Fraction (b) Figure 1.30: Conduction band edge ( E C ) , light-hole band edge ( E L H ) , and heavy-hole band edge (EHH) for (a) In ( , . x ) Ga x As and (b) In ( i_ z ) Al z As, both of which are grown on (001) InP. The band-edge energies in (a) and (b) are plotted relative to those of Ino.53Gao.47As and Ino.52Alo.48As, respectively. There is another important property of strained layers that is used in Chapter 4 to decrease the sensitivities of C Q W structures to the compositional variations that are expected to occur during the growth process. It is obvious from Figs. 1.30(a) and (b) that 102 ~±2 % variation in composition from the nominally unstrained Ino.53Gao.47As and Ino.52Alo.48As layers wi l l have a smaller effect on the light-hole band edge than on the electron or heavy-hole band edges. This is particularly true for the Ino.53Gao.47As layer shown in Fig. 1.30(a). Therefore, i f the electrorefraction effect in a C Q W structure having Ino.53Gao.47As quantum wells is based on anticrossing the light holes, rather than electrons or heavy holes, the C Q W structure should be the least sensitive to compositional variations. Similar effects are characteristic of nearly-lattice-matched quaternary compounds. The small sensitivity of the light-hole band edge to the lattice-mismatch-induced strain is due to the opposite-sign contributions of the hydrostatic and shear components of the strain, as can be explain using the model-solid theory presented below [50], [85]. The strain-dependent electron, heavy-hole, and light-hole band edges are given as £ c = < A v + y + £ o + A i ^ , (1.63) Em = K.AW + y + A ^ a v + A ^ h h , and (1.64) ^ H = < A v + y + A ^ A V + A F L A H , (1.65) respectively, where the quantities having "0" superscripts correspond to unstrained materials. The first term in Eqs. (1.63)—(1.65) is called "the average valence-band 103 energy," as it is literally obtained as an average of the energies for the heavy-hole, light-hole, and split-off bands, i.e., F° - H H LH + C S O ^V.AV ~ • (I.OO) The average valence-band energy is used as an absolute energy level for each of the two materials across the heterojunction and, thus, determines the band-edge discontinuities. In the second term in Eqs. (1.63)—(1.65), A 0 is the spin-orbit splitting energy. The conduction band energy in Eq. (1.63) is different from the heavy-hole and light-hole band energies in Eqs. (1.64) and (1.65), respectively, because it contains the band-gap energy term (fs£), and because its hydrostatic strain contribution term =a c (2£ l l + £ 1 ) (1.67) is proportional to the conduction-band hydrostatic deformation potential, ac, rather than the valence-band hydrostatic deformation potential, ay, which is the case for the hydrostatic strain contribution to the valence bands, i.e., AE^AV = av(2En + E±). (1.68) The strain-induced change of the average valence-band energy in Eq. (1.68) changes the energies of the light-hole band edge and the heavy-hole band edge in the same manner and does not lift their degeneracy. It is the shear-strain component of the total strain, not the hydrostatic-strain component, that lifts the degeneracy. The shear-strain effect on the heavy-hole and the light-hole band edges is represented by 104 AE&—±6E* (1.69) and ^ ^ L H = + \ 6 E S H + ^ ( A ° ) 2 + A " ^ " + \ (3ES"f > (L70) respectively, where 5ES" =2b(e±-eH). (1.71) The quantity b in Eq. (1.71) is the "tetragonal shear deformation potential." As b is a negative number and ay is a positive number, AE^H and AE^ AV have opposite signs and make ELH less sensitive to strain than EHH. In this thesis, as is typical for ternary and quaternary strained layers [50], [85], the various quantities used in this theory are interpolated from their theoretically or experimentally obtained binary equivalents. 1.6 Thesis Outline The following five chapters of this thesis (Chapters 2-6) present manuscripts that have been published, have been submitted for publication, or wi l l shortly be submitted for publication in journals. More specifically, Chapter 2 (article published in IEEE Photonics Technology Letters) presents the initial design for a push-pull polarization modulator, which contains two types of InGaAlAs-InGaAlAs C Q W structures in its waveguide. One of these two structures (TS structure) mainly increases the refractive index for the T M 105 polarization, via the anticrossing of the two lowest energy light holes, while the other structure (CS structure) mainly decreases the refractive index for the T E polarization, via the anticrossing of the two lowest energy heavy holes. In Chapter 3 (article published in Journal of Vacuum Science and Technology A ) , numerical simulations show that the TS structure is very insensitive to the layer thickness and compositional variations, while the CS structure is quite sensitive to these variations. In Chapter 4 (article published in IEEE Electron Device Letters), the mechanisms responsible for the decreased sensitivity of the TS structures to the layer variations are fully recognized. The anticrossing of the two lowest energy light holes, rather than the two lowest energy heavy holes or the two lowest energy electrons, is shown to be the mode of operation that is least sensitive to the layer variations, and, therefore, the most desirable technology for future C Q W modulators. Chapter 5 (manuscript that wi l l shortly be submitted for publication) presents a design method that can be used for improving the robustness to growth of C Q W structures in general, including those that have their E A or E R effects based on the anticrossing of electrons or of light holes. In Chapter 6 (manuscript submitted to IEEE Photonics Technology Letters, moderate revisions have been requested), the robustness to growth of the initial design of the push-pull polarization modulator given in Chapter 2 is improved using the knowledge obtained in the work presented in Chapters 3-5. Here, besides several changes that are made to the original design, the CS structure is replaced by a more robust C Q W structure that mainly decreases the refractive index for the TE polarization via the anticrossing of the two lowest energy electrons, not the heavy holes. 106 Chapter 7 is the concluding chapter of the thesis and contains discussion of the thesis' findings and recommendations for the future work. R e f e r e n c e s [1] S. Ristic and N . A . F. Jaeger, "Push-pull polarization conversion using novel asymmetric coupled quantum-weil structures," IEEE Photon. Technol. Lett., vol. 18, no. 2, pp. 316-318, 2006. [2] S. Ristic and N . A . F. 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Chemla, and F. Bechstedt, "Numerical calculation of the optical absorption in semiconductor quantum structures," Phys. Rev. B, vol. 54, no. 16, pp. 11592-11601,1996. [71] R. Winkler, "Excitons and fundamental absorption in quantum wells," Phys. Rev. B, vol. 51, no. 20, pp. 14395-14409, 1995. 115 [72] Y . C. Chan and K . Tada, "Field induced optical effects coupled quantum wells," IEEE J. Quantum Electron., vol. 27, no. 3, pp. 702-707, 1991. [73] J. A . Trezza, M . C. Larson, S. M . Lord, and J. S. Harris, Jr., "Low-voltage, low-chirp absorptively bistable transmission modulators using type-IIA and type-IIB Ino.3Gao.7As/Alo.33Gao.67As/ Ino.15Gao.85As asymmetric coupled quantum wells," J. Appl. Phys., vol. 74, no. 11, pp. 6495-6502, 1993. [74] T. H . Stievater, W. S. Rabinovich, P. G. Goetz, R. Mahon, and S. C. Binari, " A surface-normal coupled-quantum-well modulator at 1.55 um," IEEE Photon. Technol. Lett., vol. 16, no. 9, pp. 2036-2038, 2004. [75] N . Susa, "Electric-field-induced refractive index changes in three step asymmetric coupled quantum wells," J. Appl. Phys., vol. 73, no. 12, pp. 8463-8470, 1993. [76] M . P. Earnshaw and D. W. E . Allsopp, "Electrooptic effects in G a A s - A l G a A s narrow coupled quantum wells," IEEE J. Quantum Electron., vol. 37, no. 7, pp. 897-904, 2001. [77] T. Suzuki et. a l , "Observation of giant electrorefractive effect in five-layer asymmetric coupled quantum wells (FACQWs) ," J. Journal Appl. Phys., vol. 43, no. 12A, pp. L1540-L1542, 2004. [78] J. P. Pang, K . Tada, Y . Nakano, and H . Feng, "Multiple-quantum-well traveling-wave-type ultra-high-speed optical modulator," in Proc. OECC'98 Tech. Digest. (Optoelectronics and Communications Conference), pp. 454-455, 1998, paper 16B1-5. 116 [79] T. Suzuki et. al., "Fabrication and optical characterization of five-layer asymmetric coupled quantum well ( F A C Q W ) , " J. Journal Appl. Phys., vol. 41, no. 4B, pp. 2701— 2706,2002. [80] H . Feng, J . P. Pang, M . Sugiyama, K . Tada, Y . Nakano, "Field-induced optical effect in a five-step asymmetric coupled quantum well with modified potential," IEEE J. Quantum Electron., vol. 34, no. 7, pp. 1197-1208, 1998. [81] H . Mohseni, H . An , Z. A . Shellenbarger, M . H . Kwakernaak, and J . H . Abeles, "Enhanced electro-optic effect in GalnAsP-InP three-step quantum wells," Appl. Phys. Lett, vol. 84, no. 11, pp. 1823-1825, 2004. [82] Reprinted with permission from W. Bardyszewski, D. Yevick, Y . L iu , C. Rolland, and S. Bradshaw, "Theoretical and experimental analysis of Mach-Zehnder quantum-well modulators," J. Appl. Phys., vol. 80, no. 2, pp. 1136-1141, 1996. Copyright 1996, American Institute of Physics. [83] N . Yoshimoto, T. Yamanaka, S. Kondo, Y . Noguchi, and K . Wakita, "Large field-induced refractive-index change on TM-polarized light in a InGaAlAs-InAlAs M Q W waveguide structure," IEEE Photon. Technol. Lett., vol. 9, no. 2, pp. 200-202, 1997. [84] S. L . Chuang, Physics of Optoelectronic Devices. New York: Wiley, 1995, pp. 707-711. [85] M . P. C. M . Krijn, "Heterojunction band offsets and effective masses in I I I -V quaternary alloys," Semiconduct. Sci. Technol., vol. 6, pp. 27-31, 1991. 117 [86] J. J. Coleman, "Strained layer quantum well heterostructure lasers," in Quantum Well Lasers, P. S. Zory, Jr., Ed. San Diego: Academic Press, 1993, pp. 367-413. \ Chapter 2 Push-Pull CQW Polarization Modulator 118 2.1 Introduction Electrorefraction (ER) modulators based on asymmetric coupled quantum-well (ACQW) structures exhibit an electric-field-induced refractive index change that can be an order of magnitude larger than in structures based on square quantum wells [1], [2]. Compared to electroabsorption modulators, A C Q W modulators, as well as E R quantum well intensity modulators, in general, are typically implemented in a directional coupler or M a c h -Zehnder configuration, and they can provide lower temperature and wavelength sensitivities, lower carrier generation rates, as well as push-pull modes of operation with © [2006] IEEE. Reprinted, with permission, from S. Ristic and N. A. F. Jaeger, "Push-Pull Polarization Conversion Using Novel Asymmetric Coupled Quantum-Well Structures," IEEE. Photon. Technoi. Lett., vol. 18, no. 2, pp. 316-318, Jan. 2006. 119 adjustable chirp [3]. In this letter, we propose a novel type of E R modulator, which, in a single optical waveguide, contains multiple repetitions of two novel InGaAlAs-InAlAs A C Q W structures, grown on (001) InP. By applying an appropriate bias field in the [001] direction, in one of the structures a change in the applied field can be made to cause an increase in the refractive index for TM-polarized light without a commensurate increase in the refractive index for TE-polarized light. In a similar manner, in the other structure the same change in field wi l l cause a decrease in the refractive index for TE light without a commensurate decrease in the refractive index for T M light. Since each structure provides unequal refractive index changes for the two polarizations, i f either structure were placed in an optical waveguide, a change in polarization state at the output relative to the state at the input, i.e., polarization conversion, could be achieved (provided that, at the input, light is coupled into both the TE and T M modes). Typically, the powers in the two modes would be matched at the output so that a 90° polarization rotation could be obtained. To achieve push-pull polarization conversion, the effective refractive index change for the T E mode should be opposite in sign to the change for the T M mode [3]. Incorporating both of our structures into an optical waveguide wil l allow for the design of new push-pull-type quantum well polarization conversion (QWPC) modulators. In addition to traditional applications such as on-off keying, these modulators would likely find applications in areas not typical for quantum well modulators [4]. 2.2 Device Basics 120 A schematic illustration of the compositions and the thicknesses of the layers in the two A C Q W structures is shown in Fig. 2.1. The figure also illustrates the direction of the applied electric field F. Each structure has Ino.53Alo.47As barriers, which are lattice-matched to the InP substrate, as well as one lattice-matched Ino.53Gao.47As well. The Ino.46Gao.52Alo.02As well in Fig. 2.1(a) is under -0.50 % of tensile strain (TS), and the Ino.58Gao.37Alo.05As well of the structure in Fig! 2.1(b) is under -0.34 % of compressive strain (CS). Because of these lattice-mismatch-induced biaxial strains, the structures of Fig. 2.1(a) and (b) wil l be referred to as the TS structure and the CS structure, respectively. Ino.53Alo.47AS -50 A lno.53Gao.47AS 72 A. Ino.53Alo.47AS 36 A Ino.46Gao.52Alo.02AS 66 A Ino.53Alo.47AS -50 A. (a) F Ino.53Alo.47AS -50 A Ino.53Gao.47AS 69 A In0.53AI0.47AS 12 A lno.58Gao.37Alo.05AS 45 A Ino.53Alo.47AS -50 A (b) F Figure 2.1: Schematic for layers (a) TS structure and (b) CS structure. Outside barriers of each structures are -50 A in order to ensure -100 A decoupling between any two neighboring structures. Direction of applied electric field F is indicated as well. 2.2.1 TS Structure 121 As mentioned above, for a well-chosen bias field, the TS structure provides a large electric-field-induced refractive index increase for T M light and a smaller electric-field-induced refractive index increase for the TE light. In E R modulators, typically, the refractive index change for T E light is somewhat larger than the one for T M light simply because the lowest energy light holes are at higher energies than the lowest energy heavy holes [2]. Although in some quantum well modulators the polarization dependence of the refractive index has been decreased by lowering the light-hole energies with a tensile strain [2], to the best of our knowledge, no other biaxially strained A C Q W structure for use in 1550 nm E R waveguide modulators has been designed to provide a dominant T M -polarization refractive index change. Fig. 2.2 shows the two lowest energy electron wave functions (*FEI and * F E 2 ) , heavy-hole wave functions (YHHI and *FHH2), and light-hole wave functions (WLHI and v PLH2) of the TS structure of Fig. 2.1(a) as well as the band edge diagram for its conduction and valence bands for F= 0, 13, and 35 kV/cm. The band gap in the figure is not to scale, and the strain-induced band edge splitting between the heavy-hole and the light-hole bands in the quantum well under tensile strain is illustrated by drawing the heavy-hole band edge with a full line and the light-hole band edge with a dashed line. The material parameters for the structures are calculated using the model-solid theory [5], [6], and the exciton 122 energies and wave functions are obtained using the effective-mass envelope-function approximation that includes intersubband mixing [7]. F = 0 kV/cm F= 13 kV/cm F = 35 kV/cm * • • (a) (b) (c) Figure 2.2: Energy band diagram of TS structure of Fig. 1(a) and two lowest energy electron, heavy-hole, and light-hole wave functions at applied electric fields of (a) 0, (b) 13, and (c) 35 kV/cm. Lowest energy electron wave function (WEI) and heavy-hole wave function (WHHI) are shown as full thick lines, whereas second lowest energy electron wave function (WE2) and heavy-hole wave function (WHH2) are shown as full thin lines. Lowest energy light-hole wave function (WLHI) is shown as dashed thick line, and second lowest energy light-hole wave function (WLH2) as dashed thin line. Light-hole band edge is indicated with dashed line. Band gap is not shown to scale. For low electric fields, T E I , WHHI, and WLHI are predominantly localized in the wider unstrained well [see Fig. 2.2(a)]. Therefore, the W E I - W L H I and the W E 2 - W L H 2 excitons are direct, and the W E i - W L H 2 and the W E 2 - W L H I excitons are indirect. Since for low fields the overlaps between the electron and the light-hole wave functions of the direct excitons are 123 large and those of the indirect excitons are small, the direct excitons wil l dominate the light-hole absorption spectra for both polarizations [1]. At -13 kV/cm, the direct and the indirect excitons start anticrossing as WLHI and WLH2 are approximately equally distributed between the two wells [see Fig. 2.2(b)]. The v P E 2 ~ v P L H I and the WE 2 - WL H 2 excitons actually anticross at a somewhat different resonant electric field than the W E I ~ W L H I and the W E I _ W L H 2 excitons. For thin middle barriers, however, these fields are close enough to be treated as one resonant electric field ( F R ) . At F R , both the direct and the indirect light-hole excitons contribute to the light-hole absorption spectra, although their total contribution is smaller than it was for low electric fields because WLHI and WLH2 are further delocalized [1]. In fact, at FR, the area underneath the absorption spectrum of the W E I - W L H I , WEI~WLH2 , WE2~WLHI , and W E 2 _ W L H 2 excitons wi l l be at its minimum for either polarization. For fields greater than F R , the W E I - W L H I and WE2~WLH2 excitons become indirect and the W E 2 - v P L H I and the W E i - W L H 2 excitons become direct [see Fig. 2.2(c)]. The area underneath the absorption spectrum of the light-hole excitons, for either polarization, wi l l increase to be approximately as large as it was for very low fields. Similar anticrossings of the heavy-hole excitons do not occur as the tensile strain in the strained well makes the well effectively shallower for W H H I - ^ H H I consequently stays localized to the unstrained well for the electric fields of interest here. 124 Considering that the heavy-hole momentum matrix element is practically zero for T M light, the area underneath the total absorption spectrum (including both the heavy-hole and the light-hole excitons) for the T M polarization wi l l change in the same manner as the area underneath the light-hole excitons [3]. Also, there wi l l be a similar, although smaller, change of the area underneath the total absorption spectrum for the T E polarization. The change is smaller because the momentum matrix element for the anticrossing light-hole excitons is one quarter of its value for T M polarization, and the heavy-hole excitons, although having a large momentum matrix element for TE polarization, remain stationary and inactive for the electric fields of interest. As the two quantum wells are narrow enough to prevent the light-hole excitons from sizable "red shifting," the change of refractive index for T M light (A«TM), at wavelengths higher than the wavelength corresponding to the absorption peak of the T E I - ^ L H I exciton (k = 1394 nm for F = 0 kV/cm), wi l l follow the change of area underneath the T M polarization absorption spectrum in a straightforward manner [1]. As the electric field is increased toward FR, the slope of A « T M wi l l be negative, and as the electric field is increased above FR, the slope wi l l be positive. The change of the refractive index for T E light (A«TE) wi l l behave similarly, following the smaller change of the area underneath the T E polarization absorption spectrum. 125 Fig. 2.3 illustrates how A « T M and A « T E of the TS structure, plotted relative to their zero-electric-field values, depend on the electric field. The two curves are plotted for operation at X = 1550 nm and are labeled AMTM(TS) and A«TE(TS), respectively. Similarly, Fig. 2.3 also depicts A « T E and A « T M of the CS structure (AHTE(CS) and A«TM(TC> respectively). 6 | 1 ' i 1 1 r -12 I < 1 1 i i • I 0 5 10 15 20 25 30 35 Electric Field (kV/cm) Figure 2.3: Refractive index changes of TS and CS structures, for both TE and T M polarizations, plotted relative to zero-electric-field values at X = 1550 nm. A«TM(TS) and A« T E(TS) are labels for refractive index changes for respective T M and TE polarizations of TS structure. Similarly, A« T E(cs) and A« T M(CSJ are refractive index changes for TE and T M polarizations of CS structure. 2.2.2 CS Structure 126 Fig. 2.4 depicts the band edge diagram and the two lowest energy electron, heavy-hole, and light-hole wave functions of the CS structure of Fig. 2.1(b), as was done for the TS structure in Fig. 2.2, now for F=0, 27, and 35 kV/cm. In this case, however, it is the F=0kV/cm F=27kV/cm F=35kV/cm (a) (b) (c) Figure 2.4: Energy band diagram of CS structure of Fig. 1(b) and two lowest energy electron, heavy-hole, and light-hole wave functions at applied electric fields of (a) 0, (b) 27, and (c) 35 kV/cm. Energy band diagrams and wave functions are depicted as in Fig. 2.2. light-hole excitons that are stationary because the compressive strain in the right-hand side well makes it effectively shallower for the light holes. The CS structure is designed so that FR ~ 27 kV/cm for the heavy-hole excitons. Considering that the heavy-hole momentum matrix element for the T M polarization is zero and that it is dominant for the T E polarization, the slopes of A « T E ( C S ) on either side of F R wi l l have larger magnitudes as 127 compared to those of A « T M ( C S ) for the electric fields of interest, as shown in Fig. 2.3. Here, the wavelength corresponding to the absorption peak of the W E I - ^ H H I exciton, for F = 0 kV/cm, is X = 1465 nm. 2.3 Low-Chirp Push-Pull QWPC Modulator Between F ~ 15 kV/cm and F ~ 25 kV/cm, the TS structure mainly contributes a positive slope of A«TM(TS) (with a much smaller positive slope of AHTE(TS)), and the CS structure mainly contributes a negative slope of A«TE(CS) (with a much smaller negative slope of AMTMCCS))- Placed together in an optical waveguide of a QWPC modulator, multiple repetitions of the two structures will cause the effective refractive index seen by the fundamental TM mode («eff,TM) to increase and the effective refractive index seen by the fundamental TE mode (« e f f ,TE) to decrease, providing a push-pull polarization conversion for input light that contains both polarizations. i Each structure's relative contribution to the changes in «Cff,TM and «eff,TE depends on the structure's overlap with the TM and TE modes, respectively. These optical overlaps can be controlled by adjusting the number of repetitions of each structure in the waveguide, by adjusting the position and order of the repetitions of the two structures, and/or by adjusting the waveguide's geometry. For example, if one were to form a symmetric slab waveguide consisting of an Ino.53Alo.47As substrate, six repetitions of the TS structure, 128 five repetitions of the CS structure, six repetitions of the TS structure, and an Ino.53Alo.47As superstate, i.e., an Ino.53Alo.47As/6xTS/5xCS/6xTS ZIno.53Alo.47As structure, in a 3.4 mm long waveguide, a 90° polarization rotation could be achieved when the electric field was increased from 20 to 25 kV/cm. Here, the magnitudes of the changes in "cf f .TM and H C ff ,TE are relatively closely matched so that only a small chirp au < 0.1 [3] should arise due to nonlinear refractive index and absorption changes in the two structures. Also, a similar result would be obtained or an Ino.53Alo.47As/lOxTS/6xCS/Ino.53Alo.47As structure. In both examples, the linear electro-optic contributions have been taken into account. R e f e r e n c e s [1] C. Thirstrup, "Refractive index modulation based on excitonic effects in GalnAs-InP coupled asymmetric quantum wells," IEEE J. Quantum Electron., vol. 31, no.6, pp. 988-996, 1995. [2] H . Feng, J. P. Pang, M . Sugiyama, K . Tada, Y . Nakano, "Field-induced optical effect in a five-step asymmetric coupled quantum well with modified potential," IEEE J. Quantum Electron., vol. 34, no. 7, pp. 1197-1208, 1998. [3] G. L . L i and P. K . L . Y u , "Optical intensity modulators for digital and analog application," J. Lightw. Technol., vol. 21, no. 9, pp. 2010-2030, 2003. 129 [4] A . H . Gnauck, et al., "6 x 42.7-Gb/s transmission over ten 200-km EDFA-amplified S S M F spans using polarization-alternating R Z - D P S K , " O F C 2004, in Proc. OFC, vol. 2, 2004. [5] M . P. C. M . Krijn, "Ffeterojunction band offsets and effective masses in I I I -V quaternary alloys," Semiconduct. Sci. Technol., vol. 6, pp. 27-31, 1991. [6] S. L . Chuang, Physics of Optoelectronic Devices. New York: Wiley, 1995, pp. 154-157. [7] I. Galbraith and G. Duggan, "Exciton binding energy and external-field-induced blue shift in double quantum wells," Phys. Rev. B, vol. 40, no. 8, pp. 5515-5521, 1989. 130 Chapter 3 Robustness of Push-Pull CQW Polarization Modulator 3.1 Introduction Studies on optimization of electro-optic [1] and nonlinear [2] properties of asymmetric coupled quantum well (ACQW) structures have been performed for many years now. Together with three-step quantum well structures [3], A C Q W structures [4] have been shown to provide electrorefractive responses that are considerably greater than the ones of square quantum wells. Recently, the authors proposed two InGaAlAs-InAlAs A C Q W structures, grown on (001) InP, which, due to their electrorefractive properties, could be used in polarization-Reprinted with permission from S. Ristic and N. A. F. Jaeger, "Influence of layer thickness and compositional variations on the electrorefractive properties of a quantum well polarization-conversion modulator," J. Vac. Sci. Technoi. A-Vacuum, Surfaces, and Films, vol. 24, no. 4, pp. 962-965, Jul./Aug. 2006. Copyright [2006], American Vacuum Society. 131 conversion as well as intensity-modulation applications [5]. In one of the two structures, which contains a quantum well that is under tensile strain, the TS structure, a change in an applied electric field wi l l increase the refractive index seen by TM-polarized light significantly more than the refractive index seen by TE-polarized light. In the other structure, which contains a quantum well that is under compressive strain, the CS structure, the same change in applied electric field wi l l decrease the refractive index seen by TE-polarized light significantly more than the refractive index seen by TM-polarized light. Because each of the structures primarily affects only one of the supported polarizations, and thereby only one of the modes, placing multiple repetitions of either structure in an optical waveguide can facilitate an electric-field-induced change in the polarization state of the output light for input light that contains both the fundamental T E and T M modes. At the output of the waveguide, the powers of the two modes must be matched in order to achieve switching between two orthogonal polarization states (i.e., to achieve 90° polarization rotation). However, i f multiple repetitions of both the TS and CS structures are placed in the optical waveguide of a modulator, the two structures can be made to provide an opposite-sign, and approximately equal-magnitude, change of the effective refractive indices for the two modes. Such a device, which we refer to as a quantum well polarization-conversion (QWPC) modulator, should provide push-pull type of polarization modulation, a mode of operation that is not typical for quantum well modulators and that wi l l find use in an ever-increasing number of polarization-modulation applications [5]-[7]. 132 Figures 3.1(a) and (b) illustrate the thicknesses and compositions of the layers in the TS and CS structures, respectively, as well as the direction of the applied electric field F [5]. By placing appropriate numbers of repetitions of these TS and CS structures, so as to give overlap factors of -0.41 and -0.22 for the fundamental T E and T M modes in a waveguide forming a Q W P C modulator, respectively, a push-pull polarization conversion giving a small chirp parameter, i.e. aH » 0.05 [8], should be possible to achieve in,a 3.0 mm long device when the electric field is switched from 20 to 25 kV/cm. This chirp would mainly be due to nonlinearities in the absorption and refractive index changes in the two structures [5]. Ino.53Alo.47AS barrier Ino.53Gao.47AS 72 A Ino.53Alo.47AS 36 A Ino.46Gao.52Alo.02AS 66 A Ino.53Alo.47AS barrier (a) F Ino.53Alo.47AS barrier fno.53Gao.47AS 69 A Ino.53Alo.47AS 12 A Ino.58Gao.37Alo.05AS 45 A lno.53Alo.47AS barrier (b) F Figure 3.1: (a) TS structure containing a quaternary quantum well which is under -0.50 % of tensile strain and (b) CS structure containing a quaternary quantum well which is under -0.34 % of compressive strain. 3.2 Influence of Layer Variations 133 As the next step in developing the TS and CS structures for use in a Q W P C modulator, we investigate how the electrorefractive properties of these structures depend on the thickness and compositional variations of their layers. The investigation reveals the trends and issues pertaining to the variations, and it indicates the design objectives for optimizing the performance of the structures. For each structure, the thickness of its unstrained quantum well, w i , and the thickness of its strained quantum well, w%, are varied by ±Aw\ and ±Au>2, respectively. The gallium mole fractions in both quantum wells, x\ and X2, are varied by the same amount, ±Ax, and the aluminum mole fraction in the quaternary, strained quantum well, zi, is varied by ±Az. Even when neglecting the compositional and thickness variations for the barriers in each structure, the remaining two thickness and three compositional variations necessitate 3 5 (=243) different simulations of each structure in order to obtain the statistical average absorptions and refractive indices of the two structures [4]. The statistical average responses, as figures of merit for the layer thickness and compositional variation effects on the TS and CS structures, are calculated as described below. If in one of the variations of a structure, say structure / , x\ is decreased by Ax, Z2 is increased by Az, and w2 is increased by Aw 2 , the probability of its occurrence wi l l be Pi=0--Pti-PldPlxPlitt ~Pli ~P'x2)Pz2- (3-1) 134 Here, p*wl and p~wX are the probabilities of w{ changing by +Awi and -Awi , respectively, and pxl is the probability of x\ changing by -Ax. Similarly, p*w2 and p*2 are the probabilities of w2 and z 2 changing by +Aw 2 and +Az, respectively, while px2 and p'x2 are the probabilities of x 2 changing by +Ax and -Ax, respectively. In the rest of the discussion, we wi l l assign the probability pw to p~wX, p* 2 , and p~2. Similarly, we wi l l assign the probability pc to pxi, p ~ , p+x2, px2, p+z2, and p~2. Also, the statistical average absorption at wavelength, A , and electric field, F, of a structure for either polarization is calculated as where apol(str) (A, F) is the absorption for the /th structure for either polarization (pol = T E or T M ) and structure (str = TS or CS) [4]. The statistical average refractive index for either polarization and structure, « p o i ( s t r ) ( A , . F ) , is obtained from a p oi( Str) (A, F) via the Kramers-Kronig relations, and both quantities are calculated relative to their values at F = 14 kV/cm, which is taken as the lower limit of the region of interest for the electric field. 243 (3.2) 3.2.1 TS Structure 135 The TS structure was first analyzed by allowing the thicknesses of its two quantum wells to change by ±5 % and allowing x in both wells and z in the strained well to change by ±0.02. At the wavelength of operation A = 1550 nm and for electric fields between F = 14 and 28 kV/cm, where the positive change of its refractive index for T M polarization A«TM(TS) [5] is larger than the positive change of its refractive index for TE polarization AWTE(TS) [5], Fig. 3.2(a) compares nTM{TS) obtained forpc = 0 andpw = 0 (n T M i T S ) ° ' ° ) with nTM(TS) obtained forpc = 0.10, 0.20, 0.33 andpw = 0.33. Similarly, Fig. 3.3(a) compares —, 9 I i i 1 1 1 1 1 | 1 1 1 1 1 1 1 | 1 1 1 r ~ — i 1 1 14 16 18 20 22 24 26 28 14 16 18 20 22 24 26 28 14 16 18 20 22 24 26 28 Electric Field (kV/cm) Electric Field (kV/cm) Electric Field (kV/cm) (a) (b) (c) Figure 3.2: nTM(TS) for (a) Aw\ = ± 5 % of w\, Aw2 = ± 5 % of w 2 , Ax = Az = ±0.02, (b) Awi = Awz = ±1 A , and Ax = Az = ±0.01, and (c) Awi = Aw>2 = ±1 A , and Ax = Az = ±0.005. 136 HTE(TS) w i m nTE{TS)0'0 for the same pc and pw values. nTM(TS)u'" and nTE(TS)v'" therefore represent the electric-field-induced refractive index changes of the nominally designed i structure, when they are adjusted to be zero for F= 14 kV/cm, as shown in the two figures. In Fig. 3.2(a), the obtained nTM(TS) curves behave similarly to nTM(TS)0'0. This is not true fo rw r £ ( 7 S ) of Fig. 3.3(a), which is seen to have a pronounced zigzag pattern. The reason for this zigzag pattern is that the confinement of the lowest energy heavy-hole _ 2.5 0,0 0 ,0 1 1 1 1 1 1 A // / - / -TE • i • i 16 18 20 22 24 26 28 14 16 18 20 22 24 26 28 14 16 18 20 22 24 26 28 Electric Field (kV/cm) Electric Field (kV/cm) Electric Field (kV/cm) (a) (b) (c) Figure 3.3: nTE(TS) for the same variations as Fig. 3.2. wave function (^ HHI) [5] wi l l be weakened for any structure variation that decreases the nominal depth or thickness of the unstrained well or increases the nominal depth or the thickness of the strained well. YHHI wi l l , therefore, transfer to the strained well for lower values of the electric field than it would in the nominal structure contributing to a refractive index change for the TE polarization [9]. Due to the large heavy-hole mass and 137 the large thickness of the middle barrier, these WHHI transfers, i.e., heavy-hole exciton anticrossings [1], [5] wi l l occur for very small electric-field increments causing the rapid changes nTE{TS) • Fig. 3.3(a) does not capture all of the features of nTE(TS) as the data in the figures have been obtained and plotted in 1 kV/cm increments in F. Similar to Figs. 3.2(a) and 3.3(a), Figs. 3.2(b) and 3.3(b) show nTM(TS) and nTE(TS) for the same values of pc and pw except that now Aw\ = Aw2 = ±1 A , Ax = ±0.01, and Az = ±0.01. Even though, for these smaller variations, the refractive index for T M -polarized light changes less from its nominal value, the variations are still not small enough to eliminate the heavy-hole exciton anticrossings that contribute to nTE(TS) • However, here, the number of variations of the structure that result in the heavy-hole exciton anticrossings for the electric fields between F = 14 kV/cm and F = 28 kV/cm is smaller than in Fig. 3.3(a) so that none of the resulting zigzag features of the nTE(TS) curves are captured. However, decreasing the compositional variations in the structure even further, [see Fig. 3.2(c) and Fig. 3.3(c)], so that Aw\ = A w 2 = ±1 A , Ax = ±0.005, and Az - ±0.005 does eliminate the zigzag pattern successfully and makes the nTM(TS) and nTE(TS) curves overlap almost entirely with the nTM(TS) ' and ^TE(TS) curves. Here, the absence of the zigzag pattern in nTE{TS) is real, and it is not an artifact of the sampling, as it was in Fig. 3.3(b). This is confirmed by simulating the most extreme variation of the structure, which is the one that gives the narrowest and shallowest unstrained well and 138 the widest and deepest strained well, and observing that *FHHI does not transfer to the strained well for the electric fields of interest. 3.2.2 CS Structure Similar to Figs. 3.2 and 3.3, Figs. 3.4 and 3.5, show nTE(CS) and nTM(CS) of the CS structure for the same well thickness and compositional variations. Compared to the TS ^ 2 '© 00 § -2 CJ Z -o A tS -6 3 a -io 0-12 PcPw 0.00, 0.00 0.10, 0.33 0.20, 0.33 0.33,0.33 i i i i i 1 1 TE i i i i \ - v ~* \ \ v \ \ v -\ \ _ \ \ • i \ \ \ .^ 4 16 18 20 22 24 26 28 14 16 18 20 22 24 26 28 14 16 18 20 22 24 26 28 Electric Field (kV/cm) Electric Field (kV/cm) Electric Field (kV/cm) (a) (b) (c) Figure 3.4: nTE(CS) for the same variations as Fig. 3.2. structure, nTE(CS) for the CS structure deviates significantly from its nominal value, even for the smallest thickness and compositional variations, [see Fig. 3.4(c)]. The CS structure has a greater sensitivity to the compositional and thickness variations because the CS structure, for its operation, relies on the heavy holes, i.e., the anticrossing of the 139 14 16 18 20 22 24 26 28 14 16 18 20 22 24 26 28 14 16 18 20 22 24 26 28 Electric Field (kV/cm) Electric Field (kV/cm) Electric Field (kV/cm) (a) (b) (c) Figure 3.5: nTM(CS) for the same variations as Fig. 3.2. heavy-hole excitons, and the TS structure on the light holes, i.e., the anticrossing of the light-hole excitons. The heavy holes, because of their greater mass, are closer to bottoms of the quantum wells and more susceptible to changes of the wells' compositions and widths. Also, the heavy-hole band edge (EHH) and the conduction band edge (Ec) change more dramatically than the light-hole band edge (Ew) when a variation in composition occurs [10]. For example, for the strained well of the CS structure, Ew wi l l change much less than £ H H and Ec when x is changed from its nominal value of 0.37. In fact, £ L H w i l l increase (toward positive energies) by less than 1 meV when x is increased to 0.39. For the same change in x, EHH wil l decrease by ~9 meV, and EQ wi l l increase by ~7 meV. It is, therefore, not surprising that nTM(TS) and nTM(CS) show better performance than nTE(TS) and nTE(CS), both of which strongly depend on the behaviour of the heavy holes. 140 3.3 Discussion In analyzing the TS and CS structures of Fig. 3.1, each of which could be used separately, or both of which could be combined to form a push-pull Q W P C modulator, the electrorefractive properties of the TS structure prove to be less sensitive to the structure variations provided that it has been designed so that no variation of the structure allows the heavy-hole excitons to anticross. The electrorefractive properties of the TS structure wil l then barely deviate from their nominal values for compositional variations that are smaller than or equal to -0.01 and for the well thickness variations o f -1 A . This structure would be useful in any device requiring that upon the application of a (001) electric field the refractive index change for TM-polarized light be significantly greater than the refractive index change for TE-polarized light. The current design of the CS structure is significantly more sensitive to small variations in thickness and composition than that of the TS structure. For example, considering the effect of a 1 A thickness and 0.005 compositional variation on the performance of a Q W P C modulator for all the different values of pc and pw used in the figures above, we find that as pc increases from 0 to 0.10, 0.20, and 0.33 (again, here, pw = 0.33 for all values of pc except for the nominal structure, where they are both zero), the length of the 141 Q W P C modulator needed to maintain a low drive electric field of 5 kV/cm increases from 3.0 mm to 3.7, 4.1, and 4.9 mm, respectively. Also, the chirp parameter corresponding to />c=0.10, 0.20, and 0.33 wi l l approximately increase to 1, 1.5, and 3, respectively. While we conclude that very tight control of the fabrication tolerances would be needed for the CS structure in a low-chirp Q W P C modulator, this study has shown that a practical TS structure should be possible and provides the motivation for further investigation into the design of the CS structure. R e f e r e n c e s [1] C. Thirstrup, "Refractive index modulation based on excitonic effects in GalnAs-InP coupled asymmetric quantum wells," IEEE J. Quantum Electron., vol. 31, no.6, pp. 988-996, 1995. [2] J. Khurgin, "Second-order nonlinear effects in asymmetric quantum-well structures," Phys. Rev. B., vol. 38, no. 6, pp. 4056-4066, 1998. [3] H. Mohseni, H . Ah , Z. A . Shellenbarger, M . H . Kwakernaak, and J. H . Abeles, "Enhanced electro-optic effect in GalnAsP-InP three-step quantum wells," Appl. Phys. Lett., vol. 84, no. 11, pp. 1823-1825,2004. [4] K . Tada, T. Arakawa, K . Kazuma, N . Kurosawa, and J.-H. Noh, "Influence of one monolayer thickness variation in GaAs/AlGaAs five-layer asymmetric coupled quantum well upon electrorefractive index change," 142 Jpn. J. Appl. Phys., vol. 40, no. 2A, pt. 1, pp. 656-661, 2001. [5] S. Ristic and N . A . F. Jaeger, "Push-pull polarization conversion using novel asymmetric coupled quantum-well structures," IEEE Photon. Technol. Lett., vol. 18, no. 2, pp. 316-318, 2006. [6] F. Heismann, "Compact electro-optic polarization scramblers for optically amplified lightwave systems," / Lightw. Technol., vol. 14, no. 8, pp. 1801-1814, 1996. [7] A . L . Campillo, "Interchannel nonlinear crosstalk in analog polarization modulated W D M systems,"/. Lightw. Technol., vol. 24, no. 3, pp. 1186-1193, 2006. [8] G. L . L i and P. K . L . Y u , "Optical intensity modulators for digital and analog application," / Lightw. Techno!., vol. 21, no. 9, pp. 2010-2030, 2003. [9] R. Lang and K . Nishi, "Electronic state localization in semiconductor superlattices," Appl. Phys. Lett., vol. 45, no. 1, pp. 98-100, 1984. [10] S. L . Chuang, Physics of Optoelectronic Devices. New York: Wiley, 1995, pp. 154-157. Chapter 4 Robust C Q W Electrorefraction Modulator 143 4.1 Introduction Similar to square quantum well (SQW) structures, coupled quantum well (CQW) structures have been studied for their use in both electroabsorption and electrorefraction (ER) modulators made in the InP and GaAs material systems [l]-[4]. Although, in E R modulators, the magnitude of the electric-field-induced refractive-index change achievable with C Q W structures is similar to that achievable with SQW structures, C Q W structures have the advantage that they exhibit no significant absorption modulation [1]. A n additional advantage of C Q W structures is that they provide the refractive-index © [2007] IEEE. Reprinted, with permission, from S. Ristic and N. A. F. Jaeger, "Robust Coupled-Quantum-Well Structure for Use in Electrorefraction Modulators," IEEE. Electron Dev. Lett., vol. 28, no. 1, pp. 30-32, Jan. 2007. 144 change for considerably lower applied electric field F (leading to much lower drive voltages in devices using them) [l]-[3]. Until now, C Q W structures have had the disadvantage that they are typically very sensitive to layer thickness and compositional variations [2], [3], [5]. Here, we introduce an InGaAs/InAlAs C Q W structure (structure #1) that is very robust to these layer thickness and compositional variations. To achieve this robustness, this structure has its E R effect based on the anticrossing of the lowest energy light-hole wave function (*FLHI) with the second lowest energy light-hole wave function (*FLH2) and, consequently, is more appropriate for use in transverse magnetic (TM) E R modulators [6]. We compare our structure to a similar InGaAs/InAlAs C Q W structure (structure #2) that has its E R effect based on the anticrossing of the lowest energy electron wave function (WEI) with the second lowest energy electron wave function QVE2), and as such, the structure is more appropriate for use in transverse electric (TE) E R modulators [1]. In our comparison, we show that structure #1 has superior robustness, because its E R effect is based on the anticrossing of the light-hole wave functions. We also offer an explanation as to why designing an InGaAs/InAlAs C Q W structure that has its E R effect based on the anticrossing of light holes rather than electrons, or heavy holes should provide for better robustness to layer thickness and compositional variations. 4.2 Influence of Layer Variations 1 4 5 In our model, the energy bands are obtained using the model-solid theory [ 7 ] . The absorption spectra are obtained using the variational method to solve the exciton equation in the effective-mass approximation [ 8 ] . The Lorentzian broadening function used in the model is described in [ 9 ] , and the refractive-index changes are obtained using the Kramers-Kronig relation [ 1 ] . Our model is very similar to that used in [ 3 ] and [ 1 0 ] except that, in our model, the refractive-index changes are obtained for a whole period of a C Q W structure and not only for the quantum well layers. Fig. 4.1(a) illustrates W E i , W E 2 , W L H I , W L H 2 , and the two lowest energy heavy-hole wave functions ( W H H I and WHH2), as well as the band-edge diagram of structure #1 for F = 0 , 1 5 , and 3 5 kV/cm. The structure has Ino.52Alo.48As barriers and Ino.53Gao.47As wells, both of which are lattice-matched to ( 0 0 1 ) InP. The thickness of each of the two external barriers is 9 0 A , while the thickness of the middle barrier is 2 7 A . The thicknesses of the left-hand side well ( Q W 1 ) and the right-hand side well ( Q W 2 ) are 7 2 and 6 0 A, respectively (i.e., structure #1 is a 9 0 - 7 2 - 2 7 - 6 0 - 9 0 structure). As shown in Fig. 4.1(a), for F = 0 kV/cm, W L H I and W E i will be mainly localized in the wider Q W 1 , while W L H 2 and W E 2 will be mainly localized in the narrower Q W 2 . As F is 1 4 6 0 kV/cm • 15kV/cm • 35 kV/cm • A A V A A V 0 kV/cm lOkV/cm • 35 kV/cm A V A H \ J A V (a) (b) Figure 4 . 1 : Energy band diagram of structure #1 and WEI and W E 2 , W L H I and W L H 2 , and W H H I and W H H 2 for F = 0 , 1 5 , and 3 5 kV/cm. Here, the size of the band gap and the thicknesses of the external barriers are not shown to scale. W H H I is shown as dashed thick line, while W H H 2 is shown as dashed thin line. W E i and W L H I are shown as full thick lines, while W E 2 and W L H 2 are shown as full thin lines, (b) Analogously, band diagram and wave functions of structure # 2 for F= 0 , 1 0 , and 3 5 kV/cm. increased to 1 5 kV/cm, W L H I and W L H 2 w i l l become approximately equally distributed between the two wells, while WE I and W E 2 wi l l become more strongly localized to the same wells as they were previously. As F is further increased to 3 5 kV/cm, W L H I wi l l switch to Q W 2 and W L H 2 wi l l switch to Q W l , while WEI and W E 2 wi l l be yet more strongly localized to their respective wells. Similar to [ 6 ] , here, the electric field at which the anticrossing of W L H I and W L H 2 occurs is called the resonant electric field for the light holes F L H (below, we wi l l also use FE for the resonant electric field for the,electrons). For structure # 1 , F L H ~ 1 5 kV/cm. Consequently, above FLH, the electric-field-induced 147 refractive-index change for the T M polarization ( A « T M I ) w i l l have a smooth positive slope and the electric-field-induced refractive-index change for the T E polarization ( A « T E I ) w i l l also have a smooth positive, although considerably smaller, slope [6] as shown in Fig. 4.2. Here, AWTEI and A « T M I (shown relative to their zero-electric-field values) are plotted for X, = 1550 nm and the energy separation between ^EI and WLHI 10 15 20 25 Electric Field (kV/cm) 30 35 Figure 4.2: Structure #1 and structure #2 refractive-index changes for T E and T M polarizations. corresponds to X. = 1402 nm for F = 0 kV/cm. Below F W , A H T M I and A « T E I wi l l have smooth negative slopes except at F ~ 4.5 kV/cm, where there is a large dip in A « T E I - The dip is due to the anticrossing of WHHI and T H H 2 at FHH - 4 . 5 kV/cm, and it does not appear in A«TMI as the heavy-hole momentum matrix element is zero for T M light. The 148 dip is very sharp because of the large effective masses of WHHI and W H H 2 and the relatively large thickness of the middle barrier [10]. Analogous to structure #1, structure #2 is a 90-51-30-54-90 lattice-matched structure, and its wave functions and the band-edge diagram are shown in Fig. 4.1(b) for F= 0, 10, and 35 kV/cm. Unlike structure #1, which is designed for modulation of T M light via the anticrossing of WLHI and W L H 2 , structure #2 is designed for modulation of T E light via the anticrossing of W E T and W E 2 (FE ~ 10 kV/cm), while all other wave functions remain localized to their respective wells for the electric fields of interest. In structures of this type, the refractive-index change for T E light is somewhat larger than the refractive-index change for T M light and the structures are, therefore, better suited for use in T E modulators [1]. Here, A « T E 2 and A « T M 2 (shown relative to their zeroelectric-field values) are also plotted for X. = 1550 nm and the energy separation between WEI and WHHI corresponds to X. = 1395 nm for F = 0 kV/cm. Note: For the purpose of comparison, structure #1 and structure #2 were designed to have similar magnitudes of A H T M I and A « T E 2 above their corresponding resonant fields and similar resonant fields for light holes and electrons and, furthermore, they were designed to have nearly equal wavelength detunings of their lowest energy anticrossing wave functions from the wavelength of operation ( X = 1550 nm), which should yield similar absorption coefficients at the wavelength of operation. 149 To compare sensitivities to the layer variations, in each structure, the thicknesses of the two quantum wells and the middle barrier are changed by ±3 A . In [3], it is shown that even such small thickness variations can significantly modify the distributions of wave functions in C Q W structures and, in turn, influence the absorption spectra and the refractive-index changes in these structures. Similarly, the InAs mole fractions in the three barriers and each quantum well are changed by ±0.02. Treating the compositional and thickness variations separately amounts to 35(= 243) compositional and 33(= 27) thickness variations of each of the two structures. Each thickness variation is assigned the same occurrence probability of 1/27, and similarly, each compositional variation is assigned the same occurrence probability of 1/243. The refractive-index changes for either polarization corresponding to each of the variations are multiplied by the probability of the variation's occurrence and then added together in a statistical average, which is, again, obtained separately for the thickness and the compositional variations. Prior to any variations of structure #1, A « T M I is plotted as A « T M I ° in Fig. 4.3(a) relative to its value at F= 25 kV/cm for electric fields between Fm and F =35 kV/cm. The average refractive-index changes for the thickness variations ( A H T M I 7 ) and for the compositional variations ( A « T M I C ) are also plotted in Fig. 4.3(a). Similarly, Fig. 4.3(b) shows the corresponding curves for A « T E 2 for structure #2 as A « T E 2 ° , A « T E 2 f , and A « T E 2 , respectively. It is shown in Fig. 4.3(b) that there wil l be certain variations (only compositional variations for this particular design of structure #2) that wi l l allow WHHI and W H H 2 to anticross in the electric-field region of interest, which creates the sharp dip 150 shown in the TE curve (the data in Fig. 4.3 are obtained and plotted in 1 kV/cm increments of F, which has resulted in loss of detail as regards the features of this dip). 8 6 U T3 C S -4 Pi 20 25 30 Electric Field (kV/cm) 35 10 15 20 25 30 Electric Field (kV/cm) 35 Figure 4.3: (a) Dependence of structure #1 refractive-index changes for TE and T M polarizations on the thickness and the compositional variations and (b) the same for structure #2. Even ignoring the heavy-hole anticrossings, it is clear from Fig. 4.3 that AMTMI is less sensitive to both thickness and compositional variations than A«TE2- Here, A H T M I ° and A « T M I C are practically indistinguishable for all electric fields, which means that structure #1 shows practically no sensitivity to the compositional variations. The same cannot be said for structure #2, as in the region of the greatest slope of A « T E 2 ° (15 kV/cm < F < 25 kV/cm), A « T E 2 ° = 7.3 x 10"4, and it is -77 % larger than A n T E 2 C Structure #1 is also far less sensitive to the thickness variations than structure #2. In the region of the greatest 1 5 1 slope of A « T M I ° ( 2 0 kV/cm < F< 3 0 kV/cm), A « T M I ° = 8 . 7 x 1 ( V 4 , and it is only ~ 1 1 % larger than A « T M i r , whereas in the region of the greatest slope of A « T E 2 ° , A « T E 2 ° is - 1 5 7 % larger than AnjE2T- Above F ~ 2 5 kV/cm, A « T E 2 does not seem greatly influenced by the variations, however, its slope in this region is considerably smaller than its maximum slope, The use of decoupled random variations of the layer thicknesses and compositions is intended to illustrate the robustness of structure #1 to such variations, but it is not intended to model any specific growth process. The main reason for the smaller sensitivity of structure #1 to thickness variations is that the barriers confining the anticrossing W L H I and WLH2 in structure #1 are significantly lower than the barriers confining the anticrossing WE I and W E 2 in structure #2 . When close to their respective resonances, W L H I and WLH 2 are thus spread out over a larger region in space than WE I and WE2 , and W L H I and W L H 2 are less influenced by small thickness variations of the quantum well layers because they overlap less with these layers. For the same reason, any compositional variations wi l l also have a smaller effect on W L H I and WLH2 in structure #1 than they wi l l on W E i and WE2 in structure #2 . Our simulations show that InGaAs/InAlAs C Q W structures based on the anticrossing of WHHI and W H H 2 [ 6 ] , [ 1 1 ] do not benefit from the small valence band discontinuity ratio to the extent that structure #1 does, because the effective masses of W H H I and WHH2 are much larger than those of W L H I and WLH2 and, thus, WHHI and W H H 2 are very sensitive to layer variations [ 5 ] . Also, even though the light 152 holes have larger effective masses than the electrons, their reduced localizations make them less sensitive to the layer variations. Another reason for the smaller sensitivity of structure #1 to thickness variations is that its quantum wells are thicker than those in structure #2 so that the ±3 A thickness variations represent smaller percentage changes to the thicknesses of the quantum wells in it. In order for structure #1 and structure #2 to have similar wavelength detunings, structure #1 has to have thicker quantum wells because the energy separation between W L H I and W E I is larger than that between *PHHI and W E i in lattice-matched C Q W structures. In addition to the benefit that small valence band discontinuity ratio has on the sensitivity of structure #1 to compositional variations, these variations wi l l have considerably smaller effects on the energy levels of the light-hole band edge than they wil l on the energy levels of the electron or the heavy-hole band edges because of the opposite-sign contributions of the hydrostatic and the shear strain components to the light-hole band edge [7]. Any wave function in a C Q W structure wi l l be far less sensitive to the layer variations when it is strongly localized to one of the two wells, as opposed to when it is approximately equally shared by both of the wells [5], which is the case when it is close to resonance and anticrossing with its counterpart of opposite symmetry. Consequently, for the electric fields of interest for structure #1, the variations wi l l have very small effects on the otherwise sensitive W E i , W E 2 , W H H I , and WHH2 as they are strongly localized. This is unlike other C Q W structures that have the disadvantage of basing their E R effects 153 on anticrossing of either W E i and or ¥ H H I and W H H 2 , rather than the insensitive W L H I and W L H 2 . R e f e r e n c e s [1] C. Thirstrup, "Refractive index modulation based on excitonic effects in GalnAs-InP coupled asymmetric quantum wells," IEEE J. Quantum Electron., vol. 31, no.6, pp. 988-996,1995. [2] J.-H. Noh, S. Hasegawa, T. Suzuki, T. Arakawa, K . Tada, and H . Asahi, "Migration-enhanced epitaxy (MEE) growth of five-layer asymmetric coupled quantum well (FACQW) and its cross-sectional S T M observation," Physica E, vol. 23, no. 3/4, pp. 482-486, 2004. [3] K . Tada, T. Arakawa, K . Kazuma, N . Kurosawa, and J.-H. Noh, "Influence of one monolayer thickness variation in GaAs/AlGaAs five-layer asymmetric coupled quantum well upon electrorefractive index change," Jpn. J. Appl. Phys., vol. 40, no. 2A, pt. 1, pp. 656-661,2001. [4] T. H . Stievater,W. S. Rabinovich, P. G. Goetz, R. Mahon, and S. C. Binari, " A surface-normal coupled-quantum-well modulator at 1.55 ^im," IEEE Photon. Technoi. Lett., vol. 16, no. 9, pp. 2036-2038, 2004. [5] R. Lang and K . Nishi, "Electronic state localization in semiconductor superlattices," Appl. Phys. Lett., vol. 45, no. 1, pp. 98-100, 1984. 154 [6] S. Ristic and N . A . F. Jaeger, "Push-pull polarization conversion using novel asymmetric coupled quantum-well structures," IEEE Photon. Technoi. Lett, vol. 18, no. 2, pp. 316-318, 2006. [7] M . P. C. M . Krijn, "Heterojunction band offsets and effective masses in I I I -V quaternary alloys," Semiconduct. Sci. Technoi, vol. 6, pp. 27-31, 1991. [8] S. L . Chuang, Physics of Optoelectronic Devices. New York: Wiley, 1995, pp. 561-566. [9] N . Susa, "Electric-field-induced refractive index changes in InGaAs/InAlAs asymmetric coupled quantum wells," IEEE J. Quantum Electron., vol. 31, no. 1, pp. 92-100, 1995. [10] T. Arakawa, K . Tada, N . Kurosawa, and J.-H. Noh, "Anomalous sharp dip of large field-induced refractive index change in GaAs/AlGaAs five-layer asymmetric coupled quantum well ," Jpn. J. Appl. Phys., vol. 39, no. 11, pp. 6329-6333, 2000. [11] S. Ristic and N . A . F. Jaeger, "Influence of layer thickness and compositional variations on the electrorefractive properties of a quantum well polarization-conversion modulator," J. Vac. Sci. Technoi. A, Vac. Surf. Films, vol. 24, no. 4, pp. 962-965, 2006. Chapter 5 Improving Robustness in CQW Structures 155 5.1 Introduction The coupled quantum well (CQW) structures used in electrorefraction (ER) and electroabsorption (EA) modulators can offer lower drive electric fields, i.e., lower drive voltages, than the square quantum well (SQW) structures typically used in these devices [1], [2]. Increasing the thickness of the middle barrier separating the two quantum wells in a C Q W structure can be used to further increase the sensitivities of the structure's electronic states to changes in the applied electric field, F, leading to lower drive voltages [3]. However, it has been shown that increasing the thickness of the middle barrier A version of this chapter will be submitted for publication. S. Ristic and N. A. F. Jaeger, "Improving robustness to layer thickness and compositional variations of coupled quantum well structures," J. Vac. Sci. Technol. A- Vacuum, Surfaces, and Films. 156 increases the sensitivity of the electronic states to any thickness and compositional variations that may occur in the quantum well layers during the growth process [4], and the operation of devices using C Q W structures can, therefore, be seriously compromised. Conversely, the middle barrier in C Q W structures used in modern E A modulators is quite thin, and the potential of C Q W structures to provide very low drive voltages is not fully realized [2]. In this article, we show how decreasing the depth of the quantum wells and simultaneously increasing the thickness of the middle barrier in a C Q W structure can be used to reduce the sensitivities of the electronic states to the layer variations without sacrificing their sensitivities to changes in F. 5.2 Robustness improvement approach The C Q W structure of interest here, structure S A , is lattice-matched and contains two 51 A Ino.53Gao.47As wells that are separated by a 30 A Ino.52Alo.4gAs middle barrier and enclosed by 90 A Ino.52Alo.4sAs external barriers. The energy difference between the wells and the barriers in the conduction band is AEc - 481 meV [5]. The electron effective mass used in the wells and the barriers is m c * / mo = 0.0427 + 0.0328 / 0.48 x z, where z represents the A l mole fraction in In(i. z . y )Al z Ga y As [6]. The ground state electron wave function, W E , in the C Q W structure is obtained by solving the one-dimensional, single-particle, time-independent Schrodinger equation using the transfer matrix method [7]. In order to make sure that the results presented here are not influenced by the 157 particular method of solving the Schrodinger equation, the results have also been confirmed using the finite-difference method [8], where S A is enclosed in a larger quantum well with infinite barriers. Fig. 5.1(a) shows S A and W E for F = 0 kV/cm before (full lines) and after (dashed lines) the thickness of the left-hand side well (tw\) has been increased by 3 A . Before the increase in tw\, W E wi l l be equally distributed between the two wells because S A is a i ; • *" — i i • - \ AEC > n ij *y tmb 90 A ?wl 90 A Afwl — (a) (b) Figure 5:1: (a) S A structure and its electron ground state wave function shown for F = 0 kV/cm before (full lines) and after (dashed lines) the thickness of the left-hand side well is increased by 3 A and (b) same for S B structure. 158 symmetric structure. After the increase in tw\, W E wi l l be more strongly localized to the wider well. One way of quantifying the localization of W E in one of the two wells is to use the "localization parameter" P [4 ] which is defined as P = (pi - pi) I (p\ + pi), where p\ and pi are the probabilities of finding the ground state electron in the left-hand side well and the right-hand side well, respectively. For example, in Fig. 5.1(a), for W E , P = 0 before the increase in tw\ and P = 0 . 6 0 after the increase in t^\. Fig. 5.1(b) shows how W E responds to a 3 A change in Avi for F= 0 kV/cm after the heights of the barriers have been decreased to A£c = 219 meV (i.e., for Ino.52Alo.24Gao.24As lattice-matched barriers), and the thickness of the middle barrier (/mt>) has been increased to 4 6 A . This new structure is labeled S B . It is clear from Fig. 5.1(b) that the localization of W E does not change as much in S B as it does in S A for the same change in tw\ for F = 0 kV/cm; now P changes from 0 to 0 . 3 7 . Furthermore, the larger thickness of the shorter middle barrier in S B causes the localization of W E in S B to have the same response to changes in F as the localization of W E in S A does, as shown in Fig 5.2. This behaviour is both interesting and useful. In Fig. 5.2, the electric-field dependences of localizations of W E in S A and W E in S B are illustrated by plotting the values of P for SAand S B , respectively. The two curves are practically indistinguishable for all values of F. In a similar fashion, it is also shown in Fig. 5.2 that W E in S B is less sensitive to a 3 A thickness variation (A? wi = + 3 A) than W E 159 •+-» 4> 03 OH a o -4—» cd N 8 -0.2 o 10 20 30 Electric Field (kV/cm) Figure 5.2: Localization parameters corresponding to structures S A and S B for a 3 A increase of the thickness and a 10 meV decrease of the depth of the left-hand side well. in S A is, for all values ofF. In addition, the remaining two curves in Fig. 5.2 show that W E in S B wi l l also be less susceptible to changes in the depth of the quantum wells. Here, the depths of the left-hand side wells in S A a n d S B have been decreased by 10 meV (Adw\ = -10 meV) by making the wells out of quaternary compounds, here Ino.52Alo.01Gao.46As wells. Compared to W E in S A , W E in S B is less sensitive to thickness and compositional variations in the quantum wells because the barriers in S B are shorter than the barriers in S A - Consequently, W E wi l l be located in the external barriers to a greater extent, see Fig 5.1(b), and it wi l l be less influenced by any changes that occur in the quantum wells, while still providing the same response to the applied field. Fig. 5.3 shows that similar changes to the thicknesses and compositions of the middle barriers in S A and S B wi l l have smaller effects on W E than the changes in the quantum 1 Electric Field (kV/cm) Figure 5.3: Localization parameters corresponding to structures S A and S B for a 3 A increase of the thickness and a 10 meV decrease of the height of the middle barrier. wells do. Here, the thicknesses of the middle barriers are increased by 3 A (A/mb = +3 A), and S B , with its thicker barrier, performs somewhat better than S A . When the heights of the middle barriers are decreased by 10 meV (Ahmb = -10 meV), the effects on W E are 1 6 1 even smaller. Here, S A performs better than S B , but only slightly, when the middle barriers are Ino.52Alo.47Gao.01 As and Ino.52Alo.23Gao.25As, respectively. Lastly, Fig. 5 . 4 shows that decreasing the height of the left-hand side external barrier by 1 0 meV (A/?ib = - 1 0 meV) wil l also have small effects on W E in S A and W E in S B . Here, S A again performs S A S B S A : A/zib = - 1 0 m e V S B : A / % = -10 me V 40 10 20 30 Electric Field (kV/cm) Figure 5 . 4 : Localization parameters corresponding to structures S A and S B for a 1 0 meV decrease of the height of the left-hand side barrier. only slightly better, and that is because the probability of finding W E in the in the external barriers is lower for S A . Nevertheless, these marginally increased sensitivities of S B to the compositional variations in the middle and external barriers by no means outweigh the benefits of the structure's decreased sensitivities to the remaining possible layer variations. 162 5.3 Discussion We have presented insights that form a basis for developing C Q W structures that are robust to growth processes. As modern growth processes are becoming increasingly capable of achieving tight control of quaternary compounds [9], the approach to the design of CQWs should, in turn, allow the design of efficient E R and E A modulators. R e f e r e n c e s [1] H . Feng, J. P. Pang, M . Sugiyama, K . Tada, Y . Nakano, "Field-induced optical effect in a five-step asymmetric coupled quantum well with modified potential," IEEE J. Quantum Electron., vol. 34, no. 7, pp. 1197-1208, 1998. [2] T. H . Stievater, W. S. Rabinovich, P. G. Goetz, R. Mahon, and S. C. Binari, " A surface-normal coupled-quantum-well modulator at 1.55 um," IEEE Photon. Technoi. Lett., vol. 16, no. 9, pp. 2036-2038, 2004. 163 [3] C. Thirstrup, "Refractive index modulation based on excitonic effects in GalnAs-InP coupled asymmetric quantum wells," IEEE J. Quantum Electron., vol. 31, no.6,pp. 988-996, 1995. [4] R. Lang and K . Nishi, "Electronic state localization in semiconductor superlattices," Appl. Phys. Lett, vol. 45, no. 1, pp. 98-100, 1984. [5] M . P. C. M . Krijn, "Heterojunction band offsets and effective masses in I I I -V quaternary alloys," Semicond. Sci. Technoi., vol. 6, no. 1, pp. 27-31, 1991. [6] S. L . Chuang, Physics of Optoelectronic Devices. New York: Wiley, 1995, pp. 710. [7] S. L . Chuang, Physics of Optoelectronic Devices. New York: Wiley, 1995, pp. 157-160. [8] K . Kawano, and T. Kitoh, Introduction to optical waveguide analysis: solving Maxwell's equations and the Schrodinger equation. New York: Wiley, 2001, pp. 253-254. [9] C. Ebert, A . Bond, J. Levkoff, J. Seiler, and C. Wanamaker, " M O C V D process modeling using in-situ reflectance test structure measurements for process control improvement," in Proc. IPRM (International Conference on Indium Phosphide and Related Materials), 2005, pp. 219-222. Chapter 6 I m p r o v e d P u s h - P u l l C Q W P o l a r i z a t i o n M o d u l a t o r 164 6.1 Introduction Traditionally associated with LiNb03 or GaAs modulators based on the linear electro-optic effect, polarization modulation has been used to improve the performance of both digital and analog optical communication systems [1]. Recently, we proposed a push-pull polarization-modulation scheme using InGaAlAs-InAlAs C Q W structures, which, compared to the traditional polarization-modulation schemes, could provide significant reductions of drive electric fields and waveguide lengths for devices based on it [2]. In this scheme, an electric field, F, is applied to the optical waveguide containing multiple A version of this chapter has been submitted for publication. S. Ristic and N. A. F. Jaeger, "Improved Push-Pull Polarization Modulation Using Coupled Quantum-Well Structures," IEEE. Photon. Technoi. Lett. 165 repetitions of two types of C Q W structure, which, in this manuscript, we wi l l refer to as structure #1 and structure #2. As a result, structure #1 provides a large, positive change of the refractive index for TM-polarized light ( A « T M I ) , accompanied by a smaller change of the refractive index for TE-polarized light (AJITEI ) - Similarly, structure #2 provides a large, negative change of the refractive index for TE-polarized light ( A « T E 2 ) , accompanied by a smaller change of the refractive index for TM-polarized light ( A « T M 2 ) -Placed together in an optical waveguide, the two structures could provide equal-magnitude and opposite-sign changes of the effective refractive indices for the fundamental TE and T M modes ( A « C f f (TE) and A « c f f ( T M ) ) and enable push-pull polarization modulation [2]. Subsequent investigation into the effect that layer thickness and compositional variations have on the two C Q W structures revealed that structure #2 is quite sensitive to these variations, which would make polarization modulators using this structure challenging to grow reliably [3]. The main reason for the poor robustness of structure #2 is that its electrorefraction is based on anticrossing of the two lowest energy heavy-hole wave functions (WHHI and W H H 2 ) , which have large masses and are thus very sensitive to the variations [3]. Here, we show how to redesign structure #2 [see Fig. 6.1(b)] to obtain a structure with dramatically improved robustness by designing its electrorefraction to be based on the anticrossing of the two lowest-energy electron wave functions (WEI and W E 2 ) , which have much smaller masses. We also show how structure #1 [see Fig. 6.1(a)], with its electrorefraction based on anticrossing of the two lowest-energy light-hole wave functions ( W L H I and W L H 2 ) , could be improved. Modeling of each 166 structure is performed for 1550 nm light and is based on the variational method to solve the exciton equation in the effective-mass approximation [4]. Ino.52Alo.48As 30 A Ino.6oAlo.4oAs 45 A Ino.52Gao.20Alo.28AS 30 A Ino.49Gao.5i A s 72 A Ino.40Gao.55Alo.05AS 66 A Ino.60Alo.40As 30 A F Ino.52Gao.20Alo.28AS 36 A Ino.49Gao.5i As 63 A lno.63Gao.27Alo.loAS 66 A Ino.60Alo.40As 45 A Ino.52Gao.20Alo.28AS 30 A Ino.52Alo.4sAs 30 A (a) (b) Figure 6.1: Schematic of layers for (a) structure #1 and (b) structure #2. 6.2 Improved Structure #1 and Structure #2 Fig. 6.2(a) illustrates the band edge diagram and the wave functions for structure #1, for F = 15 and 30 kV/cm. The left-hand side well (QW1) is made wider in order for WEI and WLHI to be mainly localized in it for F ~ 0 kV/cm. As F is increased, WLHI and W LH 2 anticross at the resonant field F L H ~ 8 kV/cm. Similar to [2], the electric field region of interest (15 kV/cm <. F s 30 kV/cm) is above the resonant field, and A M T M I has a positive slope [see Fig. 6.3(a)]. This is because ati^LH both the IS excitons corresponding to the symmetric-mode transitions ( W E I - W L H I and W E 2 - W L H 2 ) and the asymmetric-mode 167 transitions ( W E I - W L H 2 A N A " V ^ E 2 - V P L H I ) have large radii (i.e., small binding energies) and make small contributions to the TM-polarization absorption coefficient, and, in turn, via the Kramers-Kronig relation, to the refractive index [2]. AsF is increased, the refractive index also increases, as the absorption F= 15 k V / c m • F= 15 k V / c m F=2Q k V / c m •—•"J r-= l^_ . ._ . . . ^---{ • F=30 kV/cm F=25 kV/cm F=30kV/cm (a) (b) Figure 6.2: (a) Structure #l's wave functions and energy band diagram for F= 15 and 30 kV/cm. Band gap is not shown to scale. W E I , W L H I , and WHHI are shown as thick lines, while WE2, WLH2 , and WHH2 are shown as thin lines. In addition, light-hole wave functions and band edge are shown as dashed lines and (b) structure #2's wave functions and band diagram for F = 15, 20, 25, and 30 kV/cm. Here, WHH3 is shown, instead of W H H 2 , as a full thin line. coefficient is dominated by the asymmetric-mode excitons that become well confined and thus have small radii. As also shown in Fig. 6.3(a), A « T E I has a positive, although much 168 smaller, slope because the light-hole momentum matrix element is smaller for the T E polarization, and WHHI and W H H 2 do not anticross and make a negligible contribution to the electrorefraction [2]. 20 25 Electric Field (kV/cm) ( a ) 30 -a-o u 00 c CS u X a -o c o <D at ,G3' i3 " ^ T E A t o -^ - A « T E 2 r V - a - A « T E / c A « T M 2 ---©-- A / 7 T M 2 r " - + - - A " T M 2 C - B - A / 7 T M / c 1 1 15 20 25 30 Electric Field (kV/cm) (b) Figure 6.3: Effects of the layer variations on (a) A « T E I and A « T M I and (b) on A/? T E2 and A«TM2 for 1550 nm operation wavelength. Compared to structure #1 in [2], our new structure #1 offers two main improvements. First, Q W l and QW2 are under tensile strain in order to increase the energy separation between the electron and the heavy-hole wave functions. Consequently, the absorption coefficient for the T E polarization at the wavelength of operation is decreased. The compressive strain in the barriers is added to balance the total strain in structure #1 and 169 structure #2. Second, for the electric field region of interest, WHHI is localized in the right-hand side well (QW2) rather than QW1. The lowest energy W E I - W H H I transition is, therefore, indirect, which should further reduce the T E polarization absorption. Here, the W E i - W H H I energy separation corresponds to X = 1439 nm for F = 15 kV/cm, and, similarly, the W E I - W L H I energy separation corresponds to X = 1411 nm. Considering these large energy separations and the fact that the high-energy asymmetric-mode excitons dominate the optical absorption above F L H , the optical losses at 1550 nm should be small [5]. Also, localization of W H H I in QW2, rather than in QW1, offers another important benefit. For the electric fields of interest, it is easier to confine WHHI to QW2 than to QW1, so that the 3 A thickness and 1 % compositional variations considered below cannot redistribute the heavy-hole wave functions and cause sharp dips in the refractive index change curves for the T E polarization [3]. We study the effects of the thickness and compositional variations on structure #1 separately, with no intention to model any specific growth process. To do so, the thicknesses of QW1, QW2, and the middle barrier are changed by ±3 A independently (i.e., 27 variations). For both polarizations, the refractive index change corresponding to each variation is multiplied by the variation's occurrence probability of 1/27 and added in the average refractive index change for T E polarization (AMTEI 7 ) or T M polarization (AWTMI )• This is equivalent to havingpw - 1/3 in [3]. Similarly, the InAs composition in 170 all five layers is changed by ±0.01 (i.e., 243 variations) and, usingpc = 1/3 [3], we obtain the average refractive index changes for each polarization ( A « T E I C and A « T M I C ) -As illustrated in Fig. 6.3(a), structure #1 is quite insensitive to both thickness and compositional variations. Its insensitivity to compositional variations is, in part, due to the insensitivity of the light-hole band edge to these variations [3]. Also, the structure's insensitivity to both thickness and compositional variations is, in part, due to the small valence-band discontinuity ratio of the InGaAs-InAlAs material system [4]. The confinement of the anticrossing light-hole wave functions to the quantum well layers is, thus, small and the light-hole wave functions are, in turn, less sensitive to the thickness and compositional variations that occur in these layers. The band edge diagram and the wave functions for structure #2 are shown in Fig. 6.2(b). Here, it is WHHI and WH 3 that are shown in the figure as they make the main contributions to the electrorefraction (WHH2 has negligible magnitude in Q W l and a node in QW2). The function of the small Ino.52Gao.20Alo.28As barriers is to decrease the confinement of the anticrossing electron wave functions and decrease their sensitivities to the layer variations [4]. The large Ino.52Alo.48As barriers are sufficiently far from Q W l and QW2 to significantly affect the sensitivities of the wave functions, and they help to confine the wave functions and decouple neighboring C Q W structures. As shown in Fig. 6.2(b), the compressive strain in QW2 is used to make it deeper for WEI and WHHI, 171 confining them for F = 15 kV/cm, while the tensile strain in Q W l is used to make it only slightly^ deeper for W L m , confining it for F= 15 kV/cm and allowing it to move toward QW2 for larger F. The wavelengths corresponding to the W E I - W H H I and W E I - W L H I energy separations are 1391 nm and 1312 nm, respectively. As F i s increased to 30 kV/cm, WEI and WE2 become approximately equally shared by Q W l and QW2 giving the largest heavy-hole exciton radii and the smallest refractive index for the T E polarization [4] [see Fig. 6.3(b)]. Simultaneously, W L H I becomes approximately equally shared by the two wells, also giving largest exciton radii and smallest refractive index for the T M polarization. However, unlike in the C Q W structures mentioned above, here, both the conduction-band, i.e., electron, and the valence-band, i.e., light-hole, wave functions exchange wells as F is increased. Consequently, the overlaps between W E i and W L H I as well as W E 2 and WLH2 reach their maxima for F~25 kV/cm, and A « T M 2 does not have the smooth, negative slope typical for C Q W structures. Rather, it is a nonlinear function [see Fig . 6.3(b)], and a desirable large difference between A « T E 2 and A « T M 2 is achieved. As illustrated in Fig. 6.3(b), structure #2 is not very sensitive to 3 A thickness variations (pw = 1/3), but it is somewhat sensitive to the compositional variations (pc = 1/3). Since structure #2 has more layers than structure #1, most of which are also quaternary compounds, only the variations in the quantum wells are considered in order to keep the number of simulations manageable. Simulations show that the thickness and the compositional variations in the barriers, as compared to the wells, have much smaller 172 effects on the distributions of the wave functions, which justifies the approximation. Also, the effects of compositional variations are well approximated by restricting the changes of the Ga and A l mole fractions in each well to have the same magnitudes but opposite signs. Fig. 6.3(b) also shows the effect of treating the thickness and compositional variations as being coupled ( A « T E 2 T C and A n x M 2 r C ) -Fig. 6.4(a) shows the effects that the layer variations have on A M ^ T E ) and AwCffrTM) of the ID waveguide in Fig. 6.4(b). A« C f f (TE) and A« e f f (TM )are obtained using the designed values for refractive index change curves ( A H T E I , A « T M I , A « T E 2 , and A « T M 2 ) , biasing the electric field at F B = 23 kV/cm, and applying an electric field change of A F - ±3 kV/cm. The magnitudes of A« C f f (TE) and A « c f f ( T M ) are very similar, and polarization switching between two orthogonal states could be achieved for an electrode length of L = 3.2 mm. However, i f for the same F B , AF, and L we consider the effects of the layer variations on polarization modulation, Fig. 6.3 shows that the magnitudes of the effective index changes for the TE and T M polarizations (An eff(TE)' and A ^ C ^ T M ) ' ) become very different. Here, the actual variations that can be expected in the epitaxial layers are approximated by AHTEI and A « T M I in structure #1 and by A H T E I and A « T M I in structure #2. Due to the nonlinear shape of A « T M 2 , the magnitudes of the two effective index changes can be matched by changing the values of FB and A F . In addition, i f the long electrode is split into two equal-length electrodes (L\ = L2 = 1.6 mm), each having a different bias field 1 7 3 Ino.52Alo.48As cladding 6 x structure #1 (153 nm) r T E = 0.21 r T M = 0.19 10 x structure #2 (288 nm) r T E = 0.48 r T M = 0.43 Ino.52Alo.48As cladding 0 2 4 6 8 10 Electric Field Change (kV/cm) (a) (b) Figure 6 . 4 : (a) Effects of the layer variations on A M ^ T E ) and Aneff(TM) in a (b) 1-dimensional waveguide containing 6 periods of structure #1 and 1 0 periods of structure #2 . In each C Q W region, T I E and T T M are the optical confinement factors for the TE and T M fundamental modes. (FBi = 2 0 kV/cm and FBI = 2 7 kV/cm) and the same field change A F = ± 5 kV/cm, not only would the effective index changes (A« e ff(TE)" and A« eff(TM)") have closely matched magnitudes, but also the curvatures of the T M response would cancel themselves to an extent, making the T M response more linear [see Fig. 6.4(a)]. The nominal polarization switching efficiency in [ 2 ] is somewhat better than here, considering that the structures in [ 2 ] have smaller values for T I E and T T M (FTE = 0 . 4 3 and T T M = 0 . 3 7 for 1 0 periods of the old structure #1 and T I E = 0 . 1 9 and T I M - 0 . 1 7 for 6 periods of the old structure #2 , respectively) and still provide a smaller A F * ! product ( 5 kV/cm x 3 . 5 mm). However, the new structures are more robust to the layer variations and provide for the possibility of matching the magnitudes of AI^TTE) and A n e f f ( T M ) even in the presence of these variations. 174 R e f e r e n c e s [1] J. D. Bul l , et al., "40 G H z electro-optic polarization modulator for fiber optic communications systems," in Proc. ofSPIE, 2004, vol. 5577, pp. 133-143. [2] S. Ristic and N . A . F. Jaeger, "Push-pull polarization conversion using novel asymmetric coupled quantum-well structures," IEEE Photon. Technoi. Lett., vol. 18, no. 2, pp. 316-318, 2006. [3] S. Ristic and N . A . F. Jaeger, "Influence of layer thickness and compositional variations on the electrorefractive properties of a quantum well polarization-conversion modulator,"/ Vac. Sci. Technoi. A, Vac. Surf. Films, vol. 24, no. 4, pp. 962-965, 2006. [4] S. Ristic and N . A . F. Jaeger, "Robust coupled-quantum-well structure for use in electrorefraction modulators," IEEE. Electron Dev. Lett, vol. 28, no. 1, pp. 30-32, 2007. [5] W. Bardyszewski, D. Yevick, Y . Liu , C. Rolland, and S. Bradshaw, "Theoretical and experimental analysis of Mach-Zehnder quantum-well modulators,"/. Appl. Phys., vol. 80, no. 2, pp. 1136-1141, 1996. 175 Chapter 7 Summary, Conclusions, and Suggestions for Future Work 7.1 Summary It was clear in the early stages of the research presented in this thesis that E R modulators based on C Q W structures would be strong competitors to other modulator technologies in future-generation digital and analog optical links. For example, compared to SQWs, C Q W structures can provide similar refractive index changes for considerably lower drive electric fields. In addition, the exciton absorption peaks in C Q W structures exhibit smaller red shifts, so that E R modulators based on C Q W structures are expected to have smaller absorption coefficients and electric-field-induced changes of the absorption coefficients at the wavelength of operation than E R modulators based on SQWs. Consequently, E R C Q W modulators have great potential to exceed the performance of E R SQW modulators in applications requiring ultra-fast intensity modulation. The same is true for polarization modulators. Push-pull polarization modulators based on C Q W 176 structures are expected to have ultra-wide electro-optic bandwidths, low drive voltages, low differential group delays, and small voltage-dependent changes in the absorption. When compared to E A SQW modulators, E R C Q W intensity modulators are expected to have lower chirp and lower drive voltages. Also, as they operate at wavelengths that are further from the lowest energy exciton (i.e., they are more detuned) and, thus, absorb less light, the E R C Q W intensity modulators are expected to have smaller temperature and wavelength dependences and higher optical saturation powers. The thesis presents, in a chronological order, work aimed at improving the state-of-the-art of electro-optic modulators based on C Q W structures. In Chapter 2, a design for the first quantum-well-based push-pull polarization modulator was presented. The conceptual push-pull polarization modulation was achieved by placing multiple repetitions of two different InGaAlAs-InAlAs C Q W structures in the optical waveguide of the modulator. The electroabsorption in these C Q W structures, and also in all of the InGaAlAs-based C Q W structures designed in this thesis, was modeled by solving the 1S exciton equation in the effective-mass envelope-function approximation using the variational method, and the electrorefraction was obtained using the Kramers-Kronig relations. One of the two structures contains a small amount of tensile strain (TS structure) and provides a large increase of the refractive index for the T M polarization and only a small increase of the refractive index for the T E polarization. The other structure contains a small amount of compressive strain (CS structure) and provides a large decrease of the refractive index for 177 the T E polarization and only a small decrease of the refractive index for the T M polarization. The novel function of biaxial strain in each of the two structures was to achieve a large difference in the refractive index changes for the T E and T M polarizations. Placing 10 repetitions of the TS structure and 6 repetitions of the CS structure in a slab waveguide, similar-magnitude and opposite-sign changes of the effective refractive indices for the T E and T M modes were predicted for an electric-field change of 5 kV/cm, providing a 90° polarization rotation in a 3.4 mm long waveguide. In Chapter 3, the sensitivities of the TS and CS structures to thickness and compositional variations were studied. Although the wave functions that contribute to the electrorefraction effects in C Q W structures become less sensitive to the variations when the thicknesses of the middle barriers are decreased [1], doing so also decreases the slopes of the electric-field-induced refractive index changes [2]. The goal of the study in Chapter 3 was to obtain a better understanding of the TS and CS structures' sensitivities to the layer variations, and, eventually, based on the knowledge obtained from the study, minimize the effects of the layer variations without resorting to very thin middle barriers. It was found that the CS structure is very sensitive to the layer variations because its electrorefraction effect is based on the anticrossing of the two lowest energy heavy-hole wave functions and that the TS structure is quite insensitive to the layer variations because its electrorefraction effect is based on the anticrossing of the two lowest energy light-hole wave functions. The effective masses of the heavy-hole wave functions are 178 considerably larger than those of the light-hole wave functions, and distributions of heavy-holes wave functions in C Q W structures are, therefore, more susceptible to any thickness or compositional variations in the C Q W structures [1]. Although the CS structure provides a very large change of the refractive index change for the T E polarization and a very small change of the refractive index change for the T M polarization, which is very desirable in polarization modulators, it was clear from the study that major improvements of the CS structures needed to be done because of their great sensitivities to the layer variations. It was also learned from the study that the TS structure was quite insensitive to the compositional variations because the energy of the light-hole band edge depends less on the compositional variations than do the energies of the heavy-hole and electron band edges. It was anticipated that the TS structure could be made very robust to the layer variations i f slightly redesigned in order to prevent the undesirable anticrossing of the two lowest energy heavy holes, which caused undesirable dips in the refractive index response for the TE polarization. In Chapter 4, inspired by the robustness of the TS structure, a lattice-matched InGaAs-InAlAs C Q W structure (structure #1) that has it electrorefractive effect based on the anticrossing of the two lowest energy light-hole wave functions, and is, thus, more appropriate for T M modulation, was compared with a similar lattice-matched InGaAs-InAlAs C Q W structure (structure #2) that has it electrorefractive effect based on the anticrossing of the two lowest energy electron wave functions, and is, thus, more 179 appropriate for TE modulation. It was demonstrated that structure #1 is considerably less sensitive to the 3 A thickness variations and the 2 % compositional variations and that InGaAlAs-based C Q W structures that have their electrorefraction effects based on anticrossing of the light holes are a preferable technology for the future E R modulators. Besides the insensitivity of the light-hole band edge to compositional variations, it was seen in this study that there was an additional mechanism which contributed to structure #l's small sensitivity to both thickness and compositional variations. It was shown that a small valence band discontinuity ratio in InGaAlAs-based systems makes anticrossing light-hole wave functions quite insensitive to relative differences in thickness and composition between the two quantum wells because shallow quantum wells provide only weak confinement to the light-hole wave functions. Consequently, the magnitudes of the light-hole wave functions in the quantum well layers are relatively small, and so are the probabilities that the relative distributions of the wave functions between the two quantum wells can be affected by the thickness and compositional variations in the two quantum wells. In Chapter 5, building on the findings in Chapter 4, it was shown that decreasing the confinements of electron wave functions in C Q W structures could be used to improve the robustness to growth of E A and E R modulators that have their respective electroabsorption and electrorefraction effects based on anticrossing of electron wave functions. A lattice-matched C Q W structure (SA) that has ternary wells (Ino.53Gao.47As) 180 and ternary barriers (Ino.52Alo.4sAs) was compared to another lattice-matched C Q W structure ( SB ) that has ternary wells (Ino.53Gao.47As) and quaternary barriers (Ino.52Gao.24Alo.24As). The middle barrier in S B was made wider than the one in S A in order to compensate for its smaller height and provide the same sensitivities of the lowest energy electron wave functions in the two structures to the same change in the electric field. Although the lowest energy electron wave function in S B has the same sensitivity to the electric field as the lowest energy electron wave function in S A , it is considerably less sensitive to the 3 A thickness variations and only slightly more sensitive to the 10 meV potential energy variations that can be expected to occur in the quantum well and barrier layers during growth. Consequently, increasing the thicknesses of the middle barriers in C Q W structures and decreasing the confinements of the anticrossing electron wave functions by decreasing the potential energies of all barriers (or increasing the potential energies of the quantum wells) can be used as a design method for improving the overall robustness to growth in these structures. The same is true for the C Q W structures where the anticrossing of the light-hole wave functions is used. However, as was shown in Chapter 3, anticrossing of the heavy-hole wave functions should be avoided because their large effective masses make them extremely sensitive to layer thickness and compositional variations. In Chapter 6, the design method presented in Chapter 5 was used to improve the performance of the CS structure in Chapters 2 and 3. Although it is designed to achieve 181 the same effect as the CS structure, i.e., large electric-field-induced decrease of the refractive index for the TE polarization without a commensurate decrease of the refractive index for the T M polarization, the new structure (structure #2) has two important characteristics that distinguish it from the original CS structure. First, the large refractive index change for the TE polarization and the small refractive index change for the T M polarization in structure #2 are achieved via anticrossing electron wave functions, rather than heavy-hole wave functions. The small effective masses of the anticrossing electron wave functions in structure #2 make them considerably less sensitive to layer thickness and compositional variations than the anticrossing heavy-hole wave functions in the CS structure. However, the large refractive index change for the TE polarization and the small refractive index change for the T M polarization is an effect that is not easily achieved via anticrossing electron wave functions. The large negative change of the refractive index for the TE polarization is achieved by anticrossing the two lowest energy electron wave functions and preventing the heavy-hole wave functions from making any significant electric-field-induced redistribution between the two wells of structure #2, as is typically done in E R C Q W modulators [2], [3]. However, in order to achieve only a small change of the refractive index for the T M polarization, considering that the anticrossing of the electron wave functions is used, a small, controllable redistribution of the two lowest energy light-hole wave functions is allowed. The redistribution of the light-hole wave functions 182 accompanied by a lack of redistribution of the heavy-hole wave functions is facilitated by using tensile strain in the left-hand side well and compressive strain in the right-hand side well. The opposite-sign strains provide unequal potential energies for the heavy-hole and light-hole wave functions, strongly confining the heavy-hole wave functions and allowing the light-hole wave functions to slowly redistribute between the two wells. The electric-field-induced movement of the light-hole and electron wave functions does not provide the typical monotonic decrease or increase of the overlap integrals between these wave functions. Rather, the overlap integrals have an "S-shape" dependence on the electric field, which is also evident in the refractive index change for the T M polarization. Consequently, the refractive index change for the T M polarization remains small in the electric field region of interest. Second, unlike the CS structure, the new structure #2 has very small quaternary barriers in the vicinity of two quantum wells in order to decrease the confinement of the anticrossing electron wave functions and, ultimately, improve the structure's sensitivity to the layer variations. Simulations show that the heights of the barriers far from the quantum wells, where the magnitudes of the electron wave functions are small, do not affect the wave functions' sensitivities to the layer variations. Consequently, the barriers in these regions are made ternary in order to provide better decoupling between the neighboring structures and to provide confinement to the second lowest energy light-hole 183 wave function and the third lowest energy heavy-hole wave function, both of which are included in the modeling of electrorefraction. Structure #1 in Chapter 6 is not fundamentally different from the TS structure in Chapters 2 and 3. Similar to the TS structure, the main function of structure #1 is to provide a large increase of the refractive index for the T M polarization without a commensurate increase of the refractive index for the T E polarization, and this effect is achieved by anticrossing the two lowest energy light-hole wave functions. However, in structure #1, the lowest energy heavy-hole wave function is confined to the right-hand side well and not the left-hand side well. This configuration is more energetically favorable for the heavy holes, and the undesirable heavy-hole anticrossing evident in Chapter 3 is extinguished for the 3 A thickness and 1 % compositional variations used in Chapter 6. Also, confining the lowest energy heavy-hole wave function to the right-hand side well in the electric field region of interest decreases the optical absorption strength associated with the lowest energy electron-heavy-hole transition (E1-HH1) and effectively decreases the optical loss for the T E polarization at the wavelength of operation. In addition, tensile strain is used in both wells of structure #1 in order to increase the wavelength detuning of the heavy-hole excitons, which typically have lower energies than the light-hole excitons, and to further decrease the optical loss for the TE polarization. The compressive strain in the barriers of structure # 1 is used to balance the overall strain in structure # 1 and structure #2. Although structure #1 is very insensitive to the 3 A thickness and the 1 % 184 compositional variations, structure #2 still shows a certain degree of sensitivity to the compositional variations. However, it is pointed out in Chapter 6 that two inline, separately biased electrodes can be used to match the magnitudes as well as to linearize the opposite-sign refractive index changes for the TE and T M polarizations, further reducing the sensitivities of modulators based on structure #1 and structure #2 to layer variations. 7.2 Conclusions As mentioned above, C Q W structures have the potential to provide large electroabsorption and electrorefraction effects, which wil l allow devices using them to have short lengths and, consequently, ultra-wide electro-optic bandwidths, while maintaining small drive voltages. Also, unlike in SQWs, in which the electrorefraction effect is based mainly on red shifting the lowest energy excitons, in C Q W structures, the electrorefraction effect is based on changing the overlaps between the conduction and valence wave functions, and there is no appreciable red shifting that causes electric-field-induced increases of the optical absorption. Besides these two benefits of C Q W structures, the work presented in this thesis reveals at least four additional properties of C Q W structures that wi l l be important when designing future C Q W modulators. 185 First, as the electrorefraction effect in a C Q W structure is based on the electric-field-induced transfer of particles (electrons, light holes, or heavy holes) between the two wells, it is relatively easy to select which particles are allowed to transfer by manipulating the strain and geometry of the layers in the C Q W structure. Consequently, C Q W structures can provide for efficient polarization modulators because, again, electron-light-hole transitions solely contribute to the TM-polarization response and electron-heavy-hole transitions dominate the TE-polarization response. Second, in InGaAlAs-based C Q W structures, because of the very small dependence of the light-hole band edge on compositional variations in the quantum wells, because of the relatively small effective masses of light holes, and because of the small valence band discontinuity ratio, the anticrossing light-hole wave functions are far less sensitive to layer thickness and compositional variations than the anticrossing electron or heavy-hole wave functions. Consequently, an InGaAlAs-based C Q W structure can be made to be quite sensitive to the electric field, by allowing the middle barrier to be fairly thick, and still remain quite insensitive to the layer thickness and compositional variations that are expected to occur during the growth process. As the light-hole momentum matrix element for the T M polarization is considerably larger than that for the T E polarization, C Q W structures that utilize anticrossing of light-hole wave functions are suitable for T M phase and intensity modulators, such as structure #1 in Chapter 4, or polarization modulators, such as structure #1 in Chapter 6. Therefore, i f the designers of future phase 186 and intensity modulators desire to use T M light in their devices, the best choice for these devices would be simple lattice-matched, ternary InGaAs-InAlAs C Q W structures, which utilize the anticrossing of light holes (such as structure #1 in Chapter 4). In addition, i f the designers want to make very efficient polarization modulators that are not necessarily push-pull devices, the best choice for these devices would be simple strained, ternary InGaAs-InAlAs C Q W structures, which utilize the anticrossing of light holes (such as structure #1 in Chapter 6). Third, certain devices, such as surface-normal modulators used in optical interconnects, free-space optical communications, and optical links requiring large-area and/or polarization insensitive devices, do not have optical waveguides. For example, in the surface-normal electroabsorption modulator in [4], light enters the stack of InGaAs-InAlAs C Q W structures through a p-doped region at normal incidence and, thus, its electric-field vector is parallel to the planes of the quantum wells. This device is polarization-independent because the C Q W structures provide the same electroabsorption effect for all directions of the electric-field vector around the optical axis. For this device and other devices requiring modulation of plane-parallel light, the anticrossing of electrons is a more efficient mode of operation than the anticrossing of light holes because the light-hole momentum matrix element for the plane-parallel light is relatively small. O f course, the anticrossing of electrons in a C Q W structure is more efficient than the anticrossing of light holes only i f the C Q W structure can be designed so that the 187 electron wave functions are not significantly more sensitive to the layer thickness and compositional variation than the light-hole wave functions. Chapter 5 presents a method that designers of devices that modulate the plane-parallel light can use to make C Q W structures based on the anticrossing of electrons more robust to the layer variations. The method requires barriers or quantum wells in C Q W structures to be quaternary in order to weaken the confinement of the electron wave functions to the quantum well layers and allow them to tunnel to a greater extent into the external barriers. As argued in Chapter 5, the small increase of the wave functions' sensitivities to the compositional variations in the barriers is small compared to the large decrease of their sensitivities to both the thickness and compositional variations in the quantum well layers. Fourth, as mentioned above, the C Q W structures that utilize the anticrossing of light-hole wave functions, such as structure #1 in Chapter 6, can be used in polarization modulators because these structures provide large phase changes for the T M polarization, accompanied by only small phase changes for the T E polarization. On the other hand, for those applications that may require devices that provide large phase changes for the TE polarization accompanied by only small phase changes for the T M polarization, a C Q W structure similar to structure #2 in Chapter 6 could be used. Again, in Chapter 6, the structure #2 was used together with structure #1 for a particular purpose, which is a realization of a robust C Q W push-pull polarization modulator. .7.3 Suggestions for Future Work 188 In order to fully realize the potential that the C Q W structures developed in this thesis have for improving the state-of-the-art in optical modulators, additional work wil l be necessary. This future work wi l l include: (a) Growth and fabrication of C Q W modulators, (b) Systematic performance tests of these devices, (c) Improvement of their electro-optic bandwidth, and (d) Integration of C Q W modulators with other devices. Compared to the polarization modulators that could be realized using the C Q W structures in Chapter 6, TM-phase modulators that could be realized using the robust structure #1 in Chapter 4 are the simplest to build and are the most likely to provide good results in the short term. Therefore, initial future research should likely focus on T M C Q W modulators. (a) Multiple repetitions of the C Q W structure could be grown using Molecular Beam Epitaxy in a typical Mach-Zehnder p-i-n diode configuration, as illustrated in Fig. 1.17. The shape of the optical waveguide could be patterned using reactive ion etching. Each wafer can contain many Mach-Zehnder interferometers with 1-2 mm long branches. In order to avoid Fabry-Perot resonances, the optical waveguide facets could be angle-cleaved or antireflection coated. (b) Mach-Zehnder interferometers containing multiple repetitions of the C Q W structure should be tested for their basic operation, and their low-frequency electro-optic transfer 189 functions should be obtained. Extinction ratios greater than 15 dB and drive voltages lower than 1 V are expected. Should more details of the electroabsorption and the electrorefraction properties of the CQW structures be required (e.g., in order to compare the experimental results with the modeling), the photocurrent spectra of single branches of the Mach-Zehnder interferometers could be obtained. The single-branch experiments may also include more detailed phase and absorption measurements [5]. All measurements should be performed across the communication wavelength bands using a tunable laser, and the modulator is expected to show a flat response at least across the C-band. More than one modulator design will likely be needed to cover the whole telecommunication spectrum. (c) Once the basic functionality and moderately wide electro-optic bandwidths of the p-i-n devices are demonstrated, additional work should be focused on the design, fabrication, and testing of the traveling-wave electrodes that will provide ultrahigh bandwidths (-100 GHz). These electrodes will not necessarily be of the typical p-i-n diode configuration. They may be coplanar strip or coplanar waveguide electrodes with Schottky contacts [6], or the new type of substrate-removed microstrip-like electrode recently introduced [7], [8]. (d) In order for CQW modulators to be competitive in future optical links, it will also be important to eventually integrate them with other devices (e.g., lasers, photodetectors, 190 wavelength converters, optical amplifiers, etc.) on a single chip. The monolithic integration of several optoelectronic devices on a single chip reduces the cost of packaging, reduces the coupling losses between devices, reduces the power consumption, and improves the circuit reliability. How the integration will be achieved will depend on the photonic integrated circuit of interest [9]. Growth, fabrication, and testing procedures for the TE-polarization intensity modulators will be basically the same as those for the TM-polarization modulators. As CQW polarization modulators contain multiple repetitions of two types of CQW structure, each type of CQW structure should be grown, fabricated, and tested separately in procedures similar to the ones described above. Only when the electroabsorption and the electrorefraction properties of each type of CQW structure are well known, should the two structures can be put together in the optical waveguide of a polarization modulator for further optimization and testing. Also, if a polarization modulator is not intended to be used in the push-pull mode of operation, the device can be based on only one for the two structures presented in Chapter 6, e.g., structure #1. Although the CQW polarization modulators based on the strained structures presented in Chapter 6 are more difficult to realize than the TM-phase modulators based on structure #1 presented in Chapter 4, both types of device are expected to provide state-of-the-art performance predicted by the modeling. R e f e r e n c e s 191 [1] R. Lang and K . Nishi, "Electronic state localization in semiconductor superlattices," Appl. Phys. Lett, vol. 45, no. 1, pp. 98-100, 1984. [2] C. Thirstrup, "Refractive index modulation based on excitonic effects in GalnAs-InP coupled asymmetric quantum wells," IEEE J. Quantum Electron., vol. 31, no.6, pp. 988-996, 1995. [3] H . Feng, J. P. Pang, M . Sugiyama, K . Tada, Y . Nakano, "Field-induced optical effect in a five-step asymmetric coupled quantum well with modified potential," IEEE J. Quantum Electron., vol. 34, no. 7, pp. 1197-1208, 1998. [4] T. H . Stievater, W. S. Rabinovich, P. G. Goetz, R. Mahon, and S. C. Binari, " A surface-normal coupled-quantum-well modulator at 1.55 um," IEEE Photon. Technoi. Lett, vol. 16, no. 9, pp. 2036-2038, 2004. [5] D. J. Krause and J. C. Cartledge, IEEE Photon. Technoi. Lett, vol. 16, no. 8, pp. 1915-1917, 2004. [6] J. P. Pang, K . Tada, Y . Nakano, H . Feng, "Multiple-quantum-well traveling-wave-type ultra-high-speed optical modulator," mProc. OECC'98 Tech. Digest. (Optoelectronics and Communications Conference), 1998, paper 16B1-5, pp. 454-455. [7] J. H . Shin, C. Ozturk, S. R. Sakamoto, Y . J. Chiu, and N . Dagli, "Novel T-rail electrodes for substrate removed low-voltage, high-speed GaAs/AlGaAs electro-optic modulators," IEEE Trans. Microw. Theory Tech., vol. 53, no. 2, pp. 626-643, 2005. 192 [8] J. H . Shin, S. Wu, and N . Dagli, "Bulk undoped G a A s - A l G a A s substrate-removed electrooptic modulators with 3.7-V-cm drive voltage at 1.55 urn," IEEE Photon. Technoi. Lett., vol. 18, no. 21, pp. 2251-2253, 2006. [9] L . A . Coldren, J. Raring, M . Sysak, J. Barton, and L . Johansson, "Improved functionality and performance in photonic integrated circuits," in Proc. IPRM (International Conference on Indium Phosphide and Related Materials), 2006, paper P L E 1 , pp. 1-6. 

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