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Slow-wave electrode structures for III-V semiconductor based electro-optic travelling-wave modulators Lee, Zachary Ka Fai 1992

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SLOW-WAVE ELECTRODE STRUCTURES FOR^SEMICONDUCTOR BASED ELECTRO-OPTIC TRAVELLING-WAVE MODULATORS  by ZACHARY KA FAI LEE B. Sc. (Honours), University of British Columbia, 1989  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (The Department of Electrical Engineering)  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA April 1992 ©Zachary Ka Fai Lee, 1992  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of  eiec  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  A- CY1 1^19 ci  Abstract Electro-optic modulators are important devices in high-speed optical communications and signal processing. These devices are particularly useful if they are fabricated on semiconductor materials since they may be monolithically combined with other electronic and/or optoelectronic devices. Conventional travelling-wave electro-optic modulators fabricated using III-V semiconductor materials such as gallium arsenide and indium phosphide suffer from the problems created by the velocity-mismatch between the microwave modulating signal and the optical signal (or equivalently, the mismatch between the effective refractive indices of the microwave and the optical wave). These problems ultimately limit the modulator bandwidth and require large amounts of modulating microwave power to obtain useful modulation depths. It was our objective to design coplanar slow-wave electrode structures allowing the fabrication of wide-band modulators with reasonable demands of modulating power levels. This objective was achieved. In this thesis slow-wave coplanar electrode structures for use in these modulators are described. They are periodically loaded with narrow capacitive fins. Small pads may also be added on the ends of the fins to further increase the capacitance. If the fins are narrow, the capacitance per unit length can be increased substantially due to the fringing electric fields about the fins (and pads), while the effect on the inductance per unit length can be relatively small. This increase in capacitance per unit length results in the slowing of the microwave. By carefully choosing the dimensions, slow-wave electrodes having a prescribed microwave effective refractive index as well as 50 or 75 ohm characteristic impedances can be designed. A dielectric  ii  superstrate may also be used to help slow down the microwave somewhat, potentially enabling fine tuning, as well as to protect the electrodes. The electrodes may also be partially buried in the substrate to achieve some additional slowing of the microwave. Design formulas have been derived. Design curves for 50 and 75 ohm slow-wave electrode structures fabricated on gallium arsenide and indium phosphide based materials are presented. In order to verify the theory, a large number of slow-wave electrodes as well as a conventional coplanar strip electrode have been fabricated on semi-insulating gallium arsenide. The fabrication was straight forward, with the whole electrode being formed in a single layer of metallization using a single-step photo-resist patterning and lift-off technique. Measurements of the microwave effective refractive indices of the electrodes fabricated clearly indicate that these electrodes are capable of velocity-match in modulators fabricated using gallium arsenide based materials. The measured values range from 2.84 to 3.46. Good agreement was found between the measured results and the theoretical predictions. The electrodes were also found to have very low microwave losses of a few tenths of a decibel per centimetre, and very low dispersion at frequencies at least up to the 20 GHz limit of the microwave source. Slow-wave electrode structures, capable of matching the velocities of microwaves to those of optical waves in III-V semiconductor travelling-wave electro-optic modulators, that offer low loss, low dispersion, flexible dimensions, and ease of fabrication have been designed, fabricated, and tested.  iii  ^  Table of Contents Abstract ^  ii  Table of Contents ^  iv  List of Figures ^  vi  List of Tables ^  ix  Acknowledgements ^  x  Chapter 1. Introduction 1.1.^Importance of Slow-Wave Electrode Structures and Electro-optic Modulators ^ 1.2.^Organization of the Thesis  ^1 1 ^4  Chapter 2. Background ^ 2.1.^Introduction 2.2.^The Linear Electro-optic Effect ^ 2.3.^Velocity Matching, Phase Retardation, and Bandwidth 2.4.^Current Research in Electro-optic Modulators 2.5.^Coplanar Slow-Wave Electrodes in Mach-Zehnder Modulators  6 ^6 9 ^12 ^15 ^18  Chapter 3. Theory 3.1.^Introduction 3.2.^The Slow-Wave Electrode Structures 3.3.^Physics of the Slow-Wave Electrode Structures 3.3.1. Introduction 3.3.2. The Transfer Matrix Method 3.3.3. Dispersion Characteristics and Group Velocity 3.3.4. Low Frequency Approximation and Design Formulas 3.3.5. Computation of Capacitance 3.3.6. Fringing Fin Capacitance 3.3.7. Calculated Results and Design Curves 3.3.8. Attenuation Constant and Microwave Loss 3.3.9. Scaling of Dimensions ^ 3.3.10. Input Sections 3.3.11. The Effects of High Dielectric Constant Superstrates and Partially Buried Electrodes 3.4.^Conclusion  ^22 ^22 ^23 ^28 ^28 ^30 ^35 ^39 ^44 ^52 ^54 ^61 63 ^64  iv  ^64 ^65  Chapter 4. Device Fabrication 4.1. Introduction 4.2. Mask Design ^ Fabrication Procedure 4.3. 4.3.1. Cleaving GaAs Wafers 4.3.2. Cleaning GaAs Substrates 4.3.3. Photo-resist Patterning 4.3.4. Etching GaAs Substrates for Burying Electrodes 4.3.5. Evaporation of Aluminum 4.3.6. Removal of Excess Metal (Lift-off) 4.3.7. Summary of Fabrication Procedure 4.4. The Completed Electrodes ^  ^67 ^67 68 ^74 ^76 ^76 ^77 ^80 ^83 ^84 ^85 86  Chapter 5. Device Testing and Measured Results 5.1. Introduction The Measurement Equipment ^ 5.2. 5.3. The Interference Technique 5.4. The Resonance Technique ^ Measured Results ^ 5.5. Discussion of Results ^ 5.6.  ^93 ^93 94 ^95 100 105 111  Chapter 6. Conclusion and Recommendations for Future Research ^ 115 115 Conclusions ^ 6.1. 117 Recommendations for Future Research ^ 6.2. References ^  119  Appendix I. Conventional Coplanar Strips ^  127  Appendix II. Coefficients of Inverse Capacitance ^  131  Appendix III. Conditions for Resonance ^  135  v  List of Figures Figure 2.1.1.^Typical lumped element electrode electro-optic modulator. ^ 7 Figure 2.1.2.^Typical travelling-wave electro-optic modulator.  ^8  Figure 2.2.1. Principal axes of the perturbed index ellipsoid for 43m crystals  ^12  Figure 2.4.1. A p-i-n travelling-wave phase/polarization modulator.  ^16  Figure 2.5.1. A Mach-Zehnder modulator employing a coplanar strip slow-wave electrode  ^20  Figure 2.5.2 . Cross section of a Mach-Zehnder modulator employing a coplanar strip ^ slow-wave electrode and graded index optical waveguides. 21 Figure 3.2.1 . Plan view of a section of a slow-wave electrode with fins only. .. . ^26 Figure 3.2.2 . Plan view of a section of a slow-wave electrode with fins and pads. . ^27 ^ Figure 3.3.2 .1. Model for transfer matrix analysis. 34 Figure 3.3.3. 1. 6) versus B curves for the lowest two pass bands of a typical slow-wave electrode ^36 Figure 3.3.3 .2. Phase and group velocities as a function of frequency of a typical slow-wave electrode.  ^37  Figure 3.3.3. 3. Characteristic impedance as a function of frequency of a typical slow-wave electrode.  ^38  Figure 3.3.5. 1. Calculated capacitance of fins in air having S2 = 1 Am, t = 0.5 pm, 1 = 1, 2, 3, and 4 Am, and W2 ranging from 3 to 10 Am. ^50 Figure 3.3.5. 2. Assumption made in calculating the capacitance of fins with pads. .. ^51 Figure 3.3.6. 1. Fin capacitance to length ratio for fins in air having S2 = 1 pm, = 0.5 Am, / = 1, 2, 3, and 4 Am, and W2 ranging from 3 to 10 pm.  vi  ^53  Figure 3.3.7.1. Design curves for 50 and 75 ohm velocity-matched slow-wave electrodes on GaAs substrates for e s = 1, S2 = 1 Am, ^ t = 0.5 Am, and / = 1 Am 57 Figure 3.3.7.2. Design curves for 50 and 75 ohm velocity-matched slow-wave electrodes on GaAs substrates for e s = 3.5, S2 = 1 AM, t = 0.5 Am, and / = 1 Am. ^  58  Figure 3.3.7.3. Design curves for 50 and 75 ohm velocity-matched slow-wave electrodes on InP substrates for e s = 1, S2 = 1 AM, t= 0.5 Am, and / = 1 Am.  ^59  Figure 3.3.7.4. Design curves for 50 and 75 ohm velocity-matched slow-wave electrodes on InP substrates for e s = 3.5, S2 = 1 AM, t = 0.5 Am, and / = 1 Am.  ^60  Figure 3.3.10.1. Input section for slow-wave electrodes showing the various ^ dimensions. 66 Figure 4.2.1. Layout of Mask #1.  ^71  Figure 4.2.2. Layout of Mask #2.  ^72  Figure 4.2.3. Dimensions of the two input sections used in Mask #2.  ^73  Figure 4.3.1. Main steps of the fabrication procedure.  ^75  Figure 4.4.1. SEM micrograph of electrode #2 of Mask #1 (half-buried).  ^88  Figure 4.4.2. SEM micrograph of electrode #2 of Mask #1 (totally-buried).  ^89  Figure 4.4.3. SEM micrograph of electrode #1 of Mask #2 (half-buried).  90  Figure 4.4.4. SEM micrograph of electrode #5 of Mask #2 (surface deposited).  .  91  Figure 4.4.5. SEM micrograph of electrode #7 of Mask #2 (surface deposited).  .  92  Figure 5.2.1. Picture showing the network analyzer, synthesized sweeper, and directional bridge.  ^96  Figure 5.2.2. Picture showing the probe jig with probe mounted (left) and the sample platform (right).  ^96  Figure 5.2.3. Picture showing the whole setup of the equipment.  ^97  vii  Figure 5.3.1. Schematic of the setup used for the interference technique.  ^98  Figure 5.3.2a. Measured normalized reflected power in dB with the probe not touching the conventional coplanar strip electrode of Mask #1. . ^. 102 Figure 5.3.2b. Measured normalized reflected power in dB with the conventional coplanar strip electrode of Mask #1 probed. ^ 103 Figure 5.4.1. Schematic of the setup used for the resonance technique. ^ 104 Figure 5.5.1. Measured normalized reflected power as a function of frequency of electrode #1 (half-buried) of Mask #2. ^  108  Figure 5.5.2. Measured normalized reflected power as a function of frequency of electrode #2 (half-buried) of Mask #2. ^  109  Figure 5.5.3. Measured normalized reflected power as a function of frequency of electrode #5 (unburied) of Mask #2. ^  110  Figure I.1.^A coplanar strip electrode on a substrate. ^  129  Figure II.1.^Dimensions for cells oriented parallel and perpendicular to one another^  134  Figure III.1.^Equivalent circuit of an electrode having a characteristic impedance Zo , terminated by a load ZT and connected to a source Z5 with voltage V. ^  viii  137  List of Tables Table 3.3.5.1.  Capacitance of fins in air.  ^48  Table 3.3.5.2.  Capacitance of fins with pads in air  ^49  Table 3.3.7.1.  Calculated Nd s and Z' 's of some slow-wave electrodes fabricated and tested  ^56  Table 4.2.1.  Mask #1 electrode dimensions.  ^70  Table 4.2.2.^Mask #2 slow-wave electrode dimensions. Table 4.3.4. 1. Etching characteristics of various etchants used in the fabrication  70 ^83  Table 5.5.1.^Measured and theoretical Neis of the conventional coplanar strip electrode ^  106  Table 5.5.2. ^Measured and theoretical Nd s of the slow-wave electrodes (Mask #1). ^  106  Table 5.5.3. ^Measured and theoretical Neff's of the slow-wave electrodes (Mask #2). ^  107  ix  Acknowledgements I would like to thank the members of my family for their support, encouragement, and faith throughout my education. I would like to express my gratitude to my supervisor, Dr. N. Jaeger, for suggesting this project and providing continual guidance and support during my research. My thanks extend to our research engineer, H. Kato, for invaluable assistance during the fabrication of the devices. I am also grateful to all members of the solid state group and those individuals in the Electrical Engineering department who have helped me in various ways throughout my research. Finally, I would like to acknowledge the support of the Canadian Cable Labs Fund and the Natural Sciences and Engineering Research Council (NSERC) of Canada.  x  Chapter 1 Introduction 1.1. Importance of Slow-Wave Electrode Structures in Electro-optic Modulators  One of the fundamental problems associated with conventional integrated electro-optic travelling-wave modulators is the phase and group velocity mismatch between the optical wave and the modulating microwave signals, which limits the achievable bandwidth as well as modulation depth for a certain amount of modulating power [1,2:pp. 160-163]. In particular, in modulators using III-V semiconductor materials such as gallium arsenide (GaAs) and indium phosphide (InP), the microwave signals travel faster than the optical waves if conventional coplanar strip electrodes are used. These problems, however, can be solved by using slow-wave electrodes in place of the conventional coplanar strip or coplanar waveguide electrodes [3,4,5,6]. The capacitively-loaded slow-wave electrode structures introduced in this work are designed for use in integrated electro-optic modulators based on III-V semiconductor materials. They offer low loss, low dispersion, flexible dimensions, ease of fabrication, and most important of all, they allow one to engineer the velocity of the microwave signal so as to match it to that of the optical wave, thus achieving the velocity-matched condition. The huge bandwidth, high information-carrying capacity, immunity to electrical interference, and security offered by contemporary optical fibres make it very attractive to develop broadband optical modulators, with low power consumption, for use in high-speed 1  optical communications and signal processing [2:pp. 1-3]. There are generally two common ways to impress information onto the optical wave. These include the direct modulation of semiconductor laser diodes and the use of external modulators. Direct modulation of semiconductor laser diodes offers simplicity and large amplitude modulation [7]. Bandwidths in the range of 20 GHz have been obtained in some highspeed Fabry-Perot lasers [8,9]; however, these lasers produce large linewidths [9]. Distributed feedback lasers offer the advantages of single mode operation and narrow linewidths [7]; however, frequency chirping under direct modulation still broadens the linewidth [10,11], resulting in bandwidths of about 13 GHz [12]. Quantum well lasers are excellent sources for producing narrow linewidths [13,14,15]; however, the modulation frequency depends on many laser parameters including differential gain, gain saturation, and the highest possible frequency is limited by the finite trapping time of the carriers [16]. The injection-locking technique [17,18,19] results in significant reduction in linewidth and frequency chirping [20]; however, the use of two semiconductor lasers increases the complexity of the system. The use of external modulators, on the other hand, offers potentially higher performance and is not necessarily more complicated than direct modulation of semiconductor laser diodes. Two major kinds of external optical modulators are known. These are the electroabsorptive devices [21] and the electro-optic devices [22]. Electro-optic modulators offer the advantages of simplicity and ease of fabrication. Electro-optic Mach-Zehnder modulators are also immune to frequency chirping [23]. There are generally two categories of material used to implement these modulators -lithium niobate (LiNbO3 ), which is a ferroelectric insulating crystal, and GaAs and InP, which 2  are semiconductors. Although LiNbO 3 generally has superior optical qualities than the semiconductors (i.e., lower optical loss, higher electro-optic coefficients...etc.) [1], the semiconductors allow one to monolithically integrate electro-optic, electronic, and opto-electronic devices on a single substrate. For example, one can fabricate a laser source, an electro-optic modulator, as well as other control circuitry on the same GaAs or InP substrate. Moreover, the use of GaAs and InP allows one to take advantage of their low (microwave) dielectric constants. For instance, as is shown in this thesis, one is able to fabricate slow-wave electrodes on GaAs so that the velocity of a microwave can be matched to that of an optical wave, which is one of the most important criteria for achieving wide bandwidths, low modulating powers, and high modulation depths [1,2:pp. 160-163]. However, this is not easily done for LiNbO 3 since the high dielectric constant of LiNbO 3 makes the microwave travel slower than the optical wave, although some recent work involving the use of a last-wave electrode' employing a metal shield overlying the substrate has been published [24]. Motivated by the many advantages of external compound semiconductor electro-optic modulators, as well as the very high performance possible for these devices when the microwave velocity is matched to that of the optical wave, a slow-wave electrode structure, being fabricated on compound semiconductor materials, that enables one to engineer the velocity of a microwave so as to match it to that of an optical wave, is developed. This thesis provides a detailed account of the theory, design, fabrication, and measurement of such a novel slow-wave electrode structure.  3  1.2. Organization of the Thesis  In Chapter 2, the electro-optic effect and other basic principles are reviewed. The basic operation of a travelling-wave electro-optic modulator is described. The current status of research in this area is provided. The application of slow-wave electrodes to Mach-Zehnder modulators is finally given. Chapter 3 gives a detailed treatment of the theory behind the slow-wave electrode structures introduced in this thesis. Analysis using a quasi-static approximation is carried out. The basic principles of capacitive loading, the high frequency filter characteristics and dispersion, as well as structures useful in travelling-wave modulators are described. The computation of the capacitance of the loading fins using finite difference and finite element methods are also presented. Improved and simplified design formulas for slow-wave electrodes are derived. Design curves for possible electrode structures for both GaAs and InP modulators are presented. The microwave losses of the slow-wave electrodes are also calculated. Chapter 4 is dedicated to describing the fabrication of the slow-wave electrodes. The basic steps in the fabrication are given. Various anisotropic etchants are reviewed and the corresponding fabrication results presented. Chapter 5 deals with the measurements based on the Hewlett Packard HP8757C scalar network analyzer, the HP8341B synthesized frequency source, and the HP85027E directional bridge. The measurement techniques and the measured results are presented. Chapter 6 gives a conclusion as well as recommendations for the future continuation of this research. 4  Appendix I summarizes the design formulas for conventional coplanar strip electrodes that are useful in the design of the slow-wave electrodes presented in this thesis. Appendix II summarizes the equations useful for implementing a finite element program for calculating the capacitance of the loading fins. Appendix III provides a derivation of the reflection coefficient for the electrodes. As will be shown here, the rapid decrease of the reflection coefficient at resonance allows one to calculate the microwave effective refractive indices of the electrodes by measuring the frequency separations between the resonances.  5  Chapter 2 Background 2.1. Introduction  The linear electro-optic effect is the basis for most electro-optic devices including modulators. When a voltage is applied to an electrode placed over or alongside an optical waveguide in a medium having a non-zero electro-optic coefficient, such as most compound semiconductors including gallium arsenide (GaAs) and indium phosphide (InP), the electric field created in the medium causes a change in the refractive index. This index change may result in phase and/or polarization modulations. When an interferometer such as a Mach-Zehnder structure is used, intensity modulation results [1,2:pp. 174-184]. There are generally two major categories of electrodes used in these modulators--lumped element electrodes and travelling-wave electrodes [2:pp. 153-163]. The lumped element electrode modulators (Figure 2.1.1) have an electrode length of less than a quarter of an electromagnetic wavelength [2:pp. 159-160]. The induced refractive index change of the medium is substantially constant throughout the whole length of the electrode. The modulation bandwidth of this type of modulator is the smaller of the inverse of the optical transit time, or the resistance-capacitance (RC) time constant of the lumped-circuit parameters. The latter, however, is usually more restrictive. In order to achieve wide bandwidth the capacitance and therefore the electrode length should be as short as possible. A short electrode, however, results in high power being needed for a particular modulation depth [1,2:pp. 159-160]. This 6  Figure 2.1.1. Typical lumped element electro-optic modulator.  is one of the greatest disadvantages of the lumped element electrode design. Modulators of this type having a bandwidth of 4.5 GHz and a half-wave voltage of 13 V have been reported [25]. The travelling-wave electrode (Figure 2.1.2) is an extension of the transmission line which carries the modulating signal. In modulators employing this type of electrode, the optical and modulating microwave signals travel alongside each other, but generally at a different velocity. The phase of the optical signal is being modulated as it propagates along the waveguide. The 7  Figure 2.1.2. Typical travelling-wave electro-optic modulator.  modulating bandwidth in this type of modulator is not limited by the RC time constant of lumped element electrodes but rather by the velocity mismatch between the optical and microwave signals [1,2:pp. 160-163]. For broad-band operation, the velocity of the microwave should be equal to that of the optical wave. It is our objective to develop slow-wave electrode structures, being fabricated on compound semiconductor materials, to match these velocities. These slow-wave electrode structures are described in this thesis. 8  This chapter presents a brief review of the electro-optic effect associated with III-V semiconductors. The bandwidth and modulating voltage of travelling-wave modulators are given. Finally the current status of research in this area is provided.  2.2. The Linear Electro-optic Effect  The linear electro-optic effect associated with cubic crystals having 43m crystal symmetry in crystallographic system is described mathematically by the electro-optic matrix rii as [26:pp. 227-229] 0 0 0 0 0 0 0 0 0 Ti] =  r41 0 0 0 r41 0 0 0 r41.  At a free-space optical wavelength of 1 Am, r41 = - 1.4x10 1° cm/V for both GaAs and InP [2:p. -  364]. When an electric field having components Ex , Ey , and Ez is applied to the crystal, the index ellipsoid is perturbed and becomes  9  1 2  no (x y  z) r41 Ez  r41 Ez r41 Ey  1 2  no  r41 Ey r41 Ex  r41 Ex 1 2  no  where no is the refractive index of the optical wave. At a free-space wavelength of 1 Am, no = 3.43 for GaAs and no = 3.3 for InP [2:p. 364,27,28]. It is apparent that the principal axes of the index ellipsoid are in general not parallel to the x, y, and z directions with an electric field applied. By diagonalizing the matrix one can obtain the perturbed refractive indices and the principal axes corresponding to electric fields applied along different directions. The three most common orientations of the applied electric field are the [100], [011], and [111] directions. The perturbed refractive indices and the principal axes with electric fields along these directions are summarized as follows (see also [2:pp. 362-364]): (I) electric field oriented along a [100] direction (i.e., E = Ez): n[011] = n o - An n10177 = n o + An n[100] = n o  (II) electric field oriented along a [011] direction (i.e., Ex = Ey = E/i2) nbr711] = no + An nb12111 = n o - An n[01T] = no 10  (III) electric field oriented along a [111] direction (i.e., E x = Ey = Ez = E/13) nfir01 = no + An/i3 n[1127 = n o + An/^3 nil 1 1] = no -  where 1 3  A n= — n r E .  2 ° 41  Since the [100] direction is the preferred direction of epitaxial growth [2:p. 364], the orientations of electric fields in cases (I) and (II) are most commonly used. It can be seen that for electric fields applied along a [100] direction, with an optical wave propagating along a [O1T] direction, only TE-polarized optical fields will be modulated while TM-polarized fields remain unperturbed. This is because TM-polarized fields are parallel to the unperturbed principal axis (i.e., they are oriented along a [100] direction). For electric fields applied along a [011] direction, with an optical wave propagating along a [O1T] direction, both TE- and TM-polarized fields will be modulated, since the two principal axes that lie in the plane perpendicular to the direction of propagation are oriented 45° with respect to the TE and TM polarizations. Linearly polarized inputs will result in polarization rotations. The relative directions of the principal axes in these two cases are shown in Figure 2.2.1.  11  1 1o  no  -  A^no-An  no  Am  +Am  ^7 no +Am  7  -  I- //  ]^  [  Z11]  Figure 2.2.1. Principal axes of the perturbed index ellipsoid for 43m crystals.  2.3. Velocity Matching, Phase Retardation, and Bandwidth  The phase retardation Al)(1E) of a TE-polarized optical wave, with the microwave electric field oriented along the [100] direction, having travelled an interaction length L in a waveguide, 12  is given by [1,2:pp. 160-163] as sin(rriyo)  Ad:0(TE) a AC.^ (t.flf0)  (2.3.1)  2.1c An^nor4i von 4:0 „ - ^ L- ^ 1S 1  (2.3.2)  where 3  fo_  (cILNeff 1 n °/Neff )  (2.3.3)  —  f  the frequency of the modulating electric field (usually a microwave), Vo the voltage amplitude  of the microwave electric field, A the wavelength of the optical wave, S1 the gap between the electrodes (see Figure 2.1.2), Neff the effective refractive index of the modulating wave, and r is the overlap integral between the normalized optical field Eop, and the applied electric field E, as defined by [2:p. 155]  sl — vo ffElEoptdA  (2.3.4)  where the integration is over the cross-sectional area of the optical mode. The value of r for coplanar strip electrodes is typically about 0.7 [22]. For TM-polarized optical wave, the phase retardation is zero since the refractive index is not perturbed by electric fields applied along 13  [100]. For a microwave electric field oriented along [011] the phase retardations of both TE- and TM-polarized optical fields are twice that for the TE-polarized optical field when the microwave electric field is oriented along [100]. The [011] orientation results in a higher induced birefringence and a higher modulation since both TE- and TM-polarized optical fields are modulated [22]. It can be seen from equations (2.3.1) to (2.3.3) that when the velocities are matched (i.e., Neff =  n„) the total phase retardation of the optical wave is maximized. The optical wave then  sees the same voltage over the entire electrode length. When this happens, the total phase retardation is proportional to the product VoL and arbitrarily long electrodes can be used so as to reduce the drive voltage and no frequency limitation results if microwave losses can be ignored. On the other hand, if the velocities are not matched, there will be cancellations in the modulation as the optical wave travels a certain distance down the waveguide. Consequently the total phase retardation will be smaller for the same value of VoL. In fact, for a fixed L, there will be no modulation at all if the modulation frequency f is an integral multiple of fo . Moreover, if the modulation frequency is sufficiently high, the total phase retardation will approach zero. The 3dB bandwidth [1,2:pp. 160-163] of the modulator can be obtained from equation (2.3.1) and (2.3.3) by noting that 2  Ad)  is reduced to 50% of its value at f=0 when  foil  2c/Neff 1 —n o/Neff  (2.3.5) For conventional coplanar strip electrodes on GaAs, Neff. = 2.64, no = 3.43, and zifi, = 24 14  GHz.cm . For velocity-matched electrodes, like the ones described in this thesis, 24 is theoretically infinite provided microwave losses may be ignored. At very high frequencies microwave losses become the dominating factors that limit the bandwidth.  2.4. Current Research in Electro-optic Modulators  Two major electrode configurations for GaAs travelling-wave modulators have been reported [22]. These are the microstrip configurations using a p-i-n structure [29] (Figure 2.4.1) and the coplanar strip configurations using undoped epitaxial layers grown on semi-insulating substrates [30]. The slow-wave electrode structures described in this thesis are of the coplanar strip configuration. Shown in Figure 2.5.2 is the cross section of a Mach-Zehnder modulator (to be discussed in the next section) employing these slow-wave electrodes and graded index optical waveguides. Although microstrip electrodes fabricated on GaAs can be designed to give a microwave phase velocity very close to that of the optical wave, the metal strips (signal lines) have to be very close to the ground plane in order to have a high enough electric field for any useful modulation. Consequently, this requires a very thin, usually a few microns, optical guiding layer (usually semi-insulating GaAs) sandwiched between the strips and the ground plane. Unfortunately, growing GaAs on metal is not possible let alone handling such a thin layer. Therefore, a p-i-n structure with thick p and n + layers above and below the semi-insulating optical guiding layer is necessary [29]. However, the n + substrate causes very high microwave losses and dispersions, which severely limit the performance of the modulators [22]. 15  /p  1 7-1 -t\s -.  G  n  aAs  A 1X A  n+ Po"^  P\etoll izoLion Figure 2.4.1. A p-i-n travelling-wave phase/polarization modulator showing a single optical waveguide underneath the electrodes.  On the other hand, coplanar strip electrodes fabricated on semi-insulating gallium arsenide have very low loss and dispersion. However, the overlap between the optical and microwave fields is smaller (usually in the range of 70%) as compared to that in microstrip modulators (which can be almost 100%) [22]. Moreover, the phase velocity of the microwave in coplanar strip electrodes is higher because of the portion of the field travelling on the air side. Consequently, in conventional coplanar strip structures the velocity mismatch between the microwave and the optical wave limits the achievable bandwidth at moderate power levels of the modulating signal 16  [1,2:pp. 160-163]. A technique for slowing down the microwave is therefore needed in order to achieve high performance in coplanar strip modulators. Recently, a velocity-matched planar microstrip structure using an n - -i-n configuration with relatively low loss has been proposed and analyzed [31]. Nees et al. cemented a GaAs superstrate directly on top of the coplanar strip electrodes to suppress both velocity mismatch and electrical dispersion [32]. Lee et al. published a velocity-matched GaAs travelling-wave optical modulator using a modulated coplanar slow-wave electrode with periodic cross-tie overlays [33,34,35]. This type of slow-wave electrode, however, requires a multi-layer electrodedielectric structure. Walker reported a GaAs Mach-Zehnder push-pull travelling-wave modulator employing a capacitively-loaded coplanar strip electrode with segmented modulator elements [36,37], in which the loading capacitance between the coplanar strips is formed via the modulator elements and an n + epitaxial layer. Very high figures of merit have been reported for these modulators; however, they use isolation trenches, air bridges, as well as other biasing and decoupling circuitries [36]. The capacitively-loaded slow-wave electrode structures described in this thesis can be used to directly replace the coplanar strip electrodes in the modulators [3,4,5,6]. These slow-wave electrodes possess the advantages of low loss, low dispersion, flexible dimensions, ease of fabrication, and most importantly, they allow one to flexibly engineer the velocity of the microwave so as to achieve the velocity-match condition. Also, the full length of the electrodes can be used to modulate the optical wave. The electrodes do not require the use of any potentially lossy doped epitaxial layers and they can be easily formed in a single layer of metallization using a standard lift-off technique [38:pp. 115-138]. Moreover, the electrodes will 17  not introduce any more stress to the substrate than any other coplanar strip or coplanar waveguide electrodes. The microwave effective refractive indices and characteristic impedances will also be insensitive to the parameters of the substrates such as the thickness of the epitaxial layers.  2.5. Coplanar Slow-Wave Electrodes in Mach-Zehnder Modulators  The coplanar slow-wave electrodes described in this thesis are designed for use in MachZehnder modulators. A Mach-Zehnder modulator is an intensity modulator using two interfering arms [2]. There is one input and one output port for the optical wave. The optical waveguide consists of a straight input section, a 3dB power splitting Y-junction, two straight phase modulating sections, a power combining Y-junction, and an output section. Such a modulator employing the slow-wave electrodes described in this thesis is depicted in Figure 2.5.1. Figure 2.5.2 shows the cross section of a Mach-Zehnder modulator employing these slow-wave electrodes and graded index optical waveguides. Here, the microwave electric fields are in the [100] direction. The input wave is split into two equal components by the Y-branch power splitter. Each of these components then propagates and is modulated over one arm of the interferometer. Since the modulating electric fields are in opposite directions over these two arms, the phase retardation is in opposite directions. Consequently the total phase difference between the two arms is twice that over one arm. The modulating voltage is therefore halved  18  for the same interaction length L. The optical waves in the two arms eventually recombine at the output Y-junction. If the guided modes in the two arms are in phase as they recombine, then they interfere constructively and excite the lowest order mode of the output waveguide [2:pp. 182-183]. The transmitted light is a maximum. If they are 180° out of phase as they recombine, then they excite the first antisymmetric mode which is not supported by the output section. This mode is cut off and radiated into the substrate and the transmitted light is a minimum [2:pp. 182183]. In this type of modulator, it is essential that the two interferometric arms be sufficiently separated so as to prohibit evanescent field coupling between them. Since the slow-wave electrodes satisfy the velocity-match condition, the bandwidth of the Mach-Zehnder modulator will be theoretically infinite if microwave losses may be ignored. Arbitrarily long electrodes may therefore be used. Assuming r  = 0.5 (this value is lower than  that commonly found in modulators employing conventional coplanar strip electrodes due to the use of loading fins and pads), S1 = 13 Am, and A = 1 Am, a half-wave voltage, i.e., voltage needed for creating a phase difference of 7r between the two arms of the interferometer and complete turn off of the output, of 22 V for a 1 cm interaction length, or 22 V.cm, will be needed for modulating TE-polarized optical fields. This voltage, however, can be reduced by using longer interaction lengths (i.e., longer electrodes). For instance, if 2 cm interaction lengths are used, the half-wave voltage will be reduced to half, or 11 V. Since arbitrarily long interaction lengths can be used without sacrificing the bandwidths, these modulators have a theoretically infinite bandwidth to modulation voltage ratio (if losses may be ignored). It should be noted that only modulators using velocity-matched slow-wave electrodes will have theoretically infinite bandwidths that are independent of the interaction lengths. On the contrary,  19  the bandwidths of modulators employing conventional coplanar strip electrodes are inversely proportional to the interaction lengths (see equation (2.3.5)); consequently, these modulators will either have wide bandwidths at the expense of very high modulation powers, or narrow bandwidths at the gain of reduced modulation powers. These modulators typically have a bandwidth to modulation voltage ratio of around 1 GHz/V [1]. Modulators employing velocitymatched slow-wave electrodes, such as the ones described in this thesis, will have the benefits of both wide bandwidths and low modulation powers. MICROWAVE ELECTRODES^  O PT I CA L WAVEGU I DES  C)  INPUT WAVEGUIDE  Y -BRANCH PHASE MODULATOR SECTIONS  Y -BRANCH OUTPUT WAVEGUIDE  Figure 2.5.1. A Mach-Zehnder modulator employing a coplanar slow-wave electrode. Note the capacitive loading fins between the coplanar strips. 20  WI  Si  WI  El ectrodes  x .N  A 1,<Go,_■'\s  Graded Index Region  N/  x  S . I . GaAs Substrate Meta 1 1 1 zat Ion  Figure 2.5.2. Cross section of a Mach-Zehnder modulator employing a coplanar slow-wave electrode and graded index optical waveguides. The optical guiding layer is formed of AlGa i _xAs with a graded Al concentration. The capacitive loading fins in the gap between the two coplanar strips are shown.  21  Chapter 3 Theory 3.1. Introduction As already discussed in Chapter 2, the velocity mismatch between the optical wave and the modulating microwave signal in a travelling-wave electro-optic modulator limits its bandwidth and the amount of microwave power needed for a certain modulation depth is relatively high [1,2:pp. 160-163]. In III-V semiconductor based modulators employing conventional coplanar strip or coplanar waveguide electrodes, with the superstrate being air, the microwave signals travel faster than the optical signals. Therefore, a slow-wave electrode, being fabricated on III-V semiconductors such as gallium arsenide (GaAs) and indium phosphide (InP) based materials, that enables one to engineer the velocity of the microwave that travels along it, is highly desirable. Several slow-wave electrode structures have been reported. All of these structures operate on the principle that by increasing the effective capacitance per unit length the microwave is slowed. These include the metal-insulator-semiconductor (MIS) structures [39,40,41,42], the Schottky contact structures [41,42,43], and the coplanar waveguide structures using periodically doped semiconductor substrates [44]. These electrode structures, however, are inadequate for use in high-speed electro-optic modulators due to the inherently lossy nature of the doped semiconductor materials. Recently a coplanar electrode structure with periodic cross-tie overlays has been proposed and its use in a Mach-Zehnder modulator has been demonstrated [33,34,35]. This structure, however, is a multi-layer structure requiring two layers of metalization and one 22  layer of dielectric, complicating its fabrication. The slow-wave electrode structures introduced in this work not only enable one to engineer the velocities of the microwaves that travel along them, thus achieving the velocity match condition when they are used in III-V semiconductor electro-optic modulators, but they also offer the advantages of low loss, low dispersion, and ease of fabrication [3,4,5,6]. The structures do not require the use of any potentially lossy doped semiconductor material. The microwave velocity and characteristic impedance are also insensitive to the parameters of the substrate such as the thickness of epitaxial layers.  3.2. The Slow-Wave Electrode Structures  The slow-wave electrode is a coplanar strip periodically loaded with capacitive elements. There are two generations of the electrode structure. In the first generation, narrow capacitive fins extending into the inter-electrode gap region are used as the capacitive loading elements [3]. Due to the fringing electric fields about the fins, the amount by which the capacitance per unit length increases is greater than the corresponding amount by which the inductance decreases, resulting in the slowing of the microwave. The second generation electrode structure is basically the same as the first except that rectangular pads are added to the ends of the fins to further increase the loading capacitance [4]. The higher capacitance, as will be shown later in this chapter, allows one to increase the fin period as well as the width of the electrodes, thus increasing the flexibility of the electrode dimensions required as well as reducing the resistive loss and the number of loading fins and pads needed. The reduction in the number of fins and 23  pads makes possible a higher yield in the fabrication. The electrode structure of the first generation is shown in Figure 3.2.1. Here capacitive loading fins of length 1, width W2, gap width S2, and period d are added to the coplanar strips of width W1 separated by a gap S1 , and both the fins and the coplanar strips are of thickness t. In order to exploit the fringing electric fields produced by the fins, it is essential to keep the fins narrow (i.e., small length 1 as compared to the fin width W2). This is because narrow fins have a higher capacitance to fin length ratio as compared to wide fins. The decrease in inductance per unit length due to the narrow fins is also smaller. In other words, the narrower the fin, the closer it is to a purely capacitive element. The electrode structure of the second generation is shown in Figure 3.2.2. Here pads of width W' and length 1' are added to the ends of the fins. All other dimensions are defined in the same way as those of the first generation. The function of the pads is to further increase the capacitance between the signal and ground lines, thus resulting in a more effective capacitive loading. If pads of proper dimensions are used in conjunction with narrow fins, as will be discussed, it is believed that their effect on the inductance per unit length will be the same as if the pads were absent. Electrode structures of both generations may be partially buried or surface deposited. Burying the electrodes helps increase the capacitance per unit length somewhat so that a higher degree of slowing may be achieved. A high dielectric constant superstrate such as a polyimide may also be applied, as it also increases the capacitance per unit length of the electrodes. In this chapter, the analysis of the slow-wave electrode structures is given and design formulas are derived. Design curves, based on GaAs and InP substrates, are calculated for half24  buried electrodes with superstrates having dielectric constants of 1 (i.e., air) and 3.5 (e.g., a polyimide).  25  ^  V  ^ ^ d d d  _I  \^t--I  r" A#A#11 / L^/^r llo^16^lo^lo  ^7^7^7 /  / /  /  /  W2 1 lei 52 ^  / /  "77/AtAA  W2  WI  Figure 3.2.1. Plan view of a section of a slow-wave electrode showing the dimensions S 1 52 , W2 , d, and 1.  26  ,W, 1  V  N  d  d  d  N  d  W, V2  32  S1  \i/2 W1 \/  Figure 3.2.2. Plan view of a section of a slow-wave electrode with pads showing the dimensions SD W1, S2, W2, d, 1, W', and l'.  27  3.3. Physics of the Slow-Wave Electrode Structures  3.3.1. Introduction  According to the quasi-static (quasi-TEM) approximation, the phase velocity of a microwave propagating along thin conventional (unloaded) coplanar strip electrodes on an infinitely thick substrate with an infinitely thick superstrate is constant regardless of the width of the strips and the gap separating the strips [45:pp. 257-288]. This is because any change in the dimensions of the electrodes resulting in a change of the shunt capacitance per unit length C always causes a corresponding change in the series inductance per unit length L such that the  net effect is that the phase velocity vp° = (LC) 112 remains constant. With minor modifications -  of the expressions given in references [45:pp. 257-288,46:pp. 363-381], vp° is given in terms of the dielectric constants of the substrate E r , and superstrate e s as O C V =— P No  (3.3.1.1)  where c is the speed of light in vacuum and No the effective refractive index given by  ,1  N0= \ er 2+ es =c LC .  The characteristic impedance Z, is given as 28  (3.3.1.2)  7  L No C = cc •  (3.3.1.3)  If capacitance Cp in the form of narrow fins (and pads, if applicable), is added to the electrode at periodic intervals, then, the inductance per unit length will only be minimally reduced and the microwave will be slowed due to an increase in the effective capacitance per unit length. The slow-wave electrodes introduced in this work are based on this principle. In the following sections, the slow-wave electrode structures are first analyzed using a transfer matrix method. Their low frequency and high frequency behaviours are discussed. Then design formulas are derived and design curves generated. The computation of the fin capacitance is also outlined.  29  3.3.2. The Transfer Matrix Method  The transfer matrix method is useful for analyzing periodic structures [46:pp. 363381,47,48]. In simple terms, it relates the voltage and current at one section of a periodic structure to the voltage and current at an adjacent section by a 2 X 2 matrix. Here, the slowwave electrode structures are analyzed using such a method. In the analysis that follows, the fins (and pads where appropriate) are treated as pure capacitive elements; their effects on inductance are ignored. Then in section 3.3.5, their effects on inductance are approximated by introducing a weighted average. The microwave losses of the electrode structures are calculated in section 3.3.8. The model of the slow-wave electrode structure is shown in Figure 3.3.2.1. Here the electrode consists of sections each of which is subdivided into a smaller section of length d/2, a capacitive section, and another section of length d/2. Using standard network analysis, the voltage V and current I at the nth and (n+1) th sections are related by  vIn  8^. 6 cosh— Z smh— 2^°^2^1 0 =  6 '6.)C 1 ^8 —sink— cosh — f 1 Zo^2^2  6^. 0 cosh— Z smh— 2^°^2 1 . 0^0 —smh— cosh— Zo^2^2 (3.3.2.1)  where  30  =(a o +jk)d=(a 0 +j)d=(a o+j) )d  A,^V  (3.3.2.2)  0  and Zo , a o , k, and A o are, respectively, the characteristic impedance, attenuation constant, propagation constant, and microwave wavelength of an unloaded section,  cf the capacitance of  a pair of fin (and pad, where appropriate) elements, vp ° the phase velocity of an unloaded electrode, and 6) the frequency of a microwave that travels along the electrode. Note that the first and the last matrices on the right hand side of equation (3.3.2.1) are the transfer matrices for sections 'A' and 'C' respectively (see Figure 3.3.2.1); the middle matrix is the transfer matrix for the admittance ja)C produced by a pair of fin (and pad, where appropriate) elements. Equation (3.3.2.1) can be simplified to give the equivalent transfer matrix of the loaded (slowwave) electrode  f vd In  cosh()^ +j CIZ° sink() 2 (1)  ZosinhO +j(  (4 C^co C — 1 sinhe +j(—fcoshe + —J.) Zo^2^2  Z,2^2 CO^CO C f^Z, C f 2  cosh° + j  cosh°  co CIZ0 . smite 2  2  ) [Iv:+ +,1  1  (3.3.2.3) which can be written as  [ vnI n  cosh()  Z isinhO  lsinhO ZI  31  cosh  (3.3.2.4)  where N„  (I)= (a + j (3)d= (a + j^k)d  (3.3.2.5)  and a and 13 are, respectively, the attenuation and propagation constants of the slow-wave electrode; and and Z' are, respectively, the effective refractive index and characteristic impedance of the slow-wave electrode and  N,  the effective refractive index of the unloaded  electrode. Comparing equations (3.3.2.3) and (3.3.2.4) gives coC ,Z  coshcl) =cosh° +j^° sinhe 2  (3.3.2.6)  Since the attenuation constants a o and a are generally very small as compared to the propagation constants k and 13 (see section 3.3.8), they can be ignored when calculating N eff and Z'. Then in section 3.3.8 equation (3.3.2.6) will be used when the attenuation constants and microwave losses are calculated. Ignoring a o and a, equation (3.3.2.6) becomes Rd=coskd  C'/Z0 sinkd 2  - ^  (3.3.2.7)  Similarly, the characteristic impedance Z' of the slow-wave electrode can be obtained as follows:  32  z 1 zo 1.  co C1  2  1^w C^(DC —sink 1 + —f coskt1+ f  (3.3.2.8)  z 2^2 °  The effective refractive index Neil of the slow-wave electrode can be obtained using equations (3.3.2.5) and (3.3.2.7). Equations (3.3.2.7) and (3.3.2.8) provide useful information on the characteristics of the slow-wave electrodes, such as dispersion, group velocity, and high frequency filter characteristics. These are discussed in the next section.  33  Section 8 Scion A^I^Sec-Lion C 70  70  V, n  <  d/2  d/2  Figure 3.3.2.1. Model for transfer matrix analysis.  34  3.3.3. Dispersion Characteristics and Group Velocity The phase velocity vp of the slow-wave electrode, as given by &B, can be found by solving equation (3.3.2.7). The group velocity vg , as given by d&dB, can also be approximated by 4 6)/ AB. Thew versus B curve for a typical structure having d = 7.3 pm, S2 = 1 pm, W2 = 7 pm, t = 0.5 pm, C1 = 0.68 fF, Zo = 65 ohm, e r = 12.9, e s = 1, and Neff = 3.43 is shown in Figure 3.3.3.1. It is apparent that at a very high frequency when the fin spacing d becomes comparable to a wavelength the electrode begins to behave like a band-pass filter. There is an infinite number of pass-bands and stop-bands; however, the lowest stop-band frequency for a typical structure is very high (in the range of 1000's of GHz). For the intended application of the electrode only the lowest pass-band will be used. Therefore in the discussion that follows only the lowest pass-band will be considered. vp and vg are shown in Figure 3.3.3.2. The characteristic impedance Z' is shown in Figure 3.3.3.3. It can be seen that vp = vg at frequencies up to about 200 GHz before they begin to depart. At frequencies around 3000 GHz both vp and vg drop rapidly towards zero, exhibiting their filter characteristics. Z' also drops off rapidly at this range of frequency. This range of frequency will definitely be above the frequency for any foreseable future operation of such electrodes. As a result, this dispersion of the electrodes, due to filter characteristics, will not introduce any problems for their intended use and will not be considered further. However, one must keep in mind that this analysis may fail at very high frequencies at which the wavelengths approach the transverse dimensions of the electrodes.  35  STOP BAND  1 1 1 1 1 1^1 1 1  1 1^1 1 1 1  .11.  1^  1  ......^  11  1  .11.1.11,  1  ......."  2^ 3^ 4  p 1 O5 m-1) (  Figure 3.3.3.1. 6.) versus 13 curves for the lowest two pass bands of a typical slow-wave electrode. Note that the two pass bands are separated in frequency by a forbidden stop band. 36  1 0.0  8.0 (f)  F 0 6.0 v-  6 o  ti  -  4.0  Phase Velocity — — — Group Velocity V g  ^  2.0  Vp  ti  1^1^1^1^1^1 1 1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1 1 1^1^1^1  1  10^100^1000 Frequency (CHz)  Figure 3.3.3.2. Phase and group velocities as a function of frequency of a typical slow-wave electrode. Note that the two velocities begin to depart at around 200 GHz.  37  60  50  20  10  I^1^1^1 1 1 1 1 1  1  I^►^1^► 1 1 1 1 1^►^I^►^1^1 1 1 1  10^100^1000 Frequency (GHz)  1^1^1  Figure 3.3.3.3. Characteristic impedance as a function of frequency of a typical slow-wave electrode. Note that losses are not taken into account in the calculation.  38  3.3.4. Low Frequency Approximation and Design Formulas  As shown in section 3.3.3, the intended frequency of operation of the slow-wave electrode is much lower than the stop-band frequency. In other words, the fin spacing d will be much smaller than the wavelength (i.e., 13d < < 1 and kd < < 1). In this regime equations (3.3.2.7) and (3.3.2.8) can be simplified. Using cos x %--, 1 - x2/2 and sin x x for x < < 1, equation (3.3.2.7) becomes 1-  k2d2 6.)CiZokei N2 2d2 e^-1^2 2NO2^2  which can be simplified using equations (3.3.1.1), (3.3.1.3), and (3.3.2.2) as  I ).  C, \ C', NO N° \11+----"-=c L(C+— Cd^d  (3.3.4.1)  Similarly equation (3.3.2.8) can be simplified as  zi=z  1  0  1  coCf  .  kd caC coC +—+--f-k 2d2 1e Zo^4  Eliminating second order terms and using equations (3.3.1.1) and (3.3.1.3) one obtains  39  zc zo  1  C C C+ f d —  L  \  (3.3.4.2)  C+ —I. C d  It can be seen from equations (3.3.4.1) and (3.3.4.2) that in the low frequency regime, in which the electrodes are intended to be used, the introduction of the loading fins (and pads, where appropriate) is equivalent to simply changing the capacitance per unit length to (C + C/d). This property enables the equations governing the slow-wave electrodes in the regime of interest to be greatly simplified. This simplification is justified because the error involved is only second order. Moreover, the quasi-static analysis is in itself an approximation. The objective of this analysis is to find a simple yet relatively accurate mathematical description of the behaviour of the slow-wave electrode structures. For a more accurate description, a full wave analysis will be inevitable. However, the assumption that the fins (and pads, where appropriate) were pure capacitive elements as used in the analysis above was not a very valid assumption, even though the results would reflect the general trend of the behaviour. In an attempt to yield more accurate yet simple formulas, an approximation of the inductive effects produced by the fins, being based on a weighted average between the inductances of two extreme cases, is introduced. This approximation results in significantly more accurate formulas that can be easily used to design the slow-wave electrodes. With some manipulations of equations (3.3.1.1) and (3.3.1.3) the inductance per unit length L and capacitance per unit length C of a conventional (unloaded) coplanar strip electrode 40  can, respectively, be written as NZ L- ° ° c  (3.3.4.3)  and N^ Zo c  (3.3.4.4)  Note that the substrate and superstrate here are assumed to be infinitely thick. This is consistent with the normally used GaAs substrates, which are typically quite thick as compared to the electrode dimensions. A dielectric superstrate of thickness much greater than the electrode dimensions could also be easily applied. If air is used as the superstrate (e s = 1), then its thickness will automatically be infinite. Note also that the analysis used throughout this thesis assumes that the semi-insulating substrates have infinite resistivities. This is justified because semi-insulating GaAs substrates typically have resistivities greater than 10 7 ohm.cm and have very low background carrier densities (on the order of 10 8 cm 3 ) [49]. To approximate the inductance per unit length of a slow-wave electrode, the weighted average between the inductance per unit length of coplanar strips having widths IV/ and W1 +W2 is taken. The weighting is based on the fin length / as compared to the fin period d. Using equation (3.3.4.3), the weighted inductance per unit length L' becomes  41  (3.3.4.5)  where Z, and Z, are the characteristic impedances of coplanar strip electrodes having strip widths W1 and W1 + W2 respectively. Note that the approximation above assumes that the loading pads,  if used, do not contribute to the weighting. In other words, the inductive effects of fins with pads on their ends are assumed to be the same as those of fins alone. This is justified since the fins are generally narrow and the pads, due to their sizes, are mainly capacitive. The capacitance per unit length C' of a slow-wave electrode can also be written as N C . Zo c d  (3.3.4.6)  C 1 =-2-+--f  Therefore, the Neff and Z' of a slow-wave electrode become  Neff CV C I =  1Z (1 -L)+Z 11 1 Ni 0  1--  -  ci  C c No !+ ji  l  (3.3.4.7)  and Z1 = j  Li  NC  = [Zo (1- -1--)+Z  d  11 1  No Cfc Zo d  (3.3.4.8)  Multiplying and dividing the two equations above one obtains the following design formulas: 42  (3.3.4.9)  and AletZi = [Z0 (1 - 1 ) +Z,  1  N  0  0  (1 ±)N .^(3.3.4.10) -  0  The term Zyd in equation (3.3.4.10) is typically much smaller than Z (1 - 1/d) and may be 0  ignored for convenience. In designing these slow-wave electrodes, by pre-defining Neff, Z', and Cf of the fins (and pads, where appropriate) to be used, one can easily determine Z, and d by solving equations (3.3.4.9) and (3.3.4.10). The following section describes how the capacitance of the fins are calculated. Knowing their capacitance, design curves can be easily generated using the design formulas (3.3.4.9) and (3.3.4.10).  43  3.3.5. Computation of Fin Capacitance  The capacitance of half-buried fins are calculated using both finite difference [50,51,52] and finite element methods [53]. Both of these methods yield similar results. Computer programs have been written. The finite difference method requires a longer computation time and memory; consequently the finite element method was used to calculate the fin capacitance in most cases. Due to the more complicated geometry of fins having pads at their ends, as well as their high demands of computation time and computer memory, their capacitances were calculated using an approximate formula. To obtain the capacitance of a pair of half-buried fin (and pad, if applicable) elements, their capacitance in air C' is first calculated. Then it is multiplied by the factor er + es  2  (3.3.5.1)  which can be easily obtained by considering the contributions to the electric energies stored in the substrate and the superstrate. This factor can also be used for fins (and pads) that lie entirely on the surface provided they are thin. For instance, as will be shown in Chapter 5, if the thickness to gap width ratio is not too large, i.e., less than about 0.3, there will only be negligible differences between the Ne's of surface deposited electrodes and those of half-buried ones. The finite difference method solves Laplace's equation with a unit potential difference applied between the fin elements. The electric field distribution and total electric energy U in 44  the space around the fin elements are then calculated. The capacitance in air CI is obtained from C =2U. 1  The finite element method solves the integral equation k=2  =E  f G(F,F )q(e)ds 1  k=1  i  (3.3.5.2)  corresponding to a pair of fin elements (labelled k =1 and k =2) in air producing a potential 4)(6 located at a field point r, where 1 ^1 G- ^ 4 rc e 0  (3.3.5.3)  is the Green's function, c o the permittivity of free space, and q(r) the electric charge density at location r' on a fin element sk . Upon discretization of the fin surfaces into cells of area Cj the above equations can be put in the form NT  i=E f=1  (3.3.5.4)  where Ply - 1 fc f c G(P,P)dsds 1^(3.3.5.5) j C iCi  45  and NT is the total number of cells in the pair of fin elements, (D i the potential at cell C, and Qj ,  the electric charge at cell From reference [53], the Pi ; s are given in two general algebraic forms corresponding to parallel and perpendicular cells. The expressions corresponding to these two cases are given in Appendix II. The capacitance in air is obtained from Q  C  = Q/(P, where  is the total charge on a fin element and (i) the potential difference between the fin elements. The capacitance of fins in air, as calculated by the finite element method, are given in  Table 3.3.5.1 and are plotted in Figure 3.3.5.1. These values are used to calculate the design curves as given in section 3.3.7. The finite element program was written in FORTRAN. The fins were divided into over 200 cells, depending on their dimensions. The average computation time was in the range of 4 hours on a 80386 33MHz IBM PC compatible computer using a 80387 math coprocessor. Based on the comparisons made with some of the structures having similar dimensions, the results obtained here are within 5% agreement with those given in reference [54], in which the computational accuracy is given as 5%. The finite element program, however, offers more flexibility in calculating the capacitance of fins having various dimensions. To calculate the capacitance of fins with pads, the assumption depicted in Figure 3.3.5.2 is made. Here the capacitance produced by a pair of fin and pad elements having dimensions S2 , W2 , W', 1',  and / is assumed to be equivalent to the capacitance of a pair of fin elements  having S2, W2, and / plus the capacitance per unit length of a section of a pair of coplanar strips having a gap S2 and strip width W' multiplied by the length (/' - 1), or mathematically, cfr,^ cfin(s,,w2,0^ceo„,..„,,,s(svw) [1' -I] (3.3.5.6)  46  where Ceop/anar strips is the capacitance per unit length of a pair of coplanar strips, as given by [45:pp. 257-288] (see also Appendix I). This is a reasonable assumption since the fringing electric field along the edges of the pad will be very much like that produced by the fin. The extra capacitance contributed by the pad will be produced by electric field lines that run beteen the pads and which do not have any component in the length direction (i.e., /') along the pads. This extra capacitance is best approximated by the capacitance per unit length associated with a pair of coplanar strips, having the same gap and width, multiplied by the length of the pad at which the electric field has no component along the pads, i.e., (l' - 1). The capacitance of fins and pads (in air) having some chosen dimensions are given in Table 3.3.5.2. These values are used to calculate the Neis and Z' 's of the slow-wave electrodes fabricated (see section 3.3.7 for theoretical results and Chapter 5 for measured results).  47  Table 3.3.5.1. Capacitance of fins in air with S2 = 1µm and t = 0.5 Am. / = 2 Am  C1 (fF)  C2 (fF)  Cl (fF)  C7 (fF)  3  0.0612  0.0969  0.122  0.145  4  0.0721  0.111  0.139  0.164  5  0.0820  0.123  0.153  0.180  6  0.0910  0.135  0.166  0.195  7  0.0994  0.145  0.178  0.208  8  0.107  0.154  0.189  0.220  9  0.115  0.163  0.199  0.231  10  0.122  0.171  0.208  0.242  48  / = 3 Am  / = 4 Am  / = 1 gm  W2 (-1 m)  Table 3.3.5.2. Capacitance of fins with pads in air.  S2 = 4 Am, W2 = 28 Am, / = 4 Am, t = 1µm /' = 8 Am  /' = 12 Am  /' = 16 Am  W' = 7 Am  0.408 fF  0.472 fF  0.536 fF  W' = 14 Am  0.420 fF  0.496 fF  0.576 fF  49  0.25  U  0.05  I^I^I^I^I^I^I^I^I^iI^11[11111111^I^I  ,^I^I^I^I^I^iiii  4^5^6^7^8^9^10 W2 (pm)  Figure 3.3.5.1. Calculated capacitance of fins in air having S2 = 1 pm, t = 0.5 pm, / = 1, 2, 3, and 4 gm, and W2 ranging from 3 to 10 pm.  50  ^  A  A —  I ^I E31^C^18  —  I—  C  BB  I ^I  +  ^I^I 131^C^18  BBB  C  I^ I A  A  Figure 3.3.5.2. Assumption made in calculating the capacitance of fins with pads. Here, the capacitance of the pair of fin and pad elements is approximated by the sum of the capacitances of the pair of fin elements and the pair of coplanar strips on the right. The letters A, B, and C show the different areas that contribute to the total capacitance. Areas having the same letter indicate that their contributions to the total capacitance are assumed to be the same.  51  3.3.6. Fringing Fin Capacitance  The slow-wave electrodes are capable of slowing down a microwave sufficiently so that its phase velocity can be matched to that of an optical wave in a III-V semiconductor electrooptic modulator. This is because the fringing electric fields produced by the narrow fins and pads increase the capacitance to fin length ratio Cy/ significantly as compared to the capacitance per unit length associated with two coplanar strips having the same width as the fins. In order to obtain a large C://, / has to be kept as small as possible. In other words, the fins have to be as narrow as possible. The word 'narrow' here refers to a small / as compared to  W2.  Moreover, a small 1 reduces the inductive effects of the fins, resulting in a more effective capacitive loading. The narrower the fins, the more capacitive and less inductive they are. Figure 3.3.6.1 gives the values of Cii/ for fins of Table 3.3.5.1 having various 1. It is clearly seen that the narrower the fins the greater the value of Cji/. For the fins having 1 = 1 Am, Cy/ is almost twice that for fins having 1 = 4 gm. In other words, the narrower that we can make the fins, the more effective the loading. In the limiting case when / = co, Cji/ becomes the capacitance per unit length corresponding to a pair of coplanar strips. It is obvious from the figure that wide fins (large 1) will not have a very high C J/1 as compared to that of coplanar strips and consequently result in a less effective loading. In practice, 1 is chosen so that the electrodes can be fabricated reliably. Even though modern electron beam lithography techniques make it possible to fabricate sub-micron structures, fins having 1 = 1 Am are sufficient to give good results in terms of amount of slowing and ease of fabrication. The results are given in the next section. 52  0.13  0.11  6* 0.05 — --  0.03 —  0.01  Il  lif^IIIIII  IIIIIII  IfIIIIIII  IIIIIII  4^5^6^7^8^9^10  W2 (gm) Figure 3.3.6.1. Fin capacitance to length ratio C.f// for fins in air having S2 = 1µm, t = 0.5 Am, / = 1, 2, 3, and 4 Am, and W2 ranging from 3 to 10 Am. / = co corresponds to a pair of coplanar strips.  53  3.3.7. Calculated Results and Design Curves  Bearing in mind the practicalities of modern fabrication, the value of / is limited to about 0.5 gm but 1µm should be sufficient to give good results. The separation between the two coplanar strips in a typical Mach-Zehnder modulator also limits the practical values of W2 to the range of about 3 to 10 gm. For these reasons, design curves for slow-wave electrodes having these dimensions are calculated. Design curves for half-buried electrodes with superstrates having e s = 1 (i.e., air) and e s. = 3.5 (e.g., Dupont PI-2525 polyimide [55]) are calculated [3,5]. Also Z' = 50 and 75 ohm are assumed. The Neal' for electrodes designed for velocity-match in GaAs based waveguide systems is 3.43 [1,2:p. 364] for an optical free-space wavelength of 1 gm while the N eff for electrodes designed for InP based waveguide systems is 3.3 [2:p. 364,27,28]. The calculations here are for velocity-matched electrodes based on e,. = 12.9 for GaAs [38:pp. 25-26] and e r = 12.4 for InP [56]. The design curves are shown in Figures 3.3.7.1 to 3.3.7.4. As is apparent from the figures, as W2 increases d also increases while Z, decreases. This is because a larger W2 corresponds to fins having a higher capacitance, which allows a greater fin period d to obtain the same increase in capacitance per unit length. As the aspect ratio, 1/W2 , of the fins is decreased, the fins more and more nearly approximate a purely capacitive load. Therefore in the limit, the capacitance of the fins is determined solely by W2 . It is apparent that for each of the cases Z' = 50 ohm and Z' = 75 ohm, Z, approaches a limiting value as W2 is increased. This limiting value corresponds to that Z, that should be used for a purely capacitively-loaded electrode. 54  To illustrate the use of the design curves, a 50 ohm slow-wave electrode on GaAs, with e s = 3.5, Neff = 3.43, Si = 15 Am, S2 = 1 Am, and W2 = 7 Am, will have d = 8.5 Am, and  IV/ = 30 Am (this corresponds to Zo = 69 ohm). The dimension W1 is determined by using the design formulas for conventional coplanar strip electrodes [45:pp. 257-288] (see Appendix I). Another illustration is a 50 ohm electrode on GaAs with air being the superstrate (i.e., e s = 1), Neff = 3.43, Si = 15 gm, S2 = 1 Am, and W2 = 7 Am, will have d = 6 earn, and Wi = 23 Am  (this corresponds to Zo = 80 ohm). Even though the design curves are for half-buried electrodes, the fact that the electrodes are only 0.5 gm thick makes them usable for surface deposited (unburied) electrodes with little error, as the extra contribution to capacitance by half-burying the fins of this thickness is very small. In fact, as will be shown in Chapter 5, half-burying electrodes will only increase their Neff's very slightly, if not negligibly. Moreover, if thicker electrodes are to be designed, these  curves may still be used by slightly modifying the dimensions in favour of a higher Neff, such as reducing d. The Neff's and Z' 's for a number of slow-wave electrodes have been calculated based on the design formulas of section 3.3.4 and the capacitances of section 3.3.5. These electrodes have been fabricated and tested. The measured results are given in Chapter 5. The calculated N eff's and Z' 's are given in Table 3.3.7.1.  55  Table 3.3.7.1. Calculated Neff's and Z' 's of some slow-wave electrodes fabricated and tested. Si = 60 Am, Electrode  S2  = 4 Am, W2 = 28 pm, / = 4 Aim, t = 1.0µm  WI (hm)  W' (.hm)  1' (hm)  (in)  #1  72  0  4  18  3.50  49.6  #2  110  0  4  32  3.16  57.1  #3  110  7  8  32  3.28  55.0  #4  110  7  12  32  3.39  53.1  #5  110  7  16  32  3.49  51.4  #6  110  14  8  32  3.30  54.7  #7  110  14  12  32  3.44  52.4  #8  110  14  16  32  3.56  50.4  56  d  Neff  Z'  (ohm)  130  13.0 ^  —120  11.0  —110  E  9.0  -  -^0 -  —100 -0 ^ 7.0 —  -  —90  5.0 —  —80 ^ Z' = 50 0  _ _ _ Z' = 75 0  3.0 ^ I I I I I^70 3.0^4.0^5.0^6.0^7.0^8.0^9.0^10.0  vv2 (, 1-171 )  Figure 3.3.7.1. Design curves for 50 and 75 ohm velocity-matched slow-wave electrodes on GaAs substrates for E s = 1, S2 = 1 Am, t = 0.5 Am, and 1 = 1 Am.  57  -  20.0  ^  110  ../  /  -_,  16.0 —  7  ........ --^ -  .^> ___ .^ 4  E  — 100  --  _- 90 0N  7  12.0 —  <^z ,  7  7  /  .7  8.0 — —70  4.0  Z' = 50 0 _ _ _ Z' = 75 0 IIIIIIIIIjIIIIIIIIIIIIIIIII^i^i^II^i^i^60 3.0^4.0^5.0^6.0^7.0^8.0^9.0^10.0  vv 2 (gm  Figure 3.3.7.2. Design curves for 50 and 75 ohm velocity-matched slow-wave electrodes on GaAs substrates for E s = 3.5, S2 = 1 gm, t = 0.5 gm, and / = 1 gm.  58  13.0 ^ —125  — 115  105 N -— _^o  7 7 7 7  7.0  —95  0  7  —85 5.0 — —75  Z' = 50 0 Z' = 75 0  65 3.0 ^ 3.0^4.0^5.0^6.0^7.0^8.0^9.0^10.0  vv2 (,gym)  Figure 3.3.7.3. Design curves for 50 and 75 ohm velocity-matched slow-wave electrodes on InP substrates for e s = 1, S2 = 1 Am, t = 0.5 Am, and 1 = 1 Am.  59  ^ -  24.0 ^ ...  --^_  20.0 —  -,"^  -_,  -...- 95 __.^ >^_ _ ...  _1-  ^,-----,^_  E  ,  16.0 --: . ._.^_  12.0 —_  ...- — 105  N _-o — 85 „..„,  ,  _^ __ •.___.  _ —75  .  8.0 —  — 65 Z' = 50 0 _ _ _ Z' = 75 0  4.0 I i 11111111111111111111111111 I m,^55 3.0^4.0^5.0^6.0^7.0^8.0^9.0^10.0 '  Figure 3.3.7.4. Design curves for 50 and 75 ohm velocity-matched slow-wave electrodes on InP substrates for E s = 3.5, S2 = 1 pm, t = 0.5 gm, and / = 1 pm.  60  3.3.8. Attenuation Constant and Microwave Loss  Equation (3.3.2.6) can be used to calculate the attenuation constant a of the slow-wave electrodes. By equating the imaginary parts, one gets sinkd+ 63 CiZ° coskd sinhad_ ^2 sinha od^sinf3d  (3.3.8.1)  which can be simplified since ad, a od, 13d, and kd < < 1. Therefore kd+  6)Cflo 0 k 2d2 )  a ^2^2  (3.3.8.2)  a 0^f3d  Ignoring second order terms and using equations (3.3.2.5), (3.3.4.1), and (3.3.4.4), equation (3.3.8.2) can be simplified to give a . 1 {Neff + No a 0 2 [ No No  (3.3.8.3)  which is always greater than unity. This is an interesting point, as it shows that the slow-wave electrodes always have a higher loss than the conventional coplanar strips. However, since the Neff's are not much higher than No , the losses of the slow-wave electrodes and conventional  coplanar strips are comparable. Using Neff = 3.43, No = 2.64, one gets a/a o = 1.03, indicating 61  that the losses are almost the same. The attenuation (in dB, i.e., is equal to 8.68 times the attenuation constant [46:pp. 76-79]) due to conductor loss of conventional coplanar strips can be calculated using Wheeler's incremental inductance formulas [45:pp. 257-288] (see also Appendix I). The dielectric loss for electrodes on GaAs is very low [45:p. 73, pp. 257-288] as compared to the conductor loss and is therefore ignored here. For the slow-wave electrodes described by the design curves given in section 3.3.7, their losses at 20 GHz are typically around 3 dB/cm, and increase as the square root of the frequency [57:pp. 151-153] (see Appendix I). The scaled up versions of the slow-wave electrodes given in Table 3.3.7.1 have much lower losses of around 0.7 dB/cm, due to their greater dimensions. It should be noted that these calculated values may be over-estimated since the greater widths of the fin sections will reduce the ohmic losses. The losses of these slow-wave electrodes, however, are still very low as compared to those of other types [33-35,40-45].  62  3.3.9. Scaling of Dimensions  Since the capacitance of a pair of fin elements scales as their dimensions, the term Cid will remain unchanged if all the fin dimensions S2 , W2, 1, and t, and fin period d are scaled by the same factor. Moreover, as given in Appendix I, since the characteristic impedance Zo of a thin conventional coplanar strip electrode depends on the ratio S 1 /(S1 + 2W1) alone, scaling the dimensions of the coplanar strips (i.e., S1 and W1 ) will have little effect on Zo . Therefore, a thin slow-wave electrode will have the same Neff and Z' if all its dimensions are scaled by the same factor. This property is especially useful for experimental purposes since, as will be shown in Chapter 5, scaled-up versions of the slow-wave electrodes having the same Neff and Z' can be fabricated using cheaper photo-generated masks. This scaling, however, has certain limits. First, the scaled-up fin period d still has to be much smaller than the microwave wavelength in order to avoid the dispersion caused by the filter characteristics. Second, if the transverse dimensions of the electrode approach the wavelength, the quasi-static analysis might fail. Third, as given in Appendix I, if the electrode has a nonzero thickness t, Zo does not remain exactly the same for a constant S1 /(S1 + 2W1) but decreases because of the higher capacitance due to the finite thickness. These limitations, however, are often not very restrictive for most practical applications.  63  3.3.10. Input Sections  Since the width W1 and separation S 1 of the main strips of the slow-wave electrode are usually too small for direct probing and connector bonding, an input section having a larger separation or pitch is sometimes needed. The fact that the characteristic impedance depends substantially on S1 /(S1 + 2W1) alone suggests that one can use a section of coplanar strip having a gradually increasing pitch while maintaining the same value of S1 /(S1 + 2W1). It can be shown using simple geometry that this ratio is maintained throughout the whole length of the section if its two ends, each satisifying the same ratio, are joined by straight lines. This is depicted in Figure 3.3.10.1. In order to keep reflections and discontinuities as low as possible, it is desirable to keep the input sections as short as possible.  3.3.11. The Effects of High Dielectric Constant Superstrates and Partially Buried Electrodes  As mentioned in Section 3.2, a high dielectric constant superstrate such as a polyimide may be applied so as to increase the capacitance per unit length, thus resulting in higher Neff's. The electrodes may also be partially buried in the substrate to increase their Neff's. However, this may only result in a marginal increase (see Chapter 5). While it is not essential to use a superstrate with a dielectric constant higher than that of air, using it has several advantages. It increases the capacitance per unit length so that the period of the fins (and pads, if applicable) can be increased while maintaining the same N. Increasing 64  the period of the fins will reduce any unwanted interactions between them, thus allowing a more effective capacitive loading. The increase in fin capacitance allows one to use shorter fins. Therefore, the separation between the coplanar strips can be reduced, resulting in an increase of the electric field for modulation. Z, can also be reduced, resulting in wider electrodes which have lower microwave losses. Tuning of the microwave effective refractive index to achieve better velocity-match is also possible by applying superstrates having various dielectric constants. Despite of these advantages, surface deposited electrodes with air superstrates require the least number of fabrication steps and are perhaps more practical in modulator use. In Chapter 5, experimental results for both partially-buried and surface deposited electrodes are given.  3.4. Conclusion  In this chapter, slow-wave electrode structures employing narrow capacitive loading fins and pads are described. The analysis of these electrodes is given. Design curves, for both 50 and 75 ohm slow-wave electrodes, for matching a microwave's effective refractive index to an optical wave's effective refractive index in modulators fabricated using GaAs and InP based materials, are provided. The calculated microwave effective refractive indices and characteristic impedances for a number of slow-wave electrodes are presented.  65  cr) 3  I  S/( S+ 2W ) = S ' / ( S ' + 2 Ili ' ) Figure 3.3.10.1. Input section for slow-wave electrodes showing the various dimensions.  66  Chapter 4 Device Fabrication 4.1. Introduction  In order to verify the slow-wave effect as predicted by theory, slow-wave electrodes, both partially buried and surface deposited, and having various dimensions, have been fabricated on (100) semi-insulating GaAs. Since a high dielectric constant superstrate is not needed in order to achieve the velocity match condition, as already discussed in Chapter 3, air was used as the superstrate (i.e., e r = 1) due to the reduced number of fabrication steps required. The reason why partially buried (especially half-buried) electrodes were fabricated was because the fins were assumed to be half-buried when their capacitance was computed. Consequently, half-buried electrodes would more closely approximate the ideal model as analyzed in Chapter 3. Moreover, fabricating both partially buried and surface deposited electrodes allows us to compare their effects on the amount of slowing. Such comparison is made in Chapter 5. A total of twenty electrodes were fabricated, all of which had 50 ohm characteristic impedances. Among them, one was a conventional coplanar strip electrode, seven were slowwave electrodes loaded only with fins, and twelve were slow-wave electrodes loaded with fins and pads. All of these electrodes had different dimensions and some were buried. The electrodes were intended to be easily fabricated, being formed in a single layer of metallization using a standard chlorobenzene lift-off technique [38:pp. 115-138]. Due to the high cost of electron-beam masks, the dimensions of the electrodes were scaled up so that cheaper  67  photo-generated masks could be used. This is justified since, as discussed in Chapter 3, the effective refractive index and characteristic impedance of a slow-wave electrode will remain the same if all of its dimensions are properly scaled. The fabrication procedure for the different electrodes was basically the same except that the buried electrodes required that the GaAs substrates be etched. Here, wet chemical etching techniques were used. However, wet etchants result in some lateral etching which can create air spaces around the fins and pads and in turn reduces the microwave effective refractive index. As a result, different wet etchants were tried in order to reduce the lateral-etch. This problem will be discussed in section 4.3.4.  4.2. Mask Design  Two photo-generated masks were designed using the CAD package ICED by IC Editors, Inc. Both masks were fabricated by Precision Photo Mask (PPM) Inc. of St. Laurent, Quebec, on ultra-flat sodalime glass. The precision as specified by PPM was +/- 0.25 Am. The masks were dark field, i.e., the areas of the electrode patterns were transparent. There were only four electrodes on the first mask. Two of the them were conventional coplanar strip electrodes and the other two were slow-wave electrods loaded only with fins. The two conventional coplanar strip electrodes were the same except that one of them was short circuited at one end while the other was open. Similarly, one of the other two slow-wave electrodes was shorted at one end; other than that, they were exactly the same. The reason why both shorted and open electrodes were designed was because at the time of designing the masks 68  we were not certain whether electrodes shorted or open at one end would enable better and more accurate measurements to be made. However, as it turned out, electrodes shorted at one end proved to offer better results since a plane of reflection can be established very easily by a short circuit while the open end capacitance of an open circuit produces an equivalent electrode length that extends from the end. The mask layout and electrode dimensions are respectively given in Figure 4.2.1 and Table 4.2.1. Input sections were not used in these electrodes since the pitch (i.e., the separation between the two contact points) of the coplanar strip probe used in the measurements was just right for these electrodes. The second mask contained eight electrodes (#1 to #8), all of which were slow-wave electrodes having different dimensions. They were all shorted at one end, as this had been found to give better results. Electrodes #1 and #2 contained only fins while electrodes #3 to #8 had pads at the ends of the fins. Electrode #1 had a smaller fin period d, and a smaller strip width W1 , as compared to other electrodes. Dimensions of electrodes #2 to #8 are all the same except  that the pad sizes are different. This enables us to compare the effects on the  Neal  to different  pad sizes. The mask layout and electrode dimensions are given in Figure 4.2.2 and Table 4.2.2 respectively. These electrodes all had input sections to facilitate proper probing. The dimensions of the input sections used in these electrodes are given in Figure 4.2.3.  69  Table 4.2.1. Mask #1 electrode dimensions. t = 1.75µm, E r = 12.9, e s = 1.0  Electrode  S1  W1  S2  (pm)  #1  (hm) 17  133  (1/1n) N/A  #2 (slowwave)  31.5  120  #3 (slowwave)  31.5  #4  17  Shorted /Open  d  /  (pm)  (pm)  N/A  N/A  (pm) N/A  shorted  3.5  14  29  3.5  shorted  120  3.5  14  29  3.5  open  133  N/A  N/A  N/A  N/A  open  W2  Table 4.2.2. Mask #2 slow-wave electrode dimensions. S1 = 60 gm, S2 = 4 gm, W2 = 28 gm, / = 4 gm, t = 1.1µm  E r = 12.9, E s = 1.0  Electrode  W' (1-im)  #1  W1 (pm) 72  #2  l'  d  (pm)  0  (111n) 4  Input Section  18  A  110  0  4  32  B  #3  110  7  8  32  B  #4  110  7  12  32  B  #5  110  7  16  32  B  #6  110  14  8  32  B  #7  110  14  12  32  B  #8  110  14  16  32  B  70  2  Wi^ 91 Wi  cm  -  -  132 ulli^0.5 cm W2^\AA  S2—=- S I W2^WI  d  -  -->  W2^WI  *  >< 120 ul,iji^0.5 cm  --  ^S1 S2 '-- — W2^WI  0.5 cm  VA Si Wi  \./  Figure^4.2.1. Layout of Mask #1.  71  . 5 cm 1 W2  -  S S2 W2^WI -  100 um)  120  0.3 cm  \fir  2 120 u:  120  Figure 4.2.2. Layout of Mask #2.  72  0 . 3 cm  us1 9 I ..  „...  132  urn  V 120  INPUT SECTION A  15  urn  120  urn  ^30  ■ ".-.  ■,, 1 44\u m  um^---  1  132  UM  /  INPUT SECTION B  Figure 4.2.3. Dimensions of the two input sections used in Mask #2.  73  um  20  urn  4.3. Fabrication Procedure  A combination of chlorebenzene lift-off and substrate etching techniques were used in the fabrication of the electrodes [38:pp. 115-138, pp. 96-106]. After cleaning and degreasing the wafers, positive photo-resist was spun onto them and was exposed through the dark field masks. After developing the photo-resist, the areas of the electrode patterns became exposed while the other areas were covered by the photo-resist. Then, if partially buried electrodes were to be made, the exposed areas of the substrate had to be etched to a depth equal to the amount that the electrodes were to be buried. Here, for simplicity and convenience, wet etching techniques were used. Aluminum of the desired thickness was then thermally evaporated onto the wafer. This produced wafers in which the exposed areas, including the etched pits for the electrodes if the substrate had been etched, were filled with aluminum. Finally the photo-resist, along with the excess aluminum, was stripped off, leaving the metallized electrodes. The main steps of this procedure are schematically shown in Figure 4.3.1. More details are given in the following sections.  74  111111111111 22L111111:-21- 111-=111=111=1111111111111 111 111 111 111 111 111 111= -  -  -  -  -  -  (d)  a)  ( e )  ALU/A I NUIA PHOTO-RESIST GoAs SUBSTRATE  Figure 4.3.1. Main steps of the fabrication procedure showing (a) the unexposed photo-resist on a GaAs substrate; (b) the photo-resist after development; (c) the substrate after wet etch; (d) the sample after evaporation of aluminum; and (e) the half-buried electrodes after stripping the photo-resist and excess metal. The procedure for forming surface deposited electrodes is the same except that the substrate does not have to be etched [step (c)].  75  4.3.1. Cleaving GaAs Wafers  In order to make better use of the available GaAs wafers, they were cut into smaller pieces so that they would just accomodate the electrode patterns. This was easily accomplished by making a small scratch along the edge of the wafer, then pressure was applied to one side of the scratch until the wafer broke along one of its natural cleavage planes (i.e., the (011) planes). However, care had to be exercised to avoid scratching other parts of the wafer.  4.3.2. Cleaning GaAs Substrates  In order to ensure that the GaAs substrates were free from dust and grease, the substrates had to be cleaned prior to photo-resist patterning. Due to the small dimensions of the structures, which are typically on the scale of microns, any dust particle on the substrate could result in an electrode that is cut off and render it useless. Any grease on the substrate would also affect the adhesion of the photo-resist, which would very likely peel off during the wet chemical etch that follows the photo-resist patterning. Due to the brittle nature of GaAs, the use of an ultrasonic bath was avoided. Instead, manual agitation was used throughout. The wafers were first cleaned in hot (approximately 50°C) acetone for 10 minutes. This would allow the acetone to dissolve any grease on the wafers. On removing the wafers from acetone, the wafers were immediately dipped into hot (approximately 50°C) isopropanol for 5 to 10 minutes. This would remove the residue and water  76  that are usually found after removing the wafers from acetone. As soon as the wafers were removed from isopropanol, they were blown dry using a jet of compressed nitrogen. In order to avoid dust particles landing on the wafers, photo-resist was immediately spun onto them. If needed, the wafers could also be dipped in buffered hydrofluoric acid (BHF) for 1 minute, then rinsing them in deionized (D.I.) water for 5 to 10 minutes in a cascade bath, and finally baking them at 120-130°C for approximately half an hour.  4.3.3. Photo-resist Patterning  After cleaning the wafers, Shipley PR 1400-30 positive photo-resist was spun onto them at 3500 RPM for 35 seconds. This photo-resist was chosen because it could be applied in a relatively thick layer. In an attempt to ensure that the wafers were dust-free, compressed nitrogen was first used, followed immediately by spinning the wafers, for about half a minute, without applying any photo-resist. Then the photo-resist was immediately applied. This procedure ensured that the wafers were as dust-free as possible. The photo-resist was pre-baked at 70°C for 30 minutes. The samples were allowed to cool to room temperature after they had been removed from the oven. In the meantime, they had to be properly handled and covered so that they would not accumulate any dust particles. Since we were not, at this time, fabricating electro-optic modulators, the orientation of the crystal was not critical. Nevertheless, the electrodes fabricated using the first mask were aligned parallel to the [011] direction, i.e., same direction they would be if they were used in electrooptic modulators. However, in fabricating partially buried electrodes, which required a 77  subsequent etching of the substrate, this orientation produced non-vertical etched side walls, thus producing air gaps around the fins and reducing their capacitance and producing lower microwave effective refractive indices. In order to fabricate half-buried electrodes that closely approximate the ideal structure which the theory is based on, the electrodes fabricated using the second mask were oriented along the [001] direction. This direction enabled side walls that were more nearly vertical to be etched [38:pp. 96-106,58]. The masks were aligned and the photo-resist exposed using a Karl-Suss MJB3 contact mask aligner. The samples were rotated on the mask aligner so that the major flat, which corresponds to the [01T1 direction, was aligned either parallel to or at 45° with respect to the direction of the electrodes. The latter direction of alignment produced electrodes that were oriented along the [001] direction. Since very high resolution can be obtained with positive photo-resist, the standard contact setting on the mask aligner was used. After proper contact was made, the photo-resist was exposed for 25 seconds using a power density of 25 mW/cm 2 produced by the mercury lamp with a wavelength of 320 nm. The exposed photo-resist was then treated in chlorobenzene for 8 minutes so that it would be undercut on development, producing a lip overhanging at the top [38:pp. 115-138]. This would subsequently enable a successful removal of the photo-resist and the excess metal. At the end of the chlorobenzene treatment the sample was blown dry using compressed nitrogen. It was then allowed to sit for at least 15 minutes so that the chlorobenzene would evaporate completely. This was necessary as the photo-resist would not develop properly if the chlorobenzene was not completely gone. The photo-resist was then developed in cooled (below 5°C) 50% Shipley MF-312 photo78  resist developer / 50% DI water for 1 minute. Constant stirring was needed. At the end of 1 minute the pattern should be seen clearly. The sample was rinsed in D.I. water for about 1 minute so that most of the developer was gone. Then it was checked under a microscope. If the pattern did not form altogether, the sample would have to be put into the developer for another 15 seconds. This process was repeated until all of the pattern formed. In most circumstances the whole pattern should form within a total developing time of 2 minutes. If the pattern looked acceptable, the sample was to be rinsed in a D.I. water cascade bath for a least 5 minutes. The electrodes fabricated using the first mask were either half-buried or totally buried, while some of the electrodes fabricated using the second mask were half-buried and some surface deposited (i.e., unburied). Due to the fact that most wet chemical etchants attack photo-resist to some extent, all the wafers on which partially buried electrodes were to be formed (i.e., wafers that had to be etched) had to be hard-baked (or post-baked) so that water would be removed and the photo-resist become more resistant to the etchants [38:pp. 115-138]. On the other hand, the wafers on which only surface deposited electrodes were to be formed did not have to be hard-baked. In fact, hard-baking the photo-resist only made them more difficult to remove. Hard-baking was accomplished by putting the samples in a temperature controlled oven initially set at 90°C. At the end of 5 minutes the temperature was increased to 100°C. When another 5 minutes had elapsed the temperature was increased to 110°C. The temperature was further increased in this fashion (10°C every 5 minutes) until a maximum of about 130°C was reached. The samples were then removed from the oven and allowed to cool to room 79  temperature.  4.3.4. Etching GaAs Substrates for Burying Electrodes  Only the substrates on which buried electrodes were fabricated had to be etched. A buried electrode is formed by etching the exposed electrode areas of the substrate to a depth equal to the amount that the electrode is to be buried before the metal is evaporated. Due to its relative ease of use, wet chemical etching was chosen. In this section the etching characteristics of different wet etchants used in the fabrication are discussed. Wet chemical etchants tend to etch in both the vertical and lateral directions [38:pp. 96106,58-61]. The lateral etch produces an undercut beneath the photo-resist. They can result in buried fins that are surrounded by air gaps instead of the substrate. Consequently the amount of slowing of the microwave may be less than as expected due to the corresponding lower fin capacitance produced by the air gaps. Wet etchants may also yield non-vertical walls, due to the difference in etch rate between different crystallographic planes. Non-vertical walls also affect the fin capacitance and consequently the amount of slowing of the microwave. In choosing a wet etchant, the etch rate, the amount of undercutting, the relative anisotropy, and the amount of damage to the photo-resist had to be considered. Since the electrodes were relatively thin, the amount that they were to be buried in the substrate was only on the order of a fraction of a micron. Therefore, the depth of the substrate to be etched was also on the order of a fraction of a micron. Due to this small etch depth, the  80  etch rate also had to be small since a high etch rate would make the etch depth difficult to control. The amount of undercutting was the second most important consideration. A small amount of undercutting would enable the sides of the fins to be as close to the substrate as possible so that the air gaps could be reduced. A small air gap would result in a larger fin capacitance and subsequently a larger amount of slowing of the microwave. Due to the nature of wet etchants, however, it is impossible to completely eliminate the air gaps. Hence, a compromise would need be found, i.e., one would need an etchant that gives a relatively small undercut as well as an acceptable etch rate. Due to the anisotropic nature of wet etchants, the etched walls will be inclined either towards the centre of the etch pattern or away from it, depending on the orientation of the pattern. In order to obtain more or less vertical walls, the electrodes can be oriented 45° with respect to the natural cleavage planes (the (011) planes) [38:pp. 96-106,58]. Even though this is not the orientation for the ultimate use of the electrodes, it provides better fabrication results (i.e., small air gaps and nearly vertical side walls) so that the electrodes will approximate more closely to the ideal structure on which the theory is based. Consequently a more meaningful comparion between the measured results and theory could be made using the half-buried slowwave electrodes fabricated in this orientation. A somewhat less significant but important issue to be considered is whether the photoresist could stand up against the etchants. Even though the photo-resist had been hard-baked, different etchants attacked the photo-resist to varying degrees. For instance, the etchant HC1:H 2 02 :H2 0 (1:1:9) supposedly produces the least undercutting [59], yet it attacked the photo-  81  resist so severely that it could not be used. Table 4.3.4.1 summarizes the etching characteristics of some of the etchants that satisfy the criteria mentioned above [38:pp. 96-106]. Except for HCEH 20 2 :H 20 (1:1:9), all other three etchants were used in the fabrication. All etchant compositions were freshly prepared from semiconductor grade starting materials, and all etchings were performed at room temperature. In mixing the etchants, water was always added first, followed by H 2 02 and the acids or bases. This helped avoid vigorous reactions when the different compositions were mixed. The etchants were constantly stirred to ensure uniformity of etch rate across the wafers. The etch rates of the various etchants were first measured by patterning the GaAs substrates, followed by a timed etch, and finally the etch depths measured using a Tencor Alpha-Step 200 profilometer. The measured etch rates were very close to those given in reference [59]. The measured etch rates are listed in Table 4.3.4.1. Knowing the etch rates, the desired etch depths could be very accurately attained. The scanning electron microscope (SEM) micrographs of the completed electrodes fabricated using the different etchants are presented in section 4.4.  82  Table 4.3.4.1. Etching characteristics of various etchants used in the fabrication. Etchant  Measured Etch Rate (100) (Am/min)  Relative Amount of Undercut (1 = smallest 4 =largest)  Smoothness of Etched Surface  Amount of attack to photo-resist  NH 4 OH:H 2 0 2 :H2 0 (1:1:8)  1.5  4  Very smooth  Relatively little  NH4 OH:H 2 02 :H2 0 (5:2:240)  0.17  3  Very smooth  Relatively little  HC1:H 2 02 :H2 0 (1:4:40)  0.21  2  Smooth  Not too severe, acceptable  *** 1 Most severe HC1:H 2 02 :H2 0 0.20 (1:1:9) *** Not found since the photo-resist was damaged y the etc ant and the electrode pattern di not form.  4.3.5. Evaporation of Aluminum  The metal chosen to form the electrodes was aluminum. This is due to the low cost as well as the thickness of the electrodes required. The electrodes were typically between 1.1 and 1.7 Am thick (see Tables 4.2.1 and 4.2.2). For these thicknesses the most efficient way to deposit metal was thermal evaporation of aluminum in vacuum. Aluminum was evaporated using a Carl Herman & Associates (CHA) vacuum thermal evaporation system. Before loading the samples into the system, compressed nitrogen was used to remove the dust from them. Three coils of aluminum were loaded each time. This would allow approximately 0.5 Am of aluminum to be evaporated. This process was repeated until the 83  desired amount of aluminum was deposited. However, at the end of each evaporation, approximately half an hour was allowed for the evaporated aluminum to cool before letting air into the evaporation chamber. Otherwise the hot aluminum would oxidize. New coils were loaded very quickly so that the exposure of the aluminum to air would be minimized. It should be noted that the film thickness monitor (Inficon Model 321) used in our laboratory was not very accurate. It had to be calibrated to ensure that aluminum layers of the predefined thicknesses were evaporated. This was done using a Tencor Alpha-Step 200 profilometer as a reference. Aluminum was first evaporated onto a substrate and the thickness measured using the profilometer. Then the aluminum thickness as indicated by the thickness monitor was compared with that measured by the profilometer. It turned out that the film thickness monitor consistently gave a reading that was about 23% smaller than that measured using the profilometer. This error was taken into account during evaporation.  4.3.6. Removal of Excess Metal (Lift-off)  After evaporation, the photo-resist, along with the excess aluminum, was removed. This was accomplished using either heated (above 70°C) Microstrip 2001 or hot (50°C and above) acetone. On some occasions the photo-resist was difficult to remove and Microstrip 2001 was necessary. However, in most situations acetone could remove the photo-resist very quickly and effectively. In using Microstrip 2001, care had to be taken to avoid getting it too hot; if Microstrip 2001 got very hot (above 90°C), it would begin to etch the aluminum. 84  Sometimes the photo-resist (and the aluminum attached to it) would not come off, especially in the inter-electrode gap region. Blowing at it from very close using compressed nitrogen often helped. However, care was exercised so as not to scratch the pattern with the nitrogen gun. In situations when it was very difficult to remove the photo-resist, the samples, along with the beaker holding Microstrip 2001 or acetone, were put into an ultrasonic bath very briefly (less than 20 seconds), although the ultrasonic bath should be avoided if possible as it may break the samples. Every so often the pattern was examined under the microscope. When all the photo-resist had come off, the samples were rinsed in hot (70°C) D.I. water if Microstrip 2001 was used, and hot (50°C) isopropanol if acetone was used.  4.3.7. Summary of Fabricaton Procedure  The fabrication of the slow-wave electrodes was relatively easy, being accomplished by a single-step lift-off. Substrate etching is only necessary in forming buried electrodes. The main steps can be summarized as follows: (1) Photo-resist patterning, (2) wet chemical etching of substrate (for buried electrodes only), (3) evaporation of aluminum, and (4) removal of the photo-resist and excess aluminum.  85  4.4. The Completed Electrodes  Figure 4.4.1 is a SEM micrograph showing the isometric view of a pair of half-buried fin elements of electrode #2 of Mask #1 with the substrate etched using NH 4 OH:H 2 0 2 :H 20 (5:2:240). The fin elements are parallel to the [011] direction. As is clearly seen, the etched wells were over-etched, producing relatively large air gaps of over 1 Am at the ends of the fins. Here, the sizes of the air gaps were measured from other SEM micrographs of the same electrode. As explained earlier, this over-etch was caused by the undercut beneath the photoresist. Figure 4.4.2 is a SEM micrograph showing the isometric view of a pair of totally-buried fin elements of electrode #2 of Mask #1 with the substrate etched using NH 4 OH:H2 02 :H2 0 (1:1:8). The fin elements are also parallel to the [011] direction. It should be noted that even though the air gaps look very large, the amount of undercut produced by this etchant is not much larger than that produced by NH 4 OH:H2 02 :H2 0 (5:2:240) since the etch depth in this case is twice that of the above. The anisotropic nature of the etchant, manifested by the wider air gaps at the ends of the fins than along the fins, is clearly seen. Figure 4.4.3 is a SEM micrograph showing the plan view of a pair of half-buried loading fins, the gaps between them, and the wells containing them. The electrode is #1 of Mask #2. The etchant used to make the wells was HC1:H 2 02 :H2 0 (1:4:40). The fins are oriented along [010] (i.e., the electrode oriented along [001]). Nearly vertical side walls were obtained with this orientation. The use of this etchant also reduced the amount of over-etch. Here, air gaps of less than 0.5 Am were obtained. Also, the air gaps are uniform all around the fins, except 86  that they are somewhat smaller along [011]. Air gaps, caused by the undercut beneath the photo-resist, reduce the effectiveness of the capacitive loading and are believed to be in part responsible for producing microwave effective refractive indices slightly lower than those predicted by our theory. With electrodes oriented along [001] and using the etchant HC1:H 2 0 2 :H 2 0 (1:4:40), the best results were obtained. Even though [001] is not the ultimate direction to be used in making electro-optic modulators, this direction makes it possible to obtain fabrication results that most closely approximate the ideal structure that the analysis is based on. As discussed in Chapter 3, the electrodes do not necessarily have to be buried when they are applied to electro-optic modulators. Figures 4.4.4 and 4.4.5 are, respectively, the SEM micrographs showing the fins and pads of the surface-deposited electrodes #5 and #7 of Mask #2. Since the substrates were not etched in making these electrodes, over-etching was not a concern. However, the fact that the electrodes were deposited on the surface of the substrate would reduce the loading capacitances as compared to those produced by half-buried fins. Nevertheless, this reduction in capacitance should be very small due to the small thickness of the electrodes. This is evidenced by the small differences in their measured Neff's as compared to those of the half-buried electrodes as presented in Chapter 5.  87  Figure 4A.1. SEM micrograph showing the isometric view of a pair of half-buried fm elements of electrode #2 of Mask #1.  88  Figure 4.4.2. SEM micrograph showing the isometric view of a pair of totally-buried fin elements of electrode #2 of Mask #1.  89  Figure 4.4.3. SEM micrograph showing the plan view of a pair of half-buried loading fins" of electrode #1 of Mask #2.  90  Figure 4.4.4. SEM micrograph showing the fins and pads of electrode #5 of Mask #2 (surface deposited).  91  Figure 4.4.5. SEM micrograph showing the fins and pads of electrode #7 of Mask #2 (surface deposited).  92  Chapter 5 Device Testing and Measured Results 5.1. Introduction In order to experimentally verify the slow-wave effect, as predicted by the theory, the microwave effective refractive indices Neff of the fabricated slow-wave electrodes were measured. Two measurement techniques were used. The first technique was based on the interference between waves reflected from the electrodes and waves reflected from a reference short circuit. With a frequency sweep, interference maxima and minima were obtained and their separations measured. These separations allowed us to calculate the Neff's of the electrodes. The interference peaks obtained using this technique, however, were not very sharp and it was sometimes difficult to locate them precisely. The second technique was based on the resonance made possible by the probe-electrode mismatch. The N eff's were calculated by measuring the frequency separations between the peaks of the resonances. This technique enabled us to obtain very sharp resonance peaks and was, therefore, the preferred technique. In this chapter both techniques are described. The results of the measurements are given and compared.  93  5.2. The Measurement Equipment  The equipment used for the measurements for both techniques was as follows: (1) a Hewlett Packard HP 8757C scalar network analyzer, (2) a Hewlett Packard HP 8341B synthesized sweeper, (3) a Hewlett Packard HP 85027E directional bridge, (4) a Tektronix TMP9215 microwave coplanar strip probe, (5) a Merrimac CWM-6M-10G 6dB directional coupler, (6) a Hewlett Packard HP 0960-0055 SMA coaxial short, (7) a Hewlett Packard HP 9836 computer, (8) an IBM PC compatible computer with an IEEE488 interface card, and (9) a stereoscopic microscope with microscope lamp. The scalar network analyzer, synthesized sweeper, and directional bridge were used to measure the normalized reflected power, i.e., 10 log(reflected power/incident power), as a function of frequency. This system was capable of swept frequency measurements from 10 MHz up to 20 GHz. The output power of the synthesized sweeper could be automatically levelled. Power levels up to 20 dBm could be selected. However, due to the insertion loss ranging from 9 dB to 12 dB of the directional bridge, the actual power output was significantly less than the full 20 dBm. The directional bridge also carried a certified directivity of over 40 dB. The overall power resolution of the system was 0.01 dB. All connectors were of Hewlett Packard APC 3.5 (3.5 mm) standard. The network analyzer, synthesized sweeper, and directional bridge are shown in Figure 5.2.1. 94  The coplanar strip probe had a rated DC to 26.5 GHz bandwidth. Its return loss for matched loads as measured by its manufacturer was greater than 15 dB at its lowest point and over 40 dB at its highest point. Its connector was of Wiltron type K (3.5 mm) standard. The 6 dB directional coupler was used in the interference technique measurements only. It had an insertion loss of about 1 dB. Its rated bandwidth was 2 GHz to 18 GHz. Its directivity and isolation were over 12 dB and 18 dB respectively. All connectors were of SMA (3.5 mm) type. All SMA, type-K, and APC 3.5 connectors are compatible. The HP 9836 computer was used for automated system control functions while the IBM PC compatible computer was used primarily for data acquisition. Items used but not listed above were various coaxial cables and adaptors. A sample platform and a jig used for attaching the probe to an XYZ micro-positioner were also designed and built. The sample platform and probe jig with the probe mounted on it are shown in Figure 5.2.2. A number of system controller and data acquisition programs for both computers were also written to facilitate the measurements and subsequent analysis. The whole equipment setup is shown in Figure 5.2.3.  5.3. The Interference Technique  A schematic of the setup used in the interference technique is shown in Figure 5.3.1. Here the directional coupler behaves both as a power divider and power combiner. The signal leaving the output of the directional bridge is split into two by the directional coupler. The coaxial short circuit on the 'output' port of the directional coupler reflects the signal back into 95  Figure 5.2.1. Picture showing the network analyzer on top of the synthesized sweeper. The directional bridge is attached to the output of the synthesizer sweeper. a  Cy  E_ -  G^C  41  •  Figure 5.2.2. Picture showing the probe jig with probe mounted (left) and the sample platform (right). The stereoscopic microscope is also shown.  96  V •-••••• • • V  jp Ir-17„, -011•1200--  • • * 2 0  •  ,_^I 41:14 ■ •%4 atii^ 'r 1^VAI 11111111^AS,a. Ai° CIFØl. ■--  ir.,„411111  gilli% 4111L or  --  Figure 5.2.3. Picture showing the whole equipment setup. The two computers used are also shown,  97  COMPUTER  SYNTHESIZED SWEEPER  (SYSTEM CONTROLLER) HEWLETT PACKARD HP9836  HEWLETT PACKARD HP834IB RF OUT IN  DIRECTIONAL BRIDGE  SY HP% REFL  MERRIMAC CWM-8M-10G  NETWOR‹ ANALYZER HEWLETT PACKARD HPB757C  HEWLETT PACKARD HP85027E OUT IN  DIRECTIONAL COUPLER  HPI9  OUT  SYS HPIB  COMPUTER (DATA ACQUISITION) IBM PC COMPATIBLE  COAXIAL SHORT  HEWLETT PACKARD HP0960-0065SAA  CPL  COPLANAR STRIP PROBE  SLOW-WAVE ELECTRODE  TEKTRONIX TMP9215  Figure 5.3.1. Schematic of the setup used for the interference technique.  98  the directional coupler while the signal entering the electrode via the 'coupled' port will travel to the end of the electrode where it is reflected by the short. The signal reflected by the electrode then enters the directional coupler and combines with that reflected by the coaxial short circuit. The phase difference between the reflected signal at the coaxial short circuit and that at the input end of the electrode is a function of frequency. By sweeping the frequency one is able to calculate the phase difference and thus the microwave effective refractive index Neff of the electrode. By equating the phase change in the reflection between two adjacent peaks, one gets 47c 2Tc =- [6 o +Neip Af  c  (5.3.1)  where c is the speed of light in vacuum, Af the frequency separation between two peaks (or troughs), it the intrinsic phase difference due to the directional coupler, cables and probe, and L the length of the electrode. Without the probe touching the electrode equation (5.3.1) becomes 47c 2Tc =- 0 ° Ai  (5.3.2)  c  where Af is the new frequency separation between two peaks. Subtracting equations (5.3.1) and (5.3.2) one gets the expression giving the Neff N fl.='[ 1] e- 2L Af Al  99  •  (5.3.3)  Usually many peaks occur for a typical frequency sweep from 5 to 20 GHz. One can therefore take the average between a number of peaks. Equation (5.3.3) can be written as N  eff  c NI N2  2L if1 Oft  (5.3.4)  where N1 and N2 are the number of peaks corresponding to the frequency separations 4f; and 4f2 , respectively. Due to the sharpness of the peaks, this technique has an estimated accuracy for Neff of about 0.08. This technique was used to measure the Neff of the shorted conventional coplanar strip electrode of mask #1. The results are given in Figures 5.3.2a and 5.3.2b. Using equation (5.3.4) the Neff is calculated to be 2.60.  5.4. The Resonance Technique  A schematic of the setup used in the resonance technique is shown in Figure 5.4.1. Here the output of the directional bridge is connected directly to the coplanar strip probe. Due to the small impedance mismatch between the probe tip and the electrode, a portion of the energy will be reflected back into the electrode at the probe tip, leading to resonance. The condition is described in Appendix III. Using reasoning similar to that in section 5.3, Neff can be shown to be given by c „=-2L L N Af  100  (5.4.1)  where N is the number of resonance peaks in the frequency interval df. Since this technique relies on the resonance produced in the electrode, the peaks are generally very sharp. This makes locating the peaks easy and results in higher accuracy. Moreover, the elimination of the directional coupler and coaxial short removes the intrinsic phase difference inherent in the interference technique. Consequently, the accuracy improves due to the fact that only one frequency sweep is needed to obtain Neip whereas the interference technique requires two measurements: one without lowering the probe to the electrode and one with the electrode probed. Moreover, this technique enables one to calibrate the network analyzer at the probe tip so that more accurate and noise free measurements, as well as the loss of the electrodes, can be made. Due to these advantages, the error in Neff of this measurement technique is estimated to be less than 0.05. Measurements were made between 8 to 20 GHz. To calibrate the network analyzer at the probe tips, the coplanar strip probe was first shorted by lowering it onto a pad of aluminum formed on a GaAs substate. This produced a plane of theoretically perfect reflection (i.e., reflection coefficient of unity magnitude) at the probe tips. Due to the terminal capacitance at the open probe tips, a well-defined plane of reflection cannot be realized easily by just leaving the probe tips open in air. On the other hand, shorting the probe tips can easily establish a very good plane of reflection. Consequently, calibration was made with the probe tips shorted. The network analyzer was then calibrated by measuring the normalized reflected power as a function of frequency and storing the calibration data into the network analyzer's memory. By using the calibration function of the network analyzer, all subsequent measurements were automatically calibrated by subtracting the measured results by the stored calibration data. 101  o3  P  13.424 GHz  O 0  -  J  0  • 1 .-1 a  E O  z 5^8^11^14^17^20  Frequency (GHz) Figure 5.3.2a. Measured normalized reflected power in dB, i.e., 10 log(reflected power/incident power), for the setup of Figure 5.3.1 with the probe not touching the conventional coplanar strip electrode of Mask #1.  102  5  ^ ^ ^ 11^14^17 8 20  Frequency (GHz) Figure 5.3.2b. Measured normalized reflected power for the setup of Figure 5.3.1 with the conventional coplanar strip electrode of Mask #1 probed.  103  COMPUTER  SYNTHESIZED SWEEPER HEWLETT PACKARD HP63416 RF OUT IN  DIRECTIONAL BRIDGE HEWLETT PACKARD HP85027E OUT  COPLANAR STRIP PROBE  (SYSTEM CONTROLLER) SYS HPIB REFL  HEWLETT PACKARD HP9636 HPIB  NETWORK ANALYZER  SYS. HPIB  HEWLETT PACKARD HP8757C  SLOW-WAVE ELECTRODE  TEKTRONIX IMP9215  Figure 5.4.1. Schematic of the setup used for the resonance technique.  104  COMPUTER (DATA ACQUISITION) IBM PC COMPATIBLE  The network analyzer was interfaced to an IBM PC compatible computer using an IEEE488 interface card. A data acquisition program was written to transfer the measured (and calibrated) data to a disk file. Since this technique relies on the reflection at the probe tip, the magnitude of the resonance depends on how the electrodes are probed. Sometimes the resonance peaks were small; by intentionally lowering the probe off centre often resulted in larger peaks (in the ideal case when the probe was perfectly matched to the electrode, no resonance would occur). Larger resonance peaks enable one to locate the resonances more accurately and thus improve the accuracy of the measurement. It was found, as expected, that how the electrodes were probed could only affect the magnitude of the resonances, and not the frequency at which the resonances occur.  5.5. Measured Results  Due to the higher accuracy and ease of locating the peaks, all the results presented here were obtained using the resonance technique. The measured N eff's for the electrodes fabricated on semi-insulating GaAs are tabulated in Tables 5.5.1 to 5.5.3. The theoretically predicted values are also given. The normalized reflected power for slow-wave electrode #1 (half-buried) of Mask #2, electrode #2 (half-buried) of Mask #2, and electrode #5 (surface deposited) of Mask #2 are, respectively, shown in Figures 5.5.1, 5.5.2, and 5.5.3.  105  Table 5.5.1. Measured and theoretical Ne ff' of conventional coplanar strip electrode. Si  WI  (pm)  (pm)  t (pm)  17  133  1.7  Comments  Etchant NH4 OH:H 2 02 :H2 0 (5:2:240)  Neff  Neff  (theory)  (measured)  2.64  2.60  Half-buried  Table 5.5.2. Measured and theoretical Ne ' s of slow-wave electrodes (Mask #1). Si = 32 pm, WI = 120 pm, S2 = 5 pm, W2 = 13.5 pm, d = 30 pm, 1 = 3 pm, t = 1.7µm Etchant  Comments  NH 4OH:H 2 0 2 :H 2 0 (5:2:240)  Neff  Neff  (theory)  (measured)  Half-buried  2.96  2.84  HCEH 2 02 :H2 0 (1:4:40)  Half-buried  2.96  2.87  NH 4 OH:H 2 02 :H2 0 (1:1:8)  Totally-buried  ---  2.95  106  Table 5.5.3. Measured and theoretical Neff of slow-wave electrodes (Mask #2). S'1 = 60 Arn, S2 -= 4 gm, W2 = 28 Am, / = 4 Am, t = 1.1µm Electrode  WI (AM)  W'  (pm)  1' (gm)  d (gm)  #1  72  0  4  #2  110  0  #3  110  #4  Neff  Neff  Neff  (theory, halfburied)  Halfburied  Surface deposited  18  3.50  3.40  3.33  4  32  3.16  3.10  3.03  7  8  32  3.28  3.24  3.17  110  7  12  32  3.39  3.32  3.30  #5  110  7  16  32  3.49  3.43  3.38  #6  110  14  8  32  3.30  3.25  3.18  #7  110  14  12  32  3.44  3.33  3.32  #8  110  14  16  32  3.56  3.46  3.43  107  1 1 1 1 1 —12  11111[111 -  8  8.835 GHz  1  1IIIIIIIIIIIIIIIIIII  10^12^  14^16^18^20  Frequency (CHz)  Figure 5.5.1. Measured normalized reflected power as a function of frequency of electrode #1 (half-buried) of Mask #2.  108  L_ (1) 0 0_  CD N  ._ o - 12— E  1  0  6.480 GHz — — — —I  z  —15  'm i lli 8^10  II  IIII I IIII I IIII I ^ 20 14^16^18  Frecuency (G1 1z) -  Figure 5.5.2. Measured normalized reflected power as a function of frequency of electrode #2 (half-buried) of Mask #2.  109  12  -  I I L ^  —18  I I  8.758 0Hz  J  iiii i IIII I IIII I IIII I IIII I IIII 8^10^12^14^16^18 20  Frec uency (G Hz)  Figure 5.5.3. Measured normalized reflected power as a function of frequency of electrode #5 (unburied) of Mask #2.  110  5.6. Discussion of Results The Neff of 2.60 for the conventional coplanar strip electrode as obtained using the resonance technique is, within two decimal places, the same as that obtained using the interference technique. This value is slightly lower than the theoretically (quasi-static analysis) predicted value of 2.64, even though the difference is small, being less than the experimental error of 0.05. Visual examination of the SEM micrographs indicated that the electrode is, in fact, only about 1/3 buried, which would result in a lower Nes, The slow-wave electrode of Mask #1 was fabricated using various etchants (see Table 5.5.2). Two were half-buried and one was totally-buried. The measured N eff's for both halfburied electrodes are almost the same except that the one fabricated using HC1:H 2 02 :H2 0 (1:4:40) has an Neff of 0.03 higher than that fabricated using NH 4 OH:H 2 02 :H2 0 (5:2:240). Even though this difference is less than the experimental error, the small increase may in part be accounted for by the smaller air gaps produced by the former etchant. Further evidence to support this assumption comes about by considering the Neff for the totally-buried electrode of Table 5.5.2, which is about 0.1 higher than those of the half-buried ones, indicating that burying the electrode in the substrate does in fact increase its Neff. Also, all of the Neff's of the halfburied electrodes of Mask #2 are higher than those of the surface deposited electrodes. Even though we cannot conclude definitively that the half-buried electrodes have higher Neff's, as the differences in most cases are less than the experimental error, it is expected that this is so. Comparison of the results obtained for the surface deposited electrodes and the half-buried electrodes of Mask #2 indicates that half-burying electrodes with their fins (and pads, if 111  applicable) having a thickness to gap width ratio of less than about 0.3 would only result in negligible increases in Neff. Due to the extra fabrication steps involved, it may not be very practical to half-bury electrodes since velocity-matched electrodes can be easily designed without having to bury them. Moreover, the design curves of section 3.3.7 may also be used for surface deposited electrodes provided the electrodes are thin, even though they were calculated by assuming that the electrodes were half-buried. As a comparison with theory, the predicted Neff of the half-buried slow-wave electrodes of Mask #1 is 2.96 or 0.32 higher than that of the conventional coplanar strip electrode using the modified design formulas of section 3.3.5 while the measured value is 2.87 or 0.27 higher than that measured for the conventional coplanar strip electrode. On the other hand, the Neff predicted using the equations that do not take inductance into account (section 3.3.4) is 3.18 or 0.54 higher than that of the conventional coplanar strip electrode. It is apparent that the modified design formulas provide better predictions of the New In fact, if the increases in Neff .  are considered, the measured and theoretical results agree within the experimental error of 0.05. For electrodes #1 and #2 of Mask #2, the theoretically predicted Neff's are, respectively, 3.50 and 3.16 using the modified design formulas of section 3.3.5, corresponding to, respectively, increases of 0.86 and 0.52 over the conventional electrode. The measured values of 3.40 and 3.10, corresponding to, respectively, increases of 0.80 and 0.50, indicate an agreement of over 93% in the increases of Neff between the experimental results and the modified design formulas of section 3.3.5. For electrodes #3 to #8, the agreements are equally as good, despite the fact that the capacitances of the fins with pads were calculated using the approximation formulas of section 3.3.5. 112  Considering the experimental errors, the measured results are in very good agreement with theory. As discussed above, processing problems such as the over-etch of the GaAs substrate, which resulted in more air around the fins, could also be in part responsible for the lower measured microwave indices. These problems, however, may be alleviated using dry etching techniques [38:pp. 173-195]. Comparison between the various slow-wave electrodes of Mask #2 shows that the Neff of a slow-wave electrode can be increased by reducing d, and by correspondingly adjusting for the change in inductance per unit length due to the smaller fin period by changing Z0 . Adding pads to the ends of the fins while keeping the same d could also result in significant increases in the Neff, However, the dimension W' does not seem to have a significant effect on the Neff's, as  indicated by almost the same N et's between electrodes having pad widths W' = 7 pm and W' = 14 pm. This suggests that the electric fields are confined to the gap region. This is also evidenced by the small changes in the calculated capacitance of the fins with pads having different W' 's (see Table 3.3.5.2). The uniformity of the frequency separations between the resonance peaks (Figures 5.5.1 to 5.5.3) suggests that the dispersion in the slow-wave electrodes is very low. In fact, almost no dispersion was detected. Moreover, at frequencies up to 20 GHz, the normalized reflected power in all cases is only a fraction of a dB below 0 at frequencies where anti-resonances occur, i.e., when the voltage and current waves are 90° out of phase and the normalized reflected power is maximum. This indicates that the losses of the electrodes are well within 0.5 dB/cm up to 20 GHz, comparable to conventional coplanar strips having similar dimensions [45:pp. 257-288]. Moreover, the losses have little dependence on the frequency, at least up to 20 GHz. These 113  losses are smaller than the 0.7 dB/cm as calculated in section 3.3.8, which was somewhat overestimated. Even though the calculated losses of 3 dB/cm in the slow-wave electrodes that are to be used in modulators are higher due to the smaller dimensions of the electrodes, they are still quite small as compared to those reported by others [33]. Results of the measurements (in particular, electrodes #1, #5, and #8 of Mask #2) indicate that we have been successful in achieving the microwave/optical wave velocity-match condition for GaAs based electro-optic modulators using the slow-wave electrodes described in this thesis. The electrodes of Mask #2 can readily be scaled down to the dimensions needed in electro-optic modulators. III-V semiconductor travelling-wave electro-optic modulators incorporating these slow-wave electrodes are expected to be capable of providing very wide bandwidths, low modulating powers, and high modulation depths.  114  Chapter 6 Conclusions and Recommendations for Future Research 6.1. Conclusions Coplanar slow-wave electrode structures capable of matching the velocity of a microwave to that of an optical wave in III-V semiconductor electro-optic modulators have been invented, analyzed, designed, fabricated, and tested. These electrode structures slow down microwave by exploiting the fringing electric fields produced by narrow capacitive loading fins and pads. The slow-wave electrode structures also offer low loss, low dispersion, ease of fabrication, and can be designed to match to transmission lines having 50 or 75 ohm characteristic impedances. Modulators employing these velocity-matched electrode structures should be capable of achieving very wide bandwidths, low modulation powers, and high modulation depths. The slow-wave electrodes are periodically loaded with capacitive fins; rectangular pads may also be added to their ends to further increase their capacitance. If the fins are narrow, the capacitance per unit length between the electrodes can be substantially increased, whereas the inductance per unit length along the electrode will only be minimally affected, thus resulting in the slowing of the microwave. The change in inductance per unit length due to the fins can be compensated for by changing the width of the coplanar strips. By choosing the fins and pads to be used, electrodes having a prescribed microwave effective refractive index, as well as characteristic impedance, can be designed. 115  A quasi-static analysis was used to analyze the electrode structures. This was done using a transfer matrix method. High frequency dispersion characteristics representing the upper limit of operation of the electrodes were calculated. Finite difference and finite element methods were used to calculate the capacitance of the loading fins. Results of the capacitance computation show that the narrower the fins, the higher the capacitance to fin length ratio. In order to obtain a high enough degree of slowing of the microwave, a high capacitance to fin length ratio is needed. The design formulas obtained using the transfer matrix method were simplified and modified, allowing easy application as well as improved agreement with experimental results. Finally, design curves for 50 and 75 ohm electrode structures suitable for use in both GaAs and InP based electro-optic modulators are given. Aluminum electrodes, both buried and surface deposited, having various dimensions have been fabricated on semi-insulating GaAs substrates using a combination of single-step lift-off and substrate etching techniques. The fabrication process for these electrode structures was proved to be straight forward and reproducible. The microwave effective refractive indices of the electrode structures fabricated were measured using a microwave scalar network analyzer. Both interference and resonance measurement techniques were used. Good agreement was found between the measured results and the modified design formulas. Results of the measurements show that 50 ohm electrodes having microwave effective refractive indices of 3.43 (i.e., matched to the effective refractive index of the optical wave in AlGa i _ x As/GaAs/AlGa l _„As waveguide structures) have been achieved. These electrodes should enable modulators having very high performance to be 116  realized. The measurements of the slow-wave electrodes also showed that their losses and dispersions were very low. The uniformity in frequency separation of the resonance peaks indicated that the effective refractive indices of the microwave were constant up to at least 20 GHz. Since the synthesized frequency source had a 20 GHz working limit, measurements could only be made up to this limit. However, the electrodes are expected to work at frequencies much higher than 20 GHz. While the theoretical analysis presented is relatively simple, being based on the quasi-static approximation, the agreement between the predicted and measured results was quite good, especially when fabrication errors were taken into account.  6.2. Recommendations for Future Research  Although the losses indicated by the measurements were very low, it is possible to reduce them further. For example, thicker and wider electrodes can be designed so as to reduce the resistive losses; gold instead of aluminum can be used; coplanar waveguide structures, which have lower losses than coplanar strips, can also be designed using the same principles. Besides placing the fin elements (and pads) directly in line with one another, as done in this thesis, capacitive loading fin elements can also be placed side by side to each other with each of the fin elements attached to each of the coplanar strips so that a section of the pair of fin elements overlaps in the interelectrode gap region. This kind of structure is also worth investigating. 117  By computing the electric field distribution around the electrodes, the overlap between the microwave and optical fields can be found. The electrodes can then be optimized for the highest overlap while providing the highest modulating electric field, thus allowing furthur reduction of the modulating power requirements. 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Electrochem. Soc.: Solid State Science and Technology, vol. 128, no. 4, pp. 874-880,  1981. 60.  S. Adachi and K. Oe, 'Chemical Etching Characteristics of (001) GaAs,' J. Electrochem. Soc.: Solid State Science and Technology, vol. 130, no. 12, pp. 2427-2435, 1983.  61. S. H. Jones and D. K. Walker, 'Highly Anisotropic Wet Chemical Etching of GaAs Using NH 4 OH:H 20 2 :H 20,' J. of Electrochem. Soc., vol. 137, no. 5, pp. 1653-1654, 1990.  126  Appendix I. Conventional Coplanar Strips  According to the quasi-static analysis, the characteristic impedance Z, of a pair of coplanar strips having a width W, thickness t, and separated by a gap S on a substrate of thickness h, dielectric constant € ,., under a superstrate of dielectric constant  Es  (Figure 1.1) is given by  [45:pp. 257-288] as Zo -  1207E ace) Ki(ke) e K'(k  ke -k  (1-k 2)A 2W  k-  S S+2W '  (IA  g  )  where  (1.3)  A - 1.25/1 -Fin( 4n FV n^t  127  )]  ,  (1.4)  1.4(e re -1)t/S eeere ^; e,O, Er z9,t/W<0.1 , [Kl(k)IK(k)]+1.4tIS  kW er2e s [tanh(1.785log(h/W)+1.75)+— 1 0.04-0.7k+0.01[1-0.1(e r +e s -1)](0.25 -4)1 , hi  1  K(k) _ 1 in2 1+Vic ;  K'(k)  K(k) _  K'(k)  71 . 1  IC  -IA  (1.6) 0.707sks1 ,  k / 41-k2 ; Osks0.707 .  (1.8)  In  From the formulas as given above, it can be easily seen that Z, depends only on the ratio S/(S+2W) if the coplanar strips are very thin (i.e., t approaches zero) and the substrate very  thick (i.e., h approaches infinity). As discussed in sections 3.3.9 and 3.3.10, this property allows thin conventional coplanar strips and thin coplanar slow-wave electrodes to be scaled up (or down) while having little changes in the microwave effective refractive indices and characteristic impedances. This property also makes the design of input sections easy.  128  e  SupersEraLe  11^ W ^ 011 S  141^W ^01  7777 i  Er  Substrate  Figure 1.1. A coplanar strip electrode on a substrate. The dimensions S, W, t, and h are shown.  The capacitance per unit length C between a pair of coplanar strips is given as C-e o e r +e s r(k) 2 K(k)  (1.9)  where 6 0 is the permittivity of free space. Equation (1.9) is useful for calculating the capacitance of fins with pads as described in section 3.3.5. For coplanar strips on GaAs, conductor losses are much higher than dielectric losses [45:p. 73, pp. 257-288]. For the purpose of our calculations, the dielectric losses are ignored. The conductor losses a„ (in dB/unit length) can be calculated using Wheeler's incremental inductance formula [45:pp. 257-288], as given by 129  1+ 1.25 in 4Tc W, 1.25t t^TC W a cs = 17 . 34 1R 111+W ; TES^S) 1 1 +2 W + 1.254 1 +in 47c WI [^S^ITS^t ) —  (I.10)  where P, =  K(k) K (k)  fp  9  '  k P=  k 2 )(1 k2) 314 1 ^( r (k))2  (1 si 1 -  for  -  a  (1 _ k) N  0.0dcs0.707  -  K(k)  9  for  (I.12)  0.707 s ks 1.0  and Rs is the surface resistivity given by [57:pp. 151-153] as ^  (I.13)  ^  (I.14)  Rs =3.26 x 10 -7 (f (Ohm)  for aluminum electrodes and Rs =3.03 x 10 -7111 (Ohm) -  for gold electrodes, where f is in Hz.  130  Appendix II. Coefficients of Inverse Capacitance From reference [53], the PiIs of section 3.3.6 are given in two general cases for cells oriented parallel and perpendicular to each other as follows (see Figure II.1): (1) Cells oriented in parallel: k=4 m=4[ 2 2  b m-co E E (-1)k- ^a kln(a k + p)+ , ,^ 2 4ne^ o f f s s . m=1  1  a  b  a  bk 1  2 2  (IIA)  ak-cii^1 2 a ,b ^bm1n(b m +p) (b.-2cii +a k2)p bmcoa ktan-1 - m -  2^6^  pcii  where p  =  (ak2 +bm2 +cii2.) 1/2^  fa sa a l = a:,-- -' 2 2 ,  f  a Sa a2 = au + — - — , i  2 2  131  (II.2)  (II.3)  (11.4)  fa Sa + li 2 2  a = a.. F 3^ -  a4^ = a-  —  fa  '  —  So  V 2+2- '  b1  =  -  fb  bu i --  b2 = biff  b 3 = b i . 4I  b  Si,  2  ----  (II.5)  (II.6)  (II.7) ,  sb  27 ,  (II. 8)  -ri- ,  (II.9)  .  --  fb  sb  (II. 10)  132  ^ ^ ^  (2) Cells oriented perpendicular to each other:  .  r, E. ,,=,E  ^2  1 Pi^ 'I 47ce o fa fb s a s_ b -  ^2  4 2 2 i  ,,  ak c i  ( - 1 )1 + m + k + 1 2 - — 6 clin(am+P)+  (a 2^,2  bmcip 2^ 1 „ b millfr 1+ P)+ a kb mc PO k+ P ) ^3 -  ^3  ^a  ^2^2  k^1 bm c 1 bma k -1, a kC I . a kCI^-ii a kb m) tan k _..._ )^tan k _t an - ()^  ^6 ^akp^2^bmp^2^cip  where p and ak are the same as for parallel cells and  ^c  b, = bv -F-1 S ,  b2 = bii  -  (II.12)  1 ,^ )  c, = cIC e--2  (II.13)  ,  (11.14)  fc ^ ., = c -...^.,^ 2 .  (II.15)  ;;  133  Sb  Cells oriented in porollel^cij  Fa  Cells oriented perpendiculor to each other Sb  Se,  Figure II.1. Dimensions for cells oriented parallel and perpendicular to each other. 134  Appendix III. Conditions for Resonance  As discussed in Chapter 5, resonances occur in the electrodes due to impedance mismatch at the probe tip. This happens when waves are partially reflected at the electrode input and output ends. Figure III. 1 shows an electrode having a characteristic impedance Zo terminated by a load of impedance ZT and on the input end is a source V,. having an impedance 4 (this represents the impedance of the probe, cable, and directional bridge). Suppose at a particular instant a voltage wave starts to propagate along the electrode. The instantaneous voltage at the input end of the electrode will be VZ o/(Z„. + Zo) since the wave has not yet travelled to the other end and so there is no reflection. As the wave propagates to position z the voltage will become [11,20 /(4. + Zo)Je rz, where y is the propagation constant. -  When the wave reaches the load, it will be reflected by the load reflection coefficient P T ZT Zo P T— ZT+Zo  The voltage of the first reflected wave at position z then becomes VsZo  Z+Z s o  -yL^-Y(1,--z) 7  e p e  (III.2)  This first reflected wave will again be reflected at the input end by the input reflection coefficient Ps  135  Zs—Zo Ps—  (III.3)  Zs+Zo  This second-time reflected wave will have a voltage at position z equal to VsZo Zs +Zo  (III.4)  e -yL pe -yL pse -yz  Due to superposition the steady state voltage at position z is the sum of an infinite number of reflection terms, i.e., V(z  VsZ (e -yz + p 7e -2 yLe yzp + p 7.p se -2 yL + ( p 7,p se -2yL)2 + ...] .  )—^  Zs +Zo  (III.5)  Substituting (L - d) for z and simplifying one obtains V(d)-  VsZoe - YL eYd +p ie - I' d Zs +Zo  1 — p rp se -2YL  •^(III.6)  Since the reflection coefficient r of the 'resonant' electrode is related to V(d) by V(d) = V+ (1  + r), r can be obtained by noting that V + is simply the V(d) when ps and P T are zero, i.e., both source and load are perfectly matched to Zo . At the input end, d = L and so r is given by  136  r-  ^2Z0(1+p  le -2YL)^  (Zs +Zo)(1-p Tp se -2Y L)  1.  ^  (III.7)  Equation (III.7) implies resonance whenever e-2YL = 1 or -1, depending on p T and ps . In other words the return loss will be maximum (r minimum) at certain resonance frequencies separated by df  af_ c 1  (III.8)  - 2L Neff  where c is the speed of light in vacuum. As discussed in Chapter 5, equation (III.8) is useful for calculating Neff by measuring the separations Lif between the peaks of the resonances. Source  ^  Electrode^Load  ^p  Zo  d  Z  L  ki^  ^0  01  Figure III. 1. Equivalent circuit of an electrode having a characteristic impedance Zo , terminated by a load ZT and connected to a source Z, with voltage V.  137  

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