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Electrooptic mach-zehnder modulators in gallium arsenide Tsou, Benny Pen-Cheng 1993

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ELECTROOPTIC MACH-ZEHNDER MODULATORS IN GALLIUM ARSENIDEbyBENNY PEN-CHENG TSOUB.A.Sc., University of British Columbia, Canada 1988B.Sc. (Honours), University of British Columbia, Canada 1990A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIES(The Department of Electrical Engineering)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIASeptember 1993©Benny Pen-Cheng Tsou, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of ELEC TRICAL ENGINEERINGThe University of British ColumbiaVancouver, CanadaDate SEPTEt-f8ER Z&I l'Ici3DE-6 (2/88)AbstractExternal travelling-wave modulators, utilizing the electrooptic effect, are one class ofdevices currently being investigated for converting electrical signals to optical signals inapplications involving high-speed data transmission. Modulators fabricated on semiconductorsubstrates, such as GaAs and InP, are particularly attractive in that there exists the possibilityof monolithic integration of these devices with other optoelectronic components. Thebandwidths of this type of modulator are limited by the velocity mismatch between themicrowave and the optical wave, where, in these compound semiconductors, the microwavehas the greater phase velocity. To eliminate this problem, a slow-wave electrode structurehas been developed in our laboratory in which the phase velocity of the microwave is reducedto match the phase velocity of the optical wave.In this thesis, modulators, based on the integrated optics version of the Mach-Zehnderinterferometer, using GaAs substrates, were modelled and fabricated for use with these slow-wave electrodes. The variational method, effective-index method, and beam propagationmethod were used to model the propagation of light in these devices, and, hence, obtainingthe necessary design parameters for these devices.Based on these numerical simulations, modulators were designed and fabricated inAlGaAs epitaxial layers, grown on GaAs substrates by the metal organic chemical vapourdeposition technique. Optical attenuation in the waveguides was measured. Parameters suchas the half-wave voltage, intrinsic bias, extinction ratio, and microwave indices of theelectrodes were measured. The extinction ratio, for most devices, was greater than 99 % .The half-wave voltage and the microwave indices, which could be calculated theoretically,were found to be in good agreement with the measured values.111Table of ContentsAbstract ^  iiTable of Contents ^  ivList of Figures  viList of Tables ^  xAcknowledgements  xiChapter 1 Introduction ^  1^1.1^Introduction  11.2^Recent Developments in Electrooptic Modulators ^ 31.3^Organization of the Thesis ^  5Chapter 2 AlGaAs Semiconductor Waveguides and Electrooptic Modulation ^ 72.1^Introduction ^  72.2 Methods for Fabricating Semiconductor Waveguides ^ 72.3^Refractive Index of AlxGai,As ^  122.4 Metal Organic Chemical Vapour Deposition (MOCVD) ^ 142.5 Electrooptic Effect of GaAs/AlGaAs  152.6^Operating Principles of Mach-Zehnder Interferometer ^ 182.7 Bandwidth Limitations of Electrooptic Modulators  212.8 The Slow-Wave Electrodes ^  24Chapter 3 Numerical Simulation  273.1^Introduction ^  273.2^Variational Method  293.3^One-Dimensional Scalar Variational Expressions ^ 313.4 Beam Propagation Method ^  353.5^Effective-Index Method  393.6^Simulation Results  413.6.1 The Planar Waveguide  413.6.2 The Rib Waveguide ^  493.6.3 Coupling Characteristics of Parallel Rib Waveguides ^ 573.7^Calculation of r ^  643.8 Chapter Summary  69Chapter 4 Fabrication  714.1^Introduction ^  714.2 Mask Design  71iv4.3 MOCVD Epitaxial Layers ^  734.4^Etch Calibration ^  784.5^Device Fabrication  814.5.1 Fabrication of Waveguides  814.5.2 Alignment of Waveguides and Electrodes ^ 844.5.3 Chlorobenzene Lift-off ^  864.5.4 The Completed Devices  90Chapter 5 Experimental Results ^  955.1^Introduction  955.2 Planar and Channel Waveguides ^  955.3^Optical Attenuation Measurement  1015.4^Current-Voltage Characteristics of the Electrodes ^ 1035.5 Measurement of Device Parameters at Low Frequencies ^ 1065.6 Measurement of n ^the Resonance Technique  1145.7 Chapter Summary  119Chapter 6 Summary, Conclusions and Suggestions for Future Work ^ 1216.1^Introduction ^  1216.2 Summary  1216.3^Conclusions  1246.4^Suggestions for Future Work^  124Appendix A Beam Propagation Method  127Appendix B Evaluation of f.:x2e-x2f(x)dx Using Gaussian Quadrature ^ 132References ^  137vList of FiguresFigure 2.5.Figure 3.2.Figure 3.3.Figure 3.4.(a) Cross-sectional view of an heterostructure semiconductor planarwaveguide with optical mode confined only in the depth direction.(b) Cross-sectional view of a rib waveguide with an optical modeconfined in both lateral and depth directions. ^  9Refractive index of AlGaAs as a function of the aluminum molefraction and wavelength ^  13(a) Coordinate transformation for the electrooptic effect.(b) Definition of the input light polarization for the crystalorientation shown  ^19Operating principle of a Mach-Zehnder interferometer. (a) Themodes of the branches combine in phase and excite the fundamentalmode in the output section. (b) The modes combine it radians outof phase and excite a radiation mode in the output section ^ 20Plan view of the modulator showing the slow-wave electrodes andthe Mach-Zehnder interferometer^  25Cross-sectional view of the modulator  26Plots showing the functions used for approximating the optical fielddistributions of the fundamental and the next higher order modes. . . . 33Calculated fundamental mode size as a function of the refractiveindex profile width parameter d for three different values ofaluminum mole fraction in the cladding layers^ 44The Gaussian distribution of the aluminum mole fraction (dashed curve)in the planar waveguide's depth direction. The solid curves are thelinear approximations^  46Comparison of the Gaussian intensity profile (solid curve, wy=0.855,um)with the intensity profile obtained by the beam propagation method(dashed curve). The refractive index profile used by the BPMcorresponds to the linear approximations shown in Figure 3.5, butdoes not include the refractive index of the GaAs substrate. ^ 47Figure 2.1Figure 2.2Figure 2.3.Figure 2.4.Figure 2.6.Figure 3.1.Figure 3.5. Refractive index profile of the actual layer structure, i.e., the linearapproximation shown in Figure 3.3, with the refractive indices ofvithe superstrate and substrate^  48Figure 3.6.Figure 3.7.Figure 3.8.Figure 3.9.Figure 3.10.Figure 3.11.Figure 3.12.Figure 3.13.Figure 3.14.Figure 3.15.Figure 4.1.Calculation of the propagation loss, due to leakage of guided lightinto the GaAs substrate, as a function of bottom cladding layer'sthickness. The solid curve is an exponential fit of the calculateddata ^  50A plot of neff as a function of the separation between the locationof npeak and the surface of the sample. The dashed curve is calculatedusing the effective index method. The solid curve is a sixth orderpolynomial fit to the dashed curve with data near the discontinuityremoved^  51The lateral effective refractive index profile of a 4pm wide andlpm high rib waveguide (dashed curve). The solid curve is theerror function approximation, i.e., equation (3.21), with A=3.24and B=2.16  54Calculation of the fundamental mode size as a function of rib width,for three different rib heights. ^  55Comparison of the Gaussian intensity profile (solid curve, wx=2.32pm)with the intensity profile obtained by the beam propagation method(dashed curve) for the case of a 4m wide and lpm highrib waveguide. ^  59Model used for calculating the coupling characteristics of twoparallel waveguides, as a function of their separation^ 60Coupling characteristics of two parallel rib waveguides with4pm width. ^  62Coupling characteristics of two parallel rib waveguides with5m width  63Three-dimensional plots showing the coupling characteristics of two4m wide parallel channel waveguides, for a propagation length of2cm. The separation of the waveguides is (centre to centre) (a) 12pm,(b) 14p.m, (c) 16pm, and (d) 20pm ^  65Cross-sectional view of the modulator with the electrodes modelled asa pair of conventional coplanar electrodes, as described in the text. . . 68Schematics showing the various labels and dimensions of (a) the Mach-viiZehnder waveguides, (b) the slow-wave electrodes, and (c) the input/output sections. ^  74Figure 4.2. Target structure of the epitaxial layers grown by MOCVD ^ 77Figure 4.3. Etch rate of citric acid/hydrogen peroxide (10:1) mixture for -30%aluminum mole fraction. ^  82Figure 4.4. Edge profile ((01T) face) of an AlGaAs rib etched using citricacid/hydrogen peroxide (10:1) mixture. ^  83Figure 4.5. Orientation of waveguides on a Spire wafer (not to scale) . ^ 85Figure 4.6. Alignment marks on the waveguide mask (a) and the electrodemask (b). All dimensions are in Am. ^  87Figure 4.7. Chlorobenzene lift-off technique.  89Figure 4.8. SEM micrograph showing the plan view of device no. 7 (fins only). . ^ 92Figure 4.9. SEM micrograph showing the isometric view of the device shown inFigure 4.8. ^  92Figure 4.10. SEM micrograph showing the plan view of device no. 8 (fins and pads). 93Figure 4.11. SEM micrograph showing the isometric view of the device shown inFigure 4.10^  93Figure 4.12. Photograph showing the input/output section of a device with fins. . . ^ 94Figure 4.13. Photograph showing the input/output section of a device with finsand pads. ^  94Figure 5.1. Diagram showing the setup used for adjusting the polarization of thelaser.  97Figure 5.2. Diagram showing the setup used for viewing outputs from planar andchannel waveguides . ^  97Figure 5.3. (a) Image of light confined by the planar waveguide as appeared onthe IR sensor card  99(b) Image of an output spot from a channel waveguide with litm highrib ^  100(c) Image of an output spot from a channel waveguide with 1.4Amviiihigh rib. ^  100Figure 5.4.Figure 5.5.Figure 5.6.Figure 5.7.Measured optical attenuation as a function of length for 4Am widechannel waveguides with lihm high rib ^  104Measured current-voltage (I-V) characteristics of a pair of slow-waveelectrodes deposited on a layer of Si02. ^  105Diagram showing the setup used for measuring device parameters atlow frequencies. ^  107Oscilloscope image showing the modulation characteristics ofdevice no. 7.  ^110Figure 5.8. Oscilloscope image showing the modulation characteristics ofdevice no 8^  110Figure 5.9. Fitted transfer function for device no. 7^  111Figure 5.10. Fitted transfer function for device no. 8  112Figure 5.11. Diagram showing the setup used for measuring the microwaveindices of the electrodes using the resonance technique. ^ 115Figure 5.12. Measured resonance characteristics of electrode no. 7^ 117Figure 5.13. Measured resonance characteristics of electrode no. 8^ 118ixList of TablesTable 4.1.^Dimensions of the waveguides and electrodes on the masks: (a) groupI, (b) group II, and (c) group III. ^  75Table 5.1.^Measured device parameters for the Mach-Zehnder modulator. .^. 113Table 5.2.^Designed and measured n4 's of slow-wave electrodes in group Ion the electrode mask ^  116xAcknowledgementsI am most grateful to my parents for their support and encouragement throughout thiswork.I would like to thank my supervisor, Dr. N.A.F. Jaeger, for providing me with anopportunity to work in the area of integrated optics. Besides the generous financial support,the guidance and advice he has given me have been invaluable.My thanks also extend to Z. Lee for his help in designing my electrode mask, to W.Lai for allowing me to modify her computer program on beam propagation to suit my needs,and to H. Kato, L. Huang, and Dr. D. Hui for their help in the laboratory. In addition, Iwould also like to thank other members of the solid-state group and those individuals in thisdepartment who have helped me during the course of my study.Finally, I would like to acknowledge the Roger's Canadian Cable Labs Fund, the BCAdvanced Systems Institute, and the Natural Sciences and Engineering Research Council ofCanada for their generous financial support of this project.xiChapter 1Introduction1.1 IntroductionThe use of semiconductor lasers in conjunction with optical fibres to transmit signalsin applications such as telecommunication, signal processing, and instrumentation has manyadvantages over the conventional metallic wires or waveguides. Some of these are lightweight, larger transmission bandwidth, immunity to electromagnetic interference, securityfrom monitoring, and low loss [1:p. 1]. Methods currently being investigated for impressingthe electrical signal onto the optical carrier are direct modulation of the semiconductor laser'sdrive current [2:pp. 272-294] and external modulation of the semiconductor laser's output[3:pp. 203-205].The linewidth of a semiconductor laser usually broadens under direct modulation ofthe drive current. Such broadening may be attributed to the increase in the number of lasingmodes under high-speed modulation or the dynamic wavelength shift, which is due to thechange in the refractive index of the active region, as a result of the variation in carrierdensity during direct modulation [4]. Consequently, the transmission bandwidth is reduced,especially for communication systems utilizing single mode fibres. External modulators offerthe potential to achieve higher bandwidths by limiting the spectral broadening of thesemiconductor laser.The operation of an external modulator can be based on either the linear electroopticeffect or electroabsorption. The linear electrooptic effect is a change in the refractive index1of a crystal due to an applied electric field [5:p. 242]. The crystal usually remainstransparent at the working wavelength. Electroabsorption can be due to the Franz-Keldysheffect [6:p. 48] or the quantum-confined Stark effect [7]. In either case, light is absorbedin the substrates when an electric field is applied, resulting in intensity modulation. The typeof modulator discussed in this thesis uses the linear electrooptic effect.An electrooptic modulator can possess either lumped electrodes or travelling-waveelectrodes, depending on the physical length of the electrodes as compared to the wavelengthof the modulating electrical signal. For high-speed applications, travelling-wave electrodesare required. In this case, the bandwidth is limited by the phase velocity mismatch betweenthe microwave, travelling on the electrodes, and the optical wave, propagating in thewaveguides beneath the electrodes [8], as well as by the loss in the electrodes. The amountof velocity mismatch depends on the electrode structure, and on the dispersion of thesubstrate's dielectric constant as a function of frequency.Typical materials, which are currently being used as the optical substrates, includeferroelectrics, such as lithium niobate (LiNb03) and lithium tantalate (LiTa03), and thecompound semiconductors, such as gallium arsenide (GaAs) and indium phosphide (InP).Researchers have been interested in LiNb03 for several reasons. First, the opticalwaveguides are relatively easy to fabricate. Techniques such as titanium indiffusion andproton exchange are well developed [3:pp. 146-150]. Second, the optical loss of theresulting waveguides, in both the visible and the near infrared spectrum, is low. FinallyLiNb03 has relatively large electrooptic coefficients, e.g., the electrooptic effect of LiNb03is approximately six times that of GaAs. However, the one disadvantage is that, for a pair2of conventional coplanar electrodes, the phase velocity of the microwave is approximatelyone half that of the optical wave's. Therefore, an electrode structure, capable of increasingthe phase velocity of the microwave, is required to achieve larger bandwidths for modulatorsfabricated using this material.Recent advances in molecular beam epitaxy (MBE) and metal organic chemical vapourdeposition (MOCVD) have reduced the optical loss of semiconductor waveguides in the nearinfrared. This has made III-V compound semiconductors, such as GaAs and AlGaAscompounds, more competitive materials for the fabrication of electrooptic modulators. Also,the use of III-V compound semiconductors offers the possibility of monolithic integration ofthe optical components with other electronic and photonic devices. For a pair ofconventional coplanar electrodes, the microwave travels approximately 30% faster than theoptical wave, mainly due to the portion of the microwave travelling in the air above theelectrodes. In this case, an electrode structure, capable of decreasing the phase velocity ofthe microwave, to match that of the optical wave, is required.1.2 Recent Developments in Electrooptic ModulatorsResearch in LiNb03 has produced devices with 3-dB optical bandwidth in excess of10 GHz. Izutsu et al. [9], Gee et al. [10], and Korotky et al. [11] fabricated modulatorswith asymmetric coplanar electrodes on titanium indiffused waveguides with bandwidths of10 GHz, 17 GHz, and 22.5 GHz, respectively. Cross et al. [12] achieved a bandwidth of13 GHz with a modulator utilizing coplanar waveguide electrodes and an asymmetricwaveguide structure to achieve proper biasing. Dolfi et al. [13] used aperiodic phase3reversals to solve the velocity mismatch problem. Their bandwidth was approximately 10GHz and the 5-dB bandwidth was greater than 40 GHz. Kawano et al. [14], tried tominimize the phase velocity mismatch by enclosing the modulator in a metallic shield. Theyachieved a bandwidth of 20 GHz. Finally, LiNb03 and LiTa03 electrooptic modulators withproton exchanged waveguides, having a bandwidth up to 10 GHz, are now availablecommercially [15].Research into the use of semiconductor substrates has also been extensive in recentyears, especially the GaAs and AlGaAs compounds. Buchmann et al. [16] placed a pair ofconventional coplanar electrodes on optical waveguides fabricated in a doped substrate. Themodulator had a bandwidth of 4.5 GHz, which they believed was limited by a resonance inthe mount and bias circuit. Lin et al. fabricated devices using waveguides with p-i-n (GaAs-AlGaAs) structures on n+ substrates [17], and waveguides in undoped epitaxial layers onsemi-insulating substrates [18]. The p-i-n devices had bandwidths of 4 GHz. The limitationon these devices was the high microwave loss due to the n+ substrate. Consequently, theyfabricated devices on MOCVD grown epitaxial layers on semi-insulating substrates, for whichthey achieved a bandwidth in excess of 20 GHz. Nees et al. [19] attempted to reduce thephase velocity mismatch by placing a piece of GaAs on top of the electrodes to replace air.By measuring the response of a subpicosecond pulse, they estimated the bandwidth of theirdevice to be 100 GHz. Walker et al. [20] tried to slow the microwave using a segmentedelectrode structure to provide capacitive loading. The structure was complex and includedisolation trenches and air bridges. The device had a bandwidth of 26.5 GHz.In this laboratory, research into electrooptic modulators has resulted in slow-wave4electrodes being modelled and fabricated. This electrode structure was shown to be capableof slowing the microwave for modulators fabricated on compound semiconductors [21]. Thestructure is a pair of conventional coplanar strips, capacitively loaded with periodic fins, andcan be easily formed in a single layer of metallization. Here this work was continued, in thata GaAs semiconductor waveguide structure, resembling the integrated optics version of theMach-Zehnder interferometer, for use with such a slow-wave electrode structure, wasmodelled and fabricated.1.3 Organization of the ThesisIn Chapter two, the operating principles of the electrooptic modulators, fabricatedusing GaAs substrates, are given. Several methods for fabricating semiconductor waveguidesare described. The linear electrooptic effect of GaAs is reviewed. The working principleof a Mach-Zehnder interferometer is given. A discussion on the bandwidth limitations ofelectrooptic modulators is included. Finally, a slow-wave electrode structure, suitable foruse with modulators fabricated on semiconductor substrates, is described.In Chapter three, the variational method, the effective-index method, and the beampropagation method are presented. These methods were used to predict the behaviour of lightin planar and channel waveguides. From the various calculations and simulations, designparameters are given for the planar waveguides, channel waveguides and Mach-Zehnderinterferometers.In Chapter four, fabrication procedures are given. The layout of the masks areshown. The type of etchant used is described. A summary of the fabrication steps is given.5A description of the chlorobenzene lift-off technique, which was used to fabricate theelectrodes, is included.In Chapter five, verification of waveguiding in AlGaAs epitaxial layers is shown.Results of measured optical attenuation in channel waveguides are given. Device parameterssuch as the half-wave voltage, intrinsic bias, extinction ratio, and microwave indices of theelectrodes were measured and the results are presented.In Chapter six, a summary of the present work, the conclusions, and suggestions forfuture work are given.In Appendix A, the derivation of the beam propagation equation used in this work isgiven. In Appendix B, the expression for a weighing factor, used by the Gaussian quadratureformula, is derived.6Chapter 2AlGaAs Semiconductor Waveguides andElectrooptic Modulation2.1^IntroductionIn this chapter, background information is provided on semiconductor waveguides andon the operating principles of the integrated optics version of the Mach-Zehnderinterferometer. In section 2.2, several methods for fabricating optical planar and channelwaveguides on semiconductor substrates are described. In section 2.3, the refractive indexof AlxGai_xAs in the near infrared, as a function of the aluminum mole fraction x, is given.In section 2.4, a brief description of the MOCVD process for growing AlGaAs epitaxiallayers is given; the planar waveguides used in this work were grown by this technique. Insection 2.5, the linear electrooptic effect of GaAs-AlGaAs compounds is reviewed. Insection 2.6, the operating principles of the integrated optics Mach-Zehnder interferometer arepresented. The limitations on the modulation bandwidth, due to the velocity mismatchbetween the microwave travelling on the electrode and the optical wave travelling in theoptical waveguide, are discussed in section 2.7. Finally in section 2.8, a description of aslow-wave electrode structure, which is used to match the two velocities, is given.2.2 Methods for Fabricating Semiconductor WaveguidesThe formation of optical waveguides, either planar waveguides or channelwaveguides, is achieved by surrounding a region of high refractive index (guiding region)by regions having lower refractive indices (cladding regions). Planar waveguides confine7light only in the depth direction while channel waveguides confine light in both transversedirections. The number of optical modes that can be supported by a particular waveguideat a given wavelength is determined by the difference in the refractive indices of the guidingand cladding regions as well as the transverse dimensions of the waveguide.There are several methods for fabricating planar and channel waveguides on GaAssubstrates. Two commonly used methods for fabricating planar waveguides are carrierconcentration reduction [22-25] and composition variation in epitaxial layers [26-28].Channel waveguides are formed on planar waveguides by introducing a variation in therefractive index in the lateral direction. This is often achieved by etching ribs or ridges onthe epitaxial layers [29], introducing strain in planar waveguides by deposition of metal strips[30], and disordering the superlattices by thermal diffusion of impurities [31,32]. Figure2.1a shows a planar waveguide formed by having several layers of different refractiveindices. The value of n2, the refractive index of the guiding layer, is larger than n3 and nt,the refractive indices of the bottom and top cladding layers, respectively. The light isconfined only in the depth direction, and can propagate freely in the lateral direction. Figure2.1b shows a channel waveguide where a rib has been etched into the top cladding layer.The effective refractive index underneath the rib is higher than on either side. Hence lightis confined to a spot around the region underneath the rib.The presence of free carriers (n+ or p+) in a doped semiconductor reduces therefractive index from that of the undoped one [33:p. 318]. The change in the refractiveindex is usually sufficient to permit guiding of light, i.e., An in the range 10-3 to 10-2. Thisproperty has been used to create waveguides in the proximity of p-n junctions where the8Figure 2.1. (a) Cross-sectional view of an heterostructure semiconductor planar waveguidewith an optical mode confined in the depth direction. (b) Cross-sectional view of a ribwaveguide with an optical mode confined in both the lateral and depth directions.9refractive index is higher in the carrier-depleted regions and lower in the surrounding dopedregions [22,23]. The optical loss for this type of waveguide is usually large ( > 4 dB/cm atA. = 1pm [34]) due to the extension of the evanescent fields into the doped layers. This isreferred to as the free carrier absorption loss [33:p. 319].Also belonging to this class are planar waveguides formed on doped substrates byeither proton bombardment or ion implantation [24,25]. In these cases, high energy protonsor ions create defect centres in the substrates which act as deep level traps for both holes andelectrons. This results in the formation of a free carrier compensated region which has ahigher refractive index than the surrounding regions. The optical loss is high (can be up to200 dB/cm at 1. = 1.15Am [35]) due lattice damage. However the loss can be reduced toabout 2-3 dB/cm by annealing at low temperature (-500°C) [35].Another technique for fabricating optical planar waveguides is by compositionvariation [26-28]. This can be done by incorporating aluminum in the GaAs lattice throughvarious epitaxial techniques. The change in the refractive index is achieved by varying thealuminum mole fraction. Increasing the aluminum mole fraction decreases the refractiveindex and increases the bandgap [33:p. 320]. The dependence of the refractive index on thealuminum mole fraction, for several different wavelengths, in the near infrared, is shown inthe next section. Note that the increase in the bandgap shifts the fundamental band-edge toshorter wavelengths, in effect, decreasing the fundamental absorption loss.Since the lattice constants of AlAs and GaAs are almost identical, with a differenceof 0.15% at 300K (AlAsiattice =5.6611A and GaAsiattice =5.6533A) [36], continuous aluminumconcentrations between 0 and 1 can be grown on a GaAs substrate. However, for most10devices, the aluminum concentration is usually kept below 0.36, the point at which thebandgap changes from direct to indirect [37]. The indirect bandgap does not affect theoperation of the waveguides, but it does affect the efficiency of any emitters or detectors thatmay be monolithically integrated with the waveguides [35]. The epitaxial layers can begrown by either molecular beam epitaxy (MBE) or metal organic chemical vapour deposition(MOCVD). The optical loss for waveguides fabricated using these processes can be less than1 dB/cm for 10 = 1.15 to 1.52 Am [38].The waveguide fabrication techniques described thus far have been some of those forplanar waveguides. Planar waveguides, particularly those fabricated using compositionvariation, can be further processed to make channel waveguides; this can be done by severalmethods. These methods usually involve altering the refractive index distribution in thelateral direction. The simplest method is to etch ribs or ridges on the planar waveguide [29].Here, the effective refractive index is higher in the unetched regions (guiding regions) andlower in the etched regions (cladding regions) [39:p. 115]. Another method is to depositmetal strips on planar waveguides; in this case, the metal strips induce stresses in thesubstrate which in turn change the refractive indices through the photo-elastic effect [30].Recently, optical waveguides have been fabricated by thermally diffusing zinc (Zn)into GaAs-AlGaAs and GaAs-AlAs superlattice structures using silicon nitride (Si3N4) as amask [31,32]. The introduction of Zn into the superlattices promotes random alloy mixing(disorder) at relatively low diffusion temperatures (500°C - 600°C) [32]. The disorderedregions, i.e. , regions not covered by the mask, in combination with the presence of freecarriers (-1019 cm-3), have a lower refractive index then the regions which have been11protected by the mask. The disordered parts of superlattices then become the claddingregions for confining light in the lateral direction. Optical loss for these types of waveguidesis about 2 dB/cm at 10 = 1.3m [31].The planar waveguides used in this work utilized the method of composition variation;MOCVD was used here. The MOCVD-grown epitaxial layers were undoped to minimizethe free carrier absorption loss. The channel waveguide was formed by etching rib-likestructures on the epitaxial layers using a wet chemical etchant.2.3 Refractive Index of Al,paitsThe refractive index of AlxGai,As in the near infrared, as a function of the free spacewavelength 10 (in Am) and the aluminum mole fraction x, can be calculated using theempirically determined Sellmeier equation [40]1n(x,A 0) = Ft(x)+ ^B ^D(x)12,1 212,- C(X)(2.1)where A(x) =10.906-2.92x, B=0.97501, C(x)= {(0.52886-0.735x)2, x0.36; (0.30386-0.105x)2, x0.361, and D(x) =0.002467(1.41x +1). Figure 2.2 shows the dependence of therefractive index on the aluminum mole fraction for several wavelengths, calculated usingequation (2.1).For A0=1.3Ahm, the refractive index decreases almost linearly with increasingaluminum mole fraction. Using the method of least squares, the following expression canbe obtained to relate these two quantities12X0 (gm)0.850.901.061.151.303.00 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII0.00^0.10^0.20^0.30^0.40^0.50Aluminum Mole Fraction (x)Figure 2.2. Refractive index of AlGaAs as a function of the aluminum mole fraction andwavelength.13n(x) = 3.40 - 0.477x.^ (2.2)2.4 Metal Organic Chemical Vapour Deposition (MOCVD)For this work, the AlGaAs epitaxial layers in which the devices were fabricated weregrown by MOCVD. In this section, a brief account of this process is given [41].The GaAs substrates are placed on a susceptor, usually made of SiC, inside a cold-walled quartz tube reactor. The substrates are heated to a temperature in the range 650 to750°C either by RF induction or by infrared irradiation. Hydrogen gas (H2), bubblingthrough the highly volatile liquid organometallic compounds, trimethylgallium ((CH3)3Ga)and trimethylaluminum ((CH3)3A1), is used as a carrier gas to transport these compounds intothe reactor tube where they are mixed with arsine (AsH3). The organometallic compoundsthen pyrolyze above the heated substrate. The decomposition of the organometalliccompounds and the eventual formation of the AlGaAs thin film involve complex intermediatesteps. The following equation represents a very simplified irreversible net reaction [42:p.134H2(carrier)(1-x)[(CH3)3Ga] + x[(CH3)3A1] + AsH3 ^ AlxGai_xAs + 3CH4T=650 - 750°Cwhere x is the desired aluminum concentration in the final film. The composition of thealloy is determined by the relative initial partial pressures of (CH3)3Ga and (CH3)3A1 [42:p.132]. These pressures are controlled by the flow rate of H2 carrying the organometalliccompounds into the reaction tube. The uniformity of the layer is improved by rotating the14GaAs substrates during growth [42:p. 132].2.5 Electrooptic Effect of GaAs/AlGaAsThe phase difference between the interferometric arms of the integrated optics Mach-Zehnder interferometer results from the linear electrooptic effect [43]. Here, like GaAs,Al„Gai_xAs with x up to 0.36 is assumed to be a cubic crystal (the diagonal elements of thedielectric tensor have the same value, i.e., 11) and have the 743m symmetry [43:p. 83]. Thematrix for the electrooptic coefficients rij, in contracted notation, is (r41=r52=r63) [43:p. 229]000000000r41 0 00 r41 00 0 r41Here, the value of r41 is assumed to be the same as that for GaAs at 1.3 pm.When an electric field is applied, the index ellipsoid^( 1 ) x2^+ ( 1 ) y2 + ( 1 z2 =^ (2.4)^2^3(2.3)becomes15( 1 )n2x214. ( 1 )n 2v2.12+ ( 1 )n2723+ 2H1/1 2yz + 2E1-)4 n2XZ + 2(-1-)5 n2xy =61 (2.5)where x, y, and z are the optical axes (these are also the crystallographic axes [43:p. 127]).The changes in the coefficients are related to the applied field through the relation3= Er..E.Jn^j=iwhere i=1,2,..,6 and j =1,2,3 correspond to x, y, and z, respectively [44:p. 127]. Usingequation (2.3) and the fact that the crystal is cubic, the index ellipsoid with the E., Ey, andEz components applied, is [45]2 y2 Z 2—+—+— +2r41Exyz+2r41Eyxz+2r41Ezxy = 1.2^2^2no n n0^0(2.7)The presence of the mixed terms means that the axes of the perturbed index ellipsoid are nolonger parallel to the optical axes. It then becomes necessary to find a new coordinatesystem x', y', and z' such that equation (2.7) is in the formv•2X •2 Z •2+^+^= 1.2 2^2nx,^n, n.(2.6)(2.8)It will be shown shortly that the type of device discussed in this work uses only the Ez16component, i.e., the perpendicular field. An index ellipsoid in the form shown in equation(2.8) can be obtained by rotating the axes through 71-14 radians in the x-y plane.Generally the [100] direction is the preferred epitaxial growth direction. Thus GaAssubstrates with a growth plane perpendicular to this direction, i.e., the (100) plane, arecommonly used in fabricating optoelectronic devices. The following coordinate system isused to describe the linear electrooptic effect. The z axis is parallel to the [100] direction,the x axis is parallel to the [010] direction, and the y axis is parallel to the [001] direction.After rotating 7(14 radians in x-y plane, the x' axis is parallel to the [011] direction, the y'axis is parallel to the [OT1] direction and the z' remains parallel to the [100] direction.Figure 2.3a shows the coordinate system just described.The three principal refractive indices with Ez applied are then [45]wherenx, = n[011] = no - Anfly, = n[011] = no + Annz, = n[100] = no3An = 1—2 n0r41Ez.(2.9a)(2. 9b)(2.9c)(2.10)If light is propagating in the [OT1] direction, it can be seen that modulation occurs only forlight with a TE-like polarization, i.e., for light polarized parallel to the [011] direction.There is no modulation for light with a TM-like polarization, i.e., for light polarized parallelto the [100] direction. Figure 2.3b shows the nomenclature used in describing the different17polarizations. The modulator described in this thesis is based on the modulation of light witha TE-like polarization by an applied electric field in the [100] direction. Examples of electricfields applied in other directions and the corresponding electrooptic effects are given inReference [45].2.6 Operating Principles of Mach-Zehnder InterferometerThe structure of the optical modulator designed for this work is based on theintegrated optics version of the Mach-Zehnder interferometer. This structure consists of astraight input channel waveguide, a Y-branch power splitter, a phase shifting section with twoparallel channel waveguides, a power combining Y-branch, and a straight output channelwaveguide [46]. Figure 2.4 shows the structure just described. A description of itsoperation is given below.The injected light, travelling down the input section, splits into two equal componentsat the ideal input Y-branch. Each component, propagating down the channel waveguide ofthe phase shifting section, has its phase changed by the electrooptic effect, as a result of thevoltage applied to the electrodes deposited on the waveguides. When the two componentscombine in phase at the ideal output Y-branch, the fundamental mode is excited in the outputchannel waveguide and the light propagates through with minimum attenuation. However,when the two components combine with a phase difference of ir radians, since the channelwaveguide is designed to support only the fundamental mode, the optical energy is radiatedinto the substrate, resulting in a minimum output intensity. In general, the output18( b )Figure 2.3. (a) Coordinate transformation for the electrooptic effect. (b) Definition of theinput light polarization for the crystal orientation shown.19opticalnode--*direction ofpropagation--›direction ofpropagationopticalnodenIn InunoutputIntensitynode radiatesInto substrateb)nax Inunoutputintensitysingle nodewavegu ides(D Input channel waveguide(D 1-branch power splittercD phase shifting sectionpower conbining Y-branchoutput channel waveguldeFigure 2.4. Operating principle of a Mach-Zehnder interferometer. (a) The modes of thebranches combine in phase and excite the fundamental mode in the output section. (b) Themodes combine ir radians out of phase and excite a radiation mode in the output section.20intensity is a sinusoidal function of the phase difference between the optical wavespropagating in the interferometric arms.The increase or decrease in the refractive index due to the electrooptic effectaccording to equation (2.10) is usually on the order of 10-6 to 10-5, compared to the Aneff of10-3 to 10-2 necessary for the formation of single mode channel waveguides. If the channelwaveguides are properly designed, such that they are away from cutoff but are not yetmultimode, the small changes in the refractive index will have a negligible effect on theguiding properties, except in the introduction of the phase change, that accumulates after thelight beam has propagated over a significant distance.2.7 Bandwidth Limitations of Electrooptic ModulatorsThere are two types of electrode structures that can be used with the electroopticmodulators. The first uses lumped electrodes in which the electrode length is less than onequarter of the modulating field's wavelength. Here, the modulation bandwidth is limited bythe capacitance of the electrodes. The second uses travelling-wave electrodes, for broadbandoperation [47]. The bandwidth, in this case, is limited by the difference between the phasevelocity of the microwave propagating on the electrodes and the optical wave propagating inthe underlying optical waveguides, and by the loss in the electrodes [8].The output intensity of a modulator with travelling-wave electrodes is related to thetotal phase shift. Assuming the impedance of the electrodes is matched to the driving source,the expression for the total phase shift can be shown to be [48]21nn:rvpim^- 1)2+4e -casin2usm(2nft0+0)oG N^(aL)2+(2u)2(2.11)6 . tan_i(  e sin2u^tan_1( aL)e - " cos2u - 1 2u)IT fLino-nviu - ^where no is the effective refractive index of the waveguide, n ^the effective microwaveindex of the electrodes, r is the relevant electrooptic coefficient, V4 is the applied voltage,r is an overlap integral, L is the length of the modulating section, A. is the free spacewavelength, G is the interelectrode gap width, a is the loss in the electrodes, f is thefrequency of the applied signal, c is the speed of light in vacuum, and t0 is the time whenmodulation begins. The parameter r is defined as [8]r= ^2GifElE°ldA" VFL ffiki2dA(2.12)where Ei, is the applied electric field and E. is the optical field distribution. Often, tosimplify the analysis, a is assumed to be small so equation (2.11) becomes22,n —II,\ 2I 8 au' + 8 GaAs-(2.15)4)It n 3r41 V rL sinu0 ^p cosanfto +u). (2.13) XG^uIt can be seen that when no=n,„ the refractive index change, due to the modulatingvoltage, seen by a particular optical wavefront, is constant along the entire modulationlength. Therefore, an arbitrary electrode length can be used to reduce the drive voltage.However, in reality, the bandwidth is still limited by the loss in the electrodes.When noon,„ the total phase shift is reduced and even becomes zero for sufficientlyhigh frequency or long electrode lengths. A commonly used parameter is the optical 3dBmodulation bandwidth-electrode length product, in which the 0 drops to 50% from its valueat f=0. This is defined as [8]foL - ^2c n In -n I0 p(2.14)where fo is the 3dB bandwidth.For coplanar electrodes fabricated on GaAs, the microwave effective index can beapproximated, assuming the electric field travels equally in air and the substrate, aswhere cair and E GaAs are the dielectric constants of air and GaAs at microwave frequencies,respectively. Substituting e air = 1 and GaAs 12.9, ni, is equal to 2.64. This value has been6 = experimentally verified in Reference [21]. Using n0=3.40, the refractive index of GaAs at23A0=1.3pm, f0L-25 GHz•cm.Furthermore, it can be seen that no > ni,. This means that the phase velocity of themicrowave (c/ni,) is faster than the phase velocity of the optical wave (c/no). Hence, thebandwidth can be increased if a slow-wave electrode structure is used to reduce themicrowave phase velocity. Such a structure is described in the next section.2.8 The Slow-Wave ElectrodesA slow-wave electrode structure has been developed in our laboratory which issuitable for use with III-V semiconductor waveguides [21]. The structure is similar to a pairof coplanar strips except for narrow fins, or fins with pads attached at the ends, beingincorporated at periodic intervals. These additional features behave like capacitive elementsin which the increase in capacitance C per unit length is more than the decrease in theinductance L per unit length. Consequently, according to the relation vp = (LC), themicrowave phase velocity is reduced.The effective microwave indices of two electrodes fabricated in Reference [21] weremeasured to be 3.10 and 3.40, where the design values were 3.20 and 3.50, respectively.This work continues the development by fabricating a complete Mach-Zehnder modulator ofwhich slow-wave electrodes were deposited on AlGaAs channel waveguides. Figure 2.5 isa plan view of the modulator, where the electrodes are shown containing fins and pads.Figure 2.6 is a cross-sectional view of the electrodes and the underlying channel waveguides.24N 71?o' CMgO. cr)CD1-1 t■.)■-• • LAF r)', 2 i IA' N0''.(I) rD6,5 tc Daa-tiC. .CM5*-CD(4'—0wCco088.."ci).,N0.,5.,c D4RFt■.)(.111 1 I C I - 0 ave InSIO2buff erI ayerslow  waveelectrodesFigure 2.6. Cross-sectional view of the modulator.26Chapter 3Numerical Simulation3.1 IntroductionIn this chapter, numerical methods that were used in simulating the propagation oflight in AlGaAs-GaAs planar and rib waveguides are presented. These are the variationalmethod, the effective-index method, and the beam propagation method.The variational method is discussed in section 3.2. In section 3.3, variationalexpressions for the fundamental and the next higher order optical modes are given for theplanar waveguide. The beam propagation method is discussed in section 3.4. In section 3.5,equations for the effective-index method, which were used to reduce the dimension of therefractive index distribution, are given. In section 3.6, and its subsections, the variousnumerical results for the planar waveguides, the rib waveguides, and the couplingcharacteristics of two parallel rib waveguides are given. In section 3.7, an expression forcalculating the overlap of the electric field, produced by the coplanar electrodes, and theoptical field, propagating in the rib waveguides, is given. Finally, a chapter summary isgiven in section 3.8.The layer structure of the planar waveguide was designed to result in a buriedGaussian refractive index profile in the depth direction. By burying the waveguide,scattering loss, caused by the roughness of the etched walls, is reduced. This is because theetched walls are mainly in the top cladding layer. A graded refractive index profile furtherreduces the optical loss as scattering at layer boundaries is eliminated. A Gaussian profile27was used because it is both a smooth function and easily modelled.The variational method was used to obtain the parameters of the Gaussian profile; itwas intended that only the fundamental IE0 mode would be supported by the planarwaveguide. The refractive index profile would then fix the aluminum mole fraction in theepitaxial layers; here we used equation (2.2). For a particular refractive index profile, thevariational method yields an approximation to the optical field distribution which can be usedin calculating the overlap integral.Once ribs were etched on the planar waveguide, to form channel waveguides, we useda function, that was the sum of two error functions, to model the step-like effective refractiveindex profile in the lateral direction. Once again, the variational method was used todetermine the width and height of the rib so that only the fundamental mode would beguided.The beam propagation method had several applications in this work. It was used todetermine the thickness of the bottom cladding layer of the planar waveguide to ensure thatpropagation loss, due to leakage of the guided mode into the substrate, would be negligible.It was also used to calculate the coupling between two parallel rib waveguides, as a functionof the rib height and waveguide separation, for a predetermined propagation length.To increase the speed of the beam propagation method, the effective-index methodwas used to reduce the two-dimensional refractive index distribution of the rib waveguide toan one-dimensional effective refractive index distribution.Finally, an expression for the overlap integral r was obtained by integrating theproduct of the electric field, produced by conventional finite width coplanar electrodes, and28the optical field, obtained from the variational method.3.2 Variational MethodThe variational method, which has been shown capable of calculating the optical modeprofile for channel waveguides fabricated in LiNb03 [49], was used to obtain the approximateoptical field distribution for the fundamental mode and to predict the emergence of the nexthigher order mode. Here, it is assumed that x is the lateral direction, y is the depthdirection, and z is the direction of propagation.The exact optical field distribution in the transverse plane, i.e. , x-y plane, satisfiesthe vector wave equations [50:p. 595]V ,2 t + (n2(x ,y)lc - 132)E' = -V, [V,1nn2(x,y)-Ed^(3.1a)v t2ii t (n2(x,y)ko2 _ 132 )gt. (vtxiit) x vrinn2(xy)^(3.1b)where ko (= 27r/10) is the free space wavenumber, n(x,y) is the transverse refractive indexdistribution, Et and 1-4 are the transverse components of the electric and magnetic fields, Vtand Vt2 are, respectively, the transverse grad and Laplacian operators, and 13 is thepropagation constant in the z direction. In principle, either Et or Fit needs to be solved sincethe other quantity can be obtained through Maxwell's equations [51:p. 61]. Here, only thevector wave equation for the transverse electric field is considered.Using the coordinate system defined above, the mode is TE-like when the electric29field is polarized primarily parallel to the x axis and the mode is TM-like when the electricfield is polarized primarily parallel to the y axis. Hence the vector wave equation can bewritten asa2E„ a2Ex alnn2^2k-2 -+n^0^132 Ex =0^(3.2a){a2inn2l Exax 2 ay 2 ax2 ax^axfor the TE-like mode (or Ex mode) anda2Ey a2Ey {a2inn2 +alnEy ahm2 +n2k02 _ p 2} Ey=0aX2 aY2^aY2^aY aY(3.2b)for the TM-like mode (or Ey mode). The subscript on E indicates the polarization directionof the dominant electric field component.However, for weakly guiding waveguides with gradual refractive index distributions,which are commonly encountered in integrated optics, the partial derivatives involving ln(n2)can usually be assumed to be negligible in equations (3.2a) and (3.2b), yielding the scalarwave equation82* + (2211 [n2(x,y)le, - 0141 = 0ax2^ay'(3.3)where * can be either Ex or E.Premultiplying equation (3.3) by tjr* and integrating the result over the entire x-y30plane, the following expression for 82 can be obtained by applying Green's theorem11-{P112 + (A)2 - e,n2(x,y)*ax _f-J-0,4r2dxdyHere, it is assumed that * vanishes at infinity [52].Equation (3.4) has two important properties. First it is stationary with respect to firstorder variations of the function *. As a result, when * changes by 6*, which is sufficientlysmall that only the first order terms need be considered, the corresponding change in 132, i.e.6132, is zero [53:p. 156]. Equivalently, this means that the errors incurred by choosing a trialfunction, that is not an exact solution to the wave equation (3.3), are second order errors.The second property is that the exact solution to equation (3.3) gives the maximum value of132• Any other trial functions yield values that will be less than or at most equal to the onefor the exact field distribution [53]. Hence, if * is chosen to be a function of severalparameters, these parameters may be optimized to maximize 132. One then obtains the bestpossible 132, which is a lower bound to the exact 132, for the class of trial functions used.Obviously, this method produces better values for 132 if lir closely approximates the actualsolution to equation (3.3) or if a large number of optimization parameters are used [53:p.157].3.3 One-Dimensional Scalar Variational ExpressionsThe one dimensional scalar variational expression in the y direction can be readily(3.4)31obtained from equation (3.4) by setting all derivatives with respect to x to zero, i.e.,'A)2 - k2n2(y)11,2ay^°f :ip2dySuitable trial functions must be chosen to approximate the two lowest order mode profiles.For this work the trial function for the fundamental mode is chosen to be the Gaussianfunction2^2 2^•-P*(y)oc e^Y^mze(3.6)where wy is a width parameter and 13D1 is the propagation constant. The subscripts D and1 stand for the depth direction and one extremum in the field profile, respectively. Themagnitude of equation (3.6) is plotted in Figure 3.1.This function has been commonly used in approximating the fundamental mode inoptical fibres and waveguides [54-56]. It is used here since the refractive index distributionis symmetrical about a peak value that is well below the surface of the sample. Thediscontinuity of the refractive indices at the surface is assumed to have negligible effect onthe evanescent fields of the propagating mode.The planar waveguide should be single mode. Hence, a variational expression for thenext higher order mode is required to check for the onset of that mode. Two necessarydY^ (3.5)32Figure 3.1. Plots showing the functions used for approximating the optical fielddistributions of the fundamental and the next higher order modes.33criteria are that there should be two extrema in the field profile and the function should decayto zero at infinity. A function suitable for this purpose which is also orthogonal to theGaussian function, resembling the orthogonal nature of the propagating modes, is the firstasymmetric Hermite-Gaussian function1(_ — —2 2Y^W^-JP D2ZYWYThe subscript 2 stands for two extrema in the field profile. The magnitude of equation (3.7)is also plotted in Figure 3.1.Substituting equations (3.6) and (3.7) into equation (3.5) the following variationalexpressions are obtained:Y220.5^e,f :n2(y)e wY dyn 2PDI =W2 11-1 Wfor the fundamental mode, and2 -Liko2f -n2(y) .„ e wY dyII 2^1.5PD2 -W2 (Ft /2)Wy(3.7)(3.8)(3.9)34for the next higher order mode.Similarly, variational expressions in the lateral direction can be obtained by replacingy by x, wy by wx, Bm by BLI , and BD2 by 8L2, where the subscript L stands for the lateraldirection.3.4 Beam Propagation MethodThe beam propagation method is capable of simulating optical waves propagatingthrough a device structure with either longitudinally variant or invariant refractive indexprofiles. It has two applications in this work which are described below.The planar waveguide is fabricated on a GaAs substrate. Pure GaAs has the highestrefractive index (nGaAsz3.4) in the entire layer structure. Therefore, light propagating in theplanar waveguide can leak into the substrate if the bottom cladding layer is not sufficientlythick [38]. The beam propagation method can be used to calculate the amount of lightleakage into the substrate as a function of the cladding thickness.One of the design criteria for the Mach-Zehnder interferometer is that the separationof the interferometric arms should be sufficiently far apart so that light propagating in onearm will not couple into the other arm. Usually, when two parallel waveguides are separatedby a finite distance, the optical field propagating in one of the waveguides will eventuallycouple into the other waveguide; due to the evanescent fields extending across the gapbetween the two [57]. To determine the coupling characteristics, a detail knowledge of theoptical field distribution is needed. One might be tempted to use the Gaussian function tocalculate the amount of coupling. However, the Gaussian function is inappropriate for this35type of calculation since it underestimates the evanescent field of the guided mode; the actualfield decays as exp(- lx I) instead of exp(- I x12), as assumed by the Gaussian function,outside the guiding region. The beam propagation method can be used to calculate theamount of coupling between two parallel rib waveguides.The beam propagation method was introduced in 1978 by Feit and Fleck to solve lightpropagation problems in graded index optical fibres [58]. It had been used in applicationssuch as solving for propagation of laser beams through the atmosphere [59]. The method isrestricted by the condition that reflected waves can be neglected, i.e., unidirectionalpropagation is assumed, and that the refractive index perturbation is small so that the scalarwave equation is valid [58]. The beam propagation method takes into account the couplingof guided modes with the radiation mode. Recently, semivectorial beam propagation methodshave also been developed [60,61]. For this work, a beam propagation method based on theFresnel approximation to the scalar wave equation was used.In the beam propagation method, the electric field distribution, in a particulartransverse plane of a waveguide structure, is calculated from the electric field distribution ina previous transverse plane. The relationship between the electric fields at these two adjacentplanes, i.e., planes perpendicular to the z axis and separated by distance Az, is [58]E(x,y,z+Az)=exp{  jAz Vlexp{-1L' fz+Ain(x,y,z)-noldz14n0k0^2n0 zexp{ jAz Vt.} gx,Y,z) exPfinokobal + 0(Az)34noko(3.10)36where '7 is the transverse Laplacian operator, and no is the refractive index of the cladding.Derivation of equation (3.10), starting with the vector wave equation for the electric field,is given in Appendix A. Equation (3.10) is equivalent to stating that the actual continuousrefractive index distribution in the longitudinal direction has been replaced by an array ofinfinitesimally thin lenses separated from each other by Az and immersed in a homogeneousmedium with the refractive index of the cladding. As light passes through this system, it isconstantly refocussed by each lens with the appropriate phase change [62].Equation (3.10) can be implemented using the Fast Fourier Transform (FFT) whereeach propagation step requires two such operations. This implementation has severaldisadvantages. The first one is that a large amount of memory is required, i.e., the two-dimensional grid must be large and dense enough to adequately describe the evanescentfields. The second one is that when propagating over ten thousand steps or more, i.e., adevice having a length of one centimetre or more with a propagation step of lbtm, two FFToperations per step, with a large number of grid points, would cause the simulation time tobe excessively long. Consequently, an efficient and stable Split-Step Finite Difference BeamPropagation Method (FDBPM), was used [63]E(x,y,z+ Az) = D xDyexpl- yn—Lii2 (xg ) -noldzID yD xE(x,y,z)ei-jn ok oh, z)3ik°0 z+°I(3.11)37whereDx -^1+  -jA z a2^1+  JA z a2^8n0lc0ax2 8noko ay2^^and D - ^ .Y -jAz a2^1 jAz a2 1^8n0lc0 8x2 8n0lc0 ay2To further reduce the simulation time, the effective-index method can be used toconvert the two-dimensional transverse refractive index profile into a one-dimensionaltransverse profile, i.e., only the lateral direction. Hence, equation (3.11) will be a functionof x and z.One problem associated with the beam propagation method is that the reflection of theelectric field back from the grid boundaries into the solution region causes ripples in theresulting field distribution calculations [62]. One solution to this problem is to employ anabsorber at the grid boundaries. A commonly used absorber is [62] 1^IXI<IX al[1+ COSI* X 11 IX al<IXI<IX blX — Xa b0^IX bl<IXI<IXRIabsorber(x) = (3.12)where xR is the coordinate of the grid boundary, xi, is the inner edge of the absorber and xbis the outer edge.383.5 Effective-Index MethodWhen ribs are etched on the planar waveguide to provide lateral confinement, theresulting channel waveguides have two-dimensional refractive index distributions. Theeffective-index can be found by replacing each point on the sample with a planar waveguidehaving an effective refractive index. This reduces the refractive index distribution to afunction of only the lateral direction. When using this technique, it is assumed that radiationinto the top and bottom boundaries of the waveguides is negligible.The mode dispersion equation for the TE-like mode is [64]2k0fY21/ 2^2yi n (y)-neif dy + 24:01 + 242 = 2m7c (3.13)where 201 is the phase of the reflection coefficient at turning point yi , 202 is the phase ofthe reflection coefficient at turning point y2 ( > y1), and neff is the effective index for them'th mode. The turning points, which are the boundary points separating the oscillatory andevanescent regions of the fields, are defined byn2(y1) - ne2if = 0^j = 1,2.^ (3.14)Turning point y2 is usually located far below the substrate surface where the refractive39index profile is continuous where it can be shown that 202 is equal to -7//2 [64]. However,there are two possibilities for turning point yi . The first one is the unburied mode, i.e., yiis at the surface. The corresponding phase change is [64] I ^2^224= _2tan-i neff - ns)1n2 —n2\surf eff(3.15)where n8 is the refractive index of the superstrate and nsurf is the refractive index at thesurface. The second one is the buried mode, i.e., yi is below the surface. Thecorresponding formula for 201 is [65] eff_ n 2(y) di^(3.16)IT1 Vn2 _ 2 _ii2 22401= — -i —2tan1- , effn sn ern surf exp( —2k,, vne2ff_ns2 +Vneir2 _n2siGenerally, equation (3.13) cannot be solved analytically. The value of neff has to besolved numerically by the trial and error method.403.6 Simulation ResultsIn the following subsections, simulation results for determining the layer structure ofthe planar waveguide, the widths and heights for single mode rib waveguides, and therequired separation between the interferometric arms of the Mach-Zehnder interferometer aregiven for the case of 1.0=1.3Am. The programs were written in PASCAL and ran on a 33MHz 486 desktop computer. Most of the calculations could be performed quickly. Thebeam propagation method, in spite of the simplifications made, remained the mostcomputationally intensive portion; propagation of 10000 steps took approximately 45 minutes.3.6.1 The Planar WaveguideThe choice of the Gaussian function as the graded refractive index profile enablesanalytical expressions to be obtained for BLI and 42. The Gaussian refractive index profilecan be expressed asn(y) = + (npeak - n)e ‘di (3.17)using the coordinate system defined in section 3.2, where nc is the refractive index of thecladding layer, peak is the maximum refractive index in the graded region, and d is aparameter controlling the thickness of the planar waveguide. Here it is assumed that theposition of npeak corresponds with the origin of the y axis.Substituting equation (3.17) into equations (3.8) and (3.9), the following analyticalexpressions are obtained for Bin and i3i2:412^0.5^2 2^2n nk2dcDI — 2^ + ncico +WY 1/W2 +d2(3.18)and2D2 =1.5^2 2 2n Am/422 + rick +  c^°3WY 2(Wy2 +d2)(3.19)where An = npealc-nc and all other quantities have been specified previously. The onlyapproximation made in obtaining equations (3.18) and (3.19) is n2(y)2ne An exp{-(y/d)2}.The value of Ilpeak was set to 3.261, corresponding to an aluminum mole fraction of0.30. Then the width parameter wy was varied to maximize both BLI and BL2 for differentvalues of ne and d. The value of nc and hence the aluminum mole fraction of the claddinglayers needed to be chosen judiciously. The difference between the minimum and maximumaluminum mole fractions in the epitaxial layers should be sufficiently large so that they canbe differentiated during growth. The value of d should not be large, in order to keep thelayers thin. Furthermore, having a large d, a rib with a deeper etch would be needed toprovide lateral confinement; this might introduce problems during fabrication of theelectrodes.The values of ne used were 3.242, 3.237, and 3.233, corresponding to the aluminummole fractions of 0.34, 0.35, and 0.36, respectively. A plot of the fundamental mode size42(1/e intensity at full width), as a function of d, for different values of the aluminumconcentration in the cladding layer, corresponding to different values of nc, is shown inFigure 3.2. All three sets of calculations show the mode size goes through a minimum. Atthe minimum, the optical field of the fundamental mode is mostly confined within the planarwaveguide. This is desirable since, first, the evanescent tails of the optical field would havetheir minimum interaction with the electrodes and, second, a thinner bottom claddingthickness would be required to prevent leakage of guided light into the GaAs substrate. Forthis work, the parameters chosen for the Gaussian refractive index profile were II, =3.237(aluminum concentration =0.35) and d =0.70Am. The corresponding wy for the fundamentalmode is 0.855Am.The emergence of the next higher order mode was checked by setting nc =3.237 andIlpeak =3 .261, and maximizing 42 by varying wy for different values of d. It was found that81232 could not be maximized for d < 1. lm, indicating the absence of the next higher ordermode in this range. To summarize, the parameters for the refractive index profile of a singlemode planar waveguide were nc =3.237 (aluminum concentration =0.35),n--peak =3.261(aluminum concentration =0.30), and d =0.70Am. The linear relationship between therefractive index and the aluminum mole fraction assured that the aluminum concentrationprofile would also be a Gaussian function.Actually, the limited control available using MOCVD means that, in fact, only simplestructures such as a linearly-graded aluminum concentration profile can be grown.Therefore, it was necessary to approximate the Gaussian refractive index profile by severallinear segments. Figure 3.3 shows the Gaussian refractive index profile (dashed curve) being43Figure 3.2. Calculated fundamental mode size as a function of the refractive indexprofile width parameter d for three different values of aluminum mole fraction in thecladding layers.44fitted to four linear segments (solid curves), using the method of least squares. Also, in thecalculations, the Gaussian refractive index profile was assumed to extend to infinity.However, the actual epitaxial layers should have finite thicknesses. Therefore, the topcladding layer was set to 0.70Am with a constant aluminum mole fraction of 0.35. Then thealuminum mole fraction decreased from 0.35 to 0.30 in 1.3,am and increased back to 0.35in 1.3Am. Finally, the bottom cladding layer had an aluminum mole fraction of 0.35 andits thickness was determined using the beam propagation method.First the eigenmode of the planar waveguide, without including the GaAs substratein the refractive index profile, was generated by propagating an arbitrary mode profile alongthe imaginary z axis, i.e., the jz axis. This was done by replacing Az by jA z in the beampropagation expressions [66]. Consequently, the lowest order mode would growexponentially instead of oscillating in the z direction. After many propagation steps thelowest order mode would become the dominant mode.For this work, the eigenmode was obtained by propagating a Gaussian function, i.e.,the arbitrary mode profile, with a width parameter wy=2/im for 1000 steps. Both therefractive index profile and the optical field profile were discretizetl with 1000 points. Thespacing between the points was 0.1,um. The propagation was performed in 1,.m incrementalsteps. As a comparison, Figure 3.4 shows the intensity profile (wy =0.8554m, solid curve)obtained from the variational method and the intensity profile obtained from the beampropagation method (dashed curve). The curves are normalized, so that the areas beneaththe two are the same. The refractive index profile used for the FDBPM corresponded to thelinear gradients of the actual layer structure, as shown in Figure 3.5.450.32 =a)0.31 =0.30 =z0.29^- ^1111111111111111111111111111111111111^II I—4 —3 —2 —1^0^1^2^4Vertical dimension (gm)Figure 3.3. The Gaussian distribution of the aluminum mole fraction (dashed curve) inthe planar waveguide's depth direction. The solid curves are the linear approximations.46(f) 0.80 —a)=4-^0^-L_0_>,0.60 —-1--,--cnC^-a)+-,c• — 0.40 —-0a)N --..-6E 0.20 —L_0Z^-\0.00 IIIII^IIIII^iiiiiiii—6.0^—4.0^—2.0^0.0^2.0^4.0^6.0Vertical dimension (pm)I\1.00 —_Figure 3.4. Comparison of the Gaussian intensity profile (solid curve, wy =0.85511m)with the intensity profile obtained by the beam propagation method (dashed curve). Therefractive index profile used by the BPM corresponds to the linear approximations shownin Figure 3.5, but does not include the refractive index of the GaAs substrate.473.5003.400a)-o_3.3000141=000(2 3.2001n1 = 1 (air, not to scale)= 3.237 AlGaAs, x=0.35n3 = 3.261 AlGaAs, x=0.30)n2^ )n4 = 3.404 GaAs)3goirI^I^I^111111 11 11^I^IIIFF I^I I^II^In4—6 00 —4.00 —2.00^0.00^2.00^4.00^6.00Vertical dimension (gm)Figure 3.5. Refractive index profile of the actual layer structure, i.e., the linearapproximation shown in Figure 3.3, with the refractive indices of the superstrate andsubstrate.48The eigenmode obtained above was then propagated in a planar structure includingthe GaAs substrate for various cladding thicknesses. The total propagation length was 3cm,and the evolution of the optical field was stored at lcm intervals. The propagation loss wascalculated using the equation, propagation loss(dB/cm) = -101og10 (P1/1314), i = 1,2,3, where13; is the optical power at the i'th cm and 134 is the optical power at the (i-1)'th cm. Figure3.6 shows the propagation loss, averaged over the 3cm length, as a function of the claddingthickness. The solid curve is an exponential fit, indicating the propagation loss would droprapidly with increasing cladding thickness; such behaviour is well known [67]. It can be seenthat for a cladding thickness of 3.2Am, the propagation loss due to light leaking into thesubstrate should be negligible. We used a thickness of 3.2Am for the bottom cladding layer.3.6.2 The Rib WaveguideRibs etched in the epitaxial layers provided lateral confinement for the guided light.For our waveguide structure, the effect of the etch is such that the distance between the peakof the refractive index profile and the surface of the wafer is decreased, reducing theeffective refractive index from that of the unetched region. The effective index method wasused to calculate the effective refractive index along the surface of the sample as a functionof the location of the peak beneath the surface. Figure 3.7 shows a plot of the effectiverefractive index as a function of the peak index beneath the surface (dashed curve), calculatedusing equations (3.13), (3.15), and (3.16).The discontinuity is caused by the fact that, as the peak moves closer to the surface,the turning point y1 also moves closer to the surface, and the expression given by equation491 2.0 ---__(I)Cl)0 6.0 -c000_0L._^_Ci- 2.0 ---_0.0 ^0.5-i^I^1^I^i^I^1^I^'^I^1^11.0 1.5 2.0 2.5 3.0 3.5Cladding thickness (gm)Figure 3.6. Calculation of the propagation loss, due to leakage of guided light into theGaAs substrate, as a function of lower cladding layer's thickness. The solid curve is anexponential fit to the calculated data.503.248 —3.246 —3.244 —3.242 —cu3.240 —3.238 —3.236 —3.234 ^0.00^0.50^1.00^1.50^2.00^2.50^3.00Distance of npeak from surface (gm)Figure 3.7. A plot of neff as a function of the separation between the location of npeak andthe surface of the sample. The dashed curves were calculated using the effective indexmethod. The solid curve is a sixth order polynomial fit to the dashed curves with datanear the discontinuity removed.51(3.15) becomes less accurate for yi in the proximity of the surface. To circumvent thisproblem, data near the discontinuity were discarded and a polynomial of degree six was fittedto the remaining data, i.e., where they were known to be most accurate. The polynomialexpression is6n etf(1) = E ai=o(3.20)where 1 is the distance between the peak index and the surface and a0=3.24, al =-3.09 x10-2,a2=7.96 x10-2, a3=-7.38X102, a4=3.34 x10-2, a5=-7.46 x10-3, and a6=6.54 x10-4. Theunits on the ai's are such that neff remains dimensionless. Equation (3.20) is plotted as thesolid curve in Figure 3.7.The etched ribs were assumed to be sloped 54° with the horizontal. The effectiverefractive index of the sloped region was calculated using equation (3.20). It should also bemade clear that, for this work, the width of the rib waveguide is defined as the widthmeasured at half height.In order to obtain the variational expressions for the fundamental mode and the nexthigher order mode, a continuous function has to be used to model the sloping and the step-like natures of the lateral effective refractive index profile. Here, a function composed oftwo error functions was usedn(x) = nb + An f(x)f(x) = —1 terf [A(x +^erf [A(x-B)112(3.21)52where nb is the effective substrate refractive index, An is the maximum difference in theeffective refractive indices of the unetched and fully etched regions, A and B are theparameters relating to the slopes of the side walls and the widths of the ribs, respectively.The values of A and B were obtained by fitting equation (3.21) to the effective refractiveindex profile calculated using equation (3.20), using the method of least squares. Figure 3.8shows one such fit for a 4Am wide and 1m high rib waveguide. The values of A and B are3.24 and 2.16, respectively. Once the fitting was done for a rib waveguide of a certainheight, the width of the waveguide could be specified by parameter B.An analytic expression could be obtained for 13?2^0.5^22^20/1 =^+ nblco + 2nbAn lc,pfw 21 I w2 ^1 N B2 A2B2 (3.22)(see page 137 of Reference [68]). The mode size, as a function of the width for threedifferent rib heights is shown in Figure 3.9. It can be seen that for rib heights of 0.7 and1.0Am, the mode size also goes through a minimum. These plots have a smaller curvaturethan those shown in Figure 3.2. This is mainly due to the much smaller effective refractiveindex difference between the guiding and cladding regions.The expression for 132 was more complicated since an analytical expression could notbe obtained. However, the integrals are of the forms which lend themselves to be evaluatedusing Gaussian quadrature [49]. Substituting equation (3.21) into (3.9) and after somemanipulations533.246 — 3.245 —9— 3.244 —a)3.243 —3.242 ^—4.0 —3.0 —2.0 —1.0 0.0^1.0^2.0^3.0^4.0^Lateral dimension^m)Figure 3.8. The lateral effective refractive index profile of a 4ilin wide and lAm highrib waveguide (dashed curve). The solid curve is the error function approximation, i.e.,equation (3.21), with A=3.24 and B=2.16.54Ec_c-I-,13r--a)-co10.00 -9.00 =8.00 -7.00 -•6.00 - •5.00 -_ • • ••• •••4.00 - •_ • •3.00 -Rib^height2.00 - • 0.7um• 1.0,um-1.00 - • 1.2,um0.001 2 3 4 5^6 7^8Width of rib waveguide (,um)Figure 3.9. Calculation of the fundamental mode size as a function of rib width, forthree different rib heights.55linB2 2^1/VS V 712 /-^ erf ^^[\l2wx ^1e -x2dx + 8Annb 1B2 A2B22^1.5^2 2Pm = -^+ nbko2wx(3.23)_ B22^8Annbk^w2oe x+ ^f- erf(wxAx)sinh —x2B 2e _x2dxFr wx_ B2^w2^16Annic xoBe Ilni x erf (w xAx) cosh '''xWx■FIEWk^XThe two integrals can be calculated as^f-^ 2Betf(wxAx)sinh —2Bx).x2ex2dx Pi E vi eif(wxAxdsinh —x.-- (wx^•^(wx i)1andf- x erf (w xA x) cosh (x)e-x2dx -wx^1 I i eif(wilx: ) cosh ( 1/2-xi 1.The nodes xk and xk' are the roots of the generalized Laguerre polynomial LM(x2) and theHermite polynomial Hn(x), respectively. Here k is used as an index. The weights Vk andUk are56(m+ 1)!/ 42m(/7-2)!12[/,‘\„-ii2(x12,)IN 2 ) i^2/ 1 and11+InVi1 7 k — 2[H n + i(X k' )]2where Vk is taken from [49] and Uk is taken from [69]. The derivation of Vk is given inAppendix B. The values of m and n determine the accuracy of the integration. It was foundthat m and n of 40 were required to give three significant figures for the width parameter.Using equation (3.23), it was found that the width for which the next higher ordermode cut in were approximately 7, 4, and 3/2m for rib heights of 0.70, 1.0, and 1.2thm,respectively. However, waveguides with narrow widths would be difficult to control duringfabrication, therefore, the widths of the waveguides fabricated for the actual devices werechosen to be 4, 5 and 6m (the 6pm wide waveguide would possibly be multimode). Butit was hoped that the higher order mode would only be very wealdy guided.3.6.3 Coupling Characteristics of Parallel Rib WaveguidesWhen two parallel channel waveguides are separated by a finite distance, such as inthe case of the modulation arms of the Mach-Zehnder waveguide, optical energy in one arm-257will couple to the other arm and then back to the original arm with a spacial periodicity equalto the coupling length. In this section, the FDBPM was used to determine the couplingcharacteristics of two parallel rib waveguides.The first step was to obtain the effective refractive index profile for the etched ribwaveguide with sloped side walls. The spacing between the grid points was 0. lm, and1000 grid points were used. The inner edge of the absorber was at +35,um and the outeredge at +501.Lm. The effective refractive index of each grid point was calculated usingequation (3.20). The eigenmode U0(x) of a single rib waveguide was generated bypropagating a Gaussian beam (w=4/.Lm) along the imaginary z-axis. Figure 3.10 shows theeigenmode obtained for a 4Am wide, high, rib waveguide. The dashed curve is UO2 andthe solid curve is the square of the Gaussian field (wx =2.32Am) calculated from thevariational method. Again, the curves are normalized so that the areas beneath the two arethe same.Then a new effective refractive index profile consisting of two rib waveguides wasgenerated to model parallel waveguides. This effective refractive index profile was invariantin the z-direction. The waveguides were denoted as #1 and #2 and are shown in Figure 3.11.Only waveguide #1 was excited with U. at the input. The eigenmode was then allowed topropagate for 2cm, the modulation length of the Mach-Zehnder interferometer, in 1mincremental steps.The power coupled to waveguide #2 at the end of the propagation was calculatedusing [70]58[III15.01^1^1^1^I^I i^1^1^1^1^1^1^1^1^1^1^1^1^T I^I^I^1—15.0 —10.0^—5.0^0.0^5.0^10.0Lateral dimension1.00 —-cn 0.80 —a)47:2>,0.60*(7)a)0.40-0a)E 0.2000.00 ^Figure 3.10. Comparison of the Gaussian intensity profile (solid curve, wx=2.32ihm)with the intensity profile obtained from the beam propagation method (dashed curve) forthe case of a 4Am wide, 11.1m high, rib waveguide.59L wavegu 1 deI ^separation2cnInpute I gennodenef fFigure 3.11. Model used for calculating the coupling characteristics of two parallelwaveguides, as a function of their separation.H60P2 % coupling -Pi+P2P 1 = If IE(x)U:(x+ i rP2 = If IE(X)U4X- ifwhere P1 is the optical power in waveguide #1, P2 is the optical power in waveguide #2, E(x)is the sum of the optical fields in both waveguides at the end of the propagation, and s is theseparation of the optical fields measured between the centres of the waveguides.Calculations were done for the 44m and 5p,m wide rib waveguides. In each case, thecoupling characteristic of three different rib heights was investigated. The results for the4/Am waveguide are shown in Figure 3.12, and for the 5/Am waveguide in Figure 3.13. Thewaveguide separation shown is measured between the inner edges at half height. This wasdone so that the results presented here could be used to specify, directly, the dimensions onthe waveguide maskThe calculated data were fitted to a decaying exponential functions. Such behaviourhas been shown to exist in coupled slab waveguides [67]. It is clear that for small ribheights, large waveguide separations would be needed due to weak lateral confinement of themodes. It can also be seen that the waveguide separation required to achieve a certain powercoupling does not vary linearly with the rib height. Deeper etches have less effect on thewaveguide separation. This corresponds with the fact that the lateral mode size does not varysignificantly when the etch height goes from 1.0Am to 1.2/Am, as shown in Figure 3.9.611 00 —_(N^__90 —a)-0 0•— 8 —D0)0> 70- Rib height• 0.7p,m• 1.0/im• 1.2p,m00- I-)60 —50 —40 —30 —20 —10 —-I*0^ ilIIIIII^o 4^8^12^16^20^24^28Waveguide separation (gm)Figure 3.12. Coupling characteristics of two parallel rib waveguides with 4m width.62Rib height• 0.7,um• 1.0,um• 1.21.cm100-(N^_-90 -a)P•5 80 -cr)a)> 70 -C60-0+-,P 50:a)D0o30 -L_a)20-010 -W -0 o^l i^III,^I4^8^12^16^20^24Waveguide separation (pm)Figure 3.13. Coupling characteristics of two parallel rib waveguides with 5/.1m width.63Figures 3.14a, b, c, and d show the three-dimensional plots of the coupling process for twoparallel waveguides.Experience in fabrication has shown that it should be possible to deposit electrodeson top of a lpm high rib waveguide. Thus the height of the waveguides on the actualdevices was chosen to be lpm. Based on this choice, the waveguide separations on the maskwere chosen to be at least 12pm for the 4pm waveguides and 11pm for the 5p,m waveguides.According to the curves shown in Figures 3.12 and 3.13, the coupling between the twoparallel waveguides should be less than 5%.3.7 Calculation of rThe overlap integral r (see equation (2.12)), is used in calculating the half-wavevoltage. In this section, an expression for r is derived by assuming the modulating electricfield is produced by a pair of conventional coplanar electrodes of finite width, instead ofelectrodes having the capacitive loading elements. Furthermore, it is assumed that theelectrodes are flat on an imaginary plane which is flush with the surface of the Si02 bufferlayer on the rib waveguides. The optical field distribution used in the calculations is obtainedfrom the variational method described in section 3.6, with Gaussian distributions in both xand y directions. The effect of the oxide layer is taken into account by offsetting theelectrodes from the surface by the thickness of the oxide layer.6420—250 -- NV. —N.76 — 6 \_^,c., c!,,c, c, ea c- I-Ax"'" r- c' \(a)a^ .2s3a'1/...^ V:.c7., 42, (.. lj'<‘'. -- 2.^....^ 6 \_4a *c.-..e'`."‘ —tm \(b)Figure 3.14. Three-dimensional plots showing the coupling characteristics of two 4mwide parallel channel waveguides, for a propagation length of 2cm. The separationbetween the waveguides is (centre to centre) (a) 12Am, (b) 14Am, (c) 16Am, and (d)20Am.65Cif- NEP^-<a\_-e%(c)Co'•■•••••a •- ca. )t- "Figure 3.14. Continued.66The x and y components of the electric fields, produced by a pair of coplanarelectrodes of gap g and width W, can be expressed asElL -^1 Re ^1 gK(k) \1142(()1z 2 I^gijlz?g^iEl ^1 1 E= Im ^gK(k)^I^ 2 2 2^ 21 -kliZ) 1 -(-g Z)41(3.24a)(3.24b)z = x - iy ,^k-  g 2W+gwhen normalized to the applied potential [71]. E' ^the x and y components of themodulating electric field produced by the electrodes, and K is the complete elliptical integralof the first kind. The above expression is for an isotropic material, i.e., ex = ey, a class ofmaterial to which GaAs belongs. A cross-sectional view of the modulator, with theelectrodes modelled as described above, is shown in Figure 3.15.Here it is assumed that the most important contribution of the modulating electricfields is produced by those sections of the electrodes placed on top of the waveguides.Furthermore, the integrand of the overlap integral is also weighed by the optical intensityprofile which has a rapidly decreasing Gaussian nature in directions away from the centre ofthe optical mode, further diminishing the effects of the electric fields outside the waveguideregions.67Magi nary planesupporting the^electrodeelect rodeIopt i ca Inode\I'Ysgox I delayerY/--> XWFigure 3.15. Cross-sectional view of the modulator with the electrodes modelled as apair of conventional coplanar electrodes, as described in the text.68According to Figure 3.15, EE3,1, i.e., the field perpendicular to the surface of thesubstrate, was used to produce the phase changes. Therefore, r could be calculated usingX—Xs12,1 y—ys121f fe-^Y 141 -k1-2^--(-2Z)2)1 - -2 C1X4^g ^g r -n K(k)w xw y(3.25)Simpson's rule was used to evaluate the double integral in equation (3.25). To integrate overmost parts of the optical intensity profile, the limits of integration in the x direction werechosen to be 0 and 2; m (; is shown in Figure 3.15) and in the y direction, 0 and 6.5p,m.For the dimensions of the waveguides and the electrodes used in the actual modulators, whichare given in the next chapter, r is approximately in the range of 0.4 to Chapter SummaryIn this chapter, a single mode planar waveguide structure was designed using thevariational method and the beam propagation method. The layer structure is, in the orderof growth, 3.2,m of Al„Gai,As, with x=0.35, 1.3m of AlxGai,, with x decreasinglinearly from 0.35 to 0.30, 1.3Am of AlxGai,As, with x increasing linearly from 0.30 to0.35, and finally, 0.70m of AlxGai,As, with x=0.35.Then the variational method and the effective-index method were used to analyzechannel waveguides formed by etching rib-like structures on the planar waveguide. Achannel waveguide with a lihm high rib was predicted to be single mode when the width ofthe rib, measured at half height, was in the range of 4 to 5thm.69To ensure that the evanescent field coupling between the two branches of the Mach-Zehnder interferometer would not affect the performance of the modulators, the effective-index method and the beam propagation method were used to investigate the couplingcharacteristics of two parallel waveguides. For 1Am high rib waveguides, the separationbetween the centres of the 4/.Lm or 5/./m wide waveguides should be at least 16m, to resultin less than 5% coupling.70Chapter 4Fabrication4.1^IntroductionThere were two major processes involved in fabricating the modulators. First, ribwaveguides were etched into the epitaxial layers using a wet chemical etchant, then achlorobenzene lift-off technique was used to form the electrodes on top of the waveguides.A standard photolithographic process was used throughout to delineate the required patterns.In section 4.2, the various dimensions of the waveguides and the electrodes on the masks aregiven. In section 4.3, the layer structure for the planar waveguide, corresponding to thedesign given in section 3.6.1, is shown. Etchant calibration is given in section 4.4, wherethe etch rate and the etch profile are determined. Finally, a complete list of the fabricationsteps, including a description of the chlorobenzene lift-off technique, is given in section 4.5and its subsections.4.2 Mask DesignTwo masks were required to fabricate the modulators. One contained the patterns ofthe Mach-Zehnder waveguides and the other contained the patterns of the correspondingelectrodes. The masks were designed on a Sun SPARCstation IPC, Sun Microsystems,Mountain View, CA, using the VLSI CAD package, Edge, by Cadence Design SystemInc., CA. A tape cartridge containing the design files was sent to Precision Photo MaskInc., Montreal, Quebec, where a master mask was made on a quartz substrate using electron-71beam lithography. The precision was ±0. 1m for the waveguide mask and +0.25Am forelectrode mask. The masks used for the actual fabrication were copies of the master.The waveguide mask was light field, i.e., the waveguide patterns were opaque andthe rest of the mask was transparent. The electrode mask was dark field, i.e., the electrodepatterns were transparent and the remaining portion of the mask was opaque.There were three groups (group I, II, and III) of waveguides, with the correspondingelectrodes, on the two masks. Each group contained patterns for fabricating ten devices.The groups were differentiated by the effective microwave index ni, of the electrodes. Thedimensions of the electrodes in group I were designed to give n in accordance withthe optical refractive index of AlGaAs with approximately 30% aluminum mole fraction.The electrodes in groups II and III had n4=3.43 and 3.00, respectively, bracketing the valuen4=3.24. For each Mach-Zehnder waveguide, electrodes with only fins and electrodes withfins and pads, were designed. The electrodes were terminated at both ends in such a wayso they could be probed by Tektronix TMP 9215 coplanar strip microwave probes with a1502m pitch.The waveguide patterns for the Mach-Zehnder interferometers had three differentwidths: 4, 5 and 6p,m. The half angle of the branch was 0.25°. The gaps between theparallel waveguides, measured from the inner edges, were 12 and 16pm for the 4/./m widewaveguides and 11 and 15Am for the 5Am wide waveguides. Referring to Figures 3.12 and3.13, with these dimensions, the coupling between the parallel waveguides, for rib heightsgreater than 1gm, should be less than 5%. In addition, two 6m wide waveguides wereincluded on the mask; the gap for these devices was 14pm. For all these devices, the length72of the modulating section was 2cm, and the length of the entire device was 3.6cm.Two electrode patterns were designed for each Mach-Zehnder waveguide of a certainwidth and gap. The first pattern had fins only, while the second pattern had both fins andpads. The dimensions were designed to give the electrodes characteristic impedances closeto 50 ohms.Figure 4.1a gives the labels and some of the dimensions of the Mach-Zehnderwaveguides. Figure 4.1b shows the labels used for the electrodes, and Figure 4.1c showsthe dimensions of the input/output sections of the electrodes. Tables 4.1a, b, and c give thedimensions of the Mach-Zehnder waveguides and electrodes.4.3 MOCVD Epitaxial LayersThe epitaxial layers were grown by Spire Corporation, Bedford, Massachusetts, usingMOCVD. The epitaxial layers were grown on a 2 inch, undoped, semi-insulating GaAssubstrate. The orientation of the substrate was 2° off (100) towards the nearest (110) plane.The slight misorientation usually gives an better surface morphology [72: p. 290]. The targetstructure was, in the order in which the layers were grown, 3.2Am of A1.35Ga65As, as abottom cladding layer, 1.3Am of AlxGai,As, with x graded linearly from 0.35 to 0.30,1.3Am of AlxGai,As, with x graded linearly from 0.30 back to 0.35, and 0.7/.Lm ofA1.35Ga.65As, as a top cladding layer. A schematic of this structure is shown in Figure 4.2.From the data sheet provided by Spire, the total thickness of the epitaxial layers wasapproximately 5.8Am. The average aluminum mole fraction in the layers, measured byphotoluminescence, was approximately 0.316. With close inspection, the samples showed730.2 5 °(a)(b)(b)(c)Figure 4.1. Schematics showing the various labels and dimensions of (a) the Mach-Zehnderwaveguide, (b) the slow-wave electrodes, and (c) the input/output sections.74Deviceno.M-Z Waveguide Electrode ni,=3.24, 1=1.512M, S2 =2/./mwidth(jhm)gap(gm)NV1(jhm)NV2(pm)NW(gm)l'(am)d(thm)1 4 12 17.5 5.5 0 0 72 4 12 26 5.5 2.5 5 123 4 16 29.75 7.5 0 0 94 4 16 41.25 7.5 3.5 6 165 5 11 17.5 5.5 0 0 76 5 11 26 5.5 2.5 5 127 5 15 29.75 7.5 0 0 98 5 15 41.25 7.5 3.5 6 169 6 14 17.5 5.5 0 0 710 6 14 26 5.5 2.5 5 12TáN 41. a)Deviceno.M-Z Waveguide Electrode nit=3.43, 1=1.5Am, S2 =2/.Lmwidth(jhm)gap(/hm)W1(pm)W2(gm)WI(am)l'(2m)d(/hm)11 4 12 10.75 5.5 0 0 5.7512 4 12 18 5.5 2.5 5 9.513 4 16 19.5 7.5 0 0 714 4 16 29.5 7.5 3.5 6 12.515 5 11 10.75 5.5 0 0 5.7516 5 11 18 5.5 2.5 5 9.517 5 15 19.5 7.5 0 0 718 5 15 29.5 7.5 3.5 6 12.519 6 14 10.75 5.5 0 0 5.7520 6 14 18 5.5 2.5 5 9.5Table 4.1. (b)75Deviceno.M-Z Waveguide Electrode ni,=3.00, 1=1.5Am, S2 =211mwidth(jum)gap(Am)W1(pm)W2(am)WI(j/m)l'(jam)d(gm)21 4 12 33.5 5.5 0 0 1122 4 12 43.25 5.5 2.5 5 1923 4 16 53 7.5 0 0 1424 4 16 65.5 7.5 3.5 6 2525 5 11 33.5 5.5 0 0 1126 5 11 43.25 5.5 2.5 5 1927 5 15 53 7.5 0 0 1428 5 15 65.5 7.5 3.5 6 2529 6 14 33.5 5.5 0 0 1130 6 14 43.25 5.5 2.5 5 19a e . . cTable 4.1. Dimensions of the waveguides and electrodes on the masks: (a) group I, (b)group II, and (c) group III.76A I.35Ga.75As0.7pm1.3pm1.3pm3.2pmA I.35Ga.75AsA 135Ga75A s• it •A 1.30Ga.70A sA 130Ga70A s• t •A 1.35Ga75A s2 inch SIGaAs Wafer 450 pmFigure 4.2. Target structure of the epitaxial layers grown by MOCVD.77surface haze. Using the Tencor Alpha Step 200 profilometer, Tencor Instrument Inc.,Mountain View, CA, it was shown that the surface had a roughness in the range of 200-400A. The surface haze should not have a significant effect on the optical loss since thewaveguiding region was much deeper into the epitaxial layers.4.4 Etch CalibrationChemical etching of most III-V semiconductor materials is generally proceeded by anoxidation process at the semiconductor surface, followed by the dissolution of the oxidizedmaterial [73:p. 95]. Hence, the etchant usually consists of an oxidizer and an oxidedissolving agent. In this section, the characterization of an etchant, based on a combinationof citric acid (C6H807) and hydrogen peroxide (H202), is presented. In this case, hydrogenperoxide is the oxidizer and citric acid is the oxide dissolver. The advantages of using thisetchant include: sharper edges, smoother etched surfaces, and compatibility with Shipley's1400-series positive photoresist, i.e., the etchant would not severely erode the photoresist,no matter what the citric acid/hydrogen peroxide volume ratio was [74].The etch rate and the etch profile were investigated using a small piece of AlGaAs.The following procedures were used for the characterization experiment; most of the stepswere also applicable in fabricating the rib waveguides.In order for the photoresist to adhere properly to the surface, the sample was cleanedand degreased by first submerging it in acetone for 10 minutes at -50°C and then in 2-propanol for 5-10 minutes, also at -50°C; the rate at which 2-propanol evaporates is muchslower than that of the acetone, hence, giving ample time to blow dry the sample using78compressed nitrogen (N2) gas. The result was a cleaner surface. Since AlGaAs is brittle,ultrasound agitation was not used.The sample was immediately spin coated with Shipley's 1400-27 positive photoresistfrom Shipley Company Inc., San Jose, CA, at 4000 rpm for 35 seconds. The resultingphotoresist film was approximately 1.2,um thick [75]. The sample was then softbaked at90°C for 20 minutes and allowed to cool to room temperature. Softbaking removes waterand solvents from the newly applied photoresist; hence, the photoresist hardens, to improveadhesion [73: p. 129].The patterning was done on the Karl-Suss MJB3 contact mask aligner, Karl SussAmerica Inc., Waterburgh, VT. The mask used consisted of groups of straight waveguideshaving widths ranging from 3m to 10Am. The mask was aligned so that the waveguidesran parallel to the [01T] direction of the sample. After proper contact was made between themask and the sample, the photoresist was exposed for 54 seconds at the wavelength of320nm. The exposed sample was then postbaked at 90°C for 20 minutes and allowed to coolto room temperature. Postbaking offers several advantages such as improved line widthcontrol, by eliminating the standing wave effect, increased contrast, and improved adhesion[76].The patterns were developed in Shipley's MF-319 developer at room temperature(-22°C). The sample was immersed in the developer until the exposed photoresist began todissolve. Then, the sample was gently stirred in the solution until all the patterns wereclearly visible. The above process usually took between one and one and a half minutes.The sample was then removed from the developer and rinsed in deionized (DI) water and79blown dry using N2. The widths of the waveguide patterns were checked under themicroscope. If the patterns were underdeveloped, the sample was re-immersed in thedeveloper for a few seconds and rechecked under the microscope. When the patterns weresatisfactory, the sample was hardbaked for 20 minutes at 120°C, so that the photoresistdelineating the waveguides would be more resistant to the etchant [76]. The height of thephotoresist patterns was then measured on the Tencor Alpha Step 200 profilometer. Thisprovided a reference for determining the amount of material that was removed.The etchant was a two part mixture. Anhydrous citric acid crystals were dissolvedin de-ionized water with a ratio of 1 g to 1 ml. The reaction was endothermic, so thecrystals did not dissolve instantaneously; therefore, the solution was prepared at least one dayin advance. Before conducting the etching experiment, the citric acid solution wasthoroughly mixed with 30% hydrogen peroxide with a ratio of 10:1 by volume (i.e., 10 partsC6H807 to 1 part H202). The mixture was allowed to sit for 15 to 20 minutes so that thetemperature of the solution would return to room temperature (-22°C), if any changesoccurred due to the mixing.The sample was then placed in a holder and submerged in the etchant. The samplewas gently moved around in the etchant, and occasionally lifted out of the etchant. Atcertain fixed time intervals, the sample was removed from the etchant, rinsed in running de-ionized water for about 30 seconds, and blown dry using N2. The etch-depth was thenmeasured using the Tencor Alpha-Step 200 profilometer. Figure 4.3 shows a plot of theetch-depth versus the etch-time. As can be seen, the relationship is basically linear. Froma linear fit of the data, the etch rate, i.e., the value of the slope, is found to be approximately800.13gm/min, in agreement with the value of 0.12gm/min given in References [77] and [78].The experiment was stopped after approximately of material had been removedfrom the surface. The photoresist was then removed in warm acetone and 2-propanol baths.The sample was then observed under the scanning electron microscope (SEM). Figure 4.4is the SEM micrograph of an etched rib waveguide.The slope of the side walls was measured to be approximately -54° from thehorizontal, in agreement with the assumptions made in the simulations.4.5 Device FabricationIn the following subsections, a complete list of the fabrication steps is given. Insection 4.5.1, the steps for fabricating Mach-Zehnder waveguides are given; in section 4.5.2,the procedures for aligning the electrodes with the waveguides are given; and in section4.5.3, the lift-off technique for fabricating the electrodes is given.4.5.1 Fabrication of WaveguidesThe steps for fabricating the Mach-Zehnder waveguides were almost identical to theones used for preparing the sample for the etch calibration. The only additional step was thesputter deposition of a Si02 buffer layer after the waveguides were fabricated. These stepsare summarized below:a. clean the sample in acetone and 2-propanol;b. spin on Shipley's 1400-27 positive photoresist at 4000 rpm for 35 seconds;c.^softbake the photoresist at 90°C for 20 minutes and allow it to cool to room814.00--_-_'.-.3.00-_-=^_-- H = 0.129t + 1.079-_-0.00 IIIIIIIIIIIIIIIIII4^6^8^10 12 14 16 18 20Etch—time t (minutes)10Figure 4.3. Etch rate of the citric acid/hydrogen peroxide (10:1) mixture for -30%aluminum mole fraction.828. 0 k 4092 5. OFT..,Figure 4.4. Edge profile ((01T) face) of an AlGaAs rib etched using the citric acid/hydrogenperoxide (10:1) mixture.83temperature;d. expose the photoresist through the waveguide mask (waveguides are aligned parallelto the [01T] crystal direction, see Figure 4.5);e. postbake the exposed photoresist at 90° for 20 minutes and allow it to cool to roomtemperature;f. develop the photoresist in Shipley's MF-319 developer;g. hardbake the photoresist at 120°C for 20 minutes;h. measure the initial photoresist height on the Tencor Alpha Step 200 profilometer;i. etch the sample to remove li/m from the regions not covered by the photoresist;j. remove the photoresist on the waveguides using a positive photoresist stripper;k. RF sputter deposit a Si02 buffer layer onto the sample using the Perkin-ElmerSputtering System model 3140, Perkin-Elmer, Norwalk, CT. The mode should beset to "bias sputter" with a forward power of 150 watts and a reflected power of 2.5watts. The sputtering rate is approximately 2000A/hour.4.5.2 Alignment of Waveguides and ElectrodesAfter the rib waveguides were fabricated and the Si02 buffer layer was deposited,electrodes had to be deposited above the waveguides. To aid in the alignment of theelectrodes with the underlying waveguides, alignment marks were included on the waveguideand electrode masks, such as those shown in Figure 4.6. The alignment marks were placedat both ends of the waveguide and electrode patterns, at locations where they would notinterfere with the operation of the devices. Indicators were placed beside the alignment84Secondaryflat( 0 1 1 )Pr I riar yf lot-( 0 1 1 )Figure 4.5. Orientation of the waveguides on a Spire wafer (not to scale).85marks for every fifth device.On the mask aligner, the mask was held in a fixed position. The chuck supportingthe sample had linear and angular movements, and was used to bring the sample intoalignment with the patterns on the mask. The patterns were first aligned under lowmagnification (10x microscope objective on the mask aligner). The position of the samplewas adjusted until the two alignment marks, for a particular device, were aligned. Then, thehigher microscope objective (32x) was used for fine alignments. At this magnification, theimages of the rib waveguides on the sample could be seen clearly through the transparentregions on the electrode mask, and were used to aid in the aligning process. The patternsfor the electrodes could be placed to within a fraction of micron of the desired location.4.5.3 Chlorobenzene Lift -offThe electrodes were fabricated using an established chlorobenzene lift-off technique[79]. The basic premise for this technique is that, by immersing the exposed photoresist inchlorobenzene, solvents and low molecular weight resins are removed near the surface of thephotoresist film; thus, the top photoresist layers are more resistant to being developed thanthe lower layers. Windows opened in the photoresist film will develop with overhanginglips. When metal is deposited on the sample, usually by thermal evaporation, part of thedeposition will be in contact with the substrate which will later become the electrodes, andpart will rest on top of the photoresist. An organic solvent, such as acetone, is then used todissolve the photoresist, in effect carrying away the metal on top, leaving only the electrodesbehind. True success of this technique relies on the fact that the thickness of the metal86Figure 4.6. Alignment marks on the waveguide mask (a) and the electrode mask (b).All dimensions are in Am.87deposition should be less than the thickness of the photoresist. Figure 4.7 shows thesuccessive steps of this method.Since there was a layer of Si02 on the surface, Shipley's Microposit Primer, basedon Hexamethyldisilizane (HMDS), was first spun on the wafer, with the etched ribs, at 4000rpm for approximately 10 to 15 seconds. This improved the adhesion of the photoresist tothe surface. Then, Shipley's 1400-27 positive photoresist was spun on at 3000 rpm for 35seconds; at this speed, the thickness of the photoresist was approximately 1.57gm. Thesample was then baked at 700 for 20 minutes and allowed to cool to room temperature.Here, baking at high temperatures (in excess of 100°C) was avoided since the photoresistwould be more difficult to remove using acetone.Then electrode patterns were aligned with the rib waveguides following the alignmentprocedures given in section 4.5.2. The exposure time for the photoresist was extended to oneminute and 45 seconds, due to the increase in the photoresist film thickness. After theexposure, the sample was soaked in chlorobenzene for approximately 8 minutes. The samplewas then removed from the chlorobenzene and blown dry. It was then placed in a coveredpetri-dish for approximately 20 minutes, allowing for complete evaporation of thechlorobenzene.The sample was then developed in Shipley's MF-319 developer. After approximatelytwo and half minutes, the patterns were developed. When observed under the microscope,the effect of the undercut was seen as a faint outline surrounding the main patterns.Aluminum was then thermally deposited onto the sample using a thermal evaporationsystem manufactured by Carl Herman & Associates Inc., Menlo Park, CA. Approximately88Outline of a substrate with etched ribs.AvArArAwirwAvAralAr".....Substrate spin coated with photoresist.Ii^TEPhotoresist exposed, soaked in chlorobenzene,and developed with overhanging lips.Metal thermally deposited.Removal of unwanted metal by dissolving thephotoresist with solvents.Figure 4.7. Chlorobenzene lift-off technique.890.6 to 0.7/.Lm of aluminum was deposited in a chamber pressure of approximately 2 x10-6Ton. To reach this range of thickness, two tungsten coils with approximately 15 pieces of0.75" long aluminum strips hanging on them were required. The thickness of the depositionwas monitored by an Inficon model 321-50 sputtering thickness sensor, Inficon Inc., EastSyracuse, NY. The reading on the thickness monitor was usually 10% to 20% lower thanthe actual thickness, which was taken into account during evaporation.After the desired thickness had been achieved, the sample was allowed to cool down,in vacuum, for approximately 30 minutes, before being removed from the evaporationchamber. This prevented oxide from forming on the electrodes, which might prevent properelectrical contact between the microwave probes and the electrodes.The sample was then immersed in warm to hot acetone (-50°). After a few minutes,the metal deposited on top of the photoresist would start to wrinkle and peel off as theunderlying photoresist was dissolved. A spray bottle was used to spray jets of acetone ontothe sample to help remove the unwanted metal. The sample was repeatedly observed underthe microscope to check whether the unwanted metal had been completely removed. Beforeeach observation, the sample was immersed in hot 2-propanol, and then blown dry using N2.4.5.4 Completed DevicesThe wafer with the completed devices was cleaved to expose the input and output endsof the waveguides. GaAs substrates having (100) major faces have natural cleavage planesalong the [011] and [01T] crystallographic directions. Here the sample was cleaved alongthe [011] direction. The cleaving was performed by first scratching a small mark on the90sample, using a diamond scriber. The sample was then flipped over and placed on a softsurface. Gentle pressure was applied on top of the mark until the piece broke.Figure 4.8 shows a plan view of a device with fins only, and Figure 4.9 shows anisometric view of the same device. Figure 4.10 shows a plan view of a device with fins andpads, and Figure 4.11 shows an isometric view of the same device. Figures 4.12 and 13show the input/output sections of an electrode with fins and an electrode with fins and pads,respectively.91Figure 4.8. SEM micrograph showing the plan view of device no. 7 (fins only).Figure 4.9. SEM micrograph showing the isometric view of the device shown in Figure 4.8.92Figure 4.10. SEM micrograph showing the plan view of device no. 8 (fins and pads).Figure 4.11. SEM micrograph showing the isometric view of the device shown in Figure4.10.93Figure 4.12. Photograph showing the input/output section of a device with fins only.Figure 4.13. Photograph showing the input/output section of a device with fins and pads.94Chapter 5Experimental Results5.1^IntroductionIn this chapter, measured device parameters for the electrooptic modulators are given.Diagrams of the experimental setups used in making these measurements are shown. Insection 5.2, the existence of planar and channel waveguides on a Spire wafer isdemonstrated. In section 5.3, the optical loss of channel waveguides, having transversedimensions similar to those used in the modulators, measured by the sequential cleavingtechnique, is given. In section 5.4, the current-voltage characteristic of a pair of electrodes,fabricated on a 5i02 buffer layer, is shown. In section 5.5, device parameters such as thehalf-wave voltage, the intrinsic bias, and the extinction ratio, are obtained from the optical-intensity-out/voltage-in transfer functions measured at low frequencies. In section 5.6,microwave indices of the slow-wave electrodes, measured using the resonance technique, aregiven for several devices. Finally, a chapter summary is given in section Planar and Channel WaveguidesBefore fabricating the modulators, a small piece of a Spire wafer was first checkedfor the existence of a planar waveguide. This observation was important since channelwaveguides would not form if the light was not initially confined in the depth direction.Each end of this sample was cleaved to expose a clean edge; so that light could be coupledinto and out of the planar waveguide using the "end-fire" technique.95First, the laser was adjusted so that the optical field was predominantly TE-like, i.e.,the polarization of the laser output was parallel with the epitaxial layers. Figure 5.1 showsthe setup used for making this adjustment. The infrared laser was a LAS 300-1300-6 diodelaser from LaserMax Inc., Rochester, NJ. The wavelength was 1. 3m and the polarizationratio was quoted to be greater than 200:1. A polarizing beam splitter cube, model PB.9 fromNewport Corp., Fountain Valley, CA, with an extinction ratio of 1000:1, was placed in thebeam path. For the orientation of the cube shown in Figure 5.1, the TE-like component wastransmitted while the TM-like component was reflected perpendicular to the beam path. Thepower of the TE-like component was measured using a Newport 81 81R germanium infraredphotodetector connected to a Newport 835 digital power meter. For the adjustment, the laserwas rotated in its holder until the reading on the power meter was maximum. At that point,the laser was secured in place.Once the polarization of the laser was fixed, the setup shown in Figure 5.2 was usedto find the planar waveguide. The collimated laser output was focused at the edge of thesample using a Newport F-L1OB 10 x lens. This lens was mounted directly in front of thelaser which, in turn, was mounted on a three-axes translational micropositioner. A NewportF-LAO 40 x lens, mounted separately on a three-axes translational micropositioner, was usedto magnify the image, at the output of the waveguide, onto a screen. The transmission ofboth lenses at 1.3/.Lm was greater than 90%. The screen was an infrared sensor card, modelADQ-22 from Quantex, Rockville, MD. It would convert infrared radiation withwavelengths in the range of 0.7-1.6 Am to 0.64 Am. To maintain the image, the cardrequired periodic charging under visible light.96Polarizingbean-splittercube PU.9Rotate laserto maxinizeTE-Ilke mode041ii1100111!!!FserLAS 300-1300-6 DigitalpowernetNewport835TE- likeATM-likeInputpolarizationYJIWAWWAWAGaAs substrateIR photo-detectorNewport818 IRFigure 5.1. Diagram showing the setup used for adjusting the polarization of the laser.Planar wavegu I deOrchannel wavegul deI OxlensF-LiOBHR laser^1()I^r—Li4 0 xI ensF-L40xyz micro-posItIoner xyZ micro-pos it lonerIR sensorcardADO-22Figure 5.2. Diagram showing the setup used for viewing the outputs from planar andchannel waveguides.97The waveguide structure was interrogated by moving the laser downward from thesurface of the sample. As this was done, a set of fringe patterns was observed. This wascaused by the discontinuity of the refractive indices at the surface of the sample. As the lasermoved downward, a bright line could be observed, indicating that the light was confined inthe depth direction. This was the planar waveguide that was being sought. As the laser wasmoved further downward, the bright line would disappear, since the light was no longercoupled into the planar waveguide. Another set of fringe patterns could be observed. Thisset was caused by the discontinuity of the refractive indices between the bottom claddinglayer and the substrate. Figure 5.3a is a photograph showing light confined by the planarwaveguide.Once the existence of the planar waveguide was confirmed, straight channelwaveguides were fabricated using the procedures outlined in section 4.5. Two samples werefabricated. The height of the ribs on one was approximately lm, and on the other 1.4pm.The output spots of these two samples were observed using the same setup shown inFigure 5.2. Figures 5.3b and 5.3c are photographs showing the output spots of a ljm highrib waveguide and a 1.4,um high rib waveguide, respectively. For the case of a 1m highrib waveguide, both the planar waveguide mode and the channel waveguide mode werevisible. For the case of a 1.4Am high rib waveguide, only the spot from the channelwaveguide was visible; the planar waveguide mode was no longer supported. From theeffective-index calculations given in section 3.6.2, it was expected that the planar waveguidewould be cut off when approximately 1. 6m of material was etched off from the surface (thisis where the curve in Figure 3.6 ended at the left hand side, at which no solution could be98Figure 5.3. (a) Image of light confined by the planar waveguide as appeared on the IRsensor card.99Figure 5.3. (b) Image of an output spot from a channel waveguide with lmm high rib.Figure 5.3. (c) Image of an output spot from a channel waveguide with 1.4Am high rib.100found for equation (3.13)). The discrepancy between the observed and theoretical resultsmight be attributed to the fact that the thickness of the epitaxial layers was less than thedesigned specification, as mentioned in section Optical Attenuation MeasurementFactors contributing to the optical loss of semiconductor waveguides include the freecarrier absorption loss due to unintentional doping, the scattering loss due to roughness at theepitaxial layer boundaries and the etched side walls of the rib, the scattering loss due toinhomogeneity within the epitaxial layers, and the leakage loss due to light coupling from theguiding region into the substrate. The fundamental absorption loss should not be a majorfactor in this case, since the photon energy (0.95eV for 10=1.3p.m) is less than the bandgapof the semiconductor (-1.82eV).Here, no attempt was made to characterize the individual losses. Instead, the totaloptical loss of the semiconductor waveguides was measured using the sequential cleavingmethod [44:pp. 84-85]. This method essentially involved the measurement of optical powerthrough channel waveguides of decreasing length, obtained by successively cleaving thesample.Assuming an exponential loss behaviour, the input and output intensity of a channelwaveguide can be described asP our = P^ (5.1)where Pour is the transmitted optical power, PIN is the input optical power, a is the optical101attenuation constant in cm-1, and L is the length of the channel waveguide in cm. By takingthe logarithm of base 10 on both sides of equation (5.1), and multiplying the results by 10,the following equations are easily obtained10log10P0ur = 101og10Plig - 4.34aL^(5.2a)orPow. (dB) = 13 IN (dB) - adBL.^ (5.2b)Therefore, by plotting 10log10P0ur against L, the slope is the optical loss in dB/cm.The accuracy of this method was limited by the reproducibility of the coupling thatcould be obtained from one measurement to the next. Hence, results for several waveguideswere needed to obtain a good estimate for the optical loss. Fortunately, for semiconductorwaveguides, the quality of the end faces could be consistently reproduced, since cleaving wasused.A piece of AlGaAs, with lcm long straight channel waveguides, having 1,um highribs, was fabricated using the procedures outlined in section 4.6. The sample was placed ina setup similar to that shown in Figure 5.2, where the sensor card was replaced by theNewport IR photodetector, for measuring the transmitted optical power. For eachmeasurement, the optical power was maximized by adjusting the position of the laser and theoutput lens.Figure 5.4 shows the results of optical loss measurement for several 4Am wide102waveguides. The data for each waveguide were fitted to a straight line using the leastsquares method. From the slopes, the optical attenuation for a buried waveguide, fabricatedusing wet etching, was determined to be -1.6 dB/cm. Extrapolating the curves to zerolength, it can be seen that approximately 1.2 mW of optical power could be coupled into thewaveguide using the "end-fire" coupling technique.5.4 Current-Voltage Characteristic of the ElectrodesThe sputtered Si02 buffer layer served two purposes. Firstly, it was used to minimizethe absorption of the optical field by the electrodes at the sides of the etched ribs. Secondly,it was used to prevent current from being injected into the substrate. The current-voltage (I-V) characteristic of an unterminated device was measured using a HP4145A SemiconductorParameter Analyzer from Hewlett-Packard, Colorado Springs, CO. The voltage was sweptfrom -30 volts to +30 volts. The measured I-V curve is shown in Figure 5.5. Within thisvoltage range, the current flowing between the electrodes was in the range of picoamperes,indicating that the sputtered Si02 layer was providing good insulation for the electrodes.Furthermore, the leakage current only began to increase when the applied voltage hadreached approximately +30 volts. These voltages, as will be shown later, are larger thanthe half-wave voltage of the modulators, and would not be reached when performing small-signal operation at microwave frequencies.1032.00 —__-_L_^_Q) _—1.00 -0^-_0 —2.00 --4--J- --'E^__2 —3.00 -2^_1-- __—4.00 —---o Waveguide 1, 4umo Waveguide 2 4um• Waveguide 3, 4um• Waveguide 4, 4um_-—5.00 1^i^filmilIiiiiiiiii0.00 0.50^1.00^1.50^2.00Length (cm)Figure 5.4. Measured optical attenuation as a function of length for 4p.m wide channelwaveguides with 1m high rib.104250200 —150 —50 —0ci—50——100——150 I^II^II^IiI^II^III^!III!—35 —30 —25 —20 —15 —10 —5 0 5 10 15 20 25 30 35Voltage (V)Figure 5.5. Measured current-voltage (I-V) characteristics of a pair of slow-waveelectrodes deposited on a layer of Si02.1055.5 Measurement of Device Parameters at Low FrequenciesThe electrooptic modulators were first tested at low frequencies, at a few kilohertz,to obtain the optical-intensity-out/voltage-in transfer functions. The half-wave voltage, theintrinsic bias, and the extinction ratio (on/off ratio) could then be extracted from thesetransfer functions. In this frequency range, the electrodes were essentially lumped elements,and did not require 5011 terminations.The experimental setup used is shown in Figure 5.6. The optical portion of the setupis the same as before. The modulating signal was provided by a function generator, modelPM5132 from Philips, W. Germany. The function generator could produce a voltage swingfrom -15V to 15V, from a few hertz to 2 MHz. To obtain a larger voltage swing, an audiotransformer with a turns ratio of 4:1 was used to step-up the voltage.A sinusoidal signal was applied to the electrodes using two micromanipulator probesmounted separately on two 3-axes positioners, model PRO195RE from WentworthLaboratories, Brookfield, CT. The modulated optical signal was detected by a low noiseinfrared photodetector, model 1811 IR from New Focus Inc., Mountain View, CA. Thisdetector had a conversion gain of approximately 3.28 x104 V/W and a frequency responseof 125 MHz. Both the applied electrical signal and the modulated optical signal weremonitored on a HP54600A dual channel digital storage scope from Hewlett-Packard. Amicrocomputer was linked to the oscilloscope using the RS-232 port. A data acquisitionprogram, Scopelinkm, was used to transfer image and text data from the oscilloscope to thecomputer.106VoltageStep-upTransforner[==^ FunctionGeneratorPM5132ProbesIk///IR Photo-detector1811 CH1Oscillo-scopeHP54600AI^  ( SanpleIR Laser 10x^40xLens Lens[2ThCH2SerialPortIBM PCConpatibleFigure 5.6. Diagram showing the setup used for measuring device parameters at lowfrequencies.107The ideal transfer function of a Mach-Zehnder interferometer is [46]Poutn VP in {1 + COS(177—r+2(5.3)where Pout and Pin are the optical power at the output and the input of the modulator, V isthe applied voltage, V, is the half-wave voltage, and (pi is the intrinsic bias of a device. Thehalf-wave voltage is defined as the voltage required to obtain a phase difference of it radiansbetween the fundamental modes propagating in the two branches of the Mach-Zehnderinterferometer.To compensate for the fact that the transfer function is usually not ideal, i.e., theextinction ratio is not 100% as assumed in equation (5.3), a modified transfer function isoften used to fit the experimental optical-intensity-out/voltage-in data [80]( n V )Pout = Pdc + Acos — + 4)1.Kr where Pd, is the dc offset of the signal and A is the amplitude of the transfer function. Theextinction ratio is defined as (Pdc+A)/(Pde -A).Modulators were fabricated on two halves of a Spire wafer. The devices fabricatedwere mainly those with electrodes in group I on the electrode mask, with =3.24. 'Scopeimages, and the associated ASCII data, were collected for several devices. 'Scope imagesof devices no. 7(electrodes with fins) and no. 8(electrodes with fins and pads) are shown inFigures 5.7 and 5.8, respectively. In each figure, the upper trace is the applied voltage and(5.4)108the lower trace is the modulated optical signal. In these measurements, the applied voltagewas larger than the half-wave voltages of the devices. The ASCII data were fitted toequation (5.4) using MATLAB to extract Pdc, A, V,, (pi, and the extinction ratio. Theresults of the fits for devices no. 7 and no. 8 are shown in Figures 5.9 and 5.10,respectively. The dots are the collected data and the lines are the fitted transfer functions.Parameters for several other devices are listed in Table 5.1.The equation for calculating VT, taking into account the push-pull condition, andassuming the electrodes are coplanar strips of finite width isS AV - ^1 °,= 2 no310 r41(5.5)where Si is the interelectrode gap and L is the length of the modulating section; all othervariables have been defined previously. The value of no was set to 3.25, corresponding tothe average aluminum mole fraction of 0.316 in the layers, as quoted by Spire. The valueof r41 was set to 1.28 x10-12 m/V at A0=1.3p,m [81]. Here it was assumed that, due to theclose lattice-match, the electrooptic coefficient of AlGaAs with approximately 30% aluminumconcentration was the same as that for GaAs. The value of L was 2cm. The value of r foreach device was calculated using equation (3.25), where the thickness of the Si02 layer wastaken to be 4454A, as measured on the Tencor profilometer for the devices fabricated.109Vp-pC1)=40.00 V Vp-pC2)=1.969 V FreqC1)=9.050kH250.0t/ ME^fi RUN1 20.0V 2 1.00VVp-pC1)=39.38 V Vp-pC2)=1.937 V FreqC1)=9.091kHz20.0V 2 1.00V 50.0t/ mi^f1 RUNFigure 5.7. Oscilloscope image showing the modulation characteristics of device no. 7.Figure 5.8. Oscilloscope image showing the modulation characteristics of device no. 8.110V dark level2.00 --L_ 1.50 -0-1-,-o-1-, 1.00 -o_c0.50 -00.00-20.0 -10.0^0.0^10.0Applied voltage (V)20.0Figure 5.9. Fitted transfer function for device no.11120.0-10.0^0.0^10.0Applied voltage (V)V dark levelI I III I II^I I I^I^I^I^I^I2.00 -__--_^0.00 l^—20.0Figure 5.10. Fitted transfer function for device no. 8.112Deviceno.W (gm) S1(gm)r(theo.)vi (v)(theo.)v, 00(exp.)(pi(degree)on/offratio4 (fins+pads)41.25 17 0.551 23.0 24.7 -113.9 125:15 (finsonly)17.5 13 0.472 20.5 25.6 -65.9 126:16 (fins+pads)26 13 0.465 20.9 25.2 -95.7 75:17 (finsonly)29.75 17 0.546 23.2 24.1 -56.1 122:18 (fins+pads)41.25 17 0.544 23.3 25.0 -78.7 122:110 (fins+pads)26 13 0.447 21.7 21.3 -56.9 9:1Table 5.1 Measured device parameters for the Mach-Zehnder modulator.From Table 5.1, it can be seen that the calculated VT's are almost always somewhatlower in value than those that were measured. Two plausible reasons are that the electricfield underneath the actual slow-wave electrodes was not as high as that underneath aconventional pair of coplanar strips, due to the inclusion of the fins, and the fact that asimple offset was used to account for the Si02 buffer layer; hence, r would be reduced,leading to higher VT's.It can also be seen that the intrinsic bias varies from one device to another. Thisvariation is expected, as a result of small variations in the fabrication process. For most ofthese devices, the on/off ratio was good. The measurements were essentially limited by theresolution of the HP digital storage oscilloscope that we used, i.e., there were only 256quantization levels for representing data. The good on/off ratios would imply that the113waveguides of these devices were single mode. The low on/off ratio of device no. 10 wasprobably due to the fact that the waveguides were becoming multimode.5.6 Measurement of n ^the Resonance TechniqueThe microwave indices, n4, of the slow-wave electrodes were measured using theresonance technique [21]. The sample used had aluminum electrodes, from group I on theelectrode mask, deposited over ribs etched on GaAs. The thickness of the electrodes wasapproximately 0.7p.m. This sample was used because the electrodes were similar in structureto the model used in calculating the dimensions of fins and pads.A schematic of the setup for making these measurements is shown in Figure 5.11.The microwave source was a Hewlett-Packard HP8341B synthesized sweeper, capable ofsweeping from 10MHz to 20GHz. A Hewlett-Packard HP85027E directional bridge wasused to route the microwave signal from the sweeper to the devices and to route the reflectedmicrowave signal to a Hewlett-Packard HP 8757C scalar network analyzer. The microwavesignal was applied to the electrodes using a microwave coplanar strip probe, model TMP9215from Tektronix, Beaverton, OR., mounted on a three-axes translational micropositioner.The expression for ; is given by [21]c Nn --- --'1^2L, Af(5.6)where c is the speed of light in vacuum, Le is the length of the electrodes, N is the numberof resonance peaks in the frequency range A f.114SynthesIzedSweeperHP 8341BREISimenIBM PCConpatIbleScalarNetworkAnalyzerHP 8757C'CH ADirect IonalBrIclqHP 85027ECoplanarStripProbeIMP9215\--SanpleFigure 5.11. Diagram showing the setup used for measuring the microwave indices of theelectrodes using the resonance technique.115Since the electrodes fabricated for this work were not shorted at one end, as werethose in Reference [21], an open tip calibration procedure was adopted. First, the probe tipwas left open in air. The reflected power was measured and stored in the memory of thescalar network analyzer. Once this was done, the probe was lowered onto the electrodes, andmeasurements were obtained by subtracting from the actual reflected power the calibrationdata stored in the memory. A BASIC program, running on a microcomputer equipped withan 1EEE488 interface card, was used to acquire the ASCII data from the scalar networkanalyzer through the system HPIB port.All of the electrodes are considered to be surface deposited. Figures 5.12 and 5.13show the resonance curves obtained for devices no. 7 and no. 8, respectively. It can be seenthat some of the resonance peaks are sharp, and can be used to calculate ng. Microwaveindices measured for several devices are listed in Table 5.2.Device no. ni, (designed) ni, (measured)2 (fins +pads) 3.24 3.193 (fins only) 3.24 3.094 (fins +pads) 3.24 3.217 (fins only) 3.24 3.118 (fins +pads) 3.24 3.1410 (fins +pads) 3.24 3.13Table 5.2 Designed and measured ni,'s of slow-wave electrodes in group I on the electrodemask.It can be seen that the measured results are in good agreement with the design values;the differences are within 5%. There are several possible sources of error that may accountfor the 1 to 5% difference between the designed microwave indices and the measured values.116I^I^1^I^I^I^1 I^I^I^I6 10 14Frequency (GHz)1^12Figure 5.12. Measured resonance characteristics of electrode no. 7.1172^6^10^14Frequency (GHz)Figure 5.13. Measured resonance characteristics of electrode no. 8.118First, we estimated the experimental error of our measurements to be -1.5%. Second, theeffects of the end sections, the parts not including the fins, have not been taken into account.Third, the inner portions of the electrodes are on the ribs. The electric field distribution ofthis structure will be different from that of perfectly flat planar electrodes, assumed whendesigning the slow-wave electrodes. Fourth, the dimensions of the fins and pads on thedevices could be slightly different from their design values, due to the fabrication tolerances.Fifth and finally, the various approximations that we have made in modelling the electrodesmay also lead to an initial overestimation of the amount of slowing.Electrodes with loading elements, consisting of both fins and pads, i.e., devices no.2, 4, and 8, yield better results. The average microwave index for these devices is 3.15,representing an increase of 0.51 over that of conventional coplanar electrodes (ni, =2.64).The presence of ribs appears to have negligible effect on the slow-wave property of theelectrodes. This is expected since the slow-wave effect is mainly the result of the fringingfields near the gap region between the capacitive loading elements; whereas the wide stripsare used to modulate light in the optical waveguides. The resonance curves in Figures 5.12and 5.13 show that electrode loss was not very dependent on frequency in the range used forthe measurements.5.7 Chapter SummaryIn this chapter, optical waveguides were shown to exist in the MOCVD-grownepitaxial layers, having the structure designed in chapter 3. Optical attenuation was measuredto be -1.6 dB/cm for channel waveguides fabricated in these layers. The I-V characteristic119of the electrodes were measured. The half-wave voltage, intrinsic bias, and extinction ratiowere obtained for several devices from their respective optical-intensity-out/voltage-in transferfunctions. The measured half-wave voltages were within 80% of the theoretical values, andwere always somewhat larger in value. For most devices, the extinction ratios were greaterthan 99%, indicating the channel waveguides in the Mach-Zehnder interferometers wereeither single mode or very close to being single mode. As expected, there were somevariations in the intrinsic biases as a result of inherent differences between devices duringfabrication. The microwave indices of several slow-wave electrodes deposited on GaAs weremeasured. The values were between 3.09 and 3.21, with an average value of 3.15; thedesign value was 3.24. Substituting 3.24 for n. and 3.09 for ni, (the worst value measured)into equation (2.14), foL is -130 GHz•cm. The other values of ng, which are more likelyto be obtained, would put fol, in the range of hundreds of GHz•cm. Therefore, the effectof electrode loss on bandwidths should be investigated.Assuming a perfect velocity match, equation (2.11) simplifies to cp. 0, (1-e')/aL,since u=0. Here, a(dB/cm) = a0f1/2, where f is the microwave frequency in GHz and at),in dB/cmN6Trz", is a constant depending on the geometry of the electrodes [48]. A typicalvalue is 1 dB/cmNGHz, obtained from Reference [8]. Consequently, the 3dB opticalbandwidth of a modulator with 2cm long electrodes is -12 GHz. In order to achievebandwidths in excess of 100 GHz, the value of ao needs to be reduced to below 0.35dB/cmNGHz.120Chapter 6Summary, Conclusions, and Suggestions for Future Work6.1 IntroductionA summary of results for the electrooptic modulators fabricated on GaAs is given insection 6.2. The conclusions are given in section 6.3. In section 6.4, suggestions of possibledirections for the continue study of this type of device are given.6.2 SummaryA travelling-wave electrooptic modulator, based on the integrated optics version ofthe Mach-Zehnder interferometer, having a slow-wave electrode structure, has been designed,fabricated, and tested. The modulators were fabricated on GaAs substrates with MOCVD-grown AlGaAs epitaxial layers for optical confinement. A slow-wave electrode structure,consisting of a pair of coplanar strips with capacitive loading elements, was used to increasethe modulation bandwidths by reducing the problem of phase velocity mismatch between themicrowave and the optical wave, that normally exists in this type of device.Optical confinement in our devices was achieved through compositional variation inAlxGai,As (x =0.30 to x =0.35). A linear expression was used to relate the refractive indexto the aluminum mole fraction. The layer structure of a single mode planar waveguide witha Gaussian refractive index profile was designed using the variational method. Using thismethod, the profile of the fundamental TE-like mode and the cutoff condition for the nexthigher order modes were calculated. The beam propagation method was used to calculate121the light leakage into the GaAs substrate by evanescent field coupling. Consequently, asufficiently thick lower cladding thickness was used to reduce this effect. From thesecalculations, a layer structure for the planar waveguide, with the aluminum mole fractionfully specified, was obtained.Since single mode channel waveguides are crucial to devices of this type, acombination of effective-index method and variational method was used to analyze channelwaveguides formed by ribs etched on the AlGaAs epitaxial layers. From the calculations,it was found that, for our planar waveguide structure, single mode channel waveguides couldbe obtained by etching 1 Am high ribs, with the width of the ribs being -4Am.Since there should be negligible coupling between the parallel branches in a Mach-Zehnder interferometer, the beam propagation method was used to calculate the couplingcharacteristics of two parallel channel waveguides. Rib heights of 0.7Am, 1 Am, and 1.2tcmwere investigated for various waveguide separations, for both 4Am and 5Am widewaveguides. The final rib height was selected to be 1/.4m. As a result, waveguideseparations, measured at the half height between the inner edges, were chosen to be at least12Am for the 4Am wide waveguides and 11,um for the 5Am wide waveguides, to result inless than 5% coupling.Optical attenuation in straight channel waveguides, with transverse dimensions similarto those used in the actual modulators, were measured using the sequential cleavingtechnique. The measured attenuation was -1.6 dB/cm, similar to values published in theliterature.Electrooptic modulators, with structural dimensions obtained from the numerical122simulations, were fabricated on GaAs substrates with AlGaAs epitaxial layers. The ribwaveguides were formed by etching the sample in a citric acid and hydrogen peroxidemixture; for which the etch rate had been calibrated to give sufficient control over the etchdepth. The slow-wave electrodes were fabricated in a single layer of metallization using astandard chlorobenzene lift-off technique.The devices were tested at low frequencies. The half-wave voltage, intrinsic bias, andextinction ratio were measured for several devices. The average V, measured was -25 Vas compared to the average calculated value of 22 V. The extinction ratios, for most devices,were greater than 99%. There were some variations in the intrinsic phase, which wasexpected due to slight fabrication variations between devices.Microwave indices of the electrodes were measured using the resonance techniquewhich involved the use of a scalar network analyzer, a swept frequency source, a directionalbridge, and a coplanar strip probe. The average microwave index measured was -3.15, anincrease of 0.51 over that of a pair of conventional coplanar strip electrodes; this value isalso in good agreement with the design value of 3.24, which is the effective refractive indexof the optical waveguides with the aluminum mole fraction profile that we used. From thesemeasurements, several points could be observed. First, electrodes with loading elements,consisting of both fins and pads, yielded better results. Second, the presence of ribs beneaththe electrodes appeared to have negligible effect on the microwave index. Finally, it wasalso found that the electrode loss was not very dependent on frequency, in the range used forthe measurements.1236.3 ConclusionsAccording to the results obtained, the type of travelling-wave electrooptic modulatordeveloped in this work should be capable of achieving a very large bandwidth, as a result ofnearly eliminating the phase velocity mismatch problem. In a controlled fabricationenvironment, electrodes that can operate in excess of 100 GHz should have no problem beingfabricated routinely, assuming the loss in the electrodes can be made small or negligible.Therefore, investigation into reducing the electrode loss should be a continuation of thiswork.From the numerical methods presented in this and previous works, it can be seen thateach part of this modulator can be modelled quite accurately. As a result, the structure ofthe modulator could be optimized further to give performance.Furthermore, the structure of the modulator is simple. Other than growing theepitaxial layers, the entire modulator can be easily fabricated in our laboratory, usingstandard fabrication equipment and procedures.The possibility of using the MOCVD epitaxial layers as the optical substrates has theadded advantage that, should these modulators become commercialized, the ability to growlarge quantities would reduce the cost considerably. This is because, at present, the MBEprocess can only grow samples in small quantities, and the cost of running an MBE systemis high.6.4 Suggestions for Future WorkThe next step for this work is to experimentally determine the bandwidth of the124modulators. A way of measuring the optical response of the modulator, without using testinstruments and accessories at rates up to tens of gigahertz, is the counterpropagationtechnique [82]. In this measurement, the microwave and the optical wave are launched insuch a way that they propagate in opposite directions. Then the minus sign before ng in thedefinition of u in equation (2.11) is changed to a plus sign, in effect, contracting the responsefunction given by equation (2.13). From the calculated value of no and the measured valueof ng, the first minimum of the response should occur at —2.35 GHz. For a 6 GHzphotodetector, such as New Focus's model 1514 Photoreceiver, two such minima should beobserved. Since the value of ng is known accurately from the resonance technique, theexperimental value of no can be obtained. The experimental values of no and ng can thenbe used to predict the actual bandwidths of these modulators. However, at high frequencies,the bandwidths of the modulators are likely limited by loss in the electrodes.To reduce the electrode loss, gold, instead of aluminum, should be used as theelectrodes. At frequencies above 10 GHz, the current is mostly confined within the skindepth of the electrodes, which is just a fraction of a micron. Therefore, reduction of theelectrode loss by increasing the thickness and width of the electrodes should be investigated.Another possibility for reducing the electrode loss is to design slow-wave electrodes usingthe coplanar waveguide structure. This type of electrode generally has a lower loss than thecoplanar strips.An investigation into the reduction of the half-wave voltage, therefore, decreasing themicrowave drive power, should be considered. To a certain extent, increasing the rib heightreduces the necessary separation between the waveguides. This reduces the electrode gap and125decreases the half-wave voltage. However, the coupling efficiency to optical fibre isreduced, as a result of a large modal mismatch between the small mode size of thesemiconductor waveguide and the large mode size of an infrared optical fibre. But thisproblem could be solved by using waveguide tapers.Another direction to look in reducing the half-wave voltage is to increase the overlapintegral. This can be achieved by analyzing the overlap of the electric field around the slow-wave electrodes with the optical field distribution in the optical waveguides. The result isan optimum placement of the electrodes with respect to the optical waveguides.126Appendix ABeam Propagation MethodThis appendix contains the derivation of the two-dimensional beam propagationequations used in this work. The derivation starts with the vector wave equation for theelectric field, then the various assumptions made in obtaining equation (3.10) are given.The source-free vector wave equation for the electric field is [50:p. 594]V2E + V(Et.Vtlnn2) + n2k02È = 0^(A.1)where E=E(x,y,z), n=n(x,y,z) is the refractive index distribution, k. = 27r/l0 is the freespace wave number, and V and Vt are the grad and transverse grad operators, respectively.For a gradual refractive index distribution, the Vt1n(n2) term can be assumed to be negligible,giving the homogeneous Helmholtz equationVE + n2k02E = 0.^ (A.2)Assuming the beam to be propagating in the z direction, E(x,y,z) can be written as[61]E(x,y,z) = e(x,y,z)exp (--jnokoz)^ (A.3)127where no is a reference refractive index which is taken either as the refractive index of thecladding or the substrate. Substituting (A.3) into (A.2) givesa2e a2e a2e^. M 2 2 2,+ — + —az2 - 2in°k°—az + ko(n -ndK = OaX2^ay2^. (A.4)If the refractive index varies slowly in the z direction such that [61] --132 « 2n o kaz2^01 az(A.4) reduces to the Fresnel approximation82r+ al!. - 2jn k 82 + Ico2(n2-n!)r= 0.ax2^a?0 0 az (A.5)For g'=Ex, (A.5) becomesaEx - ^j  Ia2 + k2(n2az^2n0k0 ax2 (A.6)Since the formal solution of atir/az = a(z)tir is [59]11/ (Z) = III (0) exP(foza(z')dz)Ex can be written as128(  j  a2 )^2no( iko fz+Az(n2-nf)dejE(x,z).^(A.7)Ex(x,z+ez) = exp^Az exp --z2noko ax2Equation (A.7) can be written in the symmetrized split operator form, and with the samplingof the refractive index at z = zi-ez/2, the following expression can be obtained [58]Ex(x,z+ez) = exp(  f  32 Az)exp( ik[n2(y,z+ Az) nolez)exp(  i  32 Az)Ex(x,z),4nolco ax2^2n^2^4nolcoax20(A.8)This is, in fact, equation (3.10).Using the approximation [83]exp(-jA ) -a new operator symbol 13), can be defined asDx = exp( 4nj0. ko as32x2Az)P.+  -jAz a2snok, ax2-jAz a2snoko ax2(A.9)The two-dimensional beam propagation equation is then129^jkoez[ (^Az) 2Ex(X,Z +AZ) = Dxexp(- —2n0 n2 X,Z + — -no 1)DxEx(x,z)2jAz^AE(x,z+ Az) = Dxexp( ko{n1X,Z+ Z )-nl)Dxexp(-jkonoAz)E(x,z). (A.10)2no^2^°To implement (A.10), the operator Dx is evaluated implicitly by rewriting Jr = DxEoras [63]112}1 E = SE - E.jAz  328noko ax2Hence(1j6z a2 )SE = 2E8noko ax2or8E + j Az  8E1 = 2E.8noko(A.11)Using the central difference scheme to represent SE", i.e.,1308E., - 28Ei + SEi_i8E" -^j+1 ta2(A.11) can be written aslAz  SE,. 1 + [1^-12Az 1SEJ +^/Az  8E; 1 = 2E,8nok,,Ax2 '+-^8nolc„Ax 2^8nolcoAx 2 ^- ' .(A.12)Equation (A.12) defines a system of linear equations that is tridiagonal, i.e.,_8E18 E2SEN_I8EN- 2E1 -2E22EN_12ENb a 0 -a b a -... a b a- 0 a b..wherejAz a=8nolcoAx2and b - 1  J2Az 8n0k0Ax 2 .This type of linear system can be easily solved using the program given in [84:p. 48].131Appendix BEvaluation of I x2e2f(x)ch Using Gaussian QuadratureThis appendix contains the derivation of Vk given in section 3.6.2 for even integervalues of m. The derivation, for most part, was obtained from Dr. N.A.F. Jaeger [85] andReference [69].The derivation starts with the use of the Hermite interpolation formula to obtain thefollowing integration formula:flx2e-x2ftx) dx E Vk j(xk) + E 1-7-k r(Xk)k=1^k=1(B. 1)whereVk — 1^ fc*X2e—X 7t(X)  dx,(xdX - xk1 Vk Tc(xk) f X2e—x ic(X)1k(x)dx.' (B.1.1)(B.1.2)The auxiliary functions, n(x) and lk(x), are polynomials of degree m and m-1, respectively.If the nodes xl,^xm are chosen so that Vk 1= 0, (B. 1) reduces tof-x2e-x2fix) dx^Ak f(xk).k=1A formula of this type is called Gaussian quadrature formula.The auxiliary function, lk(x), can be written as(B.2)132Hence Vk is equal to zero for each odd term of lk(x) provided m is even, i.e.,= fx2e 27C (X)X ndY = 0^m even, n odd.When m and n( < m) are both even, integral I can be written as102 —I =^x2e-x21C(X)X ndx = f r2e-r(20(r)r2dr (B.3)using the transformation r=x2, where n-(x) = Q(x2), and Q(r) is a polynomial of degree 421-.The last expression of (B.3) is equal to zero ifQ(r) = L‘,,21(r)^ (B.4)2where LIN (r) is the Laguerre polynomial of degree m/2. Therefore the nodes xk are +Airkwhere rk are the roots of LTM (r).Vk is then calculated using (B.1.1), which is repeated here for convenience,- 1^ f c*X2e -X2 71 (x) dxn.^xxk) X - xkwhere(!)src (.0 = 42(x =^(x —-I,and Hm is the Hermite polynomial of degree m.133gn+1(x)(-1) 2 2nt +1( ILI )! X2XL XL XLH.+1(x)x211".+1(x) H'.+i(xk)Xkd (11„,+11dx xthenSince^- ^Xk f. x 2 e _x2 H.+1(x)^dx.Vk ^H'.+1(xk)^(x - xk) XTo calculate (B.5) the Christoffel-Darboux identity is used^H1(x) H i(xk)^H.+2(x) H.+1(xk)^H.+1(x) H.+2(xk)in4-1^X^Xk^X^Xk^X^Xk E  _1.1^Y i am+iym+/(x - xk)Am+2am+1 - A.+1Ynt+1 = f 'x2e-x2 [Hm+112dxwhere Aff,±1 is the leading coefficient of H.+ i/x. 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