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## Material Information- Title:
- Potassium-sodium ion-exchanged waveguides and integrated optical components
- Creator:
- Miliou, Amalia Nikolaos, 1961- (
*Dissertant*) Srivastava, Ramakant (*Thesis advisor*) Ramaswamy, Ramu V. (*Thesis advisor*) Bosman, Gijs (*Reviewer*) Thomson, Ewen M. (*Reviewer*) Phillips, Winfred M. (*Degree grantor*) Lockhart, Madelyn M. (*Degree grantor*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1991
- Copyright Date:
- 1991
- Language:
- English
- Physical Description:
- x, 213 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Alkalies ( jstor )
Diffusion index ( jstor ) Directional couplers ( jstor ) Electric fields ( jstor ) Electrons ( jstor ) Ions ( jstor ) Light refraction ( jstor ) Optical waveguides ( jstor ) Waveguides ( jstor ) Wavelengths ( jstor ) Dissertations, Academic -- Electrical Engineering -- UF Electrical Engineering thesis Ph. D Electrooptical devices ( lcsh ) Integrated optics ( lcsh ) Optical wave guides ( lcsh ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Abstract:
- The K+-Na+ ion exchange process is very attractive for fabrication of passive integrated optical components with assured reproducibility within specified tolerance. Main advantages are the small diffusion rates, use of pure KNO3 molten bath, and single-mode fiber-compatible mode sizes. A systematic theoretical and experimental study of the K+-Na+ ion exchange process was recently conducted in our laboratory which resulted int he correlation of the index profile with the diffusion theory, and a correlation of the index change with the melt composition. A novel method was developed to determine the mobility of K+ ions, and fiber-compatible buried channel waveguides were fabricated for the first time with a two-step process. In this work, the previous studies are extended to obtain a detailed modeling of the index change caused by ionic substitution. the modeling includes the limitations of the existing models and their applicability to the K+-Na+ ion exchange process. More over, a unique systematic method for determining the compressive stress generated in glass due to the ionic substitution is described. To obtain guidelines for fabrication of various devices, the single mode K+-Na+ ion exchange channel waveguide has been modeled using the effective index method and the multilayed step index approximation and the results are correlated with the potassium concentration measurements with the electron microprobe.The results are finally applied to device developments. As an example, symmetric directional couplers as TE/TM polarization splitter is modeled, fabricated, and tested. The spectral response of the device is measured in the 1.0-1.6um wavelength region and the power output at 1.32 um wavelength for both polarizations shows a cross power ratio of greater than -18 dB. In the course of this work, a 3 dB power splitter for TE as well as for TM polarization is also realized using a symmetrical directional coupler. The splitting ratio is 52:48 for the TE and 51:49 for the TM polarization.
- Thesis:
- Thesis (Ph. D.)--University of Florida, 1991.
- Bibliography:
- Includes bibliographical references (leaves 200-212).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Amalia Nikolaos Miliou.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 001759737 ( ALEPH )
26736922 ( OCLC ) AJH2820 ( NOTIS )
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POTASSIUM-SODIUM ION-EXCHANGED WAVEGUIDES AND INTEGRATED OPTICAL COMPONENTS By AMALIA NIKOLAOS MILIOU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1991 "For a moment there was a wild lurid light alone, visiting and penetrating all things" Edgar Allan Poe Copyright 1991 Amalia Nikolaos MIliou ACKNOWLEDGEMENTS I would like to express my deep gratitude to my advisors Dr. Ramakant Srivastava and Dr. Ramu V. Ramaswamy for all their guidance and constant encouragement throughout my study. I am really thankful to them for providing to me a well equipped Photonics Research Laboratory for completing the experimental work needed for this project and also an atmosphere conducive to research and to the interaction and exchange of knowledge among the group members. I am deeply grateful to Dr. R. Srivastava whose invaluable suggestions and critical discussions helped me immensely during the course of this research work. I would like also to express my gratitude to Dr. R. V. Ramaswamy whose challenging discussions stimulated my interest in photonics. Also I would like to extend my deep appreciation to the other members of my Ph.D committee Dr. G. Bosman, Dr. E. Thomson, Dr. U. Kurzweg and Dr. U. Das for taking time out of their busy schedule and being on my committee. Appreciations are also extended to my fellow co-workers Dr. Hsing Chien Cheng and Christopher Hussell for their assistance and stimulating discussions and also to the other present and former members of the Photonics group: Ron Slocumb, Kirk Lewis, Jamal Natour, Mike Pelczynski, Hyoun S. Kim, Sang K. Han, Dr. Young Soon Kim, Dr. Song Jae Lee, Dr. Sang Sun Lee and Dr. Chang Min Kim and fellow researchers: Huo Zhenguang and Dr. Simon Xiaofan Cao for their informative discussions in our weekly group meetings. I am also grateful to many members of the department for their assistance without which it would be very difficult for me to complete my project; I thank each of them: James Chamblee and Jim Hales for maintaining the Microelectronics Laboratory and providing assistance on various processing procedures, Allan Herlinger for machining parts necessary for this work as well as the secretaries Peggy Lee and Betty Lachowski for their help with the administrative procedures. Last, and certainly not the least, I shall always remain grateful to my family for their constant love and support in all my endeavors and especially my husband and best friend Nikola who believed in me and shared with me the difficulties and successes of this journey. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS .................................................................................. iv A B ST R A C T ..................................................................................................................ix CHAPTERS 1 INTRODUCTION.................................................................................... 1 1.1. Integrated O ptics ......................................................................... 1 1.2. K+-Na Ion Exchange.................................................................. 2 1.2.1. K-Na Ion Exchanged Waveguides........................... 3 1.2.2. K+-Na* Ion Exchanged Waveguide Devices ........... 5 1.3. Organization of the Chapters.................................... ............ 7 2 ION-EXCHANGE/DIFFUSION IN GLASS AND NUMERICAL MODELING .............................................. ........... ..... 9 2.1. In trodu action ........................................................................................ 9 2.2. Equilibrium at the Ion Source- Glass Interface............... 10 2.3. Diffusion Kinetics..................................... ................. ..... ........ 17 2.3.1. Diffusion Equation..................... ............................ 17 2.3.2. Thermal Diffusion Without Applied Electric Field................................................. ............................. 2 6 2.3.3. Diffusion With Applied Electric Field ................. 30 2.3.4. Two-Step Diffusion Process.................................... 34 2.4. Peculiarity of K Na Ion Exchange ....................................34 2.5. Solutions of the Diffusion Equation.............................. ........... 36 2.5.1. Analytical Solutions of the Diffusion Equation..... 36 2.5.2. Numerical Techniques for the Solution of the Diffusion Equation....................................................... 39 2.6. Index Change by Ion Substitution....................................... 42 2.6.1. Polarizability and Volume Change.......................... 43 2.6.2. Stress-Induced Index Change................................... 44 2.7. M ixed-Alkali Effect.............................. ....................................... 46 2.8. Space Charge Effect............................................................. ....... 49 2.9. Side Diffusion........................................................................... 50 3 GUIDED WAVES IN PLANAR AND CHANNEL WAVEGUIDES NUMERICAL ANALYSIS ......................................................................... 52 3.1. Introduction ............................................................................... 52 3.2. Guided-Wave Helmholtz Equation ......................................... 54 3.3. Numerical Analysis of 1-D Waveguides ................................ 57 3.3.1. W KB M ethod ................................................................ 58 3.3.2. Finite Difference Method (FDM)............................... 63 3.3.3. Multilayer Stack Theory ......................................... 66 3.4. Numerical Analysis of 2-D Waveguides ................................ 70 3.4.1. Effective Index Method ........................................... 70 3.4.2. Finite Difference Method........................................... 74 4 WAVEGUIDE FABRICATION ............................................................ 76 4.1. Fabrication Procedure .......................................................... 76 4.2. Planar (1-D) Waveguides 77 4.2.1. Sample Preparation.................................................... 77 4.2.2. Melt Preparation ............................... .............. 79 4.2.3. Diffusion- Ion Excahange ................... .................. 79 4.2.4. End-Face Polishing..................................................... 80 4.3. Channel (2-D) Waveguides........................................................ 81 4.3.1. Deposition of Mask Material............... ........................ 82 4.3.2. Photolithography......... ............................................. 83 4.3.3. Wet Etching Technique................................................ 85 4.3.4. Removal of Mask.................................................. 86 4.3.5. Lift-off Technique ................ .............. .............. 86 5 CHARACTERIZATION OF WAVEGUIDES........................................ 88 5.1. Introduction ............... ............................... .................. 88 5.2. Refractive Index Profile.......................................................... 88 5.2.1. Mode Index Measurements ........................................ 89 5.2.2. Inverse WKB Method............................ ........... .. 93 5.2.3. Concentration Profile Measurements ................... 95 5.3. M ode Field Profile ...................................................................... 97 5.4. Loss Measurements ............................... ......... 100 5.4.1. Fresnel Loss.................................1...... 00 5.4.2. Mode Mismatch Loss .................................... .. 102 5.4.3. Propagation Loss......................................... .............103 5.5. Spectral Response ....................................... .......................... 104 6 MODELING OF INDEX CHANGE ........................................................ 107 6.1. Introduction .............................................. 107 6.2. Index Change by Ion Exchange ................................ 110 6.2.1. Huggins and Sun (HS) Model.................................. 112 6.2.2. Appen Model............................. ............................. 114 6.3. Volume Change, Stress and Birefringence......................115 6.3.1. Volum e Change ........................................ ............... 115 6.3.2. Stress-Induced Index Change.................................120 6.3.3. Total Index Change and Birefringence...............123 6.4. M ethodology................................. ............................................124 6.5. Results and Discussion................................................... ...... 128 6.6 Sum m ary............................................................ ............................ 13 5 7 DIRECTIONAL COUPLERS AS TE/TM POLARIZATION SPLITTER AND 3 dB POWER SPLITTER.......137 7.1. Introduction ................................... ........................... 137 7.2. K Na TE/TM Polarization Splitter..................................139 7.3. TheoreticalApproach.................................................................139 7.3.1. Coupled-Mode Theory.......................................... 140 7.3.2. Normal-Mode Approach .......................... ..... 144 7.3.3. Relation between Coupled-Modes and Norm al-M odes................................................ .... 146 7.4. Modeling of the TE/TM Polarization Splitter .................. 150 7.4.1. Design Considerations.............................................150 7.4.2. Index Profile ........................ .... ............. 153 7.4.3. Numerical Method........................................................ 158 7.4.4. Numerical Simulation of the Device......................158 7.5. Fabrication .................................... ........ 171 7.6. Characterization.................. .......................175 7.6.1. Spectral Response Measurements.....................175 7.6.2. Intensity Profile Measurements .......................... 181 7.6.3. Comparison of Experimental Results with Modeling ........................ ........... 181 7.7. 3 dB Power Splitter .................................................. ..................190 7.8. Sum m ary...................................................... ..............................193 8 CONCLUSION AND FUTURE WORK.......................................... 194 8.1. Conclusion............................................................... .................. 194 8.2. Future W ork................................................ ................................ 195 8.3. K+ Na+ Ion Exchange Devices-Practical Configurations.................................. .. ......................................195 8.3.1. Asymmetric Directional Coupler............................196 8.3.2. Asymmetric Y Branch......................................197 8.4. K* Na+ Ion Exchange Devices- Novel Applications .............................................. ..................198 8.4.1. Waveguide Lasers and Amplifiers............................198 8.4.2. Nonlinear Waveguides ........................................ 199 8.4.3. Opto-Electronic Integration .....................................199 REFERENCES............................................................ ...................00 BIOGRAPHICAL SKETCH .............................. ................... 213 viii Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy POTASSIUM-SODIUM ION-EXCHANGED WAVEGUIDES AND INTEGRATED OPTICAL COMPONENTS By Amalia Nikolaos Miliou December 1991 Chairman: Ramakant Srivastava Cochairman: Ramu V. Ramaswamy Major Department: Electrical Engineering The K' -Na+ ion exchange process is very attractive for fabrication of passive integrated optical components with assured reproducibility within specified tolerance. Main advantages are the small diffusion rates, use of pure KNO3 molten bath, and single-mode fiber-compatible mode sizes. A systematic theoretical and experimental study of the K-Na+ ion exchange process was recently conducted in our laboratory which resulted in the correlation of the index profile with the diffusion theory, and a correlation of the index change with the melt composition. A novel method was developed to determine the ix mobility of K+ ions, and fiber-compatible buried channel waveguides were fabricated for the first time with a two-step process. In this work, the previous studies are extended to obtain a detailed modeling of the index change caused by ionic substitution. The modeling includes the limitations of the existing models and their applicability to the K+-Na ion exchange process. Moreover, a unique systematic method for determining the compressive stress generated in glass due to the ionic substitution is described. To obtain guidelines for fabrication of various devices, the single mode K+-Na+ ion exchange channel waveguide has been modeled using the effective index method and the multilayed step index approximation and the results are correlated with the potassium concentration measurements with the electron microprobe. The results are finally applied to device developments. As an example, symmetric directional couplers as TE/TM polarization splitter is modeled, fabricated, and tested. The spectral response of the device is measured in the 1.0-1.6 gm wavelength region and the power output at 1.32 gm wavelength for both polarizations shows a cross power ratio of greater than -18 dB. In the course of this work, a 3 dB power splitter for TE as well as for TM polarization is also realized using a symmetrical directional coupler. The splitting ratio is 52:48 for the TE and 51:49 for the TM polarization. CHAPTER 1 INTRODUCTION 1.1. Integrated Optics Integrated optical communication systems consist of "active" components such as lasers, modulators, etc. and "passive" components such as power dividers, polarization splitters, etc. These components are small and compact and their integration can lead to optical circuits with durable and reliable construction with low-power requirements. These requirements have directed the research into several areas: the development of substrate materials, fabrication processes and techniques and also the development of new devices. A variety of substrate materials such as polymers, fused silica, soda-lime silicate glass, borosilicate glass, pyrex glass, lithium niobate, and III-V semiconductors have been used. The processes of fabrication include ion implantation, sputtering, chemical vapor deposition (CVD), ion diffusion, metal indiffusion, and epitaxial layer growth. The first four methods are used to fabricate waveguides in glass, the metal indiffusion and ion exchange are used for LiNbO3 and the last one for semiconductor materials. Propagation loss of the waveguide and ease of fabrication have been the major criteria for choosing a substrate material. From these considerations, titanium indiffusion in LiNbO3 and ion exchange in glass have become the most popular techniques for waveguide fabrication, followed by a rapid development of epitaxial layer growth of semiconductor materials. The devices developed extend from simple branching structures to more complicated configurations such as high speed switches and electrooptic modulators. The majority of these devices so far has been fabricated on LiNbO3 substrates using Ti indiffusion. The electrooptic tuning applied is extremely useful in meeting the desired device specifications. However, at the receiving end of an optical communication system there is no need to modulate any information and some inexpensive "passive" material such as glass can be considered. 1.2. K Na' Ion Exchange The feasibility of using binary ion exchange in glass for making graded index optical waveguides was demonstrated almost two decades ago [Izaw72]. Since then, significant progress has been made toward understanding the ion exchange process and the role of the processing conditions on the propagation characteristics of the resulting waveguides. Ion exchange process and ionic diffusion of various cations in glasses and the resulting characteristics have been discussed in many excellent reviews [Dore69, Tera75, Find85, Rama88b]. In fact, today, passive glass waveguides are considered to be promising candidates for applications in optical communication systems principally due to compatibility with optical fibers, low cost of fabrication and low propagation losses. Since the glass index cannot be tuned by application of an external field, glass devices must be fabricated with assured reproducibility within specified tolerances. Such reproducible characteristics are more likely to be achieved with the K+-Na+ ion exchange process where small diffusion rates and the use of pure KNO3 molten bath assure better control of the process. Another advantage of the K+-Na+ exchange is the relatively small refractive index change (0.008-0.009) which is very attractive for conventional single-mode fiber-compatible mode sizes. The K+-Na+ exchanged waveguides are also characterized by relatively high birefringence (1x10-3), low losses, and negligible depolarization [Jack85]. 1.2.1. K+ Na+ Ion Exchanged Waveguides The first study of K+-Na+ exchanged waveguides was reported by Giallorenzi et al. [Gial73]. Since then, studies of diffusion profile of K+- Na+ ion exchanged planar waveguides in soda-lime silicate [Albe85, Yip85], BK7 [Gort86a], Pyrex [Gort86b], and semiconductor-doped glasses [Cull86] have been reported. Effects such as stress-induced index change and birefrigence in K+ ion exchanged waveguides have also been observed [Bran86, Tsut86, Albe87, Abou88, Tsut88]. However, in these reports, neither the waveguides nor the device structures were optimized for fiber compatibility, and a systematic study of the diffusion process and its correlation with the waveguide characteristics was lacking. Recently, we conducted a detailed study in our laboratory permitting a deeper understanding of the role of the processing conditions and of the substrate glass in influencing the index profile of planar, surface and buried channel waveguides [Mili89]. The results of this work are listed below. 1. Verification of the index profile. An electron microprobe was used to measure the K+ concentration profile and the data were correlated with the index profile derived from the mode-index characterization and with the diffusion profile calculated by solving the diffusion equation. The index profile for the case of planar waveguides was examined in detail for soda-lime silicate and BK7 glasses and was observed to be Gaussian and ERFC, respectively, in agreement with the earlier reports in similar glasses. The differences in the profiles in the two glasses were attributed to the large disparity of the mobility ratio of the two glasses. This was the first study which explained the index profiles satisfactorily. 2. Correlation of the index change with the melt composition. The surface-index change was measured as a function of the melt composition ( by varying the KNOs/NaNO3 ratio ) and was observed to vary nonlinearly with the melt concentration of K+ a result in sharp contrast to the linear behavior reported earlier in another glass [Garf68]. The observed birefringence and the surface-index change in BK7 glass have now been quantitatively explained on the basis of the stress-induced effects in the glass (see chapter 6 for details). 3. Novel method to determine the mobility of K+ ions. The mobility of the K+ ions was estimated by determining the diffusion depth of the planar surface waveguides fabricated by electromigration. A variation of the ionic current was observed (an almost exponential decrease for soda-lime glass and almost no change for BK7), and the behavior was explained for both the glasses on the basis of the mixed-alkali effect [Tera75]. 4. Fabrication of buried channel waveguides with a two-step process. Using a two-step process, buried channel waveguides for operation at 1.3 (pm were fabricated in both the glasses with nearly circular symmetric near-field intensity profiles. Finally, record low-loss waveguides with insertion loss of less than 1.0 dB in 20 mm long devices were obtained using the two-step process in BK7 glass [Mili89]. 1.2.2. K'-Na+ Ion Exchanged Waveguide Devices Passive waveguide components in glass have shown great promise in recent years for various applications needing optical circuitry. Such devices fabricated by the K+-Na+ ion exchange include wavelength division multiplexers/demultiplexers (WDM), power dividers, ring resonator and TE/TM polarization splitter at visible wavelengths. A single mode wavelength division multiplexer directional coupler was demonstrated by Corning (Europe) for operation at 0.63 (pm and 0.79 (pm [McCo87]. The measured loss was 2.1 dB at 0.63 .m for 4 p.m windows and the calculated rejection was 22 dB for a device of 2.9 mm [McCo87]. The same function was realized with a Y-branch wavelength multi/demultiplexer [Goto90] for 1.3 pim and 1.55 pim. Wavelength separation was demonstrated with rather poor extinction ratios of 5.3 and 5.7 dB at 1.3 and 1.55 p.m respectively. Power dividers have been demonstrated both in the directional coupler [Yip84] as well as in lxN star coupler configurations [Find82, Haru85, Bett90a]. For the first case, the power dividing ratio was close to unity with further control or fine tuning with the deposition of a thin dielectric film over the coupling region [Yip84]. For the second case, the results of the power division vary depending on the number of branches, the presence or absence of bends, as well as on the geometry of the junction region. The splitting ratio was observed to be wavelength and polarization insensitive over a wavelength range of 700 to 900 nm [Bett90a]. However, no reports of operation at 1.3 pIm or 1.55 |jm are available. A ring resonator was demonstrated in K+-Na+ ion exchange [Hond84] but due to fabrication problems the finesse of the device was poor. Nevertheless, using the device as a sensor, it was possible to measure wavelength deviations to 5x10-4 AO, and temperature variations to 0.01 oC. Finally, a TE/TM polarization splitter has recently been reported [Bett90b]. The device relies on the use of birefringence to create differences in the coupling coefficients for the two orthogonal polarizations in an optical directional coupler. The device was demonstrated with an operating bandwidth of 50 nm at 830 nm. The best crosstalk performance achieved was 15 dB and 12 dB in the two output channels and the propagation loss was 0.4 dB/cm. 1.3. Organization of the Chapters In this dissertation a systematic theoretical and experimental study of the K+-Na+ ion exchange process for fabrication of single mode optical waveguides is presented. Moreover, based on those guidelines a symmetric directional coupler has been modelled, fabricated, and tested for TE/TM polarization splitting at 1.3 rim. In chapter 2 the ion exchange equilibrium and kinetics are presented. The nonlinear diffusion equation is derived from the first principles. Issues such as the contributions to index change, mixed- alkali effect, space charge effect and side diffusion are discussed. Chapter 3 covers the propagation analysis of planar and channel waveguides and determination of the propagation constants of the guided modes as well as their associated transverse electric field distributions by solving the Helmholtz equation. A comprehensive review of the numerical techniques used in this work is also presented. Chapter 4 describes the fabrication procedure for the case of planar as well as surface channel and buried waveguides. The characterization of the waveguides is covered in chapter 5. Mode index measurements, concentration and index profile determination, channel waveguide mode field profile, propagation loss, and spectral response measurements are reported in this chapter. In chapter 6, we discuss the modeling of the index change caused by the ionic substitution. The modeling includes the verification of the validity of the existing models, their limitations and applicability to the K+-Na+ ion exchange case. A unique systematic method for determining the compressive stress generated in glass due to the ion substitution is also described. Chapter 7 presents the modeling, fabrication and testing of a symmetric directional coupler used as a TE/TM polarization splitter. The spectral response of the device is measured in the 1.0-1.6 pm wavelength region. The device splits the two polarizations at 1.32 |jm wavelength with a cross power ratio of greater than -18 dB. In the course of this work, 3 dB power splitters for TE as well as for TM polarization are also realized using a symmetrical direction coupler. The splitting ratio is 52:48 for the TE and 51:49 for the TM polarization. Finally the results are summarized in chapter 8 and an illustration of other applications of the K+-Na+ ion exchange process and modeling for devices useful in optical communications is also presented. CHAPTER 2 ION EXCHANGE / DIFFUSION IN GLASS AND NUMERICAL MODELING 2.1. Introduction The study of ion exchange/diffusion in glass is important since it is a simple and effective way of forming a higher index layer in glass substrates necessary for fabrication of optical waveguides. The diffusion mechanism essentially determines the device parameters such as the waveguide depth and the index profile while the ion exchange equilibrium controls the magnitude of the index change at the surface. Ion exchange in glass, by common definition, is the exchange of ions of the same sign and valence between a melt or a solution and the glass framework upon immersion. The process is reversible and stoichiometric, i.e., every ion removed is replaced by an equivalent amount of another ionic species, but the concentration ratio of the two counterions is not necessarily the same in both phases. In the last two decades significant progress has been made [Garf68, Dore69, Izaw72, Find85, Rama88b, Rama88c] toward a better understanding of the ion exchange process and the role of the processing conditions on the propagation characteristics of the resulting waveguides. In the following discussion we describe the ion exchange equilibrium and kinetics at the glass-melt interface and how they influence the boundary conditions for the cation diffusion in the glass. The nonlinear diffusion equation is also derived from first principles giving an insight of the diffusion process. Several cases are considered and it is shown that analytical solutions of the diffusion equation can be obtained only for a few specific diffusion conditions. Different available numerical methods for the solution of the diffusion equation will also be discussed and associated results will be presented. The last sections of this chapter deal with issues related to ion exchange/diffusion such as the contributions to index change, mixed- alkali effect, space charge effect and finally the side diffusion under the mask. These issues have a direct impact on the final result of the process (optical waveguide), 2.2. Equilibrium at the Ion Source Glass Interface Upon immersion of the glass in the melt, given sufficient time, an equilibrium condition is reached between the cations in the melt and in the glass surface such that there are no gradients of concentration in the exchanger. Figure 2.1 describes qualitatively the process. At equilibrium, the ions will be distributed between the two phases in a fixed ratio. This equilibrium can be represented by A+B<-A+B (2.1) BEFORE ION EXCHANGE 0 0 MOLTEN SALT GLASS SUBSTRATE AFTER ION EXCHANGE MOLTEN SALT MOLTEN SALT GLASS WAVEGUIDE Fig.2.1 Ion exchange process where the bar denotes the cations on the interface inside the glass. In a liquid-solid exchange process, the rate of ion exchange can be limited by mass transfer of the reactants to and removal of products from the reaction interface in the melt (i.e. source depletion), by the kinetics of the reaction at the interface, and by the transport of cations in the glass phase. However, if the liquid-phase diffusion can supply ample reactants and remove enough products to and from the interface, the process is not melt mass transfer limited. Also the surface kinetics are not likely to be a rate limiting factor since they are much faster than the transport process in the melt and glass phase. Transfer of cations in the melt takes place mainly via diffusion and to a somewhat reduced extent via convection. Convection is driven by density variations and is not expected to be a dominant mechanism because of isothermal operation and small exchange amounts. Convection can be enhanced by stirring the melt and thus increasing the melt mass transfer rate. However, even in the enhanced convection case, a region may exist near the glass-melt interface where no convective mixing occurs because of fluid friction at the interface. Across this boundary, all the mass transfer occurs through diffusion. The equilibrium state of Eq.(2.1) specifies the surface boundary condition for the diffusion process and the accompanying surface index change. The equilibrium state for the exchange reaction in Eq.(2.1) is governed by the equilibrium thermodynamic constant given by Garfinkel [Garf68] aA aB K = (2.2) aA aB where a's are the thermodynamic activities of the cations. The absolute value of K depends upon the reference functions and states chosen to define the activities. The value of K reflects the "selectivity" of the exchanger and it is a quantitative measure of the preference of the exchanger for one ion over another in solution with it. Many empirical equations have been proposed to describe the variation of selectivity of ion exchangers with solution concentration. The most successful of these equations is that of Rothmund and Kornfeld [Rothl8] who suggested that in the glass phase the ratio of the activities of the ions in an ion exchange system is given by NAn a-B [X-B (2.3) where NA = CA / (CA + CB), (CA is the absolute concentration of the cation A in the glass) and n = lnaiA /alnCA. For ions in silicate glasses "n" is usually of the order of unity. It is assumed that the activities of the cations in the exchanger are proportional to the nt power of their concentration [Lait57, Karr62]; thus, ai = C!. Equation (2.3) has been referred to as n-type behavior [Karr62] and, besides the cation pair depends also on the glass composition. Garrels and Christ [Garr65] have shown that this empirical relationship is equivalent to the regular solution theory [Kirk61] for intermediate glass compositions. Using the regular solution theory for the ratio of the activity coefficients in the melt and the n-type behavior in the glass phase, the relation between the melt concentration (NA) and the surface concentration in the glass (NA) can be written [Garf68] as follows: n[NA1 E (1 2NA) = n A InK (2.4) 11-NAJ RT 41-NA where E is the interaction energy of the two cations in the melt R is the gas constant (8.317 J/oK.mole) and T is the exchange temperature (oK) If the assumption of the regular solution behavior is valid, then a plot of the left-hand side of Eq.(2.4) versus In[NA / (1 NA)] should give a straight line with slope equal to "n" and intercept equal to In(l/K). So far the only work reported in ion exchange equilibrium between glass and molten salt for potassium ions is that of Garfinkel [Garf68] which gives a linear relation between NK' in the melt and NK* in the glass with n=1.2 and K=0.94. Thus, the glass exhibits almost equal preference for the two ions at T=500 oC with sodium being slightly favored for that particular composition of glass that does not contain any potassium. However, the value of K is a function of temperature and composition of the glass and the melt. Therefore, in a study conducted in our laboratory, measuring the surface index change as a function of the melt concentration at T=370 oC in BK7 and soda-lime silicate glasses, we observed a nonlinear behavior, Fig.2.2. As an attempt to explain our results we compared the content of K20 and Na20 in the two glasses as well as the difference in ionic radii of the two ions, Table 2.1. Moreover, using the results of Fig.2.2 we 0.01 0.009 --- BK7 -e- Soda Lime 0.008 0.007 An 0.006 0.005 0.004 0.003 0.002 70 75 80 85 90 95 100 % KN03 Fig. 2.2 Index change vs percentage of KN03 in the melt TABLE 2.1 GLASS COMPOSITION (wt %) Composition Soda-Lime Silicate BK7 Si02 Na20 K20 MgO B203 A1203 Traces 72.25 14.31 1.2 6.4 4.3 1.2 0.34 69.6 8.4 8.4 9.9 2.5 1.2 rK = 1.28 x 10-10 m and rNa = 0.95 x 10-10 m plot Eq.(2.4) for BK7 glass, Fig.2.3. As it was expected we observe a linear behavior. From the plot we can deduce the values of n=1.3 and K=0.0004, and since K<1 the glass has a preference in retaining sodium for this particular temperature and composition. This result is in good agreement with the small diffusion coefficient and index change of the system. 2.3. Diffusion Kinetics The index profile in ion-exchanged waveguides is a replica of the diffusion (concentration) profile which can be calculated by solving the diffusion equation with appropriate boundary conditions. In general, the intruding and original ions will have different mobilities; thus as diffusion proceeds, one ion tends to outrun the other, leading to a buildup of electrical charge. However, accompanying this charge is a gradient in electrical potential that slows down the faster ion and speeds up the slower one. To preserve the electrical neutrality, the fluxes of the two ions must be equal and opposite: the electrical potential ensures this condition in spite of the difference in the mobilities of the two ions. 2.3.1. Diffusion Equation Much of the recent experimental work has been devoted to testing and confirming theoretical predictions of the Nernst-Planck model, which has replaced earlier models that employ constant inter- diffusion coefficient. The inter-diffusion system described in this work is based on the Nernst-Planck equations, which are derived by 0.5 0.4 0.3 0.2 0.1 -5.9 -5.8 -5.7 -5.6 n [ NA / 1- NA Fig. 2.3 Plot of Eq. (2.4). The slope of the line is equal to "n" and the intercept is equal to ln(1/K) -5.5 assuming that the driving force for diffusion is the electrochemical potential and that the inter-diffusion coefficient describing the system is concentration dependent. According to the Nernst-Planck equations the flux for each cation species is given by Oi = Di Vi +L i zi Ci E (2.5) alnCi where Oi: molar flux of cation i (A or B) (moles/m2. sec) Ci: concentration of cation i in glass (moles/m3) Di: self-diffusion coefficient of cation i (m2/sec) gi: electrochemical mobility (m2/V. sec) zi : electrochemical valence (zi =1 monovalent ions) and E: local electric field (V/m), which consists of the space- charge induced field distribution near the diffusion boundary, Es, which moves deeper into the substrate as time evolves, and the externally applied field EA. The space charge field is discussed later in this section. The first term in Eq.(2.5) represents the ions diffused under the chemical potential difference resulting from the concentration gradient, while the second term represents the ions driven under the electric potential difference or the drift term. The Di and ti are temperature dependent [Dore64] and are assumed not to be functions of concentration Ci and therefore independent of position and time during the diffusion; however, this oversimplification is not justifiable in the case of strong multi-alkali effect [Isar69, Day76, Char82b]. It should also be noted that the self-diffusion coefficient varies with glass composition and its temperature dependence below the glass transition temperature is given by an Arrhenius type relation, Fig.2.4, [Dore64] Di =Do exp[- (2.6) where Do: constant and AH: activation energy (J/mole). The activation energy is made up of two contributions: the Coulombic energy required to separate positive and negative charges and the energy to squeeze an ion through a restricted opening in the network. The Do and AH depend on the glass composition as well as the ion pair involved in the exchange. Further, if it is also assumed (for simplification) that the ionic transport (conduction) mechanism under an applied field is the same as that for diffusion. Then the Nernst-Einstein relation is satisfied, i.e. Di = kT = k _T iL (2.7) ni q2 q where k: Boltzmann's constant (1.38 x 10-23 J/K) ni: number of ions i per m3, i=A, B oi: ionic conductivity (mho/m) and q: electronic charge (Cb) -6.8 I I I I -7 7.2 -7.4 1 0.00151 0.001527 0.001543 0056 0.00156 1/T (OK) Fig. 2.4 Self-diffusion coefficient of potassium ion in BK7 glass as a function of temperature In most cases, however, the thermal migration of cations is thought to be slightly different from the electric field-induced transport and Eq.(2.7) is not satisfied [Beie85, Tera75]. In these cases the relation between the self-diffusion coefficient and the mobility is instead written as Di = fk i (2.8) The above relation involves a correlation factor f whose value varies from 0.1 to 1 depending upon the composition of the glass [Beie85, Tera75]. It is convenient to assume that the exchange is strictly one-to- one at all times and thus the space charge can be neglected, i.e., CD << Co However, there are two dissimilar species of unequal mobilities involved in the exchange process. As a result, local imbalance in the charge distribution (space-charge) is created in the glass giving rise to an electrochemical potential. The local field thus created assists the movement of the slower ions (K+) while impeding the progress of the faster ions (Na+), making the flow rates of the two cations equal and maintaining the charge neutrality condition, outside the space charge region. From the condition of electrical neutrality for the glass, it is required that the total ionic concentration be constant. CA + CB + CD = Co (2.9) where Co : concentration of ion B in glass before diffusion. and CD : concentration of the net depleted mobile ions (space charge. The diffusion equation for the incoming ion can be derived as follows; the electric field E in Eq.(2.5) is determined via the Poisson equation V. (eE) = p (2.10) where the local space charge density, p, is given by p = q (CA+ CB Co) =-q CD (2.11) For a weakly guiding case, the change in the dielectric constant, E, due to ion substitution is very small and Eq.(2.10) can be replaced by V. (E) = (CA + CB Co) 0 (2.12) In most cases of interest the space-charge effect can be neglected, because the space charge density corresponding to the induced local field is usually quite smaller compared to the mobile charge density. Thus, the flux of the incoming ions into an elemental volume *e does not represent the dielectric constant at optical frequencies, which determines the refractive index of the medium and depends on the composition of the glass. It is rather a low frequency dielectric constant of the isotropic substrate. 24 approximately balances the flux of the outgoing ions, i.e., we assume CA + CB = Co, where Co is constant. Thus, aCA aCB (2.13) (2.13) at at Since the number of ions is conserved, the continuity equations are applicable aCA -- V. )A at and (2.14) aCB -- V. tB at Hence V.((A+ )B) = 0 (2.15) Substituting Eq.(2.5) for the two ions in Eq.(2.15), and assuming constant self-diffusion coefficients and mobilities, the divergence of the field E can be expressed as 2- 2- . n (DA VCA + DB V 2CB) E.(A VCA + B VCB) ( V. E- (2.16) 1A CA + B CB The rate of change of the concentration of ion A can be derived substituting Eqs(2.5) and (2.16) into Eq.(2.14) aCA 2- S=n DA VCA [A E.VCA --A CA V.E at DA 2C- (DA/DB) JLA CA + 9B CB = n DAV CA CA CA + PB CB A LB(CA + CB) A E. VCA -CA+C (2.17) 9A CA + 9B CB In terms of normalized concentration NA and NB (mole fractions) Eq.(2.17) can be rewritten as aNA 2- (DA/DB) -A NA + B NB = n DAV NA at 9A NA + B NB 9A 9B E. VNA -- AB (2.18) -A NA + tB NB or A =DV2NA- E. VNA (2.19) at where D is the inter-diffusion coefficient defined [Dore64] as Sn D (DA/DB) 9A NA + B NB 2.20 D = n DA (2.20) DA NA + lB NB and g is the inter-mobility coefficient defined as 9L= IAB 9A A (2.21) A NA + LB NB 1 ( 1 ) NA 1- a NA 9B where a= ( A) (2.22) tB Applying the Einstein relation, Eq.(2.7), we see that DA/ 9IA = DB / PB and Eq.(2.20) can be simplified as p = n DADB nDA nDA (2.23) DA NA + DB NB 1 ( 1- A NA 1- a NA tLB However, it is well known that the Einstein relation does not always hold and a modification is needed. The modified Einstein equation, Eq.(2.8) can be applied to obtain Eq.(2.23), assuming that the correlation factor fA = fB. From the definition of the inter-diffusion coefficient Eq.(2.20) and inter-mobility-coefficient Eq.(2.21) it is clear that D and R are concentration dependent for a NA # 0. Equation (2.19) is the nonlinear diffusion equation of the system. Numerical techniques, described in section 2.5.2, have been employed to solve the nonlinear differential equation Eq.(2.19). However, explicit analytical solutions can also be obtained for specific cases as shown in, section 2.5.1. 2.3.2. Thermal Diffusion without Applied Electric Field The absence of an electric current (EA = 0) gives (2.24) OA + OB = 0 27 In this case, E = Es, the local field due to space charge is given by [Lili82] Es = (DA DB)VNA (2.25) A NA + LB NB or Es =-fkT VNA (2.26) q (1 a NA) Then the nonlinear diffusion equation can be rewritten as NA D A +NA ~.V NA)1 = V. (D VNA) (2.27) 5t (1 a NA) Several cases are considered where simplifications of the 3-D nonlinear equation (2.27) occurs as follows [Cran75]: (a) Two dimensional case: Equation (2.27) can be reduced to DNA n DA ANA a n DA aNA ( + -( (2.28) at ax 1 a NA ax ay 1 a NA ay The analysis of the 2-D case with concentration dependent inter- diffusion coefficient is complex due to lack of boundary conditions when masks are present [Wilk78]. Numerical solutions are usually used for the solution of Eq.(2.28), described in section 2.5.2. (b) One dimensional case: Further reduction of Eq.(2.27) yields aNA D n DA aNA NA ( n DA N) (2.29) at 3x 1 a NA 3x (c) a=0, i.e.; 9A = 9B The two ions are equally mobile. Thus Eq.(2.28) and Eq.(2.29) become NA n DA 2NA + NA (2.30) at )x2 y2 and NA- n DA (2.31) at ax2 respectively. Crank [Cran75] expects a complementary error function solution as "a" approaches zero. An example is given in Fig.2.5 which presents the K+ concentration profile in the BK7 glass. (d) NA << 1 When the concentration of the incoming ion is much smaller than the host alkali ion in the glass, Eq.(2.28) and Eq.(2.29) are reduced to Eq.(2.30) and Eq.(2.31) respectively. This approximation is not valid when diffusion is carried out from concentrated melts, K+-Na+ process from pure KNO3 melt, but can be used when Ag+-Na+ exchange is performed from dilute melts [Rama88b]. o e-Microprobe ...- ERFC SA Diffusion Eq. 0.8- a = 0.01 No = 0.90 0.59- N(x) 0.39 0.18 ' 0 5 10 15 20 25 x [rim] Fig. 2.5 Concentration profile of BK7 glass fitted to the solution of the Diffusion Eq. and to a ERFC profile (e) = 1, i.e., 4A << B (DA << DB) Equations (2.28) and Eq.(2.29) are reduced to Eq.(2.32) and Eq.(2.33) respectively aNA ( n DA NA) a n DA aNA ) + -( ___) (2.32) at ax 1- NA x By 1 NA ay and aNA a ( n DA NA2.33) at ax 1- NA ax Crank [Cran75] expects a step-like profile as "a" approaches unity. For intermediate values of "a" between 0 and 1 the profile obtained is Gaussian-like as in the case of K+ Na+ exchange in soda-lime silicate glass, Fig.2.6. 2.3.3. Diffusion with Applied Electric Field The diffusion of incoming ions can be enhanced by applying an external electric field across the substrate and this decreases the processing time and temperature and modifies the index profile. In the case of an applied electric field the total ionic flux, 0o, corresponding to the electrical current density, J = F Io is A +B = 0 OO (2.34) where F is Faraday's constant (96,485 Cb/mole). Using Eq.(2.5), the total field can be written as 0.8 0 0 e-Microprobe 0.7 0 -- auian 0 A Difuon Eq 0.6 a 0.98 N 0.8 0.5 N(x) 0.4 0 0 0.3 0 0.2 0.1 O O 0 05 10 15 20 25 30 x (Gm) Fig. 2.6 Concentration profile of soda-lime glass fitted to the solution of Diffusion Eq. and a Gaussian profile E=Es+EA -f kT aVNA +fkT (OA + B) q (1 a NA) q BCO ( NA) (2.35 where the applied electric field is given by f k T (A + OB) EA = fk (-PA+4~B (2.36) qEA C (1 a NA) ( The space charge is generally nonzero in the case of field-assisted diffusion if "a" is not zero. Simplification of Eq.(2.19) with applied electric field for the case of NA << 1 or a = 0 yields aRA 2- S= n DA V A E VNA (2.37) St Abou-el Leil and Cooper [Abou791 have analyzed the problem of electric field induced ion exchange in detail from the viewpoint of strengthening the glass and compared the results with the experimental data. Recently numerical solutions of Eq.(2.37) have been obtained and compared also with the experimental results in the cases of Ag+ Na+ [Chen87] and K+ Na+ waveguides [Mili89, Albe90]. Figure 2.7 presents a typical step-like index profile (a = 0) when an external electric field is applied. 1.518 ' A Mode Indices (InWKB points) Diffusion Eq. 1.5164 E2= 100 V/mm t2= 82 sec 1.5148 1.5132 1.5116 1.51 I I I 8 12 16 20 24 28 x [im] Fig. 2.7 Index profile of soda-lime glass with applied field fitted to the solution of the Diffusion Equation 2.3.4. Two-Step Diffusion Process In order to increase the symmetry of the index profile (thereby improve the fiber-waveguide coupling) and reduce the scattering losses caused by the proximity of the glass surface to the waveguiding region, a second-step diffusion is necessary in absence of cation A in the melt. The diffusion equation is solved numerically in this case with the first-step index profile as the initial condition [Chen87]. In the case of planar waveguides, the width of the diffused guides varies as a square root of the second-step diffusion time t2 and the depth Xpeak, to which the guide is buried, is proportional to the product of the applied field E2 and the time t2. Moreover, in order to achieve symmetric index profile electric field must be applied in the second- step. An attempt to model the two-step process for a single-mode buried channel waveguide has been made by Albert et al [Albe90]. However, the model does not include the mixed-alkali effect, stress, and the effect of side diffusion which have significant contribution to the K Na ion exchange case. Thus, a detailed simulation of the two- step process and comparison with experimental data remains a challenge. 2.4. Peculiarity of K+ Na' Ion Exchange When the sizes of the exchanging ions are quite different, e.g. the K' Na+ exchange, large stresses are built up in the glass. These stresses slowly relax if the exchange temperature is not far below the glass transition temperature, and this relaxation can change the short- range structure of the glass and the mobilities of the ions. However, if the ion exchange is conducted at sufficiently low temperatures, the structure of the glass cannot relax, thereby creating internal stresses arising from swelling compressivee stress). Under these conditions we are no longer dealing with a system in mechanical equilibrium. Thus the basic transport equation Eq.(2.5), in addition to chemical and electrical potential terms, includes a third term deriving from the partial stress of the penetrant [Cran75]. In this case, the thermodynamics of the irreversible processes leads to an expression for the flux given by i = -Di VCi + +lizi Ci E + Ci vs (2.38) alnCi where vs is the velocity of the ions moving under strain. The velocity Vs can be expressed as a function of the created stress. Thus on making the assumption that the stress is proportional to the total uptake of the penetrant ions Eq.(2.38) becomes aln1in i -Di VCi + Ji zi Ci E + L V. o (2.39) alnCi where a is the stress tensor of the penetrant ions [Fris69, Wang69, Sane78]. Moreover, the generalized diffusion equation, Eq.(2.19), is substituted by aNA 2 2 NA = D V2NA E. VNA [ V2 (2.40) at Thus in cases as K_ Na+ exchange where the swelling is predominant, Eq.(2.40) must be used instead of Eq.(2.19). Some researchers as will be explained in chapter 6 often include the stress effect in the diffusion term by defining a polarization dependent self- diffusion coefficient [Yip85]. This approach is unrealistic since the stress induced term is not proportional to the gradient of the ionic concentration. However, the index change depends on polarization due to anisotropic stress [Tsut88]. 2.5. Solutions of the Diffusion Equation Analytical, close form solutions are only possible for the case of one-dimensional diffusion equation with or without an applied external field in the glass and under special conditions. However, in the two- dimensional case, there is an abundant number of different numerical techniques that can be used for the solution of the diffusion equation. 2.5.1. Analytical Solutions of the Diffusion Equation (a) Solution of Eq. (2.31) Analytical solutions are available for 1-D Eq.(2.31) which is valid in the case that no external field is applied and one of the following holds: (i) equal mobility ions (i.e., no space charge); (ii) unequal mobilities but low concentration (i.e., small space charge); these belong to cases (c) and (d) in the previous section. A complete solution of Eq.(2.31) requires knowledge of the initial and boundary conditions. The concentration NA at the glass surface is constant during the diffusion; therefore, At the surface, NA(x,t=0) = 0, NA(x=O, t) = NAo , for x > 0 for t 2 0 (2.41b) At the other boundary for infinite glass thickness, NA(-, t) = 0, for t 2 0 (2.41c) The solution of Eq.(2.31) [Dore64, Cran75] is NA(x,t) = NAo erfc( X Weff (2.42) where erfc( x / Weff) is a complementary error function profile defined as ex/Weff erfc( x / Weff) = 1 exp(-s2) ds Jo where Weff = 2 n DAt (2.43) is called the effective depth of diffusion and corresponds to that distance from the waveguide surface (x=0) where NA / NAo = erfc(1) = 0.157. The 1/e width of this profile is given [Rama 86] by Wi/e = 0.64 Weff. Such index profiles have been observed (2.41a) in dilute Ag Na+ exchange [Lagu86a], and K Na process in BK7 glass [Mili89], as shown in Fig.2.5. In general, the K Na+ ion exchange is carried out from concentrated melt solutions (infinite diffusion source) and ion exchange equilibrium results near the surface shows that NAo -- 1. For finite diffusion source the concentration at the surface decreases if the ions are not replenished during the diffusion and the index profile becomes closer to Gaussian due to the difficulty in terminating the diffusion exactly at the point of exhaustion. (b) Solution of 1-D in the presence of an external field For one dimension Eq. (2.37) is reduced to aNA n DA 2A- AE aNA (2.44) at ax2 ax Approximate analytical solutions are obtained for the following specific cases: (i) For the special boundary conditions NA = NAo, NA0 for x = (2.45a) ax and NA = for x -oo (2.45b) ax the solution of Eq.(2.44) is [Abou79] NA(X) = + exp[vNAo +expvN Ao a (x- v t)]-1 (2.46) \n DA with the concentration-dependent diffusion front velocity v = vo (1 a) / (1 a NAo) and Vo depending on the current density caused by the applied field. (ii) For low concentrations (NA << 1) For the conditions given by Eq.(2.41) an analytical solution [Cuch61, Malk61, Lili82] is obtained as NA = NAo erfc(u u') + exp(4uu') erfc(u + u')} (2.47) 2 where u = and u' AE t Weff is defined by Eq.(2.43) and t is Weff Weff the time in seconds. For large electric fields such that u' > 2.5 it can be shown [Rama86] that the contribution of the second term in Eq.(2.47) can be neglected and diffusion does not play significant role in the index profile. The index profile is step-like with the diffusion depth given by Vo t [Abou79], where Vo is the velocity of the diffusion front. 2.5.2. Numerical Techniques for the Solution of the Diffusion Equation The 1-D and 2-D Diffusion equations can be solved numerically by a number of well known methods: the finite difference method (FDM) [Cran75], finite element method (FEM), the Green's function approach [Chen87, Rama88a] etc.. In this section we describe briefly the finite difference method (FDM) used in this work. To illustrate the FDM technique let us consider 1-D diffusion. We first let the range in x be divided into equal intervals 6x and the time into intervals 6t, so that the region x-t is covered by a grid of rectangles of sides 6x, 6t. Let the coordinates of the representative grid point (x,t) be (i6x, j8t), and also denote the value of NA at the above point as NA(i,j) where i and j are integers, Fig.2.8. Fig.2.8 Uniform discretized grid for FDM scheme By using the Taylor's series in the x direction and keeping t constant, we can write S2 x2- NA(i+l,j) = NA(ij) + 6x (A) + 1 (x)2 (a NA ax x,,tj 2 ax2 x,,tj + 0 (5x)2 DNA NA(i-l,j) = NA(i,j) 6x ( A) Sx x,t, + 1 (5x)2 (aNA) 2 aX2 x,,tj + 0 (5x)2 On adding we find S2NA NA(i+1,j) 2 NA(ij) + NA (i- ,j) + (5x)2 a-2x2 ,tj (8x2) (2.48) (2.49) (2.50) 41 Similarly, by applying Taylor's series in the t direction, keeping x constant, we have 8N82- _NA-A 1 (8t)2 (NA NA(ij+1) = NA(i,j) + t (aNA) + x t 2 (t2 + -.. (2.51) from which it follows that 4NA = NA(ij+1)- NA(ij) + (2.52) at xtj 8t where the O (8x)2 and O (6t) are the terms that have been neglected (error terms). By substituting Eq.(2.50) and Eq.(2.52) into Eq.(2.31) we find after slight rearrangement NA(ij+l) = NA(ij) + r (NA(i-,j) 2 NA(i,j) = NA(i+l,j) (2.53) where r = n DA6t / (5x)2 We can use Eq.(2.53) with a chosen value of "r" to calculate the values of NA at all points along the successive time rows of the grid provided we are given some initial starting value at t=0, and some conditions on each of the boundaries. This method is called "explicit method" since one unknown value can be expressed as a function of known values. Crank and Nicolson [Cran75] proposed a variation of this method, called the "implicit method", by replacing a2NA / ax2 by the mean of its finite-difference representation on the jth and (j+l)th time rows. Under this approximation Eq.(2.31) becomes -r NA(i-lj+l) + (2 +2r)NA(ij+l) r NA(i+l,j+l) = r NA(i-l,j) + (2 -2r) NA(ij) + r NA(i+1,j) (2.54) Thus if there are N internal grid points along each time row, then for j=0 and i=1,...,N Eq.(2.54) gives N simultaneous equations for N unknown values along the first time row expressed in terms of the known initial values and the boundary values at i=0 and N+1. The strong advantage of this method is its convergence and stability. However, this method can not be used for solving the nonlinear diffusion equation since it is impossible to convert a nonlinear equation to a matrix which is linear in nature. In this case the first approach is more appropriate. 2.6. Index Change by Ion Substitution The substrate of the glass is a network of glass former, the most common of which are SiO2 and B203, modified by other components of the glass composition. In the binary ion exchange process, both the cations which exchange with each other are network modifiers. As a result, the basis structure of the glass is left unchanged and only the refractive index of the glass is modified. The net index variation depends on three major physical changes; namely, (1) ionic polarizability, (2) molar volume due to the different ionic radii and (3) 43 stress created by the size mismatch upon substitution since the ion exchange takes place below the stress relaxation temperature of the glass. 2.6.1. Polarizability and Volume Change The electric field of the light propagating through the glass interacts with its polarizable ions causing displacement of the electronic charges with respect to their nuclei, creating dipoles. Such an interaction lowers the phase velocity of light by a factor n, the index of refraction of the glass. For an isotropic medium consisting of a large number of molecules of various species, the polarization is given by P Ni Pi = Ni ai Ei (2.55) i i where Pi is the dipole moment of the i molecule, Ni is the average number per unit volume and ai is the molecular polarizability. The Eoc0 represents the local field, given by the summation of the macroscopic electric field E with an internal field, created by the closely packed molecules, i.e., -1oc = E + 1 (2.56) 3eo The dielectric constant of the medium is defined as =D EoE+ P (2.57) EoE EoE where eo is the free space permittivity and D is the electric displacement. Combining Eq.(2.55)-Eq.(2.57) we get e 1 = Ni ai (2.58) e + 2 3 Eo Equation (2.58) is known as the Clausius-Mossotti relation. At optical frequencies, Re(e) = n2, where n is the index of refraction, the equation is called the Lorentz-Lorentz [Lore80a, Lore80b] relation n2 1 Ni 1i R (2.59) n2 + 2 3 eo V where R is defined as the molar polarizability and V the molar volume. The molar volume determines the density of the polarized ions. Several models exist in order to estimate the refractive index of glass or its changes as a function of composition. These are all based on the fact that the refractive index is an additive quantity and they do not take into account the multi-alkali effect (see section 2.7). These models are described in details in chapter 6. 2.6.2. Stress-Induced Index Change The theory which relates the changes in the index of refraction to the state of stress is based on the photoelastic effect which couples the mechanical stress to the optical index of refraction. The state of stress and stain in an elastic body (solid) is characterized by second- order tensors. When stress is applied the dielectric tensor changes and in the first approximation these changes are linearly related to the stress and strain components [Dall65] n i (2.60) where A -I is the index of refraction tensor, and Pij is elements of the stress optic tensor. medium) Pij is given by [Dall65] Pij= P11 P12 P12 P12 P11 P12 P12 P12 P11 0 0 0 0 0 0 0 0 0 0 0 0 P44 0 0 0 0 0 0 0 0 0 0 P44 0 0 P44- oJ is the stress tensor For glasses isotropicc (2.61) where 2P44 = (P11 P12). Combining Eq.(2.60) and Eq.(2.61) we get P11 ox + P12 (Gy + Tz) A P11 n2 Pi Gy + P12 (Gx + aGz) (2.62) oz + P12 (Gx + ay) A '= Ln2J]x Forming the equation of the index of ellipsoid in the presence of stress we get the new refractive indices and also the index differences given by [Dall65] Anx = C11 x + C12 (ay + oz) Any = C11 (y + C12 (Ox + z) (2.63) Anz = C11 oz + C12 (Ox + y) where Ci = n3 P, is known as the photoelastic coefficient and n is the refractive index of the unstressed medium. Equations (2.63) show that stress transforms an isotropic medium into an anisotropic one, causing strained glass to exhibit birefringence. This birefringence can be easily observed in K+-Na+ exchanged. 2.7. Mixed Alkali Effect The "mixed-alkali effect" is the nonlinear behavior of the material properties on substitution of one alkali (i.e. K+) ion for another (i.e. Na+) in alkali containing substrates, such as occurs in the ion-exchange process. Properties most affected are those associated with alkali ion movement such as electrical conductivity, alkali diffusion and alkali self-diffusion coefficient [Isar69, Day76, Neum79] which typically exhibit either a minimum or a maximum, with the substitution of a second alkali, at a composition close to that of the crossover of the individual alkali properties Fig.2.9(a) and Fig.2.9(b). The mixed alkali effect increases with the difference in size between the two alkali ions. The alkali exchange process in glass leads to inter-diffusion of the monovalent cations in the rigid glass matrix. As the two ions of different mobilities inter-diffuse, a potential gradient is set up to maintain electrical neutrality. When ionic activity coefficients in the glass are constant, inter-diffusion of ions A and B can be treated in terms of an inter-diffusion coefficient D defined in Eq.(2.20). When the glass is first immersed into the molten salt the concentration of the second alkali in the glass (NA) is very small and D = n DA, i.e. the exchange process is controlled by the mobility of the second alkali which is smaller than that of the first alkali. It is important to note that although the self-diffusion coefficient of the alkali in the glass DB (Na+) is larger than DA (K+), its value is initially unimportant to the exchange process, Fig.2.9(a). Ideally, the maximum initial exchange rate occurs when the two alkali ions have the same mobility. However, compositional changes that increase DA (K+), will increase the exchange rate, since increasing the mobility for the less mobile ions is more important than lowering the mobility of the more mobile ions by the same amount. A general consequence of adding a second alkali is a significant reduction in the diffusion coefficient of the original alkali ion, Fig.2.9(a). The alkali ratio corresponding to the crossover point, is usually not found at equal concentrations of the alkali ions but is a function of: (i) the type of alkali ions present (ii) the total alkali concentration and (iii) the glass former. The change in alkali diffusion 48 DNa -8 - -10 o (a) -11 S-12 0 0.2 0.4 0.6 0.8 1.0 Alkali ratio K/K+Na 100 O O 4 10 - o 0 I 1 \ 0 0.2 0.4 0.6 0.8 1.0 Alkali ratio K/K+Na Fig.2.9 (a) Alkali diffusion coefficient and (b) electric conductivity vs alkali ratio coefficient provides a basis for interpreting also the change in the electrical conductivity. Because of the different compositional dependence for the two alkali diffusion coefficients, the minimum in the conductivity occurs closer to the concentration where CA = CB, Fig.2.9(b). Several theories have been presented in the literature trying to explain the mixed-alkali effect emphasizing the structural features of the glass and electrodynamic interaction between ions but none of them has been successful in explaining the phenomenon fully [Day76]. In general, however, the selection of alkali system is very important since the diffusion coefficients vary with the overall chemical composition. It should be possible to minimize the difference between the the two diffusion coefficients by compositional changes which increase the diffusion coefficient of the less mobile ion. Such optimization of the glass composition has indeed been undertaken recently [Ludw87]. Finally, the alkali ratio in the molten salt bath can be controlled so that the alkali ratio at the surface does not exceed that value where D is a maximum. 2.8. Space Charge Effect As discussed previously, since the substrate glass is rich in cations B, 9A ( [B when the substrate is immersed in the melt. Cation B migrates faster than cation A, causing the space charge effect. As a consequence an electric potential develops balancing the diffusion process [Char80]. The induced field affects the movement of both the ions, A (K+) and B (Na+), opposing the chemical potential driving B (Na) and aiding that of A (K+). As a result the out-diffusion of the sodium ions is slowed down and the movement of the potassium ions into the glass is accelerated until an ionic equilibrium is reached. The extent to which space charge effect modifies the index profile depends upon the range of variation of MA and JB in the two extreme concentrations resulting in index profiles from complimentary error function to step index. 2.9. Side Diffusion In the case of channel waveguide fabrication by ion-exchange, side diffusion under the mask is created by an electric field generated by a potential difference between the nitrate bath and the metal mask used in the process, due to metal cations (e.g. A13+) ions going into the solution and leaving electrons behind [Walk83b]. Thus, the mask is at negative potential with respect to the melt. If the negative potential of the mask is greater than the negative potential of the glass with respect to the melt then the in-diffused ions will be accelerated and the out-diffused ions will be retarded. However, since the ion exchange is strictly one-to-one the refractive index increase must also be retarded and all excess in-diffused ions not balanced by the out- diffusion will be deposited beneath the mask edge. The side diffusion increases the waveguide's width and influences its index profile. Moreover, in the case of Ag*- exchange it was found that the induced local field at the edges of the mask reduce the silver ions into metallic form in that region, increasing the loss of the waveguide [Walk83b]. Anodization of the aluminum diffusion mask has been proven to be useful and simple technique for the production of dielectric masks [Walk83b]. Moreover, the use of a sufficiently thick (300 AO) A1203 layer of low ionic conductivity may be used to isolate the metal from the melt, as described in chapter 4, and thus reduce the side diffusion effect as well as the silver ion reduction. Finally, a more complicated technique can be used to avoid side diffusion by placing "guard rings" at the areas underneath the mask where the accumulation of undesired ions is taking place [Clin86]. CHAPTER 3 GUIDED WAVES IN PLANAR AND CHANNEL WAVEGUIDES NUMERICAL ANALYSIS OF PROPAGATION 3.1. Introduction The field of integrated optics is based primarily in the fact that light waves can be guided in very thin layers (optical waveguides) of optically transparent materials. Combining these waveguides together and shaping them into appropriate configurations, a large variety of components that perform a wide range of operations on optical waves have been realized especially for applications in the areas of optical communications and signal processing. The understanding of the waveguide characteristics of both planar and channel waveguides is, therefore, essential in the design of high performance devices. Propagation analysis of a dielectric waveguide with a given index profile (determined by the diffusion process) consists mainly of solving the Helmholtz wave equation. This process yields the number of modes supported by the waveguide, their propagation constants, and their associated transverse electric field distributions. This analysis is only exact for a selected number of refractive index distributions i.e., quadratic and exponential. However, in most practical cases the index profile may approach a Gaussian or a complementary error function and it is necessary to use numerical techniques. Some of the widely used methods include (a) WKB Method [Whit76, Chia85, Sriv87a] (b) Matrix Method [Chil84, Ghat87] (c) Multilayer Stack Theory [Thur86] (d) 1-D and 2-D Finite Difference Method (FDM) [Ster88a, Ster88b] (e) Variational Approach [Mats73, Mish86, Haus87b] (f) Variational Approach with Finite Element Method (FEM) [Yeh79, Maba81, Kosh85a, Kosh85b, Lagu86b] (g) Perturbation Technique [Kuma83] (h) Spectral Index Method [Mcil89, Mcil90, Ster90], and (j) Effective Index Method (EIM) [Rama74, Hock77, Chia86, Veld88]. Generally in all of the above numerical techniques we assume that the index profile is invariant along the z-direction (propagation direction). However, some integrated optical devices involve evolution of the propagating wave along the z-axis i.e., taper velocity couplers, cross couplers, etc. In these cases numerical techniques such the beam propagation method (BPM) [Feit78, Feit79, Feit80], the local normal mode approach [Marc74] and the step transition model [Marc70] are applicable. A comprehensive review of numerical methods is presented by Kogelnik [Koge88]. The first part of this chapter deals with the derivation of the Helmholtz equation while the remainder discusses some of the most used numerical approaches for both one and two dimensional waveguides. 3.2. Guided-Wave Helmholtz Equation Wave propagation in a source-free, linear dielectric medium is governed by the vector Helmholtz equations, known also as the wave equations. Assuming a z-propagating wave with its time and z dependence expressed as exp(jot jpz) and E, No and P independent of z, the wave equations become [Koge88] Vt2 Et + Vt (Et Vt InE) + (W2 Epo p2) Et = (3.1) and Vt2 Ht + (Vt Ine) x (Vt x Ht) + (02 E to 2) Ht = 0 (3.2) where Et: transverse component of the electric field Ht : transverse component of the magnetic field co : angular frequency S: permittivity of the medium to : permeability of the medium and 3 : propagation constant The Et, Ht and E depend only on the transverse coordinates x and y and they are also independent of time. The transverse Laplacian is a2 92 defined in the Cartesian coordinates as Vt2 + T h e ax2 ay2 corresponding z-components of the fields are derived from V. H = 0 and V. (E E) = 0 and are given in terms of the transverse components as follows: j Ez = Vt. Et + Et. Vt Ine j 3 Hz = Vt. Ht (3.3) (3.4) The e, Ez and Hz are also independent of time and the z coordinate. In each of the three regions of a step index waveguide the term Vt Ine is equal to zero. It can also be neglected for a slowly varying index V= << 1, and the gradient terms (second medium, Vt Ine = V Ine = 1, and the gradient terms (second e term) in Eq.(3.1) and Eq.(3.2) can be ignored. equations above can be rewritten as Vt2 Et + (02 e to p2) Et = 0 Vt2 Ht + (0c2 E o 32) Ht = 0 Thus, the four (3.5) (3.6) (3.7) j P Ez = Vt. Et j 1 Hz = Vt. Ht (3.8) We only need to solve the above equations for either Et or Ht. The other field vector can be calculated by using Maxwell's curl equations VxE= -j o Lo0 H and (3.9) VxH=j coE (3.10) For a two-dimensional surface waveguide, with the y-direction parallel to the surface of the substrate there are two types of modes; the quasi- TE modes with the electric field primarily polarized in the y-direction (i.e., Ex / Ey is nearly zero), and the quasi-TM modes with the magnetic field primarily polarized in the y-direction (i.e., Hx / Hy is nearly zero). Thus, the scalar Helmholtz equation is expressed as Vt2 + (0o2 Eo p2) F = 0 (3.11) where F represents Ey for quasi-TE modes or Hy for quasi-TM modes. In the above equation, F and e are functions of x and y only. Equation (3.11), the scalar Helmholtz equation, can be rewritten as D2F 22F + + + (n2 2) F = 0 (3.12) ax2 Oy2 where ( = ko N N : effective mode index ko : free space wave vector, ko = o2 Eo Lo and n : refractive index = In the one-dimensional waveguide (no confinement in the y- direction), the above equation can be further simplified as -2+ k2(n2 N2)F = (3.13) ax2 Furthermore, normalization of Eq.(3.13) might be useful in applications with arbitrary index profiles [Hock77] a2F + V2 [f(x') b] F = 0 (3.14) ax'2 where x' = x / W W : 1 /e width of the waveguide V : normalized frequency, V = ko W (nf n )1/2 b : normalized propagation constant, b = (N2 n3)/(n2 nQ) nf: surface refractive index ns: substrate refractive index and f(x') : normalized diffusion profile. The f(0) is always 1.0 and f(1) is zero for linear and quadratic profile, 0.157 for complementary error function profile and represents the 1/e value for exponential and Gaussian profiles [Lagu83]. 3.3. Numerical Analysis of 1-D Waveguides Two approaches are usually used to characterize a one- dimensional waveguide; the ray optics approach and the wave optics. The ray optics or geometrical optics approach is an approximation that is valid as long as the waveguide dimensions are much larger than the wavelength of the light. On the other hand, the wave optics approach is always valid. Since we are mostly interested in graded index waveguides where the refractive index n(x) varies gradually over the cross-section of the waveguide from maximum change to zero, we will emphasize the field distribution of the guided waves. However, the ray approach is simpler and has a direct application in the WKB method, thus it will also be discussed in the following subsections. 3.3.1. WKB Method The well-known WKB method was first used to obtain approximate solutions of the Schridinger equation. It can also be used to solve the wave equation Eq.(3.13) for the modes in a multimode graded index waveguide with slowly varying index profile n(x). Once the effective mode indices have been determined, Eq.(3.13) becomes an ordinary differential equation. The fields can be obtained by solving the equation either analytically or by using a numerical technique. Consider the ray propagation in a graded-index waveguide as shown in Fig.3.1. According to the WKB method, the characteristic equation of the mth order mode is given by [Gord66] 2 ko [n2(X) Nm]1/2 dx -2 0s 2 0c = 2 m 7 (3.15) where xm : turning point of the mth order -2 Os : phase shift at the turning point and 2 c : phase shift at the cover-guide interface Equation (3.15) is called the dispersion relation. The left hand side of Eq.(3.15) represents the total transverse phase shift the ray Total Internal Reflection (c = n/2 N(Xm)=Nm n(x) Turning Point 0s = n/4 Fig.3.1 Ray diagram for the graded-index waveguides. The index profile is shown in the left. experiences in one round trip across the waveguide. This must equal 2min in order to achieve constructive interference, a condition for the light to be guided along the length of the guide. At the turning point Xm, the wave is propagating in the z direction with the propagation constant equal to Pm, i.e., ko n(xm) = pm = ko Nm (3.16) The phase shift at the cover-guide interface equals c = tan1 r N -n (3.17) L V n? N_ while, the phase shift at the turning point is given by s = tan- r n (3.18) where nf: surface index ns: substrate index nc: cover index and r : constant, r=l for TE modes and n? / n? > 1 for the TM modes. Since at x=xm, the ray approaches grazing incidence with the index discontinuity approaching zero, the phase shift for both TE and TM modes, in this limiting case equals (- n/2) [Hock77]. It must be pointed out, however, that the approximation -2 Os = n/2 is valid only in the well guided regime. It has been argued [Sriv87a] that as the mode approaches cutoff, Ps --0, and the above approximation gives erroneous results. The accuracy of the WKB method depends on how accurate we can estimate the two phase shifts if no approximation is used. For the case of a three-layered slab, step index waveguide, the approximations s = -C / 4 and Os = n / 2 are not accurate enough and the exact expressions for the phase shifts must be used. For TE modes, Eq(3.15) becomes 2 koW /n? Nm 2 cs 2 Oc= 2 m (3.19) where W is the guide width. No analytical solutions for the above equation are available. In dealing with waveguides it is convenient to write the dispersion relations, Eq.(3.19), in terms of the normalized frequency V and normalized propagation constant b [Koge74] as |o V Vf(x') b dx' = s + 0c + m 7 (3.20) The normalized parameters V, b and f(x') were defined in the previous section. In general, for practical cases of interest, the index difference between the cover and the surface is rather large and it can be represented by the asymmetry parameter "a", defined as aTE = (n3 n@) / (nF n3) for the TE modes and aTM = (n / nf) aTM for TM modes. Figure 3.2 shows the b versus V for TE modes and for a= . ..... .......I........r ...... f ............. .... ............ ..... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . ..... .. .. . . . . .. . V Fig. 3.2 The normalized mode index (b) vs the normalized frequency (V) characteristics of 3-layer step-index waveguides for m=0,1,2 and 3. 0, 1, 10, and o for a 3-layered step index waveguide. The b value is zero for modes close to cutoff and one for well-guided modes. There is no cutoff for the fundamental mode when a=0. The effective mode indices for TE modes are always larger than the TM modes with the same mode number m in the case of an isotropic medium. Exceptions occurs in birefringent media such as LiNbO3 or glasses where, due to the stress induced by the ion exchange, the effective mode index of the TM mode is larger than that of the corresponding TE mode. 3.3.2. Finite Difference Method (FDM) As described earlier, the nonlinear diffusion equation can be converted into a finite difference form and solved by iterations. Similarly, the scalar Helmholtz equation can be converted into a finite difference expression. However, the Helmholtz equation Eq.(3.13) involves two unknowns; the propagation constant of the mode and the corresponding mode field. Using the finite difference method, the b-V characteristics and the modal fields are simultaneously determined by solving an eigen- value equation. The high accuracy of the method has been tested by calculating the b-V curves and the guided mode field profiles in the cases of step and exponential profiles for which exact analytical solutions are also available. The FDM is described as follows: In FDM, the domain of interest [Xo, Xp+1] is divided in p equal elements of length Ax' = h. The points x' are called the grid points or nodes and h is the grid spacing. Let us consider the first term in Eq.(3.13) which can be represented in the form of Eq.(2.51). Thus, at x=xi F(i+1) 2 F(i) + F(i-1) 2 2 F (i+l ) 2 F(i) + F(i- + k [n2(i) N2 F(i) = 0 (3.21) (Ax)2 or F(i-1) + (k2 n2 Ax2 2) F(i) + F(i+l) = k N2 Ax2 F(i) (3.22) where F(i) = F(x=xi). It is known that the guided field is zero away from the waveguide region. Therefore, it is reasonable to assume that F(0) = 0 and F(p+l) = 0 at the calculation boundaries, as shown in Fig.3.3. For i=l to p Eq.(3.22) can be expressed as al 1 0 -FI F la2'-. F2 F2 1 = (3.23) 0 1 ap Fp_ Fp_ where ai = ki n2 (Ax)2 and Y = k0 N2 (Ax)2. The eigen-vectors of this matrix are the mode fields of the waveguide. The eigen-values and vectors can be solved by commercially available subroutines. Since we assume that the fields vanish at the calculation boundaries, only the guided modes can be solved with this method while for modes close to cutoff the fields may extend away from the guided region and more points should be taken. Compared to other methods, the finite difference method is easy and quick and gives accurate results as long as an appropriate 0) o =0 Fp+l =0 Xo Xi-2 Xi-1 Xi Xi+l Xi+2 Xp+l Fig. 3.3 1-D field profile with the assumption Fo and Fp+1 =0 using FDM. F scientific subroutine package is available. However, the algorithm above does not distinguish between the polarization of the modes, and therefore the method is not suitable to simulate devices that are birefringent without modification. 3.3.3. Multilaver Stack Theory The multilayer step index approximation was presented first in [Thur86] for multichannel step index waveguide structure and later used successfully in [Huss89] for multichannel graded index structures. According to this method we divide the space into p regions including two semi-infinite regions. Each region is represented with a step index, Fig.3.4, with interfaces between constant index regions at xi, i=1, ... ,p-1. If F represents the normal mode propagating along + z, then F(x,z) = F(x) exp( i pz) (3.24) where F satisfies the wave equation in each slab of constant index ni, 2Fi Fi+ kh [n (x) Neff] Fi = 0 (3.25) ax2 The wave equation (3.25) has trigonometric solutions if Nff < ni and hyperbolic or exponential if Nff > n,. I I I I I I I I I I I I I I I I I I I I I I I Semi- Infinite Region ni n2 n3 xl X2 I I I I nn-1 1n Xn-1 Fig. 3.4 Multilayer stack theory Semi- Infinite Region Sai cos (ui) + bi sin(ui) I ai cosh (ui) + bi sinh (ui) Neff < ni (3.26) Neff > ni where ui = ki ( x- xi-1) and ki = ko VNeif-n? i=2, 3 ..., p Now we apply the boundary conditions (BC). The fundamental BC is that the field must go to zero at infinity. The second BC is that F (Ey and Hz for TE and Hy and Ez for TM mode) and DF/ax are continuous along the boundaries at any x=xi. Using these boundary conditions we can evaluate the a and b coefficients in relation to the corresponding coefficients of the previous slice. Thus for TE modes we get: f ai cos (ui) + bi sin(ui) I ai cosh (ui) + bi sinh (ui) - ai ki sin (ui ) + bi ki cos (ui) bi+1 ki+1= = [ ai ki sinh (ui) + bi ki cosh (ui) also al= a2 Neff < ni Neff > ni Neff ni (3.27) (3.28) Neff > ni (3.29a) (3.29b) al ki = b2 k2 (3.29c) ai = bi an= bn (3.29d) For TM modes, since continuity of Ez implies continuity of (1/n2) DHy/Dx in Eqs. (3.27)-(3.29), each ki needs to be replaced by (ki /n2). For convenience in the calculations we assume ai = 1 until such time as all the ai, bi are redetermined by normalization of the eigen- functions. The process is as follows: we start with an initial guess of N and Eq.(3.27) and Eq.(3.28) enable us to determine ai and bi. Then these values are checked with Eq. (3.29d) and if they are not satisfied a new guess of Neff is made. The limitation of this method is that it is difficult to find the number of roots and the initial guesses of the Neff. A good solution to this problem is the use of bisection method [Hage88] to determine the mode indices or, even better, a combination of bisection and Newton's method [Hage88] offering a fast convergence and highly accurate results. The main advantage of the multilayer step index approximation is that one can determine the field profile and the mode indices very accurately within specified tolerance. The method was tested for several profiles against different numerical methods. In the case of step index profile the method gives exact results while using other profiles the results are as good as any other method's. However, for steep profiles, in order to get accurate results, there is a need to use a large number of slices, which adds to the round off error accumulated from the calculation of successive coefficients. The way to approach the problem is by using non- uniform slicing or Aiken's method [Hage88]. 3.4. Numerical Analysis of 2-D Waveguides Most guided-wave optical devices demand the solution of a two- dimensional channel waveguide. Only few of them have explicit analytical solutions [Marc79]. Thus, numerical methods are very important for an understanding of 2-D guided-wave behavior in rectangular geometry. Two methods are introduced in this section. The first is the effective index method and the second is the finite difference method. 3.4.1. Effective Index Method The effective index method, first suggested by Knox et al [Knox70] and later by Ramaswamy [Rama74], is relatively easy to implement for 2-D channel waveguides. This method is mainly used to calculate the propagation constants of the waveguides. By converting a 2-D index profile into its 1-D equivalent profile, the propagation constants of the various guided modes can be evaluated using the numerical or analytical methods introduced in the last section for the solution of the 1-D planar waveguide. In general, the 1- D equivalent index profile is much less cumbersome to solve than the 2-D index profile. The method is briefly described below : If we consider the evolution of one transverse component of the electromagnetic field in time and space of the form F = F(x,y) exp[ j (pz cot)] (3.30) then the scalar wave equation can be written as F(x,y)+ 2F(x + k2 [n2(x,y) N2] F(x,y) = 0 (3.31) aX2 ay2 where F is the transverse field and it is represented by Ex and Hx for the quasi-TE and the quasi-TM modes respectively, ko = 2n: / X is the wavenumber in vacuum and N is the mode index of the waveguide. In order to find a method to determine the mode index, we express the modal field F(x,y) as F(x,y) = Fi(x,y) F2(y) (3.32) where Fl(x,y) is a slowly varying function of y so that a2F(x,y) a2 a2F27) -[F2Fx(x,y) F2(y)= F1(x,y) 2F2(y(3.33) -y2 ay2 ay2 Defining an effective index profile neff (y) such that Lt + k2 [n2(x,y) n2ff{y)] F1 = 0 (3.34) ax2 the original two-dimensional scalar wave equation can be reduced into a one-dimensional scalar wave equation a2F2 2+ ko2 [nff(y) N2] F2 = 0 (3.35) 3y2 This means that once the effective index profile neff(y) has been determined by solving Eq.(3.34), the mode index N in the original two dimensional waveguide can be obtained from a one dimensional waveguide with index profile neff(y), by solving Eq.(3.35). To determine the effective index profile we consider its definition relation. At each point yi, neff(yi) is obtained by solving numerically the one dimensional asymmetrical three layer problem for a particular index profile. The implementation of the effective index method is shown in Fig.3.5. The channel waveguide is sliced along the the x- direction in many slices. The thickness of the slice should be very small in order to accommodate index variations especially in the case where the profile is very steep, i.e., exponential and ERFC. For those profiles a non-uniform slicing would be recommended with increasing thickness away from the center of the waveguide. Each slice is now considered as a three layer problem, and the 1-D waveguide problem can be solved using one of the methods described in section 3.3. The profile constructed now from the effective indices for each slice represents the 1-D profile used in the y-direction to calculate the mode index N with one of the available numerical techniques. Although there are restrictions in the derivation of the effective index method, that is, F1(x,y) should be a slowly varying field, the algorithm 73 Effective Index Method TM mode I I I I I I S I i I I I S I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I Y1-I Yl Yi+l I TE mode Effective index profile Mode index Fig. 3.5 The procedure in the effective index method for evaluating the mode index of 2-D guides is still valid for the first two normal modes of a weakly coupled system provided both the modes are away from the cutoff. The effective index method is known to fail near the cutoff [Chia86]. An important point that needs to be emphasized is that the quasi-TE mode of a channel waveguide has its polarization parallel to the surface of the guide (y-direction). Therefore in Eq.(3.35) F2 represents the transverse E field. However, in order to maintain the polarization in the same direction, TM mode is used in Eq.(3.34), where F1 represents the transverse H field. Similar considerations apply for the TM modes. 3.4.2. Finite Difference Method The one dimensional finite difference method can be easily extended to the two dimensional method. In this case Eq.(3.23) becomes di 1 0 F1 F1 1 d2 F2 F2 =T (3.36) 1 0 1 dpy _Fpxpy _Fpxpy where di is a matrix, given by a(i-1)px+l 1 di = 0 1 a(1. 1)p,+2 0 1 1 a(i-)p,+px (3.37) Special care should be taken for modes that are close to cutoff where the boundaries need to be extended and the number of points must be increased to maintain the accuracy. Nonuniform finite difference method in this case may be appropriate since it allows to take more points in the areas needed without increasing the total number. CHAPTER 4 WAVEGUIDE FABRICATION 4.1. Fabrication Procedure Several commercial glass substrates are available for ion exchange such as soda-lime silicates and borosilicate glasses. For the experimental work reported in this study we used BK7 borosilicate glass commercially available from Schott. The BK7 glass was chosen due to its high optical homogeneity, excellent transmission properties at the wavelength of interest (- 1.3 gm) and the ability to withstand the attack of molten salt, specifically the nitrate salts. Moreover, the sodium and potassium contents of the glass are sufficient to achieve the desirable index change (An < 0.1 for Ag+-Na+ exchange and 0.01 for K+-Na+ exchange), suitable for single-mode application. Moreover, the transition temperature of the glass is also higher than the temperature required for the potassium or silver ion exchange process. The process of fabricating high-performance, low-loss optical waveguides is described in details in the following sections. 4.2. Planar l(-D) Waveguides The procedure for fabricating one dimensional (1-D) waveguides is rather simple. It does not require any masking process since they are fabricated on planar substrates. One-dimensional waveguides are used for the characterization of the diffusion and waveguide parameters such as the self-diffusion coefficient, activation energy, mobility ratios, and maximum index change. These parameters are then used for the design and simulation of various devices. The fabrication process involves the sample and melt preparations and diffusion/ion-exchange. 4.2.1. Sample Preparation Glass substrates are first cut by a wafer saw to appropriate sizes which depend on the different application at hand. Orientation of the glass substrate is not important, as in the case of LiNbO3, since it is an isotropic material. Sample cleaning is carried out in a class 100 clean room and includes the following steps. First the sample is scrubbed with a mildly abrasive detergent and washed with DI water. Next, the sample is immersed sequentially into warm methanol, acetone, TCA (1,1,1-Trichloroethane), acetone and finally methanol and dried out with a N2 gun. The samples are then hard-baked at 150 o C for three hours before diffusion to remove any moisture remaining at the surface. Fig. 4.1 The diffusion setup used in the ion-exchange process 4.2.2. Melt Preparation The preparation of the melt depends on the ions used for the exchange. The desirable characteristics which influence the choice of the salt for a given ion are its melting point and the dissociation temperature. Nitrate salts have some of the lowest melting temperatures and exhibit reasonable stability. For the case of K+ Na+ ion exchange, pure KNOs is used since the maximum index change achieved is of the order of An 0.01, suitable for single-mode applications. Other ions can also be used for exchange such as Ag where it was found experimentally that the optimum melt concentration in order to achieve approximately the same index change is 0.333 MF KNOa + 0.666 MF NaNO3 + 0.001 MF AgNO3 [Bran86]. Both ions have been proven to be an excellent choice for ion-exchange since they provide very low loss waveguides. Great care must be taken in order to keep the melt free of contaminations since any contamination can alter the exchange conditions and reduce the performance of the waveguide. Therefore only ACS grade chemicals are used. A frequent change of the salt bath and etching of the aluminum recipient after each use is also necessary. 4.2.3. Diffusion-Ion Exchange The diffusion is performed in an aluminum vessel because the oxidized surface layer on the vessel is inert to the nitrate melt and Al cannot easily diffuse into the sample. The experimental set-up is shown in Fig.4.1. The sample, after preheating for a few minutes right above the surface of the melt, is placed at the bottom of the interior vessel with a thermocouple above it to record the temperature of the melt near the surface of the sample. Since the temperature variation of the hot plate may be quite large in order to assure temperature uniformity ( 1 OK) we place the Al vessel in a large salt bath that contains KN03 + NaN03 and use a stirrer to keep the temperature of the bath constant. A temperature control and a precise temperature setting are very important since the waveguide characteristics strongly depend on the diffusion temperature. A timer is used to monitor the diffusion. Inhomogeneities in the waveguides during the diffusion process can be avoided by assuring the substrate surface to be free of air bubbles. The diffusion temperature was optimized (for best control of the diffusion rate) at 370 oC and the diffusion times depending on the application range from 1-48 h. After diffusion, the sample is allowed to cool and the residue of the solidified salt on the glass surface is removed with warm DI water. The surface of the waveguide is then cleaned with warm methanol and dried with a N2 gun. 4.2.4. End-Face Polishing An important step for end-fire coupling of light to the waveguide, is the end-face polishing. The end surface of the sample should be smooth, flat, and scratch free and the edges of the sample should be sharp since the waveguide is close to the substrate surface, confined to few pm in depth. To prevent rounding of the waveguide edges during polishing, two pieces of the same substrate material are attached at the edges of the waveguide surface to be polished, using a high temperature wax. The waveguide ends, sandwiched between the pads, are mounted in a holder with a low temperature wax, making it easy to remove the waveguide from the holder after polishing. The edge of the sample/pads extends 2-3 mm beyond the edge of the holder. The polishing procedure consists of grinding the sample edges in four different compounds with progressively decreasing particle sizes. First, the sample is lapped using a 400 grit silicon carbide powder and a 5.0 p.m aluminum oxide powder tracing "figure eight". Subsequently the sample is polished using diamond paste (1 pm and 0.25 gm) on a nylon cloth, using a mechanically rotating plate until all scratches from previous steps have disappeared. After polishing, the wax is removed from the sample by placing it on a hot plate in a beaker with acetone and heating it at 60 o C for 10 minutes. 4.3. Channel (2-D) Waveguides The fabrication process of surface channel (2-D) waveguides is quite similar to the planar (1-D) waveguides described earlier but it includes additional steps such as the choice of mask material and mask deposition, photolithography, etching or lift-off and the removal of the mask material after diffusion. Moreover, for the fabrication of buried (2-D) waveguides the two step process is needed. The second step involves the diffusion of the sample in molten salt bath in the absence of ion A (i.e. potassium) with or without the application of an external field. 4.3.1. Deposition of Mask Material The masking of the substrate is a crucial step in the process of creating the desired waveguide pattern. The choice of the mask material is dictated by the following; good bond to the substrate, tolerance to the chemical attack by nitrate salt and immune to reaction with the exchanged ions without affecting the diffusion process, withstand high temperatures, not contaminate the melt, dissolve easily after the diffusion and finally being able to open windows with sharp edges with wet etching or lift-off technique employed in our laboratory. Taking into consideration all of the above criteria the best choice for mask is 1500 Ao Aluminum (Al) with a thin layer of Aluminum Oxide (A1203) 300 Ao between the substrate and the Aluminum, Fig.4.2(a). The Aluminum Oxide is used in order to minimize the side diffusion in the waveguide [Walk83b] since a dielectric mask is reported to have smaller electrochemical gradient effect discussed in chapter two. For the evaporation of A1203 and Al we use a Veeco E-Beam evaporator at a vacuum of about 10-5 torr. The thickness is measured by a quartz crystal thickness monitor. 4.3.2. Photolithography After the deposition of the mask material, the mask is patterned as illustrated in Fig.4.2. The substrate glass is spin coated at 4000 rpm for 30 seconds with a layer of positive photoresist (Shipley AZ 1400-17) to a thickness of approximately 0.5 pm, Fig.4.2(b). The photoresist is subsequently hardened by soft-baking the samples at 90 0 C for 30 minutes in N2 atmosphere. The photoresist patterns are delineated by exposure to UV light (- 365 nm wavelength, intensity of Mask Evaporation (a) Aluminum -Gla A s203 Glass substrate Photoresist Spinning (b) UV Exposure Glass substrate SPhotoresist 1400-17 Dark-Field <- Photomask --Exposed area Glass substrate Development (d) Glass substrate Mask Etching (e) Photoresist Removal (f) Diffusion (g) Fig. 4.2 Photolithographic procedure using wet etching technique ~LL~ $U ~-k~ 16.5 mW/cm2) using a Karl-Seuss mask aligner with provision for constant intensity control. The dark field photomasks used in this experiment were designed using a workstation VIA-100 and generated by an electron beam exposure system (EBES) with 0.25 pm resolution. The photomask has several bands of the same pattern separated by 100 pm openings. Each band has twenty straight channels ranging from 0.5 p.m to 10 [pm width in steps of 0.5 pm with 75 ptm separation between them. The optimum exposure time for the above intensity of the light is approximately 4.5 seconds, Fig.4.2(c). After the exposure, the exposed photoresist is developed for 50-60 seconds using Shipley 351 developer diluted 5:1 with DI water, and immediately rinsed in DI water for 2 minutes to stop the action of the developer, Fig.4.2(d). The exposure and development of the pattern are very critical steps in the fabrication of high quality reproducible waveguides and precise control of the time is absolutely necessary in order to avoid over or under exposure or development of the pattern. The quality of the pattern is inspected for the straightness of the windows, the sharpness of the edges and the openings are measured under an optical microscope. Finally, the substrate is hard-baked at 110 o C for 10 minutes in order to improve the adhesion and chemical resistance of the image. An important factor in the whole process is the cleanliness of the substrate and its maintenance throughout the process. Any dirt or micron-size dust particles can cause poor bondage between the substrate and the photoresist patterns, making the sample unusable. 4.3.3. Wet Etching Technique The etching process is used to dissolve the Al and A1203 through the opening defined by the photoresist in order to open a window for the ion-exchange to take place. The PAE aluminum etchant used is commercially available and consists of 16 parts phosphoric acid, two parts water, one part nitric acid and one part acetic acid. Etching time may vary from sample to sample depending on the thickness variations along the substrate during the E-Beam evaporation. To assure reproducibility, we etch for an additional minute after the 100 jim channels are approximately 80 % transparent, Fig4.2(e). The PAE etchant is also able to etch the alumina; however, the etching rate is much slower. In order to estimate the etching time several samples were tested and measured with the Sloan Dektak IIA surface profilometer. After etching, the samples are rinsed in DI water to stop the action of the etchant, dried with N2 gun and inspected again under the microscope. Over-etched samples tend to exhibit larger widths than the mask openings and rougher edges while under-etched samples have dark spots or lumps in the etched regions. After etching, the photoresist is removed with acetone, Fig.4.2(f), and the samples are cleaned again by the standard cleaning procedure and prebaked to harden the pattern and remove the moisture from the sample before diffusion. Diffusion is carried out in the same way as for planar waveguides with the ion- exchange occurring through the open windows, Fig.4.2(g). 4.3.4. Removal of Mask After diffusion the aluminum mask is removed by immersing the sample in the PAE etchant at 70 0 C for 15-20 minutes. The alumina can also be removed by the same etchant, however, it is preferable to use dilute nitric acid for 30 minutes. Aluminum or alumina residue can increase the waveguide losses substantially Subsequently the samples are rinsed in DI water, dried and cleaned again by the standard cleaning procedure before proceeding to polishing and characterization. 4.3.5. Lift-off Technique Figure 4.3 describes briefly the lift-off technique. It is an alternative technique to wet etching which usually provides better resolution and reproducibility of the mask pattern than wet etching. However, there are distinct disadvantages with the lift-off procedure which makes it quite difficult to implement in our laboratory. First, of all, exposure and development times must be carefully controlled. However, since both the parameters are temperature dependent this is difficult to implement. Secondly, thorough cleaning of the photoresist is necessary before evaporation of aluminum because photoresist residue can cause breaks in the Al film, making the sample unusable. Finally, the need of using HMDS to stick the photoresist to the glass and chlorobenzene to help create the mushroom structure [West87] for the break away of aluminum in the exposed areas adds more steps in the already complex procedure. Photoresist Spinning (a) UV Exposure (b) o Photoresist lasssbs t -- 1400-17 ////////////////// HMDS Glass substrate +Y Clear-Field Photomask [ // ////////f/"." J////i/A Glass substra Development (c) Mask Evaporation (d) Unexposed te area Glass substrate <- Aluminum Glass substrate Photoresist Removal (e) Diffusion (f0 Glass substrate Glass substrate Fig. 4.3 Photolithographic procedure using lift-off technique J | i ! I CHAPTER 5 CHARACTERIZATION OF WAVEGUIDES 5.1. Introduction The ion-exchanged optical waveguides are characterized by the refractive index profile, the propagation constants of the guided modes (mode index), the mode field profile and the propagation losses. In addition, since the characteristics of the waveguide depend on the wavelength, a spectral response measurement is performed to determine the cut-off wavelength of the modes and thus control the fabrication conditions in order to produce a single-mode waveguide over a desirable wavelength range. For the spectral response measurements a high power tungsten lamp is used as a source for the monochromator. All the measurements are done with a 1.3 gim semiconductor laser as the source. A He-Ne laser (0.6328 gm) is used for the visual alignment of the system. Descriptions of the different methods used to characterize the waveguides are presented in the next sections. 5.2. Refractive Index Profile Two different approaches are used for determining the refractive index profile. The first one is based on the optical measurements of the mode indices of the guided modes [Tien69, Tien70, Ulri70]. The InWKB, [Whit76, Chia85] is then used to construct the refractive index profile from the mode indices as well as estimate the diffusion coefficient of the waveguide. The waveguides used in this experiment must support at least 4 or 5 guided modes, a condition that assures higher accuracy in the constructed index profile, Section 5.2.3. The second approach is based on the fact that the concentration profile of the substituting ion is analogous to the index profile and one employs analytical tools such as electron or ion microprobe [Gial73], scanning electron microscope [Lagu86a], secondary ion mass spectroscopy (SIMS), and atomic absorption spectrophotometry [Chlu87] for measuring the concentration profile directly. The two methods employed in this study are the mode index measurements and the concentration measurements using electron microprobe, described in the following sections. 5.2.1. Mode Index Measurements Measurements of the mode indices are made by the prism coupling technique shown in Fig.5.1 [Tien69, Tien70, Ulri70]. A He- Ne laser or a 1.3 pm semiconductor laser beam, (either TE or TM), is incident on a prism whose refractive index is higher than that of the film region of the waveguide. The polarization of the beam is controlled by a Babinet-Soleil compensator to choose either the TE or TM polarization. The beam is focused by a large focal length lens such that the beam waist coincides with the coupling point at the prism base. The prism is placed on an XYZ translation stage that is mounted on a precision (<1 min of arc) rotation table. The prism used was Aperture ixuUc .s ' prism Chopper HeN / 13 Compensator | He-Ne /1.3 pJm Laser Power Lens Power Lens meter Fig. 5.1 Prism coupler measurement setup |

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