Gallium Arsenide-Gallium Aluminum Arsenide Distributed Feedback and Distributed Bragg Reflector Lasers Citation Yen, Huan-wun (1976) Gallium Arsenide-Gallium Aluminum Arsenide Distributed Feedback and Distributed Bragg Reflector Lasers. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/vc0r-rk03. https://resolver.caltech.edu/CaltechTHESIS:11252025-193026356 Abstract This work describes theoretical and experimental studies of GaAs/GaAlAs distributed feedback and distributed Bragg reflector lasers. These lasers are strong candidates as the light source in integrated optical circuits and optical communication systems. A coupled-mode formalism is used to study the propagation of electromagnetic waves in a dielectric waveguide with periodic surface corrugation. The reflection and transmission characteristics of both passive and active periodic waveguides are found as a function of wavelength. These results are used to derive the oscillation conditions of two different laser structures: (1) the distributed feedback laser - where a corrugated active waveguide section is the basic structure, (2) the distributed Bragg reflector laser - where an active region is flanked by two sections of passive periodic waveguides. The procedure of determining the lasing wavelength is outlined. The merits and disadvantages of various laser structures are compared and discussed. Experimental results on fabrication and measurements of GaAs/GaAlAs distributed feedback and distributed Bragg reflector lasers are presented and compared with the theory. Various fabrication and measurement techniques developed during the course of the investigation are described in some detail. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Electrical Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Electrical Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Yariv, Amnon Thesis Committee: Unknown, Unknown Defense Date: 11 May 1976 Funders: Funding Agency Grant Number International Business Machines Corporation UNSPECIFIED NSF UNSPECIFIED Office of Naval Research (ONR) UNSPECIFIED California Institute of Technology UNSPECIFIED Record Number: CaltechTHESIS:11252025-193026356 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:11252025-193026356 DOI: 10.7907/vc0r-rk03 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17774 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 02 Dec 2025 18:52 Last Modified: 02 Dec 2025 19:07 Thesis Files PDF - Final Version See Usage Policy. 47MB Repository Staff Only: item control page

GALLIUM ARSENIDE - GALLIUM ALUMINUM ARSENIDE DISTRIBUTED FEEDBACK AND DISTRIBUTED BRAGG REFLECTOR LASERS Thesis by Huan-wun Yen In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1976 (Submitted May 11, 1976) To my de.a.Jr. pa.Jr.e.n.t..6 -iiiACKNOWLEDGMENT I would like to express my sincere gratitude to my research advisor, Professor Amnon Yariv, for his excellent guidance and constant encouragement throughout the course of this work. It has been a very pleasant and fruitful five years for me to be part of the quantum electronics group. I enjoyed every minute of it. During this study we were very fortunate to enjoy the cooperation from the Central Research Laboratory of Hitachi, Tokyo, Japan. In particular I would like to thank Dr. Michiharu Nakamura for his technical advice and helpful discussions. and friendship are greatly treasured. His contributions Also I would like to acknow- ledge Dr. Hugh Garvin of Hughes Research Laboratories in Malibu, California. His professional assistance and stimulating discussions on the fabrication of corrugations are most helpful. indebted to Dr. Elsa Garmire for introducing me I am also to the Laboratory work and epitaxial crystal growth. It is a pleasure to acknowledge the skillful technical assistance of Desmond Armstrong with the experimental apparatus. Special thanks go to Ms. Dian Rapchak and Mrs. Ruth Stratton for doing such a wonderful job of typing the manuscript. Financial support received from the International Business Machines Corporation, the National Science Foundation, the Office of Naval Research, and the California Institute of Technology is greatly appreciated. Finally my thanks go to my wife, Julie, for her love and support. -ivABSTRACT This work describes theoretical and experimental studies of GaAs-GaAlAs distributed feedback and distributed Bragg reflector lasers. These lasers are strong candidates as the light source in integrated optical circuits and optical communication systems. A coupled-mode formalism is used to study the propagation of electromagnetic waves in a dielectric waveguide with periodic surface corrugation. The reflection and transmission characteristics of both passive and active periodic waveguides are found as a function of wavelength. These results are used to derive the oscillation conditions of two different laser structures: (l) the distributed feedback laser - where a corrugated active waveguide section is the basic structure, (2) the distributed Bragg reflector laser - where an active region is flanked by two sections of passive periodic waveguides. The procedure of determining the lasing wavelength is outlined. The merits and disadvantages of various laser structures are compared and discussed. Experimental results on fabrication and measurements of GaAsGaAlAs distributed feedback and distributed Bragg reflector lasers are presented and compared with the theory. Various fabrication and measurement techniques developed during the course of the investigation are described in some detail. -vTABLE OF CONTENTS Page CHAPTER 1 ~ INTRODUCTION 1 1-1 Integrated Optics and Optical Communications l 1-2 GaAs -GaAlAs System for Monolithic Integrated Optics 4 1-3 Distributed Feedback and Distributed Bragg Reflector Lasers 6 1-4 Experimental Techniques 10 Chapter l References 11 CHAPTER 2 - COUPLED~MODE THEORY IN PERIODIC WAVEGUIDES 2-1 Introduction 13 13 2-2 Coupled-Mode Formalism 20 2- 3 The Coupling Constant 23 2-4 Solutions of the Coupled-Mode Equations 36 2-5 Reflection and Transmission Characteristics of a Periodic Waveguide 2-6 Periodic Waveguide with External Reflector 42 46 Chapter 2 References 51 CHAPTER 3 - DISTRIBUTED FEEDBACK LASERS 52 3-1 Introduction 52 3-2 Amplification and Oscillation in Periodic Waveguides 57 3-3 Alternative Derivation of the Oscillation Condition 63 -vi- Page 3-4 Determination of the Lasing Frequency and the 66 Threshold Gain 3-5 The Effect of Distributed Feedback on Spontaneous Emission Spectrum 72 3-6 Comparison of GaAs Distributed Feedback Lasers and Fabry-Perot Lasers 77 3-7 Design Factors in GaAs Distributed Feedback Lasers 80 3-8 Optically Pumped GaAs Distributed Feedback Lasers 83 3-9 GaAs-GaAlAs Distributed feedback Injection Lasers 98 Chapter 3 References CHAPTER 4 DISTRIBUTED BRAGG REFLECTOR LASERS 4-1 Introduction 107 110 110 4-2 Distributed Bragg Reflector Laser with Lossless Reflect ors 112 4-3 Determination of Mode Structure 117 4-4 The Effect of Lossy Reflectors 118 4-5 Comparison of Distributed Feedback and Distributed Bragg Reflector Lasers 128 4-6 Experiments with Optically Pumped GaAs Distributed Bragg Reflector Lasers 130 Chapter 4 References 138 -viiCHAPTER 5 - EXPERIMENTAL TECHNIQUES 5-1 Introduction 5-2 Liquid Phase Epitaxy in GaAs-GaAlAs System Page 140 140 141 5-3 Grating Fab r ication by Holographic Photolithography 5-4 149 Ion Beam Milling and Chemical Etching of GaAs Gratings 157 5-5 Optical Measurements 167 Chapter 5 References 179 -1CHAPTER l INTRODUCTION 1-1 Integrated Optics and Optical Communications From the beginning of radio communications i t has been the goal of electrical engineers to explore higher carrier frequency ranges for better signal quality and larger information capability. The evolution started with AM radio in the KHz range, proceeded to FM transmission in MHz range, and on to microwaves in the GHz range. It was not until about 1960 with the advent of lasers that communications in the optical frequency regime began to be considered seriously . Some of the potential advantages of optical communication systems are : (a) extremely large bandwidth and therefo re high data rates , (b) very wide spectral ranges , (c) small system components. Optical communication , however, did not receive serious attention since propagation throµgh the atmosphere involves high transmission losses and low loss optical waveguides did not exist. During the last few years the situation has changed radically . Techniques for fabricating very low loss waveguide - optical fibers were developed, and transmission losses as low as 2 db/km at near infrared wavelengths were achieved(l). This technological breakthrough stimulated once again the interest in optical communications, in particular the communications through optical fibers . Besides their large bandwidth potential , the fiber communication system has the merits of small size and light weight , no ground loop problem exists, and the channels are essentially free of any -2- interference and pick-up problems. It is expected that fiber channels will provide both short range and long range high data density communication. For these applications we will need terminals at both ends and repeaters along the line to complete the system. Fig. 1-1 shows a simplified diagram of such a link. At the terminals signals are either generated and modulated for launching or detected, amplified and demodulated for processing. In repeaters the signals are detected , amplified, and used to modulate another source for relaunching. These terminals and repeaters should be reliable and have dimensions comparable to ~hose of the fibers. While most of the conventional optical systems in use today are bulky and extremely sensitive to alignment, it is essential that a new kind of technology be developed for this purpose. Optics ' 2 ) comes in. 1 /\.nd this is where 11 Integrated ( The technology of integrated opt i cs centered around the study of optical dielectric waveguides and devices made using such waveguides. These devices include grating filters, Bragg reflectors, grating couplers, taper couplers, lenses, prisms, directional couplers , lasers, modulators, polarizers and detectors, all in planar form. It is thus conceivable that one could fabricate on a common substrate all the necessary components which \!Jill be interconnected by waveguides to form a small, rugged integrated optical circuit in very much the same way as electronic integrated circuits. Most of the work done so far in the area of integrated optics, however , involves the demonstration of individual components and the technology of integration is only in its very beginning . There is Input Information tFig. 1-1 Fibers output Information ! Station Receiving Block diagram of an optical fiber communication link. Fibers Station Station Station Fibers Repeoter Repeater Transmitting I I w -4no apparent obstacle, however, for achieving such an integration in the near future. There remains the difficult problem of inter- connecting the fibers and the optical circuits which will be addressed in Chapter 4. 1-2 GaAs-GaAlAs System for Monolithic Integrated Optical Circuits Many different materials were used in fabricating optical components during the past few years. Each particular device requires the optimization of certain parameters which dictate the choice of material . 1 ' In the integration process one could use the hybrid approach, in which several different materials are incor1 : porated in one circuit so that each individual component performance can be optimized . In practice this approach poses a major problem, which is that of interfacing different components. The coupling efficiency between components of different material is usually low unless some precautions are taken in designing and manufacturing. This process can become very complicated even for a simple circuit . Hence it is important to search for materials suitable for monolithic optical circuits for simplicity and reliability . First let us examine some of the basic functions to be performed by an optical circuit. These include (a) light generation , (b) light detection, (c) modulation and swi t ching , (d) waveguiding , (e) coupling. The versatility of GaAs in terms of electrical and optical properties makes it a very strong candidate for the basic material of integrated optics( 3 , 4) . Let us examine GaAs against the basic requirements listed above : (a) GaAs is the only material to -5date from which small size cw lasers that operate at room temperature can be made. (b) A reverse l y biased p-n junction or an ion implanted region in GaAs are known to be reasonably good detectors( 5 , 5). (c) GaAs possesses one of the largest electro-optic and acousto-optic figures of merit and has been used in making efficient modulators(?). (d) GaAs can be alloyed with AlAs to form Ga .-x Al XAs . The index of 1 refraction of this material varies with x, the mole fraction of Al. The larger x the smaller the index of refraction. Hence it is convenient to form dielectric waveguides with layers of GaAs and Ga 1_i\As. (e) Directional couplers__(B), grating couplers( 9) and taper couplers(lO) have been fabricated with high quality in GaAs. Moreover GaAs technology such as liquid phase, vapor phase , and molecular beam(ll) epitaxial crystal growth, ion implantation , ion beam etching, chemical etching, diffusion , doping , ohmic contacting Schottky barriers , etc. are being pursued and developed to a very high technical quality necessary for making optical circuits. Another important factor that favors GaAs is that GaAs injection lasers emit light with wavelength in the region of 0.8 - 0.9 µm which coincides with the low loss window of the fiber transmission spectrum(l) . Recently several new techniques in liquid phase epitaxy have been developed. These include the growth of tapered waveguides(lO) and the selective growth through masks( 12 •13 ). These two techniques are essential to the integration of components . It is the author's opinion that the successive selective epitaxial growth is going to play a vital role in integrated optics as the selective diffusion -6- does in electronic integrated circuits. 1-3 Distributed Feedback and Distributed Bragg Reflector Lasers The most important component in a transmitting terminal or a repeater of the fiber communication system is the light source. The conventional miniature light source is the GaAs-GaAlAs injection laser. The reflection feedback in these lasers is usually obtained by cleaving the crystal along a pair of parallel crystal planes. Because of the dielectric constant difference between GaAs and the air finite reflectivity is achieved on both ends. This fabrication process is discrete and therefore not compatible with the planar technology. A novel 11 mirror-less 11 laser structure was suggested by Kogelnik and Shank(l 4 ) which utilizes the spatial periodic modulation of the properties of the lasing medium to cause coupling between waves going in opposite directions. Such a feedback mechanism is not localized but rather distributed along the length of the medium and is referred to as distributed feedback . A periodic surface corrugation of a waveguide can be viewed as a periodic modulation of the waveguideis effective index of refraction . If a wave propagates in the waveguide with a guide wavelength Ag= 2A then the backward scattered wave from the corrugations will add up in phase as depicted in Fig . 1-2. We shall call this phenomenon backward Bragg scattering (or reflection) and the condition A= Ag/2, the Bragg condition since it is analogous to x-ray scattering by crystal planes. A section of a waveguide which is corrugated can thus be used as a mirror (Bragg reflector) with reflectivity which is I I I I I I I ---,4------J------~-----~ : I I 3- .A.=2A Backward Bragg scattering in a periodic waveguide. I I Fig. 1-2 I I r e-l 6 Tr -+- - L - - - - - _ ._ - - - - - 1- - - - - - -1I . f. L ' I I -t4TT ~-~------L-------, - - - -' - - - - - - - ... - - - - - ~ re I j , i I I I • I ,. __ - - __I - - -_~ _______ - - - - - - +.-l r e-L2JT .,_ r . _____ i . It-A~ I --..J I -8- a strong function of the frequency of the incident wave. If this type of periodic structure is incorporated into a conventional GaAs injection laser it results in reflection feedback without a need for cleaved end mirrors. this feedback are shown in Fig. 1-3. Two approaches to achieve In (a) the corrugation (or grating) extends over the whole length of the active region. structure is called distributed feedback (DFB) laser . This In (b) the corrugations are present on both sides of the active region and serve as mirrors . Such a structure is referred to as the distributed Bragg reflector (DBR) laser(l 5). The advantages of using DFB and DBR lasers are four-fold: (a) The fabrication process is compatible with planar technology. (b) Better laser longitudinal and transverse mode control(l 4 ,l 5 ) result from the frequency selective nature of the feedback . (c) Better frequency stability against temperature variation(l?) . (d) The presence of the gratings makes possible new schemes of coupling laser output into fibers(lB) or other optical circuits. There is still the reliability (operating lifetime) problem which needs to be solved before any practical applications of these lasers are possible. In Chapters 2, 3, and 4 we will describe the theoretical and experimental studies of DFB and DBR l asers and their potential applications in integrated optical circuits . -9- p-GaAs (a ) n -Ga As Substrate -GaAs -GaAs Cb ) n-Ga,.xAlxAs n -GaAs Substrate Fig. 1-3 Schematic diagram of dou bl e-hetero structure GaAs lasers. (b) (a) Distributed feedback laser Distributed Bragg reflector laser. -10- 1-4 Experimental Techniques Because of the small dimensions of the optical circuits and because of the strict requirement of edge smoothness of certain components special fabrication techniques are needed. In order to produce reliable long - life, low loss devices it is important that one knows the limitations of each technique. In Chapter 5 we will describe some of the techniques used during the course of studying GaAs-GaAlAs DFB and DBR lasers . . These include: (a) Liquid phase epitaxial crystal growth -- probably the most important single technique - in fabricating GaAs lasers, waveguides, and other related components . (b) Grating fabrication -- for use in DFB and DBR lasers , wire - grid polarizers , couplers, and filters. Grating period ranges from 0. 1 µm to l µm. (c) Ion beam etching and sputtering -- combined with photo- lithography is capable of fabricat in g complex circuits on a single substrate. (d) Optical measurements -- photoluminescence measurements, laser spectroscopy, and waveguide parameter measurements are used for sample and device characterization. (e) Other device fabrication techniques such as diffusion , ohmic contact fo rmation , chemical etching, etc . _,, _ CHAPTER l REFERENCES (l) D. B. Keck, R. D. Maurer, and P. C. Schultz, "On the ultimate lower limit of attenuation in glass optical waveguides," Appl. Phys. Lett.££., 307 (1973). (2) "Special issue on integrated optics and optical waveguides,'' IEEE Trans. Microwave Theory Tech. 23, January (1975) . (3) A. Yariv, "Active integrated optics, 11 Fundamental and Applied Laser Physics , Proceedings of the Esfahan Symposium, John Wiley & Sons, New York, p. 897 (1971). ( 4) V. Evtuhov and A. Yari v, 11 GaAs-GaA lAs devices for integrated optics," IEEE Trans. Microwave Theory Tech . 23, 44 (1975). (5) F. K. Reinhart, 11 Electroabsorption in Al Ga As-Al Ga As y 1-y X 1-X double heterostructures, 11 Appl. Phys. Lett.££., 372 (1973). (6) H. Stoll, A. Yariv, R. G. Hunsperger, and G. L. Tangonan, "Proton-implanted optical waveguide detectors in GaAs, 11 Appl. Phys. Lett.~, 664 (1973). (7) F. K. Reinhart and 8. I. Miller, "Efficient GaAs-Al Ga X 1-X As double-heterostructure light modulators," Appl. Phys. Lett. 20, 36 (1972) . (8) S. Somekh, E. Garmire, and A. Yariv, "Channel optical waveguiding directional couplers," Appl. Phys. Lett. 22, 46 (1973). (9) F. K. Reinhart , R. A. Logan, and C. V. Shank, "GaAs-Alia,_ls injection lasers with distributed Bragg reflectors," Appl. Phys. Lett. ?l_, 45 (1975). (10) J. L. Merz, R. A. Logan, \,J. Wiegmann, and A. C. Gossard, "Taper couplers for GaAs-Al Ga X 1-X As waveguide layers produced by liquid -12phase and molecular beam epitaxy," Appl. Phy s . Lett. ?_~, 337 (1 975). (11) A. Y. Cho, "Film deposition by molecular-beam techniques," J . Vac . Sci. Technol . ~' S31 (1971) . ( 12) T. Tsukada, "GaAs-Ga 1-X Al As buried heterostructure injection X 1asers ,'1 J. App 1. Phys. 45, 4899 ( 197 4). ( 13) I. Sarni d, C. P. Lee, A. Gover, and A. Vari v, "Embedded hetero structure epitaxy: A technique for two dimensional thin-film definition , " Appl. Phys. Lett. 'll_, 405 (1975) . (14) H. Kogelnik and C. V. Shank, "Coupled-wave theory of distributed feedback lasers / 1 J. Appl. Phys . 43, 2327 (1972). (15) S. Wang, "Principles of distributed feedback and distributed Bragg-reflector lasers , " IEEE J. Quantum Electron. l.Q_, 413 (1974). (16) S. Sornekh , "Transverse mode control in a distributed feedback semiconductor laser," Proc. IEEE (Lett.), 62, 277 (1974). (17) M. Nakamura, K. Aiki, J . Umeda, A. Katzir, A. Yariv, and H. W. Yen, il GaAs-GaAlAs double-heterostructure injection lasers with distributed feedback," IEEE J . Quantum Electron. lJ_, 436 (1975). (18) H. W. Yen, W. Ng, I. Samid , and A. Yariv, "GaAs distributed Bragg reflector lasers , 11 to be published in Optics Communication . -13- CHAPTER 2 COUPLED-MODE THEORY IN PERIODIC WAVEGUIDES 2-1 Introduction The name 11 periodic waveguide'' refers to a waveguide with parameters which are periodically modulated along the length of the guide. Before treating periodic waveguides let us review some of the basic properties of an ordinary slab waveguide as shown in Fig. 2-l(a). The waveguide consists of a thin layer of thickness t and an ' index of refraction n2 sandwiched between two media of indices of refraction n1 and n3 . Assuming that there is no ~ariation in the y-dimension, i.e. a/ay = 0, one can show that(l) such a structure can guide a finite number of confined TE modes with field components Ey , Hx, H2 and TM modes with components Hy, Ex, and E2 • There is also a continuum of radiation modes associated with this structure. These radiation modes are referred to as unguided modes because they are not confined to the inner layer. We shall disregard the radiation modes for the time being and proceed with the discussions on guided modes. Let us consider TE modes first. obeys the wave equation 9 2E Y n. = 1 The field component EY(x,z,t) 2 a2E y c2 at2 ( 2- l) where i = l ,2,3 indicates the three different regions with indices n , n , and n respectively. 3 1 2 We take EY(x,z,t) in the form (2-la) -14- The transverse function ~y(x) is taken as ?'Y (x) = C exp(-qx) C[cos(hx) (q/h)sin(hx)J C[cos(ht) + (q/h)sin(ht)Jexp[P(x+t)J 0 $ X < 00 -t $ X < 0 - oo $ X $ (2-lb) -t Substituting (2-la) and (2-lb) in (2-1) for regions 1 ,2,3 yields h = (n22k2 - 82)1/2 q = (B2 _ nl2k2)1/2 (2- lc) p = (B2 _ n 2k2)1/2 3 where k = w/c. The solutions for ~y (x) and J(2 (x) =(i/wµ) 3 ~Y(x)/3x must be continuous at both x = 0 and x = -t. By imposing these continuity conditions we get from (2 -lb ) tan(ht) = ~(p+g) h - pq (2-ld) At a given frequency (i.e. a given k), the eigenvalue equation (2-ld) can be satisfied only at a finite number of B values. For each such B we solve using (2-lc) for the corresponding p, q and hand therefore for the field components in (2-lb). Each field configuration resulting from a given eigenvalue B corresponds to a guided mode of the waveguide. The arbitrary constant C appearing in Equation (2-lb) can be defined such that the field ~y(x) corresponds to a power flow of one watt (per unit width in they-direction) in the mode. for which Ey = A~y(x) will thus correspond to a power flow of !Ai 2 Watts/m. A mode -15For TM modes the field components are Hy(x,z,t) = jfy(x)ei(wt-Sz) . aH 1 Ex(x,z,t) = -uE _y_ az . (2-2) aH 1 E2 (x,z,t) = - -ws _y_ ax where the function J y (x) is taken as 1(y(x) = -C[~ cos(ht) + sin(ht)Jexp[p(x+t)J X < -t C[- ~ cos(hx) + sin(hx) J -t $ X < 0 h C(- q)exp(-qx) x> 0 (2-2a) The continuity of HY and E2 at x = 0 and x = -t requires that where + g) tan(ht) = h(p 2 -h - pq 2 2 n2 n2 p= p and q = - 2 q n 2 3 nl (2-2b) (2-2c) The normalization constant C is chosen so that the field ~(x) represents, as in the case of TE roodes, a power flow of one watt (per unit width in they-direction) in the mode. The constants C for TM mode and TE mode are different . In an ideal waveguide, that is one with homogeneous media and smooth boundaries, the guided modes once launched will propagate down the guide independently of each other. In other words there will be no energy transfer among the guided modes. The situation is quite -16- different once a perturbation is introduced. The perturbation in general will cause scattering or coupling of the original guided modes to the other guided modes and (or) the unguided radiation modes . If the perturbation is introduced along the guide and arranged in such a way that the scattering or coupling effect adds up coherently, then such a waveguide section acts as reflector or coupler . This situation is possible when the perturbation is periodic. The perturbation can consist of a periodic variation of the index of refraction or the gain (or loss) coefficient( 2) of the guiding structure, or of the waveguide height. In our study we will confine. ourselves to the latter case. In a real device this is achieved by a periodic corrugation of one of the waveguide interfaces as shown in Fig. 2-l(b). If we define the dielectric function of the unperturbed waveguide [Fig. 2-l(a)J as £(r) = £1 X > 0 £2 0 > X > -t £3 -t > X then the dielectric function of the perturbed waveguide [Fig. 2-l(b)] can be written as where (for a square wave corrugation) -17 - n. (e,) X=O (CA.) n2 t E2) 'X=-t n3 <~J) (b) a-=O J= L I I ?( =0 I • -----1-. 1\=-0..---- -. --•• n,. (E2) 7(=-t----------------- Fig. 2-1 Schematic diagrams of (a) perfect slab waveguide (b) slab waveguide with square wave corrugation. -180 X > 0 1l 00 (El -E2) + . £TI srn- n/ exp(i 2 ~£ z) 0 > X > -a £=-oo £10 X < -a 0 and A is the period of the corrugation perturbation . To study the properties of wave propagation in the structure described above we can use the coupled-mode formalism( 3 ) which leads to a set of coupled differential equations which describe the rate of change of the amplitudes of the modes involved. In most of our analysis we are interested in the interaction of two waves ; one is the forward going wave and the other is the backward going wave of the same mode number. Let -i s z E~(r) = B(z) ~~(x)e m and m m Er(r) = A(z) ~y(x)e i Smz be these two waves , where e,~(x) is the normalized m-th TE mode of the unperturbed waveguide and A(z) and B(z) are the complex mode amplitudes. The coupled-mode equations dA _ . B -i2L'IBz dz - 1 K e (2-3) dB _ - l. K Ae i 2L'l 8z dzcan be derived th rough a perturbation onalysis(Z, 3) where (2-4) -19- According to equation (2-3) the rate of change of A along z is proportional to the value of Batz and vice versa. constant K is called the 11 coupling coefficient." The proportional In general K depends on the shape of the periodic surface corrugation and the modes E~(z) and E;(z). It can be calculated from the overlap integral -oo where m designates them-th waveguide mode and£ denotes the i-th order harmonic of the corrugation function that is responsib1e for the coupling. Hence K is the quantity that measures the strength of interaction between the two opposite going waves. One can solve equation (2-3) with the proper boundary conditions to obtain Ei(z) and Er(z). It can be shown that when a waveguide mode with 68 = 0, i.e., the mode with wavelength Ag= 2A (twice the corrugation period) propagates down the guide and enters the corrugated section it will be strongly coupled to the backward going mode. So the incident wave will evanesce as it propagates down the waveguide. At the same time there is a build-up of the reflected wave. Thus effectively a corrugated waveguide section acts as a reflector or 11 11 "mirror" whose reflectivity can be varied by adjusting the corrugation height and length. This behavior of periodic waveguide makes possible the realization of DFB and DBR semiconductor lasers. It should be noted that the gratings not only couple the forward going guided modes to the backward going guided modes but also couple the guided modes to the radiation modes. This will be regarded as -20- radiation loss( 4 ) if the periodic waveguide is used as a reflector or laser. If the periodic waveguide is used as an input-output coupler(S, 6 ) the coupling between guided and radiation modes is a useful phenomenon. In this chapter we shall study the properties of the coupledmode equations, derive the coupling constant for waveguides with periodic surface corrugation and apply the coupled-mode formalism to study the reflection and transmission characteristics of a section of periodic waveguide. 2-2 Coupled-mode Formalism The problem of electromagnetic wave propagation in periodic structures has been studied extensively(?). One of the most common methods used in deriving useful analytical expressions is the coupledmode formalism. We shall outline the procedures in obtaining the coupled mode equations and the overlap integral, equation (2-4), that determines the coupling constant K for a periodic waveguide. Fig. 2-l(a) shows a regular slab waveguide with dielectric constants s 1 , s 2 , and s 3 in three different regions. We define X > 0 E ( r) = 0 > X > -t -t > X as the dielectric function of this unperturbed waveguide. (2-5) Fig. 2-l(b) shows a slab waveguide with square wave perturbation at the x = 0 boundary . If we let -210 X > 0 0 > X > -a 0 (2-6) -a > X where a is the height of the square wave perturbation , then the dielectric function of the pertu~bed waveguide is given by (2-7) Now the field vector D(r,t) becomes B(r,t) = €'(r)E(r,t) = [€(r) + 6£(r)JE(r,t) = € 0 E(r,t) + P(r,t) = £ 0 E(r,t) + P0 (r,t) + 6P(r,t) where r (r,t) = [€(r) 0 and £ 0 JE(r,t) 6P(r,t) = 6€(r)E(r,t) ( 2.-8) Equation (2-8) is then used in the wave equation to obtain (2-9) This is the wave equation that describes the propagation of electromagnetic waves in a slab waveguide with boundary perturbation represented by 6P(r,t) which is given by equation (2-6) and (2-8). -22- The field E(r,t) of the periodic waveguide can be expanded in terms of the modes of the unperturbed smooth waveguide. as an example the case of TE modes. Consider, They-component of E(r,t) can be written as l i(wt-Smz) EY(r,t) = 2 I Am(z) ~m(x)e + complex conjugate m Y (2-10) (m) i(wt-smz) where ~Y (x)e is they component of the electric field of them-th eigenmode of the unperturbed waveguide. We substitute equation (2-10) into (2-9), and with the help of (2-6) and (2-8), and limiting ourselves to the case of coupling between the positive and negative going m-th mode we obtain (2-11) where (2-12) -co .R:rr i'.\S = Sm - A = Sm - So and i is the order of the corrugation (or grating) harmonics responsible for the coupling. The electric fields associated with the coupled incident and reflected modes are E~(r) = Bm(z)~;(x)e i(wt-S z) m -23- and E~(r) = Am(z)~;(x)e respectively. i(wt+s z) m A simple calculation using equation (2-11) shows that This is merely a description of the conservation of power since the waveguide is assumed to be lossless and 1Bm1 2 , 1Am1 2 are proportional to the power (per unit width) carried by the forward and backward going waves respectively. 2-3 The Coupling Constant Equation (2-12) defines the coupling constant K which appears in the coupled-mode equation (2-11). In a square wave corrugation 6E:(r) is X> 0 0 n2 t::.dx) = l n2 2 0 Q > X > -a -a> X So the integral reduces to K = -a Using the eigenmode function of them-th TE mode(B) we have 2 . £TT k sin 2 K= $TT£ 2 2 2 0 hm (n 2 -n 1 ) J q 2 2 [cos(hmx) - hm sin(h x)]2dx teff(hm +qm )ra m m If we carry out the actual integration K becomes -24K sin ; i a 2 2 2 = nismteff {2 (n2 -n, )k sin 2hma 2 2 qm + ---.4"""h_m__ (hm -qm) + 2 (1-cos 2hma)} (2-13) where p, q, h, 8 are given by equations (2-lc), (2-ld), and teff = t + 1/pm + 1/qm is the effective waveguide thickness for them-th mode of the unperturbed waveguide. If a<< t we can find an approximate expression for K to first order in a as 2 hm a t = 1 ,3,5, ... !Kl = nt8mteff (2-14) It is seen that K is different for different transverse modes m. In Fig. 2-2 we plot IKI as a function of waveguide thickness t for several transverse modes. We see that for each m, K reaches a maximum fort slightly above its cutoff value and then decreases rapidly as t increases. In a thick waveguide higher order modes have larger coupling constants than the lower order modes as evident from the figure. Equation (2-14) is valid only in the case of square wave corrugation where the x and z dependence of 6E(x,z) are separable . In corrugations with shapes other than squarewave we have to expand 6E(x,z) in a Fourier series with x dependent coefficients and then use the and pick out the proper term corresponding to the t -th order grating operation. The details of this procedure have been worked out by Streifer et al( 9 ). Up to this point we limited our discussion to TE modes. The -25- m=0 2 m= I m=2 IXIO-I 8 6 4 \ 2 1x10- 2 n 1 =I 8 6 i 0 Ln..J7..n.I'Ta=I000A n2=3.59 T 4 2 1x10-3 L - _ _ , __ 0 __..._ _ _ . _ _ . . . l , , _ _ . . . _ _ _ J_ 0.2 0.4 0.6 0 .8 1.0 Fig. 2-2 _,__-L.._....&..-_""'--.., 1.2 1.4 1.6 1.8 2.0 t (,um) Coupling constant versus waveguide thickness for different transverse TE modes. -26- procedure for finding Kin the case of TM modes is similar except the calculation of the integral is considerably more complicated(lO)_ In the rest of this section we will introduce a different approach to derive the coupling constants which will help us understand better the origin of the coupling. From the coupled-mode equation (2-11), if set 6S = 0 then so physically K is the amplitude reflection coefficient per unit interaction length. Let us apply this simple idea to the case of wave propagation through an infinite periodic dielectric medium whose index of refraction is described by the function n(z) as 2TT z n(z) = n + n1cos A (2-15) We are not going to solve this problem directly but rather consider the special case shown in Fig. 2-3. The medium consists of alternating layers of index of refraction n-n 1 and n+n 1 with a period A. incident wave with propagation constant S = 2TTn/\ An = TT/A will undergo a reflection at each boundary between the n-n 1 and n+n 1 layers given by (n+n ) - (n-n ) n - (n+n ) + (n-n ) '\., n 1 1 where we have assumed that n1 << n. 1 'v - 1 1 r - ~-~~-~ Within each period we have two reflections, one is in the interface between the n+n 1 and n-n 1 medium, the other is between the n-n1 medium and the n+n 1 medium. Although these two reflection coefficients have opposite sign, the wave propagation phase delay between these two reflected waves is exactly TT at the Bragg wavelength, hence they add -27- 0 -1.A. 4 -SA 4 n{z) - ·3A -A .A. I I 5./L 3A. 4 4 4 4 4 I 'l.A. II.A. 4 4 4 --------- Fig . 2-3 An infinite periodic medium consists of layers of index of refraction n-n 1 and n+n 1 interlaced with a period A. I> 7.A.. n + N=O +1 , +2 . j - I -28- up in phase. The coupling constant K is thus given by 2r 2n, K =-, =-A 1. n Since A= 2\ n we can rewrite K as 4n 1 (2-16) Ksquare = T This is the coupling constant of a medium with square wave index of refraction modulation under fundamental operation. The index of refraction of such a medium can be described as n(z) = n + Lin(z) n, {4N- l).A < z < ( 4N+ l}.A. 4 4 -nl (4N+ 1 }.A. < z < {4N+3}A 4 4 Lin(z) = N = 0 ,± l ,±2, • • • so n(z) can be expanded in a Fourier series. Let us pick out the fundamental harmonic term 4nl 2n n(z) = n + ----;- cos A z 4n 1/n can be regarded as the effective sinusoidal modulation amplitude of the square wave modulation. So to find the coupling constant in a medium with an index of refraction n(z) given by equation (2-15) all we have to do is multiply Ksquare by n/4, or 4n 1 n nn 1 Ksinusoidal = ->-- x 4 = - \- (2 -17) This is identical to the result given by Kogelnik and Shank( 2). Next we shall apply this method to a periodic waveguide . Again we consider a square wave corrugation first . The surface corrugation -29on a waveguide is effectively a modulation of the waveguide height. Each discontinuity along the guide will cause a partial i~eflection of the incident wave. Fig. 2-4 shows a step discontinuity of waveguide thickness from t to t+a. discontinuity point. For our purpose here we choose z =Oat the The waveguide at z < 0 with thickness tis referred to as waveguide l, the one at z > 0 with thickness t+a as ... ; 81 z i 81 z waveguide 2. Let E1 (x)e be the incident wav~ and rE 1 (x)e -18 z be the reflected wave in waveguide l and tE 2(x)e 2 be the transmitted wave in waveguide 2. Neglecting radiation losses, we can write down the field continuity equations at z = 0 as and (l+r)E 1(x) = TE 2(x) (2-18) 8 (1-r)E 1(x) = 82TE (x) 2 1 (2-19) where rand Tare the amplitude reflection and transmission coefficients respectively. Taking the ratio of (2-18) and (2-19) we find or (2-20) If we define 82-8 1 = 88 then _ -68 -08 ( 08) r - 28 l +08 ~ ~ l - ~ 28 · µl 1 where 68 is due to a small step change a in waveguide height t. 11 11 The next task is to find a relation between 68 and a. This is done by taking the implicit derivative of equation (2-ld)with respect tot. We can rewrite equation (2-ld) for TE modes as Waveguide l t 1 a-=o I ~· rE,ooei~,i:I I I Waveguide 2 ----,: ~ t E2oo e-i~2J- E,CX) e-,p,i : . a I I-- - - - _'4£ ' -- l t+a from t to t+a at z = 0. Fig . 2-4 Schematic diagram shows a step change of waveguide thickness n3 n2 n, 1 I 0 w -31- = f( B) Hence as _ h at - ~t +cos2ht f'(S) and n f•(s) = S(p+g)(~2+g2)(h2+p2) hpq(h -pq) 2 After simplification we have and S(t+a) can be expanded into a Taylor''s series as :. 6S = S(t+a) After substituting this result into equation (2-21) we have r -_ - h2a 2 2S teff For a periodic waveguide with square wave corrugation the coupling constant can be found as (2-23) -32The first order term (in a) is This is identical to equation (2~14) with 1 = l. The effective sinusoidal modulation coefficient of the square wave surface corrugation with height a can be found to be n2a cos A21r z Hence a sinusoidal surface corru~ation with peak to peak height "a:' will have a coupling constant The extension of this method to corrugations of arbitrary profiles is obvious. All we need to do is to calculate the appropriate Fourier coefficient and compare with that of the square wave corrugation. Then the coupling constant is given by 1 Carb 1 1 = 1,3,5, ... (2-25) Csquare where c!rb is the amplitude of the 1-th Fourier co~ponent of the arbitrary corrugation and c!quare is that of the square wave corrugation. For 1 even this method fails because there are no even Fourier components in a square wave. But we can always approximate an arbitrary tooth shape by a series of step functions and obtain the reflectivity r of one period A and calculate K from the ratio r/A. As an example let us calculate the first order (1=1) and second order (1=2) coupling ,..33., constants of a corrugation as shown in Fig. 2-5(a). For first order coupling or = 1T 8/1. Refer to Fig. 2-S(b), the reflectivity summed up at z = 0 for one period (from z = 0 to z = A) is r = r 1 + r 2e -i -1T2 + (-r 2)e-in + (-r )e 1 3n -i 2 where = rl 8 t - B t+a 2 B t + B t+a/2 r 2 = B(t+a 2) - B(t+a B t+a/2 + B t+a and 2S(t+ a/2) [S(t+a) - S(t)J = [S(t) + B(t+a/2)][S(t+a/2) + S(t+a)J where oS = S(t+a) - S(t) , I I.A. = /2 h2a " K 2s 2t ,Q,=l eff !::, _ h2a - 8t eff A It is educational to double check this result by finding the fundamental Fourier coefficient of this corrugation c1 A 4 = A [ A/8 3/1./8 O A/8 J a cos A2n zdz + f j a 211 _ 12a 2 cos --;1 zdz - -T1- ~ a T ~~1 ..!. a 2 ..l a I I _..,. 1 I 1 I <b > 'I ....... I I I I -· I r. 2 +-' r2 I I I 1 I I -+II I 1 -ti I I +.-: -r2 I +-'I -r ' I -r-A z=o z=A Fig. 2-5 Schematic diagram of a periodic waveguide with surface corrugations being represented by step functions. - 35- Using equation (2-25) we obtain 2 h a /2 a TI IKI£=1 = 1rSte·f -f x x 2a = A- rr h2a 12 1rSteff We thus tested the validity of this 11 reflectivity summing" procedure. For second order coupling between the same forward and backward modes we double the length of the period, i.e., SA= 21r Referring to Fig. 2-5(b) the phase shift associated with each step is double that of the first order case and the sum reflectivity at z = O is and The tooth shape shown in Fig. 2- 5(a) can be taken as a very rough approximation of either a symmetrical triangular or a sinusoidal corrugation. For these functions the second order coupling constants 9 have been found( ) to have the functional dependence a2 . This is also A seen in the expression for jKj£= 2 . Hence the results of this method are at least qualitatively correct . To obtain quantitatively correct results we have to increase the number of step functions which are used to approximate the true function. The method described above can be applied to TM modes as well because equation (2-20) holds also for TM modes. The only -36- modification is that the characteristic equation is now tan ht= p+ § h(l- ~) h where n2 2 p=-2-P, - n3 One can easily show that as) _ (at TM where n 2 2 -yn3 p is the effective waveguide thickness for TM modes. 2 • IK ITM -- TI8t~ff ha •• (2-26) for a square wave corrugation in first order (t =l) operation. Equation (2-26) and (2-24) are identical in form except we have to use the respective values of h2 , S, and teff for TE and TM modes. 2-4 Solutions of the Coupled-mode Equations The coupled-mode equations (2-11) are rep roduced here: dA . Be-i26Sz dz=- lK (2 - 27) dB_ . A i26Sz dz - -1K e Let us look at the special case 6S = 0 first. The equations reduce to -37dA dz = iK B (2-28) dB dz = -iK A whose solutions are A(z) = c1sinh KZ + c2cosh KZ B(z) = -i(C 1 cosh KZ + c2sinh KZ) Let us consider an infinitely long waveguide with a corrugated section between z = 0 and z = L. Ei(O) = l. The boundary conditions are Er(L) = 0 and Since E.(z) = B(z)e-iSz l and then conditions on A and Bare A(L) = 0 and B(O) = l. Under these conditions the solution becomes K(L-z) A(z) = -i sinh cosh KL B(z) = cosh K(L~z) cosh KL Note that 1Ei(z)l 2 = IB(z)l 2 and 1Er(z)l 2 = IA(z)i 2 . If we carry out the calculation we will find which is a constant independent of z. This means that the net power flowing in the waveguide is the same everywhere. The behavior of IEi(z)l 2 and 1Er(z)l 2 is plotted in Fig. 2-6 for two values of KL, KL= 4.0 and 1.0. A large KL represents strong coupling and the decay -38- z=0 z= L I I I I waveguide KL= 1.0 0 L/4 L/2 3L/4 1Ej(0)l 2 KL= 4.0 1Er(0)l 2 L z I I I I I I I I I I I llEi (L)l 2 0 1- i q. ? -6 L/4 L/2 3L/4 2 2 Tile l;ehavior of JEi(z)J and 1Er(z) ! L in ,, periodic waveguide with KL= 1.0 and 4.0 (66= 0). z -39- rate of the incident wave is fast. This is evident from the figure. We can rewrite E.(z) as l = cash K(L-z) -i Bz cash KL e - i f3Z = e [eK(L - z) + e-K(L-z)J 2cosh KL eKL [e- KZe-isz + e-2KLeKZe-isz] 2cosh KL Hence the dominant behavior of IE;(z)l 2 is = 2 eKL 2 -2KZ ( ) IE. z 1 ~ 2 h L e l COS K We define Leff as the effective length of the corrugation :-: uch that l KLeff = l or Leff= K At z = Leff the intensity of the incident wave drops down to~ e -2 of its original value. The reflectivity of a section of periodic waveguide at the exact Bragg condition (6S = 0) is given by IRl 2 Er(O) 2 = E. ~ (0) 2 (2-30) = t an h KL l 2 It can be shown that IRI (6S=O) is the maximum reflectivity a periodic waveguide section can provide. Fig. 2-7 is a plot of equation (2~30) which shows that lr1 2 rises very fast from zero to unity for KL between 0 and 3. Suppose 6(3 f Owe have to solve equation (2-27) with the same boundary condition A(L) = O, B(O) = l to obtain Ei ( z ) -= B( z ) e-i Sz = [i6Ssinhy(L-z) + ycoshy(L-z)Je i6SsinhyL + ycoshyL -is 0 z -40- IRl 2 1.0 ------------::..,;;;;a,,,--------- 0.9 0.8 0.7 0.6 IRl 2 = tan h2 KL 0.5 (6/3 = 0) 0.4 0.3 0.2 0.1 2 3 4 5 6 Fi g. 2- 7 Maximum reflectivity from a periodic waveguide IRl plotted as a function of KL. 2 KL -41- and = A(z)eiSz E (z) r is 0 z -iKsinhy(L~z)e = i6Ssinhyl + ycoshyl where - TT So = Ji y 2 = K 2 6S = S - S0 ' - 6S ( 2-32) and 2 Again we can find the reflection coefficient as Er(O) -iKsinhyL R = Ei(O) = i6Ssinhyl + ycoshyl (2-33) and the transmission coefficient -iS L e o = itSsin1yL+ycoshyL (2-34) Since 6S = S-S 0 = w n n 6S is directly related to the C eff - A' frequency w once neff is known. neff = cS/w will be assumed as given since we can always solve the waveguide characteristic equation to determines. Both Rand Tare thus frequency dependent complex numbers and can be written as (2-35) (2-36) -422-5 Reflection and Transmission Characteristics of a Periodic W~veguide By straightforward calculation one can show that 1/2 l r(w) == - - -2~.- - (2-37) l + 2 2 K sinh yL cp ( w) == ; + tan- 1 (~f3 tanhyL) + mTT (2-38) and t(w) - l ~l/2 2 . 2 { l + yK2 srnh yl (2-39) iµ(w) ::: S0 L + tan-l (y66 tanhyL) + m,r (2-40) where m is an integer I6f3 I m == 0 2 $ [(;L) + (K)2] l /2 We note first of all that both rand tare even functions of is, i.e. , r(6S) == r(-6f3), t(6S) == t(-M3). for all 6f3. Also This again is a statement of the conservation of energy. - The behavior of r(w) can be seen from equation (2-37). 6f3 < K, y is real. When So r(w) is always larger than zero and approaches unity when KL is large . As a matter of fact r(w) reaches a maximum -43- value of tanhKL at 6S = 0 as described earlier. is imaginary. When 6S > K, y \~e can replace y by ir then r(w) becomes r(w) =f, 1 2 + . r K2sin 2rL } 1/2 The denominator of r(w) becomes infinite whenever sinfl = O. Hence the zeros of r(w) appear at 6S =-;!: [ / + (~)2]1/2 The location of local maxima of r(w) is determined by finding the minima of rL 2 f(rl) = ( sin ·rL) The first few roots of f 1 (x) = 0 are x = 4.493, 7.725, 10.904, 14.066, 17.221, etc. The corresponding values of 6S are calculated from 6S = t G2 + (r) i71/2 J X Reflectivity plots are shown in Fig. 2-8, where we plotted r 2(w) and ¢(w) for KL= 2.0 and KL= 5.0. It is seen that as K increases the central high reflectivity band also widens. The curve of t 2 (w) can be determined by computing 1 - r 2 (w). Let us go back to equation (2-31) and (2-32) and factor out the z-dependence of Ei(z) and Er(z). We will find that they consist of waves with propagation constant (2-41) so that in the case of an incident wave with frequency w such that -44- r2 (a) KL= 5.0 KL=2.0-- 1.0 I I ,, -, I I ' \ \ \ \ \ I I 0.5 I I I \ \ \ \ "'\ II -10 -8 -6 -4 \ I -2 r 2 4 6 8 10 6/3L 0 2 4 6 8 10 6/3L 0 ( b) 37T 21T -10 -8 -6 -4 -2 -21T -37T Fig . 2-8 Plots of reflectivity R = re-i¢ of two periodic waveguides with different Kl . (a) Intensity reflectivity r2 as a function of 6[3L . (b) Phase ¢ as a function of !\BL. -45tiB(w) = ~ n c eff - :I. < K A ' ,· the effective propagation constant _is a complex number. The range of frequencies where B' is complex is called the "stop band" or 11 forbidden region. 11 For frequencies in this region the wave will have evanescent behavior in the periodic waveguide as shown earlier in Fig. 2-6. The stop band will correspond to the high reflectivity part of Fig. 2-8. The width of this stop band is 6w = 2KC/neff and the maximum value of the imaginary part of B' is K as derivable from equation (2-41) directly. We find that a section of periodic waveguide can be used as a 1 ' band rejection reflectivity. H filter or a reflector with frequency sensitive Also, because of the coupling between two opposite traveling waves inside such a structure, if enough gain is provided oscillation can occur. This is the principle of distributed feedback lasers and will be discussed in more detail in Chapter 3. We can also use two sections of periodic waveguide to form a very high Q optical resonator . The laser based on this configuration is called a distributed Bragg reflector laser and will be treated in Chapter 4. -462-6 Periodic Waveguide with External Reflector So far in solving the coupled-mode equations we have been using the boundary condition A(L) = 0. This is true only if the region beyond z =Lis of infinite extension and, except for the corrugation, similar to the periodic waveguide. reflection at z = L. In most real situations we have non zero Regardless of the origin of this reflection we can represent it by an equivalent reflection coefficient pat the z = L plane looking to the right. The boundary condition instead of using the condition Er(L) = 0 we now have Er(L) =p Ei(L) If this condition is used together with Ei(O) = 1 to solve equation (2-27), one obtains the reflectivity as -i2f3 OL -iKsinhyL + pe (ycoshyL-i6f3sinhyL) (i6f3sinhyL+ycoshyL) + (iKsinhyL)pe-i 2f3oL If there are integral number of periods in L, i.e. =e -i2· :i!_ ,NA = e -i2Nn = 1 A The above expression reduces to -iKsinh cosh L-i6Ssinh L RI = ~----e---,----~.....,_.--_.__......,...---'--,,......,..--,,...,......__.. i6Ssinhy +ycos yl + p iKsinhyL (2-42) It may be instructive to solve the same problem by a different approach. As shown ~n Fig. 2-9, we can treat the whole problem as a periodic waveguide with no external reflector plus a discrete reflector. The periodic waveguide has a reflection coefficient Rand a transmission coefficient T as given by equations (2-33) and (2-34), and the -47- Z=Q Z=L R,T e e 1 R T er ()2RT2 t'T eRT 2 e RT 2 2 (> R T e3 R~T 2 PRT 2 3 2 p3R3T • • . . Fig. 2-9 Model used in solving the problem of periodic waveguide with external reflector. -48- discrete reflector has reflectivity p. Assume that the incident wave has unity amplitude, then there will be a reflected wave of amplitude Ratz= 0 and a transmitted wave of amplitude Tat z = L due to the periodic waveguide. The wave Tat z =Lis then reflected by the reflector and becomes a left-going incident wave of amplitude pT This wave in turn will have a reflected wave pRT at z = L and a transmitted wave pT 2 at z = 0. If we continue this process we at z = L. will end up with infinite partial waves at z = 0 and z = L. The reflectivity is then found by summing up the amplitude of the partial waves going to the left at z = 0. Hence R' = R + pT 2 + p2RT 2 + p3R2T2 + = R + pT 2 (1+ R+p 2R2+p 3R3+ · ··) = R + ~ = R + p(T2-R2) l-pR l - pR (2-43) Substituting the expressions for Rand T we have 2 - i2S L 2 . h2 L . . h L + (Ye o +K sin y) R' = -lKSln Y P i6SsinhyL+ycoshyL (i6Ssinhyl+ycoshyl) + p(i KsinhyL) Again if L = NA, e-i 2Sol= l and R' = -i Ksinh L + -i6Ssinh L+ cash L i6SsinhyL+ycoshyL + p i KsinhyL This is identical to equation (2 -42) which was obtained by solving the boundary value problem. When we put p = 0 in (2 -43} i.e. no reflection, R' reduces to R 2 as it should. Also when R = 0 (i.e. !Tl= l) IR'! 2 = !r:i \ . In Fig. 2-10 we plotted !R' !2 as a function of 6BL for KL= 2.0 and -49 .-s KL= 2.0 p = 0.5 e-,1r 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 -10 -8 -6 -4 -2 Fig . 2-10 0 2 4 6 8 10 Reflectivity spectrum of a periodic waveguide (KL= 2. 0) with some external reflector (o = 0. 5 e-i n). 6{3L -50p = 0. 5 e -in It is evident that R~ is no longer a symmetrical 2 function of frequency . The maximum of IR'·J is shifted to 6SL ~ -0.6 with a value of ~ 0.962 (as compared to 0.93 at 6SL = 0 for p = 0). For 6SL > 0 the sidelobes oscillation amplitude is bounded by IPl 2 = 0.25 while for 6SL < 0, we have a lower bound for IR' 1 2 at 0. 25. This behavior is a direct consequence of the sign of p. p = 0.5 If we use instead, then the side lobe oscillation will have a lower bound for 6S > 0 and an upper bound for 6S < 0. .-51CHAPTER 2 REFERENCES (l) D. Marcuse, Theory of Dielectric Optical Waveguide, Academic Press, Inc. New York, 1974. (2) H. Kogelnik and C. V. Shank, "Coupled-wave theory of distributed feedback 1asers, 11 J. App 1. Phys. 43, 2327 ( 1972). (3) A. Yariv, ,;Coupled-mode theory for guided-wave optics, 11 IEEE J. Quantum Electron.2_, 919 (1973). (4) D. Marcuse, "Mode conversion caused by surface imperfections of a dielectric slab waveguide," Bell Syst. Tech. J. 48, 3187 (1969). (5) M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, "Grating coupler for efficient excitation of optical guided waves in thin fi 1ms," Appl. Phys. Lett. 1§_, 523 ( 1970) . (6) H. Kogelnik and T. Sosnowski, "Holographic thin film couplers, 11 Bell Syst. Tech. J . 49, 1602 (1970). (7) See for example L. Brillouin, Wave Propagation in Periodic Structures, McGraw-Hill, New York (1946) . (8) See for example, A. Yariv, Quantum Electronics, 2nd edition, John Wiley & Sons, Inc. New York, p. 512 (1975). (9) W. Streifer, D. R. Scifres and R. D. Burnham, "Coupling coefficients for distributed feedback single- and doubleheterostructure diode lasers," IEEE J . Quantum Electron. l!_, 867 (1975). (10) W. Streifer, D. R. Scifres, and R. D.Burnham , "TM-mode coupling coefficients in guided-wave distributed feedback lasers," IEEE J. Quantum Electron . .:!1_, 74 (1976). -52- CHAPTER 3 DISTRIBUTED FEEDBACK LASERS 3-1 Introduction A conventional laser oscillator, as shown in Fig. 3-l(a), consists of two major parts: medium. the optical resonator and the laser The laser medium which is pumped by some external agent pro- vides the gain. The resonator is usually formed by two (or more) mirrors, outside the gain medium, which provide the necessary feedback. This type of feedback is localized at the two mirrors and is completely separated from the gain medium. In 1971 Kogelnik and Shank(l) suggested and demonstrated a new type of laser structure called the "distributed feedback" laser in which the feedback mechanism is distributed along the length of the laser and integrated with the gain medi um as illustrated in Fig. 3-l(b). It utilizes the backward Bragg scattering of the optical waves in a periodic structure as the feedback mechanism . A simplified picture of this phenomena is illustrated in Fig. 3-2. In Fig. 3-2(a) we show an incident wave which travels at an angle e to the normal of the periodic planes. When the 11 Bragg condition 11 211.cose = m\ (m = 1,2,3 ... ) is satisfied there will be a constructive reflection of the incident wave . \ is the wavelength and A is the period of the spatial modulation. Fig. 3-2(b) shows the special case when e = 0, i.e., normal incidence. The Bragg condition now becomes A= m\/2 and constructive backward scattering or reflection results without the benefit of external reflectors. If we introduce a periodic modulation of the parameters of the lasing medium and provide sufficient gain , laser oscillation -53- (a) Excitation D, Output ¢J D, .(J, Laser Medium Mirror (b) Mirror 2 Excitation Laser medium Substrate Fig . 3 -l Schematic diagrams of (a) a conventional laser (b) a distributed feedback laser . -54- (0) 2 Acose = m A Bragg Condition A. <b > Bragg Condition : 2 A= m >-.. Fig. 3-2 A simplified picture showing the Bragg reflection for (a) incident with an angle e (b) normal incidence. -55is possible in this kind of st ructure. But because of the frequency sensitivity of the Bragg reflection process, laser oscillation is limited to a very narrow spectral range centered around w0 = mnc/Aneff (m = 1 ,2,3, .. . ). A coupled-wave analysis of one dimensional distributed feedback lasers was first given by Kogelnik and Shank( 2 ). They derived the oscillation condition and determined the mode structure. Their results were later extended to multi-layer dielectric waveguide configurations by several researchers( 3- 5)_ Also the effect of external reflectors on the longitudinal mode structure was discussed( 6 , 7). And schemes of transverse mode control in this kind of laser were suggested(S). Because of the possibility of longitudinal and transverse mode control, the lasing frequency stabilization by the period of modulation, and the compatibility of fabrication process with planar technology , the distributed feedback laser has attracted a great deal of interest in the field of integrated optics. Researchers are striving to realize a practical distributed feedback laser as the light source for integrated optical circuits . Distributed feedback laser oscillation was first observed i n organic thin films doped with Rhodamine 6G dye where a periodic spatial gain modulation was created by pumping the film by the interference pattern produced by two laser be ams{l) . It is possible to chan9e the angle of the two incident beams and vary the period of the modulation to achieve tuning i n this ki nd of laser (9-11) . It was pointed out that a periodic corrugation of a waveguide surface is in effect a periodic modulation of the effective refractive -56,- index of the waveguide modes. Laser oscillation was demonstrated in both liquid dye(l 2 ) and organic thin films doped with dye(l 3) on corrugated substrates. Similarly one can also corrugate the surface of a GaAs waveguide and achieve distributed feedback oscillation. A series of experiments(l 4-l 5 ) were carried out at Caltech which clearly demonstrated the feasibility of GaAs distributed feedback laser for the first time. A theoretical analysis by Nakamura and Yariv(lB) indicated that GaAs heterostructure injection lasers with corrugation feedback can be fabricated with thresholds comparable to those in ordinary injection lasers . Since then ·GaAs distributed feedback injection lasers of various structure have been demonstrated at low temperatures(l 9- 2l). It is not until the adoption of separate confinement double heterostructure that room temperature cw operation was possible( 22 , 23 ). There remain, however, additional problems to be solved before these lasers can be used as practical devices. Among them is the reliability (operating lifetime) problem. In this chapter we will apply the coupled-mode fonnalism established in the previous chapter to study the characteristics of an amplifying periodic waveguide. The distributed feedback laser oscillation condition is derived by two different approaches. The general properties of distributed feedback lasers and Fabry Perot lasers are compared . Also we describe the design of GaAs distributed feed back lasers. Finally expe r imental results on GaAs lasers are given and compared with theory . -573-2 Amplification and Oscillation in Periodic Waveguides The propagation properties of a periodically corrugated passive optical waveguide were considered in Chapter 2. We found that in the Bragg regime (A% A/2) the behavior of the waveguide is characterized by strong evanescence of the incident wave due to backward coherent scattering (reflection). In this section we shall consider the propagation of waves in an amplifying periodic waveguide. Assume that the total field inside the guide is given by E(z) = E.(z) + E (z) = A(z)eiSz-az + B(z)e-isz+az l r where a is the amplitude gain coefficient of the uniform waveguide. If we follow a procedure similar to that leading to equation (2-11) we obtain the coupled-mode equations dA _ iKBe-i2(6S+ia)z dz- (3-1) dB _ -iKAei2(6S+ia) z dzThese equations are of a form identical to that of equation (2-27) provided we perform the substitution 68 ➔ 66 + ia With this substitution we can then use (2-31) and (2-32) to obtain the solutions for the incident wave E.(z) = B(z)e-isz+az and the reflected l wave E (z) = A(z)ei Gz-az within a section of waveguide of length Las r -58- = Ei(O) [(a-i6S)sinhy(L-z) - ycoshy{L-z)Je (a-i6S)sinhyl - ycoshyl -is 0 z (3-2) is 0 z = E (O) iKsinhy(L-z)e i (a-i6S)sinhyl - ycoshyl where (3-3) • )2 y2 = K2 + ( a-16S 2 and IE (z)i 2 for KL= 1.0, Fig. 3-3 shows the behavior of IE.(z)l 1 . r al= 1.0 and 6SL = 0. We see that the incident wave no longer decays along the whole length but after a certain distance it starts to grow exponentially. Also the reflected wave grows more rapidly and exceeds the incident wave at z = 0 so that the device acts as an amplifier. This is fundamentally different from that of a passive waveguide which is also shown for comparison. We can define, as before, the amplitude reflection and transmission coefficient as R=-Er_(~O-) =~-----i_K_s_in_h~y_L_~_ Ei(O) (a-i6S)sinhyl - ycoshyl (3-4) -ye -iS oL T = Ei(O) = (a-i6S)sinhyl - ycoshyl (3-5) E.(L) 1 If the condition (a-i6S)sinhyl = ycoshyl is satisfied, it follows from equation (3~4) and (3-5) that both R and T become infinite. The device acts as an oscillator since it yields finite output fields Er (0) and E.(L) with no input field 1 -59- z=0 z =L I I waveguide I I I KL= 1.0 al= 1.0 I I I I I I 11Ej(L)l 2 I 0.5 L/4 L/2 3L/4 L z I KL= 1.0 al=0 I I I I :,Ei(L)l 2 I (z )1 2 0 Fig . 3-3 L/4 L/2 3L/4 L 2 and IE (z)l 2 in a periodic The behavior of !E.(z)i 1 r waveguide (a) with gain al= 1 .0 (b) without gain <1L = 0. z -60 - (Ei(O) = 0) . Hence equation (3 ~6) is the oscillation condition of a distributed feedback laser . For frequencies very near the Bragg frequency (w0 = nc/Aneff)' where condition (3~6) can be nearly satisfied with small gain coefficient a the device acts as a high gain amplifier. The amplified output is available either in reflection with amplitude gain R or in transmission with amplitude gain T. In Fig. 3-4 and 3-5 we plot the intensity reflection gain 1~r(O)/Ei(O)l 2 and the transmission gain IEi(L)/Ei(O)l 2 as functions of 6SL and al for KL= 0.4. Each plot contains four infinite gain singularities at which the condition (3-6) is satisfied. The plots are symmetrical with respect to 6SL. The coordinates of the singularities correspond to the oscillation frequency and threshold for different longitudinal modes of the laser as will be discussed in more detail in Section 3-4. It should be noted that Fig. 3-4 and 3-5 were plotted using linear analysis which means that saturation effects were not taken into consideration. In a real device the maximum amplification is always limited by saturation . It can be treated by a nonlinear analysis with numerical computation as was done by Hill and Watanabe( 24 )_ Ei (0) 3t--...... 0 .2 2 - 4 I • 0.1 - - + - - - - - - -· - - ~ I 5 . - ; - - - -- 6 I : - + -- - 7 8 ,1 - - - - . - - - . - -- -- - r- -~ KL= 0.4 9 3000 10 11 \ 12 13 6/3L 14 - - - ,..----------r------..----------r------..--------, Fig. 3-4 The reflection gain as a function of 6BL and al for KL= 0.4. 3 ...--1 \1f£_'\)I! I ----+ I 4 . . . _ _ - - + -- 5 6 1 , - - - -~ - -- . - - -- - - , - - - - , - -- 2 1,---- al Er(0)12 0.1 I 0 .2 10 100 3000 2000 1000 600 4000 I I O'I --' I + 6 I IEi (0) I I I ----+-- 20-r I I~ 80 1 ~ 1 °0 1 KL= 0.4 I I I I 0 ~ \ I ol 2 3 1 4I / I 2 I I J I I f ' \ . : I I , , I ooo' .... r u u 1 , I 000 I ~1 - -_, \I \ '-~-/ I .,{ I 7"' -----= I I 20 4 5 6 •7 8 9 10 II 12 Fig . 3-5 The transmission gain as a function of 6SL and al for KL= 0. 4. 3 13 ~/3L 14 4-+--t--+--+--r--~-~~r----,-----+---+--+4 0 1000 ,1500..,, I ~1000 600 ¥I i °f--1 I~ I l I I I F: f~SJ9, - 'ie 0 I I 5MT I =F1l ~~~~iio al Et(L) i2 I I ~ -633-3 Alternative Derivation of the Oscillation Condition In an ordinary laser oscillator the oscillation condition is dete rmined by following a wave generated inside the cavity for one complete round trip. By setting this round trip gain equal to unity one can obtain the condition for oscillation . In this section we shall use this recipe in a distributed feedback laser . As shown in Fig . 3-6 the periodic waveguide section with gain a extending from z = 0 to z = L. Let us pick a random reference plane, say z = t , and find the reflection coefficients at this plane looking into the two divided waveguide sections. respectively . Denote them by R(t) and R(L-t) Using equation (3-4) we can write R( £) and R(L-£) directly as R( £) = iKsinhyt (a- i6S)sinhy£ -ycoshyt and And the round trip gain in the periodic waveguide is R(£)R(L- t ) = l Substituting the expressions for R(i) and R(L-£) we have i Ksinh £ [ iKsinhy(L-£) ] =1 [ a-i 6S sinhyt - ycoshyi L(a-i6S)sinhy{L-t) - ycoshy{L-t)j After simplification it becomes (a-i 6S )sinhyL - ycoshyl = O This is the same as relation (3-6) obtained earlier . Note that the oscillation condition is independent of the choice of £ as it should be. As a matter of fact we can take t to be infinites i mally small, -64- z=L z=O Waveguide I )1 I I E1=1 ~ I ~ E2- . R( L- U I E3=R(l)-➔, I I I I I Oscillation Condition: RCl)R{L-l)=l Fi'g. 3-6 Schematic drawing shows the derivation of the distributed feedback laser oscillation condition. -65.Q,=E, and write ~!~ R( E)R(L~E) = l Since!!~ R(E) + 0 and~!~ R(L-E) + R(L) we must have R(L) = oo. This is the condition used to derive equation (3-6). As a special case 1 let us take .Q, = L/2, then the oscillation condition can be expressed as L L -i¢(~) R( ) = r( )e = ±1 2 where (3-7) 2 where y = [K 2+(a-i6B) 2J112 = yr + iy l. When 68 = 0, yi = 0 and ¢(L/2) reduces to TT/2 so equation (3-7) cannot be sat4sfied. This means no laser oscillation is possible at the exact Bragg frequency. If the laser is connected to other circuit components there will be some external effect that cannot be neglected. But we can always \ represent these outside effects as an external reflector and calculate the composite reflectivity as was done in section 2-6 . Then the oscillation condition is derived with the new reflection coefficients. -66- Let us assume that the external reflectivity at z = 0 is pl and at z =Lis p2 . (Refer to Fig. 3-6.) Then the composite reflectivities are -[p (a-i6S)-iK]sinhy£ -p ycoshy£ [ (a-i 6S) -i p1K]s i nhy£ -ycoshy£ = . . . . .1. - - - - : - - - . . - - - - c , - - - - ~ .1 --~-.,.-- and RI -[p (a-i6S)-iK]sinhy{L-£)-p ycoshy(L-£) (L-£) = - r -2- - ~ - - ~ - . . . . . . -......... 2 [ (a-i~S) ~i p2KJS inhy (L-£ )-ycoshy ( L-£) .----~'--T- Then the oscillation condition becomes Again this condition is independent of£ and it reduces to equation (3-6) when both pl and p2 vanish. Note that if either pl or p2 is zero the equation can be greatly simplified. The effect of external reflectors on the dsitributed feedback laser mode structure is very complicated and we must resort to numerical examples to see what is going on. A special case of pl= p2 was treated by Chinn( 6 ) and the more general case pl; p2 was calculated by Streifer et al(?). 3-4 Determination of the Lasing Frequency and the Threshold Gain The oscillation condition of a uniform distributed feedback laser with no external reflectors was found to be (a-i6S)sinhYL = ycoshyL 2 2 where y = K + (a-i6S) 2 . -67In general y is a complex number y = yr + iy.l where and So the oscillation condition (a-i6S)sinh(y r+iy.)L , = (y r+iy.)cosh(y 1 r +iy.)L , becomes F(a,6S) =0 (3-10) This is the eigenvalue equation of the distributed feedback laser which can be solved numerically or graphically to obtain an infinite set of eigenvalue pairs (6Sn, gn). Each such pair determines the oscillation frequency 6Sn and the threshold gain gn of then-th longitudinal mode of the laser. It can be shown easily that F(a,6S) = F(a,-6S) which means that the location of the longitudinal modes are symmetric about the Bragg frequency (6S=0). reasoning requires that a be positive. Also simple physical In Fig. 3-7 we plot the solu- tions of F(a,6S) = 0 for various values of KL. It is evident from the figure that the laser always oscillates outside the stop band, i.e., ltsnl > K for all n. The dependence of al on KL can be seen 0 D 6. I 2 \ \ \ h \ • \ \ *\ \ D '' 3 \ ''' "O..... 4 ·., 5 \ \ \ 6 n=I 6. \ r\ I 'Q \ \ ' n=2 ' 'o 7 8 9 \ ~ '\ ♦ I \ I I I I I I *I I * KL= 1.0 KL= 2.0 KL= 4.0 • KL= 0.5 * KL= 0.25 \ \ b \ 10 \ 'q 11 n=3 Fig. 3-7 Threshold gain al versus frequency deviation 6BL for disributed feedback lasers with different values of KL. 00 r 2~ 31- 4~ al ♦ I en ~/3L I co -69from the dotted curves in Fig. 3-7 for different longitudinal modes. We can also derive the dependence of threshold gain a on the laser length L for a fixed KL. This result is shown in Fig. 3-8. In general the process of solving for the eigenvalues is tedious. It is desirable to obtain some approximate analytic solutions instead. Let us look into the case when K is small and 68 is large. Recall the expression for yr (3-9) If K is small and 68 is large, the laser will need a large gain to oscillate. Under these conditions Yr becomes a( l+o) ( 3-11) and consequently yi = - a~8 ~ - M3 ll r L where The oscillation condition (3-6) can be written as y - a-i68 2yL = -l y + a-i68. e (3-13) Use (3-11) and (3-12) in (3-13), we have (a+i68 . e2[a(l+o)]L e-i268(1-o)L = _1 2 a.-168 ( 3,. 14) -- 70- a K 80 = IOcm- 1 ----60 I E u ---C ·0 (.9 40 '"O 0 ..c (/) a.> ~ ..c .,._ 20 n=3 n=2 n=I 0 0 0.5 1.0 1.5 2.0 Length (mm) Fig. 3~8 A plot of threshold gain a vs. length of the laser L for different longitudinal modes. L -71Equating the phases of both sides of (3-14) we have where n = 0,±l ,±2,±3, ... If an >> 6Sn the oscillation frequencies will be given approximately by and the frequency Sc ( S +6B ) - n - o n c = w + (n + }) TTc L O wn - neff - neff neff (3-15) Once again we see that no laser oscillation can exist at w = w0 (6Sn = 0). The longitudinal mode spacing in this limiting case is TIC 6 w = wn+l - wn = n eff L This result is identical to that of a Fabry-Perot laser with a length L. Once 6Sn is found we can write or Hence (3-16) where -72- Thus each longitudinal mode is represented by a straight line with l/2A n slope in the neff-A plane. We can also plot the waveguide dispersion curve n (A)= A8(A) eff 2TT on the same plane as shown in Fig. 3-9. The intersections of this curve with the family of lines represented by equation (3-16) give the lasing wavelengths of the different longitudinal modes. If the waveguide can support several transverse modes the sitaution is more complicated. As discussed in Chapter 2, different transverse modes experience different coupling constants so that for each transverse mode we have a different dispersion curve and a different set of lines like equation (3-16) to detennine the lasing wavelengths. 3-5 The Effect of Distributed Feedback on Spontaneous Emission Spectrum In this section we study the spectral properties of a distributed feedback laser operating below threshold. This problem is treated by using multi-reflection approach to calculate the output of spontaneous emission from a section of amplifying periodic waveguide. As depicted in Fig. 3-10 the waveguide has a length L with amplitude gain coefficient a. We pick an arbitrary plane z0 and consider a small volume of the medium. Because of the spontaneous emission process this small volume emits electromagnetic wave of amplitude E isotropically. We are interested in the direction perpendicular to the periodic planes {parallel to the z-axis). - 73- 0 ni=l A=ll82A ~ i,..- n~It=0.6µ.m n3 =3.50 _ _ _ _ _ _ _ _____.._____,#---------- m= I 3.55 \ n eff = M3(x) 21T 3.50 8350 8450 8400 0 Wavelength (A) Fisi 3-9 Graphical method for determining the lasing wavelength of a distributed feedback laser. -74- Z=Q Z=L z=z 0 I Goin: Waveguide T', R' d,,. T , R I I ( 1+ R)T'E ; +--i I RE~ I I r-+ Cl +R)R'E I Cl+R)T'RR'E: +---, I I :Cl +R)TR'E i---+ ( 1+R)RR'E ~ I 2 :(1 +R>TRR' E ~ I I Fig. 3-10 A schematic drawing shows the processes of amplified spontaneous emission in an amplifying periodic v,1avegui de. -75- We could follow the radiation emitted from the differential volume to the output and then sum the contributions from the rest of the volume to find the total output at z = L. The section of the waveguide to the right of the z = z0 plane with length L - z0 is characterized by the transmission and reflection parameters T and R. And the other section with length z0 is characterized by T1 and R1 • We sum up the partial waves due to the repeated reflections and obtain Eout =TE+ (l+R)RlTE = (l+R')TE l~RR ~ 1... RR 1 Expressions for R,R 1 ,T,T 1 can be obtained by slight modification of equation (3-4) and (3-5). It can be shown by direct substitution that ycoshyz 0 -[ a-i(6B- K)]sinhyz 0 2 2 2 IE out I -- j Ej ycoshyl - (a-i 6B )sinh yl l The intensity output at z =Lis obtained by integrating jEoutl 2 over z (we sum the intensities since spontaneous emissions 0 from different sections are assumed to be incoherent). L I(z=L) = f jEouti 2dz ( 3-17) 0 0 Equation (3- 17) is plotted in Fig . 3-11 as a function of frequency for various values of al and KL= 0.4. is evident from the curves . Spectral narrowing For KL= 0.4 the lowest order longitudinal mode of this device will oscillate at 6BL = ±2. 2 when al= 2.9 . The absolute output powers at ASL= 2.2 and 6SL = -2.2 are plotted as a function of the gain coeffi cient al in Fig . 3-l2(a) . At each of -80 -6.0 -5.0 -4 .0 -3.0 -2 .0 6/3L ~0 .0 -1.0 0.1 0.2 0.3 ==10 Power 2.0 3.0 4.0 5.0 6.0 KL= 0.4 7.0 KL= 0.4. The amplitude gain coefficient al is used as a parameter. Fig. 3-11 The emission spectrum of amplified spontaneous emission from a periodic waveguide with -70 I 1033.7 X7 6038 04 0.5 0.6 0.7 0.8 09 1.0 Norma I ized Output 8.0 I 0) I -..J -77.- these two 6BL values the output power increases rapidly as the threshold (al= 2.9) is approached. Although these two modes have equal threshold gains their output powers are not identical. A plot of the spectral width of the peak (6B = -2.2) below threshold as a function of 1/Pout is shown in Fig. 3-l2(b). It is almost a straight line, implying l cS (6B) a: - - pout a relation common to all amplified spontaneous emission processes. Similar results were also obtained by Chinn and Kelley( 25 ). If the waveguide gain coefficient exceeds the threshold of oscillation the above analysis is not valid. A nonlinear analysis should be used to include saturation effects. 3-6 Comparison of GaAs Distributed Feedback Lasers and Fabry-Perot Lasers As discussed in section 3-4 the lowest order longitudinal mode of distributed feedback lasers with large KL lase at 66 ~ ±K. And since B0 = TT/A is of the order of 30 µm -1 in GaAs, while typical value of K is ~10- 2µm- 1 , so generally speaking a distributed feedback laser oscillates at a frequency B0 c or w~w = - - = ~ 0 neff (3-18) where neff is the effective index of refraction of the laser mode in the waveguide . If the £-th order corrugation is used for the feedback equation (3-18) is modified to -78- 20 - 15 ..d lo- 10 6,8L = -2.2 ~,BL= 2.2 +- C :J 0 .......... ~ ::> 0 a.. 5 0 ...___---'-_ _ 0 --1,.__ _1.,___ __L_ __L__ _~ 3 al 2 8(6,BL) 3 2 2 Fig . 3-:- 12 3 4 pl (orb. unit) OUT Plots of (a) The dependence of output power on gain (b ) The relation between spectral width and the reciprocal of the output power. -79(3-19) This is a very simple expression which determines roughly the lasing wavelength once neff and A are known. And it is obvious that one can design a waveguide structure with known neff' and select a proper A for a desired wavelength~. for ordinary injection lasers. This is not true in general The gain spectrum of GaAs usually spans several hundred angstroms, the cavity is formed by two parallel cleaved crystal planes which provide almost uniform reflectivity over a large frequency range. Hence there is no strong mechanism to provide longitudinal mode discrimination. Usually several modes oscillate simultaneously with comparable thresholds and span typically a spectral region of ~30A or more. This nonmonochromatic radiation presents a problem in fiber communication systems. In distributed feedback lasers we have a built-in longitudinal mode discrimination so it is not difficult to obtain single mode operation with linewidths less than one angstrom. Another advantage of distributed feedback lasers over Fabry~ Perot GaAs lasers is that the temperature stability of lasing wavelength is improved. The temperature variation of lasing wavelength in distributed feedback lasers is due to the change of the effective index of refraction (neff) with temperature as evident from equation (3-18). Hence aneff aT If we use an ff -4 (26) aneff d>. a~ ~ 3 x 10 /deg and neff - >- a>. ~ 4.5, dT out to be~ 0.58i/deg. turns In ordinary GaAs lasers the band gap energy varies with temperature thus causing both the spontaneous emission spectrum and the laser wavelength to shift with temperature. This 0 shift has a rate of~ 2A/deg which is about 4 times that of distributed feedback lasers. There is, however, a penalty which accompanies this improved wavelength stability. An ordinary laser can operate at any temperature if threshold is attainable. But in distributed feed- back lasers the choice of A 11 clamps 11 the lasing wavelength for a certain temperature and since the temperature coefficients of >-DFB and the center of gain spectrum are different at some temperatures ADFB falls outside the gain spectrum co~pletely and no oscillation is possible. 3-7 Design Factors in GaAs Distributed Feedback Lasers In this section we address ourselves to the problem of designing a GaAs distributed feedback laser. We treat the problem in a general way without going into details of specific structures. First of all we have to specify the desired lasing wavelength A and the operating temperature T. Next we choose a proper waveguide structure which can be either a homojunction structure, or single heterostructure, or double heterostructure, etc. From the given structure we can calculate the number of guided transverse modes and their respective effective indices of refraction. In order to obtain single mode operation, however, it is desirable to use a single mode waveguide with definite -81The necessary corrugation period A is calculated through neff at A, the equation A= A/2neff for fundamental Bragg coupling and A= iA/2neff for the i-th order coupling. The choice of A has to be such that at the particular operating temperature the medium can provide sufficiently large gain at that wavelength. The typical 0 photoluminescence spectrum of GaAs is about 150-200A wide. At 77°K the peak occurs at ~8450A and at 300°K it shifts to 8900A . For temperatures in between, the luminescence peak can be estimated roughly by the formula 0 A(T) ~ [8450 + 2x(T~77)]A The luminescence spectrum will change slightly by varying carrier concentrations and dopants. It can also be varied by incorporating into the active region a small amount of Al. As a matter of fact checking the luminescence spectrum is one of the methods of obtaining a rough estimate of the amount of Al in a GaAlAs layer, For example in GaAs typical numbers at 77°K are\= 8450~, n ~ 3.59, then A= 1177A is required for i = l fundamental operation. This small period is difficult to fabricate and i = 3 is used in most 0 of our experiments, which results in A= 3531A. The corrugation height can be measured by using SEM (scanni~g electron microscope) pictures. This quantity is then used to calculate the coupling coefficient. From a curve similar to those in Fig. 3-8 we can obtain, given K, an optimal length for our laser. The laser oscillation threshold can be estimated from -82- where a~h(K,L) is that found by solving the eigenvalue equation (3-10) F(a,6S) = 0 and a.bulk is the bulk material loss coefficient at the lasing wavelength. If a third order corrugation is used there exist losses due to the coupling of the guided laser mode to the unguided radiation modes through the first and the second order Bragg scattering processes. These losses can be represented by a distributed loss coefficient arad( 27 , 2a) which must be added to the other losses in the medium . The laser threshold condition in this case becomes (3-21) Typical value of a.rad for a third order corrugation with tooth height 500A'ii is ~lQcm -1 . In most of our optical pumping experiments we start out with slabs of GaAs-GaAlAs waveguides. Through photoluminescence measurements we determine the peak of the gain spectrum. By making Fabry-Perot lasers from part of the sample and measuring the longitudinal mode spacings we obtain a good estimate of neff' The corrugation period A is then determined from the condition A - ;\ - 2neff where;\ corresponds to the central wavelength of the luminescence spectrum. 3-8 Optically Pumped GaAs Distributed Feedback Lasers Optical excitation is a very convenient way and a very powerful tool for studying the spontaneous and stimulated emission processes in semiconductors. This method of pumping obviates the need for electrical contacts which greatly simplifies the sample preparation and makes it possible to explore new materials and structures without first solving the contact problem. Furthermore, it is nondestructive in the sense that sample surface will not be damaged unless excessive optical power density is used. In this section we shall describe some of the experiments on optically pumped GaAs distributed feedback lasers. The first attempt to demonstrate distributed feedback lasing action in GaAs was done by optically pumping a slab of a GaAs crystal with periodic surface corrugation. A schematic drawing of the laser is shown in Fig. 3-13. The GaAs \'1afer was n-type (Si doped) with a carrier concentration of about 10 18 cm- 3 . The top surface (100 plane) was polished and chemically etched. The corrugation was produced by ion milling through a photoresist mask generated by holographic photolithography as will be described in Chapter 5. The heiqht of the grooves was estimated from the SEM picture to be around 5ooE. The period A was 0.35 µm which corresponds to t = 3 in the Bragg condition 2B where s is the propagation constant of the guided mode. The experimental set-up is shown in Fig. 3-14. The gain was pro- vided by optical pumping using a Q-switched ruby laser (A ~ 0.6943 µm). Each individual pumping pulse had a duration of ~20 nsec and the peak power was attenuated to ~10 KW. A cylindrical lens was used to pump a Output ~ rvQ.35µ ~ I ~ 00 I . l~soo A .T Fig . 3-13 Schematic structure of a GaAs distributed feedback laser. GaAs Pumping Beam • Fig. 3-14 Photomultiplier Filter Filter I feedback surface l asers. □TI 1----0----) ___.J,,,,~ I Q-switched Ruby Laser Experimental set-up for optical excitation of GaAs distributed Photomultiplier Oscilloscope Monochromator © Sample ~ Red filter ~Lens ,,, I ii Cylindrical lens Copper block I 0::, I u, -86rectangular strip of 3x0.5 mm. Samples were attached to a copper block heat sink with vacuum grease and mounted inside a liquid nitrogen dewar. Special care was taken to make sure that the pumping beam did not fall across the whole length of the sample . Also the output face of the crystal was lapped so that it was not parallel to the grooves and to the other ·end face. This was done to minimize the reflection feedback . The system was carefully aligned with a He-Ne laser. The pumping beam passed through the front window of the dewar and struck the sample surface perpendicularly. The output from the GaAs sample passed through the side window and was collected by a lens and fed into a monochromator . The signal was subsequently amplified by a photomultiplier (S-1) and displayed on a memory scope. The pumping pulse was also monitored on the scope. The oscillation threshold of such lasers at 77 °K was found to be ~2 xl0 5W/cm2 . A typical emission spectrum of a sample excited above threshold is shown in Fig. 3-15. Stimulated emission is indicated by the narrow resolution limited peak. Also shown in the same figure is the spectrum of a sample without corrugation under simililr pumping conditions . This sample displays only the broad (~180~) spontaneous emission feature. The stimulated emission peak at \ = 0.832 µm cor- responds, using equation (3-19), to an index of refraction n = 3.6 at 77 °K for GaAs . Figure 3-16 shows plots of the emission power as a function of the pumping intensity for a corrugated and an uncorrugated sample . The 2 "brea k" in the curve of the corrugated sample near 2.5 x 10 5w;cm coin cides well with the first appearance of the narrow spectral peak in • -87- 15r---------------------. Corrugated Sample --+- Uncorrugated Sample --o- ---~ C -.j ~ ~ . 10 ..0 ~ -0 ~ <l) 3 0 Q_ C 0 Cf) Cf) 5 E w I 0'----------------------------0.83 0.82 0.84 Wavelength \ (µ,) Fig. 3-15 The emission spectrum of a corrugated and an uncorrugated sample. 5 X 10 5 vJ/cm2. Pumping intensity is - 88- -0- Corrugated Sample ---e- Uncorrugated Sample 10 -C :::> .Ii '- 0 '- ~ ~ 5 C .Q Cl) .!:!? E w o..,___.....___.....____.._____..____ 0 2 4 6 6 5 Pumping intensity ( I o w/cm 2) Fig . 3- 16 Emission powe r as a function of pumpin g in t ensi ty at A = 0.832 µm . -89- Fig . 3-15. The characteristics of the uncorrugated sample, however, remain linear up to the highest pumping power employed . For pumping power exceeding 10 6W/cm 2 surface damage was found on both corrugated and uncorrugated samples. This very first experiment demonstrated that a GaAs crystal with surface corrugation can be made to lase if enough gain is provided. The threshold, however, was very high. confinement. This is largely due to poor optical In a bulk crystal the optical confinement comes from the "inverted layer" which is not a strong effect. The threshold pumping level is expected, as in the case of injection lasers, to depend strongly on the optical confinement. Realizing this we repeated the experiment with epitaxial GaAs dielectric waveguides. Two different types of dielectric waveguides, illustrated by Fig. 3-17, were used in the experiment. The first structure consisted of an epitaxial GaAs layer which in different experiments varied between 1 and 3 µmin thickness with a carrier concentration of n ~ 6 x 10 16 cm -3 . The substrate was a GaAs crystal with n ~ 2 x 1018 cm- 3 . Due to the carrier concentration difference the epitaxial layer has a larger index of refraction than the substrate and the condition for dielectric waveguiding are thus satisfied. The second structure consisted of GaAs and Ga 0 _7Al 0 . 3As double layers on a GaAs substrate. The larger the Al concentration, the smaller the index of refraction of the layer. were produced as described above. The surface corrugations Due to the improved confinement, the pumping threshold intensity was reduced by lOX and a different pumping source could be used. This is shown in Fig. 3-18. A repetitively pulsed -90- Ga As epitaxial layer ( n "'10 16 cm-3) (a) Ga As substrate (n"' 10 18 cm-3) Ga As epitaxial layer Gao_7Alo_3As epitaxial layer (b) GaAs substrate Fig. 3-17 Cross sections of GaAs waveguide structure distributed feedback lasers. L- I --~ Ill ,,,,,, <¢.> Amplifier Boxcar Integrator Adjustable slit Cylindrical lenses lb_ I Osc i 11 oscope (v - x-y recorde r ' Filter Dye Laser N2 Laser - · - Beam limiter Fig. 3-18 The optical pumping set-up using an N?-laser pumped dye laser . L Photomultiplier ~ SpectrometerI"' Dewar Copper block I I \.,:) --' -92nitrogen laser pumped a dye laser (Rhodamine B) whose output tuned to ~o.63 µm was used to pump the GaAs dielectric waveguide . Each indivi- dual pumping pulse had a duration of ~7 nsec and a peak power of up to 2 KW . Cylindrical lenses and an adjustable slit were used to pump a rectangular strip 0, 3 mm wide and of a variable length . The output was collected by a lens and guided into a spectrometer whose acceptance wavelength was scanned to record the emission spectrum. When the first sample was pumped the output power from the corrugated region was about two orders of magnitude smaller than that from the uncorruqated region of the same wafer. It was found that ion mill- ing introduced defects in the GaAs layer which reduces the carrier recombination efficiency drastically. Since the laser emission was re- stricted by dielectric waveguiding to the vicinity of the surface the effect of this damage on threshold was severe. Annealing in a hydrogen atmosphere at 450°C for ~30 min removed most of the defects and made lasing possible at threshold pumping intensity ~10 4w;cm2 , A typical output spectrum from these waveguide lasers is shown in Fig . 3- 19 displaying both the spontaneous and the stimulated emission peaks . The stimulated emission peak was found to be stable against excitation level while the spontaneous emission peak moved toward longer wavelength as the excitation level was increased. A number of waveguides were prepared with different corrugation periods . The measured oscillation wavelengths were .plotted against the corrugation period A in Fig . 3-20. plot of the equation The straiqht line is a theoretical -93 - --+- C Ga As Ga Al As Ga As :) _Q ~ --0 >, -+(/) C Q.) -+- C 1-i 8270 8290 8310 0 Wavelength (A) Fig. 3-19 Em i ssion spectrum of an optically pumped waveguide laser with surface corrugation. -94- 8340...---- - - - - - - - - - - - - - - - - - -- - 8320 _... o<I: .__,, n = 3 . 59 I 8300 i I .c -+- Ol C (1) - 8280 • (1) > 0 ~ 8260 8240 822 0 ' - - - - - - ' - - - - - - - - - a . - -- ------3440 3450 3460 3470 3480 0 Per iod A (A) Fig . 3~20 Os cillation wave l ength vs . period of corr ugat ion . -95- with neff = 3.59 , It is seen that a tuning range of by varying the corrugation period from 345oE to 347G~. 45R was spanned This is a clear in di cation t hat the laser feedback is indeed caused by the corrugation. And the corruqation acts as a built-in grating filter leading to a stabili zation of the output wavelength. As a chec k on the theory we calculated the coupling constant K for on e of the samples and determined it to be ~1 .93 cm-l. The laser threshold was measu red for various pumping lengths of the same waveguide. The result is shown in Fiq. 3-27. The solid curve is a plot of equation (3-21) using the data of this sample with abulk+ arad ~ 15 cm-l as determined by experiments . This aspect of the experiment The fair agreement between the experimental is described in Chapter 5. results and the theoretical curve in Fig. 3-21 lends support to the existence of distributed feedback action in this sample. GaAs waveguides with a corrugation period of 0. 115 wm were fabricated and pumped optically. This period corresponds to the fundamental ( 9, = l) Bragg reflection in equation (3-19) . Single mode as well as multi - longitudinal mode oscillation were observed at different pumping levels and pumping lengths as shown in Fig. 3-22. longitudinal mode spacing 6\ In (b) the measured is ~ 1R which agrees with the theoreti- cal value calculated from 8n) 2L ( n - \ a\ where \ is the vacuum oscillation wavelength and L is the laser -96- 1000 theory • ,,__ I experiment E u C ·0 CJ) 100 --0 0 ..c en Q) ~ ..c r- I0 L--- - 0 ~ - - ~ - - _ _ , _ _ __ 0.5 I 1.5 2 Length (mm) Fig, 3-21 Threshold gain of a distributed feedback laser plotted against the laser length. -97- ...,~ (a) L= 150µ.m I - = I.I Ith -C ~ ..0 \... 0 ....__.. - >- (/) -.fJ.- ( b) C Q) L=700µ.m C 1--4 - I = 1.4 Ith 8230 8240 8250 0 Wavelength (A) Fig . 3~22 Oscillation spectrum of GaAs distributed feedback laser with fundamental corrugation A~ 0.115 µm. -98- length~ 700 µmin our example. The use of fundamental corrugation was expected to reduce the radiation loss and to increase the magnitude of coupling constant (as compared to the 3rd order corrugation with the same corrugation height) . The expected reduction in threshold was not realized, however, in our experiment . The threshold pumping intensity was comparable to that obtained with a 3rd order grating . This is believed to be due to the fact that the quality of the 0.115 µm corrugation was not as good as that of the 0.35 µm corrugation . In fact it was founrl difficult to produce a 0.115 µm corruga- tion with tooth height larger than ?ooE while for 0,35 µm corrugation we can easily fabricate tooth heights larger than 1500~. This point will be considered further in Chapter 5. 3-9 GaAs-GaAlAs Distributed Feedback Injection Lasers Although optical pumping is convenient in laboratory studies, it is not practical in actual applications . Since the technology of ordinary GaAs injection lasers has been very successful it was felt that GaAs distributed feedback lasers with electrical injection could also be fabricated . A theoretical analysis by Nakamura and Yariv(lB) of GaAs injection lasers with a corrugated interface indicated the possibility of a large reduction of the threshold current density as well as of frequency and mode discrimination. We thus started experi - menting with the fabrication of a do uble heterostructure distributed feedback laser diode. Figure 3-23 shows a schematic drawing of a double-heterostructure GaAs laser with internal corrugations. The fabrication process went as contact Fi g. 3-23 Cross section of a GaAs -GaAl As double-heterostructure distributed feedback l aser diode . Au-Ge - Ni n - Ga As Substrate n - Ga 0 .7 Al 0 _3 As p - GaAs ¢=1 p- GaAs p - Ga 0 . 7 Al 0 _3 As contact output Au - Cr I \.0 \.0 I -100follows. An n- Ga 0 _7Al 0 _3As layer ('v 4 µm) doped with Sn and a p-GaAs active layer ('v 1.5 µm) doped with Ge were grown on an n- GaAs substrate by liquid phase epitaxy . The process is then interrupted for the fabrication of corrugations . The thickness of the active layer was etched down to 0.5-1 . 3 µm before fabricating corrugations with a period of 'v0, 34 µm and a depth of 'v900~. Next a p-Ga 0 _7Al 0 _3As layer ('v 3 µm) and a p-GaAs layer ('v l µm), both doped with Ge, were grown on the corrugated surface of the p-GaAs layer by liquid phase epitaxy. Special care was taken in this step to prevent the meltback of the corrugated surface during the epitaxial growth to be described in Chapter 5. The laser had a mesa-stripe geometry so that the current injection was limited to a rectangular region . The width of the stripe was 'v5Qµm. Meta 11 i c con tacts to the di odes were made by evaporating Cr and Au on the p-side and Au-Ge-Ni on then-side. was varied from 150 to 700 µm. The length of the Au-Cr contact A lossy unpumped waveguide with a length 2.5-3 mm was contiguous to the current injecting area . This was done in order to minimize the optical feedback from the end face . The output was taken f rom the front face as indicated in Fig. 3-23. A pulser which provided current pulses of 'v50 nsec duration with variable repetition rates was used . The diode characteristics were measured under var ious temperatures. lasers is shown in Fig . 3-24. A typical spectrum from these 0 The corrugation period is 3416A and the threshold current density is 9 KA/cm 2 . The output spectrum consists of two main components . The spontaneous emission, which has a broad peak centered around 8135g and, at injection current above threshold (2.6A), -1 01 - 3 82 °K A!50-2-4P ~I-- -tllt100 -- 3.0A 2 0 C 0.3 A 50 :::> ...0 I,.. 0 .._, . - >-. C l) 0 1 8110 8 ll2. 8U4 C (1) C 2.6A 0 r...c,......- - - - - - - - - - - - - - -- - - - - - - - - - 8200 8100 Wavelength (A) Fig. 3-24 The emission spectra of a typical laser with a corrugation period of 3416~ and a corrugation height of 900A. -102a narrow peak at s112E due to stimulated emission. the stimulated emission is usually less than 1E. The line width of From the lasing wavelength and the period we calculated the effective index of refraction for the lasing mode in our diode structure to be ~3.56. The inset of Fig. 3-24 shows the detail of the stimulated emission peak. It shows that the diode lased in a single longitudinal mode whose wavelength was stable against the change of excitation level. The laser output was polarized with the electric field parallel to the junction plane. Lasers in which the length of the unexcited waveguide section was less than 1 mm long exhibited the coexistence of two types of laser oscillation. In one of these , the feedback was due to the corrugation. In the other the feedback was due to the two end faces. These two oscillation spectra are separable by their temperature dependence as was discussed in section 3-6. When the unexcited waveguide length ex- ceeded 2 mm no effect of the end face was observed and the laser spectrum became very simple. The lowest threshold obtained in this kind of laser was ~ 2.5 kA/cm 2 at 80°K . It was found that the surface damage caused by ion -milling severely reduced the radiative recombination efficiency of the carriers in the active region. This caused the threshold of this type of laser to become too high for oscillation at the higher temperatures. This problem was alleviated somewhat by using chemical etching for fabricating rather than ion etching the corrugations. We were then able to operate lasers at lower thresholds and at temperatures up to -103~ 145°K. Figure 3-25 shows the lasing wavelength as a function of temperature for both distributed feedback and Fabry-Perot lasers. It is seen that the distributed feedback lasing wavelength shifts at a rate of 0.5~/degree and that of the Fabry-Perot lasers shifts ~ 2A/degree which agrees well with the results given in section 3-6. The improvements in laser performance which resulted from chemical etching were not sufficient to obtain room temperature operation. This is believed due to the large number of nonradiative recombination centers introduced by the etching. This problem was solved, partially, by adopting a separate-confinement heterostructure (SCH). A schematic drawing of the cross section of a laser diode is shown in Fig. 3-26. The first two layers were grown in a manner similar to that of doubleheterostructure. The gratings were not fabricated on the p-GaAs layer, instead two additional layers were grown directly, one was a p-Ga 0 _83 A1 0 _07 As layer(~ 0. 1 µm) and the other is a p-Ga 0 _93Ala.O?As layer(~ 0.2 µm). The corrugation was fabricated on the last mentioned layer. This was followed by a layer of p-Ga 0 _7Al 0 _3As layer(~ 2 µm) and a layer of p~GaAs (~ l µm). In this structure the electrons injected from the n-Ga 0 _7A1 0 _3As layer 11 see 11 a potential barrier presented by the p-Ga 0 _83 A1 0 _17 As layer and thus are prevented from reaching the corrugated region where nonradiative recombination centers exist . The thickness (~ 0.1 µm) of this layer is too small, however, to affect the confinement so that the wave extends and sees 11 the corrugation. The optical mode confinement is sketched on the left side of Fig . 3-26. The need for the p-Ga 0 _93 A1 0 _07 As layer is due 11 to the fact that the p-Ga 0 _83 A1 0 _17As layer oxidizes when exposed to air -104- 8400 O DFB laser • - Fabry-Perot laser 12.5-3-~P 8300 o<( - ...c C) 122-4-6'P C - Q) Q) > 0 3 8200 ~ A 50 - 2 - 4 P 8100 ~;f--,L._ _.____.__ _.____..._ 80 100 120 __.___...____ Tempera tu re ( °K) Fig . 3-25 The lasing wavelength as a function of temperature. 140 p-Gao.1A.io.3 As (,....,2µ.m) Au-Ge-Ni contact p-GaAs (rv0.3µ.m) distributed feedback injection laser. Cross section of a separate confinement heterostructure GaAs n-GaAs substrate n-Gao.7 A.io.3 As(,....,2µ.m) p-Gao.83 A.£0.11 As (rvO. I µ.m) I ~ ~ ~ ~ ~ ~ ~ ~ 1:--p-G □ o.93 A..eo.07 As(~0.2µ.ml Fig . 3-26 Optical Confinement i p-GaAs (rvlµ.m) Au - Cr contact ·1 U7 I 0 --' -106- and subsequent epitaxial growth is impossible. The corrugations are thus fabricated on the oxidation resistant p-Ga 0 _93Al 0 _ As layer. 07 Diodes with this structure were found to lase with extremely low thresholds. A= 36ooE 80°K. For example, one of the lasers with length L = 500 µm, lased with threshold current density Jth = 500A/cm2 at Another laser with L = 500 µm, A= 3770E has Jth ~ 3 KA/cm 2 at 300°K. This value is low enough for CW operation if proper heat sinking is provided as reported by Nakamura et al _( 23 ) So far most of the experiments in GaAs distributed feedback lasers have been limited to third order corrugations for which the radiation loss is not negligible. Hopefully some day we will have con- venient CW UV source with wavelength less than 2000R so that corrugations with period around 0.1 µm can easily be made and used in GaAs lasers to further reduce the threshold. With the room temperature GaAs distri- buted feedback lasers available the next question is how to incorporate the laser with other optical circuit components. This is an interesting current research area because the result could lead to the first monolithic integrated optical circuit. Meanwhile an equally important problem that has to be solved is the laser lifetime. The lifetime of the laboratory GaAs distributed feedback lasers ranged from ten minutes to twenty or thirty hours . In order to be a practical device, lifetime on the order of 10 6 hours is necessary. So there remains a lot of research to be done in this area. -107..Chapter 3 (1) - References H. Kogelnik and C. V. Shank, "Stimulated emission in a periodic structure," Appl. Phys. Lett.~' 152 (1971). (2) H. Kogelnik and C. V. Shank, "Coupled-wave theory of distributed feedback lasers," J. Appl. Phys. 43, 2327 (1972). ( 3) S. Wang, "Proposal of periodic layered waveguide structures for distributed lasers," J. Appl. Phys. 44, 767 (1972). (4) A. Yariv, "Coupled mode theory for guided wave optics," IEEE J. Quantum Electron.~, 919 (1973). (5) R. Shubert, "Theory of optical-waveguide distributed feedback lasers with non-uniform gain and coupling," J. Appl. Phys. 45, 209 (1974). (6) S. Chinn, "Effects of mirror reflectivity in a distributed feedback laser, 11 IEEE J. Quantum Electron. ~' 574 (1973). (7) W. Streifer, R. D. Burnham, and D. R. Scifres, "Effect of external reflectors on longitudinal modes of distributed feedback lasers," IEEE J. Quantum Electron . .L!.., 154 (1975). (8) S. Somekh, "Transverse mode control in a distributed feedback semiconductor laser, 11 Proc. IEEE (Lett.) g, 277 ( 1974). (9) C. V. Shank, J.E. Bjorkholm, and H. Kogelnik, "Tunable distributed feedback dye laser, 11 Appl. Phys. Lett. ~, 395 ( 1971). (10) R. L. Fork, K. R. German, and E. A. Chandross, "Photo-dimer distributed feedback laser," Appl. Phys. Lett. £Q_, 139 (1971). (11) J.E. Bjorkholm and C. V. Shank, "Higher-order distributed feedback oscillators," Appl. Phys. Lett. £Q_, 306 (1972). -108(12) P. Zory, "Laser oscillation in leaky corrugated optical waveguides," Appl. Phys. Lett. 22, 125 (1973) . (13) D. P. Schinke , R. G. Smith, E. G. Spencer, and M. F. Galvin, "Thin film distributed feedback laser fabricated by ion milling," App 1 . Phys. Lett. ~' 494 ( 1972). (14) M. Nakamura , A. Yariv , H. W. Yen, S. Somekh, and H. L. Garvin , "Optically pumped GaAs surface laser with corrugation feedback," Appl. Phys. Lett. 22 , 515 (1973). (15) M. Nakamura, H. W. Yen, A. Yariv, E. Garmire, S. Somekh, and H. L. Garvin, "Laser oscillation in epitaxial GaAs waveguides with corrugation feedback," Appl . Phys. Lett.~, 224 (1973) . (16) H. W. Yen, M. Nakamura , E. Garmire, S. Somekh, A. Yariv, and H. L. Garvin, "Optically pumped GaAs waveguide lasers with a fundamental 0.11µ corrugation feedback," Optics Commun. 9, 35 (1973). (17) C. V. Shank, R. V. Schmidt, and B. I. Miller, "Double-heterostructure GaAs distributed-feedback laser," Appl . Phys. Lett. 25, 200 ( 197 4). (18) M. Nakamura and A. Yariv, "Analysis of the threshold of double hetero-junction GaAs - GaAlAs lasers with a corrugated interface," Optics Commun . l!.., 18 (1974) , (19) R. D. Bu r nham , D. R. Scifres , and W. Streifer, "Single-heterostructure distributed feedback GaAs diode lasers," IEEE J . Quantum Electron . l!.., 439 (1975) . -109(20) M. Nakamura , K. Aiki, J . Umeda, A. Katzir, A. Yariv, and H. w. Yen , 11 GaAs-GaA l As double - heterostructure injection lasers with distributed feedback, 11 IEEE J . Quantum Electron. l!_, 436 (1975). (21) H. M. Stoll and D. H. Seib, 11 Distributed feedback GaAs homojunction injection laser, 11 Appl. Opt . ..l1_, 1981 (1974). (22) K. Aiki, M.' Nakamura , J. Umeda, A. Yariv, A. Katzir, and H. W. Yen , 11 GaAs - GaAlAs distributed feedback diode lasers with separate optical and carrier confinement," Appl. Phys. Lett.'?]_, 145 (1975) . (23) M. Nakamura, K. Aiki, J . Umeda, and A. Yariv, "CW operation of distributed feedback GaAs-GaAlAs diode lasers at temperatures up to 300°K, 11 Appl. Phys. Lett.?]_, 403 (1975). (24) K. 0. Hill and A. Watanabe , "Envelope gain saturation in distributed feedbac k lasers ,S' Appl . Opt. .Ji, 950 ( 1975). (25) S. R. Chinn and P. L. Kelley, 11 Analysis of the transmission, re- flection and noise properties of distributed feedback laser amplifiers , " Optics Commun . .!Q_, 123 (1974) . (26) J . Zo r oofchi and J . K. Butler , "Refractive index of n-type gallium arsenide , " J . Appl. Phys. 44, 3697 (1973). (27) D. Marcuse , "Radiation losses of tapered dielectric slab waveguide," Bell Syst. Tech. J. 49, 273 (1970). (28) P. Zory, "Corrugated grating coupled devices and coupling coefficients , " Integrated Optics Conference--Technical Digest, Salt Lake City , Utah , January 1976 . -110 - Chapter 4 Distributed Bragg Reflector Lasers 4-1 Introduction As described in Chapter 3 the GaAs-GaAlAs distributed feedback laser has attracted a great deal of interest in the field of integrated optics. This is due, in part, to the frequency selective nature of the feedback which makes possible longitudinal and transverse mode control. Another important advantage is the compatibility of the fabrication process with planar technology. One disadvantage revealed in the course of experiments on such lasers(l) is the degradation of the recombination efficiency in the active region by the steps used to obtain the corrugation. Another un- resolved problem is the effect of the corrugation on the laser operating lifetime. These problems can be alleviated somewhat by schemes such as separate optical and carrier confinement in which the recombining carriers are kept away from the corrugati on as described in the previous chapter . An alternative to the distributed feedback laser is one where the Bragg coupling between the laser forward and backward waves is achieved in two corrugated waveguide sections which are contiguous with the active (amplifying) region . It has been suggested theoretically( 2 , 3) and . (4-6) that sue h sections can ac t as h'19 h redemonstrated experimentally flectance "mirrors" for waveguide modes at frequencies within the "stop band" where the Bragg condition is nearly satisfied. Such Bragg reflector lasers are expected to possess the main advantages of the -111- distributed feedback lasers, i.e . , frequency and mode selectivity while avoiding the problem of degradation of the recombination efficiency . The theoretical analyses of distributed Braqg reflector (DBR) lasers to date( 2 , 3) employ a generalization of the analysis of the distributed feedback laser(?) and are consequently complicated and not easily amenable to intuitive physical interpretation. In addi- tion, a variation of the basic configuration from that of the original model requires that the complicated boundary value problems be solved anew. In this chapter we present a formally equivalent solution of the distributed Bragg reflector laser. In this approach we replace, in the analysis, the Bragg reflectors; i . e., the physically corrugated sections of the waveguide adjacent to the amplifying region, by fictitious reflectors with complex reflectances r 1 , 2(w)exp[-i~ 1 , 2 (w)J. The complex reflectance of each reflector (denoted by subscript l or 2) is obtained from the solutions of Bragg reflectors and is assumed known . The derivation of the laser oscillation condition now becomes identical to that of a conventional two-reflector laser. It is ob- tained by requiring that the round-trip complex gain of the laser field amplitude including two reflections be equal to unity. The oscillation condition can then be solved numerically or graphically to determine the lasing frequency and threshold. It is shown that distributed Bragg reflector lasers have similar frequency and modal control as the distributed feedback lasers. The effect of - 112- reflector losses, which include bulk absorption loss and radiation loss, on the lasinq characteristics is also treated. The losses in the periodic section are found to have little effect on the lasing frequency but can cause the threshold to go up considerably. Experimental results on optically pumped GaAs waveguide distributed Bragg reflector lasers were given at the end of the chapter. termined and compared with theory. Lasing characteristics were de- Also some potential applications of this kind of laser in fiber communication systems are discussed. 4-2 Distributed Bragg Reflector Laser with Lossless Reflectors We have shown in Chapter 2 that a section of lossless periodic waveguide behaves as a band rejection filter or reflector. In general, the reflectivity of such a device can be expressed as a complex quantity R(w) ( 4- l ) = The spectral behavior of r 2 (w) in Fig. 2 - 8(0.). When close to unity . When .6.S < K , .6.S > K, for a lossless waveguide has been shown 2 r ( w) is large , and for KL > 3. 0 is r 2(w) has several side lobes with pro- gressively decreasing reflectivities . Figure 4- l(o.) shows a schematic drawing of a distributed Bragg reflector waveguide laser . It consists of an amplifying region of length L flanked on each side by corrugated sections of length L1 and L3• 2 We use the equivalent reflectors method described in the last two chapters to obtain the equivalent laser shown in Fig. 4-1 (b). The corrugation s are replaced , at their respective input planes , by two conventional mirrors with properly assigned complex reflectivities. The laser -113- ~ L 1--,► .-.1~4-- L 2 --------~1----- L 3 ~ Gain: g ~Ai.- (a) Substrate I◄ .. Net Gain: g l ( b) Fig. 4-1 I L2 (a) Schematic diagram of a distributed Bragg reflector laser. (b) Equivalent cavtty of a distributed Bragg reflector laser, -114cavity is of length L2 and consists of a medium with net gain g at the wavelength region of interest. The oscillation condition is then obtained by setting the round trip complex gain equal to unity , i . e. ' (4-2) = After using equations(2-33) and (4-1) the last equation becomes (4-3) where y 2 = K2- 68 2 , 8 is the propagation constant of the laser mode and 68 ~ 8 - ~ . Equation (4-3) is the eigenvalue equation of a distributed Bragg reflector laser a~d it can be solved numerically or graphically to obtain an infinite set of eigenvalues (68n,gn). such set gives the oscillation gain gn 11 frequency 11 M3 n Each and the threshold of the nth longitudinal mode of the laser. The extension of equation (4-3) to the case of two different corrugated sections (i .e. , Kl f K2 , 68 1 f 68 2 , etc.) is obvious . Equation (4-2) can be written as r 1 (w) r 2 (w) e 2gl2 e -i(¢1+ ¢2 + 268Lz + 28ol2) = 1 Hence the oscillation frequency (i.e. , 68 at oscillation) and threshold gain are obtained by equating the phase and amplitude on both sides of the last equation (4-4a) ,-115(4-4b) where n = 0 ,± l ,±2, • .. . In a given distributed Bragg reflector structure we can use equation (2-32) to find r 1 , r 2 , ¢1 and ¢2 as functions of 66 . The round trip phase delay can thus be plotted as a function of 6S . The intersections of this curve with the horizontal lines ¢ = 2nTT determine the set of oscil- lation frequencies 6Sn . The corresponding the threshold gain gn . This procedure is illustrated in Fig. 4-2 r 1(6Sn) and r 2(6Sn) are then used in equation (4-4a) to find the corresponding values of L1= L3= L, L2= 2L, KL= 3,0 and s0 L = 2NTT. 2 Under these conditions equations (4-4a) and (4-4b) reduce to for the special case l g = 2[ in r1 (4-4c) (4-4d) ¢ = ¢ + 26SL and ¢ = nTT are plotted in Fig. 4-2(~) The intersec- tions are at 6SL = ±0,67, ±2.00, ±3,22, ±4.14, etc. The correspond- ing reflectivities are obtained from Fig. 4-2(b) and used in equation (4-4c) to calculate the threshold gains and 1 .2629, etc. 2gl = 0,0050, 0.0127, 0.0830, It is seen that although the effective reflectivity of the modes at 6SL = ±2,00 (0 .974) is only slightly smaller than that of the modes at 6SL= ±0.67 (0.989) the threshold for the modes at 6SL= ±2.00 is more than twice as big as that of the modes at -116- KL=3.0 (a) -5 -4 -3 -2 5 6/3L -21r -3rr KL=3.0 (b) 1.0 0.5 -6 -5 -4 -3 -2 -I Fig. 4-2 0 2 3 4 5 6 Graphical method of determining oscillation condition of a distributed Bragg reflector laser. 6/3L -117- 68 = ±0,67. This clearly illustrates the longitudinal mode selecti- vity of distributed Bragg reflector lasers. 4-3 Determination of Mode Structure To obtain the actual oscillation wavelength from the solutions of 6Sn we recall that M3 n = B - Bo therefore (4-5) So each longitudinal mode is represented by a straight line with slope 2i- in the neff- A plane. We can also plot the waveguide mode disn persion relation = on the same diagram. The intersections of this curve with the lines represented by equation (4-5) give the oscillation wavelengths of different longitudinal modes. This procedure is very similar to that described in section 3-4 for distributed feedback lasers. The longi- tudinal mode spacing of distributed Bragg reflector lasers, as expected , in the limit of L2 » L1 ,L 3 is inversely proportional to L2 as in the Fabry-Perot lasers. Bragg reflectors can determine the longitudinal wave vector of the laser but do not affect directly the propagation vectors in the other two dimensions . This could result in oscillation of more than -118- one transverse mode. Control of modes whose transverse K vectors lie in the plane of the junction can be achieved by controlling the width of the pumped area . The quenching of transverse modes (k vectors normal to the junction plane) can be obtained by controlling the height of the laser waveguide. For small enough heights the effect of mode dispersion causes the wavelength at which the Bragg condition is satisfied by a higher order transverse mode to be appreciably shorter than that of the fundamental transverse mode. It can thus be shifted far enough from the peak of the gain profile so as to exercise a substantially lower gain. To obtain the oscillation frequencies and threshold gains of a given transverse mode we repeat the procedure described above but use the appropriate dispersion neff(\) curve of the mode in equa- tion (4-5) and recall that, in general, the coupling constant will be different for different modes. We have described above how a judicious choice of the thickness of the active layer and the corrugation period could be used to obtain a single transverse mode operation. By carefully choosing the corru - gation period we can cause the oscillation wavelength of the lowest order longitudinal mode to occur near the peak of the gain profile. Since this mode was found to possess the lowest threshold gain gn , it is clear that the external pumping threshold for this mode will be lowest and one can obtain single mode operation . This kind of modal control is also similar to that of distributed feedback lasers. 4-4 The Effect of Lossy Reflectors Up to this po i nt we have considered lossless reflectors only. .-119- In many real applications the reflector sections are made of the same material as the active region but are unpumped. As a result the reflectors are lossy. In this section we are going to examine the property of a section of lossy periodic waveguide and its effect on the lasing characteristics of a laser with Bragg reflectors made out of this kind of lossy waveguide . We start with equations (3-1) dA dz = . Be -i2(6°+ia)z µ lK dB = -lK . Ae i2(6B+ia)z dz These are the coupled-mode equations describing propagation in an amplifying periodic waveguide with amplitude gain a. by -a so that a a Now if we replace represents the amplitude loss coefficient we ob- tain dA = i KB e-i2(6B-ia)z dz (4-6a) dB (4-6b) dz i a)z = - 1. K A e i2( 6°µ The solutions of equation (4-6) can be obtained from equations (3-2) and (3-3) if we replace a by -a. They are Ei ( z) = B(z) e- iBz - az = E.(O)[(a+i6B)sinhy(L-z) +ycoshy(L-z)]e 1 (a+i 6B ) sinhyl + ycoshyl -i B0 z (4- 7) and E ( z) r = A(z) e ; Bz+az iB z -i Ksinhy(L-z)e 0 = Ei(O) (a+i lB )sinhYL + YcoshYL (4-8) -120- For 6S = 0 , we have IE (z)l2 = lasinhy(L-z) + ycoshy(L-z) 12 i asinhy[ + ycoshyl (4-9) and 2 I Ksinhy{L-z) 1 IEr(z) I = asinhy[ + ycoshy[ 2 ( 4-10) Equation (4-9) and (4-10) are plotted in Fig, 4-3 for KL= 8.0, al= 0 and al= 2 . We see that the power distribution along the i,,rnveguide carried by the incident wave does not vary much for K >> a from that in the lossless (a=0) case. The reflected wave, on the other hand, is seen to undergo attenuation. The intensity re- flectivity in this case is reduced from~, .0 at a=0 to roughly 2 . For KL= 8.0, al= 2.0 the reflectivity is down to ~0.64. .......!:S....1 a+K l The effective distance penetrated by the wave into the waveguide taken as the l/e 2 point of the forward power is ~L/8 for our lossy waveguide. ation. This is due to the strong coupling rather than the attenu2 2 (Note that } =~) . . In Fig . 4-4 we plot 1Ei(z)l , 1Er(z)l for KL= 2.0 with al= 0 and al= 2.0 . Since now KL and al are comparable we would expect the effective penetration depth to be L ~ l eff ✓ 2 = 0,354 L 2 K + a This is easily verified from the figure . Also, the maximum intensity reflectivity is reduced to around K - ....- --_----'1 / 2+ a2 a +\K 2 = 0. 172 -121- 1.0 KL= 8.0 al=0 1Er(z)l 2 1Ei(z)l 2 OL__ _L__ _:::]~==--~--L.--J.._--L-_ z=0 L/8 L/4 3L/8 L/2 __,J,_ _~ 5L/8 3L/4 7L/8 z=L 1.0 KL= 8.0 al= 2.0 1Ei(z)l 2 1Er(z)l 2 0L.__--1..,,:~L..a.ij~-.....-___,JL...---L.._ z=0 L/8 Fig. 4~3 L/4 3L/8 L/2 __J_ _....,L__ __J~ 5L/8 3L/4 7L/8 z=L 2 2 Plots of I E; ( z)I and !Er(z)I for KL= 8.0 and ( a ) ,JL = 0 ( b ) etl = 2.0. -122- 1.0 KL= 2.0 aL=0 0.5 z= L 1.0 KL= 2.0 al= 2.0 0.5 9=o L/8 Fig. 4-4 L/4 3L/8 L/2 5L/8 3L/4 7L/8 Plots of IE 1(z)l 2 and 1Er(z)l 2 for KL= 2. 0 and (a) ,d_ :c: 0 ( b) ,'il = 2 . 0. z=L -123The third case a>> K is plotted in Fig . 4-5 with KL= 0. 5 and al= 0 and 2.0. The maximum reflectivity is now ~ li:I 2 = 0.016 The effective penetration length is Leff ~ -a1 = 0.5L For the general case of 6S r O, we can write the reflectivity as -iKsinhyL -i¢ (a+i6S)sinhyl + ycoshy[ = re A plot of ( 4- 11 ) r2 for KL= 3.0, al= 1.0 is shown in Fig. 4-6(0..) We also plot the curve with al= 0 for comparison . The magnitude of the reflectivity in the presence of waveguide loss is substantially lower than that of the loss-free case. "oscillation" is smoothed out. In addition, the sidelobe In this case the reflectivity is still a strong function of the incident wave frequency. only changes the magnitude of the reflectivity r the phase ¢ The loss not but also changes This is shown in Fig . 4-6Cb). For al = l the behavio r of ¢ is such that it almost follows the curve for al= 0 when 6SL is sma 11 . As 6SL increases, ~ gradually falls off the 'l'al=l al= 0 curve and stays nearly constant. Now consider the case where two lossy sections serve as the reflectors of a laser. The oscillation condition (4-3) is modified to -124- 1.0 KL= 0.5 al=O 0.5 z=O L/8 L/4 3L/8 L/2 . 5L/8 3L/4 7L/8 z=L 1.0 1<. L= o.5 oe.L= 2.0 0.5 IE/z)l 2 - z =O L/8 L/4 3L/8 L/2 5L/8 3L/4 7L/8 z = L Fi g. 4-5 Plots of ! E; (z) I ( a) ,xL = 0 2 and I Er(z) I ( b ) r_y_L = 2 . 0 . 2 for KL = 0.5 and -125- r2 KL= 3.0 al= 0 al= I 1.0 --', --I -- / - 0.8 I I I 0.6 ' 'I -6 -4 -2 \ \ I I I -10 -8 \ I I 0.2 " 0 2 ct> KL= 3.0 al=0 - - 6 a L= I 4 -8 -6 -4 Cb) / / 6 4 / / / I I 7 L. I I / / / / 8 ' 10 6{3L 8 6{3L / 2 0 2 4 6 -2 -4 -6 Fi ~. 4-6 Plots of (a) the intensity reflectivity (b) the phase of the reflectivity of a Bragg reflector with and without loss. -1262 2(g-iS)L 2 = [(a+i68) + ycothyL J[(a+i68) + ycothyL ] -Ke 1 3 where y2 = (a+i6S) 2 + K2 . (4-12) Let us consider, as an example, the special case L1 = L3= L, L2 = 2L, KL= 3. 0, S0 L2 = 2NTT considered previously, but take al= l .0. The phase shift diagram is shown in Fig. 4-7(~). The lasing frequencies are at 68L = ±0.67, ±2.00, ±3.36, ±4,85, etc. We note that as compared to the lossless case described in section 4-2 the lasing frequencies of the lower order modes hardly change . Because of the substantial reduction of the reflectivities, the thresholds however, are much higher. They are 2gl = 0.3367, 0.4077, 0.6813, 1.0814, etc. In a GaAs distributed Bragg reflector laser the unpumped corrugated sections will absorb laser radiation. Also, if we use higher order corrugations to provide the feedback ;\ (A=~ 2n , ~=2,3,···) eff there will exist radiation loss due to the coupling between the guided modes and the radiation modes through the lower harmonics of the corrugation. Theoretically, the use of fundamental mode (A= n:\ ) should make it impossible to couple to radiation modes. 2 eff Deviations from perfect periodicity, however, will always introduce some scattering loss. We can account for these losses in the analysis by a distributed loss coefficient arad so the loss factor a that appears in equation (4-11) consists of two parts "" - ""bulk + a rad IV - IV A typical value of a as estimated in our experiments is'\, 15 cm -1 . • -127- cp KL=3.0 al= 1.0 (a) I -1T 2 3 4 5 2 3 4 5 r2 KL =3.0 al= 1.0. 1.0 (b) -6 -5 -4 -3 -2 -I Fig. 4-7 0 6 ~/3L Graphical method of determining oscillation condition of a distributed Bragg reflector laser with lossy reflectors (al = l. 0). .-1284-5 Comparison of Distributed Feedback and Distributed Bragg Reflector Lasers The distributed Bragg reflector lasers and the distributed feedback lasers are very similar in several respects. They both utilize corrugations to provide the feedback and thus have the same degree of modal control as discussed earlier. The main difference between them is that in distributed feedback lasers the gain and the feedback region are not separated while in the distributed Bragg reflector lasers the gain and the feedback regions can be separated. One important consequence is that in distributed Bragg reflector lasers it is possible to have oscillation at or very near the center of the ' stop band where the reflectivity is maximum leading to very 1 11 low threshold. While in distributed feedback lasers no oscillation can occur for l6SI < K . is lower. For 6S outside the stop band the reflectivity Therefore distributed feedback lasers possess higher thresholds. The threshold gain of a distributed Bragg reflector laser, from equation (4-4a), is and the loss coefficient that is used to obtain r 1 and r 2 is It is clear that the effect of radiation loss in this kind of laser can be reduced by making L2 large, i.e . , to use a longer active region. This is to be, compared with the threshold of a distributed feedback laser [equation (3-21)] -129- DFB a.th = a~h (K,L) + ao + a.rad • t o increase • DFB by an amoun t equa 1 to h th e e ff ec t of a.rad 1s were a.th a.rad to compensate for the radiation loss . Another major advantage of distributed Bragg reflector lasers over distributed feedback lasers is that the corrugations are separated from the active region so that there will be no degradation of the carrier recombination efficiency as observed in distributed feedback lasers. And as described in section 3-9 the fabrication of GaAs-GaAlAs distributed feedback laser diodes requires that the liquid phase epitaxy process be interrupted in order to make the corrugation. And then the second growth process is carried out on the corrugated surface, which is a difficult procedure. In distributed Bragg reflector lasers, on the other hand, it is possible to perform liquid phase epitaxial growth in one step as in the fabrication of ordinary injection lasers. A disadvantage of distributed Bragg mirror lasers is that in order to have the corrugation close to the optical wave we need to etch the wafer so that a mesa structure is obtained as shown in Fig. l-3(b). The corrugations are then fabricated on both sides of the mesa. It is more difficult to obtain a high quality corrugation on this kind of 11 nonsmooth 11 surface especially at the region immediately next to the mesa. -130.- 4-6 Experiments with Optically Pumped GaAs Distributed Bragg Reflector Lasers Fig. 4-8 shows the cross section of a typical optically pumped GaAs waveguide distributed Bragg reflector laser used in our experiment . Two layers of GaAlAs and GaAs were grown on an n-GaAs substrate by liquid phase epitaxy to form a waveguide structure . Photoresist grating mask was then laid over the whole surface of the wafer by holographic photolithography. Part of the grating was covered by a stripe mask so that subsequent chemical etching through the photoresist grating resulted in a gap of length L2 between the two corrugated end sections. A= 3\ /2neff were used. Third order gratings The wafers were then saw-cut at one end for access to the laser output. Typical dimensions are L1 ~ 250 µm, L2 ~ 500 µm, L3 ~ 3 mm and L4 ~ 3 mm. The relatively large lengths of sections L3 and L4 which are lossy at the laser wavelength is to ensure that there is only negligible reflection from the far face of the sample . Samples were cooled to liquid nitrogen temperature (77°K) and pumped optically by a repetitively pulsed dye laser tuned to ~6300~ . The pumping beam was focused by a cylindrical lens onto the sample . During the pumping the corrugations could be totally or partially covered in order to study the effect of lossy reflectors . It was found that with the corrugations pumped the laser threshold decreased by a factor of 2- 3 while the spectrum did not change significantly. Both observations are in agreement with the theory. ~L1 E •+• L2-+ L3 GaAs Substrate +-L4--j I 1 ...... w ...... 11~3µm :.~3µ.m t+ Arv0.355 fl-m GaAs ~ Gao.7AIQ3~ ~ Fig , 4-8 Cross section of an optically pumped GaAs distributed Bragg reflector laser. OUTPUT <=i 0 DU Pumping Beam ( A"'6300A) 0 ~132Fig. 4~9 is a spectrum from one of the lasers which shows single mode operation up to two times the threshold pumping intensity. this case the corrugations were not pumped. In In waveguides of larger height the mode selectivity, dtscussed in section 4-3, decreases and it is possible to have several transverse modes oscillating simultaneously. An example of 'such a spectrum is shown in Fig. 4~10. Three peaks are shown which belong to different transverse modes. They are separated by 12~ and 24A, and have TE polarization (electric field parallel to the junction plane). In some lasers, with a somewhat 0 thinner top layer, transverse mode spacings as large as 40A were observed. Because of the coexistence of several transverse modes in our laser there is a possibility of intermode coupling leading to hybrid laser modes(B, 9 ). For example, the right~going m=O mode can couple to the left-going m=l mode and vice versa. This kind of hybrid mode will oscillate at a wavelength roughly halfway between the m=O and m=l mode wavelength. Fig. 4-ll shows one example of this phenomena where we have m=O, m=l, m=2 modes and the hybrid mode between m=O and m=2 designated as (0,2). The distributed Bragg reflector structure also offers a possible solution to the difficult problem of coupling a waveguide laser output into fibers. It is suggested that one can fabricate a Bragg reflector laser with fundamental grating on one side of the active region and a second order grating on the opposite side. It is well known that a second order grating couples light in first order in a direction perpendicular to the grating plane(lO,ll ,l 2). We -133- BM-3 ~~ ~ ~ ~ +- (/) C Q.) +- C 1---4 Q.) > +- -0 8493 8495 8497 Q.) 0:: 8415 8435 8455 8475 8495 8515 8535 0 Wavelength (A) Fig. 4-9 Single mode oscillation spectrum of an optically pumped GaAs distributed Bragg reflector laser. -134I I I I l I I BM-9 - >. ~~ - +- (/) C - - > - - Q) +- C ~ Q) +- 0 • Q) 0::: - I ..... - J \. " " I I I 8370 8380 8390 I I 8400 8410 I I - 8420 8430 0 Wavelength (A) Fig. 4-10 Multi-transverse mode oscillation spectrum of a GaAs waveguide distributed Bragg reflector laser. -135- BM-3 ~~ m=O m= I >"'I +- (/) C Q) +- C t--t Q) > +- 0 m = (0, 2) Q) 0::: m=2 8440 8450 8460 84 70 8480 8490 8500 0 Wavelength (A) Fig. 4-11 Spectrum shows hybrid mode oscillation in a multi-transverse mode distributed Bragg reflector laser. -136- can fix one end of a fiber to the grating and thus couple the laser light into the fiber as illustrated in Fig. 4-12. The coupling eff iciency can be made large by adjusting the corrugation depth so as to obtain an effective radiating aperture with cross sectional dimensions which approximate those of the fiber. A modulator section can be inserted between the grating and the active region to modulate the output intensity . Some preliminary experimental results on this scheme were obtained . This was done by fabricating lasers with one cleaved end face and one second order Bragg reflector, optically pumped at both 77°K and 300°K. One end of the fiber was held against the corrugation next to the pumped region , the other end was held against the entrance slit of a spectrometer. Although the measured output power f rom the fiber was low , it demonstrated the feasibility of this scheme. More work on the optimal design of this type of structure is needed to improve the coupling efficiency . , Substrate GaAIAs GaAs ~12Al4- D D D ~DD~ / Fig . 4-12 Sketch of t he proposed scheme of coupling laser light into optical fiber. ~nnnn ~~ Arv0.12µm Pumping r//J Core / Opt ica I Fiber I " w I _. Cladding -138- CHAPTER 4 REFERENCES (1) M. Nakamura, K. Aiki, J. Umeda, A. Katzir, A. Yariv, and H. w. Yen, "GaAs-GaAlAs double-heterostructure injection lasers with distributed feedback," IEEE J. Quantum Electron. l!.., 436 (1975). (2) R. Shubert, 11 Theory of opticahwaveguide distributed lasers with nonuniform gain and coupling/' J. Appl. Phys. 45, 209 (1974). (3) S. Wang, "Principles of distributed feedback and distributed Bragg-reflector lasers, 11 IEEE J. Quantum Electron. l.Q_, 413 (1974). (4) F. K. Reinhart, R. A. Logan, and C. V. Shank, "GaAs-Al Ga X 1-X As injection lasers with distributed Bragg reflectors , 11 Appl. Phys. Lett. 27, 45 (1975). (5) H. W. Yen, A. Yariv, and W. Ng, "Optically pumped GaAs waveguide lasers with Bragg reflectors, 11 Integrated Optics ConferenceTechni cal digest, Salt Lake City, Utah, Jan. 1976. (6) W. T. Tsang and S. Wang, "GaAs-Ga _xA\As double heterostructure 1 injection lasers with distributed Bragg reflectors, 11 Integrated Optics Conference - Technical digest (post~deadline paper) Salt Lake City, Utah , Jan. 1976. (7) H. Kogelnik and C. V. Shank, "Coupled-:-wave theory of distributed feedback 1asers, u J. Appl . Phys. 43, 2327 ( 1972). (8) P. Zory, "Laser oscillation in leaky corrugated optical wave guides, 11 Appl. Phys . Lett. g, 125 (1973). (9) H. Kogelnik, C. V. Shank , and J . E. Bjorkholm, 11 Hybrid scat- tering in periodic waveguides," Appl. Phys. Lett. g, 135 (1973). -139 .. (10) Zh. I. Alferov, S. A. Gurevich, R. F. Kazarinov, M. N. Mizerov, E. L. Portnoi , R. P. Sei syzn, and R. A. Suri s, "Semi conductor laser with extremely low divergence of radiation, 11 Sov. Phys.Semicond. i, 541 (1974). (11) R. D. Burnham, D. R. Scifres, and W. Streifer, "Low divergence beams from grating-coupled composite guide heterostructure GaAlAs di ode lasers, 11 Appl. Phys. Lett. 26, 644 ( 1975). (12) P. Zory, and L. D. Comerford, "Grating-coupled double-heterostructure Al GaAs di ode lasers, 11 IEEE J. Quantum Electron. ll, 451 ( 1975). CHAPTER 5 EXPERIMENTAL TECHNIQUES 5-1 Introduction Due to the short wavelength and the small size of optical fibers, the dimensions of the optical circuit components which need to interface with them are typically on the order of a few microns. The requirements on the precision and definition of devices are thus very strict. For example~ in fabricating a dielectric waveguide, the smoothness of the sidewalls and the uniformity of the guiding layer thickness will determine whether the guide is lossy or not. This is due to the fact that the nonuniformity of the waveguide structure causes mode scattering, radiation losses, and other undesirable consequences(l). Generally speaking - the tolerance on the waveguide sidewall smoothness should be much smaller than the guided wavelength. Another example is the fabrication of optical gratings for filters( 2 , 3), polarizers( 4), Bragg reflectors etc . where the period can be as small as 0.1 µm and is required to be uniform across a considerable length. Conventional photolithography is not suitable for this purpose. Hence new technique had to be developed to meet these requirements. Another requirement regarding the fabrication of optical circuits is that the fabrication processes be canpatible with planar technology. This will be important in integrating several optical components on one common substrate. In this chapter we describe some of the important techniques used and developed during the course of studying GaAs distributed feedback and distributed Bragg reflector lasers. These include -141- liquid phase epitaxial crystal growth, holographic photolithography, chemical and ion beam etching of GaAs crystals, and some optical measurements. 5-2 Liguid Phase Epitaxy in GaAs~GaAlAs System Liquid phase epitaxy (LPE) by definition, is the precipitation from a liquid phase of a crystalline layer onto a parent substrate where the crystallographic orientation of the layer is determined by that of the parent substrate. It is a very common method of growing semiconductor thin films for optical·, acoustical, and microwave devices. The basic principle is described as follows. The solution and the substrate are kept apart in the growth apparatus (a graphite boat) prior to the growth. The solution is saturated with the growth material and heated up to a prescribed temperature. Then the solution is brought into contact with the substrate surface and allowed to cool down at a constant rate for an appropriate length ·of time depending on the thickness of the layer desired. If the substrate is single crystalline and the lattice constant of the precipitating material is the same or nearly the same as that of the substrate, the precipitating material forms a layer on the substrate surface which is an extension of the single crystal body of the substrate. For example in the GaAs-GaAlAs system a GaAs single crystal wafer is used as substrate and layers of GaAs or Ga 1_xA1xAs are grown on it. Because of the close match of the lattice constants between AlAs and GaAs, it is possible to grow layers of Ga 1-X Al XAs on GaAs -142and vice versa with relatively low defect density. A typical liquid phase epitaxial growth set-up is shown in Fig . 5- l(a), which consists of three major parts ; a furnace, a quartz tube, and a graphite boat . The graphite boat consists of a housing unit and a slide. There are recessed areas in the slide for holding the substrates . In the housing unit there are several wells for the different melts . As an example, to fabricate a GaAs-GaAlAs double heterostructure injection laser as shown in Fig. 5-l(b) we have to grow four layers successively on an n-GaAs substrate. The wells in the graphite boat are filled with melt materials in the following sequence. Well no. l contains Ga, GaAs seeds, Al, and Sn. Well no. 2 contains Ga, GaAs, and Ge. Al and Ge. Well no. 3 contains Ga, GaAs, Well no. 4 contains essentially the same materials as well no . 2 except for a larger quantity of Ge. The amount of each element in the wells depends on the composition and the carrier concentration of the layer grown from it. Before the growth, the boat is assembled, the substrate and the melt materials are loaded. The boat is then inserted inside the quartz tube roughly at the center of the furnace. A quartz "pushing rod" is connected to the slide so that the latter can be moved from the outside. After purging the tube thoroughly with hydrogen the furnace is turned on and heated up to around 825°C. A constant flow of hydrogen is maintained through the tube at all times . As the temperature stabilizes and the melts are in thermal equilibrium states a biasing circuit is turned on to slowly lower the temperature of the boat cb > Ca) I V I 11 ? s;; < }7 2' }( ; 6 b by LPE. A double heterostructure GaAs injection laser structure grown (b) j - Pushing rod Thermocouple ~slide ! '""-' __ A typical liquid phase epitaxial crystal growth system using horizontal three-zone furnace and sliding graphite boat. n- GaAs : Si - Al ~ I !=~:~lGe I 1 c >< A Fig. 5-l(a) Substrate H~ Grap boa r" f H2 I ....J I w ~ -144- (which is monitored by a thermocouple placed underneath the graphite boat). The cooling rate ranges from a few tenth of a degree per minute to a few degrees per minute . prescribed value, As the temperature drops to a for instance 820°C, (this is called the contact temperature) the substrate is pulled into position under the first melt. It is kept there for a period of time appropriate for the thickness of the first layer to be grown and then is moved to the next solution, and so on until the multilayer structure is grown. The substrate is then moved away from the last melt and the boat is cooled down to room temperature. One of the main difficulties involved in crystal growth is keeping oxygen out of the growth system when the heating stage begins . At high temperatures the surface of the melts and the substrate can be easily oxidized which would prevent the growth. Also the preparation of the substrate and the melt materials has to be done carefully . Any undesirable impurity will degrade the growth. Successful crystal growth is probably the main key to the fabrication of long life cw GaAs injection lasers and other electronic devices based on GaAs epitaxy . To fabricate a double heterostructure distributed feedback injection laser as shown in Fig. 3-23 we have to corrugate the surface of the p~GaAs active layer before growing the p-GaAlAs layer and the p-GaAs top layer . This involves the challenging problem of growing an epitaxial layer over a corrugated surface without destroying the corrugations. It has been pointed out that during epitaxial growth there is always some meltback of the substrate . -145This presents a serious problem if growth over shallow corrugations is desired. We thus carried out a series of experiments by growing a layer of GaAlAs on corrugated GaAs substrates with different growth temperatures and cooling rates to determine the dependence of the amount of meltback on the growth conditions. After growth, part of the GaAlAs layer was etched away by HF and the GaAs substrate surface was examined by using scanning electron microscope (SEM) and by laser beam diffraction experiment to see whether the corrugation survived the growth. Table 5-1. The results are summarized in It was found that the epitaxial GaAlAs layer grown above 700°C on the corrugated substrate had a mirrorlike finish. A great deal of improvement in reducing the amount of meltback was achieved by lowering the contact temperature from 820°C to 700°C. A small improvement was also obtained by increasing the cooling rate from l to 5°C/min . The depth of the corrugation can be directly measured from the SEM pictures . It can also be estimated by measuring the relative intensity of the various orders of diffracted light from the corrugation after removing the epitaxial layer . In our work a He-Ne (6328A) laser was used for this purpose . The diffraction efficiency, i.e., the ratio of the first order diffraction intensity to that of the input intensity in a number of samples grown under varying conditions is listed in Table 5-1 . Fig . 5-2 is an SEM picture of sample A-5 after partial removal of the GaAlAs layer . The corrugation survived almost intact with only minimal meltback . The irregular edge and the small holes were caused by the penetration of HF through imperfect photoresist mask. Fig . 5-3 shows the 5 5 820 - 810 750 - 720 700 - 670 A-2 A-4 A-5 Table 5-1 Before Growth 1 820 - 810 A-1 ', 580 500 400 100 < 100 Final Corrugation Height Peak to Peak (.fl.) Phase Epitaxy on Corrugated Substrate. Summary of the Experimental Results of Liquid 5 Cooling Rate (°C/min) Growth Temperature Range (°C) Sample Number 1 x ,0- 1 8 X 10- 2 4 X 10- 2 1 X 10- 3 5 X 10 - 4 Diffraction Efficiency I O"I I ~ ....... -147- Fig. 5-2 An SEM picture of sample A~5 with the top GaAlAs layer partially etched away. -148- p-GaAs p-GaAlAs p-GaAs ( active region) n-GaAlAs n-GaAs substrate Fig. 5~3 Cross section of a double heterostructure QaAs~qaAlAs distributed feedback laser grown by LPE, -149cross section of a completed double heterostructure distributed feedback injection laser which shows excellent growth interfaces. 5-3 Grating Fabrication by Holographic Photolithography Small period (<l µm) gratings have been widely used in integrated optics as couplers, filters, and reflectors. For these applications the grating period ranges from ~0.1 µm to~, µm. It is difficult for ordinary _photolithography to fabricate this kind of pattern because of the tight spacing. Hence a maskless exposure process--holographic photolit~6graphy( 5) was developed for this purpose and is described in what follows. Two plane waves of the same frequency and of amplitudes E1 and E2 are incident upon a smooth surface at an angle 28 as shown in Fig. 5-4. Taking the complex amplitudes of the waves across the surface as E _ Ae-ik(xsine-ycose) l - E 2 = RAe""ik(-xsine-ycose)-i¢ respectively we find that the intensity on the surface is IE 1+E 2 1 2 = IAl 2{(1-R) 2 + 4R cos 2 (kxsine-¢)} y=O = IAl 2 {l + R2 + 2R cos2rkxsin8-¢]} The intensity is thus modulated in the x direction with a period 2TI A A= 2ksine = =2h-s-i~n~e (5-1) -150where n is the index of refraction of the medium in which the waves meet over the surface and A is the vacuum wavelength of the waves. Equation (5 - l) shows that as long as¢, the phase difference between E1 and E2 is constant (not a function of time) the interference pattern will remain stationary in space . Furthermore if R = l, i.e., E1 and E2 have equal intensities, then \ This means that the intensity modulation is 100%. If Rt 1 there .. will be a lift-up of the nodes plus a reduction of the peak to valley intensity difference. Fig. 5-4(b) shows the intensity distribution of the interference pattern with different values of R. If we place a sample coated with photoresist at they= 0 plane, the photoresist will be exposed by the interference fringes whose intensity is shown in Fig . 5-4(b) . The intensity distribution associated with R = 1 is ideal for this purpose. After developing the exposed photoresist we obtain a grating pattern . So far we have not considered the effect of the substrate. Most of the substrates used in practice are highly reflective. As a result we have to consider the total field of two incident waves and their reflected waves . Assuming the substrate has 100% reflectivity, for incident waves with electric field polarized in the z direction the total intensity is given by( 6) . 2(2TTncosey) I a sin A cos 2(2TTncosex) A y (a) X --R= 1.0 ·-·-·-· R= o.s Cb> -- - - - - - - - - --4.0 -3 Fig. 5-4 -2. -1 0 2 3 4 (a) The coordinate system that describes the interference of two plane waves. (b) Plots of intensity distribution IE 1+E 2 [!=o along x. The parameters are taken as ¢ = 0, e = 11°, ~ = 0. 4416 µm, and !Al = 1. <pm) -152- Which consists of a standing wave in they-direction, in addition to that in the x direction, with a period A 2ncose This periodic intensity variation in they-direction has an important effect on the exposure time of the photoresist. If the photoresist 5urface is close to one of the nodal planes, there will be no fringes recorded on the surface of the photoresist. ment will not give a grating pattern. The subsequent develop- Ideally the photoresist thickness should never exceed Otherwise it is difficult to develop the photoresist down to the substrate surface. Fig. 5- 5 is a schematic diagram of the experimental set-up used to fabricate the gratings. The output of a He-Cd laser is divided into two beams by a dielectric beam splitter. Each beam is reflected from a mirror and passes through a spatial filter. The mirrors are set so that the laser beams intersect at an angle 20 at the sample holder. A spatial filter is inserted after each mirror. The spatial filter consists of a quartz lens and a 12.5 µm diameter pinhole . The beam is focused by the lens and made to pass through the pinhole which is located at the focal point of the lens . The purpose of the spatial filters is to ensure that the beams coming out of the pinholes possess a high degree of spatial coherence. (This is achieved by limiting the transmission to essentially the lowest order Gaussian -153- He-Cd Laser Mirror Spatial Filter / / / Spatial Filter Fig. 5.5 A schematic diagram of the grating exposure set-up. -154transverse component). divergent. The beams emerging from the pinholes are In most experimental situations the distance between the pinholes and the sample is large and the sample area is small so that at the sample surface the beams have essentially planar wavefronts. The advantage of this arrangement is that it does not require very high quality mirrors since mirror defects are cleaned up by the 11 11 spatial filters . This is especially important at shorter laser wavelengths where high quality mirrors do not exist. A drawback, however, is that each time that the angle 28 is changed the spatial filters need to be realigned. If spatial filters are placed in front of the mirrors the problem of angle changes is alleviated but high quality mirrors must be used. The key for obtaining high quality grating patterns is good optical alignment (i.e . equal incidence angles for both beams). This can be checked by putting a highly reflective test sample in the sample holder. If the system is well aligned, the reflected beam of one of the incident beams from the test sample will coincide with the other incident beam and vice versa. 0 As an example, to make a grating of A= 3500A the angle e is determined from equation (5~2). If we use the blue output of an He-Cd laser A is 4416~ hence e = 39.1°. Theoretically the smallest grating period that can be made by this interference method is A = ½· when 8 = 90°. The shortest wavelength cw laser available to date is A= 0 3250A from an He-Cd laser. 0 Thus the smallest A obtainable is 1625A. In section 3~7 we showed that for the first order distributed feedback oscillation in GaAs a period A= 0.1184 µmis required. obtainable period A= 0.1625 µm The shortest is thus too big for our purpose. However the above limitation exists only for exposures performed in air (or vacuum). A is If one uses a medium with an index n then the shortest A/2n. For example one can use a quartz block or a bath of high index liquid( 7 , 3 , 9). Fig. 5-6(a) shows a configuration in which a quartz block is used for producing 0.11 µm period grating. The quartz prism is attached to the sample with the help of some index matching fluid (immersion oil, xylene, etc) of the prism. so that the light can couple out The index of refraction " of the quartz nq ~~ 1.5. Using A= 3250~, e = 70° we have A= 1153A. is shown in Fig. 5-6(b). A variation of this method Here the sample is immersed in a tank filled with xylene (n ~ 1.5). The tank has two low loss optical flats, one on each side, for the access of the incident beams. Before depositing the grating photoresist mask on GaAs wafers we first have to thoroughly clean the surface of the GaAs samples. This is usually done by rinsing in warm trichloroethylene (TCE) acetone, 2-propanol, and deionized water. The samples are subsequently etched slightly in H2so 4 :H 20:H 2o2(4:l :1) solution. After dry blowing the samples are kept in a dust free dish and baked in an oven (90°C) for about 10 minutes. Shipley photoresist (AZ 1350B) ·ts diluted (one part photoresist to two parts thinner) and coated on the samples by spinning. The thickness of the photoresist layer is estimated to be ~1500A. The photoresist~coated samples are then prebaked for about 10 minutes at 90°C. In our exposure system the typical optical intensity at -156- He-Cd laser 3250A Quartz prism (Cl) n ~ 1.s Sample He-Cd laser Xylene n ~ 1.s Sample Fig. 5-G Experimental configurations for producin~ ~ratings ~,ith periods from O. 1 µm to O. 16 JJnl. -157The exposure time is around 15 seconds. the sample is ~4 mw per beam. The developing time is fixed at 10 seconds. In some cases the photo- resist gratings are slightly over-exposed in order to develop the fringe patterns all the way to the substrate surface. This leads to better result during the subsequent chemical etching or ion beam etching. Grating quality can be determined by visual inspection of the film color or by measuring the efficiency as a diffraction grating. 5-4 Ion Beam Milling and Chemical Etching of GaAs Gratings We employed two methods of transferring a photoresist grating onto a GaAs substrate. process. One is a dry process while the other is a wet In the dry process ion beJm etching(S,lO,ll •12 ) is used and in the wet process chemical etching is used. ion beam etching is as follows. The basic principle of An inert gas (e.g. Argon) is fed into a high vacuum chamber and ionized through an electrical discharge. These heavy ions are collimated and accelerated to impinge on the sample surface. The high energy (typically 0.5 keV - 5 keV) particles cause sputtering ejection of the sample material. The samples can be masked to allow the etching of different patterns. If the masking material has a lower etching rate than the substrate material, patterns can be transferred from the mask to the substrate. The relative etching rates of photoresist and GaAs have been studied by several investigators(lO,ll,l 2). Typical values reported are around a 1 to 3 ratio. With this differential etching rate it is possible to etch patterns into GaAs using photoresist as a mask. -158Photoresist masks are usually baked prior to the etching process to increase their strength. The samples are mounted on a stainless steel plate using Ni clips or vacuum grease. During etching, a motor is used to provide a translational motion of the samples across the ion beam for uniform etching. With an ion current density of "vl mA/cm2 and an accelerating voltage of "vQ.7 kv the typical etching time for GaAs gratings is 3-5 minutes. The photoresist mask subjected to the ion beam etching sometimes becomes very hard and difficult to dissolve by the conventional immersion in acetone. However, it can _be removed by scrubbing with a swab soaked with acetone or by immersing in a mixture of AZ-303 developer (~20 c.c.) and KOH (a few drops). Typical SEM photographs of GaAs gratings fabricated by ion milling through photoresist masks are shown in Fig. 5-7 and 5-8. Fig. 5-7 shows a surface corrugation with a period "vQ.35 µm. It has well defined patterns with a corrugation depth estimated to be around 0 700A. In Fig. 5~8 we show a grating with a period "vQ.115 µm produced by ion milling through a photoresist mask. The poor surface quality is probably due to the nonuniformity in the photoresist mask after development. In general, the results of ion beam etched gratings are quite satisfactory. There are some methods that one can use to increase the depth of the corrugation. For example one can "shadow'' the photoresist grating by evaporating a thin film of some low etching rate metal at a shallow angle. mask. This can increase the endurance time of the -159~ Fig . 5~7 An SEM picture of a GaAs surface corrugation produced by ion beam etching through a photoresist mask. The period is~ 0!35 µm, :·"if ~ ')' ..-.;_.~. -~ ~ : .: ~ ~ ";'t ~~~ •• ..' ~~".:- ,~ · :t-~ • t. _.~ ..:..:.. l. "'~ ~ f,Vi-~ ?: ' '' ,.s.,~~ i ; ·1' .ilf ;J,a,::.~~~ ~}-•::, :J~.-.,. - ,- -~·-,,.,._,' - ,,,~'<;"" - .•, ·"''."',.ii!" - I 1 0 m -" -161- In ion milling the removal of material is achieved by bombardment with energetic particles. As a result the process introduces a high density of defects in the samples. These defects cause an increase in the threshold current of the GaAs double heterostructure distributed feedback lasers as described tn Chapter 3. As an alternative approach we investigated the possibility of using chemical etching. The selective chemical etching properties of GaAs crystal have been studied extensively(l 3 ,l 4 ). It has been found that the Br2-cH 30H (1:1000 in volume) system gives good results in preferential etching I of GaAs. If a slot mask is deposited on a GaAs crystal {100} plane and aligned parallel to one of the two cleavage faces ({0lf} or {Oii}) the etched pattern can have one of two different types of groove profiles depending on the orientation of the mask. This is illustrated in Fig. 5-9(a) . In these two types of grooves the side walls are all A{lll} crystal planes of GaAs . This is because the etching rate of this plane is much lower than any other low Miller indices planes. As a result the v-shaped groove is essentially self-terminating while the other kind of groove diverges. For the purpose of grating fabrication we prefer the v-groove profile(l 5). The direction of the mask slots for this groove can usually be determined by etching a jcross 11 pattern in a small part of the sample . The A{lll} planes make 1 an angle of 54°44 1 with the {100} planes. Hence in making gratings by chemical etching the maximum depth achievable is roughly or h= i tan 54°44' h = 0. 711\. -162- (Cl) {oTT} • I i I ,. / 54°44' ·-·-·- Fig. 5-9 (a) The preferential etching patterns on the {100} face of a GaAs crystal. (b) A schematic diagram shows the relation between the grating period A and the V-shaped groove maximum depth hmax· -1630 For A= 0,35 µm, this comes out to be ~2500A and for A= 0.11 µm, hmax is ~780~ as illustrated in Fig. 5-9(b). This is based on the assumption that we can get a symmetrical triangular profile. Some of the corrugations fabricated by this method are shown in Fig. 5-10 and 5-11. Fig. 5-10 is an SEM picture of a GaAs grating with a period of ~o.345 µm and depth of ~0.18 µm. It shows clearly the crystal planes and the etched v-shaped profile. Fig. 5-11 is a picture of a grating with period ~0.11 µm which shows good surface quality as compared to the ion-milled grating {Fig. 5-8). The depth of this grating is estimated to be ~500~. Alt~ough Br 2-cH 30H system attackes photoresist, it is still possible to etch gratings into GaAs with deep grooves without a special treatment. One can also shadow the photoresist for better results(l 5). Other etching solutions that do not attack photoresist can also be used. Regardless of the etching method employed it is important that the photoresist grating mask be developed down to the surface of the sample for best result. It has been .explained in section 5-3 that because of the large index of refraction of GaAs the standing wave pattern in they-direction [see Fig. 5-4{a)] has a minimum at the surface of the substrate. As a result the photoresist cannot be developed away completely and a thin film is left between fringes as illustrated in Fig. 5-12(a). If the grating period is large (e.g. A~ 3500A) we can over develop the photoresist slightly to obtain a mask as shown in Fig. 5-12(b). In fabricating the very short 0.11 µm gratings this procedure is difficult to control. However we can use ion beam milling to shape the photoresist grating prior to chemical ,, ' ,,).~~·:•1;·~ ~f~~· · ..r. -.~ ' -166- ((U Original Photo resist Grdting ( 0. 3 1:,1m) lb) Fu rt her Development . C'\ C'\ C'\ C'\ C'\ (Cl Original Photoresist Grating ( 0.1 1:,1m} Cd) After Ion Milling 0, ,,. ,0 ,,. Ct, c:::,. 0, t::) t:) c:,, Fig. 5-12 Schemes of shaping photoresist gratings mask for chemical etching. ,., 167- etching. This is illustrated in Fig. 5-12 (c) and (d). 5-5 Optical Measurements In this section we describe a few important measuring techniques used in our study of GaAs distributed feedback and distributed Bragg reflector lasers. These include the characterization of GaAs epitaxial layers as an active device material, the determination of the aluminum content in a GaAlAs layer, the detennination of GaAs distributed feedback laser parameters, and the measurement of grating period by laser beam diffraction. Photoluminescence measurement is a very common technique for material characterization in semiconductors. By measuring the spectral location of the luminescence peak we can obtain information concerning the energy band structure of the sample. Since the bandgap energy of GaAlAs is a function of the mole fraction of Al in the material we can thus apply the luminescence measurement results to obtain an estimate of the amount of Al in the material. This process is important because a precise control of the Al content in an epitaxial layer is difficult. An example of detennining the composition of a multi-layer structure is shown in Fig . 5-l3(a) . We first etch a small part of the sample so as to obtain a "staircase'' as in Fig. 5.-13(b). to one of the layers. Each step corresponds We next measure the photoluminescence spectrum from each step in a system similar to that shown in Fig. 3.-18 . The pumping beam is focused by a convex lens to a small spot on the sample surface which is tilted at an angle of ~45° with respect to the pumping beam. The output from the surface of the sample is collected -168- GaAs Substrate <b) ij ~ Pumping Beam a-----------'--~ n GaAs Substrate CC> f-Pumping Beam GaAs Substrate ' · ..... ......... ·:..... 8 _ _ _ _ _ _ _ _ _ _ _ _ ___,J ________ ......... Fig. 5--13 Schematic diagrams showing two different ways of preparing samples for photoluminescence measurement. by a lens and guided into a spectrometer. If the composition profile of each layer is needed we can perform angle lapping of the sample so that the thickness of the layer is "amplified'' by a factor of 1/sine where e is the angle of lapping. This is illustrated in Fig. 5... 13(c). The small pumping spot is scanned from z = 0 to z = L and the corresponding luminescence spectra are used to obtain the con~ centration of Al as a function of z. This information is then used to obtain the profile of Al content withtn each layer. The dependence of peak luminescence wavelength on the mole fraction of Al is given(l 6 ) in Fig. 5-14. As mentioned in section 3~7, given a GaAs waveguide prior to fabricating a distributed feedback laser we need first to take a luminescence measurement in order to locate the center of the gain spectrum. At the same time, by measuring the magnitude of the luminescence output intensity we obtain an indication of the relative internal quantum efficiency of the sample . A low radiative recombination efficiency usually means that the sample contains a high density of defects which act as traps of the carriers. Thus by scanning the pumping spot across the surface of a sample we can map the relative defect density of the sample. In the next paragraph we will describe how the optical pumping method can be used to determine some distributed feedback laser parameters. Recall in section 3-7 that we gave the threshold gain of a distributed feedback laser as -1700 ,----,~---.---.----T"---,.---.--......---.--.....---- 4000 A 3.0 Direct band-gap Indirect band-gap -> a, _, 25 0 SOOOA C) w a. C C) >C) ... a, C 0 w 6000A 2.0 ·----·~ .-· -· • 7000A 300°K 0 8000A 1.5 0 8500A 0 o.s GaAa X 1.0 AIAs Fig. 5-14 Band-gap energy of Ga 1_xAlxAs against composition. -171- where a~h{K,L) is the intrinsic threshold gain of a distributed feedback laser, abulk and arad are the amplitude absorption and radiation coefficients respectively. In an experiment we measure ath as the ''true 11 threshold of our laser. In order to compare with the theory we have to know abulk and arad {at the lasing wavelength). As shown in Fig. 5-lS{a) we optically pump a corrugated GaAs waveguide with a beam of constant width and of fixed length £0 . We then measure the output power Pas a function of£, the distance between the crystal edge and the pumped region. The output power is related to the internally generated power P0 through the relation P = P0 e -2acorr £ where acorr is the overall amplitude loss coefficient of a corrugated waveguide and can be expressed as {5-3) acorr -- a bulk + a rad The magnitude of abulk is estimated in a similar manner except that an uncorrugated region is used so that arad = 0. This is shown in Fig. 5-lS{b) . Another method of determining abulk is by manufacturing GaAs Fabry-Perot lasers of various lengths from the same waveguide and measuring their threshold pumping intensity. Since in a Fabry Perot laser the threshold gain is given by 1 < 1 1 th = abulk + [ in R where Lis the length of the laser and R is the intensity reflectivity of the cleaved mirrors. If we plot ath as a function of 1/L we obtain - 172- : .: UII : UIII : .t Cbl I F/lllA 'f- (C) Fabry-Perot loser 0 I ( l\Yb . unit) I 2. L Fig. 5-15 De te rminati on of t he bul k absorp t ion and ra dia ti on loss coeff ici ent s of a co rrugated GaAs sample . -173- a straight line with slope £n ~ and an intercept with the ath axis gives abulk as illustrated in Fig. 5-l5(c). Once abulk is found we can use it in equation (5-3) to determine a ra d" These methods are only practical with optical excitation where the changes in the excitation conditions are easily affected. It is very important to determine accurately the period of the grating in a distributed feedback laser. This can be appreciated from the lasing wavelength equation 0 where 2neff ~7.2. A. change of lOA in the grating period, say, will result in a shift of the lasing wavelength by nearly 72~. Such a big shift can easily move the lasing wavelength from the center of the gain spectrum to the tail where the gain is low, thus increasing the threshold beyond any practical pumping level. Hence we need to determine whether the holographic exposure system is set to give the required period before exposing the GaAs samples. This is accomplished by first exposing a test sample and using a laser beam diffraction to measure its period . The sample is mounted on a rota ry table and a laser beam is directed to pass through the rotating axis of the table as shown in Fig . 5-16(a) . A reference angle e0 is detennined by letting the inci:dent beam and the reflected beam coincide at normal incidence. The grating is then rotated until the Littrow reflection takes place~ that is when the diffracted beam and the incident beam directions coincide. This occurs when the condition -1740 (a.) He - Cd Laser A = 441 & A --------------------------------A A / -''- = 2 S;n e n I ~ v- (b) 4000 3800 ~ 80 I \ 3&00 o<( '--' qo \. \ \ <}:=, ro o.la.. CD ~ \ \ 3400 60 _.... ~ 3lOO )>o 50-........... a.. \ -0 0 ·>- (b \ 3000 ~ 40 "" It) \ tt) \ cf 2800 ' '. A 2600 de 10 ~o 30 40 An5le Fig. 5-16 . c:¢> 20 . ' ·,·,. 2400 0 30 '''. -d.A - ·-·-·- 2too -....J 50 60 70 80 10 qo e (degree) (a) Diagram shows the geometry of the grating period measuring system. (b) Plots of equation (5-4) and (5-5). -175- 2/1. sine= "A (5-4) is satisfied, where e is as indicated in Fig. 5-16(a). reading at this point is e1 , then e = Je 1-e 0 J. If the angular Let us calculate the sens i ti vi ty of fl.with respect to a sma 11 change in e ~~ = - fl.cote (5-5) Equations (5-4) and (5-5) are plotted in Fig. 5-l6(b). The dependence of d/1./de on eis such that it approaches infinity when the angle is small and decreases to zero when e is near 90°. As a typical example consider a grating with a period of 3500~ using the blue line 0 of an He-Cd laser (4416A). The required e can be determined from Fig. 5-l6(b) to be 39.11°. At this angle d/1./de is ~75.13A/degree hence an error of l minute in reading the angle will result in an error of l.25~ in fl. which is easily tolerated. A good rotary table can yield angular accuracies considerably in excess of the l minute figure used above. The measurement of fundamental gratings is not as straightforward. For a grating with a period fl.~ O.ll µm we cannot observe 0 diffraction using "A= 3250A in air. Thus we have to pass the UV laser beam through a quartz prism first as shown in Fig. 5-17(a). The prism which has a rectangular base with index of refraction n is mounted on a rotary table. A sample with 0.11 µm grating is attached to one of the prism faces with xylene or immersion oil. The prism is first set so that the incident beam incident on one of the prism faces is reflected backward retracing the incident beam direction. The -176(O.) He-Cd Laser - - - - - - _ _ _ _ _ ll; __ _ --' e / .,.✓ i ..... ,,,,. n = 1.54415 (b) 1330 A 1300 8 -de ·-·-·-· dA. 1i10 - • <( - 7 1a4o _,,,...-.,. /" . / ~ 1i10 ""O 0 ' .'\ / . I Q) ,,so / 11io «> 4 \ \ I \ \ \ 1oo0 0 -.....,/ 3 \ i \ \ .I \. i \ 10 20 30 40 Angle Fig. 5-17 50 60 70 80 ~ lb I\) . I I . I ~ e:::::t> I ro<to ........... Q.. \ / )>o 5 \ / 1180 ~~ ,--... \ / '- 0.. .,,,,. 6 qo 0 9 (degree) (a) Schematic diagram of the system for measuring grating period less than 0.16 µm. (b) Plots of equation (5-6) and (5-7). angular reading at this position is denoted as e . 0 The prism is then rotated until the diffracted beam from the grating retraces the incid~nt Let the corresponding angular reading be e , which 1 means that the prism is rotated an angle of e= Ie -e I . Using the beam direction. 1 O geometry shown in Fig. 5-17(a) we can find the period of the grating as /1. " (5-6) = ---- 21n 2-sin 2e 0 If >- = 3250A is used we have fl.= 1625 (~) /n 2.. sin 2e where e is the measured angle of rotation. In this formula the index of refraction of the prism n enters, so we have to known as accurately as possible. A rough estimate of n can be obtained by a simple laser beam refraction experiment . A more accurate value is obtained by using 0 a grating with a larger period (A > 1700A) which is measurable in air using>-= 3250~. This same grating is then attached to the prism and one looks for its diffraction in higher orders. 0 if fl.= 2771A is measured in For example air, using the rough estimate of n ~ 1.5 we find that the Littrow reflection angles inside the prism will be¢~ 61.5° for the first order beam and¢~ 17.4° for the second order beam. The first order reflected beam is not observable in the forward direction because Sino= nsin¢ > l -178However we will be able to see the second order reflected beam. Assume, for example, that the measured angle e is 35°27 1 , the index of refraction is then determined to very high precision as n = 1.54415 (at ~ = 3250A). This n is used in equation (5-6) to calculate the grating periods with A smaller than 0,16 µm. The grating period measured by this method is less sensitive to the accuracy of angular readings than the previous method. This is seen from Equation (5-7) and equation (5~6) are plotted in Fig. 5-17(b) for n = 1.54415. -179CHAPTER 5 REFERENCES (1) D. Marcuse, uMode conversion caused by surface imperfections of a dielectric slab waveguide, 11 Bell Syst. Tech. J. 48, 3187 ( 1969). (2) D. C, Flanders, H. Kogelnik, R. V. Schmidt, and C. V. Shank, MGrating filters for thin-film optical waveguide, 11 Appl. Phys. Lett. 24, 194 (1974). (3) R. V. Schmidt, D. C. Flanders, C. V. Shank, and R. D. Standley, 11 Narrow'"band grating filters for thin-film opti ca 1 waveguide/' P.ppl. Phys. Lett. 25, 651 (1974). (4) H. L. Garvin, J. E. Kiefer, and S. Somekh, "Wire grid polarizers for 10.6 µm radiation,u Conference on Laser Engineering and Applications~ Digest of Technical Papers, 100 (1973). (5) H. L. Garvin, E. Garmire, S. Somekh, H. Stoll, and A. Yariv, 1' Ion beam micromachining of integrated optics components, 11 Appl. Opt. l?.., 455 (1973}. (6} L. P. Boivin , ''Standing wave effect in grating production by 1aser beam interference on res tst coated highly reflecting substrates, 11 Opt. Commun . ~' 206 (1973) . (7) C. V. Shank and R. V. Schmidt, l 1 0ptical technique for producir.g 0, 1"."µ peri odic surface structures," Appl. Phys. Lett.~' 154 (1973). (8) H. W. Yen , M. Nakamura , E. Garmire, S. Somekh, A. Yariv, and H. L. Garvin, ~optically pumped GaAs waveguide lasers with a fundamental (1973). o.11µ corr ugati'on feedback/' Opt. Commun. 2,, 35 -180(9) W. T. Tsang and S. Wang, ''Simultaneous exposure and development technique for making gratings on positive photoresist, 11 Appl. Phys. Lett. 24, 196 (1974). (10) E. G. Spencer and P, H. Schmidt, "Ion-beam techniques for device fabrication,u J. Vac. Set. Technol. ~' S52 (1971). (11) Per G. Gloersen, 11 Ion beam etching, 11 J. Vac. Sci. Technol. -12, 28 (1975). .. (12) H. L. Garvin, 11 High resolution fabrication by ion beam sputtering,u Sol. State . Tech. 31 (1973). (13) Y. Tarui, Y. Komiya, and Y. Harad.a, "Preferential etching and etched profile of GaAs,'·1 J. Electrochem. Soc. 118, 118 (1971). (14) S. Iida and K. Ito, "Selective etching of gallium arsenide crystals in H so -H 2o2~Hi system, 11 J. Electrochem. Soc. 2 4 118, 768 (1971) .• (15) L. Comerford and P. Zory, "Selectively etched diffraction gratings i'n GaAs," Appl . Phys. Lett . ~' 208 (1974) . (16) M. B. Panish, 11 Heterojunction injection lasers," IEEE Trans. Microwave Theory Tech . £1, 20 ( 1975) .