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Modeling of Distributed Feedback Semiconductor Lasers

Permanent Link: http://ufdc.ufl.edu/UFE0041962/00001

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Title: Modeling of Distributed Feedback Semiconductor Lasers
Physical Description: 1 online resource (109 p.)
Language: english
Creator: Shih, Meng-Mu
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: coefficient, coupling, distributed, electrical, emitting, feedback, infrared, laser, magnetic, modeling, quantum, semiconductor, te, tm, transverse, waveguide
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This work demonstrates the multi-parameter modeling processes of calculating coupling coefficients of the optical waveguide structures for various distributed feedback (DFB) semiconductor lasers. These lasers operate on Near/Mid-Infrared and TE/TM modes. Substrate-emitting DFB quantum cascade laser analyzed and performance improvement are discussed.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Meng-Mu Shih.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Zory, Peter S.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041962:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041962/00001

Material Information

Title: Modeling of Distributed Feedback Semiconductor Lasers
Physical Description: 1 online resource (109 p.)
Language: english
Creator: Shih, Meng-Mu
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: coefficient, coupling, distributed, electrical, emitting, feedback, infrared, laser, magnetic, modeling, quantum, semiconductor, te, tm, transverse, waveguide
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This work demonstrates the multi-parameter modeling processes of calculating coupling coefficients of the optical waveguide structures for various distributed feedback (DFB) semiconductor lasers. These lasers operate on Near/Mid-Infrared and TE/TM modes. Substrate-emitting DFB quantum cascade laser analyzed and performance improvement are discussed.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Meng-Mu Shih.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Zory, Peter S.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041962:00001


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1 MODELING OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASERS By MENG MU SHIH A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010

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2 2010 Meng-Mu Shih

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3 To my parents Mr. Shi -Chen Shih and Mrs. Ai Chiao Chen Shih and those who have helped me

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4 ACKNOWLEDGMENTS First and foremost, I sincerely appreciate my academic adviso r, Prof. Peter Zory for his guidance and encouragement. His two classes of photonics and laser electronics initiated my strong interest and fundamental background for my Ph. D. dissertation. He provides the physics background with intuitive but deeper ins ights for my mathematical modeling. Dr. Zory is also my life mentor. His experiences in engineering and management in enterprises and academia has shaped into a unique life of philosophy. I have cultured from his unique advantages which have helped me to think and solve academic and life problems from multi -facet ed perspectives. I really enjoy extensive and intensive discussions with Dr. Zory from research topics to life of philosophy, not only solving my problems wisely but also help ing me explore views m ore. With his patient and enthusiastic efforts for improving both my written and oral communication skills, I can pass the oral exam and also optimistically face unexpected challenging life issues. I would like to thank all the other members of my superv isory committee: Dr. Toshikazu Nishida, Dr. Huikai Xie, and Dr. Angela Lindner. Dr. Xie taught me the semiconductor physical electronics and provided many suggestions during my oral proposal based on his optic s specialty. Dr. Nishida and Dr. Angela also pr ovided substantial suggestions. Some U niversity of Florida staff have work ed behind the scene. Dr. Linder, as Associate Dean of Students Affairs, helped me deal with life issues. Ms. Debra Anderson at International Center has provided extensive help far beyond tha t which I can express. Ms. Shannon Chillingworth, my departmental graduate advisor, has

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5 directed me to follow requirement details. Mr. Keith Rambo provides labs and equipments. I would like to thank Dr. Zorys previous students Dr. Horng-Jye Luo for his help in the C hapter 3 of my dissertation. His mathematical modeling and manipulation techniques inspired my desire for further modeling process. I also thank colleagues from Interdisciplinary Microsystems Group: Dr. Lei Wu and Sean under Dr. Xie. They helped me conduct experiments on semiconductor lasers and quantum cascade lasers. This work was partly funde d by the Defense Advanced Research Projects Agency ( DARPA contract no. 00062850) co nducted by Dr. Dan Botez in University of Wisconsin. Friends from different countries have enriched my life in this college town. I have priceless opportunities to interact with people from different cultures. Through them, I comprehend the world of diversity. Dr. Wang, my high school classmate, has encouraged me since my graduate study in the U.S although he has been quite busy in performing long -hour surgeries D r. Along and Dr. Shiau provide me with good suggestions about doing research. Mr. Chu provides friendly support. Mr. Ray Mr. Justin and Mr. Toby ofte n find interesting topics to broaden my view toward American culture. Last but definitely not the least, I want to thank my parents for their endless support and deepest love throughout the life journey, sometimes quite challenging to me. No matter where I would be, they would be with me.

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6 TABLE OF CONTENTS page ACKNOWLEDGMENTS ...................................................................................................... 4 LIST OF TABLES ................................................................................................................ 8 LIST OF FIGURES .............................................................................................................. 9 ABSTRACT ........................................................................................................................ 12 CHAPTER 1 INTRODUCTION ........................................................................................................ 13 1.1 Study Moti vation ................................................................................................ 13 1.2 Basic Semiconductor Laser Concepts .............................................................. 14 1.3 Thesis P roblem Background ............................................................................. 17 1.4 Dissertation Organization .................................................................................. 20 2 FUNDAMENTAL OF WAVEGUIDES ......................................................................... 22 2.1 Planar Waveguide Concepts ............................................................................. 22 2.2 Corrugated Four -Layer Waveguides ................................................................. 35 2.2.1 Diffractive Waves in Corrugated Waveguides ....................................... 35 2.2.2 First -Order Distributed Feedback in Corrugated Waveguides .............. 36 2.2.3 Second-Order Distributed Feedback in Corrugated Waveguides ......... 37 3 TM EIGENMODE EQUATIONS FOR PLANAR AND DISTRIBUTED FEE DBACK CORRUGATED SENICONDUCTOR WAVEGUIDES WITH SHINY METAL CONTACTS ...................................................................................... 41 3.1 Floquet -Bloch Formalism for Corrugated Waveguides .................................... 41 3.2 Derivation of the Eigenmode Equation for Planar Waveguides by Truncated Floquet -Bloch Formalism ................................................................. 45 3.3 Derivation of the Eigenmode Equation for First order DFB Corrugated Waveguides by Truncated Floquet -Bloch Formalism ....................................... 48 3.4 Derivation of the Eigenmode Equation for Secondorder DFB Corrugated Waveguides by Truncated Floquet -Bloch Formalism ....................................... 50 4 CALCULATION METHODOLOGY AND RESULTS .................................................. 54 4.1 Methodology ....................................................................................................... 54 4.2 Computation ....................................................................................................... 58 4.3 Numerical Results and Discussions .................................................................. 62 5 MODIFED MODELS CONSIDERING REAL METALS ............................................. 65

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7 5.1 Full Floquet -Bloch Formalism for Real Metals .................................................. 65 5.2 Numerical Results and Discussions .................................................................. 68 6 SUBSTRATE-EMITTING DFB -QCL .......................................................................... 76 6.1 Multi Layer Structure of DFB -QCL .................................................................... 76 6.2 DFB Coupling Coefficients ................................................................................ 78 7 CONCLUSION ............................................................................................................ 80 APPENDIX A MAXWELLS EQUATIONS AND GUIDED ELECTROMAGNETIC WAVES ............ 83 B D ERIVATION OF EQUATION 3 32 FOR PLANAR WAVEGUIDEDS IN TM MODE .......................................................................................................................... 91 C DERIVATION OF EQUATION 3 41 FOR FIRSt-ORDER DFB WAVEGUIDES IN TM MODE ................................................................................................................... 98 D EIGENMODE EQUATION FOR PLANAR WAVEGUIDES IN TM MODE .............. 102 E EIGENMODE EQUATION FOR FIRST -ORDER DFB WAVEGUIDES IN TM MODE ........................................................................................................................ 103 F COUPLING -COEFFICIENT SENSITIVIY TO GEOMETRIC PARAMETERS FOR WAVEGUIDES IN TM MODE ......................................................................... 104 G FIELD INTERACTION AT DIELECTRICMETAL INTERFACE FOR PLANAR WAVEGUIDES IN TE AND TM MODES ................................................................. 105 LIST OF REFERENCES ................................................................................................. 106 BIOGRAPHICAL SKETCH .............................................................................................. 109

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8 LIST OF TABLES Table page 2 -1 Comparison of the fir st order DFB and second order DFB .................................. 40 F -1 Coupling -coefficient sensitivity to geometric parameters ................................... 104

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9 LIST OF FIGURES Figure page 1 -1 Semiconductor laser A) Schematic structure of semiconductor laser B) Power vs. current diagram ..................................................................................... 14 1 -2 Structure and energy band diagram of diode lasers ............................................. 15 1 -3 Structure and energy band diagram of the basic unit in quantum cascade lasers ...................................................................................................................... 16 1 -4 A schematic diagram of semiconductor waveguide with metal grating ................ 17 1 -5 Schematic structure of a substrateemitting DFB -QCL [5] ................................... 18 2 -1 Basic 3 layer semiconductor laser onedimensional waveguide .......................... 22 2 -2 Basic two -dimensional semiconductor laser waveguide ...................................... 23 2 -3 Schematic plot of 2 2 ,1,0jyjnHx vs. x for the fundamental TM mode .............. 31 2 -4 Calculated plot of 2 2 ,1,0jyjnHx vs. x for the fundamental TM mode ............. 32 2 -5 Bouncing ray in a guided mode waveguide .......................................................... 33 2 -6 Wave propagation in the fundamental mode ........................................................ 34 2 -7 Diffraction caused by corrugated waveguide. A) Diffractive waves due to gratings B) Real picture of diffraction .................................................................... 35 2 -8 First order DFB. A) Ray optics picture B) Wavevector diagram .......................... 36 2 -9 Secondorder DFB. A) Ray optics picture B) Wave vector diagram for the second order diffractive wave C) Wave vector diagram for the first order diffractive wave ....................................................................................................... 38 2 -10 Schematic diagrams for A) First order B) Second order DFB .............................. 39 3 -1 The shiny contact DFB laser structure .................................................................. 41 4 -1 The coupling coefficient ( ) and reflection ........................................................... 54 4 -2 Floquet -Bloch formalism for calculating of metal/dielectric corrugated structures ................................................................................................................ 55 4 -3 dispersion diagram for DFB ............................................................................ 57

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10 4 -4 Flow chart to compute coupling coefficient .................................................... 59 4 -.5 An example showing the value of eigenmode equations versus propagation constant when ka=0.5 ............................................................................................ 60 4 -6 Soluti on search to compute coupling coefficient when ka=0.5 ..................... 61 4 -7 Coupling coefficient vs. corrugation amplitude (a) ......................................... 62 4 -9 Coupling coefficient vs. active layer thickness (t) .......................................... 64 5 -1 The DFB laser structure with real metal contact ................................................... 65 5 -2 Coupling coefficient vs. corrugation amplitude (a) ......................................... 69 5 -3 Coupling coefficient vs. buffer thickness (t) ................................................... 70 5 -4 Coupling coefficient vs. buffer thickness (t) ................................................... 71 5 -5 Coupling coefficient vs. active layer thickness (t) .......................................... 72 5 -6 Coupling coefficient vs. active layer thickness (t) .......................................... 73 5 -7 Coupling coefficient vs. corrugation amplitude (a) for different metals ......... 75 5 -8 Coupling coefficient vs. corrugation amplitude (a) for different metals ......... 75 6 -1 Substrateemitting DFB -QCL [5]. A) Sketch showing laser beam emission from substrate B) SEM picture showing corrugation prior to device fabrication .. 76 6 -2 Schematic waveguide structure (zoom -in cross-sectional view at the eight layer waveguide alo ng the longitudinal z direction) from the substrateemitting DFB -QCL in Figure 6 1 ............................................................................ 77 6 -3 Schematic approximated four -layer waveguide structure for the eight layer waveguide............................................................................................................... 78 6 -4 Coupling coefficient vs. corrugation amplitude (a) for simplified 4-layer waveguide............................................................................................................... 79 A-1 Three polarizations in the slab waveguide operating on TM modes .................... 90

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11 G -1 Field inter action at the dielectric metal interface in TE and TM modes. TM mode has larger interacti on. ................................................................................ 105

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12 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MODELING OF DISTRUBUTED FEEDBACK SEMICONDUCTOR LASERS By MengMu Shih August 2010 Chair: Peter Zory Major: Electrical and Computer Engineering This work demonstrates the multi parameter modeling processes of calculati ng coupling coefficients of the optical waveguide structures for various distributed feedback (DFB) semiconductor lasers. These lasers operate on Near/Mid-IR and TE/TM modes. Substrateemitting DFB quantum cascade laser analyzed and performance improvement are discussed.

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13 CHAPTER 1 INTRODUCTION 1 .1 Study Motivation Semiconductor lasers were first demonstrated in 1962 [ 1 ] [2] Since these devices have built -in p-n junctions, they are called diode lasers. In the 19 60s and 19 70s, most applications were defense related and only small volumes of diode lasers were required. They became high volume products in the 19 80s and 19 90s with the advent of a pplication areas such as fiber optic communications and information technology In many of these application areas the diode lasers used have built -in diffraction gratings that provide narrow band spectral output. This type of diode laser is now called a distributed feedback (DFB) laser. In 1994, a new type of semiconductor laser that did not have a built in pn junction was demonstrated [ 3 ]. These lasers, called Quantum Cascade Lasers (QCLs) operate on intersubband transitions and utilize electron tunn eling to achieve population inversion in the conduction band of the semiconductor material. These QCLs are now starting to replace cryogenic diode lasers and gas lasers that have been used for many years in molecular spectroscopy applications in the mid t o far infra -red region of the spectrum [ 4 ] To date, the narrow -band spectral output required in such applications has been achieved using diffractiongrating techniques in resonator configurations external to the semiconductor laser chip. In order to reduce the cost of such systems, there has been considerable research activity in the last few years to develop QCLs with integrated diffraction gratings (DFB -QCLs). Since narrow band operation can be achieved using first -order DFB, most of the research act ivity in this area has been concentrated on t his type of DFB -QCL. In order to eliminate the expensive optics required to capture and

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14 collimate the high divergence beams emitted from these first order DFB -QCLs while retaining narrow band output, a more com plex design incorporating s econd order DFB is now being researched [5 ] 1. 2 Basic Semiconductor Laser C oncepts The multi layer d ielectric slab waveguide (WG) structure shown in Figure 1 -1 (a) is the basic structure used for all semiconductor laser s (eith er diode or QCL). Current I through the laser chip produces optical gain in the active region. Cleaved facets on the longitudinal edges provide optical feedback Fig ure 1 1 (b) shows a typical output power 0P vs. I diagram When I is greater than the threshold current thI laser action is initiated in the semiconductor chip. Figure 11. Semiconductor laser A ) Schematic structure of semiconductor laser B) Power vs. current diagram Fi gure 12 shows the schematic structure and energy -band diagram. If we rotate the laser chip shown in Figure 1-1 (a) clockwise by 90 degrees and expand the area in the vicinity of the active region, we see that current I is equivalent to e lectrons moving to the positive metal contact and the active region is a quantum well (QW) designed to trap electrons. In the energy band diagram, conduction band ( CB ) electrons e are trapped in the QW and recombine with valence band ( VB) holes Dur ing this process,

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15 photons are produced with energy 11 CHhvEE where 1 CE and 1 HE represent certain energy states in the CB and VB respectively. This process whereby electrons move from the CB t o the VB is called an i nterband t ransition. Figure 1 2. Structure and energy band diagram of diode lasers

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16 Figure 1 3 shows the schematic structure and energy -band diagram of the basic unit in a quantum cascade laser (QCL). In a typical QCL, there will be about 20 to 30 basic units in the active region. Electrons are attracted to the positive metal contact as in the diode case but the basic process used here to achieve gain is quite different. Electrons in the CB states of the injector region tunnel in to 2 CE states in the QW These electrons then m ake transitions to 1 CE states in the QW and produce photons with energy 21 CChvEE Since these transitions occur within the CB of the QW rather t han between the CB and VB of the QW, they are called i ntersubband transitions. Figure 1 3. Structure and energy -band diagram of the basic unit in quantum cascade lasers

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17 1. 3 Thesis Problem Background As discussed in Section 1.1, the method use to obtain n arrow band spectral width and low divergence beams from semiconductor lasers is to incorporate a diffraction grating into the multi -layered structure. A schematic diagram of this type of structure where the grating is defined by a corrugated metal/dielect ric interface is shown in Figure 1 -4. Figure 14 A schematic diagram of semiconductor waveguide with metal grating If the corrugated waveguide shown in Figure 1 -4 is designed correctly, light waves with a specific wavelength traveling to the right an d left inside the guide will be coupled to each other by backward diffraction. This mechanism is called distributed feedback (DFB) and, if sufficient gain is provided to the light in the guide, laser action at that specific wavelength takes place. Two p arameters associated with the metal layer that are required for low current density operation are the reflectivity of the metal at the lasing wavelength and the electrical resistance at the metal -dielectric interface. Reflectivity should be high in order to minimize absorption of the beam in the metal and

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18 electrical resistance should be low in order to minimize heat due to electrical power dissipation. Since these two requirements are sometimes difficult to achieve with the same metal, the corrugation is o ften defined in the interior of the epitaxial structure where one obtains a dielectric/dielectric interface. In order to achieve this result, a complex epitaxial re-growth step is required after grating fabrication. This option has been successfully util ized in the production of first order DFB diode lasers used in fiber optic communication applications. The corrugation fabrication technique that will ultimately be used in the production of DFB -QCLs has yet to be decided. For first order DFB operation, the grating period must be about 1/6 the vacuum wavelength of the laser light. If secondorder DFB is desired, then the grating period must be about 1/3 the laser light wavelength. If the laser operates based on facet reflection or first order DFB, the laser beam is emitted from the cleaved facet(s) of the laser chip. If the laser operates on secondorder DFB, the low divergence beam of interest is emitted from the chip through the surface adjacent to the grating (the epi side) or through the surface on the other side of the chip (the substrate side). A sketch of a laser chip operating in the substrate emission mode is shown in Figure 15 [5] Fi gure 1 5 Schematic structure of a substrate emitting DFB -QCL [5]

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19 If laser beam absorption in the subst rate material is small, substrate emission is preferred over epi -side emission because active region temperature can be minimized in this configuration. This is particularly important in QCLs where the efficiency for converting electrical power to laser p ower in the active region is about 10%. In 2006, it was realized here at UF that a DFB -QCL operating in the substrate emission mode had never been reported in the literature. As a consequence it was decided to see if a device of this type could be made. By early 2007, a successful prototype was demonstrated and the work published in that year [ 5 ]. While narrow spectral band and low divergence beams were obtained, the devices did not operate at room temperature. Since room temperature operation is highly desirable for most applications, we asked the question what could we change in the design to make this possible? One obvious possibility was to improve the material so that more optical gain was obtained for a given current density. Another was the po ssibility to increase the DFB coupling coefficient In designing the prototype device, was determined using a rough estimate since this parameter is very difficult to compute exactly. This difficulty arises because one component of the electric field in QCL beams in the waveguide configuration shown in Figure 14 is constrained to be perpendicular to the plane of the layers. In this case, the laser beam intensity builds up at the metal interface rather than going to zero as it does in conventional diode lasers operating with their electric field par allel to the plane of the layers. The reason that the beam must have an electric field component normal to the plane of the layers in QCLs is due to the fact that the intersubband photon transitions required for optical gain can only be stimulated by elec tric fields normal to the plane of the active quantum wells within these layers [ 3 ] [6]

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20 In the semiconductor laser literature, lasers of this type are said to have transverse magnetic (TM) polarization because the only magnetic field component in the beam is transverse to the beam direction. If the only electric field component in the beam is parallel to the plane of the layers, the laser is said to have transverse electric (TE) polarization. 1. 4 Dissertation Organization The goal of this work is to di scuss in detail the method(s) used to determine the DFB coupling coefficients for light traveling on the fundamental TM and TE modes of metal/dielectric corrugated waveguides of the type shown in Figure 1-4. In the first part of Chapter 2 and Appendix A, electromagnetic wave theory is used to explain the concept of modes in planar (non-corrugated) multi layer waveguide structures and provide the foundation for the more complex mathematical treatments in later chapters. In the second part of Chapter 2, the ray optics and wave vector pictures of mode propagation in planar guides are introduced and then used to explain the basics of first and second order DFB. In Chapter 3, the planar or zero order DFB model used in Chapter 2 is extended to first and second order DFB models using the Truncated Floquet -Boch Formalism. It is shown that by satisfying the magnetic and electric field tangential boundary conditions at the waveguide interfaces, one obtains a linear algebra problem involving the product of two matri ces. One of these matrices is expressed in terms of the amplitudes of the magnetic field in the various layers. The other matrix is expressed in terms of the coefficients of the magnetic field amplitudes. The derivation of the eigenmode equations associat ed with setting the determinant of the coefficient matrices equal to zero are given in Appendices B and C. In Chapter 4, the computational techniques used to find the DFB coupling coefficients are discussed and

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21 the numerical results for various waveguide configurations presented. In Chapter 5, the model used in Chapter 3 (ModA) is extended to a more exact model (ModB) in which the electromagnetic fields in the metal are no longer assumed to be zero. Comparisons between Mod A and Mod B are shown using various figures Explanations and discussion s about the figures are included Chapter 6 discusses the DFB -QCL shown in Figure 15 and ModB is used to determine the DFB coupling coefficient Th is work is summarized in Chapter 7 and suggestions for future wor k will be discussed

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22 CHAPTER 2 FUNDAMENTAL OF WAVEG UIDES In this chapter, key concepts and terminology about planar and corrugated waveguides will be defined and discussed. The three-layer planar waveguide will serve as an example. Maxwell equations a re used to derive the wave equations for TE and TM fundamental guided modes. The beam intensity profile which relates to Poynting vector will be derived and numerically plotted. The second part of this chapter will discuss the basic concepts of corrugated waveguides. Two special cases: first order distributed feedback and secondorder distributed feedback will be further discussed. 2.1 Planar Waveguide Concepts A basic three -layer semiconductor laser planar waveguide is shown in Figure 21. In general, the optical properties of each layer are specified by a complex refractive index jn where the imaginary part determines the gain or loss coefficient of an electromagnetic wave traveling in that layer. If the real part of an is greater than bn and dn light traveling in the z -direction can be trapped in the vicinity of the active layer by total internal reflection at the a-b, a d interfaces. If the imaginary part of an is such that light traveling in the z direction sees optical gain, the light will be amplified and laser action becomes possible. Figure 21. Basic 3 -layer semiconductor laser one-dimensional waveguide

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23 To actually form a laser beam in a waveguide, there must be a variation in the refractive index in the y -direction (lateral direction) as well as the x direction (transverse direction) shown in Figure 21 One such structure producing a lateral refractive index variation is shown in Figure 2-2 where the stripe layer defines the section of layered material in which the active layer underneath has optical gain. If the effective refractive index above the stripe layer is greater than the effective refractive indices on either side, then light traveling in the z direction will be trapped in the vicinity of the region above the stripe layer as shown in Figure 2-2 Figure 22. Basic twodimensional semiconductor laser waveguide The electric field amplitude associated with the elliptical beam cross section (transverse section) sketched in Figure 22 can be defined in general by a complex vector function ,,xyxyE where xy defines the relative phase of each point on the beam wave front. Since this possible phase variation has no bearing on the concepts to

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24 be discussed, it will be assumed in further discussion that xy is zero and in general that ,,xyxy E = xy E is given by: ,,,, xyzxyExyiExyjExyk E (2 -1a) The corresponding magnetic field function ,,xyxy H = xy H is given by: ,,,, xyzxyHxyiHxyjHxyk H (2 -1b) where i j k are the unit vectors in x, y and z directions. Assuming the laser beam is traveling in the + z direction, the general expressions for the two vector functions defining the whole beam, electric field ,,, xyzt E and magnetic field ,,, xyzt H can be written in the form: ,,,,exp xyztxyizt E=E (2 -2a) ,,,,exp xyztxyizt H=H (2 -2b) As shown in detail in Appendix A, each of four transverse components of the beam functions, ,xExy ,yExy ,xHxy and ,yHxy can be expressed as linear combinations of the spatial derivatives of the longitudinal components, ,zExy and ,zHxy As a consequence, the general transverse field of a beam in a waveguide can be written as a linear combination of a transverse electric (TE) beam and a transverse magnetic (TM) beam. As shown in equations A in Appendix A, the TE beam has in general 4 nonzero terms involving spatia l derivatives of ,zHxy and the TM beam has in general 4 nonzero terms involving spatial derivatives of ,zExy

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25 Experimentally, the beams from semiconductor lasers are either TE or TM. Our major interest, quantum cascade semiconductor lasers (QCLs), always generate TM beams because of the selection rules governing intersubband radiative transitions [ 6 ]. As a consequence, the only nonzero transverse field terms are those involving the spatial derivatives of ,zExy viz. 2 z x tE i E kx (2 -3) 2 z y tE i E ky (2 4) 2 z x tE i H ky (2 5) 2 z y tE i H kx (2 6) The beam cross -section sketched in Figure 22 represents what is called the fundamental TM mode of the waveguide. Although in principle a QCL can oscillate on many TM modes simultaneously, modes other than the fundamental are usually suppressed in such waveguides by making the active layer thickness d and stripe width w sufficiently small. In the remainder of this work, we will assume that the laser mode under discussion is a fundamental TM mode. It is customary to compare the near field measurement at the laser output facet to the calculated timeaverage of the Poynting vector xy S in the propagation or z direction. Since S is a function of E and H its time dependence is sinusoidal as shown in Equations 22a and 22b and the expression for zaveS is [ 7 ]

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26 *,1 ReEH 2zaveSk (2 -7) To the TM case, Equation 2 -7 can be reduced: ** 1 Re 2 0zave xyz xyijk SEEEk HH (2 8) The effective refractive index method shows that computations can be greatly simplified without loss of accur acy by setting 0 y in the above equations representing xE yE xH and yH There are two reasons for this assumption. The aspect ratio of the elliptical beam at the laser output facet is usually 5:1, as shown in Figure 22, so the field variation in y direction is relatively small compared with the variation in x direction. The small effective refractive index, in lateral direction, results in we ak lateral beam confinement. In TM modes, yE in equation 2-4 and xH in equation 2-5 vanish, the only non zero wave components in equation 2-8 are. xE zE and yH 1 Re0 2 00zave x z yijk SEEk H *1 Re 2xy EH (2 -9) Substituting Equations 2 -3 and 2-6 into Equation 29, zaveS becomes: 2 01 Re 2zave yy jS HH n

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27 2 2 01 2y jH n (2 -10) where jn is the refractive index in each layer j To calculate the zaveS in Equation 210, yH needs to be derived in the following. zE can be obtained from Equation A -16 in Appendix A. 1 y zH E ix (2 -11) Take the derivative of Equation 211 2 21 y zH E xix (2 -12) From equation 2-6, replacing zE x with 2 2 yH x from equation (212) to obtain the wave equation in each lay er j for TM modes 2 222 2,0 1 ,00 yj j yjHx kniHx xi (2 -13a) If 0 for each dielectric layer, Equation 213a will become: 2 222 2,0 ,00 yj j yjHx knHx x (2 -13b) Solving the differential Equations 213b for the waveguid e in Figure 2-1, we obtain the following form for yH in each layer [7][20] : ,exp cossin,0 exp0,0d d yja x x b bGqxdxd H HHFkxGkxxd H Fpxx (2 14)

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28 where 2222xaknk 2222 bpnk 2222 dqnk are transverse wave constants. F G bF and dG are constants. In the next section, we will use wave vector diagram to show the relationship between the transverse wave constants and For TM modes, the tangential component of the magnetic field ( yH ) in Equation 214 and the tangential component of the electric field ( zE ) in Equation 211 must be continuous at the following two layer interfaces at: (1) x=0; (2) x=d, as shown in Figure 2 -1. (1) At x = 0, application of the boundary conditions ,,00yb yaHH and ,,00zb zaEE leads to: 0bFF (2 -15) 0x b bak p FG (2 -16) (2) At x = d, application of the boundary conditions ,, ya ydHdHd and ,, za zdEdEd leads to cossin 0xxdkdFkdGG (2 -17) sin cos 0xx xxd aadkk q kdFkdGG (2 -18) Equations 215 to 218 constitute a linear homogeneous matrix system with four variables bF F G dG and can be written in the following matrix:

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29 1100 00 0 0cossin1 0sincosb a x b xx d a xxxx dF pk F G kdkd G kkdkkdq (2 -19) The number of equat ions is equal to the number of variables. To have a nontrivial solution for the four variables bF F G dG the determinant of the 4 x 4 coefficient matrix sh ould be zero. 1100 00 0 0cossin1 0sincosa x b xx a xxxx dpk kdkd kkdkkdq (2 -20) Manipulating Equation 2-20 leads to the following transcendental equation: 2tanaa x db x aa x dbqpk kd kpq (2 -21) It is customary to specify the refraction index of each layer ( jn ) in the se multi -layer structures. As a consequence, the medium permittivity j in Equation 2 -21 is replaced by 2 jn where is the vacuum permittivity 22 22 22 2 22tanaa x db x aa x dbnn qpk nn kd nn kpq nn (2 -22) 0 0

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30 In order to show explicitly t hat only certain values of xk are allowed by Equation 222 thereby giving rise to the modal nature light propagation in waveguides, one uses the trigonometric identities to obtain the following dispersion relationship expressed by t he phase angles and mode number M: 22 11 22tan tan aa x dx bxnn qp kd M nknk (2 23) Take the three-layer waveguide shown in Figure 2 1 for example. The active layer v For this near -inf rared (NIR) wavelength range, the material combination of GaAs for active layer and 1 xxAlGaAs for cladding and buffer layers are commonly used. The buffer layer refractive index is depends of the mole fraction x or the AlAs compostion in 1 xxAlGaAs and this relationship is shown in equation 2 -24 [ 8 ]. 2()3.5900.7100.091bnx xx (2 -24) The GaAs has refractive index about 3.6 and A l As has refractive index close to 3.0. The lower fraction of AlAs will make t he cladding and buffer layer have lower refractive indices The ratio of a bn n become larger, the more energy of propagating wave will be confined in the active layer and the confinement factor will become larger. However, t hi s compound alloy has direct bandgap when the fraction x of AlAs is less than 0.45 and its corresponding refractive index is about 3.3 [ 8 ] Semiconductor with indirect bandgaps are not efficient light emitters for applications [ 9 ]. Other factors and more ad vanced opto-physical effects which will cause the change of refractive index are discussed in [ 10 ] and [ 11]. I n this dissertation, t he refractive indices for

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31 general GaAs/AlGaAs waveguides are an = 3.6 and bn = dn = 3.4. The effective refractive index en is 3.4266 can be numerically solved from Equation 2 -22. Figure 23 shows the schematic plot of 2 2 ,1,0jyjnHx vs. x for TM mo des. Since the active layer has higher refractive index so the profile of the plot will be lower. Figure 23. Schematic plot of 2 2 ,1,0jyjnHx vs. x for the fundamental TM mode From Equation 214, ,0yHx in each layer j is calculated. Figure 2-4 shows the calculated result of 2 2 ,1,0jyjnHx vs. x for the relative quantity pointing vector deriver in Equation 2 -10. Figure 24 is particularly interesting for semiconductors lasers operating in a TM modes since the ,0zSx function is discontinuous due to the different refractive indices in each layer. The measured beam profile of QCLs utilizing TM mode also have such unsmooth at the a-b and a d interfaces [12].

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32 Figure 2 4. Calculated plot of 2 2 ,1,0jyjnHx vs. x for the fundamental TM mode As discussed in the Introduction section, the main interest in this work is to determine the coupling coefficients responsible for laser oscillatio n and output power in corrugated waveguides. In order to understand the behavior of guided waves in waveguides of this type, it is useful to introduce the ray optics or bouncing ray picture of wave propagation. Since the light waves travel in the z direct ion but are trapped in the vicinity of the active layer by total internal reflection at the a-b and ad interfaces, wave propagation can be depicted as shown in Figure 2 -5. The angle associated with the zigzag motion of the light ray is called the mode b ounce angle. -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 x [ m] Normalized Poynting Vector (H2 / n2 ) nIII=3.4 nII=3.6 nI=3.4

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33 Figure 25 shows the bouncing ray in a three-layer waveguide in the fundamental mode. The incident rays come from the left side and propagate in the direction as the arrow indicated. The critical angels [13] at upper interface and lower inter faces are: 1sind d an n (2 -25 ) 1sinb b an n (2 -26 ) If max(,) 2bci the propagating light will be confined inside the active layer by the total internal reflection. This light will propagate along the z direction in a zigzag path as shown in Figure 2-5. 0 is the bounce angle for the incident wave 0P Figure 2 5. Bouncing r ay in a guided -mode waveguide Figure 26 [ 14 ] shows the relationship among the propagation constant in the z direction in the fundamental mode, 0 the effective incident angle 0 and wave vector k The relationships can be expressed as follows: 00sinzaekknkn (2 -2 7a) 222 00cosxaakknnk (2 2 7b)

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34 In the above equation, 02 k is the wave vector in vacuum. 0 is the wavelength in vacuum. en is considered to be the effective refractive index From Figure 26, for a fixed magnitude of incident wave vector ank its longitudinal component, 0 zk and its transverse component, xk will determine the propagation behavior i n this waveguide. The larger the real value of 0 is, the more propagation along the z direction. If the component 0 is too small and xk is large, the waveguide may have less propagat ion in the z direction and radiation out of from either buffer layer or clad later in the x direction, called radiation mode [ 7 ]. Figure 26. Wave propagation in the fundamental mode These two key parameters, mn and will help derive and analyze mathematical models in Chapter 3 and 4.

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35 2.2 Corrugated F our -L ayer W aveguides 2.2.1 Diffractive W aves in C orrugated W aveguides Figure 27 shows the corrugated grating will generate diffracted waves. Under the buff er layer, a corrugated metal layer with refractive index cn is added. The incident wave in terms of power can be expressed as 0P the first order diffractive wave 1P and the secondord er diffractive wave 2P A B Figure 27 Diffraction caused by corrugated waveguide. A) Diffractive waves due to gratings B) R eal picture of diffraction

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36 2.2.2 First O rder D istributed F eedback in C orrugated W aveguides Next, two special diffraction cases in such corrugated waveguides will be discussed. If the first order diffractive wave 1P happens to be the reverse direction of 0P after one backward grating shift, it is called t he first order distributed feedback (DFB). Figure 28 (a) shows the ray picture for this case. A B Figure 28. Fi rst o rder DFB A) Ray optics picture B) Wave vector diagram Figure 28 (b) shows the following relationship. 11 010 02 (2 -2 8) In Equation 2 -2 8, the grating vector 1 12 where 1 is the grating period [ 15 ] for the first order DFB.

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37 Substituting Equation 2-7a into Equation 2-8, the grating period 1 for the fir st order DFB is 12en (2 2 9) Substituting Equation 2-7a into Equation 2-2 8, the bounce angle 1 for the first order DFB has the same magnitude with 0 1 10 010sin sinaakn kn (2 3 0) 2.2. 3 SecondO rder D istributed F eedback in C orrugated W aveguides If the second order diffractive wave 2P happens to be the reverse direction of 0P after two grating shifts, it is called the secondorder dist ributed feedback (DFB). Figure 2 -9 (a) shows the ray picture for this case. Figure 29 (b) shows the following relationship for the secondorder DFB. 22 020 02 (2 31) Substituting Equation 2-27(a) into Equation 231, the bounce angle 2 for the second order DFB has the same magnitude with 0 2 20 020sin sinaakn kn (2 32) Figure 29 shows the following relationship for the first order diffractive waves. This first order feedback waves 1P after one backward grating shift, diffract out perpendicularly to the z direction (i.e. 10 ) so it does not have the propagation component along z direction. 22 01 0sin00akn (2 33)

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38 Equations (23 1) and (233) show the conditions for secondorder DFB match with each other. Substituting Equation (227a) into Equation either 231 or 2-33, the grating period 2 for the secondorder DFB is 2en (2 -34) A B Figure 29. Secondorder DFB. A) Ray optics picture B) Wave vector diagram for the second order diffractive wave C ) Wave vector diagram for the first order diffractive wave Compare Figure 2-8 and Figure 2-9, we can notice the second order DFB will ha ve smaller beam divergence angle [5] than the first order DFB. Equation 2 -15 shows

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39 the divergence angle D is proportional to wavelength and inversely proportional to the beam emission aperture D [ 16 ][ 17]. DD (2 -35) Figure 210 (a) shows first order DFB lasers emit beams from the edge of active layers and directions of beams are normal to the edge cross -section of active layers, also called edgeemitting lasers The emission apertures 1D are equal to the active layers thickness (d) which is usually in microns and about 1,000 times smaller than the emission apertures of substrateemitting lasers. This small aperture will greatly increase th e beam divergence and limit the practical applications in long wavelength operation. A B Figure 210. Schematic diagrams for A) F irst order B) S econd order DFB

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40 Figure 210 (b) shows the second order gratings have smaller beam divergence angle 2 due to its larger emission aperture 2D which is equal to laser length (L), than that 1 of the first order DFB grating [ 5 ]. The second order DFB lasers emit beams from the substrates and the directions of beams are normal to the substrate surfaces, also called substrate emitting lasers. The divergences of beams are inversely proportional to the laser length, which are usually in mini meters in the direction of longitudinal direction. Applications, such as detecting explosives [ 18 ] or drugs [ 19 ], require wavelength 300 m or 0.3 mm with high frequency in terahertz (THz, 10 to the order of 12). For extensive infrared applications such as gas sensing, optical communication [21] night vision, ther mograph/ thermal tracking [22] the required wavelength range is from 0.7 m up to1 mm. Second order DFB lasers can help maintain the beam divergence in a smaller range in different applications. Table 1 summarizes the first order DFB and secondorder DF B Table 2 1 Comparison of the first order DFB and secondorder DFB First order DFB Second order DFB Beam direction Edge emitting Substrate emitting Grating period 12en 2 en Divergence angle 1 11~10 1 m Dm 2 21~10 1000 m Dm

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41 CHAPTER 3 TM EIGENMODE EQUATIO NS FOR PLANAR AND DI STRIBUTED FEEDBACK CORRUGATED SENICONDUCTOR WAVEGUIDES WITH SHINY METAL CONTACTS Floquet -Bloch formalism was used to derive expressions yielding the TE0 mode back metal contacts [Luo 93]. In this chapter, we will derive similar expressions for TM0 modes in the same types of waveguides. As mentioned previously, TM0 modes are of interest sinc e quantum cascade lasers (QCLs) only operate on such modes. 3.1 Floquet Bloch Formalism for Corrugated Waveguides Figure 31 shows a semiconductor laser structure with a perfect metal sinusoidally corrugated contact layer. Figure 31. The shiny contact DFB laser structure TM modes in such structure have their magnetic field pointing in the y direction. The coordinate system chosen has x = 0 at the interface of metal and semiconductor to simplify equations and computations. The Floquet -Bloch formalism re quires that the magnetic field in each layer, cladding layer ( dH ), active layer (aH), buffer layer ( bH ), and metal layer ( cH ), be expanded in a plane wave series as follows:

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42 (,)exp exp dmm m mHxzAqxtdiz (3 -1) (,)cos sin exp a mmmm m mHxzBxtCxtiz (3 -2) (,)exp expexp b mmmmm mHxzDpxEpxiz (3 -3) (,)0 cHxz (3 -4) 002mmm (3 -5) 222 mamnk (3 -6) 222 mmbpnk (3 -7) 222 mmdqnk (3 -8) 2 k (3 -9) In the above equations, is the wave propagation constant, m is the index for the grating diffraction order, is the grating period, is the vacuum wavelength, k is the wave vector and 15 ]. For TM modes, the tangential component of the magnetic field ( yH ) and the tangential component of the electric field ( zE ) must be continuous at the layer interfaces The zE continuous condition can be converted to the condition that yH x is continuous using the following Maxwell Equation [7] : E H t (3 -10) The result for zE is y z jH i E x (3 -11)

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43 where j is the medium permittivity in each layer j and is the circular frequency. Therefore the tangential component boundary conditions are equivalent to saying yH and yH x must be continuous at the following three interfaces: (1) x=t ; (2) x= t+d; (3) x = 0. (1) At x = t, the equations of interest are Equations 3-2 and 33. Application of the boundary conditions leads to: exp expexpexp expexp,mmmmmmm mm mmmmmBizDptEptiz BDptEptm (3 12) exp expexpexp expexp ,mm mm mmmmm mm ab mm mmmmm abp Ciz DptEptiz p CDptEptm (3 -13) (2) At x = t + d, the equations of interest are Equations 31 and 3 -2. Application of the boundary conditions leads to: exp cos sinexp cos sin,mmmmmmm mm mmmmmAizBdCdiz ABdCdm (3 -14) exp sin cosexp sin cos,mm mm mmmmm mmda mm mmmmm daq AizBdCdiz q ABdCdm (3 -15) (3) At x = 0, it is necessary to specify an equation that describe the grating interface. In this case, the equation is:

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44 coscos2 z xfzaKza (3 -16) where 2a is the grating depth. Since the metal contact is perfect metal, bH vanishes at x = 0. Using Equation 3-3, this condition leads to: exp()exp exp0mmmm m mDpfzEpfziz (3 -17) By Fourier series expansion, the exponential terms in Equation (317) become [23] : exp expcos expm m nm npfzpaKzIpaiKzn (3 -18) where nIu is the modified or (hyperbolic) Bessel function of the first kind of order n. This function has the following properties: 1n nn nnIuIu IuIu (3 -19) Substituting Equation 3-18 and 3-19, Equation 3 -17 beco mes: expmnm mnDIpaiKzn 1 expexp0n m nm m nEIpaiKzniz (3 -20) By substituting Equation 35, Equation 320 becomes: (3 -21) expmnm mnDIpaimnKz 01 exp exp0n m nm nEIpaimnKziz

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45 Since the coefficients of exp inKz terms for each n should be zero, we obtain: 1 0,mn mmnm mmnm mDIpaEIpan (3 -22) From the above equations, we can obtain a liner homogeneous matrix system with variables mA mB mC mD mE The number of equa tions is equal to the number of variables. To have a nontrivial solution for this system, the determinant of the coefficient matrix should be zero [ 2 4 ] [2 5 ] In the next few sections, we will discuss three special cases: planar, first order DFB and secondorder DFB waveguides for TM modes In section 3.2, the eigenmode equation for the planar waveguides (grating depth 2a= 0) is derived by setting m=0 in Equations from (312) to (315) and in Equation (322), modified Bessel functions are equal to 1 due to 2a= 0. These five equations form a 5 by 5 matrix. In section 3.3, the eigenmode equation for first order DFB waveguides is derived by setting m= 0 and m= 1 in the above equations. This is valid because all the other m waves are evanescent. In this case, t hese ten equations form a 10 by 10 matrix. In section 3.4, the eigenmode equation for secondorder DFB waveguides is derived by setting m= 0, m= -1, and m= 2 in the above equations. This is valid because all the other m waves are evanescent. In this case, these fifteen equations form a 15 by 15 matrix. 3.2 Derivation of the Eigenmode Equation for Planar Waveguides by Truncated Floquet Bloch Formalism A grating depth 2a = 0 is equivalent to an unperturbed, planar four -layer waveguide with a perfect metal contact. In this case, Equations 313, 3-14, 3-15, 316 and 3 -17 reduce to Equations 323, 324, 3-25, 3-26 and 3-27 respectively: 00000expexp0 BDptEpt (3 -23)

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46 00 0 00000exp exp0ab bpp CDptEpt (3 -24) 00000cossin0 ABdCd (3 -25) 00 0 00000sin cos0da aq ABdCd (3 -26) 0000000 DIEI (3 -27) wh ere 1,0 0 0,0nn I n (3 -28) Next we will manipulate the above 5 equations into a closed-form expression. Equation 326 divided by B3 25 and use trigonometric identity [ 26 ] : 0 0 0000 1 0 00 0 0 00000 0 0 0tan sincos / tan /cossin 1tand aC d BdCd q BC d C BdCd B d B 1 00 00 00/ tantan /d aCq d B (3 -29) Equation 324 divided by 3 -23: 0000 00 0 000000expexp / /expexpb aDptEpt Cp BDptEpt 0 0 00 0 0 0 0 0/ 1 / exp(2) / 1 /a b a bDp pt E p (3 -30) Rearrange Equation 327:

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47 0 000000 00 010 D IDEIE E (3 -31) For a non-trial solution, 000,0 DE from Equation 3-31, we obtain: 0 010 D E (3 -32) Compare Equations 3 30 and 332: 0 0 00 0 0 0 0 0/ 1 / exp(2) 1 / 1 /a b a bDp pt E p 00 00 00/exp(2)1 coth /exp(2)1a bpt pt p pt 0 00 0/ coth /b appt (3 -33) Combine Equations 3-29 and 3 -33: 1 0 00 00 0 0 00// tantan coth //db aaCqp d pt B 1 000 00 0/ cothtantan /d ba apq pt d (3 -34) It is customary to specify the refraction index of each layer ( jn ) in these multi -layer structures. As a consequence, the medium permittivity j in Equation 3 -31 is replaced by 2 jn where is the vacuum permittivity. 0 0

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48 2 1 000 00 22 2 0coth tantana ba dp qn pt d nnn (3 -35a ) Equation 335 can be expressed in the explicit form of mode number N. 22 11 00 00 22 00tan tancothaa dbqnnp d ptN nn (3 -35b) If N=0 in, the waveguide will have s ingle laser beam spot and this is the fundamental mode for this waveguide [ 27 ] For precise applications, we discuss the fundamental mode with single spot in this dissertation. If both 2 21d an n and 2 21a bn n then Equatio n 3-35 will become the form of TE eigenmode equation, as derived in [Luo 90]. However, the derivation process for TE modes is based on electric fields while the derivation process for TM modes is based on magnetic fields. As mentioned at the end of section 3.1, Equations 323 to 327 constitute a linear homogeneous matrix system and can be written in matrix form and the matrix size is 5 by 5. To have an nonzero solution for variables 0A 0B 0C 0D 0E in the above equations, the determinant of the coefficient matrix must be zero. By using matrix expansion, we can obtain the same Equation 335. The details of this manipulation process by using matrix expansion are given in Appendix B. 3.3 Derivation of the Eigenmode Equation for First -order DFB Corrugated Waveguides by Truncated Floquet Bloch Formalism In most practical DFB lasers utilizing first order diffraction, the condition ka<<1 is satisf ied. As a consequence, a truncated Floquet -Bloch formalism [23] is sufficient to compute the backward coupling coefficient ( ). In this case, there are only two types of

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49 traveling waves: the fundamental forward wave (m=0, propagati on constant 0 ) and the backward diffractive wave (m= 1, propagation constant 0 ). All other m waves are evanescent along the z direction, and are neglected in the following calculation. The first -order DFB co ndition implies the following: 0002 (3 -33) As mentioned previously, ten equations are needed to describe first order DFB. Five equations are for m = 0, and five equations are for m= 1. Four of the five m = 0 equations are Equations 3 23 to 326 and four of the 5 m = -1 equations are Equations 3 -34 to 3-37. Ten equations are listed together in the following for the completeness of this first order DFB model: 00000expexp0 BDptEpt (3 -23) 11111exp exp0 BDptEpt (3 -3 4) 00 0 00000exp exp0ab bpp CDptEpt (3 -24) 11 1 11111exp exp0ab bpp CDptEpt (3 -35) 00000cossin0 ABdCd (3 -25) 11111cos sin0 ABdCd (3 -36) 00 0 00000sin cos0da aq ABdCd (3 -26) 11 1 11111sin cos0da aq ABdCd (3 -37)

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50 Since the Bessel functions of small argument are fast decaying with increasing order, it is sufficient to choose just the n = 0 and n = -1 for Equation 3 -22. 000000 1111110 DIpaEIpaDIpaEIpa (3 -38) 010010 101 1010 DIpaEIpaDIpaEIpa (3 -39) Following the procedure in section 3.2, the above ten linear homogeneous equations will only have a non-trivial solution if the determinant of the coefficient matrix is zero. Manipulating the determinant leads to the following eigenmode equation: 0001 01 1011IpaIpa IpaIpa (3 -40) where 22 11 221 tanhtanhtantanhi ia id ii ib iapn n d pt n qn (3 -41) The details of the manipulation process are given in Appendix C. If both 2 21d an n and 2 21a bn n then Equation 3-41 will become the TE eigenmode equation, as derived in [ 23 ]. Since we consider shallow grooves the Bessel functions in Equation 3-40 can be approximated as the following polynomials: 0 11 2 Iu u Iu u<<1 (3 -42) 3.4 Derivation of the Eigenmode Equation for Second-order DFB Corrugated Waveguides by Truncated Floquet Bloch Formalism In most practical DFB lasers utilizing secondorder diffraction, the condition of shallow groove, ka<<1, is satisfied. Consequently, a truncated Floquet -Bloch formalism

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51 is sufficient to compute the backward coupling coefficient ( ). In this case, there are three types of traveling waves: the fundamental forward wave (m=0, propagation constant 0 ), the upward surface emitting wave (m= 1, propagation constant = 0) and the backward diffractive wave (m= 2, propagat ion constant 0 ). All other higher order m waves are evanescent along the z -direction, and are neglected in the following calculation. The secondorder DFB condition implies the following: 0002 (3 -43) With the similar procedures discussed in Section 3.3, fifteen linear homogeneous equations are needed to describe this secondorder DFB. Five equations are for m = 0, five equations are for m= 1 and five equations are for m= -2. Four of the five m = 0 equations are Equations 3 23 to 326, four of the five m = 1 equations are Equations 3 34 to 337 and four of the five m = -2 equations are 344 to 347. Fifteen equations are listed together in the following for the completeness of this second order DFB model: 00000expexp0 BDptEpt (3 -23) 11111exp exp0 BDptEpt (3 -34) 22222exp exp0 BDptEpt (3 -44) 00 0 00000exp exp0ab bpp CDptEpt (3 -24) 11 1 11111exp exp0ab bpp CDptEpt (3 -35) 22 2 22222exp exp0ab bpp CDptEpt (3 -45)

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52 00000cossin0 ABdCd (3 -25) 11111cos sin0 ABdCd (3 -36) 22222cos sin0 ABdCd (3 -46) 00 0 00000sin cos0da aq ABdCd (3 -26) 11 1 11111sin cos0da aq ABdCd (3 -37) 22 2 22222sin cos0da aq ABdCd (3 -47) Since the Bessel functions of small argument are fast decaying wi th increasing order, it is sufficient to choose just the n = 0, n = 1 and n= 2 for Equation 322. 000000 111111DIpaEIpaDIpaEIpa 2222220 DIpaEIpa (3 -48) 010010 101 101DIpaEIpaDIpaEIpa 2122120 DIpaEIpa (3 -49) 020020 111 111DIpaEIpaDIpaEIpa 202 2020 DIpaEIpa (3 -50) These fifteen equations form a 15 by 15 matrix and leads to the following eigenmode equation: 0100022022IpaIpaIpaIpaIpa 1111002220IpaIpaIpaIpa

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53 1200202IpaIpaIpa (3 -51) where i is shown in equation (341) and i 0, -1, -2. The details of the manipulation process are much more vigorous but similar to Appendix C If both 2 21d an n and 2 21a bn n then Equation 3-51 will b ecome the TE eigenmode equation, as derived in [ 23 ]. Equation 351 is a neat closed-form expression for second order DFB simulation. Since we consider shallow grooves, the Bessel functions in Equation 3-51 can be approximated as the following polynomials: 2 0 1 2 21 4 2 8 u Iu u Iu u Iu u < <1 (3 -52)

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54 CHAPTER 4 CALCULATION METHODOLOGY AND RESULTS 4 .1 Methodology In modeling corrugated DFB semiconductor lasers, the backward diffraction coupling coefficient kappa ( ) replaces the end mir ror reflectivity used to determine the threshold current requirement in conventional lasers. Figure 4-1 shows waveguide with corrugated grating. The incident power (P) comes into and propagates along this corrugated structure. During this process, the inc ident power will generate transmitted power (TP) and reflected power (RP) with reflection coefficient 2tanh RL [ 28]. Fig ure 4 1. The coupling coefficient ( ) and reflection For dielectric/dielectric gratings, coupled mode perturbation theory is normally used to calculate using the overlap integral of the product of the square of the index perturbation n with the square of the electric field mode function E(x) : 22()() nxExdx [ 29 ]. However, this method cannot be used for metal/dielectric gratings because of the very large index perturbation at the metal/dielectric interface and the uncertainty on how to choose the unperturbed waveguide [ 30] [31 ]. In addition, since the electric field is essentially zero in the metal, the overlap concept cannot even

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55 be defined. Figure 4-2 shows a method that can be used to calculate for metal/dielectric gratings is called the Floquet -Bloc h formalism [ 2 3]. Figure 4 2. Floquet -Bloch formalism for calculating of metal/dielectric corrugated structures In the Floquet -Bloch formalism, one expands the eigenmode solutions to the waveguide equation in each layer of the waveguide into an infinity numbers of forward going plane and backwardgoing plane waves. Since most of the waves in the expansion are evanescent and decay rapidly in the wave direction, one only needs to consider the traveling wave terms to get good acc uracy for in the DFB problem. This approach is called truncated Floquet Bloch formalism (TFBF), by choosing the most significant terms and neglecting the fast -decaying terms in order to have a well balanced tradeoff of consideri ng between effectiveness and efficiency during modeling process. This approach has been used to construct models for the fundamental transverse electric waveguide mode (TE0) in planar, first order DFB and secondorder DFB waveguides [ 2 3]. In this work, t he truncated Floquet -Bloch formalism is extended to the more complex transverse magnetic waveguide mode (TM0) problem since QCLs only lase on this type of mode [ 3 ].

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56 The procedures of truncated Floquet -Bloch formalism include two major parts. The first pa rt is to u se Maxwells equations to derive a wave equation for the transverse magnetic field in a planar metal -clad waveguide This part includes: solving the wave equation and obtaining the solutions in each layer, apply ing boundary conditions at the lay er interfaces and finally d etermine the eigenmode equation and solve for the propagation constant 0 for the fundamental TM mode ( 0TM ). The second part is to c onvert the planar waveguide into a corrugated w aveguide by applying a sinusoidal corrugation at the metal dielectric interface This part includes: u sing the 0TM mode of the planar waveguide problem to form an infinite plane wave expansion for the transverse magnetic field in each l ayer of the corrugated waveguide, s elect the type of DFB (first order or second order), k eep ing the terms in the plane wave expansion associated with the type of DFB and ignore the evanescent wave terms a pply ing boundary conditions at the layer -interfaces and obtain a set of linear homogeneous equations where the coefficients are functions of using linear algebra to transform the equation set into a coefficient matrix describing this corrugated waveguide system using the coeff icient matrix equation to derive a transcendental equation for and finally solving the coefficient matrix equations numerically to find 0 i where is the imaginary part of conjugat ed complex propagation constant and is also the DFB coupling coefficient Figure 43 shows the dispersion diagram for DFB corrugated waveguides. The period of the DFB corrugate d waveguide introduces the stop band at the Bragg frequency [23]. Then the propagation constant becomes complex and has a

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57 imaginary part. This concept is similar to energy wavevector (E k) diagram [ 32 ] for the electron energy states in the periodic lattice of semiconductors [33 ] Fig ure 4 3. dispersion diagram for DFB This stop band of this dispersion curve can show the interaction between the periodic corrugation and optical mode. The magnitude of this stop band has the following relationship between and by coupled mode theory [ 34]: 2en c (4 -1) where c is the velocity of light in vacuum and en is the effective refractive index for waveguides. Coupling coefficient in Equation 4 1 can be expressed by the imaginary part of complex propagation constant for the corrugated waveguides:

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58 12 0011111 222 2222 11 Im ImIm 22e e een f f n nkn cc (4 -2) where 12 fff 11 fc and 22 fc Coupling coefficient in Equation 4 2 can be expressed by the function of wavelengths: 21 12 12 2 2 2111 22 2 2 22 1 4 ee e eenn n nn (4 -3) where 21 22 and 12 4.2 Computation In Chapter 3, we developed mathematical models for planar waveguides and corrugated waveguides. Figure 44 shows the flow chart to compute c oupl ing c oefficient Flow chart is a useful tool [ 35] to organize the logic of computer codes to iteratively search the complex propagation constant and the complicated optical coupling calculation. M ATLAB sof tware [ 36] with numerical concepts [ 37 ] is used to write the computer codes to find the complex propagation constant and plot the figures The imaginary part of this complex propagation constant is the coupling coefficient.

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59 Fi gure 4 4. Flow chart to compute c oupling c oefficient Figure 45 shows an example of plotting the value of the determinant of coefficient matrix or the eigenmode equation 341 versus complex propagation constant for a corrugated waveguides with normalized corrugation amplitude ka=0.5 on TE mode.

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60 The v alue on the z axis equal to zero will be the solution for the eigenmode equation341. An algorithm to search a solution in a zigzag way on complex plan of propagation constant is not very efficient. Figure 45 can visualize how the equations varied with the propagation constant so that we can narrow down to the possible neighborhood where the solution point is. Fi gure 4 .5 An example showing the value of eigenmode equations versus propagation constant when ka=0.5 Figure 46 shows the zoom -in plot for t he point with zero value on the z axis in Figure 45. In Figure 4-6, the x z plan shows the eigenmode equation has zero value when the real part of complex propagation constant has the value 25.3109. Figure 46,

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61 the y z plan shows the eigenmode equation h as zero value when the imaginary part of complex propagation constant has the value 0.0146 [1/ m] or 146 [1/cm]. The x y plane is the complex plan for propagation constant. Fi gure 4 6. Solution s earch to compute c oupling c oefficient when ka=0.5

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62 4.3 Numerical Results and Discussions Figure 4 7 shows the coupling coefficient versus the corrugation amplitude for 1st order DFB for 0TM and 0TE modes The curve with circle in this figure is named as TM 1 (ModA), which means TM mode first order DFB (Model A by using models in Chapter 3). The waveguide use the following parameters : 850 nm active layer thickness 100 dnm buffer layer thickness 300 tnm For lasers made of GaAs/AlGaAs materials, they have typical values of active layer refractive index 3.6an and buffer/cladding layer refractive index 3.4bdnn Larger corrugation amplitude will cause l arger perturbation so the coupling effect will become larger. The TM mode will have more interaction with corrugated metal so its coupling coefficient is larger than TE coupling coefficient. Fi gure 4 7. C oupling c oefficient vs c orrugation a mplitude (a) 0 20 40 60 80 100 120 140 0 50 100 150 200 250 300 Corrugation Amplitude (a) [nm]Coupling Coefficient ( ) [1/cm] TM-1 (ModA) TE-1 (ModA)

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63 Figure 4 8 shows the coupling coefficient versus the buffer layer thickness for 1st order DFB for 0TM and 0TE modes The waveguides use the following parameters : the normalized corrugation amplitude ka=0.2, 850 nm active layer thickness 100 dnm the refractive indices are 3.6an and 3.4bdnn If we reduce the buffer thickness, the mode will interact with corrug ation more and the coupling coefficient would become larger. After reaching the maximum coupling coefficient, there is a roll over due to the mismatch of the mode. After the roller over the curves will have a cut off point on their left -most end when the e ffective refractive index reaches 3.4 in this case. Fi gure 4 8. C oupling c oefficient vs. buffer thickness (t ) 0 100 200 300 400 500 600 101 102 Buffer Thickness (t) [nm]Coupling Coefficient ( ) [1/cm] TM-1 (ModA) TE-1 (ModA)

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64 Figure 4 8 shows the coupling coefficient versus the active layer thickness for 1st order DFB for 0TM and 0TE modes The waveguides use the following parameters : the normalized corrugation amplitude ka=0.2, 850 nm active layer thickness 100 dnm buffer layer thickness 300 tnm The refractive indices are 3.6an and 3.4bdnn The curves in Figure 4-9 for active layer thickness variation has similar trend with the curve in Figure 4-8 for buffer layer variation. Fi gure 4 9. C oupling c oefficient vs. active layer thickness (t ) 0 50 100 150 200 250 300 0 10 20 30 40 50 60 Active Layer Thickness (d) [nm]Coupling Coefficient ( ) [1/cm] TM-1 (ModA) TE-1 (ModA)

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65 CHAPTER 5 MODIFED MODELS CONSIDERING REAL METALS 5.1 Full Floquet Bloch Formalism for Real Metals In Chapter 3, we assume the magnetic field is zero inside the perfect metal layer. To be more precisely model different real metal, the fields inside the metal is not zero. Figure 51 shows a semiconductor laser structure with a perfect metal sinusoidally corrugated contact layer. The metal layer has a complex refractive index cn Figure 51. The DFB laser structure with real metal contact TM modes in such structure have their magnetic field pointing in the y direction. The coordinate system chosen has x = 0 at the interface of metal and semiconductor to simplify equations and computations. The Floquet -Bloch formalism requires that the magnetic field in each layer, cladding layer ( dH ), active layer (aH), buffer layer ( bH ), and metal layer ( cH ), be expanded in a plane wave series as follows: (,)exp exp dmm m mHxzAqxtdiz (5 -1) (,)cos sin exp a mmmm m mHxzBxtCxtiz (5 -2) (,)exp expexp b mmmmm mHxzDpxEpxiz (5 -3)

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66 (,)exp expcmmm mHxzFqcxiz (5 -4) where 002mmm (5 -5) 222 mamnk (5 -6) 222 mmbpnk (5 -7) 222 mmdqnk (5 -8) 222 mmcqcnk (5 -9) 2 k (5 -10) In the above equations parameters are defined in Chapter 3, is th e wave propagation constant, m is the index for the grating diffraction order, is the grating period, is the vacuum wavelength, k is the wave vector and grating vector. The field is not zero for real metals [38][39]. For TM modes, the tangential component of the magnetic field ( yH ) and the tangential component of the electric field ( zE ) must be continuous at the layer interfaces. The zE continuous condition can be converted to the condition that yH x is continuous using the following Maxwell Equation: E H t (5 -10) The r esult for zE is y z jH i E x (5 -11) where j is the medium permittivity in each layer j and is the circular frequency.

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67 Therefore the tangential component boundary conditions are equivalent to saying yH and yH x must be continuous at the following three interfaces: (1) x=t ; (2) x= t+d; (3) x = 0. By using the similar manipulation process to satisfy the above boundary conditi on, we can obtain a liner homogeneous matrix system with variables: mA mB mC mD, mE mF The number of equations is equal to the number of variables. To have a nontrivial solution for this system, the determinant of the coefficient matrix for magnetic field should be zero. W e will discuss three special cases: planar, first order DFB and secondorder DFB waveguides for TM modes T he eigenmode equation for the planar waveguides (grating depth 2a= 0) is derived by setting m=0 in Equations 5 -1 to 5 10. These six equations form a 6 by 6 matrix as shown in Appendix D T he eigenmode equation for first order DFB waveguide s is derived by setting m= 0 and m= -1 in the above equations. This is valid because all the other m waves are evanescent. In this case, these t welve equations form a 12 by 12 matrix as shown in Appendix E T he e igenmode equation for second order DFB wave guides is derived by setting m= 0, m= -1, and m= 2 in the above equations. This is valid because all the other m waves are evanescent. In this case, these eighteen equations form a 18 by 18 matrix. In the next section, we will also compare results (curves ) of TM with results of TE. To derive TE models, electric fields are used in planar wave expression by using TFBT [23]. However, [23] did not consider the electric field by using TFBT. In this work, we consider real metals have decay electric fields from t he corrugation interface [38] Then, the matrix size for TE modes would become 6 by 6 for planar, 12 by 12 for first order DFB, 18 by 18 for second order waveguides.

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68 5.2 Numerical Results and Discussions For convenience, we will use Model B (ModB) for t he models in Chapter 5 using the nonzero fields inside the metal layer. In Chapter 3, the models are named Model A (Mod A). Figure 5 2 shows the coupling coefficient versus the corrugation amplitude for first -order DFB for 0TM and 0TE modes The curve with diamond points in this figure is named as TM -1 (ModB), which means TM modefirst order DFB (Model B by using models in Chapter 5). The waveguide use the following parameters : 850 nm active layer thickness 100 dnm buffer layer thickness 300 tnm for GaAs/AlGaAs materials, active layer refractive index 3.6an and buffer/cladding layer refractive index 3.4bdnn Large r corrugation amplitude will cause larger perturbation so the coupling effect will become larger. The TM mode will have more interaction with corrugated metal so its coupling coefficient is larger than TE coupling coefficient. The coupling coefficient of T M -1 (ModB) is larger than TM -1 (ModA). Because TM Model B consider the nonzero magnetic field at the corrugated interface between the dielectric buffer layer and metal layer so the interaction at this interface is bigger than TM Model A which approximate the magnetic field inside the metal as zero. For the curve TE 1 (ModA) means the electric field inside the metal is zero but it is the exact situation to perfect metal. The curve named TE -1 (ModB) considers the real metal with electric field inside the metal. Since the real metal has lower diffraction efficiency at this corrugated interface, the field interaction at this interface is smaller than the perfect metal with higher diffraction efficiency. Thus, the coupling coefficient of TE -1 (ModA) is larger than TE -1 (ModB).

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69 Fi gure 5 2. C oupling c oefficient vs. c orrugation a mplitude (a) Figure 5 3 shows the coupling coefficient versus the buffer layer thickness for 1st order DFB for Model A and Model B, operating on 0TM modes The waveguides use the following parameters : the normalized corrugation amplitude ka=0.2, 850 nm active layer thickness 100 dnm the refractive indices are 3.6an and 3.4bdnn If we reduce the buffer thickness, the mode will interact with corrugation more and the coupling coefficient would become larger. After reaching the maximum coupling coefficient there is a roll over due to the mismatch of the mode. After the roller over the curves will have a cut off point on their left -most end when the effective refractive index reaches 3.4 in this case. 0 20 40 60 80 100 120 140 0 50 100 150 200 250 300 350 400 450 Corrugation Amplitude (a) [nm]Coupling Coefficient ( ) [1/cm] TM-1 (ModB) TM-1 (ModA) TE-1 (ModA) TE-1 (ModB)

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70 The coupling coefficient outside the roll over region of TM -1 (ModB) is larger than the coupling coefficient of TM -1 (ModA). Because TM Model B consider the non-zero magnetic field at the corrugated interface between the dielectric buffer layer and metal layer so the interaction at this interface is bigger than TM Model A which approximate the magnetic field inside the metal as zero. ModA and B has close value of maximum coupling coefficient. Coupling coefficient calculated by ModA and ModB have close change rate with respect to the buffer thickness Fi gure 5 3. C oupling c oefficient vs. b uffer thickness (t ) Figure 5 4 shows the coupling coefficient versus the buffer layer thickness for 1st order DFB for Model B, operating on 0TM and 0TE modes The waveguides use the 0 100 200 300 400 500 600 0 20 40 60 80 100 120 140 160 180 Buffer Thickness (t) [nm]Coupling Coefficient ( ) [1/cm] TM-1 (ModB) TM-1 (ModA)

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71 following parameters : the nor malized corrugation amplitude ka=0.2, 850 nm active layer thickness 100 dnm the refractive indices are: 3.6an and 3.4bdnn Fi gure 5 4. C oupling c oefficient vs. buffer thickness (t ) Figure 5 5 shows the coupling coefficient versus the active layer thickness for 1st order DFB for Model A and B, operating on 0TM modes The waveguides use the following parameters : the normalized co rrugation amplitude ka=0.2, 850 nm active layer thickness 100 dnm buffer layer thickness 300 tnm The refractive indices are 3.6an and 3.4bdnn 0 100 200 300 400 500 600 0 20 40 60 80 100 120 140 160 180 Buffer Thickness (t) [nm]Coupling Coefficient ( ) [1/cm] TM-1 (ModB) TE-1 (ModB)

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72 The curves i n Figure 5-5 for active layer thickness variation has similar trend with the curve in Figure 5-3 for buffer layer variation. Coupling coefficient calculated by ModA and ModB have close change rate with respect to the active layer thickness. However, the ma ximum coupling coefficient calculated by ModB are larger than that by ModA. Fi gure 5 5. C oupling c oefficient vs. active layer thickness (t ) Figure 5 6 shows the coupling coefficient versus the active layer thickness for 1st or der DFB for Model B, operating on 0TM and 0TE modes The waveguides use the following parameters : the normalized corrugation amplitude ka=0.2, 850 nm active layer thickness 100 dnm buffer layer thickness 300 tnm The refractive indices are 3.6an and 3.4bdnn 0 50 100 150 200 250 300 0 10 20 30 40 50 60 70 80 90 100 Active Layer Thickness (d) [nm]Coupling Coefficient ( ) [1/cm] TM-1 (ModB) TM-1 (ModA)

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73 The curves in Figure 5-6 for active layer thickness variation has similar trend with the curve in Figure 5-4 for buffer layer variation. However, the maximum coupling coefficient calculated by TM 1 ModB is larger than TE 1 ModA. Fi gure 5 6. C oupling c oefficient vs. active layer thickness (t ) From Figure 52 to Figure 5 -6, versus geometric parameters (a, t, d) in ModB are discussed. Next, versus a physical parameter will be discussed. Figure 5 7 s hows the plot of coupling coefficient versus corrugation amplitude for different metal material at the corrugated interface by using TM -1 (ModB). The perfect metals theoretically have a complex refractive index i However, i n the real world there is no such material. However, gold has the refractive index 0.165.3 i is considered to be close to a perfect metal. We try another imaginary metal, refractive index 12.8 i closer 0 50 100 150 200 250 300 0 10 20 30 40 50 60 70 80 90 100 Active Layer Thickness (d) [nm]Coupling Coefficient ( ) [1/cm] TM-1 (ModB) TE-1 (ModB)

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74 to a perfect metal than gold. Another imaginary metal, refractive index 20.1 i is less a perfect metal than gold is and should be considered more like a dielectric material instead of a metal In TM cases, the magnetic fields have big amplitude outside the perfect metal ; while TE case, the electric fields are almost zero just outside the per fect metal. In TM case, this big amplitude will cause big interaction at the interface and cause larger coupling coefficient if the metal at the grating is more like a perfect metal. T he waveguide with metal of refractive index 12.8 i h as a larger coupling coefficient than the waveguide with gold. When the corrugation amplitude becomes larger, this metal effect and interaction will become even larger and the coupling coefficient curve becomes super -linear. Vice versa, the waveguide with metal layer replaced by a material of refractive index 20.1 i has a smaller coupling coefficient than the waveguide with gold. Its coupling coefficient curve is smaller than the curve f rom TM1 (ModA) which consider s such metal effect t o be very small. Figure 58 shows the plot of coupling coefficient versus corrugation amplitude for different metal material at the corrugated interface by using TE -1. The perfect metals theoretically have the complex refractive index i .Gold with a refractive index 0.165.3 i is considered to be close to a perfect metal. The metal with refractive index 12.8 i is closer to a perfect metal than gold is. The waveguide with metal of refractive i ndex 12.8 i has a larger coupling coefficient than the waveguide with gold. In TE case, the perfect metal has zero electric field inside the perfect metal. Other imperfect metals can allow electric field penetrate into the metal surface and this penetrate depth is call skin depth [ 38 ]. The less perfect metals will have worse diffraction efficiency [Luo 90] and the coupling coefficient will be smaller.

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75 Fi gure 5 7. C oupling c oefficient vs. corrugation amplitu de ( a ) for different metal s Fi gure 5 8. C oupling c oefficient vs. corrugation amplitude ( a ) for different metals 0 20 40 60 80 100 120 140 0 100 200 300 400 500 600 700 Corrugation Amplitude (a) [nm]Coupling Coefficient ( ) [1/cm] TM-1 (ModB), 0-12.8i Linear baseline for -12.8i TM-1 (ModB), gold (0.16-5.3i) TM-1 (ModA) TM-1 (ModB), 2-0.1i 0 20 40 60 80 100 120 140 0 50 100 150 200 250 300 Corrugation Amplitude (a) [nm]Coupling Coefficient ( ) [1/cm] TE-1 (ModA) TE-1 (ModB), 0-12.8i TE-1 (ModB), gold (0.16-5.3i)

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76 CHAPTER 6 SUBSTRATE-EMITTING DFB -QCL 6.1 Multi-L ayer Structure of DFB QCL Figure 61(a) is reproduced from Figur e 15 and Figure 61(b) shows a scanning electron micrograph (SEM) picture of the corrugated QCL material prior to device fabrication [5] A B Fi gure 6 1. Substrate emitting DFB -QCL [ 5 ] A) S ketch showing laser beam emission from substrate B ) SEM picture showing corrugation prior to device fabrication T he corrugation or grating shown in Figure 6 -1 is fabricated on top of an epitaxial layer structure grown on an InP substrate by a crystal growth technique usually referred to as M OCVD. In Figure 6-2, the relatively thick layers of this epitaxial structure are shown with the InP substrate (not shown) at the top of the figure. The active region in the middle of the structure is composed of 30 stages and each of those stages is a mu ltilayer structure containing typically about 20 very thin layers. One of these very thin layers in each stage is a quantum well for electrons and its in these layers where optical gain is created by stimulated emission between energy states in the conduction band.

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77 Fi gure 6 2. Schematic waveguide structure (zoom -in cross-sectional view at the eight layer waveguide along the longitudinal z direction) from the substrateemitting DFB -QCL in Figure 6 1 As mentioned in Chapter 1, the main reasons why the DFB -QCLs made from this material did not operate at room temperature are likely to be insufficient gain from the 30 QWs in the active region and/or a too small value of TM 2. Because of the past experience in this lab in calculating TE -1 values in metal -clad, corrugated, we decided to study the possible TM -2 problem. In this case, the detailed structure of the active region can be ignored and the only layer parameters required are active layer thickness (d) and its refractive index an As shown in Figure 62, there are 8 layers that we need to consider when computing the DFB coupling coefficients. In reality, the bottom layer called the Ti -Au, bi metal layer is actually composed of two layers; a thin titanium layer with complex refractive index 6.36.60633Tini for optical wavelength 5.1 m and thickness

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78 10Titnm and a much thicker gold layer with 3.736.0849Auni and thickness 100nm. The thin Ti -layer is normally used in fabricating semiconductor lasers because it adheres better than gold to semiconductor materials like InP and GaAs. As shown in the previous chapters, when the Floquet -Blo ch Formalism is used to solve waveguide problems, the introduction of each new layer into the structure creates two more infinite summations and correspondingly bigger matrices and determinants. In this case, the problem can be solved by replacing the 8l ayer with a 4layer structure that has the same complex effective refractive index as the 8 layer structure and the same laser mode intensity profile. In this case, the waveguide parameters for the replacement guide are: 5.1, m 3.38,an 3.10,bdnn 4.027.5975,cni 1.5, dm 1.7, tm and the complex effective refractive index for both guides is 3.20.0003eni The replacement 4-layer guide is shown in Fi g ure 63 Fi gure 6 3. Schematic approximated four layer waveguide structure for the eight -layer waveguide 6.2 DFB Coupling Coefficients The parameters associated with the corrugation used in [ L ya07] and sketched in Figure 63 are: 5.13.2471.57,enmm corrugation amplitude 0.3 am

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79 and its normalized corrugation amplitude is 25.10.30.37 ka The first order coupling coefficient is about 112[] cm and secondorder coupling coefficient is about 13[] cm From the design criteria 1 L based on coupledmode theory [ 40 ] the longitudinal device length is about 3.3 mm. The device length is 2.5mm [ 5 ]. Figure 6 -4 shows the coupling coefficient v ersus corrugation amplitude (a) by using TM Model B. The first order coupling coefficient is about 116[] cm for normalized corrugation amplitude ka=0.5. Fi gure 6 4. Coupling c oefficient vs. c orrugation a mplitude (a) for simpli fied 4-layer waveguide 0 50 100 150 200 250 300 350 400 450 0 2 4 6 8 10 12 14 16 Corrugation Amplitude (a) [nm]Coupling Coefficient ( ) [1/cm] TM-1 (ModB) TM-2 (ModB)

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80 CHAPTER 7 CONCLUSION In this dissertation, we start from the introduction of semiconductor history in Chapter 1. Chapter 2 with Appendix A introduces the fundamental concepts of optical waveguides needed to establish our models. C hapter 3 demonstrat es the modeling process and Chapter 4 shows the numerical results of DFB semiconductor waveguides with corrugated metal contacts operating on fundamental TM modes Chapter 5 modifies the above modes and develops more universal models considering the real metals, operating on TE and TM modes. Our calculation results in Chapter 4 and 5 are compared with labs previous work [2 3] and have reasonable results to valid ate our models. Then we apply to our model to our labs previous D FB -QC L, operating on longer wavelength [ 5 ] In Chapter 3, although Model A provides a quick approximation, it has limits and only provides better approximation for waveguide with smaller corrugation amplitude, thick buffer or thick active layer. These thick -layer cas es have smaller coupling coefficients but have closer value to Model B in Chapter 5. Model B can deal with wider cases. If the buffer thickness is decreased, the coupling coefficient will increase. This thin buffer will also help heat dissipation from acti ve layer to metal contact, which serve s as the heat sink for hot active region in laser waveguides. Besides, Model B can consider different real metals with higher accuracy. The thinner buffer layer finally will reach a peak with maximum coupling coefficie nt. After that peak, the thinner buffer layer will have a deep roll over and reach a cut off point quickly due to the mismatch of the mode interaction with the corrugation. We should avoid this roll over region in design. Coupling coefficient has similar r esponse to the variation of active layer.

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81 The coupling coefficient is more sensitive to the variation of corrugation amplitude than the variation of buffer or active layer as shown in T able F -1 in Appendix F Bigger corrugation amplitude will cause direct and stronger interaction at the corrugated interface However, the layer thickness cause s the mode profile change and then the tail of mode profile indirectly change s the interaction at the corrugation interface. The TM waveguides usually have bigger coupling coefficient due to the stronger field interaction at the corrugation interface than TE waveguides. Appendix G shows the difference of field interaction at the dielectric -metal interface for planar waveguides between TM and TE modes Although thi s TM field interaction increase seems to be small compared with the peak of the entire field profile, this interaction increase cause s the significant change of coupling coefficient since the coupling coeffici ent is related to the imaginary part of complex propagation constants U sua lly the imaginary part of the propagation constant is much smaller than the real part of the propagation constant. Thus, t he small amount of interaction change in the interface will cause the small amount but comparatively large percent age of change in coupling coefficient. However, the TM has thicker cut off buffer layer so the heat dissipation to metal may become a problem. Lasers using multi layer in the active region will have bigger thickness in such active region. The TM curves of describing the relati onship between coupling coefficient and active layer thickness usually is the right -shit with respect to the TE curves. Such TM curves usually have larger thickness when the maximum coupling happens and when the cut off happens. The secondorder DFB may have smaller coupling coefficient than the first order DFB does. The second order DFB has looser grating period, so the interaction at this

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82 corrugation would be not as intense as the first order DFB. The secondorder DFB also has first order diffracted waves which carry some mode and emit away from the corrugation interface, so the coupling coefficient would be decreased. In Chapter 6, we discuss the design consideration based on fabrication and material and then apply our models to calculate the DFB -QCL. The coupling coefficient of this DFB -QCL is comparatively smaller than the modeling case in Chapter 5. This DFB -QCL have longer wavelength 5.1 m than the 0.85 m cases in Chapter 5. Considering the grating period about 1/6 of the wavelength for the first order DFB and 1/3 of the wavelength for the second DFB, the grating period of MidIR DFB -QCL is much larger than that of near -IR lasers so the interaction at the corrugated grating would be weaker and the couplin g coefficient would be smaller One way to increase the coupling coefficient is to increase the corrugation amplitude to compensate the density of grating. The other factor of decreasing the coupling coefficient is that this DFB -QCL active layer thickness is very large but the active layer thickness is needed for the multiple quantum wells. Due to the complexity of material control and optical loss of material this calculated coupling coefficient may be theoretically larger than the actual coupling coeffic ient of the real devices.

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83 APPENDIX A MAXWELLS EQUATIONS AND GUIDED ELECTROMA GNETIC WAVES In this appendix it is shown that, in general, an electromagnetic wave whose beam shape is independent of propagation direction (guided wave) can be specified by a li near combination of a transverse electric (TE) vector beam function and a transverse magnetic (TM) vector beam function. Assume an electromagnetic wave is traveling in a medium characterized by permeability dielectric constant and conductivity If the net charge density in the medium is zero, Maxwells equations for the electric field ,,, xyzt EE and magnetic field ,,, xyzt HH are [ 41 ]: 0 E (A-1) 0 H (A-2) t H E (A-3) t E HE (A-4) If the wave travels in the +z -direction and has a finite extent in the x y plan e that is independent of z (a guided wave traveling in the +z -direction) then E and H can be expressed in the following general form [ 41 ]: ,exp xyizt E=E (A-5) ,exp xyizt H=H (A-6) where is the propagation constant (longitudinal wave constant), is the circular frequency and the vector beam functions xy E and xy H are given by:

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84 ,,,, xyzxyzxyExyiExyjExykEiEjEk E (A-7) ,,,, xyzxyzxyHxyiHxyjHxykHiHjHk H (A-8) In equations A 7 and A 8, the symbols ij and k represent unit vectors in the x,y and z directions respectively. It should be noted while the v ector beam functions xy E and xy H defined in equations A 7 and A 8 are independent of z as required by the guided wave assumption, they do in general have a k vector or longitudinal component. In other words, w e are not assuming here that the guided waves are transverse. Note also that bold, capitalized symbols in this appendix represent vector functions while their nonvector aspects are represented by non -bold, capitalized-italicized symbols. Substituting the electric field wave expressions in equations A -5 and A -7 into Maxwells electric field divergence equation A -1 leads to: 0y x zE E iE xy (A-9) Substituting the magnetic field wave expressions in equations A 6 and A 8 into Maxwells magnetic field divergence equation A 2 leads to: 0y x zH H iH xy (A-10) Substituting the electric and magnetic field wave expressions in equations A 5 through A -8 into Maxwells electric field curl equation A -3 leads to three equations: z yxE iEiHy (A-11) z xyE iEiH x (A-12)

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85 y x zE E iH xy (A-13) Substituting the electric and magnetic field wave expressions in equations A 5 through A -8 into Maxwells magnetic field curl equation A 4 leads to three equations: z yxH iHiE y (A-14) z xyH iH iE x (A-15) y x zH H iE xy (A-16) The transverse field functions ,xExy ,yExy ,xHxy and ,yHxy can be expressed in terms of derivatives of the longitudinal field functions ,zExy and ,zHxy by manipulating equations A -9 through A 16 as shown below. The expression for ,xExy is obtained by eliminating yH from equations A 12 and A 14 and introducing a parameter tk called the transverse wave constant. 222tki (A-17) 2 zz x tEH i E kxy (A-18) The expression for ,yExy is obtained by eliminating xH from equations A -11 and A 15. 2 zz y tEH i E kyx (A-19)

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86 The expression for ,xHxy is obtained by eliminating yE from equations A -11 and A 15. 2 zz x tEH i H kyx (A-20) The expression for ,yHxy is obtained by eliminating xE from equations A -12 and A 14. 2 zz y tEH i H kxy (A-21 ) Using equations A -18 and A 19, one can write down a general expression for the transverse electric vector field of a guided electromagnetic wave, 22 ,,, txy zzzz ttxyExyiExyj EHEH ii ij xyyx kk E (A-22) Using equations A -20 and A 21, one can write down a similar expression for the transverse vector magnetic field of a guided electromagnetic wave, 22 ,,, txy zz zz ttxyHxyiHxyj EHEH ii ij yxxy kk H (A-23) The linear combination of equations A -22 and A 23 is the total transverse vector field distribution (vector beam function) for a guided electromagnetic wave ,,, tttxyxyxy FEH (A-24) The eight terms in F,txy from equations A 22 and A -23 can be arranged into two groups, one group of four containing no longitudinal electric field terms, called transverse electric ( TE) and the other group containing no longitudinal magnetic field

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87 terms, called transverse magnetic (TM). This grouping in F,txy ca n be expressed by zE and zH in the following form: ,,,,(0)(0) t tzztTEz tTMzxyEHE H FFFF (A-25) where ,(0) tTEzE F is given by 22 (0) zz zz tTEz ttHHHH ii Eij yxxy kk F (A-26) and ,(0) tTMzH F is given by 22 (0) zzzz tTMz ttEEEE ii Hij xyyx kk F (A-27) Experimentally it is found that the beams emitted from QCLs are polarized perpendicular to the plane of the waveguide layers (x -direction in our notation). This is due to the fact that the intersubband transitions responsible for providing optical gain in a thin quantum well active layer can only be excited by an elect ric field normal to the layer [Fai 94]. The condition that 0 xE for QCLs implies that 0 yE When used in equation A 22, one obtains the following: zzHE xy (A -28a) zzEH yx (A -28b) Substituting equation A -28a into equation A 26: 2 22 2 (0) zz zz tTEz tt zz tHEEH ii Eij yyyy kkHH i ij yy k F (A-29)

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88 Substituting equation A -28b into equation A 27: 2 22 2 (0) zz zz tTMz tt zz tEHHE ii Hij xxxx kk EE i ij xx k F (A-30) If we use the assumption 0 y for the s lab waveguide described in Chapter II and substituting 0 y into equation A 29, the transverse field operating on TE modes, ,tTEF ,will vanish. The transverse filed operating on TM modes, ,tTMF in equation A-30 depends on zE and x. From equations A 24 and A -25, tTMF can be further expressed by transverse electric field and transverse magnetic field, both operating on TM modes: ,,,,,, tTM tTM tTMxyxyxy FEH (A-31) Substituting 0 zH and 0 y into equation A -22, ,,tTMxy E is given by: ,, 2 z tTM xTM tE i xyi x k EE (A-32) Substituting 0 zH and 0 y int o equation A 23, ,,tTMxy H is given by: ,, 2 z tTM yTM tE i xyj x k HH (A-33) Similarly to equation A -31, the total longitudinal vector field distribution (vector beam function) ,,lTMxy F for a guided electromagnetic wave operating on TM modes is the linear combination of longitudinal electric filed and longitudinal magnetic field, both are operating on TM modes:

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89 ,,,,,, lTM lTM lTMxyxyxy FEH (A-34) Substituting 0 zH and 0 y into equation A -16, ,,lTMxy E is given by: ,,1 y lTM z zTMH xyEk k ix EE (A-35) Substituting 0 zH and 0 y into equation A -13, ,,0 lTMxy H Therefore, the total longitudinal field operatin g on TM modes, ,,lTMxy F only has a non zero term ,,lTMxy E as shown in equation A 35. Figure A 1 summaries the three polarizations in the slab waveguide operating on TM modes. From the derivation in Chapter II, we would use one of these three polarizations: the only one magnetic field yH obtained from equation A -33, to describe the other two polarizations. These two polarizations are electric fields: ,,xTMxy E derived from equation A-32 and ,,zTMxy E derived from equation A -35. In this figure, we remove subscripts TM and use yH xE and zE to represent the above three polarizations. In brief, for slab waveguides operating on transverse magnetic modes, only the nonzero magnetic fields yH (no electric field) exist in the transverse y -direction.

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90 Figure A 1. Three polarizations in the slab waveguide operating on TM modes

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91 APPENDIX B DERIVATION OF EQUATI ON 3 32 FOR PLANAR WAVEGU IDEDS IN TM MODE Re -list equations (323) through (3-27) as follow: 00000expexp0 BDptEpt (3 -23) 00 0 00000exp exp0ab bpp CDptEpt (3 -24) 00000cossin0 ABdCd (3 -25) 00 0 00000sin cos0da aq ABdCd (3 -26) 0000000 DIEI (3 -27) where 1,0 0 0,0nn I n (3 -28) Equations 323 to 327 constitute a linear homogeneous matrix system and can be written in matrix form as follows: 00 0 000 00 0 0 00 0 00 0 00 001 0exp exp 00 expexp 0 1cos sin 0 0 sin cos 0 0 00011abb da aptpt A pp ptpt B C dd D q dd E (B-1) To have a nonzero solution for 0A 0B 0C 0D 0E in equation B -1, the determinant of the coefficient matrix must be zero [ 24 ][25 ]

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92 00 000 00 00 00 0 0001 0exp exp 00 expexp 0 1cos sin 0 0 sin cos 0 0 00011abb da aptpt pp ptpt dd q dd (B-2) To simplify equation B 2, we used the matrix expansion of this determinant [Bro 89] by minors of the fifth row. This 5 by 5 matrix is reduced into two 4 by 4 matrices as follows: 0 00 0 00 00 00 0001 0 exp 00 exp 1 1cos sin 0 // 1sin cos 0 //ab aa ddpt p pt dd dd qq 0 00 0 00 00 00 0001 0 exp 00 exp 1 1cos sin 0 // 1sin cos 0 //ab aa ddpt p pt dd dd qq =0 (B-3) (i ) From the first term in equation B 3, the first column is chosen for expanding this 4 by 4 matrix into two 3 by 3 matrices: = 0 00 0 00 00 001 0 exp 10 exp // sin cos 0 //ab aa ddpt p pt dd qq

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93 0 00 0 001 0exp 10 exp cossin 0abpt p pt dd (B-4) From the first term in equation B 4, the first row is chosen for expanding this 3 by 3 matrix into two 2 by 2 matrices: = 00 0 0 0 0exp 1 / cos 0 /ab a dp pt d q 0 0 00 00 000 exp // sin cos //a aa ddpt dd qq = 2 0 00 00 00 00/ / expcos expsin /a a dbp ptd ptd qq (B-5) From the second term in equation B -4, the first row is c hosen for expanding this 3 by 3 matrix into two 2 by 2 matrices: = 00 0 0 0 0 00exp 0 1 exp sin 0 cossinab ap pt pt d dd = 0 00000expsin expcosbp ptdptd (B-6) (ii) From the second term in equation B -3, the first column is chosen for expanding this 4 by 4 matrix into two 3 by 3 matrices:

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94 = 0 00 0 00 00 001 0 exp 10 exp // sin cos 0 //ab aa ddpt p pt dd qq 0 00 0 001 0exp 10 exp cossin 0abpt p pt dd (B-7) From the first term in equation B 7, the first row is chosen for expanding this by 3 matrix into two 2 by 2 matrices: = 00 0 0 0 0exp 1 / cos 0 /ab a dp pt d q 0 0 00 00 000 exp // sin cos //a aa ddpt dd qq = 2 0 00 00 00 00/ / expcos expsin /a a dbp ptd ptd qq (B-8) From the second term in equation B -7, the first row is chosen for expanding this 3 by 3 matrix into two 2 by 2 matrices: = 00 0 0 0 0 00exp 0 1 exp sin 0 cossinab ap pt pt d dd = 0 00000expsin expcosbp ptdptd (B-9)

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95 (iii) Combine th e above expanded equations into a more condensed formula. Add equations B -5 and B -8 to obtain = 00 000 0/ cosexpexp /a dbp dptpt q 2 0 000 0/ sinexpexpadptpt q (B10) Add equations B -6 and B -9 to obtain = 0 000sinexpexpbp dptpt 0 000cosexpexpadptpt (B-11) Combine equations B -10 and B 11 to obtain the full value of the original 5 by 5 determinant in equation B 2. 00 00 00 0/ expexp cossin /a bdp ptpt dd q + 00 00 00 0/ expexp sincos /a adptpt dd q =0 (B-12) Divide equation B 12 by 2 and use t rigonometric formula [ 26 ] to obt ain: 00 0 00 0/ cosh cossin /a bdp pt dd q 00 0 00 0/ sinh sincos /a adpt dd q =0 (B-13) Divide equation B 13 by 0cos t 00 00 0/ cosh tan /a bdp pt d q

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96 00 00 0/ sinh tan1 /a adpt d q =0 (B-14) Divide equation B 14 by 0sinh pt 00 00 0/ coth tan /a bdp pt d q 00 0 0/ tan1 /a add q =0 0 0 0 0 0 0 0 0 0/ 1tan / / coth / / tan /a d a b a dd q pt p dq 0 0 0 0 0 00 0/ tan/ / / / 1tan /d a a b d aq d p q d (B-15) Use t rigonometric formula [ 26] to simplify equation B -15: Let 1 00 00// tan tan //dd aaqq (B-16) Substitute equation B -16 into B -15 0coth pt 0 0 0 0tantan / / 1tantana bd p d 0 0 0 0tantan / / 1tantana bd p d 1 00 0 00// tantan //ad baq d p (B-17)

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97 It is customary to specify the refraction index of each layer ( jn ) in these multi layer structures. As a consequence, the medium permittivity j in equation B 17 is replaced by 2 jn where is the vacuum permittivity 22 1 00 00 22 00/ coth tantan /aa bdn qn pt d pn n (B-18) Thus, equations 331 and 3-32 Q.E.D. 0 0

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98 APPENDIX C DERIVATION OF EQUAT ION 3 41 FOR FIRS T -ORDER DFB WAVEGUIDES IN TM MODE Rearrange equation 3 -23 through 3-26 as follow: 00000cossin ABdCd (C -1) 00 0 00000sin cosda aq ABdCd (C -2) 00000expexp BDptEpt (C -3) 00 0 00000exp expab bpp CDptEpt (C -4) Equation C 1 divided by C 2 and use t rigonometric formula [ 26 ] to obtain: : 0 0 0000 1 000 0 0 00000 0 0 0tan cossin / tantan /sincos tan1adB d BdCd CB d B qBdCd C d C 1 00 00 00/ tantan /a dB d Cq (C -5) Equation C 3 divided by C 4: 0000 00 0 000000expexp / /expexpa bDptEpt B CpDptEpt 0 0 00 0 0 0 0 0/ 1 / exp(2) / 1 /b ab ap Dpt p E (C -6)

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99 Similarly, from equations 3-34 through 337: 1 1 1 1 1 1 1 1 1/ 1 / exp(2) / 1 /b a b ap D pt p E (C -7) Rearrange equations 338 through 339 as follow: 00001111IpaDEIpaDE (C -8) 10000111IpaDEIpaDE (C -9) Equation C 8 divided by C 9: 0 1 0001 0 1 01 1011 1 01 1 1 1 D D IpaIpa E E DD IpaIpa E E (C -10) Substi tute equation C -6 into C -10 and use t rigonometric formula [ 26 ] to manipulate: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0/ 1 / 1exp(2) / 1 1 / / 1 1 / 1exp(2) / 1 /b a b a b a b ap pt p D E D p E pt p 00 000 00 00 000 00// 1exp(2) 1 // // 1exp(2) 1 //bb aa bb aapp pt pp pt

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100 0 000 0 0 000 0/ 1exp(2)1exp(2) / / 1exp(2)1exp(2) /b a b ap pt pt p pt pt 0 00000 0 0 00000 0/ exp()exp()exp()exp() / / exp()exp()exp()exp() /b a b ap ptptptpt p ptptptpt 0 00 0 0 00 0/ tanh1 / / tanh /b a b ap pt p pt 1 0 00 01 / tanhtanh /b ap pt 0 11 00 00 001 tanhtanhtantanhba adp d pt q (C -11) Similarly, substitute equation C -7 into C -10: 1 1 1 11 1 D E D E 1 11 11 11 111 tanhtanhtantanhba adp d pt q (C -12) Combine equations C -11 and C 12: 111 tanhtanhtantanhi ib ia ii ia idp d pt q (C -13)

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101 It is customary to specify the refraction index of each layer ( jn ) in these multi layer structures. As a consequence, the medium permittivity j in equation C -13 is replaced by 2 jn where is the vacuum permittivity. 22 11 221 tanhtanhtantanhi ia id ii ib iapn n d pt n qn (C -14) Substitute equations C 11, C 12 and C -14 into C 10: 0 1 0001 0 1 01 01 1011 1 01 1 1 1 D D IpaIpa E E DD IpaIpa E E Thus, equations 340 and 3-41 Q.E.D. 0 0

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102 APPENDIX D EIGENMODE EQUATION FOR PLANAR WAVEGUIDES IN TM MODE From equation 5-1 through 5 -11 with boundar y conditions, we obtain the following linear homogeneous matrix form : 00 00 0 0 00 22 2 0 00 0 000 00 0 222 0 0 00 2221cos sin 0 00 sin cos 0 00 01 0exp exp 0 00 expexp0 000111 000 0da a abb bbcdd q A dd nnn B ptpt C pp ptpt D nnn E F pp qc nnn CX (D -1) In equation D 1, X is the magnetic field amplitude matrix. T o have a non-trivial solution for the linear homogeneous matrix equat ion D 1 the determinant of the coefficient matrix 66C needs to be zero: 00 00 0 00 22 2 00 000 00 222 000 2221cos sin 0 00 sin cos 0 00 01 0exp exp 0 0 00 expexp 0 000111 000da a abb bbcdd q dd nnn ptpt pp ptpt nnn p pqc nnn (D -2)

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103 APPENDIX E EIGENMODE EQUATION FOR FIRST -ORDER DFB WAVEGUIDES IN TM MODE From equation 5-1 through 5 -11 with boundary conditions, we obtain a linear homogeneous matrix form To have a non-trivial solution for the linear homogeneous matrix equation, the determinant of the coefficient matrix 1212C needs to be zero: 00 00 0 00 00 000 00 00 00 00 11 11 11 000 0001cos sin 0 0 000 0 0 0 0 sin cos 0 0 000 0 0 0 0 01 0exp exp 000 0 0 0 0 00 expexp 000 0 0 0 0 000 000 000da a abbdd q dd ptpt pp ptpt IpaIpaIqca IpaIpaIqca pIpapIpaq 000 111111011 11 11 1 11 11 111 11 10000 00 0 0 0 01cos sin 0 0 0 00 0 0 0 0 sin cos 0 0 0 00 0 0 0 001 0 exp exp 0 00 0 0 0 000 exp exp 0 000da a abbcIqca pIpapIpaqcIqca dd q dd ptpt pp ptpt Ipa 10 10 01 01 01 010 010 010 101 101 1010 000 000 000 IpaIqca IpaIpaIqca pIpapIpaqcIqca pIpapIpaqcIqca Where 2222 00 11 01010000,,,,,,,aabbddcc bbccp qc p qc ppqcqc nnnn .

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104 APPENDIX F COUPLING -COEFFICIENT S ENSITIVIY TO GEOMETRIC PARAMETERS FOR WAVEGUIDES IN TM MODE Coupling -coefficient sensitivity Sp to geometric parameters (P ) can be defined according to the slope s of curve s plotted i n the figures of coupling coefficient versus paramet e rs: SSlope = p p (F -1) Table F 1 C oupling -coefficient sensitivity to geometric parameters Parameter Model Groove depth (a) Buffer thickness (t) Active thickness (d) t=0.3, d=0.1 a<0.13 m k a=0.2,d=0.1 t<0.3 m k a=0.2,t=0.3 d<0.3 m TM 1 (ModB) 34 5 3.3 TM 1 (ModA) 22 5 3.1 TE 1, (ModA) 22 8 2.1 TE 1 (ModB) 19 8 1.9 All ratios in Table F 1 are in 106 [ cm2] The curve s of coupling coefficient versus groove depth have the largest slope s than those of coupling coef ficient versus layer thickness Thus the groove depth affect s the coupling coefficient more than the layer thickness.

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105 APPENDIX G FIELD INTERACTION AT DIELECTRICMETAL INTERFACE FOR PLANAR WAVEGUIDES IN TE AND TM MODES Fi gure G -1. Field inter action at the dielectric metal interface in TE and TM mode s TM mode has larger interaction.

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106 LIST OF REFERENCES [1 ] R. N. Hall et al, Coherent l ight e mission from GaAs j unctions, Phys. Rev. Lett. vol. 9, pp. 366368, 1962. [ 2 ] R. N. Hall, Injection l asers, IEEE J. of Quantum Electron. vol. 23, no. 6, pp. 674-678, 1987. [3 ] J. Faist et al, Quantum cascade laser, Science vol. 264, no. 5158, pp. 553556, 1994. [4 ] M. Pushkarsky,et al.," Sub-parts per billion l evel d etection of u sing r oom temperature q uantum c a scade l asers," Proc. Natl. Acad. Sci. U.S.A. v ol. 103, pp. 1963019634, 2006. [ 5 ] A. Lyakh et al., Substrateemitting distributed feedback quantum cascade las er, App. Phys. Lett 91, 181116, 2007. [ 6 ] A. Lyakh, High -power Bipolar and Unipolar Quantum Cascade Lasers Ph.D. Dissertation, University of Florida, 2007. [7 ] D. Marcuse, Theory of Dielectric Optical waveguides Second Edition. San Diego, CA: Academic Press, 1991. [8 ] S. M. Sze, Physics of Semiconductor Devices Second Edition. New York: John Wiley & Sons, 1981. [9 ] B. G. Streetman and S. Banerjee, Solid State Electronics Devices Upper Saddle River, NJ: Prentice Hall, 2000. [10 ] M. Bass, et al. Handbook of Optics New York: McGraw -Hill, 1995. [11 ] K. Okamoto, Fundamentals of Optical Waveguides Second Edition. San Diego, CA: Elsevier, 2006. [12 ] N. Yu, Near -field Imaging of quantum cascade laser Transverse Modes, Optics Express vol. 15, no. 20, pp. 1322713235, 2007. [13 ] J. J. Wilson and J. Hawkes, Optoelectronics: An Introduction Thir d Edition. New York: Prentice Hall, 1989. [14 ] N. Nishihara et al., Optical Integrated Circuits New York: McGraw -Hill, 1982. [15 ] R. J. Noll and S. H. Macomber, Analysis of grating surface emitting lasers, IEEE J. of Quantum Electron ., vol. 26, no. 3, pp.456466, 1990. [16 ] D. L. Lee, Electromagnetic principles of Integrated Optics New York: John Wiley & Sons.1986.

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107 [17 ] S.Fathololoumi et al., Beam p attern i nvestigation of t erahertz q uantum c ascade l asers Progress In Electromagnetics Research Symp osium Hangzhou, China, pp. 401404, 2008. [ 18 ] Y. C. Shen, Detection and identification of explosives using terahertz pulsed spectroscopic imaging, Appl. Phys. Lett vol. 86, 241116, 2005. [ 19 ] K. Kawase et al., Non-destructive terahertz pulsed imaging of illicit drugs using spectral fingerprints, Optics. Express, 11, 25492554, 2003. [20 ] K. Iizuka, Elements of Photonics: for Fiber and Integrated Optics. New York: John Wiley & Sons, 2002. [21 ] C. K. Kao, Optical Fiber Systems: Technology, Design, a nd Application New York: McGraw -Hill, 1982. [22 ] G. Gaussorgues, Infrared Thermograph London, Great Britain: Chapman & Hall,1994. [23 ] H. J. Luo, Shiny Contact Distributed Feedback Semiconductor Lasers Ph.D. Dissertation, University of Florida, 1993. [24 ] D. M. Bloom, Linear Algebra and Geometry New York: Cambridge University Press, 1979. [25 ] J. G. Broida and S. G. Williamson, A comprehensive Introduction to Linear Algebra. New York: Addison-Wesley, 1989. [26 ] R. Barnett, Analytic Trigonometry with Applications Third Edition. Belmont, CA: Wadsworth, 1984. [27 ] H. Yonezu et al., A 1 xxGaAsAlGaAs double heterostructure planar stripe laser, Japanese J. of Appl. Phys vol. 12, pp1585 1587, 1973. [28 ] L. A. Coldren and S. W. Corzine, Di ode Lasers and Photonics Integrated Circuits New York: John Wiley & Sons, 1995. [ 29 ] W. Streifer et al., Coupling coefficients for distributed feedbacks single and double heterostructure diode lasers, IEEE J. Quantum Electron., vol. 11, pp. 867-873, 19 75. [ 30 ] P. Correc, Coupling coefficients for trapezoidal gratings, IEEE J. Quantum Electron ., vol. 24, pp. 810, 1988. [ 31 ] Y. Shani et al., 11/xx xyyxPbSnSePbEuSnSe Corrugated diode laser, IEEE J. Quantum Electron., vol. 25, pp. 1828-1844, 1989. [32 ] S. S. Li, Semiconductor Physical Electronics New York: Plenum Press, 1993.

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108 [33 ] C. Kittel, Introduction to Solid State Physics Eighth Edition. Hoboken, NJ : John Wiley & Sons, 2005. [34 ] A. Yariv. Optical Electronics Third Edition. Orlando, FL: Holt, Rinehart and Winston, 1991. [35 ] H. H. Yee, Novel Self -consistent method for Determining optical coupling coefficients and grating losses of Distributed Bragg Reflector Lasers, Opt. eng. vol. 41, no. 5, pp1077-1083, 2002. [36 ] D. Hanselman and B. Little field, Mastering MATLAB 6 : A Comprehensive Tutorial and Reference. Upper Saddle River NJ: Prentice Hall, 2007. [37 ] R. J. Schilling and S. L. Harris, Applied Numerical Methods for engineers Using MATLAB and C Pacific Groove, CA: Brooks/Cole, 1999. [ 38 ] F. T. Ulaby, Fundamentals of Applied Electromagnetics Upper Saddle River NJ: Prentice Hall, 1997. [39 ] J D. Jackson, Classical Electrodynamics. Third Edition. New York: John Wiley & Sons, 1999. [40 ] H. Kogelnik and C. V. Shank, Coupledwave theory of distributed feedback lasers, J. Appli. Phy ., vol. 43, no. 4, pp.23272335, 1972. [41 ] R. K. Wangsness, Electromagnetic Fields New York: John Wiley & Sons. 1986.

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109 BIOGRAPHICAL SKETCH MengMu Shih was born in Taiwan. He received B achelor of Science (B.S.) and M aster of Science (M.S.) degrees in mechanical engineering from National Taiwan University (NTU). He received two M.S. degrees : one in electrical engineering from the College of Engineering, and the other in management from the Warrington Colleg e of Business Administration, both at the University of Florida (UF). Later, he joined the Photonics Research Laboratory under Professor Zory, for working toward his Doctor of Philosophy ( Ph. D. ) degree. Mr. Shih has been partly funded by the Defense Advan ced Research Projects Agency (DARPA) for quantum cascade lasers (QCLs). He has been involved in the modeling, design, physics, characterization and measurement of semiconductor lasers and light emitting devices He interests in areas including system model ing, product and design methodology, nanotechnology, electromagnetism, semiconductor devices, optics, photonics, optoelectronics, and optical distributed feedback waveguides with corrugated metal gratings.