MSU RETURNING MATERIALS: Place in book drop to LIBRARIES remove this checkout from -_. your record. FINES W‘iH be charged if book is returned after the date stamped be1ow. on] ’ti 2%? LEE 0 0 ma AH IHTEGRAL-OPERATOR APPROACH TO THE ELECTROHAGHETICS OP INTEGRATED OPTICS Br Hark Stephen Viola A DISSERTATION Subaitted to Hichigan State University in partial fulfillaent of the require-ent- for the degree of DOCTOR OF PHILOSOPHY Departeent of Electrical Engineering and Syeteae Science 1988 In ABSTRACT An INTEGRAL-OPERATOR APPROACH To THE ELECTROHAGHETICS FOR IHTEGRATED OPTICS By Hark Stephen Viola There is an increasing interest in the study of optical and elec- tronic circuits iasersed in a layered dielectric surround. Conventional differential-operator forsulations for the fields vithin these circuit devices are ineffective due to the inseparability of the applicable boundary conditions for structures having practical shapes. An inte- gral-operator forsulation, based on the identification of equivalent polarization currents, circusvents this difficulty. An electric field integral equation (EPIE) is developed for the integrated systes consisting of electrically heterogeneous dielectric regions esbedded vithin a tri-layered substrate/fila/cover background environnent. Uniqueness of the solution to this EFIE is established. Special consideration is given to axially unifors integrated dielectric vave guiding systess. It is observed that the longitudinal invariance of the integrated vaveguiding systes renders the integral-operator of the EFIE convolu- tional in the axial variable. Use of the Paltung theores is prospted and a Fourier transfors-dosain EPIE is obtained. Analysis of solutions to this transfors-dosain EPIE in the cosplex-plane of the transfers variable facilitates the identification of a propagation-soda spectrus. Surface vave nodes, cosprising a discrete spectrus, are associated vith pole singularities. Regines of purely-guided and leaky surface-save are detersined. A continuous spectrus of radiation-nodes arises fros solu- tions to the transfors-dosain EFIE along an appropriately chosen branch cut contour. An asysptotic fora of the transfors-dosain EFIE is presented and is applied to the study of dielectric saveguides capable of supporting a surface-vave node at lisitingly low frequency. Exasples are given to support the validity of this asysptotic EPIE. An iterative schese is devised to generate solutions to the forced transfors-dosain EPIE. Applications of this sethod to the graded-index asyssetric slab and graded circular dielectric vaveguide indicate that one iteration can provide a reasonably good approxisation for high-fre- qency spectral cosponents of the continuous spectrua when the refractive contrast is small. Copyright by MARK STEPHEN VIOLA 1988 TO HY WIPE, LAURA NY Je 1'9 ACKHOHLEDGEHEHTS The author would like to express sincere thanks to Dennis P. quuist for his inspiration and guidance. Special thanks are given to Jes Asnussen, K.H. Chen, and Byron Drachsan for their support in this research. vi LIE LIST OF FIGURES ................. 1. TABLE OF CONTENTS ELECTROHAGIETICS OF LATERED DIELECTRICS ............ 2.1 IuTRODUCTION .0.QI0.0.0.0.0....QO.Q....OOQOOQOOOOQOOQOQO... 2.2 HERTZIAN POTENTIAL GREEN'S DVAD .......... 2.2.1 PRIHARY GREEN'S DYAD ............................... 2.2.2 REFLECTED GREEN'S DYAD FOR SOURCES IN THE COVER .... 2.2.3 REFLECTED GREEN'S DYAD FOR SOURCES IN THE FILH ..... 2.3 2.4 SUHHARY ..... ELECTRIC DYADIC GREEN'S FUNCTION .......................... 2.3.1 DERIVATIVES OF THE HERTZIAH POTENTIAL .............. 2.3.2 DEVELOPHENT OF THE PRINCIPAL DYAD .................. 2.3.3 EQUIVALENCE OF PRINCIPAL VOLUHES ...... .QQOQQOQOOOOQOQOOOOQQOO. AI INTEGRAL-OPERATOR APPROACH TO INTEGRATED OPTICS ............. 3.1 3.2 3.3 30‘ IuTRODUCTION 0..OOOOQOQQOOOOQOQOOQQOQO.Q.O ELECTRIC FIELD INTEGRAL EQUATION .......................... 3.2.1 EQUIVALENT SOURCE IDENTIFICATION ................... 3.2.2 CONSTRUCTION OF THE INTEGRAL EQUATION .............. 3.2.3 UNIQUENESS OF SOLUTION ............................. AXIALLY-UNIFORH "AVEGUIDES ssseeeessseesseasssssesssssesees 30301 EFIE FOR THE TRAnSVERSE FIELD esssssssssesaesesss SUHHARY ..... vii 6 6 10 13 15 18 21 21 24 28 31 31 4. 5. 6. mpmmTI.-m9m0000000000000 ..... OOOOOOQQOOOQOCOQ 4.1 IRTRODUCTION0.00.00.00.00...00......QOQOQOOOQOOQOQOQOQOQOO 4.2 COHPLEX l-PLANE ANALYSIS .................................. 4.2.1 GREEN'S DYAD Z-PLANE SINGULARITIES ................. 4.2.2 ELECTRIC FIELD l-PLANE SINGULARITIES ............... 4.2.3 CONTOUR DEFORHATION ................................ 4O 3 THE DISCRETE SPECTRU“ O Q Q Q O O O O O O Q 0 I Q O O Q 0 O O O O O O O O O O O O O O O O O O O 4. 3. 1 DETERHIRIRG THE RESIDUE O O O O O Q 0 O O O O O O O O O 0 O O O O O O O O O O O 4. 3. 2 DETERnIuATION 0P POLE ORDER 0 O O O O Q O O O O O O O O O O Q O O O O O O Q 4. 3. 3 SURFACE'uAVE LEAKAGE O Q Q C O O O O O O O I O O O O O O O O O O O O Q 0 O O O O O 4. ‘ THE couTIuUOUS SPECTRUfl Q Q C O O O O O O O O Q 0 O O Q 0 O O O O O O O O O O O O O O O O 0 O ‘05 SUHHARY00......000......QQOOOOOQOQOOOOOQOOOOOOOQQQ00...... “WICEIEQOQOOOOQQ. ..... 0.00.00...OOQOQOQOQOQOOOOOOOOO 5.1 INTRODUCTION .QOQOOOOQOIOQIQOCOOQ..0...COOOOOOQQOOOQOOOOOOQ 5.2 THE GRADED-INDEX ASYHHETRIC SLAB WAVEGUIDE ................ 5.2.1 EFIE FOR THE ASYHHETRIC SLAB ....................... 5.2.2 TE HODE AEFIE FOR THE ASYHHETRIC SLAB .............. 5.2.3 TH HODE AEFIE FOR THE SYHHETRIC SLAB ............... 5.3 AGENERAL AEFIEI.QOCOOIOOOQOOQOOOOOOOQOOQOOOQOOOQOOOOOOOOO 5.4 AEFIE OF THE CIRCULAR FIBER ........... .................... 5.5 SUHHARY 0.00.00.00.00...00......Q..0.COCOOOOOOOOOOOOOOOOOOQ APPROXIHATIOI OF THE CONTINUOUS SPECTRUH ....................... 6.1 IuTRODUCTIon.00...OOOOCOQOQOQOOOOQOQOQO0.0000QOOQOOOOOIOQ. 6.2 ITERATIVE HETHOD .......................................... 6.2.1 ERROR ANALYSIS ..................................... 6.2.2 NEUNANN SERIES THEOREH ............................. 6.2.3 RELATIVE ERROR OF ONE ITERATION .................... 6.2.4 REHARK ............................................. 6.3 ANALYSIS OF THE ASYHHETRIC SLAB ........................... 6.3.1 TE HODE OPERATOR ANALYSIS .......................... 6.3.2 TE HODES OF THE STEP-INDEX SYNHETRIC SLAB .......... 6.3.3 TH OPERATOR ANALYSIS ............................... 6.3.4 TH HODES OF THE STEP-INDEX SYHHETRIC SLAB .......... viii 52 52 53 53 55 61 71 72 74 74 75 81 84 APP APP APP APP 6.4 ANALYSIS OF THE CIRCULAR FIBER ........ 6.4.1 FOURIER SERIES EXPANSION ........................... 6.4.2 IHPRESSED FIELD .................................... 6.4.3 SPECIALIZED EFIE .......... 6.4.4 EXACT FIELD .......... 6. 6. 6. 4.5 FIRST ITERATE ........ 4. 6 COHPARISON O C O O O O I O I O O O O O O O O O O O O O O O O O O C O Q C Q Q I O O C O O O I 4. 7 RESULTS 0 O O O O O Q 0 O O O O O O O O O O Q 0 O O O O O O O O O O O O O O O O O O O 0 O O O O 6.5 SUHHARYOOQQQOQOOOOQQOOOOOOOOOOOOO..QQOOOQQOOOOOOQQQIOOOOOC 7. COICLUSIEUB AID REEEHHEIDATIOIS ....... ”Pmr“.0.00......QOOOOOOOOOOOOQOQOQOO ‘PP’I‘a0.0.0.0...QOOOOQOOOOIOOO... APPENDIX C ........ . ........... . ..... ............... ‘PPmIXDOOOQCQOOOOQOQOOO ‘PPmI‘EOOOOOQOOQOQOOOOC LIST OF REFERENCES ......... ix 115 126 129 130 131 132 134 136 136 148 151 153 161 163 167 170 Fl 10. 11. LIST OF FIGURES Figure 1. A typical integrated optical circuit............................ 2. Tri-layered structure used as the background environsent ....... for integrated optical and electronic circuits. 3. Examples of practical optical, silliseter-wave, and elec-....... tronic integrated circuits. (a) Hicro-strip waveguide. (b) Hilliaeter-wave dielectric strip waveguide. (c) Op- tical dielectric strip waveguide. (d) Optical dielectric channel waveguide. 4. Tri-layered structure with (a) sources exclusively in the....... cover: (b) sources exclusively in the file. 5. Tri-layered structure with currents issersed exclusively ....... in the cover region. 6. Tri-layered structure with currents issersed exclusively ....... in the fill region. 7. Four paths with different y-dependent phases. (a) -y-y' ....... (b) y+y'¢2t. (c) y-y'+2t. (d) -y+y'+2t 8. A 'slice' principal voluse excluding the singularity point r ... of the electric dyadic Green's function; closed surface 56 is the boundary of the slice voluse. 9. A general optical device issersed within a tri-Iayered ......... dielectric surround. 10. (a) The physical systea of an optical device vithin an in; ... integrated surround. (b) An equivalent systes in which P accounts for all effects of the inhosogeneous dielectric ob- stacle. 11. Geoaetry for field behavior at the boundary surrface S of an .. optical device isseresed in a unifora surround. Page 2 7 9 11 16 19 20 25 32 34 38 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27, 28. 29. 30. 31. 32. 33. 34. 3S. 36. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. A longitudinally-invariant waveguide deposited within the ..... cover of an integrated background environsent. Complex n-plane with contour Cn Complex Z-plane singularities of the transform-domain.......... electric field. Closed contour C in and on which the transfors-dosain.......... electric field is analytic except at Z 8 In. Detersination of the proper branch for each Y1................. Hyperbolic branch cuts..... coalesced brunet, cuts...O.............QOOOOOOOOOOOOO0.0.0000... Hyperbolic branch cuts in the cosplex t-plane in the lisit..... of low loss. The graded-index asyssetric slab waveguide..................... Geosetry of the step-index circular fiber... ....... ............ complex z-plane vith branCh out cbOQOOOOOOOOOOOQOOOOOOOOOOOO... Relative error vs. 6c for TE aodes............................. Relative Relative Relative Relative Relative Relative Relative Relative Relative Relative Relative Relative Relative field field field error error error field field field field field field error amplitudes for TE node (6C a 2.0)............... asplitudes for TE node (QC aaplitudes for TE node (QC 500).... ....... ...... 10.0)...00000000000 2 VB. (ac/kc) for T” we ‘m a 0032)....QOOOQOOO 2 vs. (QC/kc) for TH node (An .203)............ 2 vs. (QC/kc) for TH lode (An = .520)............ 2 asplitudes for TH sode (An = .032)............. asplitudes for TH anplitudes for TH amplitudes for TH aspiitudes for TH asplitudes for TH node node lode lode node 2 (An (An (An (An (An 0032)....OOOQOOOOO 0203,0000000000000 0203,0000000000000 0520)....OOOOOOOOQ 0520,0000000000000 2 vs. (QC/kc) for n = 0 node (An = .032)......... xi 42 44 56 57 60 62 63 70 76 89 102 108 109 110 111 116 117 118 119 120 121 122 123 124 137 37. 38. 39. 40. 41. 42. 43. 44. Relative Relative Relative Relative Relative Relative Relative Relative error error field field field field field field vs. (O /k ) for n c c vs. (0 lk ) for n c c 2 0 node (An 8 .203)......... 2 0-Ode (m =0362)essssesee asplitudes for n = 0 node asplitudes for asplitudes for asplitudes for aaplitudes for asplitudes for n = 0 n = O xii lode lode node sode node 2 (An (An (An (An (An (An 0032)....OOQOOO .032)... 0203,0000000000 0203)....OOO... 0362’ssses .362). 138 139 140 141 142 143 144 145 Il‘. 1m 0P be a1 vi 80 £0 11 pt H II (I INTRODUCTION There is an increasing interest in the study of optical circuits issersed in an integrated dielectric surround. Typically, an integrated optical circuit (Figure 1) consists of a layered substrate/fils/cover background in which circuit devices (represented by regions of electric- al heterogeneity) are issersed. This dissertation is intended to pro- vide a sethod of analysis appropriate for electrosagnetic phenosena as- sociated with this structure. In particular, this sethod is specialized for, and applied to the study of integrated dielectric waveguides. Forsulating a general analytic description of the electrosagnetic fields within integrated dielectric waveguides is cosplicated by the peculiar geosetry of the background/waveguide structure. In fact, it is precisely the inseparability of boundary conditions which render conven- tional differential-operator forsulations ineffective for devices having practical shapes. Approxiaate solutions have been obtained fros a dif- ferential-equation approach for the step-index rectangular strip guide (see for exasple Harcatili [1]) but this sethod is inaccurate near cut- off. The elaborate sode-satching approach of Peng and Oliner [2.3] yields an exact description for this class of structures, but it used a discretized radiation spectrus. Recently, integral-operator forsulations have been used. Notably. the boundary-elesent sethod has been used by a nusber of investigators [4.5.6]. This sethod is a generalization of the surface forsulation COVER OVERLAY core n( / __ ..{L __ 3w /»-m or ////A Figure l. A typical optical circuit. 2 ue Ci ac CC ic tl' de vi e] it de II t1 t1 1: 01 t: f: tc 1: used by Chang and Harrington [71 and is based on field equivalence prin- ciples at surfaces. Although conceptually exact. this sethod does not account for inhosogeneously-graded structures and fails to address the continuous spectrum. This dissertation uses an integral-operator approach, based on the identification of equivalent polarization currents, which circusvents the inadequacies of the aforesentioned sethods. This forsulation was developed by Johnson and quuist [81 and advanced by Bagby [9] and pro- vides a conceptually exact description of the electric field within an electrically heterogeneous region of arbitrary shape eabedded in an integrated surround. All waveguiding phenosena are unified by this description, thus furnishing a powerful sodel. The text is divided into seven chapters. In Chapter two, a Green's function for the tri-layered background is developed. Subtleties in the appropriate electric dyadic Green's function for this structure are thoroughly discussed. This dyad is used in Chapter three to construct the integral equation which describes the electric field. A transfors- donain version of this integral equation, suitable for the study of longitudinally-uniform waveguides, is developed and fares the basis for all subsequent analyses. In Chapter four, the propagation-lode spectrus of these guides is identified. A discrete spectrus is found to be asso- ciated with surface waves, while superposition of the continuous spec- true yields the radiation field. A specialized integral equation, ap- propriate for the study of waveguides operating at asysptotically low frequency, is developed in Chapter five. Application of this equation to the asynnetric slab [101 and the circular fiber [11) provide asysp- totic expressions which are shown to concur with [101 and (11]. An tr Cl 81 80 PF‘ F1: 18: em Und. 81.1 An iterative schese is devised in Chapter six to approxisate those spec- tral cosponents of the continuous spectrus having high spatial frequen- cies. First iterative approxisations to the graded-index asyssetric slab and the circular fiber were sade. Cosparisons to exact well known solutions are given. Finally, conclusions and recounendations are provided in Chapter seven. Sose of the notation which is used liberally should be clarified. First, vector quantities appear bold faced. Sisilarly, dyads are bold faced and are overstruck with a double bar. Unless stated otherwise, integrals with no specific lisits are to ieply an integration over the entire doeain. Finally, throughout this dissertation the following assuaptions are effected: (1) All aedia are linear and isotropic; (2) All of space is sagnetically hosogeneous such that the sagnetic-induction field B is related to the sagnetic field H by the constitutive equation B 8 pl. (3) Electrically heterogeneous regions are characterized by persittivity ((r) and conductivity c(r) such that the electric displacesent D and conduction current density Jc are related to the electric field by the constitutive relations D I (E. and Jo 8 0E respectively. Jot (4) The tine dependence is harsonic (e ) and is suppressed. Under these assusptions, Haxwell's equations in H-H-S units are greatly sisplified and say be written as: V's E = p ...Bauss' law (Is) VkE = -JuuH ...Faraday's law (lb) VkH = J + Jos.E ...Asperes' law (Is) V-H = 0 ...Absence of aagnetic sonopoles (1d) where s. = s + c/ju is the coaplex persittivity. The coaplex persittiv- 2 ity say be written as e.(r) = eon (r) where n is the coaplex refractive index and so is the vacuus persittivity. L! ur ov fr Al Re ELECTRONAGNETICS OF LAYERED DIELECTRICS 2.1 INTRODUCTION In this chapter, the electrosagnetics of layered dielectric struc- tures is investigated. Description of the fields in a layered environ- sent say be furnished by integrating the inner product of an appropriate Green's dyad with the electric source density saintaining the fields. Hnovledge of this dyad is of particular isportance since it is used throughout this dissertation as the kernel in the integral equation for the electric field in integrated optical waveguides. Analysis of electrosagnetic fields in a layered environaent was first aade by Soaserfeld [12] in 1909. Fields produced by electric di- poles oriented norsal or tangential to an air-earth interface were con- sidered. Integral-transfors techniques were used to obtain integral representations for these fields. These integral expressions were of generic fors and have since been categorized as Sosserfeld integrals. Sosserfeld integrals appear in the forsulation of fields in layered sedia for sore cosplicated situations. Attention is focused on the tri-layered structure depicted in Fig- ure 2. A file layer of thickness t and refractive index nf is deposited over a substrate region (y < -t) which is characterized by index of re- fraction n.. The region (y > O) is the cover with refractive index nc. All dielectrics are assused to possess lisitingly ssall dissipation with Re(n£) > Re(n.) > Re(nc), where Re( ) designates the real part of the **)D~< region 1: y > 0 n (cover) c V ’30 i=4! region 2: -t < y < O -o e nt (filn) J V y--t region 3: y < -t as (substrate) Figure 2. Tri-layered structure used as the background environsent for integrated optical and electronic circuits. 7 quac tair havj atic of 1 uitt diel gro‘ intl lay pro den r911 are de; Ole int int the Dr Pol quantity within the braces. Issersed electric current density J sain- tains electroaagnetic fields in all three regions. Although the ensuing analysis say be generalized for a structure having any nusber of dielectric layers with eabedded currents, the situ- ation in Figure 2 provides a useful sodel for the background environsent of practical electronic, silliseter-wave and optical integrated circ- uits. Several exasples of these structures are shown in Figure 3. A dielectric substrate is used for integrated optics, while a conducting ground plane replaces the substrate for silliseter-wave and electronic integrated circuits. In the next section, the electric Hertzian potential 0 for the tri- layered structure is expressed as a superposition integral of the inner product of the appropriate Green's dyad S with the ispressed current density J. Green's dyads are quantified for situations in which cur- rents are esbedded exclusively in either the cover or the file region. In section 3, two equivalent representations of the electric field are given. A spectral representation is used to identify a natural depolarizing dyad L which is relevant to the Green's dyad as for the electric field. Finally, the electric field is expressed as a voluse integration of the inner product of as with the electric current source J, sodified by a correction tern in which i appears. The voluse of integration extends over the support of the current density but excludes the singularity point of G9. The excluding region is identified as the 'principal volune' which corresponds to the preferred choice of the de- polarizing dyad. n( CC Figu nctcover) conducting HI/I/l/l/ll n£(fils) Ill////////////l/l/l/l/l/l/l/l conducting ground plane (a) nc(cover) guiding I n(r) I nf(fils) n.(substrate) (c) n (cover) c guiding 1w) ] n£(fils) Illl/l/l/l/l/l/lll/l/l/l/l/l/l conducting ground plane (b) n (cover) c guiding 922! n(r) l nf(fils) n.(substrate) (d) Figure 3. Exasples of practical optical, silliseter-wave, and elec- tronic integrated circuits. (a) Hicro-strip waveguide. (b) Hilliseter-vave dielectric strip waveguide. (c) Op- tical dielectric strip waveguide. (d) Optical dielectric channel waveguide. 2.2 HERTZIAN POTENTIAL GREEN'S DYAD For the reader who is unfasiliar with the definition of the Hertz- ian potential, a review is provided in Appendix A. The relationship of the potential to the electric field, along with the Helsholtz equation which the potential satisfies, are given. A general developsent of the Hertzian potential Green's dyad C for layered dielectrics has been discussed by Bagby and Nyquist [131. Based on the classical developsent of Sosserfeld [14], the Hertzian potential dyadic Green's function was shown to have scalar cosponents represented by tvo-disensional spectral integrals. In the subsequent developsent, the analysis in [13] is altered slightly so that identification of a natural depolarizing dyad i, corresponding to the Green's dyad &? for the electric field, say be sade. Consider the situations shown in Figures 4(a) and 4(b). Electric current density J, issersed in the ith region (i8c(f), for cover(fils) ) of Figure 4a(b), produces Hertzian potentials in each region of the tri- layered structure. The Hertzian potential subject to the Lorentz gauge satisfies the Heleholtz equation 2 2 (v + kg)!!! = -J/Jss1 m in each region (18s,f,c for substrate, file, cover). Note that in (1), J 8 O for L 8 i. Forsal operation on (i) with the two-disensional Fourier transfors Pm = H m 9’3”” dxdz A A vhere A 8 xi + z!, reduces equation (1) to the ordinary differential equation 2 2 2 (a lay - pl)n£(xsy) 8 -J(A.;y)/Jes1 (2) 10 P1 3>~< region 1: y > O J nc (cover) V Y‘O -————:2_!§__ region 2: -t < y < 0 wet- "°° nf (file) v=-t region 3: y < -t na (substrate) (a) ‘--%>~< region 1: y > O n (cover) c v=0 region 2: -t < y < 0 -¢ 9 n! (file) ys-t region 3: y < -t n. (substrate) ssed as a spectral inte- through the reflected dyad Figure 4. Tri-laye: cover: /./ 12 9“ ‘b F1! )>~< region 1: y > 0 J nc (cover) V region 2: -t < y < 0 -u e nf (file) y=-t region 3: y < -t nB (substrate) (a) Y region 1: y > O n (cover) c y=0 .___,_£ region 2: -t < y < O -o e nt (file) v ys-t region 3: y < -t n. (substrate) (b) Figure 4. Tri-layered structure with (a) sources exclusively in the cover: (b) sources exclusively in the file. 11 Th gr 2 2 2 2 where n 8 F(flz), J 8 F(J), and p£8 t + Z - k Solution of (2) is L l' elesentary, and say be written as a sus of particular and cosplesentary solutions. Physically, the particular solution represents the prisary wave which radiates in the hypothetical unbounded region of space, while the cospleaentary solution represents the reflected wave saintained by the surface polarization current at the interfaces of dielectric discon- tinuity. This deco-position for the transfors-dosain potentials is ., P..M.}‘PY”PY n£(k,y) 611%.IV g (A,y,r ) dV 9 I£(A) e L + I£(A) e 1 30:1 where gp(A:y,r') = e-JAer erpily-y I/2p1 and 61‘ is a Kronecker delta. ... The coefficients I; are detersined by satisfying the appropriate bound- ary conditions [13] across the dielectric interfaces and as yefu. These boundary conditions and their isplesentation are given in Appendix B for currents isaersed exclusively in the file. The result for currents in the cover is discussed in Section 2.2.2. Solution to the probles with sources in the substrate say be obtained free the solution to the prob- les with sources in the cover by interchanging n. with nc and saking the coordinate transforsation y 8 -t - y. Inversion of the transfora-dosain potentials say be perforsed and yields the solution to (1) with the potential in the ith region given by n m . 1 2 H 6’2"” gp(A3y,r') '73—’- dV' } am: (3) 1 (2!!) v j": + I Gr(rlr')- Jig-l'dV'. V Jae1 The principal portion of potential HP is expressed as a spectral inte- gral and the reflected part fl? is described through the reflected dyad Or. The reflected dyad say be written as 12 vher diel nent CUII mil of late 2.2. sore [15, 3 pr The int: thll P. 4: the. int, 'he] r r 36 36 - A A A A A A A A Gr(rlr') 8 xGix + y (Txc x # (32y + 82‘: z) 0 26,2 where G: (6:) yields cosponents of potential tangential (norsal) to the dielectric interface saintained by tangential (norsal) current cospo- nents. 6: accounts for the coupling between tangential cosponents of current with the norsal cosponent of potential. The scalar cosponents of Gr have spectral integral representations which are elaborated in later sections. 2.2.1 PRIHARY GREEN'S DYAD The principal part of the Hertzian potential in (3) say assuse a sore faeiliar fora by use of Fubini's theorea for isproper integrals [15, p.473). Although the integrand of the spectral integral in (3) is a proper volume integral, the principal part of (3) say be written ‘ H 3"" {I " ° ' mu '} [Pm (2"): e (1:38 v_vg<).,y.u->JM1L dV dtdz (4a) = (2:): (1138 H .52" {I‘M gp(A;y,r') %dV'} dzdz. (4b) The voluse v is any voluse which excludes the singular point r'8r. The interchange of the lisit with the spectral integral is justified since this integral converges uniforsly [15, p.473). Use of Abel's test [15, p.472] for unifors convergence of isproper integrals along with Fubini's theores for isproper integrals allows the spatial integration to be interchanged with the spectral integration. Therefore, (4b) becoaes J(r') .— as1 'p I I upcr) . ‘1'“ I‘M, s (rlr ) av 8p up i p z where the principal dyad C is given by 6 8 G . I is a unit dyad and 13 the pri An alte transfo: loving : There I 'here J “no 1: Jo Inun. lay be 1 t1cular e f°rl “crate: the principal Green's function Gp is given by dIdZ. (5) I l , Gp(rlr') = %II ’p1 y Y .jA0(r-r ) (2n ) 2’1 An alternative integral representation of 6p can be obtained free (5) by transforsing the double integral to polar coordinates. Haking the fol- lowing substitutions ! 8 A case I 8 A sine 2 H 2 p1 8 (A - k1) A dOdA n. M n N u x - x' 8 lt-t'l case 2 - z' 8 lt-t'l sine 2 2 where lt-t'l 8 [(x-x') o (z-z') 1”, equation (5) becoses a -p ly-y'l 2n _ , _ Gp(rlr') = 1 a! L—é— H 9"" ‘ '°°"° ” dO}MA (6s) (2n) 0 pi 0 . '9 'Y’Y" . z = —1 9—1— Jo(A[(x-x') +(z-z') 1") MA. (6b) 2n 0 2p1 where Jo is the zeroeth order Bessel function of the first kind. In going froa (6a) to (6b), use was sade of an integral representation of Jo found in (16, p.360]. Evaluation of the integrals in (5) and (6b) say be perforsed, but need not be when it is realized that HP is a par- ‘ticulsr solution to (1) with conditions on its asysptotic behavior at y 8 ts. Uhence, the fasiliar free space Green's function obtains as -Jk1lr-r I P . 2_________. G (rlr ) 8 4nlr-r'I . (7) The fore of (7) is not a convenient representation of Sp for use in in- tegrated optics. For practical applications, the spectral 14 Tepre tenti where 2.2.: site the the disc 8“99 Phas 'her Ulat duQ line representations (5) or (6b) are recon-ended. Finally, the Hertzian po- tential in (3) say assuse the standard fora _ - , J(r') n1(r) - $15 IV-v G(rlr )- juei dV' 3 3 8p =1- where G is given by G 8 G + G . 2.2.2 REFLECTED GREEN'S DYAD FOR SOURCES IN THE COVER In this section, the reflected Green's dyad Sr is detailed for the situation shown in Figure 5. Source and field points are situated in the cover region. The reflected wave, which is illustrated, represents the grand sun of all waves reflected fros the interfaces of dielectric discontinuity which travel in the positive y direction. Intuitive appeal suggests that the y dependent part of the reflected wave should have a phase of y * y'. In fact, the scalar cosponents of Sr are given as G:(rlr') Rt(A) -P (v*y') _ . 2 Gr(rlr') = If a (A) 9——9-—————'o3*"’ ’ ’ d A n n 2 2(2n) pc G;(rlr') C(A) where dzA 8 dldl. The reflection coefficients Rt and Rn as well as the coupling coefficient C have been derived in [9, pp.163-l72] and are tab- ulated below. Coaputation of the reflection and coupling coefficients is tedious due to their cosplicated dependence on environsental paraseters. Out- lined below is a sieple procedure for obtaining these coefficients. 1. Calculate tangential reflection and transsission coeffi- cients associated with the cover-file interface as 15 Pigu )PN: region 1: y > O n (cover) c reflected .rimar V wave 1’ wave y ’30 V==8hlfi region 2: -t < y < O -a e nf (file) region 3: y < -t n. (substrate) Figure 5. Tri-layered structure with currents issersed exclusively in the cover region. 16 6. t t R1c ' ‘pc ’1”‘Pc’p1’ Rc1 (P1 pc”‘pc’p1’ t 2 t -2 ch 2Nfcpf/(pc*pf) ch : 2ufcpcupc.pf) where Nfc 8 (nf/nc). Calculate tangential reflection and transaission coefficients associated with the file-substrate interface as at = (p -p )/(p 8p ) Tt . 2N-zp /(p +p ) sf f s f s fs sf f f s where Nsf 8 (nB/nf). Calculate the noraal reflection and transaission coefficients associated with the cover-file interface as n 2 2 n 2 Rfc - (Nfcpc pf)/(Nfcpc.pf) ch 2pc/(ufcpcIPf) Tn - 2N2 /(n2 + ) fc - fcpf fcpc pf ° Calculate the norsal reflection and transaission coefficients associated with the file-substrate interface as n 2 2 n 2 R31 ("sfpf pa”‘"s1p1’ps’ T13 2’1""¢1p1'93’° Calculate interaediate expressions t _ _ t t ~2p t D - 1 Rchsfe f n _ n n -2p t D - 1 1 RfcRsfe f . Calculate the interaediate coupling coefficients as 2 2 n (n -1)Tt (1 . at. 2p1t t C1 ‘ fc fc cf sf. ”D ‘"1cpc’p1 C a u2 (n ~1>Tt Tt /Dt(N p +p ). 2 sf sf of fs .1 1 Finally, the reflection and coupling coefficients are evaluated 17 2.2.1 regi dete Poin IlBE adj; tov: tie: in ‘ Obs: is . Pat: V2( SCa l I _ t t t t -2p t t 3 t t -2p t t : Rt Rfc + chchR.fe f ID ‘(Rfc O R.fe f )/D : I l 8 n n n n -2p t n 8 n n -2p t n : Rn Rfc + chchR.£e f ID (Rfc 1 R.£e f )/D : I I _ n n -2 -2p t n : C - C1 + ch(R.focC1 + C2)e f ID : g n -2p t n -2p t n : [C1(1 . R.fe 1 > . Cszce 1 ]/0 : 2.2.3 REFLECTED GREEN'S DYAD FOR SOURCES IN THE FILH Consider the situation shown in Figure 6. Electric current density J, iaaersed in the file region, saintains electroaagnetic fields in each region. Again, the y dependence of the reflected wave say be correctly deterained by use of a physical picture. As seen in Figure 6, a source point at y' produces a prisary disturbance in the file region. Trans- aission and reflection of the priaary wave occur at the interfaces of adjacent regions. A wave which is reflected froa one interface travels toward the other interface where it experiences transaission and reflec- tion. Figure 7 shows that there are four fundasentally different ways in which a wave froa a source at y' aay arrive, via reflection, at the observation point at y. The y dependence of the reflected Green's dyad is coaprised of four teras with phases associated with these distinct paths. Using Figure 7, the phase path lengths are 91(y,y',t) 8 -y-y',_ a2(y,y',t) 8 y+y'+2t, 03(y,y',t) 8 y-y'+2t, and a“(y,y',t) 8 -y+y'¢2t. Scalar coaponents of Sr say be written as C’u-Ir') Rt(A) t 4 1 e-pf(01(y'y"t)l jANr-r') 2 G’u-Ir') = H r. R"(A) —— e d A D 1 2 i=1 2(2n) pf G’u-Ir') C (A) c 1 , 18 re: re Figure A region 1: y > O n (cover) c 7 ' 0 22¢ reflected .rimary wave wave region 2: -t < y < O -a .- ‘ '. a n1 (file) V reflected primary y : 't wave wave region 3: y < -t n. (substrate) Figure 6. Tri-layered structure with currents iaaersed exclusively in the file region. 19 -.+ ‘u‘. F19Ur «o1- (a) -m e -a e ~< n O -n 8 Figure 7. (d) Four paths with different y-dependent phases. (a) -y-y'. (b) yvy'+2t. (c) y-y'02t. (d) -y0y'92t 20 The 2.3 GTE p01 9V! ti where RE, R2, and C1 are derived and detailed in Appendix B. 2.3 ELECTRIC DYADIC GREEN'S FUNCTION In this section, calculation of the electric field in the tri-lay- ered environsent is aade. The electric field is related to the Hertzian potential as given in equation (A.5) by E 8 (k2 + VVW)H. Obviously, evaluation of derivatives of n are required to calculate E. The evalua- tion of these derivatives desands the use of strict aatheaatical rigor. 2.3.1 DERIVATIVES OF THE HERTZIAN POTENTIAL Each of the reflection and coupling coefficients appearing in the reflected dyad is a bounded function of A as lAlee. Pole singularities are present in the coaplex A-plane and are associated with surface-wave phenosena. Under the assuaption that all aedia have liaitingly saall dissipation, these surface wave poles are located off the real axis in the coaplex A-plane. Due to the decaying exponential tera in the inte- grand of the spectral integrals of the scalar coaponents of Sr, these integrals converge uniforaly and absolutely for all r in the doaain over which they are integrated. Hence, derivatives of the reflected part at of the Nertzian potential are obtained by foraally differentiating under the values integral. In fact, the second order derivatives of Or are given by au'___(_r) 'I a_____§’(rlr') 1m; dV' axaaxb V 3x ”8x jasi Special attention is required in deteraining derivatives of the princi- pal part of n. The principal part up of the Nertzian potential is represented by 21 the I C tha pact by It Spli diff Tang brac deri Pria the 'hic the . the spectral integral on the right side of (3). It is shown in Appendix C that under the assuaption that J and VbJ are continuous and of cos- pact support in V, derivatives up to second order of up say be obtained by foraally differentiating under the spectral integral. Therefore, 12” W.I emu-[Iv gery. rogue") 1 dV'ledA. p Vveflp(r) 8 (2n)2 Splitting the spatial integral into regions in which g is continuously differentiable yields I” 11:7.Ie.1"""[Iy q 9pm,, '13—'-;:1’ale¢12 . II w-IJ""IIy,>y gp(A;y,r') 3):: av'IIazA). Tangential derivatives (i.e. derivatives with respect to x and z) of the JA-r bracketed terns above operate only on e . However, perforaing the VVOflp(r) 8 (2n )2 derivatives with respect to y deaands additional considerations. Appro- priate use of Leibnitz's rule (15, pp.321-325] for differentiation under the integral sign reveals that §_, I p . , J(r') , I p . , J(r') , 8y I y,y g (A,y,r ) Jae dV 1 1 ._. 3_ P . . J‘t') . I 3_ P . Jtr') . Iy (y 3y (A, y,r ) “:1 av . y ,y 3’ (A:y,r ) 5"1 av I -3)”. .‘P1(Y’Y ) {Pity #0 J__(r') ' 2’1:1 291 188 y' =7 p(A,y,r ') J" ’ av' +I a p(A;y,r ') J—L-av' a_ IY' '<7 3? we1 Y"? 37 $351 which is the result obtained by foraally passing the derivative under the integral sign. However, perforaing a subsequent differentiation 22 vith r thick tern. 'her The the DOE with respect to y shows 2 a I p J(r') , I p r’ 1( “z' -- , (A; ,r') dV + , (A: ) dV' 2 2 a 1 §__ ( '1 , = J‘Y'y 2 °p(x'y'r I, 1%)— dv 8y “i 3y “i W” -jA-r a__e 91‘: V ’ a_e p1‘V V’ J_(r') a , --- --'- x dz' 37 291 i881'v'8Y 2 2 a p J(r') 3 p J( 5') , =I g (A:y,r r') av' . I , g (A:y,r r') av 2 2 y 'v a, 3881 II J(x’121z') .-jA8r' dx'dz'. Jae1 vhich is altered free the foraal result by the presence of a correction tera. Finally, evaluation of VVODP yields Wollp(r) 8 II [Iv' 9(A3r,r r')o 3‘2) 1 av J a A y J1(x' ,y,z' ) _ , e VT“- dx'dz' ] e1»: dzA (2n )2 II [ Iv ;(A:r,r')o l££-'dV' ] dzA - ieJ(r)/J.¢i (8) “1 AA where L 8 yy and the dyad ; is given by the expression - ' - - V 2 “71.3”” r ’ e "1‘V V ’18" Pi). WW 8 g(A:r,r') 8 JA8(r-r ) .-p1(y -y) 2 W [e /8n p11, y'>y. The tera L-J was extracted froa exploitation of the Fourier inversion theorea [17, p.315], and is found to correspond exactly with that ex- posed in [18) for a 'pillbox' principal voluae. The fora of 5 suggests 23 that the 'slice' exclusion in Figure 8 sight be a sore natural principal voluae pertaining to i. This assertion is verified below. 2.3.2 DEVELOPHENT OF THE PRINCIPAL DYAD Using (8), the principal part of the electric field say be written as Ep(r) = -jqu II{.IV 3'(x;r.r')-J-J(r') dV' - i-J1r113ue1 (10) 6 where V6 is the slice in Figure 8 excluding the singularity of 3’ at r, and E9 is given by . -p (7-y') 2 - 2 H (i + W/ki) 25"" r ’ L-L—y—d x. Y'y. i 2(2n) p1 Note that the differential operator in (11) say be passed outside of the spectral integration. In fact, the slice principal voluae is equivalent to a pillbox when the differentiation is perforaed lastly. A classical developaent is used below to establish this equivalence. 2.3.3 EOUIVALENCE 0F PRINCIPAL VOLUNES Starting with the cannon representation for the free space Green's function v(rlr') 8 e-3ki'r-r lMnlr-r'l, it is shown in Appendix D that for a slice principal voluae. the correction tera Ec(r) for the electric dyadic Green's function for field points in the source region is l jusi c Q a m - I V'w(rlr')n'-J(r') d5' (12) £35 55 where 56 is shown in Figure 8. The correction tera above is now shown to correspond to the correction terss appearing in equations (9) and (10). The surface integral tera in (12) is split into integration over 26 S1 and 52 (planes at yté respectively). This yields for (12) c - _ 1 _ I . . . . . E (r) - 30:1 #16 < 51? w(rlr ”y“ '7’5 Jy(x' ,y*6,z )dx dz ’ Is As 6+0, S1 approaches S2 and Jy(x'.yt6,z') approaches Jy(x'.y,z') due to (13) V'w(rlr')ly, J (x' ,y-6,z')dx'dz'). , '-y- 6 y the saoothness of the boundary of V and the continuity of J at y'Iy. Thus, (13) siaplifies to y'syes Jy(x'.y,z')dx'dz' (14) c , _l__ I . . E (r) 10¢ m S V (n(rlr ) ' i y 8y-6 where S extends over the x'-z' plane. Expressing V"w in Cartesian fora .‘Jkin A V"w(rlr') = (-jk1-l/R) ----'tx(x' -x) o y(y' -y) + z(z' -z)] 4nRz where R=lr-r'l, it is found that v'=y*6 ,‘Jkiks . V'Mrlr') 8 (-Jk1-1/R) —-T- y26 (15) y'8y-6 QNRG 2 2 2 x where R6 = [(x-x') + (z-z') o 6 J . Substitution of (15) into (14) yields (3‘1“; Ec(r) - -——— fi§$fl (y Is 25(- -1k -1/Rs) ----'Jy(x ,y,z )dx'dz 1. (16) 4nR 6 The integral in (16) say be deco-posed into the sun of integrals , over S - Cv and CV. Cv is a circle centered at (x,z) with radius v. As 660. integration over S-Cv vanishes. If v is chosen sufficiently saall, then Jy(x',y,z') 8 Jy(r) and e 3k cRG 8 1 so that (16) becoaes 1 Jk1+1/36 EC (r) 8 - 3--'ny (r) £15 (6 IC— 2 6dx dz ) (17a) ‘1 v 2"R6 27 Jk +1/r6 2n = - ny (r) (6 I0 do I: —pdp) (17b) 36:1658 2nrz 6 1kaiV1/r6 = - j“ _pdp) (17c) 1 6 2 2 where r6 = (p +6 )V. In going froa (17a) to (17b), integration over Cv has been transforaed to polar coordinates. Perforling the angular inte- gration is trivial and yields (17c). Noting that the integrand in (17c) is a perfect differential, the tera in braces becoaes 3k +1/r6 v 2 2 x 2 2 -5 6 I: —pdp 6 (3&1 1n(p +6 1 - (p o6 1 1 6 2 2 x 2 2 -x 6 (Jki [ln(v *6 ) - 1n6] * [1/6 - (v *6 ) I) 1 (as 620). (18) Finally, substitution of (18) into (17c) yields A Ec(r) = - -;— y J (r) which is precisely the saae correction tera appearing in (10). There- fore, the correction tera for the slice exclusion is identical to that for a pillbox. Hence, this establishes the equivalence of these princi- pal voluaes. 2.4 SUHHARY In a tri-layered dielectric configuration, the Hertzian potential u in a current carrying region decoaposes into principal and reflected parts. The principal wave is that wave which propagates directly froa the source to the point of observation. Surface polarization currents, which are induced at the boundary of adjacent regions by the priaary 28 wave, account for the reflected part of the disturbance. Integral representations for n say be expressed in either spectral form as n (r) = 1 H .Jx-rH gp().;y,r'1 J" ’ dv'}dtdz (191 1 (21112 V 3"1 +I Era-(ru- £53- dV', V jusi or in a sore standard fora as - , J(r') 01(r) $55 IV-v E(rlr ). 10:1 dV' (20) where E 8 Ep + er and v is any voluae which excludes the singularity point r'=r of Sp. Scalar coaponents of a are represented as two dimensional spectral integrals. The spectral fora (19) is useful in practical applications, while the standard fore in (20) say be sore suitable for theoretical purposes. The electric field corresponding to the Hertzian potential is given by E 8 (k2 + vv~1n. It is found that use of the spectral repre- sentation for the principal part of u yields a natural depolarizing dyad i 8 ;§ in the foraulation of E. Integral representations E are ex- pressible in either spectral fora as s 2 s E(r) 8 -Jqu II ‘{IT g.(A;r.r')-J(r') dV'} d A - L8J(r)/Jus1 (21) 2 I + (k + w-1 Iv E(rlr'h Maw i 30:1 or in the standard fora 2 I em = (1: + mm 115! Ecrur'h J—‘Ll dV' (22.1 i v V-v Jae1 29 =9 ' I O 8 = -jqu 115 Iv-vss (rlr )-J(r 1 dV - L8J(r)/Jusi (22b) +I (13+ w-1 (Ya-Irw- M dV' v 1 Jssi where 69 is given in (11) and V is a the slice exclusion in Figure 8. 6 As with the Hertzian potential, the spectral fora (21) of E is useful for nuaerical analysis, while equations (22a) and (22b) are appropriate for theoretical use. 30 AI IITEBRAL-GPEEATOR APPROACH TO IITEBRATED OPTICS 3.1 INTRODUCTION hany probleas in aatheaatical physics, which are described by a differential equation subject to particular boundary conditions, say be expressed alternatively by an integral equation. In an integral equa- tion, an unknown function appears as part of an integrand. There are several advantages to using an integral equation over its differential counterpart. First, when placed in the context of linear operator theory, integral operators often have desirable properties (e.g. boundedness) which are absent in the differential problea. Conse- quently, powerful analytic theoreas say be used to generate, and study properties of, solutions to the problea. Second, an integral equation relates an unknown function to its values throughout an entire region, including its boundary. Therefore, boundary conditions are incorporated naturally in an integral equation. In fact, when boundary conditions are inseparable, a differential foraulation is highly iapractical. With these considerations in aind, study of optical circuits iasersed in an. integrated surround say proceed. Figure 9 illustrates the configuration which is to be investigated. A dielectric obstacle, characterized by refractive index n(r), is eabed- ded in a voluae V within the cover of the tri-layered structure of Chap- ter two. Electric current J, iaaersed in the source voluae V., is the source of electroaagnetic fields. Unfortunately, none of the eleven 31 region 1: y > O V nc (cover) ' source V voluae n(r) Vgo _L-=—_i_’£ region 2: -t < y < O -n‘. an nf (film) y=-t region 3: y < -t n. (substrate) Figure 9. A general optical device iaaersed within a tri-layered dielectric surround. 32 orthogonal coordinate systems, for which the Helmholtz equation is sepa- rable, match the boundaries associated with this optical system. Hence, development of a Hertzian potential Green's dyad for this system is at best, intractable. An integral-operator formulation, based on identifi- cation of equivalent voluae polarization currents, circumvents this dif- ficulty. In the next section, this equivalent polarization source is identi- fied. The electric field integral equation (EFIE) for the field within the obstacle is constructed. Solution to the EFIE is shown directly to satisfy Haxwell's equations along with the appropriate boundary condi- tions. In section 3, the EFIE is specialized for axially invariant wave- guides in the integrated surround. A transform-domain integral equation is introduced which is used extensively throughout this dissertation. Transverse field components are shown to satisfy an integral equation which is independent of the longitudinal components. 3.2 ELECTRIC FIELD INTEGRAL EQUATION Consider the physical system depicted in Figure 10(a). The tri- layered structure of Chapter two is perturbed by introducing an optical device, of refractive index n(r), into a voluse V of the cover. System excitation is provided by an impressed current source J, within a volume V., which maintains an impressed field E1. Scattering of the impressed field occurs due to the contrast 6n2(r) 8 n2(r) - n: of refractive indi- ces between the optical device and the unifora cover. The scattered. field E8 superposes with the iapressed field so that at any point the total field E is given by E 8 E1 8 El. 33 V V source n (cover) c volume 1'80 -a + n: (film) a). ys-t n. (substrate) (a) V nc (cover) V source volume 180 -o 6 n1 (film) 9 a y--t n. (substrate) (b) Figure 10. (a) The physical system of an optical device within an tegrated surround. (b) An equivalent system in which R accounts for all effects of the inhomogeneous dielectric ob- stacle . 34 The electric field of the optical circuit in Figure 10(a) remains unaffected by replacing the dielectric obstacle with an equivalent po- larization source Peq as shown in Figure 10(b). This equivalent polar- ization radiates in the uniform cover surround and is the source for the scattered field E'. Ampere's law is used below to identify Peq. 3.2.1 EOUIVALENT SOURCE IDENTIFICATION For the physical system in Figure 10(a), Ampere's law within V is given by 2 vun(r1 8 jueon (r)E(r) (1) while for the equivalent system in Figure 10(b), Ampere's law is eq 2 VhE(r) 8 J»? (r) + JusOnCE(r) (2) where juPeq 8 Jeq. Subtracting (1) from (2), and solving for Peq yields 2 P°q(r1 = so6n (r)E(r) (31 where Peq is the excess induced polarization which augments the polariz- ation existing in the cover background. With the Green's function of the integrated surround known, construction of the EFIE for the field within the optical device may now be accomplished. 3.2.2 CONSTRUCTION OF THE INTEGRAL EQUATION Impressed field E1 is the field maintained by J in the unperturbed integrated surround. Therefore, use of equation (2.22s) yields J(r') is: C 2 31m = (1c‘= . W8) Iv 6(r1r'1- dV' (41 s where the limit on the improper integral has been omitted since the 35 excluding region is shape independent. Similarly, scattered field EB, maintained by Peq' may be expressed as q . 2 - EB(r) = (k + vv~1 I E(rlr'1- E—--‘-£-1-dv' (5.1 c V s C 2 6n2( '1 = (kc + w-1 Iv ——2’— C(rlr')-E(r') dV' (51:11 c where use of (3) has been made in going from (5a) to (5b) . Using linear superposition, the total field may be written as 2 2 I E(r) = (kc . Wu Iv 5—"-—‘—"—-’-é(r1r'1-1-:(r'1 dV' + aim (6) 2 n c so that transposition of the scattered field to the left side of (6) yields 2 2 0 E(r) - (kc + 9V8) IV é"Liz-"1'1§a(rlr')-E(r') dV' 8 Ei(r). (7) C It should be remarked that (7) is a valid expression for all r such that y > O. Nowever, to the extent that knowledge of E within V determines the field everywhere throughout the region y > O, (7) is an integral equation for E with domain r E V. hore precisely, (7) is an integro- differential equation for the unknown electric field within the optical device. Although use of (1.17b) allows (7) to be converted into a pure integral equation, the degree in singularity of the resulting kernel is much greater than that of 5. Hence, use of (7) is preferred for most analytical purposes. 3.2.3 UNIQUENESS OF SOLUTION It is well known that the solution to (7) is unique if it satisfies Naxwell's equations along with the appropriate boundary conditions. By 36 construction, solution of (7) satisfies haxwell's equations. Therefore, to prove uniqueness of solution, it is sufficient to show that the field has the proper behavior at the boundary of the optical device. Figure 11 assists in the discussion of the applicable boundary conditions. Observation point r lies on the boundary S of the optical device. Without loss of generality, r- (r’) is interior (exterior) to V and displaced a distance 6 from r along a linear path defined by the surface normal 3. Field behavior across 5 may be determined by evaluat- ing the field at r. and r- and forming the difference AE given by AE 115 (E(r’) - E(r'11 638{ [sin-’1 - 21(r'1]o[£'(r'1 - E'(r'1]}. . (31 Consider now, each bracketed term in (8) separately. First, it is shown that the impressed field is continuous at r. Under the assumption that V and V. are disjoint, the volume integration in (4) never passes through the singularity point r of 6. Hence, 6 and its partial deriva- tives are continuous for all r'6 V and r E V, whereby a standard theorem of advanced calculus [15, p.322) guarantees continuity of E1 at r. Thus, the first bracketed term in (8) vanishes as 640. Next, examine the scattered field E. as given by (5a). Since the reflected dyad fir and its partial derivatives are continuous throughout the cover, that portion of the scattered field originating from proper- ties of the layered surround is also continuous. However, analysis of the principal part of the scattered field reveals a discontinuity in the component of field which is normal to the boundary of the optical de8 vice. This claim is asserted below. Using (5a), the principal part of the scattered field may be 37 5) Figure 11. Geometry for field behavior at the boundary surface S of an optical device immersed in a uniform surround. 38 written as E8(r) p 811 2 I (k + w-1 I 69(1-(r'1- L—" ’ dV' (91 c V juec Il(r) + 12(r) where the integrals I1 and I2 are given by 2 I 11(1-1 = 1: I Ep(rlr')- k-dv' (10111 c V jssc q . I (r) . w-I §p(r1r'1- m dV' (101:1 2 V jsec Under the assumption that Jeq and V'OJ.q are continuous and of compact support in V, use of (0.4) allows (10b) to be written 12m = (- .[v V'Gp(rlr') v'oJ’q(r'1 dV' 1 jus c (111 A 1 IS n'OJ.q(r') V'Gp(rlr') dS'). In general, Jeq will not be zero at the boundary surface 5. Therefore, unlike in (0.6), the surface integration in (11) must be retained. Continuity of II in (10s) as well as the volume integral in (11) may be established by arguments used by Kellogg (19, pp.150-151] in showing continuity of static potentials and thus will not be given here. It is in fact precisely the surface integral of (11) which leads to dis- continuity in the component of electric field which is normal to S. Thus, the difference AE becomes - A AE - 3:: 115 Is [v'GPu-‘uru - v'GPu- lr') ] n'-J°“(r'1 dS' C 1 I p ’3’. ‘ 521 . V'G (rlr') n'8 (r') dS'. jsec 6&8 S r8r Decomposing this surface integral into the sum of integrals over S-Sv 39 and Sv yields 1 p r8r ‘ eq AE 8 jue 6&5 IS-S VG (rlr) _ n 8J (r) dS c v r8r (12) Q r8r k + Is v'sPuu-n _ n'-J°q(r'1ds' V r8r where 5v is a subset of S such that r 6 Sv and v is the maximum chord of S . v The first integral in (12) vanishes as 640 since V"Gp is a bounded and continuous function for r'6 S - Sv as 6+0. Evaluation of the inte- gration over Sv may be simplified by an appropriate choice of Sv. Let Cv be a circle of radius v and center r lying in the tangent plane to S A at r. Choose 8V to be the projection along n of Cv onto S. For v suf- “.88.‘8q- ficiently small, n 8J (r ) 8 n8J (r ) and the surface integral over Sv may be approximated by integration over Cv' Therefore (12) reduces to 1 I"? jusc AB: 5 - n-J’qu 1 115 Ic V'Gp(rlr') V r=r A Aligning the unit normal n along the y-axis by an appropriate coordinate rotation, it is found that O r8r .-jch6 , V'Gp(rlr') _ = (qua-um —— (- n26). r8r 4nR 6 If v is chosen so small that e-chR6 8 1, then AE becomes jk +1/R A - a: - nB-fqu- 1 m (6! —°—§ dx'dz') (131 jus C 2 c v 2"R6 which is essentially the same quantity (with n8y), modulo sign, as that in (2.17s) for the correction term Ec to the electric Green's dyad for a slice principal volume. Using this result, (13) becomes 40 AA - nnd'qu ) A: g jue c 2 whereupon substitution of jeso 6n E for JVq reveals + - “ 6n2(r-) “ - E(r ) - E(r ) 8 n -—-——-' (n8E(r )) c from which the relations 9 - Et.n(r ) 8 Et.n(r ) 2 6 2 _ - n E (r ) 8 n (r )E (r ) c norm norm A A where Et 8 an and E 8 nOE, are found to agree with the well an norm known boundary conditions for the electric field at the interface of dielectric discontinuity. This establishes uniqueness to the solution of (7). 3.3 AXIALLY-UNIFORH WAVEGUIDES Unless the geometry and electrical characteristics of the optical device discussed above are specified, there is not much that can be said about solution to (7). However, there are several interesting simplifi- cations which can be made when this device is a longitudinally-uniform waveguide. Figure 12 depicts a typical integrated optical waveguide having infinite extent along the z-axis and a general cross sectional shape CS in the transverse (x,y) plane. This guide is assumed to be optically dense with refractive index n(p) > n1, where p 8 xx 8 yy designates the transverse position vector. As a consequence of the axial independence of the index of refraction, the integral operator in (5b) is rendered convolutional in z with 8 88E 8 I é(plp';z-z')8E(p',z') dz' 41 ‘1 J region 1: y > O V nc (cover) CS ' source volume y 8 O vh——-————:i:£1 region 2: -t < y < O -uf 90 nt (film) Y3": region 3: y < -t n. (substrate) Figure 12. A longitudinally invariant waveguide deposited within the cover of an integrated background environment. 42 thereby prompting use of the Faltung theorem (16, p.1020l. Formal oper- ation on (7) with the one dimensional Fourier transform -sz r (-1 = I (-1 e dz 2 leads to the transform-domain EFIE 2 2 ~~ l 8 e(p,Z) - (kc 8 W8) JCS 50—;L’ g:(plp')8e(p',?.) dS' 8 e1(p,l) (l4) nc - i g u 8 A . A .3— A .3.- where e - F2(E), e Fz(E1), V’ V; 8 sz, and V; x 3x 8 2 Oz . The transform-domain Green's dyad a: decomposes as before into principal and reflected parts with ;z(plp') 8 52(plp') 8 ;;(plp'). The principal dyad a: is given by 8:8p '3 19: where the principal transform-domain Green's function g: is given by -p IY-y'I . p , _ I e c j£(x-x ) ( I ) - ----—-'e dt. (15) 9: P P 4npc An alternative integral representation of 9: can be obtained from (15) by considering a mapping of the real l-line into a contour Cn in a com- plex n-plane by letting t 8 Yc sinhn (where Yc 8 ((z- k;)*). Figure 13 shows the contour Cn which is determined by requiring Im(yb sinhn) 8 O, Re(yc) > O, and Im(yc) > O. Invoking these restrictions, it is found that 8. 8 tan-1(Im(ye)/Re(yc) < n/2. After transforming the spatial variables to polar coordinates, extensive complex-plane analysis shows that (15) can be transformed into u-jn p , 8 1 -jy lp—p'lsinhn gz(plp ) I; I- e c dn (16a) 43 :1111 <~ n/2 -¢+JO. ------------------- -1- O. a) "r -9. '11- ------:----------m ...-,0 <1— -“/2 Figure 13. Complex n-plane with contour C". 44 _1_- (2)_ . _ . 4n ( jn) Ho ( jyclp-p I) ...larg( jyclpvp |)l < "/2 (16b) .1. . 2" Ko(yclp-p I) ... O < arg(yc) < n/2 (16c) 2 where H; ) is the Hankel function of order zero and No is a modified Bessel function of order zero. In going from (16a) to (16b), use was ) 2 made of an integral representation of H; found in (16, p.360] while 2 the relationship between K0 and H: ) [16, p.375] was used to obtain (16c). It can be shown that (16c) is valid for -n/2 < arg(yc) < "/2. The reflected dyad a: has the form r 8r ‘ r ‘ ‘ ach ‘ r ‘ r ‘ ‘ r ‘ 9: ‘ “9:1” ’ V ax “ ' gzn V ' nglc z ’ zgztz' where the scalar components of a: are represented as spectral integra- tions with r g (plp') R (A) It t -p (Y*V') r , e c j£(x-x') g2n(plp ) 8 I Rn(k) --:;;—-——'e d! c r o ng(plp ) C(A) where as before p: 8 (28 Zz- k: and the reflection coefficients Rt and Rn as well as the coupling coefficient C are given by the expressions tabulated in the preceding chapter. There are several advantages to using the transform domain EFIE in (14) over the EFIE in (7). First, numerical approximation to the solu- tion of (7) for each I is more readily accomplished since both spatial and spectral integrations have been reduced in dimension. Second, dif- ferentiation with respect to z transforms to multiplication by the transform variable Z. Third, as shown in the next section, field 45 components which are transverse to z satisfy an integral equation which is independent of the longitudinal component. Fourth, and perhaps most important, identification of a propagation-mode spectrum may be effected by a study of solutions to (14) in the complex Z-plane. Chapter 4 is devoted to a detailed discussion of this topic. 3.3.1 EFIE FOR THE TRANSVERSE FIELD It is possible to formulate a transform domain EFIE which uncou- ples the longitudinal component from the transverse components. Devel- opment of this transverse EFIE relies upon use of the Fourier transform of Gauss' law (1.1a) ~ 2 V2 (n O) = 0. (17) Writing e 8 e A t 8 zez, (17) may be written as 2 2 v;- (n et) + jZn e2 = o. (18) Solving (18) for jZe2 yields 2 jlez 8 - -%'VE8 (n st) (19a) n 2 Vin 8-Vt8et-[ 2 ]..t (19b) n where use of the vector identity 9%- (aA) 8 QWE'A 8 V¥e8A has been made. AA Operating on (14) with 228 yields 2 ‘ - 2 £2.12Ll “ . . . ‘ . zez(p,Z) - k‘= JCS 2 (28gz(plp )8e(p ,z))z (is C 2 Ann I A 8 jle8 ICS “—i—p-l' gl(plp')8e(p',Z) dS' 8 ze:(p,£). (20) n c Subtracting (20) from (14) yields 46 2 2 6n‘ ') g I I ‘ ‘ ' I I I 0t(Pv7~) = kc ICS 72— (gz(plp )-e(p ,Z) - ztz-gz(plp he”: .01) d5 C 2 " 6n (2') 8 , , , i + vtv- I05 “2 gz(plp )°e(p ,7.) d5 + et(p.?.) c 2 2 6n ( O) . I I A ‘ 3 I I I = kc ICS 7L (gz(plp he”: ,7.) - ztz-gz(plp )-e(p .2”) d5 2 (21) 6n ( ') ~ I , , , i + t JCS —;9— V-(gz(plp He”: ,0) d8 + et(p,l). c The inner products appearing in (21) may be expanded as s s s ‘ “z" ' 'z"t ’ 'z'z'z : . e o [A P + Q r o ‘3! r 19 (220) 'z 't ’9: “at 7 “to z A A g A A 3 A . A ztz-gz-e] = z( z-gz-et + z-gz-zez ) = $( g”. + gr e ) (22b) Z 2 It 2 643‘... . (rt-[36.1 . Jé-Sz-o 8 8 ‘ 5 8 h ' A 8 Vt.['l..t + gz-zez] + j“ z-gz-et 0 z-gz-zez ) r 39 3 8 p r (C Vt"°z"t] ’ <9: ’ “at ’ ay ) J": r = [V J; Joe + 9" + g’ + 39": jZe (22c) t I t I It 3y 2 so that substitution of equations (22) into (21) leads to 3 6n2(g') I * . o o r o I o et(p,l) kc ICS n2 (gl(plp )oet(p ,Z) 4 ygzc(plp )Jlez(p .ZH dS c i + et(p,Z) (23) 53¢ ') —L— . I I I I 0 + t I65 “2 ((Vt-gt(plp )l-et(p ,Z) 0 g1(plp)j?.ez(p ,ZH dS c 47 where the function 91 is given by r p r 39 Zc Oz‘P'P" ' 9: ’ glt ’ ay ° Using (19b), JZez may be eliminated from (23) resulting in 2 _ 1 2 I 6n (2') 8 , . , , et(p,l) - et(p,Z) + kc CS "2 gz(plp ) et(p ,7.) d5 (24) c 6n2( ') —L 3 o o o r t ICS 2 [Oz-gz(plp )l-et(p ,2) d8 c 2 v' 2w) n _ ‘ 2 §fl__(_2'_) r I o o _t‘___ 0 } o ykc I05 2 gzc(plp ) {Vt-stun ,Z) + 2 . 0 et(p ,Z) dS n n (p ) c v' 1: '> t" p 2 - M . . . Vt I05 "2 gl(plp ) {Vt-eta) ,Z) 0 c It is desirable to eliminate the terms in (24) which contain Vt..t by 0 e (p',Z)}dS'. , t n (p ) integrating by parts. This is accomplished by using the two-dimensional divergence theorem A Jcs VtOMp) as . [1. amp) dl A where 05 has boundary contour r and n is the outward unit normal vector. Let g represent either ago or g1 in (24). Using the vector identity V%'{Q‘} 8 m 951 * Vb-A, it is discovered that 2 3 , V"n (p') éfl-%2-l'g(plp'){V;-et(p',l) 9 -£;--'0 et(p',l)} nc n (p') 6n2( ') 6n2( ') = ”Log—3L- g(plp')et(p',l)} - ——29— et(p',l)-Vég(plp') n n c 2 ° (25) VZn (p') - g(plp')e (p'.Z)-'----. t 2 n (p') Application of (25) along with the two dimensional divergence theorem 48 allows (24) to be written 2 3 1 2 6n (2') ' p p I et(p,Z) et(p,l) o kc 165 n: gz(plp )-e (p ,Z) d5 (26) t c t L356u :2 )tvl'9z(P'P )J'et(p' ,l) d5 c .___1L__ , r , ' ' c v' ’c u n p _t__ , ,- , , . JCS nz(p') . .t(p .Z) glc‘p'p ’ d5 A ' L- 541—249;: (9'9') Ot(p',l)0n' up C M— I I I I * Vt {I65 "2 Vtg1(plp )0et(p ,l) dS c 2 V"n (p') -t_— I I I + JCS 2 ' 0 et(p ,Z) gl(plp ) d5 n (p ) 2 - Mn!- ' I ." 0 Ir a: g1(plp ) et(p ,0 n dl C Finally, after combining terms and performing a considerable amount of algebraic manipulation, (26) simplifies to \ et(p,l) . e:(p,l) . k: Ics 6n2(p') at1(plp')°st(p',l) dS' (27) Vinz(p') . ' ' 0 I08 7:: 0 et(p ,l) gt2(plp ) dS - Ir a, gt2(plp ) et(p ,n-n d1 2 M ' . . . 0 165 2 gt3(plp )Oet(p ,l) dS c 49 where the quantities at1' 't2' and at3 are given by r 8 sip+;r;+‘ ro—agzc‘ “t1 9: “at ’ “an ’ agr ‘ 3 r p r to gt2 . ykoala . 9t (a! + alt . 8y ) r . 'v {‘Llr , r ‘39:.“ °t3 t ’ay “an “It 8y ' It should be noted that while atl and ;t3 are dyadic functions, gt2 is simply a vector quantity. 3.4 §UHHARY Due to the inseparability of boundary conditions for the general integrated optical circuit, development of a Hertzian potential Green's dyad for this system is intractable. An integral equation, based on identifying an equivalent source system, circumvents this difficulty. An electric field integral equation (EFIE) for a generally heterogeneous dielectric, immersed within the cover of the tri-layered background, may be written as 2 z I E(r) = (kc + vvw) Iv éfl—fii-l-é(rur')os et(p,l)0n d1 0 2 .6_n__<2'_> = . . . . Ics a gt3(plp )Oet(p ,Z) dS where the dyads ;t1 and Eta, as well as the vector gt2 are given by r 3 ip ArA A r agzc A “t1 ' 9: ’ ”92t“ ’ V “an ' ay V 39 ‘ A 2 r p to 't2 yl‘c:"z¢: ’ V1: (“a “It ay °t3.vt{ ’aylgzn “zt ay H“ 51 THE PIOPABATIOI-IODE SPECTIUI 4.1 INTRODUCTION As was stated in Chapter three, the transform-domain EFIE (3.29) may be used to identify the propagation-mode spectrum of longitudinally invariant integrated dielectric waveguides. Analysis of solutions to (3.29) in the complex Z-plane facilitates this identification. Use of Cauchy's theorem for contour integrals (17, pp.218-220] allows the ini- tial real-line inversion-integral of the transform-domain electric field to be deformed. An appropriate choice of contour deformation reveals that the field decomposes into two types of modes. Before identification of the propagation-modal types may be made, location of Z-plane singularities of the transform-domain field must be determined. Due to the complicated nature of the integro-differential operator appearing in (3.29), firm mathematical statements regarding the locality of these singularities do not exist. nevertheless, heuristic arguments which have intuitive appeal may be made to determine the placement of these singularities. These arguments are supported by situations for which solutions (or numerical approximation to solutions) are known (e.g. [20, pp.485-5081). Existing singularities in the Z-plane are either isolated (e.g. poles) or branch points with associated branch cuts. It is shown that at simple poles, modal fields satisfy the homogeneous transform-domain EFIE. These modes comprise a discrete spectrum which is associated with 52 surface waves supported by the waveguide. A continuous spectrum arises from solutions to the forced EFIE (3.29) along an appropriately chosen branch cut. It is argued that superposition of this continuous spectrum leads to the radiation field of the waveguide. 4.2 CDHPLEX Z-PLANE AHALXSIS Inversion of the unknown transform-domain electric field in (3.29) can be obtained as the Cauchy principal value [17, p.317] _ 1 R jlz E(r) - his ER'IR e(p,Z) e d2. (1) Deformation of this real-line integration requires knowledge of the singularities of e in the complex z-plane. Since e is described in (3.29) through the Green's dyad al' it makes sense to discuss Z-plane singularities associated with at“ It is argued that any l-plane branch point of a: is also a branch point of e. After locating these branch points, a discussion of the appropriate branch cuts is given. Finally, Cauchy's theorem for contour integrals is used to determine the appro- priate contour used in identifying the propagation spectrum. 4.2.1 GREEN'S DYAD Z-PLANE SINGULARITIES As seen in the previous chapter, scalar components club of a! are represented by spectral integrals having the generic form e-pclyty ' .Jt(x-x') 4np d!. (2) gzab(plp') = I 'afl‘x) c The integrand of (2) has a complicated functional dependence on the 2 2 2 wavenumbers p1 8 (t + Z - k1)“. The signs of these square roots are chosen to satisfy the physical constraints which require waves to decay 53 and propagate outwardly. These conditions are satisfied when Re(p1) > O and Im(pi) > 0. Note that integration in (2) passes through I 8 O, whence p1 8 Y1 where Y1 8 ((z- ki)“. In order to ensure that p1 remain single-valued, it is necessary that 71 is single-valued with Re(y1) > O and Im(y1) > 0. Therefore, a: has branch points at fk in the complex t-plane. Branch 1 points at 1k are removable singularities, hence not implicated, since I all integrands for the components of a: are even functions of pi. 4.2.2 ELECTRIC FIELD Z-PLANE SINGULARITIES An indirect method of proof may now be used to show that e shares the branch points of 32' Assume that e is an even function of Yc (Y.’ so that the singularities at l 8 the (l 8 1k.) are removable. Then the inner product of a: with e, which appears in the integral operator for the scattered field of (3.29), yields a function which has branch points at the (2kg). Since both the scattered and impressed fields have these singularities, so does their sum. The sum of the scattered and im- pressed fields is the total field e which was assumed to have removable singularities at Z 8 the (Z 8 tkg). This is the desired contradiction which establishes that e must have branch points at fkc (tk'). In addition to the branch point singularities, e may have a finite number of isolated singularities. The existence of a pole singularity depends on the cutoff characteristics associated with the waveguide. Chapter five is devoted to a detailed discussion of this topic. For the present, it is assumed that a finite number 2" of simple poles exist at Z 8 it“ (n81....,N). The possibility that e has poles of higher order is discussed. 54 Figure 14 shows the location of the singularities of e in the com- plex l-plane. Real and imaginary parts of Z are designated {r and 21 respectively. Under the assumption that the optical system has limit- ingly small lose, all singularities reside in quadrants two and four and are infinitesimally displaced off the real Z-axis. Simple poles are confined to the region such that lit)“" > HZnH'.x > lk.l where 'k'max is the maximum value of k 8 n(p)ko. Branch cut lines originate from each branch point and extend to the point at infinity. At this stage, these cuts may be chosen arbitrarily so long as they do not intersect the initial inversion integral. Subsequent analysis involving contour deformation demands a particular choice for the branch cuts. 4.2.3 CONTOUR DEFORHATION Calculus of residues provides a powerful analytic technique for evaluating certain types of definite integrals. Specifically, the resi- due theorem (17, p.275) may be used to deform the initial real-line in- version integral in (1). The details of this deformation are given below. Consider the closed contour C in the complex Z-plane as shown in Figure 15. The line segment -R < Zr < R is closed in the upperhalf (lowerhalf) plane with a contour consisting of the upperhalf (lowerhalf) C of the circle Ill 8 R which detours around the branch cuts C Since R b' e(p,l) exp(jlz) is analytic inside and on C except at the points In (ith), the residue theorem guarantees 112 322] L: e(p,l) e d! 8 2113 E Resnte(p,?.)e where C is taken in the positive sense and Res"! 1 denotes the residue 55 Figure 14. Complex z-plane singularities of the transform-domain electric field. 56 Figure 15. Closed contour C in and on which the transform-domain electric field is analytic except at Z 8 In. 57 of the bracketed quantity at In (flu) within the closed contour C. It is now seen that R I e(p,Z) e -R 322 sz dz = 2n) )2 Res (“and“) - I e(p,!) e dz. (3) n Cb+C If the contours Cb of the branch cuts are chosen so that the integration over CR vanishes as R+w, then the space-domain electric field as given by (1) may be expressed as 1!: E(r) a —1(2nj t Resn(e(p,l)ejzz) - 1c e(p,Z) e an. (4) 2n b It is now apparent that the electric field decomposes into the sum of two fundamentally different spectral superpositions. This decomposition is recognized as a sum of a discrete spectrum and an integration of a continuous spectrum. The considerations necessary to determine the appropriate branch cuts Cb, for which (4) is valid, are subsequently shown. First, it must be decided in which half-plane the contour C is to be closed. Without loss of generality, this may be accomplished by assuming that the space-domain impressed source is a surface current located at 2' such that J(p',z) 8 J(r') 6(2-2') dz' where 6(2-2') is a Dirac delta. Then the impressed field e1 is of the form 1 : ’jIZ'Jl 2 u. T m LL .’ s s e (p, Z) e CSnu‘c. W ) gz(plp ). Juc dS dz where CS" is the infinite transverse cross section. Obviously, e1 is .. I proportional to e 1!: . Exploiting the linearity of (3.29), the total - I field e must also be proportional to e 3!: . Hence, the exponential 32(2-2') factor e appears as part of the integrand in (3). In the upper- half (1owerhalf) plane, this exponential factor is decaying for 2-2' > O 58 (z-z' < 0) while it increases exponentially for 2-2' < 0 (2-2' > 0). Hence, C must be closed in the upperhalf (lowerhalf) plane when 2 > z' -Jl(z-z') (z < 2') so that on C e vanishes as R+o. R' Second, the branches for each Y1 must be chosen so that the inte- grand of (2) represents a decaying and outward-propagating wave. This requires that Re(y1) > O and Im(y1) > 0 along the initial real-line inversion contour. Figure 16 is helpful in determining these branches. H 5 Writing Yi 8 (Z - k1) (z + k1) , it is seen that the arguments of each factor satisfy the inequalities o < arg(Z - k1)“ < n/2 -n/2 < arg(! + k1),‘ < 0 since 0 < 6; < n and -n < e; < O. A careful examination shows that the sun of these arguments satisfies 0 < 8. + 8- < n. Hence, the argument 1 i of Yi lies in the interval 0 < arg(y1) < n/2. Thus, on the proper branch, the positive root of Y1 must be chosen. Finally, consider the behavior of e along CR. Inasmuch as e is described in (3.29) through the Green's dyad 5:, it is intuitively ex- pected that as IZlen, e should exhibit asymptotic properties similar to those of 5:. It can be seen that for Z 6 CR' 3 increases without limit (approaches zero) as Rea for Re(yi) < O (Re(yi) > 0). Hence, 322' *5: max le(p,l) e l 8 O. (5) In order to ensure that CR remains on the proper branch for which Re(yi) > O, the branch cut emanating from k must be the boundary which i separates the proper and improper branches. Thus, Cb must be the con- 2 tour defined by Re(yi) 8 0. Observe that when Re(y1) 8 0, Y1 satisfies 59 Figure 16. Determination of the prOper branch for each Y1- 60 r 2 2 2 both (i) Im(y1 ) 8 O and (ii) Re(y1 ) < 0. Writing k1 8 ki- Jki, 71 may be written as 2- [(z2 :2) (k'2 k'2 Yi - r I I 1 1 1 )1 + 32 [art . k k 1 i i i from which it can be seen that condition (1) is satisfied if and only if =-" Zrli k1 1. (6) 2 -2 2 2 2 2 Along the hyperbolas defined by (6), Re(yi ) 8 Ir (Zr- k; )(lr8 k; ). In order to satisfy condition (ii), Zr must satisfy the inequality - k1< Zr < hi. (7) Conditions (6) and (7) describe the portions of the hyperbolas shown in Figure 17. A decrease in the losses associated with the cover and sub- strate implies a decrease in k; and k;. In the limit of zero loss, the hyperbolic branch cuts emanating from k. and kc coalesce resulting in the contour depicted in Figure 18. In either the case of moderate loss or limitingly low loss, these branch cuts guarantee that for all I 6 CR, Re(y1) > 0. Use of Jordan's Lemma [17, pp.303-3OSJ along with (5) as- sure that 3" dz = o *1: ICRNp, l) e assuring the validity of (4). With the appropriate contour deformation determined, analysis of the discrete and continuous spectrums proceeds. 4.3 THE DI§QRETE SPECTRUH A discrete-mode propagation spectrum has been shown to arise from evaluation of the residues of e at poles in the complex l-plane. In practice, it is an extremely difficult task to determine these residues 61 k. 7|- , .i. Figure 17. Hyperbolic branch cuts. 62 F1Cure 18. Coalesced branch cuts. 63 due to the unknown order of the pole singularity. In this section, a recursive procedure for obtaining the residue through knowledge of the order of the pole is presented along with a method to determine the pole order. Finally, a discussion of the regimes of surface-wave leakage and purely-guided waves is given. 4.3.1 DETERHIHIHG THE RESIDUE Under the assumption that e has no essential singularities, the principal part of its Laurent series terminates in some neighborhood of Zn. Assuming that e has a pole of order H at Zn, its Laurent series may be written e_u(p) e_1(p) a . e(p,Z) 8 -————————-+ ... 8 ----- 8 E e (p) (Z - Z ) (8) H 1 m n (Z - Zn) (Z - Zn) m8O for O < IZ - Zn) < Z for some Z. Substituting (8) into (3.29) yields a m a m i E e.(p) (Z - Zn) 8 L°p{ E e.(p) (Z - Zn) } 8 e (p, Z) (9) m8-H m8-H where the linear operator top is defined by 2 2 ““ 6n ( ') 8 , . , nop( 1 (1:; W» Ics —-2—n2 gz(plp m r as . (10) c By choosing Z sufficiently small, the impressed field e1 and the opera- tor ‘op are analytic functions of Z for all IZ - Zn) < Z and may be expressed by the Taylor series at Zn e1(p,Z) . 1: 3(9) (z - z )" (11s) I n m8O C 2b (o: - 2 2b {-1 (z - an)' (11b) p m8O pm where the coefficients in the series are given by 64 8'e1(p,Z ) i n e.(p) 8 :T‘--—7:—-- 3: n 3'; (-) ‘ (., . LT.___22___ 0pm -' az' Iz-zn. Substitution of expressions (11a) and (11b) into (9) reveals O Q U 1 z e (p) (z - an )' = z to : ._tp) (: - an )' (z - an) m8-H 1:0 °91 m8-H " i m 8 E e (p) (Z - Z ) I n m8O N G 1* D 1 ' = E 2 2b (e.(p)) (z - an) ' + E e.(p) (z - an) m8-H 1:0 Pi m8O (12) By making the change of indices q8i8m and appropriately changing the order of summations, the double series may be converted into a single series so that (12) becomes a u [flog ] . z e (p) (z - an )' . 2 t z (e - (9)) (z - z > (13) us-n' m8-H 1:0 °91 ' 1 n ” 1 + E e (p) (z - z )'. m n m8O Uniqueness of series representation allows coefficients to be equated term by term from which the following relations are obtained: H8m e. (p) .150‘0 p1(e._1(p)) ... -H s m < O (14) H8m 1 e. (p) 815020 p1(e..1(p)) 8 e.(p) ... O S m. As can be seen from (14), e-n(p) satisfies the homogeneous form of the transform-domain EFIE (3.29) 2 use ._Hcp) - (kc8 vgvgo) Icsu :; ’3‘“ (plp )8e _l(p) d5 = o (15) 65 where 6; and 3‘ are respectively the del-operator and the transform- n domain Hertzian potential Green's dyad evaluated at Z8Zn. After solving (15) for e_ (14) may be used to obtain the forced EFIE ”I e1_n(p) - E (el_n(p)) 8 lbp1(e_n(p)). opo A recursive relation results and the residue e_1 satisfies the EFIE H-l e_1(p) - lbpo(e_1(p)) sifllbp1(e-1_1(p)). In the special case where H 8 1, the residue satisfies the homogeneous transform-domain EFIE 2 2 ~ ~ 6n ( ') = , , . '-1“” (kc8 vnvno) Ics —L-n2 gzn(plp )8e_1(p) as o. (16) c In order for this recursive procedure to be useful, knowledge of the order of the pole is required. A method for establishing the order of a pole singularity is now given. 4.3.2 DETERHIHATIOH OF POLE ORDER As was shown in the previous section, solution to the homogeneous transform-domain EFIE (15) yields a function proportional to the leading term in the Laurent expansion of the solution to the forced EFIE. Oper- ating on each side of (3.29) with the inner product operator defined by 2 . 6n (2) . <.-H' ICS d8 "2 e_a(p) c leads to the equation i 8 . (17) As asserted by Bagby, quuist and Drachman [21), the reciprocity rela- tion for the electric Green's dyad as stated by Collin [20) is valid for the transform-domain Green's dyad. Therefore, (17) can be written }> 8 (18) 8 8 E (Z - Z )' 8 . Use of (15) along with the commutative property m of the inner product allows (19) to be written Q E (Z - Zn)I 8 - (20) m81 m Substituting the Laurent series (8) for e into (20) and manipulating the resulting double series reveals a H8m z t 2 1 (Z - Zn). 8 - . (21) Since the right side of (21) is not singular at Z8Zn, uniqueness of series representation implies that either H 8 1 or H8m 151‘.._1"°p1‘.-n’> 8 0 a a a for .tl-u' 2-". a a a p -10 (22) Role order H may be deduced from (22) by determining the number of val- ues m for which l-H S m s -1. However, in all of the waveguiding struc- tures which have thus far been investigated, all poles have been simple [22]. In this case, H81 and the amplitude coefficient an of the modal function e_1 can be extracted from (20). Let e(p) 8 ane_1(p) in (20). 67 Equating the leading terms of each series in (20) and solving for an yields a 8 - /. n - - op - 1 O l 1 1 By exploiting the reciprocal property of the electric Green's dyad and using (15) with H81, an assumes the familiar form g _ 1 -jZ 2 an Jute 1V J(r)8e_1(p) e n dV / which agrees, modulo the form of the normalization, to that given by Collin [20, pp.483-485]. 4.3.3 SURFACE-WAVE LEAKAGE The physical phenomenon of surface-wave leakage from open-boundary integrated dielectric waveguides was identified Peng and Oliner [2,3] through use of an elaborate mode-matching method. A leaky wave is a surface wave which propagates in a direction deviating from the wave- guiding axis. As stated in [2,3], leakage may actually be a desirable effect in certain applications of novel devices such as a leaky-wave directional coupler. However, the waveguide is usually used as a compo- nent in an optical or millimeter-wave integrated circuit, and this leak- age can cause unwanted coupling between circuit devices. It is there- fore essential to know which modes, if any, of a particular guiding structure are leaky. An examination of the homogeneous transform-domain EFIE is used to predict whether a mode is leaky or purely guided. Suppose that e has a simple pole at Z 8 Zn. Then the residue e‘_1 satisfies the homogeneous transform-domain EFIE (16). Scalar components of the reflected Green's dyad appearing in (16) are of the form r (gz )afl n 68 -p (A)(y8y') , r , g e c jZ(x-x ) (gzn)afi(plp ) I Ww(A) _——‘""c”" e dt (23) where W is representative of Rt' Rn and C. An alternative form for ca (23) may obtained by deforming the initial real-line integral in the complex Z-plane. The Z-plane singularities of the integrand of (23) 2 H and 1(Zn- k.)* along with associ- 2 2 ated branch cuts and a finite number of poles at ZP 8 2(Ap- Zn)“ where 2 consist of branch points at 2(Zn- kc) A.p corresponds to the poles of W If the branch cuts are taken along ”A the familiar infinitesimal-loss hyperbolic contours shown in Figure 19, then the analysis of section 4.2.3 applies to the deformation of the initial real-line integral in (23) and the scalar components of g: may n be written 'p (A )(YTY') o r ' a e c p j! lx-x I (gz )ag(plp ) E ReleaB(A)J 4np (A ) p n p c P o'pc‘x""’ ’ jle-x'l -I w m Cb ab 4npc(A) d! (24) where evidently (9; )a9 decomposes into the sum of a discrete spectrum n and integration of a continuous spectrum. Observing that A.p is confined in the interval k. < A.p < kf whereas Z lies within the interval k < Z < k , it is clear that either Z > n s n max n 2 2 A or Z < A are possible. If Z > A , then Z 8 (A - Z )H must be P n P n P P P n purely imaginary and the x-dependent function appearing in the residue series of (24) is exponentially decaying. In this case, the time-aver- aged transverse power flow is zero and the surface wave is purely bound. On the other hand, if Zn < Ab, then Zp is purely real and power is transported transversely. This leaky wave propagates in the direction 69 I p W branch points at 2 2 2(Z - k )” and 1(Z - k )H n c n s Figure 19. Hyperbolic branch cuts in the complex Z-plane in the limit of low loss. 70 A -1 which makes an angle 6 8 tan [Re(Zp)/Re(Zn)l with the waveguiding axis. This agrees with the result given in [2,3]. 4.4 THE COHTIHUOUS PECTRUH A continuous-mode propagation spectrum has been shown to arise from solutions to the forced transform-domain EFIE (3.29) at each point along the hyperbolic branch cut C shown in Figure 18. An examination of the b spatially dependent functions which appear in the integrand of ;Z re- veals that a: is a spectral superposition of oscillatory y-dependent waves. Hence, the continuous spectrum is appropriately called the radia- tion spectrum. Each spectral component of the radiation spectrum may be regarded as a superposition of solutions to the transform-domain EFIE with elem- entary (point source) excitation. Consider the form of the impressed field e1 as given by i g 2 "“ 8 , 1(2'IZ) . e (p, Z) ICSu‘kc. W8) gl(plp )8 j.¢c dS : e . M s Icsa‘ I 19(plp mm 3"c d! ) dS where Ig(plp';Z,Z) is the integrand of the spectral representation of (k;8 668) ;z(plp'). A unit point source of current located at p' 8 p'. and polarized along 3 (c‘ 8 2, §. or 2) produces a component of impressed field given by [Ig(plp';Z,Z)8 61/1020. This elementary impressed field maintains the elementary radiation spectral component Ru(plp';Z,Z) which satisfies the transform-domain EFIE 71 2 2 ”“ 6n ( ') 8 R (plp ,Z,Z) (kc8 VV’) lCS n2 gz(plp )8 I (p lp ,Z,Z) d5 [3 A + tig(pIp-;t,z)- alljuec. Defining a radiation spectral dyad i by A i(plp':Z,Z) 8 E Ra(plp';Z,Z) c the total radiation field ER is given by 3-4 3‘2 a. M m ER(r) 2" ch dZ e I dZ Icsn2(p)p .t,:)- 3"c dS which verifies the conjecture [23] that the total radiation field may be obtained by a two dimensional spectral integration. 4.5 SUHHARY Identification of the propagation-mode spectrum of axially uniform integrated dielectric waveguides may be made by analyzing solutions to (3.29) in the complex Z-plane. By appropriately deforming the initial real-line inversion integral of the transform-domain electric field, the space-domain field E may be expressed as E(r) = —1- (2nJ ): Res (“p.113“) - I e(p,Z) a“: dZ) (25) 2n n Cb where Cb is the hyperbolic branch out shown in Figure 17. It can be seen from (25) that E decomposes into a superposition of two types of modes. A discrete-mode spectrum arises from evaluation of the residues of e at poles in the complex Z-plane while spectral components of the continuous spectrum are solutions to (3.29) along the branch out Cb. The residue e_ of e at a pole Zn is obtained by a recursive proce- 1 dure. Under the assumption that e has no essential singularities, the 72 principal part of its Laurent series terminates. The leading term in the Laurent series e_“ (H21) is shown to satisfy the homogeneous EFIE 2 e_n(p) - (kc8V “nV’8) ICSM :; gZn (plp ) 8 e _n(p) dS - O. c If H > 1, then higher order terms are solutions to the forced EFIE H8m e (p) 8 E l (e _ (p)) ... 1-H S m S -1 (26) m 1'0 op1 m i with top given by (10). In order for (26) to be useful, it is necessary to determine H. This is accomplished by deducing the number of values m such that H8m i=1 8 O ... for m81-H, 2-H,..., -1 is satisfied. Surface-wave leakage is the phenomenon in which a surface wave propagates in a direction which deviates from the axis of the guiding structure. Through an analysis of the homogeneous transform-domain EFIE, regimes of leaky and purely-bounded waves are identified. If A.p is a pole of the integrated background, then a purely-guided wave has a pole Zn < Ab. If Zn > Ab, then the surface wave is leaky and propagates in the direction which makes an angle 6 8 tan-:[Re(Zp)/Re(Zn)J with the waveguiding axis. 73 CHAPTER FIVE AN ASYHPTOTIC EFIE 5.1 INTRODUCTION In this chapter, an asymptotic form of the transform-domain EFIE (henceforth designated an AEFIE) is developed which is appropriate to the study of dielectric waveguides capable of supporting a surface-wave mode with no low-frequency cutoff. It is a conjecture that such a sur- face wave can exist on any axially-invariant dielectric waveguide im- mersed in a uniform surround. Although it is beyond the scope of this discussion to prove this claim, examples are given which provide evi- dence to support it. By the simple example of the graded-index asymmetric slab wave- guide, it is shown that unless the guiding surround is uniform, the ex- istence of a surface-wave at limitingly low frequency cannot be guaran- teed. Allowing the surround to become uniform, the asymptotic form of the fields and propagation numbers for the TE and TH modes are computed directly from the integral equation formulation. Specializing the situ- ation to the step-index symmetric slab waveguide, these limiting forms are shown to agree with the small argument approximation of the well known results as given by Harcuse [10, pp.13,16l. Next, the step-index circular fiber is examined. Again, both the asymptotic field and propagation number are extracted from the AEFIE. These results are shown to concur with a small argument approximation to the well known results as given by Johnson [24]. 74 Aside from the academic purpose of proving the aforementioned hypo- thesis, the (AEFIE) provides a favorable alternative to numerically im- plementing the non-asymptotic EFIE in the investigation of surface waves propagating in the regime where the phase constant approaches the wave- number of the surround. While no numerical results are given here, this is suggested as possible research for the future. 5.2 THE BEADED-INDEX AS!H!ETRIC SLAB NAVEQUIDE The geometry of the graded-index asymmetric slab is shown in Figure 20. A guiding region of thickness t and refractive index n(y) is depos- ited over a substrate region (y < 0) which is characterized by index of refraction n.. The cover region (y > t) has refractive index nc. Obvi- ously, the background of this guide is a specialization of the tri-lay- ered surround with nf 8 n.. It is assumed that the excitation of this system is uniform along x, thereby rendering the total electric field e independent of x. The x-invariance of e allows a specialized form of the transform- domain EFIE (3.30) for the transverse field to be formulated from which a pair of uncoupled integral equations for ex and ey are readily devel- oped. This latter derivation is lengthy but the resulting equations simplify the subsequent analysis. The EFIE for the x-component of field is used to show that dielectric waveguides immersed in a non-uniform . surround can be incapable of supporting a surface wave which has no low- frequency cutoff. For the special case in which the background is uni- form, the uncoupled integral equations are used to obtain asymptotic (7680) forms of the propagation wavenumbers for two independent modal types which agree with well known results for step-index guides. 75 >~< n (cover) c 7‘0 ‘L———-_-:e_8. guiding region n(y) y=-t na (substrate) Figure 20. The graded-index asymmetric slab waveguide. 76 5.2.1 EFIE FOR THE ASYHHETRIC SLAB Since both the refractive index n and the field are not functions of x, the vector gt2 and the dyads atl and at3 are the only functions appearing in equation (3.30) which depend on x'. Hotice that these quantities are composed of functions gt(plp') which have the generic form 1) wt)!" . , 8 e c jZ(x-x ) gt(plp ) I Wt(A) --:-;--'e dZ. Therefore, carrying out the x'-integration in (3.30) involves integrals of the type -p lyfy'l e c jZ(x-x') I 9 (PIP') dx' 8 I { I W (A) --- --'e d! I dx'. (1) t t 4npc Exploiting the Fourier inversion theorem [17, p.317], (1) simplifies to e-vclytr'i Igt(plp ) dx = wtusm ZYc (2) where it is seen that each of these integrals is independent of x. Use of equation (2) reduces (3.30) to the specialized EFIE t 1 2 2 I t I I I et(y,Z) 8 et(y,Z) 8 ko Io 6n (y ) ht1(yly )8et(y ,Z) dy (3) t 2n'( ') 8 I ‘———x——'h (yly') ey(y',Z) dy' o n(y') t2 6n2(z') ""t h (yly') e (y',Z) 2 t2 y n y'80 c t 6n2(y') 8 I 2 ‘t3(yly')8et(y',Z) dy' 0 n C where the dyads h and i as well as the vector ht are given by the t1 t3 2 expressions 77 3hr . p A r A A {C A "u in: +xh ataac—+y(:‘£n . ay )y (4.) ‘ an?“ (4b) bt2 8 y kc %h 8 3y ah ‘ §__ r r lc ‘t38y{ az:[hz:y+h::: h::]}y. (46) Y p r r r Scalar functions hz, hlt' hln' and hZc may be written as ‘Y |y-7" hp(YIY" 3 .__C____ I 2yc r I hzt(yly ) rt 'Y (7*? > hr (yly') . r ° Zn n 27° h:c(yly') c where the reflection coefficients rt and rn as well as the coupling co- efficient c can be calculated by letting n1 8 n. and £80 in the expres- sions for Rt' Rn' and C tabulated in Chapter two. Effecting these spe- cializtions, it is found that “2 Yc Ys z u Yowe 2m2 1) Yge 2 (fl ycoy.)(yb¢y') 2 2 where N 8 (n./nc) and as before y18 (Z - 1:1),‘ with 18s,c. As stated in harcuse [10. pp.7-8], the invariance along x allows modes of the asymmetric slab to be classified as TE (transverse-electric with e280) or TH (transverse-magnetic with hz80). Examination of lax- A well's equations reveals that the TB modes have s 8 xex. while the Th 78 A A modes have e 8 yey+ zez. Through a careful inspection of EFIE (3) and equations (4a)-(4c), this uncoupling of the transverse field coaponents becomes apparent and is manifested by a further decomposition of this EFIE into two independent equations for the x and y-components of field. TE modes are described entirely by an EFIE for the x-component of field, while the Th modes can be characterized through an EFIE for the y-compo- nent. By writing the x-component of equation (3), the EFIE for the TB modes is found to be given by 2 t 2 I I I I 1 ex(y,l) 8 k0 Io 6n (y ) hxx(yly ) ex(y ,l) dy + exty,l) (5) where the function hxx is defined as p,hr hxxw'y ) 8 hi (t e-chy-y I , rt.'Yc"” » 27¢ ' (6) Similarly, the Th modes arise froa the y-component of equation (3) and are described by the EFIE 2 l' 6n ( ') 1 7 n C t 2n'(y') , , , + Io n(y') ht2(yly ) ey(y ,Z)dy 6n2( '> "'t - —1—h (yly')e(y',?.) 2 t2 y , n y 80 c vhere h and the function h are defined as t2 1?? 2: r an 2 31: h (yly') 81:2 upo—it-o kzoi— [bro—‘39] (8a) lyy c l a 2 c 2 Zn 8y Y a? ,. h(l')8k2hr+a—-hp+hr+§£ (8b) t2” c an ay 2 It ay ' 79 Elementary forms for (8a) and (8b) can be established through insightful administrations of the following relationships: ( r l V r 1 ”It th . g_ r ' _ r 3,4 “In > ybi hln r (9.) r r {h Zc) 3‘ch 1 2 2 y e k I: I (9b) C C Z2 (9 ) c Ycrt Ycrn' c Performing the required algebra is straightforward and thus the details have been omitted. Using equations (9a)-(9c) allow (8a) and (8b) to be reduced to h < I ') = k2 (hp . h” ) lyy Y y c Z In 2 e-vcty-y I , , ,-vct (13) 2 2 where <6n > designates the average of Sn and is expressed as 2 2 <6n > 8 i’I 6n (y) dy. For nc # n., the right side of (13) may be multiplied and divided by the quantity Yc - Y. so that Y ' Y 1 k0 <6n >t 2 - 2 Yb Y. <6n2>t '3r-—-;'(yb - Y.). (14) n - n s c It is apparent that (14) is inconsistent with the hypothesis that 7140 unless n. 8 nc thereby establishing the desired contradiction. That is 82 to say that unless the guiding surround is uniform, there is no TE sur- face waves which exist at limitingly low frequency. A direct result of this development is that determination of the asymptotic propagation wavenumber for the dominant TE mode of the graded-index symmetric slab is facilitated. Letting nc 8 n. in (13) the asymptotic form of ye is found to be 2 2 Yo ~ ko (Sn >t/2. (15) It should be emphasized that in order for (15) to be valid, it is not necessary to assume that kodo. Letting nc 8 n. in (11). the only as- sumption which needs to be effected is that the exponential may be ap- proximated by the leading term in its haclaurin series. This demands that yet is asyaptotically small from which (15) implies that the pro- duct k: <6n2>tl2 tends to zero. Therefore, subsequent developments do not impose the restriction that ho must be small but rather assume that the product of ye with the maximum chord of the guide approaches zero. The asymptotic expression (15) may be specialized for step-index guides. If the index of refraction n of the guiding region is a con- stant. then the contrast of refractive indices is designated Ana. The asymptotic form of Yc for the step-index symmetric slab becomes 2 Yo “ Ak t/2 (16) where AI:2 = Anzkz. This expression is now shown to agree with the small argument approximation of the well known result given in (10, p.13]. It is shown in [10, p.13] that the mode of the step-index symmetric slab which exhibits no low frequency cutoff has as its characteristic equation 83 K tan(Kt/2) 8 Yc 22 2 where x = (n k0 - an)”. Approximating tan(Kt/2) by the leading term of its haclaurin series, it is discovered that 2 (Kt/2) " yet/2. 2 2 2 2 Using the fact that K + Yb 8 AR , it is easily seen that Yc “ Ak t/2 which is precisely the relationship given by (16). 5.2.3 Th hODE AEFIE FOR THE SYHHETRIC SLAB Deriving the AEFIE for the TH modes of the graded-index symmetric slab is slightly more involved than the corresponding development for the TB aodes. Again, surface-wave modal fields are solutions to the homogeneous version of (10). In addition, the symmetry of the back- ground implies rn 8 0. Then the small Yc approximation for hyy becomes h < I '> ~ 1/2 ,, y Y Yo while the derivative of gyy has as its limiting form -1/2 eae forY)Y' h (yly') " i 3’ yy1/2 ... for y < y' Effecting all of these simplifications results in the AEFIE 2 ( ) k° I 6 2( > < > d (17) e ~ __ n p e o I Y Y 2Y o Y Y Y Y t n'(y') { I:m n(y' ) . y‘y ) dy - I, n(y') .y(y ’ dy I 1 2 2 - -; 6n (t) e (t) + 6n (0) e (O) . 2n y Y C 84 A closed form solution to this AEFIE exists and can be obtained by dif- ferentiating (17) with respect to y. This transforms (17) into the first-order ordinary differential equation -2 which has the well known solution ey(y) c n (y). A substitution of this solution into (17) leads to 2 2 _2 ko t 6n (y) 4. _2 y _2 t n (y) “ §-’I 2 dy + 2 n (y') - n (y') Yc 0 n (y) 0 y 2 2 1 6n (t) . 6n (0) } 2 2 2 2n n (t) n (O) c from which the asymptotic form for Yc emerges as 2 k t ~-9-I 6n2()(n/n())2d YC 2 o Y C y Y' If the index of refraction n of the guiding region is a constant, then the limiting form of Yc simplifies to 2 2 y “ (n In) Ak t/2. c c This expression is now shown to agree with the small argusent approxima- tion of the well known result expressed in [10, p.16]. It is shown in (10, p.16] that the characteristic equation for the dominant TH mode is written 2 (n In) K tan(Kt/2) 8 y c c 2 2 2 x where as before K 8 (n ko - In) . A similar development to that for the TB characteristic equation shows that the asymptotic behavior of ya can 2 2 be expressed as Yc “ (nc/n) 4k t/2. This agrees with the result 85 obtained above. 5.3 A GENERAL AEFIE Attention is now directed towards obtaining an asymptotic form of the transform-domain EFIE (3.30). Since the example of the asymmetric slab demonstrates that an integrated dielectric waveguide can be incap- able of supporting a surface wave at arbitrarily low frequency, an AEFIE is developed only for those guides placed in a uniform surround. In the absence of a layered background, EFIE (3.30) becomes et(p,l) 8 e:(p) + k: JCS 6n2(p') g§(p|p') et(p') dS' (lB) Vin2(p') p * JCS -;;:;;:'0 ettp') ngz(plp') dS' -Imv p(1') ('Zh‘dl' r 2 tgzp" 'tp' " c where the principal Green's function and its gradient can be expressed in teras of the KO Bessel function as p . . .1. . gz(plp ) 2" KOWC'P’P l) p . g .1. . . .22. Vtgz(plp ) 2" Kowclp-p I) Yc lp-p'l . (19) In order to derive the AEFIE which pertains to the homogeneous ver- sion of (18), an examination of the behavior of g: and Vigg as yer-40 (r' is the maximum chord of the waveguide) is appropriate. Approxi- mating “0 by the limiting form for small argument as found in [16, p.375] shows that the asymptotic expression for the principal Green's function is p I ~_— I gz(plp ) 2 log(yclp~p I). (20a) " -'- log(ycr.) (20b) The small argument approximation for the gradient of g: can be obtained by using either the small argument approximation of (19) or by directly formulating the gradient of the right side of (20a). Taking the latter approach, it can be seen that the asymptotic form of Vig: becomes p I ~ _ A £L Vtgz(plp ) 2" lp-p'l . (21) Finally, simple substitution of equations (20b) and (21) into (18) re- sults in the general AEFIE ~ _ .l. 2 2 et(p) n log(ycr-) kO JCS 6n (9) et(p) d8 (22) 2 I I A , .1. Ir ale; _lz:1>_z 't‘P"'“' up. n lp-p'l c This AEFIE provides a formulation which affords a description of field behavior in the regime where the propagation phase constant is limiting- ly close to the wavenumber of the background. It was stated in the introduction that no attempt will be made to prove that a solution to ' (22) exists for all possible guiding structures. A general existence proof seems precarious at this time due to the lack of available theory on coupled systems of integral equations. In all practicality, numeri- cal investigation of (22) for specific waveguides may be pursued and is usually required. An exact analysis of the step-index circular fiber is 87 possible and is the topic for the next section. 5.4 AEFIE OF THE CIRCULAR FIBER Figure 21 illustrates the geometry of the step-index circular fi- ber. A guiding region of radius a and refractive index n is centered at the origin of coordinates. As usual, the cover region is characterized by index of refraction no. A natural choice of the polar coordinates (p,¢) is invoked so that AEFIE (22) can be appropriately specialized. Outward unit normal is the unit radial vector 3 where 3 is given in terms of cartesian unit vectors as 3 8 xcose + ysine. Application of (22) to the circular fiber is greatly assisted by use of the following expressions: 2 'P’P" = p r p - pr'coo1 ( ) n nc no a Yo. 09 yes 2 2 where the product of yes and log(yca) vanishes with Yes. flence, 2( 2+ 2) ~ 2 2 an: 2 1 ( > n nc nc a 09 yea whereby sisple division yields the asysptotic expression 2 2 n 0 nc 1 log(yca) " - 2 2 2 n OR a c which is precisely the the relationship given by (27). 5.5 §QHEARY It is postulated that all axially-invariant dielectric waveguides issersed in a unifors surround are capable of supporting a surface-wave 92 node with no low-frequency cutoff. The exaaple of the asysaetric slab shows that unless the background is unifora. an arbitrary guiding struc- ture cannot be assured to support such a surface wave. For the x-invariant sodes of the graded-index asyaaetric slab, EFIE (3.30) can be greatly sisplified. This independence along x allows sodes to be classified as TE or TH. This uncoupling is aanifested by a decoaposition of the EFIE into two independent equations for the trans- verse field coaponents. The TE nodes are characterized by an EFIE for the x-coaponent of field: ( z) 1: a) + 22 It 6n2( ') h ( l ') ( ' a: d ' (31> exY' 9x7: 00 Y ”‘77 QXY. Y where the function hxx is defined as e’Yc'Y‘Y I , rte'Yc‘Y" ) hxx(yly ) = . (32) 2yb The asysptotic fora of the hoaogeneous version of (31) has no solution unless the guiding surround is unifora. Allowing the background to be unifora. the asyaptotic field is a constant and the liaiting fors for the wavenuaber Yc is given by ~ 2 2 Yc ko (an >t/2 where <6n2> is the average value of an. In the case of step-index waveguides, this asyaptotic expression agrees with the ssall argusent approxiaation of the well known result given in [10. p.13] The TH aodes are described entirely by an EFIE for the y-coaponent of field: 93 e ( z) . .1: ' z) . k2 It sn'I ') h ( I ') e ( ' a: d ' (33) Y v. y y . o o y yy Y 7 y y . y t gafCY') Bhyy(yly ) ' ' + , e (y ,Z) dy o n(y ) 3y y 1 ~ 0 2 6n ( ,) ahnyny ) ' y -t - -———1- e (y ,l) 1 37 v n y'OO c where h and h are given by Y? Y? .‘Yc'Y'Y I , ’n .-chv*7 ) h ( I ') t 4- (34a) Y? Y y 27¢ ~ .‘Yc'Y'Y I _ ’n .‘Yb(7*7 I h ( I ') = - . (34b) 22 y ’ 2vc For guides in a unifors surround, the asyaptotic fore of the hosogeneous 2 versions of (33) has a solution ey " l/n . The liaiting fora for ye is established as 2 k t *3} afl)Innqud YO 2 o y c Y Y° For step-index waveguides, this asyaptotic fors concurs with the liaiting fora of the well known results in [10. p.16]. For dielectric waveguides isaersed in a unifors surround. EFIE (3.30) becoaes . 1 a! z I p 0 O 0 et(p,?.) et(p) 0 k0 CS 6n (p ) gz(plp ) et(p ) d5 (35) 2 V"n (p') . I -£——-—-—— CS 2 , n (p ) 0 p I 0 0 et(p ) Vtgz(plp ) d5 2 - If éfl-ée-l'Vigg(plp') etIP'.z)'n dl' n C 94 where the principal Green's function and its gradient can be expressed in terss of the Ko Bessel function as P ...l . gzIpIp ) 2" KOIYCIp-p I) p . , .1. . . .22; VtgzIplp ) 2" Kochp-p I) Yc 'P’P" An asysptotic fors of this transfors-doaain EFIE (designated as an AEFIE). which is appropriate for the study of surface waves propagating in the regiae where the product of ye with the saxisua chord of CS is approaching zero. is expressed by ~ _ .1. 2 2 (p) log(ycr') ko ICS 6n (p) e (p) d8 (36) ' t 2 Wu (9') - if _t 2 2" CS n (p') Ip-p'l Application of (35) to the step-index circular fiber allows the asysp- totic field and propagation wavenusber to be detersined. It is found that the field asysptotically becoaes constant while the liaiting fore for Yc is expressed as 2 2 n 9 nc 1 loegca) " - (: 2 V) 2 2 . nc Ak a These results to agree with a ssall argusent approxiaation to the well known results as given in [24]. 95 CHAPTER 81‘ APPROXIIATIOI 0F TIE COITIIUDUS SPECTRUI 6.1 INTRODUCTION In Chapter four, coaplex-plane analysis was used to identify the coaplete propagation-lode spectrus of longitudinally invariant integrat- ed dielectric waveguides. It was found that spectral cosponents of a continuous spectrus are solutions to the forced transfora-dosain EFIE (3.29) at each point along the faailiar hyperbolic branch cut. Due to the cosplexity of (3.29), nuaerical approxiaations of solutions to this EFIE are required for all but a few guiding structures. Standard nuaerical techniques such as the sethod of sosents (fish) [26] say be used to approxiaate spectral coaponents along a finite por- tion of the branch out. As the spatial frequency increases, the corre- sponding spectral cosponent of field oscillates sore rapidly. This in- creasingly oscillatory behavior deaands that an extensive nusber of bas- is functions be used in iapleaenting the hon. Accoapanying aatrix sizes becose so large that the hon technique is rendered ineffective due to liaitations on coaputation. Therefore, an alternative sethod is sought to approxiaate spectral coaponents of high spatial frequencies. In the next section, an iterative scheae is devised which say be used to generate solutions to (3.29) for high spatial frequencies. The aforesentioned inadequacies of the hon are not present in the iterative sethod, since each iterate is a closed-fora analytic function of Z. In section 3, the iterative sethod is applied to the graded-index 96 asyasetric slab. Error estiaates of the nth iterate along with conver- gence criteria and are established for both the TE and Th nodes. These criteria are shown to differ fundasentally fros one another due to the absence of charge in the TE case. Finally, the graded-index circular fiber is exaained. Fourier series analysis is perforaed and angular nodes are shown to uncouple by exploiting orthogonality. The first iterate of the angularly-invariant node is calculated for the step-index fiber which is excited by an axi- ally-polarized point source of current. A coaparison of the first iter- ate to the exact solution is sade. 6.2 ITERATIVE HETHOD The sethod of successive approxiaations is a standard aatheaatical technique which has a wide variety of applications. This technique is used to generate a sequence of iterates which converges to the solution of the prescribed problea. Heuristic arguaents are given below to soti- vate the application of the iterative sethod towards obtaining spectral coaponents of the continuous spectrus. Consider the transfora-dosain EFIE (3.29) which is expressed in operator fora as e - 2 (e) . e1 (l) 0P where the operator ‘op is given by 2 - 6n2(2') 20pm = (II; W0) Ics n2 gz(plp No) as . c 2 If 6n is in soae sense ssall, then it is plausible to clais that s can be approxisated by the iapressed field. hence. the initial approxisa- 0 tion is denoted e‘ ) 8 e1. A better approxiaation is found by 97 ' o substituting e‘ ) for e in the right side of (1). This yields for the l 0 first iterate e‘ ) = lbp(e( )) 2 e1. Continuing in this fashion, a se- quence of iterates is generated and is given by (0) i Q 3. I t e (2) e1 + 2 I2 )‘I.1). op s=l Forsally. this sequence converges to solution of (1). Error analysis sakes discussion of convergence rigorous. 6.2.1 ERROR ANALYSIS It is natural to inquire how closely .(n) approxiaates e. Typical- ly, this accuracy is seasured by the absolute error (In) which is de- fined as e‘“’ 2 Ile - e‘“’II where II-II is a suitable nora chosen to sake coaplete a vector space of coaplex-valued functions (coaplete norsed vector spaces are called Ban- ach spaces [27]). Foraing the difference e - .(n) by subtracting (2) . froa (1) reveals eIn) g "‘ ‘. _ .(n-l) op )II which by inductive reasoning becoses 0 e‘"’ = |I(l )“Ie - e‘ ’III on 98 . IIu. )“Ie - e1)“ op . IIu. )“’1I.)II. (3) Op If lis e(n) 8 0, then lis .(n) 8 e with convergence being in the nors. Sufficient conditions under which the sequence of iterates converges to solution of (l) are sade precise below by borrowing a theores fros lin- ear operator theory [27, pp.86-87]. 6.2.2 REUHARR SERIES THEOREH Suppose that top is a bounded linear operator on a Banach space B such that the operator nors, defined by Ill (flll O llfll , (f i 0) Ill.o II a sup " fEB satisfies the inequality Illbpll < 1. Then solution to (l) is unique and is given by the Reusann series [27, pp.86-B7J 1 "’ 1 e8e +E(£°)n(e) I4) n81 p with convergence of the series being in the nors. 2 For sufficiently ssall I6n 'sax' it is expected that lllbpll < 1. Unfortunately, the notoriously cosplex nature of the top sakes forsulat- ing a rigorous proof of this statesent rather dubious. Therefore. nus- erical investigation of conditions for which the iterative process suc- cessfully approxisates the continuous spectrus is usually isperative. 6.2.3 RELATIVE ERROR OF ONE ITERATION Since each successive iterate obviously becoses sore difficult to generate than the preceding one. subsequent analysis involves the detersination of the accuracy of one iteration. A useful gauge of this 1 accuracy is the relative error 6‘ ) which is defined by rel I (I) IIe-e‘ ’II erel ‘ IIeII ' ‘5’ An estiaate of the relative error can be obtained in terss of the opera- tor nors as follows. First, substituting the right side of (3) (with n = l) for the nuserator of (5) yields 2 (1) IIIz.o ) (eIII trel : IleII (6, “ext, sultiplying and dividing the right side of (6) by Illbplelll and invoking the definition of the operator nors, it is found that Ill (l (eIIII Ill (eIII 6(1) a on op op rel Illbplelll IIeII S lll lI Ill II 0P OP 2 IIl II . (7) Op Therefore, the relative error of the first iterate is bounded by the square of the operator nors. Inductive reasoning can be used to show that the relative error of the nth iterate is bounded by Illbplln.1. 6.2.4 REhARK As a final note, an observation on the consequence of the defini- tion of the operator nors is given. Whenever an upper bound of the quantity IllbpleIII/Ilell is found, it sust be greater than or equal to the operator nors. That is to say, that if a nusber c is found by sose lethod such that Ill lelll S a llell. then Ill II S c. op op 100 6.3 ANALYSIS OF THE ASYHHETRIC SLAB In this section, the iterative schese is applied to the graded- index asyssetric slab of Chapter five. Estisates of operator norss and :2; for both the TE and Th sodes are obtained on the space of bounded and continuous cosplex-valued functions equipped with relative errors s the sup nors where the sup nors is defined by IIeaII” 8 sup Iea(y,Z)I, (u8x,y). It is well known [27, pp.21-22J that this space is a Banach space, whence the theores of the previous section applies. Spectral coaponents of the continuous spectrus for the TE and Th sodes are solutions to the forced equations (5.31) and (5.33) respec- tively, at adjacent points on each side of the branch cut Cb shown in Figure 22. noting that the Green's functions appearing in (5.31) and (5.33) depend explicitly on (live), analysis is focused on approxisating spectral coaponents of field on the portion of Cb where IRe(Z)I < kc. In this regise, both Yc and y- are purely isaginary with Yc' 10° YD: Jul 5 2 2 and O 8 (k - Z ) . s s 2 2 where O = (k - z ) c c 6.3.1 TE hODE OPERATOR ANALYSIS The EFIE which describes the x-invariant TE radiation sodes of the asyssetric slab is expressed in operator fors as e 8 lTEIe ) + e1 x op x x 101 Figure 22. Cosplex Z-plane with branch cut C b. 102 where the operator 2:: is defined by ‘TEI l ' R2 It 6n2I ') h ( l ') ( ' I) d ' op ex - 0 o y xx y y 9x y ' y for Z 6 Ch. Along that portion of C function hxx specializes to b where IReIZII < kc, the Green's .-JOcIy-y I ’ rt.-JOc(y*y ) hXXIny ) 8 230a and the tangential reflection coefficient r becoses the negative and t real-valued function of I given by An estisate for the operator nors is obtained fros the following relations. TE t 2 2 Ilbp(ex)l a 1:0 Io 6n (y ) hxx(yly ) ex(y ,l) dy ' S R2 It 6nz( ') h ( I ') ( 'l d ' o 0 Y ”‘77 9"? Y 2 2 S k0 <6n >t [sup lh l] IIe ll . , xx x a Yvy Using the triangle inequality for cosplex nusbers [17, p.205], the su- presus of 'hxx' is seen to be bound by (l 0 Ir I)/20c. This provides t the following estisate for the operator nors. 2 - 2 TE (an >(ko) IllprI s _ (1 + Irtl) (8) 20c 2 - 2 - <6n >(ko) 20 : s 26 6 . 6 c s c where k°8 kot and ac: Oct are norsalized wavenusbers. 103 It is apparent fros (B) that the operator nors becoses less than 2 unity for either sufficiently ssall <6n > or sufficiently large 06. Hence, convergence of the iterative sequence can be guaranteed with the relative error of the nth iterate satisfying the inequality 2 - 2 - n81 <6n >(k ) 20 (n) 0 g 20 O + O O I C Actual error say be considerably less than the upper bound predicted by (9). An investigation of the accuracy of one iteration for the step- index syssetric slab excited by an infinite line source is given below. Equivalence of the first iterate and a first-order expansion in An2 of the exact solution is directly established. Cosparison between actual 1 relative error ‘rel and its estisate in (9) is sade. 6.3.2 TE HODES OF THE STEP-INDEX SYHHETRIC SLAB Excitation of the x-invariant TE radiation sodes of the syssetric slab is provided by an x-independent current source which is directed along the x-axis. Without loss of generality, the source density Jx is taken to be Jx(y,z) 8 I 6(y-y') 6(z-z') with y' > t. The ispressed field e: in the guiding region is given by i 2 ~~ FZIJI ex(y,l) 8 8 {(kc + 9V5) ICS gz(plp')8 1 us dS'} c .-CJO (Y "Y’ 'JKZ' 3 ko ”I 23°C (I/jsco) 6(y'-y') e dy' 8 C(Z) edocy - I .. I where C(Z) 8 - (opal/20c) e Jocy e 1‘2 is a spatial constant. The resulting total field ex can be found in closed fors by any of several 104 sethods (e.g. [9, Appendix BJ) and can be written as ex(y,Z) = AIZ) 930’ . BIZ) e'Ja’ (10) 2 2 2 x where O 8 (n ko- Z ) and the coefficients AIZ) and BI!) are given by Oc(0 + 0c) ejoct A(Z) 8 888*: 2 C(Z) (11a) ZOCO cosOt # J(Oc+ O ) sinOt Oc(O - 0c) ejoct B(Z) 8 -------- 2 6(1). (11b) 2 2060 cosOt 0 J(Oc0 O ) sinOt A first iterative approxisation to (10) is generated by the proce- dure discussed in the previous section. Specializing l;: for the step- index syssetric slab, the first iterate becoses t -JO Iy-y'I 1 2 2 . e( )(y,Z) 8 C(Z) .jOcy 4 k I An 27 94, .jOcy dy' Y 0 o 2,10c An2k2 An2k2 o 8 C(l) 1 - .JOcy - j( )(t - y) .jOcy 20 400 c 2k: (12) o - 0 2 e jOcy . 40 The inequality in (9) isplies that the relative error of the first iter- ate is of the order (An2)2. In fact, a first-order expansion of ex in powers of An2 is identical to the first iterate. A direct proof of this statesent is given below. Approxisation of ex by the first two terss in its Haclaurin series 2 in An is greatly assisted by use of the following relations. 2 2 22 a 2 a6 + (kc/20c) An + O((An ) I (13a) .thI 230°: 2 2 2 2 8 [l : J(kot/Zflc) An I e 0 OIIAn ) I (13b) 105 2 2 2 2 sinOt 8 sinOct + [(kot/ZOC) An I cosOct + 0((An ) ) (13c) 2 2 2 2 cosOt 8 cosOct - [(kot/20c) An 1 sinOct + 0((An ) I (13d) where O(-) notates 'big 0' [15, pp.141-142J of the ters in braces. How, 2 AI!) and BI!) can be approxisated to first order in An by substituting 2 (13a)-(13d) into (lla)-(1lb), retaining terss up to order An , and sanipulating the resulting expressions. Thus, 2 22 2 20 ejoct (1 + An k0/4O I A(Z) . 2 889 9 59 t 2 2 2 20 e c (1 + (l 0 JD t)(An kol20 )I c c c Cl!) 22 An k0 . 1 - (1 + J20ct) 2 40 c 2 2 2 2 20 ejoct (An kO/QOC) ,9 8(2) ' 2 30 t 2 2 2 20 e c (1 * (1 * JO t)(An ko/zo )1 c c c C(Z) 22 An k0 40 c C(l). These expressions along with (13b) with t 8 y yield the first order ap- proxisation of the field 2 2 2 2 An ko An ko JO y eXIy,Z) 8 CI!) 1 - (l + J20ct) 2 l + J 20 y e c 40 c c Anzk2 Anzkz o e 2 1 - J 20 y e JOcy 40 c c Anzk2 Anzk2 o o . C(Z) 1 - a ejacy - 3 (t - y) ejac’ 40 c c An’k' o , e 306) 40 c 106 which is precisely the first iterate. A plot of the relative error versus 06 is shown in Figure 23. The cover is assused to be vacuus with nc8 1.0. In order to guarantee sono- aode surface-wave propagation, Anti: sust be less than n). If k0 is chosen such that ko8 .57kco, where kco is the cutoff wavenusber of the first higher order surface-wave sode, then dunk: 8 3.25. It is clear that actual relative error follows the curve of its estisate closely. Figures 24 through 26 show relative field asplitudes ex/C(Z) and e(1)/C(l) versus the noraalized y-coordinate. In Figure 24, Oct 8 2 and a relative error of 0.3 results. Observation of Figure 25 indicates clearly that for Oct 8 5, field asplitude is very closely approxisated by the first iterate. Finally, Figure 26 has Oct 8 10 and the first iterate is nearly indistinguishable fros the exact field. The absence of charge in the TE sode situation affords the luxury of isproving ac- curacy of one iteration to any desired tolerance by serely increasing Dc. However, in sost instances, charge is present and it is the con- trast of refractive indices which sets a lisit on the relative error. This effect can be seen fros exasination of the TH sodes and is the top- ic of the next section. 6.3.3 TH OPERATOR ANALYSIS Analysis of the TH sodes proceeds in the sase sanner as that of the TE sodes. A cosplete description of the TH sodes is provided by an EFIE for the y-cosponent of field. In operator fora, this EFIE is written as e 8 lFHIe ) 0 e1 7 OP Y Y where the operator lg: is defined by 107 1.00 '- 0.80 L .L,4L I 1 0.60 L4_1 I 0.40 Relative error L Li I 0.20 a 0.00 TE—mode Actual error -—-— Upper bound jllllTlTlllllllllTl] 0.00 2.50 5.00 7.50 10.00 Normalized wavenumber (QC t) Figure 23. Relative error vs. 0° for TE sodes. 108 1.50 _. TE—mode - Exact field .. ——— First iterate —h l\) U1 1.00 0.75 Relative electric field amplitude 0.50 IIII1IIT1IITIrIIIII| 0.00 0.25 0.50 f 0.75 1.00 Normalized y—coordinate (y/t) Figure 24. Relative field asplitudes for re sode I6c - 2.0). 109 1.10 b 01 1.00 0.95 Relative electric field amplitude 0.90 TE—mode Exact field —-- First iterate IIIIITIIIIIIIIIIIII] 0.00 0.25 0.50 0.75 1.00 Normalized y—coordinate (y/t) Figure 25. Relative field asplitudes for TE sode (0c 5.0). 110 1.02 ._.1 O 1.00 0.99 Relative electric field amplitude 0.98 L I izzJ l TE—mode Exact field —-— First iterate TllljllllllTlTllllI] 0.00 0.25 0.50 0.75 1.00 Normalized y—coordinate (y/t) Figure 26. Relative field asplitudes for TE sode (0c 8 10.0). 111 2"“) k2!t&z(')h(l') ('Z)d' 8 e op y o o 7 ,y y Y y y . y t 22'(Y') ahyy(Yly ) . (’2 z) dy' o n(y') 3y y ' 2 ~ ' '8 6n (y') ahvy(y'y ) , Z) y t ' 2 3y '2‘, ' nc y'80 for Z 6 Cb. Again, the iterative schese is applied to spectral cospo- nents along that part of Cb where IRe(Z)I < kc. Then, the Green's functions h and h specialize to YY Y7 .-jOcIy-y I . rn.-JOc(y¢y ) h I I ') = — V? y y 2506 ~ e-JOCIy-y I _ r .-JOc(y¢y ) n h I I ') = -: yy 7 ’ 210c and the norsal reflection coefficient rn becoses a real-valued function of Z given by 2 H O - O c s r 8 -;——-—- n u a + a c s where H 8 (n /n ). s c Obtaining an estisate for the operator nors is achieved fros the following relations. llFHIe )I 8 k2 It 6n2( ')I1 ( I ') e ( ' I) d ' op y o a Y yy Y 1 y Y I Y . t 2n'( ,) ahyy(yly ) ' . ——’—, e (y ,7.) dy o n(y ) By y 2 , ah I I ') '8t - én_%z_l_rv ’ Y . (y.'z,}, y I n 3’ y'so C 112 s k2 It en'I '11) I I ') I a) d I . ' ' o o y y, y y y y . r t 2n'(y') 8h”(y|y ’ n(y') 3y O I ( ' Z) d ' 0 ’7' VI 2 , a; (yly') . I { 6n :2 ) yy n 7"11 3y I e (y',t)} ’ 7"0 2 2 ~ 5 { k <6n >t Isu Ih I] o lsu Iah 13 I] o p y, 2 p y, Y YIY' YIY’ 2 2 2 ~ + IIan (t) + 6n (0))l2n 1 [sup Ian layll } IIe II. c y',. y) r . where very crude estiaates have been used. Upper bounds for the supresa of hyy and the derivative of E)? can be obtained by using the triangle inequality. These provide the following estisate for the operator nors. 2 - 2 2 2 TH <6n >(ko) 6n (t) 0 6n (0) IIlb II S _ O + 2 (1 0 IrnI) (14) p 20 Zn c c 2 - 2 2 2 2- <6n >(ko) 6n (t) 9 6n (0) 2H 0. S _ 0 9 2 -;:--:- 20 Zn H 0 O 0 c c c s which does not vanish as DC tends to infinity. Hence, this result is fundasentally different fros the TE case. Indeed, if the contrast an is not sufficiently ssall, then the iterative solution sight fail. Hev- ertheless, an upper bound for the relative error of the nth iterate can be obtained by using (14). For the special case of the step-index sys- setric slab, the relative error of the first iterate satisfies the inequality 2 An'Ii )2 II) An. .....JL_. ‘rel S 2 0 _ . n 20 c c In the next section, a cosparison of the actual error with this upper 113 bound is sade. 6.3.4 TH EODES OF THE STEP-INDEX SYUHETRIC SLAB Excitation of the x-invariant Th radiation nodes of the syssetric slab is provided by an x-independent current source which is polarized along either the y-axis or the z-axis. For sisplicity, the source density is taken to be J(y,z) I Q I 6(y-y') 6(2-2') with y' > t. Then, the y-cosponent of ispressed field is given by 1 A 2 ~~ . F213) ey(y,Z) = y-{ (kc + W0) JCS“ gz(plp )0 .10“, d5 } t -JO (y'-y) 2 e c _ , -jlz' , o In 230 (I/Jsco) 6(y y ) e dy c = 0(2) .jOcy .. I - I where 0(2) = (ZI/Zuec) e Jocy e 3!: is a spatial constant. The re- sulting total field ey is given by ony,z> = XI!) cJOV + §III e'307 (15) 22 2 ~ n- where as before 0 8 (n ko- Z )X and the coefficients n(l) and 8(1) are ~2 ~ Oc(0 0 n 0:) eJOct A(() 8 ~ ~‘ 2 2 0(2) Zn 060 cosOt o J(n 06+ 0 ) sinGt ~2 ~ Oc(0 - n no) ejoct 8(l) 8 D(l) 2n ace cosOt . jIn‘o:+ oz) sinOt with K - (n/nc) a noraalized refractive index. A first iterative approxisation to (15) is generated by the proce- dure discussed in the section 6.2. Specializing 1;: for the step-index syssetric slab, the first iterate becoses 114 {C10 Ir 7" . e;1)(YIZ) - 0(1) : .jflcy + ha “I '---————- .Jflcy dY' 23°C _:nz,( -cJO H" y" 10 7' y8t)} n: C n0 7.0 A“ 2": -99.: 50 “a“: :I =DIzI 1 2 z ec’-32° Itonc’ ' 40¢ 2n c c (16) An “I: 2 o m .-Jocy } a 40 2n 6 c As with the TB sodes, the first iterate is equivalent to a first order expansion of ey in powers of Ana. The proof of the statesent is virtu- ally identical to that given in section 6.3.2 and is therefore ositted here. Three plots of relative error versus Oc/kc are given in Figures 27 through 29. Again, net 1.0 and the operating wavenusbers are chosen so that only the doainant surface-wave sode propagates. It is clear that the actual relative errors are such less than the upper bounds obtained through the rather crude estiaates used in section 6.3.3. The explana- tion of this result can be seen by exaaining the relative field aspli- tudes shown in Figures 30 through 35. It can be seen that estisating the field at y I 0 and y 8 t by the saxisus value of field can be a gross over estisate. However, the analysis provides a starting point froa which sore detailed studies can proceed. 6.4 ANALYSI§ OF THE CIBCULAR FIBER In the preceding section, the Banach space of continuous coaplex- valued functions equipped with sup nora provided a space on which upper 1 bounds for relative errors C:.i could be obtained analytically. The 115 TM—mode Exact field --— First iterate 1.00 e n=1.016 kCa=10.0 ‘ I 0.80 - l j I _ i E; 0.60 - I a J I <1) - 1 .3 4 I .9 0.40 — a) — a: 0.20 - (3.C)C) 'I I I I | I I I 'I [* -,'~; __' " -—- __ I ‘Ffi-T—I-T-i 0.00 0.50 1.00 1.50 2.00 4 Normalized wavenumber (QC/kc) 2 Figure 27. Relative error vs. (Dc/kc) for Th sode (An 8 .032). 116 Tlvl—mode Exact field -——— First iterate n=1.097 kCa=4.0 '0 CD 1 .1._J__J l 0.80 0.60 - 0.40 - \ Relative error I 0.20 - (3.()C) 1*1”1 I 1 1’1 I I [*1*I I I 1 1‘1 I I 1 I I I I 1 0.00 1.00 2.00 3.00 4.00 5.00 Normalized wavenumber (QC /kC) 2 Figure 28. Relative error vs. (Dc/kc) for Th sode (An 8 .203). 117 TM—mode Exact field --—— First iterate 1,00 _ \ n=1.233 kCa=2.5 ‘ l 1 4 \ 0.80 - \ r \ 2 \ s. 1 \ g 0.60 -I \ 33 1 \ a) J \ .2 q \ \ :52 0.40 - ‘ ~ z x (v _ s - a: 0‘20 _ w 4 0.00 *1 j l I l 1 l I I I l 1 0.00 2.00 I 1 I 1 4.00 6.00 8.00 Normalized wavenumber (QC /kC) 2 Figure 29. Relative error vs. (Dc/kc) for Th sode (An 8 .520). 118 1.10 '0 01 '0 O 0.95 0.90 Relative electric field amplitude L I l I 1.1 Isl l .L.L J I I l I _I l I l I I I J.J TM - mode Exact field —-— First iterate n=1.016 kca=10.0 §a=—0.95kca \\ .1 0.85 IITTTTUTIIIIIIIFIIT 0.00 0.25 0.50 0.75 1.00 Normalized y—coordinate (y/t) 2 Figure 30. Relative field asplitudes for Th sode (An 8 .032). 119 0.97 Relative electric field amplitude 0.96 TM—mode -— Exact field ——- First iterate 1 §a=0.0 .J - /\ .. /\ ...] \ f a I \ / \ :/ ‘ /\ ’I 3’ l / \ l1 - ‘ I \ I 1 _ \ / \ I 1 . c \ ’ \ 1 \ / j / fl .. llll]llll]ll'T1TfiIl| 0.00 0.25 0.50 0.75 1.00 Normalized y—coordinate (y/t) 2 Figure 31. Relative field asplitudes for T! sode (An 8 .032). 120 1.25 b 01 0.85 0.65 Relative electric field amplitude 0.45 0.00 Normalized y—coordinate (y/t) TM-4mode Exact field —-- First iterate n=l.097 k a=4.0 La=-—0.9kcaC \ \ \ \ \ \ \ \ \ \ \ \ III1IIIIIIIITI| 0.50 0.75 1.00 2 Figure 32. Relative field asplitudes for TH sode (An 8 .203). 121 TM—mode --- Exact field ——— First iterate n=‘| ..097 kC a=4.0 1.00 .1 €0=12.0kCO "'I \ /‘ I I I x I \ cu 0.95 - I ’ I ‘ 3 ~ I l I i: a I ‘ I 'a 1 I \ I \ I ‘ E - l ‘ l l I 1 C) CL£9() “ 1 1 1 32 ‘ i 1 1 £13 I I I’ I ~ I l ‘5 0.85 J , I , 0 “ 1 1 £2 .. l \ l ‘D - \ I l I i’ " J \ a: 0.80 - / O .— —03 _ a q 0.75 1 I 1 I1 I I I I 11 I I I 1 TI I I 1 0.00 0.25 0.50 0.75 1.00 Normalized y—coordinate (y/t) 2 Figure 33. Relative field asplitudes for Th sode (An 8 .203). 122 0.90 .0 oo 01 .0 00 O .0 \l 01 0.70 0.65 Relative electric field amplitude I J I I l I I I I 1 III I I l I 1.1 1.1 I I I I III 1.4 I_J 0.60 TM—mode —— Exact field -—— First iterate n=1.233 kCa=2.5 €0=—O.5kCO *1 1* I I 1 I‘T 1’ I 1 I I I’I ‘1 I 1 I I 1 0.00 0.25 0.50 0.75 1.00 Normalized y—coordinate (y/t) 2 Figure 34. Relative field asplitudes for TH sode (An 8 .520). 123 TM—mode —— Exact field --—- First iterate n=12233 kca=2.5 41.00 __ /\ §0=15.OkCO _ I : \ \ I \ A I 1 ‘ ' I 0.90 -I I l , e . ' I .2 j I l I I 2.: —I 1 1 o. l E 0.80 -_+ 1 l l o .. i l . ‘o * I ‘ l 1 L ’ ‘63 j l I I l I ;,-_ 0.70 _ l l 1 1 l g) 4 ‘ i ‘ 1 I i c — I I I l I 45 J i 1 . l gam— I, I! lI 1 ‘D ‘ l - l l l a) l 1 l .2 I I , I I l I 1 E 0.50 :7 \. t \/ V g .I (3.11C) I I I 1’ 1 I I I I‘1* I 1* I I 1 I I 1’ I 1 0.00 0.25 0.50 0.75 1.00 Normalized y—coordinate (y/t) 2 Figure 35. Relative field asplitudes for Th sode (An 8 .520). 124 relative sisplicity of the operators allowed sose rather crude estiaates of operator norss to lead to useful results. Attespts to estisate oper- ator norss of sore cosplicated operators by sisilar sethods often fail. An exasination of the operator for the radiation sodes of the circular fiber exposes this difficulty. Consider the circular fiber of Chapter five sodified by allowing the refractive index to have radial grading with 6nz(p) 8 anlp). Then, specialization of (5.35) leads to the coupled pair of integral equations e ( I) 8 e ( Z) (17a) 9" pp’ 39 2 d": [e (p',“ gp(plp p'n1 “C ’N p l 2 i _ §_ a6n (a) P '8 M p I D 0 0 Mp.) [1:8lp.1)az s t e < > 25"” 9 me P e (p) = E e (p) .Jnu Jnv n M 0 e (p) (p) e Jnv e (p) = E e (p) e CF‘DP’C where the cue-ation extends over all integer valued n. Note that the Z- dependence of e and e has been suppressed. Substitution of these ser- ies into (17a) and (17b), with the suasing notation isplied, yields a 3n. . i jnu _ §_ a6n (a) epn(p) e em(p) e 8p ——n2 c I a— 20" ') I I I l * 3p 0 '37391"opn(p ’ "n(p.p :v) 9 do epn(a) hn(p,a:e) (18a) a 2 a e to]. 6n (p') em(p') fl:(p,p';e) p'dp' o a 2 2 g + k I 6n (p') e (p') I (p p'-e) p’dp' o o m n O 0 126 O a jnu i Jnv _ %.§_ 2§£;L21.. (.) ”n(p'.3.) (18b) e ( ) e 8 e ( ) e on p on p v n c . 2n'(p') - (p') Hn(p.p':v) p'dp' ,lLI p 3v o n(p'>_ an a 2 2 I I c I I I 9 k0 Io 6n (p ) .an“, ) lin(p,p :e) p dp z . 2 I I . I I I - k0 Io 6n (p) em(p) “n(p,p :9) p dp where the quantities Hi, fig. and an abbreviate the integrals H:(p,p';efl cos(9-o') “ I H:(p,p':e)) ' I sin(e-e') e3". g:(plp') do'. -n “n(mfm)J 1 flaking the change of variable 6 = u - 0' gives h:(p,p':a) cose‘ n s , ' - 1, Jne I -jn0 (1) fln(p.p 3e) ‘ e ." sinO ) e "0 (0°80) d6 (19) ln) (2) (1) n_1(O p J (O p ) fl (Ocp>) I .. , , 2 Jay - “n(p,p ,9) 4 e (J (O p<) H c >) “.1 c ‘ n+1 n-l c (3) .. g _ in. luv Hn(p,p ,Q) 2 e Jn(Ocp<) "n (Ocp>) After substituting the expressions above into equations (18a) and (18b), the orthogonality of the cosplex exponentials are used again allowing Fourier coefficients to be equated ters by ters. This results in the following pair of coupled integral equations. 15 6n2(a) 2 a n c a _ 1n 3 I 2n'(g') , (1) , , 2 5; o n(p') epn(p ) Jn(°cp<’ H" (Ocp)) p dp 1 e (p) = .1 (p) . [e (a) (a a) a‘ ’(0 a)! J'(O p) (21.) n pn pn c n c n c p a - in 3 I 1 . . (1’ . . ‘ kg 0 6n (p ) epn(p ) Jn_1(Ocp<) fln_1(Ocp>) p dp a - in. 1 I 3 . . (1’ . . ‘ he 0 6n (p ) epn(p ) Jn.1(ch<) an.1(Ocp>) p dp ( ) , , l c < n-l‘ocp>) p dp I on: k2 I. 6n2( ') ( ') J (a ) a 2 o o p 0.“ p n_ p (2) , , (Ocp() "n+l(°cp>) p dp +5k'r6n'w) (u: 4 0 o p .en p n+1 128 2 e (p) - ei (p) - 599- ‘“ m [e m n‘ ’(a a)! J(O p) (211:) on on 2 p n2 pn n c n c c . 5,5. 2n'( ') , (2) , , . 2 p In —-Ln(p,) em(p ) Jn(Ocp<) an (06(3)) p dp .Ek2 I. an” ') (')J (0 ”(mm ) 'd ' 4 0 o p .pn p n-l cp< n-l cp> p p -Ekz I. 6n’( ') (')J (a )n‘3’m ) 'd' T o o p .pn p n01 cp< n91 op) p p 111:1 I. Ain't ') (w J (a > am”: > 'd ' 4 0 o p .en p n-l cp< n-l cp> p p 11k: I. 6n2( ') (')J (9 ”(mm ) 'd' 4 ° 0 p .wn p n+1 cp< nol cp> p p It can be seen that the estiaates used in the previous section are not adequate to obtain analytic bounds for the operator nora. nevertheless, a single iteration sight still provide a 'good' approxisation for wave- guides with sufficiently ssall contrast of refractive indices for those spectral coaponents of sufficiently high spatial frequency. As a sisple exasple, the n 8 O sode of a step-index fiber is investigated. Excita- tion is provided by an axially-polarized point source and a cosparison of one iteration with the exact solution is sade. 6.4.2 IHPRESSED FIELD Consider an axially-polarized point source of current located at (p',z') where p' > a. Then, J I ; J 6(pvp') 6(2-2'). As usual the iapressed field is given by F (J) jeec 1 s 2 an. F ' e (p) (kc. W0) Ics.gz(plp ) dS'. 129 The radial cosponent of field can be written 1 = 2— -1 (2) I "jZZ' a- g o ep(p) 3: 3p I65: ‘> no (OOIp-p I) (J/Jsec) e 6(9 p ) as fros which use of (20) allows the Fourier coefficient eta to be extract- ed. Then, e (p) = - JJ—(zo )umm p') (3‘2' o""" J '(o p) C n c n c i pn 406° and for n = 0 i . jJ ( ) (2) , -jlz' epo(p) ‘.‘c 20° "0 (Cap ) e J1(Ocp) . Ml) Jimcp) (22) (2) where A(Z) 8 (jJ/4oe )(ZO )"o c c -Jzzl (Ocp') e is a spatial constant. Sisilarly, the e-cosponent of field is expressed as i 8 _1) , -jlz' e.(p) 31- p '2‘, Icsfl‘i' no (Belg-p n (J/jeec) e 6(p'-p') dS' so that e1 becoses en i . J (2) , -jzz' .-Jne' ew(p) ("‘c z)an (Gap ) e (3% (0 Op). Therefore, the e-cosponent of the n 8 0 radiation sode is not excited by a z-directed current source. 6.4.3 SPECIALIZED EFIE lhen equations (21a) and (21b) are specialized for the n 8 O sode, an uncoupling of the transverse coaponents results. To see this. invoke (2 ) (3) the relations J 8 - J and H 8 - flI found in (16, p.360). Then, -1 (21a) becoses 130 a (p) o .i _ :5 6n (a) (1) p0(p) no [epo(a) (Pea) no (Oca)l Jo(Ocp) c ‘90 a - 15.2..I Zniieil . (') . . 2 3p 0 n(p') epo(p ) Jo(0cp‘) Ho (Ocp>) p dp a - H II 2 o o (I) o o 2 ko 0 6n (p ) epo(p ) Jo(0cp<) no (Gap)) pldp which for the step-index guide sisplifies to - 12.9n. (1) (p) 2 n' tepo(a) «no.» no (ac-)1 Jo(0°p) (23) c (p) 8 e1 ”no 90 p - 1.! 2 I a I I I I I (1) 2 ko 0 6n (p ) epo(p ) Jomcp ) p do "a (Pop) - 1. " k2 I. 6nz( '> < ') n"’ 1cppdp2 1cp Ocp Oc°cpicp' The second integral is evaluated in Appendix E (5.9) and is given by 3 m L m m I J1(Ocp) a, (flap) pdp - 2 [11(Ocp) u1 (Ocp) . Jomcp) no (cap): (23) _ 2_ (2) ac J1(Ocp) Ho (ch). Substituting (27) and (28) into (25) gives first iterate (1) 111A."— e (p) 8 AH.) :Jlmcp) - 2 2 ((Oca) JIHOJ J:(0cp) 2 2 - m 2 [9. 2 m L 2 m 2 a: 2 Jomcp) uI (Ocp) + 2 J1(Ocp) uI (Gap) (2 0 2 2 "— J°(Ocp) J‘mcp) a: )(Ocp) . L J‘mcp) a ) 0 0 (Cap) c c 2 2 & (3) - L 2 (2) 2 Jomcp) JI(Ocp) no (cap) 2 J‘mcp) II1 (Gap) 2 a. - .I. 2 (Jan0 . J‘ng Jimcp) °c (Jlflo) Jl(0cp)]} 9 where Bessel functions without specified argusents are assused to have argusent Oca. Judicious factorizations and use of Uronskian derived in Appendix E (8.8) sisplify the first iterate to 2 . HAL 3 (p) A(Z){J1(Ocp) . 2 n, u: a/OOU‘ROJ Jlmcp) (29) c .(1) p0 0 2 ((Ocp) Jamcp) - 2J‘(ch)l - m 2 = 4 a: a (Julio . Jill!) J!(Ocp) . 133 6.4.6 COMPARISON As with the sodes of the slab waveguide. the first iterate (29) is 2 directly shown to be equivalent to a first-order expansion in An of the exact solution (24). The following expansions facilitate this develop- sent. 2 2 n - n; + An (31.) 2 AR , 2 2 Jump) 8 Jomcp) ¢ 2 (Cap) Jomcp) + OHAn ) l c Ak2 2 2 8 Jo(0cp) - - ’(Ocp) J‘(Ocp) 9 O[(An ) 1 (31b) c Ak2 2 2 31(0p) 8 J:(Ocp) + 2 (Cap) J;(Dcp) + O[(An ) l c Aka 8 J1(Dcp) + 2 ((Ocp) Joocp) - J1(Ocp)l (31c) c 2 2 + O[(An ) 1 flow, the quantity U appearing in (24) can be approxisated by using equa- 2 tions (31a) through (31c). retaining terss of order An and sanipulating the resulting expression. This reveals 2 2 A): (on . nc °c”1{~'o . a ) 2 (2 nc a 10:0.) 3‘ [Jo - (Oca)JI)} 2 29 1:20.150.) sz’ton-n’oa J ,g:_<_9_J ocean -J). c c c c0 1 202 hi 1 c O 1 c c Forsing the difference in these expressions yields (2 2 (2) 2 ) I 8 n O Jo(Oa) a! (O a) - n O J‘(Oa) “o (0 a) c c c c 134 2 2 A!_ 2 2 2 w - nc canon1 - JIHO) . 20’ {ac ochon‘ . J:fl°) - nc Oca(J°Ho . J18!) c 2 2°C (32) ' 2 Jiflo ko Use of Uronskian (8.8) shows that Jul-l1 - J‘Ho 8 21/(nOca), whereby (32) becoses 2 Ah: 2 2 2 U 8 nc 06(J2lnoca) + 20; "coc (2J1Ho 9 (jZ/nOca)) - nc Casual!o + Jlflx) c ._2 ' 2 Jlflo} ko 2 2° 2 2 l 0 a 2 AR 1n 3 ___§_ 8 (Jch/na) :1 + (l 0 (Des) (Jolio 8 J13!) - in k2 J1B°)}. c c 2 - Uith e 8 A(Z) (Jch/na) H 1 J!(Op), the first order approxisation is p0 given by 2 Aka 1! 2 I Oca epo(p) = M1) 1 - 27(1 . 2 (Des) (Julio . Jinx) - 1!! a We c c 2 Ah x (J‘(Ocp) o 2 [Gap JOOcp) - 11(0cp)l) . Am J‘mcp) . m 2 2 [(Z a/Oc)Jlflol J‘(Ocp) a” u '9” 03 2 [Cop J°(Ocp) - 21‘(Ocp)l 2 2 a: a (John . Jinx) J, 3 I \ '3 - I \ ’ .9 " I \ I Q) .— 0: a l 1 (J.C)C) ”I I I I 1 I I ‘r’I’ r I I I ‘r I I I Irizrl 0.00 0.25 0.50 0.75 1.00 Normalized radius (p/a) 2 Figure 41. Relative field aaplitudea for n 8 0 aode (An 8 .203). 142 0°50 Exact field ———-— First iterate ,~ n=1.097 kca=4.0 F3 C) h) 45 CD CD 1 l 1 1 1 1 l 1 [,1 1 1 1 l 1 1 1 1 | 1 1 I 1 I 1 l 1 L_l Relative electric field amplitude (3.()() I I I I I I I I I I I I I I I I I 0.00 0.25 0.50 0.75 1.00 Normalized radius (P/a) I I l I Figure 42. Relative field aaplitudea for n 8 0 node (An28 .203). 143 —- Exact field 080 .11 -——— First iterate : n=1.167 kCa=3.0 : §O=0.0 (D E // \\ '8 0.60 -; / \\ 22:). Z ’ \ E : ,’ \\ C) : \ :9 : \ 1.9:) 0.40 : \\ O 3 \ *3 : Q) _ 713 : .Z .- ‘6 : YB _ 0: I 0.00%]llllTlll|lll1rllll| 0.00 0.25 0.50 0.75 1.00 Normalized radius (p/a) 2 Figure 43. Relative field aaplitudee for n 8 0 node (An 8 .362). 144 Relative electric field amplitude 0.50 .0 .p. O .0 CH O 0.20 0.10 0.00 ; —— Exact field : ——— First Iterate ;~ n=1.167 kca=3.0 .: I" £0=J5.OkCO I I l 3 l 3 l q l j l 1 l I l : ‘ '3 ... l \ T. I \ 2 I I \ 1 l \ \ Z \ \ \ \ Z l l \ \ I I l \ I d l \ : l I ‘ I I' ‘ I I I I I I I l I l I I I 1 I 1 T I Ij I 0.00 0.25 0.50 0.75 1.00 Normalized radius (p/a) 2 Figure 44. Relative field aaplitudee for n 8 O aode (An 8 .362). 145 approxiaations avoids these deficiencies. Transfora-doaain EFIE (3.29) say be expressed in operator fore as i e 8 t (e) * e (33) 0P vhere the operator ‘op is given by 2 2 - 6n ( ') - , , loph) Ikco w-I Ics —9—n gz(plp I-I-I dS . Heuristic reasoning suggests that the total field should be close to the 2 iapressed field provided that 6n is adequately ssall. An iterative se- quence is generated by the recursive foraula o - e‘ ) = e1 .(n) 8 e1 0 lbple‘n 1’), (n21). A theores froa functional analysis guarantees that this sequence con- verges to solution of (33) if the operator nora lllbpll defined by Ill (flll O llfll , (I i O) lllb II a sup 9 f6! is less than one. Usually in practice, only one iteration is feasible. The relative error of the first iterate is defined by I m IIg-o‘ ’II t ' rel llell and is bounded above by the square of the operator nora. Therefore, an estisate of the operator nora provides an estisate of the relative er- ror. As a consequence of the definition of the operator nora. any esti- sate of the fore Illbplelll S a llell isplies that lllbpll S a. Analyses of the slab waveguide and the circular fiber indicate that one iteration can provide a 'good' approxiaation to solution of (33) for 146 2 2 sufficiently high spatial frequency provided that (6n ) << 1. The results quantify the accuracy of the first iterate. 147 COICLUSIOIS AID REEMHHHUIATIIIE An integral-equation foraulation, providing a conceptually exact description of the electroaagnetic field within a fairly general class of integrated optical circuits, has been presented. This foraulation affords an extreaely powerful aethod of analysis for both theoretical and practical applications. Apparently. the integral-operator approach is particularly useful in the investigation of axially-invariant wave- guides. A detailed discussion of the electric dyadic Green's function for a tri-layered substrate/fila/cover dielectric structure was given in Chapter two. Knowledge of this dyad was of utaost iaportance since it was used throughout this dissertation as the kernel in the EFIE for integrated optics. The developaent of the electric Green's dyad deaand- ed a aatheaatically rigorous treataent and revealed a natural depolariz- ing dyad appropriate for the Soaaerfeld-integral representation of dyad- ic coaponents. Again, rigorous analysis effected the subsequent identi- fication of the corresponding principal voluae. This developaent is be- lieved to be new and is the subject of a paper which is currently in the review cycle [29]. In Chapter three, a transfora-doaain EFIE was developed to facili- tate the study of axially-invariant dielectric waveguides. A straight- forward derivation resulted in an EFIE for the transverse field, thereby uncoupling the axial field coaponent. This derivation is presented in a 148 forthcoaing paper [30] as a potentially new contribution. Coaplex-plane analysis effected the identification of the propaga- tion-sode spectrus for axially-unifors integrated dielectric waveguides in Chapter four. Pole singularities of the transfora-doaain field were observed to coaprise a discrete spectrus associated with surface waves, while hyperbolic branch cuts corresponded to a continuous spectrus. Re- giaes of purely-guided and leaky wave poles were also identified. An asymptotic EFIE, appropriate to the study of dielectric wave- guides capable of supporting a surface-save sode with no low-frequency cutoff, was derived in Chapter five. It was conjectured that such a surface wave can exist on any axially-invariant dielectric waveguide is- aersed in a unifors surround. Applications of the AEFIE to the graded- index asyaaetric slab and the step-index circular fiber provided evi- dence to support this conjecture. Aside froa the acadeaic purpose of proving the aforeaentioned hypothesis, the AEFIE provides a favorable alternative to nuserically iapleaenting the non-asyaptotic EFIE in the investigation of surface waves propagating in the regiae where the phase constant approaches the wavenusber of the surround. Finally, standard nuaerical techniques were observed to be ineffec- tive in approxiaating spectral coaponents of the continuous spectrus having high spatial frequencies. An iterative schese was proposed which avoids deficiencies in such aethods. Analyses of the slab waveguide and the circular fiber indicated that a single iteration can provide an ade- quate approxiaation to high-frequency spectral coaponents when the con- trast of refractive indices is sufficiently ssall. These analyses were perforsed on the space of continuous functions equipped vith sup nora. This space is quite restrictive and future endeavors are encouraged to 149 investigate other noraed vector spaces (e.g. the Hilbert space of square integrable functions). At this tiae, there seeas to be little hope in successfully approxiaating high spectral coaponents for guides having high refractive-index contrast. Research in this area is also recoa- aended. 150 APPENDICES APPENDIX A APPENDIX A ELECTRIC NERTZIAI POTENTIAL The Hertzian potential is developed by observing the nonexistence of aagnetic aonopoles through (l.ld). Borrowing a theores froa vector calculus (15. pp.640-643], the aagnetic field R say be expressed as the curl of a vector potential. In an electrically hoaogeneous aediua. I say be expressed in terss of the Nertzian potential 0 as N = Jae van. (1) Substitution of (1) into Faraday'a law (l.lb) yields 92¢: - kzn) . o (2) where k2 = sap: is the wavenusber in the aediua. Under suitable conditions (see [15, p.639]), (2) isplies a . k’n - v» (a) where a is a suitable scalar field. Use of (1) and (3) into Aapere's law (1.1c) yields (\72 + k2”! = -J/Jss . Viv-n up) (4) where use of the vector identity vxvxr 8 WOF - VzF has been aade. Ac- cording to a well knovn theores froa potential theory [31, pp.66-67l, a vector field is uniquely detersined through knowledge of its curl and divergence. Therefore, at this stage, n is arbitrary since its diver- gence is unspecified. Choosing Vin 8 -e uniquely detersines n. Thus (4) siaplifies to 2 2 (v + k )1] = '3/106 151 which is the Helaholtz equation for the Hertzian potential subject to the Lorentz gauge vwn = -e. Use of this gauge in (3) yields 2 B = (k 0 vv-)n (5) which relates the electric field to the Hertzian potential. 152 APPENDIX B APPEHDIX B IEBTZIAI POTEITIAL flflflfll's DYAD The boundary conditions for the scalar coaponents of the transfora- dosain Hertzian potential are given in [13) and say be expressed as "co = “fc sf“ ... c = x,y,z (1a) an an cc 8 2 fa g g -3;- "fc -3;- ... c x,z at y 0 (1b) anc an: _2 -5;1 - .5;1 = ("fc - 1) (1E "cx # )1 "c2, (lc) - n2 - (1d) "fa - .1 n.“ ... a - x,y,z an an fc : 2 so = _ _ 3y ".1 8y ... c x,z at y t (le) an: an. 2 -5;1 - .5;Z = - (N.f - 1) (1t "sx ¢ J! "32) (1f) where "fc I (nilnc) and fl.‘ = (“./°f)° Additionally, the potential sust vanish as Iyl +m. These boundary conditions are isplesented below for the situation in which currents are isaersed exclusively within the file region of the tri-layered dielectric structure. 153 As shown in Chapter two, the transfora-doaain potentials decospose into reflected and principal parts. Enforcing the boundary condition at y = to, the potentials for sources in the file are given by w an) 2'ch y > o "ccu‘w c 3 + pr ’ 'PY - nmmy) unmet ovum. 1 on:(x,y) O>y> t w an) epsy -t > y "san’w s where the principal part a: of the transfors-doaain potential is -p ly-y'l J (r') _ e f ~5A9r' a , «3(ky) - 1v pr e in, dV . It will be useful to abbreviate the following quantities. Let uu(x) and va(A) be defined as P uu(x) : na(A.0) - fl. -Jk°r’ig(r_’ . - V 2 . Jss dV Pt 1 g P - vu(A) - na(h. t) - .-p£(t+y ) -jA0r' Ja(r ) ' - «‘7— ‘7'" pi in: Then, it is easy to see that anp(A,y) a = - p u (1) 3y ly-O f a anp(l,y) __g_____ 8 p v (A) 3y 'ys-t f c How, ispleaenting the boundary conditions (la)-(lf), the following sys- 154 ten of equations is generated. w n2 (w' w' + ) ( ) (2a) 8 O 3 co: fc fa fa uc1: a x,y,z 2 - p H 8 N H c cc fc(pf ua) ...(c 8 x,z) (2b) + - fa ' pfwfc ' pf - p H u ) 8 (N * - 2 8 cy ' (pf'fy ' Pf'fy ' P; , fc ' 1) (J! Vex + J: wcz) (2c) + -p t - p t - 2 -p t - "fa e f + "fa e f + va - N.f H.“ e s ...(c - x,y,z) (2d) + -p t _ - p t 2 -p t pf'fa e f pfflfu e f # pfva 8 N.f p.U.a e s ...(c 8 x,z) (2e) -pt + -p t _ - p t _ pf('fy e f '1’ e f + vy) p.fl.y e s (2f) t 2 - - = - (n - 1)(3z w e P- P t 3 IX o 3! v.2 e s ). f Tangential coaponents of potential (c 8 x,z) satisfy the boundary conditions (2a), (2b). (2d), and (2e). These relations provide a systes of eight linear equations in eight unknowns. Solving (2a) and (2b) sis- ultaneously for H and V’ in terss of V and u yields cc a fa c f w . 1‘ (w' + u ) (3.) cc fc fa c o t - "fa 8 Rcf('fa o ua) (3b) where the tangential reflection and transsission coefficients associated with the cover-file interface are t t 2 Rcf ’ ‘Pt' pc”‘”£’ Pc’ ch zufcpf/(pf. Pc" Sisilarly, solving (2d) and (2e) sisultaneously for If“ and H.“ in terss of H‘ and va yields fa - 3 t + -2p t -p t "fa Rsf('fc e f * Va e f ) (4a) 3 t + (p - p )t p t sc ng(ufa e s f + v“ e s ) (4b) 155 where the tangential reflection and transsission coefficients associated with the file-substrate interface are given by t _ _ t -2 Rsf ' (pf ps”‘p£’ ’3’ Tfs ‘ ZN-tpt"pt’ Pa" Finally, solving the resulting equations (3a), (3b), (4a) and (4b) sis- ultaneously yields w = (7‘ u . It at e'Ptt v )lDt (5.) ca fc c fc sf c + t t t -p t t 'fc 8 (Rcf “a * Rch.fe f va)/D (5b) w‘ = (R‘ at e'zptt u + at e-pft v not (5c) fa cf sf a sf a H 8 (Rt Tt e(ps- pf)t u + Tt epst v )IDt (5d) so of fs a fs a where the quantity Dt is defined as Dt 2 l - REfR:fe-2pst. Noraal coaponents of potential (a 8 y) satisfy the boundary condi- tions (la). (1c), (ld). and (1f). These relations provide a systea of four linear equations in four unknowns. Solving (2a) and (2c) sisultan- eously for U. in terss of H; . u , H , and U yields y y cx as f? -2 w’ - - a“ w‘ - R" - -!£9—:—1- { z w + z w ) (6) fy - fc fy fc uy N2 . J cx 3 c2 fcpc pf where the norsal reflection and transaission coefficients associated with the cover-fill interface are R“ (n2 )lth + ) T“ . zuz Im2 . ) fc fcpc 9: fcpc 9: fc fcpf fcpc 9: ° Sisilarly, solving (2d) and (2f) sisultaneously for HEY in terss of 'fy' v . V , and H yields y sx sz 156 N (N - l) w' = a" e’2P1t w‘ + R“ e p1t v + -5£——5£—-—- (3: w + J! I 1 fy sf fy sf y "2 . sx s2 sfpf ps (7) x e.(psO pf)t where the reflection and transsission coefficients associated with the file-substrate interface are n 2 2 n 2 R31 ‘"-1p1 ps”‘"s1p1’ pa’ T1: 291""a1p1’ ps” flow, equations (6) and (7) can be solved sisultaneously for 'fy and Hg, revealing n n n -p t -2 R R R e f N - l "fy = - -§E “y - -£E_£§___-—|VY - 3 £9 n (it 'cx . JZ 'cz’ D D (flfcpco pf)D (8) N2 (N2 1) - .i of R" .'(P.* pf)t (j! H t J! H 1 (N p * p )Dn fc sx sz sf f s n n -2p t n -p t -2 _ U- _ RfcRsfe f Rsfe f l‘fc 1 n -2p t fy - - n uy . n vy - 2 n Rsfe f D D ‘nfcpc. pf)D (9) a: (N2 1) x (j! w + J! u ) + -5§—-5£———-——-e"P-’ "1’t (1: w 1 3! w 1 ex oz (K p ’ p )Dn sx s2 sf f s where the quantity DD is defined as Dn s l + R2c82fe-2pft. Now, the reflected part of the potential in the file can be calcu- lated by inverse Fourier transforaing the transfora-dosain potentials. First, use equations (Sa)-(5d), (8) and (9) to express tangential cospo- nents as 8 t t t -p t t p y «fa(x.y) nz¢x.y> + ((Rcf uu + acfn.fo 1 va)/D 1 e 1 (10) + («at Rt {2P1t u + Rt {’1t v )lDtl 9‘91’. of s a f c sf 157 Then, application of the inverse Fourier-transfers operator r"(-) a 1 If (a) 93"” dzx 2 (2n) to (10) yields the tangential coaponents (a 8 x.z) of potential J (r') a P . . nfa(r) 8 lg! IV-v G (rlr ) 1": dV J (r') Jaef r , . + IV Gttrlr ) dV . Here, GP is the principal dyad discussed extensively in Chapter two, and r t is expressed as the the tangential cosponent of the reflected dyad G spectral integral 9'91”: .jA0(r-r') 2 4 G:(rlr') = If t R:(x) d x. (11) 181 2(2n) p1 The tangential reflection coefficients Rt are given by 1 t t t 31 - Ref/D at . at xnt 2 'f (12) t t t t a, 36:3.f/n t t 3‘ - a, and the phase lengths .1 are given in Section 2.2.3. Calculation of the norsal cosponent of potential is slightly sore elaborate due to their coupling with the tangential coaponents of cur-_ rent. Therefore, define "fyy as that part of "fy produced by Jy' Then. the tential n becoses p° fry J (r') . p . _z____. . flfyy(r) $33 IV-v G (rlr ) 5": dV J (r') . I Gr(rlr') -1-———-dv'. V n Just 158 The norsal cosponent of the reflected dyad G: is expressed as the spec- tral integral 4 -p 9 - . G:(rlr') . If 2 32(2) -3-—£;8-e3*"’ ’ ’ dz). (13) 1-1 212") p1 n 1 are given by where the norsal reflection coefficients R n f ID n sf n n n RfcRsf/D In“ c an ll (14) was) an: a): up: u¢+ Likewise, define "fyc as that part of "(y produced by tangential current Ju' Then, the potential nfyc becoses a r J (r') nfyc(r) 8 IV 5;;’Gc(rlr ) just dV where the coupling coaponent of the reflected dyad G; is expressed as the spectral integral e-pf’i eJA9(r-r') z 4 G:(rlr') a If t c1¢x) 2 d A. (15) i8l 2(2n) pi and the coupling coefficients C1 are given by l llfc - 1 t "sf("sf - 1) t n t -2p t C1 8 - -;----'T + -;-----'R R T e f DtDn N p . p fc I p . p cf fc fs fc c f sf f s u” 1 n2 (x' 1) C a _ l __fg__ Rt Rn .rt .-2pft _ sf sf Tt 2 t n sf sf fc fs 2 2 "fcpc. pf N.fp£+ ps n" 1 a: ' fc _ 2 2 nfcpc. pf l.£p£0 ps u" 1 n2 ( 1 fc t t . sf 2 2 "1cpc’ p1 "-191’ P- (16) O I I ... a :I '4 ('0' . H . H 1 a) ('0' ... 4+ 159 Finally, the potential in the file can be expressed as 3 13.4..1 .1(_r_). nf(r) 113 IV-v G (rlr ) - dV v firtrtr ) - J i av where the principal dyad is IGP and the reflected dyad is written in terss of scalar coaponents as r r 86 36 ' 8 A rA A c A ta 6 A A rA érulr ) thx 1 y <—ax x 0 Gay + 32 z)+ zGtz. The scalar coaponents of the reflected dyad are calculated by using equations (ll)-(16). 160 APPENDIX C APPENDIX C DIFFERENTIATION HUBER SPECTRAL INTEGRALS It is now shown that the differentiation under the spectral integ- ral of (2.3) is a legitiaate operation. Without loss of generality, Justifying this interchange of operations for the following is suffic- ient. e 2 J! w-Ie“ :- Ivgp(A;y.r') J(r') dV' ] d A (1) r In (1), kr is the real part of kc and evaluation of pc is sade on the Riesann sheet with Relpc) > O. Assuaing that J and VWJ are continuous and have cospact support in V, use of the vector identity Vclu) 8 e VOA 0 WOA along with the div- ergence theores on (1) yields XII w-[efl’r Ivgpmym') J(r') dV' l d2). 3.“ VP)” Ivv"‘""’ r r p 2 x g (A1y,r') dV'ld A -p ly-y'l . all vlejx" I F{V'oJ(r')) '—;;—— dy'] d1). (2) c 1" where Riv-J) is the Fourier transfora of V-J as defined in Chapter two. Next, use pc 8 pr + in' where pr and p1 are the real and isaginary parts of pc respectively. The exponential .-prly-y'l is of constant sign for all y. Using the generalized fora of the first aean value theores for integrals (15, p.117], the right side of (2) say be written 161 sax e-prly-y l 41 Vlejxw Re{F(V'-J(x'.n)z'))e-in'y-nl } 2p r sin dy'] d2). (3) C Vh-a*< Y - - . .JJl vlen-r 1-{r1v'thx'.e,z'))o"p1'7'°'H." . p )y y I dy'] d2)“ 1‘ 2)) ysin c where yain‘ n.e < y.‘x (J 8 O for all y < y.1n. y > y..x). The spatial integral in (3) is trivial and leads to 5»: ma ly-nI "pr", 2 I Vle Re{F(V'°J(x'.n.z'))e i }——l d A (4) A 2prpc , _ _ “P :7) +3 vle“ ’ Is{F(V'-J(x',9,z'))e 3’1" 9' } ——’-—l :13. 2prpc r p ( -pr(y-Yain’- r ysax.y)). Since VWJ is contin- where e(prsy) 8 (2- e e- uous and of cospact support in V, VWJ s L2 (i.e. the space of square integrable functions). In particular, for each y, VMJ is an 1.2 function in the x-z plane. Using a standard theores froa Fourier transfors the- ory [17, pp.310-313], the 2-D Fourier transfora of VMJ is an L2 function in the :-z plane. Thus. F(V-J) . 0051") as (no. ¢>0). The 1htograhd in (4) is dosinated in sagnitude by a function which is independent of r and 0(x'2"). The Ueierstrauss fl-test (15, p.470] guarantees that the integral in (4) converges uniforaly. A standard theores fros advanced calculus [15, p.474] justifies the interchange of differentiation and spectral integration. 162 APPENDIX D APPENDIX D ELECTRIC DYADIC fiRflflN'S FUNCTION IN THE SOURCE REEION A classical developsent of the electric dyadic Green's function for observation points within the source region proceeds as follows. The Hertzian potential saintained by electric current sources J in an un- bounded sediua is given by , J(r') , n(r) 8 118 IV-v e(rlr ) Jae dV (1) where v is any voluse containing r'8r. and -Jklr-r'l 0 —— V"" ’ ‘ 4nlr-r'l is the faailiar representation for the free-space Green's function. Since (1) is independent of the shape of v, it is often written without the lisit. The electric field E is related to n by E 8 (k2+ vv-)n. Evidently, calculation of E requires evaluation of the derivatives of n. Fikioris [32] has shown that first derivatives of n say be obtained by forsally differentiating under the integral in (1) provided that J is continuous in V. Therefore Vbn(r) 8 1:: IV VW