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' ‘- w 31;? .‘ . ; 1“ . .- i ”M l 52“” {‘1‘ :5’9‘u' f2”): ”if“. an r 24 202L045} 222323;: 22222222 22.2 2222222 2 :22222222222 ‘ 16 O.- 6098 UWQESSW ”A“??? 2g This is to certify that the thesis entitled TE WAVE EXCITATION AND SCATTERING ON ASYMMETRIC PLANAR DIELECTRIC WAVEGUIDE presented by Boutheina Kzadri has been accepted towards fulfillment of the requirements for ___M.£.__degree in M c- 2' Major professzrj Date July 19, 1989 0-7 539 MS U is an Affirmative Action/Equal Opportunity Institution PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before duo due. DATE DUE DATE DUE DATE DUE Grammy 348 --Q W."__—‘—_| l MSU Is An Affirmative Action/Equal Opportunity Institmlon TE WAVE EXCITATION AND SCATTERING 0N ASYHHETRIC PLANAR DIELECTRIC WAVEGUIDE By Boutheina Kzadri A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Electrical Engineering and Systems Science 1989 Cvo+o3c2l ABSTRACT TE HAVE EXCITATION AND SCATTERING ON ASYHNETRIC PLANAR DIELECTRIC WAVEGUIDE By Boutheina Kzadri An asymmetric planar dielectric waveguide is formed by the tri- layered substrate/film/cover environment, typical of integrated circuits for millimeter and optical wavelengths. The structure supports surface waves when the film-layer guiding region has positive index contrast relative to its surround. An electric Green’s function (believed new) is constructed for the TB field maintained in the film layer by currents immersed in that region. Using a direct complex analysis approach. the Green’s function is expanded in the discrete and continuous propagation spectrum components for the asymmetric planar waveguide. The electric Green’s function is exploited to study scattering of TE surface waves by dielectric obstacles in the film layer. If the y-axis is normal to the layer interfaces and the waveguiding z-axis is parallel to them, then an x-invariant TE field, having only an x component, is excited by the similar component of current. Spectral analysis in the axial transform domain leads to Ex(y,z) =.[ Glyly’;z-z’) Jx(y’,z’) dy’dz’ LCS where LCS designates the longitudinal cross section of the source region and the Green’s function has a spectral integral representation. Subsequent to complex transform plane analysis, G(yly’;z-z’) is decomposed into the superposition of a discrete surface wave, arising from pole singularities, and a radiative component arising from integrations about substrate/cover branch cuts. If a dielectric discontinuity having index contrast 6n2=n:(y,z)-n2 f film layer, an excess polarization current is excited and maintains a is immersed in the scattered field. This current is proportional to the product of the induced field and the refractive index contrast. Within the obstacle 3-31 235 the total field - + consists of the impressed field of an incident wave augmented by the scattered field. Rearranging leads to the EFIE Ex(y,z)—jweo J 5n2(y’,z’) G(yly’;z-z’)Ex(y,z)dy’dz’ = E:(y,z) LCS A pulse-Galerkin’s solution leads to the induced field, from which scattering coefficients are calculated. Extensive numerical results for various obstacle configurations will be presented. TDNYPARENTS iv ACKNONLEDGNENTS The author wishes to express special thanks to Dennis P. Nyquist for his help and guidance in this research. TABLE OF CONTENTS LIST OF FIGURES 1. 2. 3. 4. INTRODUCTION .................................................... 1 ELECTROMAGNETICS OF ASYMMETRIC LAYERED DIELECTRICS .............. 3 2. 1 INTRODUCTION ................................................ 3 2.2 HERTZIAN POTENTIAL GREEN’S FUNCTION ......................... 4 2.2.1 Axial Transform Domain Field Equations .................. 6 2.2.2 Green’s Function Decomposition; Primary Component ........ 8 2.2.3 Reflected Green’s Function for Sources in the Film Layer 10 2.2.4 Reflected Green’s Function for Sources in the Cover Layer 15 2.3 A PHYSICAL INTERPRETATION ................................... 19 2.4 SUMMARY ..................................................... 21 TE PROPAGATION MODE SPECTRUM OF ASYMMETRIC PLANAR WAVEGUIDE ..... 24 3.1 INTRODUCTION ................................................ 24 3.2 COMPLEX C-PLANE ANALYSIS .................................... 25 3.2.1 Green’s Function C-plane Singularities .................. 25 3.2.2 Contour Deformation ..................................... 28 3.3 THE DISCRETE SPECTRUM ....................................... 37 3.3.1 TE Surface Wave Poles ................................... 38 3.3.2 Pole Integral Evaluation ................................ 40 3.4 CONTINUOUS SPECTRUM ......................................... 44 3.5 COMPLEX PLANE ANALYSIS FOR THE CASE OF SOURCES IMMERSED IN THE COVER .................................................... SS 3 6 SUMMARY ..................................................... 56 APPLICATION TO SCATTERING BY OBSTACLES ALONG ASYMMETRIC SLAB WAVEGUIDE .................................................. 59 4.1 INTRODUCTION ................................................ 59 vi 4.2 EQUIVALENT POLARIZATION DESCRIPTION OF DISCONTINUITY REGION 60 4.3 FIELDS MAINTAINED BY IMPRESSED AND INDUCED CURRENTS ......... 64 4.4 ELECTRIC FIELD INTEGRAL EQUATION FOR AN UNKNOWN FIELD INDUCED IN THE DISCONTINUITY REGION ................................. 67 4.4.1 TE Surface Waves Supported planar layered background .... 68 4.5 MOMENT-METHOD SOLUTION OF THE EFIE .......................... 71 4.5.1 Pulse Galerkin’s Solution ............................... 73 4.5.2 Scattering Coefficients ................................. 77 4.5.3 Numerical Results ....................................... 79 4 6 SUMMARY .................................................... 80 6. CONCLUSIONS ..................................................... 90 APPENDIX A ......................................................... 91 APPENDIX B ......................................................... 93 APPENDIX C ......................................................... 98 APPENDIX D ......................................................... 101 APPENDIX E ......................................................... 103 LIST OF REFERENCES ................................................. 107 vii Figure 1. Tri-layered structure used as the background environment for integrated circuits ....................................... 2. Primary and scattered waves in the tri-layered structure with current immersed in the film ................................... 3. Primary and scattered waves in the tri-layered structure with current immersed in the cover ................................. 4. The four different path lengths ............................... 5. Complex (-plane singularities of the transform domain Green’s function with arbitrary branch cut ............................ 6. Determination of the proper branch of each pc ................. 7. Deformation contour C’ on which the transform domain Green’s function is analytic .......................................... 8. Hyperbolic branch cuts ........................................ 9. Coalesced branch cuts ......................................... 10. Branch cut contour partially cancels for low loss limit ...... 11. Evaluation of sign of p to the left and right of the cut and along upper and lower side of the cut ........................ 12. Scattering (reflection, transmission, and radiation) of an incident TE surface-wave mode by a dielectric-slice discontinuity along a planar-slab waveguide of arbitrary shape 13. Scattering (reflection, transmission, and radiation) of an incident TE1 surface-wave mode by a dielectric-slice discontinuity along a planar-slab waveguide of rectangular shape 14. Scattering parameters Vs uniform refractive index of the slice discontinuity with t/A=.5 1/t=.5 nf=3.2(GaAs) ns=3.04 nc=2.88 15. Scattering parameters Vs uniform refractive index of the slice discontinuity with t/A=1.2 l/t=.75 nf=1.5(Glass) ns=1.425 n =1.35 ...................................................... c 16. Scattering parameters Vs normalized length along z-axis with LIST OF FIGURES t/A=0.5 n =3.2(GaAs) n =3.04 n =2.88 n =1.4S ................. f s c d viii Page 12 16 20 27 3O 31 35 36 46 48 61 63 81 82 83 17. 18. 19. 20. 21. 22. 23. 24. Scattering parameters Vs normalized length along z-axis with t/A80.5 nf=3.2(GaAs) ns=3.04 nc=2.88 nd=1.(air) .............. 84 Scattering parameters Vs normalized length of the slice discontinuity with t/A81.2 nf=1.5 ns=1.425 nc=1.35 nd=3 ...... 85 Scattering parameters Vs normalized length of the slice discontinuity with t/A=1.2 nf=1.5 ns=1.425 nc=1.35 nd=1 ...... 86 Distribution of field Ex(y,z) excited in dielectric discontinuity region with TE1 incident mode wave n =1.5 n =1.425 n =1.35 t/A=1.2 l/t=.5 nd=1 ........................................... 87 Distribution of field Ex(y,z) excited in the dielectric discontinuity region with TE incident mode wave and with n =3.2 1 f n =3.04 n =2.88 n =1. t/A=.5 l/t=0.5 ......................... 88 s c d Complex n-plane .............................................. 96 Evaluation of integral around the pole ....................... 102 Symmetric slab specialization leading to § = y + t/2 ......... 106 ix Chapter One INTRODUCTION The subject of planar integrated optical circuits is of increasing interest. An asymmetric planar dielectric waveguide is formed by a tri-layered substrate/film/cover environment. typical of integrated circuits for millimeter and optical wavelengths. This thesis is intended to construct an electric Green’s function (believed new) for the TE field maintained by currents immersed either in the film or the cover layer. The electric Green’s function is exploited to study scattering of TE surface waves by dielectric obstacles in the film layer. Discontinuities in dielectric waveguides are assuming increasing importance in the design and development of optical and millimeter wave components. A discontinuity problem arises in the splicing of two dielectric waveguides and is relevant to inter-device coupling in millimeter and optical integrated circuits. Various methods have been presented recently by several authors for the analysis of discontinuity problem in slab waveguides [1.2,3,4,S,6]. Analysis of longitudinal discontinuities in dielectric slab waveguides was treated by Uzunoglo [5]. His approach was relevant to the symmetric slab waveguide. Moreover, he exploited Sommerfeld integrals to formulate the electric Green’s function for the TE field in the film layer. This thesis uses a direct complex analysis approach to construct the Green’s function which is represented by a 1-D spectral integral. The second chapter contains a general statement of the equations governing fields and Hertz potentials in the asymmetric tri-layered environment of Figure 1. A Hertz potential Green's function for the tri-layered background is developed for sources immersed in either the cover or the film layer. It will be shown that the Hertz potential Green’s function decomposes into a reflected part augmented by a primary component. Due to the x-invariance of the fields, this Green's function is represented by a 1-D spectral integral instead of a 2-D integral [6]. The propagation mode spectrum is treated in chapter three. A discrete spectrum is found to be associated with surface waves, while superposition of the continuous spectrum yields the radiation field. A polarization EFIE description of slice discontinuities along the asymmetric slab guiding region is developed in chapter four. Method of moment (MoM) numerical solutions are obtained for the discontinuity field, leading to scattering coefficients and radiated power. Some words about notation here might be helpful. As a convention, upper case letters denote space domain quantities, while their transform domain counterparts are designated by lower case letters. The symbol 1 denotes the elementary imaginary number, while J denotes the current density in the transform domain. Finally, the following assumptions are valid throughout the thesis: (1) All media are linear and isotropic unless otherwise specified. (2) An exp(jwt) time dependence is assumed for the electromagnetic fields and is suppressed. (3) All media are non-magnetic with permeability ”o' Chapter Two ELECTRONAGNETICS OF ASYNNETRIC LAYERED DIELECTRICS 2.1 INTRODUCTION This chapter is devoted to the evaluation of electromagnetic fields in the tri-layered environment of Figure 1. The electric field in the system is expressed in terms of the electric source density maintaining the fields, integrated into an appropriate Green ’5 function. Details of the development of the Green ’3 function for the layered structure of Figure 1 will be established. In fact, knowledge of this specialized Green’s function is of primary importance, since it will be used later in this thesis to formulate the integral equation for the electric field within discontinuities immersed in the layered dielectric environment. Analysis of electromagnetic fields in a layered environment was first made by Sommerfeld [7] in 1909. The first problem attacked by Sommerfeld was that of electric dipoles oriented normal or tangential to an air-earth interface. Integral-transform techniques were used to obtain integral representations for the fields produced by former dipoles. These integral expressions are known as Sommerfeld integrals. The tri-layered environment, typical of integrated circuits at millimeter and optical wavelengths, is depicted in Figure 1. A uniform dielectric guiding region (film layer) of refractive index n occupies f the region -t < y < 0 ;it is immersed between a substrate region (y < -t) with refractive index nS, and a cover surround which fills the space y > 0 and is characterized by a refractive index nc. All dielectric media are assumed to possess limitingly small dissipation with Re{nf} > Re{ns} > Re{nc), where Re{'} designates the real part of the quantity within the braces. The electric current density immersed in the film region is parallel to the x-axis so it only maintains TE polarized electromagnetic fields in all three regions. In the next section, the electric Hertzian potential D for the tri- layered structure is formulated as a convolution of the impressed current density 3 with an appropriate Green’s function. The development of the Green’s function will be detailed. In section 3, a physical interpretation will be given to explain the y dependence of all the terms present in the expression for the Green’s function. 2.2 HERTZIAN POTENTIAL GREEN’S FUNCTION Details of the relationship of the Hertz potential to the electric field, along with the Helmholtz equation for the potential, is reviewed in Appendix A. Development of general dyadic electric Green’s functions for layered structures has been presented by Bagby and Nyquist [8]. The electric field Green’s dyads are found in terms of associated Hertzian potential Green’s dyads, developed by an extension of Sommerfeld’s classic method [9]. The Hertzian potential dyadic Green’s function was shown to have scalar components expressed as 2-D spatial frequency integrals of the Sommerfeld type. In the subsequent development, the analysis in [8] is specialized for the tri-layered structure of Figure 1. Using the TE symmetry where the fields are invariant with respect to x, the Green’s function will have only one component expressed in 1-D region 1: y>0 nc(cover) region 2: -t0 o 3 n (cover) c =0 xo 1L; 2 y region 2: -t 0 and Im{p£} > 0. The appropriate boundary conditions are adapted from Sommerfeld’s [7] development of Hertzian potential boundary conditions. Enforcing continuity of tangential 3 at the interfaces requires 2 nxc(o’c) Nfcuxf(o'<) and 2 uxf(-t'§) stuxs(-t’c) where Nicand N: are the ratio of the film permittivity to the cover permittivity and the ratio of the substrate permittivity to the film ‘permittivity, respectively. In a similar fashion, continuity of 11 y=—t region 1: y>O transmitted wave nc(cover) xe > z :egionm2: -t0 [Jprimary nc(cover) x6 > 2 region 2: -t 0 (3.3) Im{p£} > 0 The integrand in (3.1) is a multivalued function of C [11] because of the two branches of the function.pl. Hence, to ensure that the integrand is analytic, the complex C-plane must be cut by branch lines emanating from branch points and extending to a point at infinity. The branch points occur at C = tkz. Branch points at §= tkf are removable singularities and the branch cuts emanating from them are not implicated, since the integrand in (3.1) is an even function of pf. In addition to the branch point singularities, the integrand has a finite number of isolated pole singularities. In fact, the reflection (2) + o 1)and Rf?) have simple poles associated with coefficients Ri“, R R: surface-wave modes supported by the tri-layered environment. Figure 5 shows the location of the singularities of the integrand in the complex c-plane. Real and imaginary parts of q are designated gr and C1» respectively. Practical dielectric media exhibit small losses which move the poles and the branch points off the real C-axis. It is subsequently assumed that k = k + jk l Zr £1 ; 2r Hence, all singularities reside in quadrants two and four. At this stage, branch cuts are chosen arbitrarily as long as they do not cross the initial real-line contour C. One possible construction for the branch cut is shown in Figure 5. Subsequent analysis involving contour 26 A55: 4:: -€p 4:. 4r. . . ”t, Figure 5: Complex C-plane singularities of the transform domain Green’s function with arbitrary branch cuts. 27 deformation demands a particular choice for the branch cuts. 3.2.2 Contour Deformation Cauchy’s theorem provides a powerful analytic technique for evaluating certain types of definite integrals. Specifically, Cauchy’s theorem for contour integrals [10] may be used subsequent to deforming the initial real line path for inversion integrals. Consider for now an arbitrary closed contour C’ in the complex C-plane. The correct contour is determined by considering several constraints. First, the branches for each pt must be chosen so that the integrand of (3.1) represents decaying and outward-propagating waves. As far as the contour C’ is concerned, the branch cuts may be chosen quite arbitrarily as long as they do not intersect the contour C’. Second, it must be decided in which half-plane the contour C’ is to be closed. Therefore, as C is deformed into C’ we have JC(z-z’)d G(y|y’;z-z’) = g; [ g(y|y’;§)e C (3.4) CI As was stated earlier, the radiation conditions are satisfied when Re{p£} > 0 and Im{p£} > 0. Writing p£= : I/C‘kt 1/§+k£, then for any paint C along the real C-axis, the phase angle of the two factors (C-kl) and (C+k£) lies in the range 9 O < n < 0 0 < arg(§-k£) -n < arg(§+k£) Nihi- as Cr ranges from -m to +m. Those angles were determined from Figure 6. 28 A careful examination shows that the sum of these arguments satisfies 0 < a: + a; < x . Hence the phase angle of +V§2-k: is always greater than zero but less than 3. In order to satisfy the specific requirements on p2 as in eqn (3.3), the positive root or branch must be chosen. Hence Pa = “(CZ-k: (3.5) With this branch, the convergence of the integral (3.4) at infinity is ensured. Secondly, the exponential factor ej§(z-z ) appears as part of the integrand in (3.4). Writing C = Cr+j§‘ the above exponential factor will be e1C(z-z’) = e-§‘(z-z’)e)§r(z-z’) Therefore, in the upper half C-plane (lower half Q-plane) this exponential is decaying for 2-2’ > 0 (2-2’ < 0) while it increases exponentially for z-z’ < 0 (2-2’ > 0). Hence, when z>z’ (zz’, the contour must be closed in the upper half C-plane. Since the branch out cannot be crossed, the contour C’ must come back in from infinity on one side of the branch cut, encircle the branch point at C = -k£, and recede out to infinity again along the opposite side of the cut. This contour is illustrated in Figure 7 (dashed) and denoted i for the branch out by Con for the semicircle at infinity and by Cb integral. 29 5C. Figure 6: Determination of the proper branch of each pt. 30 Figure 7: Deformation contour C’ on which the transform domain Green’s function is analytic. 31 Since a branch cut was introduced, the integrand in (3.4) is single valued and Cauchy’s theorem applies [11]. Moreover, by deforming the i b integrand in (3.4) will be analytic and Cauchy’s theorem for contour closed contour C’ = C + C0° + C around the surface wave poles (tcp), the integrals leads to J g(y|y';c)e’§‘z‘z"dg = 0 CI Note that the closed contour C’ contains the contour around the surface- + wave pole c; such that + + C’ = C + C + C‘ + C’ m b p + + where the plus (minus) sign in CB and CB refers to the contour being in the upper half plane (lower half-plane). The original integration along C is thus replaced by J (....) = - J (....) - J +(....) - J + (....) (3.6) C C C; C- If the integral along the semicircle Cco in (3.6) vanishes, then the original integral is equal to the branch out integral plus the surface- wave pole integral. Hence the space domain Green’s function as given by (3.1) may be expressed as 32 JC(z-z’)d 8(yly’:<)e C C(YIY';z-z’) = $5 {- I C _L The integrand of G(y|y’;z-z’) is an even function of C except for 1C(Z-2’) O‘l+ 3(YIy’;C)e’c(z'z')d< } (3.6.a) 131+ proportionality to e , hence the lower half-plane closure can be combined with the upper half-plane closure. In fact, since we have performed an upper half-plane closure for z>z’ and a lower half-plane closure for zz’(upper half-plane) and z 0. It was shown earlier that eJC(z-z’) vanishes at Con by choosing the correct half-plane closure. Hence, at Can the integrand in (3.4) vanishes provided Re{pt} >0. In order to ensure that Can remains on the proper branch for which Re{p£} > 0, the branch out emanating from branch point ikl has to separate the proper branch of pl for which pbr > 0 from the improper branch for which plr < 0. The correct branch out lies along the boundary line between plr > O and ptr < O and is defined by the line leading to Re{p£} = 0 or larg(p£)| = g. E 2 arg(p:) = n or (i) Im(pi) = 0 and (ii) Re{p:) < 0. Writing Observe that when pl satisfies Iarg(p£)l = , p: satisfies 2 kl klr+Jk£i’ pe may be written as 2 _ 2 _ 2 _ 2 _ 2 _ pi - (Cr Ci ) (klr kfli) + J2(crci kirkli) from which it can be seen that condition (i) is satisfied if and only if (3.8) which defines a pair of hyperbolas. Along the hyperbolas defined by (3.8) we have 34 1c, 2}... Figure 8: Hyperbolic branch cuts. 35 1°C. Figure 9: Coalesced branch cut. 36 In order to satisfy condition (ii), Cr must satisfy the inequality —k£r < Cr < kzr (3.9) Conditions (3.8) and (3.9) describe the portions of the hyperbolas shown in Figure 8. A decrease in the losses associated with the cover and substrate implies a decrease in kC and k8 In the limit of zero 1 1' loss, the branch out emanating from tks cancels with part of the branch out emanating from ikc resulting in the cut depicted in Figure 9. In either case of moderate loss or limitingly low loss, these branch cuts guarantee that for all C e Cm, Reipt} > 0 and hence the integration along Cco vanishes. Thus the validity of (3.7) is ensured. Now that we have determined the right contour deformation, we proceed with the analysis of the discrete and continuous spectrum. 3.3 THE DISCRETE SPECTRUM A discrete surface wave mode has been shown to arise from evaluation of the pole integral in the complex C-plane. The integrand in (3.4) has poles whenever the reflection coefficients Ril),R:23 (1 R_ ) ’and 11:2 become infinite. First, identification of those poles will be performed and then evaluation of the pole integral is presented. We let Gpole denote the discrete part of the Green’s function in equation (3.7) , then we have 37 c;po l e 411p 4’ + f C P (yly’;z-2’) 3 _ [ 1 [R(1)e-pf(y+y +212) + R‘f’2)e-pf(y-y 421;) + R:i)epf(y-y -2t) + R:2)epf(y+y ) eJC|z-z | dC (3.10) Note that we are integrating around the pole in the upper half plane, that is around C = —Cp . 3.3.1 TE Surface Wave-Poles Recall the expressions for the reflection coefficients Ril),ll R”), R12) from Chapter Two _ P t (1) _ (pf + pc)(pf ps) e f R - + 2 2cosh(pft)[(pf + pcps)tanh(pft) + pf(ps+pc)] - - p t R(2) = (pf ps)(pf pc) e f + 2 2cosh(pft)[(pf + pcpsltanh(pft) + pf(ps+pc)] R(1) = R(2) - + (p + p )(p -p ) ep:t R(2) = f s f c 2cosh(pft)[(p: + pCpS)tanh(pft) + pf(ps+pc)] + (3. (3. (3. (3. Note that all the reflection coefficients have the same denominator (1) hence they are associated with the same simple poles. R+ becomes 38 (2) 11a) 11b) llc) 11d) and infinite when - 2 — 2(C) - (pf +pcps)tanh(pft) + pf(ps+pc) - O (3.12) which leads to TE surface-wave poles at C = :Cp . We define the following parameters = ,/§2- k2 _._ ”4:-8 = ,x (3. 12a) pf f _ 2_ 2 = PC - C kc z (3.12b) _ 2_ 2 = ps — C kS 6 (3.12c) The condition that 7 of (3.12b) as well as K of (3.12a) are both real quantities limits the range of C to the following interval [61 kc < ks < cp < kf which identifies the region that discrete values of the propagation constant for guided modes can occupy. Using (3.12a) through (3.12c), 2(C) becomes 3(75 - x2) tanKt = -;K(7 + a) Rearranging leads to the well-known eigenvalue equation for TE modes of the asymmetric slab K(7+6) tanKt = (3.13) KZ- 75 39 We now proceed to evaluate the pole integral in eqn(3.10) 3.3.2 Pole Integral Evaluation The function of C that leads to TE surface-wave poles at C = :Cp of eqn(3.12) can be approximated by a Taylor's series of first degree near C = th. Hence we can write d2 2(C) 3 2(th) +.__ (CFCp) dc C=tcp since 2(2Cp) vanishes, 2(C) reduces to z ’ _ ' 3. 2(C) Z (+ Cp) (C+Cp) ( 14) where 2’ denotes the derivative of 2(C) with respect to C. (2) + (1) Rewriting the reflection coefficients Ril), R , R_ and R12)in equations (3.11a) through (3.11d) as follows A1(c) - P t + 27:?” :0 l A (C) _ 2 p t _ (1) R+ ‘ 21:)— e f ‘ R- A3(C) = p t R_ met. the discrete part of the Green’s function in (3. 10) becomes 40 dC - I - - ’ (YIY';Z'Z') = 'J 4upr C [A1(C)e pf(y+y +t) + A2(C)e pf(y y +t) + C P G pole + A2(C)epf(y-y’-t) + A3(C)9pf(y+yl+t)] ejClz-z’| (3.15) Making use of the approximation for 2(C) in eqn(3.14), integrating the first term of Gpole around the upper half plane surface—wave pole leads to _ A1(C) e-pf(y+y’+t) eJClz-z’l dC C+ 4npf2 (-cp)(c+cp) P _ _ , _ _ . dC = A1(€p) e pf(Cp)(y+y +t) e JCPIZ z I J C+C (3.16) 4npr (‘Cp) C+ p P dC The integral -:—— is evaluated in Appendix D and found to be C, c cp P dc = -2u , c+cp ’ C the minus sign in front of an is related to the contour around C= -Cp loeing directed in the clockwise direction. Hence equation (3.16) becomes 41 + 4npf2’(-Cp)(C+Cp) P _ J A1(C) e-pf(y+y’+t) e-ijlz-z’| dC C = 31(Cp) e-pf(Cp)(y+y’+t) e-Jcplz-z’| where 31(Cp) is the amplitude of surface-wave mode contribution from (1) + R and is expressed as jA1(Cp) 2pf(Cp)Z’(-Cp) B1(Cp) = In the same fashion, the amplitudes of surface-wave mode contributions (2) (1) from R+ , R_ and RiZ) are established as follows 1A2(Cp) 82(Cp) = 2pf(Cp)Z'(-Cp) 1A (C ) _ 3 p B3(Cp) - 2pf(€p)2’(-Cp) Hence, Gole can be written as p _ -p (c )(y+y’+t) -p (c )(y-y’+t) pole - { 31(Cp) e f p + 82(Cp) e f p + B (C ) epf( -kc, whereas only pS changes sign in crossing the branch cut lying between -ks and -kc. In the latter case, pc has the same sign on both sides of the cut. The branch cut contour C+ can now be divided to include a real line b contour along Cr and an imaginary line contour along C1 as follows 'k 0 m c J + dC = J dC r+ I dC r+ J ij 1 (3.19) Cb -k -k 0 S C First, we have to determine the sign of'pQ above and below the branch out along Cr and to the right and to the left of the out along jCi. 45 A )9 (a) Branch cut contour for the case of some losses'. (b) kti —-> 0 Figure 10: Branch cut contour partially cancels for low loss 1 imit. 46 pi = VCz-k: can be written.as II 0 ....along C 42 2 :- Pe=tl -C =t1 ....along C1 ll 0 Figure 11 helps in determining the argument of p! along the branch out. Along the right and lower sides of the upper half plane branch out the TI argument of pc is equal to 2 and hence pl is positive; along the left and upper sides of the cut, the argument of pl is equal to -% showing that pl is negative there. Second, we have to determine the behavior of the coefficient R£1L (2) + 9 (1) R_ ) R and R12 in crossing the branch out where C > -kc. Along this part of the cut, each of pi is purely imaginary. Hence, we define the following parameters 2 _ _ 2 = _ = pc - C kc 9 hence pc in 2 _ 2_ 2 = _ 2 = ps C kS 1 hence pS 31 (3.20) 2 2 2 2 _ Pf - C kf - c hence pf - 5c Since the argument of each pl is g , they are located along the right and lower sides of the upper half plane branch out (C > -kC). Exploiting the expression for p! as in (3.20) in writing the coefficient Ri“ as in (3.11.a) leads to 02+0‘( -1:)- ‘l.’ (1) = p p 20(T+p) coszot +2(O‘2+p1.’) sin2 (rt +21( -kc), each pl will change sign leading to pc= ‘JP. PS= -jr and pf= -Jo. Using the form of pl along the left and upper side of the cut in the expression for R11) .1 shows that Ri” changes to Ril) in crossing the branch cut, where the (2) asterisk refers to complex conjugates. In the same fashion , R+ , R:})and R12) are transformed to their complex conjugate in crossing the branch out from right to left or from lower to upper side. Consider now the branch cut lying between Cr = -ks and CI- = -kc. Along this part of the cut, pc is real whereas pS and pf are purely imaginary and expressed as in (3.20). pc has the same sign on both sides of the cut. Therefore, pc is defined as follows p = C -kC = 7 hence pc = 7 = real The expressions for RF), R12), Rim and R12) are changed to their complex conjugates in crossing the lower side of the cut to the upper side. With this in mind, we proceed to evaluate the branch out integral using (3.19). As can be seen from equation (3.18) the principal wave component of the radiation Green’s function does not depend upon pS or pc. Therefore, the only implicated branch out is that associated with pf. We have to note that the sum of all the different components of the integrand in GR is an even function of pf, whereas each individual component is not. Hence, in integrating along the branch out contour as in equation (3.19), pf changes sign (the pf branch out is crossed). We choose a convenient sign of pf since pc and pS have constant signs on 49 each different side of the pf branch out. Consequently, we can write the branch out integral of the principal wave as -p ly-y’l O m I e f eJClz-z | dC = I C+ 411pf b 5 Note that the integral from CI. = -ks to CF = -kC and the integral from CI. = -kC to Cr = 0 are combined now since the integrand is the same in both regions. Choosing the sign of pf to be positive along the right and the lower sides of the cut and negative along the left and upper side of the cut, we can write -p ly-y’l _ . I e f ejC|z z | dC + 4np Cb f O 2 2 , 2_ 2 _ I _ 1 e-Jng-Cr ly-y I + e JVEf Cr [V y I )Clz-Z’I _ __ - e dC 4n 2 2 2 2 r ks J f-Cr -’ f-Cr e—CilZ-z’liji m 2 2 I 2 2 I 1 I [- e'JV£f+C1 ly-y | + 61V£f+ci Iy-y I 0 My: -)V{:+c: (3.21.a) The first integral has two terms in the integrand because we are considering the lower and upper side of the cut. The minus sign in front of the first integral accounts for the integration being performed in the opposite direction of the branch cut. Now, we want to separate the terms arising from substrate and cover radiation from those arising SO from substrate radiation only. After some manipulation equation (3.21.a) becomes -p ly-y’l _ . I e f ejC|z z | dC C + 4np b f -k c: 42 2 , J I COS[ f-Cr I Y‘Y I] J< |z_zl I = 2— , e r dC n 2 2 r 1‘s f-Cr 0 ‘4; . J 1 cos[ g-C: ly’y I) it lz-Z’l + __ e r dC 2n 2 2 r kc f-Cr e-C1|z-z’| dCi (3.21b) 1 J” cos[¢£:+C: ly-y’l] 2n 7 0 “Q“: In the region where -ks< Cr < -kc, only substrate radiation is present. We define the parameter 7 as before where 7 = VCZ-k: (3.22) I‘ and by convention 1 is a positive real quantity. From this we can write Cralong the negative real axis of the upper half plane branch out as 51 -k c The first integral J (---) dCrcan be conveniently rewritten by -k s changing the integration variable to 7 . From equation (3.22) dCr can be written as 7 dC =—d7 r Cr Also, from equation (3.22) we can see that when Cr = -ks and Cr = -kc, 7 is equal to k6Vn:-n: and zero, respectively . Similarly, in the region C > -kC both substrate and cover radiation are present. We define the parameter p as before where 2 2 = kC Cr....alongC1 0 2 c 2 2 2 p =k—c = k 2 - +Ci....alongCr-0 (3.23): The latter two integrals in equation (3.21.b) can be conveniently combined. The integration variables are changed from Cr and C1 to p in both cases. From equation (3.23) when CI. is equal to zero and -kc, p is equal to kC and zero respectively. Hence we have - kz- for O S S k c p .... p c j p -k ....for kc s p (a With all this in mind, rearranging equation (3.21.b) leads to 52 -p ly-y’l _ . e f4 ejC|z z | dC c+ "pf o "2'“: cos[c( )( - ’)] f 7 y y e1C(7)|z-z’| 7 d7 2n 0(1) Q 7 O m cos[o(p)(y-y')] jC(p)|z-Z’I p ’ Mp) e T): p d" o where C(p) = -V72+k: (3.24) in the first integral term and - -p ....O S p s kc C(p) = ,——— +J Pz'k: ....kc s p < w (3.25) in the second integral term. Also, since 02 = ki-C2 we have 0(7) = ka:(n:-n:)-12 ....first integral 0‘: q(p) = ka:(n:-n:)+p2 ....second integral (3.26) The reflected wave component of the radiation Green’s function is analyzed in the same fashion as for the principal component, taking into account that the different reflection coefficients are transformed to 53 their complex conjugates as the branch out is crossed. Hence, GR can be expressed as GR(YIy’;z-z’) = - I [ e'PfIY'Y I + R:1)e-pf(y+y +2t) + Cb I I I JCIZ’ZII (2) -p (y-y +2t) (1) p (y-y -2t) (2) p (y+y ) e dC 4' R+ e f + R_ e f + R _ e f W 2 2 n -n o s c I + %_ [ { cos 6(7)(Y'Y') + Re{ R(1) e 10(7)(y+y +2t)} n + 0 + + Re{ R’Z) ;)0(7)(y~y’+2t) } + Re{ R:1) ejc(7)(Y'Y’-2t)} + Re{ R(2) e10(7)(y+y’)} } eJC(7)|z-z’|¢ 77 277 m + J { cos 6(P)(y-Y’) + Re{ R:1)e-J¢(p)(y+y +2t)} O , Re{ R:2) ;)o(p)(y-y’+2t) } + Re{ R11) ejG(p)(y-y’-2t)} + Re{ R:2) e)o-(p)(y+y’)} } e)C(p)|z—z'|c pp gpp ] (3.27) which establishes the final form of the space domain Hertz potential radiation Green’s function. The parameters C(7), C(p), 0(7) and 6(p) are expressed as in equations (3.24) through (3.26). 54 3.5 COMPLEX PLANE ANALYSIS FOR THE CASE OF SOURCES IMMERSED IN THE COVER Recall form Chapter Two, the expression of the spectral representation of the Hertz potential Green’s function with sources immersed in the cover layer Q G(YIYISZ‘Z’) = é—il 5% { e-pcly-yll + R(C) e'Pc(Y+y') } eJC(z-z’ )dc C m where the reflection coefficient R(C) is expressed as 2 R(C) a pf(pc-ps) + (pops-pf) tanh(pft) 2 pf(ps+pc) + (pspc+pf) tanh(pft) Note that the second term of the spectral integral representation of the Hertz potential Green’s function involving R(C) contributes surface wave modes associated with simple poles of R(C). Radiation modes associated with branch cuts of pc and pS are contributed by both terms. Branch cuts associated with pf are not implicated since the integrand of G(y|y’;z-z’) is an even function of pf. Deformation of the real line integration path leads to the same contour C’as when sources were immersed in the film layer (Figure 7). In fact, the radiation condition is the same in both cases. Equating the denominator of R(C) to zero leads to the well known eigenvalue equation for TE modes of the asymmetric slab as in equation (3.13). Evaluating the pole integral in complex C-plane gives the discrete part of the Green’s function as 55 (yly’;z-z') = R(Cp) e-pc(Cp)(y’y')e'J§p|z‘z'l G pole where 2 JPCPSPf B(Cp) = 2 2 Cp(kf-kcl(pc+ps+pcpst) The evaluation of the branch out integral is similar to the case of the sources immersed in the film region. R(C) changes to its complex conjugate in crossing the upper half plane branch out from right to left or from lower to upper side. The expression for the radiation component of the Green’s function is then formulated as V nZ-n2 _ ( ) , s c e 1C 1 |z-z | O , , - - ( + ’) GR(y|y :z-z ) = g [ Im{R(C) e 7’ Y Y } cm d7 0 °° ‘JC(p) lz-z’ I - I { cosp(y-y’) + Re{R(C) 3’p(y’y ) }} e {(P) dp ] o where C(7) = V 72+k: ..... in the first integral term and 2 2 k -p ....for 0 s p s kC C C(p) = J kfi-p’ = '1V pZ-k: ....for kc s p < m ..... in the second integral term 3.6 SUMMARY Identification of the propagation mode spectrum of asymmetric planar slab waveguides may be made by analyzing solutions to (2.20) or 56 (2.24) in the complex C-plane. By appropriately deforming the initial real-line inversion integral of the transform domain Green’s function, the space domain Green’s function may be expressed as G(yly':z-z’) = ‘ 25 I 8(y/y’;C) eJCIZ’Z'I at + C P ’ 23 I g(y/y’:<) e’clz'z I at (:4: b + b branch out in the upper half plane respectively. where c; and C are the contour around the pole and the hyperbolic From eqn (2.25) it can be seen that the electric field decomposes into a superposition of two types of modes. A discrete mode spectrum arises from integrating around the surface-wave pole in the complex C-plane while spectral components of the continuous spectrum are given by (2.20) or (2.24) along the branch out contour c;. 57 e l...’ _ I._I I,_I Gf(y|y ,z z ) jkozo [ Gpol°(y|y ,z z ) + GR(y|y ,z z ) ] - -p (C )(y+y’+t) -p (C )(y-y’+t) 1k020[{B1(Cp) e f p + B2(Cp) e f p + 82(Qp) epf(Cp)(y-Y"t) + 83(Cp) ePf(Y+YI+t)} e‘JCplz-Z’I + n -n o s c . + 2? [ { cos 0(7)(Y'Y') + Re{ R(l) e 10(1)(y+y +2t)} 0 + Re{ R12) e‘Jo(1)(y-y’+2t)} + Re{ Ria) ejo(7)(y-Y’-2t)} + Re{ R12) e’°(7)(Y+Y’)} } eJC(1)|2‘Z'| 7 d1 0(7) C(7) m + I { cos 0(P)(Y‘Y') + Re{ Ril)e-’o(p)(y+y +2t)} 0 + Re{ R:2) e-10(p)(y-y’+2t)} + Re{ R:2) e)0(p)(Y’Y’-2t)} + Re{ R12) e10(p)(y+y’)} eJC(p)|z-z’| 9 d9 0(p) C(p) (3.28) This electric Green’s function will be used in next chapter to evaluate the unknown electric field inside a discontinuity in the film region. 58 Chapter Four APPLICATION TO SCATTERING BY OBSTACLES ALONG ASYMMETRIC SLAB WAVEGUIDE 4.1 INTRODUCTION When the geometry of the waveguide is perfect, and if we can neglect losses in the dielectric material itself, the guided modes will travel without change and without attenuation. It is, however, impossible to build dielectric waveguides to such a perfection. Studying the modes of a perfect waveguide as was done in the previous chapters is an important first step of determining its properties. In order to be able to evaluate the performance of a realistic waveguide, it is necessary to study its behavior if departures from the perfect geometry occur. The imperfections of dielectric waveguides occur in many forms: (1) losses of the dielectric material, (2) departure from perfect straightness, (3) inhomogeneities of the dielectric material, (4) departure of the core/cladding interface from a perfect plane in slab waveguides, and (5) many others. Of all imperfections mentioned, the influence of inhomogeneities of the dielectric material in the core region will be analyzed. Waves scattered by such a discontinuity along the guiding structure can be quantified on the basis of a polarization electric field integral-equation (EFIE) description of the discontinuity field. Various methods have been presented by several authors for the analysis of discontinuity problems in slab waveguides. Among those is 59 Marcuse’s treatment [12,13] which dealt with the abrupt junction between two dissimilar guides and the interaction of surface waves with small, distributed surface irregularities. The most rigorous analysis for step discontinuities is Rozzi’s investigation [1,14] based on a two- dimensional integral equation formulation for the fields in transverse discontinuity planes. A polarization EFIE description of slice discontinuities along a symmetric-slab waveguide was first exploited by Nyquist and Hsu [2,3]. Subsequent applications of the integral-operator description [4,5] were based on different representations of the Green’s function kernel and expansions of the unknown field. The EFIE formulation in [2] and [3] was generalized [6] to include discontinuities having arbitrary shapes and complex refractive index profiles along open boundary dielectric waveguides. In this chapter, a polarization EFIE description of the discontinuity region along an asymmetric-slab guide is developed. Method of moment (MOM) numerical solutions were obtained for the discontinuity field, leading to scattering coefficients and the fractional radiated power. 4.2 EQUIVALENT-POLARIZATION DESCRIPTION OF DISCONTINUITY REGION An equivalent polarization description of the dielectric discontinuity region is obtained in terms of the contrast of its refractive index against that of the unperturbed surface waveguide. Figure 12 indicates a dielectric discontinuity along an open-boundary 60 y=0 y=-t \ I is cover ,z" //) n. g M) _ x _ _ 147 > 2 film V nf i d 5 ——>3 .3 ——)i.’ eq E8 e—-— nd :ubstrate \\.,\~j;:;:;::INUIrv "__I a s (// REGION 28 Figure 12: Scattering (reflection, transmission, and radiation) of an incident TE1 surface-wave mode by a dielectric -slice discontinuity along a planar-slab waveguide of arbitrary shape. 61 dielectric waveguide. When a surface-wave mode is incident upon the discontinuity, it is subsequently scattered; i.e, it is reflected, transmitted and radiated. Let nu(?) denote the refractive index of the unperturbed surface waveguide with the decomposition ns(r) ...at points in the substrate n = f(r) ...at points in the waveguide core u nc(?) ...at points in cladding The discontinuity region V with refractive index nd(?) is that region d where the refractive index differs from nu(?). The incident wave E1 induces an equivalent polarization distribution in region V d’ and this polarization excites and maintains the scattered field ES. We know from Ampere’s law at any point in the system v x fi(?) = 39(?) + jw€(?) E(?) where C(?) denotes either the unperturbed permittivity eu(?) in the unperturbed region or cd(?) = n:(?)co in the discontinuity region. je(?) is the impressed electric current which maintains an impressed incident field E1. An equivalent polarization current is obtained [15] by adding and subtracting the displacement current of the unperturbed current in the Ampere’s law Maxwell equation. We obtain v x fi(?) = je(?) + jwco(n2-n3) E(?) + jwcon: E(?) je(?) + jeq(?) + jwcon: E(?) A "Mr V II 2 2 a jwco(n nu) E(r) is the equivalent induced polarization current which describes the 62 cover nc xe > 2 film Vd nf i s ——)E e3 —:1.’ eq “d substrate z=-l72 z=l/2 nS rectangular gs discontinuity Figure 13: Scattering (reflection, transmission, and radiation) of an incident TE1 surface—wave mode by a dielectric -slice discontinuity along a planar-slab waveguide of rectangular shape. 63 discontinuity region V and excites the scattered field ES. The induced d current, non vanishing only in the discontinuity region Vd, is expressed in terms of the total field E(?) in that region as a _ 2 9 a jeq(r) - jwcoan (r) E(r) where 6n2(?] = n§(?)-n:(?) is the refractive index contrast. We specialize this result to a rectangular discontinuity placed in the film region as in Figure 13. Since the unperturbed refractive index nu is now equal to n we have f, jeq (3) = jweo(n:-n:) E(3) = choén?(3) E(3) (4.1) which expresses the excess current of the discontinuity in the film region and where 3 = 9y + 22 is the 2-D position vector. 4.3 FIELDS MAINTAINED BY IMPRESSED AND INDUCED CURRENTS Total electric field E along the open-boundary surface waveguide is excited by an effective current 3 = 3e + jeq consisting of both primary impressed and equivalent induced components. The effective current 3 is oriented in the Q direction in order to excite TE type modes. The total field along the perturbed waveguide system can be expressed as ___ e I, _ I I I I I Ex(y,z) LéstWIy ,z z ) Jx(y ,2 ) dy dz which illustrates the E field maintained by j in terms of the electric 64 Green’s function expressed as in chapter three. G:(y|y’;z-z’) is decomposed into discrete and continuous spectral contributions as follows e I,_I=e I,_I e I,__I Gf(y|y ,z z ) Gpole(y|y ,z z ) + GR(y|y ,z z ) (4.2) where G; and G:°l°are expressible as in Chapter Three. Equation (4.2), a representation of the electric Green’s function, is constructed from complex analysis on the spectral integral representation of the Hertz potential Green’s function. Some of the terms in the expression for G: can be combined to emphasize the reciprocity of the electric Green’s function. In fact, G:(y|y’;z-z’) can be written as 65 -1K(y+y’+t) Kt G:(y|y’;z-z’) = -]koZO {3,‘Cp’ e +232‘Cp) cosK(y-y’)e-’ + 83(Cp) e’K(y+y’+t)} e_jcp|z-z I 2 2 n -n J o s c I + 2H [ [k { cos 0(7)(Y'Y ) O + Re{ R:1) e-jo(7)(y+y +2t)} + 2Re{ R12) e-’o(7)(2t)cosc(7)(y-y’)} + Re{ R12) e10(1)(y+)")} eJC(7)|z-z’| 7 d7 0(7) C(w) m + [ { COS 0(p)(y-y’) + Re{ R:1)e-JG(P)(Y+Y +2t)} O + 2Re{ RiZ) e-’0(p)(2t)coso(p)(y-y’)} + Re{ R(2) e1c(p)(y+y’)}} ejC(p)|z-z’| p dp 0(9) CW (4.2a) 66 4.4 ELECTRIC FIELD INTEGRAL EQUATION FOR AN UNKNOWN FIELD INDUCED IN THE DISCONTINUITY REGION The field excited along an open-boundary dielectric waveguide and within the discontinuity region by impressed and induced currents is _. e I, _ I e I I I I I Ex(y.z) -J Gf()'|y .2 z ) [wa .2 ) + Jeqw .2 )] ds LCS _ i s - Ex(y,z) + Ex(y.z) (4.3) which represents the electric field at any point in the system. E:(y,z) is the impressed field maintained by a primary impressed current J: and E:(y,z) is the scattered field maintained by excess polarization current Jeq induced in the discontinuity region. Rearranging leads to Ex(y,z) - sin/,2) = E:(y,z) for all mm 6 LCS (4.4) where E:(y,z) = jwcoJ 5n2(y’,z’) G:(y|y’;z-z’) Ex(y’,z’) ds’ (4.5) LCS is the scattered field maintained by an induced current in the discontinuity region with longitudinal cross section LCS. Expressing E: in equation (4.4) in terms of the total field EX within the discontinuity region by using equation (4.5) leads to the EFIE 67 N' 0’? E (y,z) - J .[ 6n2(y’,z’)Ge(y|y’;z-z’)E (y’,z’)dy’dz’ = E1(y,z) x I£S f x x O for all (y,z) e LCS (4.6) where k0 = 1.101060)“2 is the free space wavenumber and 20 = (#080)1/2 is the associated intrinsic impedance. It is assumed that J: is a remote source that maintains impressed field E: consisting of a single principal TE surface wave mode in the region of interest. 4.4.1 TE Surface Waves Supported by Planar Layered Background We assume that the film thickness t in Figure 13 is chosen such as to support only the TE1 principal surface-wave mode. The basic equations for TE-mode guided waves are [13] 2 2 2 _ V hz + (kt-C ) hZ - 0 t 9 _ 2B A 9 (4.7) et C (ZXht) "=_J_C_ ht vthz 2 2 (kl C ) G where l = s,f,c (substrate/film/cover) and lower case 3 and 3 denote the transverse fields. Since the slab waveguide has no field variation in the x direction, which we express symbolically by the equation (4.8) ml OJ X I] O and V2 become then the operators Vt t 68 V = -—— and V = f... N n- ‘<> 03' Q) ‘< We have shown in Chapter three that the discrete values of the propagation constant C for guided modes is limited to the following range kc < ks < gp < kf which has prompted the definition of the parameters 6, K and 1 as 7': 0) II [I W n W 3‘ O N H, 'ON 3 k 1 = {p- 0 Solutions of equation (4.7) provides the expressions for hz, hy and ex in each region (substrate/film/cover). Taking into account the radiation condition as y—e to the fields in the cover region region are expressed as n (y) = A e""y Z y>0 1C -1y h = —— A e y(y) 1 efl 'W ex(y) 7 A e which consist of waves attenuating in the upward direction. Similarly, in the film region (-t all z’. The transmitted wave E; is defined as the incident wave augmented by the forward scattered wave. The reflection coefficient is expressed as the back scattered wave divided by the incident wave, all evaluated at the input plane along the z-direction of the discontinuity, i.e at z = -£/2. Thus we have as E (y,z) R- x - —I—_———- [4.13) The transmission coefficient is expressed as the transmitted wave 77 evaluated at the output plane (zed/2), divided by the incident wave evaluated at the input plane of the discontinuity _ E;(y,z)|z c/z T E:(y,z)|2 -1/2 Since the surface-wave scattered field E3" depends on the discrete electric Green’s function, we expect that Gjolohas the same functional dependence on y as the incident field. This latter derivation is lengthy and the result is given as 000 A jk.Z N [ 0 e ., _ . = Giol°(y|y ,z z ) p Z sinKy’-cosKy][ Z sinKy-cosKy] e.’cp|z—Z I K K (4.14) where A0 and No are defined as _ 2 2_ 2 2_ 2 A0 - 4Cpk01/(nf nS)(nf nc) (1+6+16t) 41(K‘75+K27531 N _ Vk‘+1262+K2(12+62) O Substituting the pulse function expansion for the discontinuity field Ex(y,z) and using (4.14) for the discrete Green’s function, in the expression for the forward and backseattered field leads to the final expression for the reflection and transmission coefficients as follows 78 -4k N g o 0 -JC l y A2 2 I _ -jC 2 R AoEocpK e ‘p sin(Ké§)sin(Cp—§) An ; an(K sinKyn cosKyn)e p n T = e—JC t 1- -EE:EB— e-JC tsin(KéZ)sin(C 93) Anal a (1 sinK -cosK ) p Aosogpx p 2 p 2 n n K yn yn e J cp211] 4.5.3 Numerical Results A study of scattering parameters (reflection, transmission and power radiated) permits to know the behavior of the waveguide in the presence of the discontinuity region. The waveguiding structure is chosen such that there is a 5% refractive index contrast between film and substrate, and a 10% contrast between film and cover regions. The normalized width t/A of the guiding region is chosen such that only the TE1 surface-wave mode is excited; all other modes are at cutoff. This specific value of t/A is extracted from the dispersion curve of the waveguide. Curves of scattering parameters versus the discontinuity refractive index are illustrated by Figure 14 and 15. To ensure mono-mode surface-wave propagation, t/A must be equal to 0.5 and 1.2 for the case of GaAs film region (nf = 3.2) and glass (nf =1.5), respectively. It is apparent that for small contrast between the film and the discontinuity region, we obtain small reflection (less than 20%). As the contrast An2 gets higher, the reflection coefficient gets bigger (up to 60%). Figures 16 through 19 show the scattering parameters versus the normalized length of the discontinuity along the z-axis. It appears that when the discontinuity gets longer the transmission coefficient gets lower. This is in agreement with our physical intuition. 79 In Figures 20 through 21, relative field amplitude [Ex(y’Z)/Emax| versus normalized length y/t are shown. We can see that the field distribution is asymmetric with respect to y a -0.5t axis. This is expected since we are dealing with an asymmetric slab waveguide. 4.6 SUMMARY A polarization EFIE description of the discontinuity region along an asymmetric slab waveguide is developed. An equivalent polarization description of the dielectric discontinuity is obtained in terms of the contrast of its refractive index against that of the unperturbed surface waveguide. Hence we have a = 2 a a 3eq(r) jwcodn (r) E(r) The fields excited within the discontinuity consist of the impressed field of an incident wave augmented by the scattered field maintained by the equivalent current E (y z) = Es(y z) + E1(y z) x ’ x ’ x ’ Rearranging leads to the EFIE k 0 2IIe I,_I II II=1 Ex(y,z) 1 2;.Ll36n.(y ,z )Gf(y|y ,z z )Ex(y ,z )dy dz Ex(y,z) where E:(y,z) consists of a single forward propagating TE1 surface-wave mode and expressed as 80 + transmission coeff a reflection coeff o * power radiated 0.?0 .111 [11.1 0.40 1 11111 scattering parameters 0 (:0 0.20 ‘ - o Q 1111111111111111111111111111111] 0[.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 discontinuity refractive index Figure 14: Scattering arameters Vs uniform re ractive index of the slice discontinuity with t/A=.5 |/t=.5 n,=3.2(GaAs) ns=3.04 nc=2.88 81 + transmission coeff o reflection coeff 0 =1 power radiated 0.50 0.40 scattering parameters 020 060 141114111111141111 11111111111111111] .00 1.50 2.00 2.50 3.00 3.50 discontinuity refractive index .0.00 Figure 15: Scattering parameters Vs uniform refractive index of the slice discontinuity with t/A=1.2 |/t=0.75 nf=‘l.5(glass) n31.425 nc=l.35 82 * power radiated a reflection coeff 1,00 .4 + transmussuon coeff 0.80— (f) .4 L (D -i .1.) Q) -1 €0.60— 0 .4 L. O .— Q d 0‘ "t u : £0.40— L (D -‘ o +J .H d O 0 .1 (f) -1 0.20—1 A 0'00 [[[IIII[ll—[IIIIIIIIITIIIIIIPI] 0.00 0.50 1.00 1.50 normalized length I/t Figure 16: Scattering parameters Vs normalized length along z—axis with t/A=0.5 nf=3.2 (GaAs) ns=3.04 nc=2.88 nd=].45 83 + transmission coeff o reflection coeff o * power radiated 0.810 0.60 1111J1111 0.40 1 lJlJ scattering parameters 0.20 i l l O o. TIIIIITIIIIIIIIIITTIIITTTTVIIITTW] c0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 normalized length along z-axis I/t Figure 17: Scattering parameters Vs normalized length of the slice discontinuity with t/A=.5 nf=3.2%GaAs) ns=3.04 nc=2.88 nd=1. air) 84 + transmission coeff o reflection coeff 0 =1 power radiated 0.80 LJ 141 i 1 0.60 1 [All 0.40 l 111] scattering parameters 0.20 1 l l J 0. WFIIFITTIIFFFTTTITITTTI]lIIFFTTIl| ‘000 0.25 0.50 0.75 1.00 1.25 1.50 1.75 normalized length along z—axis l/t Figure 18: Scattering parameters Vs normalized length of the slice discontinuity with t/A=1.2 nf=1.5 ns=1.425 nc=1.35 nd=3 85 + transmission coeff o reflection coeff * power radiated .20 l l 111 l .00 l 0.80 1 [11 #1 0.60 0.40 11111111114111 scattering parameters 0.210 L111 .00 1111111111111111111111111111111111] C0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 normalized length along z-axis l/t Figure 19: Scattering parameters Vs normalized length of the slice discontinuity with t/A=1.2 nf=1.5 ns=1.425 nc=1.35 nd=1. 86 0.80 A N. >\ _ v _. X [4.] .. H— —1 031 (1)0- '0 q 3 t -i "g.- *— 0C5— Z/|=0.15 8 _ g 1 082 §0_ z/|=O.45 O _. C _ o- QirirTrriiliiiiliriri <21.00 —0.75 -0.50 -0.25 0.00 normalized length along y—axis (y/t) Figure 20 : Distribution of field Ex(y,z) excited in dielectric discontinuity region with TE, incident mode wave nf==1.5 ns=1.425 nc=1.35 t/A=1.2 [/t=.5 nd=1. 87 1.00 1111J 0.60 0.80 L 1 1 1 1 l 1111 normalized amplitude of Ex(y,z) O *— o‘z z/l=0.133 g_ z/l=0.4 0‘. .1 O 911F1111111111111111] 91.00 —0.75 —0.50 -0.25 0.00 normalized length along y-axis (y/t) Figure 21 : Distribution of field Ex(y,z) excited in dielectric discontinuity region with TE, incident mode wave nf=3.2 ns=3.04 nc=2.88 t/A=0.5 [/t=.5 nd=]. 88 1 = z - -i< Ex(y,z) E°(K sinKy cosKy)e p The EFIE is solved using the standard method of moments technique. The unknown discontinuity field is expanded into a pulse function expansion. The EFIE is reduced to an MOM matrix equation. A pulse Galerkin’s implementation is used to establish the MOM matrix. 89 Chapter Five CONCLUSIONS A detailed development of the electric Hertz potential Green’s function for a tri-layered substrate/film/cover dielectric structure was presented in chapter two. The electric Green’s function is a constant multiple of the Hertz potential Green’s function due to the x-invariance of the fields. This development demanded a mathematically rigorous treatment and revealed that the electric Green’s function is represented by 1-D spectral integral, which is alternatively evaluated by contour deformation. In chapter three, complex-plane analysis applied to the spectral integral representation of the electric Green’s function leaded to the identification of the propagation mode spectrum of the asymmetric planar dielectric waveguide. A discrete mode spectrum was shown to arise from integrating around the surface-wave poles, while hyperbolic branch cuts corresponded to a continuous spectrum. This electric Green’s function was specialized for the case of symmetric slab. It was found that it agrees completely with Rozzi’s result [1,14]. In fact, after shifting the axis and some tedious manipulations, our specialized Green’s function reduces to the TE even Rozzi’s Green’s function for the symmetric slab waveguide. This is detailed in appendix E. Finally, a polarization EFIE description of a discontinuity region along an asymmetric slab was developed. Standard method of moments(MOM) numerical solution were obtained for discontinuity field, leading to scattering coefficients and the fractional radiated power. 90 APPENDICES APPENDIX A APPENDIX A Electric Hertzian potential Using V-H = 0 (non existence of magnetic monopoles) the magnetic field H can be expressed as the curl of a vector potential. In an electrically homogeneous medium, H may be expressed in terms of the Hertzian potential H as E = jwc vxfi (1) From Faraday’s law we have VxE = ~10“ H Substituting (1) into Faraday’s law yields vXu‘a‘ - 1311) = 0 (2) where k2= wgpc is the wavenumber in the medium. Equation (2) implies if = szl — vo (3) where p is a suitable scalar field. Using (1) and (3) into Ampere’s law VxH = j + jch yields 2 2 _-3 . (v +k)fi-m+v1vfi+o) (4) where we used the vector identity VxVxx = vv-X - V22. Since a vector field is uniquely determined through knowledge of its curl and divergence, by choosing V-fi = -p uniquely determines H. Thus (4) simplifies to 91 2 2 (V +k)fi3fi which consists of the Helmholtz equation for the Hertzian potential subject to the Lorentz gauge V-H = -p . Use of this Gauge in (3) yields E = (k2+ VV-fii which relates the electric field to the Hertz potential. 92 APPENDIX B APPENDIX B Primary Green’s Function The principal Hertzian potential satisfies the following Helmholtz equation azup -j x1 2 X 2 ‘ Pi“:1‘Y'<’ ‘ 355 6y i "ii can be written in terms of a primary Green’s function in transform domain +m Jx(y.C) 11:1(YoC) =I -—JJ€T- 8‘:(Y|y 3C) dY where g§(y|y’;C) satisfies 628? 2 - p181; = -6(y-y’) (1) 6y2 We perform a Fourier transformation on the y-axis. we define the Fourier transform pair f(y) e——9 f(n). Hence g§(n.§) and g§(y,C) are defined as +00 I g§(y.c1 e'”y dy ~P 81(D.C) and +m p 1 ~p iny 81(y.c) - 5; I 81(D.C) e dn (2) Substitute (2) into (1) yields 93 +0 2 [ 9—- - pf ] g; I g§(n.c1 emy dn = —6(y-y') 2 By a +00 _ -1 in(y-Y’) - 2; I e dn 0 Rearranging the above equation yields +00 I { [n2+ Pi] £§tn.<) - e’my } e’"y dn = o (3) (m We can see that (3) is the inverse Fourier transformation of the quantity in brackets. Since 9;‘{---} = 0 e {...} = 0 -iny’ Hence §§(H.C) = -§-——5— (4) n + p 1 Use of (2) and (4) Yields to the primary Green’s function in axial transform domain dn +m JD(Y’Y') p 1 e 2 2 n +P1 dn (5) +m I 1 I ein(y-y ) m (n-1p1)(n+ipii We apply a deformation of this real line integration and apply Cauchy’s theorem J(...) d1, = O C + + where C = C0 + c; + c; is the deformation contour. Co is the initial + real line contour along the Retn) axis. c; is the pole contour. Note 94 that the integrand in (5) has two simple poles at n = 11p1 i = f,c for sources immersed in the film and cover region respectively. We specialize our solution for y’= 0 since the final result can be shifted to any y’¢ 0. Hence we can write +00 iny g‘i’imi = 712—1: I 9 dn (6) a (n-Jp1)(n+ip1) JHY The exponential factor e appears as part of the integrand in (6). Writing n = nr+jn1 the above exponential factor will be eJDY = e‘DIY eJDrY Therefore to ensure convergence of the integrand, we perform an upper half closure for y > 0 and a lower half closure for y < 0. This is shown in Figure 22. J (°-°) dn = 0 since the integrand converges + C- cm Hence, for y > 0: +O einy einy d” = 211:] W 2 2 TI _ then 'P Y p = e i 21(y.C) 291 -y > 0 and for y > 0 95 {\ at "- t.- ~ / I \ \ + x I l \ Cm ’ it ’Y>0 \ ’ \ - *‘ 11:pr C; \‘ _. .g .. P l ‘ -: Co L i . T v r I ' Dr ‘ C- V, x \— - J \ P —- \ \1L-,pr.- \ y y’ near y = O, that is all field points are greater than source points. We define Vy(C) as follows 98 +"Jx(y’.() epr’ Vy(C) = jwc 2p dy’ f f Hence boundary condition (1) yields 2 t r+ r- _ N w - w - "x: - Vy(§) (5) cf xc xf In the same fashion boundary condition (2) yields P _ P N2 w" 3: (w r - w”) = —£ V (C) (6) cf xC p xf xf p y c c Near y = -t interface all y < y’, hence boundary condition (3) at y = -t interface yields wr+ epft - wr- e-pft + szf' e-pst = e-pft W (C) (7) X! xf If x- y up" where Uy(€) is defined as + coJx(y’.c) e-pr’ W (C) a we 2 dy’ Y J f Pf 0 Similarly, boundary condition (4) gives pf r+ p t r- -p t 2 t. -p t pf -p t ‘—— (w e f - w e f ) + N it e s = —— e f U (Q) (8) xf X! at x: ps y S First, we eliminate Hie between (S) and (6) leading to P _ P P w'* (1- —§ ) + u' (1+ —5 ) = v (c) ( —5 - 1) (9) xr pC xr pc y pc Secondly, we eliminate w; between (7) and (8) leading to P _ P _ P _ w'*(-£+1)eprt+w’ (1-i)eprt=w(c) (i-1)eprt(1o) xf S xf ps y ps Combining (9) and (10) and after some manipulations, the 99 4. _ expressions for w'‘- and "it are as follows X r+ _ (1) -2p t (2) -2p t "x: --R+ e f Hy(€) +R+ e f Vy(§) (11) r- _ (1) -2p t (2) "x: - R_ e f Hy(C) + R_ Vy(§) (12) where 31‘? Biz? R11)and R12)are defined below Pft Rl1)= (pt+pc)(pf-p') e 2 2cosh(pft) [(pf+p§p.)tanh(pft)+pf(ps+pc)l _ _ P t R(2)= (pf P'pr pc) 8 f 2 Zcosh(pft) [(pf+p€p')tanh(pft)+pf(p.+pc)] (1)_ (2) R_ — R+ (p +p )(p -p ) eptt R(2)_ f s f c 2cosh(pft) [(p:+pcp.)tanh(pft)+pf(p'+pc)l Substitute the expressions of Vy(§) and Wy(€) in (11) and (12) gives the final expressions for w;: and w:; as follows Xf 2506 p + + y +onj(,c) r+ = I x y ’ (R}1)e-pf(y’+2t)+ R(Z) epf(Y'-2t)) d r r “N and +me(y’.C) ,. = (1) -p (y’+2t) (2) p y’ . w I—leep (R_ef +R_ ef )dy xf f 1‘ -co 100 APPENDIX D APPENDIX D We would like to evaluate J —9S— . That is, we want to perform +c+cp P C integration around the upper half plane surface wave pole. change of variables such that <+< ’¢ II 0 CD = J¢ hence dc ’8 e d¢ We make a where the contour around g = -Cp, 8 and w are shown in Figure 23. Therefore we have *5 O + W n 'U I d§ _ _ )8 ejw + c ejv -2nj 101 Figure 23: Evaluation of integration around the pole. 102 APPENDIX E APPENDIX E Symmetric Slab Specialization In order to recover Rozzi’s result, we must shift the y-axis as shown in Figure 24. It will be shown that both the discrete and the continuous part of the Green’s function specialize to Rozzi’s for the symmetric slab waveguide. For this case ps - pc, hence the coefficients 81(Cp), 82(Cp) and B3(C) in the expression of the discrete Green’s function in Chapter Three become - ‘17 81(cp) ’ 4§p12+7t) 83(Cp) 2 2 p 4§p12+1t5 From Figure 24, we define § such that y = § - t/2. We also let t/Z = d. By doing so, the discrete Green’s function becomes pol, = 'ZJI‘OZO [31(Cp) + 32(Cp)(c082Kd-Jsin21(d)] [cosKy cosKy cost - cosK§ sinKy’ sian] + [82(c052Kd-jsin2Kd) - Bl][sinKy sinKy’ cost + sinKy cosKy’ sian] e-Jcplz-Z I We use the eigenvalue equation as in Chapter Three specialized for the 103 symmetric slab -27K 7-K2 tan(2Kd) = (1) to evaluate cos(2Kd) and sin(2Kd). We found K2- 2 cos(2xd) = 7 (2) 2 2 kf-kc and sin(2Kd) = 315———— (3) 2 2 k -k f c Using (1) and (2) in the expression of Gmflo and the fact that §' = y’ + t/Z, we have R Z 7 , = o 0 ~ ~, -j§ |z-z | Gmne 2gp 1+7d cos(Ky) cos(Ky ) e p (4) In Rozzi’s result, the discrete Green’s function is expressed as G (§|§'-z-z') = -A2e (37) e (37') e'JCplz'z'I (5) pole ’ X0 X0 where exo(y) = A cos(Ky) and 2 R020 A = 2 ch [ cos7(xd) + d + sinéZKd) ] Using (1) and (3), cosz(Kd) is found to be K2 72+K2 cosz(Kd) = (6) By substituting (3) and (6) in the expression for A, we have 104 k 2 1 A2 = o o 2§p(1+7d) With the above expression for Ag, Gpole in (4) will be the exact replica of (5). Similarly, the radiation Green’s function of Rozzi’s is the exact replica of our specialized continuous Green function. In fact, Rozzi’s results are N 2 (p) ~ ~,, _ , _ _ n: ~ ~. J<(p)lz-2’| GR(y|y ,2 z ) - I 4 exo(y.p) exo(y ,p) e dp 0 Q 2 (p) - TE " ~. JC(p)|z-z’| J 4 exE(y.p) ex£(y .p) e dp 0 where e (§ p) = A cos(0§) x: ’ C e (§,p) = A sin(c;) xo ~ C and A =J27; a-(p) = /v2+p2 V n -n2 k 2 f c 0 V C(p) = \/1+(v/p)zsinza~d C(p) = \/1+(v/p)2coszo~d -k 2 Z := o 0 TE C(PI C(p) is defined as in chapter three for the second integral term. 105 “y cover x y= o > 2 film _________________________ o y=-t/2 y=-t substrate Figure 24: Symmetric slab specialization leading to § = y + t/Z. 106 LIST 0!" REFERENCES FT __"1 [1] [2] [3] [4] [5] [6] [7] [a] \\\,/ (9] [10] [11] [12] LIST OF REFERENCES T.E. Rozzi,“Rigorous analysis of the step discontinuity in a planar dielectric waveguides,"IEEE MTT Trans.,vol.26,pp.738-746 oct.1978. S.V. Hsu and D.P. Nyquist,”Integral-operator formulation for scattering from obstacles in dielectric optical waveguides,“ in Digest of the U.S National Committee/International Union of Radio Science Meeting (National Academy of Sciences, Washington, D.C.,1979),p.90. S.V. Hsu and D.P. Nyquist,“Integral-equation formulation for mode conversion and radiation from discontinuity in open boundary waveguide,”in Digest of the U.S National Committee/International union of Radio Science Meeting (National Academy of Sciences, Washington, D.C,1980),p.62. N.K. Uzunoglu,"Scattering from inhomogeneities inside a fiber waveguide,"0pt.Soc.Am.,vol.71,No.3,March 1981. P.G. Cottis and N.K. Uzunoglu,"Ana1ysis of longitudinal discontinuities in dielectric slab waveguides,"0pt.Soc.Am.,vol.1, No.2,Feb.1984. T.G. Livernois and D.P. Nyquist,"Integral-equation formulation for scattering by dielectric discontinuities along open boundary dielectric waveguides,“0pt.Soc.Am,vol.4,pp.1289,July 1987. A. Sommerfeld,"Ueber die Ausbreitung der Wellen in der drahtlosen Telegraphic,“Ann.Physik,vol.28,pp.665,1909. 3.3. Bagby and D.P. Nyquist,“Dyadic Green’s function for integrated electronic and optical circuits,"IEEE MTT-S Tran.,vol.MTT-35, pp.206-210,Feb 1987. A. Sommerfeld, Partial differential Equations in Physics,New york: Academic Press,pp.236-265,1965. H.F. Weinberger, A First Course in Partial Differential Equations, New york: John Wiley and Son,Inc.,196S. R.E. Collin, Field Theory of Guided Waves, New york: Mc Graw Hill, pp.485-488,1960. D. Marcuse, Light transmission Optics, Princeton, N.J: Van 107 Nostrand Reinhold,chap.9. [13] D. Marcuse, Theory of Dielectric Optical Waveguides, New York: Academic,chap.3,1974. [14] T.E. Rozzi and G.H. In’t Veld,“Fie1d and network analysis of interacting step discontinuities in planar dielectric waveguides," IEEE MTT Tans.,vol.MTT-27,pp.303-309,1979. [15] R.F. Harrington, Time Harmonic Electromagnetic fields, New York: No Graw Hill,pp.125-128,1961. ’ 108 "‘(IWL'TII‘II‘IIITIMIIIIITB