DYADIC GREEN'S FUNCTIONS FOR LAYERED GENERAL ANISOTROPIC MEDIA AND THEIR APPLICATION TO RADIATION OF DIPOLE ANTENNAS

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1 Sracse Universit SURFACE Dissertations - ALL SURFACE Ma 4 DYADIC GREENS FUNCTIONS FOR LAYERED GENERAL ANISOTROPIC MEDIA AND THEIR APPLICATION TO RADIATION OF DIPOLE ANTENNAS Ying Hang Sracse Universit Follow this an aitional works at: Part of the Engineering Commons Recommene Citation Hang, Ying, "DYADIC GREENS FUNCTIONS FOR LAYERED GENERAL ANISOTROPIC MEDIA AND THEIR APPLICATION TO RADIATION OF DIPOLE ANTENNAS" (4. Dissertations - ALL. Paper 99. This Dissertation is broght to o for free an open access b the SURFACE at SURFACE. It has been accepte for inclsion in Dissertations - ALL b an athorize aministrator of SURFACE. For more information, please contact srface@sr.e.

2 ABSTRACT In this issertation, the aic Green s fnctions (DGFs for nbone an laere anisotropic meia, with no restriction impose on the meim propert, are erive. Utilizing the obtaine DGFs, the raiation problems of a Hertzian ipole an a microstrip antenna in the presence of an anisotropic sbstrate are solve. After a brief introction, the eigenvector aic Green s fnctions (E-DGFs for an nbone general anisotropic meim throgh the eigen-ecomposition metho are erive. The E-DGFs of a laere anisotropic geometr are then constrcte base on the erivation of the nbone E-DGFs sing two ifferent approaches. One is throgh the smmetrical propert of the DGFs an the other is throgh the irect constrction metho. Rigoros proof an etaile erivation of the formlation for the E-DGFs are presente. The sage an limitation of each approach as well as the relationships between the corresponing E-DGFs are iscsse. Appling the metho of stationar phase to the associate E-DGFs, we formlate the raiation fiels of an arbitraril oriente Hertzian ipole locate either above or insie the laere anisotropic meim. The important new finings incle the analsis of the raiation fiel in terms of the reflection coefficients as a fnction of incience angle, an the se of the biasing magnetic fiel to improve the broasie irectivit for a z-irecte sorce when a groelectric meim is involve. In aition to solving the raiation of a Hertzian ipole in the presence of a laere anisotropic meim, the laere E-DGFs erive here are also tilize to solve the more practical problem of a microstrip ipole printe on an anisotropic sbstrate. A metho of

3 moment soltion is formlate with the E-DGF in the spectral omain. To emonstrate the feasibilit of this metho applicable to a general anisotropic meim, the crrent istribtion, inpt impeances, an raiation patterns are nmericall calclate for a microstrip ipole printe on varios anisotropic sbstrates. Frthermore, a etaile parametric st of the effect of freqenc, an irection an magnite of the biasing magnetic fiel is provie for a ipole printe on a groelectric sbstrate. The parametric analsis in this issertation ma lea to a metho whereb the aitional freeom introce b the groelectric meim can be tilize effectivel to ajst the resonant length an raiation pattern of a printe ipole antenna.

4 DYADIC GREEN S FUNCTIONS FOR LAYERED GENERAL ANISOTROPIC MEDIA AND THEIR APPLICATION TO RADIATION OF DIPOLE ANTENNAS b Ying Hang B.S., Nanjing Universit, 5 M.S., Nanjing Universit, 8 Dissertation Sbmitte in partial flfillment of the reqirements for the egree of Doctor of Philosoph in Electrical an Compter Engineering in the Graate School of Sracse Universit Ma 4

5 Copright Ying Hang 4 All Rights Reserve

6 ACKNOWLEDGEMENTS Foremost, I wol like to express m sincere gratite to m avisor Professor Ja Koon Lee for the continos spport of m PhD st an research, for his patience, motivation, enthsiasm, an immense knowlege. His giance helpe me in all the time of research an writing of this issertation. I col not have imagine having a better avisor an mentor for m PhD st. I wol also like to thank Professor Ercment Arvas, Professor Qi Wang Song, Professor Amit Agrawal, Professor Prasanta Ghosh, an Professor Thong Qoc Dang for serving as members of the oral examination committee. I wol like to acknowlege the spport from the Graate Eiting Center of Sracse Universit an its staff for proofreaing the chapters. I also thank m colleage Dr. Jennifer Graham for spening her time an effort in proofreaing Chapter 3. throghot. Last, bt b no means least, I thank m famil for their spport an encoragement v

7 CONTENTS Abstract..i Acknowlegements v List of Figres xi List of Tables..xxii INTRODUCTION.... Anisotropic Materials an Applications.... Previos Work Daic Green s Fnctions an Raiation of a Hertzian Dipole A Printe Dipole in the Presence of an Anisotropic Sbstrate Chapter Otlines... DYADIC GREEN S FUNCTIONS FOR AN UNBOUNDED GENERAL ANISOTROPIC MEDIUM Eigen-ecomposition of DGF for a General Anisotropic Meim DGFs of an Unbone Uniaxial Meim DGFs of an Unbone Grotropic Meim vi

8 .3. DGFs of a Groelectric Meim DGFs of a Gromagnetic Meim DYADIC GREEN S FUNCTIONS FOR HALF-SPACE AND TWO-LAYER GEOMETRIES Daic Green s Fnctions with Sorce insie the Isotropic Region DGF for the Region of z z Moifie Smmetrical Propert for (, G ( r, r of the Region z z an z z Proof of Moifie Smmetrical Propert of (, G ( r, r DGF for the Region of z z Moifie Smmetrical Propert for (, G ( r, r an (, G ( r, r Proof of the Moifie Smmetrical Propert Usage an Limitation of the Moifie Smmetrical Propert DGFs of a Two-Laer Geometr with a Sorce insie the Anisotropic Region Direct Constrction Metho to Obtain the DGFs for All the Regions Discssion of DGF (, G ( r, r Obtaine Using Two Approaches RADIATION OF A HERTZIAN DIPOLE IN THE PRESENCE OF A LAYERED ANISOTROPIC MEDIUM vii

9 4. Raiation of a Hertzian Dipole for a Half-Space Problem The Sorce Embee in the Isotropic Region The Sorce Embee in the Anisotropic Region Raiation of a Hertzian Dipole for a Two-Laer Problem The Sorce Embee in the Isotropic Region The Sorce Embee in the Anisotropic Region Analtic an Nmerical Valiation Case I: Self-check of (, G an (, G Case II: Unbone Isotropic Meim an Laere Biaxial Meim The Anisotropic Region Fille with Groelectric an Gromagntic Meia Raiation of a Hertzian Dipole in the Presence of Groelectric Meim Raiation of a Vertical Dipole on Top of a Half-space Groelectric Meim Parametric Effect of Freqenc an Biasing Magnetic Fiel Relation of Directive Raiation with Total Internal Reflection The Two-Laer Problem with a Sorce above the Anisotropic Region The Two-Laer Problem with a Sorce Embee in the Anisotropic Region Raiation of a Horizontal Dipole insie a Grone Isotropic Plasma Slab Raiation of a Horizontal Dipole insie the Grone Groelectric Slab Raiation of a Vertical Dipole insie the Grone Groelectric Slab viii

10 5 RADIATION OF A MICROSTRIP DIPOLE PRINTED ON AN ANISOTROPIC SUBSTRATE Formlation of Metho of Moment (MOM Basis Fnction Excitation Smmetr Analsis for Impeance Matrix Zim an Z mi Smmetr of Integran for Z im Integration Path Sampling Rate an Integration Range Properties of the Daic Green s Fnction Integration Path to Avoi the Singlarit of (, G Srface Wave Pole A Grone Isotropic Slab A Grone Uniaxial Slab A Grone Biaxial Slab A Grone Groelectric Slab Crrent Distribtion of a Microstrip Dipole Crrent Distribtion of a Microstrip Dipole on a Grone Biaxial Slab Crrent Distribtion for a Microstrip Dipole on a Grone Groelectric Slab... 4 ix

11 5.4 Antenna Parameters Formlations of Inpt Impeance an Raiate fiel Inpt Impeance Raiation Behavior Antenna Gain Nmerical Valiation with Printe Dipoles on Varios Meia A Printe Dipole on an Isotropic Slab A Printe Dipole on a Grone Biaxial Slab A Printe Dipole on a Grone Ferrite Slab A Printe Dipole on a Grone Groelectric Slab Inpt Impeance an Resonant Length Raiation Behavior CONCLUSIONS... 3 BIBLIOGRAPHY VITA x

12 LIST OF FIGURES Fig. -: Two ifferent polarizations of plane waves in niaxial meim: (a Tpe I (orinar wave an (b Tpe II (extraorinar wave Fig. -: Arbitrar biasing magnetic fiel in the xz coorinate Fig. 3-: Geometr of (a half-space an (b two-laer problem Fig. 3-: Amplite vectors of waves in the two-laere geometr Fig. 3-3: Two ifferent geometries for the proof of moifie smmetrical propert Fig. 3-4: Two ifferent geometries constrcte here for the proof of smmetrical propert when the sorce are in ifferent regions Fig. 3-5: Geometr of the two-laer problem with sorce insie the anisotropic region Fig. 4-: Geometr for the half-space problem with a Hertzian ipole place at a certain istance awa from the bonar separating the isotropic an anisotropic regions. (a Dipole in Region (isotropic region an (b ipole in Region (anisotropic region Fig. 4-: Two-laer geometr: (a The ipole is locate in Region (isotropic meim an (b The ipole is locate in Region (anisotropic meim Fig. 4-3: Fig. 4-4: Geometr of Case I with (a DGF (, G (b DGF Raiate fiels calclate sing (a DGF of (, G (, G an (b DGF of (, G Fig. 4-5: Geometr of nbone isotropic meim Fig. 4-6: Raiation pattern of the x, an z-irecte Hertzian ipole in nbone meim in the XZ plane... 9 xi

13 Fig. 4-7: E vs. in the 45 o plane for a z-irecte Hertzian ipole in Region (free space with h. an x,,3, x =, 5, Fig. 4-8: E vs. h in the 45 o plane for a z-irecte Hertzian ipole in Region with o o o (,, (,8,4, (,, (3,,... x z Fig. 4-9: E vs. in the 9 o plane for varios vales of h for a z-irecte Hertzian ipole positione in region of a two-laer problem. Region is biaxial with (,, (,8,4, (,, (,, an slab thickness is.4... x z Fig. 4-: E vs. in the o plane for varios vales of (rotation angle an an x- irecte Hertzian ipole positione in Region of a two-laer problem. Region o o o is biaxial with (,, (,8,4, (,, (,7, an with x z h.3, Fig. 4-: Geometr of the grone groelectric slab in [46].... Fig. 4-: E vs. in the plane of o for a z-irecte Hertzian ipole locate in Region, with h.55 awa from the interface of free space an groelectric slab. (a Reslt from W [46] an (b Raiate fiel sing the formla in this chapter Fig. 4-3: Geometr of a two-laer problem fille with gromagnetic meim o o Fig. 4-4: E an E vs. in the plane of, 9 with the sorce in Region an the fiel in Region xii

14 Fig. 4-5: E vs. in the plane of o for a z-irecte Hertzian ipole locate in Region with h awa from the interface of free space an groelectric meim. The biasing magnetic fiel is along the (a z an (b irections Fig. 4-6: Fiel pattern for a z-irecte Hertzian ipole locate in Region with h awa from the interface of free space an groelectric meim. The biasing magnetic o o o o o fiel is along, 4, 6, 8, an the freqenc of the incience B B wave is.. (a E an (b E vs. in the plane of o ; (c E an ( E vs. in the plane of 9 o... 9 Fig. 4-7: Normalize wave vector srface of an isotropic meim an these for Tpe I an Tpe II waves of a groelectric meim vs. in the plane of o... 3 Fig. 4-8: Magnite an phase of reflection coefficients R, R vs. ifferent incience angles for an incience plane of o Fig. 4-9: (a E an (b E vs. in the plane of o for a z-irecte Hertzian ipole locate in region with h awa from the interface of free space an groelectric meim with the irection of biasing magnetic fiel along o. vv hv B 9 p, b.9 p,., b b p Fig. 4-: (a E an (b E vs. in the plane of o for a z-irecte Hertzian ipole locate in Region with h awa from the interface of free space an a xiii

15 groelectric meim with. an the biasing magnetic fiel along o o 9, B B Fig. 4-: (a E an (b E vs. in the plane of 9 o for a z-irecte Hertzian ipole in Region with h awa from the interface of half-space groelectric o o meim with the biasing magnetic fiel along 9, Fig. 4-: Normalize raiate fiel E for an x-irecte ipole locate at the interface of the grone groelectric slab an air in the xz-plane for ifferent freqenc regions of the groelectric meim Fig. 4-3: Normalize raiate fiel patterns E for an x-irecte ipole locate at an airgroelectric meim interface of a grone slab as a fnction of ifferent biasing magnetic fiel irections Fig. 4-4: Geometr of a horizontal ipole locate insie a grone groelectric slab with a istance h awa from the air-slab interface Fig. 4-5: Power ensit (B of (a component an (b component as a fnction of the slab thickness Fig. 4-6: Power ensit (B of component as a fnction of the slab thickness. B B h.5 an f. f p Fig. 4-7: Power ensit of (a component an (b component raiate in broasie irection b a nit crrent moment -irecte Hertzian ipole place at the center xiv

16 of a grone groelectric slab as a fnction of the slab thickness with b.p, f. f p Fig. 4-8: (a Power ensit an (b phase ifference of P ( an P ( for a nit crrent moment -irecte Hertzian ipole place in the mile of a grone groelectric slab as a fnction of the slab thickness with b p, f. f p.. 46 Fig. 4-9: Normalize raiate fiel for a -irecte ipole locate in the mile of the isotropic plasma slab in the XZ-plane as a fnction of observation angle which is measre from the z-axis Fig. 4-3: Raiation patterns for a -oriente ipole locate in the mile of the anisotropic groelectric slab vs. ifferent slab thicknesses Fig. 4-3: Geometr of a vertical ipole locate insie the grone anisotropic slab with a istance of h awa from the air-slab interface Fig. 4-3: Normalize fiel pattern of E vs. in the plane of o with vertical ipole locate in the mile of the grone groelectric slab of thickness of.5. 5 Fig. 4-33: (a The angle of the maximm raiation of the vertical ipole insie the grone groelectric slab vs. ifferent freqencies. (b The ifference of the raiate power along the maximm raiation irection of the Hertzian ipole insie the slab an in free space vs. ifferent freqencies Fig. 4-34: Normalize fiel pattern of E an E (xz plane for a vertical ipole locate in the mile of a grone groelectric slab of thickness.5 (free space xv

17 wavelength with ifferent irection of the biasing magnetic fiel p, b.8 p,. p Fig. 4-35: Normalize fiel pattern of E (XZ plane for the vertical ipole locate in the mile of the grone groelectric slab with the biasing magnetic fiel perpeniclar to the observation plane as a fnction of ifferent thickness p, b.8 p,. p Fig. 4-36: Normalize fiel pattern of E (XZ plane for a vertical ipole locate in the mile of a grone isotropic slab of vs. ifferent slab thickness Fig. 5-: Geometr of the microstrip ipole problem Fig. 5-: Locations of the sbomain basis fnctions Jx( x an J ( along x-irection Fig. 5-3: Integration in the kx kplane. Area over (a the entire kx kplane, (b the pper half-space, an (c the first qarant of kx kplane Fig. 5-4: Smmetr relation for the aic Green s fnction elements of (, xx x, x G k k for a reciprocal meim Fig. 5-5: Integration regions in the kx - k plane: (a Region (b Region an (c Region Fig. 5-6: Fig. 5-7: Region integration paths in the kx an k planes: (a kx plane, (b k plane Region A integration paths in the k x an k planes: (a kx plane, (b k plane. 77 xvi

18 Fig. 5-8: Region B integration paths in the kx an k planes: (a kx plane, (b k plane.. 77 Fig. 5-9: Region C integration paths in the kx an k planes: (a kx plane, (b k plane. 78 Fig. 5-: Effect of the magnite of on the calclation of.5k, (b.6k Re G (, xx in the region A. (a.,, 5,3,4,,, o, o, o x z, an h Fig. 5-: Geometr of the grone isotropic slab Fig. 5-: Graphical etermination of kz for (a TM an (b TE moes Fig. 5-3: Srface wave propagation constants for a grone ielectric slab with r =.55, for / to Fig. 5-4: Geometr of the grone niaxial slab with the optic axis along z-irection.. 84 Fig. 5-5: Graphical metho to obtain the srface wave moes for a grone niaxial slab z The slab thickness is (a.63 an (b of Fig. 5-6: The vale of (a Re ( G xx an (b the eterminant I R R vs. k for the propagation angle of tan k kx 45 o... 9 Fig. 5-7: The eterminant I R R vs. k for the ifferent propagation angles of the srface wave from o to 9 o tan (k / k x xvii

19 Fig. 5-8: Srface wave poles an branch point loci for a grone biaxial slab of x,, z 3,5,4 with.3. (a in all the for qarants of the kx - k plane, an (b in the first qarant of the kx - k plane Fig. 5-9: The propagating constant k x, sw (a (with k an (b k, sw (with kx verss the thickness of a biaxial slab for varios srface wave moes present in a conctor backe biaxial slab with,, 3,5, x z Fig. 5-: k x, sw obtaine with k verss the thickness of the groelectric slab for varios srface wave moes present in a conctor backe groelectric slab with the biasing magnetic fiel along (a -irection, an (b z-irection.... Fig. 5-: k, sw obtaine with kx verss the thickness of the groelectric slab for varios srface wave moes present in a conctor backe groelectric slab with biasing magnetic fiel along (a -irection, an (b z-irection.... Fig. 5-: Crrent istribtions of a ipole on (a a nroaste biaxial meim of x,, z 5,3,4 an (b a rotate biaxial meim with,, o, o,3 o Fig. 5-3: Crrent istribtion of Re ( J x for a /ipole on groelectric meim with an the biasing ra / s, p ra / s, b.5 p o o magnetic fiel along x-irection ( 9,. h.,w.. N=6, B N= an N= B xviii

20 Fig. 5-4: Crrent istribtion of Re J( x an Imag ( J x for a / ipole on a 9 9 groelectric meim with.59 ra / s, ra / s,.5 an the biasing magnetic fiel along (a the z-irection ( o b p o o an (b the -irection ( 9, 9. h.,w.. N= Fig. 5-5: (a Real an (b imaginar parts of the inpt impeance of a printe ipole on an isotropic sbstrate of,, 3.5,3.5,3.5 with the ipole with of x z.. The height of the slab is Fig. 5-6: Inpt impeance for gap-fe ipole with with of. printe on an isotropic. The height is sbstrate with,,.45,.45,.45 x z Fig. 5-7: (a Real an (b imaginar parts of the inpt impeance vs. ipole length of a microstrip ipole on an nrotate biaxiall anisotropic meim of x,, z 5,3,4 B. The the ipole with is Fig. 5-8: Effect of rotation angle β to the inpt impeance an the resonant length with α = o an γ= o. (a an (a are from Pettis [37] Fig. 5-9: Effect of rotation angle α to the inpt impeance an the resonant length with B β= o an γ= o. (a an (a are from Pettis [37] p B Fig. 5-3: Directive gain of a resonant ipole on a conctor backe slab fille with an x z nrotate biaxial meim of,, 5,3,4. (a from Pettis [37]... 8 xix

21 Fig. 5-3: Center-fe ipole resonant length verss freqenc of printe ipole on a ferrite sbstrate of.6, H.5 T, M s.5 T an an isotropic r sbstrate of.6. The irection of the biasing magnetic fiel is along the r o o irection of the ipole axis, i.e., x-irection 9, Fig. 5-3: Wave vector srfaces of Tpe I an Tpe II waves for the groelectric meim 9 9 with.59 ra / s, ra / s,, an the biasing p b p magnetic fiel is along z-irection ( o... Fig. 5-33: (a Real an (b imaginar parts of the inpt impeance of the printe ipole on groelectric sbstrate with ifferent irections of the biasing magnetic fiel ra / s, p ra / s, b p, B the ipole with is B B. an the sbstrate thickness is Fig. 5-34: (a Real an (b imaginar parts of inpt impeance vs. varios ipole length for a microstrip ipole printe on a groelectric sbstrate with grofreqenc.5,.5, an... 5 b p b p b Fig. 5-35: Co-polarize an cross polarize raiation pattern of printe ipole for (a E- plane an (b H-plane with the irection of the biasing magnetic fiel along x- irection Fig. 5-36: Co-polarize an cross polarize raiation pattern of printe ipole for (a E- plane an (b H-plane with the irection of the biasing magnetic fiel along - irection p xx

22 Fig. 5-37: Co-polarize an cross polarize raiation pattern of printe ipole for (a E- plane an (b H-plane with the irection of the biasing magnetic fiel along z- irection Fig. 5-38: (a Co-polarize fiel pattern an (b cross-polarize fiel pattern in E-plane with grofreqenc.5,.5,,when the biasing magnetic fiel b p b is along x irection Fig. 5-39: (a Co-polarize fiel pattern an (b cross-polarize fiel pattern in H-plane with p grofreqenc.5,.5,, when the biasing magnetic b p b p fiel is along x irection b b p p xxi

23 LIST OF TABLES Table 4-: Freqenc bans with ifferent propagating waves for a groelectic meim.. 6 Table 5-: k verss the thickness of the slab for varios srface wave moes in a grone niaixal slab with (a,,.55,.55,.55, (b z.5,.5,.55, (.55,.55,.5, (c.55,.55, Table 5-: Propagating moes for ifferent choices of permittivit... 9 Table 5-3: Wave vector srfaces of all the freqenc regions for Case I with the choice of b p Table 5-4: Wave vector srfaces of all the freqenc regions for Case II with the choice of b p Table 5-5: (a E-plane an (b H-plane irectivit for a center-fe ipole on a ferrite sbstrate.... Table 5-6: Resonant lengths for microstrip ipoles on a groelectric sbstrate xxii

24 INTRODUCTION. Anisotropic Materials an Applications Dielectric an magnetic anisotropic effects often exist within man materials that are se as the sbstrate for integrate microwave circits an printe-circit antennas []. Two ifferent tpes of anisotropic effects are sall observe: reciprocal an non-reciprocal anisotropies. Sapphire an boron nitrie [-4] are niaxiall anisotropic, while PTFE cloth an glass cloth [5-7] are biaxiall anisotropic. Both niaxial an biaxial properties belong to reciprocal anisotropies. Some of these reciprocal anisotropic effects are nintentional an occr natrall in the material, while others are introce ring the manfactring process. Non-reciprocal anisotropic effects (magnetic or ielectric are sall introce b appling the external magnetic fiel to the ferrite, plasma or semiconctors. The freqenc-epenent permeabilit of ferrite reveals interesting applications in antennas, incling the ajstment of raiation patterns an the rection of raar cross-sections [8-]. The freeom of controlling the non-reciprocal behavior of the ferrite, b appling the external magnetic fiel, ncovers wie applications for microwave evices sch as isolators, circlators an phase shifters [-4]. However, there are some limitations for the ferrite material. First, relative banwith ecreases in the operating freqenc beon 4 GHz e to material limitations. Seconl, it is ifficlt to achieve the fll integration for the microwave integrate circits (MIC e to the incompatibilit of the ferrite an semiconctor processing technolog [5-6]. To overcome these rawbacks of ferrite, semiconctor evices base on the

25 groelectric propert escribe b the tensor permittivit have been extensivel researche in the last ecae. Uner the inflence of a constant magnetic fiel applie to the semiconctors, the groelectric effect arises from the cclotron motion of nearl free electrons, in accorance with the Dre moel. The behavior of magnetize semiconctors can then be characterize b a freqenc-epenent permittivit tensor [7], as illstrate in the eqation below. j ( j where p( j pc p,, j c j c j is the grofreqenc (or cclotron freqenc calclate sing eb / m, where m represents c the mass of each electron with charge e (a negative nmber an B is the constant magnetic fiel applie to the meim. is the static ielectric constant of the material. Collision freqenc / is incle in the expression of the ielectric tensor, which moels the losses in the semiconcting material. is the momentm relaxation time, which is eqivalent to the mobilit of the semiconctor. p is the plasma freqenc ecie b carrier concentration as / N e m, where N is the nmber of free electrons per nit volme. Common semiconctor materials associate with magnetoplasma evices are InSb, Te, GaAs, Si, an Hg-xMnxTe. The properties of these materials are provie in Table of [8].

26 3 For example, a high-qalit, moeratel-ope GaAs with a carrier concentration of n 5 3. cm is eqivalent to a groelectric meim with plasma freqenc p 3 ra/s. With a magnetic fiel of 38 G, the cclotron freqenc is given as ra/s. c De to the non-reciprocal groelectric characteristics of the semiconctor, wie application of the semiconctors to circlators [9-4] an resonators [5] are evelope. Propagation for semiconctor wavegies has also been of interest for researchers. A mltilaer grotropic thin-film semiconctor wavegie comprising S-I GaAs/AlAs/n GaAs/AlGaAs in a static magnetic fiel of.5 T has been analze over the freqenc range of GHz [6]. Srprisingl, while extensive research is performe on the microwave evices, ver few works is fon for the application of the groelectric meim (especiall magnetize semiconctors to the printe ipole antenna. In aition to the anisotropic materials presente above, recent avances in material technolog allow the proction an control of material with the anisotropic propert that oes not natrall occr. The so-calle metamaterial is gaining more an more attention e to its novel characteristics. A significant amont of research has been one for applications to antennas, wavegie miniatrization [7-8], an transmission lines [9]. Especiall, the strong grotrop [3-33], introce b the artificial materials, has starte to attract attention as a potential caniate for applications in microwave an optoelectronic evices. As presente above, an anisotropic effect either occrs natrall or is intentionall introce for specific applications. In both cases, accrate analsis of ifferent tpes of anisotropic effects is reqire for the fll nerstaning of the electromagnetic wave s interaction with the anisotropic meim. For the planar laere geometr commonl se in

27 4 toa s transmission an raiation applications, one of the well-establishe tools for the analsis of electromagnetic propagation, raiation an scattering problems is the metho of Green s fnction [34]. In this issertation, the approach of the Green s fnction is tilize to analze the raiation properties of a Hertzian ipole an printe ipole on top of an anisotropic sbstrate. In particlar, the research objectives primaril incle the following: To formlate Green s fnction soltions with the sorce place insie an nbone general anisotropic meim with no restriction impose on the propert of the meim. To formlate Green s fnction soltions for the cases with the sorce place either above or below the interface of a single-laer or two-laer geometr fille with a general anisotropic meim. 3 To evelop nmericall efficient asmptotic techniqes for the analsis of the raiate fiels of a Hertzian ipole place above an below the interface of a single-laer or twolaer geometr fille with a general anisotropic meim. Especiall, etaile nmerical analsis is focse on the laere geometr fille with a groelectric meim. 4 To appl Green s fnction solving for the properties of microstrip ipole antennas printe at the interface of a conctor-backe, general anisotropic slab. Especiall, etaile nmerical analsis here is focse on the laere geometr fille with a groelectric meim.. Previos Work In this section, we will briefl review the previos work on the Green s fnction involve with the anisotropic meim an the raiation of a Hertzian ipole in the presence of an

28 5 anisotropic meim. Also, previos work is smmarize here on the application of the Green s fnction to solve the raiation of a printe ipole on top of an anisotropic sbstrate. Finall, throgh the complete literatre srve on the previos work, it is emonstrate how the major contribtions are mae b achieving or research objectives... Daic Green s Fnctions an Raiation of a Hertzian Dipole Several ifferent approaches have been propose to obtain the Green s fnction of laere strctre, incling Forier transform metho, which is eqivalent to the eigen-ecomposition metho [35-4]; the transition matrix metho propose b Krowne [4]; the eqivalent bonar metho b Mesa et al. [4]; an the clinrical vector wave fnction metho b Li et al. [43-44]. The transmission line metho, propose in [45], obtains the Green s fnction of an isotropic meim base pon the ecomposition of fiels into TE an TM moes. This metho can actall be treate as a special case of the eigen-ecomposition metho. The eigen-ecomposition metho was first introce b Lee an Kong [35] to obtain the Green s fnction of laere geometr fille with a niaixal meim with an arbitraril oriente optic axis. The same metho was then extene to obtain the aic Green s fnction of a laere biaxial anisotropic meim [36-37], an nbone groelectric meim [38-39], an a gromagnetic meim [4]. The basic featre of the eigen-ecomposition metho is base on the calclation of the corresponing eigenvectors of the electric fiel from the ajoint matrix of the electric wave matrix. This matrix is erive from the secon-orer ifferential eqation of the electric fiel in the spectral-omain b appling a 3D Forier transform. The eigenvectors can be calclate either analticall [35-4] or nmericall. Different from the eigen-ecomposition metho, the transition matrix metho applies D Forier transform an tilizes a matrix

29 6 exponential approach metho. It reces the 6 Maxwell s eqations in the Cartesian coorinate to for eqations in which tangential electric an magnetic fiels are expresse in terms of the normal electric fiel an magnetic fiel components. For the sake of brevit, the E-DGFs correspon to the DGFs obtaine sing the eigen-ecomposition metho, while the T-DGFs refer to the DGFs obtaine sing the transition matrix metho throgh the issertation. The thir metho emploe for the mlti-laer strctre is the eqivalent bonar metho (EBM [4]. One of the main featres of the EBM is its objective to obtain not the biimensional spectral DGF, bt its inverse. It is claime that the EBM metho leas to a compact an stable algorithm. The EBM metho is sitable for the calclation of the propagation characteristics of the slot line, parallel plate wavegie, an the coplanar wavegie since this metho is constrcte for a geometr with pper-electric an lower-electric walls. Another metho to constrct the DGF is the clinrical vector wave fnction metho. The scattering aic Green s fnction for each laer is constrcte in terms of the clinrical vector wave fnctions b appling the metho of scattering sperposition. This metho has been applie to fin the DGF of the laere isotropic meim [43] an the nbone grotropic [44] meim. Ot of so man ifferent methos, the E-DGFs an the T-DGFs are the two most commonl se techniqes to constrct the aic Greens fnction for the planar stratifie anisotropic geometr. Both of the DGFs are sitable for the propagation an raiation problems. Comparing the E-DGFs with the T-DGFs, each has its merits. Derivation of the T-DGFs are more complex in mathematics, an onl the elements of the DGFs relating the tangential crrents an tangential sorces can be obtaine irectl from the soltion to a 4 b 4 linear eqation. However, the E-DGFs are expresse in terms of coorinate tensors an therefore make

30 7 the integral of the electromagnetic fiel straightforwar an sccinct. In aition, all the nine elements of the E-DGFs which correlate the electromagnetic fiels with an arbitrar crrent istribtion are obtaine one time with no frther work reqire. The T-DGFs are sitable for the problem with tangential crrent at the interface an are wiel applie for the nmerical analsis of transmission lines an patch antenna problems. The E-DGFs are sitable for problems with arbitrar crrent istribtion. With the DGFs obtaine, the raiate fiels of a Hertzian ipole can then be formlate. In the last few ecaes, researchers have extensivel stie the raiation of a Hertzian ipole in the presence of an nbone an laere anisotropic sbstrate. With the assmption of the biasing magnetic fiel along the z irection, the raiation of a ipole in nbone an laere groelectric meia is treate in [46-47]. The raiation of a Hertzian ipole involve with a gromagnetic meim is consiere in [48-49]. Extensive work for the raiation of the ipole over an insie the laere niaxial slab can be fon in [5-54] e to the simple tensor form of the niaxial meim. For the more complicate case of biaxial slab, formlation an analsis are available in [37]. It is note here that raiate fiels available in [37, 54] are erive from the E-DGFs for the corresponing geometr an meim. The raiate fiels of a ipole for the general anisotropic laers characterize b arbitraril oriente axes of anisotrop are consiere in [55-57], which is closel relate to the erivation of the T- DGFs. Unlike the expressions of a raiate fiel erive from the T-DGF (as in [55-57], which appl to general anisotropic geometr, formlations for the raiate fiel erive from the E- DGFs in previos work are sall solve for onl one specific tpe of an anisotropic meim. The raiate fiel for the general anisotropic geometr is not crrentl available. Accoring to

31 8 the literatre srve above, the reason for this is e to the lack of the E-DGFs for a laere geometr fille with a general anisotropic meim. To overcome this limitation of former E- DGFs an also to fill the gap, it is essential to evelop the E-DGFs for a laere general anisotropic geometr, which reqires the availabilit of the E-DGFs for an nbone general anisotropic meim. This irectl correspons to the first an secon research objectives of the crrent st, as propose in Section.. The thir research objective is to formlate the raiate fiels of a Hertzian ipole in the presence of a general anisotropic meim with no restrictions impose on the permittivit or permeabilit. This goal can be achieve sing the E-DGFs available after the first research objective is accomplishe. It is also inicate in the crrent literatre srve that nmerical reslts for the raiate fiel of a Hertzian ipole above or insie the groelectric slab with an arbitrar biasing magnetic fiel are not available. One special case investigate in [46-47] is abot the raiation of a Hertzian ipole in the presence of a groelectric meim with the biasing magnetic fiel along the z-irection. However, the effect of the biasing magnetic fiel on the raiation is not taken into accont. Ths, in aition to eveloping the general formlation, sting the raiation behavior of the ipole in the presence of a groelectric meim with arbitraril irecte biasing magnetic fiel becomes the secon part of the thir research objective... A Printe Dipole in the Presence of an Anisotropic Sbstrate Crrentl, there is an increasing interest in complete monolithic sstems which combine antenna elements or antenna arras on the same sbstrate as the integrate RF/IF front en network. One of the most poplar antenna elements is the printe antenna e to its

32 9 characteristics of low-cost, low-profile, conformabilit, an ease of manfactring. Tpical methos se to analze the printe circits incle the metho of moments (MoM, the finite ifference time omain metho (FDTD, an the finite element metho (FEM. The initial research emploing the metho of moments (MoM tilize a combination of spatial an spectral techniqes in orer to analze microstrip ipole an microstrip patch antennas on a single laer grone isotropic meim [58] [64]. In [58-6], the Green s fnctions are formlate in the spectral omain an the reaction formlation is applie in the spatial omain. Soon after, athors sch as Deshpane an Baile [6] following Itoh an Menzel s analsis techniqe [6] applie the reaction formlation entirel in the spectral omain. Man athors, incling Pozar [63-64], followe this approach. Later on, this approach was extene to solve the raiation problem of the microstrip ipole on the grone anisotropic sbstrate, which incle niaxial [4], biaxial [37] an ferrite [8-9] sbstrates. It is worth noting here that most crrent nmerical analses of transmission lines an patch antenna problems on anisotropic sbstrates [4, 8-9, 37] are involving the application of the T- DGFs, while ver little research has been performe on the nmerical application of the E-DGF. One relevant an available st is [65], which applies the E-DGFs to solve the microstrip antenna on an arbitraril oriente biaxiall anisotropic meim. However, it is again note here that the E-DGF se in [65] restricts its application to a laere biaxial meim onl. The srve of microstrip antennas above natrall leas to the forth research objective propose in Section., that is, to appl the E-DGFs solving for the properties of microstrip ipole antennas printe at the interface of a conctor-backe, general anisotropic slab. Since the E-DGFs are evelope in the spectral omain, this issertation will follow Itoh an Menzel s analsis techniqe [6] appling the reaction formlation entirel in the spectral omain to

33 etermine the crrent istribtion of microstrip antennas on a single laer, arbitraril oriente general anisotropic meim. The singlarit analsis of the E-DGF is also performe to accratel calclate the nmerical integral for the printe ipole on a laere biaxial slab. With the crrent istribtion obtaine, all other antenna parameters (incling resonant length, inpt impeance, an raiation patterns can be obtaine. In particlar, etaile nmerical analses will be focse on the printe ipole on a laere groelectric meim since there is ver little sch work fon from literatre srve..3 Chapter Otlines The major contribtions of this issertation incle the evelopment of the E-DGFs for a laere general anisotropic meim an the nmerical applications of the corresponing E-DGFs to the raiation of a Hertzian ipole an a printe ipole on a laere geometr. With extensive nmerical reslts an etaile analses, it emonstrates the feasibilit an valiit of appling the E-DGFs to a general laere anisotropic meim with no restriction impose on the propert of the meim. This issertation is organize as follows accoring to the research objective otline in Section.. In Chapter, the eigenvector aic Green s fnctions (E-DGFs for an nbone general anisotropic meim incling both reciprocal an non-reciprocal meia are erive sing the eigen-ecomposition metho. The E-DGFs for the nbone niaxial an grotropic meia are presente in Section. an Section.3, respectivel. It is iscovere that moification to the initial E-DGF of an nbone groelectric meim erive in [38] is reqire to fll represent the non-reciprocal behavior of the meim.

34 Chapter 3 presents the complete formlations of the E-DGF of each region for the halfspace an two-laer anisotropic problems. The E-DGFs with a sorce insie the isotropic region is presente in Section 3.. In Section 3., the moifie smmetrical propert is erive to obtain the DGFs above the sorce point when the sorce is insie the isotropic region. Particlarl, it is pointe ot that original smmetrical propert, from which the E-DGFs above the sorce point can be obtaine from E-DGFs below the sorce point, nees moifications when the anisotropic region is fille with a non-reciprocal meim. In Section 3.3, the moifie smmetrical propert for the DGFs with sorce an fiel points in two ifferent regions is erive for a laere nonreciprocal meim. The sage an limitation of this moifie smmetrical propert is iscsse. In Section 3.4, the irect constrction metho is propose to obtain the complete E-DGFs with a sorce locate in the anisotropic region. A comparison of the E-DGFs obtaine throgh these two ifferent approaches (moifie smmetrical propert an irect constrction metho is presente an interesting relationship is iscovere. In Chapter 4, the E-DGF constrcte in Chapter 3 is applie to solve the problem of raiation of a Hertzian ipole locate above an insie the anisotropic region sing the metho of steepest escent. Compare with the reslts obtaine in [55-57], the concise expressions for the raiate fiels consisting of the reflection an transmission coefficients provie a straightforwar phsical insight to the raiate fiel when the ipole is locate either above or insie the anisotropic laer. In Section 4., the focs is on the half-space geometr with a sorce locate either in an isotropic or an anisotropic region. In Section 4. the formlation of the raiate fiel is presente for a Hertzian ipole embee either insie an isotropic or an anisotropic region of two-laer geometr. In Section 4.3, the explicit formlas obtaine in previos sections are valiate nmericall with the available reslts from crrent literatre. In

35 Section 4.4, nmerical analsis is presente in etail for the raiation of a Hertzian ipole place both above an insie the laere groelectric slab, an a potential application for the enhance raiation of a ipole sing the grone groelectric slab is propose. In Chapter 5, the problems of the microstrip ipole on a grone sbstrate fille with varios anisotropic meia are solve sing the Forier transform omain metho of moment emploing Galerkin s metho. Section 5. presents the formlation of metho of moment. In Section 5., the properties of E-DGFs are stie in etail to facilitate the nmerical integration obtaine from Section 5.. The crrent istribtions of a printe ipole over the ifferent grone sbstrates, incling niaxial, biaxial, ferrite (gromagnetic an groelectric meia, are presente in Section 5.3. The inpt impeance, resonant length, an raiation patterns of the microstrip ipole over a grone isotropic slab, a grone biaxial slab, an a grone ferrite slab are calclate an compare with the previos reslts in Section 5.4. Also, in this section, nmerical iscssions will be presente in etail to illstrate the effect of the magnite an irection of the biasing magnetic fiel to the inpt impeance, resonant length, an raiation pattern of the microstrip ipole on a groelectric meim. Finall, conclsions are rawn an iscsse in Chapter 6. As emonstrate in the previos chapters, since the E-DGF evelope in this issertation is for the general anisotropic meim with no restrictions impose on the tpe of the meim, it has wie applications for a Hertzian ipole or a printe ipole on general anisotropic sbstrates. Ftre works are also briefl iscsse.

36 3 DYADIC GREEN S FUNCTIONS FOR AN UNBOUNDED GENERAL ANISOTROPIC MEDIUM In this chapter, the approach in [35] is extene so it can be applie to obtain the eigenvector aic Green s fnctions (E-DGFs of an nbone general anisotropic meim with no restriction impose on the propert of the meim. This approach will be calle the eigen-ecomposition metho. The basic featre of the eigen-ecomposition metho is base on the calclation of the corresponing eigenvectors from the specific ajoint wave matrix, which is obtaine from the secon-orer ifferential eqation in the spectral omain. This chapter is organize as follows. In Section., the eigen-ecomposition metho is applie to obtain the electric tpe E-DGF of a general anisotropic meim e to an electric crrent sorce. In Section., the eigen-ecomposition metho is applie to obtain the analtic formla of the E-DGF for a niaxial meim an an isotropic meim. Exact agreement is obtaine with the previos reslts. In Section.3, E-DGFs of the grotropic meia are erive sing the same metho. Particlarl, it is iscovere that the formlations of the E-DGFs for the nbone non-reciprocal meim an reciprocal meim are ifferent an moifications to the formlas propose in [38] are reqire.. Eigen-ecomposition of DGF for a General Anisotropic Meim A meim is consiere anisotropic when its electrical an/or magnetic properties epen pon the irections of fiel vectors. The relations between fiels can be written in the following form.

37 4 D E B H r r (.- In the eqation above, an are the free space permittivit an permeabilit, while r an are the relative permittivit an permeabilit tensors. For a general anisotropic r meim, the permittivit an permeabilit tensors are of the following form. xx x xz r x z zx z zz xx x xz r x z zx z zz (.- For reciprocal meia sch as niaxial an biaxial meia, r an r are smmetric matrices. In the principal coorinate sstem (where the coorinate axes are aligne with the principal axes of the permittivit tensor, onl the iagonal elements of r are non-zero. For non-reciprocal meia, sch as groelectric an gromagnetic meia, the matrices for r an r are antismmetric. Even in the principal coorinate sstem (where the coorinate axis is aligne along the irection of the biasing magnetic fiel, the off-iagonal elements of the permittivit an permeabilit matrices are non-zero an are complex conjgate to each other for the lossless meia. The complete set of DGFs for a general anisotropic meim with the electric an magnetic crrent sorces ( J an M locate at z z in the nbone region has been erive in [39]. In this issertation, the electric tpe E-DGF which correlates the electric crrent sorce an electric fiel as in Eq. (.-3 is onl tilize. 3 (, ( (.-3 E G r r J r r v

38 5 Ths, this section will present the eigen-ecomposition metho onl for the electric tpe E-DGF. Throgh the analsis, time variation of i t e is assme. Eq. (.-4 shows the electric tpe DGF for an nbone anisotropic meim erive in [39], which is expresse as a 3D Forier transform of ajw E (ajoint of the electric wave matrix W E. i i ajw G( r, r W e k or G( r, r e k ik ( rr 3 E ik ( rr WE E (.-4 kz k W E kr k k r, k kz kx, k xk x k zk z (.-5 k kx k x, k an kz are the x,, z components of the wave vector in the Cartesian coorinate sstem. The application of the eigen-ecomposition metho starts with recing the 3D integration of Eq. (.-4 to a D integration over k x an k b integrating the integran of G( r, r over k. z To aress how this is performe, the analsis of the eterminant of electric wave matrix W E is reqire. It is eas to erive that W E can be expane as a forth orer polnomial in terms of k z with the coefficients of the polnomial as the fnctions of k x an k. Generall, with given k x an k, there exist for ifferent vales of k z to let the eterminant of wave matrix (e.g., W E to be zero, which appear as for poles of the integran of DGF G( r, r. Denoting the for soltions for kz to the eqation of WE as k ( p I, II; q,, the q zp eterminant W E can be written as Booker qartic eqation below.

39 6 WE a4 ( kz kzi ( kz kzi ( kz kzii ( kz kzii (.-6 where a 4 is a constant epenent on the permittivit an permeabilit matrix. For a groelectric meim with z-irecte biasing magnetic fiel, a 4 is erive in [38]. The notations of q k zp ( p I, II; q, are explaine here. Accoring to the location of q k zp on the wave vector srface, the sbscript p inicates the tpe of the characteristic wave for a general anisotropic meim with p I an p II corresponing to the Tpe I an Tpe II waves, respectivel. The sperscript q inicates the irection of power flow. For propagating waves, the sperscripts an inicate the waves carring power awa from the interface in the pwar an the ownwar irections, respectivel. For evanescent waves (where there exists no power flow awa from the interface in either pwar or ownwar irections or where the z-component of time average Ponting vector is zero, the sperscripts of an inicate the wave amplites ecaing awa from the interface along the pwar an ownwar irections, respectivel. For a lossless meim (for which permittivit an permeabilit matrices are hermitian matrices, the for istinct soltions of k z can be categorize into the following three cases: two pairs of complex conjgate soltions, two real an two complex conjgate soltions or for real soltions. The propagating waves are characterize with k z being the real soltions an evanescent waves are characterize with k z being complex or pre imaginar soltions. It nees to be note here that the wave with a positive real nmber of q k zp sall inicates an pwar propagating wave. However, this is not alwas the case. De to the existence of the backwar wave, the irection of power flow an phase avance can be opposite to each other.

40 7 As iscsse earlier, soltions of k, k, k, k pose singlarit to the integran of zi zi zii zii G( r, r shown in Eq. (.-4. These singlarit poles can be extracte from the integration b integrating over k z sing resie theorem an the raiation bonar conition at infinit. With the assmption of the meim to be slightl loss, i.e., Im k Re k, Im k, Im k Re k, Im k, the following DGF in the D integral form is obtaine b zii zii performing the contor integration over k z. zii zi zi zi For z z ajw ( E k zi iki ( rr e a4( kzi kzi ( kzi kzii ( kzi kzii G( r, r x k k ( ajwe kzii ikii ( rr e a4( kzii kzi ( kzii kzi ( kzii kzii (.-7 For z z G( r, r ( kzi e( kzi e( kzi e a ( kzi kzi ( kzi kzii ( kzi kzii ( kzii e( kzii e ( kzii e a4( kzii kzi ( kzii kzi ( kzii kzii 4 iki ( rr ikii ( rr kxk (.-8 where the wave vectors are efine as k xk k zk k xk k zk ; k xk k zk k xk k zk I x x zi I x x zi II x x zi II x x zii Appling the eigen-ecomposition techniqe, the ajoint electric wave matrix q ajw ( k ( p I, II; q, in the integran of E zp G( r, r shown in Eq. (.-7 an Eq. (.-8 can be ecompose into a single a. Details will be presente below.

41 8 It is shown b H. Chen [66, p. 9] that if the eterminant of a matrix is zero an the ajoint matrix is a non-zero matrix, this matrix ma be expresse as a sm of two as an the ajoint matrix as a single a. Ths, for an electric wave matrix, if k z is taken as the vale sch that the eterminant of electric wave matrix W E is, then the ajoint electric wave matrix ajw E can alwas be ecompose into the form of a single a as follows. ajwe v (.-9 For now, is chosen as the eigenvale of ajw E an no constraint has been pt on the vectors of an v in the above aic form. It will be shown below that the matrix of ajw E has onl one non-zero eigenvale as long as WE. As for the choice of vectors of an v, it is epenent on the propert of the ajoint electric wave matrix ajw. The propert is affecte b the soltions for kz of the Booker qartic eqation for specific k x an k, since k z can be E taken as an vale e to the integration over the infinite range of k x an k. For a non-magnetic general anisotropic meim ( r I shown in Eq. (.-, the electric wave matrix is expresse as follows. with the permittivit matrix k k k k k k k k k xx z x x xz x z E r x x z x z z k zx kxkz k z kkz k zz kx k W kk k k k k k k k k k k (.- With the anisotropic meim being either reciprocal or non-reciprocal, the propert of the electric wave matrix is ifferent. Ths, to emonstrate the ifference in the formlation of DGF sing the eigen-ecomposition metho, these two cases are consiere here.

42 9 A. Reciprocal Meim For a lossless reciprocal meim, the permittivit tensor r is a smmetric matrix. It is straightforwar to erive from Eq. (.- that W E is alwas a smmetric matrix. Ths, the ajoint matrix of W ( ajw is also a smmetric matrix. If k z is real, then E smmetric matrix, an if kz is not real, then E ajw E is a real ajw E will be a complex smmetric matrix. kz is the soltion to the Booker qartic eqation for given k x an k. The following analsis will show that ajw ( k has onl one non-zero eigenvale when k k, k, k, k E z z zi zii zi zii eigenvales, the characteristic eqation for ajw ( k is first erive. E z. To obtain the f ( I ajw ( k E z tr( ajw ( k tr( ajaj( W ( k ajw ( k 3 E z E z E z (.- where tr stans for the trace of a matrix. Appling the following ientities, [65, p. 3] ajw E E W (.- aj( ajw W W E E E (.-3 when k k, k, k, k, W, the characteristic eqation of Eq. (.- reces to the z zi zii zi zii E following form. f ( tr( ajw ( k 3 E z (.-4 Ths, the eigenvales for ajw ( k are E z

43 tr( ajw ( k, E z 3 (.-5 It is known that the nitar iagonalization exists for a real smmetric matrix aic ecomposition takes the form as [67]. ajw E an ajw E (.-6 Since onl one non-zero eigenvale exists, then ajw ( k q q e q e q, p I, II; q, (.-7 E zp p p p q where is the non-zero eigenvale of the matrix ajwe ( kzp an e q p is the corresponing q p eigenvector. A comparison of Eq. (.-7 with Eq. (.-9 reveals that if the matrix ajw E is real an smmetric, then both of the vectors v, will be taken as the eigenvectors corresponing to the onl one non-zero eigenvale. then However, if k k, k, k, k is complex (incling the case of being pre imaginar, z zi zii zi zii ajw E is a complex smmetric matrix. It will be shown below a complex smmetric matrix can still be ecompose into a a compose of two ientical vectors. A non-efective matrix is a matrix which has a fll linearl inepenent set of eigenvectors. Since showing that iagonal form. ajw E is a non-efective matrix [67, p. 86], which can be prove b WE has three istinct eigenvales, ajw E can be factorize in the following

44 ajw ( k X X E z v 3,, X X v 3 v 3 (.-8 j is the j th colmn of X, which correspons to an eigenvector of iagonal entr of, which is the associate eigenvale. v j ajw E an j is the j th is the j th row of X. It is prove earlier that ajw E has onl one non-zero eigenvale when k z k q zp ; ths, q q ( q q ajw k e v E zp p p p (.-9 where v q p satisfies q q p p,, ;, v e p I II q (.- An inspection of Eq. (.-9 with Eq. (.-9 reveals that vector is actall the q eigenvector of ajw ( k which correspons to the non-zero eigenvale, an the vector v has E zp to satisf the conition v. Combining Eq. (.-7 an Eq. (.-9 shows the complete q eigen-ecomposition for the matrix ajw ( k accoring to the ifferent ranges of E zp q k zp. q If k is real, ajw is a real smmetric matrix. zp E q q ( q q ajw k e e, p I, II; q, E zp p p p q If k is complex, ajw is a complex smmetric matrix. zp E q q q q q q ajw ( k e v p, v p e, p I, II; q, E zp p p p (.-

45 q q q Especiall if the eigenvector is taken sch that epx ep epz, then it is seen that v q p e q p. Ths, for a reciprocal meim (both for lossless an loss cases, the ajoint electric wave matrix can alwas be ecompose into the form of a single a which is compose of the eigenvector corresponing to the onl one non-zero eigenvale. However, this is not the case if the meim is non-reciprocal. B. Non-reciprocal Meim If the meim is lossless an non-reciprocal, which inicates that the ielectric permittivit tensor is a hermitian matrix, then it is seen from Eq. (.- that, if an its ajoint matrix ajw ( k are hermitian matrices, an if E z q k zp is real, the matrix WE q k zp is complex, the matrix W E an its ajoint matrix ajw ( k are non-hermitian matrices. It is known that the nitar E z iagonalization can be applie to a hermitian matrix. For a non-hermitian matrix, similar iagonalization can be applie if the matrix is non-efective. With ajw ( k being a hermitian E z matrix or a non-hermitian an non-efective matrix ner the ifferent conitions of k q zp, the corresponing eigen-ecomposition for a non-reciprocal meim is shown as follows. q If k is real, ajw is a hermitian matrix. zp E * q q ( q q ajw k e e, p I, II; q, E zp p p p q If kzp is complex, ajwe is a non-hermitian an non-efective matrix. ( q q q q q ajwe kzp p ep v p, q v p ep, p I, II; q, (.-

46 3 If the meim is loss an non-reciprocal, which inicates that the matrix hermitian matrix even if alwas takes the form of W is not a E q k zp is real, an then the eigenvale ecomposition for a loss meim ( q q q q, q ajw q E kzp pep vp vp ep, p I, II; q, (.-3 Discssion above shows that Eq. (.-4 or the statement in [68] b A. Erogl that for nbone groelectric meim the ajoint electric wave matrix in the aic Green s fnction q of D integration form ( ajw ( k ( p I, II; q, can alwas be ecompose into a E zp single a form as follows b assming ajw is a hermitian matrix is not correct. * E ajw ( k q q e q e q, p I, II; q, (.-4 E zp p p p q q where is the non-zero egienvale of ajw ( k. p E zp It has been shown that that ajw E is alwas a linear matrix an can be ecompose into a single a form. A carefl treatment is reqire to obtain the aic form of DGFs when the eigen-ecomposition metho is se if the meim is grotropic or a loss anisotropic meim. Detaile iscssions abot sing the eigen-ecomposition metho to obtain the DGFs of niaxial an grotropic meia will be presente in the following sections.. DGFs of an Unbone Uniaxial Meim DGFs of niaxial an isotropic meia are consiere in this section. To simplif the analsis, we consier the electric tpe Green s fnction for a niaxial meim with the optic axis

47 4 aligne along the z axis, which is characterize b the relative permittivit an permeabilit tensors of the following form. r I, r z (.- As shown in previos section, to obtain the electric tpe DGF, it reqires solving the inverse of electric wave matrix, which can be obtaine sing W ajw E E. Sbstitting Eq. W E (.- into Eq. (.-5, the eterminant of W E can be expresse in terms of kx, k, k z as follows. W ( ( E kk k k z kz kzi kz k (.- zii k k k k k k (.-3 zi zii z where k k k We note that x. k k, k k zi zi zi zi k k, k k zii zii zii zii (.-4 The ajoint electric wave matrix is given as A A A ajwe A A A A A A (.-5 The elements of the matrix in Eq. (.-5 are liste as follows.

48 5 A ( k k k k k ( k k k 4 z x z x z z A ( k k k k k z z x A ( k k k k k 3 z x z A A ( k k k k k z z x A ( k k k k k ( k k k 4 z z z z A ( k k k k k 3 z z A A ( k k k k k 3 3 z x z A A ( k k k k k 3 3 z z A ( k k k k ( k k k 4 33 z z z (.-6 Appling Eq. (.- to Eq. (.-4 an then Eq. (.-4 can be written as follows. i ik ( rr G( r, r e k k k ( ( ajwe 3 kz kz kzi kz k zii x z (.-7 It is seen from Eq. (.- that for each specific k, k, for poles of the integran exist when k k, k, which are given in Eq. (.-3 an Eq. (.-4. If the optic axis of the z zi zii meim is not aligne along the z-axis, all the nine elements of the permittivit matrix are nonzero. In this case, for istinct poles will exist, an Eq. (.-7 can be written in a more general form as follows. x i ik ( rr G( r, r e k k k ( ( ( ( ajwe 3 kz kz kzi kz kzi kz kzii kz k zii x z (.-8 Assming the meim to be slightl loss, (i.e., Im k Re k,im k, zi zi zi Im k Re,Im zii kzii kzii, when performing the contor integration over k z sing the resie theorem an the raiation bonar conition at infinit, the DGFs for the regions above an below the sorce point can be written as follows.

49 6 For z z For ajw ( E k zi iki ( rr e kz ( kzi kzi ( kzi kzii ( kzi kzii G( r, r x k k ( ajwe kzii ikii ( rr e kz ( kzii kzi ( kzii kzi ( kzii kzii z z ajw ( E k zi iki ( rr e kz ( kzi kzi ( kzi kzii ( kzi kzii G( r, r x k k ( ajwe kzii ikii ( rr e kz ( kzii kzi ( kzii kzi ( kzii kzii (.-9 For a niaxial meim with the optic axis aligne along the z-irection, appling Eq. (.-3 an Eq. (.-4 into Eq. (.-9, then it can be simplifie as follows. For z z ajw ( ( ( E kzi iki r r ajwe k zii ikii ( rr G( r, r e e k xk 8 kzkzi ( kzi kzii kzkzii ( kzii kzi For z z (.- ajw ( ( ( E kzi iki r r ajwe k zii ikii ( rr G( r, r e e k xk 8 kzkzi ( kzi kzii kzkzii ( kzii kzi The DGFs given b Eq. (.- can also be represente in a aic form b fining the eigenvales an eigenvectors of the ajoint matrix ajw. For the niaxial meim with the optic axis aligne along the z-axis, the eigenvales are obtaine as follows. E

50 7 kk( (.- I I z k k ( k ( (.- 4 II II z z We also note that k (( k ( k k (( k ( k k k ( (.-3 o z zi zii o z zi zii z It is seen from Eq. (.- an Eq. (.- that for both Tpe I an Tpe II waves, the eigenvales of pwar an ownwar waves are the same. It will be shown below that eigenvectors are the same for Tpe I pwar an ownwar waves. However, for Tpe II waves, eigenvectors are ifferent for pwar an ownwar waves. When sbstitting Eq. (.- an Eq. (.- into Eq. (.-, the DGFs are obtaine as follows. For z z For G( r, r 8 z z k k e e IIe z ( II ik I r r ( e Ie Ie z k xk kzi k k zii ikii ( rr (.-4 G( r, r 8 k k e e IIe z ( II i I r r ( e I e Ie z k xk kzi k k zii iii ( rr The eigenvectors can then be obtaine from the characteristic eqation with the corresponing eigenvales sbstitte into the eqation. The etaile erivation is given below. First, it is shown that the ajoint matrices for Tpe I pwar an ownwar waves are the same an take the following form.

51 8 A A ajwe ( kzi ajwe ( kzi A A (.-5 where A k ( k, A A k ( k k, A k ( k z z x z x The ajoint matrix can be simplifie b extracting k ( as follows. z k kxk ajwe ( kzi ajwe ( kzi k ( z kxk kx (.-6 Also, it is known from Eq. (.- that the eigenvales are the same for both the pwar an ownwar ajoint wave matrices. The eigenvectors e I e I can be solve from the following eqation an it is easil seen that the eigenvectors for pwar an ownwar propagating waves are the same. Step : ajw ( k e I e E zi zi I Step : k kxk e Ix e Ix k ( z kxk kx ei kk ( z ei e Iz e Iz Step 3: k k, x e, Ix ei eiz k k (.-7 When enoting k xk k zk, k xk k zk, it is easil verifie that the I x zi I x zi vectors of Tpe I waves agree with the formlations of vectors shown in [35]. e e e ( z k ( z k z k I z k I I, I I I I (.-8

52 9 It s eas to observe from Eq. (.-8 that the Tpe I wave is polarize along the irection that is perpeniclar to the optic axis an the wave vector, which is enote as an orinar wave. Now, let s consier the Tpe II wave. Different from the Tpe I wave, the ajoint matrices for the pwar an ownwar Tpe II waves exhibit two ifferent forms, thogh the eigenvales for both waves are the same as seen from Eq. (.-. The ajoint matrices are first erive here for both of the Tpe II pwar an ownwar waves. ( ( ( A ( kzii A ( kzii A3 ( k zii ( ( ( ( ajwe ( kzii A ( kzii A ( kzii A3 ( kzii ( ( ( A3 ( kzii A3 ( kzii A33 ( kzii where A ( k A ( k ( k k z zii zii zii x z A ( k A ( k ( k k z zii zii zii z A k A k k k 33( zii 33( zii ( ( z z A k A k A k A k k z ( zii ( zii ( zii ( zii ( ( zii z A ( k A ( k A ( k A ( k ( k k k 3 zii 3 zii zii zii x zii z A ( k A ( k A ( k A ( k ( k k k 3 zii 3 zii 3 zii 3 zii zii z kk x (.-9 Then the eigenvectors for the pwar an ownwar Tpe II waves erive from the ajoint matrix are shown below.

53 3 kxk zii kxk zii k ( kzii k / z k ( kzii k / z e IIx e IIx kk zii e II eii, e kk zii II e II k ( kzii k / z e IIz e k ( kzii k / z IIz k k z ( kzii k / z z ( kzii k / z (.- It is easil observe from Eq. (.- that the eigenvectors for the pwar an the ownwar propagating waves satisf the following relations when the optic axis of the niaxial meim is along the z-axis. e e, e e, e e (.- IIx IIx II II IIz IIz Eq. (.- agrees with the reslt obtaine in [35]. Also, when enoting wave vectors k xk k zk an k xk k zk, mltipling the wave vectors II x zii II x zii k II an k II for the pwar an ownwar waves with the corresponing electric fiel vectors of Eq. (.- shows that, e II k II e II k II e II k II e II k II (.- As shown in Eq. (.- that the isplacement vector D instea of the electric fiel is perpeniclar to the wave vector for Tpe II wave, which is enote as an extraorinar wave. The polarizations of both orinar (Tpe I an extraorinar (Tpe II waves are shown in Fig. -. For Tpe I wave (orinar wave, the E fiel is polarize perpeniclar to the plane forme b the optic axis an the wave vector as shown in Fig. -(a. For Tpe II wave (extraorinar

54 3 wave, the E fiel is polarize in the plane forme b the optic axis an the wave vector as shown in Fig. -(b. Orinar wave Optic axis ks ( Extraorinar wave s k Optic axis E H B D E (a H B D (b Fig. -: Two ifferent polarizations of plane waves in niaxial meim: (a Tpe I (orinar wave an (b Tpe II (extraorinar wave. Limiting Case of Isotropic Meim: If, then the niaxial meim reces to an isotropic meim. From the z ispersion relation of the isotropic meim, it is easil seen that the z component of the wave vector will satisf the following eqation. k k k, k k k zi zii z zi zii z (.-3 When k xk k zk, k xk k zk (.-4 x z x z the ajoint matrix for an isotropic meim becomes zero: ajw E (.-5 Ths, the aic Green s fnction can no longer be obtaine from the ajoint matrix irectl. However, it can be obtaine b letting into the aic Green s fnction of Eq. (.-4 for the niaxial meim. The can be rewritten for the isotropic case as follows. z

55 3 For z z e I e I e II e II ik ( rr (.-6 G( r, r k xk e 8 kz k z For z z e I e I e II e (.-7 II ik ( rr G( r, r k xk e 8 kz k z In the eqations above, the eigenvectors e I, e I, e II, e II are efine below. where e I e I z k ( z k, e II xe e ze e II xe e ze, IIx II IIz IIx II IIz kk kk e e e e e e x z z IIx IIx, II II, IIz IIz kk kk k k (.-8 Defining z k kx xk h h( kz k k k x h ( kz k h ( kz k v v ( kz, v v( kz k k (.-9 it is easil verifie that e e h h, v e, v e I I II II (.-3 As shown in the eqation above, the E fiel of Tpe I wave is perpeniclar to the plane of incience inicating horizontall polarize wave in an isotropic meim, an the E fiel of

56 33 Tpe II wave is parallel to the plane of incience an perpeniclar to the wave vector, inicating verticall polarize wave in an isotropic meim. It is note here that in this case D an E are in the same irection. The aic Green s fnction for an isotropic meim can be obtaine as follows with h an v efine in Eq. (.-9. This reslt is consistent with the reslt shown in [69], which is repeate in Eq. (.-3. For For z z z z (, G r r k xk h he v v e 8 k z ik ( r r ik ( rr (, G r r k xk h he v v e 8 k z ik ( r r ik ( rr (.-3 The erivation in this section analticall shows that the eigen-ecomposition of ajw E into a single a hols for the niaxial meim. The as in the DGFs are compose of two eigenvectors corresponing to the orinar an extraorinar waves, respectivel. As state in Section., ajw E can alwas be ecompose into a single a compose of the eigenvector corresponing to the non-zero eigenvale of ajw E since the ajoint electric wave matrix is alwas smmetric for an nbone niaxial meim or biaxial meim with an arbitraril rotate optic axis. Ths, the DGFs are expresse in Eq. (.-3. In Eq. (.-3, ( q an q ( stan for the eigenvale an eigenvectors of the ajoint matrix ( q ajw k, k zp ek zp where q,, p I, II. a 4 stans for the coefficient of the forth orer polnomial of kz in E zp the expansion of W. It is expecte that for a biaxial meim ( an ( correspon to E ek zi ek zi

57 34 the pwar an ownwar a waves, an ( an ( correspon to the pwar an ownwar b waves in [36-37]. ek zii ek zii For z z For z z G( r, r G( r, r ( kzi e( kzi e( kzi e a ( kzi kzi ( kzi kzii ( kzi kzii ( kzii e( kzii e ( kzii e a4( kzii kzi ( kzii kzi ( kzii kzii 4 4 iki ( rr ( kzi e( kzi e( kzi e a ( kzi kzi ( kzi kzii ( kzi kzii ( kzii e( kzii e ( kzii e a4( kzii kzi ( kzii kzi ( kzii kzii ikii ( rr iki ( rr ikii ( rr kxk kxk (.-3.3 DGFs of an Unbone Grotropic Meim.3. DGFs of a Groelectric Meim If the meim is characterize b the relative permittivit tensor in the following form, it is calle a groelectric meim. where b ( I b b i ( b I b b g // shows the irection of the applie constant (c magnetic fiel (.3- When the biasing magnetic fiel irection is along z irection, i.e., b tensor form can be written in the following matrix form. z, the above

58 35 ig p i g // (.3- For example, if the meim is col plasma which is a groelectric meim, then the relative permittivit tensor parameters are given as,, eb b m p b p p g // b ( b N e, p m (.3-3 b is calle the grofreqenc or cclotron freqenc an p is calle the plasma freqenc. N shows the nmber of free electrons per nit volme, an m represents the mass of each electron with charge e (a negative nmber. z B Biasing magnetic fiel b x B Fig. -: Arbitrar biasing magnetic fiel in the xz coorinate However, if the biasing magnetic fiel is rotate b B with respect to the z-axis an with respect to the x-axis in the Cartesian coorinate sstem as shown in Fig. -, the permittivit tensor will have nine non-zero elements. The matrix elements ( m, n x mn,, z are erive accoring to the transformation matrix as follows. B

59 36 xx x xz sinb cosb c T pt x z, T cos B cosb cos B sinb sin B zx z zz sinb cosb sinb sinb cos B (.3-4 The DGFs for the two ifferent cases one with the biasing magnetic fiel along the z- irection an one with an arbitrar irection of B an B are erive in this section. A. Z-oriente Biasing Magnetic Fiel First, we consier the case when the biasing magnetic fiel is along the z-axis. Then, the relative permittivit takes the form of Eq. (.3-3. The eterminant of the electric wave matrix is written as follows. k k k k k i k k k o z x g o x z E x g o o x z z kxkz kkz ko // kx k W k k i k k k k k k (.3-5 Expansion of W E leas to the forth orer eqation in kz as follows. W ak bk c 4 E z z a k o // b k ( k k ( k 4 o x // o // c k ( k ( k k ( k ( k k 6 4 o // g o x g // o x (.3-6 When the biasing magnetic fiel is along the z-axis, two istinct sets of k z exist to make the eterminant of the electric wave matrix to be zero as shown in Eq. (.3-7. k zi a a b b 4 ac 4, k b b ac zii (.3-7 Then the eterminant of the electric wave matrix can be written as follows.

60 37 W a( k k ( k k k ( k k ( k k (.3-8 E z zi z zii // z zi z zii Following the same procere as escribe in Section., the DGF for the nbone groelectric meim can be expresse as the a form which is obtaine from the eigenvale an eigenvectors of the ajoint matrix aj( W E as follows. A A A , 3 ajw E kzi A A A3 A A A (.3-9 A k k k ( k k ( k k k 4 I x x // x z // A k k k k i ( k k k k ik 4 I x g x // x g // A3 ki kxkz k kxkz igk k z A k k k k i ( k k k k ik 4 I x g x // x g // A k k k I 4 ( kx k // ( k kz k // A3 ki k kz k k kz igkxk z A3 ki kxkz k kxkz ixk k z A3 ki k kz k k kz igkxk z 4 A33 ki kz k ( kx k kz k g In Eq. (.3-9, k k k an it represents the wave nmber of the Tpe I wave. I zi Sbstitting k I with k II will give the ajoint wave matrix for Tpe II wave. It nees to be note here that A A alwas hols as long as k x an k are real. However, relation of * * * A3 A3, A3 A3 oes not necessar hol an it epens on the choice of k z. Accoring to Section., the ajoint wave matrix can be ecompose into the following a form if the matrix is a hermitian matrix.

61 38 * * ajw E kzi I e ni ( kzi e ni ( kzi, ajw E kzii II e nii ( kzii e nii ( kzii k k k (3 k k ( k ( 4 4 I I I // zi // g // 4 4 II kii kiik (3 // kziik ( // k ( g // (.3- Derivation of the eigenvector is shown below. A A A3 e e A A A e e 3 A3 A 3 A 33 e 3 e 3 ( A e A e A e ( a 3 3 A e ( A e A e ( b 3 3 A e A e ( A e ( c (.3- Eliminating e 3 from (a an (b in Eq. (.3-, we obtain p e p e, p ( A A A A, p A A ( A A, ( Eliminating e from (b an (c in Eq. (.3-, we obtain q e q e, q ( A A A A, q A A ( A A ( Eliminating e from (a an (c in Eq. (.3-, we obtain s e s e, s ( A A A A, s A A ( A A ( For ifferent cases shol be consiere here in calclating e, e, e 3. Case I: p & p e s e, e3 s (.3-5 Case II: p & p e q e, e3 q (.3-6

62 39 Case III: Case IV: p & p & q & q e p e3, e p p & p & q & q p s e, e, e3, p s (.3-7 (.3-8 Sbstitting p, p, s, s in Eqs. (.3- an (.3-4 gives the normalize e ni ( kzi for case IV below. e ( k e ni ( k, e ( k I zi 3 3 I 3 zi I zi norm( ei ( kzi A3A A3I A3A A A A A A A A A A A A A A3 A3A A3I A3A A3 3 3 I 3 I (.3-9 norm ( ei ( k conj ei ( k ei ( k is the sqare root of the inner proct of the vector ei ( k an itself. zi zi zi zi Then the ajoint electric wave matrix can be expresse in terms of single a below. ajw e npv p, p I, II (.3- E p If k zp is real, then ajwe is a hermitian matrix. e * np v p (.3- It is note here that Eq. (.3-9 an Eq. (.3- agree with the reslts obtaine in Erogl an Lee [38]. However, the reslts obtaine in [38] state that the DGF for a groelectric meim

63 4 alwas take the form of Eq. (.3-, which is not tre. As iscsse in etail in Section., if k zp is real, then the following form. ajwe is no longer a hermitian matrix, an the vector v p in Eq. (.3- takes A A v [ v v v ], v, v, v 3 p p p p3 p p p3 pep pep pep A (.3- Sbstitting Eq. (.3-8 into Eq. (.-4 gives i ik ( rr 3 G( r, r e k (.3-3 ( ( ajwe 3 k // kz kzi kz k zii Appling the resie theorem an the raiation bonar conition at infinit, the 3D integration can be rece to D integration b integrating over k z. Ths in each region, the DGF is obtaine as follows. For z z * I e ni ( kzi e ni ( k zi iki ( rr e k zi G( r, r k xk 8 k * // ( kzi kzii II e nii ( kzii e nii ( kzii ikii ( rr e k zii For z z G( r, r 8 k k x * I e ni ( kzi e ni ( k zi i e kzi k * // ( kzi kzii II e nii ( kzii e nii ( kzii e kzii I ( rr iii ( rr (.3-4 In the above formlas, the wave propagation vectors are efine as follows.

64 4 k xk k zk xk k zk k xk k zk xk k zk I x x zi I x x zi II x x zii II x x zii (.3-5 It is seen from above formla that if the biasing magnetic fiel is along the z-axis, the Tpe I pwar an ownwar waves will have z component of wave vectors of the same magnite an opposite sign. If the biasing magnetic fiel is oriente along an arbitrar irection, then this will be no longer the case. B. Arbitraril Directe Biasing Magnetic Fiel Assming the biasing magnetic fiel is along an arbitrar irection of B an B, the permittivit takes the form of Eq. (.3-4. W E can then be expane in terms of kz as follows. W a k a k a k a k a 4 3, E 4 z 3 z z z a k ( sin cos 4 B // a k ( ( k cos k sin sin cos 3 // a k ( [ k cos ( k sin k cos sin ] B x B B B //[( kx cos B k sin B sin B ] B x B B b B [ k ( k ]sin ( k k ]cos g // o B // o B a k [( k ( k ]( k cos k sin sin cos // g // o x B B B B a k ([( k ( k ]( k cos k sin sin // g // o x B B B ( k k [ k ( k ] // o x o (.3-6 For istinct soltions of k, k, k, k exist for a given set of kx an k. The wave zi zii zi zii vectors with k, k correspon to pwar Tpe I an Tpe II waves, while k, k correspon zi zii to ownwar Tpe I an Tpe II waves. Then the eterminant of the electric wave matrix is written as zi zii

65 4 W k ( sin cos ( k k ( k k ( k k ( k k (.3-7 E o B // B z zi z zii z zi z zii With kz obtaine, the electric wave matrix is given below. k k k k k k k k k o xx z x o x x z o xz * E x o x o x z z o z W k k k k k k k k k k k k k k k k k k * * x z o xz z o z o zz x (.3-8 The ajoint matrix of the electric wave matrix takes the following form. A A A ajwe A A A A A A (.3-9 where 4 * * A ( k kz kx k zz ( kx kz ( kx k k ( zz z z kkzk ( z z * * 4 A ( k kz kxk k x ( kx k zzkxk k ( kxkz k z kz xz k xz k z x zz A3 ( k kz kxkz k xk kz kxk z k z ( kxk xk k xz ( kx kz k * * * 4 * A ( k kz kxk k x ( kx k zzkxk k ( kxkz z kkz xz k xz z k x zz 4 * * A ( k kz k k zz ( k kz xx( kx k k ( xx zz xz xz kxkzk ( xz xz * 4 * * A3 ( k kz kkz k xxkkz xkxk z k ( xzkxk z ( k kz k ( xz x xx z * * * * A3 ( k kz kxkz k xk kz kxk z k z ( kxk xk k xz ( kx kz k * * 4 * A3 ( k kz kkz k xxkkz xkxk z k ( xzkxk z ( k kz k ( xz x xx z 4 * * A33 ( k kz kz k ( xx kz xxkx k k ( xx x x kxkk ( x x Appling Eq. (.3-7 to the DGF of Eq. (.-4 gives the 3D integration form of the DGF. i aj( W E ik ( rr G( r, r e k k k ( c( k k ( k k ( k k ( k k ck 3 ( sin cos o B // B z zi z zii z zi z zii x z (.3-3 Integrating Eq. (.3-3 over k z an appling resie theorem gives the DGF.

66 43 For z z ii ( rr aj( WE ( k e zi ( ( k k k k zi zi zi zii x i ( // II rr o B B zi zii E zii G( r, r k k 4 k ( sin cos ( k k aj( W ( k e ( k k ( k k zii zi zii zii (.3-3 For z z iki ( rr aj( WE ( k e zi ( ( k k k k zi zi zi zii x ik ( // II rr o B B zi zii E zii G( r, r k k 4 k ( sin cos ( k k aj( W ( k e ( k k ( k k zii zi zii zii k xk k zk xk k zk ; k xk k zk xk k zk I x x zi I x x zi II x x zi II x x zii (.3-3 Again the ajoint matrix can be represente in the following aic form if k q zp is real. * * ajw E k ni ( ni (, E ni ( ni ( zi I e k e k ajw k zi zi zii II e kzii e k zii * * ajw E k ni ( ni (, E ni ( ni ( zi I e kzi e k zi ajw k zii II e kzii e k zii q q q q 4 p ( k ( kzp ( k ( kzp k // k3 p k // k g // k k sin cos k sin sin k cos q q 3 p x B B B B zp B (3 ( ( ( (.3-33 The eigenvectors are shown below. q q e ( kzp A3A A3 p A3A q q e n( kzp, e( kzp q q norm( e ( kzp A3A A3p A3A q q A A 3A A3p A3A p A q A3 A 3A A3p A3A A3 p I, II, q,, (.3-34 It is note here that the above eigenvectors are vali onl for the case IV iscsse previosl.

67 44.3. DGFs of a Gromagnetic Meim A gromagnetic meim is the meim characterize b the relative permittivit of r an the relative permeabilit tensor r in the following form. r ( I b b ig b I // b b (.3-35 When the biasing magnetic fiel irection is along z irection, i.e., b z the above tensor form can be written in the following matrix form of p, which inicates the relative permeabilit in the principal coorinate sstem. ig r p ig //,,, M, H m m g // m (.3-36 m is efine as the Larmor precession freqenc of the electron in the applie magnetic fiel H an is efine as the resonant freqenc. M is the satrate magnetization vector an is in the same irection as the applie magnetic fiel H. is the gromagnetic ratio an its 5 ra m vale is given as.*. If the biasing magnetic fiel is along an s Atrns arbitrar irection, the permeabilit matrix will take the following form. xx x xz sinb cosb r T pt x z, T cosb cosb cosb sinb sin B zx z zz sinb cosb sinb sinb cos B (.3-37 With the permeabilit matrix of Eq. (.3-37, the electric wave matrix is erive below.

68 45 E W k k k I r r a a a3 kz k xx x xz kz k a a a 3 kz kx x z kz kx k r I a 3 a3 a 33 k kx zx z zz k kx (.3-38 The eterminant of the electric wave matrix a 4 r B // B W E can then be expane in terms of k z. 4 3 WE a4kz a3kz akz ak z a (.3-39 k ( sin cos a k ( ( k cos k sin sin cos r 3 r // x B B B B a k { [ k cos ( k sin k cos sin ] [( k cos k sin sin ] r B x B B B // x B B B [ k ( k ]sin ( k k ]cos } x // o r B // o r a k [( k ( k ]( k cos k sin sin cos r // g // o r x B B B B a k {[( k ( k ]( k cos k sin sin r // z // o r x B B B ( k k [ k ( k ]} // o r g o B It is easil seen that Eq. (.3-39 has for istinct soltions of k, k, k, k. zi zii zi zii WE ko r ( sin B // cos B( k k ( k k ( k k ( k k zi zii zi zii (.3-4 r Comparison of the coefficients of the ispersion eqation [Eq. (.3-39] for gromagnetic meim with the coefficients of the ispersion eqation for a groelectric meim [Eq. (.3-6] shows that except the factor r (eterminant of the relative permeabilit matrix being ae to each term, alit relation exists between Eq. (.3-39 an Eq. (.3-6. Sbstitting each term of r in Eq. (.3-6 with the corresponing matrix elements of r gives Eq. (.3-39.

69 46 Then following the same procere se to obtain the DGF for the groelectric meim, the electric tpe aic Green s fnction for a gromagnetic meim is obtaine as follows. For z z G( r, r iki ( rr aj( WE ( k e zi r ( k k ( k k zi zi zi zii kxk k ik ( // II r r o r B B k zi k zii aj WE k e zii 4 ( sin cos ( ( ( ( k k ( k k zii zi zii zii (.3-4 For z z G( r, r ii ( rr aj( WE ( k e zi r ( k k ( k k zi zi zi zii kxk k i ( // II r r o r B B k zi k zii aj WE k e zii 4 ( sin cos ( ( ( ( k k ( k k zii zi zii zii (.3-4 Again the ajoint matrix can be represente in the following aic form if k q zp is real. * * ajw E k ni ( ni (, E ni ( ni ( zi I e k e k ajw k zi zi zii II e kzi e k zii * * ajw E k ni ( ni (, E ni ( ni ( zi I e kzi e k zi ajw k zii II e kzii e k zii q q ( k ( kzp ( k ( kzp q k (3 // p q 4 ( k3 p k ( // k ( g // (.3-43 where q q k3 p kx sinb cosb k sinb sinb kzp cos B The eigenvectors can be obtaine sing Eq. (.3-34 with an appropriate sbstittion. The DGF of Eq. (.3-43 agrees with what s obtaine in Park [4]. However, it is note here that the DGF of the nbone gromagnetic meim obtaine in [4] is onl vali for the case when

70 47 the ajoint wave matrix is a hermitian matrix with k q zp being a real nmber. Combining all the special cases consiere in this chapter, the DGFs of an nbone general anisotropic meim can be smmarize in the following form. For z z For z z ( k ( ( zi e kzi v k zi iki ( rr e i a4( kzi kzi ( kzi kzii ( kzi kzii G ( r, r x k k ( k ( ( zii e kzii v kzii ikii ( rr e a4( kzii kzi ( kzii kzi ( kzii kzii ( k ( ( zi e kzi v k zi iki ( rr e i a4( kzi kzi ( kzi kzii ( kzi kzii G ( r, r x k k ( k ( ( zii e kzii v kzii ikii ( rr e a4( kzii kzi ( kzii kzi ( kzii kzii (.3-44 (.3-45

71 48 3 DYADIC GREEN S FUNCTIONS FOR HALF-SPACE AND TWO-LAYER GEOMETRIES In this chapter, the electric tpe eigenvector aic Green s fnctions (E-DGFs for the half-space an two-laer problems with a sorce locate either in the isotropic or anisotropic region are obtaine. In Section 3., the DGFs for the regions below the sorce point when it is locate in the isotropic region are erive. In Section 3., the moifie smmetrical propert applicable to the DGFs with sorce an fiel points in the same region (Region is erive to obtain the DGFs above the sorce point when the anisotropic region is non-reciprocal. In Section 3.3, the moifie smmetrical propert for the DGFs with sorce an fiel points in two ifferent regions is erive for a laere geometr with a non-reciprocal meim. The sage an limitation of the moifie smmetrical propert is iscsse. To overcome the limitation of the moifie smmetrical propert in Section 3.3, the irect constrction metho is propose to obtain the complete DGFs with a sorce locate in the anisotropic region in Section 3.4. A comparison of the DGF for the isotropic region with a sorce locate in the anisotropic region erive sing two ifferent approaches is presente an interesting relationship is iscovere. 3. Daic Green s Fnctions with Sorce insie the Isotropic Region In this section, the DGFs of half-space an two-laer problems are given for each region below the sorce point when a sorce is locate in the isotropic region. The half-space an twolaer geometries are shown in Fig. 3-(a an (b, respectivel. For both half-space an twolaer geometries, Region is an isotropic region with relative permittivit an permeabilit

72 49 enote as r an r, while Region is an electricall anisotropic region with relative permittivit tensor r an relative permeabilit r. For a two-laer geometr, the thickness of the anisotropic region is enote as, an Region is an isotropic region with relative permittivit an permeabilit of r an r. For both problems, the sorce is locate at in Region. z z z z z z z z z= z= z= Fig. 3-: Geometr of (a half-space an (b two-laer problem. 3.. DGF for the Region of z z For the half-space problem shown in Fig. 3-(a, the DGFs of interest are (, G ( r, r an (, G ( r, r. The first an secon inices of sperscript of the DGFs are se to inicate the region of the fiel an sorce points, respectivel, while r an r inicate the fiel an the sorce points. G (, ( r, r refers to the DGF of Region with sorce in Region, an (, G ( r, r correspons to the DGF of Region with sorce in Region. When consiering the two-laer geometr shown in Fig. 3-(b, the DGF for Region is enote as G (, ( r, r in aition to the DGFs of (, G ( r, r an (, G ( r, r. With DGFs for each region of a two-laer

73 5 problem assigne, the Maxwell s eqations for the DGFs of the two-laer geometr are written below. (, I k r r G ( r, r I ( r r (, I kr r G ( r, r (, I k r r G ( r, r (3.- The continit conitions of the tangential electric an magnetic fiels reqire the twolaer DGFs to satisf the following bonar conitions. At z (, (, z G ( r, r z G ( r, r (, (, z G ( r, r z G ( r, r r r (3.- At z (, (, z G ( r, r z G ( r, r (, (, z G ( r, r z G ( r, r r r (3.-3 It is note here that for a half-space problem, the DGFs have to satisf Eq. (3.- onl. B appling the same matrix metho se in [35] to the DGFs of the nbone anisotropic region obtaine in Chapter, the DGFs for each region of a half-space problem can be expresse in the aic form below.

74 5 For zz i r h ( kz e ik r R hh h ( kz e h ( k z ik r (, i Rhv v ( kz e G ( r, r k xk 8 k i r z v ( kz e ik r Rvh h ( kz e v ( kz ik r Rvv v ( kz e i r e (3.-4 For z i I r X e I ( kzi e hei h( k z i II r (, II ( i X e k he zii e II G ( r, r k xk e 8 k z i I r X e I ( kzi e vei v ( kz i II r X e II ( kzii e ve II i r (3.-5 It is note here to be consistent with the notations of DGFs erive in [37]; the constant coefficient i is extracte ot from Green s fnction of Eq. (3.-4 an Eq. (3.-5. Throgh all the following iscssions, i is not incle in the DGFs. In Eq. (3.-4 an Eq. (3.-5, k an enote the wave vectors of pwar an ownwar propagating (or ecaing waves along the z-irection in Region, respectivel. k xk k zk, xk k zk x z x z (3.-6 The components of the wave vector in Eq. (3.-6 have to satisf the ispersion relation of the Region fille with the isotropic meim as follows. k k k k, k x z r r (3.-7

75 5 h ( kz an v ( kz correspon to the two ifferent polarize electric fiels in Region with the propagating irection being either pwar or ownwar inicate b the sign of k z been iscsse in Chapter that h ( kz enotes the wave polarize perpeniclar to the. It has incience plane, while v ( kz enotes the wave polarize parallel to the incience plane an perpeniclar to the wave vector. For convenience, the for ifferent polarize electric fiels in the isotropic regions for a two-laer geometr are simplifie as h n, h n v n, v n with the sbscript n = corresponing to Region an n = corresponing to Region. The expressions of the electric fiels are given below. z k k n x xk, (3.-8 hn hn k k x k k k k x h ( x nz nz ( x n k xk k k k z k k nz k n v n k k k k n nr nr x h ( x nz nz ( x n k xk k k k z k k nz n v n k k k k n nr nr x (3.-9 (3.- k an enote the wave vectors of Region for pwar an ownwar propagating m m characteristic waves of Tpe m ( m I, II, respectivel. k xk xk zk, k xk k zk I x zi II x zii xk k zk, xk k zk I x zi II x zii (3.- As shown in Chapter, the for ifferent vales k ( m I, II an n=,, inicating the n zm z component of wave vector for Region, can be obtaine b setting the eterminant of electric wave matrix to be zero with given tangential components of wave vector.

76 53 The normalize electric fiel vectors for the pwar an ownwar characteristic waves of Tpe m are enote as e m( k an ( e m k ( m I, II zm zm, respectivel. As escribe in Chapter, these vectors can be obtaine sing the eigen-ecomposition metho. The terms of R an X inicate the half-space reflection an transmission coefficients when the wave is incient from Region to Region. The first an secon inices of the sbscript inicate the polarizations of the incient wave an transmitte wave, respectivel. For a two-laer problem, the aic Green s fnctions in each region can be expresse in aic forms as follows. For zz For z i r h ( kz e ik r R hh h ( kz e h ( k z ik r (, i Rhvv ( kz e G ( r, r k xk e 8 k i r z v ( kz e ik r Rvh h ( kz e v ( kz ik r Rvv v ( kz e I i r AheI e I ( kzi e ik I r BheI e I ( kzi e II AheII e II ( kzii e (, B e ( k e G ( r, r k k k A e k e I BveI e I ( kzi e AveII e i II ( kzii e BveII e II ( kzii e i r ik II r i heii II zii x 8 i I r z vei I ( zi ik r ik r i r h ( kz e v ( kz II r II i r (3.- (3.-3

77 54 For z i r X hh h ( kz e h ( k z (, i r i X v ( k e G ( r, r k xk e 8 k i r z X vh h ( kz e v ( kz i r X vv v ( kz e hv z i r (3.-4 In Eq. (3.-, R stans for the two-laer reflection coefficients of the reflecte wave with polarization from the incient polarize wave. In Eq. (3.-3, A an B stan for the coefficients of the ownwar an pwar waves with polarization converte from incient wave with polarization insie the anisotropic meim. In Eq. (3.-4, X stans for the two-laer transmission coefficients of the transmitte wave with polarization from the incient polarize wave. a b B A coefficients can be significantl simplifie b ecomposing the two-laer problem into two half- z= z= A B c z= z= Fig. 3-: Amplite vectors of waves in the two-laere geometr. The total 6 reflection an transmission coefficients for the two-laer problem can be calclate appling the bonar conitions at z= an z=. The procere of obtaining the

78 55 space problems as shown in [35]. This approach is briefl reviewe here since the similar approache will be tilize to obtain complete DGFs of all the regions when the sorce is locate in the anisotropic region in Section As shown in Fig. 3-, a stans for the amplite vectors of the ownwar waves generate b the sorce in Region. b stans for the amplite vectors of the total reflecte waves along the pwar irection e to the bonar at z= in Region. A stans for the amplite vectors of the total ownwar waves in Region an B stans for the amplite vectors of the total pwar waves in Region. c stans for the amplite vectors of the total transmitte ownwar waves in Region. From the iscssion in Chapter, it is known that in an anisotropic meim, two ifferent polarize characteristic waves exist along a specific irection escribe b kx, k, k z. The two polarizations in Region are assigne as the Tpe I an Tpe II polarize waves, which can be obtaine sing the eigen-ecomposition metho escribe in Chapter. For an isotropic meim as in Region an Region, the two polarizations correspon to the h- an v- polarize waves as shown in Eq.( Eq. (3.-. Ths, all the waves (incling the incience wave a an reflecte wave b in Region, the ownwar an pwar propagating waves ( A an B in Region, an the transmitte wave c in Region can all be expresse as a b vectors, which correspons to the coefficients for the normalize electric fiel vectors of two ifferent polarizations in the given region. Then, all the waves in each region can be relate throgh the half-space reflection an transmission matrices as follows. b R X a B R, A A B c X R X (3.-5

79 56 In Eq. (3.-5, pq R (p, q=,, is a b matrix which gives the half-space reflection coefficients of waves incient from Region p to Region q. X pq is a b matrix which gives the half-space transmission coefficients of waves incient from Region p to Region q. The halfspace reflection an transmission matrices are expresse in the form below. R X R R R X X hh vh hh vh, X Rhv Rvv X hv X vv R exp eiei i kzi kzi ReIIeI exp i kzii kzi ReIeII exp i kzi kzii ReIIeII exp i kzii kzii ei, I eii, I ei, II e II, II X exp eih i kzi kz R exp eiih i kzii kz ReIv exp i kzi kz ReIIv exp i kzii kz XI, h XII, h XI, v X II, v (3.-6 (3.-7 (3.-8 It is note here that R eiei is reflection coefficient with reference plane at z=, the phase shift at z= is taken into accont b the mltiplication of the exponential terms as shown in Eq. (3.-7. Rearranging Eq. (3.-4 gives the following matrix relationships. b Ra, R R X R ( I R R X A Da, D ( I R R X B Ua, U R ( I R R X c Xa, X X ( I R R X (3.-9

80 57 Sbstitting the half-space reflection an transmission coefficients into Eq. (3.-9 gives all the coefficients of the DGFs shown in Eq. (3.- Eq. (3.-4 as follows. R R X ei (e I,I L e II,I M X ei X ei (e I,II L e II,II M X eii X eii (e I,I L e II,I M X ei X eii (e I,II L e II,II M X eii A ei A eii L X ei L X eii M X ei M X eii (3.- B n (e I,n L e II,n M X ei X ( X I, L X II, M X ei (e I,n L e II,n M X eii ( X I, L X II, M X eii L L I R R M M (3.- where,, I, II stan for the polarizations of the wave. The procere escribe in this section provies the DGFs for all the regions below the sorce point when the sorce is locate in Region regarless of Region being either a reciprocal or non-reciprocal meim. The DGF above the sorce point can be irectl obtaine from the transpose of the DGF below the sorce point if Region is fille with a reciprocal meim as shown in [37]. However, existing smmetrical propert oes not work for the case with Region being a non-reciprocal meim, which is of the main interest in the next section. 3. Moifie Smmetrical Propert for (, G ( r, r of the Region z z an z z In this section, a moifie smmetrical propert is propose to correlate the DGFs for the regions above an below the sorce point when Region is fille with a non-reciprocal meim.

81 58 Appling the moifie smmetrical propert, the DGF above the sorce point is frther obtaine to facilitate the formlation of the raiate file which will be emonstrate in Chapter Proof of Moifie Smmetrical Propert of (, G ( r, r In this section, it is proven that when the sorce point an fiel point are both locate in Region, the smmetrical propert of the aic Green s fnction in Eq. (3.- is alwas vali, even for the anisotropic region fille with a non-reciprocal meim. (, (, G ( r, r G ( r, r r r (3.- r z= r z= z= z= Fig. 3-3: Two ifferent geometries for the proof of moifie smmetrical propert. Two ifferent geometries as shown in Fig. 3-3 are consiere here to facilitate the proof of the moifie smmetrical propert. In both Fig. 3-3(a an (b, Region an Region are enote as the isotropic regions. The relative permeabilit an permittivit for Region an Region are enote as r, r an r, r. However, the meia in Region are ifferent for Fig. 3-3(a an (b. Region is enote as the anisotropic meim with relative permeabilit an permittivit of for the geometr shown in Fig. 3-3(a, while in Fig. 3-3(b, Region is, r Tr enote as the anisotropic meim with relative permeabilit an permittivit of. In, r r

82 59 aition to the ifferent meia assigne for Region, the sorce locations are also ifferent for Fig. 3-3(a an (b. In Fig. 3-3(a, the sorce is locate at r an in Fig. 3-3(b, the sorce is locate at r in Region. In orer to prove Eq. (3.-, the aic-aic Green s theorem of the secon kin given b Tai [34] is se, which is shown below. V Q P Q P v S T n Q P Q n P s (3.- Step : Let (, (, P G ( r, r, Q G ( r, r It is note here Q an P represent the DGFs of Region for the geometries shown in Fig. 3-3(a an (b, respectivel. Ths, for the region of z, eqations. Q an P satisf the following wave Q k Q I ( r r Q k Q I ( r r r r r r P k P I ( r r P k P I ( r r r r r r (3.-3 Sbstitting Eq. (3.-3 into Eq. (3.- gives V k rrq I ( r r P v s (3.-4 S Q k rrp I ( r r Q n P n Q P T The volme integral is for Region an the close srface integral is for the srface boning the volme of Region. With Q an P sbstitte, Eq. (3.-4 is eqivalent to

83 6 V (, (, k r rg ( r, r I ( r r G ( r, r v (, (, G ( r, r k r rg ( r, r I ( r r S (, (, n G ( r, r G ( r, r (, T s (, G ( r, r ng ( r, r (3.-5 Simplifing Eq. (3.-5 gives (, (, n G ( r, r (, (, G ( r, r G ( r, r G ( r, r s (, T (, S G ( r, r ng ( r, r (3.-6 The close srface integral can be rece to a srface integral abot the z= plane b emploing the raiation bonar conition as z. In this case, Eq. (3.-6 reces to (, (, z G ( r, r (, (, G ( r, r G ( r, r G ( r, r s (, T (, z plane G ( r, r z G ( r, r (3.-7 It is note that the negative sign is e to the fact that n z. The continit conitions of the electric fiel an magnetic fiel at z for the geometr shown in Fig. 3-3(a reqire the DGFs to satisf the following bonar conitions. (, (, z G ( r, r z G ( r, r, (, (, z G ( r, r z G ( r, r r r (3.-8

84 6 The continit conitions of the electric fiel an magnetic fiel at z for geometr shown in Fig. 3-3(b reqire the DGFs to satisf the following eqations. (, (, z G ( r, r z G ( r, r, (, (, z G ( r, r z G ( r, r r r (3.-9 The following ientit is also sefl. z I z I (3.- Appling the bonar conitions of Eq. (3.-8 an Eq. (3.-9 together with the ientit of Eq. (3.- to Eq.(3.-7, it reces to (, (, z G ( r, r (, (, G ( r, r G ( r, r G ( r, r s (, T (, z plane G ( r, r z G ( r, r z plane r (, (, G ( r, r z I G ( r, r s (, T (, G ( r, r z I G ( r, r (, (, G ( r, r z G ( r, r s T (, (, z plane G ( r, r z G ( r, r r z plane (, (, G ( r, r z G ( r, r s (, (, G ( r, r z G ( r, r T (3.-

85 6 Step : Let (, (, P G ( r, r, Q G ( r, r It is note here Q an P represent the DGFs of Region for the geometries shown in Fig. 3-3(a an (b, respectivel. With Q an P sbstitte in Eq. (3.-, the volme integral in Eq. (3.- is for Region an the close srface integral is for the srface boning the volme of Region. Ths for Region, Q an P satisf the following wave eqations. T T rr rr Q k Q Q k Q P k P P k P r r r r (3.- Sbstitting Eq. (3.- together with efine Q an P into Eq. (3.- an replacing the close srface integral with a srface integral abot the z= plane gives (, (, k r r G ( r, r G ( r, r v (, (, (, (, V G r r k r r G r r (, (, (, r k G r r r G ( r, r (, (, G ( r, r k r r G ( r, r v V (, (, z G ( r, r G ( r, r s (, T (, z plane G ( r, r z G ( r, r z plane (, (, z G ( r, r G ( r, r s (, T (, G ( r, r z G ( r, r (3.-3

86 63 It shol be note that n z for the z plane an n z for the z plane. Appling the ientit of Eq. (3.-, Eq. (3.-3 can be written as (, (, z G ( r, r G ( r, r s (, T (, z plane z plane G ( r, r z G ( r, r z plane z plane (, (, G ( r, r z I G ( r, r s (, (, G ( r, r z I G ( r, r (, (, G ( r, r z G ( r, r s (, (, T z plane z plane G ( r, r z G ( r, r T (3.-4 Step 3: Let (, (, P G ( r, r, Q G ( r, r It is note here Q an P represent the DGFs of Region for the geometries shown in Fig. 3-3(a an (b, respectivel. Ths, for the region of z, Q an P satisf the following wave eqations. Q k Q Q k Q r r r r P k P P k P r r r r (3.-5 Sbstitting Eq. (3.-5 into Eq. (3.- gives V k rrq P v s S Q k rrp Q n P n Q P T (3.-6

87 64 The volme integral is for Region an the close srface integral is for the srface boning the volme of Region. With Q an P sbstitte, Eq. (3.-6 is eqivalent to V S (, (, k rrg ( r, r G ( r, r v (, (, G ( r, r k rrg ( r, r (, (, n G ( r, r G ( r, r (, T s (, G ( r, r n G ( r, r (3.-7 Simplifing Eq. (3.-7 gives (, (, n G ( r, r G ( r, r s (, T (, S G ( r, r ng ( r, r (3.-8 The close srface integral can be rece to a srface integral abot the z plane b emploing the raiation bonar conition as z. In this case, Eq. (3.-6 reces to (, (, z G ( r, r G ( r, r s (, T (, z plane G ( r, r z G ( r, r (3.-9 The continit of the electric fiel an magnetic fiel at z Fig. 3-3(a reqires the DGFs to satisf the following bonar conitions. for the geometr shown in

88 65 (, (, z G ( r, r z G ( r, r, (, (, z G ( r, r z G ( r, r r r (3.- The continit of the electric fiel an magnetic fiel at z Fig. 3-3(b reqires the DGFs to satisf the following bonar conitions. for geometr shown in (, (, z G ( r, r z G ( r, r, (, (, z G ( r, r z G ( r, r r r (3.- Appling the bonar conitions of Eq. (3.- an Eq. (3.- together with the ientit of Eq. (3.- to Eq. (3.-9, it reces to (, (, z G ( r, r G ( r, r s (, T (, z plane G ( r, r z G ( r, r z plane r r z plane (, (, G ( r, r z I G ( r, r s (, (, G ( r, r z I G ( r, r (, (, G ( r, r z G ( r, r s T (, (, z plane G ( r, r z G ( r, r (, (, G ( r, r z G ( r, r s (, (, G ( r, r z G ( r, r T T (3.- Eq. (3.- is eqivalent to

89 66 z plane (, (, G ( r, r z G ( r, r s (, (, G ( r, r z G ( r, r T (3.-3 Sbstitting Eq. (3.-33 into Eq. (3.-4 leas to z plane (, (, G ( r, r z G ( r, r s (, (, G ( r, r z G ( r, r T (3.-4 With Eq. (3.-4 sbstitte into Eq. (3.-, the following smmetr propert is obtaine. (, (, G ( r, r G ( r, r (, (, (, (, r r G r r G r r ( DGF for the Region of z z The DGFs for the region of z z in Region of the half-space problem is alrea obtaine in Section 3.., which is repeate below. For z z G (, ( r, r i r ik r h ( kz e Rhh h ( kz e h ( k z ik r i R v ( k e k xk e 8 k i r ik r z v ( kz e Rvh h ( kz e v ( kz ik r Rvv v ( kz e hv z i r (3.-6 Using the smmetrical propert erive in Eq. (3.-5, the DGF for the region of z z is obtaine as follows.

90 67 For z z (, (, G ( r, r G ( r, r i 8 i r ik h ( kz e Rhh h ( kz e i r h ( kz e ik r Rhv v ( kz e kxk k z i r v ( kz e Rvh h ( kz e i r v ( kz e Rvv v ik r ( kz e r ik r (3.-7 In orer to make the above eqation have more clearl phsical meaning, the following sbstittion is applie to the above eqation, k k, k k (3.-8 x x Uner this transformation, the fiel vectors in the isotropic Region become as follows. h ( k, k k h ( k, k, k, v ( k, k v ( k, k (3.-9 x z x z x x Sbstitting Eq. (3.-9 into Eq. (3.-7, the DGF for the region above the sorce point ( z z is written as follows. ik r h ( kz e ik r i r h ( kz e Rhh ( kx, k h ( kz e i r (, (, i Rhv kx k v ( kz e G ( r, r x 8 k k k z v ik r ( k z e ik r i v ( kz e r Rvh ( kx, k h ( kz e i r Rvv ( kx, k v ( kz e (3.-3 It can be shown nmericall that the following relation hols for a biaxial meim.

91 68 R ( k, k R ( k, k hh x hh x R ( k, k R ( k, k hv x vh x R ( k, k R ( k, k vh x hv x R ( k, k R ( k, k vv x vv x (3.-3 For the two-laer geometr, the aic Green s fnction for the region above the sorce point ( z z can also be obtaine sing the smmetrical propert of Eq. (3.-5. ik r h ( kz e ik r i r h ( kz e Rhh( kx, k h ( kz e i r (, (, i Rhv kx k v ( kz e G ( r, r x 8 k k k z v ik r ( k z e ik r i r v ( kz e Rvh ( kx, k h ( kz e i r Rvv ( kx, k v ( kz e (3.-3 Same as the half-space reflection coefficients, it can also be shown nmericall that the twolaer reflection coefficients satisf the following relation for a two-laer problem with the anisotropic region being a biaxial meim. R ( k, k R ( k, k hh x hh x R ( k, k R ( k, k hv x vh x R ( k, k R ( k, k vh x hv x R ( k, k R ( k, k vv x vv x (3.-33 It nees to be note here no sch kin of relationship hols for both the half-space problem an two-laer geometr fille with a non-reciprocal meim. The DGF of (, G ( r, r for the region z z obtaine here has no restriction to the propert of the meim. The anisotropic

92 69 meim col be either reciprocal or non-reciprocal meim. It is crcial to obtain the DGF of G (, ( r, r for the region of z z to calclate the raiate fiel e to a crrent sorce in the presence of a laere geometr with arbitrar anisotropic meim involve, which will be shown in etail in Chapter Moifie Smmetrical Propert for (, G ( r, r an (, G ( r, r In orer to obtain the far fiel in Region when the sorce is locate in Region (anisotropic region, the DGF for region z> is sall neee. It is known from [37] that if the sorce is locate in Region (anisotropic meim instea of Region (isotropic meim, the DGF for the region above the sorce point can be obtaine from the DGF for the anisotropic region below the sorce point sing smmetrical propert if the meim is reciprocal. What if the anisotropic region containing the sorce is non-reciprocal with the permittivit or permeabilit as a hermitian matrix instea of a smmetric matrix, can the smmetrical propert still be applie? The reciprocit relationships for the nbone grotropic meia is consiere in [7]. However, smmetrical propert for a laere geometr with a non-reciprocal meim involve is not investigate previosl, which is of primar interest in this section Proof of the Moifie Smmetrical Propert It will be shown in this section that (, G ( r, r can be obtaine from (, G ( r, r which correspons to the DGF of Region with the sorce locate in Region for a laere geometr as erive from Section 3.. In particlar, the following relationships hol for a laere geometr with the anisotropic region fille with a reciprocal meim.

93 7 Region is reciprocal: G ( r, r G ( r, r (, (, r r (3.3- Eq. (3.3- states that for the half-space problem or two-laer geometr fille with a reciprocal meim, the DGF in Region (isotropic region when sorce is locate insie the anisotropic region can be obtaine from the transpose of the DGF of Region with sorce in Region. If the sorce is embee insie the anisotropic region of a non-reciprocal meim, the moifie smmetrical propert is given in Eq. (3.3-. Region is non-reciprocal: G r r G r r (, (, r (, ( (, region r ( region r r (3.3- Eq. (3.3- states that the DGFs of Region for the both the half-space an two-laer problems can still be obtaine from the DGFs of Region with sorce locate in Region ner the conition that Region is fille with the meim whose permittivit tensor is the transpose of that of the original meim of Region. For the grotropic meim, this conition is eqivalent to reversing the irection of the biasing magnetic fiel of Region. Observing Eq. (3.3- an Eq. (3.3- reveals that Eq. (3.3- is actall a special case of Eq. (3.3- as the relationship of r r hols for a reciprocal meim. Ths, the proof of Eq. (3.3- is onl presente. Daic-aic Green s theorem of the secon kin (Eq. (3.-, given b Tai [34] is again essential in this proof of the moifie smmetrical propert. To facilitate the proof of the moifie smmetrical propert, two ifferent geometries are consiere as shown in Fig. 3-4.

94 7 r z= r z= z= (a Region z= (b Fig. 3-4: Two ifferent geometries constrcte here for the proof of smmetrical propert when the sorce are in ifferent regions. In both Fig. 3-4(a an Fig. 3-4(b, Region is enote as the isotropic region with the relative permittivit an permeabilit of r, r, an Region is enote as the isotropic region with the relative permittivit an permeabilit of r, r. However, the sorce locations an the meia for Region are ifferent for the two geometries. In Fig. 3-4(a, Region is fille with the T meim characterize b r, r an the sorce is locate at r in Region, while in Fig. 3-4(b, the sorce is locate at r in Region with Region fille with the meim of r, r. Step : Q an P are assigne as follows. (, (, Q G ( r, r, P G ( r, r (3.3-3 In Eq. (3.3-3, Q represents the aic Green s fnction of Region when the sorce is Region for the geometr shown in Fig. 3-4(a, an P represents the aic Green s fnction of Region when the sorce is in Region for the geometr shown in Fig. 3-4(b. Ths, for Region the DGFs of Q an P satisf the wave eqations shown below. (, (, rr G ( r, r k G ( r, r I ( r r (, (, rr G ( r, r k G ( r, r (3.3-4

95 7 Sbstitting Eq. (3.3-3 into Eq. (3.- iels V (, (, G ( r, r G ( r, r v (, (, G ( r, r G ( r, r (, (, n G ( r, r G ( r, r s (, T (, S G ( r, r ng ( r, r (3.3-5 In the above eqation, the volme integration is over Region an the close srface integration is for the srface boning the volme of Region as shown in Fig Sbstitting Eq. (3.3-4 into Eq. (3.3-5 reslts in V (, (, k rrg ( r, r I ( r r G ( r, r v (, (, G ( r, r krrg ( r, r (, (, n G ( r, r G ( r, r (, T s (, S G ( r, r ng ( r, r (3.3-6 The close srface integral in Eq. (3.3-6 can be rece to a srface integral at z= plane sing the raiation conition at infinit, which is shown below. (, (, z G ( r, r (, G ( r, r G ( r, r s (, T (, z plane G ( r, r z G ( r, r (3.3-7 Step : Q an P are assigne as follows.

96 73 (, (, Q G ( r, r, P G ( r, r (3.3-8 In Eq. (3.3-8, Q represents the aic Green s fnction of Region when the sorce is in Region for the geometr as shown in Fig. 3-4(a, an P represents the aic Green s fnction of Region when the sorce is in Region for the geometr as shown in Fig. 3-4(b. Ths, for Region the aic Green s fnctions satisf the wave eqations shown below. (, (, r r G ( r, r k G ( r, r (, (, rr G ( r, r k G ( r, r I ( r r (3.3-9 Sbstitting Eq. (3.3-8 into Eq. (3.- iels V (, (, (, (, G ( r, r G ( r, r G ( r, r G ( r, r v (, (, (, T (, n G ( r, r G ( r, r G ( r, r n G ( r, r s S (3.3- Sbstitting Eq. (3.3-9 into Eq. (3.3- gives V (, (, k r r G ( r, r G ( r, r v (, (, G ( r, r k rr G ( r, r I ( r r S (, (, n G ( r, r G ( r, r (, T s (, G ( r, r ng ( r, r (3.3- In Eq. (3.3-, the volme integral is for Region an the close srface integral is for the srface boning the volme of Region. Simplifing Eq. (3.3- gives

97 74 V (, (, k r G ( r, r ( r r G ( r, r v (, G ( r, r I ( r r (, (, n G ( r, r G ( r, r s (, T (, S G ( r, r ng ( r, r (3.3- This irectl leas to (, (, n G ( r, r G ( r, r (, G ( r, r s (, T (3.3-3 (, S G ( r, r ng ( r, r Again the close srface integral consists of the planes locate at z= an z= plane, therefore, Eq. (3.3-3 can be expane as G (, ( r, r (, (, z G ( r, r G ( r, r s (, T (, z plane G ( r, r z G ( r, r z plane (, (, z G ( r, r G ( r, r s (, T (, G (, r r z G ( r, r (3.3-4 It is note that the normal to the srface at z= is in the positive z irection an the normal to the srface at z= is in the negative z irection. Step 3: Q an P are assigne as follows.

98 75 (, (, Q G ( r, r, P G ( r, r (3.3-5 In Eq. (3.3-5, Q represents the aic Green s fnction of Region when the sorce is in Region for the geometr as shown in Fig. 3-4(a, an P represents the aic Green s fnction of Region when the sorce is in Region for the geometr as shown in Fig. 3-4(b. Ths, for Region the aic Green s fnctions satisf the wave eqations shown below. (, (, rr G ( r, r k G ( r, r (, (, rr G ( r, r k G ( r, r (3.3-6 Sbstitting Eq. (3.3-5 into Eq. (3.- leas to V (, (, (, (, G ( r, r G ( r, r G ( r, r G ( r, r v (, (, (, T (, n G ( r, r G ( r, r G ( r, r n G ( r, r s S (3.3-7 Sbstitting Eq. (3.3-7 into Eq. (3.3-7 iels V (, (, krrg ( r, r G ( r, r v (, (, G ( r, r krrg ( r, r (, (, n G ( r, r G ( r, r (, T s (, S G ( r, r n G ( r, r (3.3-8 Inspecting Eq.(3.3-8 shows that

99 76 S (, (, n G ( r, r G ( r, r s (, T (, G ( r, r ng ( r, r (3.3-9 The close srface integration is for the srface boning the volme of Region an the close srface integral can be rece to a srface integral at z= plane sing the raiation conition at negative infinit, which is as follows, (, (, z G ( r, r G ( r, r s (, T (, z plane G ( r, r z G ( r, r (3.3- At z = plane, the continit of the tangential electrical fiel an tangential magnetic fiel reqires the DGFs to satisf the following bonar conitions given in Eq. (3.3- an Eq. (3.3-, corresponing to the geometries shown in Fig. 3-4 (a an Fig. 3-4 (b, respectivel. (, (, z G ( r, r z G ( r, r (, (, z G ( r, r z G ( r, r r r (3.3- (, (, z G ( r, r z G ( r, r (, (, z G ( r, r z G ( r, r r r (3.3- Sbstitting the bonar conitions of Eq.(3.3- an Eq. (3.3- into Eq. (3.3- gives (, (, (, (, T z G ( r, r G ( r, r G ( r, r z G ( r, r s z plane which can be written in the form below.

100 77 T (, (, (, (, G ( r, r z I G ( r, r G ( r, r z I G ( r, r s z plane Appling the ientit of z I z I, we obtain (, (, (, (, T G ( r, r z G ( r, r z G ( r, r G ( r, r s z plane Again sbstitting the bonar conitions of Eq.(3.3- an Eq. (3.3-, it is obtaine G r r z G r r z G r r G r r s (, (, (, T (, r (, (, (, (, r z plane which is eqivalent to (, (, z G ( r, r G ( r, r r (, T (, r z plane G ( r, r z G ( r, r s (3.3-3 Comparing the secon integral in Eq. (3.3-4 an Eq. (3.3-3, it is seen that this secon term is an Eq. (3.3-4 can be rewritten as follows. (, (, z G ( r, r (, G ( r, r G ( r, r s (, T (, z plane G ( r, r z G ( r, r (3.3-4 At z= plane, the continit of the tangential electrical fiel an tangential magnetic fiel reqires the DGFs to satisf the following bonar conitions given in Eq. (3.3-5 an Eq. (3.3-6, corresponing to the geometries shown in Fig. 3-4(a an Fig. 3-4(b, respectivel.

101 78 (, (, z G ( r, r z G ( r, r (, (, z G ( r, r z G ( r, r r r (, (, z G ( r, r z G ( r, r (, (, z G ( r, r z G ( r, r r r (3.3-5 (3.3-6 Making se of Eq.(3.3-5 an Eq. (3.3-6 in Eq. (3.3-4 an following a similar procere, it is obtaine (, (, z G ( r, r G ( r, r (, r G ( r, r s (, T (, r z plane G ( r, r z G ( r, r (3.3-7 Comparing Eq. (3.3-7 with Eq.(3.3-7 reveals the following relation. G r r G r r (3.3-8 (, (, r (, ( (, region r ( region r r If the anisotropic region is fille with a reciprocal meim with the permittivit satisfing r r, then the above smmetrical propert reces to the following form. G ( r, r G ( r, r (3.3-9 (, (, r r It is note here that the smmetrical propert is erive base on the two-laer geometr, however it also applies to the half-space anisotropic geometr, which is treate as one special case of the laere geometr.

102 Usage an Limitation of the Moifie Smmetrical Propert Accoring to the iscssion presente in Section 3.3., to obtain the DGF of Region (isotropic region above the sorce point with sorce locate insie Region (anisotropic region, the moifie smmetrical propert of Eq. (3.3-8 can be se for a laere geometr with the anisotropic region being non-reciprocal meia as a grotropic meia, an smmetrical propert of Eq.(3.3-9 can be se for a laere geometr with the anisotropic region fille with reciprocal meia sch as niaxial an biaxial meia. In this section, the smmetrical propert of Eq. (3.3-8 is applie to obtain the DGFs of Region with sorce locate in Region for both the half-space an two-laer problems with the anisotropic Region fille with either the reciprocal or non-reciprocal meim. In aition to the sage of the smmetrical propert, the limitation of the moifie smmetrical propert is also briefl reviewe in this section. First, the moifie smmetrical propert of Eq. (3.3-8 is applie to the half-space DGF of G (, (, r r obtaine in Eq. (3.-5, an the aic Green s fnction for the region of z z with sorce in Region is obtaine as follows (assming non-magnetic meim. (, (, G ( r, r G ( r, r i 8 i I r X hei e I ( kzi e i r h ( kz e i II r X hei e II ( kzii e kxk k z i I r X I ( i r vei e kzi e v ( kz e i II r X veii e II ( kzii e (3.3-3 In orer to have the DGF shown in Eq. (3.3-3 above give more straightforwar phsical meaning, the transformation shown in Eq.(3.3-3 is applie to the above formla. k k, k k (3.3-3 x x

103 8 Then, it s straightforwar to get For z (, G ( r, r i x 8 k k k z i h ( k, k, k e v ( kz, kx, k e i I r X hei ( kx, k ei ( kzi, kx, k e r z x i II r X hei ( kx, k e II ( kzii, kx, k e i r X vei ( kx, k e i I I ( kzi, kx, k e X veii ( kx, k e II ( kzii, kx, k e r i II r (3.3-3 It can be shown that the propagation vector an fiel vector of the isotropic region (Region satisf the following relation. ( k, k, k k ( k, k x z x h ( k, k k h ( k, k, k, v ( k, k v ( k, k x z x z x x ( Then Eq. (3.3-3 reces to the following form. For z (, G ( r, r i x 8 k k k z he ve ik r ik r X k k e k k k e X k k e k k k e i I ( kx, k r hei ( x, I ( zi, x, i II ( kx, k r hei ( x, II ( zii, x, i X vei ( kx, k e I ( kzi, kx, k e X veii ( kx, k e II ( kzii, kx, k e I ( kx, k r i II ( kx, k r (3.3-34

104 8 It is note here that for half-space reciprocal meim, the coefficients, wave vectors an eigenvectors shown in Eq. ( are calclate for the anisotropic meim of r, while for the half-space non-reciprocal meim, the are calclate for the anisotropic meim of r. If Region is fille with a biaxial meim, the propagation vectors an fiel vectors of Region satisf the following relation, which is prove in Appenix M of [37]. T ( k, k, k k ( k, k, ( k, k, k k ( k, k I x zi I x II x zii II x e I ( k, k e I ( k, k, e II ( k, k e II ( k, k x x x x ( It nees to be note here that the relationship between the eigenvectors of Region are chosen sch that it is consistent with the notation se b Pettis [37]. Sbstitting Eq. ( into Eq.( gives For z G i 8 (, ( r, r ik I r X (, ( hei kx k e I kzi e ik r he X hei ( kx, k e II ( kzii e kxk k z X vei ( kx, k e I ( kzi e ik r ve X veii ( kx, k e II ( kzii e ik II r ik I r ik II r ( It was given in Eq of [37] that the following relationships hol for a laere biaxial meim.

105 8 X ( k, k ( k, k, X ( k, k hei x x eih x X ( k, k ( k, k, X ( k, k vei x x eiv x X ( k, k ( k, k, X ( k, k heii x x eiih x ( X ( k, k ( k, k, X ( k, k veii x x eiiv x It is claime in [37] that the exact fnctional relationship of ( k, k,, ( k, k, is x x nknown. Here we show that ( k, k,, ( k, k, are relate with the amplite x x coefficients of the a in Green s fnction of the nbone meim fille in Region. This will be shown in Section 3.4., where the irect constrction metho of aic Green s fnction is iscsse when the sorce is locate in Region. Smmetrical propert is also applie to the aic Green s fnction of the two-laer problem in Eq. (3.-3 as follows. (, (, G ( r, r i G ( r, r x 8 k k k z i h ( kz e v ( kz e r i r i I r ik I r AheI e I ( kzi e BheI e I ( kzi e A e k e B e k e i II r ik II r heii II ( zii heii II ( zii i I AveI e r ik I r I ( kzi e BveI e I ( kzi e II AveII e II ( kzii e BveII e II ( kzii e i r ik II r ( Appling the transformation as shown in Eq. (3.3-3 an Eq.(3.3-33, Eq. ( can be written as follows.

106 83 (, G ( r, r i x 8 k k k z h ( k e v ( k e A k k e k k e ik I ( kx, k r hei ( x, I ( x, ik I ( kx, k r BheI ( kx, k ei ( kx, k e ik r z ik II ( kx, k r AheII ( kx, k e II ( kx, k e BheII ( kx, k e II ( kx, k e A k k e k k e ik II ( kx, k r ik I ( kx, k r vei ( x, I ( x, ik I ( kx, k r BveI ( kx, k ei ( kx, k e ik r z ik II ( kx, k r AveII ( kx, k e II ( kx, k e B veii ( k, k e ( k, k e x ik II ( k x, k r II x ( Eq. ( can be applie for a two-laer problem fille with either a reciprocal or a nonreciprocal meim. It is note here that for a two-laer reciprocal meim, the coefficients, wave vectors an eigenvectors shown in Eq. ( are calclate for the anisotropic meim of r, while for a two-laer non-reciprocal meim, the are calclate for the anisotropic meim of T r. As was shown in the half-space case, if Region is fille with a reciprocal meim, the propagation vectors an fiel vectors satisf the relation shown in Eq. ( Then, the above formla of Eq. ( for the DGF reces to a simplifie form shown below.

107 84 (, G ( r, r i x 8 k k k z ik I r AheI ( kx, k e I ( kzi e ik I r B (, I ( ik hei kx k e kzi e r h ( kz e ik II r AheII ( kx, k e II ( kzii e ik II r BheII ( kx, k e II ( kzii e I ik r AveI ( kx, k e I ( kzi e ik I r B (, I ( v ik vei kx k e kzi e r ( kz e ik II r AveII ( kx, k e II ( kzii e ik II r BveII ( kx, k e II ( kzii e (3.3-4 Eq. (3.3-4 can onl be applie to the geometr fille with a reciprocal meim, an the amplite coefficients of the formla will have the same relationship of Eq. ( as for the half-space problem if the reciprocal meim is a biaxial meim, which will be iscsse in etail in Section In the integran of Eq. (3.3-4, the wave vectors k, I, II, the Region fiel vectors h, v an the characteristic fiel vectors for the anisotropic region, I, II e are obtaine for a specific set of (k x,k. However, the amplite coefficients A, B are calclate from the halfspace reflection an transmission coefficients that are calclate for the incience wave with tangential wave vector (k x,k at the interface. This is ifferent from Eq. (3.3-39, which shows that except the fiel vectors h, an v for the isotropic Region, all other fiel vectors, wave vectors an amplite coefficients are calclate for given (k x,k with the permittivit of the matrix of anisotropic meim as the transpose of the permittivit matrix of the original meim.

108 85 As shown in this section, appling the moifie smmetrical propert can give the DGF (, G ( r, r irectl from a known DGF of (, G ( r, r. Similarl, the DGF of G (, ( r, r can also be obtaine b appling the moifie smmetrical propert to the DGF of G (, ( r, r. This propert is extremel sefl if the meim is reciprocal as it oes not reqire an aitional calclations, as long as the DGFs of a laere anisotropic geometr with the anisotropic meim characterize b r are alrea solve. However, if the meim is non-reciprocal, even if the DGFs of a laere anisotropic geometr with the anisotropic meim characterize b r is alrea provie, to appl the moifie smmetrical propert, aitional calclations of the DGFs of a laere anisotropic geometr with the anisotropic meim characterize b r is navoiable. In aition, if the T DGF of (, G ( r, r is of interest, then this smmetrical propert will not help. In this case, to get the complete set of DGFs for the laere geometr with the sorce locate in the anisotropic region, the irect constrction metho nees to be se. It will be shown in Section 3.4 that in aition to proviing the complete set of DGFs of all the regions incling the anisotropic region where the sorce is locate, a straightforwar phsical insight can be obtaine from the DGFs erive sing the irect constrction metho. 3.4 DGFs of a Two-Laer Geometr with a Sorce insie the Anisotropic Region As emonstrate in the previos section, appling the moifie smmetrical propert to the DGF ofg (, ( r, r with the sorce insie Region is onl to provie the DGF of (, G ( r, r.

109 86 To obtain the DGFs of all the regions for a laere geometr with a sorce insie the anisotropic region, the irection constrction metho is iscsse in etail in this section Direct Constrction Metho to Obtain the DGFs for All the Regions The first step to tilize the irect constrction metho is to obtain the DGFs for the nbone anisotropic meim. As presente in Section., the aic Green s fnctions for an nbone anisotropic meim are given in Eq. (.-3, which are repeate here again with i extracte ot. For For z z ( k ( ( zi e kzi v k zi iki ( rr e i a4( kzi kzi ( kzi kzii ( kzi kzii G ( r, r x k k ( k ( ( zii e kzii v kzii ikii ( rr e a4( kzii kzi ( kzii kzi ( kzii kzii z z ( k ( ( zi e kzi v k zi iki ( rr e i a4( kzi kzi ( kzi kzii ( kzi kzii G ( r, r x k k ( k ( ( zii e kzii v kzii ikii ( rr e a4( kzii kzi ( kzii kzi ( kzii kzii In Eq. (3.4- an Eq. (3.4-, a 4 inicates the coefficient for the forth orer term of kz (3.4- (3.4- in the expansion of the polnomial in terms of kz for the electric wave matrix W E. For example, if the anisotropic meim is a groelectric meim with the biasing magnetic fiel along an arbitrar irection of B an B, then a k ( sin cos. The eigenvale ( 4 o B 3 B an eigenvectors ( an ( are efine in Section.3.. ek zi vk zi k zi

110 87 For the comparison of aic Green s fnction obtaine sing the irect constrction metho an the aic Green s fnction obtaine sing the moifie smmetrical propert as i presente in the previos section, the common term k 8 z is extracte prposel from the aic Green s fnction, an Eq. (3.4- an Eq. (3.4- are rewritten as follows. For z z I iki ( rr ci ( k ei v e zi i G( r, r k xk 8 k z ikii ( rr cii ( k e II vii e zii (3.4-3 For z z I i I ( r r ci ( k e I v e zi i G( r, r k xk 8 k z iii ( rr cii ( k e II vii zii e (3.4-4 where c k k k c k ( ( ( ( ( ( z I z II I (, ( zi II zii a4 k k k k k k a zi zii zi zi zi zii 4 k k k k k k zii zi zii zi zii zii k k c k c k ( ( z I z II I ( zi, II ( zii a4( kzi kzii ( kzi kzi ( kzi kzii a4( kzi k zii kzii kzi kzii kzii The secon step of the irect constrction metho is to appl the matrix metho to the nbone DGFs obtaine from the eigen-ecomposition metho. The matrix metho has alrea been se in Section 3. to obtain the DGFs of a two-laer problem with a sorce in the isotropic meim. In that case, the irect wave was the ownwar incient wave in the isotropic meim. However, in this section, the irect pwar an ownwar waves from the sorce in Region as shown in Fig. 3-5 are se to constrct the DGF.

111 88 z= z= Fig. 3-5: Geometr of the two-laer problem with sorce insie the anisotropic region. In Fig. 3-5, the a an b stan for the amplite coefficient matrices of the nit vectors corresponing to the pwar an ownwar waves generate b the sorce locate insie the anisotropic region (Region bone b z= an z=. A an B represent the amplite coefficient matrices for the nit vectors of the total pwar an ownwar waves existing in Region e to the mltiple reflections of the a an the b waves at both the bonaries of z= an z=. C an D represent the amplite coefficient matrix for the nit vectors of the transmitte waves in Region an Region, respectivel. As inicate in Fig. 3-5, all the waves in each region incle two ifferent polarizations. Insie the anisotropic region, the waves A an B incle Tpe I an Tpe II polarizations, while the transmitte waves C an D insie the isotropic regions incle the h an v-polarizations as escribe in Section.3. The two-laer geometr ner consieration can be ecompose into two half-space problems with one corresponing to the reflection an transmission at the bonar z = (separating Region an Region an the other one corresponing to the reflection an transmission at the bonar z= (separating Region an Region. The waves in each

112 89 region can then be relate throgh the half-space reflection an transmission coefficient matrices below. A R ( b B, B R ( a A, C X ( a A, D X ( b B (3.4-5 where pq R an X pq are the half-space reflection an transmission coefficient matrices with wave incient from Region p to Region q efine in Eq. ( Eq. (3.-8. Rewriting Eq. (3.4-5 sch that A, B, C, an D are expresse in terms of the irect waves generate b the sorce in Region, we obtain a b a b a b a b A A a A b B B a B b C X a X b D T a T b a R R b R R e I ei eii ei ei ei eii ei A ( I R R R R, A ( I R R R R R R R ei eii eii e II ei eii eii e II a R R ei ei eii e I B b R R ei ei eii ei ( I R R R, B ( I R R R R R R ei eii eii e II R R ei eii eii e II a X X b X X e I h eii h ei h eii h X X ( I R R, X X ( I R R R X X X X ei v ei v ei v eii v a T T ei h eii h b T T ei h eii h T X ( I R R R, T X ( I R R T T T T ev I eii v ei v eii v (3.4-6 Using the coefficients above, the aic Green s fnctions in each region when a sorce is locate in the anisotropic region can be constrcte. The constrction of the DGF in Region is first consiere an it nees special attention. The anisotropic meim is separate into two regions with one corresponing to the space above the sorce point (z an the other corresponing to the space below the sorce point. For region above the sorce point, the irect wave incles the pwar wave onl, an for the region below sorce point, the irect wave

113 9 incles ownwar wave onl. With the DGFs for the nbone anisotropic region (Eq. (3.4-3 an Eq. (3.4-4 an coefficients obtaine from Eq. (3.4-6, we have for z z, For z z G (, i 8 k ( r, r z k k x c ( k e I v I e c ( k e II v II e iki ( rr ikii ( rr zii iki r ikii r R ci v I e R cii v II e e I ei eii ei ii r i R ci v I e R cii v II r II e e I ei eii ei iki r ikii r R ci v I e R cii v II e ei eii eii eii I II R ci v I e R cii v II e e I eii eii eii iki r ikii R ci v I e R cii v r II e e I ei eii ei ii r iii r R ci v I e R cii v II e ei ei eii ei iki r ikii r R ci v I e R cii v II e e I eii eii eii I II R ci v I e R cii v II e e I eii eii eii I zi II i I ee r I iii r e IIe i r i r ik I ee r I ikii r e IIe i r i r (3.4-7 Similarl, for the region below the sorce point, we have for z z,

114 9 For z z G (, i 8 k ( r, r z k k x c k e v e c k e v e I II II zii I R ci v I e R cii v II e e I ei eii ei I R ci v I e R cii v II e e I ei eii ei I R ci v I e R cii v II e ei eii eii eii I I I II II e I eii eii eii iki r ik R ci v I e R cii v II e e I ei eii ei I R ci v I e R cii v II e ei ei eii ei I R ci v I e R cii v II e e I eii eii eii I I I II II e I eii eii eii i ( r r i ( rr I ( I I ( II II zi ik r ikii r i I ee r I i r iii r ik r ikii r iii r e IIe i r iii r R c v e R c v e II r ik I ee r I i r iii r ik r ikii r ikii r e IIe i r iii r R c v e R c v e (3.4-8 Observation of Eq. (3.4-7 an Eq. (3.4-8 inicates that the first two terms insie the integral represent the irect waves e to the sorces, which are obtaine from the DGFs of the nbone anisotropic region as shown in Section.. All the other terms represent the pwar ( A an ownwar ( B propagating waves reflecte at the two bonaries. The tangential electric fiel an magnetic fiel mst satisf bonar conition at z=. The Green s fnction for Region ( z when a sorce is locate in the anisotropic region can be expresse in terms of coefficients from Eq. (3.4-6, which are obtaine for the transmitte wave C as

115 9 For z G (, ( r, r k k x i 8 k z X h e X v e c k v e eih X h e X v e c k v e eii h i X h e X v e c k ei h X h e X v e c k v e eii h ik r ik r ( I e zi Iv I ik r ik r II ( zii II e II v ik r ik r I ( ei v zi v I e ik r ik r II ( zii II e II v iki r ikii r I r iii r (3.4-9 Similarl, the DGF for Region (isotropic region below the anisotropic slab can be erive from the coefficients corresponing to the transmitte wave D. Ths we have For z G (, ( r, r k k x i 8 k z T h e T v e c k v e eih T h e T v e c k v e eii h i T h e T v e c k ei h T h e T v e c k v e eii h ir i r ( zi I e Iv I ir i r II ( zii II e II v ir i r I ( ei v zi v I e ir i r II ( zii II e II v iki r ikii r I r iii r (3.4- It is note here that, the vectors of v I an v II as the latter parts of the as in Eq. ( Eq. (3.4- are taken from the reslts of Eq. ( Eq. (3.4-4 for the DGFs of the nbone anisotropic meim.

116 Discssion of DGF (, G ( r, r Obtaine Using Two Approaches In this section, the aic Green s fnctions of Region with sorce locate in Region (, G ( r, r erive sing the smmetrical propert an the irect constrction metho are compare. It shows nmericall that both of the two formlations give the consistent reslts of DGF for meim. (, G ( r, r with anisotropic region fille with either reciprocal or non-reciprocal For a reciprocal meim, Eq. ( gives the coefficients of the a in the DGF of (, G ( r, r compte via the moifie smmetrical propert. The coefficients in Eq. ( are obtaine from Eq. (3.- with the tangential wave vector ( k, k sbstitte. The coefficients of the DGFs sing the irect constrction metho are given in Eq. (3.4-9 an obtaine irectl from Eq. (3.4-6 with the tangential wave vector of ( k, k. The coefficients of the DGF obtaine sing the two ifferent approaches are repeate here in Eq. (3.4- an Eq. (3.4- for convenience. x x A hei AveI D ( I R R X AheII A veii BheI BveI U R ( I R R X BheII B veii a X X eh I e II h X X ( I R R X X ei v eii v b X X ei h eii h X X ( I R R R X X ei v eii v (3.4- (3.4-

117 94 The coefficients of the DGF obtaine sing the moifie smmetrical propert as in Eq. (3.4- are compose of the half-space transmission matrix X for the waves incient from Region to Region. However, it oes not represent the actal phsical scenario of the problem of interest since the sorce is locate insie Region. On the other han, the complete coefficients of the a obtaine via the irect constrction metho, given b the RHS of Eq. (3.4-, provie more phsical insight to the problem. It is verifie nmericall that the following relationship hols for Eq. (3.4-, AheI ( kx, k AheII ( kx, k AveI ( kx, k AveII ( kx, k ci ( kx, k X ( kx, k cii ( kx, k X ( kx, k ei h eii h ci ( kx, k X ( kx, k cii ( kx, k X ( kx, k ei v eii v BheI ( kx, k BheII ( kx, k BveI ( kx, k BveII ( kx, k ci ( kx, k X ( kx, k cii ( kx, k X ( kx, k ei h e II h ci ( kx, k X ( kx, k cii ( kx, k X ( kx, k ei v eii v (3.4-3 (3.4-4 For a half-space problem, onl the pwar irect wave will be transmitte throgh the bonar of Region an Region, ths, the (, G ( r, r of a half-space problem are onl associate with the terms of pwar irect waves with the coefficients satisfing Eq. ( Comparing Eq. (3.4-3 with Eq. ( leas irectl to the following relation. c ( k c ( k I zi II zii (3.4-5 The relation given b eqation (3.4-5 reveals that nknown constants of an (as inicate in [37] actall has their own phsical meanings. These constants represent the

118 95 amplite of the pwar irect waves generate b the sorce in Region. This is a new an interesting iscover. In the case of a laere geometr, both the pwar an ownwar irect waves generate b the sorce will contribte to the DGF of (, G ( r, r. As seen in Eq. (3.4-3 an Eq. (3.4-4, each term in the LHS matrix is a proct of the coefficients c an X. The coefficients of c represent the amplites for the irect waves generate b the sorce in the nbone anisotropic meim. The vectors of c, c, c, c correspon to the Tpe I pwar wave, Tpe II pwar wave, Tpe I ownwar wave an Tpe II ownwar wave. The coefficients X of Eq. (3.4- correspon to the two-laer transmission coefficients in Eq. (3.4-6 for waves incient from Region (where the sorce is to Region. Each term of Eq. (3.4-6 has its own phsical interpretation. Since the sorce is locate insie the bone anisotropic slab, the irect wave generate b the sorce will experience mltiple reflections at both bonaries. The total pwar wave from the accmlation of all the reflections is inicate b( I R R. The transmission of the total pwar wave from Region to Region is characterize b the half-space transmission matrix I II I X. Appling the concept of the eigen-ecomposition an the matrix metho to obtain the coefficients for the laere geometr, the DGF for the laere problem with a general anisotropic meim when the sorce is locate insie the isotropic region can be obtaine. If the sorce is locate insie the anisotropic region an the anisotropic region is a reciprocal meim, the DGF of Region above the sorce point can be obtaine b appling the smmetrical propert. If Region is a non-reciprocal meim, then the conventional smmetrical propert nees to be moifie. It is state that for a non-reciprocal meim sch as a grotropic meim, II

119 96 an interchange of the sorce an the observation points in the two regions necessitates a reversal of the c biasing magnetic fiel to calclate the corresponing DGF. The moifie smmetrical propert of DGF simplifies the constrction of the DGF. However, appling the moifie smmetrical propert cannot provie the complete set of DGFs for all the regions when the sorce is locate insie the anisotropic slab. Also, the available smmetrical propert oesn t appl to the meim with magnetic anisotrop. A new metho to constrct the DGFs of laere meim with arbitrar anisotrop irectl from the characteristic waves in each region sing the eigen-ecomposition an matrix metho propose here. This metho can easil be extene to calclate the DGFs for a mltilaere geometr fille with general anisotropic (electric or magnetic meim with a sorce locate in an region. Frther, the DGF obtaine via irect constrction metho is compare with the DGF obtaine sing the smmetrical propert for the reciprocal meim case. An interesting relationship for the coefficients of the a in the DGFs obtaine throgh two ifferent methos is observe an iscsse. Ths, a straightforwar phsical insight to the DGFs is reveale from the irect constrction metho when the sorce is locate insie the anisotropic region as compare to the reslts obtaine sing the smmetrical propert. The DGFs obtaine here have wie applications in the scattering an raiation problems with arbitraril shape 3D objects locate insie the anisotropic region.

120 4 RADIATION OF A HERTZIAN DIPOLE IN THE PRESENCE OF A LAYERED ANISOTROPIC MEDIUM 97 In the last few ecaes, the raiation of a Hertzian ipole in the presence of a laere anisotropic sbstrate has been extensivel stie b the researchers. Accoring to the literatre srve in Chapter, no nmerical reslts have been given for the raiate fiel of a Hertzian ipole in the presence of a half-space groelectric meim. Also, the raiate fiel of a Hertzian ipole above or insie the groelectric slab with an arbitrar biasing magnetic fiel is not fon from the crrent literatre. To fill this gap, the raiate fiels of a Hertzian ipole for the above cases are solve an iscsse in this chapter. This chapter is organize as follows. In Section 4., the formlation for the raiate fiel of a Hertzian ipole is presente for half-space geometr with a sorce locate in both the isotropic an anisotropic regions sing the metho of stationar phase. In Section 4., the analsis of the raiate fiel of a Hertzian ipole embee insie either the isotropic or the anisotropic region of two-laer geometr is presente. In Section 4.3, the explicit formlations from previos sections are nmericall valiate with the available reslts from crrent literatre. In Section 4.4, etaile iscssions are presente on the raiate fiel of a ipole on top of the half-space groelectric meim an insie the laere groelectric slab. 4. Raiation of a Hertzian Dipole for a Half-Space Problem As shown in Fig. 4-, Region of the half-space geometr is enote as the isotropic region an Region is enote as the anisotropic region. r an r are the permittivit

121 98 an permeabilit for the isotropic region. r an r are the permittivit an permeabilit tensors for the anisotropic region. an are the free space permittivit an permeabilit. It s note here that in the following iscssion, r an r, while r an r for Region take the following form for a general anisotropic meim. xx x xz r x z zx z zz xx x xz r x z zx z zz (4.- An arbitraril oriente Hertzian ipole is locate at a istance h awa from the interface separating the isotropic an anisotropic regions. The ipole can be locate either in Region as shown in Fig. 4-(a or Region as shown in Fig. 4-(b. x z Region Region h (a Region Region h (b Fig. 4-: Geometr for the half-space problem with a Hertzian ipole place at a certain istance awa from the bonar separating the isotropic an anisotropic regions. (a Dipole in Region (isotropic region an (b ipole in Region (anisotropic region. The Hertzian ipole sorce is given b J( r ( Il ( x ( ( z h (4.-

122 99 The negative sign shows that the ipole is locate in Region above the interface, while the positive sign inicates that the ipole is locate in Region below the interface. The irection of the Hertzian ipole is inicate b the normalize nit vector. 4.. The Sorce Embee in the Isotropic Region To obtain the raiate fiel in Region for the geometr shown in Fig. 4-(a, the aic Green s fnction for the region of z z when the sorce is locate in Region is reqire. The DGF of G (, ( r, r has been erive in Chapter 3, which is repeate here for convenience. For z z G i 8 (, ( r, r ik r h ( kz e ik r i r h ( kz e Rhh ( kx, k h ( kz e i r Rhv ( kx, k v ( kz e kxk k z ik r v ( kz e ik r i v ( kz e Rvh ( kx, k h ( kz e i Rvv ( kx, k v ( kz e r r (4.-3 The terms of R ij ( i, j,, an p, q h, v in Eq. (4.-3 inicate the half-space pq reflection coefficients. The first inex of the sperscript i inicates the region of incient wave, an the first inex of the sbscript inicates the polarization of the incience wave. The secon inex of the sperscript inicates the region of transmission, an the secon inex of the sbscript inicates the polarization of the reflecte wave. For example, Rhh inicates the halfspace reflection coefficient of the wave incient from Region to Region with both the incient an reflecte waves being h-polarize waves.

123 It ma be shown nmericall that the half-space reflection coefficients at the interface of the isotropic an anisotropic regions satisf the relation in Eq. (4.-4, when the anisotropic region (Region is fille with a niaxial or biaxial meim. R ( k, k R ( k, k, R ( k, k R ( k, k hh x hh x hv x vh x R ( k, k R ( k, k, R ( k, k R ( k, k vh x hv x vv x vv x (4.-4 If Region is fille with a non-reciprocal meim, the above relation is not vali between the reflection coefficients for incient wave with ( k, k an the reflection coefficients for incient wave with ( k, k. However, the above relationship is still vali if the LHS an RHS of x x the above formlas are calclate for the half-space problem with the anisotropic region of an, respectivel. If Region is fille with a niaxial or biaxial meim, sing Eq. (4.-4, the DGF of region z z in Eq. (4.-3 reces to the following form. For z z G i 8 (, ( r, r ik r h ( kz e ik r i r h ( kz e Rhh ( kx, k h ( kz e i r Rhv ( kx, k v ( kz e kxk k z ik r v ( kz e ik r i r v ( kz e Rvh ( kx, k h ( kz e i r Rvv ( kx, k v ( kz e (4.-5 It is known that the raiate fiel is relate with the crrent sorce an the aic Green s fnction in the following form.

124 V (, E ( r i vg ( r, r J( r (4.-6 Ths, sbstitting Eq. (4.- an Eq. (4.-5 into Eq. (4.-6 gives the electric fiel in terms of half-space reflection coefficients an eigenvectors in free space in Eq. (4.-7. E( r ik r i r ik r h e Rhh h e he i r Rhv v e i i V k ( xk J r 8 k z ik r i r V v ik e R r vh h e ve i r Rvv v e Il 8 k k x k z ikzh ikzh h e R hh h e ik r he ik zh Rhv v e ikzh ikzh ik r v e Rvh h e ve ik zh Rvv v e (4.-7 In general, the integral in Eq. (4.-7 cannot be obtaine analticall. However, the far fiel raiation pattern can be approximatel obtaine sing the metho of stationar phase [7]. This ik techniqe assmes that the phase term e r is oscillating so rapil that on average the contribtions to the integral are mostl from the point which correspons to the minima of the phase fnction k r, an it is assme that the a an kz is almost constant aron this stationar point. Extracting all the aic elements an the slowl varing scalar fnction from the integran gives ik zh ik zh ik zh h he Rhh he Rhv v e Il ik r E ( r kxk e 8 k z ik zh ik zh ik zh v v e Rvh he Rvv v e (4.-8

125 The stationar phase point can be obtaine b taking the partial erivatives of the phase term k r with respect to k, k an setting the erivatives eqal to zero. The stationar phase point x occrs when the propagation vector is aligne with the fiel point vector [7, p.43]. The final expressions of the raiate fiel for a Hertzian ipole over the half-space geometr fille with reciprocal an non-reciprocal meia are given in Eq. (4.-9 an Eq. (4.-, respectivel. For a half-space reciprocal meim, the raiate fiel in the region of z z is ik zh ik zh ik zh h he Rhh he Rvh v e Il ikr E ( r e 4 r ik zh ik zh ik zh v v e Rhv he Rvv v e (4.-9 For a half-space non-reciprocal meim, the raiate fiel in the region of z z is ikzh ikzh he R (, hh kx k h e h ik zh hv ( x, Il R k k v ikr e E ( r e 4 r ikzh ikzh v e Rvh ( kx, k he v ik zh Rvv ( kx, k v e ( The Sorce Embee in the Anisotropic Region For the region above the sorce point ( z z, the raiate fiel of a Hertzian ipole correlates with the crrent sorce throgh the aic Green s fnction of following form. (, G ( r, r in the V (, E ( r i vg ( r, r J( r (4.-

126 3 (, G ( r, r is the aic Green s fnction of Region when the sorce is locate in Region. As erive in Chapter 3, the aic Green s fnction of (, G ( r, r can be obtaine throgh the irect constrction metho as well as throgh appling the smmetrical propert to the DGF of (, G ( r, r. These two approaches give the consistent aic Green s fnctions. Since the raiate fiels of a Hertzian ipole in the presence of both laere reciprocal an non-reciprocal meia are of interest, it s more straightforwar to tilize the DGF formlate with the irect constrction metho. For convenience, the DGF of irect constrction metho is repeate below. (, G ( r, r obtaine throgh the ikr ik r ( zi I e Iv I ikr ik r II ( zii II e II v iki r (, X h e X v e c k v e i eih G ( r, r k xk 8 k z ikii r X h e X v e c k v e eii h (4.- where the coefficients are efine as X X X X ei h eii h X ei v eii v c c k ( ( ( z I I ( k zi a4 k k k k k k zi zii zi zi zi zii k ( ( ( z II II ( k zii a4 k k k k k k zii zi zii zi zii zii (4.-3 In Eq. (4.-3, a 4 is the coefficient of the forth orer term of kz in Booker qartic eqation, which is the forth orer polnomial of the eterminant of the electric wave matrix expane in terms of k z. k q ( p I, II; q, are the for roots of the Booker qartic eqation. The terms of zp ij X ( i, j,; p e, e ; q h, v inicate the half-space transmission coefficients. The pq I II first inex of the sperscript i inicates the region where the wave is incient from, an the first

127 4 inex of the sbscript p inicates the polarization of the incience wave. The secon inex of the sperscript j inicates the region where the transmitte wave exists, an the secon inex of the sbscript q inicates the polarization of the transmitte wave. For example, X inicates the eh I half-space transmission coefficient for a wave incient from Region to Region, while the incient an transmitte waves are Tpe I an h-polarize, respectivel. Appling the metho of stationar phase as in Section 4.., it iels the raiate fiel for the isotropic region of Region as follows. ikzi h X h X v c k v e eih Il ik r E ( r i e 4 r ikzii h X h X v c k v e eii h ( zi I e Iv I II ( zii II e II v ( Raiation of a Hertzian Dipole for a Two-Laer Problem In this section, the raiate fiels of a Hertzian ipole locate above an insie an anisotropic slab are formlate. The two-laer geometries with a Hertzian ipole sorce place in Region an in Region (anisotropic are shown in Fig. 4-(a an (b, respectivel. (a (b Fig. 4-: Two-laer geometr: (a The ipole is locate in Region (isotropic meim an (b The ipole is locate in Region (anisotropic meim. In both cases, it is assme that the Hertzian ipole is positione at a istance of h awa from the interface separating Region (anisotropic an Region (isotropic. For a two-laer

128 5 problem here, Region is isotropic region with relative permittivit an permeabilit of r, r, Region is enote as the anisotropic region with relative permittivit an permeabilit tensors,, an Region is isotropic region with relative permittivit an permeabilit of of r r,. The thickness of anisotropic region (Region is. r r 4.. The Sorce Embee in the Isotropic Region The first case consiere here is the raiation of a Hertzian ipole when it is locate in the isotropic region as shown in Fig. 4-(a. The half-space problem with sorce insie the isotropic region as presente in Section 4.. can be treate as one special case of a two-laer problem. In orer to calclate the raiate fiel for the region of z z, the DGF of the corresponing region for the two-laer geometr is reqire. The etaile erivation of the DGF of the two-laer geometr has been iscsse in Section 3.4. For convenience, the DGF of the two-laer geometr for region of z z is repeate below. For z z G i 8 (, ( r, r ik r h ( kz e ik r i r h ( kz e Rhh( kx, k h ( kz e i r Rhv ( kx, k v ( kz e kxk k z ik r v ( kz e ik r i r v ( kz e Rvh ( kx, k h ( kz e i r Rvv ( kx, k v ( kz e (4.- In Eq. (4.-, the two-laer reflection coefficients can be expresse in terms of half-space reflection coefficient matrices as follows.

129 6 Rhh Rhv R X R ( I R R X Rvh R vv (4.- If Region is fille with a biaxial meim, the relation in Eq. (4.-3 hols for two-laer reflection coefficients as Eq. (4.-4 for half-space reflection coefficients. It is note here that no rigoros proof is given for Eq. (4.-3. This relation is verifie nmericall onl. R ( k, k R ( k, k hh x hh x R ( k, k R ( k, k hv x vh x R ( k, k R ( k, k vh x hv x R ( k, k R ( k, k vv x vv x (4.-3 Using Eq. (4.-3, the DGF of Region as in Eq. (4.- for a two-laer problem with Region fille with a biaxial meim can be simplifie to Eq. (4.-4. ik r h ( kz e ik r i r h ( kz e Rhh( kx, k h ( kz e i r (, (, i Rhv kx k v ( kz e G ( r, r x 8 k k k z v ik r ( k z e ik r i r v ( kz e Rvh ( kx, k h ( kz e i r Rvv ( kx, k v ( kz e (4.-4 If Region is fille with a non-reciprocal meim, sch as a grotropic meim, then the relation Eq. (4.-4 no longer hols. DGF of Eq. (4.- mst be se for the calclation of the electric fiel in Region. The electric fiel in Region is expresse in terms of the aic Green s fnction of Region as follows. V (, E ( r i vg ( r, r J( r (4.-5

130 7 To obtain the raiate fiel for the region of z h, Eq. (4.-5 still applies with the DGF of G (, ( r, r in Region replace with the DGF of (, G ( r, r for Region ( z h. The etaile erivation abot how to obtain the DGF of G (, ( r, r is presente in Section 3.4, an it is repeate here for convenience. G (, ( r, r i r i r i X hh h ( kz e X hvv ( kz e h ( kz k xk e 8 k i r i r z X vh h ( kz e X vvv ( kz e v ( kz i r (4.-6 The two-laer transmission coefficients are given in the following form. Xhh Xhv X X ( I R R X Xvh X vv (4.-7 Sbstitting Eq. (4.- an Eq. (4.-7 into Eq. (4.-3 gives the raiate electric fiel i E ( r i v kxk k 8 V z ik r i r he R hh( kx, k h e ik r he i r Rhv ( kx, k ve v e Rvh ( kx, k he ik r ve i r Rvv ( kx, k v e ik r i r Jr ( (4.-8 When the ipole sorce is locate at z h, the above eqation reces to Eq. (4.-9

131 8 Il E ( r v k xk 8 V k z ikzh ikzh he R (, hh kx k h e ik r he ik zh Rhv ( kx, k v e ikzh ikzh v e Rvh ( kx, k he ik r ve ik zh Rvv ( kx, k v e (4.-9 Appling the metho of stationar phase as iscsse in Section 4.., the raiate fiels of a Hertzian ipole in the isotropic region of Region are expresse in Eq. (4.- an Eq (4.-. ipole place over a laere reciprocal meim ik zh ik zh ik zh h he Rhh he Rvh v e Il ikr E ( r i e, 4 r ik zh ik zh ik zh v v e Rhv he Rvv v e (4.- ipole place over a laere non-reciprocal meim Il 4 r ikr E ( r i e z he R (, hh kx k h e h ik zh Rhv ( kx, k v e z v e Rvh ( kx, k he v ik Rvv ( kx, k v zh e ik h ikzh ik h ikzh (4.- The raiate fiel for Region ( z both the laere reciprocal an non-reciprocal meia as follows. below the sorce point takes the same form for Il ik ( r ik zh ik zh E r i e X hhh X hvv h e X vhh X vvv v e 4 r (4.- In Eq. (4.- - Eq. (4.- above, the wave vectors, the h- an v-polarize waves in Region an are efine as follows.

132 9 k xk k zk, xk k zk n x nz n x nz ( z kn hn knz hn ( knz, k h ( n knz kn hn ( knz n v n( knz, v n( knz, n or, k k n n (4.-3 The h- an v-polarize waves can be expane as follows. k kxk nz kxk nz k kk knk k k x k nz kk nz hn hn, v n, v n, k knk knk k k k n k n k k k x k k k k k nz n x n nr nr n or, (4.-4 where k x, k are the tangential components of the wave vector for a specific irection. Observing Eq. (4.- shows that the total electric fiel in the far fiel region incles two parts. ik zh ik zh h h e v v e (4.-5 ikzh ikzh R (, (, hh kx k h e Rvh kx k h e h v ikzh ikzh Rhv ( kx, k v e Rvv ( kx, k v e (4.-6 The first part shown in Eq. (4.-5 inicates the waves generate b the sorce in the isotropic region of Region in the absence of the anisotropic meim. The secon part shown in Eq. (4.-6 has for terms that inicate the reflecte fiel from the bonar of isotropic an anisotropic regions. Comparing Eq. (4.- with Eq. (4.-9 reveals that the raiate fiels of two-laer an half-space problems take the same form with ifferent coefficients of the a.

133 It can also be observe from Eq. (4.-6 that in a given irection onl one elementar wave, having the same incience angle as the observation angle, contribtes significantl to the reflecte fiel. The terms associate with reflection coefficients of R ( k, k an hv x R ( k, k inicate that the reflecte waves at the interface of the Region an Region vh x have excite both the polarizations of E v an E h. This is e to the cross-copling of the two characteristic waves insie the anisotropic region. The cross-copling generall reslts in an ellipticall polarize wave instea of a linearl polarize wave that occrs in free space. When the ipole is oriente in the x -irection, inicating x, an then Eq. (4.- can be written in Eq. (4.-7 ecompose into the horizontal an vertical components as follows. E( r Eh( r Ev( r (4.-7 Il k ikr ik zh ik zh k xk ( (, z ik zh E (, h r i e e Rhh kx k e Rhv kx k e h 4 r k kk Il ikr kxk k z ik zh ik zh ik zh Ev( r i e e Rvv ( kx, k e Rvh ( kx, k e v 4 r k k k (4.-8 When the ipole is oriente in the irection, as eqivalent to, then the raiate fiel can be simplifie to Eq. (4.-7 with the horizontal an vertical components shown below. Il ikr k kk x ik zh ( ik zh z (, ik zh E (, h r i e e Rhh kx k e Rhv kx k e h 4 r k kk Il kk ikr z ik zh ik zh k x ik zh Ev( r i e e Rvv ( kx, k e Rvh ( kx, k e v 4 r k k k (4.-9

134 When the ipole is oriente in z irection, impling z, then Eq. (4.- can be written in the form of Eq. (4.-7 with the horizontal an vertical polarize components shown below. Il k E h( r i e h Rhv ( kx, k e 4 r k ikr ikzh Il k E r i e v e R k k e 4 r k ikr ik zh ik zh v( vv ( x, (4.- It is observe from Eq. (4.-8 an Eq. (4.-9 that for a horizontall oriente ipole, both h an v polarize waves are excite from the reflection of the irect waves at the interface, inicating both the co-polarize an cross-polarize reflections contribte to the reflecte waves. However, for a verticall oriente ipole, as shown in Eq. (4.- onl the v polarize irect wave can be excite while the h polarize irect wave cannot be excite. Ths, the h polarize component of the total raiate fiel comes onl from the cross-polarize reflection e to the the v polarize irect wave. The v component of the total raiate fiel is from the irect wave of the v polarization an the co-polarize reflection with the v polarize incience wave onl. No cross-polarize reflecte wave exists, since no irect h polarize wave can be excite for z -irecte ipole. It is note here that in the spherical coorinate sstem, the h component of the far fiel correspons to the component an the v component of the far fiel correspons to the component.

135 4.. The Sorce Embee in the Anisotropic Region If the sorce is embee insie the anisotropic region instea of the isotropic region, the following aic Green s fnctions obtaine in Section 3.4 are reqire to calclate the raiate fiel. For z For z X h e X v e c k v e eih X h e X v e c k v e ik r ik r ( I e zi Iv I ik r ik r II ( zii II e II v ik r ik r I ( ei v zi v I e ik r ik r II ( zii II e II v (, i eii h G ( r, r k xk 8 k z X h e X v e c k ei h X h e X v e c k v e eii h iki r ikii r ii r T h e T v e c k v e eih ir i r ( zi I e Iv I i r II ( zii II e II v ir i r I ( ei v zi v I e ir i r II ( zii II e II v iii r iki r i r ikii r (, T h e T v e c k v e i eii h G ( r, r k xk 8 k z ii r T h e T v e c k ei h T h e T v e c k v e eii h iii r (4.- (4.- where the coefficients are efine as k k c k c k ( ( ( ( ( ( z I z II I (, ( zi II zii a4 kzi kzii kzi kzi kzi kzii a4 kzii kzi kzii kzi kzii kzii k k c k c k ( ( z I z II I ( zi, II ( zii a4( kzi kzii ( kzi kzi ( kzi kzii a4( kzii kzi kzii kzi kzii kzii X X X X X a X ( I R R, X X ( I R R R X X X X ei v eii v ei v eii v T T T T ei h eii h ei h eii h T a X ( I R R R, T b X ( I R R T T T ei v eii v T ei v eii v e I h eii h ei h eii h b

136 3 Appling the metho of stationar phase iels For z For z iile E( r 4 r iile E( r 4 r ikr ikr X h X v c k v e eih X h X v c k v e eii h X h X v c k v e ei h X h X v c k e eii h ( I e Iv I zi ( zii II e II v II I ( zi I e I v II ( eii v zii v II ( I e Iv I zi ( zii II e II v II I ( zi I e I v II ( eii v zii v II ikzi h ikzii h ikzi h T h T v c k v e eih T h T v c k v e eii h T h T v c k v e ei h T h T v c k e eii h ikzii h ikzi h ikzii h ikzi h ikzii h (4.-3 (4.-4 Observing the formlas above shows that when the sorce is locate insie the anisotropic region, the raiate fiel in Region consists both h -polarize an v -polarize transmitte waves. Each of these waves is e to the transmission of both pwar waves characterize b e I, e II an ownwar waves characterize b, e I e II. Comparing the transmission matrix X b (as it correspons to the transmission of the ownwar waves e I, e II with transmission matrix X a (as it correspons to the transmission of the pwar waves e I, e II reveals that X b X ar, which implies that the ownwar wave is first reflecte at the bonar of Region an Region an then transmitte into Region.

137 4 The raiate fiel in Region consists of the h an v waves transmitte from both the pwar an ownwar waves. Observing the transmission matrix T a shows that the ownwar waves a are transmitte irectl. On the other han, the pwar waves are first reflecte at the bonar of Region an Region an then transmitte throgh the bonar of Region an Region as inicate b T b. 4.3 Analtic an Nmerical Valiation In orer to valiate the raiation fiels obtaine in the previos sections, three cases are consiere in this section. Case I correspons to the self-check of the analtical reslts obtaine in the previos two sections. Case II presents the raiation of a Hertzian ipole in the presence of a biaxial slab as shown in [37]. Case III correspons to the two-laer slab fille with groelectric an gromagnetic meia with the sorce locate in the free space Case I: Self-check of (, G an (, G The geometr consiere here is shown in Fig z= z= z= (a z= (b Fig. 4-3: Geometr of Case I with (a DGF (, G (b DGF (, G.

138 5 The ipole is place right at the interface between Region an Region. Region is assme to be isotropic having the same permittivit as Region with. Region is assme to be a gron plane with. The thickness of the slab is assme to be. The raiate fiels of a vertical ipole place at the interface of h are solve sing (, G an (, G, respectivel. Since Region an Region are assme to be the same, it is expecte to obtain the exact same expressions for raiate fiels erive from either (, G or (,. G First, the general expressions for the raiate fiels of a z-irecte ipole obtaine sing (, G as in Eq. (4.- are repeate here in Eq. (4.3- for convenience. Il k E h( r i e h Rhv ( kx, k e 4 r k ikr ikzh Il k E r i e v e R k k e 4 r k ikr ik zh ik zh v( vv ( x, (4.3- When Regions an Region are the same meia, the half-space reflection an transmission matrices at the interface of Region an Region take the following forms. R R, X X (4.3- Then, the two-laer reflection coefficient matrix of the geometr reces to Eq. ( Rhh Rhv R X R ( I R R X R Rvh R vv (4.3-3 R can be calclate as follows with =.

139 6 R ikzi ikzi ik z e e e ik ik zii z ikzii e e e (4.3-4 The above expression of Eq. (4.3- for the z-irecte ipole reces to the following form. ik z ( Il ikr e k Eh( r, Ev i e v 4 r k (4.3-5 When the same problem is solve sing (, G, the raiate fiel for the region above the sorce point can be obtaine sing Eq. (4.-3, which is repeate here for convenience. iile E( r 4 r ikr X h X v c k v e eih X h X v c k v e eii h X h X v c k v e ei h X h X v c k e eii h ( I e Iv I zi ( zii II e II v II I ( zi I e I v II ( eii v zii v II ikzi h ikzii h ikzi h ikzii h (4.3-6 Since Region an Region are the same, the transmission coefficients rece to the following form. X a X X e I h eii h X X X ei v eii v (4.3-7 X X i k z ei h eii h e X b X R R ik z X X e ei v eii v (4.3-8 The characteristic wave vectors of v I, v I, v II, v II in Region rece to h, h, v, v an the amplite coefficients of c, c, c, c rece to since the I I II II Region an in Region are the same isotropic meia. Ths, Eq. (4.3-7 reces to Eq. (4.3-9.

140 7 ik r X h X v h X h X v v i ei h ei v eii h eii v Ile E ( r z 4 r X h X v h X h X v e v I h ei v eii h eii v ikr iile X h X v X h X 4 r eii h eii v eii h eii v k v k (4.3-9 Sbstitting the transmission coefficients of Eq. (4.3-7 an Eq. (4.3-8 back into Eq. (4.3-9 gives the following expression of the far fiel in Region. ik r i k z i ( Ile e k Eh( r, Ev( r v 4 r k (4.3- Inspecting Eq. (4.3- an Eq. (4.3-5 shows that the raiate fiel of the z -irecte ipole calclate sing either (, G or (, G of the two-laer geometr gives the exactl same reslts. Assming Region an Region are both free space with the thickness for Region being h =.5, an Region being PEC or same as Region (eqivalent to an nbone problem, the raiation patterns of a z-irecte Hertzian ipole place at the interface of Region an Region are plotte in Fig Fig. 4-4(a plots the raiate fiel of a vertical ipole calclate sing DGF of (, G, while Fig. 4-4(b plots the raiate fiel of a vertical ipole calclate sing DGF of (,. G 3 z-irecte ipole 33 inc = o E-PEC E-free 3 z-irecte ipole 33 inc = o E-PEC E-free (a 7 9 (b Fig. 4-4: Raiate fiels calclate sing (a DGF of (, G an (b DGF of (,. G

141 8 Inspecting Fig. 4-4(a an Fig. 4-4(b shows that raiation patterns calclate sing both the DGFs give the exactl same reslts. Appling the same procere, the raiate fiels of a horizontal x -irecte ipole at the interface of a laere geometr with Region an Region being ifferent meia are erive. It is shown in Eq. (4.3- that a horizontal ipole s raiate fiels erive from both the DGFs of (, G an (, G give the same reslts. E r i Ile k R k k ikr h( hh( x, 4 r k ikr iile kxk z Ev( r Rvv ( kx, k 4 r kk (4.3- It is note here that if both regions of the laere geometr are fille with ifferent meia, the raiate fiel for a vertical ipole place at the interface of Region an Region, obtaine sing (, G an (, G, are sall not the same becase the normal component of the electric fiel is not continos across the bonar. For this reason, Region an Region are assme to be the same in the erivation of the raiate fiel for a vertical ipole Case II: Unbone Isotropic Meim an Laere Biaxial Meim In this section, we show that the formlas of raiate fiels obtaine in Section 4. when the sorce is locate in Region an Region are consistent with a nmber of known cases. A. Unbone Isotropic Meim The first problem analze is shown in Fig. 4-5, where both Region an Region are isotropic with permittivit, i.e., it is an nbone problem. The far fiel raiation pattern was calclate in the plane with a Hertzian ipole oriente along x, an z irections.

142 9 Fig. 4-5: Geometr of nbone isotropic meim. It is known that the theoretical normalize raiate fiel pattern for a z-irecte ipole in nbone isotropic meim is as follows. f ( sin (4.3- For an x-irecte ipole, the normalize raiate fiel pattern is given b z fx (, sin cos (4.3-3 while for a -irecte ipole, the normalize raiate fiel pattern is f (, sin sin (4.3-4 The calclate an exact reslts for the x, an z-irecte Hertzian ipoles are shown in Fig. 4-6(a, (b an (c. Excellent agreement between the nmerical soltion an the exact soltion is observe for all three cases. 7 3 x-irecte ipole E-calc E-exact 9 3 -irecte ipole E-calc E-exact.5 (a 7 9 (b 7 9 (c 3 z-irecte ipole E-calc E-exact Fig. 4-6: in the XZ plane. Raiation pattern of the x, an z-irecte Hertzian ipole in nbone meim It is note here that to calclate the raiation fiel of a Hertzian ipole with Region being isotropic meim sch as the case of the free space, the three iagonal elements of the

143 permittivit tensor for Region are chosen as,,., so the anisotropic meim reces to the isotropic case. Choosing,,. instea of,, is becase the ajoint wave electric matrix is alwas zero for an isotropic meim, an the eigenecomposition metho is no longer applie. The etail has been explaine in Chapter. B. Biaxial Meim Now we consier the raiation of a Hertzian ipole in the presence of a biaxial slab. The raiations of a Hertzian ipole, locate in an isotropic or biaxiall anisotropic region for the halfspace an two-laer problems, have been thoroghl analze b Pettis [37]. To verif with the reslts in [37], raiation patterns for the following for ifferent cases are given. First, let Region an Region as shown in Fig. 4- are the same meim; then, the twolaer problem reces to a half-space problem. The raiation patterns of the Hertzian ipole when it is locate in the isotropic region an biaxial region are shown in Fig. 4-7 an Fig. 4-8, respectivel. E for z-irecte ipole x = x =5 x =9 h Fig. 4-7: E vs. in the 45 o plane for a z-irecte Hertzian ipole in Region (free space with h. an x,,3, x =, 5, 9.

144 h =.5 h =.5 h =.5 h 7 9 Fig. 4-8: E vs. h in the 45 o plane for a z-irecte Hertzian ipole in Region with o o o (,, (,8,4, (,, (3,,. x z Now let Region be an isotropic meim, then the raiate fiel can be calclate sing the formla erive in Section 4., an the reslts are isplae in Fig. 4-9 an Fig. 4-, respectivel. E for z-irecte ipole h =. h = h =.3 h Fig. 4-9: E vs. in the 9 o plane for varios vales of h for a z-irecte Hertzian ipole positione in region of a two-laer problem. Region is biaxial with (,, (,8,4, (,, (,, an slab thickness is.4. x z = o =.5 o =45 o =67.5 o h 7 9 Fig. 4-: E vs. in the o plane for varios vales of (rotation angle an an x- irecte Hertzian ipole positione in Region of a two-laer problem. Region is biaxial with o o o (,, (,8,4, (,, (,7, an with h.3,.4. x z

145 Fig. 4-9 is consistent with the reslt of Fig shown in [37]. Fig. 4- is consistent with reslt of Fig shown in [37]. As shown in Fig. 4-9, the raiation pattern of a Hertzian ipole is smmetric when the ipole is place in the isotropic region, while it is not when the ipole is insie the biaxial slab. The nmerical reslts valiate the se of the Green s fnction obtaine in Chapter 3. Since the prpose here is for valiation onl, etaile iscssion abot the biaxial anisotropic effect on the raiation of a Hertzian ipole is omitte here The Anisotropic Region Fille with Groelectric an Gromagnetic Meia Now let s consier the raiation of a Hertzian ipole in the presence of a non-reciprocal slab fille with grotropic meia. Fig. 4-: Geometr of the grone groelectric slab in [46]. A. Groelectric Meim The first case shown here is for a nmerical verification with the work of W [46]. The ipole is place in the isotropic region over a grone groelectric slab as shown in Fig. 4-. The nmericall obtaine raiation patterns of the ipole over a grone slab of ifferent thickness are plotte in Fig. 4-. The istance of the ipole awa from the interface between Region an Region is h.55 an the thickness of the slab is chosen as.5,.3 an.55. The relative freqenc for the groelectric meim is chosen as b., p.6.

146 (a Fig. 4-: E vs. in the plane of o for a z-irecte Hertzian ipole locate in Region, with h.55 awa from the interface of free space an groelectric slab. (a Reslt from W [46] an (b Raiate fiel sing the formla in this chapter z-irecte ipole inc = o slab=.5 slab=.3 slab=.55 free-space ipole 3 (b Fig. 4-(a isplas the nmerical reslts obtaine from [46], while Fig. 4-(b isplas the raiation pattern calclate sing the formlas erive in this chapter. As observe in Fig. 4-, ver goo agreement is obtaine between or reslts an the reslts from [46]. B. Gromagnetic Meim The secon case will establish consistenc between the nmerical reslts obtaine here an these of Tsalamengas an Uznogl [54]. The geometr of the problem is shown in Fig The ipole is locate in Region along the vector x z at a istance of 6cm from the ferrite slab whose thickness is 5mm an the raiate fiel in Region is of interest. Region is fille with a ferrite meim, which is electricall isotropic with r 4.8 an magneticall grotropic 9 with.8 ra / s, 9 m. ra / s. The incient freqenc is GHz. Fig. 4-3: Geometr of a two-laer problem fille with gromagnetic meim.

147 4 Normalize raiation patterns of E an for the XZ-plane are shown in Fig. 4-4 (a, (b, while normalize raiation patterns of E an for the YZ-plane are shown in Fig. 4-4(c, (. The raiation patterns obtaine from [54] are isplae in Fig. 4-4(a-(. E E E for x-irecte ipole = o =4 o =6 o =8 o (a (a E for x-irecte ipole = o =4 o =6 o =8 o (b (b E for x-irecte ipole = o =4 o =6 o =8 o (c (c E for x-irecte ipole = o =4 o =6 o =8 o ( ( o o Fig. 4-4: E an E vs. in the plane of, 9 with the sorce in Region an the fiel in Region. Each sbplot shows for crves, corresponing to the raiate fiels of an x-irecte ipole on the laere ferrite slab with for istinct orientations of the biasing magnetic fiel: o o o o o o o o,,5, 4,5, 6,5, 8,5. It is seen that the raiation patterns of

148 5 the transmitte fiel for Region obtaine sing the E-DGF evelope here (Eq. (4.- give ver goo agreement with what is obtaine in [54]. 4.4 Raiation of a Hertzian Dipole in the Presence of Groelectric Meim In this section, we will investigate the raiation of a Hertzian ipole in the presence of a groelectric meim. First, the raiation of a Hertzian ipole above a half-space groelectric meim is presente in Section Then the raiations of a Hertzian ipole when it is above an insie a two-laer groelectric slab are presente in Sections 4.4. an 4.4.3, respectivel Raiation of a Vertical Dipole on Top of a Half-space Groelectric Meim Since the groelectric meim is ispersive, the meim propert is epenent on the freqenc of the incient wave. Accoring to the specific wave tpe that can propagate insie the groelectric meim, the whole freqenc is ivie into eight ifferent freqenc bans [7]. For convenience, the eighth freqenc bans are repeate in Table 4-. It is seen from Table 4- that Regions,, 3 an Region 8 alwas exist. If b p, Region 4 an Region 5 cannot exist. If, Region 6 an Region 7 cannot exist. b p Both characteristic waves propagate in Region an Region 6 an o not epen on the irection of propagation. In Region 3 an Region 7, the Tpe I wave alwas exists, while the Tpe II wave exists onl for certain propagation irections with respect to the biasing magnetic fiel. In Region, the Tpe I wave exists for all the propagation irections. In Region 4, tpe II wave exists for all the propagation irections, while in Region 8, the Tpe II wave exists onl for

149 6 certain propagation irections with respect to the biasing magnetic fiel. There is no propagation in Region 5. Freqenc Ban Wave Propagating Region Tpe I, Tpe II Alwas exist Region Tpe I Alwas exist Region 3 max( b, p Tpe I, Tpe II Alwas exist Region 4 max( b, 3 p Tpe II Exist onl if b p p Region 5 b 3 No propagation Exist onl if b Region 6 p b Tpe I, Tpe II Exist onl if b p p Region 7 3 min( b, p Tpe I, Tpe II Exist onl if b Region 8 min( b, 3 Tpe II Alwas exist b b b b p, b p, 3 p, 3 p, 4 4 Table 4-: Freqenc bans with ifferent propagating waves for a groelectric meim. The far fiel of a Hertzian ipole place on top of a half-space groelectric meim for ifferent incient freqencies is analze in this section sing the formla propose in the previos sections. The Hertzian ipole is z-oriente ( the interface is. In this case, the fiel expression reces to z ; an the istance from the ipole to

150 7 h Rhv ( kx, k v z Il ikr E( r i e 4 r v vz Rvv ( kx, k v z (4.4- where k k sin cos, k k sin sin x obs obs obs obs Spherical components of the far fiel are obtaine as follows. Il ikr E ( r i e Rhv ( obs, obs sinobs 4 r (4.4- Il ikr E i e Rvv ( obs, obs sinobs 4 r (4.4-3 Then, the normalize fiel pattern for the z-irecte ipole is obtaine as follows. Rvv ( obs, obs f (, sin max R (, obs obs obs vv obs obs Rhv ( obs, obs f (, sin max R (, obs obs obs hv obs obs (4.4-4 It is seen from Eq. (4.4-4 that when the z -oriente Hertzian ipole is place at the interface of free space an a groelectric meim, both co-polarize an cross-polarize fiel components exist if the cross-polarize reflection coefficient R (, is non-zero. It is also seen that hv obs obs the co-polarize fiel pattern f is the free space pattern sin mltiplie b the co-polarize reflection factor Rvv ( obs, obs, an the cross-polarize fiel pattern f is the free space pattern sin mltiplie b the cross-polarize reflection factor R (,. It nees to be hv obs obs note here that sch simple relations onl hol for a z-irecte ipole when it is locate at the interface with h. The raiation pattern of a z -irecte ipole is first stie for a half-space

151 8 groelectric meim with ifferent freqencies an a biasing magnetic fiel along ifferent orientations Parametric Effect of Freqenc an Biasing Magnetic Fiel A. Effect of Freqenc Since the groelectric meim is ispersive, the first parameter of interest is the freqenc of the incient wave. The following nmerical reslts an iscssions investigate the effect of the freqenc of the incient wave to the raiation with 9 p an b.9. Consiering a z-irecte Hertzian ipole in Region at the interface of the half-space groelectric meim, the raiation patterns E in the plane of o are isplae in Fig Fig. 4-5(a an (b correspon to the cases with the biasing magnetic fiel of a groelectric meim along z-irection an -irection, respectivel. It is seen in Fig. 4-5 that the main beam irection varies with freqenc of incience wave. For the case of a biasing magnetic fiel along the -irection, the main beam will become narrower with a sie lobe also appearing as the freqenc is increase. p z-irecte ipole =. =. =.4 z-irecte ipole =. =. = inc = o o (a Fig. 4-5: E vs. in the plane of o for a z-irecte Hertzian ipole locate in Region with h awa from the interface of free space an groelectric meim. The biasing magnetic fiel is along the (a z an (b irections. 9 7 inc = o =9 o 9 (b

152 9 B. Effect of the Biasing Magnetic Fiel Direction Then, the effect of the biasing magnetic fiel irection to the raiation pattern is stie. Fig. 4-6(a an (b show the co-polarize fiel pattern E an cross-polarize fiel pattern E in the XZ plane, while Fig. 4-6(c an ( show the co-polarize an cross-polarize fiel patterns E an in the YZ plane. In each sbplot of Fig. 4-6, there exist for crves E which correspon to the raiation patterns of a Hertzian ipole on the groelectric sbstrates with for ifferent biasing magnetic fiels. 3 E for z-irecte ipole B = o B =4 o B =6 o B =8 o 3 E for z-irecte ipole B = o B =4 o B =6 o B =8 o 7 9 (a 7 9 (b o E for z-irecte ipole E for z-irecte ipole B = o B =4 o B = o B =4 o B =6 o B =6 o.5 B =8 o.5 =8 o 7 9 (c 7 9 ( Fig. 4-6: Fiel pattern for a z-irecte Hertzian ipole locate in Region with h awa from the interface of free space an groelectric meim. The biasing magnetic fiel is along o o o o o, 4, 6, 8, an the freqenc of the incience wave is.. (a E an B o B = o (b E vs. in the plane of o ; (c E an ( E vs. in the plane of 9o. When the biasing magnetic fiel is in the XZ plane ( the co-polarize fiel patterns in the XZ plane ( o B o, it is seen in Fig. 4-6(a that are alwas smmetric for ifferent o o o o irections of, 4, 6, 8 ; frther, the main beam of the co-polarize fiel for the B

153 3 XZ plane is constant an aron 7 egrees. However, the co-polarize fiel pattern in the YZ plane ( 9 o is not smmetric, as shown in Fig. 4-6(c, an the irection of the main beam varies with B. As B increases, the cross-polarize fiel pattern in the XZ plane increases, as shown in Fig. 4-6(b, an the cross-polarize fiel pattern of YZ plane ecreases, as shown in Fig. 4-6( Relation of Directive Raiation with Total Internal Reflection As shown in the previos section that the irective raiation exists. In this section, the relation of the irective raiation with the total internal reflection associate with the characteristic waves of the groelectric meim is reveale. To be consistent with the iscssion of previos section, the plasma freqenc an grofreqenc for the groelectric meim are chosen as 9 p an b.9 p, sch that both Tpe I an Tpe II waves exist [7]. b b When the freqenc of the incient wave is in Region as., p 4 an the biasing magnetic fiel is along the z-irection, the relative permittivit an permeabilit of the groelectric meim are given b i.496i.485,.893 r r (4.4-5 The st of the raiation pattern starts with the analsis of the reflection coefficients as it is seen from Eq. (4.4-4 that the co-polarize an cross-polarize reflection factors of R (, an R (, are essential to the corresponing fiel patterns of f an vv obs obs hv obs obs f. Total internal reflection is one of the most interesting phenomena analze here.

154 3 It is known that when the wave is incient from ense meim pon less ense meim (assming both meia are isotropic, there exists one critical angle beon which total internal reflection occrs with the magnite of the reflection coefficient to be. If the wave is incient from an isotropic meim pon an anisotropic meim, there sall exist two critical angles since there are two tpes of waves in an anisotropic meim. To emonstrate the concept of the total internal reflection, the magnite of the wave vector srface for waves propagating in the isotropic meim an the groelectric meim characterize b Eq. (4.4-5 is plotte in Fig Fig. 4-7: 5 k for XZ plane in Region.5 Normalize wave vector srface of an isotropic meim an k -isotropic these for Tpe I an Tpe II waves of a groelectric meim vs. in the plane of k for XZ plane in Region o. k -tpe I wave k -tpe II wave k -isotropic k -tpe I wave k -tpe II wave It is note here that the magnites of the wave vector for each case have been renormalize to the wave nmber of the isotropic meim. It is observe from Fig. 4-7 that since the wave vector srfaces for both Tpe I an Tpe II waves are within that of the isotropic meim, the total internal reflection will exist. It is eas to etermine the critical angle at the interface of two isotropic meia sing k sin t c k i, where k i an kt are the wave nmbers of the incient an transmitte regions. However, nlike a wave vector srface of an isotropic meim, the magnite of the wave vector in an anisotropic meim varies along ifferent propagation

155 3 irections. Ths, it is important to get the magnites of the wave vectors for both two tpes of waves along the irection at which the total reflection occrs, enote as k I an k II. Assming total reflection occrs for a wave incient from an angle of,, it leas to the o transmitte wave of the specific tpe propagating along 9,.Then the angle t t in between the propagation irection of the transmitte wave an the irection of the biasing magnetic fiel, of the groelectric meim can be calclate b B B in in cos sin B cos ( in B (4.4-6 The wave nmbers k, k can be expresse in terms of the permittivit of the meim an angle an have been erive as follows. I II g sin // cos 4 k I g // g // k sin // cos sin // cos g sin // cos sin 4 cos 4 k II g // g // k sin // cos sin // cos sin 4 cos (4.4-7 (4.4-8 The tpical eqations ci k sin I k, sin II k cii i k i can be se to calclate the critical angles for Tpe I an Tpe II waves in a groelectric meim. It is note here that the above critical angles are erive base on the assmption that both Tpe I an Tpe II waves can exist in a groelectric meim. Actall it is known that since a groelectric meim is ispersive, one tpe of wave or both tpes of waves ma not exist in the groelectric meim epening on the freqenc of the incient wave. Eq. (4.4-4 inicates that the raiate fiel is closel relate with the reflection coefficients R (, an R (,, which are the vv obs obs hv obs obs

156 Magnite Phase 33 co-polarize an cross-polarize reflection coefficients when the incience angles are obs an Ths, the magnite an phase of the reflection coefficients of R (, an obs. vv obs obs R (, are plotte with respect to ifferent incience angles in Fig The hv obs obs relative permittivit of the groelectric meim is given b Eq. (4.4-5, with b b o., p, an the biasing magnetic fiel along z-irection B. 4.8 R hv R vv 8 9 R hv R vv.6 cii = o.4. =. inc = o ci =65 o Incience Angle/in egree -9-8 =. inc = o Incience Angle/in egree Fig. 4-8: Magnite an phase of reflection coefficients R, R vs. ifferent incience angles for an incience plane of o. vv hv It is observe from Fig. 4-8 that 65 o an o correspon to the critical angles ci of Tpe I an Tpe II characteristic waves, respectivel. It is known that for transverse waves with the propagation irection perpeniclar to the biasing magnetic fiel, the Tpe I wave correspons to an orinar characteristic wave with a linear polarization, while Tpe II wave is an extraorinar wave with an elliptical polarization. This is consistent with the phenomenon that onl co-polarize reflection coefficient exists at 65 o with the cross-polarize reflection cii coefficient to be, while at o both co-polarize an cross-polarize reflection coefficients exist with the phase ifference of 9 o. As expecte, a linearl polarize raiate fiel is

157 34 observe at 65o an an ellipticall polarize wave (circlar polarization is possible epening on the propert of the meim is observe at o. In aition, since the biasing magnetic fiel is along the z-irection, all of the co-polarize an cross-polarize reflection coefficients are inepenent of obs an smmetric with respect to obs as shown in Fig Ths, the normalize fiel pattern for a z-irecte ipole locate at the interface is also inepenent of the angle an smmetric with respect to the z-axis. z-irecte ipole z-irecte ipole E -gro E -free (a 7 9 = E. (b 9 = E -gro E -free 33 E (c -. ( (a E an (b E vs. in the plane of o for a z-irecte Hertzian ipole Fig. 4-9: locate in region with h awa from the interface of free space an groelectric meim with the irection of biasing magnetic fiel along B o. p 9, b.9 p,., b b 4 p. Renormalizing the fiel components of the raiate fiel of the Hertzian ipole for both half-space an free-space to the same maximm vale of max( E half gro, the D patterns of

158 35 E an E of o are shown in Fig. 4-9(a an (b, while the 3D patterns are shown in Fig. 4-9(c an (. It is seen from Fig. 4-9(a that the co-polarize fiel pattern is not smmetric an more irective compare to the raiation of the ipole in free space. Also, the cross-polarize pattern occrs as shown in Fig. 4-9(b. The co-polarize fiel pattern E has a maximm raiation angle aron 65 o, which correspons to the total internal reflection angle for the Tpe I wave. This can be verifie with the magnite of the reflection coefficient R as vv shown in Fig. 4-8(a. Meanwhile, the cross-polarize fiel pattern E has a nll aron 65 o, which correspons to the minimm of the magnite of the reflection coefficients of R hv as isplae in Fig. 4-8(a. Consier a z-oriente ipole place at the interface of the isotropic an groelectric meia o o with the biasing magnetic fiel along the -irection of 9, 9. The raiate fiel patterns for the XZ plane ( o an YZ plane ( 9 o are plotte in Fig. 4- an Fig. 4-, respectivel. B B 33 E-gro E-free 3.75 z-irecte ipole E-gro E-free (a Fig. 4-: (a E an (b E vs. in the plane of o for a z-irecte Hertzian ipole locate in Region with h awa from the interface of free space an a groelectric meim o o with. an the biasing magnetic fiel along 9, B B.5 9 (b Fig. 4-(a presents the co-polarize raiation pattern of E, an Fig. 4-(b isplas the cross-polarize raiation pattern of E. It is seen in Fig. 4-(a that when the biasing

159 36 magnetic fiel is along the -irection, the maxim raiation occrs at aron the angle of o o 5,. Also, it shows that the co-polarize raiation pattern E is no longer smmetric with respect to z-axis in the XZ plane. As expecte, for the observation plane of the XZ plane, which is perpeniclar to the biasing magnetic fiel, no-cross polarize fiel exits, which is obvios in Fig. 4-(b E-gro E-free E-gro E-free (a (b Fig. 4-: (a E an (b E vs. in the plane of 9 o for a z-irecte Hertzian ipole 7 9 in Region with h awa from the interface of half-space groelectric meim with the o o biasing magnetic fiel along 9, 9. B B However, as shown in Fig. 4-(a an (b, both the co-polarize an cross-polarize patterns are smmetric with respect to the z-axis in the YZ plane. This is becase when the observation plane is parallel to the irection of the biasing magnetic fiel, the reflection coefficients satisf the smmetric conition, which leas to smmetric raiation patterns. Also, it is observe in Fig. 4-(b that the maximm cross-polarize fiel is in the same orer as the copolarize fiel. The maximm raiation along the peak irections in both patterns are e to the total internal reflections of the Tpe I an Tpe II waves of a groelectric meim The Two-Laer Problem with a Sorce above the Anisotropic Region In realit, raiation of a ipole with a grone sbstrate is a more practical problem. Ths, the raiation of a Hertzian ipole locate over the grone groelectric slab is iscsse in this section. It is known from the image theor that the raiate fiel of a horizontal ipole

160 37 over a gron plane significantl reces with the ecrease of the istance between the ipole an the gron plane. For a horizontal ipole over a gron plane to raiate effectivel, a minimm istance of qarter wavelength awa from the gron is neee. In this case, the raiate fiel along the broasie irection will be oble the fiel of ipole in free space. Here we wish to show that even with an ltra thin grone groelectric slab, enhance raiation for a horizontal ipole over the slab is still possible. The following assmptions are mae for the analsis. The irection of the biasing magnetic fiel is along the z-irection with o B an the ipole is locate at the interface of h.the operating freqenc range of the groelectric meim is change b varing the plasma freqenc while keeping the freqenc of 9 the ipole as an the ratio of grofreqenc to the plasma freqenc as a constant of.5. b p Fig. 4-: h grone isotropic Freq Region Freq Region3 Freq Region4 nbone isotropic Normalize raiate fiel E for an x-irecte ipole locate at the interface of the grone groelectric slab an air in the xz-plane for ifferent freqenc regions of the groelectric meim. With the choice of the plasma freqenc as the mile freqenc point of each existing freqenc region as shown in Table 4-, the normalize co-polarize raiation patterns of a horizontal Hertzian ipole over a grone groelectric slab with thickness of.5 are plotte in Fig. 4- for the plane of o. Usall there exist 8 ifferent freqenc regions. However,

161 38 9 for a groelectric meim with the choice of,.5 here, Region 6 an Region 7 on t exist. Also, no enhance raiation is observe for the raiation patterns in Region, 5 an 8, ths the raiation patterns in these regions are not consiere. b p It is seen from Fig. 4- that when an x-irecte ipole is place.5 awa from the gron, the maximm magnite of the raiate fiel along the broasie irection (soli line is less than two thirs of the magnite of the raiation for a Hertzian ipole in free space (ash line. However, enhance raiation is still obtaine sing the groelectric slab when it is operate in freqenc Region 3 (otte line. Compare with the raiation of a Hertzian ipole over the gron plane in the absence of the slab (grone isotropic case, the maximm raiation power is increase b two times (which is aron 6B an the maximm raiation is aron 45 o. Fig. 4-3: 7 3 E-x-irecte ipole B = B =3 o B =6 o B =9 o free space Normalize raiate fiel patterns E for an x-irecte ipole locate at an airgroelectric meim interface of a grone slab as a fnction of ifferent biasing magnetic fiel irections. In aition to changing the operating freqenc region, aitional freeom of tning the biasing magnetic fiel for the groelectric meim helps to tne the irection of the maximm raiation. Parametric analsis of the effect of the biasing magnetic fiel for the raiation of an x- irecte ipole over the grone groelectric slab operating in freqenc Region 3 with

162 39 thickness of h.5 is shown in Fig It is seen that the irection an the magnite of the maximm raiation are epenent on the choice of the irection of the biasing magnetic fiel. Abot a times (6B increase in maximm raiation power relative to the raiation of a ipole in free space is obtaine along o when 6 o B. It is expecte that maximm raiation along the broasie irection is potentiall achievable with proper tning of the irection of the biasing magnetic fiel. It has been shown in this section that a horizontal ipole place over an ltrathin groelectric slab of.5 can still raiate effectivel with the maximm raiation aron two times (6B higher than the raiation of a ipole in free space. In the next section, focs will be on the raiation of the ipole embee insie the grone groelectric slab The Two-Laer Problem with a Sorce Embee in the Anisotropic Region It has been shown in [73-75] that the enhance raiate fiel along the broasie irection can be realize b placing a horizontal ipole insie a grone isotropic plasma slab with permittivit ver close to zero. In the previos work [73-75], the analsis of the irective raiation is consiere for a grone metamaterial slab base on the isotropic ispersive moel. Here, the anisotropic effect to the raiation of a Hertzian ipole introce b appling the biasing magnetic fiel is consiere. It is shown that tilizing the anisotropic effect, the broasie raiation can be obtaine for a vertical ipole that is oriente perpeniclar to the interface. In aition, it is emonstrate that appling the biasing magnetic fiel helps to rece the reqire thickness of the isotropic plasma slab for the enhance broasie raiation. This section is organize as follows. First, the raiation featres for a horizontal ipole insie a grone isotropic slab are stie an compare with the reslts available in [75]. Then the isotropic slab is extene to an anisotropic slab, an the effect of the anisotrop to the

163 4 raiation pattern of the ipole is stie. Finall, the raiation of a vertical ipole insie an anisotropic slab is stie Raiation of a Horizontal Dipole insie a Grone Isotropic Plasma Slab h Air Groelectric Fig. 4-4: istance h Geometr of a horizontal ipole locate insie a grone groelectric slab with a awa from the air-slab interface. The geometr of interest is shown in Fig A horizontal ipole is locate at the mile of a grone groelectric slab with the orientation of the ipole esignate as. The ipole is h awa from the air-slab interface, an the irection of the biasing magnetic fiel for the groelectric sbstrate is esignate with an B. The permittivit of the groelectric B meim with an arbitraril biasing magnetic fiel is provie in Eq. (.3-4. In the absence of the biasing magnetic fiel, which is eqivalent to the choice for the grofreqenc of b =, the groelectric meim reces to an isotropic meim. Setting the plasma freqenc being GHz an the operating freqenc of the sorce being. GHz as shown in Eq. (4.4-9, the ielectric constant of the groelectric meim is given in Eq. (4.4-. ra s (4.4-9 p 9 9 ra / s,. p. /.736 p (4.4-

164 4 The first case consiere here is where a horizontal ipole oriente along the -irection ( is embee at the center of the slab. Accoring to Baccarelliet et al. [73], there exists an optimize freqenc to obtain the maximm broasie raiation for a fixe thickness of the slab fille with the isotropic plasma. For the fixe freqenc of the ipole, the broasie raiation varies perioicall with the thickness of the slab. The observation istance is r 4, where is free space wavelength an efine as in Eq. (4.4-. / c c p, 654mm f f (4.4- It is known that the two general characteristic waves insie the isotropic plasma are h- polarize an v-polarize. When the ipole is locate insie the slab, the Tpe I wave is assigne as an h-polarize wave an the Tpe II wave is assigne as a v-polarize wave in the general expression of the raiate fiel. If the incience plane is an XZ-plane ( oriente along the -irection, the general expression reces to the following form. o an the ipole is iile E( r 4 r ikr X h X v c k v e eih X h X v c k v e eii h X h X v c k v e ei h X h X v c k e eii h ( I e Iv I zi ( zii II e II v II I ( zi I e I v II ( eii v zii v II ikzi h ikzii h ikzi h ikzii h (4.4- where the coefficients are given in Eq. (4.4-3 an Eq. ( X X X X X ( I R R, X ( I R R R X X X X ei v eii v ei v eii v e I h eii h ei h eii h (4.4-3 For the special case of an isotropic meim, it is straightforwar to erive that

165 P ( P ( (B (B 4 k c c c c, k k k k k (4.4-4 z I I II II z z z kz Then, the raiate fiel in Region can be expresse as ikr iile k z ikzi h ikzi h E( r X e X e h (4.4-5 eih ei h 4 r k z For the fiel along the broasie irection ( k, kz kz k an the fiel can be written as ikr ik h ik o iile ik h e e E ( e h ik 4 r ( e (4.4-6 It is eas to see that if h /, corresponing to the case of the ipole locate in the o mile of the slab, the perio of E( as fnction of the slab thickness is, as shown in Eq. (4.4-. The raiation power ensit along the broasie irection for ifferent slab thicknesses is shown in Fig. 4-5 with a soli line with slab free space with slab free space Fig. 4-5: slab/ slab thickness. (a slab/ Power ensit (B of (a component an (b component as a fnction of the (b

166 P ( (B 43 It is note here that to plot Fig. 4-5, the ipole is alwas place in the mile of the grone slab ( h / for ifferent slab thickness an the permittivit of the slab is characterize b Eq. (4.4-. It is seen that for an isotropic plasma slab no cross-polarize raiate fiel E exists. As shown in Fig. 4-5, the power ensit along the broasie irection for the ipole insie the slab has been increase b 4B compare with the raiation of a Hertzian ipole in free space (otte line in Fig It can also be observe that P ( is a strictl perioic fnction of the slab thickness an shows maxima when the thickness of the slab is an o integer mltiple of /. This reslt is consistent with the reslt obtaine in [73] with slab h =/4 free space slab/ Fig. 4-6: Power ensit (B of component as a fnction of the slab thickness. h.5 an f. f p. o If h /4, then it is eas to erive from Eq. (4.4-6 that the perio of E( as a fnction of the slab thickness is /. This is verifie b plotting the raiation power ensit along the broasie irection verss the thickness of the slab with h.5 in Fig It is observe that the maximm broasie raiation occrs when the thickness of the slab is an integer mltiple of /. The maximm raiation has been increase b 4B compare to the

167 44 raiation of a Hertzian ipole in free space, which is the same as the case of the ipole locate in the mile of the slab. Inspection of Fig. 4-5 an Fig. 4-6 shows that the location of the Hertzian ipole will not change the irectivit of the ipole when it is embee insie the slab. However, the perio of the slab thickness for the maximm raiation along the broasie irection is affecte b the location of the Hertzian ipole. This is e to the ifferent phase avance introce b the ipole location. This reslt obtaine above is consistent with the conclsion in [73]. It shol be note here that in [73] onl the line sorce is consiere. In the next section, the anisotropic effects introce b the biasing magnetic fiel for the raiation of horizontal an vertical ipoles are stie. It will be shown that with the proper choice of the grofreqenc an the thickness of the grone groelectric slab, irective emission along the broasie irection is achievable Raiation of a Horizontal Dipole insie the Grone Groelectric Slab In this section, the anisotropic effect introce e to the biasing magnetic fiel is stie for the raiation patterns of a horizontal ipole. The analsis starts with the minor anisotropic effect b setting.. Then, the off-iagonal elements are introce into the b permittivit matrix of the groelectric meim as follows. p p.736.8i.8i (4.4-7

168 P ( P ( (B (B 45 It is note here that e to the choice of extremel small., all the iagonal elements of the permittivit matrix appear to be same. The power ensit raiate b the nit crrent moment Hertzian ipole place in a grone slab is plotte in Fig b p with slab free space with slab free space slab/ (a slab/ (b Fig. 4-7: Power ensit of (a component an (b component raiate in broasie irection b a nit crrent moment -irecte Hertzian ipole place at the center of a grone groelectric slab as a fnction of the slab thickness with b.p, f. f p. It can be observe from Fig. 4-7 that e to the off-iagonal element, the significant o cross-polarize raiate component P ( has been introce an has maxima ever / o in a perioic wa verss the thickness of the slab. On the other han, P ( is not mch affecte an still behaves in the same wa as the raiation of the horizontal ipole in the grone isotropic plasma slab. If the grofreqenc b is frther increase, then the negative elements will be introce into the iagonal elements of the permittivit matrix. Letting b p, f. f p, the permittivit matrix of the groelectric meim is given b

169 P( (B Arg(E -Arg(E 46 p i 4.39i (4.4-8 The raiation power ensities of both an components along the broasie irection for the horizontal ipole in the mile of the grone slab with p shown in Eq. (4.4-8 are plotte in Fig P ( slab/ (a Fig. 4-8: (a Power ensit an (b phase ifference of P ( an P ( for a nit crrent moment -irecte Hertzian ipole place in the mile of a grone groelectric slab as a fnction of the slab thickness with b p, f. f p. P ( free space slab/ (b It is observe that the amplites of an components are eqal an overla each other an their phase ifference is 7 egrees. Ths, the raiate fiel along the broasie irection is now a circlar polarize wave for the special case of b p. Also, it is seen in Fig. 4-8 that the first maximm raiation along the broasie irection occrs when the slab thickness is.75 an maxima repeat when the thickness is increase b ever.55. This thickness is onl half of the optimm thickness to achieve the maximm raiation when the ipole is place insie an isotropic plasma slab. Compare to the maximm raiation of power ensit of the

170 47 Hertzian ipole in free space, shown as the straight line in Fig. 4-8(a, the raiation of the Hertzian ipole insie the slab has been increase b.5b for both the an components. It has been observe from Fig. 4-5 an Fig. 4-6 that in orer to achieve a high irectivit along the broasie irection for a horizontal ipole insie the grone isotropic plasma slab, the thickness of the slab is sall in the orer of free space wavelength at the operating freqenc. Consiering the case of the isotropic plasma slab with 9 f p Hz, f. f p, the optimm thickness for the slab to achieve the maximm raiation along the broasie irection is / c p mm. f (4.4-9 To reveal the negative impact of the thinner isotropic slab, the raiate fiels E verss angles in XZ plane are shown Fig. 4-9 for a -irecte ipole insie sch a grone isotropic plasma slab with ifferent thicknesses of.5,.5 an.. It is seen that when the thickness of the isotropic plasma slab is rece from the optimm thickness of. to.5, the maximm raiation irection maintains aron the broasie irection. However, the amplite of the maximm raiation is ecrease. When the thickness of the isotropic plasma slab is rece to.5, the raiation power ecreases continosl across the whole anglar range. Along the broasie irection, the raiation power is 5B lower than the raiation power of a Hertzian ipole in free space. Onl the component of the raiate fiel contribtes to the total raiate power since the horizontal ipole is oriente along the -irection which is perpeniclar to the observation plane.

171 E (B h=.5 h=.5 h=. free space Fig. 4-9: Normalize raiate fiel for a -irecte ipole locate in the mile of the isotropic plasma slab in the XZ-plane as a fnction of observation angle which is measre from the z-axis (in egree If the anisotropic groelectric slab with b p is se instea of an isotropic slab, a similar kin of behavior can be observe, as shown in Fig The raiation power ensities P ( an P ( for a groelectric slab of thicknesses.,.5, an.5 are shown in Fig. 4-3(a, (b an (c. The raiation power ensit of the ipole in the free space is given with a black otte line in each figre. As shown in Fig. 4-3(a, in aition to the enhance raiation along the broasie irection, extra three peaks of raiation occr at the angles of 58 o, 7 o, an 8 o. The same phenomenon can be observe for the slab thickness of.5, as shown in Fig. 4-3(b. However, when the thickness of the slab is rece to.5, no sch strong raiation peaks occr an more as shown in Fig. 4-3(c. Actall, when the thickness of the slab reces to.5, the horizontal ipole can no longer raiates effectivel since the image crrent of the horizontal ipole over PEC cancels ot each other. For all the observation angles in the XZ plane, the raiation power of a Hertzian ipole in sch a slab has alwas been lower than the

172 Power (in B Power (in B Power (in B P ( 49 Hertzian ipole in free space. However, it is a ifferent case if the ipole is verticall oriente (zoriente. Detaile iscssion is given in the next section. 5 4 h= h=.5 h= (in egree P P free space (a - P P P P -3 free free space space (b (in egree 4 3 h= P -3 P free space (in egree (c Fig. 4-3: Raiation patterns for a -oriente ipole locate in the mile of the anisotropic groelectric slab vs. ifferent slab thicknesses Raiation of a Vertical Dipole insie the Grone Groelectric Slab Isotropic h z Anisotropic Fig. 4-3: istance of h Geometr of a vertical ipole locate insie the grone anisotropic slab with a awa from the air-slab interface.

173 5 If the ipole insie the slab is oriente to be perpeniclar to the interface of the slab as shown in Fig. 4-3, then the thickness of the slab can be chosen ver thin an ver high irectivit can still be achieve along the broasie irection. The parametric st of the effects of freqenc, biasing magnetic fiel irection an the slab thickness is presente in this section to show how the irective raiation along the broasie irection will be achieve with a vertical ipole insie a grone groelectric slab. A. Effect of Freqenc Assming the gro-freqenc.8, the effect of the freqenc to the raiation b p featres of a vertical ipole insie the groelectric slab is first stie. Normalize fiel patterns E of a vertical ipole insie the grone groelectric slab with ifferent freqencies of.9 f p,.95 f p,.5 f p, an.f p are plotte with a soli line, ashe line, otte line, an a ash-ot mixe line in Fig f=.9f p f=.95f p f=.5f p.5 f=.f p 7 9 Fig. 4-3: Normalize fiel pattern of E vs. in the plane of o with vertical ipole locate in the mile of the grone groelectric slab of thickness of.5. It can be seen from the normalize fiel patterns of E that the beam is smmetric with respect to the z-axis, an the maximm raiation occrs at aron o for all for cases.

174 max (in egree P max -P max (free space in B 5 However, the raiations along 7 o for the freqencies of.95 f p an.5 f p are mch smaller than the raiation for the freqencies of.9 f p an.f p. To gain better nerstaning of the effect of the freqenc of the excitation, the maximm raiation irection an the irectivit with respect to the raiation of the ipole in free space vs. the excitation freqenc of the sorce are shown in Fig. 4-33(a an (b, respectivel. 8 5 P 7 P P 5 P Fig. 4-33: (a (a The angle of the maximm raiation of the vertical ipole insie the grone groelectric slab vs. ifferent freqencies. (b The ifference of the raiate power along the maximm raiation irection of the Hertzian ipole insie the slab an in free space vs. ifferent freqencies f/f p f/f p (b As shown in Fig. 4-33(a, the maximm raiation irection is aron o when f.9f p an ecreases to aron o when the freqenc increases to the plasma freqenc of f p. It is note here that the maximm raiation irection of o varies with the change of the biasing magnetic fiel. It is seen in Fig. 4-33(b that onl when the freqenc is above.9f p the maximm raiation power of the Hertzian ipole insie the slab is stronger than that of the ipole in free space.

175 5 Also, the maximm raiation power of both an components in the XZ observation plane increases when the freqenc of the Hertzian ipole approaches the plasma freqenc. When the excitation freqenc of the vertical Hertzian ipole ecreases or increases from the plasma freqenc, the ifference between the maximm raiation power e to an components ecreases as shown in Fig. 4-33(b. When the freqenc reces to the grofreqenc.8, the maximm raiation power for an components are the same b p thogh the maximm raiation of component occrs at aron 75 o an the maximm raiation power of component occrs at aron 35 o as shown in Fig. 4-33(a. For now, onl the raiation of a vertical Hertzian ipole insie a grone groelectric slab with a z-oriente biasing magnetic fiel is consiere. The effect of ifferent irections of the biasing magnetic fiel is presente in the following iscssion. B. Effect of Biasing Magnetic Fiel Direction Normalize fiel patterns for the ifferent biasing magnetic irections are shown in Fig When the biasing magnetic fiel is along the x-axis (e.g., 9,, which is parallel to the XZ plane, the normalize fiel pattern E is smmetric with respect to the z- axis. When the biasing magnetic fiel is perpeniclar to the observation plane (e.g., B B 9, 9, onl the co-polarize fiel component E exists an the cross-polarize B B fiel component E oes not exist, becase the two tpes of characteristic waves of the groelectric meim are ecople, one of the waves corresponing to the h-polarize wave in free space, an the other wave corresponing to the v-polarize wave in free space. It is note

176 53 here that in Fig. 4-34, all the fiel patterns are normalize to the maximm vale of both E an E E B =6 o B = o B =6 o B =3 o B =6 o B = 6 o B =6 o 6 o B =9 o 3 33 E B =6 o B = o B =6 o B = o B =6 o B = o B =6 o B = o.5 free space.5 B =6 o B = o 7 9 (a 7 9 (b E B =9 o B = o B =9 o B =3 o B =9 o B =6 o B =9 o B =9 o free space.5 Fig. 4-34: Normalize fiel pattern of E an E (xz plane for a vertical ipole locate in the mile of a grone groelectric slab of thickness.5 (free space wavelength with 9 ifferent irection of the biasing magnetic fiel.,.8,.. (c E p b p p ( 9 B =9 o B = o B =9 o B =3 o B =9 o B =6 o B =9 o B =9 o free space C. Effect of Slab Thickness Assming the biasing magnetic fiel is along the -axis (perpeniclar to the incience plane, the raiation patterns of a vertical ipole in the presence of the groelectric slab with ifferent thicknesses are shown in Fig It is known that since the biasing magnetic fiel is perpeniclar to the observation plane (XZ plane, the cross-polarize fiel component oes not exist ( E for both the grone groelectric slab an the grone isotropic slab. Ths, onl E is isplae here. It is seen from Fig. 4-35(a an (b that a z-oriente ipole can raiate along the broasie irection if it is locate insie a grone groelectric slab instea of a grone isotropic slab.

177 E =.5 =. =.5 free space (a Fig. 4-35: Normalize fiel pattern of E (XZ plane for the vertical ipole locate in the mile of the grone groelectric slab with the biasing magnetic fiel perpeniclar to the 9 observation plane as a fnction of ifferent thickness.,.8, p b p p =.5 =.5 = free space (b For the prpose of comparison, the normalize fiel patterns of E (XZ plane for a vertical ipole locate in the mile of a grone isotropic slab with ifferent thicknesses are shown in Fig =.5 = = free space 7 9 Fig. 4-36: Normalize fiel pattern of E (XZ plane for a vertical ipole locate in the mile of a grone isotropic slab of vs. ifferent slab thickness. With the increase of the thickness of the slab, the irection of the maximm raiation shifts from o to the broasie irection, an the maximm raiation power also increases. The optimm height to raiate along the broasie irection is aron one.5, the raiate fiel along the broasie irection is now almost 8.5 B higher than the raiation of the ipole in free space. This height is mch less than the optimm height of. for the enhance raiation of a horizontal ipole insie a grone isotropic slab with the same choice of the plasma freqenc. When the thickness of the slab increases beon one qarter of free space wavelength, the

178 55 maximm raiation irection again shifts awa from the broasie irection. It is to be note here that the vertical ipole can raiate effectivel along the broasie irection onl for the biasing magnetic fiel perpeniclar to the slab. When the biasing magnetic fiel is along the z- axis, no broasie raiation occrs. In this chapter, the raiate fiel of an arbitraril oriente Hertzian ipole locate either above or insie a laere anisotropic meim is obtaine b appling the metho of stationar phase to the corresponing E-DGFs. Nmerical analsis for the raiation of an elementar ipole is presente for three ifferent cases incling the ipole locate over a half-space groelectric meim, above a grone laere groelectric slab, an immerse insie the slab. The raiation for a vertical ipole on top of a half-space groelectric meim is first presente. Throgh the analsis of reflection coefficients as a fnction of incience angle, it is reveale that the maximm raiation irection for a vertical ipole on top of a half-space groelectric meim is closel relate to the critical angles of the characteristic waves of the groelectric meim. The raiation of a Hertzian ipole in the presence of a grone groelectric slab is frther analze. The analsis inicates that a grone groelectric slab can be se to achieve the irective raiation sing two ifferent mechanisms. One is throgh the reflection b placing the ipole over the slab, an the other is throgh the transmission b placing the ipole insie the slab. When a horizontal ipole is place over a grone groelectric slab, the parametric st of the effect of the freqenc range an the biasing magnetic fiel of the groelectric meim inicates that the irective raiation is still achievable even for an ltra-thin slab. A maximm 6 B (twice the raiate fiel of a ipole in free space increase in the raiation is observe for a

179 56 ipole place above the groelectric slab of.5, when the irection of the biasing magnetic fiel is 6 o an the freqenc is operate in Freqenc Region 3. The maximm raiation irection for the ipole over a groelectric slab can be frther change throgh the ajstment of the biasing magnetic fiel. Even higher raiation is possible b placing a Hertzian ipole insie a groelectric slab an changing its orientation from horizontal to vertical. In particlar, almost 8.5 B enhance raiation along the broasie irection, compare to the raiation of the ipole in free space, is obtaine with the proper ajstment of the thickness of the slab an the magnite an irection of the biasing magnetic fiel. Withot the biasing magnetic fiel, the groelectric meim reces to isotropic plasma, an the optimm thickness reqire to achieve the irective emission along the broasie irection with a horizontal ipole insie the slab is significantl larger. This analsis ma lea to a metho whereb the volme of the raiator can be rece simpl b changing the orientation of the magnetic fiel an sing a z-irecte raiator. This size rection ma make it possible to create a miniatrize antenna, which is the goal of most antenna manfactrers.

180 5 RADIATION OF A MICROSTRIP DIPOLE PRINTED ON AN ANISOTROPIC SUBSTRATE 57 Crrentl, there is an increasing interest in complete monolithic sstems which combine antenna elements or antenna arras on the same sbstrate as the integrate RF/IF front en network. One tpe of the most poplar antenna elements is the printe antenna e to its characteristics of low-cost, low-profile, conformabilit, an ease of manfactring. The literatre srve from Chapter shows that existing works mainl st printe antennas on an isotropic sbstrate [57-59, 63] or on a specific tpe of anisotropic sbstrates sch as niaxial meim [4], ferrite meim [8], an biaxial meim [37, 65]. To emonstrate the general feasibilit an valiit of the eigenvector aic Green s fnctions evelope in previos chapters, we appl the evelope E-DGFs with the metho of moments (MOM to solve a microstrip ipole printe on a general anisotropic sbstrate in this chapter. Particlarl, previos literatre inicates that there exist few reslts on the application of a groelectric meim to the printe antenna. To fill the gap, etaile analsis on the raiation behavior of a printe ipole on a groelectric sbstrate will be presente tilizing the methos evelope in this chapter. This chapter is organize as follows. First the formlation of the metho of moment is presente. Since the DGF iscsse in the previos chapter is obtaine sing the eigenecomposition metho, an it applies to the general anisotropic meim with no restrictions impose on the propert of the meim, the formlation of MOM is applicable to a printe ipole on a general anisotropic sbstrate. In the secon section, the nmerical reslts of a printe ipole over ifferent grone sbstrates are presente for the prpose of valiation. The inpt

181 58 impeance, resonant length, an raiation patterns of a microstrip ipole over a grone isotropic slab, a grone biaxial slab, an a grone ferrite slab are calclate an compare with the previos research. As will be mae clear, the comparisons sggest that the crrent st is in agreement with previos research. In the thir section, nmerical reslts an iscssions will be presente in etail to illstrate the effect of the magnite an irection of the biasing magnetic fiel to the crrent istribtion, inpt impeance, resonant length, an raiation pattern of a microstrip ipole on a groelectric sbstrate. It will be shown that tnable resonant length an raiation patterns of a printe ipole are achievable to a certain egree b changing the material properties throgh ajsting the biasing magnetic fiel of the groelectric meim. 5. Formlation of Metho of Moment (MOM Region Region,, r z= z= Fig. 5-: Geometr of the microstrip ipole problem. Geometr of a microstrip ipole is shown in Fig. 5-. The ipole is printe at the interface of an isotropic meim an an anisotropic meim. It is assme that the geometr extens laterall to infinite in both regions. The length of the ipole antenna is enote as L, an the ipole is assme to be oriente along the x-axis. The with of the antenna is W, an it is measre along the -axis. Frthermore, it is also assme that the with of the antenna is mch

182 59 less than the wavelength. Hence, transverse crrents are neglecte, an crrent is assme to flow onl in the x-irection. Metho of moment formlation for the microstrip ipole is presente in this section. The formlation presente in this section is similar to the approach in [59], which analze the raiation properties of the microstrip ipoles tilizing the Greens fnctions in the spectral form b the methos of moment. 5.. Basis Fnction If the with of the ipole is a small fraction of wavelength, the crrent istribtion along the x-axis is assme. It is also assme that the crrent is separable in its x an epenence. J ( x, J ( x J (, n=x or (5.- n n n For a thin ipole, crrent is floating exclsivel in the x-irection an ipole crrent is J ( x, J ( x J ( (5.- x x x To approximate arbitrar crrent istribtions, sbomain basis fnctions instea of entire omain basis fnctions are se here as the crrent basis fnctions in the longitinal irection. Particlarl, trianglar basis fnctions instea of sinsoial expansion fnction [59] are se. a Jx( x a, x xp a xp a x xp xp a x, xp x xp a (5.-3 where a L / N, L is the length of the ipole, N is the nmber of sbivisions se along x- irection, x p is the x-coorinate of the center of the basis fnction. For the analsis along the length of the ipole, the Forier transform of the trianglar basis fnction is given as

183 6 ik xxp J x( kx ae sin kxa kxa (5.-4 For the transverse crrent istribtion, the plse fnction is se as the basis fnction. M W, p Jx( W M, else (5.-5 where p is the -coorinate of the center of the basis fnction, an W is the with of the ipole. M is the nmber of sbivisions se along -irection. The Forier transform of the transverse crrent istribtion is as follows. M ik W k p J ( sin,, x k e k J x( k Wk M (5.-6 In the following analsis, it is assme that an arbitrar nmber of plses are presente in the transverse irection. However, for exceeingl narrow microstrip ipole, the with of the ipole is sall eqal to the with of the basis fnction. This implies that onl a single transverse plse is se for the analsis. We orer the location of the centers of the basis fnctions in the following convention. Total nmber of basis fnction is MN.,... m N M. Fig. 5-: Locations of the sbomain basis fnctions Jx( x an J ( along x-irection. x