《光波导理论与技术 Optical Waveguides Principles and Technologies》课程教学资源（参考文献）Coupling Characteristics of Two Rectangular Dielectric Waveguides Laid in Different Layers

IEEE JOURNAL OF QUANTUM ELECTRONICS.VOL.QE-22,NO.6.JUNE 1986 959 Coupling Characteristics of Two Rectangular Dielectric Waveguides Laid in Different Layers KAZUHITO MATSUMURA,MEMBER,IEEE,AND YOSHIRO TOMABECHI,MEMBER,IEEE Abstract-The coupling characteristic of rectangular dielectric waveguides which lay in different,parallel layers and one waveguide crossed over the other one are studied.Step-like approximation and the variational method are used for calculating the coupling coefficient which varies with axial distance z.Electromagnetic field expression for the dielectric waveguide mode of rectangular cross section is newly in- troduced and used.Calculated coupling characteristics are compared OWG-F with experimental results carried on 50 GHz band.The principle of crossed waveguides leads us to the new design concept of"multilay- ered integrated circuit." 1.INTRODUCTION TN recent years much research has been directed towards Fig.I.Coupled rectangular dielectric waveguide and their coordinate sys- Ithe use of dielectric waveguide (DWG)for the milli- tems. meter,submillimeter and optical wave transmission.To realize millimeter through optical wavelength integrated circuits,several ideas for the waveguiding mechanism and tems.Each guide lays in different layers (parallel to x-z circuit component have been proposed [1]-[4]. plane)and are separated by a distance d in the y-direction. The projection of the guides axes in the x-z plane makes The coupling mechanism of dielectric waveguides has been analyzed [5]and the results applied to the design of small angle 0.This dielectric waveguide structure is a typ- directional couplers [6],[7].Relating to these applica- ical model of circuit elements or waveguide components in the multilayered integrated circuit.In this paper,we tions,couplings of nonparallel dielectric waveguide have also discussed [8],[9]. assume that the each dielectric waveguide supports only one fundamental mode. In most of the previous work,the axes of the coupled dielectric waveguides are coplanar. In this paper,we treat the coupling mechanism of rect- B.Power Transfer Between Two Waveguides Coupled angular dielectric waveguides which lay in the different, to Each Other with Varying Coupling Coefficient parallel,layers.One waveguide crosses over the other. To analyze the power transfer phenomena between This crossed waveguide will conduct us to the new design crossed-over guides,we consider first that the coupling concept of"'multilayered integrated circuit.'' coefficient C varies with axial distance z.In the coupled An electromagnetic field expression for the guided dielectric waveguide system,let us assume that the trans- mode along the rectangular dielectric waveguide is intro- mitted mode in each guide have the same propagation duced [10]and used to analyze the coupling mechanism. constant.This is true if the guides are separated enough Experimental studies are carried out on the 50 GHz band. With the mode A launched on the dielectric wave- Measurements of coupled powers between two guides are guide-1(DWG-1)at z=zo,it is required to find the power also shown and described. coupled to the mode 42 in the DWG-2 as a function of z. Coupling phenomenon between the two forward mode II.COUPLING CHARACTERISTICS OF CROSSED OVER A1(z)and A2(z)is,as well known,described approxi- DIELECTRIC WAVEGUIDES mately by the coupled wave equations [11]. A.Structure of the Coupling Dielectric Waveguides Fig.I shows the crossed over rectangular dielectric waveguide (RDWG)structure and their coordinate sys- dA =-jBA()+C(A() d Manuscript received September 24.1985;revised January 28.1986. dA②=C)A(a-jBA2 (1) The authors are with the Faculty of Engineering,Utsunomiya Univer- dz sity.Utunomiya 321.Japan. IEEE Log Number 8608179. where A(z)and A2(z)are the amplitudes of the guided 0018-9197/86/0600-0959Ss01.00©1986IEEE

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-22, NO. 6, JUNE 1986 959 Coupling Characteristics of Two Rectangular Dielectric Waveguides Laid in Different Layers Abstract-The coupling characteristic of rectangular dielectric waveguides which lay in different, parallel layers and one waveguide crossed over the other one are studied. Step-like approximation and the variational method are used for calculating the coupling coefficient which varies with axial distance z. Electromagnetic field expression for the dielectric waveguide mode of rectangular cross section is newly in￾troduced and used. Calculated coupling characteristics are compared with experimental, results carried on 50 GHz band. The principle of crossed waveguides leads us to the new design concept of “multilay￾ered integrated circuit.” I I. INTRODUCTION N recent years much research has been directed towards the use of dielectric waveguide (DWG) for the milli￾meter, submillimeter and optical wave transmission. To realize millimeter through optical wavelength integrated circuits, several ideas for the waveguiding mechanism and circuit component have been proposed [ 11-[4]. The coupling mechanism of dielectric waveguides has been analyzed [5] and the results applied to the design of directional couplers [6], [7]. Relating to these applica￾tions, couplings of nonparallel dielectric waveguide have also discussed [8], [9]. In most of the previous work, the axes of the coupled dielectric waveguides are coplanar. In this paper, we treat the coupling mechanism of rect￾angular dielectric waveguides which lay in the different, parallel, layers. One waveguide crosses over the other. This crossed waveguide will conduct us to the new design concept of “multilayered integrated circuit. ” An electromagnetic field expression for the guided mode along the rectangular dielectric waveguide is intro￾duced [lo] and used to analyze the coupling mechanism. Experimental studies are carried out on the 50 GHz band. Measurements of coupled powers between two guides are also shown and described. 11. COUPLING CHARACTERISTICS OFROSSED OVER DIELECTRIC WAVEGUIDES A. Structure of the Coupling Dielectric Waveguides Fig. 1 shows the crossed over rectangular dielectric waveguide (RDWG) structure and their coordinate sys￾Manuscript received September 24, 1985; revised January 28, 1986. The authors are with the Faculty of Engineering, Utsunomiya Univer￾IEEE Log Number 8608 179. sity, Utunomiya 32 1, Japan. Fig. 1. Coupled rectangular dielectric waveguide and their coordinate sys￾tems. tems. Each guide lays in different layers (parallel to x-z plane) and are separated by a distance d in the y-direction. The projection of the guides axes in the x-z plane makes small angle 6. This dielectric waveguide structure is a typ￾ical model of circuit elements or waveguide components in the multilayered integrated circuit. In this paper, we assume that the each dielectric waveguide supports only one fundamental mode. B. Power Transfer Between Two Waveguides Coupled to Each Other with Varying Coupling Coeficient To analyze the power transfer phenomena between crossed-over guides, we consider first that the coupling coefficient C varies with axial distance z. In the coupled dielectric waveguide system, let us assume that the trans￾mitted mode in each guide have the same propagation constant. This is true if the guides are separated enough. With the mode AI launched on the dielectric wave￾guide-1 (DWG-1) at z = zo, it is required to find the power coupled to the mode A, in the DWG-2 as a function of z. Coupling phenomenon between the two forward mode A,(z) and A2(z) is, as well known, described approxi￾mately by the coupled wave equations [ 111. where A ,(z) and A2(z) are the amplitudes of the guided 0018-919718610600-0959$01 .OO O 1986 IEEE

960 IEEE JOURNAL OF QUANTUM ELECTRONICS,VOL.QE-22.NO.6.JUNE 1986 modes in the DWG-I and-2,respectively.C(z)is a cou- pling coefficient. Solving these equations with initial conditions WG- A1(zo)=1, A2(20）=0, we obtain A1(z)=cos (2) In a lossless system,mode power P(z)and P2(z)are reduced to Pi()=A1(z)Ai(2)=cos (Gcon P2(z)=A2(z)A(z)=sin (3) Fig.2.Step-like approximation for the coupled dielectric waveguide sys- tem. From these equations,it is clear that the power distri- bution along the guides depends both on coupling coeffi- cient C(z)and coupling length. R(z) DWG-I C.Coupling Coefficient C(z)of the Waveguide System To determine the coupling coefficient C(z)of the DWG-1 crossed over dielectric waveguide system(Fig.1),step- like approximation is introduced.The DWG-2,in Fig.2, is assumed to be constructed by large number of wave- guide segments.Each of the segment is very short length (AZ),and is separated /(z)from the DWG-1 in the pro- Fig.3. Cross section of the approximated waveguide system. jection plane (x-z plane).Let us introduce an imaginary dielectric waveguide DWG-i at the location of this seg- ment which is parallel with the DWG-1,and consider the component of e(x,y)and en(r,y)with longitudinal coupling coefficient between DWG-1 and the imaginary phase constant B and B2,respectively.When the guide guide DWG-i is identical to C(z)at this location. separation [d and/or /(z)]becomes small,coupled trans- From the above considerations,the problem of finding mission modes arise in the system. C(z)is reduced to determining the coupling coefficient C The transverse field component of the coupled mode P between two rectangular parallel dielectric waveguides. is assumed to be given by a linear combination of each Fig.3 shows a cross section of the parallel waveguides guided mode component [5] system. er=men mzen (4) By calculating the C value,step by step,for every waveguide segment separated /(z)in the projection plane, where m and m2 are constants which are determined un- the coupling coefficient C(z)along the z-axis should be der the condition that the phase constant of this coupled obtained. mode takes a stationary value.We substitute (4)into a variational expression [12]of the phase constant ce, -2e ds 82 (5) x e)2-wee?ds The coupling coefficient C between the DWG-I and the By differentiating with respect to m:and m2,two equa- DWG-i is proportional to a difference of phase constant tions d82/dm=0 and d82/dm2=0 permit the evaluation of the coupled transmission modes (symmetric and anti- of the ratio m/mz that substituted in (5)yields the sta- symmetric modes)on the system. tionary value 82[5].In the case that the DWG-I and-2 If the two guides are separated enough,each guide sup- have same physical dimensions and dielectric constant, ports only one mode.The modes have transverse field m/mz should be reduced to +1.Two values of phase con-

960 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-22, NO. 6, JUNE 1986 modes in the DWG-1 and -2, respectively. C(z) is a COU￾pling coefficient. Solving these equations with initial conditions A,(zo) = 1, MZO) = 0, we obtain In a lossless system, mode power Pl(z) and P2(z) are reduced to / PZ \ From these equations, it is clear that the power distri￾bution along the guides depends both on coupling coeffi￾cient C(z) and coupling length. C. Coupling Coeflcient C(z) of the Waveguide System To determine the coupling coefficient C(z) of the crossed over dielectric waveguide system (Fig. 1), step￾like approximation is introduced. The DWG-2, in Fig. 2, is assumed to be constructed by large number of wave￾guide segments. Each of the segment is very short length (AZ), and is separated Z(z) from the DWG-1 in the pro￾jection plane (x-z plane). Let us introduce an imaginary dielectric waveguide DWG-i at the location of this seg￾ment which is parallel with the DWG- 1 , and consider the coupling coefficient between DWG-1 and the imaginary guide DWG-i is identical to C(z) at this location. From the above considerations, the problem of finding C(z) is reduced to determining the coupling coefficient C between two rectangular parallel dielectric waveguides. Fig. 3 shows a cross section of the parallel waveguides system. By calculating the C value, step by step, for every waveguide segment separated Z(z) in the projection plane, the coupling coefficient C(z) along the z-axis should be obtained. Fig. 2. Step-like approximation for the coupled dielectric waveguide sys￾tem. La-I I Fig. 3. Cross section of the approximated waveguide system. component of erl(x, y) and e&, y) with longitudinal phase constant PI and P2, respectively. When the guide separation [d and/or Z(z)] becomes small, coupled trans￾mission modes arise in the system. The transverse field component of the coupled mode Pt is assumed to be given by a linear 'combination of each guided mode component [5] e, = mleIl + m2er2 (4) where ml and m2 are constants which are determined un￾der the condition that the phase constant of this coupled mode takes a stationary value. We substitute (4) into a variational expression [12] of the phase constant - w Eel - w - (v . cer)2] ds E 21 The coupling coefficient C between the DWG- 1 and the DWG-i is proportional to a difference of phase constant of the coupled transmission modes (symmetric and anti￾symmetric modes) on the system. If the two guides are separated enough, each guide sup￾ports only one mode. The modes have transverse field By differentiating with respect to ml and m2, two equa￾tions dP2/drnl = 0 and dp2/dm2 = 0 permit the evaluation of the ratio m,/m2 that substituted in (5) yields the sta￾tionary value [5]. In the case that the DWG-1 and -2 have same physical dimensions and dielectric constant, ml/m2 should be reduced to + 1. Two values of phase con-

MATSUMURA AND TOMABECHI:TWO RECTANGULAR DIELECTRIC WAVEGUIDES 961 20 个 11 22 17 12 23 18 13 d-2.2 mm 77777☑ 24 19 14 10 25 20 1151 10 5 Fig.4.25 subregions for calculation. stant for the coupled transmission modes are determined and named here B,and B_.Then,the couling coefficient 850 250 C of this coupled dielectric waveguide system(Fig.3)is Fig.5.Examples of calculated coupling coefficient C(z) C=94-B sions;DWG-1 and-2 have relative dielectric constant e, 2 (6) =2.01,cross section a×b=4×4 mm and step dis- where tance△z=0.75mm. 8=-1干N2 Examples of calculated coupling coefficient C(z)are (7) shown in Fig.5.From these results,it is evident that, D1土D2 near crossing center,the C(z)increases rapidly.In these examples,crossing angle 6 is assumed to be small because Dg=J【,×ea,×e-ueaeds of using the step-like approximation. (8) Variations of the transfer power P2(z)which calculated from (3)by numerical integration,are shown in Fig.6(a)- N=∫，×7，×ea-ea) (g).The initial conditions are taken to be A1(2o)=1,A2(2o)=0 (10) (V:x V:x er w'uee) at a point zo =-450 mm. -w2uee(V,·ea),·e】d i,j=1,2. III.EXPERIMENTAL RESULTS AND DISCUSSIONS (9) Variation of field intensity along the coupled dielectric To find the coupling coefficient C(z),it is necessary to waveguide is measured on the 50 GHz experimental determine the transverse field expressions e,er2 and model.The rectangular dielectric waveguide is made from phase constants B,B2 of the rectangular dielectric wave- tetrafluoroethylene plastics (e,=2.01)and has dimen- guides.But,unfortunately,rigorous expressions for the sions,as same as the theoretical calculation,of 4 x 4 mm transverse field distribution and phase constant of guided in cross section,900 mm in length. modes have not been found.Here,in this paper,we apply The experimental set up is shown in Fig.7.The DWG- an approximate field expression resulting from the crossed 2 is mounted on the rotatable table which makes us the dielectric slab method (CDSM)[10].The CDSM field change of 6 easy.The waveguides are separated by air. expressions are given in the appendix.For the calculation Field intensity (E,)on the DWG-2(in correspond to the of phase constant of the guided mode,the generalized ef-transfer power P2)is picked up by a mono-pole antenna fective dielectric constant method(GEDCM)[7]is ap-which can be moved along the z'-axis smoothly.Mea- plied with some modifications. sured results are shown also in Fig.6(a)-(g). From these results,we can see that the measured power D.Numerical Calculations transfer to an adjacent guide has somewhat slower vari- The coupling coefficient and transfer power between the ation with the z'axis than that of the theoretical calcula- rectangular dielectric waveguide-1 and-2,crossed over tion,particularly for the case that the crossing angle a each other,are calculated. and/or guide separation d are small.The result seems to At first,the coupling coefficient C(6)should be cal-come from the field distribution of the coupled DWG sys- culated as a function of axial distance z.The Dyj,Niy in tem.When the DWG-1 and-2 have a same physical di- (8)and(9)are obtained by numerical integration taking mensions,the ratio of m/m2 takes +1 as described in into account the e and ey;are function of z.As the trans- Section II-D.Then,we assume in this analysis the field verse electromagnetic field around the guide is expressed distribution of the coupled guide system may be expressed separately in nine divided regions (as shown in Fig.8),by a linear combination of the field distribution of each the above integrations [(8)and(9)]are performed on every guides [see (4)].But,when the guides approach each 25 subregions.The subregions are shown in Fig.4. other,the coupled field should not be expressed at all by In these calculations,we take the following dimen- the linear combination of each guide's field.This is an

MATSUMURA AND TOMABECHI: TWO RECTANGULAR DIELECTRIC WAVEGUIDES 96 1 I II I stant for the coupled transmission modes are determined and named here 0, and 6-. Then, the coding coefficient C of this coupled dielectric waveguide system (Fig. 3) is c= IP+ - P-l 2 where D.. = lJ S [(v, X e,;) - (v, X etj) - u2pceti * eti] ds (8) N.. V = S [(v, X V, X eri - w2pce,,) (v~ X V, X ed - u2pced) - w2pce(V, e,,)(V, * e,.)] ds i, j = 1, 2. (9) To find the coupling coefficient C(z), it is necessary to determine the transverse field expressions e,l, er2 and phase constants PI, P2 of the rectangular dielectric wave￾guides. But, unfortunately, rigorous expressions for the transverse field distribution and phase constant of guided modes have not been found. Here, in this paper, we apply an approximate field expression resulting from the crossed dielectric slab method (CDSM) [lo]. The CDSM field expressions are given in the appendix. For the calculation of phase constant of the guided mode, the generalized ef￾fective dielectric constant method (GEDCM) [7] is ap￾plied with some modifications. D. Numerical Calculations The coupling coefficient and transfer power between the rectangular dielectric waveguide-l and -2, crossed over each other, are calculated. At first, the coupling coefficient C (6) should be cal￾culated as a function of axial distance z. The Dij, Nil in (8) and (9) are obtained by numerical integration taking into account the e,; and etj are function of z. As the trans￾verse electromagnetic field around the guide is expressed separately in nine divided regions (as shown in Fig. S), the above integrations [(8) and (9)] are performed on every 25 subregions. The subregions are shown in Fig. 4. In these calculations, we take the following dimen- 2o i -+ Z’ (mm) 250 Fig. 5. Examples of calculated coupling coefficient C(z). sions; DWG-1 and -2 have relative dielectric constant E, = 2.01, cross section a X b = 4 X 4 mm and step dis￾tance Az = 0.75 mm. Examples of calculated coupling coefficient C(z) are shown in Fig. 5. From these results, it is evident that, near crossing center, the C(z) increases rapidly. In these examples, crossing angle % is assumed to be small because of using the step-like approximation. Variations of the transfer power P2(z) which calculated from (3) by numerical integration, are shown in Fig. 6(a)- (8). The initial conditions are taken to be A, (zo) = 1, A2(ZO) = 0 (10) at a point zo = -450 mm. 111. EXPERIMENTAL RESULTS AND DISCUSSIONS Variation of field intensity along the coupled dielectric waveguide is measured on the 50 GHz experimental model. The rectangular dielectric waveguide is made from tetrafluoroethylene plastics (E, = 2.01) and has dimen￾sions, as same as the theoretical calculation, of 4 X 4 mm in cross section, 900 mm in length. The experimental set up is shown in Fig. 7. The DWG- 2 is mounted on the rotatable table which makes us the change of % easy. The waveguides are separated by air. Field intensity (E,) on the DWG-2 (in correspond to the transfer power P2) is picked up by a mono-pole antenna which can be moved along the ?-axis smoothly. Mea￾sured results are shown also in Fig. 6(a)-(g). From these results, we can see that the measured power transfer to an adjacent guide has somewhat slower vari￾ation with the z‘ axis than that of the theoretical calcula￾tion, particularly for the case that the crossing angle % and/or guide separation d are small. The result seems to come from the field distribution of the coupled DWG sys￾tem. When the DWG-1 and -2 have a same physical di￾mensions, the ratio of ml/m2 takes -t 1 as described in Section 11-D. Then, we assume in this analysis the field distribution of the coupled guide system may be expressed by a linear combination of the field distribution of each guides [see (4)]. But, when the guides approach each other, the coupled field should not be expressed at all by the linear combination of each guide’s field. This is an

962 IEEE JOURNAL OF QUANTUM ELECTRONICS,VOL.QE-22.NO.6,JUNE 1986 0 -10 -20 Measurement 20 Calculation 250 0 250 Z'(mm (d）d2.2nm,6=5° Z'mm) (a)d=2.2mm,80 Cal. -10 -10 -20 -20 250 3050 250 Z'm n) Z(m) (b）d=2.2mm,=1° e】d1.2m题， 0 VMeas: 人Meas Cal. Cal. -10 号.10 心 -20 ~2950 0 250 -3950 250 Z mm (r）d-1.6mm,8-3 Z'(mm (c)d=2.2mm,62° 01 Meas. -10 sz -20+ Cal 250 Z'(mm》 (g）d2.2mm,6=3 Fig.6.Examples of calculation and measurement result of coupling power P(z)on the dielectric waveguide 2. (a. IV.CONCLUSIONS DWG-1 Coupling characteristics of two rectangular dielectric ABSORBING DW6-2 9 TERMINAL(A,T。) waveguides laid in different layers are analyzed.The cou- pling coefficient C(z)is determined as a function of guide CROSSING MONO-POLE ANTENNA separation d and crossing angle 0.In this analysis,the CENTER ON THE MOVABLE MOUNT approximated electromagnetic field expression of the rect- M.M.WAVE SOURCE angular dielectric waveguide is introduced and used.The WITH MODE LAUNCHER experimental results,carried out on 50 GHz band,are also Fig.7.Experimental set-up. described.From the results of calculation and measure- explanation of the discrepancy between theoretical and ment,we can conclude for the case0>2°，d>2mm experimental results.An improved field expression of the that the calculation method by using the variational prin- coupled system is currently under investigation by the au- ciple,described in this paper,is effective for the analysis thors.The improved expression will give to more precise of coupled power in the crossed dielectric waveguide sys- results of the coupled power. tem

962 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-22, NO. 6, JUNE 1986 0- \ - --lo 8 - a N, -20 - Jf Meas. i Cal. -30-7 ' - 750 0 250 o Z' (mm) (b) d=2.2mm, 0=1 -30~""'" -750 250 0 Z' (mm) (c) d=2.2mm, 0=2 -30. -250 3 0 250 ( d ) d = 2.2 mm, e = 5O Z' (mm) OT -+ Fig. 6. Examples of calculation and measurement result of coupling power P2(z) on the dielectric waveguide 2. ABSORBING TERMINAL (A.T.) DWG-2 WITH MODE LAUNCHER Fig. 7. Experimental set-up. explanation of the discrepancy between theoretical and experimental results. An improved field expression of the coupled system is currently under investigation by the au￾thors. The improved expression will give to more precise results of the coupled power. IV. CONCLUSIONS Coupling characteristics of two rectangular dielectric waveguides laid in different layers are analyzed. The cou￾pling coefficient C(z) is determined as a function of guide separation d and crossing angle 0. In this analysis, the approximated electromagnetic field expression of the rect￾angular dielectric waveguide is introduced and used. The experimental results, carried out on 50 GHz band, are also described, From the results of calculation and measure￾ment, we can conclude for the case 0 > 2", d > 2 mm that the calculation method by using the variational prin￾ciple, described in this paper, is effective for the analysis of coupled power in the crossed dielectric waveguide sys￾tem

MATSUMURA AND TOMABECHI:TWO RECTANGULAR DIELECTRIC WAVEGUIDES 963 REFERENCES [1]E.A.J.Marcatili,Dielectric rectangular waveguide and directional coupler for integrated optics,"Bell Syst.Tech.J..vol.48.no.9.pp 2071-2102,Sep.1969. Er [2]W.V.McLevige et al.,'New waveguide structure for millimeter- wave and optical integrated circuits,IEEE Trans.Microwave The- ory Tech.,vol.MTT-23.pp.788-794,Oct.1975. [3]T.Yoneyama and S.Nishida,Nonradiative dielectric waveguide for Fig.8.Nine regions in the cross section of a rectangular dielectric wave- millimeter wave integrated circuits,IEEE Trans.Microwave Theory guide. Tech.,vol.MTT-29.Pp.1188-1192,Nov.1981 [4]T.Itanami and S.Sindo,"'Channel dropping filter for millimeter wave integrated circuits,"IEEE Trans.Microwave Theory Tech..vol. The result of this work will be applied to a new design MTT-26,pp.759-769,0ct.1978. concept for the"multilayered integrated circuit''for mil- 15)M.Matsuhara and N.Kumagai.''Coupling theory of open type trans- limeter,submillimeter and optical waves. mission lines and its application to optical circuits,''Trans.1.E.C.E Japan,vol.55-C,no.4,pp.201-206,Apr.1972. (6]K.Solbach,'The calculation and measurement of the coupling prop- APPENDIX erties of dielectric image lines of rectangular cross section,IEEE Trans.Microwave Theory Tech.,vol.MTT-27,pp.54-58,Jan.1979. A.Transverse Field Expression of a Rectangular [7]T.Trinh and R.Mittra,''Coupling characteristics of planar dielectric Dielectric Waveguide Mode waveguides of rectangular cross section.''IEEE Trans.Microwave The transverse field expression of a rectangular dielec- Theory Tech..vol.MTT-29,pp.875-880,Sept.1981. [8]T.Findakly and C.L.Chen,"Optical dielectric couplers with vari- tric waveguide mode led by the crossed dielectric slab able spacing,Appl.Optics,vol.17,no.5,pp.769-773,March. method is as follows.The field is divided into nine parts 1978. which correspond to nine regions indicated in Fig.8. [9]I.Anderson,"'On the coupling of gegenerate modes on non-parallel dielectric waveguides,Microwave,Opt.Acoust.,vol.3,no.2.pp 56-58.Mar.1979. Region 7:- BA cos (k,)cosk》 (A.1) [10]Y.Tomabechi and K.Matsumura,"'Radiation loss caused by steps WEDEr in rectangular dielectric waveguide,"'Tech.Group,I.E.C.E.Japan. vol,MW84.no.1,pp.1-8,Apr.1984. 111]S.E.Miller,Coupled wave theory and waveguide application," Regions 2 and 3:- 一Acoskx2 cos (ky) Bell Syst.Tech.J.,vol.33,no.3,pp.661-719.May 1954. 112]K.Kurokawa,"'Electromagnetic waves in waveguides with wall impedance,IRE Trans.Microwave Theory Tech.,vol.MTT-10,pp. 314-320,Scpt.1962. (A.2) BA cos (cos Kazuhito Matsumura (S'62-M'67)was born in Regions 4 and 5:- Yamanashi,Japan,on August 11,1938.He re- ceived the B.S.degree in electrical engineering from the Yamanashi University in 1961.He also received the M.S.and Dr.of engineering degrees (A.3) in communication engineering from the Tohoku University.Sendai.Japan,in 1964 and 1967.re- spectively. Regions 6,7,8,and 9: From 1967 to 1970,he was a Research Assis- tant,and from 1971 to 1972 an Associate Profes- sor at Tohoku University.Since 1973 he has been -A cos an Associate Professor at Utsunomiya University,Utsunomiya,Japan From 1976 to 1977.he was with the Technische Hochschule Darmstadt, Federal Republic of Germany on a temporary basis as a Guest Professor. exp A.4) His research has been concerned with wave-guiding mechanisms and cir- cuit components on the millimeter,submillimeter,and optical wave re- gions. where k,is a transverse (x-direction)phase constant of an Dr.Matsumura is a member of the Institute of Electronics and Com- even TEo mode in y-extended slab dielectric waveguide munication Engineers of Japan. of relative dielectric constant erey.Similarly,k is a trans- verse (y-directed)phase constant of even the TMo mode in an x-extended slab dielectric waveguide whose relative Yoshiro Tomabechi (M'83)was born in Mo- dielectric constant is erer.The erey and erer are derived rioka,Japan,on August 16,1948.He received the B.S.,and M.S.degrees in electrical engineering from the generalized effective dielectric constant method from the Yamagata University.Yonezawa,Japan, [7].1/Yr and 1/Yy are the penetration depth in the x-and in 1971 and 1973,respectively. y-direction,respectively.A is an amplitude factor. Since 1973,he has been a Research Assistant of Utsunomiya University.His present research activity has been directed toward the analysis of ACKNOWLEDGMENT wave-guiding mechanisms,especially the rectan- The authors wish to thank S.Toratani and K.Kikuchi gular dielectric waveguide. Mr.Tomabechi is a member of the Institute of for helpful discussions and technical assistance. Electronics and Communication Engineers of Japan

MATSUMURA AND TOMABECHI: TWO RECTANGULAR DIELECTRIC WAVEGUIDES 963 I I 9 I5 (1 I Fig. 8. Nine regions in the cross section of a rectangular dielectric wave￾guide. The result of this work will be applied to a new design concept for the “multilayered integrated circuit” for mil￾limeter, submillimeter and optical waves. APPENDIX A. Transverse Field Expression of a Rectangular Dielectric Waveguide Mode The transverse field expression of a rectangular dielec￾tric waveguide mode led by the crossed dielectric slab method is as follows. The field is divided into nine parts which correspond to nine regions indicated in Fig. 8. R Region I: - - A cos (k,x) cos ‘(k,y) WEOE,, P P WEoErex Regions 2 and 3: - ~ P Regions 4 and 5: - - A cos (k,x) cos WE0 Regions 6, 7, 8, and 9. where k, is a transverse (x-direction) phase constant of an even TEo mode in y-extended slab dielectric waveguide of relative dielectric constant Similarly, k,, is a trans￾verse (y-directed) phase constant of even the ‘TM, mode in an x-extended slab dielectric waveguide whose relative dielectric constant is erex. The irey and E,, are derived from the generalized effective dielectric constant method [7]. lly, and lly, are the penetration depth in the x- and y-direction, respectively. A is an amplitude factor. ACKNOWLEDGMENT The authors wish to thank S. Toratani and K. Kikuchi for helpful discussions and technical assistance. REFERENCES [I] E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J., vol. 48, no. 9, pp. 2071-2102, Sept. 1969. [2] W. V. McLevige er al., “New waveguide stmcture for millimeter￾wave and optical integrated circuits,” IEEE Trans. Microwave The￾ory Tech., voll MTT-23, pp. 788-794, Oct. 1975. [3] T. Yoneyama and S. Nishida, “Nonradiative dielectric waveguide for millimeter wave integrated circuits,” IEEE Trans. Microwave Theory Tech., vol. MTT-29, pp. 1188-1192, Nov. 1981. [4] T. Itanami and S. Sindo, “Channel dropping filter for millimeter wave integrated circuits,” IEEE Trans. Microwave Theory Tech., vol. 151 M. Matsuhara and N. Kumagai, “Coupling theory of open type trans￾mission lines and its application to optical circuits,” Trans. I.E. C.E. Japan, vol. 55-C, no. 4, pp. 201-206, Apr. 1972. 161 K. Solbach, “The calculation and measurement of the coupling prop￾erties of dielectric image lines of rectangular cross section,” IEEE Trans. Microwave Theory Tech., vol. MTT-27, pp. 54-58, Jan. 1979. [7] T. Trinh and R. Mittra, “Coupling characteristics of planar dielectric waveguides of rectangular cross section,” IEEE Trans. Microwave Theory Tech.. vol. MTT-29, pp. 875-880, Sept. 1981. [8] T. Findakly and C. L. %hen, “Optical dielectric couplers with vari￾able spacing,” Appl. Optics, vol. 17, no. 5, pp. 769-773. March, 1978. 191 I. Anderson, “On the coupling of gegenerate modes on non-parallel dielectric waveguides,” Microwave, Opt. Acoust., vol. 3, no. 2, pp. 56-58, Mar. 1979. [lo] Y. Tomabechi and K. Matsumura, “Radiation loss caused by steps in rectangular dielectric waveguide,” Tech. Group, I.E.C.E. Japan, vol. MW84, no. 1, pp. 1-8, Apr. 1984. [ll] S. E. Miller, “Coupled wave theory and waveguide application,” Bell Syst. Tech. J., vol. 33, no. 3, pp. 661-719, May 1954. I121 K. Kurokawa, “Electromagnetic waves in waveguides with wall impedance,” IRE Trans. Microwave Theory Tech., vol. MTT-IO, pp. 314-320, Sept. 1962. MTT-26, pp, 759-769, Oct. 1978. Kazuhito Matsumura (S’62-M’67) was born in Yamanashi, Japan, on August 11, 1938. He re￾ceived the B.S. degree in electrical engineering from the Yamanashi University in 1961. He also received the M.S. and Dr. of engineering degrees in communication engineering from the Tohoku University, Sendai, Japan, in 1964 and 1967, re￾spectively. From 1967 to 1970, he was a Research Assis￾tant, and from 1971 to 1972 an Associate Profes￾sor at Tohoku University. Since 1973 he has been an Associate Professor at Utsunomiya University, Utsunomiya, Japan. From 1976 to 1977, he was with the Technische Hochschule Darmstadt, Federal Republic of Germany on a temporary basis as a Guest Professor. His research has been concerned with wave-guiding mechanisms and cir￾cuit components on the millimeter, submillimeter, and optical wave re￾gions. Dr. Matsumura is a member of the Institute of Electronics and Com￾munication Engineers of Japan. Electronics and Commu Yoshiro Tomabechi (”83) was born in Mo￾rioka, Japan, on August 16, 1948. He received the B.S., and M.S. degrees in electrical engineering from the Yamagata University, Yonezawa, Japan, in 1971 and 1973, respectively. Since 1973, he has been a Research Assistant of Utsunomiya University. His present research activity has been directed toward the analysis of wave-guiding mechanisms, especially the rectan￾gular dielectric waveguide. Mr. Tomabechi is a member of the lnstitute of lnication Engineers of Japan