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-