CHIANG:COUPLED-ZIGZAG-WAVE THEORY FOR GUIDED WAVES 1385 TABLE I APPROXIMATE DISPERSION RELATIONS FROM THE PRESENT THEORY FOR THE GUIDED MODES OF AN ARRAY CONSISTING OF N EQUALLY SEPARATED IDENTICAL WAVEGUIDES AND THE CORRESPONDING PROPAGATION CONSTANTS OBTAINED FROM THE COUPLED-MODE THEORY. Mode Class Dispersion Relations Propagation Constants F=0 F+1/2=0 十x F-1/2=0 3-K F+2/2112=0 .+21/2 F=0 3 F-21/2c1/2=0 3-21/2 y F+[3+/22=0 元+[3+5/22。 2 F+3-/2j2n=0 +[3-同/22 F-[3-)/22=0 -(3-同/2s 4 F-3+⑤/2]2112=0 -3+/22x 1 F+31/2c72=0 3。+3/2 2 F+2=0 ,+ F=0 F-c1f2=0 一N 5 F-321/2=0 3.-3/2x V=tn)1/2,the normalized propagation constantbis the array can be described as a set of coupled zigzag waves that calculated,which is defined by b=(B/)2-n2 /(n2-n2). propagate in the individual waveguides.These zigzag waves satisfy transverse resonance conditions simultaneously and The numerical results are shown in Fig.3 where b is plotted as a function of V for the first two TE modes (even and odd couple only to the adjacent ones through the coupling of the phase shifts at the boundaries of the guiding slabs.This new modes)of the array for two values of relative waveguide sep- model has led to an exact dispersion relation,which is simple aration d/t.As expected,the accuracy of the weak-coupling approximation (40)and the coupled-mode theory improves in form and easy to solve.The close correspondence between this coupled-zigzag-wave theory and the well-known coupled- with increasing V at a given d/t,or with increasing d/t at a given V.When the array is used as a directional coupler, mode theory for weakly coupled waveguides has also been demonstrated with examples.The analytical results presented the power transfer between the two waveguides is determined by the beating effect of these two modes and,therefore,is in this paper should facilitate the design of slab waveguide characterized by the coupling factor C,which is given by C= arrays,and,when used in conjunction with the effective-index V(b-b)/4,where be and bo are the normalized propagation method [24],should also facilitate the study of arrays of rectangular waveguides. constants for the even and the odd modes,respectively.In Fig.4,coupling factor C is plotted as a function of V for several values of relative waveguide separation.When the APPENDIX:DERIVATION OF (35) coupling is weak,both (40)and the coupled-mode theory are Eliminating 6Ri from (27)and (28),we have accurate.When the coupling is not weak (at small values of V and d/t),it appears from Fig.4 that the coupled-mode theory 0Li+1= c4.a+1(Y+1+Z+161i=1,2,,N-1 provides a better estimate for C than (40),despite the fact that W+1-X2+16L (A1) (40)may actually give a better estimate for the normalized where propagation constant as shown in Fig.3.This simply means that the errors in be and 6o,calculated by the coupled-mode Wi+1 Fi(1-ci.i+ipRipLi+1)+ci.i+iPLi+i (A2) theory,are largely canceled. It should be mentioned that the coupled-mode solutions in Figs.3 and 4 are based on a first-order theory [1]-[81. Xi+1=1-CG,i+1PL1+1Z:+1 (A3) Recently,several other formulations of the coupled-mode theory have been developed [17]-[21].For the TM modes, the recent formulations can produce improved results,but for the TE modes,the difference is not significant [22].[23].It is Yit=1-PRiF (A4) not the purpose of this paper to assess the accuracy of these and theories.The early version of the coupled-mode theory is used here for comparison simply because of its simplicity. Zi=PRi+F (A5) VI.CONCLUSION We denote An alternative way of understanding the guided wave in a 6zi+1=G+1B+ i=1,2.…,N-1 (A6) slab waveguide array has been introduced.The guided wave in 4+1'I I CHIANG: COUPLED-ZIGZAG-WAVE THEORY FOR GUIDED WAVES 1385 TABLE 1 APPROXIMATE DISPERSION RELATIONS FROM THE PRESENT THEORY FOR THE GUIDED MODES OF AN ARRAY CONSISTING OF .\- EQUALLY SEPARATED IENTICAL WAVEGUIDES AND THE CORRESPONDING PROPAGATION CONSTANTS OBTAINED FROM THE COUPLED-MODE THEORY. s Mode Class Dispersion Relations Propagation Constants 1 2 3 4 5 1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 V = tk(nf - n:)l'*, the normalized propagation constant b is calculated, which is defined by b = (/3/kf - n:] / (nf - n:). The numerical results are shown in Fig. 3 where b is plotted as a function of V for the first two TE modes (even and odd modes) of the array for two values of relative waveguide sep￾aration d/t. As expected, the accuracy of the weak-coupling approximation (40) and the coupled-mode theory improves with increasing V at a given dlt, or with increasing dlt at a given V. When the array is used as a directional coupler, the power transfer between the two waveguides is determined by the beating effect of these two modes and, therefore, is characterized by the coupling factor C, which is given by C = V(b, - b0)/4, where be and bo are the normalized propagation constants for the even and the odd modes, respectively. In Fig. 4, coupling factor C is plotted as a function of V for several values of relative waveguide separation. When the coupling is weak, both (40) and the coupled-mode theory are accurate. When the coupling is not weak (at small values of V and d/t), it appears from Fig. 4 that the coupled-mode theory provides a better estimate for C than (40), despite the fact that (40) may actually give a better estimate for the normalized propagation constant as shown in Fig. 3. This simply means that the errors in be and bo, calculated by the coupled-mode theory, are largely canceled. It should be mentioned that the coupled-mode solutions in Figs. 3 and 4 are based on a first-order theory [1]-[8]. Recently, several other formulations of the coupled-mode theory have been developed [17]-[21]. For the TM modes, the recent formulations can produce improved results, but for the TE modes, the difference is not significant [22], [23]. It is not the purpose of this paper to assess the accuracy of these theories. The early version of the coupled-mode theory is used here for comparison simply because of its simplicity. [ VI. CONCLUSION An alternative way of understanding the guided wave in a slab waveguide array has been introduced. The guided wave in the array can be described as a set of coupled zigzag waves that propagate in the individual waveguides. These zigzag waves satisfy transverse resonance conditions simultaneously and couple only to the adjacent ones through the coupling of the phase shifts at the boundaries of the guiding slabs. This new model has led to an exact dispersion relation, which is simple in form and easy to solve. The close correspondence between this coupled-zigzag-wave theory and the well-known coupled￾mode theory for weakly coupled waveguides has also been demonstrated with examples. The analytical results presented in this paper should facilitate the design of slab waveguide arrays, and, when used in conjunction with the effective-index method [24], should also facilitate the study of arrays of rectangular waveguides. APPENDIX: DERIVATION OF (35) Eliminating 6~~ from (27) and (28), we have where