LOVE AND RIESEN:SINGLE-.FEW-.AND MULTIMODE Y-JUNCTIONS 307 B B 1 0.9 0.8 Fundamental 7 Mode 6 -Fundamental Mode in A . oSecond Mode in A -x-Second Mode in B -Third Mode in B Fundamental 0.3 Second(First-Odd) Mode Mode 0.2 0.1 0 (a) (b) 0.5 1.5 2 2.5 3.5 Fig.8.Evolution of(a)the FM and (b)the second(first-odd)mode between the Output Arm A Width(Micrometers) stem and arms A and B of the asymmetric two-mode Y-junction,respectively. Fundamental Mode in A Second Mode it 300 An example of a more complex modal evolution is simulated in Fig.9,where the third,or second-even mode,propagates alone through the three-mode 7 um wide stem of the asym- 1 metric Y-junction.If the width of arm A is sufficiently narrow that it supports just the FM with an effective index that roughly matches that of the third mode in the stem,then the third mode 2 is totally transformed by the Y-junction into the arm's FM.This is shown by Fig.9(a)for an arm width of 1 um.Similarly,if the width of arm A is large enough to support both the fundamental and second (or first-odd)modes,then the third mode in the stem is transformed into the arm's second mode,provided that there X (um] is a sufficiently close match between their respective effective indexes.The simulation in Fig.9(b)shows this evolution for (a. (b) an arm width of 4 um.These transformations can be correlated Fig.9.Evolution of the third mode of the stem into the left hand arm A of the with the plots of effective indexes in Fig.10 for the three modes asymmetric Y-junction,showing (a)transformation into the FM and(b)trans- involved.For an arm width of I um and a stem width of 7 um, formation into the second or first-odd mode. only the first and third mode effective indexes will match,while for the arm width of 4 um,only the second and third mode ef- 1.475 x-TEO fective indexes equate. 147 F Multimode Asymmetric Y-Junctions 萝 1.465 TM2 Unlike symmetric Y-junctions,the behavior of few-mode 1.46 asymmetric Y-junctions(and multiarm Y-junctions)is approx- 1.455 imately independent of the phases of the individual modes provided it is sufficiently adiabatic and asymmetric,and each 1.45 mode shows a strong preference for a particular output arm. 1445 This last requirement is difficult to achieve in practice for 144 few-mode asymmetric Y-junctions,and even more so for 0.5 1.5 2.5 3.5 4.5 5.5 6.5 multimode asymmetric Y-junctions.Furthermore,in the case Waveguide Core Width(Micrometers) of a highly multimode asymmetric Y-junction,the behavior Fig.10.Variation of the effective indices of the first three TE/TM modes as a becomes approximately independent of the phases of the indi- function of the Y-junction stem or arm widths. vidual modes. The asymmetry of highly multimode asymmetric Y-junctions can,then,be tailored to provide arbitrary tap-off fractions [7]. second,and third modes evolve into respective combinations The split in optical power is determined by ray tracing as the of the FMs in the three output arms.The output power in each ratio of the number of modes in each arm,provided that the arm,in general,will depend on the relative phase difference mode number is large enough. accumulated between the three modes in the stem. In the case of the asymmetric three-arm Y-junction,each of III.MULTIARM Y-JUNCTIONS the three stem modes can be transformed into the FM of a par- The basic properties of a two-arm symmetric or asymmetric ticular arm and vice versa,provided that the arm widths are all Y-junction with a two-mode stem and two single-mode arms sufficiently different.Restrictions on the level of asymmetry do, can be generalized to Y-junctions with three-or-more modes in however,arise,as is the case for Section II,because of the even- the stem,and three-or-more output arms. tual excitation of higher order modes in the output arms.Under In the case of the symmetric three-arm Y-junction with all this arrangement,the FM of the stem evolves through the junc- arms single mode,and with three modes in the stem,the first, tion into the FM of the widest arm and its field shape is retainedLOVE AND RIESEN: SINGLE-, FEW-, AND MULTIMODE Y-JUNCTIONS 307 Fig. 8. Evolution of (a) the FM and (b) the second (first-odd) mode between the stem and arms A and B of the asymmetric two-mode Y-junction, respectively. An example of a more complex modal evolution is simulated in Fig. 9, where the third, or second-even mode, propagates alone through the three-mode 7 m wide stem of the asym￾metric Y-junction. If the width of arm A is sufficiently narrow that it supports just the FM with an effective index that roughly matches that of the third mode in the stem, then the third mode is totally transformed by the Y-junction into the arm’s FM. This is shown by Fig. 9(a) for an arm width of 1 m. Similarly, if the width of arm A is large enough to support both the fundamental and second (or first-odd) modes, then the third mode in the stem is transformed into the arm’s second mode, provided that there is a sufficiently close match between their respective effective indexes. The simulation in Fig. 9(b) shows this evolution for an arm width of 4 m. These transformations can be correlated with the plots of effective indexes in Fig. 10 for the three modes involved. For an arm width of 1 m and a stem width of 7 m, only the first and third mode effective indexes will match, while for the arm width of 4 m, only the second and third mode ef￾fective indexes equate. F. Multimode Asymmetric Y-Junctions Unlike symmetric Y-junctions, the behavior of few-mode asymmetric Y-junctions (and multiarm Y-junctions) is approx￾imately independent of the phases of the individual modes provided it is sufficiently adiabatic and asymmetric, and each mode shows a strong preference for a particular output arm. This last requirement is difficult to achieve in practice for few-mode asymmetric Y-junctions, and even more so for multimode asymmetric Y-junctions. Furthermore, in the case of a highly multimode asymmetric Y-junction, the behavior becomes approximately independent of the phases of the indi￾vidual modes. The asymmetry of highly multimode asymmetric Y-junctions can, then, be tailored to provide arbitrary tap-off fractions [7]. The split in optical power is determined by ray tracing as the ratio of the number of modes in each arm, provided that the mode number is large enough. III. MULTIARM Y-JUNCTIONS The basic properties of a two-arm symmetric or asymmetric Y-junction with a two-mode stem and two single-mode arms can be generalized to Y-junctions with three-or-more modes in the stem, and three-or-more output arms. In the case of the symmetric three-arm Y-junction with all arms single mode, and with three modes in the stem, the first, Fig. 9. Evolution of the third mode of the stem into the left hand arm A of the asymmetric Y-junction, showing (a) transformation into the FM and (b) trans￾formation into the second or first-odd mode. Fig. 10. Variation of the effective indices of the first three TE/TM modes as a function of the Y-junction stem or arm widths. second, and third modes evolve into respective combinations of the FMs in the three output arms. The output power in each arm, in general, will depend on the relative phase difference accumulated between the three modes in the stem. In the case of the asymmetric three-arm Y-junction, each of the three stem modes can be transformed into the FM of a par￾ticular arm and vice versa, provided that the arm widths are all sufficiently different. Restrictions on the level of asymmetry do, however, arise, as is the case for Section II, because of the even￾tual excitation of higher order modes in the output arms. Under this arrangement, the FM of the stem evolves through the junc￾tion into the FM of the widest arm and its field shape is retained