306 JOURNAL OF LIGHTWAVE TECHNOLOGY,VOL.30,NO.3,FEBRUARY 1,2012 B B B Second Even +Odd Supermodes Mode Fundamental 100%Radiation 100%Radiation Mode (a. (b) Fig.5.Evolution of the second mode in arm A through a two-mode Y-junction. Fig.6.Evolution of the FM between the stem and (a)the wider arm and (b)the narrower arm of an asymmetric single-mode Y-junction. split from the stem to both arms for the second mode,but trans- 0.9 mitting no power in the opposite direction. 08 -Output Arm A:FM 0.7 Output Arm B:FM C.Multimode Symmetric Y-Junctions In a multimode symmetric Y-junction,the power splitting 0.5 will depend on the phase differences between the stem modes 0.4 that correspond to same-order pairs of supermodes across the 0.3 output arms.As mentioned,all other pairs of modes split equally 0.2 as they are independent of phase.Since these phase differences 0.1 are random,they will,therefore,approach a continuum in the 0 0.75 1.151.351.55 1.75 case of a highly multimode symmetric Y-junction.Since on av- 0.95 1.952.15 Output Arm A Width(Micrometers) erage the splitting will be 1:1,a highly multimode symmetric Y-junction will behave as a 3 dB splitter.Furthermore,the split Fig.7.Relative power output between arms A and B as a function of the width of arm A. is also wavelength independent,does not depend explicitly on the specific number of modes,and is also independent of the index contrast between the core and the cladding. exits through arm B,gives an equal 3 dB split when the arm The symmetric Y-junction is necessarily a reciprocal device,widths are equal,and predominantly exits arm A when its width and therefore,if all N modes are excited in one of the arms of an is larger. N-mode Y-junction,only the first N/2-order modes (rounded down)can be accommodated in the stem and the remaining E.Few-Mode Asymmetric Y-Junctions higher order N/2(rounded up)modes must radiate away. Now consider the case of a two-mode,asymmetric Y-junc- Highly multimode asymmetric Y-junctions can be analyzed tion where the stem supports the first two modes and each arm using ray tracing and have previously been described in some supports just the FM.The symmetric FM in the stem exits as the detail [7]. FM in the wider output arm A in Fig.8(a),as was the case for the single-mode Y-junction.This is again because of the close D.Single-Mode Asymmetric Y-Junctions proximity of the effective indices in the stem and output arm. In the case of an asymmetric Y-junction where the two arms The second or first-odd mode in the stem exits as the FM of differ sufficiently in their widths(or refractive indices),the first-the narrower arm B,as shown in Fig.8(b),because its effective even and first-odd local supermode fields simplify and become index better matches the effective index of the FM in the nar- the FM fields of the wider and narrower arms,respectively [4]. rower arm.In other words,the second mode is transformed into Thus,in the case of a single-mode Y-junction with asymmetric the FM as it transits the junction,with no loss of power. arms,the FM in the stem evolves through the junction into the Conversely,if the FM of the narrower arm propagates in FM of the wider arm,as shown in Fig.6(a).This transition oc- the opposite direction,it is transformed by the junction into curs because the effective index of the FM in the stem is closer the second(first-odd)mode of the stem without loss of power. to that of the FM in the wider arm. In other words,this type of Y-junction acts as a reciprocal In the reverse direction,the FM in the wider arm evolves into mode transformer.This modal behavior again results from the the FM of the stem,while the FM in the narrower arm,having matching of modal effective indexes [10].As a mode in the the smaller effective index,evolves into the unguided second stem propagates through the Y-junction,it evolves into the mode of the single-mode stem and therefore all its power is mode of the output arm with the closest effective index [13]. radiated,as shown in Fig.6(b). This allows for the careful design of the output arms to ensure If the relative widths of the two arms are changed continu-almost arbitrary proportions of mode separation. ously such that the sum of the arm widths remains constant, The performance of a 2-D Y-junction with two modes in the then Fig.7 shows the output power split from the FM in the stem and two output arms can be described quantitatively using stem.When the width of arm A is small,power preferentially the mode conversion factor proposed by Burns and Milton [5].306 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 30, NO. 3, FEBRUARY 1, 2012 Fig. 5. Evolution of the second mode in arm A through a two-mode Y-junction. split from the stem to both arms for the second mode, but trans￾mitting no power in the opposite direction. C. Multimode Symmetric Y-Junctions In a multimode symmetric Y-junction, the power splitting will depend on the phase differences between the stem modes that correspond to same-order pairs of supermodes across the output arms. As mentioned, all other pairs of modes split equally as they are independent of phase. Since these phase differences are random, they will, therefore, approach a continuum in the case of a highly multimode symmetric Y-junction. Since on av￾erage the splitting will be 1:1, a highly multimode symmetric Y-junction will behave as a 3 dB splitter. Furthermore, the split is also wavelength independent, does not depend explicitly on the specific number of modes, and is also independent of the index contrast between the core and the cladding. The symmetric Y-junction is necessarily a reciprocal device, and therefore, if all modes are excited in one of the arms of an -mode Y-junction, only the first -order modes (rounded down) can be accommodated in the stem and the remaining higher order (rounded up) modes must radiate away. Highly multimode asymmetric Y-junctions can be analyzed using ray tracing and have previously been described in some detail [7]. D. Single-Mode Asymmetric Y-Junctions In the case of an asymmetric Y-junction where the two arms differ sufficiently in their widths (or refractive indices), the first￾even and first-odd local supermode fields simplify and become the FM fields of the wider and narrower arms, respectively [4]. Thus, in the case of a single-mode Y-junction with asymmetric arms, the FM in the stem evolves through the junction into the FM of the wider arm, as shown in Fig. 6(a). This transition oc￾curs because the effective index of the FM in the stem is closer to that of the FM in the wider arm. In the reverse direction, the FM in the wider arm evolves into the FM of the stem, while the FM in the narrower arm, having the smaller effective index, evolves into the unguided second mode of the single-mode stem and therefore all its power is radiated, as shown in Fig. 6(b). If the relative widths of the two arms are changed continu￾ously such that the sum of the arm widths remains constant, then Fig. 7 shows the output power split from the FM in the stem. When the width of arm A is small, power preferentially Fig. 6. Evolution of the FM between the stem and (a) the wider arm and (b) the narrower arm of an asymmetric single-mode Y-junction. Fig. 7. Relative power output between arms A and B as a function of the width of arm A. exits through arm B, gives an equal 3 dB split when the arm widths are equal, and predominantly exits arm A when its width is larger. E. Few-Mode Asymmetric Y-Junctions Now consider the case of a two-mode, asymmetric Y-junc￾tion where the stem supports the first two modes and each arm supports just the FM. The symmetric FM in the stem exits as the FM in the wider output arm A in Fig. 8(a), as was the case for the single-mode Y-junction. This is again because of the close proximity of the effective indices in the stem and output arm. The second or first-odd mode in the stem exits as the FM of the narrower arm B, as shown in Fig. 8(b), because its effective index better matches the effective index of the FM in the nar￾rower arm. In other words, the second mode is transformed into the FM as it transits the junction, with no loss of power. Conversely, if the FM of the narrower arm propagates in the opposite direction, it is transformed by the junction into the second (first-odd) mode of the stem without loss of power. In other words, this type of Y-junction acts as a reciprocal mode transformer. This modal behavior again results from the matching of modal effective indexes [10]. As a mode in the stem propagates through the Y-junction, it evolves into the mode of the output arm with the closest effective index [13]. This allows for the careful design of the output arms to ensure almost arbitrary proportions of mode separation. The performance of a 2-D Y-junction with two modes in the stem and two output arms can be described quantitatively using the mode conversion factor proposed by Burns and Milton [5]