《光波导理论与技术 Optical Waveguides Principles and Technologies》课程教学资源（参考文献）Single-, Few-, and Multimode Y-Junctions

304 JOURNAL OF LIGHTWAVE TECHNOLOGY,VOL.30,NO.3,FEBRUARY 1,2012 Single-,Few-,and Multimode Y-Junctions John D.Love and Nicolas Riesen Abstract-The theory of modal propagation through symmetric and asymmetric 2-D weakly guiding Y-junctions is extended to cover few-mode,multimode,and multiarm Y-junctions.A concep- B A B tual approach based on the evolution of modal effective indexes and composite supermodes is used to determine the qualifative func- First-Even First Even +Odd tionality of these devices,with quantification being determined nu- Supermode Supermodes merically using the beam propagation method. Index Terms-Asymmetric Y-junctions,mode-division multi- plexing,mode separation,power splitting,symmetric Y-junctions. 3dB Attenuation Fundamental Mode I.INTRODUCTION Fundamental Mode HE majority of fiber-or waveguide-based light-manip- (a). (b) ulating devices that are used to modify the amplitude or Fig.1.(a)Forward and (b)backward modal propagation through a symmetric, phase of the fundamental mode(FM)rely on the basic optical single-mode Y-junction. phenomena of interference,coupling,or reflection for their functionality.These devices include,for example,a wide va- riety of couplers,interferometers,and wavelength multiplexers past [2]-[6].Asymmetric Y-junctions where the arms have and demultiplexers.In addition to these devices,there is another differing cross sections (or differing refractive indexes)have class of devices whose functionality is based solely on their received far less attention.Nonetheless they are sometimes geometrical design.This class includes,for example,tapered used as polarization splitters,mode combiners,and mode single-mode fibers or waveguides that modify their transverse splitters in optical switches [7]-[9].They can also be used as modal field distribution via cross-sectional variations along wavelength multiplexers [10],and as variable power splitters their length.Provided that such a change is undertaken suffi- when designed with an adjustable gap region [11].The use ciently slowly,i.e.,approximately adiabatically,there will be a of asymmetric Y-junctions for mode-division multiplexing of negligible loss of modal power through radiation or coupling few-mode waveguides for high-capacity data transmission has to other modes [1]. also been suggested [12]. A second category of devices that rely solely on geomet- The beam propagation method(BPM)has been used to an- rical design for their functionality are Y-junctions.The basic alyze numerically the behavior of weakly guiding 2-D Y-junc- Y-junction has a stem and two diverging arms,and its light split- tions.In the case of weak guidance,where the core-cladding ting and light combining properties can be generalized to mul-index difference is small,the TE and TM modes are degenerate tiarm Y-junctions.Like tapers,the cross-sectional evolution of and,hence,exhibit similar behavior [13]. Y-junctions,needs to occur sufficiently slowly with axial dis- The simulations in this paper assume typical parameters such tance to ensure approximate adiabatic propagation of modes as a pure silica cladding,small index contrast,and a 1.55 um through the device.This ensures that minimal power is trans- source wavelength.The simple Y-junction structures simulated ferred between the modes or to the radiation field.In practice, in general have stem widths equal to the sum of the output arm this is readily achieved by making the divergence/taper angle widths.It should,however,be noted that the behavior of the between the two arms sufficiently small. Y-junction is largely insensitive to the specific geometry of the These approximately adiabatic devices can be subdivided taper,provided a sharp junction vertex is maintained [1]. into two categories:symmetric and asymmetric.Symmetric single-mode Y-junctions are wavelength-independent,equal II.TWO-ARM Y-JUNCTIONS 3 dB splitters and have been studied to some extent in the A.Single-Mode Symmetric Y-Junctions The working mechanism of single-mode symmetric Y-junc- Manuscript received August 16,2011;revised December 04,2011;accepted tions can be described by considering normal local even and odd December 09,2011.Date of publication December 15,2011;date of current version January 25,2012. supermodes covering the two output arms A and B as shown in The authors are with the Physics Education Centre,The Australian National Fig.1[4]. University,Canberra,A.C.T.0200,Australia(e-mail:john.love@anu.edu.au; This description is possible when the branching angle,0 is nicolas.riesen@anu.edu.au). Color versions of one or more of the figures in this paper are available online approximately adiabatic (1).The power of the FM in the at http://ieeexplore.ieee.org. stem of the Y-junction will split equally into FMs in the output Digital Object Identifier 10.1109/JLT.2011.2179976 arms as shown in Fig.1(a)just by symmetry.This can also be 0733-8724/$26.00C20111EEE

304 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 30, NO. 3, FEBRUARY 1, 2012 Single-, Few-, and Multimode Y-Junctions John D. Love and Nicolas Riesen Abstract—The theory of modal propagation through symmetric and asymmetric 2-D weakly guiding Y-junctions is extended to cover few-mode, multimode, and multiarm Y-junctions. A concep￾tual approach based on the evolution of modal effective indexes and composite supermodes is used to determine the qualitative func￾tionality of these devices, with quantification being determined nu￾merically using the beam propagation method. Index Terms—Asymmetric Y-junctions, mode-division multi￾plexing, mode separation, power splitting, symmetric Y-junctions. I. INTRODUCTION T HE majority of fiber- or waveguide-based light-manip￾ulating devices that are used to modify the amplitude or phase of the fundamental mode (FM) rely on the basic optical phenomena of interference, coupling, or reflection for their functionality. These devices include, for example, a wide va￾riety of couplers, interferometers, and wavelength multiplexers and demultiplexers. In addition to these devices, there is another class of devices whose functionality is based solely on their geometrical design. This class includes, for example, tapered single-mode fibers or waveguides that modify their transverse modal field distribution via cross-sectional variations along their length. Provided that such a change is undertaken suffi- ciently slowly, i.e., approximately adiabatically, there will be a negligible loss of modal power through radiation or coupling to other modes [1]. A second category of devices that rely solely on geomet￾rical design for their functionality are Y-junctions. The basic Y-junction has a stem and two diverging arms, and its light split￾ting and light combining properties can be generalized to mul￾tiarm Y-junctions. Like tapers, the cross-sectional evolution of Y-junctions, needs to occur sufficiently slowly with axial dis￾tance to ensure approximate adiabatic propagation of modes through the device. This ensures that minimal power is trans￾ferred between the modes or to the radiation field. In practice, this is readily achieved by making the divergence/taper angle between the two arms sufficiently small. These approximately adiabatic devices can be subdivided into two categories: symmetric and asymmetric. Symmetric single-mode Y-junctions are wavelength-independent, equal 3 dB splitters and have been studied to some extent in the Manuscript received August 16, 2011; revised December 04, 2011; accepted December 09, 2011. Date of publication December 15, 2011; date of current version January 25, 2012. The authors are with the Physics Education Centre, The Australian National University, Canberra, A.C.T. 0200, Australia (e-mail: john.love@anu.edu.au; nicolas.riesen@anu.edu.au). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JLT.2011.2179976 Fig. 1. (a) Forward and (b) backward modal propagation through a symmetric, single-mode Y-junction. past [2]–[6]. Asymmetric Y-junctions where the arms have differing cross sections (or differing refractive indexes) have received far less attention. Nonetheless they are sometimes used as polarization splitters, mode combiners, and mode splitters in optical switches [7]–[9]. They can also be used as wavelength multiplexers [10], and as variable power splitters when designed with an adjustable gap region [11]. The use of asymmetric Y-junctions for mode-division multiplexing of few-mode waveguides for high-capacity data transmission has also been suggested [12]. The beam propagation method (BPM) has been used to an￾alyze numerically the behavior of weakly guiding 2-D Y-junc￾tions. In the case of weak guidance, where the core-cladding index difference is small, the TE and TM modes are degenerate and, hence, exhibit similar behavior [13]. The simulations in this paper assume typical parameters such as a pure silica cladding, small index contrast, and a 1.55 m source wavelength. The simple Y-junction structures simulated in general have stem widths equal to the sum of the output arm widths. It should, however, be noted that the behavior of the Y-junction is largely insensitive to the specific geometry of the taper, provided a sharp junction vertex is maintained [1]. II. TWO-ARM Y-JUNCTIONS A. Single-Mode Symmetric Y-Junctions The working mechanism of single-mode symmetric Y-junc￾tions can be described by considering normal local even and odd supermodes covering the two output arms A and B as shown in Fig. 1 [4]. This description is possible when the branching angle, is approximately adiabatic . The power of the FM in the stem of the Y-junction will split equally into FMs in the output arms as shown in Fig. 1(a) just by symmetry. This can also be 0733-8724/$26.00 © 2011 IEEE

LOVE AND RIESEN:SINGLE-,FEW-,AND MULTIMODE Y-JUNCTIONS 305 Stem Modes:Output Arm 0.9 0.1：A 0.1B ie.0,1:A 0.8 0.7 0.2：A.日 First-Odd 03：A.B Supermode 入in 0.6 12:A,B 0.5 0.4 0.3 0.2 Second(First-Odd) 0.1 2,3:B 2,3:A Mode 0 0 % 100 150200 250300 350 400 Phase Difference(Degrees) Fig.3.Power in the output arms of the symmetric Y-junction as a function Fig.2.Evolution of the second stem mode into the arms of a few-mode sym- of the phase difference between pairs of modes from a few-mode stem(modes metric Y-junction. indexed from 0). understood as the excitation of the first-even supermode across B the output arms. In the reverse direction,exciting the FM in arm A(or B)is Second-Even Second-Odd equivalent to exciting a superposition of both the first-even and Supermode Supermode first-odd supermodes with equal amplitudes,as shown in Fig. 1(b)[4].The stem is single mode,allowing the even supermode from the arms to evolve into the FM of the stem.whereas the Third(Second-Even Fourth(Second-Odd) odd supermode evolves into the second or first-odd mode of the Mode stem.Since the stem is single mode,this mode is radiated away from the stem,as shown in Fig.1(b).Therefore,as required by (a) (b) the reciprocity ofthis adiabatic device,the Y-junction attenuates Fig.4.Evolution of(a)the third mode and (b)the fourth mode,from the stem the signal from the arm by 3 dB. through a few-mode Y-junction. Similarly,injecting in-phase equal amplitude FMs into both arms is equivalent to the excitation of only the first-even super- mode,and results in only the FM being excited in the stem with equally into the same second (first-odd)modes of the arms but no radiation loss [4].Generally,the output in the stem,however, with a x phase difference between them,as shown in Fig.4(b). depends on both the phase difference and amplitude difference As with the two-mode Y-junction,if both the third and fourth between the two incident modes in the arms.With equal ampli- modes are excited equally in the stem,the power distribution tude and a phase difference,i.e.,the first-odd supermode,all of the second mode outputs in the arms(superposition of the power leaks from the stem [4]. second even and odd supermodes)depends on the relative phase difference accumulated between the modes in traversing the B.Few-Mode Symmetric Y-Junctions Y-junction.This can vary from 0%to 100%and 100%to 0% in each output,respectively,given an ideal adiabatic Y-junc- These principles are extended to higher order modes of few-tion.More generally,the behavior of the Y-junction depends on mode symmetric Y-junctions.For these Y-junctions,the second the phase difference between the same-order pairs of even and (first-odd)mode in the stem splits its power equally into two odd supermodes (or between the corresponding pairs of stem equal amplitude FMs in the output arms but with a phase modes).The behavior of all other pairs of supermodes(or stem difference between them (i.e.,first-odd supermode),as shown modes)traversing a junction is independent of phase.This im- in Fig.2. plies that the behavior of the first odd and even supermodes(and If both the fundamental and second modes are excited equally the associated fundamental and second modes in the stem)is in the stem,the power output in each arm will depend on the phase dependent when traversing the junction,as is the case for accumulated relative phase difference between the modes in the second odd and even supermodes(and the third and fourth traversing the Y-junction.The power in the output arms can,modes in the stem),and so on,as demonstrated in Fig.3. therefore,vary from0%to 100%and 100%to 0%,respectively, An interesting consequence of the evolution of the third and as quantified by the 0,1:A and 0,1:Bcurves in Fig.3.In practical fourth modes concerns the symmetric,two-mode Y-junction devices,slight radiation losses and mode coupling due to nona- where the stem and arms all support just the first and second diabatic propagation can modify the total power in the output modes.If the second mode is excited in either or both arms. arms.These losses are dependent on the specific modes in ques-then none of the power is guided into the stem because the stem tion. cannot support the third and fourth modes into which it would The third (second-even)mode in the stem splits its power evolve,as shown qualitatively in Fig.5.In others words,the equally into identical second(first-odd)modes in the arms,as two-mode Y-junction acts as a three-port optical isolator (i.e., shown in Fig.4(a).The fourth(second-odd)mode also splits a linear unidirectional device),allowing propagation and a3 dB

LOVE AND RIESEN: SINGLE-, FEW-, AND MULTIMODE Y-JUNCTIONS 305 Fig. 2. Evolution of the second stem mode into the arms of a few-mode sym￾metric Y-junction. understood as the excitation of the first-even supermode across the output arms. In the reverse direction, exciting the FM in arm A (or B) is equivalent to exciting a superposition of both the first-even and first-odd supermodes with equal amplitudes, as shown in Fig. 1(b) [4]. The stem is single mode, allowing the even supermode from the arms to evolve into the FM of the stem, whereas the odd supermode evolves into the second or first-odd mode of the stem. Since the stem is single mode, this mode is radiated away from the stem, as shown in Fig. 1(b). Therefore, as required by the reciprocity of this adiabatic device, the Y-junction attenuates the signal from the arm by 3 dB. Similarly, injecting in-phase equal amplitude FMs into both arms is equivalent to the excitation of only the first-even super￾mode, and results in only the FM being excited in the stem with no radiation loss [4]. Generally, the output in the stem, however, depends on both the phase difference and amplitude difference between the two incident modes in the arms. With equal ampli￾tude and a phase difference, i.e., the first-odd supermode, all power leaks from the stem [4]. B. Few-Mode Symmetric Y-Junctions These principles are extended to higher order modes of few￾mode symmetric Y-junctions. For these Y-junctions, the second (first-odd) mode in the stem splits its power equally into two equal amplitude FMs in the output arms but with a phase difference between them (i.e., first-odd supermode), as shown in Fig. 2. If both the fundamental and second modes are excited equally in the stem, the power output in each arm will depend on the accumulated relative phase difference between the modes in traversing the Y-junction. The power in the output arms can, therefore, vary from 0% to 100% and 100% to 0%, respectively, as quantified by the 0,1:A and 0,1:B curves in Fig. 3. In practical devices, slight radiation losses and mode coupling due to nona￾diabatic propagation can modify the total power in the output arms. These losses are dependent on the specific modes in ques￾tion. The third (second-even) mode in the stem splits its power equally into identical second (first-odd) modes in the arms, as shown in Fig. 4(a). The fourth (second-odd) mode also splits Fig. 3. Power in the output arms of the symmetric Y-junction as a function of the phase difference between pairs of modes from a few-mode stem (modes indexed from 0). Fig. 4. Evolution of (a) the third mode and (b) the fourth mode, from the stem through a few-mode Y-junction. equally into the same second (first-odd) modes of the arms but with a phase difference between them, as shown in Fig. 4(b). As with the two-mode Y-junction, if both the third and fourth modes are excited equally in the stem, the power distribution of the second mode outputs in the arms (superposition of the second even and odd supermodes) depends on the relative phase difference accumulated between the modes in traversing the Y-junction. This can vary from 0% to 100% and 100% to 0% in each output, respectively, given an ideal adiabatic Y-junc￾tion. More generally, the behavior of the Y-junction depends on the phase difference between the same-order pairs of even and odd supermodes (or between the corresponding pairs of stem modes). The behavior of all other pairs of supermodes (or stem modes) traversing a junction is independent of phase. This im￾plies that the behavior of the first odd and even supermodes (and the associated fundamental and second modes in the stem) is phase dependent when traversing the junction, as is the case for the second odd and even supermodes (and the third and fourth modes in the stem), and so on, as demonstrated in Fig. 3. An interesting consequence of the evolution of the third and fourth modes concerns the symmetric, two-mode Y-junction where the stem and arms all support just the first and second modes. If the second mode is excited in either or both arms, then none of the power is guided into the stem because the stem cannot support the third and fourth modes into which it would evolve, as shown qualitatively in Fig. 5. In others words, the two-mode Y-junction acts as a three-port optical isolator (i.e., a linear unidirectional device), allowing propagation and a 3 dB

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]

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 retained

LOVE 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

308 JOURNAL OF LIGHTWAVE TECHNOLOGY,VOL.30,NO.3,FEBRUARY 1,2012 0.9 NHU Wavelength (Micrometers) 0.6 ×1.00 0.5 -1.11 01.22 9 14 11.55 0.3 1.66 0 0.2 17 0.1 201000 X (um) X (um) X (um) 2.5 4 (a). (b) (c). 3.5 4.5 Output Arm A Width (Micrometers) Fig.11.(a)(c)Separation and transformation of the three stem modes into Fig.12.Effect of variation in the source wavelength on the output power in FMs of the three output arms of a three-way Y-junction. arm A,for the asymmetric two-mode Y-junction with the FM excited in the stem. with only a change in amplitude to ensure power conservation with the change in core width.This transition is simulated in 0.9 Fig.11(a). 0.8 The second mode (first-odd)in the stem is transformed adi- 0.7 Index Contrast abatically into the FM of the next widest arm,as shown in 06 北之一三三之字空“x -￥-0.015 Fig.11(b),and likewise the third(second-even mode)of the 0.5 0.064 -0-0.112 stem is transformed into the FM of the third and narrowest arm, 04 -—0.209 as shown in Fig.11(c).The ordering of the output arms is not -0.258 0.3 -0.306 significant in a sufficiently adiabatic device because each mode 0.2 -0.355 in the stem will exit from the arm whose FM effective index it 0.1 best matches. The modal behavior in Fig.11 is reversible,so that one,two, 1.5 2 2.5 3 3.5 4.5 or all three modes in the stem can be excited through the ap- Output Arm A Width (Micrometers) propriate arms.Furthermore,the wavelengths of the FMs in Fig.13.Effect of variation in the index difference on the output power in arm the three arms are arbitrary,provided that the effective index A,for the asymmetric two-mode Y-junction with the FM excited in the stem. matching and single-mode conditions are still satisfied. The principle of mode sorting can be extended to multiple- output Y-junctions.The length of the device,however,scales power provided by the stem and arms,since the waveguide approximately as N4 with the number of modes N [1].This V-parameter decreases with increasing wavelength. limits the number of modes separable with a practical Y-junc- B.Dependence on Core-Cladding Index Difference tion to about four or five [1].More modes can,however,be sep- arated if restrictions on device length and insertion losses can Considering the same Y-junction as addressed earlier in Sec- be relaxed [1]. tion IV-A,increasing the index difference between the core and cladding generates a similar effect to wavelength changes,as IV.PARAMETRIC DEPENDENCE shown in Fig.13. This can also be explained in terms of the waveguide V-pa- In this section,we examine the effect of variations in the rameter of the Y-junction which decreases with decreasing source wavelength,the core and cladding index difference and index difference.The dependence of the output power-split on the Y-junction taper/divergence angle on the results presented the index difference rapidly decreases with higher values. in Section II. C.Dependence on Y-Junction Taper Angle A.Dependence of Power Split on Source Wavelength The conservation of total modal power propagating through The precise power split in each of the examples of Y-junc- a Y-junction depends critically on the taper angle between the tions studied in Section II will show some variation with source arms of the Y-junction.In the stem,the waveguide cross section wavelength.For example,in the case of the asymmetric,two- is constant and,therefore,modes propagate without loss,but be- mode Y-junction in Fig.8(stem width of 5 um),the split of FM yond this the cross section of the Y-junction increases linearly, power between the two arms varies,as shown in Fig.12,as a thereby enabling the guided modes to couple with one another function of wavelength. and also with the radiation field.Intuitively,the larger the taper As the source wavelength increases,the change in output angle,the more significant the losses. power split with increasing width of arm A (and hence de- This loss is quantified in Fig.14(blue Mode Sorting curves) creasing arm B width)becomes more rapid.This accelerated for the asymmetric two-mode Y-junction in Fig.8(stem width change can be attributed to the decreased confinement of modal of 5 um),where only the FM is excited in the stem.As the taper

308 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 30, NO. 3, FEBRUARY 1, 2012 Fig. 11. (a)–(c) Separation and transformation of the three stem modes into FMs of the three output arms of a three-way Y-junction. with only a change in amplitude to ensure power conservation with the change in core width. This transition is simulated in Fig. 11(a). The second mode (first-odd) in the stem is transformed adi￾abatically into the FM of the next widest arm, as shown in Fig. 11(b), and likewise the third (second-even mode) of the stem is transformed into the FM of the third and narrowest arm, as shown in Fig. 11(c). The ordering of the output arms is not significant in a sufficiently adiabatic device because each mode in the stem will exit from the arm whose FM effective index it best matches. The modal behavior in Fig. 11 is reversible, so that one, two, or all three modes in the stem can be excited through the ap￾propriate arms. Furthermore, the wavelengths of the FMs in the three arms are arbitrary, provided that the effective index matching and single-mode conditions are still satisfied. The principle of mode sorting can be extended to multiple￾output Y-junctions. The length of the device, however, scales approximately as with the number of modes [1]. This limits the number of modes separable with a practical Y-junc￾tion to about four or five [1]. More modes can, however, be sep￾arated if restrictions on device length and insertion losses can be relaxed [1]. IV. PARAMETRIC DEPENDENCE In this section, we examine the effect of variations in the source wavelength, the core and cladding index difference and the Y-junction taper/divergence angle on the results presented in Section II. A. Dependence of Power Split on Source Wavelength The precise power split in each of the examples of Y-junc￾tions studied in Section II will show some variation with source wavelength. For example, in the case of the asymmetric, two￾mode Y-junction in Fig. 8 (stem width of 5 m), the split of FM power between the two arms varies, as shown in Fig. 12, as a function of wavelength. As the source wavelength increases, the change in output power split with increasing width of arm A (and hence de￾creasing arm B width) becomes more rapid. This accelerated change can be attributed to the decreased confinement of modal Fig. 12. Effect of variation in the source wavelength on the output power in arm A, for the asymmetric two-mode Y-junction with the FM excited in the stem. Fig. 13. Effect of variation in the index difference on the output power in arm A, for the asymmetric two-mode Y-junction with the FM excited in the stem. power provided by the stem and arms, since the waveguide V-parameter decreases with increasing wavelength. B. Dependence on Core-Cladding Index Difference Considering the same Y-junction as addressed earlier in Sec￾tion IV-A, increasing the index difference between the core and cladding generates a similar effect to wavelength changes, as shown in Fig. 13. This can also be explained in terms of the waveguide V-pa￾rameter of the Y-junction which decreases with decreasing index difference. The dependence of the output power-split on the index difference rapidly decreases with higher values. C. Dependence on Y-Junction Taper Angle The conservation of total modal power propagating through a Y-junction depends critically on the taper angle between the arms of the Y-junction. In the stem, the waveguide cross section is constant and, therefore, modes propagate without loss, but be￾yond this the cross section of the Y-junction increases linearly, thereby enabling the guided modes to couple with one another and also with the radiation field. Intuitively, the larger the taper angle, the more significant the losses. This loss is quantified in Fig. 14 (blue Mode Sorting curves) for the asymmetric two-mode Y-junction in Fig. 8 (stem width of 5 m), where only the FM is excited in the stem. As the taper

LOVE AND RIESEN:SINGLE-,FEW-,AND MULTIMODE Y-JUNCTIONS 309 RefeREnceS 0.9 [1]J.D.Love,R.W.C.Vance,and A.Joblin,"Asymmetric,adiabatic 0.8 Output Arm A multipronged planar splitters,"Opt.Ouantum Electron.,vol.28,no.4, Mode Sorting: 0.7 ·-Output Arm B Pp.353-369,1996. [2]H.Sasaki and I.Anderson,"Theoretical and experimental studies Power Splitting: Fundamental Mode on active Y-junctions in optical-waveguides,"IEEE J.Quantum 0.5 Second Mode Electron.,vol.14,no.11,pp.883-892,Nov.1978. 0.4 [3]A.G.Medoks,"Theory of symmetric waveguide Y-junction,"Radio Eng.Electron.P,vol.13,no.1,p.106,1968. 0.3 [4]M.Izutsu,Y.Nakai,and T.Sueta,"Operation mechanism of the 0.2 single-mode optical-waveguide-Y junction,"Opt.Lett.,vol.7,no.3, 0.1 I51 w.K.Bumns and A.F.Milton."Mode conversion in planar-diele 0 0.5 2.5 4.5 6.5 8.5 10.5 12.5 separating waveguides,"IEEE J.Quantum Electron.,vol.QE-11,no. Taper Angle(Degrees) L,pp.32-39,Jan.1975 [6]W.Y.Hung,H.P.Chan,and P.S.Chung,"Novel design of wide-angle single-mode symmetric Y-junctions,"Electron.Lett.,vol.24,no.18. Fig.14.Power outputs from the two arms of the asymmetric two-mode Y-junc- pp.1184-1185,1988. tion as a function of taper angle,with the FM excited in the stem (blue curves) [7]W.M.Henry andJ.D.Love,"Asymmetric multimode Y-junctionsplit- and the power in the first-even and first-odd supermodes of a symmetric two- ters,"Opt.Quantum Electron.,vol.29,no.3,pp.379-392,1997. mode Y-junction resulting from independent excitations of the first and second [8]N.Goto and G.L.Yip,"A TE-TM mode splitter in LINBO by proton- modes in the stem,respectively (red curves). exchange and TI diffusion,"J.Lightw.Technol.,vol.7,no.10,pp. 1567-1574,0ct1989. [9]J.Vandertol and J.H.Laarhuis,"A polarization splitter on LINBOa using only titanium diffusion,"J.Lighne.Technol.,vol.9,no.7,pp angle increases,the output of arm A decreases and that from 879-886.Jul.1991. arm B increases due to mode conversion at the junction,but as [10]J.D.Love and A.Ankiewicz,"Purely geometrical coarse wave- length multiplexer/demultiplexer,"Electron.Lett.,vol.39,no.19,pp. radiation loss begins to dominate with further increasing taper 1385-1386.2003. angle,the power in both arms eventually decreases. [11]K.Shirafuji and S.Kurazono,"Transmission characteristics of optical Similarly,the combined power in the first two supermodes of asymmetric-Y junction with a gap region,"J.Lighnw.Technol.,vol.9, the symmetric two-mode Y-junction output arms also decreases n0.4,Pp.426-429,Apr.1991. [12]N.Riesen and J.D.Love,"Dispersion equalisation in few-mode fi- at larger angles because of radiation loss as shown in Fig.14 bres,"Opt.Quantum Electron.,vol.42,no.9,pp.577-585,2011. (e.g.,red Power Splitting curves). [13]J.E.Baran and D.A.Smith,"Adiabatic 2 x 2 polarization splitter on LINBOa,"Photon.Technol.Lett.,vol.4,no.1,pp.39-40,1992. [14]J.D.Love and A.Ankiewicz,"Photonic devices based on mode con- version,"in Proc.Australian Conf.Opt.Fibre Technol,Sydney,Aus- tralia,2001,pp.80-81. V.CONCLUSION John D.Love was born in the U.K.in 1942.He received the M.A.and M.Maths degrees in mathematics from the University of Cambridge,Cambridge,U.K. In this paper,the conceptual framework for understanding and the M.A.,D.Phil.,and D.Sc.degrees in mathematics from the University of propagation through symmetric and asymmetric single-mode, Oxford.Oxford.U.K. few-mode,and multimode weakly guiding Y-junctions has Since 1973,he has been with The Australian National University,Can- berra,A.C.T.,Australia,researching theoretical aspects of fiber optics,planar been presented,with the functionality of these devices being waveguides,and associated light processing devices.He co-authored Optical confirmed using BPM simulations.The paper also demon- Waveguide Theory (Berlin,Germany:Springer,1983)and Silica-based Buried strates the properties of asymmetric Y-junctions that make Channel Waveguides and Devices (London,U.K.:Chapman Hall,1996). His research activities encompass both academic and industrial problems.In them suitable for low-loss wavelength-and phase-independent teaching,he is the Convenor for the Master of Photonics degree and teaches mode sorting.Foreseeable applications for asymmetric Y-junc- several undergraduate and masters courses in photonics. tions include mode-division multiplexing/demultiplexing for Dr.Love is an honorary Life Member of the Australian Optical Society. high-capacity data transmission or high-resolution distributed sensing Nicolas Riesen was born in Canberra,A.C.T.,Australia,in 1987.He received the B.Sc.degree majoring in physics and the B.Eng.degree (first class Hons.) in systems engineering from The Australian National University,Canberra,in ACKNOWLEDGMENT 2011,where he is currently working toward the Ph.D.degree at the Research School of Physics and Engineering (in collaboration with the Commonwealth Scientific and Industrial Research Organization). The authors would like to thank Dr.S.Madden and the Laser His main research interests include mode-division multiplexing and dis- Physics Centre,The Australian National University for the use tributed fiber sensing. Mr.Riesen received the University Medal from The Australian National Uni- of their software. versity.He is the recipient of an ANU research scholarship

LOVE AND RIESEN: SINGLE-, FEW-, AND MULTIMODE Y-JUNCTIONS 309 Fig. 14. Power outputs from the two arms of the asymmetric two-mode Y-junc￾tion as a function of taper angle, with the FM excited in the stem (blue curves) and the power in the first-even and first-odd supermodes of a symmetric two￾mode Y-junction resulting from independent excitations of the first and second modes in the stem, respectively (red curves). angle increases, the output of arm A decreases and that from arm B increases due to mode conversion at the junction, but as radiation loss begins to dominate with further increasing taper angle, the power in both arms eventually decreases. Similarly, the combined power in the first two supermodes of the symmetric two-mode Y-junction output arms also decreases at larger angles because of radiation loss as shown in Fig. 14 (e.g., red Power Splitting curves). V. CONCLUSION In this paper, the conceptual framework for understanding propagation through symmetric and asymmetric single-mode, few-mode, and multimode weakly guiding Y-junctions has been presented, with the functionality of these devices being confirmed using BPM simulations. The paper also demon￾strates the properties of asymmetric Y-junctions that make them suitable for low-loss wavelength- and phase-independent mode sorting. Foreseeable applications for asymmetric Y-junc￾tions include mode-division multiplexing/demultiplexing for high-capacity data transmission or high-resolution distributed sensing. ACKNOWLEDGMENT The authors would like to thank Dr. S. Madden and the Laser Physics Centre, The Australian National University for the use of their software. REFERENCES [1] J. D. Love, R. W. C. Vance, and A. Joblin, “Asymmetric, adiabatic multipronged planar splitters,” Opt. Quantum Electron., vol. 28, no. 4, pp. 353–369, 1996. [2] H. Sasaki and I. Anderson, “Theoretical and experimental studies on active Y-junctions in optical-waveguides,” IEEE J. Quantum Electron., vol. 14, no. 11, pp. 883–892, Nov. 1978. [3] A. G. Medoks, “Theory of symmetric waveguide Y-junction,” Radio Eng. Electron. P, vol. 13, no. 1, p. 106, 1968. [4] M. Izutsu, Y. Nakai, and T. Sueta, “Operation mechanism of the single-mode optical-waveguide-Y junction,” Opt. Lett., vol. 7, no. 3, pp. 136–138, 1982. [5] W. K. Burns and A. F. Milton, “Mode conversion in planar-dielectric separating waveguides,” IEEE J. Quantum Electron., vol. QE-11, no. 1, pp. 32–39, Jan. 1975. [6] W. Y. Hung, H. P. Chan, and P. S. Chung, “Novel design of wide-angle single-mode symmetric Y-junctions,” Electron. Lett., vol. 24, no. 18, pp. 1184–1185, 1988. [7] W. M. Henry and J. D. Love, “Asymmetric multimode Y-junction split￾ters,” Opt. Quantum Electron., vol. 29, no. 3, pp. 379–392, 1997. [8] N. Goto and G. L. Yip, “A TE-TM mode splitter in LINBO by proton￾exchange and TI diffusion,” J. Lightw. Technol., vol. 7, no. 10, pp. 1567–1574, Oct. 1989. [9] J. Vandertol and J. H. Laarhuis, “A polarization splitter on LINBO using only titanium diffusion,” J. Lightw. Technol., vol. 9, no. 7, pp. 879–886, Jul. 1991. [10] J. D. Love and A. Ankiewicz, “Purely geometrical coarse wave￾length multiplexer/demultiplexer,” Electron. Lett., vol. 39, no. 19, pp. 1385–1386, 2003. [11] K. Shirafuji and S. Kurazono, “Transmission characteristics of optical asymmetric-Y junction with a gap region,” J. Lightw. Technol., vol. 9, no. 4, pp. 426–429, Apr. 1991. [12] N. Riesen and J. D. Love, “Dispersion equalisation in few-mode fi- bres,” Opt. Quantum Electron., vol. 42, no. 9, pp. 577–585, 2011. [13] J. E. Baran and D. A. Smith, “Adiabatic 2 2 polarization splitter on LINBO ,” Photon. Technol. Lett., vol. 4, no. 1, pp. 39–40, 1992. [14] J. D. Love and A. Ankiewicz, “Photonic devices based on mode con￾version,” in Proc. Australian Conf. Opt. Fibre Technol., Sydney, Aus￾tralia, 2001, pp. 80–81. John D. Love was born in the U.K. in 1942. He received the M.A. and M.Maths. degrees in mathematics from the University of Cambridge, Cambridge, U.K., and the M.A., D.Phil., and D.Sc. degrees in mathematics from the University of Oxford, Oxford, U.K. Since 1973, he has been with The Australian National University, Can￾berra, A.C.T., Australia, researching theoretical aspects of fiber optics, planar waveguides, and associated light processing devices. He co-authored Optical Waveguide Theory (Berlin, Germany: Springer, 1983) and Silica-based Buried Channel Waveguides and Devices (London, U.K.: Chapman & Hall, 1996). His research activities encompass both academic and industrial problems. In teaching, he is the Convenor for the Master of Photonics degree and teaches several undergraduate and masters courses in photonics. Dr. Love is an honorary Life Member of the Australian Optical Society. Nicolas Riesen was born in Canberra, A.C.T., Australia, in 1987. He received the B.Sc. degree majoring in physics and the B.Eng. degree (first class Hons.) in systems engineering from The Australian National University, Canberra, in 2011, where he is currently working toward the Ph.D. degree at the Research School of Physics and Engineering (in collaboration with the Commonwealth Scientific and Industrial Research Organization). His main research interests include mode-division multiplexing and dis￾tributed fiber sensing. Mr. Riesen received the University Medal from The Australian National Uni￾versity. He is the recipient of an ANU research scholarship