## Abstract

The theory of mode-sorting in bimodal asymmetric Y-junctions is extended to multimode asymmetric Y-junctions with multiple output arms. This theory allows for the optimization of these mode-sorting planar structures. Asymmetric Y-junctions provide unique opportunities for spatial mode division multiplexing (MDM) of optical fiber. Spatial MDM is considered paramount to overcoming the bandwidth limitations of single-mode fiber. The design criteria presented in this paper facilitate their design.

© 2012 Optical Society of America

## 1. Introduction

#### A. Overview and Applications

Unlike most optical devices, Y-junctions rely purely on geometry for their functionality. The slow cross-sectional variation of these devices allows for the approximate adiabatic propagation of the modes through the junction [1]. Y-junctions can be categorized as either *symmetric* or *asymmetric*. Symmetric Y-junctions are suitable for equal power-splitting and have been studied extensively [2–5]. Asymmetric Y-junctions are less common, and are characterized by dissimilar output arms in terms of cross-sectional width or refractive index. They are sometimes used as polarization-splitters, mode-combiners, and mode-splitters in optical switches [6,7]. They can also be used as wavelength multiplexers [8], or as variable power-splitters [9]. More recently, asymmetric Y-junctions have been proposed as candidates for the spatial-multiplexing of few-mode optical fiber [10].

Spatial mode division multiplexing (MDM) is considered as one of the final frontiers for increasing optical fiber capacity. The potential increases in capacity arise from the extra degrees of freedom provided by several propagating modes. This approach however requires the independent excitation and detection of the spatial modes, which remains a challenge, especially for more than two modes [10]. The unique mode-sorting characteristics of asymmetric Y-junctions therefore provide invaluable opportunities for the spatial-multiplexing/demultiplexing of optical fiber. Asymmetric Y-junctions have for instance been proposed for the spatial-multiplexing of polarization-maintaining optical fiber for ultra-high capacity data transmission [10]. This paper presents a theory of mode-sorting in asymmetric Y-junctions that could aid in their design.

#### B. Qualitative Descriptions of Asymmetric Y-Junctions

The working mechanism of asymmetric Y-junctions [1,4,5], and their ability to separate several modes at the junction [1,11] have previously been reported.

In the case of a two-mode asymmetric Y-junction with two output arms (Fig. 1(a,b)), the fundamental mode in the stem exits as the fundamental mode in the wider of the two arms as shown in Fig. 1(a) [5]. This results from the matching of effective modal indices, as the fundamental mode has the largest effective index of all modes in the stem, and the wider output has the higher effective index of the two arms [8]. Similarly, the second ( first-odd) mode in the stem exits as the fundamental mode in the narrower of the two output arms as shown in Fig. 1(b). In other words, as a mode propagates through a Y-junction, it evolves into the *mode of the output arm with the closest effective index* [11]. This can however only occur when the cross-sectional variation at the junction is sufficiently small so as to ensure approximate adiabatic behavior. This can be achieved using a small branching/divergence angle between the arms.

The principle of mode-sorting can be extended to multi-arm Y-junctions with multiple modes in the stem (structure shown in Fig. 1(c)) [1]. The length of such a device however scales approximately as ${N}^{4}$ with the number of modes $N$. This limits the number of modes separable with a practical Y-junction to about four or five [1]. More modes can however be separated if restrictions on device length or insertion losses are relaxed [1].

#### C. Quantitative Descriptions of Mode-Sorting

Quantitative descriptions of the performance of mode-sorting have only been developed for the case of two-mode, two-arm asymmetric Y-junctions [5]. Burns and Milton proposed a mode conversion factor (MCF) for two-dimensional structures to this effect. In this paper we extend this theory to multi-arm Y-junctions with multiple modes in the stem, whilst providing criteria for their optimal design. The behavior of simple three-dimensional Y-junctions can be inferred. More specifically, the theory developed in this paper characterizes the level of mode-sorting in multi-arm asymmetric Y-junctions. The theory allows for simple numeric optimization of multi-arm, mode-sorting asymmetric Y-junctions in terms of key design parameters.

## 2. Analysis of Asymmetric Y-Junctions

#### A. Two-Mode Mode-Sorting Y-Junctions

The approximate behavior of a two-dimensional Y-junction with two modes in the stem and two output arms [Figs. 1(a) and 1(b)] can be described quantitatively using the *Mode Conversion Factor* (MCF) given below [5]. The MCF is based on a propagating mode analysis as well as coupled mode theory. It can be thought of as a measure of Y-junction asymmetry, with larger levels of asymmetry generally resulting in better separation of the two stem modes. The MCF also demonstrates that, unlike equal power-splitters, mode-sorters are suited to small divergence angles between the arms. Since radiation losses increase with divergence angle, low-loss mode-sorters are therefore easier to design.

Mode-sorting occurs when the MCF is greater than approximately 0.43, providing a transition boundary between mode-separation and power-splitting [5]. The MCF depends on the difference in propagation constants of the fundamental modes in the output arms. Therefore it assumes that no higher-order modes are excited, which can however occur if the effective index of the second-mode in an arm closely matches that of the second-mode in the stem. This can be avoided by ensuring that the asymmetry of the output arm widths is restricted. This is a particularly important consideration for multi-arm Y-junctions. Similar restrictions arise when the asymmetry is attributed to different refractive indices of the output arms.

#### B. Multimode Mode-Sorting Y-Junctions

We extend the principle of the MCF to multimode, multi-arm Y-junctions (as in Fig. 1(c)), whilst maintaining the assumption of only fundamental modes exiting the output arms. The theory assumes that the behavior of a multi-arm Y-junction can be described in terms of all pairs of output arms. This is an appropriate assumption considering that these are linear devices that operate purely by *phase-matching*.

The MCF of a specific pair of output arms, $i$ and $j$, is then given by:

*statistical combinations*of the MCF, which we refer to as the

*Multiple Output Factor*(MOF):

*first fundamental design criterion*for mode-sorting is therefore to minimize the MOF for a given value of $\theta $. This yields the propagation constants for the mode-sorter output arms, ${\beta}_{1},\dots ,{\beta}_{N}$,

This forms the *second fundamental design criterion* and should be equally weighted in the optimization. The excitation of higher-order modes in the output arms is avoided provided that the *largest value of the minimum mismatch between stem modes and the fundamental mode in a particular output arm, is significantly smaller than the smallest mismatch between a mode in the stem and a higher-order mode in an output arm*. This statement is a consequence of the phase-matching behavior of an asymmetric Y-junction. In other words, the propagation constant of each stem mode must closely match the fundamental mode of an output arm, and not an unwanted higher-order mode. Mathematically this second design criterion is represented as follows:

The propagation constants in Eqs. (6) and (7) are determined by solving the following set of transcendental equations. For the odd and even TM modes of a step-index slab waveguide [12]:

In summary, provided an asymmetric Y-junction is sufficiently long and adiabatic, such that minimal mode coupling/crosstalk and radiation losses are present [13], the condition for optimal mode-sorting with only the fundamental modes excited in the output arms is given by

#### C. Asymmetric Y-Junction Dimensional Space

The design of an $N$-arm asymmetric Y-junction with $N$ modes in the stem can also be aided by considering an $N$-dimensional space. This provides a means of interpreting the design criteria of Eq. (11). Assuming Cartesian coordinates with axes that denote the widths (or refractive indices, etc.) of each output arm, the assumption that no higher-order modes are excited in the arms is valid for a small $N$-dimensional region centered at the point of a symmetric Y-junction. Within this region, even smaller $N$-dimensional sub-regions represent successful mode-sorting.

In the simplest of Y-junctions, where the sum of the output arm widths is equal to the stem width, the space can be reduced to $N-1$ dimensions as the width of the ${N}^{\mathrm{th}}$ arm is determined from the previous $N-1$ arms. An example space for an $N=2$ Y-junction with arms labeled by A and B is given in Fig. 2. In this case the MOF reduces to the reciprocal of the MCF as shown. As mentioned, the fundamental mode (FM) exits the wider arm, whereas the second mode (SM) exits the narrower arm due to the matching of effective indices [5,13]. Note that the bold black lines signify the excitation of higher-order modes in either of the arms. Also shown are the sections ($*$) which appropriately satisfy both of the design criteria given in Eqs. (6) and (7).

An example of a typical $N=3$ Y-junction dimensional space is shown in Fig. 3. This example shows the regions (shaded red) which appropriately satisfy both of the design criteria given in Eqs. (6) and (7). The region satisfying only Eq. (7), where no higher-order modes exit the arms, takes the form of a triangle (shaded blue). The un-shaded regions show the preferred output arms for the fundamental (FM), second (SM) and third (TM) modes in the stem, as verified using the *Beam Propagation Method* (BPM).

It follows from Eq. (7) that in the case of an $N=4$ Y-junction, a polyhedron centered at the point of Y-junction symmetry represents the region sufficiently far from the onset of higher-order modes in the arms. This polyhedron will contain three-dimensional sub-regions having low MOF, analogous to Fig. 3.

## 3. Evaluation of Design Criteria

The design criteria of Eq. (11) are demonstrated for the cases of three- and four-arm asymmetric Y-junctions, respectively. We first assume a weakly guiding Y-junction with three output arms, a three-mode stem of width 6 μm, a source wavelength of 1.55 μm, and core and cladding indices of 1.4746 and 1.4446, respectively. The design criteria predict that suitable output arm widths for mode-sorting (with only fundamental modes exiting the arms), are 2.5 μm, 2 μm and 1.5 μm. These values have been optimized to within 0.1 μm and Fig. 4 confirms optimal mode-sorting for these predicted arm widths.

In the case where the design criteria are not satisfied either higher-order modes are excited in one or more output arms, or modes are split amongst the output arms. The restriction of only having fundamental modes exit the output arms and the requirement of strong mode-sorting are relatively difficult to achieve simultaneously, especially when considering more than two or three modes.

Assuming a four-arm Y-junction with a stem width of 8.8 μm and having the same parameters as above, the design criteria predict optimal arm widths of 3.5 μm, 2.7 μm, 1.7 μm and 0.9 μm. These values have been optimized to within 0.2 μm and Fig. 5 confirms the optimal mode-sorting of these output arm widths.

Since the length of the Y-junction scales approximately as ${N}^{4}$ [1], the number of modes separable with a practical Y-junction is however quite limited [1]. This length can be minimized by careful optimization of the mode-sorting, as demonstrated in this section.

## 4. Conclusion

In this paper, a quantitative description of the performance of mode-conversion in bimodal Y-junctions has been extended to $N$-arm Y-junctions with $N$ modes in the stem. This generalization allows for the design of mode-sorting Y-junctions. The design process can also be facilitated by considering an $N$-dimensional space. The design criteria developed for optimal mode-sorting, have been demonstrated for the case of three- and four-arm asymmetric Y-junctions. The separation of more than three or four modes is however limited in practice due to restrictions on device length and insertion losses. Nonetheless multi-arm Y-junctions have potential use for the spatial MDM of few-mode waveguides for ultra-high capacity data transmission.

The authors would like to thank Dr. Steve Madden and the Laser Physics Centre at The Australian National University for the use of their software. N. Riesen is the recipient of both ANU and CSIRO Research Scholarships.

## References

**1. **J. D. Love, R. W. C. Vance, and A. Joblin, “Asymmetric, adiabatic multipronged planar splitters,” Opt. Quantum Electron. **28**, 353–369 (1996). [CrossRef]

**2. **H. Sasaki and I. Anderson, “Theoretical and experimental studies on active Y-junctions in optical-waveguides,” J. Quantum Electron. **14**, 883–892 (1978). [CrossRef]

**3. **A. G. Medoks, “The theory of symmetric waveguide Y-junction,” Radio Engineering and Electronic Physics-USSR **13**, 106 (1968).

**4. **M. Izutsu, Y. Nakai, and T. Sueta, “Operation mechanism of the single-mode optical-waveguide-Y junction,” Opt. Lett. **7**, 136–138 (1982). [CrossRef]

**5. **W. K. Burns and A. F. Milton, “Mode conversion in planar-dielectric separating waveguides,” J. Quantum. Electron. **QE-11**, 32–39 (1975). [CrossRef]

**6. **W. M. Henry and J. D. Love, “Asymmetric multimode Y-junction splitters,” Opt. Quantum Electron. **29**, 379–392 (1997). [CrossRef]

**7. **W. Y. Hung, H. P. Chan, and P. S. Chung, “Novel design of wide-angle single-mode symmetric Y-junctions,” Electron. Lett. **24**, 1184–1185 (1988). [CrossRef]

**8. **J. D. Love and A. Ankiewicz, “Purely geometrical coarse wavelength multiplexer/demultiplexer,” Electron. Lett. **39**, 1385–1386 (2003). [CrossRef]

**9. **K. Shirafuji and S. Kurazono, “Transmission characteristics of optical asymmetric-Y junction with a gap region,” J. Lightwave Technol. **9**, 426–429 (1991). [CrossRef]

**10. **N. Riesen, J. D. Love, and J. W. Arkwright, “Few-mode elliptical-core fiber data transmission,” IEEE Photon. Technol. Lett. **24**, 344–346 (2012). [CrossRef]

**11. **J. D. Love and A. Ankiewicz, “Photonic devices based on mode conversion,” in Proceedings of Australian Conference on Optical Fibre Technology (ACOFT, 2001), pp. 80–81.

**12. **A. W. Snyder and J. D. Love, *Optical Waveguide Theory*(Chapman and Hall, 1983).

**13. **J. D. Love and N. Riesen, “Single-, few-, and multimode Y-junctions,” J. Lightwave Technol. **30**, 304–309 (2012). [CrossRef]