The coupling behaviour of two-core mode-selective couplers (MSC) depends on the spatial-orientation of the asymmetric higher-order modes. This restricts their use for mode de-multiplexing in few-mode fibre networks. The use of three-core MSC's is presented as a solution

Love, John D

Riesen, Nicolas

English

The Australian National University

Nicolas Riesen and John D. Love Physics Education Centre The Australian National University, Canberra ACT 0200, Australia   Abstract Summary The coupling behaviour of two-core mode-selective couplers (MSC) depends on the spatial-orientation of the asymmetric higher-order modes. This restricts their use for mode de-multiplexing in few-mode fibre networks. The use of three-core MSC's is presented as a solution.  Keywords- few-mode fibres; mode-selective couplers; mode-splitting; spatial-mode-division-multiplexing. I. INTRODUCTION The ever-increasing demand for high-bandwidth optical fibre communications is destined to challenge the limits of single-mode fibre. This has led to a renewed interest in the use of few-mode fibre to exploit the extra degrees of freedom provided by several propagating modes. Few-mode fibre avoids the severe modal crosstalk experienced in highly-multimode fibre, thus allowing for each mode to be considered an independent data channel. The independent excitation and detection of the modes however presents major challenges. Currently there are no simple techniques available for multiplexing and de-multiplexing more than two orthogonal spatial modes. The multiplexing/de-multiplexing of two (and possibly more) modes could however in principle be achieved using simple waveguide structures such as mode-selective couplers (MSC) [1–3], asymmetric Y-junctions [4–6] or multimode interference devices (MMI) [7]. The use of asymmetric Y-junctions has recently been delineated [8]. Asymmetric Y-junctions allow for mode-division-multiplexing of elliptical and highly-birefringent fibre. As for multimode interference devices, they are typically limited to merely separating (or combining) the asymmetric from the symmetric planar modes [7].  This paper is concerned with the use of mode-selective couplers. More specifically, a novel multiplexing/de-multiplexing technique is presented that uses simple three-core mode-selective fibre couplers (MSC's). Mode-selective fibre couplers involve a higher-order mode in one fibre coupling to the fundamental modes of one or more other fibres, and vice versa.1This situation differs from a traditional fibre coupler in which the mode-order remains unchanged. Mode-selective couplers rely on the matching of the modal propagation constants (e.g. using non-identical fibres). However if the                                                            N. Riesen is the recipient of both ANU, Canberra and CSIRO, Lindfield research scholarships.  higher-order mode is asymmetric (i.e. non-axisymmetric or having non-zero azimuthal number), the coupling behaviour  also depends on the spatial-orientation of its electric field. This poses somewhat of an issue if MSC's are to be used in fibre mode de-multiplexers. This is because the spatial-orientations of the asymmetric modes are inherently unstable in any sizeable length of circularly-symmetric fibre. In the case of a two-core MSC the power transfer can range anywhere from 0 to 100% depending on the spatial-orientation [2–3].  This paper however demonstrates that certain configurations of three-core MSC's can have 100% power transfer regardless of the spatial-orientation of the higher-order mode. This implies that three-core fibre MSC's can be used for mode-division-multiplexing of circularly-symmetric fibre. This is an obvious advantage over planar devices (i.e. asymmetric Y-junctions or multimode interference devices) which can only de-multiplex modes of fixed spatial-orientation (e.g. the modes of highly-birefringent or elliptical-core fibre). The main drawback of mode-selective couplers are their inherent wavelength dependence. II. COUPLED-MODE THEORY This section provides an analysis of three-core mode-selective fibre couplers, highlighting their ability to de-multiplex modes of arbitrary spatial-orientation. The general three-core mode-selective fibre coupler used in the analysis is shown in Fig. 1. The fibre cores are assumed to be both circularly-symmetric and weakly-guiding.         Figure 1. Three-core mode-selective fibre coupler (with LP11 shown in fibre 1)  Three-Core Weakly-Guiding Mode-Selective Fibre Couplers A higher-order linearly-polarized mode (LPlm) in core 1 is assumed to have the same propagation constant β, as the fundamental modes of two identical outer cores 2 and 3. Furthermore, the electric field of the higher-order mode in core 1 is assumed to have a cosine azimuthal dependence. This means that for the LP11 mode, for instance, the line defined by the angle α is perpendicular to the zero line of the anti-symmetric mode's field. So in other words if α = 0 the lobes of the LP11 mode, for instance, are bisected by a horizontal line.       Referring to Fig. 1, the angle between cores 2 and 3 relative to core 1 is given by ϕ. The coupling coefficients between cores 1 and 2, and cores 1 and 3 are given by C and C̄ , respectively. The coupling coefficient between cores 2 and 3 is given by D. It is assumed that D << C ⋃ C̄  and to a first order approximation D = 0. The assumption is accurate for sufficiently large values of ϕ. Note that even if D ≠ 0, the coupled-mode equations could still be solved analytically. The assumption of D = 0 is however necessary to ensure a truly α-independent MSC (i.e. 100 % power decoupled irrespective of the spatial-orientation, α). In the case of finite values of D, coupling between the outer cores will occur at length scales of π/D. This length scale should be much larger than the coupler length scale π/CR.  In this paper it is also assumed that the mode-selective coupler is weakly-coupled (i.e. coupling length per unit length is small) such that, |C| << |β| and |C̄ | << |β|. This means that the distance d, between core 1 and either of the outer cores is assumed to be sufficiently large such that there is a small overlap of the modal fields. In this case the fields of the fibre modes can be considered in isolation [2], [9]. The fields of the modes are also assumed to be orthonormal. Now, the z-dependences of each mode of the three fibre cores can be expressed as [9],                                .3,2,1)()( == iezbza zjiiβ  (1)  where the amplitudes b1, b2 and b3 satisfy the following coupled-mode equations,           .;; 1312321 bCjdzdbjCbdzdbbCjjCbdzdb==+=  (2)  Note that only the forward-mode equations are considered, which are assumed to be decoupled from the backward-mode equations. This is a valid assumption for weakly-coupled MSC's that are also phase-matched (i.e. β1 = β2 = β3 = β). Assuming core 1 initially contains all the power (i.e. b1(0) = 1 and b2(0) = b3(0) = 0), the modal powers along the three cores are solved analytically,                               )(cos)()( 2211 zzbzP Ω==  (3)                           )(sin)()( 222222 zCzbzP ΩΩ==  (4)                           )(sin)()( 222233 zCzbzP ΩΩ==  (5)  where P1(z)+P2(z)+P3(z) = 1, and Ω2 = C2+C̄ 2. If CR denotes the radial dependence of the coupling coefficients then [2],                )cos())(cos( αφα lCClCC RR −=−−=  (6)  where l is the azimuthal number of the higher-order mode. Note that since the coupler is assumed to be phase-matched, the coupling behaviour  is also reciprocal. From (6),                        .)(cos))((cos 22 αφα llCR +−=Ω    (7)  This implies that when the outer cores are at angle ϕ = π/2l + nπ/l, with n = 0,1,2,3,..., we have that Ω = CR. Therefore the power in each core is given by,                                  )(cos)( 21 zCzP R=  (8)                            )(sin)(sin)( 222 zClzP Rα=  (9)                            ).(sin)(cos)( 223 zClzP Rα=  (10)  The power decoupled from core 1 to the outer cores is therefore,                            )(sin)()( 232 zCzPzPP RT =+=  (11)  where the coupling length is given by zc = π/2CR.  Therefore all the power of a higher-order mode can be decoupled regardless of the azimuthal orientation of its field α, when using a three-core MSC with outer cores at angle ϕ = π(n+1/2)/l. In the special case where l = 0, Ω = √2CR but CR becomes CR/√2 so P2,3(z) = sin2(CRz)/2. In the case of l = 0 only a two-core MSC is of course needed for total power transfer, although a three-core structure could be used to reduce the coupling length. The radial dependence of the coupling coefficient (i.e. for coupling between core 1 and core 2, 3) depends on the fibre and source parameters and was previously derived [2],   )()()()/(22)1(212111112/11,32112/32,2122wKwKwKdwKnvvnuukClllcocolR+−∆−=ρρρ  (12)  where, k is the wave-number, ρi is the core radius, Δi is the relative index difference, ui and wi are the core and cladding modal indices, nco,i is the core index, K is the modified Bessel function of the second kind, d is the spacing between the axes of cores 1 and 2 (or 1 and 3), and vi is the normalized frequency. As before, subscripts denote the higher-order mode fibre (i = 1) or one of the two outer cores (i = 2). The coupling coefficient (12) assumes weak-coupling and weak-guidance. Note that in the case where l = 0, CR becomes CR/√2. III. NUMERICAL SIMULATIONS The following numerical Beam Propagation Method (BPM) simulations confirm the predicted α-independent behaviour  of the three-core fibre MSC's. The parameters common to both simulations are: nco,1,2 = 1.4446, Δ1,2 = 6.85×10-3 and λ = 2π/k  = 1550 nm. The simulations shown in Fig. 2, demonstrate the α-independent decoupling of an LP11 mode when using a three-core MSC with two outer cores that have an angular offset ϕ = π/2 with l = 1 and n = 0. Here, ρ1 = 4.338 μm, ρ2 = 2 μm and the coupler length is zc ≈ 3.0 mm. Note also that the power in the two outer cores could be combined in practice in order to avoid a power penalty.                      (a).                          (b).        (c).                          (d).       Max.              Min.  Figure 2.  The decoupling of an LP11 mode (i.e. see insets) from a two-mode core to two outer cores which have an angular offset ϕ = π/2. The total decoupled power is independent of the mode's spatial-orientation α, as demonstrated for (a) α = 0, (b) α = π/8, (c) α = π/4 and (d) α = π/2.   The simulations of Fig. 3 also demonstrate the α-independent behaviour  of three-core MSC's for the case of the LP21 mode. In this example ϕ = 3π/4 with l = 2 and n = 1. The parameters for this simulation are ρ1 = 6.48 μm, ρ2 = 1.992 μm and the coupler length is zc ≈ 4.1 mm.                           (a).                             (b).     (c).                             (d).          Max.               Min.   Figure 3. The decoupling of an LP21 mode (i.e. see insets) from a four-mode core to two outer cores which have an angular offset ϕ = 3π/4. The total decoupled power is again independent of the mode's spatial-orientation α, as demonstrated for (a) α = 0, (b) α = π/16, (c) α = π/8 and (d) α = π/4.   IV. MSC MULTIPLEXERS/DE-MULTIPLEXERS FOR FEW-MODE FIBRE NETWORKS This paper has shown that three-core mode-selective fibre couplers allow for the de-multiplexing of any given asymmetric spatial-mode (i.e. l > 0) regardless of its spatial-orientation. As suggested, two-core mode-selective couplers are nonetheless sufficient for de-multiplexing the symmetric spatial-modes (i.e. l = 0). Therefore a linear sequence of two- and three-core couplers constituting the de-multiplexer, could allow for the separation of all modes at the end of a few-mode fibre. A linear sequence of two-core mode-selective couplers would be sufficient for the multiplexer, since the spatial-orientation of the modes excited at the start of the fibre is irrelevant. V. CONCLUSION In summary this paper has presented and numerically verified, theory describing the behaviour  of three-core mode-selective fibre couplers. The paper has shown that these simple devices could be used for de-multiplexing all the power of any given fibre mode, regardless of its spatial-orientation. A series of inline two- and three-core MSC's could therefore be used as the mode demultiplexer of a few-mode fibre network, whereas a series of inline two-core MSC's could be used as the multiplexer. Difficulties are expected to arise when multiplexing/de-multiplexing modes with closely-spaced propagation constants. This problem can be avoided by limiting the number of modes guided by the fibre, so as to avoid a 'continuum' of propagation constants. The wavelength dependence also limits the level of wavelength division multiplexing (WDM) possible when using mode-selective couplers.  Major practical challenges concerning MSC's include their fabrication as well as minimizing their dependences on external variables [1], [10]. The principles presented in this paper also apply to other waveguide structures, such as buried-channel waveguides and possibly also microstructured optical fibres. ACKNOWLEDGMENT The authors would like to thank John W. Arkwright and Jong H. Chow for useful suggestions and discussion. REFERENCES [1] K. Y. Song, I. K. Hwang, S. H. Yun, and B. Y. Kim, “High performance fused-type mode-selective coupler using elliptical core two-mode fiber at 1550 nm,” IEEE Photonic. Tech. L., vol. 14, pp. 501–503, April 2002. [2] N. Riesen and J. D. Love, “Weakly-guiding mode-selective fiber couplers,” IEEE J. Quant. Electron., vol. 48, pp. 941-945, July 2012. [3] J. D. Love and N. Riesen, “Mode-selective couplers for few-mode optical fiber networks,” unpublished. [4] J. D. Love and N. Riesen, “Single-, few-, and multimode Y-junctions,” J. Lightw. Technol., vol. 30, pp. 304–309, February 2012. [5] N. Riesen and J. D. Love, “Design of mode-sorting asymmetric Y-junctions,” Appl. Optics, vol. 51, pp. 2778–2783, May 2012. [6] J. D. Love, R. W. C. Vance, and A. Joblin, “Asymmetric, adiabatic multipronged planar splitters,” Opt. Quant. Electron., vol. 28, pp. 353–369, July 1996. [7] J. Leuthold, R. Hess, J. Eckner, P. A. Besse, and H. Melchior, “Spatial mode filters realized with multimode interference couplers,” Opt. Lett., vol. 21, pp. 836-838, June 1996. [8] N. Riesen, J. D. Love, and J. W. Arkwright, “Few-mode elliptical-core fiber data transmission,” IEEE Photon. Technol. Lett., vol. 24, pp. 344-346, March 2012. [9] A. W. Snyder and J. D. Love, Optical Waveguide Theory, London, U.K.: Chapman & Hall, 1983, pp. 542–546. [10] K. Y. Song and B. Y. Kim, “Broad-band LP02 mode excitation using a fused-type mode-selective coupler,” IEEE Photon. Technol. Lett., vol. 15, pp. 1734–1736, December 2003. 

Three-core weakly-guiding mode-selective fibre couplers

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Three-core weakly-guiding mode-selective fibre couplers

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