a Schematic representation of the structure composed of identical cylindrical waveguides arranged in a diamond chain configuration. Each unit cell j hosts three waveguides sj≡Aj,Bj,Cj forming a triangle with central angle θ. The distances between waveguide centres are dAj−Bj=dAj−Cj≡d, dBj−Cj=2dsin(θ/2) and dAj−Aj+1=2dcos(θ/2). The blue arrows indicate the couplings. b Refractive index profile of the waveguides, defined by ncore = 1.548, nclad = 1.540 and waveguide radius R = 1.9 μm. Field intensity of the ℓ=0 (green) and ℓ=1 (red) modes, where βℓ is the propagation constant of mode ℓ, k0 = 2π/λ0 is the vacuum wavenumber and λ0 is the light wavelength in vacuum. c Numerically calculated coupling strengths for separation distances d = 5 μm, 5.5 μm, 6 μm, 6.5 μm, 7 μm and 7.5 μm using λ0 = 700 nm. In particular, c0 (crosses) accounts for the coupling between the ℓ=0 modes, and c1 (circles) and c2 (squares) account for the coupling between the ℓ=1 modes with the same or opposite circulation directions, respectively. The dashed and solid lines correspond to the exponential fittings of c0d≈K0exp(−κ0d), c1(d)≈K1exp(−κ1d) and c2(d)≈K2exp(−κ2d), where K0 = 387 mm−1, κ0 = 1.17 μm−1, K1 = 19.39 mm−1, κ1 = 0.52 μm−1, K2 = 56.25 mm−1 and κ2 = 0.59 μm−1. The inset in c shows c2/c1 with respect to the separation distance d
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