Modes of coupled photonic crystal waveguides

Abstract

We consider the modes of coupled photonic crystal waveguides. We find that the fundamental modes of these structures can be either even or odd. in contrast with the behavior in coupled conventional waveguides, in which the fundamental mode is always even. We explain this finding using an asymptotic model that is valid for long wavelengths.© 2004 Optical Society of America OCIS codes: 130.2790, 230.7370. 050.1960 Coupled waveguides (CWs) occur in many optical devices, For example, the key element of directional couplers consists of two waveguides that are closely spaced to allow energy exchange. CWs have been studied both in conventional guided wave structures' and in photonic crystals.I:"The latter have received much recent attention with the claim that short coupling lengths, the length over which energy couples between the guides, can be achieved, providing the promise of compact devices.An issue that has arisen is that of the bound modes of CWs. In symmetric conventional planar structures, the fundamental mode is even and the second mode is odd.' The equivalent issue for photonic crystal waveguides, which does not affect their operation as a directional coupler, is not so well understood. Boscolo et al. 3 argued that the fundamental coupled waveguide mode (CWM) is even, as in planar structures.However, here we show that for some structures the fundamental CWM is even or odd, depending on the guides' spacing. To illustrate the features of different geometries we use three examples, shown in Fig. 1. In all three cases we consider the polarization in which the electric field is orthogonal to the figure, The first [ Fig. 1(a)] is conventional planar CWs. The second geometry [ Fig. 1(b)] is CWs in a layered Bragg structure (period d). Finally [ Fig. 1(c)] we consider CWs in twodimensional photonic crystals with a square lattice of period d. The last two structures act only as waveguides and thus support only CWMs for frequencies within a bandgap of the periodic structure.The analysis of the three structures initially proceeds in a common way. The key outcome of this analysis is Eq. (8) ere ,8p = ,80 + 271' P / d are the direction sines of these orders, Xp = (k 2 -{3p2)1/2, where k is the wave number, are the associated direction cosines, and Yois a reference plane, taken to be the top dashed line in R~, representing the reflection off the semi-infinite structures surrounding the guides, is a scalar for the planar and the layered geometries and a square matrix for the two-dimensional photonic crystal. The same is true for RN and TN, indicating the reflection and transmission, respectively, of the barrier between the guides, which consists of N layers.Given these definitions, we continue the analysis by relating the fields in Fig. 1: r.-= R~Pr. + , r.+ = RNPf : -+ TNPh + , -Here P is the operator that propagates the field between the dashed lines in Fig. lover guide width h. Since the guides are uniform, the field can be written as a plane wave for the planar and the layered geometries and as a plane-wave superpo...

Complete power concentration into a single waveguide in large-scale waveguide array lenses

“…We note that negative coupling, in general, can be achieved when the fundamental waveguide mode is odd 35 . Incidentally, this is also possible in all-dielectric arrays based on photonic crystal waveguides 36 .…”

Experimental observation of intermodal dispersion in photonic crystal directional couplers

“…If a PhCDC with a separation of two rows of rods between the arms is implemented, the relative position of the even and odd supermodes can be reversed and the intermodal dispersion may change its sign. 16 This particular effect has been studied theoretically to implement an intermodal dispersion compensator. 17 The theoretical difference in the group delay ͑per unit length͒ between the two supermodes of the PhCDC as a function of frequency is shown in the inset in Fig.…”

Energy localization and transport in binary waveguide arrays

“…Moreover, binary arrays with different coupling coefficients have been considered, since they might offer a more feasible experimental framework in which to exploit a two-band structure in the linear and nonlinear regimes [14,15]. In this instance, the use of photonic crystal waveguides [16] or waveguides based on plasmonic confinement [17,18] offers a unique setting in which to exploit propagation in the so-called alternating positive and negative coupling regime [19,20]. Efremidis et al [20], in particular, studied nonlinearly uniform arrays where the coupling coefficients are equal in modulus but of opposite sign.…”