Guided modes in the plane array of optical waveguides

It is known that for the isolated dielectric cylinder waveguide, the modes with a certain angular symmetry cannot be guided ones (without loss by radiation) if their frequency is smaller than the cutoff frequency ${\omega }_{*}$, which is completely determined by the waveguide refractive index and its radius. It is shown in the paper that the infinite plane periodic array of such waveguides possesses guided modes (with the same angular symmetry) within the frequency domain which is below the frequency ${\omega }_{*}$. This is due to the inevitable interaction between the waveguides. As far as the finite array is concerned, the modes in this frequency domain are weakly radiating ones, but their quality factor $Q$ increases as $Q\left(N\right)\sim N^3$, with $N$ being the number of the waveguides in the array. This dependence is both obtained numerically, using the multiple scattering formalism, and justified within the framework of a simple analytical model.

Figure 1

The optical waveguide array. The polar coordinates of radius vector $\mathbf{r}$ relative to different waveguides.

Figure 2

The dispersion curve for the infinite array.

Figure 3

The dependence of the quality factor on the number of the waveguides $N$ for the plane array of waveguides.

Figure 4

The partial amplitude for the mode with the highest-quality factor.