## 12 Summary and Discussion  
Recently the BSCs above the light line were shown to exist in various systems of one-dimensional arrays of dielectric rods and holes in a dielectric slab [31; 32; 33; 37; 38; 44; 45; 46]. Similar acoustic BSCs called embedded trapped Rayleigh-Bloch surface waves were obtained in system of material rods [26; 29; 47; 98]. One could ask why BSCs occur in periodic dielectric structures (gratings) but not in homogeneous structures like a slab or a rod which can support guided EM modes below the light line only. Let us begin with the simplest textbook system of a dielectric slab infinitely long in the \(x,y\) plane with the dielectric constant \(\epsilon>1\). The Maxwell equations can be solved by separation of variables for scalar function \(\psi(x,y,z)=e^{ikx+ikyy}\psi(z)\) to result in bound states below the light line \(k_{0}^{2}=k_{x}^{2}+k_{y}^{2}\)[99] while all solutions above the light line are leaky [100]. The situation can be cardinally changed by replacing the continual translational symmetry by the discrete symmetry \(\epsilon(x,y,z)=\epsilon(x+p,y,z)\) where \(p=0,\pm 1,\pm 2,\ldots\). All space variables are measured in the period length.. Then the radiation continua of plane waves \(e^{ikx,nx+ikyy+ikz}\) are quantized \(k_{x,n}=\beta+2\pi n\), \(n=0,\pm 1,\pm 2,\ldots\) with the frequency \(k_{0}^{2}=k_{x,n}^{2}+k_{y}^{2}+k_{z}^{2}\). Here \(\beta\) is the Bloch wave vector along the \(x\)-axis, and the integer \(n\) refers to the diffraction continua [33]. The physical interpretation of this statement is related to that the slab with the discrete translational symmetry can be considered as a one-dimensional diffraction lattice in the \(x\)-direction. Let us take for simplicity \(\beta=0,k_{y}=0\). Assume there is a bound solution with the eigenfrequency \(k_{0,BSC}>0\) which is coupled with all diffraction continua enumerated by \(n\). Let \(k_{0,BSC}<2\pi\), i.e., the BSC resides in the first diffraction continua but below the others. Because of the symmetry or by variation of the material parameters of the modulated slab we can achieve that the coupling of the solution with the first diffraction continuum equals zero [37; 38; 44; 45; 46]. However the solution is coupled with evanescent continua \(n=1,2,\ldots\) giving rise to exponential decay of the bound solution over the \(z\)-axis. The length of localization is given by \(L\sim\frac{1}{\sqrt{4\pi^{2}-k_{0,BSC}^{2}}}\). Therefore, the evanescent diffraction continua play a principal role in the space configuration of the BSCs. Moreover, one can see from Figure 5 that in the limit \(h\to\infty\) the BSCs frequency \(k_{0 BSC}\to 0\) to leave no room for the BSCs with \(k_{0,BSC}>0\). In the present paper we chose another strategy to quantize the radiation continuum. We replace the rod with continual translational symmetry by a periodic array of dielectric spheres. Because of the axial symmetry of the array aligned along the \(z\)-axis the quantized continua are specified by two integers, \(m\) and \(n\). The first integer is the azimuthal quantum number and the second number defines discrete directions of outgoing cylindrical waves (17) given by the wave vector \(k_{z,n}=\beta+2\pi n\) in each sector \(m\) where \(\beta\) is the Bloch vector along the array. Bottoms of the particular continua with \(m=0\) and \(n=0,\pm 1\) and \(n=-2\) are shown in Figure 5. By arguments similar to those presented above for the grated slab we obtain that the BSC with \(\beta=0\) embedded into the first radiation continuum \(m=0,n=0\) is localized around the array. The symmetry of the system is also important for classifications of the BSCs which are labelled by the azimuthal number of the continuum \(m\) of cylindrical vectorial waves and the Bloch wave vector \(\beta\). The symmetry properties of the BSC play a very important role since it is difficult to provide a zero coupling even with the lowest continua \(n=0\) because of the degeneracy in polarization.