## 6 Light Guiding above the Light Line
There are three types of dipolar solutions [21], namely; longitudinal magnetic (LM), longitudinal electric (LE), and transverse electromagnetic (TEM) waves. In Figure 7 we plot the lowest frequency modes of each type in comparison against the multipole solution [24]. In all cases if the \(k-\beta\) curve is above the light line \(k=\beta\) the vacuum wave number becomes complex valued. The imaginary part of \(k\) is linked to the mode life-time through the following formula:  
\[\tau=-[\varepsilon\Im(k)]^{-1}. \tag{30}\]Two approaches are possible for description of the leaky modes; complex frequency \(\omega\)[61; 72], or complex Bloch number \(\beta\)[21; 73]. In the latter case the inverse of the imaginary part of \(\beta\) is the penetration depth into the array \(L_{\tau}=[\Im(\beta)]^{-1}\). The quantities \(\tau\) and \(L_{\tau}\) are, in fact, proportional  
\[L_{\tau}=v\tau, \tag{31}\]  
where \(v\) is the group velocity \(v=d\Re(\omega)/d\beta\). Here, we do not present the imaginary part of \(\beta\) mentioning in passing that the penetration depths for dipolar waves were analyzed in refs. [21; 73]. What is important the numerical data available so far [21; 56; 73] indicate that all dipolar leaky modes are relatively short-lived, in particular, no dipolar BSCs were found in Ref. [56]. In compliance with Ref. [21] Figure 7 demonstrates that only two lowest eigenmodes are fairly described by the dipole approximation. Now, let us consider the multipolar quasi-guided modes within the first radiation continuum [56]. The dispersion curves for a leaky mode for two different ratios \(R/a\) are plotted in Figure 8a. One can see that in contrast to the dipolar waves in Figure 7 now the solutions could be long-lived with the life-time Equation (30) growing up to infinity at the BSC points. It should pointed out that for both \(R/a\) of all leaky modes of the array we plot only one which has a Bloch BSC point \(\Im(k)=0\) at \(\beta\
eq 0\). As shown in Figure 8b the BSC exists in a wide range of parameter \(R/a\). The magnetic and electric vectors could be found in terms of Mie coefficients \(a_{n}^{m}\), \(b_{n}^{m}\). For instance, outside the spheres one has for the electric vector \(\mathbf{E}(\mathbf{r})\)[24]  
\[\mathbf{E}(\mathbf{r})=\sum_{j=-\infty}^{\infty}e^{iaj\beta}\sum_{n=m^{*}}^{ \infty}\left[a_{n}^{m}\mathbf{M}_{n}^{m}(\mathbf{r}-\mathbf{r}_{j})+b_{n}^{m} \mathbf{N}_{n}^{m}(\mathbf{r}-\mathbf{r}_{j})\right], \tag{32}\]  
where \(j\) the number of the particle in the array, m - azimuthal number, \(m^{*}=max(1,m)\), and \(\mathbf{N}_{n}^{m}(\mathbf{r}),\mathbf{M}_{n}^{m}(\mathbf{r})\) are spherical vector harmonics [1]. Only \(m=0\) Bloch BSCs were found in Ref. [56]. Our numerics indicate that for BSCs in Figure 8 the dominating term in the expansions (32) corresponds to coefficient \(a_{3}^{0}\). In the insets in Figure 8b we plot the components of the electric and magnetic vectors of the BSC solution. One can see that the electromagnetic field is localized in the vicinity of the array. The amplitude of a wave propagating along the array attenuates exponentially according to a simple formula  
Figure 7: Dispersion diagram of an infinite array of dielectric nanospheres of radius \(R\) with dielectric constant \(\epsilon=15\), \(R/a=0.4\). The array centerline is aligned with the \(x\)-axis as shown in the south-east corner of the plot. The real parts of dipolar solutions are shown by dash-dot red lines. The thick gray lines are the real parts of the full-wave solutions; negative imaginary parts \(-\Im(k)\) of the full-wave solutions are shown by blue dashed lines. The thin gray line is the light line. \[F(x)=e^{-x/L_{\tau}}, \tag{33}\]  
where \(x=ja\) is the distance. In the vicinity of a BSC the \(\omega-\beta\) dependance could be approximated as  
\[\omega-\omega_{0}=v_{0}(\beta-\beta_{0})+\mathcal{O}[(\beta-\beta_{0})^{2}], \tag{34}\]  
where \(\omega_{0},\beta_{0},v_{0}\) are the BSC eigenfrequency, Bloch number, and group velocity, correspondingly.