## 6 Light Guiding above the Light Line  
The arrays of dielectric nanoparticles can serve as subwavelength waveguides to be emploied as the key components for future integrated optics [15; 16; 17; 18; 19; 21]. They could be potentially advantageous against nanoplasmonics due to, for instance, the opportunity to control the frequencies of electric and magnetic Mie resonances by changing the geometry of high-index nanoparticles, and the absence of free carriers resulting in a high Q-factor. Arguably, the arrays of dielectric nanoparticles provide one of the most promising subwavelength set-ups for efficient light guiding [16; 22] as well as more intricate effects such as resonant transmission of light [22], and optical nanoantennas [23]. So far the major theoretical tool for analyzing the infinite arrays of spherical dielectric nanoparticles has been the coupled-dipole approximation [63; 64; 65]. In that approximation guided waves in arrays of magnetodielectric spheres were first considered by [PERSON] and [PERSON] [66; 67] who derived the dispersion relation and computed the dispersion curves for dipolar waves. Recently a more tractable form of the dispersion equations was presented by the same authors [68] with the use of the polilogarithmic functions. The dipolar waves in arrays of Si dielectric nanospheres were thoroughly analyzed in [21]. It was shown that only two lowest guided modes could be fairly described by the dipole approximation which breaks down as the frequency approaches the first quadruple Mie resonance. This limits the application of the dipolar dispersion diagrams to realistic waveguides assembled of dielectric nanoparticles. As an alternative to the dipole approximation a \"semiclassical\" approach based on the coupling of the whispering gallery modes of individual spheres could be employed to recover the array band structure [69; 70] if the wavelength is much smaller than the diameter of the spheres. The general case, however, requires a full-wave Mie scattering approach to account for all possible multipole resonances [19] involving a very complicated multiscattering picture which mathematically manifests itself in infinite multipole sums. Luckily, such an approach was recently developed by [PERSON], [PERSON], and [PERSON] who managed to obtain a multipole dispersion relation in a closed form suitable for numerical computations [24]. The above approach was used for analyzing the spectra of dielectric arrays above the line of light in Ref. [56]. It was demonstrated that under variation of some parameter such as, for example, the radius of the spheres the leaky modes dominating the spectrum can acquire an infinite life-time. In other words, the array can support bound states in the radiation continuum (BSCs) [33; 35; 37; 48; 71]. In this letter, we will address the ability of the BSCs to propagate light along the array primarily motivated by finding new opportunities for designing subwavelength waveguides. The dispersion diagram of an array of dielectric nanoparticles was obtained [57]. The dispersion curves were computed by solving the dispersion equations \(f_{d,m}(k,\beta)=0\), where \(k\) is the vacuum wave number \(k=\omega/c\), and \(\beta\) is the Bloch wave number, while the subscripts \(d,m\) designate either dipole [21; 68], or multipole [24] dispersion relations. For brevity we do not present the exact dispersion relations \(f_{d,m}(k,\beta)=0\). A mathematically inquisitive reader is referred to the above cited papers to examine the rather cumbersome expressions for \(f_{d,m}(k,\beta)\). Here we assume that the array consists of spherical no nanoparticles of radius \(R\) with dielectric constant \(\epsilon=15\) (Si) in vacuum. The centers of the nanoparticles are separated by distance \(a\). It is worth mentioning that at a given dielectric constant the dispersion is only dependent on a single dimensionless quantity \(R/a\). This allowed to scale the model for a microwave experiment [21].