Fig. 1. (a) Schematic representation of a unitary cell used for the calculation of the dispersion relations as an eigenvalue problem. The periodicity Λ is along the z axis. (b) Scheme for the calculation of the beam propagation. A unitary cell contains a finite number of nanowires along the propagation direction (z axis), and the periodicity Δ is now along the x axis including PMLs.

Fig. 2. Schemes of (a) the periodical array of gold nanowires immersed in a homogeneous dielectric medium with refractive index nd=1.5. The height of the nanowires is e=150  nm, the width is w=80  nm, and the period is Λ=130  nm. (b) The same MNW chain on a substrate of refractive index nsub=2.0, and (c) an integrated structure of MNW on a dielectric waveguide with core index nw=2.0.

Fig. 3. (a) Dispersion relations for the quadrupolar (upper branch) and dipolar transversal (lower branch) Bloch modes. The propagation distance of the (b) quadrupolar branch is shorter than that of the (c) dipolar transversal mode.

Fig. 4. Energy density maps and electric field distribution at the Bragg condition for (a) the quadrupolar Bloch mode at λ=540  nm and (b) the dipolar transversal Bloch mode at λ=916  nm. The corresponding squares show the phase distributions and orientation of the charges.

Fig. 5. (a) Dispersion relation of the MNW chain on a dielectric substrate (nsup=1.5, nsub=2.0). The top and bottom branches belong to the quadrupolar and transversal Bloch modes, respectively. The middle branch corresponds to the excitation of the SPP at the interface between the metallic nanowires and the substrate. Energy density maps and electric field distributions at the Bragg condition for (b) the quadrupolar mode at λ=562  nm, (c) the SPP-like mode at λ=655  nm, and (d) the dipolar transversal mode at λ=997  nm. The charge distribution in (c) exhibits a dipolar longitudinal interaction between the MNW only at the metal–substrate interface.

Fig. 6. Dispersion curves of the integrated structure (red lines), the isolated dielectric waveguide (blue), and the isolated MNW chain (green lines). The quadrupolar Bloch mode (inset) is coupled to the dielectric waveguide at λ=546  nm, generating antisymmetric and symmetric supermodes. The dipolar transversal mode does not cross the fundamental TM0 mode of the dielectric waveguide.

Fig. 7. Transmission, reflection, and absorption spectra for the integrated structure. In the transmission curve, the quadrupolar mode is excited at λ=559  nm, and the constructive interference of the dipolar transversal mode is positioned at λ=990  nm. The minimum at λ=1055  nm is a cavity resonance effect. In the reflection curve, Bragg reflections are located at λ=465  nm, λ=557  nm, and at λ=960  nm.

Fig. 8. Amplitude maps of the Hy component of the electromagnetic field corresponding to (a) the excitation of the quadrupolar mode (λ=559  nm) and (b) the interference of the dipolar transversal mode with the fundamental TM0 mode of the dielectric waveguide at λ=990  nm.