Bloch oscillations in arrays of helical waveguides

We study optical Bloch oscillations in the one- and two-dimensional arrays of helical waveguides with transverse refractive index gradient. Longitudinal rotation of waveguides may lead to notable variations of the width of the band of quasienergies and even its complete collapse for certain radii of the helix. This drastically affects the amplitude and direction of Bloch oscillations. Thus they can be completely arrested for certain helix radii or their direction can be reversed. If the array of helical waveguides is truncated and the near-surface waveguide is excited, the helix radius determines whether periodic Bloch oscillations persist or are replaced by the irregular near-surface oscillations.

Figure 1

(a) $\beta Z$ versus normalized Bloch momentum $k_x/\mathrm{K}$ (dispersion curves) for the 1D array of helical waveguides for different radii of the helix $R=0\mu \mathrm{m}, 2\mu \text{m},4\mu \text{m},6.05\mu \text{m},8\mu \text{m}$, and $10\mu \text{m}$. Arrow indicates the direction of increase of helix radius $R$. (b) Scaled difference $\delta \beta Z$ of quasienergies in the center and at the edge of the Brillouin zone versus helix radius $R$. The inset shows schematic illustration of the 1D waveguide array.

Figure 2

Dynamics of light propagation in the 1D array of helical waveguides with a transverse refractive index gradient $\alpha =0.2{\mathrm{m}}^{-1}$, when only one central waveguide is excited (narrow excitation). Helix radius $R=0\mu \mathrm{m}$ (a), $3\mu \text{m}$ (b), $6.05\mu \text{m}$ (c), and $10\mu \text{m}$ (d). (e) Maximal scaled width ${\text{W}}_{\mathrm{m}}/d$ of the wave packet, acquired upon propagation, versus helix radius $R$.

Figure 3

Dynamics of light propagation in the 1D array of helical waveguides with a transverse refractive index gradient $\alpha =0.2$, when multiple waveguides are excited by a broad Gaussian beam. Helix radius $R=0\mu \mathrm{m}$ (a), $3\mu \mathrm{m}$ (b), $6.05\mu \mathrm{m}$ (c), and $10\mu \mathrm{m}$ (d), respectively. (e) Maximal scaled transverse displacement of the wave-packet center $D_{\mathrm{m}}/d$, acquired upon propagation, versus helix radius $R$.

Figure 4

Propagation dynamics around the edge of truncated 1D array of helical waveguides with a transverse refractive index gradient $\alpha =0.2$ for $R=0\mu \mathrm{m}$ (a), $3\mu \text{m}$ (b), $6.05\mu \text{m}$ (c), and $10\mu \text{m}$ (d). Broad Gaussian beam centered at the edge waveguide is used for array excitation.

Figure 5

(a) Sketch of the 2D square waveguide array. Field modulus distributions at $z=Z_{\text{Bloch}}/4$ (b),(c) and $z=Z_{\text{Bloch}}/2$ (e),(f) for excitation of only one central waveguide [panel (d)] in the array for helix radius $R=0\mu \mathrm{m}$ (b),(e) and $5\mu \text{m}$ (c),(f). Maximal scaled radius of the wave packet $R_{\mathrm{m}}/d$ at $z=Z_{\text{Bloch}}/2$ as a function of helix radius $R$.