Topologically protected frequency control of broadband signals in dynamically modulated waveguide arrays

We theoretically propose a synthetic frequency dimension scheme to control the spectrum of a light beam propagating through an array of evanescently coupled waveguides modulated in time by a propagating sound wave via the acousto-optical effect. Configurations are identified where the emerging two-dimensional synthetic space-frequency lattice displays a nontrivial topological band structure. The corresponding chiral edge states can be exploited to manipulate the frequency spectrum of an incident beam in a robust way. In contrast to previous works, our proposal is not based on discrete high-Q cavity modes and can be applied to the manipulation of broadband signals with arbitrary spectra.

Figure 1

(a) Concept of the traveling sound wave used to modulate a single optical waveguide. (b) Induced 1D spectral lattice in the dispersion relation of the fundamental waveguide mode. (c) Dispersion relation in the frequency lattice, that is the longitudinal wave vector $k_z$ plotted as a function of the frequency momentum $k_f$. (d) Top (bottom) Illustration of the qualitatively different dynamics for narrowband (broadband) light beam compared to the modulation frequency ${\omega }_m$.

Figure 2

Power spectral density of light traveling in a dynamically modulated waveguide as a function of the propagation distance. (a) and (c) Diffraction in the frequency space of a narrowband (a) or broadband (c) signal with, respectively, $\mathrm{\Delta }\omega =0.3{\omega }_m$ and $\mathrm{\Delta }\omega =3{\omega }_m$. (b) and (d) Bloch oscillations in the frequency space of the same narrowband (b) and broadband (d) signals in the presence of a finite phase mismatch $\mathrm{\Delta }k=\beta /2$. (e) and (f) Propagation in the frequency space of the broadband signal injected in the waveguide, now with a time delay $\mathrm{\Delta }T=\pm \pi /\left(2{\omega }_m\right)$. For this and all the following simulations $800\mathrm{n}\mathrm{m}$ light and a $\delta n=5\times {10}^{-5}$ refractive index modulation are considered.

Figure 3

(a) Scheme of a dynamically modulated array of coupled waveguides. (b) Scheme of the two-dimensional Harper-Hofstadter lattice in the hybrid space composed of the spatial and the frequency dimensions with the corresponding hopping amplitudes. (c) Dispersion relation of a $\mathrm{\Phi }=1/4$ HH lattice. Edge states are color coded accordingly to their localization as in (b): green is for left-edge localized states and orange for right edge. (d) Topological frequency conversion on the right edge of the modulated waveguide array when up–/down-shifting topological modes are selectively excited using properly delayed pulses. Main panels show the intensity distribution in the hybrid frequency space of output light. The insets display the power spectral density during propagation on the excited waveguide. Same parameters as in Fig. 2.

Figure 4

Spectral power density distribution as a function of the propagation distance in the presence of a frequency-dependent refractive index disorder for a broadband beam propagating in a single modulated waveguide (left) or in a unidirectionally propagating edge mode of a topological array (right). Same parameters of Fig. 2; the noise is uniformly distributed with maximum amplitude $\mathrm{\Delta }n\approx \beta /k_0$.

Figure 5

Spectral power density distribution in each waveguide after a propagation of $10\mathrm{c}\mathrm{m}$ when a strong absorption, centered at $\omega /{\omega }_m=9$, is considered. (Left) Broadband absorption acts as an effective hard wall for the whole wave packet. (Right) Narrowband absorption acts as an effective hard wall only for a fraction of the wave-packet spectrum, while the rest keeps propagating undisturbed. (Insets) Spectral power density distribution as a function of the propagation distance in the leftmost waveguide of the array.

Figure 6

Schematic of the proposed experiment. (a) Front view of the device. A thin layer of a piezoelectric material is deposited on a SiO2 sample with surface waveguides. (Inset) The evanescent tail of the Rayleigh wave supported by the piezolayer leaks into the fused silica waveguide modulating the refractive index via the acousto-optic effect. (b) Top view of the device. Interdigitalized transducers (IDTs) with a fixed distance $d$ are lithographed on the sample surface.