Generation of nested frequency combs in a topological source

#The present invention generally relates to sources of radiation structured as frequency combs and, more particularly, to the generation of frequency combs configured as nested frequency combs in topological sources that include arrays of ring resonators possessing edge states, and frequency combs that remain robust against fabrication disorders.

#As known in related art, the term “frequency comb” is conventionally used to refer generally to a radiative output (and/or the frequency spectrum of such output) from a source of radiation (e.g., a laser source, as one non-limiting example) that is represented by a series of discrete, equally spaced frequency lines. Optical frequency combs, for example, can be generated by a number of mechanisms, including periodic modulation (in amplitude and/or phase) of a continuous-wave laser, four-wave mixing in nonlinear media, or stabilization of the pulse train generated by a mode-locked laser.

#Optical frequency combs find a multitude of applications in metrology, spectroscopy, and precision clocks, for example. While optical frequency combs emerge naturally in mode-locked ultra-fast lasers, using nonlinear parametric processes such as the Kerr effect in integrated photonics resonators offers a much more convenient and compact route to generate optical frequency combs and to realize, for example, on-chip transceivers for tera-bit scale wavelength-multiplexed optical communications, chip-scale light detection and ranging (LiDAR), on-chip frequency synthesizers, and optical clocks.

#The idea of the invention and wxamples of the present disclosure will be appreciated and understood by referring to the following Detailed Description of some of Embodiments in conjunction with the not-to-scale Drawings, of which:

#FIGS. 1A, 1B, 1C, 1D, 1E, 1F, and 1G illustrate the working of the topological frequency comb methodology configured according to an embodiment of the invention in comparison with the generation of the frequency comb with the use of a single ring resonator. FIG. 1A: Schematic of a single-ring resonator. FIG. 1B: power spectrum of the single-ring resonator in the linear regime. FIG. 1C: Temporal and spectral responses at the output of the single ring resonator in the regime of a single Kerr soliton. The temporal output includes a series of pulses separated by TR, which is the round trip time of the single ring resonator. The spectral output includes a series of narrow lines separated by FSR ΩR. FIG. 1D: An indicative spatio-temporal intensity distribution in the ring, showing different operating regimes as a function of input pump frequency detuning δωp from the cold-cavity resonance. FIG. 1E: A schematic of a 2D array of constituent ring resonators configured to simulate the anomalous quantum-Hall model for photons and exhibits topological edge modes at its boundary. FIG. 1F: Power spectrum of the embodiment of FIG. 1E in the linear regime showing edge state resonances (shaded) and bulk bands. The edge states extend throughout the boundary of the lattice and constitute a super-ring resonator, with longitudinal mode separation ISR. The transmission spectrum repeats every FSR ΩR of the individual ring resonators. FIG. 1G: Plots representing temporal and spectral distributions of the radiative output (the topological frequency comb) generated by the embodiment of FIG. 1E in the single nested soliton regime.

#FIGS. 2A, 2B, 2C, 2D, 2E, 2F, 2G, 2H, 2I, 2J, 2K, and 2L illustrate operation of a topological comb in the regimes of Turing rolls and chaos. Here, FIG. 2A illustrates total pump power in the super-ring resonator as a function of the input pump frequency detuning (ωp−ω0)/J, with normalized input pump field E=1.1. FIG. 2B depicts spectral power of the generated frequency comb 2κex|aIO,μ|2 in different FSRs (indexed by μ, the fast frequency). FIG. 2C: Total pump power in the super-ring resonator, for pump frequencies in one of the edge state resonances. FIG. 2D: Spatio-temporal (or fast-time) intensity distribution in the ring resonators, integrated over the rings on the edge of the lattice, Σ{m∈edge} |ãm,τ|2 as a function of the pump frequency detuning. This plot can be compared with that of a single ring resonator, shown in the inset of FIG. 1A. Analysis is performed for two different pump frequencies, δωp1=0.111 J and δωp2=0.08 J as indicated in FIGS. 2C and 2D. FIGS. 2E, 2F: Spatial intensity distribution in the lattice. FIGS. 2G, 2H: Output comb spectra at δωp1 and δωp2, in the regimes of phase-locked Turing rolls and chaos, respectively. The insets show slow frequency spectra for a given longitudinal mode μ. For plotting the comb spectra, ΩR was chosen to be 20. FIGS. 2I, 2J: Slow frequency spectra. FIGS. 2K, 2L: Slow-time temporal profiles at the output, for pump frequencies δωp1 and δωp2, respectively.

#Generally, the sizes and relative scales of elements in Drawings may be set to be different from actual ones to appropriately facilitate simplicity, clarity, and understanding of the Drawings. For the same reason, not all elements present in one Drawing may be shown in another.

#In accordance with preferred embodiments of the present invention, methods, and apparatus are disclosed for the generation of radiation of frequency combs of radiation and temporal Kerr solutions in a multidimensional array of evanescently coupled ring resonators and, in particular, of radiation with a spectrum judicially structured as a nested frequency comb.

#Specifically, the inability of single-ring resonator-based devices to generate frequency combs of radiation (e.g., optical radiation, or radiation in a THz frequency, or radio-frequency (RF) waves) outside of the limitations imposed on the spectrum of such combs by the free spectral range (FSR) of a given single-ring resonator is solved by devising a topological waveguide system that includes at least one judiciously-structured multidimensional array or lattice of mutually-coupled ring constituent resonators configured to ensure that: the frequency spectrum of an edge state of radiation (that propagates only around the peripheral portion (e.g., an edge portion) of the array of the system as a result of appropriately structured radiative pumping of such an array) includes a “supercomb” represented by a series of discrete groups of frequency combs (such groups being substantially equally spaced from one another by a frequency gap defined by the FSR of a constituent ring resonator), thereby defining a series of edge mode resonances oscillating within each FSR of each of the constituent ring resonators; the temporal profile of the radiative output produced by such an array of the system is defined by nested pulses, that is, by a series of groups of pulses of radiation (which pulses are delayed with respect to one another within a given group by a first delay time substantially equal to the round-trip time for propagation of radiation within the constituent ring resonator) while constituent groups of pulses of radiation are delayed with respect to one another by a second delay time substantially equal to the round-trip time for propagation of radiation in the edge state around the peripheral portion of the array; and such an edge state of radiation propagating through the system remains robust against any defects in the lattice of the device.