Kerr superoscillator model for microresonator frequency combs

Microresonator-based optical frequency combs (“microcombs”) have attracted lots of attention in the past few years thanks to their promising applications in telecommunications, spectroscopy, and optical clocks. The process of comb generation in microresonators can be modeled in the frequency domain using coupled mode equations and has recently been successfully described in the time domain using a nonlinear Schrödinger equation known as the Lugiato-Lefever equation. Time-domain approaches have brought many interesting insights for the understanding of microcombs. In this paper we present an intuitive frequency-domain model of microcombs that describes the overall structure of the spectra in terms of a few collective excitations of groups of neighboring comb lines, which we term “superoscillators.” This approach ties in nicely with the recently developed time-domain model based on soliton crystals and links the microcomb generation process with frequency response theory.

Figure 1

Illustration of the formation of frequency combs within microresonators and the application of the frequency response function to this process. (a) An optical microresonator is pumped with cw light, generating a frequency comb via four-wave mixing. (b) The envelope of the comb spectrum can be described as the response X(ω) of the microresonator, which is given by the force term F(ω) multiplied by the frequency response function H(ω). Shown here are the magnitudes of these three functions on a decibel (dB) (logarithmic) scale. F(ω) takes the form of a hyperbolic secant, which comes from the interaction of the Kerr nonlinearity with dispersion in the microresonator, and corresponds to the spectrum of a Kerr soliton [7]. H(ω) is a function containing poles and zeros (“positive and negative superoscillators”), characteristic of linear-response functions of complex systems with resonances and antiresonances. (c) The complex wave form circulating in the resonator causes a rapid and convoluted modulation of the instantaneous optical resonance frequency via the dynamic Kerr shift, which is depicted here as F(ω) acting on the resonance frequency via a system of masses and springs. The shifting of the resonance relative to the fixed pump frequency leads to a rapid modulation of the pump power coupled into the resonator.

Figure 2

(a) The experimental setup used to measure the combs’ phase spectra (see Refs. [21, 22] for a full description of this method). (b) Generation of superoscillators around a zero crossing of the detuning parabola for a comb in which the pump is blue-detuned from its cavity resonance. The detuning parabola represents the detuning ${\delta }_{\mathrm{rc}}$ of the various resonator modes relative to their corresponding comb modes in a resonator with pure quadratic anomalous dispersion and a pump frequency that is blue-detuned from its resonance. The parabola moves down, e.g., from the yellow to the blue curve as a result of the instantaneous Kerr shift from a passing light pattern in the microresonator. For each parabola, superoscillators are generated around the points where the detuning crosses zero as the parametric gain varies with ${\delta }_{\mathrm{rc}}$ as the Lorentzian $1/\phantom{1\left({\delta }_{\mathrm{rc}}^2+{\gamma }^2\right)}\left({\delta }_{\mathrm{rc}}^2+{\gamma }^2\right)$, where γ is the resonator's half linewidth. (c) Measurement of the amplitudes and phases of the lines of a bunched comb, showing a series of superoscillators. Each superoscillator exhibits a phase shift of π between the highest- and the second-highest-amplitude lines, exactly as predicted by the model.

Figure 3

Spectral features in the superoscillator model. The first column shows theoretical calculations of phases (green, upper panels) and power spectral densities (blue, envelope functions, lower panels) for different spectral features. The second column shows measurements of actual microcombs (red, comb lines) as well as phase measurements for sections of the comb (black crosses). The data are fitted and in excellent agreement with the superoscillator model (blue and green graphs). (a) and (b) Two positive superoscillators or poles, (c) and (d) two negative superoscillators or zeros, (e) and (f) one negative and one positive superoscillator, and (g) and (h) a negative and positive superoscillator separated by less than one comb mode spacing. The sections of the comb spectra containing the features in the left-hand column are highlighted with black rectangles. The combs (b) and (d) were generated in a silica microrod resonator and (f) and (h) in a silica disk resonator. The similarity of (d) and (h) indicates that this model applies equally well to both resonator geometries. The pump mode is the same in (f) and (h), although the coupling conditions (pump power, detuning, and taper position) are slightly different. The closest superoscillator to the pump is one-third of a free spectral range closer in (f) than in (h), which is due to the different pump detunings leading to vertically shifted detuning parabolas.

Figure 4

Derivation of the continuous envelope of a comb spectrum. For a comb resulting from a localized defect or group of defects in an otherwise regular lattice of solitons, increasing the size of the resonator while adding more solitons to the regular lattice so as to keep the lattice spacing constant means decreasing the comb spacing while keeping the same envelope. Taking this to its natural conclusion, we find that the envelope itself corresponds to the same defect in an infinite regular lattice of the same spacing. The defect shown here is an anomalous spacing of 3.01 times the lattice spacing. From left to right: 8, 18, 38, and an infinite number of solitons. The hyperbolic secant factor corresponding to the spectrum of a single soliton has been omitted from the spectra, meaning that these are in fact Fourier transforms of a series of Dirac $\delta$ functions at the centers of the solitons and that the envelope is what we have defined as the response function $H\left(\omega \right)$. The form of $H\left(\omega \right)$ here is that of Eq. (3), where $\tau$ is the spacing of the regular lattice in time and $\alpha =2.01$.

Figure 5

Positions of Dirac $\delta$ functions in the time domain that correspond to Eqs. (3) (top) and (4) (bottom) in the frequency domain. Both are an infinite lattice with spacing $\tau$ and a defect with a spacing of $\tau \left(1+\alpha \right)$. The lower graph contains an additional $\delta$ function that is offset by $\beta \tau /2$ from the center of the anomalous spacing.

Figure 6

Fits (blue, envelope functions, lower panels) of Eq. (B1), i.e., Eqs. (3) or (4) with an arbitrary frequency offset, multiplied by a hyperbolic secant with arbitrary width and frequency offset, to the peaks of the comb power spectra (red, comb lines, lower panels) shown in Fig. 3. Also shown are the phase spectra (green, upper panels) calculated from the fits and some measured comb line phases (black crosses). The fits are performed on a frequency rather than wavelength scale to ensure the perfect regularity of spacings necessary for a good fit over the entire spectrum. The combs are presented in the same order as in Fig. 3.