If the spectrum of a soliton pulse extends into regions where second-order dispersion changes sign, then radiation into a new pulse, the dispersive wave, may occur at a phase-matching wavelength [1,2]. The generation of these waves is analogous to Cherenkov radiation [3] and extends the spectral reach of optical pulses [4]. The recent ability to control dispersion in microresonators has allowed accurate spectral placement of dispersive waves relative to a radiating cavity soliton [5]. Such dispersion-engineered control has made possible 2f–3f self-referencing of frequency microcombs [6] and octave-spanning double-dispersive waves [7]. Dispersive-wave generation in optical fibers has traditionally relied upon control of geometrical dispersion in conjunction with the intrinsic material dispersion of the dielectric [4], and this same method has been successfully demonstrated in microresonators [5]. Recently, spatial mode interactions in multimode fiber have also been used for this purpose [8–10].

Here, spatial mode interactions within a microresonator are used to phase match a soliton pulse to a dispersive wave. These mode interactions often frustrate the formation of solitons [11] and, as a result, microresonators are typically designed to minimize or exclude entirely the resulting modal avoided crossings [5,12–15]. Also, while dispersive wave phase matching is normally induced by more gradual variations in dispersion, spatial-mode interactions produce spectrally abrupt variations that can activate a dispersive wave in the vicinity of a narrowband soliton. Below, the demonstration of dispersive-wave generation by this process is presented after characterizing two strongly interacting spatial-mode families. The phase matching of the dispersive wave to the soliton is then studied, including the effect of soliton frequency offset relative to the pump, as is caused by soliton recoil or by the Raman-induced soliton self-frequency shift (SSFS) [5,13,16–18]. It is shown that this mechanism enables active tuning control of the dispersive wave by pump tuning.

In the experiment, an ultrahigh-$Q$ silica microresonator (3 mm diameter) with a 22 GHz free-spectral range (FSR) was prepared [19]. Typical intrinsic quality factors were in excess of 200 million (cavity linewidths were less than 1 MHz). Mode dispersion was characterized from 183.92 THz (1630 nm) to 199.86 THz (1500 nm) by fiber-taper coupling to a tunable external-cavity diode laser and calibrating the laser frequency scan using a Mach–Zehnder interferometer [13]. Multiple mode families were observed, and their measured frequency spectra are plotted versus mode number, $\mu$, as the blue points in Fig. 1(a). In the plot, a linear dispersion term corresponding to the FSR of the soliton-forming mode family ($\mathrm{\Delta }{\omega }_{-}$) at mode number zero is subtracted so that a relative-mode-frequency is plotted. Mode zero is by convention the mode that is optically pumped to form the soliton. Three weak perturbations of the soliton-mode family dispersion are observed for $\mu <0$. The mode family associated with one of the perturbations is plotted as the nearly vertical line of blue points. A much stronger interaction occurs near $\mu =165$, causing a strong avoided mode crossing that redirects the soliton-forming branch to lower relative mode frequencies.

 

Fig. 1. Dispersive-wave generation by spatial mode interaction. (a) Measured relative mode frequencies (blue points) of the soliton-forming mode family and the interaction mode family. Mode number $\mu =0$ corresponds to the pump laser frequency of 193.45 THz (1549.7 nm). Hybrid mode frequencies calculated from Eq. (1) are shown in green, and the unperturbed mode families are shown in orange. The dashed horizontal black line determines phase matching for ${\omega }_r=D_{1A}$. (b) Measured soliton optical spectrum with dispersive-wave feature is shown. For comparison, a ${\text{Sech}}^2$ fitting is provided in red. The pump frequency (black) and soliton center frequency (green) indicate a Raman-induced SSFS [also see Fig. 2(c)]. A microwave beatnote of the photodetected soliton and dispersive wave is shown in the inset (frequency scale is offset by 21.973 GHz; resolution bandwidth is 10 kHz).

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The mode frequencies can be accurately modeled using a coupled-mode approach. Accordingly, consider two mode families ($A$ and $B$) that initially do not interact and that feature frequencies ${\omega }_{A,B}(\mu )$. An interaction between the mode families is introduced that is characterized by a coupling rate $G$. The coupling produces two hybrid mode families with upper/lower-branch mode frequencies ${\omega }_{\pm }(\mu )$ given by the following expression [20–22]:

The soliton studied here is a dissipative Kerr soliton (DKS). The formation of DKSs has recently been described in both optical fiber [23] and microresonators [5,12–15]. This type of dissipative soliton [24] forms through a double balance of second-order dispersion with the Kerr nonlinearity and cavity loss with Kerr-induced parametric gain [12]. The optical spectrum of a DKS pumped at $\mu =0$ (193.45 THz or 1549.7 nm) using a fiber laser is presented in Fig. 1(b). The soliton is triggered and stabilized using the method described in [13,25]. For comparison, the ideal ${\text{Sech}}^2$ spectral profile that would occur under conditions of pure second-order dispersion [5,12,13] [mode $A$, dashed orange curve in Fig. 1(a)] is provided as the red envelope in Fig. 1(b). A small SSFS [5,13,16–18] is apparent in the measured soliton spectrum, as indicated by the spectral displacement of the soliton spectral center relative to the pumping frequency. The perturbations to the ideal spectral envelope that are caused by both the weak modal crossings ($\mu <0$), as well as the strong avoided modal crossing, are apparent. For $\mu >0$, a dispersive-wave feature is apparent (maximum near 198.62 THz or $\mu =235$). In contrast to the weak-avoided-crossing-induced distortion for $\mu <0$, the dispersive wave results from a resonance condition (see discussion below) and the comb teeth are accordingly enhanced in strength. The coherence of the soliton and dispersive wave is verified by measuring the electrical spectrum of the detected soliton and dispersive-wave pulse train using a photodetector [inset of Fig. 1(b)].

Phase matching between the soliton and the dispersive wave occurs when the $\mu$th soliton line at ${\omega }_p+{\omega }_r\mu$ (${\omega }_p$ is the pump frequency and ${\omega }_r$ is the soliton repetition frequency) is resonant with the $\mu$th frequency of the soliton-forming mode family, i.e., ${\omega }_p+{\omega }_r\mu ={\omega }_{-}(\mu )$. As an aside, the Kerr shift for mode $\mu$ is much smaller than other terms in this analysis and is neglected in the phase-matching condition. So that it is possible to use a graphical interpretation of the phase-matching condition based on the relative mode frequency of Fig. 1(a), ${\omega }_0+D_{1A}\mu$ is subtracted from both sides of the phase-matching condition to give the following condition:

If the soliton repetition frequency equals the FSR at $\mu =0$ (i.e., ${\omega }_r=D_{1A}$), then the r.h.s. of Eq. (2) is the horizontal dashed black line in Fig. 1(a) [repeated in Fig. 2(a)]. Under these circumstances, the dispersive wave phase matches to the soliton pulse at the crossing of that line with the soliton-forming mode branch. However, while the mode dispersion profile $(\mathrm{\Delta }{\omega }_{-}(\mu ))$ is determined entirely by the resonator geometry and the dielectric material properties, the soliton repetition rate ${\omega }_r$ depends upon frequency offsets between the pump and the soliton spectral maximum. Defining this offset as $\mathrm{\Omega }$, the repetition frequency is given by the following equation [26,27]:

 

Fig. 2. Dispersive-wave phase-matching condition and Raman-induced frequency shift. (a) Relative mode frequencies for the soliton and interaction mode families are shown [see Fig. 1(a)] with dispersive-wave phase matching as dashed lines [see Eq. (4)]. The black line is the case where ${\omega }_r=D_{1A}$, and the green line includes a Raman-induced change in ${\omega }_r$. The intersection of the soliton branch with these lines is the dispersive-wave phase-matching point (arrows). (b) Soliton optical spectra corresponding to small (red) and large (blue) cavity–laser detuning ($\delta \omega$). ${\text{Sech}}^2$ fitting of the spectral envelope is shown as the orange curves. (c) Left: SSFS, $\mathrm{\Omega }$, versus $1/{\tau }_s^4$ (${\tau }_s$ is pulse width). The theoretical line is calculated with $Q=166\text{million}$ (measured) and Raman shock time 2.7 fs [18]. Right: dispersive-wave spectra with cavity–laser detuning (soliton power and bandwidth) increasing from lower to upper trace. (d) Measured dispersive-wave peak frequencies (red points) and soliton repetition rate (blue points) are plotted versus SSFS. The dashed blue line is a plot of Eq. (3). The dashed red line uses Eq. (4) to determine the dispersive-wave frequency ($\approx {\mu }_{\mathrm{DW}}{D}_{1A}+{\omega }_0$) as described in the text. The offset for the repetition rate vertical scale is $D_{1A}=21.9733\mathrm{GHz}$.

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The offset frequency $\mathrm{\Omega }$ can be caused by soliton recoil due to a dispersive wave and also by the Raman-induced SSFS [5,13,16–18]. In this work, $\mathrm{\Omega }$ is dominated by the Raman interaction, because the typical dispersive-wave power is $<0.2\%$ of the soliton power, causing a negligible dispersive-wave recoil (recoil of less than one mode). Photothermal-induced change in $D_{1A}$ is another possible contribution that will vary ${\omega }_r$ as pumping is varied [28]. However, the thermal tuning of $D_{1A}$ is estimated to be $\sim -4.5\mathrm{kHz}/\mathrm{mW}$ (by measurement of resonant frequency photothermal shift of $\sim -40\mathrm{MHz}/\mathrm{mW}$). With total soliton power of less than 1 mW [13], this photothermal-induced change in repetition frequency is negligible compared with that caused by the Raman self-frequency shift (see below).

Combining Eqs. (2) and (3) gives the following phase-matching condition:

The frequency shift, $\mathrm{\Omega }$, repetition frequency, ${\omega }_r$, and the dispersive-wave frequency were measured for a series of soliton powers that were set by controlling the cavity-pump detuning frequency ($\delta \omega$) using the method in Refs. [13,25]. ${\omega }_r$ was measured using an electrical spectrum analyzer after photodetection of the resonator optical output. The offset frequency $\mathrm{\Omega }$ was measured on an optical spectrum analyzer by fitting the center of the optical spectrum [see Fig. 1(b)] to determine the spectral maximum and then measuring the wavelength offset relative to the pump. This same spectral fitting also allows determination of the soliton pulse width, ${\tau }_s$ [13]. Once the soliton pulsewidth is known, the pump–resonator frequency detuning operating point can be inferred using $\delta \omega \approx D_2/2D_1^2{\tau }_s^2$ [13]. $\delta \omega /2\pi$ ranged between 7.8 and 21.1 MHz during the measurement. As an aside, a plot of $\mathrm{\Omega }$ versus $1/{\tau }_s^4$ in Fig. 2(c) (left) verifies that $\mathrm{\Omega }$ is dominated by the Raman self-frequency shift [18].

The soliton repetition rate is plotted versus $\mathrm{\Omega }$ in Fig. 2(d) and is fitted using Eq. (3). The intercept closely agrees with $D_{1A}$, and the slope allows determination of $D_{2A}/2\pi =14.7\mathrm{kHz}$ [in good agreement with 15.2 kHz from fitting to the measured dispersion curve in Fig. 1(a)]. The dispersive-wave frequency is also plotted in Fig. 2(d) versus $\mathrm{\Omega }$ and compared with a calculation using Eq. (4). In this calculation, $\mathrm{\Delta }{\omega }_{-}(\mu )$ is approximated using a linear expansion in $\mu$ near $\mu =200$. Also, a $-60\mathrm{kHz}$ offset is added to $D_{1A}/2\pi$ in $\mathrm{\Delta }{\omega }_{-}(\mu )$ due to the calibration uncertainty ($\sim \pm 100\mathrm{kHz}$) of FSR [29]. No other free parameters are used in the plot.

Spatial mode interactions provide a way to phase match a DKS to a dispersive wave. The degree to which these interactions can be engineered and controlled is an active area of investigation. Geometrical control of dispersion over broad spectral spans using microfabrication methods [30] could be applicable for dispersive-wave control.

It has been shown theoretically and through measurement that the dispersive-wave frequency can be actively tuned because of coupling of the soliton repetition rate to the offset frequency $\mathrm{\Omega }$. In the silica microcavities tested here, this offset is dominated by Raman-induced SSFS, and the dispersive wave is predicted and observed to tune to higher frequencies with increasing soliton power and bandwidth. As a further test of the theory, the dependence of repetition frequency on SSFS was combined with measurement to extract resonator dispersion parameters, which compared well with direct measurements based on resonator dispersion characterization. The dispersion induced by modal interactions in the tested device has been measured and accurately modeled using a coupled-mode formalism. While the limitations imposed upon the spectral proximity of the dispersive wave to the soliton are under investigation, it seems possible that more complex resonator designs could not only engineer the placement of these crossings, but also locate multiple avoided crossings near a soliton so as to induce multiplets of dispersive waves.

Note: During submission of this work, Matsko et al., reported on Cherenkov radiation by avoided mode crossings in microresonators [31].