1. INTRODUCTION

The development of the optical frequency comb (OFC) based on femtosecond pulsed mode-locked lasers [1–3] in conjunction with nonlinear spectral broadening constituted a dramatic simplification over large-scale harmonic frequency chains [4]. In the frequency domain, OFCs give equidistant optical lines, where the frequency of each component obeys $f_n=n·f_{\text{rep}}+f_0$. The spacing between the lines is determined by the pulse repetition rate of the laser $f_{\text{rep}}$ ($n$ being an integer). Knowledge of the comb’s overall offset frequency $f_0$ (also referred to as the carrier envelope offset frequency) along with $f_{\text{rep}}$ and $n$ allows linking the optical frequency $f_n$ to the electronically countable frequency $f_{\text{rep}}$ and $f_0$ in the radio frequency or microwave domain. In this way the measurement of $f_{\text{rep}}$ and $f_0$ corresponds to counting the cycles of light. Self-referencing of the comb, i.e., a self-contained measurement of $f_0$ and $f_{\text{rep}}$, has been achieved using nonlinear interferometers, whereby the comb’s bandwidth needs to be broadened to encompass typically two-thirds of or a full octave [5–8]. This measurement is a key prerequisite for many applications of OFCs. For example, referencing one of the comb lines of an OFC to an atomic frequency standard [9] allows the comb to function as a “gearbox,” realizing the next generation of atomic clocks based on optical transitions. Self-referenced frequency combs can also be used for optical frequency synthesis, and have enabled the most accurate frequency measurements [10,11]. Frequency combs with large mode spacings ($\ge 10\mathrm{GHz}$), which are challenging to obtain via mode-locked lasers [12], can be used in a growing number of applications, including astronomical spectrometer calibration [13], dual-comb coherent Raman imaging [14], high-speed optical sampling or coherent telecommunications [15]. In each of these applications, having a line spacing $\gtrsim 10\mathrm{GHz}$ is either beneficial or even required.

Different from mode-locked lasers, microresonator frequency combs (MFCs) [16,17] are generated using parametric frequency conversion [18,19], of a continuous-wave (CW) laser. This approach exhibits several attractive features that have the potential to extend further the use of OFCs to new areas in precision optical measurement, spectroscopy, astronomy, telecommunications, and industrial applications. Fundamentally different from mode-locked laser systems, MFCs offer high repetition rates ($>10\mathrm{GHz}$), compact form factors, broadband parametric gain that can be generated in wavelength regimes ranging from the visible [20] to the mid-infrared [21,22], CMOS-compatible microresonator platforms, and high power per comb line. Some unique characteristics of MFCs are that the pump laser constitutes a comb line and that there is no active laser gain medium in the system. These properties have encouraged in recent years the intense investigation of microresonator-based frequency combs with the ultimate goal of realizing a self-referenced system. Progress in recent years includes new microresonator platforms in crystalline materials [23], fused silica microtoriods [16], and photonic chips (based on silicon nitride [24,25], aluminum nitride [26], Hydex, [27,28] and diamond [29]). In addition, MFCs without self-referencing have been used for coherent telecommunications [15], compact atomic clocks [30], stabilized oscillators [31], and optical pulse generation [32–34]. The dynamics of microresonator frequency combs have been investigated, and regimes with low noise frequency comb operation have been identified based on intrinsic low phase noise regimes, or via tuning mechanisms such as $\delta -\mathrm{\Delta }$ matching [35], parametric seeding [36], and injection locking [36,37], or via the observation of phase locking [33]. Moreover recently, low noise frequency combs have been generated via temporal dissipative soliton formation [34,38–40] and numerical tools to simulate comb dynamics emerged based on the Lugiato–Lefever equation [41–43] and the coupled-modes equation [34,44].

However, despite the rapid progress in understanding, simulations, applications, and platforms, there is a milestone that has not yet been reached: a self-referenced microresonator system, capable of phase coherently linking the microwave and the optical frequency domain. So far, knowledge of the absolute frequency of all comb lines of a MFC has been achieved only by using an auxiliary self-referenced fiber laser frequency comb as a Ref. [45], as well as via locking of two comb teeth to a rubidium transition [30]. Self-referencing, however, has never been achieved. One reason for this is the high repetition rate of the microresonators, leading to correspondingly low pulse peak intensities. This makes external spectral broadening using nonlinear fiber, which is widely employed in mode-locked lasers, difficult to apply. While octave spanning combs [46,47] have been attained directly from MFCs, they have not been suitable for self-referencing techniques due to excess noise associated with subcomb formation [35].

Here, we demonstrate a coherent microwave-to-optical link using temporal dissipative soliton formation in a crystalline microresonator in conjunction with external spectral broadening, measuring simultaneously both $f_{\text{rep}}$ and $f_0$, which are necessary for linking the optical to the radio frequency domain. Our approach uses the newly discovered class of temporal dissipative cavity solitons in microresonators [34,39,48,49] and marks the first time that the carrier envelope offset frequency of such a soliton has been measured, to our knowledge. Our results demonstrate that MFCs generated by soliton formation are suitable for absolute frequency measurements. In particular, the pulse-to-pulse timing jitter is sufficiently low to enable precise measurement of the carrier envelope offset frequency via self-referencing. Although not demonstrated here, it has already been shown that microresonator comb parameters $f_{\text{rep}}$ and $f_0$ can be stabilized by controlling the pump laser frequency and by actuating the resonator free spectral range (FSR) by heating or mechanical stress [45,50].

2. EXPERIMENT AND RESULTS

Optical microresonators support different azimuthal optical whispering gallery modes (WGMs). The FSR between modes of a particular mode family is determined by material and geometric dispersion. It has been shown that WGMs can have quality factors exceeding ${10}^{10}$ [51,52]. When a CW pump laser is coupled to a WGM, the resonator’s Kerr nonlinearity can lead to efficient nonlinear frequency conversion. The resonator used in this work is a crystalline ultrahigh-$Q$ resonator made by polishing crystalline ${\mathrm{MgF}}_2$ [53–55] (see Fig. 1), and can support WGMs [51] confined in one of the protrusions that extend around the circumference of the resonator. The mode used has a quality factor of $\sim {10}^9$ and a FSR of 14.0939 GHz. Light from a CW fiber laser at 1553 nm with $\sim 150\mathrm{mW}$ can be coupled into and out of the resonator via evanescent coupling using a tapered optical fiber [56]. To form the solitons in the cavity, the pump laser’s frequency is scanned over the resonance and stopped when the appropriate conditions are met [34]. The duration of the pulse inside the resonator depends mainly on the coupling and detuning of the pump laser to the resonator mode, and can be estimated from the bandwidth of the spectrum shown in Fig. 1, to be $\sim 130\mathrm{fs}$. The optical spectrum generated by the soliton is not yet sufficiently broad for self-referencing; however, the spectrum can be broadened via supercontinuum generation [57].

 

Fig. 1. Crystalline ${\mathrm{MgF}}_2$ microresonator and temporal dissipative soliton generation. (a) Optical image of the employed ultrahigh-$Q$ crystalline whispering gallery optical microresonators on a magnesium fluoride pillar with a diameter of several millimeters. The ultrahigh-$Q$ whispering gallery optical modes are confined in the fabricated protrusions that extend around the circumference. The top resonator was used in the experiments and the mode of interest has a FSR of 14.0939 GHz and a loaded $Q\approx {10}^9$. (b) The hyperbolic-secant shaped spectrum (fit: red dotted line) of the single temporal soliton produced inside the resonator by the CW pump laser. The inset shows the ability to resolve the microresonator comb lines on a grating-based spectrometer.

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The experimental setup after the resonator is shown in Fig. 2. A portion of the pulse train produced by the resonator is sent to an optical spectrum analyzer (OSA). The rest of the pulse train then has the CW background, which consists of the residual pump laser [34] minimized by fiber optic filters. Then another small amount is sent to a photodetector and an electronic spectrum analyzer (ESA) to detect $f_{\text{rep}}$. The pulse is then preamplified with an erbium-doped fiber amplifier (EDFA). The main type of fiber in the experiment is SMF28, which has anomalous group velocity dispersion in the wavelength range of the soliton. Before being sent into a high power EDFA, the soliton pulse is prechirped using dispersion compensating fiber (DCF), which has normal group velocity dispersion, to minimize nonlinear effects in the EDFA. In this way the average power of the pulse train is increased to 2 W. After the EDFA, the pulse is recompressed using SMF28 fiber via the cut-back method to a duration of $\sim 300\mathrm{fs}$ with an energy of $\sim 150\mathrm{pJ}$. This pulse is subsequently sent through approximately 2 m of highly nonlinear fiber (HNLF) (Menlo Systems), where supercontinuum generation occurs. The resulting spectrum can be seen in Fig. 3. The blue trace shows the soliton spectrum after the optical microresonator, and the red after the HNLF fiber. The latter spectrum exceeds two-thirds of an octave, which is sufficient for self-referencing via a $2f-3f$ interferometer [5–8]. Importantly, the broadened spectrum is coherent. This is verified by using a heterodyne beat with additional external lasers at the two ends of the comb, as detailed below. Figure 3(b) shows a zoom into the spectrum taken after the HNLF fiber, where the individual comb lines are clearly visible, even with the limited resolution of the OSA (0.02 nm).

 

Fig. 2. Experimental setup and $2f-3f$ microresonator self-referencing scheme. (a) The frequency domain picture showing the relevant frequency components used to self-reference the comb and to determine the carrier envelope offset frequency ($f_0$). (b) The simplified experimental setup used for self-referencing. A portion of the solitons that are outcoupled from the resonator are sent to an OSA to measure the spectrum, then residual pump light is filtered out using fiber optic filters before a portion is picked off and sent to a photodetector (PD) to measure the repetition rate on an ESA. The pulse is first preamplified in an EDFA and then prechirped to temporally broaden it with a DCF before being amplified by a high-power EDFA. The pulse is subsequently recompressed and coupled into a HNLF where the coherent supercontinuum is generated. A fraction of the spectrum is mixed with light from the 1908 nm thulium fiber laser and sent through a 1908 nm bandpass filter to a PD and ESA to measure ${\mathrm{\Delta }}_{1908}$. The same is done with the 1272 nm external cavity diode laser to measure ${\mathrm{\Delta }}_{1272}$ and a servo loop is used to phase lock the laser to the optical frequency comb and fix ${\mathrm{\Delta }}_{1272}$ where a signal from an atomic clock is used as a reference. To create the $2f-3f$ interferometer light from the 1272 nm laser is frequency doubled in a periodically poled lithium niobate crystal (PPLN) to produce light at 636 nm. Light from the 1908 nm laser is frequency doubled to 954 nm in a PPLN crystal, and subsequently combined with 1908 nm and sent through a PPLN crystal phase matched for sum frequency generation, creating light at 636 nm. The generated visible light is optically heterodyned on a PD with the frequency doubled light from the 1272 nm laser, permitting us to measure ${\mathrm{\Delta }}_{2f-3f}$ on an ESA. This offset frequency is fixed by phase locking the 1908 nm laser via the $2f-3f$ interferometer. With this scheme the carrier envelope frequency is measured by recording ${\mathrm{\Delta }}_{1908}$.

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Fig. 3. Optical spectrum of the microresonator before and after external broadening. (a) The blue trace shows the optical spectrum generated in the crystalline optical microresonator by the temporal dissipative soliton state. The large central spike originates from residual light from the pump laser. The spectrum after the supercontinuum generation is denoted in red. It is composed of data taken from two different OSAs as result of the limited bandwidth of the individual instruments. (b) Zoom into the broadened spectrum revealing the widely spaced comb lines. (c) Heterodyne beatnote of a laser at 1272 nm with the broadened comb demonstrating a SNR exceeding 40 dB in the resolution bandwidth (RBW) of 300 kHz. (d) Heterodyne beatnote at the long wavelength end of the comb at 1900 nm (RBW 100 kHz).

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Self-referencing is achieved by measuring $f_{\text{rep}}$ and $f_0$ of the generated frequency comb. By picking off a small portion of the light after it leaves the resonator and sending it to a photodetector, $f_{\text{rep}}$ can be directly measured (see Fig. 2). The broadened spectrum is sufficiently wide and allows employing a $2f-3f$ interferometer [5] to determine $f_0$. This is traditionally implemented by frequency tripling a component of the low-frequency part of the spectrum $f_L=n·f_{\text{rep}}+f_0$ using a combination of second-harmonic and sum-frequency generation to give $3f_L=3n·f_{\text{rep}}+3f_0$, where $n$ is an integer. In addition, a component of the higher frequency part of the spectrum $f_H=m·f_{\text{rep}}+f_0$ is frequency doubled using second-harmonic generation to give $2f_H=2m·f_{\text{rep}}+2f_0$, where $m$ is an integer. Mixing the doubled and the tripled light and detection on a photodetector gives the offset frequency $3f_L-2f_H=f_0$, granted $3n=2m$, i.e., necessitating a spectrum that covers two-thirds of an octave. Here a scheme involving two transfer lasers is implemented, which has the advantage of allowing independent verification of the coherence of the supercontinuum generation at the two ends of the spectrum. The coherence of the generated broadband OFC is verified by optically heterodyning the two reference lasers at $\sim 1272\mathrm{nm}$ (external cavity diode laser) and $\sim 1908\mathrm{nm}$ (thulium fiber laser) with the supercontinuum and detecting the optical heterodyne beat signal on a photodetector (see Fig. 3). The beatnote frequencies can be written in terms of the frequency comb parameters and an offset as

 

Fig. 4. Repetition rate and carrier envelope frequency signals of the self-referenced microresonator comb. (a) The offset frequency ($f_0$) of the optical microresonator frequency comb divided by three as described in Eq. (7). The measured optical heterodyne beat frequency has a center frequency of 3.543 GHz and exhibits a SNR that exceeds 30 dB in a 100 kHz RBW. (b) The repetition rate $f_{\text{rep}}$ of the soliton in the optical microresonator with a center frequency (CF) of 14.0939 GHz and a SNR $>60\mathrm{dB}$ measured in a resolution bandwidth of 100 Hz. The large SNRs are sufficient for accurate phase tracking of the two microwave signals.

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It should be noted that the determination of the comb line index with a mode-locked laser frequency comb with repetition rates $<1\mathrm{GHz}$ can be challenging. Here the much denser comb spectrum does not normally permit resolving the comb line with a grating-based OSA, and moreover, the sensitivity to drifts in repetition rate is significantly higher, making it necessary to fully phase stabilize the comb in order to determine the comb line index. In contrast, the comb line index can be obtained in the present work with a low-resolution wavemeter without requiring locking of either $f_{\text{rep}}$ or $f_0$. In this way, the comb line number of the pump laser was determined to be $n_{\text{pump}}=13,696$. Self-referencing, along with knowledge of the comb line numbers, implies that the absolute laser frequency of the pump laser ($f_{\text{pump}}$, as well as any other comb teeth) can be directly determined via the repetition rate beatnote $(f_{\text{rep}})$ and the recorded beatnote $({\mathrm{\Delta }}_{1908})$, since the absolute laser frequency is related to the two quantities via

3. CONCLUSIONS

Our results constitute the first phase-coherent link from the microwave to the optical domain using a microresonator, by demonstrating measurement of the carrier envelope frequency of a temporal dissipative soliton in a microresonator. These results demonstrate that microresonator-based frequency combs can provide accurate and precise absolute optical frequency standards for a wide range of applications in optical frequency metrology, optical atomic clocks, optical frequency synthesis, or low-noise microwave generation by frequency division. In terms of soliton dynamics, this demonstration constitutes the first measurement of the carrier envelope offset frequency of a temporal dissipative Kerr cavity soliton, to our knowledge. This opens a new route to studying nonlinear dynamics of solitons. A further important step in the future will be to phase lock both $f_{\text{rep}}$ and $f_0$ to an external radio frequency reference, and thereby achieve a phase-stabilized self-referenced microresonator frequency comb. This can be achieved by controlling both $f_{\text{rep}}$ and $f_0$ independently via changing the pump frequency detuning along with either the pump power [45] or by applying a stress to the resonator with a piezoelectric crystal [50]. The crystalline-microresonator-based microwave-to-optical link can be made more compact with almost all optical components being fiber optic based, and aside from the optical microresonator, the non-fiber-based components (filters and sum-frequency generation stages) can be replaced with fiber-based components in the future. Finally, the external broadening stage itself, required in the present case, can, in suitably dispersion engineered optical microresonators, be avoided when making use of soliton-induced higher order spectral broadening [42,47,58]. Eventually, this provides a path to counting the cycles of light using chip-scale microresonators.