1. Introduction

The ability of Kerr microresonators to generate high-quality optical frequency combs has recently been the subject of significant attention [1,2]. Potential applications of these novel chip-scale sources have been demonstrated across diverse fields that range from medical diagnostics [3] to optical telecommunications [4]. For an important class of comb applications, most notably spectroscopy [5–8], imaging [3,9] and ranging [10,11], two independent frequency combs operated in a Vernier configuration are often desired. Such dual-combs allow the beat signals of adjacent comb pairs to be unambiguously mapped to the radio-frequency (RF) domain enabling high-speed, high-resolution measurement [12]. In a series of recent reports, researchers have been able to demonstrate dual-comb operation in optical microresonators, raising the tantalizing prospect of chip-scale dual-comb sources. The realisations reported to date include both dual-combs produced from separate microresonators fabricated onto the same chip [6,9], as well as dual-combs generated within a single microresonator [7,10,13–19].

For microresonator dual-combs, as for dual-combs in general, the ability to set both the frequency offset, and the difference in optical line-spacing, between the two combs is key to fully optimised operation. These parameters respectively set the RF comb’s center-frequency ($\upsilon _0$) and line-spacing ($\delta \upsilon$) and allow a user to choose the desired balance between measurement resolution and acquisition time. The use of two independent microresonators to realise a dual-comb enables direct control over both these comb variables. However, this approach does not permit common-mode operation of the two combs, thus introducing the potential for unwanted excess noise. Conversely, single resonator systems have been demonstrated to successfully generate stable dual-combs, via either the simultaneous excitation of co-propagating combs occupying different resonator spatial modes [14], or counter-propagating combs occupying the same spatial mode with offset driving powers [20] or detunings [21]. In these single resonator systems, both combs follow a strictly identical optical path ensuring excellent common-mode noise rejection. Unfortunately, however, this comes at the expense of considerably less flexibility in setting the RF comb parameters. Specifically, dual micro-combs realized through driving different spatial modes of the same resonator yield fixed values of $\upsilon _0$ and $\delta \upsilon$ set by the respective effective-indices of the two modes used. Likewise, dual micro-combs generated by counter-propagating pumps in the same spatial mode, have to date only been demonstrated with pumps driving the same azimuthal mode of the resonator. In this configuration, the RF comb is typically centred close to zero, and only small tunability in $\upsilon _0$ and $\delta \upsilon$ is possible by varying the ratios of drive powers or detunings for the two combs [20,21].

In this work, we present a new dual micro-comb configuration in which two counter-propagating combs are driven at different azimuthal mode frequencies; that is, different longitudinal modes of the same spatial mode family. This additional degree of freedom provides a simple mechanism that enables the discrete tunability of the comb’s center-frequency and RF line-spacing. Further fine-tuning is also demonstrated by varying the relative detuning of the two pumps. Experimentally, we use a magnesium fluoride (MgF$_2$) micro-disk resonator to generate counter-propagating dual micro-combs formed from either two combs occupying the same mode family, or two combs in different mode families. We then show that it is possible to excite counter-propagating combs different by two distinct pump frequencies separated by between 0 and 80 free-spectral-ranges (FSR). This allows us to realise a fine discrete tunability in both the RF comb’s center frequency and line-spacing. Experimentally, when driving two combs counterpropagating in the same mode family, we realise a tunable range of $\delta \upsilon$ between 100 kHz and 1 MHz (adjustable in steps of $\sim$10 kHz). We are also able to demonstrate a further $\sim$100 kHz of continuous tunability in $\delta \upsilon$ by adjusting the relative detunings of the two combs to alter the relative amount of nonlinear phase shift they experience [20].

2. Theory - counterpropagating combs in the same spatial mode

The generation of coherent frequency combs in a Kerr optical microresonator is realized through the excitation of localized dissipative structures known as temporal cavity solitons (CS), or alternatively, dissipative Kerr solitons [1,2]. We analyze first the case of two counter-propagating CSs occupying the same mode family of a single Kerr resonator. We consider a clockwise (CLK) propagating pump with a frequency detuning $\delta \omega _1$ from the $m$th azimuthal mode of the mode family of interest that generates the clockwise CS comb. A counter-clockwise (CCLK) pump with a frequency detuning $\delta \omega _2$ from the $(m+p)$th azimuthal mode of the same mode family produces the counter-clockwise CS comb. The fact that these two combs counter-propagate ensures negligible nonlinear interaction between them, and enables the excitation of two spectrally offset counter-propagating CS combs as shown in Fig. 1.

 

Fig. 1. Comb structure of the clockwise circulating comb (green) and the counter-clockwise circulating comb (orange) offset by $p$ azimuthal modes.

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We wish to control the offset in comb-line spacing between the two counter-propagating combs. This offset, key to the operation of Vernier dual-combs, arises from the combined influence of the dispersive and nonlinear properties of the resonator. Considering first the dispersive shift, we approximate the resonant frequencies of the mode-family of interest around the $m$th azimuthal mode via a quadratic expansion, ignoring higher-order dispersion: $\omega _{m+\mu }\approx \omega _m+D_1\mu +D_2\mu ^2/2$, where $\mu$ is the relative mode index, $D_1/2\pi = FSR_m$ is the free-spectral-range of the driven mode family evaluated at mode $m$ and $D_2 = dD_1 / d\mu$. The absolute frequency of the $n$th comb line of the clockwise soliton comb is thus located at an optical frequency,

By contrast, the counter-clockwise comb, spectrally offset from the first comb by $p$ azimuthal modes, will experience a different FSR and thus a different comb-spacing: $D_1'=2\pi FSR_{m+p} = D_1 + D_2p$. The absolute frequency of $n$th comb line of this comb can thus be written as

Equations (1) and (2) can then be used to evaluate the absolute RF frequencies arising from beating between adjacent lines of the two combs to yield the location of the $n$th RF comb line: $\upsilon _{n}=\upsilon _0+n\delta \upsilon _D$, where $\upsilon _{0}$ is the RF beat frequency between the center frequency of the counter-clockwise comb and the $p$th comb line of the clockwise comb,

Equations (3) and (4) thus explicitly show that as we vary the difference in azimuthal mode separation between the two combs ($p$), the line-spacing of the resultant RF comb will vary linearly with $p$, and the center frequency will vary as $p^2$. This is the key result of this paper, and shows that both these parameters are amenable to fine adjustment by adjusting the azimuthal mode number of the driving mode of either of the two combs.

In addition to these dispersive shifts, we note that there is also a smaller continuously tunable shift in comb line-spacing offset due to the difference in self-phase-modulation (SPM) experienced by the two counter-propagating solitons. The peak power of each CS is set by the frequency offset of its driving field from the closest resonator mode [22],

3. Theory - counterpropagating combs in different spatial modes

The expressions given above are valid for the case where the CLK and CCLK fields occupy a single spatial mode family. We now consider the case where the two fields occupy different spatial mode families, each described by their own set of dispersion parameters. The second comb thus acquires an additional constant frequency offset from the first comb $dD_0$, typically of the order of a fraction of the FSR, and possesses a free spectral range at mode $p$, $D_{12}'(p)=D_{12}+D_{22}p$, where $D_{ij}$ is the $i$th order of dispersion for the $j$th mode family. It follows that Eq. (3) can be rewritten:

4. Experiment

We experimentally realise the proposed dual micro-comb architecture in a monolithic MgF$_2$ micro-disk. The resonator is cut from a crystalline blank using single-point diamond turning and then hand polished to a finesse of 30000. The major diameter of the resonator is $\sim$1.25 mm, yielding an FSR of $\sim$55 GHz. A micron-diameter fiber taper is used to couple three optical pump waves to the resonator. The first two pumps are generated by amplified external cavity C-band lasers, tunable between 1525 and 1580 nm. These pumps act as the CLK and CCLK soliton pumps respectively. They are frequency-offset by an integer number of FSRs to couple to different azimuthal modes of the spatial mode families of interest. The third pump is derived from an amplified L-band external cavity laser operating at 1585 nm. This pump is combined with the CLK soliton pump using a fiber wavelength-division-multiplexer and coupled to a different resonator mode to act as an auxiliary beam to compensate for thermal effects. [23]. When correctly detuned this auxiliary field counteracts the thermal shifts induced by the CLK and CCLK pumps enabling access to the desired comb states simply by tuning the respective pump into the soliton resonance. We note that the use of two independent free-running lasers to drive the CLK and CCLK solitons in our setup will result in an increase in the phase noise of the resultant RF comb when compared to the use of two phase-coherent pumps. Whilst this does not invalidate the proof-of-concept experiments we present below, for realisations where optimium noise performance is desired, phase-coherent pumps should be used. These could be realised, for example, via a single laser source driving an electro-optic comb generator, with two teeth of this comb selected, filtered and amplified for use as the two soliton pumps.

Experimentally, we are able to observe three different mode families in this resonator that support soliton states. As above, we first consider the operation of this system with the CLK and CCLK pumps coupled to a single spatial mode family, and set to wavelengths around 1534 nm and 1561 nm, respectively. This represents a frequency shift of 61 azimuthal modes between the two pumps. In Fig. 2(a) we plot the comb spectra obtained when the CLK (green) and CCLK (red) pumps are separately tuned into a single soliton state whilst the other pump remains off-resonance. The RF spectra of these combs (right-hand panel) show the typical low-noise characteristics of a CS state, with only a single narrowband RF tone arising from the beating between the soliton comb and the off-resonance pump. These RF spectra are obtained with identical optical power incident on the photodiode and so can be directly compared. This is the case for all RF spectra presented in this paper. Next, both pumps are tuned into their respective soliton states and the two combs are combined using a fiber coupler. This combined spectrum is shown in Fig. 2(b) and is seen to match the sum of the two individual comb spectra of Fig. 2(a), with no evidence of any additional nonlinear interactions present. The combined spectrum is then filtered by a band-pass filter [transmission $\sim$ 1545 to 1558 nm, shown as black dashed lines in Fig. 2(b)] to retain only the section of comb spectrum between the two pumps. The RF spectrum of this dual-comb signal is then measured (right-hand panel) and seen to display the expected RF comb structure of an optical dual-comb. The RF line-spacing is measured to be 640 kHz which allows us to make an initial estimate of the mode-family dispersion of $D_2/2\pi \sim 10.5$ kHz.

 

Fig. 2. (a) Individual spectra of CLK (green) and CCLK (red) propagating soliton combs, (b) combined spectrum of CLK and CCLK propagating soliton combs in dual-comb operation. The right-hand panels show the associated low-frequency RF spectrum in each case.

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Next, we verify the operation of this dual micro-comb setup as the frequency shift between the two pumps is varied. Figures 3(a–c) show the optical and RF spectra measured for dual-soliton combs with frequency detunings of 44, 61 and 72 azimuthal modes respectively. As predicted, the RF beat frequencies of 420, 640 and 780 kHz are observed to increase linearly with the frequency detuning of the pumps. This is further confirmed in Fig. 3(d) where we plot the measured RF comb line-spacing over a wider range of pump separations. The solid blue circles show the measured RF line-spacing obtained when the two pumps are both positioned at the center of their respective soliton detuning ranges. A linear fit to this data (red line) confirms the predictions of the theory presented in the last section, and allows us to obtain a more accurate estimate of the second-order dispersion of this mode family ($D_2/2\pi =13$ kHz). The fact that we restrict our RF measurement to only optical comb lines located between the two pumps (see the dashed black lines in Fig. 2) means that it is difficult for us to easily track the center frequency of the RF comb, and so we are unable to directly confirm the predicted quadratic dependence of the RF comb’s center frequency with $p$ (Eqn. 3).

 

Fig. 3. Spectra of combined CLK and CCLK single soliton combs observed at pump separations of (a) 44, (b) 61 and (c) 72 azimuthal modes. The right-hand panels show the measured low-frequency RF spectrum in each case. (d) Measured RF comb line-spacing as a function of pump frequency separation (blue circles). The yellow line shows a linear fit to this data. The linear fit does not account for locking that may occur at small pump separations, however this behaviour was not observed for separations as low as 15 azimuthal modes. The green squares indicate the maximum and minimum values of RF line-spacing achievable by adjusting the CLK and CCLK detunings at a pump separation of 61 FSRs.

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In addition to the dispersive shift in RF line-spacing, we are able to continuously tune $\delta \upsilon$ by controlling the relative amount of SPM experienced by the two solitons. The green squares plotted in Fig. 4 show the accessible range of this nonlinear tuning, with the largest (smallest) shift obtained with the CLK propagating comb’s detuning set to the lowest (highest) value that can maintain a stable CS, and the CCLK comb’s detuning similarly set to the highest (lowest) value that can maintain a stable CS. The total tuning range accessible via this nonlinear shift was measured to be $\sim$100 kHz. We note that previous work has shown that when counter-propagating solitons are driven by two pumps that are very close in frequency (driving the same azimuthal mode) it is possible for the repetition rates of the two soliton to lock [13,20,21]. This locking is enabled by optical scattering inducing a weak coupling between the CLK and CCLK propagating fields. For the experiments presented above, the two pump frequencies separations are never less than 15 FSRs. Under these conditions the repetition rates of the two soliton combs are sufficiently different that we do not see any evidence of locking behaviour. Experimentally such an effect would be easily detected, as it would result in the disappearance of the RF comb.

 

Fig. 4. (a) Individual spectra of CLK (green) and CCLK (red) propagating soliton combs from different mode families. (b) Combined spectrum of CLK and CCLK propagating soliton combs in dual-comb operation. The right-hand panels show the associated RF beat spectrum around 6.24 GHz.

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We next consider the configuration in which the CLK and CCLK pumps are coupled to different spatial mode families. An indicative example of the comb spectra is shown in Fig. 4. The soliton driven by pump P2 now resides in a second mode family. This is evident from the change to the optical spectrum of the P2 comb. Similar to Fig. 2, we find that when each comb is excited separately, the RF spectrum remains in a low noise state (Fig. 4(a)), and when the two combs are excited simultaneously, the RF comb structure can be observed (Fig. 4(b)). Due to the frequency offset between different spatial modes ($dD_0$), we find the RF comb has been shifted to a considerably higher RF frequency ($\sim$6.24 GHz). Here, the dual-comb signal is free from technical noise present at lower frequencies, leading to a much clearer signal.

The RF comb characteristics are then measured as the pump separation is varied. Figure 5 shows the measured RF comb frequency as a function of azimuthal mode detuning for: (i) the original single mode family dual comb presented in Fig. 3(d) [gray trace], (ii) the two mode family dual-comb shown in Fig. 4 with an RF comb measured around $\sim 6.24$ GHz [blue trace], and (iii) a second two mode family dual-comb where the P2 soliton is excited in another mode family yielding an RF comb measured around $\sim 21.0$ GHz [orange trace]. In the latter two cases, the P2 soliton is fixed whilst the azimuthal mode of the P1 soliton is varied. As a result, all three curves depend on the dispersion of the P1 mode family ($D_{21}$) only and thus exhibit identical slopes. It is also interesting to note that, as predicted by Eq. (7) for the two-mode dual-combs, the RF comb line-spacing is not zero at $p = 0$ but rather set by $\Delta D_1$. This offset provides a useful measure of the difference in FSRs between the two mode families of the dual-comb. For the mode families examined here, we find this offset to be $2.48$ and $4.22$ MHz respectively.

 

Fig. 5. Measured RF comb line-spacing as a function of pump frequency separation for three different spatial mode combinations. The gray trace corresponds to the original single mode family dual comb presented in Fig. 3, the blue trace to the two mode family dual-comb shown in Fig. 4 with a measured $\upsilon _{0}\sim 6.24$ GHz, and the orange trace to a different two mode family dual-comb where the P2 soliton is excited in the third soliton mode family yielding a measured $\upsilon _{0}\sim 21.0$ GHz.

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Finally, we demonstrate a prototypical application of the microresonator dual-comb configurations presented in this work. Using the same mode families presented in Fig. 4, the CCLK pump (P2) is excited in a single soliton state, while the CLK pump (P1) is excited into a series of multi-CS states. In Figs. 6(a–c), we show the corresponding optical and RF measurements made with the P1 comb excited into a state with two bound CS, a two-CS crystal (where the two CS are at antipodal positions in the cavity), and a three-CS crystal respectively [24,25]. In each case, the measured RF spectrum can be clearly seen to reproduce the optical spectra of the P1 soliton, exactly as required for a dual-comb spectral measurement. From the two- and three- soliton crystal states, we expect to be able to measure slight differences in the comb line-spacing, however the comb line-spacing also has contributions from the detunings of the two counter-propagating soliton pumps (see Eqn. 6). These detunings changed from realisation to realisation, masking any systematic changes in the RF combs shown in Fig. 6.

 

Fig. 6. Spectra of combined CLK and CCLK soliton combs, with the CLK soliton combs corresponding to (a) two bound solitons, (b) a two-soliton crystal, (c) a three-soliton crystal. The right-hand panels show the associated RF beat spectrum around 6.24 GHz in each case.

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5. Conclusion

In conclusion, we have demonstrated that microresonator dual-combs driven by frequency offset pumps enable fine tuning of the output RF comb’s parameters. Using a magnesium-fluoride microdisk resonator that supports several different soliton mode families, we are able to demonstrate soliton dual-combs where the two solitons counter-propagate in both the same and different resonator mode families. We show that in both configurations, fine tuning of the RF comb parameters is possible by varying the azimuthal modes of the two driving fields. Experimentally we are able to demonstrate $\sim 1$ MHz of discrete tunablity in the RF comb line-spacing as the frequency offset between the two pumps is stepped from 0 to 80 FSR. A further continuous fine tuning $\sim 100$ kHz can be accessed via the difference in SPM the two soliton combs experience as a result of their relative pump detunings. This configuration provides a simple and robust method to generate a dual-comb source from a single microresonator device with tunable RF-comb characteristics that do not require any post-processing of the resonator itself.