Ring resonators made with materials having cubic nonlinearity produce optical frequency combs when pumped with continuous wave (cw) light of sufficient power and certain frequency [1,2]. These frequency combs, dubbed Kerr combs or microcombs, can be as broad as an octave [3,4]. The Kerr combs may operate in the mode-locked regime, corresponding to a train of short optical pulses in time domain [5–7]. The mode-locked Kerr combs have a well-defined spectral shape, or envelope, described by a hyperbolic secant function. However, in the majority of experimentally observed Kerr combs, the spectral line corresponding to the pump frequency significantly exceeds the Kerr comb envelope. When such combs are recorded with a photodetector, the strong pump line leads to a large constant shift of the electrical signal (DC shift), making it hard to study various Kerr frequency comb regimes. The DC shift reduces the contrast of the detected optical pulses, degrades the signal to noise ratio, and can saturate the detector. The small comb contrast (amplitude ratio of the comb envelope to the pump frequency comb harmonic) occurs in part due to the geometrical mismatch between the pump light and the mode of the cavity. In addition, the conventional mode-locked Kerr combs have low conversion efficiency due to saturation of the nonlinear resonant process [8,9]. This happens because the pump laser frequency must be detuned from the resonator mode for soliton generation. Most of the pump power is reflected from the resonator under such detuning. Frequency combs with an ideal envelope free of the excessive pump harmonic are useful for many applications including radio frequency (RF) photonic filters and oscillators, fiber telecommunications, and spectrometers.

The add–drop coupler configuration, consisting of an input coupler (add port) and an output coupler (drop port), allows reducing the pump spectral line by the mode anti-crossing approach in normal dispersion Kerr combs [10]. However, the pump line generally exceeds the harmonics of the mode-locked Kerr comb retrieved from the drop port rather significantly [11,12]. Importantly, the Kerr comb theory predicts the existence of a regime with high unity contrast (amplitudes of pump harmonic and comb envelope are equal), which has been considered an artifact of an incomplete theory and has not been observed in experiments so far. One of the main results of our study is validation of the theory and observation of this high contrast regime in a properly designed ring resonator. We show that the high contrast comb is possible at the drop port when the intracavity amplitude of the comb envelope exceeds amplitude of the intracavity pump line. Our Letter is organized as follows. We first derive the condition for the ideal intracavity comb envelope amplitude to exceed the pump line amplitude. We then validate the analytical result numerically and present the Kerr frequency comb with a suppressed pump line experimentally generated with a ${\mathrm{MgF}}_2$ photonic belt resonator. We compare the experimental results with numerical simulations and provide an explanation of why pump harmonic suppression is not possible in resonators supporting large numbers of modes.

We consider a mode-locked Kerr frequency comb as a train of $N$ identical temporal dissipative Kerr solitons circulating in the cavity. The envelope amplitude of the pulses can be found from the damped driven nonlinear Schrödinger equation, also known as the Lugiato–Lefever equation (LLE). We introduce the dimensionless parameters for the detuning of the pump line $\omega$ from the frequency of the corresponding resonator mode ${\omega }_0$, ${\zeta }_0=({\omega }_0-\omega )/\gamma$, and for the pump power $f^2=(2g\eta P_{\mathrm{in}})/({\gamma }^2\hslash {\omega }_0)$. Here $P_{\mathrm{in}}$ is the input power, $g=(\hslash {\omega }_0^{2}c{n}_2)/(n^2{V}_{\mathrm{eff}})$ is the nonlinear coupling constant, which is also the Kerr frequency shift per photon [13], $V_{\text{eff}}$ is the mode volume, $n$ and $n_2$ are linear and nonlinear refractive indices, respectively, $c$ is the speed of light, $\hslash$ is the reduced Planck constant, and $\eta =1-{\gamma }_0/\gamma$ is the coupling efficiency parameter with $\gamma$ and ${\gamma }_0$ corresponding to total and intrinsic resonance half-linewidths. The resonator group velocity dispersion (GVD) is characterized by $d_2=({\omega }_{+1}+{\omega }_{-1}-2{\omega }_0)/(2\gamma )$, where ${\omega }_{\pm 1}$ are the frequencies adjacent to the pumped mode. We approximate the solution of the LLE as [6,14,15]

Considering that the width of the solitons is small ($2\pi b\ll 1$), we find the cw component of the field:

An example of a graphical solution of the above equation obtained by varying the relative detuning ${\zeta }_0$ and using $f=8.5$, $d_2=0.05$ is shown in Fig. 1. This also includes the simulated single-soliton Kerr comb spectra corresponding to several values of ${\zeta }_0$. Numerical simulations can be carried out using the LLE, but we found the mode decomposition approach to be more convenient in our case [13,16,17]. By exploring multiple parameters of the system, we found that the carrier suppression is always possible if the pump power is high enough.

 

Fig. 1. Numerically evaluated expression for the apparent carrier suppression for $N=1,2,3$ solitons in the resonator mode and the single-soliton Kerr frequency combs ($N=1$, μ–mode number, inset) for four different detunings ${\zeta }_0$. The pump harmonic is suppressed (b and d) at certain detunings given by Eq. (4).

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To validate the theoretical prediction, we fabricated the crystalline resonator with only one mode per free spectral range (FSR) [18], and coupled this resonator in add–drop configuration [10–12] (Fig. 2). It is important to have the resonator with only a single, fundamental mode family to avoid undesirable perturbations from high-order modes. We pumped this resonator with 1561 nm light and observed generation of Kerr frequency combs with the suppressed pump harmonic (Fig. 3).

 

Fig. 2. (a) Schematics of the experimental setup. Light emitted by a tunable fiber laser is amplified with an erbium-doped fiber amplifier (EDFA) and coupled into a photonic belt resonator (PBR) using a mode-matched angle-polished fiber coupler (APC). Light exiting the resonator through the first APC is detected by a photodiode (PD). Light from another APC (drop port) is measured with an optical spectrum analyzer (Yokogawa AQ6319). (b) Picture of the cylinder that holds the PBR structures. Several belt resonators are made on the side surface of the crystalline cylinder by single point diamond turning. Location of the belt from (c) is shown approximately with arrows. (c) Profile of the PBR taken with an optical interferometric profilometer (Veeco Wyko NT9300). Belt width and height is approximately 10 μm. (d) Coupling of two APC (add and drop ports) with the PBR.

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Fig. 3. Numerically simulated optical spectra (shown in red) compared with the experimental results (shown in blue). (a) Incoherent spectrum. To obtain the smooth spectrum envelope we averaged the time-dependent chaotic spectrum over several cavity ring-down times. This procedure resembles the operation of a conventional optical spectrum analyzer [20]. (b)–(d) Kerr comb soliton spectra. The simulations are performed for $f=8.5$ and $d_2=0.05$. This corresponds to the parameters of our ${\mathrm{MgF}}_2$ comb generator: $P=19\mathrm{mW}$, $Q={\omega }_0/2\gamma =0.47\times {10}^9$, $\eta =1/2$, $n=1.37$, $\lambda =1.561\mathrm{\mu m}$, $n_2=0.9\times {10}^{-20}{(\mathrm{m}/\mathrm{V})}^2$, and $V_{\mathrm{eff}}=2.8\times {10}^{-13}{\mathrm{m}}^3$.

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The experimental setup is shown in Fig. 2(a). Three photonic belt resonators (PBRs) are fabricated on a ${\mathrm{MgF}}_2$ polished cylinder with 2675 μm diameter [Fig. 2(b)]. The fabrication accuracy is better than a few micrometers, which allows us to create a resonator with a predictable spectrum. These resonators have typical cross sections of the order of $10\mathrm{\mu m}\times 10\mathrm{\mu m}$ and support dispersion engineering [Fig. 2(c)] [18]. Angle polished evanescent field SMF-28 fiber couplers were used for add and drop ports. The coupling efficiency is measured as the maximum achievable dip of transmission when the distance between the coupler and the PBR is varied. This minimum transmission corresponds to critical coupling, where the resonator’s coupling loss equals the intrinsic loss. The critical coupling can be less than 100% if the resonator’s output beam is not well matched to the coupler transmission. Here, all measurement were done under the critical coupling setting and the transmission dip was over 60%. The drop port coupling was only enabled during comb measurements. The additional loss to the drop port results in the shift toward the under-coupled regime [19]. We measured the $Q$-factor of a specific PBR at 470 million (critically coupled) by recording its resonance with a Koheras 1561 nm laser and using phase modulation sidebands for frequency calibration. While single-mode by design, this resonator still supports a couple of higher order modes with poor coupling efficiency and low $Q$. These modes are well separated from the fundamental mode frequency in the vicinity of the pump wavelength.

We excited the frequency comb in this resonator by pumping it with 19 mW of light as measured at the add port. We observed the step-like multistability on the resonance curve recorded from the add port, which is typical of solitonic combs [6,7,17]. The laser frequency was first tuned to the blue slope of the resonance and we observed generation of the first pair of sidebands at 11 FSRs of the resonator ($\mathrm{FSR}=25.78\mathrm{GHz}$). The laser was then tuned closer to the mode center and a comb typical of a non-coherent state was observed [Fig. 3(a)] [20].

We then used the frequency jump method to excite single and multiple soliton comb states [Figs. 3(b)–3(d)]. In this method, the laser frequency is tuned abruptly to the expected solitonic states within the nonlinear resonance curve. As predicted by our calculations, we found that the pump harmonic was suppressed in all of these states and merged with the envelope of the drop port comb. No tuning of experimental parameters was required to observe spectra in Fig. 3. In our case the solitonic states were not very stable due to temperature drift of the resonator after the frequency jumps. The slight asymmetry of the recorded spectra may be due to higher order dispersion and to the drift of the laser detuning during acquisition of a single spectrum (10–30 s). Controlling the laser parameters after the frequency jumps can be used to stabilize the comb.

There is a very good agreement between the experimental data and the results of our numerical simulations, in which actual experimental parameters were used, including the anomalous GVD (Fig. 3). The evaluation of fields in the 255 resonator modes started at a chosen initial detuning with a seeded analytical soliton train and propagated in time with slow pump frequency tuning until the boundary of soliton existence was reached. The detuning of the pump frequency was adjusted until the simulated and experimental comb widths were matched. It was assumed that the intracavity field is sampled using the drop coupler, so the power of all harmonics of the frequency comb outside of the resonator is proportional to the intracavity power.

We found that some experimental spectra correspond to two optical pulses confined in the resonator. We determined the angular distance between the solitons ${\varphi }_2-{\varphi }_1$ from the autocorrelation of the field given by the Fourier transform of the experimental spectra [7]. This angular distance was used to generate the analytical expression for the soliton train, which was then used as an initial condition in simulations. We found that the combs corresponding to the small detunings (corresponding to longer solitons) found in, e.g., Fig. 3(d) were not stable. The solitons interact and move with respect to each other until a stationary two-soliton cluster is formed. This observation explains the experimental difficulties with the observation of these particular comb states, as well as the small dissimilarity of the comb wings observed experimentally and simulated numerically [Fig. 3(d)].

Let us now discuss some of the reasons that the pump harmonic is not always completely suppressed in other published results, when the add–drop port configuration is used and a broadband mode-locked Kerr comb is generated. In this regime, the conversion efficiency is saturated and the intracavity pump frequency field is clamped above the comb threshold. Most of the pump light is reflected from the resonator. The reflected light doesn’t reach the drop port if the resonator has a single mode family. Therefore, the pump harmonic is expected to be suppressed in a single mode resonator. However, the insufficient pump power is one of the reasons that it does not always happen. Indeed, let us note that the condition of high contrast solitons $\sqrt{d_2}{\zeta }_0/2f>1$ [which follows from Eq. (2) and $s/2b\ge c_0$] may be cast as $\sqrt{d_2}f\gtrsim 1$ if we take the soliton existence condition into account: ${\zeta }_0\le {\pi }^2{f}^2/8$ [6,15]. This means that the pump power needed to suppress the pump harmonic is inversely proportional to $Q^3$ of the mode (since $\sqrt{d_2}f\gtrsim 1\equiv d_2{f}^2\gtrsim 1$, and, by definition, $d_2\sim Q$ and $f^2\sim P_{\text{in}}{Q}^2$). Thus, in the majority of experiments involving low-$Q$ resonators the required pump power is too high to observe this phenomenon. With the exception of a few weakly coupled higher order low-$Q$ modes, the resonator used in this work has the highest $Q$-factor of any single mode family resonators studied to date, to the best of our knowledge.

Another reason for lack of published experimental evidence of pump harmonic suppression is that combs have been generated in multimode resonators. In this case, there are multiple channels (modes) for the pump light to leak to the drop port. The leakage through the higher order modes in the vicinity of the pump frequency is suppressed by mode matching of the couplers. However, this suppression usually does not exceed 20 dB. Some pump light enters the high-order modes and leaves the resonator without participating in the nonlinear frequency conversion. As a result, the drop port power at the pump frequency increases. To illustrate this process, we have measured the modal spectra of two different resonators in the linear regime. The spectrum of an 18 GHz FSR wedge-shaped resonator is shown in Fig. 4(a). The resonator supports many high-order modes that create a $-20\mathrm{dB}$ background in the spectrum. Moreover, there is a mode that nearly overlaps with the pump mode, resulting in interference and sharp dispersion features (Fano resonances [21,22]). We generated solitonic combs with this resonator under conditions similar to those described above (see also Fig. 7 in [23] for the resulting comb spectra). Up to 10% of the pump light reached the drop port, leading to the prominent pump line observed above the comb envelope in the drop port transmission. We believe that this mechanism explains the large pump line on top of Kerr combs presented in other published studies. On the other hand, there is only one channel for the pump light to reach the drop port [Fig. 4(b)] in the PBR resonator [18,24]. Thus, the drop port coupler samples only the field participating in the nonlinear process. Finally, it is worth noting that the leakage light from the higher order modes can result in multi-path interference noises in the detected beatnote signals.

 

Fig. 4. (a) Linear optical spectra of a sharp edge multimode and (b) a single mode photonic belt resonator. The spectra are taken in the add–drop port configuration.

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In conclusion, we have studied the Kerr frequency combs in a nonlinear ring resonator pumped with cw coherent light under conditions optimized for increased comb contrast. We have shown that resonators with reduced density of modes can be used to generate Kerr combs with a suppressed pump line. An increase of the mode density results in a decrease of the Kerr comb contrast due to leakage of the pump light through the high-order modes, which are not involved in the nonlinear process. The observation is important for multiple practical applications, as large cw background produced by the pump harmonic masks optical pulses leaving the resonator and increases the noise of the comb-based oscillators.