\section{Introduction}

Kerr frequency combs have revolutionized several fields in optical sciences since they were first proposed in monolithic microresonators \cite{kippenberg0}. To date, they have been successfully applied to a vast variety of state-of-the-art technologies and highly sophisticated measurement techniques as ultra-high resolution spectroscopy \cite{swann,hansch}, massively parallel coherent optical communications \cite{koos1}, light detection and ranging (LIDAR) \cite{koos2}, optical-frequency synthesizers \cite{scott} and microphotonic astrocombs \cite{benedick,herr}. Frequency combs in optical microresonators are generated by the interaction of {either, a continuous-wave (cw) pump laser or pumped optical pulses \cite{herr2},} with the resonator modes via parametric four-wave mixing (FWM), assisted by the Kerr non-linearity of the material. This parametric process fulfills the energy conservation and is enhanced when the side bands created by FWM coincide with the resonator frequencies. Since the intensity-dependent refractive index induces a frequency shift on the modes, an increasing free spectral range (FSR) with frequency (anomalous dispersion) is required to compensate this effect and effectively produce equidistant spectral lines suitable to support the cascaded FWM \cite{kippenberg5}. When a high number of mode-locked modes are involved in this non-linear interaction and dissipation cannot be neglected, the complex non-linear dynamics may give rise to disipative Kerr solitons (DKS), which arise as a double balance between non-linearity and dispersion (preserving their shape), as well as dissipation and parametric gain (preserving their amplitude) \cite{kippenberg4}. {Moreover, multiple-soliton formations have become very interesting in the context of localized-state interactions \cite{tlidi,rivas2,torner}, breather solitons \cite{gaeta1,kippenberg6,kippenberg7} and soliton crystals \cite{cole,karpov}.} DKS are notably relevant because their corresponding spectra exhibit highly coherent frequency combs with perfect single-FSR spacing between side bands \cite{kippenberg1,gaeta5}, and dispersive waves in presence of high-order dispersion \cite{erkintalo,skryabin,rivas,kippenberg2}. The latter is particularly important because it allows a further spanning of the frequency combs, highlighting the important role of dispersion engineering on parametric gain, and consequently, on comb generation \cite{gaeta2}. Such task has been successfully achieved in non-linear microresonators by varying the ring geometrical parameters in order to tailor the dispersion for specific comb functionalities \cite{michel,gaeta3,yu}. Nevertheless, this geometry has a limited parameter space, thus restricting the choice of materials, operation wavelength and size of the final device. Recently, Fabry-Perot resonators have been proposed as appealing candidates for frequency comb generation since different methods to reshape the cavity dispersion can be applied \cite{lugiato,papp}, however, advanced dispersion engineering still remains challenging. 

In this {work}, we propose a coupled-cavity waveguide (CCW, also known as coupled-resonator optical waveguide or CROW) as a new candidate to efficiently generate low-threshold Kerr frequency combs and DKS. Similar to ring resonators, CCWs define a discrete spectrum of propagating modes which may trigger cascaded parametric FWM in the anomalous dispersion regime, and lead to a highly coherent comb if the waveguide dispersion is conveniently optimized. In fact, since their appearance two decades ago \cite{yariv}, CCWs have shown to be extremely flexible for advanced dispersion engineering and enhancement of non-linear phenomena via slow-light \cite{fan,blair,krauss,notomi1,notomi2,momchil1}. {We show in Fig.~\ref{fig3}(a) a schematic representation of the CCW studied in this work. The periodic waveguide is formed by a staggered distribution of cavities where first (dashed red) and second (dashed green) neighbor cavity-cavity coupling is assumed. The system is pumped with a cw source and a DKS is propagating along the waveguide direction $y$.}

\begin{figure}[t!]
\centering
\includegraphics[width=0.47\textwidth]{fig3v2.eps}
\caption{{(a) Schematic representation of a CCW system with first and second neighbor cavity coupling. The system is pumped with a cw source and a DKS propagates along the waveguide direction $y$.} (b) Photonic crystal CCW formed by coupled L3 cavities where the red and green holes are allowed to vary in size to tune the first and second neighbor coupling, respectively, while the out-of-plane losses of the waveguide Bloch modes is optimized by varying the position and size of the white holes. The intensity profile of the displacement field [$\mathbf{D}(\mathbf{r})=\epsilon(\mathbf{r})\vec{{\cal E}}(\mathbf{r})$] is shown at the boundary of the Brillouin zone.}\label{fig3}
\end{figure}

\begin{figure*}[t!]
\centering\includegraphics[width=0.97\textwidth]{fig4v3.eps}
\caption{(a) Dispersion relation, (b) second order dispersion, and (c) group index in the Brillouin zone of a photonic crystal CCW of 400 L3 cavities. Red and blue curves are where the dispersion is normal and anomalous, respectively. The waveguide is pumped at $a\alpha_0=1.2742$. (d) Threshold (red curve) and bistability boundaries (blue curves) determined by the external pump intensity $|{\cal F}_{\alpha_0}|^2$ as a function of the laser detuning $\sigma_{\alpha_0}$. Dashed curves are for $\beta_{\rm TPA}=0$. (e) Averaged intra-waveguide power of the CCW in the steady state as a function of $\sigma_{\alpha_0}$ for $|{\cal F}_{\alpha_0}|^2=6|{\cal A}_{\alpha_0}|^2_{\rm th}$. Hysteresis arises in the yellow region {$\sigma_{\alpha_0}<-\gamma_{\alpha_0}\sqrt{3}\rho(\kappa)/2$} with $\rho(\kappa=0.2236)=1.84$, and the inset corresponds to a zoom of the rectangular gray area where the discrete steps, signature of switching between soliton states, appear. The corresponding frequency combs at the marked points I-IV in (e) are respectively shown in (f) supercritical Turing pattern of 40-FSR (355.2~GHz) repetition rate, (g) supercritical Turing pattern of 41-FSR (363.3~GHz) repetition rate, (h) soliton molecule of two pulses with single FSR (9.1~GHz) repetition rate, and (i) soliton pulse with single FSR repetition rate. All power quantities are given in {dB units relatively to the} threshold {$|{\cal A}_{\alpha_0}|_{\rm th}^2$}, while detunings are in $\gamma_{\alpha_0}$ units.}\label{fig4}
\end{figure*}

\section{Non-linear photonic crystal CCW}

{The system of Fig.~\ref{fig3}(a) is realized with} the waveguide of coupled L3 photonic crystal slab cavities shown in Fig.~\ref{fig3}(b), where we have plotted the intensity profile of the displacement field [$\mathbf{D}(\mathbf{r})=\epsilon(\mathbf{r})\vec{{\cal E}}(\mathbf{r})$] at the boundary of the Brillouin zone. The photonic crystal is formed by a hexagonal lattice of holes, pitch $a=400$~nm and hole radii $r=0.25a$, etched in a silicon membrane of thickness $d=0.55a$, { and the L3 cavity is introduced by removing three holes along the $\Gamma K$ direction of the lattice}. This CCW configuration allows us to tune first and second neighbor coupling between the L3 cavities by varying the size of the red and green holes, respectively, and optimize the out-of-plane losses by varying the position and size of the white ones. The waveguide of Fig.~\ref{fig3}(b) has been previously proposed as a compact CCW with outstanding figures of merit obtained via automated global optimization \cite{momchil1} and successfully measured experimentally \cite{mohamed,mohamed2}. Here, we adopt the design with largest averaged group index and small second order dispersion reported in Ref.~\citenum{momchil1} with parameters $(\Delta r_1,\Delta r_2, \Delta r_3, \Delta x)=(-0.0049,-0.0340,-0.1016,0.2204)a$, where $r_1=r+\Delta r_1$, $r_2=r+\Delta r_2$ and $r_3=r+\Delta r_3$ are the radii of the red, green and white holes, respectively, and $\Delta x$ is the outward displacement of the latter ones. The dispersion relation $\nu_\alpha$ of the waveguide and decay rates $\gamma_\alpha$ of the Bloch modes of momentum $\alpha$ are calculated with the guided mode expansion method (GME) \cite{gme} for a system length of $M=400$ cavities (400 normal modes within the Brillouin zone of the CCW) with period $l=\sqrt{3}a$ (waveguide length $L=Ml$), while the non-linear mode volume is computed with a commercial FDTD solver \cite{lumerical} and found to be $V_c=0.345~\mu$m$^3$. {This sample length represents a good compromise between total number of normal modes [avoiding finite-size effects in the coupled mode equations Eq.~(\ref{cme7})] and robustness to fabrication disorder \cite{song,mohamed2}.} Silicon parameters at telecom frequencies are considered for the material, namely, dielectric constant $\epsilon=12.04$, Kerr coeficient $n_2=5.52\times10^{-18}$~m$^2/$W and two-photon absorption (TPA) coefficient $\beta_{\rm TPA}=1\times10^{-11}$~m$/$W \cite{jalali}. The GME dispersion relation, second order dispersion and group index are shown in Fig.~\ref{fig4}(a)-\ref{fig4}(c), respectively, for which red and blue segments correspondingly highlight the regions of normal and anomalous dispersion. {Notice that these dispersion curves are valid for a straight waveguide with periodic boundary conditions, nevertheless, we expect them to describe very well a close loop of coupled cavities as bending losses is usually very small, with respect to out-of-plane losses, in photonic crystal geometries \cite{zhang}}. The system is pumped at $a\alpha_0=1.2742$, where $\nu_{\alpha_0}=193.39$~THz, $(ac)^{-1}(d^2\nu/\alpha^2)_{\alpha_0}=8.63\times10^{-4}$, $n_{g,\alpha_0}=119.34$ and the normal mode quality factor is $Q_{\alpha_0}=5.72\times10^4$. This sets an internal mode threshold given by
\begin{equation}\label{Thm}
\left|{\cal A}_{\alpha_0}\right|_{\rm th}^2=\frac{\epsilon V_c}{2ln_{g,\alpha_0}n_2Q_{\alpha_0}}f(\kappa)=138~\mbox{mW},
\end{equation}


\noindent where ${\cal A}_{\alpha_0}$ is the slowly-varying amplitude of the Bloch mode $\alpha_0$, and $f(\kappa)=(\sqrt{1+\kappa^2}+2\kappa)/(1-3\kappa^2)$, with $\kappa=c\beta_{\rm TPA}/(2n_2\omega_{\alpha_0})$, is a function of the material parameters only which gives $f(\kappa=0.2236)=1.73$ for silicon at $\omega_{\alpha_0}/2\pi=\nu_{\alpha_0}=193.39$~THz, and reduces to $f(\kappa=0)=1$ for $\beta_{\rm TPA}=0$. This result is particularly remarkable because, even by considering that TPA increases the threshold by a factor of 1.73, $|{\cal A}_{\alpha_0}|_{\rm th}^2=138~$mW is still one order of magnitude smaller than in typical mm-size crystalline non-linear ring resonators \cite{chembo2}. Equation~(\ref{Thm}) undoubtedly strengths the potential capabilities of CCW structures for low-threshold comb generation. While the factor $V/Q$ also enters in the expression for the internal mode threshold of ring resonators \cite{chembo0,chembo1,chembo3}, CCWs can drastically decrease this minimal power in the slow-light regime, i.e., for $n_{g,\alpha_0} \gg \sqrt{\epsilon}$. Such structural effect allows to access the rich physics of DKS in CCWs at much lower powers than in their monolithic counterparts even in presence of strong TPA, which is the main source of non-linear losses in silicon structures at telecom wavelengths \cite{gaeta4,agrawal}. It is worth emphasizing that, since we are not specifying any particular coupling architecture with the external pump, we are referring to the internal mode threshold which scales as $1/Q$, and not to the external power threshold which is found to scale as $1/Q^2$ for ring resonators at critical coupling with the external source \cite{kippenberg3}. {While the latter is an important parameter to set the excitation power in an experimental setup, the former quantity, also called the minimum intracavity power for comb generation \cite{chembo2}, gives the corresponding effective power into the specific normal mode of the system.} We plot in Fig.~\ref{fig4}(d) the driven intensity $|{\cal F}_{\alpha_0}|^2$, in $|{\cal A}_{\alpha_0}|_{\rm th}^2$ units, evaluated at threshold (continuous red lines) and at the bistability boundaries (continuous blue curves), as a function of the laser detuning $\sigma_{\alpha_0}=\Omega_0-\omega_{\alpha_0}$ in units of $\gamma_{\alpha_0}$, where the dashed lines are the corresponding curves for $\beta_{\rm TPA}=0$. The effects of TPA on the threshold are clearly seen in this plot, moreover, the hysteresis region (where the system has bistable states) is red-shifted and the area between the bistability boundaries is significantly reduced. Hence, the system needs to be pumped stronger and the laser frequency has to be further decreased to see Kerr frequency combs and DKS in presence of TPA. 

\begin{figure}[t!]
\centering\includegraphics[width=0.47\textwidth]{fig5v4.eps}
\caption{Steady state envelope functions at the marked points I-IV in Fig.~\ref{fig4}(e). (a) Supercritical Turing pattern with 40 rolls. (b) Supercritical Turing pattern with 41 rolls. (c) Soliton molecule of two pulses. (d) Single soliton pulse. {$|\psi^s|^2$ is given in units of the power threshold $|{\cal A}_{\alpha_0}|_{\rm th}^2$.}}\label{fig5}. 
\end{figure}

\section{Frequency comb and DKS solutions}

In order to find the possible steady state solutions determined by the non-linear CCW dynamics, we carry out a frequency scan at $|{\cal F}_{\alpha_0}|^2=6|{\cal A}_{\alpha_0}|_{\rm th}^2$ in Fig.~\ref{fig4}(d) within the $\sigma_{\alpha_0}/\gamma_{\alpha_0}$ interval $[-3,1]$. The dynamical non-linear coupled-mode equations Eq.~(\ref{cme7}) are thus propagated in time for each $\sigma_{\alpha_0}$ value along this trajectory using an explicit Runge-Kutta integrator and fast Fourier transform (see Ref.~\citenum{hansson}) until the steady state is reached. Our simulation takes into account the momentum-dependent decay rate $\gamma_{\alpha}$ of the Bloch modes, and the initial condition of the integrator is a sharp Gaussian pulse $\psi(y,0)=\exp[-0.5(y/l)^2]$, where $\psi(y,t)=\sum_\alpha{\cal A}_\alpha(t)e^{i\sigma_\alpha t}e^{-i(\alpha-\alpha_0)y}$ is the envelope function along the CCW direction. Results of this analysis are presented in Fig.~\ref{fig4}(e) where we show the averaged waveguide power in the steady state $\langle|\psi^s|^2\rangle=\int_L|\psi^s(y)|^2dy$, in units of threshold, as a function of the laser detuning. When the laser frequency is red-shifted the intra-waveguide power start to increase until a clear series of discrete steps (see inset), decreasing the average power, are seen in the yellow area, which corresponds to the region where there is hysteresis in the system and the soliton solutions are expected to appear \cite{kippenberg4,chembo4}. In fact, these steps have been previously measured experimentally in the transmission of non-linear ring resonators, and they are associated to the formation of different soliton states within the system, in excellent agreement with the predictions of the non-linear coupled mode equations and the Lugiato-Lefever formalism \cite{kippenberg1}. We show in Figs.~\ref{fig4}(f)-\ref{fig4}(i) the respective frequency combs at the four representative points marked in Fig.~\ref{fig4}(e). Figures~\ref{fig4}(f) and \ref{fig4}(g) displays a sequence of primary combs separated by 40-FSR and 41-FSR, respectively, which correspond to supercritical Turing patterns as they are excited above threshold. In Figs.~\ref{fig4}(h)-\ref{fig4}(i), both combs are subcritical (excited right below threshold) with a single FSR spacing, and they are the signature of soliton complexes. Notably, because of the presence of non-trivial high order terms in the dispersion relation of the photonic crystal CCW, these frequency combs {have} signatures of dispersive wave formation or soliton Cherenkov radiation {at the linear phase matching condition $\omega_\alpha-[\omega_{\alpha_0}+\zeta_1(\alpha-\alpha_0)]=0$, where $\zeta_1$ represents the group velocity \cite{kippenberg2}. Particularly, the dispersive peak clossest to the zero comb line is predicted around $a\alpha\simeq1.63$, which is in perfect agreement with our numerical simulations}. The corresponding steady state intensity of the envelope functions are shown in Fig.~\ref{fig5}. For the I (40 FSR repetition rate) and II (41 FSR repetition rate) states, we see 40 and 41 Turing rolls, respectively, in agreement with the Lugiato-Lefever theory of Kerr frequency combs in the anomalous dispersion regime \cite{chembo4}. Moreover, for the III and IV states we clearly identify, respectively, a soliton molecule composed of two pulses and a single soliton pulse, propagating along the waveguide direction while keeping their shape and amplitude. It must be said that because these soliton structures are subcritical (pumped below threshold), they are extremely sensitive to the initial conditions, and therefore, the system may follow different trajectories in Fig~\ref{fig4}(e) when slightly changing the Bloch mode amplitudes at $t=0$ \cite{chembo4,kippenberg1}. {Furthermore, the characteristic width of the soliton pulse is much smaller than the waveguide length thus ruling out finite-size effects in the basis expansion.} {While results presented in Fig.~\ref{fig4} are strictly 
valid for a CCW system with periodic boundary conditions, we still expect highly coherent combs in finite waveguides where non-negligible corrections to the non-linear interaction are predicted by the theory of frequency combs in Fabry-Perot resonators \cite{lugiato}. Nevertheless, a rigorous quantitative analysis is out of the scope of this paper, and will be focus of a future work.}  

\begin{figure}[t!]
\centering\includegraphics[width=0.47\textwidth]{fig6.eps}
\caption{Frequency spectrum of the DKS in Fig.~\ref{fig5}(d), where the dashed vertical lines represent the normal modes of the photonic crystal CCW. A small repetition rate of 9.1~GHz is obtained.}\label{fig6}
\end{figure}


Figure~\ref{fig6} shows the frequency spectrum of the DKS in Fig.~\ref{fig5}(d), where the unloaded modes (normal modes of the CCW) are represented by the vertical dashed lines. As expected from energy conservation, the spectral lines are equally spaced with a repetition rate of 9.1~GHz. Such rates have been previously achieved in non-linear ring resonators of 7 mm diameter \cite{vahala}, in contrast to our system length of $\sim277~\mu$m. { Notice that the minimal external power for comb generation may significantly vary depending on the material, geometry and more importantly, the coupling architecture with a bus waveguide \cite{chembo2}, however,} this important result notably highlights the potential capabilities of $\mu$m-scale low-threshold CCW systems to generate low-rate frequency combs, which are desirable in high-precision comb applications \cite{kippenberg4}. 

{It is important to stress that the specific choice of photonic crystal CCWs is mainly motivated by their characteristic diffraction-limited mode volumes \cite{hughes1} which play an important role on decreasing the threshold power and final device size. Nevertheless, the results presented in this section only rely on the dispersion relation of the CCW and are therefore expected for any CCW system, e.g. coupled microrings, displaying anomalous dispersion with, more generally, $n$-th neighbor coupling.} 

\section{Conclusions}

In conclusion, we have presented results on Kerr frequency combs and DKS in CCWs where the main source of non-linear loses is given by TPA. We found that the internal mode threshold for comb generation depends on the main CCW figures of merit, namely, cavity volume, normal mode quality factor and group index. While TPA losses has the main effect of increasing the threshold power and inducing a red shift in the optimal laser detuning, structural slow-light plays an important role on reducing the minimal power to trigger FWM phenomena between the CCW normal modes, evidencing the capabilities of CCW systems for low-threshold frequency combs. Specifically, we have demonstrated the possibility of DKS in a realistic dispersion-engineered silicon photonic crystal CCW at telecom wavelengths, where highly-complex combs were seen with signatures of soliton Cherenkov radiation. Repetition rates of few gigahertz in a $\sim277~\mu$m-length photonic crystal CCW have been obtained into the soliton regime, which are commonly achieved in $\sim7$~mm size ring resonators, thus demonstrating the potential of CCW systems for high-precision comb applications in ultra-compact devices. Finally, although the spectral spanning of the combs generated in CCW is limited to the waveguide bandwidth, our results open the way to Kerr frequency comb and DKS generation via advanced dispersion engineering and slow-light non-linear enhancement.


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