Abstract
Dispersion engineering of microring resonators is crucial for optical frequency comb applications, to achieve targeted bandwidths and powers of individual comb teeth. However, conventional microrings only present two geometric degrees of freedom – width and thickness – which limits the degree to which dispersion can be controlled. We present a technique where we tune individual resonance frequencies for arbitrary dispersion tailoring. Using a photonic crystal microring resonator that induces coupling to both directions of propagation within the ring, we investigate an intuitive design based on Fourier synthesis. Here, the desired photonic crystal spatial profile is obtained through a Fourier relationship with the targeted modal frequency shifts, where each modal shift is determined based on the corresponding effective index modulation of the ring. Experimentally, we demonstrate several distinct dispersion profiles over dozens of modes in transverse magnetic polarization. In contrast, we find that the transverse electric polarization requires a more advanced model that accounts for the discontinuity of the field at the modulated interface. Finally, we present simulations showing arbitrary frequency comb spectral envelope tailoring using our Fourier synthesis approach.
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Introduction
Frequency combs based on integrated nonlinear microresonators are a powerful tool to bring metrology outside the lab. They allow for low power consumption and portabilty^{1} while maintaining metrological quality^{2} while in the dissipative Kerr soliton (DKS) regime. Although octavespanning frequency combs—needed for carrierenvelope stabilization through selfreferencing of the comb—have been demonstrated^{3,4}, reaching beyond an octave is particularly interesting so that the strong pump can be doubled in selfreferencing schemes. Yet, it is extremely challenging, especially at short wavelengths towards the visible. Materials that are used for microcomb generation, including Si_{3}N_{4}^{5,6,7,8,9}, AlN^{10}, and LiNbO_{3}^{11}, present increasingly large normal dispersion the shorter the wavelength is^{12,13,14}. Modal confinement of the light in resonators with wavelengthscale crosssections adds a geometrical component to the dispersion, which in many cases is enough to compensate for the normal material dispersion. However, a simple rectangular crosssection microresonator does not offer enough degrees of freedom to achieve broad enough anomalous dispersion (needed for bright DKS states) to tackle goals such as spectral bandwidths wellbeyond an octave while extending well into the visible. Alternative approaches have been proposed. Among them, multipumped DKS^{8} and pulsedpump resonators^{15,16} have been successful in realizing spectral bandwidths beyond that of conventional DKS microcombs. Yet, these solutions often increase the complexity of the setup required for field deployment of these microcombs. It is thus necessary to create additional methods through which one can engineer the dispersion of a microring resonator. Approaches with multilayer material stacks^{17}, complex ring crosssections^{18} and concentrical rings^{19}—each relying on avoidedmode crossings—have generated much more complex dispersion profiles. However, fabrication has been challenging and may be incompatible with topdown foundrylike mass fabrication processes^{20}. In addition, broad bandwidth microcombs based on these approaches have not yet been demonstrated. Nevertheless, the concept of an avoidedmode crossing, which relies on modecoupling, can be harnessed in different fashions, for example, by coupling the clockwise (CW) and counterclockwise (CCW) directions of the same transverse optical mode. Such CW/CCW coupling has been demonstrated by Lu et al., where modulation of the microresonator sidewall creates a photonic crystal that frequency splits a targeted mode without impacting the nearest neighbors, determined by the number of photonic crystal periods within the ring circumference^{21}. Interestingly, nonlinear states such as optical parametric oscillation^{22,23,24} and DKSs can be created in this system^{25}. Moreover, this photonic crystal ring concept has been expanded to modulation amplitudes far beyond a simple perturbation, where a full band gap of hundreds of gigahertz to several terahertz is resolved, impacting the band structure among several neighboring modes^{26}. More importantly, for this work, it has been shown that it is possible to introduce multiperiod photonic crystal patternings that create controlled frequency splittings for a few (up to 5) targeted modes^{27}. In particular, sinusoidal ring width modulations corresponding to multiple single modes targeted simultaneously (i.e., with different amplitudes and modulation periods) can be summed with limited impact to the other modes. In summary, ref. ^{27} lays out a methodology based on Fourier synthesis for microring dispersion engineering^{28}, where a spectral profile of mode coupling and corresponding shift of several individual modal frequencies follows a discrete Fourier transform of the ring width modulation.
Yet, whether a vast number of modecouplings through Fourier synthesis – which remains a perturbative approach—can be implemented in a predictive manner that yields precise and efficient dispersion engineering consistent with the needs of, for example, broadband microresonator frequency combs is still unclear. In this work, we propose to answer this question with an indepth study of photoniccrystalmediated microring dispersion engineering in the limit of large (tens of gigahertz) spectral shifts. We demonstrate that the previously utilized simple analysis that links the modal frequency shifts directly to ring width modulation is inaccurate in our regime of interest. In contrast, modulation of the ring effective refractive index, which is mapped (nonlinearly) to a ring width modulation, is a better approach. We also show that the polarization considered greatly impacts the validity of the perturbative approach with which this modebymode dispersion engineering is predicted using our Fourier synthesis model. This limit arises from the boundary conditions on the dominant electric field components at discontinuous boundaries. Consequently, the transverse magnetic polarization is more suited for predictive dispersion engineering using our straightforward Fourier synthesis approach, while the transverse electric polarization may require an approach based on full threedimensional numerical simulations of Maxwell’s equations in modulated microring structures. Using our technique, we fabricate silicon nitride photonic crystal microrings in which dozens of resonances are shifted in a controlled fashion by up to 50 GHz, compatible with the integrated dispersion mitigation needed for broadband (e.g., octavespanning) combs, and in good agreement with simulations. Finally, we use coupled LugiatoLefever equation modeling to predict the spectral behavior of microcombs that can be generated using our dispersion technique, and in particular, the possibility of considerably extending their bandwidth and the possibility of creating multicolor Bragg solitons.
Results
Towards a broad and flat microcomb spectrum: motivating photonic crystal dispersion engineering
Under the right conditions of pump power and detuning, microcombs based on thirdorder optical nonlinearity can support DKS states^{29}. The usual way of studying the dynamics of such a system is through the LugiatoLefever equation^{30} which is essentially a dissipative nonlinear Schrödinger equation^{31} that takes into account the microring resonator’s periodic boundary conditions while operating under a slow varying envelope (meanfield) approximation^{32}. The single DKS solution in the anomalous dispersion regime follows the wellknown hyperbolic secant (sech) spectral envelope (sech^{2} for spectral intensity). To quantify the resonator dispersion, the community usually opts for a Taylor expansion and defines a quantity termed the integrated dispersion \({D}_{{{{{{{{\rm{int}}}}}}}}}={\sum }_{k\ > \ 1}\frac{{D}_{k}}{k!}{\mu }^{k}={\omega }_{{{{{{{{\rm{res}}}}}}}}}({\omega }_{0}+{D}_{1}\mu )\), with D_{1} being the linear repetition rate at the pumped mode with resonant frequency ω_{0}, and μ the azimuthal mode number referenced to the pumped mode. The higher order dispersion terms D_{k} are of great importance as they drive the shape of the integrated dispersion and, ultimately, the properties of the microcomb. The sech^{2} comb envelope width is inversely proportional to D_{2}, which must be positive for anomalous dispersion. The odd terms D_{2k+1} drive the recoil of the soliton, resulting in the drift of its repetition rate away from the linear one. The even terms D_{2k} are responsible for symmetric zero crossings of D_{int} and yield dual dispersive waves (DWs) as comb teeth become resonant at these modal frequencies. The experimental demonstrations of DWs^{5} have fundamentally changed the landscape of microcombs by bypassing the sech^{2} width driven by D_{2} and expanding it to octave span^{3,4}. In this work, we will refer to the frequency span between the D_{int} = 0 frequencies as the ‘dispersive wave span’ of the microcomb [Fig. 1a, b]. Although these DWs have been the key enabler for broadband microcombs, the power in these modes relies on the available power in the sech^{2} soliton envelope. Therefore, if D_{2} is too large, leading to a sharp comb envelope close to the pump, the power available at the DW locations will be insignificant for resonant enhancement. This encapsulates the dispersion engineering challenge for broadband integrated frequency combs: getting as broad as possible the D_{int} zerocrossings while keeping the D_{2} > 0 portion as flat as possible.
Guidedwave photonics results in wavelengthdependent light confinement: the longer the wavelength, the larger the mode and less confined it is. This unique feature allows one to tailor the dispersion beyond the material dispersion. For a microring resonator, there are essentially three userdefined input parameters: ring radius (RR), ring width (RW), and thickness (H) [Fig. 1b]. The RR mainly acts on D_{1} and typically has little influence on the higher order dispersion terms. Therefore, most dispersion engineering efforts focus on RW and H. However, with only these two parameters available, the DW span and D_{2} increase together [Fig. 1c]. Although increasing the comb width is the ultimate goal, increasing D_{2} reduces the power available at the D_{int} zerocrossings, ultimately preventing useful DWs from forming. Thus, with current dispersion engineering approaches, an apparent tradeoff exists.
In recent years, a modified microring resonator has been developed, where modulation of the ring width allows for coherent backscattering between the clockwise (CW) and counterclockwise (CCW) traveling wave modes^{21,27}. It has been demonstrated, using a perturbative approach theoretically and verified experimentally, that a single harmonic modulation of the ring width results in a single azimuthal mode of a given transverse spatial mode family experiencing this coupling^{21}. These two coupled modes hybridize into symmetric and antisymmetric modes, creating a mode splitting proportional to the amplitude of the modulation [Fig. 1d]. In the unmodulated ring, the necessity to match the field phase after one round trip results in a set of azimuthal modes described by a mode number \(M\in {\mathbb{Z}}\). When the modulation is applied, a new spatial period πRR/M_{0} ≪ 2πRR becomes important. Here, the modulation consists of 2M_{0} periods around the ring circumference. In the language of photonic crystals (PhCs)^{33}, the frequency splitting created by coupling of the CW and CCW azimuthal modes at ± M_{0} is a frequency band gap, with the Brillouin zone folded at these points as well. These modulated devices can thus be referred to as photonic crystal (PhC) microrings, and in the limit of strong modulation, large band gaps of several terahertz have been demonstrated while maintaining high optical quality factor (Q)^{26}. In addition, smaller modulation PhC microrings have been used to change the dynamics of DKS formation^{25}. For smaller modulations, it has also been shown that the splitting is linearly proportional to the modulation amplitude employed, and that the CW/CCW coupling is nearly absent for all modes near the targeted mode M_{0}^{27}. Therefore multiple mode splittings (PhC modulations) can be implemented on a single ring. Following ref. ^{27}, the design rules then become simple: a straightforward sum of individual modulation patterns gives the microring ring width modulation to implement [Fig. 1d], such that \(R{W}_{{{{{{{{\rm{mod}}}}}}}}}={\sum }_{m}A(m)\cos \left(2m(\theta +\varphi )\right)\) with A(m) the modulation intensity of each targeted mode m and θ the microring azimuthal angle. Therefore, the total ring width modulation in this scheme is, per definition, the discrete inverse Fourier transformation (DFT) of the modal coupling envelope A(m).
The frequency shift created by the CW/CCW coupling can be used to modify the dispersion of the resonator locally. The integrated dispersion is then shifted by half the resonance splitting at each coupled mode, creating two bands. One band is pushed toward higher D_{int}, and one reduces the value of D_{int}. The latter can overcome the aforementioned tradeoff in large D_{2} and large potential comb bandwidth if the Fourier synthesis design approach allows for predictive mode shifts in the tens of gigahertz range to counterbalance the maximum D_{int} value of an octavespanning comb. By carefully engineering the coupling, one could create flatter integrated dispersion while obtaining a large comb width [Fig. 1e]. The resulting microring resonator undergoes a local ring width modulation consistent with the inverse DFT [Fig. 1f]. Such a system is compatible with standard single device layer fabrication as only the ring width as a function of angle is modified, and no other fabrication steps or materials are added.
Limitations of a direct implementation of the ring width modulation
In this section, we seek to verify the above approach for predictive dispersion engineering. We fabricate devices with H = 440 nm thick silicon nitride (Si_{3}N_{4}), nominal ring width RW = 1450 nm, and radius RR = 60 um on top of silicon dioxide (SiO_{2}) without any top cladding other than air. We implement a targeted Gaussian modal envelope (in mode number space) defined by \(R{W}_{{{{{{{{\rm{mod}}}}}}}}}(\mu )={A}_{0}^{{{{{{{{\rm{RW}}}}}}}}}{\exp }^{{(\mu /\sqrt{2}\sigma )}^{2}}\), where μ = M − M_{0} is the mode number relative to a centered one set at M_{0} = 392 [Fig. 2a], and \({A}_{0}^{{{{{{{{\rm{RW}}}}}}}}}\) is chosen based on the desired maximum frequency splitting. At the moment, only the fundamental transverse electric mode (TE_{0}) will be considered, which we have determined through finite element method (FEM) simulation to be at 193.6 THz at M_{0}. In the layout, we vary the maximum amplitude of the Gaussian A_{0} and its width σ while other parameters are fixed. The resulting summed RW modulation profile follows, as expected, the inverse DFT profile (Fig. 2b). The envelope of the RW modulation is therefore also a Gaussian, and the mapping between frequency splitting of a given mode and realspace RW modulation amplitude is linear, based on the simple perturbative analysis from ref. ^{27}. After summing these different modulations, the maximum realspace modulation of the ring width is \(\max [R{W}_{{{{{{{{\rm{mod}}}}}}}}}(\theta )]={A}_{0}^{{{{{{{{\rm{RW}}}}}}}}}\sigma \sqrt{2\pi }\), and is much larger than \({A}_{0}^{{{{{{{{\rm{RW}}}}}}}}}\), which is the maximum of the modal Gaussian profile [Fig. 2a, b], and the consequences of which will be discussed shortly. The maximum RW modulation could be reduced if a phase shift is applied to each modal modulation. Yet, the symmetric and antisymmetric modes arising from CW/CCW coupling are intrinsically standing waves, with their spatial mode profiles along the azimuthal direction locked by the node and antinode positions of the modulation. Applying such a phase shift might not be desirable as it could reduce nonlinear interaction between modes. Thus, we will only study the extreme modulation case where φ = 0 for Fourier synthesis dispersion engineering, which if demonstrated to work should be applicable for arbitrary φ. A zoomin scanning electron microscope image of a fabricated device, created using highresolution electronbeam lithography, is shown in Fig. 2c.
We proceed with transmission spectroscopy measurements between 185.18 THz and 198.6 THz (i.e., 1510 nm and 1620 nm) of the ring resonators with different modulation amplitude and width, where each resonance is spaced by a free spectral range of 398 GHz. A stackedup representation of the transmission around each resonance is represented in Fig. 2d, highlighting the spectral profile of the modal coupling obtained (i.e., mode splitting vs. mode number). However, while overlapping this experimental data with the originally designed coupling from the direct inverse DFT, it is evident that none of the experimental data match. In particular, multiple behaviors are reported with the “collapsing” of the splitting around μ = 0 for small modulation amplitude A and/or small width σ, while at large A and large σ, the modal coupling becomes much more unpredictable. The maximum coupling for each design has been calibrated with a single mode coupling at μ = 0. It is worth reporting that \({A}_{0}^{{{{{{{{\rm{RW}}}}}}}}}=12.5\) nm and σ = 8 yields \(\max [R{W}_{{{{{{{{\rm{mod}}}}}}}}}(\theta )]=250\) nm, which is far from the upper range of what has been previously experimentally verified and confirmed to match predictions based on a perturbative approach^{27}.
We propose that the discrepancy between designs and experiments comes from two fundamental interactions in the PhC ring, which we highlight through finitedifference timedomain (FDTD) simulations. Here, we consider a simple FDTD simulation whose intent is to provide a qualitative assessment of these complicating factors. To highlight qualitative features while retaining a small enough structure to avoid convergence and/or simulation time issues, we consider a waveguide containing two highly modulated regions, as shown in Fig. 2e. First, under a total modulation where the RW becomes small enough, a cavitylike effect will occur with boundaries localized at the local modulated sections of the ring [Fig. 2e (i)]. Essentially the propagating modes – either CW or CCW – will also be coupled to this cavity mode, making the mode splitting much more complex. We believe such an effect is mostly seen in the higher modulation case, as highlighted in \({A}_{0}^{{{{{{{{\rm{RW}}}}}}}}}=50\) nm and σ = 8. The second effect that comes into play is the multimode interference [Fig 2e (ii)]. The RW modulation, in particular reducing the ring width, allows for a nonzero overlap between the higherorder transverse spatial modes of the unperturbed ring. Similar to an unoptimized racetrack resonator^{34}, this yields a coupling between these modes, which adds up to the CW/CCW one and creates a much more complex pattern of mode splitting. This multimode interference effect is also associated with higher radiation losses and quality factor asymmetry, as can be seen for μ > 5 in the case of \({A}_{0}^{{{{{{{{\rm{RW}}}}}}}}}=50\) nm and σ = 8 in Fig. 2d.
Effective index approach: improved Fourier synthesis and polarization considerations
We have demonstrated that direct Fourier synthesis for predictable dispersion engineering using a simple RW modulation approach is inherently flawed when modulation amplitudes are sufficiently large. However, given that PhC band structures are intrinsically related to the modal n_{eff}, in this section, we consider whether we can produce a better model for dispersion engineering with DFT design.
The modal effective refractive index is not linear with the RW, given that cutoff occurs in our asymmetrically airclad system under a small enough RW and will grow asymptotically toward the silicon nitride index with large RW. Thus it is more suitable to apply the inverse DFT on the effective index modulation rather than the RW modulation [Fig. 3a]. The mapping between effective index n_{eff} and RW is easily achieved through finite element method simulations. Of course, this function will also depend on thickness and ring radius, yet we assume they are fixed throughout this study. From the inverse DFT n_{eff} profile along the resonator angle, it is then possible to map the actual RW modulation that we have to fabricate using this simulated calibration of n_{eff} [Fig. 3b]. It is instructive to notice the difference between the modulation profile obtained by the previous direct RW modulation method and the one described in this section [Fig. 3c]. In the former case, the maximum RW modulation is close to the nominal one, creating a local RW close to zero, which is responsible for the cavitylike and higherorder mode coupling presented earlier. It can be understood that the local variation of n_{eff} is much stronger than expected, creating a local cutoff that leads to strong Bragg reflection at that point. However, in the n_{eff} modulation case, the RW modulation accounts for the nonlinear dependence of n_{eff} with the ring width, in particular, the narrower it becomes. Therefore, the ring does not present the same bottleneck effect, which reduces the two spurious effects partially responsible for the mode splitting not following the intended design. In addition, the “teeth” of the PhC are much sharper than in the simple sine modulation case, which we believe also reduces the PhC cavity mode within the Bragg grating. This also suggests that the upper modulation limit is not the unperturbed ring width, but is rather limited by either the disk n_{eff} (i.e., RW → RR) or the cutoff RW. This maximum modulation can be tailored with the nominal RW, but a tradeoff must be found to allow for operating in the correct dispersion regime of the resonator for a given application (e.g., weak anomalous dispersion for a bright, broadband soliton microcomb).
Based on this new inverse DFT process, we proceed to fabricate four types of modal coupling designs (Fig. 3d) made around the same microring parameters presented in the previous section. We implement the same Gaussian profile, as well as a Lorentzian one defined by \({n}_{{{{{{{{\rm{eff}}}}}}}}}^{({{{{{{{\rm{Lor}}}}}}}})}(\mu )={A}_{0}^{{{{{{{{\rm{neff}}}}}}}}}\frac{1}{1+{(\mu /\sigma )}^{2}}\), a sinc pattern defined by \({n}_{{{{{{{{\rm{eff}}}}}}}}}^{({{{{{{{\rm{sinc}}}}}}}})}(\mu )={A}_{0}^{{{{{{{{\rm{neff}}}}}}}}}\frac{\sin \left(\mu /\sigma \right)}{\mu /\sigma }\), and a linear profile \({n}_{{{{{{{{\rm{eff}}}}}}}}}^{({{{{{{{\rm{lin}}}}}}}})}(\mu )=2{A}_{0}^{{{{{{{{\rm{neff}}}}}}}}}\frac{\mu }{\sigma }\) with μ ∈ [ − σ, σ] and no coupling outside this range for this profile [Fig. 3e]. Measuring the TE_{0} modes, we retrieve the mode splitting for each resonance, where once again, experimental data and the modal coupling design do not match [Fig. 3e], except for the sinc profile.
To understand the origin of this discrepancy, it is important to recall some basic boundary conditions put forth in Maxwell’s equation, that is, that the electric field is discontinuous at the orthogonal interfaces of the polarization. While in this work we have used a simple perturbative expression for the coupling term between the CW and CCW modes (albeit while taking into account the waveguide effective modal index dependence on width), more generally, the coupling term between the CW and CCW modes can be semianalytically determined by assuming a onedimensional Bragg grating with two coupled counterpropagating waves. The transverse magnetic (TM) polarization – with the dominant electric field component along the thickness direction of the ring – does not experience a discontinuity at the modulated interface, which is also true for the magnetic field in this case along the radial direction [Fig. 3f]. Therefore, the coefficient between CW/CCW coupling can be expressed as \(J\propto {\int}_{r}\frac{1}{\varepsilon }H(r)H{(r)}^{* }{{{{{{{\rm{d}}}}}}}}r\) with H(r) the magnetic mode profile along the radial direction of the ring. However, the TE mode experiences a discontinuity along the modulated ring wall. Thus, one needs to account for a correction term in the coupling such that \(J\propto \int\varepsilon \left(E(r)E{(r)}^{* }{\partial }_{r}E(r){\partial }_{r}E{(r)}^{* }\right){{{{{{{\rm{d}}}}}}}}r\)^{35}. This translates into different behaviors of the bandgap (i.e., splitting amplitude) between the two polarizations, where the TM mode exhibits a continuously increasing band gap while the TE mode undergoes a band gap closing with the increase of the refractive index contrast of the PhC (Fig. 3f, band gap obtained from FEM simulation). This difference in behavior can help explain the mismatch between design and experimental data in the TE polarization. Interestingly, the sinc profile effectively is a single targeted modulation M_{0} that is truncated to match the sinc modal width. Therefore, much less effect due to the polarization will occur and the behavior is closer to the single mode splitting case, explaining the good agreement with the designed modal mode splitting profile. In addition, the maximum n_{eff} modulation for a given σ and A_{0} is much smaller in the case of the sinc profile, which makes the system behave in the linear regime of band gap opening with modulation amplitude (e.g., as shown in Fig. 3f).
The above explanation suggests that the TM polarization should not suffer from such a theoryexperiment disagreement for any profile. We measured the same microrings in the TM mode, where M_{0} = 392 is resonant at 204.6 THz [Fig. 3g]. As expected from the absence of band gap closing due to the field continuity with the RW modulation, the TM mode follows the intended modal coupling design for all of the profiles implemented. We note that the maximum obtained modesplitting reached up to 100 GHz (i.e., coupling J ≃ 50 GHz), which is compatible with integrated dispersion mitigation for ultrabroad microcomb generation. Nevertheless, it is fair to assume that larger coupling strength can be easily implemented. The quality factor for both TE and TM is not particularly impacted in the spectral region where the coupling happens and remains at the >5 × 10^{5} level (see Methods Fig. 5). However, we believe that substantial losses will be introduced at shorter wavelengths. First, approaching the mode 2M_{0}, the band folding will not occur between the two contrapropagative modes but instead with the Γ direction in the reciprocal space of the PhC (i.e., close to the surface normal direction). Therefore, the light will be extracted almost vertically. This is relevant for optical angular momentum generation^{36,37}, while for our frequency comb application, it will generate considerable loss given the strength of the PhC. Secondly, at short wavelengths, the RW modulation will not be subwavelength anymore, and will essentially act as a scattering element which is likely to cause losses.
Going forward, we note that with accurate modeling of the TE coupling between CW and CCW modes and the experimental demonstration of the band gap closing  which has already been observed in a large modulated PhC ring^{26}  along with accurate predictions of the effective modal refractive index, it may be possible to effectively develop Fourier synthesis dispersion engineering accurately in this polarization, though it may also be the case that full numerical solution of Maxwell’s equations within a modulating microring might be needed for accurate prediction of the dispersion. This work is outside of the scope of this manuscript; instead, in the next section, we focus on how understanding how the demonstrated Fourier synthesis dispersion engineering technique can impact microresonator frequency comb generation.
Broadband microcomb dispersion engineering and Bragg multicolor solitons using the Fourier synthesis approach
In this section, we numerically investigate the potential that the inverse DFT dispersion design could bring to ultrabroadband microcombs and the new kinds of microcombs that could be generated using this technique.
To accurately model the system, we use a set of linearly coupled LugiatoLefever equations (LLEs) that account for the CW and CCW interaction in the PhC ring. To account for the crossphase modulation, we assume that the contrapropagative light will travel at twice the speed of light, given that the LLE is intrinsically in the moving frame of the light. Therefore, the crossphase modulation can neglect the fast oscillations of the light and simply use the average of the contrapropagative mode power^{38}:
where a_{↑} and a_{↓} are the envelope field in the CW and CCW mode, respectively, \(\dot{a}=\partial a/\partial t\) the temporal derivative, A_{↑,↓}(μ, t) = FT[a_{↑,↓}(θ, t)], the Fourier transform of the field in the azimuthal direction, J(μ) is the coupling between CW/CCW mode μ arising from the PhC, which is mode dependant as previously designed and exhibits a minus sign because of the π phase shift induced by the reflection, κ/2π = 755 MHz and \({\kappa }_{{{{{{{{\rm{ext}}}}}}}}}=\frac{1}{2}\kappa\) are the total loss rate and the coupling loss rate respectively (corresponding to a coupling and intrinsic quality factor Q_{i} = Q_{c} = 7.50 × 10^{5} at 283 THz, following Refs. ^{13,39}), δω is the pump detuning relative to the pumped mode μ = 0, L = 2π × 2 × 23 μm is the resonator circumference, γ = 2.3 W^{−1}.m^{−1} is the effective Kerr nonlinearity, P_{↑} = 150 mW is the input pump power in the CW mode, and δ_{↑} is the Kronecker delta function.
In the case of interest in this manuscript, we will only focus on Fourier synthesis outside of the pumping region to tailor the integrated dispersion. Thus, our model assumes that any azimuthal component μ presenting J(∣μ∣ ≫ 0) ≠ 0, namely where the presented CW/CCW engineered mode coupling occurs, is far from the pump mode. Therefore, at the pump, J(μ = 0) = 0 and only one propagative mode is excited in the LLE model. As stated earlier in Fig. 1a, the current stateoftheart for dispersion engineering is to design both ring width and thickness to achieve the desired dispersion regime and comb width, which ultimately leads to a tradeoff between bandwidth and flatness. Using our Fourier synthesis technique, one could use a very asymmetric D_{int} where the pump frequency and potential short dispersive wave position should be harmonic [Fig. 4a] corresponding to an oxideclad device with ring width RW = 1125 nm and thickness H = 770 nm. However, the D_{int} barrier being too large and steep, no power is transferred through Cerenkov radiation to create an appreciable DW. Using a simple Gaussian spectral coupling profile, which can be added up around different modes to effectively decrease this barrier, it is not only possible to allow for sufficient energy transfer to create an appreciable DW [Fig. 4a], but also reach a level of flatness through the different pockets of anomalous dispersion that have been created through Fourier synthesis dispersion engineering. The azimuthal profile of the pulse remains similar to a DKS, while the CCW light, which only exists where the coupling is not zero, exhibits a pulsetrainlike behavior.
In addition, the concept that CCW light exists only at frequencies where coupling happens is of great interest for metrology application onchip, where usually complex interposers have to be created to filter, guide, and interfere different portions of the microcomb spectrum^{40}. In the case of the Fourier synthesis microcomb, one could imagine simplifying such an interposer greatly by harnessing the CW/CCW nature of the intracavity field [Fig. 4b]. Using a second drop bus waveguide allows only extracting the short wavelength where the CW/CCW coupling occurs, simplifying the filtering and routing of the downstream interposer.
Finally, we investigate the properties of the generated solitons in the presence of these new pockets of anomalous dispersion. As expected from previous studies^{18,41}, additional anomalous spectral windows can facilitate the exchange of energy through Cherenkov radiation—which is similar to the DW coupling to the soliton^{42,43} – as long as their relative dispersions overlap the same frequency grid. We investigate by applying the same Gaussian Fourier synthesis coupling to a regular integrated dispersion profile from a previously studied microring resonator^{39}, corresponding to RW = 1060 nm and H = 440 nm for the airclad Si_{3}N_{4} microring, and only the coupling amplitude J is modified [Fig. 4c]. Once the coupling is strong enough that the integrated dispersion in this spectral region becomes close enough to zero, Cherenkov radiation is permitted, allowing for the soliton to exchange energy with this other color and ultimately leading to a multicolor soliton. Interestingly, the second color of the soliton is entirely driven by the PhC and the Bragg reflection nature of the ring, creating a multicolor Bragg soliton^{44,45}. Increasing the coupling to reach a negative value of the integrated dispersion in the CW/CCW coupling region still allows for a multicolor soliton to exist. However, it is interesting to notice a new set of DWs on the other spectral side of the pump. These DWs – or idlers—are signs of two solitons traveling at the same speed (i.e., repetition rate) through crossphase modulation coupling^{46}, yet offset from one another, which has been observed in photonic molecules^{47,48} and multipump DKS^{8} where such frequency interleaving might be harnessed for metrology applications.
Discussion
We have demonstrated a thorough approach for Fourier synthesis dispersion engineering of microring resonators. By summing the modulation at different amplitudes targeting several azimuthal modes, the azimuthal ring width variation can be obtained by an inverse discrete Fourier transform. We note that photonic molecules^{47,49} (i.e., created by the coupling of two ring resonators) provide another mechanism to significantly alter dispersion compared to single ring resonators. In contrast to photonic molecules, where the coupling between the two rings’ modes is dependent on the physical gap between them and is not dispersive enough to tailor modal frequencies on a modebymode basis, our approach allows for a complete arbitrary spectral envelope (i.e., ondemand dispersion) by addressing modes individually. We further demonstrated that the simple ring width modulation approach is insufficient for the predictive design of avoided mode crossings between CW and CCW modes created by the photonic crystal. Instead, an effective refractive index approach inverse DFT, which is then mapped onto the ring width modulation, provides a much better accuracy for the mode splitting design. In addition, we discuss the implication of the polarization of the modes, where the transverse electric modes are intrinsically more complex to Fourier synthesize than the transverse magnetic modes due to electric field discontinuity along the ring width modulation. We demonstrated that using TM polarization, one can accurately reproduce a coupling design experimentally over twenty modes, following any arbitrary spectral envelope. Finally, we discuss through numerical simulations how such Fourier synthesis dispersion engineering can profoundly impact ultrabroadband microcombs and lead to interesting new physics with multicolor Bragg solitons. Although our work has focused on microcomb dispersion engineering using an effective refractive index profile that follows a sine modulation, it can be extended to other types of approach. First, any other periodic modulation profile, such as beadlike^{50}, rectangular^{51,52}, or spikelike^{53} profiles studied previously, can be used in a Bloch expansion and hence are amenable to our Fourier synthesis approach. Moreover, in our photonic crystal rings, the Brillouin zone edge remains below the light cone, yet coupling to the light cone can happen with double the modulation period and can find application in the ejection of orbital angular momentum beams^{54} and the creation of optical vortexgenerating microcombs^{53,55}. Our work can, for example, allow for multiplexing multiple OAM states for each individual comb line of a microcomb.
Methods
Additional data on mode splitting measurements
The modesplit and contrapropagative coupling strengths we present throughout the manuscript are extracted from linear measurements such as the one presented in Fig. 5a. In the case of the TE mode spectroscopy ranging from about 185 THz to 198 THz, the continuously tunable laser (CTL) is calibrated by simultaneously probing a trigas cell. Using the NIST calibration of H_{12}CN, ^{13}CO, and ^{13}CO absorption lines we are able to accurately retrieve the wavelength of the transmission scan. We define the normalized mode number μ relative to the centered modesplitting, which is retrieved with a singlemode splitting device (i.e., sine modulation of the ring width), and each mode is then indexed from this mode, allowing to retrieve the coupling from the mode splitting [Fig. 5b]. From this linear spectroscopy data, we also extract the intrinsic and the coupling quality factors through a simple coupled modetheory model^{56}. In Fig. 5c, we present the average intrinsic quality factor for the four modal coupling envelopes presented in Fig. 3 for the TE polarization, with the nominal spectral size σ for each envelope, with the error bar representing the standard variation of Q over the different modes measured. The intrinsic quality factor does not seem to be impacted and remains close to a million with little variation over the different modes, although the modulation of the ring width is extremely local (see the following section). The excess radiation losses induced by the photonic crystal modulation can therefore be concluded to be minimal in our system.
Fabrication details
The microresonators are made of a 440 nm thick stoichiometric Si_{3}N_{4} grown via low pressure chemical vapor deposition with a 7:1 ammonia/dichlorosilane gas ratio, on top of thermal SiO_{2}. The layout presented in Fig. 6 is patterned using electronbeam lithography. The ring and waveguide are then etched using a reactive ion etching method. The facets are clad with SiO_{2} while the rings are still airclad thanks to a liftoff process of the SiO_{2}. The chips are then diced and polished for testing.
Data availability
The data that supports the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.
Code availability
The LLE code to study the microcomb dynamics is a modified version of pyLLE^{58} which can be made available upon reasonable request.
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Acknowledgements
The authors thank Sashank Shridar and Yi Sun for their valuable input. The authors acknowledge funding from the DARPA APHI and NISTonachip programs. While preparing the manuscript, we have been made aware of ref. ^{57} which has some similarities to our approach. Our work goes beyond the regime of relatively small mode splittings to theoretically and experimentally treat the ≈100 GHz shifts needed for broadband dispersion engineering and multicolor soliton formation
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G.M. led the project, designed the ring resonators, conducted the experiments, and helped develop the theoretical framework. X.L. and J.S. helped with the design, D.W. fabricated the devices. G.M. and K.S. wrote the manuscript with input from all authors. All the authors contributed and discussed the content of this manuscript.
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Moille, G., Lu, X., Stone, J. et al. Fourier synthesis dispersion engineering of photonic crystal microrings for broadband frequency combs. Commun Phys 6, 144 (2023). https://doi.org/10.1038/s42005023012536
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DOI: https://doi.org/10.1038/s42005023012536
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