The invention of optical frequency combs, which serve as a bridge between optical and radio frequencies, has revolutionized the field of optical frequency metrology [1–3]. While the initial demonstration of frequency combs was achieved using Ti:sapphire laser systems in conjunction with photonic crystal fibers [4], the subsequent development of compact frequency comb sources through the use of the Kerr effect in microresonators (microcombs) has substantially advanced the potential for miniaturizing comb sources [5,6]. Recent research on CMOS-compatible microresonators has set the stage for cost-effective comb generation in photonic integrated circuits, potentially leading to widespread adoption in various scientific and technological domains. The ability to produce frequency combs that are not only turnkey [7] but also operable on battery power [8] highlights the transformative potential of this technology.

Later, the demonstration of dissipative Kerr solitons highlighted the impact of balancing dispersion and nonlinearity on the spectral and temporal characteristics of microcombs [9,10]. In the temporal domain, the nature of the soliton—whether bright [9], dark [11], or soliton molecules [12]—is determined by group velocity dispersion (GVD). In the spectral domain, expanding the bandwidth of the generated comb requires flat and close to zero dispersion, which facilitates the overlap of comb lines with cavity resonances [13].

The dispersion profile of a resonator can be quantified by its integrated dispersion Dint. This is exemplified by envisioning an equidistant comb source centered around the pump resonance frequency, represented as ω0. For each resonator mode, indicated as ωμ, where μ is the mode index symmetrically arrayed around ω0, Dint is defined by the frequency offset between ωμ and its associated comb line. This concept is illustrated in Fig. 1(a). The mathematical formulation of Dint is as follows [14]: Dint=ωμ−ω0−D1μ,(1)Dint=D2μ22!+D3μ33!+…,(2)where D12π denotes the resonator free spectral range (FSR) at the pump mode (ω0). Additionally, D22π and D32π correspond to the second- and third-order dispersion terms, respectively.

A resonator’s dispersion profile is predominantly influenced by its material and geometric properties. Changing the cross-section of the waveguide is a simple way for engineering the dispersion profile [15]. More advanced and precise tailoring can be achieved through computational techniques like inverse design [14,16]. Additionally, methods involving coupled resonators [17–20] and the use of stacked materials [21] have been employed to modify the dispersion profile. However, although adjustments to waveguide geometry and material selection can effectively shape the overall dispersion profile across a wide frequency range, these modifications can inadvertently alter nonintended parts of the spectral range when targeting specific spectral regions. These inherent limitations underscore the need for innovative approaches to achieve comprehensive customization of dispersion characteristics in resonators.

Recently, a novel approach in tailoring microresonator dispersion with engineered backscatterers has attracted growing interest [22–25]. In high-quality-factor resonators, the existence of a backscattering point can couple clockwise and counterclockwise propagating modes. This strong coupling creates hybrid modes with distinct resonance frequencies. As a result, the initially degenerate modes split into two distinct modes in the frequency domain [22,26,27]. The amount of mode splitting is directly related to the backscattering rate, which governs the strength of coupling between the counterpropagating modes within the resonator [22,28,29].

As shown conceptually in Fig. 1(b), precise control of backscattering and the resulting mode splitting at each resonance enable the correction of deviations between the cavity resonances and their respective equidistant comb lines. This method could ensure proper alignment and overlap of the cavity resonances with the comb lines, thus facilitating the generation of a broader comb. As noted in earlier studies [22,23], the mode splitting profile of a given mode is related to the Fourier transform of the distortion in its effective refractive index. A resonator, fully modulated with a constant modulation period (Pmod) and modulation amplitude (Amod) [as shown in the right of Fig. 2(a)], facilitates the back-reflection of light at the wavelength λlight=2neffPmod. Here, neff represents the effective refractive index of the optical mode, specifically the transverse magnetic (TM) mode in our study. According to the Fourier transform, fully modulated resonators exhibit a narrow backscattering profile, resulting in mode splitting in only one mode [22]. Consequently, dispersion engineering across several modes requires superposing various fully modulated profiles, each with distinct modulation amplitudes [Amod(μ)] and periods [Pmod(μ)] [23]. This technique could require creation of intricate and small structures on the resonator’s edge, posing stricter requirements for the design and fabrication process.

Here, we propose dispersion engineering in ring resonators by modulating only a portion of the resonator’s inner edge, rather than its entire circumference. Having a step function as the envelope of modulation adds new frequencies to the Fourier transform of the sidewall modulation and consequently partially modulated resonators exhibit a broader mode splitting profile, as illustrated in the left of Fig. 2(a). By setting the splitting profile of these partially modulated resonators as the unit functions for dispersion engineering, one can reach a sufficiently flat dispersion profile over a wide spectral range with only a few of these functions. We present a thorough analysis of the mode splitting in partially modulated resonators. Our examination covers how mode splitting varies with changes in the modulation amplitude (Amod), the number of modulations (nmod), and modulation period (Pmod). The importance of the geometrical position of the reflectors is highlighted, particularly when their mode splitting profiles overlap. We present a proof of concept showing improvement in the dispersion profile across a spectral range spanning more than 100 nm. This improvement is achieved by integrating only four modulated sections, each with varying Pmod(i) and Amod(i).

The primary experimentally investigated resonators are silicon nitride (SiN) ring resonators, with a height of 400 nm, a width of 1800 nm, and an outer radius of 100 μm. Details on the resonator’s fabrication process can be found in our previous work [30,31]. Additionally, a section of its inner sidewall is modulated in order to form a Bragg “reflector” R(Pmod,Amod,nmod,θmod). In this notation, Pmod, Amod, nmod, and θmod represent the modulation period, the modulation amplitude, the number of modulations (unit cells), and the angular position of the reflector’s center in the ring, respectively, as shown in Fig. 2(a). When the resonator incorporates multiple segments with different modulation parameters, this is described as having multiple reflectors (Ri).

To ensure consistency throughout the study, triangular modulation has been chosen for all the results presented. Alternative modulation shapes, such as sinusoidal or square functions, could also be used as the unit cell for modulation without significantly altering the response. Each unit cell in a reflector has the same modulation amplitude and period. Figures 2(a) (top left) and 1(c) provide both a schematic representation and a scanning electron microscope (SEM) image of a resonator with a single reflector.

We first examine the impact of various parameters associated with partially modulated reflectors on the mode splitting profile, such as Amod, nmod, and Pmod. To characterize the samples, we measure and calibrate the resonance spectra of SiN resonators using a reference fiber cavity and a hydrogen cyanide gas cell, spanning a wavelength range from 1510 nm to 1630 nm [32].

We explore the relationship between mode splitting and Amod by comparing two samples: Sample01 with small modulation amplitudes (below 125 nm) and Sample02 with large modulation amplitudes (125–1000 nm). Figure 2(b) depicts the variations in mode splitting for these two samples. The maximum mode splitting increases with modulation amplitude linearly for smaller Amod. However, this increase begins to level off for larger Amod. In the inset of Fig. 2(b), the mode splitting profile for large Amod is shown. The sinc function response can be attributed to the Fourier transform of the modulation envelope (which is a step function) [33].

Figure 2(c) illustrates how the mode splitting profile varies with nmod. According to Fourier transform principles, increasing the number of modulations narrows the spectral reflection window of the reflectors and affects fewer modes. Additionally, the peak of mode splitting increases with the addition of more unit cells. Notably, a fully modulated resonator with the selected modulation period (Pmod=475nm) results in an odd number of unit cells at the edge of the resonator. As a result, the peak of the reflection window does not align with any of the resonator modes, and consequently the peak of the mode splitting profile is absent [34,35].

The modulation period (Pmod) of reflectors determines the optical wavelength (λ0) that experiences the maximum mode splitting. Figure 2(d) illustrates the variation of the mode splitting profile as a function of Pmod for the reflectors with nmod=100 and Amod=125nm in the design.

After characterizing individual reflectors, our study explores the behavior of resonators with two identical reflectors, placed at varying positions within the resonator. Figures 3(a)–3(c) display schematics for three configurations, each with a total of 200 modulations (nmod=200) of equal modulation amplitude (Amod=125nm) and modulation period (Pmod=475nm), but in different arrangements. In Fig. 3(a), all modulations are positioned at the angular position of 270°. In Figs. 3(b) and 3(c), however, the modulations are divided into two groups, positioned at relative angular distances of 90° and 180°, respectively. The corresponding mode splitting profiles are shown in Fig. 3(d). It illustrates the importance of the relative position of the reflectors when there is an overlap in their mode splitting profiles. For instance, when the two reflectors are positioned at relative angular difference of 180°, the splitting is reduced by a factor of more than six. This outcome corroborates previous theoretical studies that establish a direct relation between the mode splitting profile and the Fourier transform of perturbations applied to the mode’s effective refractive index [33,36]. This insight is essential for broadband dispersion engineering with high accuracy, requiring placement of multiple reflectors within the resonator and consideration of their interactions.

Figure 4(a) illustrates the mode splitting profile over a 100 nm range. The combination of reflectors induces mode splitting in most modes. We assume that, for each mode, the splitting occurs symmetrically around the initial mode, resulting in a frequency shift relative to the initial mode that is half the splitting value for each split mode. Figure 4(b) compares the intrinsic quality factor (Qint) of an unmodulated resonator and the modulated resonator. As expected, the sidewall modulation can increase the resonator losses by outward scattering of photons, leading to a lower quality factor. However, for many modes the quality factor remains above one million. An adiabatic increase in the amplitude of sidewall modulation could help improve the quality factor.

Although the measured frequency shift in Fig. 4(a) is several GHz for most modes, it remains insufficient to offset the strong negative Dint/2π of a 400-nm-thick resonator, which can reach tens of GHz in the given spectral range, as indicated by the green dots in Fig. 4(c). However, the significance of our method becomes clearer when applying such frequency shifts to SiN resonators with increased thickness. We conduct a theoretical investigation assuming the same mode splitting profile is applied to a resonator with the same waveguide width and ring radius, but increased SiN thickness. The measured mode splitting profile in Fig. 4(a) is fitted with a fourth-order polynomial and the results are incorporated into the simulated dispersion profile of a 600-nm-thick SiN resonator with the same waveguide width and ring radius. The Dint/2π profiles for both the unmodulated and modulated resonators are illustrated in Fig. 4(c). The modulated resonator exhibits a flatter Dint/2π, which can lead to more efficient comb generation. To explore this enhanced comb generation, we conduct a numerical simulation by incorporating up to fourth-order Dint/2π into the Lugiato–Lefever equation (LLE) [37,38]. Figure 4(d) shows more efficient dark comb generation in the modulated resonator compared to an unmodulated resonator, with some comb lines enhanced more than 30 dB. This result highlights the simplicity and effectiveness of our approach, where incorporating just four modulated sections leads to broadband enhancement in comb generation over a range of more than 100 nm.

We propose and demonstrate a method for on-chip microresonator dispersion engineering by modulating different sections of the inner edge of ring waveguides. Shorter modulation sections influence a wider spectral range for dispersion engineering compared to a modulation on the entire inner edge. We investigate the effects of different modulation parameters, including modulation amplitude, modulation period, and the number of modulations on the mode splitting profile. Additionally, we examine the role of the relative angular distance between two identical reflectors in the mode splitting profile.

To demonstrate the practicality of our approach in dispersion engineering, we employed four modulated sections, each with distinct modulation amplitudes and periods, to reduce the dispersion across a spectral range exceeding 100 nm. In addition, we numerically investigate the significant power enhancement of microcomb generation in the dispersion engineered resonators. We believe our results offer a simple yet powerful approach for achieving broad comb generation.

Acknowledgment. We thank the Max Planck Society for its support. M.K. appreciates the support from the Max Planck School of Photonics and fruitful discussions with Vahid Sandoghdar. M.K. also acknowledges the Technology Development and Service Unit at the Max Planck Institute for the Science of Light for providing access to, training for, and management of the institute’s cleanroom facilities, where the samples fabrication took place. L.H. and P.B. acknowledge support from the Ministère des Relations internationales et de la Francophonie, of Québec, through its Appel à projets coopération Québec-Bavière, which aims to develop close collaborations between the Québec-Bavière partners.