1. INTRODUCTION

With the rapid development of nanofabrication techniques, nonlinear- and quantum-based applications are being realized in chip-scale devices. In order to operate with high efficiency, third-order parametric nonlinear processes must satisfy phase-matching (PM) conditions largely governed by the group-velocity dispersion (GVD). In photonic waveguides, the GVD has contributions from the material and from the waveguide confinement. The waveguide GVD can be tuned by changing the structure dimensions [1], which for four-wave mixing requires anomalous or near-zero GVD. This is critical for applications such as Kerr frequency comb generation (KCG) [2,3] and photon-pair generation (PPG) [4]. For all photonic materials [e.g., silicon nitride (SiN), silica, etc.] the GVD becomes highly normal at shorter wavelengths, which makes it impossible to satisfy the PM conditions with conventional waveguide dispersion engineering. Overcoming this obstacle is essential to developing chip-based photonics in the visible and near-visible regimes.

In this paper, we perform dispersion engineering by utilizing the high-order-modes of chip-based waveguides to create anomalous GVD across a large range of wavelengths in the near-visible and visible regimes. We demonstrate the power of this dispersion engineering by generating broadband Kerr combs near 784 nm, which represents the first soliton comb at this wavelength regime using a monolithically integrated chip-based platform. The low wavelength side of the comb reaches visible regime ($<740\mathrm{nm}$; the definition of visible regime may vary), which are the shortest wavelength components generated directly by a Kerr frequency comb. We also show that this dispersion engineering enables a silicon chip-based narrowband photon-pair source at near-visible wavelengths and is compatible with rubidium (Rb) quantum memories (operating near 795 nm). We also show through simulation that combs deep in the visible are possible, which overlap ${\mathrm{Hg}}^{+}$ and Yb transitions.

2. DISPERSION ENGINEERING USING HIGH-ORDER MODES

The refractive indices of most materials can be modeled by a simple two-level system [5], which yields the Sellmeier equation. Such a model yields a refractive index $n(\omega )$ that scales as ${({\omega }_0^2-{\omega }^2)}^{-1/2}$, where ${\omega }_0$ is the frequency of the material resonance. For visible light, the dominating resonance frequency corresponds to the bandgap energy, which creates strong normal GVD as the light frequency shifts towards it.

To illustrate the power of dispersion engineering using high-order waveguide modes, we start with normalized parameters [6] for a step-index slab waveguide [inset, Fig. 1(a)]. Our results capture the behavior of rectangular waveguides since their effective refractive index can be approximated by two slab waveguides with the effective index method [6]. The two essential parameters are

 

Fig. 1. (a) Normalized GVD. Inset, structure of the slab waveguide. (b) Simulated GVD for a $730\times 1330\mathrm{nm}$ microring resonator with a bending radius of 22 μm. (c) Effective refractive index of the ${\mathrm{TE}}_{00}$ mode of a straight waveguide and ${\mathrm{TE}}_{10}$ mode of a ring. Efficient mode conversion is realized when the two modes have the same effective indices. (d) Device schematic. Inset, measured mode profile. (e) Measured resonance.

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Designing a waveguide that compensates for the large normal GVD of photonic materials in the near-visible and visible regimes requires access to the large negative part of $B_2$. This can be achieved by using high-order modes and small waveguide structures. In this work, we base our devices on the SiN platform. We use a finite-element method (FEM) mode solver with a well characterized Sellmeier equation for our SiN thin film [9] to accurately model the total GVD of rectangular waveguides and microresonators. Although stronger anomalous GVD can be realized with a smaller waveguide size, the reduction in size also introduces additional losses due to an increase in scattering at the core–cladding interface. For the current experiments, we choose a waveguide cross section of $730\mathrm{nm}\times 1330\mathrm{nm}$ and a ring resonator radius of 22 μm, with a width fabrication uncertainty of $\pm 40\mathrm{nm}$. The simulated GVD is plotted in Fig. 1(b). We operate at the ${\mathrm{TE}}_{10}$ mode, whose zero-GVD point is near 760 nm. We design the bus-ring coupling region so that the ${\mathrm{TE}}_{00}$ mode of the bus excites the ${\mathrm{TE}}_{10}$ mode of the ring [10]. This is done by matching the effective refractive index of the two modes, which leads to bus dimensions of $730\times 568\mathrm{nm}$ [Fig. 1(c)]. In this way, we ensure that the input and output modes are fundamental even though the resonating modes are in higher order, making this device easy to interface with any other on-chip component. To verify that the ${\mathrm{TE}}_{10}$ mode is indeed excited, in selected devices we include a drop port that has the same cross section as the ring. The mode profile in the ring can be imaged at the output of the drop port [top inset, Fig. 1(d)] with a microscope configuration and a camera. These drop-port devices, however, are not used for KCG or PPG, since the extra coupling loss reduces the $Q$. The final schematic of the device is shown in Fig. 1(d). We characterize the $Q$ of our devices using a wavemeter-calibrated frequency scan and a Lorentzian fit. The full width at half-maximum (FWHM) is measured to be 501 MHz, which corresponds to a loaded Q of $7.7\times 10{}^5$ with an extinction ratio of 74%.

3. KERR COMB GENERATION

The first application we demonstrate using this high-order-mode nonlinear microresonator is KCG reaching the visible regime. Kerr frequency combs provide a pathway towards fully integrated comb sources that could find applications across numerous areas of science and engineering [2,3,11,12], including spectroscopy [13–16], ranging [17,18], frequency synthesizers [19], and coherent communication [20]. Moreover, extending KCG to near-visible and visible wavelengths enables additional applications in optical clockwork [21,22], astronomical spectrograph calibration [23–25], and biological imaging [26,27]. Additionally, these wavelength regimes are compatible with low-cost high-quality Si-based detectors and cameras. Various schemes have been proposed and tested in the past, including direct generation [28–31] and frequency conversion from telecom combs [32,33], but to date, mode-locking has only been demonstrated by Lee et al. [34] near 778 nm using a silica WGMR. While SiN has proven to be highly promising platform in the near-IR, at near-visible wavelengths, SiN has a material GVD that is more than 5 times higher than that of the silica used in [34]. To date, the lowest wavelength achieved by a mode-locked Kerr comb on this platform is demonstrated by Yu et al. near 770 nm [35] by generating a dispersive wave fed by a main soliton pumped at 1 μm. The dispersive wave process is governed by the higher-order dispersion coefficients, making it highly sensitive to fabrication variations [36]. It is thus advantageous to generate a soliton centered in this wavelength regime, since it also can provide higher comb power as compared to dispersive waves.

We generate a soliton mode-locked comb using the device described in the previous section with the thermal tuning method as demonstrated by Joshi et al. [37]. Specifically, we fix the pump laser frequency to the blue side of the resonance and decrease the heater voltage, which blue-detunes the resonance. At the output, we record the comb spectrum and power. The output power curve [Fig. 2(b)] is composed of four segments that correspond to four different dynamical processes leading to the final mode-locked comb, namely, subthreshold state, Turing pattern state, chaotic state, and soliton state. The heater voltage is slightly increased after the transition into soliton state (known as the “soliton step”) to compensate for the temperature difference owing to an abrupt change of intracavity power. In Fig. 2(c), we show the spectrum of a four-soliton state where the solitons are equally spaced inside the ring [inset, Fig. 2(c)]. This is the preferred state, since it has an intracavity power that is close to that of the chaotic state, making it experience the least amount of thermal backlash after the soliton step, which we find to be more pronounced than in the near-IR. For pumping at 782 nm, which is one free spectral range (FSR) away from 784 nm, the five-soliton state is found to be most stable, indicating that a mode interaction near 792 nm is the dominant factor in determining the pattern of the soliton state [38]. Limited by the dynamic range of the optical spectrum analyzer ($\approx 60\mathrm{dB}$), we obtain the spectrum in two shots under the same experimental conditions. We observe comb lines down to 710 nm, which represents the shortest wavelength generated by a mode-locked Kerr comb. We then fit our experiment spectrum with a ${\mathrm{sech}}^2$ envelope which yields a 3 dB bandwidth of 9.9 THz.

 

Fig. 2. (a) Voltage pattern used to modulate the on-chip heater. (b) Evolution of comb power corresponding to the pattern in (a). (c) Experimental comb spectrum calibrated for collection losses. Inset, simulated intracavity four-soliton state. (d–f) Optical spectra corresponding to point A (minicomb formation), B (chaotic state), and C (soliton state) in (b), respectively. (g–i) RF noise corresponding to (d)–(f).

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We further characterize the noise properties of the generated soliton state with a radio-frequency spectrum analyzer (RFSA). We gradually tune the pump laser into resonance and record the noise at three different stages of the KCG process [indicated in Fig. 2(b)], namely, the (A) minicomb state, (B) chaotic state, and (C) soliton state. As observed previously [37], the minicomb stage has a low noise floor with sharp peaks, the chaotic state has broadband and high noise, and the soliton state has a flat noise spectrum and the lowest RF noise of all three stages.

One appealing property of the generated near-visible combs is that they span several atomic transitions that are commonly used for time and frequency metrology. For example, the transitions allow for full stabilization of the comb by locking two of the comb lines to two atomic transitions [22]. Here, we show the interaction between our comb and Rb atoms by scanning a comb line near 795 nm across the well-known Rb D1 transition. To generate such a comb line from our four-soliton state, we increase our pump power from 220 to 270 mW. At large pump-cavity detuning, we observe weaker comb lines appearing between the strong comb lines [Fig. 3(a)]. This is caused by the disruption of the perfect symmetric soliton locations due to high power, which is well modeled in our simulations [Fig. 3(b)]. We use a bandpass filter to select the comb line near 795 nm and send it through a 2 cm-long ${}^{85}\mathrm{Rb}$ pure isotope cell held at 53.7°C. As a result of soft thermal locking [39], this comb line can be tuned by up to 10 GHz by simply tuning the pump wavelength. In our experiment, we tune the pump laser by 5 GHz through piezoelectric tuning and record the transmission of the 795-nm comb line [Fig. 3(c)]. We also fit our experimental data to the theoretical calculations based on published properties of Rb [40]. The fitting parameters are the atomic density, the location of the zero detuning (defined by the $F=3$ to $F^{\prime }=2$ transition), and the scanning range. We observe good agreement between theory and experiment, indicating that the comb lines are strongly interacting with atoms. The slightly lower transmission at the two extremes of detuning is caused by residual ${}^{85}\mathrm{Rb}$ atoms in the ${}^{85}\mathrm{Rb}$ cell.

 

Fig. 3. (a) Asymmetric four-soliton state with denser comb line spacing. (b) Simulated spectrum of (a). (c) Rb D1 transition probed by a comb line of our near-visible comb. The Doppler-free transitions and their relative strengths are shown as vertical sticks.

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4. PHOTON-PAIR GENERATION

Nonlinear photonic devices in the near-visible and visible wavelength regime are important for quantum applications. First, nonclassical states can only be generated through nonlinear processes. Secondly, the existence of high-quality avalanche photodetectors (APDs) and well-studied atomic transitions at this wavelength regime makes it extremely attractive. As an example of near-visible quantum optics in our high-order-mode nonlinear microresonator, we show correlated narrowband PPG in this wavelength regime, which has important application in Rb or cesium memory-based quantum information networks [41–43]. Previously, such photon pairs have been generated via spontaneous four-wave mixing (SFWM) in warm alkali vapor [44] or via spontaneous parametric downconversion (SPDC) in a bulk lithium niobate WGMR [45] and a lithium niobate Fabry–Perot cavity [46]. PPG in SiN microresonators is promising for quantum information applications due to its scalability and low loss [47]. Moreover, the flexibility in nanofabrication provides ways for better controlling of the photon states [48–50]. Lu et al. [51] demonstrated the correlation between one visible and one telecom photon generated from a SiN microresonator [35]. However, for applications that do not require long-distance fiber communication, such as free-space quantum communication and memory-assisted quantum computing [52], it is desirable to generate both photons at near-visible wavelengths, which has lower design and fabrication complexity. Moreover, due to their similarity, both photons can be further manipulated with the same devices and be detected with room-temperature high-efficiency silicon-based APDs.

We generate near-visible photon pairs by driving our microresonator at pump powers below the parametric oscillation threshold. We set our pump at 784.7 nm and use bandpass filters to select photons at 794.8 nm (signal) and 774.9 nm (idler). We verify the photon correlation by measuring the coincidence function $g_{\mathrm{si}}^{(2)}(\tau )$ between the signal and idler. We observe a clear coincidence peak at zero relative delay [Fig. 4(a)]. We also observe a significant amount of uncorrelated noise photons, which is proportional to the pump power in the bus waveguide (see Supplement 1). This is due to SiN used in our samples being slightly Si-rich, which generates broadband fluorescence when pumped below 1.1 μm. This can be overcome by refining the fabrication process to reduce the Si concentration [53]. Since the FWHM of each resonance is 501 MHz, the theoretical FWHM of $g_{\mathrm{si}}^{(2)}$ is 238 ps [54], which is comparable to the timing jitter of our single-photon counting module (SPCM), and thus cannot be readily resolved in our $g_{\mathrm{si}}^{(2)}(\tau )$ measurement. We further characterize the generation rate by measuring the coincidence at different pump power levels. The generation rate is proportional to the square of the pump power, as all data points fall on a straight line with a slope of 2 in the log–log plot [Fig. 4(b)]. We fit the measurement and achieve a PPG efficiency of $7.6\times {10}^4$ pairs ${\mathrm{s}}^{-1}{\mathrm{mW}}^{-2}$ for these narrowband photons, which is comparable to many other cavity-enhanced PPG demonstrations [55].

 

Fig. 4. (a) Normalized coincidence between signal and idler $g_{\mathrm{si}}^{(2)}(\tau )$. (b) PPG rate scaling. The plot is corrected for collection and detector losses. (c) Photon interaction with Rb D1 transition.

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Our photon source is also easily tunable by jointly controlling the pump wavelength and the on-chip heater. We show the tunability by sending the signal through an Rb vapor cell and tuning it across the D1 transition. Figure 4(c) shows the normalized coincidence rate versus the pump detuning. For stable operation, we blue-detune the pump to operate in the soft thermal locking regime [39]. Since this is comparable to the FWHM bandwidth of the resonance, it must be incorporated into the photon model. The photon spectrum has the shape

5. DISCUSSION

We can take advantage of higher-order mode dispersion engineering for KCG at even shorter wavelengths. The state-of-the-art mode-locked Ti:sapphire oscillator has been shown to produce spectral components below 600 nm [56], enabling the interaction with more species of clock atoms. Our simulations show that this can also be achieved by higher-order mode Kerr combs with realistic parameters. Following the previous argument, a tighter mode confinement is required to push the anomalous GVD region to shorter wavelengths. In Fig. 5(a), we show FEM simulation of a $400\mathrm{nm}\times 1000\mathrm{nm}$ SiN waveguide cladded by ${\mathrm{SiO}}_2$, with a bending radius of 100 μm. The ${\mathrm{TE}}_{20}$ mode has a zero GVD point at 587 nm. We simulate the corresponding soliton spectrum using the Lugiato–Lefever equation [57], where we assume a pump of 300 mW at 595 nm and a resonance FWHM of 700 MHz [Fig. 5(b)]. We see the clock transitions of ${\mathrm{Hg}}^{+}$ (563 nm) and Yb (578 nm) atoms are covered by sufficiently strong comb lines [58]. While anomalous GVD can be achieved at even shorter wavelengths with even higher-order modes, the increased cavity losses caused by the use of these modes at shorter wavelengths, as well as the onset of two-photon absorption, pose significant challenges to Kerr comb generation below 550 nm via this approach.

 

Fig. 5. (a) Simulated dispersion of a microresonantor with $400\mathrm{nm}\times 1000\mathrm{nm}$ cross section and 100 μm bending radius. (b) Simulated Kerr comb spectrum for the ${\mathrm{TE}}_{20}$ mode in (a).

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Even though in principle PPG can occur regardless of PM, in practice PM is required to achieve a high generation rate and to avoid spurious processes. In our model for PPG with dispersion (see Supplement 1), we also find that PM contributes to the narrowness of the photon bandwidth. Notably, a blue-detuned pump balances the phase mismatch that comes from a small anomalous GVD so that both high generation rate and narrow bandwidth can be achieved in this regime.

6. CONCLUSION

In conclusion, we have shown that dispersion engineering through high-order waveguide modes can compensate for the strong normal material GVD at near-visible and visible wavelengths to realize small or anomalous GVD that is not possible with the fundamental modes. With this approach, we designed a high-order-mode nonlinear microresonator that generated the first Kerr comb from a near-visible pump on a monolithically integrated chip-based platform. This dispersion engineering capability enables potential applications such as compact atomic clocks, astrocombs, and high-quality bioimaging light sources. In addition, we used the same nonlinear microresonator to demonstrate, to the best of our knowledge, the first silicon-chip-based narrowband near-visible photon-pair source that could enable applications in memory-based quantum information networks.